Membrane Process Design Using Residue Curve Maps

MEMBRANE PROCESS DESIGN USING RESIDUE CURVE MAPS MEMBRANE PROCESS DESIGN USING RESIDUE CURVE MAPS MARK PETERS DAVID GL...

Author: Mark Peters | David Glasser | Diane Hildebrandt | Shehzaad Kauchali

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MEMBRANE PROCESS DESIGN USING RESIDUE CURVE MAPS

MEMBRANE PROCESS DESIGN USING RESIDUE CURVE MAPS MARK PETERS DAVID GLASSER DIANE HILDEBRANDT Centre of Material and Process Synthesis (COMPS) University of the Witwatersrand Johannesburg, South Africa

SHEHZAAD KAUCHALI School of Chemical and Metallurgical Engineering University of the Witwatersrand Johannesburg, South Africa

Copyright Ó 2011 by John Wiley & Sons. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at 877-762-2974, outside the United States at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Membrane process design using residue curve maps / Mark Peters ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-0-470-52431-2 (cloth) 1. Membrane separation 2. Diffusion. 3. Pervaporation. 4. Gas separation membranes. I. Peters, Mark, 1981TP248.25.M46M44 2011 660’ .28424–dc22 2010019515 Printed in the United States of America oBook ISBN: 978-0-470-91003-0 ePDF ISBN: 978-0-470-91002-3 ePub ISBN: 978-0-470-92283-5 10 9 8

7 6 5 4

3 2 1

CONTENTS PREFACE

xi

ACKNOWLEDGMENTS

xiii

NOTATION

xv

ABOUT THE AUTHORS

xix

1 INTRODUCTION

1

2 PERMEATION MODELING

7

2.1 Diffusion Membranes 2.1.1 Gas Separation 2.1.2 Pervaporation 2.2 Membrane Classification

8 8 11 13

3 INTRODUCTION TO GRAPHICAL TECHNIQUES IN MEMBRANE SEPARATIONS 3.1 A Thought Experiment 3.2 Binary Separations 3.3 Multicomponent Systems 3.3.1 Mass Balances 3.3.2 Plotting a Residue Curve Map

15 15 16 20 21 23 v

vi

CONTENTS

4 PROPERTIES OF MEMBRANE RESIDUE CURVE MAPS 4.1 4.2 4.3 4.4 4.5

29

Stationary Points Membrane Vector Field Unidistribution Lines The Effect of a-Values on the Topology of M-RCMs Properties of an Existing Selective M-RCM 4.5.1 Case 1: When the Permeate Side Is at Vacuum Conditions (i.e., pP 0) 4.5.2 Case 2: When the Permeate Pressure Is Nonzero (i.e., pP > 0) 4.6 Conclusion

36 38

5 APPLICATION OF MEMBRANE RESIDUE CURVE MAPS TO BATCH AND CONTINUOUS PROCESSES

41

5.1 Introduction 5.2 Review of Previous Chapters 5.3 Batch Membrane Operation 5.3.1 Operating Leaves in Batch Permeation 5.3.2 Material Balances 5.3.3 Permeation Model 5.3.4 Operating Regions: Nonselective Membranes 5.3.5 Operating Regions: Selective Membranes 5.4 Permeation Time 5.5 Continuous Membrane Operation 5.5.1 Nonreflux Equipment 5.5.2 Reflux Equipment 5.6 Conclusion 6 COLUMN PROFILES FOR MEMBRANE COLUMN SECTIONS 6.1 Introduction to Membrane Column Development 6.1.1 Relevant Works in Membrane Column Research 6.2 Generalized Column Sections 6.2.1 The Difference Point Equation 6.2.2 Infinite Reflux 6.2.3 Finite Reflux

29 30 31 32 34 34

41 44 45 45 46 48 48 50 52 54 54 58 64

65 66 67 68 70 71 74

CONTENTS

6.2.4 CPM Pinch Loci 6.3 Theory 6.3.1 Membrane Column Sections 6.3.2 The Difference Point Equation for an MCS 6.3.3 Permeation Modeling 6.3.4 Properties of the DPE 6.4 Column Section Profiles: Operating Condition 1 6.4.1 Statement 6.4.2 Mathematics 6.4.3 Membrane Residue Curve Map 6.5 Column Section Profiles: Operating Condition 2 6.5.1 Statement 6.5.2 Mathematics 6.5.3 Column Profile 6.5.4 Analysis 6.5.5 Pinch Point Loci 6.5.6 Further Column Profiles 6.5.7 Direction of dT 6.5.8 Direction of Integration 6.5.9 Crossing the MBT Boundary 6.6 Column Section Profiles: Operating Conditions 3 and 4 6.6.1 Statement 6.6.2 Mathematics 6.6.3 Column Profile 6.6.4 Pinch Point Loci 6.6.5 Analysis of Column Profile 6.6.6 Pinch Point 6.6.7 Further Column Profiles 6.6.8 Variations in XD and rD 6.7 Applications and Conclusion 7 NOVEL GRAPHICAL DESIGN METHODS FOR COMPLEX MEMBRANE CONFIGURATIONS 7.1 Introduction 7.2 Column Sections 7.2.1 Definition

vii

76 80 80 81 82 84 85 85 85 85 87 87 87 88 89 93 94 96 96 97 97 97 97 98 99 100 102 102 104 105

107 108 110 110

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CONTENTS

7.3

7.4

7.5

7.6 7.7

7.8

7.2.2 The Difference Point Equation 7.2.3 Vapor–Liquid Equilibrium and Permeation Flux 7.2.4 Column Profiles Complex Membrane Configuration Designs: General 7.3.1 Overview 7.3.2 Petlyuk Membrane Arrangement 7.3.3 Material Balances Complex Membrane Configuration Designs: Operating Condition 1 7.4.1 Statement 7.4.2 Mathematics 7.4.3 Column Profiles 7.4.4 Requirements for Feasibility 7.4.5 Analysis and Behavior of Column Profiles 7.4.6 Feasible Coupled Columns Complex Membrane Configuration Designs: Operating Condition 2 7.5.1 Statement 7.5.2 Mathematics 7.5.3 Column Profiles 7.5.4 Feasibility Complex Membrane Configurations: Comparison with Complex Distillation Systems Hybrid Distillation–Membrane Design 7.7.1 Overview 7.7.2 Material Balances 7.7.3 Feasibility Conclusion

8 SYNTHESIS AND DESIGN OF HYBRID DISTILLATION–MEMBRANE PROCESSES 8.1 Introduction 8.2 Methanol/Butene/MTBE System 8.2.1 Design Requirements 8.3 Synthesis of a Hybrid Configuration

111 113 113 114 114 114 116 117 117 117 119 120 121 124 132 132 132 133 134 138 138 138 140 141 150

151 152 153 155 156

CONTENTS

8.4 Design of a hybrid configuration 8.4.1 Column Sections of Hybrid Configuration 8.4.2 Degrees of Freedom 8.4.3 Generating Profiles for Hybrid Columns 8.4.4 Comparing Feasible Design Options 8.4.5 Attainable Region 8.5 Conclusion 9 CONCLUDING REMARKS 9.1 Conclusions 9.2 Recommendations and Future Work 9.3 Design Considerations 9.3.1 Processes for Which Membrane Separations Are Particularly Suitable 9.3.2 Processes for Which Membrane Operations Are Unsuitable 9.3.3 Pressure Difference as a Design Consideration 9.3.4 Effect of Reflux in Membrane Columns 9.4 Challenges for Membrane Process Engineering

ix

159 159 161 163 164 164 167 169 170 171 172 172 173 174 175 176

REFERENCES

177

APPENDIX A: MemWorX USER MANUAL A.1 System Requirements A.2 Installation A.3 Layout of MemWorX A.4 Appearance of Plots A.5 Step-by-Step Guide to Plot Using MemWorX A.6 Tutorial Solutions

183 183 184 184 186 186 192

APPENDIX B: FLUX MODEL FOR PERVAP 1137 MEMBRANE

201

APPENDIX C: PROOF OF EQUATION FOR DETERMINING PERMEATION TIME IN A BATCH PROCESS

203

APPENDIX D: PROOF OF EQUATION FOR DETERMINING PERMEATION AREA IN A CONTINUOUS PROCESS

207

x

CONTENTS

APPENDIX E: PROOF OF THE DIFFERENCE POINT EQUATION E.1 Proof Using Analogous Method to Distillation E.2 Proof Using Mass Transfer

209 209 213

INDEX

217

PREFACE

It is the intention of this book to introduce the reader to new, exciting and novel methods of designing and synthesizing membrane-based separation processes. Initially developed by the authors in fulfillment of postgraduate (PhD) research, these methods provide a unique way of analyzing membrane systems. This book is a monograph (of sorts) documenting the various aspects of the research done. Some of this work has already appeared in reputable scientific journals and has also been presented at numerous conferences. By capturing this work in the form of a book, rather than a thesis only, it is hoped that the research presented here will generate interest, planting a seed that will grow. While selective membranes are useful in that they are able to perform highpurity separations, they can be very costly to fabricate. However, the work displayed here shows that basic, nonselective membranes are also a viable option for achieving useful separations. What’s more, the nonselective membranes are more robust—making them cheaper and allowing for a reconsideration of the design procedure. The contents of this book are aimed at undergraduate and postgraduate students, research academia, engineers and scientists in industry involved in the process design, as well as membrane specialists. The book explains the ideas and conceptualizes them by incorporating tutorials and worked examples. A computer-aided program, entitled MemWorX, written by the authors and fellow postgraduate students, is included to assist the reader with the

xi

xii

PREFACE

contents of the book. MemWorX is not intended as a design tool, but rather a learning aid, and should be used in a supplementary manner with the book material. MARK PETERS DAVID GLASSER DIANE HILDEBRANDT SHEHZAAD KAUCHALI Johannesburg, South Africa

ACKNOWLEDGMENTS

We, the authors, wish to thank the following postgraduate students for their efforts and significant contribution to the preparation of the book and CDROM: Craig Griffiths, Neil Stacey, Chan Yee Ma, Aristoklis Hadjitheodorou, Ronald Abbas, Nik Felbab, and Daniel Beneke. Their inputs have been extremely valuable, enhancing the overall outcome of this work. A special thanks to Darryn van Niekerk for his contribution in preparing the cover artwork. Several organizations have supported this work, both directly and indirectly, and we are grateful to them. These include the University of the Witwatersrand, Johannesburg, South Africa; The Academic and Non-Fiction Authors’ Association of South Africa (ANFASA); The National Research Foundation (NRF) of South Africa; and Sasol Technology for their financial contributions. MARK PETERS DAVID GLASSER DIANE HILDEBRANDT SHEHZAAD KAUCHALI

xiii

NOTATION Scalar quantities represented in italics. Vector quantities represented in bold italics. SYMBOLS A A0 An B c D F J J Ji ki L n p0i P P0i P_ R

Membrane area Dimensionless membrane area Normalized area Bottoms flow rate Number of components Distillate flow rate Feed flow rate Membrane flux Vector of fluxes Flux of component i through the membrane Parameter value for component i Liquid flow rate Number of theoretical stages in a distillation CS Saturation pressure of component i Permeate flow rate (continuous) Permeability of component i Permeate removal rate (batch) (continuous) Retentate flow rate

m2 — mol/s mol/s — mol/s mol/s mol/sm2 mol/sm2 mol/sm2 (see Appendix A) mol/s — Pa mol/s molm/sm2Pa mol/s mol/s xv

xvi

R r rD S S sr t TF V x x XD y y

NOTATION

(batch) Retentate holdup Ratio of pressures (flux model) Reflux ratio Separation vector Side-draw flow rate Split ratio (hybrid design—see Section 8.4.2) Time Feed temperature Vapor flow rate Residual fluid molar composition Retentate composition Difference point Permeate composition Vapor phase molar composition

mol — — — mol/s — s C mol/s — — — — —

GREEK LETTERS aD ij aM ij b(A) D d d ci pP pR t

Relative volatility for distillation Ratio of permeabilities, or membrane selectivity Ratio of RT to R(A) Net molar flow in a column section Thickness of the membrane Difference vector Liquid phase activity coefficient for component i Permeate (low) pressure Retentate (high) pressure Dimensionless time

SUBSCRIPTS Symbol Acc B D F i j M P R T

Designates Accumulated amount Bottom Distillation Feed Component i Component j Membrane separation Permeate Retentate Top

— — — mol/s m — — Pa Pa —

NOTATION

SUPERSCRIPTS Symbol D M O

Designates Distillation Membrane separation Initial conditions Local composition

ABBREVIATIONS AR CPM CS DCS DE DPE D-RCM MBT MCS M-RCM MTBE NRTL RCM SP VLE

Attainable region Column profile map Column section Distillation column section Differential equation Difference point equation Distillation residue curve map Mass balance triangle Membrane column section Membrane residue curve map Methyl tertiary-butyl ether Nonrandom two liquid Residue curve map Stationary point Vapor–liquid equilibrium

xvii

ABOUT THE AUTHORS MARK PETERS obtained his undergraduate degree in Chemical Engineering cum laude in 2003 and his PhD in 2008 from the University of the Witwatersrand, Johannesburg, South Africa. The topic of his thesis, entailing the development of graphical tools for membrane process design, has formed the basis of this book. Mark has spent time at the University of Illinois at Chicago (UIC), Chicago, Ilinois, USA as a research student. He has authored four scientific articles in the field of membrane separation and has presented work at numerous internationally recognized conferences. He previously worked as a research process engineer at Sasol Technology, focusing on Low Temperature Fischer–Tropsch (LTFT) Gas-to-Liquids (GTL) conversion. He is currently a Separations Consultant and Research engineer at the Centre of Material and Process Synthesis (COMPS), based at the University of the Witwatersrand. DAVID GLASSER is a personal Professor of Chemical Engineering and director of the Centre of Material and Process Synthesis (COMPS) at the University of the Witwatersrand, Johannesburg, South Africa. He obtained his BSc (Chemical Engineering) from the University of Cape Town and his PhD from Imperial College in London. Along with Diane Hildebrandt, he pioneered the work in the Attainable Region (AR) approach for process synthesis and optimization. He has been awarded an A1 rating as a scientist in South Africa, by the National Research Foundation, the central researchfunding organization for the country. He has been awarded the Bill Neale–May Gold Medal by the South African Institution of Chemical xix

xx

ABOUT THE AUTHORS

Engineers (SAIChE), as well as the Science for Society Gold Medal of the Academy of Sciences of South Africa for his research work. He has also been awarded the inaugural Harry Oppenheimer Gold Medal and Fellowship. He has authored or coauthored more than 100 scientific papers and was editor-in-chief of the new book Series on Chemical Engineering and Technology, published by Kluwer Academic Publishers of The Netherlands. He has authored a chapter published in Handbook of Heat and Mass Transfer Volume 4, Advances in Reactor Design and Combustion Science. He has also served as President of the South African Institution of Chemical Engineers. He has worked in a very wide range of research areas, including optimization, chemical reactors, distillation, and process synthesis. DIANE HILDEBRANDT is the codirector for the Centre of Material and Process Synthesis (COMPS) at the University of the Witwatersrand, Johannesburg, South Africa. She obtained her BSc, MSc, and PhD from the University of the Witwatersrand. She has authored or coauthored over 70 scientific papers and has supervised 40 postgraduate students. She has been both a plenary speaker and invited speaker at numerous local and international conferences. In 1998, Diane became the first woman in South Africa to be made a full professor of Chemical Engineering when she was appointed as the Unilever Professor of Reaction Engineering at the University of the Witwatersrand. In 2003, she became the first woman professor of chemical technology in The Netherlands when she was appointed as a part time Professor of Process Synthesis, University of Twente, The Netherlands. In 2005, she was recognized as a world leader in her area of research when she was awarded an A rating by the National Research Foundation. She has been the recipient of numerous awards, including the Bill Neale–May Gold Medal from the South African Institute of Chemical Engineers (SAIChE) in 2000, and Distinguished Women Scientists Award presented by the Department of Science and Technology (DST) South Africa. Most notably, in 2009, she was the winner of the African Union Scientist of the year award. She has worked at Chamber of Mines, Sasol and the University of Potchefstroom and has spent a sabbatical at Princeton. SHEHZAAD KAUCHALI obtained his PhD at the School of Chemical and Metallurgical Engineering, University of the Witwatersrand, Johannesburg, South Africa. He is currently a full time senior academic and the director of the Gasification Technology & Research Group at the university. Shehzaad’s thesis developed process synthesis tools for reactor and separation networks. He has developed expertise in the areas of reactor network synthesis, hybrid membrane–distillation design and gasification technologies. Shehzaad has spent ten months at Carnegie Mellon

ABOUT THE AUTHORS

xxi

University in Pittsburg, Pennsylvania USA as a research scholar. Shehzaad completed a sabbatical, in the capacity of visiting professor, at the Indian Institute of Technology (IITB), Powai, Mumbai, India. Shehzaad has coauthored over ten publications in the field of chemical engineering and has cosupervised some theses for masters and doctoral degrees. Shehzaad has been a consultant with the Centre of Material and Process Synthesis (COMPS) and has consulted for Sasol, AECI, de Beers Diamonds, Element Six, Pratley, and the Paraffin Association of Southern Africa.

Figure 6.5. Extended RCM. MBT in blue, line of discontinuity in green.

Figure 6.6. Column profile map (CPM) for XD ¼ [0.2, 0.3, 0.5] and rD ¼ 5. MBT is shown in blue, the “moved” triangle is shown in red.

Figure 6.7. CPM for (a) XD ¼ [1.2, 0.3, 0.5] and rD ¼ 5; (b) XD ¼ [1.2, 0.3, 0.1] and rD ¼ 5; and (c) XD ¼ [0.4, 0.2, 0.8] and rD ¼ 5.

Figure 6.8. “Moving” triangles for varying rD with a fixed XD ¼ [0.3, 0.3, 0.4]. (a) Positive rD values of 3 (green), 6 (orange), 12 (red), and þ 1 (blue, MBT). (b) Negative rD values of 3 (green), 6 (orange), 12 (red), and 1 (blue, MBT).

Figure 6.9. Pinch point loci for XD ¼ [0.3, 0.3, 0.4].

Figure 6.10. All sets of different pinch point loci for fixed XD in the six other regions.

Figure 6.14. Entire M-RCM (MBT in blue, line of discontinuity in green).

Figure 6.16. Shifted M-RCMs. (See text for full caption.)

Figure 6.17. Tangency of curves to the column profile. Each curve corresponds to a different b(A) value and is allowed to run to its stable pinch point.

Figure 6.18. Pinch point loci for dT ¼ [ þ , , ] and dT ¼ [, þ , þ ].

Figure 6.19. Column profiles for selected values of (a) RT with xT ¼ [0.3, 0.3] and (b) xT with RT ¼ 100 mol/s. Figure .16(a) has been superimposed.

Figure 6.21. Pinch point loci for XD ¼ [0.3, 0.3].

Figure 6.22. Shifted M-RCM. The blue triangle represents the MBT, the red shows the transformed triangle. XD ¼ [0.3, 0.3], xT ¼ [0.3, 0.2], rD ¼ 6. Actual membrane profile (see Figure 6.20) is tangent to the bold profile at the point indicated.

Figure 6.23. Tangency of curves to the column profile. Each curve corresponds to a different rD value and is allowed to run to its stable pinch point.

Figure 6.25. Effect of the sign of rD on the trajectory of the profile. Each profile is influenced by its associated stable pinch point locus.

Figure 7.4. Coupled membrane column sections.

Figure 7.5. Retentate (solid) and permeate (dotted) column profiles for CS1 (red) and CS2 (blue). xT ¼ [0.2, 0.3], XD ¼ [0.3, 0.3], rD1 ¼ 5, and rD2 ¼ 5.

Figure 7.6. (c) Pinch point loci for XD ¼ [0.3, 0.3]. (d) Tangency of curves to a column profile commencing at xT ¼ [0.3, 0.2]. Each curve corresponds to a different rD value and is allowed to run to its stable pinch point (see Chapter 6).

Figure 7.8. (a) Example of retentate profile (solid) passing through the pinch point loci (crosses), resulting in the permeate curve (dotted) intersecting with the retentate. (b) Zoomed-in version of (a), showing the mass balance lines (dashed).

Figure 7.9. Region of xT locations for feasibility with XD ¼ [0.3, 0.3] in the membrane coupled column sections.

Figure 7.10. Feasible xT regions for (a) XD ¼ [, þ , ] and (b) XD ¼ [ þ , , þ ].

Figure 7.11. Retentate (solid) and permeate (dotted) profiles for CS1 (red) and CS2 (blue), commencing at xT ¼ [0.45, 0.4], with XD ¼ [0.3, 0.3] operating at (a) rD1 jT ¼ 10 ¼ rD2 jT , (b) rD1 jT ¼ 16:037 ¼ rD2 jT , and (c) rD1 jT ¼ 20 ¼ rD2 jT .

Figure 7.12. Retentate (solid) and permeate (dotted) column profiles for CS1 (red) and CS2 (blue). (See text for full caption.)

Figure 7.13. Retentate (solid) and permeate (dotted) column profiles for CS1 (red) and CS2 (blue). (See text for full caption.)

Figure 7.15. Coupled distillation and membrane column sections.

Figure 7.16. Commencing the retentate profiles at various xT values. The corresponding permeate profile eventually always approximates its pinch point curve.

Figure 7.18. Liquid pinch point locus within the MBT.

Figure 7.19. The permeate (purple) and liquid (green) pinch point loci defining an initial region of feasibility.

Figure 7.20. CPM for the DCS at rDD ¼ 5 and XD ¼ [0.3, 0.3]. MBT in blue, shifted triangle in red.

Figure 7.21. Region of possible xT locations, for the operating conditions chosen, that would result in a feasible arrangement.

Figure 7.22. (a) Column profiles for the membrane and distillation sections with xT in the feasible region. (b) Zoomed in version of (a).

Figure A.4. Full set of plots on same axes.

CHAPTER 1

INTRODUCTION

Separation processes are fundamentally important in the chemical industry. It is inevitable that during any chemical process, be it continuous or batch, the need for effective separation will arise. There are a variety of separation options available. However, distillation has proved to be the most effective and commonly used method, especially for the separation of mixtures containing compounds with relatively low molecular weights, such as organic substances. Other efficient separation techniques are absorption and liquid–liquid (solvent) extraction. In recent decades, membrane permeation has come to the fore as a successful method of separating mixtures, both gaseous and liquid. One would like to have available a technique to select the appropriate method of separation to achieve the required product specifications. This chapter will begin to address this need, laying the foundation for the rest of the book. Membranes have been developed for various separation applications. Examples of these include, among others, reverse osmosis, electrodialysis, pervaporation, and gas separation. Rautenbach and Albrecht (1989) discuss each of these. The aim of this book is not to reiterate what numerous texts have discussed previously. Rather, the reader is referred to this, and other texts (such as Geankoplis, 1993; Hoffman, 2003; Drioli and Giorno, 2009), which give Membrane Process Design Using Residue Curve Maps, First Edition. Mark Peters, David Glasser, Diane Hildebrandt, and Shehzaad Kauchali. Ó 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 1

2

INTRODUCTION

more detailed appraisals of each individual membrane application. In order to demonstrate design and synthesis techniques, only diffusion membranes (e.g., gas separation and pervaporation membranes) will be considered in this book, but the method developed can be adapted and applied to the other kinds of membranes. In diffusion membrane separation, a high-pressure fluid mixture comes into contact with a membrane, which preferentially permeates certain components of the mixture. The separation is achieved by maintaining a lower pressure (sometimes vacuum) on the downstream, or permeate, side of the membrane. The remaining high-pressure fluid is known as the retentate. Figure 1.1a,b depicts basic batch and continuous diffusion membrane separation units, respectively. A more detailed discussion of membrane process operation is given in the book where appropriate. Gas separation involves the diffusion of a gaseous mixture, whereas pervaporation is a separation process where one component in a liquid mixture is preferentially transported through the membrane and is evaporated on the downstream side, thus leaving as a vapor. These processes are discussed in more detail in Chapter 2. The conventional way of analyzing membrane separators is to ask what permeate composition can be achieved for a particular feed, as in the experiments conducted by Van Hoof et al. (2004) and Lu et al. (2002). Furthermore, the flux of a particular component through the membrane is also reported as a function of the feed in these and similar experiments. However, it must be remembered that the flux of any of the components may not necessarily remain the same and may vary as permeation proceeds down the length of the membrane (continuous operation). Therefore, the conventional information, although accurate, is insufficient, especially when it comes to designing industrial-scale membrane separators, as well as sequencing of such equipment. When examining how the other, more established, separation processes are analyzed, it can be seen that the methods used for membranes are somewhat ineffective. In distillation, as well as single-stage flash separations, one never reports how either the top or bottom products are related to the feed, but rather

Figure 1.1. (a) Batch and (b) continuous diffusion membrane units.

INTRODUCTION

3

how they are related to each other. A similar kind of analysis is conducted when designing solvent extraction circuits—one requires the equilibrium data that relates the aqueous phase to the organic phase. Relating the two product streams to each other allows one to design multiple flash units, or cascades, as well as countercurrent liquid-extraction circuits and distillation columns. The design of such reflux cascades would be impossible if either one of the product compositions were related to the feed entering any of the units within the cascade. It is necessary to analyze membrane separators in a similar manner—one needs to investigate how the permeate composition is related to that of the retentate. The relationship of the permeate to the retentate has been modeled mathematically. For example, Eliceche et al. (2002) model a pervaporation unit for a binary separation—the analysis was carried out by considering the flux of each component through the membrane and solving simultaneous mass and energy balance equations. Stephan et al. (1995), on the other hand, describe the permeation by a simple dual-mode transport model, and make use of Henry’s law to relate the permeate composition to the composition of the retentate. A thermodynamic and more fundamental approach is given in the article by Wijmans and Baker (1995). In this review, the concentration and pressure gradients in the membrane are described using chemical potentials as the fundamental starting point. By appropriately modeling the chemical potentials, and by making use of Fick’s law of diffusion, Wijmans and Baker (1995) are able to model the permeation for the various types of membranes, including gas separation and pervaporation. The details of the resulting equation for gas separation are discussed later in Section 2.1.1. Figure 1.2 gives a basic sketch of membrane permeation. Although the information in these models is correct, it is somewhat difficult to interpret and utilize them for design purposes. It is therefore the aim of this book to formulate a graphical technique that can incorporate the appropriate models in order to interpret, analyze, and design membrane separators in a convenient and efficient manner. Conventionally, it was believed that residue curve maps (RCMs), and their binary equivalent (i.e., x–y plots), were suitable only for equilibrium-based separations and could not be used for the representation of kinetically based processes (Fien and Liu, 1994). This is discussed further in Chapter 3. However, as will be shown, the differential equations that describe a residue curve are merely a combination of mass balance equations. Because of this, the inherent nature of RCMs is such that they can be used for equilibriumbased, as well as nonequilibrium-based processes. This now allows one to consider kinetically based processes, such as reactive distillation (Barbosa and Doherty, 1988; Doherty and Malone, 2001; Huang et al., 2004) as well as membrane separation processes.

4

INTRODUCTION

Figure 1.2. Membrane permeation. Differing permeabilities provide the driving force for separation.

This book guides the reader through the development of graphical tools for nonreacting membrane systems. It is necessary to mathematically describe the flux of material through a membrane, as detailed in Chapter 2. The models derived and discussed are used throughout the text. Chapter 3 introduces the concept of membrane plots for various systems, and from Chapter 4 onwards the various applications of these maps are explored. The MemWorX Package A mathematical computer program, coded in MatlabÒ , entitled MemWorX, was especially developed for this book. A CD-ROM containing the MemWorX program is available for the reader and is to be used to aid understanding of the material covered in this book. Throughout the book, where appropriate, references are made to MemWorX, giving basic steps on how to generate plots shown in the book and, where necessary, allow users to produce their own plots. For details on how to install and run MemWorX, the reader is referred to Appendix A, which also includes a step-by-step guide to producing plots using MemWorX.

INTRODUCTION

5

The majority of figures displayed throughout the book can be reproduced using MemWorX, and, where appropriate, the reader is prompted to attempt to reproduce the figure being referred to (or similar) using MemWorX. The symbol, shown alongside, will indicate when MemWorX should be used. Should any problems be encountered, Table A.4 in Appendix A lists the MemWorX parameters used to produce each figure, according to the tutorial number as listed in the book. This table can be regarded as the solutions to each of the tutorials involving the MemWorX package.

CHAPTER 2

PERMEATION MODELING

In order to generate graphical representations of membrane separation, it is necessary to have mathematical equations that describe how various chemical components are transported through the membrane material. In short, a relationship between the retentate composition, x, and that of the permeate, y, is needed. The methods and ideas discussed in this book rely on the use of a flux model, which, in turn, is specific to the type of membrane used—without it, graphical representation would be impossible. However, as will be shown, these methods are general, and they are not developed for any particular set flux model, but can easily be adapted for any membrane, provided there is a reliable flux model associated with it. It is not the aim of this book to illustrate how one would go about determining these models, or fitting parameters to them. The aim is to show the reader what design and synthesis capability can be achieved once a model is available. In this chapter, flux models for two types of diffusion membranes are discussed. These models will be used throughout the book.

Membrane Process Design Using Residue Curve Maps, First Edition. Mark Peters, David Glasser, Diane Hildebrandt, and Shehzaad Kauchali. Ó 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 7

8

PERMEATION MODELING

2.1 DIFFUSION MEMBRANES As mentioned in the introductory chapter, this book focuses on diffusion membranes. These include gas separation and pervaporation, and rely on partial pressure differences to bring about the driving force for separation. 2.1.1 Gas Separation Gas separation involves the diffusion of a gaseous mixture through the membrane material. In order to obtain a general flux model for a gas separation membrane, the fundamental thermodynamic approach incorporating the solution–diffusion model, as derived by Wijmans and Baker (1995), is used: Ji ¼

P0i ðpR xi pP yi Þ d

ð2:1Þ

where Ji is the flux of component i through the membrane (mol/sm2), P0i is the permeability of component i (which is equal to the product of solubility and diffusivity through the membrane material) (molm/sm2Pa) pR is the (high) retentate pressure (Pa), pP is the (low) permeate pressure (Pa), xi is the mole fraction of component i in the retentate phase, yi is the mole fraction of component i in the permeate phase, and d is the effective thickness of the membrane (m). Note: The symbol p has been used for total pressure (unlike the usual symbol of P) to avoid confusion. This has been done since P (and many of its derivatives) has been or will be used for other parameters. This is a general flux expression for describing the behavior of a gas separation membrane and is used for demonstration purposes. Since the flux of any component is the amount of material that permeates through the membrane per unit time per unit area, one may also write Ji ¼

P yi A

where P is the permeate flow (mol/s), and A is the effective membrane area (m2).

ð2:2Þ

DIFFUSION MEMBRANES

9

Tutorial 2.1 Equations (2.1) and (2.2) are two different expressions that describe the same parameter (i.e., flux, Ji). Using these equations (Hint: for components i and j), show that yi ðrxi yi Þ P0i pR M ¼ aM ¼ and r ¼ ; where a ij ij 0 ðrxj yj Þ yj Pj pP

ð2:3Þ

where aM ij is known as membrane selectivity, or the relative permeability of i with respect to j, and r is known as the ratio of pressures. An effective separation is one in which the flux of material is in the direction from the retentate to the permeate. Thus, for an effective separation to occur, pR xi > pP yi or

yi yB, and xC > yC. In the smaller region: xA < yA, xB > yB, and xC < yC. It is clear that on either side of the boundary, the distributions of A and B remain the same, with the permeate being richer in A and the retentate richer in B in both regions. Component C, on the other hand, is richer in the retentate phase than the permeate in the larger region, but in the adjacent region, the opposite occurs. Such information is useful when one needs to identify which phase, for example, component C would ultimately report to. If one wants a permeate phase rich in A, then operating in the larger region would yield this since the retentate will be richer in components B and C. Operating in the smaller region would allow one to obtain a B-rich retentate, with most of the A and C leaving in the permeate. Furthermore, the maximum amount of C that can be obtained in either phase is indicated by the point where the appropriate residue curve (that the batch experiment follows) intersects the unidistribution line. Such information is very difficult to extract from mathematical equations, yet it is quite simple to visualize with the use of the M-RCM. 4.4 THE EFFECT OF a-VALUES ON THE TOPOLOGY OF M-RCMs The location of nodes and/or saddles as well as the curvature of the lines is collectively known as the topology of a residue curve map. The rate at which the various components permeate through the membrane is what will ultimately determine the topology of an M-RCM. Thus, the topology of an M-RCM is dependent on the relative permeabilities. The relative

THE EFFECT OF a-VALUES ON THE TOPOLOGY OF M-RCMs

33

permeabilities, themselves, depend on the inherent properties of the membrane and the material from which the membrane is made. The order of permeability (from highest to lowest) of the M-RCM shown in Figure 3.5 is A–C–B. If the values of the relative permeabilities are changed, while keeping the order the same, the location of the stationary points will not be altered. However, the curvature of the curves will be changed. The larger the difference in the values of the relative permeabilities, the easier the separation becomes, resulting in the residue curves being less curved. If the order of the relative permeabilities is altered, the topology of the map is rearranged. For example, if the order of permeabilities is changed from A–C–B, as in Figure 3.5, to A–B–C, the locations of the stationary points are changed. An example of such a plot is shown in Figure 4.2(a), where the pure B component point is now the saddle point, and C has now become the stable node since it now has the lowest relative permeability. There are many different scenarios that can occur, all of which depend on the order of the relative permeabilities of the involved substances. Furthermore, within each case, it is also possible to analyze the curvature (while keeping the order the same), as was done for the A–C–B case. A “special case” arises when two of the relative permeability values are equal to each other. Physically, this means that two of the three components are permeating through the membrane at exactly the same rate, resulting in a selective membrane. Mathematically, it results in the M-RCM consisting of curves that are essentially straight lines. These lines originate from (or terminate at) the pure component node of the substance that permeates at a higher (or lower) rate than the other two, which are permeating at the same rate. Figure 4.2(b)

Figure 4.2. The effect of changing the order of the relative permeabilities (Peters et al., 2006a): (a) A—unstable node, B—saddle point, C—stable node. (b) Components A and B have equal permeabilities, whereas that of C is lower.

