Biological Reaction Engineering: Dynamic Modelling Fundamentals with Simulation Examples

I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Biological Reaction Engineering Dynamic Modelling Fundamentals with ...

Author: Irving J. Dunn | Elmar Heinzle | John Ingham | Jiri E. Prenosil

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I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil

Biological Reaction Engineering Dynamic Modelling Fundamentals with Simulation Examples Second, Completely Revised Edition

Biological Reaction Engineering. Second Edition. \. J. Dunn. E. Heinzle, J. Ingham, J- E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheitn ISBN: 3-527-30759-1

Also of Interest Ingham, X, Dunn, I. J., Heinzle, E., Pfenosil, J. E.

Chemical Engineering Dynamics An Introduction to Modelling and Computer Simulation Second, Completely Revised Edition 2000, ISBN 3-527-29776-6

Irving J. Dunn, Elmar Heinzle, John Ingham, Jifi E. Pf enosil

Biological Reaction Engineering Dynamic Modelling Fundamentals with Simulation Examples Second, Completely Revised Edition

WILEYVCH WILEY-VCH GmbH & Co. KGaA

Dr. Irving J. Dunn ETH Zurich Department of Chemical Engineering CH-8092 Zurich Switzerland Professor Dr. Elmar Heinzle University of Saarland Department of Technical Biochemistry P.O. Box 15 11 50 D-66041 Saarbrucken Germany Dr. John Ingham University of Bradford Department of Chemical Engeering Bradford BD7 1DP United Kingdom Dr.JiriE.Prenosil ETH Zurich Department of Chemical Engineering CH-8092 Zurich Switzerland

This book was carefully produced. Nevertheless, authors and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

First Edition 1992 Second, Completely Revised Edition 2003

Library of Congress Card No.: Applied for. British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library.

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

© 2003 WILE Y-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printing: Strauss Offsetdruck, Morlenbach Bookbinding: GroBbuchbinderei J. Schaffer GmbH & Co. KG, Griinstadt Printed in the Federal Republic of Germany. Printed on acid-free paper. ISBN 3-527-30759-1

Table of Contents

TABLE OF CONTENTS PREFACE

V XI

PART I PRINCIPLES OF BIOREACTOR MODELLING

1

NOMENCLATURE FOR PART I

3

1

9

MODELLING PRINCIPLES

1.1 FUNDAMENTALS OF MODELLING 9 7.7.7 Use of Models for Understanding, Design and Optimization of Bioreactors 9 1.1.2 General Aspects of the Modelling Approach 10 1.1.3 General Modelling Procedure..... 72 1.1.4 Simulation Tools 75 7.7.5 Teaching Applications 75 1.2 DEVELOPMENT AND MEANING OF DYNAMC DIFFEREOTTAL BALANCES 16 1.3 FORMULATION OF BALANCE EQUATIONS ..21 7.5.7 Types of Mass Balance Equations 27 1.3.2 Balancing Procedure 23 1.3.2.1 Case A. Continuous Stirred Tank Bioreactor 24 1.3.2.2 CaseB. Tubular Reactor 24 1.3.2.3 Case C. River with Eddy Current 25 1.3.3 Total Mass Balances 33 1.3.4 Component Balances for Reacting Systems 34 1.3.4.1 Case A. Constant Volume Continuous Stirred Tank Reactor 35 1.3.4.2 Case B. Semi-continuous Reactor with Volume Change 37 1.3.4.3 Case C. Steady-State Oxygen Balancing in Fermentation 38 1.3.4.4 Case D. Inert Gas Balance to Calculate Flow Rates 39 7.5.5 Stoichiometry, Elemental Balancing and the Yield Coefficient Concept.. 40 1.3.5.1 Simple Stoichiometry 40 1.3.5.2 Elemental Balancing 42 L3.5.3 Mass Yield Coefficients 44

VI

Table of Contents 1.3.5.4 1.3.6 1.3.6.1 1.3.6.2 1.3.7 1.3.6.3

2

Energy Yield Coefficients Equilibrium Relationships General Considerations Case A. Calculation of pH with an Ion Charge Balance Energy Balancing for Bioreactors..... Case B. Determining Heat Transfer Area or Cooling Water Temperature

BASIC BIOREACTOR CONCEPTS 2.1 INFORMATION FOR BIOREACTOR MODELLING... 2.2 BIOREACTOR OPERATION 2.2.7 Batch Operation 2.2.2 Semicontinuous or Fed Batch Operation..... 2.2.3 Continuous Operation 2.2.4 Summary and Comparison

3

45 46 46 47 49 52 55

,

BIOLOGICAL KINETICS

.....55 .....56 57 ....58 60 63 67

3.1 ENZYME KINETICS 68 3.1.1 Michaelis-Menten Equation 68 3.1.2 Other Enzyme Kinetic Models 73 3.1.3 Deactivation 76 3.1.4 Sterilization 76 3.2 SIMPLE MICROBIAL KINETICS 77 3.2.1 Basic Growth Kinetics 77 3.2.2 Substrate Inhibition of Growth 80 3.2.3 Product Inhibition 81 3.2.4 Other Expressions for Specific Growth Rate 81 3.2.5 Substrate Uptake Kinetics 83 3.2.6 Product Formation 85 3.2.7 Interacting Microorganisms ....86 3.2.7.1 Case A. Modelling of Mutualism Kinetics..... 88 3.2.7.2 Case B. Kinetics of Anaerobic Degradation 89 3.3 STRUCTURED KINETIC MODELS ..........91 3.3.1 Case Studies 93 3.3.1.1 Case C. Modelling Synthesis of Poly-B-hydroxybutyric Acid (PHB) 93 3.3.1.2 Case D. Modelling of Sustained Oscillations in Continuous Culture 94 3.3.1.3 Case E. Growth and Product Formation of an Oxygen-Sensitive Bacillussubtilis Culture 97 4

BIOREACTOR MODELLING 4.1 GENERAL BALANCES FOR TANK-TYPE BIOLOGICAL REACTORS 4.1.1 The Batch Fermenter. 4.1.2 The Chemostat 4.1.3 The Fed Batch Fermenter 4.1.4 Biomass Productivity 4.1.5 Case Studies

101 101 103 104 1 06 109 109

Table of Contents 4.1.5.1 Case A. Continuous Fermentation with Biomass Recycle 4.1.5.2 Case B. Enzymatic Tanks-in-series Bioreactor System 4.2 MODELLING TUBULAR PLUG FLOW BIOREACTORS 4.2.1 Steady-State Balancing 4.2.2 Unsteady-State Balancing for Tubular Bioreactors 5

MASS TRANSFER

VII 110 112 113 113 775 117

5.1 MASS TRANSFER IN BIOLOGICAL REACTORS 117 5.7.7 Gas Absorption with Bioreaction in the Liquid Phase 777 5.1.2 Liquid-Liquid Extraction with Bioreaction in One Phase 778 5.1.3 Surface Biocatalysis 778 5.7.4 Diffusion and Reaction in Porous Biocatalyst 779 5.2 INTERPHASEGAS-LIQUID MASS TRANSFER 119 5.3 GENERAL OXYGEN BALANCES FOR GAS-LIQUID TRANSFER 123 5.3.1 Application of Oxygen Balances 725 5.3.1.1 Case A. Steady-State Gas Balance for the Biological Uptake Rate 125 5.3.1.2 Case B. Determination of KLa Using the Sulfite Oxidation Reaction 126 5.3.1.3 Case C. Determination of Kjâ by a Dynamic Method 126 5.3.1.4 Case D. Determination of Oxygen Uptake Rates by a Dynamic Method 128 5.3.1.5 Case E. Steady-State Liquid Balancing to Determine Oxygen Uptake Rate.. 129 5.3.1.6 Case F. Steady-State Deoxygenated Feed Method for KJJI 130 5.3.1.7 Case G. Biological Oxidation in an Aerated Tank 131 5.3.1.8 Case H. Modelling Nitrification in a Fluidized Bed Biofilm Reactor 133 5.4 MODELS FOR OXYGEN TRANSFER IN LARGE SCALE BIOREACTORS 137 5.4.1 Case Studies for Large Scale Bioreactors 7 39 5.4.1.1 Case A.Model for Oxygen Gradients in a Bubble Column Bioreactor 139 5.4.1.2 Case B.Model for a Multiple Impeller Fermenter 140 6

DIFFUSION AND BIOLOGICAL REACTION IN IMMOBILIZED BIOCATALYST SYSTEMS 145 6.1 EXTERNAL MASS TRANSFER 6.2 INTERNAL DIFFUSION AND REACTION WITHIN BIOCATALYSTS ..... 6.2.1 Derivation of Finite Difference Model for Diffusion-Reaction Systems. 6.2.2 Dimensionless Parameters from Diffusion-Reaction Models 6.2.5 The Effectiveness Factor Concept. 6.2.4 Case Studies for Diffusion with Biological Reaction 6.2.4.1 Case A. Estimation of Oxygen Diffusion Effects in a Biofilm 6.2.4.2 Case B. Complex Diffusion-Reaction Processes (Biofilm Nitrification)....

7

AUTOMATIC BIOPROCESS CONTROL FUNDAMENTALS 7.1 7.2

ELEMENTS OF FEEDBACK CONTROL TYPES OF CONTROLLER ACTION 7.2.7 On-OffControl 7.2.2 Proportional (P) Controller 7.2.3 Proportional-Integral (PI) Controller

146 149 151 754 755 757 157 157 161 161 163 163 764 765

VIII

Table of Contents

7.2.4 Proportional-Derivative (PD) Controller 7.2.5 Proportional-Integral-Derivative (PID) Controller 7.3 CONTROLLER TUNING 7.3.1 Trial and Error Method 7.3:2 Ziegler-Nichols Method. 7.3.3 Cohen-Coon Controller Settings 7.3.4 Ultimate Gain Method 7.4 ADVANCED CONTROL STRATEGIES 7.4.1 Cascade Control 7.4.2 Feed Forward Control 7.4.3 Adaptive Control 7.4.4 Sampled-Data Control Systems 7.5 CONCEPTS FOR BIOPROCESS CONTROL 7.5.7 Selection of a Control Strategy 7.5.2 Methods of Designing and Testing the Strategy REFERENCES REFERENCES CITED IN PART I RECOMMENDED TEXTBOOKS AND REFERENCES FOR FURTHER READING PART II

8

166 167 169 769 769 170 777 172 772 173 774 774 175 776 7 78 181 181 184

DYNAMIC BIOPROCESS SIMULATION EXAMPLES AND THE BERKELEY MADONNA SIMULATION LANGUAGE. 191

SIMULATION EXAMPLES OF BIOLOGICAL REACTION PROCESSES USING BERKELEY MADONNA

193

8.1 INTRODUCTORY EXAMPLES 193 8.7.7 Batch Fermentation (BATFERM) 793 8.7.2 ChemostatFermentation (CHEMO) 799 8.1.3 Fed Batch Fermentation (FEDBAT) 204 8.2 BATCH REACTORS 209 8.2.7 Kinetics of Enzyme Action (MMKINET) 209 8.2.2 Lineweaver-Burk Plot (LINEWEAV) .....272 8.2.3 Oligosaccharide Production in Enzymatic Lactose Hydrolysis (OLIGO) 215 8.2.4 Structured Model for PHB Production (PHB) ....279 8.3 FED BATCH REACTORS 224 8.3.1 Variable Volume Fermentation (VARVOL and VARVOLD) 224 8.3.2 Penicillin Fermentation Using Elemental Balancing (PENFERM) 230 8.3.3 Ethanol Fed Batch Diauxic Fermentation (ETHFERM) 240 8.3.4 Repeated Fed Batch Culture (REPFED) 245 8.3.5 Repeated Medium Replacement Culture (REPLCUL) 249 8.3.6 Penicillin Production in a Fed Batch Fermenter (PENOXY) 253 8.4 CONTINUOUS REACTORS 257 8.4.7 Steady-State Chemostat (CHEMOSTA) 257 8.4.2 Continuous Culture with Inhibitory Substrate (CONINHIB) 267 8.4.3 Nitrification in Activated Sludge Process (ACTNITR) 267

Table of Contents

IX

8.4.4 Tubular Enzyme Reactor (ENZTUBE) 272 8.4.5 Dual Substrate Limitation (DUAL) 275 8.4.6 Dichloromethane in a Biofllm Fluidized Sand Bed (DCMDEG) 280 8.4.7 Two-Stage Chemostat with Additional Stream (TWOSTAGE) 286 8.4.8 Two Stage Culture with Product Inhibition (STAGED) 290 8.4.9 Fluidized Bed Recycle Reactor (FBR) 295 8.4.10 Nitrification in a Fluidized Bed Reactor (NITBED)... 299 8.4.11 Continuous Enzymatic Reactor (ENZCON) 305 8.4.12 Reactor Cascade with Deactivating Enzyme (DEACTENZ) 308 8.4.13 Production ofPHB in a Two-Tank Reactor Process (PHBTWO) 314 8.5 OXYGEN UPTAKE SYSTEMS 318 8.5.1 Aeration of a Tank Reactor for Enzymatic Oxidation (OXENZ) 318 8.5.2 Gas and Liquid Oxygen Dynamics in a Continuous Fermenter (INHIB) 321 8.5.3 Batch Nitrification with Oxygen Transfer (NITRIF) 327 8.5.4 Oxygen Uptake and Aeration Dynamics (OXDYN) 331 8.5.5 Oxygen Electrode for Kia (KLADYN, KLAFIT and ELECTFIT) 335 8.5.6 Biofiltration Column with Two Inhibitory Substrates (BIOFILTDYN). 342 8.5.7 Optical Sensing in Microtiter Plates (TITERDYN and T1TERB1O) 349 8.6 CONTROLLED REACTORS 354 8.6.1 Feedback Control of a Water Heater (TEMPCONT) 354 8.6.2 Temperature Control of Fermentation (FERMTEMP) 358 8.6.3 Turbidostat Response (TURBCON) 363 8.6.4 Control of a Continuous Bioreactor, Inhibitory Substrate (CONTCON)367 8.7 DIFFUSION SYSTEMS ....371 8.7.1 Double Substrate Biofilm Reaction (BIOFILM) 377 8.7.2 Steady-State Split Boundary Solution (ENZSPLIT).... 377 8.7.3 Dynamic Porous Diffusion and Reaction (ENZDYN).... 383 8.7.4 Oxygen Diffusion in Animal Cells (CELLDIFF) 388 8.7.5 Biofilm in a Nitrification Column System (NITBEDFILM) 393 8.8 MULTI-ORGANISM SYSTEMS ..400 8.8.1 Two Bacteria with Opposite Substrate Preferences (COMMENSA) 400 8.8.2 Competitive Assimilation and Commensalism (COMPASM) 406 8.8.3 Stability of Recombinant Microorganisms (PLASMID) 411 8.8.4 Predator-Prey Population Dynamics (MIXPOP) 417 8.8.5 Competition Between Organisms (TWOONE) 422 8.8.6 Competition between Two Microorganisms in a Biofilm (FILMPOP). 425 8.8.7 Model for Anaerobic Reactor Activity Measurement (ANAEMEAS).... 433 8.8.8 Oscillations in Continuous Yeast Culture (YEASTOSC) 441 8.8.9 Mammalian Cell Cycle Control (MAMMCELLCYCLE) 445 8.9 MEMBRANE AND CELL RETENTION REACTORS 451 8.9.1 Cell Retention Membrane Reactor (MEMINH) 451 8.9.2 Fermentation with Pervaporation (SUBTILIS) 455 8.9.3 Two Stage Fermentor With Cell Recycle (LACMEMRECYC) 464 8.9.4 Hollow Fiber Enzyme Reactor for Lactose Hydrolysis (LACREACT). 470 8.9.5 Animal Cells in a Fluidized Bed Reactor (ANIMALIMMOB) 477

X

Table of Contents

9

APPENDIX: USING THE BERKELEY MADONNA LANGUAGE.. 483 9.1 9.2

A SHORT GUIDE TO BERKELEY MADONNA SCREENSHOT GUIDE TO BERKELEY MADONNA

483 488

10 ALPHABETICAL LIST OF EXAMPLES

497

11 INDEX

499

Preface Our goal in this textbook is to teach, through modelling and simulation, the quantitative description of bioreaction processes to scientists and engineers. In working through the many simulation examples, you, the reader, will learn to apply mass and energy balances to describe a variety of dynamic bioreactor systems. For your efforts, you will be rewarded with a greater understanding of biological rate processes. The many example applications will help you to gain confidence in modelling, and you will find that the simulation language used, Berkeley Madonna, is a powerful tool for developing your own simulation models. Your new abilities will be valuable for designing experiments, for extracting kinetic data from experiments, in designing and optimizing biological reaction systems, and for developing bioreactor control strategies. This book is based on part of our successful course, "Biological Reaction Engineering", which has been held annually in the Swiss mountain resort of Braunwald for the past twenty five years and which is now known, throughout European biotechnology circles as the "Braunwald Course". More details can be found at our website www.braunwaldcourse.ch. Modelling is often unfamiliar to biologists and chemists, who nevertheless need modelling techniques in their work. The general field of biochemical reaction engineering is one that requires a very close interdisciplinary interaction between applied microbiologists, biochemists, biochemical engineers, engineers and managers; a large degree of collaboration and mutual understanding is therefore important. Professional microbiologists and biochemists often lack the formal training needed to analyze laboratory kinetic data in its most meaningful sense, and they may sometimes experience difficulty in participating in engineering design decisions and in communicating with engineers. These are just the very types of activity required in the multi-disciplinary field of biotechnology. Chemical engineering's greatest strength is its well-developed modelling concepts, based on mass and energy balances, combined with rate processes. Biochemical engineering is a discipline closely related to conventional chemical engineering, in that it attempts to apply physical principles to the solution of biological problems. This approach may be applied to the measurement and interpretation of laboratory kinetic data or as well to the design of large-scale fermentation, enzymatic or waste treatment processes. The necessary interdisciplinary cooperation requires the biological scientists and chemical engineers involved to have at least a partial understanding of each other's field. The purpose of this book is to provide the mathematical tools necessary for the quantitative analysis of biological kinetics and other biological process phenomena. More generally, the mathematical modelling

XII

Preface

methods presented here are intended to lead to a greater understanding of how the biological reaction systems are influenced by process situations. Engineering science depends heavily on the use of applied theory, quantitative correlations and mathematics, and it is often difficult for all of us (not only the biological scientist) to surmount the mathematical barrier, which is posed by engineering. A mistake, often made, is to confuse "mathematics" with the engineering modelling approach. In modelling an attempt is made to analyze a real and possibly very complex situation into a simplified and understandable physical analog. This physical model may contain many subsystems, all of which still make physical sense, but which now can be formulated as mathematical equations. These equations can be handled automatically by the computer. Thus the engineer and the biologist are freed from the difficulties of mathematical solution and can tackle complex problems that were impossible before. Models, however, still have to be formulated and one of the most important tools of the biochemical engineer, in this operation, is the use of material balance equations. Though it may not be easy for the microbiologist to fully appreciate the importance of differential equations, mass balance equations are not so difficult to understand, since the first law of conservation, namely that matter can neither be created nor destroyed, is fundamental to all science. Mass balances, when combined with kinetic rate equations, to form simple mathematical models, can be used with very great effect as a means of planning, conducting and analyzing experiments. Models are especially important as a means of obtaining a better understanding of process phenomena. A rational approach to experimentation and design requires a considerable knowledge of the system, which can really only be achieved by means of a mathematical model. This book attempts to demonstrate this by way of the many detailed examples. The contribution made to biotechnology by the biochemical engineering modelling approach is especially important because the basic procedure can be developed from a few fundamental principles. An aim of this book is to demonstrate that you do not have to be an engineer to learn modelling and simulation. The basic concepts of the material balance, combined with biological and enzyme kinetics, are easily applied to describe the behavior of well-stirred tank and tubular fermenters, mixed culture dynamics, interphase gas-liquid mass transfer and internal biofilm diffusional limitations, as demonstrated in the computer examples supplied with this book. Such models, when solved interactively by computer simulation, become much more understandable to non-engineers. The Berkeley Madonna simulation language, used for the examples in this book, is especially suitable because of its sophisticated computing power, interactive facility and ease of programming. The use of this digital simulation programming language makes it possible for the reader, student and teacher to experiment directly with the model, in the classroom or at the desk. In this way it is possible to immediately determine the influence of changing various

Preface

XIII

operating parameters on the bioreactor performance - a real learning experience. The simulation examples serve to enforce the learning process in a very effective manner and also provide hands-on confidence in the use of a simulation language. The readers can program their own examples, by formulating new mass balance equations or by modifying an existing example to a new set of circumstances. Thus by working directly at the computer, the no-longer-passive reader is able to experiment directly on the bioreaction system in a very interactive way by changing parameters and learning about their probable influence in a real situation. Because of the speed of solution, a true degree of interaction is possible with Berkeley Madonna, allowing parameters to be changed easily. Plotting the variables in any configuration is easy during a run, and the results from multiple runs can be plotted together for comparison. Other useful features include data fitting and optimization. In our experience, digital simulation has proven itself to be absolutely the most effective way of introducing and reinforcing new concepts that involve multiple interactions. The thinking process is ultimately stimulated to the point of solid understanding.

Organization of the Book The book is divided into two parts: a presentation of the background theory in Part I and the computer simulation exercises in Part II. The function of the text in Part I is to provide the basic theory required to fully understand and to make full use of the computer examples and simulation exercises. Numerous case studies provide illustration to the theory. Part II constitutes the main part of this book, where the simulation examples provide an excellent instructional and self-learning tool. Each of the more than fifty examples is self-contained, including a model description, model equations, exercises, computer program listing, nomenclature and references. The exercises range from simple parameter-changing investigations to suggestions for writing a new program. The combined book thus represents a synthesis of basic theory and computerbased simulation examples. Quite apart from the educational value of the text, the introduction and use of the Berkeley Madonna software provides the reader with the considerable practical advantage of a differential equation solution package. In the appendix a screenshot guide is found concerning the use of the software. Part I: "Principles of Bioreactor Modelling" covers the basic theory necessary for understanding the computer simulation examples. This section presents the basic concepts of mass balancing, and their combination with kinetic relationships, to establish simple biological reactor models, carefully presented in a way that should be understandable to biologists. In fact, engineers may also find this rigorous presentation of balancing to be valuable.

XIV

Preface

In order to achieve this aim, the main emphasis of the text is placed on an understanding of the physical meaning and significance of each term in the model equations. The aim in presenting the relevant theory is thus not to be exhaustive, but simply to provide a basic introduction to the theory required for a proper understanding of the modelling methodology. Chapter 1 deals with the basic concepts of modelling, the basic principles, development and significance of differential balances and the formulation of mass and energy balance relationships. Emphasis is given to physical understanding. The text is accompanied by example cases to illustrate the application of the material. Chapter 2 serves to introduce the varied operational characteristics of the various types of bioreactors and their differing modes of operation, with the aim of giving a qualitative insight into the quantitative behavior of the computer simulation examples. Chapter 3 provides an introduction to enzyme and microbial kinetics. A particular feature of the kinetic treatment is the emphasis on the use of more complex structured models. Such models require much more consideration to be given to the biology of the system during the modelling procedure, but despite their added complexity can nevertheless also be solved with relative ease. They serve as a reminder that biological reactions are really infinitely complex. Chapter 4 is used to derive general mass balance equations, covering all types of fermentation tank reactors. These generalized equations are then simplified to show their application to the differing modes of stirred tank bioreactor operation, discussed previously and which are illustrated by the simulation examples. Chapter 5 explains the basic theory of interfacial mass transfer as applied to fermentation systems and shows how equations for rates of mass transfer can be combined with mass balances, for both liquid and gas phases. A particular extension of this approach is the combination of transfer rate and material balance equations to models of increased geometrical complexity, as represented by large-scale air-lift and multiple-impeller fermenters. Chapter 6 treats the cases of external diffusion to a solid surface and internal diffusion combined with biochemical reaction, with practical application to immobilized biocatalyst and biofilm systems. Emphasized here is the conceptual ease of handling a complex reaction in a solid biocatalyst matrix. The resulting sets of tractable differential-difference equations are solved by simulation techniques in several examples. Chapter 7 describes the importance of control and summarizes control strategies used for bioreaction processes. Here the fundamentals of feedback control systems and their characteristic responses are discussed. This material forms the basis for performing the many recommended control exercises in the simulation examples. It also will allow the reader-simulator to develop his or her own control models and simulation programs.

Preface

XV

Part II, "Dynamic Bioprocess Simulation Examples and the Berkeley Madonna simulation language" comprises Chapter 8, with the computer simulation examples, and Chapter 9, which gives the instructions for using Madonna, Each example in Chapter 8 includes a description of its physical system, the model equations, that were developed in Part I, and a list of suggested exercises. The programs are found on the CD-ROM. These example exercises can be carried out in order to explore the model system in detail, and it is suggested that work on the computer exercises be done in close reference to the model equations and their physical meaning, as described in the text. The exercises, however, are provided simply as an idea for what might be done and are by no means mandatory or restrictive. Working through a particular example will often suggest an interesting variation, such as a control loop, which can then be programmed and inserted. The examples cover a wide range of application and can easily be extended by reference to the literature. They are robust and are well tested by a variety of undergraduate and graduate students and by also the 350 participants, or so, who have previously attended the Braunwald course. In tackling the exercises, we hope you will soon come to share our conviction that, besides being very useful, computer simulation is also fun to do. For the second edition, the text was thoroughly revised and some of our earlier, less relevant material was omitted. On the other hand, a number of new examples resulting mainly from the authors' latest research and teaching work were added. There was also an opportunity in this new edition to eliminate most of the past errors and to avoid new ones as much as possible. Most importantly, the examples have been rewritten in Berkeley Madonna, which all of our reader-simulators will greatly appreciate. Our book has a number of special characteristics. It will be obvious, in reading it through, that we concentrate only on those topics of biological reaction engineering that lend themselves to modelling and simulation and do not attempt to cover the area completely. Our own research work is used to illustrate theoretical points and from it many simulation examples are drawn. A list of suggested books for supplementary reading is found at the end of Chapter 6, together with the list of cited references. The diversity of the simulation examples made it necessary to use separate nomenclature for each. The symbols used in Chapters 1 - 6 are defined at the end of Part I. The authors' four nationalities and three mother tongues, made it difficult to settle on American or British spelling. Somehow we like "modelling" better than "modeling". We are confident that the book will be useful to all life scientists wishing to obtain an understanding of biochemical engineering and also to those chemical and biochemical engineers wanting to sharpen their modelling skills and wishing to gain a better understanding of biochemical process phenomena. We hope that teachers with an interest in modelling will find this to be a useful textbook for undergraduate and graduate biochemical engineering and biotechnological courses.

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Acknowledgements A major acknowledgement should be made to the excellent pioneering texts of R. G. E. Franks (1967 and 1972) and also of W. L. Luyben (1973), for inspiring our interest in digital simulation. We are especially grateful to our students and to the past-participants of the Braunwald course, for their assistance in the continuing development of the course and of the material presented in this book. Continual stimulus and assistance has also been given by our doctoral candidates, especially at the Chemical Engineering Department, ETH-Zurich, as noted throughout the references. We are grateful to and have great respect for the developers of Berkeley Madonna and hope that this new version of the book will be useful in drawing attention to this wonderful simulation language.

Part I

Principles of Bioreactor Modelling

Biological Reaction Engineering, Second Edition, I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30759-1

Nomenclature for Part I

Symbols A A a a b b B c C CP

CPR D D d d DO E E ES f f F G G hi H AH I I j K K KD

Units Area Magnitude of controller input signal Specific area Constant in Logistic Equation Constant in Luedeking-Piret relation Constant in Logistic Equation Magnitude of controller output signal Fraction carbon converted to biomass Concentration Heat capacity Carbon dioxide production rate Diffusivity Dilution rate Differential operator and diameter Fraction carbon converted to product Dissolved oxygen Enzyme concentration Ethanol Enzyme-substrate concentration Fraction carbon converted to CO2 Frequency in the ultimate gain method Flow rate Gas flow rate Intracellular storage product Partial molar enthalpy Henry's Law constant Enthalpy change Inhibiting component concentration Cell compartment masses Mass flux Mass transfer coefficient Constant in Cohen-Coon method Acid-base dissociation constant

m2 various m2/m3 1/h 1/h m3/kg h various

kg/m3, kmol/m3 kJ/kg K, kJ/mol K mol/h m2/h 1/h -, m

g/m3, 9 air sat. g/m3 kg/m3 g/m3 1/h m3/h and m3/s m3 kg/m3 kJ/mol bar m3/kg kJ/mol or kJ/kg kg/m3 kg/m3 kg/m2h, mol/m2h 1/h various

Biological Reaction Engineering, Second Edition. I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30759-1

Nomenclature

k Kca kGa KI KLa kLa

KM

Kp

KW

KS

L M m N N n n OTR OUR P P P

Q

q

R R R

r fi r

i/j

RQ rx S S s T T T L t Tr U V V v

max

w

Constant Gas-liquid mass transfer coefficient Gas film mass transfer coefficient Inhibition constant Gas-liquid mass transfer coefficient Liquid film mass transfer coefficient Michaelis-Menten constant Proportional controller gain constant Dissociation constant of water Monod saturation coefficient Length Mass Maintenance coefficient Mass flux Molar flow rate Number of mols Reaction order Oxygen transfer rate Oxygen uptake rate Pressure Product concentration Output control signal Total transfer rate Specific rate Ideal gas constant Recycle flow rate Residual active biomass Reaction rate Reaction rate of component i Reaction rate of component i to j Respiration quotient Growth rate Concentration of substrate Slope of process reaction curve Stoichiometric coefficient Temperature Enzyme activity Time lag Time Transfer rate Heat transfer coefficient Volume Flow velocity Maximum reaction rate Wastage stream flow rate

various 1/h 1/h kg/m3, kmol/m3 1/h 1/h kg/m3, kmol/m3 various kg/m3 m kg or mol 1/h kg/m2 h mol/h — _ mol/h and kg/h mol/h and kg/h bar kg/m3 and g/m3 various kg/h and mol/h kg/kg biomass h bar m3/ K mol m3/h kg/m3 kg/m3h, kmol/m3h kg i/m3h kg /m3h, kmol/m3h mol CO2/mol ©2 kg biomass/m3 h kg/m3, kmol/m3 various — CorK kg/m3 h, min. or s h, min and s mol/m3 h kJ/m2 C h m3 m/h kmol/m3 h m3/h

Nomenclature

w X Y Yi y Z

Mass fraction Biomass concentration Yield coefficient Yield of i from j Mol fraction in gas Length variable

kg/m3 kg/kg kg i/kg j,mol i/mol j m

Greek 8

a 8 A O

V

p

Z T T T

Controller error Partial differential operator Concentration difference quantity Difference operator Thiele Modulus Effectiveness factor Specific growth rate Maximum growth rate Stoichiometric coefficient Density Summation operator Residence time Controller time constant Electrode time constant

various kg/m3

1/h 1/h

kg/m3 h and s s s

Indices *

Refers to equilibrium concentration Refers to initial, inlet, external, and zero order 1 Refers to time ti, outlet, component 1, tank 1, and first order 2 Refers to tank 2, time t2 and component 2 1,2,..., n Refers to stream, volume elements and stages A Refers to component A, anions and bulk ARefers to anions Refers to ambient a Ac Refers to acetoin and acetoin formation aer Refers to aerobic agit Refers to agitation anaer Refers to anaerobic Refers to apparent app ATP/S,Ac Refers to ATP yield from reaction glucose --> Ac ATP/S,CO2 Refers to ATP yield from glucose oxidation ATP/NADH Refers to ATP produced from NADH ATP/X Refers to consumption rate ATP «> biomass

0

Nomenclature

avg B Bu CO2 d D D E E f G H+ i I I inert K K+ L m m max n NH4 NC>2 NOs O and O2 P PA PB Q Q/O2 Q/S R r r,S s S SL Sn tot X X/i X/S

Refers to average Refers to component B, base, backmixing, and surface position Refers to butanediol Refers to carbon dioxide Refers to deactivation and death Refers to derivative control Refers to D-value in sterilization Refers to electrode Refers to energy by complete oxidation Refers to final Refers to gas and to cellular compartment Refers to hydrogen ions Refers to component i and to interface Refers to inhibitor Refers to integral control Refers to inert component Refers to cellular compartment Refers to cations Refers to liquid Refers to maximum Refers to metabolite Refers to maximum Refers to tank number Refers to ammonium Refers to nitrite Refers to nitrate Refer to oxygen Refers to product Refers to product A Refers to product B Refers to heat Refers to heat-oxygen ratio Refers to heat-substrate ratio Refers to recycle stream Refers to reactor Refers to reaction of substrate Refers to settler Refers to substrate and surface Refers to liquid film at solid interface Refers to substrate n Refers to total Refers to biomass Refers to biomass-component i ratio Refers to biomass-substrate ratio

Nomenclature

Refers to difference between cations and ions Bar above symbol refers to dimensionless variable

1

Modelling Principles

1.1

Fundamentals of Modelling

1.1.1

Use of Models for Understanding, Design and Optimization of Bioreactors

An investigation of bioreactor performance might conventionally be carried out in an almost entirely empirical manner. In this approach, the bioreactor behavior would be studied under practically all combinations of possible conditions of operation and the results then expressed as a series of correlations, from which the resulting performance might hopefully be estimated for any given set of new operating conditions. This empirical procedure can be carried out in a very routine way and requires relatively little thought concerning the actual detail of the process. While this might seem to be rather convenient, the procedure has actually many disadvantages, since very little real understanding of the process would be obtained. Also very many experiments would be required in order to obtain correlations that would cover every process eventuality. Compared to this, the modelling approach attempts to describe both actual and probable bioreactor performance, by means of well-established theory, which when described in mathematical terms, represents a working model for the process. In carrying out a modelling exercise, the modeller is forced to consider the nature of all the important parameters of the process, their effect on the process and how each parameter can be defined in quantitative terms, i.e., the modeller must identify the important variables and their separate effects, which, in practice, may have a very highly interactive combined effect on the overall process. Thus the very act of modelling is one that forces a better understanding of the process, since all the relevant theory must be critically assessed. In addition, the task of formulating theory into terms of mathematical equations is also a very positive factor that forces a clear formulation of basic concepts. Once formulated, the model can be solved and the behavior predicted by the model compared with experimental data. Any differences in performance may Biological Reaction Engineering, Second Edition. I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30759-1

10

1 Modelling Principles

then be used to further redefine or refine the model until good agreement is obtained. Once the model is established it can then be used, with reasonable confidence, to predict performance under differing process conditions, and it can also be used for such purposes as process design, optimization and control. An input of plant or experimental data is, of course, required in order to establish or validate the model, but the quantity of experimental data required, as compared to that of the empirical approach is considerably reduced. Apart from this, the major advantage obtained, however, is the increased understanding of the process that one obtains simply by carrying out the modelling exercise. These ideas are summarized below. Empirical Approach: Measure productivity for all combinations of reactor operating conditions, and make correlations. - Advantage: Little thought is necessary - Disadvantage: Many experiments are required. Modelling Approach: Establish a model, and design experiments to determine the model parameters. Compare the model behavior with the experimental measurements. Use the model for rational design, control and optimization. - Advantage: Fewer experiments are required, and greater understanding is obtained. - Disadvantage: Some strenuous thinking may be necessary.

1.1.2

General Aspects of the Modelling Approach

An essential stage in the development of any model, is the formulation of the appropriate mass and energy balance equations (Russell and Denn, 1972). To these must be added appropriate kinetic equations for rates of cell growth, substrate consumption and product formation, equations representing rates of heat and mass transfer and equations representing system property changes, equilibrium relationships, and process control (Blanch and Dunn, 1973). The combination of these relationships provides a basis for the quantitative description of the process and comprises the basic mathematical model. The resulting model can range from a very simple case of relatively few equations to models of very great complexity. Simple models are often very useful, since they can be used to determine the numerical values for many important process parameters. For example, a model based on a simple Monod kinetics can be used to determine basic parameter values such as the specific growth rate (JLI), saturation constant (Ks), biomass yield coefficient (Yx/s) and maintenance coefficient (m). This basic kinetic data can be supplemented by additional kinetic factors, such as oxygen transfer rate (OTR), carbon dioxide production rate (CPR), respiration quotient (RQ) based on off-gas analysis and related quantities, such as specific oxygen

1.1 Fundamentals of Modelling

11

uptake rate (qo2)> specific carbon dioxide production rate (qco2)> may also be derived and used to provide a complete kinetic description of, say, a simple batch fermentation. For complex fermentations, involving product formation, the specific product production rate (qp) is often correlated as a complex function of fermentation conditions, e.g., stirrer speed, air flow rate, pH, dissolved oxygen content and substrate concentration. In other cases, simple kinetic models can also be used to describe the functional dependence of productivity on cell density and cell growth rate. A more detailed "structured kinetic model" may be required to give an adequate description of the process, since cell composition may change in response to changes in the local environment within the bioreactor. The greater the complexity of the model, however, the greater is then the difficulty in identifying the numerical values for the increased number of model parameters, and one of the skills of modelling is to derive the simplest possible model that is capable of a realistic representation of the process. A basic use of a process model is thus to analyze experimental data and to use this to characterize the process, by assigning numerical values to the important process variables. The model can then also be solved with appropriate numerical data values and the model predictions compared with actual practical results. This procedure is known as simulation and may be used to confirm that the model and the appropriate parameter values are "correct". Simulations, however, can also be used in a predictive manner to test probable behavior under varying conditions; this leads on to the use of models for process optimization and their use in advanced control strategies. The application of a combined modelling and simulation approach leads to the following advantages: 1. Modelling improves understanding, and it is through understanding that progress is made. In formulating a mathematical model, the modeller is forced to consider the complex cause-and-effect sequences of the process in detail, together with all the complex inter-relationships that may be involved in the process. The comparison of a model prediction with actual behavior usually leads to an increased understanding of the process, simply by having to consider the ways in which the model might be in error. The results of a simulation can also often suggest reasons as to why certain observed, and apparently inexplicable, phenomena occur in practice. 2. Models help in experimental design. It is important that experiments be designed in such a way that the model can be properly tested. Often the model itself will suggest the need for data for certain parameters, which might otherwise be neglected, and hence the need for a particular type of experiment to provide the required data. Conversely, sensitivity tests on the model may indicate that certain parameters may have a negligible

12


effect and hence that these effects therefore can be neglected both from the model and from the experimental program. 3.

Models may be used predictively for design and control. Once the model has been established, it should be capable of predicting performance under differing sets of process conditions. Mathematical models can also be used for the design of relatively sophisticated control algorithms, and the model, itself, can often form an integral part of the control algorithm. Both mathematical and knowledge based models can be used in designing and optimizing new processes.

4.

Models may be used in training and education. Many important aspects of bioreactor operation can be simulated by the use of very simple models. These include such concepts as linear growth, double substrate limitation, changeover from batch to fed-batch operation dynamics, fedbatch feeding strategies, aeration dynamics, measurement probe dynamics, cell retention systems, microbial interactions, biofilm diffusion and bioreactor control. Such effects are very easily demonstrated by computer, as shown in the accompanying simulation examples, but are often difficult and expensive to demonstrate in practice.

5. Models may be used for process optimization. Optimization usually involves considering the influence of two or more variables, often one directly related to profits and one related to costs. For example, the objective might be to run a reactor to produce product at a maximum rate, while leaving a minimum amount of unreacted substrate.

