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 Kj^a 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

1 Modelling Principles

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

1.1 Fundamentals of Modelling

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

1 Modelling Principles

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.

1.1 Fundamentals of Modelling

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

1 Modelling Principles

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

1 Modelling Principles

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

1 Modelling Principles

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

1 Modelling Principles

( 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 ^in 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

1.3 Formulation of Balance Equations

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

1 Modelling Principles

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

1.3 Formulation of Balance Equations

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

1 Modelling Principles

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<&y^

Figure 1.9. A multi-tank model for the complex river flow system.

//.

Identify system

the transport streams which flow across the boundaries

Having defined the balance regions, the next task is to identify all the relevant inputs and outputs to the system. These may be well-defined physical flow rates (convective streams), diffusive fluxes, and also interphase transfer rates. It is important to assume a direction of transfer and to specify this by means of an arrow. This direction might reverse itself, but will be accomodated by a reversal in sign. Out by diffusion Convective flow out

Convective flow in

In by diffusion Figure 1.10. Balance region showing convective and diffusive flows in and out.

///.

Write the mass balance in word form

This is an important step because it helps to ensure that the resulting mathematical equation will have an understandable physical meaning. Just

27

1.3 Formulation of Balance Equations

starting off by writing down equations is often liable to lead to fundamental errors, at least on the part of the beginner. All balance equations have a basic logic as expressed by the generalized statement of the component balance given below, and it is very important that the mathematical equations should retain this. Thus: ' Rate of accumulation of mass of component the system )

f Mass flow ^\ of the component into \the system^

f Mass flow ^ of the component out of ^the system^

/

Rate of production of the component by \ the reaction /

This can be abbreviated as, (Accumulation)

=

(In) -

(Out) + (Production)

IV. Express each balance term in mathematical form with measurable variables A. Rate of Accumulation Term This is given by the derivative of the mass of the system, or the mass of some component within the system, with respect to time. Hence: (Rate of accumulation of mass of component i within the system) =

dMi

where M is in kg or mol and time is in h, min or s. Volume, concentration and, in the case of gaseous systems, partial pressure are usually the measured variables. Thus for any component i dMj dt

_ d(CjV) dt

=

where, Q is the concentration of i (kmol/m3 or kg/m3), and pi is the partial pressure of i within the gas phase system. In the case of gases, the Ideal Gas Law can be used to relate concentration to partial pressure and mol fraction. Thus, piV= niRT where R is in units compatible with p, V, n and T. In terms of concentration,

28

1 Modelling Principles _ = ni c

i T

=

Pi

yiP

RT = "RT

where yi is the mol fraction of the component in the gas phase and p is the total pressure. The accumulation term for the gas phase can be written as, /piV dMj _ d(CjV) _ d(QV) _ . d _ For the total mass of the system: dM _ d(p V) dt = dt

with units m3

s

B. Convective Flow Terms Total mass flow rates are given by the product of volumetric flow multiplied by density. Component mass flows are given by the product of volumetric flow rates times concentration. Mass \ (VolumeJ kg s

m 3 kg - s m3

Total mass flow = F p Component mass flow MI = F Q A stream leaving a well-mixed region, such as a well stirred tank, has the same properties as the system volume as a whole, since for perfect mixing the contents of the tank will have uniform properties, identical to the properties of the fluid leaving at the outlet. Thus, the concentrations of component i both within the tank and in the tank effluent are equal to Qi, as shown in Fig. 1.11.

29

1.3 Formulation of Balance Equations

lilf

Figure 1.11. Convective flow terms for a well-mixed tank bioreactor.

C. Diffusion of Components As shown in Fig. 1.12, diffusional flow contributions can be expressed by analogy to Pick's Law for molecular diffusion Ji = -°i dZ

where jt is the flux of any component i flowing across an interface (kmol/m2 h or kg/m2 h) and dQ/dZ (kmol/m) is the concentration gradient as shown in Fig. 1.12.

Figure 1.12. Diffusion flux j j driven by concentration gradient (Qo - Cji) / AZ through surface area A.

In accordance with Pick's Law, diffusive flow always occurs in the direction of decreasing concentration and at a rate proportional to the concentration gradient. Under true conditions of molecular diffusion, the constant of proportionality is equal to the molecular diffusivity for the system, Dj (m2/h). For other cases, such as diffusion in porous matrices and turbulent diffusion, an

30

1 Modelling Principles

effective diffusivity value is used, which must be determined experimentally. The concentration gradient may have to be approximated in finite difference terms (Finite differencing techniques are described in more detail in Sec. 6.2). Calculating the mass rate requires the area, through which diffusive transfer occurs. ( Massrate of (^component i

( Diffusivity YConcentrationY Area ^ ( of gradient perpendicular =-DJ (^component ij\^ ofi J^ to transport J

kg sm 2

2 m

_ m 2 kg 2 _ kg " s m4 m " T

D. Interphase Transport Interphase mass transport also represents a possible flow into or out of the system. In bioreactor modelling applications, this is most frequently represented by the case of oxygen transfer from air to the liquid medium, followed by oxygen taken up by the cells during respiration. In this case, the transfer of oxygen occurs across the gas liquid interface, which exists between the surface of the air bubbles and the surrounding liquid medium, as shown in Fig. 1.13.

Figure 1.13. Transfer of oxygen across a gas-liquid interface of specific area "a" into a liquid phase of volume V.

Other applications may involve the supply of oxygen to the bioreactor by transfer from the air, across a membrane and then into the bulk liquid. Where there is interfacial transfer from one phase to another, the component balance equations will need appropriate modification to take this into account. Thus, an oxygen balance for the well-mixed gas phase, with transfer from the gas to the liquid, can be written as,

31

1.3 Formulation of Balance Equations

/

Rate of \ Rate \ /Mass flow\ /Massflow\ / of interfacial accumulation of the of the of the mass transfer oxygen oxygen = - from the gas mass of oxygen into the from the in the gas phase into Vgas phase ) ^ gas phase> V the liquid s V phase system J This form of transfer rate equation will be examined in much more detail in Chapter 5. Suffice it to say here that the rate of transfer can be expressed in the form shown below: Rate Of \ ( Mass ^ ^Area peA /Concentration^ /SystemA Uass transferJ= tra"sP°rtf ^ volume ) ^driving force ) VvolumeJ V coefficient. f

= KaACV where, a is a specific area for mass transfer, A/V (m2/m3), A is the total interfacial area for mass transfer (m2), V is the liquid phase volume (m3), AC is the concentration driving force (kmol/m3 or kg/m3) and, K is the overall mass transfer coefficient (1/s). Mass transfer rate expressions are usually expressed in terms of kmol/s, and can be converted to mass flows (kg/s), if desired. The units of the terms in the equation (with appropriate mass quantity units) are: kg 1 kg mj Production Rate This term in the component balance equation allows for the production or consumption of material bv by reaction and is incorporated into the component balance equation. Thus, Rate of >\ accumulation of mass of component the system )

/ Massflow\ of the component into \the system /

/ Massflow\ of the component out of the system /

/

Rate of production of the component by v the reaction /

Chemical production rates are often expressed on a molar basis and, as in the case of the interfacial mass transfer rate expressions, can be easily converted to mass flow quantities (kg/s). The production rate can then be expressed as

32

1 Modelling Principles

f Mass rate \ production of ^component Ay

/^Reaction rate\ = rA V = ^ per volume ) (Volume of system)

kg

__

kg m3 s m3

Equivalent molar quantities may also be used. The quantity r^ is positive when A is formed as product, and TA is negative when a reactant A is consumed. The growth rate for cells can be expressed in the same manner, using the symbol rx- Thus, / Mass rate of ^ Vbiomass production^

/^Growth rate^ = rx V = ^per vo lume) (Volume of system) kg

_

kg mj s m3

The consumption rate of substrate, r$, is often directly related to the cell growth rate by means of a constant yield coefficient YX/S, which has the units of kg biomass produced per kg substrate consumed. Thus, ( Ma<» rate \ U>nsumptionJ

=

kg s m3

V.

growth rate V 1 \ er M> volume ABiomass-substrate yield) (Volume)

m

~

=

kg biomass kg substrate kg biomass s m3

Introduce other relationships and balances such that the number of equations equals the number of dependent variables

The system mass balance equations are often the most important elements of any modelling exercise, but are themselves rarely sufficient to completely formulate the model. Other relationships are therefore needed to supplement the material balance relations, both to complete the model in terms of other important aspects of behavior and to satisfy the mathematical rigor of the modelling, such that the number of unknown variables must be equal to the number of defining equations.

1.3 Formulation of Balance Equations

33

Examples of this type of relationships which are not based on balances, but which nevertheless form an important part of any model are:

-

Reaction rates as functions of concentration, temperature, pH Stoichiometric or yield relationships for reaction rates Ideal gas law behavior Physical property correlations as functions of concentration Pressure variations as a function of flow rate Dynamics of measurement instruments as a function of the instrument response time Equilibrium relationships (e.g., Henry's law) Controller equations Correlations of mass transfer coefficient, gas holdup volume, and interfacial area, as functions of system physical properties and degree of agitation or flow velocity

How these and other relationships are incorporated within the development of particular modelling instances are shown later in the cases given throughout the text and in the simulation examples.

VI. For additional insight with complex problems, draw an information flow diagram Information flow diagrams can be useful in understanding complex interactions (Franks, 1966). They help to identify missing relationships and provide a graphical aid to a full understanding of the interactive nature of system. An example is given in the simulation example BATFERM.

1.3.3

Total Mass Balances

In this section the application of the total mass balance principles will be presented. Consider some arbitrary balance region, as shown in Fig. 1.14 by the shaded area. Mass accumulates within the system at a rate dM/dt, owing to the competing effects of a convective flow input (mass flow rate in) and an output stream (mass flow rate out).

34

1 Modelling Principles Mass flow rate out Mass flow rate In

Figure 1.14. Balancing the total mass of an arbitrary system. The total mass balance is expressed by,

=

dM

( Mass flow \ fMass flow out\ U .he system) - Uhe system J

= Mass rate in - Mass rate out

or in terms of volumetric flow rates, F, densities, (p), and volume, V d(p V) system

3t

= F

oPo-FiPi

When densities are equal, as in the case of water flowing in and out of a tank, dV

dT = F O-FI

The steady-state condition of constant volume in the tank (dV/dt = 0) occurs when the volumetric flow in, FQ, is exactly balanced by the volumetric flow out, FI. Total mass balances therefore are mostly important for those bioreactor modelling situations in which volumes are subject to change.

1.3.4

Component Balances for Reacting Systems

Each chemical species can be described with a component balance around an arbitrary, well-mixed, balance region, as shown in Fig. 1.15.

35

1.3 Formulation of Balance Equations Species i outflow

Species i inflow

Figure 1.15. Component balancing for species i.

Thus for any species i, involved in the system, the component mass balance is given by: /

Rate of \ accumulation of mass of component i

i in tnp cvct<=»m

=

i

'Mass flow of^ component i into v the system ^

/^MassflowoA f Rate of N component i production of out of component! ^ the system ) \ by reaction

}

Expressed in terms of volume, volumetric flow rate and concentration, this is equivalent to: ~ = (F 0 C i0 )-(F 1 C il )+(r i V) with units of mass/time:

,3*6. m~

_

m^

m3 m-

1.3.4.1

Case A.

ill 3

_ —

Constant Volume Continuous Stirred Tank Reactor

A constant volume, continuous, tank reactor with reaction A —> 2B is considered here, as shown in Fig. 1.16.

36

1 Modelling Principles

FQ

C AO

F

1 C A 1 C B1

C BO

>f '.; '•.??•: 5: ' ' :•; f'.<$ 'XI •' ;:?-'?:;':- ''M

I!

i l l ilili

;|i

Figure 1.16. Continuous stirred tank reactor with reaction A —> 2B.

Component A is converted to component B in a 1 to 2 molar ratio. The component balances for A and B are: d(VC A1 ) dt

= F0 CAO -

d(VC B i) dt

=

F0CBo -

Here it is convenient to use molar masses, such that each term has the units of kmol/h. Under constant volume conditions: d(VCA) = VdC A d(VCB) = VdC B and in addition FQ = FI. Thus the two model equations, then simplify to give: dCAi

=

F

V

- CAI) +

and

dCBi ~dT~

=

F V ( c BO~C B i)

In these two balances there are four unknowns CAI, Q*i, rAl an(^ rB l 8 kinetics are assumed to be first order, as often found in biological systems at low concentration. Then: r Al = According to the molar stoichiometry,

37

1.3 Formulation of Balance Equations

rfil = -2r M = + 2 k C A ! Together with the kinetic relations there are 4 equations and 4 unknowns, thus satisfying the conditions necessary for the model solution. With the initial conditions, CAI and CBI at time t = 0, specified, the solution to these two simultaneous equations, combined with the two kinetic relations, will give the resulting changes of concentrations CAI and CBI as functions of time. The simulation example ENZCON, is similar to the situation of Case A.

1.3.4.2

Case B.

Semi-continuous Reactor with Volume Change

The chemical reaction and reaction rate data are the same as in the preceding example, but now the reactor has no effluent stream. The operation of the reactor is therefore semi-continuous. AO

*

2B

Figure 1.17. A semi-continuous reactor example.

The kinetics are as before: t ^ rA = -kC A

moles —T— m s

In terms of moles the stoichiometry gives, rB = - 2 rA = + 2 k CA

The component balances with no flow of material leaving the reactor are now:

= FC AO + r A V d(V C ) —at— = IB v B

38

1 Modelling Principles

The number of unknowns is now five and the number of equations is four, so that an additional defining relationship is required for solution. Note that V must remain within the differential, because the volume of the reactor contents is now also a variable and must be determined by a total mass balance. Assuming constant density p, this gives the defining equation as: dt

= FF

With initial conditions for the initial molar quantities of A and B, (VGA, and the initial volume of the contents, V, at time t = 0 specified, the resulting system of equations can be solved to obtain the time varying quantities VCA(t), VCs(t), V(t) and hence also concentrations CA and CB as functions of time. Similar variable volume situations are found in examples FEDBAT, and VARVOL.

1.3.4.3 Case C.

Steady-State Oxygen Balancing in Fermentation

Calculation of the oxygen uptake rate, OUR, by means of a steady-state oxygen balance is an important application of component balancing for fermentation. In the reactor of Fig. 1.1.8, the entering air stream flow rate, oxygen concentration, temperature and pressure conditions are shown by the subscript 0 and the exit conditions by the subscript 1. Gas

>

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 Flowrate^i ( Flowrate"\ /^Rate of O2 uptake A ( ulationofO 2 J~ [ofO 2 in J~(ofO 2 out J~ I by the cells J

1.3 Formulation of Balance Equations

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

1 Modelling Principles

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+ <± C3H6O3 + NAD+ Pyruvic Acid Lactic Acid

1.3 Formulation of Balance Equations

41

This relation indicates that 1 mol of pyruvic acid reacts with 1 mol of NADH to produce 1 mol of lactic acid. Another example of stoichiometry is that of the oxidative decarboxylation of pyruvic acid to yield acetyl-CoA C3H4O3 + CoA-SH + NAD+ -> 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 Modelling Principles

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

1.3 Formulation of Balance Equations

04

Substrate H

b2 Oc2 Nd2

N Source

C34 Hb4 Oc4 Nd4

Figure 1.19. Flow inputs into a system.

The system is represented here in terms of the various flow inputs, where f is the corresponding flow vector () = <]>

<1>

O

O

O

<E>

<5

The steady state balance for the system is then represented by: <|) • E = 0 where E is the elemental composition matrix

E = a 4 b4 c 4 d 4 O4 0 0 2 0 O5 1 0 2 0 O6 0 2 1 0
The combination of 7 unknown quantities and 4 elemental balance equations) leave 3 quantities are independent. Thus assuming fluxes <E>1 (biomass), O2 (substrate) and <J>3 (product) are known, the unknown fluxes 04, 05, Og and 7 can be obtained by methods of linear algebra and which are detailed by Roels (1983).

44

1 Modelling Principles

In more complex cases with growth and product formation, more information is needed. The introduction of the concept of the degree of reduction is useful. For organic compounds this is defined as the number of equivalent available electrons per gram atom C, that would be transferred to CC>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, f^O 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:

1.3 Formulation of Balance Equations

..

* 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

1 Modelling Principles

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.

1.3 Formulation of Balance Equations

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

1 Modelling Principles

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

1.3 Formulation of Balance Equations

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 ^Energy ,

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 ^agitation ^

50

1 Modelling Principles

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).

1.3 Formulation of Balance Equations

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

1 Modelling Principles

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.

1.3 Formulation of Balance Equations

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 <* "| accumulation V of substrate J

(

=

( Substrate \ ( Substrate >| (f^d [n) _ consumption \ rate )

Under controlled conditions, in which the substrate concentration is maintained constant or kept small, the accumulation term in the above equation will also be small, with the result that the feed rate of substrate into the reactor will balance the rate of consumption by reaction.

60

2 Basic Bioreactor Concepts

One other balance equation, however, is also necessary, i.e. the total mass balance, f Rate of accumulation of ^ V mass in the reactor /

=

( Mass flow rate of feed ^ V to the reactor )

which for constant density conditions reduces to (Rate of change of volume) = (Volumetric rate of feeding) Further extensions of fed batch operation are possible, such as the cyclic or repeated fed batch, which involves changing volume with a filling and emptying period. The changing reactor concentrations repeat themselves with each cycle. This operation has similarities with continuous operation and approaches most closely to continuous operation, when the amount withdrawn is small and the cycle time is short. The simulation examples FEDBAT, Sec. 8.1.3 and in Sec. 8.3 (VARVOL, PENFERM, PENOXY, ETHFERM, REPFED) allow detailed investigations of fed batch performance to be made on the computer.

2.2.3 Continuous Operation In continuous operation fresh medium is added continuously to the bioreactor, while at the same time depleted medium is continuously removed. The rates of addition and removal are such that the volume of the reactor contents is maintained constant. The depleted material, of course, contains any products that have been excreted by the cells and, in the case of suspended-cell culture, also contains effluent cells from the bioreactor. Continuous reactors are of two main types, as indicated in Fig. 2.5, and these may be considered either as discrete stages, as in the continuous, stirred-tank bioreactor, or as differential devices, as represented by the continuous tubular or column reactor.

Continuous tank bioreactor

Continuous tubular bioreactor

Figure 2.5. The two main types of continuous reactors.

61

2.2 Bioreactor Operation

As shown later, these two differing forms of continuous reactor operation have quite different operational characteristics. Both however are characterized by the fact that after a short transient period, during which conditions within the bioreactor change with time, the bioreactor will then achieve a steady state. This means that operating conditions, both within the bioreactor and at the bioreactor outlet, then remain constant, as shown in Fig. 2.6. Concentration

Startup period

Steady state

time Figure 2.6. Startup of a continuous reactor.

Continuous reactors, however, have found little use as biological reactors on a production scale, although there are a few important examples (Id's single-cell protein air lift process, wastewater treatment and the isomerization of corn sugar to fructose syrup). Frequent use is made of continuous reactors in the laboratory for studying the kinetics of organism growth and for enzyme reaction kinetics. This is because the resulting form of the balance equation, leads to an easy method for the determination of reaction rate, as discussed in Ch. 4. The behavior of the two differing forms of continuous reactor, are best characterized by their typical concentration profiles, as shown in Fig. 2.7. In this case, S is the concentration of any given reactant consumed, and P is the concentration of any given product. So

Tank

So

Tube

Cone.

Cone.

distance

distance

Figure 2.7. Profiles of substrate and product in steady state continuous tank and tubular reactors.

As seen, the concentrations in a perfectly mixed tank are uniform, throughout the whole of the reaction vessel contents and are therefore identical to the concentration of the effluent stream. In a tubular reactor the reactant concentration varies continuously, falling from a high value at the inlet to the

62

2 Basic Bioreactor Concepts

lowest concentration at the reactor outlet. The product concentration rises from inlet to outlet. These differences arise because in the tank reactor the entering feed is continuously being mixed with the reactor bulk contents and therefore being diluted by the tank contents. The feed to the tubular reactor, however, is not subject to mixing and is transformed only by reaction, as material moves down the reactor. No real situation will exactly correspond to the above idealized cases of perfect mixing or zero mixing (plug flow), although the actual behavior of tanks and tubes tends in the limit towards the corresponding idealized model. The characteristics of continuous operation are as follows: 1. 2. 3. 4. 5.

Steady state after an initial start-up period (usually) No variation of concentrations with time Constant reaction rate Ease of balancing to determine kinetics No down-time for cleaning, filling, etc.

The balance equations at steady state for a well-mixed tank reactor have the form 0 = (Input) - (Output) + (Production) since at steady-state the rate of accumulation and therefore the rate of change is zero. This equation predicts that the reaction rate causes a depletion of substrate from the feed condition to the outlet, (the product will increase) and that the rate of production can be obtained from this simple balance: (Rate of production) = (Rate of output) - (Rate of input) For a non well-mixed reactor such as a tubular or column reactor, steady-state implies the same non-transient conditions, but now concentrations also vary with position. The same situation also applies to the case of a series of wellmixed tanks. The balance form is then: 0 = (Rate of input) - (Rate of output) + (Overall Rate of Production) Here the overall rate of reaction is obtained by summing or integrating over every part of the reactor volume. The concentration characteristics of a tubular reactor, as shown in Fig. 2.7, are well approximated by a series of tank reactors. Referring to Fig. 2.8, and moving downstream along the reactor cascade, the substrate concentration decreases stepwise from tank to tank, while the product concentration increases in a similar stepwise manner. As the number of tanks in the cascade increases, so the performance becomes more and more similar to that of a tubular reactor. In the case of a reaction, whose rate of reaction increases with increasing

63

2.2 Bioreactor Operation

substrate concentration S, the multiple tank configuration or a tubular reactor would thus have a kinetic advantage over that of a single tank. The same is true, in the case of product inhibition kinetics, in which the rate would be lowered by high product concentration, P. Substrate inhibition systems would be run preferably in single tanks, however, since then the substrate concentration is always at its lowest value. Cone.

distance Figure 2.8. Stirred tanks in series and their concentration profiles.

A calculation of the tank volume or residence time requirement involves the formulation of the tank balance equations, as before and then the application of the equations, successively from tank to tank such that the effluent from the preceding tank is the feed of the next and so on. Tanks-in-series bioreactor operations are illustrated by the simulation examples TWOSTAGE, STAGED and DEACTENZ in Sec. 8.4.

2.2.4 Summary and Comparison The operating characteristics of the various reactor modes are summarized in Table 2.1. The important bioreactor operating parameters will depend on the mode of operation. In batch operation, concentration levels can be varied by adjustment of the initial values, whereas in continuous and semi-continuous operation, the concentration levels depend on the feed rate and feed concentration. As indicated previously, the manner in which the bioreactor is operated can therefore give rise to different concentration levels and therefore differing productivities. The consequent concentration profiles depend, of course, on the reaction kinetics, which express the rate of reaction as a function of the concentrations of reactants and products.

64

2 Basic Bioreactor Concepts

Table 2.1. Summary of reactor modes. Mode of operation Advantages

Disadvantages

Batch

Equipment simple. Suitable Downtime for loading and for small production. cleaning. Reaction conditions change with time.

Continuous

Provides high production. Better product quality due to constant conditions. Good for kinetic studies.

Requires flow control. Culture may be unstable over long periods.

Fed batch

Control of environmental conditions, e.g. substrate concentration.

Requires feeding strategy to obtain desired concentrations.

