Dynamics of Environmental Bioprocesses: Modelling and Simulation

J. B. Snape, I. J. Dunn, J. Ingham, J. E.Pfenosi1 Dynamics of Environmental Bioprocesses 4b VCH The included diske...

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J. B. Snape, I. J. Dunn, J. Ingham, J. E.Pfenosi1

Dynamics of Environmental Bioprocesses

4b

VCH

The included diskette contains the ISIM simulation language as well as simulation examples. It can be run on all DOS-PC’s.

OVCH VerlagsgesellschaftmbH, D-69451Weinheim (Federal Republic of Germany), 1995 Distribution: VCH, P.0.Box 101161,D-69451Weinheim (Federal Republic of Germany) Switzerland: VCH, P. 0.Box, CH-4020 Basel (Switzerland) United Kingdom and Ireland: VCH, 8 Wellington Court, Cambridge CBl lHZ (United Kingdom) USA and Canada: VCH, 220 East 23rd Street, New York, NY 10010-4606 (USA) Japan: VCH, Eikow Building, 10-9 Hongo 1-chome, Bunkyo-ku, Tokyo 113 (Japan) ISBN 3-527-28705-1

Jonathan B. Snape, Irving J. Dunn John Ingham, Jiii E. Pfenosil

Dynamics of Environmental Bioprocesses Modelling and Simulation

+

VCH

Weinheim . New York Base1 Cambridge Tokyo

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Dr. J. B. Snape Nippon Lever Shibuya Tokyo 105 Japan

Dr. I. J. Dunn, Dr. J. E. Pfenosil Department of Chemical Engineering ETH Zurich CH-8092 Ziirich Switzerland

Dr. J. Ingham Department of Chemical Engineering University of Bradford Bradford BD7 1DP United Kingdom

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

Published jointly by VCH Verlagsgesellschaft mbH, Weinheim (Federal Republic of Germany) VCH Publishers, Inc., New York, NY (USA)

Editorial Directors: Karin Sora, James Gardiner Production Manager: Claudia Gross1

Library of Congress Card No. applied for

British Library Cataloguing-in-PublicationData: A catalogue record for this book is available from the British Library

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Dynamics of environmental bioprocesses : modelling and simulation / Jonathan B. Snape ... Weinheim ; New York ;Basel ; Cambridge ;Tokyo : VCH. Medienkombination ISBN 3-527-28705-1 NE: Snape, Jonathan B. Buch. - 1995 Diskette. - 1995

OVCH Verlagsgesellschaft mbH, D-69451 Weinheim (Federal Republic of Germany), 1995 Printed on acid-free and low-chlorine paper All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form -by photoprinting, microfilm, or any other means -nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Data conversion: Hagedornsatz, D-68519 Viernheim Printing: strauss offsetdruck GmbH, D-69509 MBrlenbach Bookbinding: GroBbuchbindereiJ. Schafer, D-67269 Griinstadt Printed in the Federal Republic of Germany

Preface The aim of this text is to utilise the twin tools of modelling and computer simulation in developing a better understanding of environmental bioprocesses and their dynamics. This is achieved via a combination of basic theory and bioprocess description, combined with sixty-five simulation examples. This latter feature is probably the most important feature of the book in providing the opportunity for direct on-line computer experimentation and computeraided self learning. The book follows from our series of post-experience short courses, held annually in the Swiss mountain resort of Braunwald, and the two resulting sister volumes "Biological Reaction Engineering" and "Chemical Engineering Dynamics", both of which are also published by VCH. Modelling is often an unfamiliar technique for many persons active in the field of environmental science as life scientists may sometimes lack the formal training needed to analyse laboratory kinetic data in its most meaningful way. An additional aim of this book is therefore to provide the mathematical tools required for a quantitative analysis of biological and chemical rate phenomena, based on the techniques of mathematical modelling and digital simulation. More generally, mathematical modelling methods should also lead to a better understanding of bioprocess environmental system behaviour. In formulating models of chemical and bioprocess operations, one of the most important tools is the material balance. Mass balances are simply representations of the first law of conservation, namely that matter can neither be created nor destroyed. The mass balance is thus fundamental to all science and, when combined with other forms of defining relationship, can be used with very great effect in forming mathematical process models, which may be quite simple in form but which nevertheless can be very powerful in describing quite complex process phenomena. Having formulated a mathematical model, the model must be solved. This is nowadays very easily effected by the computer and the modern approach of using desktop computers with easy-to-program software helps considerably in making modelling far more attractive than in the past. Modern simulation languages are now available that provide the possibility of carrying out an interactive simulation at one's own desktop. The ISIM simulation language, provided with this book, is the language we have used during the last five years of our continuing education short courses. ISIM is especially suitable owing to its sophisticated computing power, interactive ability and ease of programming. We have found that the ISIM based simulation examples enforce the learning process in a very effective manner. Readers can program their own examples, by formulating their own models and programs or by modifying an existing program to a new set of circumstances. A true degree of interaction is possible because, at the stroke of a key, any simulation run can be stopped,

VI

Preface

parameters changed, and the run restarted from the point of interruption. The programs also provide a very convenient graphical output of the computed results. Runs can be repeated with new parameters, and the combined results from multiple runs easily plotted. In our experience, digital simulation has proven itself to be absolutely the most effective way of introducing and reinforcing new concepts involving multiple interactions. We hope you will enjoy working on the simulation examples and agree with us that simulation is both useful and enjoyable.

Organisation of the Book This book is divided into three chapters covering, respectively, principles of modelling, environmental bioprocess descriptions and simulation examples. The aim of the first chapter is to introduce the basic theory necessary to understand the simulation examples presented in Chapter 3. The chapter opens with a discussion of the need for models in environmental technology and highlights their usefulness and limitations. This is followed by a brief overview of model classification and modelling procedure, including a comparison of available simulation programs. The software provided with this book, ISIM, is discussed in detail and its use is illustrated via the step-by-step development of a simple programming example. The formulation of mass and energy balances is a central theme of this book and a rigorous presentation of the necessary procedures is included. The physical meaning and significance of each term in the model equations are explained and the text is amplified by several examples of relevance to environmental bioprocesses. Our aim here is to present the material in a way that can be understood by biologists who rarely receive any formal training in modelling, and to demonstrate to engineers that they can apply their knowledge to systems outside their normal field of study. In addition to mass and energy balances, other relationships and balances are required in any model formulation. Transport phenomena often need to be modelled and diffusion and interface transport are included. Reactor process technology is discussed, and it is shown how this can be applied both to wastewater treatment plants and to natural water bodies. An understanding of microbial kinetics is essential to environmental bioprocess modelling and an introduction to microbial kinetics is presented in the following section. Monod kinetics are introduced, and then more complex kinetics involving different types of inhibition and interactions, as are more frequently encountered in real life situations, are discussed. The second chapter provides background information on various environmental bioprocesses. It is not our aim to give a comprehensive review of this large and ever-expanding field, but to give the reader a feel for the types of process that can be modelled and how one can set about a modelling problem. Inevitably the areas covered are biased towards our own research

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VII

interests and experiences, but we hope the material presented will be of interest to the majority of readers. This chapter first covers wastewater, its characteristics, analyses and treatment. The section on the treatment-of-wastewater gives an overview of the major primary, secondary and tertiary processes used to treat wastewater. The secondary processes are divided into aerobic and anaerobic processes and also biofilm and floc processes. The modelling of biofilms and flocs is also covered. The next section is concerned with water pollution modelling. The eutrophication of lakes and reservoirs is discussed, including the major factors causing eutrophication, its consequences and how it can be reversed or prevented. An overview of types of eutrophication model is given. Next the discharge of pollutants into rivers and streams, based on the Streeter-Phelps oxygen sag model, is discussed. Modifications to the basic theory and the ways in which these can be modelled are dealt with. Following on from this the topical area of modelling groundwater pollution is discussed with reference to leaching of landfill sites. Some basic hydrology necessary for modelling ground water pollution is introduced, as well as the fundamental transport and adsorption relationships. The final section deals with the treatment and disposal of solid waste and covers composting, landfill sites and anaerobic digestion. The third and final chapter comprises 65 simulation examples, all of which are on the diskette and which can be run with the included ISIM software. Each employs the theory covered in Chapter 1 to model the processes describ-ed in Chapter 2. Each simulation example is self contained and includes a model description, the model equations, nomenclature, references and suggest-ed exercises. The exercises vary in complexity from very simple parameter changing to quite complex suggestions for new programs, and are intended to stimulate the reader to gain a greater understanding of the system under study. Due to limitations of space, it has not been possible to reproduce all the program listings, but these can be viewed very easily on the PC. However, when some new program technique is used for the first time, this is mentioned in the program text and the relevant listing is shown. It was not our aim to present very large unwieldy programs that may not easily be understood by the novice programmer, but to present simple, easy to understand models that perhaps cover one or two of the most important processes. This is perhaps best exemplified by the section on activated sludge processes. Here we have presented models that simulate the complex ecological interactions, the sludge settler, processes within an individual floc, temperature effects and different reactor configurations and process strategies. A complete model of an activated sludge process would take into account all these processes (as well as others not considered here), but the size and speed of execution of the program would make it useless as a teaching aid. By splitting the process into these subprocesses it becomes manageable. We hope that after reading this book, the readers will feel motivated to apply what they have learnt in their own specialist field.

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In researching the literature for suitable examples to include in this text, it became obvious that some models presented in the literature are incomplete, in that values of parameters are not given or even equations are missing, so that the process could not be simulated. We would like to thank, therefore, the authors of the work we have cited in the text for publishing their comprehen-sive models, and apologise to them for occasionally simplifying and modifying their models to meet the requirements of this text. The source reference is always given at the end of each example, and the reader is referred to the original papers for more complete explanations and the detailed models.

ISIM Simulation Software The ISIM software is made available only for the purposes described in this book, and its features are restricted to these examples. An advanced simulation language, ESL, is highly recommended and is also available. Users wishing to purchase the latest menu-driven version of ISIM or ESL should contact ISIM International Simulation directly. User manuals for ISIM may also be purchased for &40from ISIM International Simulation. ISIM International Simulation Limited, Technology House, Salford University Business Park,Lissadel Street, Salford M6 6AP, England, (Tel: +44-(0)61-745 7444; Fax: +44-(0)61-7377700).

Acknowledgements A major acknowledgement should be made to the pioneering texts of Franks (1967, 1972) for inspiring our interest in digital simulation. Interest in environmental engineering came from our research, and with it an awareness of the important contribution chemical engineering can make to this field. We are especially grateful to all the participants and course collaborators of our post-experience courses, for their assistance in the development of the material presented in this book. Continual stimulus and assistance has also been given by a sequence of students, at the ETH-Zurich. One of us (J. B. S . ) is grateful to the ETH for a postdoctoral fellowship which allowed course material on which this book is based to be developed and also to Dr. M. Nakajima of the National Food Research Institute, Japan for providing word processing facilities and support for this project. Our special thanks are again due to Professor John L. Hay of ISIM International Simulation Limited for his agreement to release the ISIM digital simulation programming language, for use with this book. We hope that the

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IX

book will be useful in drawing attention to his advanced simulation language, ESL, for which we are happy to include a programmed example and an advertisement. This book was written on an Apple Macintosh, using Microsoft Word and Claris MacDraw Pro. Scans of the computer outputs were done with Adobe Photoshop. The Linotype film was made from the Word and Photoshop files. Special thanks are due to Albert Ochsner for his able work on the text and drawings. Marc Deshusses read the entire text and suggested many improvements. We are grateful to VCH for giving financial assistance for the word processing and especially wish to thank Louise Elsam, James Gardiner, Claudia Gross1 and Karin Sora of VCH for correcting the text so carefully and for their many useful discussions and cooperation.