1.1.3

General Modelling Procedure

One of the more important features of modelling is the frequent need to reassess both the basic theory (physical model) and the mathematical equations, representing the physical model (mathematical model) in order to achieve the required degree of agreement, between the model prediction and actual plant performance (experimental data). As shown in Fig. 1.1, the following stages in the modelling procedure can be identified: (i) The first stage involves the proper definition of the problem and hence the goals and objectives of the study. These may include process analysis, improvement, optimization, design and control, and it is important that the aims of the modelling procedure are properly defined. All the relevant theory must then be assessed in combination with any practical experience with the process,

13


Physical Model

LJt

Revise ideas and equations

New experiments

Mathematical Model

Solution: C = f(t)

Experimental Data

NO Comparison OK? YES

Use for design, optimization and control

Figure 1.1. Information flow diagram for model building.

and perhaps alternative physical models for the process need to be developed and examined. At this stage, it is often helpful to start with the simplest possible conception of the process and to introduce complexities as the development proceeds, rather than trying to formulate the full model with all its complexities at the beginning of the modelling procedure. (ii) The available theory must then be formulated in mathematical terms. Most bioreactor operations involve quite a large number of variables (cell, substrate and product concentrations, rates of growth, consumption and production) and many of these vary as functions of time (batch, fed-batch operation). For these reasons the resulting mathematical relationships often consist of quite large sets of differential equations. The thick arrow in Fig. 1.1 designates both the importance and the difficulty of this mathematical formulation. (iii) Having developed a model, the model equations must then be solved. Mathematical models of biological systems are usually quite complex and highly non-linear and are such that the mathematical complexity of the

14


equations is usually sufficient to prohibit the use of an analytical means of solution. Numerical methods of solution must therefore be employed, with the method preferred in this text being that of digital simulation. With this method, the solution of very complex models is accomplished with relative ease, since digital simulation provides a very easy and a very direct method of solution. Digital simulation languages are designed specially for the solution of sets of simultaneous differential equations using numerical integration. Many fast and efficient numerical integration routines are now available and are implemented within the structure of the languages, such that many digital simulation languages are able to offer a choice of integration routine. Sorting algorithms within the structure of the language enable very simple programs to be written, having an almost one-to-one correspondence with the way in which the basic model equations were originally formulated. The resulting simulation programs are therefore very easy to understand and also to write. A further major advantage is a convenient output of results, in both tabulated and graphical form, that can be obtained via very simple program commands. (iv) The validity of the computer prediction must be checked and steps (i) to (iii) will often need to be revised at frequent intervals during the modelling procedure. The validity of the model depends on the correct choice of the available theory (physical and mathematical model), the ability to identify the model parameters correctly and the accuracy of the numerical solution method. In many cases, owing to the complexity and very interactive nature of biological processes, the system will not be fully understood, thus leaving large areas of uncertainty in the model. Also the relevant theory may be very difficult to apply. In such cases, it is then often very necessary to make rather gross simplifying assumptions, which may subsequently be eliminated or improved as a better understanding is subsequently obtained. Care and judgement must also be used such that the model does not become over complex and so that it is not defined in terms of too many immeasurable parameters. Often a lack of agreement between the model and practice can be caused by an incorrect choice of parameter values. This can even lead to quite different trends being observed in the variation of particular parameters during the simulation. It should be noted, however, that often the results of a simulation model do not have to give an exact fit to the experimental data, and often it is sufficient to simply have a qualitative agreement. Thus a very useful qualitative understanding of the process and its natural cause-and-effect relationships is obtained.


15

1.1.4 Simulation Tools Many different digital simulation software packages are available on the market for PC and Mac application. Modern tools are numerically powerful, highly interactive and allow sophisticated types of graphical and numerical output. Most packages also allow optimisation and parameter estimation. BERKELEY MADONNA is very user-friendly and very fast. We have chosen it for use in this book, and details can be found in the Appendix. With it data fitting and optimisation can be done very easily. MODELMAKER is also a more recent, powerful and easy to use program, which also allows optimisation and parameter estimation. ACSL-OPTIMIZE has quite a long history of application in the control field, and also for chemical reaction engineering. MATLAB-SIMULINK is a popular and powerful software for dynamic simulation and includes many powerful algorithms for non-linear optimisation, which can also be applied for parameter estimation.

1.1.5

Teaching Applications

For effective teaching, the introduction of computer simulation methods into modelling courses can be achieved in various ways, and the method chosen will depend largely on how much time can be devoted, both inside and outside the classroom. The most time-consuming method for the student is to assign modelling problems to be solved outside the classroom on any available computer. If scheduling time allows, computer laboratory sessions are effective, with the student working either alone or in groups of up to three on each monitor or computer. This requires the availability of many computers, but has the advantage that pre-programmed examples, as found in this text, can be used to emphasize particular points related to a previous theoretical presentation. This method has been found to be particularly effective when used for short, continuing-education, professional courses. By use of the computer examples the student may vary parameters interactively and make program alterations, as well as working through the suggested exercises at his or her own pace. Demonstration of a particular simulation problem via a single personal computer and video projector is also an effective way of conveying the basic ideas in a short period of time, since students can still be very active in suggesting parametric changes and in anticipating the results. The best approach is probably to combine all three methods.

16

1.2


Development and Meaning of Dynamic Differential Balances

As indicated in Section 1.1, many models for biological systems are expressed in terms of sets of differential equations, which arise mainly as a result of the predominantly time-dependent nature of the process phenomena concerned. For many people and especially for many students in the life sciences, the mention of differential equations can cause substantial difficulty. This section is therefore intended, hopefully, to bring the question of differential equations into perspective. The differential equations arise in the model formulation, simply by having to express rates of change of material, due to flow effects or chemical and biological reaction effects. The method for solution of the differential equations will be handled automatically by the computer. It is hoped that much of the difficulty can be overcome by considering the following case. In this section a simple example, based on the filling of a tank of water, is used to develop the derivation of a mass balance equation from the basic physical model and thereby to give meaning to the terms in the equations. Following the detailed derivation, a short-cut method based on rates is given to derive the dynamic balance equations. Consider a tank into which water is flowing at a constant rate F (m3/s), as shown in Fig. 1.2. At any time t, the volume of water in the tank is V (m3) and the density of water is p (kg/m3).

Figure 1.2. Tank of water being filled by stream with flow rate F.

During the time interval At (s), a mass of water p F At (kg) flows into the tank. As long as no water leaves the tank, the mass of water in the tank will increase by a quantity p F At, causing a corresponding increase in volume, AV. Equating the accumulation of mass in the tank to the mass that entered the tank during the time interval A t gives, pAV = p F A t Since p is constant, At

-F h -

1.2 Development and Meaning of Dynamic Differential Equations

17

Applying this to very small differential time intervals (At —> dt) and replacing the A signs by the differential operator "d", gives the following simple first order differential equation, to describe the tank filling operation, dV dT = F

What do we know about the solution of this equation? That is, how does the volume change with time or in model terms, how does the dependent variable, V, change with respect to the independent variable, t? To answer this, we can rearrange the equation and integrate it between appropriate limits to give,

t

or for constant F,

o

= F Jtf l l dt= F(ti-to) o

Integration is equivalent to summing all the contributions, such that the total change of volume is equal to the total volume of water added to the tank, IV = IF At

For the case of constant F, it is clear that the analytical solution to the differential equation is, V = F t + constant In this case, as shown in Fig. 1.3, the constant of integration is the initial volume of water in the tank, VQ, at time t = 0.

dt

Vo

Figure 1.3. Volume change with time for constant flow rate.

18


Note that the slope in the variation of V with respect to t, dV/dt, is constant, and that from the differential equation it can be seen that the slope is equal to F. Suppose F is not constant but varies linearly with time. F = Fo-kt The above model equation applies also to this situation. Solving the model equation to obtain the functional dependence of V with respect to t, JdV = |F dt = J(F0 - k t)dt = FO Jdt - k Jt dt Integrating analytically, V = FO t -

kt"

+ constant

The solution is,

v = FO t - kt' — + V0

."t Figure 1.4. Variation of F and V for the tank-filling problem.

Note that the dependent variable starts at the initial condition, (Vo), and that the slope is always F. When F becomes zero, the slope of the curve relating V and t also becomes zero. In other words, the volume in the tank remains constant and does not change any further as long as the value of F remains zero. Derivation of a Balance Equation Using Rates A differential balance can best be derived directly in terms of rates of change. For the above example, the balance can then be expressed as: /The rate of accumulation^ V of mass within the tank )

=

/The flow rate of mass^ \ entering the tank )

1.2 Development and Meaning of Dynamic Differential Equations

19

Thus, the rate of accumulation of mass within the tank can be written directly as dM/dt where the mass M is equal to p V. The rate of mass entering the tank is given by p F, where both sides of the equation have units of kg/h. =

PF

and

d(oV) = pF Thus this approach leads directly to a differential equation model, which is the desired form for dynamic simulation. Note that both terms in the above relationship are expressed in mass quantities per unit time or kg/h. At constant density, the equation again reduces to,

dV dT = F which is to be solved for the initial condition, that at time t = 0, V=Vo and for a variation in flow rate, given by, F = F0-kt which is valid until F = 0. These two equations, plus the initial condition, form the mathematical representation or the mathematical model of the physical model, represented by the tank filling with an entering flow of water. Thus this approach leads directly to a differential equation model, which is the desired form for simulation. This approach can be applied not only to the total mass but also to the mass of any component. We have seen an analytical solution to this model, but it is also interesting to consider how a computer solution can be obtained by a numerical integration of the model equations. This is important since analytical integration is seldom possible in the case of real complex problems. Computer Solution The numerical integration can in principle be performed using the relations:

dV

20


where t - to represents a very small time interval and V - VQ is the resulting change in volume of the water in the tank. As before, the flow is assumed to decrease with time according to F = FQ - k t. This integration procedure is equivalent to the following steps: 1) Setting the integration time interval. 2) Assigning a value to the inlet water flow rate at the initial value, time t = to3) The term involving the water flow rate, F-kt, is equal to the derivative value, (dV/dt), at time t = to. 4) Knowing the initial value of V and the slope dV/dt, enables a new value of V to be calculated over the small interval of time, equivalent to the integration time interval or integration step length. 5) At the end of the integration time interval, the value of V will have changed to a new value, representing the change of V with respect to time from its original value. The new value of V can thus be calculated. 6) Using the new value of V, a new value for the rate of change of V with respect to time, (dV/dt), at the end of the integration time interval can now be calculated. 7) Knowing the value of V and the value of dV/dt at the end of the integration time interval, a new value of V can be estimated over a further step forward in time or integration time interval. 8) The entire procedure, as represented by steps (2) to (7) in Fig. 1.5 below, is then repeated with the calculation moving forward with respect to time, until the value of F reaches zero. At this point the volume no longer increases, and the resulting steady-state value of V is obtained, including all the intermediate values of V and F, which were determined during the course of the calculation.

Figure 1.5. Graphical portrayal of numerical integration, showing slopes and approximated values of V at each time interval.

1.3 Formulation of Balance Equations

21

Using such a numerical integration procedure, the computer can thus be used to generate data concerning the time variations of both F and V. In practice, more complex numerical procedures are employed in digital simulation languages to give improved accuracy and speed of solution than illustrated by the above simplified integration technique.

1.3

Formulation of Balance Equations

1.3.1

Types of Mass Balance Equations

Steady-State Balances One of the basic principles of modelling is that of the conservation of mass, which for a steady-state flow process can be expressed by the statement,

(

Rate of mass flow^ into the system J

f Rate of mass flow^j ^ out of the system J

Dynamic Total Mass Balances Many bioreactor applications are, however, such that conditions are in fact changing with respect to time. Under these circumstances, a steady-state mass balance is inappropriate and must be replaced by a dynamic or unsteady-state mass balance, which can be expressed as:

(

Rate of accumulation of ^ mass in the system J

( Rate of ^ ^mass flow inj

( Rate of ^ ^mass flow out J

Here the rate of accumulation term represents the rate of change in the total mass of the system, with respect to time, and at steady-state this is equal to zero. Thus the steady-state mass balance represented earlier is seen to be a simplification of the more general dynamic balance, involving the rate of accumulation. At steady-state:

22


( Rate of ^ = 0 = (Mass flow in) - (Mass flow out) I accumulation of mass . hence, when a steady-state is reached: (Mass flow in) = (Mass flow out) Component Balances The previous discussion has been in terms of the total mass of the system, but most fluid streams, encountered in practice, contain more than one chemical or biological species. Provided no chemical change occurs, the generalized dynamic equation for the conservation of mass can also be applied to each component. Thus for any particular component: Rate of accumulation of mass of component in the system

Mass flow of A the component _ into the system J

(

Mass flow of > the component out of the system ,

(

Component Balances with Reaction Where chemical or biological reactions occur, this can be taken into account by the addition of a further reaction rate term into the generalized component balance. Thus in the case of material produced by the reaction: ' Mass flow ^ ^ Mass flow ^ Rate of ^ accumulation of the of the = of mass component - component + of component into out of ^the system, ^the system, ^ in the system, '

Rate of production of the component by the reaction>

and in the case of material consumed by the reaction: Rate of ' accumulation of mass = of component în the system,

Mass flow ' Mass flow of the of the component - component into out of ^the system^ v the system,

Rate of consumption of the component by the reaction


23

Elemental Balances The principle of the mass balance can also be extended to the atomic level and applied to particular elements. Thus in the case of bioreactor operation, the general mass balance equation can also be applied to the four main elements, carbon, hydrogen, oxygen and nitrogen and also to other elements if relevant to the particular problem. Thus for the case of carbon: ' Rate of accumulation^! (Mass flowrate of "\ ( Mass flow rate of \ of carbon in = carbon into the carbon out the system J ^ the system J V of the system ) Note the elemental balances do not involve reaction terms since the elements do not change by reaction. The computer example PENFERM, is based on the use of elemental mass balance equations for C, H, O and N which, when combined with other empirical rate data, provide a working model for a penicillin production process. While the principle of the mass balance is very simple, its application can often be quite difficult. It is important therefore to have a clear understanding of both the nature of the system (physical model), which is to be modelled by means of the mass balance equations, and also of the methodology of modelling.

1.3.2

Balancing Procedure

The methodology described below outlines six steps, I through VI, to establish the model balances. The first task is to define the system by choosing the balance or control region. This is done using the following procedure: I.

Choose the balance region such that the variables are constant or change little within the system. Draw boundaries around the balance region

The balance region may be a reactor, a reactor region, a single phase within a reactor, a single cell, or a region within a cell, but will always be based on a region of assumed constant composition. Generally the modelling exercises will involve some prior simplification. Often the system being modelled is usually considered to be composed of either systems of tanks (stagewise or lumped

24


parameter systems) or systems of tubes (differential systems), or even combinations of tanks and tubes, as used in Case C, Sec. 1.3.2.3.

1.3.2.1

Case A. Continuous Stirred Tank Bioreactor

AO Total mass = pV Mass of A = C VA

Balance region

Figure 1.6. The balance region around the continuous reactor.

If the tank is well-mixed, the concentrations and density of the tank contents are uniform throughout. This means that the outlet stream properties are identical with the tank properties, in this case CA and p. The balance region can therefore be taken around the whole tank. The total mass in the system is given by the product of the volume of the tank contents V (m3) multiplied by the density p (kg/m3), thus Vp (kg). The mass of any component A in the tank is given as the product of V times the concentration of A, CA (kg of A/m3 or kmol of A /m3), thus V CA (kg or kmol).

1.3.2.2

Case B. Tubular Reactor Balance region

'AO

A1

Figure 1.7. The tubular reactor concentration gradients.

In the case of tubular reactors, the concentrations of the products and reactants will vary continuously along the length of the reactor, even when the reactor is operating at steady-state. This variation can be regarded as being equivalent to

25


that of the time of passage of material as it flows along the reactor and is equivalent to the time available for reaction to occur. Under steady-state conditions the concentration at any position along the reactor will be constant with respect to time, though not with position. This type of behavior can be approximated by choosing the balance regions sufficiently small so that the concentration of any component within a region can be assumed to be approximately uniform. Thus in this case, many uniform property subsystems (well-stirred tanks or increments of different volume but of uniform concentration) comprise the total reactor volume.

13.2.3

Case C. River with Eddy Current

For this example, the combined principles of both the stirred tank and differential tubular modelling approaches need to be applied. As shown in Fig. 1.8 the main flow along the river is very analogous to that of a column or tubular process, whereas the eddy region can be approximated by the behavior of a well-mixed tank. The interaction between the main flow of the river and the eddy, with flow into the eddy from the river and flow out from the eddy back into the river's main flow, must be included in any realistic model. The real-life and rather complex behavior of the eddying flow of the river, might thus be represented, by a series of many well-mixed subsystems (or tanks) representing the main flow of the river. This interacts at some particular stage of the river with a single well-mixed tank, representing the turbulent eddy. In modelling this system by means of mass balance equations, it would be necessary to draw boundary regions around each of the individual subsystems representing the main river flow, sections 1 to 8 in Fig. 1.9, and also around the tank system representing the eddy. This would lead to a very minimum of nine River

Eddy

Figure 1.8. A complex river flow system.

26


component balance equations being required. The resulting model could be used, for example, to describe the flow of a pollutant down the river in rather simple terms. 1

>

„ *

1

', i

1

2 2

', i

1

3 3

I

1

1

4 4

i

1

, i

>k V

5 5

1

, i

I

66

1

, i

1

7 7

1

, i

>

8 8

fc

C

r

Flow interaction

f

fitSliM^B

:i

iK&^'ffK

F

i>yi> T i> PI

Air F

0'yO'T0'P0

Figure 1.18. Entering air and exit gas during the continuous aeration of a bioreactor.

Writing a balance around the combined gas and liquid phases in the reactor gives, f Rate of accum-^j_ f Flowrateî ( Flowrate"\ /^Rate of O2 uptake A ( ulationofO 2 J~ [ofO 2 in J~(ofO 2 out J~ I by the cells J


39

At steady-state, the accumulation terms for both phases are zero and Flow of O2 in - Flow of C>2 out = Rate of C>2 uptake. For gaseous systems, the quantities are often expressed in terms of molar quantities. Often only the inlet air flow rate FQ and the mol fraction of 62 in the outlet gas, yi9 are measured. It is often assumed that the total molar flow rate of gas is constant. This is a valid assumption as long as the number of carbon dioxide mols produced is nearly equal to the number of oxygen mols consumed or if the amounts of oxygen consumed are very small, relative to the total flow of gas. Converting to molar quantities, using the Ideal Gas Law, pV = nRT or in flow terms: pF = N R T where N is the molar flow rate, R is the gas constant and F is the volumetric flow rate. Thus, for the inlet gas flow: P

°

where NO is molar flow rate of the oxygen entering. Note that the pressure, po, and temperature, TO, are measured at the point of flow measurement. Assuming NO = NI, then measurement of NO gives enough information to calculate oxygen uptake rate, OUR, from the steady-state balance. Thus, 0 = yo NO - yi NI - ro2 VL

OUR = ro2 VL = yo NO - yi NI If NO is not equal to NI, then this equation will give large errors in oxygen uptake rate, and NI must be measured, or determined indirectly by an inert balance. This is explained in the Sec. 1.3.4.4 below.

1.3.4.4

Case D.

Inert Gas Balance to Calculate Flow Rates

Differences in the inlet and outlet gas flow rates of a tank fermenter can be calculated by measuring one gas flow rate and the mol fraction of an inert gas in the gas stream. Since inert gases, such as nitrogen or argon, are not consumed or produced within the system (rinert = 0), their mass rates must

40


therefore be equal at the inlet and outlet streams of the reactor, assuming steady-state conditions apply. Then for nitrogen /Molar flow of\ V nitrogen in )

=

/Molar flow of\ ^ nitrogen out /

=

NI yi

and in terms of mol fractions, NO YO inert

inert

From this balance, calculation of NI can be made on the basis of a combination of measurements of NO and the inert gas partial pressures (yinertX at both inlet and outlet conditions. N,1 =

y i inert

Since the inlet mol fraction for nitrogen in air is known, the outlet mol fraction, yi inert' must be measured. This is often done by difference, having measured the mol fraction of oxygen and carbon dioxide concentration in the exit gas.

1.3.5

Stoichiometry, Elemental Balancing and the Yield Coefficient Concept

Stoichiometry is the basis for any quantitative treatment of chemical and biochemical reactions. In biochemical processes it is a necessary basis for building kinetic models.

1.3.5.1

Simple Stoichiometry

The Stoichiometry of chemical reactions is used to relate the relative quantities of the different materials which react with one another and also the relative quantities of product that are formed. Most chemical and biochemical reactions are relatively simple in terms of their molar relationship or Stoichiometry. For single reactions stoichiometric coefficients are clearly defined and may usually easily be determined. Some examples are given below: C3H4O3 + NADH + H+ CH3CO-S-CoA + CO2 + NADH + H+ Pyruvic Acid Acetyl-CoA Stoichiometry relations also describe more complex pathways and can be written with exact molar relationships, like the pentose-phosphate pathway below. Glucose + 12 NADP+ + ATP + 7 H2O -> 6 CO2 + 12 (NADPH + H+) + + ADP + Pi where 1 mol of glucose reacted, consumes 7 mol of water and produces 6 mol of carbon dioxide. Here the molar quantities of NADPH and ATP produced and consumed, respectively, are shown. For many complex biological reactions, however, not all the elementary reactions and their contributions to the overall observed reaction stoichiometry are known (Roels, 1983; Bailey and Ollis, 1986; Moser, 1988). Thus the case of a general fermentation is usually approximated by an overall reaction equation, where Substrate + Nitrogen source + O2 -> Product + CO2 + H2O v N H3(t)NH 3 + v02(t)O2

>

VC02(t) CO2 + VH2o(0 H2O

where the i-th product, such as metabolites or biomass, is given by a general formula. In the case above, the generalized elemental formulae are used for substrate, biomass and products, but the nitrogen source is given simply as ammonia. The stoichiometric coefficients, v, for each component are taken relative to that of substrate and their coefficients may vary as a function of time as a result of changing fermentation conditions. Some indication as to the relative magnitudes of the stoichiometric coefficients can be obtained from elemental balancing techniques, but in general the problem is so complex that other concepts, such as the more approximate yield coefficient concept, are used to relate the relative proportions of materials undergoing conversion during the fermentation.

42


1.3.5.2

Elemental Balancing

The technique of elemental balancing can be represented as follows: Taking the general case of CHmOi + a NH3 + b 02 —> [substrate]

c CHpOnNq + d CHrOsNt + e H2O + f CO2 [biomass] [product]

where c, d and f are the fractions of carbon converted to biomass, product and CO2, respectively. Elemental balances for C, H, O and N give C H O N

1 m+3 a l+2b a

= c +d +f = c p + dr + 2e =cn + ds + e + 2f =cq+dt

In this general problem there are too many unknowns for the solution method to be taken further, since the elemental balances provide only four equations and hence can be solved for only four unknowns. Assuming that the elemental formulae for substrate, biomass and product and hence 1, m, n, p, q, r, s and t are defined, there still remain six unknown stoichiometric coefficients a, b, c, d, e and f and only four elemental balance equations. Thus the elemental balances need supplementation by other measurable quantities such as substrate, oxygen and ammonia consumption rates (assuming controlled pH conditions), and carbon dioxide or biomass production rates, such that the condition is satisfied that the number of unknowns is equal to the number of defining equations. In principle the problem then becomes solvable. In practice, there can be considerable difficulties and inaccuracies involved, although the technique of elemental balancing can still provide useful data. The application of so-called macroscopic principles (Roels, 1980, 1982 and 1983; Heijnen and Roels, 1981) introduces a more strict systematic system of analysis. This is depicted in Fig. 1.19.

43


2, H2O and NH3 upon oxidation. Taking charge numbers: C = 4, H = 1 , O = -2, and N = -3, reductance degrees (y) can be defined for substrate (S) biomass (X) product (P)

ys = 4 + m - 2 1 yx = 4 + p - 2 n - 3 q yp = 4 + r - 2 s - 3 t

The reductances for NHs, fÔ and CC>2 are of course zero. Often the elemental composition of the substrate is not known and then the reductance method may be supplemented by the following regularities, which apply to a wide variety of organic molecules. Qo2 = 27 J per g equivalent of available electrons transferred to oxygen Yx = 4.29 g equivalent of available electrons per equivalent 1 g atom C in biomass GX = 0.462 g carbon / g dry biomass

1.3.5.3

Mass Yield Coefficients

Yield coefficients are biological variables, which are used to relate the ratio between various consumption and production rates of mass and energy. They are typically assumed to be time-independent and are calculated on an overall basis. This concept should not be confused with the overall yield of a reaction or a process. The biomass yield coefficient on substrate (Yx/s) is defined as: Y v x/S = rs

In batch systems, reaction rates are equal to accumulation rates, and therefore

/dX\ Y

IdTj

dX

X/S = - TdST = - dS" IdTj

After integration from time 0 to time t the integral value is obtained:


..

* x/s

=

45

amount of biomass produced total amount of substrate consumed X(t)-X(t=0) S(t=0)-S(t)

For a steady-state continuous system the mass balances give rs

SQ-S!

where index 0 and 1 indicate feed and effluent values, respectively. In the literature, yield coefficients for biomass with respect to nutrients are most often used (e.g. Dekkers, 1983; Mou and Cooney, 1983; Roels, 1983; Moser, 1988). In many cases this is very useful because the biomass composition is quite uniform, and often product selectivity does not change very much during an experiment involving exponential growth and associated production. Some useful typical values are given in Table 1.1.

1.3.5.4

Energy Yield Coefficients

Energy yield coefficients may be defined similarly to mass yield coefficients. In terms of oxygen uptake, TO amount of heat released Yq/02 = — = — 7 TQ2 amount or oxygen consumed In terms of carbon substrate consumed, v

- rQ rs

amount of heat released amount of carbon source consumed

46


Table 1.1. Typical mass and energy yield values (Roels, 1983; Atkinson and Mavituna, 1991). Type of yield coefficient

Dimension

Value

Y

c-mol / c-mol c-mol / c-mol c-mol / mol c-mol / mol kJ / mol kJ / mol kJ / c-mol kJ / c-mol

0.4-0.7 0.1-0.2 1-2 0.35 380-490 460 325-500 120-190

X/S,aer Yx/S,anaer

Yx/02 (Glucose) YX/ATP Y

Q/O2 YQ/C02

YQ/x,aer (Glucose) Yq/x,anaer

Note: The molecular weight of biomass is taken here as 24.6 g/C-mol The yield coefficients are usually determined as a result of a large number of elementary biochemical reactions and it can easily be understood that their values might vary depending on environmental and operating conditions. A detailed description of some of these dependencies is given in the literature. Despite this inconsistency, measured yield coefficients are often very useful for practical purposes of process description and modelling.

1.3.6

1.3.6.1

Equilibrium Relationships

General Considerations

In many biological systems, processes with large ranges of time constants have to be described. Usually it is important to start with a simplification of a system, focusing on the most important time constant or rate. For example, if the growth of an organism is to be modelled with a time constant of the order of hours, it is very useful to ignore all aspects of biological evolution with time constants of years. Also fast equilibrium reactions or conformational changes of proteins having time constants below milliseconds should be ignored. Fast reactions can, however, be very important when considering allosteric activation or deactivation of proteins or simply pH-changes during biochemical reactions.


47

pH changes can have dramatic effects on the enzyme and microbial activity but can also strongly influence absorption and desorption of carbon dioxide. A typical equilibrium reactions is the dissociation of a receptor-protein ligand complex, RL, into the free protein, L, and the receptor protein, P

This reaction is characterized by the corresponding dissociation equilibrium constant KD

CLP

k_j

In most cases such relationships can be used to express the concentration of all concentrations in explicit form using, e.g. a protein balance. r1 _i_ c* C Ptot — ~ ^P ~r ^LP

^

^ KD - ^ptot Cr + K

D

The total concentration, Cptot, is then included in a material balance equation and the concentrations of the free receptor and the receptor-ligand complex are determined by the equilibrium relationship. This is also true for a simple acidbase equilibrium relationship. In more complex cases with interactions of various receptors or with a buffer system containing several components, it is not possible to express the concentrations in explicit forms and a non-linear algebraic equation has to be solved during the simulation. The implementation of such problems into BerkeleyMadonna is shown below with the example of pH calculation

1.3.6.2

Case A.

Calculation of pH with an Ion Charge Balance.

Modelling systems with variable pH requires modelling of acid-base equilibria, whose reactions are almost instantaneous. Production of acids or bases causes a variation of pH, which depends on the buffer capacity of the system. pH also influences the biological kinetics. It has been shown that only the undissociated acid forms are kinetically important substrates in anaerobic systems. The

48


concentration of these species is a function of the pH as can be seen in the equilibrium equation Base' + H+

Acid •£ with dissociation constant

CBase- H+ CAcid

where CAcid is the concentration of the undissociated acid and CBase" is the concentration of the corresponding base (salt). An ion charge balance can be written £ (cations * charge)

= £ (anions * charge)

In the pH range of interest (usually around pH = 7) all strong acids and strong bases are completely dissociated. Moderately strong acids and bases exist in both the dissociated and non-dissociated forms, In the usual pH range the sum of the cations are much larger than the H+ ions.

ICK+»CH+ where ]£CK+ is the total cation concentration. Negative ions originate mainly from strong acids (e.g. Cl% SO42') but also arise from weak acids (Ac", Pr, Bu~, HCO3'). The concentration of CO32' is always much smaller than that of The ion balance reduces to

KBj

KW

C

C

Btot,i+£ K+ =

V1

KAi

where KAI are the acid dissociation constants (e.g. KAC); KBI are the base dissociation constants (e.g. KNHS); KW is the dissociation constant of water; Cfitot,i are the total concentrations of base i; CAtot,i are the total concentrations of acid i and EC An" is the sum of the anions. The pH can be estimated from the above equation for any situation by solving the resulting non-linear implicit algebraic equation, provided the total concentrations of the weak acids, CAtot,i> weak bases, CBtot,i» cations of strong bases, CK+, and anions of strong acids, CAIT> are known.

49


It is convenient to use only the difference between cations and anions

After neglecting any ammonia buffering effect, it is useful to rearrange the above equations in the form,

The example ANAMEAS, Sec. 8.8.6 includes this ion balance for pH calculation. This equation represents an algebraic loop in a dynamic simulation which is solved by iteration at each time interval until 8 approaches zero. This is accomplished with the root-finding feature of Berkeley Madonna. If there is pH control, then strong base or acid would be usually added. The addition of strong alkali for pH control would cause an increase in £CK+ » which in accordance with the above equation would result in a decrease of CH+. An alternative approach, which avoids an algebraic loop, is to treat the instantaneous equilibrium reactions as reactions with finite forward and backward rates. These rates must be adjusted with their kinetic constants to maintain the equilibrium for the particular system; that is, these rates must be very fast compared with the other rates of the model. This approach replaces the algebraic loop iteration with a stiff er and larger set of differential equations. This could be an advantage in some cases.

1.3.7

Energy Balancing for Bioreactors

Energy balances are needed whenever temperature changes are important, as caused by reaction heating effects or by cooling and heating for temperature control. For example, such a balance is needed when the heat of fermentation causes a variation in bioreactor temperature. Energy balances are written following the same set of rules as given above for mass balances in Sec. 1.3. Thus the general form is as follows: Accumu-^ lation rate of Ênergy ,

Rate of^ 'Rate of ^ 'Rate of" energy energy energy out by out by in by ^flow , flow

'Rate of > energy generated ^by reactiony

'Rate of > energy added by âgitation ^

50


The above balance in word form is now applied to the measurable energy quantities of the continuous reactor shown in Fig. 1.20. AL H » agit

\ M)'

PO

P1»

' '

U,A,T S

l! Figure 1.20. A continuous tank fermenter showing only the energy-related variables. An exact derivation of the energy balance was given by Aris (1989) as,

S

- h n ) ) + U A (Ta - TI) + rQ V + AHagit agi

"dT =

where ni is the number of moles of component i, cpi are the partial molar heat capacities and hi are the partial molar enthalpies. In this equation the rate of heat production, TQ, takes place at temperature TI. If the heat capacities, cpi, are independent of temperature, the enthalpies at TI can be expressed in terms of heat capacities as

hn = hio + Cpi (TI -TO)

and with S

2>i0 1=1

S

- 2X 1=1

= vp

Thus with these simplifications, Vpc p

L

=

F0 p cp (T0 - TI) + U A (Ta - TI) +rQ V + AHagi

The units of each term of the equation are energy per time (kJ/h or kcal/h).


51

Accumulation Term Densities and heat capacities of liquids can be taken as essentially constant. dT VpcPdF

has units: m3 (kg/m3) (J/kg K) K s

_ ~

kJ s

Here (p cp T) is an energy "concentration" term and has the units, /jnass\ / energy \ _ /energy \ Vvolumey Vmass degree^ VaegreeJ - Vyolume,/ Thus the accumulation term has the units of energy/time (e.g. J/s) Flow Terms The flow term is F p CP (T0 - TI)

with the units, /energy i^nn^J\ /volume\ \ctisr) = /energy \ This term actually describes heating of the stream entering the system with TO to the reaction temperature TI. It is important to note here that this term is exactly the same for a continuous reactor as for a fed-batch system. Heat Transfer Term The important quantities in this term are the heat transfer area A, the temperature driving force or difference (Ta-Ti), where Ta is the temperature of the heating or cooling source, and the overall heat transfer coefficient, U. The heat transfer coefficient, U, has units of energy/time area degree, e.g. J/s m2 °C. The units for U A AT are thus, (heat transfer rate) = U A (Ta - TI) energy ~B55~

=

energy area time degree (area) (degree)

The sign of the temperature difference determines the direction of heat flow. Here if T a > TI heat flows into the reactor. Reaction Heat Term The term rq V gives the rate of heat released by the bioreaction and has the units of

52


energy _ energy volume time ( volume ) - time The rate term TQ can alternatively be written in various ways as follows: In terms of substrate uptake and a substrate-related heat yield, rq = rs YQ/S

In terms of oxygen uptake and an oxygen-related heat yield, rq = ro2YQ/Q2 In terms of a heat of reaction per mol of substrate and a substrate uptake rate, rQ = AHr,s rs Here rs is the substrate uptake rate and AHr?s is the heat of reaction for the substrate, for example J/mol or kcal/kg. The rs AHr>s term therefore has dimensions of (energy/time volume) and is equal to TQ. Other Heat Terms The heat of agitation may be the most important heat effect for slow growing cultures, particularly with viscous cultures. Other terms, such as heat losses from the reactor due to evaporation, can also be important.

1.3.6.3

Case B.

Determining Heat Transfer Area or Cooling Water Temperature

For aerobic fermentation, the heats of reaction per unit volume of reactor are usually directly related to the oxygen uptake rate, ro2Thus for a constant-volume batch reaction with no agitation heat effects, the general energy balance is /Accumulation rate^ V of energy )

/Energy out^ ~ ~ \ by transfer J

+

/Energy generated\ V by reaction )

where YQ/Q2 often has a value near 460 kJ/mol ©2, as given in Table 1.1.


53

If T is constant (dT/dt = 0): UA(Ti-Ta) = r 0 2YQ/o 2 V (heat transfer rate) = (rate of heat release) Using this steady-state energy balance, it is possible to calculate the cooling water temperature (Ta) for a given oxygen uptake rate and cooling device. Thus, _-ro 2 UA Alternatively this same relation can be used in other ways: 1) To calculate the additional heat transfer area required for a known increase in cooling water temperature. 2) To calculate the biomass concentration allowable for a given cooling system, knowing the specific oxygen uptake rate (kg O2 / kg biomass h). 3) To calculate the cooling area required for a continuous fermenter with known volume inlet, temperature, flow rate and biomass production rate.

2

Basic Bioreactor Concepts

2.1 Information for Bioreactor Modelling Both physical and biological information are required in the design and interpretation of biological reactor performance, as indicated in Fig. 2.1. Physical factors that affect the general hydrodynamic environment of the bioreactor include such parameters as liquid flow pattern and circulation time, air distribution efficiency and gas holdup volume, oxygen mass transfer rates, intensity of mixing and the effects of shear. These factors are affected by the bioreactor geometry and that of the agitator (agitator speed, effect of baffles) and by physical property effects, such as liquid viscosity and interfacial tension. Both can have a large effect on gas bubble size and a corresponding effect on both liquid and gas phase hydrodynamics. The biokinetic input involves such factors as cell growth rate, cell productivity and substrate uptake rate. Often this information may come from laboratory data, obtained under conditions which are often far removed from those actually existing in the large scale bioreactor. Although shown as separate inputs in Fig. 2.1, there are, in fact, considerable interactions between the bioreactor hydrodynamic conditions and the cell biokinetics, morphology and physiology, and one of the arts of modelling is to make proper allowance for such effects. Thus in the large scale bioreactor, some cells may suffer local starvation of essential nutrients owing to a combination of long liquid circulation time and an inadequate rate of nutrient supply, caused by inadequate mixing or inefficient mass transfer. Agitation and shear effects can affect cell morphology and hence liquid viscosity, which will also vary with cell density. This means that the processes of cell growth affect the bioreactor hydrodynamics in a very complex and interactive manner. Changes in the cell physiology, such that the cell processes are switched from production of further biomass to that of a secondary metabolite or product, can also be affected by selective limitation on the quantity and rate of supply of some essential nutrient in the medium. This can in turn be influenced by the bioreactor hydrodynamics and also by the mode of the operation of the bioreactor. The overall problem is therefore very complex, but as seen in Figure 2.1, when all the information is combined successfully in a realistic and well founded Bioreactor Model, the results obtained can be quite impressive and Biological Reaction Engineering, Second Edition, I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30759-1

56

2 Basic Bioreactor Concepts

may enable such factors as cell and product production rates, product selectivities, optimum process control and process optimization to be determined with some considerable degree of confidence. Physical Aspects (flow patterns, residence time, mass transfer)

Biokinetics (order, inhibition,pH, temperature)

Production rate Selectivity Control Figure 2.1. Information for bioreactor modelling.

2.2

Bioreactor Operation

The rates of cell growth and product formation are, in the main, dependent on the concentration levels of nutrients and products within the bioreactor. The concentration dependencies of the reaction or production rate are often quite simple, but may also be very complex. The magnitude of the rates, however, depend upon the level of concentrations, and it will be seen that concentration levels within the bioreactor depend very much on its type and mode of operation. Differing modes of operation for the bioreactor can therefore lead to differing rates of cell growth, to differing rates of product formation and hence to substantially differing productivities. Generally, the various types of bioreactor can be classified as either stirred tank or tubular and column devices and according to the mode of operation as batch, semi-continuous or continuous operation.

57

2.2 Bioreactor Operation

2.2.1 Batch Operation Most industrial bioreactors are operated under batch conditions. In this, the bioreactor is first charged with medium, inoculated with cells, and the cells are allowed to grow for a sufficient time, such that the cells achieve the required cell density or optimum product concentrations. The bioreactor contents are discharged, and the bioreactor is prepared for a fresh charge of medium. Operation is thus characterized by three periods of time: the filling period, the cell growth and cell production period and the final emptying period as depicted in Fig. 2.2. It is only during the reacting period, that the bioreactor is productive. During the period of cell growth, strictly speaking, no additional material is either added to or removed from the bioreactor, apart from minor adjustments needed for control of pH or foam, small additions of essential precursors, the removal of samples and, of course, a continuous supply of air needed for aerobic fermentation. Concentrations of biomass, cell nutrients and cell products thus change continuously with respect to time, as the various constituents are either produced or consumed during the time course of the fermentation, as seen in Fig. 2.3.

Filling

Reacting

Emptying

Cleaning

Figure 2.2. Periods of operation for batch reactors.

concentration

A

ubstrate

biomass

product

time Figure 2.3. Concentration-time profiles during batchwise operation.

58

2 Basic Bioreactor Concepts

During the reaction period, there are changes in substrate and product concentration with time, and the other time periods are effectively lost as regards production. Since there is no flow in or out of the bioreactor, during normal operation, the biomass and substrate balances both take the form, (Rate of accumulation within the reactor) = (Rate of production) This will be expressed in more quantitative terms in Ch. 4. Batch reactors thus have the following characteristics: 1) 2) 3)

Time-variant reaction conditions Discontinuous production Downtime for cleaning and filling

2.2.2 Semicontinuous or Fed Batch Operation In semi-continuous or fed batch operation, additional substrate is fed into the bioreactor, thus prolonging operation by providing an additional continuous supply of nutrients to the cells. No material is removed from the reactor, apart from normal sampling, and therefore the total quantity of material within the reactor will increase as a function of time. However if the feed is highly concentrated, then the reactor volume will not change much and can be regarded as essentially constant.