Table 2.2 lists the main operating parameters for the three differing modes of bioreactor operation. Table 2.2. Operating variables for batch and continuous bioreactors. Batch Continuous Semicontinuous Initial medium composition Inlet medium and inoculum composition

Feed and initial substrate composition

Temperature, pressure

Temperature, pressure Temperature, pressure

pH if controlled

pH if controlled

pH if controlled

Reaction time

Liquid flow rate (residence time)

Liquid flow rate (residence time)

Aeration rate

Feeding rate and control program

Aeration rate Stirring rate Stirring rate Aeration rate Stirring rate

65

2.2 Bioreactor Operation

The foregoing discussion of the varying characteristics of the different reactor types and their concentration profiles allows a qualitative comparison of the volume requirements for the different types of reaction, according to the particular kinetics. For this it is first necessary to consider the qualitative nature of the basic forms of kinetic relationship: zero order, first order, product and substrate inhibition. The detailed quantitative treatment of these kinetic forms is dealt with in Ch. 3. The rate of a zero order reaction is independent of concentration. Many bioreactions at high substrate concentrations follow zero order kinetics and are therefore insensitive to concentration and to the effects of concentration gradients. From the kinetic viewpoint, therefore, any reactor type would be equally suitable. First order reaction rates are directly proportional to concentration. Bioreactions at low concentration are generally first order, and this would favor operation in either a batch or a tubular/column type reactor. This is because reactant concentrations in such reactors are generally high overall and hence the overall rates of reaction are also consequently high. Hence the reactor volume required for a given duty would generally be small. (In the case of a batch reactor, this of course neglects the time lost for filling, emptying and cleaning.) A reaction with substrate inhibition would be best run in a tank at low substrate concentration, since the concentration would be low throughout the whole of the tank contents. Conversely, product inhibition would be more pronounced in tank reactors, since product concentration would be at its highest. In this case, a tubular type reactor or batch reactor would be preferred. Table 2.3. Kinetic considerations for reactor choice. Continuous Reaction Batch Tank Tanks-inContinuous Kinetics Series or Single Tank Tubular Zero order OK OK OK First order

Best

Best

Substrate inhibition

Low initial concentration

Low tank concentrations

Product inhibition

Best

Best

Production triggered by shift in environment

OK for temp- Possible erature-shift

Low conversion only Best

Fed Batch Low conversion OK

Best

Low conversion only

Low conversion only

Not suitable

Best for concentrationshift

66

2 Basic Bioreactor Concepts

Table 2.3 compares the performance of batch tanks, continuous tubular or tanks-in-series reactors and single continuous tank reactors. As discussed in Sec. 4.2.1, batch tank concentration-time profiles are exactly analogous to the steady state concentration-distance profiles obtained in continuous tubular reactors. In terms of performance, therefore, the batch reactor would be the same as a tube, when compared on the basis of equal batch time in the tank and time of passage through the tube. Tanks-in-series reactors, as shown in Fig. 2.8, involve step wise gradients, which in the limit are very similar to those of continuous tubular reactors, hence, making their performance similar to that of a tubular reactor. Owing to the high degree of mixing which leads to a uniform concentration, the performance of the single continuous stirred tank reactor is very different to that of the other reactor types. An exact quantitative comparison can be made using the mass balance equations developed in Ch. 4 for each reactor type.

Biological Kinetics

As explained in Sec. 2.1, a realistic bioprocess model will usually require the input of kinetic rate data. In the case of even simple chemical reactions, this data has to be obtained by laboratory experiment. Since biochemical reactions are controlled by enzymes, it is appropriate to start with a consideration of simple enzyme kinetics (Sec. 3.1), In the case of modeling the behavior of enzyme reactors, knowledge of the enzyme reaction kinetics is most important. The sheer complexity of the biological reactions, occurring in a living cell, seem to imply an almost impossible task in obtaining meaningful rate data for biological modelling applications. Fortunately this is not the case and, as shown in Section 3.2, a quite reasonable overall description of cell growth rate data is possible, based on an overall empirical relationship, the Monod Equation, which has been found to give a good fit to many general observations of cell growth. This overall view, based on the net result of many simultaneously occurring and highly interacting biochemical reactions, of course represents an incredible oversimplification of the actual situation. Fortunately it seems to work in many instances and can also be easily modified to allow for the uptake of substrate by the cells and to include such additional effects, as substrate limitation, multiple substrate limitation and product inhibition. It is interesting, that the basic enzyme rate equation, or MichaelisMenten equation, based on the theory indicated in Sec. 3.1, is of the same basic form as the empirically-based Monod equation for the growth of microorganisms. When used in this manner, the cell kinetics are completely devoid of any mechanistic interpretation and constitute what is known as an unstructured kinetic model (Fig.3.1 A). In other cases, it may be necessary to look in some detail at individual cell processes and reactions, in order to obtain a more realistic description, thus leading on to the use of structured kinetic models (Fig. 3.IB) as described in Sec. 3.3. In a most simple case, the cells are composed of a catalytic part comprising proteins, RNA, DNA and other cellular compounds and of a storage part, e.g. poly-hydroxy-alkanoic acids (PHAs) or inclusion bodies of recombinant proteins. A most simple segregated model considers different stages of cells and therefore a distribution of cell stages in a culture without structuring the cell composition (Fig. 3.1C). In the most realistic, but most complex situation the model is structured and segregated 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

68

3 Biological Kinetics

(Fig. 3.ID). For the purpose of this book, the differences of these models can be best described by their different balance regions. Non-structured

Structured

B

3 CO

t o

£

8

Figure 3.1. Types of kinetic models for cells. Balance regions: A Total cell biomass, B Cell parts, C biomass parts, D Biomass and cell parts.

3.1 3.1.1

Enzyme Kinetics Michaelis-Menten Equation

The rate of reaction catalyzed by a soluble enzyme can usually be described by the Michaelis-Menten kinetic equation. This equation can be derived from the accepted Briggs-Haldane mechanism for a simple enzyme reaction, which is very similar to that for conventional chemical heterogeneous catalysis. Thus,

3.1 Enzyme Kinetics

69

>

S 4- E

<

ES

ES

£-» P + E

where E is enzyme, S is substrate, P is product and ES is an enzyme-substrate complex. For a batch reaction, the balances for S and ES are written in terms of the mechanism as, JQ

— = - ki S E + k_i (ES) dt

dt

= ki S E - (k_i + ki) (ES)

with initial conditions at t = 0:

(ES) = 0

S = S0

The concentration changes for a batch reactor are shown qualitatively in Fig. 3.2. While the enzyme concentration is usually much lower than that of the substrate, most of the enzyme is present during the reaction in the form of the enzyme-substrate complex, ES. Analytical solution is then possible by assuming a quasi-steady state for the enzyme-substrate complex, ES, d(ES) -= 0 dt This assumption is valid for E « SQ. Using the total enzyme mass balance, EO = E + (ES)

the above equations can be solved for the unknown concentrations E and ES to give,

and ki S E 0 ~ k_i .+ k2

E = E

k.i + k2 ° k_i + k2

= E

70

3 Biological Kinetics

o

1 Q) O

8

-^

time

Figure 3.2. Concentration changes of the reaction species for a simple enzymatic reaction taking place in a batch reactor.

Substituting for E and ES the substrate balance becomes,

dS _ dt ~

k2SEo

giving the Michaelis-Menten equation,

where the parameters in terms of the mechanistic model are for the maximum reaction rate (kmol/m3min): v max = k 2 E 0

and for the Michaelis-Menten constant (kmol/m3): KM -

k-i + k2 ki

The Michaelis-Menten equation exhibits three distinct regions for the reaction rate. At very low and very high substrate concentrations the rs versus S curve is essentially linear, as seen in Fig. 3.3.

71

3.1 Enzyme Kinetics

vm

rS Michaehs-Menten region vm/2""

0

KM

10 KM

5 KM

15 KM

Figure 3.3. Reaction rate versus substrate concentration for the Michaelis-Menten equation.

The low concentration region can be approximated by first-order kinetics. The Michaelis-Menten equation becomes for S « KM, _ VmaxS

For high substrate concentration (S » KM) the relation approaches zero-order, r

S = v max

and the rate of reaction is thus independent of substrate concentration and is constant at the maximum value. In the intermediate substrate concentration range, 0.1 KM < S < 10 KM, the full Michaelis-Menten equation must be used to guarantee an accuracy for rs greater than 10 %. The parameters vmax and KM can be determined from experimental data, either graphically following a linearization of the MichaelisMenten equation or, better, numerically by using nonlinear parameter estimation techniques. Graphical determination of KM and vmax is based on rearrangement of the Michaelis-Menten equation into a linear form,

Inversion and rearrangement give,

72

3 Biological Kinetics

1 rs

KM I v

max S

max

A graphical representation of this equation is called the Lineweaver-Burk diagram (Fig. 3.3) from which the kinetic parameters vmax and KM may be determined.

Figure 3.3. Lineweaver-Burk double-reciprocal plot.

Typical values of the enzyme kinetic constants are given in Table 3.1. It is interesting to note that the rate of formation of the enzyme-substrate complex can be extremely fast, with the constant ki approaching 1 x 1010 L/mol s. This is the maximum value for a rate constant of a reaction that is limited by diffusion of a small substrate molecule in aqueous solution. Table 3.1. Typical values of the constants of the Michaelis-Menten equation. Constant

Value range

KM

105 - 109 L/mol s 10- 104 1/s 1 - 106 1/s 10-6 „ 10-l mol/L

3.1 Enzyme Kinetics

73

The simulation example MMKINET enables a computer study of the basic characteristics of the Michaelis-Menten equation to be carried out, and LINEWEAV simulates the study of the Lineweaver-Burk plot for a batch enzyme reaction.

3.1.2

Other Enzyme Kinetic Models

The reaction mechanism of enzyme catalysis can be very complex, resulting in complicated kinetic equations that are treated in specialized textbooks, as given in the reference section. Some of the more readily used forms of the modified Michaelis-Menten kinetics are presented here. Double Michaelis-Menten Kinetics This refers to the case when two substrates are involved in the reaction: v

r

max Si 82

_

s - (KMi + Si) (KM2 + S2)

Inhibition Inhibition occurs when a substance, inhibitor (I), reduces the rate of an enzyme-catalyzed reaction, usually by the inhibitor binding to the enzyme active site. Three simple types of reversible inhibition kinetics are given in Tab. 3.2. Table 3.2. Enzymatic inhibition kinetics. Mechanism

Inhibition

Rate equation, rs v

I

maxS

competitive

KM (1 + I/KI) + S

E+S«ES-»P c c El «• ESI

non-competitive

(i + I/KI) (KM + S)

ES +1 «=> ESI

uncompetitive

vmax

74

3 Biological Kinetics

Usually, the substance I is the substrate or the product, and the reaction kinetics are known as substrate or product inhibition, respectively. Allosteric Kinetics A simple model to describe allosteric inhibition is given, in which the enzyme can bind to more than one substrate molecule. Thus: nS + E

z

ESn

)

nP + E

when n is the number of substrate molecules. The resulting kinetic expression is referred to as Hill kinetics, Vmax S n s

" K M n + Sn

As shown in Fig. 3.5, for values of n > 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

3 Biological Kinetics

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

3 Biological Kinetics

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^.

3.2 Simple Microbial Kinetics

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

3 Biological Kinetics

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

3.2 Simple Microbial Kinetics

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

3 Biological Kinetics

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.

3.2 Simple Microbial Kinetics

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

3 Biological Kinetics

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.

3.2 Simple Microbial Kinetics

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

3 Biological Kinetics

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

3.2 Simple Microbial Kinetics

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

3 Biological Kinetics

(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

3 Biological Kinetics

— -

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

3.3 Structured Kinetic Models

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

3 Biological Kinetics

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

3.3 Structured Kinetic Models

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

3 Biological Kinetics

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 Structured Kinetic Models

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

3 Biological Kinetics

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

3.3 Structured Kinetic Models

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~

=

( ^Ac/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

3 Biological Kinetics

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.

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

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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.

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4 Bioreactor Modelling

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

4.1 General Balances for Tank-Type Biological Reactors

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.

4.1 General Balances for Tank-Type Biological Reactors

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.

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4 Bioreactor Modelling

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.

4.1 General Balances for Tank-Type Biological Reactors

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

4 Bioreactor Modelling

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

4.1 General Balances for Tank-Type Biological Reactors

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

4 Bioreactor Modelling

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

4 Bioreactor Modelling

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

4 Bioreactor Modelling

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 K^a 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

5.3 General Oxygen Balances for Gas-Liquid Transfer

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); K^a (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, K^a, 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

5.3 General Oxygen Balances for Gas-Liquid Transfer

The classical dynamic KLa method assumes that K^a 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 ^a 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

5.3 General Oxygen Balances for Gas-Liquid Transfer

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

5.3 General Oxygen Balances for Gas-Liquid Transfer

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 K^a by parameter estimation (Shioya et al., 1978). Deoxygenated C liquid LO

t

/•"V—X

"N^

-^^^

r-^—^-~\_/~\^^x

nM&^/m^M:WM.

Xiiiiiiiiiiji ^^^SRSSSS

^^^iO^wiBii O

Wiiiiilmiiiiiii Air

Figure 5.12. Steady-state dissolved oxygen difference measurement for Kj^a.

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 K^a 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.

5.3 General Oxygen Balances for Gas-Liquid Transfer

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.

5.3 General Oxygen Balances for Gas-Liquid Transfer

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 Models for Oxygen Transfer in Large Scale Bioreactors

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<~Ln * - £1 »r c * C H r Gn or CLn

P

02

-77rl

and rn = Qo2m |

For simplification only oxygen is assumed limiting and growth is not considered; however the biomass concentration is contained in the maximum oxygen uptake Qo2m- The dynamics of the oxygen transfer and uptake processes are obtained by solving these differential equations simultaneously for each stage. The resulting solution then gives CLH and CGn > 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

5.4 Models for Oxygen Transfer in Large Scale Bioreactors

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-<

-CGI) -KLa(CLi -CLI)VL f\

\

-rr-

//~1

^

/"I

\A 7

where the plug flow nature of the gas is partially accounted for by

142

5 Mass Transfer

and

CLI

= - Qo2r

Stage 2: dCL2 VL -gf = (FL+ FB) CLI + FBCL3 - (FL+ FB) CL2 - FB CL2 K L a(C L 2*-C L 2)V L

= FG (CGI - CG2) - KLa (CL2* - CL2)VL where

CL2. . CL2 n r2 = -Qo2mK 0 + CL2

and

Stage 3: dCL3

K L a(C L 3*-CL3)V L + r 3 V L V

- CG3) - KLa (CL3*- CL3) VL

G

where

and

CL3. . CL3 = -QO2n K0 + CL3

The above equations describe the dynamic oxygen concentrations in the multiimpeller continuous bioreactor. Note that the liquid phase balances for the two end stages 1 and 3 differ from that of the intermediate stage 2, owing to the

5.4 Models for Oxygen Transfer in Large Scale Bioreactors

143

absence of any backmixing flow contribution exterior to the column. A batch reactor would be described by setting the liquid flow, FL, equal to zero. Since the biomass balance and growth kinetics are not included here, the solution would be valid at only one time during the fermentation, corresponding to the assumed value of Qo2m> which is proportional to the value of X existing at that time. Variations in X are, however, easily incorporated into the model by adding cell and substrate balance equations.

Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

The retention and immobilization of enzymes and cells usually requires the presence of an additional solid carrier phase or flocculant cell mass. As illustrated in Fig 6.1, in order to reach a reaction site, substrate S must first be transported by convection from the bulk liquid to the exterior stagnant film (point A). Then transport by diffusion must occur through the film (from A to B) to the surface of the carrier (point B), where surface reaction can take place. If further reaction sites are available within the carrier matrix, an additional internal diffusion path (from B to C) is then also required. Similarly product P, formed within the carrier matrix, must diffuse out of the matrix towards the surface, and then away from the surface via the external mass transfer laminar film to the bulk liquid. Concentration

Diffusion film

i

Bulk liquid

Solid carrier

A

B

B

Figure 6.1. Concentration profiles for a biocatalyst immobilized on a solid carrier.

The stagnant film and the immobilization matrix constitute mass transfer resistances which may slow the overall reaction rate, since reaction cannot proceed at a rate greater than the rate at which substrate is supplied by the mechanism of diffusion. The diffusional mass transfer process via the external film is referred to as external mass transfer. Since the reaction site may often be located within a gel, a porous solid, biofilm or biofloc, the transfer of substrate or substrates from the exterior surface of the biocatalyst to reaction sites, located within the internal structure of the carrier, is also usually necessary. This process is therefore referred to as internal mass transfer or intraparticle transfer. In what follows, external transport and internal transport 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

146

6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

are considered separately, although, of course, the two effects can exert a combined effect in reducing the effective reaction rate, compared to that which would be obtained if there were no diffusional limitations.

6.1

External Mass Transfer

Fig. 6.2 illustrates the substrate concentration profile, existing in the very near region of an immobilized biocatalyst surface, supported on a non-porous carrier. Also shown is the idealized concentration profile, as represented by the film theory. As previously discussed in Sec. 5.2, the "film theory" assumes the presence of a stagnant layer of liquid to exist at the solid-liquid interface. This stagnant region is termed the diffusion film or Nernst-diffusion film and constitutes the external resistance to mass transfer. It thus determines the rate of supply of substrate to the surface, for subsequent reaction.

Substrate cone.

SA A

Figure 6.2. External diffusion model of substrate transport to a reactive enzyme immobilized on a solid surface.

The rate of supply of substrate to the surface is defined by mass transfer considerations, such that the mass flux to the catalyst surface is given by, JS = ks L (SA-S B )

147

6.1 External Mass Transfer

where, js is the mass flux (mol/m2 s), ksL is the mass transfer coefficient (m/s) and SA, SB are the substrate concentrations for the bulk and surface conditions [mol/m3], respectively. The steady-state balance can be written for the transport-reaction process, (Rate of supply by diffusion)

= (Surface reaction rate)

k

SL (SA - SB) = ks SB = rapp

In the following treatment, the surface reaction is assumed to be first-order, such as found for a biocatalytic reaction with Michaelis-Menten kinetics and S « KM- The apparent reaction rate per unit surface area, r app (mol/m2 s), is equal to the rate of both processes. Solving the equation, for the surface concentration, SB,

SB =

-SA

and hence

= ksSB=U^ l k S+ k SL

Two extreme conditions can be identified: 1) For ks/ksL » 1, SB approaches zero, and the reaction is completely mass transfer controlled, with rapp = 2) For ks/ksL «.!» SB approaches SA, and the reaction is kinetically controlled, with an apparent rate equal to that defined by the reaction kinetics, with rapp = ks SAThe intermediate situation is given by the full equation, for which the apparent reaction rate is influenced by both the true kinetic rate constant ks and by the diffusional mass transfer coefficient ksLFor a zero-order reaction: k S L(S A -S B ) = ks where ks is now a zero-order kinetic rate constant. The concentration at the reaction surface SB is thus,

SB = S A -ks/k S L

148

6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

which indicates that the ratio of the magnitudes of the kinetic rate constant to that of the mass transfer coefficient determines SB- If the reaction is zeroorder, the overall order of reaction rate is not influenced by diffusional considerations, but the effective rate will still be reduced, owing to the lowered concentration SBFor Michaelis-Menten kinetics, which encompass the range between effective zero and first order reaction kinetics, the relation between rate of supply and the rate of reaction becomes, kSL (SA - SB) = : > - = r app After rearrangement, the resulting quadratic equation can be solved for SB, with the solution indicating that, in general, the magnitudes of all the coefficients can influence the overall reaction rate and also that the external transfer can change the overall observed reaction kinetics. Thus they no longer follow the Michaelis-Menten form with respect to bulk concentration, and the apparent kinetics can differ substantially from the intrinsic true reaction kinetics. Under these conditions, it is no longer correct to equate the Michaelis-Menten constant, KM, to the substrate concentration at which the observed reaction rate is equal to the half of the maximum observed rate. This can be most easily seen from the above equation; when SB is low, the effective surface rate reduces to the form, rapp = vm SB/KM- The overall reaction rate then becomes, = app

(v m /K M )S A (v m /K M k S L ) + l

showing that the apparent rate of reaction depends on the magnitude of the mass transfer coefficient It is only possible to measure true reaction kinetics, by operating experiments in a truly kinetic regime, such that any influence of the external diffusional mass transfer is negligible. This can be achieved by ensuring that the ratio of vm/ksL is sufficiently low. Under these conditions,

which are the intrinsic Michaelis-Menten kinetics. The ratio can be made low by increasing the mass transfer coefficient, k$L, and by increasing the mass transfer rate enhancing parameters, such as flow velocity and stirring speed. Conversely those factors affecting the maximum reaction rate, vm, should be decreased, for example enzyme loading and temperature. The regimes of possible external mass transfer influence on the observed kinetics are summarized in Table 6.1, together with the important parameters.

6.1 External Mass Transfer

149

Table 6.1. Characteristics of overall reaction rate influenced by external transfer. Regime of operation

Reaction parameter having an influence on overall rate

Transfer control

Temperature (slight influence due to viscosity and diffusion rate). Stirring speed in tank. Flow in packed and fluidized beds. S in bulk liquid.

Kinetics control

S in bulk liquid. Enzyme loading on surface. Temperature.

Intermediate regime

All of above.

6.2

Internal Diffusion and Reaction within Biocatalysts

Reactions with enzymes and whole cells entrapped or immobilized in a porous solid matrix will be subjected to a mass transfer influence. Example systems are whole cells immobilized in alginate, enzymes adsorbed on ion-exchange resins, or naturally occurring biological films on surfaces or flocculated biomass. In the case of a biological film attached to an impermeable solid, the substrate can enter from only one surface, as shown in Fig. 6.1, through the diffusion layer A-B and into the biocatalyst B-C. In the case of an alginate bead, a biofloc or its two-dimensional approximation, substrate can enter from opposing directions, as shown below in Fig 6.3. In this case, the diffusion will result in a symmetrical, steady state concentration profile. The case of complete penetration of substrate through the biofloc is shown by the solid line. Whereas an incomplete penetration, as shown by the dashed line, results in the center of the film being completely ineffective, in terms of reaction capability. Note that in this case, the effects of external diffusion are neglected. The uptake of substrates within solid material requires transport by a diffusional process. The driving force for diffusion is a gradient in concentration, and the diffusional flux is given by Pick's law,

150

6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

jA = -D A

dCA dZ

with j having units of kg/m2 s.

'AO 'AO

Diffusion

Diffusion

'AO

Figure 6.3. Internal concentration profiles in a symmetrical rectangular biocatalyst matrix.

If a reaction occurs within the matrix, a concentration gradient will be established as a result of the simultaneous diffusion and reaction processes. The reaction rate at each position, being usually a function of concentration, will vary, and the overall or apparent reaction rate per unit volume of matrix (kg/s m3), rapp, will be determined by the transfer rate at the surface (kg/s), /Apparent rate\ Vin bulk liquid)

=

/Rate of substrate^ /Net rate of reaction^ \entering matrix ) = v within matrix )

Vr a p p = (j|z=o)A = r a v g A L where the units of each term are kg/s. Here ravg represents an average value in the matrix, which will increase with higher internal substrate concentrations. Regarding the influence of diffusion for a particular situation, it is possible to arrive at some quantitative guidelines without considering any mathematical details. Obviously the concentration profiles are caused by a competition between reaction and diffusion. The ratio of the maximum intrinsic reaction rate (not influenced by transfer) to maximum diffusion rate provides a useful dimensionless parameter,

6.2 Internal Diffusion and Reaction within Biocatalyst r

max A L

r(C 0 ) A L

JA

D (Co/L) A

~

151

maximum reaction rate maximum diffusion rate

For first order reaction, r = k CQ, this dimensionless group becomes k L2/D, and for zero order reaction, r = k, it is k L2/D CQ. Therefore for any kinetic form of equation, the distance coordinate or length of diffusion path, L, plays an important role since the ratio of maximum diffusion rate to maximum reaction rate varies according to L2. The higher the value of this ratio, the greater in magnitude are the substrate gradients. With this qualitative feeling for diffusion-reaction phenomena, more quantitative aspects can be considered.

6.2.1

Derivation of Finite Difference Model for Diffusion-Reaction Systems

Diffusion with biological reaction can be treated by mathematical modelling, and from this it is possible to develop equations describing changes of concentration, with respect to both time and position. The same technique of finite differencing is used as in the modelling of the dynamic behavior of tubular bioreactors, Sec. 4.2.2. Consider the case where the substrate varies from a concentration SQ in the bulk liquid to some concentration, at the position L (a wall or the center of a symmetrical particle). At the center by symmetry or at a wall, owing to the absence of diffusion into the wall, the concentration gradient must be zero. The actual continuous concentration profile, through the slab, may be approximated by a series of increments, as indicated in Fig. 6.4 and by a series of biocatalyst matrix elements as shown in Fig. 6.5.

