Biological Reaction Engineering (Second Edition)

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

Biological Reaction Engineering

Dynamic Modelling Fundamentalswith Simulation Examples

Second, Completely Revised Edition

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

Also of Interest

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

Chemical Engineering DynamicsAn Introduction to Modelling and Computer SimulationSecond, Completely Revised Edition2000, ISBN 3-527-29776-6

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

BiologicalReaction EngineeringDynamic Modelling Fundamentalswith Simulation Examples

Second, Completely Revised Edition

WILEY-VCH

WILEY-VCH GmbH & Co. KGaA

Dr. Irving J. DunnETH ZurichDepartment of Chemical EngineeringCH-8092 ZurichSwitzerland

Professor Dr. Elmar HeinzleUniversity of SaarlandDepartment of Technical BiochemistryP.O. Box 15 11 50D-66041 SaarbruckenGermany

Dr. John InghamUniversity of BradfordDepartment of Chemical EngeeringBradford BD7 1DPUnited Kingdom

Dr.JiriE.PrenosilETH ZurichDepartment of Chemical EngineeringCH-8092 ZurichSwitzerland

This book was carefully produced. Nevertheless,authors and publisher do not warrant the informa-tion contained therein to be free of errors. Rea-ders are advised to keep in mind that statements,data, illustrations, procedural details or otheritems may inadvertently be inaccurate.

First Edition 1992Second, 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 fromthe British Library.

Bibliographic information published by Die Deut-sche Bibliothek. Die Deutsche Bibliothek lists thispublication in the Deutsche Nationalbibliografie;detailed bibliographic data is available in theInternet at <http://dnb.ddb.de>.

© 2003 WILE Y-VCH VerlagGmbH & Co. KGaA, Weinheim

All rights reserved (including those of translationinto other languages). No part of this book may bereproduced in any form - by photoprinting, micro-film, or any other means - nor transmitted ortranslated into a machine language without writ-ten permission from the publishers. Registerednames, trademarks, etc. used in this book, evenwhen not specifically marked as such, are not tobe considered unprotected by law.Printing: Strauss Offsetdruck, MorlenbachBookbinding: GroBbuchbinderei J. SchafferGmbH & 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 V

PREFACE XI

PART I PRINCIPLES OF BIOREACTOR MODELLING 1

NOMENCLATURE FOR PART I 3

1 MODELLING PRINCIPLES 9

1.1 FUNDAMENTALS OF MODELLING 97.7.7 Use of Models for Understanding, Design and Optimization of Bioreactors 91.1.2 General Aspects of the Modelling Approach 101.1.3 General Modelling Procedure..... 721.1.4 Simulation Tools 757.7.5 Teaching Applications 75

1.2 DEVELOPMENT AND MEANING OF DYNAMC DIFFEREOTTAL BALANCES 161.3 FORMULATION OF BALANCE EQUATIONS ..21

7.5.7 Types of Mass Balance Equations 271.3.2 Balancing Procedure 23

1.3.2.1 Case A. Continuous Stirred Tank Bioreactor 241.3.2.2 CaseB. Tubular Reactor 241.3.2.3 Case C. River with Eddy Current 25

1.3.3 Total Mass Balances 331.3.4 Component Balances for Reacting Systems 34

1.3.4.1 Case A. Constant Volume Continuous Stirred Tank Reactor 351.3.4.2 Case B. Semi-continuous Reactor with Volume Change 371.3.4.3 Case C. Steady-State Oxygen Balancing in Fermentation 381.3.4.4 Case D. Inert Gas Balance to Calculate Flow Rates 39

7.5.5 Stoichiometry, Elemental Balancing and the Yield Coefficient Concept.. 401.3.5.1 Simple Stoichiometry 401.3.5.2 Elemental Balancing 42L3.5.3 Mass Yield Coefficients 44

VI Table of Contents

1.3.5.4 Energy Yield Coefficients 451.3.6 Equilibrium Relationships 46

1.3.6.1 General Considerations 461.3.6.2 Case A. Calculation of pH with an Ion Charge Balance 47

1.3.7 Energy Balancing for Bioreactors..... 491.3.6.3 Case B. Determining Heat Transfer Area or Cooling Water Temperature 52

2 BASIC BIOREACTOR CONCEPTS 55

2.1 INFORMATION FOR BIOREACTOR MODELLING... .....552.2 BIOREACTOR OPERATION .....56

2.2.7 Batch Operation 572.2.2 Semicontinuous or Fed Batch Operation..... ....582.2.3 Continuous Operation , 602.2.4 Summary and Comparison 63

3 BIOLOGICAL KINETICS 67

3.1 ENZYME KINETICS 683.1.1 Michaelis-Menten Equation 683.1.2 Other Enzyme Kinetic Models 733.1.3 Deactivation 763.1.4 Sterilization 76

3.2 SIMPLE MICROBIAL KINETICS 773.2.1 Basic Growth Kinetics 773.2.2 Substrate Inhibition of Growth 803.2.3 Product Inhibition 813.2.4 Other Expressions for Specific Growth Rate 813.2.5 Substrate Uptake Kinetics 833.2.6 Product Formation 853.2.7 Interacting Microorganisms ....86

3.2.7.1 Case A. Modelling of Mutualism Kinetics..... 883.2.7.2 Case B. Kinetics of Anaerobic Degradation 89

3.3 STRUCTURED KINETIC MODELS ..........913.3.1 Case Studies 93

3.3.1.1 Case C. Modelling Synthesis of Poly-B-hydroxybutyric Acid (PHB) 933.3.1.2 Case D. Modelling of Sustained Oscillations in Continuous Culture 943.3.1.3 Case E. Growth and Product Formation of an Oxygen-Sensitive Bacillus-

subtilis Culture 97

4 BIOREACTOR MODELLING 101

4.1 GENERAL BALANCES FOR TANK-TYPE BIOLOGICAL REACTORS 1014.1.1 The Batch Fermenter. 1034.1.2 The Chemostat 1044.1.3 The Fed Batch Fermenter 1 064.1.4 Biomass Productivity 1094.1.5 Case Studies 109

Table of Contents VII

4.1.5.1 Case A. Continuous Fermentation with Biomass Recycle 1104.1.5.2 Case B. Enzymatic Tanks-in-series Bioreactor System 112

4.2 MODELLING TUBULAR PLUG FLOW BIOREACTORS 1134.2.1 Steady-State Balancing 1134.2.2 Unsteady-State Balancing for Tubular Bioreactors 775

5 MASS TRANSFER 117

5.1 MASS TRANSFER IN BIOLOGICAL REACTORS 1175.7.7 Gas Absorption with Bioreaction in the Liquid Phase 7775.1.2 Liquid-Liquid Extraction with Bioreaction in One Phase 7785.1.3 Surface Biocatalysis 7785.7.4 Diffusion and Reaction in Porous Biocatalyst 779

5.2 INTERPHASEGAS-LIQUID MASS TRANSFER 1195.3 GENERAL OXYGEN BALANCES FOR GAS-LIQUID TRANSFER 123

5.3.1 Application of Oxygen Balances 7255.3.1.1 Case A. Steady-State Gas Balance for the Biological Uptake Rate 1255.3.1.2 Case B. Determination of KLa Using the Sulfite Oxidation Reaction 1265.3.1.3 Case C. Determination of Kjâ by a Dynamic Method 1265.3.1.4 Case D. Determination of Oxygen Uptake Rates by a Dynamic Method 1285.3.1.5 Case E. Steady-State Liquid Balancing to Determine Oxygen Uptake Rate.. 1295.3.1.6 Case F. Steady-State Deoxygenated Feed Method for KJJI 1305.3.1.7 Case G. Biological Oxidation in an Aerated Tank 1315.3.1.8 Case H. Modelling Nitrification in a Fluidized Bed Biofilm Reactor 133

5.4 MODELS FOR OXYGEN TRANSFER IN LARGE SCALE BIOREACTORS 1375.4.1 Case Studies for Large Scale Bioreactors 7 39

5.4.1.1 Case A.Model for Oxygen Gradients in a Bubble Column Bioreactor 1395.4.1.2 Case B.Model for a Multiple Impeller Fermenter 140

6 DIFFUSION AND BIOLOGICAL REACTION IN IMMOBILIZEDBIOCATALYST SYSTEMS 145

6.1 EXTERNAL MASS TRANSFER 1466.2 INTERNAL DIFFUSION AND REACTION WITHIN BIOCATALYSTS ..... 149

6.2.1 Derivation of Finite Difference Model for Diffusion-Reaction Systems. 1516.2.2 Dimensionless Parameters from Diffusion-Reaction Models 7546.2.5 The Effectiveness Factor Concept. 7556.2.4 Case Studies for Diffusion with Biological Reaction 757

6.2.4.1 Case A. Estimation of Oxygen Diffusion Effects in a Biofilm 1576.2.4.2 Case B. Complex Diffusion-Reaction Processes (Biofilm Nitrification).... 157

7 AUTOMATIC BIOPROCESS CONTROL FUNDAMENTALS 161

7.1 ELEMENTS OF FEEDBACK CONTROL 1617.2 TYPES OF CONTROLLER ACTION 163

7.2.7 On-OffControl 1637.2.2 Proportional (P) Controller 7647.2.3 Proportional-Integral (PI) Controller 765

VIII Table of Contents

7.2.4 Proportional-Derivative (PD) Controller 1667.2.5 Proportional-Integral-Derivative (PID) Controller 167

7.3 CONTROLLER TUNING 1697.3.1 Trial and Error Method 7697.3:2 Ziegler-Nichols Method. 7697.3.3 Cohen-Coon Controller Settings 1707.3.4 Ultimate Gain Method 777

7.4 ADVANCED CONTROL STRATEGIES 1727.4.1 Cascade Control 7727.4.2 Feed Forward Control 1737.4.3 Adaptive Control 7747.4.4 Sampled-Data Control Systems 774

7.5 CONCEPTS FOR BIOPROCESS CONTROL 1757.5.7 Selection of a Control Strategy 7767.5.2 Methods of Designing and Testing the Strategy 7 78

REFERENCES 181

REFERENCES CITED IN PART I 181RECOMMENDED TEXTBOOKS AND REFERENCES FOR FURTHER READING 184

PART II DYNAMIC BIOPROCESS SIMULATION EXAMPLES ANDTHE BERKELEY MADONNA SIMULATION LANGUAGE. 191

8 SIMULATION EXAMPLES OF BIOLOGICAL REACTIONPROCESSES USING BERKELEY MADONNA 193

8.1 INTRODUCTORY EXAMPLES 1938.7.7 Batch Fermentation (BATFERM) 7938.7.2 ChemostatFermentation (CHEMO) 7998.1.3 Fed Batch Fermentation (FEDBAT) 204

8.2 BATCH REACTORS 2098.2.7 Kinetics of Enzyme Action (MMKINET) 2098.2.2 Lineweaver-Burk Plot (LINEWEAV) .....2728.2.3 Oligosaccharide Production in Enzymatic Lactose Hydrolysis (OLIGO) 2158.2.4 Structured Model for PHB Production (PHB) ....279

8.3 FED BATCH REACTORS 2248.3.1 Variable Volume Fermentation (VARVOL and VARVOLD) 2248.3.2 Penicillin Fermentation Using Elemental Balancing (PENFERM) 2308.3.3 Ethanol Fed Batch Diauxic Fermentation (ETHFERM) 2408.3.4 Repeated Fed Batch Culture (REPFED) 2458.3.5 Repeated Medium Replacement Culture (REPLCUL) 2498.3.6 Penicillin Production in a Fed Batch Fermenter (PENOXY) 253

8.4 CONTINUOUS REACTORS 2578.4.7 Steady-State Chemostat (CHEMOSTA) 2578.4.2 Continuous Culture with Inhibitory Substrate (CONINHIB) 2678.4.3 Nitrification in Activated Sludge Process (ACTNITR) 267

Table of Contents IX

8.4.4 Tubular Enzyme Reactor (ENZTUBE) 2728.4.5 Dual Substrate Limitation (DUAL) 2758.4.6 Dichloromethane in a Biofllm Fluidized Sand Bed (DCMDEG) 2808.4.7 Two-Stage Chemostat with Additional Stream (TWOSTAGE) 2868.4.8 Two Stage Culture with Product Inhibition (STAGED) 2908.4.9 Fluidized Bed Recycle Reactor (FBR) 2958.4.10 Nitrification in a Fluidized Bed Reactor (NITBED)... 2998.4.11 Continuous Enzymatic Reactor (ENZCON) 3058.4.12 Reactor Cascade with Deactivating Enzyme (DEACTENZ) 3088.4.13 Production ofPHB in a Two-Tank Reactor Process (PHBTWO) 314

8.5 OXYGEN UPTAKE SYSTEMS 3188.5.1 Aeration of a Tank Reactor for Enzymatic Oxidation (OXENZ) 3188.5.2 Gas and Liquid Oxygen Dynamics in a Continuous Fermenter (INHIB) 3218.5.3 Batch Nitrification with Oxygen Transfer (NITRIF) 3278.5.4 Oxygen Uptake and Aeration Dynamics (OXDYN) 3318.5.5 Oxygen Electrode for Kia (KLADYN, KLAFIT and ELECTFIT) 3358.5.6 Biofiltration Column with Two Inhibitory Substrates (BIOFILTDYN). 3428.5.7 Optical Sensing in Microtiter Plates (TITERDYN and T1TERB1O) 349

8.6 CONTROLLED REACTORS 3548.6.1 Feedback Control of a Water Heater (TEMPCONT) 3548.6.2 Temperature Control of Fermentation (FERMTEMP) 3588.6.3 Turbidostat Response (TURBCON) 3638.6.4 Control of a Continuous Bioreactor, Inhibitory Substrate (CONTCON)367

8.7 DIFFUSION SYSTEMS ....3718.7.1 Double Substrate Biofilm Reaction (BIOFILM) 3778.7.2 Steady-State Split Boundary Solution (ENZSPLIT).... 3778.7.3 Dynamic Porous Diffusion and Reaction (ENZDYN).... 3838.7.4 Oxygen Diffusion in Animal Cells (CELLDIFF) 3888.7.5 Biofilm in a Nitrification Column System (NITBEDFILM) 393

8.8 MULTI-ORGANISM SYSTEMS ..4008.8.1 Two Bacteria with Opposite Substrate Preferences (COMMENSA) 4008.8.2 Competitive Assimilation and Commensalism (COMPASM) 4068.8.3 Stability of Recombinant Microorganisms (PLASMID) 4118.8.4 Predator-Prey Population Dynamics (MIXPOP) 4178.8.5 Competition Between Organisms (TWOONE) 4228.8.6 Competition between Two Microorganisms in a Biofilm (FILMPOP). 4258.8.7 Model for Anaerobic Reactor Activity Measurement (ANAEMEAS).... 4338.8.8 Oscillations in Continuous Yeast Culture (YEASTOSC) 4418.8.9 Mammalian Cell Cycle Control (MAMMCELLCYCLE) 445

8.9 MEMBRANE AND CELL RETENTION REACTORS 4518.9.1 Cell Retention Membrane Reactor (MEMINH) 4518.9.2 Fermentation with Pervaporation (SUBTILIS) 4558.9.3 Two Stage Fermentor With Cell Recycle (LACMEMRECYC) 4648.9.4 Hollow Fiber Enzyme Reactor for Lactose Hydrolysis (LACREACT). 4708.9.5 Animal Cells in a Fluidized Bed Reactor (ANIMALIMMOB) 477

X Table of Contents

9 APPENDIX: USING THE BERKELEY MADONNA LANGUAGE.. 483

9.1 A SHORT GUIDE TO BERKELEY MADONNA 4839.2 SCREENSHOT GUIDE TO BERKELEY MADONNA 488

10 ALPHABETICAL LIST OF EXAMPLES 497

11 INDEX 499

Preface

Our goal in this textbook is to teach, through modelling and simulation, thequantitative description of bioreaction processes to scientists and engineers. Inworking through the many simulation examples, you, the reader, will learn toapply mass and energy balances to describe a variety of dynamic bioreactorsystems. For your efforts, you will be rewarded with a greater understanding ofbiological rate processes. The many example applications will help you to gainconfidence in modelling, and you will find that the simulation language used,Berkeley Madonna, is a powerful tool for developing your own simulationmodels. Your new abilities will be valuable for designing experiments, forextracting kinetic data from experiments, in designing and optimizingbiological reaction systems, and for developing bioreactor control strategies.

This book is based on part of our successful course, "Biological ReactionEngineering", which has been held annually in the Swiss mountain resort ofBraunwald for the past twenty five years and which is now known, throughoutEuropean biotechnology circles as the "Braunwald Course". More details canbe found at our website www.braunwaldcourse.ch. Modelling is oftenunfamiliar to biologists and chemists, who nevertheless need modellingtechniques in their work. The general field of biochemical reaction engineeringis one that requires a very close interdisciplinary interaction between appliedmicrobiologists, biochemists, biochemical engineers, engineers and managers; alarge degree of collaboration and mutual understanding is therefore important.Professional microbiologists and biochemists often lack the formal trainingneeded to analyze laboratory kinetic data in its most meaningful sense, andthey may sometimes experience difficulty in participating in engineeringdesign decisions and in communicating with engineers. These are just the verytypes of activity required in the multi-disciplinary field of biotechnology.Chemical engineering's greatest strength is its well-developed modellingconcepts, based on mass and energy balances, combined with rate processes.

Biochemical engineering is a discipline closely related to conventionalchemical engineering, in that it attempts to apply physical principles to thesolution of biological problems. This approach may be applied to themeasurement and interpretation of laboratory kinetic data or as well to thedesign of large-scale fermentation, enzymatic or waste treatment processes. Thenecessary interdisciplinary cooperation requires the biological scientists andchemical engineers involved to have at least a partial understanding of eachother's field. The purpose of this book is to provide the mathematical toolsnecessary for the quantitative analysis of biological kinetics and otherbiological process phenomena. More generally, the mathematical modelling

XII Preface

methods presented here are intended to lead to a greater understanding of howthe 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, whichis posed by engineering. A mistake, often made, is to confuse "mathematics"with the engineering modelling approach. In modelling an attempt is made toanalyze a real and possibly very complex situation into a simplified andunderstandable physical analog. This physical model may contain manysubsystems, all of which still make physical sense, but which now can beformulated as mathematical equations. These equations can be handledautomatically by the computer. Thus the engineer and the biologist are freedfrom the difficulties of mathematical solution and can tackle complex problemsthat were impossible before. Models, however, still have to be formulated andone 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 themicrobiologist to fully appreciate the importance of differential equations, massbalance equations are not so difficult to understand, since the first law ofconservation, namely that matter can neither be created nor destroyed, isfundamental to all science. Mass balances, when combined with kinetic rateequations, to form simple mathematical models, can be used with very greateffect as a means of planning, conducting and analyzing experiments. Modelsare especially important as a means of obtaining a better understanding ofprocess phenomena. A rational approach to experimentation and designrequires a considerable knowledge of the system, which can really only beachieved by means of a mathematical model. This book attempts todemonstrate this by way of the many detailed examples.

The contribution made to biotechnology by the biochemical engineeringmodelling approach is especially important because the basic procedure can bedeveloped from a few fundamental principles. An aim of this book is todemonstrate that you do not have to be an engineer to learn modelling andsimulation. The basic concepts of the material balance, combined withbiological and enzyme kinetics, are easily applied to describe the behavior ofwell-stirred tank and tubular fermenters, mixed culture dynamics, interphasegas-liquid mass transfer and internal biofilm diffusional limitations, asdemonstrated in the computer examples supplied with this book. Such models,when solved interactively by computer simulation, become much moreunderstandable to non-engineers.

The Berkeley Madonna simulation language, used for the examples in thisbook, is especially suitable because of its sophisticated computing power,interactive facility and ease of programming. The use of this digital simulationprogramming language makes it possible for the reader, student and teacher toexperiment directly with the model, in the classroom or at the desk. In this wayit is possible to immediately determine the influence of changing various

Preface XIII

operating parameters on the bioreactor performance - a real learningexperience.

The simulation examples serve to enforce the learning process in a veryeffective manner and also provide hands-on confidence in the use of asimulation language. The readers can program their own examples, byformulating new mass balance equations or by modifying an existing exampleto a new set of circumstances. Thus by working directly at the computer, theno-longer-passive reader is able to experiment directly on the bioreactionsystem in a very interactive way by changing parameters and learning abouttheir probable influence in a real situation. Because of the speed of solution, atrue degree of interaction is possible with Berkeley Madonna, allowingparameters to be changed easily. Plotting the variables in any configuration iseasy during a run, and the results from multiple runs can be plotted togetherfor comparison. Other useful features include data fitting and optimization.

In our experience, digital simulation has proven itself to be absolutely themost effective way of introducing and reinforcing new concepts that involvemultiple interactions. The thinking process is ultimately stimulated to the pointof solid understanding.

Organization of the Book

The book is divided into two parts: a presentation of the background theory inPart I and the computer simulation exercises in Part II. The function of the textin Part I is to provide the basic theory required to fully understand and to makefull use of the computer examples and simulation exercises. Numerous casestudies provide illustration to the theory. Part II constitutes the main part of thisbook, where the simulation examples provide an excellent instructional andself-learning tool. Each of the more than fifty examples is self-contained,including a model description, model equations, exercises, computer programlisting, nomenclature and references. The exercises range from simpleparameter-changing investigations to suggestions for writing a new program.The combined book thus represents a synthesis of basic theory and computer-based simulation examples.

Quite apart from the educational value of the text, the introduction and useof the Berkeley Madonna software provides the reader with the considerablepractical advantage of a differential equation solution package. In the appendixa screenshot guide is found concerning the use of the software.

Part I: "Principles of Bioreactor Modelling" covers the basic theorynecessary for understanding the computer simulation examples. This sectionpresents the basic concepts of mass balancing, and their combination withkinetic relationships, to establish simple biological reactor models, carefullypresented 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 anunderstanding of the physical meaning and significance of each term in themodel equations. The aim in presenting the relevant theory is thus not to beexhaustive, but simply to provide a basic introduction to the theory required fora 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 ofmass and energy balance relationships. Emphasis is given to physicalunderstanding. The text is accompanied by example cases to illustrate theapplication of the material.

Chapter 2 serves to introduce the varied operational characteristics of thevarious types of bioreactors and their differing modes of operation, with theaim of giving a qualitative insight into the quantitative behavior of thecomputer simulation examples.

Chapter 3 provides an introduction to enzyme and microbial kinetics. Aparticular feature of the kinetic treatment is the emphasis on the use of morecomplex structured models. Such models require much more consideration tobe given to the biology of the system during the modelling procedure, butdespite their added complexity can nevertheless also be solved with relativeease. They serve as a reminder that biological reactions are really infinitelycomplex.

Chapter 4 is used to derive general mass balance equations, covering all typesof fermentation tank reactors. These generalized equations are then simplifiedto show their application to the differing modes of stirred tank bioreactoroperation, discussed previously and which are illustrated by the simulationexamples.

Chapter 5 explains the basic theory of interfacial mass transfer as applied tofermentation systems and shows how equations for rates of mass transfer can becombined with mass balances, for both liquid and gas phases. A particularextension of this approach is the combination of transfer rate and materialbalance equations to models of increased geometrical complexity, asrepresented by large-scale air-lift and multiple-impeller fermenters.

Chapter 6 treats the cases of external diffusion to a solid surface and internaldiffusion combined with biochemical reaction, with practical application toimmobilized biocatalyst and biofilm systems. Emphasized here is theconceptual ease of handling a complex reaction in a solid biocatalyst matrix.The resulting sets of tractable differential-difference equations are solved bysimulation techniques in several examples.

Chapter 7 describes the importance of control and summarizes controlstrategies used for bioreaction processes. Here the fundamentals of feedbackcontrol systems and their characteristic responses are discussed. This materialforms the basis for performing the many recommended control exercises in thesimulation examples. It also will allow the reader-simulator to develop his orher own control models and simulation programs.

Preface XV

Part II, "Dynamic Bioprocess Simulation Examples and the Berkeley Madonnasimulation language" comprises Chapter 8, with the computer simulationexamples, and Chapter 9, which gives the instructions for using Madonna, Eachexample in Chapter 8 includes a description of its physical system, the modelequations, that were developed in Part I, and a list of suggested exercises. Theprograms are found on the CD-ROM. These example exercises can be carriedout in order to explore the model system in detail, and it is suggested that workon the computer exercises be done in close reference to the model equationsand their physical meaning, as described in the text. The exercises, however, areprovided simply as an idea for what might be done and are by no meansmandatory or restrictive. Working through a particular example will oftensuggest an interesting variation, such as a control loop, which can then beprogrammed and inserted. The examples cover a wide range of application andcan easily be extended by reference to the literature. They are robust and arewell tested by a variety of undergraduate and graduate students and by also the350 participants, or so, who have previously attended the Braunwald course. Intackling 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 ourearlier, less relevant material was omitted. On the other hand, a number of newexamples resulting mainly from the authors' latest research and teaching workwere added. There was also an opportunity in this new edition to eliminatemost of the past errors and to avoid new ones as much as possible. Mostimportantly, the examples have been rewritten in Berkeley Madonna, which allof our reader-simulators will greatly appreciate.

Our book has a number of special characteristics. It will be obvious, inreading it through, that we concentrate only on those topics of biologicalreaction engineering that lend themselves to modelling and simulation and donot attempt to cover the area completely. Our own research work is used toillustrate theoretical points and from it many simulation examples are drawn. Alist of suggested books for supplementary reading is found at the end ofChapter 6, together with the list of cited references. The diversity of thesimulation 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. Theauthors' four nationalities and three mother tongues, made it difficult to settleon 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 toobtain an understanding of biochemical engineering and also to those chemicaland biochemical engineers wanting to sharpen their modelling skills andwishing to gain a better understanding of biochemical process phenomena. Wehope that teachers with an interest in modelling will find this to be a usefultextbook for undergraduate and graduate biochemical engineering andbiotechnological courses.

XVI Preface

Acknowledgements

A major acknowledgement should be made to the excellent pioneering texts ofR. G. E. Franks (1967 and 1972) and also of W. L. Luyben (1973), forinspiring our interest in digital simulation.

We are especially grateful to our students and to the past-participants of theBraunwald course, for their assistance in the continuing development of thecourse and of the material presented in this book. Continual stimulus andassistance has also been given by our doctoral candidates, especially at theChemical Engineering Department, ETH-Zurich, as noted throughout thereferences.

We are grateful to and have great respect for the developers of BerkeleyMadonna and hope that this new version of the book will be useful in drawingattention to this wonderful simulation language.

Part I Principles ofBioreactor Modelling

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

Nomenclature for Part I

Symbols

AAaabbBcCCP

CPRDDddDOEEESffFGGhiHAHIIjKKKD

AreaMagnitude of controller input signalSpecific areaConstant in Logistic EquationConstant in Luedeking-Piret relationConstant in Logistic EquationMagnitude of controller output signalFraction carbon converted to biomassConcentrationHeat capacityCarbon dioxide production rateDiffusivityDilution rateDifferential operator and diameterFraction carbon converted to productDissolved oxygenEnzyme concentrationEthanolEnzyme-substrate concentrationFraction carbon converted to CO2Frequency in the ultimate gain methodFlow rateGas flow rateIntracellular storage productPartial molar enthalpyHenry's Law constantEnthalpy changeInhibiting component concentrationCell compartment massesMass fluxMass transfer coefficientConstant in Cohen-Coon methodAcid-base dissociation constant

Units

m2

variousm2/m3

1/h1/hm3/kg hvarious

kg/m3, kmol/m3

kJ/kg K, kJ/mol Kmol/hm2/h1/h-, m

air sat.g/m3, 9g/m3

kg/m3

g/m3

1/hm3/h and m3/sm3

kg/m3

kJ/molbar m3/kgkJ/mol or kJ/kgkg/m3

kg/m3

kg/m2h, mol/m2h1/hvarious

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

Nomenclature

kKcakGaKIKLakLaKMKp

KWKSLMmNNnnOTROURPPPQqRRRrfiri/jRQrxSSsTTTLtTrUVVvmaxw

ConstantGas-liquid mass transfer coefficientGas film mass transfer coefficientInhibition constantGas-liquid mass transfer coefficientLiquid film mass transfer coefficientMichaelis-Menten constantProportional controller gain constantDissociation constant of waterMonod saturation coefficientLengthMassMaintenance coefficientMass fluxMolar flow rateNumber of molsReaction orderOxygen transfer rateOxygen uptake ratePressureProduct concentrationOutput control signalTotal transfer rateSpecific rateIdeal gas constantRecycle flow rateResidual active biomassReaction rateReaction rate of component iReaction rate of component i to jRespiration quotientGrowth rateConcentration of substrateSlope of process reaction curveStoichiometric coefficientTemperatureEnzyme activityTime lagTimeTransfer rateHeat transfer coefficientVolumeFlow velocityMaximum reaction rateWastage stream flow rate

various1/h1/hkg/m3, kmol/m3

1/h1/hkg/m3, kmol/m3

various

kg/m3

mkg or mol1/hkg/m2 hmol/h—_mol/h and kg/hmol/h and kg/hbarkg/m3 and g/m3

variouskg/h and mol/hkg/kg biomass hbar m3/ K molm3/hkg/m3

kg/m3h, kmol/m3hkg i/m3hkg /m3h, kmol/m3hmol CO2/mol ©2kg biomass/m3 hkg/m3, kmol/m3

various—C o r Kkg/m3

h, min. or sh, min and smol/m3 hkJ/m2 C hm3

m/hkmol/m3 hm3/h

Nomenclature

wXYYiyZ

Mass fractionBiomass concentrationYield coefficientYield of i from jMol fraction in gasLength variable

kg/m3

kg/kgkg i/kg j,mol i/mol j

m

Greek

8

a8AO

V

pZT

T

T

Controller errorPartial differential operatorConcentration difference quantityDifference operatorThiele ModulusEffectiveness factorSpecific growth rateMaximum growth rateStoichiometric coefficientDensitySummation operatorResidence timeController time constantElectrode time constant

various

kg/m3

1/h1/h

kg/m3

h and sss

Indices

*

0121,2,..., nAA-aAcaeragitanaerappATP/S,Ac

RefersRefersRefersRefersRefersRefersRefersRefersRefersRefersRefersRefersRefersRefers

ATP/S,CO2 RefersATP/NADH RefersATP/X Refers

to equilibrium concentrationto initial, inlet, external, and zero orderto time ti, outlet, component 1, tank 1, and first orderto tank 2, time t2 and component 2to stream, volume elements and stagesto component A, anions and bulkto anionsto ambientto acetoin and acetoin formationto aerobicto agitationto anaerobicto apparentto ATP yield from reaction glucose --> Acto ATP yield from glucose oxidationto ATP produced from NADHto consumption rate ATP «> biomass

Nomenclature

avg Refers to averageB Refers to component B, base, backmixing, and surface positionBu Refers to butanediolCO2 Refers to carbon dioxided Refers to deactivation and deathD Refers to derivative controlD Refers to D-value in sterilizationE Refers to electrodeE Refers to energy by complete oxidationf Refers to finalG Refers to gas and to cellular compartmentH+ Refers to hydrogen ionsi Refers to component i and to interfaceI Refers to inhibitorI Refers to integral controlinert Refers to inert componentK Refers to cellular compartmentK+ Refers to cationsL Refers to liquidm Refers to maximumm Refers to metabolitemax Refers to maximumn Refers to tank numberNH4 Refers to ammoniumNC>2 Refers to nitriteNOs Refers to nitrateO and O2 Refer to oxygenP Refers to productPA Refers to product APB Refers to product BQ Refers to heatQ/O2 Refers to heat-oxygen ratioQ/S Refers to heat-substrate ratioR Refers to recycle streamr Refers to reactorr,S Refers to reaction of substrates Refers to settlerS Refers to substrate and surfaceSL Refers to liquid film at solid interfaceSn Refers to substrate ntot Refers to totalX Refers to biomassX/i Refers to biomass-component i ratioX/S Refers to biomass-substrate ratio

Nomenclature

Refers to difference between cations and ionsBar above symbol refers to dimensionless variable

1 Modelling Principles

1.1 Fundamentals of Modelling

1.1.1 Use of Models for Understanding, Design andOptimization of Bioreactors

An investigation of bioreactor performance might conventionally be carriedout in an almost entirely empirical manner. In this approach, the bioreactorbehavior would be studied under practically all combinations of possibleconditions of operation and the results then expressed as a series ofcorrelations, from which the resulting performance might hopefully beestimated for any given set of new operating conditions. This empiricalprocedure can be carried out in a very routine way and requires relatively littlethought concerning the actual detail of the process. While this might seem tobe rather convenient, the procedure has actually many disadvantages, since verylittle real understanding of the process would be obtained. Also very manyexperiments would be required in order to obtain correlations that would coverevery process eventuality.

Compared to this, the modelling approach attempts to describe both actualand probable bioreactor performance, by means of well-established theory,which when described in mathematical terms, represents a working model forthe process. In carrying out a modelling exercise, the modeller is forced toconsider the nature of all the important parameters of the process, their effecton the process and how each parameter can be defined in quantitative terms,i.e., the modeller must identify the important variables and their separateeffects, which, in practice, may have a very highly interactive combined effecton the overall process. Thus the very act of modelling is one that forces abetter understanding of the process, since all the relevant theory must becritically assessed. In addition, the task of formulating theory into terms ofmathematical equations is also a very positive factor that forces a clearformulation of basic concepts.

Once formulated, the model can be solved and the behavior predicted by themodel compared with experimental data. Any differences in performance may


10 1 Modelling Principles

then be used to further redefine or refine the model until good agreement isobtained. Once the model is established it can then be used, with reasonableconfidence, to predict performance under differing process conditions, and itcan 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 toestablish or validate the model, but the quantity of experimental data required,as compared to that of the empirical approach is considerably reduced. Apartfrom this, the major advantage obtained, however, is the increasedunderstanding of the process that one obtains simply by carrying out themodelling exercise.

These ideas are summarized below.Empirical Approach: Measure productivity for all combinations of reactoroperating conditions, and make correlations.- Advantage: Little thought is necessary- Disadvantage: Many experiments are required.

Modelling Approach: Establish a model, and design experiments to determinethe model parameters. Compare the model behavior with the experimentalmeasurements. 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 theappropriate mass and energy balance equations (Russell and Denn, 1972). Tothese must be added appropriate kinetic equations for rates of cell growth,substrate consumption and product formation, equations representing rates ofheat and mass transfer and equations representing system property changes,equilibrium relationships, and process control (Blanch and Dunn, 1973). Thecombination of these relationships provides a basis for the quantitativedescription of the process and comprises the basic mathematical model. Theresulting model can range from a very simple case of relatively few equationsto models of very great complexity.

Simple models are often very useful, since they can be used to determine thenumerical values for many important process parameters. For example, amodel based on a simple Monod kinetics can be used to determine basicparameter values such as the specific growth rate (JLI), saturation constant (Ks),biomass yield coefficient (Yx/s) and maintenance coefficient (m). This basickinetic data can be supplemented by additional kinetic factors, such as oxygentransfer 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 bederived and used to provide a complete kinetic description of, say, a simplebatch fermentation.

For complex fermentations, involving product formation, the specificproduct production rate (qp) is often correlated as a complex function offermentation conditions, e.g., stirrer speed, air flow rate, pH, dissolved oxygencontent and substrate concentration. In other cases, simple kinetic models canalso be used to describe the functional dependence of productivity on celldensity and cell growth rate.

A more detailed "structured kinetic model" may be required to give anadequate description of the process, since cell composition may change inresponse to changes in the local environment within the bioreactor. The greaterthe complexity of the model, however, the greater is then the difficulty inidentifying the numerical values for the increased number of model parameters,and one of the skills of modelling is to derive the simplest possible model thatis capable of a realistic representation of the process.

A basic use of a process model is thus to analyze experimental data and touse this to characterize the process, by assigning numerical values to theimportant process variables. The model can then also be solved withappropriate numerical data values and the model predictions compared withactual practical results. This procedure is known as simulation and may beused to confirm that the model and the appropriate parameter values are"correct". Simulations, however, can also be used in a predictive manner to testprobable behavior under varying conditions; this leads on to the use of modelsfor process optimization and their use in advanced control strategies.

The application of a combined modelling and simulation approach leads tothe following advantages:

1. Modelling improves understanding, and it is through understanding thatprogress is made. In formulating a mathematical model, the modeller isforced to consider the complex cause-and-effect sequences of theprocess in detail, together with all the complex inter-relationships thatmay be involved in the process. The comparison of a model predictionwith actual behavior usually leads to an increased understanding of theprocess, simply by having to consider the ways in which the model mightbe in error. The results of a simulation can also often suggest reasons asto why certain observed, and apparently inexplicable, phenomena occurin practice.

2. Models help in experimental design. It is important that experiments bedesigned in such a way that the model can be properly tested. Often themodel itself will suggest the need for data for certain parameters, whichmight otherwise be neglected, and hence the need for a particular type ofexperiment to provide the required data. Conversely, sensitivity tests onthe model may indicate that certain parameters may have a negligible


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

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

4. Models may be used in training and education. Many important aspectsof bioreactor operation can be simulated by the use of very simplemodels. These include such concepts as linear growth, double substratelimitation, changeover from batch to fed-batch operation dynamics, fed-batch feeding strategies, aeration dynamics, measurement probedynamics, cell retention systems, microbial interactions, biofilm diffusionand bioreactor control. Such effects are very easily demonstrated bycomputer, as shown in the accompanying simulation examples, but areoften difficult and expensive to demonstrate in practice.

5. Models may be used for process optimization. Optimization usuallyinvolves considering the influence of two or more variables, often onedirectly related to profits and one related to costs. For example, theobjective might be to run a reactor to produce product at a maximumrate, 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 toreassess both the basic theory (physical model) and the mathematical equations,representing the physical model (mathematical model) in order to achieve therequired degree of agreement, between the model prediction and actual plantperformance (experimental data).

As shown in Fig. 1.1, the following stages in the modelling procedure can beidentified:

(i) The first stage involves the proper definition of the problem and hence thegoals and objectives of the study. These may include process analysis,improvement, optimization, design and control, and it is important that the aimsof the modelling procedure are properly defined. All the relevant theory mustthen be assessed in combination with any practical experience with the process,


Physical Model

LJt Revise ideasand equations

Mathematical Model

Newexperiments

NOExperimental Data

Solution: C = f(t)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 developedand examined. At this stage, it is often helpful to start with the simplest possibleconception of the process and to introduce complexities as the developmentproceeds, rather than trying to formulate the full model with all its complexitiesat 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 andproduction) and many of these vary as functions of time (batch, fed-batchoperation). For these reasons the resulting mathematical relationships oftenconsist of quite large sets of differential equations. The thick arrow in Fig. 1.1designates both the importance and the difficulty of this mathematicalformulation.

(iii) Having developed a model, the model equations must then be solved.Mathematical models of biological systems are usually quite complex andhighly non-linear and are such that the mathematical complexity of the


equations is usually sufficient to prohibit the use of an analytical means ofsolution. Numerical methods of solution must therefore be employed, with themethod preferred in this text being that of digital simulation. With thismethod, the solution of very complex models is accomplished with relative ease,since digital simulation provides a very easy and a very direct method ofsolution.

Digital simulation languages are designed specially for the solution of setsof simultaneous differential equations using numerical integration. Many fastand efficient numerical integration routines are now available and areimplemented within the structure of the languages, such that many digitalsimulation languages are able to offer a choice of integration routine. Sortingalgorithms within the structure of the language enable very simple programs tobe written, having an almost one-to-one correspondence with the way in whichthe basic model equations were originally formulated. The resulting simulationprograms are therefore very easy to understand and also to write. A furthermajor advantage is a convenient output of results, in both tabulated andgraphical 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 modellingprocedure. The validity of the model depends on the correct choice of theavailable theory (physical and mathematical model), the ability to identify themodel parameters correctly and the accuracy of the numerical solution method.

In many cases, owing to the complexity and very interactive nature ofbiological processes, the system will not be fully understood, thus leaving largeareas of uncertainty in the model. Also the relevant theory may be verydifficult to apply. In such cases, it is then often very necessary to make rathergross simplifying assumptions, which may subsequently be eliminated orimproved as a better understanding is subsequently obtained. Care andjudgement must also be used such that the model does not become overcomplex and so that it is not defined in terms of too many immeasurableparameters. Often a lack of agreement between the model and practice can becaused by an incorrect choice of parameter values. This can even lead to quitedifferent trends being observed in the variation of particular parameters duringthe simulation.

It should be noted, however, that often the results of a simulation model donot have to give an exact fit to the experimental data, and often it is sufficient tosimply have a qualitative agreement. Thus a very useful qualitativeunderstanding of the process and its natural cause-and-effect relationships isobtained.


1.1.4 Simulation Tools

Many different digital simulation software packages are available on the marketfor PC and Mac application. Modern tools are numerically powerful, highlyinteractive and allow sophisticated types of graphical and numerical output.Most packages also allow optimisation and parameter estimation. BERKELEYMADONNA is very user-friendly and very fast. We have chosen it for use inthis book, and details can be found in the Appendix. With it data fitting andoptimisation can be done very easily. MODELMAKER is also a more recent,powerful and easy to use program, which also allows optimisation andparameter estimation. ACSL-OPTIMIZE has quite a long history ofapplication in the control field, and also for chemical reaction engineering.MATLAB-SIMULINK is a popular and powerful software for dynamicsimulation 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 intomodelling courses can be achieved in various ways, and the method chosen willdepend largely on how much time can be devoted, both inside and outside theclassroom. The most time-consuming method for the student is to assignmodelling problems to be solved outside the classroom on any availablecomputer. If scheduling time allows, computer laboratory sessions areeffective, with the student working either alone or in groups of up to three oneach monitor or computer. This requires the availability of many computers,but has the advantage that pre-programmed examples, as found in this text, canbe used to emphasize particular points related to a previous theoreticalpresentation. This method has been found to be particularly effective whenused for short, continuing-education, professional courses. By use of thecomputer examples the student may vary parameters interactively and makeprogram alterations, as well as working through the suggested exercises at his orher own pace. Demonstration of a particular simulation problem via a singlepersonal computer and video projector is also an effective way of conveyingthe basic ideas in a short period of time, since students can still be very active insuggesting parametric changes and in anticipating the results. The bestapproach is probably to combine all three methods.


1.2 Development and Meaning of DynamicDifferential Balances

As indicated in Section 1.1, many models for biological systems are expressedin terms of sets of differential equations, which arise mainly as a result of thepredominantly time-dependent nature of the process phenomena concerned.

For many people and especially for many students in the life sciences, themention of differential equations can cause substantial difficulty. This sectionis therefore intended, hopefully, to bring the question of differential equationsinto perspective. The differential equations arise in the model formulation,simply by having to express rates of change of material, due to flow effects orchemical and biological reaction effects. The method for solution of thedifferential equations will be handled automatically by the computer. It ishoped that much of the difficulty can be overcome by considering thefollowing case. In this section a simple example, based on the filling of a tankof water, is used to develop the derivation of a mass balance equation from thebasic 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 toderive the dynamic balance equations.

Consider a tank into which water is flowing at a constant rate F (m3/s), asshown in Fig. 1.2. At any time t, the volume of water in the tank is V (m3) andthe 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 increaseby 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 tankduring the time interval A t gives,

pAV = p F A tSince p is constant,

- FAt - 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 firstorder differential equation, to describe the tank filling operation,

dVdT = F

What do we know about the solution of this equation? That is, how does thevolume 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 canrearrange the equation and integrate it between appropriate limits to give,

or for constant F,to

= F f l l dt= F(t i - to)Jto

Integration is equivalent to summing all the contributions, such that the totalchange 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 thedifferential equation is,

V = F t + constant

In this case, as shown in Fig. 1.3, the constant of integration is the initial volumeof water in the tank, VQ, at time t = 0.

Vodt

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


Note that the slope in the variation of V with respect to t, dV/dt, is constant, andthat 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 -k t

The above model equation applies also to this situation.Solving the model equation to obtain the functional dependence of V withrespect to t,

JdV = |F dt = J(F0 - k t)dt = FO Jdt - k Jt dt

Integrating analytically,

The solution is,

V = FO t -kt"

+ constant

kt'v = FO t - — + V0

."tFigure 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 theslope is always F. When F becomes zero, the slope of the curve relating V and talso becomes zero. In other words, the volume in the tank remains constantand 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^ /The flow rate of mass^V of mass within the tank ) = \ 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 asdM/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.

= P F

andd(oV)

= pF

Thus this approach leads directly to a differential equation model, which is thedesired form for dynamic simulation. Note that both terms in the aboverelationship are expressed in mass quantities per unit time or kg/h.

At constant density, the equation again reduces to,

dVdT = F

which is to be solved for the initial condition, that at time t = 0, V=Vo and for avariation in flow rate, given by,

F = F 0 -k twhich is valid until F = 0.

These two equations, plus the initial condition, form the mathematicalrepresentation or the mathematical model of the physical model, representedby the tank filling with an entering flow of water. Thus this approach leadsdirectly to a differential equation model, which is the desired form forsimulation. This approach can be applied not only to the total mass but also tothe mass of any component.

We have seen an analytical solution to this model, but it is also interesting toconsider how a computer solution can be obtained by a numerical integrationof the model equations. This is important since analytical integration is seldompossible in the case of real complex problems.