Table of Contents Preface

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V

Organisation of the Book . . . . . . . . . . . . . . . . . . . VI ISIM Simulation Software . . . . . . . . . . . . . . . . . . . VIII Acknowledgements . . . . . . . . . . . . . . . . . . . . . VIII

Nomenclature for Chapters 1 and 2 1

Modelling Principles

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1

The Role of Modelling in Environmental Technology . . . . . 1 3 General Aspects of the Modelling Approach . . . . . . . . Model Classification . . . . . . . . . . . . . . . . . . 4 Deterministic Models . . . . . . . . . . . . . . . . . . 4 Stochastic Models . . . . . . . . . . . . . . . . . . . 5 Steady-State Models . . . . . . . . . . . . . . . . . . 5 Dynamic Models . . . . . . . . . . . . . . . . . . . . 6 General Modelling Procedure . . . . . . . . . . . . . . 6 Simulation Tools . . . . . . . . . . . . . . . . . . . . 7 ISIM . . . . . . . . . . . . . . . . . . . . . . . . 10 Introductory ISIM Example: WASTE . . . . . . . . . . . 11 Formulation of Dynamic Balance Equations . . . . . . . . 14 Mass Balance Procedures . . . . . . . . . . . . . . . . 15 Case A . Continuous Stirred-Tank Reactor . . . . . . . . . 17 Case B . Tubular Reactor . . . . . . . . . . . . . . . . 18 Case C . River with Eddy Current . . . . . . . . . . . . . 18 Rate of Accumulation Term . . . . . . . . . . . . . . . 21 Convective Flow Terms . . . . . . . . . . . . . . . . . 22 Production Rate . . . . . . . . . . . . . . . . . . . . 23 Diffusion of Components . . . . . . . . . . . . . . . . 24 Interphase Transport . . . . . . . . . . . . . . . . . . 25 Case A .Waste Holding Tank: Total and Component Mass Balance Example . . . . . . . . . . . . . . . . . 26 1.8.1.10 Case B . The Plug-Flow Tubular Reactor . . . . . . . . . . 28 1.8.1.11 Case C . Biological Hazard Room . . . . . . . . . . . . . 30 1.8.1.12 Case D. Lake Pollution Problem . . . . . . . . . . . . . 35 1.8.2 Energy Balancing . . . . . . . . . . . . . . . . . . . 40

1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.5 1.6 1.7 1.8 1.8.1 1.8.1.1 1.8.1.2 1.8.1.3 1.8.1.4 1.8.1.5 1.8.1.6 1.8.1.7 1.8.1.8 1.8.1.9

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1.8.2.1 Case A . Determining Heat Transfer Area or Cooling Water Temperature . . . . . . . . . . . . . . . . . . . 1.8.2.2 Case B. Heating of a Filling Tank . . . . . . . . . . . . 1.9 Chemical and Biological Reaction Systems . . . . . . . . . 1.9.1 Modes of Reactor Operation . . . . . . . . . . . . . . 1.9.1.1 Batch Reactors . . . . . . . . . . . . . . . . . . . . 1.9.1.2 Semi-Continuous or Fed-Batch Operation . . . . . . . . . 1.9.1.3 Continuous Operation . . . . . . . . . . . . . . . . . 1.9.2 Reaction Kinetics . . . . . . . . . . . . . . . . . . . 1.9.2.1 Chemical Kinetics . . . . . . . . . . . . . . . . . . . 1.9.2.2 Biological Reaction Kinetics . . . . . . . . . . . . . . . 1.9.2.3 Simple Microbial Growth Kinetics . . . . . . . . . . . . 1.9.2.4 Substrate Uptake Kinetics . . . . . . . . . . . . . . . . 1.9.2.5 Substrate Inhibition of Growth . . . . . . . . . . . . . . 1.9.2.6 Additional Forms of Inhibition . . . . . . . . . . . . . . 1.9.2.7 Other Expressions for Specific Growth Rate . . . . . . . . 1.9.2.8 Multiple-Substrate Kinetics . . . . . . . . . . . . . . . 1.9.2.9 Structured Kinetic Models . . . . . . . . . . . . . . . . 1.9.2.10 Interacting Micro-Organisms . . . . . . . . . . . . . . . 1.10 Modelling of Bioreactor Systems . . . . . . . . . . . . . 1.10.1 Stirred Tank Reactors . . . . . . . . . . . . . . . . . . 1.10.2 Modelling Tubular Plug-Flow Reactor Behaviour . . . . . . 1.10.2.1 Steady-State Balancing . . . . . . . . . . . . . . . . . 1.10.2.2 Unsteady-State Balancing . . . . . . . . . . . . . . . . 1.11 Mass Transfer Theory . . . . . . . . . . . . . . . . . . 1.11.1 Phase Equilibria . . . . . . . . . . . . . . . . . . . . 1.11.2 Interphase Mass Transfer . . . . . . . . . . . . . . . . 1.11.2.1 Case A . Steady-State Tubular and Column Modelling . . . . 1.11.3 Case Studies . . . : . . . . . . . . . . . . . . . . . 1.11.3.1 Case A . Aeration of a Tank of Water . . . . . . . . . . . 1.11.3.2 Case B . Biological Oxidation in an Aerated Tank . . . . . . 1.11.3.3 Case C. Determination of Biological Oxygen Uptake Rates by a Dynamic Method . . . . . . . . . . . . . . . 1.11.4 Gas-Liquid Phase Transfer Across a Free Surface . . . . . . 1.12 Diffusion and Biological Reaction in Solid Phase Biosystems . 1.12.1 External Mass Transfer . . . . . . . . . . . . . . . . . 1.12.2 Finite Difference Model for Internal Transfer . . . . . . . 1.12.3 Case Studies for Diffusion with Biological Reaction . . . . . 1.12.3.1 Case A . Estimation of Oxygen Diffusion Effects in a Biofilm . . . . . . . . . . . . . . . . . . . . . . . 1.12.3.2 Case B . Biofilm Nitrification . . . . . . . . . . . . . . 1.13 Process Control . . . . . . . . . . . . . . . . . . . . 1.14 Optimisation. Parameter Estimation and Sensitivity Analysis . 1.14.1 Case A . Estimation of Bioreaction Kinetic Parameters for Batch Degradation Using ESL and SIMUSOLV . . . . .

43 44 45 45 45 47 48 50 50 52 53 56 57 59 59 61 62 62 65 65 67 67 68 70 70 70 72 73 73 75 77 78 79 82 83 86 86 87 90 94 96

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2

Environmental Bioprocess Descriptions

2.1 2.1.1 2.1.2 2.1.2.1 2.1.2.2 2.1.2.3 2.1.2.4 2.1.3 2.1.3.1 2.1.3.2 2.1.3.3 2.1.3.4 2.1.3.5 2.1.3.6 2.1.4 2.1.4.1 2.1.4.2 2.1.4.3 2.2 2.2.1 2.2.2 2.2.3 2.2.3.1 2.2.3.2 2.2.3.3 2.2.3.4 2.2.4 2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.2 2.3.2.1 2.3.2.2 2.3.2.3 2.3.3 2.3.3.1 2.3.3.2 2.3.3.3 2.3.3.4 2.3.3.5

Wastewater Treatment Processes . . . . . . . . . . . . . 103 Wastewater Characteristics . . . . . . . . . . . . . . . 103 Physical Characteristics and Analyses of Wastewater . . . . . 106 Total Suspended Solids . . . . . . . . . . . . . . . . . 106 Volatile Suspended Solids . . . . . . . . . . . . . . . . 106 Temperature . . . . . . . . . . . . . . . . . . . . . 107 Other Physical Parameters . . . . . . . . . . . . . . . 107 Chemical Characteristics and Analyses of Wastewater . . . . 108 Biochemical Oxygen Demand . . . . . . . . . . . . . . 108 Chemical Oxygen Demand . . . . . . . . . . . . . . . 109 Total Organic Carbon . . . . . . . . . . . . . . . . . . 110 Nitrogen Analyses . . . . . . . . . . . . . . . . . . . 111 Kjeldahl Total Nitrogen Test . . . . . . . . . . . . . . 111 Phosphorus . . . . . . . . . . . . . . . . . . . . . . 112 Biological Characteristics and Analyses of Wastewater . . . . 112 Pathogenic Micro-Organisms . . . . . . . . . . . . . . 113 Pollution Indicator Organisms . . . . . . . . . . . . . . 113 Micro-Organisms Responsible for Biological Treatment . . . 113 Primary Treatment Processes . . . . . . . . . . . . . . 114 Equalisation . . . . . . . . . . . . . . . . . . . . . 114 Neutralisation . . . . . . . . . . . . . . . . . . . . . 115 Sedimentation . . . . . . . . . . . . . . . . . . . . . 115 Discrete Settling . . . . . . . . . . . . . . . . . . . . 115 Flocculent Settling . . . . . . . . . . . . . . . . . . . 116 Zone Settling . . . . . . . . . . . . . . . . . . . . . 117 Coagulating Agents . . . . . . . . . . . . . . . . . . . 117 Flotation . . . . . . . . . . . . . . . . . . . . . . . 117 Secondary Treatment Processes . . . . . . . . . . . . . . 118 The Activated Sludge Process . . . . . . . . . . . . . . 118 Biology of the Activated Sludge Process . . . . . . . . . . 119 Process Analysis . . . . . . . . . . . . . . . . . . . . 119 Modifications of the Activated Sludge Process . . . . . . . . 120 Aerobic Fixed Film Processes . . . . . . . . . . . . . . 121 Trickling Filters . . . . . . . . . . . . . . . . . . . . 122 Fluidised Sand Beds . . . . . . . . . . . . . . . . . . 122 Rotating Biological Contactors . . . . . . . . . . . . . . 124 Anaerobic Treatment Processes . . . . . . . . . . . . . . 124 Reactions and Stoichiometry in Anaerobic Digestion . . . . . 125 Modelling Anaerobic Reactors . . . . . . . . . . . . . . 125 Response Dynamics of Anaerobic Reactors . . . . . . . . 129 Control of Anaerobic Reactors . . . . . . . . . . . . . . 130 Anaerobic Reactor Design . . . . . . . . . . . . . . . . 134

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XIV 2.3.4 2.3.4.1 2.3.4.2 2.3.4.3 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.5 2.5.1 2.5.1.1 2.5.1.2 2.5.1.3 2.5.1.4 2.5.2 2.5.2.1 2.5.2.2 2.5.2.3 2.5.2.4 2.5.3 2.5.3.1 2.5.3.2 2.5.3.3 2.6 2.6.1 2.6.1.1 2.6.1.2 2.6.2 2.6.2.1 2.6.2.2 2.6.2.3 2.6.2.4 2.6.3 2.6.3.1 2.6.3.2 2.6.3.3 2.6.4 2.6.4.1 2.6.4.2 2.6.5 2.6.5.1 2.6.5.2

Table of Contents

Biofilms and Flocs . . . . . . . . . . . . . . . . . . Formation of Biofilms . . . . . . . . . . . . . . . . Modelling Biofilms . . . . . . . . . . . . . . . . . . Bioflocs . . . . . . . . . . . . . . . . . . . . . . . Tertiary Treatment Processes . . . . . . . . . . . . . . Grass Plots . . . . . . . . . . . . . . . . . . . . . . Lagoons . . . . . . . . . . . . . . . . . . . . . . . Filters . . . . . . . . . . . . . . . . . . . . . . . . Microstrainers . . . . . . . . . . . . . . . . . . . . . Membrane Technology . . . . . . . . . . . . . . . . Water Pollution Modelling . . . . . . . . . . . . . . . Eutrophication of Lakes and Reservoirs . . . . . . . . . Factors Influencing Lake Productivity . . . . . . . . . . Consequences of Eutrophication . . . . . . . . . . . . Eutrophication Models . . . . . . . . . . . . . . . . Prevention and Reversal of Eutrophication . . . . . . . . Discharge of Pollutants into Rivers and Streams . . . . . . Modifications to the Streeter-Phelps Theory . . . . . . . Surface Reaeration . . . . . . . . . . . . . . . . . . River Parameters . . . . . . . . . . . . . . . . . . Photosynthesis and Respiration of Green Plants and Algae . Groundwater Pollution . . . . . . . . . . . . . . . . Sources of Groundwater Pollution . . . . . . . . . . . . Modelling Groundwater Pollution . . . . . . . . . . . . Nitrates in Groundwater . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Waste Treatment and Disposal Sources of Solid Wastes . . . . . . . . . . . . . . . . Municipal Solid Waste . . . . . . . . . . . . . . . . Waste Sludge . . . . . . . . . . . . . . . . . . . . . Sludge Processing and Disposal . . . . . . . . . . . . Summary of Disposal Methods . . . . . . . . . . . . Sludge Thickening and Dewatering . . . . . . . . . . . Use of Sludge on Agricultural Land . . . . . . . . . . . Dumping at Sea and Pipeline Discharges . . . . . . . . . Composting . . . . . . . . . . . . . . . . . . . . . . Composting Processes . . . . . . . . . . . . . . . . Compost Ecology . . . . . . . . . . . . . . . . . . Process Factors . . . . . . . . . . . . . . . . . . . . Disposal of Municipal Solid Waste . . . . . . . . . . . . . . . . . . . Microbiology of Landfill Gas Production Landfill Gas Production . . . . . . . . . . . . . . . . Anaerobic Digestion . . . . . . . . . . . . . . . . . Reactor Conditions . . . . . . . . . . . . . . . . . Comparison with Landfill Sites . . . . . . . . . . . . .

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135 136 138 139 140 140 141 141 142 142 146 146 147 151 153 155 157 158 159 160 161 161 163 165 168 169 169 169 170 171 171 172 172 172 173 173 174 175 177 177 179 181 181 181

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2.9

References for Chapters 1 and 2 . . . . . . . . . . . . Recommended Textbooks and References for Further Reading . . . . . . . . . . . . . . . . . . . . . . . Glossary . . . . . . . . . . . . . . . . . . . . . . .

3

Simulation Examples of Environmental Bioprocesses

3.1 3.1.1 3.1.2

Introductory Examples . . . . . . . . . . . . . BASIN - Dynamics of an Equalisation Basin . . . . OXSAG - Classic Streeter-Phelps Oxygen Sag Curves

3.2

Basic Biological Reactor Examples . . . . . . . . . . . BATREACT . Batch Growth and Substrate Uptake . . . . CONTANK . Continuous Tank Reactor Startup and Operation . . . . . . . . . . . . . . . . . . . . . . Semi-Batch Reactor with Batch Startup . . . SEMIBAT . UPTAKE . Substrate Uptake with Monod Kinetics . . . . OXIBAT .Oxidation of Substrate in an Aerated Tank . . . RESPMET . Oxygen Uptake Experiment with Respirometer . . . . . . . . . . . . . . . . . . . . . REPEAT .Repeated Fed-Batch Culture . . . . . . . . . CONTI . Continuous Reactor with Substrate Uptake . . . FEEDINH . Control of Inhibitory Substrate Feed Rate to a Continuous Reactor . . . . . . . . . . . . . . . CONINHIB . Continuous Bioreactor with Inhibitory Substrate . . . . . . . . . . . . . . . . . . . . . . .

