EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS E N G I N E E R I N G - 11
COMPUTER-AIDED CHEMICAL ENGINEERING Advisory Editor: R. Gani Volume Volume Volume Volume
1: 2: 3: 4:
Volume 5: Volume 6: Volume 7: Volume 8: Volume 9:
Distillation Design in Practice (L.M. Rose) The Art of Chemical Process Design (G. L. Wells and L.M. Rose) Computer Programming Examples for Chemical Engineers (G. Ross) Analysis and Synthesis of Chemical Process Systems (K. Hartmann and K. Kaplick) Studies in Computer-Aided Modelling. Design and Operation Part A: Unit Operations (1. Pallai and Z. Fony6, Editors) Part B: Systems (1. Pallai and G.E. Veress, Editors) Neural Networks for Chemical Engineers (A.B. Bulsari, Editor) Material and Energy Balancing in the Process Industries - From Microscopic Balances to Large Plants (V.V. Veverka and F. Madron) European Symposium on Computer Aided Process Engineering-10 (S. Pierucci, Editor) European Symposium on Computer Aided Process Engineering-11 (R. Gani and S.B. J~rgensen, Editors)
COMPUTER-AIDED CHEMICAL ENGINEERING, 9
EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS E N G I N E E R I N G - 11 34 * European Symposium of the Working Party on Computer Aided Process Engineering ESCAPE- 11, 27-30 May, 2001, Kolding, Denmark
Edited by
Rafiqul Gani Sten Bay Jorgensen CAPEC, Technical University of Denmark, Department of Chemical Engineering, Building 229, DK- 2800 KGS, Lyngby, Denmark
2001 Elsevier Amsterdam
- London
- New York-
Oxford
- Paris - Shannon
- Tokyo
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
9 2001 Elsevier Science B.V. All rights reserved.
This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 1DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail:
[email protected]. You may also contact Global Rights directly through Elsevier's home page (http://www.elsevier.nl), by selecting 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.
First edition 2001 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.
ISBN:
0-444-50709-4
The paper used in this publication meets the requirements of ANSUNISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.
Preface This book contains papers presented at the 11th European Symposium of Computer Aided Process Engineering (ESCAPE-11), held in Kolding, Denmark, from May 27-30, 2001. The ECSAPE series started in 1992 at Elsinore, Denmark, on a strong foundation of 23 events of the European Federation of Chemical Engineers (EFCE) Working Party on Computer Aided Process Engineering (CAPE). The first event on computer applications was organized by the CAPE working party in Tutzing, Germany in 1968. ESCAPE-11 is the 34 th event of the CAPE Working Party. The most recent symposia were organised in Florence, Italy 2000 (ESCAPE10), Budapest, Hungary 1999 (ESCAPE-9) and Brugge, Belgium 1998 (ESCAPE-8). The ESCAPE series serves as a forum for bringing together scientists, researchers, managers, engineers, and students from academia and industry, who are interested in CAPE. The scientific aim of the symposium is to present and review the latest developments in CAPE or Systems (Process/Product) Engineering. This research area bridges fundamental chemical and biological sciences with various aspects of process and product engineering. The objective of ESCAPE-11 is to highlight the use of computers and information technology tools, that is, the traditional CAPE topics as well as the new CAPE topics of current and future interest. The theme for ESCAPE-11 is Process and Tools Integration with emphasis on hybrid processing, cleaner and efficient technologies (process integration), computer aided systems for modelling, design, synthesis, control (tools integration) and industrial case studies (application of integrated strategies). The papers at ESCAPE-11 are arranged in terms of the following themes: computer aided modelling, computer aided design/synthesis, computer aided control/operations, computer aided manufacturing, process & tools integration, and, new frontiers in CAPE. A total of 188 papers, consisting of 5 keynote and 183 contributed papers are included in this book. All the papers have been reviewed and we thank the members of the international scientific committee for their evaluations, comments and recommendations. It was a very difficult task since we started with more than 450 submitted abstracts. The selection process involved review of abstracts, review of manuscripts and final selection of the revised manuscript. We hope that this book will serve as a valuable reference document to the scientific and industrial community and that it will contribute to the progress in computer aided process and product engineering. Rafiqul Gani Sten Bay JCrgensen
vi
International Scientific Committee
R. Gani (Denmark, Co-chairman) S. B. JCrgensen (Denmark, Co-chairman) J. Aittamaa D. Bogle B. Braunschweig D. Cameron M. Doherty Z. Fonyo U. Gren X. Joulia B. Kalitventzeff Z. Kravanja D. R. Lewin T. Malik Y. Naka
(Finland) (United Kingdom) (France) (Norway) (USA) (Hungary) (Sweden) (France) (Belgium) (Slovenia) (Israel) (United Kingdom) (Japan)
Y. Natori C. SCrlie S. Pierucci P. Pilavachi E. N. Pistikopoulos M. Pons L. Puigjaner H. Schmidt-Traub S. Skogestad E. L. S~rensen J. Van Schijndel J. Vinson
(Japan) (Norway) (Italy) (Belgium) (United Kingdom) (France) (Spain) (Germany) (Norway) (Denmark) (The Netherlands) (USA)
National Organizing Committee
R. Gani (CAPEC-DTU, Co-chairman) S. B. JCrgensen (CAPEC-DTU, Co-chairman) B. Christensen N.-J. Friis E. Hansen S. W. Jensen G. Jonsson L. B. J Crgensen C. Koch
(Kemira Agro Oy) (Cheminova Agro) (I~vens Kemiske Fabrik) (Novo Nordisk) (KT-DTU) (Danisco Sugar) (Statoil Raffinaderiet)
H. K. Nielsen M. B. Sommer E. L. SCrensen L. Wiebe M. R. Eden P. M. Harper
(Novo Nordisk) (H. Lundbeck) (Haldor TopsCe) (Danisco Ingredients) (CAPEC-DTU) (CAPEC-DTU)
Conference Secretariat
Computer Aided Process Engineering Center (CAPEC) Department of Chemical Engineering Building 229, Technical University of Denmark DK-2800 Kongens Lyngby, Denmark Phone: +45 4525 2800, Fax: +45 4588 2258 E-mail: capec @kt.dtu.dk URL: http://www.capec.kt.dtu.dk
vii
Contents Keynote Papers Samad, T., Cofer, D. Autonomy in automation: trends, technologies, tools Pantelides, C.C. New Challenges and Opportunities for Process Modelling Cordiner, J.L. Use of Prediction and Modeling in Early Evaluation of Process Options Ng, K.M. A Multiscale-Multifaceted Approach to Process Synthesis and Development Stephanopoulos, G., Schmitt, W.A. System Biology: an Emerging Theme in Biological Research
15 27 41 55
Contributed Papers Computer Aided Systems for Modeling Alexandridis, A.P., Siettos, C.I., Sarimveis, H.K., Boudouvis, A.G., Bafas, G.V. Modelling of Nonlinear Process Dynamics using Kohonen's Neural Networks, Fuzzy Systems and Chebyshev Series Barnard, J.P., Aldrich, C. A Systematic Methodology for Empirical Modeling of Non-linear State Space Systems Barnard, J.P., Aldrich, C. Modelling of Air Pollution in an Environmental System by use of Non-linear Independent Component Analysis Batres, R., Aoyama, A., Naka, Y. A Life-cycle Approach for Model Reuse and Exchange Baur, R., Taylor, R., Krishna, R. Dynamics of a Reactive Distillation Column for TAME Synthesis Described by a Non-equilibrium Stage Model Berezowski, M., Jacobsen, E.W., Grzywacz, R. Dynamics of Heat-integrated Heterogeneous Tubular Reactors with Axial Heat Conductivity in Reactor Wall Bj6rn, I.N., Gren, U., Svensson, F. Simulation and Experimental Study of Intermediate Heat Exchange in a Sieve Tray Distillation Column Cameron, D.B., Odegaard, R.J., Glende, E. On-line Modelling in the Petroleum Industry: Successful applications and future perspectives Charles, A.S., Azzaro-Pantel, C., Domenech, S., Pibouleau, L., Floquet, P., Jaume, D., Tilhac, F. Implementation of a Failure Model Validation Technique using a Discrete-event Batch Simulator: Application to semiconductor manufacturing
69 75 81
87 93 99 105
111
1t 7
viii Costa Jr., E. F., Vieira, R. C., Secchi, A. R., Biscaia Jr., E. C. Automatic Structural Characterization of DAE Systems Eliceche, A.M., Corvalfin, S.M., Ortiz, I. Steady State Analysis of Membrane Processes for the Treatment of Industrial Effluents F araoni, V., Mancusi, E., Russo, L., Continillo, G. Bifurcation Analysis of Periodically Forced Systems via Continuation of a Discrete Map Galen, O., Palazoglu, A., Romagnoli, J.A. Modelling and Optimisation of a High Density Fermentation Process using Multilinear Models: An Application to a Bench Scale Bioreactor Garea, A., Marqu6s, J.A., Hechavarria, T.L., Irabien, A. Simulation of the FGD In-duct Injection Technology using Complex Kinetic Models Harding, S.T., Floudas, C.A. EQUISTAR: Reliable Software for Design of Nonideal and Reactive Systems Hjertager, L.K., Hjertager, B.H., Solberg, T. CFD Modeling of Fast Chemical Reactions in Turbulent Liquid Flows K6hler, R., Rieber, J., Zeitz, M. Symbolic Discretization of Distributed Parameter Process Models on Self-adaptive Moving Grids Kohout, M., Schreiber, I., Kubicek, M. Computational Tools for Nonlinear Dynamical and Bifurcation Analysis of Chemical Engineering Problems Kosek, J., Stepanek, F., Novak, A., Grof, Z., Marek, M. Multi-scale Modeling of Growing Polymer Particles in Heterogeneous Catalytic Reactors Kr~xner, S., Gesthuisen, R. Semi-Batch Emulsion Copolymerisation: A Gereral Model for a Copolymer Formed from n Monomer Units Kristensen, N.R., Madsen, H., Jorgensen, S.B. Computer Aided Continuous Time Stochastic Process Modelling Lakner, R., Hangos, K.M., Cameron, I.T. Assumption Retrieval from Process Models Lim, Y.I., Le Lann, J.M., Meyer, X.M., Joulia, X. Dynamic Simulation of Batch Crystallization Process by using Moving Finite Difference Method Liu, Y., Jacobsen, E.W. Effective Model Reduction for Analysis of Distributed Parameter Systems Lucia, A., Yang, F. Global Terrain Methods for Chemical Process Simulation Lovik, I., R~nnekleiv, M., Olsvik, O., Hertzberg, T. Estimation of a Deactivation Model for the Methanol Synthesis Catalyst from Historic Process Data Mancusi, E., Maffettone, P.L., Gioia, F., Creseitelli, S. Nonlinear Analysis of an Industrial Ammonia Reactor with Heterogeneous Model Marchetti, M., Rao, A., Vickery, D. Mixed Mode Simulation - Adding Equation Oriented Convergence to a Sequential Modular Simulation Tool
123
129
135
141
147 153 159
165
171
177 183 189 195 201 207 213
219 225
231
ix Martinez, E.C., Lopez, G.D. Adaptive Optimal Operation of the Fenton's Batch Process for Industrial Wastewater Treatment Moharir, A.S., Shah, S.S., Gudi, R.D., Devereux, B.M., Vanden Bussche, K., Venimadhavan, G. Generalized Reactor Model: An Object Oriented Approach to Reactor Modelling Morton, W., Kozel, L., Lim, P.P.S., Douglas, D. Step Restriction for a Bounded Newton's Method Paloschi, J.R. Improving Robustness using Homotopy as an Open Solver in a Dynamic Simulation Package Preisig, H.A. Using Wavelets in Process Identification: A New Link to the State Space Rouzineau, D., Meyer, M., Prevost, M. Evaluation of Coupled Reactive Distillation Performances by Means of a Rigorous Simulation Procedure Sakizlis, V., Bansal, V., Ross, R., Perkins, J.D., Pistikopoulos, E.N. An Adjoint-Based Algorithm for Mixed Integer Dynamic Optimization Santana, P.L., Vasco de Toledo, E.C., Meleiro, L.A.C., Scheffer, R., Freitas Jr., B.B., Maciel, M.R.W., Maciel Filho. R. A Hybrid Mathematical Model for a Three-phase Industrial Hydrogenation Reactor Schneider, R., Kenig, E.Y., G6rak, A. Complex Reactive Absorption Processes: Model Optimisation and Dynamic Column Simulation Shacham, M., Brauner, N., Cutlip, M.B. A Web-based Library for Testing Performance of Numerical Solvers for Solving Nonlinear Algebraic Equations Siepmann, V., Haug-Warberg, T., Mathisen, K.W. Analysis and Consistency of Process Models with Application to Ammonia Production Teoh, H.K., Sorensen, E., Tumer, M., Titchener-Hooker, N. Dynamic Modelling of Chromatographic Processes: A Systematic Procedure for Isotherms Determination Tolsma, J.E., Barton, P.I. Process Simulation and Analysis with Heterogenenous Models Vale Lima, P., Saraiva, P.M. A Structured and Selective Framework for Hybrid Mechanistic-Empirical Model Building Wolf-Maciel, M.R., Soares, C., Barros, A.A.C. Validations of the Nonequilibrium Stage Model and of a New Efficiency Correlation for Nonideal Distillation Process through Simulated and Experimental Data Yiagopoulos, A., Yiannoulakis, H., Morris, J., Kiparissides, C. Simulation of an Industrial Olefin Polymerization FBR Operating under Condensed Mode Zhuang, H., Chiu, M.-S. Extended Self-Organizing Map with Application to the Modelling of Pulse Jet Fabric Filters
23 7 243 249
255 261 267 273 279 285 291 297 303 309 315
321 327
333
Computer Aided Systems for Synthesis and Design Agrawal, R., Herron, D.M. Feed Pretreatment for Binary Distillation Efficiency Improvement Bayer, B., Weidenhaupt, K., Jarke, M., Marquardt, W. A Flowsheet-Centered Architecture for Conceptual Design Bertok, B., Friedler, F., Feng, G., Fan, L.T. Systematic Generation of the Optimal and Alternative Flowsheets for Azeotropic Distillation Systems Bfihner, C., Schembecker, G. Reactor Selection and Design for Heterogeneous Reaction Systems Caballero, J.A., Grossmann, I.E. Generalized Disjunctive Programming Model for the Synthesis of Thermally Linked Distillation Systems Camarda, K.V., Sunderesan, P., Siddhaye, S., Suppes, G.J., Heppert, J. An Optimization Approach to the Design of Value-Added Soybean Oil Products Cismondi, M., Brignole, E.A. ECOFAC- Computer Aided Solvent Design and Evaluation in Environmental Problems, Based on Group Contribution Methods with Association Csukas, B., Balogh, S. Evolutionary Synthesis of Almost Closed Conservational Processes Eliceche, A.M., Hoch, P.M., Ortiz, I. Analysis of Azeotropic Distillation Columns combined with Pervaporation Membranes G~imt~s, Z.H., Floudas, C.A Nonlinear Bilevel Programming: A Deterministic Global Optimization Framework Hostrup, M., Balakrishna, S. Systematic Methodologies for Chemical Reaction Analysis Ierapetritou, M.G. An Efficient Approach to Quantify Process Feasibility based on Convex Hull Jim6nez, L., Costa, J. Design, Sizing and Modeling of a Reactive Extractive Distillation Unit and Solvent Recovery System Kahn, D., Plapp, R., Modi, A. Modeling a Multi-Step Protein Synthesis and Purification Process: A Case Study of a CAPE Application in the Pharmaceutical Industry Ko, D., Kim, M., Moon, I., Choi, D.-K. New designed TSA bed with cooling jacket for purification and regeneration of benzene and toluene Kr6ner, A., Kronseder, Th., Engl, G., v. Stryk, O. Dynamic Optimization for Air Separation Plants Li, X.-N.L., Rong, B.-G., Kraslawski, A. TRIZ-Based Creative Retrofitting of Complex Distillation Processes- An Industrial Case Study Liu, F., Hallale, N. Retrofit of Refinery Hydrogen Systems
339 345 351 357
363 369 375 381 387 393 401 407 413 419
427 433 439 445
xi Marcoulaki, E.C., Kokossis, A.C., Batzias F.A. Computer- Aided Synthesis of Molecular Mixtures and Process Streams Marechal, F., Kalitventzeff, B. A Tool for Optimal Synthesis of Industrial Refrigeration Systems Omota, F., Dimian, A.C., Bliek, A. Design of Reactive Distillation Process for Fatty Acid Esterification Pajula, E., Seuranen, T., Hurme, M. Selection of Separation Sequences by using Case-based Reasoning Patsiatzis, D.I., Papageorgiou, L.G. Optimal Multi-floor Process Plant Layout Reneaume, J.-M., Niclout, N. Plate Fin Heat Exchanger Design using Simulated Annealing Reyes-Labarta, J.A., Grossmann, I.E. Optimal Synthesis of Liquid-Liquid Multistage Extractors Rigopoulos, S., Linke, P., Kokossis, A. Development of Novel Process Designs for Simultaneous Oxidation and Denitrification of Wastewaters Rodriguez-Donis, I., Gerbaud, V., Joulia, X. Middle Vessel Heterogeneous Batch Distillation of an Azeotropic Mixture Samanta, A., Jobson, M. Optimisation of Heat Integrated Distillation Sequences in the Context of Background Process Samanta, A., Jobson, M. A New Heat Integration Model for Streams of Variable Temperature and Constrained Matches Sanchez Daza, O., Perez-Cisneros, E., Bek-Pedersen, E., Hostrup, M. Tools for Reactive Distillation Column Design: Graphical and Stage-to-Stage Computation Methods Schroer, J.W., Wibowo, C., Ng, K.M., O'Young, L. Development of Software Tools for Crystallization System Synthesis Sobocan, G., Glavic, P. Optimization of Ethylene Process Design Steinbach, W., Friedl, A., Hofbauer, H. Optimization of an Acidic Chlorine Scrubber with a Rate-based Simulation Engine Strouvalis, A.M., Heckl, I., Friedler, F., Kokossis, A.C. An Accelerated Branch-and-Bound Algorithm for Assignment Problems of Utility Systems Suh, M.-h., Friedler, F., Park, S., Lee, T.-y. Retrofit Design of Chemical Processing Networks under Uncertainties: Application to Petrochemical Industry Szitkai, Z., Lelkes, Z., Rev, E., Fonyo, Z. Optimisation of an Industrial Scale Ethanol Dehydration Plant. A Case Study Takano, K., Gani, R., Kolar, P., Ishikawa, T. Multi-Level Computer Aided System for the Design and Analysis of Processes with Electrolyte Systems Torres Alvarez, M.E., Martini, R.F., Wolf-Maciel, M.R. Characterization and Simulation of the Pervaporation Process for Separating Azeotropic Mixtures
451 457 463 469 475 481 487
493 499
505
511 517 523 529 535 541
547 553
559
567
xdi
Uerdingen, E., Fischer, U., Hungerbtihler, K. A Screening Method for Identifying Economic Improvement Potentials in Retrofit Design Vasquez-Alvarez, E., Pinto, J.M. MILP Models for the Synthesis of Protein Purification Processes Wang, Y.P., Achenie, L.E.K. A CAPD Approach for Reaction Solvent Design Wasylkiewicz, S.K., Castillo, F.J.L. Automatic Synthesis of Complex Separation Sequences with Recycles
573 579 585 591
Computer Aided Systems for Control and Operation
Aartovaara, M. Model-based Temperature Control of an Exothermic Semi-batch Reactor Akay, B., Ertunc, S., Bursali, N., Hapoglu, H., Alpbaz, M. Adaptive General Predictive Controller for a Nonlinear Bioreactor Aziz, N., Mujtaba, I.M. Optimal Control of Semi-batch Reactors Blanco, A.M., Figueroa, J.L., Bandoni, J.A. Feedback Control Design by Lyapunov's Direct Method Bonn6, D., Jorgensen, S.B. Batch to Batch Improving Control of Yeast Fermentation Coffey, D.P., Ydstie, B.E., Andersen, T.R., Jorgensen, S.B. Distillation Control Using Passivity Dechechi, E.C., Meleiro, L.A.C., Maciel Filho, R. A Novel Adaptive Multivariable DMC Controller: Application to an Industrial Reactor Ender, L., Scheffer, R., Maciel Filho, R. Computer Design of a New Predictive Adaptive Controller Coupling Neural Networks and Kalman Filter Eo, S.Y., Chang, T.S., Lee, B., Shin, D., Yoon, E.S. Function-Behavior Modeling and Multi-Agent Approach for Fault Diagnosis of Chemical Processes Gehan, O., Farza, M., M'Saad, M., Binet, G. Robust Predictive Control Combined with Nonlinear Observation for Monitoring (Bio)chemical Processes Govatsmark, M.S., Skogestad, S. Control Structure Selection for an Evaporation Process Grosman, B., Lewin, D.R. MPC using Nonlinear Models Generated by Genetic Programming Huang, Y., Reklaitis, G.V., Venkatasubramanian, V. Wavelet Shrinkage Based Coariance Matrix Estimation from Process Measurements Jordache, C., Temet, D., Brown, S. Efficient Gross Error Elimination Methods for Rigorous On-line Optimization
597 603 609 615
621 627 633 639 645 651 657 663
669 675
xiii Kint, E., Samyudia, Y., de Jong, P. A Combined Data and Gap Metric Approach to Nonlinear Process Control
Macias, J.J., Feliu, J.A. Dynamic Study of Inferential Sensors (Neural Nets) in Quality Prediction of Crude Oil Distillation Tower Side Streams M6ndez, C.A., Cerd~i, J. An Efficient MILP Continuous-Time Formulation for the Optimal Operation of General Multipurpose Facilities Michiel Meeuse, F., de Deugd, R.M., Kapteijn, F., Verheijen, P.J.T., Ypma, S.M. Increasing the Selectivity of the Fischer Tropsch Process by Periodic Operation Mourikas, G., Seferlis, P., Morris, A.J., Kiparissides, C. On-line Optimal Operating Policy and Control of a Batch Free Radical Polymerization Process Nagy, Z., Agachi, S., Allgower, F., Findeisen, R., Diehl, M., Book, H.G., Schloder, J.P. Using Genetic Algorithm in Robust Nonlinear Model Predictive Control Preuss, K., Le Lann, M.-V. Inferential Control of Microbial Batch Culture Rovaglio, M., Manta, D., Cortese, F., Mussone, P. Multistability and Robust Control of the Ammonia Synthesis Loop Ruiz, C., Basualdo, M.S., Molina, A., Jim6nez, L., Parisse, B., Richalet, J. Predictive Functional Control (PFC) Applied to an Unstable System- An Industrial Application Schmidt, H., Jacobsen, E.W. Selecting Control Configurations for Performance Shah, S.S., Madvadhan, K.P. Design of Controllable Batch Processes Silva, D.C.M., Oliveira, N.M.C. Optimization and Nonlinear Model Predictive Control of Batch Polymerization Systems Silva-Beard, A., Flores-Tlacuahuac, A., Fernandez-Anaya, G. Interval Matrix Robust Control of a MMA Polymerization Reactor Simon, L., Karim, N.M. Model Predictive Control of Apoptotis in Mammalian Cell Cultures Singer, A.B., Bok, J.-K., Barton, P.I. Convex Underestimators for Variational and Optimal Control Problems Skotte, R., An, W., Lenz, D.H., Baptiste, D.R., Lapham, D.S., Lymburner, C.J., Kaylor, J.M., Pinsky, M., Gani, R., Jorgensen, S.B. Modeling, Simulation and Control of an Industrial Electrochemical Process Smets, I.Y.M., Van Impe, J.F.M. Generic Properties of Time and Space Dependent Optimal Control of (Bio-) Chemical Processes Szederk6nyi, G., Kristensen, N.R., Hangos, K.M., Jorgensen, S.B. Nonlinear Analysis and Control of a Continuous Fermentation Process Torgashov, A.Yu. Nonlinear Process Model-based Self-optimizing Control of Complex Crude Distillation Column
681 687 693 699 705 711 717
723 731 737 743 749 755 761 767 773 781 787 793
~dv
Tousain, R.L., Michiel Meeuse, F. Closed Loop Controllability Analysis of Process Designs: Application to Distillation Column Design Tresmondi, A., Domingues, A., Maciel Filho, R. Online Optimization Integrated with Online Analyzers and Multivariable Predictive Controller in Industrial Airlift Reactors Verdijck, G.J.C., Lukasse, L.J.S., Preisig, H.A. A Control Methodology for Product Quality Control in Climate Controlled Operations involving Agro-materials Xaumier, F., Ettedgui, E., Le Lann, M.-V., Cabassud, M., Casamatta, G. A Model-Based Supervisory Control Routine for Temperature Control of Batch Reactors: Experimental Results Zeaiter, J., Gomes, V.G., Barton, G.W., Romagnoli, J.A., Gilbert, R.G. Strategies for Optimisation and Control of Molecular Weight and Particle Size Distributions in Emulsion Polymerization
799 805 811
817
823
Computer Aided Systems for Manufacturing Aoyama, A., Batres, R., Naka, Y. Process Safety Management for Batch Process Operation Badell, M., Ruiz, D., Puigjaner, L. Dynamic Cross-functional Factory-to-Business Links in the Batch Industry Canton, J., Afonso, A., Graells, M., Espuna, A., Puigjaner, L. Ad-hoc Scheduling~Planning Strategies in Distributed Computing Systems: An Application to Pipeless Batch Plants Castro, P., Barbosa-P6voa, A.P.F.D., Matos, H., Duarte, B. Dynamic Modelling and Scheduling of an Industrial Batch Digester Cooking System Dash, S., Kantharao, S., Rengaswamy, R., Venkatasubramanian, V. Application and Evaluation of Linear~Restricted Nonlinear Observers to a Nonlinear CSTR Gabbar, H.A., Suzuki, K., Shimada, Y. Design Considerations of Computer-Aided RCM-based Plant Maintenance Management System Gatica, G., Shah, N., Papageorgiou, L.G. Capacity Planning under Clinical Trials Uncertainty for the Pharmaceutical Industry Gupta, A., Maranas, C.D. Multiperiod Planning of Multisite Supply Chains Under Demand Uncertainty Harjunkoski, I., Grossmann, I.E. Combined MILP-Constraint Programming Approach for the Optimal Scheduling of Multistage Batch Processes Henning, G.P. Development of Interactive Facilities in a Knowledge-based Scheduling Framework H6treux, G., Perret, J., Pingaud, H. Computer Aided System for Short-term Scheduling of Batch Processes based on Hybrid Simulation Kim, D., Lee, Y., Moon, I., Lee, Y., Yoon, D. Automatic Accident Scenario Generation for Petrochemical Processes
829 835 841 847 853 859
865 871 877 883 889 895
XV
Mockus, L., Vinson, J.M., Luo, K. The Integration of Production Plan and Operating Schedule in a Pharmaceutical Pilot Plant Ortiz-G6mez, A., Rico-Ramirez, V., V/tzquez-Rom/m, R. Mixed-Integer Multiperiod Model for the Planning of Oilfield Production Sequeira, S.E., Graells, M., Puigianer, L. Decision-making Framework for the Scheduling of Cleaning~Maintenance Tasks in Continuous Parallel Lines with Time-decreasing Performance Strouvalis, A.M., Kokossis, A.C. A Conceptual Optimisation Approach for the Multiperiod Planning of Utility Networks Venkatasubramanian, V., Zhao, J., Viswanathan, S., Zhao, C., Mu, F., Harper, P., Russel, B.M. An Integrated Environment for Batch Process Development- From Recipe to Manufacture Zhao, J., Viswanathan, S., Zhao, C., Mu, F., Venkatasubramanian, V. Knowledge-based Management of Change in Chemical Process Industry Zhu, X.X., Majozi, T. A Novel Continuous Time MILP Formulation for Multipurpose Batch Plants Integrated Planning, Design and Scheduling
901 907 913
919
925 931 937
Process and Tools Integration
Alfadala, H.E., Sunol, A.K., E1-Halwagi, M.M. Retrofitting of Mass Exchange Networks with Temperature Effects Azzaro-Pantel, C., Davin, A., Pibouleau, L., Floquet, P., Domenech, S. Implementation of Multiobjective Optimisation for Multipurpose Batch Plant Planning Banares-Alcantara, R., Fraga, E.S., Perris, T. Concurrent Process Engineering & The Implications for CAPE Bansal, V., Perkins, J.D., Pistikopoulos, E.N. A Unified Framework for Flexibility Analysis and Design of Non-Linear Systems via Parametric Programming Belaud, J.-P., Alloula, K., Le Lann, J.-M., Joulia, X. Open Software Architecture for Numerical Solvers." Design, Implementation and Validation Bildea, C.S., Dimian, A.C., Iedema, P.D. Multiplicity and Stability of CSTR-Reactor-Separator-Recycle Systems Dua, V., Bozinis, A., Pistikopoulos, E.N. A New Multiparametric Mixed-Integer Quadratic Programming Algorithm Dunn, R.F., Wenzel, H. A Process Integration Design Method for Water Conservation and Wastewater Reduction in Industry Fraser, D.M., Harding, N., Matthews, C. Retrofit of Mass Exchange Networks Georgiadis, M.C., Schenk, M., Gani, R., Pistikopoulos, E.N. The Interactions of Design, Control and Operability in Reactive Distillation Systems
943 949 955 961 967 973 979 985 991 997
xvi
Grancharova, A. General Strategy for Decision Support in Integrated Process Synthesis, Design and Control Henriksen, J.P., Russel, B.M. Static and Dynamic Optimisation of CAPE Problems using a Model Testbed Hertwig, T.A., Xu, A., Nagy, A.B., Pike, R.W., Hopper, J.R., Yaws, C.L. A Prototype System for Economic, Environmental and Sustainable Optimization of a Chemical Complex Kheawhom, S., Hirao, M. Decision Support Tools for Process Design and Selection Kwok, Y.-Y., Hui, C.-W. Site-wide Energy Optimization with Steam Pressure Changes Li, H.-Q., Chen, H.-Z., Li, Z.-H., Li, B.-H., Yao, P.-J. A Combined Approach for the Overall Energy Integration and Optimal Synthesis of Low-Temperature Process Systems Lid, T., Skogestad, S. Implementation Issues for Real Time Optimization of a Crude Unit Heat Exchanger Network Ma, K., Bogle, I.D.L. An Approach to Controllability and Economic Design Analysis of Nonlinear Systems with Multiplicity Minet, F., Heyen, G., Kalitventzeff, B., Di Puma, J., Malmendier, M. Dynamic Data Reconciliation of Regenerative Heat Exchangers coupled to a Blast Furnace Okada, H., Shirao, T. New Chemical Process Economic Analysis Methods Pingen, J. A Vision of Future Needs and Capabilities in Process Modelling, Simulation & Control Schlegel, M., Binder, T., Cruse, A., Oldenburg, J., Marquardt, W. Component-based implementation of a dynamic optimization algorithm using adaptive parameterization. Sequeira, S.E., Graells, M., Puigjaner, L. Integration of available CAPE Tools for Real Time Optimization Systems Shang, Z., Kokossis, A.C Design and Synthesis of Process Plant Utility Systems under Operational Variations Shethna, H.K., Jezowksi, J., Castillo, F.J.L. Near-independent Subsystems in Heat Exchanger Networks Design Sorsak, A., Kravanja, Z. Simultaneous MINLP Synthesis of Heat Exchanger Networks Comprising Different Exchanger Types Subramanian, D., Pekny, J.F., Reklaitis, G.V. SIM-OPT: A Computational Architechture to Address Valuable Business Aspects of Research & Development Pipeline Management Szitkai, Z., Lelkes, Z., Rev, E., Fonyo, Z. Solution of MEN Synthesis Problems using MINLP: Formulations of the Kremser Equation
1003 1009
1017 1023 1029 1035
1041 1047 1053 1059 1065 1071 1077 1083 1089 1095
1101 1109
xvii Wang, K., Clark, G.A., Chung, P.W.H., Rossiter, D. Modelling Interface for Chemical Plantwide Process Design Yang, A.-D., von Wedel, L., Marquardt, W. An Open Thermodynamics Server for Integrated Process Engineering Environments
1115 1121
New Frontiers in CAPE
Elgue, S., Cabassud, M., Prat, L., Le Lann, J.M., Casamatta, G., Cezerac, J. Optimisation of Global Pharmaceutical Syntheses Integrating Environmental Aspects Garg, S., Achenie, L.E.K. Genome Wide Functional Annotation using Mathematical Programming Georgiadis, M.C., Kostoglou, M. On the Optimisation of Drug Delivery Devices Halim, I., Palaniappan, C., Srinivasan, R. An Integrated Framework for Developing Inherently Safer and Environmentally Benign Processes Hill, P.J., Ng, K.M. Particle Size Distribution by Design Maranas, C.D. Optimization in Molecular Design and Bioinformatics Smith, R.L., Mata, T.M., Young, D.M., Cabezas, H., Costa, C.A.V. Designing Efficient, Economic and Environmentally Friendly Chemical Processes Xie, X., Hua, B., Chen, Q., Liang, R., Zeng, M. Study on Lifecycle and Agility of Process Industry Yang, T.C.-K., Lin, H.-S., Wang, S.-F., Cheng, M.-C. Dynamic Assessment of Induced Thermal Stresses on the Semiconductor Packaging Substrates in a Radiation Belt Furnace by Computational Fluid Dynamics AUTHOR INDEX
1127 1133 1139
1145 1151 1157 1165 1171 1177 1183
This Page Intentionally Left Blank
European Symposiumon ComputerAided ProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsewerScience B.V. All rights reserved.
Autonomy in automation: trends, technologies, tools Tariq Samad and Darren Cofer Honeywell Laboratories, 3660 Technology Drive Minneapolis, MN 55418, U.S.A. tariq, samadldarren.cofer @ honeywell.com We focus on the topic of autonomy in automation and control systems. The trend toward increasing autonomy is discussed and illustrated with examples from multiple domains. Economics, performance, and human safety are highlighted as key considerations driving research into autonomous systems. We note that autonomy implies an ability to react appropriately to unforeseen situations, and identify two technical concepts that are instrumental for realizing this ability: multimodels and dynamic resource management. The need for new tools is discussed in this context with particular emphasis on integrating diverse aspects of models and systems. Finally, we speculate that some degree of "consciousness" will be required of automation systems if they are to become truly autonomous. 1. I N T R O D U C T I O N There are many perspectives we can take in examining the progress of automation in complex engineering systems such as process plants. In this paper we focus on the topic of autonomy. This is perhaps not an obvious principal concern today in the process industries, but we maintain that it has always been at least an implicit consideration in the development of automation and control technology in general and one that is increasingly capturing the interest of the process engineering research community. By autonomy here we mean the substitution by automated tools of functions that are or were performed by people. Complete automation of any significant system is not feasible now and will not be feasible in the foreseeable future. Increased autonomy thus implies essentially that the role of the human is shifted from lower to higher level tasks. What used to be accomplished with a large team may now be done with a smaller team, or with one person. Our discussion will largely be focused on the autonomous operation of systems. (Automated system design is another related topic and one that is well advanced in terms of tools available in many domains.) However, we do not limit our discussion to process engineering, although we frequently refer to it. The trend toward autonomy spans many industries and there are numerous points of similarity between parallel developments in different areas. The following section illustrates the trend toward autonomy with some examples, drawn from different domains. In Section 3, we briefly note three reasons why autonomy is being sought: performance improvement, economics, and human safety. Next, we note that autonomy implies an ability to respond appropriately to unforeseen situations, and we highlight two technical concepts that are critical for this ability: multimodels and dynamic
resource management. We discuss the types of new tools that are needed to facilitate the development of autonomous control systems in Section 5. Before concluding with a summary we reflect on the topic of consciousness vis-~.-vis autonomy. Some of the ideas discussed here are further elaborated in (Samad and Weyrauch, 2000). 2. T O W A R D A U T O N O M Y m E X A M P L E S
A couple of examples may help underscore the trend toward autonomous operation. In the process industries, metrics such as "loops per operator" are being used and there is continuing emphasis on increasing this quantity. In fact, operator employment has been shrinking. From 1980 to 1998, the number of production workers involved in petroleum refining in the United States shrank from 93,000 to 60,000--even as total refinery production increased from 5.3 to 6.2 billion barrels (U.S. Bureau of the Census, 1999). Analogously, a little over fifty years ago Lockheed introduced what was then a revolutionary new aircraft, the Constellation. It required a cockpit crew of five: a pilot, a copilot, a radio operator, a flight engineer, and a navigator. Because of improvements in navigation tools and avionics, the newest airliners today operate with just a pilot and copilot. The trend toward the replacement of human operation by automated systems can also be discerned through the levels of automation in engineering systems. Our engineered systems are multilevel, complex entities. From the lowest levelqsingle-loop control--to the highestqenterprise-wide optimization--concepts of feedback, dynamics, and adaptation are relevant. The degree of autonomy can be related to the levels through which operations are largely automated. Requirements for knowledge of the system's dynamics, regulation through feedback control, and adaptation to changing conditions can, in principle, be fulfilled through manual or automatic means. In all industries and application domains, we can see steady advances in levels of automation. In the early days of flight, control automation as we know it today was nonexistent. The pilot received sensory inputs directly (through visual and vestibular channels) and operated the actuators. The first step toward automatic control was the development of what we now refer to as inner-loop flight controllers. These allowed pilots to specify higher level commands to the aircraft, such as the desired pitch, roll, and yaw, with the controller responsible for the subsecond-scale feedback loop for sensing and actuation. The next step was the development of controllers that could take as input a heading command and automatically produce a sequence of desired states based on predefined "handling qualities." Today, all commercial aircraft have flight management systems (FMSs) on board that remove even this level of responsibility from the pilot (under normal conditions). A sequence of flight _ route h..I
_~
[
new heading .... management ~l system
desired
Flight i an~l"Handling laif .... t i state ~1
I
qualities" controller I
]
control i ] [ surface aircraft ~l "Inner- [ movements state l~176flight I ~lAi .... f,i ..~ controller structure
I
feedbackof aircraft state (via sensors)
Figure 1. Control loops in commercial aviation today.
waypoints can be entered through the console of the FMS and the aircraft can automatically fly the route with no further manual assistance (Figure 1). Similar advances in the level of complexity under investigation have also occurred in the process industries. In a meeting to identify future needs for process automation with some aluminum processing industry representatives a few years ago, the discussion centered not on regulatory or multivariable control of individual processes, but on the management of the enterprise as a whole (Figure 2). Each of the processes shown in the figure will have several control loops associated with it, but it is automated solutions for the end-to-end system, from the bauxite ore as input to refined alumina as product, that are now desired. Increased autonomy, and the technological infrastructure that has enabled it, also implies that larger-scale systems are now falling under the purview of automation. Sensors, actuators, processors, and displays for entire facilities are now integrated through one distributed computing system. The control room in an oil refinery can provide access to and/or manipulate 20,000 or more "points" or variables. One of the largest integrated control system implementations, the Munich II international airport building management system, can be used to control everything from heating and cooling to baggage transport. The system controls more than 100,000 points and integrates 13 major subsystems from nine different vendors, all distributed over a site that includes more than 120 buildings (Ancevic, 1997).
3. W H Y A U T O N O M Y ? What is motivating the development of increasingly autonomous systems and the enabling technology for them? We note three key drivers. Performance. Automation can provide superior performance compared to manual operation. This can be due to the reduced sensing and actuation latencies of automated systems, their greater calculation prowess, greater memory fidelity in some cases, etc.
Ore [ GrindingI Caustic ~ [ ~ Steam Water~ ~ [ Digestion,[~Wash~ Water ~,........ I ,I Europe, I HeatExchl
i so,i] I Cool I
t--I Pr~pit~ion ~[ I ~'~n~'i~ I Alumina
Figure 2. Processes involved in alumina refining. Performance means different things in different applications. Response times, settling times,
disturbance rejection, and setpoint tracking are some common measures for low-level control. At higher levels, yield, throughput, environmental impact, and energy efficiency are important parameters for many processes. Economics. We can also find many examples of a preference for an autonomous solution even where the manual alternative is no worsemor is even better~in terms of performance. Typically, in these cases there is an additional cost associated with manual operation that the associated performance improvement cannot compensate for. Our favorite example of this is the now obsolete traffic policeperson whose job consisted solely in standing in the middle of an intersection, generally on a raised and somewhat protected platform, and through hand signals directing roadway traffic. He or she could respond in an intelligent, context-sensitive manner to a wide variety of situations and incidents. From the point of view of performance a traffic light is a poor substitute, but it is considerably cheaper. H u m a n Safety. In some cases autonomy is sought because the task to be performed is dangerous for people. Examples include toxic material handling, bomb disposal, and combat. Considerable research is underway focusing on the development of uninhabited vehiclesm aerial, terrestrial, and undersea~for such applications. We note that both economics and performance are often considered supercriteria, in the sense that any other factor (including economics for performance and vice versa) can be subsumed by them. We are using these terms here in more specific senses. 4. T E C H N O L O G I E S FOR AUTONOMOUS SYSTEMS The discussion above has presented, as evidence of the trend toward autonomy, several examples of the advances in automation. Autonomy, however, is much more than automation. Today's engineered systems may be highly automated, but they are brittle and capable of "hands-off.' operation only under more-or-less nominal conditions. As long as the system only encounters situations that were explicitly considered during the design of its operational logic, the human element is dispensable. As soon as any abnormal situation arises, control reverts to the human. A pump fails in an oil refinery, or there is excessive turbulence on the route being flown by a commercial airliner, or the paper being manufactured in a paper machine consistently exceeds its caliper tolerancemin all these cases the human is immediately put back in the loop. Automation is part of the solution currently, but its role is in support of the operatormusually by supplying some relevant information about the state of the system. Ultimately the intelligence required to fix the problem resides in the operator. Autonomy, as distinct from automation, is not characterized by an ability to handle a complex system without human assistance in normal operating regimes. An autonomous agent must be capable of responding appropriately to unforeseen situationsmthat is, situations unforeseen by its designers. Some degree of circumscription of a system's operating space will always exist, since survival under every environmental extreme is inconceivable, but "precompiled" behaviors and strategies are not sufficient for effective autonomy. What does it mean to be able to react appropriately to unforeseen situations? To be capable of exhibiting behaviors that are not precompiled? We highlight two technical concepts in this section, multimodels and dynamic resource management.
Table 1 Types of models ......Characterizing dimension ....
. . . . . . . . . . . . . . . .
4.1
,,
Examples First-principles, heuristic, and statistical models Temporal and algebraic models Operationally and componentially localized models
Knowledge source Complexity Domain of competence H
Multimodels
Domain knowledge is a central requirement for control and automation. Current control solutions fulfil this requirement in various specific ways, but generally they incorporate their designers', implementers', and users' understanding about the application domain. Automation is a multifarious proposition, and its different aspectsmfeedback and supervisory control, control system design, prognostic and diagnostic procedures, operator display, etc.m demand different types of knowledge. Even within one area, the knowledge required often needs to be differentiated; thus system dynamics often vary drastically with the operating regime of a physical system. Increasing automation requires increasing articulation and representation of domain knowledge within the automation system. Today, when aircraft can fly from point A to point B with no human interaction, it is in large part because we now have available explicit representations (models) of the dynamics of the aircraft under different conditions: the rate of fuel burn at different altitudes, speeds, and payloads; terrain topography; locations of navigation aids; special use airspace regions; etc.
4.1.1 Multimodel Types and Mechanisms In the process industries, model-predictive control technology, which integrates a predictive model of the system to be controlled (e.g., a distillation column in an oil refinery), is now well established. These models are still limited in various waysmthey are generally linear and valid only over a small operating regime. Higher fidelity models are also sometimes available, but not for online control. Instead, their use is generally limited to offline activities, such as operator training and process design. Just as linear, dynamic, highdimensional multivariable models used to be beyond the capability of process control platforms a decade or two ago, we similarly foresee that more complex models will be widely used in real-time. The trend is clear: increasing amounts of knowledge, in the form of models, will be incorporated within automation systems and tools. This should not evince much surprise given the objectives of increasing autonomy. If detailed and accurate models are not accessible online, the generation of appropriate responses to unforeseen situations is impossible. Control and automation of complex systems cannot be based on any single, unitary concept of model. Thus, a fundamental research need in modeling for autonomous systems is active multimodeling: the online use of a variety of model types. Some examples of different types of models, characterized by different dimensions, are listed in Table 1. The knowledge embedded in a model can result from a first-principles understanding of the physics and chemistry involved in the operation of the system, from the heuristic understanding of expert personnel, and from empirical data. Models can be more or less complex both algebraically
Input ~l9 i Inpu!~
]
fl
] "1
Firstprinciples model
~
N.... ! network model (a)
r u .v sor I
ut n Weight. . ~
i (b)
i
-
2
-
(c) Figure 3. Selected multimodeling mechanisms. (linear to increasingly nonlinear) and temporally (static versus dynamic). Finally, a model can be localized to a particular operating regime and/or it can describe the behavior of a particular component or subsystem. Approaches for integrating multiple models have become an increasingly popular topic. Figure 3 shows some common examples. Figure 3a shows a superposition of a first principles model and an empirical model (in this case a neural network); the neural network is fit to predict the error between the actual data and the first principles model prediction (Su et al., 1992). In Figure 3b the outputs of different individual models are blended through a supervisory function (Murray-Smith and Johansen, 1997). A simple hybrid dynamical system model (Lemmon, 2000) is shown in Figure 3c. The ellipses represent different operational modes with customized models. The arcs represent mode transitions triggered by conditions. A diagram such as this one could represent the operation of a simple batch chemical reactor, distinguishing between reactor charging, heating, and product draw, with potential for reversion to recharging or reheating during product draw. A final interesting example of a multimodeling approach is multiscale and multiresolution models. For example, the wavelet transform of a signal can be seen as providing a multiplicity of separate yet consistent representations of the signal.
4.2
Dynamic Resource Management
Until recently, only highly constrained algorithms could be implemented and executed on real-time control system platforms 9 PID controller calculations and PLC ladder logic were about the limit of online computational complexity. Multivariable control, optimization, estimation, and process monitoring were implemented on nonreal-time computers in the plant, such as VAXes, with limited interfaces to DCSs and other real-time platforms. A PID algorithm executes periodically, deterministically, and in constant time: When installing the code realizing the algorithm on the control platform, engineers know how frequently the code needs to be executed and a good upper bound on its execution time, assumed constant. With online processing consisting entirely of such tasks, the execution profile or schedule for the real-time system can be determined in advance and can be assumed to hold indefinitely 9 Capabilities for different operational modes may be available, but even these provide essentially just a small finite set of processing options 9 While today's control
systems are more capable than yesterday's, and can allow, for example, iterative MPC calculations, they still require static, offline-determined task scheduling. The control system infrastructure, as it currently exists, does not permit dynamic adaptive resource management and thereby severely limits the "intelligence" that can be incorporated within automation. With human operators available to handle abnormal conditions, this limitation has by and large not caused much concern. As we attempt to endow automation systems with the algorithmic capabilities to autonomously manage complex enterprises, the lack of dynamic resource management technology is becoming a serious bottleneck.
4.2.1 Algorithmic Complications The need for dynamic resource management becomes evident when we consider the computational support needed for anytime algorithms. Anytime algorithms are fexible, scalable methods to solve particular problems, whether related to regulation, optimization, system health management, or other functions. They are characterized by their ability to use profitably as much time as is available. That is, the longer an anytime algorithm is run, the better will be its computed result. Depending on the current operational objective, the state of the system, and other computational requirements, an anytime algorithm can be given more or less CPU cycles. In addition, anytime algorithms are not limited to periodic execution; they can be event-driven and aperiodic as well. All of this stands in sharp contrast to the algorithms that are currently employed on real-time platforms, where rates of execution, priorities, and computational requirements can be fully characterized offline. Examples of anytime algorithms include: Data-centric forecasting. Online-accessible databases provide a new approach for modeling and prediction (Kulhav3), Lu, and Samad, 2001). Instead of relying on a first principles or empirical model generated offline as a computationally static entity, relevant operational data can be dynamically accessed and used for "just-in-time" modeling. In such a scenario, the number of samples accessed can vary dramatically, depending on the frequency with which the particular operational regime of interest has occurred in the past as well as on accuracy and response time requirements. Computational complexity will vary accordingly. Evolutionary Computing. Randomized optimization algorithms inspired, however loosely, by biological evolution are increasingly being explored for high-dimensional, analytically untractable problems. The quality of solution obtained with these algorithms is related to the number of iterations performed. Genetic algorithms are the most familiar example, but several other methods also exist (Fogel, 1995). Multiresolution Modeling. For high-performance applications and for failure tolerance, online modeling and identification will often be necessary. Depending on circumstances and resource availability, it may be appropriate to undertake model development or model execution at different levels of detail.
4.2.2
Optimizing Performance by Control Task Adaptation
The challenge to resource scheduling arises when we contemplate the need to run a large number of processing tasks in a finite amount of time where these tasks can include anytime and fixed-time; periodic, aperiodic, and event-driven; and deterministic and nondeterministic tasks. In addition to any control optimizations being performed, the allocation of available computing resources must be actively adapted to optimize the performance of the overall system. The relationships between the overall application goals, computing resource models,
and control models can be thought of as follows (Figure 4). Based on the computed or observed system state, task criticalities and computing requirements are assigned. Computing
system& ,~ ~'P~~iti~ce,~ . environment/ ~ "~taskcriticality
Figure resources are made available to tasks based on criticality, pending requests, and a schedulability analysis. Control tasks then execute within their allotted time. These tasks must adapt to meet the application constraints (deadlines, accuracy, etc.). Active models will be configured as schedulable tasks with a variety of service requirements that must be satisfied by the execution environment. Models will specify their requirements as ranges of values within which the execution environment may optimize overall vehicle performance. For example: 9 A model may be able to run at a range of rates, within a given minimum and maximum, and adapt its execution time correspondingly. Example: servicing a message queue. 9 A model may be able to run at several discrete rates. Example: a control loop or state estimator that can use different tables of values for gains and time constants. 9 A collection of models may provide the same data but by using algorithms of varying fidelity and execution time. Example: process models for fault detection and identification may include a simple static gain model, an approximate empirical mode, and a full dynamic model. The set of currently executing models and tasks will have to adapt to both internal and external triggers to make optimal use of available computational resources. Mode changes will cause tasks to start and stop, and external influences (new targets, feedstock changes, etc.) will cause changes in task criticality and computing loads.
5. THE NEED FOR TOOLS This paper has generally been concerned with articulating concepts for the autonomous operation of complex engineering systems. As such, we have not specifically discussed the need for new tools. However, there are some important implications of our discussion for process engineering tools.
5.1
Multimodei Tools
Most modeling and identification tools available today assume a unitary notion of model. Even if a tool permits more than one type of model to be developed, provisions for
compositions of models into effective multimodel architectures are not likely to be available. Multimodeling tools, not just modeling tools, are needed. In this context it is important to note that the model-centric nature of complex automation solutions requires that tools be developed that are informed by a suitably rich notion of model. Thus we should distinguish between, and integrate within tools, three "facets" of models (Figure 5): 9 A model is a set of mathematical formulae that capture some important characteristics of the behavior of a physical system. This is the typical control science and engineering perspective, and most tools limit themselves to it. 9 Models are computational entities with processing, memory, and communication requirements. For online execution of multiple models, these requirements must be satisfiedmgenerally a nontrivial undertaking given resource constraints of control system platforms. This computational facet needs to be provided by tools addressing real-time applications. 9 Models can be viewed as objects, with inputs, outputs, and functional capabilities abstractly specified. This view is especially useful for tools that purport to facilitate model composition.
5.2
Tools Integrating Performance and Computation
A related gap in the tools available today concerns design, simulation, and evaluation of automation and control systems that integrate performance and resource usage aspects. The incorporation of anytime algorithms within real-time platforms complicates design and assessment. Designers need to be concerned not only with the performance of a particular algorithm under a fixed resource availability, but they must consider and effect tradeoffs between performance and computing requirements--given multiple complex algorithms, bounds on computational resources, and diverse scenarios. We have recently developed a prototype simulation and evaluation tool as a first attempt to address this gap. This development is an outcome of an ongoing project focusing on the autonomous operation of uninhabited aerial vehicles (UAVs) (Godbole, Samad, and Gopal, 2000). This tool combines real-time scheduling and execution of control algorithms with aircraft and environmental models running in simulated time (see Figure 6). This framework can be populated with a variety of control and computational models and algorithms. The system permits control tasks (active models) to be run as real executable code (not simulations) that have real-time deadlines. The aircraft and environmental simulations can be based on high fidelity models since they are not required to execute in real time. Each aircraft is created as a separate process that can later be transferred to separate
Figure 5. Facets of a model. hardware to more closely approximate a real multi-aircraft application.
Real time
10 performance data for the control algorithms and scheduling infrastructure can be tracked independently of the (nonreal-time) simulation. Execution time can be scaled to simulate the improved hardware performance expected for future platforms. More details are available in (Agrawal et al., 2000). Simulation and evaluation tools such as the UAV-specific one discussed above are also needed for the process industry domain. While the details will differ considerably, the overall framework of Figure 6 is, we believe, still relevant.
Figure 6. Multimodel simulation tool for a UAV application. The separation of the nonreal-time execution of system models with the simulated and scaled real-time execution of control models and algorithms, the allowance for dynamically managed computational loads among different tasks, and possibly the componentwise integration for assessing system-level automation solutions (e.g., multiunit integration for plantwide automation) should all be of interest in this case. 6. A PHILOSOPHICAL SIDEBAR: AUTONOMY AND CONSCIOUSNESS The notion of developing engineered sensors or actuators, or even low-level models of computation, that are based on biologically gleaned principles is uncontroversial. Embodying higher-level cognitive capabilities in computational systems, however, is another matter. Some researchers argue that such capabilities cannot even in principle be realized by the sorts of machines we are contemplating. The levels of autonomy, intelligence, and adaptability exhibited by humans are thereby excluded (the argument goes) from realization in engineered systems. At the center of this controversy lies the indeterminate notion of consciousness. There is no accepted precise definition of the term, but it is generally held that it is a key to human (and possibly other animal) behavior and to the subjective sense of being human. Consequently, any attempt to design automation systems with humanlike autonomous characteristics requires designing in some elements of consciousness: in particular, the
11 property of being aware of one's multiple tasks and goals within a dynamic environment and of adapting behavior accordingly. The nexus between consciousness and computation is a favorite topic of some philosophers and neuroscientists. There are two theoretical limitations of formal systems that are driving much of the controversymthe issue under debate is whether humans, and perhaps other animals, are not subject to these limitations. First, we know that all digital computing machines are "Turing-equivalent"--they differ in processing speeds, implementation technology, input/output media, etc., but they are all (given unlimited memory and computing time) capable of exactly the same calculations. More importantly, there are some problems that no digital computer can solve. The best known example is the halting problem--we know that it is impossible to realize a computer program that will take as input another, arbitrary, computer program and determine whether or not the program is guaranteed to always terminate. Second, by G6del's proof, we know that in any mathematical system of at least a minimal power there are truths that cannot be proven. The fact that we humans can demonstrate the incompleteness of a mathematical system has led to claims that G6del's proof does not apply to humans. In analyzing the ongoing debate on this topic, it is clear that a number of different critiques are being made of what we can call the "computational consciousness" research program. In order of increasing "difficulty," these include the following: 9 Biological information processing is entirely analog, and analog processing is qualitatively different from digital. Thus sufficiently powerful analog computers might be able to realize autonomous systems, but digitally based computation cannot. Most researchers do not believe that analog processing overcomes the limitations of digital systems; the matter has not been proven, but the Church-Turing hypothesis (roughly, that anything computable is Turing-Machine [i.e., digitally/algorithmically] computable) is generally taken as fact. A variation of this argument, directed principally at elements of the artificial intelligence and cognitive science communities, asserts that primarily symbolic, rule-based processing cannot explain human intelligent behavior. [] Analog computers can of course be made from non-biological material, so the above argument does not rule out the possibility of engineered consciousness. Assertions that the biological substrate itself is special have also been proposed. Being constructed out of this material, neural cells can undertake some form of processing that, for example, silicon-based systems cannot. Beyond an ability to implement a level of self-reflection that, per G6del, is ruled out for Turing machines, specifics of this "form of processing" are seldom proposed, although Penrose's hypothesis that the brain exploits quantum gravitational effects is a notable exception (Penrose, 1989). (It is worth noting that no accepted model of biological processing relies on quantum-level phenomena.) [] It has also been argued that intelligence, as exhibited by animals, is essentially tied to embodiment. Disembodied computer programs running on immobile platforms and relying on keyboards, screens, and files for their inputs and outputs, are inherently incapable of robustly managing the real world. According to this view, a necessary (not necessarily sufficient) requirement for an autonomous system is that it undertakes a formative process where it is allowed to interact with the real world. 9 Finally, the ultimate argument is a variation of the vitalist one, that consciousness is something extra-material. For current purposes this can be considered a refrain of the Cartesian mind/body dualist position. Contemporary explorations of this theme include
12 Chalmers (1995)--an article that also includes a rebuttal by Christof Koch and Francis Crick. Consciousness is a multifaceted phenomenon. Reflective, deliberative decision making is an important element, although admittedly not the only one. Thus the technical concepts discussed earlier~multimodels, anytime algorithms, dynamic resource allocation--which, we have argued, are essential for high-performance autonomous behavior, are by the same token necessary correlates of consciousness. (Our observation of) our own conscious processing supports this contention--we dynamically allocate cognitive resources as appropriate for an unforeseen situation, scale the precision and resolution of our processing accordingly, and rely on our knowledge of the various systems and phenomena that constitute our environment. 7. SUMMARY Any but the most indomitable technophile today would, we expect, refuse to be a passenger in a pilotless airliner or have an uninhabited yet operational refinery in her neighborhood. But, regardless, the trend toward reducing human involvement in the operation of complex engineering systems, driven as it is by considerations of economics, performance, and safety, appears inexorable. Further substantial improvement in process automation, however, will require more than evolutionary technological advances. Our focus in this paper has been on autonomy, a property absent in today's automation systems. In order to satisfy the demands of industry and society, we will need to make our automation solutions autonomous--they will need to be able to respond appropriately to unforeseen situations, not just limited to precompiled behaviors. We have noted two research directions that are central for engineering autonomy. First, diverse representations of knowledgemof process dynamics, failure modes, the environment, and other factorsmwill be needed and will need to be integrated so that unanticipated situations can be explicitly reasoned about. Models are widely recognized as principal determiners of control system performance; we must broaden our attention now to multimodels. Second, dynamic resource management technology must be developed to allow multiple, competing, heterogeneous computational tasks to execute on real-time platforms under hard and soft deadlines and resource constraints. The full space of relative priorities that could be faced by an autonomous system cannot be predicted; tradeoffs among resource allocations for different tasks will need to be made online and adaptively, not through static scheduling. Engineering autonomy will require maturing concepts into tools. Tools that can truly enable multimodeling and dynamic resource management are unavailable today. A key, general differentiating attribute that distinguishes the tools that are needed from those that are currently in use is the level of integration. Whether it is models of different subsystems, or knowledge from disparate sources, or performance and resource usage aspects of computational tasks, tools for autonomous systems will require integration of diverse phenomena, features, and representations. Finally, we suspect that our pursuit of autonomy in automation and control will bring the topic of consciousness~at least in the limited sense of an agent's awareness of its
13 environment and context and its deliberative adaptation of its information processing priorities--into the foreground of discussion.
Acknowledgement: This research is supported in part by the U.S. Defense Advanced Research Projects Agency (DARPA) under contract number F33615-98-C-1340. REFERENCES 1. 2. 3. .
5. 6. .
8. .
10. 11. 12.
Agrawal, M. et al. (2000). Real-time adaptive resource management for multi-model control. Submitted for publication. Ancevic, M. (1997) Intelligent building system for airport. ASHRAE Journal, November, pp. 31-35. Chalmers, D. (1995). The puzzle of conscious experience. Scientific American, pp. 8086, December. Fogel, D.B. (1995). Evolutionary Computing: Toward a New Philosophy of Machine Intelligence, IEEE Press, Piscataway, N.J., U.S.A. Godbole, D., T. Samad, and V. Gopal (2000). Active multi-model control for dynamic maneuver optimization of unmanned air vehicles. Proc. IEEE Int. Conf. on Robotics and Automation, San Francisco. Kulhav3~, R., J. Lu, and T. Samad (2001). Emerging technologies for process optimization. To appear in Proceedings of CPC-VI, Tucson, Ariz., U.S, January. Lemmon, M. (2000). Supervisory hybrid control systems. In Perspectives in Control Engineering: Technologies, Applications, and New Directions, T. Samad (ed.), IEEE Press, Piscataway, N.J., U.S.A. Murray-Smith, R. and T.A. Johansen (eds.) (1997). Multiple Model Approaches to Modelling and Control Taylor & Francis Ltd., London. Penrose, R. (1989). The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford Univ. Press. Samad, T. and J. Weyrauch (eds.) (2000). Automation, Control and Complexity: An Integrated View, John Wiley & Sons, Chichester, U.K. Su, H.-T., et al. (1992). Integrating neural networks with first principles models for dynamic modeling. Preprints of the IFAC Symp. on DYCORD+, College Park, Md. U.S. Bureau of the Census (1999). Statistical Abstract of the United States: 1999, 118th edition, Washington, D.C.
This Page Intentionally Left Blank
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
15
New challenges and opportunities for process modelling Constantinos C. Pantelides abt aCentre for Process Systems Engineering, Imperial College of Kingdom bprocess Systems Enterprise Ltd., 107a Hammersmith Bridge Road, London W6 9DA, United Kingdom Over the past decade, process modelling has made substantial progress with respect to both methodological advances and applications in engineering practice. This paper discusses some of the major issues that need to be addressed for this kind of progress to be sustained over the next decade. Particular emphasis is focused on three areas, namely the modelling of complex distributed systems, the construction of validated models, and multiscale modelling. For each of these areas, recent progress is reviewed and some key outstanding problems are identified. 1. I N T R O D U C T I O N Process modelling has made substantial progress over the past decade. We are now capable of building mathematical models with a degree of detail and predictive accuracy that would be almost unimaginable in the early 1990s. In some cases, this reflects our improved understanding of process physics. In other cases, the main driver for recent progress has been the advent of generic modelling concepts and software that permit this knowledge to be transformed into practically usable descriptions. Perhaps more impressively, we are actually able to solve many of these models and to perform various other manipulations based on them. For some types of problem (e.g. steady-state simulation), the underlying algorithms and codes existed well before the 1990s; although some improvements have undeniably taken place since then, much of the credit for progress has to be given to improved computer speed and memory availability. However, in other areas (such as the simulation of hybrid discrete/continuous processes and dynamic optimization), the past decade has indeed witnessed much advancement in both mathematical understanding and numerical algorithms. The above progress has provided the driving force behind the increasingly central role played by process modelling within most process engineering activities. However, this increased usage has also exposed certain obstacles and bottlenecks to the wider deployment of this technology. This paper identifies three of the major issues in process modelling that need to be addressed if further significant progress is to be achieved over the next decade. These are the modelling of complex distributed systems (section 2), the construction of validated models (section 3) and multiscale modelling (section 4). In all cases, the paper is primarily concerned with process modelling methodology. Naturally, substantial improvements in physical understanding contFAX: +44-20-7594 6606, email: c. pantelides@ ic. ac. uk
16 tinue to be a major requirement in several areas of importance to process engineering. However, this topic is beyond the scope of the present paper.
2. MODELLING OF COMPLEX DISTRIBUTED SYSTEMS As a significant part of the process industries moves away from commodity chemicals towards higher value-added products, there is an increasing requirement for the accurate prediction and control of the quality of these products. This had led to a shift in emphasis from traditional measures of process performance, such as yield and conversion, to other measures that directly affect product quality, such as selectivity. As far as process modelling is concerned, the above developments have often required the abandonment of "perfect mixing" simplifications and the direct consideration of both spatial and temporal variations within process equipment. This is necessary as many of the properties of interest are affected by mixing inhomogeneities. From the mathematical point of view, this leads to distributed (as opposed to lumped) systems described by mixed sets of integral, partial and ordinary differential, and algebraic equations (IPDAEs). Many of the higher value-added products mentioned above involve materials, such as polymers and crystalline solids, that are more complex than the relatively simple liquids and gases that have been the mainstay of the process industries in the past. The design and operation of such processes requires the prediction of product quality in terms of properties that are of interest to the end-user. These properties are often determined by complex characteristics of the product such as the distributions of crystal sizes and shapes (for crystalline solids) or of chain lengths and degree of branching (for polymeric materials). Thus, we have distributed processes, the mathematical modelling of which again requires the introduction of IPDAE systems. In view of the above, it is not surprising that a decisive factor in the wider use of generalpurpose process modelling tools during the past decade has been their ability to describe and solve some classes of distributed systems [1, 2, 3]. Interestingly, this feature has made such tools useful not only to engineers concerned with describing entire process flowsheets but also to specialists who are concerned with the detailed modelling and design of a single item of processing equipment (e.g. chemical reactor or separation device). The use of common tools for modelling both individual equipment items and the entire process has been a highly desirable recent development. It has to be recognised, however, that the distributed modelling capabilities of general-purpose process modelling tools are still limited in some important ways. One restriction is that the domains that can be described by current technology must be expressed as the cartesian product of one or more line segments. This limits the applicability of these tools to relatively simple geometries (e.g. rectangles, cylinders, spheres) and presents a serious obstacle to the accurate modelling of processing equipment of irregular shapes. A second limitation concerns the reliability and efficiency of numerical solution. For example, although it is possible to describe 3-dimensional models of fluid flow within the latest generation of process modelling tools, the resulting models cannot easily be solved using generic numerical codes of the type typically incorporated in these tools. The two limitations mentioned above are already addressed quite well by a different type of technology which has been evolving along quite separate lines to general process modelling tools. More specifically, Computational Fluid Dynamics (CFD) tools (see, for example, [4])
17 can describe equipment of arbitrarily complex geometry. They also employ specialised solution techniques that exploit the nature of the underlying equations to deal with numerical complexity. These characteristics have allowed CFD tools to make important contributions to process modelling in recent years [5]. On the other hand, they are limited to a specific set of equations and, consequently, lack the wide scope of general-purpose process modelling tools (e.g. with respect to describing complex multicomponent mass transfer phenomena) and the capabilities they offer in terms of making a common process model available to diverse types of application. Moreover, the specialised numerical methods that they employ cannot always deal effectively with highly nonlinear and/or stiff systems, such as those involving very fast reactions. Recent work has sought to combine the capabilities of CFD tools with those of more conventional modelling technology. Work along these lines has been reported for the modelling of industrial crystallisation [6], gas/liquid bubble column reactors [7] and exothermic batch reactors [8]. Despite the diversity of the systems studied in these papers, a number of common features can be discerned: 9 The combined model encompasses a "simple" model (e.g. a network of well-mixed regions in [6], a set of 1-dimensional vertical zones of gas or liquid in [7], and a perfectly mixed tank in [8]) coupled with a CFD model. 9 Both the CFD model and the simple model describe the same spatial domain. However, the CFD model incorporates only phenomena that are directly related to fluid dynamics while all other phenomena (e.g. crystal nucleation and growth in [6], mass transfer and reaction in [7], reaction in [8]) are incorporated in the simple model. 9 The CFD model is used to compute parameters that are necessary for the simple model, being treated as fixed constants by it (e.g. the mass fluxes between adjacent regions and the turbulent energy dissipation rate (which affects crystal nucleation) in [6], the lateral flows between the gas and liquid zones in [7] and the heat transfer coefficient between the process fluid and the cooling jacket in [8]). 9 The solution of the CFD model requires knowledge of properties that depend on the solution of the simple model (e.g. the effective density and viscosity of a pseudo-homogeneous fluid in [6], the bubble size and gas-liquid mass trasfer rate in [7] and the liquid density and viscosity in [8]). 9 In view of the last two points, an iteration between the simple and the CFD models is necessary to obtain a converged and consistent solution. In all cases, a successive substitution approach has been used. The publications mentioned above, as well as other unpublished material, indicate that this approach can lead to striking improvements in predictive accuracy for models of industrialscale equipment. However, much remains to be done to transform it into a tool that can be used routinely for process modelling. First, we need a set of genetic concepts for deciding the partitioning of the process physics between the CFD and the simple model, and for mapping the physical domain into each of the two models. Secondly, any procedure that involves repeated solutions of CFD models (some of which may take several hours or days) may lead to serious efficiency concerns. Finally, the successive substitution procedure may not be the most reliable
18 way of combining the two models - especially in cases where there is strong two-way coupling between the fluid mechanical and the other (e.g. reaction) phenomena. More fundamentally, for strongly coupled systems, the loss of accuracy that is inherent in the construction of any simplified model may be unacceptable in some applications. In these cases, a simultaneous solution approach that treats all phenomena together appears unavoidable. Such an approach would require a certain degree of convergence between general-purpose process modelling technology on one hand and CFD technology on the other. Some of the issues that have to be addressed in this context (e.g. the description of irregular geometries) are primarily of a software nature, and satisfactory solutions already exist and can be borrowed from the CFD area. Extending the numerical solution techniques so that they can deal with general IPDAE systems will be more challenging. On past experience, however, the ever increasing computer power may well diminish the importance of using solution techniques specifically designed to solve particular classes of problems, and widen the range of applicability of more general methods ~. However, the real difficulties lie at the fundamental mathematical level. Our current understanding of the mathematical properties of IPDAEs (including such basic questions as wellposedness and validity of initial and boundary conditions) is arguably at approximately the same level as our understanding of DAEs was in the mid-1980s. Even for specific well-studied PDE systems such as the Navier-Stokes equations of fluid mechanics, the formulation of a correct set of boundary conditions has been the subject of intense study and debate (see, for instance, [9-12]). For more general systems, it is only very recently that ao index classification of PDAE systems has been introduced [13] and that this has been related to the specifications (e.g. initial or boundary conditions) that may independently be imposed on the system along a particular co-ordinate direction [14, 15]. Moreover, our understanding of the relation between the mathematical properties of IPDAE system and the methods that may be applied for its numerical solution is also at a preliminary stage (see, for instance, [16]). !n summary, much theoretical progress is still required before we can have practical tools that are able to guide the modeller towards the correct formulation and solution of complex distributed models. 3. CONSTRUCTION OF VALIDATED PROCESS MODELS The previous section identified some outstanding issues that relate to the modelling of complex distributed systems. However, there are other important problems that affect almost all process models, even some relatively simple ones. Almost all non-trivial models involve parameters that describe various thermodynamic and kinetic phenomena. In fact, the unavailability of values for these parameters is one of the most frequently cited obstacles to the wider use of process modelling in industrial practice. The problem has been exacerbated by the move of the process industry towards processes involving newer and/or more complex materials and reactions that are not yet well characterised, and towards higher fidelity modelling that often requires information relating to different types of phenomena (e.g. parameters for modelling finite rates of mass transfer) beyond those that were routinely modelled in the past (e.g. phase equilibria). The use of rigorous optimisation techniques for the estimation of model parameters from ~Analogous developments took place in numerical methods for nonlinear algebraic equations and differentialalgebraic equations (DAEs) in the 1970s and 1980s.
19 measured data is now well established. The objective functions primarily used are based on maximum likelihood or nonlinear least squares criteria for both steady-state and dynamic models. Progress in this area has benefited from general developments in numerical optimisation algorithms and codes, often adapted to exploit the special form of the objective function in the parameter estimation problem. For example, dynamic optimisation techniques have formed the basis for fitting parameters to data from dynamic experiments [ 17] while, more recently, global optimisation techniques have been used to ensure that the correct (globally optimal) solution of the parameter estimation problem is obtained [18, 19]. Beyond point estimates of the parameter values, the computation of measures of statistical significance of these estimates has also received attention, especially for the case of nonlinear models for which standard linear measures may be misleading [20]. In principle, the parameter estimation techniques mentioned above apply equally well irrespective of whether the measured data used come from laboratory-scale experimentation or from measurements carried out in pilot or, indeed, industrial-scale plants. In particular, fitting models to match their predictions to observed plant behaviour has been, and still is, common practice. However, an interesting consequence of our increasing ability to handle complexity in process modelling, especially where this arises because of inhomogeneities in mixing, is that we can now often afford to adopt a more systematic approach towards the development of high fidelity models. As always in process modelling, we have to start with the identification of all important phenomena that may occur in the process. Ideally, this is followed by the quantitative characterisation of each such phenomenon in isolation. This is achieved using laboratory experiments specifically designed for this purpose; for example, homogeneous reaction kinetics can be measured under conditions of high mixing intensity, thereby removing any mixing effects. Such a complete decoupling may not always be possible, especially in heterogeneous systems where mass transfer and reaction phenomena may be difficult to separate. Nevertheless, there is a definite advantage in minimising the number of phenomena that are involved in any individual experiment so as to achieve a better statistical characterisation. Once all phenomena of interest are characterised in this manner, they can be combined within a single plant model. The latter can often adequately describe the plant behaviour with little, if any, need for additional adjustment. Where successful, the application of the above approach can lead to some important benefits. Since diverse unit operations within the same or different processes involve combinations of the same phenomena, the physical knowledge regarding these phenomena may be re-used, thus reducing the overall cost of process modelling. Moreover, models that have been built from the combination of phenomena characterised in isolation are likely to predict plant behaviour accurately even when changes in operating conditions affect the relative importance of these different phenomena (e.g. when significant changes in flow and mixing shift the balance between mass transfer and chemical reaction). Of course, laboratory experimentation is often an expensive and time consuming activity itself. The need to carry out such activities in the most effective and efficient manner possible has led to increased interest in systematic techniques for the design of experiments during the last decade. Asprey and Macchietto [21] have recently outlined a comprehensive scheme for constructing validated models via a combination of laboratory experimentation and model-based optimisation techniques. The scheme starts with one or more candidate models, each involving
20 one or more unknown parameters. The models are first tested for parametric identifiability, i.e. the ability to determine unique values for the parameters appearing in them given a set of quantities that can be measured experimentally. Models that are deemed to be identifiable in principle are then tested against the results from experiments specifically designed to discriminate between competing models. Once the best model is selected, further experiments are designed to allow the accurate estimation of the parameters in it. The above scheme comprises a number of computational steps, each involving the solution of an appropriate optimisation problem. The step that is perhaps best understood at present is the final one, namely that of designing experiments for optimal parameter estimation. Dynamic experiments are particularly important in this context as they have the potential of generating a large amount of information in a single run. The design of a dynamic experiment typically involves several decisions, including its initial conditions and duration, the variation of any available control variables over the latter, and the times at which various measurements are to be taken. Several alternative measures for the information content of an experiment have been proposed in the literature [22, 23, 24]. The determination of an optimal experiment design involves the solution of a dynamic optimisation problem. Thus, like their counterparts for parameter estimation, experiment design techniques have also benefited from the significant progress in dynamic optimisation algorithms and codes over the past decade. A more fundamental problem is that the optimisation is carried out using a model that involves parameters, the values of which are still subject to significant uncertainty. Consequently, experiment design is an iterative process: once an experiment is designed, it is executed in the laboratory, and the data collected from it are then used to re-estimate the model parameters. A new experiment design can then be determined on the basis of this improved model, and the process is repeated. An attempt to accelerate this iteration by designing "robust" experiments that take direct account of the parametric uncertainty in the underlying models has recently been reported [25]. In summary, the last decade has witnessed some methodological progress for the construction of validated models, with emphasis shifting away from plant-level data fitting and towards laboratory-scale experiments targetted at the characterisation of individual physical phenomena. There may still be scope for further decomposition in certain applications such as those involving complex reaction networks. For example, it would be beneficial to determine systematically a set of distinct experiments, each of which involves only a (hopefully small) subset of the species and reactions in the network. We now also have a clearer idea of an integrated approach to this problem, and the various computational and experimental components that it may comprise. However, much remains to be done in terms of both the mathematical formulation and the numerical solution of some of these computational components. On the other hand, both hardware and software for automated laboratory experimentation have achieved significant progress in recent years (see, for instance, [26, 27]), partly driven by the demanding needs of the pharmaceutical and biotechnological industries. This ultimately opens the way for the seamless integration of the experimental components with their computational counterparts, with substantial potential savings in both the time and the cost required for the development of validated process models.
21 4. MULTISCALE M O D E L L I N G The previous section considered the use of laboratory experimentation for the characterisation of the fundamental phenomena that need to be described in process models. An increasingly viable alternative is provided by techniques of computational chemistry that attempt to model matter at the molecular, atomic or sub-atomic levels based on classical Newtonian mechanics, quantum mechanics, or combinations of the two. At the simplest level, computational chemistry techniques may be used to generate "pseudoexperimental" points that can then replace or complement the results of real experiments. This type of approach is particularly feasible for simple thermodynamic and transport properties, for which the predictive accuracy of computational chemistry has improved significantly in recent years. Moreover, it has the advantage that it does not actually require any changes to standard methodologies and tools of either computational chemistry or process modelling. A more challenging, but also potentially much more powerful, mode of utilisation of computational chemistry techniques is in their direct incorporation within process models. This kind of approach is best understood within the wider context of the multiscale nature of phenomena and operations of interest to process engineering [28], ranging from the nanoscale of molecules, atoms and sub-atomic particles (involving distances of O(10 -1~ m and times under O(10 -12) s) to the megascale of global supply chains (with distances of O(107) m and times of O(108) s). Intermediate scales include the microscale of particles, eddies and bubbles, the mesoscale of process equipment items, and the macroscale of process plants. It should be stressed that, albeit commonly used, neither the terminology introduced above nor the delineation of the boundaries between different scales are unique. In fact, sometimes the same terms are used to describe different delineations- for example, material scientists allocate quantum mechanical and classical mechanical phenomena to different scales, with the high end of their classification reaching up to what was called "microscale" above (see, for example, [29]). What cannot be doubted is the widely different scales of both space and time involved in process engineering, and the challenges that these pose to our attempts to encode our knowledge in terms of mathematical models. The traditional way of addressing multiscale complexity can, perhaps, best be described as scale decoupling. This simply focuses scientific and engineering endeavour on each individual scale with the aim of building the best possible understanding and description of the phenomena taking place at that scale. For example, at the nanoscale, such descriptions may take the form of models of molecular and intra-molecular motion; at the microscale, we have descriptions of mixing and fluid flow of the type mentioned in section 2 of this paper; mesoscale models involve detailed descriptions of dynamic behaviour of processing equipment; at the macroscale, we have models of the type used to schedule the operation of complex multipurpose plants; and at the megascale, there are the models used to analyse the dynamics of supply chains. Of course, the different scales are not truly independent of each other. After all, our mesoscale models of dynamic unit operation models invariably require some description of the behaviour of both the materials involved and of fluid flow, which are precisely the objects of study at the nanoscale and microscale respectively. The traditional approach in dealing with such interactions has largely been based on scale aggregation, leading to simplified descriptions of behaviour at each scale in terms of quantities that are directly relevant to higher scales. For example,
22 9 the nanoscale's detailed descriptions of the behaviour of matter are aggregated into equations of state and relatively simple kinetic laws so that they can be used in higher-level models; 9 the complexities of fluid flow at the microscale are hidden behind the well-mixed region (or networks of well-mixed regions) approximations used for modelling process equipment; 9 the details of the dynamic behaviour of batch processing equipment studied at the mesoscale are replaced by the simple concept of a task with a finite duration and fixed demands on resources, of the type that can be used for plant scheduling; 9 the large networks of interacting resources and tasks used for modelling multipurpose plants at the macroscale are replaced by a few simple linear constraints describing overall production capacity for the purposes of modelling supply chains involving interacting manufacturing and distribution operations. The scale aggregation approach has proven very successful in handling the inherent complexity of process engineering, representing a pragmatic trade-off between the predictive accuracy of a model and its complexity at both the conceptual and the computational levels. It has to be recognised, however, that any aggregation operation involves an inherent approximation. As our demands for model accuracy at a given scale (e.g. the mesoscale of processing equipment) become more stringent while the predictive accuracy of models at lower scales improves, there often comes a point at which the loss of useful information involved in the aggregation of lowerlevel behaviour becomes unacceptable. At this point, we are forced to consider an alternative to scale decoupling, namely scale integration. Scale integration involves the use of descriptions at different scales within the same model. At present, the most common ways of achieving this are the so-called serial and parallel integration strategies [29]. A serial integration strategy is one in which the finer scale model is simply used to generate some of the parameters or data required by the higher-scale one. This is not very different from the scale decoupling approach except that the aggregate description (e.g. the equation of state) is formally derived from the lower-scale model (rather than, for instance, being determined empirically by fitting to experimental data). On the other hand, a parallel integration strategy involves the simultaneous use of descriptions at different scales applied to the same computational domain. The results of one description form inputs to the other, and vice versa. Thus, an iteration between the two models is normally required to achieve a consistent overall description. The combination of CFD with conventional process models described in section 2 of this paper is a good example of such a parallel strategy. In addition to the serial and parallel strategies mentioned above, we could also envisage a third hierarchical integration strategy in which the finer-scale model is formally embedded within the higher-scale model to represent a set of relations among macroscopic quantities occurring in the latter. Finally, there is a simultaneous strategy in which the higher-scale model is formed completely from finer-scale descriptions. This is currently possible only in certain applications, mostly towards the higher scales of interest to process engineering. For example, advances in numerical methods for dynamic simulation and increased computing power now routinely allow us to build dynamic (macroscale) models of entire plants simply by assembling
23 detailed dynamic (mesoscale) models of individual equipment items without the need for any simplification at this stage. Also (megascale) models used for short-term scheduling of combined production/distribution operations can be constructed from detailed (macroscale) models of each individual manufacturing site; mathematical decomposition techniques can then be used for the solution of these very large problems [30]. Multiscale modelling is key to the effective utilisation of computational chemistry techniques for process modelling applications. A recent example of such an approach is the work by Rodgers and Jensen [31] on modelling of chemical vapour deposition devices. Although the bulk of the fluid can be described by standard continuum equations, the rate of reaction at the solid surface is determined by the latter's inhomogeneities, the size of which is small compared with the mean free path of molecules in the gas phase. Consequently, the continuum hypothesis does not apply to this part of the system; instead, a molecular simulation approach is used to estimate the nett flux from the gas to the solid phase. This flux then forms the boundary condition for the continuum model describing the gas phase. Of course, the results of the molecular model depend on the gas phase composition and temperature. Consequently, the two models are coupled and an iteration between them is necessary. Another important example of multiscale modelling is in studying the rheology of complex fluids. The standard equations of fluid dynamics are an asymptotic approximation based on the assumption that the time scales of molecular and intramolecular motion are much shorter than those of the fluid flow [32]. While this is a reasonable assumption for most simple fluids, it is certainly not true for complex molecules such as polymers. The conventional way of dealing with this effect is via the use of constitutive equations for stress that attempt to account for the history of deformation of the fluid. In the general context of multiscale modelling outlined above, these can be thought of as aggregate descriptions of molecular behaviour. However, the predictive accuracy of such models is fundamentally limited since the state of the system at any particular time is determined by a much larger number of variables (i.e. those describing the distribution of molecular conformations) than is encompassed in these constitutive equations. Moreover, the additional computational complexity involved in the introduction of these constitutive equations within the standard Navier-Stokes equations is far from negligible. In view of these difficulties, an increasingly attractive alternative is provided by the direct combination of the continuum conservation equations with a stochastic simulation of molecular motion [33]. Ultimately, detailed atomistic models may be necessary to describe certain areas of the flow such as those adjacent to solid walls. A review of these and other important issues in computational rheology has recently been presented by Keunings [34]. As evidenced by the references mentioned above, much of the work in multiscale modelling to date has been carried out by pioneers who are interested in solving particular types of problems. Looking at it from the viewpoint of process systems engineering, it is clear that a major aim should be to use these examples as the basis for the development of generic methodologies for multiscale modelling, and for identifying and solving the underlying mathematical problems. These essential pre-requisites for the wide and reliable use of these approaches become increasingly important as we move towards strategies in which the models at different scales are more tightly coupled. Consider, for instance, a hierarchical integration strategy in which a fine-scale model is embedded within a coarse-scale one. An important question is whether the results of the solution of the former model correspond to a well-defined function of its inputs. Moreover, depending
24 on the mathematical techniques used for solving the coarse-scale model, the continuity and differentiability of this function become important issues. It must be stressed that it should not be taken for granted that techniques (e.g. for CFD or molecular modelling) that have evolved over long periods of time for "stand-alone" use automatically possess the properties that are necessary for their integration within wider computational schemes. Indeed, our experience has been that the contrary is often true and that significant theoretical and algorithmic effort may have to be invested in ensuring that these additional requirements are met. An illustration of these issues for the case of molecular dynamics is provided in [35-37]. 5. CONCLUDING REMARKS
This paper has focused on issues that are directly related to process modelling methodology and which, in the author's opinion, represent significant, but hopefully attainable, challenges for the next decade. These issues must be seen in the context of several other important problems in the related areas of mathematical solution methods (e.g. for obtaining globally optimal solutions of optimisation problems [38, 39] and for the optimisation of hybrid processes [40]) and of software developments (e.g. the move towards open system architectures for process modelling software [41 ]). Regarding the first of the three issues considered in this paper, namely that of modelling complex distributed processes, it is likely that significant progress can be made in the relatively short term by the combination of CFD with conventional process modelling technology. Longer-term progress is likely to involve a certain degree of convergence between the two technologies. However, a better mathematical understanding of the properties of general PDAE systems will be required before this can be achieved. Addressing the second issue considered, that of the construction of validated process models, in a satisfactory manner will require the solution of several novel types of optimisation problem beyond those that have already been widely recognised in the literature. Some of these new problems will require significant developments in global optimisation techniques, especially for the solution of time-dependent problems (cf. [18]). Finally, multiscale modelling is an essential pre-requisite for making full use of advances in scientific understanding within engineering applications of practical interest. Albeit sometimes regarded as primarily a problem of software integration, multiscale modelling in fact involves fundamental questions of at least three different kinds: 9 conceptual issues (e.g. the optimal way of partitioning the descriptions of the relevant physical phenomena among the different scales considered); 9 mathematical issues (e.g. regarding the precise definition and well-posedness of the mathematical problem that is actually posed by the multi-scale model); 9 numerical issues (e.g. regarding the reliability and efficiency of the overall solution algorithm). Multiscale modelling is still in its infancy as far as general systematic methodologies are concerned. It is the author's belief that process systems engineering can make valuable contributions to this field, providing some of the concepts and techniques that are needed to bind the different scales together.
25
Acknowledgements The author's work on process modelling is partially funded under grant GR/N08636 of the United Kingdom's Engineering and Physical Sciences Research Council (EPSRC). The author is indebted to Dr Steven Asprey and Professor Sandro Macchietto for many useful discussions on some of the material appearing in this paper.
REFERENCES 1. M. Oh, Modelling and Simulation of Combined Lumped and Distributed Processes, PhD thesis, University of London (1995). 2. M. Oh and C.C. Pantelides, Comput. chem. Engng., 20 (1995) 611. 3. S.E. Zitney and D.M. Laing, paper 259g, AIChE Annual Meeting, Los Angeles, CA, November 2000. 4. J.D. Anderson Jr., Computational Fluid Dynamics, McGraw-Hill, New York, 1995. 5. A.D. Gosman, Chem. Engng. Res. Des., 76 (1998) 153. 6. Z. Urban and L. Liberis, in Proc. Chemputers'99 Conference, Dtisseldorf, Germany, October 1999. 7. M. Bauer and G. Eigenberger, Chem. Eng. Sci., 54 (1999) 5109. 8. E Bezzo, S. Macchietto and C.C. Pantelides, Comput. chem. Engng., 24 (2000) 653. 9. J.C. Strikwerda, Comm. Pure Appl. Math., 30 (1977) 797. 10. E Dutt, SIAM J. Numer. Anal., 25 (1988) 245. 11. T. Miyauchi, M. Tanahashi and M. Suzuki, JSME Int. J., Series B, 39 (1996) 305. 12. E Moin and K. Mahesh, Ann. Rev. Fluid Mech., 30 (1998) 539. 13. S.L. Campbell and W. Marszalek, Math. Comp. Model. Dynam. Systems, 5 (1999) 18. 14. W.S. Martinson, Index and Characteristic Analysis of Partial Differential Equations, PhD thesis, Massachusetts Institute of Technology (2000). 15. W.S. Martinson and EI. Barton, SIAM J. Sci. Comput., 21 (2000) 2295. 16. B.-M. Pfeiffer and W. Marquardt, Math. Comput. Simul., 42 (1996) 617. 17. I.B. Tjoa and L.T. Biegler, Ind. Eng. Chem. Res., 30 (1991) 376. 18. W.R. Esposito and C.A. Floudas, Ind. Eng. Chem. Res., 39 (2000) 1291. 19. C.Y. Gau and M.A. Stadtherr, Comput. chem. Engng., 24 (2000) 631. 20. J.S. Albuquerque and L.T. Biegler, AIChE J., 43 (1997) 986. 21. S.E Asprey and S. Macchietto, Comput. chem. Engng., 24 (2000) 1261. 22. D.M. Espie and S. Macchietto, AIChE J., 35 (1989) 223. 23. L.C. Zullo, Computer-Aided Design of Experiments - An Engineering Approach, PhD thesis, University of London (1991). 24. I. Bauer, H.G. Bock, S. K6rkel and J.E Schl6der, J. Comput. Appl. Math., 120 (2000) 1. 25. S.E Asprey, S. Macchietto and C.C. Pantelides, in Proc. of ADCHEM'2000 Conference, L.T. Biegler, A. Brambilla and C. Scali (eds.) Vol. II, IFAC (2000) 869. 26. H. Du, L.A. Corkan, K. Yang, EY. Kuo and J.S. Lindsey, Chemometr. Intell. Lab. Syst., 48 (1999) 181. 27. D.G. Cork, T. Sugawara, J.S. Lindsey, L.A. Corkan and H. Du, Lab. Robot. Autom., 11 (1999) 217. 28. J. Villermaux, Chem. Engng. Res. Dev., 73 (1995) 105. 29. D. Maroudas, AIChE J., 46 (2000) 878.
26 30. A.D. Dimitriadis, Algorithms for the Solution of Large-Scale Scheduling Problems, PhD thesis, University of London (2000). 31. S.T. Rodgers and K.E Jensen, J. Appl. Phys., 83 (1998) 524. 32. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. 33. K. Laso and H.C. Ottinger, J. Non-Newtonian Fluid Mech., 47 (1993) 1. 34. R. Keunings, in Proc. 13th Intl. Congress on Rheology, D.M. Binding et al. (Eds), British Society of Rheology, Glasgow, 1 (2000) 7. 35. J. Stefanovi6 and C.C. Pantelides, in Proc. 5th Intl. Conf. on Foundations of ComputerAided Process Design, M.E Malone and J.A. Trainham (eds.), CACHE Publications (2000) 236. 36. J. Stefanovi6, B. Fouchet and C.C. Pantelides, in Proc. FOMMS 2000: Foundations of Molecular Modeling and Simulation, P. Cummings (ed.), Keystone, Colorado, July 2000. 37. J. Stefanovi6 and C.C. Pantelides, "On the Mathematics of Molecular Dynamics", Parts I, II, and III, to appear in Molecular Simulation, (2001). 38. C.A. Floudas, J. Process Control, 10 (2000) 125. 39. P.M. Pardalos, H.E. Romeijn and H. Tuy, J. Comput. Appl. Math., 124 (2000) 209. 40. C.C. Pantelides, M.P. Avraam and N. Shah, in Scientific Computing in Chemical Engineering, E Keil, W. Mackens and H. Voss (eds.), Springer Verlag (1999) 62. 41. B.L. Braunschweig, C.C. Pantelides, H.I. Britt and S. Sama, in Proc. 5th Intl. Conf. on Foundations of Computer-Aided Process Design, M.E Malone and J.A. Trainham (eds.), CACHE Publications (2000) 220.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
USE OF PREDICTION AND MODELLING OF PROCESS OPTIONS
27
IN EARLY
EVALUATION
JL Cordiner Technology and Projects, Syngenta (Formally Zeneca Agrochemicals), Leeds Road, Huddersfield, England HD2 1FF Often in the chemical industry modelling tools are used to design unit items for processes that have already been fixed. Predictive techniques and modelling tools can be used at route and process selection stages to facilitate rapid evaluation of process options. The tools can be used to maximise the return on experimental time required in the development of alternative chemical routes and processes. In addition making the tools available to and useable by both chemical engineers and chemists gives a common language, facilitating joint working and enabling them to bring their different insights to select and design processes. This work has proven to be very successful in Syngenta where there are many examples where alternative processing options have been considered and evaluated. 1. I N T R O D U C T I O N Fine Chemicals Manufacturing is increasingly looking at reducing the time to market, this means that decisions about the process are pushed further and further back the decision train. These decisions are then required when less and less of the apparently required information is available. Conventional wisdom needs to be tested to consider what information is really needed and what level and quality of decision is required at each stage. In some cases, for example pharmaceuticals, the process route needs to be decided very early for registration reasons. The choice of the route can have large implications on the costs of production and capital requirement. It is then advantageous to have methods to challenge the normal route selection and development processes. This paper considers how early evaluation tools have been used and developed to address these issues within Syngenta. In addition the paper demonstrates benefits of wide spread use and the format of the tools best suited for the wide range of potential users working on development of processes.
2. THE D E V E L O P M E N T PROCESS Delivering a plant with an agrochemical ready for market involves a complex process as described in figure 1 which is a schematic of the overall development process from a research route.
28
Research route
"•
\
Activity VPC / margin Capital
/
/ /
ROUTE SELECTION
/
THE KEY DECISION POINT
FF&P=andformulation, fill pack
I Outcosts hnef&/sI
//
Selection criteria
SHEimpact ~ . . Market requirements ~ I (quality; tox)
Generate route options
C eve,oe, Process
"~
/
[~ FF&P development
Small manufacture ~
Product
[ f~176 I / y / J
sPfeomCr i~~ki! busines: targets
cDetegs~imf/"St~e /
( I .
J
Ongoing F deve,o en
Decision to invest
Field trials" registration
Figure 1. Schematic of the development process for an agrochemical product (after Carpenter
[1,21) One can see that major impact of early evaluation tools are at the key stage of route selection, ensuring the best overall route is chosen taking into account issues such as manufacturing safety, health and environment, meeting business requirement for initial and (and possibly longer term) business requirements. This is discussed further by Carpenter [ 1,2]. 3. G E N E R A T I N G A N D R E V I E W I N G A L T E R N A T I V E R O U T E S Clearly the synthetic routes from Research are usually not practical for a manufacturing setting. The chemist and engineer need to work together to consider how all the routes for consideration will be operated at the manufacturing scale desired by the business. At this stage it is vital the Early Evaluation Tools are able to aid this process in generating processes that can be radically different from conventional wisdom. Each chemical route can be operated at a manufacturing scale in a number of different ways and these need considered in any route evaluation. In addition the Early Evaluation tools are required to enable comparison of routes and processes to enable the most practical options to be chosen. Clearly the level of information on each route will be sparse at this stage and therefore the tools must allow quality decision to be taken on the limited data. It is therefore important to remember that comparison requires the data to be consistent but not necessarily accurate at this stage. As it is important to consider the whole supply chain in route selection one should use the tools alongside experience from different professionals rather than expecting the tools to do the whole job. A trap one can fall into is trying to develop a tool to perform a large complex remit. This becomes unworkable particularly by the non-frequent or non expert user. A tool with a smaller remit could perform a useful task and considerably speed up the total process and enable novel ideas.
29 4. C U S T O M E R S / U S E R S O F E A R L Y E V A L U A T I O N T O O L S Very many synthetic routes will be considered at the early stages and reduced down to a small number for further investigation until a single route is chosen and moves into development (process, formulation and packing). Plant (3) has reviewed the number of active molecules on average coming from discovery to reach the end of development, in a typical agrochemical business (table 1). Phase
Activity
Time Number per year ..... (years) .required for 1.produc ~Phase 1-Invention ...........investigates interesting activity 5 - 1 0 12 .... Phase 2 - Evaluation Clarifies the preferred candidate 2 4 Phase 3 - Development Answers critical questions 4 1 Table 1" A typical agrochemical development chain (after Plant [4]) '" This means that often a diverse and large number of people are involved in many different processes. Most of these people are non-specialists and in-frequent users of CAPE tools. It is imperative therefore that these people have access to the tools to enable them to benefit from early evaluation. This brings the benefit of the expertise of the specialist to the largest numbers of processes. Such people need training to use the early evaluation tools to help them make informed decisions for route selection or design of experiments etc. The in-house software SMSWIN was developed to provide early evaluation tools for all chemists and engineers working in process and product development. The large number of processes, speed required and typically batch operation means that large simulations of a process are often not appropriate and certainly not at the early stages where route selection takes place. Much consideration was given to the format and training the tool would have as described below. The in-house training delivered along with the launch of SMSWIN included fundamental thermodynamics, phase equilibria, solvent selection, short cut separation selection, crystallisation etc. This training used case studies to show the use and power of the tools. The training was also developed into Intranet Expert Systems. This allows rapid access for the personnel, where they can get assistance in following the evaluation process, selecting a separating method or making use of the tools. Training has been given to chemists and engineers together from all facets of the product process. This enables common language and mutual understanding of the issues that can be considered using the tools. These training sessions also give feedback on the content using the tools to facilitate improvements and usability. The tools and training allow the chemists and engineers to work more closely together to look at the routes. This helps focus their efforts and time on the key issues by rapidly evaluating each, looking for potential issues. The teams are them able to plan their experiments, make more detailed models and work programmes to concentrate on these issues. This leads to more cocurrent development, again shortening the development time.
30 It can be beneficial to have relatively trivial calculations (to a person experienced in thermodynamics or CAPE) available in very visual tools or tables. This can speed the work of the developers and allows other disciplines to carry out the calculations or work alongside the chemical engineer. Examples of this can be seen later in the paper. Giving the user the choice of discrete calculations rather than just an expert system that is set up for the whole problem aids understanding of the process options. In addition the discrete calculations can be used for process improvements of parts of the process either at route selection or at any time of the process life.
5. CHALLENGES FOR THE EARLY EVALUATION TOOLS. The traditional simulation tools are extremely comprehensive. The tools are very daunting and difficult for the in-frequent or non-expert user who would have to go through many forms and make a number of selections. The tools therefore need to be user friendly, robust and easy to use. In particular the tools need to be as intuitive as possible for the infrequent user, minimising the number of forms to be filled in or clicks required. This can be seen in setting the most commonly used information at very easy and fast reach as shown in figure 2.
Figure 2 Property selection From SMSWIN Where ever possible an expert system to select items or calculation methods needs to be employed in such a way that it is easy for the non specialist, whilst still giving enough information and guidance to give some training/understanding whilst the tool is being used. For example the physical property method for early evaluation and setting up of this method needs to be made very easy. This can be demonstrated by pre-setting the groups for UNIFAC (and it's
31 variants) for as many molecules and frequently used building blocks for molecules as possible as is done typically for UNIQUAC for molecules. Many of the developers will need help in selecting a property method, where an expert system would be beneficial. Such a tool has been developed by Gani and O'Connell [4]. A decision tree (as shown in figure 3), written for typically encountered processes, is also a useful tool for the non expert. This allows rapid selection and points to when further advice from the expert system is required.
I Method Zeneca Selection. Property
~
N
o
~
~ yes
~
no
AC [(.Asp.U. . N. I FSMSWIN)
yes
4,
(need some
e x p e r i m e n t a l
Use EOS Seek Advice I f only moderate Use N RTL-HOC o r Wllson-HOC pNRTI-SRK, ....... or Wilson-SRK (S . . . I. . . . for 2 liquid . phases Wilson/ NRTL)
yes
[
o
1
I
~
I yes
data)
U1 R e g r e s s
data
using
iAnt~oPa~n ~ s a s w | N
. . . . . . T.
'
.
.
.
.
I I
I
.
T
' '
Use Wllson-HOC for Carboxyllc
I I
.
.
.
.
I w,,~~
]
I
acids
I
N RTL l
~1"
*
I
'
yes
Use I ................HF/H20 ! Carboxylic ENRTL-HF
acids for
Property Method Selectlon-"x~ ~ ee Notes onJoanPhysical Cordiner (PSG) Hudd ext 6O84 J or
Contact
Figure 3 Decision Tree The tools should also be as visual as possible. Many visualisation tools have been provided in SMSWIN (in-house programme) to help developers rapidly access route options. For example, residue maps, eutectic diagrams, solubility plots, and scatter diagrams for evaluating crystallisation's feasibility - some of these are discussed later. Having diagrammatic ways of presenting the same data can aid understanding of a process. For example, looking at ternary VLE data in a triangular diagram shows two liquid phase regions well. A pseudo 2 component diagram shows what and how much entrainer or solvent breaks or sufficiently moves azeotropes. A residue diagram shows feasible separation regions.
6. FACETS OF ROUTES NEEDING CONSIDERATION There are many issues that need to be considered from materials handling, where solids as well as fluids need to be handled with issues relating to toxicity, environmental impact etc. through to the issues of packing and formulation. A variety of tools, for example, databases and
32 expert systems, aid the professional in decision points by having the commonly used information easily at hand. It is useful to have a tool that allows a rapid means of reviewing each research process assessing the manufacturing process options. The chemist or chemical engineer can carry out a rapid assessment using diagrams and tables (some of which are shown later in the paper) seeing the potential issues easily and quickly. The reaction chemistry can have a large impact on the choice of route depending on the yield and selectivity obtained. Typically yield is considered most important given that materials are often not recycled due to complexity of separation. Where recycling is possible the best route for this needs to be considered and yield is therefore not such an issue. Often the reaction chemistry can be highly altered both in products and in rate via the choice of solvent. This is graphically demonstrated in the aromatic nucleophilic substitution of the azide ion with 4fluoronitrobenzene (Cox (5,6)), where the rate changes by 6 orders of magnitude depending upon the solvent as shown in figure 5. Clearly choice of solvent is very important. F
N3
ks
+
N3
+ NO 2
F
NO 2
.
.
.
.
.
.
.
.
.
.
.
.
Solvent H20 MeOH M e 2 S O 1 1.6 113"i0 ~ ks]kH2o Figure 5: Effect of solvent on rate constant
kH20 = 4.0* 10 -8 M -is -1 ,,,,
HCONMe2 413"i04 '
. . . . .
(Me2N)3PO 210-*-i06 ........................
The solvent chosen potentially can be used for the separations required or indeed may need to be separated. In most fine chemical processes there tends to be more than one stage of chemistry required and often different solvent may be chosen for each stage with initial manufactures requiring purification and separation at each stage. This can mean a large number of different solvents are used throughout the process which involves much separation leading to yield losses. Selecting the solvents for each reaction stage carefully to minimise solvent swaps can be very cost effective and can also increased yield. Clearly any tool which aids solvent selection can radically reduce the capital and operating costs of any route. The tools can lower the experimentation time required by reducing the number of solvents to be tested in the laboratory. One can look at using the tool as doing the experiments quicker. Reducing the experimentation time and hence aid faster development enables more radical processes to be tried. The techniques can then also be used to look at selecting a stage wide or process wide solvent rather than having a number of solvents and solvent swaps through the process. The tools used, to aid solvent selection for reaction, range from using taxonomies of solvent properties and effects, the use of principle component analysis to develop further taxonomies for example by Chastrette [7] as shown in figure 5, through to using optimisation techniques to evaluate potential solvents for a given set of constraints.
33
Figure 2. Solvent Taxonomy These classifications solvents are however restricted to known or commonly used solvents. This needs to be challenged with increasing environmental issues and legislation driving use away from the problematic traditional solvents as discussed by Odele and Macchietto [8], Pistikopoulos and Stefanis [9], Hostrup et al. [10] and Harper [ 11 ]). Therefore, tools to generate novel solvent options are particularly valuable. The tools being developed take a number of formats all making use of optimisation MINLP or Global Optimisation. These allow the generation of molecules which fit selection criteria for the solvent. The generation of the molecules allows novel solvents to be considered. The criteria to set the objective function and constraints being based on a number of requirements for a successful solvent. The tool developed by Gani et al at CAPEC Consortium of the Danish Technical University (ProCamd) makes use of UNIFAC for solvent property prediction have been used successfully and integrated with in-house software tools. In order to test out these methods a process wide solvent selection for the nitric acid oxidation of anthracene to anthraquinone was carried out using solvent selection tool Pro-Cared, an inhouse programme SMSWIN and Global Optimisation in collaboration with Dr C Adjiman and P Bavishi [12] of Imperial College. The techniques looked at solvent effect on reaction, solubilizing the starting material, solubility of nitric acid, recovery of product, separation scheme for recovery of solvent, recovery of nitric acid, boiling point and melting point, vapour pressure, price, safety and toxicity factors etc.
34 Some of the techniques can be used independently in SMSWIN being discrete calculations and visualisations as shown later, however the optimisation selection tools 'Pro-CAMD' and Global optimisation were set covering as wide a range of the selection criteria as possible. The solubility of the starting material and the product can be visualised with SMSWIN as shown in the example in figure 6.
Figure. 6. Solid Solubility This allows selection of solvent with highest solubility of the starting material, it can also be used to select a solvent with the highest recovery of product. The scatter graph shows how the solubility of solute varies with solvents that SMSWIN has selected. The Productivity Index is the product of solubility at the higher temperature and % Recovery hence the best solvents for recovery of product are in the top right hand corner, the names appearing as required. The crystallisation of the product can then be considered using a eutectic diagram as shown in figure7. SMSWIN allows the selection of crystallisation temperature and calculates the recovery of Product into the solid phase.
35
Figure 7: Crystallisation calculations. Alternatively if a drown out crystallisation is used, the solvent swap or drown out can be visualised in SMSWIN very quickly allowing a graph of composition versus recovery at different temperatures or a table allowing for multiple easy recalculations as shown in figure 8. In selecting the solvent recovery system, due to the presence of water, solvents were required to be immiscible in water as one alternative. A means of looking at miscibility ranges over a range of solvents is therefore useful or a calculated table which allows sorting of the various solvents data to find the immiscibility range required. This could also be seen by use of Ternary diagrams for the non experienced personnel. Where extractive distillation is considered pseudo binary or ternary data can also be used to look at minimum entrainer usage possible (Knapp and Doherty [ 13]). Further separations can be evaluated by use of residue and distillation curve diagrams shown in figures 9 and 10. as proposed by Doherty and Perkins [14,15,16,17], Dongen and Doherty [18] and Stichlmair [19,20]. Where the early evaluation tool allows rapid access to such diagrams, a number of distillation can be considered for each process and compared, looking for potentially difficult or prohibitive separations.
36
_Figure 8 Solvent Swapping and Drownout Calculations._
Figure 9 Example of Residue Curve Map
37
Figure 10. Example of Distillate Curve Map
Figure 11. Example of Short cut method taken from training programme.
38 This can be further developed by rapid, or short cut tools ,as shown in figure 11, to allow the developer to look at the scale of equipment that might be required, although trivial calculations, the rapid access can be very useful for the developer whether they be a chemist or an engineer.
7. FUTURE CHALLENGES AND REQUIREMENTS FOR EARLY EVALUATION TOOLS Many of the typical processes contain very complex molecules of which there is little information. These complex molecules have many functional groups and be in the presence of similar molecules which are produced as by products or as pre or post stage products. Indeed many final molecules are required in a particular enantiomer. Some typical molecules are shown in figure 12 (from Carpenter [2]). The selection of the separation task therefore becomes complicated. It is important therefore to have good predictive tools for the important physical properties and the ability to improve these predictions with as much known information as possible. This sort of tool has been developed by the CAPEC Group of the Technical University of Denmark. The tools however are limited in their range of applicability. The large complex molecules which are common in the fine chemicals industry cannot be reliably modelled using the existing group contribution techniques. There are however ways forward by using as much information as available from the molecule and similar molecules to give some guidance. This is where using the tools along side experience and experiment can work very well. BF
01"1_m"~- ~Me
c~
0
0
0
+ 0
N\\
>=o 0
NH
/
A synthetic pyrethroid insecticide
A substituted diphenyl ether used as an herbicide
o
A green azo dyestuff for dying polyester
Figure 12. Typical Molecules. A joint project with the CAPEC group looked at a Gap Analysis of the physical properties required (Cordiner and Nielsen [21] and what Gaps existed in the available technology. This showed that predictive ability in complex multi-functional molecules, along with electrolytes, salt forms of complex molecules, surfactants and surface properties were key areas. In many of
39 these areas it is common to use molecular modelling and dynamics to obtain properties of interest. The physical property "gaps" were prioritised by demand, difficulty to resolve and ease of measurement. The simple to measure properties are best measured where possible. It is common in many processes to have by-products and intermediates that are very similar in structure to the product, indeed it is also common to have enantiomers where one is the active compound and all other enantiomers inactive. This makes the separation selection and also the prediction of the properties more difficult. Measurement of the required physical properties can also be problematic due to the difficulty of producing a pure sample of any by-product. There is therefore a substantial gap in the currently available property prediction methods to be filled. The optimisation techniques mentioned for solvent selection need to be further developed to take account of wider solvent issues and could also be widened to route selection including formulation of active products i.e. surfactant selection etc. In addition visualisation tools along with optimisation that allow selection of separation scheme taking into account efficiency of separation are being developed by the CAPEC group (Bek-Pedersen et al.[22])and others and will prove very useful. Solvent selection tools will also be greatly improved when reaction effects are better predicted. 8. CONCLUSION Early Evaluation tools are proving very useful in improving route selection practise, bringing chemical engineers and chemist together and facilitating co-current development that is focussed much earlier reducing the necessary experimentation and development time-scales. The paper considered the benefits of making the tools available to the numerous and wide range of people involved in the many development processes that are required to facilitate the manufacture of a new successful product.
REFERENCES 1. Carpenter, K.J., 2000, Chemical engineering in Product Development- the application of engineering science, Entropic 223,4 2. Carpenter ,K.J. 16th International Symposium on Reaction Engineering (ISCRE 16), to be published in Chem.Eng.Sci., 2001 3. Plant. P.,July 1999, Internal Zeneca Agrochemicals, The development chain. 4. Gani, R. and O'Connell, J. P., Computers Chem Engng, 13(4/5), pp397-404, 1989. 5. Cox, B. G., 1994, Modern liquid phase kinetics, Oxford Chemistry Primer Series 21, Oxford University Press 6. Cox B.G. and Parker, A.J,1973, J. Am. Chem. Soc., 95,408 7. Chastrette JACS Vo1107 No 1 1-11 1985 8. Odele O. and Macchietto S. Fluid Phase Equilibria. Vol 82, pp 57-54, 1993 9. E.N. Pistikopoulos ,S. K. Stefanis Computers Chem. Engng 1998 Vol 22, pp 717-733, 10. Hostrup, M., Harper, P.M., Gani, R., Comput Chem Eng, 23, pp 1394-1405, 1999. 11. Harper, P.M., 'A Multi-Phsae, Multi-Level Framework for Computer Aided Molecular Design, Ph.D.-thesis, 2000 CAPEC Danish Technical University 12. Bavishi P, MEng Thesis-2000, Imperial College supervised by Adjiman C. 13. Knapp and Doherty AIChE Journal Vol.40 No2 p243-268 1994
40 14. 15. 16. 17. 18. 19. 20. 21. 22.
Doherty M. and Perkins J. Chemical Engineering Science1978 Vol.33 281-301 Doherty M. and Perkins J. Chemical Engineering Science1978 Vol.33 569-578 Doherty M. and Perkins J. Chemical Engineering Science 1979 Vol.34 1401-1414 Doherty M. and Perkins J. Chemical Engineering Science 1982 Vol.37 381-392 Dongen and Doherty M. Chemical Engineering Science 1984 Vol.39 No.5,883-892 Stichlmair Chem.Eng.Prog. 1989 63-69 Stichlmair AIChE Journal 1992 Vol.38 No.10, 1523-1535 Cordiner JL, Nielsen TL, "Gap analysis" Syngenta Internal Report 1999 Bek-Pedersen, E., Gani, R., Levaux, O., Comput Chem Eng. 24 (2-7) pp253-259, 2000
ACKNOWLEDGEMENTS Permission to publish from Syngenta is gratefully acknowledged. Thanks to a great many friends and colleagues for advice and information, especially: Dr Keith Carpenter and Dr. Alan Hall, Dr Will Wood of Syngenta Technology and Projects and James Morrison Consultant.
European Symposiumon ComputerAided ProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
41
A Muitiscale-Multifaceted Approach to Process Synthesis and Development K a M . Ng Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong This paper presents a strategy for the synthesis and development of chemical manufacturing processes. By viewing an undertaking from various length scales and its different facets, this approach is expected to lead to an improved process with reduced time and effort. Some of the issues, methods and tools for this approach are illustrated with processes involving crystallization and solids processing steps. 1. I N T R O D U C T I O N The global chemical processing industries (CPI) are undergoing profound changes. With the increasingly free flow of capital, technology, and human resources, the CPI in developing countries are rapidly expanding. For example, Asian countries with a large internal/regional market and low labor costs are witnessing an annual growth rate of over 10%. While a large fraction of production is destined for import substitution for countries such as China, other countries such as Singapore and Thailand are poised to become significant net exporters. In response to the increased competition, the companies in the developed countries have been locating some of their chemical plants overseas as well as repositioning themselves through divestitures, mergers and alliances. They are also intensifying their effort to innovate and to refocus on high-value-added (HVA) chemicals in search of a higher return. The allure is clear when one compares the 8% profit margin in a typical chemical firm to the 20% figure of a pharmaceutical company in the U.S. [ 1]. All of this has led to new challenges to the CPI (Table 1). One is the accelerated pace of process design and development [2]. For commodity chemicals, it used to take eight years Table 1. Challenges to Process Synthesis and Development Accelerated pace of process design and development Increasing product complexity Increasing process complexity Cost reduction from a product life viewpoint Environmental considerations Technology transfer among people, laboratories and sites Expanding project scope to entire business chain
42 or so to build a large-scale grassroots plant starting from basic chemistry [3]. Now the desirable time is 3 to 4 years. For pharmaceuticals, rapid process development is essential in order to produce enough materials for clinical trials and to prolong effective patent lives. Due to the shift to HVA chemicals, product specifications are becoming more complex. Rather than just purity, particle size distribution, particle shape, morphology and polymorphism are specified product attributes. This is accompanied by an increase in process complexity, involving multiple processing steps and complex chemistry. The economic pressure has also greatly intensified. Whether the objective is to reduce cost or improve profit, it is critical that process development be viewed over the entire product life. For example, the regeneration of monomers from PET is becoming increasingly common for environmental reasons. An accelerated pace of development demands an even closer collaboration among traditionally different functional units such as chemistry, chemical engineering and business. If one thinks it is challenging to coordinate the efforts of a chemist and a chemical engineer in close proximity, it is now necessary to coordinate efforts of people with different cultures in laboratories all over the world. All these technology transfer issues need to be resolved effectively. To take advantage of any possible synergism and leverage, the project scope is also expanding. It is desirable for a company to capture a coherent set of technologies and supply chain for the entire enterprise or product line in order to gain a competitive advantage. 2. M U L T I S C A L E - M U L T I F A C E T E D APPROACH These multiple demands necessitate a multifaceted approach to process synthesis and development (Table 2). To demonstrate how some of these facets come together as a coherent whole, we consider a specific area-the synthesis and development of processes where crystallization and solids processing play an important role. This is particularly relevant to HVA chemical and pharmaceutical manufacturing processes, many of which have solids as an intermediate or the final product.
43 Table 2. A Multiscale-Multifaceted Approach Facet
Conventional Approach
New Strategies
Length scale
Individual scale one at a time
Multiscale
Scope
Single plant
Enterprise
Financial goal
Cost reduction / Improved profit
Product life economic analysis
Knowledge base
Scattered
IT-based integration
Process vs. Product
Process engineering
Product-based processing
Technical focus
Science and Engineering
Science/Engineering aided by design rules and workfiow
Experiments and Design
Separated
Integrated
Chemists / Chemical Engineers
Limited collaboration
An integrated team
Scaleup
Rule-based
Fundamental models
Timeline
Sequential
Concurrent
Procedure
Ad-hoc
Systematic, hierarchical
The foremost element of this approach is a multiscale perspective. The time and length scales for a solids plant is shown in Figure 1. Similar concepts have been advanced by Villermaux [4], Sapre and Katzer [5], Lerou and Ng [6], and Grossmann and Westerberg [7].
44 10-16
10-10
_//J//I
10-6
10 -4
10-2
10 0
10 2
I
,
,
, . . . . . . . . . . .
108m
10 4
,
r //
109S
4
iEnterprise Plant i.............,..................z ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reaction Chemistry
1
:................
!
............... r........... :ji Crystallizer
J
--
10 4
--
10 2
--
100
_
10-2
--
10-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fluid Dynamics and Transport
I ii
1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~ . . . . . . .
Particle Nucleation i and Growth ........ ~................................... ~........... i'
i
i
i i i i
/ / /
-
10-14
Molecular / Electronic e i
10-16
Figure 1. The length and time scales covered in the multiscale-multifaceted approach for a crystallization-based plant. The boundaries for each region are not meant to be exact. Contiguous regions signify potential interactions and synergism between them. The more the regions overlap, the more the scales interact. For example, there is a significant overlap between the enterprise and the plant. Indeed, a chemical plant is never designed in isolation. It is absolutely critical to consider the entire global enterprise, that is, the product line both upstream and downstream of the business chain. Make-buy decisions, toll manufacturing, and alliances can have an enormous impact on profitability. The interface between the plant and crystallizer (equipment scale) is the traditional area of process synthesis of a single plant [8]. The effect of reaction, and transport on equipment design has been a rich and fruitful research domain. Molecular considerations are critical for reaction chemistry, etc. but there is limited benefit in linking them to enterprise level issues. As stressed in Table 2, the scope of work in process systems engineering (PSE) has expanded tremendously. Driven by competitive pressures, in addition to our traditional tasks in process research and pilot plants (Figure 2), we have to redouble our efforts on technology transfer, and manufacturing and operations [9, 10]. Building on its strengths in manufacturing, the Huntsman Corp. has risen to the fourth place in terms of sales among all U.S. chemical producers in 1999 [ 11 ].
45
Process Research & Development
Pilot Plants
Technology Transfer
~] Manufacturing & Operations
9 Processchemists, Process synthesis engineers,
9 Processengineers
9 Processengineers, technicians and 9 Testingand validation operators
9 Processengineers, technicians and operators
Financial analysts 9 Laboratory experiments
9 Redesignand adaptation
9 Adaptationat manufacturing sites
9 Benchmarkingand best practices
9 Trainingand documentation
9 Process improvements
9 Full-scalevalidation Figure 2. Synthesis and development of manufacturing processes. In considering the business issues, it is imperative to conduct product life economic analysis to obtain maximum return. It is also important to tie together product and process design [12]. Product attributes are defined based on customer demands. The materials with the proper properties are identified and the process to transform the materials into the final desired product is then synthesized. As we proceed to the equipment scale and below, we begin to take advantage of the relevant basic sciences. However, there is a limit as to where basic sciences can take us in process development. The use of design rules can significantly reduce the search space, thus shortening the time required for process synthesis and optimization. It is equally important to closely integrate process synthesis and experiments. One has to identify the critical experiments to perform, and the regions of parameter and composition space where the data can have the highest impact [13]. For example, while reaction data focusing on high yield are useful, selectivity data at a low conversion can be more important because of the recycle of unconverted reactants. In this regard, the integration of chemistry and chemical engineering is essential to create a high quality process in the shortest possible time. This sentiment was echoed in a discussion of the experiences, barriers, benefits and the path forward in building an integrated team of chemists and chemical engineers in process development [ 14]. Because of time pressure or technological limitations, scaleup tests are performed by necessity. While some scaleup rules are based on experimental observations or simply experience, we need to fathom the fundamental reasoning behind the scaleup rules whenever possible. For example, it has been shown that the feed addition time should be considered in scaling up a semibatch liquid phase reactor in the micromixing regime although this is not commonly practiced. Also, the use of constant agitator tip speed is not a good scaleup rule [ 15]. For a typical process development project, the performance of a process in terms of economics, safety, operability, etc. generally increases while the uncertainty of the expected performance decreases with the total amount of effort (Figure 3). However, one has to recognize that there is a technological limit beyond which our additional effort has a rapidly diminishing return. For example, in the design of a new large-scale multiphase reactor with complex reactions, there is always a degree of uncertainty until the commercial reactor is up and running for a period of time. To minimize time, it is essential that development tasks be performed concurrently instead of sequentially. Obviously, there is an upper limit to effective
46 effort/time (i.e., concurrent effort) in that, depending on the nature of the problem and the available resources, some tasks have to be performed sequentially.
Total Effort = Time x Concurrent Effort i.e.
Effective Effort T-~me )
Figure 3. The dependence of process performance and uncertainty on the total amount of effort invested in the process development project. This is exactly why systematic, hierarchical design procedures can play an important role in process synthesis and development. Before delving into the details of a development project, a preliminary plan can be drawn (Table 3):
47
Table 3. Preliminary Planning in Process Development 1. Identify and prioritize the project objectives and decision points. 2. Identify the tasks to meet those objectives. 3. Identify the technical expertise, time and resources required for achieving those objectives. 4. Identify the relevant in-house functional units and the possibility of outsourcing R&D. 5. Organize the workflow among the functional units. 6. Identify the achievables and non-achievables based on the available time, technological limits, and both human and monetary resources. 7. Identify the time horizon for the entire project (Be Realistic). Such design procedures can help us anticipate the key decision points and the data and tools to achieve the project objectives. It also helps us to devise the workflow to integrate the contributions from disparate functional units. Admittedly, some of the stumbling blocks are hard to foresee even for experts in systematic design, particularly for completely new processes. Yet this preliminary exercise, if properly executed, is expected to provide a roadmap to expedite the entire project. 3. SYNTHESIS AND DEVELOPMENT OF CRYSTALLIZATION AND SOLIDS PROCESSES To be more concrete, let us examine a few selected issues of the multiscale-multifaceted approach in the context of crystallization and solids processing [16]. Enterprise and molecular scale issues will not be considered. Technical Focus
Table 4 shows a taxonomy of the issues in the design of solids processes as viewed from four length scales-plant, unit operations (or equipment), continuum and particle. Tools from business, management and basic sciences are required for a full consideration of a process development project.
48 Table 4. Subjects Examined at each Length Scale, and the Typical Problems and Tools Scale Plant
Unit Operation
I
Subjects Process synthesis Process simulation Process optimization Process control
Typical Problems Multicomponent crystallization Solid-liquid separation downstream of crystallizers Granulation / tableting operations
Equipment performance Equipment sizing Equipment costing
Mixing-Demixing Blender, hydrocyclone, filters Size Change Processes Crystallization Comminution, granulation
Heat & mass transfer Kinetics Discretized population equations CFD
SLE phase diagrams Flow in filter cakes Fluidization Pneumatic conveying
Thermodynamics Continuum mechanics
Solvent effect on crystal shape Nucleation and growth Particle breakage Interparticle forces
Solid-state physics Quantum mechanics Solid mechanics Statistical mechanics Colloid and interface science
Continuum SLE, Solubility Flow of powders and slurries Particle
Particle Attributes Composition, PSD, density, strength, shape, chirality, polymorphism
i
Tools Hierarchical design Optimization tools Simulation tools Process economics
At the plant level, the subjects to be considered are those of the entire chemical plant. Hierarchical design procedures and optimization techniques are some of the tools for addressing these problems. Process synthesis is impossible without the necessary knowledge about the units. At the unit operation scale, we are concerned with equipment selection, sizing, operations, control and innovations. There are two classes of solids processing equipment, depending on whether the particle size changes or not. In addition to the fundamentals of transport, population balance equations are essential for the problems at this scale. Computational fluid dynamics (CFD) tools are now routinely used in process development. Description of the continuum behavior necessitates both thermodynamic considerations such as solid-liquid equilibrium (SLE) and dynamic considerations such as flow of powders and slurries. At the particle scale, we focus on how to obtain the various desirable solids attributes such as size, shape, chirality, polymorphism and density for which tools such as solid-state physics, and solid mechanics are needed. Recently, separation of enantiomers by means of crystallization [17] and chromatography [18], and prediction of crystal shape [ 19] have received much attention.
49
Preliminary Planning Workflow is devised based on the problems and tools thus identified. Specifically, let us consider a process development project where an active pharmaceutical ingredient (API) is recovered from a multicomponent mixture by crystallization. After filtration, washing, dewatering and drying, we obtain the cut with the acceptable particle size distribution (PSD) through bulk solids processing. Then, the API is blended with various excipients before tableting and coating. Figure 4 shows the overall workflow for this project. The central column shows the tasks or problems to be considered. The rectangles on the left indicate the data required for the corresponding problems. Experiments are necessary because the state of the art simply does not allow reliable predictions of these quantities [20]. While it is possible to predict SLE for ideal simple eutectic systems, it is difficult for complex molecules. And it is impossible for systems with compound formation. Washing efficiency is simple to measure but it can have considerable impact on cost because the wash solvent is recycled through the distillation system for purification. Similarly, breakage depends not only on the material properties, but also on the processing history of the particles. Obviously, some of these measurements can be and should be carried out concurrently. SLE data, and nucleation and growth kinetics were often not measured in a project because of time constraints. However, in the past several years, a number of industrial laboratories have been set up to measure the relevant quantities on a routine basis. A similar situation existed for reaction kinetics although systematization of those measurements are much farther along. In comparison, except for a few companies, measurements for bulk solids processing are less systematic, with the work relegated to specialists in individual unit operations, who may overlook some of the systems problems. The rounded rectangles on the right indicate the techniques, design procedures and basic sciences required for the corresponding tasks. Let us examine some of the methods and procedures although, within the scope of this article, it is not possible to get into the myriad workflow details required for coordinating all the functional units.
50
SLE Experimental Data
\ ,~/ Crystallization ~ ] Separation "~ Scheme /
Filter Cake Resistance; Washing Efficiency
~/,-\,, DownstreamCrystaliZerproces~sing
(Visualizationof High-D'~ PhaseDiagrams; | Movementsin | ~.. Composition Space )
Solid-Liquidsynthesis ProcedurePr~ 1
Nucleation and Growth Kinetics
~/
Crystallizer "~ Design and ~ "~ Operating Policy ,/
(Heat & Mass Transfer;'~ Crystallizer Types & ~ ~. OperationModes /
Specific Rate of Breakage; Work Index
~/ "~
BulkSolids ~_~Synthesis Procedure to~ Processing / " ~MeetPSD Requiremen~
r
Hamaker Constant, Dielectric Constant, Unconfined Yield Stress
(\ Operating Issues ~ - ~ ~ in Solids Processing
V~ and Effects 1 Acting on Particles
Figure 4. Overall workflow for the synthesis and development of a process with crystallization and solids processing steps. Systematic Design Procedures Plant Scale
There exist a wide variety of crystallization techniques, ranging from fractional crystallization [21-25], extractive crystallization [26, 27], reactive crystallization [28, 29], and drowning-out crystallization [30]. It can be shown that these crystallization processes can be represented by four basic crystallization-relatedmovements in composition space, namely, cooling/heating, stream combination/split, solvent addition/removal and MSA (mass separating agent) addition/removal [31]. Depending on the relevant features on the phase diagram, this synthesis procedure generates the feasible schemes without being restricted to a specific separation technique. It is applicable to multicomponent mixtures by using cuts and projections to visualize high-dimensional systems (i.e., 4 or higher) [32, 33]. This approach is also applicable to other unit operations such as distillation, and membrane processes [34]. However, the crystallization system does not exist in isolation. It is closely linked to the downstream processing system in which dry crystals are produced. A systematic procedure is available for the synthesis of flowsheet alternatives for treating the crystals as well as the crystallization, wash, and recrystallization solvents [35]. The PSD of the dried crystals can be one of the specified product requirements. For example, in a pharmaceutical tableting
51 process, the PSD of the API and that of the excipients have to be controlled for a number of reasons. The relevant synthesis procedure as well as the simulation method for the entire plant based on discretized population equations has been proposed [36-37].
Equipment Scale We have developed a hierarchy of solids processing models. The basic ones are heat and mass balances. The intermediate ones consider kinetics and mass transfer while the detailed ones account for changes in PSD. Particle size can change due to breakage, agglomeration, nucleation, growth and dissolution (Figure 5). Each mechanism has been modeled separately [38-41]. An equipment unit with any combination of these mechanisms can be modeled by selecting the corresponding mechanisms in the population balance equation, as shown below:
dNi-'dNi (L_~_i_)Agglomerafion+ dNi (L_~)Growth& (--dT-)Breakage +dNi
+dNi Qout (L'~)Dissolution+Qin V, miin +--(-, m i
Equipment Unit
Ts
I Mechanisms
Breakage Agglomeration ~.~ Nucleation and Growth "Dissolution
Figure 5. Modeling by functionality.
Particle Scale Solids plants are plagued by operational problems. To identify the potential problems and resolve them effectively, we have to consider both the continuum and particle scales. As Figure 6 indicates, the combination of material characteristics, particle attributes, equipment design and operations determines the bulk mechanical properties of the powder as well as the forces acting on the particles. Whether an operational problem such as particle breakage, adhesion, segregation, arching (as in a hopper) occurs or not depends on the causal effect and the effect that opposes it. For example, encrustation on equipment walls can be a problem if the adhesion force is greater than the forces that entrain the particle. A systematic procedure has been proposed to examine these problems [42]. 4. S O F T W A R E TOOLS To facilitate the execution of this approach for process synthesis and development, we have been developing a number of software tools such as those for the visualization of phase diagrams, for tracking PSD around a chemical plant, etc. While related, these codes are modular in nature. They are based on Visual Basic or Java, and can be linked to other codes (Figure 7). In addition to computations, each program includes design rules to identify decision points and shows how to make those decisions, parameter maps for typical model parameter values, and cause-effect tables for showing the likely consequence of an action.
52
Material Characteristics
~
es~ article Attribut
Hamakerconstant Dielectricconstant Young'smodulus
[ ] L
PSD 1 Shape 1 Composition/
~
gl quipment Desi
(' /
[ Geometry l [ Constituentparts 1 L Materialproperties/
Operating ~ 1 Conditions /
| Speedofmovingparts I 1 Temperature [ L Humidity )
Performance of Individual and Interconnected Units Figure 6. Interrelationships of factors affecting the performance of a solids processing unit (after Wibowo and Ng, 2001).
Visual Basic / Java User supplied codes in Fortran, C++
LL
9 Calculates and plots high-D phase diagrams. ~ Tracks PSI) around the plant. o Back-calculates model parameters.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Public codes such as Excel, GAMS
OLE
9 Things-to-consider tables. ~ Parameter maps. ~ Cause-effect tables. ~
I
I
J
"
Figure 7. Modular computer codes with built-in knowledge-base and graphical output for process synthesis and development. 5. CONCLUSIONS Process systems engineering is an evolving subject [43]. As the conventional part of chemical engineering as a discipline matures, PSE is destined to expand in both scope and
53 depth. This trend is captured in this multiscale-multifaceted approach for process synthesis and development. The enterprise is viewed scale by scale in a hierarchical manner. In conjunction with other facets of this approach, it is expected to help exploit more fully the synergism within the enterprise, thus leading to an improved process with reduced time, effort and money. To put this approach in practice, the perspective is merely a starting point. It serves as a framework to which methods, procedures, techniques, etc. need to be added so that the user can tackle a product-based process synthesis and development project with an efficient workflow. Some of the design procedures and methods, in conformity with this approach, for the synthesis of processes with crystallization and solids processing steps are discussed. Similar methods have been developed for reaction systems [44-49]. It should be emphasized that this approach does not replace the need for experts in various sub-fields yet. For example, segregation of solids is a complex subject and demands careful treatment if it is encountered in process development. Similarly, mixing and reaction, particularly those in multiphase systems, can still be a challenging problem. However, technology integration relentlessly marches forward, providing new computer tools for both traditional tasks and innovations. While not illustrated in this article, business issues, detailed workflow, technology transfer, and molecular considerations are an integral part of this approach. Contributions from researchers in these areas and industrial sectors other than solids processing are needed to realize the full potential of this approach. We anticipate an expanding role of PSE in an increasingly integrated global CPI in the coming years. 6. ACKNOWLEGMENT The financial support from the National Science Foundation (Grant No. CTS-9908667) is gratefully acknowledged. The author also likes to acknowledge the invaluable impact of the industrial collaborations on my thinking. Specifically, I would like to thank George Stephanopoulos, Yukikazu Natori, and Lionel O'Young of Mitsubishi, Jan Lerou formerly of DuPont, Prabir Basu of Pharmacia, and Alan Leviton of Rohm and Haas for their advice and guidance. REFERENCES
1. Arora, A., Landau, R., and Rosenberg, N., Chemicals and Long-Term Economic Growth: Insights from the Chemical Industry, Wiley, New York, NY, 1998. 2. Pisano, G. P., The Development Factory: Unlocking the Potential of process Innovation, Harvard Business School Press, Boston, MA, 1997. 3. Vogel, H., "Process Development," Ullman's Ency. Ind. Chem., 5th ed. Vol. B4, B. Elvers, S. Hawkins, and G. Schulz (eds.), VCH Verlagsgesselschaft, Weinheim, p. 438, 1992. 4. Villermaux, J., Trans. IChemE, Part A, 73, 105 (1995). 5. Sapre, A. V., and Katzer, J. R., Ind. Eng. Chem. Res., 34, 2202 (1995). 6. Lerou, J. J., and Ng, K. M., Chem. Eng. Sci., 51, 1595 (1996). 7. Grossmann, I. E., and Westerberg, A. W., AIChE J., 46, 1700 (2000). 8. Douglas, J. M., Conceptual Design of Chemical Processes, McGraw-Hill, New York, 1988.
54
9. Technology Vision 2020. The U.S. Chemical Industry, ACS, Washington, D.C., 1996. 10. The Process Development Division of the American Institute of Chemical Engineers, www.pd-aiche.com. 11. Chem. Eng. News, June 26, 2000. 12. Westerberg, A. W., and Subrahmanian, E., Comput. Chem. Eng., 24, 959 (2000). 13. O'Young, L., Natori, L., Pressly, T. G., and Ng, K. M., Comp. Chem. Eng., 21, $223 (1997). 14. Chemists and Chemical Engineers: An Integrated Team for Process Development, www.pd-aiche.com. 15. Samant, K. D., and Ng, K. M., AIChE J., 45, 2371 (1999). 16. Rajagopal, S., Ng, K. M., and Douglas, J. M., Comput. Chem. Eng., 16, 675 (1992). 17. Schroer, J. W., Wibowo, C., and Ng, K. M., AIChE J., in print (2001). 18. Migliorini, C., Mazzotti, M., Zenoni, G., Pedeferri, M., and Morbidelli, M., AIChE J., 46, 1530 (2000) 19. Winn, D., and Doherty, M. F., AIChE J., 46, 1348 (2000). 20. Basu, P. K., Mack, R., and Vinson, J. M., Chem. Eng. Prog., 95(8), 82(1999). 21. Dye, S. R., and Ng, K. M.,AIChE J., 41, 2427 (1995). 22. Cistemas, L. A., and Rudd, D. F., Ind. Eng. Chem. Res., 32, 1993 (1993). 23. Berry, D. A., and Ng, K. M., AIChE J., 42, 2162 (1996). 24. Ng, K. M., Separations Tech., 1, 108 (1991). 25. Cesar, M. A. B., and Ng, K. M. 1rid. Eng. Chem. Res., 38, 823 (1999). 26. Rajagopal, S., Ng, K. M., and Douglas, J. M., AIChE J., 37, 437 (1991). 27. Dye, S. R., and Ng, K. M.,AIChEJ., 41, 1456 (1995). 28. Berry, D. A., and Ng, K. M.,AIChEJ., 43, 1737 (1997). 29. Kelkar, V. V., and Ng, K. M., AIChE J., 45, 69 (1999). 30. Berry, D. A., Dye, S. R., and Ng, K. M., AIChE J., 43, 91 (1997). 31. Wibowo, C., and Ng, K. M., AIChE J., 46, 1400 (2000). 32. Samant, K. D., Berry, D. A., and Ng, K. M.,AIChEJ., 46, 2435 (2000). 33. Samant, K. D., and Ng, K. M., AIChE J., in print (2001). 34. Pressly, T. G., and Ng, K. M.,AIChEJ., 45, 1939 (1999). 35. Chang, W. C., and Ng, K. M., AIChE J., 44, 2240 (1998). 36. Wibowo, C., and Ng, K. M., AIChE J., 45, 1629 (1999). 37. Hill, P. J., and Ng, K. M.,AIChEJ., 43, 715 (1997). 38. Hill, P. J., and Ng, K. M.,AIChEJ., 41, 1204 (1995). 39. Hill, P. J., and Ng, K. M.,AIChEJ., 42, 727 (1996) 40. Kumar, S., and Ramkrishna, D., Chem. Eng. Sci., 52, 4659 (1997). 41. Hill, P. J., and Ng, K. M., AIChE J., 42, 1600 (1996). 42. Wibowo, C., and Ng, K. M., AIChE J., in print (2001). 43. Perkins, J., Comput. Chem. Eng., 24, 1367 (2000). 44. Kelkar, V. V., and Ng, K. M., AIChE J., 44, 1563 (1998). 45. Kelkar, V. V., and Ng, K. M., AIChE J., 46, 389 (2000). 46. Samant, K. D., and Ng, K. M., AIChE J., 44, 1363 (1998). 47. Samant, K. D., and Ng, K. M.,AIChEJ., 44, 2212 (1998). 48. Samant, K. D., and Ng, K. M., AIChE J., 44, 2689 (1998). 49. Samant, K. D., and Ng, K. M., AIChE J., 45, 1808 (1999).
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
55
Systems Biology: an Emerging Theme in Biological Research Gregory Stephanopoulos and William A. Schmitt Chemical Engineering Department, Room 56-469 Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307
The sequencing of genomes is rapidly altering the landscape of biotechnological and biomedical research. This is due to the direct access to gene-coding and intergenic sequence information, as well as genomics-based and other technologies that allow high throuput measurement of important classes of biological molecules. This information has the potential of elucidating gene regulation mechanisms and identifying genes implicated in disease, or perfecting industrially important high performance strains. To fully utilize this wealth of information, raw data must be appropriately upgraded through a variety of computational methods. A review of the computational issues associated with the utilization of genomic and physiological data is provided in this paper. These methods aim to integrate such data in a physiologically coherent framework and also provide the basis for a quantitative description of cell function. The introduction of computational methods to integrate and quantify biological data is an important step in the development of the emerging field of systems biology.
1. INTRODUCTION We have entered a period of rapid change in biological research. These changes are transforming traditional approaches in biological studies and are also generating new opportunities in biotechnology and biomedical research. At the core of these changes are recent developments in genomics and the associated need for integrating in a coherent framework diverse measurements of biological function. This integration underlies the emergence of systems biology as a field that aims to provide a global and quantitative understanding of cellular function. In order to materialize this vision a number of problems need to be solved, both experimental and computational. The objective of this review is to identify and describe in some detail the computational challenges before us. There are four driving forces that provide the impetus for these changes. The first is genomics: specifically the sequencing of genomes of many species. At least 50 species had been sequenced by the end of 2000 and it is expected that the genome of all industrially important organisms and many biomedically relevant species will be sequenced within the next 1-2 years (http://www.ncbi.nlm.nih.gov/). The second is the extensive monetary investment in the life sciences from both the public and private sectors. It is estimated that during the past 20 years more than 200 billion dollars have been invested in the life sciences in the US alone. Although most of the research activity has been aimed towards health-
56 related problems, the resulting fundamental advances in the life sciences are equally applicable to other fields. There are presently molecular biological methods available that can be applied on a routine basis to introduce controls at the genetic level and to construct optimal genetic backgrounds for potential medical and industrial applications. The enormous opportunities created by these fundamental advances are the third driver for change. Besides obvious applications in the field of medicine, other areas include production of chemicals (cell factories), pharmaceuticals and specialty chemicals (in particular, molecules of specific chirality), materials with special properties, and environmental applications [1]. The final driver of change is the development of technologies that probe the expression, proteomic, and metabolic phenotypes. The development of DNA microarrays allowed the genome-wide measurement of gene expression in cells [2, 3, 4]. This advance has sparked an interest in the development of other high-throughput technologies for the measurement of other important classes of cellular variables. Ongoing research is investigating the possibility of measuring on a cell-wide basis proteins (proteome) [5, 6, 7] and metabolites (metabolome) [8, 9]. The integration of such measurements could provide a rigorous understanding of cell physiology in its entirety (physiome).
Fig. 1. Probing cellular function. A schematic of cellular processes initiated by a receptorligand binding event that leads, through signal transduction to transcription (mRNA), translation, and ultimately metabolic and other cellular reactions, mRNA transcript populations and fluxes can be presently assessed through developing tools such as DNA microarrays and metabolic flux analysis techniques. Other methods aiming at the high throuput measurement of proteins and metabolites are under development (from M. Klapa). Figure 1 demonstrates that, as important as such measurements are, they do not individually provide a complete picture of cellular function. Instead, all available information must be considered simultaneously to understand the complex interactions of cellular function. To achieve this goal, these measurements must be integrated in a coherent framework of cell physiology and quantified to the extent that this is possible. A number of computational problems arise from this integration effort. These problems can be classified
57 as those associated with the utilization of sequence information, the measurements of cellular parameters, the mechanistic analysis of cellular black-box analysis of cell-wide measurements. All of these problems deal of biological information that defines, in essence, the field of bioinformatics
upgrade of raw data, and finally with the upgrade [ 10].
2. SEQUENCE-DRIVEN COMPUTATIONAL PROBLEMS The raw data generated by sequencing projects consist of base-pair sequence lists of individual DNA fragments obtained from restriction digests of the entire genome. Before any systematic sequence analysis is undertaken, these individual DNA sequences must be organized in a contiguous and coherent sequence that is also complete and unique. Additionally, absent in these data are the location and function of the various genes and other genomic factors. Therefore, key computational challenges in the use of raw sequence data include [11, 12, 13, 14]: a) Integrating sub-sequences together to form the entire genome sequence by uniquely matching the prefixes and suffixes of a large number of individual sub-sequences. As this sequence reconstruction may yield non-unique answers, multiple genome coverage is often desired and determining the exact extent of coverage is an interesting problem in its own fight b) Identifying open-reading frames (genes) once the entire genome sequence has been reconstructed c) Identifying gene splicing sites and intron location in eukaryotic cells d) Determining gene function (gene annotation) e) Determining sequence patterns of regulatory sites that are important in understanding gene regulation and expression in normal and diseased tissues f) Hypothesizing evolutionary relationships between organisms by construction and analysis of sequence-based phylogenetic trees Although quite diverse in nature, many of the above questions can be answered by solving a few generic types of problems including [11, 12]: a) Given two sequences align them optimally and determine the extent of homology between them (sequence alignment) b) Solve the above problem (a) for many sequences (multiple sequence alignment) c) Discover patterns characteristic of sequences that belong to the same family by virtue of the fact that they code for the same type of gene in different genomes (pattern discovery in sequences) d) For the case of protein sequences (see below), discover small characteristic sequences (motifs) that are common in proteins of the same family or by virtue of appearance in a protein database at significantly higher than expected frequency Similar problems need to be solved in the analysis of protein sequences. Such problems include a) identification of functional motifs in protein sequences, b) correlation of protein sequence to protein structure and function, and c) extensive determination of protein sequence homologies among proteins of similar function [ 15, 16, 17]. These problems have been the subject of intense investigation in recent years. Methods such as BLAST, FASTA, ORFinder (http://www.ncbi.nlm.nih.gov/gorf/gorf.html), TIGR's Glimmer (http://www.cs.jhu.edu/labs/compbio/glimmer.html), and IBM Research's Teriesias (http://www.research.ibm.com/bioinformatics/) have been developed and are extensively
58 used for sequence comparisons, homology analyses, gene identification, and the development of function discovery techniques of varying success. It is important to note that concepts familiar in the systems theoretic and optimization literature, such as dynamic programming [11, 18], Markov chains [19], and other statistical methods [20, 21] provide the basis of these sequence analysis approaches. These methodologies and algorithms make use of extensive genomic and proteomic databases, many of which are in the public domain and easily accessible through the web (http://www.ncbi.nlm.nih.gov/, http://www.expasy.ch/). The continued efforts of researchers in this area are certain to lead to both improved and novel tools for upgrading the content of the immense sequence libraries being generated.