Computer Solution

The numerical integration can in principle be performed using the relations:

dV


where t - to represents a very small time interval and V - VQ is the resultingchange in volume of the water in the tank. As before, the flow is assumed todecrease 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 = to-3) 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 theintegration time interval or integration step length.

5) At the end of the integration time interval, the value of V will havechanged to a new value, representing the change of V with respect to timefrom 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 withrespect to time, (dV/dt), at the end of the integration time interval cannow be calculated.

7) Knowing the value of V and the value of dV/dt at the end of theintegration time interval, a new value of V can be estimated over a furtherstep 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 longerincreases, and the resulting steady-state value of V is obtained, includingall the intermediate values of V and F, which were determined during thecourse of the calculation.

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

1.3 Formulation of Balance Equations 21

Using such a numerical integration procedure, the computer can thus be usedto generate data concerning the time variations of both F and V. In practice,more complex numerical procedures are employed in digital simulationlanguages to give improved accuracy and speed of solution than illustrated bythe 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^ f Rate of mass flow^j

into the system J ^ out of the system J

Dynamic Total Mass Balances

Many bioreactor applications are, however, such that conditions are in factchanging with respect to time. Under these circumstances, a steady-state massbalance is inappropriate and must be replaced by a dynamic or unsteady-statemass balance, which can be expressed as:

(Rate of accumulation of ( Rate of ^ ( Rate of ^

mass in the system J ^mass flow inj ^mass flow out J

Here the rate of accumulation term represents the rate of change in the totalmass 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 asimplification of the more general dynamic balance, involving the rate ofaccumulation.

At steady-state:


( Rate of ^

I accumulation of mass .= 0 = (Mass flow in) - (Mass flow out)

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, butmost fluid streams, encountered in practice, contain more than one chemical orbiological species. Provided no chemical change occurs, the generalizeddynamic equation for the conservation of mass can also be applied to eachcomponent. Thus for any particular component:

Rate ofaccumulation of mass

of componentin the system

( Mass flow of Athe component _into the system J ( Mass flow of >

the component outof the system ,

Component Balances with Reaction

Where chemical or biological reactions occur, this can be taken into account bythe addition of a further reaction rate term into the generalized componentbalance. Thus in the case of material produced by the reaction:

Rate ofproduction

of thecomponent

by the reaction>

Rate ofconsumption

of thecomponent

by the reaction

' Rate of ^

accumulation

of mass

of component

^ in the system,

=

' Mass flow

of the

component

into

^the system,

-

^ Mass flow ^

of the

component

out of

^the system,

+

and in the case of material consumed by the reaction:

Rate of '

accumulation

of mass

of component

în the system,

=

Mass flow

of the

component

into

vthe system,

-

' Mass flow

of the

component

out of

^the system^

-


Elemental Balances

The principle of the mass balance can also be extended to the atomic level andapplied to particular elements. Thus in the case of bioreactor operation, thegeneral mass balance equation can also be applied to the four main elements,carbon, hydrogen, oxygen and nitrogen and also to other elements if relevantto 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 outthe system J ^ the system J V of the system )

Note the elemental balances do not involve reaction terms since the elements donot change by reaction.

The computer example PENFERM, is based on the use of elemental massbalance equations for C, H, O and N which, when combined with otherempirical rate data, provide a working model for a penicillin productionprocess.

While the principle of the mass balance is very simple, its application canoften be quite difficult. It is important therefore to have a clear understandingof both the nature of the system (physical model), which is to be modelled bymeans of the mass balance equations, and also of the methodology ofmodelling.

1.3.2 Balancing Procedure

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

I. Choose the balance region such that the variables areconstant or change little within the system. Drawboundaries around the balance region

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


parameter systems) or systems of tubes (differential systems), or evencombinations 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 contentsare uniform throughout. This means that the outlet stream properties areidentical with the tank properties, in this case CA and p. The balance region cantherefore be taken around the whole tank.

The total mass in the system is given by the product of the volume of thetank contents V (m3) multiplied by the density p (kg/m3), thus Vp (kg). Themass of any component A in the tank is given as the product of V times theconcentration 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

' A O A1

Figure 1.7. The tubular reactor concentration gradients.

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


that of the time of passage of material as it flows along the reactor and isequivalent to the time available for reaction to occur. Under steady-stateconditions the concentration at any position along the reactor will be constantwith respect to time, though not with position. This type of behavior can beapproximated by choosing the balance regions sufficiently small so that theconcentration of any component within a region can be assumed to beapproximately uniform. Thus in this case, many uniform property subsystems(well-stirred tanks or increments of different volume but of uniformconcentration) 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 anddifferential 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 ortubular process, whereas the eddy region can be approximated by the behaviorof a well-mixed tank. The interaction between the main flow of the river andthe eddy, with flow into the eddy from the river and flow out from the eddyback 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 (ortanks) representing the main flow of the river. This interacts at some particularstage 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 benecessary to draw boundary regions around each of the individual subsystemsrepresenting the main river flow, sections 1 to 8 in Fig. 1.9, and also around thetank system representing the eddy. This would lead to a very minimum of nine

River

Eddy

Figure 1.8. A complex river flow system.


component balance equations being required. The resulting model could beused, for example, to describe the flow of a pollutant down the river in rathersimple terms.

>„*

1 1 1 1 I I 1

1 ' 2 ' 3 1 4 1 5 1 6 1 7 1 81 , 2 , 3 4 , 5 , 6 , 7 , 8

i i i i i i i

V

>

f

kFlow interaction

fitSliM^B

:iiK&^'ffK<&y^

>fc Cr

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

//. Identify the transport streams which flow across thesystem boundaries

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

Out by diffusion

Convective flowin

Convectiveflow out

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 resultingmathematical equation will have an understandable physical meaning. Just


starting off by writing down equations is often liable to lead to fundamentalerrors, at least on the part of the beginner. All balance equations have a basiclogic as expressed by the generalized statement of the component balancegiven below, and it is very important that the mathematical equations shouldretain this. Thus:

' Rate ofaccumulation

of massof component

the system )

f Mass flow ^\of the

componentinto

\the system^

f Mass flow of the

componentout of

^the system^

/ Rate ofproduction

of thecomponent by

\ the reaction /

This can be abbreviated as,

(Accumulation) = (In) - (Out) + (Production)

IV. Express each balance term in mathematical form withmeasurable variables

A. Rate of Accumulation Term

This is given by the derivative of the mass of the system, or the mass of somecomponent within the system, with respect to time. Hence:

dMi(Rate of accumulation of mass of component i within the system) =

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 _ d(CjV)dt = dt

where, Q is the concentration of i (kmol/m3 or kg/m3), and pi is the partialpressure of i within the gas phase system. In the case of gases, the Ideal GasLaw can be used to relate concentration to partial pressure and mol fraction.

Thus,p i V = n i R T

where R is in units compatible with p, V, n and T.In terms of concentration,


_ ni Pi y i Pci= T = RT = "RT

where yi is the mol fraction of the component in the gas phase and p is the totalpressure.

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

B. Convective Flow Terms

m3 s

Total mass flow rates are given by the product of volumetric flow multiplied bydensity. Component mass flows are given by the product of volumetric flowrates times concentration.

( Mass \VolumeJ

kg m3 kgs - 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 sameproperties as the system volume as a whole, since for perfect mixing thecontents of the tank will have uniform properties, identical to the properties ofthe fluid leaving at the outlet. Thus, the concentrations of component i bothwithin the tank and in the tank effluent are equal to Qi, as shown in Fig. 1.11.


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 byanalogy 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 hor kg/m2 h) and dQ/dZ (kmol/m) is the concentration gradient as shown inFig. 1.12.

Figure 1.12.surface area A.

Diffusion flux j j driven by concentration gradient (Qo - Cji) / AZ through

In accordance with Pick's Law, diffusive flow always occurs in the direction ofdecreasing concentration and at a rate proportional to the concentrationgradient. Under true conditions of molecular diffusion, the constant ofproportionality 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


effective diffusivity value is used, which must be determined experimentally.The concentration gradient may have to be approximated in finite differenceterms (Finite differencing techniques are described in more detail in Sec. 6.2).Calculating the mass rate requires the area, through which diffusive transferoccurs.

( Massrateof

(^component i

( Diffusivity Yof

(^component ij\^

ConcentrationY Area ^ (gradient perpendicular =-DJ

ofi J^ to transport J

kg 2 _ m2 kg 2 _ kgsm2 m " s m4 m " T

D. Interphase Transport

Interphase mass transport also represents a possible flow into or out of thesystem. In bioreactor modelling applications, this is most frequentlyrepresented 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, thetransfer of oxygen occurs across the gas liquid interface, which exists betweenthe surface of the air bubbles and the surrounding liquid medium, as shown inFig. 1.13.

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

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


/ Rate of \accumulation

of themass of oxygen

in the gasV phase system J

=

/Mass flow\of the

oxygeninto the

Vgas phase )

-

/Mass flow \of the

oxygenfrom the

^ gas phase>

-

/ Rate \of interfacialmass transferfrom the gasphase into

V the liquid s

This form of transfer rate equation will be examined in much more detail inChapter 5. Suffice it to say here that the rate of transfer can be expressed in theform shown below:

f Rate Of \ ( Mass Ûass transferJ= tra"sP°rt

fV coefficient.

Ârea peA /Concentration^ /SystemA^ volume ) ^driving force ) VvolumeJ

= K a A C V

where, a is a specific area for mass transfer, A/V (m2/m3), A is the totalinterfacial area for mass transfer (m2), V is the liquid phase volume (m3), AC isthe concentration driving force (kmol/m3 or kg/m3) and, K is the overall masstransfer coefficient (1/s). Mass transfer rate expressions are usually expressedin 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 kgmj

Production Rate

This term in the component balance equation allows for the production orof material bv reaction and is incorporated into the component

This term in the component balance equation allows for the production orconsumption of material by reaction and is incorporated into the componentbalance equation. Thus,

Rate of >\accumulation

of massof component

the system )

/ Mass flow \of the

componentinto

\the system /

/ Mass flow \of the

componentout of

the system /

/ Rate ofproduction

of thecomponent by

v the reaction /

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


f Mass rate \production of

^component Ay

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

kg __ kgs m3 m3

Equivalent molar quantities may also be used. The quantity r^ is positive whenA 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 thesymbol rx- Thus,

/ Mass rate of ^ /^Growth rate^Vbiomass production^ = rx V = ^per volume) (Volume of system)

kg _ kgs m3 mj

The consumption rate of substrate, r$, is often directly related to the cell growthrate by means of a constant yield coefficient YX/S, which has the units of kgbiomass produced per kg substrate consumed. Thus,

( Ma<» rate \ growth rate V 1 \U>nsumptionJ = M>er volume ABiomass-substrate yield) (Volume)

kg ~ kg biomass kg substrates m3 m =

s m3 kg biomass

V. Introduce other relationships and balances such that thenumber of equations equals the number of dependentvariables

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


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

Reaction rates as functions of concentration, temperature, pHStoichiometric or yield relationships for reaction ratesIdeal gas law behaviorPhysical property correlations as functions of concentrationPressure variations as a function of flow rate

- Dynamics of measurement instruments as a function of the instrumentresponse 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 ofagitation or flow velocity

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

VI. For additional insight with complex problems, drawan information flow diagram

Information flow diagrams can be useful in understanding complexinteractions (Franks, 1966). They help to identify missing relationships andprovide a graphical aid to a full understanding of the interactive nature ofsystem. 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 bepresented. Consider some arbitrary balance region, as shown in Fig. 1.14 bythe shaded area. Mass accumulates within the system at a rate dM/dt, owing tothe competing effects of a convective flow input (mass flow rate in) and anoutput stream (mass flow rate out).


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,

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

dM= Mass rate in - Mass rate out

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

d(p V) system3t = F o P o - F i P i

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

dVdT = FO-FI

The steady-state condition of constant volume in the tank (dV/dt = 0) occurswhen the volumetric flow in, FQ, is exactly balanced by the volumetric flow out,FI. Total mass balances therefore are mostly important for those bioreactormodelling 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 anarbitrary, well-mixed, balance region, as shown in Fig. 1.15.


Species iinflow

Species ioutflow

Figure 1.15. Component balancing for species i.

Thus for any species i, involved in the system, the component mass balance isgiven by:

/^MassflowoA f Rate of N

component i production of/ Rate of \

accumulationof mass

of component ii in tnp cvct<=»m i

=

'Mass flow of^component i

intov the system ^

out of component!^ the system ) \ by reaction }

Expressed in terms of volume, volumetric flow rate and concentration, this isequivalent to:

~ = (F0Ci0)-(F1Cil)+(riV)

with units of mass/time:

,3*6.m~_ m^ m3

m-3 _ill —

1.3.4.1 Case A. Constant Volume Continuous Stirred TankReactor

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


FQ C AO C BO

> f

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

I!;|iillilili

F1 CA1 CB1

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(VCA1)dt

d(VCBi)dt

= F0 CAO -

= F0CBo -

Here it is convenient to use molar masses, such that each term has the units ofkmol/h.Under constant volume conditions:

d(VCA) = VdCA

d(VCB) = VdCB

and in addition FQ = FI. Thus the two model equations, then simplify to give:

dCAi F= V

and- CAI) +

dCBi F~dT~ = V (cBO~CBi)

In these two balances there are four unknowns CAI, Q*i, rAl an(^ rBl8

kinetics are assumed to be first order, as often found in biological systems atlow concentration. Then:

rAl = -

According to the molar stoichiometry,


rfil = -2rM = + 2 k C A !

Together with the kinetic relations there are 4 equations and 4 unknowns, thussatisfying the conditions necessary for the model solution. With the initialconditions, CAI and CBI at time t = 0, specified, the solution to these twosimultaneous equations, combined with the two kinetic relations, will give theresulting changes of concentrations CAI and CBI as functions of time. Thesimulation example ENZCON, is similar to the situation of Case A.

1.3.4.2 Case B. Semi-continuous Reactor with VolumeChange

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

AO

* 2B

Figure 1.17. A semi-continuous reactor example.

The kinetics are as before:t ^

rA = -kCA

In terms of moles the stoichiometry gives,

moles— T—m s

rB = - 2 rA = + 2 k CA

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

= FCAO + rAV

d(V CB)—at— = IB v


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

Fdt = F

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 resultingsystem 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, andVARVOL.

1.3.4.3 Case C. Steady-State Oxygen Balancing inFermentation

Calculation of the oxygen uptake rate, OUR, by means of a steady-state oxygenbalance is an important application of component balancing for fermentation.In the reactor of Fig. 1.1.8, the entering air stream flow rate, oxygenconcentration, temperature and pressure conditions are shown by the subscript0 and the exit conditions by the subscript 1.

> F i > y i > T i > PIGas

F 0 ' y O' T 0 'P 0

Air

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 reactorgives,

f Rate of accum-^j_ f Flowrateî ( Flowrate"\ /^Rate of O2 uptake A

( ulationofO2 J~ [ofO2in J~(ofO2out J~ I by the cells J


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 molarquantities.

Often only the inlet air flow rate FQ and the mol fraction of 62 in the outletgas, yi9 are measured. It is often assumed that the total molar flow rate of gas isconstant. This is a valid assumption as long as the number of carbon dioxidemols produced is nearly equal to the number of oxygen mols consumed or ifthe amounts of oxygen consumed are very small, relative to the total flow ofgas.

Converting to molar quantities, using the Ideal Gas Law,pV = nRT

or in flow terms:pF = NRT

where N is the molar flow rate, R is the gas constant and F is the volumetric flowrate. 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 tocalculate 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 oxygenuptake rate, and NI must be measured, or determined indirectly by an inertbalance. 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 becalculated by measuring one gas flow rate and the mol fraction of an inert gasin the gas stream. Since inert gases, such as nitrogen or argon, are notconsumed or produced within the system (rinert = 0), their mass rates must


therefore be equal at the inlet and outlet streams of the reactor, assumingsteady-state conditions apply. Then for nitrogen

/Molar flow of\ /Molar flow of\V nitrogen in ) = ^ nitrogen out /

and in terms of mol fractions,

NO YO inert = NI yi inert

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

N, =1 yi 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 measuredthe mol fraction of oxygen and carbon dioxide concentration in the exit gas.

1.3.5 Stoichiometry, Elemental Balancing and theYield Coefficient Concept

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

1.3.5.1 Simple Stoichiometry

The Stoichiometry of chemical reactions is used to relate the relative quantitiesof the different materials which react with one another and also the relativequantities of product that are formed. Most chemical and biochemical reactionsare relatively simple in terms of their molar relationship or Stoichiometry. Forsingle reactions stoichiometric coefficients are clearly defined and may usuallyeasily be determined. Some examples are given below:

C3H4O3 + NADH + H+ <± C3H6O3 + NAD+

Pyruvic Acid Lactic Acid


This relation indicates that 1 mol of pyruvic acid reacts with 1 mol of NADH toproduce 1 mol of lactic acid.

Another example of stoichiometry is that of the oxidative decarboxylation ofpyruvic 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 bewritten with exact molar relationships, like the pentose-phosphate pathwaybelow.

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 molof carbon dioxide. Here the molar quantities of NADPH and ATP producedand consumed, respectively, are shown.

For many complex biological reactions, however, not all the elementaryreactions and their contributions to the overall observed reaction stoichiometryare known (Roels, 1983; Bailey and Ollis, 1986; Moser, 1988).

Thus the case of a general fermentation is usually approximated by anoverall reaction equation, where

Substrate + Nitrogen source + O2 -> Product + CO2 + H2O

vNH3(t)NH3 + v02(t)O2 >

VC02(t) CO2 + VH2o(0 H2O

where the i-th product, such as metabolites or biomass, is given by a generalformula.

In the case above, the generalized elemental formulae are used for substrate,biomass and products, but the nitrogen source is given simply as ammonia. Thestoichiometric coefficients, v, for each component are taken relative to that ofsubstrate and their coefficients may vary as a function of time as a result ofchanging fermentation conditions. Some indication as to the relativemagnitudes of the stoichiometric coefficients can be obtained from elementalbalancing techniques, but in general the problem is so complex that otherconcepts, such as the more approximate yield coefficient concept, are used torelate the relative proportions of materials undergoing conversion during thefermentation.


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 —> c CHpOnNq + d CHrOsNt + e H2O + f CO2

[substrate] [biomass] [product]

where c, d and f are the fractions of carbon converted to biomass, product andCO2, respectively.

Elemental balances for C, H, O and N give

C 1 = c + d + fH m + 3 a = c p + dr + 2eO l + 2 b =cn + ds + e + 2fN a = c q + d t

In this general problem there are too many unknowns for the solution methodto be taken further, since the elemental balances provide only four equationsand hence can be solved for only four unknowns. Assuming that the elementalformulae for substrate, biomass and product and hence 1, m, n, p, q, r, s and t aredefined, there still remain six unknown stoichiometric coefficients a, b, c, d, eand f and only four elemental balance equations. Thus the elemental balancesneed supplementation by other measurable quantities such as substrate, oxygenand ammonia consumption rates (assuming controlled pH conditions), andcarbon dioxide or biomass production rates, such that the condition is satisfiedthat the number of unknowns is equal to the number of defining equations. Inprinciple the problem then becomes solvable. In practice, there can beconsiderable difficulties and inaccuracies involved, although the technique ofelemental balancing can still provide useful data. The application of so-calledmacroscopic 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.


<P2 Substrate

C32 Hb2 Oc2 Nd2

04 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 thecorresponding flow vector

() = <]> <1> O O O <E> <5

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

E=a 4 b4 c4 d4 O4

0 0 2 0 O5

1 0 2 0 O6

0 2 1 0 <D7

(C) (H) (O) (N)

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

<I>7 can be obtained by methods of linear algebra and which are detailed byRoels (1983).


In more complex cases with growth and product formation, moreinformation is needed. The introduction of the concept of the degree ofreduction is useful. For organic compounds this is defined as the number ofequivalent available electrons per gram atom C, that would be transferred toCC>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) ys = 4 + m - 2 1

biomass (X) yx = 4 + p - 2 n - 3 qproduct (P) yp = 4 + r - 2 s - 3 t

The reductances for NHs, fÔ and CC>2 are of course zero.Often the elemental composition of the substrate is not known and then thereductance method may be supplemented by the following regularities, whichapply to a wide variety of organic molecules.

Qo2 = 27 J per g equivalent of available electrons transferred to oxygenYx = 4.29 g equivalent of available electrons per equivalent 1 g atom C in

biomassGX = 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 ratiobetween various consumption and production rates of mass and energy. Theyare typically assumed to be time-independent and are calculated on an overallbasis. This concept should not be confused with the overall yield of a reactionor a process. The biomass yield coefficient on substrate (Yx/s) is defined as:

vYx/S = rs

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

/dX\IdTj dX

YX/S = - TdST = - dS"IdTj

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


.. amount of biomass produced* x/s =

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 biomasscomposition is quite uniform, and often product selectivity does not changevery much during an experiment involving exponential growth and associatedproduction. 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 releasedYq/02 = — = — 7

TQ2 amount or oxygen consumed

In terms of carbon substrate consumed,

v - rQ - amount of heat releasedrs amount of carbon source consumed


Table 1.1. Typical mass and energy yield values (Roels, 1983; Atkinson andMavituna, 1991).

Type of yield coefficient Dimension Value

YX/S,aerYx/S,anaerYx/02 (Glucose)YX/ATPYQ/O2YQ/C02YQ/x,aer (Glucose)Yq/x,anaer

c-mol / c-molc-mol / c-molc-mol / molc-mol / molkJ / molkJ / molkJ / c-molkJ / c-mol

0.4-0.70.1-0.21-20.35380-490460325-500120-190

Note: The molecular weight of biomass is taken here as 24.6 g/C-molThe yield coefficients are usually determined as a result of a large number ofelementary biochemical reactions and it can easily be understood that theirvalues 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 forpractical purposes of process description and modelling.

1.3.6 Equilibrium Relationships

1.3.6.1 General Considerations

In many biological systems, processes with large ranges of time constants haveto 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 thegrowth of an organism is to be modelled with a time constant of the order ofhours, it is very useful to ignore all aspects of biological evolution with timeconstants of years. Also fast equilibrium reactions or conformational changesof proteins having time constants below milliseconds should be ignored. Fastreactions can, however, be very important when considering allosteric activationor deactivation of proteins or simply pH-changes during biochemical reactions.


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

This reaction is characterized by the corresponding dissociation equilibriumconstant KD

CLP k_j

In most cases such relationships can be used to express the concentration of allconcentrations in explicit form using, e.g. a protein balance.

C — r1 _i_ c*Ptot ~ P ~r ^LP

^ ^ KD- ^ptot

Cr + KD

The total concentration, Cptot, is then included in a material balance equationand the concentrations of the free receptor and the receptor-ligand complex aredetermined by the equilibrium relationship. This is also true for a simple acid-base equilibrium relationship.

In more complex cases with interactions of various receptors or with a buffersystem containing several components, it is not possible to express theconcentrations in explicit forms and a non-linear algebraic equation has to besolved during the simulation. The implementation of such problems intoBerkeleyMadonna is shown below with the example of pH calculation

1.3.6.2 Case A. Calculation of pH with an Ion ChargeBalance.

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


concentration of these species is a function of the pH as can be seen in theequilibrium equation

Acid •£ Base' + H+

with dissociation constantCBase- H+

CAcid

where CAcid is the concentration of the undissociated acid and CBase" is theconcentration 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 strongbases are completely dissociated. Moderately strong acids and bases exist inboth 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' isalways much smaller than that of

The ion balance reduces to

KBj V1 KAiKW

CBtot,i+£CK+ =

where KAI are the acid dissociation constants (e.g. KAC); KBI are the basedissociation 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 concentrationsof acid i and EC An" is the sum of the anions.

The pH can be estimated from the above equation for any situation bysolving the resulting non-linear implicit algebraic equation, provided the totalconcentrations of the weak acids, CAtot,i> weak bases, CBtot,i» cations of strongbases, CK+, and anions of strong acids, CAIT> are known.


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

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

The example ANAMEAS, Sec. 8.8.6 includes this ion balance for pHcalculation. This equation represents an algebraic loop in a dynamic simulationwhich is solved by iteration at each time interval until 8 approaches zero. This isaccomplished with the root-finding feature of Berkeley Madonna.

If there is pH control, then strong base or acid would be usually added. Theaddition 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 theinstantaneous equilibrium reactions as reactions with finite forward andbackward rates. These rates must be adjusted with their kinetic constants tomaintain the equilibrium for the particular system; that is, these rates must bevery fast compared with the other rates of the model. This approach replacesthe 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, ascaused by reaction heating effects or by cooling and heating for temperaturecontrol. For example, such a balance is needed when the heat of fermentationcauses a variation in bioreactor temperature. Energy balances are writtenfollowing the same set of rules as given above for mass balances in Sec. 1.3.Thus the general form is as follows:

Accumu-^

lation

rate of

Ênergy ,

Rate of^

energy

in by

^flow ,

'Rate of ^

energy

out by

flow

'Rate of"

energy

out by

<transfer>

'Rate of >

energy

generated

^by reactiony

'Rate of >

energy

added by

âgitation ^


The above balance in word form is now applied to the measurable energyquantities of the continuous reactor shown in Fig. 1.20.

A HL » agit

\M)' PO

' '

U,A,TS

l!

P1»

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"dT = - hn) ) + U A (Ta - TI) + rQ V + AHagiagit

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

hn = hio + Cpi (TI -TO)

and withS2>i01=1

Thus with these simplifications,

S- 2X

1=1= v p

Vpcp 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 5 1

Accumulation TermDensities and heat capacities of liquids can be taken as essentially constant.

dTV p c P d F

has units:m3 (kg/m3) (J/kg K) K _ kJ

s ~ 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 TermsThe flow term is F p CP (T0 - TI)

/energy \ /volume\ /energy \with the units, i^nn^J \ctisr) =This term actually describes heating of the stream entering the system with TOto the reaction temperature TI. It is important to note here that this term isexactly the same for a continuous reactor as for a fed-batch system.

Heat Transfer TermThe important quantities in this term are the heat transfer area A, thetemperature driving force or difference (Ta-Ti), where Ta is the temperature ofthe heating or cooling source, and the overall heat transfer coefficient, U. Theheat 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 energy~B55~ = area time degree (area) (degree)

The sign of the temperature difference determines the direction of heat flow.Here if Ta> TI heat flows into the reactor.

Reaction Heat TermThe term rq V gives the rate of heat released by the bioreaction and has theunits of


energy _ energyvolume 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 thesubstrate, for example J/mol or kcal/kg. The rs AHr>s term therefore hasdimensions of (energy/time volume) and is equal to TQ.

Other Heat TermsThe heat of agitation may be the most important heat effect for slow growingcultures, particularly with viscous cultures. Other terms, such as heat lossesfrom the reactor due to evaporation, can also be important.

1.3.6.3 Case B. Determining Heat Transfer Area or CoolingWater Temperature

For aerobic fermentation, the heats of reaction per unit volume of reactor areusually directly related to the oxygen uptake rate, ro2-

Thus for a constant-volume batch reaction with no agitation heat effects, thegeneral energy balance is

/Accumulation rate^ /Energy out^ /Energy generated\V of energy ) ~ ~ \ by transfer J + V by reaction )

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


If T is constant (dT/dt = 0):

UA(Ti-Ta) = r02YQ/o2V

(heat transfer rate) = (rate of heat release)

Using this steady-state energy balance, it is possible to calculate the coolingwater temperature (Ta) for a given oxygen uptake rate and cooling device.Thus,

_-ro2

UA

Alternatively this same relation can be used in other ways:

1) To calculate the additional heat transfer area required for a knownincrease in cooling water temperature.

2) To calculate the biomass concentration allowable for a given coolingsystem, knowing the specific oxygen uptake rate (kg O2 / kg biomass h).

3) To calculate the cooling area required for a continuous fermenter withknown 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 andinterpretation of biological reactor performance, as indicated in Fig. 2.1.Physical factors that affect the general hydrodynamic environment of thebioreactor 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 thebioreactor 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 onboth liquid and gas phase hydrodynamics. The biokinetic input involves suchfactors as cell growth rate, cell productivity and substrate uptake rate. Often thisinformation may come from laboratory data, obtained under conditions whichare 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, considerableinteractions between the bioreactor hydrodynamic conditions and the cellbiokinetics, morphology and physiology, and one of the arts of modelling is tomake proper allowance for such effects. Thus in the large scale bioreactor,some cells may suffer local starvation of essential nutrients owing to acombination of long liquid circulation time and an inadequate rate of nutrientsupply, caused by inadequate mixing or inefficient mass transfer. Agitation andshear effects can affect cell morphology and hence liquid viscosity, which willalso vary with cell density. This means that the processes of cell growth affectthe bioreactor hydrodynamics in a very complex and interactive manner.Changes in the cell physiology, such that the cell processes are switched fromproduction of further biomass to that of a secondary metabolite or product, canalso be affected by selective limitation on the quantity and rate of supply ofsome essential nutrient in the medium. This can in turn be influenced by thebioreactor hydrodynamics and also by the mode of the operation of thebioreactor.

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 wellfounded Bioreactor Model, the results obtained can be quite impressive and


56 2 Basic Bioreactor Concepts

may enable such factors as cell and product production rates, productselectivities, optimum process control and process optimization to bedetermined with some considerable degree of confidence.

Physical Aspects(flow patterns, residence

time, mass transfer)

Biokinetics(order, inhibition,pH,

temperature)

Production rateSelectivity

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 onthe concentration levels of nutrients and products within the bioreactor. Theconcentration dependencies of the reaction or production rate are often quitesimple, but may also be very complex. The magnitude of the rates, however,depend upon the level of concentrations, and it will be seen that concentrationlevels within the bioreactor depend very much on its type and mode ofoperation. Differing modes of operation for the bioreactor can therefore leadto differing rates of cell growth, to differing rates of product formation andhence to substantially differing productivities.

Generally, the various types of bioreactor can be classified as either stirredtank or tubular and column devices and according to the mode of operation asbatch, semi-continuous or continuous operation.

2.2 Bioreactor Operation 57

2.2.1 Batch Operation

Most industrial bioreactors are operated under batch conditions. In this, thebioreactor is first charged with medium, inoculated with cells, and the cells areallowed to grow for a sufficient time, such that the cells achieve the requiredcell density or optimum product concentrations. The bioreactor contents aredischarged, and the bioreactor is prepared for a fresh charge of medium.Operation is thus characterized by three periods of time: the filling period, thecell growth and cell production period and the final emptying period asdepicted in Fig. 2.2. It is only during the reacting period, that the bioreactor isproductive. During the period of cell growth, strictly speaking, no additionalmaterial is either added to or removed from the bioreactor, apart from minoradjustments needed for control of pH or foam, small additions of essentialprecursors, the removal of samples and, of course, a continuous supply of airneeded for aerobic fermentation. Concentrations of biomass, cell nutrients andcell products thus change continuously with respect to time, as the variousconstituents are either produced or consumed during the time course of thefermentation, as seen in Fig. 2.3.

Filling Reacting Emptying Cleaning

Figure 2.2. Periods of operation for batch reactors.

concentration

Aubstrate biomass

product

time

Figure 2.3. Concentration-time profiles during batchwise operation.


During the reaction period, there are changes in substrate and productconcentration with time, and the other time periods are effectively lost asregards 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) Time-variant reaction conditions2) Discontinuous production3) 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 thebioreactor, thus prolonging operation by providing an additional continuoussupply of nutrients to the cells. No material is removed from the reactor, apartfrom normal sampling, and therefore the total quantity of material within thereactor will increase as a function of time. However if the feed is highlyconcentrated, then the reactor volume will not change much and can beregarded as essentially constant.

Figure 2.4. Fed batch bioreactor configuration.


Semi-continuous operation shares the same characteristics as pure batchoperation, in that concentration levels generally change with time and that somedowntime occurs during the initial charging and final discharge period at theend of the process.

The ability to manipulate concentration levels within the bioreactor by anappropriate controlled feeding strategy confers a high degree of flexibility tofed batch or semi-continuous operation, since differing concentration levels canbe utilized to manipulate the rates of reaction. In Fig. 2.4, both the volumetricfeeding rate, F, and the feed substrate concentration SQ, may be constant or mayvary with time, giving the possibility of such feeding strategies as:

1. Slow constant feeding, which can be shown to result in linear growthof 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 keycomponent concentration.

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

1. Extension of batch growth or product production by additionalsubstrate feeding.

2. Possibility of operating with separate conditions for growth andproduction 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 batchoperation, but since additional substrate feeding to the reactor now occurs, thesubstrate balance takes the form:

( Rate<* "| ( Substrate \ ( Substrate >|accumulation = (f^d [n) _ consumption

V of substrate J \ rate )

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


One other balance equation, however, is also necessary, i.e. the total massbalance,

f Rate of accumulation of ^ ( Mass flow rate of feed ^V mass in the reactor / = 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 orrepeated fed batch, which involves changing volume with a filling andemptying period. The changing reactor concentrations repeat themselves witheach cycle. This operation has similarities with continuous operation andapproaches most closely to continuous operation, when the amount withdrawnis 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 thecomputer.

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 ofaddition and removal are such that the volume of the reactor contents ismaintained constant. The depleted material, of course, contains any productsthat 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 thesemay be considered either as discrete stages, as in the continuous, stirred-tankbioreactor, or as differential devices, as represented by the continuous tubularor column reactor.

Continuous tank bioreactor Continuous tubular bioreactor

Figure 2.5. The two main types of continuous reactors.


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

ConcentrationStartupperiod

Steady state

time

Figure 2.6. Startup of a continuous reactor.

Continuous reactors, however, have found little use as biological reactors on aproduction scale, although there are a few important examples (Id's single-cellprotein air lift process, wastewater treatment and the isomerization of corn sugarto fructose syrup). Frequent use is made of continuous reactors in thelaboratory for studying the kinetics of organism growth and for enzymereaction kinetics. This is because the resulting form of the balance equation,leads to an easy method for the determination of reaction rate, as discussed inCh. 4.

The behavior of the two differing forms of continuous reactor, are bestcharacterized by their typical concentration profiles, as shown in Fig. 2.7. Inthis case, S is the concentration of any given reactant consumed, and P is theconcentration of any given product.

So

Cone.

Tank So

Cone.

Tube

distance distance

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

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


lowest concentration at the reactor outlet. The product concentration rises frominlet to outlet. These differences arise because in the tank reactor the enteringfeed is continuously being mixed with the reactor bulk contents and thereforebeing diluted by the tank contents. The feed to the tubular reactor, however, isnot subject to mixing and is transformed only by reaction, as material movesdown the reactor.

No real situation will exactly correspond to the above idealized cases ofperfect mixing or zero mixing (plug flow), although the actual behavior oftanks and tubes tends in the limit towards the corresponding idealized model.The characteristics of continuous operation are as follows:

1. Steady state after an initial start-up period (usually)2. No variation of concentrations with time3. Constant reaction rate4. Ease of balancing to determine kinetics5. No down-time for cleaning, filling, etc.

The balance equations at steady state for a well-mixed tank reactor have theform

0 = (Input) - (Output) + (Production)

since at steady-state the rate of accumulation and therefore the rate of change iszero.

This equation predicts that the reaction rate causes a depletion of substratefrom the feed condition to the outlet, (the product will increase) and that therate 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-stateimplies the same non-transient conditions, but now concentrations also varywith position. The same situation also applies to the case of a series of well-mixed 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 overevery 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, andmoving downstream along the reactor cascade, the substrate concentrationdecreases stepwise from tank to tank, while the product concentration increasesin 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


substrate concentration S, the multiple tank configuration or a tubular reactorwould 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 byhigh product concentration, P. Substrate inhibition systems would be runpreferably in single tanks, however, since then the substrate concentration isalways 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 theformulation of the tank balance equations, as before and then the application ofthe equations, successively from tank to tank such that the effluent from thepreceding tank is the feed of the next and so on. Tanks-in-series bioreactoroperations are illustrated by the simulation examples TWOSTAGE, STAGEDand DEACTENZ in Sec. 8.4.

2.2.4 Summary and Comparison

The operating characteristics of the various reactor modes are summarized inTable 2.1.

The important bioreactor operating parameters will depend on the mode ofoperation. In batch operation, concentration levels can be varied by adjustmentof the initial values, whereas in continuous and semi-continuous operation, theconcentration levels depend on the feed rate and feed concentration. Asindicated previously, the manner in which the bioreactor is operated cantherefore give rise to different concentration levels and therefore differingproductivities. The consequent concentration profiles depend, of course, on thereaction kinetics, which express the rate of reaction as a function of theconcentrations of reactants and products.


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

Batch Equipment simple. Suitable Downtime for loading andfor small production. cleaning. Reaction

conditions change withtime.

Continuous Provides high production. Requires flow control.Better product quality due Culture may be unstableto constant conditions. over long periods.Good for kinetic studies.

Fed batch Control of environmental Requires feeding strategy toconditions, e.g. substrate obtain desiredconcentration. concentrations.

Table 2.2 lists the main operating parameters for the three differing modes ofbioreactor operation.

Table 2.2. Operating variables for batch and continuous bioreactors.Batch Continuous Semicontinuous

Initial medium composition Inlet medium Feed and initial substrateand inoculum composition composition

Temperature, pressure Temperature, pressure Temperature, pressure

pH if controlled pH if controlled pH if controlled

Reaction time Liquid flow rate Liquid flow rate(residence time) (residence time)

Aeration rateAeration rate Feeding rate and control

Stirring rate programStirring rate

Aeration rate

Stirring rate


The foregoing discussion of the varying characteristics of the different reactortypes and their concentration profiles allows a qualitative comparison of thevolume requirements for the different types of reaction, according to theparticular kinetics. For this it is first necessary to consider the qualitative natureof the basic forms of kinetic relationship: zero order, first order, product andsubstrate inhibition. The detailed quantitative treatment of these kinetic forms isdealt with in Ch. 3.

The rate of a zero order reaction is independent of concentration. Manybioreactions at high substrate concentrations follow zero order kinetics and aretherefore insensitive to concentration and to the effects of concentrationgradients. From the kinetic viewpoint, therefore, any reactor type would beequally suitable.

First order reaction rates are directly proportional to concentration.Bioreactions at low concentration are generally first order, and this would favoroperation in either a batch or a tubular/column type reactor. This is becausereactant concentrations in such reactors are generally high overall and hencethe overall rates of reaction are also consequently high. Hence the reactorvolume required for a given duty would generally be small. (In the case of abatch reactor, this of course neglects the time lost for filling, emptying andcleaning.)

A reaction with substrate inhibition would be best run in a tank at lowsubstrate concentration, since the concentration would be low throughout thewhole of the tank contents. Conversely, product inhibition would be morepronounced in tank reactors, since product concentration would be at itshighest. In this case, a tubular type reactor or batch reactor would be preferred.

Table 2.3. Kinetic considerations for reactor choice.

ReactionKinetics

Zero order

First order

Substrateinhibition

Productinhibition

Productiontriggered byshift inenvironment

Batch Tank

OK

Best

Low initialconcentration

Best

OK for temp-erature-shift

ContinuousTanks-in-Series orTubularOK

Best

Low tankconcentrations

Best

Possible

ContinuousSingle Tank

OK

Low con-version onlyBest

Low con-version only

Not suitable

Fed Batch

Low con-versionOK

Best

Low con-version only

Best for con-centration-shift


Table 2.3 compares the performance of batch tanks, continuous tubular ortanks-in-series reactors and single continuous tank reactors. As discussed inSec. 4.2.1, batch tank concentration-time profiles are exactly analogous to thesteady state concentration-distance profiles obtained in continuous tubularreactors. In terms of performance, therefore, the batch reactor would be thesame as a tube, when compared on the basis of equal batch time in the tank andtime 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 ofcontinuous tubular reactors, hence, making their performance similar to that ofa tubular reactor. Owing to the high degree of mixing which leads to a uniformconcentration, the performance of the single continuous stirred tank reactor isvery different to that of the other reactor types. An exact quantitativecomparison can be made using the mass balance equations developed in Ch. 4for each reactor type.

Biological Kinetics

As explained in Sec. 2.1, a realistic bioprocess model will usually require theinput of kinetic rate data. In the case of even simple chemical reactions, thisdata has to be obtained by laboratory experiment. Since biochemical reactionsare controlled by enzymes, it is appropriate to start with a consideration ofsimple enzyme kinetics (Sec. 3.1), In the case of modeling the behavior ofenzyme 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 forbiological modelling applications. Fortunately this is not the case and, asshown in Section 3.2, a quite reasonable overall description of cell growth ratedata is possible, based on an overall empirical relationship, the MonodEquation, which has been found to give a good fit to many generalobservations of cell growth. This overall view, based on the net result of manysimultaneously occurring and highly interacting biochemical reactions, ofcourse represents an incredible oversimplification of the actual situation.Fortunately it seems to work in many instances and can also be easily modifiedto allow for the uptake of substrate by the cells and to include such additionaleffects, as substrate limitation, multiple substrate limitation and productinhibition. It is interesting, that the basic enzyme rate equation, or Michaelis-Menten equation, based on the theory indicated in Sec. 3.1, is of the same basicform as the empirically-based Monod equation for the growth of micro-organisms.

When used in this manner, the cell kinetics are completely devoid of anymechanistic interpretation and constitute what is known as an unstructuredkinetic model (Fig.3.1 A). In other cases, it may be necessary to look in somedetail at individual cell processes and reactions, in order to obtain a morerealistic 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 arecomposed of a catalytic part comprising proteins, RNA, DNA and other cellularcompounds and of a storage part, e.g. poly-hydroxy-alkanoic acids (PHAs) orinclusion bodies of recombinant proteins. A most simple segregated modelconsiders different stages of cells and therefore a distribution of cell stages in aculture without structuring the cell composition (Fig. 3.1C). In the mostrealistic, but most complex situation the model is structured and segregated


68 3 Biological Kinetics

(Fig. 3.ID). For the purpose of this book, the differences of these models canbe best described by their different balance regions.

Non-structured Structured

3CO

t

B

o

£8

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

3.1 Enzyme Kinetics

3.1.1 Michaelis-Menten Equation

The rate of reaction catalyzed by a soluble enzyme can usually be described bythe Michaelis-Menten kinetic equation. This equation can be derived from theaccepted Briggs-Haldane mechanism for a simple enzyme reaction, which isvery 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-substratecomplex.

For a batch reaction, the balances for S and ES are written in terms of themechanism as,

JQ

— = - ki S E + k_i (ES)dt

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

with initial conditions at t = 0:

S = S0 (ES) = 0

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 thesubstrate, most of the enzyme is present during the reaction in the form of theenzyme-substrate complex, ES.Analytical solution is then possible by assuming a quasi-steady state for theenzyme-substrate complex, ES,

d(ES)dt

-= 0

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 togive,

andki S E k.i + k2

E = E0 ~ k_i .+ k2 = E° k_i + k2


o

1Q)O

8

- time

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

Substituting for E and ES the substrate balance becomes,

dS _dt ~

k 2 SEo

giving the Michaelis-Menten equation,

where the parameters in terms of the mechanistic model are for the maximumreaction rate (kmol/m3min):

vmax = k2E0

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

k-i + k2KM - ki

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


vm

rS

vm/2""

Michaehs-Menten region

0 KM 5 KM 10 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. TheMichaelis-Menten equation becomes for S « KM,

_ VmaxS

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

rS = vmax

and the rate of reaction is thus independent of substrate concentration and isconstant at the maximum value.

In the intermediate substrate concentration range, 0.1 KM < S < 10 KM, thefull Michaelis-Menten equation must be used to guarantee an accuracy for rsgreater than 10 %. The parameters vmax and KM can be determined fromexperimental data, either graphically following a linearization of the Michaelis-Menten equation or, better, numerically by using nonlinear parameterestimation techniques.

Graphical determination of KM and vmax is based on rearrangement of theMichaelis-Menten equation into a linear form,

Inversion and rearrangement give,


1rs

KM Ivmax S max

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

Figure 3.3. Lineweaver-Burk double-reciprocal plot.

Typical values of the enzyme kinetic constants are given in Table 3.1. It isinteresting to note that the rate of formation of the enzyme-substrate complexcan be extremely fast, with the constant ki approaching 1 x 1010 L/mol s. Thisis the maximum value for a rate constant of a reaction that is limited bydiffusion 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 s10- 104 1/s1 - 106 1/s10-6 „ 10-l mol/L


The simulation example MMKINET enables a computer study of the basiccharacteristics of the Michaelis-Menten equation to be carried out, andLINEWEAV simulates the study of the Lineweaver-Burk plot for a batchenzyme reaction.

3.1.2 Other Enzyme Kinetic Models

The reaction mechanism of enzyme catalysis can be very complex, resulting incomplicated kinetic equations that are treated in specialized textbooks, as givenin the reference section. Some of the more readily used forms of the modifiedMichaelis-Menten kinetics are presented here.

Double Michaelis-Menten KineticsThis refers to the case when two substrates are involved in the reaction:

vmax Si 82 _rs - (KMi + Si) (KM2 + S2)

InhibitionInhibition occurs when a substance, inhibitor (I), reduces the rate of anenzyme-catalyzed reaction, usually by the inhibitor binding to the enzymeactive 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

I vmaxScompetitive KM (1 + I/KI) + S

E+S«ES-»Pc c non-competitive (i + I/KI) (KM + S)El «• ESI

ES +1 «=> ESI uncompetitive vmax


Usually, the substance I is the substrate or the product, and the reaction kineticsare known as substrate or product inhibition, respectively.