Figure 2.4. Fed batch bioreactor configuration.

2.2 Bioreactor Operation

59

Semi-continuous operation shares the same characteristics as pure batch operation, in that concentration levels generally change with time and that some downtime occurs during the initial charging and final discharge period at the end of the process. The ability to manipulate concentration levels within the bioreactor by an appropriate controlled feeding strategy confers a high degree of flexibility to fed batch or semi-continuous operation, since differing concentration levels can be utilized to manipulate the rates of reaction. In Fig. 2.4, both the volumetric feeding rate, F, and the feed substrate concentration SQ, may be constant or may vary with time, giving the possibility of such feeding strategies as: 1.

Slow constant feeding, which can be shown to result in linear growth of the total cell biomass.

2.

Exponential feeding to maintain constant substrate concentration and, resulting in unlimited, exponential cell growth.

3.

Feedback control of the feed rate, based on monitoring some key component concentration.

The important characteristics of fed batch operation are therefore as follows: 1.

Extension of batch growth or product production by additional substrate feeding.

2.

Possibility of operating with separate conditions for growth and production phases.

3.

Control possibilities on feeding policies.

4.

Development of high biomass and product concentration.

For fed-batch operation, the cell balance follows the same form as for batch operation, but since additional substrate feeding to the reactor now occurs, the substrate balance takes the form: Rate 1 an S-shaped function results. KM is the substrate concentration with r$ = vmax/2. The simulation example PHB employs this kinetic form. Temperature and pH Influence Rates of biological reactions, including growth rates, exhibit a maximum when plotted versus temperature or pH. The maximum point is referred to as the temperature optimum or pH optimum for the system. The term temperature optimum must be used with caution because the curve is a result of two temperature dependent processes, the enzyme catalysed reaction and the enzyme deactivation reaction, respectively.

75

3.1 Enzyme Kinetics

Figure 3.5. Michaelis-Menten and Hill kinetics: v max = 1; Km = 5; n = 1, 2, 3, 5.

At temperatures well below the optimum the enzyme deactivation may be neglected and the temperature influence on the reaction rate described by the Arrhenius equation. At higher temperatures both enzymatic reaction and enzyme deactivation rate equations must be solved together with their respective kinetic constants expressed in terms of the Arrhenius equation. Most enzymes exhibit a distinct pH optimum. This can be explained by dissociation of acidic and basic groups of the enzyme, especially of its active center. The following equation is a useful description of this.

r =

K

S,H+CH+CH/ K I,H

76

3.1.3

3 Biological Kinetics

Deactivation

Biocatalysts in reactors usually undergo irreversible conformational changes generally known as denaturation or deactivation. This often causes an exponential decrease of activity with time and can be described by a first-order reaction rate process: rd = -kdE Considering that for a batch reactor, dE/dt = r^, the integrated form can be written as E = E 0 e- kdt Substitution in the Michaelis-Menten equation yields =

MOJL

kdt

KM+S This equation suggests an exponential decrease of reaction rate regardless of substrate concentration. The simulation example DEACTENZ, Sec. 8.4.12 illustrates this. Engineering models for the kinetics of deactivation are given by Prenosil et al. (1987).

3.1.4

Sterilization

Similar to enzyme deactivation, sterilization kinetics can be viewed as a process of inactivation or the removal of viable organisms or cells from the system. Inactivation can be achieved by using heat, radiation or chemicals. It is a statistical process, with the rate of killing being usually proportional to the number of the organisms at any time. Therefore it can be described again by first-order kinetics: rd = - k d X where For a batch reactor, dX

= - k d X =-k d =k 0 e- E a / R T X

which upon integration gives,

3.1 Enzyme Kinetics

77

x

_ ,,-kdt

where XQ is the initial live biomass concentration, X is the viable biomass after the treatment time t, and kd is the specific deactivation constant (1/s). The sterilization time will depend on the initial level of contamination. For this purpose the D-value is defined as the treatment time required to reduce the population by a factor of ten. This time is related to the rate constant by 2.3

3.2 3.2.1

Simple Microbial Kinetics Basic Growth Kinetics

Under ideal conditions for growth, when a batch fermentation is carried out, it can be observed experimentally that the quantity of biomass, and therefore also the concentration, increases exponentially with respect to time. This phenomena can be explained by the fact that all cells have the same probability to multiply. Thus the overall rate of biomass formation is proportional to the biomass itself This leads to an autocatalytic reaction, which is described by a first order rate expression as where rx is the rate of cell growth (kg cell/m3 s), X is the cell concentration (kg cell/m3) and k is a kinetic growth constant (1/s). For a batch system, this is equivalent to,

where dX/dt is the rate of change of cell concentration with respect to time (kg cell/m3 s). The analytical solution of this simple, first-order differential equation is of the form In X = k t + In X0 or, = ekt

where XQ is the initial cell concentration at time t = 0.

78


Plotting experimental growth data in the form of the natural logarithm of cell concentration versus time will often yield a straight line over a large portion of the curve, as shown below in Fig. 3.6. InS

Limitation \Stationary InX

X

Death

Exponential Lag

time Figure 3.6. Biomass and substrate concentrations during batch growth.

In the range from ti to t2 the logarithmic curve is linear, and this is the region of exponential growth. Three other regions can be identified: between t = 0 and ti, there exists a period of cell adaptation or lag phase, and before t2 there is a region where the growth is limited by the lack of a particular substance, which is known as the limiting substrate. The slope of the linear part of the curve between ti and t2 is the growth rate per unit mass of cells or specific growth rate and is given the symbol \i. =

1 dX X" "dT

=

= specific growth rate = |i

In many processes cells begin to die (after ts), because of lack of nutrients, toxic effects or cell aging. This process can typically be described by a first order decay, rd = - kd X where rd is the death rate and k^ is the specific death rate, with the same dimensions as the specific growth rate. This expression is identical with sterilization kinetics, Sec. 3.1.4. The exponential and limiting regions can be described by a single relation, that sets JLI equal to a function of substrate concentration. It is observed experimentally that |a is at a maximum when the particular limiting substrate concentration S is large, and for low concentration ja is proportional to S. Over the whole range from low to high S, |i is described by the following Monod equation.

79

3.2 Simple Microbial Kinetics

Thus |i varies with S in the same fashion as does the enzymatic rate of Michaelis-Menten kinetics. Again, this is a two-parameter equation involving two constants, the maximum specific growth rate |im and the saturation constant KS. It is best considered to be an empirical relation, but since it has the same form as the Michaelis-Menten enzyme kinetics equation, it is sometimes taken to be related to a limiting enzymatic step. Although very simple, it often describes experimental data for growth rates very well. The form of this relation is shown in Fig. 3.7. M

Monod Relation

Figure 3.7. Specific growth rate versus limiting substrate concentration according to the Monod relation.

The important properties of this relationship are as follows:

S -» 0,

S = KS,

Urn Jl = ~

The first introductory simulations in Sec. 8.1 are based on Monod kinetics. When two substrates can be limiting, it is often the case that a double Monod type relationship can be used, as given in Sec. 3.2.4 and as shown by the simulation examples NITRIF, Sec. 8.5.3, and BIOFILM, Sec. 8.7.1.

80

3.2.2


Substrate Inhibition of Growth

Many substrates can be utilized by organisms at low concentration, but at high concentrations they can also act as toxic growth inhibitors. The |i versus S curve may then appear in the form shown in Fig. 3.7, and can be described by the relation: _ |imS

^ " (KS + s + s2/KO

whose shape depends on the values for KS and KI. This is a modified Monod relation to allow for the inhibitory effects of high substrate concentration. As shown in Fig. 3.8, the inhibition term (S2/Ki), which is small in magnitude at low values of S, increases dramatically at high values of S and causes a decrease in \i. Note that high values of KI correspond to a decreasing effect of substrate inhibition. It is seen that larger values of KS shift the left side of the curves towards the right, while increasing values of KI raise the right side of the curves. Thus a wide range of shapes can be achieved by varying the three parameters, but a maximum value of (I is always obtained at some intermediate value of S. 1.0 -, 0.8 -

K

I

0.6 -

0.0

Figure 3.7. Substrate inhibition kinetics for various values of KS and Kj. The parameters used are as follows: For all curves |im = 1.0 1/h. Curve A: KS = 1 and KI = 10, Curve B: KS = 0.1 and KI = 10; Curve C: KS = 1 and KI = 20; Curve D: KS = 0.1 and KI = 20. The units of KS, KI and S are g/m^.


81

The substrate inhibition kinetic curve has the following characteristics, which result from the kinetic equation: 1) When S = Ks

_ - - 2 + Ks/Kj 2) When S = KI - 2 + Ks/Ki 3) The maximum occurs at S = (Ks Ki)°-5 and

M- =

3.2.3

Mm 2 (Ks/Ki)°-5 + 1

Product Inhibition

When the formation of product inhibits the rate of cell growth, the basic Monod equation can be modified, by the addition of a product inhibition term P/Kj. Thus,

3.2.4

Other Expressions for Specific Growth Rate

The Modified Monod Form MmS

M- - KS S0 + S shows the influence of initial concentration, which is sometimes observed if other components are limiting. The Teisser Equation ILL = |Li m (l-e-S/k) relates |Li to S exponentially.

82


The Contois Equation

- KX + S expresses the effective saturation constant as being proportional to the biomass concentration X. At high X, |i is inversely proportional to X. This is sometimes used to represent a diffusion limitation in flocculating or immobilized biomass. The Logistic Equation M, = ( a - b X ) encompasses exponential growth and the levelling off to zero growth rate at high X. For a batch fermentation the biomass balance is, = aX-bX2 Thus when X is small, growth is exponential and given by

dX When X is large,

At steady state or zero growth rate,

0 = a X - b X2 and thus

X = a/b Multiple-Substrate Monod kinetics can be used to describe the influence of many substrates, which for two substrates takes the form, Si ^ f S2 A l + SiJ ^2 + 827 In this way either substrate may be limiting under conditions when the other substrate is in excess. Note that the multiplicative effect gives for S\ = K\ and


83

82 = K2, ILL = |LLm/4. An example of such kinetics is the simultaneous requirement of glucose and oxygen by aerobically growing organisms. Double-Monod kinetics can also be written for two substrates as parallel reactions, according to lSi

k2S2 V

1

This form gives an additive, fractional contribution for each substrate. Thus for Si = KI and 82 = K2, the result is |Li = |Lim/2. For the case Si = 0 and 82 large, then ji = |iim k2/(ki+k2). Each substrate thus allows a different maximal growth rate. If both Si and 82 are large then |ii = |iim. Note that the flexibility of this kinetic form requires twice as many kinetic parameters as the simpler double Monod kinetics. An example of this kinetics is the parallel use of alternative substrates, such as various types of sugars. Diauxic Monod Growth can be modelled for two substrates by the relation

K 2 + S2 + S / K ! in this way the consumption of substrate 82 will be inhibited until Si is exhausted, for suitably low values of Kj. Diauxic growth can be observed in many organisms. An example is E. Coli, where the uptake of lactose is repressed in the presence of glucose. The simulation example SUBTILIS, Sec. 8.9.2 uses this kinetic form.

3.2.5

Substrate Uptake Kinetics

The rate of uptake of substrate by micro-organisms is generally considered to be either related to that of growth or to that required for cell maintenance. This can be expressed as: ~rx v rs = ?»5 " m X where rs is the rate of substrate uptake by the cells (kg substrate/m3 s). As explained in Section 1.3.5, YX/S is the stoichiometric factor or yield coefficient and relates mass cells/mass substrate.

84


The maintenance factor, m, gives the (mass substrate/mass cells time) required for non-growth functions. The total substrate utilization for cell maintenance is, of course, taken to be proportional to the total quantity of cells, and therefore for a batch reactor it is proportional to cell concentration, X. Often the uptake and production rates of substances are expressed in terms of the particular quantities related to unit mass of cells and are then known as specific cell quantities. Thus: For the specific growth rate (1/h) rx » = X

For the specific substrate uptake rate (kg S/kg biomass h), rs qs = x

For the specific oxygen uptake rate (kg C>2/kg biomass h), r

O2

qo2 = — For the specific carbon dioxide uptake rate (kg CCVkg biomass h),

qco2 = "IT" For the specific product production rate (kg P/kg biomass h), qp = £

Note that qx = rx/X = |JL is the specific biomass production rate. Specific rate quantities may take simple or complicated forms, for example, for the case of substrate: rs = y^" - m X

then,

qs = where |i is also a function of S.


85

By necessity, in wastewater treatment systems the substrate concentration, S, is taken often as total dissolved organic carbon, rather than considering a specific substance, such as glucose. The biomass concentration, X, also must be related to the total of all microbial species present. Naturally a gross simplification of such a complex system results. In wastewater treatment systems, biomass growth is immeasurably slow, whereas the substrate uptake can usually be measured fairly accurately. Under such circumstances it is then more useful to base the kinetics on the more measurable rate and to express r$ as a separate rate equation, which is independent of rxFor example, this can be done using an expression, analogous in form to that of the Monod equation, VmS

where the constant vm is proportional to the quantity of biomass in the system and is the maximum rate of substrate consumption, observed at high S.

3.2.6

Product Formation

The kinetics of product formation can be exceedingly complex. Product is sometimes formed during growth and sometimes after all growth has stopped. A useful equation for the rate of product formation, flexible enough to find frequent application, is that of the Luedeking-Piret model: X

where rp is the rate of product formation (kg product/m3 s). YX/P is a yield factor (kg cell produced/kg product produced), relating the growth and product stoichiometry in the "growth associated" term of the Luedeking-Piret equation, and b is a non-growth related term and is important for cultures which produce product independent of growth. Often both coefficients of the above equation (YX/P and b) are not constant but are functions of substrate or product concentration. When little is known about the detailed kinetics of product formation, a more general expression rp = qp X is used, where qp will usually vary with culture conditions and concentrations.

86

3.2.7


Interacting Microorganisms

Multiple-organism populations will involve interactions in which one species of organisms will exert some influence on another. Such interactions and their models are described below. Considering two microbial species, A and B, three types of interactions on each other are possible; a positive beneficial effect (+), a negative detrimental effect (-), or a neutral effect (0). The resulting interaction possibilities are given in Table 3.3. The different types of interactions can also be described by a graphical representation. Thus the growth kinetics can be described by a solid arrow connecting the substrate to the product, where the organism involved is shown above the arrow. A solid arrow from one substrate symbol to the same symbol in another growth path indicates that the product from one organism acts as a substrate for another. Substrate or product inhibition is indicated by a dashed arrow connecting the component to the inhibited organism. The symbols +, or 0 at the right hand side of the diagram indicate whether or not the organism involved has benefited by the interaction. This is made clear by the examples below: Table 3.3. Definition of microbial interaction types. Interaction-type Neutralism Commensalism Mutualism Predator-prey Predator-prey Amensalism Amensalism Competition

A 0 0

Organisms B 0 +

0 0

Predator-Prey Kinetics Organism A consumes substrate S, and organism B consumes organism A.

The simulation example MIXPOP demonstrates this type of system.


87

Commensalism Organism A uses substrate 82 to produce product P; organism B uses substrate Si to produce product 82, which benefits organism A since product 82 acts as its substrate. A

81

The following processes with the compound 82, shown in brackets, involve this form of commensalism: nitrification (NC>2~ ) anaerobic digestion (organic acids) methanogenation (H2, €62) as found in simulation example ANAMEAS, Sec. 8.8.7. Commensalism with Product Removed Organism A utilizes a substrate 82, which inhibits the growth of B on substrate Si.

s 2 - - -> p 2

0

This effect may be found in the removal of toxic wastes in mixed cultures with multiple carbon sources. An example is found in ANAMEAS in which the hydrogen substrate of the methanogens (A) inhibits the acetogenic organisms (B). Mutualism with Product Removed Organism A utilizes substrates 82 to produce product P. Organism B utilizes substrate S\ to produce 82, which inhibits organism B.

An example of this would be found in anaerobic digestion for hydrogen gas (see ANAMEAS, Sec. 8.8.7)

88


Mutualism with Products Used Mutually as Substrates Both organisms benefit from each other's products.

3.2.7.1

Case A. Modelling of Mutualism Kinetics

Organism A utilizes the product from organism B, which also helps B because PB inhibits its growth.

An example of kinetic modelling is presented for this case, in which the growth of the two organisms, A and B, takes place in a batch reactor with initial substrate concentrations SIQ and 820- The growth rate is expressed by Monodtype kinetics and constant yield factors are used to express the substrate uptake and product formation rates. Substrate Si balance, dt

_ -

dt

_ -

Substrate 82 balance.

Product PB balance, dPB IT

A PB

Species A balance, dXA "3f" =

Species B balance,

BS2


89

dXB =

~3T

^B XB

The kinetics are given by Monod-type relations, with a double form of Monod equation employed for species A and a product inhibition term employed for B, Si PB

HA - MmA KI + Si

B = K2 + S2 + (PB/KI) Other examples of interacting microorganism effects are given in the simulation examples ACTNITR (neutralism), Sec. 8.4.3, COMPASM (competitive assimilation and commensalism), Sec. 8.8.2, MIXPOP (predatorprey population dynamics), Sec. 8.8.4 and TWOONE (competition between organisms), Sec. 8.8.5.

3.2.7.2

Case B. Kinetics of Anaerobic Degradation

Anaerobic degradation is a very complex multi-substrate, multi-organism process, as is depicted in Fig. 3.9. It is shown here how its modelling can be approached using Monod-type kinetics. This problem is of interest because overloading of an anaerobic reactor causes accumulation of intermediates (organic acids, hydrogen) and consequent inhibition of the final methanogenic step (Gujer and Zehnder, 1983). In a first step, polymer materials (carbohydrates, proteins or lipids) are hydrolyzed to yield the monomer compounds (amino acids, sugars and fatty acids). In a second acidogenic step, these compounds are fermented to organic acids (mainly acetic, propionic and butyric acid) and hydrogen. In the third acetogenic step, organic acids with more than three atoms of carbon per molecule are converted to acetic acid and hydrogen. In a last methanogenic step, the intermediates, acetic acid and hydrogen and carbon dioxide are converted to methane. Five different groups of organisms accomplish these reactions as shown in Fig. 3.9. According to thermodynamic calculations (Archer, 1983) the oxidation of propionic to acetic acid should only be possible at a hydrogen partial pressure °f PH2 < 10~4 bar. From a series of observations it seems evident that disturbances in the methanogenic step, which is generally considered to be the most sensitive one, lead to the accumulation of H2- Additionally the state of an anaerobic reactor may be characterized by its volatile fatty acid levels based on the CH4/CO2 ratio and the total gas production rate.

90


(0

'55

CO

g0) TJ

o>

"5

W '(0 0)

0) D) O I O

Figure 3.9. Reaction scheme of anaerobic degradation. The symbols are: Poly - polymer material (proteins, fats, hydrocarbons, etc.); XJJY - Biomass hydrolyzing Poly; Mono monomeric materials from hydrolysis of Poly; XAG ~acid generating biomass; HPr - propionic acid; Pr" - propionate; Xpr - biomass growing on propionate; HBu - butyric acid; Bu" - butyrate; XBU - biomass growing on butyrate; HAc - acetic acid; Ac" - acetate; XAC - biomass growing on acetate; XH - biomass growing on hydrogen and carbon dioxide. Dashed arrows indicate gaseous compounds transfer to the liquid phase. T - gaseous compounds transferred to liquid gas phase.

The respective reaction rates rj for the production of biomass Xi, for the consumption of substrate Si and for the formation of product Pi in each step are: rxi = Hi Xi - kdi Xi

3.3 Structured Kinetic Models

91

Hi Xj rpi =

where k^ is the specific death rate, including maintenance metabolism, and the specific growth rates take the Monod form, Mimax Si

W - KSi + Si

or its modified form in the case of substrate inhibition by acetate, Mimax Si

These kinetics can then be combined with the mass balances as discussed in Ch. 4 for each component, Si and Pi, and for the biomass balances for each organism type, Xi» Following this approach a model was established (Denac et al., 1988) and combined with particular control algorithms to simulate a continuous anaerobic digestor with feed rate control. This included a gas phase balance, thermodynamic equilibrium constraints and acid-base equilibria using an ion charge balance (Sec. 1.3.6.2). The simulations were used to adjust the control parameters, which were employed on laboratory reactors (Heinzle et al., 1992). The simulation example ANAEMEAS, Sec. 8.8.7, gives details concerning this model.

33

Structured Kinetic Models

In many cases the characterization of biological activity by simply the total biomass concentration is insufficient for a true model representation (Fredrickson et al., 1970; Roels, 1983; Moser, 1988). A variation in the biomass activity per unit biomass concentration may be caused by: Loss of plasmids (Imanaka and Aiba, 1981). See also the simulation example PLASMID. Induction and repression of genes. Variation of RNA content of the cells (Harder and Roels, 1982; Furukawaet al., 1983). — Variation of enzyme content of the cells. — Post-translational modification of proteins, e.g. phosporylation. — Signaling networks.

92


— -

Membrane transport. Accumulation of storage materials (Heinzle and Lafferty, 1980; Heinzle et al., 1983). See example PHB, Sec. 8.2.4. Morphological changes, e.g., branching of filamentous organisms, volume to surface ratio of yeast cells (Fig. 3.10), Furukawa et al., 1983.

-

Such variations in biomass activity and composition require a more complex description of the cellular metabolism and a more structured approach to the modelling of cell kinetics. Structured models are based on a compartmental description of the cell mass as shown in Fig. 3.1. In general it is very difficult experimentally to obtain sufficient mechanistic knowledge about the cell metabolism for the development of a "realistic" structured model. Parameter estimation may be very difficult, and the application of complex numerical methods may easily lead to physically meaningless results. Often the verification of even simple unstructured models is not possible owing to experimental difficulties. This problem becomes much more significant with increasing complexity of the model. For this reason, structured models are seldom used for design or control. Structured models may be useful to model transient behavior of a biological system. Such models also may be required if a wide range of changes of environmental conditions have to be described with one model and one set of parameters (Moes et al., 1985 and 1986). Changes of cellular composition as functions of steady-state growth rate are well known. For example, in long-term experiments the cellular composition of lipids, carbohydrates, protein, RNA and DNA in Baker's Yeast were found to change as a function of dissolved oxygen and dilution rate (Furukawa et al., 1983). In this work the yeast shape and specific area also changed with dissolved oxygen concentration (Fig. 3.10).

L/d; S/V x 10 2.0

[1/m]

L/d

1.5 S/V

1.0 0.01

0.1

1.0

D0[g/m3]

Figure 3.10. Dissolved oxygen concentration, DO, influenced the shape of yeast in continuous culture as given by the ratios of length/diameter (L/d) and surface/volume (S/V).

93


In what follows three cases involving structured kinetics models will be discussed briefly. Case C describes a simple, two-compartment model with storage material. Case D gives an example of a complex, structured threecompartment model. Here the biomass contains storage material and an enzyme that degrades the storage material. In this example the problem of model discrimination is discussed briefly. Case E describes the application of ATP balancing, a method of linking stoichiometry of various pathways in complex models.

3.3.1

3.3.1.1

Case Studies

Case C. Modelling Growth and Synthesis of Poly-Bhydroxybutyric Acid (PHB)

Fig. 3.11 represents the process of cell growth and the synthesis of intracellular product PHB. Residual biomass (R) is the difference between total cell dry mass (X) and product PHB (P). Synthesis of PHB occurs with a single limiting substrate NH4+ (S) and constant dissolved gas concentrations of H2, C>2 and CO2 (So). Mass flows are indicated by solid lines, and regulatory mechanisms are symbolized by dashed lines.

inhibiting inhibiting Figure 3.11. Schematic diagram of growth and synthesis of the intracellular product PHB (P) with constant concentrations of dissolved gases H2, O2, and CO2 (SG)« X is the total biomass (X=R+P).

94


It can be seen that the catalytically active biomass, R, is produced from both S and SQ. During exponential growth the PHB content is constant, and thus the rate of intracellular product formation is proportional to the rate of formation of the residual biomass. On this basis, the basic mass balance equations for a batch process can be formulated as shown in the simulation example PHB, Sec. 8.2.4. This model was used successfully in describing experimental batch growth and the PHB product formation (Heinzle and Lafferty, 1980), as shown in Fig. 3.12. S [g/L]

X,P,R [g/L]

P/X [-]

3 -

2 -

1 -

Figure 3.12. Comparison of simulation results from the structured PHB model with experimental data (Heinzle and Lafferty, 1980).

3.3.1.2

Case D.

Modelling of Sustained Oscillations in Continuous Baker's Yeast Culture

Oscillations of continuous culture of baker's yeast have been observed by a number of researchers. An example of such sustained oscillations is shown in Fig. 3.13 (Heinzle et al., 1983).

95


15

10

10

0.5

X [g/L]

°'1 DO [mg/L] 0.05 0.0 0.5

0

E [g/L]

2

S [mg/L]

0 100

2.0 50

1.5

RQ

q X [mmol/h L]

1.0 0

10

20

t[h] Figure 3.13. Oscillating profiles from a continuous culture of S. cerevisiae. (Heinzle et al., 1983). Symbols used are X (total biomass), DO (dissolved oxygen), E (ethanol), S (glucose), QCO2 and QO2 (specific gas reaction rates), and RQ (respiratory quotient).

One possible mechanism to explain the observed results is that a storage material (G) having the same oxidation state as glucose (S) must cyclically be produced and consumed. This storage material was identified as trehalose and glycogen. Under conditions of high glucose uptake rate and high degradation

Figure 3.14. Structured model to describe Baker's Yeast oscillations.

96


rate of G, ethanol, E, is accumulated in the medium and can be later oxidized to yield biomass, R. From this and additional information on the activity of the enzyme T, which catalyzes the degradation of G, the reaction scheme shown in Fig. 3.14 was developed and used as a basis for model formulation. Here R is the biomass, not including the intracellular storage material, G. The enzyme, T, is not considered to contribute significantly to the total biomass and was therefore neglected in the total biomass balance. The detailed kinetic model for continuous culture is given in the simulation example YEASTOSC, Sec. 8.8.8. Many parameters could be determined from independent experiments. Some were taken from the literature, and some, especially those describing the enzyme T activity had to be based on simulation results. The model leads to oscillations (Fig. 3.15), whose existence and dependency on operating conditions agree qualitatively with experimental observations.

(g/L)

10

Time (h)

r

15 Time (h)

D=0.05 h'1

I ' ' ' ' I 10 15 Time (h)

Figure 3.15. Simulation of the Baker's yeast model (simulation example YEASTOSC, Sec. 8.8.8, showing oscillations of all the components, Q.

97


3.3.1.3 Case E. Growth and Product Formation of an OxygenSensitive Bacillus subtilis Culture This example shows how knowledge of the biochemical pathways, when combined with experimental data, can lead to model development. In this research an oxygen-sensitive culture was to be used for mixing studies, and it was important to establish the kinetic model (Moes et al, 1985, 1986) in order to describe the batch profiles as shown in Fig. 3.17. Since it was not possible to describe the growth behavior by simple Monodtype models, an ATP balance was used to establish the available energy for biomass synthesis. This was possible because the biochemical pathways (Fig. 3.18) for the fermentation and their associated chemical energy production and consumption steps were known. Gl

Ac, Bu (g/L)

X (9/L) - 3

10 - 2

- 1

Figure 3.17. Growth and product formation of Bacillus subtilis at constant DO. Gl - glucose, X - biomass, Ac - acetoin, Bu - butanediol.

The formulation starts with a steady state ATP balance, which assumes that all energy-producing steps are balanced by those that consume energy. The form of this balance is as fpllows: i A ' I' U

dt

= 0 = [ qs/co2 YATF/s,co2 + qs/Ac YATP/S,AC + u - qBu/Ac) YATP/BU - qATp/x ]X

98


In this equation q$/GO2 is the rate at which sugar S is converted to CC>2 by respirative growth. The rate of the sugar conversion to acetoin is qs/Ac- The rates of conversion of acetoin to butanediol and the reverse reaction are given by qAc/Bu, and QBu/Ac- Energy is required for growth, and the ATP consumption rate is qATP/X- The corresponding ATP yields Y convert these rates to ATP equivalents. The nomenclature at the end of this example defines all symbols in detail. Here knowledge of the yields of ATP for each step is important. Rate expressions for the reaction pathways, as given in Fig. 3.18, were thus established for each step in terms of the participating reactants. Batch mass balances for all species were then written in terms of the individual rates of production and consumption of each relevant component. NADH Cells Glucose

>*

6CO 2

2Pyruvate

NADH ATP

Acetoin ^

^ NAD+

NADH ADP

r

>

Butanediol

NAD"1"

^ ATP

Figure 3.18. Biochemical pathways for the production of acetoin and butanediol.

Using the steady-state approximation that the ATP level does not vary significantly, allows setting the condition that dATP/dt = 0. The steady state ATP balance is then solved for qATP/X, which is the rate of ATP available for growth. The required yields can be calculated from the reactions given in Fig. 3.17. In these calculations it was assumed that at high oxygen concentration 1 mol NADH was converted to 3 mol ATP in the respiratory chain. At low oxygen concentrations, the conversion equivalent was assumed to be a function of oxygen as determined by parameter estimation, based on the experimental data. The glucose substrate balance can be written in terms of the rates at which substrate was consumed for complete oxidation (qs/GO2)» f°r biomass (qATP/X YX/ATP / YX/S) and for product formation (qs/Ac)'

dS

,.

qATP/x YX/ATP

x

X


99

It was observed experimentally that the reaction stoichiometry for the formation of metabolites was almost constant and was given by Yp/S = 0.57 mol (Ac + Bu)/mol glucose. Thus, another balance can be written for the substrate: _ qS/Ac X dt Y P/S These two equations were used to determine dS/dt and qs/GO2The biomass balance is, dX

=

qATP/x YX/ATP X

The metabolite balances are, dAc "dt" = ( qS/Ac ~ qAc/Bu + qBu/Ac )

and, dBu ~dt~

=

( Âc/flu ~ qBu/Ac ) X

In the above balances, all specific rate terms, q, are in the units (mol/g biomass h). All concentrations (ATP, S, Ac, Bu) are in mol/L units except X (g/L). All yield coefficients Y are in mol/mol units except when involving biomass, e.g. YX/S is in units of g/mol. Empirical Monod-type kinetic relationships, not given here, were established to calculate the rate of glucose to acetoin, qs/Ac» an(i the reversible acetoin to butanediol rates, qAc/Bu and qBu/Ac» as a function of reactant concentrations for glucose, S, for acetoin, Ac, for butanediol, Bu, and for dissolved oxygen, DO. Additional empirical kinetic terms were needed to fit the following experimental observations: 1) 2) 3) 4) 5)

Growth stopped when S approached zero. qs/Ac was limited by C>2 qBu/Ac was promoted by high C>2 qAc/Bu was promoted by low C>2 The influence of DO with S = 0 was negligible.

The many kinetic parameters were determined partly by direct experiments and partly by fitting the data using a parameter estimation computer program. The influence of oxygen was determined using data from experiments at controlled oxygen conditions and determining the best values of the oxygen sensitive rates by parameter estimation procedures. A simple graphical procedure then allowed determination of the appropriate constants.

100


The quantities YATP/NADH and YX/ATP are linearly dependent on each other and could therefore not be determined from experimental data. The maximum value of YATP/NADH was arbitrarily fixed at the maximum theoretical value of 3.0, which has a direct influence on the estimation of YX/ATP (here 5.7 g/mol). Good agreement of the batch curves with the model at constant DO was achieved as shown in Fig. 3.18. From these results it is seen that the model predicts the metabolite and biomass profiles. The model was quite versatile and reasonably accurate, considering the large differences in biomass formation at high DO (X = 3.4 g/L) and low DO (X = 2.5 g/L), as well as the variation of the butanediol formation at high DO (Bu = 0.2 g/L) and low DO (Bu = 2 g/L), X(g/L)

GI(g/L)

t(h) Figure 3.19. Comparison of simulation results with the Bacillus subtilis fermentation (Moes et al., 1986). X=biomass, Ac=acetoin, Bu=butanediol, Gl=glucose.

The simulation example SUBTILIS, Sec. 8.9.2, employs a more conventional approach to the kinetics but also makes use of the biochemical pathways.

4

Bioreactor Modelling

4.1

General Balances for Tank-Type Biological Reactors

Fermentation systems obey the same fundamental mass and energy balance relationships as do chemical reaction systems, but special difficulties arise in biological reactor modelling, owing to uncertainties in the kinetic rate expression and the reaction stoichiometry. In what follows, material balance equations are derived for the total mass, the mass of substrate and the cell mass for the case of the stirred tank bioreactor system as shown in Fig. 4.1. ' X0! F0

1f

F1

Figure 4.1. The variables for a tank fermenter.

In this generalized case, feed enters the reactor at a volumetric flow rate FQ, with cell concentration, XQ, and substrate concentration, SQ. The vessel contents, which are well-mixed are defined by volume V, substrate concentration Si and cell concentration X\. These concentrations are identical to those of the outlet stream, which has a volumetric flow rate FI.


102

4 Bioreactor Modelling

General Balance Form As shown previously, the general balance form can be derived by setting: (Rate of accumulation) = (Input rate) - (Output rate) + (Production rate) and can be applied to the whole volume of the tank contents. Expressing the balance in equation form, gives: Total mass balance:

d(Vp)

—— = P(FO-FI) Substrate balance: d(VSi) —gj— = F o S o - F i S i + r s V Organism balance:

d(V Xi)

J J t • = FO XQ - FI X j + rx V

where the units are: V (m3), p (kg/m3), F (m3/s), S (kg/m3), X (kg/m3) with rs and rx (kg/m3 s). The rate expressions can be simply:

= KS + S! and using a constant yield coefficient,

-*x fs =

Yxl

but other forms of rate equation may equally apply. The above generalized forms of equations can be simplified to fit particular cases of bioreactor operation.

4.1 General Balances for Tank-Type Biological Reactors

4.1.1

103

The Batch Fermenter

A batch fermenter is shown schematically in Fig. 4.2.

Jiiilii

Figure 4.2. The batch fermenter and variables.

Starting from an inoculum (X at t = 0) and an initial quantity of limiting substrate, S at t = 0, the biomass will grow, perhaps after a short lag phase, and consume substrate. As the substrate becomes exhausted, the growth rate will slow and become zero when substrate is completely depleted. The above general balances describe the particular case of a batch fermentation if V is constant and F = 0. Thus, Total balance:

£-»

Substrate balance:

dSi V^r = r s V Organism balance: V

T = rxV

Suitable rate expressions for r$ and rx and the specification of the initial conditions would complete the batch fermenter model, which describes the exponential and limiting growth phases but not the lag phase. The simulation example BATFERM, Sec. 8.1.1, demonstrates use of this model.

104


4.1.2 The Chemostat A chemostat, as shown in Fig. 4.3, normally operates with sterile feed (Xo = 0) at constant volume steady state conditions, meaning that dV/dt = 0, d(VSi)/dt = 0, d(VXi)/dt = 0. In addition constant density conditions can be taken to apply

UN

Figure 4.3. The chemostat and its variables.

The total mass balance simplifies to,

o = FQThe dynamic component balances are then, Substrate balance VdSi

= F(S 0 -Si)

Cell balance VdXi

= F(X0-Xi)

where F is the volumetric flow through the system. These dynamic equations are used in the simulation example CHEMO, Sec. 8.1.2. At steady state, dSi/dt = 0 and dXi/dt = 0, Hence for the substrate balance: 0 = F (S0 - Si) + rs V


105

Chemostats normally operate with sterile feed, XQ = 0, and hence for the cell balance, 0 = - F X! + rx V Inserting the Monod-type rate expressions gives: For the cell balance, FXi

= rx = Ji X!

hence F H = v =

D

Here D is the dilution rate and is equal to 1/T, where i = V/F and is equal to the tank mean residence time. For the substrate balance, ]

from which: Xi = YX/S (So-Si) Thus the specific growth rate in a chemostat is controlled by the feed flowrate, since [I is equal to D at steady state conditions. Since |Li, the specific growth rate, is a function of the substrate concentration, and |Li is also determined by dilution rate, the flow rate F then also determines the outlet substrate concentration Si. The last equation is, of course, also simply a statement that the quantity of cells produced is proportional to the quantity of substrate consumed, as related by the yield factor YX/SThe curves in Fig. 4.4 represent solutions to the steady-state chemostat model as obtained from the simulation example CHEMOSTA, Sec. 8.4.1, with KS = 1.0. The variables Xi, Si, as well as the productivity DXi are plotted versus D. Thus as flow rate is increased, D also increases and causes the steady state value of Si to increase and the corresponding value of Xi to decrease. It is seen when D nears |im, Xi becomes zero and Si rises to the inlet value SQ. This corresponds to a complete removal of the cells by flow out of the tank; a phenomenon known as "washout". Fig. 4.4 shows washout occurring for D well below |jm, which is caused by the large value of KS. . When D is nearly zero (low flow rates) then Si approaches zero, and Xi approaches YSo The productivity of biomass, DXi, (kg X/m3 h) passes through a maximum rather close to the washout region.

106

4 Bioreactor Modelling X1,S1,DX1 10.0 T

5.0 • -

0.25

1.0

D (1/h)

Figure 4.4. Variation of the steady state variables in a chemostat with Monod kinetics as a function of dilution rate.

Chemostat applications are largely found in research laboratories. Microbial physiology studies can be made conveniently, since the cells can be controlled by the flow rate to grow at a particular value of specific growth rate \i. Kinetic studies can be made by measuring the concentration of the limiting substrate for a range of |u (=D) values, permitting the kinetics, |j = f(Si), to be determined. The yield coefficient can be determined by steady state measurements of substrate, biomass and product. The influence of any substrate in the culture can be tested by adding it to the medium at various concentrations.

4.1.3

The Fed Batch Fermenter

As shown in Fig 4.5 the outlet is zero for a fed batch fermenter, and the inlet flow, FO, may be variable. As a result the reactor volume will change with time.


107

S«.F, o1 ' o

Figure 4.5. The fed batch fermenter and its variables.

The balance equations then become for constant density, dT

=

T7 0

F

d(VSi) = F0 S0 + rs V dt d(V XQ = rxV dt Expanding the differential terms, which are products of the two variables V and Si and V and Xi, respectively, and substituting for dV/dt = FQ gives: VdSi

= F 0 (S 0 -Si)

The above equations are identical to those for a constant volume chemostat reactor. It can be shown by simulation that a quasi-steady state can be reached where dXi/dt = 0 and fi = F/V (Dunn and Mor, 1975) as seen in the Fig. 4.7. Since V increases, therefore n must decrease, and thus the reactor moves through a series of changing steady states for which |a = D, during which Si and p decrease, and Xi remains constant. This is analogous to a constant volume reactor with slowly decreasing F. These phenomena are demonstrated by the simulation examples FEDBAT, Sec. 8.1.3, and VARVOL, Sec. 8.3.1.

108


A fed batch fermenter, in which the inlet feed rate is very low, will not exhibit a large increase in volume and will not reach a quasi-steady state, unless X is very high. Assuming V to be approximately constant, the general equations can be integrated analytically for the case of |j = constant, giving an exponential increase in X. The constant |u condition is maintained by constant Si, which can be obtained via exponential feeding. Another phenomenon can be proven from these equations for the case of constant feed rate and essentially constant V; this is the linear growth situation, where X increases linearly with time. As shown in Fig. 4.6, the slope of the curve is related to the feed rate and the yield coefficient. If V changes as a consequence of dilute feed, then the total quantity of biomass (VX) will increase linearly.