34

PROPERTIES OF MEMBRANE RESIDUE CURVE MAPS

shows an example of an M-RCM when the permeation rate of A and B are equal to each other, and that of C is lower than A and B. This means that all the curves terminate at pure C, making that point the only stationary point in the entire map. In this particular case, that point is a stable node. If, however, the relative permeability of pure C is higher than that of A and B, then the stationary point would be an unstable node, and all curves would move in an outward direction from C. Tutorial 4.1 Try this for yourself: Using the MemWorX program, reproduce Figures 4.2(a) and 4.2(b) by using a ¼ [3,1.5,1] and a ¼ [3,1,1] respectively. Remember to check “Plot within MBT only” for now. (Hint: The plot type is “Full CPM” using the nonconstant (membrane) flow assumption.) Try producing other similar plots with any a-values that you choose, and note how the topology is affected. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. 4.5 PROPERTIES OF AN EXISTING SELECTIVE M-RCM As discussed before, the membrane residue curve equations (Equation (3.4)) are general and are not limited to the constant relative permeability case; furthermore, they are not limited to only gas separation. Provided an appropriate and accurate method of calculating the flux of each component through the membrane is employed (the choice of which is left to the designer), one is able to generate the plot for the chosen membrane. As an example of how M-RCMs are useful, consider the following real system for the separation of methanol from a ternary mixture of methanol/butene/MTBE, as discussed in Section 2.1.2. The system of DEs is solved for two cases: one where a vacuum is maintained on the permeate side, and another where the permeate pressure is nonzero. Each case is discussed in more detail. 4.5.1 Case 1: When the Permeate Side Is at Vacuum Conditions (i.e., pP 0) This is a very common mode of operation for membrane systems. In this case, the flux equation for methanol (refer to Equation (2.15)) is reduced to J1 ¼ k1 p01 ðc1 x1 Þ2

ð4:3Þ

PROPERTIES OF AN EXISTING SELECTIVE M-RCM

35

Figure 4.3. M-RCM for methanol/butene/MTBE with pP 0. (Peters et al., 2006a).

Using this flux, along with the fluxes of the other two components (Equations (2.16) and (2.17)), the M-RCM is plotted as shown in Figure 4.3. As stated before, the membrane is selective to methanol permeation, resulting in the permeate having a high concentration of methanol (this is not seen in Figure 4.3, since the M-RCM shows the change in retentate composition only). Because the retentate is rapidly depleted of its methanol, the residue curves move out of the pure methanol node, making this point the unstable node. Tutorial 4.2 Try this for yourself: Using the MemWorX program, reproduce Figure 4.3 by making use of the NRTL model. (Hint: The plot type is “Full CPM” using the nonconstant (membrane) flow assumption.) Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. The flux equations for MTBE and butene (Equations (2.16) and (2.17)) do not depend on the pressure of the permeate, but do, however, depend on the flux of methanol. As soon as the net flux of methanol equals zero, the fluxes of the other two components go to zero. This results in the liquid (retentate) composition not changing with time, reaching an arheotrope. The arheotrope will occur when x1 ¼ 0 (refer to Equation (4.3)). Since this is the only condition for arheotropism in this system, the entire MTBE/butene binary (vertical) line is a locus of arheotropes. This means that each residue curve

36


originates from the pure methanol (unstable) node and terminates when it reaches a point on the MTBE/butene binary line. The curves do not move into either the MTBE or the butene node, but rather terminate at whatever binary composition the residual retentate is at when x1 ¼ 0. Only curves moving along the methanol/MTBE binary (horizontal) or methanol/butene binary (diagonal) will terminate at either pure MTBE or butene, respectively. This phenomenon of a locus of arheotropes seems to be unique to membrane systems, since the inherent properties of the membrane allow for selective permeation. Such a phenomenon does not occur in distillation processes, for example, since boiling will continue no matter what components are present. The locus of arheotropes is dependent on the membrane used and may or may not occur. If the membrane is not selective to which species it allows through, then it will more than likely not occur. In this case, however, the membrane is highly selective to methanol permeation. When methanol completely permeates, separation no longer occurs, and a locus of arheotropes exists. 4.5.2 Case 2: When the Permeate Pressure Is Nonzero (i.e., pP > 0) As established in Case 1 above, permeation will stop when the flux of methanol goes to zero. Solving Equation (2.15) for J1 ¼ 0, and remembering that x1 > 0: x1 ¼

pP p01 c1

ð4:4Þ

Equation (4.4) tells us that there is, in fact, some value of x1 > 0 that will make the flux of methanol go to zero. From this we can conclude that, at pP > 0, it is not possible to deplete the retentate of all its methanol, unlike when pP 0. Such an occurrence is due to the properties of the membrane. Using Equation (4.4), it is possible to find the locus of arheotropes for a set permeate pressure. In order to simplify the calculation, an isothermal process of 80 C was assumed. Figures 4.4(a)–4.4(c) show the resulting M-RCMs for chosen permeate pressures of 0.5, 1.0 and 1.5 bar. In Figure 4.4(a), it can be seen that the locus of arheotropes has moved from the MTBE/butene binary into the MBT region, as represented by the dotted line. Operation in the unshaded region to the right of the locus will result in the permeate being effectively stripped of its methanol. However, when the associated residue curve reaches the locus of arheotropes, permeation will cease. Using the diagram, it is possible for one to easily calculate how much methanol can be removed from the retentate, as well as to determine the amount of methanol remaining in the retentate. Such a

PROPERTIES OF AN EXISTING SELECTIVE M-RCM

37

Figure 4.4. M-RCM for methanol/MTBE/butane (Peters et al., 2006a). (a) pP ¼ 0.5 bar, (b) pP ¼ 1.0 bar, and (c) pP ¼ 1.5 bar. Unshaded region ¼ from retentate to permeate (effective separation); shaded region ¼ from permeate to retentate (ineffective separation).

38


calculation would prove to be difficult to solve without the use of the M-RCM and its associated equations. In the shaded region, which lies to the left of the locus of arheotropes, the methanol concentration in the retentate is such that its flux is negative. This results in the fluxes of the other two components being negative, too. Physically, this means that permeation through the membrane has now been reversed. That is, material is moving from the low-pressure vapor permeate side to the high-pressure liquid retentate, and the liquid is now being enriched with methanol. Since the membrane was designed to remove methanol from the retentate, rather than enrich it, it can be said that this region represents a region of ineffective separation. But as before, when the curves in this region intersect the locus of arheotropes, there is no more permeation in either direction. Figures 4.4(b) and 4.4(c) graphically verify the fact that, as the permeate pressure increases, the driving force for effective separation decreases. This results in the effective separation region reducing in size as the permeate pressure increases. Although it is agreed that it is unlikely for one to operate at such high permeate pressures, Figures 4.4(b) and 4.4(c) have been included for demonstration purposes. They serve to enlighten the reader about how the permeate pressure affects the degree of separation that is obtainable. This phenomenon can be acquired from the flux equations themselves but is easily visualized and understood through the M-RCM. Using the M-RCM, it is also easier for the designer to choose an appropriate permeate pressure at which to operate, no matter how small. Essentially, it is only methanol that permeates through the membrane—the other components move through in relatively small amounts, and do so only when methanol permeates. Therefore, for vaporization to occur, the permeate pressure has to equal the saturation pressure of methanol. This is only possible when pure methanol is permeating. Although permeation will occur, it is not very useful, and furthermore, the composition will not change as permeation proceeds. Thus, the locus of arheotropes collapses into a single point at the pure methanol node, and the entire MBT becomes a region of ineffective separation. Such a result can be confirmed by analyzing Equation (4.4). For permeate pressures greater than the saturation pressure of methanol, the phenomenon of arheotropism is no longer in the MBT, and the entire MBT is in a state of ineffective separation.

4.6 CONCLUSION This chapter and the previous one serve as the theoretical background to M-RCMs. A residue curve map for membrane permeation systems has been

CONCLUSION

39

produced. Since the equations that describe a single residue curve are merely a combination of mass balances, it can be concluded that an RCM could be used for kinetically based processes and is not restricted to equilibrium-based processes, as was previously assumed. The curves that exist within the map trace the composition of the retentate in a batch separation process, as discussed in Section 3.3. At any time during batch permeation, the permeate composition is related to that of the retentate at that particular time, thereby allowing a membrane separation vector to be defined as the vector difference between the two compositions. An M-RCM is produced for a real system, using flux data obtained from literature. The resulting plots give valuable insight into the operation of the particular membrane. The qualitative understanding of how the permeate pressure influences the effective separation is confirmed, and the result can easily be seen by examining the appropriate M-RCM. Furthermore, using the M-RCM, the concentration of methanol remaining in the retentate as well as the amount of methanol removed from a particular feed can be determined with ease. In Chapter 5, it will be shown how an M-RCM can be used in the conceptual design of both batch and continuous membrane operations.

CHAPTER 5

APPLICATION OF MEMBRANE RESIDUE CURVE MAPS TO BATCH AND CONTINUOUS PROCESSES

Membrane residue curve maps (M-RCMs), as discussed in Chapters 3 and 4, plot the change, over time, of the retentate composition in a batch still. In this chapter, the relevance of M-RCMs to both batch and continuous processes is investigated. This work originally appeared in a prior publication (Peters et al., 2008). A constant relative permeability model is used for demonstration purposes, and the theory is then extended to a real system of methanol/ butene/methyl tertiary butyl ether (MTBE). It has been shown that the differential material balances over a continuously operated membrane unit are mathematically equivalentto those thatdescribe the M-RCM.Thetimevariable in the batch setup is analogous to the spatial variable in continuous units. The retentate composition in a nonreflux continuous unit, for example, will follow the residue curve that passes through the initial feed composition. The M-RCM, in conjunction with the necessary flux equations, allows a designer to determine permeation area (or time) required in a continuous (or batch) process. Membrane columns operating at total and infinite reflux are discussed. 5.1 INTRODUCTION Throughout the past several decades, membrane-based processes have received a substantial amount of interest as an attractive alternative to Membrane Process Design Using Residue Curve Maps, First Edition. Mark Peters, David Glasser, Diane Hildebrandt, and Shehzaad Kauchali. Ó 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 41

42

APPLICATION OF MEMBRANE RESIDUE CURVE MAPS TO BATCH

conventional separation methods. Not only are membranes an economically viable option, but they also offer unique separation behavior. Being a kinetically based process, membrane separation is not limited by equilibrium constraints, and allows for effective “breaking” of azeotropes (Rautenbach and Albrecht, 1985a). However, only a few membrane units, both continuous and batch, have found their place in industrial practice (Clement et al., 1992), either as stand-alone or as part of a hybrid arrangement. One of the reasons for this is the lack of high-performance membranes for various applications. Another reason is that the design of membrane separators is complex and not as well understood and formulated as traditional separation methods, such as distillation or liquid–liquid extraction. These customary separation units, such as distillation or absorption, are equilibrium-based processes. Membrane separators, on the other hand, are rate-based operations and are conventionally analyzed and assessed in the same manner as other kinetic processes. Traditionally, it was understood that the graphical technique of residue curve maps (RCMs) was appropriate only for equilibrium-based separations (Fien and Liu, 1994). However, the equations describing any single residue curve are purely a combination of material balance equations. With this said, RCMs can be used for both equilibrium- and rate-based processes. This now enables one to consider kinetically based processes, of which membrane permeation is an example, in the same manner as equilibrium processes. There is a broad spectrum of membranes that have been developed for various applications (Rautenbach and Albrecht, 1989; Wijmans and Baker, 1995). The work displayed in this chapter is not limited to a single type of membrane, but is general and can be applied to any membrane. However, only diffusion membranes (i.e., gas separation and pervaporation) will be considered here, so as to demonstrate the theory presented. The work can then be extended further and applied to other membrane types. The driving force for separation in diffusion membranes is the difference in partial pressures for each component on either side of the membrane. A vast amount of research into the development of membranes has been carried out. Numerous works have focused on the modeling of permeation fluxes through various membranes, as well as the relationship between permeate and retentate phases. These have been formulated both theoretically and experimentally. Wijmans and Baker (1995) provide a review on the solution-diffusion model. They describe general flux models for a wide range of membrane operations, making use of chemical potentials as the starting point for derivations. Stephan et al. (1995), on the other hand, employ Henry’s law for describing the permeate composition as a function of the retentate composition. Mathematical modeling of a membrane’s

INTRODUCTION

43

performance using the appropriate transport equations has been researched (Rautenbach and Albrecht, 1985a,b; Marriott et al., 2001; Marriott and Sørensen, 2003a). A membrane can be utilized in numerous ways in order to achieve a required outcome—either production of desired products or to provide data for modeling and scale-up design. A batch membrane operation, in the form of bench-scale or pilot tests, is often used to select a membrane best suited for a required purification. Such work has been done by Feng and Huang (1992) and Qiao et al. (2005), among others. Similar tests are also used to determine suitable selectivity and flux models of a particular membrane (e.g., Clement et al., 1992). Work by Stoica-Guzun et al. (1996) provides an example that batch membrane systems have found a substantial place in the biomedical field, too. While the batch experiments are necessary for continuous process design and scale-up, operating a membrane in a batch process may have very little commercial value. Continuously operated membranes, in contrast, have shown their economical value, especially in the last twenty years or so. Different modes of operation have been researched and modeled accordingly. Nonreflux membrane equipment is designed such that a particular feed mixture undergoes a single-pass separation without being recycled back. Due to the area required as well as energy considerations, a series of such membranes with interheaters may be necessary to achieve a desired separation (Bausa and Marquardt, 2000). Ideal flow patterns of perfect mixing, cross flow, countercurrent flow, and cocurrent flow, and their associated models, have been documented (Rautenbach and Albrecht, 1989). Blaisdell and Kammermeyer (1973) discuss the Weller–Steiner and Naylor–Backer models for both co- and countercurrent continuous gas permeations, whereas Marriot et al. (2001) aim to derive a generalized model, irrespective of flow regime and simplifying assumptions. Various methods and techniques for the modeling, design, and optimization of membrane systems have been the topic of numerous works in recent years (Tessendorf et al., 1999; Marriott and Sørensen, 2003b; Chang and Hou, 2006). A wide range of reflux units have previously been researched. A membrane column was first identified by Hwang and Thorman (1980). The column is filled with hollow-fiber membranes, and a high pressure is maintained on one side in order to allow permeation through the membrane to the low-pressure side. To allow the recycle streams to flow, a valve and compressor are needed. It is claimed that complete separation of a binary mixture into pure products is achievable in a finite area (Hwang and Thorman, 1980). Variations of the membrane column have also been mentioned and examined, such as the recycle permeator (Yoshisato and Hwang, 1984) and two-membrane column (Matson et al., 1983).

44


5.2 REVIEW OF PREVIOUS CHAPTERS Peters et al. (2006a) proposed a novel method to analyze and understand membrane permeation systems. Consider the previously discussed “thought experiment”: a high-pressure fluid mixture is allowed to permeate through a membrane in a batch still to a maintained low-pressure side. One could continually analyze the change in retentate composition over time, with differing initial feed compositions. Similarly, one can write the appropriate unsteady-state material balances. As was shown in Chapter 3 (Section 3.3.1), the combination of the overall and component mass balances resulted in the membrane residue curve equation: d ðxi Þ ¼ ðxi yi Þ; i ¼ 1; 2; . . . ; c1 dt

ð5:1Þ

where xi is the mole composition of component i in the retentate phase, yi is the mole composition of component i in the permeate phase, c is the number of components present in the initial charge, and t is a dimensionless time, defined by

Rð0Þ t ¼ ln RðtÞ

ð5:2Þ

where R(0) is the initial charge of material to the batch still (mol), and R(t) is the amount of retentate present at time t (mol). In order to demonstrate the graphical representation of the membrane residue curve equation, a simple gas permeation model, with constant relative permeabilities, was used to relate yi to xi. This model is discussed in more detail in Section 2.1.1. Figure 5.1 shows a membrane residue curve map (M-RCM) for the relative permeabilities indicated. In this chapter, the applicability of M-RCMs to the synthesis and design of batch and continuous processes is discussed. The graphical technique allows one to understand and evaluate membrane systems in an expedient and efficient manner. The batch experiment, although somewhat straightforward, provides valuable insights into continuous operation, both reflux and nonreflux. As is shown in Section 5.5.1, the differential balance over a nonreflux separator is mathematically analogous to the residue curve equation (Equation (5.1)). Some of the results obtained and displayed in this chapter may appear to mirror those arrived at in the distillation field. Remember, however, that the techniques being shown are an innovative way of understanding membrane

BATCH MEMBRANE OPERATION

45

Figure 5.1. Membrane residue curve map (M-RCM) for an ideal A–B–C system with aM AB ¼ 3 and aM CB ¼ 1:5. (Peters et al., 2006a).

separation, which is a kinetically based process. While similarities do arise, there are subtle yet distinct differences between distillation and membrane permeation. These are discussed in the chapter where necessary and will be dealt with in more detail in later chapters (see Chapters 6, 7, and 8). Several authors have focused on binary systems due to the ease of the equations that result. It is worth noting that the methods shown and discussed in this chapter are carried out for ternary systems but are not limited to threecomponent systems. The conclusions arrived at can be applied to binary, ternary, and higher order systems, even when graphical visualization becomes difficult. The following section of this chapter deals extensively with the M-RCM and the batch process. Operation regions and obtainable products are discussed. An equation to calculate the time needed for permeation is also derived. In Section 5.5, the applicability of the M-RCM to various modes of continuous membrane operation is detailed—both nonreflux as well as total and infinite reflux. Where pertinent, the similarities to the batch process are included.

5.3 BATCH MEMBRANE OPERATION 5.3.1 Operating Leaves in Batch Permeation Peters et al. (2006a) proposed a thought experiment, as detailed in Section 3.1, whereby a fluid mixture was allowed to permeate through a batch still, with the

46


Figure 5.2. Simple batch membrane separation, with accumulating permeate. (Peters et al., 2008).

permeate being removed as soon as it was formed. Now, consider the following extension to the experiment: The permeate is stored and accumulated in a separate container. Figure 5.2 shows this addition. 5.3.2 Material Balances One can write time-dependent overall and component mass balances across the entire still/storage container system: _ Rð0Þ ¼ RðtÞ þ PDt

ð5:3Þ

_ i;acc ðtÞDt Rð0Þxi ð0Þ ¼ RðtÞxi ðtÞ þ Py

ð5:4Þ

where R(0) is the initial charge of material to the batch still (mol), xi (0) is the initial retentate composition of component i, R(t) is the amount of retentate present at time t (mol), xi (t) is the composition of component i in the retentate at time t, P_ is the permeate removal rate (mol/s), t is the time (s), and yi,acc(t) is the accumulated permeate composition of component i. Only a differential material balance across the still has previously been written. As discussed in Section 5.2, Equation (5.1) is known as the membrane residue curve equation, and a full derivation is provided in Section 3.3.


47

Combining Equations (5.2), (5.3), and (5.4) in an appropriate manner, it can be shown that yi;acc ðtÞ ¼

xi ð0Þet xi ðtÞ 1et

ð5:5Þ

As t ! 1, yi,acc ! xi(0). This result is expected and will be discussed later in this section. Using the differential mass balance over the still (Equation (5.1)), for a particular value of t, there is a corresponding xi -value—that is, the composition of the retentate left in the batch still after the experiment has been allowed to run for a dimensionless time of t. At that particular time, the accumulated y-value can also be determined using Equation (5.5). Tutorial 5.1 A batch membrane still initially contains 2 mol of ternary mixture with composition [0.7, 0.1]. After 1.272 mol has permeated, the composition of the retentate becomes [0.532, 0.179]. Find the accumulated permeate composition, yacc. Solution 5.1 From the definition of t, t ¼ ln

Rð0Þ RðtÞ

But by mass balance RðtÞ ¼ Rð0ÞPðtÞ So

t ¼ ln

2 21:272

¼ 1:0099

Using Equation (5.5) and substituting, we can get yA;acc ¼

0:7e1:0099 0:5321 ¼ 0:796 1e1:0099

48


yB;acc ¼

0:1e1:0099 0:1789 ¼ 0:0548 1e1:0099

yC;acc ¼ 1yA;acc yB;acc ¼ 0:149

&

5.3.3 Permeation Model In order to solve the set of differential equations in Equation (5.1), it is necessary to incorporate a suitable flux model that describes the permeation rate of each component through the membrane. This was the topic of Chapter 2, and the reader is referred to that chapter for a detailed discussion of the various permeation models, as well as the different types of membranes. 5.3.4 Operating Regions: Nonselective Membranes For an initial retentate (feed) composition for a ternary mixture, plots of the following can be generated: (1) The change in the retentate composition, x (referred to as a membrane residue curve), (2) The local (or instantaneous) permeate composition, y (3) The accumulated permeate composition, yacc These are produced by allowing permeation in the batch still to occur for a certain period. An example is shown in Figure 5.3(a). Tutorial 5.2 Try this for yourself: Using the MemWorX program, reproduce Figure 5.3(a) by using a ¼ [3,1,1.5]. Remember to check “Include permeate profile”. (Hint: The plot type is “Single profile” (using the initial x-value shown in Figure 5.3) and the nonconstant (membrane) flow assumption.) Try producing other similar plots with any x-values that you choose. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. Equations (5.3) and (5.4) are straight line mass balances and can be plotted for every retentate composition (and corresponding permeate fraction). The very first mass balance line that is set up during an experiment is the one that


49

Figure 5.3. Initial retentate composition x(0) ¼ [0.4, 0.3] (Peters et al., 2008). (a) (i) Residue curve, (ii) permeate curve, and (iii) accumulated permeated curve. (b) Operating region or “leaf.” The region is bounded by the curves (i) and (iii), as well as mass balance lines (A) and (B).

includes the initial permeate composition, y(0). The local permeate, y(t), as well as the accumulated permeate, yacc(t), curves begin at the same point— namely, the compositional value of the very first amount of material that diffuses through the membrane. This is also the permeate composition that will contain the highest fraction of the most permeable component (A, in this case). The time lapse to produce the first amount is essentially zero, and by analyzing Equation (5.4), it is obvious that the retentate composition has changed only ever so slightly. The straight line connecting the feed, initial permeate, and retentate composition can be plotted, as shown in Figure 5.3(b) (refer to line (A)). The other important mass balance line is the one that exists when the last drop of retentate is about to diffuse through the membrane. This would happen after allowing permeation to occur for an infinite length of dimensionless time (refer to Equation (5.2)). Equation (5.4) shows that, by this stage, the accumulated permeate composition is essentially equal to the initial retentate charge. This result is expected since, by this stage, all the material has permeated through the membrane and is being accumulated in storage. This particular mass balance is also shown in Figure 5.3(b) (see line (B)). The entire region(s) is bounded by these two mass balance lines as well as the residue and accumulated permeate curves in the region of operation for the batch still. This so-called operation leaf shows graphically the region of all possible products attainable in a batch membrane permeation with an initial charge (feed) of x(0). The interior points within the bigger region can be obtained by commencing operation at x(0), permeating for a fixed time, then allowing mixing of the residual material with some of the original charge. Likewise for the smaller region—mixing occurs between the accumulated permeate and the initial charge.


50

The relevance of an operating leaf is that it informs one of all the possible products obtainable from a feed of x(0) using a batch membrane separation process. It yields, in a graphical manner, the retentate and accumulated permeate compositions. Furthermore, a knowledge of the initial amount of retentate charged to the still easily allows one to calculate the respective amounts of the final products obtained. Also, embedded within the calculations is the actual time it would take to achieve the desired separation (refer to Section 5.4). In a batch still that utilizes a nonselective membrane that has different permeation rates for all three components, the operating leaf shows graphically some important results: . .

.

Only the species with the lowest permeation rate can be obtained as pure component (component B, in this case). In order to obtain this pure component, however, the amount of remaining retentate would be only as much as a single drop (refer to the mass balance line (B) in Figure 5.3(b)). The composition of the most permeable component (A, in this case) would be the highest when permeation initially begins, after which its composition tends toward that of the initial feed, as previously explained.

Although some of these results may be somewhat intuitive, the method is unique in that one can rapidly deduce such outcomes from the graph. One can generate operating leaves for any given x(0). The plot obtained will be similar to the one displayed in Figure 5.3(b), and the result is as discussed above.

5.3.5 Operating Regions: Selective Membranes The results obtained for a nonselective membrane do not hold true for membranes where one component is preferentially transported through the membrane, and/or where the composition of the permeate is essentially pure in that component. An example of this type of membrane is PERVAP 1137, discussed earlier. The membrane has a very high selectivity for methanol, resulting in the permeate having a mole fraction of methanol always higher than 0.98 (Bausa and Marquardt, 2000). This means both the permeate and the accumulated permeate compositions will always have a value greater than 0.98. For simplifications, assume that these values represent a pure fraction of methanol. Analysis of Equations (5.3) and (5.4) yields that x(t), x(0), and yacc(t) must all lie on the same straight line, and that the locus of x(t) is the same straight line. Hence, the operating leaf collapses into a single straight


51

Figure 5.4. Operating line in the methanol/butene/MTBE system: (a) pP 0 and (b) pP > 0. (Peters et al., 2008).

line, as shown in Figures 5.4(a) and 5.4(b). Furthermore, the “operating line” happens to be the residue curve passing through x(0). The reader is referred to Chapter 4 (Peters et al., 2006a) for a full discussion on the properties of this particular map. The residue curves for this particular membrane depend on the permeate pressure (pP) (see Section 4.5). An arheotrope is a stationary point in the membrane vector field where there is no change in both permeate and retentate compositions with time. A locus of arheotropes was found to exist, as well as to migrate from the binary butene/MTBE line (pP ¼ 0) through to the pure methanol node (pP ¼ saturation pressure of methanol). This results in the operating line extending from the methanol node, through x(0), and terminating at the locus of arheotropes. The operating line, as before, reveals some interesting results. These results may seem somewhat trivial, but the operating line enables one to identify them quickly and easily: .

.

.

It is still only ever possible to obtain a pure composition for one component only, namely, methanol. That particular result is expected, since the membrane is designed to do such a separation. Unlike the nonselective membrane, this particular membrane will allow a sufficient amount of methanol through, yielding a good recovery of it. Depending on the feed composition and the permeate pressure, it may even be possible to recover all the methanol initially present, in a finite period of time. If the permeate pressure is such that the feed composition lies in the ineffective separation region (i.e., the gray region in Figure 5.4(b)), then the retentate will become enriched with methanol. This is said to be


52

ineffective since the membrane was not designed to enrich the retentate with methanol, but rather deplete it.

5.4 PERMEATION TIME Suppose it is desired to separate a fluid mixture of components A, B, and C in a batch still incorporating a Knudsen membrane (i.e., constant relative permeabilities) until we achieve a specified final purity of x(f). The initial charge of material, R(0), is known, as well as its composition, x(0). When designing such an experiment or, more importantly, when using such apparatus for commercial production of a particular product, it is vital to know the following: . .

How long the separation from x(0) to x(f) will take. What final amount of material, R(f), of composition x(f) will be available.

The desired product may be the permeate. In this case, the amount of material accumulated can then be calculated further by mass balance. Even though the M-RCM only informs one of the changes in the retentate composition, it is possible to extract the required information from it, in conjunction with the relevant permeation model and material balances. Consider, for demonstration purposes, the following example: A mixture of x(0) ¼ [0.7, 0.1] needs to reach a final purity of x(f) ¼ [0.023, 0.68]. The separation will be done in a batch still using a gas separation membrane with constant relative permeabilities (refer to Section 5.2). The residue curve and corresponding (local) permeate curve are displayed in Figure 5.5(a). Now, from the residue curve Equation (5.1), we see that xiððf Þ

t¼ xi ð0Þ

1 dxi xi yi ðxÞ

ð5:6Þ

This is the familiar Rayleigh equation (King, 1971). It shows that the area under the graph of gi ¼

1 versus xi xi yi ðxÞ

between the limits of xi (0) and xi (f ) would yield the dimensionless time required to achieve the separation. This is graphically displayed in Figure 5.5 (b) for component B. It is worth noting that t is in fact a function of x, and t is

PERMEATION TIME

53

Figure 5.5. (a) Residue (solid) and permeate (dashed) curves. (b) Inverse of separation vector (gi) versus xB. (c) Graph of hi versus xB. (Peters et al., 2008).

the same for all i. This now enables one to quantify the amount of residual fluid remaining in the still since, from Equation (5.2); Rðf Þ ¼ RðtÞ ¼ Rð0Þ et

ð5:7Þ

It is evident that as t increases, the amount of retentate available reduces since a large t would correspond to a retentate with a high purity of the least permeable component. In order to determine the actual time needed for separation, consider the overall mass balance over the still: dR ¼ P_ dt

ð5:8Þ

If one were to assume a constant permeate removal rate, then t¼

Rð0Þ ð1et Þ P_

ð5:9Þ

However, it is not likely that the permeation rate will remain constant throughout the entire separation, since it is a function of the retentate

54


composition, among other things. Thus, by integrating Equation (5.8), differentiating Equation (5.6), and using the Knudsen model for permeation with a vacuum on the permeate side, the following equation results (refer to Appendix C for proof): d t ¼ Rð0Þ A P;i pR

xiððf Þ

xi ð0Þ

yi ðxÞ et 1 dxi xi yi ðxÞ xi

ð5:10Þ

Feng and Huang (1992) arrive at a similar expression. Once again, we note that by computing the area under the graph of hi ¼

yi ðxÞ et 1 versus xi xi yi ðxÞ xi

between xi(0) and xi(f) would yield a scaled time. Figure 5.5(c) illustrates this for component B of the example mentioned earlier. Multiplying by the relevant constants displayed in Equation (5.10) would give the real time needed to achieve the required separation. One could calculate t for every component, but the result would remain the same. It is worth noting that the area under the graph of hi versus xi (which yields the scaled time) will not be the same for each component. The ratio of the area for component i to that of j will be equal to aM ij . Furthermore, the permeation time depends on a number of physical parameters. As one increases the membrane area, A, or retentate pressure, pR, the time needed for permeation will decrease. The time would increase with an increase in membrane thickness since the permeating material now has a longer distance to travel through the membrane.

5.5 CONTINUOUS MEMBRANE OPERATION 5.5.1 Nonreflux Equipment A single-stage nonreflux membrane module is the most common unit used for continuous operation. An example of such a unit is shown in Figure 5.6. A feed stream enters the unit at an appropriate chosen pressure (pR), where it comes into contact with the membrane. A permeate pressure (pP) is maintained on the downstream side of the membrane, resulting in material moving through the membrane. Material permeates in the direction normal to the membrane. The actual shapes of the module and membrane may vary from unit to unit. The unit could be spiral wound, or a plate module, or even consist

CONTINUOUS MEMBRANE OPERATION

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Figure 5.6. Single-stage nonreflux continuous membrane operating under a cocurrent flow regime. (Peters et al., 2008).

of hollow fibers. Any shape allows for an effective membrane area (A) for separation to occur. There are a number of ideal flow patterns that can occur within the membrane module, shown in Figure 5.6. The important ones are shown in Figure 5.7 and include (a) complete mixing, (b) cross flow, (c) cocurrent flow, and (d) countercurrent flow. The method displayed here is applied for cocurrent flow but is applicable to all the flow patterns, and any combination of patterns. It is important to note that when talking about the material that permeates, we are referring only to the material resulting from the local flux at each point

Figure 5.7. Ideal flow patterns in continuous membrane operation. In all diagrams, the lower chamber is at high pressure (pR), and material flows to the upper chamber, which is at low pressure (pP): (a) complete mixing, (b) cross flow, (c) cocurrent, and (d) countercurrent.

56


down the length of membrane area. Once that material enters into the bulk permeate stream, it will mix with the already present material, resulting in a mixed composition. This mixed, or accumulated, composition will be discussed later in this section, and also in more detail in later chapters (refer to Chapters 6, 7, and 8). (Note that a superscript of * will be used to denote compositions that relate to the local composition; for example, the composition of the permeate as it emerges from the membrane will be denoted as y. Any bulk compositions will be written without a superscript.) Material Balances Consider the cocurrent flow regime displayed in Figure 5.6. When one operates a membrane module in such a manner, the composition of the retentate changes down the length of the membrane, until it reaches the product composition at the end of the module. The compositional change occurs in a plug-flow manner. An overall and component mass balance over the strip of area size DA can be written, assuming steady-state, as rjA ¼ p DA þ rjA þ DA

ð5:11Þ

ðr xi ÞjA ¼ p y*i DA þ ðr xi ÞjA þ DA

ð5:12Þ

where r is the retentate flow rate (mol/s), p is the local permeate flux (mol/m2 s), xi is the bulk retentate composition, y*i is the local permeate composition, and A is the effective membrane area (m2). Since permeation occurs continually down the length of the membrane, rather than at discrete points, it is obvious that the limit DA ! 0 should be imposed. Combining the resulting two equations, the following is obtained: dxi ¼ xi y*i ; i ¼ 1; 2; . . . ; c1 0 dA where A0 is the dimensionless membrane area, defined by F 0 A ¼ ln RðAÞ

ð5:13Þ

ð5:14Þ

with F and R(A) being the total molar retentate flow in the separator at A ¼ 0 (i.e., initially) and at area A, respectively. Thus, A0 is the natural logarithm of the reciprocal of fraction of retentate remaining to obtain a desired purity.


57

Note that Equation (5.13) can be applied to membranes of different shapes (tubular, sheet, etc.). For example, in a flat sheet membrane module, dA0 ¼ w dz0 , where w is the width of the sheet and z is its length. The appropriate relationship can be written for other types of membrane shapes. Bausa and Marquardt (2000) give a similar proof of this equation for this mode of operation. In their paper, they analyze hybrid distillation–membrane systems and use a result similar to the one shown in Equation (5.13) for modeling the membrane module. They also provide the relevant energy balance equation, since there is a temperature drop down the length of the membrane. This is especially true in pervaporation systems where the heat of vaporization is withdrawn from the liquid as it permeates and ultimately vaporizes. Due to the scope of this report, the energy balance, although useful, is not necessary here and will therefore not be discussed further. The derivation given here is for a cocurrent flow pattern, but can be performed for all other common flow regimes. These are left to the reader and are not shown here, but the result obtained will be as given in Equation (5.13). Relevance of the Batch Experiment It can be seen that Equation (5.13) is mathematically equivalent to the residue curve Equation (5.1). The time variable in the batch operation corresponds to a position (spatial) variable in the continuous operation. Clement et al. (1992) arrive at the same result, referring to it as a “surface area–time duality.” Thus, the composition profile of the retentate down the length of the membrane will follow a residue curve, as the retentate continually becomes enriched in the least permeable species. The exact residue curve followed depends on the feed composition, since it corresponds to the initial retentate composition. (Refer to the M-RCM shown in Figure 5.1.) This implies, in principle, that a membrane can be analyzed for its performance using a simple batch experiment. Local Permeate Mixing with the Bulk Stream Permeation occurs continually down the length of the membrane. Local permeation involves material permeating at every point through the membrane. As each local permeate leaves the membrane it is mixed with the previously permeated material. In this case, the composition of the permeate also changes in a plug-flow manner. The graphical representation, using an M-RCM, for a particular feed to a nonreflux module is as shown for the batch process in Figure 5.3(a). The feed composition to the module is x(0), and the lines represented by (i), (ii), and (iii) retain their original description. Lines (i) and (iii) are the compositional change of the retentate and bulk permeate, respectively.