Allosteric KineticsA simple model to describe allosteric inhibition is given, in which the enzymecan bind to more than one substrate molecule. Thus:

nS + E

ESn z ) n P + E

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

Vmax Sn

s " KMn + Sn

As shown in Fig. 3.5, for values of n > 1 an S-shaped function results. KM isthe substrate concentration with r$ = vmax/2. The simulation example PHBemploys this kinetic form.

Temperature and pH InfluenceRates of biological reactions, including growth rates, exhibit a maximum whenplotted versus temperature or pH. The maximum point is referred to as thetemperature optimum or pH optimum for the system. The term temperatureoptimum must be used with caution because the curve is a result of twotemperature dependent processes, the enzyme catalysed reaction and theenzyme deactivation reaction, respectively.


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

At temperatures well below the optimum the enzyme deactivation may beneglected and the temperature influence on the reaction rate described by theArrhenius equation. At higher temperatures both enzymatic reaction andenzyme deactivation rate equations must be solved together with their respectivekinetic constants expressed in terms of the Arrhenius equation.

Most enzymes exhibit a distinct pH optimum. This can be explained bydissociation of acidic and basic groups of the enzyme, especially of its activecenter. The following equation is a useful description of this.

r =KS,H+CH+CH/KI,H


3.1.3 Deactivation

Biocatalysts in reactors usually undergo irreversible conformational changesgenerally known as denaturation or deactivation. This often causes anexponential decrease of activity with time and can be described by a first-orderreaction rate process:

rd = -kdE

Considering that for a batch reactor, dE/dt = r^, the integrated form can bewritten as

E = E0e-kdt

Substitution in the Michaelis-Menten equation yields

= MOJL kdtK M + S

This equation suggests an exponential decrease of reaction rate regardless ofsubstrate concentration. The simulation example DEACTENZ, Sec. 8.4.12illustrates this.

Engineering models for the kinetics of deactivation are given by Prenosil etal. (1987).

3.1.4 Sterilization

Similar to enzyme deactivation, sterilization kinetics can be viewed as a processof inactivation or the removal of viable organisms or cells from the system.Inactivation can be achieved by using heat, radiation or chemicals. It is astatistical process, with the rate of killing being usually proportional to thenumber of the organisms at any time. Therefore it can be described again byfirst-order kinetics:

rd = -k dX

where

For a batch reactor,dX

= -kdX =-kd=k0e-E a /R TX

which upon integration gives,


x _ ,,-kdt

where XQ is the initial live biomass concentration, X is the viable biomass afterthe treatment time t, and kd is the specific deactivation constant (1/s).

The sterilization time will depend on the initial level of contamination. Forthis purpose the D-value is defined as the treatment time required to reduce thepopulation by a factor of ten. This time is related to the rate constant by

2.3

3.2 Simple Microbial Kinetics

3.2.1 Basic Growth Kinetics

Under ideal conditions for growth, when a batch fermentation is carried out, itcan be observed experimentally that the quantity of biomass, and therefore alsothe concentration, increases exponentially with respect to time. This phenomenacan 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 itselfThis leads to an autocatalytic reaction, which is described by a first order rateexpression 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 isequivalent 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 differentialequation is of the form

In X = k t + In X0

or,

= ekt

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


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

InS

InX

Limitation \Stationary

X

Exponential

Lag

Death

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 regionof exponential growth. Three other regions can be identified: between t = 0and ti, there exists a period of cell adaptation or lag phase, and before t2 thereis 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 rateper 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 firstorder decay,

rd = - kd X

where rd is the death rate and k^ is the specific death rate, with the samedimensions as the specific growth rate. This expression is identical withsterilization 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 observedexperimentally that |a is at a maximum when the particular limiting substrateconcentration S is large, and for low concentration ja is proportional to S. Overthe whole range from low to high S, |i is described by the following Monodequation.

3.2 Simple Microbial Kinetics 79

Thus |i varies with S in the same fashion as does the enzymatic rate ofMichaelis-Menten kinetics.

Again, this is a two-parameter equation involving two constants, themaximum specific growth rate |im and the saturation constant KS. It is bestconsidered to be an empirical relation, but since it has the same form as theMichaelis-Menten enzyme kinetics equation, it is sometimes taken to be relatedto a limiting enzymatic step. Although very simple, it often describesexperimental data for growth rates very well. The form of this relation is shownin Fig. 3.7.

M Monod Relation

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

The important properties of this relationship are as follows:

S -» 0,

UrnS = KS, 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 Monodtype relationship can be used, as given in Sec. 3.2.4 and as shown by thesimulation examples NITRIF, Sec. 8.5.3, and BIOFILM, Sec. 8.7.1.


3.2.2 Substrate Inhibition of Growth

Many substrates can be utilized by organisms at low concentration, but at highconcentrations they can also act as toxic growth inhibitors. The |i versus Scurve may then appear in the form shown in Fig. 3.7, and can be described bythe 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 highvalues of S and causes a decrease in \i. Note that high values of KI correspondto a decreasing effect of substrate inhibition. It is seen that larger values of KSshift the left side of the curves towards the right, while increasing values of KIraise the right side of the curves. Thus a wide range of shapes can be achievedby varying the three parameters, but a maximum value of (I is always obtainedat some intermediate value of S.

1.0 -,

0.8 -

0.6 -

0.0

KI

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


The substrate inhibition kinetic curve has the following characteristics, whichresult 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

MmM- = 2 (Ks/Ki)°-5 + 1

3.2.3 Product Inhibition

When the formation of product inhibits the rate of cell growth, the basic Monodequation 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 FormMmS

M- - KS S0 + S

shows the influence of initial concentration, which is sometimes observed ifother components are limiting.

The Teisser EquationILL = |Lim(l-e-S/k)

relates |Li to S exponentially.


The Contois Equation

- KX + S

expresses the effective saturation constant as being proportional to the biomassconcentration X. At high X, |i is inversely proportional to X. This issometimes used to represent a diffusion limitation in flocculating orimmobilized biomass.

The Logistic EquationM, = (a -bX)

encompasses exponential growth and the levelling off to zero growth rate athigh X. For a batch fermentation the biomass balance is,

= a X - b X 2

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 thusX = a/b

Multiple-Substrate Monod kineticscan be used to describe the influence of many substrates, which for twosubstrates takes the form,

Si ^ f S2 Al + SiJ ^2 + 827

In this way either substrate may be limiting under conditions when the othersubstrate is in excess. Note that the multiplicative effect gives for S\ = K\ and


82 = K2, ILL = |LLm/4. An example of such kinetics is the simultaneous requirementof glucose and oxygen by aerobically growing organisms.

Double-Monod kineticscan also be written for two substrates as parallel reactions, according to

lS i k2S2 V 1

This form gives an additive, fractional contribution for each substrate. Thus forSi = 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 growthrate. If both Si and 82 are large then |ii = |iim. Note that the flexibility of thiskinetic form requires twice as many kinetic parameters as the simpler doubleMonod kinetics. An example of this kinetics is the parallel use of alternativesubstrates, such as various types of sugars.

Diauxic Monod Growth can be modelled for two substrates by the relation

K2 + S2 + S / K !

in this way the consumption of substrate 82 will be inhibited until Si isexhausted, for suitably low values of Kj. Diauxic growth can be observed inmany organisms. An example is E. Coli, where the uptake of lactose isrepressed 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 tobe either related to that of growth or to that required for cell maintenance. Thiscan be expressed as:

~rx vrs = ?»5 " m X

where rs is the rate of substrate uptake by the cells (kg substrate/m3 s). Asexplained in Section 1.3.5, YX/S is the stoichiometric factor or yield coefficientand relates mass cells/mass substrate.


The maintenance factor, m, gives the (mass substrate/mass cells time)required for non-growth functions. The total substrate utilization for cellmaintenance 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 termsof the particular quantities related to unit mass of cells and are then known asspecific cell quantities. Thus:

For the specific growth rate (1/h)rx

» = X

For the specific substrate uptake rate (kg S/kg biomass h),

rsqs = x

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

rO2qo2 = —

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, forthe case of substrate:

rs = y^" - m X

then,

qs =

where |i is also a function of S.


By necessity, in wastewater treatment systems the substrate concentration, S, istaken often as total dissolved organic carbon, rather than considering a specificsubstance, such as glucose. The biomass concentration, X, also must be relatedto the total of all microbial species present. Naturally a gross simplification ofsuch a complex system results.

In wastewater treatment systems, biomass growth is immeasurably slow,whereas the substrate uptake can usually be measured fairly accurately. Undersuch circumstances it is then more useful to base the kinetics on the moremeasurable rate and to express r$ as a separate rate equation, which isindependent of rx-

For example, this can be done using an expression, analogous in form to thatof the Monod equation,

VmS

where the constant vm is proportional to the quantity of biomass in the systemand 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 issometimes formed during growth and sometimes after all growth has stopped.A useful equation for the rate of product formation, flexible enough to findfrequent 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 yieldfactor (kg cell produced/kg product produced), relating the growth and productstoichiometry in the "growth associated" term of the Luedeking-Piret equation,and b is a non-growth related term and is important for cultures which produceproduct independent of growth. Often both coefficients of the above equation(YX/P and b) are not constant but are functions of substrate or productconcentration.

When little is known about the detailed kinetics of product formation, a moregeneral expression rp = qp X is used, where qp will usually vary with cultureconditions and concentrations.


3.2.7 Interacting Microorganisms

Multiple-organism populations will involve interactions in which one species oforganisms will exert some influence on another. Such interactions and theirmodels are described below. Considering two microbial species, A and B, threetypes of interactions on each other are possible; a positive beneficial effect (+),a negative detrimental effect (-), or a neutral effect (0). The resultinginteraction possibilities are given in Table 3.3.

The different types of interactions can also be described by a graphicalrepresentation. Thus the growth kinetics can be described by a solid arrowconnecting the substrate to the product, where the organism involved is shownabove the arrow. A solid arrow from one substrate symbol to the same symbolin another growth path indicates that the product from one organism acts as asubstrate for another. Substrate or product inhibition is indicated by a dashedarrow connecting the component to the inhibited organism. The symbols +, -or 0 at the right hand side of the diagram indicate whether or not the organisminvolved has benefited by the interaction. This is made clear by the examplesbelow:

Table 3.3. Definition of microbial interaction types.

Interaction-type

NeutralismCommensalismMutualismPredator-preyPredator-preyAmensalismAmensalismCompetition

OrganismsA B0 00 +

00

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

The simulation example MIXPOP demonstrates this type of system.


CommensalismOrganism A uses substrate 82 to produce product P; organism B uses substrateSi to produce product 82, which benefits organism A since product 82 acts asits substrate.

A

81

The following processes with the compound 82, shown in brackets, involve thisform 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 RemovedOrganism A utilizes a substrate 82, which inhibits the growth of B on substrateSi.

s2 - - - > p2 0

This effect may be found in the removal of toxic wastes in mixed cultures withmultiple carbon sources. An example is found in ANAMEAS in which thehydrogen substrate of the methanogens (A) inhibits the acetogenic organisms(B).

Mutualism with Product RemovedOrganism A utilizes substrates 82 to produce product P. Organism B utilizessubstrate 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)


Mutualism with Products Used Mutually as SubstratesBoth 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 becausePB inhibits its growth.

An example of kinetic modelling is presented for this case, in which the growthof the two organisms, A and B, takes place in a batch reactor with initialsubstrate concentrations SIQ and 820- The growth rate is expressed by Monod-type kinetics and constant yield factors are used to express the substrate uptakeand product formation rates.

Substrate Si balance,

Substrate 82 balance.

_dt -

_dt -

BS2

Product PB balance,

Species A balance,

Species B balance,

dPBIT A PB

dXA"3f" =


dXB~3T = ^B XB

The kinetics are given by Monod-type relations, with a double form of Monodequation employed for species A and a product inhibition term employed forB,

Si PBHA - MmA KI + Si

B = K2 + S2 + (PB/KI)

Other examples of interacting microorganism effects are given in thesimulation examples ACTNITR (neutralism), Sec. 8.4.3, COMPASM(competitive assimilation and commensalism), Sec. 8.8.2, MIXPOP (predator-prey population dynamics), Sec. 8.8.4 and TWOONE (competition betweenorganisms), Sec. 8.8.5.

3.2.7.2 Case B. Kinetics of Anaerobic Degradation

Anaerobic degradation is a very complex multi-substrate, multi-organismprocess, as is depicted in Fig. 3.9. It is shown here how its modelling can beapproached using Monod-type kinetics. This problem is of interest becauseoverloading of an anaerobic reactor causes accumulation of intermediates(organic acids, hydrogen) and consequent inhibition of the final methanogenicstep (Gujer and Zehnder, 1983).

In a first step, polymer materials (carbohydrates, proteins or lipids) arehydrolyzed to yield the monomer compounds (amino acids, sugars and fattyacids). In a second acidogenic step, these compounds are fermented to organicacids (mainly acetic, propionic and butyric acid) and hydrogen. In the thirdacetogenic step, organic acids with more than three atoms of carbon permolecule are converted to acetic acid and hydrogen. In a last methanogenicstep, the intermediates, acetic acid and hydrogen and carbon dioxide areconverted to methane. Five different groups of organisms accomplish thesereactions as shown in Fig. 3.9.

According to thermodynamic calculations (Archer, 1983) the oxidation ofpropionic 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 thatdisturbances in the methanogenic step, which is generally considered to be themost sensitive one, lead to the accumulation of H2- Additionally the state of ananaerobic reactor may be characterized by its volatile fatty acid levels based onthe CH4/CO2 ratio and the total gas production rate.


(0'55

CO

o>

g0)

TJ"5

W'(00)

0)D)O

IO

Figure 3.9. Reaction scheme of anaerobic degradation. The symbols are: Poly - polymermaterial (proteins, fats, hydrocarbons, etc.); XJJY - Biomass hydrolyzing Poly; Mono -monomeric materials from hydrolysis of Poly; XAG ~ acid generating biomass; HPr - propionicacid; 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 onacetate; XH - biomass growing on hydrogen and carbon dioxide. Dashed arrows indicate gaseouscompounds 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 theconsumption of substrate Si and for the formation of product Pi in each stepare:

rxi = Hi Xi - kdi Xi

3.3 Structured Kinetic Models 9 1

Hi Xjrpi =

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

Mimax SiW - 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 eachorganism type, Xi»

Following this approach a model was established (Denac et al., 1988) andcombined with particular control algorithms to simulate a continuous anaerobicdigestor with feed rate control. This included a gas phase balance,thermodynamic equilibrium constraints and acid-base equilibria using an ioncharge balance (Sec. 1.3.6.2). The simulations were used to adjust the controlparameters, which were employed on laboratory reactors (Heinzle et al., 1992).The simulation example ANAEMEAS, Sec. 8.8.7, gives details concerning thismodel.

33 Structured Kinetic Models

In many cases the characterization of biological activity by simply the totalbiomass concentration is insufficient for a true model representation(Fredrickson et al., 1970; Roels, 1983; Moser, 1988). A variation in thebiomass 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.


— 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 complexdescription of the cellular metabolism and a more structured approach to themodelling of cell kinetics. Structured models are based on a compartmentaldescription of the cell mass as shown in Fig. 3.1.

In general it is very difficult experimentally to obtain sufficient mechanisticknowledge about the cell metabolism for the development of a "realistic"structured model. Parameter estimation may be very difficult, and theapplication of complex numerical methods may easily lead to physicallymeaningless results. Often the verification of even simple unstructured modelsis not possible owing to experimental difficulties. This problem becomes muchmore significant with increasing complexity of the model. For this reason,structured models are seldom used for design or control. Structured modelsmay be useful to model transient behavior of a biological system. Such modelsalso may be required if a wide range of changes of environmental conditionshave 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 arewell known. For example, in long-term experiments the cellular compositionof lipids, carbohydrates, protein, RNA and DNA in Baker's Yeast were found tochange as a function of dissolved oxygen and dilution rate (Furukawa et al.,1983). In this work the yeast shape and specific area also changed withdissolved oxygen concentration (Fig. 3.10).

L/d; S/V x 10 [1/m]

2.0 -

1.5

1.0

L/d

S/V

0.01 0.1 1.0 D0[g/m3]

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

3.3 Structured Kinetic Models 93

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

3.3.1 Case Studies

3.3.1.1 Case C. Modelling Growth and Synthesis of Poly-B-hydroxybutyric Acid (PHB)

Fig. 3.11 represents the process of cell growth and the synthesis of intracellularproduct PHB. Residual biomass (R) is the difference between total cell drymass (X) and product PHB (P). Synthesis of PHB occurs with a single limitingsubstrate NH4+ (S) and constant dissolved gas concentrations of H2, C>2 andCO2 (So). Mass flows are indicated by solid lines, and regulatory mechanismsare 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).


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

S [g/L]

3 -

X,P,R [g/L] P/X [-]

2 -

1 -

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

3.3.1.2 Case D. Modelling of Sustained Oscillations inContinuous Baker's Yeast Culture

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


15

10

X [g/L]

E [g/L]

0.5

0

100

50

q X [mmol/h L]

10

0.5

°'1

0.05

0.0

2

0

2.0

1.5

1.0

DO [mg/L]

S [mg/L]

RQ

0 10 20t[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 storagematerial (G) having the same oxidation state as glucose (S) must cyclically beproduced and consumed. This storage material was identified as trehalose andglycogen. Under conditions of high glucose uptake rate and high degradation

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


rate of G, ethanol, E, is accumulated in the medium and can be later oxidizedto yield biomass, R. From this and additional information on the activity of theenzyme T, which catalyzes the degradation of G, the reaction scheme shown inFig. 3.14 was developed and used as a basis for model formulation.Here R is the biomass, not including the intracellular storage material, G. Theenzyme, T, is not considered to contribute significantly to the total biomass andwas therefore neglected in the total biomass balance. The detailed kinetic modelfor continuous culture is given in the simulation example YEASTOSC, Sec.8.8.8.

Many parameters could be determined from independent experiments. Somewere taken from the literature, and some, especially those describing theenzyme T activity had to be based on simulation results. The model leads tooscillations (Fig. 3.15), whose existence and dependency on operatingconditions agree qualitatively with experimental observations.

(g/L)

Time (h)

D=0.05 h'1

r10 15

Time (h)

I ' ' ' ' I10 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.


3.3.1.3 Case E. Growth and Product Formation of an Oxygen-Sensitive Bacillus subtilis Culture

This example shows how knowledge of the biochemical pathways, whencombined with experimental data, can lead to model development. In thisresearch an oxygen-sensitive culture was to be used for mixing studies, and itwas important to establish the kinetic model (Moes et al, 1985, 1986) in orderto describe the batch profiles as shown in Fig. 3.17.

Since it was not possible to describe the growth behavior by simple Monod-type models, an ATP balance was used to establish the available energy forbiomass synthesis. This was possible because the biochemical pathways (Fig.3.18) for the fermentation and their associated chemical energy production andconsumption steps were known.

Gl Ac, Bu (g/L)

10

X (9/L)

- 3

- 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 allenergy-producing steps are balanced by those that consume energy. The formof 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


In this equation q$/GO2 is the rate at which sugar S is converted to CC>2 byrespirative growth. The rate of the sugar conversion to acetoin is qs/Ac- Therates of conversion of acetoin to butanediol and the reverse reaction are givenby qAc/Bu, and QBu/Ac- Energy is required for growth, and the ATPconsumption rate is qATP/X- The corresponding ATP yields Y convert theserates to ATP equivalents. The nomenclature at the end of this example definesall symbols in detail.

Here knowledge of the yields of ATP for each step is important. Rateexpressions for the reaction pathways, as given in Fig. 3.18, were thusestablished for each step in terms of the participating reactants. Batch massbalances for all species were then written in terms of the individual rates ofproduction and consumption of each relevant component.

NADH

Cells 6CO2

Glucose >* 2Pyruvate

NADH

ATP Acetoin ^ > Butanediol

NADH ^ NAD+ r NAD"1"

ADP ^ ATP

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

Using the steady-state approximation that the ATP level does not varysignificantly, allows setting the condition that dATP/dt = 0. The steady stateATP balance is then solved for qATP/X, which is the rate of ATP available forgrowth. 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 1mol NADH was converted to 3 mol ATP in the respiratory chain. At lowoxygen concentrations, the conversion equivalent was assumed to be a functionof oxygen as determined by parameter estimation, based on the experimentaldata.

The glucose substrate balance can be written in terms of the rates at whichsubstrate 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 xX


It was observed experimentally that the reaction stoichiometry for theformation of metabolites was almost constant and was given by Yp/S = 0.57mol (Ac + Bu)/mol glucose. Thus, another balance can be written for thesubstrate:

_ qS/Acdt X

YP/S

These two equations were used to determine dS/dt and qs/GO2-

The biomass balance is,dX = qATP/x YX/ATP X

The metabolite balances are,

dAc"dt" = ( qS/Ac ~ qAc/Bu + qBu/Ac )

and,dBu~dt~ = ( Âc/flu ~ qBu/Ac ) X

In the above balances, all specific rate terms, q, are in the units(mol/g biomass h). All concentrations (ATP, S, Ac, Bu) are in mol/L unitsexcept X (g/L). All yield coefficients Y are in mol/mol units except wheninvolving biomass, e.g. YX/S is in units of g/mol.

Empirical Monod-type kinetic relationships, not given here, were establishedto calculate the rate of glucose to acetoin, qs/Ac» an(i the reversible acetoin tobutanediol rates, qAc/Bu and qBu/Ac» as a function of reactant concentrations forglucose, S, for acetoin, Ac, for butanediol, Bu, and for dissolved oxygen, DO.Additional empirical kinetic terms were needed to fit the followingexperimental observations:

1) Growth stopped when S approached zero.2) qs/Ac was limited by C>23) qBu/Ac was promoted by high C>24) qAc/Bu was promoted by low C>25) The influence of DO with S = 0 was negligible.

The many kinetic parameters were determined partly by direct experiments andpartly by fitting the data using a parameter estimation computer program. Theinfluence of oxygen was determined using data from experiments at controlledoxygen conditions and determining the best values of the oxygen sensitive ratesby parameter estimation procedures. A simple graphical procedure thenallowed determination of the appropriate constants.


The quantities YATP/NADH and YX/ATP are linearly dependent on each other andcould therefore not be determined from experimental data. The maximumvalue of YATP/NADH was arbitrarily fixed at the maximum theoretical value of3.0, 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 wasachieved as shown in Fig. 3.18. From these results it is seen that the modelpredicts the metabolite and biomass profiles. The model was quite versatile andreasonably accurate, considering the large differences in biomass formation athigh DO (X = 3.4 g/L) and low DO (X = 2.5 g/L), as well as the variation of thebutanediol 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 (Moeset al., 1986). X=biomass, Ac=acetoin, Bu=butanediol, Gl=glucose.

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

4 Bioreactor Modelling

4.1 General Balances for Tank-Type BiologicalReactors

Fermentation systems obey the same fundamental mass and energy balancerelationships as do chemical reaction systems, but special difficulties arise inbiological reactor modelling, owing to uncertainties in the kinetic rateexpression and the reaction stoichiometry. In what follows, material balanceequations are derived for the total mass, the mass of substrate and the cell massfor 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, withcell concentration, XQ, and substrate concentration, SQ. The vessel contents,which are well-mixed are defined by volume V, substrate concentration Si andcell concentration X\. These concentrations are identical to those of the outletstream, which has a volumetric flow rate FI.


102 4 Bioreactor Modelling

General Balance FormAs 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 Xj + 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,

-*xfs = Yxl

but other forms of rate equation may equally apply.

The above generalized forms of equations can be simplified to fit particularcases of bioreactor operation.

4.1 General Balances for Tank-Type Biological Reactors 103

4.1.1 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 limitingsubstrate, S at t = 0, the biomass will grow, perhaps after a short lag phase, andconsume substrate. As the substrate becomes exhausted, the growth rate willslow and become zero when substrate is completely depleted. The abovegeneral balances describe the particular case of a batch fermentation if V isconstant and F = 0. Thus,

Total balance:

Substrate balance:

Organism balance:

£ - »dSi

V^r = r s V

VT = rxV

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


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 betaken to apply

UN

Figure 4.3. The chemostat and its variables.

The total mass balance simplifies to,

o = FQ-

The dynamic component balances are then,

Substrate balance

Cell balance

VdSi

VdXi

= F(S0-Si)

= F(X 0-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


Chemostats normally operate with sterile feed, XQ = 0, and hence for the cellbalance,

0 = - F X! + rx V

Inserting the Monod-type rate expressions gives:

For the cell balance,FXi

= rx = Ji X!hence

FH = v = D

Here D is the dilution rate and is equal to 1/T, where i = V/F and is equal to thetank 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 determinesthe outlet substrate concentration Si. The last equation is, of course, alsosimply a statement that the quantity of cells produced is proportional to thequantity of substrate consumed, as related by the yield factor YX/S-

The curves in Fig. 4.4 represent solutions to the steady-state chemostatmodel as obtained from the simulation example CHEMOSTA, Sec. 8.4.1, withKS = 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 valueof Si to increase and the corresponding value of Xi to decrease. It is seenwhen D nears |im, Xi becomes zero and Si rises to the inlet value SQ. Thiscorresponds to a complete removal of the cells by flow out of the tank; aphenomenon known as "washout". Fig. 4.4 shows washout occurring for D wellbelow |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 Theproductivity of biomass, DXi, (kg X/m3 h) passes through a maximum ratherclose to the washout region.


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 afunction of dilution rate.

Chemostat applications are largely found in research laboratories. Microbialphysiology studies can be made conveniently, since the cells can be controlledby the flow rate to grow at a particular value of specific growth rate \i. Kineticstudies can be made by measuring the concentration of the limiting substratefor a range of |u (=D) values, permitting the kinetics, |j = f(Si), to bedetermined. The yield coefficient can be determined by steady statemeasurements of substrate, biomass and product. The influence of anysubstrate in the culture can be tested by adding it to the medium at variousconcentrations.

4.1.3 The Fed Batch Fermenter

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


S«.F,o1 'o

Figure 4.5. The fed batch fermenter and its variables.

The balance equations then become for constant density,

T7dT = F0

d(VSi)dt

d(V XQdt

= F0 S0 + rs V

= r x V

Expanding the differential terms, which are products of the two variables Vand Si and V and Xi, respectively, and substituting for dV/dt = FQ gives:

VdSi= F0(S0-Si)

The above equations are identical to those for a constant volume chemostatreactor. It can be shown by simulation that a quasi-steady state can be reachedwhere 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 movesthrough a series of changing steady states for which |a = D, during which Siand p decrease, and Xi remains constant. This is analogous to a constantvolume reactor with slowly decreasing F. These phenomena are demonstratedby the simulation examples FEDBAT, Sec. 8.1.3, and VARVOL, Sec. 8.3.1.


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

FS0YX/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 substrateinhibition kinetics. Two cycles of operation are shown. The dimensionless variables aredefined in the simulation example VARVOL, Sec. 8.3.1.


Other important types of operation described by the general balances arecyclic fed batch (Keller and Dunn, 1978). An example of cyclic fed batchoperation 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 = DYx/s(So-Si) = DYx/s

The conditions for the maximum value of DXi as shown in Fig. 4.4, can beobtained 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 biomassproduction rate is equal to Xf/tf.

Comparing the continuous biomass production rate to the batch biomassproduction rate, it is found that the continuous fermenter will have a 2 - 3 timeslarger maximum biomass production rate. This is because the batchfermentation starts at the inoculum value of X and has correspondingly lowinitial growth rates.

Biomass production rates for fed batch fermenters are calculated by takingthe total mass produced over the time of operation, or (VX)f - (VX)o/tf . Forcyclic operation the biomass production has been shown to depend on thestarting and final volume ratio (Keller and Dunn, 1978). This and morecomplex questions regarding product productivity for particular kinetics can beanswered by making suitable changes in the simulation example REPFED, Sec.8.3.4. The productivity of a repeated fed batch as compared to chemostatoperation will depend on the operating variables, as well as the kinetics (Dunnet al., 1979).


4.1.5 Case Studies

4.1.5.1 Case A. Continuous Fermentation with BiomassRecycle

The retention of active biomass is a means of improving the reactorproductivity or efficiency for substrate uptake. The biomass separation couldbe performed by any suitable process (flotation, sedimentation, membranefiltration, or centrifugation). The cell recycle stream has a volumetric flow rateR and a biomass recycle concentration XR, where X R > X I and Xi is thebiomass concentration in the stream leaving the fermenter.

Consider the operation of a biological reactor with cell recycle as shown inFig. 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: Biomassaccumulates both within the reactor of volume Vr and also within the separationunit with volume Vs. Assuming that biomass leaves only in the wastage streamand that growth occurs only in the reactor, the balance is then

dXi= 0 - W X R

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

0 = -


Thus, the wastage rate of biomass must equal its production rate, otherwise Xiwill change. Wastage rate is an important control parameter in wastewatertreatment, 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 inletand outlet mass flows. The efficiency of the continuous biomass separationdetermines 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 = R X R - ( F + R)X!+rxV r

Cell separation and recycle lead to high cell concentrations in the reactor,which, when neglecting the contribution by growth, would be XR/(F+R). Sincethe 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 meansthat the specific growth rate is decreased from the chemostat value, D. This isdue to the reduction in substrate concentration, Si, which is caused by thehigher biomass concentration, resulting from the cell recycle. Washout isimpossible due to the complete biomass retention, and for this reason flow ratesgreater than in a chemostat are possible.

The substrate balance gives

JQ

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

which shows that at steady state


This would also be the case without cell recycle, since the substrate is assumedto 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. Thisgives a theoretical basis to the F/M concept, which is well known for the controlof 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 ofenzyme using a membrane, and SUBTILIS, Sec. 8.9.2, involves the retention ofbiomass.

4.1.5.2 Case B. Enzymatic Tanks-in-series BioreactorSystem

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

illi11

81

>.

•S;^;:-);'::;HaKS

: :'•;: y~: •: :w> Vv

1111

S 2

>„

illiFigure 4.9. Tanks-in-series bioreactor.

For the first tank,dSijp = F(S0-Si)

SQ-SIdividing by

— =

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 inthese three equations. Note that different tank sizes may be accounted for bydiffering values for the tank residence times TI, 12 and 13.

If the kinetic terms r$ are only dependent on S, then the above equations canbe solved without any further balances. It is often the case that enzymatic rateequations 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 bysimulation 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 additionalbalance equations for these components must be included in the model. Whencombined with equations for the complete kinetics description (rates as afunction of all the influential concentrations), the model can be solved to obtainthe dynamics of the system and also the final steady state values. It can beshown that a three tanks-in-series reactor system will provide a goodapproximation to the performance of a corresponding tubular reactor, exceptfor very high conversions.

4.2 Modelling Tubular Plug Flow Bioreactors

4.2.1 Steady-State Balancing

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


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 mannerwhich is analogous to the time-varying concentrations that occur in a batchreactor.

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 directanalogy to that of a simple batch reactor. Thus using the batch reactorsubstrate balance (p = constant),

dSdF = fS

The flow velocity, v, for the liquid is defined as,

dZv = dF

or,

dt =dZ

where v = F/A and F is the volumetric flow rate through the tube with cross-sectional area A. Thus substituting for dt,

dS_ rs_dZ = 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 stateconcentration profile of S. Additional component balances would be requiredfor more complex kinetics. This is demonstrated by the simulation examplefor a tubular enzyme reactor ENZTUBE, Sec. 8.4.4.

4.2.2 Unsteady-State Balancing for TubularBioreactors

If dynamic information is needed for tubular or column systems, then changeswith respect to both length and time must be considered. In order to achievethis, the reactor can be considered by dividing the volume of the reactor into Nfinite-differenced axial segments (Fig. 4.12), and treating each segmenteffectively 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,

dSnAV "dT = Sn-l Fn-l - Sn Fn + rSn AV

Constant density gives Fn_i = Fn = F. Dividing by AV,


dSn = A(SF)dt AV

+ rsn

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

38 1 d(SF)- = — -- x J_at A az

When the volumetric flow, F, is constant,

At steady state,

as _F asat ~ ~ A az + rs

and

—V dZ + fS

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 betweentwo or more phases. Usually one of the reactants is transferred from one phaseinto a second phase, in which the reaction takes place. The following cases areexamples of biological systems.

5.1.1 Gas Absorption with Bioreaction in the LiquidPhase

The gas phase is dispersed as gas bubbles within the liquid phase. Mass transferoccurs across the gas-liquid interface, out of the gas into the liquid, where thereaction occurs. The typical example is aeration of the bioreactor broth andthe 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.


118 5 Mass Transfer

5.1.2 Liquid-Liquid Extraction with Bioreaction inOne Phase

An immiscible liquid phase is dispersed in a continuous liquid phase. Masstransfer of a reactant takes place across the liquid-liquid interface, shown here(Fig. 5.2) from the continuous phase into the dispersed phase, where reactionoccurs. An example might be the transfer of a substrate in an oil phase to anenzyme 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 andB diffuse from the liquid to the reaction sites on the surface of the solid, wherereaction occurs. The product C must similarly be transferred away from thesolid reaction surface, as shown in Fig. 5.3. Examples are found withimmobilized enzyme and cell systems. In Sec. 6.1 the modelling aspects of thistype 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. Substratesdiffuse into the porous internal structure of the biocatalyst support and react.Similarly, the products must diffuse away from the reaction sites within thesolid 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. Actualconcentration 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 froman interaction between the mass transfer process and the local fluidhydrodynamics, which change gradually from stagnant flow, close to theinterface, to perhaps fully-developed turbulence within each of the bulk phases.

According to the Two-Film Theory, the actual concentration profiles, asrepresented in Fig. 5.5 can be approximated by linear gradients, as shown inFig. 5.6.

A thin film of fluid is assumed to exist at either side of the interface. Awayfrom 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 theconcentrations, CG and CL, are uniform throughout each relevant phase. At thephase interface itself, it is assumed there is no resistance to mass transfer, andthe interfacial concentrations, CGI and CLI, are therefore in local equilibriumwith each other. All the resistance to mass transfer must, therefore, occur withinthe films. In each film, the flow of fluid is assumed to be stagnant, and masstransfer 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 moleculardiffusion of some component A is given by,

dZ

5.2 Interface Gas-Liquid Mass Transfer 121

GasInterface

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 steadystate concentration gradient (mol/m3). Thus applying the same concept to masstransfer across the two films,

JA = DG ZG ZL

where DG and DL are the effective diffusivities of each film, and ZG and ZL arethe respective thicknesses of the two films.

The above equations can be expressed in terms of mass transfer coefficientskc and kL (m2/s) for the gas and liquid films,

JA = kG(CG-CGi) = kL(CLi-CL)

The total rate of mass transfer, Q (mol/s), is given by,

Q = J A A = jA(aV)

where "A" is the total interfacial area available for mass transfer, and "a" isdefined as the specific area for mass transfer or interfacial area per unit liquidvolume (m2/m3). Thus for the total rate of mass transfer:

In terms of the total interfacial area A,

Q = kGA(CG-CGi) = kLA(CLi-CL)

In terms of a and VL,

Q = ko a (Co - CGi) VL = kLa(CLi-CL)VL

122 5 Mass Transfer

Since the mass transfer coefficient, k, and the specific interfacial area, a, dependon the same hydrodynamic conditions and system physical properties, they arefrequently combined and referred to as a "ka value" or more properly a masstransfer capacity coefficient.

In the above theory, the interfacial concentrations CGI and CLI cannot bemeasured, and are therefore of relatively little use, even if the values of the filmcoefficients are known. For this reason, by analogy to the film equations,overall mass transfer rate equations are defined, based on overall coefficients ofmass transfer, KG and KL, and overall concentration driving force terms, where:

Q = KGA(CG-CG*) = KLA(CL*-CL)

Here, CG* and CL* are the respective equilibrium concentrations, correspondingto the bulk phase concentrations, CL and CG, respectively, as shown in Fig. 5.6.

Equilibrium relationships for gas-liquid systems, at low concentrations ofcomponent A usually obey Henry's law, which is a linear relation between gaspartial pressure, PA, and equilibrium liquid phase concentration, CLA*:

PA=

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

C L C *

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

For gases of low solubility, e.g., oxygen and carbon dioxide in water, theconcentration gradient through the gas film is very small, as compared to thatwithin 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 greaterresistance to mass transfer in the liquid film. The main resistance to masstransfer is predominantly within the liquid film. This causes a large change inconcentration (Cy - CL), since the resistance is almost entirely on the liquidside of the interface.

At the interface, the liquid concentration, Cy, is in equilibrium with that ofthe gas, CGI, and since CGI is very close in magnitude to the bulk gasconcentration, CLI must then be very nearly in equilibrium with the bulk gasphase concentration, CG- This is known as liquid film control and correspondsto the situation where the overall resistance to mass transfer resides almostentirely within the liquid phase. The overall mass transfer capacity coefficientis KLa. Hence the overall mass transfer rate equation used for slightly solublegases in terms of the specific area is

Q = KLa (CL*-CL)VL

where CL* is in equilibrium with CG, as given by Henry's law,

CG= HCL*,

Mass transfer coefficients in fermentation are therefore generally spoken of asKL values or Kâ values for the case of mass transfer capacity coefficients.

5.3 General Oxygen Balances for Gas-LiquidTransfer

In order to characterize aeration efficiency, to predict dissolved oxygenconcentration, or to follow the biological activity it is necessary to developmodels, which include expressions for the rate of oxygen transfer and the rateof oxygen uptake by the cells. Well-mixed phase regions, in which the oxygenconcentration can be assumed uniform, can be described by simple balancingmethods. Situations in which spatial variations occur require more complexmodels, as described in Sec. 5.4. The following generalized oxygen balanceequations are derived for well-mixed phases, using the well-mixed tankconcept. In the situation in Fig. 5.8, both the liquid and gas phases are definedby distinct well-mixed regions and by the total volumes of each phase, VL andVG.

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 ofaccumulation

of oxygenin gas ;

Flow of >

oxygen ininlet gas streamy

f Flow of \_ oxygen out _

Vin exit stream/

( Rate of ^transfer

of oxygenv from gas ,

Thus, for the gas phase,

dCGiKLa(CLi*- CLi)VL

where VQ represents the volume of gas in the dispersed phase, or the gasholdup.

For the liquid phase,

Rate of âccumulationof oxygen

^ in liquid

/Flow ofoxygenin inletliquid

V stream J' Rate ofconsumption

of oxygenin liquid

( Flow ofoxygenout inexit

Rate oftransfer

of oxygen^ from gas

Rate of oxygen consumption = -rO2 = -qo2


Thus for the liquid phase,

dCLi- LiCL i + KLa(CL1*-CLi)VL -

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

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

5.3.1 Application of Oxygen Balances

5.3.1.1 Case A. Steady-State Gas Balance to Determine theBiological Uptake Rate

The convective terms in the generalized liquid balance equation can usually beneglected, owing to the low solubilities of oxygen in water (about 8 g/m3). Thisgives the steady state liquid balance, dCL/dt=0, relation as:

KLa(CLi*- CLI) = qo2Xi

Thus at steady-state, the oxygen transfer rate is effectively equal to the oxygenuptake 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 theentire two-phase system. It shows that the biological oxygen uptake rate can becalculated 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 permitsthe 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 SulfiteOxidation Reaction

If a chemical reaction, classically the oxidation of sodium sulfite, is used to takeup the oxygen from solution, then the term qo2 X VL in the liquid phasebalance may be replaced by the chemical reaction term, ro2 VL- At steady-state,

KLa(CLi*-CLi) = r02

Usually ro2 is obtained by taking samples and titrating for the fractionalconversion of sulfite, which can be related by stoichiometry to the oxygenconsumption. Since the chemical reaction causes the liquid dissolvedconcentration CLI to fall essentially to zero and with CLI* calculated from theoxygen concentration in the exit gas, the value of the overall mass transfercapacity coefficient, Kâ, can be estimated. An improved method uses the gasbalance instead of titration to obtain ro2 in the manner outlined above forqO2 X and also provides a check on the sulfite measurements. The sulfitemethod is useful for comparing aeration systems, but the values are difficult torelate to actual fermentation conditions owing to the very different physicalconditions (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 ofthe dissolved oxygen will increase with time, from zero to effectively 100% airsaturation at the end of the experiment. The exact form of the response curveobtained depends on the values of KLa, the driving force, (CLI* - CLI), and themeasurement dynamics of the dissolved oxygen electrode. The liquid balance,for the unsteady state batch aeration condition, gives:

= KLa(CLi*-CL i)VL


The classical dynamic KLa method assumes that Kâ and CLI* are constant.Under these conditions, the differential equation can be integrated analyticallyto give the relationship:

( C * ^r * Lr = KLatCL - CL\}

Plotting the natural logarithmic concentration function on the left side of theequation versus time, should, in principle, give a straight-line relationship, withâ as the slope.

Usually deoxygenation is accomplished with nitrogen, so that initially the gasphase consists of nitrogen, which is gradually displaced and mixed with air.Under these conditions, CLI* is n°t constant, and a gas balance must beemployed to calculate the variation in CGI versus t. Since the liquid phaseconcentration, CLI, is measured by means of a membrane covered oxygenelectrode, 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 measurementsignal.

The fractional response of the electrode for a step change in CL wouldappear as shown in Fig. 5.9.

time

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

Note that TE corresponds to the time for the electrode to reach 63 % of the finalresponse. 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 shownbelow:

time

Figure 5.10. Response of the gas, the liquid and the electrode measurement during a dynamicKLa 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 simulationexample KLADYN, Sec. 8.5.5.

5.3.1.4 Case D. Determination of Oxygen Uptake Rates by aDynamic Method

Low oxygen uptake rates, as exist in slow growing systems (plant and animalcell cultures, aerobic sewage treatment processes, etc.), cannot easily bemeasured by a gas balance method, since the measured difference between inletand outlet oxygen gas phase concentrations is usually very small. Due to thelow solubility of oxygen in the liquid media, quite small oxygen uptake rateswill 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 ina small chamber or by turning off the reactor air supply, according to theliquid balance equation

dCLi


Dissolved oxygen concentration decreases linearly and is equal to qo2 X asshown 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 islarge, as compared to the electrode time constant, the method is quite accurateand no correction for the electrode measurement dynamics is required (Monaet al., 1979). If the response is too fast the sample can be diluted. This methodis illustrated by the simulation example OXDYN, Sec. 8.5.4. A similarsimulation example, ANAMEAS, Sec. 8.8.7, illustrates dynamic measurementsin anaerobic systems.

5.3.1.5 Case E. Steady-State Liquid Balancing to DetermineOxygen 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 bymeasuring the liquid flow rate and the difference in dissolved oxygen in theliquid inlet and outlet flow streams, according to the following steady-stateliquid 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 oxygenuptake by the cells. This method provides a very sensitive way of measuringlow oxygen uptake rates (Keller et al., 1992, Tanaka et al., 1982). The case-study 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. Steady-State Deoxygenated Feed Method forKLa

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* - CLi)VL

Knowing the flow rate L, the oxygen liquid concentrations CLO and CLI and theoutlet oxygen in the gas phase (to determine CLI*) permits the calculation of


KLa. Another variation of this would be to gas with oxygen-enriched air orwith nitrogen, which would avoid the difficulty of producing a continuoussource of deoxygenated liquid. A similar steady state method has beenemployed to obtain steady oxygen concentration profiles in column (Meister etal., 1980), and tubular bioreactors (Ziegler et al., 1977). A suitable steady statemodel for the tubular reactor then allows calculating the unknown Kâ byparameter estimation (Shioya et al., 1978).

Deoxygenatedliquid CLO

t/•"V—X "N^ -^^^

nM&^/m^M:WM.Xiiiiiiiiiiji

r-^— -~\_/~\^^x

^^^SRSSSSO iO^wiBii

WiiiiilmiiiiiiiAir

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

5.3.1.7 Case G. Biological Oxidation in an Aerated Tank

A batch reactor liquid is aerated with a continuous flow of air to support abiological 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 oxygenconcentration. Since oxygen is relatively insoluble (approximately 8 g/m3

saturation for air-water) the transfer rate is important to maintain a highdissolved oxygen concentration CL. The batch oxygen balance for the liquidphase is then:

f Rate of \accumulation of =

V 02 in liquid )

dCL

, f f N(Transfer rate o f \l02 mto the hqmdj

/ Uptake rate of^ by the cells

= KLa(CL*-CL)VL - kCLVL

A steady-state can be reached for which the mass transfer rate is equal to theoxygen uptake rate by reaction:

giving for CL

0 = KLa (CL* - CL) - k CL

KLaCL*CL = KLa

Using this equation, the reaction rate constant, k, can be determined if CL ismeasured and Kâ is known or measured. The equilibrium value, CL*, can becalculated from the gas phase concentration, and if there is little oxygendepletion it can be calculated from the inlet gas conditions.

If the oxygen depletion in the gas phase is appreciable, then the molefraction of oxygen in the exit may not be the same as in the inlet, and a gasphase balance must be applied to determine CL*:

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


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

/ Rate of O2 \ / Rate of O2 \ / Transfer rate of \0 = V in by flow ) ~ Vout by flow ) ~ \ C>2 to the liquid )

Using the nomenclature in Fig 5.13,

0 = yoN 0 -y iNi -K L a(CL*-CL)VLwhere

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

CL = k~" 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 ) Reaction rate control applies for low values of k/KLa, when re approachesk CL*, and CL approaches CL*

2) Diffusion control applies for high values of k/KLa, when re approachesKL& CL* and CL approaches 0.