FS

0 Y X/S

Figure 4.6. Linear growth under conditions of feed limitation with constant volume.

Figure 4.7. Repeated fed batch operation in terms of dimensionless variables for substrate inhibition kinetics. Two cycles of operation are shown. The dimensionless variables are defined in the simulation example VARVOL, Sec. 8.3.1.


109

Other important types of operation described by the general balances are cyclic fed batch (Keller and Dunn, 1978). An example of cyclic fed batch operation in which the quasi-steady state can be observed is shown in Fig. 4.7. Similar results can be obtained from the simulation example REPFED.

4.1.4

Biomass Productivity

The specific biomass production rate for a chemostat, DXi, (kg biomass/m3 h) can be calculated by applying the above model equations. Thus, D X i = DY x /s(So-Si) = DYx/s

The conditions for the maximum value of DXi as shown in Fig. 4.4, can be obtained by setting d(DXi) dD

=

°

or by running the simulation example CHEMOSTA, Sec. 8.4.1. The production rate for a batch culture can be calculated by dividing the biomass concentration by the time for the culture. Thus, the batch biomass production rate is equal to Xf/tf. Comparing the continuous biomass production rate to the batch biomass production rate, it is found that the continuous fermenter will have a 2 - 3 times larger maximum biomass production rate. This is because the batch fermentation starts at the inoculum value of X and has correspondingly low initial growth rates. Biomass production rates for fed batch fermenters are calculated by taking the total mass produced over the time of operation, or (VX)f - (VX)o/tf . For cyclic operation the biomass production has been shown to depend on the starting and final volume ratio (Keller and Dunn, 1978). This and more complex questions regarding product productivity for particular kinetics can be answered by making suitable changes in the simulation example REPFED, Sec. 8.3.4. The productivity of a repeated fed batch as compared to chemostat operation will depend on the operating variables, as well as the kinetics (Dunn et al., 1979).

110

4.1.5


Case Studies

4.1.5.1 Case A.

Continuous Fermentation with Biomass Recycle

The retention of active biomass is a means of improving the reactor productivity or efficiency for substrate uptake. The biomass separation could be performed by any suitable process (flotation, sedimentation, membrane filtration, or centrifugation). The cell recycle stream has a volumetric flow rate R and a biomass recycle concentration XR, where X R > X I and Xi is the biomass concentration in the stream leaving the fermenter. Consider the operation of a biological reactor with cell recycle as shown in Fig. 4.8. S^X^F+R

Cell recycle Figure 4.8. A bioreactor with cell separation and recycle.

The mass balances around the entire system are as follows: Biomass accumulates both within the reactor of volume Vr and also within the separation unit with volume Vs. Assuming that biomass leaves only in the wastage stream and that growth occurs only in the reactor, the balance is then dXi

where W is the wastage flow rate. At steady state, 0 = -

=0-WXR


111

Thus, the wastage rate of biomass must equal its production rate, otherwise Xi will change. Wastage rate is an important control parameter in wastewater treatment, where the separator is usually a sedimentation tank. For the substrate, which is consumed only in the reactor section, dSi

dSR

At steady state, 0 = F So — FI Si — W Sj + r$ Vr

Here it is seen that the uptake rate is equal to the difference between the inlet and outlet mass flows. The efficiency of the continuous biomass separation determines XI/XR, which controls the recycle ratio, R/(F+R). Considering the fermentation tank only, the balances are as follows: For the biomass, At steady state, 0 =RXR-(F + R)X!+rxVr Cell separation and recycle lead to high cell concentrations in the reactor, which, when neglecting the contribution by growth, would be XR/(F+R). Since the rates are proportional to Xi, an increase in reactor efficiency is obtained. Assuming, rx = |n Xi gives,

MX, where, XR > Xi , and D = F/V is the nominal dilution rate. This equation means that the specific growth rate is decreased from the chemostat value, D. This is due to the reduction in substrate concentration, Si, which is caused by the higher biomass concentration, resulting from the cell recycle. Washout is impossible due to the complete biomass retention, and for this reason flow rates greater than in a chemostat are possible. The substrate balance gives JQ

Vr =±= F S0 + R Si - (F + R) Si + rs V dt

which shows that at steady state

112


This would also be the case without cell recycle, since the substrate is assumed to pass unchanged in concentration through the separator. The above equation can be written as =

where the right hand side of this equation is the F/M (food/biomass) ratio. This gives a theoretical basis to the F/M concept, which is well known for the control of waste treatment plants. The simulation example ACTNITR, Sec. 8.4.3, enables the main operating characteristics of cell recycle systems to be studied. A related simulation example MEMINH, Sec. 8.9.1, considers the retention of enzyme using a membrane, and SUBTILIS, Sec. 8.9.2, involves the retention of biomass.

4.1.5.2

Case B.

Enzymatic Tanks-in-series Bioreactor System

A three tanks-in-series system is used for an enzymatic reaction, as shown in Fig. 4.9.

illi

11

: :'•;: y~: •:

81

:

w>

Vv

S2

>

.

1 1 1 1

•S;^;:-);'::;HaKS

>„

illi

Figure 4.9. Tanks-in-series bioreactor.

For the first tank,

dSi jp = F(S 0 -Si) dividing by SQ-SI

— =

where i\ = Vi/F and is the mean residence time of the liquid in tank 1. The balances for tanks 2 and 3 have the same form except for the subscripts:

4.2 Modelling Tubular Plug Flow Bioreacrors

113

For known flow rate, F, and known tank volumes, there are six unknowns in these three equations. Note that different tank sizes may be accounted for by differing values for the tank residence times TI, 12 and 13. If the kinetic terms r$ are only dependent on S, then the above equations can be solved without any further balances. It is often the case that enzymatic rate equations of the form given below can be used for each tank n = 1, 2 and 3:

This gives now six equations and six unknowns, and the problem is solvable by simulation methods. If the situation is more complex, such that r$ depends on other components, for example P, in the case of product inhibition or biomass X, then additional balance equations for these components must be included in the model. When combined with equations for the complete kinetics description (rates as a function of all the influential concentrations), the model can be solved to obtain the dynamics of the system and also the final steady state values. It can be shown that a three tanks-in-series reactor system will provide a good approximation to the performance of a corresponding tubular reactor, except for very high conversions.

4.2 4.2.1

Modelling Tubular Plug Flow Bioreactors Steady-State Balancing

The tubular reactor can be modelled for steady state conditions by considering the flow as a series of fluid elements or disks of liquid, which behave as a batch reactor during their time of passage through the reactor. This can be understood by considering a pulse of unreacting tracer in Fig. 4.10 that passes from entrance to exit unchanged without mixing.

114


tracer pulse

response

Figure 4.10. Plug flow idealization of the tubular reactor with no axial mixing.

A reaction will cause a steady state axial concentration profile as shown in Fig. 4.11. Thus at steady state, the concentrations vary with distance in a manner which is analogous to the time-varying concentrations that occur in a batch reactor.

Concentration

Figure 4.11. Axial profiles of steady-state concentrations in a tubular reactor.

This means that steady-state tubular reactor behavior can be modelled by direct analogy to that of a simple batch reactor. Thus using the batch reactor substrate balance (p = constant), dS dF = fS The flow velocity, v, for the liquid is defined as, v

dZ = dF

or, dt =

dZ

where v = F/A and F is the volumetric flow rate through the tube with crosssectional area A. Thus substituting for dt, dS_ dZ

=

rs_ v

4.2 Modelling Tubular Plug Flow Bioreacrors

115

This is the steady state tubular reactor design equation. With the kinetics model, rs = f(S), the equation can be integrated from the inlet, at position Z = 0, to the outlet, at Z = L, to obtain the steady state concentration profile of S. Additional component balances would be required for more complex kinetics. This is demonstrated by the simulation example for a tubular enzyme reactor ENZTUBE, Sec. 8.4.4.

4.2.2

Unsteady-State Balancing for Tubular Bioreactors

If dynamic information is needed for tubular or column systems, then changes with respect to both length and time must be considered. In order to achieve this, the reactor can be considered by dividing the volume of the reactor into N finite-differenced axial segments (Fig. 4.12), and treating each segment effectively as a separate stirred tank.

)))))> Figure 4.12. Finite-differencing the tubular reactor.

Figure 4.13. Balancing the difference segment n for the tubular reactor.

Formulating the substrate balance for S over a single segment n of volume AV = A AZ: f Accumulation^ ( input \ { output \ ^production rate\ V rate of S ) = Vrate of S) ~ \ rate of S) + ^ of S by growth) The balances have the same form. Thus for segment n, AV

dSn "dT = S n-l F n-l - Sn Fn + rSn AV

Constant density gives F n _i = Fn = F. Dividing by AV,

116


dSn dt

=

A(SF) + rsn AV

Setting AV —» AdZ and AS —> dS gives the partial differential equation, which describes changes in time and distance, as, 38

-

at

=

1 d(SF)

— -- x

A az

J_

When the volumetric flow, F, is constant,

At steady state,

as _F as + rs at ~ ~ A az

and V

— + fS dZ

which is the steady state equation derived earlier in Section 4.2.1. To model the dynamics by simulation methods, the partial differential equation must be written in difference form as, AV ^- = F (Sn.j - Sn) + rsn AV or

dt

AV/F

where AV/F = x , the residence time of the liquid in a single segment.

Mass Transfer

5.1

Mass Transfer in Biological Reactors

Multiphase reaction systems usually involve the transport of material between two or more phases. Usually one of the reactants is transferred from one phase into a second phase, in which the reaction takes place. The following cases are examples of biological systems.

5.1.1 Gas Absorption with Bioreaction in the Liquid Phase The gas phase is dispersed as gas bubbles within the liquid phase. Mass transfer occurs across the gas-liquid interface, out of the gas into the liquid, where the reaction occurs. The typical example is aeration of the bioreactor broth and the supply of oxygen to the cells as shown in Fig. 5.1.

Figure 5.1. Absorption of oxygen from an air bubble to the liquid medium.

Biological Reaction Engineering, Second Edition, I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30759-1

118

5.1.2

5 Mass Transfer

Liquid-Liquid Extraction with Bioreaction in One Phase

An immiscible liquid phase is dispersed in a continuous liquid phase. Mass transfer of a reactant takes place across the liquid-liquid interface, shown here (Fig. 5.2) from the continuous phase into the dispersed phase, where reaction occurs. An example might be the transfer of a substrate in an oil phase to an enzyme in the droplet aqueous phase.

Figure 5.2. Liquid-liquid extraction plus reaction.

5.1.3

Surface Biocatalysis

In this case, a liquid phase is in contact with solid biocatalyst. Substrates A and B diffuse from the liquid to the reaction sites on the surface of the solid, where reaction occurs. The product C must similarly be transferred away from the solid reaction surface, as shown in Fig. 5.3. Examples are found with immobilized enzyme and cell systems. In Sec. 6.1 the modelling aspects of this type of system are considered in detail.

5.2 Interface Gas-Liquid Mass Transfer

119

Figure 5.3. Reaction of two substrates on a solid biocatalyst surface.

5.1.4

Diffusion and Reaction in Porous Biocatalyst

Here a porous biocatalyst sphere is suspended in a liquid medium. Substrates diffuse into the porous internal structure of the biocatalyst support and react. Similarly, the products must diffuse away from the reaction sites within the solid to the outer surface, where they are then transported into the liquid. Detailed modelling of this process is treated in Ch. 6.

Figure 5.4. Reaction within a solid biocatalyst.

5.2

Interphase Gas-Liquid Mass Transfer

Concentration gradients are the driving forces for mass transfer. Actual concentration gradients (Fig. 5.5) in the very near vicinity of the gas-liquid

120

5 Mass Transfer

interface, under mass transfer conditions, are very complex. They result from an interaction between the mass transfer process and the local fluid hydrodynamics, which change gradually from stagnant flow, close to the interface, to perhaps fully-developed turbulence within each of the bulk phases. According to the Two-Film Theory, the actual concentration profiles, as represented in Fig. 5.5 can be approximated by linear gradients, as shown in Fig. 5.6. A thin film of fluid is assumed to exist at either side of the interface. Away from these films, each fluid is assumed to be in fully developed turbulent flow. There is therefore no resistance to mass transfer within the bulk phases, and the concentrations, CG and CL, are uniform throughout each relevant phase. At the phase interface itself, it is assumed there is no resistance to mass transfer, and the interfacial concentrations, CGI and CLI, are therefore in local equilibrium with each other. All the resistance to mass transfer must, therefore, occur within the films. In each film, the flow of fluid is assumed to be stagnant, and mass transfer is assumed to occur only by molecular diffusion and therefore to be Interface Gas

Figure 5.5. Concentration gradients at a gas-liquid interface.

described by Pick's law, which says that the flux JA (mol/s m2) for the molecular diffusion of some component A is given by,

dZ

121

5.2 Interface Gas-Liquid Mass Transfer

Gas

Interface

Liquid

Figure 5.6. Concentration gradients according to the Two-Film Theory.

where D is the molecular diffusion coefficient (m2/s) and dC/dZ is the steady state concentration gradient (mol/m3). Thus applying the same concept to mass transfer across the two films,

JA = DG

Z

G

Z

L

where DG and DL are the effective diffusivities of each film, and ZG and ZL are the respective thicknesses of the two films. The above equations can be expressed in terms of mass transfer coefficients kc and kL (m2/s) for the gas and liquid films, JA = k G (C G -C G i) = k L (C Li -C L ) The total rate of mass transfer, Q (mol/s), is given by, Q = JAA = jA(aV) where "A" is the total interfacial area available for mass transfer, and "a" is defined as the specific area for mass transfer or interfacial area per unit liquid volume (m2/m3). Thus for the total rate of mass transfer: In terms of the total interfacial area A, Q = k G A(C G -C G i ) = k L A(C L i -C L ) In terms of a and VL, Q = ko a (Co - CGi) VL = k L a(C L i -C L )V L

122

5 Mass Transfer

Since the mass transfer coefficient, k, and the specific interfacial area, a, depend on the same hydrodynamic conditions and system physical properties, they are frequently combined and referred to as a "ka value" or more properly a mass transfer capacity coefficient. In the above theory, the interfacial concentrations CGI and CLI cannot be measured, and are therefore of relatively little use, even if the values of the film coefficients are known. For this reason, by analogy to the film equations, overall mass transfer rate equations are defined, based on overall coefficients of mass transfer, KG and KL, and overall concentration driving force terms, where: Q = K G A(C G -C G *) = K L A(C L *-C L ) Here, CG* and CL* are the respective equilibrium concentrations, corresponding to the bulk phase concentrations, CL and CG, respectively, as shown in Fig. 5.6. Equilibrium relationships for gas-liquid systems, at low concentrations of component A usually obey Henry's law, which is a linear relation between gas partial pressure, PA, and equilibrium liquid phase concentration, CLA*:

PA= where HA (bar m3/kg) is the Henry's law constant for component A in the medium. Henry's law is generally accurate for gases with low solubility, such as the solubility of oxygen in water or in fermentation media. Thus from this relation, as shown in Fig. 5.7, the corresponding equilibrium concentrations can be easily established.

CLC* Figure 5.7. Equilibrium concentrations based on Henry's law.

For gases of low solubility, e.g., oxygen and carbon dioxide in water, the concentration gradient through the gas film is very small, as compared to that within the liquid film, as illustrated in Fig. 5.6. This results from the relatively

5.3 General Oxygen Balances for Gas-Liquid Transfer

123

low resistance to mass transfer in the gas film, as compared to the much greater resistance to mass transfer in the liquid film. The main resistance to mass transfer is predominantly within the liquid film. This causes a large change in concentration (Cy - CL), since the resistance is almost entirely on the liquid side of the interface. At the interface, the liquid concentration, Cy, is in equilibrium with that of the gas, CGI, and since CGI is very close in magnitude to the bulk gas concentration, CLI must then be very nearly in equilibrium with the bulk gas phase concentration, CG- This is known as liquid film control and corresponds to the situation where the overall resistance to mass transfer resides almost entirely within the liquid phase. The overall mass transfer capacity coefficient is KLa. Hence the overall mass transfer rate equation used for slightly soluble gases in terms of the specific area is Q = KLa (C L *-C L )V L where CL* is in equilibrium with CG, as given by Henry's law, C G = HCL*, Mass transfer coefficients in fermentation are therefore generally spoken of as KL values or Kâ values for the case of mass transfer capacity coefficients.

5.3

General Oxygen Balances for Gas-Liquid Transfer

In order to characterize aeration efficiency, to predict dissolved oxygen concentration, or to follow the biological activity it is necessary to develop models, which include expressions for the rate of oxygen transfer and the rate of oxygen uptake by the cells. Well-mixed phase regions, in which the oxygen concentration can be assumed uniform, can be described by simple balancing methods. Situations in which spatial variations occur require more complex models, as described in Sec. 5.4. The following generalized oxygen balance equations are derived for well-mixed phases, using the well-mixed tank concept. In the situation in Fig. 5.8, both the liquid and gas phases are defined by distinct well-mixed regions and by the total volumes of each phase, VL and

VG.

124

5 Mass Transfer

Gas

CQO» GO

Figure 5.8. The balance regions for well-mixed gas and liquid phases in a continuous reactor.

For the gas phase the oxygen balance can be developed as follows: Rate of accumulation of oxygen in gas

( Rate of ^ > Flow of f Flow of \ transfer oxygen in _ oxygen out _ of oxygen Vin exit stream/ ; inlet gas streamy v from gas ,

Thus, for the gas phase, dCGi

K L a(C L i*- C L i)V L

where VQ represents the volume of gas in the dispersed phase, or the gas holdup. For the liquid phase, Rate of ^ accumulation of oxygen ^ in liquid

/Flow of ( Flow of oxygen oxygen in inlet out in liquid exit V stream J ' Rate of consumption of oxygen in liquid

Rate of oxygen consumption = -rO2 = -qo2

Rate of transfer of oxygen ^ from gas


125

Thus for the liquid phase, dCLi

- LiC L i + K L a(C L 1 *-C L i)V L -

The above equations include accumulation, convective flow, interphase transfer and biological oxygen uptake terms. Here CLI* is the equilibrium solubility of oxygen corresponding to the gas phase concentration, CGI , and is calculated by Henry's law, according to the relationship:

Typical units are as follows: CG and CL (kg/m3); G and L (m3/s); Kâ (1/s); VG and VL (m3); qO2 (kg/kg s); X (kg/m3). In the next sections, the general equations, given above, will be applied to important special situations.

5.3.1

5.3.1.1

Application of Oxygen Balances

Case A. Steady-State Gas Balance to Determine the Biological Uptake Rate

The convective terms in the generalized liquid balance equation can usually be neglected, owing to the low solubilities of oxygen in water (about 8 g/m3). This gives the steady state liquid balance, dCL/dt=0, relation as: K L a(C L i*- CLI) = qo2Xi Thus at steady-state, the oxygen transfer rate is effectively equal to the oxygen uptake rate. Even during batch fermentations this is approximately true. Substituting this relationship into the steady state gas balance gives,

0 = GO CGO - GI CGI - qo2 X VL The above equation can also be derived from a steady state balance around the entire two-phase system. It shows that the biological oxygen uptake rate can be calculated from knowledge of the gas flow rates and the gas concentrations.

126

5 Mass Transfer

This application is very important in fermentation technology, since it permits the on-line monitoring of the rate of fermentation, by gas balancing methods (Heinzle and Dunn, 1991, Ingham and Dunn, 1991).

5.3.1.2

Case B. Determination of Ki,a Using the Sulfite Oxidation Reaction

If a chemical reaction, classically the oxidation of sodium sulfite, is used to take up the oxygen from solution, then the term qo2 X VL in the liquid phase balance may be replaced by the chemical reaction term, ro2 VL- At steadystate, K L a(C L i*-C L i) = r02 Usually ro2 is obtained by taking samples and titrating for the fractional conversion of sulfite, which can be related by stoichiometry to the oxygen consumption. Since the chemical reaction causes the liquid dissolved concentration CLI to fall essentially to zero and with CLI* calculated from the oxygen concentration in the exit gas, the value of the overall mass transfer capacity coefficient, Kâ, can be estimated. An improved method uses the gas balance instead of titration to obtain ro2 in the manner outlined above for qO2 X and also provides a check on the sulfite measurements. The sulfite method is useful for comparing aeration systems, but the values are difficult to relate to actual fermentation conditions owing to the very different physical conditions (coalescence, aeration rates) (Ruchti et al., 1985).

5.3.1.3

Case C. Determination of Ki,a by a Dynamic Method

If water is initially deoxygenated and is then re-aerated, the concentration of the dissolved oxygen will increase with time, from zero to effectively 100% air saturation at the end of the experiment. The exact form of the response curve obtained depends on the values of KLa, the driving force, (CLI* - CLI), and the measurement dynamics of the dissolved oxygen electrode. The liquid balance, for the unsteady state batch aeration condition, gives: =

K L a(CLi*-C L i)V L

127


The classical dynamic KLa method assumes that Kâ and CLI* are constant. Under these conditions, the differential equation can be integrated analytically to give the relationship: CL* ^ = K L at r * CL - rCL\}

(

Plotting the natural logarithmic concentration function on the left side of the equation versus time, should, in principle, give a straight-line relationship, with â as the slope. Usually deoxygenation is accomplished with nitrogen, so that initially the gas phase consists of nitrogen, which is gradually displaced and mixed with air. Under these conditions, CLI* is n°t constant, and a gas balance must be employed to calculate the variation in CGI versus t. Since the liquid phase concentration, CLI, is measured by means of a membrane covered oxygen electrode, the dynamics of measurement method usually cannot be neglected. The dynamics of the measurement electrode can be described, approximately, by a first-order lag equation, dCE

where TE represents the electrode time constant, and CE is the measurement signal. The fractional response of the electrode for a step change in CL would appear as shown in Fig. 5.9.

time Figure 5.9. Response of electrode for a step change in CE from zero to 100 % saturation according to a first-order lag model.

Note that TE corresponds to the time for the electrode to reach 63 % of the final response. The overall process dynamics involves thus the gas phase, the liquid

128

5 Mass Transfer

phase and the electrode response. The three responses might appear as shown below:

time Figure 5.10. Response of the gas, the liquid and the electrode measurement during a dynamic KLa experiment.

The values of three individual time constants determine the process response. These are TG = VG/G, (representing the dynamics of the gas phase), 1/KLa (representing the dynamics of the liquid phase mass transfer process), and IE (representing the measurement dynamics). This is illustrated in the simulation example KLADYN, Sec. 8.5.5.

5.3.1.4

Case D. Determination of Oxygen Uptake Rates by a Dynamic Method

Low oxygen uptake rates, as exist in slow growing systems (plant and animal cell cultures, aerobic sewage treatment processes, etc.), cannot easily be measured by a gas balance method, since the measured difference between inlet and outlet oxygen gas phase concentrations is usually very small. Due to the low solubility of oxygen in the liquid media, quite small oxygen uptake rates will cause measurably large changes in the dissolved oxygen concentration. Thus it is possible to measure qo2 X either by taking a sample and placing it in a small chamber or by turning off the reactor air supply, according to the liquid balance equation dCLi


129

Dissolved oxygen concentration decreases linearly and is equal to qo2 X as shown in Fig. 5.11.

Figure 5.11. Oxygen uptake rate determined by a dynamic method.

When the time required for an appreciable decrease in dissolved oxygen is large, as compared to the electrode time constant, the method is quite accurate and no correction for the electrode measurement dynamics is required (Mona et al., 1979). If the response is too fast the sample can be diluted. This method is illustrated by the simulation example OXDYN, Sec. 8.5.4. A similar simulation example, ANAMEAS, Sec. 8.8.7, illustrates dynamic measurements in anaerobic systems.

5.3.1.5

Case E. Steady-State Liquid Balancing to Determine Oxygen Uptake Rate

If the biomass is immobilized or retained by membranes within the reactor, oxygen can be supplied to the cells by means of a circulating liquid supply, which is aerated in a separate unit, external to the reactor.

130

5 Mass Transfer

CL1

CLO Figure 5.12 Oxygen uptake rate determined by a steady-state liquid balance.

It then becomes possible to determine the oxygen uptake rate, simply by measuring the liquid flow rate and the difference in dissolved oxygen in the liquid inlet and outlet flow streams, according to the following steady-state liquid phase balance equation: 0 = L(CLo - CLI) - qo2 Xi VL Thus the rate of oxygen supply via the liquid is equal to the rate of oxygen uptake by the cells. This method provides a very sensitive way of measuring low oxygen uptake rates (Keller et al., 1992, Tanaka et al., 1982). The casestudy H in Sec. 5.3.1.8 is an example of this use for an experimental reactor. The simulation example FBR, Sec. 8.4.9, also demonstrates this method.

5.3.1.6

Case F. KLa

Steady-State Deoxygenated Feed Method for

Feeding a deoxygenated liquid continuously to an aerated tank (Fig. 5.11) allows the oxygen transfer rate to be determined by difference measurement. Thus the liquid phase balance becomes 0 = L (CLO - CLI) + KLa (CLi* - C L i)V L Knowing the flow rate L, the oxygen liquid concentrations CLO and CLI and the outlet oxygen in the gas phase (to determine CLI*) permits the calculation of

131


KLa. Another variation of this would be to gas with oxygen-enriched air or with nitrogen, which would avoid the difficulty of producing a continuous source of deoxygenated liquid. A similar steady state method has been employed to obtain steady oxygen concentration profiles in column (Meister et al., 1980), and tubular bioreactors (Ziegler et al., 1977). A suitable steady state model for the tubular reactor then allows calculating the unknown Kâ by parameter estimation (Shioya et al., 1978). Deoxygenated C liquid LO

t

/•"V—X

"N^

-^^^

r-^—^-~\_/~\^^x

nM&^/m^M:WM.

Xiiiiiiiiiiji ^^^SRSSSS

^^îO^wiBii O

Wiiiiilmiiiiiii Air

Figure 5.12. Steady-state dissolved oxygen difference measurement for Kjâ.

5.3.1.7

Case G. Biological Oxidation in an Aerated Tank

A batch reactor liquid is aerated with a continuous flow of air to support a biological reaction, as shown in Fig. 5.13. air

air Figure 5.13. A batch bioreactor with continuous aeration.

132

5 Mass Transfer

The biological reaction in the liquid phase is first-order in oxygen concentration. Since oxygen is relatively insoluble (approximately 8 g/m3 saturation for air-water) the transfer rate is important to maintain a high dissolved oxygen concentration CL. The batch oxygen balance for the liquid phase is then: f

Rate of \ accumulation of V 02 in liquid ) dCL

=

,^ f (Transfer rate of f N\ l02 mto the hqmdj

/ Uptake rate of ^ by the cells

= K L a(CL*-C L )VL - k C L V L

A steady-state can be reached for which the mass transfer rate is equal to the oxygen uptake rate by reaction: 0 = KLa (CL* - CL) - k CL

giving for CL KLaCL*

CL = K L a

Using this equation, the reaction rate constant, k, can be determined if CL is measured and Kâ is known or measured. The equilibrium value, CL*, can be calculated from the gas phase concentration, and if there is little oxygen depletion it can be calculated from the inlet gas conditions. If the oxygen depletion in the gas phase is appreciable, then the mole fraction of oxygen in the exit may not be the same as in the inlet, and a gas phase balance must be applied to determine CL*:

Figure 5.14. Inlet and outlet oxygen mole fractions and total gas molar flow rates.


133

From the ideal gas law as shown before, assuming a well-mixed gas phase in steady state, N = (p / RT) F, where NO is the molar flow rate of air and F is the air volumetric flow rate. 0

=

/ Rate of O2 \ / Rate of O2 \ / Transfer rate of \ V in by flow ) ~ Vout byflow) ~ \ C>2 to the liquid )

Using the nomenclature in Fig 5.13,

0 = yoN0-yiNi-KLa(CL*-CL)VL

where

Assuming NO = NI, these equations can be solved to obtain yi and CLSolving for CL gives,

k~" CL*

CL = or for the apparent reaction rate, k re = -

k~ CL

,

Thus it is possible to distinguish between two different regimes for this system, transfer control and reaction control: 1) 2) 3)

Reaction rate control applies for low values of k/KLa, when re approaches k CL*, and CL approaches CL* Diffusion control applies for high values of k/KLa, when re approaches KL& CL* and CL approaches 0. If KLa = k, then rc = - (k/2) CL*, and CL approaches (1/2) CL*.

5.3.1.8

Case H. Modelling Nitrification in a Fluidized Bed Biofilm Reactor

Nitrification is a two-step microbiological process, in which the ammonium ion is oxidized to nitrite ion and further to nitrate ion as shown: NH4+ -» NO2- -» NO3-

134

5 Mass Transfer

This reaction is important in waste water treatment because of the toxicity of ammonia and its large oxygen demand. Several known organisms can gain energy from either of the two oxidation steps, but most commonly Nitrosomonas and Nitrobacter are responsible for steps (1) and (2), respectively. These organisms grow very slowly, obtaining their carbon from dissolved carbonate. Due to the very slow grow rates, it is of interest to retain the biomass within the reactor. One possibility considered here is to immobilize the biomass as a natural biofilm on a fluidized bed of sand (Tanaka and Dunn, 1982). The stoichiometric relations for the reaction steps (1) and (2) are: , 3 , NH4+ + j O2 -> NO2" + H2O + 2 H+ O2 -> NO3"

Summing the above steps (1) and (2) gives NH4+ + 2O 2 The reactor of volume, Vr, consisted of a conical sand bed column, which was fluidized by the liquid recycle stream flowing up through the bed. The recycle stream was oxygenated in a separate, baffled, tank contactor of volume VT, with turbine impeller and air or oxygen sparging. The reactor and oxygenator were thus separate parts of a recycle loop configuration. This could be operated batchwise or with a continuous feed and effluent stream flow to and from the system. When operating at high recycle rates, the whole system acted effectively as one well-mixed tank system. The reactor-oxygenator recycle loop can be analyzed as a total system or broken down into its individual components as shown in Fig. 5.14. These include liquid phase balance over the reactor and combined phase, liquid phase and gas phase balances over the oxygenator and over the total system.


135

Figure 5.15. Mass balancing regions for the fluidized bed reactor nitrification system.

The mass balances to be considered are those for oxygen and the nitrogencontaining reactants and products. The oxygen balance taken over the total system can be simplified by neglecting the accumulation terms and the liquid flow terms, that will be small compared to the gas rates and the consumption by reaction, owing to the relatively low solubility of oxygen in the liquid medium. Thus the oxygen balance becomes, 0 =

Here Vr is the volume of the reactor column. The nitrogen (N) components, NH4+, NCV, and NOs', in the liquid phase can be balanced around the total system by considering the accumulation, flow, and reaction terms for each of the N-containing components. For the total system each component equation has the form, VdCp = F(C N i-C N 2 ) dt When the reactor is operated as a batch system, F = 0, and when used as a continuous steady state reactor, dCN2/dt = 0. This equation can be used in column systems for very low single-pass conversion, when the differences in local reaction rate at the reactor inlet and outlet are not large. Although the reactions actually occur in the solid phase, because of the high solid-liquid interfacial area, the system is treated here as being quasi-homogeneous. The gas-liquid interfacial mass transfer area will often be small enough to be important for the overall process, and it is therefore useful to consider the gas

136

5 Mass Transfer

and liquid phases as separate balance regions. The absorption tank can be described by the oxygen balances for the liquid phase: 0 = F R (C L4 -C L3 ) + K L a(C L *-C L3 )V T and for the gas phase: 0 = G(CGi - CG2) - KLa (CL* - CL3) VT The liquid phase oxygen balance for the total system is 0 = KL

where ro2 is the oxygen uptake rate by the reaction. These equations, which assume ideally mixed phases, are useful in designing the gas absorber according to the required oxygen transfer coefficient. Balancing the oxygen around the reactor gives 0 =

Since CL4 at the reactor outlet is usually very low, then, FR CL3 = - ro2 Vr

which says that the oxygen uptake rate by reaction must be equal to the supply rate from the oxygenation tank. This is the condition of reaction-rate limitation by the oxygen transfer in the absorber. From the stoichiometry, the relationships between the molar reaction rates (rNH4> rO2» rH* r2,NO2» an^ fNO3) can be found. Thus, for example, the first nitrification step gives TNH4 =

2 T r l , O 2 = ~ri,NO2

and the total rate for 02 is given by the sum of the rates for steps (1) and (2). r

O2 = r l,O2 + r2,O2

From the measured concentration dependency of these rates, the reaction kinetics of the individual steps can be determined. The dependency of these rates on the individual concentrations can then be used to establish the reaction kinetic model. This model is the basis of the simulation example NITRIF, Sec. 8.5.3. A similar type of recycle, fluidized-bed reactor is the theme of simulation examples FBR, Sec. 8.4.9 and DCMDEG, Sec. 8.4.6.

5.4 Models for Oxygen Transfer in Large Scale Bioreactors

5.4

137

Models for Oxygen Transfer in Large Scale Bioreactors

Large-scale industrial fermenters can generally be expected to exhibit deviations from the two idealized flow conditions of perfect mixing or perfect plug flow. Thus the assumption of completely mixed gas or liquid phases may not be valid. Little experimental information is available on concentration inhomogeneities or concentration gradients within large bioreactors. Residence time distribution information, from which a physical and mathematical model could be established, is also generally not available. Convection currents within the liquid phase of a bioreactor are usually caused by the mechanical energy inputs of agitation and aeration. It is often reasonable to assume that slowly changing quantities, such as biomass concentration, substrate concentration, pH and temperature are uniform within the whole mass of bioreactor liquid. Oxygen must be considered, however, as a rapidly changing substrate, owing to its low solubility in fermentation media. It is therefore necessary to consider that differences in oxygen transfer and uptake rates will create oxygen concentration gradients throughout the reactor. Buoyancy forces carry the gas from the lower gas inlet point up to the top liquid surface. In the absence of mechanical agitation, the gas phase might move from the bottom to the top of the reactor in an approximate plug flow manner, with very little backmixing. If the stirring power supplied to the fermenter, however, is sufficient to create liquid velocities, that are greater than the free rise velocity of a bubble (about 26 cm/sec) then the bubbles will circulate around the fermenter, before eventually escaping. Very high power inputs can cause the smaller bubbles to circulate many times within the vessel and spend an appreciable time before reaching the surface. Under such conditions, if no bubble coalescence occurs, the gas phase would contain a fraction of small bubbles, depleted of oxygen but with a large surface area. Obviously any well-mixed phase assumption becomes difficult to justify. The gas phase flow conditions in large scale industrial fermenters usually lie somewhere between the extreme cases of idealized plug flow and perfect mixing. Experimental residence-time distribution information, obtained by helium tracer techniques under actual operating conditions, are then necessary to characterize the gas phase flow. Unfortunately very little experimentation on industrial scale equipment has been reported. Hydrostatic pressure gradients in tall fermenters will cause large differences in the oxygen solubility, CL*, with regard to the depth position in the tank. In a 10m tall reactor, the oxygen solubility for a given gas composition will be twice that at the bottom of the tank as compared to the top surface, since the total pressure is effectively doubled. This is seen by Henry's law which can be written as:

138

5 Mass Transfer

d* = y°H2p where yo2 is the mole fraction of oxygen in the air and p is the total pressure at some point in the tank. The possibility that oxygen gas compositions, dissolved oxygen concentrations, oxygen solubilities, gas holdup volumes, bubble sizes and other transfer parameters can vary with depth in a tall fermenter introduces a much greater degree of complexity to the problem of modelling the reactor. This makes it difficult to obtain data on oxygen mass transfer coefficients. Although it is impossible to give specific recommendations that apply to any particular situation, a further discussion of possible models and their underlying assumptions may help to define the problem. Incorporated into the more complex models, discussed below, are such factors as gas and liquid phase flow pattern, gas composition gradients and the effects of hydrostatic pressure. Great caution and wisdom must be exercised to avoid creating a model that is too complex to verify by experimentation. Experienced engineers will say "Keep it simple!" and "Avoid too much model!". All large scale reactors, whether multi-impeller tanks or column fermenters, will display some axial dissolved oxygen concentration gradients. The most general method for modelling is to represent the reactor using balances in a series of sections or stages. Mass balances in multi-stage process are easy to formulate, since both the liquid and gas phases may be assumed to be wellmixed, for any given stage of the cascade.

Figure 5.16. A single gas-liquid stage with backmixing of the liquid phase.

The formulation of the mass balances for a single stage, as shown in Fig. 5.16, follows closely that described previously, except that now the reactor is made up of many stages which are interconnected by the flows of gas and liquid between stages and by diffusive mass transfer mechanisms.

139


5.4.1.

Case Studies

5.4.1.1

Case A. Model for Oxygen Gradients in a Bubble Column Bioreactor

The application of the stagewise modelling approach is shown below, where a bubble column reactor is modelled as a five-stage reactor system. The reactor will be assumed to operate cocurrently, as would be also the case for the riser of an airlift bioreactor. Exit Gas

Exit Liquid

A CQB A CLS Gas

Gas Feed

ill

Liquid Feed GO

Figure 5.17. Stagewise model of a bubble column bioreactor.

L'

LO

140

5 Mass Transfer

The oxygen balance equations for the gas and liquid phases of each stage are as follows: if

= FG(C G n-l-C Gn ) - K L a(C Ln *-C Ln )V L

f\(~^

VL-dT = F L(C L n-i-C Ln ) + K L a(C L n*-C L n)VL - rn VL where,

r for each stage as functions of time and also yields the resulting final steady state values. Note that the biomass concentrations Xn are assumed constant, otherwise biomass balance and growth kinetics equations would have to be added to the model. Using simulation methods, other effects, such as the effect of hydrostatic pressure on CG or on bubble size could be included. The simulation example DCMDEG, Sec. 8.4.6, demonstrates some aspects of the stagewise modelling approach.

5.4.1.2

Case B. Model for a Multiple Impeller Fermenter

Mixing in a tank reactor is complex, and it would be necessary to consider liquid flow in both directions. It is generally assumed, however, that the intensity of mixing is such that no radial variations occur. Fig. 5.16 represents a multiple impeller reactor with well-mixed liquid zones in the region of each impeller. The reactor can be described approximately by means of a threestage model. Mixing of the liquid in a direction which is directly opposite to that of the main flow liquid (here upwards) can be incorporated into the model, by the assumption of a backmixing stream, with flow rate FB- This backmixing stream accounts for a flow interaction between the mixing zones and for deviations from ideal stage mixing. To determine FB, a tracer experiment

141


would need to be performed to obtain the necessary information regarding the degree of backmixing actually existing in the reactor. Exit Gas

Exit Liquid

c

FG3 A

A C L3

d

\_^

Gas L2

L3

G2

Gas 'LI

G1

FL+FB Gas

I

F

G- C GO

\\>cu>

Liquid Feed

Inlet Gas

Figure 5.18. Stagewise approximation for stirrer regions in multi-stirrer tank.

To model this system, the liquid-phase impeller zones are assumed to be wellmixed, and the plug-flow gas is described by a series of well-mixed phases, together with an arithmetic-mean, concentration-driving-force approximation. Here the flow rates and mass transfer coefficients are assumed constant.

Stage 1: dCLi VL -ar = FLCLO + FBCL2 - (FL + FB) CLI + + K L a(CLi*-C L i)VL + riV L V

G

dCGl

"•""'

xx-
=0.0

The semicolon or curly brackets are used for comments. INIT specifies the initial conditions. XQ, SQ and PQ are used here for the initial conditions, or the values at time=0. The form X' designates the time derivative or d/dt(X) can be used. Most models are conveniently structured in terms of mass balances and kinetics. Any result quantity on the left of the equal sign is stored for further calculations or for use in graphing. Usually concentration versus time is of interest, but rates versus concentrations make very useful plots for understanding the kinetics. The five integration methods require specifying time intervals, such as DT, DTMIN and DTMAX. This requires a bit of experience. Care must be taken to see that the same results are obtained by two different methods or for at least two different DT values. As is seen in the Appendix, Berkeley Madonna provides many possibilities to change the parameters and graph new runs. These include the following: changing parameters with the parameter window and making overlay plots; changing parameters with sliders; using the Batch Runs facility.