2.7 2.8

3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8 3.2.9 3.2.10

3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.3.8

.

182 187 192

. . . . .

201

. . . . 201 . . . . 201 . . . . 206 . 210

.

210

. . .

214 218 221 223

. .

226 229 233

.

235

'

239

Activated Sludge Wastewater lleatment Processes . . . . . . 244 ANDREWS . Model of a Batch Activated Sludge Process . . . 244 ASCSTR . Continuous Stirred Tank Reactor Model of Activated Sludge . . . . . . . . . . . . . . . . . . . 247 Plug-Flow Model of an Activated Sludge Process . 250 ASPLUG . CURDS . Curds' Model of Sludge Ecology . . . . . . . . 254 FLOCl and FLOC2 Diffusion and Reaction in a Sludge Floc . 260 ASTEMP -Temperature Gains and Losses in an Activated Sludge Process . . . . . . . . . . . . . . . . . . . . 264 STEPFEED . Step Feed Activated Sludge Process with Structured Kinetics . . . . . . . . . . . . . . . . . . 270 SETTLER . Solids-Liquid Separation in a Continuous Settler . 275

XVI

Table of Contents

3.4 3.4.1 3.4.2 3.4.3

Fixed Film Reactors for Wastewater Tkeatment . . . . . ROTATE - Model of a Rotating Biological Disc Reactor TRICKLE - Model of a Trickle Filter . . . . . . . . BIOFILM - Biofilm Tank Reactor . . . . . . . . .

3.5 3.5.1

Nitrification of Wastewater , , . . . . . . . . . ACIlVITR - Nitrification in a Single-Stage Activated Sludge System . . . . . . . . . . . . . . . . AMMONOX - Continuous Nitrification with Immobilised Biomass . . . . . . . . . . . . . AMMONFED - Fed-Batch Nitrification with Immobilised Biomass . . . . . . . . . , . . . NITBED - Nitrification in a Fluidised Bed Reactor

3.5.2 3.5.3 3.5.4

3.6 3.6.1 3.6.2

. . . .

. . . .

280 280 . . 284 . . 288

. . . . .

292

. . . . .

292

. . . . . . . , . . . . . . . . . . . .

Primary Watment of Wastewater SEDIMENT - Removal of Solids in a Sedimentation Tank LAGOON - Aerated Lagoon for the Treatment of Industrial Wastewaters . . . . . . . . . . . . . .

. .

296

. . . .

300 302

. . . .

307 307

. . .

311

3.7 3.7.1

Tertiary Water 'keatment Processes . . . . , . FILTER -Tertiary Water Treatment by Filtration

. . . . . , . . , . . .

315 315

3.8 3.8.1

Sludge Disposal and Processing . . . . . . . . COMPOST - Microbial Kinetics in a Continuous Compost Reactor . . . . . . . , . . , . . WINDROW -Batch Windrow Compost Process

. . . . .

.

319

. . . . . . . . . . . .

319 329

3.8.2

3.9.4

Biodegradation Processes . . . . . . . . . . . . . . PCPDEG and PCPDEGCF - Batch and Continuous Biodegradation of Pentachlorophenol by Mixed Cultures . BIOFILTl and BIOFILT2 - Biofiltration Column for Removing Ketones from Air . . . . . . . . . . . . . DCMl and DCM2 - Airlift Biofilm Sandbed for Dichloromethane-Waste Air Treatment . . . . . . . . . FBR - Biofilm Fluidised Bed with External Oxygen Supply

3.10 3.10.1 3.10.2

Anaerobic Digestion Processes . . . . . . . . . . . ANAEROBE - Andrew's Model of Anaerobic Digestion WHEY - Model for the Anaerobic Degradation of Whey

3.9 3.9.1 3.9.2 3.9.3

. .

332

. .

332

. .

335

. . . .

341 347

. . . , . . . . .

350 350 357

XVII

Table of Contents

Anaerobic Fixed Film Processes . . . . . . . . . DENITRIF - Denitrification of Drinking Water in a Fluidised Bed Reactor . . . . . . . . . . . . . MOLASSES - Anaerobic Degradation of Molasses in a Packed Bed Reactor . . . . . . . . . . . . . .

. . . . 366

3.12 3.12.1 3.12.2 3.12.3

Microbial Interaction Kinetics . . . . . . . . . MIXPOP - Predator-Prey Population Dynamics . TWOONE -Competition Between Organisms . . COMPETE - Lotka-Volterra Model of Competition

. . . .

3.13 3.13.1 3.13.2

Ecological Population Studies . . . . . . . . . . . . . . 380 BLOWFLY - Cycling Populations of the Australian Blowfly . . 380 BUDWORM - Dynamics of the Growth of the Spruce Budworm in Canada . . . . . . . . . . . . . . . . . . 382

3.14 3.14.1

River and Stream Modelling . . . . . . . . . . . . . . RIVER - Dissolved Oxygen and BOD Dynamic Profiles along a River . . . . . . . . . . . . . . . . . . . . . STREAM - One-Dimensional Steady-State Model of Aeration and Degradation in a Stream . . . . . . . . . . DISCHARG - Dissolved Oxygen and BOD Steady-State Profiles along a River . . . . . . . . . . . . . . . .

. 387

Lake and Reservoir Modelling . . . . . . . . . . . . . NCYCLE - Nitrogen Cycles in a Reservoir in Slovakia . . . PCYCLE - Phosphorus Cycles in a Lake . . . . . . . . ALGAE -Algal Growth in a Deep Lake in Canada . . . . EUTROPH - Eutrophication in a Shallow Lake in Hungary . METAL -Transport of Heavy Metals in Water Column and Sediments . . . . . . . . . . . . . . . . . . . . .

3.11 3.11.1 3.11.2

3.14.2 3.14.3

3.15 3.15.1 3.15.2 3.15.3 3.15.4 3.15.5 3.16 3.16.1 3.16.2 3.16.3

3.17 3.17.1 3.17.2 3.17.3

. . . .

. . . .

363

. . . . 363

. . . 373 . . . 373 . . . 376 . . . 378

387

.

391

.

396

. . . . .

400 400 408 414 421 429

Land Pollution Modelling . . . . . . . . . . . . . . . . LANDFILL - One-Dimensional Transport of Pollutant from a Landfill Site . . . . . . . . . . . . . . . . . . LEACH - One-Dimensional Transport of Solute Through Soil . SOIL - Bioremediation of Soil Particles . . . . . . . . . . Miscellaneous Examples . . . . . . . . . . . . . . . . DEADFISH - Distribution of an Insecticide in an Aquatic Ecosystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GAIA - The Parable of Daisyworld CABBAGE - Structured Growth of the White Cabbage . .

433 433 437 443

. 449

. .

449 452 456

XVIII

Table of Contents

Appendix: Instructions for Using ISIM

. . . . . . . . . . . . . . .

5

ISIM Installation Procedure . . . . . . . . . . . . . . Programming with ISIM . . . . . . . . . . . . . . . . Getting Started . . . . . . . . . . . . . . . . . . . . Reading Files From Disk . . . . . . . . . . . . . . . Running Simulations . . . . . . . . . . . . . . . . . Interacting with ISIM Simulations . . . . . . . . . . . . Editing ISIM Files . . . . . . . . . . . . . . . . . . Writing ISIM Models . . . . . . . . . . . . . . . . . ISIM Statements and Functions . . . . . . . . . . . . output . . . . . . . . . . . . . . . . . . . . . . . . Useful Sequences of Statements . . . . . . . . . . . . Further Information . . . . . . . . . . . . . . . . . Summary of ISIM Commands . . . . . . . . . . . . . COMMAND and INPUT modes . . . . . . . . . . . . ISIM Commands . . . . . . . . . . . . . . . . . . . . ISIM Program Statements . . . . . . . . . . . . . . . ISIM Error Messages . . . . . . . . . . . . . . . . . Monitor and Syntax Failures . . . . . . . . . . . . . . Compilation Failures . . . . . . . . . . . . . . . . . Execution Phase Errors . . . . . . . . . . . . . . . . . . . . . Quick Reference to Common ISIM Commands

Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 3 3.1 3.2 3.3 4 4.1 4.2 4.3

461

. .

461 462 462 . 462 . 462 . 463 . 463 . 464 . 466 467 . 469 . 470 . 470 . 470 470 . 474 . 477 . 478 . 479 . 480 . 481 483

Nomenclature for Chapters 1 and 2 Symbol

Description

Units

a a A A, b

Specific interfacial or surface area Constant in Logistic equation Surface or cross-sectional area Cross-sectional area Constant in Logistic equation Molar heat capacity Drag coefficient Anion concentration Cation concentration Equilibrium concentration in liquid phase Equilibrium concentration in gas phase Concentration Particle diameter Depth Diffusion coefficient Hydrodynamic dispersion coefficient Dispersion coefficient Activation energy Eddy diffusivity Gravitational constant Gradient Volumetric gas flow rate Hydraulic head Partial molar enthalpy Henry's coefficient Depth of river, lake etc. Enthalpy change Hydraulic gradient Concentration of inhibitor Flux Total mass flux; Mass transfer rate Dispersion flux Rate constant Hydraulic constant

m2 m-3 h-I m2 mL m3 kg-' h-1 kJ kg-I K-I mole L-1 mole ~ - 1

cP

CD CA CC CL* CG* C d D D Dhd DS E ED g grad G h h H H AH 1

I j J JS

k k

mg ml-1; mmole L-1 mg ml-1; mmole L-1 mg ml-I; mmole L-1 m m m2 s-1 m2 s-1 m2 s-1 J mole-I m2 s-1 m s-2 m- 1 m3 h-I m kJ kg-1 bar m3 kg-' m kJ mole-I; kJ kg-I mg ml-1; mmole L-1 kg m-2 h-l; mole m-2 h-1 kg S - I ; mole h-1 kg m-2 h-l; mole m-2 h-' various

xx

Nomenclature for Chapters 1 and 2

~

kC kC kCL

kd kF kSL

K K Ka KI KLa KM KP KS L L m m m M n n N N

P P P P q02

Q r rQ '02

R R R Re Ra S Sh t

Reaction rate constant Concentration parameter Saturation kinetics constant Death rate coefficient Flow parameter Mass transfer coefficient Constant in Monod equation Mass transfer coefficient Atmospheric reaeration coefficient Inhibition coefficient Mass transfer coefficient Constant in Darcy's Law Proportional gain Saturation constant Volumetric liquid flow rate Length; Eddy length; Biofilm thickness Equilibrium constant Maintenance coefficient Constant in Freundlich equation Mass Hydraulic constant Number of moles Total number of segments Molar flow rate Partial pressure Surface oxygen production Total pressure Product concentration Specific oxygen uptake rate Volumetric flow rate Reaction rate Rate of heat production Rate of oxygen uptake Universal gas constant Surface respiratory demand Total reaction rate or rate Reynolds number Rate of oxygen transfer (reaeration) Substrate concentration Sherwood number Time

hkg m-3 kg m-3 hm3 h-' m s-1 kg m-3; mole m-3 m h-l hkg m-2 h-l h-1 m s-1 mg ml-1; mmole L-' m3 h-' m various h-

mole h- 1 bar g 0 2 m-2 h-1 bar mg ml-1; mmole ~ - 1 g 0 2 g biomass - l h-l m3 h-1. m3 s-1 kg m-3 h-l; kmole m-3 hJ s-l g s-1; mmole s-1 J mole-l K-' g 0 2 m-2 h-l kg h-l; kmole h-' g m-3 h-l mg m1-1; mmole ~ - 1 various

XXI


T TI12 Urms

U v, v Vmax VS

V Vm VM W X

XA

X Y Y

Y Z Z Z

Temperature Radioactive half life Root mean square eddy velocity Heat transfer coefficient Velocity Maximum rate of substrate consumption Sedimentation velocity Volume Groundwater microscopic flow rate Groundwater macroscopic flow rate Width of river, lake, etc. Mole fraction Fractional conversion of A Cell concentration Mole fraction in gas phase Variable name Yield coefficients Axial distance Frequency factor Ionic charge

Greek

a

P

A

6 E E

77

h

P P Pmax V @C

@m

P

c z z

Reaction order, constant Reaction order, constant Difference operator Partial differential operator Control error Fractional gas holdup Efficiency Constant in Moser equation Dynamic viscosity Specific growth rate Maximum specific growth rate Stoichiometric coefficient Sludge age Sludge residence time Density Summation operator Controller time constant Residence time

OC or K h m s-1 J ,-1 m-2 OC-I m s-1 g m-3 h-' m s-1 m3 m h-l m h-l m kg m-3

XXII z ZE

TG


Tortuosity (hydrology) Electrode time constant Gas dynamics time constant

Indices

* 0 1 2 , 3,...., n a, b, ..., f a amb air agit aPP A Ac B Bu d d D f G hY d i, j in I lim L max n out P P phot Pr 4/02 Q/S r resp R S

set

Equilibrium concentration Initial; Inlet; External values Component 1; Outlet conditions Components 2, 3, ...., n Constants Adsorption Ambient conditions Air conditions Agitation Apparent value Component A Acetic acid Bulk conditions; Component B Butyric acid Death Desorption Derivative control Final; Outlet conditions Gas phase Hydrodynamic Components i, j Inlet conditions Inhibition; Integral control Limiting value Liquid Maximum value Segment; Reaction order Outlet conditions Particle; Pollutant Product; Proportional control Photosynthesis Propionic acid Heat yield per unit oxygen uptake Heat yield per unit substrate uptake Reaction Respiration Recycle Surface value Set point