Liquid

Figure 6.4. Finite differencing a solid, showing concentration gradient approximation.

152

6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

In

jn-1

n-1

> ,

n

I n+1

n+1

Figure 6.5. Series of volume elements connected with diffusion fluxes.

A magnified view of element n is shown in Fig. 6.6, where the flux, jn, depends on the local concentration gradient, and the reaction rate, rsn, depends on the local concentration in element n.

Jn-1

AZ Figure 6.6. A single element n of volume AV and thickness AZ, showing the diffusion fluxes.

A component mass balance is written for each segment and for each component as /Accumulation^ V rate )

=

A AZ

/Diffusion^ /Diffusion\ / Production \ \ rate in ) ~ \ rate out ) + \rate by reaction/ dSn ~dT = Jn-1 A - jn A + rSn A AZ

Using Pick's law in the difference form,

(Sn-i ~ Sn) °s

and similarly for Jn gives, n-l ~ Sn)

+rSnAAZ

6.2 Internal Diffusion and Reaction within Biocatalyst

153

Dividing by A AZ, dSn T

(S n ,j -2S n + S n+ i) s

5

+ rsn

The equivalent partial differential equation is,

as

a2s

+ rs

The finite-differenced forms of the model equations, however, are especially suitable for simulation programming. Thus, N equations are obtained, one substrate balance equation for each element, and these are solved simultaneously. Note that the boundary conditions, for elements 1 and N, must be described separately. For the above case, the boundary conditions are dS/dZ = 0 at Z = L and S = SQ at Z = 0. Thus the equations for the first and last elements must be written accordingly, as shown in simulation example BIOFILM, Sec. 8.7.1. Note also that it would be also possible, in principle, to include external diffusion effects, by formulating a boundary condition, balance for the first element as: /Accumulation^ V rate /

=

/External trans- A /Diffusion^ f Production \ V port rate in ) ~ \ rate out ) + Vrate by reaction/

where the external transport rate through area A is, Q = k S L (So-Si)A where SR is the bulk reactor concentration.

Coupling the Biocatalyst Matrix to the Reactor Liquid. The biocatalyst diffusion model can be combined with a well-mixed tank model, as shown in Fig. 6.7. The bulk liquid-phase component balances take the form: ^= |(S F -SO)-Jsa| z = 0 where a=A/V, , A dS , Js a| z=o = - DS v dZ I z=o

where,

154

6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

dS dZ I z=o =

The direction of the positive diffusion is into the biofilm, since dS/dZ is negative and (SR - Si) /AZ is positive. Here SR corresponds to So in Fig. 6.4, and Si is the concentration in the first element. The boundary condition is S = SR at Z = 0.

F,SF

F,SO

Figure 6.7. Coupling the biofilm model to the continuous tank model.

In this way it is possible to simulate immobilized biocatalyst performance in a single tank or in a column by using a tanks-in-series model, The simulation example BIOFILM, Sec. 8.7.1, demonstrates this approach.

6.2.2

Dimensionless Parameters from DiffusionReaction Models

There are several advantages of formulating model equations in dimensionless form. The number of variables in the model is reduced, thus reducing the number of experiments or simulations required to investigate all combinations. It is also possible, on the basis of the numerical values of the parameters, to access the relative importance of certain terms. Finally, the dimensionless form makes the solution much more generalized because the units of the individual quantities are no longer important. The governing dimensionless parameters can be obtained by re-examining the defining model equations and arranging them such that the variables range

6.2 Internal Diffusion and Reaction within Biocatalyst

155

only between the values of zero and unity. Thus new dimensionless variables, S=S/S 0 , Z = Z/L and dimensionless time, f = t / ( L I D ) , can be defined. Substituting these new variables into the diffusion-reaction, partial differential equation, for the case of a first order biochemical reaction, gives,

as S

2

° (L /D8)at

=

as

_

or _

at - az2 " DS

_ s

Thus the solution depends only on the value of [lq L2/DsL which is a dimensionless diffusion-reaction parameter. For zero-order reaction the equation becomes,

as a2s2 at " az where ko L2/Ds SQ is the governing parameter. It is seen that the dimensionless parameters in the model have the same form and significance as was derived from the qualitative reasoning presented earlier. For heterogeneous reaction systems this dimensionless group is known as the Damkohler Number, and its square root is called the Thiele Modulus. In the above equations, all the terms, excepting that of the reaction term, have dimensionless parameters of unity. On this basis, it can be said that if the reaction parameter for a first order reaction, [ki L2/DsL has a value of 1.0 or greater, then the reaction will have a large effect on the solution, that is, on the concentration gradients. Similarly, ko L2/D$ SQ will govern a zero order reaction. Such "order of magnitude analysis" is important for physical understanding and also to obtain information from differential equations without having to actually develop an analytical solution. Dimensionless formulation of equations is also explained in the simulation example VARVOL, Sec. 8.3.1, and KLADYN, Sec. 8.5.5.

6.2.3

The Effectiveness Factor Concept

The relative influence of diffusion on biochemical reaction rate, can be expressed by means of an effectiveness factor, T|, where,

156

6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

T|

=

actual apparent rate rate at bulk liquid concentration

Solutions of the above diffusion reaction model are available in the literature for simple reaction orders (Satterfield and Sherwood, 1963). In Fig. 6.8 the values of t| for zero, first and second reaction order have been plotted against the Thiele Modulus ,, where, O = '\/L 2 kSo n - 1 /D This figure shows that a zero order reaction is not influenced by concentration gradients until the substrate falls to zero in the matrix, corresponding to O > \2 . The other reaction-types are influenced by low concentrations, as the curves for T| indicate. It is seen, for example, that for a first order reaction a value of O = l corresponds to t| = 0.8.

1.0

-

0.8

-

0.6

-

0.4

-

0.2

-

Incomplete penetration at O >V~2~

A A = zero-order B = first-order C = second-order

0

0 = L(kSo n ' 1 /D) 172 Figure 6.8. Effectiveness factor TJ versus the dimensionless reaction/diffusion parameter (Thiele Modulus O ) for reactions in a flat film with diffusion from one side (after Satterfield and Sherwood, 1963).

6.2 Internal Diffusion and Reaction within Biocatalyst

157

6.2.4

Case Studies for Diffusion with Biological Reaction

6.2.4.1

Case A.

Estimation of Oxygen Diffusion Effects in a Biofilm

For a biofilm or floe, whose oxygen uptake might be taken as a constant (zero order), the corresponding group would be [L2 qo2 Xbi0fiim/D Co2L where qo2 Xbiofiim corresponds to the rate constant k and the oxygen concentration in the outside liquid phase is Co2- Note that Xbiofiim is the biomass per unit of biofilm volume and is not easy to measure. Substituting values obtained from an aerobic biofilm nitrification experiment gives, 2 0

L2 qQ2 X = D C02

=

(0.01 mm2) (80 mg O2/L min) (0.1 mm2/min) (8 mg O2/L)

=

l

For this order-of-magnitude analysis, the value of 1.0 can be used to separate the regions of reaction and diffusion dominance. Thus it is seen if L = 0.1 mm, then the dimensionless group will have a value of 1.0, and it could therefore be expected that a film or floe thickness greater than 0.1 mm would be oxygen limited. From the exact solution, as seen in Fig. 6.8, gradients would appear for a zero order reaction at a value of <&2 = 2.0, instead of 1.0. This example shows how the Thiele Modulus can be used to make useful estimates for diffusion reaction problems, providing rate and diffusion data are available.

6.2.4.2

Case B.

Complex Diffusion-Reaction Processes (Biofilm Nitrification)

Nitrification reactions, considering only the substrate conversion reactions and ignoring the slow organism growth processes, the reactions can be written as, NH4+ + 3/2O 2 ->

N02- + 1/2 02 -> N03

158

6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

The oxygen requirements for the first and second steps can be related to the nitrogen content of NH4+ and NC>2~. These values are si = 3.5 mg 62 / mg NH4+ - N and 82 = 1.1 mg 62 / mg NCV - N. The low yields and low growth rates make it unnecessary to consider growth requirements and kinetics. In previous work (Tanaka and Dunn, 1982) the intrinsic substrate uptake kinetics for the two steps were shown to have a double Monod form for the first step, rNH4 = v m i

. K NH4

and for the second step, rN02 = vm2 where v m i and vm2 represent the maximum rates for a particular biomass concentration and the chemical symbols represent concentrations. Considering the diffusion phenomena in the biofilm to be represented by one-dimensional diffusion with quasi-homogeneous reaction, differential balance equations can be written for all reactants and products to describe the concentration profile in the film. Proceeding as described in Sec. 6.2.1, a component mass balance is written for segment n and for each component: (Accumulation^ _ (Diffusion^ _ (Diffusion^ ^ rate ) ~ \ rate *n ) ~ \ rate out J

( Production ^ Vrate ^ reacti°n J

and the equivalent partial differential equation is obtained by letting AZ approach zero as 3S 32S

Applying this to each component gives the following balances: For NH4+, a2NH4+

3NH4+ = DNH4

For NO2",

aNQ2~3r

= D

NO2

^2

-wo

- TNH4

+ rNH4 - TNO2

6.2 Internal Diffusion and Reaction within Biocatalyst

159

For NO3%

32N03

3N03 For O2,

3202

" s l rNH4 ~

The stoichiometric oxygen requirements for the first and second reaction steps are given by si and s2. The boundary conditions used represent the bulk liquid phase or reactor concentrations and the zero gradient at the biofilm- solid interface, as discussed earlier. These equations can be written as differential-difference equations using the finite-differencing technique (Sec. 6.2.1). Thus for each of N increments, four component balances will be needed. Three simulation examples, BIOFILM, ENZDYN, CELLDIFF in Sec. 8.7, demonstrate this approach. This system was also analyzed in terms of dimensionless variables. A comparison of the resulting dimensionless NH4+ and O2 balances reveals that, when the second reaction is neglected, the equations are identical if DNH4 = Do2 and if, Q2R where the subscript R refers to the concentrations in the bulk reactor liquid. Under these conditions, to a good approximation, the penetration distances of O2 and NH4+ would be the same. The ratio O2R/NH4+R, which can be varied according to the reactor operating conditions, can thus be used as a criterion to evaluate whether NtLj."1" or O2 might be penetration-limiting. The O2R/NH4+R criterion indicates which component can be limiting, O2 if the ratio is less than 3.5 or NH4+ if the ratio is greater than 3.5. These conditions are not sufficient for limitation, but indicate which component would be limiting. Simulation results from a model that was developed using finite-differencing demonstrates this phenomenon. The profiles in Fig. 6.9, are for the case O2R/NH4+R = 0.07.

160

6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

NH 4 ,NO3,N0 2 (mg/L) 100

0 2 (mg/L)

H

20

80 60 NO

10

40 20

Figure 6.9. Steady state profiles for constant bulk concentrations showing incomplete oxygen penetration.

Coupling the liquid and biofilm for a batch nitrification reactor as explained in Fig. 6.7, gave the results in Fig. 6.10. Here the influence of oxygen limitation caused the oxygen in the midpoint of the film to rise as the nitrogen substrates were successively consumed. NH4

0 2 [mg/L]

NO 3, NO2" (mg/L)

100

80

80

64

60

48

8

40

32

6

20

16

0

0

12 10

4

2 0 120

t (min) Figure 6.10. Simulated biofilm nitrification profiles in a batch reactor. The N-component concentrations are in the bulk liquid. O2 is in the midpoint of the biofilm and indicates limitation during the first 60 minutes.

7

Automatic Bioprocess Control Fundamentals

The purpose of automatic process control is to maintain time-dependent changes of the relevant process variables (deviations, errors), within prescribed limits and without a direct action of an operator. Process control may be considered as a corrective action involving three steps: 1. 2. 3.

Measuring the variable to be controlled (controlled variable) Comparing the measurement with the desired value (set point) Adjusting some other variable (manipulated variable) that has a direct effect on the controlled variable, until the set point is reached.

A number of advantages or reasons for process control may be listed, which include uniform and higher quality products, safety, increase of productivity, minimization of waste, optimization, freeing the labor force from drudgery and danger, and decrease of labor costs. Obviously, process control is highly dynamic in nature, and therefore its modelling requires the solution of sets of differential equations, and it is therefore highly suited to solution by digital simulation. A brief introduction to the basic principles of process control required for solution of simple simulation examples is given here.

7.1

Elements of Feedback Control

The simple temperature control of a fermenter shown in Fig. 7.1 illustrates the essential idea of any automatic control system that the process and the controller form a closed loop, which usually functions in a feedback fashion.

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

162

7 Automatic Bioprocess Control Fundamentals

Controller mechanism

Desired value

I Measured I value I

Thermocouple

Fermenter Figure 7.1. Simple feedback control system.

The components of such a control system can be best understood using a generalized block diagram (Fig. 7.2). They are the process itself, the measuring element (thermometer), the controller (including a comparator), the final control element (automatic control valve) and the transmission lines. The information on the measured variable, temperature, taken from the system is used to manipulate the flow rate of the cooling water in order to keep the temperature at the desired constant value, or set point. Controller mechanism

Load Actuator

1 Com)jarator Controller _ i'

$~

Desired+c value | *A/

1 1 Measured value

J

1 K

1 1

Process

-*U

Measuring element k

A A

A A

A A

A A

A

^A^A^AAA^A

Figure 7.2. Block diagram of the feedback control system in Fig. 7.1.

Controlled variable

7.2 Types of Controller Action

163

Similar temperature control systems are given for a simple water heater, TEMPCONT, in Sec. 8.6.1 and for a batch fermenter, FERMTEMP, in Sec. 8.6.2

7.2 7.2.1

Types of Controller Action On-Off Control

The most common and simplest type of control is an on-off or two position action, sometimes called discontinuous control (Fig. 7.3). An example is a contact thermometer, which closes or opens the heater circuit. The controller changes the value of the controller output, or the manipulated variable, from one extreme to the other, when the controlled variable moves above or below the set point. This leads to oscillations that could become very fast, depending on the speed of response of the process. The real on-off controller has therefore a built in feature called a differential gap or a dead zone. It is a small interval on either side of the set point, within which the controller does not respond. When the controlled variable moves outside the dead zone, the manipulated variable goes on or off. This is illustrated in Fig. 7.3. Such shifts from the set point are known as offset. Such a controller is simple and inexpensive, but the oscillatory nature of the action and the offset make it very imperfect. The usefulness of this type of control was demonstrated for a biological, sequential batch process by Hediger and Prenosil, (1985). More sophisticated function control modes consider the magnitude and time behavior of the control error. Three principal functional modes of control generally employed for process control are proportional (P), integral (I) and derivative (D) control.

164

7 Automatic Bioprocess Control Fundamentals

100 Manipulated variable 0

Control variable

Xl___4/_ .Set point

Differential gap

Figure 7.3. On-Off controller with differential gap or dead zone.

7.2.2

Proportional (P) Controller

The produced output signal P is proportional to the detected error, e, according to P = PO + Kp 8 where Kp is the proportional gain, and P0 is the controller output for zero error. An example of a level control is shown in Fig. 7.4. The action of this type of controller is shown in Fig. 7.5 and 7.6.

Figure 7.4. Response of proportional-mode controller to sinusoidal error input.

165

7.2 Types of Controller Action

Systems with proportional control often exhibit pronounced oscillations and for sustained changes in load the controlled variable attains a new equilibrium, steady-state position, or control point. The difference between this point and the set point is the offset (Fig. 7.5) Integral and derivative modes are used mostly in combination with the basic proportional control mode. The simulation Example INHIB, Sec. 8.5.2 includes the application of this simple control mode in the recommended exercises. F + AF

Setpoint

Figure 7.5. An example of proportional-mode level controller illustrating offset.

7.2.3

Proportional-Integral (PI) Controller

Pi-controller, sometimes called automatic reset, produces an output signal related to the error by Kpe+

P =

K

or for e = constant dP

for t=Ti dP

dF = K P £

l

?

TI

166

7 Automatic Bioprocess Control Fundamentals

where i\ is the integral time constant or reset time. This is the time required to enable the controller to repeat the initial proportional output action (Fig. 7.6). The integral part of the control mode eliminates the offset and it is especially useful for correction of very small errors because the controller output P will continue to change as long as an error persists. This can be understood by considering a constant error, which would cause P to increase linearly (Fig. 7.6) at a rate proportional to the error. This type of controller is found most often and the simulation examples TEMPCONT, FERMTEMP, TURBCON and CONTCON in Sec. 8.6 demonstrate the use of this control mode. The examples also show the ease by which the programming of the PI controller equations is made using the simulation language.

2Kp

Controlled output

PI action

Kp P action

Error signal

Figure 7.6. Response of a proportional-integral controller to a unit step change in error.

7.2.4

Proportional-Derivative (PD) Controller

A controller with derivative function projects the error in the immediate future and the controller output is proportional to the current rate of the error change. The output signal varies only if the error is changing.

167

7.2 Types of Controller Action

d£ P = P0 + Kp e + Kp TD gj-

where TD is the derivative time constant. A PD-controller output is compared with pure proportional mode in Fig. 7.7.

P (alone)

t Figure 7.7. Response of a PD-Controller to a constant rate of decrease in error. Comparison with P and PID modes.

The drawbacks of the derivative control mode standing alone are that a constant error (e ^ 0) gives no response at all, since de/dt = 0, and therefore an unnecessarily large response might occur as a result of small but fast error changes.

7.2.5

Proportional-Integral-Derivative (PID) Controller

In industrial practice it is common to combine all three modes, sometimes termed as Proportional-Reset-Rate-Control. The action is proportional to the error (P) and its change (D) and continues if residual error is present (I):

168

7 Automatic Bioprocess Control Fundamentals

KD r

de

P = P0 + Kp e + —^ J £dt+ Kp TD -gjTI

o

This combination gives the best control using conventional feedback equipment. It retains the specific advantages of all three modes: proportional correction (P), offset elimination (I) and stabilizing, quick-acting character, which is especially suitable to overcome lag presence (D). The action of a PIDcontroller as a response to a ramp function is shown in Fig, 7.7. The performance of the different feedback control modes can also be seen in Fig. 7.8. Controlled variable / Uncontrolled response

Figure 7.7. Response of controlled variable to a step change in error using different control modes.

The selection of the best mode of control depends largely on the process characteristics. Further information can be found in the recommended texts listed in the reference section. Simulation methods are often used for testing control methods.

7.3 Controller Tuning

7.3

169

Controller Tuning

The purpose of controller tuning is to choose the controller constants, such that the desired performance is obtained. This usually means that the control variables should be restored in an optimal way, following either a change in the set point or as a result of an input disturbance to the system. Thus the controller constants can be set by experimentation. A rational basis for such experimental tuning is given in what follows. Other methods for tuning combine process dynamic experimentation with theoretically-based control methods; some of the standard methods are also described below. The simulation example TEMPCONT, Sec. 8.6.1, provides exercises for controller tuning using the methods explained below.

7.3.1

Trial and Error Method

Controllers can be adjusted by changing the values of gain Kp, reset time i\ and derivative time ID- By experimentation, either on the real system or by simulation, the controller can be set by trial and error. Each time a disturbance is made the response is noted. The following procedure might be used to test the control with small set point or load changes: 1. Starting with a small value, Kp can be increased until the response is unstable and oscillatory. This value is called the ultimate gain KPQ. 2. K p is then reduced by about 1/2. 3. Integral action is brought in with high i\ values; they are reduced by factors of 2 until the response is oscillatory, and tj is set at 2 times this value. 4. Include derivative action, increase ID until noise develops and set ID at 1/2 this value. 5. Increase Kp in small steps to achieve the best results.

7.3.2

Ziegler-Nichols Method

This method is an empirical open-loop tuning technique, obtained by uncoupling the controller. It is based on the characteristic curve of the process response to a step change in manipulated variable, equal to A. This response is called a process reaction curve, whose magnitude is B. The two parameters

170

7 Automatic Bioprocess Control Fundamentals

important for this method are given by the normalized slope of the tangent through the inflection point, S = slope/A and by its intersection with the time axis (lag time TL), as determined graphically in Fig. 7.9. The actual tuning relations which are based on empirical criteria for the "best" closed-loop response are given in Table 7.1.

B

Manipulated Variable

X Slope

Time

Figure 7.9. Process reaction curve as a response to a step change in manipulated variable.

7.3.3

Cohen-Coon Controller Settings

Cohen and Coon observed that the response of most uncontrolled (controller disconnected) processes to a step change in the manipulated variable was a sigmoidal shape curve. This can be modelled approximately by a first order system with time lag TL, as given by the intersection of the tangent through the inflection point with the time axis. The theoretical values of the controller settings obtained by the analysis of this system are summarized in Table 7.1 The model parameters for a step change A to be used with Table 7.1 are calculated as follows: K = B/A T = B/S where B is from Fig. 7.9 and S is the slope at the inflection point/A.

7.3 Controller Tuning

171

Table 7.1. Controller Settings Based on Process Responses Controller

Kp

Ziegler-Nichols

1/TLS 0.9/TL S 1.2/TLS

P PI PID

3.33TL 2TL

TL/2

Co/ten-Coon P

T

TL \

PI 4

30 + 3TL/T

TL \

32 + 6TL/T

Ultimate Gain

P

PI PID

7.3.4

0.5 Kpo 0.45 Kp0 0.6 Kpo

l/1.2fp0 l/2fpo

l/8fpO

Ultimate Gain Method

The previous tuning transient response methods are sensitive to disturbances because they rely on open-loop experiments. Several closed loop methods were developed to eliminate this drawback. One of them is the empirical tuning method, ultimate gain or continuous-cycling method. The ultimate gain, Kpo, is the gain which brings the system with the proportional control mode to sustained oscillations (stability limits) of the frequency fpo, where l/f p o is called the ultimate period. It is determined experimentally by increasing Kp from low

172

7 Automatic Bioprocess Control Fundamentals

values in small increments until continuous cycling begins. The controller settings are then calculated from Kpo and f p o according to the tuning rules given in Table 7.1. While this method is very simple it can be quite time consuming in terms of number of trials required and if the process dynamics are slow. In addition, it may be hazardous to experimentally force the system into unstable operation. Simulation methods can be very useful if a suitable model is available as was shown by Heinzle et al. (1992) for a one and two stage anaerobic system, using a kinetic model similar to the simulation example, ANAMEAS, Sec. 8.8.7.

7.4 7.4.1

Advanced Control Strategies Cascade Control

In control situations with more then one measured variable but only one manipulated variable, it is advantageous to use control loops for each measured variable in a master - slave relationship. In this, the output of the primary controller is usually used as a set point for the slave or secondary loop. This may be relevant for some wastewater treatment plants where the high concentration of some substrate may be toxic for the microorganisms. For example, the simulation Example TURBCON, Sec. 8.6.3, could be easily adapted to this situation if the substrate concentration were subject to significant changes, as shown schematically in Fig. 7.10.

173

7.4 Advanced Control Strategies

m

| 1

!

lilllll :

';§S:'|||::||:|-

XI . .. N /6oncentrationV/Concentriilonl Controller )«H f errand x. ,s \ transmitter / ^

c

Biomass controller

^

V^

illlll

ill lit:

J_ [ Turbidometer

Figure 7.10. Cascade control of a fermenter with toxic substrate.

The simulation example TURBCON, Sec. 8.6.3. could be modified similarly by adding biomass as a measured variable. An interested reader may try to implement the cascade control strategy in these simulation programs.

7.4.2

Feed Forward Control

Feedback control may never be perfect as it only reacts to the disturbances which are measured in the system output. The feedforward method tries to eliminate this drawback by another approach. Rather than using the process output as the measured variable, this is taken as the measured inlet disturbance and its effect on the process is anticipated by means of a process model. Thus action is taken on the manipulated variable by the model, which relates the measured variable at the inlet, the manipulated variable and the process output. The success of this control strategy depends largely on the accuracy of the model prediction. For this reason sometimes an additional feedback loop is used. Many of the continuous process simulation examples in Ch. 8 could be altered in this fashion. It would be interesting to program an example using simple kinetics for the feedforward control and to describe the "actual" system with more complex kinetics. The discrepancies between the "simple" model

174

7 Automatic Bioprocess Control Fundamentals

prediction and the more complex "actual" process kinetics could then be taken care of by a feedback control loop.