3) 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 BedBiofilm Reactor

Nitrification is a two-step microbiological process, in which the ammonium ionis 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 ofammonia and its large oxygen demand. Several known organisms can gainenergy from either of the two oxidation steps, but most commonlyNitrosomonas and Nitrobacter are responsible for steps (1) and (2),respectively. These organisms grow very slowly, obtaining their carbon fromdissolved carbonate. Due to the very slow grow rates, it is of interest to retainthe biomass within the reactor. One possibility considered here is toimmobilize the biomass as a natural biofilm on a fluidized bed of sand (Tanakaand 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+ + 2O2 -

The reactor of volume, Vr, consisted of a conical sand bed column, which wasfluidized by the liquid recycle stream flowing up through the bed. The recyclestream was oxygenated in a separate, baffled, tank contactor of volume VT, withturbine impeller and air or oxygen sparging. The reactor and oxygenator werethus separate parts of a recycle loop configuration. This could be operatedbatchwise or with a continuous feed and effluent stream flow to and from thesystem. When operating at high recycle rates, the whole system actedeffectively as one well-mixed tank system.

The reactor-oxygenator recycle loop can be analyzed as a total system orbroken down into its individual components as shown in Fig. 5.14. Theseinclude liquid phase balance over the reactor and combined phase, liquid phaseand gas phase balances over the oxygenator and over the total system.


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 nitrogen-containing reactants and products. The oxygen balance taken over the totalsystem can be simplified by neglecting the accumulation terms and the liquidflow terms, that will be small compared to the gas rates and the consumption byreaction, 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 totalsystem each component equation has the form,

VdCpdt

= F(CNi-CN2)

When the reactor is operated as a batch system, F = 0, and when used as acontinuous steady state reactor, dCN2/dt = 0. This equation can be used incolumn systems for very low single-pass conversion, when the differences inlocal reaction rate at the reactor inlet and outlet are not large. Although thereactions actually occur in the solid phase, because of the high solid-liquidinterfacial area, the system is treated here as being quasi-homogeneous.

The gas-liquid interfacial mass transfer area will often be small enough to beimportant 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 bedescribed by the oxygen balances for the liquid phase:

0 = FR(CL4-CL3) + KLa(CL*-CL3)VT

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, whichassume ideally mixed phases, are useful in designing the gas absorberaccording 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 supplyrate from the oxygenation tank. This is the condition of reaction-ratelimitation 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 firstnitrification step gives

2TNH4 = T r l ,O2 = ~ri ,NO2

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

rO2 = rl,O2 + r2,O2

From the measured concentration dependency of these rates, the reactionkinetics of the individual steps can be determined. The dependency of theserates on the individual concentrations can then be used to establish the reactionkinetic 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 ofsimulation examples FBR, Sec. 8.4.9 and DCMDEG, Sec. 8.4.6.

5.4 Models for Oxygen Transfer in Large Scale Bioreactors 137

5.4 Models for Oxygen Transfer in Large ScaleBioreactors

Large-scale industrial fermenters can generally be expected to exhibitdeviations from the two idealized flow conditions of perfect mixing or perfectplug flow. Thus the assumption of completely mixed gas or liquid phases maynot be valid. Little experimental information is available on concentrationinhomogeneities or concentration gradients within large bioreactors. Residencetime distribution information, from which a physical and mathematical modelcould be established, is also generally not available.

Convection currents within the liquid phase of a bioreactor are usually causedby the mechanical energy inputs of agitation and aeration. It is oftenreasonable to assume that slowly changing quantities, such as biomassconcentration, substrate concentration, pH and temperature are uniform withinthe whole mass of bioreactor liquid. Oxygen must be considered, however, as arapidly changing substrate, owing to its low solubility in fermentation media. Itis therefore necessary to consider that differences in oxygen transfer anduptake rates will create oxygen concentration gradients throughout the reactor.

Buoyancy forces carry the gas from the lower gas inlet point up to the topliquid surface. In the absence of mechanical agitation, the gas phase mightmove from the bottom to the top of the reactor in an approximate plug flowmanner, with very little backmixing. If the stirring power supplied to thefermenter, however, is sufficient to create liquid velocities, that are greater thanthe free rise velocity of a bubble (about 26 cm/sec) then the bubbles willcirculate around the fermenter, before eventually escaping. Very high powerinputs can cause the smaller bubbles to circulate many times within the vesseland spend an appreciable time before reaching the surface. Under suchconditions, if no bubble coalescence occurs, the gas phase would contain afraction 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 liesomewhere between the extreme cases of idealized plug flow and perfectmixing. Experimental residence-time distribution information, obtained byhelium tracer techniques under actual operating conditions, are then necessaryto characterize the gas phase flow. Unfortunately very little experimentation onindustrial scale equipment has been reported.

Hydrostatic pressure gradients in tall fermenters will cause large differencesin the oxygen solubility, CL*, with regard to the depth position in the tank. In a10m tall reactor, the oxygen solubility for a given gas composition will be twicethat at the bottom of the tank as compared to the top surface, since the totalpressure is effectively doubled. This is seen by Henry's law which can be writtenas:

138 5 Mass Transfer

d* = y°2pH

where yo2 is the mole fraction of oxygen in the air and p is the total pressure atsome point in the tank.

The possibility that oxygen gas compositions, dissolved oxygenconcentrations, oxygen solubilities, gas holdup volumes, bubble sizes and othertransfer parameters can vary with depth in a tall fermenter introduces a muchgreater degree of complexity to the problem of modelling the reactor. Thismakes it difficult to obtain data on oxygen mass transfer coefficients.Although it is impossible to give specific recommendations that apply to anyparticular situation, a further discussion of possible models and theirunderlying assumptions may help to define the problem. Incorporated into themore complex models, discussed below, are such factors as gas and liquid phaseflow pattern, gas composition gradients and the effects of hydrostatic pressure.Great caution and wisdom must be exercised to avoid creating a model that istoo 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 mostgeneral method for modelling is to represent the reactor using balances in aseries of sections or stages. Mass balances in multi-stage process are easy toformulate, since both the liquid and gas phases may be assumed to be well-mixed, 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 ismade up of many stages which are interconnected by the flows of gas andliquid between stages and by diffusive mass transfer mechanisms.


5.4.1. Case Studies

5.4.1.1 Case A. Model for Oxygen Gradients in a BubbleColumn Bioreactor

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

Exit Gas

Gas Feed

Exit Liquid

Liquid Feed

A CQB A CLS

Gas ill

GO L ' LO

Figure 5.17. Stagewise model of a bubble column bioreactor.

140 5 Mass Transfer

The oxygen balance equations for the gas and liquid phases of each stage areas follows:

if = FG(CGn-l-CGn) - KLa(CLn*-CLn)VL

f\(~^

VL-dT = FL(CLn-i-CLn) + KLa(CLn*-CLn)VL - rn VL

where,r * - £1 r »r c * - P02<~Ln H CGn or CLn -77-

rland

rn = Qo2m |

For simplification only oxygen is assumed limiting and growth is notconsidered; however the biomass concentration is contained in the maximumoxygen uptake Qo2m- The dynamics of the oxygen transfer and uptakeprocesses are obtained by solving these differential equations simultaneouslyfor each stage. The resulting solution then gives CLH and CGn > for each stageas functions of time and also yields the resulting final steady state values.

Note that the biomass concentrations Xn are assumed constant, otherwisebiomass balance and growth kinetics equations would have to be added to themodel. Using simulation methods, other effects, such as the effect ofhydrostatic pressure on CG or on bubble size could be included. Thesimulation example DCMDEG, Sec. 8.4.6, demonstrates some aspects of thestagewise 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 considerliquid flow in both directions. It is generally assumed, however, that theintensity of mixing is such that no radial variations occur. Fig. 5.16 representsa multiple impeller reactor with well-mixed liquid zones in the region of eachimpeller. The reactor can be described approximately by means of a three-stage model. Mixing of the liquid in a direction which is directly opposite tothat 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 backmixingstream accounts for a flow interaction between the mixing zones and fordeviations from ideal stage mixing. To determine FB, a tracer experiment


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

Exit Gas Exit Liquid c G3 A A C L3Fd \_^Gas

G2L3

Gas

G1

L2

Gas

'LIFL+FB

Inlet Gas Liquid FeedI F G- C GO \\>cu>

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 well-mixed, 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:

VG

dCLiVL -ar = FLCLO + FBCL2 - (FL + FB) CLI +

+ KLa(CLi*-CLi)VL + riVL

dCGl "•""' xx-< f\ \ -rr- //~1 ^ /"I \A 7-CGI) -KLa(CLi -CLI)VL

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

142 5 Mass Transfer

and

= - Qo2rCLI

Stage 2:

dCL2VL -gf = (FL+ FB) CLI + FBCL3 - (FL+ FB) CL2 - FB CL2

KLa(CL2*-CL2)VL

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

CL2. .

and n CL2

r2 = -Qo2mK0 + CL2

Stage 3:

dCL3

KLa(CL3*-CL3)VL+ r3VL

where

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

CL3. .

and = -QO2nCL3

K0 + CL3

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


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

Diffusion and Biological Reaction inImmobilized Biocatalyst Systems

The retention and immobilization of enzymes and cells usually requires thepresence of an additional solid carrier phase or flocculant cell mass. Asillustrated in Fig 6.1, in order to reach a reaction site, substrate S must first betransported by convection from the bulk liquid to the exterior stagnant film(point A). Then transport by diffusion must occur through the film (from A toB) 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 additionalinternal 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 thesurface, and then away from the surface via the external mass transfer laminarfilm to the bulk liquid.

Diffusion film

iConcentration

Bulkliquid

Solidcarrier

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 transferresistances which may slow the overall reaction rate, since reaction cannotproceed at a rate greater than the rate at which substrate is supplied by themechanism of diffusion. The diffusional mass transfer process via the externalfilm is referred to as external mass transfer. Since the reaction site may oftenbe located within a gel, a porous solid, biofilm or biofloc, the transfer ofsubstrate or substrates from the exterior surface of the biocatalyst to reactionsites, located within the internal structure of the carrier, is also usuallynecessary. This process is therefore referred to as internal mass transfer orintraparticle transfer. In what follows, external transport and internal transport


146 6 Diffusion and Biological Reaction in Immobilized Biocatalyst Systems

are considered separately, although, of course, the two effects can exert acombined effect in reducing the effective reaction rate, compared to that whichwould 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 nearregion of an immobilized biocatalyst surface, supported on a non-porouscarrier. Also shown is the idealized concentration profile, as represented by thefilm theory. As previously discussed in Sec. 5.2, the "film theory" assumes thepresence of a stagnant layer of liquid to exist at the solid-liquid interface. Thisstagnant region is termed the diffusion film or Nernst-diffusion film andconstitutes the external resistance to mass transfer. It thus determines the rate ofsupply 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 immobilizedon a solid surface.

The rate of supply of substrate to the surface is defined by mass transferconsiderations, such that the mass flux to the catalyst surface is given by,

JS = ksL(SA-SB)

6.1 External Mass Transfer 147

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)

kSL (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 andS « KM- The apparent reaction rate per unit surface area, rapp (mol/m2 s), isequal to the rate of both processes.

Solving the equation, for the surface concentration, SB,

SB = -SA

and hence

= ksSB=U^lkS+kSL

Two extreme conditions can be identified:

1) For ks/ksL » 1, SB approaches zero, and the reaction is completely masstransfer controlled, with rapp =

2) For ks/ksL «.!» SB approaches SA, and the reaction is kineticallycontrolled, with an apparent rate equal to that defined by the reactionkinetics, with rapp = ks SA-

The intermediate situation is given by the full equation, for which the apparentreaction rate is influenced by both the true kinetic rate constant ks and by thediffusional mass transfer coefficient ksL-

For a zero-order reaction:kSL(SA-SB) = ks

where ks is now a zero-order kinetic rate constant. The concentration at thereaction surface SB is thus,

SB = SA-ks/kSL


which indicates that the ratio of the magnitudes of the kinetic rate constant tothat of the mass transfer coefficient determines SB- If the reaction is zero-order, the overall order of reaction rate is not influenced by diffusionalconsiderations, but the effective rate will still be reduced, owing to the loweredconcentration SB-

For Michaelis-Menten kinetics, which encompass the range between effectivezero and first order reaction kinetics, the relation between rate of supply andthe rate of reaction becomes,

kSL (SA - SB) = : > - = rapp

After rearrangement, the resulting quadratic equation can be solved for SB, withthe solution indicating that, in general, the magnitudes of all the coefficientscan influence the overall reaction rate and also that the external transfer canchange the overall observed reaction kinetics. Thus they no longer follow theMichaelis-Menten form with respect to bulk concentration, and the apparentkinetics can differ substantially from the intrinsic true reaction kinetics. Underthese conditions, it is no longer correct to equate the Michaelis-Mentenconstant, KM, to the substrate concentration at which the observed reaction rateis equal to the half of the maximum observed rate. This can be most easilyseen from the above equation; when SB is low, the effective surface rate reducesto the form, rapp = vm SB/KM- The overall reaction rate then becomes,

= (vm/KM)SAapp (vm/KMkS L) + l

showing that the apparent rate of reaction depends on the magnitude of themass transfer coefficient

It is only possible to measure true reaction kinetics, by operatingexperiments in a truly kinetic regime, such that any influence of the externaldiffusional mass transfer is negligible. This can be achieved by ensuring thatthe 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 flowvelocity and stirring speed. Conversely those factors affecting the maximumreaction rate, vm, should be decreased, for example enzyme loading andtemperature.

The regimes of possible external mass transfer influence on the observedkinetics 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 externaltransfer.

Regime of operation Reaction parameter having an influenceon overall rate

Transfer control Temperature (slight influence due to viscosity anddiffusion rate). Stirring speed in tank. Flow in packedand 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 withinBiocatalysts

Reactions with enzymes and whole cells entrapped or immobilized in a poroussolid matrix will be subjected to a mass transfer influence. Example systemsare whole cells immobilized in alginate, enzymes adsorbed on ion-exchangeresins, or naturally occurring biological films on surfaces or flocculatedbiomass. In the case of a biological film attached to an impermeable solid, thesubstrate can enter from only one surface, as shown in Fig. 6.1, through thediffusion layer A-B and into the biocatalyst B-C.

In the case of an alginate bead, a biofloc or its two-dimensionalapproximation, substrate can enter from opposing directions, as shown below inFig 6.3. In this case, the diffusion will result in a symmetrical, steady stateconcentration profile. The case of complete penetration of substrate throughthe biofloc is shown by the solid line. Whereas an incomplete penetration, asshown by the dashed line, results in the center of the film being completelyineffective, in terms of reaction capability. Note that in this case, the effects ofexternal diffusion are neglected.

The uptake of substrates within solid material requires transport by adiffusional process. The driving force for diffusion is a gradient inconcentration, and the diffusional flux is given by Pick's law,


jA = -DA

with j having units of kg/m2 s.

dCA

dZ

'AO

Diffusion'AO

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 beestablished 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\ /Rate of substrate^ /NVin bulk liquid) = \entering matrix ) = v

et rate of reaction^within matrix )

Vrapp = (j|z=o)A = r a v g AL

where the units of each term are kg/s. Here ravg represents an average value inthe matrix, which will increase with higher internal substrate concentrations.

Regarding the influence of diffusion for a particular situation, it is possibleto arrive at some quantitative guidelines without considering any mathematicaldetails. Obviously the concentration profiles are caused by a competitionbetween reaction and diffusion. The ratio of the maximum intrinsic reactionrate (not influenced by transfer) to maximum diffusion rate provides a usefuldimensionless parameter,

6.2 Internal Diffusion and Reaction within Biocatalyst 151

rmax A LJ A

r(C0) A LD (Co/L) A

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 lengthof diffusion path, L, plays an important role since the ratio of maximumdiffusion rate to maximum reaction rate varies according to L2. The higher thevalue of this ratio, the greater in magnitude are the substrate gradients. Withthis qualitative feeling for diffusion-reaction phenomena, more quantitativeaspects can be considered.

6.2.1 Derivation of Finite Difference Model forDiffusion-Reaction Systems

Diffusion with biological reaction can be treated by mathematical modelling,and from this it is possible to develop equations describing changes ofconcentration, with respect to both time and position. The same technique offinite differencing is used as in the modelling of the dynamic behavior oftubular bioreactors, Sec. 4.2.2.

Consider the case where the substrate varies from a concentration SQ in thebulk liquid to some concentration, at the position L (a wall or the center of asymmetrical particle). At the center by symmetry or at a wall, owing to theabsence of diffusion into the wall, the concentration gradient must be zero.The actual continuous concentration profile, through the slab, may beapproximated by a series of increments, as indicated in Fig. 6.4 and by a seriesof biocatalyst matrix elements as shown in Fig. 6.5.

Liquid

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


n-1jn-1

>, nIn

n+1I 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, dependson the local concentration gradient, and the reaction rate, rsn, depends on thelocal 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 eachcomponent as

/Accumulation^ /Diffusion^ /Diffusion\ / Production \V rate ) = \ rate in ) ~ \ rate out ) + \rate by reaction/

dSnA AZ ~dT = Jn-1 A - jn A + rSn A AZ

Using Pick's law in the difference form,

(Sn-i ~ Sn)

and similarly for Jn gives,°s

n-l ~ Sn) + r S n A A Z


Dividing by A AZ,

dSn (Sn,j -2Sn + Sn+i)T s 5 + rsn

The equivalent partial differential equation is,

as a2s+ rs

The finite-differenced forms of the model equations, however, are especiallysuitable for simulation programming. Thus, N equations are obtained, onesubstrate balance equation for each element, and these are solvedsimultaneously. Note that the boundary conditions, for elements 1 and N, mustbe described separately.

For the above case, the boundary conditions are dS/dZ = 0 at Z = L andS = SQ at Z = 0. Thus the equations for the first and last elements must bewritten accordingly, as shown in simulation example BIOFILM, Sec. 8.7.1.

Note also that it would be also possible, in principle, to include externaldiffusion effects, by formulating a boundary condition, balance for the firstelement as:

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

where the external transport rate through area A is,

Q= kS 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 tankmodel, as shown in Fig. 6.7. The bulk liquid-phase component balances takethe form:

^= |(SF-SO)-Jsa|z=0

where a=A/V,, A dS ,

Js a| z=o = - DS v dZ I z=o

where,


dS -dZ I z=o =

The direction of the positive diffusion is into the biofilm, since dS/dZ isnegative 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 isS = SR at Z = 0.

F,SFF,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 asingle tank or in a column by using a tanks-in-series model, The simulationexample BIOFILM, Sec. 8.7.1, demonstrates this approach.

6.2.2 Dimensionless Parameters from Diffusion-Reaction Models

There are several advantages of formulating model equations in dimensionlessform. The number of variables in the model is reduced, thus reducing thenumber of experiments or simulations required to investigate all combinations.It is also possible, on the basis of the numerical values of the parameters, toaccess the relative importance of certain terms. Finally, the dimensionless formmakes the solution much more generalized because the units of the individualquantities are no longer important.

The governing dimensionless parameters can be obtained by re-examiningthe defining model equations and arranging them such that the variables range


only between the values of zero and unity. Thus new dimensionless variables,S=S/S0 , Z = Z/L and dimensionless time, f = t / ( L I D ) , can be defined.Substituting these new variables into the diffusion-reaction, partial differentialequation, for the case of a first order biochemical reaction, gives,

asS° (L2/D8)at =

oras _ _ _at - az2 " DS

s

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

as a2sat " az2

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 theDamkohler Number, and its square root is called the Thiele Modulus.

In the above equations, all the terms, excepting that of the reaction term, havedimensionless parameters of unity. On this basis, it can be said that if thereaction parameter for a first order reaction, [ki L2/DsL has a value of 1.0 orgreater, then the reaction will have a large effect on the solution, that is, on theconcentration gradients. Similarly, ko L2/D$ SQ will govern a zero orderreaction. Such "order of magnitude analysis" is important for physicalunderstanding and also to obtain information from differential equationswithout having to actually develop an analytical solution.

Dimensionless formulation of equations is also explained in the simulationexample 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 beexpressed by means of an effectiveness factor, T|, where,


T| =actual apparent rate

rate at bulk liquid concentration

Solutions of the above diffusion reaction model are available in the literaturefor simple reaction orders (Satterfield and Sherwood, 1963). In Fig. 6.8 thevalues of t| for zero, first and second reaction order have been plotted against

the Thiele Modulus ,<f>, where,

O = ' \ /L2kSon-1 /D

This figure shows that a zero order reaction is not influenced by concentrationgradients until the substrate falls to zero in the matrix, corresponding to O > \2. The other reaction-types are influenced by low concentrations, as the curvesfor T| indicate. It is seen, for example, that for a first order reaction a value ofO = l corresponds to t| = 0.8.

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0

Incomplete penetration at O >V~2~

AA = zero-orderB = first-orderC = second-order

0 = L(kSon'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 andSherwood, 1963).


6.2.4 Case Studies for Diffusion with BiologicalReaction

6.2.4.1 Case A. Estimation of Oxygen Diffusion Effects in aBiofilm

For a biofilm or floe, whose oxygen uptake might be taken as a constant (zeroorder), the corresponding group would be [L2 qo2 Xbi0fiim/D Co2L where qo2Xbiofiim corresponds to the rate constant k and the oxygen concentration in theoutside liquid phase is Co2- Note that Xbiofiim is the biomass per unit ofbiofilm volume and is not easy to measure. Substituting values obtained froman aerobic biofilm nitrification experiment gives,

2 L2 qQ2 X (0.01 mm2) (80 mg O2/L min)0 = D C02

= (0.1 mm2/min) (8 mg O2/L) = l

For this order-of-magnitude analysis, the value of 1.0 can be used to separatethe 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 beexpected that a film or floe thickness greater than 0.1 mm would be oxygenlimited. From the exact solution, as seen in Fig. 6.8, gradients would appear fora zero order reaction at a value of <&2 = 2.0, instead of 1.0. This exampleshows how the Thiele Modulus can be used to make useful estimates fordiffusion 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 andignoring the slow organism growth processes, the reactions can be written as,

NH4+ + 3/2O2 ->

N02- + 1/2 02 -> N03


The oxygen requirements for the first and second steps can be related to thenitrogen content of NH4+ and NC>2~. These values are si = 3.5 mg 62 / mgNH4+ - N and 82 = 1.1 mg 62 / mg NCV - N. The low yields and low growthrates make it unnecessary to consider growth requirements and kinetics. Inprevious work (Tanaka and Dunn, 1982) the intrinsic substrate uptake kineticsfor the two steps were shown to have a double Monod form for the first step,

rNH4 = vmi.KNH4

and for the second step,

rN02 = vm2

where vmi and vm2 represent the maximum rates for a particular biomassconcentration and the chemical symbols represent concentrations.

Considering the diffusion phenomena in the biofilm to be represented byone-dimensional diffusion with quasi-homogeneous reaction, differentialbalance equations can be written for all reactants and products to describe theconcentration profile in the film. Proceeding as described in Sec. 6.2.1, acomponent mass balance is written for segment n and for each component:

(Accumulation^ _ (Diffusion^ _ (Diffusion^ ( Production ^^ rate ) ~ \ rate *n ) ~ \ rate out J Vrate reacti°n J

and the equivalent partial differential equation is obtained by letting AZapproach zero as

3S 32S

Applying this to each component gives the following balances:

For NH4+,

3NH4+ a2NH4+

= DNH4 -wo - TNH4

For NO2",aNQ2-~3r = DNO2 ^2 + rNH4 - TNO2


For NO3%3N03 32N03

For O2,3202

" sl rNH4 ~

The stoichiometric oxygen requirements for the first and second reaction stepsare given by si and s2. The boundary conditions used represent the bulk liquidphase or reactor concentrations and the zero gradient at the biofilm- solidinterface, as discussed earlier.

These equations can be written as differential-difference equations using thefinite-differencing technique (Sec. 6.2.1). Thus for each of N increments, fourcomponent 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. Acomparison of the resulting dimensionless NH4+ and O2 balances reveals that,when the second reaction is neglected, the equations are identical ifDNH4 = 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 variedaccording to the reactor operating conditions, can thus be used as a criterion toevaluate whether NtLj."1" or O2 might be penetration-limiting. The O2R/NH4+Rcriterion indicates which component can be limiting, O2 if the ratio is less than3.5 or NH4+ if the ratio is greater than 3.5. These conditions are not sufficientfor limitation, but indicate which component would be limiting. Simulationresults from a model that was developed using finite-differencing demonstratesthis phenomenon. The profiles in Fig. 6.9, are for the case O2R/NH4+R = 0.07.


NH4,NO3,N02 (mg/L)

100

80

60

40

20

H02 (mg/L)

20

NO10

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

Coupling the liquid and biofilm for a batch nitrification reactor as explained inFig. 6.7, gave the results in Fig. 6.10. Here the influence of oxygen limitationcaused the oxygen in the midpoint of the film to rise as the nitrogen substrateswere successively consumed.

NH4 NO 3, NO2" (mg/L)

100 80

02[mg/L]

80

60

40

20

0

64

48

32

16

0

12

10

8

6

4

2

0

120t (min)

Figure 6.10. Simulated biofilm nitrification profiles in a batch reactor. The N-componentconcentrations are in the bulk liquid. O2 is in the midpoint of the biofilm and indicateslimitation during the first 60 minutes.

7 Automatic Bioprocess ControlFundamentals

The purpose of automatic process control is to maintain time-dependentchanges of the relevant process variables (deviations, errors), within prescribedlimits and without a direct action of an operator. Process control may beconsidered as a corrective action involving three steps:

1. Measuring the variable to be controlled (controlled variable)2. Comparing the measurement with the desired value (set point)3. 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, whichinclude uniform and higher quality products, safety, increase of productivity,minimization of waste, optimization, freeing the labor force from drudgery anddanger, and decrease of labor costs.

Obviously, process control is highly dynamic in nature, and therefore itsmodelling requires the solution of sets of differential equations, and it istherefore highly suited to solution by digital simulation. A brief introduction tothe basic principles of process control required for solution of simplesimulation examples is given here.

7.1 Elements of Feedback Control

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


162 7 Automatic Bioprocess Control Fundamentals

Thermo-couple

Controllermechanism

Desiredvalue

I MeasuredI valueI

Fermenter

Figure 7.1. Simple feedback control system.

The components of such a control system can be best understood using ageneralized block diagram (Fig. 7.2). They are the process itself, themeasuring 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 systemis used to manipulate the flow rate of the cooling water in order to keep thetemperature at the desired constant value, or set point.

Controllermechanism

Load

1 Com)

Desired+cvalue | *A/

1 J1

Measuredvalue

jarator

$~Controller '_ i1

K11

Actuator

-*UProcess Controlled

variable

Measuringelementk A A A A

A A A A A

ÂÂÂ A AÂ

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

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 Types of Controller Action

7.2.1 On-Off Control

The most common and simplest type of control is an on-off or two positionaction, sometimes called discontinuous control (Fig. 7.3). An example is acontact thermometer, which closes or opens the heater circuit. The controllerchanges the value of the controller output, or the manipulated variable, fromone extreme to the other, when the controlled variable moves above or belowthe set point. This leads to oscillations that could become very fast, dependingon the speed of response of the process. The real on-off controller hastherefore a built in feature called a differential gap or a dead zone. It is a smallinterval on either side of the set point, within which the controller does notrespond. When the controlled variable moves outside the dead zone, themanipulated variable goes on or off. This is illustrated in Fig. 7.3. Such shiftsfrom the set point are known as offset. Such a controller is simple andinexpensive, but the oscillatory nature of the action and the offset make it veryimperfect.

The usefulness of this type of control was demonstrated for a biological,sequential batch process by Hediger and Prenosil, (1985). More sophisticatedfunction control modes consider the magnitude and time behavior of thecontrol error. Three principal functional modes of control generally employedfor process control are proportional (P), integral (I) and derivative (D) control.


100

Manipulatedvariable

0

Controlvariable

X l _ _ _ 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, accordingto

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 ofcontroller is shown in Fig. 7.5 and 7.6.

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


Systems with proportional control often exhibit pronounced oscillations andfor sustained changes in load the controlled variable attains a new equilibrium,steady-state position, or control point. The difference between this point andthe set point is the offset (Fig. 7.5) Integral and derivative modes are usedmostly in combination with the basic proportional control mode. Thesimulation Example INHIB, Sec. 8.5.2 includes the application of this simplecontrol 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 signalrelated to the error by

P = K p e +K l

?TI

or for e = constantdP

for t=TidPdF = KP£


where i\ is the integral time constant or reset time. This is the time required toenable 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 especiallyuseful for correction of very small errors because the controller output P willcontinue to change as long as an error persists. This can be understood byconsidering 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 oftenand the simulation examples TEMPCONT, FERMTEMP, TURBCON andCONTCON in Sec. 8.6 demonstrate the use of this control mode. Theexamples also show the ease by which the programming of the PI controllerequations is made using the simulation language.

2Kp

Controlledoutput

Kp

Errorsignal

PI action

P action

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 futureand the controller output is proportional to the current rate of the error change.The output signal varies only if the error is changing.


d£P = P0 + Kp e + Kp TD gj-

where TD is the derivative time constant. A PD-controller output is comparedwith 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. Comparisonwith P and PID modes.

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

7.2.5 Proportional-Integral-Derivative (PID)Controller

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


KD r deP = P0 + Kp e + —^ J £dt+ Kp TD -gj-

TI o

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

Controlledvariable

/ Uncontrolledresponse

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

The selection of the best mode of control depends largely on the processcharacteristics. Further information can be found in the recommended textslisted in the reference section. Simulation methods are often used for testingcontrol methods.

7.3 Controller Tuning 169

7.3 Controller Tuning

The purpose of controller tuning is to choose the controller constants, such thatthe desired performance is obtained. This usually means that the controlvariables should be restored in an optimal way, following either a change in theset point or as a result of an input disturbance to the system. Thus thecontroller constants can be set by experimentation. A rational basis for suchexperimental tuning is given in what follows. Other methods for tuningcombine process dynamic experimentation with theoretically-based controlmethods; some of the standard methods are also described below. Thesimulation example TEMPCONT, Sec. 8.6.1, provides exercises for controllertuning 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 bysimulation, the controller can be set by trial and error. Each time a disturbanceis made the response is noted. The following procedure might be used to testthe control with small set point or load changes:

1. Starting with a small value, Kp can be increased until the response isunstable and oscillatory. This value is called the ultimate gain KPQ.

2. Kp 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 thisvalue.

4. Include derivative action, increase ID until noise develops and set ID at1/2 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 byuncoupling the controller. It is based on the characteristic curve of the processresponse to a step change in manipulated variable, equal to A. This response iscalled a process reaction curve, whose magnitude is B. The two parameters


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

B

ManipulatedVariable

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 (controllerdisconnected) processes to a step change in the manipulated variable was asigmoidal shape curve. This can be modelled approximately by a first ordersystem with time lag TL, as given by the intersection of the tangent through theinflection point with the time axis. The theoretical values of the controllersettings obtained by the analysis of this system are summarized in Table 7.1The model parameters for a step change A to be used with Table 7.1 arecalculated 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

P 1/TLSPI 0.9/TL S 3.33TL

PID 1.2/TLS 2TL TL/2

Co/ten-Coon

P T

TL \ 30 + 3TL/TPI

4 TL \ 32 + 6TL/T

Ultimate Gain

PPIPID

0.5 Kpo0.45 Kp00.6 Kpo

l/1.2fp0

l/2fpo l/8fpO

7.3.4 Ultimate Gain Method

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


values in small increments until continuous cycling begins. The controllersettings are then calculated from Kpo and fpo according to the tuning rulesgiven in Table 7.1.

While this method is very simple it can be quite time consuming in terms ofnumber of trials required and if the process dynamics are slow. In addition, itmay be hazardous to experimentally force the system into unstable operation.

Simulation methods can be very useful if a suitable model is available as wasshown by Heinzle et al. (1992) for a one and two stage anaerobic system, usinga kinetic model similar to the simulation example, ANAMEAS, Sec. 8.8.7.

7.4 Advanced Control Strategies

7.4.1 Cascade Control

In control situations with more then one measured variable but only onemanipulated variable, it is advantageous to use control loops for each measuredvariable in a master - slave relationship. In this, the output of the primarycontroller is usually used as a set point for the slave or secondary loop.

This may be relevant for some wastewater treatment plants where the highconcentration of some substrate may be toxic for the microorganisms. Forexample, the simulation Example TURBCON, Sec. 8.6.3, could be easilyadapted to this situation if the substrate concentration were subject to significantchanges, as shown schematically in Fig. 7.10.

7.4 Advanced Control Strategies 173

m1

!XI . . . N

/6oncentrationV/ConcentriilonlController )«H f errandx. ,s \ transmitter /

|

:';§S:'|||::||:|-

^

lilllllilllll

ill lit:

^ V

c J_Biomasscontroller

[ Turbidometer

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

The simulation example TURBCON, Sec. 8.6.3. could be modified similarly byadding biomass as a measured variable. An interested reader may try toimplement 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 disturbanceswhich are measured in the system output. The feedforward method tries toeliminate this drawback by another approach. Rather than using the processoutput as the measured variable, this is taken as the measured inlet disturbanceand its effect on the process is anticipated by means of a process model. Thusaction is taken on the manipulated variable by the model, which relates themeasured variable at the inlet, the manipulated variable and the process output.The success of this control strategy depends largely on the accuracy of themodel prediction. For this reason sometimes an additional feedback loop isused.

Many of the continuous process simulation examples in Ch. 8 could bealtered in this fashion. It would be interesting to program an example usingsimple kinetics for the feedforward control and to describe the "actual" systemwith more complex kinetics. The discrepancies between the "simple" model


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

7.4.3 Adaptive Control

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

7.4.4 Sampled-Data Control Systems

When discontinuous measurements are involved the control system is referredto as sampled-data. Concentration measurements by chromatography wouldrepresent such a case.

Controlledvariable

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 to0.5 times the minimum period of oscillation. When the sampling time satisfiesthe above criteria, the system will behave as if it were continuous. Details of thisand other advanced control topics are given in specialized process controltextbooks, 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 thebiocatalyst environment. In a natural environment microorganisms and cells ofhigher organisms very rarely produce large amounts of products. Inbiotechnological processes organisms are usually kept in a completely"unnatural" environment. Control is therefore often necessary to induce themto produce substances in economically important amounts.

Process optimization is closely linked with control. The objectives ofoptimization and control may be to maximize productivity, final concentration,yield or to minimize effluent concentration and energy costs. Although thecriteria for optimal processes differ widely, all bioprocesses need control andautomation to run under optimal conditions. The selection of control variablesstrongly depends on the process and the final goal to be achieved.

Information about the dependency of biological rates, yields and selectivitieson environmental conditions is usually required, as given in Ch. 3. Allbiological reactions have distinct temperature and pH optima, and all respondto substrate concentrations. Therefore it is common to control these variables.

Heat is produced in all biological reactions and therefore temperaturecontrol is necessary. In large-scale production, heat removal capacity may bethe limiting factor. It is important to maintain the temperature at an optimumlevel. 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 belowthe optimal temperature, which means that the growth rates can be expected toincrease exponentially with increasing temperature. Above the optimumtemperature, further temperature increases usually cause a dramatic decrease inactivity, mainly due to inactivation of enzymes. Also, temperature shocks maybe important since enzyme formation may often be induced by a shock at theend of the exponential growth phase.

The variable pH has certain similarities with temperature because thereusually exists an optimum for biological activity; it can be relatively easilymeasured and is often controlled. The biological rates also exhibit a maximumat the optimal pH value, which is usually in the neutral pH 7 region. Againcontrol 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.


Biological rates usually depend on substrate concentration (e.g.: sugar,mineral salts, oxygen, precursor, etc.) though in many cases kinetics are ofzero-order type above certain concentration levels. In the latter case control isnot important, since high concentrations will guarantee maximum rates. Thesituation is more complex if process selectivity changes with substrateconcentration. The most well-studied process of this kind is Baker's Yeastproduction, where high glucose concentration (> 100 mg L"1) and oxygenlimitation causes undesired ethanol formation. Substrates (e.g. mineral salts,components of wastewater), precursors (e.g. in antibiotica production ortransformation processes) or products (e.g. ethanol) may be inhibiting or eventoxic at higher concentration levels. In such processes it is necessary to controlthe 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. Simpleexamples are the set-point control of constant temperature, pH, substrate andprecursor concentration. Table 7.2 gives examples of methods and strategiesfor control of biological reactors. Much of the difficulty in control lies infinding a suitable sensor. Calculated values using indirect measurements can bevery useful, e.g. measuring oxygen uptake to control substrate level. Oftenvariables such as pH or dissolved oxygen (DO) control can be used toindirectly 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 Discrete RQ Glucoseproduction control (Gas analysis) feed rate

Baker's yeast Feedback RQ Glucoseproduction control (Gas analysis) feed rate

Baker's yeast Two point DO Feed rate,production control (electrode) agitation speed

aeration rate


Table 7.2. (Continued).Process

Ethanolproduction

Wastewatertreatment

Diversefermentations

Bacillus subtilisat low DO

Penicillinproduction

a-Amylase byBacillus amylo-liquefaciens

Fed-batchpenicillin

Baker's yeast

Cephalosporin Cproduction

RecombinantE. coli

Phenol oxidation

Methodand strategyPID

Adaptivecontrol

Various

Cascade

Set-points(growth andproduction)

Feed profile

Feed profile

Feed profile

Profiled pHand temperaturecontrol

Conventionalwith addedglucose

Adaptive-questing

Controlledvariable(s)Sucrose cone, byenzyme thermister

DO(electrode)

DO(electrode)

DO(electrode)

Growth rate(CO2 rate)

Feed rate(off-line)

Substrate andbiomass cone.

RQ

pH, temp.(electrode,thermister)

Growth rate(pH)

Phenol uptake(O2 uptake)

Manipulatedvariable(s)Feed rate

Aeration rate

Stirring speed,gas composition,aeration rate,pressure

Gas flow,valve setting

Feed rate

Feed rate

Feed rate

Feed rate

Alkali feed rate,cooling water rate

Glucose and alkalirate

Flow rate

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


growth phase followed by production phase) control parameters of a linearcontroller may be inadequate to control the process. It is therefore necessary toadapt control parameters (e.g. proportional gain) according to the processrequirements. Minimization of an objective function can be used to guide theadaptive tuning of the controller. A rather simple method uses a somewhatempirical adjustment mechanism, which is driven by a secondary measurement.This was used for DO control by Heinzle et al. (1986), in which the oxygenuptake 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 fromoff-line simulation and optimization. An example is found in the exponentialfeeding profiles that can be calculated from the models for fed batchfermenters in Ch 4.

If no suitable dynamic model is available and the process changes inunpredictable ways, then on-line adaptive optimizing control may be useful.This would however require measurements of the key inputs and outputs of theprocess. An example is the optimization of a continuous anaerobic process byRyhiner et al. (1989) in which the methane and organic acids output rates werecorrelated with the input flow rate. The optimization involved a compromisebetween high methane rates and low organic acid concentration.

7.5.2 Methods of Designing and Testing theStrategy

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 wellunderstood. If possible, it is useful to use dynamic models to select a controlstrategy, and to use it for testing and tuning. An example with anaerobicdigestion is given by Heinzle et al. (1992). In Fig. 7.12 are shown the results ofa simulation and a corresponding experiment for the control of the propionicacid concentration by manipulation of the feed flow rate.


2

I

oCDO"E

0 1 2 3 4

6(

5(

4(

3( -

20-

1(2 3

Time [h]

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

1

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

iV2 I

Figure 7.12. Control of anaerobic digestion of whey wastewater. Simulation (A) andexperiment (B) of control after step change (Heinzle et al. (1992). Here the controlled variablewas 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 inMethanogenic Fermentation of Soluble Wastes. Enzyme Microb. Technol. 5,570-577.

Aris, R. (1989) Elementary Chemical Reactor Analysis. Butterworths,Boston.

Atkinson, B. and Mavituna, F. (1991) Biochemical Engineering andBiotechnology 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 BiochemicalEngineering. Adv. Biochem. Eng, 3, 127-165.

Dekkers, R.M. (1983) State Estimation of a Fed-batch Baker's YeastFermentation., 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 onthe Anaerobic Degradation of Molasses Wastewater. Biotechnol. Bioeng. 31, 1-10.

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 BatchFermentation Processes. Annals N.Y. Acad. ScL 326, 127-139.


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 Modelsfor Fermentation Processes. Adv. Appl. Microbiol. 13, 419-465.

Furukawa, K., Heinzle, E., and Dunn, IJ. (1983) Influence of Oxygen on theGrowth of Saccharomyces cerevisiae in Continuous Culture. Biotechnol.Bioeng. 25, 2293-2317.

Gujer, W., Zehnder, A.J.B. (1983) Conversion Processes in AnaerobicDigestion. Water Sci. Technol. 15, 127-167.

Harder, A., Roels, J.A. (1982) Application of Simple Structured Models inBioengineering. Adv. Biochem. Eng./Biotechnol. 21, 56-107.

Hediger, T. and Prenosil, I.E. (1985) Microprocessor Automated SequentialBatch Process, Biotechnol. Progr.l, 216-225.

Heinzle, E. and Lafferty, R.M. (1980) Continuous Mass SpectrometricMeasurement of Dissolved H2, O2, and CO2 during Chemolitho- autotrophicGrowth 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) ExperimentalMethods for On-line Mass Spectrometry in Fermentation Technology.Bio/Technology 1, 14-16.

Heinzle E. and Dunn I. J. (1991) Methods and Instruments in FermentationGas 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 Researchand 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 forAnaerobic Wastewater Treatment, Adv. Biochem. Eng., Springer Verlag.


182 References

Imanaka, T., Aiba, S. (1981) A Perspective on the Application of GeneticEngineering: Stability of Recombinant Plasmid. Ann. N. Y. Acad. Sci. 369, 1-14.

Ingham, J., and Dunn, I.J. (1991) "Bioreactor Off-Gas Analysis", in "Bioreactorsin Biotechnology", Ed. A. Scragg, Ellis-Horwood, Chichester, 195-220.

Keller, R. and Dunn, I.J. (1978), Computer Simulation of the BiomassProduction 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 forAnimal Cell Culture, in preparation, Biotechnol. Bioeng.

Luyben, W.L. (1973) Process Modeling, Simulation, and Control for ChemicalEngineers. McGraw-Hill.

Meister, D., Post, T., Dunn, I.J. and Bourne, J.R. (1979) Design andCharacterization 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) AMicrobial Culture with Oxygen-sensitive Product Distribution as a Tool forCharacterizing Bioreactor Oxygen Transport. Biotechnol. Bioeng. 27, 482-489.

Moes, J., Griot, M., Heinzle, E., Dunn, I.J., and Bourne, J.R. (1986) A MicrobialCulture 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 ProcessDynamics 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 throughComputer-aided On-line Mass Balancing in a Fed-batch PenicillinFermentation. 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, 489-545.

Roels, J.A. (1983) Energetics and Kinetics in Biotechnology. ElsevierBiomedical Press, Amsterdam.

Ruchti, G., Dunn, I.J., Bourne, J.R. and v. Stockar, U. (1985) PracticalGuidelines for Determining Oxygen Transfer Coefficients with the SulfiteOxidation Method. Chem. Eng. J. 30, 29-38.

Russell, T.W.F., Denn, M.M. (1972). Introduction to Chemical EngineeringAnalysis. Wiley, New York.

Ryhiner, G., Dunn, IJ, Heinzle, E, Rohani S., (1992) Adaptive On-line OptimalControl of Bioreactors: Application to Anaerobic Degradation, J. BiotechnoL.22, 89-106.

Ryhiner, G., Heinzle, E, Dunn, IJ. (1991) Modelling of Anaerobic Degradationand Its Application to Control Design: Case Whey, in: Dechema BiotechnologyConferences Vol.3, (Ed.: Behrens, D. and Driesel, A.J.), 469-474.

Saner, U., Bonvin, D., Heinzle, E. (1990) Application of Factor Analysis forElaboration of Stoichiometry and its On-line Application in Complex MediumFermentation 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 FermenterModeling: A Comparison for Pressure Effects. Chem. Eng. Sci., 33, 1025 -1030.

Tanaka, H. and Dunn, IJ. (1982). Kinetics of Biofilm Nitrification. BiotechnoLBioeng., 24, 669 - 689.

Tanaka, H., Uzman, S., Dunn, IJ. (1981). Kinetics of Nitrification Using aFluidized 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). TheTubular 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 andBiotechnology 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 PressLimited: London.

Glick, B.R., Pasternak, J.J. (1995) Molekulare Biotechnolgie; Spektrum,Heidelberg.

Grady, C. P. L. and Lim H. C. (1980) Biological Wastewater Treatment, MarcelDekker.

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 andEconomics. Wiley, New York.

Lee, J. M. (1992) Biochemical Engineering. Prentice Hall.

Moo-Young, M., Ed. (1985) Comprehensive Biotechnology, Vols. 1-4Pergamon Press, Oxford.

References 185

Moser, A. (1988) Bioprocess Technology.Springer, N.Y.

Nielsen, J., Villadsen, J. (1994) Bioreaction Engineering Principles. PlenumPress.

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. ElsevierBiomedical 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 ofPrimary 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; Springer-Verlag, 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 andModelling. Open University Press, Milton Keynes.

Schugerl, K. (1987) Bioreaction Engineering.Vol.1, John Wiley, Chichester.