3. COMPUTATIONAL ISSUES IN UPGRADING RAW MEASUREMENTS At the present time, there are three classes of biological measurements which can be performed at a relatively high-throughput rates: mRNA transcript levels using DNA microarrays, protein levels using 2-D gel electrophoresis and HPLC-MS, and metabolic flux measurements using NMR spectroscopy and/or mass isotopomer measurements by GS-MS. We describe below problems associated with the derivation of the relevant biological parameters from the raw measurements. 3.1. DNA microarrays The acquisition, cleaning, adjustment, and analysis of micro-array data are all important steps in the determination of reliable mRNA transcript measurements requiring customized designs for each individual system. Filtering the raw fluorescence intensities is important in removing spurious signals. Replicate spots on the same array provide measurement multiplicity that can be used for error analysis, background noise and signal to noise ratio calculations, and determination of confidence levels. Ratio adjustments (normalization) have a large effect on the true and false positives but a very small effect on discovered classes from clustering techniques [22, 23] typically used for microarray expression analysis [24]. Typically, gene specific data are normalized by the average or total signal for each fluorophore. This procedure is based on the assumption that differences among microarrays in brightness and total RNA added for each fluorophore will be corrected by such normalization. This normalization strategy, however, alters the data such that values are now reported as fractions of the overall RNA pool. While this is inconsequential when the overall RNA population per cell is constant, under conditions such that total RNA population is undergoing dramatic changes on a per cell basis a more robust normalization basis is required. This is particularly true in the increasingly popular cases of genome-subset (or partial) arrays, where the specific genes probed by the array are only those genes expected to change under the experimental conditions evaluated. Furthermore, by definition total mRNA levels are expected to differ at the different conditions studied raising questions about a normalization that is based on the total RNA. Depending on the goal of the study, care should be taken to normalize and adjust data in a manner consistent with the experimental questions under consideration. Specifically, overall conclusions about cell transcriptional profiles gene induction or repression and associated cell physiology are strongly influenced by the technique of normalization and can lead to inconsistent conclusions about cellular states.
59 3.2. Proteomic data Although 2-D gel electrophoresis effectively separates most proteins, the analysis of individual spots to determine the identity and amount of the protein represented in each spot poses experimental and computational challenges. Mass spectroscopy (MS) analysis typically is applied to peptide protein fragments resulting from the trypsinization of the protein(s) in each spot of the 2-D gel. Such peptides are identified by their molecular weights and then the original protein is determined through an elaborate reconstruction process that makes use of extensive databases of protein sequence and molecular weights. MS is applied routinely for this purpose, however, MS readings can be complicated by the presence of bound sugars (in glycosylated proteins), phosphate groups (phosphorylated proteins), or other posttranslational modification steps. Additionally, the possibility that more than one proteins may be present in a single spot adds an additional dimension to the reconstruction problem. This problem is similar to the problem of reconstructing the entire genome sequence from the sequences of individual DNA fragments discussed in a previous section. Multiple coverage of a protein, resulting from replicate trypsinizations and analyses of the resulting protein fragments enhances the accuracy of reconstruction and reliability of protein identification. [25].
3.3.
Isotopic tracer analysis
The determination of metabolite fluxes through metabolic networks is a critical step in the description of cellular function and physiology. Fluxes are the actual rates at which metabolites and proteins are processed through pathways or signaling networks and, as such, they represent the actual cellular physiological state. Macroscopic fluxes can be calculated from overall metabolite balances and extracellular metabolite measurements [26, 27], however, more detailed flux determination relies on the ability of the researcher to observe individual reactions within the network. The existence of reversible pathways, splitting pathways and metabolic loops complicates this process, requiring additional information beyond the rates of cellular inputs and outputs. Isotopic tracers allow us to mark specific carbon atoms of the input metabolites that are subsequently distributed throughout the cell's metabolic network. In general, the distribution of tracer depends on the fluxes that one wishes to determine. Measurements of these tracer distributions can be obtained from the degree of label enrichment and mass isotopomer measurement of secreted or intracellular metabolites, made possible by 13C NMR and (LC) GC-MS analysis [1, 26, 27, 28]. Metabolic network fluxes are then determined by solving the inverse problem. This process of tracer data upgrade for flux determination is depicted in Figure 2.
60
Labeled Tracer
r-
luxe
~ Isotopomer~ ~ A , istributi ,,,......_.....~ ~ D o~
~
Isotopomer Measurements (NMR, GC-MS)
""............................. Informationupgrade ............................"'" Fig. 2. Schematic of metabolic flux determination. A labeled tracer is introduced and distributed among intracellular metabolites by metabolic reactions. The latter are mapped onto tracer dependent measurements, such as label enrichment or mass isotopomer fractions that can be obtained by NMR spectroscopy or GC-MS. Fluxes are determined by inverting this mapping thus upgrading the isotopic data information content.
4. M E C H A N I S T I C LEARNING FROM DATA Learning from experimental data can be accomplished in two separate forms: mechanistic and black-box learning. Black-box learning, discussed in Section 5, refers to those methods that do not seek to directly associate measurements with known cellular mechanisms, but rather use all available information to uncover data structure. Mechanistic learning, on the other hand, generates insight into the mechanics of cellular function and usually involves the calculation of important variables from primary measurements. Current challenges in mechanistic learning are discussed below. 4.1. Reconstruction of regulatory networks One generic problem that arises in connection with differential gene expression data (obtained from DNA microarrays) is to use such data to uncover the correlational structure of their expression patterns and then elucidate the gene regulatory mechanisms. We illustrate this problem by means of an example from the transcriptional control of a well-understood system, the lac operon. As depicted in Figure 3-A, transcription of the structural genes of the operon lacZ, lacY, and lacA (coding, respectively, for ~-galactosidase, permease and galactoside transacetylase), is controlled by a repressor protein coded by lacI. In the absence of lactose, this repressor protein (indicated by a star in Fig. 3-A) binds the operator region, thus physically hindering the binding of RNA polymerase at the promoter region. When lactose is present, it binds on specific sites of the repressor molecule. This binding allosterically diminishes the repressor's DNA binding affinity with the lac operator and allows RNA polymerase to carry out its function of transcribing the polycistronic mRNA. Besides the above negative control by lactose, the lac operon also has a positive control by glucose mediated by CAP (catabolite activator protein). When the energy level (ATP) is low or cAMP is high, CAP binds to cAMP and the resulting CAP-cAMP complex binds the CAP binding site of the lac promoter, an event that promotes helix destabilization and RNA polymerase binding. For illustrative purposes we have also added in Figure 3-A a positive control of cap and lacl expression by sigma factor c~3 under growth conditions, and negative controls of the same genes by sigma factors ch and ~2 in the presence of a rich medium.
61
Fig. 3. Illustrative example of the probing of gene regulation through differential expression analysis. Dark spots represent upregulated genes, light spots downregulated genes. Figure 3-B shows a schematic of simulated gene expression patterns that would have been obtained for the genes listed on the left column under the growth conditions indicated at the top row. For simplicity only three levels of gene expression are shown: induced, basal and repressed levels indicated by black, gray, and white dots, respectively. For illustrative purposes a set of maltose inducible genes (mal) has also been added with an assumed regulation similar to that by lactose. The data of Figure 3-B are representative of the type of data that will be generated from differential gene expression experiments aiming at the elucidation of transcriptional control mechanisms similar to that of Fig. 3-A. This means that one is called to solve the inverse problem of determining the regulatory structure of gene expression from differential gene expression data obtained under a variety of experimental conditions. An important difference between the simulated example of Figs. 3-A and 3-B and a real situation is that, in the latter case, one is dealing not with a well defined and small set of genes such as those of
62 Fig. 3-B but with the expression profile of the entire genome. As the latter typically encompasses a few thousand genes, identifying those that contribute in a meaningful way to the observed cellular physiological state presents a challenge that must be addressed with further research. As more and more expression data accumulate, a characteristic pattern of gene expression is bound to emerge. Determining such a pattern will allow for the identification of the key players in a particular genetic regulatory scheme. The most straightforward approach to finding these mechanistic structures is to search for correlation [20, 22, 29, 30] between gene expression patterns, but often issues of observability hinder this approach. For example, if all intermediate players are not included in the study, it is impossible to determine whether a correlation between an experimental condition and a particular gene's expression is a direct effect or a secondary relationship [31 ]. Thus, we are currently limited in many cases to understanding the overall connectivity of genetic regulatory networks and not all of the intermediate players [7, 32]. Furthermore, while the illustrative example shows a static system, many interactions will be transient [33] and therefore the frequency of sampling becomes critical to ensure such interactions are observed [29]. As costs decrease, however, we can expect access to time-scales at finer and finer intervals. The potential of this quantity of data, combined with well-designed experiments to ensure quality of observations, will allow maximum insight into the cell's regulatory structure.
4.2. Flux determination from isotopomer data Intracellular fluxes cannot be measured directly, but they are rather estimated indirectly from measurements of extracellular metabolite consumption and production rates, as well as those of the isotopic distribution of extracellular and/or intracellular metabolites after the introduction of labeled substrate(s). This indirect estimation is possible because unknowns (fluxes) and measurements are related through mass (extracellular rates) and isotopomer (isotopic distribution) balances [26, 27, 35, 36]. Positional isotopomers are the different labeling patterns of a molecule. A molecule of n carbon atoms has 2 n isotopomers. Figure 4 shows the balance equations around the isotopomers of metabolite D in the depicted artificial pathway. The latter converts a four carbon molecule A to B and then to C, D, E, and F through metabolic reactions v l, v2, v3, v4 and v5. Metabolite carbon atoms are shaded to indicate how atoms are distributed from a substrate to a product. In reaction 2, for example, the top two carbon atoms of molecule B partition into F while the bottom two carbon atoms form product molecule D. Similar balances can be written for all isotopomers in the metabolic network. The important point is that all these isotopomer balances are intrinsic functions of the individual metabolic fluxes, vj, that we wish to determine, and the distribution of the input and measured isotopomers.
63
Fig. 4. A bioreaction network schematic depicting carbon atom distribution among network metabolites. The fluxes of the metabolic reactions dictate the fate of each carbon atom (ovals) in the substrate A. Balances can be drawn around the species within the network such as the one shown for metabolite D. Even though the isotopomer fractions cannot be measured directly, they are linearly connected with quantities measurable by Nuclear Magnetic Resonance (NMR) Spectroscopy or Mass Spectrometry (MS). Therefore isotopomer balances provide a framework for the estimation of the unknown fluxes using measurements from isotopic tracer techniques. Compared with carbon enrichment analysis, which allows the use of label enrichment measurements only for the estimation of metabolic fluxes, isotopomer analysis provides a more extended framework for flux determination. It allows the use of all possible measurements from isotopic tracer techniques, label enrichments, fine structure of NMR spectra, and mass isotopomer fractions for the estimation of metabolic fluxes. This integration can be accomplished because all of the above mentioned measurable quantities are related to the isotopomer fractions. The ability to integrate such varied types of biological measurements maximizes the insight that can be drawn from an experiment.
4.3.
Signal transduction
A final application of mechanistic learning from high throughput biological data is the analysis of signal transduction pathways. Signal transduction pathways are the means by which extra-cellular conditions are communicated to the interior of the cell. Signaling occurs
64 via consecutive phosphorylation-dephosphorylation steps whereby the phosphorylated (active) form of an intermediate protein acts as a catalyst (kinase) for the phosphorylation of the subsequent step. The final outcome of a signaling pathway is often the activation of a transcription factor that, in turn, initiates gene [37]. To date, signal transduction pathways have been investigated in isolation from one another. It has become abundantly clear, however, that there is a great degree of interaction (cross-talk) of signal transduction pathways for the simple reason that they share common protein intermediates [38]. This introduces the possibility that one ligand may effect the expression of more than one gene or that the expression of a single gene may be effected by more than one ligand. Again, the network features of signaling provide a fertile ground for the application of concepts from network analysis in conjunction with expression and, in particular, proteomic data. The key here is to recognize that the main function of signaling pathways is propagation of information instead of molecular inter-conversions. As such, conservation equations, like the ones used in metabolic network analysis, are not available. This fact that complicates quantitative network analysis and the correct formulations and applicable principles that take this fundamental difference into consideration are yet to be developed.
5. BLACK-BOX LEARNING Black-box learning refers to those techniques which are based purely on statistics and not on fundamental biological mechanisms about the system. These techniques are advantageous because they are not limited by lack of knowledge about the underlying system. Because mechanistic knowledge is especially incomplete in many biological situations, these tools can be used to solve problems such as characterization and classification based purely on the experimental data on hand. If this data is sufficient in its coverage of the system in question, then differentiating characteristics and structure will be uncovered with very low probability of detecting accidental or insignificant relationships.
5.1. Data visualization through dimensional reduction Because of the high dimensional nature of high-throughput data, visualization of the results is impossible using traditional plots. This is particularly true of DNA microarray data, where thousands of genes may be represented in dozens of experiments, creating a tangle of information requiring innumerable 2-dimensional plots to untangle. However, the genes in these experiments are not expressed independently, but rather are correlated to each other through the underlying gene regulatory network. The method of principal component analysis (PCA) [18, 21, 39] allows for a simplification of the data to eliminate these redundancies. Essentially, PCA uses eigenvalue (or singular value) decomposition to express experiments as linear combinations of the variables (genes) present. Those linear combinations that describe the largest portion of the data are retained, while redundant information is easily eliminated. In this way an experiment may be reduced to a few principle components (PCs) which then may be plotted more easily on a typical 2- or 3-D plot. This allows the researcher to quickly see if there is underlying structure in the data and use this information to proceed in the analysis. A related technique is that of canonical discriminant analysis (CDA) [18, 21] which uses the same concept to instead identify combinations of genes that are most suited to the distinction of classes under investigation. Consider Fig. 5, which shows the distinction
65
between 3 sub-types of leukemia with CDA (40, data from 30). CDA identifies two canonical variables (CVs) which are of primary importance to this discrimination. CV1 separates T-ALL from the other two disease states, while CV2 separates B-ALL from AML samples. Use of this visualization technique shows that the data easily supports the three classes, and that further analysis into the nature of their distinction is warranted. If this visualization fails to show any underlying structure, then the experiment should be reconsidered to determine how better data can be obtained.
Fig. 5. Canonical discriminant analysis of leukemia using gene expression data
5.2.
Sample characterization
Samples from different physiological conditions ranging from normal to diseased cells or wild type versus super producing industrial strains in optimized media, will have a characteristic "fingerprint" of genes that are induced or repressed. Quick identification of those genes whose expression are characteristic of a certain physiological difference is possible through careful selection and use of appropriate statistical tools. A large number of samples (> 30 for each class) for the types under consideration must be taken to accurately establish an expression distribution, and frequently such data is not available. Therefore techniques which make justifiable assumptions and are reasonably robust to small sample size are a key consideration. Statistical tests, such as 2-tailed t-tests [20] and mean hypothesis tests [41 ], adjust distributions according to sample size and are therefore generally reasonable for such purposes, although they are limited to the consideration of two classes at a time. Other techniques such as the Wilk's lambda criteria [21] and misclassification rate can be used to identify discriminatory genes for experiments involving 3 or more sample types, such as the situation shown in Fig. 5. However, when large numbers of samples are present, simpler correlation techniques [30] will provide robust results, so it is always advisable to ensure many replicates of any important experiment.
66
5.3. Classification The complimentary problem of how to use these "fingerprints" of gene expression in different samples has also been explored. The direct application lies in the field of classification or diagnosis, where the expressions of all available genes define a "fingerprint", and thus offer a broad (but incomplete) identifier for the physiological state of a sample. Traditionally, simple clustering has been applied on samples to identify which samples from known classes have a characteristic expression pattern closest to samples from unknown classes [42]. A more statistically-oriented perspective can also be obtained through the generation of a discriminant function which will classify new samples. This approach has been successfully applied to distinguish disease samples for diagnostic purposes [30]. Because DNA microarrays provide such a wealth of data in only a few experiments, wellconstructed classifiers can make very fine distinctions between disease states that would otherwise be difficult to distinguish, with obvious clinical applications. There is still a great body of modeling and classification knowledge in engineering that can be applied to improve the accuracy of these models, including the use of adaptive neural nets and other more advanced techniques. 5.4. Pattern discovery When the distinction of two or more classes of samples is known a priori, the problem of finding discriminatory features can be approached directly. However, in the case of exploratory research, it is not necessarily known which states are most similar and why. For this reason work in pattern recognition should be developed to tackle the problem of uncovering recurring themes or structure in biological data. Even experiments designed with a specific hypothesis in mind are bound to contain biological patterns that will remain hidden in the data unless data mining tools are employed for their discovery. The discovery of patterns and underlying structure in data will assist researchers in developing a new set of hypotheses which will guide further research.
6. CONCLUDING REMARKS We've arrived at a turning point in the study of biological systems, as data generation becomes less of a problem and the thorough analysis of data comes into the limelight. There are several goals of such analyses, as pointed out in this review. One goal of particular importance is to organize the data such as to uncover and highlight the relationships that exist between measures of cellular phenotype (such as gene expression, metabolite concentrations, protein levels) and measures of cellular function such as fluxes, resistance to antibiotics, production capacity, etc. This linkage, depicted in Figure 6, will help elucidate the role of individual genes and proteins in bringing about a specific biological result. Additionally, the coordination required among different cellular players will become better understood through the suggested linkage between phenotypic markers and functional metrics. As such relationships are unveiled they will undoubtedly generate specific hypotheses that can be tested experimentally. The important difference will be that such hypotheses are generated by data instead of critical analysis of the state of the knowledge about a particular system. We believe that such data driven hypotheses will shape to a significant extent future biological research with far reaching implications about the future organization of the research enterprise. This underlines the importance of developing and deploying rigorous
67 computational methods in mining as much knowledge as possible from the ever expanding data fields.
Fig. 6. Linking phenotypic markers such as expression data with metrics of cellular function such as metabolic fluxes. REFERENCES
1. Stephanopoulos, G. N., Aristidou, A. A., Nielsen, J. Metabolic Engineering: Principles and Methodologies (Academic Press, San Diego, 1998). 2. Schena, M., Shalon, D., Davis, R. W., Brown, P. O. Science 270, 467-470 (1995). 3. DeRisi, J. L., Iyer, V. R., Brown, P. O. Science 278, 680-686 (1997). 4. Spellman, P. T. et al.,Molecular Biology Of the Cell 9, 3273-3297 (1998). 5. Kahn, P. Science 270, 369-70 (1995). 6. Anderson, N. L., Anderson, N. G. Electrophoresis 19, 1853-61 (1998). 7. Hatzimanikatis, V., Choe, L. H., Lee, K. H. Biotechnology Progress 15, 312-318 (1999). 8. Tweeddale, H., Notley-McRobb, L., Ferenci, T. J Bacteriol 180, 5109-16 (1998). 9. Raamsdonk, L. M. et al., Nat Biotechnol 19, 45-50 (2001). 10. Stephanopoulos, G. Metab Eng 2, 157-8 (2000). 11. Setubal, J., Meidanis, J. Introduction to Computational Molecular Biology (PWS Publishing Company, Boston, 1997). 12. Baxevanis, A. D., Ouellette, B. F. F., Eds., Bioinformatics: A Practical Guide to the Analysis of Genes and Proteins (Wiley-Interscience, New York, 1998). 13. Baldi, P., Brunal, S. Bioinformatics: The Machine Learning Approach (MIT Press, 1998). 14. Misener, S., Krawetz, S. A., Eds., Bioinformatics: Methods and Protocols (Humana Press, 2000). 15. Robson, B., Gamier, J. Introduction to Proteins and Protein Engineering (Elsevier Science Publishers, 1988).
68 16. Branden, C.-I., Tooze, J. Introduction to Protein Structure (Garland Publishing Inc., 1999). 17. Durbin, R., Ed., Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids (Cambridge University Press, 1999). 18. Rao, S. S. Engineering Optimization: Theory and Practice (John Wiley & Sons, Inc., New York, 1996). 19. Rabiner, L. R. Proceedings of the IEEE 77, 257-286 (1989). 20. Dillon, W. R., Goldstein, M. Multivariate Analysis, Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1984). 21. Johnson, R. A., Wichern, D. W. Applied Multivariate Statistical Analysis (Prentice Hall, Englewood Cliffs, New Jersey, 1992). 22. Eisen, M. B., Spellman, P. T., Brown, P. O., Botstein, D. Proceedings Of the National Academy Of Sciences Of the United States Of America 95, 14863-14868 (1998). 23. Tamayo, P. et al., Proceedings Of the National Academy Of Sciences Of the United States Of America 96, 2907-2912 (1999). 24. Personal communication, Gill, Ryan T. 2001. 25. Pandey, A., Mann, M. Nature 405, 837-46 (2000). 26. Vallino, J. J., Stephanopoulos, G. Biotechnology and Bioengineering 41,633-646 (1993). 27. Klapa, M. I., Stephanopoulos, G. in Bioreaction Engineering Schugerl, K., Bellgardt, K.H., Eds. (Springer-Verlag, Heidelberg, 2000). 28. Stephanopoulos, G. Metab Eng 1, 1-11 (1999). 29. Arkin, A., Ross, J. Journal Of Physical Chemistry 99, 970-979 (1995). 30. Golub, T. R. et al., Science 286, 531-537 (1999). 31. Ren, B. et al., Science 290, 2306-+ (2000). 32. Thieffry, D. Bioessays 21, 895-899 (1999). 33. Gardner, T. S., Cantor, C. R., Collins, J. J. Nature 403, 339-342 (2000). 34. McAdams, H. H., Shapiro, L. Science 269, 650-656 (1995). 35. Klapa, M. I., Park, S. M., Sinskey, A. J., Stephanopoulos, G. Biotechnol Bioeng 62, 375391 (1999). 36. Park, S. M., Klapa, M. I., Sinskey, A. J., Stephanopoulos, G. Biotechnol Bioeng 62, 392401 (1999). 37. Lauffenburger, D. A., Linderman, J. J. Receptors: Models for Binding, Trafficking, and Signaling (Oxford University Press, New York, 1993). 38. Roberts, C. J. et al., Science 287, 873-880 (2000). 39. Alter, O., Brown, P. O., Botstein, D. Proceedings Of the National Academy Of Sciences Of the United States Of America 97, 10101-10106 (2000). 40. Work performed at corresponding author's laboratory, currently in submission under the title Defining Physiological States from Microarray Expression Measurements. 41. Kamimura, R. T. Ph.D. Thesis, Massachussetts Institute of Technology (1997). 42. Hughes, T. R. et al., Cell 102, 109-126 (2000).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Modelling of nonlinear process dynamics using Kohonen's Networks, Fuzzy Systems and Chebyshev Series
69
Neural
A.P. Alexandridis, C.I. Siettos, H.K. Sarimveis, A.G. Boudouvis and G.V. Bafas* Department of Chemical Engineering, National Technical University of Athens, Zografou Campus, Athens 15780, Greece This paper introduces a new approach to the problem of nonlinear system identification with the aid of neural networks, fuzzy systems and truncated Chebyshev series. The proposed methodology is of general use and results in both a linguistic and an analytical model of the system under study. The method was successfully used for identifying certain operating regions of a Continuous Stirred Tank Reactor (CSTR) where highly nonlinear phenomena, such as limit cycles and multiple steady states appear. 1. INTRODUCTION Mathematical models, which can describe efficiently the dynamics of the system under study, play an essential role in process analysis and control. However, most of the real-world processes are complicated and nonlinear in nature, making the derivation of mathematical models and/or subsequent analysis formidable tasks. So far, many approaches based on nonlinear time series [1], Poincare maps [2] and Lyapunov exponents [3] have been applied in nonlinear system modelling and analysis. During the last decade, a considerable amount of work has been published on the dynamic modelling of nonlinear systems using neural networks [1, 4] and/or fuzzy logic methodologies [5, 6]. Neural networks (NN) have proven to be very powerful tools, still providing only a black-box representation of the system dynamics. On the other hand, fuzzy logic can incorporate expertise and a-priori qualitative knowledge of the system, but due to the complexity of nonlinear processes, it is rather difficult to construct a proper fuzzy rule base. Moreover, the lack of analytical models remains the major drawback for both methodologies. This paper presents a new systematic methodology that facilitates the development of both linguistic and nonlinear analytical models with the aid of Kohonen's self-organizing neural networks, fuzzy logic and truncated Chebyshev series. The result is a model structure, which contains only few essential polynomial terms that are suitable to capture both the qualitative and the quantitative characteristics of the system dynamics. The methodology is applied to the identification of a CSTR, which, depending on the operating conditions may exhibit multiple steady states and limit cycles. The results demonstrate that the produced model captures the qualitative dynamic behaviour of the process and, furthermore, offers a satisfactory quantitative approximation. For comparison purposes, the performance of the proposed approach is compared with two other identification methods: one based on feedforward neural networks and one based on normal form theory. *Corresponding author
70 2.
O V E R V I E W OF THE I D E N T I F I C A T I O N A L G O R I T H M
The proposed identification algorithm can be split into the following basic steps: 9 The output space is clustered using a Kohonen's Self Organizing Map (SOM) network [4]. Self-organizing maps form a special class of neural networks that use an unsupervised learning approach to partition a finite set of data into a certain number of natural subsets based on similarity measures. 9 Based on the clustering of the output space, a fuzzy dynamical model is constructed to describe qualitatively the process dynamics. 9 The fuzzy system is then approximated by truncated Chebyshev series [7], resulting in low-order model structures suitable for vital tasks such as stability analysis and model predictive control applications. Assuming the availability of N training examples [xi Yi], i=l,...,N, where xi, are vectors containing the values of the input variables and yi are the values of the output variable at time point i, the proposed algorithm can be summarized as follows: Step 1. Implement a Kohonen's self-organizing map to cluster the output data: a) Select the number n of fuzzy sets, which will be used to describe the output variable. b) Set the number of neurons of the self-organizing map equal to n. Initialize the synaptic weights wj-,j = 1,...,n of the self-organizing map. c) Select randomly a value yi and find the best matching neuron e i.e., the one that minimizes the distance between yi and wj:
w =m}nly,- al,
j = l , 2 .....
n
(1)
d) Update the synaptic weights of each neuron according to wj (t + 1) = w~ (t) + rl(t)ha. (t)(y i - wj (t)) , t = 0,1,2.....
(2)
where ~/(t) is the learning parameter which decays exponentially with time and h(t) is the neighborhood function, which decreases exponentially with time and with distance between We and wj. Go back to step 1c until the synaptic weights converge. Step 2. Construct a fuzzy dynamical model as follows: a) Define fuzzy sets for each input variable. b) Set the center values of the output fuzzy sets equal to the synaptic weights that are determined by the SOM algorithm. c) For every Yi compute the membership function/z(wj,yi) of each fuzzy set j:
:IE
d) For each pair [xi, yi] find the fuzzy sets with the greatest membership in the fuzzy input and fuzzy output space, respectively. e) Construct a fuzzy rule using as the rule-antecedent the Cartesian cross product of the fuzzy sets in the input space (from step 2d) and as the rule-consequent the fuzzy set in the output space (from step 2d). f) The rules that are most frequently activated enter the rule base.
71
Step 3. Derive analytical models based on truncated Chebyshev series: a) Set a maximum order m for the Chebyshev polynomials. Use the roots of a m-th order Chebyshev polynomial as input values to the fuzzy system derived in step 2 to numerically calculate the input-output mapping in the normalized interval [-1, 1], using common fuzzification, inference and defuzzification techniques [6]. b) Use the least squares method (LS) to calculate the Chebyshev polynomial coefficients that best fit the fuzzy input-output mapping. c) Perform an analysis of variance to select the polynomial terms that mostly contribute to the variation in the data. d) Rearrange the model using only the selected terms. Fit the reduced model to the process trajectories within the operating region of interest by the LS method. If the approximation is unsatisfactory go back to step 3a and choose a larger m. 3. CASE STUDY: IDENTIFICATION OF THE NONLINEAR DYNAMICS OF A CSTR The adequacy of the methodology described above will be demonstrated by the identification of a CSTR that exhibits rich nonlinear dynamics. The dynamic behavior of the reactor is described by the following set of dimensionless nonlinear differential equations [8]: 2) exp(x 2) -
YcI = - x I + D a ( 1 - x 1 ) e x p ( x ic 2 = - x 2 + B D a ( 1 - x l )
fix, 2
(4)
where xlandx2represent the dimensionless conversion and temperature inside the reactor, respectively, D a is the Damk6hler number, B is the dimensionless heat of reaction and fl is the dimensionless heat transfer coefficient. The identification of the reactor dynamics is very challenging in two operating regions: one containing two stable steady states, say region I (figure 1a) and another with a periodic state (limit cycle), i.e. Hopf bifurcation, say region II (figure l b). The aim is to approximate the process dynamics in the operating region of interest, by extracting analytical models based on input-output data observations. Using the state variables and x~, x 2 as inputs, and their time derivatives ~l and 2 2 as outputs the fuzzy dynamical model will be formulated as follows:
Ri: If x~ is
FA, '
and is x 2 FA,' Then is J?l Fc,, and is 22 Fc2' .
3.1 Identification of the operating region with multiple steady states. A set of 1300 input-output data is used for the identification. For demonstration, figure 2a shows the derived fuzzy rule base, which describes the behaviour of kl with respect to the input variables. Seven fuzzy sets have been used for each variable assignment : Very Small (VS), Small (S), Medium Small (MS), Medium (M), Medium Big (MB), Big (B), and Very Big (VB). The fuzzy model was utilized to generate 1600 input-output data. That is, 40 points for xs and 40 points for x2 have been used to construct the output surface, applying the maxmin fuzzy inference operator and the centroid defuzzification method [6]. The truncated Chebyshev series approximation of the derived fuzzy model produces the following analytical model:
72 12 10
8 x
J
12
10
8
6
6
4
4
2
J
2
$1
0
0
0.0
0.2
0.4
0.6
0.8
Xl (a) Da=0.072, B=8.0, 13=0.3
1.0
0.0
0.2
0.4
0.6
0.8
Xl (b) Da= 0.32, B =11.0,13=3.0
Figure 1. Process phase portraits: (a) Two steady states, Sl and
S2.
(b) A limit cycle, LC.
Xl = 3.04539 + 0.23041 x 2 + 0.00278 (4x3-3x2 2) - 5.61076 x 1 - 0.58220 x~ xz + 0.02868x, (2x 2 _ 1) + 2.97509 (2x I2 - 1 ) -0.74373 (4x 3 - 3 x 1) - 0.00357(4x 3l - 3 x l ) (4X32 - 3 X 2) + 0"26440(16 Xl5 _ 20x 31 + 5Xl) + 0"00031(16 Xl5 _ 20x 3~+5X~ )(4x 32 _ 3x 2 ) k2 = 5.693x2 1.589(2x22 _ 1) + 0.255(4x32 - 3x 2) - 1.314x I - 9.508XlX 2 + 1.868x 1(2x 2 - 1 ) 0.264 x I (4x32-3x2) + 2.892 (2x 1= - 1 ) + 0.984 (2x 12 _ 1 ) x 2 - 1.814 ( 4 x ~ - 3 x ) 1 -
-
The selected polynomial terms cover the 95% of the total variance in the data. The phase portrait of the model given by the above equations is shown in figure 2b. Comparing the resulting phase portrait with the original shown in figure l a, makes clear that the produced model captures qualitatively the dynamic behavior of the process but also offers a very good quantitative approximation.
3.2 Identification of the operating region containing a Hopf bifurcation. For the operating region containing the Hopf bifurcation, a total number of 700 inputoutput data have been used for identification. Figure 2c shows the derived fuzzy rule base for describing the behavior of :~2 with respect to the input variables. The application o f the Truncated Chebyshev series methodology results in the following analytical model: k 1 = 1.2753 - 2.2542x 2 + 0.7002 (2x~ - 1) - 0.0005(4x32 - 3x2) - 1.46764 x I + 2.07313XlX 2 0.6522x, (2x22 - 1 ) - 0.9777(2x~ - 1 ) - 0.0307(2x~ -1)(2x 2 _ 1) + 0.2608(4x~ - 3x,)x 2 0.2602(16Xls _ 20x 31 + 5 X l ) - 0.0027(16x] - 2 0 x ~ + 5x,)(Zx~ - 1 ) 5:2 =9.390 -14.401x 2 +9.210 (2x22-1) -0.011(4x32-3x2)+2.556 x 1 -9.456 x I (2x22-1)8.961(2x~ - 1) + 8.062(2x~ - 1)x 2 + 0.652(2x~ - 1)(2x~ - 1) + 0.030(4x~ - 3x,)x z The resulting phase portrait of the model is shown in figure 2d. A comparison with the phase portrait of figure 1b, shows that the two-phase portraits are almost identical.
3.3 Comparison with other identification techniques: neural networks and normal form The proposed algorithm was compared with two other identification schemes, one based on feedforward neural networks and another based on normal form theory [8]. Neural networks are black-box models where no qualitative a-priori information can be incorporated. The normal form is the representation of a specific nonlinear phenomenon within the operating region of interest, using the simplest possible class of equations.
73
(c) Fuzzy rule base for :~2 for region II
(d) Model phase portrait for region II
Figure 2. Resulting fuzzy rule bases and model phase portraits. A four-layer neural network with five nodes in each of the two hidden layers was used for the prediction of each output variable. The neural network structure was selected so that the system dynamics are adequately approximated, while avoiding undesirable phenomena such as data overfitting. In the case of the two stable steady states region, training was based on 1130 input-output data, while 457 data were used for the identification of the Hopf bifurcation. Training was achieved using the Levenberg-Marquardt algorithm [1 ]. The phase portraits for the two stable steady states and the Hopf bifurcation region are shown in figures 3a and 3c, respectively. As it is clearly shown neural networks approximate well enough the process dynamics within both operating regions. The phase portraits produced for the two stable steady states region and the Hopf bifurcation region, using normal forms that qualitatively describe the dynamics of a system in the operating region of interest [8] are given in figures 3b and 3d, respectively. As it is shown, both normal forms describe the process dynamics only in a qualitative way at best. In fact the phase portrait given in figure 3b depicts clearly that the corresponding model fails to approximate the process dynamics in the left part of the region. 4. CONCLUSIONS In this work a new systematic methodology for the identification of nonlinear systems was proposed, by integrating neural and fuzzy techniques in a common framework. The proposed method results to two models: a fuzzy model, which gives a linguistic description of the process behaviour and a truncated Chebyshev series model suitable enough to represent accurately the process dynamics within the operating region of interest. The applicability of the methodology was demonstrated by means of approximating the nonlinear dynamics of a
74
(c) NN phase portrait for region II
(d) Normal form phase portrait for region II
Figure 3. Phase portraits of the neural networks and normal forms. CSTR within two operating regions: one containing multiple steady states and one containing a Hopf Bifurcation. A comparison with two other nonlinear identification methods, one based on neural networks and one based on normal forms revealed the effectiveness of the proposed approach. REFERENCES
1. J. Sjoberg, Q. Zhang, L. Ljung, A. Benvensiste, B. Deylon, P. Glorennec, H. Hjalmarsson and A. Juditsky, Nonlinear Black-box Modeling in System Identification: a unified overview, Automatica 31 (12) (1995) 1691-1724. 2. M. Henon, On the numerical computation of Poincare maps, Physica D 5 (1982) 412-414. 3. A. Wolf, J. B. Swift, Swinney, H. L. and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D 16 (1985) 285-317. 4. Haykin S., Neural Networks, 2nd Ed.,Prentice Hall, 1999. 5. R. Babuska and H. B. Verbruggen, An overview of fuzzy modeling for control, Control Eng. Practice 4(11) (1996) 1593-1606. 6. H. J. Zimmermann, Fuzzy set theory and its applications, 3rd Ed., Kluwer, 1996. 7. Rivlin, T. J., An introduction to the approximation of functions, Dover Publications, Inc., 1969. 8. N. K. Read and W. H. Ray, Application of nonlinear dynamic analysis in the identification and control of nonlinear systems-I. Simple dynamics, Journal of Process Control 1 (1998) 115.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
75
A systematic methodology for empirical modeling of non-linear state space systems J.P. Barnard and C. Aldrich Department of Chemical Engineering, University of Stellenbosch, Private Bag X1, Matieland, Stellenbosch, South Africa, 7602. Email:/
[email protected] In this paper the authors formulate a theoretical framework for the empirical modelling of non-linear state space systems. The classification of non-linear system data, selection of model structure and order, system parameterisation, stationarity of the data, handling of outliers and noise in the data, parameter estimation and model validation can all be addressed with established, though loosely associated numerical techniques, often referred to as nonlinear process modelling. Relatively few researchers in system identification are comfortable with the application of these numerical techniques, such as time series embedding, surrogate data methods, non-linear stationarity, Lyapunov exponents for chaotic processes and nonlinear predictability. The authors reinterpret some of the above non-linear empirical concepts against the established background for linear state space system identification. Hereby we lay a basis for a systematic methodology to address empirical modelling of non-linear process dynamics, which can be implemented in a non-linear system identification toolbox. In particular, we apply surrogate data methods for the classification of data as stochastic or deterministic. For deterministic data, we embed the individual observations of the process and separate the embedding variables by non-linear factor analysis to arrive at a state space parameterisation of the system. The separation function makes no prior assumptions about the probability distributions of the observations and is robust against dynamic and measurement noise. An ensemble learning technique is used to estimate the parameters of the separation function. After parameterisation of the system a multiple-layer perceptron neural network maps the time evolution of the state vector onto the observations, one sample step ahead. In this manner, the dynamics of the process are captured. Model order is established against the Schwarz information criterion, formulated for multidimensional observations as a function of the model order and modelling error. Model validation is performed against the R 2 statistic, as well as in terms of free-run prediction performance. 1.
INTRODUCTION This paper presents a formal methodological framework for empirical modeling of nonlinear multivariate dynamic systems that can be parameterised as state space systems. Identification is based on multiple time series observations. The methodology addresses classification of observations, using surrogate data techniques, parameterisation of the system by way of multiple time series embedding and prediction of the time series by using multiplelayer perceptron neural network or other suitable models.
76 I D E N T I F I C A T I O N OF N O N - L I N E A R S T A T E SPACE SYSTEMS System identification is well defined for linear dynamic systems and described in severa comprehensive publications (Ljung, 1987, Norton, 1986, Eykhoff, 1974). Amongst parametrit system identification methods, state space methods are generally regarded as superior to othe methods and therefore forms the basis of our methodology. In this section we treat mode selection, data classification and model validation.
2.
2.1. Model selection No single comprehensive mathematical model sufficiently represents all classes o dynamic systems. Thus we are interested in the class of deterministic, non-linear dynamica systems that can be represented mathematically by a state equation in a number of stat~ variables. The dynamics of a non-linear state space system are interpreted as follows. Starting fron some initial conditions, the system's state vector follows a trajectory with time, that i: confined to some bounded subspace of the total available state space. The dynamic attractor to which the trajectory thus converges, is a smooth, non-linear manifold in this state space am defines the true dynamics of the system (Thompson et al., 1995). In mathematical terms fo: discrete-time non-linear systems, the state equation is: xt+l - f[xt, ut]
(1
where x is the state vector, u the input vector of independent variables and f the stat{ transition function that maps the temporal evolution of xt to xt+l. The output vector o dependent variables of the system is defined as yt = g[xt, ut] (21 where g(.) is a non-linear function that projects xt and ut onto the output vector yr. In the first part of system identification, the evolution of xt is reconstructed from the observec system outputs yt. The remaining steps of system identification focus on approximating gof at g(.)" [xt, ut]--~yt+l as well as validating the model. 2.1.1
Parameterisation
Parameterisation plays a critical role in the ability of any model structure to estimate non. linear dynamics. For the class of non-linear state space models, parameterisation introduce,' the concept of state space reconstruction by embedding. We have previously proposed ar extension to Takens' embedding theory in which we approximate the unknown original stat~ space by a phase space, constructed by time-series embedding of each observation componen and combining these subspaces (Barnard and Aldrich, 2000). In this paper we adapt thi,, methodology by applying singular spectrum analysis and non-linear factor analysis to th~ initial combined embedding space. According to Takens (1981), one can reconstruct an equivalent representation of th~ system state space from a one-dimensional time series observation, y~ 9~n, under the conditior that the observation function h(.) is smooth. Such a reconstruction is called an embedding o: the observed time series by way of delay co-ordinates (equivalent to phase variables). Th~ number of these co-ordinates is the embedding dimension, m and the time delay, k (ir multiples of the sample period) is the delay between each co-ordinate. Optimal delay ensure,, linear, statistical independence among delay co-ordinates - a fundamental requisite of phas~ variables and thus also of delay co-ordinates. The optimal time delay between the delay co-ordinates is usually determined by th~ average mutual information (AMI) criterion of Frazer and Swinney. (1986), while the optima number of co-ordinates is typically calculated using the method of false nearest neighbour,
77 (Kennel et al., 1992). However, since inferring k from AMI is confounded by noise, we have chosen to apply singular spectrum analysis instead to determine the initial phase space. In other words, for the embedding of each observation component, yi use a default delay of k = 1 and determine the embedding dimension, mi as the linear decorrelation point in the autocorrelation sequence ofyi: mi = min[first min(y.yt_k), first(y.yt_k= 0)]i (3) The individual embedding spaces are then concatenated columnwise and separated initially by singular value decomposition. The first q significant eigenvectors are selected to span the resultant phase space. A = embed(Y, m, k), Y ~ 9~ n x p , A ~ ~l](n'J 0) x Xmi (4) where j0 = max(m/- 1) + 1, 1< i (Phase-System ?X) (forall ?X (Exists (has-material-component ?X ?A))))
(?A) (And (Material-Component
?A)
/\
Meta-ModeI-Prlmdlve
I
connected-via--~
I Model-Interface
A
I
descnbes-behav~oroof
I Process-Behavmr-Port I MatenaI-Entlty@PhyslcalD A o~s-]
I BehavIor-TherrnaI-Energy-ConnectlJon
I E..... Po~l
Behavlor-lnformatlon-Connectlon]
" ........ ^ I'" !\
[ Thegn ergl-::~c~]ss'M~et:::b~e :~s-p accomo~lates1 I Fac,,,y-Ent~@P'hy .... D [
I Information-PortI
I
connected-to
[
/ ~ I L. ...........
/~
[ Behawor-MatenaI-Connect=onI
[ ............. ~1~
owns---~
[ Equation-Model] ~r I ProtoVlodelt~ Is-described-by-equation-model1" is-u~lmately-ma de-up-of
...... < | / /
............. @........ I
I B=.... -vo~u.~ ]
MATERIAL ONTOLOGY IN THE (PHYSICAL DIMENSION, PRODUCT LIFE-CYCLE)
Material-Entity~-ow.s-~ Material-PortI /~ ~/ IMateriaI-Pr~
l
>lP....... ]>lTemperatureI
!'e'r
has-material-----~ M t al ~-has-material-component-properties //~ ] I Ph....I System I
Material-Properties
+ [
Mi~ure,
I
I
IMat.... ,-Piasel P...... Material 9
is-made-up-of
Material.Phase_Phlsical.Stare
f
has-phase-information
~1 + Set-Of-Material-Phases
I
has-component
-in-mi. .tu. .+p . . p rties
[ ...... Mass_Fracti ............ ont-" [ Molar_Fraction ] Solid" Material-Physical-State[ Material-Physical-State I vap...... Gas "Material-Physical-StateI
I
[Liquid
,"
[ Material-Component I
[Material-Physical-State]
Fugaclby Specific_Entropy Specific_Enthalpy Mass_Density Molar_Density Mass_Flow_Rate
Mass_Inventory Volumetric-Flow-Rate Volume
Molecular-Weight Molar-Heat-Capacity
Specific_Heat Other material~'-~ Surface Tension Th.... I--Conductivity snou proP,leo,~.e Des Viscosity Iadded Z_Factor
I
Figure 1. Taxonomies of the process behavior and material ontologies. The ontologies in the behavior dimension comprise definitions for physicochemical behavior that comes out as a result of the interaction between the physical dimension and exogenous factors. Ontologies of the physical dimension specify concepts for the representation of the plant structure such as classes of equipment and plant entities or for the representation of the product structure (material). The operations and management ontology defines plans and actions needed for controlling, operating and manipulating objects of the physical dimension. This work emphasizes on the use of the process behavior ontology for model representation and reuse. Each dimension has multiple views with mappings between them and across dimensions. For example, plant structure objects are defined from abstract, mechanical, functional and operational perspectives. Facilities in the abstract view are mapped one to one to equipment item instances in the mechanical view. Metamodels in the process behavior ontology and equipment facilities in the structure dimension of the plant life-cycle are defined as separate but related objects, which allows software interoperability among plant and process life-cycles. In the design of the ontologies the following aspects about behavior models were considered: 1) easiness of the information sharing and exchange between intelligent CAD systems and process
89 simulation sotLvare tools; 2) simulation model exchange with varying level of detail and coarseness; 3) unnecessary structural modifications to the existing flowsheet and the re-entering of new instances of "unit operations"; 4) modeling of undesired behaviors (e.g. leakages) without affecting the flowsheet topology; 5) exploration of operation alternatives without modifications to the simulation models; 6) the ability to associate more than one kind of behavior associated to a particular equipment item; 7) the representation of matter, energy, and information flows that do not flow through defined pipes or channels, such as a vessel leakage or sunshine heating over the equipment. 2. T H E P R O C E S S B E H A V I O R O N T O L O G Y Behavior in this ontology refers to the physicochemical phenomena that are manifested through changes in the properties of the material that is processed in the plant. This definition implies the existence of physicochemical behavior models that can be defined independently of where the modeled phenomena occur. For a given piece of equipment, engineers can combine multiple metamodels to represent proposed, assumed or actual behaviors as well functional information (roles or expected behaviors). Consequently, instead of a single flowsheet diagram, a process topology diagram is worked out in parallel to the development of 2D or 3D plant representations (such as PFD, P&ID or 3D diagrams). The behavior that takes place in a piece of equipment is formulated using metamodel behavior entities. Metarnodels can be interconnected via ports for transfen~g energy, information and mass. In this context, connecting ports is comparable to combining sets of equations. Two important properties of metamodels are aggregation and abstraction (hierarchical decomposition). Hierarchical decomposition allows a metamodel to be decomposed into a number of other metamodels. Aggregation is the property of combining and linking instances of metamodels. Metamodels that are not composed of other metamodels are called protomodels. Protomodels represent material and energy balance equations as well as thermodynamic models that are independent of the geometry or other features of a particular equipment item. Definitions about the properties of the material are defined in an ontology that belongs to the physical dimension of the product. 3. A P P L I C A T I O N OF T H E P R O C E S S B E H A V I O R O N T O L O G Y In order to illustrate the use of metamodels, an example based on the boiler model presented in [13 ] is provided. Steam Instances of top-level metamodels describe the behavior that Feed takes place in one equipment item. Figure 2 shows a simplified diagram of a fired-tube boiler in which products of combustion pass through tubes surrounded by water. In this process, water is evaporated to produce steam. The fuel is bumt with a mixture of air to provide the energy for producing the steam, but this part is excluded for space reasons. The top level metamodel associated to the boiler is shown in Figure 3. At this level an instance of balance- Figure 2. Boiler diagram (facility) volume is used to define the region of interest that is to be modeled. The energy flow between the fired section of the boiler and the tank is represented as a thermal-energy-connection. From an implementation perspective, each facility can be assigned a window upon which metamodels reside. In this modeling scenario, the modeler (a modeling person or a program) decides to construct models for each physical phase, assuming the presence of vapor and liquid. These are represented by the balance volumes fiquid_balanee_volume and vapor_balance_volume.
90 Parameters such as volume of the vessel, cross-section of the tank, etc. are available from a provision-model that is associated to the tank. In general, a provision-model supplies information about the geometry and function of the equipment item. As a model it can incorporate assumptions that facilitate the description of the physical aspects of the equipment. For example, while the cross-section of a vertical tank can be considered as a parameter for the calculation of the liquid level, an expression for the cross-section of a spherical-tank may require the value of the level. In this example, the provision model tank_geometry_model is used to represents the constraint that the volume of the tank imposes over the volume of the material of each phase. In a lower level, the modeler (probably with the assistance of a modeling wizard) selects from a model library a model for the accumulation of material (liquid_inventory_model) and a model that describes evaporation phenomena (evaporation_model). Figure 3. Metamodels associated to a boiler Ports are automatically created when connections are drawn from the ports of liquid_balance_volume to the metamodel boxes. Alternatively, ports can be interactively assigned to the metamodels. At this level, metamodels are not decomposed further. These metamodels are referred to as protomodels. Protomodels have associated equation models based on physicochemical principles such as conservation laws and chemistry. At this point, the model is incomplete, as no equations have been defined so far. 3.1 Use of Modelica as a model representation language Modelica is a model representation language that is being developed in an intemational effort that aims at providing an approach for model reuse and exchange between simulation systems. Modelica code is non-causal, which means that equations are declared in a mathematical sense and not as a computational procedure. Thus models can be formulated independently of how they are to be solved. The Dymola modeling and simulation environment was used to edit and validate the models in Modelica. Ontology definfions such as process material, metamodel parent class, and ports were defined as records, classes, and connectors. Classes were also used to define the metamodels of the example. Modelica code for liquid_inventory and evaporation metamodels is shown in Figure 4 and Figure 5 respectively. Once metamodels are instituted, Dymola checks the syntax and attempts to generate a combined set of equations from the lower level equation models and connectivity information. Subsequently, degrees of freedom are verified, the Modelica code is translated, compiled and simulated. In a combined design and simulation environment, once metamodels for each equipment item are defined, equation-models for the entire plant can be generated using the topological information of the interconnected facilities. class liquid_inventory e x t e n d s metamodel;
SIunits. Pressure P;
91 SIunits.Temperature T; SIunits.Mass M; SIunits.Volume VL; S I u n i t s . D e n s i t y D; SIunits.Length L "Level"; SIunits. SpecificEnthalpy H "Acumulated Heat"; parameter S I u n i t s . S p e c i f i c H e a t C a p a c i t y Cpliq "Specific Heat Capacity"; parameter S I u n i t s . T e m p e r a t u r e TR=273 "Reference Temperature"; parameter S I u n i t s . M a s s F l o w R a t e R R = I "Evaporation reflux parameter"; annotation ...
equation
P = material_input.material info.mdoom_pressure; material_output.material_info~mdoom_massflowrate = material_input. massflowrate; material _ info.mdoom der(M) = m a t e r i a l _ r e c y c l e _ i n p u t . m a t e r i a l i n f o . m d o o m _ m a s s f l o w r a t e ; H = Cpliq*(T - TR)*M; der(H) = material_input.material_info.mdoom_enthalpyflow material_recycleinput.material_info.mdoom_enthalpy_flow material_output.material_info.mdoom_enthalpy_flow; material_output.material_info.mdoom_temperature = T; material_output.material_info.mdoom_specific_enthalpy = Cpliq*(T - TR); material_output.material_info.mdoom_enthalpy_flow = Cpliq*(T - TR)* material_output.material_info.mdoom_massflowrate; material_output.material_info.mdoom_pressure = P; material_output.material_info.mdoom_density = material_input.material _ i n f o . m d o o m _ density; D = material_input.materialinfo.mdoom_density; VL = M/D; geometry_port.volume = VL; L = VL/geometry_port.area; end l i q u i d _ i n v e n t o r y ;
Figure 4. Modelica code for the liquid-inventory metamodel of Figure 2.
4. C O N C L U S I O N S Metamodels are specified for each piece of equipment without necessarily defining a network of metamodels for the whole plant. Metamodels are also useful in the creation of libraries of reusable models that can be exchanged between simulation tools. The property of abstraction together with the separation of behavior from the equipment information, enable a metarnodel to be replaced by another metamodel of different fidelity or coarseness without modifying the plant structure and operational information. This contributes to the consistency of the information that is exchanged among simulation tools and CAD systems. The development up to date of the metamodel concepts shows that there is a promising potential for developing modeling techniques and software tools adequate to support engineering activities along the life-cycles of the plant, processes and products. An example has been shown to illustrate the use of the metamodel concepts and to identify functional requirements of modeling tools. The example explores the use of Modelica as a model representation language as seen from the metamodel perspective. It shows how aggregation and abstraction of behavior models are both supported by Modelica. Four different categories of Modelica code were identified: 1) Modelica code for data structures; 2) automatically generated code; 3) building-blocks created by the modeler for immediate use; 4) building blocks created by the modeler to be stored in a model library. Additions to the Modelica language to explicitly distinguish among the categories would be beneficial. A number of issues are to be addressed, including the management of the level of detail in the models, determining whether the attributes of provision-models are to be fixed in the ontologies or defined by the modeler, and more importantly the development of a modeling methodology in detail based on physical principles. One aspect that the authors are exploring is the use of the same behavior models for both batch and continuous processes.
92 class e v a p o r a t i o n extends m e t a m o d e l ;
S I u n i t s . T e m p e r a t u r e T; SIunits. P r e s s u r e P; parameter S I u n i t s . S p e c i f i c E n e r g y l a m b d a "Heat of v a p o r i z a t i o n " ; parameter SIunits. S p e c i f i c H e a t C a p a c i t y C p l i q " L i q u i d S p e c i f i c Heat C a p a c i t y " ; parameter S I u n i t s . T e m p e r a t u r e TR " R e f e r e n c e T e m p e r a t u r e " ; parameter SIunits. D e n s i t y D " L i q u i d D e n s i t y " ; annotation (..);
equation
material_input.material_info.mdoom_massflowrate = material_recycle_port.material_info.mdoom_massflowrate + vapor_output.material _ info.mdoom_ massflowrate; material recycle_port.material_info.mdoom_enthalpy_flow = material_recycle_port.material_info.mdoom_massflowrate*Cpliq*(T
- TR);
material_input.material_info.mdoom_enthalpy_flow = -energy_input.mdoom_heat_flow vapor_output.material_info.mdoom_enthalpy_flow; vapor_output.material_info.mdoom_enthalpy_flow = vapor_output.material_info.mdoom_massflowrate*(lambda + C p l i q * ( T - TR)); vapor_output.material_info.mdoom_specific_enthalpy = vapor_output.material_info.mdoom_enthalpy_flow / vapor_output.material_info.mdoom_massflowrate; material_recycle_port.material_info.mdoom_specific_enthalpy = C p l i q * ( T - TR); vapor_output.material_info.mdoom_temperature = T; material_recycle_port.material_info.mdoom_temperature = T; material_recycle_port.material_info.mdoom_pressure = vapor_output.material_info.mdoom_pressure; material_recycle_port.material_info.mdoom_density = D; // The A n t o i n e c o r r e l a t i o n is u s e d for the v a p o r p r e s s u r e c a l c u l a t i o n P = exp(16.5362 - 3985.44/(T - 38.997))'1000; vapor_output.material_info.mdoom_pressure = P; end e v a p o r a t i o n ;
+
Figure 5. Modelica code for the evaporation metamodel.
REFERENCES 1. J. Bieszczad, A. Koulouris and G. Stephanopoulos, Proc. of the FOCAPD'99 Conference, Breckenridge, Colorado, USA. (1999) 2. A. A. Linninger, Proc. of the FOCAPD'99 Conference, Colorado, USA (1999) 3. N. Shah, N. J. Samsatli, M. Sharif, J. N. Borland, and L. G. Papageorgiou L. G., Proc. of the FOCAPD'99 Conference, Breckenridge, Colorado, USA. (1999) 4. P. Gawthrop and L. Smith, Metamodelling: For bond-graphs and dynamic systems, Prentice-Hall International, UK (1996) 5. H. Elmqvist, S. E. Mattsson and M. Otter, IEEE Symposium on Computer-Aided Control System Design, CACSD'99, Hawaii, August 22-27(1999) 6. W. Marquardt, Proc. of the FOCAPD'99 Conference, Breckenridge, USA. (1999) 7. R. Batres and Y. Naka, Proc. of the FOCAPD'99 Conference, Colorado, USA. 8.13. Bayer, R. Schneider, W. Marquardt, 7th International Symposium on Process Systems Engineering, PSE 2000, Keystone, Colorado USA, July 16-21 (2000) 9. R. 13atres and Y. Naka, AIChE Symposium Series No. 323, Vol. 96 (2000) 10. R. Batres, M. L. Lu and Y. Naka, Joumal of Concurrent Engineering, Vol. 7, No. 1 (1999) 11. G. Booch, I. Jacobson, J. Rumbaugh, The Unified Modeling Language User Guide. Addison-Wesley (1999). 12. M. R. Genesereth, R. E. Fikes, KIF Version 3.0 Reference Manual, [Online] http://logic.stanford.edu/papers/kif.ps (1992) 13. W. F. Ramirez, Computational Methods For Process Simulation, 2nd ed., 13utterworthHeinemann (1997)
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
93
Dynamics of a reactive distillation column for TAME synthesis described by a nonequilibrium stage model R. Baur a, R. Taylorb and R. Krishna a aDepartment of Chemical Engineering, University of Amsterdam, Nieuwe Acthergracht 166, 1018 WV Amsterdam, The Netherlands bDepartment of Chemical Engineering, Clarkson University, Potsdam, USA
The dynamics of a reactive distillation column for synthesis of tertiary amyl ether (TAME) has been analysed using a rigorous nonequilibrium (NEQ) stage model, employing the Maxwell-Stefan equations for mass transfer between fluid phases. Both pseudo-homogeneous and heterogeneous reaction models show the possibility of multiple steady states. Starting at the low-conversion steady state, a feed composition perturbation is shown to lead to a transition to the high conversion steady state, in conformity with the experiments of Mohl et al. (Chem. Eng. Sci. 54, 1029-1043, 1999). The liquid holdup within the catalytic reaction zone is seen to have a significant influence on column dynamics. 1. INTRODUCTION The design and operation issues for reactive distillation (RD) systems are considerably more complex than those involved for either conventional reactors or conventional distillation columns. The introduction of an in-situ separation function within the reaction zone leads to complex interactions between vapour-liquid equilibrium, vapour-liquid mass transfer, intracatalyst diffusion (for heterogeneously catalysed processes) and chemical kinetics. Such interactions have been shown to lead the phenomenon of multiple steady states (MSS) and complex dynamics, which have been verified in experimental laboratory and pilot plant units; for a comprehensive literature survey, see Taylor and Krishna (2000). In earlier publications we had developed rigorous nonequilibrium (NEQ) models to describe the steady-state and dynamics of RD columns (Baur et al., 2000a; Baur et al., 2000b; Higler et al., 2000). The major objective of this paper is to use our developed NEQ models to rationalise the recently published experimental results of Mohl et al. (1999) for TAME synthesis. 2. MULTIPLE STEADY STATES FOR TAME SYNTHESIS We first examine the steady-state multiplicity characteristics of the TAME process using both the pseudo-homogeneous and heterogeneous descriptions of the catalytic reactions. TAME is formed by an acid-catalysed equilibrium reaction from 2-methyl-l-butene, 2methyl-2-butene and methanol. The reaction kinetics is described by a LangmuirHinshelwood rate expression in terms of the liquid phase activities (Oost et al., 1995). We did not incorporate the isomerization reaction of the C5-olefins in the model. Furthermore, we assumed the acid group concentration per Raschig ring volume to be 0.9 eq(H+)/L and the
94 zomplete catalyst volume to be 1.2 L. Fig. 1 (a) schematically depicts the column zonfiguration of the TAME process presented in Mohl et. al. (1999). The column has an inner diameter of 76 mm. The top section of the column is packed with catalytic and the bottom section with inert (glass) Raschig rings. Both sections are 0.5 m high. The feed is located in the middle of the column between the reactive and inert packing. As specified by Mohl et. al. (1999) in their experiments the feed rate was chosen to be 0.96 kg/h. The column operates at a pressure of 0.25 MPa. In order to evaluate the vapour-liquid equilibrium at the interface the Wilson equation was used for calculating the liquid activity coefficients. Furthermore, a total condenser and total reboiler is employed. The reflux ratio was specified at 15, whereas the reboiler heat duty was used as a homotopy parameter. Each section has been divided in 20 slices. Simulations with the pseudo-homogenous model show that the overall performance correspond to an efficiency model with 10 stages and an efficiency of 0.7 proposed and experimentally validated by Mohl et. al. (1999). The column decomposition was subsequently maintained for both pseudo-homogenous and heterogeneous model. The pseudo homogeneous model replicates the steady state multiplicity experimentally observed by Mohl et. aL (1999) for a reboiler heat duty of 340 W; see Fig 1 (b) and (c). Mohl et. al. (1999) explained the steady state multiplicity with kinetic instabilities of the TAME reaction. The two steady states differ in their temperature profile and TAME purity in bottoms flow rate. As high conversion steady state is referred the steady state with higher temperatures and higher TAME purity in contrast to the low conversion steady. The TAME mole fraction in the bottoms flow rate is about 0.22 for the high conversion and 0.18 for the low conversion steady state, respectively; see Fig 1 (c). Fig 1 (b) also shows that the column is predicted not to operate in thermal equilibrium. Liquid temperature profiles, denoted by bold lines in Fig 1 (b), and vapour temperatures are not equal, except in the reactive section when operating at the low conversion steady state. This behaviour is not uncommon for small laboratory columns such as the one under consideration. It is caused by the differences in dew and bubble point temperatures when the liquid bottom mixture is vaporised in the total reboiler. For comparison purposes we also carried out simulations of the steady-state behaviour using a more detailed model additionally accounting for mass and heat transfer resistances at the liquid/solid interface and intra-catalyst diffusion (Dusty Fluid Model) and reaction; see our earlier paper by Higler et al. (2000) for details. The catalyst parameters used are: a)
f
-r ~"
0.24
~
0.22
',-'g 1 00
i
0.20
~3 0.75
~
o.18
~ m
016
- ~ n-pentane
c)
b) + methanol
Feed:
o~
15o '~
o".
='~_
E" 1.25
iso-olefins methanol n-pentane
g
E
0.50 O 25 325
t-
~ 330
n-pentane + TAM~E 9 o
Exp data (high con.) Exp.data (low con.)
-
~
-
~ o o 335 340
, 345
~ ~ ~' 350 355
' 360
250
200
Liq. temperature (high con ) Liq. temperature (low con.)
- - - Vap. - -
Vap.
temperature (high con.) temperature (low con.)
300
350
400
450
500
Reboiler heat duty / [W]
Temperature / [K] --
Pseudo homogenous model
Fig. 1 (a) Schematic depiction of the column configuration, (b) temperature profiles of the high and low conversion steady state at a reboiler load of 340 W, (c) bifurcation diagram for the pseudo homogenous model.
95
w
a)
....
" 0.24
Pseudo homogenous model
Base Case
b) 0.230
/ ~. 0.22 o
~'~ ~"
//
0.225
~= 1.5 (base case)
/
T=I.0
/ /
E
g
0.20
/.
-/ 7 /
E
/
/
10000m'~/m 3
interfacialarea
o.22o
o
uJ
:~
0.1800,
"T'
0.24
~' o~
,-,~, S 350
............. 400 450
500
c)
0.22
" "~ - - ~ ' - ' - " .
.
.
.
....~
~
~
J 500
9 o22.
, u , s e n 0..us.v..y (= 5 x average diffusivity)//
0,222
' '~'x, ~
-,-' '6
m
o
._ ~
', , - ' = , , , I , , , , 400 450
Knudsen diffusivity ignored (base case)
d)
Catalyst thickness: . lmm ~ '
=~
:~
0.215 . . . . . . . . 300 350
0.220
...
0.20
~o
~ .
0 is the shape factor which determines the time-dependent character ([3 < 1 for bum-in, [3 = 1 for stationary or [3 > 1 for wear-out) and the parameter I"1>0 is the characteristic time which represents the time scale of the effect. The expectation of the distribution represents the Mean Time Between Failure (MTBF). The time to repair also follows a statistical law (the lognormal distribution) in which the expectation of the distribution is the Mean Time To Repair (MTTR); the repair time includes the waiting time for a maintenance technician, the diagnosis time and the repair time. 4. INTEGRATION OF BREAKDOWN PHENOMENA M O D E L L I N G MAINTENANCE ACTIVITIES IN A DISCRETE EVENT SIMULATION
AND
4.1 Discrete Event Simulation principles To model the wafer fab in a high degree of detail, Discrete Event Simulation (DES) techniques have been implemented. The simulation tool called MELISSA C++ allows determining the exact chronology of discrete events occurring in the facility. In our formulation, time evolution occurs by "event jump", i.e. from an event to the following one. Typical events taken into account and managed in the simulation core have been widely presented in Navarre et al. (1998).
119
4.2 Downtime and maintenance assumptions Every equipment item has its own macroscopically failure model. Three types of maintenance procedures are considered: corrective and preventive operations performed by a technician as well as Total Productive Maintenance (TPM) actions (Pimor, 1991). These are usually performed by operators and represent simple maintenance actions of supervision and cleaning. Their frequency and duration are known. They are modelled independently of the other maintenance actions. - Corrective m a i n t e n a n c e actions can only be characterised by the MTBF and the MTTR since they occur randomly. Our model stipulates that every time a breakdown occurs, the corresponding corrective maintenance action is one of replacement that brings the system to its initial state, thus the intervention is based on an AGAN (As Good As New) assumption. - The two types of p r e v e n t i v e m a i n t e n a n c e actions (the classical ones and the TPM) are performed in accordance with a schedule. Although several types of scheduling for the preventive maintenance interventions can be used (Pellegrin, 1997), this work is focused on the classical periodic replacement model, in which preventive maintenance actions are of the AGAN type. Since the equipment set has been generally used for production for several years, a decrease in the probability of breakdown occurrence is practically observed, due to preventive replacement. Consequently, the collected failure frequencies (TBF, Time Between Failure) reflect directly the level and quality of preventive policy carried out in the wafer fab. This explains why a link between preventive and corrective actions exits intrinsically through the maintenance parameters. The data set necessary, on the one hand, to describe the workshop structure, operator/technician shifts and products and, on the other hand, the corresponding collecting mode are presented in table 1. The order of magnitude concerning the required number of parameters is also proposed, about 6000, for a workshop with 200 operating steps, 100 equipment units, 6 product families and shifts of 20 operators/5 technicians. Table 1 : Data collection mode and evaluation of parameter number for a wafer fab Data set Collecting mode Magnitude of parameter number Workshop description Available 1000 (number of operating steps, of zones, parallel units, capacity,...) 100 By production managers Production/maintenance teams (shifts, number, polyvalence...) Manufacturing recipes Operating time, Manufacturing specifications 3600 Load/unload time Measured by chronometers Production planning 72 By production managers Reliability data set (Breakdown time, By use of maintenance tuning preventive maintenance, repair time..) software 1200 At given times specified by the user during a simulation run, MELISSA-C++ records information about the workshop status. This information is particularly useful for case studies and concerns production level, inventory or WIP (Work In Process), cycle time, OEE (Overall
120 Equipment Effectiveness) which quantifies the different assignments of equipment (waiting for products, operators or technicians after a failure/for preventive maintenance, load/unload, stop for corrective/preventive/TPM actions, production). The OEE and its equivalent for operators or technicians is therefore an important indicator estimating the equipment and crew utilisation in production (Murphy et al., 1996). 5. I D E N T I F I C A T I O N M E T H O D S F O R R E L I A B I L I T Y AND M A I N T A I N A B I L I T Y LAWS: USE OF M A X I M U M L I K E H O O D E S T I M A T I O N F O R W E I B U L L L A W The available data set related to maintenance aspects includes the date and duration of both failures and predictive maintenance activities. A same methodology has been followed for carrying out a statistical data treatment in order to identify reliability and maintainability laws. The approach is illustrated for reliability law identification, for which Weibull law has been assumed. Two methods have been applied in this study : a practical one based on an empirical estimation of Weibull law hazard function using data graphical representation on the so-called Allan-Prait paper and a method based on maximum likehood estimation. For the sake of illustration, this latter is presented in what follows. As previously mentioned, the AGAN assumption has been put forward for all maintenance operations. Under this hypothesis, the TBF are easy to compute by difference between either an AGAN state (the last preventive or corrective operation) and a failed state or an AGAN state and a non-failed state (preventive operation). Let us note that the times are right censored data (Newton, 1991 ;Tan and Kramer, 1995; Pellegrin, 1997). The likehood function L is constructed as the product of the probabilities for the events k
that have occurred: L = 1-I
Pi 9Under an AGAN assumption, Pi for a failure event is the
i=l
probability density function f(t)dt normalised by the integral over the part of f(t) from which oo
~f(t)dt= 1-F(t, ), where tLi is the model time after the last ,L, maintenance (or beginning component time if no maintenance). F represents the cumulative distribution function for component failure time. Note that for an AGAN model tLi- tRi -- 0, thus cti = 1. (tRi is the reference time for event i). Likewise, for right censors (maintenance or survival without failure) Pi can be represented by R(t)/oq = (1 - F(t))/ctl In the same way, o~i = 1 under an AGAN hypothesis. Under these conditions, the likehood function is
the failure is sampled a i -
L - ~-I
flf,"tiq,, ,a-1"e-[-~) ( t,~a H m ~-~-qj
i=l
.1. 1 . i=l
1-e -I~] p
The search for the maximum value of the likehood function gives the model parameters with the highest probability. The maximisation of Ln L then gives relations satisfied by 13 and q. Using the formulation of (Newton, 1991), the following expressions are obtained : 1
~tiPlnti+s , , ~-[ tifl + s c,p 1 1
~lnts L , - fl- n
_ o
~
(1)
17=
~ c'P.l~ (2) t,,+ -n
121 The 13parameter is computed by the Newton iterative method, with 130as initialisation value. The 11 parameter is deduced from equation (2). Determination of Weibull law parameters involves reliability data collection (see table 1) so that the TBFs and censor times can be computed. The Weibull parameters are thus estimated using the aforementioned method. The MTBF is given by Frl, where F is the second species eulerian function of (1+1/13). Due to a relatively low number of data, the test of Little, Mc Clave et Often belonging to the Kolmogorov-Smirnov type) tests has been adopted for quality adjustment. 6. INDUSTRIAL VALIDATION 6.1 Validation on a manufacturing workshop
A reference state (an empty state) is considered for the workshop at simulation beginning and the corresponding simulation parameters are collected. For the following campaigns, the indicators reflecting the workshop behaviour are computed and also measured empirically to be compared. Figure 1 represents the evolution of the inventory state 1 (or WIP) vs. number of campaigns and shows that a good quantitative agreement is observed. The discrepancy (< 10% from 15th campaign) can be yet explained : first, the simulation model does not take into account the so-called "engineering" batches produced for investigations on new or improved processes as well as on new products; second, some batches may require rework actions. They both represent in average 2 to 5% of the total batches. Since WIP and product cycle time are strongly linked, it can be expected that predicted product cycle times will be slightly lower than those collected. Figure 2 shows that cycle times are 10% lower than those measured empirically, which can be explained as for WIP. Note that downtime equipment modelling involves a more important WIP level (multiplied by a factor 4) than in an ideal case with no production interruption by maintenance or breakdowns. A fluctuating evolution is also observed due to the random nature of occurring failures. Cycle time
Number of products
Averaoe cvde time (e.xpefimentall
ExperimentalWIP t
..........
20%
l
Product
1 (withbreakdown modelling) Product I (withoutbreakdownmodelling)
~WIP (withoutbreakdownmodelling) ......
!
:
0
5
Campaign,
number 10
, 15
Figure 1 WIP vs. campaign number
0
--
.....-.~
~ .........................................
5
10 Campaign
- ..................
number
15
20
Figure 2 Cycle time v s Campaign number
Another validation has been performed with the OEE indicator for technical distinct equipment (individual or collective wafer treatment, long or short operating time, with or without operator intervention ...) located at different manufacturing steps. Typical results obtained are presented in Table 2 and confirm that a good agreement is observed for experimental and computed values. 6.2 Validation on a "probe" workshop
The same operation was conducted in a "probe" workshop, dedicated to wafer test. Figure 3 represents the evolution of cycle time for a given product in function of campaign number. i For confidentiality reasons, the figures presented exhibit a "blind" scale in the Y-axis.
122 It can be seen that an equipment item has been stopped during the 10 th campaign, thus strongly affecting the product flow, as observed in the following campaigns. Table 2 9Comparison between experimental and computed O.E.E. Production Preventive Breakdown Values Waitingfor Operator
Technician
Product
Experimental
42%
4%
5%
25 %
3%
21%
Computed
39.7%
3.9%
5.3 %
25.1%
2.8%
23.2%
Relative discrepancy
6%
-2.5 %
5.5 %
1%
7%
10%
Figure 3 shows than the computed cycle times, although slightly inferior, are in good agreement with those measured in practice, which also means that some random phenomena have been neglected. The cycle times are much more fluctuating and dispersed than in a manufacturing workshop due to a limited number of equipment. Cycle time
~.~
0
i
2
"'---"pr%per~medntaluV:lue
i
4
t
6
/
i
8
Campaign number
i
10
12
Figure 3 9Evolution of product cycle time vs. campaign number (Probe workshop) 7. CONCLUSIONS This study has confirmed that reliability plays a pivotal role in production system manufacturing. Actually, plant modelling without taking into account equipment downtimes may actually misrepresent cycle times, throughput rates and inventory state. The use of reliability models, for which data collection must not be neglected, is essential for quality improvement of a production system behaviour. Reliability modelling, embedded in a discrete-event simulation tool for representing the workshop behaviour, provides a framework to investigate design, operating conditions and maintenance strategies in a batch plant. All these aspects have been largely studied in following works (Charles, 2000). REFERENCES
1. Charles A.S., 2000, INPT, PhD Thesis 2. Murphy, R., Saxena, P., Lewinson W., 1996, Semiconductor International, 125 3. Navarre, D., Pantel, M., B6rard, F., Pantel, C., Pibouleau, L, 1998, 17th IASTED Int. Conference, MIC'98, Grindelwald Switzerland 4. Newton D.W., 1991, Reliability Engng and System Safety, 34, 7 5. Pellegrin, C., 1997, Fondements de la ddcision de la maintenance, Economica 6. Pimor, 1991, TPM, La maintenance productive, Masson, Paris 7. Tan, J.S., Kramer, M.A., 1995, Reliability Engineering and System Safety, 49, 155-169
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
123
Automatic Structural Characterization of DAE Systems E.F.Costa Jr. b, R.C.Vieira b, A.R.Secchi a and E.C.Biscaia Jr. b a Departamento
de Engenharia Quimica- UFRGS - Porto Alegre - RS - Brazil
b Programa de Engenharia Quimica- COPPEAJFRJ - Caixa Postal 68.502 CEP: 21945 - 970 Rio de Janeiro, RJ - Brazil -
[email protected] The characterization of the DAE system is unavoidable for the construction of generalpurpose robust codes for numerical integration of DAEs. Such information is needed for index reduction and consistent initialization of DAE systems. In the present contribution, it is proposed the utilization of a graph theoretical approach associated with automatic differentiation tools for the automatic generation and resolution of the consistency system. All differentiation required is performed exactly and at an affordable computational cost, and the index reduction prior to numerical resolution is performed automatically. The resulting code from this project has been incorporated to the numerical integration code DASSLC. Several examples illustrate the potentiality of the approach. Encouraging results were achieved on both traditional benchmarks for initialization procedures and more realistic problems. 1. INTRODUCTION When solving DAEs one must concem about the index of the system and about the consistency of the initial conditions. The index of a DAE is the number of times that all or part of the system must be differentiated with respect to time in order to convert it into an explicit set of first order ODEs. Index 1 systems may be solved with modified ODE methods, while higher index systems (systems with index 2 or greater) require specific methods. Generally, higher index systems are rewritten in an index 1 formulation and solved with standard integration codes like DASSL [ 1] or DASSLC [2], written in FORTRAN and C. During the index reduction, some extra algebraic equations are obtained which generally correspond to derivatives of the original algebraic equations. Those hidden algebraic equations along with the original DAEs compose the extended system. Consistent initial values must satisfy not only the DAE system but also the underlying extended system [3]. Several rigorous methods for initialization have been proposed in the literature, and they all depend to some extent on the characterization of the DAE system. In other words, in order to perform the numerical integration of a DAE with standard integration codes, it is necessary to: (i)perform the characterization of the DAE, in order to determine its index and, if required, the set of equations to be differentiated; (ii)perform the differentiation; (iii)check for feasibility of the initial states arbitrarily set; (iv)solve the consistency system for a consistent initial state. In this contribution it is discussed an extension to the numerical integration code DASSLC. The code developed aims the preprocessing of the DAE in order to reduce its index and/or
124 determine consistent initial conditions. In w the DAE characterization issue is addressed. Existing code and algorithms are discussed and their potentiality and limitations are shown. In w the approach proposed in this contribution is detailed. Some numerical examples presented in w show the potentiality of the proposed approach and explain some of the main features of the code. In w some perspectives of the research in the area and the next developments of the code are discussed. 2. CHARACTERIZATION OF DAE SYSTEMS Consistent initial conditions for DAEs must satisfy not only the original equations in the system but also their differentials in respect to time up to some order. Whether or not this additional requirement actually imposes extra constraints on the initial values depends on the particular problem. Pantelides [4] derived a criterion for determining whether differentiation of a subset of the DAE system provides further constraints to be satisfied by initial conditions. A graphtheoretical algorithm was proposed to locate those subsets which need to be differentiated. Unger et al. [5] implemented two systematically different structural approaches as tools for the analysis of DAEs, in the codes PALG and ALGO. In structural computations, only hard zeros (0) and nonzeros (*) are distinguished, and a DAE system is merely represented by the pattern of its jacobian. Since the computation of the index and of the number of degrees of freedom require (repeated) differentiation, this operation has to be defined in the structural sense. Unger et al. [5] defined the structural differentiation to be of the type linear if the derivative of f(z) depends only on z'. On the contrary, if the derivative of f(z) is a fimction of z and z' the differentiation is said to be of the type highly nonlinear. Generally, neither way of structural differentiation yields the correct pattern of the Jacobian of the differentiated function. The linear and highly nonlinear definitions yield the patterns with minimum and maximum nonzeros entries, respectively. The structural computation as implemented in PALG and ALGO codes present two main drawbacks: (i) as the structural differentiation yields the wrong pattern for the derived equations, it impacts on the feasibility check of an arbitrary initial set; (ii) a purely structural gaussian elimination can not identify a set of equations that is structurally regular but actually singular. A pure numerical approach for the initialization of DAEs with known index v was proposed in [6]. The authors construct the consistency equations with all total differentials of the DAE with respect to time up to order v and extra user specifications. Then a family of forward finite difference approximations of these equations is constructed and the resulting system is solved in a least square sense. The solution of this rank-deficient algebraic problem poses a nontrivial numerical task, and existence and uniqueness of the solution can only be proven for linear constant coefficients DAEs. If only the necessary differentiation was performed in the previous algorithm, the resulting non-linear system would have a full rank Jacobian, and could be in principle solved exactly. Kr6ner et al. [7] proposed an algorithm that consists of three steps: (i) a structural analysis of the problem employing PALG/ALGO codes, in order to determine the index of the DAE, the minimum set of equations to be differentiated and the number of degrees of freedom; (ii) the numerical differentiation of these equations as in Leimkuhler's algorithm; (iii) the solution of the non-linear algebraic system. It must be stressed that steps (i) and (ii) are not exact, and
125 hence the resulting algebraic system may present severe numerical problems. The authors pointed out that the solution of the algebraic system was the most difficult step of the algorithm, and recommended restrictive damping strategies if there are steep initial gradients. Murata and Biscaia Jr. [8] developed the symbolical code INDEX for characterization of DAEs based on symbolical computation and MAPLE software. As output the code identifies general systems of index one or higher and Hessemberg forms of index two and three, besides checking for linearity and solvability of DAEs with certain properties. As an auxiliary task INDEX supplies the patterns of the jacobian matrices to ALGO/PALG structural analysis codes. In spite of the ever known tendency of computer algebra systems to use much memory space and computing time, the authors report success on the characterization of several examples. However, the code can not be coupled with a numerical integration software developed in a standard programming language (as FORTRAN or C), and most of all presents severe limitations on the size of the problems it can handle. 3. THE EXTENDED DASSLC CODE In this contribution, it is discussed the characterization/initialization module recently added to DASSLC. Based on Pantelides' algorithm, it uses the proposed graph-theoretical approach to identify the minimum set of equations to be differentiated in order to generate an index one formulation of the original DAE. Through AD code ADOL-C [9], the differentiation is performed automatically at an affordable computational cost, and, most important of all, the differentiation technique employed does not incur on truncation error. The structural pattern of the jacobian matrices required are computed via numerical perturbation of the original equations and of the exact extended system. The user must supply the DAE ..system F(z,z')=0 according to DASSLC standards. The code will apply Pantelides' algorithm, construct the extended system and inform the number r of degrees of freedom of the system. The user is prompted to give r arbitrary initial conditions, on which the code will perform a structural check for feasibility. It must be stressed that structural feasibility is a necessary but not sufficient condition for feasibility. That comes from the fact that structural computation can not determine if a structurally regular matrix is not in fact a singular matrix, as previously discussed. In the PALG/ALGO codes, structurally feasibility was neither necessary nor sufficient, as the pattern of the derived equations was computed in an approximated sense. In the present code, when an initial arbitrary set is found to be structurally feasible, the code will carry out the resolution of the extended system by a Newton type method. The extended system was constructed via automatic differentiation, and hence is exact. The initial conditions obtained from its resolution can be fed directly to the numerical integrator DASSLC. If there is a failure in any of these stages the user is warned and prompted to supply another arbitrary set. 4. NUMERICAL EXAMPLES 4.1. Condenser The model of a single component condenser presented by Pantelides et al. [10] is frequently adopted in literature for illustrative purposes. Although being a contrived example with non rigorous modeling, the widespread use of this model has transformed it into a kind of "benchmark" for initialization procedures [4,5,11 ].
126
iV = F - L PV=NRT
(la) (lb)
N C p l ~ = FCp (Tin- T)+ LAH + UAs(Tc- T) P = A exp(- B/(T + C))
(lc) (ld)
The index 2 formulation shown in Equations (1) was analyzed. The automatic index reduction was successfully performed, and the number of degrees of freedom (r = 1) was correctly determined. Equations (1 b) and (1 d) have been automatically differentiated in order to uncover the extended system. All variables are feasible arbitrary conditions, and the Newton code was capable to perform the numerical resolution of the extended system. 4.2. The Unavoidable Pendulum
The well known index 3 pendulum problem [4,5,8] is used to illustrate the superiority of the developed code over reported implementations of Pantelides' algorithm. This system presents two degrees of freedom (r=2). =G j, = Vy
(2a) (2b)
~x = ~,x
(2c)
~,y = ~ y - 1
(2d)
x 2 + y2 = 1
(2e)
PALG and ALGO codes, due to the inexact differentiation used, can not determine whether the set [Vx, Vy] is feasible or not. Depending on the type of the differentiation used, the codes reached to conflicting solutions. The code reported herein has been capable to correctly determine r and identify all feasible sets. Equations (2a) and (2b) have been differentiated once, and (2e) twice. Additionally, sets leading to singular consistency systems and/or impossible initial values have been eliminated by the resolution of the extended system with the Newton type algorithm implemented. 4.3. Batch Distillation Column
The following example illustrate the utilization of the developed code on a more realistic problem. The start-up of a batch distillation column [12] is typically a high index DAE system with highly nonlinear state equations and equilibrium relationships. Additionally, it can easily become a large scale system as mass and energy balances must be solved for every plate and there are several components involved. In this contribution a column with np plates and nc components is simulated, what leads to system with approximately (np+2).(2 nc+6) equations.
127
13/0 = V/+I Yi+l,i. + Li- ni-l'J
(3a)
Li -ni'j ~ -- V~ Yi,j
N, = ~--,~c ,,,j
(3b)
N i h i = Vi+1 Hi+ ~ + Li_ ~ hi_ 1 - Z i h i -I'~ H i + Qi
(3c)
yi,j = x i j K j ( T ~ ) ,
(3d)
Ni hi
tic = Zj=ll"li,j hj(Ti)
H i = Z~c=lYi,j Hj(T~) ~.c i=l Yi,j = 1 where:
(3e) (3f) i=0, 1..... np+l
j=l,2,...,nc
Hp,i(T)=~:=oe,,kZk~
(3g)
Ky(T):T~3k=oaj,kTk~
ej-Z~clYi,jep,i(rj) In the start-up of the column, the fbllowing relations between liquid and vapor fluxes hold: Vi+ 1 = L i
i: 0,1,..., np
(4)
In previous equations, x and y represents molar fractions on liquid and vapor phases, h and H are enthalpies on liquid and vapor phases, T is temperature, i represents plate and j represents component. All parameter used can be found in [12, pp.211-212]. With the extended DASSLC code discussed herein, it is possible to integrate the high index system directly. All index reductions and manipulation in order to achieve consistent initialization are performed automatically. The user should only give the original DAEs according to DASSLC standards, determine which are the arbitrarily set initial variables and provide initial values for those variables. Characterization results for an example with 5 components and 12 plates are shown in Table 1. It should be emphasized that other sets could be determined arbitrarily. Table 1: Characterization results for example 4.3. nc=5, np=12 Index 2 Equations (3b), (3d) (3e) and (3g) must be differentiated once Degrees of freedom: 70 [ nc (np+2) ] Feasible arbitrary initial set: moles of component i on plate j (nij) 5. CONCLUSIONS
It is presented the extension of the code DASSLC that permits its utilization for the numerical integration of high index problems. The characterization/initialization module added preprocesses the original DAE systems and converts it into an index one problem with consistent initialization. Based on Pantelides' criteria and on the Automatic Differentiation
128 tool ADOL-C, the exact and affordable computation of derivatives required is one of the key features. With the exact calculation of derivatives, structural feasibility became a necessary condition for feasibility, and this criteria is used to perform the first search for consistent arbitrary values. A Newton-type method is then used to eliminate singular but structurally regular subsets and to compute the consistent initial values. The condenser and the pendulum problems have been analyzed. These "benchmarks" have shown that the results achieved with the proposed approach are either equivalent to of better than previously reported characterization results. Additionally, the batch distillation column example has shown that the methodology proposed is specially suited to the highly nonlinear large scale models often faced in the simulation of chemical engineering processes. REFERENCES
1. 2.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Petzold, L. DASSL code, Computing and Mathematics Research Division, Lawrence Livermore National Laboratory, L316, PO Box 808, Livermore, CA 94559 (1989) Secchi., A.R. "DASSLC: User's Manual, a Differential-Algebraic System Solver", Technical Report at Chemical Engineering Department, UFRGS, Porto Alegre, RS, pp.49 (1992), http://www.enq.ufrgs.br/enqlib/numeric.DASSLC Brenan, K., Campbell, S. and Petzold, L. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, New York, Elsevier Sc. Publ. Co (1989) Pantelides, C.C. SIAMJ. Sci. Stat. Comp. 9 (1988), 213-231 Unger, J., Kr6ner, A. and Marquardt, W. Comp. Chem. Eng. 19 (1995), 867-882 B.Leimkuler, L.Petzold and C.Gear, SIAMJ. Num. Anal. 28 (1991), 205-226 Kr6ner, A., Marquardt, W. and Gilles, E. Comp. Chem. Eng. 16 (1992), S 131 Murata, V. and Biscala, E. JR., Comp. Chem. Engng. 21 (1997) $829-$834 Griewank, A., Juedes, D., Mitev, H., Utke, J., Vogel, O. and Walther, A., ACM TOMS 22 (1996) 131-167 - Algor. 755 Pantelides, C.C., Gritsis, D., Morison, K. and Sargent, R.W.S. Comp. Chem. Engng. 12(1988), 449-454 Kr6ner, A., Marquardt, W. and Gilles, E. Comp. Chem. Eng. 21 (1997) 145-158 Holland, C. and Liapis, A.I. Computer Methods for Solving Dynamic Separation Problems, McGraw Hill Publishers, 1983
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
129
Steady state analysis of membrane processes for the treatment of industrial effluents A. M. Eliceche a, S. M. Corval~n a and I.
Ortiz b
"PLAPIQUI-CONICET, Chem. Eng. Dept., Universidad Nacional del Sur, 8000 Bahia Blanca, ARGENTINA. bDpto.de Quimica, Universidad de Cantabria, 39005 Santander, SPAIN. e-mail: meliceche~0lapiqui.edu.ar
The feasibility of operating a new membrane technology for the treatment of industrial effluents in a continuous mode is analysed. The selection of the operating mode needs to be addressed at the conceptual design stage. The semicontinuous operation has been studied previously and the optimum operating conditions were reported by Eliceche et al. (2000). In this work a steady state configuration is proposed, the corresponding modelling equations are developed and a methodology for the selection of the operating conditions is presented. A given compound can be removed from the effluent in the extraction module and simultaneously it can be recovered in the stripping module, for recycling and re-use in the plant that generates the effluent. Environmental applications of this technology will reduce the amount of contaminant disposed off finally into the environment, leading to a cleaner technology. This fact motivates the need to promote its industrial application. The removal and recovery of Cr(VI) from surface treatment effluents is studied. 1. INTRODUCTION Membrane extraction processes using hollow fibres are of particular interest because of their versatility. In addition, the membrane process not only removes the required compound from the effluent but it can also recover and concentrate it simultaneously as a product. This product can be recycled and re-used in the same process that generated the effluents or in some other application. Extensive studies on dispersion-free solvent extraction with the use of microporous membranes have been carried out by D'Elia, Dahuron and Cussler (1986) and Prasad and Sirkar (1988) among other authors. Ho and Sirkar (1992) presented a general review of the NDSX technology and its applications. The simulation of a Non-Dispersive Solvent Extraction process, in batch and semicontinuos operation, was carried out by Alonso and Pantelides (1996) in gPROMS. An analysis to validate the models with the performance of a NDSX pilot plant is reported by Alonso et al. (1999). The optimisation of the semicontinuous mode of operation was implemented by Eliceche et al. (2000) in gPROMS and gOPT.
130 In this work, the steady state operation of NDSX membrane processes and the optimal operating conditions are found. As a case study, the removal and recovery of Cr(VI) from wastewater of a surface treatment plant is presented and the main numerical results reported. 2. NONDISPERSIVE SOLVENT E X T R A C T I O N PROCESS The main components of the process are two hollow fibre modules, the extraction and the stripping modules, as well as two tanks for the organic and stripping phases. A schematic diagram of the process, with both membrane modules operating in a continuous and co current mode, is shown in Fig. 1. In the semicontinuos mode of operation, the stripping stream runs in a batch mode. The stripping solution is concentrated in the stripping tank, until the required composition for recycling and re-use is reached.
Fig. 1 - Schematic diagram of the steady state operation In the membrane modules, the aqueous phases run through the lumen of the hollow fibre and the organic phase flows on the shell side. The organic solution wets the porous wall of the hydrophobic fibre. Organic phase dispersion into the aqueous phase is prevented by a slight overpressure of the aqueous stream. The organic phase, Fo, extracts the contaminant from the effluent in the extraction module. The contaminant is back- extracted from the organic phase and concentrated in the aqueous phase, in the stripping module. The organic stream is the carrier between the extraction and stripping modules. The organic stream leaving the extraction module is fed into the stripping module and the organic stream leaving the stripping module is recycled to the extraction module via a buffer tank. The stripping aqueous solution, Fs, takes the solute from the organic stream and is fed to the stripping tank for further recycling to the stripping module. An aqueous stream, Fp, free of contaminant enters the stripping tank and leaves it concentrated with the contaminant. The recovered contaminant, extracted from the effluent, leaves the plant in the purge stream, for recycling and re-use in the same plant that generated the effluent or other alternative industrial process that can re-use that stream.
131 3. STEADY STATE M O D E L OF A SELECTIVE M E M B R A N E S E P A R A T I O N PROCESS
The component mass balances in the organic and aqueous phases of the extraction, stripping modules and stirred tanks, including the mass transport through the membrane are described by the following set of ordinary differential and algebraic equations. For the development of the mathematical model it was assumed that: i) The main resistance to the solute transport lies in the micro porous membrane, therefore mass transfer resistances at both the aqueous and organic bulk phases were neglected. ii) The reactant species are in equilibrium in the whole interface. EXTRACTION MODULE Aqueous Stream 0
-
-
dCe A +__Km dz FeZ
E
(CoE - C o)
C e (Z=0)
--
Ce,in
(1)
Organic Stream __
dCE dz
+
A FoE
K m (foE
CEin ,
C E
-
o)
--
C S.... t
(2)
CT
(3)
STRIPPING MODULE Aqueous Stream 0
-
--
dCs A s + K m (CSo-Coi ) dz FsL
Cs,in
"--
Organic Stream. 0
-
--
dCS~ A --+ K dz FoL
s
m (CSo - C o i )
s
Co,in
--
E
(4)
C .... t
STRIPPING TANK 0
"--
Us ( C .... t - f T )
-
(5)
Up CT
Equations (1) to (5) represent the steady state behaviour of the extraction and stripping membrane modules. The solution of the ordinary differential equation (ODE) system allows the steady state simulation of the process that has been implemented in gPROMS (2000). 4. F O R M U L A T I O N OF T H E OPTIMIZATION P R O B L E M The objective function proposed is the maximisation of the effluent treated Fe. The algebraic
and
differential
equations
(f(:~,x,v)
=
0)
and
their
initial
conditions
( I(x, v) = 0 ), equations (1) to (5), that represent the steady state model of the NDSX process are formulated as equality constraints in the optimisation problem P1. Separation
132 specifications related to maximum composition allowed in the effluent for final disposal U < C .... t ) and minimum concentration in the stripping phase for further recycling and
(C .... t
re-use (CV > CV' c ) are formulated as inequality constraints g (x,v) in problem P 1. The vector v of the optimisation variables includes the flowrate of the organic, stripping and purge streams. Upper bounds on the optimisation variables v u correspond to maximum pump capacities. The non-linear programming problem is formulated as follows: Max
F e (v)
v
f ( ~ , x , v) I(x, v)
=
0
= 0
(Pl)
g(x, v) < 0 v c
_
0
;
m={w
1,W 2 , . . . , W N }
where Yk, r is the desired ethanol concentration, Yk is the actual ethanol concentration, ,;Ck , and y k are constant positive weights (chosen as 2, k = 10 and y ~ = 1 ), and w k are bounded inputs in the range 0 _< wk _< 130 g / L . The solution of the constrained optimization problem for the determination of the optimal sucrose concentration profile is shown in Fig. 3.
4. DISCUSSION OF RESULTS The fermentation process exhibits nonlinear behavior with strong interactions among the variables (Fig. 2). High-density batch fermentations present sucrose inhibition at early stages of the process and low conversion to ethanol. The way to overcome this drawback was by supplying sucrose to the fermentation following an optimal feed rate policy. The fermentation process was modeled as a combination of local models identified along the process trajectory. Using physical and experimental information (i.e. Fig. 2), 7 different operating regimes were identified. The parameters associated with the local models were estimated using dynamic optimization, which is a least-squares Levenberg-Marquart algorithm, coupled with a Runge-Kutta method.
i''000("
5
_1
@
.0 II1
@
0
10 20 30 time, hours
5OI 0
40
8o
I~
2'0
time, hours
3'0
4O
5
-J 6o ~40
pH
9
.r 20 UJ C 0
9
10 20 30 time, hours
40
1o
10 20 30 time, hours
Figure 3. Experimental profiles for the fed-batch operation with optimal sucrose profile. The process objective was to optimize a function of the sucrose concentration to track the ethanol concentration along of a desired profile (Fig. 2). The decision variable (sucrose concentration) was bounded in the range 0 to 130 g/L in order to avoid substrate inhibition
146 observed in batch fermentations. A subsequent optimization was performed to obtain the optimal feed flow rate without exceeding the pump and the reactor capacities, 0.1 L/min and 3.0 L respectively. It is important to mention that the sucrose concentration in the feed and the initial volume for a fed-batch process are two extra degrees of freedom that the control designer can manipulate to satisfy process requirements. Figure 3 shows the experimental profiles. The implementation of the optimal profile was open-loop; therefore it was difficult to track the system along the desired trajectory. The fedbatch operation exhibited high sucrose to ethanol yield in comparison with the batch operation (Fig. 2). REFERENCES
1. D. Bonvin, J. Proc. Cont., 8 (1998) 355. 2. G.P. Casey, C.A. Magnus, W.M. Ingledew, Applied and Environmental Microbiology, 48 (1984) 639. 3. F.H. White, Proceedings of the 15th Convention of the Society of Institute Brewing (Australia and New Zealand), (1978) 133. 4. F.S. Wang, and J.W. Sheu, Chem. Eng. Sci., 55 (2000) 3685. 5. O. Galfin, J.A. Romagnoli, A. Palazoglu, Chem. Eng. Sci., 55, (2000) 4435. 6. R. Murray-Smith, R., and T.A. Johansen (eds.), Multiple Model Approaches to Modeling and Control, Taylor & Francis, London, England, 1997.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) ~q-)2001 Elsevier Science B.V. All rights reserved.
147
Simulation of the FGD In-Duct Injection Technology using complex kinetic models A. Gareaa, J. A. Marqu6s a, T.L. Hechavarriaband A. Irabiena. a Dpt.
Quimica, E.T.S.I.I.y T., Universidad de Cantabria. Avda. los Castros, s/n, 39005 Santander, Spain.
b Facultad de Ingenieria Quimica. Universidad de Oriente, Santiago de Cuba, Cuba.
The aim of this work is the computer aided simulation of an entrained flow reactor operating under typical conditions of FGD in-duct injection at low temperatures. The modelling was performed considering the gas-solid reaction in a single particle of sorbent that is moving as well as the gas phase at any axial distance of the duct. The overall reaction rate is calculated from the transport resistance through the ash layer and the chemical reaction taking place at a sharp moving interface in the radial axis of the particle. The experimental profiles of the SO2 concentration in the extemal gas phase obtained in a pilot plant of In-Duct-Injection of Ca(OH)2 were introduced to the simulation in order to estimate the parameters of the reaction under study, working at different Calcium/Sulfur molar ratios. The proposed model allows to describe the SO2 removal and the solid conversion levels when varying the residence time in the duct, in good agreement with the experimental values.
1. INTRODUCTION Three main categories of Flue Gas Desulfurization (FGD) technologies are considered for controlling the emissions of sulfur dioxide in large coal power plants: dry sorbent injection, semi-dry and wet processes. The first option, Dry Sorbent Injection, provides a low-cost retrofit alternative for existing plants to meet emissions regulations, compared to the semi-dry and wet systems that require additional equipment and the subsequent sludge treatment in the case of wet processes. The FGD In-Duct-Injection technology at low temperatures involves the injection of a dry sorbent, typically Ca(OH)2, in conjunction with sprayed water in the ductwork ahead of the particulate collector [ 1-3]. The residence time of gas and solid phases in the duct is up to 3 seconds typically. The modelling of the desulfurization process at in-duct conditions is of practical relevance for the design and operation in large scales. It is required to describe the SO2 removal and the sorbent utilization levels at different operational conditions, being the Calcium to Sulfur molar ratio (Ca/S) the most important parameter.
148 The experimental results from the In-Duct Injection process at a pilot plant scale show that the coupling between gas and solid profiles can not be explained using a simplified model that only takes into account macroscopic balances for gas and solid [4,5]. Attemps to the modelling of the gas-solid reaction between the SO2 and calcitic sorbents at a microscopic level included the shrinking core model as well as the grain model in order to model experimental results from reaction systems based on fixed bed or thermobalance in laboratory scale [6-8]. The objective of this work is a better understanding of the gas-solid reaction that takes place in the duct section and the establishment of the desulfurization model, as a necessary tool for the design and optimization of the FGD In-duct injection process at low temperatures. The basis of the shrinking core was applied to a moving particle exposed to different SO2 concentration depending on its axial location of the duct section for describing the SO2 removal level and the sorbent utilization at the corresponding residence time. 2. M O D E L L I N G Since the pioneering work of Yagi and Kunnii in 1955, several particle structures were considered for the non-catalytic gas-solid reactions, such as the sharp interface or shrinking core model for solids assumed to be nonporous, and the grain/pore picture for porous structures [9]. The development of models involving solid reagents in some porous form was focused in the description of the structure evolution, in these cases, the effective diffusivity of gas in the ash layer and the surface kinetic constant may depend on the microstructure changes with conversion. Krishnan and Sotirchos pointed out for the direct sulphation of limestones that the reaction rate decreased faster during the reaction period than the predicted by the shrinking core model. It was required the modification of the efective diffusivity as an exponential function of the distance from the external surface of the particle [6]. Analogous conclusions were reported from other studies with calcium based sorbents, treating the effective diffusivity of gas in the product layer as dependent on solid conversion [10-12]. Taking into account the previous studies, the formulation of a model for the gas and solid behaviour in the in-duct injection process was based on the application of the shrinking core model to a single particle at any axial position in the duct. It is important to remark that the particle is in contact with a gas phase of variable SO2 concentration related to its position. The following assumptions were considered for modelling: An average particle diameter of the Ca(OH)2 sorbent. Isothermal conditions in the duct. Negligible external mass transfer resistance, being exposed the initial surface of the particle to the SO2 concentration in the external gas phase. - Steady-state profiles of SO2 concentration over the distance in the reacted layer, due to the shrinkage of the unreacted core is slower than the flow rate of SO2 toward the unreacted core. - The chemical reaction described as a simple first order reaction at the sharp interface which is moving inward toward the center of the particle during the reaction progress. -
-
-
149 Under these assumptions, the mass balance on the gaseous reactant
10[
OCt]: Or J
r 2 Or r 2 D e
(802)yields (1)
0
being concentration at any radial distance (r) in the reacted core; mol m 3. C~ the The superscript z accounts for any axial position in the duct. De the effective diffusivity in the reacted layer; m E S "1 .
$02
For providing the increase of the mass tranfer resistance with the progress of reaction, the diffusional parameter, D e , was also considered variable with the conversion of the particle as well as variable with the distance from the external surface of the particle. The boundary conditions required for solving equation (1) are - At the extemal surface of the particle: r=Ro
c t =C2
(2)
- At the reaction from: r=Rc
-Oe~
ac t r = rs
(3)
working with an average outer radius of particle Ro =2.75 10 .6 m. The chemical reaction rate per unit of surface area (rs) was defined as first order related to the SO2 concentration (Cd) corresponding to the radius of the unreacted core, Re r, : k, c~
(4)
The core of particle, R c , is shrinking with time as a result of the overall reaction rate that include both chemical reaction and mass transfer, being calculated as follows
- -dt
:
U so 2
ro . . . .
(5)
ll
where ps is the solid density (2.24 10"3 kg m -3) , Ms the molecular weight (80 g mol "1 of commercial Ca(OH)2 sorbent), and v the stoichiometric coefficients. The overall reaction rate equation is given by the following equation, in terms of mol
S "1
c~
ro. . . .
ll :
1
- - . + 4~rR2c k s
Ro-R~ 4z~R o R c De
(6)
150 The conversion of the particle is related to the variable Rc by the equation:
Xs= 1_ l R c l 3 t, Ro)
(7)
The simulated 802 concentration in the extemal gas phase at any axial position (Coz) becomes from the material balance x,
SR = 1 -
C---L~
(8)
C in.dua being introduced the operational parameter SR defined as the Calcium to Sulfur molar ratio (Ca/S stoichiometric ratio). The system of equations were transformed in order to work with the corresponding dimensionless variables of SO2 concentration, radius of the particle and total lenght of the duct. The parameters of the proposed model are the effective diffusivity in the reacted layer, De, and the surface reaction rate constant, ks. A commercial software, gPROMS (PSE Ltd.), was used for solution and the estimation of the parameters was based on the criteria of least squares minimization related to the experimental data of SO2 concentration in the gas phase at different locations in the axial axis of the duct (experimental values of Co). The experimental data were obtained in a pilot plant scale of own design detailed elsewhere [5]. The flow reactor was provided by different gas phase samplying points in order to analyze the SO2 concentration trend up to 3 seconds of residence time in the reactor, as the maximum value within the typical conditions of the in-duct injection process. The operating conditions were fixed to 1000 ppmv of SO2 concentration in the inlet gas phase, temperature of 60~ and relative humidity of 60%, varying the Ca/S parameter between the ratio of 4 to 18 in order to achieve a wide range of SO2 removal levels. 3. SIMULATION RESULTS The effects of mass transfer and chemical reaction on the overall reaction rate can be discussed from the obtained simulation results for three series of experimental data corresponding to the Ca/S molar ratios: 4, 9 and 18. The values of the estimated parameters, De and ks, in terms of time constants (DIRo 2) and (k/Ro), are shown in Table 1, with the values of the standard deviation of the estimation from each Ca/S ratio (crn_l). The estimation of the parameters for the whole range of CaJS was also performed and included in Table 1 as Global. The standard deviation values corresponding to the simulation of each serie of data Ca/S with the obtained parameters from the global estimation procedure were identified by O'n-l(global). The fitting to the experimental data is shown in Figures 1 and 2, being represented the trends of SO2 concentration in the gas phase and solid conversion up to the total residence time in the reactor or duct (3 s).
151 Table 1. Simulation results for diffusion and chemical reaction: estimated values of the effective diffusivity De and the kinetic constant k~, for Ca/S ratios 4, 9 and 18.
DIFFUSION AND CHEMICAL REACTION De~o 2 ($'9 k ~ o ( S ' 9 0"n-1[(Tn-l(global) Dan
DIFFUSIONAL CONTROL De/Ro 2 (s-t) Crn.l / O'n.l(globaO Ca/S 4 9 18 Global
83.39 46.35 36.76 48.98
0.060 / 0.094 0.084 / 0.092 0.063 / 0.087 0.089
93.48 47.05 36.54 48.55
4.31 10 4 1.54 107 4.23 10 6 1.3510 7
0.060 / 0.103 0.084 / 0.092 0.063 / 0.088 0.089
4.6 102 3.3 105 1.2 10 s 2.8 lO s
The dimensionless number DamkOeler 11, that accounts to the ratio of the chemical reaction velocity (k~(CoZ)n'l and the diffusion velocity in the particle (De/Ro)
Da• = ks (cz~)n-] Ro De
(9)
was calculated and included in Table 1 in order to quantify the relationship between these two mechanisms.
*n* e case of oo ,
rocess is con,
en, i ely
mass transfer
1~ 0 ,
that is, the concentration at the surface approaches zero. For the reaction order n=l and Dall > 102, implies that this dimensionless concentration is < 10-2 at the surface. This fact can be observed in Figure 3 showing the SO2 concentration profile in the reacted layer for different reaction times, which is consistent with the slight decrease of the unreacted core, represented in Figure 4.
{
1.0 0.8 o
. ~ ~ ' ~ ' ~ ~
0.6
j
SR4exp SR4est 9 StLOexp SR9est X SR18exp SR18est
O.4 0.2 0.0
r
0
0.5
i
1
i
1.5
i
t
2
2.5
3
x (s)
Fig. 1. S O 2 concentration in the gas phase in the duct for Ca/S: SR=4, 9, 18.
0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
9 9 II
0
0.5
;~
1
1.5
x (s)
~
2
Fig. 2. Solid conversion in the duct for SR = 4, 9, 18.
..
2.5
3
152 1
0.8 .
.
.
.
3s /
/
.
0.6
.-2"...'2".---
3 0.4
0.99 o 0.98 ~0.97 0.96
0.2 0
,
0
0.2
i
0.4
i
r
0.6
i
0.8
Fig. 3. $ 0 2 concentration profile in the reacted layer at 0.1, 1.5, 3 s for Ca/S =4.
0.95 0
0.5
1
1.5
2
2.5
3
l:(s) Pig. 4. Unreacted core during reaction time (up to 3 s) for Ca/S =4.
4. CONCLUSIONS A computer simulation procedure has been developed to solve the differential equations of the proposed model allowing to fit experimental results from the pilot plant that reproduces the conditions of the In-duct injection process. The model was based on the overall reaction in a shrinking core of a particle located at any axial position in the duct section. From the estimation of parameters, De the effective diffusivity in the reacted layer of the particle, and ks, the surface reaction constant, it can be concluded with the diffusional resistance as the controlling step, being the surface reaction rate much faster. The simulated SO2 concentration and the solid conversion at any axial distance in the reactor under diffusional control were in a good agreement with the experimental data for the range of Ca/S molar ratio under study (4-18), given by a standard deviation lower than 10%. The comparison of the fitted parameter, De = 3.7 101~ (m 2 S'l), with the effective diffusivity of SO2 at the same temperature, 4.85 10-6 (m 2 s-1) shows, that the desulfurization process under study may not be described by gas diffusion in the particle. It can be related to a solid state diffusion process or to the SO2 diffusion in the microporosity of the small grains of sorbent in the particle. REFERENCES 1. A.L. Fenouil and S. Linn, Ind. Chem. Eng. Res., 35 (1996) 1024. 2. M.R. Stouffer, H. Yoon and F.P. Burke, Ind. Eng. Chem. Res., 28 (1989) 20. 3. R.W. Rice and G.A. Bond, AIChE J., 36 (1990) 473. 4. J.A. Marqu6s, J.L. Herrera, A. Garea and A. Irabien, Clean Air'99, (1999) Lisboa, Portugal. 5. J.A. Marqu6s, A. Garea and A. Irabien, CHISA 2000 (2000) Praga, Czech Republic. 6. S.V. Krishnan and S.V. Sotirchos, Can. J. Chem. Eng., 71 (1993) 734. 7. M. Maalmi, A. Varma and W.C. Strieder, Ind. Chem. Eng. Res., 34 (1995) 1114. 8. A.B.M. Heesink, W. Prins and W.P.M. Van Swaaij, Chem. Eng. J., 53 (1993) 25. 9. J.J. Carberry and A. Varma (eds.), Chemical Reaction and Reactor Engineering, Marcel Dekker, Inc., New York, 1987. 10. A.F. Shaaban, Thermochim. Acta, 180 (1991) 9. 11. A. Garea, J.R. Viguri and A. Irabien, Chem. Eng. Sci., 52 (1997) 715. 12. I. Fernfindez, A. Garea and A. Irabien, Chem. Eng. Sci., 53 (1998) 1869.
European Symposiumon ComputerAidedProcess Engineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
EQUISTAR:
153
R e l i a b l e S o f t w a r e for D e s i g n o f N o n i d e a l a n d R e a c t i v e S y s t e m s
S. T. Harding and C. A. Floudas I Department of Chemical Engineering, Princeton University Princeton, NJ 08544, USA
1
Introduction
Many commercial packages for process design and thermodynamic calculations are available today. Typically, these packages allow a broad variety of design problems to be solved and a large number of thermodynamic models from which to choose. Many design alternatives can be screened relatively quickly using such applications, and they greatly reduce the effort needed to design complicated processes. However, the solution methods employed by these packages frequently fail for systems that exhibit complex behavior. The reason for this failure is that local solution techniques are used to solve the nonlinear equilibrium equations that arise in the problem formulations. In this paper, a new tool for robustly and efficiently solving process design and thermodynamics problems is presented. This new tool is called EQUISTAR, which stands for E Q U i l i b r i u m Solution Toolkit for Azeotropic and Reactive Distillation Design. EQUISTAR addresses the need for a computational tool that can be reliably applied to highly nonideal and reactive systems and can solve the most commonly occurring thermodynamics problems. In section 2, the capabilities and design of EQUISTAR are presented. The global optimization algorithms that EQUISTAR employs are outlined in section 3.
2
O v e r v i e w of E Q U I S T A R
Local solution approaches have two features that make them attractive, especially for commercial implementations: they are fast, and they are relatively easy to implement. In order to achieve the kind of reliability that global optimization methods provide, one usually pays a premium in speed. In addition, global approaches are substantially more difficult to implement both in terms of the analysis of the mathematical equations, and in the actual programming effort. However, recent developments in global optimization approaches take a large step toward the practical implementation of global methods for process design. By analyzing the structure of the mathematical equations, it is possible to identify properties that can be exploited to greatly reduce the computational effort for guaranteed reliability. 2.1
EQUISTAR
Capabilities
EQUISTAR is a versatile package that incorporates global optimization methods in order to provide reliable solutions of many thermodynamic equilibrium problems that arise in the design and simulation of chemical processes. In addition, EQUISTAR can use this "toolkit" of robust equilibrium solution techniques to determine the design specifications for reactive and nonreactive distillation columns. The user is not forced to use the rigorous global optimization approach for any of the problems that EQUISTAR solves. In some cases, one may wish to do 1Author to whom all correspondence should be addressed.
154 a quick local search for a solution. EQUISTAR gives the user the option of specifying whether the problem is to be solved to global optimiality, or local optimality, and the user can specify the number of local searches that are performed. EQUISTAR can solve the following thermodynamic equilibrium problems: 1) phase and chemical equilibrium, 2) Gibbs free energy minimization, 3) phase stability (through Gibbs tangent plane distance minimization), 4) finding all homogeneous reactive and non-reactive azeotropes, 5) finding all heterogeneous reactive and non-reactive azeotropes, 6) isothermal or reactive flash calculation, 7) reactive or nonreactive bubble point calculation, and 8) reactive or nonreactive dew point calculation. The equilibrium conditions in each of these problems are only necessary conditions for the global equilibrium solution. Therefore, the solution that is obtained may correspond to a thermodynamically unstable system. EQUISTAR allows the user to specify whether or not to check the stability of each equilibrium solution that is obtained. The stability check is performed by solving the tangent plane distance minimization problem to global optimality, or until a negative tangent plane distance is located. In addition to solving stand-alone thermodynamics problems, EQUISTAR incorporates its solution algorithms into the capability of solving reactive and non-reactive distillation design problems. EQUISTAR provides the user a choice of algorithms for reactive or non-reactive distillation design algorithms, 1) a modification of the Inside-Out Algorithm, and 2) a modification of the Bubble-Point algorithm. EQUISTAR allows the user to choose from a wide range of thermodynamic models for representing the system's physical behavior. A number of equations of state axe available: 1) the Redlich-Kwong equation, 2) the Soave-modified Redlich-Kwong equation, 3) the PengRobinson equation, and 4) the van der Waals equation. In addition, several activity coefficient equations are available: 1) the Wilson equation, 2) the NRTL equation, 3) the UNIQUAC equation, and 4) the UNIFAC group-contribution method. 2.2
EQUISTAR
Software Design
The EQUISTAR program primarily consists of optimization problem formulations and highlevel optimization algorithms. The program is written in C. These formulations and algorithms are based on novel analysis by several authors and are described in section 3. EQUISTAR automatically generates problem formulations in the c~BB problem format. c~BB is a global optimization approach for solving general twice-continuously differentiable nonlinear programming problems developed by [2, 1]. c~BB is based on a branch-and-bound framework coupled with novel convex underestimators and the program manages the variable branching and the formulation of the upper and lower bounding problems. In order to solve the actual optimization problems, MINOPT is called. MINOPT is a Mixed-Integer Nonlinear OPTimization modeling language and solver developed by [10]. MINOPT converts the formulation of the optimization problem into a format that can be sent to an equation solver. MINOPT has interfaces with a number of linear, nonlinear, mixed-integer linear, mixed-integer nonlinear, and differential and algebraic equation solvers. Depending upon the type of problem that is passed, MINOPT converts the formulation into the correct format, sends it to the appropriate solver, and passes the solution back to the program that called it. Through this structure, the implementation of EQUISTAR and the formulation of its problems are independent of the local optimization method. The algorithms within EQUISTAR possess their own interdependencies, as shown in figure 1. The middle layer of the figure are the basic global optimization algorithms for solving
155
equilibrium problems. Note that each of these algorithms may call the phase stability problem to verify the stability of the solution. These algorithms call c~BB as the general global optimization solver. At the top of the figure are the highest level algorithms: distillation design and phase and chemical equilibrium. Each of these top level algorithms require the repeated solution of the equilibrium algorithms.
Figure 1: Relationship between EQUISTAR components
3
R e v i e w of S o l u t i o n M e t h o d s
Each of the problem types addressed by EQUISTAR has its own solution algorithm. section provides a summary of each of the algorithms. 3.1
This
Gibbs Free Energy Minimization
The minimization of the Gibbs free energy of the system is a fundamental approach for determining the equilibrium state of a system. A necessary and sufficient condition for equilibrium is that the Gibbs free energy of a system at constant temperature and pressure be at its global minimum. The general problem referred to as the Gibbs Free Energy Minimization Problem (GMIN) is defined as follows" Given N components with initial moles {sT1, sT,... ,n T } participating in up to P potential phases and R chemical reactions at constant temperature and pressure, find the mol vector n that minimizes the value of the Gibbs free energy function and satisfies the material balance constraints. The algorithm that EQUISTAR uses to determine the global minimum Gibbs free energy is based on the approach developed by [8] and is outlined below. 1. The user provides the system temperature and pressure and overall composition and the number and type of phases. 2. The GMIN problem is solved locally to generate an upper bound. 3. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable.
156
4. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than the current best upper bound. 5. Return to Step 2 and repeat until the best upper and best lower bounds converge. 6. The solution of the problem provides the composition of each phase.
3.2
Tangent Plane Distance Minimization
[3] and [11] have proved that a necessary and sufficient condition for a candidate equilibrium solution to be the true equilibrium solution is that the tangent plane distance function be nonnegative for all phases in the candidate solution. The tangent plane distance function is defined as the distance between the Gibbs free energy surface for the new phase, and the tangent plane to the Gibbs energy surface constructed at the candidate equilibrium solution. Based on the work of [9] and [4], the tangent plane distance minimization problem (TED) is solved in EQUISTAR using the following algorithm: 1. The user candidate 2. The T P D 3. Check the
provides the system temperature and pressure, and the composition of the phase. problem is solved locally to generate an upper bound. current best upper bound. I f it is less than zero, then stop the algorithm
because the candidate phase is unstable. 4. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 5. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than the current best upper bound. 6. Return to Step 2 and repeat until the best upper and best lower bounds converge or the best upper bound becomes negative. 7. The solution of the problem determines the stability or instability of the candidate phase. Enclosing All Azeotropes
3.3
The phenomenon of azeotropy occurs in many industrial applications. Azeotropes restrict the amount of separation of a multicomponent mixture that can be achieved by distillation. The ability to predict whether a given mixture will form one or more azeotropes and to calculate the conditions and compositions of each azeotrope is essential if one wants to model separation processes. An azeotrope is defined as a liquid mixture that boils at a constant temperature where the composition of the vapor phase is identical to the composition of the boiling liquid. When the boiling liquid contains a single phase, this phenomenon is called a homogeneous azeotrope. If the liquid consists of two or more phases it is classified as a heterogeneous azeotrope. Azeotropes may also occur in systems where one or more chemical reactions are occurring. These are called reactive azeotropes, and may be classified as homogeneous reactive azeotropes or heterogeneous reactive azeotropes, depending upon the number of liquid phases.
3.3.1
Enclosing All Homogeneous Azeotropes
The algorithm presented below for the location of all homogeneous non-reactive and reactive azeotropes is based on the work of [6]. 1. The user provides the system pressure.
157
2. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 3. In each new domain a convex lower bounding problem is solved and t h e d o m a i n is d i s c a r d e d if the solution is greater t h a n zero. 4. Return to Step 2 and repeat until all domains have been eliminated, or the size of the remaining domains are within a given tolerance. 5. The solution of the problem determines the temperature and composition of all homogeneous azeotropes in the system.
3.3.2
Enclosing All Heterogeneous Azeotropes
The algorithm presented below for the location of all heterogeneous non-reactive and reactive azeotropes is based on the work of [5]. 1. The user provides the system pressure. 2. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 3. I n e a c h n e w d o m a i n , t h e p o s s i b i l i t y of a t r i v i a l s o l u t i o n is c h e c k e d . If a t r i v i a l solution is possible, the d o m a i n is k e p t , b u t S t e p 4 is skipped for the d o m a i n . 4. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than zero. 5. Return to Step 2 and repeat until all domains have been eliminated, or the size of the remaining domains are within a given tolerance. 6. The solution of the problem determines the temperature and composition of all heterogeneous azeotropes in the system. 3.4
Flash Calculation
Unlike Gibbs free energy minimization, the flash calculation takes an equation-solving approach to the determination of phase and chemical equilibria. The solution of the isothermal flash and reactive flash problems in EQUISTAR provides all compositions that satisfy the flash equations. The algorithm is based on the approach for finding all solutions to nonlinear systems of equations developed by [7]. 1. The user provides the system temperature and pressure and feed rate and composition. 2. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 3. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than zero. 4. Return to Step 2 and repeat until all domains have been eliminated, or the size of the remaining domains are within a given tolerance. 5. The problem solution specifies the composition and flowrate of the vapor and liquid phases. 3.5
Bubble
Point and Dew Point Calculation
The calculation of bubble point temperatures and dew point temperatures is a natural extension of the flash calculation. These calculations are commonly encountered in the design and simulation of distillation columns. The bubble and dew point calculations are phase a n d / o r chemical equilibrium calculations.
158 In these formulations, the phase equilibrium condition is represented as the equality of chemical potentials for all components. The difference between the bubble and dew point problems and the flash problem is that the composition of only one phase is being determined, and the temperature of the equilibrium state is being determined. 1. The user provides the system pressure and liquid (vapor) composition. 2. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 3. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than zero. 4. Return to Step 2 and repeat until all domains have been eliminated, or the size of the remaining domains are within a given tolerance. 5. The solution of the problem determines the bubble (dew) temperature and the composition of the vapor (liquid) phase.
4
Conclusion
Based on significant developments in global optimization approaches developed over the past several years, EQUISTAR provides a suite of algorithms for the reliable and efficient solution of process design and thermodynamic equilibrium problems.
References [1] Adjiman C.S., Androulakis I.P., and Floudas C.A., 1998b, A global optimization method, aBB, for general twice-differentiable N L P s - II. Implementation and computational results. Comput. Chem. Eng. 22, 1159-1179. [2] Adjiman C.S., Dallwig S., Floudas C.A., and Neumaier A., 1998a, A global optimization method, aBB, for general twice--differentiable N L P s - I. Theoretical advances. Comput. Chem. Eng. 22, 1137-1158. [3] Baker L., Pierce A., and Luks K., 1982, Gibbs energy analysis of phase equlibria. Soc. Petro. Eng. J. (p. 731). [4] Harding S. and Floudas C., 2000a, Phase stability with cubic equations of state: A global optimization approach. AIChE J. 46, 1422-1440. [5] Harding S. and Floudas C., 2000b, Locating all heterogeneous and reactive azeotropes in multicomponent systems. I~EC Res. 39, 1576-1595. [6] Harding S.T., Maranas C.D., McDonald C.M., and Floudas C.A., 1997, Locating all azeotropes in homogeneous azeotropic systems. ISJEC Res. 36, 160-178. [7] Maranas C.D. and Floudas C.A., 1995, Finding all solutions of nonlinearly constrained systems of equations. Journal of Global Optimization 7, 153-182. [8] McDonald C. and Floudas C., 1994, Decomposition based and branch and bound global optimization approaches for the phase equilibrium problem. Journal of Global Optimization 5, 205-251. [9] McDonald C. and Floudas C., 1995a, Global optimization for the phase stability problem. AIChE J. 41, 1798. [10] Schweiger C.A. and Floudas C.A., 1998c, MINOPT A Modeling Language and Algorithmic Framework for Linear, Mixed-Integer, Nonlinear, Dynamic, and Mixed-Integer Nonlinear Optimization. Kluwer Academic Publishers, in preparation. [11] Smith J., Missen R., and Smith W., 1993, General optimality criteria for multiphase multireaction chemical equilibrium. AIChE J. 39, 707.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jargensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
159
CFD Modeling of Fast Chemical Reactions in Turbulent Liquid Flows L.K. Hjertager, B.H. Hjertager and T. Solberg Chemical Engineering Laboratory, Aalborg University Esbjerg, Niels Bohrs vej 8, DK-6700 Esbjerg, Denmark. Many industrial processes involving chemical reactions happen in turbulent flow. For fast reactions the reaction rate is limited by the micromixing rate which is controlled by turbulence. Micromixing directly influences the reaction and may lead to changes in both conversion and selectivity. This paper will discuss and present results from various models including the so-called eddy dissipation concept (EDC) and the presumed probability distribution (PDF) models. The simulation results will be compared to experimental data from chemical reactions in liquid systems. 1. INTRODUCTION Many industrial processes involving chemical reactions happen in a turbulent flow. For infinitely fast reactions the reaction rate is limited by the micromixing rate which is controlled by the turbulence. Micromixing directly influence the reaction and can lead to changes in both conversion and selectivity [ 1]. Pohorecki and Baldyga [2] have performed experiments in a tubular reactor where they found the conversion length for an acid-base neutralisation at different Reynolds numbers. Hannon et. al [3] have tested two different models for chemical reactions; the finite rate combustion model and a presumed PDF multi-scale mixing model. The results from these numerical simulations were compared to the experimental results by Pohorecki and Baldyga [2]. Hannon et. al [3] showed that the finite rate combustion model was not able to predict the conversion length, and argued that this was because the finite rate combustion model contains no information on mixing at scales where viscous/diffusive effects are important. They also pointed out that the model defines the covariance of concentration fluctuations on reaction rate as a direct function of local k and ~ values [3]. They found that the multi-scale mixing model with a beta-PDF (instantaneous concentrations) could predict the length within a reasonable accuracy [3]. The Eddy dissipation concept (EDC) was developed for prediction of gaseous combustion reactions in turbulent flows. It is based on the assumption that the reaction time scales can be related to the dissipation of turbulent eddies which contains the reactants and products [4]. There are two main differences between mixing of reactants in a gas phase and a liquid phase, which are of importance when reacting flows are to be modelled. The first difference is that the coefficient of molecular diffusion is much higher in gases than in liquids, meaning that the Schmidt number in the gas phase is much smaller (So-l) than in the liquid phase (Sc>>I). The second difference results from the density variation of the gas phase and the resulting sensitivity of the gas phase density to pressure and temperature variations [5].
160 It is the objective of this paper to present results from various models including the EDCmodel and PDF-models. The simulation results will be compared to experimental data from chemical reactions in liquid systems. 2. REACTION MODELS
2.1 The Eddy Dissipation Concept The EDC model was developed for combustion reactions. These reactions can often be treated as a single-step irreversible reaction with finite reaction rate [4]: 1 kg A + s kg B -~ (1 +s) kg C
(1)
This simple reaction scheme results in mixture composition being determined by solving for only two variables, the mass fraction of species A, YA, and the mixture fraction, f. These equations read:
at (p. Y~). +
.
a
.
a
at (p "f) + ~(P
(p .u, Y~. )
.
~
" U' " f ) = cgx, L
.
Sc~ ax,
as] ax,
+R a
(2)
(3)
Here RA is the time mean reaction rate, PT is the turbulent viscosity and Sc~ is the turbulent Schmidt number. The basis for this to be valid is that the turbulent Schmidt numbers are equal for all species, an approximation, which is often found to be valid in turbulent flows. A transport equation for the mass fraction of species A is solved (2), where the reaction rate of species A is taken as the smallest of the turbulent dissipation rates of species A, B and C. R A = - A . P . - k . min Y A , - - , B . s
(4)
The constant A in equation (4) is found to be 4 and B is 0.5 for gaseous combustion reactions [4]. 2.2 Scalar mixing/dissipation theory Since the EDC-model is stricly valid only for Sc-~l, an algebraic expression for the scalar dissipation timescale could be used to take the effect of higher Sc-numbers (Sc>>l) into account. From Fox [6] the scalar dissipation rate is given as:
. . . ?' ~'2 /~ \r
.
2
.
2c
+-ln(Sc) 2
(5)
Here % is the dissipation rate of a scalar variable ~ and /~;~). If we use the expression in (5), we get the modified time mean reaction rate for chemical species A (EDCSDT) as:
.min YA,
R A =-A.p.
(6)
In the standard EDC-model the dissipation of the product C is also included, this is because combustion reactions are strongly exothermic reactions. For liquid reactions, which are isothermal the dissipation of the product C does not have to be considered. The above mentioned scalar time scale is valid for a fully developed turbulent structure. It has been found that the development in liquid mixtures takes longer time. The multiple time scale turbulent mixer model of Baldyga [7] takes account of these processes.
2.3 Multiple-time-scale turbulent mixer model (MTS) In liquid mixtures (Sc>> 1), the local value of the concentration variance O'S2, Can be divided into three parts according to the scale of segregation and the related mechanism of mixing [7]: 2
2
2
o-s = o-~ + o-2 + o'3
(7)
where cr 12, (y22 and (Y32 is the variance in the inertial-convective, the viscous convective and the viscous diffusive subrange, respectively. The inertial convective variance is produced from the macroscopic inhomogeneity of the mixture fraction, f, as a result of velocity fluctuations. Turbulent diffusivity is expressed as: DT = ~VT ; Scr
vr
= ~jUT p
(8)
Engulfment parameter, E, is given by: E = 0.058
(9)
Decreasing the scale of segregation by viscous deformation increases the wave numbers and conveys the variance into the viscous-diffusive subrange of the concentration spectrum where mixing on the molecular scale occurs by molecular diffusion in deforming slabs [7]. G ~ (0.303 + 17050/Sc). E
(10)
162 The evolution of the complete crl2, or22and era2 [7]:
variance (~s2) becomes, when summing up the equations for
COx,J
(11)
Another alternative for the time mean reaction rate presented in part 2.2 is to take account of the Multiple-time-scale turbulent mixer model given by Baldyga [7] and described above. Since for infinitely fast reactions the reaction is limited by the micromixing rate (pGcr3e) the modified form of the EDC-model (EDC-MTS) will be expressed as: RA = - A - p - G . o - 3
min/,
(12)
2.4 Presumed PDF methods
Presumed PDF's are the probability density function of the passive scalar concentration distribution. The concentration may be expressed by the mixture fraction, f [5]. Transport equations are solved for the time-mean of the square of the concentration fluctuations, (rs2, and from the calculated values of that quantity and assumptions regarding the shape of the instantaneous concentration-time profile, a hypothetical distribution of the instantaneous concentrations with time is derived [8]. Two versions of the presumed PDF will be described, namely the battlement and the beta PDF. 2.4.1 Battlement probability density function The instantaneous concentrations at a point is assumed to follow "battlement shaped time variation" with allowance for )max>f>)mi. when )m~x and )minare the maximum and minimum possible values of the instantaneous concentrations, respectively [8]. Based on this the time average of the mass fraction may be calculated by the following expression:
: a. r;, (z+)+ o-
rrA 0"_)
(13)
2.4.2 The beta probability density function The beta function written as a PDF has the following form [9]:
=
B(v, w)
Based on this the time average of the mass fraction may be calculated by the following expression: 1
0
This model was proposed and validated by Hannon et. al [3].
(14)
163
3. NUMERICAL CONFIGURATION The various reaction models will be tested against the experimental data of Pohorecki and Baldyga [2], where they found the reaction-zone length for 95 % conversion of species A (base), for a simple acid-base neutralisation expressed as: A + B ~ C. The tube arrangement that was simulated had an outer diameter of 0.04 m, and an inner tube with a diameter of 0.0052 m. Acid was fed into the outer tube which had a length of 2 m, and the base was fed into the inner tube which was 1 m long. The simulations were performed at three different Reynolds numbers; 13000, 20000 and 25000, which is the same as the ones used in the experiments by Pohorecki and Baldyga [2] and the numerical simulations of Hannon et. al [3]. The simulations were performed with the standard EDC-model, the EDC-model with an algebraic expression for the scalar dissipation time scale (EDC-SDT), and the EDC-model with the multiple-time-scale turbulent mixer model (EDC-MTS). Simulations were also performed with the use of two PDF models, the battlement PDF and the beta PDF. The Schmidt number was set to 800. The reaction rate constant, A, was set to 4 in the first cases and calibrated to 1 in the EDC-MTS case. Both the standard k-~-model and the RNG-k F u n c t i o n [ { k } , (k-l) / (kmax-l) ], -> "kmax", -> { S p a c e S m o o t h i n g -> 1.0, TimeSmoothing -> 0.i, MonitorStates -> {x[z,t] } } ]
Calling the automatic discretization of SYPPROT transforms the MDS input file into a CG input file by performing MOL discretization, DAE transformation and translation of MDS into CGlanguage, shown in Fig. 1. These three preprocessing steps are performed by executing a few SYPPRoT commands within a usual MATHEMATICA session, see [3] for a more details. 5
B E N C H M A R K RESULTS In Fig. 3, simulation results for the countercurrent adsorber model (10-13) are depicted. Characteristic for this process is a steep front moving from z = 0 to z = 1. This behavior is reflected in the trajectories of moving grid nodes on the fight in Fig. 3. Nodes crowd together at the position of the front to achieve a small spatial discretization error. For evaluation of the numerical performance, benchmark simulations on equidistant grids and moving grids are compared. The solutions obtained on an equidistant grid of 1000 nodes and on a moving grid of 80 nodes with parameters ~ = 1 and ~- = 0.1 show no significant differences. The improved performance is obvious, regarding the CPU-times on a SUN Ultra 60 machine of 2.0s and 10.0s for the moving grid and equidistant grid simulations, respectively.
170
Fig. 3: Simulation results of a countercurrent adsorber on a self-adaptive moving grid with kmax = 80, t~ = 1, r = 0.1, and considering only x(z, t) within the monitor function (7). Other benchmark models like a convective transport problem, a flame propagation model, and a circulation-loop reactor also confirm the increased perfomance for moving grid nodes instead of equidistant grids. Comparisons with equidistant grids show that only 2-20% of the grid nodes are required to compute results of the same accuracy in 15-50% of CPU-time. 6 CONCLUSIONS In the symbolic prepocessing tool SYPPRoT of the simulation environment DIVA, the automatic generation of DAEs for distributed parameter models has been extended by a moving grid technique. The grid movement is controlled by equidistribution of an arc-length monitor and is regularized by two smoothing parameters. Parametrization and change between provided grids and discretization schemes are very simple and allow a rapid test of various MOL approaches without model reimplementation. Benchmark simulations show the increased performance with steep moving spatial fronts. REFERENCES [1 ] W.E. Schiesser. The numerical method of lines: Integration of PDEs. San-Diego, 1991. [2] R. Ktihler, et al. Symbolic Preprocessing for Simulation of PDE Models of Chemical Processes. Special Issue Method of Lines in Journal of Math. and Comp. in Sim. (accepted). [3] R. Krhler, et al. Method of lines within the simulation environment DIVA for chemical processes. In A. Vande Wouwer, P. Saucez, W. Schiesser, editors, Adaptive Method of Lines. CRC Press, 2001. [4] E.A. Dorfi and L. O'C. Drury. Simple adaptive grids for 1-d initial value problems. J. Comp. Phys., 69:175-195, 1987. [5] L.G. Verwer, et al. A moving grid method for one-dimensional PDEs based on the method of lines. In J.E. Flaherty, P.J. Paslow, M.S. Shepard, and J.D. Vasilakis, editors, Adaptive Methods for Partial Differential Equations, 160-175. SIAM, Philadelphia, 1989. [6] A. Krrner, et al. DIVA - An open architecture for dynamic simulation. In R. Eckermann, editor, Computer Application in the Chemical Industry, 485-492. VCH, Weinheim, 1989. [7] S. Li, L. Petzold, and R. Yuhe. Stability of moving mesh systems of partial differential equations. SlAM J. Sci. Comput., 20(2):719-738, 1998.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
171
Computational tools for nonlinear dynamical and bifurcation analysis of chemical engineering problems M. Kohout, I. Schreiber and M. Kubf6ek Department of Chemical Engineering, Department of Mathematics, Center for Nonlinear Dynamics of Chemical and Biological Systems Prague Institute of Chemical Technology, Technick~i 5, 166 28 Prague 6, Czech Republic We present a program package CONT for modelling and analysis of nonlinear problems in general, and chemical engineering problems, such as chemical reactors or adsorption columns, in particular. Model should be in the form of ordinary differential equations, spatially distributed systems described by partial differential equations may also be treated upon transformation to ODEs by a built-in discretization. A basic method used in the program is continuation of a steady state, periodic solution or another boundary value problem with respect to a parameter which provides a solution diagram. Simultaneous stability analysis helps to identify bifurcation points in such diagrams and these may be continued with respect to another parameter to obtain a bifurcation diagram. Additionally, the program performs a direct integration of ODEs, calculates Poincare orbits and Lyapunov exponents, which are useful for analysis of complex dynamics. 1. CONTINUATION, BIFURCATIONS AND NONLINEAR DYNAMICS Dynamical systems are commonly represented by differential equations defined on a finite or infinite dimensional state space. Let us consider the finite dimensional case, leading to a set of n ordinary differential equations (ODEs)
dx d t = f(x;[t, ~),
x E 9~n,
(I)
where x is the vector of dependent variables, t is time and tx, 13are scalar parameters. Nonlinear dynamical systems described by Eq. (1) can be explored by a large number of methods [1--4] ranging from finding steady states and determining their local stability to an analysis of complex nonperiodic solutions involving calculation of a fractal dimension of a corresponding attractor (geometric complexity), Lyapunov exponents (temporal instability) and various other measures of complexity. Of particular interest for engineering purposes are special solutions to Eq. (1) subject to various constraints, such as boundary conditions, and variations of these solutions with parameter(s). The relevant method here is that of numerical continuation with respect to a parameter [1,4-7]. One-parameter families of solutions may contain bifurcation points, where phase portrait undergoes a qualitative change; such points can be localized and traced by continuation when another parameter is added. Although this procedure can be repeated again, providing
172 thus bifurcations of codimension higher than two, in practice it is often sufficient to define two distinct parameters, say o~ and ~, and construct either one-parameter solution d i a g r a m s or two-parameter bifurcation diagrams. To accomplish some of the tasks outlined we created a software computational tool CONT whose functionality is described below and illustrated by simple problems from reaction kinetics and adsorption. The program CONT can perform the following tasks: 1. one-parameter continuation of steady states or periodic orbits and determination of local stability 2. two-parameter continuation of local bifurcation points for steady states or periodic orbits (such as a Hopf bifurcation, limit and branch points, period doubling and torus bifurcation) 3. one- or two-parameter continuation of solutions subject to general nonlinear boundary conditions (other than periodic ones) 4. two-parameter continuation of homoclinic and heteroclinic orbits (i.e. a special boundary value problem) 5. direct numerical integration of ODEs providing general orbits or Poincare-discretized orbits including automatic search for steady states/periodic orbits and repeated incrementation of parameter(s) 6. Lyapunov exponents for general orbits including repeated incrementation of parameter(s) All the functions (when they are relevant) can also be applied to periodically forced ODEs (via a smooth function or pulses) and discrete iterated maps. Also, a built-in discretization scheme for partial differential equations on an interval makes it possible to apply the analysis to PDEs, provided that sufficient computational power is available. In our previous work [8-10] and below we provide some examples clarifying the functions of CONT and showing how the results can be presented. Comparison with similar software is made. 2. E X A M P L E 1 - AN AUTOCATALYTIC B I O C H E M I C A L R E A C T I O N Redox bioconversion of thiol groups (2 SH ~ S-S) in proteins within cytosol may be attributed to a variety of biological functions such as biological clocks, cell division and carcinogenesis [ 11 ]. A simple biochemical mechanism describing the kinetics in terms of dimesionless concentrations x (of the reduced "SH" proteins) and y (of the oxidized "S-S" proteins) is as follows: dx v0 + x ~t d t = f ( x , y ) = ct 1 +-----~ - x - x y ,
dy d t -- g(x,y) - ~ x + x y - S y
(2)
where tx, 8, v0 > 0 and 13,T > 1 are parameters. We fix v0 = 0.01, [3 = 1.5 and ), = 3 and use tx and 8 as bifurcation parameters. The variable x represents the autocatalyst and tx is the rate coefficient of the autocatalytic step while y represents the inhibitor and 8 is the rate coefficient of the degradation of the inhibitor.
2.1. Periodic pulsed forcing of the homogeneous reaction system Assuming for simplicity that any transport within the cytosol is effectively fast compared to chemical processes Eqs. (2) describe a spatially homogeneous system equivalent to a CSTR. From outside the cellular environment the SH ~ S-S dynamics can be controlled by periodic signalling which may cause sudden pulse-like increase/decrease of the concentration of the
173
autocatalyst or inhibitor. For instance, a signalling agent can rapidly consume the autocatalyst. This situation would correspond to adding a periodic delta function x(e -a - 1) ~, 8(t - kT) to f ( x , y ) in Eqs. (2), where A > 0 defines a forcing amplitude and T is the forcing period. Choosing tx = 25 and 8 - - 1.6, Eqs. (2) possess a unique steady state which is, however, excitable with respect to a pulsed removal of the autocatalyst. Applying repeatedly the pulses we obtain a temporal dynamical pattern of alternating small- and large-amplitude responses. These firing patterns may be periodic or not, depending on the choice of A and T. For periodic responses with period qT, q -- 1,2, 3, 4, CONT-generated branches denoted Pq for fixed A = 0.5 and varying T including stability changes and associated bifurcation points are plotted in Fig. 1a. Simultaneously, in Fig. lb the maximum Lyapunov exponent within the same range of T is plotted, indicating chaotic dynamics of responses (positive values) fitting the interval of unstable periodic orbits in Fig. 1a. Finally, taking the bifurcation points on the period one branch (i.e. q = 1) from Fig. l a and performing continuations in A - T plane generates the amplitude-period bifurcation diagram in Fig. 2, delineating the regions of multiple period one orbits (bounded by limit point curves), a central region of no stable period one orbits (bounded partly by period doubling curves and partly by limit point curves) and regions of stable orbits (bounded in part by torus bifurcation curves, and the above two other kinds of curves). As suggested by Figs. 1a,b, the central region is filled with periodic orbits with q larger than one (these can be further studied by continuation) mingled with chaotic orbits.
~
6
~
.J,# P,'~/;:" ,/
m
~,
4
2
p~/.,;,.~..,:
..--'" -"
00
,
1
2
.--,-----
o
,,~ M .
(1)
rp - ke [,,~ M.] [M]
If simplifications such as the mean number of radicals per particle (~) and the number of particles (NT) are used and the rate of formation equation is extended to n monomers, the rate can be expressed as in eq. 2. For the formation of a polymer a probability (P) that the monomer hits a certain polymerradical needs defining. This is done using the concentrations of the endgroups: Ri-
kejiPj [M i] p -~Nr
NA
j=l
with
Pj =
(2)
i e [1..n]
~
E ['" Mi'] p
i=1
Using the Quasi Steady State Approximation the concentrations can be identified. The rate of formation of a polymerradical can be expressed as such: dt
= -
kpji [" MJ'] p [Mi] p Qss=A0
kpij ["-' Mi'] p [MJ] p +
j=l
jTAi
Vi E [1..n]P,j r i (3)
j=l
jT~i
The resulting equation system can be solved generally for [~ Mi'] p and the solution can be shown to be correct. Such, the probabilities can be defined. An example solution for three monomers is given as:
[,',-'M 2-] [~ M 3.]
--
[M23]~(kp12kp31[M1] p + kp12kp32 [M2] p -~-kp13kp32 [M 3] ~) [M ] (kp13kp21 [Me] p at- kp12kp23 [M2] p at- kp13kp23 [M3] )
(4)
185
2.2. Phase equilibrium The monomers are distributed in all three phases of the system. To determine the concentrations of the monomers in the phases a phase distribution algorithm used by (Urretabizkaia and Asua, 1994) is reduced to water und particle phase. It employs constant phase partition coefficients (#//) for the equilibrium between the phases, where i defines the considered monomer, j the phase. The phase distribution algorithm calculates the volumes of the monomers in the phases and the total volume of each monomer in the reactor. The interested reader is referred to Urretabiskaia et al. for a detailed description of the algorithm. This approach gives good results and converges quickly. 2.3. Material Balances Using the above derivation and balances for monomer and initiator are given as:
dM i -- -Ri - VWkpii[Mi]W[R]w-+-hMi i, j E [1..n] dt dI w -- hi -- klI - h i - fkd[l]WV w where M i = [Mi]VR dt
(5)
(6)
I w -- [l]Wy w
2.4. Particle Balances The following particle balance can be developed. It covers particle formation by exceeding the critical chain length, agglomeration, termination and micellar absorption, if the number of micelles is known:
dNr dl
~
--
i= 1
iw w kam[R]WNm kpii[M ] Rjcrit +
N2 jcri, Jcri, T .+_NA VW-kT E [RJ] w E [ei]w - kagl NA VR j-- 1 i--jcrit-j
(7)
A micelle balance is necessary. Emulsifier is not fed to the system and such micelles are only depleated by particle formation and particle stabilisation.