7.4.3

Adaptive Control

This control system can automatically modify its behavior according to the changes in the system dynamics and disturbances. Especially systems with nonlinear and unsteady characteristics call for use of this control strategy. There are a number of actual adaptive control systems. Programmed or scheduled adaptive control uses an auxiliary measured variable to identify different process phases for which the control parameters can be either programmed or scheduled. The "best" values of these parameters for each process state must be known a priori. Sometimes adaptive controllers are used to optimize two or more process outputs, by measuring these and fitting the data with empirical functions, as employed on anaerobic treatment process, by Ryhiner, et al. (1992).

7.4.4

Sampled-Data Control Systems

When discontinuous measurements are involved the control system is referred to as sampled-data. Concentration measurements by chromatography would represent such a case.

Controlled variable

Figure 7.11. Sampled control strategy.

Here a special consideration must be given to the sampling interval T (Fig. 7.11). In general the sampling time will be short enough if the sampling

7.5 Concepts for Bioprocess Control

175

frequency is equal to 2 times the highest frequency of interest or T is equal to 0.5 times the minimum period of oscillation. When the sampling time satisfies the above criteria, the system will behave as if it were continuous. Details of this and other advanced control topics are given in specialized process control textbooks, some of which are listed in the reference list.

7.5

Concepts for Bioprocess Control

Bioprocess control consists of establishing a strategy for the management of the biocatalyst environment. In a natural environment microorganisms and cells of higher organisms very rarely produce large amounts of products. In biotechnological processes organisms are usually kept in a completely "unnatural" environment. Control is therefore often necessary to induce them to produce substances in economically important amounts. Process optimization is closely linked with control. The objectives of optimization and control may be to maximize productivity, final concentration, yield or to minimize effluent concentration and energy costs. Although the criteria for optimal processes differ widely, all bioprocesses need control and automation to run under optimal conditions. The selection of control variables strongly depends on the process and the final goal to be achieved. Information about the dependency of biological rates, yields and selectivities on environmental conditions is usually required, as given in Ch. 3. All biological reactions have distinct temperature and pH optima, and all respond to substrate concentrations. Therefore it is common to control these variables. Heat is produced in all biological reactions and therefore temperature control is necessary. In large-scale production, heat removal capacity may be the limiting factor. It is important to maintain the temperature at an optimum level. This is the theme of the simulation example, FERMTEMP, Sec. 8.6.2. As discussed in Ch. 3, bioreaction rates usually follow the Arrhenius' Law below the optimal temperature, which means that the growth rates can be expected to increase exponentially with increasing temperature. Above the optimum temperature, further temperature increases usually cause a dramatic decrease in activity, mainly due to inactivation of enzymes. Also, temperature shocks may be important since enzyme formation may often be induced by a shock at the end of the exponential growth phase. The variable pH has certain similarities with temperature because there usually exists an optimum for biological activity; it can be relatively easily measured and is often controlled. The biological rates also exhibit a maximum at the optimal pH value, which is usually in the neutral pH 7 region. Again control is often required since in almost all biological reactions acids (e.g. lactic, pyruvic acid) or bases (e.g. NH3) are either produced or consumed.

176

7 Automatic Bioprocess Control Fundamentals

Biological rates usually depend on substrate concentration (e.g.: sugar, mineral salts, oxygen, precursor, etc.) though in many cases kinetics are of zero-order type above certain concentration levels. In the latter case control is not important, since high concentrations will guarantee maximum rates. The situation is more complex if process selectivity changes with substrate concentration. The most well-studied process of this kind is Baker's Yeast production, where high glucose concentration (> 100 mg L"1) and oxygen limitation causes undesired ethanol formation. Substrates (e.g. mineral salts, components of wastewater), precursors (e.g. in antibiotica production or transformation processes) or products (e.g. ethanol) may be inhibiting or even toxic at higher concentration levels. In such processes it is necessary to control the concentration within certain limits.

7.5.1

Selection of a Control Strategy

The first step in controlling a process is to choose a control strategy. Simple examples are the set-point control of constant temperature, pH, substrate and precursor concentration. Table 7.2 gives examples of methods and strategies for control of biological reactors. Much of the difficulty in control lies in finding a suitable sensor. Calculated values using indirect measurements can be very useful, e.g. measuring oxygen uptake to control substrate level. Often variables such as pH or dissolved oxygen (DO) control can be used to indirectly keep substrate concentration constant Table 7.2. Examples of methods and strategies for the control of bioprocesses (Heinzle and Saner, 1991). Process Method Controlled Manipulated and strategy variable(s) variable(s) Baker's yeast production

Discrete control

RQ (Gas analysis)

Glucose feed rate

Baker's yeast production

Feedback control

RQ (Gas analysis)

Glucose feed rate

Baker's yeast production

Two point control

DO (electrode)

Feed rate, agitation speed aeration rate

177

7.5 Concepts for Bioprocess Control

Table 7.2. (Continued). Process Method and strategy Ethanol PID production

Controlled Manipulated variable(s) variable(s) Sucrose cone, by Feed rate enzyme thermister

Wastewater treatment

Adaptive control

DO (electrode)

Aeration rate

Diverse fermentations

Various

DO (electrode)

Stirring speed, gas composition, aeration rate, pressure

Bacillus subtilis at low DO

Cascade

DO (electrode)

Gas flow, valve setting

Penicillin production

Set-points (growth and production)

Growth rate (CO2 rate)

Feed rate

a-Amylase by Bacillus amyloliquefaciens

Feed profile

Feed rate (off-line)

Feed rate

Fed-batch penicillin

Feed profile

Substrate and biomass cone.

Feed rate

Baker's yeast

Feed profile

RQ

Feed rate

Cephalosporin C Profiled pH pH, temp. production and temperature (electrode, control thermister)

Alkali feed rate, cooling water rate

Recombinant E. coli

Growth rate (pH)

Glucose and alkali rate

Phenol uptake (O2 uptake)

Flow rate

Conventional with added glucose

Phenol oxidation Adaptivequesting

For constant value or set-point control usually constant control parameters are used. Because of non-linearities or varying process dynamics (e.g. exponential

178

7 Automatic Bioprocess Control Fundamentals

growth phase followed by production phase) control parameters of a linear controller may be inadequate to control the process. It is therefore necessary to adapt control parameters (e.g. proportional gain) according to the process requirements. Minimization of an objective function can be used to guide the adaptive tuning of the controller. A rather simple method uses a somewhat empirical adjustment mechanism, which is driven by a secondary measurement. This was used for DO control by Heinzle et al. (1986), in which the oxygen uptake rate measurement was used to adjust the controller gain. In addition to constant value control, optimal profile control may be applied. The predefined optimal profile is then followed, which may be calculated from off-line simulation and optimization. An example is found in the exponential feeding profiles that can be calculated from the models for fed batch fermenters in Ch 4. If no suitable dynamic model is available and the process changes in unpredictable ways, then on-line adaptive optimizing control may be useful. This would however require measurements of the key inputs and outputs of the process. An example is the optimization of a continuous anaerobic process by Ryhiner et al. (1989) in which the methane and organic acids output rates were correlated with the input flow rate. The optimization involved a compromise between high methane rates and low organic acid concentration.

7.5.2

Methods of Designing and Testing the Strategy

Selection of a control strategy and its parameters (e.g. for a PID controller) may be difficult, since the process and controller dynamics are often not well understood. If possible, it is useful to use dynamic models to select a control strategy, and to use it for testing and tuning. An example with anaerobic digestion is given by Heinzle et al. (1992). In Fig. 7.12 are shown the results of a simulation and a corresponding experiment for the control of the propionic acid concentration by manipulation of the feed flow rate.

179

7.5 Concepts for Bioprocess Control

Propionic acid [mg/l] Feed flow [ml/min]

2

1 I 0

1

2

3

4

6(

Propionic acid [mg/l] Feed flow [ml/min]

5( 4( o

CD O

i

3( -

"E

V2

20-

I

1( 2

3

Time [h]

Figure 7.12. Control of anaerobic digestion of whey wastewater. Simulation (A) and experiment (B) of control after step change (Heinzle et al. (1992). Here the controlled variable was the propionic acid concentration, and the manipulated variable was the feed flow rate.

References

References Cited in Part I

Archer, D.B. (1983) The Microbiological Basis of Process Control in Methanogenic Fermentation of Soluble Wastes. Enzyme Microb. Technol. 5, 570-577. Aris, R. (1989) Boston.

Elementary Chemical Reactor Analysis. Butterworths,

Atkinson, B. and Mavituna, F. (1991) Biochemical Engineering and Biotechnology Handbook. 2nd. Ed., Stockton Press, New York. Bailey, J.E. and Ollis, D.F. (1986) Biochemical Engineering Fundamentals. 2nd. Ed., McGraw-Hill, N.Y. Blanch, H.W. and Dunn, I.J. (1973) Modelling and Simulation in Biochemical Engineering. Adv. Biochem. Eng, 3, 127-165. Dekkers, R.M. (1983) State Estimation of a Fed-batch Baker's Yeast Fermentation., in: Modelling and Control of Biotechnological Processes, (Ed.: A.Halme) Pergamon Press, Oxford, 73. Denac, M., Miguel, A., Dunn, I.J. (1988) Modeling Dynamic Experiments on the Anaerobic Degradation of Molasses Wastewater. Biotechnol. Bioeng. 31, 110. Dunn, I.J. and Mor, J.R. (1975) Variable Volume Continuous Cultivation. Biotechnol. Bioeng. 17, 1805-1822. Dunn, I.J., Shioya, S. and Keller, R. (1979) Analysis of Fed Batch Fermentation Processes. Annals N.Y. Acad. ScL 326, 127-139. 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

References

181

Franks, R.G.E. (1966) Mathematical Modeling in Chemical Engineering. Wiley, New York. Franks, R.G.E. (1973) Modeling and Simulation in Chemical Engineering. Wiley, New York. Fredrickson, A.G., Megee, R.D., Tsuchita, HM. (1970) Mathematical Models for Fermentation Processes. Adv. Appl. Microbiol. 13, 419-465. Furukawa, K., Heinzle, E., and Dunn, IJ. (1983) Influence of Oxygen on the Growth of Saccharomyces cerevisiae in Continuous Culture. Biotechnol. Bioeng. 25, 2293-2317. Gujer, W., Zehnder, A.J.B. (1983) Conversion Processes in Anaerobic Digestion. Water Sci. Technol. 15, 127-167. Harder, A., Roels, J.A. (1982) Application of Simple Structured Models in Bioengineering. Adv. Biochem. Eng./Biotechnol. 21, 56-107. Hediger, T. and Prenosil, I.E. (1985) Microprocessor Automated Sequential Batch Process, Biotechnol. Progr.l, 216-225. Heinzle, E. and Lafferty, R.M. (1980) Continuous Mass Spectrometric Measurement of Dissolved H2, O2, and CO2 during Chemolitho- autotrophic Growth of Alcaligenes eutrophus strain H 16. Eur. J. Appl. Microbiol. Biotechnol. 11, 8-16. Heinzle, E., Furukawa, K., Dunn, I.J., and Bourne, J.R. (1983) Experimental Methods for On-line Mass Spectrometry in Fermentation Technology. Bio/Technology 1, 14-16. Heinzle E. and Dunn I. J. (1991) Methods and Instruments in Fermentation Gas Analysis, in Biotechnology, Vol 4, (Ed.: H.-J.Rehm and R.Reed). VCH, Weinheim, 27-74. Heinzle, E. and Saner, U. (1991) Methodology for Process Control in Research and Development, Ed. Pons, M.-N., in Bioprocess Monitoring and Control., Hanser, Munich. 223-304. Heinzle, E., Dunn, IJ. and Ryhiner, G. (1992) Modelling and Control for Anaerobic Wastewater Treatment, Adv. Biochem. Eng., Springer Verlag.

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

182

References

Imanaka, T., Aiba, S. (1981) A Perspective on the Application of Genetic Engineering: Stability of Recombinant Plasmid. Ann. N. Y. Acad. Sci. 369, 114. Ingham, J., and Dunn, I.J. (1991) "Bioreactor Off-Gas Analysis", in "Bioreactors in Biotechnology", Ed. A. Scragg, Ellis-Horwood, Chichester, 195-220. 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, 508-514. Keller, R. and Dunn, I.J. (1978) Fed Batch Microbial Culture: Models, Errors, and Applications. J. Appl. Chem. Biotechnol. 28, 508-514. Keller, J., Dunn, I. J. and Heinzle, E. (1991). A Fluidized Bed Reactor for Animal Cell Culture, in preparation, Biotechnol. Bioeng. Luyben, W.L. (1973) Process Modeling, Simulation, and Control for Chemical Engineers. McGraw-Hill. Meister, D., Post, T., Dunn, I.J. and Bourne, J.R. (1979) Design and Characterization of a Multistage, Mechanically Stirred Column Absorber. Chem. Eng. Sci., 34, 1376. Moes, J., Griot, M., Keller, J., Heinzle, E., Dunn, I.J., and Bourne, J.R. (1985) A Microbial Culture with Oxygen-sensitive Product Distribution as a Tool for Characterizing Bioreactor Oxygen Transport. Biotechnol. Bioeng. 27, 482489. Moes, J., Griot, M., Heinzle, E., Dunn, I.J., and Bourne, J.R. (1986) A Microbial Culture as an Oxygen Sensor for Reactor Mixing Effects. Ann. N. Y. Acad. Sci. 469, 482-489. Mona, R., Dunn, I.J. and Bourne, J.R. (1979) Activated Sludge Process Dynamics with Continuous TOC and Oxygen Uptake Measurements. Biotechnol. Bioeng. 21, 1561-1577. Moser, A. (1988) Bioprocess Technology, Springer, N.Y. Mou, D.G. and Cooney, C.L. (1983) Growth Monitoring and Control through Computer-aided On-line Mass Balancing in a Fed-batch Penicillin Fermentation. Biotechnol. Bioeng. 25, 225-255.

References

183

Prenosil, J., Dunn, I.J., and Heinzle, E. (1987) Biocatalyst Reaction Engineering, in: Biotechnology Vol.7a, (Ed.: HJ.Rehm and R.Reed) VCH, Weinheim, 489545. Roels, J.A. (1983) Energetics and Kinetics in Biotechnology. Elsevier Biomedical Press, Amsterdam. Ruchti, G., Dunn, I.J., Bourne, J.R. and v. Stockar, U. (1985) Practical Guidelines for Determining Oxygen Transfer Coefficients with the Sulfite Oxidation Method. Chem. Eng. J. 30, 29-38. Russell, T.W.F., Denn, M.M. (1972). Introduction to Chemical Engineering Analysis. Wiley, New York. Ryhiner, G., Dunn, IJ, Heinzle, E, Rohani S., (1992) Adaptive On-line Optimal Control of Bioreactors: Application to Anaerobic Degradation, J. BiotechnoL. 22, 89-106. Ryhiner, G., Heinzle, E, Dunn, IJ. (1991) Modelling of Anaerobic Degradation and Its Application to Control Design: Case Whey, in: Dechema Biotechnology Conferences Vol.3, (Ed.: Behrens, D. and Driesel, A.J.), 469-474. Saner, U., Bonvin, D., Heinzle, E. (1990) Application of Factor Analysis for Elaboration of Stoichiometry and its On-line Application in Complex Medium Fermentation of B. subtilis, in: Dechema Biotechnology Conferences Vol.3, (Ed.: Behrens, D. and Driesel, A.J.), 775-778. Satterfield, C.N. and Sherwood, T.K. (1963) The Role of Diffusion in Catalysis. Addison-Wesley, N.Y. Shioya, S., Dang, N.D.P. and Dunn, IJ. (1978). Bubble Column Fermenter Modeling: A Comparison for Pressure Effects. Chem. Eng. Sci., 33, 1025 1030. Tanaka, H. and Dunn, IJ. (1982). Kinetics of Biofilm Nitrification. BiotechnoL Bioeng., 24, 669 - 689. Tanaka, H., Uzman, S., Dunn, IJ. (1981). Kinetics of Nitrification Using a Fluidized Sand Bed Bioreactor with Attached Growth. BiotechnoL Bioeng., 23, 1683 - 1702. Ziegler, H., Meister, D., Dunn, IJ., Blanch, H.W., Russell, T.W.F. (1977). The Tubular Loop Fermenter: Oxygen Transfer, Growth Kinetics and Design. BiotechnoL Bioeng. 19, 507.

184

References

Recommended Textbooks and References for Further Reading

Biochemical

Engineering

Aiba, S., Humphrey, A.E. and Millis, N.F. (1973) Biochemical Engineering. Academic Press, N.Y. Atkinson, B. and Mavituna, F. (1991) Biochemical Engineering and Biotechnology Handbook., 2nd. Ed., Stockton Press, New York. Bailey, I.E. and Ollis, D.F. (1986) Biochemical Engineering Fundamentals. 2nd. Ed., McGraw-Hill, N.Y. Blanch, H.W., Clark, D.S. (1996) Biochemical Engineering; Marcel Dekker, N.Y. Bu'lock, J. and Kristiansen Eds (1987) Basic Biotechnology Acad. Press, London 1987. Doran, P. M. (1995) Bioprocess Engineering Principles; Academic Press Limited: London. Glick, B.R., Pasternak, J.J. (1995) Molekulare Biotechnolgie; Spektrum, Heidelberg. Grady, C. P. L. and Lim H. C. (1980) Biological Wastewater Treatment, Marcel Dekker. Hastings, A. (1997) Population biology. Concepts and models; Springer, N.Y. Heinrich, R., Schuster, S. (1996) The Regulation of Cellular Systems; Chapman & Hall, New York. Klefenz, H. (2002) Industrial Pharmaceutical Biotechnology; Wiley-VCH. Ladisch, M.R. (2001) Bioseparations Engineering: Principles, Practice and Economics. Wiley, New York. Lee, J. M. (1992) Biochemical Engineering. Prentice Hall. Moo-Young, M., Ed. (1985) Comprehensive Biotechnology, Vols. 1-4 Pergamon Press, Oxford.

References

185

Moser, A. (1988) Bioprocess Technology.Springer, N.Y. Nielsen, J., Villadsen, J. (1994) Bioreaction Engineering Principles. Plenum Press. Pirt, S. John (1975) Microbe and Cell Cultivation. Blackwell Scientific Publ., Oxford. Rehm H.J. and Reed R. Eds (1988) Fundamentals of Biochemical Engineering, in Biotechnology, Vol. 2, VCH, Weinheim, Roels, J.A. (1983) Energetics and Kinetics in Biotechnology. Elsevier Biomedical Press, Amsterdam. Schiigerl, K., Bellgardt, H. (Eds.) (2000) Bioreaction Engineering. Springer, Berlin, Heidelberg. Schiigerl, K. (1987) Bioreaction Engineering. Wiley, New York. Schiigerl, K. (1994) Solvent Extraction in Biotechnology : Recovery of Primary and Secondary Metabolites. Springer, Berlin. Shuler, M. L., Kargi, F. (2002) Bioprocess Engineering. Basic Concepts; Prentice-Hall. Spier, R.E., Griffiths, J.B. (1985) Animal Cell Biotechnology. Vol. 1-3, Academic Press. Wang, D.I.C., Cooney, Ch.L., Demain, A.L., Dunnill, P., Humphrey, A.E., Lilly, M.D. (1979) Fermentation and Enzyme Technology. Wiley, New York. Wingard, L.B., Jr., Katschalski-Katzir and L. Goldstein Eds (1976-1983) Applied Biochemistry and Bioengineering, Vols. 1-4 Acad. Press, London.

Bioreactor Design and Modelling Asenjo, J.A., Merchuk, J.C. (1995) Bioreactor system design; Marcel Dekker, N.Y. Hannon, B., Ruth, B. (1997) Modeling Dynamic Biological Systems; SpringerVerlag, New York.

186

References

Scragg, A.H. (1991). Bioreactors in Biotechnology. Ellis Horwood. Sinclair, C.G., Kristiansen, B., Bu'Lock, J.D. (1987) Fermentation Kinetics and Modelling. Open University Press, Milton Keynes. Schugerl, K. (1987) Bioreaction Engineering.Vol.1, John Wiley, Chichester. Subramanian, G. (1998) Bioseparation and Bioprocessing. A Handbook Volume II: Processing, Quality and Characterization, Economics, Safety and Hygiene, Wiley-VCH, Weinheim. van't Riet, K., Tramper, J. (1991) Basic Bioreactor Design. M. Dekker, New York. Vieth, W.R. (1994) Bioprocess Engineering, J. Wiley & Sons, N.Y. Webb, C., Black, G.M., and Atkinson, B. (1986) Process Engineering Aspects of Immobilized Cell Systems.Pergamon Press Ltd., Oxford.

Enzyme Engineering and Kinetics Bisswanger, H. (2002) Enzyme Kinetics. Principles and Methods; Wiley-VCH. Buchholz, K., Kasche, V. (1996) Biokatalysatoren und Enzymtechnologie, VCH.Weinheim. Cornish-Bowden, A. (1979) Fundamentals of enzyme kinetics, Butterworth, London. Drauz, K., Waldmann, H. (1995) Enzyme Catalysis in Organic Synthesis. Volume I. VCH Weinheim. Drauz, K., Waldmann, H. (1995) Enzyme Catalysis in Organic Synthesis. Volume II. VCH, Weinheim. Fessner, W.-D. (1999) Biocatalysis - From Discovery to Application, Springer, Berlin. Godfrey, T., West, S. (1996) Industrial enzymology, Macmillan Press, London. Hayashi, K.and Sakamoto, N. (1986) Dynamic Analysis of Enzyme Systems, Japan Sci. Soc. Press, Tokyo, Springer Verlag, Berlin.

References

187

Kennedy J. F., Ed. (1987) Enzyme Technology, in Biotechnology Vol. 7a, VCH, Weinheim, Liese, A., Seelbach, K., Wandrey, C. (2000) Industrial Biocatalysis, Wiley-VCH. Scheper, T. (1997) Advances in Biochemical Engineering Biotechnology. New Enzymes for Organic Synthesis, Springer. Segel, I.H. (1975) Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-state Enzyme Systems. Wiley, New York.

Metabolic Engineering Lee, Papoutsakis (1999) Metabolic Engineering, Marcel Dekker: New York, Basel. Stephanopoulos, G. N., Aristidou, A. A., Nielsen, J. (1998) Metabolic Engineering. Principles and Methodologies, Academic Press: USA.

Chemical Reaction Engineering Aris, R. (1989) Elementary Chemical Reactor Analysis. Butterworth Publ., Stoneham. Fogler, H. S. (1992) Elements of Chemical Reaction Engineering, PrenticeHall. Hagen, J. (1993) Chemische Reaktionstechnik, VCH: Weinheim. Ingham, J., Dunn, I.J., Heinzle, E., Prenosil, I.E. (2000) Chemical Engineering Dynamics: An Introduction to Modelling and Computer Simulation, VCH Verlagsgesellschaft mbH: Weinheim, Germany,. Levenspiel, O. (1999) Chemical Reaction Engineering. John Wiley & Sons, New York. Massart, D. L., Vandeginste, B. G. M., Buydens, L. M. C., de Jong, S., Lewi, P. J., Smeyers-Verbeke, J. (1997) Handbook of Chemometrics and Qualimetrics: Part A, Elsevier: Amsterdam.

188

References

Richardson, J. F., Peacock, D. G. (1994) Coulson & Richardson's Chemical Engineering. Volume 3: Chemical & Biochemical Reactors & Process Control, Pergamon, Trowbridge. Satterfield, C.N. and Sherwood, T.K. (1963) The Role of Diffusion in Catalysis, Addison-Wesley, New York.

Modelling and Simulation Basmadjian, D. (1999) The Art of Modeling in Science and Engineering, Chapman & Hall/CRC: Boca Raton. Deaton, M. L., Winebrake, J. J. (1999) Dynamic Modelling of Environmental Systems, Springer: New York. Dodson, C. T. J., Gonzalez, E. A. (1995) Experiments in Mathematics using Maple, Springer-Verlag: Berlin, Franks, R.G.E. (1966) Mathematical Modeling in Chemical Engineering. Wiley, New York. Franks, R.G.E. (1972) Modeling and Simulation in Chemical Engineering. Wiley, New York. Russell, T.W.F., Denn, M.M. (1972). Introduction to Chemical Engineering Analysis. Wiley, New York. Ruth, M., Hannon, B. (1997) Modeling Dynamic Economic Systems, Springer Verlag, New York.