58


The material balance across the module from the feed to any point down the length of the membrane can be written. It informs us that the feed composition, as well as the corresponding permeate and retentate compositions at any point, will lie on a straight line in the mass balance triangle (MBT). Additionally, the feed and retentate compositions will also lie on the same residue curve. It is therefore possible to display, graphically, the region of operability for a nonreflux membrane module. The operation leaf is as shown in Figure 5.3(b) for the ideal system, and in Figures 5.4(a) and 5.4(b) for the methanol/butene/MTBE system. Furthermore, the results obtained for the batch process are directly applicable to the nonreflux case. These results are listed in Sections 5.3.4 and 5.3.5. Permeation Area An essential design parameter for membrane permeators is the area required for separation. As discussed earlier in this section, the area (or length) variable in the continuous operation corresponds to the time variable in the batch setup. Therefore, Equation (5.10) becomes applicable when one calculates the real area needed, remembering that t becomes A, and t becomes A0 . More importantly, however, is that previously in the batch experiment the membrane area was constant. In continuous operation it changes down the length of the unit. So, taking this into account, Equation (5.10) needs to be altered (refer to Appendix D for proof): A2 ¼ F 2

xiððf Þ

xi ð0Þ

0 yi ðxÞ eA 1 dxi xi yi ðxÞ J

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u xiððf Þ u u d yi ðxÞ eA0 1 u A ¼ t2 F ; dxi Pi pR xi yi ðxÞ xi

ð5:15aÞ

ð5:15bÞ

xi ð0Þ

Equation (5.15a) is general and applicable to any flux model. Equation (5.15b) is the result for Knudsen permeation. The methods described in Section 5.4 can be adjusted appropriately to facilitate the determination of the area. 5.5.2 Reflux Equipment Total Reflux Consider altering the conditions of operation of the nonreflux separator shown in Figure 5.6 as follows: The flow pattern is changed to countercurrent, and


59

Figure 5.8. Membrane column: (a) total reflux and (b) infinite and finite reflux. (Peters et al., 2008).

material is recycled back to the column at either end. This is diagrammatically shown in Figure 5.8(a). The unit shown in Figure 5.8(a) is not of any practical use since it has no feed or products. However, the theory and understanding it provides one with useful insights into both infinite and finite reflux operation. By writing the material balances over a differential segment of the column, and taking the limit as DA ! 0, it can be shown that the differential equation (DE) in Equation (5.13) holds true for the reflux equipment. As before, we note that the same DE results as for the batch process. The dimensionless time variable, t, in the batch operation (Equation (5.1)) becomes the dimensionless space (or length) variable in the equation for the continuous reflux process. The path the retentate would trace out would be the appropriate residue curve on the M-RCM (Figure 5.1) that passes through the point xT. If pP 0, then permeation is monodirectional, that is, from retentate phase to permeate phase only. Therefore, both P and R are continually changing down the length of the column. They will either increase or decrease monotonically, depending on the flow pattern. In the countercurrent regime displayed in Figure 5.8(a), both P and R are decreasing in magnitude from the top of the column down. If the column operates under total reflux conditions, then all the material is recycled back to the column. This results in P ¼ R at every point normal to the membrane, remembering that P and R are still decreasing down the column. Furthermore, by using the material balances, one can prove that xi ¼ yi for all i

60


at any cross-sectional point down the length of the column. These compositions refer to that of the bulk in both streams. The process shown in Figure 5.8(a) is a membrane column, operating under total reflux. The continuous membrane column was first conceptualized by Hwang and Thorman (1980), as discussed in the Introduction (Section 5.1). The next section of this chapter deals with the addition and removal of feed and products to and from the column. Rautenbach and Albrecht (1989) discuss reflux cascades of membranes, as well as the membrane column. They conclude that the membrane column “can be regarded as a reflux cascade with an infinite number of stages.” What is interesting to note is that Rautenbach and Albrecht (1989) undergo a thought process to arrive at the membrane column, while in this chapter it has been shown that the continuous membrane column, when operating at total reflux, is equivalent to permeating in a batch still. Infinite Reflux When operating under the conditions of infinite reflux, there is a feed to the unit, as well as products being drawn off, such as the schematic shown in Figure 5.8(b). However, the magnitudes of the feed and product flows are extremely small compared to the flows that exist within the column. Therefore, one can assume that the column approximates the behavior of a total reflux column. It is then obvious that an infinite reflux membrane unit exhibits the same mathematical and graphical behavior on the M-RCM as the total reflux separator does. Bow-Tie Regions in Ideal M-RCMs The mass balance over a simple finite or infinite reflux membrane unit dictates the feed and top and bottom products should lie on the same straight line, known as the mass balance line. This line can be determined by the well-known “lever-arm rule.” Furthermore, the top and bottom products have to lie on the same residue curve when operating at total or infinite reflux (Fien and Liu, 1994). This result is intuitive from the discussions given in previous sections. Using these phenomena, it is now possible to identify two important mass balance lines in membrane separations. (1) A direct split mass balance line connects the feed composition and the unstable node in a region, indicating a sharp split with the highest permeating component as the pure top product of a membrane. (2) An indirect split mass balance line connects the feed composition and the stable node, indicating a sharp split with the lowest permeating component as the pure bottom product of a membrane column (or the retentate of the membrane module).


61

Figure 5.9. Bow-tie region bounded by direct and indirect mass balance lines. (Peters et al., 2008).

Graphical representations of these sharp splits are shown in Figure 5.9. A region of feasible combinations of permeate and retentate products exists for a specified feed composition. In other words, there is a range of material balance lines that makes the column feasible. All these material balances are referred to as nonsharp splits. The region of nonsharp splits is known as a bowtie region. It is bounded by the direct and indirect split mass balance lines. Any single residue curve has to pass through both parts of the region in order for feasibility to be ensured. The reader is referred to Fien and Liu (1994) for further insight. The bow-tie region for a membrane column is shaded in Figure 5.9. Sequencing In order to separate an ideal three-component system using a single-feed, two-product continuously operated membrane—which can be either reflux or nonreflux—a minimum of two units is necessary to obtain essentially pure components. Note that this discussion is about reflux separators, but the principle is exactly the same for nonreflux membrane modules. Furthermore, any particular module may consist of several single-stage membranes in a nonreflux cascade in order to achieve the required purity. Nonselective Membranes A nonselective membrane is one that allows all components to pass through, but at different rates. For the M-RCM shown in Figure 5.1, the order of permeability from highest to lowest is A–C–B. There are three possible ways of sequencing units at infinite reflux in order to achieve a separation of all three components. The first method requires two

62


units, the first of which performs the direct split to produce pure A, as shown in Figure 5.10(a), and then the second separates the binary mixture of B and C. Figure 5.10(b) follows a similar sequence but incorporates the indirect split, resulting in the top product needing to be further purified, as shown. In the last configuration, Figure 5.10(c), the first membrane follows neither the direct nor indirect split, but rather performs a nonsharp split within the bow-tie region, resulting in both the top and bottom products of this column requiring further purification in order to obtain the pure products. It is worth noting that in each of the three cases examined, the membrane material used in each unit is the same—that is, every membrane has the same relative permeabilities for each component. Every unit will still be able to

Figure 5.10. Sequencing of membrane columns using (a) direct split, (b) indirect split, and (c) nonsharp split. (Peters et al., 2008).


63

achieve the desired separation since the membrane is not selective to any component; each component moves through the membrane at different rates. Selective Membranes Since the residue curves in the selective system have essentially no curvature and are effectively straight lines, the feed, permeate, and retentate compositions all lie on the same straight line, which happens to be the residue curve passing through the feed point. Thus, for a single unit operating with a vacuum on the permeate side, the permeate product would be a pure methanol stream, whereas the retentate would be the binary butene/MTBE mixture on which the appropriate residue curve ends. Figure 5.11(a) shows a schematic representation of such a separation. Due to the nature of the membrane material, it is possible to define only the direct split mass balance. The membrane is selective to methanol permeation, with the butene and MTBE moving through at essentially the same rate. The separation of the MTBE from the butene would not be possible if one uses the same membrane. For the case where the permeate pressure is nonzero, a representation of the separation is shown in Figure 5.11(b). In this case, the permeate is still a pure methanol stream, but the retentate is a mixture still containing methanol, since it is not possible to remove all the methanol from the feed under such operating conditions. Finite Reflux When one operates the column shown in Figure 5.8(b) at finite reflux, the internal flows are no longer equal to each other, as was assumed for the total and infinite reflux cases. This results in a different, more general, DE that describes the retentate composition change with time. However, the DE for such a scenario is not simple to obtain or to apply, since the internal flows are

Figure 5.11. Sequencing of cascades of membrane modules for methanol separation at (a) pP 0 and (b) pP > 0. (Peters et al., 2008).

64


themselves also changing with position. Although it may seem trivial to some extent, these constraints greatly affect the DE and resulting retentate profile. This work has been intriguing and fascinating to the authors and will form the basis of Chapter 6, where a more detailed understanding and appreciation will be provided.

5.6 CONCLUSION This chapter serves to furnish the reader with the novel graphical methods developed in analyzing and designing membrane permeating systems. RCMs, traditionally believed to be suitable for equilibrium-based processes only, have shown their validity and capability for membrane separations—a kinetic-based process. It has been shown how several common continuous membrane structures are mathematically related to the simple batch experiment. The time variable in the batch is analogous to the spatial variable in continuous operation. This now enables one to extract more valuable information from batch experiments when attempting to scale them up. It has been demonstrated how the retentate composition in a nonreflux continuous unit will follow the residue curve that passes through the initial (feed) composition. A similar result is arrived at for membrane columns operated under the assumptions of either total or infinite reflux, with the residue curve passing through both the top and bottom composition points. Sequencing columns at infinite reflux have been discussed. Finite reflux, being the more useful mode of operation, is not so straightforward due to the complexities of the situation. Chapter 6 will present detailed discussions on the profiles for finite reflux membrane columns. The M-RCM, in conjunction with the appropriate flux equations, allows a designer to determine fundamental design parameters such as permeation area (or time) required in a continuous (or batch) process. The details of determining these values have been discussed in this chapter.

CHAPTER 6

COLUMN PROFILES FOR MEMBRANE COLUMN SECTIONS

In the previous chapter, the applicability of M-RCMs to infinite (or total) reflux membrane columns was presented. This chapter builds on this idea and details the profiles related to membrane columns operated at finite reflux. Here, the development of a novel graphical method of analyzing continuous membrane separation systems is shown. The method is applicable to all counter- and cocurrent membrane modules—single-stage, cascade, and membrane columns. All configurations can be broken down into “column sections,” no matter how complex the arrangement. Each column section, which resembles a typical single-stage separator, is modeled using the difference point equation. The difference point equation tracks the change in the retentate composition down the length of the column section. For demonstration purposes, a simple constant relative permeability flux model is employed to determine the permeate composition as well as the continually changing retentate flow. All possible operating conditions of a column section are explored. Each condition is a result of the relationships between the flows and compositions at the top of a column section. Using the difference point equation, column profiles for each condition can be plotted. The behavior of the profiles is discussed both mathematically and graphically. This innovative way of investigating membrane processes provides a unique way of synthesizing and designing them. Membrane Process Design Using Residue Curve Maps, First Edition. Mark Peters, David Glasser, Diane Hildebrandt, and Shehzaad Kauchali. Ó 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc. 65

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6.1 INTRODUCTION TO MEMBRANE COLUMN DEVELOPMENT Novel separation methods and techniques have been a focus for engineers in recent decades. Although traditional separation procedures have established their industrial value, innovative methods have been emerging in recent years. These fresh approaches have come about in the pursuit of more economical and environmentally attractive processes. Membrane-based separation procedures are one of many alternatives proving to be a viable option. A vast array of different types of membranes, ranging from reverse osmosis to ultrafiltration, have been developed for various applications (Rautenbach and Albrecht, 1989; Wijmans and Baker, 1995). An immense amount of research into permeation modeling has been completed (Rautenbach and Albrecht,1985a,b; Marriott et al., 2001; Marriott and Sørensen, 2003a). As will be evident, the nature of the work given in this chapter is general and can be applied to any membrane, and its associated flux model. For demonstration purposes, however, only diffusion membranes (i.e., gas separation and pervaporation) are discussed here. The driving force for separation in diffusion membranes is the difference in partial pressures for each component on either side of the membrane. The work can then be extended further and applied to other membrane types. Consider some of the unit operations used for continuous membrane separation. A conventional separator, shown in Figure 6.1, consists of a single feed (F) mixture entering a membrane unit. The mixture is separated into two product streams—namely, the permeate (P) and retentate (R). Numerous possibilities in flow regimes have been proposed for such a separator, with the countercurrent option being the most efficient (Tsuru and Hwang, 1995). Various external flow arrangements, using a single unit, are also possible, such as recycling the retentate to the feed, or even to the permeate side (not shown). In some cases, a single unit may not be sufficient: In order to achieve a desired degree of separation, several single-stage modules connected in series are often advantageous (McCandless, 1994). This is known as cascading. There are numerous configurations that can be employed when cascading, but it can be shown that the simple countercurrent recycle cascade is most

Figure 6.1. Example of a single-stage nonreflux continuous membrane unit, operating under a countercurrent flow regime.

INTRODUCTION TO MEMBRANE COLUMN DEVELOPMENT

67

Figure 6.2. (a) Continuous membrane column. (b) Column section breakdown.

effective (McCandless, 1994). Such an arrangement of units is what is typically used in solvent (liquid) extraction circuits. Furthermore, the same configuration of interconnected flash stages is equivalent to a distillation column (Hoffman, 2003). It is therefore logical that this same analogy be applied for membrane systems. A continuous membrane column, displayed in Figure 6.2(a), was first identified by Hwang and Thorman (1980). The column is filled with hollow-fiber membranes, resembling the design of a shell-tube heat exchanger. A high pressure is maintained on one side in order to allow permeation through the membrane to the low-pressure side. To allow the recycle streams to flow, a compressor is necessary. It is claimed that complete separation of a binary mixture into pure products is achievable in a finite area (Hwang and Thorman, 1980). The original proposition of a membrane column did not include a retentate recycle; however, it has been reported that the inclusion of the recycle has assisted separation in some cases (Tsuru and Hwang, 1995). 6.1.1 Relevant Works in Membrane Column Research Since their inception, membrane columns have been researched extensively. They were originally designed for gas separations, but Hoover and Hwang (1982) demonstrated how they can be used for pervaporation systems, too. Various authors have also compared the membrane column with other design configurations: Kao et al. (1989) evaluated the performances of the membrane column and two strippers in series, whereas McCandless (1990) compared the column with countercurrent recycle cascades. Vorotyntsev and Drozdov (2001) analyzed the ultrapurifying of gases with the use of membrane columns. An optimization technique was proposed by Chang and Hou (2006).

68


Most publications, however, assume binary mixtures for separation. This assumption is not likely to be justified in industrial practice. Kothe et al. (1989) studied the separation of ternary gas mixtures using membrane columns, and Blaisdell and Kammermeyer (1973) introduced various models for co- and countercurrent gas separation. Agrawal (1996) proposed a procedure to draw membrane cascades for multicomponent gas separation, analogous to distillation. However, there is a lack of a generalized method for the synthesis and design of separation of ternary, and higher order, component mixtures with the use of membrane columns, or any membrane-based process for that matter. Peters et al. (2006a) began to address this need by introducing membrane residue curve maps (M-RCMs)—this is the work discussed in Chapters 3 and 4. Although graphically produced for ternary systems, the ideas and methods are applicable to higher order mixtures. Furthermore, the theory developed is not restricted to one type of membrane, but is applicable to all types, provided the design engineer incorporates the relevant flux model. Applications of these maps to both batch and continuous processes (Peters et al., 2008) are introduced in Chapter 5. As well as being valid for nonreflux equipment and hybrid processes involving membranes (see Chapter 8), the maps have shown their worth in the design of reflux separators, including membrane columns. Chapter 5, however, looks only at total and infinite reflux membrane columns. The aim of Chapter 6 is to furnish the reader with a graphical method of analyzing and visualizing profiles for membrane separators, including the membrane column. With this, one is able to synthesize and design any membrane separation circuit, no matter how complex in its arrangement.

6.2 GENERALIZED COLUMN SECTIONS Consider a separation column, such as the membrane column given in Figure 6.2(a). Analyzing the entire column as a whole limits one’s possibilities to only that particular configuration. This is not useful when it comes to synthesizing a separation sequence, especially if it is intended to be optimal. Therefore, a different approach is taken: Any general column or cascade can be divided into column sections (CSs). A generalized CS is defined as “a length of column between points of addition and removal of material and/or energy” (Tapp et al., 2004). This definition is general and applicable to most, if not all, counter- and cocurrent applications, such as distillation, absorption, and stripping. Although unique in its operation, membrane permeation is no exception to this. Figure 6.2(b) shows an example of a CS breakdown for the membrane column given in Figure 6.2(a). Similar breakdowns can be done for any other arrangement. Analysis of each of the column sections in Figure 6.2(b) reveals that each one

GENERALIZED COLUMN SECTIONS

69

resembles a single unit (refer to Figure 6.1). Thus, provided the assumptions of operation are clear and defined, a CS introduces a novel way of investigating membrane separations. Note: Although termed “column sections,” it does not imply that they are pertinent only to separation columns, as they are pertinent to any general separator. CS theory was originally developed for distillation systems, but, as will be shown later, can be adapted for other forms of separation, including membrane permeation. In this section, the original theory is briefly presented for any general separation system, and then later adapted to membrane separation specifically. The general theory is developed with the assumption that material flows within a CS are constant (as was done by Tapp et al. (2004)). This assumption is not necessarily valid in membrane systems and will be relaxed accordingly (refer to Section 6.3). However, the theory derived for the general system remains applicable, as will be shown. Consider the general countercurrent CS shown in Figure 6.3. The nomenclature used in this figure is taken from distillation but is arbitrary; that is, L and V are not necessarily liquid and vapor flows, respectively, but can be regarded as general material flows into and out of a CS. In Figure 6.3, the following nomenclature applies: L and V are passing streams in a general CS (mol/s) x is the composition of stream L (–), y is the composition of stream V (–),

Figure 6.3. A generalized separation column section (countercurrent flow).


70

n is a representation of position down the column length (–), and subscript T refers to quantities at the top of the column section. The following assumptions have been applied to Figure 6.3: .

.

All flows and compositions at the top (T) of a column section are known and constant. This assumption can be altered accordingly, if necessary. Flows (L and V) are constant throughout the CS. This is equivalent to equimolar overflow in distillation. This particular assumption is often justified in distillation, but cannot always be employed in membrane systems; this will be discussed later (see Section 6.3).

6.2.1 The Difference Point Equation A component material balance can be written (in vector form) around the top of the column and any point down the CS length (refer to Figure 6.3): V yn þ 1 þ L xT ¼ V yT þ L xn

ð6:1Þ

For completeness, it should be noted that X

xi ¼ 1 and

i

X

yi ¼ 1

ð6:2Þ

i

The difference point equation was introduced by Tapp et al. (2004) for modeling how x changes as a function of position, n: dx 1 1 ¼ þ 1 ½xyðxÞ þ ½XD x dn rD rD

ð6:3Þ

where D is the net molar flow within the CS, and for a countercurrent CS is defined as D ¼ VL

ð6:4aÞ

XD is the net molar composition within the CS, and for a countercurrent CS is defined as XD ¼

VyLx D

ð6:4bÞ


71

XD is more commonly referred to as the difference point and is the vector of the net molar flow rates for all components divided by the total net molar flow. rD is the reflux ratio and is defined as rD ¼

L D

ð6:4cÞ

The relationship between the compositions of both streams is assumed to be as follows: ai xi yi ¼ P ai xi

ð6:5Þ

ai gives the separability of component i relative to a reference component. This is equivalent to relative volatility in distillation, and relative permeability in membrane separations. The values of a chosen were [3, 1, 1.5], and they were assumed constant. 6.2.2 Infinite Reflux Under the conditions of L ¼ V and xT ¼ yT, the difference point equation (DPE) (Equation (6.3)) collapses to the following: dx ¼ xyðxÞ dn

ð6:6Þ

This form of the DPE is mathematically identical to the residue curve equation (refer to Equation (3.4)). Therefore, column profiles at infinite reflux follow residue curves. For any given initial condition, x0, it is possible to integrate the DPE in both the positive (n ! þ 1) and negative (n ! 1) directions, as was done in Section 3.3.2 (see Figures 3.4 and 3.5). As previously discussed in Section 3.3.2, the mass balance triangle (MBT) or Gibbs triangle represents the region of physically achievable profiles in ternary systems. One may define this triangular region mathematically as 0 xi 1 for all i. Although describing a real process, the DPE at infinite reflux (or residue curve equation) is merely a mathematical equation. This equation is not bound by any physically relevant initial conditions, and it is possible to evaluate it at initial values of x with entries greater than 1 and less than 0 (i.e., outside the MBT). If an arbitrary initial condition of x0 ¼ [0.6, 0.6, 0.2] is chosen and integration is performed in both the positive and negative directions, the additional profile in Figure 6.4 results. If the entire space surrounding the MBT is populated in this way, on integration in both positive and negative directions, Figure 6.5 results.

72


Figure 6.4. Solutions of the DPE at arbitrary initial conditions inside and outside the MBT.

Figure 6.5 can be divided into seven different regions. The behavior in these regions represents all possible profile solutions for a constant relative separability system. Solutions of the DPE generated by populating the entire space are referred to as column profile maps (CPMs). The RCM is a special form of the CPM and occurs only at infinite reflux. The pinch points or nodes of the CPM can be

Figure 6.5. Extended RCM. MBT in blue, line of discontinuity in green. (See insert for color representation).


73

categorized in the same way as those of the RCM. At infinite reflux, the light pure component corresponds to an unstable node, the heavy pure component to a stable node, and the intermediate pure component to a saddle point (refer to Section 4.1). While one could graphically use the direction of movement of curves with respect to a particular node to determine its type (stable, unstable, saddle, etc.)—as is done in Section 4.1—there is a more fundamental and mathematically accepted procedure (Holland et al., 2004b): Liapounov’s first theorem states that the nature of a singular point is topologically similar to the linearized form of the DPE. The resulting square Jacobian matrix is constant, and not a function of n. Thus, the eigenvalues of the Jacobian, which can be determined, characterize the type of singularity. The corresponding pair of eigenvectors associated with each point determines the asymptotic direction of the trajectories in the neighborhood of the singular point. Tutorial 6.1 Try this for yourself: Using the MemWorX program, reproduce Figure 6.5 by using a ¼ [3,1,1.5], and plotting a Full CPM. Use the “Constant–distillation” flow assumption and the first equation option. To plot the discontinuity, select this as the Plot option. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. Discontinuity A discontinuity in the constant relative separability system arises because of the form of the equation relating x to y. For an ideal ternary system, yi ðxÞ ¼

ai xi aA xA þ aB xB þ aC xC

ð6:7Þ

One can see from Equation (6.7) that the function is indeterminate when the denominator is zero. It is therefore possible to determine the discontinuity by setting this denominator to zero. aA xA þ aB xB þ aC xC ¼ 0

ð6:8Þ

xC ¼ 1xA xB

ð6:9Þ

However,

74


Therefore, xB ¼

ðaC aA ÞxA aC aB aC

ð6:10Þ

The discontinuity for this system is a straight line, intersecting the xB-axis at [aC/(aB aC)], and the xA-axis at [aC/(aC aA)]. Because aC (the separability of the intermediate) is always larger than aB (the least separable component), the xB-intercept is always positive and larger than 1; hence, only profiles of regions 2, 5, 6, and 7 in Figure 6.5 will be affected by the discontinuity. In general, because of the separation models describing the system, most, if not all, real systems will contain discontinuous regions. For example, thermodynamic models such as Wilson, NRTL, and Unifac are indeterminate at certain compositions. These discontinuities are inherent to the model, and their positions depend on the model used. Discontinuous regions cannot be moved, and they always occur outside the MBT. 6.2.3 Finite Reflux At finite reflux, V 6¼ L, resulting in the CS being mathematically modeled using the DPE (Equation (6.3)). Once values for XD and rD are chosen, the DPE can be solved and profiles generated. If the difference point and reflux are arbitrarily set (say, XD ¼ [0.2, 0.3, 0.5] and rD ¼ 5), the space can be populated with composition profiles or trajectories in exactly the same way that the RCM was produced, by integrating the DPE at arbitrary initial conditions (as n ! þ 1 and 1). Both MBT and “outside” space can be populated with profiles. From Figure 6.6, it is apparent that the CPM solutions have the same geometry as the RCM (Figure 6.5). All of the original singularities are present but have been shifted in the composition space. A section of the topology from region 2 (see Figure 6.5), representing physically irrelevant composition profiles (at infinite reflux), has been shifted into the MBT, and these solutions are valid composition profiles. That is to say, any CS operating at a reflux ratio of 5 with this difference point (or net molar composition) could in fact operate on one of these trajectories. The CPM is a simple linear transform of the RCM. For constant relative separability systems, the boundaries of the MBT (representing particular residue curve solutions) are also transformed at finite reflux, maintaining their straightness to form a “transformed” triangle. This fact has led to the phenomenon being described as “transformed” or “moving” triangles; this will be discussed and used in more detail in Sections 6.5 and 6.6.


75

Figure 6.6. Column profile map (CPM) for XD ¼ [0.2, 0.3, 0.5] and rD ¼ 5. MBT is shown in blue, the “moved” triangle is shown in red. (See insert for color representation).

Figures 6.7(a)–6.7(c) illustrate CPM solutions for difference points in other regions at a reflux of 5. Under very extreme conditions, however, the singularities of the system merge and the transformed triangle collapses. Under these conditions, the entire topology of the system changes. It is interesting to note that the resulting trajectories inside the MBT follow very different paths, depending on the position of the difference point. The XD in Figure 6.7(c), for instance, shifts a saddle and an unstable node inside the MBT; this changes the path and the directions of the profiles dramatically. All profiles around the singularities are running either toward the node (stable node in the top vertex) or away from it (saddle node in the bottom left corner). It is important to note that the discontinuity present in the RCM does not move and is unaltered by the linear transformation of the DPE, as explained in Section 6.4.2. Tutorial 6.2 Try this for yourself: Using the MemWorX program, duplicate Figures 6.6 as well as Figures 6.7(a)–6.7(c) by maintaining a ¼ [3,1,1.5], and plotting a Full CPM using the parameters as indicated on the figures. Use the “Constant–distillation” flow assumption and the third equation option (full DPE). Try producing other

76


Figure 6.7. CPM for (a) XD ¼ [1.2, 0.3, 0.5] and rD ¼ 5; (b) XD ¼ [1.2, 0.3, 0.1] and rD ¼ 5; and (c) XD ¼ [0.4, 0.2, 0.8] and rD ¼ 5. (See insert for color representation).

CPMs for yourself using different XD and rD values. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. 6.2.4 CPM Pinch Loci In separation, a pinch point may be defined as any point where the driving force for separation has diminished to zero. The location of pinch points can be determined mathematically by equating the DPE (Equation (6.3)) to zero, dx/dn ¼ 0. Pinch points correspond to nodes or stationary points on the CPM. For a fixed XD, there exists a unique set of pinch points for a specific rD (refer to Figure 6.6). As rD changes, so do the locations of the pinch points (for a fixed XD). Pinch point loci have been defined as the loci of all pinch points obtained by varying the reflux ratio, rD, for a set XD (Tapp et al., 2004). Pinch loci help describe the path that the triangles take as rD is varied because the nodes for every CPM have to lie on the respective pinch curve. In this section, we will show how the behavior of the pinch point loci affects the topology of CPMs. Tapp et al. (2004) give a full explanation on the development of pinch point loci.


77

Figure 6.8. “Moving” triangles for varying rD with a fixed XD ¼ [0.3, 0.3, 0.4]. (a) Positive rD values of 3 (green), 6 (orange), 12 (red), and þ 1 (blue, MBT). (b) Negative rD values of 3 (green), 6 (orange), 12 (red), and 1 (blue, MBT). (See insert for color representation).

As shown in Figure 6.8(a), holding XD constant and varying rD (for rD > 0) allows one to draw a set of transformed triangles, the vertices of which correspond to the pinch points associated with the values of rD chosen. The same can be done for negative values of rD, as shown in Figure 6.8(b). Tutorial 6.3 Try this for yourself: Using the MemWorX program, reproduce Figures 6.8(a) and 6.8(b) by using a ¼ [3,1,1.5] and plotting “Moved Triangle” for the indicated rD-values. Use the “Constant– distillation” flow assumption and the DPE (third equation) option. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. The paths followed by these transformed triangles are the pinch point loci defined above. Figure 6.9 plots these loci for XD ¼ [0.3, 0.3, 0.4]. It can be seen that large, positive values of rD result in pinch points closer to the vertices of the MBT. This is an intuitive result since the infinite reflux case is identical to the RCM, which has nodes on the vertices of the MBT. Tutorial 6.4 Try this for yourself: Using the MemWorX program, plot Figure 6.9 using a ¼ [3,1,1.5]. Plot “Pinch Point Loci” for the indicated XDvalue. Use the “Constant–distillation” flow assumption and the DPE (third equation) option. Refer to Appendix A for more information on

78


Figure 6.9. Pinch point loci for XD ¼ [0.3, 0.3, 0.4]. (See insert for color representation).

how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. Because XD does not need to lie inside the MBT, pinch point loci show vastly different behavior depending on the position of the difference points in the xA–xB space. XD can be placed in any of the seven different regions (refer to Figure 6.5), resulting in different topologies of pinch point loci. These are shown in Figure 6.10. These pinch point loci allow one to predict the behavior of the transformed triangles. This is useful because, by modifying the reflux ratio to produce transformed triangles, one alters the topology of the column profiles within the MBT. This allows one to achieve column profiles that would not otherwise be feasible. In the infinite reflux case the column profile is limited to the trajectory of the residue curve within the MBT. Finite reflux makes a number of alternative trajectories available by modifying the topology of the system. Thus, for a given XD, manipulation of rD allows one to select from a locus of possible trajectories for the column profile. Tutorial 6.5 Try this for yourself: Using the MemWorX program, plot Figures 6.10(a)–6.10(f) using a ¼ [3,1,1.5]. Plot “Pinch Point Loci” for the indicated XD-values. Use the “Constant–distillation” flow assumption and the DPE (third equation) option. Refer to Appendix A for


79

Figure 6.10. All sets of different pinch point loci for fixed XD in the six other regions. (See insert for color representation).

more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. The section that follows adapts and extends these concepts to the specific case of a membrane column section (MCS). It is crucial to note that the specific case of the MCS differs from the general case presented here in that the flow rates are not constant in an MCS. This means that rD is not constant down the length of the column. All CPMs drawn up to this point have made use of the constant flow assumption, and the following section will discuss the changes that arise in the absence of this assumption.


80

6.3 THEORY 6.3.1 Membrane Column Sections A schematic representation of a countercurrent MCS is shown in Figure 6.11. In Figure 6.11, the following nomenclature applies: R is the retentate flow rate (mol/s), P is the permeate flow rate (mol/s), x is the composition of the retentate phase (), y is the composition of the permeate phase (), A is the dimensionless membrane area () pR is the retentate (high) pressure (Pa), pP is the permeate (low) pressure (Pa), and subscript T refers to quantities at the top of the column section. Throughout the remainder of this chapter, it is assumed that .

.

All flows and compositions at the top (T) of a column section are known and constant. This assumption can be altered accordingly, if necessary. The permeate pressure is set sufficiently close to vacuum (pP 0); thus, permeation will be monodirectional.

Figure 6.11. A generalized membrane column section (countercurrent flow).

THEORY

.

81

Permeation occurs through the membrane only in a direction that is perpendicular to the flow of the retentate stream, as indicated in Figure 6.11.

The actual area of the membrane is not used directly but is rather embedded within the flux modeling (see later). A dimensionless area is incorporated for locating a position within the column, without being restricted to exact dimensions. However, the dimensionless area is related to the real area. As a boundary condition, the membrane area is zero at the top of the column. It increases down the length of the column, until the CS is terminated. 6.3.2 The Difference Point Equation for an MCS Overall and component material balances can be written (in vector form) around the top of the column and any point down the membrane length (refer to Figure 6.11): PðAÞ þ RT ¼ PT þ RðAÞ

ð6:11Þ

PðAÞ yðAÞ þ RT xT ¼ PT yT þ RðAÞ xðAÞ

ð6:12Þ

For purposes of completeness, it should be noted that X

xi ¼ 1 and

i

X

yi ¼ 1

ð6:13Þ

i

The retentate composition, x, can be modeled using the difference point equation (DPE)—refer to Appendix E for the proof: dx D D þ 1 ½xyðxÞ þ ½XD x ¼ dA RðAÞ RðAÞ

ð6:14Þ

where D is the net molar flow within the CS, and for a countercurrent CS is defined as D ¼ PðAÞRðAÞ

ð6:15Þ

XD is the net molar composition within the CS, and for a countercurrent CS is defined as XD ¼

PðAÞ yðAÞRðAÞ xðAÞ D

ð6:16Þ

82


XD is more commonly referred to as the difference point and is the vector of the net molar flow rates for all components divided by the total net molar flow. It is worth noting that the direction of permeate flow, P(A), is taken to be the positive direction. Additionally, the resulting DPE for an MCS is mathematically similar to the general DPE form derived in Section 6.2.1, as expected. The DPE (Equation (6.14)) mathematically models how the retentate composition (x) varies as a function of position (A) in a general membrane CS. 6.3.3 Permeation Modeling In order to solve the DPE in Equation (6.14), it is necessary to understand how y is related to x. There are numerous methods of describing this relationship. It was shown, in Section 2.1.1, that for a gas separation membrane operating at a permeate pressure sufficiently close to vacuum (pP 0), that yi xi ¼ aM ij yj xj

ð6:17Þ

where aM ij is known as relative membrane permeability: aM ij ¼

P0i P0j

ð6:18Þ

Using appropriate simplifications, one can rearrange Equation (6.17), to obtain aM ij xi yi ¼ P M i aij xi

ð6:19Þ

It is agreed that the permeabilities of the components may not be constant but rather depend on a number of factors, such as the retentate and/or permeate composition(s), and mass transfer through the membrane can be quite complex. However, for demonstration purposes, a simple point of view will be taken. For this reason, constant relative permeabilities have been assumed, resulting in what is known as a Knudsen membrane (Huang et al., 2004). In order to solve the DPE, it is further required to model how the retentate flow is changing as a function of position, since material is being transferred monodirectionally (from R to P only). Consider a strip membrane area, as shown in Figure 6.12, where J is a vector containing the fluxes of all the components.