S S


S tot W X XL a 9

P

Solid; Steam; Total number of components Total value Wall conditions Biomass Biomass loss Orders of reaction

XXIII

Dynamics of EnvirOnmental Bioprocesses: Modelling and Simulation

Jonathan B. Snape, Irving J. D u n , John Ingham & Jiii E. pkenosil OVCH Verlagsgesellschaft mbH, 1995

1

Modelling Principles

1.1

The Role of Modelling in Environmental Technology

Understanding the environment we live in and man's interaction with the environment, protecting the environment from damage and rectifying the damage already caused are essential to the long term survival of the planet. The range of environmental problems is immense and covers all aspects of our daily lives, and their study embraces many traditional subject areas, in which the environmental engineer plays a very positive role. In this chapter, the basic engineering modelling concepts relevant to environmental issues are introduced and their applications are illustrated via the use of simplified case examples. Engineers have had a direct influence on many of the environmental problems we face today; for instance, the pollution caused by chemical industries, car exhaust fumes, mining and smelting and ecological damage caused by the construction of reservoirs and roads. The engineer nowadays has an important role to play in designing processes that help to improve the environment - for instance, better wastewater treatment plants, air filters, the production of biodegradable or reusable products - and in minimising the environmental impact for any new project. The modern concept of waste minimisation in process design is an important tool in this area. Engineering principles can also be applied to the study of natural processes such as the flow of water in rivers and lakes, the transport of solutes through soil and the mixing of water bodies by wind, etc. Environmental bioprocesses consist of complex interactions between physical, chemical and biological processes. The most important of these can be expressed in engineering terms using the scientific and engineering techniques of mass and energy balances, microbial and population kinetics, thermodynamics, transport processes, chemical and biochemical reactions, mass and heat transfer and stoichiometry. These aspects of the basic modelling process constitute the major part of this chapter. In addition many other factors may influence the process such as climate, human intervention, geography, natural disaster, etc. It is difficult to synthesise this large volume of material mentally and almost impossible to predict how even the simplest process will behave under different conditions. Mathematical models can be used to tie together this material into a more unified and understandable package which can be used for the purposes of prediction, process control, design, education and management.

2

1 Modelling Principles

A good mathematical model is a compromise between accuracy, applicability and clarity. It is often desirable to be able to predict future events with a high degree of accuracy which consequently requires a rather complex model. The greater the complexity of the model, however, the greater will be the difficulty in determining appropriate values for the model parameters and the less generally applicable the model will be. For instance, a model may be developed to predict the influence of an industrial effluent discharge on the dissolved oxygen concentrations in a river. An accurate prediction may be obtained by one particular model for the river in question, but this might give unsatisfactory results when applied to another river under different conditions. An alternative model could give more generally applicable results but at the cost of a loss of accuracy. In addition, an increase in complexity can lead to a reduction in understandibility. If the model is to be used by others, then it needs to be presented in a clear and understandable way if it is not to remain a mathematical curiosity. When developing a model it is therefore essential that the proposed use of the model is clearly understood and kept in mind during the model development. One final point: mathematical modelling can not replace sound experimental techniques and data, but it can prove to be an excellent way of presenting complex ideas in an efficient form. The approach in this book is to concentrate on a simplified treatment of the dynamic modelling and simulation of environmental engineering applications. Nevertheless, quite realistic process phenomena can often be described by relatively simple models, as exemplified by this text. Often a simplified approach can be very useful in clarifying preliminary ideas before tackling the full real life problem in all its complexity. No attempt is made here to deal with large-scale environmental problems, such as climatological, meteorological or geographical effects. Complex modelling software packages to solve largescale problems, such as chemical spills into rivers, estuaries or at sea, gaseous discharges from stacks, the effects of automobile emissions in urban centres or discharges into multimedia environments, are very much the concern of specialist agencies and are generally not suitable for teaching purposes - the main concern of this book. The four basic tenets of mathematical modelling, shown in Fig. 1.1, are very much a matter of common sense (Kapur 1988) and can be summarised as follows: 1. The mathematical model can only be an approximation of real-life processes which are often extremely complex and often only partially understood. Thus models are themselves neither good nor bad but should satisfy some previously well defined aim.

2. Modelling is a process of continuous development in which it is generally advisable to start off with the simplest conceptual representation of the process and to build in more and more complexities as the model develops.

3

1.1 The Role of Modelling in Environmental Technology

Starting off with the process in its most complex form often leads to confusion.

3. Modelling is an art but also a very important learning process. In addition to a mastery of the relevant theory, considerable insight into the actual functioning of the process is required. One of the most important factors in modelling is to understand the basic cause and effect sequence of individual processes.

4. Models must be both realistic and robust. A model which predicts effects which are quite contrary to common sense or to normal experience is unlikely to be received with confidence.

I

Start simple Build In comDlexlties later

I

I

. Use the model to learn

I ~~

Models are there to be applled

I

Figure 1.1. The basic principles of model building.

1.2

General Aspects of the Modelling Approach

An essential stage in the development of any model is the formulation of appropriate mass and energy balance equations. To these may be added additional relationships representing: rates of chemical reaction, rates of heat and mass transfer, system property changes, phase equilibria and control. The combination of these relationships provides a basis for the quantitative description of the process and comprises the basic mathematical model. The resulting model can range from a simple case of relatively few equations to models of great complexity. The greater the complexity of the model, however, the greater is then the difficulty in identifying the increased number of

4


parameter values. One of the skills of modelling is thus to derive the simplest possible model, capable of a realistic representation of the process. The application of a combined modelling and simulation approach leads to the following advantages: 1. Modelling improves understanding. 2. Models help in experimental design.

3. Models may be used predictively.

4. Models can be used in training. 5. Models can be used to improve processes.

1.3

Model Classification

Many different approaches to the modelling of environmental bioprocesses have been adopted. A distinction can be drawn between deterministic models and stochastic models, and between steady-state and dynamic models.

1.3.1

Deterministic Models

Models in which all the parameters have defined values are termed deterministic. The majority of models presented in this course are deterministic in that they do not take into account random environmental fluctuations and assign constant values to the model parameters such as growth rates, flow rates, volumes, concentrations and temperatures. In many processes this is a reasonable assumption and deterministic models can often give such good prediction that extra complexity introduced by assigning statistically variable parameter values is often not worthwhile. This applies especially to wastewater treatment process modelling applications where certain operating parameters usually are maintained constant.

1.3 Model Classification

1.3.2

5

Stochastic Models

Stochastic models can be used to represent random environmental fluctuations in parameters such as temperature, rainfall, effluent flow rate and concentration. Often a mean, or expected value is assigned to a parameter (this may be timedependent) and a time-dependent variation superimposed on to this value. Some parameters do not have a fixed value but show a distribution of values which may then be represented by a stochastic model. Examples of such parameters include the distributions in the age of a population, the bubble size and velocity in an aerated reactor, the frequency of sexual contact between different population groups and distributions in soil particle size. Statistical distributions, such as Gaussian, Poisson, binomial, log normal etc. can be used to describe the experimental data. Under such conditions, the model can be run repeatedly with parameter values picked at random from these distributions to generate a distribution of possible outcomes in a so-called Monte Carlo simulation. The advantage of stochastic models is that they can predict the occurrence of low frequency events which may have critical implications for the process but which are not predicted by a deterministic model. The main disadvantage is that although the mean value of a parameter may often be known with a reasonable degree of accuracy, the distribution of the values is usually not known with the same degree of accuracy, and this can lead to inaccuracies. The reader is referred to Hahn and Shapiro (1967) for a more detailed discussion on the application of stochastic models.

1.3.3

Steady-State Models

Models can also be classified as dynamic or steady state, depending upon whether conditions are changing with time. Steady-state models have the advantage that they are mathematically simpler than the equivalent dynamic model and the resulting sets of algebraic equations representing the model can often be solved analytically to derive relatively simple design and operational relationships. Many wastewater treatment processes, for example, often run at steady-state conditions and steady-state models are commonly used for their design, operation and scale-up.

6


Dynamic Models

1.3.4

All natural environmental bioprocesses are dynamic and although conditions may often approach those of steady state, an understanding of the dynamics is essential if the influence of changes in conditions is to be predicted. Even in wastewater treatment plants designed to operate at steady state, a dynamic model is often essential to be able to model startup and the influence of perturbations on the system. Dynamic models are often very complex as they are composed of sets of both algebraic and differential equations, which until the advent of modern computer-based numerical integration techniques were extremely difficult to solve. Most environmental bioprocesses are described by sets of non-linear dynamic model equations and are consequently very difficult or impossible to solve analytically. As shown in this text, the advent of digital simulation languages has, to a large extent, overcome this problem.

1.4

General Modelling Procedure

One of the more important features of modelling is the frequent need to reassess both basic theory and the mathematical equations in order to achieve a realistic outcome. As shown in Fig. 1.2, the following stages in the modelling procedure can be identified: - The

first involves the proper definition of the problem.

- The

available theory must then be formulated in mathematical terms.

- Having

developed a model, the equations must be solved.

-The computer prediction must be checked and if necessary the model revised. As mentioned earlier, environmental models are usually complex and numerical methods of solution must therefore be employed. The method preferred in this text is that of digital simulation.

1.4 General Modelling Procedure

7

Figure 1.2. Information flow in model building and its verification.

1.5

Simulation Tools

Many different digital simulation software packages are commercially available. Fortunately many, but not all, conform to the standard structure of a Continuous System Simulation Language (CSSL). The programming structures for all CSSL languages are very similar. In addition, all CSSL languages are adjuncts to other high level languages such as FORTRAN, PASCAL or BASIC and thus provide the programmer with all the facilities and all the power of the host language. Other interesting features of simulation tools concern user interfacing, computing power, portability to various computer systems, optimisation and parameter estimation. Some tools allow a graphical set-up of a model. Some languages, for example ACSL and ESL, a development by the ISIM-group as advertised in the back of this book, run on both PC's and workstations, and include more powerful numerical algorithms and supply many predefined building blocks, which facilitate such tasks as the modelling of control systems.

8


MATLAB and MATHEMATICA are more mathematically oriented software packages allowing matrix and vector operations, numerical integration and optimisation. MATHEMATICA additionally allows algebraic integration and differentiation as well as calculation with user defined precision. These two programs are available for PC (IBM, Apple), workstations and larger machines up to supercomputers (e.g., Cray). One useful application of modelling and simulation is that of optimisation. ESL, MATLAB and SIMUSOLV include powerful algorithms for non-linear optimisation, which can also be applied for parameter estimation. For this latter purpose SIMUSOLV is an excellent tool, especially for practical work, owing to its flexibility in handling experimental data (Heinzle and Saner, 1991). SIMUSOLV uses ACSL as a subset, adding very user friendly optimisation and parameter estimation routines to it, but it is only available on workstations and larger machines. STELLA may be called a simulation environment and uses a rather different approach. Running on Macintosh computers, the setup of a model uses interactive graphics. Formal consistency of the model is checked immediately. The model may also be seen in equation form. STELLASTACK gives a graphical user interface allowing interaction with the model. STELLA can be recommended for non-experienced computer users because of its graphical, non-equation approach for developing models. Some characteristics of the various simulation programs are given in Table 1.1. Recent surveys of the differing packages now available include those of Matko et al. (1992) and Wozny and Lutz (1991). Digital simulation languages are designed specially for the solution of simultaneous differential equations, by numerical integration. Many fast and efficient numerical integration routines are now available, and many digital simulation languages are able to offer a choice of integration routine. The structure of the language enables very simple programs to be written, with an almost one-to-one correspondence to the basic model equations. The resulting simulation programs are therefore very easy to understand and also to write. A further major advantage is a convenient output of results, in both tabulated and graphical form, obtained via very simple program commands.

9

1.5 Simulation Tools

Table 1.1. Characteristics of available simulation programs. Program

Computer

Characteristics

ISIM

PC

Used for simulation examples in this book, highly interactive, moderate computational power, easy-to-use.

ESL

All types

Graphical interface. Less easy to use. Allows parameter estimation from data and the use of icon block programming. See example in Sec. 1.14.1.

ACSL

All types

Powerful mathematics, building blocks especially for control purposes.

MODEL-WORKS PC and Mac

Pascal-based simulation environment, very flexible, public domain.

SIMNON

PC and others

Non-CSSL standard but very interactive and allowing both continuous and discrete subsystems.

STELLA

Mac

Graphical set up of model, very interactive but rather poor numerical methods.

SIMUSOLV

VAX, IBM, SUN

Simulation with ACSL, user-friendly optimisation and parameter estimation methods.

MATLAB

All types

Powerful mathematical package especially for linear problems, includes optimisation routines.

SIMULINK

Workstations, Mac and PC

Based on MATLAB but with improved integration routines, model building blocks and graphical interface.

MATHEMATICA

All types

Mathematical software, includes analytical and numerical integration.

SPEEDUP

Workstations, and larger

Dynamic and steady-state simulation of large processes with data base.