Dynamics and Control Astrom, K.J. and Wittenmark, B. (1989). Adaptive Control. Addison-Wesley, Reading. Coughanowr D. R. and Koppel L. B. (1965) Process System Analysis and Control. McGraw-Hill, New York.

References

189

Fish, N. M., Fox, R.L, and Thornhill, N.F. (1989) Computer applications in fermentation technology: Modelling and control of biotechnological processes. Elsevier, London. Halme, A. (1983) Modelling and Control of Biotechnical Processes. Pergamon Press, Oxford. Johnson, A. (1986) Modelling and Control of Biotechnological Processes. Pergamon Press, Oxford. Luyben W. L. (1973) Process Modeling, Simulation, and Control for Chemical Engineers, McGraw Hill, New York. Pons, M.-N., Ed.(1991) Bioprocess Monitoring and Control. Hanser, Munich. Snape, J. B., Dunn, I. J., Ingham, J., Prenosil, J. E. (1995) Dynamics of Environmentel Bioprocesses, VCH Verlagsgesellschaft mbH, Weinheim,. Stephanopoulos, G. (1984) Chemical Process Control: An Introduction to Theory and Practice, Prentice Hall. Weber. W. J., Jr., DiGiano, F. A. (1996) Process Dynamics in Environmental Systems, Wiley.

Part II Dynamic Bioprocess Simulation Examples and the Berkeley Madonna Simulation Language

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

Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

8.1

Introductory Examples

8.1.1

Batch Fermentation (BATFERM)

System The system is represented in Fig. 1, and the important variables are biological dry mass or cell concentration, X, substrate concentration, S, and product concentration, P. The reactor volume V is well-mixed, and growth is assumed to follow kinetics described by the Monod equation, based on one limiting substrate. Substrate consumption is related to cell growth by a constant yield factor YX/S- Product formation is the result of both growth and non-growth associated rates of production, where either term may be set to zero as required. The lag and decline phases of cell growth are not included in the model.

Figure 1. Stirred batch fermenter with model variables. 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

194

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Model Mass Balances: (Rate of accumulation)

= (Rate of production)

For cells

VTT = r xv or dX

For substrate V

dS dF

=

r

s

or dS dF = rS

For product dP V-3T = or dP dF = r?

Kinetics: rx = f i X

with the Monod relation, constant yield relation, and product formation kinetics:

tx/s rP = (ki + k2 |^) X

where ki is the non-growth associated coefficient, and k2 is the coefficient associated with growth. If the number of equations is equal to the number of unknowns, the model is complete and the solution can be obtained. The easiest way to demonstrate this is via an information flow diagram, as shown below in Fig. 2.

195

8.1 Introductory Examples x

o

Biomass Balance

4_ So

r

T*

x

Growth Rate

PO

M ——

1

Substrate Balance

4_

X ^ Monod

Kinetics

A

f'x Substrate Rate

r

s

p

Product Balance A 4

s ^

r

p

Product Rate

-^M ,-_

Figure 2. Information flow diagram of the batch fermenter model equations,

It is seen in that all the variables required for the solution of any one equation block are obtained as the products of other blocks. The information flow diagram thus emphasizes the complex inter-relationship involved in even this very simple problem. Solution begins with the initial conditions XQ, SQ and PQ at time t=0. The specific growth rate |i is calculated, enabling rs, rx and rp to be calculated, and hence the initial gradients dX/dt, dS/dt and dP/dt. At this time the integration routine takes over to estimate revised values of X, S and P over the first integration step length. The procedure is repeated for succeeding step lengths until the entire X, S and P concentration time profiles have been calculated up to the required final time.

Program The following Berkeley Madonna program solves the above fermentation problem: {BATPERM}

{Batch

growth

{Constants} UM=0 . 3 KS = 0 . 1 Kl=0.03 K2=0.08 Y=0. 8

with

product

;kg/m3 ;kgP/kgX h ;kgP/kgX h ;kg X/kg S

formation}

196

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

X0=0.01 SO = 10 P0=0

/Initial ;Initial ;Initial

biomass inoculum, kg/m3 substrate cone., kg/m3 product conc.,kg/m3

{Initial Conditions} INIT X=XO INIT S=SO INIT P=PO (Mass X'= S'= P' =

Balances}

RX RS RP

;BIOMASS BALANCE ;SUBSTRATE BALANCE ;PRODUCT BALANCE

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

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

S>=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 Introductory Examples

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.1 Introductory Examples

{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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.1 Introductory Examples

Run 3: 4000 steps in 0.0333 seconds 10

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

Run 8: 200000 steps in 1.38 seconds

2

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

204

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.1 Introductory Examples

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

{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

8.1 Introductory Examples

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 1: 5000 steps in 0.0333 seconds

\ \ 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

!

f^i 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

Run 1: 10000 steps in 0.15 seconds 160

«(

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

\"' /^i''^'

-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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.3 Fed Batch Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.3 Fed Batch Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.3 Fed Batch Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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<).0054 C16H1804N2S Penicillin 334 - 115 R3 C16H2005N2S R4 352 - 183 Penicilloic acid Oxygen 02 32 0 R5 CO2 Carbon Dioxide 44 -94 R6 NH3 Ammonia 17 - 19 R8 H2SO4 R9 Sulfuric Acid 98 - 194 Phosphoric Acid H3PO3 98 - 319 RIO Phenylacetic Acid C8H802 136 -69 RH H2O Water 18 -68 Rl2

Model

a) Elemental Balancing Knowing the composition of all chemical substances and the biomass mycelium (Table 1) allows the following steady state balances of the elements in terms of mol/h: For carbon

6 RI + R6 + 16 R3 + 8 RH + 16 RH + R2 = 0 For oxygen

6 RI + 2 R5 + 2 R6 + R12 + 4 R3 + 4 R9 + 4 R10 + 2 RH + 5 R4 + 0.52 R2 = 0 For nitrogen 0.16R 2 + 2 R 3 + 2R 4 + R8 = 0 For sulfur

0.00 46 R2 + R3 + R4 + R9 = 0

8.3 Fed Batch Reactors

233

For hydrogen 12 RI + 1.64 R2 + 18 R3 + 20 R4 + 3 R8 + 2 R9 + 3 RIO + 8 R n + 2 R12 = 0

For phosphorus 0.0054 R2 + RIO = 0 A steady state enthalpy balance gives the following - 303 RI - 28.1 R2 - 115 R3 - 183 R4 - 94 R6 - 19 R8 - 194 R9 -

- 3 1 9 R i o - 6 9 R n -68Ri 2 + rH = 0 where TH is the rate of heat of production (kcal/h). A total of 12 unknowns (Ri through R6, Rg through Ri 2 and TH) are involved with a total of 7 equations (6 elemental balances and one heat balance). The five additional equations are provided by five reaction kinetic relationships. The remaining rates can be expressed in terms of these basic kinetic equations. From the carbon balance - R6 = 6 RI + R2 + 16 R3 + 16 R4 + 8 RH

From the nitrogen balance - R 8 = 0.16R2 + 2R 3 + 2R 4 From the sulfur balance - R9 = 0.0046 R2 + R3 + R4 From the phosphor balance - R i o = 0.0054 R2 From the hydrogen and nitrogen balances - R5 = -6 RI - 1.044 R2 - 18.5 R3 - 18.5 R4 - 9 R n

From the enthalpy balance rH = - 669 RI - 110.1 R2 - 1961 R3 - 1961 R4 - 955 RH

234

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

To complete the model, equations for glucose uptake rate (-Ri), biomass formation rate (R2), rate of penicillin formation (Rs), precursor consumption rate (-Rn), and rate of penicillin hydrolysis (R4) must be known. Note that the reaction rates are defined with respect to total broth weight, since the process is the fed-batch type and broth weight is variable with respect to time.

b) Formulation of the Kinetic Equations Substrate (Glucose) Uptake Rate: A MONOD type equation for the uptake of sugar by P. Chrysogenum is used. -QlCiM2 Biomass Formation Rate: A linear relationship between the glucose consumption rate and growth rate of biomass is assumed. Hence, 1 - Rl = y^ ^2 + m M2

where Y2 is the maximum growth yield and m is the maintenance rate factor (mol glucose/mol mycelial biomass h). Some sugar is used in the formation of the product. Hence, - Rl = Yj R2 + m M2 + YJ (R3 + R*)

where ¥3 is the conversion yield for glucose to penicillin (mol penicillin/mol glucose). The total rate of biomass formation equals the net rate of formation, corrected for the amount transformed to penicilloic acid. Therefore, R2 = - Y 2 R i - Y 2 m M 2 - yf (Rs + «4) Precursor Conversion Rate It is assumed that the precursor is only used for penicillin synthesis. Thus

- R l l = R 3 + R4 where - RH is the precursor consumption rate.

8.3 Fed Batch Reactors

235

Rate of Penicillin Synthesis The specific rate of penicillin synthesis is assumed not to be a function of specific growth rate. So that R 3 = Q3 M2 - R4

where Q3 is the maximum specific rate of penicillin synthesis (mol/mol h), Equation for the Rate of Penicillin Hydrolysis The hydrolysis of penicillin takes place by a first-order reaction.

R 4 = K 3 M3

c) Balance Equations Total Mass Balance The individual feed rates of glucose, sulfuric acid and ammonia are adjusted to equal their molar consumption rates. Water lost by evaporation is neglected. The change in mass due to gas uptake and production is neglected. The mass flow rates are calculated from the molecular weights, the uptake rates and the mass ratio compositions. Feed rate of glucose stream (kg/h) F

l =

F

180 500

F 2.78

=

where F = mol glucose /h. Feed rate of NH3 stream (kg/h) F8 = R 8 25Q

= T4JT

Feed rate of £[2804 stream (kg/h) 18 F9 - R95QO

R9 ~ 2.55

The total mass in reactor G (kg/h) changes with time according to dG

F

"dT = Tn

+

R9

235"

Component Balances Expressed in mol/h the dynamic balances are,

+

Rg

TTTT

236

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Glucose ~3T = R l + F

Biomass dM2

Penicillin

~ar = R2 dM3

„.

Penicilloic acid dM4

The concentrations in mol/kg are as follows:

M2

Tr M3

M4

c4 = — where the masses MI, M2, M3 and M4 are in mol units.

d) Metabolism Relations The various metabolic relationships are given from Specific growth rate for cells R2

* = Ml Respiration quotient R

Re

Q = R7

Oxygen uptake rate OUR = -R 5

CO2 production rate CPR = R6

Fraction of N2 in mycelium

8.3 Fed Batch Reactors

237 R2

f 2 = 0.16 R| N2 fraction in penicillin

f3 = 1-F 2 Fraction of sulfur used for mycelium

R2 f 4 = 0.046 R| Sulfur fraction used for penicillin

f5 = 1 - F4 Fraction of glucose for cell growth =

R2

Fraction of glucose for penicillin R3 + R4

S3 = - Y3 R! Fraction of glucose for maintenance

M2 g4 = -M R^-

Program The Madonna program covers a fermentation time of 200 h starting from the initial conditions of 5500 mol glucose, 4000 mol biomass, 0 mol penicillin and 0.001 mol penicilloic acid in an initial broth weight of IxlO 5 kg. The program is on the CD-ROM.

Nomenclature Symbols a, b C CPR F

Flow rate variables Component concentration Carbon dioxide production rate Feed rate

various mol/kg mol/h kg/h

238

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

h fl f5 G 82 g3 g4 Kl

K3 M m OUR Q R RQ rq Y

Fraction of nitrogen in mycelium Nitrogen fraction in penicillin Fraction of sulfur used for mycelium Fraction of sulfur used for penicillin Mass in reactor Fraction of glucose for cell growth Fraction of glucose for penicillin Fraction glucose for maintenance Saturation constant Hydrolysis rate constant Mass of individual components Maintenance rate factor Oxygen uptake rate Maximum specific rates Conversion Respiration quotient Heat production rate Respiratory quotient Yield coefficient Specific growth rate

Indices 0 1 2 3 4 5 6 8 9 10 11 12

Exercises

initial glucose biomass penicillin penicilloic acid oxygen carbon dioxide ammonia sulfuric acid phosphoric acid phenylacetic acid water

kg

mol/kg 1/h mol mol/(mol h) mol/h mol/(mol h) mol/h kcal/h 1/h

239

8.3 Fed Batch Reactors

Results The results of Fig. 2 show the substrate MI to pass through a maximum, while the penicillin M2 develops linearly, for this constant feeding situation. Increasing the feeding linearly with time (F = 500 + 5* time) gave the results in Fig. 3, where it is seen that maintenance accounts for about 70 % of glucose consumption at the end of the fermentation. Run 1:215 steps in 0 seconds

0

20

40

60

80

100

120

140

160

180

200

TIME

Figure 2. Penicillin fed batch fermentation with total masses of glucose (M]) and biomass (M2).

240

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 1:215 steps in 0 seconds

0.9-, 0.80.7-

8

0.6-

r- 0.40.30.20.1-

020

60

80

100

120

TIME

Figure 3. Linear increase of feeding with time F = 500 + 5*T.

Reference Heijnen, J., Roels, J. A., and Stouthamer, A.H. (1979). Application of Balancing Methods in Modeling the Penicillin Fermentation. Biotechnol. and Bioeng., 21, 2175-2201.

8.3.3

Ethanol Fed Batch Diauxic Fermentation (ETHFERM)

System Yeast exhibits diauxic behavior with respect to the glucose and ethanol in the medium as alternative substrates. In addition, the glucose effect, when glucose levels are high, will cause fermentation, instead of respirative oxidation, to take place, such that the biomass yields are much reduced (Fig. 1). In this example the constant a designates the fraction of respiring biomass and (1 - a) the fraction of biomass that ferments. The rates of the process are controlled by three enzymes.

8.3 Fed Batch Reactors

24 1

^^

C02 + X

Glucose ^^^*-

Ethanol + X

Figure 1. Pathways of aerobic ethanol fermentation.

Model The rates of the processes are as follows: Respirative oxidation on glucose, R, =

Glu+K sl

Fermentation to ethanol, R2 = ——— K2 (1 - a) X Glu + KS2

Conversion of ethanol to biomass,

Enzyme activation for the transformation of ethanol to biomass is assumed to involve an initial concentration of starting enzyme EQ, which is converted to enzyme £2 and which catalyzes growth on ethanol through an intermediate enzyme EI. Thus, the production rate of enzyme EI is inhibited strongly by glucose, R4 = - -rXEo K S4 +Glu 3

and the production rate of enzyme £2 controlling the conversion of biomass to ethanol depends on EI, R5 = K 5 X E i The mass balances for the biomass, substrates and enzymes are those for a fed batch with variable volume.

242

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

For the total mass balance with constant density,

dt The component balances are written by separating the accumulation term, noting that

d(VC) _ VdC CdV _ VdC —- + - —dt dt dt dt

-

Thus, dt

V f

^o dt

=

_«

E0Q V

f -*-«.-*

Program Note that the program on the CD-ROM is formulated in terms of C-mol for the biomass. This is defined as the formula weight written in terms of one C atom, thus for yeast CHL667Oo.5No.i67-

243

8.3 Fed Batch Reactors

Nomenclature

Symbols C

E EtOU Glu K Q R V X Y a

Component concentration Enzyme concentration Ethanol concentration Substrate feed concentration Rate constants Feed flow rate Reaction rate Reactor volume Biomass concentration Yield coefficient Fraction of respiring biomass

Indices 0 1 2 3 4 5

Exercises

Refers Refers Refers Refers Refers Refers

to feed to reaction to reaction to reaction to reaction to reaction

1 2 3 4 5

mol/m3 mol/m3 mol/m3 mol/m3 various m3/h mol/m3 h m3 C-mol/m3 mol/mol

244

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results Seen in Fig. 3 are the simulation results giving the concentrations (glucose, ethanol and biomass) during the fed batch process. In Fig. 4 the maximum in ethanol concentration as a function of feedrate is given from a Parameter Plot. Run 1: 605 steps in 0.0167 seconds

30

25

60

Figure 3. Batch yeast fermentation.

245

8.3 Fed Batch Reactors

Run 2:12100 steps in 0.333 seconds 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 4. Influence of flowrate on the maximum ethanol concentration.

Reference This example was contributed by C. Niklasson, Dept. of Chemical Reaction Engineering, Chalmers University of Technology, S - 41296 Goteborg, Sweden.

8.3.4

Repeated Fed Batch Culture (REPFED)

System A single cycle of a repeated fed batch fermentation is shown in Fig. 1. In this operation a substrate is added continuously to the reactor. After the tank is full, the culture is partially emptied, and the filling process is repeated to start the next fed batch. The operating variables are initial volume, final volume, substrate feed concentration and flow rates of filling and emptying.

246

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Figure 1. One cycle of a repeated fed batch.

Model The equations are the same as given in the example FEDBAT (Section 8.1.3), where the balances for substrate and biomass are written in terms of masses, instead of concentrations. The only difference is that an outlet stream is considered here to empty the fermenter at the end of the production period.

Program Since in a Madonna program, the initial conditions cannot be reset, an outlet stream is added. The inlet and outlet streams are controlled by conditional statements as shown below. The full program is on the CD-ROM. {Statements to switch the feed and emptying streams) Fin=if time> = 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

8.3 Fed Batch Reactors

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

Refers Refers Refers Refers Refers Refers

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

Run 1: 5004 steps in 0.15 seconds

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

8.3 Fed Batch Reactors

Run 1: 5004 steps in 0.35 seconds

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.3 Fed Batch Reactors

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

Refers Refers Refers Refers

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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 Fed Batch Reactors

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

8.3 Fed Batch Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.4 Continuous Reactors

Run 3: 2021 steps in 0.15 seconds

-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).

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

258

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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)

8.4 Continuous Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.4 Continuous Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.4 Continuous Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.4 Continuous Reactors

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.

Run 10: 2000 steps in 0.0167 seconds

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

1•

3:2 (0.5) 3:3 (0.5857) _ - - 3:4(0.6714) 3:5(0.7571) — —3:6(0.8429) 3:7(0.9286) 3:8(1.014) 3:9(1.1)

1

t**

/

/

J

j* S ^

f f / t / ,f

rf'^T—

:' / / * / Jf*'^ * \ f 1 C & (

!

"^ -s^^-

K

"^"^>*i'^' ** -^

^^^*^^$^S? •** ^^*^^ **

0.5

Figure 5. Phase plane plot of influence of the initial biomass Xi from 0.5 to 1.1 for Steady states upper left and lower right.

= 0.0.

Run 20:2000 steps in 0.0167 seconds

Figure 6. Influence on the inhibition function made by varying KI between 1 and 3.

Reference Edwards, V.H, Ko, R.C. and Balogh, S.A. (1972). Dynamics and Control of Continuous Microbial Propagators Subject to Substrate Inhibition Biotechnol. and Bioeng. 14, 939-974.

267

8.4 Continuous Reactors

8.4.3

Nitrification in Activated Sludge Process (ACTNITR)

System Nitrification is the process of ammonia oxidation by specialized organisms, called nitrifiers. Their growth rate is much slower than that of the heterotrophic organisms which oxidize organic carbon, and they can be washed out of the reactors by the sludge wastage stream (Fs). In an activated sludge system (Fig. 1) when the organic load (F So/V) is high, then the high biomass growth rates require high waste rates. Nitrification will not be possible under these conditions because the concentration of nitrifiers (Ni) will become very low.

O,F O

2, F4

Reieto*

2, F3 Figure 1. Configuration and streams for the activated sludge system.

Model The dynamic balance equations can be written for all components around the reactor and around the settler. The settler is simplified as a well-mixed system with the effluent streams reflecting the cell separation. Organic substrate balance for the reactor: = F 0 So + F 2 S 2 -

268

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Ammonia substrate balance in the reactor: = FQ AQ + F2 A2 - FI A

R2Vt

Reactor balance for the heterotrophic organisms: pj7

= p2 O2 — FI QI + RI Vj_

Reactor balance for the nitrifying organisms: l

= F 2 N 2 - F i N i + R2Vi

di

Organic substrate balance in the settler: V 2 dS 2

—j^—

= F i S i - F3S2 - F4S2

Ammonia substrate balance for the settler: V 2 dA 2

—3t—

= F A

I I -

- F4A2

Balance for heterotrophic organisms in the settler: V 2 d0 2 dt

= Fl °l - F3 02

Balance for nitrifying organisms in the settler:

V2 dN2 —34— = F i N i - F 3 N 2 The equations for the flow rates are given below. Recycle flowrate:

F2 = F 0 R where R is the recycle factor. Reactor outlet flow: Flow of settled sludge:

FI = F2 + F0 = F O R + FO

8.4 Continuous Reactors

269

where C is the concentration factor for the settler. Flow of exit substrate:

F4 = FI - F3> Flow of exit sludge wastage: F5 = F3 - ?2.

Note that C and R must be chosen so that F5 is positive. Monod-type equations are used for the growth rates of the two organisms.

R

R

l

=

l^2max 2 = ^Ni =

Program The program is given on the CD-ROM.

Nomenclature Symbols A C F Fo-5 KI K2 N O R

Ammonia substrate concentration kg/m3 Concentrating factor for settler Flow rate m3/h Flow rates, referring to the figure m3/h Saturation constant of heterotrophs kg/m3 Saturation constant of nitrifying organisms kg/m3 Concentration of nitrifiers kg/m3 Concentration of heterotrophs kg/m3 Recycle factor -

270

Rl R2

s

V Y Hi

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Growth rate of heterotrophics kg/m3h Growth rate of nitrifying organisms kg/m3h Organic substrate concentration kg/m3 Volumes m3 Yield coefficients kg/kg Specific growth rate of heterotrophs 1/h Specific growth rate of nitrifying organisms 1/h

Indices Flow and concentration indices referring to Fig. 1 are as follows: 0 Refers to feed and initial values 1 Refers to reactor and organic oxidation 2 Refers to settler and ammonia oxidation 3 Refers to recycle 4 Refers to settler effluent 5 Refers to sludge wastage m Refers to maximum

Exercises

271

8.4 Continuous Reactors

Results The results in Fig. 2 demonstrate the influence of flow rate on the effluent organics 82- The ammonia in the effluent A2 is seen, in Fig. 3, to respond similarly to FQ, but for a very high value of FQ = 1000 m3/h the nitrification ceases, and A2 becomes the same as the inlet value AQ. This corresponds to washout of the nitrifiers, which would be seen by plotting NI versus time. Run 4: 405 steps in 0.0167 seconds 0.9 •, 0.8 -I

,/"

°M if If J II / It

0.3- rr •J II " 0-1 JM

02.

|

°" . 6

2

4

6

8

10

12

82:1(20) 82:2(180) 82:3(340) 82:4(500)

14

16

18

20

TIME Figure 2. Transient of S2 at various flow rates F0 (20 to 500m3/h, bottom to top).

Run 4:405 steps in 0.0167 seconds 0.1 -I

0.090.08-

3 0.06 -|-_005 J* 1

A2:1(20) — — A2:2(180) -_A2:3(340) I A2:4(5QO)

S.

0.040.03-

0.02J 0

2

4

6

8

10 TIME

12

14

16

18

20

Figure 3. Ammonia in the effluent (A2) at various flow rates F0 (5 to lOOOm^/h, bottom to top).

272

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

8.4.4

Tubular Enzyme Reactor (ENZTUBE)

System A tubular, packed-bed, immobilized-enzyme reactor is to be investigated by simulation. The flow is assumed to be ideal plug flow. The distribution of the enzyme is not uniform and varies linearly from the inlet to higher values at the outlet, as shown in Fig. 1.

Enzyme concentration Enzyme distribution

Distance along reactor, Z Figure 1. Distribution of enzyme along the tubular reactor.