Subramanian, G. (1998) Bioseparation and Bioprocessing. A HandbookVolume II: Processing, Quality and Characterization, Economics, Safety andHygiene, Wiley-VCH, Weinheim.

van't Riet, K., Tramper, J. (1991) Basic Bioreactor Design. M. Dekker, NewYork.

Vieth, W.R. (1994) Bioprocess Engineering, J. Wiley & Sons, N.Y.

Webb, C., Black, G.M., and Atkinson, B. (1986) Process Engineering Aspects ofImmobilized 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. NewEnzymes for Organic Synthesis, Springer.

Segel, I.H. (1975) Enzyme Kinetics: Behavior and Analysis of RapidEquilibrium 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) MetabolicEngineering. 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, Prentice-Hall.

Hagen, J. (1993) Chemische Reaktionstechnik, VCH: Weinheim.

Ingham, J., Dunn, I.J., Heinzle, E., Prenosil, I.E. (2000) Chemical EngineeringDynamics: An Introduction to Modelling and Computer Simulation, VCHVerlagsgesellschaft 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 ChemicalEngineering. 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 EnvironmentalSystems, Springer: New York.

Dodson, C. T. J., Gonzalez, E. A. (1995) Experiments in Mathematics usingMaple, 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 EngineeringAnalysis. Wiley, New York.

Ruth, M., Hannon, B. (1997) Modeling Dynamic Economic Systems, SpringerVerlag, 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 andControl. McGraw-Hill, New York.

References 189

Fish, N. M., Fox, R.L, and Thornhill, N.F. (1989) Computer applications infermentation technology: Modelling and control of biotechnological processes.Elsevier, London.

Halme, A. (1983) Modelling and Control of Biotechnical Processes. PergamonPress, Oxford.

Johnson, A. (1986) Modelling and Control of Biotechnological Processes.Pergamon Press, Oxford.

Luyben W. L. (1973) Process Modeling, Simulation, and Control for ChemicalEngineers, 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 ofEnvironmentel Bioprocesses, VCH Verlagsgesellschaft mbH, Weinheim,.

Stephanopoulos, G. (1984) Chemical Process Control: An Introduction toTheory and Practice, Prentice Hall.

Weber. W. J., Jr., DiGiano, F. A. (1996) Process Dynamics in EnvironmentalSystems, Wiley.

Part II Dynamic BioprocessSimulation Examplesand the BerkeleyMadonna SimulationLanguage


8 Simulation Examples of BiologicalReaction Processes Using BerkeleyMadonna

8.1 Introductory Examples

8.1.1 Batch Fermentation (BATFERM)

System

The system is represented in Fig. 1, and the important variables are biologicaldry mass or cell concentration, X, substrate concentration, S, and productconcentration, P. The reactor volume V is well-mixed, and growth is assumedto follow kinetics described by the Monod equation, based on one limitingsubstrate. Substrate consumption is related to cell growth by a constant yieldfactor YX/S- Product formation is the result of both growth and non-growthassociated 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.


194 8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Model

Mass Balances:

(Rate of accumulation) = (Rate of production)For cells

VTT = rxvor

dX

For substratedS

VdF = rsor

dSdF = rS

For productdP

V-3T =or

dPdF = r?

Kinetics:rx = f iX

with the Monod relation, constant yield relation, and product formationkinetics:

tx/s

rP = (ki + k2 | ) X

where ki is the non-growth associated coefficient, and k2 is the coefficientassociated with growth.

If the number of equations is equal to the number of unknowns, the model iscomplete and the solution can be obtained. The easiest way to demonstrate thisis via an information flow diagram, as shown below in Fig. 2.

8.1 Introductory Examples 195

xo

So

PO

BiomassBalance

4_SubstrateBalance

4_ProductBalance

A4

rx

rs

rp

T *GrowthRate

1f 'x

SubstrateRate

ProductRate

M

— —

-^M

,-_

MonodKinetics

A

X^

s^

p

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 equationblock are obtained as the products of other blocks. The information flowdiagram thus emphasizes the complex inter-relationship involved in even thisvery simple problem. Solution begins with the initial conditions XQ, SQ and PQat time t=0. The specific growth rate |i is calculated, enabling rs, rx and rp tobe calculated, and hence the initial gradients dX/dt, dS/dt and dP/dt. At this timethe integration routine takes over to estimate revised values of X, S and P overthe first integration step length. The procedure is repeated for succeeding steplengths until the entire X, S and P concentration time profiles have beencalculated up to the required final time.

Program

The following Berkeley Madonna program solves the above fermentationproblem:

{BATPERM}

{Batch growth with product formation}

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

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


X0=0.01 /Initial biomass inoculum, kg/m3SO = 10 ;Initial substrate cone., kg/m3P0=0 ;Initial product conc.,kg/m3

{Initial Conditions}INIT X=XOINIT S=SOINIT P=PO

(Mass Balances}X'= RX ;BIOMASS BALANCES'= RS ;SUBSTRATE BALANCEP' = RP ;PRODUCT BALANCE

{Kinetics}RX = U*X ; BIOMASS RATE EQUATION, kg/m3 hU = UM*S/(KS + S) ;MONOD EQUATION, 1/hRS = -RX/Y ; SUBSTRATE RATE EQUATION, kg/m3 hRP= (K1 + K2*U) *X /PRODUCT RATE EQUATION, g/m3 h

Limit 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 initialconditions, or the values at time=0. The form X' designates the time derivativeor d/dt(X) can be used. Most models are conveniently structured in terms ofmass balances and kinetics. Any result quantity on the left of the equal sign isstored for further calculations or for use in graphing. Usually concentrationversus time is of interest, but rates versus concentrations make very useful plotsfor understanding the kinetics. The five integration methods require specifyingtime intervals, such as DT, DTMIN and DTMAX. This requires a bit ofexperience. Care must be taken to see that the same results are obtained by twodifferent methods or for at least two different DT values.

As is seen in the Appendix, Berkeley Madonna provides many possibilities tochange 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

Nomenclature

Symbols

k ] and k2

KSPrSVXY

H

Product formation constantsSaturation constantProduct concentrationReaction rateSubstrate concentrationReactor volumeBiomass concentrationYield coefficientSpecific growth rate

1/h and kg/kgkg/m3

mg/m3

kg/m3 h and kg/m3 hkg/m3

m3

kg/m3

kg/kg1/h

197

Indices

2mPSX

Refers to non-growth association rateRefers to growth-association rateRefers to maximumRefers to productRefers to substrateRefers to biomass

Exercises

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

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

3. Plot the rates versus the concentrations.


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. Theresults 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

......."l"""'

0 5 1 0

"*"'.^_

— -S:2

P:2— -S:3

P:3

-rp^vtf*EV

15

TIME

/

**« /\ /*

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

-'7^'^.**&'*' \

fm \ ..

20 25 3(

-10

•9

-8

• 7

•6

.5 cn

-4

•3

•2

• 1

•0

)

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


8.1.2 Chemostat Fermentation (CHEMO)

System

A continuous fermenter, as shown in Fig. 1, is referred to as a chemostat. Atsteady 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) belowthe maximum specific growth rate (|um). Washout of the organisms will occurwhen D > (a. The start-up, steady state and washout phenomena can beinvestigated by dynamic simulation.

D,SF

Figure 1. Chemostat with model variables.

Model

S,X

The program BATFERM may be easily modified to allow for chemostatoperation with sterile feed by modifying the mass balance relationships toinclude the inlet and exit flow terms. The corresponding equations are then:

For cellsdX

.= -DX + rx


For substratedS3f = D (SF - S) + rs

For productdPdf = -DP + rp

where D is the dilution rate and Sp the concentration of the limiting substrate inthe 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. Startup asbatch reactor until time=tstart}

{Constants}UM=0.3 ; 1/hKS = 0.1 ; kg/m3Kl = 0.03 ; kgP/kgX hK2=0.08 ; kgP/kgXY = 0.8 ; kg X/kg SX0 = 0.01 ; Initial biomass inoculum, kg/m3S0=10 ; Initial substrate cone., kg/m3P0=0 ; Initial product conc.,kg/m3SF = 10 ; Feed cone. ,kg/m3Dl = 0.25 ; Dilution rate, 1/ht start = 5 ; Start time for the feed

(Initial Conditions}Init X=XOInit S=SOInit P=PO

{Mass Balances}X'=-D*X+RX ; BIOMASS BALANCE EQUATIONS • =D* (SF-S) +RS ; SUBSTRATE BALANCE EQUATIONP'=-D*P+RP ; PRODUCT BALANCE EQUATION


{Kinetics}RX = U*X ; BIOMASS RATE EQUATION, kg/m3 hU = UM*S/(KS + S) ; MONOD EQUATION, 1/hRS=-RX/Y ; SUBSTRATE RATE EQUATION, kg/m3 hRP= (K1 + K2*U) *X ;PRODUCT RATE EQUATION, kg/m3 h

{Conditional equation for D}D=if time>=tstart then Dl else 0

Prod=D*X /Productivity for biomass, kg/m3 h

Nomenclature

Symbols

DkiKSPr

SXYILL

1

andDilution rateProduct formation constantsSaturation constantProduct concentrationReaction rate

Substrate concentrationBiomass concentrationYield coefficientSpecific growth rateTime lag constant

1/h1/h and kg/kgkg/m3

mg/m3

kg/m3h andkg/m3

kg/m3

kg/m3

kg/kg1/h

Indices

F Refers to feedMONOD Refers to Monod kineticsP Refers to productS Refers to substrateX Refers to biomass


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 tochemostat operation with D < |jm. Does this reduce the time to reachsteady state? Is the exact time of switchover important?

7. Include maintenance requirements to the substrate uptake kineticsusing RS = - ( U / Y + M ) * X . Remember to add a value of themaintenance coefficient M to the constants. Investigate the influence ofthe value of M on the steady state biomass concentration.

8. Using a Parameter Plot, obtain steady state values of X and S for arange of Dl.

9. Rapidly-changing dynamic fermentations do not followinstantaneous Monod kinetics. Modify the model and the program witha dynamic lag on jo, such that d|j /dt= (nMonod - l-O/t- Compare theresponse to step changes in D for suitable values of the time lag constantt.

Results

The graphical output in Fig. 2 shows three startups of the fermenter underinitially batch growth conditions, using three values for Dl . The break in theconcentration-time dependency as feeding starts is quite apparent, and the newtransient then continues up to the eventual steady state chemostat operatingcondition or washout in the case of one run. For the results of Fig. 3 theprogram was changed by adding the line PROD = X*D, and the final, steadystate value of production rate was plotted versus Dl for twenty runs, using theParameter Plot feature of Madonna.



10

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


2

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


8.1.3 Fed Batch Fermentation (FEDBAT)

System

In this case the model equations allow for the continuous feeding of sterilesubstrate, 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 importantcharacteristics of quasi-steady state, linear growth, and use of alternative feedstrategies.

F,SF

V

X

sp

Figure 1. Fed batch fermenter with model variables.

Model

For fed batch operation, the equations become as follows:

Total balancedV

dT = F

For cells

For substrate


For product

where F is the volumetric feed rate, Sp is the feed concentration and V is thevolume 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 toobtain the concentration terms required for the kinetic relationships. Thekinetics are taken to be the same as in BATFERM.

Program

The "IF" statement in the program causes the continuous feed to start when timereaches tfeed, at which point batch operation stops and the fedbatch starts.

(FEDBAT)

{Fermentation with batch start up}

{Flow rate is initially zero and is turned on attime=tfeed.}

{ Constants}UM=0.3KS = 0 . 1Kl = 0.03K2 = 0.08Y = 0.8X0 = 0 .01S0 = 10P0 = 0SF = 10Pl-1. 5tfeed=22.5

; 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 conc.,kg/m3; Feed flow rate, m3/h; Start time for the feed

{Initialinit V=linit VX=V*XOinit VS=V*SOinit VP=V*PO

Conditions}


{Mass balances, kg/h}d/dt(V)=Fd/dt(VX)=RX*Vd/dt(VS)=F*SF+RS*V

d/dt(VP)=RP*V {kg/h}

{Calculation of concentrations}X=VX/VS=VS/VP=VP/V

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

D=F/V {nominal dilution rate, 1/h}

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

Nomenclature

Symbols

D Dilution rate 1/hF Flow rate m3/hKS Saturation constant kg/m3

ki, k2 Constants in product kinetics 1/h and kg/kgM Maintenance coefficient kg/kg hP Product concentration kg/m3

r Reaction rate kg/m3 hS Substrate concentration kg/m3

X Biomass concentration kg/m3

V Reactor volume m3

Y Yield coefficient kg/kg|i Specific growth rate 1/hT Time delay constant h


Indices

FPSX

Refers to feedRefers to productRefers to substrateRefers to biomass

Exercises

Results

Operation begins under initial batch conditions, and feeding of substrate isstarted at tfeed=22.5 h. In Fig. 2, the break in the batch growth transient, assemi-batch feeding starts is very apparent, with the transient continuing to anapparent "quasi" steady state operating condition. Under these conditions thebiomass concentration becomes constant, while the substrate concentration (notshown) is below the KS value and decreases very slowly. As seen in the zoomof Fig. 3, the values of D (= F/V) also decrease since V increases due to theincoming 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 amountof substrate that has been consumed, which is approximately equal to theamount in the reactor initially plus the amount added during the feedingperiod. During the quasi-steady state, the total biomass will increase linearlywith time if, as in this case, the feeding flow rate is constant. This is a "linear


growth" situation in which the growth rate is limited by the feeding rate. InFig. 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 ratedepends 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.

0.4.

0.35-

0.3-

0.25.3

Q 0.2-

CO0.15-

0.1-

0.05-

0 -

-~, IV

-J.T


27 28 29 30 31 32 33 34 35 36 37

TIME

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

8.2 Batch Reactors 209

8.2 Batch Reactors

8.2.1 Kinetics of Enzyme Action (MMKINET)

System

The intermediate enzyme-substrate complex is the basis for the simplest formof 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 batchreactor are:

-— = ki ES - k2ESdt

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 ~ KM + S


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

Program

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

Nomenclature

Symbols

EESk

KMPSVmax

Enzyme concentration mol/m3

Enzyme-substrate complex concentration mol/m3

Reaction rate constants variousMichaelis-Menten constant mol/m3

Product concentration mol/m3

Substrate concentration mol/m3

Maximum velocity mol/m3 h

Indices

0123SMm

Refers to initial valuesRefers to reaction 1Refers to reaction 2Refers to reaction 3Refers to substrateRefers to Michaelis-Menten

Exercises


Results

Figs. 2 and 3 give the results of the full model and the Michaelis-Mentensimplification, respectively

Run 1:119 steps in 0.0167 seconds

0.009.

0.008-

0.007-

0.006 •tn^0.005-LU

0.004-

0.003-

0.002 •

0.001 -

Lx"""" "~\ f'\ /:. fv

riLt ~mf \

• \1

*i* \.i fc*v. "i«%^

""""•%-», ^*"'..._

..]

...— 8:1ES:1

-•- P:1

.0.9

-0.8

•0.7

•0.6

a•0.5 *

to•0.4

•0.3

•0.2

• 0.1

.0

10 20 30 40 50

TIME70 80 90 100

Figure 2. Results from the full model



\\

10 20 30 40 50 60 70 80 90 100TIME

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-Mentenkinetics, of the form,

rs = 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 Kmvalue, demonstrates the importance of data in the Km region and the difficultyof obtaining this in a batch reactor. It is useful to make specially-scaled graphsin the KM region.


Figure 1. Lineweaver-Burk plot to determine vm and

Model

The model is that of a batch reactor with Michaelis-Menten kinetics.

dSdF = ~rs

Program

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

Nomenclature

Symbols

KMrS

Michaelis-Menten constantReaction rateSubstrate concentration

kg/m3

kg/m3

kg/m3


SiV

Inverse substrate concentrationReaction velocity or rateInverse reaction velocity or rate

m3/kgkg/m3 hm3 h/kg

Indices

0mS

Refers to feedRefers to maximumRefers to substrate

Exercises

Results

The results are shown in Fig. 2 (rates and concentrations versus time) for arange of Michaelis-Menten constants KM and in Fig. 3 the correspondingLineweaver-Burk plots.



'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).


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 EnzymaticLactose Hydrolysis (OLIGO)

System

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


purposes. One approach to simplify them is to neglect formation of enzyme-substrate complexes altogether and to deal only with overall reactions of thereact ants to products.

An example of such a reaction is the enzymatic lactose hydrolysis, acomplex process involving a multitude of sequential reactions leading to highersaccharide (oligosaccharides) intermediates. The mechanistic model is rathercomplex even when only trisaccharides are considered (Fig. 1).

La + E ^ ^ LaE - Ga + GI + E

Ga E + La ^ ** E + Tr

GaE + H2O ^ E + Ga

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

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

1 aL.a

LafcaCl

K!

-i- Ga

to Ga

K1

K2

. fîT V3II

^ Tr• i

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

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

Model

This simple batch reactor model is equivalent to the Michaelis-Menten productinhibition model.


dLa-gj-

dGa

- 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 = 0Range 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 orderto bring the simulation into agreement with the experimental data. Theprogram is on the CD-ROM.

Nomenclature

Symbols

GaGl

K2

LaTr

Galactose concentration mmol/LGlucose concentration mmol/LReaction rate constant (La —> Ga + Gl) 1/min

Reaction rate constant (La + Ga -> Tri) L/(mmol min)Reaction rate constant (Tri -> La + Ga) 1/minLactose concentration mmol/LTrisaccharide concentration mmol/L

Indices

0 Refers to initial concentration


Exercises

Results

The outputs in Figs. 3 and 4 show the influence of KI, KI and Lao on the sugarconcentration profiles.

100.

90.

80.

70

60.

. 50.

40

30

20

10

0

Run 1:10000 steps in 0.05 secondsr100

20 100TIME

180 200

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



160

80 100 120 140 160 180 200

«( 80'

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 ofOligosaccharides during an Enzymatic Lactose Hydrolysis Process", Parts I andII: 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 thebatch culture of Alcaligenes eutrophus under chemolithoautotrophic growthconditions, as discussed in Case C, Sec. 3.3.1. Growth and storage of PHB aredescribed as functions of limiting substrate S (NH4+), residual biomass R andproduct P (PHB) concentrations.


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 mainparts, namely PHB (P) and residual biomass (R), where R is calculated as thedifference between the total cell dry weight and the concentration of PHB (R =X - P). R can be considered as the catalytically active biomass, includingproteins and nucleic acids. With constant concentrations of the dissolved gases,two distinct phases can be recognized: growth and storage. During the growthphase there is sufficient NH4+ to permit protein synthesis. When the limitingsubstrate NH4+ (S) is exhausted, the protein synthesis ceases, and theproduction rate of PHB is increased. During the storage phase only PHB isproduced. The limiting substrate NH4+ (S) is essential to produce R and limitsits synthesis at low concentrations.

For the batch process,

dRdF = rR = M R

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 inthis example.

This is based on the postulate that there are two different mechanisms for theassimilation of NH4+ in procaryotes. This formulation is not a mechanistic one,


since in reality the enzyme system, using energy to assimilate NH4+, isrepressed by high concentrations of NH4+.

For the substratedS 1dF = rs = -YR/S **

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

dPdf = rp = rPj + rP,2

where rPj = YP/R rRThe 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 theproduct P. When the PHB content in the cells is high, the rate of synthesis of Pis decreased, which can be formally described as an inhibition.

Program

The program is found on the CD-ROM.

Nomenclature

Symbols

KI Inhibition constant, for (NH^SC^ kg/m3

KS Saturation constant kg/m3

n Hill CoefficientP Product concentration (PHB) kg/m3

R Residual biomass concentration kg/m3

rp Rate of synthesis of PHB kg/m3

TR Rate of synthesis of R kg/m3

rs Rate of substrate uptake kg/(m3 h)


XYP/RYR/S

Limiting substrate concentration kg/m3

NHj as (NH4)2S04

Biomass concentration kg/m3

Yield coefficient kg/kgYield coefficient, kg/kgSpecific rate of synthesis of R (rR/R) 1/hSpecific rate of synthesis of P (rp/P) 1/h

Indices

12m

Refers to reaction 1Refers to reaction 2Refers to maximum

Exercises


Results

Run 1: 416 steps in 0.0167 seconds4-1

3.5-

3-

2.5-

of 2-

1.5-

1 -

0.5-

0 -

/ ^"~'"~*" "

f -»*'

/ / ISi$%£•-.^ T M* ".•"T.-'V/T

'"U--b / /

*"'•" I -''"" J '*"y *y "» '/ " f/ '•. .'

^^ V— *" _j__»— *%

-16

•14

•12

-10

-8 a.

-6

•4

-2

_n

0 5 10 15 20 25 30 35 40TIME

Figure 2* Profiles of residual biomass concentration R, substrate S and product P in the batchfermentation.

Run 4: 41 6 steps in 0.01 67 seconds

35-

30-

25-

20-

a15-

10-

5 •

0 .

•—...„'"'V^

. .-'v\ / .._.. 3:3(2,3)\ / ~- P:3(2.3)

1 / — P:4 (5)

\«-*. \ /

"""-, \ f .*""r-'"

"\ \ i'' '"" "' /^

^ ";:

•5

•4.5

-4

•3.5

•3

-2.5 (/)

-2

-1.5

-1

-0.5

-n

0 5 10 15 20 25 30 35 40TIME

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

References

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


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 andVARVOLD)

System

Semi-continuous or fed batch cultivation of micro-organisms is common in thefermentation industries. The fed batch fermenter mode is shown in Fig. 1 andwas also presented in the example FEDBAT. In this procedure a substrate feedstream is added continuously to the reactor. After the tank is full or thebiomass concentration is too high, the medium can be partially emptied, andthe filling process repeated. Since the variables, volume, substrate and biomassconcentration change with time, simulation techniques are useful in analyzingthis 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,

Substrate,

dv

dT = FO


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 andVARVOLD)

System

Semi-continuous or fed batch cultivation of micro-organisms is common in thefermentation industries. The fed batch fermenter mode is shown in Fig. 1 andwas also presented in the example FEDBAT. In this procedure a substrate feedstream is added continuously to the reactor. After the tank is full or thebiomass concentration is too high, the medium can be partially emptied, andthe filling process repeated. Since the variables, volume, substrate and biomassconcentration change with time, simulation techniques are useful in analyzingthis 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 = FOSubstrate,


8.3 Fed Batch Reactors 225

- = F0S0

Biomass,d(VX)

dt = rxThe kinetics are

rx = MX

|LimS

** - (Ks + S)and

rxrs = -Y

The dilution rate is defined as

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

Vv =

XX< =

ss =F =

- _ JLMm

tf- F°

t' = t


Expanding the derivatives gives

d(V S) = V dS + S dVand

d(V X) = V dX + X dV

Substituting, the dimensionless balances now become:

VolumedV'

BiomassdX'dt'

SubstratedS'dt = ( l - S ) D - j i X

The Monod equation is:

KS +S

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


Quasi- steady

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

Programs

The program VARVOL is based on the model equations with normaldimensions. The program VARVOLD is based on the dimensionless equationsas derived above. Both are on the CD-ROM.

Nomenclature

Symbols

DF

KS

rSV

Dilution rateFlow rateSaturation constant

Reaction rateSubstrate concentrationReactor volume

1/hm3/hkg/m3

kg/m3 hkg/m3

m3


XY

Biomass concentrationYield coefficientSpecific growth rate

kg/m3

kg/kg1/h

Indices

0fmSX

Refers to feed and initial valuesRefers to finalRefers to maximumRefers to substrateRefers to biomassRefers to dimensionless variables

Dimensionless Variables

S'VX1

t'

Dimensionless flow rateDimensionless saturation constantDimensionless substrate concentrationDimensionless volumeDimensionless biomass concentrationDimensionless timeDimensionless specific growth rate

Exercises


Results

During the quasi-steady state, \l becomes equal to D, and this requires that Smust decrease steadily in order to maintain the quasi-steady state as the volumeincreases (Fig. 3). Increasing flow rates from 0.01 to 1.0 causes a delay in theonset of linear growth and causes the final biomass levels to be higher (Fig. 4).


4.5

Figure 3. Fed batch concentration and growth rate profiles, showing quasi-steady state.


5

2.5 C/>

10


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 BiomassProduction Rate of Cyclic Fed Batch Continuous Culture. J. AppL Chem.Biotechnol. 28, 784.

8.3.2 Penicillin Fermentation Using ElementalBalancing (PENFERM)

System

This example is based on the publication of Heijnen et al. (1979), andencompasses all the principles of elemental balancing, rate equationformulation, material balancing and computer simulation. A fed batch processfor the production of penicillin as shown in Fig. 1 is considered withcontinuous feeding of glucose. Ammonia, sulfuric acid and o-phosphoric acidare the sources of nitrogen, sulfur and phosphorous respectively. O-phosphoric acid is sufficiently present in the medium and is not fed. Oxygenand carbon dioxide are exchanged by the organism. The product of thehydrolysis of penicillin, penicilloic acid, is also considered, thus taking the slowhydrolysis of penicillin-G during the process into account.


Glucose Carbon dioxide

Oxygen

PrecursorPhenylacetic 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.


Table 1. Component properties and rateCompound

GlucoseMycelium

PenicillinPenicilloic acidOxygenCarbon DioxideAmmoniaSulfuric AcidPhosphoric AcidPhenylacetic AcidWater

Chemical formula

C6H1206

CHi.64Oo.52No. 16So.0046P<).0054C16H1804N2SC16H2005N2S02CO2NH3H2SO4

H3PO3

C8H802

H2O

designations.Mol wt.(Daltons)

18024.52

334352324417989813618

Enthalpy(kcal/mol)

- 303- 28.1

- 115- 183

0-94- 19- 194- 319-69-68

Conversionrate (mol/h)

RlR2

R3R4

R5R6R8R9

RIORHRl2

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 ofmol/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

0.16R2 + 2R3 + 2R4 + R8 = 0

For sulfur

0.00 46 R2 + R3 + R4 + R9 = 0


For hydrogen

12 RI + 1.64 R2 + 18 R3 + 20 R4 + 3 R8 + 2 R9 + 3 RIO + 8 Rn + 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 -

-319Rio-69Rn -68Ri2 + rH = 0

where TH is the rate of heat of production (kcal/h).A total of 12 unknowns (Ri through R6, Rg through Ri2 and TH) are involved

with a total of 7 equations (6 elemental balances and one heat balance). Thefive 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 + 2R3 + 2R4

From the sulfur balance

- R9 = 0.0046 R2 + R3 + R4

From the phosphor balance

-Rio = 0.0054 R2

From the hydrogen and nitrogen balances

- R5 = -6 RI - 1.044 R2 - 18.5 R3 - 18.5 R4 - 9 Rn

From the enthalpy balance

rH = - 669 RI - 110.1 R2 - 1961 R3 - 1961 R4 - 955 RH


To complete the model, equations for glucose uptake rate (-Ri), biomassformation rate (R2), rate of penicillin formation (Rs), precursor consumptionrate (-Rn), and rate of penicillin hydrolysis (R4) must be known. Note that thereaction rates are defined with respect to total broth weight, since the process isthe 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.

- Q l C i M 2

Biomass Formation Rate:A linear relationship between the glucose consumption rate and growth rate ofbiomass 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/molglucose).

The total rate of biomass formation equals the net rate of formation,corrected for the amount transformed to penicilloic acid. Therefore,

R2 = -Y 2Ri - Y 2 mM 2 - yf (Rs + «4)

Precursor Conversion RateIt is assumed that the precursor is only used for penicillin synthesis. Thus

-Rll = R3 + R4

where - RH is the precursor consumption rate.


Rate of Penicillin SynthesisThe specific rate of penicillin synthesis is assumed not to be a function ofspecific growth rate. So that

R3 = Q3 M2 - R4

where Q3 is the maximum specific rate of penicillin synthesis (mol/mol h),

Equation for the Rate of Penicillin HydrolysisThe hydrolysis of penicillin takes place by a first-order reaction.

R4 = K3 M3

c) Balance Equations

Total Mass BalanceThe individual feed rates of glucose, sulfuric acid and ammonia are adjusted toequal their molar consumption rates. Water lost by evaporation is neglected.The change in mass due to gas uptake and production is neglected. The massflow rates are calculated from the molecular weights, the uptake rates and themass ratio compositions.

Feed rate of glucose stream (kg/h)

180 FFl = F500 = 2.78

where F = mol glucose /h.Feed rate of NH3 stream (kg/h)

F8 = R825Q = T4JT

Feed rate of £[2804 stream (kg/h)18 R9

F9 - R95QO ~ 2.55

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

dG F R9 Rg"dT = Tn + 235" + TTTT

Component BalancesExpressed in mol/h the dynamic balances are,


Glucose

~3T = Rl + F

BiomassdM2~ar = R2

PenicillindM3 „.

Penicilloic aciddM4

The concentrations in mol/kg are as follows:

M2

TrM3

M4c4 = —

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 cellsR2

* = MlRespiration quotient

ReRQ = R7

Oxygen uptake rateOUR = -R5

CO2 production rateCPR = R6

Fraction of N2 in mycelium


R2f2 = 0.16 R|

N2 fraction in penicillinf3 = 1-F2

Fraction of sulfur used for myceliumR2

f4 = 0.046 R|

Sulfur fraction used for penicillin

f5 = 1 - F4

Fraction of glucose for cell growth

= R2

Fraction of glucose for penicillinR3 + R4

S3 = - Y3 R!

Fraction of glucose for maintenanceM2

g4 = -M R^-

Program

The Madonna program covers a fermentation time of 200 h starting from theinitial conditions of 5500 mol glucose, 4000 mol biomass, 0 mol penicillin and0.001 mol penicilloic acid in an initial broth weight of IxlO5 kg. The programis on the CD-ROM.

Nomenclature

Symbols

a, b Flow rate variables variousC Component concentration mol/kgCPR Carbon dioxide production rate mol/hF Feed rate kg/h


h

flf5G82g3g4KlK3MmOURQRRQ

rqY

Fraction of nitrogen in myceliumNitrogen fraction in penicillin -Fraction of sulfur used for mycelium -Fraction of sulfur used for penicillin -Mass in reactor kgFraction of glucose for cell growthFraction of glucose for penicillinFraction glucose for maintenanceSaturation constant mol/kgHydrolysis rate constant 1/hMass of individual components molMaintenance rate factor mol/(mol h)Oxygen uptake rate mol/hMaximum specific rates mol/(mol h)Conversion mol/hRespiration quotient -Heat production rate kcal/hRespiratory quotient -Yield coefficient -Specific growth rate 1/h

Indices

012345689101112

initialglucosebiomasspenicillinpenicilloic acidoxygencarbon dioxideammoniasulfuric acidphosphoric acidphenylacetic acidwater

Exercises


Results

The results of Fig. 2 show the substrate MI to pass through a maximum, whilethe penicillin M2 develops linearly, for this constant feeding situation.Increasing the feeding linearly with time (F = 500 + 5* time) gave the results inFig. 3, where it is seen that maintenance accounts for about 70 % of glucoseconsumption at the end of the fermentation.


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).



8

0.9-,

0.8-

0.7-

0.6-

r-- 0.4-

0.3-

0.2-

0.1-

0-20 60 80 100

TIME120

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 BalancingMethods 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 themedium as alternative substrates. In addition, the glucose effect, when glucoselevels are high, will cause fermentation, instead of respirative oxidation, to takeplace, such that the biomass yields are much reduced (Fig. 1). In this examplethe constant a designates the fraction of respiring biomass and (1 - a) thefraction of biomass that ferments. The rates of the process are controlled bythree 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+Ksl

Fermentation to ethanol,

R2 = — — — K2 (1 - a) XGlu + KS2

Conversion of ethanol to biomass,

Enzyme activation for the transformation of ethanol to biomass is assumed toinvolve an initial concentration of starting enzyme EQ, which is converted to en-zyme £2 and which catalyzes growth on ethanol through an intermediateenzyme EI.

Thus, the production rate of enzyme EI is inhibited strongly by glucose,

R4 = - -rXEoKS4+Glu3

and the production rate of enzyme £2 controlling the conversion of biomass toethanol depends on EI,

R5 = K 5 XEi

The mass balances for the biomass, substrates and enzymes are those for a fedbatch with variable volume.


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

ô = _« E0Qdt 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-


Nomenclature

Symbols

CEEtOUGluKQRVXYa

Component concentrationEnzyme concentrationEthanol concentrationSubstrate feed concentrationRate constantsFeed flow rateReaction rateReactor volumeBiomass concentrationYield coefficientFraction of respiring biomass

mol/m3

mol/m3

mol/m3

mol/m3

variousm3/hmol/m3 hm3

C-mol/m3

mol/mol

Indices

012345

Refers to feedRefers to reaction 1Refers to reaction 2Refers to reaction 3Refers to reaction 4Refers to reaction 5

Exercises


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 inethanol concentration as a function of feedrate is given from a Parameter Plot.


30

25

60

Figure 3. Batch yeast fermentation.


Run 2:12100 steps in 0.333 seconds30

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 ReactionEngineering, 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 thisoperation 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 thenext fed batch. The operating variables are initial volume, final volume,substrate feed concentration and flow rates of filling and emptying.


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 isconsidered 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 outletstream is added. The inlet and outlet streams are controlled by conditionalstatements 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 0else if time> = 21 then 0.5 else if time> = 20 then 0else 0.5Fout= if time>=33 then 0 else if time>=32 then 5.39else if time> = 21 then 0 else if time> = 20 then 5.39else 0


Nomenclature

Symbols

D Dilution rate

F Flow rateKl and K2 Product kinetic constants

KS Saturation constant

P Product concentration

S Substrate concentration

X Biomass concentration

V Reactor volumeV0 Initial volume of liquidVX Biomass in reactorVS Substrate in reactorY Yield coefficient

|i Specific growth rate

1/hm3/hvarious

kg/m3

g/m3

kg/m3

kg/m3

m3

m3

kgkgkg/kg

1/h

Indices

SX0 (zero)initialinout

Refers to substrateRefers to biomassRefers to initial and inlet valuesRefers to initial valuesRefers to inletRefers to exit

Exercises


Results

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


60 -|

50-

40-

30-

20-

10-

0-

A/ I

/ I/ I

f I/ I

/ I\l

0 5 10 15 20

.-80

pr Ti /-70|— "vsi| , go

/"( / .50

' • J -40

/ \

/ i /-V/ I / ^ \

X V ,* \ -10

- - • * ' " - ' L -Q25 30 35 40 45 50

TIME

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



10

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

References

Dunn, I.J., Mor, J.R., (1975) Variable Volume Continuous CultivationBiotechnol. Bioeng. 17, 1805.

Keller, R., Dunn, I.J. (1978) Computer Simulation of the Biomass ProductionRate 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 andhormones 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 ofrepeated replacement culture is shown in Fig. 1. In this procedure part of themedium volume (with cells) is removed after a certain replacement time andreplaced with fresh medium. Each cycle operates as a constant volume batch in


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

Replacement

Final Conditions VX VS

Initial Conditions

Figure 1. One cycle for medium replacement culture.

Model

The equations are those of batch culture, where for convenience the totalmasses are used.

dVS"dT = r s V

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 thetotal volume, V, by the equations,

* VR

f = —

VX = (1 - £) VXF


VS= ( l - f )VS 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 tovary the biomass and substrate concentrations corresponding to thereplacement of a fraction F of the culture medium. The time for each batch isthe value of INTERVAL.

Nomenclature

Symbols

Df

KS

sXVvovxvsVRY

Dilution rateFraction of volume replacedSaturation constantSubstrate concentrationBiomass concentrationReactor volumeInitial volume of liquidBiomass in reactorSubstrate in reactorVolume replacedYield coefficientSpecific growth rate

1/h

g/m3

g/m3

g/m3

m3

m3

kgkgm3

1/h

Indices

FSX0

Refers to final values at end of the cycleRefers to substrateRefers to biomassRefers to initial and inlet values


Exercises

Results

Fig. 2 shows how the biomass increases, until after six cycles the time profilesbecome almost identical.

TIME= 19.29 X = 1.26

10 20 30 40 50 60 70TIME

90 100

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


8.3.6 Penicillin Production in a Fed BatchFermenter (PENOXY)

A fed batch process is considered for the production of penicillin, as describedby Muttzall (1), The original model was altered to include oxygen transfer andthe 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

Biomass:

Substrate:

Product:

dt

d(MassX)dt

d(MassS)dt

= Vr-X

= FS f+Vr s


d(MassP) _ .= V fp

dt p

Dissolved oxygen, neglecting the content of the inlet stream is calculated from

d(MassO)dt

= KLa*(Osat-0) + Vr0

The influence of biomass concentration on the oxygen transfer isapproximated here by

K X +X

The concentrations are calculated from

_MassX MassS MassP MassO1\. — """"""""""""" , »J — , L — , U —

V V V V

The growth kinetics take into account the oxygen influence

o

The substrate uptake kinetics includes that amount used for growth, for productand for maintenance

J*o ~ Jft o _/V^ V V" ^YXS YPS

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

Oxygen uptake includes growth and maintenance

=-TTYxo


Program

The program is on the CD-ROM.

Nomenclature

Symbols

FKLa

KoKS

KXMassmoms

Osat

SfV'maxYPSYXOYXSH-max

Feed flowrateOxygen transfer coeff.Monod constant for oxygenMonod constant for glucoseConstant for biomass effect onComponent massMaintenance coeff. for oxygenMaintenance coeff. for glucoseSaturation for oxygenFeed cone, of glucoseVolumeMaximum volumeYield product to substrateYield biomass to oxygenYield biomass to substratemax.specific growth rate

m3/h1/hkg/m3

kg/m3

kg/m3

kgkg O/kg X hkg S/kg X hkg/m3

kg/m3

m3

m3

kg/kgkg/kgkg/kg1/h

Exercises


II!

References

K. Mutzall, "Modellierung von Bioprozesses", Behr's Verlag, 1994.

Program and model developed by Reto Mueller, ETH Zurich.

Results


0.008

20 40 60 80 100 120 140 160 180 200

Figure 2. Dynamics of the fed batch reactor.

8.4 Continuous Reactors 257

120

100

80

- 60

40

20


-0.008

! Ii I: I

L i II

•0.007

•0.006

•0.005

•0.004 O

-0.003

-0.002

-0.001

00 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 Continuous Reactors

8.4.1 Steady-State Chemostat (CHEMOSTA)

System

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



120

100

80

- 60

40

20

^

\». ll

'-, 1\ \

1 1I

\!

1 1III 1

! 1. i i

v i|\ i

i ' \ *\! \l

•\\ \

-0.008

-0.007

-0.006

-0.005

-0.004 O

-0.003

-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 Continuous Reactors

8.4.1 Steady-State Chemostat (CHEMOSTA)

System

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



D,S

Figure 1. Chemostat fermenter with model variables.

Model

The dynamic balance equations may be modified to apply only to the steadystate by setting the time derivatives equal to zero. The corresponding equationsare then:

For biomass,

For substrate,

Growth kinetics,

0 = - D X + rx

0 = D (S0 - S) + rs

rx = ^X

Substituting into the biomass balance gives

\i = Dwhere S is determined by the kinetics

The Monod relation results in,\i = f(S)

S =

The substrate balance gives,X = Y(S0 -S)


The productivity of the reactor for biomass is X D.

The above equations represent the steady state model for a chemostat withMonod 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 thekinetic parameters.

Program

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

Nomenclature

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

Exercises


Results

The steady state curves of X, S, and XD versus D are given Fig. 2. The resultsin Fig. 3 were obtained by varying K$ in each run. An interesting effect can be

observed on the position of the washout point.


10 -

9-

8-

7-

6-

v- 5"

4-

3-

2-

1 -

0-

*S**~\ I\ !^r Y.-I 1 iim ~s;i \!

f*r —mm .

S" 1

// /{

-4

•3.5

-3

-2.5

•2 S

•1.5

• 1

•0.5

•0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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



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 ratebelow that predicted by the Monod equation. The inhibition function may beexpressed empirically as

where KI is the inhibition constant (kg/m3).

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


to two possible steady states, as defined by the steady state condition JLI = D andas 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 beunstable. 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,S0

+- F,S,X

Figure 2. Model variables.


Cell material balance,

VdXjj- = ^ i V X - F X

or,

where D is the dilution rate = F/V.

Substrate material balance,

VdS ^ V X— = F (S0-S) -~^f—

or,dS M « XdT = D (S0 - S) - —

where Y is the yield factor.

Program

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

Nomenclature

Symbols

D Dilution rate 1/hKI Inhibition constant kg/m3



SSmaxXY

Substrate concentration kg/m3Maximum in S for inhibition function kg/m3Biomass concentration kg/m3

Yield coefficient kg/kgSpecific growth rate 1/h

Indices

0Im

Refers to inletRefers to initial valueRefers to maximum

Exercises


Results


1-1-2

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

10 15 20 25 30 35 40TIME


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 Runswith overlay.


Run 9: 2000 steps in 0 seconds

5 -

4.5-

4 -

3.5-

3 •

</) 2.5-

2 •

1.5-

1 •

0.5-

0 •

X,

^^^>*" t**1

• ^ *t %%

\ * * C^1 v"| "% % V

• \ ^ x• i ! l

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)

/ / / ; ,--^,f / / / ^' — . *x/ / J j * S " -s^^-

f f / t / ,- rf'^T— K" " >*i' ' ** -^:' / / f * / Jf*'^ ^^ * ^$^S? •**! * \ f 1 C & ( ^^*^^ **

0.5

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

= 0.0.


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 ofContinuous Microbial Propagators Subject to Substrate Inhibition Biotechnol.and Bioeng. 14, 939-974.


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 heterotrophicorganisms which oxidize organic carbon, and they can be washed out of thereactors 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 growthrates require high waste rates. Nitrification will not be possible under theseconditions because the concentration of nitrifiers (Ni) will become very low.

O,FO 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 thereactor and around the settler. The settler is simplified as a well-mixed systemwith the effluent streams reflecting the cell separation.Organic substrate balance for the reactor:

= F0So + F2S2 -


Ammonia substrate balance in the reactor:

R2V t= FQ AQ + F2 A2 - FI A

Reactor balance for the heterotrophic organisms:

pj7 = p2 O2 — FI QI + RI Vj_

Reactor balance for the nitrifying organisms:

ldi = F2N2 - FiNi + R2Vi

Organic substrate balance in the settler:

V2dS2—j^— = FiSi - F3S2 - F4S2

Ammonia substrate balance for the settler:

V2dA2

— 3t — = FIAI -

Balance for heterotrophic organisms in the settler:

- F4A2

V2d02

dt = Fl °l - F3 02

Balance for nitrifying organisms in the settler:

V2 dN2— 34— = FiNi - F3N2

The equations for the flow rates are given below.

Recycle flowrate:F2 = F0R

where R is the recycle factor.

Reactor outlet flow:FI = F2 + F0 = FOR + FO

Flow of settled sludge:


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.

Rl =

l^2maxR2 = ^Ni =

Program

The program is given on the CD-ROM.

Nomenclature

Symbols

A Ammonia substrate concentration kg/m3

C Concentrating factor for settler -F Flow rate m3/hFo-5 Flow rates, referring to the figure m3/hKI Saturation constant of heterotrophs kg/m3

K2 Saturation constant of nitrifying organisms kg/m3

N Concentration of nitrifiers kg/m3

O Concentration of heterotrophs kg/m3

R Recycle factor -


RlR2sVY

Hi

Growth rate of heterotrophicsGrowth rate of nitrifying organismsOrganic substrate concentrationVolumesYield coefficientsSpecific growth rate of heterotrophs

kg/m3hkg/m3hkg/m3

m3

kg/kg1/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 values1 Refers to reactor and organic oxidation2 Refers to settler and ammonia oxidation3 Refers to recycle4 Refers to settler effluent5 Refers to sludge wastagem Refers to maximum

Exercises


Results

The results in Fig. 2 demonstrate the influence of flow rate on the effluentorganics 82- The ammonia in the effluent A2 is seen, in Fig. 3, to respondsimilarly to FQ, but for a very high value of FQ = 1000 m3/h the nitrificationceases, and A2 becomes the same as the inlet value AQ. This corresponds towashout of the nitrifiers, which would be seen by plotting NI versus time.


0.9 •,

0.8 -I ,/"

°M ifIf JII /It

0.3- rr• J 82:1(20)

02. II 82:2(180)" 82:3(340)

0-1 JM | 82:4(500)

°" .6 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

TIME

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


0.1 -I

0.09-

0.08-

3 0.06 -|-_- A2:1(20)— — A2:2(180)

005 J* -_A2:3(340)1 S. I A2:4(5QO)

0.04-

0.03-

0.02J

0 2 4 6 8 10 12 14 16 18 20TIME

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


8.4.4 Tubular Enzyme Reactor (ENZTUBE)

System

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

Enzymeconcentration

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 1dZ = ~ v

Kinetics,

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


FV7 = ~L 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 thereactor length coordinate Z. The program is given on the CD-ROM.