Nm = as(ST -CMC) - a p
ap -- (367~)2 (VP) 2/3
(8)
Am
2.5. Radical Balances A differential equation model developed by (Li and Brooks, 1993b) based on (Smith and Ewart, 1948) will be used for the radical balance in the particle phase. d~ = kap[R] w _ -kdn-n -- tp
dt
kr
~2
where
tp-
( -
2(2kap[R] w -k-_ kdn) ( -s 2kap[R] w q- kdn -k- -
_
VPNA/NT )
(9)
The initiator-radicals in the water phase are developed as:
d[Rl]W = 2 fkd[I] _ ~ kpii[Mi]W[Ri]W_-~T[Ri]W[R]W_ (kam Nm NT ) dt NA V w + kap NA VR [RI]W (1 O) i=1 Radicals made of one monomeric unit are given as type separated equations:
d[R~ ]w dt
--
-i -nNT kdnNAVW
(
Nm
NT )
kamNAVW -~- kaPNAV R
[R~]w -~- kpii[Rl]W[Mi]
- ~ kpij[MJ]W[g~] w - kTii[R~]W[R] w
j=l
(11)
186 For radicals of more than one monomeric units, a type separation is not made:
IvANmw + kap ,Vnr. AvRNT kpii[Mi]w([ej-1] w - [Rj] w) - -w kT[gj]w[e] w - ( kamnr.V ~L ) [Rj] w
d[ej]w i=1
VjE[2;3;..;jcrit]
and
[R]W-E[Ri]W+E[R~]W+[RI] i
and
Rw [R]W=NAVW
(12)
i
2.6. The Gel Effect A detailed physical model for the gel effect has been developed by (Soh and Sundberg, 1982) and simplified by (Li and Brooks, 1993a). This model- extended to copolymerisation by a probability and fractional conversion weighted average approach- is used here. The interested reader is referred there. 2.7. Volume Balance A total volume balance is necessary to account for all reactions and find the relevant concentrations in the material balances and the phase algorithm.
dVn dt
nI "MI + PI
i:1
DMi
--VW E i=1
--
RpiMmi i=1
kpii[Mi]W[R]WMMi
PMi 1 PMi
01)) (13)
2.8. Parameters Most parameters can be found in the cited publications. Propagation constants (kpii) , initiator decomposition (f, kd), micellar surface area and absorption and particle surface coverage (Am, kam, as, CMC, Ap), termination coefficients (kTii, kT), agglomeration and critical chainlength (kagl, jcrit) are literature values for the relevant monomers. Desorption coefficients have been found by an extension of the approach presented by (Asua et al., 1989). 3. SIMULATION Figure 1 shows simulation results. The results for Vinyl Acetate and Butyl Acrylate show good agreement with published experimental data shown by (Dimitratos, 1989). The gel effect can be seen in a very pronounced manner in all simulations, namely when conversion increases strongly. This is caused by the decrease in the termination rate leading to an increase in the average number of radicals per particle. As can be seen in the first three subfigures of figure 1, the gel effect parameters need further adjustment, as (Soh and Sundberg, 1982) published it for bulk polymerisation. Although literature data for the simulated systems could not always be identified, it can be stated that the shown curves qualitatively depict expected results. 4. SUMMARY, CONCLUSION UND O U T L O O K A complete monodisperse model for a semi-continuous emulsion copolymerisation for n monomers has been developed. Different runs show that the model copes with ter-, co- and homopolymerisation of different species. The runs have all been simulated in the same manner, the monomers are added with a constant flowrate. At a certain point in time (where the discontinuity in the curve can be seen), dosage is stopped. The gel effect- using parameters
187 from (Soh and Sundberg, 1982) - has a strong effect at a high conversion level. For emulsion polymerisation it is thought that the parameters will need adjustment with experimental data. 5. NOTATION m
[.]
Concentration
p R
Density Reactor CMC Critical micelle conc. jcrit Critical chainlength kag l Agglomeration
M
NA Pj rp
Monomer Avogadro's Number Probability Propagation rate
kt
Averagetermination rate contant P Particlephase A Surfacearea f Efficiencyfactor kdn Radicaldesorption kai Absorption rate
M i Monomer i Nm Number of micelles R Sr
Radical Surfactantconcentration
w
as I kpi j
kd Mi Nr Ri V
Average number of radicals per particle Water phase Surfactant area parameter Initiator Propagation rate constant Initiator decomposition rate constant Mol. mass monomer i Number of particles Rate of reaction M i Volume
REFERENCES Asua, J. M., E. D. Sudol and M.S. E1-Aasser (1989). Radical desorption in emulsion polymerization. Journal of Polymer Science: Part A: Polymer Chemistry 27, 3903-3913. Dimitratos, Y. N. (1989). Modeling and Control of Semicontinuous Emulsion Copolymerisation. PhD thesis. Lehigh University. Forcada, J. and J. M. Asua (1990). Modeling of unseeded emulsion copolymerisation of styrene and methyl methacrylate. Journal of Polymer Science: Part A: Polymer Chemistry 28, 9871009. Li, B. and B. W. Brooks (1993a). Modeling and simulation of semibatch emulsion polymerization. Journal of Applied Polymer Science 48(10), 1811-1823. Li, B. and B. W. Brooks (1993b). Prediction of the average number of radicals per particle for emulsion polymerization. Journal of Polymer Science: Part A: Polymer Chemistry 31, 2397-2402. Min, K. W. and W. H. Ray (1974). On the mathematical modeling of emulsion polymerization reaction. Journal of Macromolecular Science - Reviews of Macromolecular Chemistry Cl1(2)(177), 177-255. Richards, John R., John P. Congalidis and Robert G. Gilbert (1989). Mathematical modeling of emulsion copolymerization reactors. Journal of Applied Polymer Science 37, 2727-2756. Smith, W. V. and R. Ewart (1948). Kinetics of emulsion polymerization. The Journal of Chemical Physics 16(6), 592-599. Soh, S. K. and D. C. Sundberg (1982). Diffusion-controlled vinyl polymerization. I. to IV. Journal of Polymer Science 20, 1299-1371. Series of 4 articles. Urretabizkaia, A. and J.M. Asua (1994). High solids content emulsion terpolymerization of vinyl acetate, methyl methacrylate, and butyl acrylate. I. Kinetics. Journal of Polymer Science, Part A: Polymer Chemistry 32(9), 1761-1778.
188 Vinyl Acetate
Butyl Acrylate
Conversion
0.02 O Dimitratos [ Model
= 0.8 o =~0.6
8
0.
.~0.015
0.4
0
5000
0.01 ~0. r..)
~
%
0.2
o=0.
~
O
0.005
10000
0
15000
0.
0
5000
Triple (VAC, BA, MMA)
10000
5000
Pair (VAC, MMA)
1
0.7
I
---- VAC I BA MMA
=00.8
15000
Pair (MMA, STY)
0.5
0.6
0.4
~0.8
0.4
0.3
0.6
o
0.2 a~ 0.2 0
0
1
2
0
3 X 104
0.2 0
0
1
2
3
4
"" ~k___ 0
5000
10000
15000
0
50;0
~o/~o
15000
x 104
0.8
0.8
o= 0.6
=o 0.6
=o 0.6
o= 0.4
"i 0.4
0.2
0.2
2
r..)
0.4 0.2
i
3 x 104
Homopol., STY
2
;
0
4 x 104
Homopol., VAC 2.5
1
=o0.8
.o
"~
Homopol., MMA t
.S~
2
~.o 1.5
0.6 o "~ 0.4
i
[m 1,_
2.
,,'~.
Conversion I CONCVAC.~
i
~
.~
i
1
---- ConversiOnSTY[ o 0.5
rjo 0.2 o
MMA STY ]
1
0.8
1
[~
0.4
/
0.1
A
1.2
o=9 =~
~
15000
1.4
,- vA ]
0.6
10000
o
1
2
time(s)
3 x 104
0
I i I m ~ i ii
0
5000
10000
time(s)
15000
0
5000
10000
15000
time(s)
Fig. 1. Simulation run summary: VAC-Vinyl Acetate, BA-Butyl Acrylate, MMA-Methyl Methacrylate, STY-Styrene
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
189
Computer Aided Continuous Time Stochastic Process Modelling Niels Rode Kristensen a, Henrik Madsen b and Sten Bay JCrgensen a aComputer Aided Process Engineering Center (CAPEC), Department of Chemical Engineering bSection for Mathematical Statistics, Department of Mathematical Modelling Technical University of Denmark, DTU, DK-2800 Lyngby, Denmark A grey-box approach to process modelling that combines deterministic and stochastic modelling is advocated for identification of models for model-based control of batch and semi-batch processes. A computer-aided tool designed for supporting decision-making within the corresponding modelling cycle is presented. 1. INTRODUCTION With the development and increasing number of possible applications of advanced model based process control schemes, e.g. model predictive control (MPC), more and more rigorous demands are placed on the quality of available dynamic process models. Model quality measures the ability of the model to predict the future evolution of the process, so in order to obtain good prediction performance, these models must be able to capture the inherently nonlinear behaviour of many process systems, such as batch and semi-batch processes. Furthermore these models must be able to provide predictions in the presence of noise, i.e. process noise due to approximation errors, unmodelled inputs and plant-model mismatch and measurement noise due to imperfect measurements. Meeting both demands with the same model is difficult, so there is a tendency in litterature to use either a deterministic approach or a stochastic black-box approach to process modelling. The deterministic approach is based on using first engineering principles to derive ordinary differential equation (ODE) models. These models are well-suited for describing nonlinear behaviour, but they lack the desired predictive capabilities in the presence of noise, because they do not encompass a noise model and because unknown parameters are estimated in an output error (OE) setting, which tends to emphasize the pure simulation capabilities of the model instead of the predictive capabilities, cf. Young (1981). The stochastic black-box approach, on the other hand, is based solely on using time series data for identifying a model, usually in the form of a discrete time transfer function model. These models usually have very nice predictive capabilities because of their inherent noise model and because unknown parameters are estimated in a prediction error (PE) setting, cf. Young (1981). Unfortunately these models are not equally well-suited for describing nonlinear behaviour, especially not outside the (possibly narrow) operating region, within which the time series data for identification is obtained.
190 In this paper an alternative grey-box approach to process modelling is advocated. This approach combines the deterministic approach and the stochastic black-box approach in a way that seeks to combine their respective strengths, i.e. from ODE models the intuitive appeal of their derivation from first engineering principles and their ability to describe nonlinear behaviour, and from stochastic black-box models the nice predictive capabilities and their ability to handle both process and measurement noise. The aim of this paper is to describe the grey-box approach and outline its advantages. This is done in Section 2, where a computer aided tool that aims to support decision-making within this approach is also presented. In Section 3 a small example is given to illustrate one of the advantages of this approach and the conclusions are presented in Section 4. 2. A GREY-BOX APPROACH TO PROCESS MODELLING
A very appealing way of combining the deterministic and the stochastic approaches to process modelling is to use stochastic differential equation (SDE) models as shown by Astrrm (1970). The grey-box approach advocated in this paper is therefore based on SDE models in the It6 sense or, to be more specific, on the continuous-discrete stochastic state space model dxt = f (xt , ut,t, O)dt + ~(t, O)dcot Yk = h(xk, uk, tk, 0) + ek
(1) (2)
where t E ~ is time, xt E X c Nn is a vector of state variables, ut c U C Nm is a vector of input variables and Yk E y C R l is a vector of measurements, xk = Xt=tk and uk = Ut=tk. 0 C 19 C RP is a vector of parameters, and f(.) C R n, or(.) C ]t~n x q and h(-) E R l are nonlinear functions, cot is a q-dimensional standard Wiener process and ek C N (O,S(tk, O)) is an/-dimensional white noise process.
Fig. 1. The modelling cycle for control which constitutes the core of the grey-box approach to process modelling.
Figure 1 shows a modelling cycle based on this model, which describes the grey-box approach, and by means of which some of its advantages can be outlined. 9 The continuous time system equation (1) allows the initial structure of the model to be determined from first engineering principles in the form of an ODE model, which is intuitively appealing, since any prior physical knowledge can be included and because
191 the parameters of the model can easily be given a physical interpretation. Furthermore, most chemical and process systems engineers are familiar with this way of constructing a model. 9 When subsequently determining unknown parameters of the model from a set of data, the continuous time system equation (1) and the discrete time measurement equation (2) make the model flexible by allowing varying sample times and missing observations. 9 The model provides a separation between process and measurement noise, which along with the stochastic nature of the model allow the parameters to be estimated in a PE setting using a statistically sound method, e.g. m a x i m u m likelihood (ML). 9 For the same reasons statistical tests and residual analysis can subsequently be applied in a systematic manner to validate the model, and if it is found that the model is not valid these tools also provide information on how to alter the model to improve its quality. In the following the individual elements of the modelling cycle are explained in more detail. Once a model structure has been determined from first engineering principles, unknown parameters of the model can be estimated from a set of data. Nielsen et al. (2000) have recently reviewed the state of the art with respect to parameter estimation in discretely observed It6 SDE's and found that only methods based on nonlinear filtering provide an approximate solution to the full problem of determining ML estimates of the parameters of the continuous-discrete stochastic state space model. Unfortunately, applying nonlinear filtering is difficult, so in order for the grey-box approach to be feasible, extended Kalman filtering (EKF) is used instead as shown in the following. Determining ML estimates of the parameters means finding the parameters 0, including the initial conditions x0, that maximize the likelihood function with respect to 0 given a set of measurements Yo, yl . . . . . y~ . . . . . YN. By introducing 9~ = [Yk,Yk-1, . . . ,Yl ,YO] and ~klk_ 1 = E { y k l ~ k - 1 , 0}, Rklk-1 = V { y k l ~ k - 1 , 0 } and ek -- Yk -- 33klk-1 and by assuming that the conditional probability densities are Gaussian, the likelihood function becomes
L(YNI 0 )
--
fip(ykl~k_l,O) k=l
p(yolO)
--
f i exp (--
e'k P~kk-
l13k) l
p(yolO
)
(3)
k=l v/det (Rk k-l) (V/~)
where, for given parameters 0, ek and Rklk-1 can be computed by using a continuous-discrete EKF. If prior information is available in the form of an a priori probability density function p(0) for the parameters, Bayes rule can provide an improved estimate of the parameters by forming the posterior probability density function, i.e. (4) p(0[YN) -- L(YNIO)p(O) o, L(YNlO)p(O) P(YN) and subsequently finding the parameters that maximize this function, i.e. by performing maximum a posteriori (MAP) estimation. By assuming that the prior probability density of the parameters is Gaussian, and by introducing/1o = E{O}, E0 = V{O} and e0 = 0-/1o the posterior probability density function becomes p(OiYN)~
1-NI exp( - l e r O - 1 l
k=l v/det (Rk k- 1) (V/~)
p(yolO)
p
v/det (E0) ( v ~ - )
(5)
192 If, instead of a single set of measurements, several consecutive, but yet separate, sets of measurements, i.e. y11, y22 ..... YNi ..... ys s, possibly of varying length, are available, a similar estimation method can be applied by expanding the expression for the posterior probability density function to the general form p(OIY)~
l~I i=1
exp(--l(e~)T(R~lk-1)-l(eik)) k=l ~//det(Rikk_l)(V/~) l fi
P(y~lO)
e x p ( - 12 e T~ 1 -1 76176 v/det ( l ~ 0 ) ( v ~ ) p
(6)
I
where Y - [yll, y22,... , Y/vi,'", ySs]" Finding the estimates of the parameters 0 is now a matter of further conditioning on Y o - [yl,y2,... ,rio,..., ySo] and applying nonlinear optimisation to find the minimum of the negative logarithm of the resulting posterior probability density function, i.e. t~ - a r g m ~ - In (p(01Y , Yo)) (7) ttEt~
With this formulation it is possible to perform MAP estimation on several data sets of varying length, but as special cases it is also possible to perform ML estimation on several data sets (with p(0) uniform), MAP estimation on a single data set (with S - 1) and ML estimation on a single data set (with p(0) uniform and S = 1). When the unknown parameters of the model have been found using one of the above estimators, statistical tests and residual analysis can be performed. First of all, since the estimators are all asymptotically Gaussian the parameter estimates and their standard deviations can be used to perform marginal t-tests for parameter significance, i.e. to test if the parameters are significantly different from zero. This is particularly important for the process noise parameters, because parameters that are significantly different from zero indicate that the model structure is not perfect, i.e. that there may be approximation errors, unmodelled inputs or plant-model mismatch. It is an inherent assumption of the above methods for estimation of parameters that the conditional probability densities are Gaussian, and for nonlinear systems this assumption is only likely to hold when small sample times are used, so the validity of this assumption should also be tested by performing a test for Gaussianity. Finally it is possible to test if the model is correct by performing a goodness of fit test as shown by B a k e t al. (1999) and by performing residual analysis. For the latter purpose both standard linear methods and nonlinear methods based on nonparametric modelling are available, cf. Nielsen and Madsen (2001). For supporting decision-making within the modelling cycle a computer aided tool, CTSM, has been developed, cf. Kristensen and Madsen (2000). Within this tool a number of program units corresponding to the individual elements of the modelling cycle have been or will be implemented, including a graphical user interface for setting up the model structure and algorithms for estimating parameters and performing statistical tests and residual analysis. Altogether these program units aid the chemical or process systems engineer when setting up models. 3.
-
EXAMPLE
The following is an example, which illustrates an important feature of the grey-box approach the possibility of determining whether a given model structure is correct from estimates of the
193
process noise parameters. The process considered is a simple fed-batch fermentation process described by an unstructured model, i.e. ds
yS
-
=
\y~,]
8 )
/-~(s)x/
S/
CO 2 4- H 2 0 *;
Partial substrate oxidation S +OH" - - + I + H 2 0
(2)
Previous research on Fenton's oxidation mechanisms for different types of organic substrates emphasizes the multi-step first-order kinetic behavior regarding both IOn'] and organic compounds 4. Assuming a first-order kinetic behavior for both s and I in the lumped reactions (2) and using pseudo-kinetic parameters kl and k2 that are dependent not only on the reactor temperature but also on the p H and reagent' s addition policy. The overall dynamics of the oxidation process can be expressed as
a[I__]]=
_ d[S___J = (k 1 + k2 ) [ S ] ;
dt
k2 [S]
dt
(3)
Integrating this pair of equations from the initial condition [s]0 = Voc]0 and [I]o = 0 for t= 0 and bearing in mind that [s]+[I]=Voc] at any time t, it only takes elementary calculus to derive [TOC] [TOC]o
k2
k1
kl + k 2
kl + k 2
= ~
+
x exp[-(k
I +
k2).t ]
(4)
Equation (4) provides a compact representation of the time dependent TOC concentration in terms of parameters that indirectly account for reagent addition policy along with temperature and pH evolutions. The kinetic parameters can be easily estimated by fitting experimental data (TOC) to the model (4) using any nonlinear regression technique such as a neural network. For optimal operation of the Fenton's process, the key issue to be solved is how to adapt on-line the reagent feeding strategy upon estimation of kl and k2. 4. ADAPTIVE OPTIMAL OPERATION Variable amounts and types of organic pollutants in the waste stream imply not only batchto-batch variations but also time-varying rates of oxidation within the same batch. Consequently, on-line estimation of rates kl and k2, and subsequent adjustments to the reagent's feeding strategy through dynamic optimization are necessary to minimize treatment time 5. The only problem is that direct TOC measurements are both expensive and delayed for on-line optimization. In this work, we propose to use instead two inferential measurements that can be readily available on-line at low cost: oxidation-reduction potential (ORP) and the production of carbon dioxide in the reactor. The ORP reading of the reactor content for a given operating condition is directly related to how much refractory to oxidation by the hydroxyl radicals are the organic pollutants present. In terms of the kinetic parameters of the lumped model, this can be simply stated as:
1
O R P oc ~ k1+ k2
(5)
This inverse proportionality relationship explains why ORP steadily increases whilst oxidation is progressing as only the more refractory compounds are left in the reactor. Thus, ORP has a strong correlation with the overall rate of oxidation. To differentiate between partial and total oxidation another measurement is needed. From equation (2) it is clear that kl is strongly correlated with the [CO2] in the outlet stream.
240 Since the kinetic constants kl and k2 are affected by the Fenton's addition policy, temperature and pH, the estimation is based on a rather short time series of these highly relevant variables, named inputs hereafter for short. A functional relationship ,p(.) that relates kl and k2 to the chosen inputs is learned off-line using prediction errors of infrequent and sparse measurements of TOC as a fitting criterion. To compactly represent the relationship q~(.), a feed-forward neural network is used. An initial set of training examples is generated off-line and later enlarged incrementally as more batches are treated. Fig. 1 summarizes the key points of the proposed on-line estimation block. The optimization problem can then simply stated as: "finding the minimum feed rate of the Fenton's reagent that maximizes the rate of pollutant degradation" k1 M i n Max. F
• exp[-(k 1+ k 2) .At]
k1+ k2
At
(6)
Subject to: {k,,k2} = qo(F,-) O E ( f 0 - gl), where ~ is a small number which we set to the equations convergence tolerance, to ensure finite norm reduction. Even if this option is active, step rejection is considered only if there is strict linear norm reduction: gl < f0. Hence in VC, and to a lesser extent HFN, NR is applied partially, but PLP gives linear norm decrease allowing consistent enforcement of NR. 3. TEST P R O B L E M S Most tests were performed on problems described in [ 1]. We tested their problems 1 - 12, 15 - 24 and 26 - 33 but did not try pipe network (13 - 14) or distillation problems (34) or a contrived algebra problem (25). The problems are small, having at most 14 variables, but some are relevant to reaction engineering or equilibrium separations. Our approach has also proved effective on substantial flowsheeting problems involving distillation, heat exchanger networks and utility cogeneration systems. For this study we added a non-ideal dewpoint problem for acetone and water on which, unusually, the VC method performed poorly. This has 10 variables and uses Antoine and Wilson models for vapour pressure and activity coefficients. We gave three starting guesses: one good (the solution with ideal VLE), one with ballpark unscaled values and one poor, with all variables guessed as 1.0. (Pressures were in mm Hg). Bounds were essentially 'LP', but variables in log arguments had small positive lower bounds and Celsius temperature had a negative lower bound large enough to prevent division by zero in the Antoine equations. Upper bounds were large, except for mole fraction variables (1.0). 4. RESULTS Our test set comprised 33 problems, separately counting three versions of a catalysis problem with different parameter sets (problem 29 in [1]). Guesses were taken from [1], and other
253
guesses were tried. For many problems and guesses we gave alternative bounds, usually by trying the bounds in [1] and also the less constraining 'LP' and 'LPN' cases. There were 99 guesses and 159 guess & bound cases. Full results are given in [3]. First we ran the guess and bound cases with hard bounds but no TR. 71 of the cases converged with no step restriction, having an average iteration count of 13.4. Another 54 cases converged without a TR for all SR methods but encountered hard bounds. From these we can compare the efficiency of the SR methods: SR method HFN VC PLP - SPK1 PLP - MCMR
Average iters 12.5 12.2 16.4 16.5
SD of difference - from HFN
- from SPK 1
1.6 10.7 2.5
6 of these cases had singular guesses but the resulting steps were not affected by bounds. No method shows a significant advantage, though there is a hint that PLP is slightly slower. Another 6 cases with singular or severely singular guesses converged for all SR methods, but they were not counted because, without a TR, iteration count depended strongly on the default step size. Both SES and PLP easily coped with all singularities, provided SES was given a non-zero default step at severely singular points. In 16 cases without a TR, at least one SR method failed but at least one converged. HFN failed on 11 of the 16, VC on 7, and PLP on 3 with MCMR or 1 with SPK1 reordering. PLP is apparently the most robust method. The remaining 12 cases could not be solved by any SR method without a TR. However, 8 of these converged if PH or LP bounds were relaxed to LPN, albeit to an unphysical solution in some cases, e.g. with negative concentrations. We also tried trust regions on all cases. The TR often reduced iteration count, especially from poor guesses when no hard bounds constrained the steps. However, the opposite could also happen and in a few cases, introducing a TR destroyed convergence by trapping the path close to a local minimum of Ilhll. In all the cases where no SR method converged without a TR, we always found a TR of some initial size and at least one SR method which converged to a solution within physical or BB bounds. A successful TR may need to be small, especially when without a TR the path terminates on bounds. The TR appears to play a similar role to the bound-shifting recovery procedure in [ 1], though it works differently and is not guaranteed to succeed. It can be important in some cases to prevent the path reaching hard bounds early on. Especially with HFN, it is easy to estimate from a failed run without TR the maximum 91 which avoids an initial step to bounds from an interior guess. In difficult problems, a value smaller than this was usually successful. HFN and PLP are the most robust step methods with a TR, each working better on different problems. VC is seldom superior to both and is often worse, because decoupling of variables from the equations and the Newton step is harmful. NR enforcement seldom improved iteration count or robustness despite its theoretical importance. Iteration counts tended to increase, most markedly when nonlinear equations had large residual scales and the TR was small. Once the solution approached such an equation, TR increase was prevented by norm increase (fl > f0) even after quite modest steps. This effect was most noticeable with PLP step restriction, which does the most effective job at reducing residual
254 norms early on. It was also seen for poorly scaled problems without NR enforcement, albeit less severely. All our test runs were unscaled: scaling is discussed in [3]. The only real use we found for NR enforcement was to terminate runs with a TR which otherwise cycled endlessly near a minimum of llh I1. 5. PROPOSED METHODOLOGY
Based on the test results in [3], we propose a solution strategy given an NLAE problem and a starting guess. 1. Apply bounds to prevent calculation errors. If desired, let the bounds be tight enough to isolate physical solutions. Try Newton with no TR using HFN or PLP with SPK1 reordering, which is likely to be the most robust SR method. 2. At a severe singularity, apply a non-zero default step when using SES. 3. If 1 and 2 are unsuccessful, and the solution path is trapped on bounds, try relaxing the bounds. Try with PLP and HFN step restriction. 4. If 3 fails or gives an unphysical solution, try a TR. The HFN run from 1 or 2 provides a maximum Pl which keeps the first step away from bounds [3]. Try with a smaller TR using HFN. Repeat with smaller TRs to prevent trapping on bounds, or to find other solutions. If HFN fails try PLP. 5. If the TR gives endless fairly small steps use NR to terminate the path, probably near a minimum of[lh II. In this case rescaling of the equations may help, as proposed in [1 ]. This approach at first omits the TR, to avoid trapping the path near a minimum of I lhll. However, we saw this in only a few cases. In many others, a TR reduced the iteration count by keeping the solution path to reasonable values. REFERENCES
[1] Bullard, L.G. and Biegler, L.T., Iterative Linear Programming Strategies for Constrained Simulation, Computers and Chemical Engineering, 15 (1991) 239-251. [2] Wayburn, T.L. and Seader, J.D., Homotopy Continuation Methods for Computer Aided Process Design, Computers and Chemical Engineering, 11 (1987) 7-25. [3] Morton, W. and Douglas, D., A NLAE Solver incorporating Bounds, Partial Singular Equation Solution and Trust Regions, in preparation (2001). [4] Fletcher, R., Practical Optimization (2nd ed), Wiley, New York (1987). [5] Rodriguez-Toral, M.A., Morton, W. and Mitchell, D.R., Using New Packages for Modelling, Equation Oriented Simulation and Optimization of a Cogeneration Plant, Computers and Chemical Engineering, 24 (2000) 2667-2685. [6] Chen H_S. and Stadtherr M.A., Sparse Matrix Methods for Equation-based Chemical Engineering Flowsheeting, I - Reordering phase, Computers and Chemical Engineering, 8 (1984) 9-18.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
255
Improving Robustness using Homotopy as an Open Solver in a Dynamic Simulation Package Jorge R. Paloschi AspenTech Ltd. Sheraton House, Castle Park, Cambridge, United Kingdom Email:
[email protected] We present a homotopy implementation for improving the robustness of nonlinear equation solving in the commercial dynamic simulation package Aspen Custom Modeler TM. The implementation is realized by making use of open solver interfaces available in the product. The code implementing the homotopy solver is presented as a Dynamic Load Library and communicates with the simulator only via the interfaces. The implementation is tested on an industrial distillation column, containing 50 plates and about 4500 equations. The internal nonlinear equation solver fails to converge when a poor initial estimate point is used, while the homotopy solver achieves convergence. 1. INTRODUCTION Robustness in terms of solving problems from bad initial points is a desirable feature in a process simulator. This feature is particularly important when an entirely new simulation is attempted, where the true location of the solution may be completely unknown (or even the fact of whether there is a solution). In the context of steady state or dynamic simulation of chemical processes, robustness is mainly related to the capacity of solving a set of nonlinear algebraic equations (NLAE's) which arises either from the steady state equations or the initialization of the set of differential algebraic equations (DAEs) in the dynamic case. In the dynamic case, robustness is also needed while solving the DAEs after the initialization stage, but since in this case we have the solution at the previous time as an initial guess, the robustness of the nonlinear solver is less important. Traditionally, solvers based on Newton's method have been used to solve NLAEs. Convergence properties for this class of methods are only local, requiring the initial point to be in a neighborhood of the solution. However, in practical terms, convergence can be achieved from points not too close to the solution by using heuristic algorithmic details devised to enlarge the convergence domain (see Dennis and Schnabel, 1983). But, there is no guarantee that convergence will be achieved.
256 There have been several attempts to devise global methods, that is, methods which ensure convergence from initial guesses far from the solution. Ideally, global methods will guarantee convergence from any point in the domain under consideration, but for NLAEs satisfying this condition is sometimes difficult (or impossible) to ensure. The earliest attempt to develop a global solution method can be traced back to Lahaye's (1934) use of homotopies or continuation methods for single nonlinear equations. This technique consists of replacing a system of NLAEs f(x) = 0
(1)
h(x,t) = 0
(2)
by a homotopy function
where f : ~tt" --+ 9t" and h : ~n+l ~ ~n. The homotopy function h is such that h(x,O) = 0 has a known solution (or it is easy to find), while the solution of h(x,1) = 0 coincides with that for equation (1). Equation (2) defines a solution path x(t), and because of the conditions on h, we can, in theory, start with x(O) and follow the continuation path until x(1) is reached, which is the solution to (1) we are seeking. The main idea behind using (2) instead of (1) is that we can attempt solving h(x,t)=O from a previous value of the parameter t, close enough to ensure convergence. The actual implementation of these methods is not as simple as it looks, since a robust implementation implies transforming this problem further into a set of ordinary differential equations, and not solving it as stated here. We refer the reader to the excellent book by Allgower and Georg (1990) for details on how this is achieved. Alternatives to continuation do exist and they share the same aim, that is, enhancing the domain of convergence. One of them is the interval method, where instead of working with numbers, we work with intervals. Hence, if we know the solution belongs to a bounded domain, we could, theoretically, use an interval method based on the domain to guarantee convergence. See the book by Neumaier (1990) for details on these methods. Another alternative, is to use global optimization methods, which have received particular attention in recent years and look very promising. See the discussion by Floudas(1999) for an overview of these methods in the context of Chemical Engineering. Each global methodology has its own advantages and drawbacks, and there is scope for all of them in the wide range of problems to be solved. All of them have a common factor, they are very expensive in execution time (because of long homotopy paths and the combinatorial nature of the other approaches), which is the price to pay for global convergence. An advantage for the homotopy methods is that they can be implemented using no more than the residual and Jacobian information that is already available for an implementation of Newton type methods. They can even be implemented without the availability of the Jacobian (using efficient sparse finite difference perturbations). Interval and global optimization methods, on the other hand, require the availability of the analytical residuals, but they provide much better global convergence properties.
257 In this paper, we will show how homotopy methods can be implemented in a commercial simulation package, using open solver interfaces available in Aspen Custom Modeler. These interfaces will be briefly discussed in Section 2, while Section 3 will be devoted to presenting the algorithmic details of the homotopies implemented. Section 4 will provide an example of use, followed by conclusions in Section 5. 2. OPEN SOLVER INTERFACES IN ASPEN CUSTOM M O D E L E R T M The Aspen Open Solver set of interfaces (AOS) have been introduced in the commercial dynamic simulator Aspen Custom Modeler (ACM) in order to allow plugging external solvers to solve NLAEs. They are available in ACM 10.2, which is the release used for the results in this paper. Once an external solver is plugged-in, it can be used as an alternative to the internal solvers available. We have chosen to implement homotopies in ACM using these interfaces, which allows the homotopies to become completely independent of the product itself. An advantage of this approach is that as AOS interfaces become implemented in other products, a solver implementation based on the interfaces can then be plugged-in without needing any adaptation or modification. The AOS set of interfaces basically consists of an Equation System Object (ESO) interface and a Services interface, both implemented and exposed by ACM. The ESO interface allows the solver to fetch properties of the equation set including residuals, Jacobian, variable names, etc. A third interface is the open solver interface, which is used to implement the homotopies. This interface receives a pointer to an AOS ESO interface on creation, which provides the problem data needed for the solution process. ACM provides a "plug" which accepts a solver implementing the AOS solver interface. The interfaces are presented in Paloschi et al (2000). Similar interfaces were proposed by the Cape Open initiative, with the main difference being that they were designed for COM or Corba implementations, while the AOS set of interfaces is based on pure C++ abstract classes for increased performance. 3. IMPLEMENTED HOMOTOPIES
The homotopy we have implemented is the sparse bounded homotopy proposed by Paloschi(1996):
h~.h (x,t) = P ( x ) h ( x , t ) + v(x) - v(x h )
(3)
which is based on a standard homotopy h(x, t). See the original paper for a detailed description of P(x) and x b. In this paper, we have used h(x,t)=f(x)-(1-t)f(x ~ while v(x)=F ~ x, where F ~ is the diagonal of F'(x~ the Jacobian off(x) evaluated at x ~ The roles of P(x) and x b are to keep the homotopy path within a bounded domain Y~. P ( x ) - i within Y~ and 0 < P(x) < 1 outside, while xb=x inside ~ . Therefore, inside ~ , h.,.h - h .
All the involved functions are smooth
(hence differentiable), plus their derivatives are easily defined in terms of those for h(x, t) and
f(x). This homotopy can be implemented based only on the residuals and the Jacobian, as shown in the original paper. The main advantage of (3) is that it can be ensured that, if x is within a
258 bounded domain f2, the homotopy path x(t) will be bounded within ~ . This is particularly important when solving chemical engineering problems involving physical property calculations because the properties package will refuse to evaluate outside the valid domain. As the tool to track the homotopy path, we have used the code PITCON from Rehinboldt and Burkardt(1983). Because the use of homotopies is a very expensive procedure, the algorithm first tries to use a nonlinear solver to converge the problem, and only resorts to using the homotopy if it fails to converge. In this way, we can use the homotopy algorithm as a general nonlinear solver without incurring in additional cost, unless it is absolutely necessary. The way we do this is by having an AOS Solver plug in the homotopy solver itself, which enables the use of any solver which implements the AOS solver interface. The homotopy solver is implemented as a DLL exposing the AOS solver interface. The following diagram shows a sketch of how it looks:
Homotopy] Solver [
DLL ~ / ,......." \\
Pitcon
&
@OSSo,vow) ES ) @OSSo,e)
\
ACM
~
....A..OlSSolver implementation
DLL
The ovals denote the AOS interfaces. The one in the middle corresponds to the ESO interface, implemented by ACM and used by the homotopy solver. The one on the left corresponds to the AOS solver interface, implemented by the homotopy solver and used by ACM. The last on the right, is again an AOS solver interface, implemented by an external solver dll and used by the homotopy solver. In our particular case, we have used the internal ACM nonlinear solver, which is also implemented in a dll exposing an AOS solver interface. 4. EXAMPLE OF USE As an example showing the use of the solver implementing the homotopy, we have chosen a distillation column simulation. It consists of a dynamic industrial distillation model instantiating a column with 50 plates, with all the necessary controls. It provides a very detailed simulation of a real column posed in Aspen Plus TM and exported for use in ACM. Once instantiated in ACM, the problem has 5418 variables, 4495 equations and 14926 nonzeros. By using an internal tearing strategy, it is decomposed into 1653 blocks, many of which are nonlinear. The problem converges without difficulties using the internal Newton-type solver in ACM, if the initial point provided is used. If the initial point is changed to one where all the variables are set to their default values, the internal solver does not converge anymore. In this case, the
259 initial point is very bad since there are no profiles at all in the column and all variables for each plate in the column are set to default values. By using the homotopy solver described here, having plugged it into ACM using the AOS interface available, convergence is achieved. Of all the nonlinear blocks to be solved, 32 do not converge with the internal solver, but converge using the homotopy. Figure 1 presents the homotopy path for all 20 variables in the first block where the homotopy was used. All the equations involved in the block belong to the last stage immediately before the reboiler. Because the variables have a wide range of values, we have scaled them in the interval [ 1-10], so we can show the homotopy path for all of them in a single graph. The horizontal axis corresponds to the homotopy parameter t, while the vertical one is common for all the variables. As we can see from Figure 1, some variables have a simple homotopy path (almost a straight line), others start and come back to the same value, while others have a more complex path. As a reference, we also present in Table 1, the starting and final values for all variables in the homotopy path, for the first block. As you can see from Table 1, some variables start from initial guesses which are far from the solution. In Table 2, we show the starting values for some of the variable types. Table 1 9Start and end values of homotopy path for variables in first block t
0
xl
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11 x12 x13 x14
x15
x16
x17
x18
x19
x20
0.04 0.50 1.00 0.00 0.00 25.00 0.50 0.50 0.50 0.50 0.50 1.00 2.50 2.50 32.04 0.01 23.63 832.82 10.00 10.00
1 0.05 0.97 1.48 0.00 0.00 86.84 0.03 0.50 0.50 0.66 0.34 0.05 3.39 0.09 32.02 0.01 24.03 769.45 0.14
1.77
Table 2 9Initial values for some variable types
Iv.rT, ool
L'01
Initvalue I 5971.15
o' a01 ow assJ 'ow o' I ,owVo, J Fracl He,0u0Mol 14330.8
I
1000
I
10
116.66710.51
2.5
[olOI:YJllil,,,I~I:l~J
Although the use of the homotopy is expensive, as compared with a standard Newton solver, it is not prohibitively expensive. For this problem, it takes about 50 seconds of cpu time to solve it, while it takes about 5 seconds to be solved from a good initial guess (i.e., not needing the homotopy). Hence, the price we pay is not really that much considering the alternative with the internal solver (which is not being able to solve the problem). The cpu times given above were obtained using a Pentium I1266Mhz PC. 5. C O N C L U S I O N S We have presented an implementation of a homotopy solver in the commercial simulation package, Aspen Custom Modeler. The implementation has been done using open solver interfaces available in ACM, which are the only way in which the homotopy solver communicates with ACM. The homotopy solver itself uses the interfaces to plug in an external nonlinear solver. The path following code PITCON is used as an internal tool for path tracking. The homotopy solver is implemented as a DLL and plugged into ACM to be used as an alternative to the internal Newton-type solvers.
260 Its use in the package is illustrated with a large distillation column example. Using a poor initial point, the homotopy solver obtains convergence, while the internal ACM nonlinear solver fails to converge, This shows how we can use homotopy techniques to improve the domain of convergence and robustness of a dynamic simulator. The methodology used in the implementation shown in this paper requires no more information than is already available for a Newton type method, that is, residuals and derivatives. No analytical information on the residuals is needed.
REFERENCES
Allgower E.L. and Georg K., Numerical continuation methods-An introduction, Springer Verlag,(1990) Dennis, J.D. and Schnabel R.B., Numerical methods for unconstrained optimization and nonlinear equations. Prentice Hall, (1983) Floudas C., Recent advances in global optimization for Process Synthesis, Design and Control: Enclosure of all solutions, Computers Chem.Engng., ppS963-$973, (1999) Lahaye E., Une methode de resolution d'une categorie d'equations transcendantes. C.R.Acad.Sci.Paris 198, pp1840-1842,(1934) Neumaier A., Interval methods for system of equations. Cambridge University Press, (1990) Paloschi J.R., Bounded homotopies to solve systems of sparse algebraic nonlinear equations.
Computers Chem.Engng.,21,5,pp531-541,(1996) Paloschi J.R., Laing D.M., Zitney S.E, Open Solvers Interfaces for Process Simulation, AIChE Annual Meeting, Los Angeles, November 12-17,(2000) Rheinboldt W. and Burkardt J., A locally parameterized continuation process. A CM Trans.Math.Software 9,pp215,235,(1983)
.
.
.
.
Figure 1 :Homotopy paths for variables in first block
1.2
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
261
Using wavelets in process identification: A new link to the state space Heinz A. Preisig a aTechnical University Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands Spline-type modulating functions were derived based on state-space considerations: they are the result of solving an observer problem. Spline wavelets are derived based on their properties by the signal processing community. That the two are the same has not been recognised by either of the two communities. In fact, the spline-type modulating functions are multi-wavelets covering a wider range of functions than this is the case with ordinary wavelets. The resulting link gives new and interesting insights. 1. I N T R O D U C T I O N The project, reported in the three papers [3-5], was on the identification of models for an industrial reactor using the spline-type modulating functions introduced by Maletinsky [ 1]. The first of the three papers [3] reviews the history, discusses structural issues and the basics of the methodology as it was known in 1990, respectively. The second paper [4] describes the implementation for on-line and off-line applications and the third [5] is devoted to the application. Spline-type identification uses B-Splines as state variable filters, which are known to counter the bias problem in estimators. At the time this work was done (1978-84) [2], "wavelets" were not yet in fashion. Whilst the wavelets boomed shortly thereafter, the modulating function method never really made it into the ranks of the primary identification methods as the small and scattered literature body gives clear testimony [3]. The two literature bodies of modulating functions and wavelets remained essentially separated though even a simple visual inspection of the modulating functions suggests that B-spline modulation functions are indeed wavelets. The reason for this separation lies in the fact that the two theories come from two different scientific communities, which, in earlier years were connected, but lost each other out of view on the way. This contribution uncovers some of the links, namely the view of the systems-oriented group and the signal-processing group. 2. B-SPLINE M O D U L A T I N G FUNCTIONS The B-spline modulating functions, as they were first derived by Maletinsky [1 ], are obtained by solving an observer problem for unknown initial conditions in SISO systems. The elimination of these initial signal derivatives yields an integral transformation, which compensates for the unknown quantities by a filtered piece of signal history. Maletinsky [1] used the oberver canonical form to derive this result, whilst in [4] parts of the derivation are done in the Laplace
262 domain. Given the SISO model in transfer function form
~aiDiy
:-- ~ i
i
biDiu,
(1)
elimination of the unknown initial conditions of the n integrators, except the last one which is directly measured by the output, transforms the model into
n E
s ai < (~n,i,y >
i
:--
bi < t~n,i,u >
(2)
9
i
The functions (~n,0 are the modulating functions, B-splines of the order n, the first index, and 0-times differentiated, the second index. Some of the important characteristics of the B-spline functions are: (P1) Compact base
t~a,(~(t) =
(P2) Derivative I Integral
Oa,a-1 (t) =
(P3) Stem function of order n
0
for r [0,nT];
(3)
Oa,a(t)dt;
(4)
,n,n(t)-- T-n ~ (-1)i (;) 8(t-iT);
(5)
i:--0
(P4) Zero Boundaries
d~a,(~(O)-- d~a,a(nT) --0.
(6)
{~n,jlJ "-
The family of modulating functions ~n,0 form together with their derivatives an upper triangular function matrix shown in Box 1.
---
Box 1: Generating the function matrix
The upper triangle of functions is generated stepwise. The first row are the Cholesky factors, the rows of the Pascal triangle with alternating signs. The Pascal triangle can be generated by consecutive convolution: ~a,a*~b,b = ~a+b,a+b thus convoluting ~1,1 n-times with itself ((~l,1)*n) generates the first row. The other elements of the triangle can be generated by successive integration (property P2) or, equally well, through convolution. It is easy to show with property P2 that the following relation holds: ~a,o~rf~b,~
:="
~)a+b,t~+~-
(7)
----,--
, ' -
~-,,
r~
~
t
|
,
I
" l
V1
1,1 2,2 3,3 4,4 5,5 6,6 1,0 2,13.2 4,3 5,4 6,5 2,03,14,25,36,4 3,04,15,26,3 4.05,16,2 5,0 6,1 order,derivative
I
i
[-1 rn~
V
w
~ _
0, ..,n}
@
~
263 3. M O D U L A T I N G FUNCTIONS VS WAVELETS: T H E LINK Strang [6], p250 writes: "Splines are piecewise polynomials, with a smooth fit between the pieces. They are older than wavelets. The 'two-scale equation' or dilation equation was at first not particularly noticed." Indeed, the relation has not been noticed and neither has Maletinsky's derivation. With the latter derivation being based on a signal generating system, Strang's statement "Almost every formula in this book comes out neatly and explicitly for splines." (p250) does not come much of a surprise. The two-scale relation is readily found in the above triangle of spline functions. The "straight" wavelet theory of B-splines uses triples of functions from the function triangle, high pass filter (~n,n, scaling function (~n,0 and wavelet (~2n,n. The respective wavelet is then f~2n,n
--- (~n,n *On,O ,
(8)
which is a direct use of Equation (7).
The wavelet community derives its functions based on requesting certain properties such as symmetry, orthogonality and norming. The analysis and development is driven by properties of the signals. The required tools are heavy gear and it is by no means trivial to enter the field and work your way to the front. The results are impressive and the information is very complete. In contrast: the derivation of the spline-type modulating function is driven by a viewpoint of the system (plant) generating the signal from a base (input) signal. The approach is simple and elegant, uses a minimum of the mathematical tools and yields surprisingly much of information [21, [1], [3]. Benefit can be drawn from a cross-fertilization. (i) The modulating function approach yields the view that the filters in a column yield average consecutive derivatives. The modulating function apply a weighted average, whereby the weight is the modulating function and thus corresponds to the convolution with the scaling function in one time scale of the wavelets. "Normal" spline wavelets use thus filtered zero-order derivative information (scaling) and n th derivative of the 2 n-order filtered signal, where, when looking at a single time scale, the wavelets use the same compact base. (ii) The wavelets pick three elements in the function matrix. (iii) The main filter characteristics is the time between two impulses of the high-pass filters, the characteristic time T of the modulating function theory [1], [3]. (iv) The modulating functions are derived from a state-space representation of a signal transforming system: There is a state. (v) The modulating functions can also be multi-scaled and implemented on-line [4]. The remaining part of this paper is devoted to some of the lesser-documented points of the above list. We shall have a look at what appears to be most spectacular, namely the combination of the multi-resolution wavelet thought and the thought of differentiation in the modulation sphere. 4. T I M E - S C A L E ANALYSIS USING B-SPLINE MODULATION The above thoughts triggered a project in which we attempt to seek time-scale information of the signal generating system - our plant of interest - in identification, using multi-scale modulation. The thought is naturally also motivated by the singular-perturbation approach as it is used in time-scale-based model reduction. This type of problem is well known and overlaps with our research on modelling theory [8].
264 As an illustration we use a third-order model, composed of three first-order systems in series, each with a different time constant (1, 0.1, 0.01). As an input a periodic pulse function was used, which starts after a delay used to initialise the modulation filters. In view of the limited extent of this paper, we present only some results of two analysis: (i) continuous wavelet analysis using B-spline wavelets and (ii) filtering of the input and output signals using a multi-scale implementation of the B-splines [4] generating all the signals {< d~n,j,U >, < ~n,j,Y > [j :-O, 1 , . . . , n - 1}.
Fig. 1. modulated input and output signals
The top row in Figure 1 shows the original signal and the 3-rd-order-modulated input signal, thus the modulated signal and the first and second derivative of it. The bottom row shows the corresponding output signal. Time is shown pointing to the right and different-length (resolution) modulation of order 3 towards the back with a scale indicating the filter constant T. Modulation is done "very hard", that is, a lot of the original signal is filtered out. Most of the tipples, caused by the high-frequent input signal, are ironed out, and as the filtering increases, the remaining ridge gets smaller. This hard filtering was done because we sought the dominating time constant, which indeed can be extracted easily by assuming a first-order model (equation (1)) and simply solving for the only parameter at every point. The result (not shown) is a pretty-much fiat plane at the dominating time constant. The general procedure is described in [5]. Also the effects one can expect when undermodelling the dynamics [4]. The B-spline modulating functions can be readily introduced into the Matlab wavelet toolbox. Applying the B-splines of order 1,2 & 3 yields the three graphs shown in Figure 2. The left graph shows that more of the additional terms are used to reconstruct the signal in the first phase. In the middle graph, this is significantly reduced and in the fight is essentially nonexistent. The difference between the middle and the fight, is very small such that one could conclude that the signal could live in the B-spline wavelet space of order 2 just as well as in the one of order 3. Clearly it is not as well represented in the space of order 1. The wavelet analysis can thus be used to determine an appropriate order.
265
Fig. 2. modulated input and output signals
5. A FEW THOUGHTS ON THE PARAMETER IDENTIFICATION TASK The high-pass filter is not very suitable as a modulating function, because it cuts out all pieces of the signal between the impulses. One thus uses a modulating function that is at least an order bigger than the minimal. How many of the derivatives one needs, depends on the number of parameters in the model and precisely on how one would like to perform the identification task. In a transfer function model, the maximum number of parameters, related to the eigenvalues of the state-space realisation and the gain, is 2n + 1. The complete state space representation, {A,b_,c_T,d}, has n 2 + 2n + 1 parameters, which is n 2 more than for the transfer function representation, n being the dimension of the state-space model. The n 2 parameters are for the similarity transformation matrix transforming the canonical representation back to the "meaningful" state-space representation. The number of parameters thus depends on the order of the model and the representation. Having determined on how many parameter need to be identified, the user has still a number of choices on how to extract these parameters from the filtered signals {< ~n,j,Y > (k), < ~n,j,u > (k) I J "= 0, . . . , n - 1,k "= 1,2, ..... ,number of samples}, which in general are available as discrete signals sampled with some rate determined by the application and realisation of the filter. In our current implementation the minimal shift time is equal to the sampling time of the original signal. Thereafter any multiple can be set as one of the method parameters. For constant model order one may choose to allow the parameters to change over the length of the experiment and use higher-order derivative information by simply differentiating the model equations sufficiently many time as to generate enough equations to solve for the parameter vector.
a+, a T [ ~0 y2~
9
]
-
-
Ua+
1
9
]
,
(9)
where the vector ~q collects the derivatives p , . . . , q of the respective modulated signal. Obviously, the order of the modulation must be chosen accordingly. Alternatively, if the parameters do not change quickly, one may choose another approach and select consecutive filtered values along the signal. In this latter approach, one needs to be aware of the fact that the modulations overlap, thus extract correspondingly identical pieces of information from the signal. In [4] it is argued that Shannon's sampling theorem suggests that the modulations should be at least T/2 shifted, which is equivalent to the discrete wavelet shift. In practice, one should though use excess information and augment the above procedure with a method such as minimal-sumof-squares, to minimize the effect of remaining noise components in the modulated signals. In the past, we used the same signal interval as one would obtain when shifting only T/2 but
266 overlapped at least 5 times more and then used a minimisation to determine the parameters. The latter also provides a measure for the remaining variance mapped into the estimation of the parameters. 6. CONCLUSIONS (i) B-splines modulating functions are multiwavelets. (ii) "straight" B-spline wavelets are defined by the triplet high pass filter ~n,n, scaling function ~n,0 and wavelet q~2n,n. (iii) The 2-scale relation operates along the horizontal axis of the B-spline function matrix. (iv) The modulating function approach operates along the column of the same matrix. (v) The use of differentiated signals in identification has become acceptable with the use of wavelets in signal processing, without being notice, though. (vi) On the sideline: the detection of events using wavelets, as being advertised in the discrete event dynamic community, can also be interpreted as finding discontinuities in higher-order derivatives on different time scales. (vii) Identification is still very much an art. The discussed relation, though, enriches the "bag of tricks" with an interesting new one. An extension to fractional B-splines has been published by [7] yet another variation of the theme. REFERENCES
1. V. Maletinsky, 1-i-P Identifikation kontinuierlicher dynamischer Prozesse. PhD Thesis, ETH-Ztirich No 6206, 1978, 143 p. 2. H A Preisig, On the Identifcation of Structurally Simple Models of Energy Distribution in Industrial-Scale Reactor Equipment. PhD Thesis, ETH-Ztirich No 7616, 1984. 3. H A Preisig, Theory and Application of the Modulating Function Method-I. Review and Theory of the Method and Theory of the Spline-Type Modulating Functions. Comp & Chem Engl7, No 1(1993), 1-16. 4. ibid.-II. Algebraic Representation of Maletinsky's Spline-Type Modulating Functions. Comp & Chem Engl7, No 1(1993), 17-28. 5. ibid.-III. Application to Industrial Process, A well-Stirred Tank Reactor. Comp & Chem Engl7, No 1(1993), 29-38. 6. G Strang, T Q Nguyen, Wavelets and Filter Banks. Wellesley-Cambridge Press, 0-96140887-11996490, 1996. 7. Unser M, Blu, Fractional splines and wavelets. SIAM Review 42, No 1(2000), 43-67. 8. Westerweele M R, Preisig H A, Weiss M, Concept and design of Modeller, a computeraided modelling tool. ESCAPE-99 Budapest Hungary Comp & Chem Eng. (1999).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
267
Evaluation of coupled reactive distillation performances by means of a rigorous simulation procedure D. Rouzineau, M. Meyer, M. Prevost INP/ENSIACET, LGC Equipe S6paration Gaz Liquide 18 Chemin de la Loge, 31078 Toulouse Cedex 4, France A non equilibrium model (NEQ) for multi component separation processes, including liquid and/or gas reactions, is developed in this paper. This model includes the finite mass transfer rate described by Maxwell Stefan equations. It is assumed that the bulk of both the vapour and liquid are perfectly mixed and that the resistance to mass and heat transfer is located in two films at the liquid/vapour interface (film theory). There are no restrictive hypotheses as to the nature and the localisation of the chemical reactions. The phase non ideally and the diffusion coefficient variation are taking into account. The original formulation of mass transfer is described and solved in order to avoid the bootstrap problem. The resulting DAE system is then solved. We describe the resolution and focus on two keys points : the coherent initial state and the reduction of differentiation index. With several examples, a comparison between equilibrium stage model (EQ) and the NEQ model is effected and a sensitivity analysis is done to highlight the importance of the film thickness. 1. I N T R O D U C T I O N Reactive distillation is a unit operation of great industrial interest enabling active process intensification. It can reduce capital and production costs by combining two units into one, and this unit can improve selectivity, heat integration, conversion, etc... Simulation and design of multi component reactive distillation is usually carried out using the equilibrium stage model. The limitation of conventional equilibrium stage efficiency calculations is discussed by many authors [1-5]. These authors assume that the generalised nonequilibrium model should be preferred for the simulation of a column for reactive distillation to the equilibrium model, because an accurate prediction of individual Murphee tray efficiencies (or HEPT for packing) is very difficult in the case of simultaneous multi component separation and reaction. The non equilibrium model seems to be better because the model takes into account the technological characteristics of the column (type of plate, packing...), nearer to reality. So, the NEQ model is developed by numerous authors. But this author include the Maxwell Stephan approach which is delicate to solve. Indeed, this formulation involve the bootstrap problem mentioned by Taylor & Krishna [4]. Moreover, the non equilibrium model requires parameters which can be as difficult to find as an efficiency. In this paper, the first part deals with of NEQ model; we focus on the diffusional layer near the interface, where a complete multi component reactive mass and heat transfer model is described. The numeric resolution, avoiding the bootstrap problem, is discussed in the second part. Finally, after the model requirement list, some examples will be discussed to illustrate the non equilibrium simulation in practice.
268 2. NONEQUILIBRIUM MODEL THEORY A schematic representation of the non equilibrium model (NEQ) is shown in Fig.1. This NEQ stage may represent a tray or a section of packing. It is assumed that the bulk of both the vapour and liquid are perfectly mixed and that the resistance to mass and heat transfer are located in two films at the liquid/vapour interface according film theory [6,7].
Figure 1 : The nonequilibrium model representation of stage j 2.1. Mass and heat transfer model A novel model is used to compute heat and mass transfer through the diffusion layer. Indeed, the fluid is considered as an NC component reactive non ideal mixture. The balance equations for simultaneous heat and mass transfer are written in steady state, taking into account the reactions. Fick formulation is limited to two components mixtures because it does not take the interactions between the different components into account; moreover the non ideality for the driving force is not considered. So, for mass transfer, the Maxwell Stefan diffusion law is used, in a novel formulation. Neither the diffusion coefficients, nor the molar flux due to the reaction, are considered to be constant. No assumption is made on the type or the number of reactions, thus they can be controlled by kinetics or equilibrium. The complete formulation for mass transfer, for NC components in non ideal mixture, is :
t)Ni 3z
M a s s continuity.
NRC @
EagijRj j=l
(
NRE "1- E '[)ij'~j j=l
t}Inyi
Diffusion law"
~
Equilibrium equation-
Kj : H a ?
8iJ "~"Xi
0X.1 T,P
-0
)~Xi= )/)z - ~ ( x iN j-xiN,) }=1
c-tDil
for/= 1 toNC
(1)
for i=ltoNC
(2)
for j= 1 to NRE
(3)
NC i=1
The dimension of system (I) formed by equations (1),(2) and (3)is 2NC+NRE. In a traditional model, only NC-1 equations of (2) are considered because of equation dependency. Indeed, equation (4) is obtained by summation of the NC terms in equation (2) : Olny; i=l l-l-Xi
0xt ~
" + 2 1+
~X--~T,p) ~z'~ i=1 ""
i=l
__~z= _ ~ - ' ~ gi ~Xn T,P
i=l j=l
___ ctDij
(4)
269
Taking into account the Gibbs Duhem relation
xi i=l
Dij-Dji,
equation (4) becomes:
~)ln~i
= 0 and the coefficient symmetry
~X j
(5)
' =0 i=l ~Z
In our model, equation (5) is never used also NC equations of type (2) are usable. Consequently, no additional equation is needed, as proposed by Taylor & Krishna [4], to obtain directly the molar fluxes Ni. For the heat transfer, the Dufour and Soret effects are neglected and the heat diffusion rate is evaluated by Fourier's law. The whole system is called HMT model and involves differential algebraic equations (DAE).
2.2. Phase equations The phase equations are classically the mass balances and energy balance in the bulk phase for each stage [4]. These equations take into account reactions, and there are no restrictive hypotheses as to the nature and the localisation of the chemical reactions. The temperature of the vapor and the liquid phases are not assumed to be equal. The entire column is taken as a sequence of a number of such stages. We consider a N stages column where stage 1 can be a total or partial condenser and stage N a reboiler. The modelling leads to a system of differential and algebraic equations (DAE).
2.3. Interface equation Physical equilibrium is assumed at the vapor liquid interface for each component. Moreover, mass and energy continuity is described by mass and energy balances. 3. N U M E R I C R E S O L U T I O N
3.1 DAE integration of HMT model Firstly, the HMT model is solved. No analytic solution is available, so, the DAE system is solved by a Gear method. To be efficient, a such integrator has need of a coherent initial state and a differentiation index as small as possible.
Initial conditions A robust procedure, leading to a coherent initial state ( ie. all algebraic equations must be satisfied) before starting the integration, has to be used. The following equations are solved by Newton method" Mass balance
9
Fox~ - Fxi + Z o ~ j - o NC
D,,3
Equilibrium equation 9
Kj
Summation equation:
1 - ~ xj = 0
= H ai' i=1
for i : 1 to n
(6)
forj = 1 to NRE
(7)
(8)
j=l
With this resolution, we obtained the molar fraction at the interface in agreement with the equilibrium reaction constraint.
270
Differentiation index An automatic substitution procedure is used to reduce the number of mass balances in order to take into account the chemical equilibrium constraints and to eliminate the ~ variables. Matrix formulation of equation (1) is"
-1
~ -~Z dN1
O1,NRC R1
~)1,1 ...................
~)I,I
...............
~)I,NRE
"
+
~
dN. -1 ~
=0
+
~.I).,,..................V,,,NRC
NRC
E
With a pivot strategy, canonic form of equation (1) is derived: 0 0 9
dz
[ 0
and
b~- - l - 1 0
for LkEi < 0 else.
(3)
%,
The key idea is to define system boundaries of minimal exergy throughput, which still include all relevant process information in the described unit.
300
EcCld,in EcSld,in E~St,in EhC~
,, 1 ~ i i.
.
.
.
.
EcCold,out E~old,out E~St,out EhCt,out S
r
v
~ .
.
.
.
.
.
.
.
.
.
Fig. 2. Converted and transiting exergy parts in a heat exchanger
4.2. Mixer efficiency Eq. (3) is directly applicable for a mixer. In this case, there is only one material system, but exergy can be converted from chemical to sensible form: AE
--
EoSut-~
EiS,j
E~,Ct-~ EiC,j
j=l
N: number of mixer inlets
(4)
j=l
The chemical contribution will always be negative and appear in the denominator of Ti. The sensible part can be zero, positive or negative, and a nonzero efficiency occurs if AE s > 0. An example of this is nitric acid neutralization by ammonia in fertilizer plants, provided that the heat of mixing is used. 4.3. Heat exchanger efficiency Heat exchangers contain two separate material systems. Both have sensible and chemical exergy, so that:
Egot ]
E
-
level, hence AE = \E~Sld/
E~~176 E sh~ \EcSld,out
=:ff TI -- EcSld'~
EcSld'in
(5)
E(~ot,in -- E~St,out
0
E~Sld,in/
4.4. Exergy efficiency of the purge gas cool down section Figure 3 gives a more detailed view of the purge gas cooling section. The indicated relative efficiencies give suitable information about the units, but they are not sufficient to calculate the overall efficiency of the process section unless the transiting contribution is also known. In that case, a modified flow-sheet can be drawn as shown in Figure 2 for a single heat exchanger. Transiting parts are led around the unit block. Using this technique, a unit-based efficiency can also be calculated for the observed process section. For instance, the chemical exergy of the fuel is - apart from the m i x i n g - not affected, so it is led around the flow-sheet. The resulting value of r I = 71.2 % is therefore more suitable to compare two similar process sections.
5. SENSITIVITY ANALYSIS The model of the primary reformer section (Figure 1) is implemented in C+ +. It is made use of a new developed object oriented data type called Dfloat, which evaluates all partial derivatives with respect to other variables runtime. In contrast to automatic differentiation, this approach using operator overloading and lazy evaluation techniques - handles nested optimization loops
301
cold comb. air
cold ee x cold proc. air
I
HP steam
c~ 829c~
/
@
H2
88.8% H4
@
hot purge gas ]
] [
t
t
88.3% ~ (Sll] ] ~ f
0%~ ~~~-
~_~ ( ~
.~ H6 I 74.6%I @
, S
0 / r
[H7 154.3%
t-~" hOt cOmb alr ~ HP steam hot feed mix hot proc. air Fig. 3. Detailed model of the purge gas cool down section and relative exergy efficiency of unit operations ref. steam
'
80.4%
~
I
and allows specification of the required derivatives runtime. In this work, Dfloat is used inside the unit operation algorithms to perform second order optimization. Furthermore, it is applied to obtain sensitivity of exergy efficiency with respect to heat exchanger specifications. The dependency of the (UA)-product of H4 is included into the flow-sheet calculation. It can be found that for instance
~TIt~
--
5.5 x 10 .4 KkW -1
~(UA)IJ4
and
OTIH1
O(UA)H4
=
- 3 . 3 x 10 .4 KkW -1
(6) "
Similar derivative information can easily be used to perform an outer optimization loop.
6. CONCLUSION The primary reformer section of an ammonia plant has been modelled as a system of four black box models. Based on a suitable reference state, both energy and exergy efficiency calculations are integrated into the simulation. Afterwards, one part of the process section is simulated in detail, and additional information is used to obtain exergy efficiencies of single operation units defined in an intuitive but generic way. Extending this concept to agglomerated sets of units, suitable exergy efficiencies are obtained. The capabilities of the process model can be extended to cope with sensitivity analysis tasks. With very little programming effort, derivative information is obtained automaticly and gives a useful insight into a complex network of interacting units.
302 SYMBOLS
A b Cp E G H h
heat exchanger surface input - output vector heat capacity exergy Gibbs energy enthalpy molar enthalpy
[m 2] [-] [J/(molK)] [J] [J] [J] [J / mol]
n p S s T U
mole-number [mol] pressure [Pa] entropy [J] molar entropy [J/(molK)] temperature [K] heat transfer coefficient [J/(m2K)]
rI
efficiency
[-]
/~
chemical potential
^ * id s f
regarding energy analysis standard state (298.15 K, 105 Pa) ideal sensible formation
0 res c
regarding exergy analysis reference state residual (non-ideal) chemical
[J/moll
REFERENCES
1. G. Wall, Exergy flows in industrial processes, Tech. Rep. 83-11, Physical Resource Theory Group, Chalmers Univ. of Technology and Univ. of Grteborg (Jul. 1986). 2. D.R. Morris, J. Szargut, Standard chemical exergy of some elements and compounds on the planet Earth, Energy (Oxford) 11 (8) (1986) 733-755. 3. T. Haug-Warberg, Exergy analysis of industrial processes. I Basic theory, submitted to Energy Convers. Mgmt. 4. T. Haug-Warberg, Exergy analysis of industrial processes. II Calculated results, submitted to Energy Convers. Mgmt. 5. M. Sorin, J. Lambert, J. Paris, Exergy flows analysis in chemical reactors, Chem. Eng. Res. Des. 76 (A3) (1998) 389-395. 6. K. Wark, Advanced Thermodynamics for Engineers, McGraw-Hill. Inc., 1995. 7. M.W. Chase, C. A. Davies, J. R. Downey, D. J. Frurip, R. A. McDonald, A. N. Syverud, JANAF thermochemical tables. Third edition. Part I, A1-Co, J. Phys. Chem. Ref. Data, Suppl. 14 (1) (1985) 1-926. 8. M.W. Chase, C. A. Davies, J. R. Downey, D. J. Frurip, R. A. McDonald, A. N. Syverud, JANAF thermochemical tables. Third edition. Part II, Cr-Zr, J. Phys. Chem. Ref. Data, Suppl. 14 (1) (1985) 927-1856. 9. J.W. Tester, M. Modell, Thermodynamics and Its Applications, 3rd Edition, Int. Series in the Physical and Chemical Engineering Sciences, Prentice Hall PTR, 1997. 10. A. W. Culp, Principles of Energy Conversion, 2nd Edition, Series in Mechanical Engineering, McGraw-Hill. Inc., 1991. 11. W. R. Dunbar, N. Lior, R. A. Gaggioli, The component equations of energy and exergy, J. Energy Resour. Technol. 114 (1992) 75-83. 12. M. Sorin, J. Paris, Integrated exergy load distribution method and pinch analysis, Comput. Chem. Eng. 23 (1999) 479-507. 13. M. Sorin, A. Hammache, O. Diallo, Exergy based approach for process synthesis, Energy 25 (2000) 105-129.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
303
Dynamic Modelling of Chromatographic Processes: A Systematic Procedure for Isotherms Determination H.K. Teoh a, E. SCrensen a*, M. Turnerb and N. Titchener-Hookerb aDept, of Chemical Engineering, University College London, Torrington Place, London, WC1E 7JE, United Kingdom bDept, of Biochemical Engineering University College London, Torrington Place, London, WC1E 7JE, United Kingdom Due to the non-linear and dynamic nature of large-scale chromatographic processes, these processes are difficult to design. The accuracy of chromatography models is particularly dependent on the description of the relevant component isotherms. Identifying the correct isotherms, and determining the corresponding parameters, is a major obstacle. In this paper, we present a simple but efficient method for isotherm parameter estimation based upon the individual elution profiles of the components in the mixture. The approach requires minimal experimentation and yields a significant reduction both in terms of the time and the effort involved in finding the most suitable isotherm. The approach is demonstrated through a case study involving a binary mixture. 1. I N T R O D U C T I O N
Successful application of process-scale chromatographic separation processes is widespread in the pharmaceutical and biotechnology industries. Traditionally, the design and realisation of a particular process-scale chromatography process is based on detailed experimental work at analytical or preparative-scales with significant efforts involved in isotherm determination. This procedure is both tedious and time consuming. Hence, a better understanding of the process dynamics and equilibria of chromatographic separations is needed in order to reduce the time and effort from process conception to actual process realisation. The main objectives of this paper are: 1) to develop a systematic procedure for isotherm determination, 2) to determine isotherm parameters for the candidate isotherms by estimation methods based on a minimal set of experimentally obtained elution profiles and 3) to identify the most appropriate isotherm model which can accommodate the displacement and competitive effects that characterise the non-linear chromatography of realistic feed stocks. When modelling chromatographic processes, a significant number of parameters must be determined experimentally a priori, as the transport and equilibria relationships are too complex to model directly from first principles. Model accuracy is particularly dependent on the isotherm description, which relates the solute concentrations in the mobile and stationary phases [5]. In *Authorto whomcorrespondenceshouldbe addressed: Fax: +44 20 7383 2348; Phone: +44 20 7679 3802; email:
[email protected] 304 the case of multi-component mixtures, an additional complexity results from the competition between the different components as they interact with the stationary phase. The amount of a component adsorbed at equilibrium is a function of the concentration of this component, as for single component isotherms, but also of the concentration of all the other components present in the solution which are adsorbed by the stationary phase [2].
2. ISOTHERM DETERMINATION Traditionally, isotherms are determined experimentally by static or dynamic methods. Static methods are time consuming batch methods, in which the adsorbate concentration in the fluid phase is monitored by gravimetry or infrared adsorption. Dynamic methods can describe the fast mass transfer kinetics and near equilibrium behaviour of the phase system utilising a chromatographic column, eg. frontal analysis (FA), frontal analysis by characteristic points (FACP), elution by characteristic points (ECP) etc. [ 1]. For the dynamic methods, the isotherm is determined by tedious fraction analysis of the elution profile from the column. It has been demonstrated that of all the methods only FA supplies accurate single component and competitive isotherm data, and only as long as the mass transfer kinetics is not too slow [2]. Unfortunately, FA methods are time consuming and require a significant amount of relatively expensive pure chemicals. A faster and simpler way of determinating such parameters utilising a numerical parameter estimation technique, coupled with minimal experimentation, is proposed in this study. This procedure can then be employed to identify more efficiently an appropriate isotherm with which to model the process equilibria. The proposed systematic procedure for isotherm determination is as follows: Step 1: The elution profiles of the individual components of a mixture as well as for the whole mixture must be generated experimentally (The individual elution profiles are for parameter estimation whilst the components elution profiles are for isotherm validation). In order to capture the competitive and displacement effects of the mixture under consideration, the experimental conditions must be such that sufficient degree of overlapping between the components is achieved. Step 2: The number of theoretical plates, Ne,i, the height equivalent to a theoretical plate, Hp,i and the apparent dispersion coefficient, Dap,i, of component i can be calculated directly from the experimental elution profile according to the following equations [2] [5]:
Np,i= 5.54(tg'i) 2
(1)
L Hp,i = Np,i
(2)
ui.L Dap,i = 2 "Np,i
(3)
where tn,i is the retention time for component i, Ai is the peak width of component i at half height, L is the column length and ui is the mobile phase velocity for component i.
305 Step 3: An UV calibration curve which relates the column outlet concentration, Gout, i to the UV absorbance, UV~ for component i must be constructed based on the Beer-Lambert Law [6]:
(4)
UVi :- Ei " Cout,i
where Ei is the Beer-Lambert Law's coefficient for component i. Provided there is no interaction between the various components, the Beer-Lambert's Law is still valid for a solution containing more than one adsorbing species in the following form [6]: n
UVtotal --"
u g i = E E i "Cout,i i=1
(5)
i=1
where Ugtota l is the total UV absorbance. This relationship holds in the linear region of the UV absorbance which is the normal operational region of chromatographic separation. Step 4: An appropriate isotherm model must be selected. Examples include: Langmuir isotherm, competitive Langmuir isotherm, bi-Langmuir isotherm, Fowler isotherm, Freundlich isotherm etc. [1] [2]. Step 5: Parameter estimation is conducted to determine the parameters of the selected isotherms. Step 6: The determined parameters can then be employed to validate the elution profiles of the mixture as a function of the operating conditions using the components elution profiles obtained in Step 1. 3. CASE STUDY A case study is presented to verify the approach outline above. The separation of a binary mixture of naphthalene and fluorene was considered. The mixture was separated in an HPLC column (4.6 X 150 mm) at different flowrates (0.8, 1.0, 1.2 and 1.5 mL/min). Different sample volumes of 10, 20, 30 and 50 lzL with 0.1 g/L feed concentration for each of the aromatic compounds were injected into the column. The mobile phase was 90 % acetonitrile and 10 % water. The stationary phase consisted of Jupiter 15 t~m C18 particles (Phenomenex, Macclesfield, Cheshire, United Kingdom). The separation was carried out under isocratic conditions. The response of the UV detector was found to be linear in the range of experiments undertaken. 4. PARAMETER ESTIMATION Both Langmuir (Equation 6) and competitive Langmuir (Equation 7) isotherms are considered for this separation:
aiCi qi = 1 + b/G
(6)
aiCi qi -- 1 +
ETbjC j
(7)
where qi and Ci are the solute concentrations of component i in the mobile phase and stationary phase respectively, ai and bi are the isotherm parameters for both the competitive Langmuir isotherm and the single-component Langmuir isotherm of that component.
306 (a)
(b)
0.5 rE
(~o 0 . 4 t'M
ur~
30.3
1.2 1
~ o.8 N 0.6 o
-8 0 . 4 ~
0.1 0
~:~ 0 . 2
0
-
1
2 Time,
3 min
4
--
5
0
1
1.5
I
II 2 Time,
21,5 min
i
3
j_
3.5
Fig. 1. Simulated and real elution profiles for naphthalene (a) F = 0.8 mL/min & Vinj - 20 pL (b) F = 1 mL/min & Vinj - - 5 0 / / L (Continuous line = simulated elution profile and 'x' = exp. elution profile)
The chromatographic column section was modelled using the well-known equilibrium-dispersive model, coupled with appropriate boundary and initial conditions [2] [5]. The single isotherms parameters, ai and bi, for both the components were estimated using gEST within the gPROMS package [4]. Although a large number of experimental data sets were available, only the experimental data for a flowrate (F) of 1 mL/min with sample volumes (Vinj) equal to 10 and 50111-, were employed for the parameter estimations. An orthogonal collocation finite element method was applied to solve the dynamic model. The computational load required for the parameter estimations was 11179 and 12781 CPUs for naphthalene and fluorene, respectively. Running on an IBM RISC System 6000 (43P-140), each parameter estimation took around 3 to 4 hours which is insignificant compared to the time and effort required to construct the detailed experimental work for isotherm determination. Further saving can be incurred in terms of reduced operating cost, raw material, overhead etc. The parameters were then used to predict the single elution profiles for both the individual component separately at different flowrates (0.8 and 1.0 mL/min) and different sample volumes (10, 20 and 30 pL). This was done to assess the goodness-of-fit of the parameter estimations. Excellent agreement was obtained between the simulated and experimental elution profiles at different operating conditions. Some of the examples are shown in Figures 1 and 2. 5. RESULTS AND DISCUSSION The isotherms parameters found for the individual components were used in predicting the elution profiles for a 1:1 binary mixture (both naphthalene and fluorene) at different flowrates (0.8 and 1.0 mL/min) with 10 #L sample volume utilising either the Langmuir or the competitive Langmuir isotherms. This was done to determine the most appropriate isotherm to describe the process equilibria of the case study investigated. Figures 3 and 4 show the simulated and experimental elution profiles for a 10 !11-, binary mixture (1:1) at 0.8 and 1.0 mL/min assuming a Langmuir isotherm and a competitive Langmuir isotherm, respectively. For both the isotherms, good agreement in terms of peak positions was obtained for both the flowrates considered. When the flowrate was 0.8 ml_,/min, slight
307 (a)
(b)
0.5 E 0.4 E i.o (.o cM
E t.,.o 0 . 4 r t-M 0.3 t.-
~ o.e o .-o "~ o . 1
-eL 0 . 2 o -.o ~ o.1 0
2
.......
3
Time,
min
illl
4
5
2
Time,
min
3
Fig. 2. Simulated and real elution profiles for fluorene (a) F = 0.8 mL/min & Vinj = 10 pL and (b) F = 1.0 mL/min & V~,,j = 10/A_, (Continuous line = simulated elution profile and 'x' = exp. elution profile)
(a)
(b)
0.7
0.7
0.6
~
0.5
@ o.5
8 o.4 r-
8o.4 e-
E r-
~
E
0.6
0.3
~
0.3
0.2
~
0.2
::3 0 . 1 0
~0.1 0
1
2 Time,
3 min
4
I
." : 5
0
0
.......
1
III
Time,
I 2 min
3
Fig. 3. Simulated and real elution profiles for the binary mixture assuming a Langmuir isotherm with a sample volume of 10 pL; (a) F = 0.8 mL/min (b) F - 1 mL/min (Continuous line = simulated elution profile and 'x' = exp. elution profile)
differences in terms of peak heights were observed whilst when the flowrate was 1 mL/min, good agreement in terms of peak heights was obtained. This is as expected due to the fact that the experimental data sets when the flowrate was 1 mL/min were used for the parameter estimation. Both the Langmuir and competitive Langmuir isotherms predicted the elution profiles reasonably well in these cases. However, from a computational load point of view, the Langmuir isotherm will be preferred due to its simple mathematical nature. Some oscillation in the numerical solutions was observed for both the isotherms considered due to the inherent characteristic of the orthogonal collocation finite element method employed. This can be reduced by using a higher number of finite elements though this is at the expense of longer computational times, q
308 (b)
(a) E
t--
0.7
~
E
0.6
0.7 0.6
o.~
~ o.5
80.4 I:::
8r-- o.e
~
~
0.3
~
0.2
-nO.1 0
0
......
1
N
2 3 T i m e , rain
4
I
I-
5
~
o.a
~
0.2
~
o.1 0
0
..........
1
2 Time, min
3
Fig. 4. Simulated and real elution profiles for the binary mixture assuming a competitive Langmuir isotherm with a sample volume of 10/zL; (a) F - 0.8 mL/min (b) F = 1 mL/min (Continuous line = simulated elution profile and 'x' = exp. elution profile)
6. CONCLUSION A systematic procedure for determinating isotherm parameters utilising numerical parameter estimation and requiring minimal experimentation effort has been developed. The most appropriate isotherm which can describe the equilibria relationship for a particular purification process can easily be identify using this method. Significant reduction both in terms of time and effort can be then realised. For the case study considered, good agreement between the experimental data and the simulated elution profiles was obtained under different operating conditions. Both the Langmuir and the competitive Langmuir isotherms captured successfully the displacement and competitive effects in non-linear chromatography for the case considered. Future work will consider the application and verification of this isotherm determination procedure at a larger scale of operation. REFERENCES
1. Bellot J.C. and J.S. Condoret, Selection of competitive adsorption model for modelling displacement chromatography, J. of Chrom. A, 657, 305-326, 1993. 2. Guiochon G.,S. Golshan-Shirazi and A.M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston, 1994. 3. James E, M. Sepulveda, E Charton, I. Quinones and G. Guiochon, Chem. Engng. Sci., 54, 1677-1696, 1999. 4. Process Systems Enterprise Ltd., gPROMS Advance User's Guide: Release 1.8, London, 2000. 5. Teoh H.K., M. Turner, N. Titchener-Hooker and E. S0rensen, ESCAPE 10, 8, 193-198, 2000. 6. Thomas, M., Analytical Chemistry by Open Learning: Ultraviolet and Visible Spectroscopy, 2nd Edition, John Wiley & Sons, 1997.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
309
Process Simulation and Analysis with Heterogeneous Models John E. Tolsma a and Paul I. Barton a aDepartment of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA This paper describes how symbolic techniques can be applied to general Fortran code in order to perform automatically many tedious, time-consuming, and error prone tasks required when using state-of-the-art algorithms for numerical calculations. Using this new approach, external models written in Fortran can be properly incorporated into an equation-oriented modeling environment. This allows the modeler to create heterogeneous models using both Fortran and the high level input language of the process simulator, depending on which is more suitable for the particular facet of the overall model. 1. INTRODUCTION AND MOTIVATION For several decades, modeling and simulation has played an important role in the design and analysis of chemical processes. The development of increasingly sophisticated processes, tighter environmental constraints, and the necessity to become more competitive will certainly make this impact even larger in the future. One reason for the widespread use of modeling and simulation in the chemical process industries is the availability of advanced process modeling environments, such as modular simulators (e.g., Aspen Plus, HYSYS, and Pro/II) and equationoriented simulators (Aspen Custom Modeler, gPROMS, and ABACUSS II). These tools provide an environment where the modeler can concentrate on constructing a correct model and not have to worry about the myriad of additional details associated with performing efficient and correct calculations. Unfortunately, the features that make these modem modeling environments convenient and user-friendly also make them inflexible. For example, modem modular simulators provide a library of unit operation models that can be assembled together (usually with a user-friendly graphical environment) to construct the flowsheet of interest. However, the modeler is effectively limited by the relatively small number of unit operation models provided in the library. Proprietary models developed in-house can be incorporated into the library, however, this is often a very difficult task, requiring a great deal of expertise in the modular simulator employed. In contrast, most modem equation-oriented simulators provide a high level, declarative input language with which the modeler constructs the mathematical model of the process. Although custom or proprietary models can be coded with this input language, problems occur when attempting to use existing legacy models, usually in the form of Fortran subroutines. (In this paper, the Fortran programming language is emphasized due to its popular and historical use for mathematical modeling. However, the ideas presented are directly applicable to any programming language employed.) One option is to re-write the existing Fortran model in the input language of the process simulator. However, this is time-consuming, error
310 prone, and sometimes not possible due to the limitations of the input language. Alternatively, most equation-oriented process simulators provide the option for linking external models as "black-boxes" into an overall flowsheet. This is problematic for the following reason. Modern numerical algorithms require substantially more information other than simply the numerical values of the equation residuals. Accurate partial derivatives, obtained efficiently, can often dramatically improve the performance of many numerical algorithms (in particular, parametric sensitivity analysis). If the model is sparse, significant improvements can be realized by exploiting sparsity for memory savings and speed increases. In addition, discontinuities within the external model must be handled explicitly to ensure efficient numerical integration [2] and correct parametric sensitivity analysis [1,4]. These discontinuities may result from nonsmooth intrinsic functions such as MIN and MAX in addition to the more obvious IF statements. By coding the model in the input language of the equation-oriented simulator, this additional information is readily extracted and exploited. In contrast, if the model is linked as a "black-box" then derivatives are computed using finite differences, sparsity is not exploited, and discontinuities are hidden, resulting in a degradation of performance and possibly incorrect results. In the following section, we comment on the advantages of process modeling using low level programming languages. We then present an approach for automatically extracting the additional information required for proper and efficient numerical calculation and demonstrate how external models can be readily and properly incorporated into a modern equation-oriented modeling environment. This allows the modeler to combine the power and flexibility of low level languages such as Fortran (as well as leveraging the large amount of existing legacy code) with the convenience of a modern equation-oriented environment.
2. HETEROGENEOUS MODELING
Even with the emergence of modern process modeling environments, the importance of Fortran models should not be underestimated. The obvious reason is that large amounts of proprietary or classified legacy code currently exists and embodies a great deal of knowledge and understanding of the process. In addition, complex physical property models are often only available as Fortran subroutine libraries. However, the importance of Fortran code is not limited to the exploitation of existing models. For example, if the modeler intends to perform a dynamic simulation with a flowsheet containing a new or proprietary model, the input language of the equation-oriented simulator may not be adequate to represent the model. Using Fortran not only offers the modeler more flexibility, but also allows the modeler to embed custom-tailored solution algorithms for solving the new model. As stated in the previous section, modern modeling environments allow the user to incorporate external models as "black-boxes" into an overall flowsheet. We present an alternative approach where external Fortran code is incorporated into the modeling environment in a manner identical to native models (i.e., models provided with the modeling environment or written in the input language of the environment). Using symbolic and code transformation techniques, the external Fortran code is analyzed and new Fortran code is generated providing all of the additional information required when performing numerical calculations using state-of-the-art algorithms. The ideas described above are implemented in a software library called DAEPACK [3]. The DAEPACK library is divided into two main parts: components for analyzing code and generating new information, and numerical components that exploit the automatically gen-
311 crated information. The symbolic library is described in this paper. The symbolic operations consist of two phases: the translation phase and code generation phase. During the translation phase, the source code for the external model is read and a symbolic representation is constructed in computer memory. During the code generation phase, the symbolic representation of the model is analyzed and new code is constructed which can be compiled and linked to the application to provide the additional information, including analytical derivatives, sparsity patterns, and discontinuity information (this is described in more detail in [3]). Figure 1 illustrates this process. In this figure, the user provides the source code for the external model and a specification file which describes what code is to be generated, the independent and dependent variables, etc. DAEPACK then generates automatically new code for computing analytical derivatives, sparsity pattern, and discontinuity-locking information. However, the code generated is not limited to just these examples. Other codes that may be generated can return the interval extension or convex estimator of a given model. Also, what is important to emphasize is that the external model does not have to be a simple set of assignments but may embed sophisticated solution algorithms, for example a root finding procedure for an equation of state. Not only does the approach described above allow legacy models to be properly uti-
User Supplied
DAEPA CK Generated l1
A SUBROUTINE KVAL_AD (NC, K, X, Y, . . .'I
DKDX,NE, IROW,JCOL)
/ IMPLICITNONE User specification file defining independent and dependent variables and code generation options. ~
. . . . . . . . . . .
- ~ _
~ ~ "7 n A i : D A ~ . i ( .......
f S~BRO~INE KVA~,NC,K,X,Y.... , ~ / / IMP~.ICI~NONE /'/
INTEGER NC
/
END
! ~ ' ~ ~
~
/ "~
~ "1
/
/
|
/ J
/,
I SUBROUTINE KV~._SPone,K,X,Y.... '1 I 1 ~, XROW JCOLI , /
hi I ~ r X C X T NONE "1 INTEGE~we
~
~
I I
J
~
_
1
NSTATES, NDISCONS, DISCONS)
Fig. 1. Automatic generation of additional code using DAEPACK.
lized, it also allows c o n s i s t e n t heterogeneous models to be formulated. The best language (e.g., Fortran, C, or the input language of the modeling environment) can be selected for different portions of the model based on the expressive power of the language, what is being modeled, and the skill of the modeler. This approach has been implemented in the equation-oriented process simulator ABACUSS II (http://yoric.mit.edu/abacuss2/abacuss2.html). ABACUSS II, like other equation-oriented environments, provides a high level declarative input language. However, ABACUSS II also employs DAEPACK for properly incorporating portions of an overall flowsheet model represented as Fortran code. ABACUSS II translates the portions of model in the form of ABACUSS II input files and constructs data structures that may be analyzed and
312 manipulated to provide the necessary symbolic information for performing a numerical calculation. In addition, ABACUSS II calls DAEPACK to construct subroutines automatically that return the same symbolic information from portions of the model represented by Fortran code. Rather than calling this external code as a "black-box", all portions of the model are treated in a correct, consistent manner. Figure 2 demonstrates how a heterogeneous model evaluation is performed. The left side of Figure 2 shows the symbolic representation of the equations con-
Fig. 2. Evaluating the residuals of a heterogeneous model and incorporating discontinuity information for proper hybrid discrete/continuous simulation.
structed from the ABACUSS II input files. Residual values are obtained by interpreting these equations and assigning the values to the appropriate position of the residual vector. In contrast, the right side of this figure, shows the portions of the model that are computed with external Fortran code. During a residual evaluation, the external code is called to compute values of the dependent variables given the current values of the independent variables, and these values are inserted into the appropriate location of the overall residual vector. This process is the same as other equation-oriented simulators that allow the user to link in "black-box" external code. What distinguishes ABACUSS II is that DAEPACK is used to construct new code from these external models which extract the hidden discontinuities so that they may be exploited during hybrid discrete/continuous calculations. This is shown at the bottom of Figure 2. Figure 3 shows how a heterogeneous Jacobian evaluation is performed. The symbolic equations constructed from the ABACUSS II input files are shown on the left side of this figure. The gradients of these equations (rows of the Jacobian matrix) are accumulated from symbolic representation and inserted into the overall Jacobian matrix. The rows of the Jacobian matrix corresponding to the external portions of the model are evaluated by calling the derivative code
313 generated automatically by DAEPACK. This has the following advantages: the derivatives are exact (up to round-off error), they are evaluated efficiently, and any sparsity of the external model is preserved in the overall Jacobian matrix. In contrast, if the external code is called as a "black-box" then the corresponding portions of the Jacobian matrix would be evaluated using finite differences and any sparsity would be lost.
Fig. 3. Evaluating the Jacobian matrix of a heterogeneous model.
The discussion above describes how external Fortran code can be incorporated into an equationoriented simulator. However, there may be situations where the reverse may be desirable, that is, creating Fortran code from ABACUSS II input files. ABACUSS II provides the option for generating Fortran code from the portions of the model described by ABACUSS II input files and integrating this output with other portions of the model originally available as Fortran code. The resulting Fortran model can then be processed by DAEPACK to generate all of the additional information described above (e.g., analytical derivatives, sparsity patterns, and hidden discontinuities). This process is desirable when the model is to be used in applications where speed is crucial (e.g., if the model is embedded within another application), or when the model is to be used with a custom or proprietary numerical algorithm. ABACUSS II can be used to construct, debug, and validate the model, and then Fortran code can be generated, compiled, and linked into an application requiting fast model and derivative evaluations. Thus, the features described in this paper provide the modeler with full inter-operability between Fortran and ABACUSS II. 3. COMMENTS ON OTHER INITIATIVES The CAPE Open committee has developed a set of interface specifications based on CORBA and COM for unit operation models, physical property models, numerical solvers, and graph analysis tools. By using components adhering to this standardization, a modeler can assem-
314 ble a collection of components, developed in-house and/or purchased from several third-party vendors, to solve a wide variety of problems. Unfortunately, substantial effort is required by the modeler in order to make the model CAPE Open compliant. This is particularly true when the model is used for dynamic simulation, where the current CAPE Open standard requires the modeler to provide not only residual values, but also sparsity patterns, analytical derivatives, and the state task network corresponding to the discontinuous equations in the model. Fortunately, the ideas described in this paper can also be used to generate automatically the additional information required by the CAPE Open standard, simplifying the creation of CAPE Open compliant components. It should be noted that the CAPE Open committee has recognized the need for assisting the user in component development and has started AD-CAPE, an initiative for incorporating automatic differentiation into the standardization. However, as described in this paper, derivatives are only a piece of the puzzle when performing numerical calculations properly. By creating standard interfaces, the CAPE Open compliant process modeling environment will be able to solve heterogeneous models, provided the model components also adhere to the standard. Although these and similar interface standards will increase the flexibility of process modeling environments, the communication overheads associated with the component architectures may impair performance. These approaches may be better suited for non-communication intensive applications distributed across multiple platforms. These additional communication overheads are not present when binary-compatible compiled code is linked directly into an application. Thus, the ideas presented in this paper provide both an enabling and alternative technology to the heterogeneous modeling approach advocated by CAPE Open. 4. CONCLUSION Using the symbolic and code generation components of DAEPACK, external Fortran code can be properly incorporated into a modem process modeling environment. Like modem equation-oriented environments that can generate all of the information necessary for robust, efficient, and correct numerical calculation from a model written in the simulator's input language, DAEPACK can generate new code which computes the same information from the original Fortran code. This allows heterogeneous models to be developed where the modeler selects different languages for different portions of the flowsheet model depending on the expressive power of the language, what is being modeled, and the skill of the modeler. Acknowledgments - This research was supported by the EPA Center for Airborne Organics at MIT and ABB Research Limited.
REFERENCES 1. 2. 3. 4.
Santos Gal~in, Willian E Feehery, and Paul I. Barton. Parametric sensitivity functions for hybrid discrete/continuous systems. Applied Numerical Mathematics, 31:17--47, 1999. TaeshinPark and Paul I. Barton. State event location in differential algebraic models. ACM Transactions on Modeling and Computer Simulation, 6(2):137-165, 1996. John E. Tolsma and Paul I. Barton. DAEPACK: An open modeling environment for legacy models. Industrial and Engineering Chemistry Research, 39(6): 1826-1839, 2000. John E. Tolsma and Paul I. Barton. Hidden discontinuities and parametric sensitivity calculations. in preparation, 2000.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
315
A Structured and Selective Framework for Hybrid Mechanistic-Empirical Model Building Pedro Vale Lima and Pedro M. Saraivaa a Department of Chemical Engineering, University of Coimbra Pinhal de Marrocos, 3030-290 Coimbra, Portugal phone: 351-239798700, fax: 351-239798703 e-mail: eq3pvl @eq.uc.pt, eq lpas @eq.uc.pt
In order to address issues related with process operation, diagnosis, optimization, improvement and control, among other tasks, several kinds of models have been used. They usually fall under the scope of two distinct paradigms: mechanistic first-principles and empirical approaches. Both have been adopted but very few frameworks were developed so far in order to combine and integrate features from each one of them into hybrid models, which share mechanistic and empirical components. In this article we describe a methodology for overcoming this lack of integration efforts, through an algorithm that leads to the construction of process models that contain mechanistic and localized empirical elements, achieved by exploring symbolic manipulation of the first-principles model equations. This new framework was tested and evaluated by application to a simulated CSTR case study. 1. INTRODUCTION The construction of mathematical models to forecast and understand the behavior of chemical processes forms the basis for a countless number of tasks (planning, optimization, improvement, fault diagnosis, control, etc). Depending on the nature of the specific process and its desired goals, several kinds of models have been developed, differing namely in scope, level of detail, and underlying structures. However, for many practical situations two separate schools of thought have emerged: on one hand, we have fully mechanistic approaches, where models are built based upon first-principles equations; on the other hand, and specially for operation analysis and improvement at existing plants with complex transformations, fully empirical techniques have also been suggested and applied, relying in operators knowledge extraction, data analysis based upon machine learning [1 ] or statistical tools. But very few efforts have been done in the past to combine and integrate both of the above paradigms, although they are conceptually believed to be complementary to each other: empirical components will in general lead to better local prediction capabilities through the full exploration of all information that is available, while mechanistic elements make it possible for one to get a better understanding of the underlying physico-chemical phenomena, predict the values for unmeasured state variables, provide additional trust, reliability and extrapolation characteristics. In the limited previous work that has been done in order to build hybrid models [2,3], one may find different perspectives: choice among alternative first-principles models followed by some parameter fitting to existing data;
316 combination of the available mechanistic equations with an empirical model adjusted to the residuals associated with first-principles predictions. However, none of them do take into account the detailed fine structure of the available underlying mechanistic model, neither do they allow for partial localized and selective introduction of empirical elements. In this article we present a framework for building hybrid models that provides these capabilities.
2. ALGORITHM FOR HYBRID MODELING Our approach for selectively building localized hybrid models covers six basic steps: 1) 2) 3) 4) 5) 6)
Choose an initial mechanistic model. Perform symbolic reformulation of the mechanistic set of equations. Solve parameter estimation problem for the initial mechanistic model. Perform best equation structure change analysis (BESCA). With the resulting structure, calculate prediction error over testing data set. Change the model structure and go back to step 4 until stopping criteria based upon overall data adjustment quality are met.
In the forthcoming paragraphs we will describe some of the above steps in more detail.
2.1. Symbolic Reformulation We conduct a symbolic reformulation of the original set of equations, provided by the user, which correspond to the available initial process mechanistic model. This symbolic reformulation is intended to transform a generic equation, of arbitrary complexity, into a set of linear equations and special algebraic structures, such as the bilinear type xy, power xy or other atomic functional forms f ( x ) . For instance, if we start with equation 2 x e 4/y - z : 0, the introduction of additional variables wi originates the equivalent formulation: 2Wl -- Z ----0 W3 -- 4W4
Wl = XW2
W2 ~ e w3
W4 - - y
-1
(1)
All the symbolic decomposition and reformulation steps mentioned are performed automatically, by means of storing in the computer memory the initial set of equations, expressed as nested lists [4]. As a result of this symbolic reformulation, we obtain a set of linearized initial equations and additional ones, related with the wi auxiliary variables definition. Both groups taken together are equivalent to the original mechanistic model.
2.2. Parameter Estimation Within the scope of this paper a model has the form of a set of algebraic equations f ( 0 , z) - - 0
(2)
where 0 stands for p unknown parameters, z represents n process variables, with k output variables, and f covers q equations. The error variables ej are defined by ej - ~ j - y j
j -
1,... ,m
(3)
where 33j represents the set of predicted values for the process output variables and y j* stands for the j case from a set of m training records of data.
317 Considering maximum-likelihood estimators, by solving the optimization problem rn
minL-- ~ 0
rn
ejTV-lej- ~_,Lj
j=l
(4)
j=l
where L is the likelihood function and V the covariance matrix for the output variables, we obtain the desired adjusted 0 values.
2.3. Best Equation Structure Change Analysis We now consider h, the subset of both linear equations (with length s) and the extended set of variables v - - z U w that includes z and w (set of t additional variables resulting from the symbolic reformulation performed). For each variable v, in each equation h, a new model structure is generated through the addition of a Tv term, where y is a new free parameter added to 0 (table 1).
Table 1 Terms used to generate new structures
fl
Zl
...
Zn
W1
...
Wt
'YZl
...
~Zn
"~1
...
~4~t
~Zl
...
'YZn 'yw1
...
'~t
. o .
fs
Then, optimization problem (4) is solved for each case, resulting in a matrix with values of the merit function L for each alternative structural change, and the model structural change introduced corresponds to the best value found.
2.4. Stopping Criteria The model resulting from a particular BESCA iteration is used and evaluated over a testing data set with r records, leading to a prediction performance provided by: etest = ~ e j T v - l e j j=l
(5)
When no significant performance improvements over the previous iteration are being achieved, we stop the algorithm and use the corresponding structure as our final model set of equations. 3. CASE STUDY To evaluate the above framework we will now describe its application to a known case study [5]. The plant model represents, with some simplifications, a steady state adiabatic CSTR with irreversible first order reaction (A ~ B) and a bypass stream. Simulated operating data for this unit were obtained using different input values for inlet temperature, To, and concentration of A, A0, and obtaining the associated outputs, outlet temperature, T, concentration of A, and B, from the following equations (that we assume to provide a perfect fit to real plant data and use to generate simulated operating values):
318 (1 - 5)Ao - A s - kl'CAs = O
kl -- O1 eO2(1-800/r)
- B s + kl'CAs = 0
A = 8,40 + (1 - 8)As
T,(--Z~r)
To- T + ~ k l A s - O
(6)
B--(1-8)Bs
pcp
where "c stands for the residence time in the reactor (100s), ~ r is the heat of reaction p the mixture density (1 g / l ) , Cp its heat capacity of the mixture ( 4 . 1 8 J / g K ) , 5 the bypass ratio (0.05), 01 and 02 are kinetic parameters (0.017719s -1, 12.483). Three data sets were built this way: the first (training) has 30 points distributed over the input space A0 E [0.8, 1.01], To E [425,544]; the second (testing) has 64 points forming a complete grid over the same space; the third (for extrapolation evaluation) has 120 points distributed in a grid over the space A0 E [0.74, 1.07], To C [391,544]. Since the previous model (6) was used just for the sake of generating simulated operating data, we must now consider an initial mechanistic approximate model for the plant, whose complete detailed first-principles equations are assumed not to be known. For that purpose, we took the basic CSTR model equations (without bypass) as our initial mechanistic plant model, which, after symbolic reformulation, corresponds to: (-4180J/mol),
Ao - A - 10001w4 = 0
w2 + 80002Wl - 02 = 0
- B + 10001w4 = 0
w3 = e wE
Wl - 1 / T
To - T + 106 01w4 ~- 0
w4 = Aw3
(7)
where the values of 01 and 02 were adjusted to the training data. Then, our selective structural hybrid modeling approach was employed in order to improve the underlying initial model set of equations fitting performance (for A, B and T). The first iteration BESCA matrix suggested the addition of T1 w4 to the first equation, while the second iteration BESCA matrix suggested the addition of 72 w3 to the fourth equation (table 2), thus leading to the following final model structure: Ao - A - (100 01 - T1) w4 = 0
W4 ~-- A e wE
- B + 10001 W4 -- 0
w2 - 02(1 - 800/T) - 72 e w2
(8)
TO-- T + 10601 w4 = 0
Table 2 Likelihood values and parameters for different hybrid model stages Stage 0 1 2
L
01
02
~1
~2
0.3534 8.459• 10 -4 1.871• -8
0.01888 0.01785 0.01867
12.530 12.296 12.486
0.08907 0.09330
0.09653
We compared the performance of this model also with the one obtained with a purely empir-
319
ical approach, consisting of an artificial neural network (ANN) trained and tested over the same datasets (table 3). As an illustrative example, the prediction error for the output temperature is plotted as function of the distance from the center of the input variables scaled space (figure 1). It can thus be seen that our hybrid model not only provides excellent fitting capabilities but clearly outperforms both of the altemative approaches considered (mechanistic with adjusted parameters and empirical).
Table 3 Prediction mean square error for the (M)echanistic (with adjusted parameters), (H)ybrid and (A)NN models as predictors of T, A adn B. Interpolation Extrapolation T A B T A B M 1.77 x 10 -1 1.14 • 10 -4 1.78 • 10-7 2.26 • 10-1 1.43 • 10 -4 2.25 • 10 -7 H 1.90 • 10 -7 1.71 • 10 -13 1.89 • 10 -13 7.92 • 10-6 7.92 • 10 -12 7.19 x 10 -12 A 1.88 x 10 -5 6.74 • 10 -3 2.88 • 10 -4 1.47 • 10-1 9.98 • 10 -4 1.04 x 10 -4
1
e.
J
interpolation :1'
9
|
I
I 0.5
i
n
o
B
,.,. a8
[]
I
I
0
0.1
0.2
|
|
: t"o2 B a ..B d
I
!,a o
8 8o ~ o8
I
I
0.3
0.4
o
~d.
,
~
.
Oo
o
a
a8
[]
o~ a;~
i
|
,~ 8 a m ~
[]
I
-0.5
0
extrapolation
o
.. 0"[] .
,
og
0 8
o
-
.
o
o~
"
.,o
, o
I
I
I
0.5
0.6
0.7
Fig. 1. Temperature prediction error with distance from center of the (D M e c h a n i s t i c , / k Hybrid, o ANN).
(Ao,To) grid
One should point out the small number of structural changes that were needed here, as opposed to the other formula building techniques, such as genetic programming [7,6]. Here, we did conduct our search over a full space of model atomic structural changes. In more complex problems, the efficiency of the algorithm can be improved by excluding some elements of the search space, based upon available process knowledge.
320 A careful analysis of the generated final model structure could be used to discover process features that were not included in the initial mechanistic model, thus enlarging our firstprinciples knowledge, although in this particular case study that was not possible, namely because the model structure used for generating simulated data falls outside from the search space considered for hybrid model construction. 4. CONCLUSIONS We have introduced and developed a framework that is supported by the symbolic reformulation of a set of first-principles equations, in order to derive hybrid models, which opens new possibilities regarding the integration of empirical data-driven with mechanistic knowledge components for model building. The use of this model reformulation strategy results in a set of atomic equations that allow for empirical elements to be added selectively and locally. A simple case study shows that such an approach may lead to quite interesting and competitive results. Since this sort of approach is able to identify and deal with simpler substructures that result from a decomposition of the original set of equations, it is likely to provide more and more added value as the complexity of the original first-principles model increases, because it is able to guide selectively the incorporation of empirical elements to specific zones of its structure. At present we are examining extensions of the approach to address noise, large scale problems and performance comparisons with a number of alternative methodologies. ACKNOWLEDGMENTS The authors acknowledge financial support provided by research grants FCT POCTI/1999/EQU/32647 and PRAXISXXI/BD/15518/97. REFERENCES
1. P. Saraiva, chapter in Stephanopoulos and Han (Eds.), Intelligent Systems in Process Engineering, Academic Press (1996) 377. 2. M. Thompson, M. Kramer, AIChE J. 40 (1994) 1328. 3. R. Oliveira, PhD Thesis, Martin-Luther Uni., Halle-Wittenberg, 1998. 4. E. Smith, PhD Thesis, Imperial College, London (1996). 5. I. Kim, M. Liebman, T. Edgar, AIChE J. 36(7) (1990) 985. 6. K. Bettenhausen, P. Marenbach, GALESIA'95 (1995). 7. J. Koza, Genetic Programming, MIT Press, Cambridge, MA (1992).
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
321
Validations of the nonequilibrium stage model and of a new efficiency correlation for non ideal distillation process through simulated and experimental data M. R. Wolf-Maciel a*, C. Soares a and A. A. C. Barros b
-
a Laboratory of Separation Process Development - Chemical Engineering School - State University of Campinas (UNICAMP) - P.O. Box: 6066 - Zip Code: 13081-970 Campinas - SP Brazil - E-mail:
[email protected] and
[email protected] b Chemical Engineering Department- Regional University of Blumenau (FURB) B l u m e n a u SC - B r a z i l - E-mail:
[email protected] The objective of this work is to use a robust software developed for distillation process involving the nonequilibrium stage model, the equilibrium stage model with Barros & Wolf correlation for efficiency calculation, the mass and heat transfer calculations and the automatic procedure for comparison of profiles calculated and obtained from experimental data. The case study is the system ethanol/water separation The intention is to compare the performance of the equilibrium stage model with the new correlation for efficiency. 1. I N T R O D U C T I O N The equilibrium stage model for distillation process is commonly used, since it does not need a complete description of the equipment. It is supposed that the streams leaving the stage are in equilibrium to each other. The material balance, the equilibrium relationships, the summation equations and the heat balance for each stage are solved using available algorithms, obtaining the liquid and vapour flows, as well as, the temperature and composition profiles along the distillation column [1]. However, it is necessary the introduction of the efficiency concept to correct the deviations of the fact that the stages, in the practice, rarely provide the equilibrium among the phases that leave them. However, the limitations presented in the several efficiency correlations, stimulated the development of a new correlation that considers parameters of mass and heat transfers as fundamental in its evaluation. An empirical efficiency equation developed by the authors of this work will be used to evaluate the behaviour of temperature and composition profiles in a distillation column. Equation (1) was obtained based on techniques of factorial design and it is dependent on the properties of the mixture, for the calculation of plate and component efficiency. ]]i,j
"--
I-- 1-004516
~38.5309 k-L~ P-L~,j ~2
*To whom all correspondence should be addressed
(1)
322 For the determination of the component efficiencies, the same correlation is used, but as function of the pure component parameters. A software involving the equilibrium stage model with the new efficiency correlation was developed using subroutines for the calculation of all parameters present in equation (1). 2. N O N E Q U I L I B R I U M STAGE M O D E L The nonequilibrium stage model, described with details by [1,2,3] is characterised by the absence of efficiency values. Besides, the calculations are based on the simultaneous solution of the mass and heat transfer equations, written independently for each phase. The plate is assumed to be at mechanical equilibrium, the mass and energy balances are written for each phase separately and the thermodynamic equilibrium is assumed only at the interface. It is established that transfers from the vapour phase to the liquid phase are positives. The vapour and liquid mass transfer rates are calculated using the Krishna and Standart method [4]. The equations that describe a separation process [ 1, 2, 3] in the steady state in the nonequilibrium stage model are: Liquid Phase Component Material Balances, Vapour Phase Component Material Balances, Interface Component Material Balances, Liquid Phase Energy Balance, Vapour Phase Energy Balance, Interface Energy Balance, Interface Summation Equations and Interface Equilibrium Relationships. The Interface Component Material Balances are represented by the mass transfer flow relationships: RFv = Nj - NjV -- 0
where: j=l to c-1
(2)
RFL = Nj - NjL = 0
where j=l to c-1
(3)
The Krishna and Standart method based on the exact solution of Maxwell Stefan's equations is used for the calculations of the mass transfer rates. The equations in the matrix form are given by: (4)
(NV)c-1,1= [K v] c-1,c-1.a. (yV-yI)c-1,1 + N T . (yV)c - 1,1 (NL)c - 1,1= [KL] c-1,c-I .a-(XI-XL)c-1,1 +NT. (XL)c- 1,1
where N T = ~ N j j=l
(5)
The interfacial area is combined with the transfer coefficients and a correlation that gives the product of these parameters is used. The binary mass transfer coefficients on the trays are calculated using the AIChE correlation. The Interface Energy Balances is given below by the equation: BE I = e v - e L = 0
(6)
The energy transfer rate equations are obtained by the sum of a conductive heat flow and a convective contribution due to the transport of enthalpy in the interface. The energy transfer
323 coefficients are calculated for the vapour phase using the Chilton-Colbum analogy and for the liquid phase through the penetration theory: v
=h
v
.a-
-T
+
Nj .Hj
(7)
L = hL -a- (TI - TL) + 2 NjL -Hi -L j=l
(8)
j=l
The nonequilibrium stage model is described by (5c+3) equations and variables for each stage. The condenser and the reboiler are described by the thermodynamic equilibrium relationships. The equations are simultaneously solved according to the method described by Naphtali and Sandholm [5] and by Newton-Raphson method. The mass and energy transfer rates together with the total and partial enthalpies and the equilibrium constant which are calculated using the model UNIQUAC for the activity coefficient are determined separately. 3. METHODOLOGY Using the equilibrium stage model with the efficiency correlation of Barros & Wolf and the nonequilibrium stage model described, a program was developed for the simulation of distillation columns in steady state, enabling, also, that results from simulations be compared with experimental data. The developed program uses Fortran language. All necessary physical properties (binary diffusion coefficients, viscosity, density, heat capacity, thermal conductivity) for pure components and mixtures, the enthalpies of the vapour and liquid streams entering and leaving a stage, the partial molar enthalpies of the bulk vapour and liquid phases are calculated rigorously using proper subroutines. Both concepts, the equilibrium and nonequilibrium, have the same number of degrees of freedom and, therefore, the initial estimates and the specifications are the same ones. To the program are supplied all the critical properties of the components, the location and the conditions of the feed, type of condenser, number of plates of the column, among other data. For the nonequilibrium stage model are necessary, besides the mentioned parameters, the column design parameters. It was observed that the computing time involved in the simulation of the nonequilibrium stage model is larger than the equilibrium stage model; that is evident, since the number of equations in the nonequilibrium stage model (5c+3) is higher than in the equilibrium stage model (2c+ 1) [6]. For validating the comparison of the obtained results using the equilibrium and nonequilibrium stage models, experimental data obtained in a distillation column made of stainless steel was used. The column with 140 mm diameter has eight sieve plates and in all the test runs made in this investigation, the column was operated with the ethanol-water system at total reflux. At the steady state condition, the liquid samples and the temperature were taken along the distillation column and the required data were recorded. 4. SIMULATION AND RESULTS Tanking into account that the computing time of the nonequilibrium stage model is quite large for on line control application, it is very important to carry out the evaluation of the equilibrium stage model using the new efficiency correlation in relation to the more realistic
324 nonequilibrium stage model and also with experimental data. Ethanol-water system was used and the temperature and composition profiles were compared. The temperature were taken along the distillation column, except in the condenser and in the reboiler. In the figures, stage 1 corresponds to the reboiler and stage 10 to the condenser. For the simulations, the specifications used were the reboiler duty (0.194E8 J/s) and the mole fraction of the ethanol on the top of the column (77.9%). The ethanol mole fraction in feed was 7%. 100 362
o Experimental ,~ Equilibrium N o n e-qou-i l i b r i u m 6 ~ __~
~,~ 360 358
o
60
~ 40 ,~
~ 356 [,.
0
i
0
354 1
2
3
4
5
6
7
8
9
~"(~-~- O ~ . x ~ e r
~/
~ 2o
l
1 2
i
i
i
~
~
i
3
4
5
6
7
8
t
i
9 10 1
Stage N u m b e r
Stage Number
o Experimental Lx Equilibrium o Nonequilibrium
Fig. 1. Comparison of experimental predicted temperature profiles
and Fig.2. Comparison of experimental predicted composition profiles
and
In Figure 1, the temperature profiles of the equilibrium (using Barros & Wolf correlation for efficiency) and nonequilibrium stage model and experimental are compared. The temperature of the liquid phase for the equilibrium and nonequilibrium stage model coincide in all stages, except for stage 2, a stage very close to the reboiler. A comparison of the predicted mole fraction of the ethanol and water with the measured mole fraction along the distillation column is shown in Figure 2. In the figures, it can be observed that the mole fractions are practically coincident in the upper part of the column, and that in the lower part, the nonequilibrium is more coincident with the experimental data. Based on results obtained in Figures 1 and 2, the behaviours of the heat and mass transfer coefficients along the distillation column were calculated by the software (Figures 3, 4, 5 and 6). It was observed that the mass transfer coefficients for the liquid phase increase in direction to the bottom of the column, and that the values for both pairs are very close (Figure 3). It can be said that the coefficients tend to increase, increasing the gradients of concentration among the phases, which happen from the top to the bottom of the column. The results obtained for the liquid phase were used in the simulation, which allowed to get the behaviour of the parameters in the vapour phase. It is observed that the binary mass transfer coefficients in the liquid phase are larger than the ones in the vapour phase (Figure 4). This behaviour shows that the resistance to the mass transfer is larger in the vapour phase. Furthermore, these coefficients increase in an opposite way.
325
200
35
=o ~
o Ethanol-Water rn Water-Ethanol
.N
34
o Ethanol-Water 190 [] Water-Ethanol
o
O O
[]
8 ~ [-
~
32
k
~
18o
[] [] []
31
~
8 8 ~ 8 8 ~
i 1
i
2
3
t
i
i
i
i
i
4
5
6
7
8
9
I0
~
Stage Number
16o 1
2
3
4 5 6 7 8 Stage N u m b e r
9
10
Fig. 3. Liquid phase mass transfer coefficients Fig.4. Vapour phase mass transfer coefficients profiles profiles ~' r~ a~. ~ ~ ~ ~ ~ ~
19000 18000 17000 16000 150OO
~
14000
~
75
a ~
73
O
O
1
2
3
4
5
6
7
Stage N u m b e r
O t
i
8
9
~e~o
69
~
67
65 10
1
i
i
t
i
i
i
i
i
2
3
4
5
6
7
8
9
10
Stage N u m b e r
Fig. 5. Liquid phase heat transfer coefficients Fig. 6. Vapour phase heat transfer coefficients profiles profiles For expressing the heat transfer coefficients in the liquid phase, the penetration theory was used, described as a function of the mass transfer coefficients in the liquid phase, the heat capacity and the mass diffusivity. For the vapour phase, the Chilton-Colbum analogy was used, which relates the average of the mass transfer coefficients in the vapour phase, the heat capacity and the Lewis number, all present in the developed software. The behaviours of the heat transfer coefficients follows the ones for the mass transfer. 5. C O N C L U D I N G R E M A R K S The software developed by the authors of this work, which includes the equilibrium and nonequilibrium stage models, were used to simulate the distillation process. Results show the prediction of the models to represent a real non-ideal process. It can be said that, using the Barros & Wolf correlation for efficiency, both stage models present profiles practically coincident and are validated with experimental data. This is an important result since one can
326 choose the most appropriate model for a particular application. Moreover, liquid and vapour phase mass and heat transfer coefficients can be calculated and analysed along the distillation column. NOTATION a BE c Cp D h
N PM RF T X
y
interfacial area (m2) energy balance function number of components average heat capacity (J/mol.K) mass diffusivity (cm2/s) partial molar enthalpy (J/mol) heat transfer coefficient (J/(cm2.s.K) multicomponent mass transfer coefficient (mol/(cm2.s) average thermal conductivity (W/cm.K) interface mass transfer rate (mol/s) average molecular weight (g/mol) mass rate relation functions temperature (K) liquid mole fraction vapour mole fraction
Subscripts i stage j component Superscripts I interface L liquid phase V vapour phase Greek Letters e interface energy transfer rate (J/s) rI Barros & Wolf efficiency P average density (g/cm3)
la
average viscosity (cP)
AKNOWLEDGEMENTS
The authors are grateful to FAPESP (Fundag~.o de Amparo 5. Pesquisa do Estado de Sgo Paulo) for the financial support for this project. REFERENCES
[ 1] R. Krishnamurthy and R. Taylor, AIChE Journal, Vol. 31, No. 3 (1985a) 449. [2] R. Krishnamurthy and R. Taylor, AIChE Journal, Vol. 31, No. 3 (1985b) 456. [3] R. Krishnamurthy and R. Taylor, AIChE Journal, Vol. 31, No. 12 (1985c) 1973. [4] R. Krishna and G. L Standart, AIChE Journal, Vol. 22, No. 2, (1976) 383. [5] L. M. Naphtali and Sandholm, AIChE Journal. Vol. 17, No. 1 (1971) 148. [6] M. H. Pescarini, A. A. C. Barros and M. R. Wolf-Maciel, Computers Chem. Engng., Vol. 20, Suppl. (1996) $279.
European Symposium on Computer Aided Process Engineering - | l R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
327
Simulation of an industrial olefin polymerization FBR operating under condensed mode A. Yiagopoulos a, H. Yiannoulakis a, J. Morris b and C. Kiparissides a a Department of Chemical Engineering and Chemical Process Engineering Research Institute Aristotle University of Thessaloniki, P.O. Box 472, Thessaloniki, Greece 540 06
b Department of Chemical and Process Engineering, University of Newcastle-upon Tyne, NE1 7RU, U.K. In the present study a comprehensive mathematical model is developed to describe the behavior of an industrial gas-phase olefin polymerization FBR operating under condensed mode conditions. Assuming complete mixing and instant vaporization of the liquid feed in the bed, detailed simulations were carried to investigate the effect of the amount of liquid in the ethylene feed and the composition of the recycle stream on the reactor temperature, spacetime yield and heat removal rate. It was shown that the introduction of a liquid feed stream in the reactor substantially broadens the safe operation window of the FBR with concomitant the increase of the space-time yield and the heat removal rate from the reactor. 1. INTRODUCTION Fluidized bed solid catalysed olefin polymerization has long been recognized as one of the main manufacturing processes for the production of polyolefins. In such a process, catalyst particles are continuously fed into a fluidized bed and react with the incoming monomers to form a particulate polymer product that is withdrawn from the bed at a point above the distributor. Heat is removed by cooling the recycle gas stream via a heat exchanger. It has been reported that cooling of the recycle stream below its dew point and the subsequent injection into the reactor of a mixed liquid-gas feed, can further improve the heat removal rate from the fluidized bed [ 1]. This is accomplished due to the higher heat capacity of the liquid fraction, lower inlet temperature of the feed stream and latent heat of vaporization of the liquid phase. Replacing noncondensable components such as nitrogen, with condensable isopentane [2] can lower the dew point of the recylce stream mixture. When the recycle stream is cooled below its dew point the fluidized bed reactor is said to operate under condensed mode. The typical level of condensation in the recycle stream is between 5-20 wt. %, although values of up to 50 wt. % have been reported [ 1-4]. The liquid is fully or partially vaporized in the bed, upon contact with the hot active polymer particles, thus removing excess heat via its latent heat of vaporization [5]. The obvious benefit of this technique is the broadening of the safe operation window of the FBR, meaning that the catalyst feed rate, and consequently the production rate, can be increased without the risk of temperature runaway. As a result, the reacrtor space-time yield can be significantly increased without increasing the size of the fluidized bed [2].
328
'~
Recycle Stream
C yc,o e
.................. Cooler
Reaction Zone
Compressor ..............
Gas Make-up Feed
>
1-Hexene Make-up Feed
Product Prepolymer "1 ~1111 Feed i,, , ,i II I I',1 i
i
~
i
i
i
i
Gas Feed
..
Flash Drum
i
i
"".,.,, Condenser
Ir Liquid Feed . j
Fig. 1. Schematic representation of an FBR operating under condensed mode conditions. In the present study a comprehensive mathematical model for a gas-phase olefin polymerization fluidized bed reactor operating under condensed mode is developed. Extensive simulations are carried out to investigate the effect of degree of condensation of the recycled stream on the dynamic and steady-state fluidized bed reactor operation. 2. F L U I D I Z E D BED R E A C T O R M O D E L
A schematic representation of an olefin polymerization fluidized bed reactor operating under condensed mode is presented in Figure 1. Liquid is injected in the bed in the form of dispersed droplets through a series of nozzles. Injection of the liquid occurs at a point above the distributor where reactor temperature is high enough to ensure adequate vaporization upon entrance in the bed. Chinh et al. [4] reported that the fluidized bed of figure 1 can be approximated by a well-mixed vessel, while 98% of the liquid entering the reactor is completely vaporized. According to Chinh et al. [4] the weight percentage of liquid that can be safelly introduced in the bed varies from 6 to 20%. Higher levels of condensation will probably lead to liquid 'weeping' in the bed and the development of a distinct three-phase gas-liquid-solid fluidization regime near the reactor' s gas distributor [7]. Based on the above observations the fluidized bed reactor can be approximated by a continuous pseudo-homogeneous phase stirred tank reactor [6]. Moreover, the liquid feed is assumed to be completely vaporized upon entrance in the bed. Since no bubble phase is included in the model, the bed voidage, abed, accounts for the overall gas volume fraction in the bed. A detailed kinetic model is used to describe ethylene and 1-hexene copolymerization [6]. Assuming uniform temperature and monomer concentration throughout the bed, the unsteady-state material and energy conservation equations can be derived. Thus, the dynamic molar balance for the "i" monomer in the reactor can be written as:
329
d[Mi]
_
FI
dt
P m l A H l~bed
[M ]in,!
U0 ([Mi Hgbe d
]in,g _ [Mi ])_ [Mi ]Q0
0 - gbed)RMi
(1)
g bed
where F1 is the liquid feed rate, Q0 is the product removal rate and u0 is the superficial gas velocity. RMi denotes the consumption rate of monomer 'T'. The dynamic molar balances for hydrogen, nitrogen, ethane and the moments of number chain length distributions as well as the mass balance for the polymer in the bed can be found elsewhere [6]. Accordingly, the dynamic energy balance equation can be written as follows:
H accum dT/dt = I2Igas,in "+-III liquid,in + fl gen -- fl gas,out
-- flprod,out
-- I[Ivap
(2)
Nm
Haccum = ( Z N i ] C p M i W [ H 2 ] C p H 2
CppolOp,- ]. +[N2]CpN2 +[C2]Cpc2 + (1 - gbed) --
(3)
gbed
i=l Nm
IiIgas,in =
U0 ( Z N i ] i n , g Hgbed
CpMi q-[H2]in CpH 2 + [N2lin CpN 2 + [C2]in,g Cpc2)(Tin - Tref)
(4)
i=l Nm
liquid,in
--
FI
(Z[Mi]in,lCpMi + [C2]in,lCpc2)(Tlin- Tref)
PmlAHgbed
(5)
i=l
Nm
(1 - ~bed) gen
--
~3bed
(Z RMiMWi)AHrxn
(6)
i=l Nm
I2Igas,out
_-- U0 ( Z N i ] C p M He bed i=l
i -I-[H2]CpH 2 +[N2]CpN 2 +[C2]Cpc2)(T-Tref)
(7)
Nm
Q0 ( Z H9prod,out - g"kl 717_1 I
N i ]CpM i at_[H2]CpH 2 +[N2]CpN 2 +[C2]Cpc 2 4-
(8)
i=l
(1 - e bed)
- g bed
yap = FI AH v a p / A H e
C ppolP p )(T - rre f ) bed
(9)
Notice that, in the derivation of the energy balance equation, the enthalpy associated with the prepolymer feed stream is considered to be negligible. In order to make a quantitative comparison between the heat removal rates from a conventional and a condensed mode FBR, the following dimensionless quantity is introduced:
Heat.emova. Factor-Hen//Has.ou.+ Hpo.ou.+ Hva )
(10)
330 To calculate the weight percentage of liquid that enters the reactor one must determine the vapor-liquid equilibrium in the flash drum. In the present study, the Soave-Redlich-Kwong (SRK) equation of state [8] was employed to calculate the amount of liquid entering the bed, based on the operating conditions in the flash drum and the composition of the recycle stream. 3. SIMULATIONS RESULTS AND DISCUSSION
To demonstrate the predictive capabilities of the present model, an ethylene-hexene-1 copolymerization FBR operating under condensed mode was considered. Numerical simulations were carried out to investigate the effect of the partial condensation of the recycle stream on the safe operation window of the FBR. In particular, the effect of the weight fraction of liquid in the feed stream and the recycle stream composition on the reactor temperature, space-time yield and heat removal rate was thoroughly examined. The reactor and flash drum operating conditions are reported in Table 1. The kinetic rate constants for ethylene-hexene-1 copolymerization can be found elsewhere [9]. Table 1: Typical reactor and flash drum operating conditions [3,4] Variable Bed dimensions, HxD (m 2) Bed voide fraction, ~;bed Condenser inlet, The Temperatures (~
Flash drum, Td Reactor liquid feed stream, Tlin Reactor gas feed stream, Tin Pressure, P (bar) Mole fractions of flash feed stream: (H2, N2, C2H4,C2H6,C6H12) Prepolymer feed rate, Fpre(g/s) Inlet active site concentration, P0,in(mol/L)
Value 16x4.85 0.5 75 55 55 55 20 (3.8, 55.4, 30.2, 1.5, 9.1) 30 0.0284
The increase in the heat removal rate from the FBR operating under condensed mode is expected to broaden the safe operation window of the reactor in comparison with the operation of conventional FBRs (e.g., 0% liquid in the bed). Figure 2 depicts the effect of the amount of liquid in the feed stream (e.g., 0, 6.4 and 12.2 %) on the reactor temperature for various prepolymer feed rates. The amount of liquid that enters the reactor can be controlled by adjusting the operating temperature of the flash (e.g., 75, 66 and 55 ~ respectively). The dew point of the feed stream was calculated to be 75 ~ In all cases, the reactor temperature increases as the prepolymer feed rate increases. Moreover, the safe operation window broadens significantly as the amount of liquid fed to the reactor increases. As an example, for a prepolymer feed rate of 20 g/s and when no liquid is fed to the reactor (see solid line) the reactor temperature is equal to 80 ~ When the liquid weight fraction in the total feed stream is 12.2 % (see dotted line), the reactor temperature is less than 40 ~ for the same prepolymer feed rate of 20 g/s. This means that the prepolymer feed rate can significantly be increased without the risk of temperature runaway. Similar behavior is observed when the amount of inert and/or active condensables in the recirculation feed stream increases. Figure 3 illustrates the effect of the hexene-1 concentration in the recycle stream on the safe operation window of the reactor. Notice that for a given prepolymer feed rate, as the amount of hexene-1 increases (e.g., from 9.12% to 11.12 %) the reactor temperature decreases. This can be explained by the
331 140
120
c..) %.-' 12o
%-11oo
~1oo 80 ~D
t..,
o
60
.ca
C6H12 mole fraction
80 ~D
iquidinFeed ~ 0
o
20
"9
r
60
O
. . . . . . . . . 6.4 -........... 1 2 . 2
0 ' 2'0' 4'0' 6'0' 8'0 '100'1 89
Prepolymer Feed Rate
//~
/I
(D k.4
iii . . iiii!1 ...... .
~176
(g/s)
Fig. 2. Effect of the amount of liquid in the feed on reactor temperature,
*~ 4o20
0
....
20 40 60
, , , , , , , .
80 100 120 140 160
P r e p o l y m e r F e e d R a t e (g/s) Fig. 3. Effect of the hexene-1 concentration in the recirculation stream on reactor temperature.
decrease of the dew point temperature of the recirculating feed stream. Thus, as the amount of condensables increases, the dew point of the mixture decreases, while the liquid weight fraction in the feed stream increases. The main benefit of the condensed mode operation in olefin polymerization FBRs is presented in Figure 4. As can be seen, for a constant reactor temperature (e.g., 85 ~ the space time yield increases linearly with the amount of liquid in the reactor feed. This can be explained in conjunction with the results of Figure 2. As the amount of liquid increases, the allowable prepolymer feed rate increases which in turn leads to an increase of the overall polymerization rate. According to the results of Figure 3, when the amount of liquid is increased from 0 to 6 %, the space-time yield increases as much as 200 %. Finally, Figure 5 shows the effect of the amount of liquid in the feed stream (e.g., 0, 6.4 and 12.2 %) on the heat removal factor given by eq. (10) for various prepolymer feed rates. As can be seen, in the case of a conventional FBR (solid line) the heat removal factor is always above unity, even at low prepolymer feed rates. On the other hand, when a mixed liquid gas feed is introduced into the bed (e.g., 6.4%, 12.2%) the heat removal factor reduces significantly (e.g., heat removal increases due to the latent heat of vaporization). Thus the FBR can operate at a higher prepolymer feed rate, resulting in a higher space-time yield.
4. C O N C L U S I O N S In the present study, a model for the simulation of an ethylene copolymerization FBR operating under condensed mode has been developed. Under the assumptions of complete mixing and instant vaporization of the liquid feed in the bed, detailed simulations were carried out in order to investigate the effect of the liquid weight fraction in the feed stream and the composition of hexene-1 in the recycle stream on the reactor temperature, space-time yield and heat removal rate. It was shown that the introduction of a liquid feed in the reactor substantially broadens the safe operation window of the FBR with concomitant the increase of
332 lOO
10t/-r------r
~" 9o
r~
~8
80
,,"
,,'
........ 6.4
,.~ 70
,,"
~6
"~ 60 ~
Liquid in Feed (% w)
~0
.
N 5o ~ ~.~
40
~
20
~4
30-
~-~10 14
Liquid in F e e d (% w) Fig. 4. Effect of the amount of liquid in feed on the reactor's space time yield (Tr = 85 ~
0
0
20
40
60
80
100
120
P r e p o l y m e r F e e d Rate (g/s) Fig. 5. Effect of the amount of liquid in the feed on the heat removal factor
the space-time yield and the heat removal rate from the reactor. These results are in qualitative agreement with findings reported in the patent literature [ 1,3-5]. REFERENCES 1. J.M. Jenkings III, R.L. Jones, T.M. Jones and S. Beret, Method for Fluidized Bed Polymerization, US Patent No. 4 588 790 (1986). 2. Y. Jiang, K.B. McAuley and J.C.C. Hsu, Ind. Eng. Chem. Res., 36 (1997) 1176. 3. M.J. DeChellis, J.R. Griffin and M.E. Muhle, Process for Polymerizing Monomers in Fluidized Beds, US Patent No. 5 405 922 (1995). 4. J.-C. Chinh, M.C.H. Filippelli, D. Newton and M.B. Power, Polymerization Process, US Patent No. 5 541 270 (1996). 5. R.J.N Bemier, R.L. Boysen, R.C. Brown, L.S. Scarola and G.H. Williams, Gas Phase Polymerization Process, US Patent No. 5 453 471 (1995). 6. H. Hatzantonis, H. Yiannoulakis, A. Yiagopoulos and C. Kiparissides, Chem. Engng Sci., 55 (2000) 3237. 7. L.-S. Fan, Gas-Liquid-Solid Fluidization Engineering, Butterworths, U.S.A., 1989. 8. R.C. Reid, J.M. Prausnitz and B.E. Poling, The Properties of Gases and Liquids, McGrawHill, U.S.A., 1988. 9. A.Yiagopoulos, Ph.D. Dissertation, Aristotle University, Thessaloniki, Greece, 2000.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
333
An Extended Self-Organizing Map with Application to the Modeling of Pulse Jet Fabric Filters Hualiang Zhuang and Min-Sen Chiu* Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Pulse jet fabric filters (PJFF) have become an attractive option of particulate collection utilities due to the feature that they can meet the stringent particulate emission limits regardless of variation in the operating conditions. The dynamics of the filter has complex nonlinear characteristics as reported in [1]. In this paper, the framework of local model networks (LMN) is employed to approximate the process dynamics of the pulse jet fabric filter that is subject to multiple operating regimes. To do so, an extended self-organizing map is employed to partition the PJFF's operating range and construct the LMN automatically. Simulation results illustrate the proposed approach and a comparison with the conventional approach is made. 1. INTRODUCTION Pulse jet fabric filters (PJFF), sometimes called bag house filters, have become an attractive option of particulate collection utilities due to the feature that they can meet the stringent particulate emission limits regardless of variation in the operating conditions. Other merits of pulse jet fabric filters are high collection efficiency, on-line cleaning applications and outside collection which allows the bag maintenance in a clean and safe environment. Despite their wide applications, investigations in the modeling and control of the pulse jet fabric filters can only be described as rudimentary. To the best knowledge of the authors, two models were reported in the literature [1,2]. However, these first principle models are too complicated to be incorporated in the controller design. Therefore there is a need to develop a simple model with reasonable accuracy for the PJFF. In this paper, a modeling methodology based on the framework of local model networks (LMN) [3] is developed to construct an accurate and simplified dynamic model of a PJFF. Despite the recent advances of LMN, a prior knowledge of the processes has to be exploited for determination of the LMN structure and the weighting functions. In this paper, an extended self-organizing map (ESOM) network, which can overcome aforementioned difficulties, is developed to construct the LMN using the input-output data. This ESOM network is a refinement of that proposed by Ge et al [4] with the improvement procedures of employing competitive learning algorithm for data clustering and using the partial least square algorithm for computing the parameters of the local models.
* To whom all correspondences should be addressed. Phone: (65) 8742223, Fax: (65) 7791936, Email:
[email protected] 334 2. L M N F O R PULSE JET FABRIC FILTERS The operation of the pulse jet fabric filter (Figure 1) can be briefly described as follows: during the filtration cycle T, influent gas is passed through a filter bag and dust cake is built up at the upstream side of the bag surface. At time T, the bag is subject to a pulse jet of air of high pressure, removing a certain fraction of the cake. The average flow-rate Q of the exhausted gas during cycle time T, i.e. the filter's filtration capacity, is the key output of the process. In this paper, Q and T are chosen to be the output and input of PJFF. Owing to the nonlinearity of the filtration process, a LMN framework is employed to tackle this problem. LMN is constructed as the weighted sum of a set of simple local models across the specified operating range. The properly that it can approximate nonlinear systems has been investigated in a great detail recently [3]. In the LMN structure, a weighting function is assigned to each local model to indicate its respective degree of validity in the operating space. To apply LMN to PJFF, the operating space 9 is decomposed into n . operating regimes ~ j . Define fl(t-1) dynamics as given by fl(t -
1) =
[Q(t -
1),..., Q ( t
as an operating point vector that characterizes the process
- nQ ) , T ( t - n d ) . . . . . T ( t - n d - n v +
where nQ and n T are integers related to the system's order; weighting function is denoted by n~ Q ( t ) : ~_~ p } ( f l ( t j=l
1))~ r ( t
pj (fl).
1)]v
na
(1)
is the process time delay. The
The LMN of PJFF can then be presented as follows, (2)
- 1)Oj
where ~(t - 1) is the regression vector and Oj is the local parameter vector as given by gt(t- 1) =
[Q(t-1),...,
Q(t-
nQ ) , T ( t -
9
n d ) . . . . . T ( t - n d - n~ +
O.j : [O.j,1 .... , Oj,nQ , Oj,no +], .., Oj,tiO +tiT' O.j,tiQ +nT +1
]r
1), 1]v
(3) (4)
In the following section, an extended self-organizing map approach is proposed to construct the LMN automatically using the input-output data. 3. E S O M A L G O R I T H M
Ge et al [4] developed an extended self-organizing map (ESOM) network that can partition the operating space of nonlinear processes automatically using the input-output data. However, their method suffers two shortcomings: (1) the computation time increases dramatically as the number of local models increases, (2) the least square algorithm employed is not reliable when the matrix to be inversed is ill-conditioned. To overcome these problems, a competitive learning method for cluster-center searching is used in the proposed ESOM network and a partial least square algorithm is also employed to compute the parameters of the local models. As shown in Figure 2, the ESOM consists of three layers" an input layer, a Kohonen layer and an output layer. The functions of these three layers are described in below.
335
Input layer: Given the input-output training data v and y, the input of this layer includes y as well as v, i.e. ~ = (v, y ) , where v ( t ) = [Q(t - 1)..... Q ( t - n Q ) , T ( t - n d ) .... , T ( t - n d - n T + 1)] T
(5)
y(t)=Q(t)
(6)
K o h n o e n Layer: The weight vector of the nodes is formatted as Wi = ( W / ' , W , Y ) , where W/' is called the master weight vector and Wiy is the slave weight vector. The following gives the self-organizing algorithm to determine a set of cluster centers f~j, j = 1,...,n o , which characterize the dynamics of the nonlinear process. S t e p 1. Initialize Wi, i = 1, 2,--., K 2. S t e p 2. At each learning step l, only the master weight vector W,", i = 1, 2,..-, K2, is used to
determine the winner node whose weight vector W,~' best matches the input v, i.e.,
where H]I denotes the Euclidean norm. S t e p 3. Update every weight vector W~ = ( W [ , W f ) in the Kohonen layer as follows:
w~ (l + 1) = w, (l) + y ( i , l ) ( ~ ( l ) - w, (l)),
1
where y ( i , l ) =
(8)
, Pi and Ps are the positions of the node i and winner node.
/(1 + l)e [pi-p''[[2 S t e p 4. Check for the convergence of W~. If not, go to step 2. Otherwise, reset learning step
l= 1 and go to Step 5. From step 5 to step 7, a competitive learning rule [5] is employed to determine a fixed number of neurons f~j, j = 1,2,-..,n~,, which are considered as the centers of the clusters of the nodes with W/', i = 1, 2,..-, K 2 in the Kohonen layer. S t e p 5. Initialize f2j, j = 1,2,..., n~,. S t e p 6. If l > K2, l = 1. Determine the neuron whose weight vector f2,. (l) best matches the
weight vector of Wtv (l), i.e., (9) Update the weight vector of the neuron f2 s as follows: ~ s (l + 1) = ~ , (l) + Z ( 0 ( g ~ (Z) -
~,
(Z))
(10)
336 where Z(I)=
, Pl and Ps are the positions of the node l and neuron ff2s. ~/(1 + l)e IIp'-p'IIz
Step 7. Check for the convergence of f2j (l) j = 1,2,...,n.. If all of them converge, stop. Otherwise, l = l + 1 and go to step 6. In relation to LMN, these cluster centers form the local models and the weighting functions Ps are chosen to be the normalized gaussian functions as follows,
(11)
Ps(~) = .. j=l
where o-s2 is a constant variance.
Output Layer Set ~3= [v, 1], we then obtain no
y(t) = ~ p j v
^T
Oj --/30
(12)
j=l
where/3 = [pl~3r
,-..,
pn.~3r ]r ~ 0 = [Or
,...,
Onr ]r , and Oj is defined in (4)
Given the input-output data {v(t), y(t), t = 1, 2,...,
K t
}, /3
is fixed after the self-organizing
process such that the output .~ is actually a linear combination of the elements of O. In this work, a partial least square method [6] is chosen to find the solution of O, which minimizes P0 -
Yll, where
P = [/3(1) ..... /3(K,)]r, y = [y(1),..., y(K, )]r.
4. S I M U L A T I O N RESULTS For the operating space under consideration, the input signal, i.e. the cleaning cycle time T, ranges from 30 seconds to 150 seconds. The corresponding steady state output Q varies from 408.3 x 10 -4 m/s to 304.1 x 10 -4 m/s. In what follows, the proposed ESOM algorithm will be performed to construct the LMN for the PJFF based on the data obtained from the nonlinear simulations of the first principle model given in [1 ]. The input-output data is generated as follows: 500 independent random signals with uniform distribution ranging from 30s to 150s are used as inputs. The corresponding plant outputs in time sequence are generated. These input-output data can be grouped in the format of (5) and (6) so that it is suitable for the ESOM learning algorithm. During the ESOM learning, a trial and error procedure is required to determine the suitable order and number of local model. In this work, a third-order is used as the order of all local models. The effect of the number of local models, n o , is also studied by selecting four
337 different values, 3, 5, 7 and 9 for comparisons. For the purpose of comparison, a single global model and conventional LMN approach are also constructed. By conventional LMN approach, we means that the operating space is uniformly partitioned with respect to the value of output variable Q. Similarly, four cases with 3, 5, 7 and 9 local models are obtained. To obtain local model at each operating point, 200 points of input-output data are generated at each operating point. To compare the predictive performances of these models, a series of step changes of input T is performed as show in Figure 3. The mean absolute errors for all nine cases are summarized in Table 1. Due to the space constraint, only the simulation results of the single model, the LMN with n o = 7 based on the conventional approach and the proposed ESOM network are illustrated in Figure 4 to Figure 6 respectively. Evidently, the ESOM method outperforms both single model and the conventional LMN. For ESOM method, the accuracy of the model increases as the number of local models increases (see Table 1). 5. CONCLUSION This paper refines the previously proposed extended self-organizing map for the LMN based modeling approach. By using the input-output data, the proposed method automatically partitions the operating space and constructs the LMN. Simulation results of a pulse jet fabric filter example show that the proposed method has better modeling accuracy as compared to both single global model and conventional LMN approach. ACKNOWLEDGMENT
The authors appreciate the grant of National University of Singapore (RP279000043112). REFERENCES
1. J. Ju, M. S. Chiu and C. Tien, J. Air & Waste Management Assoc., No. 50 (2000) 600. 2. M. D. Ravin and W. Humphries, Filtration & Separation, May/June (1988) 201. 3. T.A. Johansen and R.M. Smith, Multiple Model Approaches to Modeling and Control, Taylor & Francis, NewYork, 1997. 4. M. Ge, M. S. Chiu and Q. G. Wang, Ind. & Eng. Chem. Res., No. 39 (2000) 3778. 5. T. Kohonen, Self-Organizing and Associative Memory, Springer-Verlag, 1987. 6. P. Geladi and B. R. Kowalski, Anal. Chim. Acta., No. 185 (1986) 1.
Table 1. Mean absolute errors of validation results for three modeling methods ,,
no = 3
no = 5
no = 7
no = 9
Conventional LMN
5.23x10 -4
4.91x10 -4
4.73x10 -4
4.71x10 -4
ESOM based LMN
4.97 x 10 -4
2.03 • 10 -4
1.77 x 10 -4
1.69 x 10 -4
Single Model
9.22 x 10 -4
338
Figure 1. Schematic of a pulse jet bag filter.
Figure 3. Input sequence for model validation.
Figure 5. Validation of conventional LMN with n . = 7. ~ : m o d e l ; ---:plant.
Figure 2. ESOM architecture.