Model The equations for steady state operation are given below. Substrate balance, dS dZ

=

1 ~v

Kinetics,

The linear flow velocity is increased by the presence of the solid enzyme carrier particles according to

8.4 Continuous Reactors

273

V7 L

=

F Ae ~

The reaction velocity depends on the enzyme concentration, vm = KE

and the linear distribution of enzyme distribution given by, E = E0 + mZ

Program The model is solved by renaming the independent variable, TIME, to be the reactor length coordinate Z. The program is given on the CD-ROM.

Nomenclature Symbols A F K KM m r S vm vz Z e E

Reactor tube cross section Flow rate Rate constant Michaelis-Menten constant Enzyme distribution constant Reaction rate Substrate concentration Maximum reaction velocity Linear flow velocity Reactor length Void volume fraction of packing Enzyme concentration

m2 m3/h 1/h kg/m3 kg/m3 m kg/m3 h kg/m3 kg/m3h m/h m kg/m3

274

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Indices 0 S

Refers to inlet Refers to substrate

Exercises

Results Flow rate is the primary operating variable, along with enzyme loading and inlet concentration. In Fig. 2 the influence of F is seen in the steady-state, axial, substrate profile.

275

8.4 Continuous Reactors

Run 6:1000 steps in 0.05 seconds

7

10

12

14

16

18

20

Figure 2. Substrate profile under the influence of F (1 to 10 m^/h, bottom to top).

8.4.5

Dual Substrate Limitation (DUAL)

System In defined-nutrient growth media, one substrate can usually be made to be limiting by adjusting its concentration relative to those of the other medium components. In general, however, more than one substrate may limit the cell growth rate. In this case the yield coefficients for the various components, Yxsi> may vary depending upon the growth regime. This situation was discussed by Egli et al. (1989), who examined results at steady state with dual nutrient limitation. The present mathematical model simulates the transient behaviour of such a dual (Si -carbon, 82 -nitrogen) nutrient-limited system when carried out in a chemostat. The model assumes that the yield coefficients are each a function of the ratio 81/82, i.e. the ratio of the carbon-nitrogen substrate concentrations in the vessel. The original paper took the carbonnitrogen ratio in the feed stream as the controlling parameter. Here the concentrations in the reactor are assumed to be controlling.

Model Assuming a perfectly mixed, constant volume continuous-flow stirred-tank reactor, the mass balance equations for the cells and for the two limiting substrates are as follows:

276

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

= D

D

1

(SlFeed - Si) -

/O

O \

I »^__i^^_ I i i "V

(o2pee(j — o2) — lv YQO J ]LlA

where D = F/V. The specific growth rate is modelled as Si

V

S

2

The yield coefficients are assumed to vary with the carbon-nitrogen ratio in the reactor. Si RATIO = ^ The yield coefficients are varied according to RATIO using the following logic: Y X Sl=Yi m i n

and and

YXS2 = Y2min YXS2 = Y2max

if if

RATIO < B i RATIO > B2

where, _ Y2min Y 2max Bi = \r 1 - and Bo = 1v~,—T" Imax

- 1mm

The boundaries of the three growth regimes in Fig. 1 are defined by the quantities BI and B2.

277

8.4 Continuous Reactors

C limitation

N limitation

Double limitation

10

XSi

0.8 r

XS1

B2

S2

Figure 1. Limitation regions for carbon and nitrogen showing influence on yield.

The yield coefficients for biomass on nitrogen and carbon take maximum or minimum values when only one substrate is limiting and vary linearly with opposing tendencies in the double-limitation region.

Program Note that the programing of this example is rather more complicated than usual owing to the need to allow for the logical conditions of carbon limitation, nitrogen limitation or both substrates together causing limitation. A partial listing is seen below and the full program is on the CD-ROM. (CALCULATION

OF

YIELD

VALUES)

YXSl=if (RATIO < Bl) then YlMAX else ( if (RATIO > B2) then Y1MIN else (Y1MAX+(RATIO-B1)/(B2B1)*(Y1MIN-Y1MAX)) ) YXS2 = if (RATIO < Bl) then Y2MIN else ( if (RATIO > B2) then Y2MAX else (Y2MIN+(RATIO-B1)/(B2Bl)*(Y2MAX-Y2MIN)) )

278

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Nomenclature Symbols Bi B2 Cc Cn D F

KS R

Si S2 X Y H

Ratio of Y 2min /Yi max Ratio of Y 2max /Yi min Carbon source concentration Nitrogen source concentration Dilution rate Volumetric feed rate Affinity constant Reaction rates Carbon source concentration Nitrogen source concentration Biomass concentration Yield coefficient Specific growth rate

Indices 1 2

Exercises

Refers to carbon source Refers to nitrogen source

_ -

kg/m3 kg/m3

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

1/h

279

8.4 Continuous Reactors

Results The startup of a continuous culture is shown in Fig. 2. Note that the nitrogen level 82 in the reactor drops to a low level after 15 h and causes a change in the yield coefficients. The influence of dilution rate on the system was investigated by varying D from 0 to 1.5 as shown in Fig. 3.

Run 1: 305 steps in 0.0333 seconds

3-c

1

Figure 2. Startup of a continuous culture.

Run 4: 305 steps in 0 seconds

X 1.5

Figure 3. Variation of D from 0.1 to 1.5 (top to bottom).

280

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Reference Egli, Th., Schmidt, Ch. R. (1989). "On Dual-Nutrient-Limited Growth of Microbes, with Special Reference to Carbon and Nitrogen Substrates", in Proceed. Microb. Phys. Working Party of Eur. Fed Biotech. Eds. Th. Egli, G. Hamer and M. Snozzi, Hartung-Goree, Konstanz, 45-53. This example was developed by S. Mason, ETH-Zurich.

8.4.6

Dichloromethane in a Biofilm Fluidized Sand Bed (DCMDEG)

System The process involves the removal of dichloromethane (DCM) from a gas stream and the subsequent degradation by microbial action. The reactor consists of biofilm sand bed column with circulation to an aeration tank, into which the substrate and oxygen enters in the gas phase, or the substrate can be fed in a liquid stream, as shown in Fig. 1. The column is approximated by a series of six stirred tanks. The reaction is treated with homogeneous, double saturation kinetics with dichloromethane (DCM) inhibition. The oxidation of one mole of DCM produces 2 moles of HC1, making a hydrogen ion balance for pH important. The yield with respect to oxygen is 4.3 mg DCM/mg 62. In practice, care must be taken to prevent stripping of DCM to the air stream.

281

8.4 Continuous Reactors

C

SR6>

SRin» C jn , pH jn

Figure 1. Schematic of fluidized bed column with external aeration vessel.

Model The model does not include a gas balance on the aeration tank, since it is assumed that the gas phase dynamics are comparatively fast and hence an equilibrium with the inlet concentration of oxygen and DCM may be assumed. The biomass is assumed to grow slowly, and growth rates are therefore also not modelled. The model for pH changes does not include buffering effects. For the inlet section 1 at the bottom of the column the balances are as follows: O2 balance,

dCQ1 ^ dt

282

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

DCM balance, dC

Srl _ dt

C

Srin~ C Srl

t

+

H ion balance,

i -CHI

dCHi

2r S i 84900

Here T is the residence time of the liquid in one section of the column. The constant 84,900 converts grams to moles and includes the stoichiometry. pHi = -0.434 log |Cm| Evaluation of rates for the inlet section 1:

V

maxCSrl

-01

KI )

For the aeration tank the 62 and DCM balances are: K L a 0 2(Co2eq-Coin)

dC

•

R V

—-~ = — (CSr6 - CSrin )

at

DCM (Cs2eq -

Tr- (CSFO ~ CSrin

Program The program constants describe DCM entering the reactor in the gas stream. The DCM concentration in the liquid feed is set to zero. The program is on the CD-ROM.

8.4 Continuous Reactors

Nomenclature

oin

srin

CSFO

CSG F KI KLa KS pHn R

VR

VT Vmax YSO

Exercises •11

H+ ion concentration in section n kg mol/m3 Inlet dissolved oxygen concentration g/m3 Oxygen saturation constant g/m3 DCM saturation constant g/m3 DCM inlet concentration g/m3 DCM concentration in section n g/m3 Oxygen concentration in section n liquid g/m3 DCM concentration in feed g/m3 DCM gas concentration g/m3 Feed rate m3/h Inhibition constant g/m3 Transfer coefficients for DCM and ©2 1/h Saturation constants g/m3 pH in n section n pH units Recirculation rate m3/h Oxygen uptake rate in section n g/m3 h Substrate uptake rate in section n g/m3 h Reactor volume m3 Volume of aeration tank m3 Maximum degradation rate g/m3 h Yield coefficient for DCM/oxygen Liquid residence time in one section h

283

284

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results The concentrations in the stream leaving the top of the column (CSr6) during startup of the fluidized bed are shown in Fig. 2 for four values of F (0.5 to 10) The change of the pH for one flow rate (F = 0.5) is shown in Fig. 3.

285

8.4 Continuous Reactors

Run 4: 55 steps in 0.0167 seconds

0

0.05

0.1

0.15

0.2

0.25

0.3

0.4

0.35

0.5

0.45

TIME

Figure 2. Fluidized bed startup for four values of F (0.5 to 10, bottom to top).

Run 1:55 steps in 0 seconds 3.5

3 2.5

:

— CSR6:1 ... PH6:1

2 1.5

1 0.5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

TIME

Figure 3. Change of carbon substrate and pH in the top section 6 during startup.

Reference D. Niemann Ph.D. Dissertation 10025, ETH, 1993.

286

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

8.4.7

Two-Stage Chemostat with Additional Stream (TWOSTAGE)

System Two chemostats are arranged in series (Fig. 1) with the intention that the first operates at a relatively high rate of cell growth, while the second operates at low growth rate, but high cell density, for secondary metabolite production. Additional substrate may be fed to the second stage. , 810

X1.S!

Hi US

Figure 1. Two-stage chemostat with two feed streams.

Model The balance equations are written for each component in each reactor. Stage 1 with sterile feed,

= F[S O -S!] -

287

8.4 Continuous Reactors

Stage 2 with additional substrate feed and an input of cells and substrate from Stage 1, V2

- [F + Fi]X2

V2 ^2.= F [Si - S2] + F! [Sio - S2] dt

Yv

KS + S2

Productivity for biomass: First stage, Prodi =

V,

Both stages, Prod2 =

Program The program is on the CD-ROM.

Nomenclature Symbols F

KS Prod S V X Y

Volumetric feed rate Saturation constant Productivity for biomass Substrate concentration Reactor volume Biomass concentration Yield coefficient Specific growth rate

m3/h kg/m3 kg/m3 h kg/m3 3

kg/m3 kg/kg 1/h

288

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Indices 0 1 2 10 m

Refers Refers Refers Refers Refers

to tank 1 inlet to tank 1 and inlet of tank 2 to tank 2 and outlet of system to separate feed for tank 2 to maximum

Exercises

Results The results in Fig. 2 give biomass concentrations and productivities for both tanks during a startup with a constant feed stream to the first tank (F = 0.5). In Fig. 3 the influence on X2 of feed to the second tank Fl (0 to 1.0) with constant F is shown.

289

8.4 Continuous Reactors

Run 1: 805 steps in 0.0333 seconds •T5

35

40

Figure 2. Biomass (Xj X2) and productivities for both tanks (F = 0.5).

Run 4: 805 steps in 0.0333 seconds

5 4.5

4 3.5

3

32.5 2 1.5

1 0.5

0 10

15

20

25

30

35

40

TIME

Figure 3. Influence on X2 of feed to the second tank (Ft = 0 to 1.0, curves right to left).

290

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

8.4.8

Two Stage Culture with Product Inhibition (STAGED)

System Products may inhibit growth rates. Under such conditions a multi-staged continuous reactor as shown in Fig. 1 will have kinetic advantages over a single stage. This is because product concentrations will be lower and consequently the rates in the first tank will be higher as compared with a single tank. This effect may be conveniently investigated by simulation. Batch cultures can be expected to have similar kinetic advantages for product inhibition situations.

Figure 1. Two-stage chemostat with product inhibition.

Model The inhibition function is expressed empirically as

When product concentrations are low, the equation reduces to the Monod equation. The product kinetics are according to Luedeking and Piret, with dependence on both growing and non-growing biomass, 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

8.4 Continuous Reactors

291

rpn = (On + (3n \ln) Xn

In addition, the non-growth term, an, is assumed to be inhibited according to,

an - a"Q ~ 1i-r+rPn When product concentrations are low, a = ano. Kinetics for growth:

Kinetics for substrate consumption (neglecting consumption for product):

_ _rxn where Y is the yield factor. Mass balances: Stage 1,

j- = F[So-Si] +r S iVi jp = F[P 0 -Pi] + rp^j Stage 2 with additional substrate feed FI, dX2 V 2 -gjT- = F Xj - [F + F!]X2 + rX2V2 dS2 V2 -gj- = F [Si - S2] + FI [Sio - S2] + rS2V2 dP2 - =FPl- [F + Fi]P2 + rp2V2

Productivity for product: First stage,

292

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Prodi = Both stages,

Program The program is on the CD-ROM.

Nomenclature Symbols F KI KS P

Prod r S V X Y

a OC0

P

Volumetric feed rate Inhibition constant Saturation constant Product concentration Productivity for product Reaction rate Substrate concentration Reactor volume Biomass concentration Yield coefficient Non-growth product rate term Non-growth term with no inhibition Growth dependent product yield Specific growth rate Maximal specific growth rate

Indices n 0 1

Refers to tank n Refers to tank 1 inlet Refers to tank 1 and inlet of tank 2

m3/h kg/m3 kg/m3 kg/m3 h kg/m3 h kg/m3 m3 kg/m3 kg/kg kg P/kg X h kg P/kg X h kg/kg 1/h 1/h

8.4 Continuous Reactors

2 10

293

Refers to tank 2 and system of outflow Refers to inlet concentration of tank 2

Exercises

Results The startup and approach to steady state for the two stages is shown in Fig. 2. The influence of the inhibition can be tested by varying KI from 0.1 to 10.0, as shown in Fig. 3. The higher the KI the lower is the degree of inhibition and the greater is the product concentration P2-

294

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 1: 255 steps in 0 seconds

4

r 10

,

3.5

3 2.5

, 1.5

1 0.5

0 10

15 TIME

Figure 2. Startup and approach to steady state for the two stages. Run 4: 255 steps in 0.0167 seconds

1.4., 1.3

1.2. 1.1

I 1 0.9. 0.8 0.7 J

10

12

14

16

18

20

22

24

26

TIME

Figure 3. Product concentration P at various values of KI (1 to 5), curves bottom to top.

Reference Herbert, D. (1961). A Theoretical Analysis of Continuous Culture Systems. Soc. Chem. Ind. Monograph No. 12, London, 2L

295

8.4 Continuous Reactors

8.4.9

Fluidized Bed Recycle Reactor (FBR)

System A fluidized bed column reactor can be described as 3 tanks-in-series (Fig. 1). Substrate, at concentration SQ, enters the circulation loop at flow rate F. The flow rate through the reactor due to circulation is FR. Oxygen is absorbed in a well-mixed tank of volume VT. The reaction rate for substrate (r$) depends on both S and dissolved oxygen (CL)- The rate of oxygen uptake (ro) is related to S by a yield coefficient (Yos)- The gas phase is not included in the model, except via the saturation concentration (CLS)- The oxygen uptake rate of reactor can be determined by the difference in CL inlet and outlet values.

? So , Fn

Fluidized Bed F,S

Figure 1. Biofilm fluidized bed with external aeration.

296

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Model The model balance equations are developed by considering the individual tank stages and the absorber separately. The gas phase in the absorber is assumed to be air. Substrate balances: For the absorption tank dS

FR

dF =

For each stage n dSn FR -3T = -^(Sn-!-Sn)- rsn

Oxygen balances: For the absorption tank r

= ^(C L 3-C L )+K L a(C L s-C L ) VT

For each stage dCLn ~dT"

=

FR V

(CLn

-! ~ C Ln) ~ rOn

Kinetics for stage n:

V Tm K n +Sn K 0 +C Ln

Program The program is on the CD-ROM.

297

8.4 Continuous Reactors

Nomenclature Symbols

CL CLS F FR KLa Ks Ko r S

V VT ^m

x Y T

Dissolved oxygen concentration Saturation oxygen concentration Feed flow rate Recycle flow rate Transfer coefficient Saturation constant Saturation constant for oxygen Reaction rate Substrate concentration Reactor volume of one stage Volume of absorber tank Maximum velocity Biomass concentration Yield coefficient Inverse liquid residence time

Indices 0 l,2,3,n m O S T X

Exercises

Refers to feed Refer to the stage numbers Refers to maximum Refers to oxygen Refers to substrate Refers to aeration tank Refers to biomass

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

298

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results Note from the results below that the steady state for oxygen is reached rather quickly, compared to that of substrate.

Run 1:1003 steps in 0.0333 seconds

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 2. Oxygen concentrations in fluidized bed reactor. Top of column is the lower curve.

299

8.4 Continuous Reactors

Run 1:10003 steps in 0.4 seconds

35

tf

Figure 3. Substrate concentrations from the run as in Fig. 2.

8.4.10

Nitrification in a Fluidized Bed Reactor (NITBED)

System Nitrification is an important process for wastewater treatment. It involves the sequential oxidation of NFLt"1" to NO2~ and NC>3~ that proceeds according to the following reaction sequence: NH4+ + 1 02 -> N02- + H20 +2H+ NO2~ +

O2 -» NO3~

The overall reaction is thus NH4+ + 2O2

NO3- + H2O + 2H+

Both steps are influenced by dissolved oxygen and the corresponding nitrogen substrate concentration. Owing to the relatively slow growth rates of nitrifiers, treatment processes benefit greatly from biomass retention.

300

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

In this example, a fluidised biofilm sand bed reactor for nitrification, as investigated by Tanaka et al. (1981), is modelled as three tanks-in-series with a recycle loop (Fig. 1). With continuous operation, ammonium ion is fed to the reactor, and the products nitrite and nitrate exit in the effluent. The bed expands in volume because of the constant circulation flow of liquid upwards through the bed. Oxygen is supplied external to the bed in a well-mixed gasliquid absorber.

Model The model balance equations are developed by considering, separately, the individual tank stages and the absorber. Component balances are required for all components in each section of the reactor column and in the absorber, where the feed and effluent streams are located. Although the reaction actually proceeds in the biofilm phase, a homogeneous model with apparent kinetics is employed rather than a biofilm model, as in the example NITBEDFILM. 03.

Fluidized bed

Figure 1. Biofilm fluidised-bed recycle loop reactor for nitrification.

In the absorber, oxygen is transferred from the air to the liquid phase. The nitrogen compounds are referred to as Si, 82, and 83, respectively. Dissolved

8.4 Continuous Reactors

301

oxygen is referred to as O. Additional subscripts, as seen in Fig. 1, identify the feed (F), recycle (R) and the flows to and from the tanks 1, 2 and 3, each with volume V, and the absorption tank with volume VAThe fluidised bed reactor is modelled by considering the component balances for the three nitrogen components (i) and also for dissolved oxygen. For each stage n, the component balance equations have the form

Similarly for the absorption tank, the balance for the nitrogen-containing components include the input and output of the additional feed and effluent streams, giving

The oxygen balance in the absorption tank must account for mass transfer from the air, but neglects the low rates of oxygen supply and removal of the feed and effluent streams. This gives

For the first and second biological nitrification rate steps, the reaction kinetics for any stage n were found to be described by v r

=

r2n

=

ml Sin Qn K + S K + O

V

m2 S2n °n S K 2+ 2n O2+°n

K

The oxygen uptake rate is related to the above reaction rates by means of the constant yield coefficients, YI and ¥2, according to ron = - H n Y i -r 2 n Y 2 The reaction stoichiometry provides the yield coefficient for the first step

302

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

YI = 3.5 mg O2/(mg NNH4) and for the second step Y2 = LI mg O2/(mg NNO2)

Program The program is found on the CD-ROM.

Nomenclature

Symbols F FR Kj^a K KI K2 O Os and O* OUR r S V VA

Feed and effluent flow rate Recycle flow rate Transfer coefficient Saturation constants Saturation constant for ammonia Saturation constant for ammonia Dissolved oxygen concentration Oxygen solubility, saturation cone. Oxygen uptake rate Reaction rate Substrate concentration Volume of one reactor stage Volume of absorber tank

L/h L/h h mg/L mg/L mg/L mg/L mg/L mg/L mg/L h mg/L L L

vm Y

Maximum velocity Yield coefficient

mg/L h mg/mg

Indices 1,2,3 1,2,3 A F ij m

Refer to ammonia, nitrite and nitrate, resp. Refer to stage numbers Refers to absorption tank Refers to feed Refers to substrate i in stage j Refers to maximum

8.4 Continuous Reactors

Ol and O2 S1,S2 S and *

Exercises

Refer to oxygen in first and second reactions Refer to substrates ammonia and nitrite Refer to saturation value for oxygen

303

304

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results

Run 1: 519 steps in 0.2 seconds 6

10

15

20

25

30

35

40

45

50

Figure 2. Dynamic startup of continuous operation showing oxygen concentrations and nitrogen compounds at the top of the column.

Run 2:10386 steps in 4.83 seconds 280

P 2.5

270 260 250

1^240.

5

£,230< 220

c

M I

210 200

0.5

190 180

15

20

25

30

35

40

KLA

Figure 3. Parametric run of continuous operation showing oxygen and ammonia in the effluent versus

305

8.4 Continuous Reactors

8.4.11

Continuous Enzymatic Reactor (ENZCON)

System This example, schematically shown in Fig. 1 involves a continuous, constant volume, enzymatic reactor with product inhibition in which soluble enzyme is fed to the reactor.

EO.FE

I» S 1f P 1 § F1

Figure 1. Continuous enzymatic reactor with enzyme feed.

Model The mass balance equations are formulated by noting the two separate feed streams and the fact that the enzyme does not react but is conserved. Total flow:

FS + FE = Mass balances:

dSi

= FsSo-FiS1+rsV

306

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

r = -FiP1+rpV

Kinetics with product inhibition: r

S - - v mK M + S + (P/Ki) vm = EI K2 rP = - 2 r s

Program The program is found on the CD-ROM.

Nomenclature Symbols E F KI KM K2 P r S V vm

Enzyme concentration Flow rate Inhibition constant Saturation constant Rate constant Product concentration Reaction rate Substrate concentration Reactor volume Maximum rate

Indices 0

Refers to inlet values

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

307

8.4 Continuous Reactors

1 E P S

Refers Refers Refers Refers

to reactor and outlet values to enzyme to product to substrate

Exercises

Results Variations in the flows FE (Fig. 2) or Fs (Fig. 3) cause the product levels to change.

308

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 3: 8004 steps in 0.1 83 SeC ndS

1.8

°

1.6

--—

s

0.8 .0.6

/

0.4

tS

/

P1:2(0.2) P1:3(0.3)

,-'"

0.2

0

10

20

30

40 TIME

50

60

70

80

Figure 2. Performance for three values of FE.

Run 3: 8004 steps in 0.233 seconds

r4.5

• 3.5

x--'

•3 • 2.5 .2 -1.5

r *>"*

-^ -» **" ~ ""

I

jft

—. P1:2(1.5) .. P1:3(2)

•<*/r 0

10

20

30

40

50

60

70

80

TIME

Figure 3. Performance for three values of Fs.

8.4.12

Reactor Cascade with Deactivating Enzyme (DEACTENZ)

System Biocatalysts usually deactivate during their use, and this has to be considered in the bioreactor design. One of the methods to keep productivity fluctuations low, and hence to efficiently utilize the biocatalyst, is to use a series of reactors with biocatalyst batches having different times-on-stream in each reactor. In

309

8.4 Continuous Reactors

this example a series of three stirred tanks of a constant equal volume with biocatalyst deactivating by first order reaction kinetics is investigated (Fig. 1). After a time period ILAG? ^e biocatalyst from the tank with longest time-onstream (first tank in the cascade) is discarded and replaced by a fresh batch. The streams are switched over so that tank 1 becomes tank 3, the last reactor in the series. Other tanks are switched over correspondingly. This is equivalent to replacing the used enzyme with fresh enzyme in tank 3 and moving the used enzyme upstream from tank 3 to tank 2 to tank 1, which is easier to simulate. (3-galactosidase was taken as an example of the biocatalyst. This obeys Michaelis-Menten kinetics with competitive product inhibition, and the kinetic constants were determined with considerable accuracy. The same constants are used also in this substrate inhibition model. F,S 0

F,Si

F,S 2

F,S 3

Figure 1. Tanks in series reactor with immobilized enzyme.