Addresses: ISIM and ESL, ISIM International Simulation Ltd., Technology House, Salford University Business Park, Lissadel Street, Salford M6 6AP, UK; STELLA, High-Performance Systems Inc., 13 Dartmouth College Highway, Lyme, New Hampshire, 03768, USA; MATLAB, The Math Works, Inc., 21 Elliott Street, South Natick, MA 01760, USA; MATHEMATICA, Wolfram Res. Inc., P.O. Box 6059, Champaign, Illinois 61821, USA; ACSL, Mitchell & Gauthier Associates, 73 Junction Square Dr., Concord, MA 01742-3096, USA; SIMUSOLV, Dow Chemical Chemical Co., Midland, Michigan USA; MODELWORKS, A. Fischlin, Inst. Terrestr. Ecology, ETH, Grabenstr. 1 I , 8952 Schlieren, Switzerland; SIMNON, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden; SPEEDUP, Aspen Technology, Inc. Ten Canal Park, Cambridge, Mass. 02141, USA.

10

1.6


ISIM

The simulation examples in this book are programmed with the ISIM digital simulation language, which is supplied on the diskette. This language is a development from the Electrical Engineering Department of the University of Salford, England. ISIM is FORTRAN-based and conforms to the basic structure of a CSSL in having three regions: the INITIAL, DYNAMIC and TERMINAL regions, together with an optional CONTROL region. The INITIAL region is used for calculating and setting initial values of constants. The DYNAMIC region contains the model equations, and the TERMINAL region may be used for end-of-run calculations. During execution, data values are produced in tabular form by the command OUTPUT, and a graphical output of any one variable by the command PLOT. Post-mortem plots of all variables, stored by the command PREPARE, can be shown using the command GRAPH. The interactive feature of ISIM is particularly valuable. This allows the simulations to be interrupted during execution by pressing any key. Values of constants or parameters can be changed and the simulation continued. This means that changing process conditions can be easily effected without the need for complex programming statements. ISIM is also one of the very few digital simulation programming languages that can be used with PCs. This makes it possible for the reader, student and teacher to experiment directly with the model, in the classroom or at the desk. Probably one of the best ways to learn ISIM is to work directly on the example programs of Chapter 3, rather than by a detailed study of the manual. The introductory example in Sec. 1.7 is specially formulated to explain the main points of program structure and execution. The simulation examples enforce the learning process in a very effective manner and also provide hands-on confidence in the use of the simulation language. Readers can program their own examples, by formulating new mass balance equations or by modifying an existing example to a new set of circumstances. Thus by working directly at the PC, one can experiment directly on the system in a very interactive way. A true degree of interaction is possible with ISIM because at the touch of a few keys a simulation run can be stopped, parameters changed, and the run restarted from the point of interruption. Plotting the variables in any configuration is easy at any time during a run. Runs can be repeated with new parameters, and the results from multiple runs can be plotted for comparison. Full details of the ISIM digital simulation programming language can be found in the appendix, and by reference to the ISIM programs associated with the simulation examples of Chapter 3.

1.7 Introductory ISIM Example

1.7

11

Introductory ISIM Example: WASTE

Periodically, an effluent is contaminated by an unstable noxious waste, which is known to decompose at a rate proportional to its concentration. The effluent must then be diverted to a holding tank, prior to final discharge, as shown in Fig. 1.3.

Figure 1.3. Waste holding tank.

The model equations for this system are

where V is the volume of the contents (m3), Qo and Q1 are the inlet and outlet volumetric flow rates (m3 h-l), CAO and C A I are the inlet and outlet concentrations of the waste component (g m-3), V C A is ~ the mass of component A in the tank at any time (g), k is the first-order decomposition reaction rate constant (h-I) and t is time (h). The tank starts essentially empty, so that the outlet flow from the tank Q1 = 0. When the tank is full, Q1 = Qo, and the condition dV/dt = 0 also applies. These equations are developed in more detail in Sec. 1.8.1.9.

ISIM Program WASTE The following ISIM program, which is on the diskette, simulates the dynamics of the waste holding tank, which is influenced by a sinusoidally changing inlet concentration.

12


1:File WASTE 2:Noxious waste holding tank 3 CONSTANT VO=O.Ol :Empty tank volume, m3 4 CONSTANT VMAX=2 :Maximum tank volume, m3 5 CONSTANT QO=O.5 :Inlet flow rate, m3/h 6 CONSTANT Ql=O :Initial outlet flow rate, m3/h 7 CONSTANT CAmax=lOO:Max. amplitude in conc., g/m3 8 CONSTANT K=5 :DeCOmgOSitiOn rate constant l/h 9 CONSTANT TFIN=5 10 CONSTANT CINT-0.1 11 1 SIM; INTERACT; RESET; GOT0 1 12 INITIAL 13 v=vo :Initial volume 14 VCAl=O :Initial component mass 15 DYNAMIC 16 CAO=450.0+CAmax*SIN(2*3.14*0.5*T) :Sine inlet 17 IF(CAO.LT.O.OOl)CAO=O.O :Limiter on sine wave 18 V'=QO-Ql I :Total mass balance 19 VCAl'=QO*CAO-Ql*CAl-K*CAl*V :Component balance 20 CAl=VCAl/V :Calc. of conc. from mass 21 IF (V.lt.Vmax) Q1=0 :Logical control on overflow 22 IF (V.gt.Vmax) Ql=QO :Logical control on overflow 23 IF (CAl.lt.0.001)CA1=0.O:Limiter for conc. 24 OUTPUT T,V,CAl,Ql 25 PREPARE T, CAl,V,Ql,CAO 26 PLOT T, CAl,O, TFIN, 0,100

Line numbers, as obtained from the LIST command, are not part of the program and are used here simply to explain the program and program structure. Line number 1 is the program file name for the program. 2 is the program title. In each case the colon sign (:) is used to indicate a non-executionable, reference-only statement. 3 to 8 set the constant values for the process. 9 and 10 set the values of the control parameter, CINT, for the print interval and integration step size, and TFIN, for the end of the simulation. 11 involves the mandatory SIM statement. INTERACT and RESET cause the program to stop when T>TFIN and to reset T to zero for the next run, which is started with GO. In this way, all runs appear on the screen, as specified by the PLOT statement. 12 INITIAL specifies the start of the INITIAL region.

1.7 Introductory ISIM Example

13 and 14

15 16 to 20 21 to 23 24 25

26

13

specify the initial conditions of the variables in the differential equations. DYNAMIC specifies the start of the dynamic equation region. represent the model equations. are logical control statements for the outlet flow rate. specifies tabulated values for the OUTPUT variables, at intervals of time given by CINT. specifies the values of all the variables to be stored. All variables in the PREPARE statement can be plotted at the end of the simulation, or after an interrupt using the GRAPH command. The GRAPH T,CA 1,Q 1 command, for example, plots a combined post-mortem graph of these two variables with respect to the independent variable time. specifies an output plot during the simulation. Only one variable can be plotted during program execution. In this case, concentration, CA 1 is plotted versus time, using appropriate scaling commands of the PLOT statement.

Line 11 of the program listing makes use of a special feature in ISIM. This allows those constants and initial conditions which are declared in a CONSTANT statement to be varied, following the end of each run, by using the statement VAL. Thus, the statement 1 SIM; INTERACT; RESET; G O T 0 1 permits interaction at this point, so that constant values can be changed and the results of different runs can be compared using the GRAPH command. Graphical output obtained by the GRAPH command for the above example is shown in Fig. 1.4, for the case of three runs with different values of k.

Figure 1.4. Influence of the rate constant, k (Curve A: 5, Curve B: 1, Curve C: 3), on the concentration in the waste holding tank.

14


Formulation of Dynamic Balance Equations

1.8

The Principle of the Conservation of Mass (or matter) states that the total mass in any system is constant, or in other words, matter can neither be created nor destroyed. This is an important concept in the development of mass balance equations which are an important tool in the study of environmental processes.

Dynamic Total Mass Balances Most real situations are such that conditions change with respect to time. Under these circumstances, the total mass balance relationship is expressed by:

3

Rate of accumulation o mass in the system

(

Rate of = (mass flow in) -

(m a 2 E i f o u t )

Here the rate of accumulation term represents the rate of change in the total mass of the system, with respect to time, and at steady state this is equal to zero. For a steady-state flow process, Rate of mass flow

( into the system )

Rate of mass flow

=

(out of the system

Thus steady-state mass balance equations represent a more specific case for the more generalised dynamic mass balance equation, and for which the rate of accumulation becomes zero.

Component Balances Most process streams contain more than one component. Provided no reaction occurs, the generalised dynamic equation for the conservation of mass can also be applied. Thus for any particular component Rate of accumulation of mass

Mass flow of

in the system

into the system

Mass flow of the component out of the system

1.8 Formulation of Dynamic Balance Equations

15

Component Balances with Reaction Where a chemical or biological reaction occurs, the change due to reaction can be taken into account by the addition of a reaction rate term in the component balance equation. Thus in the case of material produced by the reaction, Rate of

Mass flow

in the system

the system

Mass flow

Rate of

out of

component

Elemental Balances The principle of the component mass balance can also be extended to the atomic level and can also be applied to particular elements. Thus for the case of carbon, in say a wastewater treatment process

[

Rate of Mass flow accumulation] = [rate carbon of into the] of carbon mass in the system the system

[

Mass flow carbon rate of the out

]

of the system

Note that elemental balances do not involve additional reaction rate terms, since the elements are not changed by chemical reaction. While the principle of the mass balance is very simple, its application can often be quite difficult. It is important therefore to have a clear understanding of both the nature of the system (physical model) which is to be modelled by means of the mass balance equations and also of the methodology of modelling.

1.8.1

Mass Balance Procedures

The mass balance equation is a mathematical expression of the Conservation of Mass and accounts for the various chemical and biological species that enter and leave a defined control volume. Several important steps need to be followed to set up a mass balance equation:

16

I.


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

The balance region may be a reactor, a reactor region, a single phase within a reactor, a single cell, or a region within a cell, but will always be based on a region of assumed constant composition. Generally the modelling exercises will involve some prior simplification. Examples of the possible control volumes for various situations are listed below: An individual micro-organism Sludge floc Gas bubble Stirred-tank reactor Plant parts (leaves, roots) Sludge settling tank Whole animals or plants Lake or reservoir Countries Planets

Figure 1.5. Two examples of balance regions.


17

It can be seen (Fig. 1.5) that there is an enormous diversity in the size of the possible control volumes. The control volume is usually continuous, e.g., a lake, tank or country, but it may be discontinuous such as leaves on a tree, bubbles in an aerated tank or micro-organisms in a bioreactor, but this makes no difference to the form of the mass balance equation. Once the control volume has been chosen, the next step is to determine what chemical or biochemical species is to be investigated. Mass balances can be performed on a wide variety of compounds, several of which are listed below: Microbial biomass Dissolved oxygen Ammonia, nitrate and nitrite Human population Pesticides Wastewater (BOD, COD, TOC) ATP Algae Carbon Often the system being modelled will be considered in terms of a conceptual system of tanks or tubes or even combinations of tanks and tubes.

1.8.1.1 Case A.

Continuous Stirred-Tank Reactor

The continuous stirred-tank reactor (CSTR) is an important concept in modelling. Consider a tank of liquid with flows entering and leaving, as shown in Fig. 1.6. If the tank is well-mixed, the concentrations and density of the tank contents are uniform throughout. This means that the outlet stream properties are identical with the tank properties, in this case concentration CA and density p. The balance region can therefore be taken around the whole tank, as shown by the shaded boundary.

Figure 1.6. The balance region around the continuous reactor.

The total mass in the system M (kg) is given by the product of the volume of the tank contents V (m3) multiplied by the density p (kg m-3), thus M=V p.

18


The mass of any component A in the tank is given in terms of actual mass or number of moles by the product of volume V times the concentration of A, CA (kg of A m-3 or kmole of A m-3), thus giving VCA in kg or kmole.

1.8.1.2

Case B. Tubular Reactor

Another important concept for modelling is the plug-flow tubular reactor (PFR). In tubular reactors the concentrations of the products and reactants vary continuously along the length of the reactor, even when the reactor is operating at steady state. For the ideal PFR case of no axial mixing, this variation can be regarded as being equivalent to that of the time of passage of material as it flows along the reactor and is equivalent to the time available for reaction to occur. Under steady-state conditions the concentration at any position along the reactor will be constant with respect to time, though not with position. This type of behaviour, obtained with tubular reactors, can be approximated by choosing the incremental volume of the balance regions sufficiently small so that the concentration of any component within the region can be assumed to be approximately uniform. Thus in this case, many uniform property subsystems (well-stirred tanks or increments of different volume but all of uniform concentration) comprise the total reactor volume. This situation is illustrated in Fig. 1.7.

Figure 1.7. The tubular reactor balance region and approximate concentration gradient.

1.8. 1. 3 Case C. River with Eddy Current For this example, the combined principles of both the stirred tank and differential tubular modelling approaches need to be applied. As shown in Fig. 1.8, the main flow along the river is analogous to that of a column or tubular process, whereas the eddy region can be approximated by the behaviour of a well-mixed tank. The interaction between the main flow of the river and the eddy, with flow into the eddy from the river and flow out from the eddy back into the river's main flow, must be included in any realistic model.


19

The real-life and rather complex behaviour of the eddying flow of the river, might thus be represented, by a series of many well-mixed subsystems (or tanks) representing the main flow of the river. This interacts at some particular stage of the river with a single well-mixed tank, representing the turbulent eddy. In modelling this system by means of mass balance equations, it would be necessary to draw boundary regions around each of the individual subsystems representing the main river flow, sections 1 to 8 in Fig. 1.9, and also around the tank system representing the eddy. This would lead to a very minimum of nine component balance equations being required. The resulting model could be used, for example, to describe, in rather simple terms, the flow of a pollutant down the river (See RIVER). River

Figure 1.8. A complex river flow system.