THEORY

83

Figure 6.12. Strip of membrane column section.

Expressing the total flux of material that permeates in the area dA (mol/m2 s) as X

Ji ¼ J

ð6:20Þ

i

allows one to write an overall differential material balance over the retentate side: dRðAÞ ¼ J dA

ð6:21Þ

dRðAÞ ¼ dPðAÞ

ð6:22Þ

It can further be shown that

For a simple gas separation membrane, with constant relative permeabilities (aM ij ), the following expression can be derived (refer to Equation (6.17)): X dRðAÞ ¼ aM ij xi dAn i

ð6:23Þ

where An ¼ A

pR P0 ref d

is a normalized form of A, and has units of mol/s. Here, d is the membrane thickness (m). For any other membrane, manipulation of the appropriate flux model should replace the right-hand side of Equation (6.23). The subscript ref denotes a chosen reference component. In this section, component B is the reference component, and the following relative permeabilities have been

84


M M assumed throughout: aM AB ¼ 3, aBB ¼ 1, and aBC ¼ 1:5. Thus, the order of permeability (from most to least) is A–C–B. So, using Equation (6.23), in conjunction with Equation (6.19), in the DPE, one can solve for and graphically interpret the retentate profile for a specified column section.

6.3.4 Properties of the DPE The DPE is derived on a mass balance basis (refer to Section 6.3.1) and is not dependent on the type of membrane used (i.e., diffusion, filtration, reverse osmosis, etc.). Whereas solving it requires knowledge of the membrane flux, the form of the DPE is general. The net flow (D) and net composition (XD) are constant throughout a CS. This can be deduced easily by overall and component mass balances (Equations (6.11) and (6.12)): D ¼ PR ¼ PT RT XD ¼

P yR x PT yT RT xT ¼ D D

ð6:24Þ ð6:25Þ

This is a very interesting result: It indicates that there remains a constant difference in flows and composition between any two points along a horizontal cross section in the CS. This occurs despite there being a continuous change in retentate and permeate flows down the length of the column. The physical implication of D is that, since the direction of flow of P(A) is taken as positive, then D > 0 results in more material moving up the CS than downward, as sketched in Figure 6.11 (i.e., P(A) > R(A)). The opposite is true for D < 0. DXD is the vector of net flows for individual components. A positive element in this vector corresponds to a net flow of that component up the CS, and vice versa. The DPE has been derived to model the retentate composition (x). An analogous DPE can be sought for the permeate composition (y) in much the same manner. Alternatively, y can simply be sought from x using a material balance over the CS (refer to Equation (6.12)). Furthermore, the equation is not limited to the countercurrent case as discussed, but can be used for other flow regimes (i.e., cocurrent). A further generality of the DPE is that knowledge of the actual shape and geometry of the membrane within the MCS is not necessary. The equation is not restricted and can be applied to any chosen shape (plate module, hollow fiber, spiral wound, etc.).

COLUMN SECTION PROFILES: OPERATING CONDITION 1

85

Table 6.1. Flow and Composition Relationships for the Various Operating Conditions in an MCS Operating Condition

Flow Relationships

1 2 3 4

PT PT PT PT

¼ RT ¼ RT 6¼ RT 6¼ RT

Composition Relationships yT yT yT yT

¼ xT 6¼ xT ¼ xT 6¼ xT

Section 6.4 6.5 6.6 6.6

In the sections that follow, all possible operating conditions for an MCS are detailed. Each condition is a result of the relationships between RT and PT, as well as xT and yT. Table 6.1 details the relationships for each condition and directs the reader to the section in which each condition can be found.

6.4 COLUMN SECTION PROFILES: OPERATING CONDITION 1 6.4.1 Statement Let us consider a special case of operation of the MCS shown in Figure 6.11, where PT ¼ RT and yT ¼ xT

ð6:26Þ

This says that both flows and compositions are the same at the top of the CS. This is equivalent to describing a total reflux column, and also sufficiently expresses an infinite reflux column, as discussed by Peters et al. (2008). 6.4.2 Mathematics The conditions of Case 1 result in D¼0

ð6:27Þ

and thus the DPE (Equation (6.14)) becomes dx ¼ ½xyðxÞ dA

ð6:28Þ

6.4.3 Membrane Residue Curve Map Equation (6.28) is the membrane residue curve equation as identified by Peters et al. (2006a) and displayed in Chapter 3. Solving this set of equations, in

86


Figure 6.13. M-RCM for an ideal A–B–C system with aA ¼ 3, aC ¼ 1:5: (Peters et al., 2006a).

conjunction with the flux model (Equation (6.19)), for a range of initial conditions (x0), allows one to produce an M-RCM as shown in Figure 6.13. From any given initial condition (x0), integration of Equation (6.28) can occur in two directions: (1) Positive integration (A ! þ 1) corresponds to determining the retentate composition from the top of the CS to the bottom (i.e., x0 ¼ xT) (2) Negative integration (A ! 1) performs the reverse direction, integrating from bottom to top (i.e., x0 ¼ xB). It is customary to indicate the direction of the positive integration on such maps (Fien and Lui, 1994), as is done in Figure 6.13. As discussed in Section , it is possible to populate the entire xA–xB space with curves. This is shown in Figure 6.14. The profiles lying outside the MBT may not be physically achievable, but the relevance of this map is very important, and will be discussed in subsequent sections. The nature and location of the stationary points in this map provide insight into the behavior of the curves (refer to Section 6.2.2). Tutorial 6.6 Try this for yourself: Using the MemWorX program, reproduce Figure 6.14 by using a ¼ [3,1,1.5], and plotting a Full CPM. Use the “Nonconstant–membrane” flow assumption and the first equation option. To plot the discontinuity, select this as the Plot option. Refer to


87

Figure 6.14. Entire M-RCM (MBT in blue, line of discontinuity in green). (See insert for color representation).

Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. Figure 6.14 closely resembles Figure 6.5 because the mathematical form of the flux model is the same as that of the model used in the general case. Furthermore, the same values of a were chosen. This will result in a number of similarities to the general CS discussed previously. However, differences will arise in the absence of the constant flow assumption. The initial similarity will serve to highlight these differences. 6.5 COLUMN SECTION PROFILES: OPERATING CONDITION 2 6.5.1 Statement Suppose we still have the CS configuration as depicted in Figure 6.11, with PT ¼ RT but yT 6¼ xT

ð6:29Þ

6.5.2 Mathematics Expanding the general DPE (Equation (6.14)), we find dx PT yT RT xT RðAÞ x ¼ PðAÞ½xyðxÞ þ ½PT RT dA PT RT

ð6:30Þ

88


Equation (6.29) implies that P(A) ¼ R(A) since D is constant (refer to Equation (6.14)). Multiplying out first and then employing the conditions of the case, it can be shown that dx ¼ ½xyðxÞ þ bðAÞ dT dA

ð6:31Þ

where RT RðAÞ

ð6:32Þ

dT ¼ yT xT

ð6:33Þ

bðAÞ ¼

dT is known as the difference vector taken at the top of the CS. Furthermore, it can be shown that X

dTi ¼ 0

ð6:34Þ

i

is a property of dT that needs to be satisfied. This can easily be proved by substituting Equation (6.33) into Equation (6.34). Since dT is a difference in two compositions, it follows that certain values (but not all) in the row vector will be of negative sign. It is also worth noting that the entries in dT change value down the length of the column since the flow rates are changing within the column: dðAÞ ¼

RT dT ¼ bðAÞ dT RðAÞ

ð6:35Þ

So d(A) is continually changing. This can easily be proved by material balance around the CS, as well as the conditions of the case. 6.5.3 Column Profile If values of the dT and RT are arbitrarily chosen (say, [0.1, 0.05] and 100 mol/s, respectively), then a membrane profile can be generated for a given initial composition. It seems apparent and obvious that this initial condition be the top retentate composition, since this point has been used as a reference. The corresponding permeate composition is easily determined through dT. Using a value of xT ¼ [0.3, 0.3] and integrating down the length of the column, the profile displayed in Figure 6.15 results.


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Figure 6.15. Membrane column profile for Case 2. Positive integration. RT ¼ 100 mol/s, dT ¼ [0.1, 0.05], xT ¼ [0.3, 0.3].

This type of curve is unique and behaves very differently from those obtained in the M-RCM (Figures 6.13 and 6.14). The behavior of the profile can be understood by analysis of the DPE for the operating conditions of this case (Equation (6.31)). Although the curve has no physical meaning once it crosses the A–B boundary, a great deal of useful information can be extracted from the trajectory it follows. Tutorial 6.7 Try this for yourself: Using the MemWorX program, reproduce Figure 6.15 by using a ¼ [3,1,1.5], and plotting a Single Profile. Use the “Nonconstant–membrane” flow assumption and the second equation option. This plot tends to require a longer computational time. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. 6.5.4 Analysis The DPE for Case 2 (refer to Equation (6.31)) is a combination of the separation vector, s ¼ x y, and the difference vector, dT ¼ yT xT. The equation is similar to the one used in Case 1 (Equation (6.28)), except now there is an additional vector term that gives rise to new topological features. If this were Case 1, then dT ¼ 0.

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b(A) is a scalar multiplier preceding dT and is a ratio of the retentate flow at the column top to the flow at a point corresponding to A. If the column were operating under the assumption of constant molar overflow, then b(A) is unity, as in distillation. However, this assumption is not true for membrane permeation, and b(A) has the following values. At the top of the column: bð0Þ ¼

RT ¼1 RT

ð6:36aÞ

At any point down the length of the column: bðAÞ ¼

RT >1 RðAÞ

ð6:36bÞ

since the retentate flow is losing material from the column top. Consider, for a moment, what topological effects would occur if b(A) were to remain constant at a value of 1. If this were the case, then the resulting map is shown in Figure 6.16(a). Note that the map obtained has similar topological features to the M-RCM (refer to Figure 6.13). However, it has been shifted, and the nodes are in different positions in Figure 6.16(a) compared to Figure 6.13. As discussed by Holland et al. (2004b), the three nodes are connected by straight lines due to the collinear nature and direction of the eigenvectors at each node, thus forming a triangle. The blue triangle indicates the MBT, as well as the location of the nodes when b(A) ¼ 0 (total reflux). The red triangle represents the “transformed” triangle, connecting the three nodes appropriately. Note that previously nonachievable profiles have moved into the physically possible (MBT) space. This shift arises from the addition of dT in the M-RCM equation. The magnitude and direction of dT affects the way in which the map is transformed—this will be discussed later. If b(A) were constant, the column profile would follow the bold curve in Figure 6.16(a), commencing at xT ¼ [0.3, 0.3]. However, b(A) is not a constant value, and it becomes larger down the length of the column. It will only have a value of 1 at the top of the column, and at no other point in the column. As soon as A > 0, then b(A) > 1, and dT essentially becomes larger (in absolute value). This implies that the trajectory of the column profile shown in Figure 6.15 will be tangent to the appropriate curve that passes through the point xT on the shifted plot (Figure 6.16(a)) when b(A) ¼ 1, namely, the bold curve in Figure 6.16(a). As permeation begins, b(A) > 1, and the shifted map shown in Figure 6.16(a) will, in fact, shift again. Since b(A) is continually changing, the map itself is


91

Figure 6.16. Shifted M-RCMs. The blue triangle represents the MBT, the red shows the transformed triangle. (a) b(A¼0) ¼ 1, (b) b(A) ¼ 1.667. dT ¼ [0.1, 0.05], xT ¼ [0.3, 0.3]. Actual membrane profile (see Figure 6.15) is tangent to the bold profile at the point indicated. (See insert for color representation).

going to constantly transform. For example, at a certain A, it will occur that b(A) ¼ 1.667. The corresponding map at this value is shown in Figure 6.16(b). The actual membrane column profile that exists (as shown in Figure 6.15) is not one of the curves shown in Figure 6.16(b), but is tangent to the curve that goes through the point when the retentate flow is such that b(A) ¼ 1.667. This point is easily determined from the data associated with the membrane column profile and is plotted in Figure 6.16(b). The corresponding curve is shown in

92


bold—it is along the one side of the transformed triangle. It is interesting to note that this particular curve was originally outside the MBT and has now been shifted in and forms part of the column profile. Tutorial 6.8 Try this for yourself: Using the MemWorX program, reproduce Figures 6.16(a) and 6.16(b) by using a ¼ [3,1,1.5], and plotting a Full CPM for each. Use the “Constant–distillation” flow assumption and the second equation option, and the stated parameters. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. Thus, it can be concluded that every point on the membrane column profile is tangent to a curve on a transformed map going through that point. The transformed map corresponds to the appropriate b(A) value at that point. Figure 6.17 shows discrete points on the membrane column profile and the associated tangent curves at the indicated b(A) values. Each of the “constant b(A)” curves is plotted to run to its stable pinch. Note that, as b(A) increases, the membrane column profile tends to maintain a similar direction to the “constant b(A)” curve for a longer period. This should be expected since, by analyzing the DPE for this case (Equation (6.31)), as b(A) ! 1, the difference vector (dT) becomes dominant, suppressing the

Figure 6.17. Tangency of curves to the column profile. Each curve corresponds to a different b(A) value and is allowed to run to its stable pinch point. (See insert for color representation).


93

effect of the separation vector (s). Thus, the left-hand side of the differential equation becomes a constant vector (for each b(A)), which has the direction of dT. This is evident from the profile obtained—as the trajectory proceeds, its ultimate direction becomes that of dT, as shown. Thus, varying the direction of the difference vector will decide the ultimate direction of the membrane column profile, provided A ! 1. 6.5.5 Pinch Point Loci Note in Figure 6.17 that there is a trend occurring with the pinch points of each “constant b(A)” curve. Due to the direction of integration, the point where each of these curves pinch would be their stable node. It is not only the stable node that moves, but also the unstable and saddle points shift. A locus of all these points can be generated for a chosen dT. Figure 6.18 shows the pinch point loci for dT ¼ [0.1, 0.05]. Tutorial 6.9 Try this for yourself: Using the MemWorX program, reproduce Figure 6.18 by using a ¼ [3,1,1.5], and plotting the Pinch Point Loci. Use the “Nonconstant–membrane” flow assumption and the second equation option. This plot tends to require a longer computational time. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial.

Figure 6.18. Pinch point loci for dT ¼ [ þ , , ] and dT ¼ [, þ , þ ]. (See insert for color representation).

94


Since dT is the difference vector, behavior of the pinch point loci can be characterized by its direction and magnitude. For a three-component system, dT has three entries, each of which can have a possibility of two signs ( þ or ). This means that there are eight possible directions that dT can have. However, dT has the property that all its entries have to sum to zero (Equation (6.34)). Therefore, the combinations where all entries are positive (or negative) are not valid directions. This leaves six directions—three unique directions and three in the opposite direction. By arbitrarily selecting dT’s in each of the three unique directions and multiplying them by 1 to obtain the other three directions, the pinch point loci can be plotted for each direction. This was done in Figure 6.18 for one of the unique directions. Similar plots can be generated for the other two unique directions of dT but are not given here (refer to Holland et al., 2004a). In Figure 6.18, the unstable node approaches the saddle point as b(A) increases, depending on the direction of dT. When the loci of each node meet, those two particular nodes become complex and the stable node is the only point that exists in the real space. Although the location of the pinch point curves has no apparent use in the current context of a profile associated with Case 2, their worth will be shown and explained in Cases 3 and 4, as well as in future work and publications on “coupled column sections” (see Chapter 7). 6.5.6 Further Column Profiles When one plots the membrane column profile in Figure 6.15, a randomly chosen xT and RT are chosen. Of course, any positive flow rate for RT is feasible, just as any point within the MBT is a possible top retentate composition. Figure 6.19(a) displays column profiles for a set xT ¼ [0.3, 0.3], varying RT as indicated. Figure 6.19(b), on the other hand, shows a range of column profiles for select xT’s at a constant RT ¼ 100 mol/s. It can be seen that, irrespective of the starting point or flow rate, all curves ultimately end up progressing in the same direction, namely, dT ¼ [0.1, 0.05]. In Figure 6.19(a), all the curves move off from the same point in the same direction. But the smaller RT is, the faster b(A) becomes larger, resulting in the dT dominating in the DPE sooner. This results in curves with small RT’s progressing in the direction of dT sooner than those with a large RT. What is interesting to note is that, by adjusting xT and RT, it is possible to obtain a profile that runs directly into a pure component. Consider the RT ¼ 20 mol/s profile in Figure 6.19(a) that passes through pure A. Interestingly enough, component A is the most permeable component and is expected to permeate the fastest. However, due to the operating conditions of Case 2, it is evident that one can synthesize any membrane column to achieve desired products.


95

Figure 6.19. Column profiles for selected values of (a) RT with xT ¼ [0.3, 0.3] and (b) xT with RT ¼ 100 mol/s. Figure 6.16(a) has been superimposed. (See insert for color representation).

In Figure 6.19(b), the initial direction of each curve is determined by the curve going through the same point on the b(A) ¼ 1 map (Figure 6.16(a)). This map has been superimposed on Figure 6.19(b). Curves commencing in the regions outside the shifted triangle, but inside the MBT, initially move in a different direction from the profiles that begin within the shifted triangle. This is due to the nature of the profiles in Figure 6.16(a) in that region (they were shifted into the space). Residue curves, such as those displayed in Case 1, cannot cross, by definition and by the nature of the equations that describe them. However,

96


it is evident from Figure 6.19(b) that membrane column profiles do intersect each other. The reason for this is the varying flow rates within the column section. Every point on a particular profile has associated with it a composition (x) as well as a flow (R and/or P), unlike a residue curve, which is related only to a composition. If one profile intersects another profile, then they share the same compositional values but will have different flows. This means that the b(A) values for each profile at the point of intersection will be different, and the curves will have different directions at the point. The result is that each curve will map out different paths. No two curves will intersect and move off in the same direction. This is possible only if they both have the same b(A) value at that point—and that is possible only if they originate from the same xT, with the same RT (i.e., the same curve). Tutorial 6.10 Try this for yourself: Using the MemWorX program, reproduce Figures 6.19(a) and 6.19(b). Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial.

6.5.7 Direction of dT It is obvious that the direction of the difference vector, dT, influences the trajectory of the profiles for Case 2. As a designer, one could adjust the magnitude and direction of dT in such a way as to ensure that the profile obtained achieves the desired products. This is not shown here but is rather left to the reader. 6.5.8 Direction of Integration When one plots the membrane column profiles, it is assumed that starting the integration from the top of the column down is appropriate. However, the starting point can be anywhere within the CS, and the calculation would include both a positive and a negative integration. If this were the case, then the positive integration would proceed as discussed before, with b(A) ! 1. However, the initial b(A) value would not be 1, but rather the b(A) value associated with the starting point of integration. The negative integration would start from the same b(A) value and diminish in magnitude. The negative integration would proceed until b(A) ¼ 1, at which point the integration must terminate since the top of the column has been reached. However, irrespective of integration direction, the curves obtained would be the same.

COLUMN SECTION PROFILES: OPERATING CONDITIONS 3 AND 4

97

6.5.9 Crossing the MBT Boundary In the discussions on the column profile thus far, we have allowed the curves to cross the MBT boundary. Of course, this is not physically possible. However, it is mathematically possible, and visualizing it has allowed one some valuable insight into the behavior of the profiles. Such information would be unattainable and misinterpreted if one were to look within the bounds of the MBT only. In reality, each profile would terminate at the boundary it intersects. When this occurs, one component equals zero, meaning that particular species has completely permeated through the membrane. We are then left with a binary mixture. The profile will then continue from the point of intersection with the boundary along the straight line boundary in the direction of the least permeable component of the two remaining species.

6.6 COLUMN SECTION PROFILES: OPERATING CONDITIONS 3 AND 4 6.6.1 Statement Cases 1 and 2 display the more theoretical operations of a CS and are unlikely to find any relevance in the commercial, industrial sector. The remaining two possible operating conditions for the CS (Figure 6.11) present situations that are of a far more practical nature. These are as follows. Case 3: PT 6¼ RT but yT ¼ xT

ð6:37Þ

PT 6¼ RT and yT 6¼ xT

ð6:38Þ

Case 4:

Case 3 would represent, for example, the terminating sections (rectifying and stripping) of an entire membrane column configuration. Case 4, alternatively, would be a general CS placed anywhere within the membrane column, or cascade. 6.6.2 Mathematics Consider the conditions of operation for Case 3. Substituting these into Equation (6.25) yields XD ¼ x T

ð6:39Þ

98


This results in the following DPE for Case 3: dx 1 1 ¼ þ 1 ½xyðxÞ þ ½xT x dA rD rD

ð6:40Þ

where rD ¼

RðAÞ D

ð6:41Þ

is known as the (local) reflux ratio. In Case 4, there is no simplification possible, and the DPE is dx 1 1 ¼ þ 1 ½xyðxÞ þ ½XD x dA rD rD

ð6:42Þ

Note that the DPEs describing the scenarios in Cases 3 and 4 (Equations (6.40) and (6.42)) are mathematically similar to each other. Each comprises the addition of two vectors: namely; the separation vector, s ¼ x y, and the mixing vector, m ¼ XD x. The only difference between the two cases is that the difference point (XD) is equal to the composition of the product being removed in Case 3. To avoid repetition, these two cases will be discussed simultaneously, due to the similar nature of their equations. Now, D is the pseudo-flow rate and does not actually exist, except in Case 3, where a product is being withdrawn from a CS. Since D is a difference in two flows, it is possible for it to have a positive or even negative sign. Similarly with the entries in XD, which is the net composition within a CS. 6.6.3 Column Profile The conditions of Case 3 compel all the entries of XD ¼ xT to be positive, that is, to lie within the bounds of the MBT. Case 4, on the other hand, allows XD to exist anywhere in the xA–xB space, including the MBT. For this reason we will select an XD within the MBT, thereby covering column profiles for both cases concurrently. Consider the following arbitrary conditions of operation (for Case 4): XD ¼ [0.3, 0.3], xT ¼ [0.3, 0.2], D ¼ 20 mol/s, and RT ¼ 120 mol/s. (Note: For Case 3, one would have to ensure that XD ¼ xT.) Integrating the DPE for Case 4 down the length of the column yields the plot shown in Figure 6.20.


99

Figure 6.20. Membrane column profile for Case 4. Positive integration.

The curve is terminated when the retentate flow equals zero, as indicated in Figure 6.20. This will be discussed in the section that follows. Furthermore, although not shown here, it is possible to determine at what A the curve pinches. Tutorial 6.11 Try this for yourself: Using the MemWorX program, reproduce Figure 6.20. Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. 6.6.4 Pinch Point Loci It is important to recognize that every point on the retentate column profile has associated with it two parameters: . .

composition, x reflux, rD.

Moreover, for a fixed flux model and XD (or xT), each rD has associated with it a set of stationary or pinch point locations. The pinch points are defined by equating the DPE to zero. As rD changes, so do the positions of the pinch points. It is therefore possible to generate pinch point loci, as shown in Figure 6.13, to visualize the movement of each node. A property of the pinch point loci, which can be proved mathematically, is that one branch will always pass through XD at rD ¼ 1. This is the case in Figure 6.21.

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Figure 6.21. Pinch point loci for XD ¼ [0.3, 0.3]. (See insert for color representation).

6.6.5 Analysis of Column Profile The behavior of the profile in Figure 6.20 can, once again, be understood by performing a mathematical analysis on the DPE for Cases 3 and 4. As mentioned before, the DPE is a combination of two vectors, s and m. Each vector is preceded by a scalar, both involving rD. Now, rD is continually changing—it has its maximum value at the top of the CS and diminishes to zero as material permeates (refer to Equation (6.41)). In the example discussed above, the maximum value of rD is rD ¼

120 ¼6 20

ð6:43Þ

The corresponding map, using this value, can be generated, as shown in Figure 6.22. The very first retentate composition, namely, xT, will have the maximum rD value associated with it. This implies that at the instant permeation begins at the top of the CS, the retentate profile is seeking the stable pinch point that is associated with rD ¼ 6 (refer to Figure 6.21). Since it is seeking this pinch point, the curve will be moving in a direction that is tangent to the profile passing through xT and that has a constant rD value of 6. However, a moment later some material has permeated, therefore changing x as well as reducing the magnitude of rD. As this happens, the retentate curve now seeks a different pinch point. Yet again, the column profile is tangent to the “constant rD” profile that passes through x, having the same rD. Figure 6.23


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Figure 6.22. Shifted M-RCM. The blue triangle represents the MBT, the red shows the transformed triangle. XD ¼ [0.3, 0.3], xT ¼ [0.3, 0.2], rD ¼ 6. Actual membrane profile (see Figure 6.20) is tangent to the bold profile at the point indicated. (See insert for color representation).

shows this tangency effect at arbitrarily chosen discrete points on the column profile. Analysis of Figure 6.23 gives the understanding necessary to explain the behavior of the profile. Each point on the profile is seeking its stable node. The stable node itself is continually moving. This implies that the column

Figure 6.23. Tangency of curves to the column profile. Each curve corresponds to a different rD value and is allowed to run to its stable pinch point. (See insert for color representation).

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profile has to adjust itself accordingly so as to move in the appropriate direction in order to seek the mobile stable node. It will continue to do this until the retentate has run out of material, at which point the profile will terminate. One can see from Figure 6.23 that the profile intersects with the pinch point locus at an rD value of about 2. However, it is evident that the profile does not pinch at this point. The reason for this is that there is still material present in the retentate for permeating. Even though there is an intersection between the profile and the locus, this is only instantaneous. A moment later (down the length of membrane), the rD value will decrease, and thus search for another stable node. Therefore, the profile does not terminate if it intersects with the pinch point locus, unless rD has gone to zero. 6.6.6 Pinch Point By definition of a pinch, dx ¼0 dA

ð6:44Þ

Thus, using Equation (6.40) or (6.42) at rD ¼ 0, we find ½xyðxÞ ¼ ½XD x

ð6:45Þ

yðxÞ ¼ XD

ð6:46Þ

Simplifying, we have

This means that when the retentate runs out of material, the very last permeate composition will be that of XD (or xT). This result is intuitive, since all the material has transferred through the membrane. One can simply use the permeation model to determine the composition of the last drop of retentate (refer to Equation (6.19)). Using the a-values selected and XD ¼ [0.3, 0.3], the pinch will occur at xpinch ¼ [0.15, 0.45]. This corresponds to the last point on the curve in Figure 6.20. The position of the pinch is fixed for a chosen XD and flux model. All curves will terminate at this point, irrespective of xT (see next section). 6.6.7 Further Column Profiles Arbitrarily chosen values for xT and rD are used in order to demonstrate a membrane column profile for Cases 3 and 4. However, any positive flow


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Figure 6.24. Column profiles for selected values of (a) various initial rD values with xT ¼ [0.3, 0.2] and (b) various xT with an initial rD ¼ 6.

rate, RT, is feasible, just as any point within the MBT is a possible top retentate composition. Figure 6.24(a) displays column profiles for a fixed xT ¼ [0.3, 0.2], varying rD as indicated. Figure 6.24(b), on the other hand, shows a range of column profiles for select xT’s at a constant rD ¼ 6. Note in Figure 6.24(a) that increasing the rD at the top of the CS does not alter the termination or pinch point location of each curve. This was discussed previously. The actual path mapped out by the profiles do, however, change. The path followed is a function of a stable pinch point locus. The larger rD is at the CS top, the closer the curve will begin to approximate an infinite reflux profile. However, as rD reduces substantially, the curve is forced to redirect its path so as to search for the mobile stable node. The profiles in Figure 6.24(b) exhibit the same properties. Irrespective of their starting point, their trajectories are dictated by the stable node pinch point locus, with all curves ultimately pinching at the same point.

104


Tutorial 6.12 Try this for yourself: Using the MemWorX program, reproduce Figures 6.24(a) and 6.24(b). Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. 6.6.8 Variations in XD and rD If one refers to Equations (6.25) and (6.41), one can deduce that rD and XD can have negative values. Consider the possible variations on XD and their effect on column profiles: .

.

.

.

Analysis and visualization of the pinch point loci associated with each XD gives insight into the trajectory followed by a profile. This is evident from the above discussions. Now, for a three-component system, XD has three entries, each of which can have a possibility of two signs ( þ or ). This means that there are eight possible directions that XD can have. However, XD has the property that all its entries have to sum to 1 (since it is a composition). Therefore, the combination where all entries are negative is not valid. So seven possibilities remain. Tapp et al. (2004) graphically identified these seven possibilities as regions within the entire xA–xB space. They showed that the pinch point loci for an XD in each region were different from that in any other region. However, they concluded that the general shape of the pinch point loci for an XD in a given region was the same, irrespective of where XD was placed in the region. The reader is referred to Tapp et al. (2004) for the graphical representation of these pinch point loci.

rD can be either positive or negative, but either way will ultimately tend toward zero as permeation proceeds. A column profile moves in such a way as to find its associated stable node. However, the stable node pinch point locus has two branches to it, one for each sign of rD (refer to Figure 6.21). Therefore, the sign of rD determines which branch the column profile will follow. An example is shown in Figure 6.25, where it can be seen that the profile with rD ¼ 6 maps out a path that follows the negative branch of the stable node locus.

APPLICATIONS AND CONCLUSION

105

Figure 6.25. Effect of the sign of rD on the trajectory of the profile. Each profile is influenced by its associated stable pinch point locus. (See insert for color representation).

6.7 APPLICATIONS AND CONCLUSION A novel graphical method of analyzing continuous membrane systems has been developed. It has been shown that any membrane configuration of membrane separators, no matter how complex in its arrangement, can be broken down into column sections. Each CS, which resembles a typical single-stage separator, is modeled using the DPE. The DPE tracks the change in the retentate composition down the length of the CS. For demonstration purposes, a simple constant relative permeability flux model was employed to determine the permeate composition as well as the continually changing retentate flow. All possible operating conditions of a column section were explored. Each condition is a result of the relationships between the flows and compositions at the top of a column section. Using the DPE, column profiles for each condition were plotted. The behavior of the profiles was discussed both mathematically and graphically. The profiles generated provide valuable insight into the operation of all membrane processes, even the simplest single-stage module. Being able to graphically visualize the change in the retentate composition assists a design engineer in making informed decisions. Although complex in their behavior, the nature of the profiles can be understood and explained. With this novel method, one is now able to synthesize and design more

106


informed and creative configurations. Being able to break down any membrane arrangement to its simplest form allows one to isolate and evaluate the individual building blocks, thereby permitting an optimal design to emerge. This method is also very useful in the design of hybrid systems involving membrane separators. This is the topic of a future publication by the authors (see Chapter 7).

CHAPTER 7

NOVEL GRAPHICAL DESIGN METHODS FOR COMPLEX MEMBRANE CONFIGURATIONS

Shortcut methods for the design and synthesis of separation systems are useful, especially in the conceptual stages of a process design. However, current approaches incorporated in these techniques result in their being suitable for traditional designs only, and unable to manage novel or complex configurations. By using column profile trajectories to describe the compositional behavior of material in both distillation and membrane sections, it will be shown how any configuration, no matter how complex, can be modeled. With this comes a deeper understanding of the operation of the chosen system. As an example, a Petlyuk-type arrangement will be considered—first incorporating membrane permeation units as the only means of separation, and then linking membrane units with distillation, thereby forming a hybrid. The feasibility of each design is discussed, and compositional regions of feasibility are established, where necessary. Although the nature of column profiles may appear to be complicated, this method allows one to quickly and easily understand complex configurations, thus saving time and money and potentially reducing energy consumption.