Model Using the stoichiometry, S —> P, the mass balances for the ith tank (i = 1, 2, 3) with the volume V can be written Substrate Product dt

Enzyme (active)

= F(P M -Pi)

310

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

=r Ei v where rate of substrate consumption (competitive)

is given by product

inhibition

s i Si = ' v max b i -7-Z~^\

r

V

K

inh

According to the molar stoichiometry R P i = -R Si The rate of enzyme deactivation is assumed to be: rEi = - kD EI

For each batch of enzyme in tank i dE

i

-i

c

This equation can be applied by changing the initial conditions for each tank when the enzyme is moved from tank to tank. Thus the final value in tank n becomes the initial condition in tank i-1. The initial conditions can also be calculated by analytical integration of the enzyme deactivation equation at times corresponding to the respective ages of the biocatalysts in the respective reactors (multiples of TLAG)- Fresh enzyme with the activity EQ is in the third tank. The other tanks start with the following enzyme activities: EI = E0 e C- (3 - i) ko TLAG]

Program In the program on the CD-ROM note that the cost calculation at the end of the program is included only as a comment but could be incorporated into the program with the corresponding values for the constants.

311

8.4 Continuous Reactors

Nomenclature Symbols COST E ECOST F ICOST kD Kinh Km

OCOST P RC m rs S T t

TDOWN TLAG v

max

Specific product costs Enzyme concentration Enzyme cost Flow rate Investment cost Deactivation constant Inhibition constant Michaelis - Menten constant Operating cost Total amount of product Reactor refill cost Reaction rate of deactivation Reaction rate of substrate Substrate concentration Residence time Time Down time Time-on-stream difference Maximum specific reaction rate

Indices 0 i

Exercises

Refers to initial, feed Refers to reactor number

$/kg kg/m3 $/kg m3/h $/kg 1/h mol/m3 mol/m3 $/kg mol $/kg kg/(m3 h) mol/(m3 h) mol/m3 h h h h mol/kg h

312

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results The results from DEACTENZ show an exponential decrease of the biocatalyst activity (Fig. 2), which causes dynamic changes in the substrate and product concentrations (Fig. 3) in all three reactors.

313

8.4 Continuous Reactors

Run 1: 50000 steps in 0.917 seconds

0.5-,\ 0.45-

-4000

\ \ %

0.4-

i**

*.

0.35-

"%

03

a '

x

.0.25-

"*

s -'

••-.

0.1-

— ^ — — -" 0

100

200

-3000 ^

-2500 3

I -2000 Q_

_-— Totalproduct:1

-1500 p

0.050-

-3500

—.— E2:1

*"">cr ^r "'"'"•-'"•*«. | j+ _/• "•• «.» -i..

0.15-

s ***

"''

x

0.2-

r*

j**

H

-1000 ••••••

—i...

• —• 300

400

500

.500 -0

600

700

800

900

1000

TIME

Figure 2. Exponential biocatalyst deactivation and total product during one run.

Run 1: 50000 steps in 0.933 seconds 120

140-i 120-

100

10080 °l C/l ^

cn

8

a-

°-

60

40 40-

20

20 0

0

100

200

300

400

500

600

700

800

TIME

Figure 3. Dynamic changes in the substrate and product concentrations.

900

1000

314

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

References Prenosil, I.E., Peter, J., Bourne, J.R. (1980). Hydrolytische Spaltung des Milchzuckers der Molke durch immobilisierte Enzyme im Festbett-Reaktor. Verfahrenstechnik 14, 392. Prenosil, J.E. (1981). Optimaler Betrieb fur einen Festbett- und einen FliessbettReaktor mit desaktivierendem Katalysator. Chimia 35, 226 . Prenosil, J.E., Hediger, T. (1986). An Improved Model for CapillaryMembrane, Fixed-Enzyme Reactors. In Membranes and Membrane Processes, Plenum, N. Y., 515.

8.4.13

Continuous Production of PHB in a Two-Tank Reactor Process (PHBTWO)

System This example considers a two-stage process for the production of PHB, a biopolymer. The kinetics of this fermentation is presented in the example PHB. The structured kinetic model involves a Luedeking-Piret-type expression and also an inhibition by the product. From this it might be expected that two stages would be better than one, and it is the goal of this example to optimize the process. The volume ratio and the feed rate are the obvious design and operating parameters. Sfeed,

> 82, F0

Figure 1. Configuration of the two-tank system.

8.4 Continuous Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.4 Continuous Reactors

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.

Run 20: 20380 steps in 5.78 seconds

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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:

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

8.5 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.

Program The program is found on the CD-ROM.

Nomenclature Symbols CL CLS>CL* G KCL ^a 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 K^a 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Indices 0

o p s

Refers Refers Refers Refers

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

8.5 Oxygen Uptake Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.5 Oxygen Uptake Systems

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.

Program The program is on the CD-ROM.

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.5 Oxygen Uptake Systems

325

326

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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 Oxygen Uptake Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

~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 )

Program The program is found on the CD-ROM.

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

8.5 Oxygen Uptake Systems

Exercises

329

330

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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 Oxygen Uptake Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.5 Oxygen Uptake Systems

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 K^a. 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

8.5 Oxygen Uptake Systems

Run 3: 10004 steps in 0.183 seconds

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 (^a) 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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 K^a 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

8.5 Oxygen Uptake Systems

337

where VG/V = TG , and K^a 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.5 Oxygen Uptake Systems

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 K^a 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.5 Oxygen Uptake Systems

Results

Run 1: 206 steps in 0 seconds

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 1: 206 steps in 0 seconds

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

8.5 Oxygen Uptake Systems

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

^^iiiriS i;|tii^|;i;||

I Transfer

^ TriT1

^•^iinS^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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

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

8.5 Oxygen Uptake Systems

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 ) ^E.

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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)

8.5 Oxygen Uptake Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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.

Run 2: 20660 steps in 25.2 seconds

• 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 Oxygen Uptake Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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.

8.5 Oxygen Uptake Systems

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 ^a 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

353

8.5 Oxygen Uptake Systems

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 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

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

8.6 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.6 Controlled Reactors

Results

Run 1: 14286 steps in 0.0667 seconds 7000

6000

3000

•1000

70

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

80

90

100

358

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.6 Controlled Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.6 Controlled Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.6 Controlled Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.6 Controlled Reactors

8 = (X-X S et)

Program The program is on the CD-ROM.

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

Refers Refers Refers Refers

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.6 Controlled Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.6 Controlled Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.7 Diffusion Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.7 Diffusion Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.7 Diffusion Systems

375

376

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results

Run 1: 5009 steps in 5.53 seconds

.1

Figure 2. Oxygen and substrate time profiles for a step change in KLA.

Run 1: 5009 steps in 5.47 seconds

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

8.7 Diffusion Systems

Run 1: 5009 steps in 5.85 seconds

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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 \ ^is 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

8.7 Diffusion Systems

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<

=^ '

P

' =^

and X' =

gives d^' dX'2

L2R' DSS0 ~

where, R' =

kES' (K M /S 0 )(1 +(S 0 F/K I ))

and, P = (1 - S')

with boundary conditions at X' = 1

X' = 0

S' = 1

dS'/dX' = 0

The catalyst effectiveness may be determined from DS Sp (dSVdX')x=l ^ L2R0 =

380

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

where RQ is the reaction rate determined at surface conditions, kES0

°~K M (i +P O /K I ) + SO Program The dimensionless model equations are used in the program on the CD-ROM. Since only two boundary conditions are known, i.e., S at X = L and dSVdX' at X' = 0, the problem is of a split-boundary type and therefore requires a trial and error method of solution. Since the gradients are symmetrical, as shown in Fig. 1, only one-half of the slab must be considered. Thus starting at the midpoint of the slab at X1 = 0, where dSVdX' = 0, an initial value for S1 is assumed (SGUESS). After integrating twice, the computed value of S is compared with the known value of SQ at X' = 1. A revised guess for S' at Xf = 1 is then made. This is repeated until convergence is achieved.

Nomenclature Symbols D E K k L P R S X

Diffusion coefficient Enzyme concentration Kinetic constant Reaction rate constant Distance from slab center to surface Product concentration Reaction rate Substrate concentration Length variable Effectiveness factor

Indices I M

Refers to inhibition Refers to Michaelis-Menten

m2/h mol / m3 kmol / m3 1/h m kmol / m3 kmol /(m3 h) kmol / m3 m

381

8.7 Diffusion Systems P S !

0 GUESS

Exercises

Refers Refers Refers Refers Refers

to product to substrate to dimensionless variables to bulk concentration to assumed value

382

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results

Run 5:1000 steps in 0 seconds

0.2

0.5

0.3

0.6

0.7

X

Figure 2. Substrate profiles generated by manual slider iterations.

Run 5:1000 steps in 0 seconds

1.86

1.84

Q.

1.82

1.8 1.78

1.76

1.74

0.1

0.2

0.3

0.4

0.5

X

Figure 3. Product profiles for the runs in Fig. 2.

0.6

0.7

0.8

0.9

8.7 Diffusion Systems

8.7.3

383

Dynamic Porous Diffusion and Reaction (ENZDYN)

System This example involves the same diffusion-reaction situation as the previous example ENZSPLIT, except that here a dynamic solution is obtained by finite differencing. The substrate concentration profile in the porous biocatalyst is shown in Fig. 1.

Model With complex kinetics a steady state split boundary problem of the type of Example ENZSPLIT may not converge satisfactorily, and the problem may be reformulated in the more natural dynamical form. Expressed in dynamic terms, the model relations become, 3S dt

ap

=

a2p

where at the center dX~dX

+R

384

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Center

Outside

S2 S3 S4

Figure 1. Finite-differencing for ENZDYN.

Using finite differencing techniques (refer to Sec. 6.2.1), these relations may be expressed in semi-dimensionless form for any given element n by dS'n W dP'n

•ar

n+l - 2S'n + S'n.f _= 2iL(*2 AX'2 L I ) f 'P r

R'n Sl

1

P' n+l« —OP ^r n _i_ "*• r n+ A ' 2

AX'

R'n I

+ S

where D S S 0 1-S L 2 R n AX'

and S'n = Sn/SI; Fn = Pn/Si andAX' = AX/L Sj is the external substrate concentration and AX is the length of the finite difference element. Boundary conditions are given by the external concentrations Sj and PI and at the slab center by setting SN+I = SN and PN+I = PN.

8.7 Diffusion Systems

385

Catalyst effectiveness may be determined according to two different methods: (a) the effectiveness factor based on the ratio of actual rate to maximum rate (here for eight segments). R R Ri + R29 + Ra3 + RA4 + RS5 + R* -J 6 + 77 + a8. R

o

(b) an estimate of the slope of the substrate concentration at the solid surface _

o D§ OQ

I c) 1~ O1

^"L^RO AX' Where the rate at the bulk conditions is kES 0

+ P 0 /K I ) + S0

The same constant values are used as in Example ENZSPLIT.

Program The numerical results of example ENZSPLIT and should be essentially the same as the steady state of ENZDYN. Both programs are on the CD-ROM.

Nomenclature The nomenclature is the same as ENZSPLIT with additional symbols and indices:

Symbols AX r|2

Increment of length Effectiveness factor based on rates

m -

386

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

111 K2

Effectiveness factor based on flux Same as KM

Indices refers to segment n

Exercises

References Blanch, H.W., Dunn, I.J. (1973) Modelling and Simulation in Biochemical Engineering in Advances in Biochemical Engineering, Eds. T.K. Ghose, A. Fiechter, N. Blakebrough, 3, Springer. Goldman, R., Goldstein, L. and Katchalski, Ch.L (1971) in Biochemical Aspects of Reactions on Solid Supports, Ed. G.P. Stark, Academic Press.

387

8.7 Diffusion Systems

Results

Run 1:1005 steps in 0.0833 seconds

1 0.9

* 0.8 B?' 0.7 0.6

— 81:1 ,.. 82:1 .. 83:1 . 84:1 - 85:1 _ _S6:1 , 87:1 _S8:1

tf" tfO.3

W0.2

0.1

0 30

50 TIME

80

Figure 2. Substrate concentrations in porous enzyme catalyst during dynamic solution.

Run 1:1005 steps in 0.0833 seconds

— P5:1 • -- P6:1

-ST-J7U. _ -P8:1 , P3:1 -P4:1

10

20

30

50 TIME

60

Figure 3. Product concentrations in porous enzyme catalyst.

70

80

90

100

388

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

8.7.4

Oxygen Diffusion in Animal Cells (CELLDIFF)

System This example treats a diffusion-reaction process in a spherical biocatalyst bead. The original problem stems from a model of oxygen diffusion and reaction in clumps of animal cells by Keller (1991), but the modelling method also applies to bioflocs and biofilms, which are subject to potential oxygen limitation. Sphere Oxygen

N

Product

I/I AV

Substrate

Rp

Rp

Figure 1. The finite differencing of the spherical bead geometry.

Diffusion and reaction takes place within a spherical bead of volume = 4/37cRp3 and area =47iRp2. It is of interest to find the penetration distance of oxygen for given specific activities and bead diameters. As shown, the system is modelled by dividing the bead into shell-like segments of equal thickness. The problem is equivalent to dividing a rectangular solid into segments, except that here the volumes and areas are a function of the radial position. Thus each shell has a volume of 4/3 n (rn3 - rn_i3). The outside area of the nth shell segment is 4n rn2 and its inside area is 4n r n _i 2 .

389

8.7 Diffusion Systems

in

Figure 2. The diffusion fluxes entering and leaving the spherical shell with outside radius rn and inside radius r n _j.

Model Here the single limiting substrate S is taken to be oxygen. The oxygen balance for any element of volume AV is given by

The diffusion fluxes are

Ar S

jn-l =

n~ S n-l Ar

Substitution gives dSn dt "'

3D

The balance for the central increment 1 (solid sphere not a shell) is 4

Since ri = Ar, this becomes

3 dS

. ,,

= J.

=

22

™™ 4 4 -

3

390

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

dS

l

_

3D

S /q

o N , p

I f - ^ S 2- S l) +R sl r

The reaction rate is expressed by a Monod-type equation RSn = - (

S

where X is the biomass concentration (cell number/m3) in the bead, OUR is the specific oxygen uptake rate (mol/cell s) and Sn is the oxygen concentration (mol/m3) in shell n.

Program As shown below, segments are programmed using the array-vector facility of Madonna, numbered from the outside to the center. The effectiveness factor, expressing the ratio of the reaction rate to its maximum, is calculated in the program, part of which is shown below. The number of elements N is called Array in the program, which is on the CD-ROM.

{Shells 2 to Array-1} a/at (S [2. . (Array-1) ] )=3*D*( ((r[i]**2)*(S[i-l]/(deltar*( (r[i]**3)-(r[i+l]**3))

Nomenclature

Symbols D KS OURmax r Rp

Diffusion coefficient Saturation constant in Monod equation Maximum specific oxygen uptake rate Radius at any position Outside radius of bead

m2/s mol/m3 mol/cell s m m

8.7 Diffusion Systems

RS S X Ar (Deltar)

391

Reaction rate in the Monod equation Oxygen substrate concentration Biomass concentration Increment length, r/N

Indices 1 2 n P S

Exercises

Refers Refers Refers Refers Refers

to to to to to

segment 1 segment 2 segment n particle substrate

mol/s m3 mol/m3 cells/m3 m

392

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results

0

10

20

30

40

50

60

70

80

90

100

Figure 3. Profiles of oxygen concentrations versus time for each shell.

Figure 4. Doubling the bead radius causes oxygen deficiency inside the bead (lower curve) as these radial profiles show.

Reference Keller, J. (1991) PhD Dissertation No. 9373, ETH-Zurich.

8.7 Diffusion Systems

8.7.5

393

Immobilized Biofilm in a Nitrification Column System (NITBEDFILM)

Nitrification is the sequential oxidation of NH4+ to NO2~ and NO3" which proceeds according to the following reaction sequence: NH4+ +1 O2 -> NO2- + H2O +2H+

NO2- + \ O2 -> NO3The overall reaction is thus NH4+ + 202 -> N0 3 -+H 2 0 Both steps are influenced by dissolved oxygen and the corresponding substrate concentration and are catalyzed by two different organism species. Since their growth rates are very low, nitrification as a wastewater treatment process benefits greatly from biomass retention. In this example, a biofilm column reactor for nitrification is modelled as three tanks-in-series with a recycle loop (Fig. 1). Oxygen is supplied only in an external contactor and circulates to the reaction column in dissolved form. This is similar to the example NITBED. However, in this case the reaction takes place within an immobilized biofilm, similar to the single tank example BIOFILM.

394

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

O3, Si3

Fluidized bed

SiA

Figure 1. Biofilm column reactor with recycle loop for nitrification.

Model The column reactor is assumed to be described by three tanks. The model balance equations for the liquid phase are developed by considering both the individual tank stages and the absorber. Component balances are required for all components in each section of the reactor column and in the absorber, where the feed and effluent streams are located. For the solid biofilm phase, where the reaction takes place, the concentrations change both with distance and time. Therefore, a descretization of the length variable is required as developed for the example BIOFILM.

395

8.7 Diffusion Systems To tank n+1

t S2ntO] S3ntO] On[0]

From tank n-1 Figure 2. Schematic of a single tank in the column.

Because of the complexity involving four components in two phases and four regions care must be taken with the nomenclature. The nitrogen compounds are referred to as Si, 82, and 83, respectively. Dissolved oxygen is referred to as O. Referring to the above figure, a single tank n is shown with the four components. [0] refers to the liquid phase in contact with the solid. Transfer to the biofilm is by diffusion to the first section, denoted [1].

Figure 3. Schematic of a single section i of biofilm in tank n.

Further diffusion brings substrate to all the biofilm sections, as shown above, for a single substrate in section i. The reactions occur in these sections. In the absorber, oxygen is transferred from the air to the liquid phase. Additional subscripts, as seen in Fig. 1, identify the feed (F), recycle (R) and the flows to and from the tanks 1, 2 and 3, each with volume V, and the absorption tank with volume VAThe fluidised bed reactor is modelled by considering the component balances for the three nitrogen components (i) and also for dissolved oxygen. For each stage n, the liquid phase component balance equations have the form dSin[0] dt

=

396

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

-J0[0] For the absorption tank, the balance for the nitrogen containing components include the input and output of the additional feed and effluent streams, giving ds iA -

_(S/QiF -SQiA )\

The oxygen balance in the absorption tank must account for mass transfer from the air, but neglects the low rates of oxygen supply and removal by the convective streams. This gives

For the first and second biological nitrification rate steps, the reaction kinetics for any stage n are given by v

=

ml Slni °ni Kl+ S lni K O i + O ni V

rs2n

=

m2 S2ni

K

2+ S 2ni

K

02+°ni

The oxygen uptake rate is related to the above reaction rates by means of the constant yield coefficients, YI and Y2, according to i -r S 2niY2 The reaction stoichiometry provides the yield coefficient for the first step YI = 3.5 mg 02/(mg N-NNH4) and for the second step Y2 = 1.1 mg O2/(mg N-NO2)

397

8.7 Diffusion Systems

Nomenclature Symbols A F FR KLa K KI K2 L N 0 Os and O* OUR r S V

VA vm Y

Specific area of film Feed and effluent flow rate Recycle flow rate Transfer coefficient Saturation constants Saturation constant for ammonia Saturation constant for ammonia Biofilm thickness Number of biofilm segments Dissolved oxygen concentration Oxygen solubility, saturation cone. Oxygen uptake rate Reaction rate per volume of biofilm Substrate concentration Volume of one reactor stage Volume of absorber tank Maximum velocity Yield coefficient

1/m m3/h m3/h h g/m 3 g/m 3 g/m 3 m g/m 3 g/m 3 g/ m3 h g/ m3 h g/m 3 m3 m3 mg/L h mg/mg

Indices 1,2,3 1,2,3 A F jn[I] m Ol and O2 S1,S2 S and *

Refer to ammonia, nitrite and nitrate, resp. Refer to stage numbers Refers to absorption tank Refers to feed Refers to substrate j in stage n in segment i Refers to maximum Refer to oxygen in first and second reactions Refer to substrates ammonia and nitrite Refer to saturation value for oxygen

398

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Exercises

Program The program is on the CD-ROM.

399

8.7 Diffusion Systems

Results Run 1: 133 steps in 13.7 seconds

5

'4.5 *C 3 A .3.5 g 4

I 60 ' j,g 5 0 -

13

£40-

•2.5

.2

a? 3 S

1.5

520.

1

55 io-

5

o60

50

70

80

90

100

TIME

Figure 4. Time profiles of the nitrogen component bulk concentrations in the first tank and the oxygen bulk concentrations in the three tanks.

Run 1: 133 steps in 13.7 seconds

4.5

4 3.5

3

0.5

0 0

10

20

30

40

50

60

70

80

90

100

TIME

Figure 5. Time profiles of the oxygen concentrations within the 10 segments of biofilm in the first tank.

400

8.8 8.8.1

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Multi-Organism Systems Two Bacteria with Opposite Substrate Preferences (COMMENSA)

System Considered here (Fig. 1) is the batch growth of a two-organism culture on two substrates, in which both species can utilize both substrates (Kim et al., 1988), but where the organisms have opposing substrate preferences. The two bacterial species involved are: Klebsiella oxytoca (XA) and Pseudomonas aeruginosa (XB). The two substrates are glucose (Y), which is preferred by K. oxytoca, and citrate (Z), which is preferred by P. aeruginosa.

XA

XB

Figure 1. Organism XA prefers substrate Y, and organism Xg prefers substrate Z. The assumptions are as follows:

- The overall individual growth rate of each species at any time is the sum of the rate of growth on glucose plus the rate on citrate. - The specific growth rate on each substrate depends on the concentration level of some key enzyme responsible for the rate-controlling step E. - The key enzyme for the preferred substrate is assumed to be constitutive. - The production of the key enzyme for the secondary substrate is subject to induction and repression by the preferred substrate. - An inhibitor I is produced from the growth of K. oxytoca on glucose and inhibits the growth of P. aeruginosa on citrate. The inhibitor is thus a growth-associated product.

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

8.8 Multi-Organism Systems

401

- The total rate of substrate consumption is the sum of the rates of consumption by each organism plus the rate of consumption of substrate for the production of inhibitor. - The oxygen uptake rate (OUR) and carbon dioxide evolution rate (CER) involve the sum of the individual contributions from each organism. - The dissolved oxygen tension in percentage air saturation (DOT) is obtained using a steady state oxygen balance.

Model The growth rates, jny, for each organism are the sums of the growth rates on glucose and citrate. The subscripts i and j have the following meaning: i refers to the organisms (K. oxytoca - A and P. aeruginosa = B) and j refers to the substrate (glucose = Y and citrate = Z). The levels of the key enzymes are denoted by E. The biomass balances for the batch system are

dXA

—

=

(MAY + MAZ) XA

dXB I The specific growth rate equations for the two organisms on each substrate are given by: A*maxAYSYEAY K

SAY + S Y

MmaxAZSZEAZ K

SAZ +S Z

A*maxBZSZEBZ I

M-BZ - —K--— S

K

I

"FT

K

SBZ+ Z V I +

The substrate balances are given by:

402

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

dS Y = —

!

a

SAY

Y

Y

I )

Y

" YSAZ

XA

SBY

dSz

dt

- - YSBZ

XB

Inhibitor (I) production is growth associated to organism A, and its decay is proportional to the cell concentration. The balance for the inhibitor is dl gj- = oc|u AY X A -pX A

The balances for the key enzymes, which control the growth on secondary substrates are given by: Sz KRAZ - -*^--rt\iu kpAZf\z^ EAZ T-T " and

dE BY - "

SY --Y

KRBY KbY

where the consecutive terms in the above equations represent induction, repression, and dilution due to cell division, respectively. Here the enzyme levels E are normalized with respect to the maximum levels (See reference). Because growth on the preferred substrates is constitutive, EAy and EBZ are equal to 1 . The oxygen uptake rate (OUR), carbon dioxide evolution rate (CER) and dissolved oxygen tension (DOT) are given by: OUR = OAY

OAZ

OBY

OBZ

CER = Y

CAY CBY

Y

CAZ CBZ

403

8.8 Multi-Organism Systems

DOT = 100 1-

OUR

K L aC 0

The cell mass fractions are given by: FA=-

X A + XB

FB = 1 - F A

Program The program is given on the CD-ROM.