8.1 Introductory Examples

197

Nomenclature Symbols k ] and k2 KS P r S V X

Y

H

Product formation constants Saturation constant Product concentration Reaction rate Substrate concentration Reactor volume Biomass concentration Yield coefficient Specific growth rate

1/h and kg/kg kg/m 3 mg/m3 kg/m 3 h and kg/m 3 h kg/m 3 m3 kg/m 3 kg/kg 1/h

Indices Refers Refers Refers Refers Refers Refers

2 m P S X

to non-growth association rate to growth-association rate to maximum to product to substrate to biomass

Exercises 1.

Vary KS, Mm separately and observe the effects in the graphs. It is useful to zoom in on regions of importance by using the zoom tool in the tool bar.

2.

Vary the product kinetics constants (Kj and K2>, and observe the effects. Observe the P versus time curve when S reaches zero.

3.

Plot the rates versus the concentrations.

198

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results The plots of X, S and P versus T in Fig. 3 show that when substrate is depleted, the growth stops, and the product continues to increase, but only linearly. The results of Fig. 4 are obtained by varying the product formation rate constants, ki in three runs using a slider, which is defined in the Parameter Menu. Run 1:1500 steps in 0 seconds .10

Figure 3. Plots of X, S and P versus time during batch growth and production.

Run 3: 1500 steps in 0.0333 seconds

^.......^

-10

"l"""'"*"'.^_

/ **«

— -S:2 P:2 — -S:3 P:3

/

\

/*

X

•7

-'7^'^ 5

1

0

15

.5

•3

•1

..

\ 20

cn

-4

•2

.**&'*' \ 0

•6

/ /' \ /s' \,"~'"

-rp^ fm vtf*EV

•9 -8

25

•0

3()

TIME

Figure 4. Plots of P and S versus time created by varying the product formation rate constant

199


8.1.2

Chemostat Fermentation (CHEMO)

System A continuous fermenter, as shown in Fig. 1, is referred to as a chemostat. At steady state the specific growth rate becomes equal to the dilution rate, |a = D. Operation is possible at flow rates (F) which give dilution rates (D = F/V) below the maximum specific growth rate (|um). Washout of the organisms will occur when D > (a. The start-up, steady state and washout phenomena can be investigated by dynamic simulation. S,X

D,S F

Figure 1. Chemostat with model variables.

Model The program BATFERM may be easily modified to allow for chemostat operation with sterile feed by modifying the mass balance relationships to include the inlet and exit flow terms. The corresponding equations are then: For cells dX

.= - D X + rx

200


For substrate dS 3f

= D (SF - S) + rs

For product dP df

= -DP

+ rp

where D is the dilution rate and Sp the concentration of the limiting substrate in the feed. The same kinetic expressions as in BATFERM will be applied here.

Program Note the conditional statement for D which allows a batch startup. {CHEMO}

(Chemostat startup and steady state. batch reactor until time=tstart} {Constants} UM=0.3 KS = 0.1 Kl = 0.03 K2=0.08 Y = 0.8 X0 = 0.01 S0=10 P0=0 SF = 10 Dl = 0.25 t start = 5

; ; ; ; ; ; ; ; ; ; ;

Startup

1/h kg/m3 kgP/kgX h kgP/kgX kg X/kg S Initial biomass inoculum, kg/m3 Initial substrate cone., kg/m3 Initial product conc.,kg/m3 Feed cone. ,kg/m3 Dilution rate, 1/h Start time for the feed

(Initial Conditions} Init X=XO Init S=SO Init P=PO {Mass Balances} X'=-D*X+RX ; BIOMASS BALANCE EQUATION S • =D* (SF-S) +RS ; SUBSTRATE BALANCE EQUATION P'=-D*P+RP ; PRODUCT BALANCE EQUATION

as

201


{Kinetics} RX = U*X U = U M * S / ( K S + S) RS=-RX/Y RP= (K1 + K2*U) *X

; BIOMASS RATE EQUATION, kg/m3 h ; MONOD EQUATION, 1/h ; SUBSTRATE RATE EQUATION, kg/m3 h ;PRODUCT RATE EQUATION, kg/m3 h

{Conditional equation for D=if time>=tstart then Dl Prod=D*X

D} else

/Productivity

0 for

biomass,

kg/m3

Nomenclature

Symbols D ki and KS P r

Dilution rate Product formation constants Saturation constant Product concentration Reaction rate

S X Y

Substrate concentration Biomass concentration Yield coefficient Specific growth rate Time lag constant

ILL 1

Indices F MONOD P S X

Refers to feed Refers to Monod kinetics Refers to product Refers to substrate Refers to biomass

1/h 1/h and kg/kg kg/m3 mg/m3 kg/m3h and kg/m3 kg/m3 kg/m3 kg/kg 1/h

h

202


Exercises 1.

Increase D interactively to obtain washout.

2.

Note the steady state values of X and S; calculate Y from these.

3.

Change SF. Does this alter S at steady state? Why?

4.

Calculate S at steady state from D. Verify by simulation.

5.

Change the program to account for biomass in the feed.

6. Operate initially as a batch reactor with D = 0, and switch to chemostat operation with D < |jm. Does this reduce the time to reach steady state? Is the exact time of switchover important? 7. Include maintenance requirements to the substrate uptake kinetics using RS = - ( U / Y + M ) * X . Remember to add a value of the maintenance coefficient M to the constants. Investigate the influence of the value of M on the steady state biomass concentration. 8. Using a Parameter Plot, obtain steady state values of X and S for a range of Dl. 9. Rapidly-changing dynamic fermentations do not follow instantaneous Monod kinetics. Modify the model and the program with a dynamic lag on jo, such that d|j /dt= (nMonod - l-O/t- Compare the response to step changes in D for suitable values of the time lag constant t.

Results The graphical output in Fig. 2 shows three startups of the fermenter under initially batch growth conditions, using three values for D l . The break in the concentration-time dependency as feeding starts is quite apparent, and the new transient then continues up to the eventual steady state chemostat operating condition or washout in the case of one run. For the results of Fig. 3 the program was changed by adding the line PROD = X*D, and the final, steady state value of production rate was plotted versus Dl for twenty runs, using the Parameter Plot feature of Madonna.

203


Run 3: 4000 steps in 0.0333 seconds 10

Figure 2. Startups of the chemostat after initial batch growth for 3 values of Dl.


2

Figure 3. Productivity in a chemostat. Steady states are shown for 20 runs using the Parameter Plot.

204


8.1.3

Fed Batch Fermentation (FEDBAT)

System In this case the model equations allow for the continuous feeding of sterile substrate, the absence of outflow from the fermenter and the increase in volume (accumulation of total mass) in the fermenter, schematically as shown in Fig. 1. Simulation of fed batch fermenters can be used to demonstrate the important characteristics of quasi-steady state, linear growth, and use of alternative feed strategies. F,SF

V X

s p

Figure 1. Fed batch fermenter with model variables.

Model For fed batch operation, the equations become as follows: Total balance dV

dT = For cells

For substrate

F

205


For product

where F is the volumetric feed rate, Sp is the feed concentration and V is the volume of the fermenter contents at time t. Thus the mass quantities, VX, VS, and VP are calculated and are divided by the volume at each time interval to obtain the concentration terms required for the kinetic relationships. The kinetics are taken to be the same as in BATFERM.

Program The "IF" statement in the program causes the continuous feed to start when time reaches tfeed, at which point batch operation stops and the fedbatch starts. (FEDBAT)

{Fermentation {Flow rate is time=tfeed.}

with

batch

initially

start zero

up}

and

is turned

on at

{ Constants} UM=0.3 ; 1/h KS = 0 . 1 ; kg/m3 ; kgP/kgX h Kl = 0.03 K2 = 0.08 ; kgP/kgX ; kg X/kg S Y = 0.8 ; Initial biomass inoculum, kg/m3 X0 = 0 .01 S0 = 10 ; Initial substrate cone., kg/m3 P0 = 0 ; Initial product conc.,kg/m3 ; Feed conc.,kg/m3 SF = 10 ; Feed flow rate, m3/h Pl-1. 5 tfeed=22.5 ; Start time for the feed {Initial Conditions} init V=l init VX=V*XO init VS=V*SO init VP=V*PO

206


{Mass balances, d/dt(V)=F d/dt(VX)=RX*V d/dt(VS)=F*SF+RS*V

kg/h}

d/dt(VP)=RP*V

{kg/h}

{Calculation

of

concentrations}

X=VX/V S=VS/V P=VP/V

{Kinetics} RX=U*X U=UM*S/(KS+S) RS=-RX/Y RP=(K1+K2*U)*X

D=F/V

{nominal

dilution

{Turning the feed on at time = tfeed} F=if time>=tfeed then Fl else 0 {batch

rate,

start

1/h}

up}

Nomenclature Symbols D F KS ki, k2 M P r S X V Y |i T

Dilution rate Flow rate Saturation constant Constants in product kinetics Maintenance coefficient Product concentration Reaction rate Substrate concentration Biomass concentration Reactor volume Yield coefficient Specific growth rate Time delay constant

1/h m3/h kg/m3 1/h and kg/kg kg/kg h kg/m3 kg/m3 h kg/m3 kg/m3 m3 kg/kg 1/h h

207


Indices F P S X

Refers Refers Refers Refers

to feed to product to substrate to biomass

Exercises

Results Operation begins under initial batch conditions, and feeding of substrate is started at tfeed=22.5 h. In Fig. 2, the break in the batch growth transient, as semi-batch feeding starts is very apparent, with the transient continuing to an apparent "quasi" steady state operating condition. Under these conditions the biomass concentration becomes constant, while the substrate concentration (not shown) is below the KS value and decreases very slowly. As seen in the zoom of Fig. 3, the values of D (= F/V) also decrease since V increases due to the incoming feed, and D eventually becomes equal to p when S falls below K$. The total biomass is determined by the yield coefficient times the total amount of substrate that has been consumed, which is approximately equal to the amount in the reactor initially plus the amount added during the feeding period. During the quasi-steady state, the total biomass will increase linearly with time if, as in this case, the feeding flow rate is constant. This is a "linear

208


growth" situation in which the growth rate is limited by the feeding rate. In Fig. 3 the values of X, S, and P are plotted versus T for a switch from batch (F = 0) to fed batch (F = 5) at time T = 20 h. The product production rate depends linearly on biomass concentration, and thus even when ja becomes very low, P will continue to increase linearly in mg/m3 amounts. TIME= 34.13 X = 12.34

.^

10-.-

10

20

30

40

50

60

70

80

90

100

Figure 2. Transients during the fedbatch fermentation.

Run 1: 5000 steps in 0.1 seconds 0.4.

0.35-

-~, I V

0.3-

0.25. 3 Q 0.2-

CO 0.15-

-J. T

0.10.05027

28

29

30

31

32

TIME

Figure 3. Zooming in on the quasi-steady state.

33

34

35

36

37

8.2 Batch Reactors

8.2 8.2.1

209

Batch Reactors Kinetics of Enzyme Action (MMKINET)

System The intermediate enzyme-substrate complex is the basis for the simplest form of enzymatic catalysis (Fig. 1): E +S

^

»»

ES

*-

E +P

k2

Figure 1. Mechanistic model for enzymatic reaction.

Model The equations for substrate, enzyme-substrate complex and product in a batch reactor are: -— = ki E S - k 2 ES dt dFS •^ = ki E S - (k2 + k3) ES

dt

Using the steady state approximation for the change of active complex,

dt

the Michaelis-Menten equation is obtained. _dS _ ~ dt ~

K

M +S

210


where vmax = k3 E0 and KM = (k2+k3)/ki.

Program The program with the detailed mechanism is on the CD-ROM.

Nomenclature

Symbols E ES k KM P S Vmax

Enzyme concentration mol/m3 Enzyme-substrate complex concentration mol/m3 Reaction rate constants various Michaelis-Menten constant mol/m3 Product concentration mol/m3 Substrate concentration mol/m3 Maximum velocity mol/m3 h

Indices 0 1 2 3 S Mm

Exercises

Refers to initial values Refers to reaction 1 Refers to reaction 2 Refers to reaction 3 Refers to substrate Refers to Michaelis-Menten

211

8.2 Batch Reactors

Results Figs. 2 and 3 give the results of the full model and the Michaelis-Menten simplification, respectively Run 1:119 steps in 0.0167 seconds

0.009. 0.0080.007-

Lx""""^"~ \ f' \ /

0.006 •

^0.005-

•0.7

f

^..] ...—

8:1

•0.6

ES:1

-•-

•0.5

P:1

riL t ~m f \

LU

0.0040.003-

•

*i*

0.001 - i

•0.4 •0.3

\

1

0.002 •

-0.8

v

:.

tn

.0.9

fc

*v.

•0.2

\. i

• 0.1

" «%^

""""•%-»,

^*"'..._

.0

10

20

30

40

50

TIME

Figure 2. Results from the full model

70

80

90

100

a * to

212



\ \ 10

20

30

40

50

60

70

80

90

100

TIME

Figure 3. Results from the Michaelis-Menten simplification.

8.2.2

Lineweaver-Burk Plot (LINEWEAV)

System This program simulates the batch uptake of substrate using Michaelis-Menten kinetics, of the form, r

s = K^TS-

The inverse rate is plotted versus the inverse concentration (Fig. 1). Comparison of this plot with the concentration-time plot together with the Km value, demonstrates the importance of data in the Km region and the difficulty of obtaining this in a batch reactor. It is useful to make specially-scaled graphs in the KM region.

213

8.2 Batch Reactors

Figure 1. Lineweaver-Burk plot to determine vm and

Model The model is that of a batch reactor with Michaelis-Menten kinetics. dS dF = ~ r s

Program To make the Lineweaver-Burk plot, the inverse values of S and rs are calculated in the program on the CD-ROM.

Nomenclature Symbols KM r S

Michaelis-Menten constant Reaction rate Substrate concentration

kg/m3 kg/m3 kg/m3

214 Si V


Inverse substrate concentration Reaction velocity or rate Inverse reaction velocity or rate

m3/kg kg/m3 h m3 h/kg

Indices 0 m S

Refers to feed Refers to maximum Refers to substrate

Exercises

Results The results are shown in Fig. 2 (rates and concentrations versus time) for a range of Michaelis-Menten constants KM and in Fig. 3 the corresponding Lineweaver-Burk plots.

215

8.2 Batch Reactors

Run 4:13710 steps in 0.133 seconds

'0.5 •0.45

.0.4 •0.35

.0.3 .0.25 £

•0.2 -0.15 •0.1

•0.05

140

160

Figure 2. Rate and concentration plots for KM = 0.2, 0.5, 1.0 and 2.0 (bottom to top curves). Run 4:13710 steps in 0.133 seconds

Figure 3. Lineweaver-Burk plots for KM = 0.2, 0.5, 1.0 and 2.0 (bottom to top curves).

8.2.3

Oligosaccharide Production in Enzymatic Lactose Hydrolysis (OLIGO)

System Some enzyme catalyzed reactions are very complex. For this reason their rigorous modelling leads to complex kinetic equations with a large number of constants. Such models are unwieldy and are usually not suitable for practical

216


purposes. One approach to simplify them is to neglect formation of enzymesubstrate complexes altogether and to deal only with overall reactions of the react ants to products. An example of such a reaction is the enzymatic lactose hydrolysis, a complex process involving a multitude of sequential reactions leading to higher saccharide (oligosaccharides) intermediates. The mechanistic model is rather complex even when only trisaccharides are considered (Fig. 1). La + E Ga E + La

^

^

^

GaE + H2O

^-

LaE

**

E + Tr

^

E + Ga

Ga + GI + E

Figure 1. Complex and simplified models for the enzymatic hydrolysis of lactose, where the symbols are La for lactose, Ga for galactose, Gl for glucose, Tr for trisaccharide and E for enzyme.

Neglecting the enzyme complexes, however, gives a simplified model (Fig. 2) requiring only three constants:

K

1L.a a

!

fî Ga T. V3II

to

K

1

^ Tr •i

La -i- Ga fcaCl

K

2

Figure 2. Simplified model for the enzymatic hydrolysis of lactose.

The simulation of this model is easy, and the constants can be adjusted to achieve good agreement with experimental data.

Model This simple batch reactor model is equivalent to the Michaelis-Menten product inhibition model.

217

8.2 Batch Reactors

dLa -gjdGa

- K! La - KI La Ga + K2 Tr = Kj La - KI La Ga + K2 Tr dTr

= KI La Ga - K2 Tr

Initial conditions: Lao =150 mmol/m3, Gao = 0, Trg = 0 Range of the kinetic constants: KI = 0.02 - 0.06 miir1, KI = 0.02 - 0.1 L/mmol min, K2 = 1 - 50 min"1.

Program It was found that K2 must be two orders of magnitude greater than KI in order to bring the simulation into agreement with the experimental data. The program is on the CD-ROM.

Nomenclature

Symbols Ga Gl

K2 La Tr

Galactose concentration Glucose concentration Reaction rate constant (La —> Ga + Gl) Reaction rate constant (La + Ga -> Tri) Reaction rate constant (Tri -> La + Ga) Lactose concentration Trisaccharide concentration

Indices 0

Refers to initial concentration

mmol/L mmol/L 1/min L/(mmol min) 1/min mmol/L mmol/L

218


Exercises

Results The outputs in Figs. 3 and 4 show the influence of KI, KI and Lao on the sugar concentration profiles. Run 1:10000 steps in 0.05 seconds 100.

r100

90. 80.

70 60.

. 50. 40 30 20 10 0 20

100 TIME

Figure 3. Sugar concentrations with Kr = 0.04, K{ = 0.05, La0 = 100.

180

200

219

8.2 Batch Reactors


«(

80

'

80

100

120

140

160

180

200

Figure 4. Sugar concentrations with Kj = 0.06, KI = 0.1 Lao = 160.

Reference Prenosil, J. E., Stuker, E. and Bourne, J. R. (1987) "Formation of Oligosaccharides during an Enzymatic Lactose Hydrolysis Process", Parts I and II: Biotechnol. Bioeng. 30, 1019-1031.

8.2.4

Structured Model for PHB Production (PHB)

System Heinzle and Lafferty (1980) have presented a structured model to describe the batch culture of Alcaligenes eutrophus under chemolithoautotrophic growth conditions, as discussed in Case C, Sec. 3.3.1. Growth and storage of PHB are described as functions of limiting substrate S (NH4+), residual biomass R and product P (PHB) concentrations.

220


Figure 1. Structured kinetic model for PHB synthesis.

Model In the model seen in Fig. 1 the whole cell dry mass (X) consists of two main parts, namely PHB (P) and residual biomass (R), where R is calculated as the difference between the total cell dry weight and the concentration of PHB (R = X - P). R can be considered as the catalytically active biomass, including proteins and nucleic acids. With constant concentrations of the dissolved gases, two distinct phases can be recognized: growth and storage. During the growth phase there is sufficient NH4+ to permit protein synthesis. When the limiting substrate NH4+ (S) is exhausted, the protein synthesis ceases, and the production rate of PHB is increased. During the storage phase only PHB is produced. The limiting substrate NH4+ (S) is essential to produce R and limits its synthesis at low concentrations. For the batch process, dR dF

= r

R = MR

where TR is the rate of synthesis of R and (j is the specific rate of synthesis of R, where S (S/Ks,2)n + S) + ^m,2 ! + (S/KS,2)n

where n is the empirical Hill coefficient (see Sec. 3.1.2), having a value of 4 in this example. This is based on the postulate that there are two different mechanisms for the assimilation of NH4+ in procaryotes. This formulation is not a mechanistic one,

8.2 Batch Reactors

221

since in reality the enzyme system, using energy to assimilate NH4+, is repressed by high concentrations of NH4+. For the substrate dS 1

dF = rs = -YR/S **

The rate of synthesis of P(rp) is assumed to be the sum of a growth associated term (rpj) and a biomass associated term (rp,2) and is given by, dP df

= rp = r P j + rP,2

where r P j = YP/R rR The non-growth associated term of the synthesis of P(rp,2) is assumed to be a function of the limiting substrate S, of the residual biomass R and of the product P. When the PHB content in the cells is high, the rate of synthesis of P is decreased, which can be formally described as an inhibition.

Program The program is found on the CD-ROM.

Nomenclature Symbols KI KS n P R rp TR rs

Inhibition constant, for (NH^SC^ Saturation constant Hill Coefficient Product concentration (PHB) Residual biomass concentration Rate of synthesis of PHB Rate of synthesis of R Rate of substrate uptake

kg/m3 kg/m3 kg/m3 kg/m3 kg/m3 kg/m3 kg/(m3 h)

222

X YP/R YR/S


Limiting substrate concentration NHj as (NH4)2S04 Biomass concentration Yield coefficient Yield coefficient, Specific rate of synthesis of R (rR/R) Specific rate of synthesis of P (rp/P)

Indices 1 2 m

Exercises

Refers to reaction 1 Refers to reaction 2 Refers to maximum

kg/m3 kg/m3 kg/kg kg/kg 1/h 1/h

223

8.2 Batch Reactors

Results Run 1: 416 steps in 0.0167 seconds

4-1

-16

3.5-

/

f

3-

•-.^

2

-

1.5-

y / " y "»

1-

/

^^ —*"^

0.50-

0

5

-10

".•"T.-'V/T

/

I J

'""

ISi$%£

M*

/ *"'•"

•12

/

T '"U--b

of

-»*'

/

2.5-

•14

^"~'"~*" "

'•. .'

*

-8 a.

-' '*"

-6

f'

•4

V

-2

_j__»— *% 10

_n

15

20

25

30

35

40

TIME

Figure 2* Profiles of residual biomass concentration R, substrate S and product P in the batch fermentation. Run 4: 41 6 steps in 0.01 67 seconds

35- •—...„

•5 '"'V^

•4.5

^.^.-

30-

-4

'v 25-

\ \

20-

a 15-

/

.._.. 3:3(2,3) ~- P:3(2.3)

/ 1

/

\

«-*.

\

— P:4 (5)

•3.5 •3

-2.5 (/)

/

-2 """-,

10-

"\ ""

5•

\

f

.*""r-'"

-1.5

\"' /î''^'

-1

-0.5

^^";:^

0.

0

5

10

15

-n 20

25

30

35

40

TIME

Figure 3. PHB formation at two different initial substrate concentrations.

References Heinzle, E., and Lafferty, R. M. (1980) Continuous Mass Spectrometric Measurement of Dissolved H2, O2, and CC>2 during Chemolitho-autotrophic

224


Growth of Alcaligenes eutrophus strain H16. Eur. J. Appl. Microbiol. BiotechnoL 11, 8.1

8.3

Fed Batch Reactors

8.3.1

Variable Volume Fermentation (VARVOL and VARVOLD)

System Semi-continuous or fed batch cultivation of micro-organisms is common in the fermentation industries. The fed batch fermenter mode is shown in Fig. 1 and was also presented in the example FEDBAT. In this procedure a substrate feed stream is added continuously to the reactor. After the tank is full or the biomass concentration is too high, the medium can be partially emptied, and the filling process repeated. Since the variables, volume, substrate and biomass concentration change with time, simulation techniques are useful in analyzing this operation. This example demonstrates the use of dimensionless equations.

Figure 1. Filling and emptying sequences in a fed batch fermenter.

Model The balances are as follows: Volume, dv

dT = FO

Substrate,

224


Growth of Alcaligenes eutrophus strain H16. Eur. J. Appl. Microbiol. BiotechnoL 11, 8.1

8.3

Fed Batch Reactors

8.3.1

Variable Volume Fermentation (VARVOL and VARVOLD)

System Semi-continuous or fed batch cultivation of micro-organisms is common in the fermentation industries. The fed batch fermenter mode is shown in Fig. 1 and was also presented in the example FEDBAT. In this procedure a substrate feed stream is added continuously to the reactor. After the tank is full or the biomass concentration is too high, the medium can be partially emptied, and the filling process repeated. Since the variables, volume, substrate and biomass concentration change with time, simulation techniques are useful in analyzing this operation. This example demonstrates the use of dimensionless equations.

Figure 1. Filling and emptying sequences in a fed batch fermenter.

Model The balances are as follows: Volume, dv

dT = FO

Substrate, Biological Reaction Engineering, Second Edition. I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30759-1

8.3 Fed Batch Reactors

225

- = F0S0 Biomass,

d(VX) dt = rx The kinetics are

rx = MX |LimS

** - (Ks + S) and rx rs = -Y The dilution rate is defined as

In order to simplify the equations and to present the results more generally, the model is written in dimensionless form. Defining the dimensionless variables: v

=

V X

X< =

s

=

s

F =

- _ JL Mm

tft' = t

F

°

226


Expanding the derivatives gives d(V S) = V dS + S dV and d(V X) = V dX + X dV

Substituting, the dimensionless balances now become: Volume dV'

Biomass dX' dt'

Substrate dS' dt

= (l-S)D-jiX

The Monod equation is: KS +S

In Fig. 2 a computer solution shows the approach to and attainment of the quasi-steady state of the dimensionless fed-batch model.

227


Quasi- steady

Figure 2. Dynamic simulation results for a fed batch culture.

Programs The program VARVOL is based on the model equations with normal dimensions. The program VARVOLD is based on the dimensionless equations as derived above. Both are on the CD-ROM.

Nomenclature Symbols D F KS r S V

Dilution rate Flow rate Saturation constant Reaction rate Substrate concentration Reactor volume

1/h m3/h kg/m3 kg/m3 h kg/m3 m3

228 X Y


Biomass concentration Yield coefficient Specific growth rate

Indices 0 f m S X

Refers Refers Refers Refers Refers Refers

to feed and initial values to final to maximum to substrate to biomass to dimensionless variables

Dimensionless Variables

S' V X1 t'

Exercises

Dimensionless flow rate Dimensionless saturation constant Dimensionless substrate concentration Dimensionless volume Dimensionless biomass concentration Dimensionless time Dimensionless specific growth rate

kg/m3 kg/kg 1/h

229


Results During the quasi-steady state, \l becomes equal to D, and this requires that S must decrease steadily in order to maintain the quasi-steady state as the volume increases (Fig. 3). Increasing flow rates from 0.01 to 1.0 causes a delay in the onset of linear growth and causes the final biomass levels to be higher (Fig. 4). Run 1:105 steps in 0 seconds 4.5

Figure 3. Fed batch concentration and growth rate profiles, showing quasi-steady state. Run 7:105 steps in 0 seconds

5

2.5

10

C/>

230


Figure 4. Influence of flow rate on growth. Flow rate increase from 0.01 to 1.0.

References Dunn, I.J., and Mor, J.R. (1975) Variable Volume Continuous Cultivation. Biotechnol. Bioeng. 17, 1805. Keller, R., and Dunn, I.J. (1978) Computer Simulation of the Biomass Production Rate of Cyclic Fed Batch Continuous Culture. J. AppL Chem. Biotechnol. 28, 784.

8.3.2

Penicillin Fermentation Using Elemental Balancing (PENFERM)

System This example is based on the publication of Heijnen et al. (1979), and encompasses all the principles of elemental balancing, rate equation formulation, material balancing and computer simulation. A fed batch process for the production of penicillin as shown in Fig. 1 is considered with continuous feeding of glucose. Ammonia, sulfuric acid and o-phosphoric acid are the sources of nitrogen, sulfur and phosphorous respectively. Ophosphoric acid is sufficiently present in the medium and is not fed. Oxygen and carbon dioxide are exchanged by the organism. The product of the hydrolysis of penicillin, penicilloic acid, is also considered, thus taking the slow hydrolysis of penicillin-G during the process into account.

231


Glucose

Carbon dioxide

Oxygen Precursor Phenylacetic acid Sulfuric acid Ammonia

Figure 1. Streams in and out of the penicillin fed batch reactor.

Table 1. lists the components and their conversion rate designation.

232


Table 1. Component properties and rate designations. Compound Chemical formula Mol wt. Enthalpy Conversion (Daltons) (kcal/mol) rate (mol/h) Glucose 180 - 303 C6H1206 Rl Mycelium 24.52 28.1 R2 CHi.64Oo.52No. 16 So.0046P = 10 then Flin else 0 {batch start up} Flin= if time> = 33 then 0.5 else if time> = 32 then 0 else if time> = 21 then 0.5 else if time> = 20 then 0 else 0.5 Fout= if time>=33 then 0 else if time>=32 then 5.39 else if time> = 21 then 0 else if time> = 20 then 5.39 else 0

247


Nomenclature Symbols D F Kl and K2 KS

Dilution rate Flow rate Product kinetic constants Saturation constant

P S X V V 0 VX VS Y |i

Product concentration Substrate concentration Biomass concentration Reactor volume Initial volume of liquid Biomass in reactor Substrate in reactor Yield coefficient Specific growth rate

Indices S X 0 (zero) initial

in out

Exercises


to substrate to biomass to initial and inlet values to initial values to inlet to exit

1/h m3/h various kg/m3 g/m3 kg/m3 kg/m3 m3 m3 kg kg kg/kg 1/h

248


Results Shown below are results of a simulation with three filling cycles.


60 -|

.-80

A pr^Ti /- 70 |— "vsi| , go / I / I /"( / .50 / I ' • J -40 f I / \ / I / i /-V / I/ I / ^\ \l XV ,* \ -10

50-

40-

30-

20-

10-

--•*'"-'

0-

0

5

10

15

20

25

30

35

L 40

45

-Q 50

TIME

Figure 2. Masses of substrate and biomass during filling and emptying cycles.

249



10

Figure 3. Concentrations of product, substrate and biomass during filling and emptying cycles. The volume is also shown.

References Dunn, I.J., Mor, J.R., (1975) Variable Volume Continuous Cultivation Biotechnol. Bioeng. 17, 1805. Keller, R., Dunn, I.J. (1978) Computer Simulation of the Biomass Production Rate of Cyclic Fed Batch Continuous Culture J. AppL Chem. Biotechnol. 28, 784.

8.3.5

Repeated Medium Replacement Culture (REPLCUL)

System Slow-growing animal and plant cell cultures require certain growth factors and hormones which begin to limit growth after a period of time. To avoid this, part of the entire culture is replaced with fresh medium. A single cycle of repeated replacement culture is shown in Fig. 1. In this procedure part of the medium volume (with cells) is removed after a certain replacement time and replaced with fresh medium. Each cycle operates as a constant volume batch in

250


which the concentration of substrate decreases, while that of biomass increases. The operating variables are replacement volume, replacement time, and substrate concentration in the replacement medium. The initial conditions for each cycle are determined by the final values in the previous cycle and the replacement volume and concentration.

Replacement

VX

Final Conditions

VS

Initial Conditions Figure 1. One cycle for medium replacement culture.

Model The equations are those of batch culture, where for convenience the total masses are used. dVS = r

"dT

sV

dvx

Monod kinetics is used. The effective starting conditions for each batch can be calculated using the final conditions of the previous cycle from the volume replaced, VR? and the total volume, V, by the equations,

* VR f= — VX = (1 - £) VXF

251


VS= ( l - f ) V S F + V R S 0 where f is the volume fraction replaced.

Program The program as shown on the CD-ROM makes use of the PULSE function to vary the biomass and substrate concentrations corresponding to the replacement of a fraction F of the culture medium. The time for each batch is the value of INTERVAL.

Nomenclature Symbols D f KS

s

X V v

o

vx vs VR Y

Dilution rate Fraction of volume replaced Saturation constant Substrate concentration Biomass concentration Reactor volume Initial volume of liquid Biomass in reactor Substrate in reactor Volume replaced Yield coefficient Specific growth rate

Indices F S X 0


to final values at end of the cycle to substrate to biomass to initial and inlet values

1/h

g/m3 g/m3 g/m3 m3 m3 kg kg m3 1/h

252


Exercises

Results Fig. 2 shows how the biomass increases, until after six cycles the time profiles become almost identical. TIME= 19.29 X = 1.26

10

20

30

40

50 TIME

60

70

90

100

Figure 2. Oscillations of biomass and substrate concentrations with replacement cycles for Interval 10 and F=0.8

253


8.3.6

Penicillin Production in a Fed Batch Fermenter (PENOXY)

A fed batch process is considered for the production of penicillin, as described by Muttzall (1), The original model was altered to include oxygen transfer and the influence of oxygen on the growth kinetics.

Figure I. Fed batch reactor showing nomenclature.

Model As explained in the example FEDBAT the balances are: Total mass

dt Biomass: d(MassX) = Vr-X dt Substrate: d(MassS) = FSf+Vrs dt Product:

254

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna d(MassP)

dt

_. = V fp p

Dissolved oxygen, neglecting the content of the inlet stream is calculated from d(MassO) = K L a*(O sat -0) + Vr 0 dt The influence of biomass concentration approximated here by

on

the

oxygen

transfer is

KX+X The concentrations are calculated from _MassX V

1\. — """""""""""""^ ,

»J —

MassS , V

L

—

MassP , V

U —

MassO V

The growth kinetics take into account the oxygen influence

o The substrate uptake kinetics includes that amount used for growth, for product and for maintenance J*o ~

^

Jft o _/V

V Y

XS

V" Y

PS

^

Product production involves two terms whose constants are turned on and off according to the value of |ii, as seen in the program.

Oxygen uptake includes growth and maintenance

=-TT Y xo

255


Program The program is on the CD-ROM.

Nomenclature

Symbols F KLa Ko KS KX Mass mo ms Osat

Sf V 'max YPS

YXO YXS H-max

Exercises

Feed flowrate Oxygen transfer coeff. Monod constant for oxygen Monod constant for glucose Constant for biomass effect on Component mass Maintenance coeff. for oxygen Maintenance coeff. for glucose Saturation for oxygen Feed cone, of glucose Volume Maximum volume Yield product to substrate Yield biomass to oxygen Yield biomass to substrate max.specific growth rate

m3/h 1/h kg/m 3 kg/m 3 kg/m3 kg kg O/kg X h kg S/kg X h kg/m3 kg/m3 m3 m3 kg/kg kg/kg kg/kg 1/h

256


II!

References K. Mutzall, "Modellierung von Bioprozesses", Behr's Verlag, 1994. Program and model developed by Reto Mueller, ETH Zurich.

Results Run 1: 2023 steps in 0.117 seconds 0.008

20

40

60

80

Figure 2. Dynamics of the fed batch reactor.

100

120

140

160

180

200

257

8.4 Continuous Reactors

Run 3: 2021 steps in 0.15 seconds -0.008

120

•0.007 100

! I i I

80

•0.006 •0.005

: I Li I I

- 60

40

•0.004 O -0.003 -0.002

20

-0.001 0 0

20

40

60

80

100

120

140

160

180

200

TIME Figure 3. Influence of initial KLa value from 100 to 160 h"^ on the S and O profiles.

8.4 8.4.1

Continuous Reactors Steady-State Chemostat (CHEMOSTA)

System The steady state operation of a continuous fermentation having constant volume, constant flow rate and sterile feed is considered here (Fig. 1).

257



-0.008

120

-0.007 100

^

1I 80

\ ». ll '-, 1

- 60

\\

40

1

-0.006

I I

1

. !i

i1

-0.005 -0.004 O

v i

-0.003

|\ i I i '\ * \ \! \l ! •\\ \

1

20

1

-0.002 -0.001 -0

20

40

60

80

100

120

140

160

180

200

TIME Figure 3. Influence of initial KLa value from 100 to 160 h"^ on the S and O profiles.

8.4 8.4.1

Continuous Reactors Steady-State Chemostat (CHEMOSTA)

System The steady state operation of a continuous fermentation having constant volume, constant flow rate and sterile feed is considered here (Fig. 1).


258


D,S

Figure 1. Chemostat fermenter with model variables.

Model The dynamic balance equations may be modified to apply only to the steady state by setting the time derivatives equal to zero. The corresponding equations are then: For biomass, 0 = - D X + rx

For substrate, 0 = D (S0 - S) + rs

Growth kinetics, rx = ^ X

Substituting into the biomass balance gives \i = D where S is determined by the kinetics \i = f(S)

The Monod relation results in, S =

The substrate balance gives, X = Y(S 0 -S)


259

The productivity of the reactor for biomass is X D. The above equations represent the steady state model for a chemostat with Monod kinetics. Using them it is possible to calculate the values of S and X, which result from a particular value of D, and to investigate the influence of the kinetic parameters.

Program In Madonna programs, time can be used as a variable which will increase from the starting time. Here it is renamed D. Thus equations will be solved for increasing values of the dilution rate. Fortunately X and S can be explicitly solved for in this problem. If not, the ROOT FINDER facility of Madonna can be used. The program is found on the CD-ROM.

Nomenclature The nomenclature is the same as the example CHEMO, Sec. 8.1.2.

Exercises

260


Results The steady state curves of X, S, and XD versus D are given Fig. 2. The results in Fig. 3 were obtained by varying K$ in each run. An interesting effect can be observed on the position of the washout point. Run 1:113 steps in 0 seconds

-4

10 9-

•3.5

87-

i

6v-

5

*S**~\ !I \1 i Y.-I

m ^r

~s;i

f*r

—mm

\!

-2.5

1

•2

.

S"

"

-3

4-

•1.5

3-

//

2-

•1

/{

•0.5

1-

•0

0-

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 2. Steady state curves of X, S and XD versus D.

0.7

0.8

0.9

1

S

261


Run 5:113 steps in 0 seconds

Figure 3. Runs obtained by varying KS from 0.2 to 1.0.

8.4.2

Continuous Culture with Inhibitory Substrate (CONINHIB)

System Inhibitory substrates at high concentrations reduce the specific growth rate below that predicted by the Monod equation. The inhibition function may be expressed empirically as

where KI is the inhibition constant (kg/m3). If substrate concentrations are low, the term S2/Kj is lower in magnitude than KS and S, and the inhibition function reduces to the Monod equation. In batch cultures the term S2/Kj may be significant during the early stages of growth, even for higher values of K[. The inhibition function passes through a maximum at Smax = (Kg Ki)°-5. A continuous inhibition culture will often lead

262


to two possible steady states, as defined by the steady state condition JLI = D and as in shown Fig. 1.

D=

Figure 1. Possible steady states for a chemostat with inhibition kinetics.

One of these steady states (A) can be shown to be stable and the other (B) to be unstable. Thus, only state A and the washout state (S = SQ) are possible.

Model A model of a chemostat with its variables is represented schematically in Fig. 2.

F,S 0

+-

Figure 2. Model variables.

F,S,X


263

Cell material balance, VdX jj- = ^ i V X - F X or,

where D is the dilution rate = F/V. Substrate material balance, VdS — = F (S 0 -S)

^VX -~^f—

or, dS dT

M«X = D (S0 - S) - —

where Y is the yield factor.

Program When the system equations are solved dynamically, one of two distinct steady state solutions are obtained, the stable condition A and the washout condition. The initial substrate and organism concentrations in the reactor will determine the result. This is best represented as a phase-plane plot X versus S. All results indicate washout of the culture when the initial cell concentration is too low; higher initial substrate concentrations increases the likelihood of washout.

Nomenclature Symbols D KI KS

Dilution rate Inhibition constant Saturation constant

1/h kg/m3 kg/m3

264 S

Smax X Y


Substrate concentration Maximum in S for inhibition function Biomass concentration Yield coefficient Specific growth rate

Indices 0 I m

Exercises

Refers to inlet Refers to initial value Refers to maximum

kg/m3 kg/m3 kg/m3 kg/kg 1/h

265


Results Run 1:2000 steps in 0 seconds 1-1-2

10

15

20 TIME

25

30

35

40

Figure 3. Time course of X, S and U.


5 i

4.5 -

4 3.5

3 e/> 2.5

2 1.5

1 0.5

0 0.5

2.5

Figure 4. Phase-plane plot of X versus with varying ST from 0 to 5 kg/m3 using Batch Runs with overlay.