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

II.

Identify the transport streams which flow across the system boundary

Having defined the balance regions, the next task is to identify all the relevant inputs and outputs to the system (Fig. 1.10). These may be well-defined

20


physical flow rates (convective streams), diffusive fluxes, but may also include interphase transfer rates. Examples include rivers flowing into lakes and wastewater flowing into treatment plants.

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

It is important to assume transfer to occur in a particular direction and to specify this by means of an arrow. Once defined, transfers in the opposite direction are accommodated by negative rate terms, which result from the driving force sign change (see Sec. 1.8.1.8).

III.

Write the mass balance in word form

This is an important step because it helps to ensure that the resulting mathematical equation will have an understandable physical meaning. Just starting off by writing down equations is often liable to lead to fundamental errors, at least on the part of the beginner. All balance equations have a basic logic, as expressed by the generalised statement of the component balance given below, and it is very important that the model equations also retain this. Thus Rate of accumulation of mass in the system

Mass flow of the component into the system

Rate of production of the

’

the system

the reaction

This can be abbreviated as (Accumulation)

=

(In) - (Out)

+

(Production)

21


IV.

Express each balance term in mathematical form with measurable variables

1.8.1.4

Rate of Accumulation Term

This is given by the rate of change of the mass of the system, or the mass of some component within the system, with changing time and is expressed as the derivative of the mass with respect to time. Hence Rate of accumulation of mass (of component i within the system)

=

dMi

(dt)

where M may be in kg or molar units and time will have any consistent time units (years, months, days, hours, minutes or seconds). Volume, concentration and, in the case of gaseous systems, partial pressure are usually the measured variables. Thus for any component i,

where Ci is the concentration of component i (kmole m-30r kg m-3). In the case of gases, the Ideal Gas Law can be used to relate concentration to partial pressure and mole fraction. Thus,

where pi is the partial pressure of component i, within the gas phase system, and R is the Ideal Gas Constant, in units compatible with p, V, n and T. In terms of concentration,

where yi is the mole fraction of the component in the gas phase and P is the total pressure of the system. The accumulation term for the gas phase can be therefore written in terms of the number of moles as

22


For the total mass of the system,

e.g., with units,

1.8.1.5

Convective Flow Terms

Total mass flow rates are given by the product of volumetric flow multiplied by density. Component mass flows are given by the product of volumetric flow rates multiplied by concentration. (Convective mass flow rate) = (Volumetric flow rate)

(Volume Mass )

for the total mass flow

M

= QP

and for the component mass flow e.g., with units

A stream leaving a well-mixed region, such as a well-stirred tank, has properties identical to the system as a whole, since for perfect mixing the contents of the tank will have spatially uniform properties, which must then be identical to the properties of the fluid leaving. Thus, the concentrations of component i both within the tank and in the tank effluent are the same and equal to Cil, as shown in Fig. 1.11.

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

23


1.8.1.6

Production Rate

The production rate term allows for the production or consumption of material by biological and chemical reaction, e.g., microbial growth rates and substrate uptakes rates, and can be incorporated into the component balance equation by an additional term where: Rate of

Mass flow

Mass flow

the system

the system

Rate of

Chemical production rates are often expressed on a molar basis but can be easily converted to mass flow quantities (kg s-l). Mass units are more common in biological systems. The mass balance equation can then be expressed as:

where RA is the total reaction rate. The units are

Equivalent molar quantities may also be used. The quantity rA is assumed here to be positive when A is formed as a product and negative when a reactant A is consumed by the reaction.

V.

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

The system mass balance equations are often the most important elements of any modelling exercise, but are themselves rarely sufficient to completely formulate the model. Thus other relationships are needed to complete the model in terms of other important aspects of behaviour in order to satisfy the mathematical rigour of the modelling, such that the number of unknown variables must be equal to the number of defining equations. Examples of this type of relationships, which are not based on balances, but which nevertheless form a very important part of any model are as follows:

24


Growth and reaction rates as functions of concentration, temperature, pH (e.g., nth-order, Arrhenius) Stoichiometric or yield relationships for reaction rates Ideal Gas Law behaviour Adsorption isotherms Physical property correlations as functions of concentration Pressure variations as a function of flow rate Light intensity as a function of water depth Dynamics of measurement instruments as a function of the instrument response time Equilibrium relationships (e.g., Henry's Law) Controller equations Correlations of mass transfer coefficient, gas holdup volume, and interfacial area, as functions of system physical properties and degree of agitation or flow velocity Relations for climatic influence, e.g.,windspeed and solar energy these and other relationships are incorporated within the development of particular modelling instances are illustrated, throughout the text and in the simulation examples.

1.8.1.7

Diffusion of Components

Environmental problems often involve diffusion, e.g., in bioflocs (FLOC 1 and FLOC2) biofilms (BIOFILM), in soil remediation (SOIL) and in pollutant transport in soil (LANDFILL). Diffusional flow contributions in engineering situations are usually expressed by Fick's Law for molecular diffusion

where ji (kmole m-2s-1 or kg m-2 s-1 ) is the flux of any component i transported, dCi/dZ (kmole m-3m-1) is the concentration gradient and Di is the diffusion coefficient (m2 s-1) of component i for the porous solid. Calculating the mass diffusion rate requires a knowledge of the area through which the diffusive transfer occurs, since

[

) (

Mass rate Diffusivity of = of component i component i

][

Concentration gradient of i

)[

Area perpendicular to transport

)

The concentration gradient can often be approximated by difference quantities, as seen in Fig. 1.12, where


25

with units for each term i.e., kg s-1. Finite differencing techniques are described in more detail in Sec. 1.12.2.

Figure 1.12. Diffusion flux ji driven by concentration gradient (Cil - Cio)/AZ through surface area A.

1.8.1.8

Interphase Transport

Interphase mass transport, which is important in aerobic environmental bioprocesses also represents a possible input to or output from the system (e.g., see DCM, NITBED, OXSAG, OXIBAT and RIVER). In Fig. 1.13 transfer of a soluble component takes place across the interface which separates the two phases. Shown here is the transfer from phase G to phase L, where the separate phases may be gas, liquid or solid.

Figure 1.13. Transfer across an interface of area A from phase G to phase L.

When there is transfer from one phase to another, the component balance equations over each phase G or L must take this into account. Thus taking a

26


balance for component i around the well-mixed phase G, with transfer of i from phase G to phase L, gives Rate of (...u;;y]

=

in phase G

-

[

Rate of interfacial mass transfer of i from phase G into phase L

]

where the rate of transfer can be expressed in the form: Rate of (mass transfer) =

transfer ( Mass coefficient ) (thAz:r%e)

( Concentration driving force

J = KAAC In terms of molar quantities the units of the transfer rate equation are kmole ~S

-

m kmole - m2 S m3 ~

where, J is the total mass transfer rate, A is the total interfacial area for mass transfer (m2), AC is the concentration driving force (kmole m-3) and K is the overall mass transfer coefficient (m s-l). It is important to note that the concentration driving force is represented as a difference between the actual concentration and the corresponding equilibrium value and is not a simple difference between actual phase concentrations. Molar transfer rates (kmole s-l) can be converted to mass flows (kg s-l), by multiplying by the molar mass of the component.

VI.

For additional insight with complex problems, construct an information flow diagram

Information flow diagrams can be useful in understanding complex interactions (Franks, 1967), and in identifying missing relationships. They provide a graphical aid to a full understanding of system interaction (see Fig. 2.7).

1.8.1.9 Case A. Waste Holding Tank: Total and Component Mass Balance Example A plant discharges an aqueous effluent at a volumetric flow rate Qo. Periodically, the effluent is contaminated by an unstable noxious waste, which is known to decompose at a rate proportional to its concentration. The effluent

27


must be diverted to a holding tank of volume V, prior to final discharge, as in Fig. 1.14.

Figure 1.14. Waste holding tank.

This situation is one involving both a total and a component mass balance, combined with a kinetic equation for the rate of decomposition of the waste component. The total mass balance is expressed by, Rate of Mass flow accumulation in the tank of mass] = (into the tank)

Mass flow out

-

( of the tank

dM dt = Mass rate in - Mass rate out

~

or in terms of volumetric flow rates, Q, densities, (p), and volume, V d(p V) system = Q O P O - Q1 PI dt When densities are equal, as in the case of water flowing in and out of a wellmixed tank,

The steady-state condition of constant volume in the tank (dV/dt = 0) occurs when the volumetric flow in, Qo, is exactly balanced by the volumetric flow out, Q1. Total mass balances therefore are mostly important for those bioreactor modelling situations in which volumes are subject to change. The component mass balance equation, based on the toxic waste component is represented by:

28

[


Rate of accumulation Rate of mass of the m a s s ] = flow of the component component in component out in the tank

Rate of

] ( ?~~~~~~ ]

[

component by reaction

The corresponding ISIM program for this case is explained in Sec. 1.7

1.8.1.10 Case B. The Plug-Flow Tubular Reactor Consider a small element of volume, AV, of an ideal plug-flow tubular reactor, as shown in Fig. 1.15.

Figure 1.15. Component balancing for a tubular plug-flow reactor.

Component Balance Equation A component balance equation can be derived for the element AV, based on the generalised component balance expression, where for any reactant, A Rate of (accurpn] =

flc)w) (Mass flow) (Rate of formation (Mass ofAiI - of A out of A by reaction +

1

29


The rate of accumulation of component A in element AV, is (AV dCA/dt), where, dCA/dt is the rate of change of concentration. The mass rate of flow of A into element AV is Q CA, and the rate of flow of A from element AV is Q CA + A (Q CA), where Q is the volumetric flow rate. The rate of formation of A by reaction is TAAV, where rA is the rate per unit volume. Substituting these quantities gives the resulting component balance equation as AV dCA dt = Q CA - (Q CA + A (Q CA)) + rAAV

The above equation may also be expressed in terms of length, since AV = A,AZ where A,, is the cross-sectional area of the reactor. Allowing AV to become very small, the above balance equation is transformed into the following partial differential equation, where

For constant volumetric flow rate, Q, throughout the reactor

where Q/A, is the superficial linear fluid velocity v, through the reactor Under steady-state conditions

-.-

and hence, at steady state dCA 1 ~dZ -

vrA

This equation can be integrated to determine the resulting steady-state variation of CA with respect to Z, when knowing the reaction kinetics, rA = f(CA) and the initial conditions, CA at Z = 0. Cases with more complex multicomponent kinetics will require similar balance equations for all the components of interest. The component balance equation can also be written in terms of fractional conversion, XA, where for constant volumetric flow conditions

30


and CAOis the inlet reactor feed concentration. Thus

The mass balance, in terms of XA, is thus given by

Reactant A is consumed, so rA is negative, and the fractional conversion will increase with Z. In terms of molar flow rates and

where NAOis the molar flow of reactant A entering the reactor. This modelling approach is applied in the simulation example ASPLUG.

1.8.1.11 Case C. Biological Hazard Room This three-part problem involves the spread and removal of an airborne contaminant within a biological hazard room.

1. Basic Problem The room is totally enclosed but ventilated by a recirculating air system, which passes through a filter prior to return to the room. At some time t=O, a flask containing a dangerous contaminant ruptures, and the contaminant is released into the air stream. It is required to determine the contaminant concentration at some later time t. The system is represented by Fig. 1.16.

31


Figure 1.16. Model flows and variables for the biological hazard room.

Assuming the recirculating air flow QR (m3 h-l) is sufficiently high, the room may be modelled as a well mixed tank with uniform concentration C, which thus defines the balance region. For the room, the component mass balance equation is given by Rate of (accumulation of mass

) = (Mass flow in)

-

(Mass flow out)

The filter efficiency, q, defines the relationship between C1 and C, e.g., c1 = C ( 1 - q )

With the initial conditions that at time t=O, the concentration C=Co, the problem is defined by the combination of one simple differential equation and one additional algebraic (efficiency term) equation. The initial concentration in the air space of the room, Co, might be determined, for example, on the assumption that the whole of the contaminant flask contents is completely vaporised into the room at time t=O. The above simple model can then be modified to include other possible complicating factors.

32


2. Additional Ventilation by Emergency Fresh Air Venting Assuming an additional air venting rate of Q (m3 h-l), the system now appears as shown in Fig. 1.17.

Figure 1.17. Addition of air venting for the biological hazard room.

Allowing for the additional inlet and outlet flow terms, the balance equation now takes the form:

For fresh air Cair = 0. Whereas the flow QR will depend on the recirculating air fan capacity, the venting flowrate, Q, will depend on the area of the vent opening and the local air flow velocity, which would need to be determined according to local circumstances.

3. Wall Adsorption It might be envisaged that the control room is coated with an adsorbent paint having a special adsorption affinity for the contaminant (Fig. 1.18). Removal of contaminant by adsorption must be expressed by an additional term in the balance equation. Rate of (accumulation)

Rate of removal

=

in) -

@Ow

Out)

-

( by adsorption


33

Figure 1.18. Addition of wall adsorption for the biological hazard room.

The rate of adsorption will be a function of both the painted surface area, A,, and the contaminant concentration, C , in the air. The exact relationship would have to be determined experimentally, but assuming a rate equation of the form Adsorption rate = k A, C" The balance equation then becomes

where the constant values k and n for the adsorption rate would have to be determined from experiment.