Nomenclature

Symbols

A Reactor tube cross section m2

F Flow rate m3/hK Rate constant 1/hKM Michaelis-Menten constant kg/m3

m Enzyme distribution constant kg/m3 mr Reaction rate kg/m3 hS Substrate concentration kg/m3

vm Maximum reaction velocity kg/m3hvz Linear flow velocity m/hZ Reactor length me Void volume fraction of packing

E Enzyme concentration kg/m3


Indices

0S

Refers to inletRefers to substrate

Exercises

Results

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



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 belimiting by adjusting its concentration relative to those of the other mediumcomponents. In general, however, more than one substrate may limit the cellgrowth rate. In this case the yield coefficients for the various components,Yxsi> may vary depending upon the growth regime. This situation wasdiscussed by Egli et al. (1989), who examined results at steady state with dualnutrient limitation. The present mathematical model simulates the transientbehaviour of such a dual (Si -carbon, 82 -nitrogen) nutrient-limited systemwhen carried out in a chemostat. The model assumes that the yield coefficientsare each a function of the ratio 81/82, i.e. the ratio of the carbon-nitrogensubstrate concentrations in the vessel. The original paper took the carbon-nitrogen ratio in the feed stream as the controlling parameter. Here theconcentrations in the reactor are assumed to be controlling.

Model

Assuming a perfectly mixed, constant volume continuous-flow stirred-tankreactor, the mass balance equations for the cells and for the two limitingsubstrates are as follows:


1= D (SlFeed - Si) -

D /O O \ I »^__i^^_ I i i "V(o2pee(j — o2) — lvYQO J ]LlA

where D = F/V.

The specific growth rate is modelled as

Si V S2

The yield coefficients are assumed to vary with the carbon-nitrogen ratio in thereactor.

SiRATIO = ^

The yield coefficients are varied according to RATIO using the following logic:

and YXS2 = Y2min if RATIO <BiYXSl=Yim i n and YXS2 = Y2max if RATIO > B2

where,_ Y2min Y2maxBi = \r - and Bo = v~, — T"1 Imax -1 1mm

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


XSi

C limitation Double limitation

rXS1

N limitation

10

0.8

B2S2

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

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

Program

Note that the programing of this example is rather more complicated than usualowing to the need to allow for the logical conditions of carbon limitation,nitrogen limitation or both substrates together causing limitation. A partiallisting 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)/(B2-B1)*(Y1MIN-Y1MAX)) )YXS2 = if (RATIO < Bl) then Y2MIN else ( if (RATIO >B2) then Y2MAX else (Y2MIN+(RATIO-B1)/(B2-

Bl)*(Y2MAX-Y2MIN)) )


Nomenclature

Symbols

BiB2

CcCnDFKSR

SiS2

XYH

Indices

12

Ratio of Y2min/Yimax

Ratio of Y2max/Yimin

Carbon source concentrationNitrogen source concentrationDilution rateVolumetric feed rateAffinity constantReaction ratesCarbon source concentrationNitrogen source concentrationBiomass concentrationYield coefficientSpecific growth rate

Refers to carbon sourceRefers to nitrogen source

_-

kg/m3

kg/m3

1/hm3/hkg/m3

kg/m3 hkg/m3

kg/m3

kg/m3

kg/kg1/h

Exercises


Results

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

3-c


1

Figure 2. Startup of a continuous culture.


X 1.5

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


Reference

Egli, Th., Schmidt, Ch. R. (1989). "On Dual-Nutrient-Limited Growth ofMicrobes, with Special Reference to Carbon and Nitrogen Substrates", inProceed. 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 SandBed (DCMDEG)

System

The process involves the removal of dichloromethane (DCM) from a gas streamand the subsequent degradation by microbial action. The reactor consists ofbiofilm sand bed column with circulation to an aeration tank, into which thesubstrate and oxygen enters in the gas phase, or the substrate can be fed in aliquid stream, as shown in Fig. 1. The column is approximated by a series of sixstirred tanks. The reaction is treated with homogeneous, double saturationkinetics with dichloromethane (DCM) inhibition. The oxidation of one mole ofDCM produces 2 moles of HC1, making a hydrogen ion balance for pH im-portant. 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.


CSR6>

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 isassumed that the gas phase dynamics are comparatively fast and hence anequilibrium with the inlet concentration of oxygen and DCM may be assumed.The biomass is assumed to grow slowly, and growth rates are therefore also notmodelled. 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


DCM balance,

H+ ion balance,

dCSrl _ CSrin~CSrldt

dCHi

t

i -CHI 2rS i84900

Here T is the residence time of the liquid in one section of the column. Theconstant 84,900 converts grams to moles and includes the stoichiometry.

pHi = -0.434 log |Cm|

Evaluation of rates for the inlet section 1:

VmaxCSrl

KI )

-01

For the aeration tank the 62 and DCM balances are:

KLa02(Co2eq-Coin)

dC • R— -~ = — (CSr6 - CSrin )at V 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 theCD-ROM.


Nomenclature

oin

srin

CSFOCSGFKIKLa

KS

pHn

R

VR

VTVmaxYSO

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/hInhibition constant g/m3

Transfer coefficients for DCM and ©2 1/hSaturation constants g/m3

pH in n section n pH unitsRecirculation rate m3/hOxygen uptake rate in section n g/m3 hSubstrate uptake rate in section n g/m3 hReactor volume m3

Volume of aeration tank m3

Maximum degradation rate g/m3 hYield coefficient for DCM/oxygenLiquid residence time in one section h

Exercises

•11


Results

The concentrations in the stream leaving the top of the column (CSr6) duringstartup 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.



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5TIME

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


3.5

3

2.5

: 2

1.5

1

0.5

0

— CSR6:1... PH6:1

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.


8.4.7 Two-Stage Chemostat with Additional Stream(TWOSTAGE)

System

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

X1.S!

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

Model

HiUS

, 810

The balance equations are written for each component in each reactor.

Stage 1 with sterile feed,

= F[SO-S!] -


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

V2 - [F + Fi]X2

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

Productivity for biomass:

First stage,

Both stages,

KS + S2

Prodi =

Prod2 =

V,

Program


Nomenclature

Symbols

FKSProdSVXY

Volumetric feed rateSaturation constantProductivity for biomassSubstrate concentrationReactor volumeBiomass concentrationYield coefficientSpecific growth rate

m3/hkg/m3

kg/m3 hkg/m3

3

kg/m3

kg/kg1/h


Indices

01210m

Refers to tank 1 inletRefers to tank 1 and inlet of tank 2Refers to tank 2 and outlet of systemRefers to separate feed for tank 2Refers to maximum

Exercises

Results

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



•T5

35 40

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


5

4.5

4

3.5

3

32.52

1.5

1

0.5

010 15 20 25

TIME30 35 40

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


8.4.8 Two Stage Culture with Product Inhibition(STAGED)

System

Products may inhibit growth rates. Under such conditions a multi-stagedcontinuous reactor as shown in Fig. 1 will have kinetic advantages over a singlestage. This is because product concentrations will be lower and consequentlythe rates in the first tank will be higher as compared with a single tank. Thiseffect may be conveniently investigated by simulation. Batch cultures can beexpected 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 Monodequation.

The product kinetics are according to Luedeking and Piret, with dependenceon both growing and non-growing biomass,



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

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

a - a"Qn ~ 1 + Pi-r rn

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] +rS iVi

jp = F[P0-Pi] + rp^j

Stage 2 with additional substrate feed FI,

dX2V2 -gjT- = F Xj - [F + F!]X2 + rX2V2

dS2V2 -gj- = F [Si - S2] + FI [Sio - S2] + rS2V2

dP2- =FPl- [F + Fi]P2 + rp2V2

Productivity for product:First stage,


Prodi =

Both stages,

Program


Nomenclature

Symbols

FKIKS

PProdrSVXYaOC0

P

Indices

Volumetric feed rate m3/hInhibition constantSaturation constant kg/m3

Product concentration kg/m3

Productivity for product kg/m3 hReaction rate kg/m3 hSubstrate concentration kg/m3

Reactor volume m3


Yield coefficient kg/kgNon-growth product rate term kg P/kg X hNon-growth term with no inhibition kg P/kg X h

Growth dependent product yield kg/kgSpecific growth rate 1/hMaximal specific growth rate 1/h

n01

Refers to tank nRefers to tank 1 inletRefers to tank 1 and inlet of tank 2


210

Refers to tank 2 and system of outflowRefers 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, asshown in Fig. 3. The higher the KI the lower is the degree of inhibition and thegreater is the product concentration P2-


4

3.5

3

2.5

,1.5

1

0.5

0


r10

,

10 15TIME

Figure 2. Startup and approach to steady state for the two stages.


1.4.,

1.3

1.2.

1.1I

1

0.9.

0.8

0.7 J

10 12 14 16 18 20 22 24 26TIME

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


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. Theflow rate through the reactor due to circulation is FR. Oxygen is absorbed in awell-mixed tank of volume VT. The reaction rate for substrate (r$) depends onboth S and dissolved oxygen (CL)- The rate of oxygen uptake (ro) is related toS by a yield coefficient (Yos)- The gas phase is not included in the model,except via the saturation concentration (CLS)- The oxygen uptake rate ofreactor can be determined by the difference in CL inlet and outlet values.

? So , Fn

FluidizedBed

F,S

Figure 1. Biofilm fluidized bed with external aeration.


Model

The model balance equations are developed by considering the individual tankstages and the absorber separately. The gas phase in the absorber is assumed tobe 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

For each stage

r = ^(CL3-CL)+KLa(CLs-CL)VT

dCLn FR~dT" = V (CLn-! ~CLn) ~ rOn

Kinetics for stage n:VTm

Kn +Sn K0 +CLn

Program



Nomenclature

Symbols

CL

CLSFFRKLa

KsKorSVVT

^m

xYT

Dissolved oxygen concentrationSaturation oxygen concentrationFeed flow rateRecycle flow rateTransfer coefficientSaturation constantSaturation constant for oxygenReaction rateSubstrate concentrationReactor volume of one stageVolume of absorber tankMaximum velocityBiomass concentrationYield coefficientInverse liquid residence time

g/m3

g/m3

m3/hm3/h1/hkg/m3

g/m3

kg/m3 hkg/m3

m3

m3

kg/m3 hkg/m3

kg/kg and g/kg1/h

Indices

0l ,2 ,3 ,nmOSTX

Refers to feedRefer to the stage numbersRefers to maximumRefers to oxygenRefers to substrateRefers to aeration tankRefers to biomass

Exercises


Results

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


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.



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 involvesthe sequential oxidation of NFLt"1" to NO2~ and NC>3~ that proceedsaccording 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 nitrogensubstrate concentration. Owing to the relatively slow growth rates of nitrifiers,

treatment processes benefit greatly from biomass retention.


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

Model

The model balance equations are developed by considering, separately, theindividual tank stages and the absorber. Component balances are required for allcomponents in each section of the reactor column and in the absorber, where thefeed and effluent streams are located. Although the reaction actually proceedsin the biofilm phase, a homogeneous model with apparent kinetics is employedrather than a biofilm model, as in the example NITBEDFILM.

03.

Fluidizedbed

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

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


oxygen is referred to as O. Additional subscripts, as seen in Fig. 1, identify thefeed (F), recycle (R) and the flows to and from the tanks 1, 2 and 3, each withvolume V, and the absorption tank with volume VA-

The fluidised bed reactor is modelled by considering the componentbalances 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-containingcomponents include the input and output of the additional feed and effluentstreams, giving

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

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

r = vml Sin Qn

K + S K + O

r2n = Vm2 S2n °n

K2+S2n KO2+°n

The oxygen uptake rate is related to the above reaction rates by means of theconstant yield coefficients, YI and ¥2, according to

ron = - H n Y i -r2nY2

The reaction stoichiometry provides the yield coefficient for the first step


and for the second stepYI = 3.5 mg O2/(mg NNH4)

Y2 = LI mg O2/(mg NNO2)

Program


Nomenclature

Symbols

F Feed and effluent flow rate L/hFR Recycle flow rate L/hKjâ Transfer coefficient hK Saturation constants mg/LKI Saturation constant for ammonia mg/LK2 Saturation constant for ammonia mg/LO Dissolved oxygen concentration mg/LOs and O* Oxygen solubility, saturation cone. mg/LOUR Oxygen uptake rate mg/Lr Reaction rate mg/L hS Substrate concentration mg/LV Volume of one reactor stage LVA Volume of absorber tank Lvm Maximum velocity mg/L hY Yield coefficient mg/mg

Indices

1,2,31,2,3AFijm

Refer to ammonia, nitrite and nitrate, resp.Refer to stage numbersRefers to absorption tankRefers to feedRefers to substrate i in stage jRefers to maximum


Ol and O2S1,S2S and *

Refer to oxygen in first and second reactionsRefer to substrates ammonia and nitriteRefer to saturation value for oxygen

Exercises


Results

Run 1: 519 steps in 0.2 seconds6

10 15 20 25 30 35 40 45 50

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

280

270

260

250

1^240.

£,230-<

220

210

200

190

180


P2.5

5 c

M I

0.5

15 20 25 30 35 40

KLA

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


8.4.11 Continuous Enzymatic Reactor (ENZCON)

System

This example, schematically shown in Fig. 1 involves a continuous, constantvolume, enzymatic reactor with product inhibition in which soluble enzyme isfed to the reactor.

EO.FE

Figure 1. Continuous enzymatic reactor with enzyme feed.

Model

I» S1f P1§ F1

The mass balance equations are formulated by noting the two separate feedstreams and the fact that the enzyme does not react but is conserved.

Total flow:

Mass balances:dSi

FS + FE =

= FsSo-FiS 1 + r s V


r = -F iP 1 + rpV

Kinetics with product inhibition:

rS - -vmKM + S + (P/Ki)

vm = EI K2

rP = -2rs

Program


Nomenclature

Symbols

E Enzyme concentration kg/m3

F Flow rate m3/hKI Inhibition constantKM Saturation constant kg/m3

K2 Rate constant 1/hP Product concentration kg/m3

r Reaction rate kg/(m3 h)S Substrate concentration kg/m3

V Reactor volume m3

vm Maximum rate kg /(m3 h)

Indices

0 Refers to inlet values


1EPS

Refers to reactor and outlet valuesRefers to enzymeRefers to productRefers to substrate

Exercises

Results

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


1.8

1.6

Run 3: 8004 steps in 0.1 83SeC°ndS

0.8

.0.6 / /

tS

s--- P1:2(0.2)— P1:3(0.3)

,-'"

0.4

0.2

0 10 20 30 40TIME

50 60 70 80

Figure 2. Performance for three values of FE.

r4.5Run 3: 8004 steps in 0.233seconds

• 3.5

•3

• 2.5

.2

-1.5

jft•<*/r

0 10

x--'

r *>"* -^ -» **" ~ ""

I—. P1:2(1.5).. P1:3(2)

20 30 40 50 60 70 80TIME

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 inthe bioreactor design. One of the methods to keep productivity fluctuationslow, and hence to efficiently utilize the biocatalyst, is to use a series of reactorswith biocatalyst batches having different times-on-stream in each reactor. In


this example a series of three stirred tanks of a constant equal volume withbiocatalyst deactivating by first order reaction kinetics is investigated (Fig. 1).After a time period ILAG? e biocatalyst from the tank with longest time-on-stream (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 inthe series. Other tanks are switched over correspondingly. This is equivalent toreplacing the used enzyme with fresh enzyme in tank 3 and moving the usedenzyme 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 obeysMichaelis-Menten kinetics with competitive product inhibition, and the kineticconstants were determined with considerable accuracy. The same constants areused also in this substrate inhibition model.

F,S0

F,Si F,S2 F,S3

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

Enzyme (active)

= F(PM-Pi)dt


=rE ivwhere rate of substrate consumption is given by product inhibition(competitive)

sirSi = ' vmaxbi - 7 - Z~^\

V Kinh

According to the molar stoichiometry

RPi= -RSi

The rate of enzyme deactivation is assumed to be:

rEi = - kD EI

For each batch of enzyme in tank i

dEi i c-

This equation can be applied by changing the initial conditions for each tankwhen the enzyme is moved from tank to tank. Thus the final value in tank nbecomes the initial condition in tank i-1. The initial conditions can also becalculated by analytical integration of the enzyme deactivation equation attimes corresponding to the respective ages of the biocatalysts in the respectivereactors (multiples of TLAG)- Fresh enzyme with the activity EQ is in the thirdtank. 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 theprogram is included only as a comment but could be incorporated into theprogram with the corresponding values for the constants.


Nomenclature

Symbols

COSTEECOSTFICOSTkD

KinhKmOCOSTPRCmrsST

t

TDOWNTLAGvmax

Indices

0i

Specific product costsEnzyme concentrationEnzyme costFlow rateInvestment costDeactivation constantInhibition constantMichaelis - Menten constantOperating costTotal amount of productReactor refill costReaction rate of deactivationReaction rate of substrateSubstrate concentrationResidence timeTimeDown timeTime-on-stream differenceMaximum specific reaction rate

Refers to initial, feedRefers to reactor number

$/kgkg/m3

$/kgm3/h$/kg1/hmol/m3

mol/m3

$/kgmol$/kgkg/(m3 h)mol/(m3 h)mol/m3

hhhhmol/kg h

Exercises


Results

The results from DEACTENZ show an exponential decrease of the biocatalystactivity (Fig. 2), which causes dynamic changes in the substrate and productconcentrations (Fig. 3) in all three reactors.


Run 1: 50000 steps in 0.917 seconds0.5-,

0.45-

0.4-

0.35-

a 0 3 '.0.25-

"* 0.2-

0.15-

0.1-

0.05-

0-

\

\\

%*.

"%xx

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

— ^ — — -" • — •

sj**

r*i**

"''— . — E2:1

_-— Totalproduct:1

-i..••••••

***

—i...

-4000

-3500

-3000

^-2500 3

I-2000 Q_

-1500 pH

-1000

.500

-0

0 100 200 300 400 500 600 700 800 900 1000

TIME

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

140-i

120-

100-

°l 8°-

C/l^ 60cn

40-

20

0

Run 1: 50000 steps in 0.933 seconds120

100

80

a-40

20

0 100 200 300 400 500 600 700 800 900 1000

TIME

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


References

Prenosil, I.E., Peter, J., Bourne, J.R. (1980). Hydrolytische Spaltung desMilchzuckers der Molke durch immobilisierte Enzyme im Festbett-Reaktor.Verfahrenstechnik 14, 392.

Prenosil, J.E. (1981). Optimaler Betrieb fur einen Festbett- und einen Fliessbett-Reaktor mit desaktivierendem Katalysator. Chimia 35, 226 .

Prenosil, J.E., Hediger, T. (1986). An Improved Model for Capillary-Membrane, Fixed-Enzyme Reactors. In Membranes and Membrane Processes,Plenum, N. Y., 515.

8.4.13 Continuous Production of PHB in a Two-TankReactor Process (PHBTWO)

System

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

Sfeed,

> 82, F0

Figure 1. Configuration of the two-tank system.


Model

The details of the structured model will not be repeated here (See PHB). Thebiomass consists of a synthesis part R and the intracellular product P. Thebiomass growth rate of R is proportional to the specific growth rate, which isgiven 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 seenin the program. The relative reactor volumes are determined by the parameterVrat. The volumetric productivities are calculated to compare the results.

Program

The program is found on the CD-ROM

Nomenclature

Symbols

FO Feed flow rate m3/hKI Inhibition constant, for (NH4>2SO4 kg/m3


n Hill CoefficientP Product concentration (PHB) kg/m3

PROD Productivity kg/(m3h)R Residual biomass concentration kg/m3

rp Rate of synthesis of PHB kg/m3

TR Rate of synthesis of R kg/m3


Sfeed

Vi and V2

XYP/RYR/S

MP

Indices

Rate of substrate uptake kg/(m3 h)Limiting substrate cone. NH4+ as

(NH4)2S04 kg/m3

Feed concentration kg/m3

Reactor volumes m3


Yield coefficient kg/kgYield coefficient, kg/kgSpecific rate of synthesis of R (rR/R) 1/h

Specific rate of synthesis of P (rp/P) 1/h

12m

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

Exercises


Results


4

90 100

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


0.2

Figure 3. Here with FO set at the optimum value of 1.24, the influence of VRAT is investigatedgiving a value for the maximum in PROD corresponding to the OPTIMIZE results. VRAT is seennot to be very important. Thus equal-sized tanks are adequate.


8.5 Oxygen Uptake Systems

8.5.1 Aeration of a Tank Reactor for EnzymaticOxidation (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 beessentially air, which eliminates the need for a gas balance for the well-mixedgas phase.

gas

•Hi

ii

Ill + 02

air

Figure 1. Schematic of the enzymatic oxidation batch reactor.

Model

The reaction kinetics are described by a double Monod relation:

_S CL

The batch mass balances lead to:


8.5 Oxygen Uptake Systems 319

dSdF = rS

dCL *— = KLa(CL* -CL) - rsYo/s

dPdF = ~rsYP/s

KLa varies with stirring speed (N) and aeration rate (G) according to:

KLa = kN3G°-5

where k = 4.78 x 10-13 with N in 1/h, G in m3/h and KLa in 1/h.

Program


Nomenclature

Symbols

CL Dissolved oxygen concentration g/m3

CLS>CL* Saturation oxygen concentration g/m3

G Aeration rate m3/hKCL Saturation constant for oxygen kg/m3

â Transfer coefficient 1/h


k Constant in Kâ correlation complexN Stirring rate 1/hP Product concentration kg/m3

r Growth rate kg/(m3 h)S Substrate concentration kg/m3

vm Maximum degradation rate g/(m3 h)Y Yield coefficient kg/kg


Indices

0ops

Refers to feedRefers to oxygenRefers to productRefers to substrate

Exercises

Results

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


8 •

7 -

6 -

5 •

3 •

2 -

0 •

C


--'/ 7s"_..-....._..-..._ f

«r *.*•** *

i

CL1 (3e+4) *CL:2 (2.1676+4) 1

--• CL:3 (1.3336+4) /-^_CL:4 (5000) 4

{ -—"-''

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 6 7TIME

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

8.5.2 Gas and Liquid Oxygen Dynamics in aContinuous Fermenter (INHIB)

System

Cell growth is limited by the oxygen mass transfer rate, and hence by thedissolved 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.


To avoid washout of cells, it is important that the reactor should never enter therange of inhibitory behavior. Schematic representation of a continuous aeratedfermenter is given in Fig. 1.

feed F, S 1 gas G, CG2

liquid F,S2, 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,

For oxygen,

dS2

dCL2

The kinetics are as follows:

L2

KS+S 2+(S 22 /K I )K 0+CL 2

rx = |a X


rs = ~

The balance for oxygen in the gas phase is:

The oxygen equilibrium relates the concentration in the gas phase to the liquidphase saturation concentration,

CL2* = MCG2The gas holdup fraction is,

VG = eVL

Proportional control of the feed rate, based on exit substrate concentration, canbe added with,

F = Fo + KpE

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

Program


Nomenclature

Symbols

CL Dissolved oxygen concentration mg/LCLS Saturation oxygen concentration nig/LE Control error g/LF Flow rate L/hG Gassing rate L/hKI Inhibition constant g/L


KLaKoKP

KSMOURrSVXYx/sYo8

H

Transfer coefficientOxygen saturation constantProportional control constantSaturation constantEquilibrium coefficientOxygen uptake rateReaction rateSubstrate concentrationReactor volumeBiomass concentrationYield coefficientMole fraction of oxygen in outlet gasgas/liquid volume ratioSpecific growth rate

1/hmg/LL/h/g/Lg/L-mg/hg/Lhg/LLg/Lg/g--1/h

Indices

012GILmOPSX

RefersRefersRefersRefersRefersRefersRefersRefersRefersRefersRefersRefers

to feedto inletto outletto gasto inhibitorto liquidto maximumto oxygento productto substrateto biomassto equilibrium

Exercises



Results


a '

8-

7.

6-

5-

4-

3-

0.

1 .

i

^. L

\

s

|N ^^™ %*%- N**

"•-""' — .*

) 1 2 3 4 5 6 7TIME

S2:1 (1)— .. CL2:1 (1)

S2:2 (4)

S2:3 (7)— — CL2:3(7)

82:4(10)— -CL2:4(10)

"™ — — -^ . ^

*-«*uT-ZZ_H.

8 9

r. — r

K

•0.1

• 0.09

-0.08

-0.07

-0.06

-0.05

•0.04

-0.03

-0.02

• 0.01

• 0)

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


.70

DJ

2 3 4 5

TIME

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


8.5.3 Batch Nitrification with Oxygen Transfer(NITRIF)

System

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

NH4+ + 02 -> N02-

O2 -> NO3-

The 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 thesecond step Y2 = 1.1 g O2/(g NO2-N).

Model

Neglecting the details of the biofilm diffusion, the apparent kinetics of thisbiofilm process can be approximately described with homogeneous kineticsthat follow a double Monod limitation:

Si CL* ±

= vm2

I + S| KOI +C

S2 CL

K2+S2

The batch balances are as follows:

ForNH4+ (Si):


For NO2~ (S2):

For NO3- (S3):

For oxygen (CL):dCL

—

~dT = - fi

dS2dt = ri - r2

dS3

~3T = r2

= -Yi r i-Y2r2 + KLa(CL*-CL)

Program


Nomenclature

Symbols

CL

CLSKKLarSiS2

S3

Vm

Indices

01,2,3Os

Dissolved oxygen concentrationSaturation oxygen concentrationSaturation constantsOxygen transfer coefficientReaction rateConcentration of NH4+ - NConcentration of NC>2~ - NConcentration of NO3~ - NMaximum degradation ratesYield coefficients

Refers to feedRefer to reaction stepsRefers to oxygenRefers to substrate

g/m3

g/m3

g/m3

1/hkg/m3

g/m3

g/m3

g/m3

g/m3hg/g


Exercises


Results

100

90

80

70

CO 60

$ 50

5) 40

30

20

10

0


+* -

X *'...X'-''

1.5TIME

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


8

2.5

Figure 3. NH4+ and dissolved oxygen in batch nitrification using three values of KLa from 20


8.5.4 Oxygen Uptake and Aeration Dynamics(OXDYN)

System

The aeration of a batch culture (with essentially constant biomass X) is stoppedand the dissolved oxygen (CL) is allowed to fall zero before re-aerating. Theslope of the CL curve is the oxygen uptake rate and it is approximated by theslope of the electrode response curve CE curve. The dynamics of the electrodeare 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 = q02X

Specific OUR,qo2m CL

°102 - KQ + CLOxygen balance,

dCL *T = KLa(CL*-CL) -OUR


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 viscousculture,

dCp CL - CpTF

and for the electrode lag,

dt =

Program

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

Nomenclature

Symbols

CKLa

Oxygen concentrationsTransfer coefficient

g/m3

1/h


KOOURQXT

Saturation constant for oxygenOxygen uptake rateSpecific oxygen uptake rateBiomass concentrationTime constants

g/m3

g/m3sg/kgskg/m3

Indices

EFLm02,0S*

Refers to electrodeRefers to liquid filmRefers to liquidRefers to maximumRefer to oxygenRefers to saturationRefers to saturation

Exercises

334 g Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Results

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

Run 2:10004 steps in 0.183 second:

8

70 80 90 100

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



8

10 20 30 40 50TIME

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

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

System

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


VQ.CQ

Hii

VG> CGO

Figure 1. Aerated tank with oxygen electrode.

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

Gas phase Liquid phase

Measurement

«CF

Electrodediffusion film

Electrode

Figure 2. Representation of the process dynamics.

Model

The model relationships include the mass balance equations for the gas andliquid phases and equations representing the measurement dynamics.

Oxygen BalancesThe oxygen balance for the well-mixed flowing gas phase is described by

VGdC

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


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

= KLa(CL*-CL)VLdt

The equilibrium oxygen concentration CL* is given by the combination ofHenry's law and the Ideal Gas Law equation where

r * RTrCL = — 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 exampleKLAFIT. It is also useful to solve the equations in dimensionless form.

Oxygen Electrode Dynamic ModelThe response of the usual membrane-covered electrodes can be described byan empirical second-order lag equation. This consists of two first-order lagequations to represent the diffusion of oxygen through the liquid film on thesurface of the electrode membrane and secondly the response of the membraneand electrolyte:

dCF_ CL-CF

dt TF

anddCg Cp ~Cg

dt " TE

Tp and TG are the time constants for the film and electrode lags, respectively. Innon- viscous water phases Tp can be expected to be very small, and the first lagequation can, in fact, be ignored.

Dimensionless model equationsDefining dimensionless variables as

C = CG C =LCGO CGO(RT/H) TG

the component balance equations then become


dt' V u' VG Hand

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- C F

dtand

dCE =C F - C B

dt' TE/^G

where Cp is the dimensionless diffusion film concentration.

CGO(RT/H)

and CE is the dimensionless electrode output

CE

CGO(RT/H)

As shown by Dang et al. (1977), solving the model equations by Laplacetransformation gives

1 ,RTVL „« = •=— + ( „ -. + 1) TO + TE + TFKLa H VG

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


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 variousexperimental parameters on the method, and is formulated in dimensionlessform. The same model, but with dimensions, is used in program KLAFIT andis particularly useful for determining Kâ in fitting experimental data of CEversus time. A set of experimental data in the text file KLADATA can be usedto experiment with the data fitting features of Madonna. All are on the CD-ROM.

The program ELECTFIT is used to determine the electrode time constant inthe first-order lag model. The experiment involves bringing CE to zero by firstpurging oxygen from the water with nitrogen and then subjecting the electrodeto a step change by plunging it into fully aerated water. The value of theelectrode time constant, TE can be obtained by fitting the model to the set ofexperimental data in the file, ELECTDATA. The value found in thisexperiment can then be used as a constant in KLAFIT.

Nomenclature

Symbols

Oxygen concentration g/m3


GHKLaRT/HtVa

Gas flow rate m3/sHenry coefficient Pa m3/molOxygen transfer coefficient 1/s(Gas constant)(Abs. temp.)/Henry coeff. -Time sReactor volume m3

Area above Cn'-time (s) curve sTime constant s

Indices

EFGL

Refers to electrodeRefers to filmRefers to gas phaseRefers to liquidPrime denotes dimensionless variables

Exercises


Results


0 20 40 60 80 100 120 140 160 180 200

TIME

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



TIME

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

References

Dang, N.D.P., Karrer, D.A. and Dunn, IJ. (1977).Oxygen Transfer Coefficientsby Dynamic Model Moment Analysis, Biotechnol. Bioeng. 19, 853.

Ruchti, G. Dunn, IJ. and Bourne J.R. (1981). Comparison of Dynamic OxygenElectrode Methods for the Measurement of KLa, Biotechnol. Bioeng., 13, 277.

8.5.6 Biofiltration Column for Removing TwoInhibitory Substrates (BIOFILTDYN)

System

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


G, C1T6

Tank6

Gas

Transfer

TriT6

L, S

rwi§fsp#KiS:Tl;niiiS?'Illllpll

G.Q1T5

II

II

TanksGas

Transfer«*

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

illliillllll

Tank4

Gas

Transfer tlwilffiIlllllll

ifankS

Gas

Transfer illlllIlllllll

Tank 2

Gas

Transfer^

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

^ iiiriSi;|tii |;i;||

ITank1

Gas

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 onlywith moist packing (biofilter). The work of Deshusses et al. (1995) investigatedthe removal of two ketones, methyl isobutyl ketone (MIBK) and methyl ethylketone (MEK), in such a biofilter. The kinetics of this multi-substrate system isespecially interesting since both substances exhibit mutual inhibitory effects ontheir rates of degradation.


Model

The model requires stagewise mass balances in the gas and liquid phases forboth components. Transfer takes place from the gas to liquid phases withreaction in the liquid phase. The symbols used for the concentrations ofsubstrates 1 and 2 in the nth tank are for the liquid phase Sixn and S2Tn andfor the gas phase CiTn and C2Tn- The reaction rates are RiTn and R2Tn and thetransfer rates are designated TriTn and Tr2Tn-

G, C1Tn, C2Tn L, S1Tn+1, S2Tn+1

G,C1Tn-1,C2Tn-1

nth

L, S1Tn, S2Tn

Figure 2. Single nm stage for the biofiltration countercurrent column.

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

Gas phase

Liquid phase

ÔL~(G(C2Tn.1-C2Tn)-Tr2Tn

-^ = - (L(SlTn+l - SlTn ) + TrlTn - rlTn VS )


1Q 1

d n = ~ (L(S2Tn+l - S2Tn ) + Tr2Tn - r2Tn VS )

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

Vs = (1 - EG - eL) Ê.

For the transfer terms

VcTr2Tn = KLa(S2EQn - S2Tn)~

For the gas-liquid equilibria:

For the reaction rate terms the following equations are used to describe themutual inhibition. Note that oxygen is assumed to be in excess.

For substrate 1 (MEK) in tank n:

rlTn = "vmi SlTn

For substrate 2 (MIBK) in tank n:Vm2 S2Tnr2Tn -

V | i i SlTn |1 +

Program

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


Nomenclature

Symbols

CGKiKLaKm

LM

N

rSTr

VC

VG

VL

vm

VS

ze

Concentration in gas phaseGas flow rateInhibition constantMass transfer coefficientMonod coefficientLiquid flowratePartition coefficient

Number of tanksReaction rateConcentration in liquid phaseTransfer rateVolume of columnVolume of gas phaseVolume of liquid phaseMaximum reaction velocityVolume of solid phaseLength or heightVolume fraction

kg/m3

m3/skg/m3

1/skg/m3

m3/s

kg/m3skg/m3

kg/sm3

m3

m3

kg/m3sm3

m

Indices

EqGinLMnTn12

Refers to equilibrium valueRefers to gasRefers to inletRefers to liquidRefers to maximumRefers to nth stageRefers to nth tankRefers to methyl ethyl ketone (MEK)Refers to methyl isobutyl ketone (MIBK)


Exercises

Reference

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


Results

0.0025-

^ 0.002-

. 0.0015-

0.001 -


1.2

0.8 ^

&0.6 -

,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+6TIME

Figure 3. Dynamic startup of the column.


• 0.1

5e-5 1e-4 1.5e-4 2e-4 2.5e-4 3e-4 3.5e-4 4e-4 4.5e-4 5e-4

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


8.5.7 Optical Sensing of Dissolved Oxygen inMicrotiter Plates (TITERDYN andTITERBIO)

System

Measurement of dissolved oxygen in microtiter plates is of potential interest forthe screening of oxygen-consuming enzymes (e.g., oxidases), aerobic cellactivities, and biological degradation of pollutants, and for toxicity tests. John etal. developed microtiter plates with the integrated optical sensing of dissolvedoxygen by immobilization of two fluorophores at the bottom of 96-wellpolystyrene microtiter plates. The oxygen-sensitive fluorophore responded todissolved oxygen concentration, whereas the oxygen-insensitive one served asan internal reference. As modelled in TITERDYN, oxygen transfer coefficientswere determined by a dynamic method in a commercial microtiter plate readerwith an integrated shaker. For this purpose, the dissolved oxygen was initiallydepleted by the addition of sodium dithionite and, by oxygen transfer from air,it increased again after complete oxidation of the dithionite. Availablecommercial readers have an intermittent operation. After a certain period ofshaking, the plate is moved around to measure dissolved oxygen concentration.During this period the plate moves more slowly and oxygen transfer rate isreduced. This may lead to oxygen depletion during the measurement process.It is essential to know the size of the errors that are introduced by thisintermittent procedure. This is evaluated by the simulation example TITERBIO.

Filter

Light

Microtiter plate

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


Model

Experiments involved measuring the oxygen uptake rate by removing theoxygen from the liquid using a chemical reaction (oxidation of sodiumdithionite). Oxygen is depleted immediately after the addition of dithionite.After the consumption of the dithionite the oxygen transfer increased theoxygen 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 andOUR 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

dCD^ 2 f

dt "" 3

In order to account for some time delay of the sensor a first order equation wasused

dCE^CL-CE

dt TE

The time constant TE was estimated to be about 1 s.

In further experiments this method was also applied to simulate a microbialcultivation in the wells of a microtiter plate. In this case the OUR value wastaken to be a constant value as measured in a larger fermentation vessel. KLavaried periodically simulating the high value during shaking and the lowervalue during the measurement period. The questions of interest are how muchthe measured OUR or KLa would differ from the actual one provided KLa orOUR were already known.


Program

Two separate programs are given on the CDROM: TITERDYN for the chemicaloxidation with re-aeration and TITERBIO for the biological oxidation and re-aeration dynamics during a cultivation in a microplate reader. For the programTITERDYN there is experimental data on the file TITERDYNDATA availableto allow fitting the value of KLa. In TITERBIO KLa during measurement is afraction of KLa during shaking and is determined by the parameter kmax. KLaduring measurement is defined as,

KLameasure=KLashaking*(kmax-l)/kmax.

In the original model settings, kmax has a value of 2. The larger kmax, thelarger 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 Dithionite concentration mMCL Oxygen concentration mMko Rate constant for dithionite reaction 1/min mMâ Transfer coefficient 1/sKQ Saturation constant for oxygen mMOUR Oxygen uptake rate mM/sQ Specific oxygen uptake rate mM/ sTE Time constant for measurement s

Indices

E Refers to electrodeD Refers to dithioniteL Refers to liquidS and * Refer to saturation


Exercises


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.


it • 'VtV'tj " *

Figure 3. Data fitting using TTTERDYN, yielding KLa=0.201, Calcfact=102 andDuration=0.48.

Reference

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


8.6 Controlled Reactors

8.6.1 Feedback Control of a Water Heater(TEMPCONT)

System

A simple feedback control system involving a stirred tank, temperaturemeasurement, controller and manipulated heater is shown in Fig. 1.

T0,F

IT*F,TR

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 orderlag

dt TQ

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


8.6 Controlled Reactors 355

u*sens .... *R ~ *sensdt 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 usingthe RANDOM function in Madonna, as explained in the HELP on the CD-ROM.

Nomenclature

Symbols

cp Specific heat kJ/(kg °C)f Frequency of oscillations 1/hF Flow rate m3/hKp Proportional control constant kJ/(h °C)Q Heat input kJ/hT Temperature °CV Reactor volume m3

8 Error °C

p Density kg/m3

TD Differential control constant hTI Integral control constant hTQ Time constant for heater h

Time constant for measurement h


Indices

CRsensset0

Refers to controllerRefers to reactorRefers to sensorRefers to setpointRefers to inlet or initial

Exercises


Results


7000

6000

3000

•1000

70 80 90 100

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


100.

80

60


-1.2e+4

1e+4

8000

6000

4000

2000

0

-2000

-4000

20 40 80 100 120

TIME

140 160 180 200

Figure 3. Response to a step change in the inlet temperature TO at 120 h. The controllerconstant 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 waterflowrate is controlled by a feedback controller. The rate of heat generation isrelated to rate of substrate uptake by a constant yield factor YQS. The coolingcoil is modelled as a well-mixed system.


Water

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

Model

The batch fermentation model is given by,

dXdf =

dS -Hdt = Y

^ = Ks+S

The energy balance equation for the reactor is,

U AdTR _ rQdt ~ VpCp (TR-Tc)


where,TQ = ^XYQ S /Y

For the well-mixed cooling coil, the energy balance equation is:

dTr F UA(Tcin - TC) + — (TR ~ Tc)

The controller is a proportional-integral type

F = F0 + KP8 +

£ =

Program

As seen on the CD-ROM and below, the control equations can be written interms of the error and its integral.

{CONTROL EQUATIONS FOR FLOWRATE}d/dt(EInt)=EF = F O + K P * E + ( K P / T I ) * E I n tlimit F> = 0E=TR-TSET

Nomenclature

Symbols

Cp Heat capacity kcal/(kg C)F Flow rate m3/hKp Controller constant m3/(h C)KS Saturation coefficient kg/m3

UA Reactor transfer-area constant kcal/kgr Rate of heat production and transfer kcal/(m3 h)V Reactor volume m3

8.6 Controlled Reactors

Indices

361

XYYQS

PT

se

H

Biomass concentrationYield coefficientHeat yield for substrateDensity

Time constant of controller

Substrate concentrationTemperature errorSpecific growth rate

kg/m3

kg/kgkcal/kgkg/m3

h

kg/m3

C1/h

CImQRSOPset

Refers to coolant and coolingRefers to integral controlRefers to maximumRefers to heatRefers to reactorRefers to substrateRefers to normal value and inlet valueRefers to proportionalRefers to setpoint (desired value)

Exercises


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 hthe reactor cools. Here Kp=0.6 and TI = 0.6.



'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 versusKp 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 ofcontinuous culture, the turbidostat may offer advantages for the investigation ofparticular problems. As shown in Fig. 1, the flow rate of the incoming substrateis controlled by the biomass concentration (more correctly, the turbidity) in thevessel. In practice, this control is usually on-off or proportional, but moresophisticated control would be simple to implement.


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_Xdt" = F(So-S) - -y~

dX F XdT = -IT

Considering product production with Luedeking-Piret kinetics:

dP FPdT = -"V" +

The turbidometer control is modelled by:

KPP ,.F = F0 + KPe + — fedts


8 = (X-XSet)

Program


Nomenclature

Symbols

ABF

FoKp

KS

PSVXYe

Growth-associated constantNongrowth-associated constantFlow rateNormal feed flow rateProportional controller constantSaturation constantProduct concentrationSubstrate concentrationReactor volumeBiomass concentrationYield coefficientError

Specific growth rate

Integral control time constant

1/hm3/hm3/hm6/h kgkg/m3

kg/m3

kg/m3

m3

kg/m3

kg/kgkg/m3

1/h

Indices

mPS and set0

Refers to maximumRefers to proportional controlRefers to setpointRefers to inlet stream


Exercises

Results


1-5

1

,3.5 c/>

5TIME

Figure 2. Startup and response of the controlled reactor.


3.5-

3.

2.5-

2 "

X"1.5.

1 •

n eVJ.O •

0-

•

,m\

r i T...— j u ' " —


_X:1

•"— F:'

... . ._..._. ._..._,,....

-5

4 5

•4

-3.5

-3 C^

-2.5

-2

-1.5

-1

4 6 8 10 12 14 16 18 20

TIME

Figure 3. Response of the controlled reactor to a step change in Xseto

8.6.4 Control of a Continuous Bioreactor withInhibitory Substrate (CONTCON)

System

The continuous fermenter is equipped with feedback control based on substratemeasurement, as shown in Fig. 1. This type of controlled fermenter has beenreferred to as an auxostat.


F,S,Feedpump

Controller

X,S

' Substratemeasurement

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 previousTURBCON model.

Kinetics:

Biomass balance,

or,

V — =dt

f

V X - F X

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


where Y is the yield factor for biomass from substrate. Also from this equationat steady state, since (j = D and dS/dt = 0, the steady-state cell concentration isgiven by

X = Y(S0-S)

A continuous inhibition culture will often lead to two possible steady states, asdefined by the steady-state condition (a = D, as shown in Fig. 2.

Control equations:£ = Sset- S

Kp rF = F0 + KP e + — I edt

•>

Program

When the system equations are solved dynamically, one of two distinct steady-state solutions is obtained, i.e., the reactor passes through an initial transient butthen ends up under steady-state conditions either at the stable operatingcondition, or at the washout condition, for which X=0. The initialconcentrations for the reactor will influence the final steady state obtained. A PIcontroller has been added to the program, and it can be used to control asubstrate setpoint below Smax. The controller can be turned on setting byKp>0. The control constants Kp, and the time delay tp can be adjusted by theuse of sliders to obtain the best results. Appropriate values of control constantsmight be found in the range 0.1 to 10 for Kp and 0.1 to 10 for TJ. Note thatthe 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


FKIKP

KS

SVXY

H(U)TITF

Flow rateInhibition constantController constantSaturation constantSubstrate concentrationVolumeBiomass concentrationYield coefficientSpecific growth rate coefficient

Controller time constantTime constant controller delay

m3/hkg/m3

kg/m3

m6/kghkg/m3

m3

kg/m3

kg/kg1/h

hh

Indices

0Im, max

Refers to inletRefers to initial valueRefers to maximum

Exercises

8.7 Diffusion Systems 371

Results

1.5


r0.3

0.25

0.2

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 ofContinuous 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 Diffusion Systems

8.7.1 Double Substrate Biofilm Reaction(BIOFILM)

System

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


Results

1.5


r0.3

0.25

0.2

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 ofContinuous 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 Diffusion Systems

8.7.1 Double Substrate Biofilm Reaction(BIOFILM)

System

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



the matrix is divided into segments, and the diffusion flux, j, from segment tosegment, is expressed in terms of the concentration difference driving force.

Solid Biocatalyst Matrix with N Segments

Liquid

j n-1 j n

n-1 ^_ n ^ n+1

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

Model

A multicomponent reaction whose reactants and products diffuse to and fromthe reaction site, for example into an immobilized enzyme or biofilm, can bedescribed by diffusion-reaction equations. The original problem in terms ofnon-linear partial differential equations, is described by a large number oftime-dependent differential-difference equations by discretizing the lengthvariable.

A component mass balance is written for each segment and for eachcomponent:

[Accumulation |^ rate J

Using Pick's law,

and dividing by A AZ,

(Diffusion^ _ (Diffusion^ ^Production rate^^ rate in J ^ rate out J v by reaction }

dSn' F = J n - l A - j n A + rS nAAZ

Sn-l ~ Snn-1 = AZ


dSn (Sn.i -2Sn + Sn+i)"dT = °s AZ2 + rs

Thus N dynamic equations are obtained for each component at each position,one for each element. The boundary conditions are for the above case dS/dZ =0 at Z = L and S = So at Z = 0. The equations for the first and last elementsmust be written accordingly.