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7.1 INTRODUCTION Feasible techniques for the synthesis and design of separation processes have long been an objective in the chemical industry. Separation processes are renowned for their high energy consumption, and hence high operating costs. Therefore, the need for better methods for designing separation systems is warranted, especially in the conceptual stages. One first needs to be mindful of what methods of separation are available. For fluid mixtures, these include the traditional processes of distillation, absorption, and solvent extraction, to mention a few. Membrane permeation, although a relatively recent addition to this list, provides a unique method of separation. Any separation circuit may contain numerous units in order to achieve a desired separation—either all of the same operation, or a combination of various methods, thereby forming a hybrid process. In the distillation field, for instance, several advances have been made to formulate shortcut methods of design (Bausa et al., 1998; Castillo et al., 1998; Pham and Doherty, 1990; Fidkowski et al., 1991; Fien and Liu, 1994; Zhang and Linninger, 2004, 2006). Several of these methods have focused on designing processes that are able to separate multicomponent mixtures, as opposed to the simple single-feed–two-product column used for binary separations. As an initial alternative to this, two or more columns can be placed in succession in order to achieve a desired degree of separation of a multicomponent mixture. Unfortunately, this method has associated with it high capital and operating costs. The idea of thermally linked columns therefore came about in an attempt to reduce the size of the equipment needed, as well as to eliminate some of the condensers and reboilers. Whereas the Petlyuk column is the most popular of these advances, other variations include a main column with a side-rectifier or side-stripper connected to it. Theoretically, these column configurations have the ability to produce the required products. However, their operation is complex, resulting in a lack of reliable design methods. Membrane permeation, although physically and chemically distinct from distillation, has undergone transformations and adaptations analogous to those of distillation. Initially, industrial membrane designs were based on a single, once-through unit, similar to a flash process. Modifications on this stand-alone unit were made to improve its performance, until the idea of cascading was suggested. This then resulted in various forms of cascade arrangements being concatenated over the years, and their efficiencies compared (Kao et al., 1989; McCandless, 1994; Vorotyntsev and Drozdov, 2001). With this, and by analogy with the distillation field, a continuous membrane column was proposed (Hwang and Thorman, 1980). This was revolutionary at the time,

INTRODUCTION

109

and it was claimed that pure products from a binary mixture were achievable. There have been numerous modifications and adjustments to the membrane column (Hoover and Hwang, 1982; Hwang and Ghalchi, 1982; Kothe et al., 1989) as well as comparisons of it with other available membrane separators (Kao et al., 1989; McCandless, 1990). Extending these ideas to multicomponent systems resulted in having two membrane columns placed in series (Hwang and Ghalchi, 1982), as well as the two-membrane permeator (Kothe et al., 1989). Agrawal (1996) suggested various cascade configurations of single-stage membrane units that would mimic some distillation column arrangements, ranging from the conventional column right through to a Petlyuk arrangement. A similar analysis can be carried out using membrane columns, as opposed to a cascade of membrane units. Distillation may be the most commonly used method for separation of fluid mixtures, but there are instances when it becomes difficult or even impossible to separate some mixtures by distillation alone. Hybrid arrangements provide an alternative solution to this. Numerous shortcut methods of design have been proposed for hybrid processes (Stephan et al., 1995; Pettersen et al., 1996; Hommerich and Rautenbach, 1998; Lipnizki et al., 1999; Bausa and Marquardt, 2000; Lelkes et al., 2000; Eliceche et al., 2002; Kookos, 2003; Daviou et al., 2004; Huang et al., 2004). However, most of these methods are either suitable only for a single set configuration, thereby limiting the design procedure as well as any optimization, or they provide very little insight into the location of feed and side-draw streams, or into the number of stages or membrane area needed. Even though the above-mentioned arrangements provide attractive ways to separate ternary, or higher order, mixtures with the use of membrane permeation and/or distillation, a reliable method of design does not yet exist. To address this problem, a more creative design approach needs to be employed. Graphical methods of design have shown their value in several separation systems, including distillation and solvent extraction. In the distillation field, graphical tools such as distillation residue curve maps (D-RCMs), operation leaves (Castillo et al., 1998), and, recently, column profile maps (CPMs) (Tapp et al., 2004) emerged to assist design engineers in analyzing such systems. Holland et al. (2004a) used the “moving triangle” phenomenon in CPMs to design complex distillation configurations. Membrane residue curve maps (M-RCMs) were proposed by Peters et al. (2006a), as discussed in Chapters 3 and 4, with the intention of formulating a graphical technique similar to that for distillation. This was done to enable one to evaluate both membrane permeation and distillation from the same viewpoint. Applications

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of M-RCMs to both batch and continuous processes have been investigated (see Chapter 5) (Peters et al., 2008). It was shown how a continuous membrane column operated at total (or infinite) reflux was mathematically equivalent both to a single stand-alone unit and to batch permeation. Finite reflux operation has been a previous topic of discussion, as displayed in Chapter 6. Column sections (CSs), initially identified by Tapp et al. (2004) for distillation systems, were adapted for membrane processes. A generalized CS is defined as a length of column between points of addition and/or removal of material and/or energy (Tapp et al., 2004). It was shown that any membrane configuration, column or cascade, can be broken down into CSs. Differential equations (DEs) were developed to describe the continually changing compositions and flow rates down the length of the CS. Using an appropriate flux model, column profiles for membrane CSs were generated. In this chapter, the ideas and phenomena related to these column profiles are used to both mathematically and graphically analyze complex membrane configurations. As an illustration, a membrane Petlyuk-type arrangement will be considered. Some simplifying assumptions and operating conditions are made, allowing one to evaluate the feasibility of such arrangements. The idea is then ultimately to extend to a hybrid arrangement of distillation and membrane permeation. The previous chapter introduced membrane column profiles and discussed their topological properties; this chapter will focus on membrane column profiles as a graphical design and analysis tool. It is not the aim of this chapter to completely solve complex membrane and hybrid arrangements, but rather to introduce the idea of how column profiles can be used to understand and design such processes.

7.2 COLUMN SECTIONS 7.2.1 Definition As mentioned in the introduction, a CS is defined as a length of column between points of addition and removal of material and/or energy (Tapp et al., 2004). Although defined for distillation sections, this statement is general and can be applied to any counter- and, in fact, cocurrent process, including membrane permeation. This was shown in Chapter 6. Figure 7.1 gives a graphical representation of a CS for each of distillation and membrane, including the nomenclature associated with each. The continual vaporization and condensation of material is inherent to distillation. This results in the transfer of material from the liquid phase to the vapor, and simultaneously in the reverse direction. Therefore, the assumption

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Figure 7.1. Generalized column sections (CS): (a) distillation and (b) membrane.

of constant molar overflow is often justified or closely approximated in distillation CSs. Operation in the membrane CS, on the other hand, relies on the differences in partial pressure on either side of the membrane in order for mass transfer to occur. A diffusion membrane is used throughout this chapter for discussion purposes. It is therefore assumed that the permeate pressure (pP) is sufficiently close to vacuum. Thus, permeation will be monodirectional and will occur from retentate to permeate, but not in the reverse direction. This results in a continual change in the flow rates within any membrane CS. As drawn in Figure 7.1(b), both the retentate and permeate will have their maximum values at the top of the CS (i.e., RT and PT). 7.2.2 The Difference Point Equation The compositional change of material down the length of any general CS is described by the difference point equation (DPE). For a distillation section (Tapp et al., 2004), the change in the liquid composition (x) with respect to position (n) can be described by the following, written in vector form: dx 1 1 ¼ þ 1 ½xyðxÞ þ ½XDD x dn rDD rDD where Table 7.1 shows the nomenclature that applies.

ð7:1Þ

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Table 7.1. Nomenclature and Symbols Used in Equations (7.1) and (7.2) a Distillation

Membrane

Subscript or superscript Column section reference Downward material flow and composition Upward material flow and composition Position variable Net flow

D DCS Liquid, L, x

M MCS Retentate, R(A), x

Vapor, V, y

Permeate, P(A), y

Theoretical stages, n DD ¼ VL

Area, A DM ¼ PðAÞRðAÞ

Difference point

XDD ¼

VyLx DD

XDM ¼

PðAÞyRðAÞx DM

Reflux ratio

rDD ¼

L DD

rDM ¼

RðAÞ DM

a

A subscript of “T” is used to refer to quantities at the “top” of a CS.

For a membrane section (refer to Section 6.3.2), the change in the composition of the retentate (x) with respect to position (A) is described analogously by dx 1 1 þ 1 ½xyðxÞ þ ½XDM x ¼ dA rDM rDM

ð7:2Þ

Refer to Table 7.1 for a summary of the nomenclature and symbols used. Note that the form of the equation for each CS is the same, but the nomenclature is different. Although the DPEs for each section (Equations (7.1) and (7.2)) share the same mathematical form, there are some subtle, yet important differences. Since each DPE describes a unique system, the way in which y(x) is modeled relates to the operation of each process—this is discussed in the next section. As mentioned earlier, the flows of the liquid and vapor streams in the DCS are constant, resulting in fixed values for DD, XDD, and rDD. Thus, the scalar terms on the right-hand side of Equation (7.1), which involve rDD, remain constant in a DCS. In the MCS, the flows of the retentate and permeate decrease in value from the top of the MCS down, but it can be shown by material balance that DM and XDM are constant (refer to Section 6.3.4). However, by definition, rDM varies in magnitude, also having its maximum value at the top of the MCS. This has to be accounted for in Equation (7.2). This means that the scalar terms in Equation (7.2) change with position (A). This is rather subtle but, as will be shown later, causes significant differences in the behavior of the column profiles for an MCS when compared to that of a DCS.

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7.2.3 Vapor–Liquid Equilibrium and Permeation Flux For demonstration purposes, simple vapor–liquid equilibrium (VLE) and permeation flux models will be used to mathematically describe the vapor and permeate compositions, respectively. A constant relative volatility VLE model is used for distillation: aD ij xi yi ¼ P D i aij xi

ð7:3Þ

where aD ij is the volatility of component i with respect to j. A constant relative permeability flux model for gas separation is incorporated for membrane sections: aM ij xi yi ¼ P M i aij xi

ð7:4Þ

where aM ij is the permeability of component i with respect to j. The form of Equation (7.4) may be the same as that of Equation (7.3), but this occurs as a result of the assumption of a vacuum being maintained on the permeate side in the MCS. Given that the retentate (and permeate) flow is changing, one can show that (Section 6.3.3) X dRðAÞ ¼ aM ij xi dAn i

ð7:5Þ

where An ¼ A pR P0 ref =d is a scaled area (mol/s). It can be seen that the retentate flow is a function of position as well as its corresponding composition.

7.2.4 Column Profiles Using the DPEs (Equations (7.1) or (7.2)) in conjunction with the corresponding equilibrium or flux models (Equation (7.3), or Equations (7.4) and (7.5), respectively) allows one to graphically interpret the compositional change of the liquid down the length of the DCS, and that of the retentate down the length of the MCS. Vapor and permeate profiles can also be generated. These are determined by material balance through XDD and XDM (refer to Table 7.1), respectively.

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7.3 COMPLEX MEMBRANE CONFIGURATION DESIGNS: GENERAL 7.3.1 Overview In this section, only configurations containing membrane operation will be considered. There are numerous configurations and cascade options for membrane units and columns. Some are relatively simple in their design and operation; others are more difficult to design. A Petlyuk-type arrangement is used as an example for illustration purposes, but there are various other configurations that can, at some stage, be analyzed in a similar way. Although we focus on only a single design here, the reader is urged to understand the relevance of this method to other arrangements. Note: Since this section of the chapter deals only with membrane units, the subscript “M” will be omitted here. It will be used again in the next section when hybrid configurations of membranes and distillation are discussed. Also, throughout this section, the following constant values for the relative M M permeabilities will be assumed: aM AB ¼ 3, aBB ¼ 1, and aBC ¼ 1:5. 7.3.2 Petlyuk Membrane Arrangement Any cascade, configuration, or arrangement of membrane permeators or columns can be broken down into a number of CSs, no matter how complex in its design. The CSs are simply identified between the points of addition and removal of material and/or energy. For the Petlyuk arrangement in Figure 7.2(a), the CS breakdown is given in Figure 7.2(b).

Figure 7.2. (a) Petlyuk-type arrangement for membranes. (b) Column section breakdown.

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115

A compressor on the main shell is needed to recycle material leaving the top of the permeate side back to the retentate side. There is some debate as to whether a similar recycle stream at the bottom of the column is necessary, shown in dotted lines in Figure 7.2(a) (Tsuru and Hwang, 1995). This ultimately is a decision of the designer and is not important in the context of this chapter. The same retentate and permeate pressures in the main column are assumed for the side fractionator. For this reason, the streams connecting the main shell and the fractionator do not require any equipment for pressure change. Now, it is assumed that the overall arrangement operates at infinite reflux. This implies that the addition of feed or removal of product will have negligible effects on the overall flows within each CS. There will still, however, be comparable differences in the flows within each CS. The process in Figure 7.2(b) is then simplified as displayed in Figure 7.3. Because any material entering the top CS in Figure 7.3 must also leave at the same point, this section performs the same operation as the compressor (see Figure 7.2(a)), mixing material from the middle two sections, and then returning it to these sections at different compositions. The same can be said

Figure 7.3. Petlyuk arrangement operating at overall infinite reflux.

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Figure 7.4. Coupled membrane column sections. (See insert for color representation).

about the bottom section behaving like a pressure-reducer (valve). This then allows one to further simplify the arrangement, as given in Figure 7.4. Although the top and bottom sections are not present in the simplified diagram (Figure 7.4), they do in fact exist. However, under the assumption of infinite reflux, the DPE for these two sections in Figure 7.3 reduces to the membrane residue curve equation (see Chapter 3) (Peters et al., 2006a). This implies that their respective profiles will follow residue curves. Therefore, the determination and behavior of the so-called coupled column arrangement in Figure 7.4 is important for the feasibility of the Petlyuk configuration. 7.3.3 Material Balances Since there is a split in the exit stream of the compressor in Figure 7.4, it should be apparent that ð7:6Þ xT1 ¼ xT2 An equal split in retentate flows (at the top of each CS) has been chosen, that is, RT1 ¼ RT2

ð7:7Þ

This is not of particular significance, but a relationship must be set in order to generate column profiles. This assumption is maintained throughout the chapter but can be relaxed where necessary—this is left to the reader. Overall

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117

and component material balances around the compressor can also be written, in vector notation: PT1 þ PT2 ¼ RT1 þ RT2

ð7:8Þ

PT1 yT1 þ PT2 yT2 ¼ RT1 xT1 þ RT2 xT2

ð7:9Þ

There are various modes of operation that can occur within the system shown in Figure 7.4. Each mode is defined by the assumptions about the retentate and permeate flows and/or compositions. In order to limit discussions, however, only two scenarios will be examined here. They will demonstrate how column profiles can be used to graphically verify the feasibility of complex membrane configurations such as the coupled membrane sections. It is worth noting, as mentioned before, that if all top flows are equal and all top compositions are equal, then the result would be that each CS would follow the same membrane residue curve (Peters et al., 2006a).

7.4 COMPLEX MEMBRANE CONFIGURATION DESIGNS: OPERATING CONDITION 1 7.4.1 Statement Consider the following general operating conditions within CS1: PT1 6¼ RT1 and yT1 6¼ xT1

ð7:10Þ

From the material balances (Equations (7.8) and (7.9)), it can be shown that the same relationships must apply for CS2. Furthermore, whereas Equations (7.6) and (7.7) still hold true, it is taken that yT1 6¼ yT2 . 7.4.2 Mathematics The DPE for membrane sections (Equation (7.2)) is employed to mathematically and graphically determine the behavior of the retentate in each CS under these conditions. It is taken that integration commences at the top of the column, where all variables are assumed to be known. Since there is equality in the top retentate compositions on either side, both column profiles will originate at the same point, namely, xT. Now, rearranging the material balance in Equation (7.8) and substituting in D (refer to Table 7.1), we have PT1 RT1 ¼ ðPT2 RT2 ÞYD1 ¼ D2

ð7:11Þ

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Note that the net flow of material in the one CS is in the reverse direction to that in the other CS. This result is intuitive, for a closed circuit system. By incorporating Equation (7.7), it can be shown that the following relationship between the reflux ratios on either side exists: rD1 jT ¼ rD2 jT

ð7:12Þ

Similarly, it can be shown that there is equality in the difference points: XD1 ¼ XD2

ð7:13Þ

Thus, the designer has the freedom of choosing the reflux ratio and difference point for one CS. The same quantities for the other CS are set by mass balance. As mentioned before, the reflux ratio in any membrane CS will have its maximum absolute value at the top of the CS, and thereafter decrease in magnitude toward zero. Table 7.2 shows how the signs of the permeate and retentate flows change as one moves down the length of the CS, operating with either a positive or negative rD value, respectively. Note: By the definition of rD, and because R has to maintain a positive value, D will share the same sign as rD. Reading Table 7.2 from left to right allows one to monitor the sign change (if any) in the flows as they would occur during permeation. The flows, like rD, also have their maximum value at the top of the CS and decrease as permeation proceeds. Therefore, any one of the flows equating to zero signifies the termination of that particular CS. From Table 7.2(a), it can be seen that the CS with rD positive will terminate at rD ¼ 0 since the retentate has been depleted of material. Table 7.2(b) indicates that the other CS will end at rD ¼ 1 due to the permeate becoming zero. These values will be Table 7.2. Signs of Permeate and Retentate Flows for a CS (a) With positive rD rD ¼ R=D D ¼ PR P R

rD > 1 þ þ þ

rD ¼ 1 þ þ þ

0 < rD < 1 þ þ þ

rD ¼ 0 þ þ 0

rD < 1 þ þ

rD ¼ 1 0 þ

1 < rD < 0 þ

rD ¼ 0 0

(b) With a negative rD rD ¼ R=D D ¼ PR P R


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maintained throughout this section of the chapter as the conditions necessary for termination. While both CSs share the same absolute reflux ratio at the top of the column (refer to Equation (7.12)), they will not exhibit the same rD value at corresponding lengths down each CS. This is because the rate of change of R with position is a function of x (refer to Equation (7.5)), which is different in either CS. This then implies that one CS will terminate, or pinch, sooner than the other, and different areas are required in each CS in order to allow them both to reach their respective termination rD’s. 7.4.3 Column Profiles Consider the following arbitrarily chosen values for CS1: . . . .

rD1 jT ¼ 5 ¼ rD2 jT XD1 ¼ [0.3, 0.3] ¼ XD2 xT1 ¼ [0.2, 0.3] ¼ xT2 RT1 ¼ 100 mol/s ¼ RT2

It is possible to generate the column profiles for both CS1 and CS2, as shown in Figure 7.5. The solid curves in Figure 7.5 represent the retentate column profiles for each CS, as indicated, and the dotted curves represent that of the permeate. Both retentate curves originate at xT, as shown. The starting point of the permeate profiles is determined by material balance across each CS.

Figure 7.5. Retentate (solid) and permeate (dotted) column profiles for CS1 (red)and CS2 (blue). xT ¼ [0.2, 0.3], XD ¼ [0.3, 0.3], rD1 ¼ 5, and rD2 ¼ 5. (See insert for color representation).

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The profiles for each CS were plotted until their respective termination rD values were reached. Tutorial 7.1 Try this for yourself: Using the MemWorX program, generate Figure 7.5 by using a ¼ [3,1,1.5], and plotting a “Single profile” for each CS. Obtain the necessary parameters from the figure. Use the “Nonconstant–membrane” flow assumption and the third equation option. Remember to check “Include permeate profile.” Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. 7.4.4 Requirements for Feasibility In order for the coupled CS configuration, shown in Figure 7.4, to be feasible, it is necessary for: . . .

Retentate profiles to intersect each other within the MBT. Permeate profiles to intersect each other within the MBT. Direction of movement of the profiles to conform to the direction of flow in the coupled CS arrangement.

Now, by the inherent operating conditions, as well as the assumption of commencing integration from the top of each CS, the retentate profiles will always intersect at xT, irrespective of its location. The intersecting of the permeate profiles, however, is not guaranteed, but would signify the end of each CS if the arrangement is feasible. It can be seen that the material exiting the pressure-reducer (valve) splits prior to being fed into each CS. This implies that the permeate compositions at the bottom of each CS are the same, thus requiring an intersection of their respective profiles. Analysis of the profiles obtained in Figure 7.5 reveals that no such intersection exists for the operating values selected. This indicates an infeasible configuration for the chosen conditions. The next question that arises is what operating conditions and values are necessary in order to obtain feasibility. One could, for example, randomly select xT values (while keeping the other conditions constant), and check for any intersections in the permeate profiles that may occur. However, this will prove to be somewhat laborious, so a more comprehensive analysis has been performed.


121

To begin with, it is essential to have an understanding of the mathematical and graphical behavior of any given single-column profile. Chapter 6 examined, in detail, the nature of column profiles for membrane column sections. A brief discussion, highlighting the essential points, is given here, but the reader is referred to Chapter 6 for further insight.

7.4.5 Analysis and Behavior of Column Profiles Consider a retentate column profile, beginning at a given xT, with a fixed flux model and XD. According to the DPE (Equation (7.2)), every point on the profile has related to it a composition (x) as well as a flow in the form of reflux (rD). Analysis of Table 7.1 reveals that rD will have its maximum value at the top of the CS, and this then diminishes toward zero as permeation proceeds since the retentate is losing material. Pinch Point Curves Now, the DPE describing this profile is said to pinch when the right-hand side is equated to zero. From this, it can be deduced that every value of rD has associated with it a set of unique stationary points in the composition space. It can further be shown that, for a “constant a” flux model, a maximum of three stationary points exist for any rD (Holland et al., 2004b). The movements of the stable, unstable, and saddle point nodes as a function of rD are graphically shown in Figure 7.6(a). The stable node is located at pure B at infinite reflux and moves away from this point in either direction as rD diminishes in size. The direction of movement is related to the sign of rD, as shown. The actual column profile does not ever terminate at (and, in fact, may not even reach) any of these stationary points, except for the point where rD ¼ 0. This is because of the perpetual change in rD. However, the profile’s direction of movement is dictated by the position and movement of the stable nodes: Every point along the column profile has a unique rD value associated with it. Furthermore, at each point there exists a “constant rD” curve that is tangent to the profile. Each of these “constant rD” curves runs to its respective stable node. But the location of the stable node is related to the value of rD on the profile. This value is continually changing, resulting in the stable node continually moving. Thus, the column profile is seeking a “mobile” stable node. Figure 7.6(b) graphically describes this phenomenon. The green curve is any arbitrarily chosen column profile. The red curve is the stable node pinch point locus for rD > 0. The dotted curves represent discrete “constant rD” curves for the values indicated.

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Figure 7.6. (a) Pinch point loci for XD ¼ [0.3, 0.3]. (b) Tangency of curves to a column profile commencing at xT ¼ [0.3, 0.2]. Each curve corresponds to a different rD value and is allowed to run to its stable pinch point (see Chapter 6). (See insert for color representation).

Profile Pinch Point A general property of any retentate column profile is that it will terminate only when rD ¼ 0, regardless of the location of xT, as is evident in Figure 7.6(b). This is obvious since it implies that the retentate flow has run out of material. Furthermore, by equating the DPE to zero and evaluating the resulting equation at rD ¼ 0, it can be shown that y xjrD ! 0 ¼ XD

ð7:14Þ


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This indicates that the corresponding permeate profile will always terminate at XD. The result is intuitive, yet quite interesting, too—the termination point of all permeate profiles are known as soon as XD is set, prior to plotting the profiles themselves. What is more, the location of the retentate pinch can be calculated using Equation (7.14) and the flux model (Equation (7.4)), and is thus also known before any profiles are generated. Location of a Profile Relative to its Pinch Point Loci Note that, in Figure 7.6(b), as rD decreases in magnitude, the column profile approximates the pinch point curve. This arises since the profile is seeking a mobile stable node. The profile initially moves in a direction to find this node and eventually does reach it at some point (at rD 2 in Figure 7.6(b)), crossing the pinch point locus. However, it does not terminate there since the retentate still has material available for permeation, thereby causing the profile to proceed further. But now the mobile stable node has moved past the curve, and the profile has to readjust its direction in order to continue searching for it. By this stage, the profile is already within close proximity to the pinch point curve and so continues to follow it closely until the retentate is depleted of material. It has previously been established that all column profiles terminate at the same point when rD ¼ 0. Furthermore, as rD ! 0, the trajectories of all profiles become increasingly similar. Thus, all profiles approach the termination point from the same location relative to the pinch point curve (stable node). This is irrespective of the position of xT relative to the pinch point loci. Consider the example of Figure 7.6(b): The column profile is found to approximate the pinch point locus, lying above it. This is found to be the case for all profiles with a positive rD. The same phenomenon occurs on the negative rD stable node branch. A reason for this occurrence is the behavior and direction of the eigenvectors associated with a profile; but bear in mind that one eigenvector will always dominate. Of course, the location of a profile relative to its loci depends on both aM values as well as XD. In the example discussed, the positive rD profile approaches termination from above the positive branch of the stable node pinch point curve, whereas the other CS profile ends up on the left-hand side of the negative branch. Permeate Profile Behavior In the description of the behavior of the retentate column profile, the pinch point loci incorporated were those relating to the retentate. The corresponding permeate pinch point loci can be generated by material balance. This would be useful when analyzing the permeate profiles. This is not done here, as the results obtained previously are directly applicable to both retentate and permeate profiles.

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Negative Reflux Ratios The above discussion considered a retentate profile with rD > 0. Again, similar conclusions are arrived at for profiles with rD < 0. The only difference is that these profiles will follow the negative branch of the stable node pinch point locus (refer to Figure 7.6(a)). However, the nature of the profiles and the pinch point locus are somewhat more complicated now: .

.

The pinch point locus for rD > 0 has a well-defined path (for the XD value chosen) such that all stable nodes with rD > 0 exist on the indicated curve shown in Figure 7.6(a). This is not true when one speaks about the pinch point loci for rD < 0. This curve appears to head off to a very large value of xB. But, as rD ! 1 there can be only a single node—this can be proved mathematically from the DPE. This means that if the pinch point loci are continuous, which they appear to be, then there must be a way in which this rD < 0 curve moves toward large xB values, yet arrives back at a known location within the MBTwhen rD ¼ 1. This introduces rather complicated topological phenomena. However, it is not felt necessary to explore its details and the effect it has on column profiles, since one is able to do sufficiently without it. But it is worth being aware of what occurs graphically. A discontinuity arises due to the nature of the flux model being a rational function (Equation (7.4)). It can be found by equating the denominator to zero. In this case, it is a straight line. The discontinuity will always lie outside the MBT. No column profiles are defined or exist on the discontinuity. Now, the only profiles that will approach the discontinuity will be those having rD < 0. But not all profiles tend toward it: it depends on their location and rD value in the vicinity of the discontinuity. If a profile does approach the discontinuity, then it will terminate before rD is allowed to reach zero.

7.4.6 Feasible Coupled Columns It is with this understanding of the nature of the profiles that one can now answer the initial question of what operating conditions are necessary to yield a feasible coupled CS arrangement. Retentate Profiles and the Pinch Point Curve It has been established that the direction of movement of a column profile is directly influenced by the stable node pinch point curve: A profile will move so as to seek its stable node, as rD ! 0. However, the sign of rD determines which branch of the stable pinch point curve the profile will follow. Due to material balance, the reflux in one CS will always be positive, while the other will


125

maintain a negative value. This inherently means that the two respective retentate profiles will seek stable nodes that exist on the opposite branches of the pinch point loci. These branches always move in the opposite direction to each other for all XD (refer to Figure 7.6(a) and Tapp et al. (2004) for all other pinch point loci with different XD locations). Therefore, the two retentate profiles will begin at the same point, namely, xT, and then diverge, not necessarily intersecting again. Furthermore, since the profiles ultimately end up on a particular side of the pinch point locus (refer to Section 7.4.5), it should become graphically and geometrically apparent that the pinch point locus then becomes a so-called boundary for the region of possible xT values. In some instances, a profile may intersect with the pinch point locus, while in others it will either never intersect or cross the pinch point locus twice. This all depends on the location of xT with respect to the side of the pinch point locus on which the profile ultimately ends up. If a retentate profile intersects with the pinch point locus, then it is guaranteed that its corresponding permeate curve (for the same CS) will itself intersect with the retentate curve. In order to explain and understand this statement, consider the permeate curves. Permeate Profiles and the Pinch Point Curve The permeate profiles were plotted by calculating the permeate composition that satisfies the material balance. Rearranging the expression for XD in Table 7.1, one can express the permeate composition, for any CS, as a function of its rD and x: yðxÞ ¼

rD 1 xþ XD 1 þ rD 1 þ rD

ð7:15Þ

By employing the familiar “lever-arm” rule, the following conclusions about the position of a permeate curve with respect to the value of rD are arrived at: .

Profiles with a positive rD: T For rD > 0, the permeate composition will lie between x and XD, on the straight line connecting them. T At rD ¼ 0, y ¼ XD, as discussed before (see Equation (7.14)). . Profiles with a negative rD: T For rD < 1, x will lie between y and XD. T At rD ¼ 1, y is undefined. This is due to the permeate having no more material (refer to Table 7.2(b)). These can all be graphically interpreted as displayed in Figure 7.7.

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Figure 7.7. Effect of the sign of rD on the location of y with respect to x and XD.

Figure 7.7 shows that the locations of the permeate profiles are on opposite sides of the retentate profiles. In order for the coupled column arrangement to be feasible, the permeate profiles must intersect. How is this possible, especially when the two retentate profiles themselves move in (generally) opposite directions? Consider, once again, the pinch point locus. The determination of the nodes on the pinch point locus is done by equating the right-hand side of the DPE (Equation (7.2)) to zero: dx 1 1 ¼0¼ þ 1 ½xyðxÞ þ ½XD x dA rD rD

ð7:16Þ

Rearranging Equation (7.16) for y(x) yields exactly the same expression arrived at in Equation (7.15). However, in this instance, the expression does not describe the permeate profile. Rather, it depicts the pinch location of y(x) for any rD. Now, it can be shown mathematically that the pinch point curve always passes through XD, irrespective of its position. Also, the location of y(x) relative to x is determined by the position of XD. Thus, when a retentate profile intersects with the pinch point locus, the straight line connecting XD, x, and y(x) is, in fact, tangent to the retentate profile on the pinch point curve. This is due to the DPE being equated to zero (Equation (7.15)). In order to obey the geometry of the situation, the location of y(x) has to change relative to the locus and XD. This thereby ensures that, when a retentate profile intersects with the pinch point curve, the permeate profile will intersect with the retentate profile for a given CS.


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Figure 7.8 serves as an example of this. The retentate and permeate profiles are shown by solid and dotted lines, respectively. For clarity, the pinch point loci are represented by crosses. The mass balance lines, described by Equation (7.15), are dashed lines. These have been drawn in at the top of the CS, and then again when the retentate profile passes through its pinch point locus. Note the tangency, as well as the resulting retentate and permeate intersection.

Figure 7.8. (a) Example of retentate profile (solid) passing through the pinch point loci (crosses), resulting in the permeate curve (dotted) intersecting with the retentate. (b) Zoomed-in version of (a), showing the mass balance lines (dashed). (See insert for color representation).

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Tutorial 7.2 Try this for yourself: Using the MemWorX program, generate Figure 7.8 by using a ¼ [3, 1, 1.5], and plotting both “Pinch point loci” and “Single profile.” Obtain the necessary parameters from the figure. (Hint: xT ¼ [0.25, 0.5].) Use the “Nonconstant–membrane” flow assumption and the third equation option. Remember to check “Include permeate profile.” Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial.

Region of Feasible Top Compositions With the understanding that a retentate profile crossing the pinch point loci results in the permeate profile intersecting the retentate, the following can be said: An xT location that causes both retentate profiles for each CS to cross the pinch point loci at least once may result in a feasible coupled column arrangement. This condition is necessary. However, as will be discussed further, it is not sufficient to ensure feasibility. For the XD value chosen, the region of possible xT locations is shown in Figure 7.9, remembering that xT must lie within the bounds of the MBT. It can be seen that in the example used initially (see Figure 7.5), the xT value was not in this region, thereby creating an infeasible arrangement.

Figure 7.9. Region of xT locations for feasibility with XD ¼ [0.3, 0.3] in the membrane coupled column sections. (See insert for color representation).


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Feasible Regions Using Other XD Directions In the entire xA xB space, there are seven possible regions in which XD can exist (Tapp et al., 2004)—this was discussed in Section 6.2.4 (refer to Figure 6.10). These were identified according the direction of XD, which is dictated by the three entries in the vector. Up to now, the determination of a feasible xT region has been discussed in detail for one of the seven directions of XD—that is, XD located within the MBT. By applying the same ideas and mathematical understanding of the profiles and pinch point loci to the six remaining directions of XD, feasible xT regions were sought, if they existed within the MBT. It was found that only two other locations of XD yielded profiles for each CS that resulted in their both crossing the pinch point loci,

Figure 7.10. Feasible xT regions for (a) XD ¼ [, þ , ] and (b) XD ¼ [ þ , , þ ]. (See insert for color representation).

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thereby creating a feasible region. Examples of these are shown in Figures 7.10(a) and 7.10(b). It was observed that only XD directions that yielded stable node pinch point loci that moved through the MBT, such as those shown in Figures 7.6(a), 7.10(a) and 7.10(b), resulted in feasible xT regions. Minimum Reflux Ratio Thus far, for a given XD, we have discussed in detail how to obtain a region of xT that may ultimately result in a feasible arrangement. However, consider the diagrams shown in Figures 7.11(a)–7.11(c). An xT value, within the feasible region, was arbitrarily chosen, and the starting magnitude of the reflux was increased, as indicated. It can be seen from the diagrams in Figure 7.11 that, for a given xT, a minimum initial (top) reflux is required in order for feasibility to exist. If the reflux is too low, as is the case in Figure 7.11(a), then no permeate intersection will occur despite xT being in the correct region. As the top reflux increases, so the permeate curves approach each other. This happens since, as the reflux increases, the profiles for both CSs initially follow a path very similar to an infinite reflux profile, or residue curve (refer to Chapter 3) (Peters et al., 2006a). There is some reflux that will eventually allow the permeate curves to tangentially meet at a point, as displayed in Figure 7.11(b), thereby creating a feasible arrangement. This is the minimum reflux needed for feasibility. At any initial reflux ratios above this minimum, the permeate curves intersect twice, as is shown in Figure 7.11(c). This indicates that multiple steady states are possible within the system. However, the degree of separation can be controlled by the amount of membrane area available in each CS for permeation. The two intersection points require different membrane areas in each CS. From the direction of movement of the profiles, it can be determined that the second intersection point requires a larger membrane area in CS1, but a smaller area in CS2. The minimum reflux value can be determined by incorporating the tangency relationship between the two permeate curves. However, since the behavior of the profiles is quite complex, a basic search algorithm was employed. For the value of xT ¼ [0.45, 0.4], it was found that the minimum reflux was 16.037. However, each xT point has associated with it a different minimum reflux. What is important to recognize, though, is that although one is able to define a feasible region of xT locations using XD and the pinch point loci, one also has to ensure that operation occurs above the minimum top reflux for the chosen starting composition.


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Figure 7.11. Retentate (solid) and permeate (dotted) profiles for CS1 (red) and CS2 (blue), commencing at xT ¼ [0.45, 0.4], with XD ¼ [0.3, 0.3] operating at (a) rD1 jT ¼ 10 ¼ rD2 jT , (b) rD1 jT ¼ 16:037 ¼ rD2 jT , and (c) rD1 jT ¼ 20 ¼ rD2 jT : (See insert for color representation).