4. Time- Varying Filter Efficiency Under normal circumstances, the filter efficiency may decline with time and may also be affected by the value of C. Although unlikely to be important within the time scale of the emergency conditions, this effect could be included in the model by means of an additional algebraic equation of the form

Again the exact form of this dependency would have to be determined by experiment. The overall model is therefore seen to be built up from the dynamic component mass balance equation, incorporating all possible flow terms and an experimentally determined "reaction rate" term plus additional algebraic equations relating to the efficiency of the filter. Further complexities could also

34


be added to the model as required and as relevent to the aim of the modelling procedure.

5. Adsorption and Desorption Kinetics for the Filter It is assumed that the filter has a limited capacity, and its efficiency may be a function of the actual concentration of the contaminant in the filter. Assuming a well-mixed gas flow in the filter, two additional balance regions around the separate gas and liquid phases in the filter must be considered, as shown in Fig. 1.19.

Figure 1.19. Modelling the kinetics of adsorption and desorption for the filter

.

The component mass balances must be formulated for both gas and solid phases as follows: For the gas phase in the room, dC V d t = QRCI-QRC For the gas phase in the filter,


35

where ra is the rate of adsorption of the contaminant onto the filter solid phase, and rd is the rate of desorption from the solid to the gas. For the solid phase of the filter,

The kinetics of adsorption and desorption may be described by equations:

The rates ra, rd and the specific area of the solid, as, are expressed relatively to the volume of solid adsorbent in the filter. The constants ka, kd and CS,lim and even the the form of the kinetic equations must be determined experimentally. This example is on the diskette and is called ROOM. A column or tubular filter could be modelled with a series of stirred tanks, and if a biofilter is to be used the modelling methods presented in BIOFILT could be consulted.

1.8.1.12 Case D. Lake Pollution Problem It is required to model the dynamics of a lake (Fig. 1.20), which is subject to

three flow inputs Qo, Q1 and 4 2 and one outlet 43,where the flows are measured as m3 h-l. The flow 4 2 occurs close to an industrial siting and pollutant can enter the flow at a concentration C2, where C2 can vary with time.

Figure 1.20. Modelling diagram for the lake pollution problem.

36


I . Simplest Case In the simplest case, the lake can be modelled as a single well mixed tank with concentration C3 identical to the outflow. The total mass balance is given by: = Qop+Qip+Q2p-Q3p

where for constant density p dV

dt = Qo + Q 1 +

4 2 - Q3

The outlet flow 4 3 will vary as some function of the depth of the lake. In the simplest case, a hydrodynamic equation of the form

could be assumed, where k is a hydraulic constant and Vo represents the lake volume at low-level, for which zero overflow occurs. The component mass balance is given by:

where for the uncontaminated inlet streams Qo and Ql, Co = C1 = 0. Hence

Variations in the inlet flow rates and concentration C2 with respect to time can be represented by appropriate algebraic relationships, e.g., sine wave functions such as t Q = kF sin and t C2 = k c sin 5

where kF and kc are appropriate flow and concentration parameters, respectively. The actual form of relationship will, however, depend on actual flow conditions. The combination of the total mass balance equation, the component mass balance equation, the hydrodynamic flow equation and the feed input variation relationships together with data on the appropriate constants and initial

37


conditions provide an albeit highly simplified, solution to the problem of modelling the lake dynamics.

2. Biological Removal of Pollutant For this, the appropriate form of kinetic relationship must be known. In this case, it is assumed that the rate of removal is directly proportional to the quantity of contaminant in the lake. Rate of removal = r, =

- k,

C3 V

where k, is the kinetic rate constant (h-l). Hence the component mass balance now is modified to:

d(\F3)

= 4 2 C2 - 4 3 C3 - k, C3 V

The proper form of the kinetics must, of course, be found by experiment, together with numerical values for the kinetic constants.

3. Oxygen Limitation The effect of oxygen limitation on the biological rate of removal can be accomplished by the inclusion of a Monod kinetics (see Section 1.9.2.3) form of relationship based on dissolved oxygen. Under these circumstances, the rate of removal becomes

where Ks is the saturation kinetics constant for dissolved oxygen. The use of the above equation in our lake model however now demands a knowledge of the lake dissolved oxygen concentration C L ~ For . this a component mass balance equation for dissolved oxygen becomes necessary. Allowing for dissolved oxygen in all feed streams and in the lake outlet, the dissolved oxygen balance becomes

where OTR is the rate of oxygen transfer from the air to the lake, and is expressed by:

38


where ko2 is the oxygen mass transfer coefficient (m h-I), A is the surface area * the dissolved oxygen saturation value (mole or kg of the lake (m2) and C L ~ is m-3). The rate of oxygen uptake by the lake organisms may be related to the rate of removal of pollutant, based on some yield factor Y,

where this relation assumes that oxygen is directly required for pollutant uptake.

4. Lake Geometry A single well-mixed tank model takes no account of the relative locations of the inlet and outlet flows. Assuming flows Qo and Q1 occur at the top end of the lake and flows Q2 and Q3 correspond to the bottom end, an extension of the single tank model to a two stage model might be appropriate (Fig. 1.21).

Figure 1.21. Additional flow complexity for the lake pollution problem.

The flow QR represents a possible interaction in the flow between the two lake sections. Separate balance equations for both pollutant and for dissolved oxygen must now be written for each lake section: For tank 1: d(VLIC) dt = QOCO+Q~CI+QRC~-(Q~+QO+QR)C-~~~VLI


39

For tank 2:

d'Vz

c3)=

d(VL2 dtcL3) =

(Qo + Qi + QR) C + 4 2 C2 - 4 3 C3 + rp2 V L ~

(Qo + Qi + QR) CL + 4 2 C L -~4 3 C L +~ OTR2 - r02,2 V L ~

where the rates of pollutant removal (re), oxygen transfer rate (OTR) and oxygen uptake rate (r02), depend on the local concentration in each section of the lake. In addition local hydrodynamic relationships for each section could be required.

5. Further Extensions These might for example include: 1. Separate the lake in different horizontal and vertical regions to reflect the actual lake geometry, i.e., the lake model now becomes a three-dimensional multi-tank arrangement. Modelling the sediment and exchanges between the lake and sediment are very important.

2. Include balance equations for the lake organisms, such as predator-prey relationships. 3. Include terms reflecting the toxicity of the pollutant on the oxygen uptake of the organisms (Fig. 1.22).

Figure 1.22. Modelling toxicity for the lake pollution problem.

4. Include terms to account for an abiotic degradation of the pollutant. 5. More complex effects of climatic variations (temperature, rainfall, sunlight, etc.) can be also added. See the simulation examples NCYCLE, PCYCLE, ALGAE, EUTROPH and METAL.

40


1.8.2

Energy Balancing

The Principle of Conservation of Energy states that the total energy in any system is constant, or that energy can neither be created or destroyed. The energy may exist in several different forms: kinetic, potential, heat, light etc. and may be transformed from one form to another. Energy balances follow from the Conservation of Energy, just as mass balances follow from the Conservation of Mass and are often very important in modelling environmental bioprocesses. For example, the light energy from the sun is used by plants for photosynthesis, microbial kinetics are temperature dependent and many microbial processes are exothermic and hence generate heat energy. An energy balance must take into account all the different forms of energy that enter and leave the control volume and account for all the changes of energy that occur within the control volume. Heat energy is often the most important form of energy in environmental applications because the heat energy determines the temperature of the system. The temperature of the system influences many different processes:

- Microbial kinetics - Stratification of lakes - Solubility of gases in liquids - Solubility of solids in liquids - Rates of chemical reaction Light energy is important as it is the energy source for photosynthesis. The main source of light energy of the light the sun. As well as the intensity and duration of the sunlight, the wavelength may also be important. Reflected light from water or snow may also strongly influence the available light energy. Kinetic energy and potential energy may be taken into consideration when modelling the flow of a liquid, of say a stream or river. Energy balances are needed whenever temperature changes are important. For example, such a balance is needed to model the startup of an activated sludge process (see ASTEMP). Energy balances are written following the same set of rules as given above for mass balances in Sec. 1.8.1. Thus the general form of energy equation is as follows: Accumu-

Rate of

Rate of

Rate of

Rate of

Rate of

energy The above balance in words is now applied to the measurable energy quantities of the continuous reactor shown in Fig. 1.23.


41

Figure 1.23. A continuous tank bioreactor showing some possible energy-related variables.

An exact derivation of the energy balance was given by Aris (1989) as,

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

and if the composition has no large influence on density then,

With the above simplifications the general energy balance simplifies to

The units of each term of the equation are energy units per unit time. The use of this general equation for variable physical properties is developed in Ingham et al. (1994).

42


Accumulation Term Densities and heat capacities of liquids can be taken as essentially constant so that the accumulation term becomes

with units: m3 (kg m-3) (kJ kg-lK-') K S

- kJ S

Flow Terms The flow term is

with the dimensions,

volume

energy

( z (volume) ) = (*)

This term actually describes heating of the stream entering the system with To to the reaction temperature Ti. It is important to note here that this term is exactly the same for a continuous reactor as for a semi-batch system.

Heat Transfer Term The important quantities in this term are the heat transfer temperature driving force or difference (Tamb-Tl), where temperature of the heating or cooling source, and the overall coefficient, U, which has units of energy/(time area degree), e.g., J The units for U A AT are thus,

area A, the Tamb is the heat transfer s-1 m-2 OC-'.

Heat transfer rate = U A (Tamb - Ti) energy time

energy - area time degree (area) (degree)

The sign of the temperature difference determines the direction of heat flow. Here if Tamb > Ti, heat would flow into the reactor.

Reaction Heat Term The term rQ V gives the rate of heat released by the bioreaction and has the units of energy volume time

energy =

time

The rate term rQ can alternatively be written in various ways as follows: In terms of substrate uptake and a substrate-related heat yield,


43

rQ = rS yQ/S In terms of oxygen uptake and an oxygen-related heat yield, rQ = r02 y Q / 0 2 In terms of a heat of reaction per mole of substrate and a substrate uptake rate,

Here rS is the substrate uptake-rate and AHr,s is the heat of reaction for the substrate, for example, J mole-l or kcal kg-'. The rS AH,,s term therefore has dimensions of (energy time -lvolume - l ) and is equal to rQ.

0ther Heat Terms The heat of agitation may be the most important heat effect for slow-growing cultures, particularly with viscous cultures. Other terms, such as heat losses from the reactor due to evaporation, solar radiation, and convective losses can also be important. The simulation examples ASTEMP, COMPOST, and WINDROW consider various heat effects.

1.8.2.1

Case A. Determining Heat Transfer Area or Cooling Water Temperature

For aerobic fermentations, the heats of reaction per unit volume of reactor are usually directly related to the oxygen uptake rate, r02. Thus for a constant-volume batch reaction with no agitation heat effects, the general energy balance is Accumulation rate ofenergy

(

)

Energy out

= -

( by transfer)

where YQ/02 often has a value near 460 kT mole-l If T is constant (dT/dt = 0) giving

+

Energy generated by reaction

(

02.

i.e., the rate of heat transfer is exactly balanced by the rate of heat release. Using this steady-state energy balance, it is possible to calculate the cooling water temperature, Tamb,for a given oxygen uptake rate and cooling device.

44


Thus,

Alternatively this same relation can be used to calculate the additional heat transfer area required for a known increase in cooling water temperature.

1.8.2.2

Case B. Heating of a Filling Tank

Consider the case depicted in Fig. 1.24, in which liquid flows continuously into an initially empty tank, containing a full-depth heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow, but heating is maintained. A total mass balance is required to model the changing liquid volume and this is combined with a dynamic heat balance equation.

Figure 1.24. Filling of a continuous stirred tank heated by an internal steam coil.

Assuming constant density, the total mass balance equation is


45

where for time t less than the filling time, V/Qo, the outlet flow Q1, equals zero, and for time t greater than V/Qo, Q1 equals Qo. The heat balance is expressed by

Assuming A0 is the total heating surface in the full tank, with volume Vo, and assuming a linear variation in heating area with respect to liquid depth, the heat transfer area may vary according to the simple relationship V A = A 0 6 More complex relationships can of course be derived, depending upon the tank geometry.

1.9

Chemical and Biological Reaction Systems

1.9.1

Modes of Reactor Operation

Biological reaction processes may occur in batch, fed-batch or continuous mode, and reactor models may be applied to a wide range of processes, including wastewater treatment and the flow in rivers, lakes and reservoirs.

1.9.1.1

Batch Reactors

Many wastewater treatment processes and some lakes and reservoirs can be treated as batch reactors. It is assumed that in the time considered there are no flows into or out of the reactor. In wastewater treatment processes the concentration of substrate will decrease with time and the numbers of microorganisms will increase. The dissolved oxygen concentration and the pH may also decrease due to the microbial activity. After the wastewater has been purified to the required level, the water is discharged, and the bioreactor is refilled with a new batch of untreated water, as depicted in Fig. 1.25.

46


Figure 1.25. Batch reactor operation.

During the reaction period, reactant concentrations and possible product concentrations will change with time, as shown in Fig. 1.26, while the other time periods are often short enough to be neglected. Biomass L

/ roduct

TIme Figure 1.26. Concentration profiles during batch bioreactor

operation.

With no flow in or out of the bioreactor during the reaction period, the component balance equation take the form: (Rate of accumulation within the bioreactor) = (Rate of production)

where ri may be influenced by various components. The simulation example BATREACT describes a batch growth process.