The kinetics used here consider carbon-substrate inhibition and oxygenlimitation. Thus,

S O

At steady state, the overall reaction rate or consumption rate of substrate can becalculated from the gradient at the outer surface. To find the resulting changeof bulk concentration, the liquid phase can be coupled with suitable massbalances. For a well-mixed, continuous-flow, liquid the resulting balanceequation would be

dS0 F SQ-SISo) - a DS ^Z

For oxygen transferred from the gas phase:

dO0= KLa(Os-0Q) - a Do AZ

Program

As seen below, the program on the CD-ROM uses the array-vector form whichpermits plotting the values at time=Stoptime versus the distance index. Also thenumber of finite-difference elements N can be varied.

a/at (s[i . .N-i] > = D S * (s[i-u -) / ( Z * Z ) + R S [ i ]

a/at (o[i . .N-I] ) = D O * (o[i-u2 * 0 [ i ] + 0 [ i + l ] ) / ( Z * Z )


Nomenclature

Symbols

aACDFjKKLaO0SR,rSVvm

Yosz

Specific area perpendicular to the flux 1/mArea perpendicular to diffusion flux m2

Concentration g/m3

Diffusion coefficients m2/hVolumetric flow rate m3/hDiffusion flux g/ (m2 h)Saturation constants g/m3

Oxygen transfer coefficient 1/hDissolved 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

01 - 10IOSnFeed

Refers to bulk liquidRefer to sections 1-10Refers to inhibitionRefers to oxygenRefers to carbon sourceRefers to section nRefers to feed

Exercises



Results


.1

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

1.6-1

1.4-


1

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.



5

Figure 4. Dynamic response of oxygen and substrate mid-points caused by a step change inKLA (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, substrateS must diffuse through the porous lattice to the reaction site, and, as shown inFig. 1, it does so from both sides of the slab. Owing to the decreasingconcentration gradient within the solid, the overall rate is generally lower thanthat at the exterior surface. The magnitude of this gradient determines theeffectiveness of the catalyst.


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 ofUubstrate into the slab

Rate at which reactant \= îs consumed by reaction^

dX

A quasi-homogeneous form for the reaction term is assumed.The boundary conditions are given by:

At X = L:

At X = 0:S = S0 , P = P0

dX dX


The external concentration is known, and the concentration profile throughoutthe slab is symmetrical.

The reaction rate is expressed by the Michaelis-Menten equation withproduct inhibition

= kESKM(I+P/KI)+S

where k, KM and K\ are kinetic constants and E and P are the enzyme andproduct concentrations. At steady state, the rate of diffusion of substrate intothe slab is balanced by the rate of diffusion of product out of the slab.Assuming the simple stoichiometry S — > P

dS dP°SdX =-°PdX

which on integration givesDS

P = (So-S)

where P is assumed zero at the exterior surface.Defining dimensionless variables

S< = ' P' = and X' =gives

d^' L2R'

dX'2 DSS0 ~where,

kES'R' =

(KM/S0)(1 +(S0F/K I))and,

P = (1 - S')

with boundary conditions atX' = 1 S' = 1

X' = 0 dS'/dX' = 0

The catalyst effectiveness may be determined from

DS Sp (dSVdX')x=l^= L2R0


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' atX' = 0, the problem is of a split-boundary type and therefore requires a trialand error method of solution. Since the gradients are symmetrical, as shown inFig. 1, only one-half of the slab must be considered. Thus starting at the mid-point 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 withthe 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

DEKkLPRSX

Diffusion coefficient m2/hEnzyme concentration mol / m3

Kinetic constant kmol / m3

Reaction rate constant 1/hDistance from slab center to surface mProduct concentration kmol / m3

Reaction rate kmol /(m3 h)Substrate concentration kmol / m3

Length variable mEffectiveness factor

Indices

IM

Refers to inhibitionRefers to Michaelis-Menten


PS!

0GUESS

Refers to productRefers to substrateRefers to dimensionless variablesRefers to bulk concentrationRefers to assumed value

Exercises


Results

Run 5:1000 steps in 0seconds

0.2 0.3 0.5

X

0.6 0.7

Figure 2. Substrate profiles generated by manual slider iterations.

1.86

1.84

Q. 1.82

1.8

1.78

1.76

1.74

Run 5:1000 steps in 0seconds

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

X

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


8.7.3 Dynamic Porous Diffusion and Reaction(ENZDYN)

System

This example involves the same diffusion-reaction situation as the previousexample ENZSPLIT, except that here a dynamic solution is obtained by finitedifferencing. The substrate concentration profile in the porous biocatalyst isshown in Fig. 1.

Model

With complex kinetics a steady state split boundary problem of the type ofExample ENZSPLIT may not converge satisfactorily, and the problem may bereformulated in the more natural dynamical form. Expressed in dynamicterms, the model relations become,

3Sdt =

ap a2p + R

where at the center

dX~dX


Outside Center

S2

S3

S4

Figure 1. Finite-differencing for ENZDYN.

Using finite differencing techniques (refer to Sec. 6.2.1), these relations may beexpressed in semi-dimensionless form for any given element n by

dS'nW

dP'n•ar

_ 2iL(*-= L2 I

n+l - 2S'n + S'n.f R'n

AX'2

'Pf « OP1 _i_ P'r n+l — r n "*• r n+

AX'2

) Sl

A R'n' + S I

where

DSS0 1-S

L2Rn AX'

andS'n = Sn/SI; Fn = Pn/Si andAX' = AX/L

Sj is the external substrate concentration and AX is the length of the finitedifference element. Boundary conditions are given by the externalconcentrations Sj and PI and at the slab center by setting SN+I = SN and PN+I =PN.


Catalyst effectiveness may be determined according to two differentmethods:

(a) the effectiveness factor based on the ratio of actual rate to maximum rate(here for eight segments).

Ri + R9 + Ra + RA + RS + R* + R7 + Ra-J 2 3 4 5 6 7 8.Ro

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

D o I c)_ § OQ 1 ~ O 1

^"L^RO AX'

Where the rate at the bulk conditions is

kES0

+ P0/K I) + S0

The same constant values are used as in Example ENZSPLIT.

Program

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

Nomenclature

The nomenclature is the same as ENZSPLIT with additional symbols andindices:

Symbols

A X Increment of length mr|2 Effectiveness factor based on rates -


111K2

Effectiveness factor based on fluxSame as KM

Indices

Exercises

refers to segment n

References

Blanch, H.W., Dunn, I.J. (1973) Modelling and Simulation in BiochemicalEngineering 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 Aspectsof Reactions on Solid Supports, Ed. G.P. Stark, Academic Press.


Results

Run 1:1005 steps in 0.0833 seconds1

0.9

* 0.8

B?' 0.7

0.6

tf"tfO.3

W0.2</)

0.1

0

— 81:1,.. 82:1

.. 83:1

. 84:1

- 85:1_ _S6:1

, 87:1

_S8:1

30 50TIME

80

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

Run 1:1005 steps in 0.0833seconds

— P5:1• -- P6:1-ST-J7U._ -P8:1

, P3:1-P4:1

10 20 30 50TIME

60 70 80 90 100

Figure 3. Product concentrations in porous enzyme catalyst.


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 inclumps of animal cells by Keller (1991), but the modelling method also appliesto bioflocs and biofilms, which are subject to potential oxygen limitation.

SphereOxygen

Product

Substrate AV

N

I/I

RpRp

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 forgiven specific activities and bead diameters. As shown, the system is modelledby dividing the bead into shell-like segments of equal thickness. The problemis equivalent to dividing a rectangular solid into segments, except that here thevolumes and areas are a function of the radial position. Thus each shell has avolume of 4/3 n (rn

3 - rn_i3). The outside area of the nth shell segment is 4n rn2

and its inside area is 4n rn_i2.


in

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

and inside radius rn_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

jn-l =

Ar

Sn~Sn-lAr

3 D

Substitution gives

dSn

dt "'

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

. , 2 ™ 4=J -4 3dS . , 2 ™ 4 3= -

Since ri = Ar, this becomes


dSl _ 3DS /q o N , pIf- ^S2-Sl)+Rsl

r

The reaction rate is expressed by a Monod-type equation

SRSn = - (

where X is the biomass concentration (cell number/m3) in the bead, OUR is thespecific 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 ofMadonna, numbered from the outside to the center. The effectiveness factor,expressing the ratio of the reaction rate to its maximum, is calculated in theprogram, part of which is shown below. The number of elements N is calledArray 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 Diffusion coefficient m2/sKS Saturation constant in Monod equation mol/m3

OURmax Maximum specific oxygen uptake rate mol/cell sr Radius at any position mRp Outside radius of bead m


RS Reaction rate in the Monod equation mol/s m3

S Oxygen substrate concentration mol/m3

X Biomass concentration cells/m3

Ar (Deltar) Increment length, r/N m

Indices

12nPS

Refers to segment 1Refers to segment 2Refers to segment nRefers to particleRefers to substrate

Exercises


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) asthese radial profiles show.

Reference

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


8.7.5 Immobilized Biofilm in a NitrificationColumn System (NITBEDFILM)

Nitrification is the sequential oxidation of NH4+ to NO2~ and NO3" whichproceeds according to the following reaction sequence:

NH4+ +1 O2 -> NO2- + H2O +2H+

NO2- + \ O2 -> NO3-

The overall reaction is thus

NH4+ + 202 -> N03-+H20

Both steps are influenced by dissolved oxygen and the corresponding substrateconcentration and are catalyzed by two different organism species. Since theirgrowth rates are very low, nitrification as a wastewater treatment process benefitsgreatly from biomass retention.In this example, a biofilm column reactor for nitrification is modelled as threetanks-in-series with a recycle loop (Fig. 1). Oxygen is supplied only in anexternal contactor and circulates to the reaction column in dissolved form.This is similar to the example NITBED. However, in this case the reaction takesplace within an immobilized biofilm, similar to the single tank exampleBIOFILM.


O3, Si3

Fluidizedbed

SiA

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

Model

The column reactor is assumed to be described by three tanks. The modelbalance equations for the liquid phase are developed by considering both theindividual tank stages and the absorber. Component balances are required forall 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 distanceand time. Therefore, a descretization of the length variable is required asdeveloped for the example BIOFILM.


tTo tank n+1

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 fourregions care must be taken with the nomenclature. The nitrogen compounds arereferred 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 fourcomponents. [0] refers to the liquid phase in contact with the solid. Transfer tothe 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, fora 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 theflows to and from the tanks 1, 2 and 3, each with volume V, and the absorptiontank with volume VA-

The fluidised bed reactor is modelled by considering the component balancesfor the three nitrogen components (i) and also for dissolved oxygen. For eachstage n, the liquid phase component balance equations have the form

dSin[0] =dt


-J0[0]

For the absorption tank, the balance for the nitrogen containing componentsinclude the input and output of the additional feed and effluent streams, giving

dsiA-

/Q Q \_(SiF-SiA)

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

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

= vml Slni °ni

Kl+Slni KO i+Oni

rs2n = Vm2 S2ni

K2+S2ni K02+°ni

The oxygen uptake rate is related to the above reaction rates by means of theconstant yield coefficients, YI and Y2, according to

i -rS2niY2

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)


Nomenclature

Symbols

AFFRKLaKKIK2LN0Os and O*OURrSVVAvmY

Indices

1,2,31,2,3AF

Specific area of filmFeed and effluent flow rateRecycle flow rateTransfer coefficientSaturation constantsSaturation constant for ammoniaSaturation constant for ammoniaBiofilm thicknessNumber of biofilm segmentsDissolved oxygen concentrationOxygen solubility, saturation cone.Oxygen uptake rateReaction rate per volume of biofilmSubstrate concentrationVolume of one reactor stageVolume of absorber tankMaximum velocityYield coefficient

Refer to ammonia, nitrite and nitrate,Refer to stage numbersRefers to absorption tankRefers to feed

1/mm3/hm3/hhg/m3

g/m3

g/m3

m-g/m3

g/m3

g/ m3 hg/ m3 hg/m3

m3

m3

mg/L hmg/mg

resp.

jn[I] Refers to substrate j in stage n in segment imOl and O2S1,S2S and *

Refers to maximumRefer to oxygen in first and second reactionsRefer to substrates ammonia and nitriteRefer to saturation value for oxygen


Exercises

Program



Results

I60'j,g 50-

£40-

520.

55 io-

o-


5

'4.5*C

4 3A

.3.5 g

13 a?3

•2.5

.2

1.5

S

1 5

50TIME

60 70 80 90 100

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

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 inthe first tank.


8.8 Multi-Organism Systems

8.8.1 Two Bacteria with Opposite SubstratePreferences (COMMENSA)

System

Considered here (Fig. 1) is the batch growth of a two-organism culture on twosubstrates, in which both species can utilize both substrates (Kim et al., 1988),but where the organisms have opposing substrate preferences. The twobacterial species involved are: Klebsiella oxytoca (XA) and Pseudomonasaeruginosa (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 ofthe rate of growth on glucose plus the rate on citrate.

- The specific growth rate on each substrate depends on the concentrationlevel 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 agrowth-associated product.


8.8 Multi-Organism Systems 401

- The total rate of substrate consumption is the sum of the rates ofconsumption by each organism plus the rate of consumption of substrate forthe 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 obtainedusing a steady state oxygen balance.

Model

The growth rates, jny, for each organism are the sums of the growth rates onglucose and citrate. The subscripts i and j have the following meaning: i refersto the organisms (K. oxytoca - A and P. aeruginosa = B) and j refers to thesubstrate (glucose = Y and citrate = Z). The levels of the key enzymes aredenoted by E.

The biomass balances for the batch system are

dXA— = (MAY + MAZ) XA

dXBI

The specific growth rate equations for the two organisms on each substrate aregiven by:

A*maxAYSYEAYKSAY+SY

MmaxAZSZEAZKSAZ + SZ

A*maxBZSZEBZ I KIM-BZ - — - - — "FTKSBZ+SZ V K I +

The substrate balances are given by:


dSY ! a

— =YSAY YI ) YSBY

dSz

dt - - YSBZ XB " YSAZ XA

Inhibitor (I) production is growth associated to organism A, and its decay isproportional to the cell concentration. The balance for the inhibitor is

dlgj- = oc|uAYXA-pXA

The balances for the key enzymes, which control the growth on secondarysubstrates are given by:

Sz KRAZ- - -- *^ -- kpAZEAZT-T " rt\iu f\z^

anddEBY SY KRBY- " - - Y KbY

where the consecutive terms in the above equations represent induction,repression, and dilution due to cell division, respectively. Here the enzymelevels E are normalized with respect to the maximum levels (See reference).Because growth on the preferred substrates is constitutive, EAy and EBZ areequal to 1 .

The oxygen uptake rate (OUR), carbon dioxide evolution rate (CER) anddissolved oxygen tension (DOT) are given by:

OUR =OAY OAZ

OBY OBZ

CER =YCAY YCAZ

CBY CBZ


DOT = 100 1-

The cell mass fractions are given by:

OUR

KLaC0

F A = -XA + XB

FB = 1-FA

Program


Nomenclature

Symbols

CERDOTFIKKLaC0*OURSXYEmaP

Carbon dioxide evolution rateDissolved oxygen tensionCell mass fractionsInhibitor concentrationSaturation and inhibitions constantsOxygen transfer rateOxygen uptake rate (normalized)Substrate concentrationBiomass concentrationsYield coefficientsLevel of key enzymeSpecific maintenance ratesYield constant for inhibitorInhibitor consumption rate constantSpecific growth rate

kg/m3 h

kg/m3

kg/m3

kg/m3 hkg/m3 hkg/m3

kg/m3

kg/kg

kg/kg hkg/kgkg/kg h1/h


Indices

ABCIMOPRSYZ

Refers to K. oxytocaRefers to P. aeruginosaRefers to carbon dioxideRefers to inhibitorRefers to maximumRefers to oxygenRefers to dilution due to cell divisionRefers to repressionRefers to substrateRefers to glucoseRefers to citrate

Exercises


Results

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


1 •

0.9.

0.8

0.7

0.6

es°-5

0.4

0.3

0.2

0.1

04 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 MixedCulture Growth of Two Bacterial Populations with Opposite SubstratePreferences. Biotechnol. Bioeng., 31, 144-159.

This example was developed from the original paper by J. Lang, ETH-Zurich.


8.8.2 Competitive Assimilation and Commensalism(COMPASM)

System

The interactions between two microbial species (Ma and Mb) in a mixedcontinuous culture are considered (Miura et al., 1980). The populationdynamics of the two microbes, is described by competitive assimilation ofsubstrate Si and commensalism, with the participation of growth factor Ga thatis excreted by microbe Ma and required by microbe Mb for growth. Mb alsoconsumes a second substrate 82 from the medium. These interactions arerepresented 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 asfollows:

Dilution rate:


Organism Ma:dXa

j- = ftia-D)Xa

Organism Mb:dXb-gj- = (jib - D) Xb

Substrates S\ and 82:

•ar = - "a

dS2 M-b Xb

-ar = - ~YT" + D (S2°"S2)

The yields for organism Mb on the two substrates are assumed here, forsimplicity, to have the same values, Yb.

The growth factor balance is

dGa Jib Xba "P 11 Y T-X f*

The mass balance for the growth factor, Ga, is formulated by assuming aformation rate, Pa |ia Xa, and consumption rate, (|Lib Xb)/Ybg. Here Xa and Xb

are the concentrations of microorganisms Ma and Mb, respectively. Theconstants Pa and Ybg are the biological yield constants for the formation andconsumption of Ga, respectively. The specific growth rates of microbes Ma andMb are expressed by:

Organism Ma:Si

K S a +SiOrganism Mb:

Si Ga

82KSb2 + S2 Kg + Ga


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 saturationconstant of Mb for growth factor Ga. Setting \imb2 = 0 corresponds to theconsumption 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 inthe vicinity of the steady state solution and determining the characteristiceigenvalues of the resultant matrix. The following relationship for co-existenceof the two microbes can be derived for the case of a single substrate.

Sio > KSa D Oima - D) +^smmaD(KSbl-KSa)

YaFaCmmbKSa -mmaKSbl)

Also, a critical dilution rate, where the maximum dilution rates of the twoorganisms cross-over can be written:

. _Cnt "Sa Sbl

Four particular cases depending on the values of the maximum specific growthrate and saturation constants of both microbes can be simulated for the singlesubstrate case (|imb2 = 0).

1- M-ma > l^mbli Ksa > K$bi: Coexistence below a certain value of D

2. |ima > Mmbi; Ksa < KSbi: No coexistence range3. |ima < |imbi; Ksa > Ksbi: Coexistence with wider range of stable focus

K$a > Coexistence at higher D and wider range of

Program



Nomenclature

Symbols

DFGKPSVXY

Dilution rateFeed rateGrowth factor concentrationSaturation constantsYield constantSubstrate concentrationReactor volumeBiomass concentrationYield coefficientSpecific growth rate

1/hm3/hg/m3

g/m3

g/m3

m3

g/m3

g/g1/h

Indices

012abblb2gm

Refers to feedRefers to substrate 1Refers to substrate 2Refers to organism aRefers to organism bRefers to organism b growing on substrate 1Refers to organism b growing on substrate 2Refers to growth factorRefers to maximum

Exercises


Results

A simulation is given in Fig. 2 for the parameters as given in the program withfeed flow rate F = 0.24 m3/h, and the feed concentrations SIQ = 500 mg/L and820 = 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.


300

250

200

150 CD

100

300

Figure 2. Competition and commensalism of two organisms (F = 0.24 m3/h, S10 = 500 mg/Land S2o = 1500 mg/L), showing the biomass concentrations.


Reference

Miura, Y., Tanaka, H., Okazaki, M. (1980). Stability Analysis of Commensaland Mutual Relations with Competitive Assimilation in Continuous MixedCulture. 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 produceimportant biochemicals by inserting a small portion of extra-chromosomalDNA (on plasmids) into the cell. These plasmids carry the genetic instructionsto produce the desired product and tend to lose their engineered propertiesduring cell division because of non-uniform plasmid distribution. Theengineered or recombinant strain usually grows more slowly than the wild-type,nonplasmid-bearing strain, so that engineered strain may be lost throughextinction. By exploiting the difference in the adaptation times of wild andengineered strains, a possibility exists of maintaining a plasmid-bearingpopulation in continuous culture by cycling the substrate feed concentration orthe dilution rate. This dynamic problem is adapted from Stephens andLyberatos (1987 and 1988), based on the concept of plasmid stability fromAiba and Imanaka (1981).

The Monod model assumes a balanced growth in which all cellularcomponents change in the same proportion at all times but does not accountfor dynamic effects. Dynamic first order lag relations are added to account forthe response of the organisms to rapid changes in the medium. It is assumedthat the time constants for the two strains are different and that the responses tochanging concentrations are therefore different. As a consequence, the strainwith the smallest time constant has the advantage when the concentration of thelimiting substrate is oscillating. The simulation model based on Fig. 1 is usedto predict the stability in the competition between wild (Xi) and engineered(X2) strains in continuous culture.


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

2 - D X2

dx2

dsg^ = D (SQ - s) - (Lli (s) Xi - |I2 (s) X2

where the time delayed specific growth rates are,

"SIT

Z2)


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 - Z I )

where p = ai/a2, and cxi and a2 are the adaptability factors or inverse timeconstants. The effect of (3 is to delay the substrate for growth in the wild andengineered organisms according to their first order time constants. For p > 1 thewild 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 isassumed, that (li^ > H2m .

The probability factor p represents the probability (or fraction) ofconversion to the wild strain during growth of the engineered strain. Thus thegrowth 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 bythe Madonna Conditional Operator, using the parameter MARK (ratio of thetime during which the function has the value 1 to the time of the completeperiod) and PER (period).

{Square wave feeding generated by conditionaloperator}SO=IF(Time/PER-INT(Time/PER))<=MARK THEN SI ELSE 0


Nomenclature

Symbols

DKMARKPERRSsXX

YPrtX

za

P

H

Indices

012dimreal

Dilution rateSaturation constantRatio controlling step functionTime periodBiomass ratioSubstrate concentrationSubstrate concentration, dimensionlessBiomass concentrationBiomass concentration, dimensionlessYield coefficientProbability factorGrowth rateDimensionless timeBiomass concentration, dimensionless

1/hkg/m3

h

kg/m3

kg/m3

kg/kg

kg/m3 h

—Delayed substrate concentration, dimensionless -Adaptability factorsRatio of adaptability factorsSpecific growth rate

Refers to inlet streamRefers to wild strainRefers to engineered strainRefers to dimensionlessRefers to 1 or 2Refers to maximumRefers to real time

1/h-

1/h


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 wavefeed concentrations, showing the engineered organism X2 being washed out. Asimilar situation for sine wave feeding is given in Fig. 3.



100 150

TIME200 250

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


100 150TIME

200

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.


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

System

Predator-Prey Population Dynamics(MIXPOP)

The growth of a predator-prey mixed culture in a chemostat can be describedwith a reaction kinetics formulation. In this growth process, the dissolvedsubstrate S is consumed by organism Xi (the mouse), while species X2 (themonster) 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 thecorresponding kinetics. The variables are given in Fig. 2.


S0 D

II

1Sl ,X - | ,X2

^*

Figure 2. Chemostat predator-prey reactor.

Substrate balance,dS

= D(So-Si) -

Species 1 (prey) balance,

^2X2

Species 2 (predator) balance,

dX2

where D is the dilution rate,

~"v

The kinetics are given by Monod relations,

- DX2

X

Program



Nomenclature

Symbols

DFKSVXY

Dilution rateFlow rateSaturation rate constantSubstrate concentrationReactor volumeBiomass concentrationYield constantsSpecific growth rate

1/hm3/hkg/m3

kg/m3

m3

kg/m3

kg/kg1/h

Indices

012m

Refers to feed streamRefers to preyRefers to predatorRefers to maximum

Exercises


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 inFig. 4 and also by the phase plane plot of Fig. 5.


•4

3.5

3

2.5«

2

1.5

1

0.550 100 150 200 250 300 350 400 450 500

TIME

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



6

450

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


Figure 5. Phase plane plot of oscillations.


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 twodifferent 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 atany value of S. For this case, in chemostat cultures with dilution rate DI, afteran 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, andonly organism B will remain in the reactor.

The situation in the right of Fig. 1 shows (IB crossing (IA- The point ofintersection can be found easily by simple algebra where:


Wnter =

Solving for S at the intersection,

Sinter =

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

Model

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

d*A „ ,

dXBjp = 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 andB competing for a single substrate.

Program



Nomenclature

Symbols

DFKS

SVXY

Dilution rateFlow rateSaturation constantsSubstrate concentrationsReactor volumeBiomass concentrationsYield coefficientSpecific growth rates

1/hm3/hkg/m3

kg/m3

m3

kg/m3

kg/kg1/h

Indices

ABM0inter

Refers to organism ARefers to organism BRefers to maximumRefers to inlet streamRefers to the intersection of the ju versus S curves

Exercises


Results


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 foran Inhibitory Substrate in a Biofilm(FILMPOP)

System

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


Results


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 foran Inhibitory Substrate in a Biofilm(FILMPOP)

System

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



Sf S0

Wall

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

Model

Fig. 1 illustrates an idealized flat biofilm with a uniform thickness Lf (m). Thebiofilm is divided into N segments for simulation purposes and each has athickness of AZ = Lf/N (m). Wastewater containing the substrate is fed to thereactor at a constant feed rate and a concentration Sf (mg/L). The bulk liquid inthe reactor is mixed throughout the tank and the substrate diffuses into thebiofilm. The substrate is transported from the bulk liquid having aconcentration S[0] (mg/L) to the surface of the biofilm having a concentrationS[l] (mg/L). A diffusion layer of a thickness Lj (m) is used to represent theexternal mass transport resistance.

Using the same approach as in the example BIOFILM, the mass balances inthe bulk liquid for the substrate and microorganisms A and B with a continuousflow are simply described as following:


dS[0] Wll _, „ S[0]-S[1]dt v f ' " DZ

m [01X fC

YA

•] mB[0]XB[0]

YB

, XA[0]-XA[1] , }- -DX A|0] - aDXA — (mALO] - bA JX A[0]

/\/1

where S is substrate concentration (mg/L). XA and XB are biomass ofmicroorganisms A and B (mg/L), respectively. Each number in the bracketsrefers 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 Aand 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 theflux (nr1), and time (day), respectively.

Reactions within the biofilm are described by diffusion reaction equations. Themass balances of the surface segment are described as following:

s[0]-sm=dt S LjAZ S AZ2

dXA[l] XA[0]-XA[1] XA[1]-XA[2]— — ~ - - -- D - —XAat LZXZ,

dXB[l] XB[0]-XB[1] n XB[1]-XB[2]=: — \j v"-p LJ YT3 ^

dt XB L,AZ XB AZ2

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


dS[i] „ 5dt DS

>[i-l]-2S[i]+S[i+]

AZ2L] MA[i]XA[i]

YA

/iB[i]XB[i]

YB

dXA[i] XA[i-l]-2XA[i] + XA[i—£t—~DXA —2

dXB[i] XB[i-l]-2XB[i] + XB[i—^t—-DXB —2

The mass balances of the boundary segment on the support wall are describedby the following equations:

dS[N] = p S[N-1]-S[N]

dt S AZ2

The "diffusion" coefficients of microorganisms, DXA and DXB» representdisplacement by cell division and by shearing off at the film boundarycontacting the bulk liquid.

Growth Kinetics Of Microorganisms

The inhibitory influence of high substrate concentration was described by theHaldane kinetics. The two types of microorganisms compete for substrate, butin the biofilm they also have to compete for the limited space available.Therefore, growth of the microorganisms was described by the Haldane kineticswith a spatial limitation term which was originally proposed as cell inhibitionkinetics by Han and Levenspiel (1988).


K

.. m_

[i]=

IA

KIB

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 maximumcapacity of total biomass of microorganisms A and B in a segment.

The formulation of the spatial limitation term used here is the most simpleone possible with non-restricted growth at zero biomass concentration and zerogrowth at maximal biomass concentration Xm.

Applying the above model it was found (Soda et al., 1999) that thequalitative behavior of the biofilm reactor is characterized by 5 regions,depending on the operating conditions, the substrate concentration in the feedand the dilution rate. In region I, both microorganisms are washed out of thebiofilm reactor. In region II, microorganism B is washed out, and in region III,microorganism A is washed out of the biofilm. In region IV, bothmicroorganisms coexist with one another. In region V, both microorganismscoexist with a sustained oscillatory behavior. Convergence to regions I-Vdepends on the initial conditions. In regions II-V, washout of either or bothmicroorganisms is also observed when the initial conditions are too far away.

Nomenclature

Symbols

a specific area perpendicular to the flux,related to bulk liquid volume m"

b biomass decay rate day"D dilution rate day"DS diffusion coefficient of substrate m /dayDx diffusion coefficient of microorganisms m /dayK! inhibition constant mg/LKS saturation constant mg/L


LILfNS[iSf

YAZ

thickness of diffusion layer mthickness of biofilm mnumber of segments in biofilmsubstrate concentration in element i mg/Lsubstrate concentration in feed mg/Lbiomass mg/Lspatial capacity of total biomassof microorganisms A and B in a segment mg/Lyield coefficientthickness of each segment m

maximum specific growth rate day "

Indices

AB

refers to microorganism Arefers to microorganism B

Numbers in brackets

01-5

refers to bulk liquidrefer to segments 1-5

Exercises


References

Soda, S, Heinzle, E,, Fujita, M. (1999) "Modeling and simulation ofcompetition of two microorganisms for a single inhibitory substrate in abiofilm 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] /YBd/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 . .nslabs-1] ) =DSB* ( X B [ i - l ] -2 * X B [ i ] ) / ( Z * Z ) + ( U B [ i ] -kdB) * X B [ i ]


Results


2 40-£ °mX

Ix 20

/ \

/\

i\

/-\,

g™™.

^^— '

---.

• '

'•••-•-.•

sr °"

XAmid:1. — . XBmid:1

Smid:1

•0.035

-0.03

-0.025

•0.02 £(A

-0.015

-0.01

-0.005

10 20 30 40 50 60 70 80 90 100TIME

Figure 2. Results corresponding to Case 2: UmB=0.4, KSB=0.1, KIB=10.


0.05

0.045

0.04

0.035

0.03

0.025 £

10 20 30 40 50 60 70 80 90 100

0.02

0.015

0.01

0.005

(A

Figure 3. Results corresponding to Case 4: UmB=1.8, KSB=0.01, KIB=0.01.


8.8.7 Model for Anaerobic Reactor ActivityMeasurement (ANAEMEAS)

System

As already discussed in Chapter 3, anaerobic processes can be described bymulti-substrate, multi-organism kinetics. As shown in Table 1, organic acidsare 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 -> nC6Hi2O6

Step 2: Acid production (example glucose)C6Hi2O6 -> CH3(CH2)2COOH + 2 H2 + 2 CO2

C6Hi2O6 + 2 H2 -> 2 CH3CH2 COOH + 2 H2O2H2O H» 2CH3COOH + 4 H2 + 2 CO2

Step 3: Acetic acid productionCH3(CH2)2COOH + 2 H2O -> 2 CH3COOH + 2 H2

CH3CH2COOH + 2 H20 -> CH3COOH + 3 H2 + CO2

Step 4: Methane productionCH3COOH -> CH4 + CO2

4 H2 + C02 -> CH4 + 2 H20

In order to evaluate the activity of an anaerobic reactor and to evaluate thecorrectness of the reactions in Table 1, an off-line measurement system hasbeen designed. This involves a small batch reactor coupled to a massspectrometer. A sample of biomass with medium is taken from the largercontinuous anaerobic reactor and put into the small batch reactor. Dissolvedgases are stripped by helium, all gas bubbles are removed and substrate isadded to start the reaction. The accumulation of organic acids, CO2, H2 andCtLj. is measured. pH is adjusted according to total acid concentration andbuffer capacity. Biomass concentration is constant throughout the experiment.


The model was developed to aid the design of the measurement system and inthe interpretation of the data.

Model

A five organism model with lumped hydrolysis and acid-generating bacteriawas established. Substrate, intermediate and product balances of the batchreactor 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 1as follows:

and the specific growth rates take the Monod form,

M-imax SiMi = 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.


Reaction Equilibrium

In the acetogenic step (Step 3 reaction in Table 1), acetic acid, hydrogen andcarbon dioxide are produced from propionic and butyric acid. Thethermodynamic equilibria for these reactions are incorporated by estimatingthe 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

KPro - 3 CEPro

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 rightside when the equilibrium condition was reached. Using the actualconcentrations, the parameters KBU* and Kpro* were estimated.

„ * CAc2CH22CH+

KBU ~~ CBu

_ 4 CAc2 CH2

3 CCQ2

- 3CPro

The ratio of these values to the true equilibrium constant, K*BU/KBU, andK*pro/Kpro will be greater than unity if the equilibrium has not yet beenreached. Using these ratios with the empirical S-shaped curve of Fig. 1, thefactor FEQ was determined and was used to modify the growth rates. Thissomewhat 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


the equilibrium is reached, but there is no reverse reaction when concentrationsof acetate, hydrogen, CO2 and H+ exceed the equilibrium values caused byother reactions. The reaction proceeds irreversibly further away from theequilibrium (K*/K > 2).

1.2

1.0-

0.8.

a 0.6-uju_

0.4-

0.2.

0.

-0.2-

K*/K

Figure 1. Equilibrium factors (FEQ) to slow the growth rates near equilibrium.

The kinetics of biomass growth butyric acid, and propionic acid were modifiedby 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 bewritten to account for the acid-base dissociation buffer effects. The ion balancerepresents an implicit non-linear equation in the dynamic model and must besolved 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 numericalsolution is not always trivial using conventional methods for non-linear


algebraic equations (e.g., Newton-Raphson, and Regula falsi). Fortunately thistype of equation can be handled conveniently by the root finding feature ofBerkeleyMadonna, as shown in the program on the CD-ROM and below.

If base is added to control pH, an additional balance for cations of strongbases (K+, Na+, ...) and anions of strong acids (Cl% SC>42~, ....) becomesnecessary as follows:

dCz= Ftitr Qtitr

Program

The program nomenclature is rather extensive and is defined within theprogram. 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-4ROOTS 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 >= 0LIMIT CHPLUS <= 1000pH=-loglO(chplus)+3

Nomenclature

The nomenclature of the program is partially defined within the program.

Symbols

C Concentration C-mol/m3

Cons Consumption rate C-mol /m3hF Stoichiometric coefficients (-)FEQ Equilibrium factor (-)Ftitr Titration flow rate m3/h


KD

KsSPProXYP/SYX/S1

Dissociation constantInhibition constantSaturation constantSubstrate concentrationProduct ConcentrationProduction rateBiomass concentrationYield coefficient, product from substrateYield coefficient, biomass from substrateSpecific rate of biomass synthesis

mol /m3

C-mol /m3

C-mol/m3

C-mol /m3

C-mol /m3

C-mol /m3 hC-mol /m3

C-mol /C-molC-mol /C-mol1/h

Indices

AcBuBufdHyiinMoPrtitrTotZ

Refers to acetic acidRefers to butanediolRefers to bufferRefers to death rateRefers to hydrogen gasRefers to reaction iRefers to initialRefers to whey substrateRefers to propionic acidRefers to titrationRefers to totalRefers 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 wheyis almost instantaneously consumed. Hy reaches a maximum very soon and isthen quickly reduced to almost zero. CH4 is produced with varying rates. CC>2reaches a maximum, which is partly caused by pH changes and byconsumption by hydrogen-consuming organisms. The peaks in the CC>2 curveoriginate from numerical inaccuracies in the stiff system. In the second graph,Fig. 3, the total concentration of volatile acids acetate (Ac), propionate (Pr) andbutyrate (Bu) are given. The thermodynamic inhibition of acetogenesis isclearly seen in the early phase of the experiment. Ac reaches a maximum muchlater than Pr and Bu, since it is produced from these two acids. In the thirdgraph, Fig. 4, the pH versus time profile is given, exhibiting an early decrease,followed by almost constant pH during the rest of the simulation.


0.06

0.05

0.04

s' 0.03

0.02

0.01


-0.3

-0.25

-0.2

0.15

-0.1

0.05

0.005 0.025 0.03

Figure 2. Dynamic profiles of substrate whey (Mo), CH4 (CH), dissolved CO2 (CO) anddissolved hydrogen (Hy). Zoomed to show the early period.


0.3

0.25

r0.2

0.15 -

0.35

-0.05

Figure 3. Total concentration of volatile acids acetate (Ac), propionate (Pr, lower curve) andbutyrate (Bu). The whey (Mo) peak is hardly visible at T = 0.


6.18

6.16

6.14

6.12

6.1

: 6.08

6.06

6.04

6.02

6

5.98

..'


-0.2

• 0.18

• 0.16

.0.14

.0.12

•0.1 £

• 0.08

•0.06

•0.04

•0.02

• 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 4. Variation of pH and acetate with time.

Exercise

0.4

References

Heinzle, E., Dunn, I.J. and Ryhiner, G. (1993) "Modelling and Control forAnaerobic Wastewater Treatment." Adv. Biochem. Eng. 48, 79-114.

Yamada, N., Heinzle, E. and Dunn, I.J. (1991) "Kinetic Studies onMethanogenic Cultures Using Mass Spectrometry." in: BiochemicalEngineering - Stuttgart (Eds. Reuss, M., Chmiel, H., Gilles).


Ryhiner, G. Heinzle, E., Dunn, I.J. (1992) "Modelling and Simulation ofAnaerobic Waste water Treatment and Its Application to Control Design: CaseWhey," 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 weremodelled 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 asfollows:

dR

dE= - D E + [ rGE(R,G,E) + rSE(S) - rEX(E) ] R

- = D (SF - S) - [ rSE(S) + rSG(S,E) ] R


dG= - D G + [ rSG(S,E) - rGE(R,G,E) ] R

dT= - D T + [ rTi(R,G,E) - rT2(R,G,E) ] R

The species in parenthesis indicate the dependencies of the rates. The kineticexpressions used in the balance equations are as follows:

rGEmE

/KGX\nl — /

SKs + S

ME)TEX = -

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 theenzyme activity (T), had to be chosen during simulations. This model leads tooscillations whose existence and dependency on operating conditionsqualitatively agree with experimental results. Also the directions in the phaseplane plot agree with the experiments.


Nomenclature

Symbols

DEGKnRrSsigTX

Dilution rate 1/hEthanol concentration kg/m3

Storage material kg/m3

Growth rate constants kg/m3

Empirical exponent in rate model -Residual biomass without G g/m3

Growth rates kg/m3 hGlucose concentration kg/m3

Rate constants. Example: sigGEm = TGEM variousEnzyme concentration g/m3


Specific growth rate 1/h

Indices

EGmSTX

Refers to ethanolRefers to storage materialRefers to maximumRefers to glucoseRefers to enzymeRefers to biomass

Exercise


Results

The influence of dilution rate is given below in plots from simulations: Fig. 2with D = 0.05 and Fig. 3 with D = 0.1. In Fig. 4 the phase plane from the runof Fig. 2 is shown.

35-

30-

25-

20-

15-

10

5-

0-M..JL.JL...

20 40 60


-0.45

0.4

0.35

0.3

0.25(

0.2

-0.15

0.1

0.05

-0

100 120 140 160 180 200TIME

Figure 2. Biomass and substrate oscillations for D = 0.05.

20


.2

40 60 80 100TIME

120

• 1.8

-1.6

1.4

1.2

-1 </>

•0.8

.0.6

.0.4

•0.2

140 160 180 200

Figure 3. Biomass and substrate oscillations for D = 0.1.



0.09-|

0.08-

0.07-

0.06-

0.05-i

0.04-

0.03-

0.02-

0.01 -

0-

12 16 18 20 22 24 26 28

X30

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 ofsustained oscillations observed in continuous culture of Saccharomycescerevisiae. 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 forunderstanding development and tumor biology. Hatzimanikatis et al. (1999)presented a model in the literature using simplified molecular mechanisms asdepicted in Fig. 1.


cycE+cdk2 CycE:cdk2-P+1 f> cycE:cdk2-P:1

Rb+E2F Rb-P+E2F

Figure 1. Schematic representation of the molecular mechanism of components andinteractions 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

— = V -Vdt ~ 2 l

dK

dKrdt


dRE

dt

^=v6>r-v6 , 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.KPI is the concentration of cyclin E-cdk2 phosphorylated complex bound toinhibitor.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 bindsto 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 andV6) and of invariant total amounts of cdk2, pRb, E2F and inhibitor gave thefollowing dimensionless equations, consisting of 3 differential and 6 algebraicequations.

— =d T~vs vd y\

dk

drp— £-= Va — VAdi 3 4

k + kp+kp j =1

r + rp + r^ = 1


=1

i + A , k P I = l

re

0I=JPJLkpi

Where c is the dimensionless concentration of cyclin, k is the dimensionlessconcentration of cdk2 and rp is dimensionless concentration of the hyperphos-phorylated form of pRb. All details about the kinetic equations and transposingthem 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 Cyclin E concentratione E2F concentrationi Concentration of cyclin E-ck ckd2 concentrationkP Phosphorylated cyclin E-cdk2 complex

concentrationkP,I Concentration of phosphorylated cyclin

E-cdk2 complex bound to inhibitorr Concentration of hypophosphorylated form

of pRbrE Concentration of hypophosphorylated form

of pRb that binds to E2FrP Concentration of hyperphosphorylated form

of pRbg Ratio of total concentrations of cdk2 and cyclin Es Ratio of total concentrations of pRb and E2F1 Ratio of total concentrations of cdk2 and inhibitort Dimensionless time


Exercises

Program


Results


0.5

Figure 2. Profiles of concentrations k, rp and c versus time as obtained from the rate constantsin the program.



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 mathematicaldescription of regulation of the Gl-S transition of the mammalian cell cycle".Biotechnol. Bioeng., 65, 631-637.

8.9 Membrane and Cell Retention Reactors

8.9.1 Cell Retention Membrane Reactor (MEMINH)

System

Consider a reactor whose outlet stream passes through a membrane that retainsonly the biomass as seen in Fig. 1.

The growth is assumed to follow substrate inhibition kinetics with constantyields. The oxygen transfer rate influences the growth at high cell densityaccording to a Monod function for oxygen.

F,S0 i Gas

illiiil

Membrane module\F"modi

Air

Figure 1. Biocatalyst retention on a continuous reactor.



Model

The reactor-membrane system is modelled as a well-mixed tank, except thatbiomass is retained in the system batchwise. The balance region is chosen toinclude both the reactor and the membrane separator, but the separator volumeis neglected.

Biomass balance:dXdT = rx

Substrate balance:dS F

-rs

Oxygen balance (neglecting the oxygen transported by flow):

dCL-gj- =KLa(CLS-CL)-ro

Kinetics:S __ CL

rx ~ ^m x

rs = rx YS/X + MS X

ro = rx YQ/X + MO X

where the maintenance coefficients are related by

MS _ YS/XM0

=

Program


8.9 Membrane and Cell Retention Reactors 453

Nomenclature

Symbols

CL

CLSFKIKLa

KoKSMrSVXY

ILL

Dissolved oxygen concentrationSaturation oxygen concentrationFlow rateInhibition constantTransfer coefficientSaturation constant for oxygenSaturation constantMaintenance coefficientsReaction rateSubstrate concentrationReactor volumeBiomass concentrationYield coefficientSpecific growth rate

g/m3

g/m3

m3/hg/m3

1/hg/m3

kg/m3

kg/(kgh), g/(kgh)kg/(m3 h)kg/m3

m3

kg/m3

kg/kg1/h

Indices

01mOSX

Refers to feedRefers to reaction 1Refers to maximumRefers to oxygenRefers to substrateRefers to biomass

Exercises


Results

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


20

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



\\.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 wellknown, as shown in Fig. 1. A kinetic model for a Bacillus subtilis strain hasbeen established from continuous culture experiments using an approachinvolving overall stoichiometric relationships and energetic considerations. Theinfluence on the culture of product removal by pervaporation was investigatedby simulation methods.

Knowledge of these pathways allowed the following overall equations to bewritten:

Respiration (reaction RI):

C6Hi2O6 — > 6CO2 + 6H2O

Formation of biomass (reaction R2):

1.2 NH3 — > 6 CHi.8O0.5No.2 + 0.3 O2 + 2.4 H2O


Formation of acetoin (reaction

C6Hi206+102 —> C4H802 + 2 C02 + 2 H2O

Conversion of acetoin to butanediol (reaction R4):

C4H8O2 + H2O —-> C4HioO2 + 0.5 O2

Complex Compounds

Biomass Carbon Dioxide(Total Oxidation)

Substrate ^ Pyruvate ^ Acetoin + CarbonSugars v i Dioxide

Lactate Butanediol

Figure 1. Reaction scheme for the acetoin - butanediol formation.

The continuous reactor was coupled to a pervaporation membrane module andwas influenced by the membrane performance, owing to the removal ofproducts and the retention of biomass. Since only the volatile products andwater can pass through the membrane, a purge stream was needed to removebiomass 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 doubleMonod function and the product inhibition by a simple inhibition kinetic term.Diauxic effects were observed, but the diauxic components were unknown, andfor simplification only one diauxic switchover was assumed. The preferred


component is referred to as Ci, and the second component is called C2- Anempirical kinetic relation was devised such that the utilization of C2 wasinhibited by the presence of Ci in amounts greater than a repression constantkRpi. The formulation then involves considering the growth as the sum of twoterms,

where,

and

Feed

Bioreactor

Recycle

C2 *pl

Membrane

Pervaporation Module

Condenser

PermeateFpeACpe , Bu

Pump

Figure 2. Bioreactor and membrane pervaporator, showing process variables.