Figure 4. Validation of single model. :single model; :~t,
Figure 6. Validation of ESOM method with n o = 7. ~ : model; - - - : plant.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
339
Feed Pretreatment for Binary Distillation Efficiency Improvement Rakesh Agrawal and D. Michael Herron Air Products and Chemicals, Inc., Allentown, Pennsylvania 18195 In this paper we discuss thermodynamic efficiency for cases where two feeds of the same composition but different enthalpy are fed to a binary distillation column. We present a quick method to determine for each of the feeds whether it should be a saturated liquid or vapor or two-phase. This study provides a process designer with a quick and reliable method for selecting thermodynamically efficient configurations through preconditioning of the feed streams and addition/removal of heat from intermediate locations of a binary distillation column. 1. I N T R O D U C T I O N
A
q>
,R,-~B
Fig. 1" Different options for adding and removing heat from intermediate locations,
Quite often, the feed to a binary distillation column originates from another distillation column or from a unit operation within a process. Thus, there is an opportunity to obtain this feed as a saturated liquid, a saturated vapor, a two-phase stream, or as multiple streams with different enthalpies. Moreover, sometimes lowpressure steam may be available as a waste heat source to allow pre-adjustment of feed enthalpy. To a process designer, if the impact of proper feed conditions on thermodynamic efficiency were known a priori, the opportunities could then be exploited to create an optimum solution through manipulation of feed conditions. Generally, preconditioning of feed is an integral part of the overall strategy to add/remove heat from an intermediate location of a distillation column [1 ]. Several options to accomplish this task are illustrated in Figure 1. The intermediate reboiler is shown as IR and intermediate condenser as IC. The feed heat exchanger FHX is optional. FHX could be a feed condenser for a vapor feed or alternatively it could be a reboiler for a liquid feed. FCON is
340 a feed condenser and FREB is a feed reboiler. When FHX is used in conjunction with FCON or FREB then the quality of feeds exiting the two heat exchangers are different. A process designer is faced with evaluating the merit of each of these five heat exchangers. In earlier work, we developed a model to systematically study the thermodynamic efficiency of binary distillation [1-3]. Through extensive computation and model development, a framework was developed that provides quick and reliable answers to the questions: (1) Is the optimum feed condition to the column saturated liquid or vapor, or twophase? (2) When does the use of an intermediate reboiler or an intermediate condenser provide a meaningful improvement in thermodynamic efficiency? (3) Which of an intermediate reboiler or condenser is more effective? (4) Where is the optimum location for an intermediate reboiler or condenser? Sometimes two feeds that are of the same composition but different enthalpy are used to improve the performance of distillation [4-5]. There are no available methods to quickly assess the proper thermodynamic state of each feed for this case. In this article, we provide a method to quickly decide for each feed whether the optimal state is all liquid, all vapor or two-phase. However, before this issue can be addressed, it is essential to briefly review the model and the optimum location for intermediate heat addition or removal. 2. BRIEF M O D E L DESCRIPTION
The analysis in this and previous work is made with the assumptions that imply that the mixture to be separated is an ideal binary mixture of constant relative volatility. The performance was expressed in terms of thermodynamic efficiency r/"
minimum work of separation total exergy loss + minimum work of separation References 1-3 provide the derivation of efficiency equations and the extensive computation through which the answers to some of the earlier posed questions were obtained. This analysis is based on the distillation of a binary mixture AB into pure products. The parameters needed to answer the questions are simplified to: mole fraction of the light component A in the feed ( ZA ), relative volatility of A with respect to B ( a ) and the quality of the feed q (liquid fraction in the feed stream). Surprisingly, the reboiler, condenser and feed temperatures are not needed to solve the final thermodynamic efficiency equations. 3. O P T I M A L LOCATION FOR INTERMEDIATE HEAT A D D I T I O N / R E M O V A L
Once it is determined that an intermediate reboiler or condenser can lead to a meaningful improvement in efficiency, the next hurdle is to find its optimum configuration and location [2-3]. The guidelines FOR selecting locations were developed based on the following two parameters [1 ]" I+Z~ aiR
"
-
~
z8
I+Z A .
alC
=
~
ZA
341 For a saturated liquid feed, the optimal location for intermediate heat addition is determined by comparing the actual relative volatility a with the value of aiR. 1. a < aiR, the highest efficiency is achieved by withdrawing a side liquid draw, subjecting it to total vaporization, then returning as shown for IR in Figure 1. As a increases, the side liquid draw location moves up the stripping section of the column. 2. a = aiR, a Z A fraction of the feed is totally vaporized and fed to the column (FREB).
3. a > aiR, FREB becomes a partial vaporizer. It is worth noting that as the value of a increases above aiR, the transition of the stream exiting FREB heat exchanger from total vapor to significantly two-phase is found to be quite gradual. Therefore, unless a is significantly greater than aiR, one would not see a significant decline in efficiency when using FREB as a total vaporizer. Similarly, for a saturated vapor feed, the relative volatility a can be compared with alC. For a < Ctic a side vapor draw with IC as a total condenser is used. When ct = ctlc, a ( Z B) fraction of the feed is totally condensed in FCON. partially condensed in FCON.
For ct >_Ct~c, a fraction of the feed is
4. TWO FEEDS OF SAME COMPOSITION BUT DIFFERENT ENTHALPY We have shown that feed pretreatment is an integral part of the strategy to add or remove heat from an intermediate location of a distillation column. These findings are now extended to deduce the optimal state for each feed when there are two feeds of the same composition and no intermediate reboiler nor condenser is to be used. Additionally, the effectiveness of an intermediate reboiler or condenser for the special case when one feed is a saturated liquid and the other a saturated vapor can also be deduced. These guidelines are shown graphically in Figure 2 and depend on relative values of a , air and Ctic. When neither intermediate reboiler nor condenser is used, the optimal feed conditions are determined by combining results of two thought experiments. In the first, the top (upper) feed is assumed to be a saturated liquid and the optimal feed condition for the bottom (lower) feed is deduced. Similarly, the second thought experiment is conducted by assuming the bottom feed is a saturated vapor. For the first thought experiment the top feed is saturated liquid. (a) a < air : Since the use of an IR is preferred over an all-vapor bottom feed, but no IR is to be used, it follows that the bottom feed to the distillation column be a vapor stream. (b)ct = aiR: Taking a Z A portion of the assumed liquid feed and completely vaporizing is preferred. Therefore the bottom feed must be saturated vapor. (c) tz > air : The bottom feed to the distillation column must be a two-phase stream. For the second thought experiment the bottom feed is saturated vapor. (d) cr < aic : The more effective method of removing heat is through the use of an IC. Since no IC is to be used, the top feed to the distillation column should be a liquid stream. (e) a = Ctic: A ( Z 8 ) portion of the assumed vapor feed should be totally condensed and fed
to the distillation column. In this case, the top feed must be saturated liquid.
342 (f) a > alC : The top feed to the distillation column must be a two-phase stream.
(~IC
~IRI
%op--2~
\t ~ 2,C=declines
~
FBOT=2 ~
iR77
/
10 ~FTop=L ~ FBOT-2(i) ~ IC=yes ~ IR=no
y
~ FTOp=2(D FBOT =v
IIR=yes C=no F op-L
FBoT=V
ICpreferred !
0.2
|
IRpreferred !
0.4 ZA 0.6
!
0.8
Fig. 2: Two-feed map for selection of optimum configuration. IC and IR recommendations are for saturated liquid and vapor feed case. 2~ = Two phase.
By combining the rules set forth by items (a) through (f) the selection of optimal feeds states can be further refined. First consider the region in Figure 2 where the feed is A-lean, i.e., Z A < 0.5. In this region air a z c , from items (c) and (f), both the feed streams should be two-phase. Similarly, optimum feed conditions for the mirror region in Figure 2, when Z~ > 0.5, can also be easily derived. For Z A = 0 . 5 , as a is increased to much higher values than 3 (where air
=arc),the
quality of the two two-phase streams approach one another. This implies that when Z A is in the neighborhood of 0.5 and relative volatilities are quite large, it will be sufficient to have one two-phase stream and the quality of this feed will be around 0.5. It is worth noting that transitions from saturated liquid or saturated vapor to two-phase across air and arc curves in Figure 2 are not sharp. This is especially true for higher values of a , i.e., for an A-rich phase (Z A > 0.5 ) when a is in the neighborhood of a~R and for an A-lean phase (Z A < 0.5) when a is in the neighborhood of a~,c . The primary reason lies in
343 the fact for an A-rich stream, as a exceeds aiR, the top feed is a two-phase feed and not a saturated liquid. As a result, its temperature would be warmer than that of the saturated liquid of the same composition. Therefore, if the optimum temperature for intermediate heat addition with a saturated liquid feed is at the saturated vapor feed temperature (for a = air ) then with a much warmer two-phase feed one would expect this optimum temperature to also shift to a temperature that is warmer than the saturated vapor feed temperature. This will require that for the A-rich feed, as a exceeds air but is in the neighborhood of c•/R, the bottom feed be saturated vapor feed and not a two-phase feed. Only when a sufficiently exceeds %R and the optimum location for intermediate heat addition has moved above the temperature of the saturated vapor feed would the bottom feed become two-phase. Therefore, in Figure 2, when c~ is in the neighborhood of a~R but exceeds aiR, it is better to use the bottom feed as a saturated vapor feed. Similar reasoning can be applied for an A-lean feed when a is in the neighborhood of a~C< -CH>C-
1 2 3 4 3 4
-OH -O-O -N< -NH-NH2
1 2 1 3 2 1
8v 5 6 6 5 4 3
Once a molecule is decomposed into its basic groups, and the atomic connectivity indices for those groups are known, then molecular connectivity indices can be computed for the entire molecule. The zeroth order molecular connectivity indices ~Z and ~ are sums over each basic group, and thus describe the identity of the groups in a given molecule: 1
~
ieG
0Z,. = ~--,~ 1 i~G
where G is the set of all basic groups in the molecule. The first order connectivity indices ~Z and 1zv are defined across the bonds of the molecule, and thus describe the internal molecular structure: iZ = ~--~~ 1 i,j~BX/-~,6,
1zV=~ i /~B
1 (~i (~j
where i and j are the groups participating in one of the bonds B of the molecule. Higher order connectivity indices can also be defined, and have been used to give a more precise description of molecular structure (Kier and Hall, 1986). However, most of the key properties of pharmaceuticals can be correlated with these four molecular connectivity indices.
372 Once the equations defining the (molecular) connectivity indices are in place, we can use these indices in empirical correlations to predict the physical properties of novel soybean-oil derived compounds. For example, the correlation derived in this work for the critical micelle concentration of a soybean-oil derived surfactant is given as 100 log (CMC) = - - ~ t[_ 1.063 ~Z +0.327 ~Z ~ +1.390 1Z - 0 . 4 3 6 1Z" + 0.828] /If
Data for this and other correlations was collected from Rosen (1978), Schick (1967), Shinoda and Friberg (1986), Shinoda et. al. (1963), and Van Os et. al. (1993). Other properties such as the hydrophilic-lipophilic balance (HLB) can also be computed using structural descriptors of this type. Since connectivity indices are defined in a very general way, they are capable of describing any molecule, and thus correlations based on them tend to be widely applicable and fairly accurate over a wide range of compounds. Correlations can also be created combining the connectivity indices of two or more different molecules, leading to equations describing the properties of mixtures. Using such correlations, an optimization problem has been formulated which has as its optimal solution a molecule derived from soybean oil which most closely matches a set of target property values for a fuel additive. 3. P R O B L E M FORMULATION The optimization problem which determines the best molecule for a given application uses an objective function which minimizes the difference between the target property values and the estimated values of the candidate molecule. This can be written as min s = ~
liP"
m~R p~cale
_ ptarget
,
I
where R is the set of all targeted properties, Pm is the estimated value of property m, Pmscale is a scale factor used to weight the importance of one property relative to another, and Pm target is the target value for property m. The molecule is represented mathematically using two sets of binary variables: a partitioned adjacency matrix with elements a(i,j,k) which are one if basic groups i andj are bonded with a k-multiplicity bond, and zero otherwise. This matrix is partitioned such that specific rows are preassigned to different basic groups, so that it can be determined a p r i o r i what 6i and 6iv values should be used for each basic group i in the molecule. Since we do not know how many of each type of group will occur in the final optimal molecule, the partitioned adjacency matrix will have many rows which do not correspond to a basic group. The binary variable wi is set to one if the ith group in the adjacency matrix exists in the molecule, and is zero otherwise. These two sets of variables provide sufficient information to compute the connectivity indices, and thus estimate molecular properties. The definition of the zeroth-order connectivity indices can be rewritten using these variables to become N
.=
ozv=
N
w
373 where N is the total number of rows in the adjacency matrix. The equations for the first order connectivity indices are also linear in this formulation, since the 8i and 6iv values are constants for a given i. Those equations become
~ ~a0~ "=
"=
]=i+l
j=i+l
v
v
j
Next, property correlations using the connectivity indices are included. Finally, structural feasibility constraints are needed to ensure that a chemically feasible molecule is derived. The valence balance for both single and double bonds can be written as i=l
N
Zaijk + ~auk = vikwi ,V i= l,...,N,k= l,2 j=l
j=i+l
where vtk refers to the number of kth multiplicity bonds which can be formed by the basic group at row i in the partitioned adjacency matrix.. Each bond which forms must also be of a single multiplicity:
~-'aok < I , V i = I , . . . , N ,
j=i+l,...,N
k=l,2
In order to guarantee that all the groups in the molecule are bonded together as a single unit, we include the constraints from a network flow problem into the formulation. These constraints require that a path through the bonds of the molecule exist between one group, called the source, and all other existent groups, called the sinks. A feasible solution to a network flow problem across the bonds of a molecule is a necessary and sufficient condition for connectedness, and the constraints required are linear and introduce no new integer variables. Other constraints include bounds on the variables and property values. The problem written in this form is an MILP, which for smaller examples can be solved directly using commercially available software. 4. E X A M P L E The example presented here produces a possible molecular structure for a fuel additive which could be synthesized from linoleic acid, the major component of soybean oil. Fixing the lipophilic portion of the molecule, the program determines the optimal structure of the surfactant head group which will give an HLB value near 8 and a low CMC, which is related to the amount of fuel additive required to give a detergent effect. An HLB value near 8 gives a sufficient lipophilic character to dissolve in hydrocarbon fuels, and yet is high enough to give the compound the capability to solubilize organic deposits. The possible basic groups in the molecule are those listed in Table 1, and the maximum number of basic groups allowed in the molecule is 35. The problem was solved using the solver CPLEX accessed through the GAMS modeling language (Brooke, et. al., 1988) on a Sun Ultra l0 workstation. The optimal structure determined is shown in Figure l, and has a predicted HLB of 7.93 and a CMC in water of 1.5xl 0-l~ . /(C H2)2~ C H-- O C H ~"-- (C H2)~--- [C H2 C H = C H]2---(C H 2)7-~ N \ O ~ (C H 2 ) 5 ~ O H
Figure 1: Structure of optimal soybean oil derivative
374 5. CONCLUSION This work has focused on the use of optimization techniques within a molecular design application to derive novel molecular structures for fuel additives derived from soybean oil. The use of connectivity indices to relate internal molecular structure to physical properties of interest provides an efficient way to both estimate property values and recover a complete description of the new molecule after an optimization problem is solved. The optimization problem has been formulated as an MILP and solution times are relatively low even for large problems. Further work will include a much larger set of physical properties and basic groups from which to build molecules, and will work toward the design of mixtures and the prediction of mixture properties via connectivity indices. REFERENCES
1. Bicerano, J. (1996). Prediction of Polymer Properties. Marcel Dekker, New York. 2. Brooke, A., Kendrick, D., and Meeraus, A. (1988). GAMS: A User's Guide. Scientific Press, Palo Alto, CA. 3. Camarda, K. V. and Maranas, C. D. (1999). Optimization in Polymer Design using Connectivity Indices. Ind. Eng. Chem. Res., 38, 5, 1884-1892. 4. Erickson, D. R., Pryde, E. H., Brekke, E., Ordean, L.. Mounts, T. L. and Falb, R. A. (1992). Handbook of Soy Oil Processing and Utilization. American Soybean Association, St. Louis, MO. 5. Gani, R., Tzouvars, N., Rasmussen, P. and Fredenslund, A. (1989). Prediction of Gas Solubility and Vapor-Liquid Equilibria by Group Contribution. Fluid Phase Equilib., 47, 2, 133. 6. Glover, F. (1975). Improved Linear Integer Programming Formulations of Nonlinear Integer Problems. Manage. Sci., 22, 4, 455. 7. Hairston, D. W. (1998) New Molecules get on the Fast Track. Chem. Eng., Sept., 30-33. 8. Kier, L. B. and Hall, L. H. (1976). Molecular Connectivity in Chemistry and Drug Research. Academic Press, New York. 9. Maranas, C. D. (1996). Optimal Computer-aided Molecular Design: A Polymer Design Case Study. lnd. Eng. Chem. Res., 35, 3403. 10. Raman, V. S. and Maranas, C. D. (1998). Optimization in Product Design with Properties Correlated with Topological Indices. Comput. Chem. Eng. 22, 6, 747. 11. Rosen, M. J. (1978). Surfactants and Interracial Phenomena. John Wiley and Sons, New York 12. Schick, M. J. (1967). Nonionic Surfactants. Marcel Dekker, New York. 13. Shinoda, K. and Friberg, S. (1986). Emulsions and Solubilization. John Wiley and Sons, New York. 14. Shinoda, K., Nakagawa, T., Tamanushi, B. and Isemura, T. (1963). Colloidal Surfactants - Some Physicochemical Properties. Academic Press, New York. 15. Trinajstic, M. (1975). Chemical Graph Theory. CRC Press, Boca Raton, FL. 16. Van Os, N. M., Haak, J. R. and Rupert, L. A. M. (1993). Physico-Chemical Properties of Selected Anionic, Cationic and Nonionic Surfactants, Elsevier Science, Amsterdam. 17. Venkatasubramanian, V., Chan K., and Caruthers, J. M. (1994). Computer-Aided Molecular Design using Genetic Algorithms. Comp. Chem. Engng., 18, 9, 833-844.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
375
E C O F A C - Computer Aided Solvent Design and Evaluation in Environmental Problems, Based on Group Contribution Methods with Association. M.Cismondi and E.A.Brignole. Planta Piloto de Ingenieria Quimica-PLAPIQUI (UNS-CONICET), Camino La Carrindanga Km 7, 8000, Bahia Blanca, Argentina Computer aided product design requires the generation of feasible compounds and the prediction of solution and pure component properties by group contribution methods. The stability of the generated chemical structures is based on the electronegativity of the group attachments. A new characterization of group combination properties and the corresponding feasibility criteria for computer aided generation of branched structures are presented. A synthesis procedure and a strategy for reducing the size of the combinatorial synthesis problem are presented. Group contribution methods for property predictions of aqueous systems in environmental problems are reported. 1. INTRODUCTION The application of group contribution methods, for prediction of pure component and mixture properties, opens the way for efficient computer aided evaluation of chemical compounds. The backward solvent selection problem was formulated as follows: "giving a set of property constraints and certain performance indexes, generate chemical structures with the desired set of physico - chemical and environmental properties"(Gani and Brignole [ 1]). For the computer aided generation and evaluation of molecular structures a combinatorial - partition strategy was proposed by Brignole et al. [2]. Several molecular design procedures based on optimization algorithms have also been proposed [3-6]. Applications have been reported for the design of polymers [5], refrigerants [3,6], product substitution [7], solvents [8,9], etc. Even though the generated molecules satisfy the neutrality condition, the chemical stability of the components is in many cases not guaranteed [5,6]. This is partly due to the way groups are defined in different group contribution methods and/or to the lack of proper combination rules for the groups. Some of the component properties of interest, from the point of view of environmental considerations, depend on mixture or solution properties. The first solvent design studies were based on solution properties derived from the UNIFAC group contribution method for computing activity coefficients [10]. Several revisions and extensions to electrolytes, polymers and equations of state, of the original UNIFAC predictive package have been presented [11]; a group contribution equation of state (GC-EOS) based on similar but more detailed group definitions, has been extended to new groups and gases [12-14]. Different group definitions have been proposed [15] for prediction of pure compound
376 properties, such as heat capacities, solubility parameters, formation energies, etc. In general, specific criteria [6] to assure the component valence neutrality and the fulfillment of the octet rule are required. These criteria are combined with optimization algorithms for the synthesis of molecules. However the generation of chemically unstable compounds is not avoided. The generation of feasible compounds can be achieved if the electronegativity of the group bonds is taken into account [2,8,9]. Three basic types of group valences, with increasing degree of electronegativity, were identified [8] in aliphatic compounds: J, L and K,. The methyl group (-CH3), even though it is a J type group, is identified as type M because of the different role it plays in the feasibility criteria analysis, with respect to the other J groups. Pretel et al. [8] proposed synthesis procedures based on intermediate and terminal groups for the generation of linear (not branched) molecules. An extension of the synthesis procedure for aliphatic branched molecules is developed in the present work. 2.
SYNTHESIS OF BRANCHED MOLECULES The extension of Pretel et al. feasibility criteria to branched structures would be:
Z i i Ki 2, 2--->1 or 2-->2 equilibrium reactions, characterized by a kinetic parameter and an equilibrium constant. All of these elementary reactions are assumed to be either first or second order ones in both directions (see Table 2). In addition to the reactions, the component transport between the mitochondria and the cytosol is described for the consumed and produced components. In the simplest model, we assume a self-controlled component transfer that corresponds to the idealized state of a developed organism. TABI~ 1. Emnlles for the passived e t m ~
Name of conla(menls citrate ( ~ ) isocitrate ( ~ ) cis-ac~tate ( ~ )
T ~
Code SOT SICT SCAC
Code SL'IT1 SOq2 SOT3 etc. SOT4 acetyl-CoA( ~ ) SACD SOT5 CoA-SH(general~ ) OEDA SCIT6 NADH~ conpx~) ~ ..~r7 v~er~ ~ ) ~-120 SOq8 etc. SICT1 citrate ~ (~) ECTS SICq2 isocitm~ d h y ~ (~) HCD SICI3 g l ~ aehyaogemse(mzyn~) SICT4 i, etc. SICT5 i, ZCrS_SOXA( ~ ) F_s SOXA S(I~I zcrs SOXA_SACO( ~ ) F_ClS SOXA SAOO S ( / t 2 F__L-~ SCIT (KDA ~cE~._SCIT_(K3OA(ad&~) ||
ii
||
2. ~ e s
for the active domnls
Fxltfilibrimn~ c a l reaction F_LTS+ SOXA=FLTS SOXA FL-TSSOXA+ SAO:)= ECIS SOXA SAO:) ECTS SOXA SAO:)=ECIS SCIT (K3OA ECHS SOT OEOA =FL-qS SCIT+OEDA SCIT= EL'qS+ ~e~-~ ECTSSCIT+ SCIT= FLqS_SCIT_SCIT(-) Fs S~)(-) zcrs + C~TP- EC~ C~Te (9 EACH+~ = EACH_Cr~ (+) EACH (SEES+SOT=EACH CF[~ SCIT EACH ~ SCIT=EACH CFES ~ A C EACH CF~ ~ C =EACH C F ~ SICT EA(]-I (SEES SICT=EACH CFES+SICT EICD+C~TP=n~_GATP(-) EICD+~=n~~(+) EICD C A I ~ + ~ = E I C D C_.exY)P(3qAD
etc.
The amounts of the consumed and produced components are the best known data that can be used for the evaluation of the synthesized alternative models. There is a considerable gap between the necessary and available data. We would need all of the initial quantities and concentrations, as well as the kinetic and equilibrium parameters for every elementary process. In the contrary, even the quantitative data about the reference measures (e.g. the amount of mitochondria) are not known exactly. We know that the global functioning of the citrate cycle corresponds to the overall Equation of AcCoA + 3 NAD + + FAD + GDP + P~ + 2 H20 --> ---> HS-COA + 3 NADH + H § + FADH2 + GTP + 2 CO2
384 It is known that this reaction produces the majority of the metabolized carbon dioxide. Consequently, from the respiratory data and from the estimated amount of the mitochondria we can calculate an overall reaction rate that is approximately 3E-4 mmol/(g mitochondria * s). Surprisingly, or rather thanks to the excellent organization and to the built-in self-control of the natural processes, the first simulations were already quite feasible. Good examples for the self-control are the inhibition of citrate synthase by the produced citrate, or the inhibition of succinyl transferase by the succinyl CoA. Remembering the difficulties, accompanying the first trials of the "artificial" reacting system, the biochemical network looks another world, where the stiffness and the other kinds of problems are less dangerous. Of course, the principle of "garbage in, garbage out" is still valid, however here the "interpretable bad" results help to refine and to tune the model. The solution of the detailed conservational model is rendered by the paradox feature that the inhibitions and promotions are described by the most detailed model, while the available knowledge belongs to the simplified gross reactions and transportation. However, we can overcome these difficulties by the transformation of the conservational model into a combined conservational / informational one. The typical subsets of the elementary reactions, describing an enzymatic process can be replaced algorithmically for the conservational process of the gross reaction and for an associated rule that models the effect of the various cofactors and effectors on the reaction rate. This transformation helps to reduce the size of the problem, and the number of parameters to be identified, as well.
2.2 The possibility space, the fitness evaluation and the run of the genetic algorithm The possible discrete and continuous features of the investigated problem are contained in the so-called possibility space. It describes the classes of the alternative properties and property sets, as well as the optional forbidden combinations of the genetic properties. For the continuous parameters, the user must declare only the upper bound, the lower bound and the suggested initial value. The genetic code is the list of the ordinal number of the properties in the individual classes, supplemented by the actual proposed values of the continuous parameters. The work of the genetic algorithm is controlled by the size of the population, as well as by the applied reproduction, selection and mutation operators. The evaluation of the variants is based on minimizing of the differences between the prescribed and calculated consumption and production of the components, participating in the gross reaction. The prescribed values come from the overall reaction equation (see above). The simulated results are "measured" in the developed pseudo-steady state condition of the simulated model. During the evolutionary run the simulator does not prepare a detailed output, however the genetic algorithm reports about the individual and average fitness values of the subsequent populations. In Fig. 2 the change of the average values obtained for the most important objectives is illustrated, as the function of the population number. Fig. 3 shows the change in the error of the carbon dioxide production of the subsequent variants. 2.3 Simulation of a proposed good enough solution (testing of the results) In the structural model based simulation we can prepare a very detailed output that shows the change of each concentrations, as well as the rate of every elementary process.
385
Figure 2. Change of the average values
Figure 3. Error of CO2 production
A characteristic part of the simulated result can be seen in Figs. 4 and 5. The concentration trends of some major components (Fig. 4) show the evolution of the almost steady state values from the arbitrary initial conditions. The rates of the elementary reactions (Fig. 5) are in accordance with the overall functioning of the citrate cycle.
Figure 4. Concentration trends
Figure 5. Process rate trends
386 3. CONCLUSIONS 3.1 Recognition of models with the generic / genetic method of Structural Modeling The generic dynamic simulation of the structural models combined with a multicriteria genetic algorithm has been successfully applied for the synthesis of a detailed metabolic model from the available and very limited data set. 3.2 Lesson of nature for engineering synthesis: Almost Closed Conservational Processes The biological systems of various levels show many examples for the case where instead of the raw materials and goal products, the appropriate intermediate materials have the keynote role. The Almost Closed Conservational Process (ACCP) means a complex network of recycles that efrom a set of easily usable and easily recoverable intermediate materials, eproduces a set of goal products, ,while these goal materials or their final derivatives after the use can be transformed into the intermediate forms, .with the a given or minimum necessary environmental input and output.
The conscious design and control of the ACCP is fundamental in the future engineering, because the global natural system is an ACCP itself, with the solar energy as the single environmental input. The large scale and long-term character of ACCP synthesis needs new methodologies and the reconsideration of the existing goal functions. It is trivial in life sciences from agriculture to medicine that human being rather controls than designs the natural processes, however this way of thinking must be adopted by all kinds of engineering in the near future. It is worth mentioning that the described methods can be extended to the simulation of the artificial evolution in the closed space of finite sources, where the evolution is limited by the conservational processes themselves. These investigations may contribute to the development of the "engineering genetics" of the artificial systems that should replace for the "diagnosis and therapy" approach of the environmental control today. REFERENCES 1. G.N. Stephanopoulos, A.A. Aristidou and J. Nielsen, Metabolic Engineering, Academic Press, San Diego, 1998. 2. J. Yin, Chem. Eng. Progress, 95 (11), 1999, pp. 65-74 3. B. Csukfis, K. Varga, R. Lakner, Hung. J. Ind. Chem., 24 (2), 1996, pp. 107-130. 4. B. Csukfis, S. Perez Uriza, Hung. J. Ind. Chem., 23(4), 1995, pp. 277-287. 5. B. Csuk~ts, E. P6zna, Hung. J. Ind. Chem., 24(1), 1996, pp. 69-80. 6. B. Csuk~is, S. Balogh, Computers in Industry, 36, 1998, pp. 181-197. 7. G. Michal (ed.): Biochemical Pathways - an Atlas of Biochemistry and Molecular Biology. John Wiley & Sons, 1999, pp. 43-44.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
387
Analysis of azeotropic distillation columns combined with pervaporation m e m b r a n e s A.M. Eliceche a , P.M. Hoch a and I. Ortiz b PLAPIQUI - CONICET, Chem. Eng. Dept., Universidad Nacional del Sur, 8000 Bahia Blanca, Argentina #
a
b Dpto. de Quimica, Universidad de Cantabria, 39005 Santander, Espafia
The main objective of this work is the analysis of azeotropic distillation columns, when a liquid side stream with the distributing non-key component is treated in a pervaporation membrane and the retentate is recycled to the column. The objective is to separate the pure distributing non-key component from the pervaporation membrane, thus helping to improve the purity in the top and/or bottom products. The operating conditions of the column such as reflux ratio, product and side draw flowrate are selected optimally. A systematic methodology for the selection of the optimum operating conditions of the azeotropic distillation column in the hybrid distillation/pervaporation system for Methyl tert-Butyl Ether production is presented. The significant reduction in the operating cost due to the optimisation of the debutanizer column is reported. 1. I N T R O D U C T I O N Debottlenecking and azeotrope breaking are fruitful fields for hybrid membrane systems. Pervaporation is an interesting membrane separation alternative, because it is generally less energy-consuming than distillation. It is not influenced by the equilibrium between components, making azeotrope breaking easier than using a sequence of distillation columns. The separation is based on a selective transport through a dense layer associated with an evaporation of the permeants. This phase changing is usually obtained by lowering the partial pressure of the permeants at the downstream side of the membranes to vacuum pressure. Recent patents propose hybrid distillation/pervaporation technologies for azeotrope breaking processes involving the separation of alcohols and ethers (Chen et al.1). Hbmmerich and Rautenbach 2 have studied the integration of pervaporation and vapour permeation into the Huels process. They analysed the # This w o r k was carried out under research grant PICT 14-04065 from A N P C y T - Argentina.
388 influence of the operating conditions in a hybrid distillation-pervaporationvapour permeation system for the Methyl tert-Butyl Ether (MTBE) production. Gonzfilez and Ortiz 3 carried out experimental work and reported a rigorous model for the pervaporation membrane to separate methanol and MTBE. However, the formal optimisation of the debutanizer column with a pervaporation membrane to treat the side stream has not been a t t e m p t e d previously. The optimum operating conditions such as reflux ratio and product flow rates are calculated solving an optimisation problem to minimise the operating cost. 2. M O T I V A T I O N This work was motivated by the possibility of revamping the Methyl tert-butyl ether (MTBE) sector of a refinery. Methyl tert-butyl ether is used as a high octane fuel additive. The production process of Methyl tert-butyl ether consists of a reaction sector, where i-C4Hlo is combined with methanol to form the ether, and a separation sector where all the MTBE must be separated from unreacted methanol and C4's. Unreacted methanol forms azeotropic mixtures with MTBE and butanes. A sequence of azeotropic distillation columns was used to break the azeotropes, thus recovering MTBE and methanol to be recycled to the reactor. A hybrid distillation-pervaporation process seems an attractive alternative, as it combines the advantages of both methods. The use of hybrid systems can improve the cost of the traditional Huels separation sequence as reported recently (HSmmerich and Rautenbach2). 3. H Y B R I D D I S T I L L A T I O N / P E R V A P O R A T I O N P R O C E S S Different configuration for the hybrid distillation/pervaporation process can be used, locating the pervaporation membrane to treat the feed, products or side stream. In this work the pervaporation membrane is located to t r e a t the side stream and remove the distributing component as a permeate, helping to improve the top and bottom purity. The separation of pure MTBE as a bottom product of an azeotropic distillation column from a mixture of C4's, methanol and MTBE is performed by means of a combined distillation column and pervaporation membrane process. The process being studied is shown in Fig. 1. The azeotrope formation of methanol with both MTBE and C4 limits the purity of the products. A high purity of MTBE is required in the bottom product (B) of the column that separates a multicomponent mixture (F1) including MTBE, C4's and methanol. A side stream (E) of the column, rich in methanol, is processed through the membrane. The membrane selectivity allows the methanol in the column sidestream to be permeated and then condensed and recycled to the reactor (Permeate liquid stream), thus helping to improve the MTBE bottom product purity. The retentate is recycled to the column (F2).
389
Figure 1: Schematic flowsheet for the distillation/pervaporation process. The objective function to be minimised is the operating cost of the process (Co). It is a s s u m e d t h a t cooling water is used to condense the distillate of the column and medium pressure steam is used to heat the bottoms of the column 4. C 0 -- Ccolumn -+- Cmemb = ( C c -F C r + C p )
[$ /
h]
(1)
C~ = Co,~Q~ Cr = Co,rQr
(2)
Cp -- C o,pQp where Qc is the heat [kJ/h] withdrew from the condenser of the column, Qr [kJ/h] the heat added to the reboiler of the column, and Q, [kJ/h] the heat withdrew in the pervaporation unit to condense the methanol. For the operation of this condenser a refrigerant fluid is needed, because the methanol condenses at very low t e m p e r a t u r e s (-5 ~ at the permeate pressure, usually 0.02 bar). The refrigeration cost to condense the permeate is a function of this temperature. 4. CASE S T U D Y The composition, temperature and pressure of the fresh feed are shown in Table 1, where C4's stands for a mixture of i-butane, n-butane, i-butene, 1butene, 2-butene trans and 2-butene cis. The true composition of C4's was used in the simulation.
390 Table 1" Feed to the column
Component
Flow rate 418.973 26.084 92.923 537.98
C4's Methanol MTBE F1 [kgmol/h] Temperature [K] Pressure [bar]
351.15 6.00
The purity and recovery of MTBE in the bottom product required is 0.99. The process was simulated with HYSYS ~. The membrane module is modelled with a splitter unit followed by heat exchangers to simulate the process of expansion and condensation of the permeate. The pervaporation takes place when a vacuum of 15 torr is maintained at the permeate side. For the start-up a vacuum pump is needed, but when steady state is reached, vacuum is maintained by the permeate condensation. The distillation column has 22 theoretical stages plus a reboiler (stage 0) and a condenser (stage 23). The HYSYS 5 selected optimisation option was the Mixed method. It starts the optimisation with the Box method using a very loose convergence. After convergence, an SQP method is then used to locate the final solution using the desired tolerance. The optimisation results are shown in table 2. A significant cost reduction of 36 % is achieved optimising the operating variables of the debutanizer column. The reflux has been reduced by 40 % at the solution point compared to its value at the initial point. Side s t r e a m E is withdrew from the plate were the liquid composition of methanol reaches its m a x i m u m value. The original locations of the fresh feed, side stream extraction and r e t e n t a t e recycle plates were modified to improve the objective function, while m a i n t a i n i n g the total number of stages fixed. There are important decisions to be made regarding the location of the fresh feed, side stream and recycle of the r e t e n t a t e stream. The systematic selection of the number of plates requires the incorporation of integer optimisation variables. The procedure developed by Hoch and Eliceche 6 treat the number of stages as continuous optimisation variables, allowing the simultaneous selection of feed and side stream locations with the flowrates in a nonlinear programming problem formulation. The implementation of this option is currently being studied.
391 Table 2: Optimisation results for the example shown. Initial point Optimum point R 2.14 1.2704 B 107.6 92.92 E 132.5 126.9 NF1 12 20 NF2 10 14 NE 18 16 X MTBE,B 0.8638 0.9907 0.9970 0.9900 Rec MTBE,B C [S/h] 69.4506 44.123 Cost r e d u c t i o n 36 %
The optimisation of the operating conditions for the hybrid distillation/pervaporation system using a rigorous model of the pervaporation membrane, following the work of Gonzalez and Ortiz 3 will be implemented. The operating conditions of the distillation column and the pervaporation membrane would then be chosen simultaneously. 5. CONCLUSIONS A systematic methodology for the selection of the optimum operating conditions of hybrid distillation/pervaporation system has been presented. The numerical robustness to find the solution of the simulation of the separation process has been greatly improved with the incorporation of the pervaporation membrane, with respect to the simulation of the distillation column on its own. This improvement allowed a successful implementation of the selection of the optimum operating conditions. Numerical results are reported for the hybrid configuration analysed for MTBE production. An important reduction of 36 % in the operating cost of the distillation/pervaporation process has been achieved, quantifying the improvement that can be expected if a systematic optimisation is carried out. The main improvement that can be expected by optimising the operating conditions of the debutanizer column are shown. Similar results can be expected in other applications of hybrid distillation/pervaporation systems. LIST OF SYMBOLS B Cc
Cp Cr
CO,C Co,p
Bottom flow rate, [kgmol/h] Condenser operating cost [S/h] Pervaporation membrane operating cost [S/h] Reboiler operating cost [S/h] Condenser operating cost coefficient [$/kJ] Pervaporation membrane operating cost coefficient [$/kJ]
392 Co,r D E F1 F2 NE NF1 NF2
Qc QR QP R Rec MTBE,B X MTBE,B
Reboiler operating cost coefficient [$/kJ] Distillate flow rate [kgmol/h] Sidestream [kgmoYh] Fresh feed flow rate [kgmol/h] Retentate flow rate [kgmol/h] Location of the side draw Location of fresh feed Location of the recycle stream (retentate) Heat withdrew from the condenser of the column [kJ/h] Heat added to the reboiler of the column [kJ/h] Refrigeration needed for the pervaporation module [kJ/h] Reflux ratio MTBE recovery in the bottom product Liquid composition of MTBE in the bottom product
REFERENCES
1 - Chen, M.S., Zionsville R. and J.L. Allentown, U.S. Patent 4774365, 1988. 2.- HSmmerich U. and R. Rautenbach. Design and optimization of combined pervaporation/distillation processes for the production of MTBE, J. Membr. Sci., 146, 53-64, 1998. 3- Gonz~lez B. and I. Ortiz. Mathematical modelling of the pervaporative separation of Methanol - Methyl tert-Butyl Ether mixtures, submitted to Ind. & Eng. Chem. Res, 2000. 4- Seider, Seader and Lewin. Chapter 10: Profitability Analysis, in Process Design Principles, John Wiley and Sons (ed.), 1999. 5 - Hyprotech, HYSYS user manual, 1999. 6- Hoch P.M. and A.M. Eliceche, "Optimal Design of non-conventional distillation columns", Process Technology Proceedings Vol. 10, Computer Oriented Process Engineering, p 369-374, 1991.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
393
Nonlinear Bilevel Programming: A Deterministic Global Optimization Framework Z. H. Gtimti~ and C. A. Floudas* Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-5263, USA A novel technique that addresses the global optimization of the general nonlinear bilevel programming problem is presented. The technique is based on a relaxation of the feasible region and a branch and bound framework, utilizing the basic principles of the deterministic global optimization algorithm, ~BB [1-4]. For problems that involve twice differentiable nonlinear functions and inner problem constraints that satisfy the linear independence condition, epsilon global optimality in a finite number of iterations is theoretically guaranteed. 1. INTRODUCTION The bilevel programming problem, BLPP, is an optimization problem that is constrained by another optimization problem. This mathematical programming model arises when two independent decision makers, ordered within a hierarchical structure, have conflicting objectives. The decision maker at the lower level has to optimize her objective f(x,y) under the given parameters x from the upper level decision maker, who, in return, with complete information on the possible reactions of the lower, selects the parameters so as to optimize her own objective F (x, y). Note that the upper level decision maker is limited to influencing, rather than controlling, the lower levels outcome. Thus, the general BLPP is formulated as follows: min
F(x,y)
(1)
x
s.t.G(x,y) l-l(x,y)
< =
0 0
min f(x, y) y
s.t.g(x,y) h(x,y) xEX
___ 0 : 0
c R nl , y E Y
c R n2
where f ,F " RnlxR n2 --+ R, g - [gl,..,gj] " RnlxR n2 -+ R J, G - [ G 1 , . . , Gj,] " RnlxR n2 -+ R J', h - [hi, ..,hi]" R n l x R n2 --+ R I, H - [H1,..,HI, ] 9R n l x R n2 --+ R I' . The BLPP model has been employed in many and diverse areas that require hierarchical decision making, including centralized economic planning problems [5], civil engineering transportation network design problems [6], and chemical engineering problems such as chemical process design with equilibrium [7-9], plant design under uncertainty [10], flexibility analysis 9Author to whom all correspondence should be addressed.
394 [11] and process design with controllability issues [12]. Given that the BLPP applications are many and diverse, effective solution algorithms are of critical importance. The linear BLPP has the favorable property that the solution is at an extreme point of the feasible set, and can be exploited by enumeration techniques. However, this does not hold for the nonlinear BLPE The conventional solution approach to the nonlinear BLPP is to transform the original two level problem into a single level one by replacing the lower level optimization problem with the set of equations that define its Karush-Kuhn-Tucker, KKT, optimality conditions. However, the KKT optimality conditions are necessary and sufficient for defining the optimum of the inner level problem only when convexity conditions and a first order constraint qualification are satisfied. When the inner problem constraints are nonconvex, the KKT conditions are only necessary, and local or suboptimal solutions may be obtained. A further difficulty arises in locating the global optimum of the resulting single level problem after the KKT transformation. The bilinear nature of complementarity conditions introduces nonconvexities even if the original problem is linear. Furthermore, when the inner problem is nonlinear, the equations that define the stationarity constraints are also nonconvex. Hence, even if the KKT conditions are necessary and sufficient for the inner problem, the global optimality of the transformed single level problem can not be guaranteed unless a global optimization algorithm is introduced. These difficulties related with the KKT-type solution approaches, which are the most efficient and widely used methods for the solution of the BLPP, confine them to the identification of only local solutions when nonlinearities are involved.
2. THEORY 2.1. KKT Optimality Conditions The KKT optimality conditions are equivalent to the inner optimization problem assuming that f, h, and g are smooth, f and g are convex, h is linear in y at fixed x for every x E X, and one of the first-order constraint qualifications, such as linear independence, Slater, KuhnTucker or weak reverse convex condition, holds in terms of x at a feasible point y*. The bilevel programming problem is transformed into a single level problem of the form: min xy
s.t.
(2)
F(x, y) G(x, y) _< 0
H(x, y) _< 0
hi(x, y ) - 0 i C I, af(x'Y) + ~x'JagJ ~=1 ahi a--V- j=, gj(x, y) + sj = O, j E J, = o, j c J,
~j, sj >_O, j E J, xEX,yCY.
(s)
(cs) (cs)
Note that the resulting single problem (2) is nonlinear and nonconvex due to the stationarity (s) and complementarity conditions (cs). If the original bilevel problem is linear, the complementarity conditions are the only nonlinearities in the single level transformed problem.
395 For the convex form of (1), solution methods in the literature generally require the following conditions at fixed x [13,7,14]: (a) f , g, h are continuous and twice differentiable functions in (x, y); (b) the linear independence condition holds at y c Y, such that the gradients of the inner problem equality and active inequality constraints, Vxgj(x,y), Vj C Ja, Vxhi(x,y) Vi C I, are independent; (c) strict complementarity condition holds at y C Y; and (d) the second order sufficiency condition holds at y C Y. Under the assumptions (a)-(d) on the functions in (1), the inducible region, IR defined by the set {(x,y) : (x,y) E ~ , y E RR(x)}. is continuous [15]. Assumptions (b) and (d) assure that the global optimum is also unique. Further, the KKT conditions are necessary and sufficient. However, the resulting single level problem is nonconvex due to the complementarity and the stationarity conditions. Optimization methods for the general nonlinear BLPP include the relaxation and active set strategy techniques [7], which also assume the conditions (a)-(d). Note that the KKT conditions can no longer guarantee global optimality of the inner problem for fixed x. This means that, even if the transformed problem is solved by a global optimization approach, global optimality of the transformed single level problem can not be guaranteed. Hence, methods for the solution of the general nonlinear BLPP that are based on the KKT optimality conditions are bound to be local. The following section presents the main concepts that we have used in order to overcome the limitations of KKT-type methods. 3. C O N C E P T U A L F R A M E W O R K To assure that KKT optimality conditions are both necessary and sufficient for obtaining the global optimum of the inner problem, the functions f and g must be convex and h must be linear at fixed x. Condition 1: If for fixed x, assumptions (a)-(d) hold, f and g are convex and h are linear in y, then the KKT optimality conditions are necessary and sufficient for obtaining the global optimum of the inner problem. If condition I does not hold, then KKT conditions are only necessary. Thus, when the nonconvex inner problem is replaced with its KKT optimality conditions, and the resulting single level problem is solved to local optimality, an upper bound on the global optimum of the BLPP is obtained, provided that the linear independence condition holds. 3.1. Underestimation for the BLPP A lower bound to the global optimum of the BLPP can be found as follows: the feasible region, ~ defined by the set {(x,y): G(x,y) _< 0 , H ( x , y ) = 0,g(x,y) _< 0, h(x, y) -- 0}, can be enlarged in such a way that the infeasible points within the convex hull are included into the feasible set. This can be done by utilizing the basic principles of the deterministic global optimization algorithm ~BB [1-4], to underestimate the nonconvex functions over the (x, y) domain (see [ 16]). Based on the underestimation of every term, a convex underestimator for any given twicedifferentiable function can be obtained through a decomposition approach. For the nonlinear functions, valid underestimators are generated by the decomposition of each nonlinear function into a sum of terms belonging to one of several categories: linear, bilinear, trilinear, fractional, fractional trilinear, convex, univariate concave, product of univariate concave or general nonconvex. After the terms are identified, a different convex underestimator is constructed for each
396 class of term, and a lower bounding function is obtained. See [16] for rigorous calculation methods of underestimators. The equality constraints of the inner problem must be linear for the KKT conditions to be necessary and sufficient. The bilinear, trilinear, fractional and fractional trilinear terms are replaced by new variables that are defined by the introduction of additional convex inequality constraints. Thus, if the equality constraint involves only this kind of variables, the resulting problem is linear. If this is not the case, and convex, univariate concave, or general nonconvex terms exist, the constraint is simply eliminated by a transformation into two inequality constraints: h(x, y) _ ~ t ( y ( t ) , z ( t ) , p , a ( p , t ) , t ) ,
to < t < t f .
(6)
To simplify notation, constant model parameters p not affected by the optimization and switching functions q are not stated explicitly. Both, however, are considered in the implemented algorithm. Functions f and ~ are introduced accordingly. Without loss of generality the objective is stated in Mayer form. Dynamic parameter identification problems result in a nonlinear least squares objective function (weighted 12 norm of deviation from measurements). This specific form of q~ is exploited in the implementation.
435
3. NUMERICAL METHODS 3.1. Dynamic optimization Using a time grid to - t h < ... < tht -- tf, the infinite dimensional path inequality constraints (6) are transcribed into nt.n h inequality constraints
0 >_ h (y(th),z(th),~,a(p, th),th) ,
i-- 1,...,nt.
(7)
The applied commercially available sequential quadratic programming (SQP) methods [ 1], [4] make use of gradient information ~d0 and dh calculated from sensitivites of the solution w.r.t. /3, i.e.,
~ff E ]~ny• r ( t ) - i)y(t)
,
s ( t ) - -~z(t) ~
c
]~nzxnp.
(8)
3.2. DAE system integration and sensitivity calculation For numerical solution of the model equations a modified BDF method with variable stepsize and order is available for semi-explicit DAEs with index 2 [5]. The sensitivities (8) at time tn+l - tn + rln+l are given by the linear system
0-
( q----n-n~+llny ctk d- fy J~ ) ( )Fn+l + ( --(rn+lqt-q-~+~?n+l) t~k + f p + fa@ ) gy
gz
Sn+ l
(9)
~p --1-~aap
~ r
D
Fn+l
which is obtained by total differentiation of the discretized model equations w.r.t./3. and r are estimates of the sensitivity r and its time derivative r at time tn+l based on k previous time steps. The coefficient t~ depends on the order k of the BDF method. Matrix D is available in decomposed form at no costs from the BDF corrector iteration.
3.3. Consistent initialization Solving DAEs numerically with arbitrary initial values y(t0), z(to) will not lead to a continuous solution. Consistent initial conditions must satisfy the original DAEs (1), (2) and, in case of semi-explicit DAEs with index greater than 1, also higher total differentials w.r.t, time of some of the equations [5]. Under certain restrictions, steady-state conditions can be used as consistent initial conditions. They are defined by the nonlinear equation system
00-
f(y(to),z(to),~,a(~,to)) ~(y(to),z(to),p,a(~,to))
(10) (11)
which originates from (4) and (5). Steady-state conditions are assumed in [to - e, to + e], e > 0. To initiate a continuous transient solution starting from a steady state, controls a(/),t) must satisfy higher order continuity conditions. Differentiation of (10) and (11) yields a linear system of equations for the sensitivities of the initial values
fz)(r(to) fp +fat~p ) ^ s(to))+(~p+~a@ gz
436
Fig. 1. Flow diagram for an air separation plant. where all functions are evaluated at t = to. After discontinuities at time tev during integration, consistent values for y(tev+), Z(tev+) and the sensitivities r(tev+), S(tev+) must also be computed. Robust and efficient methods are employed in OPTISIM | for consistent initialization of index 2 DAEs and the sensitivity equations. 4. A P P L I C A T I O N S The optimization algorithm implemented in OPTISIM | is applied to optimal load-change control of an air separation plant (ASP) and the dynamic identification of heat transfer coefficients in a batch filling process.
4.1. Load change policy for an air separation plant An ASP consists of three major parts: Feed air preparation, cooling and rectification. A standard configuration as shown in Fig. 1, comprises a high pressure column T 1, where the feed air is crudely separated into two fractions, one of which is the liquid nitrogen product (DLIN), a low pressure column T2, where highly pure nitrogen (GAN) and oxygen (GOX, LOX) are produced, and an argon column T3 with crude argon product (crude Ar). All process steps are tightly coupled through material and energy streams. The task is to decrease the load of the plant from 100 % air input to 60 %. The load change takes about one hour, the time of operation is from to = 0 [s] to tf = 6000 [s]. It is of utmost importance for stable operation and product quality that several purity restrictions are not violated during the load change. The air separation plant is modeled by a semi-explicit DAE system with index 2 consisting of about ny = 900 differential and nz = 2600 algebraic equations. The purity restrictions result in lower and upper bounds Xi,min _~ xi(t) ~_ X/,max for six state variables (cf. Tab. 1), i.e., nh = 12 in Eqn. (7). Five constraints refer to product quality. The sixth constraint is a stable operation constraint. The nu = 5 control variables describe the positions of valves a - e. Instead of a full parameterization of the control history, e.g., by piecewise polynomial functions, the optimization of already implemented control schemes which use a global parameterization
437
Table 1 Lower and upper bounds of purity constraints. description name min max oxygen fraction in liquid oxygen product 0 2 LOX 0.997 1.0 oxygen fraction in gaseous oxygen product 0 2 GOX 0.997 1.0 0 2 DLIN 0.0 5.0.10 - 6 oxygen fraction in liquid nitrogen product 0 2 GAN 0.0 5.0.10 -6 oxygen fraction in gaseous nitrogen product argon fraction in argon product Ar crude Ar 0.965 1.0 oxygen fraction in feed to argon column 0 2 ArFeed 0.90 1.0
Fig. 2. Purities and air flow for original parameter setting (values are scaled to lower and upper bounds from Tab. 1).
Fig. 3. Purities and air flow for optimized parameter setting. For legend see Fig. 2.
of u - t~(/~,t) is investigated. The controls are parameterized by np = 9 parameters. The state variable inequality constraints are discretized with a time grid of n t - - 10 equidistant points yielding n t . n h - - 120 nonlinear inequality constraints of the nonlinear programming problem (NLP). The objective is to maximize an integral term describing product gain. The Mayer form (3) is obtained by adding an additional differential equation. However, the operators are in the first place interested in finding a feasible control within the constraints for this highly complex plant. The starting values for the optimization parameters/~, for which the time histories of the relevant purities (Tab. 1) are displayed in Fig. 2, lead to a breakdown of the ASP, caused by several variables violating their bounds. A solution computed with optimized parameters is displayed in Fig. 3. All purities are now feasible within their lower and upper bounds which is most important. 4.2. Parameter identification Industrial gases, e.g., Oxygen, Nitrogen or Argon, are distributed at retail in high pressure gas bottles. At the filling station bottles are pressurized from vacuum to 200 bar with gas from a high pressure gas tank. In order to accelerate the filling process the heat balance of the bottle system must be investigated. To determine the heat flows in the system, coefficients for gas-bottle heat transfer need to be known. As no model for prediction of heat transfer coefficients in a fast pressurized gas volume is available from the literature, heat transfer coefficients must be determined from mea-
438
Fig. 4. Non-insulated bottling process, fitted and measured data for bottie (T_Bottle, T_BM) and piping (T_Pipe, T_PM) temperature.
Fig. 5. Insulated bottling process, simulated and measured data for bottle (T_Bottle, T_BM) and piping (T_Pipe, T_PM) temperature.
surements. The newly implemented algorithms are applied to a model tuning problem where constant heat transfer coefficients are identified based on data from experiments. Dynamic data reconciliation on measurements will be the scope of future work. A single depressurized gas bottle is filled from a bottle battery. The valve is opened at time zero and closed after 120 seconds when pressure equilibrium has been achieved. Temperatures at the filling pipe and at the outside of the single gas bottle are measured. Based on measurements from an experiment with a non-insulated bottle, heat transfer coefficients within the bottle are determined by dynamic parameter identification. The optimized results (solid lines) and the 8 measurements (., A) entering the optimization are shown in Fig. 4. YS_V01 denotes the valve position. The simulation model for the bottle battery, the single gas bottle, piping and valves has nx = 144 equations and an index of 1. Three heat transfer coefficients are fitted. The value of the objective function is reduced by 17%. A comparison of measurements at 9 points in time from an experiment with an insulated bottle with simulation results using the heat transfer coefficients determined above show the applicability of the fitted parameters to a related case (Fig. 5). REFERENCES
1. NAG Fortran Library Mark 18. The Numerical Algorithms Group Ltd., Oxford (1997). 2. Engl, G.; Kr/Sner, A.; Kronseder, T.; von Stryk, O.: Numerical simulation and optimal control of air separation plants. In: Bungartz et al. (eds.): Lecture Notes in Computational Science and Engineering, Springer, Berlin (1999) 221-231. 3. Eich-Soellner, E.; Lory, P.; Burr, P.; Kr/Sner, A.: Surv. Math. Ind. 7, 1 (1997) 1-28. 4. Gill, P.E.; Murray, W.; Saunders, M.A.: SNOPT: A SQP Algorithm for Large-Scale Constrained Optimization. Report NA-97-2. Dept. of Mathematics, University of California, San Diego, California (1997). 5. Brenan, K.E.; Campbell, S.L.; Petzold, L.R.: The Numerical Solution of lnitial- Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia (1996).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. J~rgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
439
TRIZ-Based Creative Retrofitting of Complex Distillation Processes --An Industrial Case Study Xiao-Ning Li, Ben-Guang Rong and Andrzej Kraslawski* Department of Chemical Technology, Lappeenranta University of Technology, P.O. Box 20, FIN-53851, Lappeenranta, Finland. *E-mail:
[email protected] An integrated approach is presented for the retrofitting of complex distillation processes. It is founded on the combination of TRIZ-based (Theory of Solving Inventive Problems) creativity support method and thermodynamic analysis. A three-step hierarchical algorithm is formulated. It has been applied to an industrial case - retrofitting of butadiene extractive distillation plant. Two optimal flowsheets are obtained by using of the proposed method. 1. INTRODUCTION The essence of process retrofit is to improve the performance of the existing plant. Process retrofit can be even more difficult than grassroot design, because of the constraints resulting from the use of the existing equipment and the combinatorial complexity of redesign. The multi-objective character of process retrofit is an additional difficulty. Traditionally, process retrofit problems are solved based on grassroot design methods like heuristics, evolutionary methods, thermodynamic analysis and mathematical programming. In order to better handle the process retrofit problems, recent research was focused on multi-objective analysis (Dantus et al., 1996) and combined approach like the thermodynamic and algorithmic one (Kovac et al., 1995). However, the aspect of innovation in process retrofit problems, manifested by handling the contradictions and the conflicts among several objectives and technical constraints, is not tackled by the existing methods. In this paper, an integrated approach combining TRIZ-based theory and thermodynamic analysis is proposed for creativity support of the process retrofitting problem solving. It is illustrated by an industrial complex distillation process - the butadiene extractive distillation plant. 2. M E T H O D O L O G Y The main idea of TRIZ method (Altshuller, 1998) is to remove the contradictions identified when solving the inventive problems. TRIZ has been based on the extensive studies of patent information. As a result, there have been extracted 39 universal characteristics identifying any technical system and 40 principles for the conflict resolution. The principles used for inventive problem solving are generic suggestions. The characteristics and principles built a contradiction matrix. The simultaneous improvement of some characteristics usually causes the deterioration of the others. It is the source of conflicts. The basic idea of TRIZ-based
440 method is to identify the emerging conflicts. The conflicts generated by any technical system could be overcome by use of the contradiction matrix. Many contradictions and conflicts are generated by trading-off the retrofitting activities and the existing constraints. The TRIZ-based approach is adopted in order to systematically identify retrofit targets and remove process bottlenecks. All the characteristics of traditional distillation systems and the principles that govern the decision-making in process design are extracted to formulate the contradiction matrix. The bottlenecks of process retrofitting can be identified and represented by the contradictions. Next, the contradictions could be removed by examining all the available principles. When the identified conflicts are solved then the nearly optimal alternatives are generated. Finally, thermodynamic analysis is implemented for the further improvement of the generated alternatives. The integrated approach, combining the TRIZ-based creativity support method and thermodynamic analysis, can handle very well the multi-objective aspects of the retrofitting process. Moreover it extends the solution space from traditional distillation schemes to new, non-traditional ones. A three-step hierarchical algorithm is formulated for applying the methodology including analysis, search and implementation phases (Fig. 1).
Fig. 1.The algorithm of problem solving using the integrated approach 3. CASE STUDY An industrial process of butadiene extractive distillation is studied with the main objectives of the improvement of the environmental impact, energy efficiency and total plant cost. 3.1. Flowsheet
The feed - Crude C4, composed of seventeen hydrocarbons, is difficult to be separated by common distillation system. The process is realised in the system (Fig. 2), composed of six traditional distillation columns: 1st stripper (DA- 102); butadiene withdrawing column (DA-104); 2 nd stripper (DA- 105); 1st common distillation column (DA- 106); 2nd common distillation column (DA-107); solvent refining column (DA-108) and two extractive distillation columns (DA101 and DA-103).
Fig.2. The flowsheet of the industrial case
441 3.2. Retrofit targets
The multi-objective analysis requires retrofit activities to be carried out in an integrated way. Not only economic criteria but also environmental sustainability, flexibility, operability and dynamics have to be considered simultaneously. Due to the complexity of the units and the intensive energy consumption, in the case under discussion, the retrofit targets are focused on reducing capital costs, optimising energy consumption and improving sustainability. TRIZ contradiction matrix for distillation processes is composed of 30 characteristics and 29 general and specific principles (Rong et al., 2000). To deal with the retrofit targets, several characteristics are chosen in the matrix: capital cost (No.24), operating cost (No.25), complexity of the system (No.23), environmental impact (No.26), and flexibility (No.28). 3.3. Generation of process retrofit alternatives
The general conflicts can be formulated by analysing and matching the above-mentioned characteristics. Four main general conflicts have been identified. Next, the conflicts are removed by the appropriate principles identified thanks to the contradiction matrix (Table 1). Table 1 The general conflicts and the suggested principles Conflicts Cells Suggested principles 1-change method, 3-change agents environmental aspect/ 26• operating cost 29-balance computer & human role, 5-mass integration 9-mutifuncional units, 4-heat integration operating cost/capital cost 25• 15-side-stream column, 7-change reflux ratio 22-decomposition, 24-simplification operating cost/flexibility 25• 23- complex design, 25-analysing design process 22-decompostion, 23-complex design complexity/flexibility 23 • 24-simplification From the suggested principles, a few most useful ones are chosen, based on the heuristic rules for the retrofitting problems (Grossmann et al., 1987). Principle 3 (change agents) and principle 4 (heat integration) are preferred for the improvement of the operational performance of the existing flowsheet. Then strategies of decomposition and process integration implied by principles 22 (decomposition) and 23 (complex design) are applied in the search for the promising flowsheet structure. A hierarchical decomposition method is formulated by applying those principles, where the heat integration problem is solved in an inner synthesis step while the process flowsheet is optimised in an outer step. When applying those principles to remove the general contradictions, twelve further contradictions are formulated at the subsequent levels. 3.3.1. Solvent evaluation
As principle 3 suggested, DMF (dimethylformamide) has been used as the solvent in the extractive distillation process. It has been compared with other solvents like ACN (acetonitrile) and NMP (N-methylpyrrolidone). The excellent solubility and selectivity among butadiene and other C4 hydrocarbons have suggested that the process operation and control would not be so difficult task. Furthermore, its aqueous solution is not corrosive and toxic. Therefore DMF is the preferred solvent from the environmental point of view.
442
3.3.2. Heat exchanger network Heat integration is aimed at generating a network that will operate at minimum cost thanks to maximum recovery of the available energy and optimal use of the existing equipment. When realising heat integration, six conflicts have emerged involving the following characteristics: flexibility, operating cost, capital cost, heat exchanger number, reboiler number, temperatures and pressures. Then the appropriate principles used to cope with these conflicts are identified in the conflict matrix (Table 2). The recycled solvent, by merging the bottom flows of two strippers, is the main heat source due to the high temperature and flow rate. Thus, the most important issue for heat matching is to maximise the heat recovery from the solvent flow. All heat sources and heat sinks of this process are shown in Table 3. Table 3 The heat sources and sinks in the existing flowsheet No. Heat sources and sinks H1 H2 H3 H4 H5 H6 H7 C1 C2 C3 C4 C5
evaporator of C4 feed 1st reboiler of 1st extractive distill, column 2ndreboiler of 1st extractive distill, column 1st reboiler of 2ndextractive distill, column reboiler of 1st common distill, column 1st reboiler of 2ndcommon distill, column 2"d reboiler of 2nd common distill, column solvent flow from the two strippers bottom flow of 2ndextractive distill, column 1st condenser of 1st stripper condenser of 2nd stripper condenser of solvent refining column
Heat load (106kcal/h) 0.929 1.904 1.385 0.483 1.183 2.4712 0.7156 7.04 0.483 0.7343 0.3511 0.1256
Initial temp. (~ 42
Final temp. (~ 50.5
78
119
70.18 48.93
80.15 49.15
58.08
61.31
162.15 144 119 156.77 152.06
40 85 85 42.35 85
Principle 17 (heat matching among columns) is adopted first for effective use of existing energy sources. Principle 6 (change pressure and temperature) and principle 7 (change reflux ratio) suggest the further considerations for adjusting operational parameters. Temperatureenthalpy (T-H) diagram is the tool for analysing and visualising the energy flows among the concerned heat and cold streams. The T-H diagram of solvent flow and its matching flows suggest that the minimum approach temperature ATmin is too high around 27~ Through decreasing ATmin to 12~ the more efficient heat matching strategy is proposed by additional use of the existing heat sources including the condensers of the 2 nd stripper and the solvent refining column. Table 2 The conflicts and the principles for heat exchanger network Conflicts Cells Principles exchanger no./reboiler no. 16• 15 20,17,19,16 reboiler no./capital cost 15• 8,9,7 reboiler no./temperature 15• 14,6 reboiler no./pressure 15• 10 14,6 operating cost/capital cost 25• 20,17,9 flexibility/exchanger no. 28• 16 4,6,11 .
.
.
.
Table 4 The conflicts and the principles integration Conflicts Cells energy flow/complexity 11 • complexity/capital cost 23• capital cost/operation cost 23• complexity/temperature 23• complexity/pressure 23• 10 flexibility/complexity 28•
for process Principles
22,23,24,28 9,15,13 4,7,16-20 3,1,2,6 1,3,2,6
22,24,23,25
443 3.3.3. Process integration In this case, the work is focused on improving the topological structure and on considering the complex columns. For this purpose, the subsequent conflicts are identified and removed by the appropriate principles via contradiction table (Table 4). As principle 25 (analysing design process) and principle 22 (decomposition) suggested, for the sake of systematic analysis, the whole process is divided into the following four modules: extractive part (the 1st and 2 nd extractive distillation columns); solvent purification part (two strippers and the butadiene withdrawing column); product purification part (two common distillation columns); solvent refining part (the solvent refining column). In the second part, it has been observed that the butadiene withdrawing column and the 2 nd stripper have the similar composition, temperature and pressure distributions based on simulation results. A complex, thermally coupled distillation column is proposed to combine two above-mentioned columns as suggested by principle 23 (complex design) and principle 20 (thermally coupled schemes). The temperature and pressure of the complex column should be changed correspondingly as implied by principle 6. This modification can not only save the number of the heat reboilers but also decrease the energy consumption. The thermodynamic approach is used to explore the potential bottlenecks of the third part by studying the Column Grand Composite Curve (CGCC). 3.4. Thermodynamic analysis and evaluation of the retrofit alternatives Thermodynamic analysis of the CGCC can provide the required information for potential column modifications. Fig. 3 shows the profile of the CGCC of the 2 nd distillation column. It is clearly seen that there is a sharp enthalpy change near the feed location, which suggests that the feed state should be changed from liquid phase to vapour one. The strategy can be implemented through changing the sequence of two common distillation columns, then the top vapour flow of the 2 nd extractive distillation column directly enters into the 2 nd c o m m o n distillation column without condensation. The heavy impurities are removed away first, then the light ones are removed in the 1st common distillation column. The modification produces another thermally coupled column and eliminates the condenser of the 2 nd extractive distillation column. The new CGCC suggests that the sharp enthalpy change is improved and the load ofreboiler is greatly reduced by 38.5% (Fig. 3). water
of condensed steam
O
heat load of reboiler (existing) D
58 L) o%,
h
,
54
C4=,
i
:ii
i
C4= . . . . .
'
i'
i
c--~
50
......~
"'!
D~
4
46
"1 426e6-2e6--
2 Zie6--
2 8e6--
3 2e6
- 3 6e6
H (kcal/h)
Fig. 3.The modification of CGCC of the 2 nd common distill, column
!
"
i O
*
...............................i...................................
DMF i ! ............................................. :.............................
i
......................................................................................... Fig. 4.The proposed optimal flowsheet
444 When getting the optimal retrofit flowsheets, heat integration is carried out for efficient use of energy in the proposed alternative flowsheets. Two optimal flowsheets have been obtained based on the different heat matching strategies. Fig.4 presents one alternative with the new structure and the heat matching strategy. Simulation results show that the saving of the consumption of steam utility for those two optimal flowsheets are of 23.7% and 27.7% respectively compared with the existing plant. Based on the above procedure, a tree structure pattern including three levels of contradictions is formulated to illustrate the TRIZ-based process retrofit as shown in Fig.5.
Extractive Distillation --~ . . . . . . . . . . Plant . . . . . . . . . . . .
~9 . . . . . . . . . . . . . . . . . . . .
environmentI loper, cost flexibility ] ]complexity oper. cost Ilcapitalcost] I ~176 flexibility [ selection
2- oc s inte.~ation
2 (DMF)
intez~ration
116xl lSx ?A
I I
9- 128x161
h~at matchTmg
thermally co led distill, column thermally coupled column instead of the 2"a strip_p_erand butadiene withdrawing column
3
g
CGCC of thermodynamicanalysis
4, changing sequence of two common distill columns
~' feed stat e changmg of 2"~ column
c~conflicts -~principles 1,2,3hierarchical steps Fig. 5.The tree-structure pattern of the TRIZ based process retrofit
4. CONCLUSIONS An integrated approach combining TRIZ idea and thermodynamic analysis is proposed for process retrofitting which is realised by a three-step hierarchical algorithm. TRIZ method is used to generate the promising alternatives by identifying the corresponding principles in order to remove the recognised contradictions and conflicts. The identified principles are used as the tools in search for the optimal flowsheets. Thermodynamic analysis could be applied to the search of potential retrofitting targets and possible solutions for further improvement. The presented approach and algorithm are applied for the retrofitting of the butadiene extractive distillation plant. A hierarchical tree-structure procedure is formulated to illustrate the integrated approach. Two optimal retrofitting flowsheets are obtained that allowed to decrease the consumption of steam utility 23.7% and 27.7% respectively. REFERENCES 1. M.M. Dantus and K. A. High, lnd. Eng. Chem. Res., 35, 4566-4578, 1996. 2. A. Kovac and P. Glavic, Comput. Chem. Engng., 12, 1255-1270, 1995. 3. G. Altshuller, 40 principle: TRIZ keys to technical innovation, Technical Innovation Center, Inc. MA, USA, 1998. 4. B. G. Rong, A. Kraslawski, L. Nystr6m, Computer-Aided Chemical Engineering, 8, 625630, Elsevier Science, 2000. 5. I.E. Grossmann, A. W. Westerberg and L. T. Biegler, Proc o f I st Int. Conf. on Found. o f Comp. Aided Proc. Oper., 403-422, Park City, Utah, July 1987. 6. X. N. Li, Study on Process Integration of Butadiene Extractive Distillation Plant, M.Sc.Thesis, Qingdao Institute of Chemical Technology, China, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
445
Retrofit of Refinery Hydrogen Systems F. Liu and N. Hallale Department of Process Integration, University of Manchester Institute of Science and Technology, PO Box 88, Manchester, M60 1QD, United Kingdom Several trends in the petroleum industry are leading to increased demand for hydrogen in oil refineries. Previous work developed a methodology for the analysis of hydrogen distribution systems. An automated approach has now been developed to include pressure issues in the design. The method is based on optimisation of a reducible superstructure. Retrofit options are decided automatically through optimisation. 1. INTRODUCTION Hydrogen is vital for oil refiners to face the trends caused by the tighter environmental requirements and heavy-end upgrading. Also a capacity increase or change in product slate of an existing refinery is often constrained by the hydrogen availability. Reducing the sulphur composition in fuel means more hydrogen is necessary for deeper hydrodesulphurisation. More hydrotreating is needed to achieve high-cetane diesel. In the meantime, lower aromatic gasoline specification will decrease the operation severity in catalytic reformers, reducing the by-product hydrogen formed. Another increase in hydrogen demand is caused by bottom-of-the-barrel upgrade. Because of stricter limitations on pollutant emissions, there has been a sharp reduction in the fuel oil market. On the other hand, market trends indicate a very large increase in diesel oil and jet fuels production. Hydrocracking can play an important role in heavy-end conversion because of its considerable flexibility and high quality of the products, but requires large amounts of hydrogen. Alves (1999) proposed a pinch-based approach for targeting the minimum hydrogen utility. A general simplified model of hydrogen consumers can identify the sinks and the sources in hydrogen systems. The flowrate and purity for sinks and sources in the network can be extracted from operating data. These can be plotted as composite curves, which will then give targets for the minimum hydrogen utility. The method is similar to energy targeting in heat exchanger networks (Linnhoff, 1993). To achieve the target from hydrogen pinch, the network can be designed by using Linear Programming. The objective function is to minimise hydrogen utility. However, the hydrogen targets from the surplus diagram may be too optimistic as they do not consider pressure. In reality, a source can only feed a sink if its pressure is sufficient. This can be accounted for by assuming new compressor installation, which however can lead to impractical design and unrealistic compression costs. In this work, we outline an automated approach for retrofit of hydrogen systems, with attention to pressure and compressor constraints.
446 2. NEW A P P R O A C H - AUTOMATED DESIGN 2.1 Network superstructure After the decomposition of hydrogen systems, we can build a superstructure including all the links between sources and sinks. A simple example in Figure 1 demonstrates the decomposition procedure. The inlet pressure of the sinks and the outlet pressure of the source are assumed fixed at their design values.
The basic mass balance constraints have been applied in equation (1) to (6). The existing compressors are decomposed into sinks and sources. Unlike the sinks and sources of the hydrogen consumers, both the flowrate and the purity of compressor sources are variables, which leads to non-linear items in the hydrogen balance equations (4) and (5). There are maximum flowrate constraints imposed on the existing compressors in equation (6). These will be equal to the design flowrate plus any spare capacity that may exist. Banning the matches between sources in lower-pressure levels and sinks in higher-pressure levels reduces the size of the superstructure and ensures that the pressure limitations are included. We can also impose practical constraints e.g. forbid links between processes for reasons of plant geography or contamination.
~., Fs~,.j = Fs,.* ,
VEr
(1)
J
~_F~,sk = Fsk*,
VSk
(2
!
(3) i
Z
-Z
j
i
VComp
(4
VComp .j
~ Fcompj < FMo~i.... Comp"
V Comp
J
Hypd/an~t~en.
(5)
i
1
1 ~i~[
,
I Fuel
I
I1 psi~
(6)
Ysr F. YSkI 200psi~ S i n k A 1600psi 1500psi~ SinkB 2200psi 1700psi~ ~ ' ~ ~ 1 6 0 0 p s i ~
200psi
2200psi~
200psi
1 6 0 0 p s i ~
1500psi
2200ps, [ ~ i (a)
~
1700psi
(b)
Figure 1. (a) A simple existing hydrogen network (b) Superstructure linking sources and sinks
447 2.2 Retrofit options and costs The insights show that it is possible to save hydrogen by piping changes, extra compression to overcome pressure difference and purification. The new compressor model can be built as in equation (4) and (5) by considering no maximum flowrate limit (other than manufacturers limitations) and variable operating pressures. The inlet pressure must be the lowest pressure over all the sources feeding the unit. The power of new compressor is decided by equation (7), which could be simplified to a linear equation once the operating pressures are fixed.
PcoMP= Y'T'R'N(Y--A~ (y--(;~] rr'N-1 I FC~
(7
A purifier can be a pressure swing adsorption (PSA) unit, membrane or cryogenic process. This work assumes that a PSA is used. With specified recovery (R), and product purity (yp), the PSA can be modelled by mass balance equations as Figure 2. The inlet pressure of PSA (Pin) must be the lowest pressure over all the sources feeding the unit. The pressure drop (AP) over a PSA is low and so the product pressure is usually very close to the feed pressure. The residue is released at very low pressure. Other purifiers will have different pressure characteristics.
.
Fi, purifier, P,
I
Feed
~ikF F
feed flowrate__~J
/'yF feed purity ~ ~P0n=min(P,)
I
] L~
Product Product flowrate F. = R. F'F-y_~ Product purity yp YP
I
]
p
I
-p
Product-- in-
I Residue Residueflowrate FR=FF_Fp "-
Residue purity y. =
~
FF-yF - F yP'..~ 9 .v ~ ~k
Figure 2 PSA model To find the optimum retrofit design automatically, the objective function has been set as minimum cost (operating and capital). The retrofit options will be selected systematically by considering the cost trade-off. The investment costs of retrofit o p t i o n s - compressors, purifiers and piping changes can be described as linear equations (8) to (10). The coefficients can depend on operating conditions (e.g., pressure and temperature) in different parts of the system. I,,,~ = (a,,,E + bp,,~" A). L (8 I p,,,= a,,,, + b,,,, F,,,,
(9
i~,o =aco~+bc,o .Pco~
(10)
The operating costs of hydrogen systems are dominated by hydrogen utility cost, compression cost and exported fuel value. The automatic retrofit design requires Mixed-Integer Non-linear Programming. The initialisation from the relaxed solution by Mixed-Integer Linear Programming (Quesada and Grossmann, 1995) can improve robustness and help to locate good optima.
448
235~ I co. I
145oo
44.35 92.00% 'k_/~
j~
75.00%
H= Plant 192.00% 0.65 5.57
~ 8.78 I.cu I
,, _., 10.66 ~
IJ
"l"
CRU catalytic reformer HCU hydrocracker
11.31 J
75.97%~
,41
!
11.29 I 75.00% ~r
3.47 J 75.00% I
4.32 / 65"000/1
I
3 47 "
*
KHT
Flows in MMscfd
,[, ~]' 12.(
kerosene hydrotreater
CNHT cracked naphta hydrotreater NHT naphta hydrotreater
12.80
NHT
6.5~ 60.00%
DHT diesel hydrotreater
HDA hydrodealkylation I Fuel
Fuel fuel gas system
Figure 3. Case study - existing hydrogen system
3. C A S E S T U D Y
An existing hydrogen system in a refinery is shown in Figure 3. Currently, 45 MMscfd of hydrogen are produced from the hydrogen plant. The objective of retrofit design is set as maximum operating cost saving. The utility prices are: hydrogen utility $2.0/MMscfd, electricity $0.03/kWh and fuel cost $2.5/MMBtu. This case study assumes that all the existing compressors have 5% spare capacity, and that a maximum of one new compressor and one new PSA (produces 99% hydrogen, recovery = 90%) will be allowed. A payback of less than two years is specified.
I H Plant l 92.00% 28.61
J 23.5
J CCR 175.00% .................................... 2:.72........
~aa >j
~31.33
""~T~, a7.1s
I.cu I -;o; ......:r
=~. ~,.,,.,o /,o
11.14
.,.o,.,
"?'~:i
_ ......
0.35j
0 04
.
.
.
.
.
.
.
.
.
~
.
.
.
.
.
!o.a5
.
.
.
.
.
.
.
.
.
.
.
.
.
75____=..-____." 00% :,,~ ....... 9 - 14 70 00% .... _::::_.-__::_:_.
i
18--18"~_..3..~8....i 2.69
6678~
0.04
*,11.53
.
.
.
.
.
.......
1~ 03 ....
-~
~=^o I N 6 37
J
,TI
99.00% = " -Ir~-; 1 } ~)U.UU70 I . ........................... ir-o~ i j E - - , ................... : 7.28-- r . . . . L_/ New I Fuel 17.33%~ Compressor
Figure 4. Case study - optimum retrofit design
449 Table 1. Cost breakdown Existing network Operating costs Hydrogen Compression Fuel export Total Investment costs Compressor PSA Piping change Total
Retrofit design $20.9 million/yr $1.87 million/yr -$6.24 million/yr $16.5 million/yr
$32.9 million/yr $1.77 million/yr -$12.2 million/yr $22.5 million/yr
$1.00 million $ 7.02 million $1.78 million $ 9.8 million
The optimum retrofit design is shown in Figure 4. To minimise the operating cost, both a new compressor and a PSA are used, as well as introducing some new pipes. The dotted lines show the piping changes and the new units. The new compressor is used to accommodate the increased recycle requirement for the NHT, as well as to compress a feed to the PSA. The cost breakdown is shown in Table 1. The possible saving of operating cost after retrofit is 6 million US$, and the payback period is 1.6 year. Often, refineries have a limited capital budget for projects. The hydrogen system shown in Figure 2 is analysed subject to the budget constraint less than 5 million US$. The possible saving of operating cost after retrofit is 3.5 million US$ which is lower because of the smaller capital investment and the payback period is 1.4 year. The retrofit design of the system (Figure 5) shows that if the capital budget for retrofit project is limited to 5 million US$, the best investment is in a PSA and not a compressor.
35.40 0.19 H2Plant J 92.00% ~2-------]~_~.-
....'~'~
I HOU[~
.o,~.9. .......
,~
[ JHT I
IONHTI
"
'
I
7500%
i
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i-*-1
11.69
0.04
J
154
"~
IDHTI
J7000% 1.b~....... ,,,,.~ .......... 6.14
_..1.0.:17 ,.
99.00% .............................................. ]PSAI 3.70--[" 19.29% { .
...............
'k_,/" 11.88 / I
....... ~_ ............
75-00%~8.65 65"00~
_--T-.-;-.--~
5.461I
37.0338.04
I CCR I75.00% 23.5
'4' ] j NHT
. . . . . . . . . . . . . . . . . .
60.00%Jr6.56
3.81
i ~Fuel Figure 5. Case study - retrofit design with limited capital budget
450 4. CONCLUSIONS Efficient hydrogen integration in refineries is very important. Previous work developed a methodology for the analysis of hydrogen distribution systems. Targets are set for hydrogen recovery and hydrogen plant production. However, the approach has neglected pressure constraints leading to an overly optimistic solution. An automated approach has now been developed to include pressure issues in the design. The method is based on optimisation of a reducible superstructure. The objective function of optimisation is set as minimum costs to consider hydrogen saving and penalty simultaneously. Multiple practical constraints can be implemented to achieve economic, realistic designs. Retrofit options (for example, additional compression, purification and piping changes) are decided automatically through optimisation. The case study demonstrates the potential of the method for cost savings with specified payback times and/or investment limits. It can also be applied to debottlenecking. NOTATION F F* ly I a,b PCOMe FpUR
flowrate flowrate as required or offered purity investment costs cost coefficients compression power feed flowrate of purifier
A L T N r/ r 7
cross-sectional area of pipe pipe length inlet temperature number of stages isentropic efficiency pressure ratio ratio of heat capacities
Sub-script i j Sr Sk Comp COMP P UR PIPE
source sink process source process sink compressor source/sink compressor purifier piping
REFERENCES 1. Alves, J. Analysis and Design of Refinery Hydrogen Distribution Systems, PhD thesis, UMIST, 1999 2. Linnhoff, B. Pinch Analysis - A State-of-the-art Overview, Transactions of IChemE (Part A), 71 (5), 1993 3. Quesada, I. and Grossmann, I. E. Global Optimisation of Bilinear Process Networks with Multicomponent Flows, Computers & Chemical Engineering, 19 (12), 1995
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
451
C o m p u t e r - Aided Synthesis of Molecular Mixtures and Process Streams E. C. Marcoulaki a*, A. C. Kokossis b and F. A. Batzias a aDepartment of Industrial Management, University of Piraeus, Karaoli & Dimitriou 80, Piraeus 185 34, Greece bDepartment of Chemical and Process Engineering, University of Surrey, Guildford GU2 7XH, United Kingdom This work considers the computer-aided design of novel solvents to facilitate difficult separation processes. These materials can be pure or within mixtures of substances that decrease the affinity between the components of the process feed. The task is formulated as an optimization problem to generate superior alternatives according to a set of objectives and constraints. The optimization process takes the form of a stochastic search among different molecular configurations and their contributions in the screened solvent blends. The search is combined with available group contribution methods (e.g. UNIFAC models) to predict the thermodynamic and environmental behavior of the designed materials. The method is employed to suggest novel blends of entrainers to replace extraction solvents commonly used in the industry. 1. INTRODUCTION Computer-aided molecular design applications assume an increasing significance over the last years. The advances on computer systems have enabled the development of powerful tools for property prediction by means of molecular and dynamic simulation. Less accurate, group-contribution (GC) methods are frequently the only available choice, when there is lack of experimental data or time. These technologies predict the thermodynamic properties of prespecified compounds, by establishing a structure-to-property link. Molecular design synthesis (MDS) tools reverse this information flow and formulate a mathematical problem for the development of molecular structures with desired behavior. This work considers the design of solvents to enhance the separation of materials in processes like 11- extraction, extractive distillation, absorption, etc. The first optimization tools for entrainer design were developed by Machietto et al. (1991) who presented a generalpurpose tool, assuming a continuous representation of the functional groups in the solution. The continuous representation enabled the formulation of the problem as an NLP, resulting though in poor interpretation of the results. Recently, Pistikopoulos & Stefanis (1998) formulated an M1NLP problem to consider the synthesis of non-toxic substituents for commonly used solvents. The technology was extended (Buxton et al., 1999) to the selection of environmentally benign blends. Hostrup et al. (1999) presented a hybrid scheme of mathematical programming assisted by heuristics to reduce the problem size. Author to whomcorrespondenceshould be addressed
452 Apparently, there are no safeguards to prevent the M1NLP technology from yielding inferior results, even one or two orders of magnitude away from the optimum. The use of smaller groups, constraints and bounds may enable applications, but, at the expense of the novelty and richness of the problem. Marcoulaki & Kokossis (1998) proposed a stochastic design tool, and reported novel molecular structures and significant improvements over other techniques. The authors (2000a) presented a generic representation of the molecular synthesis problem, along with general and systematic search strategies. Applications included large synthesis problems of solvent design using UNIFAC (2000b). The examples launched a conceptual search based on property predictions, or considered economic objectives using process simulation. The methodology was compared to conventional tools and was found powerful enough to encourage applications on even more complicated systems. Marcoulaki et al. (2000) applied the technology to design pollution-preventing alternatives of traditional chemicals for extractive distillation and liquid-liquid extraction. The approach used here adopts the stochastic technology, and extends the representation and search options to address molecular blends. 2. METHODOLOGY The effort here is to identify promising molecular configurations within a blend that satisfies certain process objectives under a set of constraints on the properties. The search strategy follows the principles of Marcoulaki & Kokossis (1999, 2000a) in order to (i) develop targets for the performance of the solvents, thus justify and assess incentives for replacing materials commonly used in the industry, and (ii) present venues to achieve mixtures of novel chemical compounds, thus explain the development of designs that are close to the targets. The search can be adjustable to preferences and arbitrary synthesis objectives and constraints. There is no need to limit the number of groups or components, nor to introduce artificial constraints just to make the optimization applicable. The variety of synthesis alternatives can only be considered using a large number of discrete and continuous variables that have to be optimized. The MDS problem is solved using stochastic optimization in the form of Simulated Annealing.
State Representation In the synthesis framework, each design instance defines a problem state. The design instance is hereby a solvent blend made up of various molecules in certain concentrations. The molecules are represented in terms of series of molecular and composition vectors of functional groups, as defined by Marcoulaki & Kokossis (2000a) for pure compounds. The optimization variables refer to the existence and the number of occurrences of each group in the molecular vectors. Additional variables account for the concentrations in the resulting component mixture. The optimization variables are subject to physical constraints on the feasibility of the representation vectors, including 9 connectivity features, so that the configurations have zero valence, the aromatic rings consist of six carbons, etc. 9 necessary features, so that the group collections conform with the representation and the property prediction models employed 9 desirable features, e.g. to specify limits on the maximum occurrences of certain groups.
453
State Perturbations A set of m o v e s is defined to generate alternatives and monitor the synthesis search. The moves are such to facilitate efficient perturbations from one feasible state to another. With respect to the state representation, the perturbations are divided into 9 b l e n d concentration moves (to alter the concentrations of the solvent constituents), and 9 m o l e c u l a r composition moves (to alter the group configurations in the constituents) The strategy adopted here enforces feasibility without resorting to trial-and-error, and consists of a p r e p r o c e s s i n g stage, a move selection stage and a p o s t p r o c e s s i n g stage. The last stage consists of minor restoration actions so that the molecules and their blends obey the feasibility constraints discussed. The preprocessing is used to set up the domains for the different possible modifications that can feasibly be applied on the molecular vectors. This stage ensures that the infeasible intermediates created during the moves can easily be restored to yield feasible new states. The move set accounts for the gross actions of 9 substitution, where an existing group is replaced by a different group, 9 contraction, where an existing group is removed from the configuration, and 9 expansion, where a new group is introduced, applied on each molecular vector. These perturbations are based on the moves developed by Marcoulaki & Kokossis (2000a), extended to address multiple molecular and composition vectors.
Implementation The algorithm is implemented as an iterative procedure, which perturbs the states based on a predefined stochastic matrix of probabilities, and follows a hierarchical scheme. The entries of the concentration vector are discretized and the random perturbations follow probability distributions similar to the scheme suggested by Marcoulaki & Kokossis (2001). This assumes a bounded continuous variable, to define a quasi-symmetric probability distribution that is biased in the vicinity of the current instance of the variable, and void outside the bounds. The molecular composition moves are applied on a randomly selected molecule of the mixture, based on a uniform probability distribution. The state is simulated to predict the physical and thermodynamic properties relevant to the process performance and the environmental behavior of the materials involved. The simulation problem involves complex nonlinear thermodynamic models (like UNIFAC), and property mixing rules most suitable for each property. The assessment can also consult process simulation components that are interfaced to the optimization tool and the group contribution models/databases. The state evolution and convergence follows the Simulated Annealing acceptance criteria, using the modified Aarts-Laarhoven cooling schedule proposed by Marcoulaki & Kokossis (1999). 3. DESIGN OF SOLVENT BLENDS TO SUBSTITUTE EXISTING SOLVENTS There are a number of favorable attributes that can be associated with an existing solvent. In replacing the solvent all of these attributes need to be considered simultaneously in the formulation of the optimization problem. These advantages may include 9 high distribution coefficient, to increase the separation driving force 9 high selectivity of the solvent towards the solute 9 high solubility of the solute in the solvent phase 9 high boiling point temperature, to allow sufficient driving force for solvent recovery
454 9 9 9 9 9
high density, to enhance immiscibility low heat capacity, to reduce the energy requirements of the entire process low viscosity, to reduce pumping and recycling costs low solvent losses, to reduce the amount of solvent wasted in the raffinate phase high decomposition temperature, to allow high temperatures in the process, etc.
Synthesis case study Consider the "sulfolane" process, an aromatic-paraffinic separation named after the solvent commonly associated with it. The separation is described as a combination of liquid extraction and extractive distillation (Meyers, 1997). A reformate of paraffins, aromatics and naphthenes is first processed to remove the heavier fractions. The remaining hydrocarbons enter the "sulfolane" unit, where the solvent extracts the paraffins and naphthenes from the aromatic raffinate. Data for the feed composition are taken from Ancheyta-Juarez & AguilarRodriguez (1994) and reformed as a solution of groups. The solute mixture of paraffins and naphthenes is represented as a single pseudo-molecule of an average group composition. The raffinate is a mixture of benzene and toluene, represented by a similar vector. Note that the averaged representation yields the correct results when UNIFAC is used, but does not necessarily agree with other GC-methods. The case study considers the design of a solvent that has all the beneficial properties of sulfolane and more. Marcoulaki & Kokossis (2000b) presented a multi-criteria scheme that simultaneously maximized the solute solubility, the solvent selectivity and the solute distribution while minimizing the solvent wasted in the raffinate. The reported solutions were considerably complex arrangements, even when the synthesis objective was replaced by a molecular complexity index (CI). This work addresses better the practicalities, and demonstrates that the use of solvent blends can yield more conventional alternatives of equally high performance. In this example the constraints are modified to include Ss > 3.3 wt./wt. (0.2902) 9 lower bound on solvent selectivity Sb > 0.15 wt./wt. (0.1291) 9 lower bound on solute solubility in solvent Sl < 10. wt. % (78.%) 9 upper bound on solvent losses to the raffinate M > 0.3 wt./wt. (0.2104) 9 lower bound on solute distribution coefficient Ps > 1000 kg/m 3 (1264.) 9 lower bound on solvent density cpL,s < 0.5 cal/g.K (0.40) 9 upper bound on solvent liquid heat capacity LC50s < 2.0 (minor) 9 upper bound on solvent toxicity 9 upper & lower bounds on solvent boiling point temperature (560.K) TBP,S > 500.K & TBP,S < 550.K The values of the constraints are based upon the relevant properties of sulfolane, given in the parentheses. Sulfolane appears very high on solvent losses, which indeed explains the need for an additional unit to purify the aromatic phase and recover the solvent. Since the constraints alone enforce the design of advanced replacements for sulfolane, the process objective is initially to minimize the molecular complexity of the solvent and its constituents. Once the complexity becomes small enough, a different objective can put into action. So the cases discussed here are considering (a) the complexity objective, with the CI formula refined for this application (b) the minimal complexity as the primal scope, while the distribution coefficient presents a secondary objective.
table 1 Results for the molecular com ~lexity objective for mixtures and pure components
CI
Itl
Ss
[9
3.330 3.786 3.786 3.787 3.787 3.586 3.704 3.964 4.036 4.085 5.568 5.569 5.675 5.720 6.348
1121 1121 1120 1119 1110 1128 1143 1150 1168 1161 1146 1299 1274 1066
S! ~
_
. _
5.957 6.014 5.865 m
5.808 g~935 8.248 8~261 8.174 71153 1.565 7~538 z~440 9305 1.100 , , , m
, .
'TBP ,, COL LCS0 504.6- .3096 506.4 .3091 501.9 .3104 5odi~ .3109 505.3 .3137 515.1 .3065 510.7 ,3036 500.8 .3033 514.7 ,2970 507.1 ,3085 515.3 3083 501.0 ,2977 519.9 2765 509.9 ,3409
Table 2 Results with a combination of minimal CI Ss p , SI % TBp 3.376 1079 9.840 501.7 3.590 1 1 0 3 9.899 500.9
vI
3.241
1060
9.678
505.1
3.402
1091
9.777
502.3
[090
9.080
509.0
M
SOLVENT DESIGN (molecules and }heir compositions)
.3726 .3726 .3727 .3727 .4068 .3911 .3949 .3967 .3763 .2612 .3300 .1973 .1959 .2622
6076 ,6075 6077 6077 6633 6376 6439 6468 6135 4259 5380 3217 3194 4275
.3495 CH3-NO2 + .6505 5ACH-AC-CH(NO2).(CH2)3Br ,2680 CH3-NO2 + .7320 5ACH-AC-CH(NO2)-CH2-Br 2582 CH3-NO2 + .7418 5ACH-AC-CH(NO2)-CH2-Br ,2836 CH3-NO2 + .7164 5ACH-AC-CH(NO2)-CH2-B_r ,2934 CH3-No2 + .7066 5ACH-AC-CH(NOz)-CHz-Br ,2117 CH3-CHz-OH + .7883 5ACH-AC-CH(NO2)-CH2-Br ,1411 CH3OH + .8589 5ACH-AC-CH _(NO2)-CH2-Br 1777 CH3COO-CH2-CH2-CH3+ .8223 5ACH-AC-CH(NO2)Br ,1998 CH3COO-CHz-CH3 § .8002 5ACH-XC-CHeqOz)Br 1275 CH3COO-CH2-CH3 + .8725.5ACH-AC-CH(NO2)Br 1.000 CHs(C=Q),CH2-CH2(C=O)-CHzsBr 1.000 CH3COO-CH2-CH2COO-CH2-CH2-Br [.000 H0-CHz-CHz(C=O)-CH2-Br [.000 Br-CH2COO-CH2-CH2-N02 [.000 C1CHz-CHzCOO-CHz-NO2 1.000 CH3COO-CH2CH(-Br)-CH2-OOCCH3.
.
.
.
.
.
.
molecular complexity and maximal distribution coefficient (M) for solvent mixtures CPL LC50 Sb M SOLVENT DE_SIGN (molecules and their compositions ) .3265 2.687 .4297 .7007 .6861 5ACH-ACCH(Br)NO2 +.3139 CH3(CH2)3-CH2 -CH2-OH .3111 2.813 .4287 .6990 .7569 5ACH-ACCH2CO-Br+.2431CH_3-(CH2)3-NO2 8092 CH3-(CH2)2-CH(Br)-CH2-NO2 3253 3.544 ,4153 6771 +. 1908_.5ACH-AC-CH(CH3)N02 9625 CH3-CH2-CH(Br)-CH2-CH2-NO2 3145 3.482 ,4124 6724 + .0375 4ACH-(ACCH3)AC..CH2-CH(CH3)2 9844 Br-(CH2)4-CH2-NO2 2984 3.048 ,4055 6611 _+ .0156 4ACH-(ACCH3)ACCH2-(CHz)2-CH3 9
3.485
[.978 1.994 [.954 [~939 [.985 [.868 2.000 1.657 1.851 1.840 1.315 1.784 1.764 1.784
Sb
,.
456 Each stochastic experiment included 20 to 30 runs. The length of the Markov chain is 150 for single molecules and 350 for mixtures, and the parameter 5 controlling the annealing speed is 0.06. The solutions are reported on Tables 1 and 2 for cases (a) and (b), respectively. The optimization results involve binary mixtures, unless the problem is restricted to single molecules (last six solutions of Table 1). The search is efficient enough to ensure the designs satisfy the synthesis constraints, and therefore present significant benefits over sulfolane. The designs of Table 1 show that the use of solvent blends yields simpler molecular arrangements. Most of the generated blends involve nitromethane (in composition 25-35% mole/mole) with aromatic nitro-bromides. Similar groups appear in the solutions for pure compounds, though here the groups are forced to reside on the same molecule and increase its complexity 9Since process performance is not included in the objective of case (a), there is no account for high or low distribution coefficients, selectivities, solubility, solvent losses, etc. The setup of case (b) uses the previous results to get a reasonable CI threshold, set here at maximum 50. Below this value a process objective is activated, like the maximization of the solute distribution coefficient (M). The first two designs of Table 2 set the M target at 0.7. The experiment also yielded slightly inferior solutions that approach single molecules with 96-98% purity. Similar to case (a) NO2 and Br groups appear in most of the optimal solutions. 4.
CONCLUSIONS
This work presents optimization technology for the design of novel molecular configurations. The technology appears an extension of the stochastic tools developed by Marcoulaki & Kokossis (2000a), to address molecular blends. The use of mixtures instead of pure molecules increases the complexity of the optimization problem and the technologies involved in the optimal search. Nevertheless, the solution space is extended, thus the final designs are better in their performance and simpler in their molecular structure. Certainly, the introduction of additional components in the solvent mixture presents an issue of concern that one needs to address at the subsequent design stages of validation and experimentation. The method is illustrated with a case study involving the design of entrainer blends for the replacement of commonly used industrial solvents. The synthesis constraints consider a series of separation-enhancing features, as well as desirable properties and environmental behavior. A hierarchical multi-objective approach is illustrated to address the complexity of the molecular designs prior to consulting a suitable process objective formula. REFERENCES
Ancheyta-Juarez and Aguilar-Rodriguez. Oil and Gas J., 91 (1994) 5, 93 A Buxton, A. G. Livingston and E. N. Pistikopoulos. AIChE J., 45 (1999) 817 M. Hostrup, P. M. Harper and R. Gani. Computers Chem. Engng, 23 (1999) 1395 S. Machietto, O. Odele and O. Omatsome. Trans. IChemE, 68 (1990) 429 E. C. Marcoulaki and A. C. Kokossis. AIChE J., 45 (1999) 1977 E. C. Marcoulaki and A. C. Kokossis. Accepted at Trans. IChemE (2001) E. C. Marcoulaki and A. C. Kokossis. Computers Chem. Engng, 22 (1998) S 11 E. C. Marcoulaki and A. C. Kokossis. Chem. Engng Sci., 55 (2000a) 2529 E. C. Marcoulaki and A. C. Kokossis. Chem. Engng Sci., 55 (2000b) 2547 E. C. Marcoulaki, A. C. Kokossis and F. A. Batzias. Computers Chem. Engng, 24 (2000) 705 9 " A. Meyers. Handbook of petroleum refining processes. 2n . Ed., McGraw Hill, NY, 199 7 E. N. Pistikopoulos and S. K. Stefanis. Computers Chem. Engng, 22 (1998) 717
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
457
A tool for optimal synthesis of industrial refrigeration systems F. Marechal I and B. Kalitventzeff 2 LASSC -University of Liege, Sart-Tilman B6a, B-4000 Liege (Belgium) A method for selecting the best refrigerants and the optimal configurations of the refrigeration system starting with the definition of the refrigeration requirements of a process is presented. The method proceeds in three steps: 1) identifying the most important temperature levels using an exergy minimisation method, 2) identifying the most suitable refrigerants to be used to generate a refrigeration system superstructure, 3) using MILP optimisation extract a set of refrigeration system configurations including compressors and heat exchangers characterisation to obtain the complete list of streams to be considered for the Heat Exchanger Network design (fourth step of the method not treated here). 1. I N T R O D U C T I O N The optimal insertion of energy saving technologies in industrial processes is a key issue for the rational use of energy in the process industry. Among the energy saving technologies, the refrigeration system is a technology area that introduces several degrees of freedom. The goal is to identify the best refrigerant(s), the optimal temperature levels and the best cycles configuration as well as the selection of the best compression technology to satisfy the refrigeration requirements of a process at minimum cost. Most of the tools developed to solve this problem use MILP (Shelton and Grossmann, 1986; Vaidyaraman and Maranas, 1999) or MINLP (Colmenares and Seider, 1989) software tools while graphical methods allow to understand the refrigeration cycle integration and deduce heuristic synthesis rules (e.g. Townsend and Linnhoff, 1983). The major difficulty related to the operational research based methods is the poor interaction between the engineer and the software: all the problem constraints should have been defined before starting the solving procedure. The heuristic based methods suffer from the combinatorial nature of the problem. In this context our goal has been to develop a method to guide the engineer from the problem statement to the solution with a list of computer aided steps leaving him the choice as much as possible of interacting with the solving procedure. 2. THE M E T H O D Starting with the definition of the process requirements defined by a Grand composite curve, the first step of the method consists in identifying the most important temperature levels to be considered in the refrigeration system (P1). Knowing these levels, the list of possible refrigerants is determined based on thermodynamic and operation criteria applied to a refrigerants data base. A superstructure of the refrigeration system is then generated systematically and is modelled using the Effect Modelling and Optimisation approach (Marechal and Kalitventzeff, 1997) that applies the heat cascade principles to model the ideal
Now with LENI-DGM, Swiss Federal Institute of Technology, EPFL, 1015 Lausanne, Switzerland, mail to:
[email protected] 2 Belsim s.a. All6e des Noisetiers, 1, B-4031 Angleur, mail to:
[email protected] 458 heat exchange system and a MILP technique to extract the optimal process configurations (P2). The model of the refrigeration system is defined in such a way that it will include the classical configurations of the cycles allowing the calculation of refrigerants and pressures cascade systems. Obviously the solution to such a problem is not unique, multiple solutions are generated and will be compared to obtain the final solution. After this step, the list of streams to considered in the heat exchanger network is known and the heat exchanger network structure will be determined (P3) using classical methods : heat load distribution, pinch design method and/or optimisation. Compared to fully automatic solvers, the proposed method allows a high level of interaction between engineer and software.
2.1. Step I : Identify the most important temperature levels of the requirement. To identify the optimal temperature levels, we first discretise the overall temperature range with a fixed precision. For each temperature in the list with nk elements, we introduce a heat contribution with a constant temperature (Tk) and an unknown heat load (qk). This contribution will be a hot stream above the pinch point and a cold stream below. The optimal heat load of the new utility streams will be computed by minimising the exergy losses in the heat transfer. In general terms, the exergy losses associated to a stream that changes its thermodynamic state from Tk+l to Tk is computed by : Aex~ =qk(1 - T ~ )
where Ttmk= T~+I-T~
Tl,,k
ln(~--~z~-)
with To the reference temperature and Tlmk the
logarithmic mean temperature of the temperature interval k. This expression simplifies in q~(1-w T--~) for the exergy losses of a heat load qk supplied at a constant temperature Tk. I k,
Introducing as constraints the heat cascade definition including the DTmin contributions of all the hot and cold streams defining the process requirements, we obtain a linear programming problem (P 1) whose objective function is the overall exergy losses of the selected heat levels. (P1)
Minimise~-' q~ 1-Rk ,Yk ,qk
~=1
subject to : heat balanceof the temperatureintervalk q, + ~ Q,k + R,+~- R~ =0
Vk = 1..... nk
i=l
Vk = 1..... nk
qmink Y~< qk < qmaxk Yk y~e{0,1} y~ < Nmax3, k =kt, +1
R~=0, R,,+~=0
y~ < Nrnax2,~y ~ < Nmax~ k=k. +1
(A1) (A2)
k=l
andR k >0
Vk=l ..... n~ +1
To the classical problem formulation, we add contraints (A1) to force any heat load qk tO be significant with respect to the total heat requirement (e.g. 10% at least); and contraints (A2) in order to limit the number of such levels in the different temperature domains. P1 is a MILP problem where n is the number of hot and cold streams; Rk the energy to be cascaded from the temperature interval k to the lower temperatures; Qik the heat load of process stream i in temperature interval k; (Qik > 0 for hot streams and < 0 for cold streams); Nmaxl the maximum number of levels to be found below ambient temperature; Nmax2 between ambient temperature and pinch temperature and Nmax3 above the pinch point; Yk is the integer
459 variable associated with the use of heat requirement qk; qmink (qmaxk) are the minimum (maximum) values accepted for the heat requirement qk. The formulation concerns the whole temperature range. It will also be applied to define the hot utility requirement and identify the most appropriate steam levels to be considered. For cold utility requirements, the values of qmink and qmaxk will be negative. In order to exploit the exergy potential of the self sufficient zones, qmink will be equal to -qmaxk for the temperatures in self sufficient zones. The result is a list of heat consumptions (above the pinch point) and heat productions (below the pinch point) that will be used to define the energy saving technology candidate. Above the pinch point (kp), the results concern the steam network and the heat pump systems, between ambient temperature (ka) and kp, these concern the organic Rankine cycles and the heat pumps, below the ambience these concern the refrigeration system. The results for the candidate levels found for a prototypical refrigeration example is given on figure 1.
Figure 1 : identification of the refrigeration levels and the possible refrigerants 2.2. Step 2 : Identify the best refrigerants to be used Knowing the best temperature levels to be considered, the next step concerns the identification of the list of refrigerants to be used. The algorithm presented on figure 2 performs a systematic examination of all the available refrigerants. The goal is a) to identify the feasible refrigerants, b) to compute the number of possible interstage levels. These values will be used to generate the refrigeration system superstructure. At this stage, the list of possible refrigerants (located on figure 1) will be proposed to the user via a web interface where he will be able to choose the acceptable ones for his process, applying additional rules based on other criteria: e.g. toxicity, safety, stability .... Simulation of the elementary refrigeration cycles Having defined the accepted refrigerants and temperatures, the goal is to determine the configuration of the refrigeration system. It means to define the list of refrigerants, condensers, evaporators and compressors to be considered, the way these are interconnected and to compute the flowrates in the different stages of the refrigeration system. For this purpose, the refrigeration system will be modeled by a set of elementary refrigeration cycles composed of one compressor, one hot (condensation) and one cold (evaporation) stream effects as shown on figure 3 for a 3 stages industrial refrigeration system.
460
O r d e r r by increasing Tboil
r
COMMENTS ON THE ALGORITHM The refrigerants are ordered according to increasing boiling point, this allows to reject all the refrigerants after the first refrigerant is rejected. The temperature list Te results form (P 1). For evaporation at Te, a new refrigerant is added if its saturation pressure Pevar(Te) is between 0.7 bar and 2 bars. For condensation at Tk, we compute the Pcondr(Tk), the saturation pressure of the refrigerant r. The condenser is introduced is accepted if the compression ratio is < 20, and if Tk is lower then Tcrit- 10~ As the major temperature levels have been computed with the minimum exergy losses method, we introduce as an option the possibility of adding temperature levels for each interstage level. Assuming that centrifugal compressors will be used, we compute the number of stages (Nstages)assuming that the speed limit of the fluid at the outlet of a stage must be lower than 90% of sound speed (or): -
I
I I
-
-
-
! I T* t = T k + D Tm i,n/2 r,k l ~ - - " ~ > ~ . . ~
~
k--k-,
/ [ ComputeN
...... ( T . = > T : )
w R zT~x withwthe N.,.,,g~= --uTand u =0.9or withc~=,l~l Pmol
}
yos update Thst ,y Add new sub-cycle r, Pcondr(Tk), Pevar(Te) ii
I
massic specific work, Pmol the molecular weight of the fluid, rhs the isentropic efficiency assumed to be 80%, C~
i
[ .....
Ce -R
Figure 2 : algorithm for refrigerant selection
Figure 3: representation of a 3 levels refrigeration cycle by elementary cycles Problem P2, as formulated here below, is a MILP problem where f,~j is the flowrate in the elementary cycle between evaporator level i and condenser level j using the refrigerant r; Cr,r(T j) the liquid fraction resulting from the flash of refrigerant r at temperature Tj from the previous temperature level of refrigerant r; AHvaPr(Tk,P(Tk))
the heat of vaporisation of
refrigerant r at temperatureTk and the equilibrium pressure at Tk; nr is the n u m b e r of refrigerants; nt the number of temperature levels identified for the refrigeration system; Qek,r the heat load of the evaporator with refrigerant r at the temperature level k; QCk,r the heat load
461 of the condenser with refrigerant r at the temperature level k; W c r the mechanical power consumption of compression for the refrigeration cycles using refrigerant r; Tk is the real temperature corresponding to the corrected temperature in the heat cascade. Therefore, Tk =T*k-DTmin/2k for the evaporator temperature and Tk =T*k+DTmin/2k for the condenser, Tk is used to compute the saturation pressure, the compression power and the head load of vaporisation. nw
Minimise ~ ( YwClw + f wC2 w) + Cel * EL i - Cel o * EL,, Rk ,Yw ,fw
(P2)
w='~"~
nr t
n
+~_. YWk * C l w r +WCr*C2w r +~.,(yck,r*ClCk,r +QCk,r*C2Ck,r)+ r=l
(Yek,r*Clek.r +Qek,r*C2ek.r k=l
k=l
subject to 9 heat balance of the temperature interval k
s163163163 w=l
r=l
Vk = 1..... n k
Rk+,-R ~ = 0 r=!
i=1
Ve = 1,..., nt Qee'r--s
~=Ik!k-O~r(Tj)llA* ,
*AnvaPr(Te'P(Te))l=O
Qe~nyee'r __ vik + vjk - 1
V i = 1 . . N - 1, j = i + 1..N, k = 1..K
(2)
zij _ ~
kvik
(5)
V i
k
3.2. Distance Constraints The single floor distance constraints presented in [5] are here extended for the multi-floor case: R i j - Lij - xi - x j
V i - 1..N-
1, j - i + 1 . . N " fij - 1
(6)
A i j - B i j - y~ - yj
V i - 1..N-
1, j - i + 1 . . N " f~j - 1
(7)
U~} - Di~ - Fh ~
k(vik - Vjk)
V i - 1 . . X - 1, j - i + 1 . . N " fij - 1
(8)
k
where the relative distance in x coordinates between items i and j, is R i j if i is to the right of j or Lij if i is to the left of j. The relative distance in y coordinates between items i and j, is A i j if i is above j or B i j if i is below j. The relative distance in z coordinates between items i and j is U/~ if i is higher than j or D~j if i is lower than j. The coordinates of geometrical centre of item i are denoted by xi, yi. Parameter fij is equal to 1 if units i and j are connected ; 0 otherwise. Thus, the total rectilinear distance, Dij, between items i and j is given by: D i j -- R i j nt- Lij + A i j + B i j + U~. + Di).
V i - 1..N-
1 , j - i + 1 . . N " f~j - 1
(9)
3.3. Equipment Orientation Constraints The length, li, and the debth, di, of equipment item i can be determined by: li -- o~ioi + ~i(1 - oi) di - c~i + ~i - li
(10)
V i
(11)
V i
where oi is equal to 1 if li=oq; 0 otherwise (i.e. li=fli).
3.4. Non-overlapping Constraints In order to avoid situations where two equipment items i and j occupy the same physical location, when allocated to the same floor (i.e zij = 1), constraints are included in the model that prohibit overlapping of their equipment footprint projections, either in x or y direction: xi -- x j + M ( 1 - zij + E l i j + E 2 i j ) >
xj - xi + M ( 2 - zij - E l i j + E 2 i j ) >_
yi - yj + M ( 2 -
zij + E l i j - E2ij) >_
li + lj
l~ + lj
di + dj
V i - 1..N-
1 , j = i + 1..N
(12)
Y i - 1..N-
1 , j - i + 1..N
(13)
V i-
1..N-
1,j-
i + 1..N
(14)
478
yj - yi + M(3 - zij - E l i j - E2ij) >_
di + dj
V i = 1 . . N - 1,j = i + 1..N
(15)
where M is an appropriate upper bound and Elij, E2~j are binary variables as used in [5]. Note that the above constraints are inactive for equipment allocated to different floors (i.e z~j = 0).
3.5. Additional Layout Design Constraints Intersection of items with the origin of axes should be avoided: xi > li -2
Yi
(16)
yi > _di -2
V i
(17)
A rectangular shape of land area is assumed to be used and its dimensions (x max, yma~) are determined by: li
xi + -~ 5 x max
g i
(18)
di _ yma~ Yi + ~
-- ~ P L q , c
"
Rmo,c + __~RLm k,o,c k=l
t if c = key component in the raffinate stream
(3)
X pLq, c >-- "l'PLq, c
9 Global mass balance: E1 + Rdef + Z PLq = Ro + Eo + Z RLk + q=l
k=l
9Mass balances in each stage: Q /~ Rj + Ej + ~ PLq,j = Rj. 1 + Ej+ 1 + Z RLk, j + ELj q=l
9Bypass mass balance:
(4)
ELi j=l
Vj~ NT
(5)
k=l
Ro
=
Ro ext + eObyp
(6) K
(7)
R def = R n + R o byp + _~ _ RLk, byp k=l
9Nonlinear equilibrium relations (Reyes et al., 1999): U?(yj,c, xj,c)-0
Vj~ NT
(8)
9 Component and total flowrate relations (bilinear terms): Fmj,r Fj.Uj,e = 0
VF e {R, E, PLq, RLk, EL, edef}, U -- {X or y} Vje NT, Vc e COMP
(9)
490 9Lateral stream balances: ~
RL k =
n
RLk,j + RLk,byp Vk ~ K
PLq = Z PLq,j
j=l
Vq ~ Q
(10)
'v'j~ NT
(11)
j=l
COMP ~ X j, c = 1
9Mass fractions:
COMP
Zyj, c - 1
c=l
c=l
The constraints in (1)-(11) involve only continuous variables, and are valid for any cascade configuration. The disjunctions in (12) are the ones associated with the discrete choice of enforcing the equilibrium relations in existing trays. This is accomplished with Boolean variable Zj, which can be true or false depending if the stage j is selected or not. The disjunction is as follows: _
-nZj Xj,c = Xj-l,c ; Yj,c = Yj+l,c [
Zj 1 Ej=Ej+l'c'Emj'c=Emj+l'c Equilibrium." trt~(Yj,c,Xj,c)=O v R j = R j _ l , c ; R m j , c = R m j _ l , c
[
Fmj,C= Fj .ujx
RLk,j = O;RLmk,jx = 0 PLq,j = 0 ; PLmq,j, c = 0 ELj =O ; ELmj, c = 0
Vj~ NT '7'C~ COMP
Vk~KVqe Q '7'FE {R,E, PLq,
RLk, EL, Rdef} zt = {x or y}
(12)
_
To avoid equivalent solutions that are due to the multiplicity of representation for a given number of trays, the following logic constraints are added: Vj~ NINT
Zj ~ Zj_ 1
(13)
The objective function involves the minimization of the total annualized cost of equipment and utilities. The capital cost is considered proportional to the sum of stages, and the operating cost of the process a function of the flowrate of the solvent feed stream, n
min
n
OF=(Eo+~_ELj).C E +~.Zj .C n j=l
(14)
j=l
In order to avoid infeasibilities in the solution of the NLP problems with fixed values of the binary variables, the design specifications are relaxed using slack variables, which are minimized in the objective function. 4. SOLUTION ALGORITHM. The proposed GDP model is solved with a modification of the Logic-Based Outer Approximation (OA) algorithm of Turkay and Grossmann (1996). This decomposition algorithm solves the problem by iterating between reduced NLP subproblems and an MILP Master Problem that consists of linear approximations in all solution points given for all the NLPs solved previously, using the big-M formulation to relax the linear approximations of the disjunctions. If we define PNLP as the set of all the previous solved NLP subproblems and NLF as the set of all nonlinear functions in the model: {bilinear terms and equilibrium relations}, the constrains in Equation (12) are replaced in the MILP master problem by the following equations:
491
ps(Knl)+Vps(~nl).(K_Knl)0 1>
simultaneous model exploits the strong interactions between the selection of operating conditions and heat recovery opportunities. Since the model can treat the temperature of both isothermal and non-isothermal streams as a continuous variable, and the problem is formulated as a MILP model, the results that are obtained are the global optimal solution. The new simultaneous heat integration model also reduces the utility cost compared to the sequential heat integration approach (Table 4). In Example 1, the same results are obtained as with the disjunctive pinch location model because the new model does not require any approximations. In Examples 2, the new model gives same result as obtained by the disjunctive pinch location model. In Example 3, the difference in utility cost compared to the disjunctive model is very small. It is to note that the number of fixed temperature intervals in Example 2 and 3, has been kept same as the Example 1. Although the new model employs slight approximations in Example 2 and 3, the difference in utility cost compared to the disjunctive model is negligible. Case B: Constrained matches Next, consider the case that constraints should be taken into account. Either the process operating conditions (stream temperatures) can be selected first, and then the constraints imposed in a sequential manner, or the selection of process operating conditions and constraints can be considered simultaneously. Three approaches to the problem are considered: 1) the sequential approach, where operating conditions are determined first and the expanded transhipment model (Papoulias and Grossmann, 1983) calculates utility requirements when constraints are imposed for these fixed operating conditions; 2) the disjunctive pinch location method, which cannot address constraints simultaneously, therefore the constraints are imposed at the fixed operating conditions; 3) the new model considers constraints and stream temperature selection simultaneously. These three approaches are used for the three examples presented in Tables 2 and 3, where heat integration is constrained to match no more than one stream or utility with any isothermal stream.
Table 5 Results for case B (constraints accommodated) Example 1 Utility cost ($/yr): 1,700,000 Optimal temperature (~ TcI1= 80 YcI2 = 110
Example2 Sequential heat integration Utility cost ($/yr): 1,700,000 Optimal temperature (~ TcI1= 80 TcI2 = 110
Example3 Utility cost ($/yr): 1,700,000 Optimal temperature (~ TcI1= 80 Tci2 = 110
Simultaneous heat integration by the disjunctive pinch location model Utility cost ($/yr): 1,745,800 Utility cost ($/yr): 1,745,800 Utility cost ($/yr): 2,189,300 Optimal temperature (~ Optimal temperature (~ Optimal temperature (~ Yci I = 80 YciI = 80 TIr~l = 180 T c I 1 = 80 TI~IN1 = 180 YcI2 = 1 4 4 . 9 5 YcI2= 144.95 TcI2 = 144.95 TOrn~l= 142
Utility cost ($/yr): 1,364,900 Optimal temperature (~ TcI1= 80 TcI2 = 148
Simultaneous heat integration by the new model Utility cost ($/yr): 1,364,900 Utility cost ($/yr): 1,451,256 Optimal temperature (~ Optimal temperature (~ Tcil = 80 TIrrNl - 180 TcI1 = 80 TIrvN1 = 180 T c I 2 = 148 Tciz = 147 TOrn~l = 155
516 The results for the constrained case are presented in Table 5. The new model considers process design (selection of operating conditions) and constrained matches simultaneously. It therefore exploits interactions between the choice of operating parameters and heat recovery opportunities. The utility costs of the resulting process are significantly lower than for the two sequential approaches. Clearly constraints can significantly influence the optimal operating conditions and heat recovery. Note that the result obtained for Example 1 using the new model is a global optimal solution because the model does not require approximations for the non-isothermal streams. In Examples 2 and 3, although the new model employs slight approximation due to the variable temperature of the non-isothermal streams, the new model obtains significantly better results compared to the disjunctive pinch location model. As the disjunctive model cannot include constraints simultaneously, it misses the benefit of exploring interactions between heat recovery opportunities and constrained matches. However, the solutions obtained by the new model may not be globally optimal. 4. CONCLUSIONS A new heat integration model has been formulated for continuously varying stream temperatures by considering individual matches between hot and cold streams. A key feature of the model is that constraints and selection of operating conditions can be addressed simultaneously. Constraints commonly arise for reasons of safety and operability. The new model uses disjunctive logic to model feasible matches and heat duties for heat integration. The temperatures of both isothermal and non-isothermal streams can be treated as continuous variables. However, the model employs a slight approximation when the temperature of a non-isothermal stream varies continuously. The model is formulated as a MINLP problem. The MINLP model reduces to a MILP model if the flowrate of nonisothermal streams are fixed and the heat duties of isothermal streams vary linearly with temperature. Several examples, involving both isothermal and non-isothermal streams, are solved to show the advantages and disadvantages of the proposed model. It is found that when constraints are to be considered, the new optimisation model generally reduces utility cost compared to the disjunctive pinch location model, which cannot include constraints simultaneously. This new simultaneous optimisation and heat integration model can be applied to the selection and optimisation of heat integrated distillation sequences, the synthesis of reaction and separation systems, etc. In these cases both flow rates and temperatures of the process streams (non-isothermal streams), and the condenser and reboiler temperature (isothermal streams), etc. are explicit optimisation variables. 5. R E F E R E N C E S
1. B. Linnhoffand J.R. Flower, AIChE J., 24 (1978) 633. 2. S.A. Papoulias and I.E. Grossmann, Comp. Chem. Engg., 7 (1983) 695. 3. J. Cerda, A.W. Westerberg and B. Linnhoff, Chem. Eng. Sci., 38 (1983) 373. 4. M.A. Duran and I.E. Grossmann, AIChE J., 32(1986) 123. 5. I.E. Grossmann, H. Yeomans and Z. Kravanja,, Comp. Chem. Engg., 22 (1998) S157.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
517
Tools for Reactive Distillation Column Design: Graphical and Stage-toStage Computation Methods Sfinchez Daza Oscar a'b, P6rez-Cisneros Eduardo a*, Erik Bek-Pedersen c, and Martin Hostrup c aDepartamento de Ingenieria de Procesos e Hidrfiulica Universidad Autdnoma Metropolitana-Iztapalapa M6xico, D. F., C. P. 09340, Fax: (52 5) 8 04 49 00. E-mail:
[email protected] bCentro de Quimica, Instituto de Ciencias de la Universidad Aut6noma de Puebla CCAPEC, Department of Chemical Engineering, Technical University of Denmark. 2800, Lyngby, Denmark Based on the element mass balance concept, a graphical design method and a stage-to-stage multicomponent design method for reactive distillation columns have been developed. For distillation columns comprising reactive and non-reactive stages, a simple design strategy based on reactive and non-reactive bubble point calculations is proposed. This strategy tracks the conversion and temperature between the feed and the end stages of the column. An illustrative example highlights the verification of the design strategy through rigorous simulation. 1. INTRODUCTION In recent years, reactive distillation has attracted attention as a highly promising hybrid process. Application of this combined reaction-separation process is considered useful only for reactive systems limited by chemical equilibrium and it has been applied with great success to the methyl acetate and methyl-tert-butyl ether (MTBE) production [ 1-2]. It has been claimed [3] recently that reactive distillation is also applicable to reaction separation processes involving complex reactive systems such as the diesel and gasoline desulfurization. The increasing interest in reactive distillation has been accompanied by development of various simulation algorithms related to the study of operation and control [4,5] of the process. Design of reactive distillation columns (RDC), however has not received the same attention. Most of the existing work related to design of RDCs has been based on the work of Doherty [6] or it has been treated as an optimization problem (MINLP) [7]. In this paper, a method based on the element composition variables that significantly reduce the dimension and complexity of the intrinsic calculation problem and thereby achieving the design goals with less computational effort is presented. The objective of the present work is to introduce a new set of graphical and multicomponent stage to stage computation methods for design of reactive distillation columns. These design methods are based on the element mass balance approach. The methods developed are similar to those typically employed for non-reactive systems. For binary element systems, which may be ternary or higher in terms of mixture compounds, a
518 simple reactive McCabe-Thiele method has been developed. For design of ternary element systems, which are usually quaternary or higher in terms of mixture compounds, a reactive stage to stage calculation method has been developed. For columns comprising reactive and non-reactive stages, the stage-to-stage procedure is used. Also the driving force approach of Gani and Bek-Pedersen [9] has been extendedto application in reactive separation systems. The methods have been tested in a systematic manner with several reactive systems. In this work, only the MTBE reactive system will be highlighted. All design examples have been verified through rigorous simulations. 2. ELEMENT BALANCES AND EQUILIBRIUM CONDITION In the equilibrium based approach, the computation of chemical and physical equilibrium (CPE) is an important step. Using the element-based approach [8], a multi-component physical-chemical equilibrium problem is transformed into a phase equilibrium problem for a mixture of elements (representing the system). In the element-based approach, the CPE problem is solved by minimizing the Gibbs energy of the reactive system NP
man G(n) = ~ fl:l
NC
~ n/#/z/#
(1)
i=l
Subject to the M constraints: NP
NC
Z Z Aj, 7 fl=l
-
0
i=l
where G(n) is the total Gibbs energy of a system containing NC species and NP phases. Equation (2) represents the M independent element mass balances, with the coefficient Aji being the number of times the reaction invariant element j is present in molecule i. The solution of this constrained optimization problem can be obtained through the Lagrange multiplier formulation where the relation between the Gibbs free energy and the Lagrange multipliers is exploited for a robust method of solution. Thus, a fully consistent thermodynamic description of a chemically equilibrated phase is obtained in terms of b, the (element) composition vector, and ~,, the corresponding element potential vector. 3. DESIGN OF REACTIVE DISTILLATION COLUMNS The design methods are divided in terms of the number of elements involved, where a graphical design method for binary element (reactive) systems while a stage-to-stage calculation method is used for ternary element (reactive) systems.
3.1 Binary Element Reactive Systems Consider a "full" reactive distillation column (all stages are reactive stages) operating under chemical and physical equilibrium conditions (see Figure 1). The feed is a mixture of two elements A and B. In the case of a binary element reactive distillation, the reactive stripping section concentrates the less-volatile element (B) in a liquid stream while the reactive rectifying section concentrates the more-volatile element (A) in the vapor stream.
519 3.2 The Reactive Equilibrium and Driving Force Curves A reactive equilibrium curve is the locus of all chemical and physical equilibrium points. For a given element liquid composition Wta, it gives the corresponding equilibrium vapor composition W~A, and vice versa. A reactive equilibrium stage p is represented as a point on the reactive equilibrium curve where WtAp and W~Ap are the liquid and vapor element compositions leaving the stage. The reactive equilibrium curve can be constructed through sequential computation of reactive bubble points. A typical reactive equilibrium curve is shown in Figure 2. A driving force diagram can be used to visualize how the operation of the reactive distillation column should be determined to achieve the best separation with the least energy. A reactive driving force diagram for the MTBE production example is given in Figure 3. On the basis of this, good preliminary estimates can be obtained for design variables such as the number of raective stages, element reflux ratio, and feed stage location. 3.3 Constant Element Molar Overflow Assumption In order to avoid energy balance calculations, it is at least approximately correct for many problems to apply the assumption of constant total element molar overflow. The change of vapor rate from stage to stage at the rectifying section can be derived by writing an energy balance as v v bp -bp+, -
bp+l(H;V+l-H*pV)-bp'(Hp' - Hp_,) *' (H; v -H;')
(3)
It follows that there are two possible conditions which will cause the total element molar vapor flow b v to be constant from stage to stage: H *v= Constant
Condition 1:
H *t-- Constant
(4)
1
Condition 2"
(H*p" - Hp 9~) 9
= - -bp+ r- 1
(5)
Rt~tifying Qc bvl ~ [
_
hF r
bBT
Fig. 1 Binary Element Reactive Distillation Column
oo
oo
02
04
o6
oa
Fig. 2 Reactive Phase Diagram g~A- grA.
1o
0,00
WA, To
IAqu)dComposilion,WA(l-Butone)
1,00
Fig. 3 Reactive Driving Force Diagram WtA- g~A-
520 If b v is constant, it follows that b t is constant. It should be noted that the enthalpies for the different phases are element-based. From condition l, it can be seen that constant element molar overflow will occur if the element molar heats of vaporization of elements A and B are identical, if sensible-heat contributions due to temperature changes from stage to stage are negligible, if there are no enthalpy-of-mixing effects (ideal liquid and vapor solutions) or if the heat of reaction is negligible.
3.4 The Reactive Operating Lines Consider the rectifying section of a simple reactive distillation column, (see Figure 1), performing an element mass balance around this section for element A, the following equation is obtained: WA - - ~ W I + -ff~-
(6)
where bdr is the total element amount distillated at the top of the reactive column when the ratio ffp/bp+l (see Eq. 5) is maintained constant. Equation (6) represents the reactive operating line for the rectifying section. At the point where the feed is introduced to the column, the element flows in the rectifying and stripping sections must change because of the feed entry. The element flows below the feed introduction (stripping section) are labeled b*l and b*V. Performing an element mass balance for element A in the stripping section gives the reactive operating line for the stripping section. b *l t bBy W ; - b ,--7 W A - ~b, v W,~B
(7)
3.5 Ternary Element Reactive Systems In order to start a stage-to-stage calculation method, it is necessary to specify a selected set of variables that satisfy the degrees of freedom completely. It is nearly always the terminal stage for which this can be done. The stage to stage approach applied in the present work for reactive systems is based on the Lewis-Matheson method. If all the element compositions at the bottom stage (Wtj,B) are known (specified), it is then possible to calculate the element compositions of the vapor phase (W~j.B) leaving the bottom equilibrium stage (re-boiler) by performing a reactive bubble point calculation. The element vapor flows are determined from, bjp - bTWjp (8) The element liquid flows from the stage above the bottom stage is then determined from an element mass balance for the bottom stage, l
v
l
l
v
l
bg,p+ 1 - bj,p+ 1 + biB (9) Alternating use of Eqs. (8) and (9) then gives successive vapor and liquid flows for stages going up the column, until a feed stage or side-stream stage is reached. At this point, Eq. 9 is replaced by Eq. 10. bj,p+ 1 - bj,p+ 1 - bid
(1 O)
521 4. APPLICATION EXAMPLE Binary Element System: MTBE production without inert. Combined Reactive Distillation Column Configuration In this example it will be shown how the two design methods are applied. It should be noted that although the reactive system is represented by two elements, the RDC configuration includes reactive as well as non-reactive stages. Therefore, the stage to stage calculation method is also required. Consider a binary element mixture that is 70 element mole percent of element A (ZA= 0.7) is to be separated into 50 element mole percent bottoms product (14/A,~= 0.5) and 99 element mole percent distillate (g/A,D= 0.99) product. The element feed flow rate is 100 moles per hour at T = 300 K and P= 1 atm. The operating pressure of the reactive distillation column is 1 atm. The reflux element ratio (RR) is 2. The physical and chemical equilibrium curve is constructed with the CPE program package [8] considering the S-R-K equation of state for the gas phase and the Wilson equation for the liquid phase. Theoretical reactive stages, a partial reactive boiler, total condenser and a chemically saturated liquid reflux has been assumed. It is desired to determine the optimum feed stage location and the number of reactive and non-reactive stages. In this case, the reaction in terms elements can be written as: Isobutylene(C4H8=)k) + Methanol(CH3OH=B) r
Methyl-Tert-Butyl-Ether(CsHl20=AB)
4.1 Design Strategy Basically, we start the design of the column considering a distillation column with only reactive stages. In this case, the design can be performed graphically with the developed graphical method. Figure 4 shows the determined number of reactive stages needed to achieve the specified separation and the optimal location of the feed stage. Table 1a shows the column design in terms of conversion to MTBE and temperature at each stage. It can be observed that the conversion at the bottom and the top of the column is negative. This means that the reaction is reversed and a decomposition of MTBE has occurred. Thus, even if the column design satisfies the given separation specifications, the design should be rejected as the conversion is not satisfactory. The next step is to introduce non-reactive stages at the bottom of the column and generate a new conversion and temperature profile. Table 1b shows that by introducing two non-reactive stages the negative conversion at the bottom of the column is eliminated (stage to stage calculation method has been used). In addition, if one non-reactive stage is introduced at the top of the column, a final design without negative conversion is obtained. It should be pointed out that the design strategy here is to track the conversion and temperature and force them to attain values such that the RDC is feasible. However, other variables such as occurrence of reactive or non-reactive azeotrope and composition of one key component could also have been selected to switch/add non-reactive and/or reactive stages. 5. CONCLUSIONS Two methods for the design of reactive distillation columns have been presented. Based on the element mass balance approach, a graphical method for binary element reactive systems is applied for the design of a distillation column with only reactive stages. When the number of elements is greater than two or when a "combined" configuration of a reactive distillation column is considered, the stage-to-stage calculation method is more appropriate. In the example presented, a design strategy for a combined reactive distillation column was
522 developed considering the conversion to MTBE and the temperature at each stage has been presented. This strategy required the use of reactive and non reactive bubble point calculations. 1.0
}
~ ......... ~A
0.6
McC~be Thlele Procedure (ELEMENTS GRAPHIC) WILSON Equat ion S-R-K E q u a ~ m o n of State
A VAP
!
i .......
! ..... ~ ' >
',
:
i
~/'
i
~
HTBE REACTIVE SYSTEM
:
j
,' / ', [ P- 1.0 T~- 30O.0 ZF}I~- 0 . 7 0 0 0 ~;ti :/........................ : I F E E D I S LIQUID-VAPOR ZF(2)-0.3000
....... ,L....... ,L__ . ~ ~ _ ~,/, i
~ ....... I 9 i
0.2
ENTER
0.4 TO
0.6
2.00
STAGE ~JLA 1 0.5000 2 0.5216 3 0.5564 4 0.6704 5 0.8919 0.8 ~JLA 1.0 6 0.9820 -
' 0.2
REFLUX=
XB(1) . . . . . 27 XB{2) = 0 0427 XB(3) = 0 9146
0.4
0
(NO-INERT)
ZF(3) = 0.0000
YD (i)- 0.9981 YD(2) 0.0000 YD(3) = 0.0019
Re~ulr~. XL(1) "YV(1) 0.0427 0.2688 0.0927 0.4741 0.2042 0.7179 0.5084 0.9216 0.8788 0.9871 0.9816 0.9981 FEED STAGE = 3
T 317.14 310.76 298.52 279.68 268.24 265.84
FINISH
Fig. 4. Design Specifications and output result for the MTBE reactive System Table 1. Conversion-Temperature Profiles for a Reactive Distillation Column Design a. Only reactive stages b. Two non-reactive stages c. Final Design Stage Conv. T (K) Stage Conv. T (K) Stage Cony. T (K) 0 320.17 l=Reb. -0.526 317.13 1=Reb. 0 320.17 1=Reb. 2 0 308.45 2 +0.048 310.42 2 0 308.45 3 +0.2908 294.33 3 +0.307 297.41 3 +0.2908 294.33 4 +0.0947 275.01 4 +0.095 275.82 4 +0.0959 275.01 0 267.03 5=Con. -0.0009 267.16 5 +0.0000 266.99 5=Con. 6=Con. -0.0038 265.55 The final design obtained has been verified with rigorous simulation and a maximum difference in temperature of five degrees between the simple and rigorous calculations was observed. Considering the speed and simplicity of the simple calculations, the results from the developed methods serve as useful first estimates. The design methods are being further extended using the driving force algorithm of Gani and Bek-Pedersen [9] to obtain as well, energy efficient designs. REFERENCES 1. Agreda V.H.; Partin L.R.; Heise W.H., (1990), Chem.Eng. Prog., 86, 40. 2. Smith, L.A., and Huddleston, M.N., (1982), HydrocarbonProcessing, 3, 121. 3. Hairston Deborah, (1999), Chemical Engineering, Vol. 106, No. 4, 32 4. Abufares A.A., and Douglas P.L., (1995), Trans Ichem.Eng., 73A, 3. 5. Monroy R, P6rez-Cisneros E., and J. Alvarez, (2000), Chem.Eng.Sci., 55, 4925. 6. Barbosa D., Doherty M., (1988), Chem.Eng.Sci., 43, 2377. 7. Gumus Z.H., and Ciric A.R., (1997), Comp.Chem.Eng., 21, $983. 8. P6rez-Cisneros, Gani R., and Michelsen M.L., (1997), Chem.Eng.Sci., 52, 527. 9. Gani, R. and Bek-Pedersen, E., (2000),AIChE J.,46 (6), p. 1271-1274
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
523
Development of Software Tools for Crystallization System Synthesis Joseph W. Schroer a, Christianto Wibowo a, Ka M. Ng a'+, Lionel O'Young b a Dept. of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, USA b MC Research and Innovation Center, Mountain View, CA 94041, USA A systematic framework has been developed as part of an effort to expedite the development of crystallization systems. The framework consists of three components: flowsheet synthesis, experimental efforts, and modeling activities. To facilitate the efforts, various software tools have been developed. These include generation of phase diagrams based on thermodynamic calculations, representation of experimental data on phase diagrams, and simulation of particle size distribution of the crystallizer product. The software tools are modular in nature so that engineers in process development can use any of the tools that they like in isolation and add their own in-house tools as appropriate. 1. I N T R O D U C T I O N In this era of globalization, there is a relentless pressure to shorten the time-to-market in the chemical processing industries. It used to take ten years or so to build a large-scale grassroots chemical plant starting from basic chemistry, while most companies are now aiming for four to five years. Systematic methods for process design and development are therefore essential for producing a reliable, optimal process while minimizing development time and effort. Indeed, approaches and techniques in systems engineering have been firmly established for the design of complete gas-liquid plants [1-3]. With the gradual shift of the chemical processing industries towards high-value-added chemicals, most of which are sold in solid form, it is highly desirable to examine solids processes, especially crystallization systems, from the perspective of process systems engineering [4]. During the last few years, we have developed a family of systematic procedures for crystallization systems synthesis [5-8]. The synthesis of downstream solid-liquid separation system to properly recover the crystalline product has also been tackled [9]. We also considered the related problems of selecting crystallizer operating policies [10], and kinetics and mass transfer effects on crystallization process paths [ 11 ]. To facilitate the systematic development of crystallization systems, various software tools have been developed. The software tools are modular in nature so that engineers in process development can use any of the tools that they like in isolation and add their own in-house tools as appropriate. They are flexible, easily updated, and are distributed on the world wide web for easy accessibility. + Present address: Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.
524 2. A SYSTEMATIC FRAMEWORK The task of crystallization system synthesis involves three key activities: separation process synthesis, downstream processing system synthesis, and crystallizer design. While in a typical scenario these three activities are performed sequentially, iteration between steps is often necessary to come up with a superior design. In each step, economic evaluation is implemented to identify the most feasible alternatives. These design activities are closely interconnected with modeling and experimental efforts, as depicted in Figure 1. Solid-fluid equilibrium (SFE) phase diagrams play a central role in our approach, since they are used as a basis for decision-making in separation process synthesis. Therefore, before beginning with this activity, it is important to generate suitable phase diagrams based on solubility data or thermodynamic information. In the absence of experimental data, the multicomponent SFE phase diagrams can be calculated by a thermodynamically consistent model. Necessary thermodynamic data include the heat of fusion and melting temperature, which are tabulated for many substances in standard references, and liquid-phase activity coefficients, which can be predicted using excess Gibbs free-energy models such as UNIQUAC. Further details on the calculation and representation of SFE phase diagram have been published by Dye and Ng [5], and Samant et al. [ 12], among others. After appropriate phase diagrams have been generated, flowsheet alternatives for the separation process are synthesized to meet the separation objectives. The number of
Figure 1. Components of the systematic framework for crystallization system synthesis.
525 crystallizers and the order of separation are determined, additional unit operations are selected. In our approach, unit operations involved in the separation process are represented on the phase diagram as movements from the feed composition points to the product composition points. The aim is to purposefully maneuver to the proper regions on the phase diagram to crystallize the desired products. Limitations such as eutectic points are bypassed using a suitable movement. A more detailed discussion as well as rules to aid decision-making is provided elsewhere [7]. Once the flowsheet alternatives have been generated, preliminary economic evaluation and feasibility checks are performed. The range of design variable values which does not violate practical or material balance constraints is determined. The crystallizer type and operating temperature are also selected. Since a crystallizer does not exist in isolation in a processing plant, the downstream processes around the crystallizer need to be synthesized. These may include unit operations such as preconcentration, filtration, washing, deliquoring, recrystallization, and drying. A systematic procedure on downstream processing systems synthesis has been proposed [9]. Necessary unit operations are selected, and destinations of the mother liquor, wash liquid, and solvents are assigned. A solvent recovery system is then considered to recover solvents and remove impurities. Economic evaluation using shortcut models are then performed to screen process alternatives. Finally, the crystallizer must be designed in such a way that the product specifications such as particle size distribution (PSD) can be met. Since PSD also has a substantial impact on downstream processing, it is crucial that the crystallizer design is done by taking into account the operation of downstream processing units [10, 13]. To some extent, population balance equation-based models can be used to predict the PSD of a crystallizer product [14]. Accurate knowledge of crystallization kinetics is critical in such an effort. Due to complexities such as the presence of secondary nucleation and inhomogeneity of the crystallizer content, laboratory experiments are essential in crystallizer design. 3. S O F T W A R E TOOLS FOR CRYSTALLIZATION SYSTEM SYNTHESIS Our objectives in developing these software tools include the requirements from both the process development community and the educational community. In process development, applications are primarily used by experts; hence flexibility and customization are necessary because processes or products may be new, requiring solutions that are often different from previous applications. Also, code reusability from project to project is important for facilitating rapid process development. Educational and informational uses present the additional requirements of ease of distribution to a large audience and robustness, because programs will be used with people unfamiliar with the codes. Rather than having complete dependence on one format or platform, our codes exist in a mixture of three common forms. Table 1 lists the forms and some of their advantages and disadvantages of the computer languages used: Visual Basic*, FORTRAN, and Java**. All of the forms support features of interoperability. FORTRAN programs can be called by Visual Basic programs or Java programs by dynamic link libraries or input/output file manipulation. MS Excel files (including plots) can be generated by Java programs developed by some software vendors.
526 Table 1. Computer languages used in our computer codes Language Visual Basic
Advantages - Integrated with MS Excel, a common and comfortable interface for engineers.
Disadvantages - Programavailable only on selected platforms. Files are large. -
FORTRAN
- Calculation speed is high. Established in engineering computations. -
Java
Platformdependent: must be recompiled for different platforms / operating systems. - User interface is not appropriate for graphical demonstrations.
-
- Low computing speed. - Flexibility: offers more functionality than Programdevelopment requires more Visual Basic or FORTRAN. expertise. Platform independence: can be run on any computer. Portability: can be distributed as applets and run by web browsers. - Encouragescomponent-based development for code reuse and ease of maintenance. -
-
-
Because these codes are part of a library of software tools for crystallization, they must be dynamically linkable, portable, and have the ability to identify themselves and what they do to other programs. For this reason, Java was selected as the language of choice for many of the programs, thus making them accessible on theoretically any computing platform. In addition, Java allows us to develop programs of much greater complexity and features that are not possible with other languages. Table 2 gives a list of some of the prototype codes and their descriptions. Most of the tools are centered around calculating and displaying SFE phase diagrams. Included are programs for calculating phase diagrams of binary, ternary, and multicomponent mixtures. In addition, some codes aid in representing experimental data in a suitable format and in calculating activity coefficients from data. The Java codes have been assembled in a package named edu.umass.ecs.ngdesign. This package is grouped into three additional packages for organization. The package edu.umass.ecs.ngdesign.demos contains demo programs to illustrate the computer codes. This packages relies on classes that reside in the remaining two packages: edu.umass.ecs.ngdesign.sfe and edu.umass.ecs.ngdesign.graphics. The former contains classes for performing solid-fluid equilibrium calculations for producing phase diagrams. The latter contains classes for rendering graphics and plotting on the computer screen. Documentation for the Java computer codes was done using the Javadoc tool. This tool produces the Application Programmer's Interface (API) in an HTML format that has a standardized look. The Javadoc output lists an object's names, arguments, and return type for each method as well as the object's fields, and allows incorporation of comments added by the programmer. This system allows for code documentation and explanation in such detail that other programmers can easily incorporate the codes into new projects.
527 Table 2. Description of some prototype computer codes. Visual Basic/Excel programs
SFE.xls BatchCC.xls ContC.xls FORTRAN
programs
csdB.exe csdC.exe Java
Contains macros for plotting ternary SFE phase diagrams in equilateral triangular coordinates. Also contains demonstrations of SFE calculations. Provides design guidelines. Plots product PSD and supersaturation profile with data from csdB. Provides design guidelines. Plots product PSD with data from csdC. Calculates the product PSD of batch cooling crystallizers for various operating policies. Calculates the product PSD of continuous crystallizers with various configurations.
programs
Package edu. umass, ecs.ngdesign. demos
BinSFEID 9 TernSFEID
9
9 PolyT3 9 Janecke4c 9 ConjugateSalt 9 Molecular
Calculates and plots binary T-x melting point diagrams for an ideal system. Calculates and plots isothermal and polythermal projection phase diagrams for a 3component ideal mixture. Calculates and plots isothermal and polythermal projection phase diagrams for a 3component nonideal mixture system. Calculates and plots polythermal Janecke projections for a 4-component nonideal system on a ternary phase diagram plot. Calculates and displays a Janecke projection of an isothermal conjugate salt pair phase diagram. Calculates all of the fixed points of an N component mixture of an nonideal molecular simple eutectic system. Returns the list of eutectic compositions, temperatures, and the adjacency and saturation variety matrices. Package edu.umass.ecs.ngdesign.sfe Package edu. umass, ecs. ngdesign. graphics
9 JSplot
Plotting routine for graphing ternary phase diagrams.