266


Run 9: 2000 steps in 0 seconds

5-

X,

4.5-

4 -

^^^>

*"

• ^ *t %%

3.5-

* * C^1 v

\ "|

3•

0 •

%

V

f/ // // /;^',--^, —. *x

2• 1.5-

0.5-

"%

• \ ^ x • i ! l

82, F0

Figure 1. Configuration of the two-tank system.


315

Model The details of the structured model will not be repeated here (See PHB). The biomass consists of a synthesis part R and the intracellular product P. The biomass growth rate of R is proportional to the specific growth rate, which is given by a two-part expression S (S/Ks,2)n (KS,i + S) -* The synthesis rate of PHB is given by a two-part expression

The term -kiP represents a product inhibition. The model requires component balances for P, R and S for both tanks, as seen in the program. The relative reactor volumes are determined by the parameter Vrat. The volumetric productivities are calculated to compare the results.

Program The program is found on the CD-ROM

Nomenclature Symbols FO KI KS n P PROD R rp TR

Feed flow rate Inhibition constant, for (NH4>2SO4 Saturation constant Hill Coefficient Product concentration (PHB) Productivity Residual biomass concentration Rate of synthesis of PHB Rate of synthesis of R

m3/h kg/m3 kg/m3 kg/m3 kg/(m3h) kg/m3 kg/m3 kg/m3

316

Sfeed

Vi and V2 X YP/R YR/S

MP


Rate of substrate uptake Limiting substrate cone. NH4+ as (NH4)2S04 Feed concentration Reactor volumes Biomass concentration Yield coefficient Yield coefficient, Specific rate of synthesis of R (rR/R) Specific rate of synthesis of P (rp/P)

Indices

1 2 m

Exercises

Refers to reaction 1 and tank 1 Refers to reaction 2 and tank 2 Refers to maximum

kg/(m3 h) kg/m3 kg/m3 m3 kg/m3 kg/kg kg/kg 1/h 1/h

317


Results Run 1:119 steps in 0.0167 seconds

4

90

100

Figure 2. A run showing the dynamic approach to steady state for X, S, P in both tanks.


0.2

Figure 3. Here with FO set at the optimum value of 1.24, the influence of VRAT is investigated giving a value for the maximum in PROD corresponding to the OPTIMIZE results. VRAT is seen not to be very important. Thus equal-sized tanks are adequate.

318

8.5 8.5.1


Oxygen Uptake Systems Aeration of a Tank Reactor for Enzymatic Oxidation (OXENZ)

System The influence of gassing rate and stirrer speed on an enzymatic, aerated reactor, as shown in Fig. 1, is to be investigated. The outlet gas is assumed to be essentially air, which eliminates the need for a gas balance for the well-mixed gas phase. gas

•Hi

Ill

+ 02

ii air Figure 1. Schematic of the enzymatic oxidation batch reactor.

Model The reaction kinetics are described by a double Monod relation:

_S

CL

The batch mass balances lead to:


8.5 Oxygen Uptake Systems

319 dS dF

dCL

—

= r

S

*

= K L a(C L * -CL) - r s Y o /s

dP dF

=

~ r s Y P/s

KLa varies with stirring speed (N) and aeration rate (G) according to: KLa = kN 3 G°- 5 where k = 4.78 x 10-13 with N in 1/h, G in m3/h and KLa in 1/h.


Nomenclature Symbols CL CLS>CL* G KCL â KS k N P r S vm Y

Dissolved oxygen concentration Saturation oxygen concentration Aeration rate Saturation constant for oxygen Transfer coefficient Saturation constant Constant in Kâ correlation Stirring rate Product concentration Growth rate Substrate concentration Maximum degradation rate Yield coefficient

g/m3 g/m3 m3/h kg/m3 1/h kg/m3 complex 1/h kg/m3 kg/(m3 h) kg/m3 g/(m3 h) kg/kg

320


Indices 0

o p s


to feed to oxygen to product to substrate

Exercises

Results The results in Fig. 2 show the influence of stirrer speed N on the dissolved oxygen level. Variations from 30,000 to 5,000 1/h were made with the Batch Run facility. Runs to obtain the results in Fig. 3 were made by varying the gas flow rate G from 25 to 5 m3/h (curves top to bottom).

321


Run 4: 1004 steps in 0.0167 seconds 8 •

7s"

--'/

_..-....._..-..._

7 -

f

* *

«r

.*•** 6 5 •

i

CL1 (3e+4) CL:2 (2.1676+4) --• CL:3 (1.3336+4) -^_CL:4 (5000)

3 •

* 1 / 4

2 -

{

-—"-''

0 • C

1

2

3

4

5

6

7

8

9

1

0

TIME

Figure 2. Influence of stirrer speed on dissolved oxygen levels. Run 5:1004 steps in 0.0167 seconds

<j 6.5

1

2

3

4

5 TIME

6

7

Figure 3. Influence of the gas flow rate on dissolved oxygen levels.

8.5.2

Gas and Liquid Oxygen Dynamics in a Continuous Fermenter (INHIB)

System Cell growth is limited by the oxygen mass transfer rate, and hence by the dissolved oxygen concentration. It is also inhibited by an inhibitory substrate S. Liquid phase balances for X, S and 62 in the liquid phase are therefore used, together with a gas phase oxygen balance to determine the rate of O2 supply.

322


To avoid washout of cells, it is important that the reactor should never enter the range of inhibitory behavior. Schematic representation of a continuous aerated fermenter is given in Fig. 1. feed F, S 1

gas G, CG2

liquid F,S 2 , X

air, G, CG1 Figure 1. Schematic of the continuous fermentation with oxygen transfer.

Model The liquid phase mass balances are as follows: For biomass,

dX

rxVL

For substrate,

dS2 For oxygen, dCL2 The kinetics are as follows: L2

KS+S2+(S22/KI)K0+CL2 rx = |a X


323

rs =

~

The balance for oxygen in the gas phase is:

The oxygen equilibrium relates the concentration in the gas phase to the liquid phase saturation concentration, CL2* = MC G2 The gas holdup fraction is,

VG = eV L Proportional control of the feed rate, based on exit substrate concentration, can be added with, F = Fo + KpE

with E = S2set ~ $2- Here the sign must be adjusted depending on the substrate region above or below the maximum kinetic rate.


Nomenclature Symbols CL CLS E F G KI

Dissolved oxygen concentration Saturation oxygen concentration Control error Flow rate Gassing rate Inhibition constant

mg/L nig/L g/L L/h L/h g/L

324 KLa Ko KP KS M OUR r S V X

Yx/s Yo 8 H


Transfer coefficient Oxygen saturation constant Proportional control constant Saturation constant Equilibrium coefficient Oxygen uptake rate Reaction rate Substrate concentration Reactor volume Biomass concentration Yield coefficient Mole fraction of oxygen in outlet gas gas/liquid volume ratio Specific growth rate

Indices 0 1 2 G I L m O P S X

Exercises

Refers Refers Refers Refers Refers Refers Refers Refers Refers Refers Refers Refers

to feed to inlet to outlet to gas to inhibitor to liquid to maximum to oxygen to product to substrate to biomass to equilibrium

1/h mg/L L/h/g/L g/L

-

mg/h g/Lh g/L L g/L g/g 1/h


325

326


Results Run 4: 10003 steps in 0.367 seconds

8-

•0.1

^ 7.

.

• 0.09 L

S2:1 (1) —.. CL2:1 (1) S2:2 (4)

\

6-

s

S2:3 (7)

5-

a'

4-

— — CL2:3(7) 82:4(10) — -CL2:4(10)

|N ^^ ™ % *%- N**

-0.08 -0.07 -0.06 -0.05 •0.04

-0.03

3"™ — — -^ .^^ 0.

1. i)

1

-0.02

—.**-«*uT-ZZ_H.r. — r • 0.01

"•-""'

2

3

4

5

6

7

8

9

•0 K)

TIME

Figure 2. Dissolved oxygen versus time at various feed rates.

Run 1:10003 steps in 0.417 seconds .70

DJ

2

3

4

5 TIME

Figure 3. Influence of the control on the reactor. The setpoint 82 is 5.0 kg/m^.


8.5.3

327

Batch Nitrification with Oxygen Transfer (NITRIF)

System Nitrification in a biofilm fluidized bed is to be modelled. oxidation of NtLj.* to NC>2~ and NC>3" proceeds according to:

The sequential

NH4+ + 02 -> N02O2 -> NO3The two steps are shown schematically in Fig. 1.

Ammonium ion -^

Nitrite ion -> Nitrate ion

Figure 1. Reaction sequence for nitrification.

The stoichiometry is for the first step YI = 3.5 g O2/ (g NPfy-N) and for the second step Y2 = 1.1 g O2/(g NO2-N).

Model Neglecting the details of the biofilm diffusion, the apparent kinetics of this biofilm process can be approximately described with homogeneous kinetics that follow a double Monod limitation: Si CL * ± I + S| KOI +C = vm2

The batch balances are as follows: ForNH 4 + (Si):

S2 K 2 +S 2

CL

328


~dT = - f i

For NO2~ (S2): dS2 dt

= ri - r2

For NO3- (S3): dS3 ~3T = r2

For oxygen (CL): dCL — = - Y i r i - Y 2 r 2 + K L a(C L *-C L )


Nomenclature Symbols

CL CLS K KLa r Si S2 S3 Vm

Dissolved oxygen concentration Saturation oxygen concentration Saturation constants Oxygen transfer coefficient Reaction rate Concentration of NH4+ - N Concentration of NC>2~ - N Concentration of NO3~ - N Maximum degradation rates Yield coefficients

Indices 0 1,2,3 O

s

Refers to feed Refer to reaction steps Refers to oxygen Refers to substrate

g/m3 g/m3 g/m3 1/h kg/m3 g/m3 g/m3 g/m3 g/m 3 h g/g


Exercises

329

330


Results

Run 1:313 steps in 0 seconds

100 +*

90 80 70 CO

-

X *'

...X'-''

60

$ 50 5) 40 30 20 10

0 1.5 TIME

Figure 2. NH4+, NO2~ and NC>3" and dissolved oxygen in a batch nitrification with KLa = 40 h"1/ Run 3: 313 steps in 0.0167 seconds

8

2.5

Figure 3. NH 4 + and dissolved oxygen in batch nitrification using three values of KLa from 20


8.5.4

331

Oxygen Uptake and Aeration Dynamics (OXDYN)

System The aeration of a batch culture (with essentially constant biomass X) is stopped and the dissolved oxygen (CL) is allowed to fall zero before re-aerating. The slope of the CL curve is the oxygen uptake rate and it is approximated by the slope of the electrode response curve CE curve. The dynamics of the electrode are known, and it is desired to investigate the lag effects as shown in Fig. 1.

Model The following equations represent the model: Oxygen uptake rate, OUR = q 0 2X Specific OUR, qo2m CL °102 - KQ + CL Oxygen balance, dCL * T = K L a(C L *-C L ) -OUR

332


C (mg/L)

6

0

20

40

60

80

time(s)

Figure 1. Typical response of the batch oxygen uptake and reaeration experiment.

Measurement dynamics for the liquid film may be important with a viscous culture, dCp CL - Cp TF and for the electrode lag,

dt

=

Program Experimental data, in the file OXDYNDATA, and the program are found on the CD-ROM.

Nomenclature Symbols C KLa

Oxygen concentrations Transfer coefficient

g/m3 1/h

333


KO OUR

Q X T

Saturation constant for oxygen Oxygen uptake rate Specific oxygen uptake rate Biomass concentration Time constants

Indices E F L m 02,0 S *

Exercises

Refers to electrode Refers to liquid film Refers to liquid Refers to maximum Refer to oxygen Refers to saturation Refers to saturation

g/m3 g/m3s g/kgs kg/m3

334

g Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results The results shown in Fig. 2 demonstrate the effect of changing Kâ. Runs varying the electrode time constant TE gave the results of Fig. 3.

Run 2:10004 steps in 0.183 second: 8

70

Figure 2. Aeration turned on at low CL for two KLa values.

80

90

100

335



8

10

20

30

40

50 TIME

Figure 3. Variation of the electrode time constant, TE from 1 to 25.

8.5.5

Dynamic Oxygen Electrode Method for KLa (KLADYN, KLAFIT and ELECTFIT)

System A simple and effective means of measuring the oxygen transfer coefficient (â) in an air- water tank contacting system involves first degassing the batch water phase with nitrogen (Ruchti et al., 1981). Then the air flow is started and the increasing dissolved oxygen concentration is measured by means of an oxygen electrode.

336


VQ.CQ

Hii

V

G> CGO

Figure 1. Aerated tank with oxygen electrode.

As shown below, the influence of three quite distinct dynamic processes play a role in the overall measured oxygen concentration response curve. These are the processes of the dilution of nitrogen gas with air, the gas-liquid transfer and the electrode response characteristic, respectively. Whether all of these processes need to be taken into account when calculating Kâ can be determined by examining the mathematical model and carrying out simulations. Measurement

«CF Gas phase

Liquid phase

Electrode diffusion film

Electrode

Figure 2. Representation of the process dynamics.

Model The model relationships include the mass balance equations for the gas and liquid phases and equations representing the measurement dynamics. Oxygen Balances The oxygen balance for the well-mixed flowing gas phase is described by

VG

dCG_ _ = G (CGO - CG) - KLa (CL* - CL) VL dt


337

where VG/V = TG , and Kâ is based on the liquid volume. The oxygen balance for the well-mixed batch liquid phase, is dt

= K L a(C L *-C L )V L

The equilibrium oxygen concentration CL* is given by the combination of Henry's law and the Ideal Gas Law equation where RT r CL * = — r CG

and CL* is the oxygen concentration in equilibrium with the gas concentration, CG- The above equations can be solved in this form as in simulation example KLAFIT. It is also useful to solve the equations in dimensionless form. Oxygen Electrode Dynamic Model The response of the usual membrane-covered electrodes can be described by an empirical second-order lag equation. This consists of two first-order lag equations to represent the diffusion of oxygen through the liquid film on the surface of the electrode membrane and secondly the response of the membrane and electrolyte: dC F _ C L -C F dt TF and dCg Cp ~Cg dt " TE

Tp and TG are the time constants for the film and electrode lags, respectively. In non- viscous water phases Tp can be expected to be very small, and the first lag equation can, in fact, be ignored. Dimensionless model equations Defining dimensionless variables as

C =

CG

CGO

CL =

C GO (RT/H)

the component balance equations then become

TG

338


dt'

V

u

'

VG H

and

Initial conditions corresponding to the experimental method are, t1 = 0 ;

C' L =C' G =0

In dimensionless form the electrode dynamic equations are C =

L-CF

dt and dC

E dt'

C =

F-CB TE/^G

where Cp is the dimensionless diffusion film concentration.

C GO (RT/H) and CE is the dimensionless electrode output

CE C GO (RT/H) As shown by Dang et al. (1977), solving the model equations by Laplace transformation gives 1 ,RTVL „

« = •=— „V -. + 1) TO + TE + TF KLa + ( H G

where oc is the area above the CE versus t response curve, as shown in Fig. 3.

339


1.0

CE'

Time (s) Figure 3. Determination of the area a above the CE' versus time response curve.

Program The program KLADYN can be used to investigate the influence of the various experimental parameters on the method, and is formulated in dimensionless form. The same model, but with dimensions, is used in program KLAFIT and is particularly useful for determining Kâ in fitting experimental data of CE versus time. A set of experimental data in the text file KLADATA can be used to experiment with the data fitting features of Madonna. All are on the CDROM. The program ELECTFIT is used to determine the electrode time constant in the first-order lag model. The experiment involves bringing CE to zero by first purging oxygen from the water with nitrogen and then subjecting the electrode to a step change by plunging it into fully aerated water. The value of the electrode time constant, TE can be obtained by fitting the model to the set of experimental data in the file, ELECTDATA. The value found in this experiment can then be used as a constant in KLAFIT.

Nomenclature Symbols Oxygen concentration

g/m3

340

G H KLa RT/H t V

a


Gas flow rate m3/s Henry coefficient Pa m3/mol Oxygen transfer coefficient 1/s (Gas constant)(Abs. temp.)/Henry coeff. Time s Reactor volume m3 Area above Cn'-time (s) curve s Time constant s

Indices E F G L

Exercises

Refers to electrode Refers to film Refers to gas phase Refers to liquid Prime denotes dimensionless variables

341


Results


0

20

40

60

80

100

120

140

TIME

Figure 4. Response of CG, CL and CE versus Ttime from KLAFIT.

160

180

200

342



TIME

Figure 5. A fit of experimental data (open circles) as dimensionless CE versus time (s) to determine KLA using KLAFIT, which gives 0.136 1/s.

References Dang, N.D.P., Karrer, D.A. and Dunn, IJ. (1977).Oxygen Transfer Coefficients by Dynamic Model Moment Analysis, Biotechnol. Bioeng. 19, 853. Ruchti, G. Dunn, IJ. and Bourne J.R. (1981). Comparison of Dynamic Oxygen Electrode Methods for the Measurement of KLa, Biotechnol. Bioeng., 13, 277.

8.5.6

Biofiltration Column for Removing Two Inhibitory Substrates (BIOFILTDYN)

System Biofiltration is a process for treating contaminated air streams. Moist air is passed thrpugh a packed column, in which the pollutants in the contaminated air are adsorbed onto the wetted packing. There in the biofilm solid phase the resident population of organisms oxidizes the pollutants.

343


L, S

r

G, C1T6

Tank6

Gas

Transfer TriT6

wi§fsp#K

iS:Tl;niiiS?' Illllpll

G.Q1T5

«* Tanks Gas

I I I I

Tank4

illlii llllll

Transfer

tlwilffi I l l l l l l l

Gas

fankS

'"';:"H? H'MK'SK

Transfer

i illlll Illlllll

Transfer

Gas

Tank 2

Transfer

:• : ' : . ; : - : ^ v \ r : : ^ -^ ;.;ji: ' ;' •';':'

^

Gas

Tank1

Gas

^îiiriS i;|tii^|;i;||

I Transfer

^ TriT1

^•îinS^p: OliiulllI

L, S1T1

Figure 1. Biofiltration countercurrent column.

Such columns can be run with a liquid phase flow (bio-trickling filter) or only with moist packing (biofilter). The work of Deshusses et al. (1995) investigated the removal of two ketones, methyl isobutyl ketone (MIBK) and methyl ethyl ketone (MEK), in such a biofilter. The kinetics of this multi-substrate system is especially interesting since both substances exhibit mutual inhibitory effects on their rates of degradation.

344


Model The model requires stagewise mass balances in the gas and liquid phases for both components. Transfer takes place from the gas to liquid phases with reaction in the liquid phase. The symbols used for the concentrations of substrates 1 and 2 in the n th tank are for the liquid phase Sixn and S2Tn and for the gas phase CiTn and C2Tn- The reaction rates are RiTn and R2Tn and the transfer rates are designated TriTn and Tr2TnG, C1Tn, C2Tn

G,C1Tn-1,C2Tn-1

L, S1Tn+1, S2Tn+1

L, S1Tn, S2Tn

m nth Figure 2. Single n stage for the biofiltration countercurrent column.

Referring to above figure, the mass balances for a single tank can be written as: Gas phase

ÔL~(G(C2Tn.1-C2Tn)-Tr2Tn Liquid phase -^ = ^- (L(SlTn+l - SlTn ) +TrlTn -rlTn VS )


345

1Q

1

n =

^~ (L(S2Tn+l - S2Tn ) + Tr2Tn -r2Tn VS )

d

Here the reaction is assumed to occur in a solid phase of volume Vs.

Vs = (1 - EG - e L ) Ê.

For the transfer terms Vc 2Tn = K L a ( S 2EQn - S2Tn)~

Tr

For the gas-liquid equilibria:

For the reaction rate terms the following equations are used to describe the mutual inhibition. Note that oxygen is assumed to be in excess. For substrate 1 (MEK) in tank n: v r

lTn = "

mi SlTn

For substrate 2 (MIBK) in tank n: V r

2Tn V

m2 S2Tn

|1+ i i SlTn |

Program The program developed by M. Waldner, ETH, is given on the CD-ROM.

346


Nomenclature Symbols C G Ki KLa Km L M N r S Tr VC VG VL vm VS

z e

Concentration in gas phase Gas flow rate Inhibition constant Mass transfer coefficient Monod coefficient Liquid flowrate Partition coefficient Number of tanks Reaction rate Concentration in liquid phase Transfer rate Volume of column Volume of gas phase Volume of liquid phase Maximum reaction velocity Volume of solid phase Length or height Volume fraction

kg/m3 m3/s kg/m3 1/s kg/m3 m3/s

kg/m3s kg/m3 kg/s m3 m3 m3 kg/m3s m3 m

Indices Eq G in L M n Tn 1 2

Refers Refers Refers Refers Refers Refers Refers Refers Refers

to equilibrium value to gas to inlet to liquid to maximum to nth stage to nth tank to methyl ethyl ketone (MEK) to methyl isobutyl ketone (MIBK)


347

Exercises

Reference Deshusses, M. A, Hamer, G. and Dunn, I. J. (1995) Part I, Behavior of Biofilters for Waste Air Biotreatment: Part I, Dynamic Model Development and Part II, Experimental Evaluation of a Dynamic Model, Environ. Sci. Technol. 29, 1048-1068.

348


Results Run 1:533 steps in 0.667 seconds 1.2

0.0025-

^

0.8 ^

0.002-

0.6

. 0.0015-

0.001 -

&-

,0.4 W

H '0.2 (0

0

5e+5

1e+6 1.5e+6 2e+6 2.5e+6 3e+6 3.5e+6 4e+6 4.5e+6 5e+6 TIME

Figure 3. Dynamic startup of the column.


• 0.1

5e-5

1e-4

1.5e-4

2e-4

2.5e-4

3e-4

3.5e-4

4e-4

4.5e-4

Figure 4. Influence of gas flowrate on the steady state fraction removed.

5e-4

349


8.5.7

Optical Sensing of Dissolved Oxygen in Microtiter Plates (TITERDYN and TITERBIO)

System Measurement of dissolved oxygen in microtiter plates is of potential interest for the screening of oxygen-consuming enzymes (e.g., oxidases), aerobic cell activities, and biological degradation of pollutants, and for toxicity tests. John et al. developed microtiter plates with the integrated optical sensing of dissolved oxygen by immobilization of two fluorophores at the bottom of 96-well polystyrene microtiter plates. The oxygen-sensitive fluorophore responded to dissolved oxygen concentration, whereas the oxygen-insensitive one served as an internal reference. As modelled in TITERDYN, oxygen transfer coefficients were determined by a dynamic method in a commercial microtiter plate reader with an integrated shaker. For this purpose, the dissolved oxygen was initially depleted by the addition of sodium dithionite and, by oxygen transfer from air, it increased again after complete oxidation of the dithionite. Available commercial readers have an intermittent operation. After a certain period of shaking, the plate is moved around to measure dissolved oxygen concentration. During this period the plate moves more slowly and oxygen transfer rate is reduced. This may lead to oxygen depletion during the measurement process. It is essential to know the size of the errors that are introduced by this intermittent procedure. This is evaluated by the simulation example TITERBIO.

Microtiter plate

Filter

Light

Figure 1. Microtiter well showing light path and sensor layer.

350


Model

Experiments involved measuring the oxygen uptake rate by removing the oxygen from the liquid using a chemical reaction (oxidation of sodium dithionite). Oxygen is depleted immediately after the addition of dithionite. After the consumption of the dithionite the oxygen transfer increased the oxygen in the liquid. The following model was used to evaluate the KLa value. dCT dt

where CL is the dissolved oxygen concentration, CL* is the saturation value and OUR is the oxygen uptake rate in mM/min. The experiment starts with high values of dissolved oxygen concentration, CL« After addition of dithionite OUR increases as calculated by OUR

= ko CL CD

As oxidation proceeds the dithionite concentration changes according to dC D ^ 2 f dt "" 3 In order to account for some time delay of the sensor a first order equation was used

dC E ^C L -C E dt TE The time constant TE was estimated to be about 1 s. In further experiments this method was also applied to simulate a microbial cultivation in the wells of a microtiter plate. In this case the OUR value was taken to be a constant value as measured in a larger fermentation vessel. KLa varied periodically simulating the high value during shaking and the lower value during the measurement period. The questions of interest are how much the measured OUR or KLa would differ from the actual one provided KLa or OUR were already known.


351

Program Two separate programs are given on the CDROM: TITERDYN for the chemical oxidation with re-aeration and TITERBIO for the biological oxidation and reaeration dynamics during a cultivation in a microplate reader. For the program TITERDYN there is experimental data on the file TITERDYNDATA available to allow fitting the value of KLa. In TITERBIO KLa during measurement is a fraction of KLa during shaking and is determined by the parameter kmax. KLa during measurement is defined as, KLameasure=KLashaking*(kmax-l)/kmax. In the original model settings, kmax has a value of 2. The larger kmax, the larger the error of KLa or OUR estimation.

Nomenclature The program TITERDYN uses minutes and TITERBIO uses seconds. Additional symbols are defined in the programs.

Symbols CD CL ko â KQ OUR Q TE

Dithionite concentration Oxygen concentration Rate constant for dithionite reaction Transfer coefficient Saturation constant for oxygen Oxygen uptake rate Specific oxygen uptake rate Time constant for measurement

Indices E D L S and *

Refers to electrode Refers to dithionite Refers to liquid Refer to saturation

mM mM 1/min mM 1/s mM mM/s mM/ s s

352

Exercises


353


Results 1:1019 steps in 0.0167 seconds '0.1

0

100

200

300

400

500

600

700

800

900

1000

TIME

Figure 2e Dynamics of biological uptake and reaeration. Program TITERBIO. Run 1: 834 steps in 0.0333 seconds

it •

Figure 3. Data fitting Duration=0.48.

using

TTTERDYN, yielding

'VtV'tj " *

KLa=0.201,

Calcfact=102

and

Reference John, G.T., Klimant, I., Wittmann, C., Heinzle, E. (2003). Integrated Optical Sensing of Dissolved Oxygen in Microtiter Plates - A Novel Tool for Microbial Cultivation, Biotechnol. Bioeng., 81, 829-836.

354


8.6

Controlled Reactors

8.6.1

Feedback Control of a Water Heater (TEMPCONT)

System A simple feedback control system involving a stirred tank, temperature measurement, controller and manipulated heater is shown in Fig. 1. T 0 ,F

IT* F,T R

ip

Figure 1. Feedback control of a simple continuous water heater.

Model The energy balance for the tank is dTR

F

Q

where Q is the delayed heat input from the heater represented by a first order lag dt

TQ

The measurement of temperature is also delayed by a sensor lag given by


8.6 Controlled Reactors

355

u*sens .... *R ~ *sens dt Tsens

A proportional-integral feedback controller can be modelled by

where the control error is given by

Program Random disturbances in flowrate or feed temperature can be generated using the RANDOM function in Madonna, as explained in the HELP on the CDROM.

Nomenclature

Symbols cp f F Kp Q T V 8 p TD TI TQ

Specific heat Frequency of oscillations Flow rate Proportional control constant Heat input Temperature Reactor volume Error Density Differential control constant Integral control constant Time constant for heater Time constant for measurement

kJ/(kg °C) 1/h m3/h kJ/(h °C) kJ/h °C m3 °C kg/m 3 h h h h

356


Indices C R sens set 0

Exercises

Refers Refers Refers Refers Refers

to controller to reactor to sensor to setpoint to inlet or initial

357


Results


6000

3000

•1000

70

Figure 2. Approach to steady state for a setpoint of 80°C.

80

90

100

358


Run 1: 28571 steps in 0.2 seconds -1.2e+4

1e+4 8000

100. 6000 4000

80

2000

60

0 -2000 -4000

20

40

80

100

120

140

160

180

200

TIME

Figure 3. Response to a step change in the inlet temperature TO at 120 h. The controller constant Kp was set higher than in the run of Fig. 2.

8.6.2

Temperature Control of Fermentation (FERMTEMP)

System Heat effects in fermentation can be important, especially on a large scale. Shown in Fig. 1 is a batch fermentation process, during which the cooling water flowrate is controlled by a feedback controller. The rate of heat generation is related to rate of substrate uptake by a constant yield factor YQS. The cooling coil is modelled as a well-mixed system.

359


Water Figure 1. Feedback control of the temperature in during a fermentation.

Model The batch fermentation model is given by, dX df = dS dt

-H = Y

^ = Ks+S The energy balance equation for the reactor is, dT dtR _ ~

rQ

UA (TR-Tc) VpCp

360


where, TQ = ^XY Q S /Y For the well-mixed cooling coil, the energy balance equation is: dTr

F

(Tcin TC) +

-

UA

— (TR ~ Tc)

The controller is a proportional-integral type F = F0 + K P 8 + £ =

Program As seen on the CD-ROM and below, the control equations can be written in terms of the error and its integral. {CONTROL EQUATIONS FOR d/dt(EInt)=E F=FO+KP*E+(KP/TI)*EInt limit F> = 0 E=TR-TSET

FLOWRATE}

Nomenclature Symbols Cp F Kp KS UA r V

Heat capacity Flow rate Controller constant Saturation coefficient Reactor transfer-area constant Rate of heat production and transfer Reactor volume

kcal/(kg C) m3/h m3/(h C) kg/m3 kcal/kg kcal/(m3 h) m3

361


X Y YQS P T

s e H

Biomass concentration Yield coefficient Heat yield for substrate Density Time constant of controller Substrate concentration Temperature error Specific growth rate

Indices C

I

m Q

R S O P set

Exercises

Refers Refers Refers Refers Refers Refers Refers Refers Refers

to coolant and cooling to integral control to maximum to heat to reactor to substrate to normal value and inlet value to proportional to setpoint (desired value)

kg/m3 kg/kg kcal/kg kg/m3 h kg/m3 C 1/h

362


Results

TIME= 3.452

S = 18.25

TR= 24.6

Figure 2. Cooling flow starts when TR > Tset (25 C); after batch growth finishes at time=4.6 h the reactor cools. Here Kp=0.6 and TI = 0.6.

363


Run 6: 9480 steps in 0.217 seconds '0.2

100

150

200

250

300

350

400

450

500

Figure 3. With Parameter Plot, the integral of the error and minimum water temperature versus Kp for a fixed value of Tj=9.

8.6.3

Turbidostat Response (TURBCON)

System Although not so widely used as the chemostatic type of operation of continuous culture, the turbidostat may offer advantages for the investigation of particular problems. As shown in Fig. 1, the flow rate of the incoming substrate is controlled by the biomass concentration (more correctly, the turbidity) in the vessel. In practice, this control is usually on-off or proportional, but more sophisticated control would be simple to implement.

364


F,S0

Feed pump X,S

Turbidometer

Figure 1. Feedback control of the biomass concentration using a turbidostat.

Model For the well-mixed tank with Monod growth:

dS

M_X

dt" = F(So-S) - -y~ dX

FX

dT = -IT Considering product production with Luedeking-Piret kinetics: dP FP dT = -"V" +

The turbidometer control is modelled by: KPP ,. F = F0 + K P e + — fedt s

365


8 = (X-X S et)


Nomenclature Symbols A B F

Fo Kp KS P

S V X Y e

Growth-associated constant Nongrowth-associated constant Flow rate Normal feed flow rate Proportional controller constant Saturation constant Product concentration Substrate concentration Reactor volume Biomass concentration Yield coefficient Error Specific growth rate Integral control time constant

Indices m P S and set 0


to maximum to proportional control to setpoint to inlet stream

1/h m3/h m3/h m6/h kg kg/m3 kg/m3 kg/m3 m3 kg/m3 kg/kg kg/m3 1/h

366


Exercises

Results

Run 1:105 steps in 0 seconds 1-5

,3.5

1

5 TIME

Figure 2. Startup and response of the controlled reactor.

c/>

367


Run 1: 205 steps in 0.0167 seconds -5

3.5-

3.

•

45 •4

2.5-

-3.5

2"

_X:1

-3

X"

•"—

1.5.

F:

C^

'

-2.5

,m

1• n e VJ.O •

0-

—j

-2

\

r

i T...

u

4

6

-1.5 ...^. ._..._. ._..._,,.... -1

' " — 8

10 TIME

12

14

16

18

20

Figure 3. Response of the controlled reactor to a step change in X se t o

8.6.4

Control of a Continuous Bioreactor with Inhibitory Substrate (CONTCON)

System The continuous fermenter is equipped with feedback control based on substrate measurement, as shown in Fig. 1. This type of controlled fermenter has been referred to as an auxostat.

368


F,S,

Feed pump

X,S ' Substrate measurement

Controller

Figure 1. Flow diagram of a feedback loop to control substrate concentration.

Model The biomass and substrate mass balances are the same as in the previous TURBCON model. Kinetics:

Biomass balance, V — = dt

VX-FX

or,

f where D is the dilution rate (= F/V). Thus steady-state behaviour, where dX/dt = 0, is represented by the conditions that |u = D. Substrate mass balance,

or,

dt

f ">-»-f


369

where Y is the yield factor for biomass from substrate. Also from this equation at steady state, since (j = D and dS/dt = 0, the steady-state cell concentration is given by X = Y(S 0 -S) A continuous inhibition culture will often lead to two possible steady states, as defined by the steady-state condition (a = D, as shown in Fig. 2. Control equations: £

=

Sset- S

Kp r F = F0 + KP e + — I edt •>

Program When the system equations are solved dynamically, one of two distinct steadystate solutions is obtained, i.e., the reactor passes through an initial transient but then ends up under steady-state conditions either at the stable operating condition, or at the washout condition, for which X=0. The initial concentrations for the reactor will influence the final steady state obtained. A PI controller has been added to the program, and it can be used to control a substrate setpoint below Smax. The controller can be turned on setting by Kp>0. The control constants Kp, and the time delay tp can be adjusted by the use of sliders to obtain the best results. Appropriate values of control constants might be found in the range 0.1 to 10 for Kp and 0.1 to 10 for TJ. Note that the control does not pass Smax even though the setpoint may be above Smax. Another feature of the controller is a time delay function to remove chatter. The program comments on the CD-ROM should be consulted for full details.

Nomenclature

Symbols D

Dilution rate

1/h

370


F KI KP KS S V X Y H(U) T

I TF

Flow rate Inhibition constant Controller constant Saturation constant Substrate concentration Volume Biomass concentration Yield coefficient Specific growth rate coefficient Controller time constant Time constant controller delay

Indices 0 I m, max

Exercises

Refers to inlet Refers to initial value Refers to maximum

m3/h kg/m3 kg/m3 m 6 /kgh kg/m3 m3 kg/m3 kg/kg 1/h h h

371

8.7 Diffusion Systems

Results Run 1: 25009 steps in 3.13 seconds r 0.3

0.25

0.2

1.5 0.15

TIME

Figure 2. A control simulation of the process with the setpoint below Sn

References Edwards, V. H, Ko, R. C. and Balogh, S. A. (1972) Dynamics and Control of Continuous Microbial Propagators Subject to Substrate Inhibition, Biotechnol. Bioeng. 14, 939-974. Fraleigh, S. P., Bungay, H. R. and Clesceri, L. S. (1989) Continuous Culture, Feedback Control and Auxostats. Trends in Biotechnology, 7, 159-164.

8.7 8.7.1

Diffusion Systems Double Substrate Biofilm Reaction (BIOFILM)

System A biocatalyst is immobilized inside a solid matrix (gel or porous solid) through which substrates diffuse and react. As shown in Fig. 1, for simulation purposes

371


Results Run 1: 25009 steps in 3.13 seconds r0.3

0.25

0.2

1.5 0.15

TIME

Figure 2. A control simulation of the process with the setpoint below Sn

References Edwards, V. H, Ko, R. C. and Balogh, S. A. (1972) Dynamics and Control of Continuous Microbial Propagators Subject to Substrate Inhibition, Biotechnol. Bioeng. 14, 939-974. Fraleigh, S. P., Bungay, H. R. and Clesceri, L. S. (1989) Continuous Culture, Feedback Control and Auxostats. Trends in Biotechnology, 7, 159-164.

8.7 8.7.1

Diffusion Systems Double Substrate Biofilm Reaction (BIOFILM)

System A biocatalyst is immobilized inside a solid matrix (gel or porous solid) through which substrates diffuse and react. As shown in Fig. 1, for simulation purposes Biological Reaction Engineering, Second Edition. I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30759-1

372


the matrix is divided into segments, and the diffusion flux, j, from segment to segment, is expressed in terms of the concentration difference driving force. Solid Biocatalyst Matrix with N Segments

Liquid

j n-1

n-1

^_ n ^

jn

^ n+1

Figure 1. Finite-differencing of the concentration profiles within the immobilized biocatalyst into segments 1 to N.

Model A multicomponent reaction whose reactants and products diffuse to and from the reaction site, for example into an immobilized enzyme or biofilm, can be described by diffusion-reaction equations. The original problem in terms of non-linear partial differential equations, is described by a large number of time-dependent differential-difference equations by discretizing the length variable. A component mass balance is written for each segment and for each component: [Accumulation | ^ rate J

(Diffusion^ _ (Diffusion^ ^ rate in J ^ rate out J

^Production rate^ v by reaction }

dSn ' F = J n - l A - j n A +r S n A A Z

Using Pick's law, n-1 =

and dividing by A AZ,

S n -l ~ Sn AZ


373

dSn

"dT = °s

(Sn.i -2S n + S n +i) AZ2

+ rs

Thus N dynamic equations are obtained for each component at each one for each element. The boundary conditions are for the above case 0 at Z = L and S = So at Z = 0. The equations for the first and last must be written accordingly. The kinetics used here consider carbon-substrate inhibition and limitation. Thus, S O

position, dS/dZ = elements oxygen

At steady state, the overall reaction rate or consumption rate of substrate can be calculated from the gradient at the outer surface. To find the resulting change of bulk concentration, the liquid phase can be coupled with suitable mass balances. For a well-mixed, continuous-flow, liquid the resulting balance equation would be dS0

F So)

- a DS

SQ-SI ^Z

For oxygen transferred from the gas phase: dO0

= K L a(Os-0 Q ) - a Do

AZ

Program As seen below, the program on the CD-ROM uses the array-vector form which permits plotting the values at time=Stoptime versus the distance index. Also the number of finite-difference elements N can be varied.

a/at (s[i. .N-i] >=DS* (s[i-u )/(Z*Z)+RS[i]

a/at (o[i. .N-I] )=DO* (o[i-u 2*0[i]+0[i+l])/(Z*Z)

374


Nomenclature Symbols a A C D F j K KLa O 0S R,r S V vm

Yos

z

Specific area perpendicular to the flux 1/m Area perpendicular to diffusion flux m2 Concentration g/m3 Diffusion coefficients m2/h Volumetric flow rate m3/h Diffusion flux g/ (m2 h) Saturation constants g/m3 Oxygen transfer coefficient 1/h Dissolved oxygen concentration g/m3 Saturation concentration for oxygen g/m3 Reaction rate g/ (m3 h) Substrate concentration of carbon source g/m3 Volume of tank m3 Maximum reaction rate g/ (m3 h) Yield for oxygen uptake Length of element m

Indices 0 1 - 10 I O S n Feed

Exercises

Refers to bulk liquid Refer to sections 1-10 Refers to inhibition Refers to oxygen Refers to carbon source Refers to section n Refers to feed


375

376


Results


.1

Figure 2. Oxygen and substrate time profiles for a step change in KLA.


1

1.6-1 1.4-

0

1

2

3

4

5

6

7

8

10

Figure 3. Oxygen and substrate distance profiles at the end of the run in Fig. 2.

377



5

Figure 4. Dynamic response of oxygen and substrate mid-points caused by a step change in KLA (as Fig. 2) followed at 3 h by a step reduction in Sfeed.

8,7.2

Steady-State Split Boundary Solution (ENZSPLIT)

System A rectangular slab of porous solid supports an enzyme. For reaction, substrate S must diffuse through the porous lattice to the reaction site, and, as shown in Fig. 1, it does so from both sides of the slab. Owing to the decreasing concentration gradient within the solid, the overall rate is generally lower than that at the exterior surface. The magnitude of this gradient determines the effectiveness of the catalyst.