Nomenclature Symbols CER DOT F I K KLaC0* OUR S X Y E m a P

Carbon dioxide evolution rate Dissolved oxygen tension Cell mass fractions Inhibitor concentration Saturation and inhibitions constants Oxygen transfer rate Oxygen uptake rate (normalized) Substrate concentration Biomass concentrations Yield coefficients Level of key enzyme Specific maintenance rates Yield constant for inhibitor Inhibitor consumption rate constant Specific growth rate

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

404

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Indices A B C I M O P R S Y Z

Exercises

Refers to K. oxytoca Refers to P. aeruginosa Refers to carbon dioxide Refers to inhibitor Refers to maximum Refers to oxygen Refers to dilution due to cell division Refers to repression Refers to substrate Refers to glucose Refers to citrate

405

8.8 Multi-Organism Systems

Results The graphical results in Fig. 2 show the dynamic changes in biomass fractions FA and FB for two values of a: 0.007 kg/kg and 0.0007 kg/kg .

Run 2: 8200 steps in 0.167 seconds

1 • 0.9.

0.8 0.7

e

0.6

s°-0.45 0.3 0.2 0.1

0 4

5

TIME

Figure 2. Dynamic changes in biomass fractions FA and FB for a = 0.007 and 0.0007.

Reference Kim, S. U., Kim, D. C, Dhurjati, P. (1988). Mathematical Modeling for Mixed Culture Growth of Two Bacterial Populations with Opposite Substrate Preferences. Biotechnol. Bioeng., 31, 144-159. This example was developed from the original paper by J. Lang, ETH-Zurich.

406

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

8.8.2

Competitive Assimilation and Commensalism (COMPASM)

System The interactions between two microbial species (Ma and Mb) in a mixed continuous culture are considered (Miura et al., 1980). The population dynamics of the two microbes, is described by competitive assimilation of substrate Si and commensalism, with the participation of growth factor Ga that is excreted by microbe Ma and required by microbe Mb for growth. Mb also consumes a second substrate 82 from the medium. These interactions are represented in the Fig. 1.

,G

Figure 1. Interaction of two organisms and two substrates in continuous culture.

Model For the chemostat shown above the unsteady-state material balances are as follows: Dilution rate:

8.8 Multi-Organism Systems

407

Organism Ma:

dXa j- = ftia-D)Xa Organism Mb: dXb -gj- = (jib - D) Xb

Substrates S\ and 82:

•ar = - "a dS2

M-b Xb + D (S2

-ar = - ~YT"

°"S2)

The yields for organism Mb on the two substrates are assumed here, for simplicity, to have the same values, Yb. The growth factor balance is dGaa

"P

11

Y

Jib Xb

T-X f*

The mass balance for the growth factor, Ga, is formulated by assuming a formation rate, Pa |ia Xa, and consumption rate, (|Lib Xb)/Ybg. Here Xa and Xb are the concentrations of microorganisms Ma and Mb, respectively. The constants Pa and Ybg are the biological yield constants for the formation and consumption of Ga, respectively. The specific growth rates of microbes Ma and M b are expressed by: Organism Ma: Si KSa+Si

Organism Mb: Si

Ga

82 K Sb 2 + S2 Kg + Ga

408

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

where K$a and K$bi are the saturation constants of Ma and Mb for substrate Si, Ksb2 is the saturation constant of Mb for substrate 82, and Kg is the saturation constant of Mb for growth factor Ga. Setting \imb2 = 0 corresponds to the consumption of only one substrate Si. A rigorous stability analysis of the system has been carried out by Miura et. al. (1980). This involves linearizing the mass balances by Taylor's method in the vicinity of the steady state solution and determining the characteristic eigenvalues of the resultant matrix. The following relationship for co-existence of the two microbes can be derived for the case of a single substrate.

Sio > 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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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.

Run 1: 3000 steps in 0.0167 seconds 300

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.

8.8 Multi-Organism Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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)

8.8 Multi-Organism Systems

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))<=MARK THEN SI ELSE

0

414

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Nomenclature Symbols D K MARK PER R S s X X

Y P r t X

z a P H

Dilution rate 1/h Saturation constant kg/m3 Ratio controlling step function h Time period Biomass ratio Substrate concentration kg/m3 Substrate concentration, dimensionless Biomass concentration kg/m3 Biomass concentration, dimensionless Yield coefficient kg/kg Probability factor Growth rate kg/m3 h Dimensionless time — Biomass concentration, dimensionless Delayed substrate concentration, dimensionless Adaptability factors 1/h Ratio of adaptability factors Specific growth rate 1/h

Indices 0 1 2 d i m real

Refers to inlet stream Refers to wild strain Refers to engineered strain Refers to dimensionless Refers to 1 or 2 Refers to maximum Refers to real time

8.8 Multi-Organism Systems

415

Exercises The stability problem can be studied by the variation of several parameters.

Results The output in Fig. 2 gives the substrate oscillations created by the square wave feed concentrations, showing the engineered organism X2 being washed out. A similar situation for sine wave feeding is given in Fig. 3.

416

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 1:2500 steps in 0.05 seconds

100

250

200

150

TIME

Figure 2. Square wave substrate feeding, causing X2 to wash out.

Run 1:40000 steps in 0.583 seconds

100

150

200

TIME

Figure 3. Sine wave feeding. Similar to Fig. 2 but with longer period and a higher b value.

References Aiba, S., Imanaka, T. (1981) in Annals of the New York Acad. of Sciences, 369, 1-15.

8.8 Multi-Organism Systems

417

Stephens, M.L., Lyberatos, G. (1988) Biotechnol. and Bioeng., 31, 464-469. Stephens, M.L., Lyberatos, G. (1987) Biotechnol. and Bioeng., 29, 672-678. Example developed by N. Mol, ETH-Zurich.

8.8.4

Predator-Prey Population Dynamics (MIXPOP)

System The growth of a predator-prey mixed culture in a chemostat can be described with a reaction kinetics formulation. In this growth process, the dissolved substrate S is consumed by organism Xi (the mouse), while species X2 (the monster) preys on organism Xi, as shown in Fig. 1.

Figure 1. Monster attacks mouse while it unsuspectingly feeds on S.

Model The model involves the chemostat balances for each species with the corresponding kinetics. The variables are given in Fig. 2.

418

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Sl,X-|,X2 ^*

S0 D

1 II

Figure 2. Chemostat predator-prey reactor.

Substrate balance,

dS = D(So-Si) Species 1 (prey) balance, ^2X2

Species 2 (predator) balance,

dX2

- DX 2

where D is the dilution rate,

~"v The kinetics are given by Monod relations,

X

Program The program is found on the CD-ROM.

419

8.8 Multi-Organism Systems

Nomenclature

Symbols D F K S V X Y

Dilution rate Flow rate Saturation rate constant Substrate concentration Reactor volume Biomass concentration Yield constants Specific growth rate

Indices 0 1 2 m

Exercises

Refers Refers Refers Refers

to feed stream to prey to predator to maximum

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

420

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results Stable steady states for the system are shown in Fig. 3 for |LLmi = 0.5 and |Lim2 = 0.11. Oscillations in the biomass populations are achieved by setting the specific growth rates nearly equal (\im\ = 0.5 and |Lim2 = 0.49) as shown in Fig. 4 and also by the phase plane plot of Fig. 5. Run 1: 5005 steps in 0.1 seconds

•4

3.5

3 2.5

« 2 1.5

1

0.5 50

100

150

200

250 TIME

300

Figure 3. Stable steady state (|iml = 0.5 and [im2 = 0.11).

350

400

450

500

421

8.8 Multi-Organism Systems

Run 1:5005 steps in 0.1 seconds

6

450

Figure 4. Oscillatory state (|iml = 0.5 and p,m2 = 0.49).

Run 1:5005 steps in 0.117 seconds

Figure 5. Phase plane plot of oscillations.

422

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

8.8.5

Competition Between Organisms (TWOONE)

System Consider organism A and organism B with their respective specific growth rates, (HA and ILLB, which both grow independently on substrate S. Assume:

S/(KSA + S) S/(KSB + S) Depending on the values of |LIM and K$, these two functions may occur in two different forms, as shown in Fig. 1.

B

M inter

B

inter

Figure 1. Comparison of growth rate curves for the competitive chemostat growth.

It is clear that the curves B and A will cross each other if (IMB < MMA and KSB < KSA- In Fig. 1, the situation on the left indicates that B will grow fastest at any value of S. For this case, in chemostat cultures with dilution rate DI, after an initial start up period, a substrate concentration S i will be reached at which (LIB = DI and for which |LIA < DI. Organism A will then be washed out, and only organism B will remain in the reactor. The situation in the right of Fig. 1 shows (IB crossing (IA- The point of intersection can be found easily by simple algebra where:

8.8 Multi-Organism Systems

423

Wnter = Solving for S at the intersection, Sinter

=

For this case a chemostat can theoretically operate stably at D = Jiinter such that both A and B will coexist in the reactor. This however is an unrealistic metastable condition, and with D < Hunter* A will wash out. With D > Hunter A will grow faster, causing B to be washed out.

Model The equations for the operation of chemostat with this competitive situation are,

d*A

„ ,

dXB jp = 0 - D XB + JIB XB

dT = D(S0 - S) In addition, the Monod relations, |IA = f(S) and |LLB = f(S), are required. Solution of these equations will simulate the approach to steady state of A and B competing for a single substrate.

Program The program is given on the CD-ROM.

424

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Nomenclature

Symbols D F KS

S V X Y

Dilution rate Flow rate Saturation constants Substrate concentrations Reactor volume Biomass concentrations Yield coefficient Specific growth rates

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

Indices A B M 0 inter

Exercises

Refers Refers Refers Refers Refers

to organism A to organism B to maximum to inlet stream to the intersection of the ju versus S curves

425

8.8 Multi-Organism Systems

Results Run 1:1004 steps in 0.0333 seconds

5

4.5

4 3.5

3 ! 2.5 ' 2 1.5

1 0.5 9

10

Figure 2. Organism A and B competing for substrate.

8.8.6

Competition between Two Microorganisms for an Inhibitory Substrate in a Biofilm (FILMPOP)

System Wastewater with toxic chemicals is often treated directly at the source with specialized microbial cultures in small-scale biofilm reactors. A model may help to understand, optimize, and control such reactors. In a paper by Soda et al. a simple biofilm model was developed to simulate the competition between two microorganisms for a common inhibitory substrate. In the model the following assumptions were made: (i) the biofilm has a uniform thickness and is composed of 5 segments, (ii) each microorganism A and B utilizes a common substrate, and the growth rates are expressed by Haldane kinetics with a spatial limitation term but is otherwise independent of the other microorganism and (iii) the diffusion of the substrate, movement of the microorganisms, and continuous loss of the biomass by shearing are expressed by Pick's law-type equations.

425

8.8 Multi-Organism Systems

Results Run 1:1004 steps in 0.0333 seconds

5

4.5

4 3.5

3 ! 2.5 ' 2 1.5

1 0.5 9

10

Figure 2. Organism A and B competing for substrate.

8.8.6

Competition between Two Microorganisms for an Inhibitory Substrate in a Biofilm (FILMPOP)

System Wastewater with toxic chemicals is often treated directly at the source with specialized microbial cultures in small-scale biofilm reactors. A model may help to understand, optimize, and control such reactors. In a paper by Soda et al. a simple biofilm model was developed to simulate the competition between two microorganisms for a common inhibitory substrate. In the model the following assumptions were made: (i) the biofilm has a uniform thickness and is composed of 5 segments, (ii) each microorganism A and B utilizes a common substrate, and the growth rates are expressed by Haldane kinetics with a spatial limitation term but is otherwise independent of the other microorganism and (iii) the diffusion of the substrate, movement of the microorganisms, and continuous loss of the biomass by shearing are expressed by Pick's law-type equations.

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

426

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Sf

S0

Wall

Figure 1. Schematic of the continuous reactor with biofilm, showing the descretization into five layers.

Model

Fig. 1 illustrates an idealized flat biofilm with a uniform thickness Lf (m). The biofilm is divided into N segments for simulation purposes and each has a thickness of AZ = Lf/N (m). Wastewater containing the substrate is fed to the reactor at a constant feed rate and a concentration Sf (mg/L). The bulk liquid in the reactor is mixed throughout the tank and the substrate diffuses into the biofilm. The substrate is transported from the bulk liquid having a concentration S[0] (mg/L) to the surface of the biofilm having a concentration S[l] (mg/L). A diffusion layer of a thickness Lj (m) is used to represent the external mass transport resistance. Using the same approach as in the example BIOFILM, the mass balances in the bulk liquid for the substrate and microorganisms A and B with a continuous flow are simply described as following:

8.8 Multi-Organism Systems

dS[0] dt

Wll v f

427

_, '

„ S[0]-S[1] " DZ

m [01X fC•] mB[0]XB[0] YA YB

, XA[0]-XA[1] , } - -DXA|0] - aDXA — (mALO] - bA JXA[0]

/\/1

where S is substrate concentration (mg/L). XA and XB are biomass of microorganisms A and B (mg/L), respectively. Each number in the brackets refers to the bulk liquid or a segment illustrated in Fig. 1 . DX, b, Y, and |Li are diffusion coefficient of microorganisms (m2/day), biomass decay coefficient (day1), yield coefficient (-), and net specific growth rate (day1). Subscripts A and B refer to microorganisms A and B. D, Ds, a, and t are dilution rate (day"1), diffusion coefficient of substrate (m2/day), specific area perpendicular to the flux (nr1), and time (day), respectively. Reactions within the biofilm are described by diffusion reaction equations. The mass balances of the surface segment are described as following:

s[0]-sm

=

dt

S

LjAZ

dX A [l]

dt

AZ2

XA[0]-XA[1]

XA[1]-XA[2]

- -- DXA -— LZXZ,

—at — ~ dX B [l]

S

=: — \j v"-p XB

XB[0]-XB[1] L,AZ

XB[1]-XB[2] n LJ YT3 ^ XB

AZ2

Component mass balances are written for each segment (i = 2, .., N-l), where:

428

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

dS[i] dt

„ 5>[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).

8.8 Multi-Organism Systems

..

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.8 Multi-Organism Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results Run 1:1017 steps in 0.967 seconds

/ \

/-

^^—' •

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 Multi-Organism Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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.

8.8 Multi-Organism Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.8 Multi-Organism Systems

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 <= 1000 pH=-loglO(chplus)+3

Nomenclature The nomenclature of the program is partially defined within the program. Symbols C Cons F FEQ Ftitr

Concentration Consumption rate Stoichiometric coefficients Equilibrium factor Titration flow rate

C-mol/m3 C-mol /m3h (-) (-) m3/h

438

KD Ks S P Pro X YP/S

YX/S 1^

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Dissociation constant Inhibition constant Saturation constant Substrate concentration Product Concentration Production rate Biomass concentration Yield coefficient, product from substrate Yield coefficient, biomass from substrate Specific rate of biomass synthesis

mol /m3 C-mol /m3 C-mol/m3 C-mol /m3 C-mol /m3 C-mol /m3 h C-mol /m3 C-mol /C-mol C-mol /C-mol 1/h

Indices Ac Bu Buf d Hy i in Mo Pr titr Tot

Z

Refers to acetic acid Refers to butanediol Refers to buffer Refers to death rate Refers to hydrogen gas Refers to reaction i Refers to initial Refers to whey substrate Refers to propionic acid Refers to titration Refers to total Refers to difference between cations and ions

Results The first of the three graphs in Fig. 2 shows dynamic profiles of substrate whey (Mo), CH4 (CH), dissolved CO2 (CO) and dissolved hydrogen (Hy). The whey is almost instantaneously consumed. Hy reaches a maximum very soon and is then quickly reduced to almost zero. CH4 is produced with varying rates. CC>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

8.8 Multi-Organism Systems

Run 1:4389 steps in 1.43 seconds

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 1:4389 steps in 1.47 seconds

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).

8.8 Multi-Organism Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.8 Multi-Organism Systems

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

Refers Refers Refers Refers Refers Refers

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.8 Multi-Organism Systems

Run 1: 40012 steps in 4.88 seconds

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.8 Multi-Organism Systems

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

=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

8.8 Multi-Organism Systems

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.

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8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 1: 50233 steps in 3.03 seconds

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.

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

452

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

Program The program is on the CD-ROM.

=

_

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

Refers Refers Refers Refers Refers Refers

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results Oxygen transfer has a pronounced influence on performance as seen in Fig. 2 for variations of K^a. The dissolved oxygen may reach values below KQ as shown in Fig. 3 for K^a values from 0.5 to 5.

Run 4: 2004 steps in 0.05 seconds

20

Figure 2. Influence of oxygen transfer coefficient (KLa = 0.1 to 1.0).

455

8.9 Membrane and Cell Retention Reactors

Run 4: 2004 steps in 0.05 seconds

\\. 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

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8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.9 Membrane and Cell Retention Reactors

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> an
i 1+ [_^_]«Ac 1+ |

and

i Bu_iaBu

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8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Formation of Acetoin and Butanediol The products were assumed to be formed when the respiration was not sufficient to cover the energy requirement for growth. While the distribution of the two products was assumed to be dependent on oxygen, through a rapid equilibrium, according to an S-shaped empirical function described by fAcBu,max (DO)2

Ac Bu

-

To avoid an algebraic loop, the kinetics for the rate of butanediol production was assumed to be dependent on the deviation from equilibrium, TBU = qX4 X = kAcBu (Ac - Bu fAcBu) The constant kAcBu was set high enough to ensure equilibrium conditions. The specific reaction rate q^3 for product formation was obtained as, ~ YR2/R1 qX,l - YR2/R4

Reaction Rates The specific reaction rates for each component were obtained from the specific reaction rates q^ and the stoichiometric coefficients, Vy (component j in reaction i) as follows, i

The volumetric rates rj [mol L"1 Ir1] for components X, S, Ac, Bu, 62 and CC>2 were related to the specific rates as, rj = qj X

The specific rate q^2 of reaction R2 was obtained from \\ as

8.9 Membrane and Cell Retention Reactors

459

V 2 , X MG X where V2,x is the stoichiometric coefficient, and MGx is the C-mol mass of biomass. The rate for the respiration reaction, q^i, was found to be influenced by the dissolved oxygen, and was described as follows: qxi =

The yield coefficients for the complex components GI and C2 , were assumed to be 1 g of each component for 1 g biomass, and the molecular weights of the components were assumed to be the same as for the biomass. The initial amounts of these components in the molasses medium were adjusted in the simulation. The corresponding rates were proportioned according to the growth rates as rri = - rv

and

rr2 =

Pervaporation Model The mass transfer in the pervaporation module was described as an equilibrium process using constant enrichment factors. Thus the concentrations of product in the reactor, Ac and Bu, were related to the concentration in the permeate, Acpe and Bupe as, Acpe = PAC Ac and Bupe = PRU Bu

Dynamic

Reactor Mass Balances

Considering volume-specific flow rates, _ FO _ Fpu _ D = y- , Dpu = -^r > Dpe =

where D, Dpu, Dpe were the flows for the feed, purge, and permeate, respectively intr 1 .

460

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

The mass balances were as follows: Total mass D = Dpu + Dpe

Biomass dx

"3T = rx - X Dpu

Sugar dS df = S0 D - S Dpu - rs

Acetoin dAc

~dT = rAc ~

Ac D

dBu ~dt~ =

Bu D U

pu ~ Acpe Dpe

Butanediol rBu

~

P

" BuPe DPe

Complex components dCi ~dT = Ci>0 D - Ci Dpu - rCi

where Q = 1 and 2. Oxygen transfer from the gas phase and the oxygen uptake rate determine the DO in percent saturation as, dDO -HT~ = KLa (100 - DO) m

100 ^—r OUR 2.34xlO"4

Here the equation is in terms of percent saturation using the DO saturation value at 30 °C of 2.34 x 10''4 mol/ L.

Nomenclature Symbols ACPE Ac ATP Bu Ci CPR D DO

Acetoin concentration in permeate Acetoin Adenosintriphosphate Butanediol Complex components, where i = 1 or 2 Carbon dioxide production rate Dilution rate Dissolved oxygen concentration

g/L g/L, mol/L g/L, mol/L g/L mol/L h 1/h % saturation

461

8.9 Membrane and Cell Retention Reactors

UcBu lAcBu.max F FM/P kAcBu

kci

KLa

KS Kprod MGX OUR P

r

J

rpm

Ri

RQ S vvm V X Yi/j

Acetoin/butanediol ratio Max. Ac/Bu ratio Flow rate Permeate / purge flow rate Kinetic const. Ac/Bu Monod-const., growth on component i Monod-const., growth depending on DO Inhibition constant product i Gas -liquid transfer coefficient for O2 Monod-const. for respiration depending on DO Repression const., complex substrate 1 on component 2 Monod-const., Growth on substrate Empirical const, for Ac/Bu-equilibrium Mol mass for biomass (1 C-mol) = 24.6 Oxygen uptake rate Product, sum Acetoin + Butanediol Specific rate of component j Specific rate reaction i, where i =lto 4 Formation or uptake rates, components j (Ac, Bu, O2, CO2, X and S) Stirrer speed Chemical reaction i, where i = Ito 4 Respiration quotient qcO2/(lO2 Substrate concentration (e.g., sugar) Gas rate per volume liquid Reaction volume Biomass concentration Yield coefficient (i formed/j used)

L/h

mol/g(acetoin) h g/L % saturation g/L 1/h % saturation g/L g/L

g/mol mol/L h g/L, mol/L mol/g(biomass) h mol/g(biomass) h mol/L h 1/min g/L, mol/L 1/min L g/L, mol/L

Greek symbols ai pi \i R,max Vi j

Exponent inhibition kinetic product i (Ac or Bu) Enrichment factor of pervaporation membrane for component i (Ac or Bu) Specific growth rate Maximum growth rate for component i Stoichiometric coeff. of component j in reaction i

1/h 1/h

462

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Indices Ac Bu

Q

CO2 i j inh 02 pe pu P Ri S X 0 Ai

Acetoin Butanediol Component i (unknown components 1 or 2) Carbon dioxide Refers to reactions, i (1 to 4) Refers to components j (Ac, Bu, C>2, 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).

Program The program is found on the CD-ROM.

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

8.9 Membrane and Cell Retention Reactors

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 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.9 Membrane and Cell Retention Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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)

8.9 Membrane and Cell Retention Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.9 Membrane and Cell Retention Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 4: 8320 steps in 1.53 seconds

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

8.9 Membrane and Cell Retention Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.9 Membrane and Cell Retention Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.9 Membrane and Cell Retention Reactors

?

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

Refers Refers Refers Refers

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results Run 1:619 steps in 53.4 seconds

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 Membrane and Cell Retention Reactors

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.9 Membrane and Cell Retention Reactors

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

Program The program is on the CD-ROM.

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

8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

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

8.9 Membrane and Cell Retention Reactors

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).

Results Run 1:1618 steps in 1.03 seconds

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 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Run 1:1618 steps in 1.13 seconds

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

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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.

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9 Appendix: Using the Berkeley Madonna Language

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

9 Appendix: Using the Berkeley Madonna Language

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 Q^ANGE 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

9.2 Screenshot Guide to Berkeley MADONNA

4J-, 4-

-9,5

3,5

9

3

8.5 -8

2.5

•2-

-7.5 7

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.

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9 Appendix: Using the Berkeley MADONNA Language

Figure 7. Part of the window to define the Sliders.

Figure 8. A graph of two slider runs, showing the Parameters Menu pulled down.

9.2 Screenshot Guide to Berkeley MADONNA

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

9 Appendix: Using the Berkeley MADONNA Language

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.

9.2 Screenshot Guide to Berkeley MADONNA

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

9 Appendix: Using the Berkeley MADONNA Language

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

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

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

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

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

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