4. C O N C L U S I O N S In developing a chemical process, we need to synthesize process altematives, simulate process performance, and generate basic and process data. Our strategy is to forge forward with minimum information; then fine tune the design with increasingly accurate models and data [ 15]. To shorten the cycle time for such an effort, the development team has to possess the right software and experimental tools, generate the right data at the right time, and to share such data and knowledge about the process. It is important that the overall development plan as well as the reasoning of any action is clearly understood among the team members. A design framework that formulates the workflow, facilitates process synthesis and analysis, guides experimental efforts, and provides an environment for data sharing is highly desirable. This article considers such a framework for the development of a crystallization system. It helps to predict SFE phase diagrams, synthesize a crystallization system flowsheet, select the solvent, the crystallizer type and the operating conditions, predict the PSD, organize and interpret experimental SFE data, and so on. However, it is not complete. For example, we considered only particle size distribution, but shape, color and other crystal attributes can be
528 equally important [ 16]. Also, other computer programs can be written. For example, we have not yet included crystallization kinetics and transport limitations in our computer codes. These are not major drawbacks, however. By creating modular codes that can be easily linked and modified, we can readily add new tools or customize existing ones to suit specific needs. This feature is useful during project execution, and allows rapid incorporation of new methods into the framework, thus speeding up the entire research cycle. In addition to process development, we hope that the prototype codes with the demos would help educators teach SFE phase diagrams and their use in process synthesis. Work is under way to expand and refine both the framework and the codes. ACKNOWLEDGMENT
We express our appreciation to the National Science Foundation, Grant No. CTS-9908667, for support of this research. Trademark or registered trademark of Microsoft Corporation. Trademark or registered trademark of Sun Microsystems, Inc. REFERENCES
1. J.M. Douglas, Conceptual Design of Chemical Processes, McGraw-Hill, New York, 1988. 2. L.T. Biegler, I. E. Grossmann and A. W. Westerberg, Systematic Methods of Chemical Process Design, Prentice Hall, Upper Saddle River, 1997. 3. W.D. Seider, J. D. Seader and D. R. Lewin, Process Design Principles: Synthesis, Analysis and Evaluation, Wiley, New York, 1998. 4. S. Rajagopal, K. M. Ng, and J. M. Douglas, Comput. Chem. Eng., 16, 675 (1992). 5. S.R. Dye and K. M. Ng, AIChE J., 41, 1456 (1995). 6. D.A. Berry, S. R. Dye and K. M. Ng, AIChE J., 43, 91 (1997). 7. C. Wibowo and K. M. Ng, AIChE J., 46, 1400 (2000). 8. J.W. Schroer, C. Wibowo and K. M. Ng, AIChE J., accepted for publication (2000). 9. W.-C. Chang and K. M. Ng, AIChE J., 44, 2240 (1998). 10. C. Wibowo and K. M. Ng,, submitted to AIChE J. (2000). 11. V. V. Kelkar and K. M. Ng, AIChE J., 45, 69 (1999). 12. K. D. Samant, D. A. Berry and K. M. Ng, AIChE J., accepted for publication (2000). 13. P. J. Hill and K. M. Ng, AIChEJ., 43, 715 (1997). 14. S. N. Tavare, Industrial Crystallization: Process Simulation Analysis and Design, Plenum Press, New York (1995). 15. L. O'Young, L. Natori, T. G. Pressly and K. M. Ng, , Comp. Chem. Eng., 21, $223 (1997). 16. R. Braatz and S. Hasebe, paper to be presented at Chemical Process Control, Tucson, AZ (2001).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
529
Optimization of ethylene process design G. Sobo6an and P. Glavi6 University of Maribor, Faculty of Chemistry and Chemical Engineering Smetanova 17, 2000 Maribor, Slovenia Process for producing ethylene from a naphtha pyrolysis gas stream has been studied. The influence of components separation was treated with two different simulators using two different sequences of distillation columns. Considerable differences in reboiler and condenser heat flow rates resulted in economic analysis of particular sequences. The same sequence gave the lowest total annual costs in spite of the use of two different simulators compared to the second best option proposed for distillation columns sequence. Use of different models resulted in different heat integrated structures and, therefore, different profits. In general, heat integration resulted in total annual cost reduction between 9 % and 19 %. Total annual costs of the best heat integrated processes were reduced for about 9 MUSD.
1. INTRODUCTION Systematic synthesis of multicomponent separation sequences is an important process design problem in chemical industry. It is concerned with the selection of a separation method and the selection of the best sequence of separators to split a multicomponent mixture into several products of relatively pure species as desired. For solving separation problems in chemical industry, distillation columns are widely employed separators. Distillation sequences can be specified by different methods: heuristic, evolutionary, algorithmic [1, 2, 3]. Columns sequencing in a multicomponent mixture separation has a decisive influence on the economics of the process. The goal of each designer is to find the sequence with the minimum total costs of separation [4]. If there is a system of distillation columns, it is usually classified as a temperature cascade, which depends on the properties of a feed stream. In a distillation column heavy components are separated from light ones. Light components are evaporated and separated as a condensate on the top of the column. Heavy components are removed from the column at the bottom. Separability of a multicomponent system depends on properties of the feed mixture, operating conditions and other additional restrictions. All these factors influence the equipment costs of a particular sequence. Utilities costs have to be taken into consideration, too. In process design a distillation column sequence usually determines the main part of the process costs with regard to the total annual costs. Utility costs depend on the type of the utility (its price) and the heat flow rate in condensers (cold heat flow rate, ~cu) and reboilers (hot heat flow rate, tPHU see Figure 1). The depreciation and the utilities costs represent the operating cost for a single sequence. The best sequence is the one with the lowest total annual costs.
530
Figure 1. Comparison of utilities influence in the system of a five-column (D l-D5) cascade. In distillation heat is supplied to a reboiler and taken away from the condenser. Heat has to be supplied at the temperature above the boiling point temperature of the vapor coming from the reboiler. The heat flow rate in the condenser is given at the temperature below the dew point temperature. In short cut process design boiling and condensation are supposed to occur at constant temperatures. We usually analyse thermally integrated distillation columns in temperature-enthalpy flow rate difference (T-A/) diagram. In the case of two-component mixtures, constant temperatures in the reboiler and the condenser of a distillation column exist and heat flow rates of reboilers and condensers are represented by horizontal lines. We have simulated ethylene production process using two different process simulators, Aspen Plus and Hysys. Two different sequences of six distillation columns were studied. Our intention was to explore whether the use of different simulators would influence the process economics more or the influence of different columns sequences would predominate. In the next step we tried to integrate each of the three variants thermally. 2. HEAT INTEGRATION OF DISTILLATION COLUMNS Distillation of multicomponent mixtures is one of the most common separation operations in the chemical industry. Because of high energy consumption, heat integration is often vital factor when considering multistage separation in distillation columns. In that case columns are coupled to form sequences. These can be specified by different methods: heuristic, evolutionary and algorithmic [1]. The goal of process design is to find the economic most favourable heat integrated sequence. Optimization of distillation columns includes different reflux ratios, pressures, side reboilers/condensers and preheating/cooling of feed mixture [5]. Heat flow rates and temperature levels of different sequences of distillation columns have to be evaluated and out of them the best combination has to be selected. The goal is to reach the lowest possible temperature difference inside the column and the lowest possible heat flow rates. That will result in a better thermal integration possibilities and lower utilities consumption.
531 In the case of a multicomponent separation the predominant utilities consumption usually takes place in distillation columns. It should be pointed out that in our research classical heat integration between distillation columns was considered. This means that total condensers and total reboilers with condenser-reboiler matches were taken into consideration. Every unit using hot or cold utilities contributes to total costs. So the main target in process integration is to make use of every available hot or cold process stream in order to substitute utilities. The more columns are connected to a heat flow cascade, the higher is the potential profit of heat integration. Heat is transferred from a condenser of a distillation column or from a process stream with a higher temperature to a reboiler of another column or to a process stream with a lower temperature. The greatest benefit is to achieve the highest possible extent of condenserreboiler matches. Maximum heat transfer between the two columns is desired. The goal of the process design is to select the structure and parameters to give the minimum sum of the depreciation and operating cost. The general methodology which leads to the best scheme is called process synthesis. We used the thermodynamic method as a first step in the synthesis of ethylene production process case study. Our research was oriented to produce ethylene and five other products of desired purity. We compared three different versions with regard to the total annual costs of the process. 3. CASE STUDY Our research was tested on a separation problem of a six-product mixture for recovery of ethylene and light products from a naphtha-pyrolysis gas stream. The composition of the mixture is shown in Table 1. The objective of the case study is to synthesise a separation train for the recovery of the nine products from steam-cracked naphtha involving about two dozen components using two different simulators. Total flowrate of the feed is 2.095 kg/s, its temperature is 333 K and the pressure is 136.5 kPa [6]. Table 1 Feed compositions and properties Key components Feed amount fraction, xi A hydrogen+ methane 0.48 B ethylene 0.28 C ethane 0.06 D propylene + propane 0.08 E 1,3 butadiene+ trans-2-butene 0.04 F n-pentane 0.06
TB ~ -161.5 -103.75 -88.6 -47/-42.1 -4.45/4 68.73
CES* 53.33 4.78 9.13 4.18 4.13 -
The ethylene separation process can be divided into two parts. In the first part the feed gas is compressed in five stages prior to cryogenic treatment. In the second part the separation of components is carried out in five distillation columns (D2 to D6). The basic process scheme is shown in Figure 2. Two different sequences of distillation columns D-2 to D-6 were studied rigorously. Figure 2 shows the option 2 from the Figure 3. They were chosen because the previous research showed that these two sequences were the most promising [7]. After each compression, the cracked gas is always cooled to 15 ~ (before another flash, Ft-1 to Ft-5). Interstage cooling is required to prevent polymerization and fouling; water and hydrocarbons, condensed at these points are separated from the pyrolysis gas in interstage separators [6].
532
L~_ ~
Ftl
(/~J Fi2 ~_~~[~Ft3 )~[~ Ft4 L)_~_~i-Ft5l ~
D1
F
Figure 2. Basic process scheme for ethylene production- version 2.
AJBCDEF A/BCDEF
....... B/C BC/DEF --~-~_..... ......... D/EF B/CDEF C/DEF-
ElF D/EF--
E/F
Figure 3. Two sequences studied in the ethylene separation process. For the computer simulation of the process process simulators Aspen Plus and Hysys were used. In our research we concentrated on blocks for simulation of distillation columns. Two different models which allow almost identical starting values to be set were used. The ASPEN module DSTWU performs a Winn-Underwood-Gilliland shortcut design calculation [8]. The Peng-Robinson thermodynamic option has been chosen to simulate ethylene separation process with both simulators. The Column Sub-Flowsheet (CSF) model of Hysys is a rigorous model with a wide range of operational parameters setting. It enables very similar definition of starting parameters as in the case of the Aspen Plus simulator [9]. The combination of the DSTWU and CSF models was taken because of the good performances of each. The process parameters obtained by the DSTWU model were then taken as starting values of reflux ratios and number of stages for the CSF model of Hysys. Particular variants were compared using total annual cost estimation. The research was limited to equipment and utilities costs. The economy of different sequences was estimated by using techniques described in literature [7, 10]. The annual operating time for the production of ethylene was 8500 h.
533 RESULTS AND COMPARISON OF MODELS Our goal was to find how simulators and different distillation columns sequences influence the economic analysis. The two different sequences were studied before and after heat integration and their total annual costs were compared. Altogether, three different simulations and economic analyses were taken into consideration. Sequences 1 and 2 (version Aspen Plus1, -2 in Figure 3) were simulated using the Aspen Plus and Hysys simulators. Finally, three versions were compared. The results of the best heat integrated variants are shown in Table 2. (CTE) represents depreciation-, (CTu) utility- and (CTA) total annual cost estimates for the heat integrated process. Version numbers 1 and 3 (Aspen Plus-1 and Hysys-1) show the same process. The CTA values obtained by Aspen Plus are lower mainly because of the lower CTU. Although different simulators were used, both show lower values for CTA than in the case of sequence-2 simulated by Aspen Plus. The CTA savings in all the three cases are very similar, from 17.5 % to 18.7 % according to the non-integrated process. The best two versions are shown in the T-A[ diagram (Figures 4a) and 4b)). Table 2 Comparison of three best versions of heat integrated processes. No. option CTE CTU CTA CTA r e d u c t i o n (S/a) (S/a) (S/a) % 1. Aspen Plus-1 19 792 000 17 868 000 37 660 000 18,73
(a) 1,4
2.
Aspen Plus-2
18 515 000
25 181 000
43 696 000
17,88
1,3
3.
Hysys-1
15 847 000
27 312 000
43 159 000
17,46
1,2
T/~ 150
/PB
hot utility
-'.//
o-I
TI~
[hot utility D-1
150"
m
/~
D-5")/
100 D-61 ' i
50
'
D.
m
D-3
,00
GCC
"
D-6
,
L
j
GCC
~4.
m
,'~ '.i
,,
,
0-4
'
:'4-
-50
-50
D-2
-100
-100
D-2
f
cold utility
cold utility
20
40
60
80
AI/kW
20
40
60
80
A//kW
a) b) Figure 4. T-A/diagram of the sequence 1" a) Aspen Plus-l, b) Aspen Plus-2. Comparison of cases 4a) and 4b) clearly shows that the main differences appear with columns D-3 and D-4 which have better properties for heat integration in the case 4a). These columns are almost totally integrated with each other but there is a great surplus of cold utility demand (columns D-3 and D-4) from external utility in the case 4b).
534 Differences between the three versions tested mainly appear because of the use of two different models (shortcut/rigorous) for simulation of distillation columns. The sequence 1 from the Figure 3 was supposed to be a better option according to CTA than the sequence 2. It had better capabilities for heat integration [7]. The same sequence (sequence 1) showed lower CTA also when using Hysys simulator (compared to Aspen Plus-2). Although the same sequence was applied in the simulation of both simulators significant differences appeared in the final CvA estimation. Differences were bigger between the Aspen Plus-1 and Hysys-1 than in the case of Aspen Plus-2 and Hysys-1. Thermodynamic models were kept the same in both simulators. Different heat flow rates and different temperatures gave different conditions for heat integration. The base process schemes in both simulators (Aspen Plus-1 and Hysys-1) were different, mostly because of the so called thermal properties. The higher CVA savings influence the longer payback period (Table 3). The Aspen Plus-1 option has the greatest CTA-reduction but also the highest investment. 4. CONCLUSIONS Process for separation of ethylene and five other products was designed with simultaneous consideration of economic and technological point of view. As the main product we wanted to obtain all the six key components at the desired purity. Heat integration between the distillation columns and their integration with other process streams is important. Energy savings achieved by this integration could exceed those obtained by mutual columns integration. The basic process scheme was simulated using two different sequences of distillation columns and two different process simulators, Aspen Plus and Hysys. Altogether, three different cases where studied. Each one was thermally integrated and studied according to the total annual costs. These were reduced by 10-19 % with total annual savings from 8.7 MUSD to 9.5 MUSD compared to the non-integrated process. This was achieved mostly by reduced areas of reboilers/condensers but the main savings showed in utilities reduction. REFERENCES
1. M. K. Kattan and P. L. Douglas, A New Approach to Thermal Integration of Distillation Sequences, Can J Chem Eng, 64/February, (1986) 162-170. 2. D. Trigueros, C. Coronado-Velasco, A. Gomez-Munoz, Synthesize Simple Distillation the Thermodynamic Way, Chem Eng, August, (1989) 129-134. 3. N. Nishida, G. Stephanopoulos and A. W. Westerberg, A Review of Process Synthesis, AIChE J., 27/3, (1981) 321-351. 4. G. Sobo6an and P. Glavi6, Optimization of Ethanol fermentation process design, App Therm Eng, 20, (2000) 529-543. 5. W. Rajah and G. T. Polley, Synthesis of Practical Distillation Schemes, Trans IChemE, 73/A, (1995) 953-966. 6. CACHE Case Study, Separation System for Recovery of Ethylene and Light Products from a Naphta-Pyrolysis Gas Stream, Camegie-Mellon University Pittsburgh (1983). 7. G. Sobo6an and P. Glavi6, A New Method for Studying Thermally Integrated Distillation Sequences, Comput Chem Eng, 20, Suppl. A, (1999) 183-188. 8. Aspen Plus, User Guide, Software Version, (1988). 9. Hysys, Reference Volume 1,2, Version 1.1, Hyprotech Ltd., Calgary (1996). 10. C. W. Hui and S. Ahmad, Total Site Heat Integration Using the Utility System, Comp Chem Eng 18/8, (1994) 729-742.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
535
Optimization of an Acidic Chlorine Scrubber with a Rate-Based Simulation Engine W. Steinbach, A. Friedl., H. Hofbauer Institute of Chemical Engineering, Vienna University of Technology Getreidemarkt 9/159, 1060 Wien, Austria email: wstein(afriedl,hhofba)@mail.zserv.tuwien, ac.at The absorption of chlorine from an exhaust gas stream into an aqueous acidic solution of ferrous chloride is modeled. Chemical reaction kinetics as well as mass and heat transfer is taken into account to determine the rate of absorption. The calculation is performed by the AspenPlus - RateFrac TM simulation engine using the ELECNRTL property method. Chemical property data is checked and several parameters based on the electrolyte NRTL activity coefficient model are regressed. A sensitivity analysis is carried out to optimize the operating conditions and the design of a random packed column with respect to the off-gas concentration of chlorine. 1 INTRODUCTION Chlorine is a poison gas, that must be removed from exhaust gases of many processes down to a legal limit of concentration. In steel industry, where HC1 is used to remove rust and scale from the metal surface, C12 can be formed while regenerating the used acid. Therefore a scrubber fed with an alkaline solution of sodium hydroxide and thiosulfate is provided. This chemicals are not needed if the iron-II containing pickling solution itself is used as washing liquid. Because reactions in electrolytic systems are usually fast and the absorption can not be calculated with an equilibrium model, the design of the column in dimension and pumparound depends on the kinetics of the process. Thus, a rate-based simulation taking into account the three terms of kinetics, mass transfer, heat transfer and chemical reaction kinetics will lead to a successful prediction of the absorption process. The scrubber is a random packed column with counter current flow as shown in Figure 1. The gaseous inlet stream can consist of typical combustion gas containing H20, N2, 02, CO2 and small amounts of HC1 and C12. The liquid inlet stream is the spent pickle liquor containing H20, HC1, FeC12 and small amounts of FeC13. Temperature is considered to be 85~ for the gas and for the liquid, the scrubber works at atmospheric pressure.
536 ILIQIN I
9
GAsouT
9
I~,ooo, J
Fig. 1: Principal flowsheet of a chlorine-scrubber 2 THERMODYNAMIC AND TRANSPORT PROPERTIES
AspenPlus TM recommends a predefined property method called ELECNRTL which was used in the simulation. The physical models contained in this property method are listed in table 1. Table 1 Summary of physical models used Common
Vapor pressure Heat of vaporization Surface tension
Vapor mixture Fugacity coefficient, Density Enthalpy, Entropy, Gibbs energy Vapor viscosity Vapor thermal conductivity Vapor diffusivity
Extended Antoine Watson/DIPPR Hakim-Steinberg-Stiel/DIPPR- Onsager-Samara Redlich-Kwong Ideal gas heat capacity / DIPPR, Barin correlation, Redlich-Kwong Chapman-Enskog-Brokaw Stiel-Thodos / DIPPR Dawson-Khoury-Kobayashi
537
Liquid mixture Activity coefficient, Gibbs energy Liquid molar volume Infinite dilution heat capacity Enthalpy, Entropy
Electrolyte NRTL, Extended Antoine, Henry's constant, Brelvi-O'Connell Rackett, Clarke Criss-Cobble Ideal gas heat capacity / DIPPR, Watson / DIPPR heat of vaporization, Criss-Cobble infinite dilution heat capacity, Electrolyte NRTL Rackett / Clarke Andrade/DIPPR- Jones-Dole Sato-Riedel/DIPPR- Riedel Wilke-Chang - Nemst-Hartley
Density Liquid viscosity Liquid thermal conductivity Liquid diffusivity
Reprinted from AspenPlus T M Handbook, 1999 As AspenPlus extends the calculation of physical properties from binary mixture to multi component mixtures the property data for the simulated system must be checked by literature [2,3]. Parameters of the electrolyte NRTL model were correlated to literature data from [3] for the systems H20-C12 and HzO-HC1-CI2. TM
100
800
E E
+,
I--.--.I
-1-
SystemH20-CI2
9
60-
80 ~ B
SystemH20-HCI-CI2
=i4o
,ooc
..
40-r 20
20-
20
i
I
I
I
40
60
80
100
0 I 0
T [~
, 4
, 8
50~
HCI [ m o l l l ]
Fig.2,3" Henry's Constant; Comparison of literature data [3] and AspenPlus results after regression.
3 ABSORPTION MODEL The overall reaction of chlorine is described as follows.
12
538
C12 + 2 Fe 2+ :=> 2 C I + 2 Fe 3+
The kinetic of this irreversible reaction is known to be in the second order (first order with respect to both C12 and Fe2+-ions) [1 ]. By analyzing Hatta's number for irreversible secondorder reaction with reasonable values, it can be found, that the absorption kinetic of chlorine is determined by mass transfer and not by reaction kinetics. Mass transfer coefficients and the interfacial area available for mass transfer in packed columns is calculated using the correlation developed by [4].
{lexpl 14 .e~ k t = 0,051. (Retw) 2/3 9
1/2
.(ap .dp) ~
(1)
"/g'/~t 1 / 3 p/ t
kg = 5,23 . (Reg )~ . (Scg,C12 )l~3 . (ap . dp ~ 2 .
(2)
a p . D g,cl2
R.rg
(3)
The heat transfer coefficient is calculated, using the Chilton-Colbum analogy [5]. kavSc2/3=
(4)
htc Cp,mix
The dissociation of HC1, FeC12 and FeC13 are defined by the following reaction equations. HC1 + H20 ~ FeC12 ~
CI + H 3 0 +
Fe 2+ + 2 CI
FeC13 ==:, Fe 3+ + 3 CI
4 OPTIMIZATION With the method of sensitivity analysis, where you vary one or more variables over a wide range of values you can get a very clear picture of the optimal design parameters and operation conditions. Because the height of a column is often limited due to construction reasons, the variables remaining for this optimization are the column diameter, the liquid pumparound and the size
539 of the packing. The packing is only available with certain sizes, thus two variables, column diameter and pumparound, are varied. The efficiency of absorption is defined by the relation of incoming to absorbed amount of chlorine. The results are shown in the figures 4 to 7. Table 2 Legend for figure 4 to 7
_Symbol
E ~
A 1 9 9
E
97 % 94 % 91% 88%
5
%" 100
4
~
,ag I1) o')
m r
8o
3
60
2
40
r~
"5 -J
"-i
0 0
Fig. 4:
20 0
,
1
Gas
3 4 Charge [kg/rn=s]
0
dv =
1 inch
Fig 5: dv = 2 inch
2
'~n 20
~ ' 6000
E
E
~15
,--, 4000
~. 10
--
.I= ~9
9~ ,
, m
--
2
3
4
3
4
5
~
~-~
o
1
Gas Charge [kglm=s]
~"
o
5
-~
2000
, m
, ~ ,
0
--
0
1
2
3
4
Gas Charge [kglmZs]
Fig. 6:
dp =
1,5 inch
5
0 0
1
2
Gas Charge [kglm=s]
Fig 7:
dp =
3,5 inch
5 CONCLUSIONS The results of the simulation show that the influence of the packing diameter, which is correlated to the specific interfacial area by equation (1), has an extremely strong influence on the performance of the absorber with a given gas charge. This significant results would not have been obtained with a equilibrium based simulation. An absorber could only be designed with a lot of experience and oversizing. The rate-based
540 simulation as shown in this practical example gives us the opportunity to design a scrubber near to its optimum. REFERENCES 1. 2. 3. 4. 5.
H. Hikita et al., Chem. Eng. Sci., 30 (1975) 607. F. Hine, et al., Bull. Chem. Soc. Jpn., 41 (1968) 71. C.C. Chen and L.B. Evans, AIChE J., 32 (1986) 444. Onda et al., J. Chem. Eng. Japan, 1 (1968) 56. F. King, M. Spiro, J. Solution Chem., 12 (1983) 65.
EuropeanSymposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B.Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
541
An Accelerated Branch-and-Bound Algorithm for Assignment Problems of Utility Systems Alexandros M. Strouvalisa, Istvan Heckl b, Ferenc Friedlerb and Antonis C. Kokossis c* Department of Process Integration, UMIST, P.O. Box 88, Manchester M60 1QD, UK. b Department of Computer Science, University of Veszpr6m, Egyetem u. 10, Veszpr6m H-8200, Hungary. c Department of Chemical and Process Engineering, School of Engineering in the Environment, University of Surrey, Guildford, GU2 7XH, UK.
a
The paper presents a methodology for integrating logic and engineering knowledge within a Branch-and-Bound algorithm with purpose to accelerate convergence. The development addresses assignment problems of utility networks with emphasis on the optimal allocation of units over periods for maintenance. The solver exploits the special structure of the problem to (i) exclude redundant combination of variables, (ii) prioritise the branching of nodes, (iii) provide bounds of nodes and (v) prune inferior parts of the binary tree. Extraction of knowledge and analysis of operations is supported by the graphical environment of the Hardware Composites. Comparisons with commercial MILP solvers demonstrate the merits of customising the solution search engine to the particular solution space. 1. INTRODUCTION The impact of Mathematical Programming and Optimisation proves significant through a variety of applications in design and operations. The Operations Research community contributed considerable part of the available optimisation tools. At their best, they epitomise general theoretical, computational and numerical knowledge in relevance to the different classes of problems considered. The result is application of general-purpose solvers designed to address formulations ranging from financial problems to chemical process design. Despite numerous efforts the proposed interfaces with solvers exhibit inferior performances as they are not capable of capturing the intricacies of the particular application. In the absence of specific knowledge, the use of general heuristics devotes a large computational effort to redundant searches and formulations that artificially expand the solution space. The importance of including logic in the modelling stage was highlighted by Raman and Grossmann (1992) who employed a combination of heuristics and logic to solve MINLP problems. The same authors later (1993) used inference logic to branch on decision variables and (1994) implemented logical disjunctions as mixed-integer constraints. Solution of MILP's is mainly addressed through application of the Branch-and-Bound algorithm. The algorithmic efficiency relies on the selection criteria for candidate problems, bounding, pruning and branching (Geoffrion et al. (1972)). As Forrest et al. (1974) mentioned, important B&B Corresponding author.
542 functions are promptly determined if the user has adequate knowledge of the physical problem. The Hardware Composites (Mavromatis and Kokossis (1998), Strouvalis et al. (1998)) are employed to reveal information and insights of utility networks that would otherwise be impractical or expensive to acquire by algorithmic approaches. The Hardware Composites not only assist in customising and tuning solvers but also analyse solution space properties and their computational impact. 2. PROBLEM DESCRIPTION AND MODEL ANALYSIS The problem considers the maintenance scheduling of turbines and boilers (assignment of tasks to periods). Objective is identification of the optimal sequence to shut-down units for inspection and maintenance with minimum disruption of the utility operation. Switching-off units imposes penalties to objective function as less efficient units or options (i.e. power purchase) are employed to compensate for the ones maintained. Optimisation has to consider demand variations over time, differences in efficiencies of units and feasibility aspects. The formulations yield MILP problems with significant number of variables. Even for moderate networks the problem can become highly combinatorial and expensive to solve. This class of scheduling problems exhibits the block angular structure with periods coupled to each other due to binary variables assigned for the ON/OFF status of units. Binary variables are present in maintenance and individual period constraints. The special model structure is exploited to set up the B&B algorithm. Especially the customised bounding and pruning criteria capitalise on decoupled combinations of maintenance scenarios while investing computational effort on options associated with linking constraints. 3. SOLVER CUSTOMISATION The customisation spans the main stages of a B&B algorithm. The incorporation of knowledge introduces: 1. Assignment of priorities for the selection of candidate subproblems and branching of variables. 2. Bounding with customised use of the LP solver. 3. Customised tests for minimising the enumerated nodes (enhanced pruning). The B&B solver is implemented in C++ with application of L I N X - a simplex-based routine collection- (Fabian 1992) as LP solver. 3.1. Assignment of Priorities As units are switched-off, the imposed to objective function penalties vary with the unit efficiency, the efficiency of the units available to replace them and the demands of the particular period. Each period is affected to a different extent and preferences/priorities are strong functions of the layout of demands. High preferences relate to minor alterations in the operation of the utility network. The period prioritisation is rigorously defined through calculation of penalties associated with the shut-down of single units in available periods. The basic priority lists are then defined by ordered sets PL(u) = (Pi, Pj..... Pn) for unit u, with period Pi assigned to higher priority than Pj,..., Pn (u switch-off in Pi contributes less increase to objective function compared to Pj,..., Pn).
543 Priorities are established among units as well. Their definition is based on a hierarchical analysis reflecting the relative importance of turbines and boilers. The visualisation of solution space by Hardware Composites offers qualitative understanding of unit efficiencies and capacity limits in view of specific periods and entire sets of demands. The unit prioritisation is included in ordered set PU - (Ua, Ub.... ,Uk), with unit Ua assigned with higher priority than Ub.... , Uk (Ua is more important to operation of the system than Ub,...,Uk). The assignment of priorities (period and unit) is referred to as the preprocessing stage and requires the solution of a number of LP's. The resources spent during preprocessing represent the computational cost of prioritising and revealing the structure of the solution space. Based on sets PL(u) and PU the selection of candidate subproblems and branching of nodes is arranged and customised. 3.2. Calculation of Lower Bounds The B&B solver performs bounding by capitalising on information acquired during preprocessing. Instead of calling the LP solver to estimate bounds at every branched node, a more refined policy of solving LP's is adopted. Enumerated nodes are classified as dependent and independent. A node is dependent if it involves the shut-down of more than one unit in a period or if this node is infeasible. Otherwise the node is termed independent. Independent nodes relate to decoupled maintenance combinations (units switched-off in different periods). These nodes qualify for having their lower bounds defined by already available and computationally inexpensive information. On the contrary, nodes of coupled operations (dependent) necessitate separate bounding. In that manner the LP solver utilisation results to reduced resources spent on calculation of bounds. 3.3. Enhanced Pruning Enhanced pruning uses the properties of dependent and independent nodes. For independent combinations tests are made using the priority lists. The tests select the combinations to enumerate and exclude a further enumeration of nodes (as having a guaranteed lower potential). Units associated with identical maintenance periods or infeasibility at a visited node are the ones justified for relaxation of the priority sequence by examining the next period(s) in preference. When a feasible independent terminal node has been reached or an independent node has been pruned then nodes involving lower periods in priority are pruned without enumeration. It is pointed out that enhanced pruning is rigorous and does not compromise on the optimality of the solution.
4. I L L U S T R A T I O N EXAMPLE The utility system of Fig. l(a) includes 9 units (steam turbines TI-Ts, gas turbine GT and boilers B1-B3) while operating horizon consists of 24 equal-length periods. Each period relates to a pair of constant demands in power and process steam as shown in Fig. l(b). Preventive maintenance imposes the shut-down of all units for one period. The optimal scheduling is expected to meet all demands and maintenance needs in the most efficient way. The problem is modelled as MILP with integer variables assigned for the status of units per period. The complete model properties are represented in Table 1.
544 Table 1: Model size for illustration example. Continuous Variables
Binary Variables
Constraints
Non-zero Elements
457
216
754
2257
Fig. 1: (a) Utility network and (b) sets of demands in power and heat. The solution of the problem is addressed in two steps: I) preprocessing of the solution space and II) application of the customised B&B solver. STEP I:
The solution space is analysed to reveal the feasible and prioritised options for the B&B solver to navigate. Preprocessing is applied in conceptual and computational terms. The location of demands on the solution space (Fig. 3) in relevance to hardware limits identifies infeasible scenarios which are excluded from optimisation. Separate LP's are solved to calculate penalties associated to the shut-down of individual units in feasible periods. Penalties in increasing sequence define priority lists PL(u) for each unit u. The penalties are furthermore used in conjunction to conceptual analysis for defining a qualitative importance of units to operations. In that manner the unit prioritisation list PU is also determined. Ordered sets PL(u) and PU formulate the solver matrix (Fig. 2) representing the prioritised solution space. The first column includes all units subject to maintenance arranged according to PU (upper elements - GT, T2, T1, T3.... - relate to higher priority units). Each element-unit of the first column associates to the corresponding period priority list PL(u) defining the rows of the matrix. STEP II:
The customised B&B searches the solution space capitalising on the structure of the solver matrix. The branching of the binary tree initiates from the first elements of the upper rows and proceeds to deeper options only if specific properties hold. The enhanced pruning effectively disregards inferior parts of the tree from enumeration. The result is acceleration of the B&B algorithm compared to the solution of the same model by OSL implemented in GAMS (Table 2).
545
-GT
5
3
15
14
16
13
6
2
7
T2
15
14
4
3
5
16
13
6
2
7
8
17
12 9
T1
5
6
4
2
3
16
13
1
7
24
12
8
17
T3
4
5
3
15
6
2
16
13
14
7
12
8
1
T5
5
18
6
11
4
1
24
2
12
7
3
23
8
BI
5
4
6
1
11
18
24
2
12
3
23
7
20
1
6
2
20
4
5
11
18
24
12
3
23
7
B3
19 20
10
9
22
21
14 17 23
8
18
15
T4
10 21
22
9
17
23
18
11
24
12
Bz
14
8
1
Fig. 2" The solver matrix reflects the B&B customisation. Table 2: Computational results for illustration example.
Nodes: Iterations: Solved LP's: CPU(sec)- 333 MHz: Objective-($/oper.horizon): (Relaxed Objective) Preprocessing Stage" 217 LP's -
Customised B&B 188 76 13.2 1,021,133.4 (1,004,690) 26.2 CPU(sec)
OSL(GAMS) 50,402 150,000 (interrupted) 50,404 1,339 1,022,899.6 (1,005,439.7)
OSL solver invested significant computational effort in searching the solution space. Even at the iteration limit of 150,000 optimality had not been reached due to the suboptimal flat profile the solver was trapped in. Alternatively, the customised B&B performed better in all aspects and identified the optimal schedule after solving (217 + 76) LP's during preprocessing and Branching-and-Bounding respectively. Optimal maintenance is performed according to vector: (GT, T2, T1, T3, Ts, B1, B2, B3, T4) = (5, 15, 6, 4, 18, 4, 1, 19,10). 5. CONCLUSIONS This work reports on the advantages observed from the customisation in optimisation applications. Experience has shown that general-purpose solvers fail to capture and profit from special properties of problems. Ad hoc inexpensive solvers prove superior to expensive commercial packages. Customised solution search engines with built-in intelligence and search technology perform better orders of magnitude. The capability to apply the basic B&B functions (branching, pruning) tailored to the structure of the solution space accelerates convergence and reduces computational cost.
546
Fig. 3: The Hardware Composites represent the solution space of the utility network. REFERENCES 1. Fabian, C.I. (1992) LINX: An interactive linear programming library. 2. Forrest, J.J.H., Hirst, J.P.H. and Tomlin, J.A. (1974) Practical solution of the large mixed integer programming problems with Umpire, Management Science, 20(5), 736. 3. Geoffrion, A.M. and Marsten, R.E. (1972) Integer programming algorithms: a framework and state-of-the-art survey, Management Science, 18(9), 465. 4. Mavromatis, S.P., and Kokossis, A.C. (1998). Hardware Composites: a new conceptual tool for the analysis and optimisation of steam turbine networks in chemical process industries Parts I & II, Chemical Engineering Science, 53(7), 1405. 5. Raman, R. and Grossmann, I.E. (1992) Integration of logic and heuristic knowledge in MINLP optimisation for process synthesis, Computers and Chemical Engineering, 16(3), 155. 6. Raman, R. and Grossmann, I.E. (1993) Symbolic Integration of logic in Mixed-Integer Linear Programming Techniques for process synthesis, Computers and Chemical Engineering, 17(9), 909. 7. Raman, R. and Grossmann, I.E. (1994) Modelling and computational techniques for logic based integer programming, Computers and Chemical Engineering, 18(7), 563. 8. Strouvalis, A.M., Mavromatis S.P., and Kokossis, A.C. (1998). Conceptual optimisation of utility networks using hardware and comprehensive hardware composites, Computers and Chemical Engineering, 22, S 175.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Retrofit Design of Chemical Processing Application to Petrochemical Industry
547
Networks
under
Uncertainties:
Min-ho Suh 1, Ferenc Friedler2, Sunwon Park 1, and Tai-yong Lee 1. 1 Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-gu, Taejon, 305-701, Korea 2 Department of Computer Science, University of Veszprem, Veszprem, Egyetem u. 10, H8200, Hungary Multiscenario retrofit design of petrochemical processing networks is addressed in this paper. The combinatorial framework developed for process network synthesis can be used to resolve the computational complexity of the retrofit design. Retrofit design of Korean petrochemical industries under product demand uncertainty illustrates the efficacy of the proposed algorithm. We obtain Pareto optimal solutions of two objectives, namely expected cost and worst-case cost. The robust optimal solution of retrofit design under uncertainty can be determined among the Pareto optimal solutions. 1. INTRODUCTION Retrofit design means addition of new units and expansion of existing units to satisfy the economic needs and product demand requirements. In retrofit design of a chemical processing network, decisions on structural variables such as process network configuration and capacity expansions, have to be made under forecasted uncertain parameters, e.g., product demand and material cost data. Since these parameters usually highly affect the profitability of the system, uncertainties should be taken into account in the design. The most common way of representing the uncertainties is to specify scenarios of the expected values of the parameters. Based on the scenario-based approach, multiscenario mathematical model can be driven by the stochastic programming framework. In comparison with the deterministic model that doesn't consider the parameter uncertainty, the stochastic model forms a large-size problem due to the scenario-dependent variables and constraints. Need for an efficient solution algorithm is emphasized in design models considering uncertainties. Moreover, we need to solve the model repeatedly to obtain the Pareto optimal solutions which is an important procedure in decision making under uncertainties. Together with process network synthesis for new process design, the retrofit design of chemical processing network has common binary decision variables of unit existence. The combinatorial framework for process network synthesis was proposed by Friedler et al [1-4]. P-graph theory and combinatorial algorithms * Corresponding author:
[email protected] 548 are rebuilt to adapt to the multiscenario retrofit design problem. Retrofit design of Korean petrochemical industries illustrates the efficacy of the proposed algorithm and robust design approach. 2. DESIGN UNDER UNCERTAINTIES
In representing the uncertain parameters, the scenario-based approach is one of the most commonly applied methods. Uncertain product demands are represented as the realizable scenarios and their probabilities in this paper. Absolute robustness concept [5] is applied to the robust optimization of retrofit design problem. The absolute robustness means the cost we accept when the worst-case scenario is realized. Consequently, the two objectives of the stochastic programming model are the expected cost and the worst-case cost. Computational aspects of multiscenario retrofit design model are addressed. 2.1 Mathematical model
In this multiscenario mathematical model, the expected cost is considered to be the objective function by constraining the required range of the worst-case cost. The objective is minimizing the expected cost. min EXCOST (1) The expected cost is calculated using p.,, the probability of scenario s.
s.t EXCOST=~p.,C,
(2)
s
Costs of all scenarios are constrained by the required worst-case cost, C w . C w >_C,, Vs
(3)
C,. is the cost of scenario s and calculated as the sum of investment cost, operating cost, material cost, and transportation cost. Indices i, j, p represent processing unit, material, plant, respectively.
i
p
i
p
"
P ~J~JRP
( j,p,p')et
T:jpp. is the transportation amount of material j from plant p to plant p ' in scenario s when (j, p, p ') is the member of the transportation allowance set t(j, p, p'). Mass balances, operating level limitations, and capacity limitations lead to the following constraints: ~" rauW~.,p i
ZraoW,.ip i
~" T.~jpp.+
~ T,jp..p > MIN.~p Vs, j ~ Jp , p
p':(j,p,p')et
p":(j,p",p)et
p':(j,p,p')et
p":(j,p",p)et
ZTdpp,+
ZT,jp,, p
~__-MAX~jpVs, j
~ Jg,p
(5) (6)
549
~-'raoW.~.ip i
~f'T~jpp.+ p':(j,p,p')et
~QT~jp..p> 0 Vs, j ~ JI~,P
(7)
p":(j,p",p)et
Equation (5) means that the amount of product j, to be produced at plant p in scenario s, should be at least MINv p . Equation (6) means that the amount of raw material j, to be used at plant p in scenario s, should be at most MAX vp. ra o is the mass balance coefficient which is negative when the material j is input to the unit i and positive when the material j is output from the unit i. Equation (8) limits the operating level W,~pto the capacity of units Qip.
W~.gp< Q,p Vs, i, p
(8)
Q~pis the sum of original capacity QOip and expansion capacity QEip. Q,p = QO~p + QE~p vi, p
(9)
QEip has lower and upper bounds which linked to the binary variable. QEL~
< QEip < QEUPipY~p Vi, p
(10)
Equations (11) and (12) represent variable conditions. Y,p ~ {0,1}
(11)
OE,p , O,p , W , , , T~jpp >- 0
(12)
2.2 Complexities in solving the multiseenario retrofit design model This multiscenario model can be driven in the form of MILP with scenario-independent structural binary variables related to the process synthesis part of the design and scenariodependent continuous variables related to the uncertainties. The large number of scenariodependent variables and constraints makes the already complex process design problem more difficult to solve. A possible way of reducing the complexity of the problem is to exploit the combinatorial properties of feasible processing networks as done in the combinatorial framework developed for process network synthesis. 3. COMBINATORIAL ALGORITHM The combinatorial framework of process network synthesis has been extended to the multiscenario retrofit design problem by adapting the combinatorial axiom system and altering the search direction on the P-graph from backward to forward direction. The basic algorithms, including the ABB (Accelerated Branch-and-Bound) algorithm have been extended to solve the multiscenario model by keeping its original efficacy.
550
NITRILE \
.
|
t
r
~
v =~
' ==
r
CAPROLACTAM
~TEREP~
PVC
,
HTHALIC
Figure 1. Maximal structure of one plant in the petrochemical processing networks.
3.1 Retrofit design feature There is no difference between the retrofit design and the process network synthesis in representing the new units. Also there is no limitation on representing two or more units of which inputs and outputs are the same in P-graph representation of retrofit design. In the retrofit design, we represent extended capacities of existing units as capacities of additional units, which have the same inputs and outputs with their corresponding existing units. 3.2 Multiplant and transportation consideration We represent the transportation routes of transportable materials as units with transportation cost. All the plants are regarded as a single processing network and the transportation of materials are represented by the above-mentioned method. The investment costs for units of transportation are nulls. 3.3 Forward direction search It is assumed that all the products should be included in the solution networks when we carry out a process network synthesis using the P-graph theory. But in retrofit design of petrochemical processing networks, we determine the profitable products among the candidate products and some of them need not be included in the solution networks. The potential product concept is adopted to represent the situation of product selection in retrofit design. As shown in Figure 1, all the petrochemical products are produced from Naphtha, the main raw material. The product-oriented search algorithm of the original P-graph theory is changed to the raw- material-oriented search algorithm.
551 3.4 Computational aspect from the multiscenario point of view The acceleration of ABB algorithm is attributable to: (i) the reduction of the initial relaxed problem to the set of combinatorially feasible structures; and (ii) the reduction in the sizes of the individual partial problems [4]. The second scheme is very effective when the ABB algorithm is applied to the multiscenario model because the reducing effect in the sizes of partial problems also is proportional to the number of scenarios. 4. RETROFIT DESIGN OF KOREAN PETROCHEMICAL INDUSTRIES
The new algorithm has been tested for Korean petrochemical industries under product demand uncertainty. In this retrofit design problem, various types of petrochemicals are produced from naphtha. The optimal solutions have been generated for the expected cost, the worst-case cost and the Pareto set of the two objectives. 4.1 Problem description There are four petrochemical complexes which are located in Inchon, Ulsan, Yosu, and Daesan in Korea. Each plant has the same maximal structure as shown in Figure 1. Some intermediate materials can be transported from one plant to another with transportation costs. All the plants have their product demands and three demand scenarios are assumed by the forecast for petrochemical product demand of domestic, China and southeastern Asian market. Retrofit period is 10 years. Scenario 1 expects 20% annual growth rate of synthetic resin (HDPE, LDPE, LLDPE, PP, PS, etc.) market and 10% annual growth rate for the rest of the products in the market with probability of 0.3. Scenario 2 expects 15% annual growth of aromatic derivatives (PS, ABS, Caprolactam, TPA, and pthalic anhydride) and 8% annual growth rate for the rest of the products with probability of 0.3. Scenario 3 expects 10% annual growth of all the products with probability of 0.4. Basic petrochemical processing networks configuration, existing capacity and demand, and material price data are imported from Bok et al [6]. 950
770
760
u~
.~_ 800
"~
8
.,... ~ 750
750
Ill "/40 - - e - - Cost of scenario 1 - . o - Cost of scenano 2 Cost of scenario 3
Expected Cost
730 850
0
2
4
6
8
10
12
Absolute Robustness Enhanced
Figure 2. Behavior of robust optimal solutions.
,
~
860
870
880
Worst-case cost (million $)
Figure 3. Pareto curve for decision making.
552 4.2 Results The problem is solved using both the general MILP solver OSL implemented in GAMS 2.25 and the proposed algorithm implemented in C++. Computation times are 942 seconds and 133 seconds, respectively, in minimizing the expected cost without the constraint of worst-case cost requirement. The computation test was carried out on a Pentium-III, 550 MHz. Robust optimal solutions are obtained by constraining the required worst-case cost as shown in Figure 2. Instead of increasing like the costs of scenario 1 and 3, the worst-case cost of scenario 2 decreases as the absolute robustness is enhanced. Figure 3 shows the Pareto curve for the expected cost and the worst-case cost. Decision maker can determine the best design solution to be invested with his or her preference to the worst-case risk and expected cost over the uncertainty.
5. CONCLUSION Combinatorial algorithm for multiscenario retrofit design of petrochemical processing networks was proposed. Retrofit design of Korean petrochemical industries was carried out using the proposed algorithm and robust optimal design method. This industrial scale problem illustrated the efficacy of the proposed method. Often the long-term design problem of chemical processing networks can be modeled as a multiperiod design model. Our future research will be focused on the extension of the algorithm to the multiperiod model with keeping the efficacy of the combinatorial framework. ACKNOWLEDGEMENT This work was partially supported by the Brain Korea 21 Projects and SK Engineering & Construction. This research was also supported in part by the Hungarian Science Foundation Grant No. T-029309. REFERENCES 1. F. Friedler, K. Tarjan, Y. W. Huang, and L. T. Fan, Graph-Theoretic Approach to Process Synthesis: Axioms and Theorems, Chem. Engng Sci., 47 (1992) 1973. 2. F. Friedler, K. Tarjan, Y. W. Huang, and L. T. Fan, Graph-Theoretic Approach to Process Synthesis: Polynomial Algorithm for Maximal Structure Generation, Computers chem. Engng, 17 (1993) 929. 3. F. Friedler, J. B. Varga, and L. T. Fan, Decision-Mapping: A Tool for Consistent and Complete Decisions in Process Synthesis, Chem. Engng Sci., 50 (1995) 1755. 4. F. Friedler, J. B. Varga, E. Feher, and L. T. Fan, Combinatorially Accelerated Branch-and-Bound Method for Solving the MIP Model of Process Network Synthesis, Nonconvex Optimization and Its Applications, State of the Art in Global Optimization, Computational Methods and Applications (Eds: C. A. Floudas and P. M. Pardalos), pp. 609-626, Kluwer Academic Publishers, Dordrecht, 1996. 5. G. Yu. On the max-min 0-1 knapsack problem with robust optimization applications, Oper. Res., 44 (1996) 407. 6. J-K. Bok, H. Lee, and S. Park, Robust investment model for long-range capacity expansion of chemical processing networks under uncertain demand forecast scenarios, Computers chem. Engng, 22 (1998) 1037.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
553
Optimisation of an industrial scale ethanol dehydration plant: A case study Z. Szitkai, Z. Lelkes, E. Rev, Z. Fonyo Chemical Engineering Department, Budapest University of Technology and Economics, H- 1521 Budapest, Hungary An industrial scale hybrid ethanol dehydration system is modelled and optimised using MINLP. The system consists of a distillation column for approaching the ethanol/water azeotrope and a pervaporation unit producing pure ethanol. The optimal design and operating parameters including number of trays, feed location, reflux ratio, number of membrane sections in series and the number of membrane modules in each section are determined. Compared to an existing plant, 12 % savings in the total annual cost can be achived by applying 32 % more membrane surface, in consequence of a radical decrease in the reflux ratio (3.3 to 1.4) in the column, and by producing less concentrated alcohol in the distillate. Sensitivity of the total annual cost to the specified ethanol yield, overall membrane surface and membrane replacement cost is studied. Although our superstructure enables partial permeate recycling, total recycling of the permeate flow proved to be optimal in all the realistic cases. 1. INTRODUCTION Distillation processes are the most widespread for ethanol dehydration, in the industrial practice. Either pressure-swing or extractive or azeotrope distillation is applied (Widagdo and Seider, 1996; Lelkes et al, 1998), high operational costs have to be faced. Pervaporation is an emerging membrane separation technology with the merit of low operational costs. A promising technology for ethanol dehydration is the distillationpervaporation hybrid system, which alloys the advantages of both distillation and pervaporation. In this article the hybrid distillation-pervaporation process is dealt with. In our MINLP formulation both the distillation and the membrane modules are rigorously modelled. Optimization of the pervaporation system is already presented by Srinivas and E1-Halwagi (1993). They used a state space model, optimised by MINLP, but investigated only membrane networks without a distillation column. Viswanathan and Grossmann (1993) optimised the distillation column with rigorous MINLP modelling. These authors did not consider capital and operation costs but optimised for the number of theoretical trays at minimum reflux ratio. Sander and Soukup (1988) experimentally determined the permeate concentration and flux in the function of feed concentration at different temperatures in a pilot plant of ethanol dehydration.
554 2.
PROBLEM STATEMENT
The aim of this article was to re-design an existing hybrid ethanol dehydration plant and investigate the possibilities of cost reduction. For this purpose our previous model for the hybrid distillation-pervaporation system, that can be represented and optimised with GAMS/DICOPT++ is used (Szitkai et al. 2000, Lelkes et al. 2000). The optimization is to be performed over the design and operating parameters including number of trays, feed location, reflux ratio, number of membrane sections in series, and the number of parallel membrane modules in each section of the membrane train. 3. S U P E R S T R U C T U R E AND M O D E L L I N G PRINCIPLES The superstructure and modelling principles applied for the hybrid system are presented in (Lelkes et al. 2000). Here we give only a short overview of the main features. The MINLP model and superstructure of Viswanathan and Grossmann (1993) has been adopted for the distillation column. Most of the column cost functions are taken from Z. Novak et al. (1996). Our membrane network superstructure considers membrane sections in series, where a membrane section consists of numerous 1/3 m 2 flat PVA membranes. In each membrane section the retentate is collected and fed to a heat exchanger for re-heating. The permeate is withdrawn as a product stream and/or recycled to the column feed. Depending on the mathematical representation of the superstructure (how binary variables are used for representing the existence of the 1/3 m E flat PVA membranes), structural multiplicity may occur that hampers the optimization. The notion of structural multiplicity and the usage of binary variables for avoiding its occurrence in this case are dealt with in (Szitkai et al. 2000). The membrane model is based on experimental data. It considers also the effect of temperature drop alongside the flat module. Using the experimentally determined characteristic pervaporation functions of Sander and Soukup (1988), the differential equations of (Neel, 1995) can be numerically integrated for the potential values of feed concentration (c0) and feed flow rate (J0). The result surfaces [J(c0, J0) ; c(co, J0)] are represented by regression in the form o f J v = 0,999. Jo - 0,031- C O and Cv =0,55"C0 "z~ . Here J and c are the retentate flow rate and concentration respectively. In our previous article (Lelkes et al. 2000) various regression and interpolation techniques are examined. Our membrane network costing assumes that the costs of the membrane network are linearly proportional to the overall membrane surface. In case of capital investment the proportionality constant is 1616.1 USD/m 2. Equations for the variable cost calculations are taken from Srinavas et al. (1993), except that the PVA membrane replacement cost was taken 775 USD/m 2 based on industrial practice. Considering 8000 annual operating hours the following utility costs were used: Table 1: Utility costs low pressure steam (160~ 5 bars) cooling water (AT= 10~ electricity permeate condenser cooling medium
135/1000 kg, 6.225 $/GJ 0.16$/GJ 0.06S/kWh 2.645/100 m 3
555 4.
INDUSTRIAL CASE STUDY Using our membrane model based on the experimental data of Sanders and Soukup (1988) the outlet streams and concentrations for the fixed industrial membrane structure and inlet stream were calculated. The calculated and measured flow rates and concentrations are shown in Table 2. The inlet stream is 1000 kg/hr, its concentration is 94 mass% EtOH. Table 2: Comparison of measured and calculated stream data of pervaporation Product flow Product conc. Permeate flow Permeate conc. (k~/hr) (Mass% EtOH) (kg/hr) (Mass% EtOH) Measured 940 99.6-99.7 60 15 Calculated 921.5 99.6 78.5 28 The relatively good correspondence gives rise to the use of the membrane data of Sanders and Soukup (1988) for the re-designing and optimising the existing hybrid ethanol dehydration plant.
4.1. Base case The hybrid ethanol dehydration plant with fixed industrial inlet stream and membrane configuration was first optimised. The results of this optimisation, that can be seen in Figure l, serves as base case for the later comparisons. It is inportant to emphasise that in the base case just the distillation column and the recycle streams were optimally designed; the membrane configuration is kept identical to the configuration of the existing plant. In the base case 97.5% ethanol yield is specified; this means that 97.5% of the total amount of inlet ethanol is withdrawn in form of absolut ethanol in the product stream. 4.2. Optimally designed hybrid system at 97.5% ethanol yield Second, optimal design for all the distillation column, pervaporation membranes and recycle streams were carried out. The result of this design are shown in Figure 2. It can be seen that the total annual cost of the plant is decreased by 12.2 %. This saving is due to the increased overall membrane surface (from the industrial 324 m 2 to 428 m2), that allows to decrease the reflux ratio in the column from 3.262 to 1.38. It is worth mentioning, that the inlet ethanol concentration to the pervaporation system dropped from 94.56 to 91.44 %. 4.3. Sensitivity analysis on overall membrane surface It has been shown that in case of 97.5% ethanol yield the total annual cost of the plant can be decreased by 12.2 % by increasing the overall membrane surface from 324 m 2 to 428 m 2. Some additional designs were carried out between the optimal and the actual plant overall membrane surface. The dependence of the TAC and the reflux ratio on the overall membrane surface is shown in Figure 3. 4.4. Influence of the specified ethanol yield on the TAC In the industrial practice (base case) 97.5% ethanol yield is set up. Depending on environmental regulations or plant conditions, however, greater or smaller ethanol yields could also be required. Because of this possibility, optimisations with fixed 95% and 99%
556 ethanol yields were also carried out. The effect of the specified ethanol yield on the TAC of the hybrid plant is illustrated in Figure 4. The increase in TAC with increasing yield is due to both changing the reflux ratio and overall membrane surface.
4.5. Partial permeate recycling The superstructure is formulated in a way to enable partial recycling of the permeate stream. In spite of this opportunity, total recycling is found in all the cases. On the onter hand, partial recycling is found optimal when the specific membrane investment cost is decreased by appr. 50 %, in case of 95 % alcohol yield. However, this option results in just 1.2% saving in the TAC compared to the optimal design with total permeate recycling. This is due to decreased mass load of the distillation column and some less diluted feed to the membrane subsystem. 4.6. Sensitivity analysis on the membrane replacement cost Optimal designs were carried out with different membrane replacement costs varying from 40% to 120% around the original price (775 USD/m2). According to the results the cost of the membranes in the investigated price interval does not considerably affect the design and operational parameters of the optimal hybrid system. This is because all in these cases the membrane inlet concentrations are in the range of 9.7 to 9.9 mass% water, which is a narrow range. It is interesting to see that this range is situated near the constraint we applied regarding the membrane's toleration of concentrated alcohol vs. its lifetime. Hence the pervaporation unit works at the same conditions, irrespectively to the cost of the membranes. refl3.~6a2io:
--t"?'~
80 ].,992.7~ ~ r theor. I 94.56 mass% stages!
feed 80 mass% EtOH
D--0.875 m
retentate (product): 920.7 kg/hr 99.7 mass % EtOH qmin=97.5~1o/
12 x 81 pieces of 1/3 m 2 fiat membranes =324 m 2 total (fixed industrial configuration) total permeate recycling
/
1175 kg/hr
I
membrane capital investment : 52,362 USD membrane replacement : 83,936 USD column capital investment : 18,05 USD column operational cost : 219,472 USD
recycled permeate ~ bottom product i ] 72 kg/hr 254.3 k g / h r TAC=373,82 USD/yr ...... 28.96 mass% EtOH 0.087 mass% EtOH Figure 1: Base case, optimised hybrid ethanol dehydration plant with fixed industrial inlet stream and membrane configuration
557
re0 i)
retentate (product): 920.7 kg/hr 99.7 mass % E t O H rlmin=97. 5%
12 x 107 pieces of 1/3 m2 flat membranes = 428 m2 total
84 1046.3 kg/hr theor. 91.44 mass% stages
feed 80 mass% EtOH
total permeate recycling
D=0.679 m
1175 kg/hr m e m b r a n e capital i n v e s t m e n t : 6 9 , 0 5 8 U S D membrane replacement : 110,758 U S D c o l u m n capital i n v e s t m e n t : 13,931 U S D : 134,377 USD column operationalcost
I ~
~ 1
recycled permeate ]bottom product 125.6 kg/hr _ [ 254.3 kg/hr 30.86 mass% E t O H v0.087 mass% EtOH
[
TAC=328 ,124 USD/yr
Figure 2: Optimally designed hybrid system at 97.5% ethanol yield 400
...................................................................
3,5
350 ,- 300
\
D cat)
D 250 -ro o
-3
\
- 2,5
o
-2
200150-
-
O
- reflux ratio
50-
)(
0 300
)(
)(
i
i
i
350
400
450
0,5 500
overall membrane surface in square meters
Figure 3: Dependence of the TAC and the reflux ratio on the overall membrane surface
558 400
"C"
............................................................
9plant membrane cost
350
9plant TAC
n 300
• optimised
membrane cost
c"-I O
)K optimised TAC
250
O optimised column cost
o 200 /x
150 100 94
4, plant column cost
O
m
m
m
i
J
i
i
i
95
96
97
98
99
100
specified ethanol yield (%)
Figure 4" Influence of the specified ethanol yield on the TAC optimised system vs. plant existing in the industry 5. CONCLUSIONS An industrial scale hybrid ethanol dehydration system is modelled and optimised using MINLP. The optimal design and operating parameters including number of trays, feed location, reflux ratio, number of membrane sections in series and the number of membrane modules in each section are determined. In our case study 12 % savings in the total annual cost can be achived by applying 32 % more membrane surface, by a radical decrease of the reflux ratio (3.3 to 1.4) in the column, and by producing less concentrated alcohol in the distillate. According to sensitivity analysis, the replacement cost of the membranes does not significantly influence the parameters of the system. In all the realistic cases total recycling of the permeate flow proved to be optimal. REFERENCES
A. Brook, et al: "GAMS. A User's Guide. Release 2.25., boyd & fraser, USA, 1992 J. Gmehling, U. Onken: Vapor-Liquid equilibrium data collection, Vol. I., Part 1, Verlag+Druckerei Friedrich Bischoff, Frankfurt, 1977 Z. Lelkes et al., AIChE J., 44, pp. 810-822, (1998) Z. Lelkes, et al.,Computers and Chemical Engineering 24, pp. 1331-1336, 2000 J. Neel, Membrane Separation Technology. Principles and Applications, Ch.5,Elsevier, 1995 Z. Novak et al. Computers Chem. Engng., 20, pp. 1425-1440, 1996 U. Sander and P. Soukup, Journal of Membrane Science, 36, pp. 63-475, 1988 B. K. Srinivas and M.M. E1-Halwagi., Computers Chem. Engng., 17, pp. 957-970, 1993 V.N.Stabnikov et al. Pishch. Prom. (Kiev) 15, 49 (1972) Z. Szitkai, et al., ESCAPE 10 proceeding, 9 Elsevier Science B.V. S. Widagdo, W.D. Seider, AIChE J., 42, pp. 96-130, 1996 J. Viswanathan and I. E. Grossmann, Computers Chem. Engng., 17, pp. 949-955, 1993
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
559
Computer Aided Design and Analysis of Separation Processes with Electrolyte Systems Kiyoteru Takano a*, Rafiqul Gani a, Petr Kolar b, Takeshi Ishikawa c aCAPEC, Department of Chemical Engineering, Technical University of Denmark DK-2800, Lyngby, Denmark bMitsubishi Chemical Corporation, Ushiodori 3-1 O, Okayama, Japan CMitsubishi Chemical Corporation, Chiyoda-ku, Tokyo 100, Japan A methodology for computer aided modeling, simulation, design and analysis, based on thermodynamic insights, for separation processes with electrolyte systems has been developed. The methodology consists of three main parts: a thermodynamic calculation part, a flowsheet design/analysis part and a flowsheet simulation part. The thermodynamic part "creates" the problem and system specific property model package, which involves pure component and mixture property models and corresponding model parameters. The flowsheet design/analysis part generates process (flowsheet) alternatives, evaluates/analyzes feasibility of separation and provides a visual operation path for the desired separation. The simulation part consists of a simulation/calculation engine that allows the validation and screening of process alternatives. In this paper, the algorithms for flowsheet design, synthesis and analysis are presented together with an illustrative case study.
1. INTRODUCTION Solution chemistry and solid-liquid (phase) equilibrium play a very important role in design, synthesis and analysis of crystallization-based separation processes involving electrolytes. The solid-liquid equilibrium (SLE)-phase diagrams can be used to identify the feasible operating paths for a desired product from a specified feed mixture. They also help to identify the separation boundaries, the temperature of operation, the list of solids that are most likely to precipitate and many more. For a computer aided system, the reliably generation of phase diagrams is an important first step that requires the use of an appropriate set of property models. Information from the generated SLE-phase diagrams such as, phase boundary data and saturation point data may be used to solve graphically, the mass balance equations related to a crystallization operation. Therefore, since the graphical solution of the mass balance equations is related to the operational paths on the phase diagrams, the temperature dependent phase boundaries and the solubility index (indicates which solid is likely to precipitate first), it is possible to simultaneously (graphically) design, analyze and simulate flowsheets with *On leave from Mitsubishi Chemical Corporation, Yokohama Research Center, 1000, Kamoshida-cho Aoba-ku, Yokohama, 227-8502
560 crystallization operations. The simultaneous graphical solution not only provides a visualization of the process operation, but also, provides very good initial estimates for future simulations with a rigorous model. Obviously, phase diagram based methodologies are limited to binary, ternary or quaternary electrolyte systems unless the dimension (component matrix) of the problem can be reduced. Since chemical and pharmaceutical processes handling electrolyte systems may involve many components, one way to achieve a reduction of the problem dimension is through a sensitivity analysis. That is, identify the key components in the component matrix. For example, when a crystallization process is operated, the product to be recovered is, in many cases, only one component. A higher dimensional problem can then be reduced to a ternary or a quaternary system by highlighting only the regions where the most sensitive components are present. If experimental data on the phase diagrams is available, the reliability of the selected property models can be verified, and if necessary, the most sensitive parameters can be adjusted to fit these data. In the developed computer aided system, the design problem is classified into three types according to what is known information and what needs to be calculated. In problem type 1, the feed compositions and temperature are considered as known while the solid that can be crystallized first, is determined. In problem type 2, the solid to be crystallized and the feed compositions are considered known, and the temperature at which the desired solid will precipitate, is determined. In problem type 3, the feed composition and the solid(s) to be crystallized are given, the operation path(s) needed to precipitate the desired solid(s) is determined. In this paper, only problem type 3 is considered in detail, as solution of this problem also requires the solution of problem types 1 and 2. A case study is used to highlight the use of the developed computer aided methodology.
2. METHODOLOGY In this paper, only the main steps of the algorithm for design of flowsheets with/without recycle using ternary SLE-phase diagrams are presented. The complete set of algorithms for the computer aided methodology can be obtained from the authors [1 ]. As shown in figure 1, the developed algorithm consists of seven main steps. In the first step, the order of recovery, which is one of the important design variables in the flowsheet design problem, is identified. In the steps 2-4, the crystallizer operations needed to obtain the desired solid products are determined together with the corresponding operation temperatures. In steps 5-7, the question of recycle is resolved. The problems solved by this algorithm may be formulated as, given, a) composition of the fresh feed stream and b) number and ID(s) of solid(s) to be recovered, determine, a) operation path to recover n products, b) amount of solids to be recovered, c) amount of solvents to be added/removed, d) composition of the mother liquor from each crystallizer and, e) composition of recycle streams if they exist. The solution of the above problem according to the algorithm of figure 1 also requires a number of sub-algorithms. For example, an algorithm-1 to identify the location of the feed point on the solid-liquid (ternary) phase diagram (step 3a), an algorithm-2 to calculate the maximum and minimum dilution/evaporation ratio (step 3b) and an algorithm-3 to calculate the maximum mixing ratio (step 6).
561 Algorithm-1 identifies the feed point location with reference to the saturation curves and the phase boundaries given on the phase diagram. Figure 2 shows a SLE-phase diagram where B1 and B2 represent the saturation curves, B3 and B4 the phase boundaries, F the feed location and P1 and P2, the limiting values of F that gives a solid product A within the 2phase region. Algorithm-2 calculates the range of dilution (evaporation) ratio necessary in order to make a crystallization operation thermodynamically feasible. Dilution (evaporation) ratio is defined as, Amount of solvent to be added(removed) Dilution (evaporation) ratio Amount of solvent in the feed Algorithm-3 calculates the composition of the mixed stream when a recycle stream is mixed with a fresh feed stream. It is an important design variable since with increase of mixing ratio, the composition of the mixed stream moves away from fresh feed point and therefore, the phase region corresponding to fresh feed location. Mixing ratio =
flowrate of recycle stream flowrate of fresh feed stream
Figure 3 shows the operation path as a result of mixing between stream 4 and feed F. The mixed composition is located at point P1. In this figure, the operational paths indicate that solid A is obtained by crystallization at 283.15 K, giving point 2 as the mother liquor. Evaporation from 2 to 3 brings the feed for the second crystallizer to point 3 from where solid B can be crystallized at 373.15 K, giving point 4 as the mother liquor. Stream 4 is then recycled and mixed with fresh feed F. Point 1 indicates the limiting composition for feed F to remain in the two-phase region. Note that no rigorous simulation or calculation is necessary to perform the simultaneous design and analysis
Figure 2: Temary SLE-phase diagram showing feed location, phase boundaries and saturation curves
Figure 3: Temary SLE-phase diagrams showing the operation paths with mixing of recycle and feed streams
562
System information *IDs of components, composition *Temperature, pressure
~
~
Productinformation *Numberof products (n) *IDs of products
I Step 1) Decide the order of recovery through Sl o r ruleof thumb. T I Step 2) Specify the temperature Ti to recover salt I (i=l,n).
I I"
" ~pnase ' Step 3-a) Compute the so1"1" la- lquia diagram at Ti and identification o the feed location together with ID(s) of solid(s) to be precipitated. YES
..
Step 3-b) Select operation. Dilution/evaporation ,.1
.
9
NO
Step 3-b) Select operation. Change Ti/pH/feed composition by stream mixing
] Step4~ Operate the crystallizer at the condition where temperature is 17.
YES
Step 5) Recycleis implemented.
NO
Step 6) Identify the range of mixing ratio and specify mixing ratio within identified range. I Step 7) Repeat step 4 f~ i=l' n"
I ,
Above operation path is converted to the continuous flowsheet/batch operation.
Figure 1: Algorithm for design of flowsheet using ternary solid-liquid phase diagrams 2.1 Validation of Generated Flowsheets
In the developed methodology, the simulation engine is mainly used for screening of processing alternatives, for verification of design and for producing the operational data needed for analysis. A generalized crystallizer model, which consists of MPS model and a decanter model, and the properties model package are implemented in the integrated computer aided system called ICAS [2], which allows the user to solve the design and analysis problems in an integrated manner. That is, once the property model package is created, it is
563 automatically available for generation of phase diagrams and for the simulation engine. Once the flowsheets are generated, the flowhseet information together with the mass balance information (from the phase diagrams) is automatically transferred to the simulation engine. In addition to the validation of the generated flowsheet, it is also possible to verify the reliability of the generated flowsheet specifications (design). Since the flowsheet design algorithm is based on the phase diagrams, the reliability of the generated flowsheet specifications is affected by the accuracy of the phase diagrams. Sensitivity analysis determines quantitatively how much the phase separation boundaries and invariant points move when the most sensitive property model interaction parameters is changed. Consequently, this sensitivity analysis implicitly helps in the design of experiments.
3. CASE STUDY In this case study, two organic salts, Glycine and Sarcosine, need to be recovered as solid products from a ternary aqueous electrolyte system. The problem description is given in Table 1.
System Products Problem
Table 1. Problem description for case study H20(1)-Glycine(2)-Sarcosine(3) Glycine, Sarcosine Flowsheet design
f Feed composition Feed temperature
H20 40g/hr, Glycine 40g/hr, Sarcosine 20g/hr 393.15 K, latm
Application of steps 1 and 2: Specify the system and problem The problem is to design a thermodynamically feasible flowsheet for the recovery of two organic salts. Here, the algorithm [3] for creation of thermodynamic property package is used. Table 2 gives the thermodynamic information needed to create the property package where the shaded parts indicate the feed components. The system includes chemical species involving 6 ionic species. Since this system is classified as an "aqueous" system, the electrolyte NRTL [4] is selected for the calculations. In principle, any other model with the necessary parameters could also have been selected. For the selected NRTL model, the necessary model parameters have been estimated using available binary solubility data. The next step is to generate the phase diagram and identify the operation paths. In this example, two flowsheets, one without any recycle streams and the other with recycle streams, have been generated considering Glycine as the solid product to be recovered first. The algorithm starts with the generation of a feasible flowsheet without any recycle.
564
Application of step 2: Specification of operation temperatures to recover Glycine The temperature of the fresh feed stream is 393.15 K. Therefore, operating temperatures should be below this value. In this example, 283.15 K is selected to recover Glycine. Application of step 3-a, 3-b: Computation of solid-liquid phase diagram at Ti and identification of the feed location to recover Glycine In Figure 1, the SLE-phase diagram is generated at the condition, where temperature is 283.15 K. First, the composition (0.4, 0.4, 0.2) of the feed stream to crystallizer 1 for recovery of Glycine is located in the two-phase region, where one liquid phase and Glycine coexist at 283.15 K. According to Rule 1 in step 3-a, neither dilution nor evaporation is required. However, to get a higher recovery rate, some of the solvent needs to be evaporated (Rule 2 in step 3-b). The algorithm calculates maximum evaporation ratio needed to keep the shifted feed point in the same region as original feed point as 0.241. Minimum evaporation ratio is zero. In this case, 0.241 is selected. After evaporating the solvent, the composition of shifted feed stream is (0.528, 0.208, 0.264). Application of step 4: Operation of crystallizer at Ti to recover Glycine First, the liquid stream, whose composition is (0.528, 0.208, 0.264), is sent to crystallizer 1 to recover Glycine. From mass balance, the composition of the mother liquor is (0.124, 0.386, 0.490), which is the feed to crystallizer 2 for recovery of Sarcosine. Repeating steps 2 and 3a, 3b, the operation temperature for recovery of Sarcosine is selected as 373.15 K. The saturation curves at 373.15 K are calculated and added to the SLE-phase diagrams. In order to crystallize Sarcosine first, however, the amount of solvent (water) needed adjustment by evaporating some of the solvent (Rule 2 in step 3-a). The algorithm calculated minimum and maximum evaporation ratio to keep the shifted feed point in the region, where only Sarcosine is crystallized. Minimum and maximum values of dilution ratio are 0.10 and 0.17 respectively. In this case, 0.17 is selected. After evaporating the solvent, the composition of shifted feed stream is (0.149, 0.261, 0.590). Extending the line joining the pure solid product (Sarcosine) and the shifted feed to the saturation curve identified the operation path and the mother liquor composition, the exit liquid stream from crystallizer 2, as (0.172, 0.310, 0.528). Application of step 5: Implementation of recycle stream From the information generated through steps 2-4, the continuous flowsheet that recovers first Glycine and then Sarcosine is generated. In the next step, recycle is considered. In this case, mother liquor from crystallizer 2 is mixed with the fresh feed stream. Application of step 6: Identification of maximum mixing ratio In this step, the maximum value of mixing ratio (defined above) was calculated to be
565 0.6786 and minimum value to be 0 (no recycle is considered) and a ratio of 0.2 was selected.
Application of step 7: Final design specification When the mixing ratio is set to 0.2, the composition of the feed stream to crystallizer 1 becomes (0.3544, 0.3802, 0.2653). Consequently, repeating steps 3-a and step 3-b gave the maximum evaporation ratio to be 0.17 (compared to 0.24 without recycle). Selecting the evaporation ratio as 0.17 gave the mixed the feed composition to be (0.427, 0.253, 0.320). Step 4 is also repeated for the two products in order to obtain the final design specifications. The composition of mother liquor from both crystallizer were found to be the same as that identified before recycle was considered. This is because invariant points corresponding to the two operation temperatures, used as target mother liquor compositions, do not change if the temperatures are not changed. Figures 4a and 4b show the operation paths and the generated flowsheet for the crystallization process. An important point to note here are that the operation paths on the phase diagrams could also be used to represent the sequence of batch operations needed to obtain the same products. Therefore, the algorithm is valid for generation of continuous flowsheets as well as sequences of batch operations.
Figure 4a: Operation paths for the recovery of glycine and sarcosine on the generated SLEphase diagram
Figure 4b: Continuous flowsheet corresponding to the operation paths shown in Figure 4a
566 4. CONCLUSION A methodology for computer aided design and analysis of separation processes based on the thermodynamic insights of electrolyte systems has been developed and highlighted through an illustrative case study. The novelty of the methodology is that all the necessary steps (from property model selection and validation) to final flowsheet validation through simulation of the process alternatives is addressed in an integrated and systematic manner using visual (graphical) computer aided tools. The methodology includes a parameter estimation technique based on the sensitivity analysis (not discussed in this paper) helps to solve problems when the number of experimental data is limited and/or number of components is larger than four [3]. The methodology includes a rigorous simulation engine option that is able to simulate the steady state behaviour of the separation processes. The sensitivity analysis in the simulation part also calculates quantitatively how the accuracy of the generated flowsheet specifications is affected by the reliability of the phase diagram, and consequently it helps the design of experiment. The identified feasible operational paths can be used to generate flowsheets for continuous operations as well as the sequences of batch operations to obtain the same products. Current and future work involves further extension of the integrated system in terms of increased application range of the process and property models, process optimization and new industrial applications.
REFERENCES
1. Takano, K., "Computer Aided Design and Analysis of Separation Processes with Electrolyte Systems", PhD Thesis, Technical University of Denmark, Lyngby, Denmark,
(2000). 2. Gani, R., Hytoft, G., Jaksland C., and Jensen, A. K., Computers & Chemical Engineering, 21, (1997) 1135. 3. Takano, K. Gani, R., Ishikawa, T., and Kolar, P., Chem Eng Res Des, 78:(A5), (2000) 763. 4. Chen, C. C., and L. B. Evans, AIChE J., 32, (1986) 1655.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Characterization and Simulation Separating Azeotropic Mixtures
of the Pervaporation
567
Process
for
M. E. Torres Alvarez, R. F. Martini and M. R. Wolf-Maciel Laboratory of Separation Processes Development (LDPS). Chemical Engineering School. State University of Campinas, (UNICAMP), CP 6066, ZIP CODE 13081-970, Campinas-SP, Brazil. E-mail:
[email protected] In this work, the characterization of a pervaporation process for separating the azeotropic mixture ethanol-water was performed based on the solution-diffusion model, with a membrane selective to water. The transport parameters were derived from experimental data found in the literature, a general expression for the separation factor as a function of the permeate component fluxes was used and the variables involved in this process were identified and analyzed. 1. INTRODUCTION Pervaporation is a separation process where a liquid mixture is vaporized through a membrane [ 1]. In pervaporation processes, the mixture to be separated is put in contact with a selective polymeric film and the permeate is removed in the vapor phase, on the opposite face of the membrane, which is condensed later. The vaporization of the permeate components takes place as a consequence of the partial pressure reduction, which happens due to the decrease in the total pressure on the permeate side; this decrease can be achieved by using a vacuum pump, for example. Pervaporation differs from other membrane processes by the change of phase on the permeate side. Another important characteristic of this process is that the permeate flow is relatively low. Thus, the pervaporation process becomes attractive when small amounts of materials are to be removed from the liquid phase. In order to accomplish this, the membrane must present high selectivity in relation to the component to be removed [2]. In this work, the characterization of a pervaporation process for the separation of azeotropic mixtures was performed. Indeed, the intention here is to increase the knowledge of the process as a whole and the complexity of the variable calculations to, later, be able to use this important separation process for other potential applications. The variables involved in this process were identified and analyzed and the general permeation equations were based on the solution-diffusion model; two assumptions were made about the pressure gradient inside the membrane (pm): flat gradient (the pressure was kept constant) and linear gradient according to [3]. With these assumptions, the permeate flux equations are expressed as a function of the feed pressure, P1, the permeate pressure, P2, the activity coefficient, y, the diffusion coefficient, D, the membrane thickness, g, the vapor pressure, Pv, the molar volume, v, the mole fraction, x, and the temperature, T. Simulation analysis was carried out using experimental data of aromatic polyetherimide membrane [4]. The effects of the downstream
568 pressure on the selectivity and on the flow were analyzed for the azeotropic system ethanolwater. 2. MODEL APPLICATION According to the solution-diffusion mechanism, which characterizes the pervaporation process, different mathematical models are proposed to describe the behavior of the permeate flux according to modifications in the variables of the process. Several empirical equations have been developed to calculate the diffusion coefficient used in such mechanism, from simple linear equations to exponential models. Those equations try to describe the relationship between the diffusion coefficient and the concentration, for instance the Fujita's theory of the free volume [5]; however, the transport equations for this theory are complex and the model parameters are quite difficult to be obtained. Moreover, the models are very sensitive to approximations in these measures [6]. In the present work, the mathematical model of the permeation equations assumes that the diffusion coefficient remains constant throughout the membrane. A software, named PERVAZ, is being developed in this work; it is based on the model presented in [3] (shown in this section) and makes use of a user-friendly platform to help the user interaction with the process. The software is written in Visual Fortran 6 language. The permeation equations adopted here consider that the pressure in the membrane varies linearly, i.e.,
Z(p, -v,)
Pm=Pl + 7
The permeate flux is a function of the chemical potential gradient, according to the expression [3,7]: Ji =
D e m x m d~ti
RT
(2)
dg
Assuming that the membrane is in contact with a liquid phase on the feed side and with a vapor phase on the permeate side, the chemical potentials in the membrane interface and in the adjacent phases must be equal. By substituting a thermodynamic expression of the chemical potential gradient in equation (2) and solving it, for the case considered here, the permeation equation becomes [3]: Dcm
L
/
s fviv"-Pl't)
vi PI - P2 - P2x2'-------~iexp R--T vi (P2 P~ ~'iX~'i Pv,i Ji = g'jtm i 1-- exp{ RT )}
(3)
RT
and, for a binary system, the total permeate flux is: J = Ji + Jj
(4)
569 where the diffusion coefficient, D, the activity coefficient in the membrane, ~tm, the concentration inside the membrane, C m, and the molar volume of the permeating component, v, are considered constant and the vapor behaves as an ideal gas. The system is assumed to be isothermal. In equation (3), the term D C m / g 7 m , called transport parameter, must be calculated from experimental data of the permeate flux and of the concentration. This methodology was used in [3], who used experimental data for an ethanol-selective membrane (theoretical values) and for a water-selective membrane (Nation and cellophane membrane), whereas in this work a water-selective membrane (polyetherimide membrane) is being used with different experimental data. Moreover, the objectives in Kataoka's work were to study the total flux in function of feed pressure and to compare pervaporation and reverse osmosis processes. For a permeation system consisting of a binary mixture, the separation factor can be defined through the following expression [8,9]:
C/,ij =
J~ .x],,
(5)
s Jj "xi, 1
Equation (5) makes possible the calculation of the separation factor at the steady state. For the calculation of the separation factors, Kataoka et al. [3] considered the permeate pressure was zero and derived expressions which were function of the transport parameters and activity coefficients, whereas, in this work, the separation factors have been calculated for values of P2 ~ 0, what means that they are functions of the permeate component fluxes and, consequently, of the permeate pressure. 3. SIMULATION AND CHARACTERIZATION Experimental results of the separation of the azeotropic mixture ethanol/water were studied using a polyetherimide membrane [4], a water-selective membrane. The experimental data of the permeate flux versus ethanol composition in the feed were presented in the form of graphs by Huang and Feng. In the present work, these experimental data were used to determine the different values of the component fluxes as a function of the ethanol composition in the feed. Such values were then used to calculate the transport parameters DC m / g7 m, for the components i andj. The resulting values were 2.326 and 9.126 (mol/mZh), respectively, for the model developed in [3]. It is important to mention that, in this work, the experimental data used were never tested with the presented model and, also, that originally the model was developed for other kinds of membranes. The influence of the downstream pressure on the rate of permeation and the effect of temperature on the flux under different downstream pressures, considering the transport parameters constant, were analyzed. Both the experimental and the calculated values of the permeated component fluxes versus the ethanol composition in the feed are presented in Figure 1. The square, sphere and triangle points represent Huang and Feng's experimental data whereas the continuous lines represent the data calculated in this work. It can be observed that the model represents quite well the experimental data for most of the ethanol composition. As the membrane is water-selective, the permeate flux of water is greater than the ethanol flux, except for very high values of ethanol in the feed. The permeate flux of ethanol increases slightly as the composition of
570
ethanol in the feed increases. This corresponds to a decrease in the permeate flux of water and, consequently, of the total flux (water + ethanol). 12 ~
Exp.
Data
This
Work
11 -I
o
A
EtOH
-
EtOH
101.]
[]
t3
Water
,
Water
o
Total
~" 9 !
Total
g
2
1 0
9
0,0
,
,
0,1
,
9
0,2
,
0,3
9
,
0,4
9
,
9
0,5
,
.
0,6
i
9
0,7
,
9
o,s
,
.
0,9
1,0
EtOH in feed (tool%)
Figure 1. Variation of the permeate flux with ethanol concentration in the feed (Pl=101.3kPa, P2 =0.1 kPa)
3.1. Pressure Influence The influence of the feed pressure on the separation of ethanol-water by pervaporation was studied in [3]. When presenting the influence of the permeate pressure (P2) on the process, they plotted the flux and selectivity versus a relative pressure (P2/Pvapor), keeping the temperature and feed composition fixed, whereas in this work, the behaviour of the permeate flux and the separation factor for different values of P2 and feed composition are shown. According to the permeation model, the behaviors of the flux and of the separation factor can be described in relation to the pressure on the permeate side and to the feed composition, as it can be observed in Figures 2 and 3, respectively.
9 8
7
~ 9
~
g ~
Pa
= 0.4
P2=0
6
~
10 9
~ . . . . . p2 = O. 1
P2=O
8
7
1
P=04
1.0
4 3 2 1 o
0,0
,
i
,
i
i
i
,
i
.
i
,
i
,
i
,
i
.
i
,
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
EtOH in feed (~
Figure 2. Variation of permeate flux versus composition of ethanol at different pressures.
0
0,0
0,1
0,2
"
"
"
~
0,4 0,5 0,6 0,7 0,8 0,9 1,0 EtOH in feed (%mol)
0,3
Figure 3. Separation factor versus composition of ethanol at different pressure
Figure 2 shows that the total flux (J) decreases as the ethanol composition increases for all permeate pressure values (P2) studied. The total flux can also be reduced by increasing the
571 permeate pressure, as can be seen in Figure 2. The same behavior can be observed for the separation factor, which decreases as the permeate pressure increases (Figure 3). This means that, the lower the pressure, the bigger the separation factor. For high concentrations of ethanol in feed (above Xl=0.8) and as P2 increases, it can be observed that the separation factor curves reach a maximum and drop, which shows that there is a complex relationship between the variables.
3.2. Influence of the temperature In pervaporation processes, due to the membrane nature, the temperature is an important factor. Therefore, a study of the temperature effect on the total permeation flux was carried out, as can be seen in Figure 4. lO 9
~====0-----~ > ~ c ~ a_~. 8 ~ < ~ ~ - . ~ ~-Az~,,
310 15 K --- 2~(P(y)) i= 1...n
(6)
Since analytic expressions for the eigenvalues of a problem (6) may be approximated as below:
s.t.
matrix are not in general available,
min z Y'~ z > 2~ax(P(y))
(7)
where 2max is an upper bound of the maximum eigenvalue of P. For a real symmetric matrix /
\
(Jennings, 1977): 2ma~(P(y))_O Mnc ]
(12) (13)
629
r cocoqlingl
1
"]Condenser ] Tray 1
~,f--1 [Tray n f
~Vn,f
coF
~coD
[
~nf-2 --
1[
~Lnf-1
a
~l Feed Tray nf ] I
~n.f + l ~n f [Tray n f + 11
~nf;2
~nf+l
I
[ Zrayn I coheqtingl "1 Reboiler [ t.
,coB
Fig. 1. Distillation column: n number of trays, n f feed tray ~)S =
OV
9
T
-.1
T
-.nc]
"'"
Fig. 2. Distillation control structure
(14)
T .~
(DT
where v is the inventories, S is the entropy and co is given in equation 14 as the derivative of entropy with respect to the inventories. The co introduced here is the same one introduced earlier as part of the driving force of the Onsager relations. To perform stability analysis the dynamic equations for the system must be substituted into the supply function (or derivative of the storage function).
dA dt
= CO,Tdv ~-=
( d s ) Tdv dvv
d---t
(15)
-(m- m*)r~
(16)
If the supply function is negative then A(t) is decreasing with time and the inequality given in equation 11 is satisfied and therefore the system is passive and stable 9 In this system there are two different view points each with a different model therefore there are two separate supply functions.
dA e dt
N
Bc (17)
i=1
k=l
j=l
630
dA l
N
Bc
-- E yfTAiffi - E - ( o y A j ~ j
dt
i=1
(18)
j=l
Equations 17 & 18 summarized the stability result from Coffey et al. (2000) and show a negative definite supply function when AA is positive semi-definite. This then proves that the simple distillation column described is open loop asymptotically stable. This result will now be extended to the design of an asymptotically stable control system. 3. DISTILLATION CONTROL A control structure will be designed using the ideas introduced by the thermodynamics and then controllers designed for asymptotic stability. From equation 14 it is shown that temperature directly effects the rate of entropy change with respect to the energy and similarly for the other intensive variables and inventories. It is with these ideas that the following control structure is designed. Temperature is used for energy control, pressure for overall mass inventory control, and concentration for component inventory control. The basic control structure will be as follows: Condenser pressure will be controlled by distillate outlet valve/pressure, condenser temperature by cooling fluid flow/temperature, condenser level by reflux flow valve. Reboiler temperature will be controlled by heating fluid flow/temperature, reboiler level by bottoms outlet pressure. Composition control of the distillate and bottoms products will be controlled in a cascade loop with the inner loop being the condenser and reboiler temperatures. The control structure is shown in figure 2. The control structure is not unique but represents a base structure designed from the fundamentals of thermodynamics. The stability of the chosen structure will now be shown. In equations 17 & 18 there are three different types of terms: boundary terms, flow terms, and difference terms from the weir equations. The terms that will be effected by control will be the boundary terms and the flow terms. Control can be implemented in two ways, either by manipulating the flow coefficient (A) or the thermodynamic conditions (X or m). Manipulating the flow coefficient is equivalent to controlling a valve. Manipulating the thermodynamic conditions is equivalent to changing the suction/output pressure of a pump or the thermal conditions of a heating or cooling fluid. In the stability proof that will follow simple proportional controllers will used. Stability of the control implemented at the boundary conditions will be looked at first. The following equation shows the form of the supply function boundary terms that will be changed by control. d A __
dt
"
.. _ ~ T ( A i o ) i - A* (.0~)
(19)
This is in a general form for either manipulation of Ai or mi and which is actually manipulated can be determined later. The proportional control law shown in equation 20 is then substituted into the supply function and results in equation 21.
-Ait.oi -4-A~ m~ dA dt
-....
(20)
-K(oi -~TK~
i
(21)
The supply function remains negative definite and therefore the system remains asymptotically stable with the introduction of the control law. The following equation shows how to manipulate
631 the flow coefficient or valve position as a function of the measured intensive variable.
Ai-
1
(A~0~;§
mi
(22)
The following equations shows how to manipulate the boundary condition and the resulting supply function (here it is assumed Ai -- A*). (0 i dA dt
-....
(23)
- K(oi_ n - (OT_nKT AiK(Oi_n
(24)
Here it should be noted that the sensor and actuator are not located at the same point. The measurement is at i - n and the manipulated variable at i. For the system to still be stable the only requirement is that the system is passive between i and i - n. Since the controller is simply adjusting the amount of dissipation, and can never become non-dissipative, then the whole system will remain dissipative and stable since we know from previous work that the system is stable. The above type of boundary condition control will be used for the condenser pressure control, reboiler level control and both the condenser and reboiler temperature control. The condenser pressure will be regulated by the distillate flow valve, reboiler level will be regulated by the bottoms flow valve, and the condenser and reboiler temperatures will be regulated by the temperature of the heating and cooling fluids. The last controller, level of the condenser, uses an internal flow with open loop supply function term of the form shown below dA dt
....
- f f ? (AiXi - A * X ; )
(25)
The control law shown in equation 26 is again stable when substituted into the supply function in equation 27. - A i X i § A~X[' dA dt
--
-KXi
....
- ffi~Kffi
(26) (27)
The internal valve position for the condenser level control is given here.
A i - (A'X* § K X i ) - ~ i i
(28)
Here the measured variable is a difference in pressures which can be used to determine the liquid level. The same type of control could have been used above for the reboiler level control. Composition control is performed by cascade control with the condenser and reboiler temperatures. The thermodynamic separation is defined by the chemical potentials differences between the condenser and reboiler. The chemical potential can be directly correlated with the temperature. This is easy to see in distillation, the nodes in composition space are the high and low boiling points within a given distillation region. The nodes of an azeotropic system can be the limits of the separation due to a minimum or maximum in the chemical potential space which will correlate with the high and low temperatures. This means when the nodes of the composition space are not the pure components, a component separation in terms of composition can
632 go through a maximum. This results in complications in designing composition control. The chemical potential is the driving force for separation and with its direct relation to temperature it is the proposal of this paper to use temperature as a means to control separation. The goal of composition control is to first control the distillate or bottom products to a reference point and then control the separation by the temperature difference. A simple example of how this would work is in binary distillation where the desired product is a pure distillate. The condenser temperature would then be set to the light key pure component boiling point and the temperature difference would be set to the desired split. If the reboiler temperature is set to the saturation temperature of the feed or below then there will be no distillate product. As the temperature of the reboiler increases (temperature difference increases) internal flows will be generated and distillate product formed. The limit would either be equipment related or of course when the reboiler temperature equals the heavy key pure component boiling point. Due to saturation conditions in the condenser and reboiler the temperature difference is related to the pressure difference. As the temperature difference increases so will the pressure difference. Separation will increase as the pressure difference increases as it is the driving force for the internal vapor flow. The temperature and temperature difference can be updates in the controllers from either concentration measurements, where possible, or from a model of VLE data. 4. CONCLUSIONS The work presented in this paper uses passivity and thermodynamics to design an asymptotically stable control structure for a general multicomponent distillation column. The structure shown in this paper is not unique but is used to illustrate the how the use of passivity and thermodynamics and can be used to better understand the distillation control problem. REFERENCES
Coffey, D. E, Farschman, C. A. and Ydstie, B. E. (2000). Distillation stability using passivity and thermodynamics, Computers Chem. Engng 24:317-322. Coffey, D. E and Ydstie, B. E. (2000). Stability of process networks, ADCHEM 2000 Pisa,
Italy. Skogestad, S. (1997). Dynamics and control of distillation columns-a critical survey, Modeling, Identification and Control 18(3): 177-217.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
A Novel Adaptive Multivariable Industrial Reactor
DMC
633
Controller. Application
to an
E. C. Dechechi a, L. A. C. Meleiro b and R. Maciel Filhob'* a p u c P R - Pontific Catholic University of Paranfi (www.pucpr.br), Curitiba- PR, Brazil. CP 16210 CEP 81611-970. Fax. +55-41-316 3033. E-mail:
[email protected] bLaboratory of Optimization, Design and Advanced Control (LOPCA). Department of Chemical Processes. School of Chemical Engineering. State University of Campinas (UNICAMP). Campinas - SP, Brazil. CP 6066 - CEP 13081-970, Fax +55-19-7887840. E-mail:
[email protected] and
[email protected] A novel adaptive control algorithm based on Dynamic Matrix Control (DMC) philosophy with adaptive features was developed and applied in an industrial hydrogenation multiphase reactor. The case study is a phenol hydrogenation reactor used to obtain cyclohexanol (an intermediate for nylon production). This is a three-phase reactor with solid catalyst in the reaction medium. This process has complex heat transfer mechanisms and fast dynamic behavior. The hydrogenation reaction is a process with distributed parameter and presents some important effects as catalyst deactivation, for instance, that are difficulty to prevent and to monitor. The motivation to use adaptive DMC controllers arose from its high potential and robustness when applied to chemical processes, since it couples prediction and adaptation behaviors in a convenient fashion. Besides, there is a lack of tests of this control strategy in the literature. The on-line internal model adaptation is carried out successfully using recursive least square method with an ARMA based model. This algorithm showed a very good performance leading the reactor to be operated safely in large range operational conditions. The results also showed the high efficiency of the developed controller when applied to this industrial process under normal operation. 1. INTRODUCTION The main focus of this paper is to present an advanced software for process control developed and tested on-line in an industrial multiphase reactor. This software, denominated D-AMPC ( D e c h e c h i - Adaptive Multivariable Predictive Control), is based mainly on predictive Dynamic Matrix Control (DMC) [10] philosophy and also on the Self-Tuning Control (STC) [14, 15] to provide adaptive capabilities with constant identification to an input-output multivariable model. Literature presents some papers dealing with the matter of coupling the predictive and adaptive advanced control strategies [7, 8, 12, 13]. However, in those papers the performance of such strategies was tested in simulations of ideal process. The aim of this work is to verify the performance of the novel proposed algorithm in an industrial multiphase reactor.
*Corresponding author (
[email protected])
634 The controller was tested in the refrigeration loop of an industrial three-phase reactor that produces cyclohexanol by phenol hydrogenation in the presence of a nickel-based catalyst. This process has complex heat and mass transfer mechanisms, fast dynamic behavior and constant load changes in all input variables (temperature, concentration of catalyst and liquid flow rate). The hydrogenation reactor is a system with distributed parameters and has some effects, as catalyst deactivation, that are difficulty to prevent and to monitor by the usual commercial software for control purposes. The development and application of this novel adaptive multivariable DMC controller (D-AMPC) involves mathematical modeling, advanced control strategies, real time optimization and connection with a real time scheduling. These strategies have been developed to be feasible in real industrial chemical plant. In this work the main objective is to present this novel multivariable DMC controller who was tested under real industrial operating conditions. 2. REACTOR DESCRIPTION Three-phase reactors can be found in several important applications and is verified in the literature a lack of works that cover these complex reactors when compared to other types of catalytic reactors [1-5]. In this case, a catalytic and exothermic reaction takes place inside several modules with concentric tubes. This industrial multiphase (slurry) catalytic hydrogenation reactor is showed in Figure 1. In this reactor, the highly exothermic reaction of phenol hydronization is developed and the heat removal happens due to both the immersion of the reactor inside a steam generator, and the coolant circulation inside the first six reacting tubes. Besides the heat removal generated by reaction, that is accomplished in the steam generator, the first six tubes have an internal refrigeration with coolant flowing in co-current with the reaction medium to maintain the temperature profile controlled inside the tubes. The coolant is composed of the condensed steam from the steam generator to which a make-up stream may be added. Coolant is circulated to each of the six first modules, and each flow rate may be manipulated. The main function of the coolant in this case is to avoid the so-called "runaway" that occurs when the inside temperature of the reactors increases exponentially. This problem can lead to unbalance thermal reactions. Runaway effect is more frequent in fixed-bed reactors, with the possibility of appearance of hot spots, however such condition cannot be discarded in slurry reactor. Besides the possibility of the runaway effect, such a temperature increase provokes an increase in the formation of by- Figure 1 - Schematic diagram of the industrial reactor products, that is, decrease in the selectivity of the reaction. The structural and operational complexity underlying to this reactor is increased by other features associated to its complexity and operational performance, such as catalyst deactivation, incrustation along the reactor (three-phase system) among others. The first six modules are identical in structure and are composed of four concentric tubes. Reactant flows from one module to another through the inner tube passage and through the annular passage
635 formed between the two external tubes. Through the other annular passages, coolant flows in order to control the temperature of the reaction of each reaction tube. The last two modules are composed of two concentric tubes and only reactant is present. These last two reaction tubes are used to prevent the total phenol conversion. In this industrial reactor is observed that in its usual operational conditions high thermal oscillations can occur. These changes are due various factors as catalyst composition in the feed, catalyst deactivation and others typical difficulties of a very complex industrial reactor. These disturbances have high impact in the selectivity, productivity as well as in the oscillations of the temperature of each tube. This complex behavior is the major challenge of the D-AMPC control developed. As such an oscillations may occur frequently in this reactor and its causes are not well understood, a robust control strategy based on predictive and adaptive features has real potential of success in a large range of operational conditions. 3. M O D E L A N D C O N T R O L L E R D E V E L O P M E N T
In this work was developed and tested in the real plant an novel advanced control strategy based on Model Predictive control, more specifically DMC [6]. This novel DMC control was specifically designed for this industrial application and has adaptive capabilities to take in account the dynamic variations that occurs in the real operational conditions. These dynamic variations occur in the input variables (load variables). This controller has a DMC philosophy and its internal model is constantly modified to take into account the dynamic variations. Figure 2. The proposed advanced control algorithm is different from the other approaches dyna~emal identilicalion described in the literature [7-12] in the way that - - the internal model is built up (with no need of step tests) as well as in controller parameter I adaptation procedure. The variations in the dynamic process are observed by a multivariable ARMAX model that is Figure 2 - Structure of the adaptiveDMC controller D-AMPC successfully identified by a Recursive Least Square "RLS" algorithm (estimator block).
i
Defining the parameters vector as O(t - 1 ) = [a,, a2,.... , a,,, b0, b],. ..... , b, b]T, and data vector as
X T( t ) - [- y(t -1), - y(t - 2), .... , - y(t - na), u(t -1), u(t - 2), ...... , u(t - n b -1)],
the
RLS
^
algorithm basically searches estimations, 0, of the unknown parameters in such a way that the cost function, J
=
q~O)'-j y ( j ) -
j
, with 0 0.50 . "5 ~ 0 a
. :50
. . ~,0 ' 6 0 ' 8 0 Time (h)
' 100' 1:20
Fig. 4: Controlled dissolved oxygen concentration for the regulator problem, perturbation in the substrate feed stream
A test was done with no measurement noise present to test the behaviour of X and to see if it would settle on a constant value or settle to zero. In Fig. 5 a comparison is made between the proposed estimation system with the self-tuning method of X and a fixed X. First it can be seen from the dissolved oxygen response (Fig. 6) with a fixed X that there is actually a need for variable X. At some set-point changes the dissolved oxygen concentration is controlled vary well, while at other set-point changes the same value of X results in a small but significant overshoot.
644 0.04 .9
m -,0
0.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.02-
;Lo = 0 0 4
.... X, = constant = 0.0
e"
/
0.01 0.00
0
#0
40
60
Time (h)
80
Fig. 5: Suppression factors without measurement noise
100
9
tD t.) t"
1:~0
0
w,v,,w,;v,-.~;--,~,;;
9
g
~I
..... SetpoJnt I o X -004 ->o~0.41 ~ X = constant = 0.03 ,n 0 20 40 .~_ ca Time (h)
60
80
Fig. 6: Control of dissolved oxygen concentration without measurement noise
The results with the proposed tuning system reveal that it is important to tune the suppression factor in such a way that no overshoot or sluggish control in all set-point changes. The first point to considered is if the minimisation criterion is a good starting point for deriving an identification system of the parameter X. X has a huge influence on the calculated movement in the minimisation criterion, but cannot only be determined of it. Therefore care has to be taken with the identification method of X which has to be suitable from the available process information. 4. CONCLUSION The proposed control algorithm has shown to be robust for the analysed disturbances, promising to have a great potential to be used in control strategies of large scale systems. Particularly, the application of the proposed algorithm in the SISO case reveals its potential for multivariable systems where the interactions and thus the suppression factors are more important. It is noted the importance of the adjustment of the suppression factor to be able to cope with change in process operations. The estimation algorithm of the suppression factor, will determine successfully the rate of change of the system due to the control action. This will result in a fast control algorithm with little or no overshoot. REFERENCES 1. Ender, L. & Maciel Filho, R. (2000), Computers & Chemical Engeneering, 24, 937-943 2. Hecht-Nielsen, R. IEEE Int. Conf. On Neural Networks San Diego (1989) v.I, p.593-605. 3. Goodwin G.C. and K.S. Sin, (1984) Adaptive filtering, Prediction and control, PrenticeHall, Englewood Cliffs, NJ 4. Rodrigues, J.A.D. and R. Maciel Filho (1996), Chemical Engineering Science June, v51, il 1 pt B, p2859-2864 5. Scheffer, R. and R. Maciel Filho (2000), in "Application of Neural Network and Other Learning Technologies in Process Engineering", ed. Mujtaba and Hussain, "Process identification of a fed-batch penicillin production process - training with the extended Kalman filter"
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Function-Behavior Modeling and Diagnosis of Chemical Processes
Multi-Agent
645
Approach
for
Fault
Soo Young Eo a, Tae Suk Chang b, Bytmgwoo Lee c, Dongil Shina and En Sup Yoon a aSchool of Chemical Engineering, Seoul National University, San 56-1, Shinlim Dong, Kwanak Gu, Seoul 151-742, Korea bSamsung SDS Co., Ltd. 707-19, Yoksam-2Dong, Kangnam-Gu, Seoul 135-918, Korea CAutomation Research Center, Pohang Univ. of Science and Technology, San 31, Hyoja Dong, Nam Gu, Pohang, Kyungbuk 790-784, Korea In this paper, we suggest a function-behavior modeling and distributed multi-agent system approach for the diagnosis of chemical processes. In the proposed system, diagnostic agents of individual units communicate by exchanging messages and try to solve the global fault diagnosis problem using collaborative diagnosis method. The benefits of the suggested collaborative problem solving approach are demonstrated by solving the diagnostic problem of a CSTR system with recycle and a first order irreversible reaction. 1. INTRODUCTION During the last two decades many diagnostic methods have been suggested, developed further and resulted into some success. However, still diagnostic systems are not widely adopted by the industry due to some difficulties remaining unsolved in spite of the progress in this area [ 1, 2]. To solve those difficulties, this research proposes and develops a fast, reliable, easy-to-understandable fault diagnostic system, which can be quickly implemented with consistency, for chemical processes. As many seen in the area of artificial intelligence, Multi-Agent System (MAS) technology is emerging to solve problems in complicated, distributed systems, such as Intemet and human bodies. Multi-agent systems have been known to provide distributed and collaborative problem solving environment [3, 4]. In the process systems engineering, there have been some efforts to apply agents in modeling and design of processes in the concept of concurrent engineering [5]. However, few efforts have been made to the application of agents as software components and agenthood in the process fault diagnosis domain. In this paper, the development of a multi-agent system for chemical process fault diagnosis and function-behavior modeling for the knowledge base of the system is presented. This system uses the diagnostic agents for each process unit and makes them communicate with each other to solve the fault diagnosis problem in a collaborative way.
646 2. MULTI-AGENT FAULT DIAGNOSIS SYSTEM
2.1 Overview of the Multi-Agent Fault Diagnosis System Comparison of the conventional, shared memory single-agent approach---centralized and integratedmand the message passing multi-agent approach--decentralized and distributed is shown in Fig. 1. Diagnostic agents are distributed and implemented for each process unit, and each autonomous agent tries to solve problems through the collaboration with others. This collaborative approach is the most prominent difference from conventional centralized approaches. The concept of distributed fault diagnosis using multi-agents in chemical processes is shown in Fig. 2. Diagnostic agents (DAs) corresponding to existing process units are placed in accordance with the topology of the process units. They have a fault detection module, knowledge bases and an inference engine which performs reasoning as well as local diagnosis. The fault detection module has the information on possible fault types, measured and unmeasured variables of individual process unit and can determine the existence of any faults. Knowledge base has the relationship between variables based on function-behavior model [6, 7]. DAs of process units are said to be connected when the corresponding units are physically connected through material/energy flow or when the units are linked by information streams, such as controller signals. In addition, DAs communicate only with other neighboring DAs. DAs of the corresponding units have their knowledge about the relations among state variables in the units represented by function-behavior models of the units. With the knowledge, DAs perform diagnosis by processing the messages from other DAs in the eventdriven way. The messages are invoked from the event such as the detection of symptoms from measured variables, reception of queries from other agents, etc. The communications between DAs are limited to directly linked DAs and this makes the MAS structure very distributed. 2.2 Function-Behavior Modeling Function-Behavior modeling of Oh [6] aimed at fast system diagnosis by simplifying the large chemical processes and resulted in process interpretation similar to the behavior modeling due to the emphasis on grouping process units by their functions. To perform the collaborative fault diagnosis based on process topology, by interpreting process units and their
~ DA: DiagnosticAgent ................~A~~'''" SA:SchedulingAgent
,roce,:
Single-AgentSystem
...
Multi-AgentSystem
Fig. 1 Comparison of single-agent and multi-agent systems.
Fig. 2 Multi-agent diagnostic syste structure.
647 functions as DAs, and their functions and status variables of the units as the contents of the DAs, a different approach from Oh's method is required. Basically, every equipment in a chemical process carries out tasks related to mass, energy and component. Mass, energy, component, etc. become objects of functions of process units; these objects are defined as keyword, and the status of the keyword, keyword status, describes the unit's behavior depending on the function of equipment. For example, the abnormal high mass flow in a pipe may be described as MASS_STATUS HIGH. All process units have such keywords, and each keyword-related behavior of units can be interpreted together with behaviors of other units via keyword. Keyword status such as MAYBE_HIGH, MAYBE_NORMAL, MAYBE_LOW describes the status estimated from that of nearby equipment when the corresponding variable of the equipment is not measured. UNKNOWN is the status that is not measured and cannot be estimated from nearby equipment. In case of keyword COMPONENT which represents component composition, it can be extended as COMPONENT1, COMPONENT2, etc. when many components are processed in the process.
2.3 Diagnostic Agent As DA is not a software component with specified requirement or format, it doesn't have regular structure. However, it has formats containing knowledge base and communication protocol. Knowledge base includes function-behavior model, which enables different local diagnostics from other units. In addition, it contains the inference engine and operation rules with of various message and functions according to the characteristics of units. All DAs share a common communication protocol while they are based on different function-behavior models. Thus, MAS shows consistent performance as communicated information is translated into a common language. Diagnosis by DAs is essentially carried out by exchanging status and local reasoning of neighboring DAs as needed. Therefore, a DA sends queries for what it wants to know, performs reasoning on its status based on the answers to queries, and notifies the user of detected faults of itself and neighboring units. Function-behavior model and activities and role of DA will be explained using a simple process, shown in Fig. 3. This system consists of three process units where input is fed into a
Fig. 3 A simple tank process for demonstration of diagnostic agenthood.
Fig. 4 MADISON for a CSTR process.
648 tank unit through a pipe unit and output comes out through another pipe unit. DA infers causal relationship through communication with neighboring DAs and fundamental selfreasoning. The detecting module of DA is always active, and DAs perform diagnosis by sending and receiving messages when events leading to faults occur. DAs of Equipment_l, 2, and 3 are named DA1, DA2, and DA3, respectively. Let us assume a leak occurs at Equipment_2, a tank. This event causes a symptom of low flow of Equipment_3 (F3) and low level of Equipment_2 (L). The symptom is expressed in the language of function-behavior model as follows: keyword status low of keyword MASS (MASS[low]) is detected in DA2, and MASS[low] is detected in DA3. Triggered by events leading to faults, DA3 asks a query to DA2 in order to verify if this symptom is propagated or intermittent. DA2 replies the answer to the query of DA3, triggered by the event of receiving a query message, and DA3 performs a local diagnosis from receiving reply in the input device. In this case, it can be concluded that the fault is propagated to unit itself based on the fact that MASS[low] is received and the status of itself is MASS[low]. In fact, a DA has many types of messages for processing problems depending on an unmeasured variable exists, a controller states a diagnosis opinion, keyword changes occur as in the heat exchanger, etc. In addition, the knowledge system may be different due to the uniqueness of process necessitated by process units. As mentioned before, multi-agent diagnosis system uses messages where an event of receiving a message triggers diagnosis. Messages will be explained more in the following section. The important functions of DA can be summed up as communication protocol of message passing, reasoning, and report of inference results. DA reports the opinion of each DA to operators by relative importance based on the results of local diagnosis. Inference characteristic of processes or process units is required in this step, and relationship between keywords and related faults should be identified in advance. How to handle physically infeasible keywords and faults in a process unit should be considered as well.
2.4 Certainty Factor (CF) Distributed fault diagnosis shows good performance in portability, expandability, and speed. However, conflicting diagnostic results may exist and should be resolved because local fault diagnosis and communication among agents are inherent in this system [8]. In this paper, possibility of fault occurrence is quantified by introducing certainty factor (CF) to resolve the conficting diagnostic results. As for the weight factor used in calculating CF, Gaussian distribution is assumed in the time domain from the start of the diagnosis to the current time. This approach is suitable for the characteristics of fault diagnosis as the latest diagnostic results take priority over the past results, and even old results are not ignored completely. 3. CASE STUDY The process used in the case study is a classic example for fault diagnosis, and has been used in many researches, including the work of Kramer and Palowitch [9]. The mathematical modeling of Sorsa and Koivo [ 10] was used for this study. There are three feedback control loops in the process, and PI controllers are used. They control the level of reactor, recycle flowrate of product, and reactor temperature, respectively. Level control of reactor is direct and no keyword change occurs, recycle flowrate control is reverse and no keyword change occurs, and temperature control of reactor is direct and
649 keyword change occurs. Therefore, different function-behavior model is required for each controller. 3.1 Multi-Agent Diagnostic System for a CSTR process We implemented a fault diagnosis system development tool for chemical processes, MADISON (Multi-Agent Diagnostic System ON-line), on G2| Expert System Shell. Fig. 4 shows a multi-agent fault diagnosis system constructed by applying MADISON for CSTR process. As process units are DAs, process topology of the real process is preserved in the diagnosis system. Fault diagnosis is carried out through local diagnosis unit-wise, and as a result, the interface is very friendly and easy to understand. A simple model to verify status is formed for PI controller used in the case study. The function of controller is to maintain the set point of controlled variable. This is not limited to PI controller, and is for almost every controller. Here lies the advantage of function-behavior model. Measured values and manipulated variables are used for this purpose. That is, manipulated variables are used to maintain the set points of controlled variables as measured values change. Hence, the relationship among measured values, and manipulated and controlled variables is important, and controller is thought to be problematic when these values and variables change out of accordance with expectation. MADISON does not consider the case of controller failure, and warns possibilities of undetected faults of nearby process units instead. There are three PI controllers in this CSTR process, and the related variables are as follows. (1) Reactor temperature controller: T (ENERGY) - CT (ENERGY-MASS)- FW (MASS) (2) Reactor level controller: L (MASS)- CL (MASS)- FP (MASS) (3) Recycle flowrate controller: FR (MASS)- CR (MASS) Proper control is said to be functioning when all the three variables show the same tendency in reactor temperature and level controllers. Recycle flowrate controller is determined as functioning when the two variables (FR and CR) show the opposite tendency. Generally, when a controlled variable shows deviation, the controller tries to compensate for the deviation error. Therefore, the controlled variable itself is not detected as a symptom, but manipulated variable shows symptom. For example, when the level L of the reactor--the controlled variable of the reactor level controller--starts to decrease, the output signal CL decreases so that the output flow (manipulated variable) of the reactor increases to maintain the level L at setpoint. In this case, LC-DA--the level controller DA--suggests that there is possible fault in the input stream to CSTR and notify it to DA-CSTR as both CL and FP decreased. DA-CSTRmthe CSTR DA--is designed to receive the notification of possible fault in the input stream and to report the another possible fault in the level of CSTR itself (which means that there might exist leak) to the operator.
3.2 Test Case: a CSTR Leak This case is when the leak of 0.5kg per second occurs in the stirred reactor after 200 second during the simulation for 500 seconds. The level could be controlled by the level controller very well and showed no symptom of deviation. After the reactor started to leak, the output signal of the level controller CL decreased and the flowrate of product FP also decreased. As the level was maintained, the temperature of the reactor did not deviate and only the flowrate of product kept decreasing. The diagnosis was performed quickly with exchanges of messages of DAs as soon as the detection of symptoms occurred.
650 Table 1 Diagnoses of CSTR leak. ID of DAs Keyword_status
Diagnosis
CF
CSTR-DA-2
MASS LOW
CSTR leak
0.570
PIPE-DA-6
MASS LOW
Pipe_blocked Pipe_leak
0.444
Control valve stuck low Pipe_blocked Pipe_leak
0.045
-
PIPE-DA-9
MASS_LOW
As a result, CSTR-DA-2 suggested the leak of CSTR as the most probable candidate of fault with 0.57 CF, which is the largest among other candidates. Note that the reactor leak was suggested as a fault regardless of no symptom of reactor level decrease as a result of collaboration and negotiation of neighboring DAs. The final results of diagnosis based on the local diagnoses and CF of each DA are summarized in Table 1. Because the leak was not so serious, the controller could maintain the normal operation conditions to some extent. Hence, the process was not so dynamic and the symptoms occurred somewhat late and the value of CF of each DA had small values around 0.5. 4. C O N C L U S I O N The multi-agent technique offers a powerful alternative for diagnostic problem solving in a lot of engineering applications. In this paper, we suggested function-behavior modeling and multi-agent diagnostic system for chemical processes where diagnostic agents of each unit communicate by exchanging messages and try to solve the global fault diagnosis problem by collaborative diagnosis method. The suggested approach was implemented on the expert system development tool, and a prototype application built showed its effectiveness. Acknowledgements
This work was partially supported by the BK 21 Program supported by the Ministry of Education and the National Research Lab Grant of the Ministry of Science & Technology. REFERENCES
1. W. R. Becraft, D. Z. Guo, P. L. Lee and R. B. Newell, PSE91, Montebello, Quebec, 1991. 2. D. Mylaraswamy and V. Venkatasubramanian, Comp. Chem. Eng., 21S (1997) 940. 3. J. M. Bradshaw (ed.), Software Agents, AAAI Press/The MIT Press., San Francisco, 1997. 4. M. N. Huhns and M. P. Singh, Readings in agents, Morgan Kaufmann, Cambridge, 1998. 5. R. Batres, S. P. Asprey and Y. Naka, Comp. Chem. Eng., 23S (1999) 653. 6. Y. S. Oh, Ph.D. Thesis, Seoul National University, Seoul, 1998. 7. S. Y. Eo, T. S. Chang, D. Shin and E. S. Yoon, Comp. Chem. Eng., 24 (2000) 729. 8. T. S. Chang, Ph.D. Thesis, Seoul National University, Seoul, 2000. 9. M. A. Kramer and B. L. Palowitch, AIChE J., 33 (1987) 1067. 10. T. H. Sorsa and N. Koivo, IEEE Trans. on Sys. Man and Cyber., 21 (1991) 815.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
651
Robust Predictive Control Combined with Nonlinear Observation for Monitoring (Bio)chemical Processes O. Gehan, M. Farza, M. M'Saad and G. Binet, Laboratoire d'Automatique de Procrdrs, L.A.P., E.A. 2611, I.S.M.R.A., Universit6 de Caen, 6 Bd Mal Juin, F-14050 CAEN cedex, France An integrated approach for the on-line monitoring of chemical and biochemical reactors is presented. While controlling the process, the proposed approach also provides on-line estimates of the reaction rates and the non measured component concentrations. The main features of this approach are detailed and illustrated in simulation through a chemical process. 1. INTRODUCTION A systematic and rigorous approach for the on-line monitoring of chemical and biochemical reactors is presented. Two main features of the resulting monitoring approach are worth to be mentioned: 9The control design is carried out using the long range predictive control culture that has been comprehensively compiled by [2]. Of fundamental interest, the design parameter specification is carried out using an iterative procedure that provides an appropriate shaping of the usual sensitivity functions, ensuring thereby the underlying robustness. 9The estimation procedure makes use of recent results on nonlinear observers' design (see e.g. [3]). The corresponding observers are synthesized on the basis of the process mathematical balance models. They do not assume or require any model for the reaction rates and are very successful in accurately estimating these variables. An outline of this paper is as follows: in the next section, we describe the main components of the robust control scheme. In section 3, we introduce (bio)chemical reactor state-space models which are the basis of the nonlinear observers synthesis. Finally, in section 4, the features of the proposed approach are illustrated through a simulation experiment. 2. THE ROBUST PREDICTIVE CONTROL APPROACH
The input-output behaviour of the plant to be controlled is assumed to be approximated using a polynomial approach by the following difference equations :
A ( q - 1 ) y ( t ) -- B ( q - 1 ) u ( t - d
- 1)+ v(t)
and
D(q-1)v(t) = C(q-1)7(t)
652 where q-1 denote the shift operator, A(q-1), B(q-1), C(q -1) and D(q -1) are polynomials in q-l; {u(t)} and {y(t)} respectively denote the control variable and the measured plant output, {v(t) } represents the external disturbances, {3t(t)} is assumed to be a sequence of widely spread pulses of unknown magnitude or random variables with zero mean values and finite variances, d+ 1 denotes the plant model delay in sampling periods.
{eu(t)} and {ey(t)} defined by" eu(t)- u(t)-~A(q-1)y*(t+d+ 1)
It has been shown in [4] that the sequences
ey(t)-- y(t)-~B(q-1)y*(t) and with A* (q-1)y* (t)=B*(q-1)u* (t) represent the input-output errors of the partial state reference model control which is motivated by the ability to specify the tracking and regulation dynamics B* (z-1) denotes the desired partial state referindependently. The pulse transfer function A*(z-1) ence model which provides the reference sequence {y*(t)} from a bounded setpoint sequence 1
{u* (t) } and 13is a scalar which is introduced to ensure a unit static tracking gain, i.e. 13-- B(0)" The underlying control design is carried out using the generalized predictive control approach proposed in [2] and leads to a linear controller that may be written as follows:
S(q-1)D(q-1)eu(t) +R(q-1)ey(t) = 0 with S(q -1) -
(1)
Sa(q-1)Wyd(q-1)Wun(q -1) R(q -1) --Ra(q-1)Wyn(q-1)Wud(q -1) and P,~a(z-1) - Ra(Z-1) '
So(Z-l)
denotes the regulator corresponding to the cascade composed by the plant together with the user defined input and output frequency weighting filters, namely
Wu(z -1) - D(z-1)Wun(Z-1) Wud(Z_ l ) and
Wyn(Z -1 ) ~y(Z -1) --- Wyd(Z_l) " The system input-output behavior is then described by:
y(t) -- ~B(q-1)y*(t) + S(q-1)Vy(t) - T(q-1)Vy(t) d- PS(q-1)(Vy(t) q-Vy(t)) u(t) -- [~A(q-1)y*(t +d+ 1)+S(q-1)Vu(t)- T(q-1)Vu(t) + RS(q-1)(Vy(t) +Vy(t)) where Vu(t) and Vy(t) respectively denote the input and the output disturbances; the terms Vu(t) and Vy(t) are the input and the output noise measurements, respectively; the pulse transfer functions
5(Z -1)
-- A(z-1)S(z-1)D(z -1) Pc(z -1 )
RS(z -1) - A(z-1)R(z -1) -pc(z-l)
T ( z - 1 ) _ z-d-ln(z-1)R(z -1) Pc(z -1 )
pS(z-1) - z-d-lB(z-1)S(z-1)D(z -1) Pc(Z-1)
will be referred to as the usual sensitivity functions. These are genuine performance and stability robustness quantifiers whenever the involved system identification is successfully performed. Their shapes may be refined by properly choosing the predictive control law parameters. The term Pc(q -1) is the characteristic polynomial of the closed loop transfer. Of particular importance, the control system is asymptotically stable if and only if the characteristic polynomial is Hurwitz.
653
3. MODELLING AND ESTIMATION IN 5BIO)CHEMICAL PROCESSES The mass balance equations associated to N components C1,..., CN, which interact through M reactions r l , . . . , rM inside a (chemical) biochemical reactor can be written as follows : (2)
C= Yr(C,t)-DC+DCin-G
where the vectors C, C/n, G E ]RN correspond to the component concentrations, the influent component concentrations and the mass outfow rates in gaseous form, respectively. The N x M matrix Y contains the yield (stoichiometric) coefficients and the vector r E IRM corresponds to the reaction rates. Finally, the scalar D denotes the dilution rate. For biochemical processes, the dynamical balance model is restricted to system (2). In the case of chemical ones, the energy balance is considered and system (2) is augmented by the following equation: TR --
--1-----~AHTr--Q-t-D(TRin-- TR) pRCpR
(3)
where TR and rRin denote the temperature inside and inlet the reactor, respectively. The row vector AH r C IRM contains the reaction enthalpies. The scalars PR and CpR denote the density and the specific heat of the reaction bulk, respectively. Finally, Q is a normalized heat flux. Now, the following system can be adopted as a unified modelling framework for chemical and biochemical reactors:
= K r - D~ + W
(4)
pRcpR Y and ~ A C;
and W =
(
DCin - GQ
DTRin -
)
for chemical reactors
K A y and W A DCin - G for bioreactors
It is well-known that the modelling of r is a difficult and hazardous task. In order to overcome these difficulties, we shall propose nonlinear observers for the on-line estimation of these variables. Such observers will also allow the estimation of the non measured component concentrations. For this aim, the following realistic conditions are considered : (C1) K, D, W are known and the temperature TR is supposed to be measured when chemical reactors are considered. (C2) The time derivatives of the reaction rates are bounded. (C3) There are M-1 component concentrations which are measured. Moreover, if ~ is partitioned into the sets of measured variables ~' and non-measured variables {"" ~ -
~,,
and
654
(K')
accordingly, K - -
K"
' then we have rank(K') -- rank(K) = M.
According to condition (C3), system (4) can be rewritten as follows:
{ ~' = K' r - D~' + W' ~"= K'lr- D~" + W"
(w,)
where
W"
(5)
denotes the partitions of W induced by the above partitions of ~ and K.
Now, on the basis of system (5), we propose the following nonlinear observer-based estimator for the on-line estimation of the rates rj and the non measured component concentrations r from the measured state variables r ,z!
-
+ W'-
30(
' -
- 1] - 302K'-1 ( ~ ' - ~')
-- _03Kt-l(~t_~ t)
(6)
It -
where 0 > 0 is a design parameter and r I denotes the first time derivative of r. 4. SIMULATION RESULTS In this section, the main features of the proposed approach are illustrated in simulation through an example dealing with one reactant chemical reaction held in a batch jacketed reaction calorimeter. It should be emphasized that this example has been chosen mainly for its simplicity and illustrative properties. Nevertheless, the whole procedure previously described can be applied to more sophisticated reaction scheme. The considered mathematical model is :
{ TR-- anr -
~
pRCpR
CA
+ Q
(7)
~" - - r
where r is the reaction rate, CA is the reactant concentration and the other terms keep the same meaning as before. Our control objective consists in tracking a desired profile for the reactor temperature. The plant input and output are the input jacket temperature Tji n and the reactor temperature respectively. While controlling the temperature, our aim also consists in obtaining on-line estimates of the reaction rate r and the component concentration CA from the sole temperature measurements. The obtained results have been compared with data issued from simulation, i.e. we have simulated the process model (7) by considering a first order reaction rate. The term Q and the jacket temperature Tj were obtained by considering the dynamics of the latter. Before being used by the control and estimation algorithms, the reactor temperature (issued from simulation) has been corrupted by an additive noise. It should be also noted that the heat production inside the reactor induced by the exothermic reaction is considered as an output perturbation. For comparison purposes, a PID controller has been synthesized using the relay method [ 1].
655 5
0
o -5
~'~Q" .
COMPLEMENTARY SENSITIVITY
"
I -10
g_
~-15
SENSITIVITY FUNCTION
-25
o~
"350
oo,F,eq,-,,..w (r.,d~)00,5
o~,2
o~
25
/
~
oi,,
0;,5
o~,. . . .
F,eq,~,.-y (~V=)
SENSITIVITY FUNCTION
2O
/
le
i --25
le
~,
g o
TION -lo
CONTROLLER
o~o5
--400
o;,
oo,5
0O25
OO2
F,eq,~rcy (~.d,~)
0005
001 0;15 Frequency (rad/I)
O0 .
.
.
.
Fig. 1. Sensitivity functions (m GPC;-- PID)
015
015
01
005
0
20
40
I
60 Time (ran)
I
80
I
100
120
_0
o51
20
40
60 Time (mn)
80
100
120
32 30
30
2e
28
2e
20
24
24
22
20
le
20
4O
"nm~mn)
O0
Fig. 2. Input-Output performances
IO0
120
18
-
20
40
60 "nine (mr,)
80
100
120
656 550
r (mol.m-3.s -I)
500 450
(mol.m -3)
400 350 30(]
25a 20C
15C 10C 5C
20
40
T~m6~mn)
80
'
~o
0
;o
~.
. . -rim, (mr,) eo
80 .
1o o
Fig. 3. Estimation of r and CA
Figure 1 shows the Bode plots of the usual sensitivity functions corresponding to the robust predictive control and PID control systems. Corresponding input-output temporal behaviors are reported in Figure 2. We remark that the performances of the robust predictive controller are better according to the input behaviour. Indeed, for the PID controller, the gain of the regulator times the sensitivity function is too high to perform well in the presence of measurement noise and unmodelled dynamics. Estimation results are reported in Figure 3 where the estimates of r and CA are compared to their 'true' values issued from the model simulation. We remark the good performances of the estimators in tracking the parameter variations and in dealing with noise rejection (estimated and simulated curves are almost superimposed). We recall that, as for the control scheme, the expression of the reaction rate introduced for simulation purposes is ignored by the nonlinear observer.
5. CONCLUSION An integrated approach for on-line control and monitoring of biochemical processes has been presented. The main features of the proposed approach have been highlighted in simulation through a typical chemical reaction. REFERENCES
1. K.J. ~str0m and T. H~igglund. PID Controllers : Theory, Design and Tuning. Instrument Society of America, 1995. 2. D.W. Clarke, C. Mohtadi, and P. S. Tufts. Generalized predictive control. Automatica, 23(2):137-160, 1987. 3. J.P. Gauthier, H. Hammouri, and S. Othman. A simple observer for nonlinear systemsapplication to bioreactors. IEEE Trans. on Aut. Control, 37:875-880, 1992. 4. M. M'Saad and G. Sanchez. Partial state reference model adaptive control of multivariable systems. Automatica, 28:1189-1197, 1992.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
657
Control Structure Selection for an Evaporation Process Marius S. Govatsmark a, Sigurd Skogestad a* aDepartment of Chemical Engineering, NTNU, N-7491 Trondheim A systematic procedure for control structure selection is applied to the evaporation process of Newell and Lee (1989). First, promising sets of controlled variables are selected, based on steady-state economic criteria. The objective is to find sets of controlled variables which with constant setpoints keep the process close to the economic optimum ("self-optimizing control") in face of disturbances and implementation errors. Second, stabilization and controllability analysis is performed for the most promising sets of economic controlled variables. 1. INTRODUCTION Control structure selection consists of selecting controlled variables, manipulated variables, measurements and links between them. A poor choice can give both dynamic and steady-state problems, such as instability, input saturation, operation outside constraints and non-optimal operation. This can be partly counteracted by using logic, model predictive control and on-line optimization, but the control system then becomes more complicated and costly than necessary. Selecting a good control structure is a precondition for getting a simple control system with good control behavior. Inspired by the many methods presented in the area of control structure selection, both from a steady-state economic point of view and from a dynamic analysis point of view, a procedure is proposed and applied to a simple, but illustrating example of the evaporation process of Newell and Lee (1989). The control structure selection is mainly based on a plant-wide control procedure presented by Larsson and Skogestad (2001). Back-off, Perkins (1998), from the nominal operation is sometimes needed in order to obtain feasible operation. We define backoff (or setpoint adjustment) as the difference between the nominally optimal setpoint and the actual setpoint, used e.g. to achieve feasible operation when there are disturbances. There are two cases: (a) Control of variables at active output constraints. The required back-off is given by the implementation (including measurement) error. (b) Control of unconstrained variables. We adjust the setpoints to achieve feasibility when there are disturbances which otherwise move the active constraints or include a new one. Note that the required back-off and corresponding economic loss depends on the selected controlled variables. Thus, the primary issue is to select the right control structure (variables), whereas the back-off is just a setpoint adjustment to deal with nonlinearities and in particular constraints. Optimal back-off can be determined by robust optimization, which minimizes the nominal steady-state economic criteria, given that the constraints are satisfied for all expected disturbances and implementation errors, see Glemmestad *e-mail:
[email protected];phone: +47-7359-4154; fax: +47-7359-4080
658 .L ~ |
~
,
,
,
',F~---,
T ~ v ,,;T:--!P ~__~
FlOO
Cooling water
| Condensate .I ~ Separator F
~ ~3
B
%oo
EvapoP'rat~~~
F
C o n d e n s a t e ~2 -~,~" Fe.Pump ---~_..
y" L Fx T
Fig. 1. Evaporation process
:~--~'I"F
08
,
F', FF,G
,
9
F 1 [~g~min]
11
12
royct F2x2T2
Fig. 2. Loss as function of feed flowrate for different alternatives with robust setpoints and simple logic
et al. (1999). 2. CONTROL STRUCTURE SELECTION PROCEDURE A short summary of methods and rules recommended in control structure selection, Larsson and Skogestad (2001): 1) Formulate a steady-state economic objective. 2) Define control objectives, steady-state and dynamic degrees of freedom, manipulated variables with allowed variations, disturbances with expected variations and possible controlled variables with expected implementation errors. 3) Steady-state optimization should be performed at the nominal point and, if possible, for extreme values of expected disturbances and implementation errors. 4) In order to reduce the number of sets of controlled variables to be evaluated in detail, use active constraint control, see Maarleveld and Rijnsdorp (1970), and the minimum singular value rule, see Skogestad (2000). 5) For the remaining combinations of controlled variables the economic loss imposed by keeping the sets of controlled variables constant (rather than at their optimal values) should be evaluated for expected disturbances and implementation errors. 6) To avoid infeasibility, try back-off from nominally optimal setpoints. 7) For stabilization select the variables to control and manipulate based on computing the poles and their associated input and output directions, see Havre (1998). 8) Perform controllability analysis for the most promising sets of economic controlled variables. Tools are provided by Skogestad and Postlethwaite (1996). 9) Design controllers and run nonlinear simulations for the most promising alternatives. 3. EVAPORATION PROCESS CASE STUDY In the evaporation process of Newell and Lee (1989) the concentration of dilute liquor is increased by a vertical heat exchanger with recirculated liquor, see figure 1. The steady-state economic objective is to minimize the operational cost (S/h) related to steam, cooling water
659 and pump work, see Wang and Cameron (1994): Y - 600F10o + 0.6F2oo -+- 1.009 (F2 + F3 )
(1)
Process constraints related to product specification, safety and design must be met: X2 > 35%
(2)
40kPa _< P2 < 80kPa
(3)
PlO0 _< 400kPa
(4)
F2oo _< 400kg/min
(5)
0kg/min < F3 _< 100kg/min
(6)
There are four manipulated variables; steam pressure, cooling water flowrate, recirculating flowrate and product flowrate: u T = [F2oo/9100F3 F2]
(7)
One degree of freedom is purely dynamic (the condenser level which needs to be stabilized), hence there are three steady-state degrees of freedom. The major disturbances are feed flowrate, feed concentration, feed temperature and cooling water inlet temperature, with expected variations about +20%: d T = IF1 Xl I"1 T20o] = [10 + 2 kg/min
5 + 1%
40 + 8~
25 + 5~
(8)
Controlled variable candidates are all possible measurements and manipulated variables: yT = [F2 F3 F4 F5 X2 T2 T3 L2 P2 F100/91o0 Qloo F200 T201 Q200]
(9)
Expected implementation error associated with each variable is based on the following rules: Flowrates (+10%), compositions (• temperature (+I~ and pressure (• The model equations are given in appendix A. 3.1. Steady-state economic evaluation and optimization The steady-state optimal values at the nominal point for the objective function is 61625/h and corresponding to the following optimal values: T
Yopt
--
[1.4 27.7 8.6 8.6 35.0 90.9 83.4 1.0 56.2 10.0 400.0 365.6 230.2 45.5 330.0]
(10)
Steady-state optimization at the nominal point and for extreme values of the disturbances yield that two of the constraints, product composition (x2) and steam pressure(P100), are always active.Active constraint control then consumes two steady-state degrees of freedom. The last degree of freedom is optimally unconstrained for most disturbances (There is one exception, for low feed flowrate the last degree of freedom is consumed by the minimum operating pressure constraint). The best economic choice for this controlled variable is related to the selfoptimizing control properties. There are 13 candidate variables. The minimum singular value rule, see Skogestad (2000), is applied to eliminate some of them. For one single input the rule is to select the controlled variable with the largest absolute process gain (IGI), when the variables are scaled with respect to the sum of the variation in their optimal value and the expected implementation error. The seven most promising combinations of economic controlled variables are
660
Table 1 Most promising alternative sets of controlled variables based on the minimum singular value rule Ys,3 IG[ Rank Alt. Ys,1 Ys,2 1 G 22 el00 T201 0.0150 2 FF X2 el00 F2oo/F1 0.0135 3 F x2 PlO0 F2oo 0.0108 4 C x2 PlOO P2 0.0044 5 A x2 Plo0 T2 0.0042 6 H x2 P10o T3 0.0042 7 B x2 /10o F3 0.0018 -
E
X2
P2
F3
-
Table 2 Average economic loss with optimal back-off to avoid infeasibility Rank Alt. Ys,! Ys,2 Ys,3 Average (X2) (P100) loss 1 F* 36 390.0 231.2 0.55% 2 G 36 390.0 48.0 0.59% 3 FF 36 390.0 20.6 0.60% 4 E* 36 57.3 41.0 0.91% 5 H 36 390.0 90.3 1.24% 6 C 36 390.0 69.6 1.24% 7 A 36 390.0 98.7 1.24% 8 B 36 181.0 1 3 4 . 4 3.24% F Infeasible E Infeasible *=With use of logic
listed in table 1. Here we have also included a feed-forward improvement of alternative F (denoted F F ) . In addition we study alternative E, proposed and used by Newell and Lee (1989). In this alternative the recirculating flowrate (F3) is not used as a manipulated variable in the basic control layer. They do not use active constraint control, because the last available manipulated variable, cooling water outlet flow (F2oo), has too small effect on the product composition (x2). The steam pressure (Ploo) is not kept on its constraints, but is used to control the product composition (x2). The cooling water flowrate (F200) is used to control the operating pressure (P2). To achieve feasibility, we compute the optimal back-off from the nominal optimum setpoints by robust optimization(Glemmestad et al. (1999)). The losses related to keeping the operation at these robust setpoints are given in table 2 and in figure 2. There exists no feasible setpoint adjustment (back-off) for alternatives E and F. Alternative F has feasibility problems only at low feed flowrates when the operating pressure (P2) gets too low. To avoid infeasibility a simple logic scheme can be introduced: If the feed flowrate becomes lower than 8.5 k g / m i n , the cooling water flowrate is reduced to avoid that the operating pressure becomes too low. For the proposed structure of Newell and Lee (1989) (alternative E) several constraints may be violated, and the logic becomes more complicated. For large feed flowrates and a high inlet cooling water temperature the cooling water flowrate should be kept constant instead of the operating pressure (P2). 3.2.
Stabilization
and
controllability
analysis
The rest of the analysis is based on a scaled linear, dynamic model. The holdup in the separator must be stabilized. Based on computing the poles and their associated input and output directions we find as expected, that the separator level must be controlled by the product flowrate. Controllability analysis is performed for the three most economic promising alternatives F, F F and G, plus alternative E. The alternatives are largely controllable. The relative gain array (RGA) is used to select links between the controlled and manipulated variables, see table 3. Alternative F F and G are shown in figure 3 and 4. Decentralized controllers were designed, and nonlinear simulations were performed to verify the controllability of the designs.
661 Table 3 Control structure used in decentralized control Alt.
Loop 1
Loop 2
E* F* FF G
x2 ~ PlOO x2 ~ F3 x2 4-~ F3 x2 ++/73
P2 ~ / 7 2 0 0 PlOO P100 P100
Loop 3
F3
F200
Qoo/FI
T201 ~-~ F200
Loop 4** L2 L2 L2 L2
* = With use of logic, ** = Stabilizing loop
++ F2 ~-~ F2 ++ Q ~ F2
(~-~t, ...."B~ " ................................................................. 17~:,,~r .. ....
:vo..........-~--~-,FC~---,, Condenser ' Cooling water
Cooling water
T ~ 2o~
T~
-F T Condensate
P '~" i ~
Steami ,~
~
Foo~oo
~ ' Condensate A A[ Separator F L'! L~,
~i o
F3"
Eva
.
Cor~densate ~ F i i ' m - ~ Feed
~ ~ ,
E vap~176
,~ .... ::
~(21
[I II I F,.
l
i.
:x
C o n d e V n s a t e ~ ,_~ihj-purnp ...........!-.......!~ Product
F~
FxT
Fig. 3. Evaporation process with control structure FF
FxT
FxT
Fig. 4. Evaporation process with control structure G
4. C O N C L U S I O N A systematic procedure for control structure selection, based on both steady-state economic evaluation and controllability analysis, has been demonstrated. The evaporation process example has been used to illustrate how important the selection of control structure is and how rewarding it may be. By putting serious effort in selecting a good control structure, it is possible to avoid significant control problems and reduce the complexity of the control system. Both alternative F F and G have small economic loss, good control behavior and no need of logic. In the end alternative G is proposed, because it is based on pure feedback control. A. M O D E L E Q U A T I O N S
20dL/dt 20dxz/dt
- - F1 - F4 - F2 -
Fix1 - F2x2
4dPz/dt =
F4 - F5
T2 = 0.5616P2 + 0.3126x2 + 48.43 T3 -- 0.507P2 -+- 55.0
(11) (12) (]3) (14) (15)
662 T100 = 0.1538P100 + 90.0
(16)
aloo = 0.16(F1 + F3) (Tloo - T2)
(17)
Q l o o - 36.6Floo
(18)
Qloo -- 38.5F4 q- 0.07Fl (T2 - T1)
(19)
Q2oo -- 6.84(T3 - 0.5 (T2oo+ T201 ))
(2o)
Q200-- 38.5F5
(21)
Q200 -- 0.07F200(T201 - T200)
(22)
Newell and Lee (1989) assumed the time lag connected to the slave controllers to be 1.2 min. This seems too large, and we use 0.1 min.
REFERENCES Glemmestad, B., S. Skogestad and T. Gundersen (1999). Optimal operation of heat exchanger networks. Computers Chem. Engng. 23, 509-522. Havre, K. (1998). Studies on the controllability analysis and control structure design. PhD thesis. NTNU. Available from http://www.chembio.ntnu.no/users/skoge/. Larsson, T and S Skogestad (2001). Plantwide control: A review and a new design procedure. Modeling, Identification and Control 22(1), 1-32. Maarleveld, A and J.E. Rijnsdorp (1970). Constraint control on distillation columns. Automatica 6(1), 51-58. Newell, R.B. and P.E. Lee (1989). Applied Process Control - A Case Study. Prentice Hall. Perkins, J.D. (1998). Plant-wide optimization: Opportunities and challenges. FOCAPO III, Snowbird, Utah pp. 15-26. Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control - Analysis and Design. pp. 246-248. John Wiley & Sons. Skogestad, Sigurd (2000). Plantwide control: The search for the self-optimizing control structure. Journal of Process Control 10(5), 487-507. Wang, F.Y. and I.T. Cameron (1994). Control studies on a model evaporation process - constraint state driving with conventional and higher relative degree systems. Journal of Process Control 4(2), 59-75.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
663
MPC using Nonlinear Models Generated by Genetic Programming Benjamin Grosman and Daniel R. Lewin PSE Research Group, Dept. of Chemical Engineering, Technion I. I. T., Haifa 32000, Israel This paper describes the use of genetic programming (GP) to generate an empirical dynamic model of a process, and its use in a nonlinear, model predictive control (NMPC) strategy. GP derives both a model structure and its parameter values in such a way that the process trajectory is predicted accurately. Consequently, the performance of the NMPC strategy, based on this model, is expected to be good. The genetic programming approach and the nonlinear MPC strategy are briefly described, and demonstrated by simulation on a multivariable process. 1. I N T R O D U C T I O N All control systems are either implicitly of explicitly model-based. They can be as simple as a PI controller, which is implicitly based on a first-order lag model of the process, or as sophisticated as a set of DAEs that model the process and are solved in a nonlinear predictivecontrol algorithm. The performance to be expected from a model-based controller is directly related to the fidelity of the model, in the frequency range where performance is sought. However, generating an accurate model is sometimes a difficult task. The development of first-principles models can be expensive and often leads to disappointing results, since it is hard to capture all relevant phenomena in the model. In contrast, empirical models are relatively easy to calibrate, but rely on model structures often selected by intuition, and may require repeated updating because of changing process conditions. Genetic programming (GP) is an evolutionary method for automatically generating nonlinear input-output models. In this work an improved Genetic Program is developed for generating models. The algorithm relies on previously published work, but involves novel methods to assess model fitness accounting both for modeling and prediction error, as well as efficient population management, and control of constant assignment for each candidate model depending on its degrees-of-freedom. These features significantly improve the predictive power of steady-state models generated by our algorithm. In this contribution, the novel GP code is used to generate empirical non-linear dynamic models for control purposes. More specifically, the models so derived are to be used for nonlinear MPC. The performance of the GP-based MPC is compared with that of a conventional linear PI control strategy, when applied to a simulated multivariable process. This paper begins by describing the GP used for model derivation, and the implementation of the nonlinear MPC algorithm. Preliminary results of testing the algorithm on a relatively simple MIMO process are given.