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Tutorial 7.3 Try this for yourself: Using the MemWorX program, generate Figures 7.11(a)–7.11(c) by using a ¼ [3, 1, 1.5], and plotting “Single profile” for each CS. Obtain the necessary parameters from the figure. Use the “Nonconstant–membrane” flow assumption and the third equation option. Remember to check “Include permeate profile.” Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. Once the correct region is located (if it exists) according to the XD, and the minimum top reflux for xT is determined, the profiles for the coupled column sections can be generated. In order to complete the design, one would need to plot the profiles for the top and bottom CSs (refer to Figure 7.3). These two CSs operate on residue curves (refer to Chapters 3, 4, and 5; or see Peters et al., 2006a, 2008) because all the permeate fed to the compressor must return as retentate (and vice versa for the valve). One can easily generate these residue curves at the top and bottom compositions of the coupled CS arrangement, thereby completing the design. These profiles would run into the unstable and stable pure component nodes. 7.5 COMPLEX MEMBRANE CONFIGURATION DESIGNS: OPERATING CONDITION 2 7.5.1 Statement Consider now the following more specific operating conditions within CS1: yT1 6¼ xT1 but PT1 ¼ RT1

ð7:17Þ

From the material balances over the compressor (Equations (7.8) and (7.9)), it can be shown that similar relationships hold true for CS2. Furthermore, it is taken that yT1 6¼ yT2 7.5.2 Mathematics By appropriately expanding the DPE for each CS (1 or 2), we have dx PðAÞ ½PT RT PT yT RT xT ¼ ½xyðxÞ þ x dA RðAÞ RðAÞ PT RT

ð7:18Þ


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By multiplying out, and then employing the conditions set in Equation (7.17), it can be shown that the equation collapses to dx ¼ ½xyðxÞ þ bðAÞ dT dA

ð7:19Þ

where RT RðAÞ

ð7:20Þ

dT ¼ yT xT

ð7:21Þ

bðAÞ ¼

dT is known as the difference vector taken at the top of the CS. Since dT is a difference in two compositions, it follows that certain values (but not all) in the row vector will be of negative sign. Furthermore, by definition, the entries in dT must sum to zero. b(A) is a scalar multiplier preceding dT and is a ratio of the retentate flow at the column top to that at a point corresponding to A. b(A) ¼ 1 at the top of each CS, while anywhere else in a CS, b(A) > 1, increasing down the length of a CS. Each CS is modeled using Equation (7.19). Solving this equation graphically yields the respective column profiles for each CS. As with Operating Condition 1 (see Section 7.4), it is taken that integration commences at the top of the column. Since there is equality in the top retentate compositions on either side, both column profiles will originate at the same point, namely, xT. From Equations (7.9) and (7.17), we have yT1 xT1 ¼ ðyT2 xT2 Þ

ð7:22Þ

dT1 ¼ dT2

ð7:23Þ

It follows that

Thus, the designer has the freedom of choosing the direction of the difference vector for one CS. However, that for the other CS is set by mass balance. 7.5.3 Column Profiles Consider the following arbitrarily chosen values for CS1: . . .

dT1 ¼ [0.04, 0.08] ¼ dT2 xT1 ¼ [0.3, 0.3] ¼ xT2 RT1 ¼ 100 mol/s ¼ RT2

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It is possible to generate the column profiles for both CS1 and CS2, as shown in Figure 7.12. Tutorial 7.4 Try this for yourself: Using the MemWorX program, generate Figure 7.12 by using a ¼ [3, 1, 1.5], and plotting a “Single profile” for each CS. Obtain the necessary parameters from the figure. Use the “Nonconstant–membrane” flow assumption and the second equation option. Remember to check “Include permeate profile.” Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. 7.5.4 Feasibility As before, the same feasibility requirements are necessary here (refer to Section 7.4.4). These include the intersections of both the retentate profiles within the MBT, as well as those for permeate profiles. This is not the case in Figure 7.12, which makes the arrangement infeasible. In order to establish whether this result is specific to the conditions chosen, or is general, one needs to consider the retentate profiles and the DPE that describes them (i.e., Equation (7.19)).

Figure 7.12. Retentate (solid) and permeate (dotted) column profiles for CS1 (red) and CS2 (blue). xT ¼ [0.3, 0.3], RT1 ¼ 100 mol/s ¼ RT2, dT1 ¼ [0.04, 0.08] ¼ dT2 are represented in green. (See insert for color representation).


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A behavioral analysis of the individual profiles for each CS can be performed in much the same manner as was done for those in Operating Condition 1 (see Section 7.4.5), but it is not necessary to explain this as the results are intuitive. The results have been detailed in Chapter 6, and the reader is referred to this for further insight. The right-hand side of the DPE for this case (Equation (7.19)) is the summation of two vectors, namely, the separation vector (s ¼ x y) and the difference vector (dT). Unlike the DPE used for Operating Condition 1 (Equation (7.2)), where both vectors were preceded by scalar quantities, only the difference vector in Equation (7.19) is preceded by the scalar quantity. This scalar is b(A) and is defined by Equation (7.20). Now, b(A) begins at a value of 1 at the top of a CS and then increases in magnitude down its length. As b(A) becomes sufficiently large, it results in dT becoming the dominant vector in the DPE. This causes the effect of the separation vector to become negligible and the retentate profile to ultimately move in the direction of dT. Now, each CS has a dT that is the opposite of the other (see Equation (7.23)). This implies that, although the profiles for each CS begin at the same point, xT, they will eventually move off from each other, going in opposite directions, dictated by their respective dT’s. The permeate profiles for each CS, on the other hand, will always differ from that of the retentate by bðAÞ dT. This is a result of the mass balance constraint and can be proved accordingly. This simply means that whereas the permeate profile for one CS will be “ahead” of its corresponding retentate profile, the other permeate profile would be “behind” its retentate profile. Figure 7.12 is an example of this. The permeate profile, although offset from the retentate profile, will ultimately follow the same direction as the retentate profile, namely, dT. Now, since the offset of the permeate profile as well as the ultimate direction of movement of the retentate and permeate profiles are dictated by dT, it appears that the position of the permeate profiles for each CS will be such that they may never intersect (refer to Figure 7.12). But is this a general result for all xT locations and dT directions? For feasibility, it is required that the permeate profiles must intersect. As the permeate profiles lie on opposite sides of their respective retentate profiles (since dT1 ¼ dT2 ), the only way that permeate profiles will meet is if they themselves cross their corresponding retentate profiles. This is not the case in Figure 7.12—the profiles for CS2 intersect, but those for CS1 do not. Consider the work of Holland et al. (2004a), where they performed a related analysis for a coupled distillation system, which is physically a very different process from membrane permeation, however mathematically similar, as has been shown. The only difference, mathematically speaking, is that in membrane permeation there is the added complexity of varying (decreasing) flow down the length of the CS, whereas in distillation it is valid to assume it is

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constant. In other words, when one describes a distillation system, b(A) would be held at a constant value of 1. For this value it was shown that regions of feasibility do exist (Holland et al., 2004a) but cannot be sought for all directions of dT. However, the direction used in Figure 7.12 does result in a region of feasibility. Consider Figure 7.13(a)—the same arbitrarily chosen values are employed, except the retentate and permeate flows are fixed. The profiles are allowed to run to their respective pinch points. All the necessary conditions for feasibility are met, as shown.

Figure 7.13. Retentate (solid) and permeate (dotted) column profiles for CS1 (red) and CS2 (blue). xT ¼ [0.3, 0.3], dT1 ¼ [0.04, 0.08] ¼ dT2 are represented in green. (a) Fixed flow rates (i.e., b(A) ¼ 1) and (b) varying flow rates with RT1 ¼ 1000 mol/s ¼ RT2. (See insert for color representation).


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This leads one to question whether or not this result is unique for the constant molar flow assumption only. Figure 7.13(b) shows the resulting profiles for varying flow when commencing with a relatively large RT value of 1000 mol/s. The required permeate intersection(s) occurs, in the correct direction, implying a feasible configuration is possible. Also, when one compares Figure 7.12 with Figure 7.13(b), one can deduce that there is some minimum RT value for feasibility. This minimum value would correspond to a single intersection between the permeate profiles and can be sought as described in Section 7.4.6. As before, this minimum flow depends on the location of xT. Tutorial 7.5 Try this for yourself: Using the MemWorX program, generate Figures 7.13(a) and 7.13(b) by using a ¼ [3, 1, 1.5], and plotting a “Single profile” for each CS. Obtain the necessary parameters from the figure. Use the “Constant–distillation” flow assumption for Figure 7.13(a) and “Nonconstant–membrane” for Figure 7.13(b). Remember to check “Include permeate profile.” Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. Using a relatively large RT value results in b(A) decreasing at a slower rate compared to commencing with a smaller RT value. Thus, the difference vector does not dominate as rapidly, which allows the separationvector to influence the trajectory of the profiles. The profiles then initially mimic the “constant b(A) ¼ 1” profiles to a certain extent (refer to Figure 7.13), until some point when the relative sizes of the twovectors swop over, causing the profiles to move in the direction dictated by dT. This changeover occurs within the vicinity of the pinch point curve (not shown). As discussed earlier (see Section 7.4.6), the crossing of the pinch point curve is what allows for intersection between the retentate and permeate profiles. This can be deduced mathematically from the equation describing each permeate curve, as well as the pinch equation. It can therefore be concluded that the same regions identified by Holland et al. (2004a) for a given dT apply to membrane systems with varying flow. There is the additional requirement that operation occurs above the minimum RT value for the chosen xT location. It is worth noting that slowing down the rate at which b(A) decreases in magnitude can be done by simply increasing RT, as discussed. There is another way that one can slow its rate by keeping RT at a fixed or similar value. Altering the physical conditions of each CS, such as the operating pressures and membrane thickness, results in a more gradual decrease in the retentate flow, and hence b(A). This result can easily be deduced from Equation (7.5).

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7.6 COMPLEX MEMBRANE CONFIGURATIONS: COMPARISON WITH COMPLEX DISTILLATION SYSTEMS Research by Holland et al. (2004a) used column profiles and the idea of “moving triangles” to model the behavior of complex distillation systems. This idea has now been extended to membrane systems. However, the results obtained for the distillation process are very different from the conclusions reached in the preceding sections about membrane systems. For example, Holland et al. (2004b) show that for the distillation equivalent of Operating Condition 1 (involving the difference vector, XD) different regions of feasibility exist for each process. The reason for this, and other differences, lies with the DPE that is used to describe each type of CS. Both DPEs involve the addition of two vectors (refer to Equations (7.1) and (7.2) for Operating Condition 1, and to Equation (7.19) for Operating Condition 2), both of which are preceded by the same scalar quantities. However, the scalar values in the DPE for the DCS are fixed for any given section, whereas for an MCS it has to vary due to ever changing flows. This results in the nodes for a DCS being stationary (and so the idea of a “moved triangle”), but mobile for an MCS. With this, and the effect of material loss, very different and unique column profile behavior occurs within membrane sections. The column profiles in a DCS move toward a single, fixed, stable node. In the MCS, the node is continually moving, and the profile has to adjust its direction to seek this node. It becomes apparent then that the required directions of movement and profile intersections are very different in each scenario, resulting in unique conclusions about feasibility for each process. If one could employ the “constant molar overflow” assumption within an MCS, the results obtained would then mimic that obtained for distillation. This, however, is possible only in extreme cases, when the membrane is selective for a component that has a very dilute concentration in the retentate. Because of these interesting differences that arise, the next section explores and introduces the design options for creating hybrids from distillation and membrane processes. 7.7 HYBRID DISTILLATION–MEMBRANE DESIGN 7.7.1 Overview In previous sections (see Sections 7.4 and 7.5) an example of a complex configuration of membrane units was suggested. Through material balance constraints, as well as the behavioral properties of the retentate and permeate profiles, it was revealed how regions of feasible configurations could be sought.

HYBRID DISTILLATION–MEMBRANE DESIGN

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One could now address the same arrangement of units, but alter the operation occurring in some of them, thereby creating a hybrid process. As a case study, a membrane module thermally linked to a distillation column will be analyzed. Distillation is equilibrium based, whereas membrane permeation is rate based, and combining the two into a hybrid should allow for each one to operate in the compositional space where it is best suited. The arrangement of the units is shown in Figure 7.2(a), except the main shell is no longer a membrane column, but rather distillation. Since the number of stages, membrane area, or feed/side-draw locations have not been specified, the arrangement can be considered to be a superstructure. In other words, there are a number of possible variations of the configuration. Some of these variations that stem from this arrangement are shown in Figure 7.14. There are a number of other possible variations not shown. Analyzing the hybrid design proposed in Figure 7.2(a) does not limit one to a set design, but rather encompasses a wide range of configuration options. In initial conceptual design stages, however, one should discern whether an arrangement of equipment is feasible or not. In the same way as before, a CS breakdown can be performed (refer to Figure 7.2(b)). Using the same assumptions of infinite reflux as a starting point, one can arrive at the same coupled sections arrangement. However, it is now a membrane section coupled with a distillation section, as shown in Figure 7.15. It is taken that the retentate side of the membrane operates at a pressure that matches that of the distillation column.

Figure 7.14. Possible variations of a membrane module thermally linked to a distillation column.

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Verifying the feasibility of this simplified form of a hybrid combination would begin to give insight and answers about the operation and possibilities of the entire design. 7.7.2 Material Balances Rearranging the material balance around the compressor/condenser in Figure 7.15, we find PT RT ¼ ðVLÞYDM ¼ DD

ð7:24Þ

This leads to the following relationship between the reflux ratios at the top of each CS: rDM jT ¼ rDD

ð7:25Þ

The reflux ratio in the MCS decreases down its length, while the reflux ratio in the DCS remains constant. By component material balance, it can be shown that there is equality in the difference points: XDM ¼ XDD

ð7:26Þ

Figure 7.15. Coupled distillation and membrane column sections. (See insert for color representation).


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Thus, the designer has the freedom of choosing the reflux ratio and difference point for one CS. The same quantities for the other CS are set by mass balance. The following operating conditions will be assumed: . . . .

rDM jT ¼ 5 ¼ rDD XDM ¼ [0.3, 0.3] ¼ XDD D D aD AB ¼ 3, aBB ¼ 1, and aCB ¼ 1:5 M M aAB ¼ 3, aBB ¼ 1:5, and aM CB ¼ 1

The respective a-values used for distillation and membrane permeation are different—it is assumed that one would use a membrane that has different separation capabilities from that of the distillation. Note that rDM > 0, while rDD < 0. Since XDM and XDD are equal by mass balance, they will both be referred to as XD for simplicity. 7.7.3 Feasibility Requirements For a feasible coupled column section arrangement, it is required that the retentate profile intersects with the liquid profile, as well as the permeate profile intersecting with the vapor profile. Furthermore, the direction of movement of the profiles, and the order of intersection must be in accordance with the flow arrangement in the coupled CS configuration. Now, it is already known that the liquid and retentate profiles will always intersect (at xT). So, all possible intersections between the vapor and permeate profiles need to be found for a feasible arrangement. Permeate Profiles Initially, consider any general permeate profile: Irrespective of the location of its starting point, its termination point is fixed at XD and occurs as rDM ! 0 (refer to Equation (7.14)). This can be confirmed by the arbitrarily plotted profiles in Figure 7.16. Also, no matter the location of xT, both the retentate and permeate profiles will move in a direction so as to seek their respective “mobile” stable nodes as rDM ! 0. This was discussed in Section 7.4.5, as well as in Chapter 6. It has previously been demonstrated that, at some point, the retentate profile would very closely follow its stable node pinch point locus. The same can be said about the permeate profile mapping out a path that mimics its respective pinch point locus. This is demonstrated in Figure 7.16. (Note that the permeate pinch point locus differs from that of the retentate by material balance and is calculated

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Figure 7.16. Commencing the retentate profiles at various xT values. The corresponding permeate profile eventually always approximates its pinch point curve. (See insert for color representation).

accordingly.) Since it is assumed that rDM > 0 for the MCS, the location and movement of the positive branch of permeate stable node pinch point locus is of interest. Remembering that it is required to find where the vapor profile intersects with the permeate profile, one can now rather search for the vapor profiles that intersect with the permeate pinch point locus. This is valid since all permeate profiles eventually follow the pinch point locus. It may be argued that the actual permeate profile follows the pinch point locus only for a short while. However, as one varies the magnitude of RT (and hence PT) the profile follows the pinch point locus for a longer period (see Chapter 6). Also, as will be revealed later, the location of xT’s that results in feasibility, as well as the relative position of the vapor and permeate to each other, is such that an intersection with the permeate pinch point locus is guaranteed. Although the permeate profile does not match the pinch point locus exactly, it is very close to the locus. Thus, using the pinch point locus is justified, especially for conceptual stages of synthesis and design, where accuracy is not of great importance.


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Tutorial 7.6 Try this for yourself: Using the MemWorX program, generate Figures 7.16(a)–7.46(c) by using a ¼ [3, 1.5, 1], and plotting first “Pinch point loci,” then using the “Cursor input” plot “Single profile” for different xT locations. Obtain the necessary parameters from the figure. Use “Nonconstant–membrane.” Remember to check “Include permeate profile.” Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. The Location of y Relative to x For both the DCS and the MCS, the respective vapor and permeate curves are determined by material balance, as described by Equation (7.15). When one uses this equation for the DCS, the rDD is held constant, while for the MCS, rDM varies, as previously discussed. It has been assumed that, for the MCS, rDM > 0. This implies that each y has to lie between XD and the corresponding x coordinate. For the DCS, on the other hand, rDD < 0 signifies that x now lies between XD and y, on the straight line that connects them. Thus, the position of the permeate with respect to the retentate is on the opposite side of the vapor compared to the liquid. Previously, when discussing the coupled membrane sections (refer to Section 7.4.6), the same phenomenon of opposite relative positions of y with respect to x occurred. It was this fact that led to an overall infeasibility in the coupled membrane sections. However, one cannot immediately say that this is the case now with a membrane section coupled to a distillation section. Previously, the profiles of both membrane sections were moving in the direction of the stable node pinch point locus—one followed the negative branch, the other the positive. This caused them to diverge. In the current scenario, on the other hand, only the one set of profiles follows the pinch point curve. The distillation profiles, alternatively, seek a single fixed node since rDD is constant. This may very well lead to the necessary intersections needed for feasibility. Vapor Profiles Up to now, it has been established what the position of the vapor profile relative to the liquid profile is. Furthermore, it is now agreed that if a vapor profile intersects the permeate stable node pinch point locus, then a feasible configuration is obtained. The location of xT ultimately determines if this will occur. It therefore needs to be established what region of xT, within the MBT, yields feasibility.

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Figure 7.17. Some liquid profiles (solid) and their corresponding vapor profiles (dashed) relative to XD.

Consider two arbitrarily chosen starting conditions for the liquid profiles, as shown in Figure 7.17. Their corresponding vapor profiles are plotted as well. It can be seen from Figure 7.17 that in one case the vapor profile lies below the liquid profile, while in another it lies above. This arises since the position of the vapor curve relative to its liquid curve is determined by XD. Of the two profiles shown, only the one with the vapor positioned below the liquid is useful, since it would pass through the permeate stable node pinch point locus. However, this is not the only profile that would exhibit such an intersection; rather, there are numerous profiles that would satisfy this condition. Since their dependency on XD is significant, all these profiles would exist in a defined region, rather than being randomly positioned. Of course, this region would have to exist within the MBT. Since the relevant vapor curves are required to pass through the permeate stable node pinch point locus, they would have to commence at a point before the locus, such that their direction of movement ensures an intersection with the locus. In this case, all profiles to the right of the locus move in the appropriate direction. Thus, the permeate stable node pinch point locus forms a boundary of the feasible region. Liquid Pinch Point Locus Figure 7.17 shows that the relative position of the vapor changes from below to above the liquid profile as xT is altered. This means that, at some starting point between these two extremes, the vapor curve, although commencing behind


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the liquid, would appear to follow the same trajectory as the liquid. It is worth noting that this overlap between the profiles would occur only up until some point when the two curves would diverge from each other, heading off to their respective pinch points. Now, in order to obtain this overlap between the profiles, the initial y value not only must lie on the straight line connecting XD and x produced, but the line should also be tangent at x. This coincides with the definition of a pinch point for the liquid—by equating the DPE to zero: ð1 þ rDD Þ½xyðxÞ ¼ ðXD xÞ

ð7:27Þ

Equation (7.27) states that the points x, y, and XD are collinear, and the straight line connecting them is tangent to the profile commencing at x. In other words, if a point on the liquid pinch point curve is used as xT, then the two profiles would overlap. This would be the case for all points on the liquid pinch point locus. Thus, the liquid pinch point locus forms another boundary of the feasible region. For the aD values assumed and the XD value chosen, the liquid pinch point curve is plotted in Figure 7.18. Only the part of the curve within the MBT is shown, since this is the only section that is of importance here. Superimposing the permeate stable node pinch point curve on Figure 7.18, and remembering that all xT values to the right of it allow for a feasible intersection, the region can be initially defined, as shown in Figure 7.19. It is worth noting that both these pinch point loci pass through and share the same XD.

Figure 7.18. Liquid pinch point locus within the MBT. (See insert for color representation).

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Figure 7.19. The permeate (purple) and liquid (green) pinch point loci defining an initial region of feasibility. (See insert for color representation).

It can be seen that the only part of the liquid pinch point locus that is now relevant to the situation is the negative branch of the unstable node. This is expected since rDD < 0, and the profiles are required to move away from the unstable node toward the stable, intersecting with the permeate pinch point locus on the way. Shifted Distillation Triangle All the liquid profiles for the DCS exist on a single map known as a column profile map (CPM). A similar map can be generated for the vapor profiles. Note that no such maps can be produced for the membrane (retentate and permeate) profiles since there are too many variables, making it impossible to show all profiles in a single map. Tapp et al. (2004) defined a CPM as a linear transform of the residue curve map (RCM). In other words, the profiles in the RCM have, graphically speaking, been shifted and possibly rotated slightly in order to produce the CPM. In doing so, the three nodes also move from the pure component locations. Holland et al. (2004b) discuss the direction of the eigenvectors at each of the shifted nodes such that one can connect the nodes with straight lines, thus forming a triangle. This is known as the “shifted” or “moved” triangle, from the original MBT. The CPM using the assumed values chosen for the DCS discussed above is plotted in Figure 7.20. Now, the permeate profiles will always move toward XD and terminate there. It should be obvious that only liquid (and hence vapor) profiles within the shifted triangle will result in a feasible arrangement. This is due to their direction of movement.


147

Figure 7.20. CPM for the DCS at rDD ¼ 5 and XD ¼ [0.3, 0.3]. MBT in blue, shifted triangle in red. (See insert for color representation).

Thus, the feasible region obtained in Figure 7.19 is further reduced in spatial size. Figure 7.21 shows the complete region. This region shows all possible xT values that will result in the coupled distillation and membrane sections being feasible, for the operating conditions assumed.

Figure 7.21. Region of possible xT locations, for the operating conditions chosen, that would result in a feasible arrangement. (See insert for color representation).

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Column Profiles Figure 7.22(a) shows the profiles obtained for an arbitrarily chosen xT value within the region. Figure 7.22(b) is a zoomed version of Figure 7.22(a). It can be seen that the vapor and permeate profiles intersect twice. This indicates that multiple steady states in hybrid processes are very possible and need to be taken into account, especially for control purposes.

Figure 7.22. (a) Column profiles for the membrane and distillation sections with xT in the feasible region. (b) Zoomed in version of (a). (See insert for color representation).


149

Tutorial 7.7 Try this for yourself: Using the MemWorX program, generate Figure 7.22 by using aD ¼ [3, 1, 1.5] and aM ¼ [3, 1.5, 1], and plotting a “Single profile” for each CS. Use various locations of xT. Obtain the necessary parameters from the figure and text. Use the “Constant–distillation” flow assumption for the DCS, and “Nonconstant–membrane” for the MCS. Remember to check “Include permeate profile.” Refer to Appendix A for more information on how to use MemWorX. Table A.4 summarizes all the input parameters required for this tutorial. Starting at any point outside this region would result in an infeasible configuration. As with the coupled membrane scenario, to complete the design, one would generate the residue curves for the top and bottom sections (Figure 7.3), commencing at the appropriate compositions and terminating at the pure components. Regions of Feasibility: General Although this region was generated for a specific set of operating conditions, the theory can be extended to any chosen values of XD, aD, aM, and rDD (or rDM). No matter what the conditions are, the region, if it exists, can be found by enclosing the area formed between: . . .

The positive branch of the pinch point locus for the stable node of the permeate The negative branch of the pinch point locus for the unstable node of the liquid The region within the moved triangle

Of course, the region has to lie within the bounds of the MBT. One could also increase the size of the region by varying the size of the reflux ratios at the top of each CS. The larger rDD is, the closer the nodes of the shifted triangle lie toward those of the MBT. This would then allow for a larger feasible region. However, there is a trade-off that the higher the reflux, the larger the equipment and operating costs. This would have to be considered in an optimization technique incorporating a cost function. The true power of using column profiles for both membrane and distillation design is that it provides important graphical insight into the behavior of the proposed configuration. Other techniques rely on trial and error and some are

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based on rules of thumb. This does not improve the designer’s understanding of the process.

7.8 CONCLUSION The use of column profiles for both membrane and distillation systems is a very powerful shortcut design technique. Any membrane arrangement, or hybrid configuration, no matter how complex, can be modeled using the CS approach. The resulting profiles allow an understanding of the behavior of the operation of each CS. One does not need to blindly design a complex system by using trial-and-error methods. Although the method shown in this chapter is useful for the conceptual stages of a design, it does provide a very good initialization point in rigorous simulation packages when more accurate and detailed results are needed. Ultimately, because there is a deeper understanding of the operation of the system, and also because the technique is relatively quick, the designer is able to make insightful decisions early on, saving money and time. It should be appreciated that this work is not applicable to membrane permeation and distillation only, but can easily be adapted for any separation method. Both constant and nonconstant flow assumptions have been dealt with here; this then equips one to use the methods displayed for the separation procedure of one’s choice. Of course, details of equilibrium/flux model, flow assumptions and directions, and so on need to be decided upon. But the method and results arrived at here would still apply.

CHAPTER 8

SYNTHESIS AND DESIGN OF HYBRID DISTILLATION—MEMBRANE PROCESSES

The techniques developed throughout this book enable one to produce column profiles for separation processes, including nonreflux membrane units and membrane column sections. Furthermore, it is possible to display a sequence of column profiles on the same map and thus to graphically interpret a system of column sections in an efficient manner. As the theory was developed for a general separation process, it is possible to do this for a CS of any separation type, offering an efficient technique for the analysis of hybrid systems. The method generates an attainable region (AR) for a chosen configuration, and also informs the designer of parameters such as membrane area and number of distillation stages required. As an example, the separation of a methanol/butene/MTBE system is investigated. Methyl tertiary butyl ether (MTBE) is the desired product and hence needs to be efficiently recovered from a reactor output. Difficulty arises when separating such a mixture with conventional distillation processes because of the binary azeotropes that exist between methanol and MTBE, as well as between methanol and butene. The driving force for separation in membrane processes differs from that in distillation, so a membrane process will not exhibit the same azeotropic behavior. The limitations on distillation processes can, however, be overcome by using a membrane unit.


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8.1 INTRODUCTION Distillation remains the most widely used of all separation processes, in spite of its relative inefficiencies and high energy consumption. However, a number of alternative separation techniques have been explored, each with its own unique characteristics. Any single type of separation process is subject to limitations that could be overcome with the use of hybrid designs. For example, a selective membrane, such as PERVAP 1137 (manufactured by Sulzer Chemtech—refer to Section 2.1.2), can obtain only a single pure component—namely, methanol. Distillation processes, on the other hand, are often restricted by the existence of azeotropes. Another difference between these two processes is that equimolar overflow, which is often assumed in distillation, results in constant flow rates down a distillation column, whereas in a membrane unit, the retentate flow is depleted as it progresses down the column, resulting in a diminishing local reflux ratio. Even in those cases where equimolar overflow is not a valid assumption, the change in the flow rates in distillation is relatively small in comparison to that which occurs in membrane separation, where any portion of the retentate flow may be permeated. This results in the two process types exhibiting different column profile behaviors. In seeking the most efficient process possible, a designer will wish to explore a wide range of feasible designs. To make this possible, an efficient method for the synthesis and assessment of any hybrid separation process has been developed. Since both processes have been analyzed using similar mathematical backgrounds, it is possible to use residue curve maps for both distillation (D-RCMs) and membranes (M-RCMs) to design hybrid systems of the two. This chapter will look at hybrid processes featuring distillation and membrane units exclusively. Even when limited to these two types of separation techniques, a wide variety of configurations is available. So one should take note that the configuration selected and analyzed is by no means an optimal process, and has merely been used to illustrate the methods. A rigorous design should consider several configurations and seek to optimize the design by taking into account considerations such as capital cost, running costs, and environmental impact. The flexibility of hybrid configurations allows a designer greater freedom in optimizing an overall process, and provides the option of overcoming deficiencies in any single process. Analysis of the properties and behavior of the D-RCM in conjunction with that of the M-RCM equips one with the necessary tools to graphically configure a hybrid process. To facilitate the construction of a hybrid design, it is important to sequence the separation units such that each unit operates in the region of the composition space where its separation ability is best suited and most effective.

METHANOL/BUTENE/MTBE SYSTEM

153

8.2 METHANOL/BUTENE/MTBE SYSTEM With the advent of M-RCMs, a novel approach to designing and synthesizing a hybrid distillation–membrane process has emerged. This section provides a brief yet explanatory overview of this innovative method. This will be demonstrated through the use of an example. Methyl tertiary butyl ether (MTBE) is a fuel additive. Its demand has increased due to an environmental drive to phase out lead-compound emissions in fuels. MTBE is produced industrially in the conventional H€uls process (Bausa and Marquardt, 2000), as shown in Figure 8.1. It is formed by the reaction between methanol and i-butene: i-butene arises from a cracking plant and arrives at the reactor as part of a C4 hydrocarbon stream. For simplicity, it is assumed that the C4 stream contains only i-butene and 1-butene. The reaction is equilibrium limited, and in order to obtain high conversions of i-butene, methanol is fed in excess (Rehfinger and Hoffmann, 1990). It is further assumed that the ratio of methanol to i-butene is such that all the i-butene is converted. The reactor product therefore consists of recently formed MTBE, unconverted methanol, and 1-butene (from now on referred to simply as butene). The reactor product needs to be efficiently separated. Figure 8.2(a) shows the D-RCM for the system at 6 bar. A two-column sequence is used, with the first column operating at a pressure of 6 bar. The distillate composition is nearly that of the butene/methanol azeotrope. This stream is sent back to the cracking plant from where the C4 stream originated. The bottoms stream is sent to a second column where pure MTBE is produced, as well as an

Figure 8.1. H€uls process for the production of MTBE. The separation is achieved by a twocolumn pressure-swing distillation circuit.

154


Figure 8.2. Methanol/butene/MTBE system: (a) D-RCM at P ¼ 6 bar and (b) M-RCM at pP ¼ 0. (Peters et al., 2008).

azeotropic mixture of methanol and MTBE. To avoid too much MTBE reporting to the azeotropic mixture, the pressure in the second column is increased to 12 bar. The MTBE is the required product, while the methanol (containing some MTBE) is recycled back to the reactor. The need for compression to 12 bar requires that substantial work be put into the process. Furthermore, the mixture has to be heated to saturation temperature at 12 bar, resulting in increased heat duty on the reboiler. Thus, the overall process is unnecessarily energy intensive, warranting the consideration of alternative processes. This chapter focuses on the use of a distillation–membrane hybrid as an alternative to the H€ uls process. Such an alternative is not something new but

METHANOL/BUTENE/MTBE SYSTEM

155

has been researched before (Hommerich and Rautenbach, 1998; Bausa and Marquardt, 2000; Lu et al., 2002). What will be proposed in this chapter, however, is a novel graphical method of designing such systems. Lu et al. (2002) designed a hybrid process using rigorous mathematical calculations, but their method does not enable them to examine other configurations. Hommerich and Rautenbach (1998) examined the feasibility of various process configurations. They designed each of these using ASPEN PLUSÒ with a FORTRAN subroutine for the membrane unit. Bausa and Marquardt (2000) utilized shortcut methods for the design of both the distillation and membrane units. This enables them to design the hybrid process graphically. However, the reflux and/or reboil ratios, as well as the number of theoretical trays needed, are not so easily accessible. They do furnish the reader with a method to obtain the minimum membrane area required. Various membranes have been developed for the separation of methanol/ butene/MTBE (Hommerich and Rautenbach, 1998; Bausa and Marquardt, 2000; Bangxiao et al. 2001; Lu et al., 2002). The chosen membrane used in this chapter has been discussed in Section 2.1.2, with its associated M-RCM shown in Section 4.5. This membrane cannot be employed as a stand-alone unit to produce pure MTBE. Using that particular membrane, it is possible to obtain only a pure methanol stream (refer to Figure 8.2(b)). As shown in Chapter 4 (Peters et al., 2006a), each of the curves in the M-RCM (pP ¼ 0) terminates when it reaches the MTBE/butene binary since it was shown that the flux of each component goes to zero at that point. Therefore, pure MTBE or butene cannot be obtained from the membrane, unless the curve followed is either the binary MTBE/methanol boundary or the butene/methanol boundary, which is very unlikely. For this reason, the membrane needs to be used in conjunction with either a different membrane (capable of removing MTBE from the mixture) or with a distillation column. 8.2.1 Design Requirements The required products from the hybrid separation are: . . .

An essentially pure MTBE stream. A pure methanol stream, which will be recycled back to the reactor. A butene-rich stream, which will be returned to the plant from which it originated.

Table 8.1 shows the compositions of the feed and product streams. The distillation residue curve map (D-RCM) for the methanol/butene/ MTBE system is shown in Figure 8.2(a). The D-RCM reveals the presence

156


Table 8.1. Compositions of the Feed and Product Streams Stream Feed (F) Distillate (D) Permeate (P) Bottoms (B)

Composition [Methanol, Butene, MTBE] [0.45, 0.09, 0.46] [0.07, 0.925, 0.005] [1,0, 0] [0.001, 107, 0.998999]

of two binary azeotropes, which create a distillation boundary that is difficult to cross. Analysis of the properties and behavior of the D-RCM in conjunction with those of the M-RCM equips one with the necessary tools to graphically configure a hybrid process. The different topologies of these two maps give a designer greater freedom in selecting column profiles, in order to achieve a desired product composition, allowing each unit to operate in the region of the composition space where its separation ability is best suited and most effective. Numerous hybrid configurations can be investigated, such as membrane placement either upstream or downstream from the column. One could also side-connect the membrane to the column in various ways, thereby creating a recycle with the column. In this chapter, two particular configurations will be examined, so as to display the application the M-RCM has in hybrid processes: . .

In the first, an example of how one can synthesize a configuration is discussed. In the second, a system is chosen and the graphical details of the design are given.