378


Biocatalyst Matrix Substrate diffusion

diffusion

X = L -*—

X = 0 —^

X=L

Figure 1. Symmetrical concentration gradients for substrate and product.

Model Under steady state conditions: f Rate of diffusion of Uubstrate into the slab

Rate at which reactant \ îs consumed by reaction^

=

dX

A quasi-homogeneous form for the reaction term is assumed. The boundary conditions are given by: At X = L: S = S0 , P = P0

At X = 0: dX

dX


379

The external concentration is known, and the concentration profile throughout the slab is symmetrical. The reaction rate is expressed by the Michaelis-Menten equation with product inhibition kES =

KM(I+P/KI)+S where k, KM and K\ are kinetic constants and E and P are the enzyme and product concentrations. At steady state, the rate of diffusion of substrate into the slab is balanced by the rate of diffusion of product out of the slab. Assuming the simple stoichiometry S —> P dS °SdX

=

dP -°PdX

which on integration gives P = (So-S)

DS

where P is assumed zero at the exterior surface. Defining dimensionless variables S
KSa D Oima - D) +

^sm ma D(K Sbl -K Sa ) Y F

a aCmmbKSa - m ma K Sbl)

Also, a critical dilution rate, where the maximum dilution rates of the two organisms cross-over can be written: . Cnt

_ "

Sa

Sbl

Four particular cases depending on the values of the maximum specific growth rate and saturation constants of both microbes can be simulated for the single substrate case (|imb2 = 0). 1- M-ma > l^mbli Ksa > K$bi: 2. |ima > Mmbi; Ksa < KSbi: 3. |ima < |imbi; Ksa > Ksbi: K$a >

Coexistence below a certain value of D No coexistence range Coexistence with wider range of stable focus Coexistence at higher D and wider range of

Program The program is given on the CD-ROM.

409

8.8 Multi-Organism Systems

Nomenclature Symbols D F G K P S V X Y

Dilution rate Feed rate Growth factor concentration Saturation constants Yield constant Substrate concentration Reactor volume Biomass concentration Yield coefficient Specific growth rate

1/h m3/h g/m3 g/m3

g/m3 m3 g/m3 g/g 1/h

Indices 0 1 2 a b bl b2 g m

Exercises

Refers to feed Refers to substrate 1 Refers to substrate 2 Refers to organism a Refers to organism b Refers to organism b growing on substrate 1 Refers to organism b growing on substrate 2 Refers to growth factor Refers to maximum

410


Results A simulation is given in Fig. 2 for the parameters as given in the program with feed flow rate F = 0.24 m3/h, and the feed concentrations SIQ = 500 mg/L and 820 = 1500 mg/L. The oscillating concentrations are given for Xa and S\ versus time. It is seen that this solution is stable and homes into a steady state, corresponding to case 1.


250

200

150 CD 100

300

Figure 2. Competition and commensalism of two organisms (F = 0.24 m3/h, S10 = 500 mg/L and S2o = 1500 mg/L), showing the biomass concentrations.


411

Reference Miura, Y., Tanaka, H., Okazaki, M. (1980). Stability Analysis of Commensal and Mutual Relations with Competitive Assimilation in Continuous Mixed Culture. Biotechnol. Bioeng., 22, 929. Example developed from the original paper by S. Ramaswami, ETH-Zurich.

8.8.3

Stability of Recombinant Microorganisms (PLASMID)

System In genetic engineering, microorganisms are used as host cells to produce important biochemicals by inserting a small portion of extra-chromosomal DNA (on plasmids) into the cell. These plasmids carry the genetic instructions to produce the desired product and tend to lose their engineered properties during cell division because of non-uniform plasmid distribution. The engineered or recombinant strain usually grows more slowly than the wild-type, nonplasmid-bearing strain, so that engineered strain may be lost through extinction. By exploiting the difference in the adaptation times of wild and engineered strains, a possibility exists of maintaining a plasmid-bearing population in continuous culture by cycling the substrate feed concentration or the dilution rate. This dynamic problem is adapted from Stephens and Lyberatos (1987 and 1988), based on the concept of plasmid stability from Aiba and Imanaka (1981). The Monod model assumes a balanced growth in which all cellular components change in the same proportion at all times but does not account for dynamic effects. Dynamic first order lag relations are added to account for the response of the organisms to rapid changes in the medium. It is assumed that the time constants for the two strains are different and that the responses to changing concentrations are therefore different. As a consequence, the strain with the smallest time constant has the advantage when the concentration of the limiting substrate is oscillating. The simulation model based on Fig. 1 is used to predict the stability in the competition between wild (Xi) and engineered (X2) strains in continuous culture.

412


S.XLX2

Figure 1. Competitive cultures x j and X2 in a continuous system.

Model The following dimensionless parameters are defined: s = S/K, P = 0,1/0.2, t = treai «2, Rmd = torn/Ok, Dd = D/a2, xi = Xi/Y K, x2 = X2 A" K. The mass balances in dimensionless form are: = (ill (zi) xi - D xi + p [12 (Z2) X2

dx2

2 - D X2

ds

g^ = D (SQ - s) - (Lli (s) Xi - |I2 (s) X2

where the time delayed specific growth rates are,

"SIT Z2)


413

and the growth rates are

In the above, z\ and Z2 represents the time-delayed substrate concentrations, and the specific growth rates of xi and X2 are taken as functions of z. Thus: dz

= P(S-ZI)

where p = ai/a2, and cxi and a2 are the adaptability factors or inverse time constants. The effect of (3 is to delay the substrate for growth in the wild and engineered organisms according to their first order time constants. For p > 1 the wild type is delayed with a shorter time constant. At high values of oci and OC2, the model describes an undelayed, instantaneous Monod growth model. It is assumed, that (li^ > H2m . The probability factor p represents the probability (or fraction) of conversion to the wild strain during growth of the engineered strain. Thus the growth rate of the engineered strain is multiplied by [1 - p].

Program In the program on the CD-ROM, the square-wave input for SQ is generated by the Madonna Conditional Operator, using the parameter MARK (ratio of the time during which the function has the value 1 to the time of the complete period) and PER (period). {Square wave feeding generated by conditional operator} SO=IF(Time/PER-INT(Time/PER))[i-l]-2S[i]+S[i+]L] AZ2

DS

MA[i]XA[i]

/iB[i]XB[i]

YA

YB

dXA[i] X A [i-l]-2X A [i] + XA[i —£t—~DXA —2

dXB[i]

—^t—-DXB

XB[i-l]-2XB[i] + XB[i

—2

The mass balances of the boundary segment on the support wall are described by the following equations: dS[N] = p S[N-1]-S[N] S dt AZ2

The "diffusion" coefficients of microorganisms, DXA and DXB» represent displacement by cell division and by shearing off at the film boundary contacting the bulk liquid.

Growth Kinetics Of Microorganisms The inhibitory influence of high substrate concentration was described by the Haldane kinetics. The two types of microorganisms compete for substrate, but in the biofilm they also have to compete for the limited space available. Therefore, growth of the microorganisms was described by the Haldane kinetics with a spatial limitation term which was originally proposed as cell inhibition kinetics by Han and Levenspiel (1988).


..

429

m_

K IA

[i]= K IB

where KI? Ks, and (im are inhibition constant (mg/L), half saturation constant (mg/L), and maximum specific growth rate (day" ). Xm (mg/L) is the maximum capacity of total biomass of microorganisms A and B in a segment. The formulation of the spatial limitation term used here is the most simple one possible with non-restricted growth at zero biomass concentration and zero growth at maximal biomass concentration Xm. Applying the above model it was found (Soda et al., 1999) that the qualitative behavior of the biofilm reactor is characterized by 5 regions, depending on the operating conditions, the substrate concentration in the feed and the dilution rate. In region I, both microorganisms are washed out of the biofilm reactor. In region II, microorganism B is washed out, and in region III, microorganism A is washed out of the biofilm. In region IV, both microorganisms coexist with one another. In region V, both microorganisms coexist with a sustained oscillatory behavior. Convergence to regions I-V depends on the initial conditions. In regions II-V, washout of either or both microorganisms is also observed when the initial conditions are too far away.

Nomenclature Symbols a b D DS Dx K! KS

specific area perpendicular to the flux, related to bulk liquid volume biomass decay rate dilution rate diffusion coefficient of substrate diffusion coefficient of microorganisms inhibition constant saturation constant

m" day" day" m /day m /day mg/L mg/L

430

LI

Lf N S[i Sf

Y AZ


thickness of diffusion layer thickness of biofilm number of segments in biofilm substrate concentration in element i substrate concentration in feed biomass spatial capacity of total biomass of microorganisms A and B in a segment yield coefficient thickness of each segment maximum specific growth rate

Indices A B

refers to microorganism A refers to microorganism B

Numbers in brackets 0 1-5

Exercises

refers to bulk liquid refer to segments 1-5

m m mg/L mg/L mg/L mg/L m " day


431

References Soda, S, Heinzle, E,, Fujita, M. (1999) "Modeling and simulation of competition of two microorganisms for a single inhibitory substrate in a biofilm reactor." Biotechnol. Bioeng., 66, 258-264. Han, K. and Levenspiel, O. 1988. "Extended Monod kinetics for substrate, product, and cell inhibition." Biotechnol. Bioeng. 32: 430-437.

Program Shown below is a portion of the program. The full program is on the CD-ROM. {BALANCES FOR BIOFILM IN 10 SEGMENTS} d/dt (S[2. .nslabs-1] )=DS* (S[i-l] 2*S[i]+S[l+l])/ (Z*Z)-UA[i] *XA[i] /YA-UB[i] *XB[i] /YB d/dt (XA[2. .nslabs-1] ) =DSA* (XA[i-l]2*XA[i] +XA[i+l] )/(Z*Z)+(UA[i] -kdA) *XA[i] d/dt ( X B [ 2 . . n s l a b s - 1 ] ) =DSB* ( X B [ i - l ] ) / ( Z * Z ) + ( U B [ i ] -kdB) * X B [ i ] 2*XB[i]

432



/ \

/-

^^—' •

sr^^°"

'

•0.035

-0.03

\, 2 40-

£ ° m /

-0.025

X

\ I x 20

•0.02

XAmid:1 . — . XBmid:1 Smid:1

£ (A

-0.015

i \

-0.01

g™™. -0.005

---. '•••-•-.• 10

20

30

40

50

60

70

80

90

100

TIME Figure 2. Results corresponding to Case 2: UmB=0.4, KSB=0.1, KIB=10.

Run 1:1058 steps in 1 seconds 0.05

0.045 0.04

0.035 0.03

0.025 £

(A 0.02

0.015 0.01

0.005

10

20

30

40

50

60

70

80

90

100

Figure 3. Results corresponding to Case 4: UmB=1.8, KSB=0.01, KIB=0.01.


8.8.7

433

Model for Anaerobic Reactor Activity Measurement (ANAEMEAS)

System As already discussed in Chapter 3, anaerobic processes can be described by multi-substrate, multi-organism kinetics. As shown in Table 1, organic acids are formed from monomeric and polymeric substrates contained in wastewater. These are then converted into hydrogen, CO2 and acetic acid. In a last step, acetic acid and H2 with CO2 form methane. Table 1. Stoichiometry of Anaerobic Reactions. Step 1:

Hydrolysis (example: carbohydrate-hexoses) (C6Hi2O6)n + n H2O -> nC 6 Hi 2 O 6

Step 2:

Acid production (example glucose) C6Hi2O6 -> CH3(CH2)2COOH + 2 H2 + C6Hi2O6 + 2 H2 -> 2 CH3CH2 COOH + 2H 2 O H» 2CH3COOH + 4 H2 + 2 CO2

Step 3:

Step 4:

Acetic acid production CH3(CH2)2COOH + 2 H2O -> 2 CH3COOH + CH3CH2COOH + 2 H20 -> CH3COOH + 3 H2 + CO2

2 CO2 2 H2O

2 H2

Methane production CH3COOH -> CH4 + CO2 4 H2 + C02 -> CH4 + 2 H20

In order to evaluate the activity of an anaerobic reactor and to evaluate the correctness of the reactions in Table 1, an off-line measurement system has been designed. This involves a small batch reactor coupled to a mass spectrometer. A sample of biomass with medium is taken from the larger continuous anaerobic reactor and put into the small batch reactor. Dissolved gases are stripped by helium, all gas bubbles are removed and substrate is added to start the reaction. The accumulation of organic acids, CO2, H2 and CtLj. is measured. pH is adjusted according to total acid concentration and buffer capacity. Biomass concentration is constant throughout the experiment.

434


The model was developed to aid the design of the measurement system and in the interpretation of the data.

Model A five organism model with lumped hydrolysis and acid-generating bacteria was established. Substrate, intermediate and product balances of the batch reactor are dSi "dT = IrSi where rsi are the rates of consumption and synthesis of S[. The respective reaction rates rj for the consumption of substrate Si and for the formation of product Pj in each step are those from the reactions in Table 1 as follows:

and the specific growth rates take the Monod form, M-imax Si Mi = KSi + Si

or modified in the case of substrate inhibition for acetate, M^maxi Si

The individual equations for each substrate Si are given in the program, Thermodynamic equilibrium constraints on the Step 3 reactions (Table 1) are also included in the model.


435

Reaction Equilibrium In the acetogenic step (Step 3 reaction in Table 1), acetic acid, hydrogen and carbon dioxide are produced from propionic and butyric acid. The thermodynamic equilibria for these reactions are incorporated by estimating the chemical equilibrium limits for butyric acid: CH3(CH2)2COO- + 2 H2O £ 2 CH3COO- + 2 H2 + H+

AGo = 48.3 kJ

From this the equilibrium constant is KBU = 2.02 x 10~3 (mol4 nr12) given by KBU

"

For propionic acid similarly, CH3CH2COO- + 2 H2O

£ 2 CH3COO- + 3 H2 + CO2 AG^ = 76.1 kJ

and a equilibrium constant of Kpro = 1.35 x 10~12 (mol4 nr12). 4 CEAc2 CEH23 CEC02 Pro 3 CEPro

K

The factor 4/3 is necessary because concentrations here are given in C-mol. An empirical approach was chosen to slow the reactions down on approaching the equilibrium, and they were not allowed to proceed to the right side when the equilibrium condition was reached. Using the actual concentrations, the parameters KBU* and Kpro* were estimated. „ * KBU

C A c 2 C H2 2 CH + ~~ CBu

_ 4 CAc2 CH23 CCQ2 3C Pro

The ratio of these values to the true equilibrium constant, K*BU/KBU, and K*pro/Kpro will be greater than unity if the equilibrium has not yet been reached. Using these ratios with the empirical S-shaped curve of Fig. 1, the factor FEQ was determined and was used to modify the growth rates. This somewhat arbitrary function starts from FEQ = 0 at K*/K < 1 and rises to FEQ = 1 at K*/K > 2. The factor FEQ causes the reaction to the right to stop when

436


the equilibrium is reached, but there is no reverse reaction when concentrations of acetate, hydrogen, CO2 and H+ exceed the equilibrium values caused by other reactions. The reaction proceeds irreversibly further away from the equilibrium (K*/K > 2). 1.2

1.00.8.

a uj u_

0.60.40.2. 0. -0.2K*/K

Figure 1. Equilibrium factors (FEQ) to slow the growth rates near equilibrium.

The kinetics of biomass growth butyric acid, and propionic acid were modified by these empirical equilibrium factors, FEQ, according to i =FEQi

Ion Charge Balance to Estimate pH As discussed in Sec. 1.3.7, in calculating the pH an ion charge balance can be written to account for the acid-base dissociation buffer effects. The ion balance represents an implicit non-linear equation in the dynamic model and must be solved by iteration for each time interval, such that 8 = 0 in the equation

I + CH+

Thus CH+ is varied iteratively until 8 becomes essentially zero. This numerical solution is not always trivial using conventional methods for non-linear


437

algebraic equations (e.g., Newton-Raphson, and Regula falsi). Fortunately this type of equation can be handled conveniently by the root finding feature of BerkeleyMadonna, as shown in the program on the CD-ROM and below. If base is added to control pH, an additional balance for cations of strong bases (K+, Na+, ...) and anions of strong acids (Cl% SC>42~, ....) becomes necessary as follows: dCz =

F

titr Qtitr

Program The program nomenclature is rather extensive and is defined within the program. The Berkeley Madonna ROOTS feature is used to calculate the pH, as shown below. The full program is on the CD. (PH> GUESS CHPLUS = le-4 ROOTS CHPLUS = KW/CHPLUS+KdBu/(KdBu+CHPLUS)*BU/4+KdPr/(KdPr+CHPLUS) *Pr/3 +KdAc/(KdAc+CHPLUS)*Ac/2 +KdC/(KdC+CHPLUS)*Cg +KdBuf/(KdBuf+CHPLUS)*BUFFER-lonen-CHPLUS LIMIT CHPLUS >= 0 LIMIT CHPLUS 2 reaches a maximum, which is partly caused by pH changes and by consumption by hydrogen-consuming organisms. The peaks in the CC>2 curve originate from numerical inaccuracies in the stiff system. In the second graph, Fig. 3, the total concentration of volatile acids acetate (Ac), propionate (Pr) and butyrate (Bu) are given. The thermodynamic inhibition of acetogenesis is clearly seen in the early phase of the experiment. Ac reaches a maximum much later than Pr and Bu, since it is produced from these two acids. In the third graph, Fig. 4, the pH versus time profile is given, exhibiting an early decrease, followed by almost constant pH during the rest of the simulation.

439



0.06

-0.3

0.05

-0.25

0.04

-0.2

s

' 0.03

0.15

0.02

-0.1

0.05

0.01

0.005

0.025

0.03

Figure 2. Dynamic profiles of substrate whey (Mo), CH4 (CH), dissolved CO2 (CO) and dissolved hydrogen (Hy). Zoomed to show the early period. Run 1:4389 steps in 1.43 seconds 0.3

0.25

r 0.2

0.15 -

-0.05

0.35

Figure 3. Total concentration of volatile acids acetate (Ac), propionate (Pr, lower curve) and butyrate (Bu). The whey (Mo) peak is hardly visible at T = 0.

440



6.18

-0.2

6.16

• 0.18

6.14

• 0.16

6.12

.0.14

6.1

.0.12

•0.1 £

: 6.08

• 0.08

6.06

•0.06

6.04 6.02

..'

•0.04 •0.02

6

•0

5.98 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 4. Variation of pH and acetate with time.

Exercise

References Heinzle, E., Dunn, I.J. and Ryhiner, G. (1993) "Modelling and Control for Anaerobic Wastewater Treatment." Adv. Biochem. Eng. 48, 79-114. Yamada, N., Heinzle, E. and Dunn, I.J. (1991) "Kinetic Studies on Methanogenic Cultures Using Mass Spectrometry." in: Biochemical Engineering - Stuttgart (Eds. Reuss, M., Chmiel, H., Gilles).


441

Ryhiner, G. Heinzle, E., Dunn, I.J. (1992) "Modelling and Simulation of Anaerobic Waste water Treatment and Its Application to Control Design: Case Whey," Biotechnol. Progr. 9, 332-343.

8.8.8

Oscillations in Continuous Yeast Culture (YEASTOSC)

System Oscillations in continuous cultures of baker's yeast have often been observed. An example of measurements is shown in Chapter 3, whose oscillations were modelled by the reaction scheme in Fig. 1.

Figure 1. Pathways of proposed model for yeast culture oscillations.

Model The balance equations for continuous culture with dilution rate D are as follows: dR dE

= - D E + [ rGE(R,G,E) + rSE(S) - rEX(E) ] R - = D (SF - S) - [ rSE(S) + rSG(S,E) ] R

442


dG dT

= - D G + [ rSG(S,E) - rGE(R,G,E) ] R = - D T + [ rTi(R,G,E) - rT2(R,G,E) ] R

The species in parenthesis indicate the dependencies of the rates. The kinetic expressions used in the balance equations are as follows:

r

GEm E /KGX\n l— /

S Ks + S

TEX =

ME)

-

1 + ( KG/G + KET/E )n rT2 = Many of the parameters were determined independently from experiments, some were taken from the literature, and some, especially those describing the enzyme activity (T), had to be chosen during simulations. This model leads to oscillations whose existence and dependency on operating conditions qualitatively agree with experimental results. Also the directions in the phase plane plot agree with the experiments.

443


Nomenclature Symbols D E G K n R r S sig T X

Dilution rate Ethanol concentration Storage material Growth rate constants Empirical exponent in rate model Residual biomass without G Growth rates Glucose concentration Rate constants. Example: sigGEm = TGEM Enzyme concentration Biomass concentration Specific growth rate

Indices E G m S T X

Exercise


to ethanol to storage material to maximum to glucose to enzyme to biomass

1/h kg/m3 kg/m3 kg/m3 g/m3 kg/m3 h kg/m3 various g/m3 kg/m3 1/h

444


Results The influence of dilution rate is given below in plots from simulations: Fig. 2 with D = 0.05 and Fig. 3 with D = 0.1. In Fig. 4 the phase plane from the run of Fig. 2 is shown. Run 1:40012 steps in 3.3 seconds -0.45

35-

0.4

30-

0.35

250.3

20-

0.25

( 0.2

15-

-0.15

10

0.1

50-

M..JL.JL... 20

40

0.05

-0 60

100 TIME

120

140

160

180

200

Figure 2. Biomass and substrate oscillations for D = 0.05. Run 1:40012 steps in 5.65 seconds .2

• 1.8 -1.6

1.4 1.2

-1 •0.8 .0.6 .0.4 •0.2

20

40

60

80

100 TIME

120

Figure 3. Biomass and substrate oscillations for D = 0.1.

140

160

180

200

445



0.09-| 0.080.070.060.05-

i 0.040.030.020.01 -

012

16

18

20

22

24

26

28

30

X

Figure 4. Phase plane giving S versus X from the run of Fig. 3. Zoomed in for detail.

Reference Heinzle, E., Dunn, I.J., Furukawa, K. and Tanner, R.D. (1983). Modelling of sustained oscillations observed in continuous culture of Saccharomyces cerevisiae. in Modelling and Control of Biotechnical Processes (ed. A.Halme), Pergamon Press, London, p.57.

8.8.9

Mammalian Cell Cycle Control (Mammcellcycle)

System Modeling of mammalian cell cycle control is of great importance for understanding development and tumor biology. Hatzimanikatis et al. (1999) presented a model in the literature using simplified molecular mechanisms as depicted in Fig. 1.

446


cycE+cdk2

CycE:cdk2-P+1 f>

Rb+E2F

cycE:cdk2-P:1

Rb-P+E2F

Figure 1. Schematic representation of the molecular mechanism of components and interactions believed to be most important in controlling the Gl-S transition. cycE- cyclin E; cdk2 - cyclin dependent kinase 2; Rb - pRb, a pocket protein; E2F - a transcription factor; P phosphate.

Model For this reaction scheme the dynamic mass balances become — = V2 -Vl dt ~

dK

dKr dt


447

dRE dt

^=v 6 > r -v 6 , f The symbols are defined as follows: V are reaction rates. C is the cyclin E concentration. K is the cdk2 concentration. KP is the phosphorylated cyclin E-cdk2 complex concentration. K P I is the concentration of cyclin E-cdk2 phosphorylated complex bound to inhibitor. R is the concentration of the hypophosphorylated form of pRb. RP is the concentration of the hyper-phosphorylated form of pRb. RE is the concentration of the hypo-phosphorylated form of pRb that binds to E2F. E is the E2F concentration. I is the concentration of the cyclin E-cdk2 complex inhibitor. The subscipts "f' and "r" denote the forward and the reverse step, respectively, of the reversible reactions. The assumption of near equilibrium operation of reversible reactions (V5 and V6) and of invariant total amounts of cdk2, pRb, E2F and inhibitor gave the following dimensionless equations, consisting of 3 differential and 6 algebraic equations.

— =v vd dT~ s

y\

dk drp di

—£-= Va3 — VA 4

k + k p + k p j =1 r + rp + r^ = 1

448


=1 i + A,kPI=l

re 0I= JPJL

kpi

Where c is the dimensionless concentration of cyclin, k is the dimensionless concentration of cdk2 and rp is dimensionless concentration of the hyperphosphorylated form of pRb. All details about the kinetic equations and transposing them into dimensionless form are given in the paper of Hatzimanikatis et al. (1999).

Nomenclature Dimensionless symbols as used in the program are listed here. c e i k kP kP,I r rE rP g s 1 t

Cyclin E concentration E2F concentration Concentration of cyclin E-c ckd2 concentration Phosphorylated cyclin E-cdk2 complex concentration Concentration of phosphorylated cyclin E-cdk2 complex bound to inhibitor Concentration of hypophosphorylated form of pRb Concentration of hypophosphorylated form of pRb that binds to E2F Concentration of hyperphosphorylated form of pRb Ratio of total concentrations of cdk2 and cyclin E Ratio of total concentrations of pRb and E2F Ratio of total concentrations of cdk2 and inhibitor Dimensionless time

449


Exercises

Program The program is given on the CD-ROM.

Results Run 1: 50233 steps in 3.08 seconds 0.5

Figure 2. Profiles of concentrations k, rp and c versus time as obtained from the rate constants in the program.

450



1 0.9 0.8 0.7 0.6

"o.5 0.4 0.3 0.2 0.1

0.32

0.33

0.34^

0.35

0.36

0.37

0.38

0.39

0.4

0.41

0.42

Figure 3. Phase plane plot of k and rp versus c.

Reference V. Hatzimanikatis, K. H. Lee, and J. E. Bailey. (1999) "A mathematical description of regulation of the Gl-S transition of the mammalian cell cycle". Biotechnol. Bioeng., 65, 631-637.

8.9

8.9.1

Membrane and Cell Retention Reactors

Cell Retention Membrane Reactor (MEMINH)

System Consider a reactor whose outlet stream passes through a membrane that retains only the biomass as seen in Fig. 1. The growth is assumed to follow substrate inhibition kinetics with constant yields. The oxygen transfer rate influences the growth at high cell density according to a Monod function for oxygen. F,S 0

i

F

Gas

\"

Membrane modi module

illiiil

Air

Figure 1. Biocatalyst retention on a continuous reactor.


452


Model The reactor-membrane system is modelled as a well-mixed tank, except that biomass is retained in the system batchwise. The balance region is chosen to include both the reactor and the membrane separator, but the separator volume is neglected. Biomass balance:

dX dT = rx Substrate balance:

dS

F

-r s

Oxygen balance (neglecting the oxygen transported by flow):

dCL -gj- =K L a(CLS-C L )-ro Kinetics: S rx

__

~ ^m

CL x

r

s = rx YS/X + MS X

r

o

= r

x YQ/X + MO X

where the maintenance coefficients are related by

MS M0


=

_

YS/X

8.9 Membrane and Cell Retention Reactors

453

Nomenclature Symbols

CL CLS F KI KLa Ko KS M r S V X Y

ILL

Dissolved oxygen concentration Saturation oxygen concentration Flow rate Inhibition constant Transfer coefficient Saturation constant for oxygen Saturation constant Maintenance coefficients Reaction rate Substrate concentration Reactor volume Biomass concentration Yield coefficient Specific growth rate

Indices 0 1 m O S X

Exercises


to feed to reaction 1 to maximum to oxygen to substrate to biomass

g/m3 g/m3 m3/h g/m3 1/h g/m3 kg/m3 kg/(kgh), g/(kgh) kg/(m3 h) kg/m3 m3 kg/m3 kg/kg 1/h

454


Results Oxygen transfer has a pronounced influence on performance as seen in Fig. 2 for variations of Kâ. The dissolved oxygen may reach values below KQ as shown in Fig. 3 for Kâ values from 0.5 to 5.


20

Figure 2. Influence of oxygen transfer coefficient (KLa = 0.1 to 1.0).

455



\\. 0

2

4

6

10

12

14

16

18

20

TIME

Figure 3. Profiles of dissolved oxygen influenced by KLB (0.5 to 5, curves bottom to top).

8.9.2

Fermentation with Pervaporation (SUBTILIS)

System The metabolic pathways for the production of acetoin and butanediol are well known, as shown in Fig. 1. A kinetic model for a Bacillus subtilis strain has been established from continuous culture experiments using an approach involving overall stoichiometric relationships and energetic considerations. The influence on the culture of product removal by pervaporation was investigated by simulation methods. Knowledge of these pathways allowed the following overall equations to be written: Respiration (reaction RI): C6Hi2O6

— > 6CO 2 + 6H 2 O

Formation of biomass (reaction R2): 1.2 NH3 —> 6 CHi.8O0.5No.2 + 0.3 O2 + 2.4 H2O

456


Formation of acetoin (reaction C 6 Hi 2 0 6 +10 2 —> C4H802 + 2 C02 + 2 H2O Conversion of acetoin to butanediol (reaction R4): C4H8O2 + H2O —-> C4HioO2 + 0.5 O2

Complex Compounds

Biomass

Substrate Sugars

^

Carbon Dioxide (Total Oxidation)

Pyruvate

^

v

Lactate

Acetoin i

+ Carbon Dioxide

Butanediol

Figure 1. Reaction scheme for the acetoin - butanediol formation.

The continuous reactor was coupled to a pervaporation membrane module and was influenced by the membrane performance, owing to the removal of products and the retention of biomass. Since only the volatile products and water can pass through the membrane, a purge stream was needed to remove biomass and salts. The recycle between the reactor and membrane module, shown in Fig. 2, was high enough to provide complete mixing.

Model Growth The sugar (S) and dissolved oxygen effects were described in terms of a double Monod function and the product inhibition by a simple inhibition kinetic term. Diauxic effects were observed, but the diauxic components were unknown, and for simplification only one diauxic switchover was assumed. The preferred

457


component is referred to as Ci, and the second component is called C2- An empirical kinetic relation was devised such that the utilization of C2 was inhibited by the presence of Ci in amounts greater than a repression constant kRpi. The formulation then involves considering the growth as the sum of two terms, where, and

C2

Membrane

Recycle

Feed

*pl Pervaporation Module

Condenser

Bioreactor

Permeate Fpe ACpe , Bu

Pump

Figure 2. Bioreactor and membrane pervaporator, showing process variables. Each product was assumed to inhibit separately,

1

1 Bu_cc Bu

and

The inhibition constants, kinh,Ac and kinh,BU> an2, CC>2, X and S) Refers to inhibition Oxygen uptake Permeate Purge Product Chemical reaction i (1 to 4) Sugar Biomass Feed or initial concentration Refers to reactions 1 to 4

Reference Dettwiler, B. Dunn I. J., Heinzle E., and Prenosil J. E. "A Simulation Model for the Continuous Production of Acetoin and Butanediol Using B. subtilis with Integrated Product Separation by Pervaporation" Biotechnol Bioeng. 41, 791 (1993).


Results The results of Fig. 3 show the influence of the FM/P ratio, which corresponds to the membrane area per unit reactor volume, on the biomass concentrations.

463


Here the enrichment factor was kept constant (pAc = 2.0), corresponding to the values found for one of the membranes. In a second set of runs the enrichment factor was varied for constant FM/P ratio = 1.0. Run 14: 2013 steps in 0.317 seconds 110

100

60

50 6

8

10

12

14

16

18

20

Figure 3. Influence of permeate/purge flow ratio on the biomass concentration. BETAAC=2, FMP=0.4, 1.0, 2.0).

(D=l,

Run 19: 2013 steps in 0.333 seconds 0.25

0.15

..p.-'*""

j- *~ "*"

0.05

Figure 4. Influence of the enrichment factor on the acetoin and butanediol in the permeate. (D=l, FMP=1 BETAAC=1, 3, 5).

464


8.9.3

Two-Stage Fermentor With Cell Recycle For Continuous Production Of Lactic Acid (LACMEMRECYC)

System This example is based on a paper (1) in which a two stage fermentor-membrane system for the continuous production of lactic acid was modelled. Membrane retention of the active biomass can be expected to increase the productivity, but the biomass concentration must be controlled in each reactor with a bleed stream. The kinetics for this process, in which glucose is converted to lactic acid by the bacteria Streptrococcus faecalls is described in the literature (2). Since the rates are inhibited by product, it can be expected that a multistage system will be advantageous.

(1-B2XB1F1+I?)

X 2 ,S 2 ,P 2

Figure 1. Two-stage membrane fermenter system for lactic acid production


465

Model The model assumes completely-mixed stages and complete cell separation. The operating parameters are feed flow rates or dilution rates and the bleed stream fractions. The bleed stream from the first fermenter is led to the second fermenter. The model equations are developed neglecting the volume of the lines and separators. Note that the retentate streams are returned to their respective reactors and can be considered as part of the well-mixed system. The kinetics is given by a product inhibition S

ii — ii

KLP

and a death rate of the bacteria caused by product is also included.

The specific glucose uptake rate is given by 1 Ys — =—+

qs

M-

The lactic acid production rate is given by

Mass Balances for the First Stage Defining the dilution rate as DI=FI/VI For the total cells ~dT~~ For the living cells

l

l

466


For the glucose

For the lactic acid l

dt

fl

l

l

l

l

Mass Balances for the Second Stage The dilution rate for the second stage depends on the bleed ratio from the first stage and the ratio of feed rates.

For the total cell mass =

BjDjXn - (8! + f )D!B2X2 +

For the active cells

^ = IBAXJ - I(B! + 00^2X2 + (ji2 - k d2 )x For the glucose fD!Sf2 +

BAS1 - ( 8 ! +f)D 1 S 2 -q 2 X 2

For the lactic acid product 1 Clt

UC

(JC

1 .

>

2+V2X2

(JC

Productivity of Lactic Acid It is assumed that all streams containing product can be recovered from all of the streams. The productivities are calculated as follows: From the first stage fti=D1(P1-Pfl)


467

From the second stage Pr 2 =-D 1 [(B 1 +f)P 2 -B 1 P 1 -fP f2 ] VAi

The total productivity Pr = -i-D 1 [(l-B 1 )P 1 +(B 1 +f)P 2 -P f ,-fP f2 ] JL ~t~ UC

Substrate Conversion From stage 1

From stage 2 Xco =1 82

(B 1+ f)S 2

--

B^+fSfz

Total glucose conversion _ Xg — I

(l-B 1 )S 1 +(B 1 +f)S 2

--

S

f 1 + fSf 2

Program In the simulation product-free feed streams are assumed with flow rates between 10-30 kg m 3 . The program is on the CD-ROM.

Nomenclature a B D f F kd

Death constant Bleed ratio Dilution rate Flowrate ratio, F2/F1 Volumetric flowrate Specific death rate Basis specific death rate

m 3 kg"1 s'1 m3 s'1 s"1 s"1

468 KP KS P Pr

qs s t

V *s

X

xt a

YP,YS 8P,8s

H V

Exercises


Product inhibition constant Saturation constant Lactic acid concentration Productivity Specific substrate uptake Glucose concentration Time Reactor volume Substrate conversion Concentration of living cells Total cell concentration Reactor volume ratio Kinetic constants Kinetic constants Specific growth rate Spec, product formation rate

kg nr3 kg nr3 kg nr3 kg nrV1 kg kg-V1 kg nr3 s m3 kgnr 3 kgm~ 3 kg kg-1 kg kg-1 s-1 kg kg'1 s-1

469


References A. Nishiwaki and I. J. Dunn, "Performance of a two stage fermentor with cell recyle for continous production of lactic acid", Bioprocess Engineering 21; 299-305, 1999. H. Ohara, K. Hiyama, T. Yoshida, "Kinetics of growth and lactic acid production in continuous and batch culture" Appl.Microbiol.Biotechnol. 38, 403-407, 1992.

Results Run 1: 405 steps in 0.0167 seconds •-30 ^—Xt1:1 ...—Xt2:1

20-

25

... S2:1 •20

P2:1

rsj 15X

gioH 10 CO

10

15

20

30

35

—1 40

v

TIME Figure 2. Results showing the steady state for biomass substrate and product in both stages.

470



160

-1

140

i

•0.9

| 120-I

B X

-

1 flf

•0.8

\

£100

-0.7 rf

80

•0.6 tfT X

60

-0.5

40

A..

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.4

0.5

B1

Figure 3. Results showing that the productivity changes slightly with bleed ratio but that the biomass concentration and the substrate conversion depend highly on bleed ratio.

8.9.4

Tubular Hollow Fiber Enzyme Reactor Module for Lactose Hydrolysis (LACREACT)

System This tubular reactor- radial diffusion model assumes a series of nine well-mixed tanks to describe a single hollow fiber module. Flow of lactose substrate passes axially through the inner region of the fiber lumen. By diffusion the substrate is transported radially outward from the lumen through the membrane and into the cylindrical porous support surrounding the membrane. The reaction takes place in this support region, where the immobilized enzyme is located. The products of hydrolysis are glucose and galactose, which diffuse back toward the liquid phase in the lumen. The parameter N can be used to adjust the number of shells required. The module is assumed to consist of a large number of identical fibers*

471


Model The model is developed by finite-differencing both the axial and radial directions. Thus there are axial stages in series, here nine plus the recycle tank, and there are multiple cylindrical sections of porous support in the radial direction. As depicted in the figures below, there is a convective liquid flow from each stage to another. Diffusional flows carry substrate and product radially from one cylindrical section of the porous support to another. Figures 1, 2 and 3 give the geometrical details.

FRfiber

LAfiber

LAtank

FRfiber

Figure 1. Hollow-fiber module showing only a single fiber as modelled by nine stages.

472


Figure 2. Details of a single well-mixed axial section.

Figure 3. Details of a finite-differencing of the porous support in the radial direction.

Fig. 3 shows a cross-section of the hollow fiber membrane showing the inner hollow fiber region (white) and the outer porous support (shaded). The finitedifference shell of volume V2 (white) is shown with diffusion fluxes of lactate JLAI entering and leaving JLAI- It is important to account for the radial variation of volumes and diffusional areas. Note that the segments are numbered from outside to inside, 1 to N. For each tank the component balances account for the accumulation, the flow in and out and the diffusion in or out from segment N of the porous membrane. For lactose in tank 1


LA

lumenl _ H ai

473

FR v v

fiber */ T A x , JlAltN]* Aj[N] T A v^^tank ' L/Mumenl ) + 77 v lumen lumen

For the external recycle tank, the flow leaving all of the fibers enters it and also the feed stream enters it. The total flow rate leaving must equal the sum of these rates. Thus for lactate Number * FRfiher tank

T

A

X

Ffeed / A

A

N

T T . LAtank ) + -Jeed_(LA feed - LA tank ) V tank

Taking the component balances for each segment in the enzyme zone account for accumulation, diffusion in and reaction. Thus for lactose in the enzyme regions of the first tank

where Aj is the area available to diffusion and TLAI!^] is the reaction rate for lactose in the ith enzyme segment of the first tank. Note that the above equations do not include the balance for segment 1. The wall condition requires that this balance contains only the rate of diffusion from segment 1 to segment 2, as seen in the program. The reaction rate is assumed, as confirmed by experiment, to have the form of *!__„

LAt[i]

Here the units are mole lactose per cm3 of porous support volume per second. Program Repeated here for the first tank section are the lactose balances, fluxes and rates as given in the program on the CD-ROM. JLA1[1. . (N-l) ]=-DLA* ( L A l E i + 1 ] -LAl[i] ) /DR

Flux of lactose flowing between segments i+1 and i JLAl E N ] =DLA* ( L A I [N] -LAlumenl) /DR

Flux of lactose into inner lumen section

474


RATELA1 [1. . N] =-vmax*LAl [i] / (LAI [i] +Km* (1+ (GA1 [i] / K i n h i b ) ) ) Reaction rate for lactose,mole/s cm3

d/dt (LAlumenl) = (FRf iber/Vlumen) * (Latank-LAlumenl ) + ( JLA1[N] *AJ[N] ) /Vlumen Dynamic balance for lactose in the inner lumen volume of the first tank d/dt (LAI [2 . .N] )-(l/V[i] ) * (JLAl[i-l] *AJ[i-l] JLA1 [ i ] * AJ [ i ] ) +RATELA1 [ i ] Dynamic balance for lactose in the segment 2 to N of the first tank d/dt ( LAI [1] )=-(l/V[l] )*(JLA1[1] *AJ[1] )+RATELAl[l] Dynamic lactose balance for the segment 1 (wall condition) of the first tank. d / d t ( L A t a n k ) = ( Number * F R f i b e r / V t a n k ) * (LAlumen9L A t a n k ) + ( F f e e d / V t a n k ) * (LAf e e d - L A t a n k ) Dynamic lactose balance for the circulation tank The geometry of the fiber is programmed such that the lumen radius and the total fiber radius are given. The number of porous segments can be varied.