664 2. G E N E T I C P R O G R A M M I N G Genetic programming (GP) is an evolutionary method for automatically generating nonlinear input-output models. The probability of a given model surviving into the next generation is proportional to how well it predicts the output data, and components of successful models are continuously recombined with those of others to form new models. The GP thus optimizes the model structure, with a lower level nonlinear least-squares algorithm harnessed to perform the associated parameter estimation. Gray et al. (1996), McKay et al. (1997) and Willis et al. (1997) have discussed the application of genetic programming to nonlinear modeling. Lakshminarayanan et al. (2000) used a composite GP-PCA approach to generate nonlinear models for product design applications. GP is based on simple rules that simulate biological evolution. Combining basis functions, inputs and constants creates an initial model population, whose complexity is controlled by the user. The models are structured in a tree-like fashion, with basis functions linking nodes of inputs and constants. Each individual model in the population is then fitted to the empirical data using nonlinear regression, and then graded according to how well it matches the data. In each generation (iteration) of the algorithm, relatively successful individuals are selected as "parents" for the next generation and form a reproduction pool. A new generation of solutions evolves, using one of three possible operations: crossover, mutation and permutation. In crossover, two individuals from the pool are selected, their tree structures are divided at a randomly selected crossover point, and the resulting sub-trees are recombined to form two new individuals. In mutation, a random change is performed on a selected individual by substitution - this can be a functional group, an input variable or a constant. In permutation, two branches of a selected individual are randomly switched. The parameters for each new individual in the new generation are regressed, the models are then graded by fitness as before, and the procedure is repeated until a stopping criteria is attained. In most cases, as in this application, the population size, npop, and the total number of generations, ncen, are decided in advance. The GP code used in this work is similar to that published by McKay et al. (1997), with some important differences: One. Overfitting-controlledfitness evaluation. The most important difference concerns how the fitness is computed, which should be a combination of the quality of the model to match the empirical data with some penalty to guard against over-complex models. In contrast with previously published work, our code preserves a portion of the data to test predictive power of the model, and uses the prediction error to dynamically limit model complexity during the course of evolution. This method successfully prevents over-fitting without requiring an arbitrary cut-off length to be specified a priori. Two. Smart control on new generation synthesis. The second difference is related to how a new generation is created from its parent population. Most codes adopt so-called patrimony, creating the next generation by keeping some fixed part of the parent population, usually the better solutions, and filling the balance from evolutionary operators applied to the reproduction pool. This approach is expensive, since no guarantee can be made that the evolved children are always superior to those parents they replace. Instead, our approach is to create npor new individuals as above, making temporarily a total of 2 npop individuals, and pick the best npop of these to constitute the next generation. To make this approach work well, it is also necessary to identify repeated solutions, and eliminate those from the gene pool. This requires tree-structure recognition capability.
66~ Three. Smart tree-structure recognition capability. To identify critical sub-tree structures and to eliminate repeated solutions from the population, it is necessary to be able recognize tree-structure components. For example, if n is the number of addition and multiplication nodes in a tree, then there are 2" equivalent trees possible. Thus, a check is performed on each newly generated model to ensure that it is not equivalent to one already existing in the population, and if so, is marked for deletion Four. Setting the number of constants dictated by model degrees-of-freedom. To ensure the parametric optimizer can exploit the degrees of freedom in a model, the code considers augmenting the model with additional constants. Good results have been obtained using our code to identify steady state models in a number of applications. The same principle used to generate steady-state models is also employed for identifying dynamic models. The main difference lies in the way the prediction and the optimization are realized. Instead of developing a conventional one-step-ahead predictive model, the complete trajectory is estimated assuming only the initial condition for the output. For this computation, pseudo-random sequences of steps in the inputs are used to excite the process, with two sets generated - one for modeling (used to optimize the model parameters) and one for prediction (used for verification of the optimized model). The weighted sum of the trajectory-matching performance obtained with both sets is used to compute fitness. 3. NONLINEAR MODEL PREDICTIVE CONTROL (NMPC) The nonlinear model predictive control strategy implemented here consists of several components: 1. Each MISO input-output model, created by the GP, which takes the general form: ~(k)= f ( .~(k -11 k),u_(k -11 k)),
(1)
where ~(k) is the predicted model output at instant k, ~(k- 11k) is a vector of the previous P predicted outputs, computed based on information available until instant k, and u_(k- I l k ) is a vector of P inputs at instant k. In principle, the approach also handles nonlinear state-space models. 2. Selected values for P, the prediction horizon that controls the predictive range of the model, and M the control horizon, which establishes the number of future moves to be optimized, noting that M __P. 3. A matrix of historical data, A, used by the model to predict the influence of previous outputs and inputs on the future outputs. The matrix has P rows and a number of columns equal to the total number of inputs and outputs, and is refreshed every sample instant. 4. A constrained quadratic objective function to be minimized, whose arguments are calculated on the basis of predicted values of inputs and outputs generated using the nonlinear GP models, resulting in a sequence of M optimal future inputs. The objective function used in this work is: (2)
666 where
~(k + l l k ) is a vector holding the estimated output trajectory P moves ahead com-
puted at instant k,
Ys,correctediS
a vector holding the corrected desired trajectory,
Au T (k + 11 k) is a vector of M future manipulated variable changes, optimized at instant k, and Q and S are tunable weights. Because of the inevitable model-plant mismatch, the actual set points used in Eq (2), YS,corrected,if left uncorrected, will lead to optimal control only in the open-loop sense. Thus, the set points implemented use current process outputs as feedback, which is equivalent to integral action, and allows the plant/model mismatch to be estimated: A
d(k)= y(k)- ~v(k),
(3)
where y(k)and ~(k)are the process and model outputs at instant k, respectively. This mismatch is assumed constant for all future predicted values, and leads to a correction to the desired set point values: Y s,corrected
(k)= Y S (k )- ~(k )
(4)
Naturally, there are upper and lower constraints on the manipulated variables, 0 < U _
~ O D e L W~WHNO DYNAMIC
I
IPP,l:I)I(:rl'l()N. ~'I.S c-.,~-~qt'J [DATASETS, SPOT VALUES
[
. . . . . .
:> [DYNAMI(? MODt:~I..I.ING F.RRORS _]
Figure 1. Usual procedures on inferential modeling
Data collection: This phase is to collect hourly averages around the time where lab samples with the proposed prediction characteristic exist. Data around this time has to be evaluated to find process instability. Some data basic statistics should be calculated like data correlation matrix, standard deviation, etc. Data Calibration and Model Generation: Several methods could be used but, in any case, attention has to be paid to data prediction statistics to be sure of the capacity of the model. Runtime model prediction: Usually, spot values are used to have the best response under process upsets.
689
Long range Model Bias update: Care should be taken if it is decided to update the model bias. We have found that in most of the cases this will only propagate lab and instrumentation errors to the model.
2.2. Proposed procedure, Dynamic PLS Algorithm The proposed methodology, schematized in figure 2, is the following: Obtain Steady State Models: It is possible to use any available calibration technology. In general, if the system is highly non-linear, and if the function that correlates inferential or output variable with the input variables is not known (and if data is good enough) then NN will be the best choice. If the range of available data is small, if data have errors and the system is highly collinear, it is probably better to use a method that has a rigid data structure and is protected against noise like PLS. Data could come from plant instrumentation or from first-principles simulators. Simulators are used more and more often due to the fact that data generated with a simulation program in steady state has not noise and is reproducible, the variation range of variables is bigger than in a real plant and, finally, the time to create the data is shorter than any method. However, real data gives the inherent characteristic of a plant than can not be found in a simulation, like process discrepancies, instrument errors, etc. Probably, the best approach is to use a hybrid methodology, using a simulator to find the best-input variables and real plant data to caliber those variables with more accurate input coefficients. ,
" ,
t:~lllhlN
,', ! t:.ti)'~ S T A l l ,
M()DI'.I,,~
_
LJ~.I.L,\I.li(,)i_:Itl.Y,,,\vi'I~,,\(.il;s ] P,I~tt .1) i ) Y N k ~ l t ( '
[
S'I'I.::\[)Y S i'A I'I: t~,,\ f~',]S
,~I~,tU I.,UtI,~)N
[ ~,11:cl ~.,,,Ni~ l
[ ()PERATIN(]
DATA
-..,,,,.. I DYNAMIC SIMULATION I
/
["tTNSTRI,rM c,NTATt.)*~ r " ( ,,r [
[
N t % l t : I . k't I" P I . A N t" l ' t t ~ ' l ,,~,Nt~ II~1-,~' t It _Ci'+'a'iy- M Xii'-M(2 - Y { i - Y i ' j )
Vi, i' e l , i < i', j e Jii'
Vi, i'e I , i < i', j e Jir
(7) (8)
9 Material balances. Material balances ensure that enough intermediate se S I is produced to run any campaign ie Ys requiring s. (i) Sinks for intermediate s produced by campaign iEFs. Intermediate s produced by campaign ie l+s is supplied to one or several sth-consuming runs featuring Fii- > 0. Qi = E Fii,
V i e I s, + se Si
(9)
i'~l 7
(ii) Sources of intermediate s required by campaign iEls. Sources of intermediate s for a run ie Ys are those campaigns i'e I+s featuring Fi.i > 0, pisQi= E F i , i
Vie Is, Se S
(10)
i'~ I +
9 Source/sink campaign matching conditions. A run ieI+s supplying intermediate s to campaign i'e I-s (i.e. Uii' -" 1 and Fii' > 0) must not start later than campaign i' Eli, X q- C 0 2 (R1) A at- 0 2 -> Y-+- C 0 2 (R2) This representation implies that oxygen consumption and production of carbon dioxide are proportional to the production of biomass. Generally both reactions take place simultaneously but R1 is predominant the beginning of a batch. The relation between reaction rates rl and r2 changes during the course of a batch run and depends on the concentration of the substrates lactic acid and amino acid. 3. G R O W T H RATE ESTIMATION At the origin of growth rate estimation is the differential equation for microbial growth with a neglected maintenance term: equation 1. dX
---~=lt
Vn,a:Yax* dX (2) , ~
*X (1)
'
Qa = ~ V n , a , d t + Q a ( t o ) : Y a , x , x to
(3)
The reaction rate vR,a of component a, which must be directly linked to growth, is defined as indicated by equation 2. Its integral Qa is equation 3. Replacing biomass X and its derivative in equation 1 by equations 2 and 3, under the assumption of a constant yield Ya, X, gives an expression, equation 4, for the growth rate:
/~(t) = vR'~
dvn,"(t)
(4)
Oa (t)
fl(t) :
at VR,a(t)
(5)
The initial quantity of component a, Qa(to), is unknown. When calculating the growth rate for a terminated batch run a posteriori, the appropriate value can be found manually. In case of starting the estimation procedure during the course of a batch run, Q~(to) can hardly be determined a priori, neither manually. Hence an additional procedure, allowing for the on-line automatic determination of Qa(to), is necessary. It relies on the alternative expression for the growth rate given by equation 5. d2X dt 2
= #*
dX --~
(6)
dvn'" Y~ x * d 2 X = dt ' dt 2
(7)
vPR
(t) = ~ a i * ti i:0
(8)
Equations 6 and 7 result from deriving equations 1 and 2 assuming that/z 4=f(t) and then introducing equations 2 and 7 in expression 6. In practice, the derivative of the reaction rate will be corrupted with noise. This problem is solved by approximating the reaction rate by a polynomial (equation 8). The data used for this approximation is collected on a gliding window of past measured values. The length of this window Atw depends on the order of the polynomial: the higher the order, the more data is required. In order to start the estimation of the growth rate quickly after the start of a batch, it is preferable to use a low order. Nevertheless the order should allow for approximating an exponential curve (microbial growth with constant growth rate). Hence n=2 appears to be a reasonable choice. Deducing the derivative
719 of the reaction rate from equation 8 is straightforward. The initialization procedure can be triggered manually or depending on a certain condition with respect to process state. It uses equations 4 and 5 to determine the initial quantity Qa(to).After initialization, only equation 4 is used for growth rate estimation. Its advantage compared to equation 5 consists in the fact that no smoothing is carried out, so that the estimation provides a quick response to variations in the process. However disturbances contained in the reaction rate are directly transferred to the estimation. For this reason the estimated growth rate must be filtered (see section 4), in order to reject disturbances, before it can be used for closed loop control. Under the assumptions that (i) the concentration of carbon dioxide in the liquid phase remains constant and (ii) the gas volume between the culture surface and the exhaust gas analyzer is zero, the reaction rate of C02 is equivalent to the exhaust rate CER. For practical implementation of the estimation procedure the reaction rate is then considered to be vR,a=CER. 4. M O D E L B A S E D F I L T E R
Disturbances contained in the reaction rate are directly transferred to the estimation. For this reason the estimated growth rate must be filtered in order to reject disturbances. The basic idea of the model based filter described in the following is that at any point of time the future evolution of the value to be filtered depends on some underlying, in the present case biological, principles. If these principles are, at least approximately, transformed into a model and if furthermore model uncertainty and not modeled effects are taken into consideration, then a dynamic range, according to the principles represented in the model, for possible evolution of the value can be calculated: ,L/fil,min (i) Crier,max(i)1 ]Llfil (i) = / ]/fil,min (i)/f r < ,L/fil,min (i) [ (10)
/u(i) else
L
J
r
(11)
at flfil,min(i)= lu~,(i-1)+(d-~-~ min(i)-ec) *At at
(12)
The bounds of the admissible range are calculated by equations 11 and 12. The positive parameter ec allows for convergence of the filtered value towards the real evolution. In these equations appears the minimum and maximum derivative of the value to be filtered. They are determined by the model mentioned above. By means of parameters e (+) and e (-) model uncertainty is taken into consideration.
+>0
d/tmax= -~
dt
l+lOOe)_)~ - - ~ -
dlt .I -~ 1- ~6-@ else
d/l.
(13)
e
.
I ( '+'/
d/~_ 100. Similar conclusions may be drown for the second basket where about 200 cells are needed since the difference between external and internal radii is greater. Details about kinetic expression and the model of the heat exchanger are reported elsewhere (Rovaglio et al., 2001). Performing the dynamic analysis of such a reactor model, an interesting result can be derived considering a temperature disturbance on the inlet flow. When the original conditions are reset, after a limit cycle is developed, the ammonia converter enters in a new limit cycle. This hysteresis pattern (Fig. 4) means that this reactor shows one stable limit cycle (the darker line) at the operating pressure of 190 bar in addition to the static attractors of high conversion (starting point) and low conversion (not reported in the figure). Therefore, it could be not possible to obtain the starting conditions by simply inverting the disturbance step on the manipulated variable as happened during the 1988 accident in Germany. A slightly higher pressure may change this dynamic pattern. As a matter of fact, at P=200 bar the converter goes back to the original steady-state con-ditions by simply switching back the inlet gas temperature to the starting values. Fig. 5 highlights the different dynamic behavior at the different pressure value. Similarly, Mancusi et al. (2000), developed a nonlinear analysis for the German reactor above described and, selecting the Fig. 4- Hysteresis pattern related to a disturbance of + 15 K pressure as a bifurcation parameter, they detected two static attractors at the design pressure. In analogy with the results here reported, they also found two static attractors and a limit cycle in a different range of the operating pressure (163 bar -166 bar). Such a different pressure range can be related to the different reactor geometry and size.
Fig. 5 - At the operating pressure of 200 bars the hysteresis disappears (both the diagrams refer to a disturbance of + 15 K)
727 3. THE SYNTHESIS LOOP Previous paragraph underlines the influence of the pre-heater on the reactor dynamics and it shows how the auto-thermal structure and the contemporary presence of an inverse response in the catalytic bed allow the presence of the limit cycle behavior to be justified. For such reason this section focuses the attention on the overall synthesis loop with the aim of addressing the following specific issues: 9 dynamic effect of more cycle integration; 9 stability impact of heat and material recycles; 9 operating conditions range affected by limit cycles and control loop scheme. The synthesis loop structure here considered was taken from literature (Nielsen, 1995) and it is sketched in fig 6. The synthesis loop examined is composed of a synthesis reactor, a cascade of heat exchangers and two separation units. As shown in the Fig. 6 there are four recycles in the adopted layout. The first one is determined by the third heat exchanger, where the effluent stream from the reactor releases thermal energy to the stream entering the reactor. This configuration is typical of the auto-thermal system where feed stream is preheated by hot outlet stream. The fifth heat exchanger is also used to preheat the recycling stream to the reactor by means of the heat exchanged with the hot stream leaving the reactor. The last recycle is determined by the flash that separates the liquid ammonia product from the unreacted gases. Indeed, hydrogen conversion per reacting-pass is approximately 30% and it is necessary to provide a recycle of the raw unconverted material to the process unit. In summary, the loop is characterized by the presence of two heat recycles used in order to configure the system as much auto-thermal as possible and by a material recycle due to the conversion needs.
Fig. 6- Layout of the ammonia synthesis loop The main model assumptions and philosophy related to the mathematical description of the loops are reported in the following while more details on the system equations are given
728 elsewhere (see Rovaglio et al., 2001). As previously mentioned, heat exchangers are used to cool the process stream exiting from the reactor and to separate, by condensation, the produced ammonia. Such exchangers are assumed instantaneously at steady-state conditions. In particular, the models proposed for the three heat exchangers encountered by the process stream, after the reactor, are written without considering the ammonia condensation since the stream temperature can be assumed to be always higher than the mixture dew point (in the range of the examined pressures and compositions). In the remaining exchangers, where the ammonia condensation takes place, the RKS equation was adopted to compute the equilibrium gas and liquid compositions (Reddy and Husain, 1982). Flash drum separators are adopted to separate the liquid ammonia from the recycle stream and the liquid ammonia from the purge gas. These units, again, are considered instantaneously at steady-state conditions because of the gas phase holdups and the corresponding inertial effects that can be assumed negligible. Data for the calculation of the equilibrium constant are taken from literature (Prausnitz et al., 1992). Purge gas and make-up are described through global mass and energy balances, neglecting the corresponding mixing enthalpy terms. The most important disturbances that have a significant impact on the plant behavior are listed in the following, namely: make-up composition, make-up temperature, variations related to the loop compressor and consequent disturbances on the loop pressure, variations on the liquid ammonia temperature in the last chiller adopted to regulate the temperature inside the flashing unit. The first two disturbances can be considered depending on the reforming section. Disturbances on the loop pressure are directly related to the working conditions of the compressor loop. Finally, disturbances on the ammonia temperature can be associated with bad operations of the refrigeration circuit. Moreover, there are also some minor disturbances that can act on the synthesis loop. In analogy with the accident reported by Morud and Skogestad (1998), the attention will be now focused on the effect of pressure disturbances on the loop behavior. In Fig. 7 it is shown the temperature transient of the outlet stream from the reactor when the considered system is the reactor alone (case a) or the overall loop (case b). While in the first case it is necessary to reduce the pressure from 195 bar to 185 bar to enter a limit cycle, in the second case a lower pressure drop down to 190 bar is enough to induce the limit cycle. This confirms the fact that integrated systems behave in a more sensitive way than the units considered separately.
Fig 7 - case (a): dynamic behavior of the reactor after two disturbance of -5 bar
Fig 7 - case (b): dynamic behavior of the w h o l e loop after a disturbance of -5 bar
Similarly, other disturbances previously mentioned can induce the same behavior. In facts, the actions modifying feed composition and/or feed temperature can induce the limit cycle
729 but, generally, even with a minor step on the forcing variable with respect to those described for the reactor alone. Therefore, since these systems are more sensitive and because of this, more potentially dangerous, it is very important to develop a robust control scheme in order to reject the disturbances mentioned above and avoid the raise of any possible oscillating condition. 4. C O N T R O L STRATEGY FOR THE SYNTHESIS LOOP From last analysis it was underlined the need of providing a robust control structure for the overall loop to avoid possible risks connected with the effects of the aforementioned disturbances. As a matter of fact, raising limit cycle behavior or undesired shutdown of the synthesis reactor can involve consistent damage of the reactor itself and/or of the other cycle units. Sustained oscillations can be dangerous due to the periodic and frequent variations of temperature that can seriously damage the catalytic material. On the other hand temperature waves that move through the heat exchangers cascade can alter the steady-state operating conditions. For example a great temperature change can induce a large variation in the boiler section and in the refrigeration circuit. Fig. 8 - A disturbance o f - 1 0 bar forces a total The main control structure for the valve closing and the raise of a limit cycle synthesis reactor and consequently for the behavior. overall synthesis loop can be based on the cold by-pass inside the reactor: a fraction of the inlet stream (cold) is separated from the main flow that passes through the heat exchanger while the by-pass flow is mixed just before entering the catalytic bed. The obtained result is a smooth thermal regulation: when the valve is progressively closed the inlet temperature becomes higher, on the other side, a valve opening involve a lower inlet temperature. The most critical condition to be maintained, for safe operations of the reactor, is the inlet temperature of the catalytic bed. Low temperatures may involve the reactor shutdown, while high temperatures may lead to the synterization of the catalyst. From the available data, the correct temperature and flow rate profiles were obtained for a valve position corresponding to a by-pass flow of 1.5 % of the total flow. Lower openings cause higher inlet temperatures while higher values lead to reactor's extinction. As a result, the control system is able to reject all the disturbances that force the valve opening, while it is evident that strong disturbances, characterized by a negative action, can easily take to the complete valve closing. This saturated condition leads to an open loop system and consequently to the possibility of reaching again a limit cycle behavior, as shown Fig. 9 - 1 ~t a n d 2 'd control l o o p
730 in Fig. 8. Therefore, it is obviously necessary to adopt another degree of freedom in order to get higher inlet temperatures. Let us now consider the part of the loop including the reactor and the first three exchangers (Fig. 9). It can be easily pointed out that manipulating the heat duty to the second heat exchanger, the inlet reactor temperature can be modified. Consequently, it is possible to help the previous controller. The by-pass valve of such exchanger is the second manipulated variable, and the corresponding value of the steady-state condition is around the 15%. The corresponding controlled variable is the inlet temperature to the second catalytic bed. In facts, varying the reactor inlet temperature does not effect the first bed inlet temperature (until the first controller works) but it modifies the inlet temperature of the second bed through the internal exchanger. A disturbance of-10 bar on the operating pressure (same as before with only one loop) may be rejected adopting the new control configuration of Fig. 9. 635
0.16
634
0.14
633
0.12 0.1 1st and 2nd 0.08 valve [%) 0.06
1st and 632 2nd bed input T 631 [K] 630
0.04
629 628
0.02 I
0
5
I
I
10 15
I
I
I
I
20 25 30 35 Time [min]
I
I
40
45
0 50
0
5
10
15 20 25 30 Time [min]
35
40
45
50
Fig.10 - Introducing a second control loop the critical disturbance is rejected
From the preliminary results it can be highlighted that the proposed control system is able to reject the main plant disturbances. Extensive simulations and analysis are still undergoing. 5. R E F E R E N C E S
Jennings J.R., "Catalytic ammonia synthesis", Plenum press, New York, (1991) Mancusi E., Merola G., Crescitelli S., Maffettone P.L., "Multistability and Hysteresis in an industrial ammonia reactor", AIChE J., 46, 824, (2000) Mizsey P., I. Kalmar, "Effect of recycle on control of chemical processes", Comp.Chem.Eng, 20, s883, (1996) Morud J., S. Skogestad, "Analysis of instability in an industrial ammonia reactor", AIChE J., 44, 888, (1998) Nielsen J.B., "Ammonia, Catalysis and manifacture ", Springer-Verlag, New York, (1995) Reddy K.V., A. Husain, "Modeling and simulation of an ammonia synthesis loop", Ind.Eng.Chem, 25,359-367 (1982) Reid R. C., J. M. Prausnitz, B. E. Poling, "The properties of gases and liquids", Mc-Graw Hill, New York, (1988) Rovaglio M., D. Manca, F. Cortese, P. Mussone "Complex dynamic behavior and control structure for the ammonia synthesis loop", submitted for publication, (2001)
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
731
Predictive functional control (PFC) applied to an unstable system. An industrial application Carlos Ruiz* a,c, Marta S. Basualdo b'c, Aldo Molina d, Laureano Jim6nez e, Benjamin Parisse f, Jacques Richalet f (a) SOTEICA S.R.L., Buenos Aires, Argentina (b) Dept. Electr6nica. FCEIyA-UNR, Rosario, Argentina (c) GIAIQ, UTN-FRR, Rosario, Argentina (d)REPSOL YPF, Refineria Plaza Huincul, Neuquen, Argentina (e) Chem. Eng. Dept. ETSEQ, Universidad Rovira i Virgili, Tarragona, Spain (0 ADERSA, Paris, France This paper describes an industrial application of Predictive Functional Control (PFC), which belongs to a family of Model Predictive Control (MBPC) developed by Richalet and coworkers during the last decades. The case presented in this work consists on controlling the bottom level and the reboiler temperature of a stripper distillation column, which belongs to the naphtha hydrotreater (Hydrobon) plant located at REPSOL-YPF Plaza Huincul refinery. The Hydrobon plant objective is to eliminate the downstream naphtha reforming (Platforming) plant catalyst. Level and temperature control by zone strategy is implemented; it involves keep the level and temperature inside a specific allowable band with respect to the set point and the corresponding controller tuning parameters are adjusted depending on that. The project consisted in three main steps, process identification, off-line controller design and on-line commissioning. The controllers were implemented in the control processor of the Distributed Control System (DCS). This paper presents the details of the identification, and off-line design, as well as the results obtained after the control strategies commissioning. 1. INTRODUCTION Several authors have published excellent reviews of MPC theoretical issues as Morari and Lee, 1999; a very good review from vendors and users of MPC point of view was presented by Qin and Badwell, 1995. In this work a combined experience between vendors, users and university researchers is presented showing industrial application results of a relative new technology of MPC philosophy such as PFC. The first industrial application of a Model Based Predictive Control (MBPC) was done in 1973 (Model Algorithmic Control). In 1975-79 the first generation appears with IDCOM (Identification and Commande, 1980-85 second generation with HIECON (Hierarchical Constraint Control), 1989 third generation with PFC (Predictive Functional Control). Since 1997, a subset of PFC called PCR (Predictive Control of Reactors) was developed for cover the gap between PID and big MBPC applications, especially focused to solve control problems where PID is not enough Author to whom all correspondence should be addressed: Alvarez Thomas 796, 3 C, 1427 Buenos Aires, Argentina. Tel.: +54 11 4555 5703, ext. 218; Fax: +54 11 4551 5701; e-mail:
[email protected] 732 and Multivariable MBPC does not apply because of expensive or high computing power requirements. However, although its name suggests applications for reactors only it can be used on other chemical units such as heat exchangers, furnaces, and distillation columns. The PCR technique resides on represent the plant with a linear impulse response model, generate the control algorithm for a coincidence point of the reference trajectory, solve it and apply the calculated input action. The later can be constrained on its maximum and minimum values and its rate of variation. The target is to slim MBPC to make it simple and easily implementable, at a very low DCS or PLC level, without losing its efficiency. The application field ranges from petrochemical (continuous) to pharmaceutical (batch) processes. In this work, an industrial application of PCR on two control loops of a distillation column implemented on a Foxboro IAS DCS in HLBL language is presented. The paper is intended to be useful to control practitioners and researchers by presenting specific theory and details of control algorithms. A detailed description of this technology can be found at Richalet, 1993, 1998. 2. PROCESS DESCRIPTION
In the Hydrobon plant the gasoline is treated in order to hydrogenate those components that are poisons for the catalyst of the downstream Platforming unit. The main contaminants are S, N and heavy metals. In Figure 1 a simplified flow diagram of the Hydrobon plant is shown. This plant consists of a reactor (R-301), where the elements mentioned above reacts with hydrogen producing SH2, NH3, H20 and the metals are kept on the reactor catalyst, a heat exchanger network for heat recovery and a distillation column (T-301) to strip out the impurities. The column bottom level needs to be controlled by manipulating the Hydrobon plant feed rate because the Platforming unit feed, the natural candidate to be used as the manipulated variable, should be maintained stable to improve the Platforming plant efficiency. Due to the long delay involved and the integrative (unstable) nature of this loop, it is not suitable to be controlled properly with a PID.
Fig. 1. Hydrobon plant flow diagram. 3. P R O J E C T PLANNING This project involves the steps of system identification, controller design and implementation in the basic library of the control hardware equipment. The IEC 1131-3 norm
733 is a convenient procedure for the embedding the new control modules in proprietary and nonproprietary environments. 3.1. Plant Test A plant test was carried on for both the column bottom level and the reboiler (H-302) temperature. For the level the difference between the set points of the feed flow rate entering and product flow rate leaving the Hydrobon plant, named "Diff"(FIC-304 - FIC-311), marked at Fig. 1 with arrows, were chosen. In this way the last one flow rate is taken into account as a direct feed-forward disturbance. At the left of Figure 2 the steps on "Diff" variable is presented, at the right the bottom level evolution of the column (LT-304) is shown. The transfer function was a first order with delay and integrator which parameters are: Kp = 0.02 %/sec./m3/h, Tp = 2.73 sec, Dp = 450 sec. Similarly, the gas valve output of the H-302 was manipulated to obtain the transfer function for the temperature. In this case, a first order plus time delay transfer function, with Kp =0.8 ~ / %, T p = 2 0 0 sec, Dp=50 sec parameters were obtained.
Fig. 2. Test plant for identification step. 3.2. Controller design The main PFC controller technology elements are: independent model approach which predicts the dynamic behavior of the plant on a certain prediction horizon; decomposition principle for unstable systems such as integrative process; exponential reference trajectory of first order closed loop response; polynomial structuration of the future manipulated variables that minimize the difference between reference trajectory and model prediction at one or most coincidence points; constraints on MV and state variables; easy tuning in time and frequency domains. 3.3. Mathematical calculations for PFC/PCR design According to the identifications done in the previous step in this section is presented as illustrative examples the procedures for designing both controllers for level and temperature loops. 3.3.1. Model Since for level control the identified transfer function was of first order plus time delay and integrator with Kp as gain, Tp time constant, D p dead time. The Z-transformed is a0.z
G p (z-')=
1-
-l
+ al.z
-2
(1 + ec ).z -l + tx.z -2 "z-q'
(1)
734 T~
with: a o = K p .ITs - Tp.(1 - or)l, ot = e T~ a~ = K p . [ -
or.T, +
T p.0 - ~)],
rp = int(
-~)'
Ts" sampling time
3.3.2. Internal model G m ( Z - l ) -- a ~
+ alm'Z-I 1 - ot ,z -i m
H
(2)
Z-' "1 -- Z -1 "z-rm
I
1
T, With:
a o m = K m .[Ts
- T m.(1 -
(~m)], am
"-
e Tm, a i m
= K m . [ - (3~m " L "+" Tm
.(1 -
a m )],
rm = int( D m )
The integrator term is decompose as is shown in Figure 3.
un, u(n)
"! I
Ii~ t
I
I I
I
H~
~
Yr(n)
I
M~
] [ . M~
I
I
Ym2(n)
Fig. 3. Decomposition of the integrator term T~ M l(Z-l)
z-l
= 1-
-l
M 2(z-l)
=
(1 - a m,).z-' been Otto1 = e v.,
CZm~.Z
(3)
1--Ctm~.Z-~
Tml is the decomposition time generally considered equal to the closed loop time response. The Z-transformed for the H 2 transfer function is H2 ( z _ l ) =
.Z -1 -2 a0m + aim .z 1 - ( a m + Ctml).Z q + a m .Otml.Z -2
(4)
Transforming to the observable state canonic equation the output of M 2 is" Ym2(n) = ~ m l "Ym2( n -
being
1) + (1--(~ml).Yp( n - 1)
(5)
yp (n- 1) the output of the plant at instant n-l, hence the internal model is written as
Ym (n) = Yml (n) + Y m2 (n) if time delay is considered (6) is transformed in (7)
ym2(n)=aml.Ym2(n-1)+(1-aml).[yp(n-1) +ym(n-1)- y,,(n-l-rm)] 3.3.3 Control algorithm For the coincidence point, C - ~p (n + h) = XH.(C- yp (n))
(6) (7)
(8)
735 3.H.Ts
Where C represents the constant set point, and ;~" = e Trbr . The predicted output is given by (9) ~p (n + h) = y p (n) + Ym (n + h) - Ym (n) The future output of the model is" Ym (n + h) = Ym~(n + h) + Ym2 (n + h) (1 O) From the linearity theorem it can be written for the model y,,~" Ym~(n + h) = Ym~(n + h).fo=ea+ Ym~(n + h).free
(11)
Yml (n + h) forzed : SF(H).u(n) ; Yml(nWh)free-SI_o(H)xom(n)+Sf_l(H)Xlm(n)
(12)
here, u(n) is the input of the plant at instant n, SL is the free response of the system which describes the time evolution from the actual time without any driving force. SF is the forced generated by external driving force and starting from zero initial conditions. The subindex "0", "1", etc. are related with the base function responses to classical inputs such as steps, ramps and so on. This allows, by considering the superposition principle, to write the forced response as a sum of base responses. Therefore, u(n) results: + (1-SI~(H))'Ym'(n)+(1-t2'~)'(Ym2(n)- yp(n)) (13) u(n)= (1-2n)'(C- yp(n))-SI-~176 SF(/4) SF(H) SF(H), SL 0(H), SLt (H) are pre-calculated in an iterative way for the coincidence point H, the state equations are calculated accounting that u(n)-I for SF(H);
u(n)-0 for SLo(H ),
SL~ (H). In addition for systems with delay u(n) is: H).(Ym2(n) - yp(n)) (1-)]H ).(C -- yp ( F / ) ) - 8/.13(/-/).X0m(F/)q- (1- SZ~(H)).ym~(n) + (1- aL; SF(H) SF(H) with yp (n) = yp (n) + Ym(n) -- Ym( n - rm) u(n) =
(14) (15)
Therefore (14) is the control algorithm, which will be used for the level. For temperature control the identified process was first order plus time delay, hence the control algorithm used is given in Eq. (16) u ( n ) = ( C - yp (n))(1 - An ). 4 y- "~ (n)
K(1-a •)
(16)
K
4. R E S U L T S PCF/PCR technologies include a Computer Aided Design (CAD) tool, which makes easier to identify and tune the controller. The parameters to be fixed or tuned for these controllers are: coincidence point (H) [sec], decomposition time: [sec] for unstable systems such as integrative, low and high closed loop time response: [TRBF, sec] applied when the controlled variable is outside the allowed control zone. By using a control zone as in the application considered here the parameter TRBF values are moving linearly between the two extremes. Hence, when the controlled variable is exactly the set point high TRBF is used. Otherwise, when it is far away from the set point TRBF decreases linearly in order to drive the controlled variable inside the zone as quickly as possible. Transition zone [%] set the allowed zone for the controlled variable expressed as + n% with respect to set point value, constraints to manipulated variable are also included. The application results are presented in Figure 4. At the right, the column bottoms level after and
736 before PCR control are compared. A dramatic control improvement is shown. At the left, the reboiler temperature controlled with PCR by manipulating the fuel gas valve output is compared with the previous PID control scheme. z a s oo 95
..Ipw .Ir,~
20 ~LoYel
9O
bofore PCR
~L~vol
wll~ PCR
19
B5
io
BO
17
'~
1 7s 70
15-
65
t4
6O
13
55
i
i
i PCR
.....
Tump S P bofo~ PC~
Tsmp SP ,,-,,h PC~
244 {30
L::
-
.., i_ .
. . . . . . Temp wllh PCR
.
.
.
.
.
.
.
.
.
.
.
243 CO
. ,
r :::
-
.
.
.
.
.
.
.
.
.
.
.
.
.
i-
.
,._
......7
....
.
12
5•1
243
11 1507
1731
1955
2219
043
307
531
755
[18130
(1200
0400
061:}0
DBDO
1000
12OO
14013
Vi,.,.
Fig. 4. Left: bottom level variations after and before predictive control implementation. Right: Temperature controlled with and without predictive control 5. CONCLUSIONS The PFC/PCR technologies turn out to be a convenient option for control loops where PID is not effective enough: unstable systems, reverse response, long time delays, constraints on manipulated and state variables, etc. Because of its cost effectiveness, practical industrial applications can be done either on large-scale plants but, also, in medium size and small industries. A significant effort has been made at 2 levels: - to make the technology easy to understand and to tune with tuning parameters that have an elementary physical significance like the Closed Loop Time response. The target is to have ground floor PID users to install controllers by themselves. A generic CAD of modeling, identification and controller design helps the local instrumentist; -to make it easy to implement into PLC's or low level control boards of DCS's, in assembly language, for safety reasons and ease of connection of the control algorithm with the overall control system. The applications cases presented in this paper are a clear demonstration of the effectiveness of the approach. Very good results were obtained compared with the previous results from classical control. Producers, operators and control experts appreciated that test case and the approach is being deployed on other sites. Moreover, it is not necessary to install new special hardware nor additional process control computers to implement this kind of predictive controllers. REFERENCES
1. M. Morari and J. H. Lee, "Model Predictive Control: Past, Present and Future", Comp. & Chem. Eng., No 23, (1999), 667-682. 2. J. Qin and T. Badgwell, "An Overview of Industrial Model Predictive Control Technology", Chemical Process Control Conference V (1996), 232-256. 3. J. Richalet, Pratique de la Commande Pr6dictive, Editorial Herm6s, Paris-France (1993). 4. J. Richalet, Pratique de l'Identification, Editorial Herm6s, Paris-France (1998).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
737
Selecting Control Configurations for Performance Henning Schmidt a, Elling W. Jacobsen a as3-Process Control, Royal Institute of Technology, S-10044 Stockholm, Sweden An important task in the design of decentralised control systems for multivariable plants is the decision of the structure of interconnections between manipulated variables and measurements. In this paper we focus on the selection of control structures with the aim of obtaining the best control performance using independent controller design. We point out the importance of the consideration of non-perfect control, contrary to the usually assumed perfect control, for this task. Furthermore we show, that not only the effect of interactions on the magnitude, but also on the phase of the subsystems should be considered. The decentralized Relative Gain Array is proposed as a simple frequency dependent tool. 1. I N T R O D U C T I O N We consider decentralized control of a general square multivariable n • n system G(s), i.e., having n inputs and n outputs. Decentralized control implies that the overall system is decomposed into a number of interacting subsystems for which individual controllers are designed, i.e., the overall controller may be written on a blockdiagonal form. Such a decomposition of the control problem is usually preferred due to the inherent robustness of such structures (both with respect to model uncertainty and sensor/actuator failures) and the ease of (re)tuning compared to the case with full multivariable controllers. However the potential cost of using a limited controller structure is reduced closed loop performance due to to the presence of interactions among the subsystems. An important task in the design of a decentralized control system is the selection of the structure of the controller. During this task, several objectives are to be considered. There is the issue of independent controller design, which is especially important, if one should be able to tune or retune a loop without having to retune the whole plant. Then there are the issues of performance of the overall system and of certain subsystems. Last, but not least, there is the issue of stability, e.g. decentralized integral controllability (DIC). The objective considered during control structure design will usually be a combination of the above mentioned objectives. Thus, there will exist a trade off between the closed loop performance, DIC and the independent tuning issue. For example we might want to choose a control structure for which independent (re)tuning of the single loops is possible and we accept to pay this with a reduced, but still acceptable, performance of the closed loop. Most results based on existing tools for the selection of control structures, like the Relative Gain Array [ 1], the Niederlinski index and the ~t-interaction measure [2] do not address performance issues, but focus merely on stability. A tool addressing the closed loop performance was introduced by Hovd and Skogestad [3]. This tool, the performance relative gain array (PRGA),
738 can aid in determining the required open loop gain (under the assumption of perfect control) in each subsystem in order to achieve a specified closed loop performance. In this paper we focus mainly on the selection of control structures with the objective to achieve the best closed loop performance using independent controller tuning. Under the assumption of independent controller design it is reasonable to assume, that a control structure resulting in a less interactive system (in terms of gain- and phase- changes) should be preferred. Therefore we introduce in the following the decentralized Relative Gain Array (dRGA), which is a useful extension of the RGA, based on the assumption of independent finite bandwidth control.
2. INTERACTIONS UNDER FINITE BANDWIDTH C O N T R O L For the case of independent controller design it is reasonable to assume, that the control structure resulting in a less interactive system will lead to a better performance of the closed loop system. Therefore we introduce in this section the decentralized RGA as an extension to the RGA for the issue of independent decentralized control. We will give examples clarifying the improved properties of the dRGA over the RGA and show, that not only the magnitude of the dRGA but also its phase information, is important in the selection of a control structure.
2.1. The RGA The RGA [1] has found widespread use as a tool for selecting control configurations. The relative gains are the ratios of the "open-loop" and "closed-loop" transfer-functions for the scalar subsystems. The common rule is to pair on subsystems with relative gains close to 1, based on the assumption that the interactions then will have a small effect on the subsystems. Since the frequency corresponding to the desired bandwidth is the most important, the focus is usually on the relative gain around this frequency. An often cited advantage of the RGA is that it is controller independent, due to the assumption of perfect control. However, this assumption is at the same time an important shortcoming of the RGA. The effect of interactions around the desired loop bandwidths is probably most important, and at these bandwidths the gain changes due to finite bandwidth control can be highly different from the changes predicted by the RGA. This is not only due to the loops having non-perfect control at this frequency, but also due to the fact that the ( n - 1) x ( n - 1) subsystems, considered perfectly controlled when calculating the RGA, in general will have a bandwidth which differs significantly from the individual loop bandwidths. This may be one reason why the RGA, while usually working well for 2 • 2 systems, is less effective for systems with n > 2. In other words: the RGA does not take into account the issue of decentralized control, while it is commonly used to decide decentralized control structures.
2.2. Example 1 Hovd and Skogestad [4] introduced the following system conventional RGA pairing rule.
1-s
G(s) -
{
-,--~-.~2 ~
(1 t J ~ }
1
-4.19
-25.96
6.19
1
-25.96
1
1
1
)
G(s)
as a counter example to the
(1)
739
(1
The RGA of the system is frequency independent and given by
A(im) =
-5 5
1 -5
5 1
(2)
Hovd and Skogestad have demonstrated, that pairing on the + 1 RGA elements results in a bad closed loop performance with a bandwidth of approx. 0.00086 rad/s and that pairing on the +5 elements leads to a much better performance with a bandwidth of approx. 0.0045 rad/s. In both cases the max. singular value of the sensitivity was restricted to be less than 2 (IISlI~ < 2). The RGA above tells us, that if we close the loops 1/11 and 2/2 perfectly, the gain in loop 3/3 will not change. While if we close the loops 1/2 and 2/3 the gain in loop 3/1 will decrease by a factor of 5. Using finite bandwidth control the gain changes are however different. For the loops mentioned above controllers were designed in an independent way, such that in each single closed loop the bandwidth was 0.001 rad/s. This is a quite low desired bandwidth compared to the RHP zero at + 1, but a slightly higher desired bandwidth (0.003 rad/s) leads to instability for the + 1 pairing, while for the +5 pairing the desired bandwidths can be increased by a factor of 100 without getting an unstable system. The resulting gain changes (open loop gain divided by the resulting gain when the other two loops are closed) in the open third loop are plotted in fig. 1-1eft. The gain in loop 3/3 doesn't remain constant, as the RGA predicts, but increases over frequency by a factor of up to 30, while the gain in loop 3/1 doesn't change very much around the desired bandwidth and above. The reason for the different gain changes for the selected pairings lies in the different performances of the controlled 2 • 2 subsystems of G(s). As mentioned above, controlling the two single loop systems up to a desired bandwidth does not imply that the 2 • 2 subsystems are controlled up to the same bandwidth. What can be seen is actually, that for the +5 pairing the performance of the 2x2 subsystems is much closer to the desired one, than for the + 1 pairings. One tool evaluating the achievable performance in subsystems, is the partial relative gain (PRG) proposed in [5]. The main shortcoming of this tool, is that the RGA of all possible subsystems down to 2 x 2 subsystems of G(s) has to be computed under the assumption of perfect control of the remaining systems. While this is still possible for 3 • 3 systems, the number of combinations to evaluate will be excessively large for n • n systems with n > 3. 2.3. Definition of the Decentralized RGA The decentralized RGA introduced below is able to predict the gain changes due to different performances of subsystems, as demonstrated above correctly. It is based on independent controller design and takes fully decentralized real controllers into account. The dRGA represents the gain changes in the diagonal elements of the transfer matrix G(s), when the remaining n - 1 diagonal elements are independently controlled using a real decentralized controller, such that the desired sensitivity in each controlled loop is given by the corresponding element in the diagonal matrix SO:
SO- ( l + 1output/input
! a ( s ) ~ ) -1
(3)
740
Fig. 1. Left: Magnitude of relative gain changes in the elements 3/3 (x) and 3/1 (o) due to finite bandwidth control of the corresponding other elements in Ex. 1. Bandwidth in single loops: f.oi = O.O01rad/s. Right: Magnitude and phase of the dRGA elements for pairings on +1 (x) and on +5 (o) elements. Bandwidth in single loops: f.l)i O.O01rad/s
=
is a diagonal matrix containing chosen desired single loop bandwidths o)i. The term A(s) results from the separation of the system (~(s) - diag (G(s)) into a diagonal allpass transfer matrix A(s) and a diagonal minimum phase system Gm(s)
(4)
G(s) =A(s)Gm(s) It can be shown, that the decentralized RGA is given by
[
[md]ii = gii gii--g re
/(1 (~)-1~-~// )-1_]_aii/-1 gCi] -1
(5)
where only the matrix f~ with the desired bandwidths has to be specified to consider different performance requirements in different subsystems of a plant (e.g. for distillation columns the performance of the composition loops is much more important than the performance of the level loops). Gm is the minimum phase term from equation (4). gii denote the diagonal elements of G, G ii the remaining ( n - 1) x ( n - 1) plant obtained by removing row i and column i in G. gri is the i-th row of G without the i-th element, gCi the i-th column of G without the i-th element. (~ is a diagonal transfer matrix containing the diagonal elements of G. K ii denotes the diagonal controller for the system ~ii. S~ is the sensitivity, when (~ is controlled using K. 2.3.1. Phase-Information from the dRGA Usually the phase information of the RGA is not used in choosing a decentralized control structure. However, the phase lag plays an equally important role for stability and performance as the change in magnitude, and it is therefore essential to consider the effect of interactions also on the phase. Because of the assumption of perfect control in the RGA its phase information around the bandwidth will not be very useful. For the example above the phase change in each element predicted by the RGA is :t:0 degrees. In terms of the dRGA, the change in phase in loop i due to interactions is given by
arg [Ad( iOl)]ii
(6)
741
Fig. 2. Left: Magnitude and phase of dRGA elements for diagonal (x) and off-diagonal (o) pairing in Example 2. Right: Diagonal elements of sensitivity for the diagonal (x) and offdiagonal pairing (o) in example 2. The dashed line shows the desired sensivitity S4.
Note that a positive phase of the dRGA implies that the phase lag increases due to interactions. Interactions may even be such, that closing loops will make zeros cross from the LHP into the RHP or vice versa (see [6]).
2.4. Example 1, continued The magnitude of the dRGA applied on the plant in Example 1 is plotted in fig. 1-right. The magnitude plot in fig. 1-right is almost equal to the one in fig. 1-1eft. It captures well the gain changes due to the behaviour of the subsystems. For low frequencies the magnitude of the dRGA is close to the standard RGA. But around and above the desired bandwidth the closed loop gains for the +1 pairing increase by approx, a factor of 25, while for the +5 pairing they increase only by a factor of approx. 1.2. Thus the pairing on the +5 RGA-elements seems to be less interactive around the desired bandwidth. The phase plot gives a similar result. The + 1 pairing seems to be especially bad, because in the frequency region, where the magnitude is significantly increased also the phase loss is large. 2.5. Example 2 Consider the 2 x 2 system
G(s) -
1 5s + 1
(
0.5 1
(s+l)2 1
)
(7)
Assume we want to design a diagonal controller which, by independent design, achieves a bandwidth of 1 for the individual loops. Fig. 2-1eft shows the dRGA elements for the diagonal and off-diagonal pairings, respectively. From the magnitude plot alone one would conclude that the interactions around the crossover is less severe for the diagonal pairing (the RGA would conclude that the pairings are equivalent). However, if one considers the phase of the dRGA one finds that the interactions will give a significant increase in the phase lag for the diagonal pairing. For the off-diagonal pairing the effect of interactions on the phase is smaller and furthermore such that the phase lag decreases. Based on this one might expect that the off-diagonal pairing
742 should be preferred, at least if independent tuning is desired. Fig. 2-right shows the diagonal elements of the sensitivity for the two pairings, when the individual open loop sensivitities are chosen with a bandwidth of 1 rad/s. Also shown is the desired Is~l. As expected from the above analysis, the interactions have by far the most severe effect on the performance if we pair on the diagonal of G. Similar results are obtained if one instead considers the overall sensitivity in terms of ~(S).
2.6. Properties of the dRGA 1. coi = o o or s -- 0 corresponds to perfect control in the subsystems and the dRGA becomes equal to the standard RGA. 2. The dRGA is pairing-dependent, which is a drawback, but probably necessary in order to get the information needed to choose the best pairing. 3. It is based on the assumption of independent controller tuning. 4. A generalization to a block dRGA is possible. 5. As an interaction based pairing rule using the dRGA one should select the control structure, for which the magnitude of the dRGA is closest to one around and above the desired bandwidth and for which the phase loss is close to 0. 6. Even if for the determination of the dRGA a certain controller is assumed, it is mainly dependent on the chosen desired closed loop bandwidths in the single loops. 3. C O N C L U S I O N S In this paper we pointed out the importance of the consideration of decentralized finite bandwidth control in the determination of the interactions between the subsystems in a decentralized control system. These interactions are especially important for the achievable closed loop performance if independent design of controllers is desired. We also showed, that not only the influences of the interactions on the magnitude of the controlled elements is important, but that the phase changes are at least equally important for the decision about the control structure.
REFERENCES 1. Edgar H. Bristol. On a new measure of interactions for multivariable process control. In IEEE Trans. Autom. Control, 1966. 2. Pierre Grosdidier and Manfred Morari. Interaction measures for systems under decentralized control. Automatica, 1986. 3. M. Hovd. Studies on control structure selection and design of robust decentralized and SVD controllers. PhD thesis, NTNU, Norway, 1992. 4. Morten Hovd and Sigurd Skogestad. Simple frequency-dependent tools for control system analysis, structure selection and design. Automatica, 1992. 5. Kurt E. H~iggblom. Partial relative gain: A new tool for control strucutre selection. In AIChE Annual Meeting, Los Angeles, 1997. 6. E.W. Jacobsen and H. Cui. Performance limitations in decentralized control. In ADCHEM 2000, Pisa, 2000.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
743
Design of Controllable Batch Processes S. S. Shah a K. P. Madhavan a* aCAD Centre, Indian Institute of Technology, Bombay, Mumbai- 400 076, India Batch process design can be more challenging than continuous process design because of the unsteady nature of the batch operation. This precludes the use of conventional controllability measures in evaluating the controllability of a given batch process design. Further the operating strategy of a batch process is characterized by trajectories of manipulated variables. Integrated approach to batch process design and control needs to address the problem of controllability of the batch process during the design phase. This is best achieved by treating the problem as a dynamic optimization problem with time invariant (design) and time variant (operating) variables. The method proposed in this paper uses the decomposition feature of Generalized Benders Decomposition (GBD) to evolve a 2-level nested optimization problem (Primal and Master), one involving time variant decision (manipulated) variables and the other involving time invariant decision (design) variables. To enhance the computational efficiency, a relaxed LP formulation of the Master problem is proposed. This variant of GBD, termed as ExGBD, is guaranteed to converge to the optimum for convex problems. ExGBD with slight modification in the iteration strategy can be shown to converge to optimum for quasi-convex problems. A simple batch reactor design problem has been chosen to demonstrate ExGBD. 1. I N T R O D U C T I O N The need to have an integrated design approach which will address concurrently issues of operability and control has been advocated in recent years. The integrated approach can be evolved with greater ease for continuous process through well defined measures of controllability and flexibility. Direct extensions of this to batch processes is not that easy. Furthermore, nonlinear dynamic models characterize the behavior of batch processes over the whole domain of their operation. The presence of uncertainty is much greater in batch processes due to short time spans available for model development. Batch process control objective is to move the process towards desired targets without violation of path constraints. Explicit incorporation of a control structure in the design problem will call for determination of time variant controller parameters. A more direct approach would be to solve the integrated design and control problem as a dynamic optimization problem, the solution of which will cover all the feasible control configurations. Such a dynamic optimization problem has to contend with two types of decision variables: time invariant decision variables related to the design and time variant decision variables related to operation (manipulated variable trajectories). *Author to whom all correspondence should be addressed to. Email:
[email protected] 744 In this paper we have restricted our attention towards the development of an integrated approach to design and control of a key unit in a batch process: the batch reactor. The objective is to develop an efficient and versatile optimization methodology, which can solve in a structured manner a host of dynamic optimization problems, involving both design and operating variables. 2. PROBLEM FORMULATION AND SOLUTION METHODOLOGIES
A general mathematical formulation of a batch process design and control problem (BP1) can be represented in the following functional form: rain J = G ( x ( t f ) , d , e , t f ) +
d,u(t)
F(x(t),u(t),d,O)dt+C(d)
(l)
Subject to,
Yc(t) - f(x(t),u(t),d, O) : O, x(to) : xo ... system dynamics g(x(t),u(t),d,O) < 0 ... path constraints h(x(tf ),d, O, tf ) 0. STEP(2) Solve the current relaxed master problem: min Yo
dED,yo
subject to, Yo >_ L* ( d j , k j , h j ,
(oj) +
0 3> L . ( d j , ~ _ j , p j , o 3 j ) +
3d
(d-dj); j
(~j,aj,dj)
~d " I(xj,u_j,dj) ( d - d j ) ; j -
-
1... ,
P
1,...1
using any LP solver. Let (d, f0) be an optimal solution; f0 is a lower bound on the optimal value of Equation 1, that is, the current lower bound is L B D - Yo. If U B D - L B D < e, then terminate, d obtained during the solution of the master problem may give rise to either feasible or infeasible primal. STEP(3a)- Feasible Primal For the new d solve the dynamic optimization (primal) problem and obtain v (d'). Update the upper bound U B D - min ( U B D , v(d') }. If U B D - L B D < e, then terminate. Otherwise, use the Lagrange multipliers to update the master problem, i.e. increase p by 1 and put ~p(t) ~ ~ ( t ) . pp(t) ~(t), top - (o. Return to STEP(2). -
-
STEP(3b)- Infeasible Primal The primal does not have a feasible solution for d = d. Solve a feasibility problem to determine the multiplier vectors ~,~, 6oand the function L, (d, ~,~, (o). Set, l = l + 1, ~-t - ~-'P-I - ~ ' - ~ - 63. U B D value remains unchanged. Return to STEP(2). Assumption about convexity of v(d) in d made to simplify the inner relaxed problem reduces the applicability of ExGBD. For problems non-convex in d, the solution using this technique may get stuck at a point which is not a local optimum. However, for certain type of non-convex problems viz. quasi-convex problems this limitation can be overcome by modifying the iteration strategy of ExGBD. The essential features of this modification are: repetitive application of
747 ExGBD (similar to the algorithm suggested by Bagajewicz and Manousiouthakis [ 1] for GBD), and its further modification developed by us. This modification introduces new constraints on Problem BP1 that gradually eliminate the non-convex portions of the function v(d). Convergence to the global optimum for quasi-convex functions has been rigorously proved [6]. 4. APPLICATION OF EXGBD FOR A BATCH REACTOR PROBLEM
ExGBD has been used to solve different types of problems, varying from simple LQR problems embedded with time invariant decision variables to a realistic case study on batch reactor design in the presence of uncertain model parameters. For the sake of brevity only a simple batch reactor problem, taken from Ray [5] (Example 3.2.2), is presented in this paper to demonstrate ExGBD as an effective tool for solving integrated design and control problems for batch processes. The problem is to obtain optimal temperature profile T*(t),for a batch reactor carrying out the reaction A --+ B --+ C, with the objective function being to maximize the concentration of B ([B]). This problem has been modified to include a time invariant decision variable; the batch time, (tf). The modified objective is to have a trade off between the two objectives namely maximize [B] and minimize the batch time, tf and is accomplished by weighted objective function involving [B] and tf.The value of these weighting factors will be dictated by the economics of the plant. The objective function is similar to that used by Ramirez [4], combining performance and minimum time requirement. The objective function of the optimization problem is given below: min [M-[B](tf) +C(tf)]
T(t),tf
where, M is some positive number ( - 1) to ensure positiveness of the objective function and C(tf) - 0.01t}. Model details are available in Ray [5]. In this problem tf is the complicating variable.For a given tf, the primal problem is solved using control vector iteration to provide optimal temperature profile T (t). The primal and master problems are solved iteratively till convergence is reached.The iteration summary of ExGBD is given Table 1. Columns 3 and 4 indicate the current upper and the lower bounds of the optimal. The problem converges to the optimum in 8 iterations. Table 1 Iteration summary for the batch reactor problem No. 1
2 3 4 5 6 7 8
fl 1.000 10.000 4.646 2.421 1.559 1.944 2.168 2.056
yo 0.399 0.399 0.399 0.378 0.377 0.373 0.373 0.373
0.000 0.162 0.307 0.363 0.369 0.373 0.373
748 5. CONCLUSIONS In this paper we have demonstrated that the integrated design and control problem for batch processes can be cast as a constrained dynamic optimization problem and the 2-level approach is a viable approach to the solution of the problem. The complexity of the 2-level design problem can be reduced through a decomposed optimization strategy based on the modification of GBD. ExGBD can solve a variety of design problems involving design parameters, initial conditions for a batch and operating variables which can be varied during a batch run. Some of the merits of ExGBD have been identified as: (a) employs a natural decomposition of the decision variables: time varying and time invariant, (b) algorithm guaranteed to converge for convex and quasiconvex functions, (c) drastic reduction in the computation of the inner relaxed problem through the formulation of LP, (d) can handle problems with uncertain model parameters 0, (e) complete iteration history is stored in the master problem, the algorithm can be stopped and restarted without loss of important information. REFERENCES
1. M.J. Bagajewicz and V. Manousiouthakis. On the generalized benders decomposition. Comput. Chem. Eng., 15(10):691-700, Oct. 1991. 2. T.K. Bhatia and L. T. Biegler. Dynamic optimization for batch design and scheduling with process model uncertainty. Ind. Eng. Chem. Res., 36(9):3708-3717, Sept. 1997. 3. A. M. Geoffrion. Generalized benders decomposition. J. of Optim. Theory Appl., 10(4):237-260, 1972. 4. W. E Ramirez. Process Control And Identification. Academic Press, Inc., London, 1994. 5. H.W. Ray. Advanced Process Control. McGraw-Hill, New-York, 1981. 6. Sunil S. Shah. Integrated approach to process design and control. PhD dissertation, Indian Institute of Technology Bombay, Systems and Control Engineering, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
749
Optimization and Nonlinear Model Predictive Control of Batch Polymerization Systems Dulce C.M. Silva and Nuno M.C. Oliveira a aDepartamento de Engenharia Qufmica, Universidade de Coimbra P61o II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal. Tel: +351-239-798700. Fax: +351-239-798703. E-mail: {dulce,nuno}@eq.uc.pt.
This paper addresses the optimization of batch polymerization systems, using a feasible path approach, with roots on Model Predictive Control (MPC) theory. The approach allows the reuse of many concepts previously developed for nonlinear MPC of continuous plants. It also provides an efficient and well integrated methodology for the optimal supervision of discontinuous chemical processes. The application of this technique in the optimization of the batch suspension polymerization of vinyl chloride is also described.
1. INTRODUCTION Due to their non-stationary nature, batch processes present interesting challenges in their control and online optimization, and create unique opportunities for the development of advanced supervision strategies. In the case of batch polymerization processes, optimal operation involves computing and accurately maintaining the optimal temperature and initiator (or coreactants) addition policies that can lead to a product with desired properties (such as molecular weight average, polidispersivity, chain length distribution) and final conversion, while minimizing the total operation time. In many cases, batch operations are still carried according to recipes based on heuristics and past experience. However, the recent availability of detailed mechanistic models, experimentally validated, provide a significant incentive for a wider use of newly developed optimization algorithms. Previous research on the determination of optimal policies for batch processes concentrated on techniques for the solution of optimization problems subject to algebraic and differential constraints. To avoid the numerical difficulties associated with the solution of a nonlinear twopoint boundary value problem, various methods based on the discretization of the differential equations have been proposed, using a simultaneous solution and optimization approach [ 1]. On the other hand, nonlinear MPC algorithms using a feasible path approach have been tested with success on continuous processes, in the presence of general constraints [2,3]. While considering different objectives, these algorithms constitute in fact general NLP optimizers, and are able to deal efficiently with differential constraints. Their use for general optimization of discontinuous processes is therefore investigated in this paper.
750 2. PROBLEM FORMULATION AND SOLUTION STRATEGY The dynamic optimization problems to be solved can be formulated as min
W(x(t),u(t))
u(t)E Y{ik
s.t.
:~ = fp (x, u, d; O)
y - gp(x; O)
(1)
Ul ~ U ~_~ Uu XI ~ _ X ~ Xu
Yl __V f ( x ) . v ,
VvER a
for which x, x + v E D.
Moreover, when v is taken as an arbitrary vector, 5 f ( x ; v ) = Vf(x) 9v, where 8f(x;v) is the G~teaux variation of f . The purpose of expressing convexity as in the above Remark is merely to illustrate a direct analogy between convexity in an Euclidean space R d and convexity in a linear space of functions 2C, the definition of which is given below.
Definition 2.2. A functional J on a set 9 C 2C is said to be [strictly] convex on 9 provided that when x and x + v E 9 then ~)J(x; v) is defined and J(x + v) - J(x) > ~)J(x; v) [with equality if and only if v=O, where (9 is the unique vector such that cO = 0x = (9, Vx E 2C, c E R]. Convexity is an important notion in conventional optimization schemes because convexity implies that any stationary point found is a global minimum (the unique global minimum for strict convexity). As expected, an analogous result exists for the minimization of convex functionals, as stated in the following proposition.
Proposition 2.3. If J is [strictly] convex on ~D C T~ then each xo E 9 for which 5J(x0; v) = 0, ~/ xo + v E ~D minimizes J on ~D [uniquely]. The proof of Proposition 2.3 is immediately evident from Definition 2.2 and is thus omitted. The interested reader is referred to Troutman [7, pp. 54-55] for the details of the proof. Note that the above Proposition illustrates the fundamental principle that minimization of convex functionals occurs for the function that makes the G~teaux variation equal to zero. The following defines convexity of functions on a subset S C R3. Specifically, functions of this form comprise the class of functions from which the integrands of functionals will be selected. Note that underlined variables are held fixed in the inequality hence only requiting the partial derivatives of f of the remaining variables.
Definition 2.4. f ( t , x , x ~) is said to be [strongly] convex on a set S C R3 if f = f ( t , x , x ~) and its partial derivatives fx and fx' are defined and continuous on this set and satisfy the inequality:
f ( t , x + v,x ~+ v') - f ( t , x , x ~) > fx(t,x,x~)v + fx' (t,x,x~)v ', V(t,x,x !) and (t,x + v,x ~+ v') E S [with equality at (t,x,x ~) only if v = 0 or v' = 0]. 3. C O N V E X U N D E R E S T I M A T O R S The existence of tight convex underestimators for real functions of special form such as univariate concave, bilinear, and trilinear functions has long been established in the field of optimization. Recently, Adjiman, Dallwig, Floudas, and Neumaier [ 1] have developed a method for deriving convex underestimators for twice continuously differentiable real functions. This section develops a rigorous analytical method for extending these known theories of convex underestimators for real functions to variational problems. At the heart of this analysis is the following fundamental theorem, which links convexity and underestimation of real functions to convexity and underestimation of integral functionals.
769 T h e o r e m 3.1. Let the functionals
F(x) --
/a
f(t,x(t),x'(t))dt
and
G(x) -
/a
g(t,x(t),x'(t))dt
be defined on the set 9 = {x E Cl[a,b]; (x(t),x'(t)) E D C R2}. /fg(t,x,x') is convex on [a,b] • D in the sense of Definition 2.4 and
g(t,x,x !) < f(t,x,x'),
V fixed t C [a,b] andx(t),x'(t) C D
then G(x) is convex on I) and G(x)
v),
which by Definition 2.2 shows that G(x) is convex. It remains to show that G(x) _ O,
~ . Then xo minimizes F(x) - fb f(t,x(t),x~(t))dt on ~) under the
t E (a,b)
t E (a,b)
3. ~,(t)~,(t,xo(t),Xlo(t)) - 0
771 Remark. Although the product ~,(t)~(t,x(t),x~(t)) may only be piecewise continuous, this poses no difficulty to integration provided there exist only finitely many points of discontinuity (cf. Rudin [4]). The following Theorem states the conditions of optimality for minimizing a bounded convex functional. Theorem 4.2. For a domain D o f R 2 suppose that f ( t , x ( t ) , x ~ ( t ) ) C Cl([a,b] • D) is convex, and we wish to minimize F(2) --
/aa
f(t,2(t),~'(t))dt
on
~) -- {2(t) C Cl[a,b]'2(a) - al,2(b) - bl}, subject to the inequality constraint g(t,2(t)) _ 0 and ~(t)g(t, xo) - O. Additionally, f o r all intervals excluding corner points the following equation must be satisfied: d -d~fx'(t,2o(t),21o(t)) - fx(t,2o(t),21o(t) -- Z(t)gx(t,2o(t)). At any corner point c, the following condition must hold: f x ' ( C - , 2 0 ( c - ) , 2 1 0 ( c - ) ) - fx,(C+,20(c+),2~o(C+)) -- kgx(t,2o(t)) f o r a constant k (for necessity, k 0, then u* (~)
-
UMIN
783 if V = 0 for z E [~i, ~i+1], then u* (~) = using(~) if ~ < 0, then u* (~) = UMAX UMIN and UMAX represent the lower and upper bound on the control input respectively. Since the Hamiltonian does not explicitly depend on the time coordinate t in the well mixed case or on the spatial coordinate z in the plug flow case, the Hamiltonian is constant when evaluated on an extremal trajectory.
2.2. Singular arc analysis On a singular interval [~i,~i+l], we have that ~ : 0. The singular control Using(~) is then obtained by repeatedly differentiating ~ with respect to ~ until u appears explicitly. Following algebraic manipulations: 9
~:0
r
d~ ~-~-0
0g[x(~)] + ~T 3f[x(~)] /1 b = 0 3x 3x J
dk T r162
r 3g[x(~)]b + ~Td -- 0 with d A Of[x(~)] b Ox Ox
d2v
02g[x( )] dx( )
9 -d-~2=0 r
~X2
dnr
d~ b + d ~ d +
n 0adx( ) ~x d~
r162[ 02g[x(~)] b ~2) x + ~ r ~ ] d d ~ ~ ) - [[3g[x(~)] +3x r
=0
~'T 0f[x(~)] d = 0 3]x
[02g[x(~)] b + ~ T ~ ] [f[x(~)] + bu] -[0g[x(~)] + ~T0f[x(~)]] d --0 ~X2 bX bX
yield the general expression for the control on the singular arc:
__
Using(~)
[0g[x(~)] ~T0f[x(~)] 02g[x(~)]b+ T3d]f[x(~)] L ~xx q" ~xx ] d - [ ~x 2 ~ ~xJ [32g[x(~)]b + b 3x 2 3xJ
(4)
In the following case studies this general relationship will be analyzed in more detail. 3. CASE STUDY #1. OPTIMIZATION OF FED-BATCH MIXED BIOREACTORS -
r)
In this first case study a fed-batch growth process is considered in which a micro-organism X [g DW] grows on one limiting substrate S [g] in a well mixed bioreactor. The input u is the volumetric flow rate F [L/h]. V [L] is the volume of the reactor, and Cs, in [g/L] is the substrate
'1~4
Cin = C(0) Tin = T(0)
C(L)
C T
/: p "~-q~
T(L)
Fig. 1. Non-isothermal tubular reactor with heat exchanger.
concentration in the feed. Following mass balance equations of the form (1) apply:
d
X
dt
V
-
+
0
u
(5)
1
x(t)
f[x(t)]
b
where cr -- P/Yx/s + m is the specific substrate consumption rate with Yx/s [g DW/g] the yield coefficient of biomass on substrate and m [g/g DW.h] the maintenance constant. The shape of the specific growth rate p [l/h] is arbitrary (i.e., monotonically increasing or non-monotonic). If the objective is to maximize the final amount of biomass (i.e., a terminal cost of the yield type)"
J[u] = -X(tf)
(6)
then Equation (4) reduces to
~CxV
t)f[x(t)] ] d - [~'T~ ] f[x(t)]
Using(t) =
[~T
~X
::=k Using(t) = [~r ~_~dx]b
Cs,in-fs
(7)
with Cx A_ X/V and Cs ~ S/V the biomass and substrate concentration respectively. It can be easily seen that singular control (7) keeps the substrate concentration Cs constant along the singular arc. A comprehensive treatment of this case can be found in, e.g., [3].
4. CASE STUDY #2. OPTIMIZATION OF CONTINUOUS PLUG FLOW REACTORS ( ~ - z) In the steady-state tubular reactor under study a chemical C is converted into a desired product in an exothermic way. The input u is the heat exchanger temperature Tw and the controlled variable is the reactor temperature T (see Figure 4). Since dispersion phenomena are neglected, the steady-state equations for the tubular exothermic reactor reduce to ordinary differential plug
785
flow equations with as independent variable the spatial coordinate z (which are of the form (1))" -
d lTlrq dz L J C
_
~-
~
T
+ ko
x(z)
-
PeP vk~ ~
-E Ce-~
4h
pCpdv
with
u
T(O)
C(O) --fin
0
v f[x(z)]
l)n
~ b
T and C are the temperature and the reactant concentration, v is the fluid superficial velocity, AH is the heat of reaction (AH < 0 for an exothermic reaction), p, Cp, ko, E, R, h, d and Tw are the density, the specific heat, the kinetic constant, the activation energy, the ideal gas constant, the heat transfer coefficient, the reactor diameter, and the heat exchanger temperature respectively. If the objective is to minimize a cost of the form (2), following expression for Using(Z) is obtained from Equation (4): /)2g[x(z)] dC ~)g[x(z)] ko ce Re_r E
Using(Z)- T +
AHdk~ c e - ~ -4- pCpdv [ 4h 4h
t)C
v
O2g[x(z)]
~)T -------T~ -
RT 2
t)Tt)C dz ]
/)g[x(z)]( E
~)-------~, R T 2
2
T/
As an illustration following cost criterion is proposed (for which a detailed motivation and analysis can be found in [2]):
J[u] --
(1 -A)[C(L)] +A f ~ [ T ( z ) - Tin]2dz
(8)
with A [-] the trade-off coefficient between terminal and integral cost and 7)n [K] the fluid inlet temperature. The terminal cost part is a measure for the process efficiency while the integral cost part accounts for the total heat loss. This results in following singular control: AHdk0... Using(Z) -- T +
4h
e
t~e Rr
(9)
It can be easily verified that this control keeps the reactor temperature constant along the singular arc. 5.
DISCUSSION
AND
CONCLUSIONS
In this paper generic properties of time and space dependent optimal control problems are discussed and illustrated with two case studies. The first case study analyzes the optimal control in time of a well mixed fed-batch reactor and the second case study focuses on the optimal control in space of a continuous tubular reactor. Both optimization problems give rise to a Hamiltonian function which is linear in the control input (i.e., the limiting substrate feed rate for the well mixed reactor and the heat exchanger temperature for the plug flow reactor) introducing the possibility of singular control. This singular control can be explicitly computed after differentiating twice the Hamiltonian. In other words, both are singular problems of order 2. In addition, the so-called costate variables can be completely eliminated from the singular control expression, i.e., a (nonlinear) law of the state variables only is obtained (see Equations (7) and (9)). Finally, depending on the specific choice of the performance index, the singular
786 control reduces to setpoint control, for which robust (adaptive) feedback algorithms can be easily designed. More specifically, if no integral cost is specified in the case of a completely mixed reactor, setpoint control of the substrate concentration in the reactor is obtained. For the plug flow reactor case, if there is no integral cost or the integral cost is no function of the chemical's concentration (i.e., bg[x(z)]/OC is equal to zero), a setpoint control of the reactor temperature results (see again Equations (7) and (9)). At the level of optimal control in time of well mixed reactors, numerous problems have been solved thus far. At the level of optimal control in (time and) space of tubular reactors, a rather limited number of solutions to control problems has been reported in literature. In this contribution, it is clearly illustrated that many problems can be casted within the same general control framework. As a result, reported solutions and their properties for mixed reactors can be transferred/adapted to solve tubular reactor control problems.
ACKNOWLEDGMENTS Author Ilse Smets is a research assistant with the Fund for Scientific Research Flanders (FWO). Work supported in part by Project OT/99/24 of the Research Council of the Katholieke Universiteit Leuven and the Belgian Program on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The scientific responsibility is assumed by its authors.
REFERENCES 1. D.E. Kirk. Optimal Control Theory: an Introduction. Prentice-Hall, Englewood Cliffs, New Jersey, 1970. 2. I.Y. Smets, D. Dochain, and J.E Van Impe. Optimal spatial temperature control of a steadystate exothermic plug flow reactor. Part I: bang-bang control & Part II: singular control. Technical Report BioTeC, 2000. 3. J. Van Impe. Optimal control of fed-batch fermentation processes. In J. Van Impe, E Vanrolleghem, and D. Iserentant, editors, Advanced Instrumentation, Data Interpretation, and Control of Biotechnological Processes, pages 319-346. Kluwer Academic Publishers, Dordrecht- Boston- London, 1998. 4. J. Van Impe and G. Bastin. Optimal adaptive control of biotechnological processes. In J. Van Impe, E Vanrolleghem, and D. Iserentant, editors, Advanced Instrumentation, Data Interpretation, and Control of Biotechnological Processes, pages 401--436. Kluwer Academic Publishers, Dordrecht - Boston - London, 1998.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
787
Nonlinear Analysis and Control of a Continuous Fermentation Process G. Szederk6nyi a, N. R. Kristensen b, K. M. Hangos a, S. B. JCrgensen b aSystems and Control Laboratory, Computer and Automation Research Institute HAS, H-1518 Budapest, Hungary, PO box 63 bDepartment of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark Nonlinear controllers of different type: feedback linearization with pole-placement controller and two versions of Hamiltonian controllers, together with an LQR controller are designed and compared for a simple fermentor near an optimal production operating point which is close to its fold bifurcation point. The performance analysis is based on simulation studies and on investigating the stability region of the controllers. The partial Hamiltonian controller is recommended for this case which has wide and predictable operating region, robustness and uses the measurable state variable only. 1. INTRODUCTION The analysis and control of nonlinear process systems is a challenging and emerging interdisciplinary field of major practical importance. Fermentation processes in particular exhibit strong nonlinear characteristics and are known to be difficult to control [3]. The investigated simple fermentation process is used as a benchmark problem for advanced nonlinear analysis and control techniques. In this paper different approaches are investigated for the stabilizing control of a continuous fermentation process. 2. MODEL ANALYSIS In order to be able to focus on the key issues in controller design and performance analysis of bio-reactors, the simplest possible bio-reactor, a perfectly stirred continuous fermentor is chosen as a test case. Despite of its simplicity, it exhibits the key properties which make bio-reactors difficult to control. 2.1. Nonlinear state-space model of the fermentor The dynamics of the process is given by the state space model dX
~-7 = u ( s ) x
aS
_
dt--
where
XF
v
_~(s)x + (S,~-S)F Y
V S
la(S) = lamax K2S2+S+r~
(1) (2) (3)
The variables and parameters of the model together with their units and parameter values are given in Table 1. The above model is in standard input-affine form with the centered state vector
788 Table 1 Variables and parameters of the fermentation process model X biomass concentration S substrate concentration F feed flow rate V volume SF substrate feed concentration Y yield coefficient Pmax, kinetic parameter K1 kinetic parameter K2 kinetic parameter
4 10 0.5 1 0.03 0.5
[{] [g-]t [~-] [/] [~] [-~] [g-]t [~]
x = [J? S-]r = IX - Xo S - So]r and manipulable input u = P = F - Fo
= f(x) + g(x)u
f(x) =
[ _t~(g+So)(X+Xo)+Xo) _ ] _,_ (SF-(S+So.))Fo Y
--
, g(x)=
[ (SF-(~+So))P v ]
V
with (Xo,So,Fo) being a steady-stateoperatingpoint.
(4) (5)
V
2.2. Stabilityand controllabilityanalysis
Stability Optimal production is achieved, when the outlet biomass mass flowrate (X. F) is at a maximum, which occurs at the point (Xo, So, F o) = (4.8907, 0.2187, 3.2088). In a neighborhood of this point the system is stable, but because the point is very close to the fold bifurcation point (X*,S*,F*) = (4.8775,0.2449,3.2128), this stability region is small. Stability analysis based on local linearization shows that the process is stable around the operating point but it does not give any information on the stability region. Controllability After calculating the Kalman-controllability matrix of the linearized model we find that the system is not fully controllable (in the linear sense) around the required operating point, because the controllability matrix has rank 1. Nonlinear controllability analysis based on the generation of controllability distributions is then used in identifying the singular points of the state space around which control of the system is problematic or even impossible. The local controllability distribution is generated incrementally in an algorithmic way [2] in two steps as follows. The initial distribution is Ao = span{g}
(6)
This is extended by Lie-bracket of f and g in the next step A1 =
Ao + span{[f, gl} = span{g, If, g]}
Ogf (x) - O ~f g (x) [f,g] (x) = ~-~
(7) (8)
The second step then gives the following A2 = A1 + [f, A1] + [g, Ax] = span{g,
If, g], [f, [f,g]], [g, If, g]]}
(9)
789
Singular points
At the point IX S-]T=[-Xo S F - So]r (X = 0 [~], S = SF) all the elements of A2 (and, of course, all the elements of Ao and A1) are equally zero. It means, that the controllability distribution has rank 0 at this point. Moreover, this singular point is a steady state point in the state-space. From this it follows, that if the system reaches this (undesired) point, it's impossible to drive the process out of it by manipulating the input feed flow rate. If 3~ = -Xo (X = 0 [~]) and S ~ SF -- SO (S ~ SF), A2 has rank 1. From a practical point of view it means that if the biomass concentration decreases to 0 [g] then it can't be increased by changing the input flow rate.
Non-singular points
At any other point in the state space including the desired operating point [3( S-]7"=[0 0] 7" the controllability distribution has rank 2, which means, that the system is controllable in a neighborhood of these points and we can apply state feedback controllers to stabilize the process. 3. CONTROLLER DESIGN AND PERFORMANCE ANALYSIS
Various controllers of different type: LQ controllers, feedback linearization based controllers and Hamiltonian controllers are compared here on the example of the simple fermentation process. The performance analysis of the controllers is performed by extensive simulation studies and by investigating the stability region of the controllers. For this purpose a simple Lyapunov function in the positive definite form of L(x) = lxrx is defined and the time derivative of this function is computed and analyzed as the function of the state variables. 3.1. LQ control The starting controller is the well-known LQ-controller. This case is used as a reference controller for comparison and tuning the nonlinear controllers. This type of controller is designed for the locally linearized model of the process and minimizes the cost function
J(x(t),u(t)) =
j~o~
(xT"(t)Qx(t) +ur(t)Ru(t))dt
(10)
where Q and R (the design parameters) are positive definite weighting matrices of appropriate dimensions. The optimal input that minimizes the above functional is in the form of a linear full state feedback controller u = -Kx. The results for two different weighting matrix selections are investigated.
Cheap control In this case the design parameters Q and R are selected to be Q = 12xe and R = 1. The results and the time derivative of the Lyapunov function for this case are shown in Fig. 1 (the desired operating point is at (x, s) = (0, 0) in the right subplot). As it is visible, the linear feedback stabilizes the system quite well. Expensive control The weighting matrices in this case were Q = 10.12x2 and R = 1. There were no significant differences in terms of controller performance compared to the previous case. 3.2. Local asymptotic stabilization via feedback linearization A nonlinear technique, feedback linearization is applied next for changing the system dynamics into linear one. Then, different linear controllers can be employed on the linearized system.
790 5
'
1 - ~*~'-~'~.'.':"..
I
0 a ~2S
. -
.
.
.
.
.
.
"
loi
" I= .lqlr~[h] 112
114
1,
118
2 " ~ ~ 4
Fig. 1. LQ control: simulation run and the time derivative of the Lyapunov function
We chose the substrate concentration S as output variable i.e. y = h(x)
(11)
-- x2 = S
To linearize the input-output behavior of the system, the following state feedback can be applied (see e.g. [2] )
u = o~(x) + ~(x)v ~(x) =
(12)
Lfh(x) 1 Lgh(x) ' ~(x) = Lgh(x)
(13)
where v denotes the new reference input and
Lfh(x) = ~x = i ) h(xf)
E0 1]f(x) = f2(x)
(14)
Lgh(x) = -~xxg(X i)h ) = [0 1]g(x) = gz(x)
(15)
It's easy to check that with this full state feedback the system becomes an integrator from an input-output point of view, i.e. 3) = v. From this point on, we can apply any well-known linear controller for stabilizing the system. To show the most important dynamic properties of the closed loop system we choose the simple pole placement controller v = -ky. The simulation results for feedback linearization control corresponding to k = I can be seen in Fig 2. As it is visible, the chosen output (the substrate concentration) shows a simple first order response with a time constant determined by k, and the biomass concentration also remains stable. Here we can arbitrarily prescribe the dynamics of the input-output behavior of the closed loop system but a major drawback of the method is the lack of robustness (because of the sensitivity of the functions ct and [3 [2] ).
791
5
35
,
,
0
i
-
-
-10
2S
1
-2 2
. ~
2
4
.
e
. e
.
.
"n'O[h]me
. ,2
.
.
,4
,e
2o
5"~
4
Fig. 2. Feedback linearization control: simulation results and the time derivative of the Lyapunov function
3.3. Nonlinear feedback based on the Hamiltonian model of the system The third approach is the design of a nonlinear full or partial state feedback that shapes the generalized energy function of the process.
Hamiltonian model of the fermentor Based on [1] it is easy to obtain the Hamiltonian equations of the fermentation process. For this, let us introduce the following state q and costate p variables p
_
_
q=
P2
] = [mx-mxo ] = [ox ]
Eqa]= q2 -
ms -- mso
16,
ms
[*]
(17)
where p = - V q denotes the component masses which are conserved. The Hamiltonian form of the model is written as [91 --
- V ( X o - ql)lJ(So - q2) Jr" (Xo - ql)F0 -k- (Xo - ql)F
(18)
P2
V(Xo-ql)lJ(So-q2) Y
(19)
--~
_ (SF-- ( S o - q 2 ) ) F o -
(SF-- ( S o - q 2 ) ) F
Then the coupling Hamiltonian Hi(q) used for feedback is calculated by simple integration. Since c-Jill
()H1
./)ql =. X0 . ql . , /)q2
(20)
(Se-(So-q2))
the natural output of the system Yl is written as Yl = Ha (q) = Xoqx - l q 2 - ( S F q 2 Z,
(Soq2- lq2)) Z,
(21)
792
4
, i
-50 -100
Fig. 3. Full state Hamiltonian control: simulation results and the time derivative of the Lyapunov function
Full state feedback using the whole natural output In this case the feedback consists of the entire calculated natural output, that is u = P = -klYl
(22)
where kl is an appropriately chosen positive constant gain. The simulation results corresponding to kl = 0.5 are visible in Fig. 3 indicating a good and balanced performance.
State feedback using only a part of the natural output It is known from the theory of Hamiltonian systems that the definiteness of the time derivative of the storage function (i.e. the stability of the closed loop system) may be influenced by using only some part of the natural output for feedback. Since the biomass concentration is quite difficult to measure in practice we chose the substrate concentration only for feedback 1 2 P = -k2"--(SFq2- (Soq2- ~q2))
(23)
This controller gives a similarly good performance as its full state counterpart. Moreover, the performance does not depend heavily on the controller gain either. A novel tuning method of the feedback gain for both cases is presented elsewhere [4].
Acknowledgement This research is partially supported by the Hungarian Research Fund through grant No. T032 479. REFERENCES 1. K.M. Hangos, J. Bokor, and G. Szederk6nyi. Hamiltonian view on process systems. AIChE Journal, accepted, 2001. 2. A. Isidofi. Nonlinear Control Systems. Springer, Berlin, 1995. 3. Ch. Kuhlmann, I.D.L. Bogle, and Z.S. Chalabi. Robust operation of fed bartch fermenters. Bioprocess Engineering, 19, 53-59. 1998. 4. G. Szederk6nyi and K.M. Hangos. Hamiltonian control of a nonlinear fermentation process. In: European Control Conference, submitted, 2001.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
793
Nonlinear process model-based self-optimizing control of complex crude distillation column A.Yu. Torgashov Process Control Laboratory, Vladivostok State University of Economics and Service, 690600, 41 Gogolya street, Vladivostok, Russia, E-mail:
[email protected] The control system based upon a nonlinear process model is presented in this paper. The self-optimizing control strategy is used for the maintenance of the optimal steady-state of a complex crude distillation column. The optimal distillation operation under unmeasurable feed composition disturbances has been considered. It has been also suggested that the stabilization of the certain relationship between pumparound streams in the sense of vector criterion provides the optimal operation of the crude distillation tower within the minimum and acceptable losses. 1. INTRODUCTION The complex crude distillation column (crude distillation tower) is a basic technological unit in the petrochemical industry for benzine, kerosene, diesel oil and mazut production. The maintaining of its optimal steady-state is very important task because at that the main economical benefit can be reached. The product quality control is the difficult problem due to the multivariable process nature and interactions between the control loops. At present, more emphasis is placed on this subject in the several papers [1-3]. Usually the principal obstacle attendant to the process control problem is the discrepancy between usable model and real process. The significant nonlinear distillation properties also hamper successful control performance using linear controllers (PI, PID etc.). In additional, the large dimension of the dynamic distillation model eliminates an application of such modem nonlinear control methods as feedback linearization, integrator backstepping and passivity based-control [4]. Thus, for the crude distillation tower the overall number of the nonlinear algebraic-differential equations is approximately equal to 3000. The using of such dynamic model is impossible under analytical nonlinear controller design. It is the author's opinion that the nonlinear process model-based (NPMB) control law [5] is more efficient method for complex distillation columns regulation. In that case the physical-chemical process nature (phase equilibrium, material and enthalpy balances equations) is taken into account in the control scheme by means of rigorous steady-state distillation model. In the previous investigation, the advantages of NPMB control strategy, as compared with another modem control modes, were found [3,5-6]. It is often necessary to provide the optimal process operation if the some degrees of freedom may be detected for optimization procedure. Among the recent techniques for solving such problem the self-optimizing control might be noted [7]. The method is simple from the practical point of view. The basic idea consists of the finding the feedback variable
794 characterizing the optimal process operation with minimum and acceptable criterion losses. In the previous work in the area of self-optimizing control the single criterion analysis was considered, whereas the complex process is often described by the vector criterion. It may be that the feedback variable does not reflect the optimal process operation in the sense of another criterion 9 This paper deals with the feedback variable reflecting the optimal steadystate of a crude distillation tower according to vector criterion. It was found that the losses of vector criterion are in several times lower compared with the losses of the individual criteria optimal values 9 2. P R O B L E M FORMULATION
2.1. Complexcrudedistillationcolumn Figure 1 shows the studied crude distillation column like the pipestill examined in the work [3]. The main purpose of this splitter is the crude separation on the products as benzine (B), kerosene (K), diesel oil (D). The crude is represented by 32 pseudo-components. The physical-chemical properties of these pseudo-components, including the coefficients for phase equilibrium, equation, enthalpies of the pure components in liquid and vapor phases with the nominal regime parameters of the column are presented in [8]. The products quality is characterized by the withdrawal temperatures of the key fractions. These temperatures are stabilized on the values located at the true boiling curve of the feed on 85 % before the end point temperature of the corresponding fraction.
2.2. The NPMBcontrollaw for crudedistillationcolumn In this work the NPMB control strategy is realized in the generic model control framework - ~ p(w)ss [5] 9 According to this control method the withdrawal temperatures of the key fractions T
are calculated from the following equation
= Tp + Kp, 1
-Tp
Kp, 2
•7.•_
i ~ ~V
R
Qvl~x-~ QP2~ ,
,
584, 582, ~" 580 ~. 578 576~ 0
-~K $2 "~
~
(I)
,
Slst ~e~
F steam
-Tp
0
N
am
M "r" ~ D
Fig. 1. Complex crude distillation column.
800 6 0 0 ~ W1(F,h'nol/h)
200
800
V V V
B~ (Kmol/h)
Fig.2(a).Surface of the first criterion.
795 Table 1 Product quality control system configuration for crude distillation tower Manipulated variable Controlled variable B T1w K D
T2 w Ta w
where 7' .p(w)ss
- set point of the withdrawal temperature; K p,1, Kp,2 "tuning coefficients; t-
current time; p=1,2,3 9 By the calculated values of -Tp(w)ss the manipulated variables are obtained with the help of inverse nonlinear steady-state distillation model. It has its origins in the Newton-Raphson method application and will be described below. The control system configuration is illustrated in the Table 1. 2.3. Criteria a n d d e g r e e s o f f r e e d o m s e l e c t i o n The crude distillation tower has two degrees of freedom in the form of flow rates WI and W2 (fig. 1) under the products purity specification as
265 1
380
800" ~ 0 0 4 0 Wl(Kmol/h) 200
0 W2 800 (Kmol/h)
Fig.2(b). Surface of the second criterion 9 2800 2600
I~ (Kmol/h)
200 ~
600 800 ~ (KmoVh) Fig.2(c) 9 Surface of the third criterion. X 109
~"2400
~ 2200 2000
1800 1600
~
~
800W16 400
(Kmol/h)
200 200
800
600~ (Kmol/h)
Fig.2(d). Surface of the fourth criterion.
800 6
~
W1 400 ~ (Kmol/h) 200
800
x,~
200
400 I~600 (Kmo]/b)
Fig.2(e) 9 Surface of the fifth criterion.
796
Tiw(sp) = (TiW~ef , i=1,2,3
(2)
where ref= reference value of the key fraction withdrawal temperature. The temperatures of the pumparound streams are constant. The flow rates W1 and W2 lie in the range from 100 Kmol/h to 800 Kmol/h. Thus, five optimization problems (five criteria) can be considered as 1. To maximize the benzine flow rate: Ji(W1,W2)=B --->max 2. To maximize the kerosene flow rate: J2(W1,W2)=K --->max 3. To maximize the diesel oil flow rate: J3(W1,W2)=D --->max 4. Minimization of the reflux flow rate: Ja(W1,W2)=R --->min 5. Minimization of the all condensers duty: Js(W1,W2)=Qc+Qpl+Qp2 --->min Fig.2 illustrates the surfaces of the selected criteria. It is obvious that the optimal criterion values will drift under the action of the unmeasurable feed composition disturbances. Therefore, it is necessary to find the structure of the control system which is insensitive to such uncertainty. 3. DEVELOPMENT OF THE SELF-OPTIMIZING CONTROL SCHEME
3.1. Control design under unmeasurable feed composition disturbances Table 2 shows the six basic cases of the unmeasurable (uncontrolled) disturbances in the form of the feed composition redistribution. The composition of each key fraction has a deviation of 20% from the nominal value. The optimal magnitudes of the criteria and manipulated variables are listed in Table 3 for these cases. The last column in Table 3 corresponds to the optimal vector criterion values in the relative units calculated in a similar manner as in the work [9]. Fig. 3 demonstrates the multiobjective optimization problem solution for one of the cases from Table 3. Finally, the intersection of the two contradictory criteria surfaces defines that solution. Notice that the optimal manipulated variables for 1, 2, 5 criteria are insensitive to uncontrolled feed composition disturbances action by virtue of the non-convexity of such criteria surfaces. Though, the optimal values of the control actions drift for third and fourth criteria (Table 3). It was found that the least changes of the optimal control actions are observed under the stabilization of the ratio W~/W2 on the optimal value in the sense of vector (fig.3). The maintaining of ratio W1/W2 as on the fig.3 provides the criterion at the ~~ least losses for the vector criterion as compared with the single optimization problem. This suggests the use of the self-optimizing control scheme integrated with the NPMB control strategy (fig.4). 3.2. Inverse steady-state distillation model for control algorithm The important element of the proposed control system (fig.4) is the inverse process model (IPM). In this section the fundamentals of the steady-state distillation calculation will be considered under specified withdrawal temperatures of the key fractions (2). The vector of manipulated variables u is determined by using the measurable outputs vector y in one procedure of the steady-state computing. Otherwise, it is necessary to solve the inverse problem y-->u without using optimization procedure as it was fulfilled in the previous work (see for example [10]). Here the Newton-Raphson method application is used as the basis in IPM and was originally discussed in [11 ]. The major modifications of that algorithm will be presented below.
797 Table 2 Feed composition redistribution Case Benzine fraction Kerosene fraction
Diesel oil fraction
1
+20%
-10%
-10%
2 3 4 5 6
-20% -10% +10% -10% +10%
+10% +20% -20% -10% +10%
+10% -10% +10% +20% -20%
Table 3 Optimization results under uncontrolled feed composition disturbances influence
J1, J2, J5 Case
W1 100 100 100 100 100 100
1 2 3 4 5 6
W2 100 100 100 100 100 100
Jl Opt 586 582.2 583.1 584 583 584
J2~ 264.3 266.2 267.1 262.1 264.1 262.4
J3 Js~ 9 5.5266 5.5267 5.5266 5.527 5.5268 5.5267
WI 105 100 100 105 100 105
W2 800 800 795 790 795 795
~opt
J4 J3~ 384 387.2 386.1 387.1 387.5 384.1
W1 295 290 305 300 295 300
W2 800 800 800 795 795 795
J4~ 1436.5 1433 1433 1434.9 1433.8 1435.3
0.572 0.571 0.568 0.57 0.571 0.574
For the studied distillation column (fig. 1) the input and output vectors are determined as u = [B K D W1 W2 ]; y=[Tl w T2TM 7"3w]. (3) uc uf The vector u in (3) consists from the two sub-vectors Uc and uf. In accordance with the work [10] the matrix component balance equations are formulated as well as the discrepancy functions by phase equilibrium (Fj) and enthalpy balances (@) forj-th separation stage. 1
~
W2)n~ g~
0.8
.dl norin
a0o W2(Kmol/h) Fig.3. Multiobjective optimization problem solution representation in the relative units.
1 (~/W2[)~
I
Fig.4. Self-optimizing system structure with NPMB control law. R(s) - transfer matrix of the robust controller.
798 But in the present paper the functions Gj are given by using the constant composition method as it was pointed by the author in the previous work [12] because the stable convergence of the computational algorithm was observed. In order to take into account the products purity specification (2), the additional discrepancy functions are proposed in the following form M~:Ti w- TiW(sP)---~O. i=1,2,3 (4) The introduction of the equations (4) is responsible for the appearance of an augmented matrices equations for every trial by the Newton-Raphson method JAX:-W, (5) where AX =[A(L/V)I...A(L/V)N,...ATI...ATN, Auc,1, Auc,2, AUc,3]T;W=[F1...FN.... G1...GN, M1, M2, M3]T; J - the augmented Jacobian matrix; Lj, Vs.- total liquid and vapor internal flow rates, respectively; N- total number of trays. With regard to the control system illustrated in the fig.4, the robust controller with transfer matrix R(s) is present before IPM. Its structure was determined using the known method of robust control design (~t - synthesis) because the block IPM+Process may be analyzed as the linear part with the nonlinear gains compensation by IPM. 4. CONCLUSION In the present paper the self-optimizing control system based upon the IPM has been proposed. Its main distinction consists in the indirect provision of the optimal crude distillation tower operation with the minimum vector criterion losses as compared to single criterion optimization. The proposed modifications of the standard Newton-Raphson method application to a distillation calculation give the possibility to reduce the computational efforts required for determination the vector of the manipulated variables under products purity specification. REFERENCES
1. L.H. Veland, J. Hoyland, C.R. Aronson, D.C. White, Hydrocarb. Proc., 73 (1987) 117. 2. K. Muske, J. Young, P. Grosdidier, S. Tani, Comput. Chem. Eng., 15 (1991) 629. 3. C.-B. Chung and J.B. Riggs, AIChE Journal, 41 (1995) 122. 4. M. Torres and R. Ortega, Proc. 14th IFAC Congress, Vol. E (1999) 403. 5. P.L. Lee (Ed.), Nonlinear Process Control: Applications of Generic Model Control, Springer-Verlag, Berlin, 1993. 6. H. Subawalla, V.P. Paruchuri, A. Gupta, H.G. Pandit, R.R. Rhinehart, Ind. Eng. Chem. Res., 35 (1996) 3547. 7. S. Skogestad, Journal of Process Control, 10 (2000) 487. 8. A.Yu. Torgashov, PhD Dissertation, Vladivostok State University of Economics and Service, (2000). (in Russian) 9. Yu.K. Mashunin, Journal of Computer and System Sciences International, 38 (1999) 421. 10. M. Fikar, A.M. Latifi, F. Foumier, Y. Creff, Canadian J. of Chem. Eng. 73 (1998) 1110. 11. C.D. Holland, Fundamentals of Multicomponent Distillation, McGraw-Hill, New York, 1981. 12. A.Yu. Torgashov and V.P. Krivosheev, Proc. 14th International Congress CHISA'2000, Set 2 (2000) 96.
European Symposium on Computer Aided Process Engineermg - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Closed loop controllability analysis Application to distillation column design
799
of
process
designs:
Rob L. Tousain t and E Michiel Meeuse * t Delft University of Technology, Mechanical Engineering Systems and Control Group, Mekelweg 2, 2628 CD Delft,
[email protected] Delft University of Technology, Department of Chemical Technology, Julianalaan 136, 2628 BL Delft The Netherlands,
[email protected] This paper presents a new approach towards the integration of process design and control. The approach compares alternative process designs based on the optimal closed-loop performance in the presence of stochastic disturbances. The most important contribution is that a clear relation is established between the degrees of freedom in the process design and a closed loop performance measure. The approach is illustrated with a case-study of a distillation column. 1. INTRODUCTION Traditionally, the design of a process control system is postponed until the process design is completed. Nowadays it is broadly accepted that this is not a desirable situation since this sequential design approach can lead to processes that are difficult to control. As a consequence, different ways to take controllability issues into account in the process design stage have been developed and are described in literature. These methods can be classified according to Lewin (1999) in methods which (i) enable to screen alternative designs for controllability, and methods which (ii) integrate the design of the process and the control system. In the first approach, the controllability of alternative designs is tested such that alternatives that might have acceptable steady-state economics but poor control performance can be rejected in an early stage of the design. The controllability is quantified using indices like the Relative Gain Array (RGA) and singular value decomposition based indices. A brief overview of available indices is presented by Lewin (1999). This method is rather easy to integrate in existing design procedures, however the indices are often calculated based on steady-state data only, and the relation between the indices and the closed loop performance is often unclear. The second class of methods is based on the simultaneous optimization of both the process and the controller, which are parameterized by means of a so-called superstructure. Alternative designs can now be compared based on, for example, the Integrated Squared Error (ISE) for specific disturbance scenarios (see e.g. Schweiger and Floudas, (1997)). Because of the computational complexity the application of this approach is limited to small case-studies with specific assumptions on the control system to be applied. Schweiger and Floudas (1997) consider for instance only SISO PI-controllers with an a priori specified pairing. An additional limitation is that the insight gained from such an approach is limited. Finally, stochastic distur-
800 bances are often not considered. To overcome the limitations of these existing approaches we propose a new method which compares alternative designs based on the optimal closed loop performance, taking into account disturbances and measurement noise. The method is based on the approach presented by Tousain et al. (2000) for the optimal sensor selection in a High Density Polyethylene reactor. This paper is organized as follows. The next section describes the modeling of the plant and the disturbances. Then Section 3 presents the proposed controllability index which is derived using Linear Quadratic Gaussian control. The use of this index will be demonstrated in the design of a distillation column as a case study in Section 4. Finally some conclusions will be drawn. 2. PLANT AND DISTURBANCE MODELS The process model In this study we will assume that a dynamic model of the new process is available in the design stage. For the sake of generality, we consider the broad class of processes of which the behavior can be described by a system of Differential Algebraic Equations (DAE):
.iCp= f (Xp, U,y, p), O=g(Xp,U,y,p),
(1)
where Xp E Rnxp, y E Nny, and u E R n" are respectively the state, algebraic and input variables, p E P = P1 x P2 x ... x Pnp are the np design parameters, where Pi, i = 1,... ,np are the parameter sets. These sets can contain real-valued or integer numbers. An example of a real-valued parameter is a column diameter, whereas an example of an integer parameter is the number of stages in a column. The operation of the plant is subject to operating constraints, 0 _ S(xp, u, p), which are assumed to be chosen such that the steady state constraints (0 = f (xp, u, y, p), 0 = g(xp, u, y, p) , 0 6CQ +6/-/20+ E, (2)
)2CO2 +2CzHsOH+E: (3)
where Er = oxidation energy of glucose (ATP+heat) = 2.816 MJ/mol glucose and Effermentation energy of glucose (ATP+heat) = 0.0820 MJ/mol glucose. The respiration rate, ro2, and fermentation rate, rf, are terms in the differential equations for O2 and CO/. The other terms are the dilution (flow divided by air volume), the loss of air from the container due to fermentation induced pressure build- up [5] with the underlying assumption Figure 2: estimated respiration level that the pressure build-up is negligible. A large number of papers is available on estimating ro2 and rf from Oz/COz-measurements in pilot scale facilities operated in either steady-state flowthrough or batch mode ([5],[6],[7]). These papers do not discuss (recursive) on-line estimation of ro2 and rf from O2/CO2-measurements under normal operating conditions. Therefore in this case study a Kalman filter based recursive estimator is developed. Figure 2 shows some of the promising initial simulation results with additive white noise.
815 3.2. Design of the controllers The research project focuses on the design of the short-term controller with given desired trajectories for the product respiration. These desired trajectories were deduced from extensive product experiments. The objective of the short-term controller is to reach and maintain the process at the desired trajectories with minimum cost. Undesired disturbances must be rejected. The controller optimises a trade-off between achieving the setpoint, inhomogeneity (distribution in both properties and spatial co-ordinates) inside the container and cost. This leads to an objective function that is written as
J = ~+I~((X -- X~e:)r W~ (x - x,e ) + AurW, Au)dt,
(4)
where W are the weighing factors that relate differences between actual and desired behaviour, including inhomogeneous conditions, to changes in the manipulable variables. The time horizon of this controller is denoted with H. Equation (4) is a quadratic objective where constraints can be added. This allows the formulation of a control problem in standard notation, e.g. a MPC type controller. A simulation study was performed using a linear control algorithm in Matlab for the short-term controller. In Figure 3 an indicative energy usage comparison is presented for four different cases resulting from a simulation study. The y-axis represents the energy usage in terms of hours of ventilation with Case 1 (a simulation with the current controllers) as reference. Case 2 represents a situation with the current controllers where airflow is controlled. Cases 3 and 4 represent situations with the supervisory controller where airflow is respectively not included as manipulable variable and is included. From these results can be concluded that for a reduction in energy usage manipulating the airflow can be interesting (as compared to current practice with continuous airflow). However, varying airflow leads to larger temperature variation inside the container and energy reduction possibilities are limited by the acceptable temperature variation. A special situation arises when, besides climate conditions, also airflow is controlled. In the simulation study this nonlinear control nonaffine problem is solved by linearisation. As such a control nonaffine problem is a typical control problem for the climate controlled processes discussed in this paper possibilities to improve the controller are investigated. The approach used for this control problem is based on the algorithm described in [3]. The steps followed to enable the use of this Figure 3: energy reduction algorithm are linearising the minor nonlinearities, the formulation of the nonlinear control nonaffine problem, calculation of the relevant matrices and to perform (iterative) control. The main idea is to use a qth order Taylor series approximation for the step response matrix, S, in the prediction equation l) = ~-~ S="
IMP(M)dM
(9)
The molecular weight polydispersity index (MWPI) was obtained from:
M W P I = <M n >
(10)
The reactor model includes the estimation of surface tension of the continuous phase using the Szyszkowski equation (Paine et al., 1995).. For our numerical simulations, the population balance equations were discretised using the stable backward finite difference method. The model was then written in gPROMS, a general process modelling software from the Centre for Process Systems Engineering, Imperial College, London.
826 3. DYNAMIC OPTIMISATION To illustrate an application of the model, an optimisation of the surfactant feed trajectory was conducted in gPROMS. The objective of the optimisation was to obtain a bimodal distribution in the PSD of the final product in addition to maintaining colloidal stability during the entire period of the reaction: The objective function to be maximised was defined as,
MFaX[
l,.~, ) = < r2 > ]
(11)
2J
The surfactant feed to the reactor (Fs) was chosen as the control variable and was specified as piecewise constant. Upper and lower bounds for the feed were set to ensure that the optimisation focused on the feasible range of operation. The time horizon was fixed to correspond with the completion of monomer feed to the reaction vessel (tf). To enable a stable emulsion, the surface tension (7) at the end of the polymerisation (an endpoint inequality constraint) was specified as indicated by the following equation: 34 dyn.cm -1 < 7 < 36 dyn.cm -1
(12)
4. RESULTS As shown in figures 1-2, the changes in temperature and monomer feedrates had major effects on the MWD. Higher temperatures resulted in shorter chains. This happens because at high temperature, chain termination becomes more important, which results in a low degree of polymerisation. On the other hand, at high monomer feedrates, the monomer concentration within a particle increases, which causes an increase in the polymerisation reaction rate. The propagation of the polymer chains becomes more important and a higher degree of polymerisation is obtained.
1:t ~
10],
8
~8
~ 7
~ 7 4
.-. a
.
.
. O
01 r
.
0.0~)0
5.0~1~
.
.
1.0E~7
. 1.5E~7
Molecular
. 2.0E~
2 1 0
. ZSE~)7
3.0E~ffr
weight
Fig. 1. Effects of monomer feedrate on MWD. temperature on MWD.
0.0E~
S.I~I~
1.0E~7
1.5E~7
,
,
Z0~lr/
Zf~t~'
Molecularweight
Fig. 2. Effects of reaction
827 As illustrated in Figures 3-4, the reaction temperature and the monomer feedrate affects the PSD. However the effect of the latter was more significant as the distribution was broader with a lower average particle size for a higher flowrate. Thus feedrate control is an important variable for obtaining desired PSD indicating the need to operate in a semi-batch mode for PSD control.
Fig. 3. PSD at different reaction temperature.
Fig. 4. PSD at different monomer feedrates.
Finally, the effects of the surfactant feed on the PSD were studied using the optimisation strategy defined previously. An optimum surfactant feed profile was developed for obtaining a maximum PSPI within the operating constraints. As a result, a bimodal distribution was obtained which is reasonable, because with the increase in surfactant feed, new micelles were formed and additional nucleation sites were created. Thus a new generation of smaller size particles was obtained as illustrated in Figure 5.
Fig. 5. Bimodal distribution of PSD under optimal surfactant feed. The surfactant feed optimal profile showed an initial increase in the surfactant feedrate after the particle nucleation stage. As particles grow in size, the interfacial area increases and hence more surfactant molecules are required to maintain colloidal stability. A subsequent
828 increase in surfactant feedrate caused secondary nucleation and a new generation of particles evolved. On completion of the nucleation stage, the feedrate rate was decreased to a new value, which ensures the stability of the emulsion. Thus surfactant feed can be manipulated in order to tailor the distribution modality.
5. CONCLUSIONS The investigation identified the important variables for the control of MWD and PSD in emulsion polymerisation. In particular, operation in semi-batch mode and manipulation of temperature, monomer and surfactant feedrates are important. An optimisation study illustrated the indicated that the PSD modality can be manipulated through surfactant feedrate control. Multi-variable optimisation can be conducted to optimise product quality and process operability by determining the required monomer and surfactant feed trajectories as well as the operating conditions (such as temperature) to obtain the desired PSD and/or MWD. REFERENCES
1. Broadhead T.O., Hamielec A.E., MacGregor J.F., Makromolekular chemie suppl, v 10/11, 1985, pp. 105. 2. Dougherty E. P., Journal of Applied Polymer Science, vol. 32,1986, pp.3051. 3. Penlidis A., Hamielec A.E., MacGregor J.F., Makromolekular chemie, Macromolecular Symposia, v 10/11, 1987, pp. 521. 4. Richards, J.R., and Congalidis, J.P., Journal of Applied Polymer Science, v37, 1989, pp.2727. 5. Li B., Brooks B.W., Journal of Applied Polymer Science, v48, 1993, pp. 1811. 6. Gugliotta L.M., Brandolini M.C., Vega J.R., Iturralde E.O., Azum J.L., Micra G.R., Polymer Reaction Engineering, v3, 1995, pp.201. 7. Liotta V., Sudol E.D., E1-Asser M.S., Georgakis C., Journal of Polymer Science, Part A: Polymer Chemistry, v36, 1998, pp.1553. 8. Arzamendi G., Asua J.M., Makromolekular chemic, Macromolecular Symposia,,v 35/36, 1989, pp. 249. 9. Sayer C., Arzamendi G., Asua J.M., Lima E.L., Pinto J.C., European Symposia On Computer Aided Process Engineering-10, 2000, pp.457. 10. Doyle F.J., Crowley T., Meadows E.S., Control of Particulate Processes VI, An International Conference, September 1999, Qld., Australia. 11. Coen E. M., Gilbert R.G., Morrison B. R., Leube H., Peach S., Polymer, vol. 39, 1998, No 26, 7099. 12. Gilbert R.G., Emulsion polymerization : A Mechanistic Approach, London : Academic Press, c 1995. 13. Paine, A.J., Wang, Z., and Rudin, A., Journal of Colloid and Interface Science, 1995, No 173,376. 14. Process Systems Enterprise Ltd. (PSE), gPROMS Advanced User Guide, Ver. 1.8, United Kingdom.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
829
Process Safety Management for Batch Process Operation Atsushi Aoyama, Rafael Batres and Yuji Naka Tokyo Institute of Technology, Frontier Collaborative Research Center Nagatsuta Midori-ku Yokohama 226-8503 Japan, Tel: +81-45-924-5137, Fax: +81-45-924-5144 Email:
[email protected],
[email protected],
[email protected] The batch process is becoming increasingly important due to a greater emphasis on low-volume, higher added value products, and the need for flexibility in a market driven environment. In the meantime, the management of process safety has become a mandatory as the public concern on environmental impacts and a safety of chemical industries heightened. In this research, a scenario oriented method of abnormal situation handling procedure design and a multi-agent based abnormal situation handling operation management scheme are proposed for batch processes. In the proposed scheme, the hazard evaluation and the abnormal situation handling procedure design are structured in three layers, inter-Unit, intra-Unit and intra-CGU levels. The proposed scheme makes it easy to design and express an abnormal situation handling procedures and simplify the task of safety operation management and seamlessly integrates those two tasks. 1. INTRODUCTION The chemical and biochemical industries face an intense pressure to improve the efficiency and the product quality. Under these circumstances, the batch process is becoming increasingly important due to a greater emphasis on low-volume, higher added value products, and the need for flexibility in a market driven environment. In the meantime, the management of process safety becomes a very important issue and its implementation has become a mandatory as the public concern on environmental impacts and safety of chemical industries heightened. The process safety management has been traditionally handled by bottom-up approaches such as the TPM (Total Preventive Maintenance) where existing process safety technologies are simply bunched together. Although these approaches are necessary, they cannot explicitly show the way to systematically integrate elements such as a process safety analysis, a process safety design and a process safety operation and handle the issue of management of changes. We have proposed a batch process operation management platform [2] in accordance with ANSI/ISA88 [ 1]. In this paper, the functions of batch process operation management platform are extended for the systematic process safety design and operation management to achieve a high level of process safety. The next section briefly revisits the batch process operation management platform. Section 3 explains a process safety design method, especially a design method of abnormal situation handling procedure. Section 4
830 explains a multi-agent based batch process management structure which executes abnormal situation handling procedure designed as described in Section 3. Section 5 briefly summarizes the advantages of the proposed method. 2. BATCH PROCESS MANAGEMNET P L A T F O R M ANSI/ISA-S88 [1] has been proposed to achieve a better batch process operation management through establishing the standard models, terminology, data structure and guidelines for language used by batch process control. The batch process operation management platform [2] is also compliant to ANSI/ISA-S88 One weak point of ANSI/ISA-S88 is that the separation of the plant information reconfiguration management, the process management and the unit management is incomplete. The operation management platform has been proposed in which those three kinds of management are clearly defined. The operation management platform is modelled as a multiple layered structure of process management, unit supervision and phase execution from the upper layer to the lower layer. "process management" executes multi-batches through the management of procedure execution and executes procedures through the management of unit procedures. "process management" also has a function to generate control recipe by reconfiguring the information contained in master recipe, schedule and plant structure. This function is corresponding to the plant information reconfiguration management. "unit supervision" is responsible for the execution of entire operation in the specific unit. "unit supervision" achieves the function of unit management. Finally, "phase execution" executes phases by sending commands to DCS. The batch process operation is executed based on a control recipe. The control recipe contains the information to produce the specific batch size of product after each operation is assigned to specific equipment. It also contains the information about when each operation should be started. The procedure defined in each recipe has layered structure, Procedure, Unit procedure, Operation and Phase. Procedure is corresponding to all the production steps to produces a batch. Unit procedure is corresponding to the production steps carried out at and around a main unit such as a reactor and a distillation column. Operation is the operation carried out at a main unit and its peripherals and Phase is a individual step to complete an operation. The activation of procedure changes the state of the structure (such as opening a valve) and the behavior comes out from such changes. The plant structure belongs to the physical dimension and refers to the description of the components of which the plant is built as well as their topological representation. The most basic component of the cell is equipment (e.g. pipes, valves). Instances of elemental-controlled-group-unit (ECGU) can be generated automatically from the equipment topology. An ECGU can be identified as an assembly of equipment that has control valves at its connection ports. On the other hand, the controlled-group-unit (CGU) and Unit could not be derived from the equipment topology only. A Unit describes an aggregation of ECGU that has a common unit such as a reactor or distillation column. A CGU is an aggregation of equipment that is actually handled as an isolated region from other parts of the cell during a operation. A procedure is an entire process necessary to manufacture a
831 batch corresponding to the Cell. A unit procedure is a process executed at a unit. An operation is a process executed at a CGU that is dynamically configured from ECGU. And phases are execution steps carried out in a CGU. 3. SAFETY DESIGN The entire safety design process is divided into three steps according to the concept of independent protection layer (IPL) [3]. At the first step, IPL1 (process design), IPL2 (basic control, process alarms, and operator supervision) and a mandatory part of IPL5 (relief devices) are designed using a rule-based approach where no specific abnormal situation is assumed. At the second step, IPL3 (critical alarms, operation supervision, and manual intervention), IPL4 (Automatic Action SIS or ESD) and IPL5 (relief devices) are designed using a scenario-based approach. Finally at the third step, the safety management activities to minimize effects to outsides of plant (IPL6, 7 and 8) are designed. This paper focuses on the design of IPL3, after the design of IPL 1, 2 and a mandatory part of IPL5 are completed.
3.1. Abnormal situation handling procedure design Abnormal situation handling procedures, requirements of the plant design modification, sensor locations are simultaneously determined at the design of IPL3 in order to satisfy regulations and social requirements and various requirements from the plant owner and the society. Although these tasks are currently done using heuristics and experiences of process designers and safety engineers, such a method is error prone and above all design rationales are not clearly recorded. It is very difficult to use design rationales in a real-time operation and an operation analysis. A cure for these problems are a scenario oriented approach where various abnormal situation scenarios are generated, risks of those scenarios are estimated and a design of IPL3 is done to contain those risks into acceptable levels. The design of IPL3 is divided into two steps, a categorization of abnormal situation by a preliminary hazard assessment and IPL3 design for each abnormal situation category (recovery, partial shutdown and total shutdown). At the first step, an initial event, which is categorized into the equipment failure, the abnormal material behavior or the mal-operation, leading to abnormal situations is identified then a preliminary hazard analysis is qualitatively and quantitatively carried out and a probability, a severity, a propagation path and an affected area are estimated. Further a propagation speed is estimated using a dynamic simulator. The risk level is computed from these estimations and whether that risk is acceptable or not is determined according to the predefined criterion. If the risk is not acceptable, abnormal situation category (recovery, partial shutdown and total shutdown) is determined and an abnormal situation handling operation procedure is designed. The abnormal situation handling procedure is evaluated using dynamic simulator. If the risk is still unacceptably high, the abnormal situation handling procedure is redesigned. This cycle is repeated until the risk is reduced to be acceptable. For each successful procedure, a plant design modification and a sensor location are designed to enable the corresponding operation procedure. However they are not immediately employed.
832 Instead, the proposals of plant design modification and sensor locations for all initial events are combined to make a final decision in order to avoid redundant design modification and sensors.
3.2. Unit and CGU based abnormal situation handling procedure expression A scenario-based method has a number of advantages such as a use of design rationale. However, it has a serious drawback, a massive number of scenarios and abnormal situation handling procedures have to be considered. In order to handle this problem, this paper proposes to divide a plant into several regions and design an abnormal situation handling procedure for each region, and construct an abnormal situation handling procedure for an entire plant as a combination of these procedures. The batch process operation platform introduces the concepts of Unit and CGU (Controlled Group Unit) and the correlation between Units, CGUs and management and operation layers. The proposed scheme uses the Unit and the CGU concept as a unit for the design and execution of abnormal situation handling. There are three ways for the Units and the CGUs to receive the effects of the initial events; from the initial event within the CGU and the Unit; through physically connected Units and CGUs, through the execution of abnormal situation handling procedures. The initial event and the abnormal situation can be categorized into the following three scenarios. 9 The effects of initial event can be contained within the CGU in which the event occurs (the initial CGU). 9 The effects of initial event exceed to the outside of the initial CGU but can be contained in the Unit in which the event occurs (the initial Unit). 9 The effects of initial event exceed to the outside of the initial Unit. In the first case, only the effects of initial events within the initial CGU are evaluated and an operation-level abnormal situation handling procedure is designed. In the second case, an operation-level abnormal situation handling procedure for the initial CGU is designed. In addition, an operation-level abnormal situation handling procedure for the other CGUs belonging to the same Unit are designed. A unit procedure level abnormal situation handling procedure also has to be designed to coordinate operation-level abnormal situation handling procedures. It is worth to point out that the deigned abnormal situation handling procedure for the other CGUs and the coordination Unit can be reused for other initial events if the effects of initial event to the outside of the initial CGU are identical. Abnormal situation handling operation procedures are designed both for the intra-CGU level (operation level) and the intra-Unit level (unit procedure level). An intra-Unit level procedure (unit procedure) coordinates all CGUs and decides the expected state of CGUs (recovery, holding, partial shut down and total shut down). The intra-CGU operation procedure brings the CGU to the expected state. For the third case, an inter-Unit level abnormal situation handling procedures are designed in addition to intra-Unit and intra-CGU level procedures. An inter-Unit level procedure (multi-task management) coordinates all Units and decides the expected state of Units (recovery, holding, partial shut down and total shut down). The three parts of procedures, inter-Unit, intra-Unit and intra-CGU level, can be independently combined to generate a new
833 abnormal situation handling procedure. It makes possible to construct a massive number of operation procedures from relatively few parts and achieve an easy maintenance of abnormal situation handling procedures. Units and CGUs are convenient regions for the purpose as these regions are isolated most of the time by valves. 4. OPERATION MANAGEMNET As described in Section 2, the dynamic reconfiguration of management structure as well as the separate handling of process management and unit management are required for the batch process operation. The multi-agent system represents a new way of analysing, designing and implementing complex software systems and are suitable to realize functions of batch process operation management because they can be generated and deleted individually. The following agents are defined for the batch process operation management. 9 Model server agent: 9 Cell (plant) agent 9 Multi-batch agent 9 Process agents 9 Unit procedure agents 9 Operation agents In addition, a model server agent, which manages databases of plant structure, manufacturing schedule and recipes and sends those data to other agents by request, is defined. How these agents work in the normal operation and in the abnormal situation handling is described in the following paragraphs. 4.1. Normal operation A cell-agent and a multi-task agent are generated when a cell enters into the operation for the first time. A cell manager gives the information about which equipments are defined as main units and which are peripherals. A cell agent obtains the information of plant physical structure from the model server agent and compares it to the inputs from the cell manager and determines the regions of Units. A cell agent also generates the control recipe and determines the CGU. A cell agent reconfigures the Units information every time the structure of cell is modified. A cell agent keeps the maintenance history of equipments and has an inventory of ongoing and previously executed processes. A multi-task agent obtains the information of manufacturing schedule and generates process agents based on it. A multi-task agent keeps track of the states of processes (non-active, active, normal, abnormal, holding etc.) and sends messages to process agents to execute, terminate and hold processes. After a process is executed, a multi-task agent deletes the corresponding process agent and sends the information held by it to the database. A process agent in turn generates unit procedure and sends them messages to execute, terminate and hold processes and keeps track of their states. A process agent pass its own status information and information sent by unit procedure agents to the multi-task agent. A unit procedure agent generates necessary operation agents and keeps track
834 of their states. A unit procedure agent also manages the regions of the Unit currently not used and the valves separating the CGU region from the non-CGU regions. An operation agent sends phase commands to DCS to execute the operation and observes the differences between the phase commands and the feedbacks from DCS. 4.2. Abnormal situation handling The operation agent searches the abnormal situation scenario database for a matching initial event when the differences between phase commands and the feedbacks from DCS exceed predefined criteria. If an initial event is not found in the abnormal situation scenarios, the operation agent alarms a cell manager or an operator for further actions. If an initial event is identified, the operation agent executes an operation (intra-CGU) level abnormal situation handling procedure and sends the identification result to the unit procedure agent. The unit procedure agent executes a unit procedure (intra-Unit) level abnormal situation handling when it receives a report of abnormality from a operation agent or identified an abnormal situation in the non-CGU region. The unit procedure agent sends a report of abnormality to the process agent, the multi-task agent, and commands the other operation agents under it to execute an operation (intra-CGU) level abnormal situation handling procedure. The multi-task agent executes an inter-Unit level abnormal situation handling procedure and triggers a rescheduling if necessary. The multi-task agent sends a report of abnormality to other process agents to execute appropriate intra-Unit and intra-CGU level abnormal situation handling procedures. An entire abnormal situation handling procedure is divided into simple procedures and executed by introducing a layered structure of the cell and the operation management. 5. CONCLUSION In this paper, a scenario oriented method of abnormal situation handling procedure design and a multi-agent based abnormal situation handling operation management scheme are proposed. The proposed scheme seamlessly integrates the safety design and the safety operation management where the abnormal situation handling procedure is constructed three levels, inter-Unit, intra-Unit and intra-CGU level. It simplifies the design and expression of abnormal situation scenarios and handling procedures increases the tractability of operation. REFERENCES 1. ANSI/ISA-S88.01-1995 Batch Control Part 1: Models and Terminology 2. Atsushi Aoyama, Isao Yamada, Rafael Batres and Yuji Naka, Development of Batch Process Operation Management Platform, European Symposium on Computer Aided Process Engineering 10 (2000) 1117 3. Center for Chemical Process Safety (CCPS), Guidelines for safety Automation of Chemical Processes, American Institute of Chemical Engineers, New York, 1993
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
835
Dynamic Cross-Functional Factory-to-Business Links in the Batch Industry Mariana Badell, Diego Ruiz, and Luis Puigjaner Chemical Engineering Department, Universitat Polit6cnica de Catalunya, Av. Diagonal 647, E-08028 Barcelona, Spain The lack of a cross-functional factory-to-business link between the shop floor and the necessary supply chain relation to e-business creates a gap. In order to bridge this gap a webbusiness-plant route is developed in a pilot plant using the TicTacToe sequencing algorithm. A web-based order management system is created to generate optimal plans taking into account the factory logistic status and detailed information of the real plant through a fault diagnosis system (FDS) with re-scheduling capabilities. 1. INTRODUCTION Organizations could no longer ensure profit improvement by cutting costs or filling plant capacities. A better coordination in the use of corporate resources, where cash must be prevalently considered, could be the way to improve production economics. The paradigm is not the same as during Taylor's era (1940-1970). Now corporations must be aware of saving and recovering the money invested during de competitive race. The new paradigm requires to use the information expensively collected to derive economic benefit by make superior business decisions, especially in manufacturing and supply functions. Integration requires the financial-production integration at the planning level in enterprise resource management (ERM) systems [1] and also precise and synchronized information on the state of the enterprise. Not considering beforehand optimal sequences in order management activities is a severe shortcoming punished by important profit discounts during the enterprise lifecycle. During the present decade the enterprise resource management systems achieved a forward development. More than 350 types of MRP-based systems are now available (Enterprise Resource Planning, ERP; Manufacturing Execution Systems, MES, etc.). From its start in the 1990s ERP business arrived approx, to a 50 billion dollars figure monopolised by leading vendors. However, limitations still remain, some of them linked to its dynamical evolution. ERP developers paid more attention outward from their customer's firms to e-business and supply chain management and less concern with connections to the shop floor layer leaving this work to MES developers and plant control systems providers. On the other hand MES developers center its focus downward to the information necessities of the plant floor including the vertical integration functionality, but without a prioritized focus on the link to ebusiness external relations and the supply chain management improvement required in these circumstances (see Figure 1).There is a special gap between the shop floor and the supply chain due to the absence of an efficient and cross-functional factory-to-business link. A disadvantage is that no standard software package can substitute a tailor-made system with appropriate cross-functional links to aid the supply chain's management. In this work a dynamic cross-functional factory-to-business link is developed in a web based order
836 management system. The created transaction-oriented approach supports realistic optimal delivery dates with price-time trade-off during the marketing management. 2. O R D E R M A N A G E M E N T SYSTEMS
Requirements of accurate on time optimal quotations during the electronic commerce are strictly placed in the new paradigm. The repertoire of optimisation techniques used to locate optimal delivery times during the firm's planning activities are frequently laborious, not always arriving to an optimum. Many practical problems were too large to the computer and hence not physically realistic. The TicTacToe algorithm is a proposed solution to those old problems and could be used for the sequence optimisation in on line web-based order entry systems. However, Internet commerce also has changed the demand on pricing from fixed to dynamic. The costing must be online estimated during the scheduling time considering, at least, updated prices of the material supplies. ERM systems are potentially capable of managing dynamically realistic delivery dates and prices during the marketing activities. The production costing data are available simultaneously with the execution of the scheduled plan making possible trade off solutions. The web-based order entry system is developed within the Common Gateway Interface (CGI). A CGI-bin or a servlet program implements a request response control structure using the manufacturing information system. Figure 2 shows the proposed structure [ 1].
Figure 1.Enterprise systems
Figure 2. A web-based autonomous order entry system.
3. SCHEDULING PROBLEM FORMULATION The timing assignment of the initial/final times to units/operations requires numerous calculations. In engineering a detailed timing is certainly necessary but during enumerative search of sequences in order management systems these calculations can be simplified.
837 Figure 3. Timing with N O T i j . Objective function formulation / result, recipes in Table 1. The goal of the scheduling problem is to find a sequence that minimizes the makespan or total production time. This can be seen as a nested decision. First, identifying the 'head' or first element of the sequence. Then, for each selection of the first position or head, finding the associated production sequence that minimizes the not-overlapping times which, in turn, guarantees the strict fulfilment of the SPT - shortest processing time - priority rule. The value of any multiproduct production sequence is given by the processing time of the head of the sequence plus the sum of the not-overlapping times of the elements of the sequence in the generated order. The minimization of the overall not-overlapping time can be formulated as a variation of the Asymmetric Travelling Salesman Problem (ATSP). The ATSP is a classical combinatorial optimisation problem that can be stated as follows: Given a directed graph G = (V, A, W) where V represents the set of nodes and A denotes the set of arcs and W 9A -> R is a cost function on the set of arcs, find a subset of arcs C c A that defines a simple directed cycle through all the nodes of the graph of minimal total cost. A variation of the problem is when a simple directed path through n-1 nodes of overall minimum cost is required. The TicTacToe algorithm is a solution method for ATSP problems [ 1]. The algorithm is an iterative heuristic procedure where each iteration consists of two phases: the constructive phase that generates feasible production plans and the reservation process, where an arc outside the current solution is selected and reserved to be part of all the solutions generated in the subsequent iterations. The Difference Matrix DM is a matrix widely used to solve ATSP with this approach: it is a tool to evaluate in a concise way the deviation from its "optimal" placement for the different tasks of the schedule. For each task, there are two minimum notoverlapping times that must be incurred: one relative to its predecessor and the other one relative to its successor in the sequence. Let Mir=Min{Mij, jr and MjC=Min{Mij, i~j} denote the minimum value in row i, respectively column j, of the MNOT matrix. Four different types of labels are used to classify qualitatively the elements of MNOX: "MM" is the label assigned to elements whose value is minimum both per row and column; i.e. Mij = Mi r and Mij = Mj c. "Mm"/"mM" are the labels assigned to elements with minimum value in their row/column but not in their column/row; and "mm" are not minimum neither for their row nor for their column. These qualitative labels constitute the r o a d m a p matrix that can be used as a p r i o r i t y d r i v e r in the construction phase (priority first to "MM", "Mm" or "mM", and "mm" last). Among the elements with the same type of label, higher priority is given to those with smaller matrix value MNOTij. The DM is a quantitative road map matrix since for each cell (i,j) its value d m v represents the increase in "time cost" of producing a product i before a product j instead of the product located in position k (minimum value in row i) before the product located in k' (minimum value in column j). The sequence is constructed in a greedy fashion. Arcs are included in the solution one at a time using an ordered list that reflects the priority relative to the considered criteria. In the first version of the algorithm the qualitative road map matrix was used as a pointer for the construction. Improved versions use the DM matrix. Let L denote the list of elements of MNOr ordered by increasing values of dmo. The construction phase can be formally represented by: PATH=k~,, Enter(i) =false i= 1..... n Leave(i)=false, i=O, .... n While L # ~ do Let (a, b) be the next element of L I f P A T H ~ (a,b) contains no subtours and Leave(a)=false and Enter(b)=false then PATH: =PA TH t~ (a, b)
838 Leave(a) =true Enter(b) =true Endif L:=Ll(a,b) End
The elemems of the ordered list are included in the sequence provided that: a) no sub-tours are created, b) each product is "preceded" by, at most, one differem product (at most, one cell per column is selected), and c) each product "precedes", at most, one different product (at most, one cell per row is selected). However, at the last step of the construction, the step, the n-1 th last element to emer the solution is not optimised since this element is pre-determined by the only position left free. For this reason it is referred as to the obliged element. The reservation phase is focused to substitute the obliged elemem by some arc (outside the currem solution) with a "higher" quality. The elemem that will substitute the obliged elemem is called reserved elemem, since in the subsequem iterations of the algorithm the elemem is preserved. Candidates to be selected as reserved elemems follow a selection process: Let (ao, bo) denote the obliged element of the currem solution. If (ao, bo) were "substituted" by an arc (ao, b), in the process to recover the feasibility of the solution the arc (Pb, b) will necessarily have to be removed (Pb denotes the predecessor of product b in the currem solution). Thus, the variation of the objective function, relative to the difference matrix, in order to recover feasibility if (ao, bo) were "substituted" by (ao, b), would be at least A (ao, b) = dmaobo+ dmpbb- 2dmaob. This variation refers to inserting the arc (ao, b) and removing the arcs (ao, bo) and (Pb, b). Similarly, if (ao, bo) were "substituted" by (a, bo), would be at least A (a, bo) = dm~obo +dm~,s~- 2dm~.bo. Now, the variation occurs for inserting the arc (a, bo) and removing the arcs (ao, bo) and (a, Sa). The list of elements that are candidates to be the reserved elemem is then defined as: CL(ao, bo) = {(ao, b) : A (ao, b) > 0 } {(a, bo) : A (a, bo) > 0}, and the reserved elemem is the elemem with a higher value of A (e.g. a positive figure means a possible contribution to the minimization of the function). That is, CL comains all the elemems with the same head or tail than (ao, bo) that, if interchanged with the obliged elemem, insure local improvemem, while the reserved elemem is the one with the largest local improvemem. The best result of the differem iterations is taken as the solution to the problem. RES=~ While not end do Apply Construction phase Apply the reservation phase and Identify the reserved element (an, be,) RES := {RES} ~ {(an, b~)] End The complexity of the construction phase is O(n 3) plus the time needed to construct DM and sort its elements (which is dominated by O(n3)). The complexity of the reservation phase is linear on the number of elements of the candidate list CL that is at most 2(n-1). Variable
depth search has been successfully used in the solution of different types of optimisation problems. This gives the procedure a higher level of flexibility. The ordered list L derived from the DM matrix provides an efficient way to solve the trouble of recovering feasibility when the set of reserved cells has increased. The inner iterations of the reservation loop are: Let Sk be the solution obtained at the constructive phase of iteration k, Order elements of Sk by decreasing values of DM and include them in an ordered list 0 t=O (t: counter of inner iterations) Let (ao, bo) be the first element of O. BuiM CL(ao, bo) stop =false (the inner iterations stop when removing more cells does not lead to an improved solution)
839 While CL(at, b~) ~ ~ a n d stop=false do Select from CL(at, b~) the element with higher value AA (aRy, bRy) Rt := Rt ~ {(aRy, bRy)} Apply constructive phase with set of reserved cells Rt Let Skt denote the new solution I f value Skt < value S/-1 t := t+l Let (at, b~) denote the next element of O else stop "= true endif Endwhile R "= Rt
Some preliminary testing has been done using the TicTacToe algorithm. Programs have been coded in C++ and run on a 400 Mhz PC, 128 RAM. A series of 36 problems has been randomly generated following a uniform distribution in [1,25]. Exact optimums have been obtained initially by a MILP formulation. The percent deviation with respect to the optimal value (value-opt) / opt for all the problems searching candidates is 1.004833 and only for 12 instances the optimal value is not found, deviation is around 1%. Also, the effort required to obtain these results is acceptable. However, the reservation phase must be refined making intelligent exploration using the inner results obtained to bound local search. 4. INTEGRATION OF THE PLANNING AND SUPERVISORY CONTROL LEVELS The proposed Fault Diagnosis System consists in a combination of a pattern recognition approach based on neural networks and an inference system based on fuzzy logic. A set of suspected faults has been identified for the multiproduct plant described in Section 5. They correspond to different faults in each of the pieces of equipment that implies a major processing time. Details of FDS implementation have been recently reported [2]. When a deviation from the predicted plan in a multiproduct batch plant is diagnosed, the FDS activates the planning module (using the TicTacToe algorithm) to minimise the effect of this deviation on the remaining plan. According to the client orders, an optimal plan is generated with the TicTacToe algorithm. The co-ordination level converts the plan in the set of control actions. Once the plan is running on the real or simulated plant, the control and supervisory system communicates to the planning system the deviations detected from the proposed plan. With this information, the planning system generates new information that is sent back to the control system. This feed-back closes the loop between the planning level and control system. 5. CASE STUDY In this case study a w e b - b u s i n e s s - p l a n t - b u s i n e s s - w e b route is developed using the TicTacToe non combinatorial sequencing algorithm to generate optimal solutions taking into account the factory logistic online status and updated financial, production and plant information. Figure 3 shows the flowsheet of the pilot plant utilized for the case study. The plant consists of three tank reactors, three heat exchangers and the necessary pumps and valves to allow changes of process configuration. Equipment of this plant is fully interconnected and the instrumentation allows configuration changes by software. The case study contemplates the use of two reactors and the respective pumps, in order to obtain ten similar products. Table 1 shows the recipes of the products and the asymmetric TSP schedule
840 formulation by the matrix of not-overlapping times MNOTof the schedules of all the possible pairs of products considered in the plant shown in Figure 4. Table 1. A batch process" Matrix MNOTobtained by using ten recipes. 0 26 26 MNOT= 26 26 26 26 26 26 26
26 0 26 26 26 26 26 26 26 26
27 27 0 27 27 27 27 27 27 27
27 27 27 0 27 27 27 27 27 27
28 28 28 28 0 28 28 28 28 28
29 29 28 28 28 0 28 28 28 28
31 31 30 30 29 29 0 29 29 29
32 32 31 31 30 30 29 0 29 29
33 33 32 32 31 31 30 30 0 30
35 35 34 34 33 33 32 32 31 0
Recipes of products I to 10. Products
i=l
i=2
i=3
i=4
i=5
i--6 i=7
i=8
i=9 i=10
Pump P0,j=I
10
10
10
10
10
10
10
10
10
10
Reactor R l, j=2
3
4
4
5
6
7
8
9
9
10
Pump PI,j=3
10
10
10
10
10
10
10
10
I0
10
Reactor R3, j=4
6
6
7
7
8
8
9
9
10
11
Pump P2, j=5
10
10
10
10
10
10
10
10
10
10
Process time, min
39
40
41
42
44
45
47
48
49
51
Figure 4. The flowsheet and optimal plan (1,4,3,2,5,6,7,9,10,8, makespan 294 min, 252 iter.) 6. CONCLUSIONS A simple strategy for a web-business-plant connection between the shop floor and a sequence optimiser in a multiproduct batch plant has been presented. When a new plan is developed to fulfil the customer orders received through the web, it is considered the actual status of the ongoing plan, which also includes the FDS amendment actions taken into account at previous time. This allows realistic and efficient plant performances to support the dynamic of the manufacturing market. Real time optimisation (RTO) problems and integrated systems are waiting for and must foster improved solutions to old, complex, and unsolved problems. New approaches with trade-off scopes must be developed to fill the gap between technology and science. The TicTacToe algorithm can be suitable for a wide range of RTO problems. Financial support from the European Community is gratefully acknowledged (projects IC18-CT98-0271, WATERMAN, and IMS 26691, Global Cape Open). REFERENCES 1. M. Badell, E. Fem/mdez and L. Puigjaner, 1st World Conf. on Production and Operations Management, Sevilla, Spain, 2000. 2. D. Ruiz, J.M. Nougu6s, J. Cant6n, A. Espufia and L. Puigjaner, Computer Aided Chemical Engineering, 8 (2000) 745.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
841
Ad-hoc Scheduling/Planning Strategies in Distributed Computing Systems: An Application to Pipeless Batch Plants Jordi Cant6n, Alexandre Afonso, Mois6s Graells, Antonio Espufia and Luis Puigjaner Chemical Engineering Department, Universitat Polit6cnica de Catalunya E.T.S.E.I.B., Avda. Diagonal 647, 08028-Barcelona, Spain*. The present work introduces a flexible scheduling framework based on a previously consolidated scheduling package. The core of the framework consists in a system co-ordinator that enables the proper actuation and communication of the different operating modules. The basic set of components should include a data module based on a hierarchical recipe representation, a timing module, a decision-making module and a series of tools for user and system interface. Over this framework, generic and/or specialised components may be incorporated, mainly in order to solve the basic decision-making associated to the scheduling problem. In the presented application, a dedicated component has been implemented and integrated in order to deal with the specific situations appearing in the scheduling of pipeless batch chemical plants, where the unloading, transport and loading of intermediates become the basic bottlenecks and their associated constraints are complex -even infeasible- to introduce in generic scheduling systems. 1. INTRODUCTION A general scheduling tool is an extremely ambitious objective very distant of the capabilities of present technology. Production facilities tend to be very particular because company policies, objectives and constraints are very specific and of different nature. Specific constraints in batch chemical processing made this case even harder. For these particular aspects, scheduling methods need to be adapted and user interfaces rebuilt each time a new application is needed. However, the modelling of batch processes may follow a common framework and share the same basic simulation procedures, user interfaces and data organisation and management. Consequently, most of the computer code developed for the general case should not be rewritten each time a new application is required, but reused. This is one of the main advantages claimed by Component Technology. Distributed components and distributed computing have the recognition of being a way to reduce software complexity and cost as well as to increase software flexibility and extensibility. * The financial support from the the European Community (project Global-Cape-Open-IMS 26691) and the Generalitat de Catalunya (through CeRTAP - Centre de Refer6ncia en Tecnologies de Producci6") are thankfully appreciated.
842 The present work introduces a flexible scheduling system based on this idea and built from previously consolidated scheduling package (Multipurpose plants Optimisation and Production Planning - MOPP) [1]. An effort has been required to migrate from the original monolithic application to a system of integrated components. However, the benefits have been readily harvest when addressing new scheduling problems. 2. STARTING POINT: THE MOPP PACKAGE In the study presented in this paper, the MOPP generic scheduling tool has been used as the basis to develop a flexible scheduling framework. The original tool included, among others, the following facilities: 9 An open modelling framework for batch / semicontinuous processes, using a continuous time representation system. 9 Optimisation algorithms to be used to calculate the optimum timing associated to a given sequence of interrelated tasks. 9 Support for tracing the use of common utilities, and alternative heuristics to detect and solve conflicts in the use of such utilities. 9 Intuitive GUI and tools for introducing manual changes based on the extensive use of the Gantt Chart representation. 3. THE RESULTING OPEN SCHEDULING F R A M E W O R K From a detailed analysis of the day-to-day operation of any scheduling system, the basic information flows have been identified, allowing the isolation of its basic functionalities. This has deal to a new architecture design, which can be identified as "generic open scheduling framework". The framework is based in a system co-ordinating the different modules: the MOPP Open Scheduling Optimisation Executive (MOSOE). This system centralises all the messages and orders and generates all decisions affecting the way of working of the different scheduler components. The basic set of components include a data module based on a hierarchical recipe representation following the ISA $88 specifications and a timing module based on the Event Operation Network (EON) model [1 ]. A graphical user interface (GUI) including an Electronic Gantt Chart allows the user to edit all the data and to use all the timing capabilities. Additional communication modules allow interaction with other pieces of software, either via ActiveX/COM interfaces, or using TCP/IP protocol. Consequently, the basic system has a great capability for being customised and for accepting new calculation features. Although the functionalities of most of these components were already incorporated in the original application, in most cases they have been completely rewritten, since the new open approach, operation procedures and framework, jointly with the new standardisation requirements, introduced radical changes in the way these functionalities should be fulfilled. 4. PIPELESS BATCH PLANTS: SPECIFIC P R O B L E M ANALYSIS Although they have received minor attention in the literature, pipeless batch plants constitute an interesting solution to minimise the amount of fixed piping, increasing production flexibility and reducing the complexity of cleaning requirements. The main idea is to execute material transportation tasks with movable vessels conveyed by AGVs (Automatic Guided Vehicles).
843 The production scheduling is a central part of pipeless batch plants operation and it is significantly different from conventional production scheduling which usually only considers the time, the lines requirements and costs. In a pipeless batch plant, the combinatorial nature of scheduling is increased by the additional problems related to AGVs assignment and routing. The plant configuration and the equipment installation become inseparable factors of the production scheduling problem. Although in principle, transport tasks can be considered in a similar way to the rest of the production tasks, they present some characteristics, which would require specific adaptation of the modelling system in order to be correctly considered during timing and scheduling: 9 Transport time: although the time required to carry out a transport task depends only on the task characteristics and the used processor, the "preparation time" (time required for arriving to the departing point) varies in a continuous way with the Processor State. Adequate formulation of this time dependency is required in order to reduce the information to be supplied to the scheduling system. 9 Special treatment should be considered to other vehicle characteristics. Among others, it deserves special consideration the existence of different vehicles, which might exhibit different characteristics in terms of speed, degree of action, manoeuvrability, materials admissible for transportation, etc. 9 Additionally, the use of Automatic Guided Vehicles imposes specific constraints that should be conveniently introduced. The most important of these are related with vehicles autonomy: Assuming that the vehicles are powered by batteries, a model of battery consumption in terms of the operation time of each vehicle should be considered in order to control battery load. Whenever low battery levels are detected, the proper charging task should be introduced in the vehicle schedule. Obviously, the models used can be easily improved to take into account, for example, the effect of the vehicle load on the battery consumption, or other effects that might be considered significant. 5. SPECIFIC SOLUTIONS FOR SPECIFIC PROBLEMS To deal with this problem, the production scheduling has been divided in two different stages, the production tasks scheduling and the transportation tasks scheduling. The production task scheduling determines the best allocation to equipment of each product and the timing of all production activities involved at each station. Once a proper solution has been devised for the production activities, the second step consists on checking the feasibility of this solution when imposing the AGVs constraints. A specific module has been developed to handle the additional constraints given by transportation tasks. Therefore, the generic system is used as the simulator module supporting the ad/hoc scheduling strategies and optimisation algorithms developed for AGV assignment and routing.