8.3 SYNTHESIS OF A HYBRID CONFIGURATION When synthesizing a hybrid process, the configuration is not fixed but is rather depicted by the objectives of the process—that is, producing the required product streams. In D-RCMs, operating leaves (Castillo et al., 1998) are defined as the region of operation required to obtain a fixed composition. An operating leaf is found by generating column profiles from a chosen fixed point—this point would be xD in the case of a rectifying section, and xB for a stripping section. These profiles are plotted with the DPE (see Equation (6.3)), with the value of XD being that of the fixed (product) composition. Profiles are drawn at all possible

SYNTHESIS OF A HYBRID CONFIGURATION

157

refluxes, commencing at infinite reflux (the residue curve through XD) and varying rD, until it tends to zero. Each profile is allowed to run from XD to its respective pinch point, thus forming the pinch point locus (for the stable node). The region enveloped by the residue curve (through XD) and the pinch point locus is known as the operating leaf. An overlap of operating leaves for adjacent CSs indicates feasibility in a design. For the system at hand, MTBE has the highest boiling point and would traditionally be expected to be the bottoms product of a distillation column, irrespective of membrane placement. A stripping section operation leaf (Castillo et al., 1998) can be generated, as shown in Figure 8.3(a). In much the same manner, a rectifying operating leaf can be plotted using the lowest boiling composition on the D-RCM, namely, the methanol/butene binary azeotrope (refer to Figure 8.3(b)). Both operating leaves have been superimposed in Figure 8.3(c). The stripping section forms a closed operating leaf, implying that all stripping profiles (at varying reflux ratios) move in the same general direction, pinching along the same pinch point locus, as can be seen in Figure 8.3(a). The rectifying leaf, on the other hand, exhibits a bifurcation, resulting in an “open” leaf. This occurs at a reflux of about 0.31. Profiles with a reflux less than 0.31 move in a general direction that is different from those generated with a reflux greater than 0.31, although commencing from the same point (xD). There is, however, an overlap of the two leaves (for reflux values greater than 0.31 in the rectifying section), indicating that a standard single feed, two-product column is feasible. For the column to be practical, however, material balance dictates that the feed composition (to the column) is to lie on the straight line connecting xD and xB. The overall feed (to the separation system) does not obey this requirement. The composition of the overall feed would probably be located as shown in Figure 8.3(c) (refer to point xF). A nonreflux membrane unit can be used prior to the distillation column so as to move the overall feed to a point on the straight line connecting xD and xB. The membrane is capable of depleting the feed of most of its methanol and is therefore best suited here. Following the curve on the M-RCM, which passes through the overall feed composition, until it intersects with the distillation column mass balance line, will map out the compositional change of the retentate. Figure 8.3(c) shows this path. Figure 8.3(d) shows the synthesized process configuration. Note how the membrane is able to cross the distillation boundary since it is not a VLE-based separation. Once the distillation feed (flow rate and composition) is obtained, the design of the column proceeds using “traditional” methods. The actual choice of reboil/reflux ratio(s) depends on an optimization process, which is outside the scope of this book.

158


Figure 8.3. (a) Stripping section operation leaf, (b) rectifying section operating leaf, (c) operating leaves superimposed, and (d) membrane profile for the process configuration. (Peters et al., 2008).

DESIGN OF A HYBRID CONFIGURATION

159

Figure 8.4. (a) A possible hybrid configuration and (b) the column sections for the hybrid system. (Peters et al., 2008).

8.4 DESIGN OF A HYBRID CONFIGURATION Consider the hybrid configuration shown in Figure 8.4(a). This particular process has been proposed to perform the required separation. Using DRCMs and M-RCMs, the feasibility of this design can be verified. Since such a design would probably entail modifying an existing plant, rather than building a new one, the operating conditions of the distillation column will be kept at its original design values. For this reason, the pressure in the column will be assumed to be 6 bar (Bausa and Marquardt, 2000). Note that this is just one of many possible configurations and is not necessarily an optimal process. It is arbitrarily chosen and is used for demonstration purposes. 8.4.1 Column Sections of Hybrid Configuration The design in Figure 8.4(a) can be divided into column sections (CSs) (Tapp et al., 2004), as shown in Figure 8.4(b). Each of the distillation column sections (DCS1–3) has its own unique profile described by the difference point equation (DPE) (the constant molar overflow assumption is employed): dx 1 1 þ 1 ½xyðxÞ þ ½XD x ¼ dn rD rD

ð8:1Þ

160


with XD ¼

VyT LxT D

ð8:2aÞ

L D

ð8:2bÞ

rD ¼

D ¼ VL 6¼ 0

ð8:2cÞ

where x is the liquid composition, n is the number of theoretical stages, y is the vapor in equilibrium with x, XD is the difference point, V is the molar vapor flow rate in a CS (mol/s), L is the molar liquid flow rate in a CS (mol/s), rD is the reflux ratio, D is the net molar flow in a CS (mol/s), D 6¼ 0, and xT and yT are the molar liquid and vapor compositions at the top of a CS. Essentially, the DPE describes how the liquid composition changes down the length of a CS. The profile for each CS depends on the XD and rD values for that particular section. With these values set, a column profile map (CPM) can be generated in much the same way as a D-RCM or M-RCM was plotted (refer to Section 6.2.3). A CPM is defined as the “linear transform of the (D-)RCM” (Tapp et al., 2004). Holland et al. (2004a) show that any distillation configuration, no matter how complex, can be modeled using CPMs. The example shown here is an extension of the work put forward by Holland et al. (2004a). While they focus on distillation only, we include the addition of a membrane unit connected to column sections. When designing a hybrid system, such as the one displayed in Figure 8.4, one would like to have available all possible concatenations for a chosen configuration. This would then allow for an easy screening process during optimization. Regardless of the configuration of the hybrid process, the objective is to meet the separation requirements. By specifying the feed and product compositions, the product flows can be determined for a set feed flow rate. It is assumed that the feed rate is 100 mol/s, and that the feed and product compositions are as set out in Table 8.1.


161

8.4.2 Degrees of Freedom In the arrangement chosen (refer to Figure 8.4), one can concatenate the various sections in numerous ways. A “top–down” method is utilized here but can easily be adapted for other methods, that is, “bottom–up.” Beginning with the top section (CS1) the net flow (D1) and difference point (XD1) can be calculated. The reflux ratio (rD1), on the contrary, is a choice of the designer—it is the first of three degrees of freedom in this design problem. It does, however, have constraints that have to be met; these will be discussed later in this section. The liquid (L1) and vapor (V1) flows can also be determined. All these values are constant throughout CS1 under the previously mentioned assumption of constant molar overflow for distillation sections. Performing the necessary material balances around the feed (F), as well as the membrane permeate (P), enables one to define all the process variables for Sections 8.2 and 8.3 in terms of rD1. The feed is assumed to be all vapor. All these values are summarized in Table 8.2. It can be seen that once an rD1 value is set, all the values in Table 8.2 are known. However, setting rD1 alone does not fully specify the entire hybrid system. Another degree of freedom is the amount of material that enters the membrane unit from the end of CS1. Only liquid is drawn off since a pervaporation membrane is incorporated. The split ratio S ð8:3Þ sr ¼ L1 is defined as the fraction of liquid from the end of Section 1 that reports to the membrane module. Although the range of sr values lies between 0 and 1, the design is feasible only over a narrower, more restricted range. This is to ensure that there is a sufficient amount of methanol being drawn off from the end of Section 8.1. Thus, it can be shown that P xP ð8:4aÞ srmin ¼ D rD1 xS srmax ¼ 1

ð8:4bÞ

Table 8.2. Design Variables in Hybrid Configurationa Section

D

XD

rD

L

V

1

D

xD

rD1

D rD1

D ðrD1 þ 1Þ

2

PþD

ðP xP þ D xD Þ PþD

D rD1 P

D ðrD1 þ 1Þ

3

B

xB

ðD rD1 PÞ PþD ðD rD1 PÞ B

D rD1 P

D ðrD1 þ 1ÞF

a

Refer to Figure 8.4. rD1 is a degree of freedom.

162


where xS is the composition of the side-draw stream. This is the same as the liquid composition leaving CS1. When operating at srmin, the amount of methanol that enters the membrane exactly matches the amount being drawn off in the permeate stream. (Note: 0 < srmin < 1.) Operating at values smaller than srmin creates an infeasible retentate composition; srmax, on the other hand, allows all the liquid leaving CS1 to be fed into the membrane module. Now, xS is a function of rD1 and of n1 (the number of stages in CS1), among others. As rD1 changes, so the profile trajectory for CS1 is altered. As n1 changes, so the termination point of CS1 moves. It can be seen that srmin is not a constant for the system, but rather depends on these factors. Another important composition is that of the liquid entering CS2: x2top ¼

D rD1 xS P xP D rD1 P

ð8:5Þ

This is calculated by mixing the retentate stream with the remaining liquid from CS1 that did not go to the membrane. Note that x2top does not depend on sr. This implies that once rD1 and n1 are set, the location of x2top is fixed. Note that 0 < x2top < 1, thereby restraining rD1 to set ranges. For this to hold true, it can be shown that rD1 >

P xP ; or D xS

rD1 < 0

ð8:6aÞ

ð8:6bÞ

This is diagrammatically represented in Figure 8.5. Inequality (Equation (8.6b)) is impossible since Section 1 operates in rectifying mode, and rD1 has to be positive by definition (Holland et al., 2004a). Therefore, rD1 has to adhere to the constraint (Equation (8.6a)) only. This would be the minimum value that rD1 can have and would necessitate x2top(methanol) ¼ 0. rD1,min is not set for the system but does depend on n1, as discussed previously for srmin. Therefore, this design has three degrees of freedom, namely: . . .

rD1 n1 sr

each having to operate within its respective bounds.


163

Figure 8.5. Feasibility ranges for rD1 (0 < x2top < 1). (Peters et al., 2008).

8.4.3 Generating Profiles for Hybrid Columns For demonstration purposes, it will initially be assumed that operation will occur at sr ¼ 0.8. In order to display the profiles obtainable for this design, rD1 and n1 values were arbitrarily chosen. Furthermore, for the design to be feasible, it is required that the profiles of each section, including the membrane, intersect in the same order and direction as that of liquid flow in the system. Figure 8.6 shows the column and membrane profiles for operation at rD1 ¼ 40 with n1 ¼ 5.

Figure 8.6. Column and membrane profiles for the hybrid design in Figure 8.4 with rD3 ¼ 40 and n1 ¼ 5. (Peters et al., 2008).

164


Table 8.3. Design Parameters for the Different Cases Displayed in Figure 8.7 Parameter n1 n2 n3 A(m2)

Option (a)

Option (b)

5 0.6 4.8 4746.6

5 5.7 3.3 4746.6

Since the feed is a vapor, an intersection of the (liquid) profiles for CS2 and CS3 signifies the end of CS2 and the beginning of CS3. This is because the composition of the liquid leaving CS2 will not be altered by the addition of a vapor feed. It is clear in Figure 8.6 that the profiles for CS2 and CS3 intersect twice (at xmethanol ¼ 0.1 and 0.16). Therefore, the termination of CS2 can occur at either of these points. Figures 8.7(a) and 8.7(b) show the possible concatenations of the sections in the hybrid. 8.4.4 Comparing Feasible Design Options To compare these two options, one would like to have the necessary design parameters easily accessible. With this novel method of design, values such as the number of stages in each CS and the membrane area can be determined with relative ease. This has been done for the two options proposed above, and the results have been tabulated in Table 8.3. The numbers of stages were calculated from the DPE (Equation (8.1)) and are theoretical. The membrane area was sought via the methods described in Section 5.5.1, using the appropriate flux model. (Note: Whereas it may appear that the membrane area is somewhat large, it is not the aim here to calculate an exact area, but rather to convey an understanding of the simple yet effective methods used.) 8.4.5 Attainable Region Figure 8.6 shows only one possible profile arrangement for the chosen hybrid design. However, there are numerous ways in which one can configure this design based on the degrees of freedom discussed above. A parametric study has been performed that allows for design flexibility. It is therefore possible to generate “attainable regions” (ARs) that integrate the degrees of freedom. It should, however, be appreciated that the ARs generated here are not the entire AR, but rather a select part of it. The complete AR can be produced but is detailed and is not done here as it is outside the scope of this chapter. For the select ARs, it was assumed that the split ratio, sr, is set at its minimum value (refer to Equation (8.4a)). Remember, the split ratio influences only the amount of material that goes to the membrane (and hence the


165

Figure 8.7. Profile options (a) and (b) for the hybrid design in Figure 8.4 with rD3 ¼ 40 and n1 ¼ 5. (Peters et al., 2008).

membrane area), but does not affect the location of any of the compositions within the MBT, except xR. So, two types of ARs can be produced: varying rD1 for a set n1, and vice versa. Figure 8.8 shows the AR for a variable rD1 and a set n1 ¼ 4. As discussed previously, rD1 has a minimum value that depends on xS (refer to Equation (8.6a)). It can be shown that, for this case, rD1,min ¼ 41.77. Although not apparent from Figure 8.8, the profile for CS1 is changing as rD1 changes. The various CS1 profiles have similar trajectories, making it impossible to distinguish on the given scale. Furthermore, there is definite intersection between the profiles for CS2 and CS3 for each rD1. We can

Figure 8.8. Attainable region for n1 ¼ 4, operating at rmin. Select rD1 values between 41.77 and 250 have been chosen. (Peters et al., 2008).

166


conclude that, for a set n1, the design is feasible provided that rD1 rD1,min. Similar plots would be obtained for other values of n1. Figure 8.9(a), on the other hand, shows the AR for a set value of rD1 ¼ 70. The number of stages in CS1 was varied from 2 to 7 in various increments. A high rD1 value was chosen to ensure that it is well above the minimum required for each n1. For clarity purposes, Figures 8.9(b) and 8.9(c) show zoomed-in versions of Figure 8.9(a), as indicated by their axes. For a set rD1, there is a set rD2 and rD3 (refer to Table 8.1). This in turn fixes the profiles for CS1 and CS3, as well as the CPM for CS2. Hence, there is only one possible profile for CS1 and for CS3, as shown. The profile for CS2 depends on x2top, which changes with each n1 (or xS). But it should be appreciated that all the profiles obtained for CS2 belong to the same CPM since both XD2 and rD2 are constant. For each chosen n1, the membrane profile intersects with that of CS1, as does CS2’s profile with the membrane profile. However, it can be seen in

Figure 8.9. (a) Attainable region for rD1 ¼ 70, operating at rmin. 2 < n1 < 7; (b) minimum n1, and (c) maximum n1 tangent to CS3 profile. (Peters et al., 2008).

CONCLUSION

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Figure 8.9(b) that the profiles for CS2 and CS3 do not intersect for all n1. For example, when n1 ¼ 2, there is no intersection, indicating an infeasible design. It has been determined that the first point of intersection occurs when n1 ¼ 2.63. This is the minimum number of theoretical stages that CS1 has to have for rD1 ¼ 70 in order for a feasible design. Note further, in Figure 8.9(c), that some profile 2 curves again do not intersect with profile 3 (for n1 ¼ 6.75 and n1 ¼ 7). This means that there is a maximum number of stages that CS1 can have as well. This maximum occurs at n1 ¼ 6.52, where profile 2 is tangent to profile 3. While it may be thought that adding an excess number of stages in a column is beneficial, this result is very interesting as it proves that adding too many stages in CS1 may result in an infeasible design. Of course, the maximum and minimum n1 values depend on rD1, but can easily be sought for each rD1 value. Similar plots to the one in Figure 8.9(a) can be generated for different rD1 values. 8.5 CONCLUSION A novel approach in synthesizing and designing hybrid distillation– membrane processes has emerged: using the M-RCM in conjunction with column profile maps (CPMs) allows one to graphically interpret hybrids in an efficient manner. With the development of M-RCMs, not only is it possible to understand the fundamentals of membrane separations more easily, but it is now possible to design such hybrid systems using a novel graphical technique. Furthermore, by focusing on the membrane in isolation, one is not limited to a single hybrid configuration, making the possibilities infinite. By looking at the M-RCM independently from the D-RCM, one can synthesize all possible combinations and indeed try to find whether to use one or the other or combinations of both. This method of synthesis has proved to be effective, as demonstrated with the methanol/butene/MTBE example. It has also been shown how the M-RCM, in conjunction with the appropriate distillation profiles, creates a method of design that can be used to quickly evaluate feasible options for a particular design. Using the graphical method not only makes the design process easier to visualize and understand, it also provides the designer with useful and necessary design parameters. The operating ranges of the reflux ratios result, as do the number of stages in each section. Using earlier discussed techniques, the membrane area can also be determined in each scenario. The method can be adapted accordingly for other process configurations, with the ultimate design chosen by a screening or optimization procedure.

CHAPTER 9

CONCLUDING REMARKS

The work contained in this book has, by introducing an innovative method, contributed to analyzing, designing, and synthesizing general separation processes, with emphasis on membrane separation. It is believed that an extremely useful and versatile technique has been developed. The method is not restricted to a single type of membrane but, with the appropriate flux equations, can easily be adapted for any membrane. Furthermore, by use of appropriate composition modeling, any separation process can be analyzed, since the techniques were devised for a general separation process. In particular, the methods developed enable one to plot column profiles for systems in which flow rates are not constant. Thus, the methods developed here offer a tool for the analysis of distillation processes for which constant molar overflow is not a valid assumption. A significant result of this work is that it has shown that high-purity separations are achievable using nonselective, Knudsen-type membranes, demonstrating that the development and manufacture of highly selective membranes is often not necessary. By offering a design and synthesis technique applicable to any particular membrane, this work opens the door for the implementation of cost-effective and robust membrane operations throughout the chemical industry.


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9.1 CONCLUSIONS Although the M-RCM was developed for a batch membrane process, it has been revealed how it is directly applicable to a wide range of continuous processes. A number of important results emerged from this, each contributing to the separations field in its own right. .

.

. .

The M-RCM can be used to explain and understand and graphically visualize the behavior of the conventional industrial membrane permeators (see Chapters 3 and 4). The same applies for total and infinite reflux membrane columns. While these are not practical applications, the theoretical results they yield are useful when explaining finite reflux arrangements (see Section 5.5.2). The profiles in the M-RCM allow one to graphically visualize the change in the retentate in either a batch still or continuous unit (see Section 3.3). Furthermore, the M-RCM also provides some necessary design parameters such as what membrane area is needed to achieve a desired separation in a continuous process, or the time taken in a batch process (see Sections 5.5.1 and 5.4).

Complex membrane columns and cascade arrangements have been researched (e.g., Hwang and Ghalchi, 1982; Kothe et al., 1989), but a shortcut method for design and synthesis for these was lacking. In this book, it has been shown how, with the identification of a membrane column section, a novel design tool is developed. Any arrangement, no matter how complex, can be broken down into column sections. These are the basic building blocks that make up any arrangement. An understanding of the behavior of each column section allows one to synthesize and concatenate sections into a configuration that achieves the desired result. The compositional change of the retentate in any column section has been modeled mathematically using the difference point equation. All possible flow cases within a column section were identified, and their corresponding profiles investigated. Some of the important results are listed: . .

.

The DPE for a column section operating at total reflux reduces to the M-RCM equation, as expected (see Section 6.4). The behavior of a column profile, operating at finite conditions, is influenced by the movement of the stable node pinch point locus (see Sections 6.5 and 6.6). A profile will continue to exist until either the retentate or permeate runs out of material. When this occurs, the profile pinches, and the column section terminates, normally in a finite area (see Section 6.6).

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Using these results, thermally linked membrane columns were examined. It was found that such an arrangement is feasible, provided that the top composition was within a region bounded by the MBT and the appropriate pinch point curve. Furthermore, there exists a minimum reflux for each composition in that region. Operation would have to occur at or above the minimum reflux (see Sections 7.4 and 7.5). One of the most important and exciting results of the work in this book was the research displayed on hybrid processes. By analyzing membrane permeation from the same mathematical viewpoint as distillation, combinations of the two processes can be performed with ease. Also, since graphical techniques were developed for membrane systems as individual processes, any configuration and design of distillation and membrane units is possible. Analyzing them simultaneously (such as was done by Huang et al. (2004)) has a shortcoming in that it limits one to a single design. However, decoupling the two unit operations from each other allows all possible configurations to emerge, thus carrying out synthesis rather than design. This was evident from the results obtained for the methanol/butene/MTBE system—an attainable region for the process was determined (Chapter 8). The technique also took on a different approach by collectively treating a hybrid arrangement as a thermally coupled arrangement between distillation and membrane units. With the use of column sections, general rules for classifying regions of feasible arrangements were developed, as was done in Chapter 7, Section 7.7.

9.2 RECOMMENDATIONS AND FUTURE WORK It should be appreciated that the work presented is not only applicable to membrane permeation and distillation, but can easily be adapted for any separation method. Both constant and nonconstant flow assumptions have been tackled here, equipping one to use the methods displayed for the separation procedure of their choice. Of course, details of equilibrium/flux model, flow assumptions and directions, and so on need to be decided upon. But the method and results arrived at here would still apply. This could be the basis of further research in the separations field. Experimental validation of the profiles in the M-RCM is another possibility that could be explored. This could then be extended to formulating a batch- or continuous-scale experiment to mimic the finite reflux membrane column section. Further theoretical and experimental work can be conducted with regard to hybrid designs and the use of column profiles to design such processes. Ongoing work of this nature is taking place at the Centre of Material and

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Process Synthesis (COMPS) at the University of the Witwatersrand, extending the theoretical understanding of membrane and hybrid separation processes, as well as developing practical applications for these technologies.

9.3 DESIGN CONSIDERATIONS The techniques developed throughout this book equip the reader with tools for the synthesis, analysis, and conceptual design of membrane and hybrid separation processes, but by no means do they constitute a rigorous and complete design process. This section serves to bring to the reader’s attention a number of design considerations and challenges involved in membrane and hybrid separation process design. The reader is advised to take note that this is not a checklist of design considerations and is not intended to be used as a design guide. Rather, it is intended to stimulate thought and discussion by putting forward ideas and identifying issues that have been anticipated by the authors. Furthermore, the book considers membrane processes as part of a separation sequence, with specified inputs and required outputs. However, like any other unit operation, membrane separators need to be considered as part of an overall process flow sheet. In this section, the authors undertake to discuss their views on how a membrane separation process can best be incorporated into an overall flow sheet, and how an overall flow sheet may be modified to accommodate a membrane operation. 9.3.1 Processes for Which Membrane Separations Are Particularly Suitable For cheap and robust nonselective membranes, the constant relative permeability model (Knudsen membrane)has beenfound to be a good approximation of actual behavior. The similarity of the constant permeability model for membrane separation to the constantrelativity modelfor distillation means that, in principle, any separation that is achievable through distillation should also be achievable through a membrane separation using a suitable membrane. However, in most cases where either process is applicable, some degree ofoptimization and costing is required before one can definitively decide on a process. Nevertheless, there exist certain circumstances in which a membrane process will clearly be more suitable. Several such cases are discussed here, but numerous others may present themselves as membrane separation technology matures. Azeotropic Systems The existence of azeotropes can make distillation highly problematic. Since M-RCMs possess different topology than D-RCMs, they will not feature the

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same separation boundaries. Thus, distillation boundaries can be bypassed by the use of membrane separation processes. The example discussed in Chapter 8 was one such case and can be considered a good illustration of the effectiveness of the hybrid process design techniques developed in this book. Gaseous Systems Not at Saturation Distillation must be carried out at a temperature and pressure such that the mixture being separated is saturated. As a result, when dealing with gaseous systems, it is often necessary to either pressurize the gas or cool it to saturation before feeding it to the column. This can result in large energy losses from the process, as well as increased equipment costs for compression or refrigeration equipment. Furthermore, if a distillation column is required to operate under high-pressure conditions, or cryogenic conditions, the material costs of that column are increased. Finally, constraining streams to saturation conditions limits the degree of freedom of a designer when setting up the overall flow sheet. Gas-diffusion membranes are not constrained to operation at saturation and can be operated at any conditions such that the mixture is a gas. In many cases, this will make it possible to operate a membrane separator at conditions matching those of its feed stream, while use of a distillation column would have required either compression or refrigeration. Systems for Which a Suitable Selective Membrane Is Available A selective membrane can easily extract a single component in high purity. If the component that is selectively permeated is also the desired product, then using a selective membrane enables one to achieve a pure product with a single separation process. Use of a selective membrane can greatly simplify the design of a separation sequence. However, it must be noted that selective membranes are often expensive to develop and manufacture, and that it will often be possible to synthesize a process using a nonselective membrane that is able to reach product specifications. 9.3.2 Processes for Which Membrane Operations Are Unsuitable It is conceivable for processes to exist for which membrane separation will be unsuitable. Various reasons for unsuitability can be identified. Low Driving Force for Separation Membrane processes may be unable to perform an effective separation. In particular, this will be the case if the relative permeabilities for the system are very low. An initial idea of the comparative separabilities offered by

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distillation and membrane separation could be gained by comparing the magnitude of the relative volatilities in the distillation process to the relative permeabilities in the membrane separation process. Note that different membranes will offer different relative permeabilities, so various membrane specifications will have to be considered. High-Maintenance Operating Conditions If operating conditions are such that the membrane is likely to be damaged or clogged, then the cost of maintaining the membrane would become excessive, and alternative methods of separation should be considered. 9.3.3 Pressure Difference as a Design Consideration The pressure difference between the retentate and permeate sides results in an unbalanced force on the membrane surface. Therefore, the membrane’s structure must be sufficiently strong to withstand this force, or membrane rupture will occur. Pressure specifications will vary between individual membranes and will be provided by the manufacturer. A designer must be aware of the pressure constraints on their selected membrane and take them into consideration throughout the design process. Selective Membranes As shown in Section 4.5, selective membrane systems display arheotropes, points on the M-RCM where the rate of flux becomes zero. The locus of the arheotrope curve is affected by pressure, and it can be seen in Figure 4.4 that it is only possible to achieve complete permeation of the permeating component if the permeate pressure is zero. Thus, effective separation in a selective membrane requires very low permeate pressure. Nonselective Membranes Although nonselective membranes rarely exhibit arheotropes in the manner of selective membranes, pressure remains a major consideration in achieving effective separations. Section 3.1 examined how the ratio of pressures affects the ease of separation in a binary system. It was shown that a high pressure ratio is desirable for effective separation (refer to Figure 3.3). An additional consideration is the rate of permeation. Since the driving force for permeation of a particular component is the difference in partial pressures of that component, it is an intuitive result that increased pressure differences will result in more rapid permeation. So, larger pressure differences will result in smaller membrane areas, while larger pressure ratios will result in more effective separation. It is for these reasons that it is considered desirable to operate a membrane with a permeate vacuum. However, if the permeate is a

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vacuum, then the retentate pressure is restricted by the allowable pressure difference. This means that, in high-pressure systems, it may be necessary to reduce the pressure of the feed stream to the retentate. This would result in energy losses if it is necessary to recompress the retentate for use downstream. The alternative—operating with a nonvacuum permeate pressure—affects the ease of separation. Thus, a designer of a membrane process must take several factors into consideration when deciding on the pressure at which to operate a membrane process. 9.3.4 Effect of Reflux in Membrane Columns Reflux in countercurrent membrane columns refers to the transfer of material either from the retenate to the permeate stream, at the bottom of the CS, or from the permeate to the retentate stream, at the top of the CS. Permeate reflux requires compression to match the pressure of the retentate stream, whereas retentate reflux must be passed through a valve or throttle before being introduced to the permeate stream. Permeate Reflux The analytical techniques developed over the course of the book make use of XD, D, and rDT as the design variables, and the resulting designs are given in terms of these variables. For a particular feed and product rate, once values for rDT and D have been chosen, the top flow rates, RT and PT, become fixed. Therefore, the permeate reflux ratio must be such that these flow rates are achieved. As a result, permeate reflux is fixed once D and rDT have been chosen. Retentate Reflux For a particular feed flow rate and a required product flow rate, the retentate product flow rate is equal to D for the bottom CS. Now, with a known RT for that CS, mass balance dictates that all material present in RT that is not discharged as product must be transferred to the permeate stream. Material can be transferred from retentate to permeate either by permeation or by way of the recycle. Since a particular quantity of material must be transferred in order to reach the flow rates specified by the mass balance, the sum of the reflux rate and the permeation rate is established. Increasing the reflux therefore reduces the total rate of permeation, therefore reducing the degree of separation and thus affecting product composition, but potentially decreasing the membrane area that is required to meet flow rate requirements. Retentate reflux rate therefore may offer benefits in terms of the capital cost of the membrane, which must be balanced against the effects on composition.

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Note: These conclusions regarding reflux are the result of the “top–down” approach to design. If a “bottom–up” approach is taken, it is the retentate reflux that becomes fixed while the permeate reflux becomes a variable. Furthermore, these conclusions result from a design approach in which the feed and product flow rates are specified before the membrane itself is designed. Also, as was mentioned earlier, the designs developed in this book make use of the assumption of vacuum permeate, which means that the composition of the permeate is neglected when determining the driving force for permeation. If permeate vacuum is not assumed, then retentate reflux may be beneficial in terms of these driving forces. The reader is referred to the work of Tsuru and Hwang (1995), which examines the effects of retentate reflux in detail.

9.4 CHALLENGES FOR MEMBRANE PROCESS ENGINEERING It is hoped that this work, in addition to providing analysis tools for membrane and hybrid processes, has made apparent the potential of these separation processes to provide versatile and efficient solutions to meet separation requirements. However, membrane technology has yet to achieve extensive proliferation in the chemical industry, in spite of its benefits. It is therefore the task of engineers in the field of membrane research to consider the challenges that must be overcome in order for membrane processes to play a full role in the chemical industry. In chemical engineering at present, the modification of existing processes is of tremendous interest, perhaps even of importance comparable to that of the development of new processes. Therefore, a new technology is of greatest use if it can readily be incorporated into an existing process. The example displayed in Chapter 8 achieves this goal admirably, integrating a membrane separation with an existing distillation process. Furthermore, the design method developed here is able to produce a design of a membrane with specified product and feed rates and compositions, making it suitable for retrofitting existing processes in which these variables are fixed.

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Appendix A

MemWorX USER MANUAL

The MemWorX application is intended as a learning tool enabling users to plot the various types of curves displayed throughout the book. Making use of simple thermodynamics and flux modeling, as discussed in the book, MemWorX is a learning aid rather than a rigorous design tool. It has been developed with the intention of allowing the user to achieve an intuitive understanding of residue curve maps, column profile maps, and pinch point loci, and how various parameters influence the behavior of these plots and, ultimately, real separation processes. At various stages within the textbook, the reader is prompted to make use of MemWorX to reproduce and modify the plots displayed within the text, to aid understanding. Note that MemWorX is intended as a companion piece to the main text, to be used only for additional enrichment. MemWorX was programmed in MatlabÒ.

A.1 SYSTEM REQUIREMENTS Windows XP or newer.


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A.2 INSTALLATION To install MemWorX: 1. Insert CD-ROM into CD-ROM drive. 2. The Auto-run feature should bring up a screen prompting you to either install or exit. If the Auto-run does not work, then access the MemWorX CD-ROM and run the file “MemWorX_Install.exe.” 3. Choose installation destination (see Figure A.1) and click “Install.” 4. The installer will extract MemWorX and automatically install the Matlab Component Runtime 7.7 (Figure A.2). MCR 7.7 is required if the machine does not have Matlab (2006 or later) installed. 5. Once installation is complete, access the folder to which MemWorX was installed. Find the program title “MemWorX.exe” and run it to start the MemWorX application. MemWorX takes a few moments to start up the first time: Please be patient.

A.3 LAYOUT OF MemWorX The general layout of the MemWorX is shown in Figure A.3. The layout of MemWorX can be subdivided into the following regions (refer to Figure A.3):

Figure A.1. MemWorX setup.

LAYOUT OF MemWorX

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Figure A.2. Matlab Component Runtime 7.7.

A—Plot type selection. Here the type of separation curve is selected, as well as the thermo or flux model to be used and the operation type. B—Input values, graph options, and equation selection. C—Plot functions. D—Displays the chosen form of the DPE. The equation form is chosen in Section B. E—Plot area.

Figure A.3. General layout of GUI.

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Figure A.4. Full set of plots on same axes. (See insert for color representation).

A.4 APPEARANCE OF PLOTS Figure A.4 shows a set of pinch point loci, a moved triangle, a full column profile map, and discontinuity all plotted together on the same set of axes. A legend is provided, indicating the appearance of the various plots shown.

A.5 STEP-BY-STEP GUIDE TO PLOT USING MemWorX Step 1: Choose a Thermo or Flux Model There are two thermo models available: “Ideal” and “NRTL.” If Ideal is chosen, constant relative separabilities (a) will be assumed. If NRTL is chosen, the thermodynamic data for the methanol/butane/MTBE system will

STEP-BY-STEP GUIDE TO PLOT USING MemWorX

187

be loaded as discussed in Chapter 2, Section 2.1.2. For each of these thermodynamic models, there is a corresponding flux model for membrane operations. The model is chosen using the drop-down menu labeled “Select Model” in Section A of Figure A.3. Step 2: Choose a Flow Rate Assumption Two options are available: “Constant–Distillation” and “Nonconstant–Membrane.” As explained in Chapter 6, flow in a column used for separation can be subject to different assumptions about the flow rate. In the case of a distillation column, flow rate in both directions can be assumed to be constant along the length of the column. In membrane separation systems, however, flow rate changes along the length of the column as material permeates from the retentate stream to the permeate, resulting in differing column profiles. The flow rate assumption is chosen using the drop-down menu labeled “Select flow rate assumption” in Section A of Figure A.3. Step 3: Select a Plot Type Over the course of the book, a number of different curves were plotted in the composition space as synthesis methods were developed. All of these plot types are available in MemWorX. These curves were all introduced from Chapter 3 onwards. The available curves are: . . . . .

Full column profile map (Note: An RCM is a CPM at infinite reflux.) Single column profile Discontinuity Pinch point loci Moved triangle

Curve type is selected using the drop-down menu labeled “Select plot type” in Section A of Figure A.3. Step 4: Select Equation Form Section B (refer to Figure A.3) is where the user selects the form of the difference point equation, inputs values for use in calculations, and sets graph options. You can switch between these three types of input by using the tabs at the top of Section B. Clicking on a tab switches the set of inputs accordingly. To select the form of the difference point equation to be used, click on the “Select Equation” tab. Figures A.5(a) and A.5(b) show the “Select Equation” block for membrane separation and distillation, respectively.

188

APPENDIX A

MemWorX USER MANUAL

Select Equation Input Values Graph Options


dx = (x – y) dA

dx = (x – y) dn

dx = (x – y) + β(A)δ dA

dx = (x – y) + δ dn

1 1 dx (X – x) = (1 + )(x – y) + rΔ rΔ Δ dA

1 1 dx (X – x) = (1 + )(x – y) + rΔ rΔ Δ dn

Select equation form

Select equation form

(a)

(b)

Figure A.5. “Select Equation” tab for (a) membranes and (b) distillation.