Nomenclature Additional symbols for the geometrical factors are defined in the program on the CD-ROM

Symbols DGA DGL DLA EO FR GAfeed GLfeeci Ki n hib

Galactose diffusivity in porous membrane Glucose diffusivity in porous membrane Lactose diffusivity in porous membrane Enzyme loading Recycle flow Galacose in feed Glucose in feed Kinetic inhibition constant

cm2/h cm2/h cm2/h mg E in each fiber cm3 mole/cm3 mole/cm3 mol/cm3

475


?

LA, GA, GL Number V

Kinetic constant Kinetic constant for vmax Length of fiber, Lactose, galactose and glucose Number of fibers Volumes Maximum rate

Indices fiber lumen tank

Exercises


to lumen to lumen to tanks or segments to recycle tank

mol/cm3 mol/mgE h cm3 cm mol/cm3 cm mol/cm3 h

476



100

200

400

500

600

TIME

Figure 4. Approach to steady state for the tank concentrations. Run 1: 619 steps in 53.4 seconds 9.4e-5

L

9.393e-5

Figure 5. Radial concentration gradients for the lactose and glucose. The left axis corresponds to the outside of the fiber.

477


8.9.5

Immobilized Animal Cells in a Fluidi/ed Bed Reactor (ANIMALIMMOB)

System This example is based on experiments with immobilized BHK cells in a fluidized bed of solid or porous carriers. The fluidized bed itself has an expanded volume of 700 mL, The complete reactor system contains a volume of 3.5 L. As seen in the figure below, the arrangement of the electrodes at the inlet and outlet of the reactor allows an accurate difference measurement of the oxygen uptake rate. Oxygen transfer takes place only in the conditioning vessel, while oxygen consumption is only in the fluidized bed column, where the cells are located. GAS OUTLET

GAS INLET FILTER

AIR

OXYGEN CARBON OXIDE

OXYGEN j (MEASUREMENT CHAMBER

SAMPLE PORTM .

VESSEL

THERMOSTAT

Figure 1. Fluidized bed for culturing animal cells on solid carriers.

The recirculation reactor is modelled by taking into account the separate aeration tank and the geometry of the column reactor. It is assumed that the reactor is not well mixed, but is described by a tanks-in-series model for the column with immobilized cells and a separate well-mixed aeration tank reactor, as shown below.

478


j3, FR

Spent medium

Figure 2. Schematic of the model structure, where Ci refers to any component concentration.

For the tanks-in-series description of the column, 3 (or more) tanks in series are used. The mass balance for one component in tank 2 is then

Here V is the volume of one tank and r is the reaction rate of the component. The circulation flowrate is FR. This balance equation form would apply to all components, but not for the biomass since it is immobilized. The kinetic model assumes the following: 1. Growth of cells is linked to the consumption of glucose and Yx/s = 0.28g biomass produced per g glucose consumed. 2. Lactate is produced in proportion to the glucose uptake rate with YiacG=2.0. 3. Oxygen, glutamine, lactate and glucose concentrations influence the rates. 4. Multiple Monod kinetics can be applied. 5. The medium is in contact with air and the solubility of oxygen is 8 mg/L. From the data in the dissertation of Keller (1) can be calculated the yield of lactate with respect to glucose, giving Yiacc = 2.0 mmol lactate/mmole glucose or 1.1 g sodium lactate/g glucose. This can be used to calculate the production rate of lactate. Also calculated from the dissertation is YgiutG=:36mg glutamine/g glucose. Thus the glucose uptake rate becomes K lac

"OX w

sox

K sglut

4-C lac


479

Note that the last term in the equation models an inhibition by lactate. The growth rates are based on the specific substrate uptake rates. Other specific rates are related by yield coefficients and biomass concentration. For example for growth rate, rx = qG X Yxg

Data of experimental values: Maximal volume of the expanded fluidized bed; 0.6 L Entire reactor system volume; 3.5 L Medium throughput; 6.5 L/day Feed glucose concentration; 3.9 g/L Glucose consumption rate; 4.7 mmol/h Oxygen uptake rate; 3.7 mmol/h Max. cell density: nonporous carriers; 2*10 cells/ml expanded bed volume porous carriers; 4*10 cells/ml expanded bed volume Ratio of inoculum cell number; approx. 5 % of final cell number Total biomass (porous carrier); 6.24 g (Approx. 2*10 cells per g biomass) Oxygen transfer coeff.; KLa 2.15 1/hr


Nomenclature Cgiutf CgiutF Cox Coxsat F FR GUR

Feed cone, for glucose Feed glutamine concentration Feed glutamine concentration Dissolved oxygen Saturation for oxygen Flowrate Circulation flowrate Glucose uptake rate

g/L mg/L mg/L mg/L mg/L L/h L/h mmol/h

480

KLa Klac K sg KSglut KSOX

OUR qCmax QOxmax

V V4 X YoiutG YLacG YoxG

YXG

Exercises


Mass transfer coeff. Inhibition constant for lactate Saturation constant, for glucose Saturation constant for glutamine Saturation constant for oxygen Oxygen uptake rate Glucose uptake rate max Oxygen uptake rate max Volume of 3 tanks in the column Aeration tank volume Biomass concentration Glutamine uptake rate Yield coefficient glucose Yield coefficient glucose Yield coefficient, biomass to glucose

1/h g/L g/L mg/L mg/L mmo 1/h g/h g cells mg/g cells h L L g/L mg glut./g glue. g lactate/g mg oxygen/g g/g

481


References Keller, J. Dissertation No. 9373, ETH, 1991. Keller, J., Dunn, I. J. and Heinzle E. "Improved Performance of the Fluidized Bed Reactor for the Cultivation of Animal Cells" in Production of Biologicals from Animal Cells in Culture, Ed. Spier, Griffiths, Meignier, ButterworthHeinemann, 10th ESACT Meeting, 513-515 (1991). Keller, J. and Dunn, I. J. "A Fluidized Bed Reactor for the Cultivation of Animal Cells", In: Advances in Bioprocess Engineering, Eds. E. Galindo and O. T. Ramirez, Kluver, pp 115-122 (1994).


11 '

10'

•6 O O

9' 8 ?

7 6 5

O

iliii X[3]:1

4 3 20

40

80 TIME

100

140

160

Figure 3. Batch run showing the difference in dissolved oxygen between the tank sections.

482



8

20

60

80

100

120

140

160

Figure 4. In this run with F=0.05 L/h first oxygen limitation develops and later glucose limitation. The immobilized biomass in the three sections increases at different rates due to the glucose gradient in the column.

Appendix: Using the Berkeley Madonna Language

9.1

A Short Guide to Berkeley Madonna

Computer Requirements Two Berkeley Madonna versions are supplied with this book on a CD, one for PC with Windows and one for the Power Macintosh. More information with downloads can be found on the following website: http://www.berkeleymadonna.com

Installation from CD The files are compressed on the CD in the same form as they are available on Internet. Information on registering Madonna is contained in the files. Registration is optional since all the examples in the book can be run with the unregistered version. Registration makes available a detailed manual and is necessary for anyone who wants to develop his or her own programs.

Running Programs To our knowledge, Madonna is by far the easiest simulation software to use, as can be seen on the Screenshot Guide in this Appendix. Running an example typically involves the following steps: Start Berkeley Madonna and open a prepared program file. Biological Reaction Engineering, Second Edition. I. J. Dunn, E. Heinzle, J. Ingham, J. E. Pfenosil Copyright © 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-30759-1

484

9 Appendix: Using the Berkeley Madonna Language

Adjust the font and size to suit. Go to Model/Equations on the menu and study the equations and program logic. Go to Parameters/Parameter Window on the menu and see how the values are set. They may be different than on the program. Those with a * can be reset to the original values. Also, if necessary, here the integration method and its parameters (DT, Stoptime, DTmax, Dtmin, Tolerance, etc.) values can be changed. Decide which plot might be interesting, based on the discussion in the text. Go to Graph/New Window and then Graph/Choose Variables to select data for each axis. All calculated results on the left side of the equations are available and can be selected. Run the program by clicking on Run. Adjust the graph by setting the legend with the legend button. Perhaps put one of the variables on the right side of the graph with Graph/Choose Variables. Possibly select the range of the axes with Graph/Axis Settings. Choose colors or line types with the buttons. Decide on further runs. It is most common to want to compare runs for different values of the parameters. This is usually done with Parameters /Batch Runs and also with Parameters/Define Sliders. If the overlay button is set then more than one set of runs can be graphed on top of the first set. Sometimes more than one parameter needs to be set; this is best done with changes done in the Parameters/Parameter Window, with an overlay graph if desired. As seen at the end of the Screenshot Guide, Parameter Plot runs are very useful to display the steady-state values as a function of the values of one parameter. For this, one needs to be sure that the Stoptime is sufficient to reach steady-state for all the runs. When running a program with arrays, as found in the finite-differenced examples, the X axis can be set with [i] and the Y axes with the variables of interest. The resulting graph is a plot of the variable values at the Stoptime in all of the array sections. For equal-sized segments, this is the equivalent of a plot of the variables versus distance. If the steady-state has been reached then

9.1 A Short Guide to Berkeley MADONNA

485

the graph gives the steady-state profile with distance. More on running programs is found in Sec. 2 of the Appendix.

Special Programming Tips Berkeley Madonna, like all programming languages, has certain functions and characteristics that are worth noting and that do not appear elsewhere in this book. Editing text The very convenient built-in editor is usually satisfactory. Also the program can be written with a word processor and saved as a text-only file with the suffix ".mmd". Madonna can then open it. Finding programming errors. Look at a table output of the variables Sometimes programs do not run because of errors in the program that cause integration problems. Some hint as to the location of the error can often be found by making an output table of all the calculated variables. This is done by going to Graph/New Window and then Graph/Choose Variables and selecting all the variables. Then the program is run and the table button is chosen. Inspection of all the values in the table during the first one or two time intervals will usually lead to an isolation of the problem for those values that are marked in red with NAN (not a number). Also, values going negative can be found easily here and often indicate an integration error. Sometimes this can be overcome with a limit function of the form, limit X>=0. Is a bracket missing? Madonna tests for bracket pairs, and a missing bracket will be indicated. Setting the axes. Watch the range of values. Remember that each Y axis can have only one range of values. This means that you must choose the ranges so that similarly sized values are located on the same axis. Are there bugs or imperfections in Madonna? Yes, there are some that we are aware of. You may find some or you may have some special wishes for improvements. The Madonna developers in Berkeley, California would be glad to receive your suggestions. See the homepage to contact them Making a pulse input to a process. This can be done it two ways: To turn a stream on and off either use the preprogrammed PULSE function or use an IF-THEN-ELSE statement.

486


Making a more complex conditional control of a program. In general the IF-THEN-ELSE conditional statement form is used, combined with logical expressions as found in the HELP. This can involve a switching from one equation to another within this statement. Another way is to use flags or constants that take values of 0 or 1 and are multiplied by terms in the equations to achieve the desired results. Nesting of multiple IF statements is possible: V = I F ( D i s k < l AND P > 1 . 9 ) THEN 0.85*KV*P/SQRT(TR+273) ELSE IF (Disk1.1) THEN KV*P/SQRT(TR+273)*SQRT(1+(1/P)*(1/P))

Parameter estimation to fit parameters to data. For fitting sets of data to one or more parameters the data can be imported as a text file and fitted by going to Parameters/Curve Fit. The Edit/Preferences/Graph Window provides the possibility of having the data as open circles. The required data format can be found in the file KLADATA. Optimisation of a variable. There is optimization available under Parameters/Optimize, but if it is something simple with one or two parameters, then sliders can also be effectively used. If the value of a maximum is sought as a function of a single parameter value, then the Parameter Plot for maximum value can be used. Finding the influence of two parameters on the steady state? A Parameter Plot choosing the "final" value can be used to find the influence of one variable on the steady state. The second parameter can be changed in the Parameter Window and additional parametric runs made and plotted with an overlay plot. Thus it is possible to obtain a sort of contour plot with a series of curves for values of the second parameter. Unfortunately, a contour plot is not yet possible. Nice looking results are not always correct. A warning! It is possible to obtain results from a program that at first glance seems OK. Always make sure that the same results are obtained when DT is reduced by a factor of 10 or when a different integration method is used. Plotting all the variables may reveal oscillations that indicate integration errors. These may not be detectable on plots of a few variables. Setting the integration method and its parameters? It is recommended to choose the automatic step-size method AUTO and to set equal values of DT and DTMAX. Run the integration once and reduce both parameters by one-half and run again. If the results are good, try to improve the speed by increasing both parameters. Finally it should be possible to set

9.1 A Short Guide to Berkeley MADONNA

487

DTMAX higher than DT, but sometimes the resulting curves are not smooth if DTMAX is too high. In most cases, good results are obtained with AUTO and DT set to about 1/1000 of the smallest time constant. If no success is found with AUTO, then try STIFF and adjust by the same procedure. Oscillations can sometimes be seen by zooming in on a graph; often these are a sign of integration problems. Sometimes some variables look OK but others oscillate, so look at all of them if problems arise. Unfortunately there is not a perfect recipe, but fortunately Madonna is very fast so the trialand-error method usually works out. Checking results by mass balance For continuous processes, checking the steady-state results is very useful. Algebraic equations for this can be added to the program, such that both sides became equal at steady state. For batch systems, all the initial mass must equal all the final mass, not always in mols but in kg. Expressed in mols the stoichiometry must be satisfied. What is a "Floating point exception"? This error message comes up when something does not calculate correctly, such as dividing by zero. This is a common error that occurs when equations contain a variable in the denominator that is initially zero. Often it is possible to add a very small number to it, so that the denominator is never exactly zero. These cases can usually be located by outputting a table of all the variables. Plotting variables with distance and time. Stagewise and finite-differenced models involve changes with time and distance. When the model is written in array form the variable can be plotted as a function of the array index. This is done by choosing an index variable for the Y axis and the [ ] symbol for the X-axis. The last value calculated is used in the plot, which means that if the steady-state has been reached then it is a steady-state profile with distance. An example is given in the"Screenshot Guide" in Sec. 2 of the Appendix and in the example CELLDIFF. Notation for the differential. In this book the differential form d/dt(x) is usual. However the x' form has the advantage that it appears in the Choose Variables menu and can be plotted. It can also be used directly in another equation. Writing your own plug-in functions or integration methods. Information on using C or C++ for this can be obtained by making contact through the BerkeleyMadonna homepage.

488

9.2


Screenshot Guide to Berkeley Madonna

This guide is intended as a supplementary introduction to Berkeley Madonna, Version 8.0.1.

(CHEMQSTATST (FILE/CHEMCr1) {Constants} UM=OJKS=OJK1 8F*10 D1=0.25 Stoptime=80 {Conditional equation for 0} !nrfX=1 ln«t P=0 {Mass Balances} ; 81OMASS BALANCE EQUATION ; SUBSTRATE QÂNGE EQUATION ; PRODUCT BALANCE EQUATION

9.2 Screenshot Guide to Berkeley MADONNA

489

Figure 1. The example CHEMO has been opened and the Menu (From left: File, Edit, Flowchart active only for flowchart programs, Model, Compute, Graph, Parameters, Window and Help) and Graph Buttons (From left: Run, Lock, Overlay, Table, Legend, Parameters, Colors, Dashed Lines, Data Points, Grid, Value Output and Zoom).

Figure 2. The Berkeley Madonna menus are shown above.

490

9 Appendix: Using the Berkeley MADONNA Language

UP AND QPEFIAT1GW 3T6RTTIME iTOPTIMI

Kl=Q.03K2=O.OBY=fl.B BF=10 D1=OJS

MTOUT LJM

0,02 0 0.3

KS Kl

0.1 0,03

DT

{Conditional equation for D} D=iftime»=tstirttnenDl

o.oa 0.8 10

01

toil 8=10 initP=G

HITX NITS N1TP

{Mass Balances}

;

EQ

;

BALANCE

0,34

5 1 10

0

Figure 3. The Model/Equations was chosen. Seen here is also the Parameter Window.

Figure 4. If a new graph is chosen under Graph/New Window then the data must be selected under Graph/Choose Variables.

491


4J-, 4-

-9,5

3,5

9

3

8.5 -8

2.5

•2-

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1.5

f-

-8.5

0,5-

0

5.5

20

40

80

SO

108

120

140

ISO

130

200

TIME

Figure 5. A graph window for variables on the left and right-side Y axis with Legend Button selected.

Run 4:10000 steps m 0.167 seconds ..........................10

7. 6-

-7

5-

•6

X 4-

•5

01

-4

3-

-3

4-2

1-

•1

0

20

40

80

100

120

140

180

!80

200

Figure 6. An Overlay Graph for three values of Dl as selected in the Parameter Window.

492


Figure 7. Part of the window to define the Sliders.

Figure 8. A graph of two slider runs, showing the Parameters Menu pulled down.


493

Figure 9. The Batch Runs window for 5 values of SF.

Figure 10. A Parametric Plot was chosen for 40 runs changing values of Dl to give the final, steady-state values. The Data Button was pressed to give the points for each run.

494


Figure 11. The Optimize Window, with the value of Dl being selected to minimize the expression -D*X. The value found was 0.27, corresponding to the Parameter Plot results.

Run 11 KOODOO steps ir, 2.53 seconds

Figure 12. Two Parameter Plots overlaid showing the effect of reducing Y from 0.8 to 0.6.


495

Figure 13. A program written in an array form allows plotting all the values versus time by choosing the variable vector, here S[ ] versus TIME for the program CELLDIFF.

Figure 14. From the same program as Fig. 13, radial profiles of three runs are plotted in an overlay plot. The [i] values can be selected in the Choose Variables. Here the parameter Radius has been changed to demonstrate the large influence of diffusion length.

496


Figure 15. Here the program file KLAFTT is run and fitted to data in the text-file KLADATA. The data consists of two columns: time and CE at equal intervals as seen by the open circles on the plot. Note that the fit variable is CE and the parameter varied to minimize the difference in least squares is KLA.

10

Alphabetical List of Examples

ACTNITR 267 ANEAMEAS 433 ANIMALIMMOB 476 BATFERM 193 BIOFILM 372 BIOFILTDYN 342 CELLDIFF 388 CHEMO 799 CHEMOSTA 258 COMMENSA 400 COMPASM 406 CONINHIB 261 CONTCON 367 DCMDEG 280 DEACTENZ 308 DUAL 275 ELECTFIT 335 ENZCON 305 ENZDYN 383 ENZSPLIT 377 ENZTUBE 272 ETHFERM 240 FBR 295 FEDBAT 204 FERMTEMP 358 FILMPOP 425 INHIB 327 KLADYN 335 KLAFIT 335 LACMEMRECYC 464 LACREACT 470 LINEWEAV 272 MAMMCELLCYCLE 445 MEMINH 450 MIXPOP 477 MMKINET 209 NITBED 299 NITBEDFILM 393

NITRIF OLIGO OXDYN OXENZ PENFERM PENOXY PHB PHBTWO PLASMID REPFED REPLCUL STAGED SUBTILIS TEMPCONT TITERBIO TITERDYN TURBCON TWOONE TWOSTAGE VARVOL VARVOLD YEASTOSC

327 275 337 378 230 253 279 374 477 245 249 290 455 354 349 349 363 422 286 224 224 447


11

Index

Absorption, 117 Absorption tank, 136 Accumulation, 125 Accumulation terms, 135 Acetogenic step, 89 Acid-base equilibria, 47 Acidogenic step, 89 Active, 110 Adaptive Control, 174 Adaptive tuning, 178 Aerated tank, 130 Aerated tank with oxygen electrode, 336 Aeration, 117, 137 Aeration efficiency, 123 Aeration rates, 126 Aeration systems, 126 Aerobic sewage treatment, 128 Agitation, 137 Air or oxygen sparging, 134 Air saturation, 126 Air supply, 128 Airlift bioreactor, 139 Algebraic loop, 49 Alginate, 149 Alginate bead, 149 Allosteric kinetics, 74 Ammonia, 134 Ammonium, 300 Ammonium ion, 133 Anaerobic degradation, 89 Analogous, 114

Analogy, 114 Analytical solution, 19 Analytically, 108 Animal cell culture, 128 Apparent reaction rate, 133 Approximation, 113, 141 Aqueous phase, 118 Arithmetic-mean, 141 Automatic process control, 161 Automatic reset, 165 Auxiliary variable, 174 Axial, 114 Axial profiles, 114 Axial segments, 115 Backmixing, 137 Backmixing flow contribution, 142 Backmixing stream, 140 Balance region, 23 Balances, 101 Batch, 57, 64 Batch aeration, 126 Batch fermentation, 11, 103 Batch reactor periods, 57 Batchwise, 134 Bed, 134 Biocatalysis, 118 Biocatalyst diffusion model, 153 Biocatalytic reaction, 147 Biofilm, 154 Biofilm, 134, 145 Biofilm nitrification, 160


500

Biofilm Reactor, 133 Biofilter, 343 Biofloc, 145 Biofloc, 149 Biological activity, 123 Biological film, 149 Biological floes and films, 388 Biological oxygen uptake, 125 Biological reaction, 145 Biological reactors, 101 Biological systems, 117 Biomass, 10, 103 Biomass recycle, 110 Biomass retention, 111, 299, 393 Biomass separation, 111 Bioprocess control, 175 Bioreaction, 117 Bioreactor, 112 Bioreactor modelling, 101 Bio-trickling filter, 343 Boundary conditions, 153 Briggs-Haldane mechanism, 68 Broth, 117 Bubble, 117, 137 Bubble coalescence, 137 Bubble column, 139 Bubble size, 140 Bulk liquid, 145 Bulk reactor concentration, 153 Buoyancy, 137 Carbon, 134 Carbon dioxide, 122 Carbon dioxide production rate (CPR), 10 Carbonate, 134 Carrier, 145 Carrier matrix, 145 Cascade, 138 Cascade control, 172 Cell concentration, 101 Cell productivity, 55 Cell recycle, 110

Index

Cells, 117 Centrifugation, 110 Chemical reaction, 126 Chemostats, 104 Circulating liquid supply, 129 Circulation time, 55 Closed-loop response, 170 Coalescence, 126 Cocurrently, 139 Cohen-Coon controller settings, 170 Column systems, 135 Commensalism, 87 Comparator, 162 Competition, 150 Competitive, 74 Completely mixed gas or liquid phases, 137 Complex diffusion-reaction processes, 157 Complex kinetics, 115 Complex models, 123, 138 Complexity, 138 Component, 122 Component Balances, 22 Components, 113 Computer Solution, 19 Concentration driving force, 122 Concentration gradient approximation, 151 Concentration gradients, 119, 137 Concentration inhomogeneities, 137 Concentration profile, 114 Conical sand bed, 134 Continuous, 64, 109, 118 Continuous Baker's Yeast Culture, 94 Continuous feed and effluent stream, 134 Continuous Operation, 60 Continuous phase, 118 Continuous-cycling, 171 Contois Equation, 82 Control, 111 Control point, 165 Control region, 23 Control strategy, 176

Index

Controlled variable, 161 Controller, 162 Controller action, 163 Controller equations, 33 Controller output, 166 Controller tuning, 169 Convection currents, 137 Convective flow, 26, 125 Convective streams, 26 Conversions, 113 Cross-sectional area, 114 Cyclic fed batch, 109 Damkohler number, 155 Data fitting, 339 Datafile, 341 Dead zone, 163 Death rate, 78 Degree of backmixing, 140 Density, 104 Deoxygenated, 126 Deoxygenated feed method, 130 Depth, 137 Derivative control, 163 Deviations, 161 Deviations from ideal stage mixing, 140 Difference form, 116 Difference segment, 115 Differential, 107 Differential control constant, 355 Differential equation, 127 Differential equations, 140 Differential gap, 163 Differential-difference equations, 159 Diffusion, 118, 119, 145 Diffusion and reaction, 388 Diffusion control, 133 Diffusion film, 146, 336 Diffusion layer, 149 Diffusion path, 145 Diffusion rate, 149 Diffusional flux, 26, 149, 389

501 Diffusional limitation, 146 Diffusional mass transfer, 145 Diffusional mass transfer coefficient, 147 Diffusion-reaction parameter, 155 Diffusion-reaction phenomena, 151 Diffusion-reaction systems, 151 Diffusive mass transfer, 138 Digital simulation, 14 Digital simulation languages, 14 Dilution rate, 105 Dimensionless, 108 Dimensionless form, 391 Dimensionless group, 151 Dimensionless parameter, 150, 154 Dimensionless variables, 155, 159, 337 Discontinuous control, 163 Disks of liquid, 113 Dispersed, 117, 118 Dispersed phase, 118 Dissociation equilibrium constant, 47 Dissolved oxygen concentration, 123, 335 Dissolved oxygen electrode, 126 Distance, 116 Distance coordinate, 151 Double Michaelis-Menten Kinetics, 73 Double-Monod kinetics, 83, 158 Driving forces, 119 Droplet, 118 D-value, 77 Dynamic, 115 Dynamic component balances, 104 Dynamic kla, 126-127 Dynamic Method, 335 Dynamic simulation, 49 Dynamics of measurement, 127 Dynamics of the liquid phase, 128 Effective diffusivity, 30, 121 Effective rate, 148 Effective reaction rate, 146 Effectiveness Factor, 155, 391 Efficiency, 110

502 Electrode measurement dynamics, 127, 129 Electrode membrane, 337 Electrode response characteristic, 336 Electrode time constant, 127, 129 Elemental balances, 23 Energy balances, 49 Entrance, 113 Enzymatic, 112 Enzyme, 112, 118 Enzyme loading, 148, 149 Enzyme reactor, 115 Enzyme-substrate complex, 69 Equations, 113 Equilibrium, 10, 122 Equilibrium oxygen concentration, 337 Equilibrium relationships, 46 Equilibrium value, 132 Errors, 161 Exit, 113 Experimental reactor, 130 Exponential, 108 Exponential and limiting growth phases, 103 External film, 145 External mass transfer, 145 External transport rate, 153 Extraction, 118 Fed Batch, 58, 64 Feed Forward Control, 173 Feedback, 161 Fermentation, 101 Fermentation media, 137 Pick's Law, 29, 120 Film coefficients, 122 Final, 109 Final control element, 162 Finite difference, 30 Finite difference Model, 151 Finite differencing, 388 Finite differencing technique, 159 Finite-differencing, 115, 153 First order, 65

Index

First order lag equation, 127, 337 First order lag model, 127 First-order, 132 First-order time lag, 354 Flocculant cell mass, 145 Flotation, 110 Flow interaction, 140 Flow velocity, 114, 148 Fluid, 120 Fluid elements, 113 Fluidized bed, 133, 299, 149 Flux, 120 Food/biomass ratio, 112 Fractional conversion, 126 Fractional response, 127 Free rise velocity, 137 Functional modes of control, 163 Gas, 117 Gas absorber, 136 Gas Absorption, 117 Gas and liquid films, 121 Gas balance, 125 Gas balance method, 128, 126 Gas bubbles, 117 Gas concentrations, 125 Gas flow rates, 125 Gas holdup, 55, 124 Gas inlet, 137 Gas phase, 117 Gas-liquid, 117, 120 Gas-liquid systems, 122 Gas-liquid transfer, 336 Gel, 145 Growth, 110 Growth rate, 32, 103 Heat of agitation, 52 Heat of fermentation, 49 Heat Transfer, 51 Henry coefficient, 340 Henry's law, 33, 122, 337 Henry's law constant, 122

Index

Heterogeneous reaction systems, 155 Hill Kinetics, 74 Hydrostatic pressure, 137, 140 Ideal Gas Law, 39 Ideal gas law, 133, 337 Idealized flow conditions, 137 Idealized plug flow, 137 Ideally mixed, 136 Immiscible, 118 Immobilization, 145 Immobilization matrix, 145 Immobilized, 129 Immobilized biocatalyst systems, 145 Immobilized enzyme and cell systems, 118 Impermeable solid, 149 Incomplete oxygen penetration, 160 Increments, 151 Industrial fermenters, 137 Information flow diagram, 33 Inhibition, 73 Inhibitory Substrate, 367 Initial conditions, 103 Initial value, 20 Inlet, 106 Inoculum, 103 Input rate, 102 Integral, 163 Integral control constant, 355 Integral time constant, 166 Integrated, 108, 115 Integration procedure, 20 Integration step length, 20 Integration time interval, 20 Intensity of mixing, 55 Intensity of mixing, 140 Interconnected, 138 Interface, 117, 118, 120 Interfacial concentrations, 122 Internal mass transfer, 145 Internal structure, 119 Interphase, 119

503 Interphase transfer, 26, 125 Intraparticle transfer, 145 Intrinsic reaction rate, 150 Ion charge balance, 47 Ion exchange resins, 149 Kinetic, 106 Kinetic control, 147 Kinetic model, 136 Kinetic rate constant, 147 Kinetic regime, 148 Kinetic relationship, 65 Kinetics control, 149 Kla, 335 Lag phase, 103 Lag time, 170 Laplace transformation, 338 Large bioreactors, 137 Large scale, 137 Length, 115 Length of diffusion path, 151 Limiting, 140, 159 Limiting substrate, 103, 106 Limiting substrate concentration, 78 Linear gradients, 120 Linear growth, 108 Lineweaver-Burk diagram, 72 Liquid, 117 Liquid balance, 125 Liquid balance equation, 128 Liquid film, 337 Liquid film control, 123 Liquid flow terms, 135 Liquid medium, 117 Liquid phase, 117 Liquid recycle stream, 134 Liquid surface, 137 Liquid velocities, 137 Liquid-liquid, 118 Liquid-phase impeller zones, 141 Logistic Equation, 82 Luedeking-Piret model, 85

504

Maintenance coefficient, 10 Maintenance factor, 84 Mammalian Cell Cycle Control, 445 Manipulated variable, 161 Mass balance equation, 16 Mass Transfer, 117, 119 Mass transfer capacity coefficient, 122, 123 Mass transfer coefficients, 121 Mass transfer control, 147 Mass transfer resistance, 145 Material balance equations, 101 Mathematical, 150 Mathematical model, 12, 137 Mathematical modelling, 151 Matrix elements, 151 Maximum, 109 Maximum observed rate, 148 Maximum rates, 158 Maximum reaction rate, 70 Measurement dynamics, 126, 128, 336 Measurement signal, 127 Measurements, 106 Measuring element, 162 Mechanical agitation, 137 Mechanical energy, 137 Medium, 106 Membrane, 112, 127 Membrane filtration, 110 Methanogenic step, 89 Michaelis-Menten constant, 70, 148 Michaelis-Menten kinetics, 148 Microbial interaction, 86 Microbial physiology, 106 Microbiological, 133 Mixing, 113 Mixing zones, 140 Mode of control, 168 Model, 109 Modelling, 113, 388 Molar flow rate of air, 133 Molar reaction rates, 136

Index

Molecular diffusion, 29 Molecular diffusion, 120 Molecular diffusion coefficient, 121 Monod equation, 67 Monod kinetics, 10 Monod-type equation, 390 Monod-type rate expressions, 105 Multiphase reaction, 117 Multiple impeller, 140 Multiple-organism populations, 86 Multiple-substrate Monod kinetics, 82 Multi-stage, 138 Mutual inhibition, 343 Mutualism, 87 Natural logarithmic, 127 Nernst-diffusion film, 146 Nitrate, 300 Nitrate ion, 133 Nitrification, 133, 299 Nitrification reactions, 157 Nitrite, 300 Nitrite ion, 133 Nitrobacter, 134 Nitrogen, 127 Nitrosomonas, 134 Non-competitive, 74 Non-porous carrier, 146 Numerical solution, 19 Objective function, 178 Offset, 163 Oil phase, 118 One-dimensional diffusion, 158 On-line adaptive optimizing control, 178 On-line monitoring, 126 On-Off Control, 163 Open-loop tuning technique, 169 Operation, 110 Order of magnitude analysis, 155 Organism balance, 102 Oscillations, 355 Oscillations of continuous culture, 94

Index

Outlet, 106 Output rate, 102 Overall mass transfer capacity coeff., 123 Overall mass transfer rate equation, 123 Overall order of reaction, 148 Overall rate of reaction, 62 Overall resistance to mass transfer, 123 Oxidation steps, 134 Oxygen, 117, 122, 388 Oxygen balances, 123 Oxygen depletion, 132 Oxygen diffusion effects, 157 Oxygen electrode, 335 Oxygen electrode dynamic, 337 Oxygen gas phase concentrations, 128 Oxygen gradients, 139 Oxygen limitation, 388 Oxygen requirements, 158 Oxygen transfer, 125 Oxygen transfer coefficient, 136, 335 Oxygen transfer rate (OTR), 10 Oxygen uptake rate, 38, 125, 128, 136, 390 OUR, 38 Oxygen-enriched air, 131 Oxygen-sensitive culture, 97 Packed, 149 Parameter, 111 Parameter estimation, 131 Partial differential equation, 116, 153 Partial pressure, 122 Penetration, 149 Penetration distance, 159, 388 Penetration-limiting, 159 Perfect mixing, 137 Perfect plug flow, 137 Performance, 113 pH control, 49 Phase interface, 120 Phases, 117 Physical model, 12, 137 Physical properties, 122

505 Plant cell culture, 128 Plug flow, 113, 141 Poly-6-hydroxybutyric Acid (PHB), 93 Porous, 119 Porous biocatalyst, 119 Porous solid, 145 Power inputs, 137 Predator-Prey Kinetics, 86 Pressure, 137 Process control, 10, 56, 161 Process dynamics, 127 Process optimization, 12, 56 Process reaction curve, 169 Process response, 128 Product, 118 Product inhibition, 113 Product inhibition kinetics, 63 Production rate, 102 Productivity, 105, 110 Programmed, 174 Proportional, 143, 163 Proportional control constant, 355 Proportional-Derivative (PD) Controller, 166 Proportional-Integral controller, 355 Proportional-Integral-Derivative Controller, 167 Proportlonal-Reset-Rate-Control, 167 Pulse, 113 Quadratic equation, 148 Quasi-homogeneous, 135 Quasi-homogeneous reaction, 158 Quasi-steady state, 107 Radial variations, 140 Rate expressions, 103 Rate of accumulation, 21, 102 Rate of fermentation, 126 Rate of mass transfer, 121 Rate of oxygen transfer, 123 Rate of oxygen uptake, 123 Rate of substrate uptake, 83 Rate of supply, 148

506 Reactants, 117 Reaction, 114, 117, 118 Reaction capability, 149 Reaction control, 133 Reaction Heat, 51 Reaction parameter, 155 Reaction rate constant, 132 Reaction rate control, 133 Reaction site, 118, 145 Reaction surface, 147 Reaction-rate limitation, 136 Reactor, 101, 138 Reactor cascade, 62 Reactor column, 135 Reactor efficiency, 111 Reactor modes, 63 Reactor operating conditions, 159 Reactor outlet, 136 Re-aerated, 126 Recycle loop, 134 Recycle loop configuration, 134 Recycle rates, 134 Recycle ratio, 111 Regimes, 133 Research, 106 Reset time, 166 Residence time, 105, 116 Residence time distribution, 137 Residual error, 167 Resistance to mass transfer, 120, 123 Respiration quotient (RQ), 10 Response, 127 Response curve, 126 Retention, 110, 145 Riser, 139 Sample, 128 Sampled data control, 174 Sampling frequency, 175 Sampling interval, 174 Sand, 134 Saturation, 127

Index

Saturation constant, 10 Scheduled adaptive control, 174 Second-order response lag, 337 Sections, 138 Sedimentation, 110, 111 Semi-Continuous Reactor, 314, 349 Separation, 110 Separator, 111 Series of tank reactors, 62 Set point, 161 Shear, 55 Simulation, 107 Simulation example, 104 Simulation methods, 140 Simulation programming, 153 Simulation programs, 14 Simulation results, 159 Simulation software, 15 Simultaneous diffusion and reaction, 150 Single stage, 138 Single-pass conversion, 135 Slab, 151 Slope, 18, 127 Solid, 118 Solid biocatalyst, 119 Solid carrier, 145 Solid phase, 135 Solid-liquid interfacial area, 135 Solubility, 122 Solution, 143 Spatial variations, 123 Specific area for mass transfer, 121 Specific carbon dioxide production rate, 11 Specific carbon dioxide uptake rate, 84 Specific death rate, 78 Specific growth rate, 10, 78, 105 Specific interfacial area, 122 Specific oxygen, 11 Specific oxygen uptake rate, 84 Specific product production rate, 84 Specific substrate uptake rate, 84

507

Index

Spherical bead, 388 Spherical shell, 389 Stages, 138 Stagewise model, 139 Stagewise modelling, 139, 140 Stagnant, 120 Stagnant film, 145 Stagnant flow, 120 Starting, 109 Startup, 61 Startup period, 61, 62 Steady state, 104 Steady state conditions, 104 Steady state tubular reactor design, 115 Steady state values, 140 Steady-state, 105 Steady-state balances, 21 Steady-state position, 165 Step change, 127 Sterile, 104 Stirring power, 137 Stirring speed, 148 Stirring speed, 149 Stoichiometric coefficients, 40 Stoichiometric oxygen requirements, 159 Stoichiometric relations, 134 Stoichiometry, 40, 126 Structured kinetic model, 11 Substrate balance, 102 Substrate concentration, 101 Substrate gradients, 151 Substrate inhibition, 65, 108 Substrate uptake rate, 55 Sulfite method, 126 Sulfite oxidation, 126 Support, 119 Surface, 118 Surface concentration, 147 Surface reaction, 145 Sustained oscillations, 94 Symmetry, 151

System, 110 Tank, 101 Tank sizes, 113 Tanks-in-series, 112 Teisser Equation, 81 Temperature, 148 Temperature measurement, 354 Theoretical basis, 112 Thiele modulus, 155 Thin film, 120 Time, 109 Time constant for heater, 355 Time constant for measurement, 355 Time constant measurement, 340 Time constants, 128 Time constants for transfer, 340 Time-varying, 114 Titration, 126 Total interfacial area, 121 Total mass balance, 102 Total system, 134 Toxicity, 134 Tracer, 113 Tracer experiment, 140 Tracer techniques, 137 Transfer control, 133, 149 Transfer parameters, 138 Transmission lines, 162 Transport of material, 117 Transport streams, 26 Transport-reaction process, 147 Trial and error method, 169 Tubular, 113 Tubular reactor, 62, 113 Turbine impeller, 134 Turbulence, 120 Turbulent flow, 120 Two position action, 163 Two-Film Theory, 120 Ultimate gain, 169 Ultimate period, 171

508 Uncompetitive, 74 Uptake rate, 111 Variable, 106 Viscosity, 149 Volumetric flow rate, 101 Washout, 105 Wastage, 110 Waste water, 111 Water, 122

Index Well-mixed, 24, 101, 123 Well-mixed gas phase, 133 Well-mixed liquid zones, 140 Well-mixed tank, 25 Whole cells, 149 Yield coefficient, 10, 32, 102 Zero order, 65, 147, 148 Zero-order kinetics, 391 Ziegler-Nichols Method, 169

Biological Reaction Engineering: Dynamic Modelling Fundamentals with Simulation Examples

Fundamentals of Reaction Engineering - Examples

Fundamentals of Reaction Engineering - Worked Examples