5.1. Transport task scheduling The specific module consists of four main steps, with can be integrated to form the basic strategies to find a transport-feasible schedule: 1- Identification of material/station transportation tasks; 2- AGV/stations assignment; 3- Routing; 4- Optimisation. Once a list of all the production operations are obtained from an initial (probably unfeasible) schedule, all the input and output operations can be identified as well as the
844 corresponding amounts of materials to be transported. Then, the number of transportation tasks (MOTT- MObile Transportation Tasks) needed to perform all the material transfers N 1L , ) being N needed can be easily found using the following expression: M O T T = ~ " ( M i + i=l 2 ' the total number of transfer operations, M i the number of material transfers to storage of operation i, and Li the number of material transfers to another operation of operation i. The AGV/stations assignment (step 2) fits each material transport task to an available AGV using different rules: 9 Simple charge constraints forcing to assign for one MOTT only AGV with a nominal capacity big enough to transport the amount of material needed. Qn > Qi
9 i = l ... n.~ o f M O T T s
In the limit, where all the nominal capacities are bigger than the charges of material to be transported, the problem is reduced to a Sequential Vehicle Rule -SVR [2]. 9 A greedy adaptative algorithm based on dispatch rules used with the SAGVs [2]; in this work three different dispatch rules were used: - Next Work Same Vehicle Unit (NWSVU) rule - Shortest Travel Time/Distance (STT/D) rule - Least Utilised Vehicle (LUV) rule The next step (step 3) is the routing of the AGVs. The time needed for the transportation tasks is determined using a distance matrix as well as the nominal speed for each AGV. A transport task is defined consisting in the travel operations (to reach the charge station and then to go to the discharge station), the charge and discharge operations and waiting times. Once all the assignment and time information for the transportation tasks is defined, the following step is to send this information back to the main schedule package and recalculate the whole timing involving the transport tasks. The information sent to the generic scheduling system includes the tasks to be performed with the times and assignments and the temporal constraints between the transport tasks and the production tasks. Each charge of a transport task is related to a discharge operation of a production task and vive-versa. Temporal constraints should be defined between these operations to assure the simultaneity of the beginning times (BT) and ending times (ET) of these operations" op
agv
These equations assure the simultaneity in the time of each pair of operations. And provides a new feasible schedule including the transport tasks. Finally, the resulting schedule can be optimised using a simple simulated annealing algorithm (SA) which explores other assignment combinations to find a best AGVs assignment that reduces the whole makespan or any other objective function defined.
5.2 Implementation A specific module has been developed to communicate to MOPP the additional constraints given by AGV assignment and routing 9 Therefore, MOPP is used as the simulator module supporting the ad/hoc scheduling strategies and optimisation algorithms developed for AGV assignment and routing 9 The module developed is written as a set of spreadsheet macros connected to the framework via a TCP/IP Client (ActiveX DLL). An initial production plan is
845 obtained from the standard decision-making tools available from the original optimisation system, and interactively processed by the client module. 6. CASE STUDY In this example two batches of two products (A and B) are produced following the flowsheets shown in Fig. 1 and Fig. 2.
y
II A
~.._ _ _
I
IIEA1
Rs
t
ORvl
l~XCl
I [-_.i-,:i-o.=, V
.
.
.
.
r___j
OllVl
7, o,~,
__..J Esquema de elaboraciOn del productoB
Fig. 1: Product A.
Fig. 3: Schedule without transport tasks.
Esquema de elaboraciOn del producto A,
l
Fig. 2: Product B.
The first step of the algorithm involves the creation of a schedule without the transportation tasks (Fig. 3). Once the assignment is performed a second feasible schedule including the transportation tasks is obtained (Fig. 4). Finally, after the application of the optimisation procedure, a best AGV tasks assignment is found improving the makespan from the initial value of 21,9 hours to 17,7 hours (Fig. 5).
7. CONCLUSIONS This work shows the potential of Component Technology in the area of Batch scheduling. The time for developing a new application has been significantly reduced by reusing general components providing basic calculation modules and focusing in the development of the code for including specific constraints and ad-hoc strategies. General components have been always kept as black boxes (complied DLLs), which seems to indicate a very interesting future to both proprietary and open component systems. The communication between all these tools is established by means of COM technology, thus constituting a hybrid distributed model of the decision-making system.
846
Fig. 4: First feasible complete schedule
The specific situation used as case study (Pipeless Batch Plant) demonstrates how a production scheduling system able to consider the most relevant characteristics associated to transport tasks has been easily developed. Once the main restrictions associated to these tasks (number, location and characteristics of available AGV, etc.) have been introduced, the system simultaneously optimises production and transport tasks schedule, taking into account the overall production policies and a common objective function. Decision making related with the transport system is then integrated and co-ordinated with the overall plant decision-making. The system is easily integrated with the rest of information, communication and control systems required to the management of the AGV system, so in case of plan deviations due to unexpected incidences, a new schedule, optimised according to the newest resulting scenario, can be on-line calculated and transferred to the plant-floor.
Fig. 5: Optimized schedule The use of this system can meaningfully improve the operation of both transport and production resources especially if both of them can be limiting. It can be also used to improve/optimise the design of the transport system. Additionally, the system can be used for planning and scheduling of actions in other cases where productive tasks are associated to moving processors. This is the case, for example, of automatic cleaning devices (including "Cleaning In Place" systems) and other maintenance services, as well as robot based collection of fruits and other products in the agricultural sector. REFERENCES
1. M. Graells, J. Cant6n, B. Peschaud and L. Puigjaner, Comp. Chem. Eng., 22S (1998) 395. 2. P.J. Egbelu and J.M.A. Tanchoco, Int. J. Prod. Res., 22 (1984), 359.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
847
Dynamic Modelling and Scheduling of an Industrial Batch Digester Cooking System P. Castro a, A. P. F. D. Barbosa-P6voa *b, H. Matos a B. Duarte c a Departamento de Engenharia Quimica, Inst. Sup. Trcnico, 1049-001 Lisboa, Portugal bCentro de Estudos de Gestao, Instituto Superior Trcnico, 1049-001 Lisboa, Portugal cCompanhia de Celulose do CAIMA - Const~ncia, Portugal
The optimal scheduling of a resource constrained four-batch digester system of an industrial acid sulphite pulp mill is addressed. This involves the development of two different models, one to model the scheduling operational problem and the other the batch digester operation - process model. The first model uses the Resource Task Network (RTN) representation leading to a Mixed Integer Linear Program (MILP), formulation. In this, the main operational limitation, steam availability, is modelled through the definition of a superstructure including all possible heating alternatives. The duration of these heating tasks was estimated through the use of the process model - a distributed heterogeneous dynamic model in gPROMS- validated with experimental plant data. The optimal schedule obtained, features a digester sequence different from the one in use in the plant and allows a higher level of productivity. 1. P R O B L E M A N A L Y S I S
The pulp cooking process considered consists of a set of four parallel batch digesters with different capacities. These process wood chips and chemicals and produce pulp in aqueous suspension. A batch on a particular digester consists of an ordered sequence of various operations: i) chip filling; ii) acid filling; iii) heating; iv) cooking; v) high pressure degassing; vi) low pressure degassing and vii) blowing the contents into a storage reservoir, blow-tank. At the plant the heating task is the process bottleneck. There is limited steam availability and as a consequence waiting periods during this task are frequent. To achieve high quality pulp, the temperature rise in the heating task is subject to some constraints, regarding the use of steam. As the amounts to use are dependent on the digester size, the final dil~ester sequence will affect the makespan. The digesters share other resources besides steam, such as the chip conveyer; acid-filling pumps; high pressure and low pressure degas tanks and the blow-tank. Thus, two different digesters cannot use one of these resources simultaneously. The scheduling model must be able to consider these common resource limitations, and determine the start and completion times of all tasks of the cooking process that minimise the makespan. 2. P R O B L E M
FORMULATION
The optimal scheduling of the cooking system in study involves the definition of two different types of models: a scheduling model and a dynamic simulation model. The latter *Corresponding author. Tel: +351-218419014. Fax: +351-218417638. E-mail:
[email protected] 848 estimates the necessary parameters to be introduced in the first model so as to obtain the optimal system scheduling.
2.1. Scheduling model The general Resource Task Network, RTN, representation (Pantelides ~) is used to model the scheduling problem. The mathematical formulation is based on a discrete representation of time (similar to the one of Kondili et al. 2) where the time horizon is divided into a number of intervals of equal and fixed duration. All system events are forced to coincide with one of the interval boundaries. The formulation gives rise to a MILP problem. 2.1.1. RTN representation The RTN representation regards all processes as bipartite graphs comprising two types of nodes, resources and tasks. Each task is an operation that transforms a certain set of resources into another set. On the other hand, the relevant resources are classified within two different types based on their functional equivalence. The common resources, shared amongst the digesters, set R; and the individual resources, representing the possible material states in the process, set S. The latter resources and the process tasks, set K, must be referred to a given digester, an element of set D. The RTN of the cooking process is represented in Figure 1. The first task (chip filling) consumes the first state (digester empty) at its beginning and produces the second state (digester filled with chips) at its end. This state is then consumed by the second task (acid filling) and so on, until the cycle is concluded by the last task (blowing), which produces the first state at its end. In Figure 1 the heating task is simplified. To describe the complex interaction between the digesters in this stage a superstructure was derived which includes all possible heating alternatives (Figure 2). Thus, task K3, from Figure l, was replaced by 4 tasks HO(i,j) and 4 tasks Hl(i,j) with ij~D. Tasks HO precede the tasks HI. Both use all the available steam, and so this resource can be considered as a discrete resource. The first stage of the heating task for the first digester in the sequence is represented by the task HO(i, i). Digester i, can then finish its heating stage, with one of the three H1 (i,j) tasks, with i,j. These tasks implicitly consider the heating of digester j with the remaining steam of digester i changing the states of both digesters i and j. As the digesters have different steam consumption, different states in digester j are produced from different digesters. Three new states are defined - $5 to $7 - representing different temperatures for the pulp suspension. After task H1 (i,j) digester j concludes the first stage of heating with task HO(j, i), producing state $4. This process continues until the last digester (/) of the sequence ends the heating requirements (in the last production cycle) with task Hl(l, l).
849
Figure 1. RTN representation of the cooking process (heating task, K3, simplified)
Figure 2. RTN representation of the heating stage. 2.1.2. Mathematical formulation
The short term scheduling of the cooking system is achieved by the following constraints: Excess resource balances: the excess amount of a resource at a given time interval t is equal
to that at the previous interval adjusted by the amount consumed by all tasks starting at the
850 beginning of t and by the amount produced by all tasks starting at a previous interval and ending at t. One set of constraints is required for each type of resource. l'k,d Rr, t = Rr, t-I + Z Z Z ] ' l k , d , r , o N k , d , keK deD O=O
'-0 V r E R , t E r
(1)
~'k,d
S ......, = S ......,-' + Z Z ~_~vk,a .....i,oNk,,,,_o Vs ~ S,i ~ D,t ~ T
(2)
keK deD 0=0
In the above equations the parameters Pk, d,r,o and Vk, d,s,i,o represent the amount of resource r (s in digester i) produced or consumed by task k of digester d at a time 0 relative to the start of task k, while the parameter rk, d represents the fixed duration of task k in digester d in time units. The excess variables Rr, t and Ss, i,t are continuous positive variables while the variables Nk, d,t are binary ones. The latter represent the extent of task k of digester d at interval t, an element of set T. Extent constraints: The number of times a given task is to be executed is dependent of the number of production cycles to be studied, Ncycles. Although the blowing task is the last task of the cooking process, we consider that each cycle ends with the heating stage of the last digester in the sequence. The following constraints apply: Nk,a, , = Ncycles Vd e D , k e K , k < 2 (3) teT
~ Nk,a, , = N c y c l e s - 1Vd ~ D,k ~ K , k __4
(4)
t~T
~.,
~ Nk,,, , = Ncycles Vd ~ D
(5)
keK,k=HO t~T
~ Nk,a, , = Ncycles Vd ~ D
(6)
k~K,k=H1 teT
Z
Z Z Nk,a., =1
(7)
k~K,k=HO(d,d) d~D t~T
Z
ZZNk,a, ,=1
(8)
k~K,k=Hl(d,d) deD t~T
Objective .function." the goal is to minimise the make-span of the given number of batch cycles. As the last cycle ends with one of the HI (d, d), diD tasks, and only one of these tasks is executed (equation 8), the objective function can be defined as: min
E
EE
Nk,a,, (T, + rk,a)
(9)
kaK,k=Hl(d,d) deD teT
where Tt is a parameter representing the absolute starting time of interval t. 2.2. Process dynamic simulation model For the scheduling model described above the duration of all tasks needs to be known before a solution can be reached. For all tasks except heating, the necessary values were collected from plant data. Since the real plant only works with one digester sequence, D 1-D4D2-D3, the heating stage parameters for the other possible sequences needed to be estimated. This was accomplished through the development of a distributed heterogeneous dynamic model for each digester/heat exchanger system, solved in gPROMS. Each model was
851 validated with real plant data, and all possible sequences were simulated, providing the heating tasks duration. Due to the lack of space this model description is omitted. 3. RESULTS The model was solved for two different scenarios, one constrained to the current sequence used in the plant and the other without this constraint. For the first scenario only the path corresponding to the actual sequence was allowed. This was accomplished by fixing the extent variables of all other heating tasks to zero, in every time interval. The solvers OSL and CPLEX solved the MILP models in a Pentium III-450 MHz for two and three production cycles. Some statistics are shown in Table 1. The results of Table 1 clearly show that the constrained model (Scenario l) is solved faster. This is due to the lower number of integer variables of the model and to its structural simplicity caused by the use of a fixed operational sequence. As the objective is to minimise the makespan, the number of time intervals to use is not known a priori. A small number of intervals turns the model infeasible while an excessive number makes it very difficult to solve. The solving strategy used was to address first the constrained model with a reasonable number of time intervals, adjust the number based on the value of the objective and then solve the unconstrained model. Scenario 2 has a better objective value than Scenario 1. In the optimum digester sequence, D 1-D2-D3-D4, the available steam is shared more efficiently amongst digesters, which allows for a faster completion of the number of production cycles studied, when compared to the actual sequence, D1-D4-D2-D3. This allows an increase of production of 1.7 %. The associated optimal schedule (Scenario 2, Ncycles=3) is presented in Figure 3. Table 1. Computational statistics and results for the scheduling model scenarios Ncycles=2 Ncycles=3 Scenario 1 Scenario 2 Scenario 1 Scenario 2 Time intervals 252 235 374 352 Integer variables 14372 17146 20968 25314 Continuous variables 26819 25017 39751 37419 Constraints 12631 11781 18731 17631 Obj. relaxed MILP (min) 1260 1255 1860 1845 Objective MILP (min) 1260 1255 1860 1845 CPU (s) (OSL/CPLEX) 19/834 93/2284 35/2263 1966/6550
852
Figure 3. Optimal schedule for the unconstrained model (Scenario 2, Ncycles=3). 4. CONCLUSIONS The problem of finding the optimal schedule of a batch digester system with an insufficient availability of steam as a heating medium was addressed. A discrete time RTN based mathematical model was developed to model the scheduling cooking process and a superstructure was created to accomplish the several heating altematives. This was coupled with a distributed heterogeneous dynamic model of the process operation that allows the determination of all the required process data describing each possible heating altemative. As final result, the optimal schedule obtained was characterised by a different digester sequence than the one in use at the real plant representing a higher production rate. This paper shows the importance of combining different types of models to describe the plant reality. This was a first approach to solve a real problem and further improvements to the scheduling model have been undertaken by the authors so as to generalise the presented model by allowing a higher flexibility to other real plant aspects. REFERENCES
1. C.C. Pantelides, Proc. 2 nd Conf. Foundations Comp. Aided Operations, (1994) 253. 2. E. Kondili and C. C. Pantelides, Comput. Chem. Eng. 17 (1993) 211.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
853
Application and Evaluation of Linear/Restricted Nonlinear Observers to a Nonlinear CSTR S. Dash a, S. Kantharao b, R. Rengaswamy b and V. Venkatasubramanian a aSchool of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA bDepartment of Chemical Engineering, liT Bombay, Powai, India Analytical model-based diagnosis exploits the functional redundancy inherent in a system model for the design of fault-sensitive systems robust to noise/disturbances. Here, the theory of two unknown input observers, one linear and the other, a restricted non-linear are outlined and built for a non-linear exothermic reactor. A scaling matrix is used to make the residuals reflect actual fault severities. We present the performances of the observers in terms of diagnostic accuracy for single/multiple fault scenarios as well as their robustness to noise. 1. INTRODUCTION Process fault diagnosis is the difficult problem of real-time diagnosis of malfunctions in a process given sensor data and process knowledge. It is important not only because of its safety impact but also process economics [1]. The increasing complexity of plants and the development of sophisticated control systems have necessitated the parallel development of fault detection and identification (FDI) systems with superior performance. Diagnostic methods vary according to the kind and form of knowledge used [2]. Redundancy-based diagnostic methods utilize hardware or functional/analytical redundancy. Hardware redundancy uses redundant sensors which may prove to be expensive or infeasible. Analytical redundancy (AR) refers to the inherent redundancy contained in the set of static/dynamic (algebraic or temporal) relationships among system inputs and measured outputs. There are a variety of methods [3] proposed to take advantage of the AR such as observers, parity-space techniques, parameteridentification approaches. While the theory of model-based diagnosis is quite well-developed for linear systems [4], application to chemical processes has been limited because of unavailability/complexity (non-linearity) of most of the models [5]. One way to use AR is to design observers that are robust with respect to disturbance/unknown inputs. This paper deals with two observers - a linear and a restricted non-linear. Section 2 provides a brief review of the key concepts in AR and also develops the theory of the two observers. In Section 3 we evaluate
854
Fig. 1. Basic Observer Scheme
the performances of both these observers on a CSTR case study. We end with conclusions in Section 4. 2. ANALYTICAL REDUNDANCY (AR) - OBSERVERS The basic idea in AR-methods is to compare the actual behavior of the system with a nominal fault-free model driven by the same inputs, using residuals. Residuals are functions accentuated by faults i.e., they represent the inconsistency between actual plant variables and the model. The two main stages in an observer design are (1) Residual Generation and (2) Residual evaluation. The first task concerns generating the residual vector under unknown fault modes, uncertain nominal model and system/measurement noise. The second stage analyzes residuals (statistical testing) in order to arrive at a diagnostic decision (fault isolation). Observers: The technique consists of reconstruction of the outputs with the aid of observers/ kalman filters and using the estimation error/innovation or a function of it as the residual. The standard structure of a linear diagnostic observer of full order is shown in Figure 1. The actual output vector y is compared with the output vector 3~of the nominal model and the difference r is fedback with the feedback gain matrix H to compensate unmatched initial conditions and provide design freedom. The residuals should (1) detect and uniquely identify different faults and (2) be robust i.e., invariant to unstructured uncertainties such as process/measurement noise and modeling uncertainties. Next we present a very simplified design of a set of unknown input observers (UIO) each sensitive to a set of faults while being insensitive to the other faults/unknown inputs. Our main aim here is to evaluate their applicability. Frank [4] presents the necessary and sufficient conditions for observer design in the general case. Linear Observer(LO): The linear observer model [4] assumes the mathematical model of the system to be given in the discrete form as Xk+1 = A x k + B U k + E d k + K f k ;
yk=CXk+Fdk+Gfk;
Xk=O=XO
(1)
where, x is the state vector, u the input vector, y the vector of measured outputs, d the unknown inputs/disturbance vector, f the fault vector and A,B,C,E,F,K,G are known matrices of appro-
855 priate dimensions. Ed models the unknown inputs to the actuators and to the dynamic process, Kf: component and actuator faults, Fd: unknown inputs to the sensors and Gf: sensor faults.
One form of the observer is given by the equations Zk+l : Rzk + Syk + Juk;
rk -- LlZk + L2Yk;
Zk=0 = z0
(2)
where Zk = TXk, iff fk : 0 after the system transients have died out and steady-state has been achieved. This observer is valid for fault detection if the residual rk has the following properties lim rk : 0 iff fk : 0 and rk ~ 0 iff fk ~ 0 'V'U,d, x0, z0
(3)
k-+oo
For the residual to have the required properties we need [4] TA-RT:SC
J-TB
LIT+LzC=0
LzF - 0
SF = 0
TE = 0
LzG ~ 0
SG ~ 0
TI~ # 0
(4)
Also, if fk ~- 0, then it can be shown that fk/rk ----L1 ( S I - R ) - I ( S G -
T K ) d- L2G. For the
residuals due to an initial mismatch in the states of the system and the observer (Xk:0 ~ Zk=0) to go tO zero, we require R to have eigenvalues in the left half of the s-plane. One way to ensure complete decoupling of the fault effects is to have (1) R diagonal, (2) L1 = ( S G - TK) -1 and (3) LzG diagonal. Restricted N o n - L i n e a r Observer(RNLO): A partially non-linear [4] observer can be devel-
oped for a certain class of non-linear systems which can be cast in the form Xk+l : Axk + B(yk, Uk) + Edk + K(Xk)fk;
Yk : CXk + G(Xk)fk
(5)
From these equations, we see that the extraneous inputs may not appear in the output equation, the dynamics equation may be non-linear only in output and input variables, and the input distribution matrix of the fault vector may only be a function of the system states. The observer is given by the equations Zk+l : Rzk + J(Yk, Uk) -~- Syk;
rk : LlZk + L2Yk;
Zk-0 = z0
(6)
where Zk = Txkiff fk = 0. As for the linear case (Equations 4), we get the same conditions except that now J(Yk, Uk) : TB(yk,Uk), TK(xk) =/= 0. Further, the transfer function fk/rk is also same except that now L1 = (SG - T K ( x ) ) - I implying it must be computed at each instant. A better performance can be expected if K(x) in Equation 5 is replaced with K(x,u) i.e., terms such as
5uiSfj
are then included in design.
S c a l e d / N o r m a l i z e d Residuals: Nonzero values of the raw residuals in the observers
signal
faults. If a suitable scaling matrix is used in the residual equations for r~ (Equations 2, 6) the scaled residuals can be made to reflect the actual fault percentage levels. Such a scaling matrix
856
can be constructed with the no-fault values of the fault variables Foi ,
as
a diagonal matrix Sc
where the Sei = -lO0/Fo,. Thus Equations (2,6) are modified to rk = Se(LlZ + L2y) where rk = % fault levels. Also, for fault detection, ri are normalized by the maximum of the values (max/
ri)
to generate normalized residuals
Ri. Ideally we want the faults to trigger the corre-
sponding Ri (:k:l) while restricting others to 0. Perfect isolation is seldom achieved in reality so we use a threshold Rth to detect faults. Thus rk estimates fault levels and Ri (along with Rth) detects faults. 3. CSTR CASE STUDY In this section we design and evaluate the observers for a CSTR case study [6]. A first-order exothermic reaction takes place in the jacketed-reactor with proportional controllers regulating the volume and temperature of the reactants. The design of the following observers exploits the large degrees of freedom, simplifying the task, thus we provide a specific design of a more general solution. Design of LO: Linearising the model equations about the steady-state values, the matrix form obtained is as follows: ~k-Ax+Bu+Kf;
y-Cx;
u-Hy
(7)
where x = [V Ca T Tj], y = x, u = [Fj F], and f = [Fo To Tjo Cao]. Note that E, F, G are all 0. Also d - 0 i.e., there are no unknown inputs. In view of these simplifications we need - L I T K to be diagonal. There are sufficient degrees of freedom to satisfy all the Equations 4. One sufficient choice would be T -- I4, R -- - I 4 . All four eigenvalues of R are chosen to be the same so that for the same fault level in the four fault variables during transients, no fault residual eclipses others. To keep L I T K diagonal, choose L1 -- K - 1 =~ J - B; S = A + I4; L 2 - - - L 1 . Thus, the linear observer with scaling matrix Sc is
r
--
- I z + Bu + (A § I4)y
-
SeK - l ( z - y )
(8)
Design of RNLO: The non-linear model of the system is to be cast in the form shown in Equations 5. The system model is found to contain products of a fault variables with other fault variables (FoCao,FoTo). Hence to separate these terms, limited linearisation about the steadystate values of the relevant variables is required. We use K(x,u) (instead of K(x,)), thus requiring linearisation of only these two terms. As in the case of LO, E - 0 G(x, u) - 0. The design parallels that of the linear observer of the previous section. As before we choose T - I4, R - - I 4 thus giving L1 -- [ - K ( x , U)] - 1 , S - A + I4, J(y, u) - B(y, u), L2 -- - L 1 and L1 - - K ( x , u) -1. Thus, the RNLO observer designed with scaling matrix Sc is
J~ --
--14z + B(y, u) + (A + 14)u
r
Sc[-K(x,u)-l(z -y)]
-
(9)
857
Fig. 2. ri for (a) LO Case 4: Quadruple Fault (b) RNLO Case 4 (filtered): Quadruple Fault
Table 1 Normalised Residuals Ri for LO and RNLO Case I No.
Faults Simulated
1
to+
2 3 4
T+T;
ror~C2o Fo+ro+rfiC2o
1
Fo-
2
To+ T)+o
3 4
ToT,loCao FoToT~C+o
R1 R2 R3 LO (5% faults, 10% noise) 0.000 1.000 -0.006 0.000 -0.170 1.000 0.000 -0.446 -0.434 0.833 1.000 -0.674 RNLO (15% faults,15% noise) -1.000 0.000 0.000 0.000 0.869 1.000 0.000 -0.723 -1.000 -1.000 -0.892 0.999
R4 0.010 0.194 -1.000
Faults Identified T+
-0.910
Tj+ [T+1 Cao [TO Tj-o] Fo+ To+ Tfo C~o
-0.004 0.095
FoTo+ T/+
-0.725 0.852
Fo TO T~+ C+
To T/o Cao
Performance Evaluation: The CSTR along with the observers were simulated in SIMULINK. A total of 24 (8 single, 8 double, 4 triple and 4 quadruple) faults are simulated with 10% (LO) and 15 % (RNLO) noises. The faults were chosen from a combination of
F~, C~o, T~, T~.
The
scaling matrix Sc (Section 2) is used to generate the fault magnitudes directly. The threshold
(Rth) is
set at •
Due to space limitations the results from all the fault simulations could not
be presented here, so only a sample of the faults are dealt with. Results for LO: 5% fault scenarios were simulated with noise level
N/S
- 0.1. No noise
filter was used so as to test the performance in the presence of noise. An example of a quadruple fault (Case 4) is shown in Figure 2(a) with good fault isolation and estimation, and the estimated magnitudes are close to the simulated values of 5%. A sample of the residuals Ri is shown in Table 1. 7 out of the 24 cases showed either a missed fault (false negative) or an absent fault (false positive) resulting in 70% diagnostic accuracy. In many of these cases the Ri
were
close
to the threshold of +0.5. The fault-related residuals are expected to have values :J:1, while all
858 the other residuals should ideally be 0. Only residual R1 shows the values expected, because the system equations include one linear equation exclusively in the fault variable Fo. In general, it is seen that we get sufficient decoupling of fault effects for faults Fo~ Ciao,while there is plenty of interaction among the fault effects for To:k Tj~. This brings to light the non-linear nature of the CSTR which cannot be taken into account completely by a linear observer. Results for RNLO: Faults with 15% magnitude and N/S = 0.15 were simulated. The increased severity of the faults is to assess the performance of this 'better' designed observer. A sample of the normalized steady state residuals Ri (no noise filter) is shown in Table 1. A perfect isolation of faults at steady state is seen for all 24 cases achieving perfect diagnostic resolution. The
Toi Tjio fault detection problem encountered in the LO case is not seen here. However, because of the increased severity of noise in this case, deterioration in transient performance (noisy residuals) may be expected. To mitigate the noise effect, a moving average discrete filter for the observer inputs is tested and an example of a quadruple fault (Case 4) is shown in Figure 2(b). 4. CONCLUSIONS Exploiting the redundancy in a process model is an useful technique for fault diagnosis. The issues of modeling errors, robustness to noise/uncertainty and perfect fault decoupling were discussed. A linear and a restricted non-linear observer were outlined, designed and evaluated for a non-linear CSTR plant. While the linear observer is simple to build, it can't compensate for the nonlinearities resulting in fault interactions manifesting in the residuals. The restricted non-linear UIO compensates partially for the plant non-linearities, and thus has superior performance. REFERENCES [1] Nimmo, I., Adequately address abnormal situation operations. Chem. Eng. Prog. 1995, 91 (9), 36 [2] Dash, S.; Venkatasubramanian, V., Challenges in the industrial applications of fault diagnostic systems. Comput. and Chem. Engng. 2000, 24(2-7), 785 [3] Gertler, J., Fault detection and diagnosis in Engineering systems. Marcel Dekker. 1998 [4] Frank, P. M., Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy - - A survey and some new results. Automatica. 1990, 26(3), 459 [5] Huang, Y.; Dash, S.; Reklaitis, G. V.; Venkatasubramanian, V., EKF based estimator for FDI in Model IV FCCU. IFAC Proceedings SAFEPROCESS2000. 2000 [6] Luyben, W. L., Process Modeling, Simulation and Control for chemical engineers. McGraw-Hill. 1990
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
859
Design considerations of computer-aided RCM-based plant maintenance management system Hossarn A.Gabbar, Kazuhiko Suzuki, and Yukiyasu Shimada Department of Systems Engineering, Okayama University 3-1-1 Tsushima-Naka, 700-8530 Okayama City, Okayama, Japan In most of the industries, Reliability-Centered Maintenance (RCM) process hasn't been fully integrated with the computerized maintenance management system (CMMS) where they run in isolation leading to outdated maintenance strategies. In this paper, the idea of integrating RCM process with the CMMS has been studied so that the maintenance strategies are dynamically changing throughout the plant lifecycle. The integrated solution has been discussed in two-folds: (1) Information, where plant design model is obtained from the plant design environment, while plant operational information is acquired from the operational systems; (2) Process logic where HAZOP, FMECA, and FTA are utilized to analyze and assess all failure types in quantitative manner, while combined Monte Carlo, Genetic algorithm, and Weibull probability distribution functions (pdf) are employed to optimize the maintenance tasks. The activity and functional models of the enhanced RCM process are developed. Modifications to the different modules of CMMS (i.e. MAXIMO TM)are proposed. 1. INTRODUCTION RCM process is intended to determine the most realistic and optimized maintenance requirements of any physical asset to continue its stated operating condition [1]. Many industries are adopting RCM technique to solve many confronted maintenance problems. Unfortunately, it didn't work as expected for many reasons: (1) RCM is a time consuming process and requires considerable amount of resources, especially for large number of assets; (2) information is not adequate to decide the suitable maintenance strategy and to optimize its cost; (3) there are non-engineering factors involved in the maintenance problems i.e. management. To achieve the expected results, it is essential to consider an integrated solution using the plant lifecycle's accumulated information. By considering quantitative asset and failure assessment techniques accurate results can be achieved. Combined HAZOP, FTA, and FMECA are used to analyze the root cause of each failure. Combined Monte Carlo simulation,
860 Genetic algorithm, and Weibull pdf are used to optimize maintenance tasks. Previously, attempts are made to integrate RCM within the design environment using fuzzy reasoning algorithms [2]. This research proposes wider vision of integration that includes plant design and operational systems, and suggests the necessary design changes to CMMS. 2. PROPOSED SYSTEM ARCHITECTURE
Fig. 1 - RCM-based integrated system architecture. Fig. 1 shows the system flowchart of the proposed integrated solution, which includes: plant design environment, operational systems, CMMS, and RCM process. The cycle starts from the plant design by developing the plant model, which includes the plant static (structure and topology), behavior, and function views. Plant asset information is extracted from the plant static model into CMMS and passed to RCM process. Failure model is developed within the plant design using HAZOP, FTA, and FMECA techniques. The failure model in the plant design domain is translated into failure hierarchies within CMMS, which is passed to RCM process for failure assessment. Assets are represented within CMMS as templates (or class definitions) and physical asset records. For example, Pumps and Valves are represented as class definitions, while the physical plant assets for pumps and valves are represented as asset records, which are associated with the corresponding asset templates. Asset information is passed to the RCM process for asset assessment practice. Maintenance strategies are initially decided during the design stage and gradually tuned throughout the plant lifecycle using feedback information from operational systems and reliability data. The data flow "D8 ") P2" is used to express the information flow from D8 (the maintenance history data within CMMS) to P2 (RCM), which
861 fortifies the utilization of the reliability data from the running CMMS. 3. RCM PROCESS MODELING
Object-oriented modeling approach has been adopted to analyze and enhance RCM process. The activity and functional models have been developed for this purpose, as below. 3.1.
RCM Activity Model
Fig. 2 - RCM process first-level activity model. The activity model has been developed using IDEFO where the RCM process has been enhanced from both business and technical views (from "As-Is" to "To-Be"). IDEFO is useful in establishing the scope of the analysis, especially for functional analysis. Fig. 2 shows the first-level activity model, which shows four major activities: (1) manage the RCM process where scope and resources are identified, (2) identify assets, functions, and performance standards where plant structure information is mapped from the plant design model, (3) analyze failure, cause, consequences, and evaluate associated risk, and (4) decide and optimize maintenance strategies and tasks. The integration on the activity model level is applied to augment the corresponding activities from CMMS and to understand the required input / output information and its originality i.e. plant design or operational systems. For example, asset identification activity has been consolidated to read the plant model from plant design environment. It creates the asset templates as the base classes for the associated assets.
862
Fig. 3 - RCM Process Function Decomposition. 3.2.
RCM Functional Modeling
Using the activity model, the function decomposition of the proposed RCM process can be constructed (as in Fig. 3). RCM can be decomposed into the following functions: assess assets, assess failures, decide maintenance strategy, decide maintenance tasks, optimize maintenance tasks, and check & validate the results. Maintenance tasks are defined as: lubrication, cleaning, inspection, replacement, repair, fabrication, and overhaul, which are used within the expert system (decision engine) to decide the suitable maintenance task based on the failure, consequence,
reliability data, accumulated
maintenance
history, and design model
requirements (including safety requirements). Maintenance task level is classified as: good-as-new (GAN), imperfect, and bad-as-old (BAO). Soft and hard life factors [3] are optimized using combined Monte Carlo simulation and Genetic algorithm (explained in details in reference [4]). Weibull function is used to model failure behavior and classify failures, which is used to construct the reliability data. Checker report using RCM knowledgebase and maintenance data is used to verify the correctness and accuracy of the obtained results (i.e. optimized maintenance tasks).
863 4. INTEGRATED SYSTEM DESIGN
Fig. 4 shows the detailed design of the integrated solution where plant model is represented in three dimensions: structure (static information about the plant i.e. Pump-1 and Valve-I), dynamic (behavioral information about the fluid, process variables changes with time i.e. fluid-pressure-out-Pump-l, which is represented by set of equations), and functional (operations performed by plant objects i.e. Valve-1 --) Open). HAZOP is used to report the possible deviations, causes, and consequences for Pump-l, and FTA reflects all the possible paths for each top-failure with the probability of failure for each minimal cut set, while FMECA defines the different failures with the different levels of details along with the criticality of each failure. Combination of these failure assessment techniques facilitates the analysis of the root cause. Assessing the root cause instead of the actual cause reduces the probability of failure and hence minimizes the associated risks.
~l
A
PlantOO ,'" u// /
, ',]
D~ Functional I
r ~ ....... r...... } I c.... q..... l / . ~ FallureModel~ ~
I
J
CMMS ~
AssetData ( Flash Point IF Operating Temperature >_ Boiling Point wf=l. 1 Else wf=l Else wf:l
(1)
Since this condition gives information to determine outcome event of accident scenarios, weighting effect follows consequences of the final accident event. Table 1. Accident Scenario Factors Accident Scenario Factors RankingValues Unit Factors UFL, UFC Process Factors PFL, PFC Equipment Factors EFL, EFC Material Factors MFC Status Factors SFL, SFC Accident Progress Factors RFL, RFC Ignition Source Factors IFL, IFC Process Condition Factors
Examples LDPE, HRDS, CDU Storage, Reaction, Distillation, Separation Reactor, Pump, Pipe Hydrogen, Benzene Startup, Normal Operation Leak, Rupture, Crack/Hole Spark, Hot Spot Temperature, Pressure, Capacity
To predict accident sequences for given input data, we compare input data to APD and decide the accident sequences depend on PCD. The potential risk of each accident scenario is also derived from the component factors to produce a numerical risk ranking and comparative analysis of those risks. The priority of generated accident scenarios is defined by ranking value.
Ranking Value = ~ Consequence, x I-Ii Likelihoodi
(2)
Once an accident scenario has been developed, for each step of the accident sequence, the factors affecting the sequence and the variable conditions could be verified by related actual accident data. The system analyzed the causes of accidents, represented the effect of failure combinations, and found the most probable dangerous scenarios in complex processes.
898
Figure 2. Flowchart of Accident Scenario Generation 4. ACCIDENT SCENARIOS FOR HYDROCRACKING PROCESS Using the CAGAS, accident scenario generation is carried out for a typical hydrocracking process, which is shown in figure 3. This process is comprised of several common process units such as heat exchanger, reactor, pump, pipe and valve. The hydrocracking reaction generates heat that increases temperature and causes the reaction rate to accelerate. Since this process is operated under high pressure and high temperature, there exist lots of potential hazards. An accident history at selected hydrocracking units in three countries is illustrated in Table 2. These accidents caused losses of property, human life, and environmental quality. Major causes of these outcomes are hydrogen blistering, erosion, overpressure, crack and partial overheating. Using the CAGAS system, various accident scenarios are visualized at different process conditions. This system generated almost three hundred accident scenarios and their risk ranks. And the system describes the accident progress sequences as well as determines root causes and contributing factors. These are summarized in figure 4 and table 3. Table 2. List of Accidents at Hydrocracking Units Unit Process
Section
Causes
Events
Loss
Location
Year
Hydrocracking Unit
Heat Exchanger
IgnitionSource
Fire
4 injuries
Korea
1997.07
Hydrocracking Unit
RecycleOil Pipe
HydrogenBlistering
Explosion
$77,000,000 Korea
1999.05
Hydrocracking Unit
Pipe(Elbow)
Erosion
Fire, Explosion
-
Korea
1997.03
Hydrocracking Unit
Separator
Overpressure
Fire, Explosion
$73,000,000
UK
1987.03
Hydrocracking Unit
Reactor
Crack
Fire
$15,000,000
USA
1980.12
Hydrocracking Unit
Reactor
Partial Overheating
Explosion,Fire
$25,000,000 USA
1970.12
899
Figure 3. Process Flow Diagram for Hydrocracker High Pressure System Reprinted from: US EPA, EPA Chemical Accident Investigation Report, 1998. The results for sequence of accident scenario in these units are as follows.
Highly Ranked Accident Sequence 1 Intemal corrosion and erosion at the piping system in a reactor causes a material weakening and rupture of a pipe. It leads to fire and explosion in the present of ignitior source.
Highly Ranked Accident Sequence 2 Relief valve trouble occurs at the separator section caused by overpressure. It leads to release of material and fire develops under high pressure and high temperature.
Highly Ranked Accident Sequence 3 Hydrocarbon is released and a fire subsequently develops in the reactor effluent pipe du~ to excessively high temperature caused by reactor temperature excursion. The highly ranked accident scenarios involve the release and auto ignition of a mixture of flammable hydrocarbons and hydrogen under high temperature and pressure as known T.A. refinery plant.
Figure 4. Major Causes of Accident for Hydrocracking Process
900 Table 3. Examples of Accident Scenario Progress No. Accident Scenario Progress 1 2 3 4 5 6 7 8 9 10
InternalCorrosion --) Weakening--)Rupture Rapid Evaporation --) Overpressure--) Rupture Changeof Condition --) TemperatureExcursion--) Rupture WrongConstruction/Design --) Rupture Erosion--) Weakening--) Rupture Poor Flow --) Changeof Condition (TemperatureExcursion) --) Rupture VentBlockage--) Overpressure--)Rupture InadequatePurification --) Runaway/SideReaction --)InternalExplosion/Fire--) Rupture WrongManual --) Mal-Operation/PoorMaintenance--) Overpressure--) Rupture InstrumentFailure --) IncorrectCondition --) Runaway/SideReaction--)InternalExplosion/Fire--)Rupture
5. C O N C L U S I O N A new method for the generation of accident scenarios was proposed for chemical process industries, and the computer aided generation of accident scenarios (CAGAS) system was developed in this study. As a result of applying CAGAS to the hydrocracking process, hazardous locations and dangerous states are found, and the system generated almost three hundred accident scenarios and their risk ranks. The highly ranked accident scenarios are fire and explosion, which are caused by material weakening from the internal corrosion and erosion in the piping system of the reactor. The current limit of this system is that it is applicable only to a domain specific process, but it will be expanded to general chemical processes by building generic libraries. If hazard analysis such as HAZOP covers most possible hazardous states, which are identified by this system during the design stage, major accidents in chemical plants could be prevented. This study proposed an approach to improve the safety of chemical plants by generating accident scenarios systematically. REFERENCES 1. F. I. Khan and S. A. Abbasi, Inherently safer design based on rapid risk analysis, Journal of Loss Prevention in the Process Industries, 11, (1998). 2. I. Moon, D. Ko, S. T. Probst and G. J. Powers, A Symbolic Model Verifier for Safe Chemical Process Control Systems, J. of Chem. of Japan, Vol. 30, No. 1 (1997). 3. S. H. Lee, J. K. Kim and I1 Moon, Safety Analysis of Boiler Process Operating Procedures using SMV, Journal of the Korean Institute of Chemical Engineering, Vol.37, No. 5 (1999). 4. H. R.Greenberg and J. J. Cramer, Risk Assessment and Risk Management for the Chemical Process Industry, New York: Van Nostrand Reinhold, 1991. 5. L.C. Cadwallader, H. Djerassi, I. Lampin and J. Rouillard, A Comparison of U.S. and European Methods for Accident Scenario Identification, Selection, and Quantification, IEEE Thirteenth Symposium (1990). 6. US EPA, EPA Chemical Accident Investigation Report, 1998. 7. NFPA, NFPA Code 325M, Fire Hazard Properties of Flammable Liquids, Gases, and Volatile Solids, National Fire Protection Association, 1991.
European Symposium on Computer Aided Process Engineering - 11 R. Gain and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
901
The integration of production plan and operating schedule in a pharmaceutical pilot plant L. Mockus, J.M. Vinson, K. Luo Pharmacia Corp., 4901 Searle Parkway, Skokie, IL 60077 In reality, planning and scheduling activities are loosely integrated. In the case of a complex supply chain, it might be too difficult to maintain both long-term and short-term information within one large model. However, in the case of a single facility and relative short time horizon this task becomes feasible. This is the case in a pharmaceutical pilot plant, where a two- or three-year planning horizon is the most one would expect due to frequent changes in the products and their demands. In earlier work, we decomposed the pilot plant planning and scheduling problem into long-term planning of resources and shortterm scheduling of operations [ 1]. The long-term plan is mainly concerned with allocation of human resource, such as operators, engineers, chemists, and analysts to projects that have uncertain timelines. The short-term schedule fixes the human resources and considers the scheduling of activities in the pilot plant itself. The algorithm resolves uncertainty in future projects by over-allocating resources so that, on average, resource constraints are met. In this paper, we continue this exploration. In particular, we explore techniques for combining the long- and short-term into one plan without creating undue burden on the solver and the planning group. 1. INTRODUCTION We present a methodology that combines a production plan and a daily operation schedule in a single model. Normally, it is unnecessary and computationally infeasible to represent detailed, everyday operational activities in the production plan. In a pilot facility, detailed activities for running a production step become finalized only a few weeks, or even a few days before the actual run. Our approach is to use general information for future campaigns and add more detail to campaigns closer to the start of the scheduling horizon. How detailed should an operation schedule be? The computational aspect is just part of the problem. The level of detail carries information such as duration and resources needed. For scheduling purposes, manpower is our major concem; clearly, the schedule should not exceed the resource limitations. Thus, we decided to model only operations with different manpower requirements, while combining those with similar manpower requirements into super-operations. For example, two charges followed by a heat operation might be combined into one operation, called "reaction preparation," if they all require the same resources. Another aspect of the proposed approach is to keep the production plan and operating schedule within one model. Currently, we create the production plan and operation schedule separately; with the operation schedule following production plan. The creation of these plans is manual and requires quite a lot of human intervention (phone calls, meetings, etc.) in
902 order to develop an acceptable schedule. Unfortunately, the two schedules can easily become unsynchronized over time, requiring continual oversight. Integration of the production plan and the operation schedule permits closer delay monitoring and faster resource conflict resolution. We automate this process by integrating the production plan and operation schedule in one model and by scheduling the resulting model with a commercial solver developed by Advanced Process Combinatorics (APC). A natural extension of this methodology gives us an operation rescheduling mechanism where operations that are already completed become frozen for the scheduler. As a result, we only have to schedule remaining operations, and thus the impact on the near-term schedule is minimal. The advantage of implementing various level of detail is two-fold. It allows keeping the model small. Second, and probably the most important benefit is that maintenance of the model is simplified in the case of daily changes. 2. BACKGROUND
Batch scheduling literature is mostly concerned with the optimal allocation of resources to tasks and optimal scheduling of tasks to meet a given demand schedule. The classic examples are from Kondili and others [2, 3] and Sahinidis and Grossmann [4]. The recent work by Honkomp and others [5] considers uncertainty prevailing in the pharmaceutical industry, but this uncertainty is just in processing time. They assume that demands are known a priori for the duration of scheduling horizon. These papers mostly reflect the manufacturing environment. In the R&D environment, this assumption might not hold primarily because the demands are dictated by clinical, toxicological, and formulation studies and are quite uncertain. In addition, production runs are unique to each campaign, so the timing and sequencing of tasks will vary from campaign to campaign. There are works addressing the problem of reacting to processing uncertainties in the chemical industry. Cott and Macchietto studied the performance of on-line scheduling with several time shift algorithms [6]. In the algorithms, the deviations between the target schedule and the actual schedule are detected at short intervals and the schedule is shifted in order to minimize the effects of processing time variations. Kanakamedala and others presented an algorithm that uses a least impact heuristic and provides significant improvement over the earliest finishing unit heuristic by emphasizing the importance of maintaining the original schedule [7]. Kim and Kim propose a mechanism for reactive scheduling where they use simulation- in conjunction with job dispatching rules - to dynamically generate a schedule for discrete manufacturing systems [8]. Knowledge based approaches involve human expertise in planning and scheduling activities (including reactive scheduling) [9]. Instead of using math program to decide the optimal schedule the human expertise is utilized by providing a suite of tools (including 'what-if analysis). Little attention is given to such issues as combining the production plan and operation schedule in a single model. In the work of Zhu and Majozi [10] the integration of planning and scheduling is being considered for the special case of multipurpose batch plants where equipment is dedicated for each process. In our pilot plant, equipment is shared across processes. As mentioned above, this work provides a methodology for tighter integration of the production plan and operation schedule. The uncertainty in the pilot plant environment is handled by continuously decreasing the level of detail for the future batches, thus focusing on the current production schedule while making allowance for the future production. One way
903 to think about it would be imposing a high penalty for the deviations from the near-term plan and continuously decreasing penalties for deviations further into the future. Penalties dynamically change as we move forward in time. In addition, the decreasing level of detail for future batches serves the purpose of a cushion to accommodate the uncertainty of future demand dates and quantities. If we plan to the lowest level of detail and something happens (equipment breakdown, rework, etc.) then much of the work (human or computational) that went into the original schedule will have to be redone. On the other hand, if a future batch is created as a single task with the duration and resources corresponding to the sum of the unit operations, then it is cushioned from uncertainties. Eventually this batch will be ready to run, the level of detail will be increased, and the reserved time slot will provide more flexibility during rescheduling. These batches can be further abstracted into a single step task, representing all the batches. The steps, too, can be rolled-up into a campaign that will act as a placeholder for the resources. 3. PROBLEM STATEMENT
In its simplest form, the planning and scheduling problem can be stated as the minimization of the sum of tardiness for all campaigns (demands) subject to constraints on allocation, sequencing, resources, and production. Allocation constraints allow only one task to be processed on a given piece of equipment at any time. Sequencing constraints give the specific order of operations for a set of tasks. In general, the recipe for the process dictates these constraints, but sometimes we link tasks artificially to ensure administrative control. For example, a task can be started only after equipment maintenance task is finished. Resource constraints limit the sum of resources used at any time by their availability. Production constraints are dictated by production process established in pilot plant. These constraints generally require zero wait between unit operations belonging to the same batch. This is required, for example, when the stability is unknown for the reaction product, as is frequently the case in R&D processes. Good Manufacturing Practices (GMP) require that equipment has to be cleaned before processing another material (equipment changeover). Cleaning and the equipment setup associated with it may take up to week in our pilot plant. Another production constraint is that no task can be started or finished at the beginning of each shift, although it may continue processing. 4. IMPLEMENTATION We have used the commercial scheduling tool VirTecs developed by Advanced Process Combinatorics (APC) for the solution of this problem. The advantage of this tool compared to other similar tools (MIMI from Chesapeake) is that a mathematical model for such a problem was implemented, whereas in MIMI one must create the model from scratch. There is a computational speed advantage as well: the scheduling engine developed by APC is tailored for pharmaceutical scheduling problems and is much faster than CPLEX (general purpose solver used by MIMI). There is a number of software to solve advanced planning and scheduling problems. Many of them are tailored to manufacturing job shop applications, rather than batch chemical manufacturing. VirTecs defines ready-to-use modeling primitives of the state-task network framework (state, equipment, resource, and task). It is quite natural to designate each unit operation as a separate task that requires equipment and operator resources. Allocation, resource, and
904 equipment changeover constraints are already defined in the model provided by VirTecs. The zero wait constraint between unit operations within the same batch is achieved by forcing the states between unit operations to have no capacity. The sequencing constraint is easily realized as a material balance constraint that is already provided by VirTecs. For example, the reaction task produces one unit of reaction product that is consumed by separation task. The only constraint that is not so trivial is the "dead" time constraint, where tasks cannot begin or finish at the shift change. The solution is to create a special resource that is always available except at the shift change. Every task requires this resource for the first and last minute of processing. There is yet another minor complication in our design. The exact sequence and timing of unit operations is taken into account only for batches that are currently running or are to be run in a few weeks. Batches that are far in future are not split into unit operations. This is done to prevent unnecessary complexity, but the main reason is that due to demand uncertainty the number of batches required to produced desired amount of product may be unknown. There is also the possibility that a clinical study may be canceled, and the entire campaign, or part of it, becomes unnecessary. On the other hand, the number and timing of studies might change, forcing a change in the demands and production targets. Another modeling issue arises with the differentiation between batches for which sequencing and timing of unit operations is known and future batches for which sequencing and timing of unit operations is unknown. In the first case, unit operations are considered as separate tasks. In the second case, the full batch is a single task. Although it is quite natural to consider all batches as single tasks, it is more convenient to split batches in close proximity into their corresponding unit operations. This comes into play when scheduling resources over a short horizon. Batches may be able to overlap at specific operations, such as starting the reaction of batch 2 while batch 1 product is in the filtration and drying operations. Similarly, one may be able to overlap batches of other steps when the specific operation resources are known. 5. CASE STUDY We compare several different schedules to test the proposed methodology. All the schedules are derived from the actual two-year production plan. Fifteen campaigns for seven different products were taken into consideration. The first schedule contains full unit operation detail for the entire two-year span. Scenarios 2, 3 and 4 are built off the base schedule, adding extra processing time to every operation. Scenarios 5-9 test the effect of our philosophy of only adding details to the near-term campaigns, while leaving information out of future campaigns. The remaining two scenarios test the effect of adding demand (new batches) to existing campaigns that are scheduled to run at the beginning of the planning horizon. The schedule for each scenario was optimized to remove resource constraints and meet its deadlines as closely as possible. Table 1 shows the results for each of the scenarios, listing tardiness as the prime indicator of schedule quality. While the base case schedule is one of the best in terms of tardiness, the effort of adding full details to every batch of every campaign is quite cumbersome. In fact, all of that effort is lost when comparing scenarios three and six. The delays of 5% in scenario three are not uncommon, while the effort to construct a schedule with only six months worth of detail is much lower (scenario six). In fact, it is rather difficult to determine the details for campaigns that are to run more than six
905 months into the future. In an R&D environment, the operations required to make a chemical can change quite frequently. This is another source of the time variability that we see when planning campaigns. Table 1 Computational results Scenario Brief Description 1 Full details in all campaigns 2 Full, 1% added to time of all campaigns 3 Full, 5% added to time of all campaigns 4 Full, 10% added to time of all campaigns 5 No detail in any campaign 6 First six months have details 7 First nine months 8 Year and a half of detail 9 Full details, increase demand in one campaign 10 Full details, increase demand in two campaigns
Tardiness (hours) 46 0 2207 5296 21399 2788 t 973 3 268 770
Increase by 1% in the duration of each activity (scenario 2) leads to the slight decrease of the total tardiness because the solver provides only "good" sub-optimal solution. The same happens for scenario 6. However, such a decrease may be treated as negligible when compared to other scenarios. The effect of slightly increased demands is quite modest, as seen in the last two scenarios. We have modeled increased demands as the addition of one batch to campaigns at the start of horizon. Adding batches at the start of horizon should maximize the total impact of the changes. The explanation of the modest effect might be the different equipment units used for those campaigns. Although the common resource shared between campaigns are operators, the optimizer manages to shift campaigns just by small amount. However, the disturbances caused by increase in demands may exceed the benefit provided by the rigor provided by detailed schedule. 6. CONCLUSIONS We have presented a methodology that combines a production plan and a daily operation schedule in a single model. This work is demonstrated on a pharmaceutical pilot plant example. The approach of combining production plan and operating schedule is tailored to the uncertainty prevailing in the pilot plant environment. The example shows that a modest increase in the processing times or demands disturbs base plan, suggesting that adding every detail throughout the planning horizon provides no benefit as far as total tardiness is concerned. In other words, the benefit of having an exact schedule disappears in the face of uncertainty. An important future research direction might be simulation of the production plan. Monte Carlo simulation is a valuable tool to estimate the robustness of the plan, as suggested in recent work by Subramanian and other for testing pipeline management philosophies [ 11 ]. In addition, we have only presented a snapshot of the schedule. Evolution of the plan over the course of several months might give a much better feeling for the uselessness of setting up campaigns with full details far into the future. Stochastic simulation of the plan will help
906 determine the quality of the schedule under known uncertainty. By introducing probability distributions for processing times, equipment breakdowns, and demand sizes one could determine how often rescheduling might be necessary, based on the level of uncertainty, and establish the desired level of plan detail. For example, the simulation might show that it is best to have a detailed schedule for three months under one set of uncertainty conditions, and six months under another. REFERENCES
1. L. Mockus, J. Vinson and R.B. Houston, Planning and Scheduling in a Pharmaceutical Research and Development. Computer Aided Process Engineering, 8, pp. 1057-1062, 2000. 2. E. Kondili, C.C. Pantelides and R.W.H. Sargent, A general algorithm for short-term scheduling of batch operations-I. MILP formulation. Computers Chem. Engng., 17, pp. 211227, 1993. 3. E. Kondili, C.C. Pantelides and R.W.H. Sargent, A general algorithm for short-term scheduling of batch operations-II. Computational issues. Computers Chem. Engng., 17, pp. 229-244, 1993. 4. N.V. Sahinidis and I.E. Grossmann, Reformulation of multiperiod MILP models for planning and scheduling of chemical processes. Computers Chem. Engng., 15, pp. 255-272, 1991. 5. S.J. Honkomp, L. Mockus and G.V. Reklaitis, A framework for schedule evaluation with processing uncertainty. Computers Chem. Engng., 23, pp. 595-609, 1999. 6. B.J. Cott and S. Macchietto, Minimizing the effects of batch process variability using online schedule modification. Computers Chem. Engng., 13, pp. 105-113, 1989. 7. K.B. Kanakamedala, G.V. Reklaitis and V. Venkatasubramanian, Reactive schedule modification in multipurpose batch chemical plants. Industrial and Engineering Chemistry Research, 33, pp. 77-90, 1994. 8. M.H. Kim and Y.D. Kim, Simulation-based real-time scheduling in a flexible manufacturing system. Journal of Manufacturing Systems, 13, pp. 85-93, 1994. 9. R. Jacobs and W. Jansweijer, A knowledge-based system for reactor selection. Computers Chem. Engng., 24, pp. 1781-1801, 2000. 10. X.X. Zhu and T. Majozi, A novel continuous time MILP formulation for multipurpose batch plants. Submitted to Industrial and Engineering Chemistry Research. 11. D. Subramanian, J.F. Pekny and G.V. Reklaitis, A simulation-optimization framework for addressing combinatorial and stochastic aspects of an R&D pipeline management problem. Computers Chem. Engng., 24, pp. 1005-1011, 2000.
European Symposiumon ComputerAided ProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
907
Mixed-Integer Multiperiod Model for the Planning of Oilfield Production A. Ortfz G6mez, V. Rico-Ramfrez* and R. Vdzquez-Rom~in Instituto Tecnol6gico de Celaya, Departamento de Ingenierfa Qufmica, Av. Tecnol6gico y Garcia Cubas S/N, Celaya, Gto., CP 38010, MEXICO We present a multiperiod optimization model for oil production planning in the wells of an oil reservoir. The problem considers a fixed topology and is concerned with the decisions involving the oil production of the wells in each period of time. We assume logarithmic behavior for the well flowing pressure (with respect to time) while calculating the oil production (Home, 1998) and consider time periods of uniform length. A numerical example has been solved through the GAMS modeling system (Brooke et al., 1998) to highlight the scope of the proposed representation.
I. I N T R O D U C T I O N Multiperiod optimization in the chemical industry has recently received considerable attention. Multiperiod planning considers a fixed topology and is concerned with the decisions involving the startup/shutdown of the operation of the process in each period of time. Oilfield operation is a multiperiod problem because the cost and the demands of oil production vary from period to period due to market or seasonal changes. An oilfield infrastructure consists of production platforms and a number of reservoirs including onshore and offshore facilities. In oilfield multiperiod models, design decisions involve the capacities of the production platforms as well as decisions regarding which production platforms and wells to install over the operating horizon (van den Heever and Grossmann, 2000). Planning decisions involve the oil production profiles in each time period. In the past, decisions regarding platforms capacities, drilling schedules and production profiles had often been made separately under certain assumptions to ease the computational burden. Simultaneous models emerged with the works of Bohannon (1970) and Sullivan (1982). Bohannon (1970) proposed simultaneous MILP models for the oilfield design and production planning whereas Sullivan (1982) developed a simultaneous MILP model for gasfield design and production planning. Also, Iyer et al.(1998) proposed a multiperiod MILP model for the planning and scheduling of investment and operation of offshore facilities. Oilfield facilities are often in operation over several decades. So, although changes in the reservoir pressure with respect to time are not significant in the short run, such changes cannot be ignored for a simulation involving future planning and investment decisions. It is known that the reservoir behavior represents a nonlinear constraint but, in all the works described above, the
* Author to whom all correspondence should be addressed
908 reservoir behavior as a function of the cumulative oil produced has been approximated by linear constraints. Recently, van den Heever and Grossmann (2000) proposed a simultaneous approach for the oilfield planning which deals with nonlinearities directly and can be solved in a reasonable computer time. In such a model it is assumed that the operating conditions are constant across the planning horizon. So, the productivity index, p, is assumed to be constant for a given period of time. The productivity index depends on the conductivity of the well and allows the calculation of the oil flow rate as a function of the pressure drop between the reservoir and the well bore: q, =p.(pr _pW) (1) where qt is the oil flow rate in period t, pr is the reservoir pressure and pW is the pressure of the well. Well analysis (Home, 1998) reveals, however, that the well flowing pressure presents a time dependent nonlinear behavior and, as a consequence, the assumption of constant operating conditions may not apply. Fig. 1 illustrates the oil extraction from a reservoir. As given by Equation (1), the driving force for the oil production from a well is the pressure difference between the reservoir and the well bore. Also notice that when the well has just been drilled (or when it has been shut in for a significant period of time), we can assume that the pressure of the well bore is the same as that of the reservoir. So, at the beginning of the operation, when the well is open to flow, oil can be extracted because of the pressure difference between the well bore and the well head. As the operation time increases, the well bore pressure decreases and that also causes an oil flow from the reservoir to the well. However, the oil flow rate from the reservoir to the well depends also on the geological properties of the well surroundings, such as permeability, thickness, porosity, etc., which determine the well production capacity. Hence, because of the resistance to the flow between the reservoir and the well bore, oil production causes the well bore pressure to decrease with time. A simple expression has been often used (Home, 1998) to represent such a behavior:
p f = pin --Cl qt [In(t)+ c2l
(2)
where cl y c2 are constants experimentally determined which result from combinations of the geological properties characterizing the well,fi in is the pressure of the well bore at the beginning of the operation (reservoir pressure) and t4 is the (final) pressure of the well bore after an operation time t. On the other hand, if the well is shut in, the well bore pressure will increase because of the effect of oil flow from the reservoir to the well. Fig. 2 shows the behavior of the well bore pressure when the oil is flowing and when the well is shut in.
Fig. 1
Extraction of Oil From a Reservoir
909 In this paper we are concerned with the short term planning of oil production in the wells of an oil reservoir. Planning decisions consist on determining the oil flow rates and the operation/ shut in times for each well of the reservoir in each period. Such decisions are based on practical considerations, which avoid the well bore pressure to decrease beyond a minimum allowable value. Also, one should remember that the oil production rate has to satisfy the oil demand for each period of time. Since we are focusing on short term decisions, we consider that the pressure of the reservoir is a constant over the time horizon. On the other hand we assume logarithmic behavior of the well bore pressure for the calculation of the oil production as given by Equation (2), and it is assumed that the values of the geological properties of the well are known.
2. PROBLEM STATEMENT This work considers the short term planning of the oil production in the wells of a reservoir over a Horizon H divided into N P time periods of length T. Hence, given the oil production demands for each period of time, planning decisions involve finding the oil flow rates and operation/shut in times of the wells. The main constraints of the problem involve avoiding the well bore pressure to decrease beyond a minimum allowable value and satisfying the oil production demands. Assumptions made in this paper include: 1) Multiple wells in a reservoir produce independently from each other. 2) Nonlinear behavior for the well bore pressure as function of the oil flow rate and time (Equation (2)). 3) The objective function is calculated in terms of cost coefficients which change for each period of time due to seasonal changes.
2.1 A Simplified Model We have developed several models of varying complexity for solving the problem described here. So, MINLP models assuming acyclic and cyclic operation for the wells which use convex envelopes for dealing with the nonlinearities are now under investigation. In this section we present a simplified model which considers time periods of uniform length. Also, in this model we assume that a well is either flowing at a constant rate or shut in across the complete time period. Such a simplification significantly reduces the problem complexity. The following sets, indices, variables, parameters and equations are defined. Sets and indices: P= set of time periods, [1...NP] W= set of wells
pUp . . . . . . . . . .
plow ~ . . . . hL
r"
t~
Fig. 2
t
t~
Time Dependent Behavior of Well Bore Pressure
t
910
i,j= indices corresponding to well and time period, iE W andjE P Continuous Variables:
qii= oil flow rate in well i and period j piniij= well bore pressure of well i at the beginning of period j //~/= well bore pressure of well i at the end of period j
Dij, Iu= pressure differential in the well bore when the well is producing and when the well is shut in, correspondingly Binary and Boolean Variables: Y~/=TRUE if well i is producing in period j (Boolean)
yij= binary associated to Yij. 1 if well i is producing in period j Wijl = 1 if the well is shut in and the well bore pressure does not go beyond the maximum allowable value
wi/2= 1 if the well is shut in and the well bore pressure reaches the maximum allowable value Parameters: a,5,T= cost coefficients
pUp,plOW=maximum and minimum allowable pressure of a well q~i - parameter for calculating the pressure increase of the well when it has been shut in T-time period length M=suitable upper limit for the residual of the equations involving pressure drop calculation
Objective:
Minimize
Z Z ruqif +Z Z 8uyuT +Z Z aoO- YO}1" i
j
i
j
i
(3)
j
Constraints: qoT > d j
Vj e P
(4)
i
Pd = Poi. - Do
v
Puy = p Ui,, + I o
v
PiiI =p~p
Vi~ W, j e P
P~7 + Iij < P:P J P~'~+ Iii > P7 Dq =qu{clOn(T)+c:~ Vie W, j e P Iij =q;{c,[ln(T)+c2]}(l- yu) Vie W, j e P q~..x {q [ln(T) + c2]}= (p~" - p:OW) max
qij < qo
Vi ~ W, j ~ P
Vie W, j ~ P
(5)
(6)
911
q,, < q~r YO + qlOWO_ YO) Vi E W, j ~ P q,i >_qlow V i e W , j e P
(7)
tn Po = P,~-~
(8)
Vi ~ W, j ~ P
The objective function of the model consists on minimizing the production cost (Equation (3)). It is assumed that there is a variable cost associated to the oil production rate (7U). Also, we assume that there is a cost associated to each well which changes depending on whether the well is open to flow (80) or shut in (o~0). The minimization is subject to the satisfaction of the demand on each period of time (Equation (4)). Equation (5) represents the behavior of the well flowing pressure. Notice that if the well is open to flow then the well flowing pressure decreases, but if the well is shut in then the pressure increases. Equation (6) establishes the upper limit for the oil flow rate in a time period so that the pressure does not decrease beyond a minimum allowable value. Equation (7) suggests that, because of operational constraints, even when a well has been shut in there is a minimum oil flow rate equal to q~OW(although in this work we use ql~ Equation (8) corresponds to the linking constraints from a time period to the next one. Finally, the disjunction in Equation (5) can be reformulated by using the Big-M representation and writing the relations among the binary variables: f
ptf -ptiJln.4-Oij > - M ( 1 - y,j) Pi~ - P,~ + Do < M O - Yo ) pti; _p:OW_Oi j > _ M O _ yu)
m
2[
>
Pij -- PiJm-- tj - - M (1- Wijl ) p,: _ p ; -I0 <MO_wo,) p; + l j - p r p <MO-w,j,) w,jz + wo2 = 1- y,j
Ptf _ pU~ ~ _M O _ wq2 ) , P,~ - P,P < MO-w,j2) Pij -bltj _pUp ~_MO_wij2)
(9)
3. RESULTS An illustrative example is presented to highlight the scope of the proposed representation. The problem has been solved by using OSL, a commercial solver available through the GAMS modeling system (Brooke et al., 1998). The system consists of a reservoir containing 8 wells, a configuration which is commonly known as the "spider" problem. The planning horizon has been divided in 6 time periods. The geological properties (well flowing pressure behavior) differ for each well of the reservoir. Furthermore, the cost coefficients vary not only for each well but also for each period of time. Seeking simplicity, here we only show the well flowing pressure profile for well 8 (Fig. 3) and the oil production (thousands of barrels per day) for each well in each period of time (Table 1). The solution time is 20.1 seconds in a Pentium III, 550 MHz. The objective function is $35.1 million dollar. 4. DISCUSSION We have developed a multiperiod optimization model for oil production planning in the wells of an oil reservoir. The problem is concerned with the decisions involving the oil production of the wells in each period of time. We consider logarithmic behavior of the well flowing pressure; however, by assuming that a well is either open to flow or shut in over a complete time period (of fixed length), the resulting model is a MILP problem. Currently, MINLP models which remove such an assumption and also consider the nonlinear behavior of the reservoir pressure are under investigation.
912
Oil Productionof Well8 T5 (thousandsof barrels~day) 8.34 53.33
t
t
~ /
34.21 I
Fig. 3
WellFlowingPressure(psia)for Well8
oo9/ oo
~ 9 7
1"~1573911~/~
v
Period
I ""15763~9 Period
Oil Production and Well flowing Pressure Profile of Well 8
Table 1. Oil Production of the wells of a reservoir (thousands of barrels/day) Period 1 2 3 4 5 6
Demand 233 150 100 100 150 200
Welll Well2 ..... 25.66 ..... 21.10 30 .......... 74.83 ..... 6.85
Well3 46.66
Well4 64.16
29.17 46.66 46.66
46.66
Well5 Well6 Well7 .......... 38.5 53.72 75.17 ..... 40.83 . . . . . . . . . . 75.17 24.79 40.83
Well8 58.34
53.33 34.21
5. A C K N O W L E D G M E N T S A. Ortfz-G6mez and V. Rico-Ramfrez would like to thank the financial support provided by the Mexican National Council for Science and Technology (CONACYT). REFERENCES Bohannon J., A Linear Programming Model for Optimum Development of Multi-Reservoir Pipeline Systems, J. Pet. Technol.22, 1429 (1970). Brooke A., D. Kendrick, A. Meeraus and R. Raman, GAMS- A User's Guide, GAMS Development Corporation, Washington, DC (1998). Home, R. N., Modern Well Test Analysis, Second Edition, Petroway Inc, Palo Alto, CA (1998). Iyer R. R., I. E. Grossmann, S. Vasantharajan and A. S. Cullick, Optimal Planning and Scheduling of Offshore Oil Field Infrastructure Investment and Operations, Ind. Eng. Chem. Res. 37, 1380 (1998). Sullivan J. A., Computer Model for Planning the Development of an Offshore Oil Field, J. Pet. Technol.34, 1555 (1982). van den Heever S. A. and I. E. Grossmann, An iterative Aggregation/Disaggregation Approach for the Solution of a Mixed-Integer Nonlinear Oilfield Infrastructure Planning Model, Ind. Eng. Chem.Res.39, 1955 (2000).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
913
Decision-making framework for the scheduling of cleaning/maintenance tasks in continuous parallel lines with time-decreasing performance Sebasti/m Eloy Sequeira, Mois6s Graells and Luis Puigjaner. Chemical Engineering Department, Universitat Polit6nica de Catalunya ETSEIB, Avinguda Diagonal 647. Pavell6 G, 2~ 08028-Barcelona, Spain
Keywords: Scheduling, Maintenance, Optimisation, Cape. The efficiency of equipment units decreases depending on the time they have been operating due to fouling, formation of subproducts, decay of conversion etc. Therefore, equipment maintenance and/or cleaning is periodically required to restore the original conditions that allow higher productivity rates. This poses a trade-off between the cost of equipment shutdown and the resulting productivity improvement. The problem of determining the maintenance frequency and the timing of the specific tasks in each equipment unit is studied a hierarchical way (planing and scheduling). A general approach is first introduced and next the implementation is presented. The solution approach proposed is based on a practical decision-making framework allowing both, the user driven what if analysis and the optimisation carried out by plugged modules. Finally, an example is shown and the results are discussed. 1. INTRODUCTION The problem addressed in this paper is quite a common one in industrial practice. The efficiency of equipment units decreases depending on the time they have been operating or the amount of different materials they have processed. Therefore, action is required to reestablish the initial performance level. The decision-making problem is determining when such actions are to be carried out, which leads to the trade-off between the cost of the action itself (resources required, production break, etc.) and the benefit it generates (productivity increase, reduction of operational costs, etc.). In the case of continuous lines, this situation is exemplified by catalysed reactors, evaporators, etc. The problem in catalysed reactors is catalyst progressive deactivation. In the case of continuous evaporation units, efficiency decreases due to different factors like fouling, deposits of insoluble materials, etc. Furthermore, the case of parallel lines introduces additional constraints such as keeping total processing rate and buffer tanks levels within certain limits. Several approaches have been proposed to cope with these problems using mathematical programming, which allows determining the optimal values for operation variables such as starting time and duration of cleaning operations. However, mathematical programming is usually a rigid approach strongly dependent on the way the models are formulated. This leads to simplified models for the physical behaviour of the equipment, the way they operate (cyclic schedules are forced, etc.) and the objective function.
914 The approach introduced in this work is the development of a general computer aided decision-making tool for the plant engineer. This tool has been conceived as a basic framework for managing different models and optimisation techniques as well as a basic user interface for addressing these kind of problems. Such a tool is designed for practical routine work including not only the optimisation procedures but also the "what if" analysis based on simulation as well as the upgrading of plant status information for subsequent rescheduling of cleaning tasks. The scheduling strategy adopted is based on the determination of the starting time and duration of the next cleaning/maintenance task (or tasks in the general case) for each processing line given the initial conditions for each one. This allows the use of the tool for scheduling/rescheduling purposes following a rolling horizon strategy. 2. GENERAL FRAMEWORK Consider an operation scheme as illustrated in Figure 1. The equipment units can be filters, catalytic reactors with decreasing activity, or any units with performance affected by time.
Figure 1: General system structure Each equipment unit will have an associated model, which represents numerically its performance as a function of time. There is also a global performance measure, which results from the individual contributions. The holdup tanks try to compensate the perturbations to and from the parallel lines, and can not be necessarily present. The problem statement is as follows: Given: 9 The material to be processed for a period of time under consideration 9 The equipment models, parameters and initial status 9 The individuals and global performances as time function Determine: 9 The frequency of cleaning 9 The material to be processed by each equipment 9 The time assignment for maintenance of each equipment" The solution of such a problem is usually presented under the form of mathematical programming [1 ]. The mathematical models include several variables and very often integer or binary variables. This kind of approach, for one hand, leads to complex models that are expensive in terms of development and calculation time (or the models are oversimplified for
915 that reason). On the other hand, the models are formulated for specific mathematical tools, which are hardly used or even understood by the common plant engineer. This work proposes to solve the problem using a hierarchical approach, dividing the problem and considering a more detailed view of its nature. Once the subproblems are identified, more practical and familiar tools can be used to address the decision-making problem. Hence, two different subproblems are identified. Determining how much material is to be processed in each unit before maintenance and which is the maintenance frequency to be adopted is a planning issue. In contrast, the timing of the specific maintenance tasks is related to scheduling. Such a problem fragmentation allows a practical and simplified problem solution. 3. IMPLEMENTATION Figure 2 shows the proposed approach. In first place the individual model is developed. It contains the equipment physicochemical behaviour and acts as a black box, returning some variable values when changes on parameters and/or decision variables occur. Secondly, the overall performance model, which evaluates the objective function using the information generated by the individual model applied in every equipment calculation. The solver is plugged then in the system for getting the best overall performance value. The same scheme is considered in both subproblems, scheduling and planning. But the input conditions for scheduling have been already optimised.
Figure 2: Proposed approach. 4. EXAMPLE The example considered consists of five parallel lines in the evaporation section of a sugar plant [2]. During the operation, the heat exchange capacity of the evaporators decreases with time due to fouling and solids deposition. Since the evaporators are fed with constant mass flow, the outlet concentration of each evaporator decreases. Therefore, system performance representing overall plant goals is assumed to be given by final concentration of the mixture in the product holdup tank. Following the guidelines of the previous sections, the different system components were developed. In first place, the evaporator model gives the output concentration according to its
916 parameters and feed conditions, the cleaning frequency and the time horizon. On second place, the objective function takes the information generated by each evaporator and calculates the overall plant performance.
Figure 3: User interface for the planing subproblem The components were written in Visual Basic for Application (VBA). Data are entered to an Excel spreadsheet [3]. The object-oriented features of VBA simplify the model development, while Excel allows entering data directly to a graph. This provides to the user an easy way for performing "what if?." analysis and interacting with tables and graphs. Additionally, a solver can be plugged to the system for solving the planing subproblem [4]. Figure 3 shows a screenshot of the spreadsheet and the colour convention used to distinguish parameters, and model inputs and outputs. Model parameters are heat exchange data (area, steam condition, etc.) and the main inputs are: 9 System feed condition (rate and composition) and fraction fed to each evaporator. 9 Time horizon and cleaning frequency for each evaporator. 9 Cleaning time required for each evaporator. The corresponding model outputs are: 9 Final outlet rate and composition for each evaporator. 9 Final rate and composition of the mixture in the product holdup tank.
917 A solution procedure may be run to obtain an improved problem output. A tailored heuristic can be used or standard solver may be plugged in the system to get an optimal solution [2] as well. In any case, the outputs are the values of the decision variables: time horizon and cleaning frequency and flow rate for each evaporator. Once the planning solution is established, the scheduling problem must be solved. Taking the time horizon, the cleaning time and the cleaning frequency for each evaporator, the time allocation for the cleaning operations is decided. The performance criterion adopted was the minimisation of the overlapping between different cleaning tasks. The reason encouraging the adoption of this criterion is double. On one hand, cleaning needs manpower and manpower overlapping is not desired. On the other hand, changes in product flow should be reduced, and a simultaneous cleaning will produce them. This problem has multiple degenerated solutions, which a solver would not distinguish. To avoid a solution depending upon the starting point, a simple heuristic (the stairway heuristic) was used to determine a most convenient solution. This heuristic assigns sequentially the cleaning tasks to the evaporators along the time horizon by starting a task as soon as the previous is finished. If overlapping can not be avoided, it is distributed in uniform way between all the maintenance tasks. The user is the one deciding which solution wants among all the degenerated ones by giving the sequence establishing how cleaning tasks must be dispatched. In addition the user may also enter the initial cleaning time if a delay must be introduced in the schedule. Figure 4 shows the user interface for the scheduling subproblem. Once the data is entered, the timing is performed automatically by the heuristic and displayed on a Gantt Chart.
Figure 4: User interface for the scheduling problem The user can also change the start and end time for the cleaning tasks, thus modifying the proposed solutions according to his own experience and information. Finally, a chart showing the resource consumption as manpower or storage needs was included as well for supporting this simulation capability. Figure 5 illustrates the level of the product holdup tank along the time horizon.
918
5. CONCLUSIONS This work introduces a practical way to solve the planning and scheduling problems of maintenance tasks in continuous parallel lines. This work shows how problems of such nature can be solved in a hierarchical way. Once the subproblems are defined, the solution procedure is simplified and tools more familiar to the engineer can be used to provide a decision-making framework for the routine work of the plant manager. An example of the application of such approach was presented, showing that this way to solve the problem offers a more practical solution to the engineers. The decision-making framework allows not only obtaining optimum solutions, but also the possibility to perform "what if?." analysis via its simulation capability. In addition, the graphical representation of information allows a fast comprehension and analysis of a problem that is formulated in a way closer to plant engineers and their every day needs. Furthermore, the practical significance of the solution procedure proposed is demonstrated by the application developed for a sugar plant. The enthusiasm the plant engineers showed for a tool they understand at the first sight shows that the tool is giving answer to a most practical question: when this evaporator should be cleaned? Finally, the modular approach adopted is easy for maintenance and for adaptability to new problems, to changes on models and objective functions, thus showing the great generality of the approach. Perfll de Navel
2 E+06
:~ 5 E+05 0E+00
-5 E+05 -1 E+06
~
j
i
~
___
1 I0 50 i '____i--i.......... _ ~ 80'____~f _ ......... -
.
,--,,
i_t .
.
.
.
.
.
.
.
Tmmpo
Figure 5: User interface for the scheduling problem Acknowledgements
Sebastian Eloy Sequeira wishes to acknowledge to the Spanish "Ministerio de Educaci6n, Cultura y Deporte" for his financial support (F.P.I. grant). References
1. V. Jain and I. Grossmann, "Cyclic scheduling of continuous parallel process units with decaying performance". AIChE Journal 44, 1623-1636, 1998. 2. S. E. Sequeira, H. Heluane, M. Colombo, M. Hernandez, M. Graells and L. Puigjaner "Scheduling of continuous parallel evaporators with decreasing heat transfer coefficient". L. A. AIChE meeting proceedings, November 2000. 3. E. M. Rosen, "A Perspective: the use of the Spreadsheet for Chemical Engineering Computations", Ind. Chem. Res., 39, 1612-1613, 2000. 4. S. E. Sequeira, M. L. Graells and L. Puigjaner, Synergic use of the available cape tools: an application to real time optimisation. L.A. AIChE meeting proceedings, November 2000.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B.Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
919
A Conceptual Optimisation Approach for the Multiperiod Planning of Utility Networks Alexandros M. Strouvalisa and Antonis C. Kokossis b* a Department of Process Integration, UMIST, P.O. Box 88, Manchester M60 1QD, UK. b Department of Chemical and Process Engineering, School of Engineering in the Environment, University of Surrey, Guildford, GU2 7XH, UK. The paper presents a paradigm of addressing large-scale optimisation problems using customised tools. Tailor-made techniques limited to specific applications are employed to effectively invest resources for solution searches. Integration of special logic is implemented at the modelling level with emphasis on the multiperiod planning of steam turbine networks. Logic extracted from conceptual preprocessing is employed to reduce combinatorial load. Logical constraints screen redundant degrees of freedom and customise the solution space to the specific application. The methodology decreases considerably the computational cost while offering a high degree of physical insights. 1. INTRODUCTION Previous efforts explain the importance of taking advantage of knowledge at the modelling stage. Floudas and Grossmann (1995) reviewed the methods for reducing combinatorial complexity of discrete problems employing logic-based models. Raman and Grossmann (1992) reported improvements in the solution of MINLP problems with a combined use of logic and heuristics. They illustrated their ideas in process synthesis problems (1993), employed inference logic for the branching of the decision variables, and studied the use of logical disjunctions as mixed-integer constraints (1994). Turkay and Grossmann (1996) extended the application of logic disjunctions to MINLP's using logic-based versions of OA and GBD algorithms. Hooker et al. (1994) applied logic cuts (as inequalities or by symbolic inference) to decrease the number of nodes visited in MILP process networks models. Integration of information with modelling is a validated contribution for improving computational efficiencies. Applications though usually comprise of algorithmic developments that lack insights and understanding. The use of logic supported primarily by conceptual analysis is alternatively propounded. The Hardware Composites (Mavromatis and Kokossis (1998), Strouvalis et al. (1998)), a graphical tool developed for the optimisation of steam turbine networks, is employed to support analysis of operations. The optimal solution space is visualised by the Hardware Composites. The graphs consist of linear segments (m and g) representing the steam turbine expansion paths. The most efficient expansion paths are first fully loaded before less efficient are used. Boiler operation is represented by zones (dashed lines). Each set of Corresponding author.
920 demands is directly traced on graphs (power output versus passout demand) as well as the optimum loading of turbines and boilers. 2. P R O B L E M DESCRIPTION
The problem considered is the multiperiod planning of utility systems with steam turbines and boilers. Given are demand patterns in power and heat, efficiencies of units and operational costs (fixed, incremental, changeover). Objective is satisfaction of demands with minimum cost. Units change modes over time to accommodate demands while allowing for efficient operations. The optimal planning is determined in terms of on/off status and modes of units (steam consumption/generation, power output). The horizon is divided into periods of equal length. The period is adequate to model start-ups or shut-downs for each turbine and boiler. Demands are considered constant over each period. The utility cost consists of unit (i) operational costs (fixed and incremental) and (//) transitional costs (start-up, shut-down costs). The optimal planning balances two trends: "coarse" profiles where each period is operated by the most efficient combination of units (stand - alone optimal modes) irrespective of preceding and succeeding periods. They usually involve several switch on and offs to accommodate the dissimilar status of units between periods. "smooth" profiles include less transitions but more operational costly modes. Changeovers are limited by employing more steady over time configurations of units at the expense of not necessarily using the most efficient unit combinations per period. 3. PROPOSED F R A M E W O R K
The proposed methodology features a blend of engineering knowledge, algorithmic methods and conceptual representations. Screening is accomplished through qualitative and quantitative assessment of operations. Options associated with promising trade-offs are embedded in operational superstructures while infeasible/inferior modes are made redundant. The operational analysis is performed according to the hierarchy:
A. Solution Space Analysis The full solution space of the Hardware Composites is reduced by selectively removing units from the graphical representations. Selection of units is justified by trade-offs between: (i) operational/transitional costs and (ii) fixed/incremental efficiencies of units. The analysis determines a customised selection of alternatives in view of the specific variation of demands.
B. Logic Extraction Demands are projected on the alternative solution spaces of Hardware Composites and their reduced versions. Periods requiring operation of specific units due to hardware limits are detected. By tracing demands over spaces, possible operating units per period and changeovers between periods are revealed. The identification of candidate configurations and transitions over time defines the connectivities of sequential modes. In that manner degrees of freedom are decreased and the operational superstructures are set up.
921
C. Integration of Logic in the Model Logic as extracted and included in superstructures is transferred to models by fixing / aggregating variables and imposing logical constraints. The model formulation is thus customised to the characteristics of the particular application in a user-defined way. The philosophy of the proposed approach is demonstrated qualitatively by Fig. 1. "Coarse" profile (i) corresponds to unit operations proposed by Hardware Composites. Each period employs the units of the domain where the set of demands locates. Depending on distribution of demands on the Hardware Composites profile (i) reflects the transitions over time. Qualified reduced Hardware Composites give rise to alternative unit combinations defined, for example, by profiles (ii), (iii), (iv) which present "smoother" planning options. Any of the four profiles or a combination of them may stand for the optimal planning.
/--..._. r-HC(in) S
r-HC0i)
/ ..._.,/ Periods
. . . Periods
.
/
I~
\ .
r-----,--~,
. . . Periods
\
%
.......
~r-HS!i2!.. l
/ .
,\
Periods
.
.
Ib
Periods
9
Fig. 1. Alternative solution spaces generate alternative planning profiles. 4. ILLUSTRATIVE EXAMPLE The network of Fig. 2(a) features 3 backpressure passout turbines (T1, T2, T3) and 3 boilers (Bl, B2, B3). Operational horizon includes 30 periods each associated with a set of demands in power and heat. Power is given in power units [PU], heat in passout units [SU] and costs in cost units [CU]. The Hardware Composites of the network are shown in Fig. 2(b). They include all units and projected demands (P1 to P30) on the full solution space. Each domain involves operation of specific units. For a transition Pt to Pt+l the units in operation and possible start-ups and shut-downs are identified. Trade-off analysis justifies the alternative reduced space which excludes turbine 3. The sole operation of Tl and T2 introduces less transitions and competitive planning profiles. The reduced Hardware Composites are presented in Fig. 2(c). The spaces under consideration generate the planning profiles for turbines and boilers shown in Fig. 3(a) and (b) respectively. They encompass only feasible modes interconnected according to permissible changeovers.
922
Fig. 2. (a) Utility network, (b) Hardware Composites, and (c) Reduced Hardware Composites for illustrative example. Transition from P 1 to P2 is presented for illustration. According to solution spaces of Fig. 2(b) and (c), P1 requires units T1 and T2 while P2 only T1. These modes are embedded in the operational superstructure of Fig. 3(a) in addition to the stand-by mode of T2 in P2 (T2 is again operational in P3). Similarly, P1 and P2 are served by boilers B1 and BE or B3 (Fig. 2(b), (c)). Alternatively P2 is operated by B1 (Fig. 2(c)). Boiler modes and their allowed interconnections are shown in Fig. 3(b). Following the same procedure the entire operational superstructures are shaped.
4.1. Computational Results Two approaches are examined: (a) full solution search with elementary implemented knowledge and (b) reduced solution search based on operational superstructures. The first experiment navigates the entire space; search is restrained only by basic feasibility constraints imposing a minimum number of operational units per period. The second experiment performs search on degrees of freedom embedded in superstructures. The sizes of the MILP models for
923 each case are presented in Table 1. The increased number of equations in (b) represents the expense of transferring logic to the model. Fixing and aggregation of variables reduces continuous and binary variables. ~
LI3
T(I-2)
B(I-2)
TI
BI 11 1 2 3 4 5 6 7 8 9 1111112131415161718192021222324252627282930 E (a)
. ~____ / -.
.
__ T(I-2)
~. - -
-. . . . . . . . . . . . . . . . . . . . . . . . .
~)
-
.
-
.......
9. . . . . . .
(b)
_ . . ... . . . .
.
~
~ ~ - ---
~
-
. . . .
0 1 2 3 4 5 6 7 8 9 1111112131415161718192(1212223242526272829311E
PERIODS
~~.Z~"~T(I-2-31 _ _ I_
Lal
. . . . .. .
O Stand-by mode
B(I-2-3)B(I.3)
-~
B(I-2)
Tl
2f~__f'f~i-'~--~-i-i~_"/i!fi:~"f-~ii-i PERIODS
. . . . . . . . . . . . . . . . . . . . . . . .
Bl 11 1 2 3 4 5 6 7 8 9 111111213141516171819211212223242526272829311 (C)
E
Ii1 2 "~ 4 5 6 7 8 9 101112131415161718192021222324252627282930 E
PERIODS
(d)
PERIODS
Fig. 3. Operational superstructures for (a) turbines, (b) boilers and optimal profiles for (c) turbines and (d) boilers. Table 1: Sizes of models for illustrative example.
Full search (a) Reduced search (b)
Continuous Variables 756 679
Binary Variables 180 44
Constraints 1,382 1,789
Non-zero Elements 4,037 5,128
Optimisation is performed with application of OSL solver of GAMS. After the limit of 1,000 CPU (sec) the Branch-and-Bound solver did not find the optimum solution. Alternatively, the solution of (b) was more efficient. Table 2 presents the computational results and Fig. 3(c) and (d) the proposed operational planning profiles. Table 2: Computational results for illustrative example. Iterations
Nodes
CPU(sec) 200 MHz PC
Full search (a)
52,629
7,787
Reduced search (b)
3,349
1,111
1,000 (interrupted) 104
Objective Value [CU/Oper. Horizon] (Relaxed Objective) 529,999.3 (381,397.9) 480,861.5 (454,842.6)
Under full solution search considerable CPU resources were spent for obtaining suboptimal operations while reduced search revealed the solution in orders of magnitude less effort. Combinatorial complexity, relaxation gaps and flat objective function profiles are major reasons
924 why full search was inefficient. The solver was trapped in suboptimal operations and had to visit many nodes to fathom the binary tree. 5. CONCLUSIONS A hybrid scheme is introduced for extracting and implementing conceptual knowledge to the solution search. The analysis identifies possible unit transitions (start-up and / or shut-down) between successive periods eliminating redundant enumerations. Operational superstructures are built involving scenarios associated with promising trade-offs. Variables are fixed and aggregated and logical constraints are imposed to adapt the search to the reduced space. The merits of the approach are not only computational. Deep insights offer overall perspective of operations. The user can study the importance of units to operations, the impact and range of trade-offs in view of alternative configurations and most important of specific demands over solution spaces. Instead of an automated algorithmic screening technology, engineering understanding is employed to drive the selection of degrees of freedom and customise the model to the particular problem. REFERENCES 1. Floudas, C.A. and Grossmann, I.E. (1995) Algorithmic approaches to process synthesis: logic and global optimisation, AIChE Symposium Series 304, Fourth International Conference on Foundations of Computer-Aided Process Design, 91, 198. 2. Hooker, J.N., Yan, H., Grossmann, I.E. and Raman, R. (1994) Logic cuts for processing networks with fixed charges, Computers and Operations Research, 21 (3), 265. 3. Mavromatis, S.P., and Kokossis, A.C. (1998). Hardware Composites: a new conceptual tool for the analysis and optimisation of steam turbine networks in chemical process industries Parts I & II. Chemical Engineering Science, 53(7), 1405. 4. Raman, R. and Grossmann, I.E. (1992) Integration of logic and heuristic knowledge in M1NLP optimisation for process synthesis, Computers and Chemical Engineering, 16(3), 155. 5. Raman, R. and Grossmann, I.E. (1993) Symbolic Integration of logic in Mixed-Integer Linear Programming Techniques for process synthesis, Computers and Chemical Engineering, 17(9), 909. 6. Raman, R. and Grossmann, I.E. (1994) Modelling and computational techniques for logic based integer programming, Computers and Chemical Engineering, 18(7), 563. 7. Strouvalis, A.M., Mavromatis S.P., and Kokossis, A.C. (1998). Conceptual optimisation of utility networks using hardware and comprehensive hardware composites. Computers and Chemical Engineering, 22, S 175. 8. Turkay, M. and Grossmann, I.E. (1996) Logic-based MINLP algorithms for the optimal synthesis of process networks, Computers and Chemical Engineering, 20(8), 959.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
925
An integrated environment for batch process development- from recipe to manufacture Venkat Venkatasubramanian, Jinsong Zhao, Shankar Viswanathan, Chunhua Zhao and Fangping Mu School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA Peter Harper, Boris Russell Computer Aided Process Engineering Center, Technical University of Denmark, Lyngby, DK Batch process development involves the process of converting a chemical synthesis into an optimum, safe, robust, and economical process for manufacturing the chemical of desired quality at the ultimate desired scale. In this paper we describe a strategy for developing a set of integrated decision support tools to facilitate this activity. The primary motivators for doing this is to perform fast, and high quality process development since there is significant economic leverage that can be gained by speeding up this activity while not compromising on quality. The application of the strategy to an industrial case study is also presented. 1. I N T R O D U C T I O N In the specialty chemicals, pharmaceuticals, agrochemicals, and advanced materials sectors the process of successfully bringing a product discovery, which usually takes the form of a new molecule or formulation, to the marketplace is complex, expensive, time-consuming, as well as information, knowledge, and computation intensive. The information flows between the various stages of this process are dynamic and voluminous, the level of uncertainty in the underlying technical and commercial data high, the technical specifications and information base which must be developed to support production very extensive, and the decisions typically highly combinatorial. Thus, it usually takes an enormous amount of time, effort and money to develop a new product. In these businesses the first to market captures the advantage of establishing a market position which is difficult for later entrants to erode. Given the limited patent life of the discovery, the shorter the time from discovery to manufacture the longer the time during which revenue generation can be protected. This is a major consideration: in the case of a successful pharmaceutical, a day gained or lost in the commercial life of a product typically represents $1 million in net revenue. Thus, the really compelling drivers, for the development process, are speed to market and getting it right the first time. As Gary Pisano[ 1] notes "Although superior process development can reduce manufacturing cost, in itself an important dimension, the power of process development more often lies in how it helps companies achieve accelerated times to market". These are therefore key motivators for developing integrated computer decision support Corresponding author. Tel: 1-765-494-4083.Email:
[email protected] 926 tools for batch process development activities because they can offer significant advantages. By increasing speed of development these tools would help focus key scientists on innovation, save resources for doing more value added activities, allow teams to work on more projects simultaneously, permit companies to wait longer if necessary to initiate a project. Integrated tools will help facilitate communication between team members in a multi-disciplinary team, integrate their knowledge by serving as bridge between different paradigms. Use of these tools would minimize "eye-lash" learning and facilitates faster integration of new engineers/chemists into teams. Integrated development tools would foster standardization of information by sharing data across the company and help standardize practices across the company from R&D through manufacturing. The aim of this work is to propose a systematic framework to provide fully automated decision support for batch process development activities from recipe to manufacture and develop a prototype integrated workbench based on this framework and demonstrate its use on an industrial case study. 2. STRATEGY The strategy that we have adopted towards developing integrated process development tools consists of the following steps: 1. Systematically identifying different stages of a process development pipeline. 2. Developing an information flow and functional interaction model between different aspects of the pipeline. 3. Identifying/developing pipeline management tools and methodologies for different stages of the pipeline. 4. Developing techniques that would allow integration of these tools based on the information and flow interaction model. The batch process development pipeline can be viewed as consisting of the following aspects - process alternatives synthesis, process alternatives evaluation, process operations development, and process operations assessment. Each of these in turn entails several sub-functions. Process synthesis involves sub-functions such as route selection, flowsheet selection, and solvent selection. Process evaluation consists of sub-functions such as preliminary scale-up, material and energy balance calculations, costing, resource planning, capacity planning, and sensitivity analysis. Process operations development entails development of batch records. Process operations assessment includes sub-functions such as process hazards analysis, vent sizing, and emission calculations. There is a lot of interaction between these activities in terms of data sharing and information transfer. Current automated systems that provide support for these activities routinely run into formulation, calculation, and analysis bottlenecks because of inadequate support for the flow and transfer of information across these activities. Also there is a predominantly sequential view of the process development activities. This is however restrictive because traditional downstream activities can provide insights that can be used to modify decisions that were made in traditional upstream activities. For e.g. results from process hazards analysis (downstream activity) can be used to modify decisions made during flowsheet or solvent selection (upstream activity). It is therefore necessary to support bi-directional flow of information to provide a fully integrated set of decision tools. The information flow and functional interaction model shown in Figure 1 captures the informational and functional interactions between the process development activities identified.
927
Figure 1 Information Flow and Function Interaction Model Table 1 Pipeline Management Tool ProSyn ProPred ProCamd DATA ReacDef TMS PAS Simul and DynSim BRIC ITOPS BatchHAZOPexpert
Tools Purpose Generation of process alternatives Pure component property prediction Design and selection of solvents and process fluids Collection of measured data for parameter estimation Definition of reaction kinetics information Thermodynamic model selection Process Analysis System Steady state and Dynamic simulation systems Simulation of batch records Intelligent Tool for Operating procedure synthesis Hazard and operability analysis
We propose a solution strategy based on total integration of processing systems that uses the two software systems - ICAS[2] and PHASuite[3,4,5,6,7]. ICAS or Integrated Computer Aided System addresses the process synthesis and process evaluation aspects of batch process development, while PHASuite addresses the operations development and assessment aspects. The different tools within these two systems and the process development aspects they address are shown in Table 1.
928 These tools share information and data to provide contiguous support during the process development lifecycle as per the function and information flow interaction model shown in Figure 2. This paradigm requires less data collection effort during the various activities and thereby ensures data consistency across the process development lifecycle. This paradigm therefore provides a better alternative to batch process development resulting in enhanced productivity. For e.g. the function and information flow interaction model dictates that iTOPS and BatchHAZOPexpert need to share both information and function as they address operations development and operations assessment aspects of process development. The detailed function and information flow across these tools is shown in Figure 2.
Figure 2 Function and Information Flow between iTOPS and BatchHAZOPexpert The critical aspects of the interaction between the two tools that need to be observed is that the overlapping information about materials, equipment, and chemistry is pulled in from a database. This ensures consistency of information and avoids redundant input of information. Also the operating procedures, another piece of input, necessary for doing safety analysis inside BatchHAZOPexpert is generated within iTOPS by invoking its functionality. Similarly safety critical information about different operations that needs to be incorporated inside the operating instructions is generated by invoking BatchHAZOPexpert's functionality and by feeding back the information to iTOPS. This illustrates how iTOPS and BatchHAZOPexpert share information and function to perform operating procedure synthesis and process hazards analysis more efficiently and effectively. This paradigm is being expanded to include the entire range of process development tools and is being tested on an industrial case study. Some of the results for this case study are highlighted in the next section.
929
3. CASE STUDY RI
R2
Toluene
Solvent_ 1
MeOH
Cat
Cat/Waste
Bot 1
Re action 1 RI+R2 Pt v i i Reaction 2 C+MeOH ->E
RI + LI ---~ A II---~C + H 2
C+II-~
Atmospheric Distillation 1
I2
Vacuum Distillation
Int _ 2
Int _ 1
12 --> C + I~
Pt --~ Pt *
H20
Solvent 2 ~ ~ . . . . Reaction 3 2E +H 20 ->D+2 MeOH
Atmospheric Distillation 2
Product
Int 3
0, this indicates that there are no plant capacities within the given ranges for which the system can be operated feasibly over the whole ranges of the uncertain supplies and demand, even with the manipulation of the control variables during operation. Four expressions are obtained for the flexibility index, corresponding to ~ l = 0, ~3 = 0, ~4 = 0 and ~7 __ 0. These are all independent of the processing capacity of plant 3, as seen in Fig. 3, which graphically illustrates the parametric expressions in the design space.
965 2 kW/K
1.2 kW/K ~< 0 ~< 1.6 kW/K
TI
563
K
323 K
Figure 4. Non-Convex Model Example.
Figure 5. ~ vs. O.
3.2. Non-Convex Problem Fig. 4 shows a heat exchanger network [9] where the heat capacity flowrate, 0, of one of the hot streams is uncertain. Solving the feasibility function problem for this example as a nonconvex mp-NLP with a tolerance of e = 0.05 leads to a number of linear expressions for ~(0). The resulting feasibility function values lead to a predicted flexibility test measure, Z = 0.7549 at the non-vertex point, 0 - 1.3831. Note that since ~ > 0 in the whole range of 0 considered, the flexibility index is zero. For this particular example, the analytical solution for ~ can be derived. Fig. 5 plots the predicted feasibility function from the application of the parametric programming framework against 0, and compares it with the actual solution. It can be seen that the parametric expressions do indeed over-estimate the real-solution to within e - 0.05 for the whole range of 0. The flexibility test measure is also over-estimated within this tolerance since the analytical value is % -- 0.7077 occurring at 0 - 1.3982. 4. E X T E N S I O N TO M U L T I - P U R P O S E P R O C E S S E S A multi-purpose system such as an in-line blending system, used to make a range of product recipes, can be described by the following model [ 10]:
hm(x,z, yz,O,d,y)
:
0,
m C M,
(3)
g l ( x , z , Yz,0,d,Y)
/
1 GetSolution0S~Interface0||| "'n> Destroy() / /~
+lowerBound/I
/~upperBound
~/ CapePublic=,Paramet .y) er(.omI +name~1~CapeStriBe.ng(.e)om
owns
~176
I§ description
+currentValue~ ~ +defautValue CapeVariant (fromBase)
DAE
ICapeNumericLASolver
ICapeNumericNLASolver SetCvgTolerance0 GetCvgTolerance0 SetMaxlterations0 GetMaxlterations0 DoNIteration0
ICapeNumericDAESolver SetRelTolerance0 GetRelTolerance0 SetAbsTolerance0 GetAbsTolerance0 AdvanceToNextEvent(
)
3. NUMERICAL SERVICES PROVIDER" Implementation Our Solver component compliant with CO version 0.9.3 is called NSP, Numerical Services Provider. According to the component scheme introduced in 2.1, this software realises the Solver package depending on the Model and Eso packages. Following the CO architecture for CORBA system, NSP acts as a numerical server through the Object Request Broker, employs the Identification Common Interface and follows the CO Error Handling strategy. It stands for the second stage (the business model) within the three-stage architecture. Obviously, this stage is a separate process that can be used on a single machine, or across the network within an internet-based enterprise business system. The NSP application combines Java and C/C++ codes as well as Fortran 77 legacy codes thanks to the wrapping technique. It supplies the LAE, NLAE and DAE objects, jointly to the configuration parameter objects. 9 The linear algebraic solver allows to solve the system A. x = b and is the result of the wrapping of the UMFPACK solver [9]. It is really efficient and offers a large choice of configuration.
971 9 The non linear algebraic solver deals with the system F(x)= 0. It relies on the Newton~F ~ k~ Raphson algorithm and profits from the linear solver for solving -~(ox j=-F(xk). 9 The differential algebraic equation solver manages the system
F t,x,-~ - 0 . It wraps
the integrator DISCo [10]. Its strategy is based on the Gear method with variable order and step. About the NSP design there are classes which implement the CO interfaces: SolverManager, Solver, LASolver, NLASolver, DAESolver, Identification, CapeSolverType and CapePublicParameter. These objects are distributed thanks to CORBA technical classes which manage the communication and put in place a delegation implementation through the tie mechanism. In order to decouple these classes fixed by the CO specification from the semantic implementation classes, the bridge design pattern is applied. Then we have in parallel the CO classification of Solver class (this "abstraction" side defines the CO higher-level methods) and our own classification of SolverImpl class (this "implementation" side provides only primitive methods). The SolverImpl hierarchy keeps the same base classes structure of Solver hierarchy in order to accelerate the development but a more detailed classification of solver could be set up. This bridging allows us to set up our own design. It decouples the objects implementing the CO interfaces and our semantic objects and improves extensibility, independently ensuring the extensions of the two hierarchies. Keeping the same approach, the model side and the solving side are fully disconnected thanks to the SystemMap concept. The resulting SystemMap objects are in charge of the appropriateness and adaptation of models into the solver formats. They manage the storage of matrices and the variables (unknowns x and independent variable t). In consequence the SolverImpl object communicates with the Model object only through a SystemMap object. The main advantage is that the SystemMap objects factorise the code guaranteeing the link with the Model and Eso components. The SolverImpl object uses an ErrorsManagement object in order to generate a CO error from an error specific to a solving algorithm. It can also produce a log file, aiming at administrating NSP. The NSP application acts as a real framework. It is a set of co-operating classes that make up a reusable design for disseminating any solver algorithm through the CO standard. It dictates the overall structure taking advantage of the CO architecture. Especially through the bridge and the map concepts, the resolution method developer (from legacy codes or not) inserts his design within the SolverImpl hierarchy and needs only to interact with the NSP objects. The CO standard life cycling has no direct effect on his new and former objects. Moreover the comprehension of the CO Numerical Solvers specification would be not required. The developer can concentrate on the specifics of his business. 4. APPLICATION USING THE NSP SOFTWARE COMPONENT: Validation
A PME application which is compliant with the CO Numerical Solvers uses the NSP in order to solve its process mathematical model. It relies on the CO architecture for CORBA
972 system. In fact it corresponds to the first stage and has to supply the Model and the Eso components to the NSP. This basic PME application illustrates the NSP component through three test cases. 9 A linear system has been generated and solved. For all those values the accuracy criterion has been satisfied. 9 An isothermal flash model has been calculated using the NSP services in order to solve the resulting non linear system. 9 The Raleigh distillation has been performed using the NSP services in order to solve its differential and algebraic equation system. 5. CONCLUSIONS The current CO Numerical Solvers specification is introduced. The first complete implementation of this standard leads to the NSP software component which can provide numerical services to any CO compliant software. Its realisation validates not only the CAPE business interfaces but also the overall architecture for a CO process simulation. GLOSSARY UML: Unified Modelling Language DCOM: Distributed Component Object Model NLE: Non Linear Equations ESO: Equations Set Object PMC: Process Modelling Components
NSP: Numerical Services Provider LAE: Linear Algebraic Equations DAE: Differential Algebraic Equations STN: State transition Networks PME: Process Modelling Environments
REFERENCES
1. CAPE-OPEN standard: www.global-cape-open.org 2. B. L. Braunschweig, C. C. Pantelides, H. I. Britt and S. Sama, Open software architectures for process modelling: current status and futures perspectives, FOCAPD, 1999. 3. C.C. Pantelides and H. I. Britt, Multipurpose Process Modeling Environments. In L.T. Biegler and M.F. Doherty (Eds.), Proc. Conf. on Foundations of Computer-Aided Process Design '94. CACHE Publications, Austin, Texas. pp. 128-141, 1995. 4. J. Rumbaugh, I. Jacobsen and G. Booch, Unified Modeling Language Reference Manual, Addison Wesley, 1997. 5. Object Management Group's CORBA/IIOP: www.omg.org 6. Microsoft's COM: www.microsoft.com 7. M. Avraam, N. Shah and C.C. Pantelides, Modelling and Optimisation of General Hybrid Systems in the Continuous Time Domain, Comput. Chem. Engng., 22S, $221-$228, 1998. 8. A.W Westerberg, , H.P. Hutchinson, R.L. Motard and P. Winter, Process Flowsheeting. Cambridge University Press, Cambridge, U.K, 1979. 9. I.S. Duff, R.G. Grimes and J.G. Lewis, User's guide for the Harwell-Boeing sparse matrix collection, Release I, Technical report, Rutherford Appleton Laboratory, 1992. 10. A. Sargousse, Noyau num6rique orient6-objet d6di6 /t la simulation des syst6mes dynamiques hybrides, PhD thesis, INPT, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
973
Multiplicity and stability of CSTR-Separator-Recycle Systems Costin S. Bildea, Alexandre C. Dimian and Piet D. Iedema University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands The nonlinear behaviour of several CSTR-Separator-Recycle systems is studied by rigorous application of the singularity theory. The plant Damkohler number (Da) is introduced as a parameter of the dimensionless balance equations. A feasible operating point exist iff Da > Da er, where the critical value Da cr corresponds to a singular point of the model equations. For one-reactant nth-order isothermal reaction, the feasible operating point is unique and stable. In other cases (second-order isothermal reaction involving two reactants; first-order adiabatic reaction), multiple steady states exist, the range of achievable conversion being limited by the instability of the low-conversion state. 1. INTRODUCTION The nonlinear behaviour of stand-alone chemical reactors, including state multiplicity, isolated solutions, and sustained oscillations, have been demonstrated by a large number of articles 1~ In all cases, the source of nonlinearity was the dependence of reaction rate on temperature, coupled with some form of energy feedback. Recently 5, we have shown that coupling chemical reactors and separation units through material recycles is an additional source of nonlinearity. This work extends previous results by including stability assessment and considering non-isothermal reactor operation. 2. DYNAMIC MODELLING For the chemical reactor, a dynamic model can be derived based on unsteady mass and energy balance. The model contains a few nonlinear differential equations, being amenable to analytic or numerical investigation. In the case of an isothermal CSTR, the mass balance for the k th chemical species has the following dimensionless form:
Z(fj'Zk,j)--fout'Zk--Da" HZ,
dZk = dt j~inlet
streams
(1)
i~reactants
where t, f a n d z are dimensionless time, flow rates and concentration, respectively. For reactors in recycle systems (for example, Figure 1), it is convenient to use the plant Damkohler number 5 which includes the reaction constant (k), reactor holdup (V), feed flow rate (F0) and concentration of the key component (CA,0): Da = kc~,~ V / F o .This definition is different from the classical one, which uses the flow rate and composition at reactor inlet (F1 and CA,l) as reference values. Dynamic modelling of the separation units is more difficult. Even a simplified dynamic distillation model might contain about one hundred differential equations. For such model, analysis and generalization of the results is not easy. For this reason, we consider a separation section where all separation units are lumped. We assume that the separation section is under
974 local control, which is achieved by manipulating internal flow rates or heat duties. This way, the composition of the outlet streams is kept constant. Then, changing the flow rate or composition of the inlet streams is reflected by a gradual change of the flow rate of outlet streams. When the recycle streams are considered and complete reactant recovery is assumed, a simple model describing the dynamic behaviour consists of a first-order differential equation: ~'kZ~
Tdf~
=
fin Zin,k -- Lut,k Zout,k
(2)
where v, f and z are dimensionless time constant, flow rate and concentration, respectively. The index k refers to the k th component, Zout,k is fixed, and fn, Zin,k are determined by the reactor performance.
3. ISOTHERMAL CSTR- S E P A R A T O R - RECYCLE 3.1. One reactant, nth-order reaction This section considers a n th-order reaction A --> B, taking place in an isothermal CSTRReactor-Separator system (Figure 1). The model has two steady state solutions: (ZA,2,f3)~ =(ZA,3,oO) and (zA,2,f3): = , x / ~ , z A , 3 . , ~ - ~ _ 1 The first solution is unfeasible, corresponding to infinite reactant accumulation. The
Da
(4) second solution is feasible (positive recycle flow rate) when z~,3 9 > 1 Figure 3 presents the conversion of the two solutions vs. Damkohler number. The stability of a steady state changes when one real eigenvalue or a pair of complex-conjugate eigenvalues crosses the imaginary axis, corresponding to one of the following conditions, respectively:
det(Js.s,2)=n(Za,3""~Sa-1)=O
(5)
~ffs " Z A,3
(6)
trace(Js.s, 2)= 0, or, equivalently, r, = -
where Js.s. is the Jacobian matrix evaluated at steady state Eqs. 5 and 6 show that the operating point satisfying Eq. 4 is stable (solid line in Figure 2). A, ZA,3
05
l_)i
ZA,3
X
o -05
Z B,4 = 1
-1 1
v_
Figure 1. One-reactant, nth-order reaction
2 D a
3
4
5
( z A3) n
Figure 2. Stability of steady solutions in CSTR-SeparatorRecycle system
9'/5
A fee_~._._~,~ ~-~1
A recycle , ZA3) ,
B recycle
{CS 1
I I
1
A recycle (f3,ZA,3) _
tcs
/
~fe~d~..,,>
_
2
(&oo..)
=
A , ZA,2, ZB,2
SP=fv,~,s
e~, ~ f ~ ~ f5, ZB,5)
Figure 3. Two-reactants, second-order reaction in CSTR-Separator-Recycle system
3.2. Two-reactants, second-order reaction This section analyses the second order reaction A + B ~ P , taking place in an isothermal CSTR-Separator-Recycle system. When the reactants are completely recycled, feasible operation is possible only if the ratio of reactants in the feed matches exactly the reaction stoichiometry. For this reason, only one reactant feed may be on flow control (fA,0=l), while the feed flow rate of the second reactant (fB,0) must be used to control its inventory. Two possible control structures 6 are presented in Figure 3: one recycle stream or the reactor effluent, respectively, on flow control. The dynamic model includes reactor and separation equations, as well as the relation for the feed flow rate of second component imposed by the control structure: CS 1:fB,0 = fRec,a -- f5
(7)
CS 2:fa,0 = fz --0 + f3 + fs - D a ' z a , 2 " z,,2)
(8)
In both cases, two steady state solutions are possible. They have complex analytical expressions, not reproduced here. The conversion of the key component, Xa, is presented in Figure 4 for the case of high purity separation (ZA,3 = ZB,5 = 1 ). For given values of the fixed flow r a t e VRec,B orj~) and separation performance (ZA,3 and ZB,5), two feasible steady states exist when the plant Damkohler number exceeds the critical value corresponding to the turning point of the D a - X A diagram. 08-
cs l]
CS 21 0.8
06-
0.6
04-
0.4
i
02- ~.=~o~,\,,,. " .... 0
[,
0
'
5
-"--:-~-..=" U..',~;...... ;..:..,;~, . . . . . . ; ....................
10 D I 5
i
0.2
29
2
0
o
,
,
5
10
D
,
,
15
20
25
Figure 4. Multiple steady states of two-reactants, second-order reaction in isothermal
CSTR-Separator-Recycle system.
The critical value D a cr represents a limit point 7 of the balance equations. Then, the following feasibility conditions (existence of steady states) can be derived:
976
CS 1" Da > Da ~r = 4
Dacr fRec,B ZA,3 9ZB,5(fRo~,B__ 1)' and CS 2: Da > -
4 ~ ZA,3"ZB,
( f2 ) 2 5
(9)
f2-1
We emphasize that a recycle system designed near Da cr (an optimisation procedure is likely to suggest this!) can suffer from serious operability problems. If the reaction kinetics is over-estimated, or the feed flow rate deviates from the nominal design value, the operating point falls at the left of the turning point in the D a - XA map, in the region where no steady state exists. In this case, infinite reactant accumulation occurs, and the plant has to be shut down. This situation was observed by dynamic simulation using a more detailed model 6. In Figure 4, the lower steady state is unstable and has an unusual behaviour: larger reactor gives lower conversion. The instability can be proven based on steady state considerations only, showing that the analogue of CSTR's slope condition is not fulfilled. Note that the lowconversion state is closed-loop unstable. Moreover, it is independent on the dynamic separation model 9Because this instability cannot be removed by control, operation is possible when the following requirements, necessary but not sufficient, are met: CS 1" X A >
1
and CS 2: X A >
ZB,5 9fR~.B + 1 '
2
(10)
2 + ZA,3"(f2 --1)
A dynamic model is needed to prove the stability of the upper solution branch, which is not guaranteed by Eq. 10. More precisely, differences in the dynamics of reactants' recycle might lead to oscillatory behaviour, because of the violation of a dynamic stability condition. Because analytical computation of the eigenvalues of the dynamic model is difficult, we had to recourse to numerical methods. For the simple separation model presented here and a wide range of parameter values, time-dependent solutions converged to the upper steady state. Numerical computation revealed negative eigenvalues of the linearised model. Direct methods for computation of Hopf bifurcation points failed to find a solution. Similar results were obtained for other dynamic separation models (for example, series of first-order elements, or time constants dependent on the feed flow rate). Although all these results indicate that the upper steady state is stable, we do not exclude, for other dynamic models of the separation section, the possibility of oscillatory behaviour on the high-conversion branch of Figure 4. The lower limit of the conversion achievable at a stable operating point decreases as the flow rates increase (Eq. 10) and Da cr increases (Eq. 9). There is, however, a lower limit of the reactor size that can be used in a recycle system, given by D a . ZA,3 "ZB,5 > 4. 4. A D I A B A T I C C S T R - S E P A R A T O R - RECYCLE
Occurrence of multiple or unstable steady states of chemical reactors can be explained by the dependence of reaction rate on temperature, coupled with some sort of heat feedback. These undesired phenomena are avoided in practice by a suitable control system, for example manipulating coolant flow rate to keep constant reactor temperature. However, many industrial reactors are operated adiabatically. In this case, although the inlet temperature is controlled, state multiplicity and instability are possible. This section considers a first-order reaction, taking place in an adiabatic CSTR (Figure 1). The steady state dimensionless model of a stand-alone reactor is2: -X+Da*.O-X).ex
p I+7.B,.x)=O
(11)
977 where the conversion X is the state variable. Activation energy (y), Damkohler number (Da*) and adiabatic temperature rise (B*) are dimensionless model parameters. Eq. 11 admits three solutions for yB* > 4(1 +B*). Because usually B* has small values, an approximate criteria for state unicity is yB* < 4. When the reactor is coupled with separation throut~h recycle, the (reactor) Damkohler number (Da*) and (reactor) adiabatic temperature rise (B) depend on the state variable X. For this reason, they have to be replaced by dimensionless parameters containing only independent variables. This can be achieved using the flow rate and concentration at plant inlet as reference values in dimensionless parameters. Hence, we introduce the plant Damkohler number (Da) and plant adiabatic temperature rise (B). Then, the following equations can be derived:
Da* = Da
X . ZA,3 . I_XI(I_zA,3) B*= B zA'3 ' l _ y . (I_ZA,3)
(12)
The steady-state model of the adiabatic CSTR-Separator-Recycle system is obtained by combining Eqs. 11 and 12. If the feed and recycle streams have the same concentration (ZA,3 = 1 ), the model reduces to:
f (X, Da, B, 7") = - X + Da . X . (1 - X). exp
B X
In addition to the trivial solution X = 0, Eq. 13 admits one or three solutions. An analytical expression is not possible, but the defining condition for a limit point singularity v, g = Og/OX = 0, has two solutions (X, Da), indicating an S-shaped X vs. Da diagram and at most three steady states. The cusp singularity 7, where the two limit points disappear, is located in the unfeasible parameter range:
(X, Da, B ) = ( - 2
y+2 y+4 y+4 / (14) ZA,aY-- 2?' --4' ZA.aYexP0' + 2)' 4 A part of the S-shaped X vs. Da diagram corresponds to negative conversion, which has no physical meaning. Hence, the number of feasible steady states changes when one of the limit points enters the region of positive conversion. This is a boundary limit point 7. Its defining condition, g = Og/OX = X = 0, has the solution:
(X, Da, B)= (O,1/z Aa,3,1/7")
(15)
Compared with the stand alone adiabatic CSTR (TB* 1, zero, two or one steady state exist for
Da < Da r Da cr < Da < 1/ZA,3, and Da > 1/Zh3, respectively. When fly < 1, a unique steady state exists for Da > 1/ZA,3 . Second diagram of Figure 5 shows the conversion vs. adiabatic temperature rise, for different values of the plant Damkohler number. When Da > 1/zA,3 , one
978 r=25 I
1
~
08
"
0.6 X 0.4
X
iiI
,
///o.,;",0
// ',,,,
02 "'" ":':'-: "[:'" ." ", 0
0.5
Da
B=0 04=11), 1
,
1.5
-0.1
-0.05
o
',,, o5
"4
0
0.05 B
".-
.....
"
0 1
0.15
. . . .
0.2
0.25
Figure 5. Bifurcation diagrams of adiabatic CSTR-Separator-Recycle system
steady state exists for any B value. When D a < 1/ZA.3 , there is a minimum value B cr of the adiabatic temperature rise for which two steady states exist. The critical values D a cr and B ~ can be obtained by finding the limit point singularity at fixed B or Da, respectively. When multiple steady states exist, the low-conversion one is unstable. If the simple dynamic model presented above describes the separation section, the high-conversion steady state is stable. When low conversion is desired (for example, due to selectivity requirements) stable operation requires that the adiabatic temperature rise does not exceed the corresponding critical value. The conservative unicity condition zB < 1 ensures also stability. Similar with the case discussed in Section 3.2., operation near the limit points is not recommended. 4. CONCLUSIONS Interaction between Reaction and Separation through material recycles generates nonlinear phenomena. In contrast with stand-alone reactors, a zero-conversion, infinite-recycle steady state always exists. This state is stable if the reactor volume is below a critical value, for a given feed flow rate and reaction kinetics. New steady states appear when the reactor volume exceeds the critical value. Two mechanisms are presented: a) occurrence of one stable steady state at a transcritical bifurcation, when the infinite-recycle state loses stability; b) occurrence of two steady states at a fold bifurcation. In this case, only one of the new states is stable. The designer has to avoid the unstable solution, which puts a lower limit on the achievable conversion. We argue that designs close to the critical points are dangerous, as disturbances or design parameter uncertainty may shift the operating point in the region where no feasible state exists. REFERENCES 1. Adoimatis, R.A. and Cinar, A., 1988, Chem. Eng. Sci, 43,887-898. 2. Balakotaiah, V. and Luss, D., 1983, Chem. Eng. Sci, 38, 1709-1721. 3. Subramanian, S. and Balakotaiah, V., 1996, Chem. Eng. Sci., 51, 401-421. 4. Uppal, A., Ray, W.H. and Poore, A.B., 1976, Chem. Eng. Sci., 31,205-214. 5. Bildea, C.S., Dimian, A.D. and Iedema, P.D., 2000, Comp. Chem. Eng., 2-7, 209-215. 6. Luyben, M.L and Luyben, W.L., 1997, Essentials of Process Control, McGraw-Hill, New York. 7. Golubitsky, M. and Schaeffer, D. 1985, Singularities and groups in bifurcation theory, Springer-Verlag, New York.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
979
A New Multiparametric Mixed-Integer Quadratic Programming Algorithm V. Dua, N. A. Bozinis and E. N. Pistikopoulos* Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College, London SW7 2BY, U.K.
A number of important engineering problems, such as mixed logical dynamical systems, which simultaneously involve process dynamics and logical constraints can be reformulated as multi-parametric mixed integer quadratic programs (mp-MIQP) by treating the control variables as the optimization variables and the state variables as parameters - the quadratic terms appear only in the objective function (typically associated with minimization of least square errors). This paper presents an algorithm for the solution of mp-MIQPs. The solution of mp-MIQPs is given by an enclosure of the nonlinear profiles of the control variables as a function of the state variables. The optimal solution is then obtained, on-line, by evaluating the profiles for a given value of the state variables and then choosing the minimum of the values corresponding to different profiles. 1. I N T R O D U C T I O N Process engineering problems usually involve uncertain parameters. These uncertain parameters arise, for example, due to variations in demand and supply for a planning problem [1]. Such problems can be addressed by using the fundamentals of parametric programming. The key advantage of using parametric programming to address process engineering problems under uncertainty is that a complete map of optimal solutions is obtained as a function of parameters, without exhaustively enumerating the entire space of these varying parameters [2-8]. On-line control problems can also be reformulated as parametric programming problems by treating control variables as optimization variables and state-variables as parameters so as to obtain the control variables as a function of state variables [9]. The on-line control problem therefore reduces to a function evaluation problem since for all the given states of the plant, the optimal control actions are available as a function of the state variables. For the case when control problems also involve logical constraints and/or discrete choices, such as the startup and shutdown of certain equipments under certain operating conditions, binary variables are introduced to formulate the logical constraints [ 11-13]. Such systems, which simultaneously involve logic, dynamics and operating constraints are known as mixed logical dynamical (MLD) systems and can be mathematically formulated as mixed-integer programs [14]. Similar to the concepts of formulating the on-line control problems as multi-parametric *Corresponding author. E-maih
[email protected], Tel.:+44 (0) 20 7594 6620, Fax: +44 (0) 20 7594 6606. The authors gratefullyacknowledge the financialsupport from DETRTETSU.
980 programs [9], MLD systems with a linear objective function can also be formulated as multiparametric mixed-integer linear programs [ 15] and solved by using the algorithms described in [3,6,8]. In this work, we address the case when the objective function in the MLD problems involves quadratic terms (such as least square errors), resulting in a multi-parametric mixed integer quadratic program (mp-MIQP). Note that mp-MIQPs are a special case of the multiparametric mixed-integer nonlinear programs (mp-MINLPs) for which we presented algorithms in our earlier work [7]. Here we present a specialized algorithm for mp-MIQPs which involves the solution of multi-parametric quadratic programs (mp-QPs) instead of general and more difficult multi-parametric nonlinear programs (mp-NLPs) (involved in the solution of mp-MINLPs). While the solution of mp-NLPs requires linear approximations, the solution of mp-QPs is given by exact quadratic profiles. The rest of the paper is organized as follows. The mathematical foundations and the algorithm for the solution of mp-MIQPs is proposed in Section 2; an illustrative example is presented in Section 3, while some concluding remarks are given in Section 4. 2. M U L T I P A R A M E T R I C M I X E D - I N T E G E R QUADRATIC P R O G R A M M I N G
2.1. Mathematical Formulation Consider an mp-MIQP problem of the following form [ 16]" z(O) -- mincTx + lxTQx
+ dTy,
s.t. Ax + Ey '4 "A I
0
.
9
"=
.
.
I
|
9
.
9
5
i
10
.
.
|
Total
|
15
Mass Transferred
Figure 4. Mass transfer composite curve for cross flow system. Figure 5. Mass transfer composite curve for Option 7. 4. R E S U L T S The methods described above were used to determine the improvement in sulphur removal for different retrofit possibilities. It was found that segregation of the streams before scrubbing (rather than mixing them all together) gave a definite increase in sulphur recovery. The best option (Option 7) was to send Stream 1 on its own to Unit 1, Streams 2 and 7 together to Unit 2, Streams 3 and 6 to Unit 3, and finally Streams 4 and 5 to Unit 4. This gave a 10.5 % increase in sulphur recovery. Using this option, the liquid split to the three stages in each unit was varied. This indicated that the present 2:1:1 split could be improved, and that, in fact, for the best segregation option, a split of 1:1:1 gave a further 2.4% increase in sulphur recovery. Another possibility that was examined was the effect of including stream 8 (the one containing phenol) in the scrubbing system (presuming the phenol had been removed). This indicated that a maximum sulphur recovery increase of 8.8% was possible. This is less than the best option without including this stream, so there does not appear to be an incentive to remove the phenol so that stream 8 can be sent to the scrubbing units. The final possibility considered was a debottlenecking of the cyclones to allow a larger gas flow through the scrubbing units. It was found that a 15% increase in throughput (for the best
996 segregation option) would result in an additional 11,6% increase in sulphur recovery. There is therefore a good incentive to increase the cyclone capacity. 5. CONCLUSIONS Two diagrams are presented for the modelling and analysis of the combined cross and cocurrent flow venturi scrubber unit examined in this study: one is the y-x diagram for a combined cross and co-current flow system, and the other is the mass transfer composite curve for such a system. These diagrams enabled us to examine a range of retrofit options to improve the sulphur recovery in these units. It was found that both segregation of the feed streams and changing the liquid feed ratios to the three stages, both of which could be implemented without major costs, would give improvements. There seems to be no incentive to include the phenol containing stream in the sulphur recovery units. Debottlenecking of the cyclones, which could be quite costly, would give a major boost in the ability of the units to recover sulphur. ACKNOWLEDGMENTS The authors wish to acknowledge the collaboration and support of Sasol Technology (Sastech), particularly Mr Daan le Roux, in the work of this project. REFERENCES
1.
E1-Halwagi, M. M.; Manousiouthakis, V. (1989) Synthesis of Mass Exchange Networks,
AIChE J., 35(8), 1233-1244. 2. 3. 4. 5. 6.
7.
8. 9. 10.
Fraser, D. M.; Hallale, N. (2000a) Retrofit of Mass Exchange Networks Using Pinch Technology, AIChE Journal 46(10), 2112-2117. Fraser, D. M.; Hallale, N. (2000b) Determination of Effluent Reduction and Capital Cost Targets Through Pinch Technology, Environmental Science and Technology, in press. Hallale, N.; Fraser, D. M. (1998a) Capital cost targets for mass exchange networks. A special case: Water minimisation, Chem. Eng. Sc., 53(2), 293-313. Hallale, N.; Fraser, D. M. (1998b) Synthesis of a cost-optimum gas-treating process using pinch analysis, Advances in Environmental Research, 2(2), 167-178. Hallale, N.; Fraser, D. M. (2000a) Capital and Total Cost Targets for Mass Exchange Networks, Part 1: Simple Capital Cost Models, Computers and Chemical Engineering, 23(11-12), 1661-1679. Hallale, N.; Fraser, D. M. (2000b) Capital and Total Cost Targets for Mass Exchange Networks, Part 2: Detailed Capital Cost Models, Computers and Chemical Engineering, 23(11-12), 1681-1699. Hallale, N.; Fraser, D. M. (2000c) Supertargeting for Mass Exchange Networks: Part I Targeting and Design Techniques, Trans IChemE (Part A), 78(Mar), 202-207. Hallale, N.; Fraser, D. M. (2000d) Supertargeting for Mass Exchange Networks: Part I I Applications, Trans 1ChemE (Part A), 78(Mar), 208-216. Treybal, R. E. (1981) Mass Transfer Operations, 3rd ed., McGraw-Hill, Singapore.
EuropeanSymposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B.Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
997
The Interactions of Design, Control and Operability in Reactive Distillation Systems Michael C. Georgiadis a , Myrian Schenk b, Rafiqul Gani c and Efstratios N. Pistikopoulos b* aCentre for Research and Technology - Hellas, Chemical Process Engineering Research Institute, P.O. Box 361, Thermi 57001, Thessaloniki, Greece. bCentre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, Prince Consort Road, London SW7 2BY, U.K. CCAPEC, Department of Chemical Engineering, Technical University of Denmark, DK-2800, Denmark. The aim of this work is to systematically study the interactions of process design, process control and operability in reactive distillation systems. Based on a rigorous and high fidelity dynamic model that has been validated against experimental data. The study is performed in a system involving the production of ethyl acetate from the esterification of acetic acid with ethanol. The problem is posed as a dynamic optimization problem under uncertainty and solved using control vector parameterization techniques. Two state-of-the-art optimization strategies are then employed. First, the design and control tasks are considered sequentially and, for the second, design and control are optimized simultaneously and the potential synergistic benefits of such an approach are investigated. In both cases, multi-loop proportional-integral (PI) controllers are used. Optimization results of the simultaneous approach indicate promising process designs and illustrate how controllability is a strong function of the process design and how careful design can result in significantly improved controllability without necessarily having to pay an economic penalty. 1. INTRODUCTION Reactive distillation systems involving reaction and separation in a single unit have the potential to greatly reduce capital and operating costs. However they are not extensively used in industry perhaps because it is perceived that the operation of a reactive distillation system will always be more difficult and will pose higher requirements on the quality of the design and control system than the conventional reactor and train of distillation columns. This can be mainly attributed to the presence of reactions coupled with the separation, which will have a significant influence on the robust operation under variations. In the light of this, the problem of designing and operating reactive distillation systems is one that would benefit from a detailed study of the interactions of process design, process control and operability. However, most of the studies that have appeared in the literature have solely concentrated on steady-state design/synthesis aspects, linear controllability analysis or on dynamic modelling and control of fixed designs [1-3]. Furthermore, in most approaches relatively simple models were used: tray hydraulics and pressure dynamic were normally Corresponding author. E-mail:
[email protected] 998 ignored [3]. On the other hand, recently the interactions of design and control issues have been systematically studied in complex distillation systems using dynamic optimisation-based approaches. The economic and operability benefits that could have been obtained through an integrated approach have been clearly demonstrated [4-5]. In this work, a reactive distillation example problem is described by a high-fidelity dynamic model in which the operability characteristics can be captured over time. The interactions of design and control are studied through the use of two strategies and the potential synergistic benefits of the integrated simultaneous approach are investigated. 2. P R O B L E M DESCRIPTION AND DYNAMIC M O D E L
The production of ethyl acetate from the esterification of acetic acid with ethanol and water is considered (see Figure 1). The saturated liquid mixture is fed at a rate of 4885 mol/h in order to produce a top product with at least 0.52% ethyl acetate composition and a bottom product of no more than 0.26% ethyl acetate. Reaction takes place in all 13 trays of the column. The objective is then to design the column and the control scheme at minimum total cost, able to maintain feasible operation over a finite time horizon of interest (24 hours); subject to (i) high-frequency sinusoidal disturbances in the acetic acid inlet composition; (ii) "slow-moving" disturbance in the cooling water inlet temperature; (iii) product quality specifications; (iv) flooding, entrainment and minimum column diameter requirements; (v) thermodynamic feasibility constraints for the heat exchangers and (vi) operating pressure limits for the column. [ " S low-tnoving" disturbance
High-frequency disturbance Zac.= a cO-496-~O.05sin(0~t) to
[
c o = 2 x / 3 6 0 0 r a d s- t
Ethy lAcetate
=
6(5~
ol/VCater
4 8 8 5 naol/h 13
Xb(sv~,a,s~icrc~wr] I~grrPe?ct~tgU:etionofp4antsl~cPti?~ mplex L opt plantconfig.J and additional
f~eP]~L~hf:~t~ ~im~ complex of plants ndividual plants and I ~r~/~ Simulabon ] ~_~1~:i~dUc~n~ ~n: fld I :,,=.u material and. ener gY.. . . . . ! . . c~a.ram, . . . ete~. oalances, . . . .rate equations, "- 3obal variables.
I [ I
:~al!bn.um relat . . . . . eny=ceJ, an~ . mermoaynamlc ;~.~ Economic Model
I
I
t \ prices, costs, mebics / "~ t/ -~
.i
/
I
SYNPHONY
.
~
101ants
X
Simulation equabons for .individual . plants . .
~
environmental design. Also, the programs
I
-
I
"
[Pm~nd~u~gne~,
chemical complex. includes
~.,'
. . . . operating -I' plant -oncdtlons'" ~
Properties
"
Iiis''*'talOn''a%i,"~'=~~ts
_..~ . . . . .
Enthalpy,~e.s~,vsco~ e~
'
~ .JTotal C o s t
..
It the
Assessment~ sustainability metrics ~ E. . . . ~c / developed by the
..................... Env'lronmantal' Graphical User Interface rProfit for comld.X [ SustainnloU~,, Optimal configuration I s ~ M ~ analysis for ~ " presenled in tables and on the I prices, costs, raw complexflowsheat Imaterials, demands SensflMly results, cornpadsons I operating con d~i(~ s with . . . . ntconflguralion l Inleracti~ changing of Input f o r l . . . . . tudles IFlowrat . . . . poseon r~,,,,,,~,,, Identification or erlvfronmental L ,.a . . . . . . . . . . Impacts from pollution Index l i { ' 7 1 ndex Indicators for sustainable use oil Source of poflutant %, ....
and
GAMS/DICOPT are used for optimal plant configuration of the
AIChE/CWRT
I/
~
/ ] ,,/
.-~generation
Sustainability M e t r i c s Working Group (Adler,
1999) and the B R I D G E S extensions (Beaver and Beloff,
2ooo).
The structure of the
Figure I Program Structurefor the Chemical ComplexAnalysisSystem
Chemical Complex Analysis System is shown in Figure 1. The system incorporates a flowsheeting component where the simulations of the plants in the complex are entered. Individual processes can be drawn on the flowsheet using a graphics program. The plants are connected in the flowsheet as shown in Figure 2. For each process material and energy balances, rate equations, equilibrium relations and thermodynamic and transport properties are entered through windows and stored in the database to be shared with the other components of the system. Also, the economic model is entered as an equation associated with each process with related information for prices, costs, and sustainablity metrics that are used in the evaluation of the Total Cost Assessment for the complex. The TCA component includes the total profit for the complex that is a function of the economic, environmental and sustainable costs and income from sales of products. Then the information is provided to either GAMS/DICOPT or SYNPHONY for solving the Mixed Integer Nonlinear Carbon Programming (MINLP) problem dioxide ,~ Ammonia for the optimum configuration of Carbon | emissions Carbon dioxide dioxide ,._1 / emissions plants in the complex. Also, the Urea I :1 ~v Urea "q Plant sources of pollutant generation Ammonium are located by the pollution index > A . . . . i,, ,~IA . . . . lum Phosphate Natural Gas I Ammonia Phosphate ._1 >i Plant Steam component of the system using Air Pl~t the EPA pollution index Hydrogen fluodde Sdlcontetraflodde methodology (Cabezas, et. al.,
t
~
emissions
Sulfur dioxide Sulfur i e m sslons i
T
1997).
Calcium
Phosphate ~_1 [
~ il~anOd~ ph~
Phosphoric
Add
Calcium Sulfate (gypsum waste) Ftgure 2
Schernahc of Agnculfural Cherncals Complex wYthRaw Matetmls, Product=, En'~=tons end Weste,~.
All interactions with the system are through the graphical user interface of the system that is written in Visual Basic. As the process flow diagram for the complex is prepared, equations for the process units and
1020 variables for the streams connecting the process units are entered and stored in the database using interactive data forms as shown on the left side in Figure 1. Material and energy balances, rate equations and equilibrium relations for the plants are entered as equality constraints using the format of the GAMS programming language that is similar to Fortran. Process unit capacities, availability of raw materials and demand for product are entered as inequality constraints. Features for developing flowsheets include adding, changing and deleting the equations that describe units and streams and their properties. Usual Windows features include cut, copy, paste, delete, print, zoom, reload, update and grid, among others. A detailed description is provided in a user's manual The system has the TCA component prepare the assessment model for use with determination of the optimum complex configuration. The AIChE/CWRT TCA program (Constable, D. et. al., 2000) is an Excel spreadsheet that has the cost in five types, as describe above. This Excel spreadsheet is an extensive listing of all possible costs. The TCA component combines these five categories of costs into three costs: economic, environmental and sustainable. Types 1 and 2 are included in economic costs, Types 3 and 4 are included in environmental costs, and Type 5 is sustainable costs. Economic costs are estimated by standard methods (Garrett, 1989). Environmental costs are estimated from the data provided by Amoco, DuPont and Novartis in the AIChE/CWRT report. Sustainable costs are estimated by the study of power generation in this report. It is an on-going effort to refine and update better estimates for these costs. As shown in Figure 1, the system will provide an option to select one of two optimization methods. Determining the optimal configuration of plants in a chemical complex is a mixed integer nonlinear programming problem where the equality and inequality constraints include material and energy balances, process unit capacities and others as described above. There are two methodologies to solve this type of optimization problem, GAMS/DCOPT and SYNPHONY. GAMS (General Algebraic Modeling System) was developed at the World Bank for very large economic models, and it can be used to determine the optimal configuration of a chemical complex by solving a M1NLP programming problem using the DICOPT solver (Kocis and Grossmann, 1989). SYNPHONY uses process graph methodology based on the work of Friedler and Fan (Friedler, Varga and Fan, 1995) to solve the MINLP problem. 3. AGRICULTURAL CHEMICAL COMPLEX EXPANSION VALUATION A major agricultural chemical company had performed a case study for expanding production of sulfuric and phosphoric acid along with heat recovery options at two locations that ten miles apart. This multiple site, multiple-plant expansion was used with the prototype system, and the results compared to the case study for validation of the prototype. In this complex, phosphate fertilizers are produced by reacting ammonia and phosphoric acid as illustrated in Figure 2. Phosphoric acid is made by digesting phosphate rock with sulfuric acid. Sulfur, air, and water are used to make sulfuric acid, and in that process, waste heat is recovered as steam to drive turbines for power generation, and to evaporate water from phosphoric acid. With excess ammoniation capacity available, the objective of the case study was to expand phosphoric acid production capacity by 28%. This requires additional sulfuric acid and steam. Since sulfuric acid can be shipped for miles and steam cannot, phosphoric acid evaporators require some steam capacity from an on-site sulfuric acid plant. When
1021 producing the sulfuric acid needed to produce phosphoric acid, the sulfuric plant produces more byproduct steam than is needed to evaporate the phosphoric acid. So, as long as the two-site sulfuric production capacity is adequate, there is some flexibility in how closely the sulfuric vs phosphoric acids production capacities have to match within each site. Also, spare power-generation capacity at a site will encourage the addition of extra heat recovery equipment to old and new plants at that site. Many U. S. fertilizer complexes have justified new power generation equipment. When a MWH sells for less than a bought MWH, the incentive drops Figure 3. Sulfuric Acid Plant Options at One of Two Plant Sites when generated power displaces the Part of Superstructure forSYNPHONY last of the site's purchased power. When utility's "avoided costs" for new construction are high, many fertilizer complexes have justified excess generating capacity to sell power to their local utility. Site power differences could make it profitable to build a sulfuric plant at one site for the steam and ship all the sulfuric acid to the other site to make phosphoric acid. More options were added to challenge the prototype, and the expansion was to be made in two stages where stage two should waste only a minimum of stage one. Stage one should still be a best choice in case stage two is never justified. Each of the two expansion stages will have one phosphoric acid expansion, and the second expansion will be at the "other" site; one sulfuric acid expansion with an option for over-sizing the first to serve as the second; and a second sulfuric acid expansion does not have to be sited away from the first expansion. Also, there are options for adding heat recovery equipment to one old and any new sulfuric plants and for adding one turbo-generator per site per stage. Enough site differences were specified to make the study interesting. The question for the prototype to answer now was what size phosphoric acid, sulfuric, heat recovery, and powergeneration expansions should be built at each site for each stage of expansion. The file input-output version of SYNPHONY was used for the optimal configuration determination. The superstructure for this demonstration had 67 different species (600 psig steam, sulfuric acid, logic switches, etc.) and 75 processing units. In Figure 3, part of the superstructure for multiple sulfuric acid units is shown for one plant site. A sulfuric plant was one unit using 8-10 species. A new turbo-generator took 10 species and 7 units to model. Two of those species were fabricated to properly couple the 7 units to work as one. Computing time for any one case was less than 15 seconds on a Pentium II PC. The results obtained with the system were consistent with the case studies done previously at the actual complexes that were modeled here, and this served to validate that the system was giving consistent and accurate results. The results of using the system gave the
1022 following evaluations. By raising the cost of shipping sulfuric acid between sites, the sites could be forced to be self-sufficient in sulfuric production capacity. This impacted steamand power-generation capacities at each site. Similarly, the cost of extra storage tanks to handle more than a minimum of sulfuric shipping could be made to limit sulfuric shipping and bias the siting of sulfuric production capacity. This happened when the cost of extra tanks overcame the energy efficiencies of specific sites. Production rate for a higheremissions, single absorption sulfuric acid plant was curtailed as expected by voluntarily limiting the two-site SO2 emissions to pre-expansion levels. With this old plant curtailment, the new sulfuric plant was built with corresponding extra capacity. The curtailed, singleabsorption sulfuric plant was converted to double-absorption for expansion stage two when the conversion cost was significantly less than the cost of a new plant and excess capacity was built in expansion stage one. However, few companies would build excess capacity in stage one without a power incentive or strong anticipation of stage two. Extra heat-recovery and power-generation equipment was justified only when longer payback periods were acceptable. Heat-recovery and power-generation equipment was installed or not installed based on installation cost and the value of the power. The value of power varied because incremental power displaced purchase at one site and added to sales at the other site. In conclusion, the prototype of a chemical complex analysis system has been demonstrated on an agricultural chemical complex expansion. 4. CONCLUSIONS A prototype of a chemical complex analysis system has been described and its capability demonstrated by duplicating and expanding an industrial case study. The system selected the best site for required new phosphoric and sulfuric acids production capacities and selected, sited, and sized the optional heat-recovery and power-generation facilities. REFERENCES
1. Adler, S. F. 1999, Sustainability Metrics Interim Report No. 1 and Interim Report No. 2 AIChE/CWRT, 3 Park Avenue, New York, NY. 2. Beaver, E. and B. Beloff, 2000, Sustainability Metrics for Industry Workshop, AIChE/CWRT and BRIDGES to Sustainability, Austin, Texas, May 17-18, 2000. 3. Cabezas, H., J. C. Bare and S. K. Mallick, 1997, Computers Chem Engr, Vol. 21, Supp $305. 4. Constable, D. et al., 2000, Total Cost Assessment Methodology; Internal Managerial Decision Making Tool, AIChE/CWRT, AIChE, 3 Park Avenue, New York, NY, February 10, 2000. 5. Friedler, F., J. B. Varga and L. T. Fan, Chem Eng Science, Vol. 58, No. 11, p. 1755. 6. Garrett, D. E., 1989, Chemical Engineering Economics, Van Nostrand Reinhold, New York, NY. 7. Johnson, J., 1998, Chemical & Engineering News, p. 34, August 17, 1998. 8. Kocis D. and I. Grossmann, 1989, Computers Chem Engr, Vol. 21, No. 7, p. 797-819. 9. Kohlbrand, H. K., 1998, Proceedings of Foundations of Computer Aided Process Operations Conference, Snowbird, Utah, July 5-10, 1998.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1023
Decision Support Tools for Process Design and Selection Soorathep H E A W H O M * and Masahiko HIRAO Department of Chemical System Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
We proposed a new tool that is capable of reducing the complexity of the process synthesis problem and analyzing a trade-off between the environmental impact, economy and robustness of the process. In addition, new efficient process robustness parameters were also proposed. Controllability was indicated by the failure probability which could be calculated with a small number of iterations. The operability was evaluated by the deviation ratio. Applicability of the method was illustrated in a case study. Three types of closed-loop toluene recovery processes, 1) membrane-based, 2) condensation-based, and 3) adsorption-based, were investigated to quantitatively compare the characteristics of each process. By the proposed methodology, it is possible to design an appropriate process with minimal environmental impact and maximal robustness at a desired economic performance. 1. INTRODUCTION The need for an environmentally benign process has been increasing due to the pressure on chemical process industries to improve their environmental performance. Thus, the environmental impact of a process must be considered and minimized at an early stage of process design. The robustness performance is also important for the selection of good designs because of the requirement for a process that is robust under given input variations. Even in a case where there are no input variations, the problem still contains uncertainties. Therefore, the robustness objectives should be considered at the design stage to ensure that a process can properly operate with an acceptable performance under these uncertainties. Conventional methods for indicating robustness are based on the feasibility function, which is a measure of the ability of a process to meet design specifications under uncertainties. However, the calculation of process flexibility parameters involves the solution of a very complex multi-extremal and non-differentiable optimization problem. A synthesis problem has more than one objective. Frequently, this problem is nonlinear and involves discrete variables. Most of the literature on decision support tools for process design focuses only on economic and environmental problems or on economic and robustness problems. Only a few studies have been on the development of methodologies for considering economic, environmental and robustness problems simultaneously. *Corresponding author. Tel.: 813-5841-6876, E-mail address:
[email protected] 1024
Fig. 1. Failure probability.