Three different forms of the difference point equation are available: 1. The infinite reflux case, where the DPE reduces to the residue curve equation (see Section 6.4: Equation (6.28)). 2. The form of the equation that arises when PT ¼ RT (see Section 6.5: Equation (6.31)). 3. The full DPE (see Section 6.6: Equation (6.42)).

Table A.1. Allowable Input Combinations Thermo Model

Flow-Rate Assumption

Ideal

NRTL

Allowable Plots

Allowable Equation Types

Constant

Full CPM Single profile Discontinuity Pinch point loci Moved triangle

1,2,3 1,2,3 1,2,3 2,3 1,2,3

Nonconstant

Full CPM Single profile Discontinuity Pinch point loci

1 1,2,3 1,2,3 2,3

Constant

Full CPM Single profile

1,2,3 1,2,3

Nonconstant

Full CPM

1


Select Equation Input Values

189

Graph Options

α

[3,1,1.5]

δ

[0.1,-0.05]

XΔ

[0.3, 0.3]

RΔ

10

X0

[1.068,-0.548]

RT

100

Δ

20

Input Values

Figure A.6. “Input Values” tab.

Which curves are available depends on the thermo model and flow-rate assumption that were selected, and which equation types can be used depends on thermo model, flow-rate assumption, and plot type. Table A.1 below lists the possible combinations of plot type and equation form depending on thermo model and flow-rate assumption. The equation type selected is displayed in region D (refer to Figure A.3). Step 5: Input Values In Section B (see Figure A.3), click on the “Input Values” tab. Figure A.6 shows an example of what should be seen. Values specific to the system considered are entered using the “Input Values” tab. Exactly which parameters are required depends on the equation form selected, the model selection, and the flow-rate assumption. All unnecessary inputs are grayed out once these are chosen. The set of possible inputs are summarized in Table A.2.

190

APPENDIX A

MemWorX USER MANUAL

Table A.2. Variables in the “Input Values” Tab

Variable

Description

a

This is the vector of relative permeabilities (see Chapter 2). A three-element row vector of positive values must be entered here. The difference vector (see Section 6.5.4). A two-element row vector must be entered here. Net molar composition (see Section 6.2.1). A two-element row vector must be entered here. Local reflux ratio (see Section 6.6). A real number must be entered here. The starting retentate/liquid composition of the column profile. A twoelement row vector must be entered here. If the nonconstant flow assumption is used, this variable is referred to as xT. The retentate/liquid flow rate at the top of the column. Net molar flow rate (see Section 6.2.1)

d

XD

rD x0

RT D

Acceptable Input Type

Example of Acceptable Input

Three-element row vector of real numbers

[1,1.5,3]

Two-element row vector of real numbers

[0.6,0.15]

Two-element row vector of real numbers

[0.4,0.2]

Real number

3.3

Two-element row vector of real numbers. Can also be input by clicking on a point in the plot area, if cursor-input checkbox is checked. Positive real number

[0.4,0.4]

Real number

20

100

Step 6: Set Graph Options Figure A.7 shows the display when the “Graph Options” tab is selected. Values that will determine the manner in which the curves will be plotted can be entered in the “Graph Options” tab, found in Section B (see Figure A.3). The options available in this tab are given in Table A.3. Step 7: Plot Curves Figure A.8 shows the “Plot” section of the MemWorX display (Section C in Figure A.3). Clicking on the “Plot” button in the bottom left corner (see Figure A.8 and Section C in Figure A.3) plots curves according to the values entered and the


191


Profile plot options Curve Density

10

Integration time

10

Axis options Min

-1

Max

2

Pinch point plot options Reflux limit

100

Reflux step

1

Misc Options Cursor input (x-initial) Include vapor profile Plot within MBT only Include lables Graph Options

Figure A.7. “Graph Options” tab.

options selected. The “Clear” button clears the plotting axes. You can navigate the plot area by making use of the “Pan On” and “Zoom On” buttons. While zoom is enabled, left-clicking on a point in the plot area will zoom in on that point, letting you view it in greater detail. Hold shift and left-click to zoom out. Enabling the “Pan On” option lets you click and drag to navigate the plot area, to view different portions of the plot area.

Pan On

Plot

Zoom On Clear

Figure A.8. “Plot” section.

192

APPENDIX A

MemWorX USER MANUAL

Table A.3. “Graph Options” Variables and Checkboxes

Option

Description

Curve density

A measure of the number of column profiles plotted on the axes for a full CPM plot selection. The larger the number, the more curves will be plotted and the longer it will take to plot. The upper limit for the integration. The larger this number, the closer the curves will approach the pinch points. The lower limit of the x-axis. The upper limit of the x-axis. The maximum value for reflux, rD, for which the pinch point loci are plotted. The difference between one set of pinch points to the next set of pinch points. The smaller the step size, the smoother the pinch point loci. Checkbox to enable/disable the input of x0 or xt via clicking on the graph. Checkbox to enable/disable the plotting of the vapor/permeate curves corresponding to the residue/retentate curves. (Only available for “Single Profile” plot option.) Checkbox to enable/disable the plotting of the vapor/permeate curves corresponding to the residue/retentate curves. Includes labels on the plot for d, XD, x0 or xT.

Integration time

Min Max Reflux limit

Reflux step

Cursor input (x-initial) Include vapor/ permeate profile

Plot within MBT only

Include labels

Acceptable Input Type

Example of Acceptable Input

Positive real number

10

Real number (for membranes, positive only)

10

Real number Real number Real number

1 2 100

Positive real number

0.5

A.6 TUTORIAL SOLUTIONS The solutions to the tutorials involving the use of the MemWorX package are given in Table A.4.

193

Select Model

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

NRTL (methanol/butene/ MTBE)

Tutorial Number (Figure Number)

3.1 (Figure 3.4)

3.2 (Figure 3.5)

4.1 (Figure 4.2)

4.2 (Figure 4.3)

Nonconstant– membrane




Full CPM

Full CPM

Full CPM

Single profile

Select Flow Select Rate Assumption Plot Type

Drop-down Menus

Table A.4. MemWorX Tutorial Solutions

1

1

1

1

N/A

Integration time ¼ 10 Curve density ¼ 10

(continued )

Integration time ¼ 10 Users can try any Curve density ¼ 20 a-values they wish as Check “Plot within MBT long as the format is only” correct (refer to Table A.2).

(a) a ¼ [3,1.5,1] (b) a ¼ [3,3,1]

To obtain a full profile, do not clear axes between performing positive and negative integrations. Other xT values can be manually entered or using the “CursorInput” graph option.

Comments

Very high curve densities Integration time ¼ 10 require large Curve density ¼ 20 computation times, Check “Plot within MBT which are unnecessary. only”

(a) Integration time ¼ 10 (b) Integration time ¼ 10 (c) Integration time ¼ 10

Graph Options (Recommended)

a ¼ [3,1,1.5]

Same as (a) and (b)

(a) a ¼ [3,1,1.5] xT ¼ [0.4,0.3] (b) a ¼ [3,1,1.5] xT ¼ [0.4,0.1]

Select Input Equation Values

Tab Options

194

Select Model

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)


5.2 (Figure 5.3(a))

6.1 (Figure 6.5)

6.2 (Figures 6.6 and 6.7)

6.3a (Figure 6.8(a))

Table A.4 (Continued )

Constant– distillation




Moved triangle

(a) Full CPM (b) Moved triangle

(a) Full CPM (b) Discontinuity

Single profile


Drop-down Menus

3

3

1

1

Integration time ¼ 10 Curve density ¼ 10 Check “Include labels”

Integration time ¼ 10 Check “Include labels”

a ¼ [3,1,1.5] XD ¼ [0.3,0.3] (a) rD ¼ 3 (b) rD ¼ 6 (c) rD ¼ 12

Axis options: Min ¼ 1 Max ¼ 2

Integration time ¼ 10 Curve density ¼ 10

Check “Include permeate profile”

Integration time ¼ 10


a ¼ [3,1,1.5] rD ¼ 5 (a) XD ¼ [0.2,0.3] (b) XD ¼ [1.2,0.3] (c) XD ¼ [1.2,0.3] (d) XD ¼ [0.4,0.2]

a ¼ [3,1,1.5]

a ¼ [3,1,1.5] xT ¼ [0.4,0.3]


Tab Options

MemWorX is not yet capable of producing the accumulated permeate profile or operating regions. Other xT values can be manually entered or using the “CursorInput” graph option.

Comments

195

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

6.3b (Figure 6.8(b))

6.4 (Figure 6.9)

6.5 (Figure 6.10)

6.6 (Figure 6.14)

6.7 (Figure 6.15)

6.8 (Figure 6.16)




Constant– Distillation



(a) Full CPM (b) Single profile (c) Moved triangle

Single profile

(a) Full CPM (b) Discontinuity

Pinch point loci

Pinch point loci

Moved triangle

2

2

1

3

3

3

Reflux limit ¼ 100 Reflux step ¼ 1 Check “Include labels” Reflux limit ¼ 100 Reflux step ¼ 1 Check “Include labels”

Integration time ¼ 10 Curve density ¼ 10 Axis options: Min ¼ 1 Max ¼ 2 Integration time ¼ 7

a ¼ [3,1,1.5] XD ¼ [0.3,0.3]

a ¼ [3,1,1.5] (a) XD ¼ [0.4,0.5] (b) XD ¼ [0.1,1.2] (c) XD ¼ [1.2;0.5] (d) XD ¼ [1.2;0.1] (e) XD ¼ [0.3,0.3] (f) XD ¼ [0.3,0.3] a ¼ [3,1,1.5]

a ¼ [3,1,1.5] d ¼ [0.1,0.05] xT ¼ [0.3,0.3] RT ¼ 100 a ¼ [3,1,1.5] xT ¼ [0.3,0.3] (a) d ¼ [0.1,0.05] (b) d ¼ 1.667[0.1,0.05]

Check “Include labels”

Integration time ¼ 10 Curve density ¼ 10 Check “Include labels”



a ¼ [3,1,1.5] XD ¼ [0.3,0.3] (a) rD ¼ 3 (b) rD ¼ 6 (c) rD ¼ 12

(continued )

This plot takes some time to compute—please be patient.

196

Select Model

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)


6.9 (Figure 6.18)

6.10a (Figure 6.19)

6.10b (Figure 6.19)

6.11 (Figure 6.20)






Single profile

Single profile

Single profile

Pinch point loci


Drop-down Menus

3

2

2

2


a ¼ [3,1,1.5]

a ¼ [3,1,1.5] XD ¼ [0.3,0.3] xT ¼ [0.3,0.2] RT ¼ 120 D ¼ 20


d ¼ [0.1,0.05] Check “Include labels” RT ¼ 100 Multiple xT’s defined by “Cursor-Input” option.


Reflux step ¼ 1 Check “Include labels”

Reflux limit ¼ 100


a ¼ [3,1,1.5] d ¼ [0.1,0.05] xT ¼ [0.3,0.3] (a) RT ¼ 120 (b) RT ¼ 100 (c) RT ¼ 80 (d) RT ¼ 50 (e) RT ¼ 20

a ¼ [3,1,1.5] d ¼ [0.1,0.05]


Tab Options



Comments

197

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

6.12a (Figure 6.24)

6.12b (Figure 6.24)

7.1 (Figure 7.5)

7.3 (Figure 7.8)

7.3a (Figure 7.11(a))




Non-constant– Membrane


Single profile

(a) Pinch point loci (b) Single profile

Single profile

Single profile

Single profile

3

3

3

3

3 Check “Include labels”


a ¼ [3,1,1.5] XD ¼ [0.3,0.3] D ¼ 20 RT ¼ 120 Multiple xT’s defined by “cursor-input” option.

(continued )

(a) Reflux limit ¼ 100 Use the “Zoom” and Reflux step ¼ 1 “Pan” buttons (b) Integration time ¼ 10 Check “Include permeate profile” Check “Include labels” Integration time ¼ 50 This plot takes some time to compute—please be Check “Include permeate patient. profile” Use the “Zoom” and Check “Include labels” “Pan” buttons.

a ¼ [3,1,1.5] XD ¼ [0.3,0.3] xT ¼ [0.25,0.5] RT ¼ 100 D ¼ 20 a ¼ [3,1,1.5] XD ¼ [0.3,0.3] xT ¼ [0.45,0.4] RT ¼ 200 (a) D ¼ 20 (b) D ¼ 20

Integration time ¼ 10 To have profiles for both Check “Include permeate CSs on the same plot, profile” do not “clear” the screen between Check “Include successive plots. labels”

a ¼ [3,1,1.5] XD ¼ [0.3,0.3] xT ¼ [0.2,0.3] RT ¼ 100 (a) D ¼ 20 (b) D ¼ 20



a ¼ [3,1,1.5] XD ¼ [0.3,0.3] xT ¼ [0.3,0.2] D ¼ 20 (a) RT ¼ 200 (b) RT ¼ 100 (c) RT ¼ 40 (d) RT ¼ 20

198




Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

7.3c (Figure 7.11(c))

7.4 (Figure 7.12)

7.5a (Figure 7.13(a))


Ideal (constant a)

7.3b (Figure 7.11(b))

Single profile

Single profile

Single profile

Single profile


Select Model

Drop-down Menus



2

2

3

3

a ¼ [3,1,1.5] xT ¼ [0.3,0.3] (a) dT1 ¼ [0.04,0.08] (b) dT1 ¼ [0.04,0.08]

a ¼ [3,1,1.5] xT ¼ [0.3,0.3] RT ¼ 100 (a) dT1 ¼ [0.04,0.08] (b) dT2 ¼ [0.04,0.08]

Check “Include permeate profile” Check “Include labels”





This plot takes some time to compute—please be patient. Use the “Zoom” and “Pan” buttons.


a ¼ [3,1,1.5] XD ¼ [0.3,0.3] xT ¼ [0.45,0.4] RT ¼ 400 (c) D ¼ 20 (d) D ¼ 20


This plot takes some time to compute—please be Check “Include permeate patient. profile” Use the “Zoom” and Check “Include labels” “Pan” buttons.

Comments



a ¼ [3,1,1.5] XD ¼ [0.3,0.3] xT ¼ [0.45,0.4] RT ¼ 320.74 (a) D ¼ 20 (b) D ¼ 20


Tab Options

199

Ideal (constant a)

Ideal (constant a)

Ideal (constant a)

7.5b (Figure 7.13(b))

7.6 (Figure 7.16)

7.7 (Figure 7.22)

(b) Single profile

(a) Pinch point loci

Single profile

(a) Nonconstant– Single profile membrane (b) Constant– distillation



3

3

2

XD ¼ [0.3,0.3] xT ¼ [0.4,0.15] (a) Membrane: a; ¼ [3,1.5,1] RT ¼ 100 D ¼ 20 (b) Distillation: aD ¼ [3,1,1.5] rD ¼ 5

Multiple xT’s defined by “Cursor-Input” option.

Integration time ¼ 20 (a) Check “Include permeate profile” (b) Check “Include vapor profile” Check “Include labels”

(a) Reflux Limit ¼ 100 Reflux Step ¼ 1 (b) Integration time ¼ 10 Check “Include permeate profile” Check “Include labels”

a ¼ [3,1.5,1] XD ¼ [0.3,0.3] RT ¼ 100 D ¼ 20



Integration time ¼ 50 This plot takes some time to compute—please be Check “Include permeate patient. profile” Check “Include labels”

a ¼ [3,1,1.5] xT ¼ [0.3,0.3] RT ¼ 1000 (c) dT1 ¼ [0.04,0.08] (d) dT2 ¼ [0.04,0.08]

Appendix B

FLUX MODEL FOR PERVAP 1137 MEMBRANE PERVAP 1137Ò (Sulzer ChemTech) has the following empirical formulas for calculating the flux of each component through the membrane (1 ¼ methanol, 2 ¼ butene, 3 ¼ MTBE) (Bausa and Marquardt, 2000): J 1 ¼ k1

pP c1 x1 þ 0 p1

c1 p01 x1 pP

ðB:1Þ

J2 ¼ k2 c2 p02 x2 J1

ðB:2Þ

J3 ¼ k3 c3 p03 x3 J1

ðB:3Þ

where ki is the transfer parameter value for component i (see below for the values), ci is the activity coefficient for component i, p0i is the saturation pressure of component i, and 1 ¼ methanol, 2 ¼ pentane, 3 ¼ MTBE.


202

APPENDIX B

FLUX MODEL FOR PERVAP 1137 MEMBRANE

Pentene is used since there was a lack of experimental data for i-butene, and also since it is chemically similar to i-butene. The parameter values, with appropriate units, are k1 ¼ 4:000 105 kmol=s m2 bar

ðB:4Þ

k2 ¼ 1:375 104 =bar

ðB:5Þ

k3 ¼ 3:250 104 =bar

ðB:6Þ

Appendix C

PROOF OF EQUATION FOR DETERMINING PERMEATION TIME IN A BATCH PROCESS

The material balance over the batch still is (refer to Figures 3.1 and 5.2, and Equation (3.1)) dR ¼ P_ dt

ðC:1Þ

Thus, RðtÞ ð

t¼

1 dR P_

ðC:2Þ

Rð0Þ

The permeate removal rate is not a constant and depends on the flux of material through the membrane: J¼

c X i¼1

Ji ¼

c P_ X P_ yi ¼ A i¼1 A

ðC:3Þ


204

APPENDIX C PROOF OF EQUATION FOR DETERMINING PERMEATION TIME

Thus, P_ ¼ JA

ðC:4Þ

resulting in 1 t¼ A

RðtÞ ð

1 dR J

ðC:5Þ

Rð0Þ

for a constant area (batch). Since J is a function of x, we need to change dR to dxi. The residue curve equation is d ðxi Þ ¼ ðxi yi Þ dt

ðC:6Þ

resulting in dt ¼

dxi xi yi ðxÞ

ðC:7Þ

or xiððf Þ

t¼ xi ð0Þ

1 dxi xi yi ðxÞ

ðC:8Þ

We see that t is a function of x. Also, from the definition of t, we have t ¼ ln R ð0Þ R ðtÞ

ðC:9Þ

RðtÞ ¼ Rð0Þ et

ðC:10Þ

1 dt ¼ dR R

ðC:11Þ

Thus,

and

APPENDIX C

PROOF OF EQUATION FOR DETERMINING PERMEATION TIME

205

which means that dR ¼

Rð0Þ et dxi xi yi ðxÞ

ðC:12Þ

So, Rð0Þ t¼ A

xð i ðtÞ

xi ð0Þ

et 1 dxi xi yi ðxÞ J

ðC:13Þ

Using the flux of component i rather than the total flux, we have Rð0Þ t¼ A

xð i ðtÞ

xi ð0Þ

yi ðxÞ et 1 dxi xi yi ðxÞ Ji

ðC:14Þ

For gas permeation using constant relative permeabilities (Knudsen membrane) and maintaining a vacuum on the permeate side (refer to equation C.15), we have d t ¼ Rð0Þ A P0i pR

xiððf Þ

xi ð0Þ

yi ðxÞ 1 et dxi xi xi yi ðxÞ

ðC:15Þ

Appendix D

PROOF OF EQUATION FOR DETERMINING PERMEATION AREA IN A CONTINUOUS PROCESS

Expanding the membrane residue curve equation (Equation (5.13)) according to the chain rule, we find dxi dA ¼ xi yi ðxÞ dA dA0

ðD:1Þ

The area required for a specific separation with constant permeate removal rate is given by F 0 ðD:2Þ A ¼ ð1eA Þ P Differentiating Equation (D.2) yields dA F A0 ¼ e dA0 P

ðD:3Þ

Substituting Equation (D.3) into Equation (D.1) and integrating, we have xið ðAÞ

dA ¼ xi ð0Þ

F A0 1 e dxi P xi yi ðxÞ

ðD:4Þ


208

APPENDIX D PROOF OF EQUATION FOR DETERMINING PERMEATION AREA

The total permeate flow rate can be related to the flux of component i by the following equation: Ji ¼

P yi ðxÞ A

ðD:5Þ

yi ðxÞ A Ji

ðD:6Þ

which can be rearranged as follows: P¼

The flux across a Knudsen membrane with vacuum pressure on the permeate side is given by Ji ¼

P0i pR xi d

ðD:7Þ

Thus, the total permeate flow rate is given by P¼

d yi ðxÞ A P0i pR xi

ðD:8Þ

Substituting Equation (D.5) into Equation (D.4) and rearranging, we find ðA A¼0

d A dA ¼ F 0 Pi pR

xið ðAÞ

xi ð0Þ

0

yi ðxÞ eA 1 dxi xi xi yi ðxÞ

ðD:9Þ

Thus, A2 d ¼ F 0 Pi pR 2

xið ðAÞ

xi ð0Þ

0

yi ðxÞ eA 1 dxi xi yi ðxÞ xi

ðD:10Þ

Appendix E

PROOF OF THE DIFFERENCE POINT EQUATION

The proof of the DPE was initially performed in an analogous manner to that done for distillation (Van Dongen and Doherty, 1985). This proof is presented in the next section.

E.1 PROOF USING ANALOGOUS METHOD TO DISTILLATION A generalized column section (distillation or membrane), shown in Figure E.1, is defined as a length of column between points of addition and removal of material and/or energy (Tapp et al., 2004). The constant molar overflow assumption is not employed here. The subscript “T” denotes a quantity at the top of the CS, while “A” denotes a quantity at any cross-sectional (dimensionless) area down the length of the column. Consider the following overall and component material balances written around the top of the column and any point down the membrane length: PA þ RT ¼ PT þ RA

ðE:1Þ

PA yA þ RT xT ¼ PT yT þ RA xA

ðE:2Þ


210

APPENDIX E


Figure E.1 A generalized membrane column section.

Adding and subtracting RA þ dA xA þ dA to Equation (E.2) and rearranging, we find PT yT RT xT ¼ PA yA RA xA þ RA þ dA xA þ dA RA þ dA xA þ dA

ðE:3Þ

Now, we define dR ¼ RA þ dA RA

ðE:4Þ

dx ¼ xA þ dA xA

ðE:5Þ

Rearranging Equations (E.4) and (E.5) and substituting into Equation (E.3), we have PT yT RT xT ¼ PA yA ðRA þ dA dRÞðxA þ dA dxÞ þ RA þ dA xA þ dA RA þ dA xA þ dA

ðE:6Þ

Multiplying out, we have PT yT RT xT ¼ PA yA RA þ dA xA þ dA þ RA þ dA dx þ dRxA þ dA dR dx þ RA þ dA xA þ dA RA þ dA xA þ dA

ðE:7Þ

PROOF USING ANALOGOUS METHOD TO DISTILLATION

211

Canceling and assuming the product dR dx is small, we find PT yT RT xT ¼ PA yA þ RA þ dA dx þ dRxA þ dA RA þ dA xA þ dA

ðE:8Þ

Substituting Equation (E.4), we have PT yT RT xT ¼ PA yA þ RA þ dA dx þ ðRA þ dA RA ÞxA þ dA RA þ dA xA þ dA ðE:9Þ and canceling, we have PT yT RT xT ¼ PA yA þ RA þ dA dx RA xA þ dA

ðE:10Þ

Substituting Equation (E.5), we find PT yT RT xT ¼ PA yA þ RA þ dA ðxA þ dA xA Þ RA xA þ dA

ðE:11Þ

Substituting in overall mass balance (Equation (E.1)), we have PT yT RT xT ¼ PA yA þ RA þ dA ðxA þ dA xA Þ ðPA þ RT PT ÞxA þ dA ðE:12Þ Rearranging, we have PT yT RT xT ¼ PA ðxA þ dA yA Þ þ RA þ dA ðxA þ dA xA Þ þ ðPT RT ÞxA þ dA ðE:13Þ By defining D ¼ PA RA

ðE:14Þ

as the net molar flow inside the column section (mol/s) and XD ¼

PAm yAm RAm xAm D

ðE:15Þ

as the difference point [], we see that, by overall mass balance (Equation (E.1)), PT RT ¼ D

ðE:16Þ

212

APPENDIX E


and by component mass balance (Equation (E.2)), PT yT RT xT ¼ DXD

ðE:17Þ

Thus, Equation (E.13) becomes DXD ¼ PA ðxA þ dA yA Þ þ RA þ dA ðxA þ dA xA Þ þ DxA þ dA

ðE:18Þ

Rearranging, we have RA þ dA ðxA þ dA xA Þ ¼ PA ðxA þ dA yA Þ þ DðXD xA þ dA Þ

ðE:19Þ

Now, using a first-order Taylor series expansion on xA , we find dx Dh xA ¼ xA þ dA dh h¼A þ dA

ðE:20Þ

where Dh is assumed to be unity for a theoretical dimensionless area step. Thus, the Taylor expansion becomes xA ¼ xA þ dA

dxA þ dA d ðA þ dAÞ

ðE:21Þ

Substituting Equation (E.21) into Equation (E.19), we have dxA þ dA ¼ PA ðxA þ dA yA Þ RA þ dA xA þ dA xA þ dA d ðA þ dAÞ

ðE:22Þ

þ DðXD xA þ dA Þ Canceling and taking the limit as dA ! 0, we find RðAÞ

dxi ¼ PðAÞ½xi yi ðxÞ þ D½XDi xi ; dA

for i ¼ 1; . . . ; c 1 ðE:23Þ

The proof of the DPE for distillation systems follows the same format and has been accepted as thus. However, there is a fundamental error when progressing from Equation (E.20) to Equation (E.21) since a step size of one is assumed. This assumption, incidentally, is justified for successive x-values close together (i.e., in the proximity of a pinch). However, this assumption becomes incorrect for large changes in x.

PROOF USING MASS TRANSFER

213

We have therefore formulated an alternative proof of the DPE, as given in the next section.

E.2 PROOF USING MASS TRANSFER Consider the membrane strip, as displayed in Figure E.1, of area DA. Figure E.2, showing the nomenclature related to this strip, gives a graphical summary of the mass transfer across a membrane. y is permeate composition at the membrane wall, while Y is the bulk permeate composition. If one considers the diffusion of permeate from y to Y, and defines J as the total transfer rate of all components, while N is defined as the total flux of all components from one phase to the other, then the total molar flux across the membrane per area is given as follows: dR dP ¼ N ¼ dA dA

Figure E.2 Mass transfer across a membrane strip of area DA.

ðE:24Þ

214

APPENDIX E


Since there is no bulk flow y to Y, mass transfer is due to diffusion only, thus N ¼ J. Hence, dR ¼ J dA

ðE:25Þ

This result can be derived from material balance over either side of the membrane. For an individual component i, dR xi ¼ Ni dA

ðE:26Þ

where Ni is the molar flux of component i. Using the fundamental mass transfer equation (Bird et al., 2007), we find Ni ¼ Ji þ xi N

ðE:27Þ

where Ji is the transfer of component i in the permeate phase. In other words, Ji is the rate at which component i moves from the membrane wall (once leaving the membrane) into the bulk phase: Ji ¼ Ky ðyi Yi Þ

ðE:28Þ

where Ky is the overall permeate phase transfer coefficient. The only way you get the same result as the initial proof (in Section E.1) is if Ky is not dependent on i. It is possible then that Ky is merely a mixing term, and independent of i. Also, N¼J

ðE:29Þ

Ni ¼ Ky ðyi Yi Þ þ xi J

ðE:30Þ

Thus, Equation (E.27) becomes

Substituting Equation (E.26) and expanding according to the product rule, we find R

dxi dR þ xi ¼ Ky ðyi Yi Þ xi J dA dA

ðE:31Þ

PROOF USING MASS TRANSFER

215

But dR ¼ xi J dA

ðE:32Þ

dxi ¼ Ky ðyi Yi Þ dA

ðE:33Þ

xi Therefore, R

Multiplying both sides by the permeate flow rate, we have P

R dxi ¼ Pðyi Yi Þ Ky dA

ðE:34Þ

Dimensionless area is defined as A* ¼

Ky A P

ðE:35Þ

dA* ¼

Ky dA P

ðE:36Þ

Therefore,

and R

dxi ¼ Pðyi Yi Þ dA*

ðE:37Þ

Adding and subtracting (Pxi þ Rxi) to Equation (E.37), we find R

dxi ¼ Pðyi Yi Þ þ Pxi Pxi þ Rxi Rxi dA*

ðE:38Þ

Multiplying out and rearranging, we have R

dxi ¼ Pðxi yi Þ þ ðPYi Rxi Þ xi ðP RÞ dA*

ðE:39Þ

216

APPENDIX E


Now we define the following terms: D ¼ PR XDi ¼

PYi Rxi D

ðE:40Þ ðE:41Þ

substituting into Equation (E.16), the final form of the DPE is obtained: R

dxi ¼ Pðxi yi Þ þ DðXD xi Þ dA*

ðE:42Þ

INDEX

Absorption, 1, 42 Activity coefficient, 12 Arheotrope, 30, 51 locus of, 35–38 Attainable region for hybrid design, 164–167 Azeoptrope, breaking of, 42 Batch operation, see Membrane, operation; Permeation time Binary systems, 16–20, 45 Bow-tie regions, 60–61 Cascades, reflux, 3 Column profiles, 85–106, 119 location relative to its pinch point loci, 123 Column profile maps, 72, 74–76 Column section (CS) breakdown, 67–68 definition, 68 distillation, 110–112

general, 68–79 membrane, 80–106, 110–112 operation finite reflux, 74–79 infinite reflux, 71–73 Complex membrane configurations, 107–150 comparison with distillation systems, 138 Continuous operation, see Membrane, operation Coupled column sections, 116 feasible profiles (example), 127, 134–136 region of feasibility, 128–129 requirements for feasibility, 120 Degrees of freedom hybrid design, 161 Difference point, 70, 75–76, 81, 84, 98, 104, 112. See also Pinch point loci coupled column sections, 118


218

INDEX

Difference point equation. See also Difference vector distillation, 111–112, 159 general, 70, 71 membrane, 81, 82, 87–88, 98, 112 properties of, 84 Difference vector, 88–97 coupled column sections, 133 Diffusion membranes description, 2, 8, 42 gas separation, see Gas separation pervaporation, see Pervaporation Dimensionless time, 21, 46–49 Discontinuity in residue curve map, 73–74 Distillation, 1, 2, 110–112. See also Equilibrium-based separations; Hybrid distillation-membrane design analogy of membrane systems to, 19 differences with membrane systems, 36 Equilibrium-based separations, 3, 42 Flow patterns in membrane equipment, 43, 55, 66 Flux, 2, 8, 12, 16, 82–83 Flux model, see Permeation modeling Gas separation, 2, 8–11 Gibbs triangle, see Mass balance triangle Hybrid distillation-membrane processes design using coupled column section approach, 138–150 design using residue curve maps, 159–167 synthesis of configurations, 156–158 i-butene, see MTBE Kinetically-based separations, 3, 42

Knudsen membrane, 11, 16, 23, 29–34, 58 Liapounov’s first theorem, 73 Liquid-liquid extraction, see Solvent extraction Mass balance triangle, 24, 71–72, 90, 97 Membrane classification, 13–14 nonselective, 13–14 selective, 13–14 operation. See also Nonreflux membrane equipment; Reflux equipment; Flow patterns in membrane equipment batch 2, 15–54, ccntinuous, 2, 43, 54–64 temperature, 12–13 types diffusion, see Diffusion membranes general, 1 Membrane area, 8, 54, 56, 83. See also Permeation area Membrane column, 43, 59–60, 66–68 Membrane column section, see Column section Membrane residue curve, 20. See also Residue curve equation Membrane residue curve map application of, 41–64 definition, 20 discontinuity, 73–74, 87 nonselective, 27, 33 plotting of, 23–27, 71–73, 85–87 properties, 29–39 selective, 33, 35, 37 Membrane selectivity, see Permeability Membrane separation, 1. See also Kinetically-based separations effective, 9, 36–38 comparison with distillation, see Distillation Membrane vector field, 30–31 Methanol, see MTBE Mixing vector, 98

INDEX

Moving triangle, see Transformed triangle M-RCM, see Membrane residue curve map MTBE, 12, 153–155. See also Selective membrane Multicomponent systems, see Ternary systems Net molar flow in a column section, 70, 81, 84, 98, 112 Net molar composition in a column section, see Difference point Nonreflux membrane equipment, 43, 54–58 Nonselective membrane, See also Knudsen membrane; Membrane, classification operating region, 48–50 sequencing, 61–63 Operating leaves, 156–158 Operating regions (membranes) batch, 45–50 continuous, 57–58, 60–63 Permeability. See also Relative permeability definition, 3 Permeate composition, 3, 15, 42, 46–54, 56–58 flow, 8, 15 pressure, 8, 9, 10–11, 15, 80 Permeate profile behavior of, 123 in relation to pinch point curve, 125 Permeation area, 58 Permeation modeling, 7–14, 34–36, 42, 82–84, 113 Permeation time, 52–54 Pervaporation, 2, 11–13 Petlyuk membrane arrangement, 114 Pinch point, 102, 122, 126. See also Pinch point loci Pinch point loci, 76–79, 93–94, 99–102, 121

219

Pressure ratio of, 9 saturation, 12 vacuum, 10, 16–18, 34–36 Reflux membrane equipment, 43, 58–64 finite, 64 infinite, 60–63 total, 58–60 Reflux ratio, 71, 98, 104, 112. See also Pinch point loci coupled column sections, 118 minimum for coupled column sections, 130 negative values, 77, 124 Relative permeability, 9, 11. See also Knudsen membrane effect on topology, 32–34 Residue curve equation, 21, 56, 59, 71, 85 derivation, 22–23 Residue curve map (D-RCM), 72 discontinuity, 73–74 Retentate composition, 3, 42 flow, 88, 90 pressure, 8 Retentate profile. See also Membrane residue curve in relation to pinch point curve, 124 Saddle point, 29, 30, 33, 34. See also Stationary points Selective membrane, 34–38. See also Membrane, classification operating region, 50–54 sequencing, 63 Selectivity, 42. See also Permeabitlity Separation vector, 31, 89, 98 Separations, processes, 1 Sequencing, 61–63 Solvent extraction, 1, 42 Solution-diffusion model, 8. See also Permeation modeling Split ratio, for hybrid designs, 161

220

INDEX

Stable nodes, 29, 30, 33, 34. See also Stationary points Stationary points, 29–30, 73. See also Pinch point Ternary systems, batch, 20, 45 mass balances, 21 Transformed triangle, 74–77, 90

Unidistribution lines, 31–32 Unstable nodes, 29, 30, 33, 34. See also Stationary points Vapor-liquid equilibrium, 113 x–y plot, 17–19. See also Binary systems

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