Each product was assumed to inhibit separately,

1 1and

Bu_ccBu

The inhibition constants, kinh,Ac and kinh,BU> an<l the exponents,were estimated from shake flask experiments.

The complete expression for specific growth rate was thus,

DO i i1+[_^_]«Ac1+|

and

Bu_iaBu


Formation of Acetoin and Butanediol

The products were assumed to be formed when the respiration was notsufficient to cover the energy requirement for growth. While the distribution ofthe two products was assumed to be dependent on oxygen, through a rapidequilibrium, according to an S-shaped empirical function described by

Ac fAcBu,max (DO)2

Bu -

To avoid an algebraic loop, the kinetics for the rate of butanediol productionwas 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 specificreaction rates q^ and the stoichiometric coefficients, Vy (component j inreaction i) as follows,

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

rj = qj X

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


V2,XMGX

where V2,x is the stoichiometric coefficient, and MGx is the C-mol mass ofbiomass.

The rate for the respiration reaction, qî, was found to be influenced by thedissolved oxygen, and was described as follows:

qxi =

The yield coefficients for the complex components GI and C2 , were assumedto be 1 g of each component for 1 g biomass, and the molecular weights of thecomponents were assumed to be the same as for the biomass. The initialamounts of these components in the molasses medium were adjusted in thesimulation. The corresponding rates were proportioned according to thegrowth rates as

rri = - rv and rr2 =

Pervaporation Model

The mass transfer in the pervaporation module was described as an equilibriumprocess using constant enrichment factors. Thus the concentrations of productin 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, respectivelyintr1.


The mass balances were as follows:

Total mass

BiomassD = Dpu + Dpe

dx

"3T = rx - X Dpu

SugardSdf = S0 D - S Dpu - rs

AcetoindAc~dT = rAc ~ Ac Dpu ~ Acpe Dpe

ButanedioldBu~dt~ = rBu ~ Bu DPU " BuPe DPe

Complex componentsdCi~dT = Ci>0 D - Ci Dpu - rCi

where Q = 1 and 2.

Oxygen transfer from the gas phase and the oxygen uptake rate determine theDO in percent saturation as,

dDO 100-HT~ = KLa (100 - DO) ^—r OUR

m 2.34xlO"4

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

Nomenclature

Symbols

ACPE Acetoin concentration in permeate g/LAc Acetoin g/L, mol/LATP AdenosintriphosphateBu Butanediol g/L, mol/LCi Complex components, where i = 1 or 2 g/LCPR Carbon dioxide production rate mol/L hD Dilution rate 1/hDO Dissolved oxygen concentration % saturation


UcBulAcBu.maxFFM/PkAcBukci

KLa

KS

KprodMGX

OURP

rJ

rpmRiRQSvvmVXYi/j

Acetoin/butanediol ratioMax. Ac/Bu ratioFlow ratePermeate / purge flow rateKinetic const. Ac/BuMonod-const., growth on component iMonod-const., growth depending on DOInhibition constant product iGas -liquid transfer coefficient for O2

Monod-const. for respiration depending onDORepression const., complex substrate 1 oncomponent 2Monod-const., Growth on substrateEmpirical const, for Ac/Bu-equilibriumMol mass for biomass (1 C-mol) = 24.6Oxygen uptake rateProduct, sum Acetoin + ButanediolSpecific rate of component jSpecific rate reaction i, where i =lto 4Formation or uptake rates, components j(Ac, Bu, O2, CO2, X and S)Stirrer speedChemical reaction i, where i = Ito 4Respiration quotient qcO2/(lO2Substrate concentration (e.g., sugar)Gas rate per volume liquidReaction volumeBiomass concentrationYield coefficient (i formed/j used)

L/h

mol/g(acetoin) hg/L% saturationg/L1/h

% saturation

g/Lg/L

g/molmol/L hg/L, mol/Lmol/g(biomass) hmol/g(biomass) h

mol/L h1/min

g/L, mol/L1/minLg/L, mol/L

Greek symbols

ai Exponent inhibition kinetic product i (Ac or Bu)pi Enrichment factor of pervaporation membrane for

component i (Ac or Bu)\i Specific growth rateR,max Maximum growth rate for component iVi j Stoichiometric coeff. of component j in reaction i

1/h1/h


Indices

AcBuQCO2

ijinh02pepuPRiSX0Ai

AcetoinButanediolComponent i (unknown components 1 or 2)Carbon dioxideRefers to reactions, i (1 to 4)Refers to components j (Ac, Bu, C>2, CC>2, X and S)Refers to inhibitionOxygen uptakePermeatePurgeProductChemical reaction i (1 to 4)SugarBiomassFeed or initial concentrationRefers to reactions 1 to 4

Reference

Dettwiler, B. Dunn I. J., Heinzle E., and Prenosil J. E. "A Simulation Modelfor the Continuous Production of Acetoin and Butanediol Using B. subtilis withIntegrated Product Separation by Pervaporation" Biotechnol Bioeng. 41, 791(1993).

Program


Results

The results of Fig. 3 show the influence of the FM/P ratio, which correspondsto the membrane area per unit reactor volume, on the biomass concentrations.


Here the enrichment factor was kept constant (pAc = 2.0), corresponding to thevalues found for one of the membranes. In a second set of runs the enrichmentfactor was varied for constant FM/P ratio = 1.0.


110

100

60

50

6 8 10 12 14 16 18 20

Figure 3. Influence of permeate/purge flow ratio on the biomass concentration. (D=l,BETAAC=2, FMP=0.4, 1.0, 2.0).


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).


8.9.3 Two-Stage Fermentor With Cell Recycle ForContinuous Production Of Lactic Acid(LACMEMRECYC)

System

This example is based on a paper (1) in which a two stage fermentor-membranesystem for the continuous production of lactic acid was modelled. Membraneretention of the active biomass can be expected to increase the productivity, butthe biomass concentration must be controlled in each reactor with a bleedstream. The kinetics for this process, in which glucose is converted to lactic acidby the bacteria Streptrococcus faecalls is described in the literature (2). Sincethe rates are inhibited by product, it can be expected that a multistage systemwill be advantageous.

(1-B2XB1F1+I?)

X2,S2,P2

Figure 1. Two-stage membrane fermenter system for lactic acid production


Model

The model assumes completely-mixed stages and complete cell separation. Theoperating parameters are feed flow rates or dilution rates and the bleed streamfractions. The bleed stream from the first fermenter is led to the secondfermenter. The model equations are developed neglecting the volume of thelines and separators. Note that the retentate streams are returned to theirrespective reactors and can be considered as part of the well-mixed system.

The kinetics is given by a product inhibition

S KPi i — i i L

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~~ l l

For the living cells


For the glucose

For the lactic acid

dt l f l l l l l

Mass Balances for the Second StageThe dilution rate for the second stage depends on the bleed ratio from the firststage 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 - kd2)x

For the glucose

fD!Sf2 + BAS1 - (8! +f)D1S2 -q2X2

For the lactic acid product

1 1 .>

2 +V 2 X 2Clt UC (JC (JC

Productivity of Lactic AcidIt is assumed that all streams containing product can be recovered from all ofthe streams. The productivities are calculated as follows:

From the first stagefti=D1(P1-Pfl)


From the second stage

Pr2=-D1[(B1+f)P2-B1P1-fPf2]VAi

The total productivity

Pr = -i-D1[(l-B1)P1+(B1+f)P2-Pf,-fPf2]JL ~t~ UC

Substrate Conversion

From stage 1

From stage 2(B1+f)S2

Xco =1 --82 B^+fSfz

Total glucose conversion

_ (l-B1)S1+(B1+f)S2Xg — I --

Sf 1 + fSf 2

Program

In the simulation product-free feed streams are assumed with flow rates between10-30 kg m3. The program is on the CD-ROM.

Nomenclature

a Death constant m3 kg"1

B Bleed ratioD Dilution rate s'1

f Flowrate ratio, F2/F1F Volumetric flowrate m3 s'1

kd Specific death rate s"1

Basis specific death rate s"1


KP

KS

PPrqsstV*sX

xta

YP,YS8P,8s

HV

Product inhibition constantSaturation constantLactic acid concentrationProductivitySpecific substrate uptakeGlucose concentrationTimeReactor volumeSubstrate conversionConcentration of living cellsTotal cell concentrationReactor volume ratioKinetic constantsKinetic constantsSpecific growth rateSpec, product formation rate

kg nr3

kg nr3

kg nr3

kg nrV1

kg kg-V1

kg nr3

sm3

-kgnr3

kgm~3

-

kg kg-1

kg kg-1

s-1

kg kg'1 s-1

Exercises


References

A. Nishiwaki and I. J. Dunn, "Performance of a two stage fermentor with cellrecyle for continous production of lactic acid", Bioprocess Engineering 21;299-305, 1999.

H. Ohara, K. Hiyama, T. Yoshida, "Kinetics of growth and lactic acidproduction in continuous and batch culture" Appl.Microbiol.Biotechnol. 38,403-407, 1992.

Results

20-

rsj 15-X

gioH


•-30

^—Xt1:1...—Xt2:1

... S2:1

P2:1

25

•20

10 CO

10 15 20TIME

30 35—1 v

40

Figure 2. Results showing the steady state for biomass substrate and product in both stages.


160

140

| 120 - I

£100BX- 80

1 60

flf 40

i

\


-1

•0.9

•0.8

A..

-0.7 rf

•0.6 tfTX

-0.5

0.4

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5B1

Figure 3. Results showing that the productivity changes slightly with bleed ratio but that thebiomass concentration and the substrate conversion depend highly on bleed ratio.

8.9.4 Tubular Hollow Fiber Enzyme ReactorModule for Lactose Hydrolysis (LACREACT)

System

This tubular reactor- radial diffusion model assumes a series of nine well-mixedtanks to describe a single hollow fiber module. Flow of lactose substrate passesaxially through the inner region of the fiber lumen. By diffusion the substrateis transported radially outward from the lumen through the membrane and intothe cylindrical porous support surrounding the membrane. The reaction takesplace in this support region, where the immobilized enzyme is located. Theproducts of hydrolysis are glucose and galactose, which diffuse back toward theliquid phase in the lumen. The parameter N can be used to adjust the numberof shells required. The module is assumed to consist of a large number ofidentical fibers*


Model

The model is developed by finite-differencing both the axial and radialdirections. Thus there are axial stages in series, here nine plus the recycle tank,and there are multiple cylindrical sections of porous support in the radialdirection. As depicted in the figures below, there is a convective liquid flowfrom each stage to another. Diffusional flows carry substrate and productradially from one cylindrical section of the porous support to another. Figures1, 2 and 3 give the geometrical details.

FRfiber LAfiber

FRfiber

LAtank

Figure 1. Hollow-fiber module showing only a single fiber as modelled by nine stages.


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 innerhollow fiber region (white) and the outer porous support (shaded). The finite-difference shell of volume V2 (white) is shown with diffusion fluxes of lactateJLAI entering and leaving JLAI- It is important to account for the radialvariation of volumes and diffusional areas. Note that the segments arenumbered from outside to inside, 1 to N.

For each tank the component balances account for the accumulation, the flowin and out and the diffusion in or out from segment N of the porousmembrane. For lactose in tank 1


LAlumenl _ FRfiber */T A T A x , JlAltN]* Aj[N]H - v v^^tank ' L/Mumenl ) + 77ai v lumen v lumen

For the external recycle tank, the flow leaving all of the fibers enters it and alsothe feed stream enters it. The total flow rate leaving must equal the sum of theserates. Thus for lactate

Number * FRfiher T A X Ffeed /T A T A N. LAtank) + -Jeed_(LAfeed - LAtank)

tank Vtank

Taking the component balances for each segment in the enzyme zone accountfor 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 forlactose in the ith enzyme segment of the first tank. Note that the aboveequations do not include the balance for segment 1. The wall conditionrequires that this balance contains only the rate of diffusion from segment 1 tosegment 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 ratesas given in the program on the CD-ROM.

JLA1[1. . (N-l) ]=-DLA* (LAlEi+1] -LAl[i] ) /DRFlux of lactose flowing between segments i+1 and i

JLAl E N ] =DLA* (LAI [N] -LAlumenl) /DRFlux of lactose into inner lumen section


RATELA1 [1. . N] =-vmax*LAl [i] / (LAI [i]+Km* (1+ (GA1 [i] /Kinhib) ) )Reaction rate for lactose,mole/s cm3

d/dt (LAlumenl) = (FRf iber/Vlumen) * (Latank-LAlumenl )+ ( JLA1[N] *AJ[N] ) /VlumenDynamic 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/dt (LAtank) = ( Number *FRfiber /Vtank ) * (LAlumen9-LAtank) + (Ff eed/Vtank) * (LAf eed-LAtank)Dynamic lactose balance for the circulation tank

The geometry of the fiber is programmed such that the lumen radius and thetotal fiber radius are given. The number of porous segments can be varied.

Nomenclature

Additional symbols for the geometrical factors are defined in the program onthe CD-ROM

Symbols

DGA Galactose diffusivity in porousmembrane cm2/h

DGL Glucose diffusivity in porousmembrane cm2/h

DLA Lactose diffusivity in porousmembrane cm2/h

EO Enzyme loading mg E in each fiberFR Recycle flow cm3

GAfeed Galacose in feed mole/cm3

GLfeeci Glucose in feed mole/cm3

Kinhib Kinetic inhibition constant mol/cm3


?LA, GA, GLNumberV

Kinetic constantKinetic constant for vmaxLength of fiber,Lactose, galactose and glucoseNumber of fibersVolumesMaximum rate

mol/cm3

mol/mgE h cm3

cmmol/cm3

cmmol/cm3 h

Indices

fiberlumen

tank

Refers to lumenRefers to lumenRefers to tanks or segmentsRefers to recycle tank

Exercises


Results

100 200

TIME


400 500 600

Figure 4. Approach to steady state for the tank concentrations.Run 1: 619 steps in 53.4 seconds

9.4e-5

L9.393e-5

Figure 5. Radial concentration gradients for the lactose and glucose. The left axis correspondsto the outside of the fiber.


8.9.5 Immobilized Animal Cells in a Fluidi/ed BedReactor (ANIMALIMMOB)

System

This example is based on experiments with immobilized BHK cells in afluidized bed of solid or porous carriers. The fluidized bed itself has anexpanded volume of 700 mL, The complete reactor system contains a volumeof 3.5 L. As seen in the figure below, the arrangement of the electrodes at theinlet and outlet of the reactor allows an accurate difference measurement of theoxygen uptake rate. Oxygen transfer takes place only in the conditioningvessel, while oxygen consumption is only in the fluidized bed column, wherethe cells are located.

GASOUTLET

GAS INLETFILTER

SAMPLEPORTM .

OXYGENj (MEASUREMENT

CHAMBER

AIR

OXYGENCARBON

OXIDE

VESSELTHERMOSTAT

Figure 1. Fluidized bed for culturing animal cells on solid carriers.

The recirculation reactor is modelled by taking into account the separateaeration tank and the geometry of the column reactor. It is assumed that thereactor is not well mixed, but is described by a tanks-in-series model for thecolumn with immobilized cells and a separate well-mixed aeration tank reactor,as shown below.


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 areused. 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 allcomponents, 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 = 0.28gbiomass produced per g glucose consumed.

2. Lactate is produced in proportion to the glucose uptake rate withYiacG=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 oflactate with respect to glucose, giving Yiacc = 2.0 mmol lactate/mmole glucoseor 1.1 g sodium lactate/g glucose. This can be used to calculate the productionrate of lactate. Also calculated from the dissertation is YgiutG

=:36mgglutamine/g glucose.

Thus the glucose uptake rate becomes

"OX K lacwsox Ksglut 4-Clac


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 specificrates 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 LEntire reactor system volume; 3.5 LMedium throughput; 6.5 L/dayFeed glucose concentration; 3.9 g/LGlucose consumption rate; 4.7 mmol/hOxygen uptake rate; 3.7 mmol/hMax. cell density: nonporous carriers; 2*10 cells/ml expandedbed volumeporous carriers; 4*10 cells/ml expanded bed volumeRatio of inoculum cell number; approx. 5 % of final cellnumberTotal biomass (porous carrier); 6.24 g (Approx. 2*10 cells perg biomass)Oxygen transfer coeff.; KLa 2.15 1/hr

Program


Nomenclature

Feed cone, for glucose g/LCgiutf Feed glutamine concentration mg/LCgiutF Feed glutamine concentration mg/LCox Dissolved oxygen mg/LCoxsat Saturation for oxygen mg/LF Flowrate L/hFR Circulation flowrate L/hGUR Glucose uptake rate mmol/h


KLaKlacKsg

KSglutKSOXOURqCmaxQOxmaxVV4XYoiutGYLacGYoxGYXG

Mass transfer coeff.Inhibition constant for lactateSaturation constant, for glucoseSaturation constant for glutamineSaturation constant for oxygenOxygen uptake rateGlucose uptake rate maxOxygen uptake rate maxVolume of 3 tanks in the columnAeration tank volumeBiomass concentrationGlutamine uptake rateYield coefficient glucoseYield coefficient glucoseYield coefficient, biomass to glucose

1/hg/Lg/Lmg/Lmg/Lmmo 1/hg/h g cellsmg/g cells hLLg/Lmg glut./g glue.g lactate/gmg oxygen/gg/g

Exercises


References

Keller, J. Dissertation No. 9373, ETH, 1991.

Keller, J., Dunn, I. J. and Heinzle E. "Improved Performance of the FluidizedBed Reactor for the Cultivation of Animal Cells" in Production of Biologicalsfrom Animal Cells in Culture, Ed. Spier, Griffiths, Meignier, Butterworth-Heinemann, 10th ESACT Meeting, 513-515 (1991).

Keller, J. and Dunn, I. J. "A Fluidized Bed Reactor for the Cultivation ofAnimal Cells", In: Advances in Bioprocess Engineering, Eds. E. Galindo and O.T. Ramirez, Kluver, pp 115-122 (1994).

Results


11 '

10'

9 '

8

? 7

6

5

4

3

iliii X[3]:1

20 40 80 100TIME

•6 OO

O

140 160

Figure 3. Batch run showing the difference in dissolved oxygen between the tank sections.



8

60 80 100 120 140 16020

Figure 4. In this run with F=0.05 L/h first oxygen limitation develops and later glucoselimitation. The immobilized biomass in the three sections increases at different rates due to theglucose gradient in the column.

Appendix: Using the BerkeleyMadonna Language

9.1 A Short Guide to Berkeley Madonna

Computer Requirements

Two Berkeley Madonna versions are supplied with this book on a CD, one forPC with Windows and one for the Power Macintosh. More information withdownloads 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 onInternet. Information on registering Madonna is contained in the files.Registration is optional since all the examples in the book can be run with theunregistered version. Registration makes available a detailed manual and isnecessary 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, ascan be seen on the Screenshot Guide in this Appendix. Running an exampletypically involves the following steps:

Start Berkeley Madonna and open a prepared program file.


484 9 Appendix: Using the Berkeley Madonna Language

Adjust the font and size to suit.

Go to Model/Equations on the menu and study the equations and programlogic.

Go to Parameters/Parameter Window on the menu and see how the valuesare set. They may be different than on the program. Those with a * canbe reset to the original values. Also, if necessary, here the integrationmethod 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 datafor each axis. All calculated results on the left side of the equations areavailable and can be selected.

Run the program by clicking on Run.

Adjust the graph by setting the legend with the legend button. Perhaps putone of the variables on the right side of the graph with Graph/ChooseVariables.

Possibly select the range of the axes with Graph/Axis Settings. Choosecolors or line types with the buttons.

Decide on further runs. It is most common to want to compare runs fordifferent values of the parameters. This is usually done with Parameters/Batch Runs and also with Parameters/Define Sliders. If the overlay buttonis 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 withchanges done in the Parameters/Parameter Window, with an overlay graphif desired.

As seen at the end of the Screenshot Guide, Parameter Plot runs are veryuseful to display the steady-state values as a function of the values of oneparameter. For this, one needs to be sure that the Stoptime is sufficient to reachsteady-state for all the runs.

When running a program with arrays, as found in the finite-differencedexamples, the X axis can be set with [i] and the Y axes with the variables ofinterest. The resulting graph is a plot of the variable values at the Stoptime inall of the array sections. For equal-sized segments, this is the equivalent of aplot 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 runningprograms is found in Sec. 2 of the Appendix.

Special Programming Tips

Berkeley Madonna, like all programming languages, has certain functions andcharacteristics that are worth noting and that do not appear elsewhere in thisbook.

Editing textThe very convenient built-in editor is usually satisfactory. Also the programcan 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 variablesSometimes programs do not run because of errors in the program that causeintegration problems. Some hint as to the location of the error can often befound by making an output table of all the calculated variables. This is doneby going to Graph/New Window and then Graph/Choose Variables andselecting all the variables. Then the program is run and the table button ischosen. Inspection of all the values in the table during the first one or two timeintervals will usually lead to an isolation of the problem for those values that aremarked in red with NAN (not a number). Also, values going negative can befound easily here and often indicate an integration error. Sometimes this canbe 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 thatyou must choose the ranges so that similarly sized values are located on thesame axis.

Are there bugs or imperfections in Madonna?Yes, there are some that we are aware of. You may find some or you may havesome special wishes for improvements. The Madonna developers in Berkeley,California would be glad to receive your suggestions. See the homepage tocontact 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 pre-programmed PULSE function or use an IF-THEN-ELSE statement.


Making a more complex conditional control of a program.In general the IF-THEN-ELSE conditional statement form is used, combinedwith logical expressions as found in the HELP. This can involve a switchingfrom one equation to another within this statement. Another way is to use flagsor constants that take values of 0 or 1 and are multiplied by terms in theequations to achieve the desired results. Nesting of multiple IF statements ispossible:V=IF(Disk<l AND P > 1 . 9 )THEN 0 . 8 5 * K V * P / S Q R T ( T R + 2 7 3 )ELSE IF (Disk<l AND P< = 1.9 AND P>1.1)

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 atext file and fitted by going to Parameters/Curve Fit. TheEdit/Preferences/Graph Window provides the possibility of having the data asopen 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 issomething simple with one or two parameters, then sliders can also beeffectively used. If the value of a maximum is sought as a function of a singleparameter 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 influenceof one variable on the steady state. The second parameter can be changed inthe Parameter Window and additional parametric runs made and plotted with anoverlay plot. Thus it is possible to obtain a sort of contour plot with a series ofcurves for values of the second parameter. Unfortunately, a contour plot is notyet possible.

Nice looking results are not always correct.A warning! It is possible to obtain results from a program that at first glanceseems OK. Always make sure that the same results are obtained when DT isreduced 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 setequal values of DT and DTMAX. Run the integration once and reduce bothparameters by one-half and run again. If the results are good, try to improvethe 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 ifDTMAX is too high. In most cases, good results are obtained with AUTO andDT set to about 1/1000 of the smallest time constant.

If no success is found with AUTO, then try STIFF and adjust by the sameprocedure. Oscillations can sometimes be seen by zooming in on a graph; oftenthese are a sign of integration problems. Sometimes some variables look OKbut others oscillate, so look at all of them if problems arise. Unfortunatelythere is not a perfect recipe, but fortunately Madonna is very fast so the trial-and-error method usually works out.

Checking results by mass balanceFor continuous processes, checking the steady-state results is very useful.Algebraic equations for this can be added to the program, such that both sidesbecame equal at steady state. For batch systems, all the initial mass must equalall the final mass, not always in mols but in kg. Expressed in mols thestoichiometry must be satisfied.

What is a "Floating point exception"?This error message comes up when something does not calculate correctly, suchas dividing by zero. This is a common error that occurs when equationscontain a variable in the denominator that is initially zero. Often it is possibleto 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 anddistance. When the model is written in array form the variable can be plotted asa function of the array index. This is done by choosing an index variable forthe Y axis and the [ ] symbol for the X-axis. The last value calculated is used inthe plot, which means that if the steady-state has been reached then it is asteady-state profile with distance. An example is given in the"ScreenshotGuide" 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 theadvantage that it appears in the Choose Variables menu and can be plotted. Itcan 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 contactthrough the BerkeleyMadonna homepage.


9.2 Screenshot Guide to Berkeley Madonna

This guide is intended as a supplementary introduction to Berkeley Madonna,Version 8.0.1.

(CHEMQSTATST(FILE/CHEMCr1)

{Constants}UM=OJKS=OJK18F*10 D1=0.25Stoptime=80

{Conditional equation for 0}

!nrfX=1

ln«t P=0

{Mass Balances}; 81OMASS BALANCE EQUATION; SUBSTRATE QÂNGE EQUATION; PRODUCT BALANCE EQUATION

9.2 Screenshot Guide to Berkeley MADONNA 489

Figure 1. The example CHEMO has been opened and the Menu (From left: File, Edit, Flowchartactive only for flowchart programs, Model, Compute, Graph, Parameters, Window and Help) andGraph 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

Kl=Q.03K2=O.OBY=fl.BBF=10 D1=OJS

{Conditional equation for D}D=iftime»=tstirttnenDl

toil 8=10initP=G

{Mass Balances}; EQ; BALANCE

3T6RTTIMEiTOPTIMI

DTMTOUTLJMKSKl

01

HITXNITSN1TP

0,0200.30.10,03o.oa0.8100,3451100

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 selectedunder Graph/Choose Variables.


4J-,

4-

3,5

3

2.5

•2-

1.5

f-

0,5-

0

-9,5

9

8.5

-8

-7.5

7

-8.5

5.520 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 Buttonselected.

7.

6-

5 -

X 4-

3-

4-

1-

020 40 80

Run 4:10000 steps m 0.167 seconds

..........................10

-7

•6

•5 01

-4

-3

-2

•1

100 120 140 180 !80 200

Figure 6. An Overlay Graph for three values of Dl as selected in the Parameter Window.


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

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


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 thefinal, steady-state values. The Data Button was pressed to give the points for each run.


Figure 11. The Optimize Window, with the value of Dl being selected to minimize theexpression -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.


Figure 13. A program written in an array form allows plotting all the values versus time bychoosing 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 anoverlay plot. The [i] values can be selected in the Choose Variables. Here the parameterRadius has been changed to demonstrate the large influence of diffusion length.


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 onthe plot. Note that the fit variable is CE and the parameter varied to minimize the difference inleast squares is KLA.

10 Alphabetical List of Examples

ACTNITR 267ANEAMEAS 433ANIMALIMMOB 476BATFERM 193BIOFILM 372BIOFILTDYN 342CELLDIFF 388CHEMO 799CHEMOSTA 258COMMENSA 400COMPASM 406CONINHIB 261CONTCON 367DCMDEG 280DEACTENZ 308DUAL 275ELECTFIT 335ENZCON 305ENZDYN 383ENZSPLIT 377ENZTUBE 272ETHFERM 240FBR 295FEDBAT 204FERMTEMP 358FILMPOP 425INHIB 327KLADYN 335KLAFIT 335LACMEMRECYC 464LACREACT 470LINEWEAV 272MAMMCELLCYCLE 445MEMINH 450MIXPOP 477MMKINET 209NITBED 299NITBEDFILM 393

NITRIFOLIGOOXDYNOXENZPENFERMPENOXYPHBPHBTWOPLASMIDREPFEDREPLCULSTAGEDSUBTILISTEMPCONTTITERBIOTITERDYNTURBCONTWOONETWOSTAGEVARVOLVARVOLDYEASTOSC

327275337378230253279374477245249290455354349349363422286224224447


11 Index

Absorption, 117Absorption tank, 136Accumulation, 125Accumulation terms, 135Acetogenic step, 89Acid-base equilibria, 47Acidogenic step, 89Active, 110Adaptive Control, 174Adaptive tuning, 178Aerated tank, 130Aerated tank with oxygen electrode, 336Aeration, 117, 137Aeration efficiency, 123Aeration rates, 126Aeration systems, 126Aerobic sewage treatment, 128Agitation, 137Air or oxygen sparging, 134Air saturation, 126Air supply, 128Airlift bioreactor, 139Algebraic loop, 49Alginate, 149Alginate bead, 149Allosteric kinetics, 74Ammonia, 134Ammonium, 300Ammonium ion, 133Anaerobic degradation, 89Analogous, 114

Analogy, 114Analytical solution, 19Analytically, 108Animal cell culture, 128Apparent reaction rate, 133Approximation, 113, 141Aqueous phase, 118Arithmetic-mean, 141Automatic process control, 161Automatic reset, 165Auxiliary variable, 174Axial, 114Axial profiles, 114Axial segments, 115Backmixing, 137Backmixing flow contribution, 142Backmixing stream, 140Balance region, 23Balances, 101Batch, 57, 64Batch aeration, 126Batch fermentation, 11, 103Batch reactor periods, 57Batchwise, 134Bed, 134Biocatalysis, 118Biocatalyst diffusion model, 153Biocatalytic reaction, 147Biofilm, 154Biofilm, 134, 145Biofilm nitrification, 160


500 Index

Biofilm Reactor, 133Biofilter, 343Biofloc, 145Biofloc, 149Biological activity, 123Biological film, 149Biological floes and films, 388Biological oxygen uptake, 125Biological reaction, 145Biological reactors, 101Biological systems, 117Biomass, 10, 103Biomass recycle, 110Biomass retention, 111, 299, 393Biomass separation, 111Bioprocess control, 175Bioreaction, 117Bioreactor, 112Bioreactor modelling, 101Bio-trickling filter, 343Boundary conditions, 153Briggs-Haldane mechanism, 68Broth, 117Bubble, 117, 137Bubble coalescence, 137Bubble column, 139Bubble size, 140Bulk liquid, 145Bulk reactor concentration, 153Buoyancy, 137Carbon, 134Carbon dioxide, 122Carbon dioxide production rate (CPR), 10Carbonate, 134Carrier, 145Carrier matrix, 145Cascade, 138Cascade control, 172Cell concentration, 101Cell productivity, 55Cell recycle, 110

Cells, 117Centrifugation, 110Chemical reaction, 126Chemostats, 104Circulating liquid supply, 129Circulation time, 55Closed-loop response, 170Coalescence, 126Cocurrently, 139Cohen-Coon controller settings, 170Column systems, 135Commensalism, 87Comparator, 162Competition, 150Competitive, 74Completely mixed gas or liquid phases, 137Complex diffusion-reaction processes, 157Complex kinetics, 115Complex models, 123, 138Complexity, 138Component, 122Component Balances, 22Components, 113Computer Solution, 19Concentration driving force, 122Concentration gradient approximation, 151Concentration gradients, 119, 137Concentration inhomogeneities, 137Concentration profile, 114Conical sand bed, 134Continuous, 64, 109, 118Continuous Baker's Yeast Culture, 94Continuous feed and effluent stream, 134Continuous Operation, 60Continuous phase, 118Continuous-cycling, 171Contois Equation, 82Control, 111Control point, 165Control region, 23Control strategy, 176

Index 501

Controlled variable, 161Controller, 162Controller action, 163Controller equations, 33Controller output, 166Controller tuning, 169Convection currents, 137Convective flow, 26, 125Convective streams, 26Conversions, 113Cross-sectional area, 114Cyclic fed batch, 109Damkohler number, 155Data fitting, 339Datafile, 341Dead zone, 163Death rate, 78Degree of backmixing, 140Density, 104Deoxygenated, 126Deoxygenated feed method, 130Depth, 137Derivative control, 163Deviations, 161Deviations from ideal stage mixing, 140Difference form, 116Difference segment, 115Differential, 107Differential control constant, 355Differential equation, 127Differential equations, 140Differential gap, 163Differential-difference equations, 159Diffusion, 118, 119, 145Diffusion and reaction, 388Diffusion control, 133Diffusion film, 146, 336Diffusion layer, 149Diffusion path, 145Diffusion rate, 149Diffusional flux, 26, 149, 389

Diffusional limitation, 146Diffusional mass transfer, 145Diffusional mass transfer coefficient, 147Diffusion-reaction parameter, 155Diffusion-reaction phenomena, 151Diffusion-reaction systems, 151Diffusive mass transfer, 138Digital simulation, 14Digital simulation languages, 14Dilution rate, 105Dimensionless, 108Dimensionless form, 391Dimensionless group, 151Dimensionless parameter, 150, 154Dimensionless variables, 155, 159, 337Discontinuous control, 163Disks of liquid, 113Dispersed, 117, 118Dispersed phase, 118Dissociation equilibrium constant, 47Dissolved oxygen concentration, 123, 335Dissolved oxygen electrode, 126Distance, 116Distance coordinate, 151Double Michaelis-Menten Kinetics, 73Double-Monod kinetics, 83, 158Driving forces, 119Droplet, 118D-value, 77Dynamic, 115Dynamic component balances, 104Dynamic kla, 126-127Dynamic Method, 335Dynamic simulation, 49Dynamics of measurement, 127Dynamics of the liquid phase, 128Effective diffusivity, 30, 121Effective rate, 148Effective reaction rate, 146Effectiveness Factor, 155, 391Efficiency, 110

502 Index

Electrode measurement dynamics, 127, 129Electrode membrane, 337Electrode response characteristic, 336Electrode time constant, 127, 129Elemental balances, 23Energy balances, 49Entrance, 113Enzymatic, 112Enzyme, 112, 118Enzyme loading, 148, 149Enzyme reactor, 115Enzyme-substrate complex, 69Equations, 113Equilibrium, 10, 122Equilibrium oxygen concentration, 337Equilibrium relationships, 46Equilibrium value, 132Errors, 161Exit, 113Experimental reactor, 130Exponential, 108Exponential and limiting growth phases, 103External film, 145External mass transfer, 145External transport rate, 153Extraction, 118Fed Batch, 58, 64Feed Forward Control, 173Feedback, 161Fermentation, 101Fermentation media, 137Pick's Law, 29, 120Film coefficients, 122Final, 109Final control element, 162Finite difference, 30Finite difference Model, 151Finite differencing, 388Finite differencing technique, 159Finite-differencing, 115, 153First order, 65

First order lag equation, 127, 337First order lag model, 127First-order, 132First-order time lag, 354Flocculant cell mass, 145Flotation, 110Flow interaction, 140Flow velocity, 114, 148Fluid, 120Fluid elements, 113Fluidized bed, 133, 299, 149Flux, 120Food/biomass ratio, 112Fractional conversion, 126Fractional response, 127Free rise velocity, 137Functional modes of control, 163Gas, 117Gas absorber, 136Gas Absorption, 117Gas and liquid films, 121Gas balance, 125Gas balance method, 128, 126Gas bubbles, 117Gas concentrations, 125Gas flow rates, 125Gas holdup, 55, 124Gas inlet, 137Gas phase, 117Gas-liquid, 117, 120Gas-liquid systems, 122Gas-liquid transfer, 336Gel, 145Growth, 110Growth rate, 32, 103Heat of agitation, 52Heat of fermentation, 49Heat Transfer, 51Henry coefficient, 340Henry's law, 33, 122, 337Henry's law constant, 122

Index 503

Heterogeneous reaction systems, 155Hill Kinetics, 74Hydrostatic pressure, 137, 140Ideal Gas Law, 39Ideal gas law, 133, 337Idealized flow conditions, 137Idealized plug flow, 137Ideally mixed, 136Immiscible, 118Immobilization, 145Immobilization matrix, 145Immobilized, 129Immobilized biocatalyst systems, 145Immobilized enzyme and cell systems, 118Impermeable solid, 149Incomplete oxygen penetration, 160Increments, 151Industrial fermenters, 137Information flow diagram, 33Inhibition, 73Inhibitory Substrate, 367Initial conditions, 103Initial value, 20Inlet, 106Inoculum, 103Input rate, 102Integral, 163Integral control constant, 355Integral time constant, 166Integrated, 108, 115Integration procedure, 20Integration step length, 20Integration time interval, 20Intensity of mixing, 55Intensity of mixing, 140Interconnected, 138Interface, 117, 118, 120Interfacial concentrations, 122Internal mass transfer, 145Internal structure, 119Interphase, 119

Interphase transfer, 26, 125Intraparticle transfer, 145Intrinsic reaction rate, 150Ion charge balance, 47Ion exchange resins, 149Kinetic, 106Kinetic control, 147Kinetic model, 136Kinetic rate constant, 147Kinetic regime, 148Kinetic relationship, 65Kinetics control, 149Kla, 335Lag phase, 103Lag time, 170Laplace transformation, 338Large bioreactors, 137Large scale, 137Length, 115Length of diffusion path, 151Limiting, 140, 159Limiting substrate, 103, 106Limiting substrate concentration, 78Linear gradients, 120Linear growth, 108Lineweaver-Burk diagram, 72Liquid, 117Liquid balance, 125Liquid balance equation, 128Liquid film, 337Liquid film control, 123Liquid flow terms, 135Liquid medium, 117Liquid phase, 117Liquid recycle stream, 134Liquid surface, 137Liquid velocities, 137Liquid-liquid, 118Liquid-phase impeller zones, 141Logistic Equation, 82Luedeking-Piret model, 85

504 Index

Maintenance coefficient, 10Maintenance factor, 84Mammalian Cell Cycle Control, 445Manipulated variable, 161Mass balance equation, 16Mass Transfer, 117, 119Mass transfer capacity coefficient, 122, 123Mass transfer coefficients, 121Mass transfer control, 147Mass transfer resistance, 145Material balance equations, 101Mathematical, 150Mathematical model, 12, 137Mathematical modelling, 151Matrix elements, 151Maximum, 109Maximum observed rate, 148Maximum rates, 158Maximum reaction rate, 70Measurement dynamics, 126, 128, 336Measurement signal, 127Measurements, 106Measuring element, 162Mechanical agitation, 137Mechanical energy, 137Medium, 106Membrane, 112, 127Membrane filtration, 110Methanogenic step, 89Michaelis-Menten constant, 70, 148Michaelis-Menten kinetics, 148Microbial interaction, 86Microbial physiology, 106Microbiological, 133Mixing, 113Mixing zones, 140Mode of control, 168Model, 109Modelling, 113, 388Molar flow rate of air, 133Molar reaction rates, 136

Molecular diffusion, 29Molecular diffusion, 120Molecular diffusion coefficient, 121Monod equation, 67Monod kinetics, 10Monod-type equation, 390Monod-type rate expressions, 105Multiphase reaction, 117Multiple impeller, 140Multiple-organism populations, 86Multiple-substrate Monod kinetics, 82Multi-stage, 138Mutual inhibition, 343Mutualism, 87Natural logarithmic, 127Nernst-diffusion film, 146Nitrate, 300Nitrate ion, 133Nitrification, 133, 299Nitrification reactions, 157Nitrite, 300Nitrite ion, 133Nitrobacter, 134Nitrogen, 127Nitrosomonas, 134Non-competitive, 74Non-porous carrier, 146Numerical solution, 19Objective function, 178Offset, 163Oil phase, 118One-dimensional diffusion, 158On-line adaptive optimizing control, 178On-line monitoring, 126On-Off Control, 163Open-loop tuning technique, 169Operation, 110Order of magnitude analysis, 155Organism balance, 102Oscillations, 355Oscillations of continuous culture, 94

Index 505

Outlet, 106Output rate, 102Overall mass transfer capacity coeff., 123Overall mass transfer rate equation, 123Overall order of reaction, 148Overall rate of reaction, 62Overall resistance to mass transfer, 123Oxidation steps, 134Oxygen, 117, 122, 388Oxygen balances, 123Oxygen depletion, 132Oxygen diffusion effects, 157Oxygen electrode, 335Oxygen electrode dynamic, 337Oxygen gas phase concentrations, 128Oxygen gradients, 139Oxygen limitation, 388Oxygen requirements, 158Oxygen transfer, 125Oxygen transfer coefficient, 136, 335Oxygen transfer rate (OTR), 10Oxygen uptake rate, 38, 125, 128, 136, 390OUR, 38Oxygen-enriched air, 131Oxygen-sensitive culture, 97Packed, 149Parameter, 111Parameter estimation, 131Partial differential equation, 116, 153Partial pressure, 122Penetration, 149Penetration distance, 159, 388Penetration-limiting, 159Perfect mixing, 137Perfect plug flow, 137Performance, 113pH control, 49Phase interface, 120Phases, 117Physical model, 12, 137Physical properties, 122

Plant cell culture, 128Plug flow, 113, 141Poly-6-hydroxybutyric Acid (PHB), 93Porous, 119Porous biocatalyst, 119Porous solid, 145Power inputs, 137Predator-Prey Kinetics, 86Pressure, 137Process control, 10, 56, 161Process dynamics, 127Process optimization, 12, 56Process reaction curve, 169Process response, 128Product, 118Product inhibition, 113Product inhibition kinetics, 63Production rate, 102Productivity, 105, 110Programmed, 174Proportional, 143, 163Proportional control constant, 355Proportional-Derivative (PD) Controller, 166Proportional-Integral controller, 355Proportional-Integral-DerivativeController, 167Proportlonal-Reset-Rate-Control, 167Pulse, 113Quadratic equation, 148Quasi-homogeneous, 135Quasi-homogeneous reaction, 158Quasi-steady state, 107Radial variations, 140Rate expressions, 103Rate of accumulation, 21, 102Rate of fermentation, 126Rate of mass transfer, 121Rate of oxygen transfer, 123Rate of oxygen uptake, 123Rate of substrate uptake, 83Rate of supply, 148

506 Index

Reactants, 117Reaction, 114, 117, 118Reaction capability, 149Reaction control, 133Reaction Heat, 51Reaction parameter, 155Reaction rate constant, 132Reaction rate control, 133Reaction site, 118, 145Reaction surface, 147Reaction-rate limitation, 136Reactor, 101, 138Reactor cascade, 62Reactor column, 135Reactor efficiency, 111Reactor modes, 63Reactor operating conditions, 159Reactor outlet, 136Re-aerated, 126Recycle loop, 134Recycle loop configuration, 134Recycle rates, 134Recycle ratio, 111Regimes, 133Research, 106Reset time, 166Residence time, 105, 116Residence time distribution, 137Residual error, 167Resistance to mass transfer, 120, 123Respiration quotient (RQ), 10Response, 127Response curve, 126Retention, 110, 145Riser, 139Sample, 128Sampled data control, 174Sampling frequency, 175Sampling interval, 174Sand, 134Saturation, 127

Saturation constant, 10Scheduled adaptive control, 174Second-order response lag, 337Sections, 138Sedimentation, 110, 111Semi-Continuous Reactor, 314, 349Separation, 110Separator, 111Series of tank reactors, 62Set point, 161Shear, 55Simulation, 107Simulation example, 104Simulation methods, 140Simulation programming, 153Simulation programs, 14Simulation results, 159Simulation software, 15Simultaneous diffusion and reaction, 150Single stage, 138Single-pass conversion, 135Slab, 151Slope, 18, 127Solid, 118Solid biocatalyst, 119Solid carrier, 145Solid phase, 135Solid-liquid interfacial area, 135Solubility, 122Solution, 143Spatial variations, 123Specific area for mass transfer, 121Specific carbon dioxide production rate, 11Specific carbon dioxide uptake rate, 84Specific death rate, 78Specific growth rate, 10, 78, 105Specific interfacial area, 122Specific oxygen, 11Specific oxygen uptake rate, 84Specific product production rate, 84Specific substrate uptake rate, 84

Index 507

Spherical bead, 388Spherical shell, 389Stages, 138Stagewise model, 139Stagewise modelling, 139, 140Stagnant, 120Stagnant film, 145Stagnant flow, 120Starting, 109Startup, 61Startup period, 61, 62Steady state, 104Steady state conditions, 104Steady state tubular reactor design, 115Steady state values, 140Steady-state, 105Steady-state balances, 21Steady-state position, 165Step change, 127Sterile, 104Stirring power, 137Stirring speed, 148Stirring speed, 149Stoichiometric coefficients, 40Stoichiometric oxygen requirements, 159Stoichiometric relations, 134Stoichiometry, 40, 126Structured kinetic model, 11Substrate balance, 102Substrate concentration, 101Substrate gradients, 151Substrate inhibition, 65, 108Substrate uptake rate, 55Sulfite method, 126Sulfite oxidation, 126Support, 119Surface, 118Surface concentration, 147Surface reaction, 145Sustained oscillations, 94Symmetry, 151

System, 110Tank, 101Tank sizes, 113Tanks-in-series, 112Teisser Equation, 81Temperature, 148Temperature measurement, 354Theoretical basis, 112Thiele modulus, 155Thin film, 120Time, 109Time constant for heater, 355Time constant for measurement, 355Time constant measurement, 340Time constants, 128Time constants for transfer, 340Time-varying, 114Titration, 126Total interfacial area, 121Total mass balance, 102Total system, 134Toxicity, 134Tracer, 113Tracer experiment, 140Tracer techniques, 137Transfer control, 133, 149Transfer parameters, 138Transmission lines, 162Transport of material, 117Transport streams, 26Transport-reaction process, 147Trial and error method, 169Tubular, 113Tubular reactor, 62, 113Turbine impeller, 134Turbulence, 120Turbulent flow, 120Two position action, 163Two-Film Theory, 120Ultimate gain, 169Ultimate period, 171

508 Index

Uncompetitive, 74 Well-mixed, 24, 101, 123Uptake rate, 111 Well-mixed gas phase, 133Variable, 106 Well-mixed liquid zones, 140Viscosity, 149 Well-mixed tank, 25Volumetric flow rate, 101 Whole cells, 149Washout, 105 Yield coefficient, 10, 32, 102Wastage, 110 Zero order, 65, 147, 148Waste water, 111 Zero-order kinetics, 391Water, 122 Ziegler-Nichols Method, 169

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Biological Reaction Engineering (Second Edition)