Fig. 2. Performance distribution.
2. E C O N O M I C INDICATOR The economic performance can be represented by a summation of fixed costs and operating costs and the subtraction of product revenues. The fixed cost must be discounted by the life spans of the units. 3. E N V I R O N M E N T A L I N D I C A T O R The Sustainable Process Index (SPI)[1] is designed to deal with various environmental objectives simultaneously. The basic concept of the SPI is to calculate the area required to sustainingly embed a process into an environment. All mass flows that the process either extracts from or emits to the environment must not influence the environment in such a way that brings natural evolution into danger. The total required area consists of the raw material area, energy supply area, infrastructure area, staff area, and product dissipation area.
4. PROCESS ROBUSTNESS INDICATOR Process robustness is based on the feasibility function, which is a measure of the ability of a process to meet design specifications under uncertainties[2]. Several methods involve solving a very complex multi-extremal and concave optimization problem, which in the worst case requires a large number of iterations. In this paper, new efficient parameters for the evaluation of the process controllability and operability are proposed.
4.1. Failure probability The controllability of a process can be evaluated using the failure probability. The calculation of this parameter requires only a small number of iterations, and it provides a picture of feasible and infeasible regions. Failure probability is defined as the probability of failure scenarios. Figure 1 is the probability density surface of two uncertain inputs, which in this case are the feed rate and the concentration of input. The volume under the infeasible region is the failure probability. The region where the process cannot operate without an adjustment of any control variables denotes the infeasible region. A smaller failure probability implies a more controllable process.
1025 Uncertain parameters identification
Attributes identification Attributes classification
MOS
construction
Sub-problem optimization
--Objectives --Goals -Hard constraints -Soft constraints
Pareto set construction Problem ] reformulation
L
Define
I
.
Deviation ratio evaluation
l
Results ] analysis N ~
MOS of deviation ratio construction
Solution [
Fig. 3. Diagram of the synthesis methodology.
Fig. 4. Flowchart of the MOS of deviation ratio construction.
4.2. Deviation ratio
Deviation ratio of parameter x -
Ox/
V/G• 1
,ox
2 + G*.t22 _~_. -]. Gin . , 2 ./ (~il . , -~-/'/i2 . , -nt-
(1) , -[- [Llin)
where gj(d,z, O)
Ref2
930 kW
~78K
257 K
272 K
606 kW
442 kW (
Refl ~/~
238 K
| verification of the heat transfer areas(A) - AL; heat transfer area which have to be within the area ranges .................... ~STOP i Fig.4. The solving scheme. for selected heat exchangers. If the proposed solution is not acceptable, the model data are updated (estimation of Ft, etc.) and the model is optimized again. . . . . . . . . .
&
.................
j",..
4. EXAMPLES
The first example involves two hot and two cold streams (Table 2.) with two-stage HEN superstructure (Fig.l). The example was solved first by the original model [1]. The model contained 12 binary variables and was solved by DICOPT++ [6] in 0.82 sec of CPU time. Only three heat exchangers, one cooler and one heater are required for the entire heat transfer (Table 3.a), with a total annual HEN cost of 362118 S/yr. The cost is underestimated because the required heat transfer area for the match between the streams H1 and C 1 at the stage 2 exceeds 200 m 2 (490.4 m 2) and therefore, requires three heat exchangers. The exact annual HEN cost increases up to 369870 S/yr.
1099
Stage t
:
Stage 2
~.L
Tel,
:
rci 1
[
-
~
!
.o
H2
~
:
~
C1
(U-tubes) I IF
~
-= !
-
Bypass
1,= 1 "! I ~ .... [ 7h~11 k=l
II ....
'
_
< ~
k-'2
cu
!
I
['r~2 -~ r~.ac2
'
The example has been solved again by the extended model (Table 3.b). Now, the model contains 31 binary variables. It has been optimized by DICOPT++ in 3.47 sec of CPU time. The topology of HEN has not been changed (Fig.5), but the annual HEN cost has been decreased to 336130 $/yr due to the selection of different exchanger types. The streams H1 and C1 are matched together within the shell & tube exchanger with Ft of 0.912 (above the estimated value, Ftest=0.9).
k=3
Fig.5. optimal solution - first example.
Table 2. First example data. F C IkW/KI a [kW/(m2K)] Hot streams HI 250 0.6 H2 25 1.0 Cold s t r e a m s C1 240 0.7 C2 13 1.0 Ft for shell & tube e x c h a n g e r t y p e estimation: 0.9 Utility streams a [kW/m2K] Tin [KI Hot (HU) 5.0 510 Cold (CU) 1.0 300 Table 3. Solution of the first example. a. Solution given by the original m o d e l M a t c h (i-j-k) A Im2] No [-1 1-1-2 490.4 3 1-2-1 73.2 1 2-1-2 60.7 1
/t!.T [K] 81.42 24.64 60.00
Tin IKI 470 450
Tout IKI 400 390
p IMPal 2.0 1.0
330 410
390 500
1.0 1.0
Tout [KI 510 321
C [ $/(kW a)l 250 21
b. Solution given by the e x t e n d e d model. At.T [K] Match(i-j-k) A [m2l No l-] 1-1-2 544.9 1 81.42 1-2-1 73.2 1 24.64 2-1-2 60.7 1 60.00
Ft l-I 0.912 1.000 1.000
Type ST DP PF
The second example contains four hot and five cold streams (Table 4.) with four-stage HEN superstructure. The stream data were taken from the HDA process case study. Since the streams contain toxic components, the selection of the plate & frame type is forbidden. The example was solved first by Yee's model [ 1]. The program contained 89 binary variables and was solved by DICOPT++ in 9.83 sec of CPU time. Table 4. Second example data. Hot streams F C lkW/Kl a [kW/(m2K)] H1 49.27 0.15 H2 27.54 0.90 H3 1088.67 0.90 H4 229.17 0.90 Cold s t r e a m s C1 38.92 0.12 C2 14.58 1.00 C3 511.33 1.00 C4 252.60 1.00 C5 236.13 1.00 Ft for shell & tube exchanger type estimation: 0.8 Tin [KI Utility streams a [kW/m2K] Hot (HU) 5.0 850 Cold (CU) 1.0 282
Tin [KI 823.20 330.85 352.32 379.90
Tout IKI 299.89 329.85 349.32 376.9
p IMPal 3.5 3.5 3.5 3.5
330.19 362.95 462.30 376.90 550.60
713.70 463.00 465.30 379.60 553.60
3.5 3.5 3.5 3.5 3.5
Tout [K] 850 290
C [ $/(kW a)] 250 21
1100
Only six exchangers and four coolers were needed (Table 5.a) for the entire heat transfer with annual HEN cost of 867102 S/yr. The cost is underestimated again, because parallel exchangers are needed for the heat transfer within some of the selected matches. The exact cost was 897990 S/yr. The example has been solved by the extended model again. Now the program contains 246 binary variables and has been optimized by DICOPT++ in 478.24 sec of CPU time. Three iterations of the synthesis/analysis scheme have been necessary to obtain the solution, which contains eight exchangers and three coolers (Table 5.b) with the annual cost of 878337 S/yr. The match between the streams H1 and C2 at stage 4 requires two heat exchangers, so the exact HEN cost is increased to 882198 S/yr. Note that the topology of the optimal HEN has been significantly changed. In order to exclude one cooler, two additional exchangers have been selected. Table 5. Solution of the second example.
a. Solution given by the original model Match (i-j-k) A Ira21 No 1-1 1-1-1 1-1-4 1-2-4 1-3-3 1-4-2 1-5-1
1269.8 544.3 154.1 130.0 25.3 49.0
11 3 1 1 1 1
b. Solution given by the extended model.
zl~nTIK1 130.89 105.97 72.59 90.49 199.00 110.81
Match(i-j-k) 1-1-1 1-1-2 1-1-3 1-2-4 1-3-3 1-4-3 1-5-2 4-1-4
A lm2l 742.6 775.3 632.6 306.2 293.0 33.5 55.2 166.8
No I-I 1 1 1 2 1 1 1 1
A~nT IKI 121.24 142.32 139.01 36.52 50.17 150.50 98.38 38.92
Ft l-I 0.953 0.831 0.952 1.000 0.998 1.000 1.000 1.000
Type ST ST ST DP ST DP DP DP
5. CONCLUSIONS Both examples clearly indicate the advantages of the proposed model, which yields not only feasible but also better HEN design with respect to utility consumption and required heat transfer area. The proposed model allows the simultaneous heat integration of HEN and selection of optimal exchanger types due to the operating limitations. NOTATION AtnT --temperature driving force [ K ] C =charge coefficient [ $/yr ] i =set of hot streams [ - ] j =set of cold streams [ - ]
k l y a
=set of superstructure stages [ - ] =set of heat exchanger types [ - ] =binary variables =conv.heat transfer coef.[kW/(m2K)]
REFERENCES
1. T.F. Yee and I.E. Grossmann, Optimization models for heat integration-II, Heat exchanger network synthesis, Computer and Chem. Engng., 14, 1165-1184 (1990). 2. G.F. Hewitt, G.L. Shires and T.R. Bott, Process heat transfer, CRC Press, 155-194 (1994). 3. K.M. Guthrie, Capital cost estimating, Chem.Engng., 76,114 (1969). 4. G. Walker, Industrial heat exchangers, Hemisphere publishing, 93-102 (1990). 5. J.J.J. Chen, Letter to the editors: comments on improvement on a replacement for logaritmic mean, Chem.Engng.Sci., 42, 2488-2489(1987). 6. J. Vishwanathan and I.E. Grossmann, A combined penalty function and outher approximation method for MINLP optimization, Computers and Chem. Engng., 14, 769-782 (1990).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1101
SIM-OPT: A Computational Architecture to Address Valuable Business Aspects of Research & Development Pipeline Management Dharmashankar Subramanian, Joseph F. Pekny*, and Gintaras V. Reklaitis School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA. (dharmash, pekny, reklaiti)@ecn.purdue.edu The R&D Pipeline management problem has far-reaching economic implications for newproduct-development driven industries, such as pharmaceutical, biotechnology, and agrochemical industries. Effective decision-making is required with respect to portfolio selection and project task scheduling in the face of significant uncertainty and an everconstrained resource pool. Recently, Subramanian et al. (2000) described the here-and-now stochastic optimization problem inherent to the management of the R&D Pipeline by viewing it as the control problem of a performance-oriented, resource-constrained, stochastic, discreteevent, dynamic system. They presented a computing architecture, Sim-Opt, which combines mathematical programming and discrete event system simulation to assess the uncertainty present in the pipeline. They introduced the concept of timelines that studies multiple unique realizations of the controlled evolution of the discrete-event pipeline system. This work demonstrates how information may be integrated across the Sim-Opt timelines to obtain insights into the dynamics of the pipeline under uncertainty. It also illustrates how such insights may be applied to plan more effectively from an operational perspective. Lastly, it discusses briefly about other business questions that can be investigated using the Sim-Opt architecture. 1. I N T R O D U C T I O N Process systems engineers are faced with a tremendous opportunity to make an impact not only to the engineering process of making a new product, but also to the business process involving product prioritization, selection and pipeline scheduling (Blau et al., 2000). The R&D pipeline management problem is one such business process. It addresses the issues of a new-product-development pipeline, where several new-product-development projects compete for a limited pool of various resource types. Each project (product) usually involves a precedence-constrained network of testing tasks prior to product commercialization. If the project fails any of these tasks, then all the remaining work on that product is halted and the investment in the previous testing tasks is wasted. It is also possible that failed projects are reconsidered as long-term investments with re-working attempted to overcome the failure. In its most general form, the deterministic R&D pipeline management problem asks the following question: Given a set of research projects, each project containing a set of activities related by generalized precedence constraints, a common pool of limited resources of various (finite) * Author to whom all correspondence should be addressed
1102 kinds and a measurement of performance, what is the best set of projects to pursue, and further, what is the best way to assign resources to activities in the chosen projects, such that the chosen measure of performance is maximized? A more realistic, and practically motivated, problem faced by new-product-pipeline decision-makers is the above question in a stochastic context, i.e., with uncertainty added in terms of task duration, task resource requirements, and task successes and task rewards. Thus, a realistic R&D pipeline management problem is a stochastic optimization problem combining the features of a project selection problem and generalized resource constrained project-scheduling problem. The additional complexities are the task success uncertainty, duration uncertainty, resource requirement uncertainty and project reward forecast uncertainty. Task success uncertainty, a predominant source of uncertainty in the R&D context, has not been adequately addressed in the literature with the noted exception of Schmidt and Grossmann (1996), Honkomp (1998), Jain and Grossmann (1999), Blau et al. (2000), and Subramanian et al. (2000). Subramanian et al. (2000) described the here-and-now stochastic optimization problem inherent to the management of an R&D Pipeline by viewing it as the control problem of a performance-oriented, resource-constrained, stochastic, discreteevent, dynamic system. They presented a computing architecture, Sim-Opt, which combines mathematical programming and discrete event system simulation to assess the uncertainty present in the pipeline. Lastly, they described three policies to for taking actions as the pipeline evolves through its states while traversing the stochastic state space subject to the resource constraints present in the system. Further details about the Sim-Opt architecture can be found in Subramanian et al. (2000). 2. INTEGRATION OF INFORMATION ACROSS TIMELINES The simulation module, in Sim-Opt, marches in time, with indeterminate number of departures to the optimizer whenever a decision-making need is encountered (Subramanian, et. al, 2000). One such controlled walk in time through the stochastic state space constitutes a timeline. The simulation thus experiences different "futures" based upon stochastic realizations, encountered in a Monte-Carlo sense, across timelines. Any single timeline contains information about the temporal coexistence of various feasible tasks, both within and across projects. This information, while being influenced by the nature of control actions exercised, is largely a function of uncertainty, and is not obtainable a priori from a single monolithic stochastic program due to uncertainties present in the task successes which are binary (succeed/fail) in nature, and those present in task processing times and task resource requirements. This coexistence information can be further processed to infer and identify resource types that are binding in the face of uncertainty, and to evaluate the worth of augmenting such resource types. It can also be used to obtain information about the relative tendencies of projects (and tasks) to crowd out other projects (and tasks) due to the associated variations in their resource needs. All the information that is obtained from the framework could be potentially incorporated in a suitable manner into the deterministic optimization formulation to bias it to recognize the same for the purposes of portfolio selection. Finally, multiple such timelines can be explored in a Monte-Carlo fashion to accumulate several unique combinations of realizations of uncertainty. The information mentioned above can be integrated across this accumulation to obtain solutions to the underlying stochastic optimization problem. This will be illustrated on an industrially motivated case study in the next section.
1103
3. CASE STUDY This section demonstrates the integration of information across Sim-Opt timelines using an industrially motivated case study. The Case Study comprises seven projects, as shown in Figure 1. There are two resource types, R1 and R2, which are required in a certain combination for feasibly carrying out any single task. The system limits for R1 and R2 are 16 units and 8 units respectively. There is significant uncertainty with respect to processing duration, resource requirement, and survival probabilities for the tasks in all seven projects, as shown in Table 1. Project reward data is shown in Table 2. Table 1. Case Study Task Data Task
12
P1
14
Duration (weeks), Custom Distribution Value Probability
R1 (units), Custom Distribution Value Probability
1 2 3 4 5 2 3 4 5 6 3 4 5 6
0.295 0.375 0.190 0.110 0.030 0.10 0.18 0.44 0.18 0.10 0.32 0.40 0.18 0.10
4 5 6 7
0.29 0.44 0.21 0.06
1 2 3 4
0.335 0.415 0.166 0.084
3 4 5 6 7
0.123 0.203 0.335 0.228 0.111 0.32 0.44 0.16 0.08 0.26 0.54 0.14 0.06 0.10 0.2O 0.36 0.28 0.06 0.32
4 5 6 7 8 10 11 12 13 14 3 4 5 6 7 4 5 6 7
0.06 0.21 0.46 0.21 0.06 0.06 0.12 0.55 0.21 0.06 0.05 0.20 0.45 0.25 0.05 0.23 0.52 0.20 0.05
10 11 12 13 2 3 4 5 4 5 6 7
0.23 0.55 0.16 0.06 0.23 0.55 0.15 0.07 0.23 0.53 0.17 0.07
12
0.225
P2
2 3 4 5 16
P3
I
I
R2 (units), Custom Distribution Value Probability
Probability Success Triangular Distribution Min
0.28 0.44 0.28
0.74
Most Max Likely . 0.80 0.86
0.22 0.56 0.16 0.06
0.7
0.75
0.8
2 3
2 i 3 4 5
0.29 0.44 J0.21 0.06
0.8
0.85
0.9
0.23 0.52 0.20 0.05
i0.7
0.8
0.85
1
4
i
1 2 3 4
0.23 0.55 0.16 0.06
0.55
0.6
0.65
2 3 4 5 1 2 3 4
0.23 0.53 0.16 0.08 0.23 0.53 0.17 0.07 0.225 0.565 0.155 0.055
0.75
0.8
0.85
0.7
0.8
0.85
0.7
0.75
0.8
0.225
0.85
i i
!
i
i
0.9
0.95
i
(JI
C~
,.~ (JI (JI (JI
~ C ~ O ~ 0 0 0 -.,I .,,,I u,) u,)
~
(.~
~'~ ~'~ . ~
U,) !
I~ h.)
-,.,! ,,..I . ~
h ~ 84
C~h ~r~ . ~
~ ) ~~
~r~ -1~ ~,)
C~ ~rl -1~ , )
~
~
-..,I --,.I u,) ~,)
0
C~
C~ ../1
0
0
0
0
C
~
C~ C~ ~rl
1105
0.284 117
P7
2 3 4 5 3 4 5 6
0.29 0.44 0.21 0.06 0.29 0.44 0.21 0.06
8 9 6 7 8 9 13 14 15
0.17 0.07 0.22 0.54 0.17 0.07 0.20 0.57 0.23
7 8 2 3 4 5 3 4 5 6
0.17 0.07 0.22 0.54 0.17 0.07 0.22 0.54 0.17 0.07
0.3
0.35
0.4
0.45
0.5
0.55
T a b l e 2. Project R e w a r d D a t a Reward $ Project P1 30,000 P2 20,000 P3 15,000 P4 40,000 P5 50,000 P6 40,000 P7 60,000
! !
(D
!
I
j
Figure 1. Case Study Activity on N o d e Graph
A p p l y i n g P o l i c y II described in S u b r a m a n i a n et al. (2000), the distribution o f r e w a r d s and the r e s o u r c e profile b e h a v i o r is as given below in Figures 2 and 3.
1106 Rewards Distribution, Policy II 800
700
15
Mean =36564.74
Average Resource Profiles for R1 and R2, Policy 2
.
.
.
.
IIFrequency
600-t
500 400
300
30[ m / -.-
200
I
System Limit I Desired |
...1
100 o L-_______.~ -2 0
2 4 6 Rewards, Dollars
8
10 x 104
0
2
4
6 8 TIME, WEEKS
10
12
14
Figure 2. Rewards Distribution From Policy II
Figure 3. Resource Profiles From Policy II The plots in Figure 3 show the dynamics of combinatorial interaction between activities across projects in the portfolio. This interaction is due to resource competition. It can be seen that with respect to resource type, R1, the average "desired profile" stays within the system limit of 16 units during the earlier time periods in the planning horizon. But the actual utilization of resource type, R1, is below the desired level, as exhibited by the "utilized profile". While this may appear counter-intuitive, it is because during the corresponding time periods, the average "desired profile" with respect to resource type R2 is well above the system limit of 8 units. This prevents effective resource utilization since the two resource types are required in the right combination. Activities in the pipeline are eligible for active processing only if the resource types R1 and R2 are available in the right combination of amounts. The plots can thus be viewed as the dynamics of interaction between resource types R1 and R2. During earlier time periods, resource type R2 is binding, while at later time periods, resource type R1 becomes binding as well. This combinatorial interaction at the current levels of availability of the resource types leads to poor utilization of the system resource levels. This knowledge can be incorporated to plan more effectively from both a design perspective and an operational perspective. The latter is illustrated with an example in the following section. 4. AN E X A M P L E OF USING THE I N F O R M A T I O N I N T E G R A T E D A C R O S S TIMELINES The resource profile information integrated across the timelines in Figure 3 revealed the under-utilization of the resource types R1 and R2 due their combinatorial interaction and how this interaction evolves in time. Having gained this insight, we can evaluate creative operational decisions, such as accumulating underutilized resource levels of a resource type from lean periods, and utilizing them for tight periods when that resource type becomes binding. This is like under-working a resource type for some time periods, in return for overworking the same resource-type for some other time periods. We implement this operational strategy in Policy IV, which is same as Policy II (Subramanian, et. al., 2000) in all other aspects. In Policy IV, accumulated resource levels are assigned to tasks along with actually present resource levels (if any), only if such an assignment fully satisfies the actual resource
1107 needs of the task for its entire processing duration. In particular, we accumulate underutilized resource levels, corresponding to resource types, R1 and R2, in units of R1-Weeks, and R2Weeks respectively. Figure 4 shows the frequency plot of rewards obtained from 15000 timelines corresponding to the same unique 15000 sets of random numbers, as used before. Rewards Distribution, Policy IV
700
5I | 1 0 I" . . . . .
600 500
Average Resource Profiles for R1 and R2, Policy IV
1
Mean =39522.62 Frequea
R1 t "
400
5
300
3O
200
20[ Rn /~
100 0 -2
0
- - System Limit - , - Desired ~- ..~..~ . . . . . . . _- - - ~ Utilized . . . ~
~'==------ " ' - - "
2
=="==" =" == ======~. == ~ == == ~ . . . . . . .
4
6
. _ . P, =,=-='.-p"- -'
=.. ,m,,=, ,P ,--
8
10
j - ~ " " ' ~ " ....
=.--==
12
-"
=..-.
14
......
.........
10 t . _ . . , . - . - - = " ' " 0
2 4 6 Rewards, Dollars
8
10 x 10"
Figure 4. Rewards Distribution From Policy IV
0
2
4
6 8 TIME, WEEKS
Figure 5. Policy IV
Resource
10
12
Profiles
14
From
CumulativeFrequencyPlotsof Rewardsfrom PolicyII and PolicyIV 1 0.9
0.8Probability
','1
0.7 0.6
0.5 0.4 0.3 0.2 0.1 0 0
2
4 6 Rewards, Dollars
8
10 x 104
Figure 6. Cumulative Frequency Plots of Rewards From Policy II and IV A Cumulative frequency plot of the rewards obtained using Policy IV is shown in Figure 6, along with that corresponding to Policy II for comparison purposes. (Note that Policy IV can be thought as Policy II implemented with an additional operational strategy as described above). Policy IV outperforms Policy II in terms of the mean as well as in terms of the cumulative frequencies. Figure 5 shows the "desired resource profiles" and the "utilized resource profiles" that are accumulated and averaged across the 15000 timelines explored using Policy IV. While the dynamics of combinatorial interaction between the resource types at the current resource levels in the system, continue to prevent effective utilization of the
1108 available resource levels, the extent of under-utilization has improved significantly over what was witnessed in Policy II. This is an example of how the insight obtained with the integration of information across timeline can be utilized effectively to influence and improve the quality of timelines that the pipeline system can witness. 5. CONCLUSIONS The concept of timelines that studies multiple, unique, realizations of the controlled evolution of the discrete-event pipeline system, has been shown to be an effective approach to obtain insights into the dynamics of the pipeline problem under uncertainty. Methods have been presented to integrate information across the timelines in terms of binding resource-types that present bottlenecks. An example has been presented that evaluates operational decisions, such as accumulating underutilized resource levels of a resource type from lean periods, and utilizing them for tight periods when that resource type becomes binding. This is like underworking a resource type for some periods, in retum for over-working the same resource-type for some other periods. Sim-Opt can be an effective architecture for integrating several different kinds of information across timelines. We can evaluate design decisions, such as, the value of acquisition of any resource type(s), the value of entering into outsourcing contracts for binding resource types and estimate the timing of these future contracts, at the here-and-now. Operational decisions such as partially satisfying the resource needs of resource-starved activities, accompanied by a proportional increase in their processing times, can also be studied with the timeline integration within the Sim-Opt architecture. Other investigations that can be carded out using Sim-Opt include analyzing the sensitivity of parameters such as resource levels, cost estimates, technical survival probabilities and commercial estimates of rewards, and answering what-if questions such as addition and generation of new projects. Sim-Opt can also be used to answer questions about the value of investment in research to improve the quality of information as well as the quality of projects and project execution, in terms of investing to improve survival probabilities and processing times. Finally, all such information can be utilized towards more effective decision-making in order to improve the quality of timelines that the pipeline system can witness. REFERENCES 1. Schmidt, C. W and Grossmann, I. E. Optimization models for the scheduling of testing tasks in new product development. Industrial & Engineering Chemistry Research. 35: (10). 3498-3510. Oct 1996. 2. Jain, V. and Grossmann, I. E. Resource-constrained scheduling of tests in new product development. Industrial & Engineering Chemistry Research. 38: (8). 3013-3026. Aug 1999. 3. Honkomp, S. J. Solving mathematical programming planning models subject to stochastic task success. Ph.D. Thesis, School of Chemical Engineering, Purdue University, 1998. 4. Blau, G. E., Mehta, B., Bose, S., Pekny, J. F., Sinclair, G., Kuenker, K. and Bunch, P. Risk Management in the Development of New Products in Highly Regulated Industries. Computers and Chemical Engineering. 24: (2-7). 659-664. July 2000. 5. Subramanian, D., Pekny, J. F. and Reklaitis, G. V., "A Simulation-Optimization Framework for Addressing Combinatorial and Stochastic Aspects of an R&D Pipeline Management Problem", Computers and Chemical Engineering. 24: (2-7). 1005 - 1011. July 2000.
European Symposmm on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1109
Solution of MEN synthesis problems using MINLP: Formulations of the Kremser equation Z. Szitkai, Z. Lelkes, E. Rev, Z. Fonyo Chemical Engineering Department, Budapest University of Technology and Economics, H-1521 Budapest, Hungary One way of solving mass exchange network (MEN) synthesis problems is to formulate the synthesis task as a mixed integer non-linear programming problem. The solution of this optimisation problem delivers the desired MEN. In most cases, the objective function of the optimisation problem is the total annual cost. Assuming linear phase equilibrium relations, capital investment calculations of staged mass exchangers are most commonly based on the Kremser equation. Discontinuous functions, such as the Kremser equation, are difficult to formulate in an MINLP model. A new method is suggested for overcoming the difficulties arising from the discontinuity of the Kremser equation. Our new method is reliable and faster than the big-M, multi-M, simple logic, and convex-hull formulations applied for this particular problem. Our method is tested on a small and a large MEN synthesis problem. The solutions for four well-known MEN synthesis problems are also presented. I. INTRODUCTION A mass exchange network (MEN) is a network of interconnected direct-contact mass-transfer units that employs mass separating agents to selectively remove certain components from different rich phases (E1-Halwagi and Manousiouthakis, 1989). Two main synthesis approaches can be employed to determine the optimal network; structure independent (Pinch analysis) and structure based (mathematical programming). In the case of mathematical programming, the synthesis of MENs is realised by the formulation and solution of a mixed integer non-linear programming (MINLP) optimisation problem (Papalexandri et al., 1994). In most cases, multistage mass exchangers are also used for constructing the mass exchange network. Assuming linear phase equilibrium relations, capital investment calculations of these units are most commonly based on the Kremser equation, which gives the required number of equilibrium stages for a given separation. Generally, discontinuous functions, such as the Kremser equation, are difficult to formulate in MINLP and may cause numerical problems in course of the solutions. Three methods are proposed in the literature for formulating discontinuous functions, namely the conventional big-M, the multi-M (Hui, 1999) and the convex hull (Balas, 1985) formulation. In this article we suggest two new formulations for the Kremser equation and compare them to formulations based on the above mentioned methods. We also discuss the MINLP solution of four MENS problems. Comparison is made between our MINLP solutions and the pinch based
1110 solutions of Hallale (1998). Computation times in this paper concern a SUN Ultra Sparc-1 workstation. 2. FORMULATIONS OF THE KREMSER EQUATION As mentioned above the the Kremser equation gives the required number of equilibrium stages for a given separation, in case of linear phase equilibrium relations. Depending on the value of the removal factor A, the Kremser equation for a given component i has two different forms. Throughout our calculations in GAMS for the case ofAr 1 we used the following form of the Kremser equation (E1-Halwagi, 1997): log
= Yi,out -- m j x j , i n
NTPA=I =
If A=I
N T P A ~ I 9log(A)
-- b j
Where: A =
(1) mjG i
Yi,~,,
- Yi,out
(2)
Y i,out -- m j x j,in -- b j
The removal factors of the units are design variables when solving the MINLP synthesis problem, their values have to be able to vary freely between their phisically imaginable bounds. A can equal 1 or can be less or greater than 1 also. Switching between the two forms of the Kremser equation in an MINLP modelling environment is not a trivial task at all. In GAMS / DICOPT for example it is not possible to build in conditional equations into the model. This is a common attribute of all the available solvers, and origins from the numerical algorithms they use. Using only the first form of the Kremser equation usually leads to a division by zero error or gives solutions that have no physical meaning. Restricting the values of A under or over 1 very likely excludes the real optimal solution from the search space. Without solving this numerical difficulty no reasonable MENs can be expected from the MINLP method. The discontinuity of the Kremser equation can be overcome in the following general way: A binary variable Y has to be defined which equals 1 in the case of A=l, and takes the value of zero when A~I. Then both equations are used to calculate the number of theoretical plates, and Y is used to select the one which corresponds to the value of A in the given mass exchanger.
NW
= r.
+ 0 - r ) . (Nr+'**, )
(3)
For calculating the binary variable Y five methods were examined. The first four formulations below divide the interval of A into three sub intervals with the boundaries of 0.01; 0.99; 1.01; 100, and switch to the formula valid for A=I when A is between 0.99 and 1.01. This interval division is arbitrary.
1111
2.1. Big-M A V >_ 0 It is convenient to write the nonlinear classification case in summation notation [ 10]. Let A be the total set of all the genes, A 1 and A 2 and let M - ml + m2 be the total number of genes. x, ~ A
,~ti =
dimensional
1
x~ e
feature
. Let t ~
be defined such that for
A 2 . The original data points x are mapped to the higher space
by
the
selected
transformation
function
#(x)" R" --> R"',n' >> n. The dot product of the original vectors xTxj is replaced by the dot product of the transformed vectors ~(x~).~(xj ). Thus the dual can be rewritten as:
min~ - , ~ 1 ' ~ ~ t i t j a i a j (qk(x).qk(x)) a / "-'I, i=l j=l
~ot,t, A4
s.t.
-
-
,=1
ai
(3)
=0
i=1
where 6 is equal to 61 and 62 for the terms corresponding to u and v, respectively. Vapnik's work on SVMs [11] allows one to replace the inner product q~(x,).O(xj ) with the inner product in the Hilbert space by the symmetric kernel function K(x, xi). The classification function is therefore dependent on the kernel function chosen. Example kernel functions are polynomial, radial basis functions and neural networks. It is interesting to note that the number of optimization variables in the dual problem remains constant irrespective of the kemel function chosen. After solving the primal (1) and dual (3) problems the resulting classification function is:
x) I where x e A 1 if f ( x ) > O, else x ~ A 2 .
1136 3. RESULTS AND DISCUSSION
The SVM model discussed above is used for annotating the budding yeast
Saccharomyces cerevisiae data set reported as a set of 2467 genes under 79 different experimental conditions. This data set was used by Eisen et. al. [1 ], for clustering and later by Brown et. al. [7]. The data was generated from microarrays using samples collected at various time points of the diauxic shift, the mitotic cell division cycle, sporulation and temperature and reducing shocks and is available on the web at http://www.rana.stanford.edu/clustering. We use the same functional class definitions made by MYGD and reported in Brown et. al. [7]. We use the SVM model to classify genes into six functional classes, namely tricarboxilic acid (TCA) cycle, respiration (RESP), cytoplasmic ribosomes (RIBO), proteasome (PROT), histones (HIST) and helixturn-helix (HTH) proteins. The first five classes have a biological basis since they exhibit similar expression profiles. HTH class is used as a control group since there is no reason to believe that the genes of this class are similarly regulated. The data has many missing values for the expression ratios. Only a small fraction (25%, 605 out of 2467) of the genes has all the 79 expression ratios reported. This smaller set of genes was used as the training set (see Table 1). A gene is referred to as a positive sample of a given class if it belongs to that class. Otherwise the gene is referred to as a negative sample. We observe that the training set has 2, 13, 49, 8, 5 and 6 positive genes for the respective functional classes. We note further that for all the functional classes, the negative sample size is a small fraction of the total negative gene set (less than 25% in all cases). Thus, the negative gene set might not be a good representation of the genes in the original data and therefore ought not to be used for training. Table 1 Training Set Data Class
PS
NS
PT
TCA RESP RIBO PROT HIST HTH
2 13 49 8 5 6
603 592 556 597 600 599
17 30 121 35 11 16
NT 2450 2437 2346 2432 2456 2451
PS%
NS %
11.76 43.33 40.50 22.86 45.45 37.50
24.61 24.29 23.70 24.55 24.43 24.44
PS: Positive Set, NS: Negative Set, PT: Total Positive Set, TN: Total Negative Set For classification and functional annotation, we employ a polynomial kemel function given as:
1+
(5)
1137 where n = 79, is the number of experimental conditions and d is the degree of the polynomial. We report the results for d = 2. The resulting classification functions are then used to classify the complete set of 2467 genes. It is observed that the support vectors generated for classification are only a small fraction of the training data set. It is also observed that the SVM based classification functions performs very well on the positive samples (see Table 2). An accuracy of nearly 100% is found for the positive samples for the first five classes and a low 81% accuracy is observed for the HTH proteins. The expression pattern within the HTH class is random and explains the low accuracy of positive sample classification. For the negative samples an accuracy rate of 56%-91% is observed. The remaining negative genes (9%-44%) are incorrectly classified as positive genes. This can be attributed to the missing values in the expression ratios and a small fraction of the negatives considered in the training set. It is to be emphasized that although the values are also missing for the positive samples, the support vectors generated are able to classify them correctly as shown Table 2. Table 2 Comparison of error rates for different classes Class
TP
TN
FP
FN
TP%
TN%
FP%
FN%
TCA RESP RIBO PROT HIST HTH
17 30 121 32 11 13
1876 1553 1315 2207 1373 1556
574 884 1031 225 1083 895
0 0 0 3 0 3
100 100 100 91.4 100 81.2
76.57 63.73 56.05 90.75 55.90 63.48
23.43 36.27 43.95 09.25 44.10 36.52
0.0 0.0 0.0 8.6 0.0 18.8
TP: True Positive, TN: True Negative, FP: False Positive, FN: False Negative 4. C O N C L U S I O N S AND O N G O I N G W O R K To improve the existing classification and the functional annotation, missing values can artificially be generated (i.e. imputed) and added to the expression ratio matrix using a data imputation technique such as hot decking, mean or median imputation, and multiple imputation based on EM algorithm [12]. However, such a technique may introduce bias into the existing data. We expect hot decking to be the most appropriate imputation technique for gene expression data imputation. This is because in hot decking, it is assumed that similar genes have similar expression ratios (i.e. similarity by homology) and therefore we can replace the missing value for a gene at the given experiment with the expression ratio of a similar gene at that experimental condition. Work incorporating hot decking imputed data is in progress. We are also currently investigating the use of the radial basis function and the artificial neural network as viable kernel functions. The feasibility of the SVM model for genome wide functional annotation of genes has been established by the present study. The results are very encouraging for the initial
1138 study in the presence of a large number of missing gene expression data. We observed that the positive samples are correctly annotated. The low percentage of true functional annotation for negative genes is attributed to missing values for the negative gene sets. We also noticed that some of the genes were assigned to more than one functional class. We expect that this multi-functional gene annotation feature of the SVM model will eventually lead to a better understanding of the underlying biological complexities in the genetic network of an organism. To the best of our knowledge this is the first study reporting the primal-dual SVM modeling strategy for functional annotation of genes based on microarray data. The ultimate goal of our research is to use this primal-dual SVM model for functional annotation of genes responsible for bone cell differentiation. This is a collaboration with bone cell researchers at the University of Connecticut Health Center researchers. REFERENCES
1. Eisen, MB, Spellman, PT, Brown, PO, Botstein D. Cluster analysis and display of genome-wide expression patterns. (1998) Proc. Natl. Acad. Sci. USA, 95, 1486314868. 2. Goffeau A, Barrell BG, Bussey H, Davis RW, Dujon B, Feldmann H, Galibert F, Hoheisel JD, Jacq C, Johnston M, Louis EJ, Mewes HW, Murakami Y, Philippsen P, Tettelin H, Oliver SG. Life with 6000 genes. (1996) Science 274, 563-567. 3. Consortium TC e S. Genome sequence of the nematode C. elegans: a platform for investigating biology. (1998) Science 282, 2012-2018. 4. Clayton RA, White O, Fraser CM. Findings emerging from complete microbial genome sequences. (1998) Curr. Opin. Microbiol. 1, 562-566. 5. Alon U, Barkai N, Notterman DA, Gish K, Ybarra S, Mack D, Levine AJ. Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays. (1999) Proc. Natl. Acad. Sci. USA, 96, 6745-6750. 6. Tamayo P, Slonim D, Mesirov J, Zhu Q, Kitareewan S, Dmitrovsky E, Lander ES Golub TR. Interpreting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation. (1999) Proc. Natl. Acad. Sci. USA, 96, 2907-2912. 7. Brown MPS, Grundy WN, Lin D, Cristianini N, Sugnet CW, Furey TS, Ares M Jr., Haussler D. Knowledge-based analysis of microarray gene expression data by using support vector machines. (2000) Proc. Natl. Acad. Sci. USA, 97, 262-267. 8. Hastie T, Tibshirani R, Eisen MB, Alizadeh A, Levy R, Staudt L, Chan WC, Botstein D, Brown P. 'Gene shaving' as a method for identifying distinct sets of genes with similar expression patterns. (2000) Genome Biology, 1(20), 1-21. 9. Burges CJC. A tutorial on support vector machines for pattern recognition. (1998) Data Mining and Knowledge Discovery, 2, 121-167. 10. Bredensteiner EJ, Bennett KP. Multicategory Classification by Support Vector Machines. (1999) Computational Optimization and Applications, 12, 53-79. 11. Vapnik, V. The Nature of Statistical Learning Theory. Springer-Verlag, New York, 1995. 12. Schafer JL. Analysis of Incomplete Multivariate Data. Chapman & Hall, London, 1997.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1139
On the Optimization of Drug Delivery Devices Michael C. Georgiadis* and Margaritis Kostoglou Centre for Research and T e c h n o l o g y - Hellas, Chemical Process Engineering Research Institute, P.O. Box 361, Thermi 57001, Thessaloniki, Greece. This work presents a novel optimization approach to achieve desired release rates in drug delivery devices using laminated layers. First, a mathematical model is presented to describe the drug release between successive layers laminated together to form matrices with different initial concentrations, drug diffusivities and thickness of each layer. First, an analytical-based optimization approach is performed and results compared with relevant work from the literature. Then, a formal optimal control approach is employed to determine the optimal initial concentration in the layers, along with their thickness and diffusivities, in order to achieve a drug release profile as close to required profile (e.g. constants release) as possible for all times. 1. INTRODUCTION The target of any controlled release system is the transfer of an active material (usually drug) from a reservoir to a target host, in order to maintain a predetermined concentration or emission level of the drug for a specified period of time or desired fraction released. In a large number of medical applications constant release rates are desired for drugs possessing a narrow range of therapeutic index. Diffusion control matrix devices have been among the most widely used drug delivery systems, mainly due to their low manufacturing cost. However, in conventional diffusion controlled devices, where the drug to be released is distributed uniformly through a polymer, the release of a dissolved drug from a homogeneous geometry inherently follows first order diffusion behaviour with an initially high release rate followed by a rapidly declining release rate. In order to achieve zero-order releases and especially to eliminate the initially high release rate, various methods have been proposed such as modification of the geometry of the device and the use of rate-controlling methods [1 ], [2]. An altemative approach which has been extensively investigated experimentally is the use of nonuniform initial concentration profiles as a mechanism for regulating drug release from diffusion controlled and surface erosion controlled matrix systems [3]. Several mathematical models have been presented to describe diffusion controlled systems containing dispersed drug [4],[5]. Other approaches presented mathematical models of simultaneous dissolution and diffusion controlled drug release from swellable viscoelastic matrices [6], [7]. However, limited work has been reported on the simulation and control of drug diffusion when its concentration is below its saturation solubility in the polymer and most of the presented mathematical models were successful in predicting release profiles from known initial concentrations [3]. * Author to whom correspondence should be addressed; email
[email protected] 1140 In general, to determine suitable initial parameters to obtain desired release behaviour requires a tedious trial-and-error simulation process. Only recently, optimal initial concentrations were calculated using optimisation techniques [8]. However, no work has been reported on the simultaneous optimisation of initial concentration, layers number and thickness and drug diffusivity (such as variations in the cross linking density) using formal mathematical methods. This can be mainly attributed to the complexity of the underlying mathematical problem. This work presents a novel optimisation approach to calculate rigorously the set of initial concentrations in a multi-layer device along with the thickness of each layer, their number and the drug diffusivities. 2. M O D E L L I N G A polymeric release systems with N layers is depicted in Figure 1. Ci, D i and 6 x i are the drug concentration, diffusion coefficient and thickness in each layer, respectively (i = 1,..,N). The disk has a thickness L and initial drug concentration Ci, o in each layer. It is assumed that drug diffusion is the rate-controlling step rather than swelling or drug dissolution.
x--0
Solventat sink conditions
dmkdiffusion IC Dill I~: Ih,,
x=L 5X1 ~X2
~xN Figure 1: Drug release
Mathematically, this problem is described using Fick's law. In a dimensionless representation the drug transfer between successive layers is described as follows:
OC i _ 1 02Ci c3-----~-(~ci) 2 Di ~0x
Vt > 0, Vi = 1,..,X
Cilx= 1 =Ci+l[x= 0
Vx~(0,1)
Vt>0, Vi=I,..,N-1
(1) (2)
1 Oi(OCiq 1.__~Oi+l(OCi+l~ Vt>0, Vi=I,..,N-1 (3) ~X--T ~-~X J x= I -- ~X i +I k ~X, ) x=O where the diffusion coefficients are assumed constant within each layer. The following boundary and initial conditions are also imposed: OC1 ~
--L-)x=0 Ci(x )
=0
= Ci, o (x)
Vt
V i,
> o, t = O,
Culx=l =0 x ~ (0,1)
Vt > 0
(4) (5)
The flux of drug is given as J = - D N ~ x1N (OCN ~, Ox ! x=l . The objective can be defined as the difference between desired drug release, J*, and the actual release rate. Cost considerations can also be incorporated to define the optimal drug release. For example increasing the
1141 number of layers one would expect to get a better release (closer to the desired one) however the overall system costs would increase due the construction of more layers. In general, there are clear trade-offs between improved release rates on the one hand and increased cost on the other. A general form of the objective function, OF, to be minimised over a total release time, t f , as follows:
OF =
(t)-
(t) dt + WL. N
(6)
o
where WL is a coefficient that represents the effect of number of layers in the objective function. 3. OPTIMIZATION The optimisation approach seeks to determine the optimal values of the available degrees of freedom, i.e. number of layers along with their thickness, initial drug concentration and diffusivities, in order to minimize the above objective function and satisfy certain constraints. A typical constraint may represent a minimum or specified fractional release. Alternatively, a fixed final release time can be imposed. However, determining the optimal values of the control variables is not a trivial tasks especially when considering their synergistic effects. Thus a formal optimisation approach needs to be employed. We consider two cases: optimisation based on an analytical approach utilizing only one control parameter and optimisation using formal optimal control numerical techniques.
3.1. Analytical Approach In the analytical approach the laminated disk is modelled as one layer where the initial drug concentration is distributed along the axial domain. The diffusion coefficient is also assumed uniform. Here, the only available control parameter is the initial concentration, C o (x). The first step is to parameterize the unknown functions by expanded each of them in a K-1 series of K known basis functions with unknown coefficients i.e. Co(x ) Z a i ~ i ( X ) =
i=O The solution of tick' s equation with the boundary conditions is: oo
C(x,t)= ~f'fjcos((j+l/2)~x)e-bjtwhere j=0 expansion coefficients K-1 1
of
f j = ~ 2a i I~oi(x)cos((j + 1/2)nx)dx. i=0
b j = ( ( j + l / 2 ) n ) 2 and
the
initial The
flux
fj
are the Fourier
distribution can
then
given be
as:
expressed
0
asJ(t) = ~ fjkje -b't where kj = n(-1)J(j + 1/2). In order to proceed our analysis one must j=0
choose the basis functions q)i. As a first attempt a set of global, orthogonal and infinitely differentiable functions are tested. The basis functions are % (x)= cos((i + 1/2)nx). After a detailed mathematical analysis the flux can be expressed as:
1142 K-1
J(t) = ~ ajkje -bit and
the
objective
function
takes
the
form
j=0 t~
K-1
F = ~(J*(t)-~ t,
ajkje-bjt)2dt.
In order to minimize F the derivatives
OF
(for
Oai
j=0
i=0,1,..K - 1) are set equal to zero. After performing some analytical integration the following 1in ear sy stem of eq uati o ns gi ves:
K-1 Z aj j=0
kj bi + bj
tl (e -(b~+b~)t' - e -(b~+bj)tl )= ~e-b'tj * (t)dt
(7)
. The above K x K s y s t e m
t,
must be solved for the K expansion coefficients ct0,al,..,aK_ 1. This is an unconstrained minimization procedure that usually give as a result an initial concentration distributions with negative values in some regions. To avoid this type of solutions a term of the following form K-1
is added to the objective function: w i[Co(x)]2 dx = w -1 /~0 a2 where w is an appropriately 0 2.= selected weighting factor. This term has as effect the reduction of oscillations of
C O(x). As w
increases, the exclusion of regions with negative values of
C O(x). Here a typical case where
the flux must be constant
t i = 0to tf = 0.5will be studied.
= 1 during the period from
This case has been numerically optimized in the literature [8]. For this particular case the integration in the right hand side of the equation (7) and objective function can be performed analytically. For K4 the initial distribution takes negative values. To overcome the problem of having an optimum distribution without physical meaning, positive values are given to w in order to reduce the oscillations. Due to space limitations the detailed analysis is not presented here. The optimum initial distributions with physical meaning for K=2, 3, 4 and 6 are depicted in Figure 2 along with the corresponding fluxes. Except for the case with K=2, the others exhibit a similar flux behaviour. It is worthwhile to note that the optimal fluxes obtained using our simple analytical approach are almost identical to the ones presented in the literature based on a complicated optimization procedure and utilizing many degrees of freedom [8].
Figure 2: Optimal initial concentration and flux profiles based on an analytical approach
3.2 Dynamic Optimization Approach. A dynamic optimisation approach based on control vector parameterisation (CVP) techniques
1143 is employed to minimize objective function (6) subject to model equations (1) to (3) along with the boundary and initial conditions (4) and (5) and a requirement for a 65% drug release. Other constraints include lower and upper bounds on the time invariant optimisation parameters (e.g. lengths, initial concentrations) are also considered. The form of the model equations after the normalization over fixed domain (from zero to one) allows the direct use of CVP techniques [9]. First, a spatial discretisation approach is employed to eliminate the independent axial distance variable. This leads to an optimisation problem described by differential algebraic equations (DAEs). Nonetheless it remain a complex, infinite dimensional nonlinear optimization problem. The infinite dimensionality arises because the state variables are a function of time rather than scalar quantities. The CVP approach converts the infinite dimensional optimization problem into a finite one and allows the use of standard non-linear programming (NLP) techniques for the solution of the finite-dimensional problem [9]. Due to space limitations the details of this approach are not presented here. The dynamic optimization problem is solved using gOPT, an implementation of the CVP approach in the gPROMS modelling system [10]. The normalized axial domain is descritized using second order orthogonal collocation on 20 finite elements. The equations that describe the model are generated by gPROMS as residual equations with symbolically generated partial derivatives (Jacobian) and used as input to gOPT. The latter employs a sophisticated integrator for the integration the DAEs and a SRQPD nonlinear programming code implementing a reduced non-linear programming code [ 11 ]. Optimal flux profiles for three and seven layers are shown in Figure (4a). For the purpose of our analysis WL has been set to zero. However, the optimal number of layers can easily be determined by solving a number of optimisation problems for different number of layers. We observe, as expected, that as the number of layers increases an improved constant release (closer to the desired release) is obtained. Figure (4b) presents a comparison for a 7-layer matrix for two cases: (i) all optimization variables are utilized and (ii) the diffusivities are kept constant equal to one. The results indicate that case (i) leads to a significantly improved released compared with the case where only two degrees of freedom are utilized. This clearly indicates that there are synergistic effects between the control variables and an integrated formal optimisation approach is clearly desirable. Comparing with the results of the analytical approach presented in the previous section and also with other results from the literature [8] it can be concluded that the proposed dynamic optimisation method favorably compares (almost one order of magnitude better value of the objective) leading to releases which are very close to the desired one. The values of optimization variables are shown in Table 1. It is clear that near to the exit of the device two very thin layers are employed to achieve tighter control of the release. The approach has also been extended to cases of non-constant releases and investigated the optimal number of layers. Due to space limitation results are not presented here. Layer Number D Co 6x
5 6 7
0.001 0.85 0.25 0.08 0.045 0.001
0.001 0.03 0.52 1.5 2.08 1.36 1.23
0.02 0.08 0.23 0.40 0.23 0.02 0.02
Table 1" Values of Optimization Variables for a 7-layer matrix
1144
2.4 2.2 20 m
~" 1.8
._ a
-
' 1.2
~IL. -
1o %~__
"-.2Q-
-.2
0.8 0.8 0.4
0.1
0.2 03 04 Time (Dimensionless)
0.5
Figure 4: (a) Flux Profiles for different layers
0
02 03 04 Time (Dmens~onless)
(b) comparison of optimization approaches
4. C O N C L U D I N G R E M A R K S
This work has considered the exploitation of optimisation approaches to achieve desired release rates in drug delivery devises. The dynamic optimisation method, formally utilizing all degrees of freedom available in the system, leads to significantly improved releases compared with an analytical approach and other simpler approaches from the literature. The proposed optimization approach can be extended to cases where the initial drug is greater than its solubility limit in the release medium and in a different geometry (e.g. polymeric hemispheres). Finally, it is worth pointing out that, because of the local optimization algorithm for the solution of the NLP, the global optimality of any solution obtained with our approach cannot normally be guaranteed. This is a common deficiency of optimization-based design methods than can only be overcome by the adoption of global optimization techniques. REFERENCES
1. E.S. Lee, S.W. Kim, J.R. Cardinal and H. Jacobs, Journal of Membrane Science, 7 (1980) 293. 2. U. Conte, L. Maggi, P. Colombo and A.L. Manna., Journal of Controlled Release, 23 (1993) 39. 3. P.I. Lee, Journal of Controlled Release, 4 (1986) 1. 4. D.R. Paul, Journal of Membrane Science, 23 (1985) 221. 5. B. Narasimhan, and R. Langer., Journal of Controlled Release, 47 (1997) 13. 6. M. Grassi, R. Lapasin and S. Pricl., Chem. Eng. Comm., 169 (1998) 79. 7. M. Grassi, R. Lapasin and S. Pricl., Chem. Eng. Comm., 173 (1999) 147. 8. S. Lu, W.F. Ramirez and K.S. Anseth., AIChE Journal, 44 (1998) 1689. 9. V.S. Vassiliadis, R.W.H. Sargent and C.C. Pantelides, Ind. Eng. Chem. Res., 33 (1994) 2123. 10. gPROMS Advanced User Guide, Process Systems Enterprise Ltd, London (2000). 11. C.L. Chen, and S. Macchietto, Technical Report, Centrefor Process Systems Engineering, Imperial College of Science Technology and Medicine, (1988).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1145
An Integrated Methodology for Developing Inherently Safer and Environmentally Benign Processes I. Halim, C. Palaniappan and R. Srinivasan* Laboratory for Intelligent Applications in Chemical Engineering, Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Stringent safety and environmental regulations, and cutthroat competition have challenged the chemical process industries to bring products to market at low life cycle costs without compromising on safety and environmental standards. This has led plant designers to consider inherent safety and waste minimization principles at early stages of the design process. Tools and methods are available for developing inherently safer process and carrying out waste minimization analysis individually without taking into account the close coupling between them. This results in an incomplete and inaccurate analysis. In this paper, we present a systematic methodology for the integrated safety and waste minimization analysis during process design. This is done using material-centric approach, which brings out the similarities between issues and prevention strategies related to inherent safety and waste minimization. The integrated methodology is discussed and illustrated on an industrial process involving acrylic acid production process. 1. INTRODUCTION Conventional design of chemical plants has been primarily driven by factors related to economics and engineering. Issues concerning safety and environment are usually addressed at the later stages of design. This approach often leads to extensive end-of-pipe treatment and add-on safety features to reduce the consequences of acute hazards and chronic effects of a release. Intense competition, demand for consistently high product quality and more stringent safety and environmental regulations have challenged process designers to develop inherently safer and environmentally benign process. The term inherently safer implies that the process is safe by its very nature and not due to the use of add-on safety systems and devices. This is normally accomplished by reducing the use of hazardous materials and unsafe operations, minimizing inventory, moderation of operating conditions and by designing a simpler plant. The concept of waste minimization incorporates any technique, process or activity, which avoids, eliminates or reduces a waste at its source, or allows reuse or recycling of the waste. Both inherently safer process development and waste minimization techniques share the same basic philosophy, i.e., eliminating undesirable traits of a process. This often leads to synergies, for example substitution of material with safer and environmentally benign material. Nevertheless, inherently safer processes are not necessarily environmentally benign even though inherent safety concepts address certain environmental issues. For example, the use of CFCs as refrigerants is inherently safer with respect to fire, explosion and acute ,
Author to whom correspondence should be addressed (email:
[email protected])
1146
toxicity hazards as compared to altematives such as propane. However, from the environmental perspective, propane is more desirable since CFCs cause ozone depletion. Thus, there are times when tradeoffs between developing inherently safer and environmentally benign process have to be reconciled in order to design an all-round better process. The need for an integrated methodology to analyze both safety and environmental issues and their interactions has been emphasized in literature [ 1,2]. Despite the growing need and obvious importance of such a design approach, its adoption into practice has been quite slow. This is mainly due to factors such as time and cost constraints during the design, conservatism in design, and lack of supporting tools. Research on developing an intelligent tool for waste minimization analysis and development of inherently safer chemical processes has been ongoing in our group. An intelligent waste minimization tool called ENVOPExpert has been developed and successfully tested on several industrial case studies [3, 4]. An inherent safety analysis tool, called/Safe, which can assist plant designers by identifying safety issues and recommending solutions is also being developed [5]. These tools highlight the issues and offer recommendations considering safety or pollution individually without taking into account interactions between them. In this paper, we present an integrated methodology for inherent safety and waste minimization analysis. The task for the integrated analysis can be defined as follows: Given the details about materials involved, process chemistry, flowsheet and the reaction and separation schemes of a chemical process plant, the goal is to identify opportunities to minimize the hazards and wastes in that process by evaluating the synergies and tradeoffs between the two. The organization of this paper is as follows: in the next section, the methodology and intelligent system for integrated analysis based on a material-centric approach is proposed. These are illustrated in Section 3 on an acrylic acid production case study; finally, in Section 4 the overall conclusion about the approach is presented. 2. M E T H O D O L O G Y FOR SAFETY & POLLUTION PREVENTION ANALYSIS The integrated methodology for developing inherently safer and environmentally benign processes is based on a material-centric view of a process. Inherent safety principles address the prevention of unintended effects of materials while waste minimization principles deal with the minimization of their release to the environment. Process materials form a focal hub and mold the hazards and wastes occurring in a process. Hazards as well as wastes can be therefore recognized and remedied by focusing on process materials. Figure 1 describes this material-centric view of safety and waste-related issues. Here, both issues have been organized based on their source as due to (1) individual materials and their properties (2) interactions between two or more materials, for example through reaction (3) interaction between material and process conditions, and (4) interaction between material and process unit. Figure l a shows some examples of safety issues that originate from these four sources. Similarly, Figure l b shows examples of pollution issues from them. The organization of safety and waste issues along this material-centric scheme brings out the essential similarities between the sources of hazards and pollution. A comprehensive inherent safety analysis would consider reduction of material inventory, substitution of material with a safer one, moderating the conditions at which a material is processed, use of simpler design to minimize the possibility of material release, etc. The material-centric view also lends itself to representing the different inherent safety principles used to develop a safer plant. Figure l c shows the five common inherent safety principles- substitution, attenuation, intensification,
1147 simplification and toleration- arranged in the material-process unit-process condition mold. A waste minimization study would focus on elimination or reduction of waste generation at the source by altering the process conditions, or substituting with an environmentally friendly material, recycling of material, etc. Thus, the waste minimization principles also can be cast in the material-centric view as shown in Figure 1d. The strong parallels between hazard and pollution prevention strategies are also clearly brought out in this material-centric view thus facilitating an integrated hazard and waste analysis. Figure 2a illustrates that the sources of environmental and safety issues can be merged into an integrated material-centric cast. The similarities in the philosophy of waste-minimization and inherent safety principles in tackling the hazards and pollution due to each of the four sources is depicted in Figure 2b.
( ~Reaction
Was(~ Ozonedepletion i generation~tential
T o x i c ~ ards I
n
~
em -- " 0
Fug.ltlve L ~ , ~
Fig l a. Safety Issues
M: Material P : Process conditions U : Process Unit
Fig lb. Environmental Issues
.
~
(•ecycle
Elimination stitution
rSeducfion urce ~ s t i t u t i o n S
Fig lc. Inherent Safety Principles
Dust Pollution
i
m
p
l
~
, i~
Fig ld. Waste Minimization Principles
Figure 1" Material-centric Approach to Identify and Resolve Safety and Pollution Issues The safety and environmental issues related to a material can be identified by evaluating its properties such as flash point, threshold limit value, ozone depletion potential, etc. Issues such as decomposition at high temperature and dust pollution due to small particle size arising from interaction of materials with process conditions are identified by evaluating changes in physical and chemical aspects of the material. Hazards and pollution arising from materialmaterial interaction such as waste generation, run away reaction, etc are identified by evaluating intended and tmintended reactions occurring in the process. Issues due to materialunit interaction such as leak of toxic chemicals from storage, fugitive emissions from flanges, etc are identified by evaluating failure modes of the equipment. Once the issues have been identified, suggestions to rectify the issues can be proposed using common keywords derived from inherent safety and waste minimization principles. Table 1
1148 shows examples of such keywords and variables applied to material, process conditions and process unit along with the suggestions derived using them. For example, in order to rectify all safety and pollution issues arising from a solid, the suggestion "Modify the particle size distribution of the solid "can be proposed. Similarly, in order to improve the absorption of useful material, the suggestion "Increase the pressure in the absorber" could be recommended. For a reactor unit that produces waste, the suggestion "Optimize the operating conditions of reactor" can be proposed. The synergies and tradeoffs from the safety and pollution perspective can then be highlighted to the designer. The synergies between the two suggestions are self-evident when making the same change made to the item (material, unit or process condition) results in improved safety and waste performance. For example, replacing benzene solvent with water would improve both safety and environmental performance. Similarly, self-evident trade-offs occur if the change has conflicting effects on different issues. The example of refrigerant CFCs versus propane described earlier falls in this category. Both the synergies and trade-offs can be found by comparing the effects of each recommendation on issues in the following order: (material, process conditions, unit), (temperature, phase) and (change, optimize, recycle) respectively. When the item and variable match, depending on whether the keywords concur or conflict, the tradeoffs and synergies can be identified. Common safety and environmental indices can be used as yardsticks for measuring inherent safeness and environmental friendliness of a process. Fig 2a. Integrated identification of Safety Pollution Issues
Material-Material interactions Material-Unit interactions
Material
condition interactions
&
Fig 2b. Integrated Inherent Safety Waste Minimization Solutions
EliminateRecycl~sS~ e
reduction
Simplify ~
ubstitute
&
Intensify
Synergies ~ ~ / ~ ~ Tradeoffs Figure 2" Commonalties between safety and environmental issues and solutions In order to automate the methodology described above, the tool for combined approach must be capable of identifying hazards and pollution issues that arise due to material and its interaction with process based on ~.nformation on materials, process chemistry, and reactionseparation scheme of a process. We have earlier developed an intelligent system called ENVOPExpertthat uses P-graph models along with digraphs and functional models for waste minimization [4]. An expert system-/Safe - that uses the same models for inherent safety analysis is currently under development [5]. In both systems, P-graph models represent the cause and effect relationship of materials, reactions and separations involved in the process. Similarly, digraph and functional models represent the relation between process variables and
1149 the issues related to safety and pollution. The knowledge bases of the two systems are being merged based on the common material-centric view and implemented in G2 as an intelligent expert system for integrated safety and waste minimization analysis. Table 1" Keywords for generating suggestions
3. CASE STUDY: ACRYLIC ACID PROCESS
We have performed a combined inherently safety and waste minimization analysis using our integrated methodology on an acrylic acid case study obtained from literature [6]. Figure 3 shows the flowsheet of the process. Acrylic acid is produced by partial oxidation of propylene in a fluidized-bed-catalytic-reactor. Reaction products are quenched immediately using cold quench recycle and off-gas is absorbed using deionized water. The quenched stream is sent to extraction column in which diisopropylether is used as solvent to separate the products from waste streams. Synergy with respect to change in operating conditions in reactor and trade-off with respect to change in operating pressure in off-gas absorber in order to improve safety and environmental performance have been identified. Two suggestions derived using our integrated approach along with the synergies and tradeoffs are shown in Table 2. 4. CONCLUSIONS Development of inherently safer and environmentally benign process is of prime importance in today's business environment. In this paper, we have discussed the need for an integrated methodology for safety and environmental impact analysis. A material-centric approach to identify hazards and pollution issues has been developed. A guideword based solution generation technique, which meshes the material-centric view with inherent safety and waste minimization principles, has also been proposed. We have discussed an intelligent system for automating the integrated analysis and illustrated the methodology using an industrial case study. REFERENCES
1. R. D. Tumey, "Designing plants for 1990 and beyond: Procedures for the control of safety, health and environmental hazards in the design of chemical plant," Trans. 2. I ChemE, vol. 68, pp. 12 - 16, 1990.
1150 3. K. Lien and Perris, T., "Future directions for CAPE research: Perceptions of industrial needs and opportunities," Computers and chemical engineering, vol. 20, pp. S1551-S1557, 1996. 4. I. Halim and Srinivasan, R., "An Intelligent System for Identifying Waste Minimization Opportunities in Chemical Processes," presented at European Symposium on Computer Aided Process Engineering - 10, 2000. 5. I. Halim and Srinivasan, R., "A Hybrid Qualitative-Quantitative Approach of Waste Minimization Analysis in Chemical Processes," presented at AIChE Annual Meeting, Los Angeles, Paper No. 233r, 2000. 6. C. Palaniappan, Srinivasan, R. and Tan, R.B.H., "An Intelligent Support System for Design of Inherently Safer Process Flowsheets," presented at AIChE Annual Meeting, Los Angeles, Paper No. 249b, 2000. 7. R. Turton, Bailie, R.C., Whiting, W.B. and Shaeiwitz, J.A., Analysis, Synthesis and Design of Chemical Processes. New Jersey: Prentice Hall, 1998.
Table 2: Synergies and Tradeoffs for Safety and Pollution for Acrylic acid Case study
European Symposium on Computer Aided Process Engineering - 11 R. Gan] and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1151
Particle Size Distribution by Design Priscilla J. Hill a and Ka M. Ng b a
Dept. of Chemical Engineering, University of Minnesota Duluth, Duluth, MN 55812, USA
b Dept. of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong This paper presents a strategy for designing solids processes to produce particles with a desired size distribution. Heuristics, and generic flowsheet structures are used to guide decision making. Computer simulations based on discretized population equations along with suitable solids processing models are used to meet the particle size distribution target. 1. INTRODUCTION It is well recognized that the product quality of many solids products depends on the product's particle size distribution (PSD). For example, the PSD of the powder in a tablet is carefully controlled in the manufacture process because the particle size has a significant effect on the dissolution and absorption rate in the gastrointestinal tract. Much work has been done on predicting PSD in individual unit operations such as crystallizers. However, the PSD may still change considerably as the crystals migrate through the downstream filters, washers, and dryers [1 ]. Similar situations are present in bulk solids systems in which the PSD changes due to breakage and granulation [2]. Thus, one must investigate the effect one unit has on other units in the entire system. To meet the need to track changes in the PSD as solids flow through a process, a modular simulation code based on discretized population balance equations was developed [3-6]. Although PSD tracking is necessary to evaluate a given process, it does not provide guidance in synthesizing a process that will produce a product with the desired product PSD. In this paper, we discuss the strategy that has been developed for process synthesis with PSD.
2. PARTICLE SIZE DISTRIBUTION BY DESIGN This hierarchical, multi-scale approach considers issues ranging from flowsheet structure to fundamentals such as the choice of the functions used to model particle breakage [5]. We start out with a general flowsheet and gradually add details as necessary. This method consists of a series of steps (Table 1). The first step is to gather all available information on the process. This includes material properties, chemical reactions, and crystallization kinetics as well as the process feed conditions, the design specifications (particularly the PSD specifications), and any design constraints.
1152 The second step is to develop a flowsheet for the process. One way of doing this is to start with a generic solids flowsheet such as the one for a crystallizer-filter-dryer train (Figure 1). Using the information gathered in the first step, the generic flowsheet is modified to produce a general flowsheet for the process. Heuristics are used to help in decision making. For example, if the crystals from the crystallization system meet the purity requirement, then obviously recrystallization is not included in the process flowsheet. Several alternatives may be generated in this step. Table 1. Strategy for particle size distribution by design Step 1. Input Information Product specifications, feed conditions, solid-liquid equilibrium data, crystallization kinetics, process constraints, etc. Step 2. Selection of Functional Structures Identification of the primary functions to be performed to meet the PSD target. Step 3. Selection of Equipment for Functional Structure Specification of equipment types and connections among the various streams. Step 4. Evaluation of Process Alternatives Based on Discretized PBEs Determination of process feasibility as well as the best process alternative. A different solids subsystem has a different generic flowsheet For example, a bulk solids system would involve screens, blenders, conveyors, crushers, granulators, etc Generic flowsheets for bulk solids have been reported [2, 7] Liquid Recycle ! T
Reaction, Extraction, Feed(s)--~Pretreatment ~-i~ and/or Dissolution
Crystallization System
Solid/Liquid Separation
1
d
Liquid Recycle ......................................................................
"J~IL
T
Recrystallization System
Solid/Liquid Separation
Postprocessing
Product with Desired PSD
Figure 1 Generic flowsheet for a crystallizer-solid/liquid separation train
1153 The third step is to select equipment or equipment items for each block on the flowsheet. This requires a knowledge of equipment capabilities and an understanding of the objective of each process operation. A short list of guidelines for choosing equipment systems is given in Table 2. Since there is more than one equipment choice for some of the steps, more process alternatives will be produced in this step. Table 2. Guidelines for selection of equipment for functional structure 1. If the desired PSD falls within the range of the PSD in the effluent of the particle generation unit (a crystallizer in this case) 9 Use hydrocyclones to remove the fines and large particles. Recycle these particles to the crystallizer feed. 9 If the amounts of fines and large particles are relatively small, leave them in the main processing train. Use screens after the dryer to obtain the desired PSD. Recycle the oversized particles to a crusher and the undersized particles to the crystallizer upstream. Alternatively, send the undersized particles to a granulator (see Figure 3). 2. 9 9 9 9 9
If particles from the particle generation unit are not sufficiently large for the desired PSD Avoid using reactive crystallization which tends to produce small particles. Change crystallizer operating conditions to produce larger particles, if possible. If recrystallization is necessary, switch to another solvent that produces larger particles. If all of the above does not work, use an agglomeration system after the dryer. Consider the use of spray dryers that may lead to larger particles.
3. If the particles are too large for the desired PSD 9 Change crystallizer operating conditions to produce smaller particles. 9 Use hydrocyclones to remove the oversized particles and return them to the crystallizer feed after dissolution. 9 Use a crusher after the dryer.
The fourth step is to evaluate the alternatives. Once the flowsheet and equipment units are chosen, simulation can be used to evaluate the altematives. There are still many decisions to be made at this stage. These include determining the significant phenomena which must by included in modeling the unit operations, the functional forms used in the population balance equations, the appropriate design variables, and the operating conditions. The particle tracking simulations are used as a tool to evaluate the altematives. From the fourth step, feasible operating conditions can be determined for each process alternative. If an economic analysis is performed, the feasible operating conditions can be evaluated to determine the more economical options. 3. SOFTWARE CONSIDERATIONS To implement this strategy, one must have the appropriate software. The computer simulation programs must include adequate methods for solving the equations. Discretized population balance equations are used in our code because they are very
1154 robust and require relatively modest calculational effort. They must also be able to incorporate the fundamental chemistry and physics needed to model the unit operations. They should also be able to include cost estimation and economic analysis. The software platform should be flexible enough to allow models from different platforms. For example, the main interface can be written in a general code which allows users to connect to commercial process simulators, commercial thermodynamics packages, spreadsheets, experimental data, and user supplied routines in FORTRAN or C. 4. CASE S T U D Y - SALT PRODUCTION
To illustrate the design procedure, the step by step development of a process for the manufacture of NaC1 salt is shown. Step 1: The objective is to crystallize NaC1 from a liquid brine at a rate of 105 ton/yr with no more than 5 mass % fines less than 160 ~tm. This process does not have any chemical reactions and recrystallization is not required for purification. Other data is given in Table 3. To meet the production rate, we will use a continuous process. Table 3. Selected input data for the salt plant Production Rate Feed Concentration Liquid Fraction in Feed Solubility of NaC1 kg NaC1/kg H20 Crystallizer Operating Temperature Agitator Speed, N Growth Rate Nucleation Rate MT 105.8 kg/m 3 Hydrocyclone cx~r 0.2 Rut* 0.15 Filter Rotation Speed, co Angle of Submergence, Vacuum Level, Ap Filter Cake Porosity, ~0
105 tons/yr = 185.4 kg/min 214.6 kg NaC1/m3 solution 8f = 1.0 SN = 0.35712 + 8.48x10 -5 T +3.17x10 6 T 2
50~ 500 rpm 6 ~trn/min 0.32 N 2 G 2 MT no./min/m 3
5 rpm 145~ 3500 N ] m 2 0.412
~rcx is the ratio of solids in the overflow to the solids in the underflow :~ Rut is the underflow to thoughput ratio Step 2: Based on Step 1 and Figure 1, we can generate a general flowsheet. Since preprocessing, reaction, and recrystallization are not needed, they are not included. Step 3: In this step we define the blocks in the generic diagram and determine the recycle stream connections. In this case we have chosen an evaporative crystallizer for removing excess water from the brine as well as crystallizing the product. The solid/liquid
1155 separation block could be represented by a single filter. However, a block is not limited to a single equipment unit; it can be a system. A second alternative is to use a hydrocyclone to separate undersized particles from the crystallizer effluent, followed by a filter. Different types of filters could be chosen for this operation including a centrifuge or a rotary vacuum drum filter. For the evaluation step, we have chosen a rotary vacuum drum filter. Another alternative to consider is the liquid stream exiting the solid/liquid separation unit. It could exit the process as a waste stream or it could be recycled to the crystallizer. Since more salt could be recovered by recycling the liquid stream back to the crystallizer, this would probably be the better option. One danger with this is that impurities could build up in the system, but this could be controlled by taking a purge stream off of the recycle stream. Step 4: Now that there are alternative flowsheets to evaluate, let us consider the flowsheet shown in Figure 2. In this flowsheet the salt production rate is P and the quantity of solids recycled from the hydrocyclone to the crystallizer is ~P, where ot is a function of the cutoff value, ds0, in the hydrocyclone. The cutoff value is the particle size where half of the particles exit in the underflow stream and the other half exit in the overflow stream of the hydrocyclone. To meet the product specifications of no more than 5 mass % fines less than 160 ~tm, we can adjust ds0. A series of simulations were performed in which ds0 was varied and the magma density was held constant at the same temperature and supersaturation. The residence time was allowed to vary to keep the magma density constant. Figure 3 shows the PSD in the actual product stream. As ds0 is increased, the fines decrease and the dominant particle size increases. A cutoff value of 141 ~tm or larger must be used to meet the fines constraint.
Water 1 Brine ~ ~
~P
I
Crystallizer
.J Filter S~t P Product Figure 2. Salt manufacturing process.
1156
85o --,~- 2 1 6 141 --~ 5 8
%
j,; 0
200
400
600
800
1000
Particle size, ~ m Figure 3. Effect of dso on product stream PSD. 5. CONCLUSIONS A four step strategy for designing solids processes to produce a specified PSD is presented. The method consists of gathering information, developing general flowsheets, specifying equipment units or systems, and using simulation with evaluation to determine feasible flowsheets and operating conditions. An accompanying computer code has been developed to facilitate such design efforts.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Chang, W.-C., and K. M. Ng, AIChE J., 44, 2240 (1998). Wibowo, C., and K. M. Ng, AIChE J., 45, 1629 (1999). Hill, P. J., and K. M. Ng, AIChEJ., 41, 1204 (1995). Hill, P. J., and K. M. Ng, AIChE J., 42, 727 (1996). Hill, P. J., and K. M. Ng, AIChE J., 42, 1600 (1996). Hill, P. J., and K. M. Ng, AIChEJ., 43, 715 (1997). Gruhn, G., J. Rosenkranz, J. Werther, and J. C. Toebermann, Computers Chem. Engng., 21, S 187 (1997).
European Symposium on Computer Aided Process Engineering - 11 R. Ganl and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1157
Optimization in Molecular Design and Bioinformatics Costas D. Maranas a aDepartment of Chemical Engineering, The Pennsylvania State University This work is an exposition on the application of optimization tools to problems in molecular design and bioinformatics. The specific areas addressed by the author include the design of polymers, surfactants, refrigerants, and enzymes. The goal is to systematically design molecules for the given application with desired performance characteristics. The performance measures of interest in polymer design are mechanical, electrical and thermophysical properties. In case of surfactants properties such as the HLB, emulsivity, detergency, and foaming stability influence the performance significantly. The performance measure in refrigerant selection and cycle synthesis is the balance between operating costs related to energy input and the investment costs. The performance measure in enzyme design is the probability of achieving a given nucleotide sequence target. The role of optimization is to "systematically" search through the alternatives. The research results in each of the applications mentioned above are presented. 1. INTRODUCTION The competitive edge and market share of many chemical industries manufacturing polymers, refrigerants, solvents, surfactants, enzymes, and biomaterials are ultimately intertwined with the identification of "new" and "better" products. Though the vast number of alternatives presents a designer with an opportunity to find a better product, it also poses the challenge of systematically searching through the alternatives. With the rapid growth in optimization theory, algorithm development and high-performance computing, exciting and unprecedented research opportunities are emerging in molecular design to assist in this endeavor. Research results in polymer design, surfactant design, refrigerant selection and enzyme design are discussed in this work. Previous work include the computer-aided design of molecular products such as polymers [9,7,5], solvents [5] and refrigerants [4,5] to name a few. The employed search algorithms include enumeration techniques, knowledge-based strategies, genetic algorithms and mathematical programming based methods. A comprehensive review of prior work can be found in Camarda and Maranas [3]. The objective is to find a molecule for a given application which optimally satisfies the desired performance targets.
2. POLYMER DESIGN In polymer design the problem of identifying the polymer repeat unit architecture so that a performance objective that is a function of mechanical, electrical and/or physicochemical properties is addressed. Since the molecular design problem is posed within an optimization
1158 framework, a quantitative representation of the molecule and a quantitative structure-property relation is required. Group contribution methods (GCM) provide popular, versatile and relatively accurate ways for estimating properties based on the number and type of molecular groups participating in a molecule or repeat unit. (GCM) are based on the additivity principle of the groups constituting the molecule under investigation and have been extensively utilized in the estimation of a wide spectrum of polymeric properties including volumetric, calorimetric, thermophysical, optical, electromagnetic and mechanical properties. An extensive compilation of these estimation methods along with the corresponding parameters can be found in van Krevelen [11]. The use of (GCM) makes adequate the molecular representation using n=(nl ,n2,... ,nN) where ni are the number of groups of type i present in the molecule. The problem of identifying the best molecule based on some measure of performance can be expressed as the following mixed-integer nonlinear optimization problem.
min
MP(pj(n)) pjL _< pj (n) _< p~
subject to ni
(OMD)
C
{nL, n L + l , . . . , n U } ,
i=l,...,N
The following two most widely used measures of performance are considered in this study [7]: (1) Minimization of the maximum scaled deviation of properties from some target values (property matching (PM)), 1
min M P - max ~ I p j ( n ) - P~I
J
Pj
where p~ is the target for property j and p} the corresponding scale. (2) Minimization/maximization of a single property j* (property optimization (PO)), min / max M P = p j. (n). To maintain structural feasibility of the molecule a number of linear constraints on n must be included in the problem (OMD). These structural feasibility constraints define the necessary conditions under which a set of molecular groups can be interconnected so that there is no shortage or excess of free attachments. The estimation of most properties pertinent to engineering design is given by the ratio of two linear expressions in ni. Though the above formulation is a mixed integer nonlinear program (MINLP) in general, the underlying mathematical functionalities of the above property estimation model are utilized to reformulate and solve the problem as a mixed integer linear program (MILP). One of the limitations of group contribution estimation is that the internal molecular structure of the polymer repeat unit is only partially taken into account. For example, both polypropylene -CHzCH(CH3)CH2CH(CH3)- and head to head polypropylene-CHzCH(CH3)CH(CH3)CH2-, have the same molecular group representation. These shortcomings are alleviated with the use of property correlations involving topological indices as structural descriptors. These indices are numerical values which uniquely identify the polymer repeat unit and contain information about the atomic and electronic structure. Specifically, Bicerano [ 1] used the zeroth- and firstorder molecular connectivity indices to correlate a wide range of polymer properties, including
1159 density, glass transition temperature, bulk modulus, and heat capacity. The functional form of the topological indices used are given in Camarda and Maranas [3]. The following additive property predictive form is utilized: (Property Prediction) = (Basic Group Contribution) + (Connectivity Indices Contribution) Though in general the above problem is a nonconvex MINLP, it is reformulated and solved as a convex MINLP utilizing the mathematical functionality of the connectivity indices. So far it has been assumed that the properties are uniquely determined by the types of groups present in the molecule and their interconnectivity. However, in reality there are discrepancies between predicted and observed values. These can be reconciled by recognizing that the parameters of the property model vary around their nominal values. This can be expressed mathematically by utilizing probability distributions to describe the likelihood of different realizations for the model parameters. The probabilistic description of performance objectives and constraints is described in Maranas [6]. This formulation involves probability terms whose evaluation for each realization of the deterministic variables requires the integration of multivariate probability density distributions. This is accomplished without resorting to computationally intensive explicit or implicit multivariate integration. This is done by transforming the stochastic constraints into equivalent deterministic ones. Furthermore, it is shown that for probabilities of interest this formulation is a convex MINLP which can be solved to global optimality using commercial packages. The objective of using this formulation is to construct a trade-off curve between performance target and the probability of meeting the target. This aids the designer in choosing the optimal level of risk in selecting the molecule. Next, the surfactant design problem is briefly discussed. 3. SURFACTANT DESIGN The design of surfactant solutions is an important problem in many industries since they are extensively utilized in diverse applications such as detergents, emulsifiers, and to ensure film coating and waterproofing. In the design of surfactant solutions the performance measures of interest are HLB, emulsivity, detergency, and foaming stability. Though this problem is also addressed within the general molecular design paradigm discussed previously, this problem presents additional unique challenges. The macroscopic properties of interest are related to structural descriptors of surfactants through fundamental solution properties such as critical micelle concentration (CMC) and area of a surfactant molecule within a micelle. Though this has the same flavor as relating property of polymers to connectivity of the molecule through topological indices there is an important difference. In polymer design, connectivity indices could be determined from the connectivity by simple evaluation. In the case of surfactants, determination of fundamental solution properties involves the minimization of free energy. Therefore the problem of identifying the molecular structure of a surfactant with optimal values for the desired macroscopic properties is posed as a two-stage optimization problem [2]. The inner stage identifies the CMC and other micellar properties by minimizing the free energy ~tg, while the outer stage optimizes over the surfactant structural descriptors. A conceptual optimization formulation of the problem is as follows: max / min subject to
f(macroscopic properties)
1160 macroscopic properties ) fundamental properties )
=
g(fundamental properties) arg min /lg (structural) descriptors
This formulation is solved using a truncated newton method. Since this problem may possess multiple local minima the problem is solved with multiple starting points. The structural descriptors include the number of carbon atoms in the surfactant tail nc, the cross-sectional area of the head ah, the charge separation for an ionic head group 5, and the dipole separation for dipolar surfactants d. These descriptors provide a concise description of the surfactant molecular topology and polarity. They are theoretically related to fundamental solution properties determining the shape, size and concentration of the surfactant micelles. The fundamental solution properties include the equilibrium area per molecule in a micelle a, the micellar shape, and the concentration at which micelles form (known as the critical micellar concentration or CMC). These properties are related through local regression models to macroscopic surfactant properties characterizing the suitability and effectiveness of the surfactant for a particular application (e.g., hydrophilic-lipophilic balance number (HLB)). Details of the functional relation of free energy to surfactant molecular structure and solution properties is given in Camarda et.al. [2]. This methodology is applied to identifying a nonionic surfactant with hydrophiliclipophilic balance (HLB) of 13.8. HLB is a widely used measure of the emulsifying ability of a surfactant. High value for HLB implies high water solubility, and suitability for detergent or emulsifier. A local regression model is constructed which relates HLB to CMC as follows: l n H L B -- 2.76 + 0.04 l n C M C
The truncated-Newton algorithm was started from a number of initial starting points, and in each case, the algorithm converged to the same optimal solution involving a head cross-sectional area of 0.54977 nm and 5.997 carbons in a straight-chain tail. The CMC for this surfactant was found to be 0.034 mM. A search over tabulated surfactant properties reveals that a surfactant with a dimethyl phosphene oxide head group and a six carbon tail is compatible with those structural descriptors. 4. R E F R I G E R A N T S E L E C T I O N AND C Y C L E SYNTHESIS The focus now shifts from designing a molecule (refrigerant) to selecting a molecule from a prepostulated set of potential candidates. This still poses a challenge when placed within the context of synthesizing refrigeration cycles. The combinatorial problem of appropriately assigning refrigerants to different locations in the refrigeration cycles requires the use of optimization tools. The problem addressed is stated as follows [ 10]: Given a set of process cooling loads, heat sinks at different temperatures and a set of available pure refrigerants, find the refrigeration cycle topology, operating conditions and refrigerants, selected from the list, that optimize a weighted sum of the investment and operating costs for the refrigeration system. The proposed model involves a superstructure representation for both the synthesis and the refrigerant selection problems. The model allows for the identification of the number of stages, their operating temperature ranges, the type of refrigerant participating in a stage, the temperature where a switch between two refrigerants occurs, the use of economizers, presaturators
1161 (31OK) (294 K) (278K) Provane Refrigerant
(263K) (247K) ( 2 3 2 K ) f----(236K) ~_
Refrigerant Switch
(218K) Ethane Refrigerant
(201K) (186K) (190K)
~
Process Stream
Fig. 1. Vertical Cascade for pure refrigerant system
or heat exchangers between intermediate stages. The objective to be optimized considers both investment and operating costs. These alternatives are compactly represented as a network. The operating temperature range of each potential refrigerant is discretized and these discretized levels are the nodes of the network. The alternatives corresponding to (i) operation of vapor compression cycle between temperature levels of a particular refrigerant (ii) heat intake from a cooling load (iii) switch between refrigerants are represented by the arcs of the network. The process configuration is obtained once the optimal energy flows in the network are identified. The optimization problem is solved as an MILE An example of the optimal configuration generated by this procedure for pumping 100kW of heat from 190K to 31 OK using a ethane-propane refrigeration system is shown in Figure 1. Examples demonstrating the advantage of simultaneous refrigerant selection and cycle synthesis over a sequential approach are given in Vaidyaraman and Maranas [ 10]. 5. E N Z Y M E DESIGN
DNA recombination techniques provide the backbone of directed evolution experiments for engineering improved proteins and enzymes. The setup of directed evolution experiments is vital to the rapid and economical production of enhanced enzymes since screening a large number of proteins for the desired property is expensive and time consuming. The goal is to develop predictive models for quantifying the outcome of DNA recombination employed in directed evolution experiments for the generation of novel enzymes. Specifically, predictive models are outlined for (i) tracking the DNA fragment size distribution after random fragmentation and subsequent assembly into genes of full length and (ii) estimating the fraction of the assembled full length sequences matching a given nucleotide target. Based on these quantitative models, optimization formulations are constructed which are aimed at identifying the optimal recombinatory length and parent sequences for maximizing the assembly of a sought after sequence target [8]. A flowchart of DNA shuffling is shown in Figure 2. First an initial set of parent DNA sequences is selected for recombination. The parent sequences undergo random fragmentation, typically by DNase I digestion. The fragment length distribution QO, which describes the fraction of fragments of length L found in the reaction mixture after fragmentation is calculated to
1162 be as follows
aO _
Pcutexp(-PcutL) for 1 < L < B - 1 exp(-PcutB) for L - B
Next, the double-stranded fragments within a particular size range (i.e., 50-200 base pairs) are isolated and reassembled by the Polymerase Chain Reaction (PCR) without added primers. This step is quantified using a fragment assembly model that tracks the fragment length distribution through a given number of annealing/extension steps. This is used to estimate how many shuffling cycles will be needed before full length genes are assembled. A sequence matching model is developed to aid in the goal of optimizing experimental parameters to maximize the probability of obtaining a desired sequence. This model quantitatively predicts the probability of having a randomly chosen full length sequence, assembled through DNA shuffling, match the given nucleotide sequence target. This model recursively calculates the probability Pi of a reassembled sequence matching the target sequence from position i to position B (length of parent sequence). The probability P1 represents assembly of the entire target sequence. The recursive expression for evaluating Pi is shown below. 1,
i>B
zxs.B
i -- B
K L-1
QO ~, AL-V,L
ei--
L=L1 •
V'-Vmin
(Ai'i+L-V-1)pi+L_V, 1< B K
In the above expression, K is the number of parent sequences, (L1,L2) are the range of fragment lengths retained for shuffling, QO is the probability that a fragment is of length L, AL-v,L is the probability that a fragment of length L will anneal with an overlap V, and Ai,i+L-V-1 is the number of parent sequences that match the target between the positions i and i + L - V - 1. The above predictive sequence matching model enables the formulation of mathematical programs for optimizing the recombinatory fragment length and parent sequence set. The objective of the desired mathematical program is the maximization of the probability of matching the target sequence (P1). Two variations of this model are considered. In the first variation, the parent sequences which are chosen occur in equal relative concentration. This problem is combinatorial in nature due to the need to choose the optimum set of parent sequences. The binary decision variables are Yk denoting if parent k is chosen and xL indicating if the recombinatory
Fig. 2. The three steps of DNA shuffling.
1163
Fig. 3. Illustrative example for the effects of fragment length and parent set selection
length is L. This results in a MILP formulation. The second variation allows all the parent sequences to be present but optimizes the relative concentration Ck of each parent sequence. This is solved as a bilinear NLP once for each L in the range being considered to find the optimal recombinatory fragment length. The importance of using the MILP and bilinear formulations are illustrated through an example. The goal is to shuffle six parent sequences each with a variety of mutations and to produce a sequence containing all twelve of the mutations. The sequences are B = 151 nucleotides long and are shown in Figure 3(a). First, when all parents are selected for recombination (achieved by fixing all Yk = 1), the optimal recombination probability P1 of 0.0105% for a fragment length L of 37 nucleotides is confirmed. However, when the complete MILP is solved for both xL and Yk, the subset of parent sequences 1, 3 and 6 is revealed to be the optimum recombinatory choice with a recombination probability of 0.0294%, an almost three-fold improvement. Note that the new optimal length is L = 70 nucleotides, almost twice the length of the previous optimum implying that the selection of the optimal fragment length L strongly depends on the selection of the parent set. A plot of P1 versus L for different parent sequence recombination sets is shown in Figure 3(b). These results suggest a surprising complexity in the shape and form of the P1 versus L plots for different parent choices. Specifically, the multimodal characteristics of these curves reveal narrow fragment length regions for which favorable recombination results are obtained. Next, the bilinear formulation is solved, producing the result shown in Figure 3(b). The optimal recombination probability is equal to 0.0297% at L - 71. The optimal parent sequence concentrations for this fragment length are C1 = 0.362, Ca = 0.339, C6 - 0.299, with all other Ck -- 0, which are fairly close to the equal relative concentration solution. These results indicate that utilizing these formulations can produce a substantial increase in recombination probability.
1164 6. S U M M A R Y
This paper discussed the application of optimization techniques to the area of molecular design and bioinformatics. Key issues in the area of polymer design, surfactant design, refrigerant selection, and enzyme design were identified. In each case it was shown how the use of optimization techniques helped in "homing in" on the desired alternatives in a systematic way as opposed to time and labor intensive trial and error approach. REFERENCES
1. J. Bicerano, Prediction of Polymer Properties. Marcel Dekker, New York, 1996. 2. K.V. Camarda, B. W. Bonnell, C. D. Maranas, and R. Nagarajan, Design of Surfactant Solutions with Optimal Macroscopic Properties. Comput. Chem. Eng., Suppl:S467, 1999. 3. K.V. Camarda and C. D. Maranas, Optimization in Polymer Design Using Connectivity Indices, Ind. Eng. Chem. Res., 38(5): 1884, 1999. 4. A.P. Duvedi and L. E. K. Achenie, Designing Environmentally Safe Refrigerants Using Mathematical Programming, Chemical Engineering Science, 51:3727, 1996. 5. R. Gani, B. Nielsen, and A. Fredenslund, A Group Contribution Approach to ComputerAided Molecular Design, AIChE Journal, 37(9):1318, 1991 6. C.D. Maranas, Optimal Molecular Design under Property Prediction Uncertainty, AIChE Journal, 43(5):1250, 1997. 7. C.D. Maranas, Optimal Computer-Aided Molecular Design: A Polymer Design Case Study, Ind. Chem. Eng. Res., 35(10):3403, 1996. 8. G.L. Moore, C. D. Maranas, K. R. Gutshall, and J. E. Brenchley, Modeling and Optimization of DNA Recombination, Comput. Chem. Eng., 24(2/7):693, 2000. 9. R. Vaidyanathan and M. E1-Halwagi, Computer-aided design of high performance polymers, Journal of Elastomers and Plastics, 26(3):277, 1994. 10. S. Vaidyaraman and C. D. Maranas, Optimal Synthesis of Refrigeration Cycles and Selection of Refrigerants. AIChE Journal, 45(5):997, 1999. 11. D. W. van Krevelen, Properties of Polymers: their correlation with chemical structure; their numerical estimation and prediction from additive group contributions, Elsevier, Amsterdam, 3rd edition, 1990.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1165
Designing Efficient, Economic and Environmentally Friendly Chemical Processes R. L. Smith a, T. M. Mata b, D. M. Young a, H. Cabezas a, and C. A. V. Costa b aNational Risk Management Research Laboratory U.S. Environmental Protection Agency 26 W. Martin Luther King Drive Cincinnati, OH 45268 USA bLaboratory of Processes, Environment and Energy Engineering Faculty of Engineering, University of Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal A catalytic reforming process has been studied using hierarchical design and simulation calculations. Approximations for the fugitive emissions indicate which streams allow the most value to be lost and which have the highest potential environmental impact. One can use this information to focus attention on these high value and/or high potential environmental impact streams. It was also found that increased recycling leads to higher potential environmental impact. The effect of larger recycle flows is an increase in fugitive emissions, which leads to larger impacts. 1. INTRODUCTION In the design of chemical processes one can consider aspects of economics, flexibility, safety, controllability, and environmental friendliness to prepare for possible future circumstances. While no one can predict the future, it is prudent to take precautions when making long-term decisions- like those involving chemical process design. The precautionary principle invokes a determination to act so that mistakes are made on the side of being too safe or too environmentally friendly. This work examines a process for which it is assumed that all of the open emissions to air, water, etc. are prevented (i.e., not released to the environment). Improving the environmental friendliness above the level of preventing open emissions means adding elements of precaution and thoroughness to a design. The critical aspect in such a case is that fugitive emissions become a dominant source of environmental impact. Therefore, this work examines the possible fugitive emissions from the catalytic reformer of a refinery, considering how one could design such a process economically and with the least potential environmental impact (PEI).
1166 2. BACKGROUND Designing chemical processes involves many people and techniques. To get an initial perspective on design alternatives the methods of Douglas (1988) can be employed. This hierarchical method leads to the synthesis of design alternatives based on various chemistries, pieces of equipment, and/or operating conditions. A hierarchical approach for waste minimization has been suggested by Rossiter and Klee (1995). The resulting flowsheets and evaluations can quickly eliminate some designs and point to others for further analysis. Once a superstructure of process elements is developed, one can use optimization methods to further specify a process (e.g., Pistikopoulos et al., 1994). One method to further analyze design alternatives is to use a process simulator (ASPEN PLUS, CHEMCAD, HYSYS, or PRO/II for example). These software programs can do a more rigorous analysis than the simpler short-cut methods often used to initially synthesize a flowsheet. The results of such an analysis can more easily include detailed reaction kinetics, separations with more complex splits, rigorous recycle streams, etc. While this further level of detailed analysis takes longer to set up and calculate, the resulting flowsheets can increase the confidence in a design. When designing processes with a view towards the environment one can use the Waste Reduction (WAR) algorithm (Young and Cabezas, 1999), which allows the user to quantify the environmental concerns of a chemical process. WAR has made available a method to simply evaluate processes with a library of approximately 1600 chemicals. (For related references, Cano-Ruiz and McRae (1998) have reviewed the environmental design literature.) The WAR algorithm applies a balance equation around the process of interest, and the potential environmental impact for a stream entering or leaving is defined as the impact of dumping that stream directly into the environment. The local impacts considered by this method include human toxicity by ingestion and by dermal/inhalation routes, terrestrial toxicity, and aquatic toxicity. Regional impacts evaluated by WAR are photo-chemical oxidation and acid rain, while global impacts include global warming and ozone depletion. Each of these categories has scores that have been normalized within the category, while weighting factors (currently all set equal to 1.0) are used to balance between the categories. Before this work, WAR has always been applied to streams that were openly emitted to the environment. These open emissions of chemicals are decreasing due to laws, penalties and increased community awareness. As a result less chemicals are allowed to enter the environment. However, fugitive emissions still escape from processes and create environmental impacts. Siegell (1997) describes the distribution of VOC's from point and non-point sources, with the majority (86%) of non-point sources being due to fugitive emissions. 3. H I E R A R C H I C A L DESIGN OF PROCESSES W I T H O U T WASTE STREAMS By controlling emissions that are openly emitted to the environment, considerable impacts are avoided. Even so, fugitive emissions do escape from processes and create environmental impacts. Here, we examine possible fugitive emissions, where an assumption is made that 0.1% of each stream becomes a fugitive emission. The expectation is that this is not a very accurate assumption, but rather something that can be refined later (e.g., by the methods described in EPA, 1993).
1167 The process under consideration is the catalytic reformer, where the reformer takes a naphtha fraction (mostly paraffins) and reacts it into higher octane gasoline (isoparaffins, naphthenes and aromatics), light gases, and hydrogen. (In some versions of the process the aromatics generated are further separated and processed.) The reactions and kinetics used in our model are described in Padmavathi and Chaudhuri (1997). Shown in Figure 1 is a diagram of the process, which consists of a reactor, a separation system (in this case a flash and distillation tower), heat exchangers, a valve for decreasing stream pressure, and sometimes a recycle stream (with an additional compressor and heat exchanger). Various operating conditions can be considered: reactor temperatures of 460-540~ reactor pressures of 500-3000 kPa, and hydrogen recycles of 0-80 percent (equivalent to defining a hydrogen to hydrocarbon molar ratio for the reactor feed). These various operating conditions are important because of their affects on the process. For instance, reactor temperature will affect the product distribution, and percent hydrogen recycled will affect the size of the recycle stream (and associated equipment).
l REACTOR
Fig. 1. The catalytic reforming process, shown without flash overhead recycle. Through spreadsheet calculations based on a hierarchical design procedure as described in Douglas (1988), a fiowsheet as shown in Figure 1 can be developed. This flowsheet is based on modeling of 26 components and 35 reactions (integrating 21 independent extents of reaction for an isothermal plug flow reactor). These reactions do not include cyclopentanes for the hierarchical design calculations. For a reactor temperature and pressure of 460~ and 500 kPa with no recycle, the results shown in Figures 2a and 2b were obtained. Figure 2a shows that the economic potential, EP, decreases with the addition of reactor and separation system costs in levels three and four of the hierarchy, respectively. (Level one is where a decision between batch and continuous operation is made.) Level two of the hierarchy considers only product values and feed and fugitive emission costs. With details added at each level, more streams are added to the flowsheet, and the amount of fugitive emissions increase. This represents an increased number of valves, flanges, and other equipment that can leak. Thus, in a manner similar to the economic potential only decreasing with additional detail, the potential environmental impact from fugitive emissions only increases with additional detail. This is a major result of this work, as it allows one to eliminate designs with too high of a potential environmental impact early in the hierarchical design procedure (i.e., with the least amount of work). In the case of Figure 2a, the potential environmental impact increases with more streams and their associated fugitive emissions, while PEI decreases with the conversion of n-paraffins at each level. The reason for the decreasing PEI with conversion is shown in Figure 2b, which
1168 shows that the reactor products and associated down-stream flows have lower PEI values due to the conversion of reactants into potentially less harmful products. Based on the economic potential of Figure 2a one would choose either low or high conversions for operation (i.e., not around 0.65), while the potential environmental impact points towards using higher conversions. Note that the increase in economic potential at higher conversions is due to an increasing octane number that leads to higher value for the products. This increase in value becomes more dramatic as the octane level increases, and so the really desirable products are made at higher conversions. The results displayed in Figure 2b depict the fugitive emissions in terms of their value (lost) and the potential environmental impacts from each stream. The figure shows that the fugitive emissions from the feed have the highest value with reactor products, tower feed and tower bottoms close in value. The tower tops (butane and smaller molecules) have less value, and the flash overhead has even less value (not depicted here). Thus, from this figure one can see that more attention should be concentrated on certain streams and less to others, unless of course, one knows that a stream has very high emissions. For example, if the tower tops leaked 10 times more than the other streams, it would be of higher significance in terms of economic potential. 1.4 l0 s-
4.0 10'
1.2 1 oS- :" - .-'-~. _.~.
3.o l O ' Economic
EP
Potential
2.o 1 0 ' -
($ /yr)
1.0 10'-
Lost
I.o 1 o ' -- -
Emissions
s . o 1o ~-
($ / yr)
6.0 1 0 ' -
......
,~
\ ' ~ , , . ~P .~'~ ,,'~- .-~,, \ ~P / ~- "~EP l
6.0 10'
I
""
-- - - _
-----
"'"
Reactor Feed
X',
-1.0 1 0 ' -
"'--.
llv Fugitive
\x,, \x,,
I
I
Reactor Product Tower Feed Tower Bottomq
4.0 lO 4-
-
-
- Tower Topq
2.0 104-
.--"
1.6 10'- " - ' ' ' "
I
\ 5.0 l O ' -
(PEI
~
~
............
Potential Environmental
Environmental Impact
~
1.2 106-
PEI 4
Potential ~
" ~. ~.
.~
~
.
fI . m rna c .
4.0 1 o' -
8 o 1o 5-
ttb
"" -- , .
PEI
/ yr)
(PEI
/ y r)
4.0 l0 s-
3.0 10'"
9" ' ' " 2.0 10' 0.00
""''!
0.20
....
......... | 0.40
PEI , 0.60
Conversion
,. .
.
. | 0.80
.
.
. 1.00
.
.
.
.
.
.
.
.
.
0.00.00
...... -
. 0.20
0.40
- .....
" .......
0.60
0.80
1.00
Conversion
Fig. 2. a) Total economic potential and potential environmental impact for various levels of the design hierarchy, b) Stream-specific economic potentials and potential environmental impacts of fugitive emissions.
1169 The potential environmental impacts by stream shown in Figure 2b follow the economic potentials rather closely. However, one can see that the feed impact is relatively more important at higher conversions, rather than the reactor products and tower feed streams which are larger at the lowest conversions. Again, this figure directs attention to the streams with potentially higher environmental impacts. 4. SIMULATION FOR E N V I R O N M E N T A L PROCESS DESIGN
After considering aspects of hierarchical design in paper and spreadsheet form as described above, it is often valuable to consider simulating the process. The simulations described offer more details of the process than from using shorthand calculations. In particular, the model of the process has been extended to 43 species and 91 reactions. For these simulations an isothermal reactor has been assumed, where normally the reactor is adiabatic with interstage heating (e.g., Rachford and Gilsdorf, 1994). A simulation at 500~ 3000 kPa, and 50% hydrogen recycle leads to a conversion of nparaffins of 0.90 and a research octane number of 96. This compares favorably to the model used in the hierarchical design to create Figure 2, where the maximum octane number was 90. This points to the need to include the various cyclopentane species in the model. Without the cyclopentane to cyclohexane route which continues on to produce aromatics, the paraffins predominantly react to form cracked products. Thus, simulating with a more complete model can be an important aspect of achieving meaningful results. An interesting result obtained from simulating identical reactors at 3000 kPa was found for various reactor temperatures and percent recycles. The results are shown in Table 1. For all temperatures the potential environmental impact increased with the amount of recycle. Since larger recycles mean larger flowrates through much of the process, it follows that the fugitive emissions are larger. With the increasing fugitive emissions the potential environmental impact increases. However, the impact does not increase proportionately to the amount of recycle. For instance, 80% recycle does not have a PEI close to four times that for 20% recycle. There are two reasons for this effect: one is that not all the process streams participate in the recycle loop, and two is that the recycle composition is mostly hydrogen which has zero environmental impact. Even with the dominance of hydrogen in the recycle, the potential environmental impact due to fugitive emissions increases with recycle. One could speculate that for other processes where the dominant recycle species is less environmentally friendly, that the affect of recycle on environmental impacts would be more substantial. Table 1. Potential environmental impact per year from fugitive emissions. Temperature (~
Percent Flash Overhead Recycled 0
20
50
80
460
6499
7352
7593
8093
500
6192
7082
7347
8169
540
6046
7115
7651
9158
1170 5. CONCLUSIONS From hierarchical design and simulation calculations performed on catalytic reformer models, fugitive emission results have been obtained. The fugitive emissions indicate that emissions from certain streams have more value than others, and also that emissions from certain streams have higher potential environmental impacts. With this knowledge one can place more attention on these streams. In addition, more recycling of chemicals through the process increases the potential environmental impact. The effect of recycle on the potential environmental impact is less pronounced when hydrogen dominates the recycle flow, although one would expect it to be more important when other chemicals dominate the recycle. REFERENCES
Cano-Ruiz, J. A. and McRae, G. J. (1998) "Environmentally Conscious Chemical Process Design," Annu. Rev. Energy Environ., 23,499-536. Douglas, J. M. (1988) Conceptual Design of Chemical Processes. New York: McGraw-Hill. EPA 453/R-93-026 (1993) Protocol for Equipment Leak Emission Estimates. Office of Air Quality Planning and Standards, Research Triangle Park, NC. Padmavathi, G. and Chaudhuri, K. K. (1997) "Modelling and Simulation of Commercial Catalytic Naphtha Reformers," Can. J. Chem. Engng., 75, 930-937. Pistikopoulos, E. N., Stefanis, S. K. and Livingston, A.G. (1994) "A Methodology for Minimum Environmental Impact Analysis," in E1-Halwagi, M. M. and Petrides, D. P. (eds.), Pollution Prevention via Process and Product Modifications, AIChE Symp. Ser., No. 303, Vol. 90, 139150. Rachford, R. H. and Gilsdorf, N. L. (1994) "Reforming Processes for Fuel Production," in McKetta, J. J. (ed.), Encyclopedia of Chemical Processing and Design, New York: Marcel Dekker, 47, 133-151. Rossiter, A. P. and Klee, H. (1995) "Hierarchical Process Review for Waste Minimization," in Rossiter, A. P. (ed.), Waste Minimization through Process Design, New York: McGraw-Hill. Siegell, J. H. (1997) "Improve VOC Emission Predictions," Hydrocarbon Processing, April, 119-121. Young, D. M. and Cabezas H. (1999) "Designing Sustainable Processes with Simulation: The Waste Reduction (WAR) Algorithm," Comput. Chem. Engng., 23, 1477-1491.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1171
Study on lifecycle and agility of process industry Xin' an Xie a, Ben Hua a, Qinglin Chen a, Ren Liangb, Mingang Zeng a aSouth China University of Technology, Guangzhou, 510641, RR.China bGuangdong College of Pharmacy, Guangzhou, 510224, RR.China Based on the characters of process industry, the life cycle of process enterprise and its agility were researched and analyzed. According to the concept of product life cycle, (a) the model for the life cycle of process enterprise was developed, (b) agile variables of process enterprise change were proposed, and (c) the process enterprise agility model oriented to the life cycle of process enterprise was presented. 1. INTRODUCTION Because of the economic globality, market dynamic characters and the intensification of market competition, the lifecycle of products and process technology in the process industry becomes shorter and shorter. To meet the changeful market demands, there are a series of problems to be solved. Such as: (a) how to speed up development of new products, new process technology and advanced equipment; (b) how to control production, so as to improve the agility of the process industry and so on. Therefor, many new concepts, thoughts and approaches in discrete manufacturing industry were put forward, like FMS(Flexible Manufacturing Systems), CIMS(Computer Integrated Manufacturing Systems) and LPS(Lean Production Systems). These technologies and ideas emphasize particularly on information integration of an enterprise, but neglect integration of information, technology and human resources. With the development of CIMS in discrete manufacturing systems, computer Integrated Process Systems (CIPS) was put forward based on the concept of CIMS. In recent years, CIPS has been succeeded in applying to many petrochemical complexes and chemical plants. The application of CIPS in current can be reduced to integrated application of information technology, it just stay at the step of integration of enterprise infra information. There is no consideration of lifecycle integration including R&D, production and marketing. [11 Base on above-mentioned causes, the concurrent engineering (CE) which the main idea is the life cycle of a product and agile manufacturing (AM) which evaluates the agility of a product in the life cycle were brought forward. Agile manufacturing is a dynamic integration of enterprises from lifecycle. It requires incorporating flexible process systems and human resources in order to get maximum benefits in the long run. The target of agile manufacturing is global optimization of social, economic, resources and environment. 2. LIFECYCLE OF PROCESS INDUSTRY ENTERPRISES Lifecycle is a new design thought for shortening time of development and production of a product, improving production flexibility.
1172 The product's lifecycle includes raw material extraction and processing, manufacturing, using, recycling, and ultimate disposal, so the every phase of a product's lifecycle should be considered from its initial conceptual design to disposition or regeneration after being used. Phases of a product's lifecycle are shown in Fig. 1. Fig.1 shows that a product come out was through many phases to be considered such as market demands, design, development, production and so on. These phases are determined by many factors in the outside circle above mentioned, like working condition, corporation's policy, utilizing resources and product's characters, etc.
Environment fmanufacture f ~ characters / " ' ~ ' ~ / "
\
,roucts:, \
-"-.~licy
~ "~.~ ~ i n n ' ~
~._ ~
-
~
.\ wor~ng "~
re.ources'/
"x,~uvv~j
] utilizing /
~oduct's ~ lifecqg.,v.deI
Fig. 1. Phases in the lifecycle of a product In virtue of the idea of a product's lifecycle in discrete manufacturing systems, the lifecycle oriented model of a continuous process industry enterprise will be set up based on the characters of continuous process industry. The raw materials and products in continuous process industry are not subjects assembled by several parts, but homogenous materials that are characterized by such chemical and physical aspects as purity and viscosity. There are many differences from discrete manufacturing systems as follows: (a) The production of continuous process industry is continuous or batch with a large scale. Material flow and energy flow is continuous and relative stable. Flowsheets are relative constant and no pause is permitted among operation units during the long production cycle. However, multi-products and discontinuous are the characteristics of discrete manufacturing systems. They have much flexibility by revising their assembling lines. (b) With waste emission during production, environment protection is a heavy duty to continuous process industry. Many units were operated under such critical environment as high temperature, high pressure, easy to flame, to explode and to poison. One unit being out of work will effect global system in no time. Safety and reliability require rigorous automated control. Therefore, fault diagnosis is very difficult but necessary. Operations in discrete manufacturing systems are usually under common environment. Fault diagnosis is relatively easy since units operated independently. (c) Continuous process industry is usually energy intensive. Especially in China, it contributes more than 50% of national energy consumption, only chemical industry, metallurgy and construction materials departments contribute 40.6%. Many products of
1173 chemical industry and metallurgy cost 1.5 or double energy in average than that in developed countries It1. Furthermore, energy intensive means high pollution and low energy-use efficiency in some tense. Therefore, energy conservation has great significance in continuous process industry and is the key to energy resources sustainability. It offers great potential profits both in economy and environment. (d) The optimization of continuous process systems aims to safety, stable, balance, long operating period, precise control and no pollution, through planning, scheduling and keeping operational parameters around optimized set points. Depending on the characteristics of continuous process enterprises, the hierarchical structure model of process industry system was put forward, as shown in the Fig.2. [21 Hard Technology
Soft Technology
~ . . . . . . . . + CapitalFlow DCS/APC~_-.-------'~ . . . . . . ~........ Maintenance Stable Pr Accidents / / . - - " ~ / / [ \Human Ware Fixed F~ed Stock I Safety _ . .- ~ ~ I ~nformation Network Product~ ] On-line M o n ~ / / ] / Flowsh]eets, and [ Fault D i a ~ o s i s ~ ~ I Enviro~lme~ Systemeval~);rrf ~.('~M~rtrmlno~",)l~urehasing | k, uperatlOn~ Debottlene~g ~ ~ ~oduct Marketing Operatipn O p i 7 " - - 7 ~ OperationO p t . ~~~~ [ t/evises Sutmlv Changl~d Feed[ Planning&ScheduyJJJ [ / Stock, k Product I Process S y n t ~ ~ l ~ " / / I / & E n v i r o n ~ Proc~/egrfit/jafi~ ...~---l.----...~J Fle~i/hitity/~ Decision ~ Expansion O p e r a ~ y ~ Making ..// Investment Reproduction: Expah~ ~ ~ -"'----'-"/ MarketPrediction Simple Sustai'na~ ~ Retrofit Expansion New Product R&D *DCS( Discrete Control Systems),APC(Advanced Process Control),IC(IntelligenceControl) Fig.2 The hierarchical structure model of process industry systems This model consists of two parts of the hardware technique and software technique of process industry systems. Two parts includes six levels such as design, operation, control and decision making, marketing, management. It shows clearly the relationship among of them and points out that process optimization is not only control optimization or operation optimization, but global optimization of integrated all activities from all levels. From the point of process industry systems lifecycle, the hierarchical structure model of process industry systems may be modified as the lifecycle oriented model of a continuous process industry enterprise. It is shown in Fig.3. Fig.3 shows that the lifecycle oriented model of a continuous process industry enterprise include an inner level and an outer level. The inner level is important in process industry systems because about 85%of the lifecycle cost is determined by the decisions made in this level [2 ]. The whole process from a plant design, control, operation to redesign forms the lifecycle of process industry enterprises. The outer level contains three phases made of management, marketing and decision making. The phases in the outer level are decision-making variables of the inner in the process system lifecycle. The change of each phase of lifecycle in process industry system is result of response to outer factors directly.
1174 Fig.3 indicates the relationship between the inner level and the outer clearly.
Fig.3 the lifecycle oriented model of a continuous process industry enterprise
Therefore, the model gives a study framework for choosing qualitative factors and quantitative factors, studying the relationship between the inner factors and the outer factors and researching the agility of a process enterprise. 3. AGILITY AND ITS MEASUREMENTS OF PROCESS ENTERPRISE
3.1. Enterprise agility Enterprise agility was put forward first in discrete manufacturing systems. Agile manufacturing is a dynamic integration of enterprises from lifecycle. It requires incorporating flexible manufacturing systems and human resources in order to get maximum benefits and responding quickly to uncertain market. The figure of two dimensions describing the enterprise agility was shown in Fig.4 based on the ability that a plant grasps chances and possesses technique innovation [3'4l.
Reactive viability
Opportunistic -Fragile
//)~ ~ g i l e ~/f Innovative
Proactive leadershin Fig.4 Figure of two dimensions describing the enterprise agility
Fig.4 shows that the reactive viability and the proactive leadership are abilities to complete agile changes in a plant, and they structure the figure of two dimensions describing the enterprise agility. Reactive viability is the abilities to manufacture products with high quality, high reliability and low price when an enterprise encounters a chance. Proactive leadership is the abilities to forecast the future tendency of market changing and develop new products through adjusting planing and designing. Assessing the enterprise agility, determining the position in Fig.4 and comparing the agility between enterprises, the enterprise agility must be measured. Studying the agile degree of each phase in a process enterprise lifecycle responding to a market, every factor impacting the agility must be found out and measured.
1175
3.2 Integration items of estimating the enterprise agility The enterprise agility is measured by four items, the items consist of cost(C), time (T), robustness(R) and scope of change (S). They are called CTRS for short [5]. The cost is the money needed that the enterprise completes once agile change, it correlates with the cost that a new product comes into market, and with the design and manufacture in the process. The time is hours spent that the enterprise completes once agile change. The robustness is obdurability and stability during completing an agile change. Scope of change is the ability to complete an agile change in a plant. The CTRS are restricted and associated. Therefore, the proportion of the CTRS must be taken into account if the global optimization is achieved in a plant. In other words, the quick change without cost is meaningless. The change without robustness is not agile for a plant under the limitation of the cost and the time. So is the scope of change. Hence the synergy among the cost, time, robustness and scope of change must be taken into account in studying the agility of process systems. 3.3 Agility variables of a process enterprise Agility variables of discrete manufacturing systems are lumped into the communication connectedness, interorganization participation, production flexibility, management involvement, and employee empowerment. These variables are called CIPME for short in discrete manufacturing systems. They stand for 5 aspects of discrete manufacturing systems, and to be used to describe the manufacturing systems agility. Continuous process systems are different from discrete manufacturing systems in many places. There are characters in continuous process systems. Therefore, variables impacting process systems agility differ from the discrete manufacturing systems. Based on the lifecycle model of process systems in Fig.3 described, agility variables of process systems are placed in the every phase in the lifecycle of process systems. Because the model in Fig.3 is continual, agility variables are not easy to be decomposed. The model in Fig.3 is modified as the "discrete" lifecycle model of process systems. It is shown in Fig.5. The basic structure of process enterprises ,,..--
Management
o i-.
o o
Marketing
Decision making
Design Control Operation
.,.,
.1
Redesign
Fig.5 the lifecycle two dimensions model with agility variables of process systems
The model is expressed with two dimensions table. These intersections (little panes) in Fig. 5 illustrate relationship of agility variables against themselves functions in each phase in process systems lifecycle. Because there are more intersections in the model i.e. there are more agility variables, it results in troublesome in setting up and solving the model. Depending on the characters of process systems, the agility variables are lumped into 6 types for solving the model easily. The agility variables of process systems are: material flow
1176 variable (M), energy flow variable (E), information flow variable (I), humanware flow variable (H), cost flow variable (C) and workpiece flow variable (W). They are called MEIHCW agility variables for short. The MEIHCW variables vary with different phase in the lifecycle of process systems, hence they are dynamic variables. 4. THE LIFECYCLE MATHEMATIC MODEL WITH AGILITY VARIABLES OF PROCESS SYSTEMS The lifecycle mathematic model with agility variables of process systems would be used to evaluate agility degree and find out the key step or the "blunt point" responded to change of process systems in different phase of whole lifecycle, so as to guide decision-making for an enterprise. Taking the MEIHCW agility variables as independent variable of measuring the CTRS synthetically, the function relationships are expressed as follows. T l=fl (M, E, I, H, C, W) T2=f2 (M, E, I, H, C, W) T3=f3 (M, E, I, H, C, W) Ta=f4 (M, E, I, H, C, W) Where flDf2Df3Dand f4r refer to the needed time, cost, robustness and scope of change respectively when the process enterprise completed an agile change. 5. CONCLUSIONS AND SUGGESTIONS This paper discusses the lifecycle and the agility in the process industry enterprise, and brings forward: (1) Lifecycle model of process industry enterprise, (2) The agility variables of process industry enterprise---MEIHCW, (3) The agility framework model of process industry enterprise, Because the concepts of lifecycle and agility were introduced in disperse manufacture, they are in studying and developing, so lifecycle model with agility variables of process systems should be more studied and developed from a few aspects as follows: (1)The lifecycle of process systems should be more researched so as to increase the meaning of the lifecycle. (2)The concepts about the material flow variable, energy flow variable, information flow variable, humanware flow variable, cost flow variable and workpiece flow variable should be analyzed clearly. The relationship among the "flows" should be made certain. ACKNOWLEDGEMENTS Financial support from the Chinese NSFC (No. 7993100) and GDSFC (No. 990638) is greatly acknowledged. REFERENCES
1. Cheng, Siwei, Proceedings of Seminar on Process System Engineering and Intensivism, Jiangxi, (1998) 1. 2. Hua, Ben, EHan, Proceedings of Seminar on Process System Engineering and Intensivism, Jiangxi, (1998)46. 3. Roger Nagel, Rick Dove. 21 st Century Manufacturing Enterprise Strategy. Iacocca institute, Lehigh University, 1991. 4. Rick Dove. Production Magazine, 9(1995). 5. Rick Dove. Production Magazine, 6(1995).
European Symposium on Computer Aided Process Engineering - 11 R. Gain and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1177
Dynamic Assessment of Induced Thermal Stresses on the Semiconductor Packaging Substrates in a Radiation Belt Furnace by Computational Fluid Dynamics Thomas C.-K. Yang *a, Hsien-Sheng Lin a , Sea-Fue Wangb, and M.-C. Cheng a a Department of Chemical Engineering,
[email protected] b Department of Materials and Resource Engineering National Taipei University of Technology, Taipei, TAIWAN This paper is studying the energy dissipation, transport, and the temperature distribution of the packaging substrates along the moving belt during the heating stage in the time basis. After
achieving
the
temperature
profiles,
coefficients
of thermal
expansion
and
temperature-dependent thermophysical properties are used as parameters for further structural analysis. Transient thermal stress, which is highly sensitive to thermal history and process time, was calculated at each grid point along the belt. As a result, the subsequent stress-time profile integrated all over the heating surface can then be used to investigate the thermal response and product quality. This information will be very useful to the process engineers and furnace designers. 1. INTRODUCTION Residual stresses of films and substrates induced during the thin film deposition or thick film fabrication play an important role in determining the reliability of electronic devices. As the temperature profiles are distributed unevenly on the heated substrate, thermal stresses will develop in the films due to a mismatch in the coefficient of thermal expansion between film and substrate. As the temperature gradient keeps increase, large stresses are generated, which may induce plastic deformation and creep deformation in the film. As a result, breakage of interconnecting line structures and short circuits between adjacent lines occur based on this stress migration [1]. These phenomena cause the defects of VLSI devices. Therefore, knowledge on the thermal stress in film structures during the heating process is very essential to understand the mechanism of formation of such faults. The radiation belt furnace has the intrinsic advantages over a conventional batch-type furnace in the aspects of a continuous operation, high ramping rates, fast response time [2,3]. Therefore, it is widely adopted for the semiconductor applications such as the curing of solder
1178 masks, thin film deposition and thick film cofiring. This paper is studying the energy dissipation, transport, and the temperature distribution of the packaging substrates along the moving belt during the heating stage in the time basis. After achieving the temperature profiles, coefficients of thermal expansion and elasticity modulus measured by the dynamic mechanical analyzer will be used as parameters for further structural analysis. The calculated stress will be compared with the maximum yield stress, which will lead the plastic deformation. This information will be very useful to the process engineers and furnace designers. 2. ANALYTICAL SYSTEM Figure 1 illustrates the firing processes of ceramic packaging substrates or thick films in a belt furnace. The sliding substrates in the entrance zone initially encounter a hot surface as it moves down the tube to the burnout zone at the temperature of 3 0 0 - 400 ~
for binder
removal. After the completeness of the polymer pyrolysis, propagating substrates were then carried into a higher temperature zone for sintering, where the densification of ceramic powders occurred. As the sintering stage completed, substrates were lead to a cooling section for annealing to room temperature [4].
Figure 1. the cross section of a belt furnace. 2.1 Grid Generation
For establishing a computation domain, an unstructured grid method (6) was used to define the geometric dimensions of the furnace including the heating elements, conveyor belt and the substrates. The furnace entrance has san inside dimension of 25.4 cm by 35.6 cm with the length of 610.0 cm. The geometry of the substrate is defined as 25.0 cm by 20.0 cm with a thickness of 0.6 cm. A thermal profile of 12 heating zones on the furnace wall is also given according to the normal operation setting programs. The inlet for the purging gases and the outlet for the evolution of organic vapors were all placed and defined in appropriate locations. Accordingly, the inlet velocity and outlet gas volumetric flow rates are given as 84.1 cm/min and 5.1x105 cm3/min. In summary, a three-dimensional flow model accompanied with radiation/convection heat transfer was imposed onto a belt furnace. About 105 grids were generated to undergo the finite-volume calculation. Various numbers of grids have been tested and no significantly different results were observed.
1179 2.2 Numerical Scheme for Moving Mesh To take the belt conveying speed into consideration, the sliding grid technique was utilized, instead of using the moving wall boundary, to generate the moving mesh for the belt and the substrate as shown in Figure 2. To facilitate the moving mesh, an unsteady state computation mode was applied [5]. The time step [~t of the system was selected so that the mesh was moved the grid spacing within that interval. This simulates the relative motion of the substrate as well as the belt to the wall. At each time step, the thermophysical properties of the leading edge of the substrate are changed from the fluid to substrate. In addition, for every one physical second, it uses 20 iterations for finite-volume computation.
Figure 2. Diagram of the computing domain. 2.3 Calculation of Thermal Stresses In order to estimate the magnitude of the heat transfer for substrates sliding along the wall, thermal conduction, force convection and thermal radiation modes were all considered This simulation uses the Discrete Ordinate method to account for the radiative property of non-gray body, where the emission energy varies with frequency (e=0.8 for 4~tm <X< 8~m, e=0 for both ~8 9m). The force convection effect was considered in the computation since the purging gasses were facilitated to remove the pyrolysis gasses during the firing process. After the temperature distributions in the solution domain were computed, thermal stresses then can be evaluated based on the transient temperature gradients. The governing equation for stress analysis is the thermoelastic equation:
Ao- = nC,,A~/(a. ~') where T is the absolute temperature, D the thermal expansion coefficient, Cp the heat capacity and M7? the thermal stresses. All the material properties are evaluated as a function of the temperature. As a result, the thermal stress at each grid point can be calculated by applying the calculated temperature gradients (such as
OT 0x
of the substrate.
aT and ~ ) with thermophysical properties @
1180 3. RESULTS AND DISCUSSIONS Figure 3 illustrates the velocity profiles and path-lines of purging gases as well as the evolving gasses. The arrows on the streamlines show the directions of the flows and obviously the chimney performs a strong suction force toward all the gasses. The expression of multicolor in the vector plots and streamlines indicates the temperature profiles. Blue line shows the temperature of 300K whereas the red line symbolizes the highest temperature in the furnace wall (1200K). Apparently, the substrates are successfully heated to l150K near the end of the furnace. This verifies that the heat transfer between substrates, furnace wall, surrounding fluid and moving belt all comes to a thermal equilibrium through the mode of conduction, convection and radiation. Figure 4 shows the temperature profile of substrates at different process time. When examined carefully in the feature figure, one can see clearly that the belt accompanied with substrates was moving continuously toward the exit direction. In addition, it takes substrates at least 50 seconds to reach the thermal equilibrium as the belt slides in a speed of 0.17 cm/s. The feature figure also shows the success usage of the sliding grid technique in the moving boundary for the dynamic system. In addition, the substrate in the last heating zone near the exit showed a gradual heating as the time elapsed and it finally goes to a stable temperature after the thermal equilibrium is achieved. Unsteady-state stress arises as a result of transient temperature gradients in a substrate as well as the mismatch between the corresponding thermal expansion and geometrically incompatible displacement within the substrate. Thermal stress and strains when below some critical values, will continuously adjust themselves to make the internal forces in the substrate self-equilibrating until a compatible displacements. However, this reversible deformation will be continuously accumulated as long as the thermal impact exists until the principal stress exceeds the mechanical limitation of the material. Figure 5 shows that the maximum thermal stress occurred at the heating zones of 10 and 11 corresponding to the highest heating temperature in the wall. The red flooded area of the substrate reads the stress intensity of 994 Mpa and the stress magnitude of the green flooded area reads to 500 Mpa. And these heavily induced stresses were built up near the four edges of the substrate. 4. CONCLUSIONS The energy dissipation, transport, and the temperature distribution of the packaging substrates along the moving belt during the heating stage was obtained. After achieving the temperature
profiles,
coefficients
of thermal
expansion
and temperature-dependent
thermophysical properties were used as parameters for further structural analysis. Transient thermal stress, which is highly sensitive to thermal history and process time, was calculated at each grid point along the belt. As a result, the subsequent stress-time profile integrated all
1181 over the heating surface can then be used to investigate the thermal response and product quality. This information will be very useful to the process engineers and furnace designers. ACKNOWLEDGEMENT Authors would like to thank Dr. Eric S.-Y. Cheng of Industrial Technology Research Institute for providing the detailed information on the belt fumace. Partial financial support form the national science council of Republic of China under the grant no.of 89-2214-E-027-003 was also appreciated. REFERENCES 1. A. G. Evans and J. W. Hutchinson, 'The Thermomechanical Integrity of the Thin Films and Multilayers', Acta Metal. Mater. 43 (1995) 2507. 2.D.K. Flattery, 'Rapid Binder Removal From Thick Film Composite', Radiant Tech. Corp., Cerritos, CA, U.S.A., 1995 3. C. R. S. Needs, 'Infrared Firing Studies of Thick Film Material System for High Volume Hybrid Microcircuits', The International Journal for Hybrid Microelectronics, 7 (1984) 1. 4. D. R. Barbour, 'Multichip Module Technology', Ceram. Eng. Sci. Proc., 9 (1988) 1549. 5. Fluent User's Guide, Fluent Inc., New Hampshire, U.S.A., 1999
Figure 3. Velocity profiles of evolving gasses and temperature distribution of the substrates as well as wall of furnace.
1182
Figure 4. Transient temperature profiles of the substrates along the moving belt.
Figure 5. Stress profiles of the substrates along the moving belt.
1183
Author Index Aartovaara, M. 597 Achenie, L.E.K. 585,1133 Afonso, A. 841 Agachi, S. 711 Agrawal, R. 339 Akay, B. 603 Aldrich, C. 75,81 Alexandridis, A.P. 69 Alfadala, H.E. 943 Allgower, F. 711 Alloula, K. 967 Alpbaz, M. 603 An, W. 773 Andersen, T.R. 627 Aoyama, A. 87,829 Aziz, N. 609 Azzaro-Pantel, C. 117,949 Badell, M. 835 Bafas, G.V. 69 Balakrishna, S. 401 Balogh, S. 381 Banares-Alcantara, R. 955 Bandoni, J.A. 615 Bansal, V. 273,961 Baptiste, D.R. 773 Barbosa-P6voa, A.P.F.D. 847 Barnard, J.P. 75,81 Barros, A.A.C. 321 Barton, G.W. 823 Barton, P.I. 309,767 Basualdo, M.S. 731 Batres, R. 87,829 Batzias F.A. 451 Baur, R. 93 Bayer, B. 345 Bek-Pedersen, E. 517 Belaud, J.-P. 967 Berezowski, M. 99 Bertok, B. 351 Bildea, C.S. 973 Binder, T. 1071 Binet, G. 651 Biscaia Jr., E.C. 123 Bj6rn, I.N. 105 Blanco, A.M. 615
Bliek, A. 463 Bock, H.G. 711 Bogle, I.D.L. 1047 Bok, J.-K. 767 Bonn6, D. 621 Boudouvis, A.G. 69 Bozmis, A. 979 Brauner, N. 291 Brignole, E.A. 375 Brown, S. 675 Biihner, C. 357 Bursali, N. 603 Caballero, J.A. 363 Cabassud, M. 817,1127 Cabezas, H. 1165 Camarda, K.V. 369 Cameron, D.B. 111 Cameron, I.T. 195 Canton, J. 841 Casamatta, G. 817,1127 Castillo, F.J.L. 591,1089 Castro, P. 847 Cerd6, J. 693 Cezerac, J. 1127 Chang, T.S. 645 Charles, A.S. 117 Chen, H.-Z. 1035 Chen, Q. 1171 Cheng, M.-C. 1177 Chiu, M.-S. 333 Choi, D.-K. 427 Chung, P.W.H. 1115 Cismondi, M. 375 Clark, G.A. 1115 Cofer, D. 1 Coffey, D.P. 627 Confinillo, G. 135 Cordiner, J. 27 Cortese, F. 723 Corvahln, S.M. 129 Costa, C.A.V. 1165 Costa, J. 413 Costa Jr., E.F. 123 CresciteUi, S. 225 Cruse, A. 1071 Csukas, B. 381
Cutlip, M.B. 291 Dash, S. 853 Davin, A. 949 de Deugd, R.M. 699 de Jong, P. 681 Dechechi, E.C. 633 Devereux, B.M. 243 Di Puma, J. 1053 Diehl, M. 711 Dimian, A.C. 463,973 Domenech, S. 117,949 Domingues, A. 805 Douglas, D. 249 Dua, V. 979 Duarte, B. 847 Dunn, R.F. 985 E1-Halwagi, M.M. 943 Elgue, S. 1127 Eliceche, A.M. 129,387 Ender, L. 639 Engl, G. 433 Eo, S.Y. 645 Ertunc, S. 603 Espuna, A. 841 Ettedgui, E. 817 Fan, L.T. 351 Faraoni, V. 135 Farza, M. 651 Feliu, J.A. 687 Feng, G. 351 Fernandez-Anaya, G. 755 Figueroa, J.L. 615 Findeisen, R. 711 Fischer, U. 573 Floquet, P. 117,949 Flores-Tlacuahuac, A. 755 Floudas, C . A . 153,393 Fonyo, Z. 553,1109 Fraga, E.S. 955 Fraser, D.M. 991 Freitas Jr., B.B. 279 Friedl, A. 535 Friedler, F. 351,541,547 Gabbar, H.A. 859 Gal6n, O. 141 Gani, R. 559,773,997
1184 Garea, A. 147 Garg, S. 1133 Gatica, G. 865 Gehan, O. 651 GeorgiaNs, M.C. 997,1139 Gerbaud, V. 499 Gesthuisen, R. 183 Gilbert, R.G. 823 Gioia, F. 225 Glavic, P. 529 Glende, E. 111 Gomes, V.G. 823 G6rak, A. 285 Govatsmark, M.S. 657 Graells, M. 841,913,1077 Grancharova, A. 1003 Gren, U. 105 Grof, Z. 177 Grosman, B. 663 Grossmann, I.E. 363,487, 877 Grzywacz, R. 99 Gudi, R.D. 243 Giimiis, Z.H. 393 Gupta, A. 871 Halim, I. 1145 Hallale, N. 445 Hangos, K.M. 195,787 Hapoglu, H. 603 Harding, N. 991 Harding, S.T. 153 Harjunkoski, I. 877 Harper, P. 925 Haug-Warberg, T. 297 Hechavarria, T.L. 147 Heckl, I. 541 Henning, G.P. 883 Henriksen, J.P. 1009 Heppert, j. 369 Herron, D.M. 339 Hertwig, T.A. 1017 Hertzberg, T. 219 Hfitreux, G. 889 Heyen, G. 1053 Hill, pJ. 1151 Hirao, M. 1023 Hjertager, B.H. 159 Hjertager, L.K. 159
Hoch, P.M. 387 Hofbauer, H. 535 Hopper, J.R. 1017 Hostrup, M. 401,517 Hua, B. 1171 Huang, Y. 669 Hui, C.-W. 1029 Hungerbfihler, K. 573 Hurrne, M. 469 Iedema, P.D. 973 Ierapetritou, M.G. 407 Irabien, A. 147 Ishikawa, T. 559 acobsen, E.W. 99,207,737 arke, M. 345 laume, D. 117 ezowksi, J. 1089 im~nez, L. 413,731 iobson, M. 505,511 [ordache, C. 675 Foulia, X. 201,499,967 Jorgensen, S.B. 189,621, 627,773,787 Kahn, D. 419 ICalitventzeff, B. 457,1053 ICantharao, S. 853 ICapteijn, F. 699 Karim, N.M. 761 ICaylor, J.M. 773 Kenig, E.Y. 285 Kheawhom, S. 1023 Kim, D. 895 Kirn, M. 427 Kint, E. 681 Kiparissides, C. 327,705 Ko, D. 427 K6hler, R. 165 Kohout, M. 171 Kokossis, A.C. 451,493, 541,919,1083 559 Kolar, P. 177 Kosek, J. 1139 Kostoglou, M. 249 Kozel, L. 183 Kr~imer, S. 439 Kraslawski, A. 1095 Kravanja, Z. 93 Krishna, R.
Ktistensen, N.R. 189,787 Kr6ner, A. 433 Kronseder, Th. 433 Kubicek, M. 171 Kwok, Y.-Y. 1029 Lakner, R. 195 Lapham, D.S. 773 201,967, Le Lann, J.M. 1127 Le Lann, M.-V. 717,817 645 Lee, B. Lee, T.-y. 547 Lee, Y. 895 Lee, Y. 895 553,1109 Lelkes, Z. Lenz, D.H. 773 663 Lewm, D.R. 1035 Li, B.-H. 1035 Li, H.-Q. 439 Li, X.-N.L. 1035 Li, Z.-H. 1171 Liang, R. 1041 Lid, T. 249 Lira, P.P.S. 201 Lim, Y.I. 1177 Lm, H.-S. 493 Linke, P. 445 Liu, F. 207 Liu, Y. 237 Lopez, G.D. 213 Lucia, A. 811 Lukasse, LJ:S. 901 Luo, K. 773 Lymbumer, C.J. 219 Lovik, I. 1047 Ma, K. 687 Macias, j.j. 279 Maciel, M.R.W. Maciel Filho. R. 279,633, 639,805 189 Madsen, H. 743 Madvadhan, K.P. 225 Maffettone, P.L. 937 Majozi, T. 1053 Malmendier, M. 723 Manca, D. 135,225 Mancusi, E. 871,1157 Maranas, C.D.
1185 Marchetti, M. 231 Marcoulaki, E.C. 451 Marechal, F. 457 Marek, M. 177 Marquardt, W. 345,1071, 1121 Marqu&, J.A. 147 Martmez, E.C. 237 Martini, R.F. 567 Mata, T.M. 1165 Mathisen, K.W. 297 Matos, H. 847 Matthews, C. 991 Meleiro, L.A.C. 279,633 M~ndez, C.A. 693 Meyer, X.M. 201,267 Michiel Meeuse, F. 699, 799 Mmet, F. 1053 Mockus, L. 901 Modi, A. 419 Moharir, A.S. 243 Molma, A. 731 Moon, I. 427,895 Morris, A.J. 327,705 Morton, W. 249 Mourikas, G. 705 M'Saad, M. 651 Mu, F. 925,931 Mujtaba, I.M. 609 Mussone, P. 723 Nagy, A.B. 1017 Nagy, Z. 711 Naka, Y. 87,829 Ng, K.M. 41,523,1151 Niclout, N. 481 Novak, A. 177 Okada, H. 1059 Oldenburg, J. 1071 Oliveira, N.M.C. 749 Olsvik, O. 219 Omota, F. 463 Ortiz, I. 129,387 Ortiz-G6mez, A. 907 O'Young, L. 523 Pajula, E. 469 Palaniappan, C. 1145 Palazoglu, A. 141
Paloschi, J.R. Pantelides, C.C. Papageorgiou, L.G.
255 15 475, 865 Parisse, B. 731 Park, S. 547 Patsiatzis, D.I. 475 Pekny, J.F. 1101 Perez-Cisneros, E. 517 Perkins, J.D. 273,961 Perret, J. 889 Perris, T. 955 Pibouleau, L. 117,949 Pike, R.W. 1017 Pingaud, H. 889 Pingen, J. 1065 pmsky, M. 773 Pinto, J.M. 579 Pistikopoulos, E.N. 273, 961,979,997 Plapp, R. 419 Prat, L. 1127 Preisig, H.A. 261,811 Preuss, K. 717 Prevost, M. 267 Puigjaner, L. 835,841,913, 1077 Rao, A. 231 Reklaitis, G.V. 669,1101 Reneaume, J.-M. 481 Rengaswamy, R. 853 Rev, E. 553,1109 Reyes-Labarta, J.A. 487 Richalet, J. 731 Rico-Ramirez, V. 907 Rieber, J. 165 Rigopoulos, S. 493 Rodriguez-Donis, I. 499 Romagnoli, J.A. 141,823 Rong, B.-G. 439 Ross, R. 273 Rossiter, D. 1115 Rouzineau, D. 267 Rovaglio, M. 723 Ruiz, C. 731 Ruiz, D. 835 Russel, B . M . 925,1009 Russo, L. 135
Ronnekleiv, M. 219 Sakizlis, V. 273 Samad, T. 1 Samanta, A. 505,511 Samyudia, Y. 681 Sanchez Daza, O. 517 Santana, P.L. 279 Saraiva, P.M. 315 Sarimveis, H.K. 49 Scheffer, R. 279,639 Schembecker, G. 357 Schenk, M. 997 Schlegel, M. 1071 Schloder, J.P. 711 Schmidt, H. 737 Schmitt, W.A. 55 Schneider, R. 285 Schreiber, I. 171 Schroer, J.W. 523 Secchi, A.R. 123 Seferlis, P. 705 Sequeira, S.E. 913,1077 Seuranen, T. 469 Shacham, M. 291 Shah, N. 865 Shah, S.S. 243,743 Shang, Z. 1083 Shethna, H.K. 1089 Shimada, Y. 859 Shin, D. 645 Shirao, T. 1059 Siddhaye, S. 369 Siepmann, V. 297 Siettos, C.I. 69 Silva, D.C.M. 749 Silva-Beard, A. 755 Simon, L. 761 Singer, A.B. 767 Skogestad, S. 657,1041 Skotte, R. 773 Smets, I.Y.M. 781 Smith, R.L. 1165 Soares, C. 321 Sobocan, G. 529 Solberg, T. 159 Sorsak, A. 1095 Srinivasan, R. 1145 Stembach, W. 535
1186 Stepanek, F. 177 Stephanopoulos, G. 55 Strouvalis, A.M. 541,919 Subramanian, D. 1101 Suh, M.-h. 547 Sunderesan, P. 369 Sunol, A.K. 943 Suppes, G.J. 369 Suzuki, K. 859 Svensson, F. 105 Szederk6nyi, G. 787 Szitkai, Z. 553,1109 Sorensen, E. 303 Takano, K. 559 Taylor, R. 93 Teoh, H.K. 303 Ternet, D. 675 Tilhac, F. 117 Titchener-Hooker, N. 303 Tolsma, J.E. 309 Torgashov, A.Yu. 793 Torres Alvarez, M.E. 567 Tousam, R.L. 799 Tresmondi, A. 805 Turner, M. 303
Uerdingen, E. 573 v. Stryk, O. 433 Vale Lima, P. 315 Van Impe, J.F.M. 781 Vanden Bussche, K. 243 Vasco de Toledo, E.C. 279 Vasquez-Alvarez, E. 579 V6zquez-Rom~n, R. 907 Venimadhavan, G. 243 Venkatasubramanian, V. 669,853,925,931 Verdijck, G.J.C. 811 Verheijen, P.J.T. 699 Vickery, D. 231 Vieira, R.C. 123 Vinson, J.M. 901 Viswanathan, S. 925,931 yon Wedel, L. 1121 Wang, K. 1115 Wang, S.-F. 1177 Wang, Y.P. 585 Wasylkiewicz, S.K. 591 Weidenhaupt, K. 345 Wenzel, H. 985 Wibowo, C. 523
Wolf-Maciel, M.R. 321,567 Xaumier, F. 817 Xie, X. 1171 Xu, A. 1017 Yang, A.-D. 1121 Yang, F. 213 Yang, T.C.-K. 1177 Yao, P.-J. 1035 Yaws, C.L. 1017 Ydstie, B.E. 627 Yiagopoulos, A. 327 Yiannoulakis, H. 327 Yoon, D. 895 Yoon, E.S. 645 Young, D.M. 1165 Ypma, S.M. 699 Zeaiter, J. 823 Zeitz, M. 165 Zeng, M. 1171 Zhao, C. 925,931 Zhao, J. 925,931 Zhu, X.X. 937 Zhuang, H. 333 Odegaard, R.J. 111