47

1.9 Chemical and Biological Reaction Systems

1.9.1.2

Semi-Continuous or Fed-Batch Operation

In semi-continuous or fed-batch operation there is an intermittent flow of liquid into the system. Semi-continuous operation shares the same characteristics as pure batch operation in that concentrations change as a function of time. In natural systems the flow of water into a lake after rainfall, for example, could be described as a semi-continuous process. A common type of semi-continuous operation is the fed-batch bioreactor as shown in Fig. 1.27.

Figure 1.27.

Semi-continuous bioreactor operation.

The ability to manipulate concentration levels and hence to manipulate the rate of reaction by an appropriate feeding strategy confers a high degree of flexibility to fed-batch or semi-continuous operation, since both the volumetric feeding rate F and the feed concentration can vary or be varied with respect to time. Examples are given in SEMIBAT, REPEAT, and AMMONFED. To describe fed-batch operation, a total mass balance Rate of accumulation (of mass in the system) =

Mass flow rate (into the system

is of course necessary. The component balance equations must now also include the input due to feeding, e.g., Rate of accumulation of substrate in the system

Mass flow

Rate of production

48


1.9.1.3

Continuous Operation

In continuous operation, fresh feed flows continuously into the reactor, while depleted material is continuously removed. This is equivalent to a lake or reservoir into which water flows in at one end and out at the other. The rates of addition and removal are such that the volume of the reactor contents is constant. Continuous reactors are of two main types, represented by well-mixed tank reactors and plug-flow tubular reactors, respectively (Fig. 1.28).

Continuous tank bioreactor

Continuous tubular bioreactor

Figure 1.28. The two main types of continuous reactors.

The two differing forms of continuous operation have quite different characteristics. Both, however, are characterised by the fact that after a short transient period, the bioreactor then achieves a steady-state operating condition, in which conditions both within the bioreactor and at the bioreactor outlet remain constant, as shown in Fig. 1.29. Steady-state period

Time Figure 1.29. Startup of a continuous reactor.

The behaviour of the two differing forms of continuous reactor, are best characterised by their typical concentration profiles, as shown in Fig. 1.30. In this case, S is the concentration of any given reactant consumed, and P is the concentration of any given product.

49


Tube SO

k CI

Q)

0

5

Distance

0

Distance

Figure 1.30. Profiles of substrate and product in steady-state continuous tank and tubular reactors.

As can be seen, the concentrations in a perfectly mixed tank are uniform throughout the whole of the reaction vessel contents and therefore identical to the concentration of the effluent stream. In a tubular reactor the reactant concentration varies continuously, falling from a high value at the inlet to the lowest concentration at the reactor outlet. The product concentration rises from inlet to outlet. These differences arise because in the tank reactor the entering feed is continuously being mixed with the reactor bulk contents and therefore being diluted by the tank contents. The feed to the tubular reactor, however, is not subject to mixing and is transformed only by reaction, as material moves down the reactor. No real situation will exactly correspond to the above idealised cases of perfect mixing (ideal tank) or zero mixing (plug flow), although the actual behaviour of tank and tubular reactors can often be approximately described by these idealised models. The balance equations at steady state for a well-mixed tank reactor have the form 0 = (Input) - (Output) + (Production)

since at steady state, the rate of accumulation and therefore the rate of change is zero. This equation predicts that the reaction rate causes a depletion of substrate from the feed condition to the outlet (the product will increase), and that the rate of production can be obtained from this simple balance: (Rate of production) = (Rate of output)

-

(Rate of input)

For a non well-mixed reactor such as a tubular or column reactor, steady-state implies the same non-transient conditions, but now concentrations vary with 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)

50


Here the overall rate of reaction is obtained by summing or integrating over every part of the reactor volume. The concentration characteristics of a tubular reactor are well approximated by a series of tank reactors, as shown in Fig. 1.31. Moving downstream along the reactor cascade, the substrate concentration decreases stepwise from tank to tank, while the biomass or product concentration increases in a similar stepwise manner. As the number of tanks in the cascade increases, so the performance becomes more and more similar to that of a tubular reactor.

Figure 1.31. Stirred tanks-in-series and their concentration profiles.

Modelling of the stirred-tank cascade involves the formulation of the tank balance equations, as before, and then the application of the equations, successively from tank to tank, such that the effluent from the preceding tank is the feed of the next and so on.

1.9.2

Reaction Kinetics

1.9.2.1

Chemical Kinetics

As shown in Sec. 1.8.1, the mass balance for a batch system becomes

Rate of accumulation

Rate of

Expressed in terms of volume V and concentration Ci, this is equivalent to

51


with units of moles per time. Here the term ri is the rate of chemical reaction expressed as the change in the number of moles of a given reactant or product per unit time and per unit volume of the reaction system. Thus for a batch reactor, the rate of reaction for component i can be defined as r i =

1 dni V dt

moles of i volume time

-__

where ni = V Ci and is the number of moles of i present at time t. The above equations also apply to a biological reaction, but mass, not molar, units are then often used. For a purely chemical system reactants and products can be related on a molar basis by a stoichiometric equation, which for the case of components A and B reacting to form product C has the form

Here vi is the stoichiometric coefficient for species i in the reaction. By convention, the value of v is positive for the products and negative for the reactants. The stoichiometric coefficients relate the simplest ratio of the number of moles of reactant and product species, involved in the reaction. The individual rates of reaction, for all the differing species of a reaction are related via their stoichiometric coefficients according to

The value of ri is therefore negative for reactants and positive for products. For the reaction A+2B

-+

3P

the individual reaction rates are therefore -rA=--

1 rg 2

' 5 1' p

Thus in defining the rate of reaction, it is important to state the particular reaction species. The reaction rate must be determined by experiment; the temperature and concentration dependencies can usually be expressed as separate functions, for example

52


The exact functional dependence of the reaction rate with respect to concentration is found by experiment, and for example may have the form rA = - k CA" CBP Here, k is the reaction rate constant, which is a function of temperature only; CA, CB are the concentrations of the reactants A, B (moles volume-l); a is the order of reaction, with respect to A; p is the order of reaction, with respect to B; (a+ p) is the overall order of the reaction. The reaction constant, k, is normally an exponential function of the absolute temperature, T, and is described by the Arrhenius equation

where E is an empirical constant called the activation energy, and R is the universal gas constant. The exponential term gives rise to highly non-linear behaviour in reactor systems which are subject to temperature changes. More complicated empirical relationships between reaction rate and temperature may need to be used; see, for example, ALGAE where the rate of photosynthesis of different alge species involve complex temperature relations.

1.9.2.2

Biological Reaction Kinetics

In many environmental problems it is necessary to be able to model the rates of growth, substrate utilisation and product formation of micro-organisms. Important examples are the micro-organisms used in wastewater treatment processes, the algae responsible for eutrophication of lakes and the soil bacteria that degrade pollutants. Many different factors can influence microbial growth as summarised by: Temperature PH Light Intensity Duration Macronutrients Carbon source (e.g., sugars) Nitrogen source Phosphorus source Micronutrients e.g., Magnesium, Manganese

53


Gases Oxygen Hydrogen Methane Carbon dioxide In many cases one component can be identified as limiting the growth of the micro-organisms and this is known as the limiting substrate. Most environmental bioprocesses involve many different micro-organisms (e.g., bacteria, protozoa, yeasts, algae, viruses, zooplankton) each of which will have differing growth rates, nutrient preferences, temperature sensitivities etc. In many cases it is sufficient to lump together all the micro-organisms into one group called biomass and use average values for the kinetic variables. In other cases, the concept of interacting micro-organisms may be important, and so more than one species of micro-organism can be included in the model.

1.9.2.3

Simple Microbial Growth Kinetics

Under ideal conditions of batch growth, it can be observed that the quantity of biomass, and therefore the concentration, increases exponentially with respect to time. This phenomena can be explained by the fact that all cells have the same probability to multiply. Thus the overall rate of biomass formation is proportional to the biomass itself. This leads to an autocatalytic reaction, which is described by a first order rate expression as rx=kX where rX is the rate of cell growth (kg cell m-3 s-l), X is the cell concentration (kg cell m-3) and k is a kinetic growth constant (s-l). For a batch system, this is equivalent to dX -= kX dt where dX/dt is the rate of change of cell concentration with respect to time (kg cell m-3 s-l). The analytical solution of this simple first order differential equation is of the form In X = k t

+ In Xo

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

54


Plotting experimental growth data in the form of the logarithm of cell concentration versus time will often yield a straight line over a large portion of the curve, as shown in Fig. 1.32.

Limitation

Stationary

t2

Tlme

Figure 1.32. Biomass concentration during batch growth.

The initial period of time up to time tl represents a period of zero growth and this is known as the lag phase. In this period the cells are synthesising enzymes and other cellular components necessary for the optimal growth of the cell under the particular environmental conditions. The time at which cell division commences varies from one organism to another so that a gradual transition from the lag phase to the exponential phase occurs. An exponential (or logarithmic) growth phase follows the lag phase and during this period the cell mass increases exponentially, growing at a constant specific growth rate, dependent on the species and the environmental conditions. The growth rate is at a maximum during this phase and the population of cells are fairly uniform with respect to chemical composition and metabolic composition but representing a population of cells at all stages of the cell cycle. The growth of micro-organisms in a batch reactor will eventually stop, owing either to the depletion of some essential nutrient or to the accumulation of a toxic product. The result is that the growth rate gradually slows down and the growth becomes nutrient limited or product inhibited. When the growth rate falls to that of the death rate no overall increase in cell numbers occurs giving rise to a stationary phase. After the stationary phase the rate of cell death exceeds that of cell division and the number of cells decreases, resulting in a death rate curve. The slope of the linear part of the curve between tl and t2is the growth rate per unit mass of cells or specific growth rate and is given the symbol p (s-l).

~dln X _ dt

-

dX _dt = _k = specific growth rate = p

55


In many processes cells may continuously die or may start dying (after time, t3) because of lack of nutrients, toxic effects or cell ageing. This process can typically be described by a first order decay relationship:

where rd is the death rate and kd (s-l) is the specific death rate coefficient. The exponential and limiting regions of cell growth can be described by a single relation, that sets p equal to a function of substrate concentration. It is observed experimentally that p is at maximum when the particular limiting substrate concentration S is large, and for low concentration p is proportional to S. Over the whole range from low to high S, the variation of p with respect to S is described by the following Monod equation.

Note that the Monod equation, like the Michaelis-Menten equation for enzyme kinetics, is a two-parameter equation involving two constants, the maximum specific growth rate pmax and the saturation constant Ks. It is best considered to be an empirical relation but, since it has the same form as the MichaelisMenten equation, it is often taken to be related to a limiting enzymatic step. Although very simple, the Monod equation often describes experimental data for growth rate very well. The form of this relation is shown in Fig. 1.33.

KS

S

Figure 1.33. Specific growth rate versus limiting substrate concentration according to the Monod relation.

The important properties of this relationship are as follows:

s -+

0,

56

1.9.2.4


s + -,

P

S = Ks,

p = -Pmax

-+ Pmax

2

Substrate Uptake Kinetics

The rate of substrate uptake by micro-organisms is generally considered to be related to that of growth and also to that required for cell maintenance and can be expressed as:

where rS is the rate of substrate uptake by the cells (kg substrate m-3 s-l). Here, YX/S (kg/kg) is the stoichiometric factor or yield coefficient, relating cell mass produced per unit mass of substrate consumed. The maintenance factor m, with the dimensions of mass substrate per mass cells per unit time, describes the substrate required for non-growth functions. The total substrate utilisation for cell maintenance is, of course, taken to be proportional to the total quantity of cells, and therefore for a batch reactor it is proportional to cell concentration, X. Often the uptake and production rates of substances are expressed in terms of the particular quantities related to unit mass of cells and these are then known as specific cell quantities. Thus: For the specific growth rate (h-l)

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

For the specific oxygen uptake rate (kg 02/kg biomass h), q02 =

r02

x

For the specific carbon dioxide uptake rate (kg C02/kg biomass h), qc02 =

rc02

x


57

For the specific product production rate (kg P k g biomass h),

Note that qx = rx/X = p is the specific biomass production rate. Specific rate quantities may take simple or complicated forms, for example, for the case: - rx - m X rS = YX/S then,

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

where the constant vmax is proportional to the quantity of biomass in the system and is the maximum rate of substrate consumption, observed at high S. Thus it is assumed that the biomass is essentially constant, where Vmax = pmaxX/YX/S. This form of kinetics is also referred to as Langmuir-Hinshelwood kinetics (see NCYCLE).

1.9.2.5

Substrate Inhibition of Growth

Many substrates can be utilised by organisms at low concentrations, but at high concentrations the substrate can also act as a toxic growth inhibitor. The p (or v) versus S curve may then appear in the form shown in Fig. 1.34 and can be described by the relation:

58


This is a modified Monod relation to allow for the inhibitory effects of high substrate concentration. As shown, the inhibition term (S2/K~),which is small in magnitude at low values of S, increases dramatically at high values of S and causes a decrease in p. Note that high values of KI correspond to a decreasing effect of substrate inhibition. It is seen that larger values of Ks shift the left side of the curves towards the right, while increasing values of KI raise the right side of the curves. This kinetic form is used in the simulation examples CONINHIB, RESPMET, and REPEAT. 1.o

1

0.4

0.2

0.0

y

I

0

10

I

S(g m-3)

20

Figure 1.34. Substrate inhibition kinetics for various values of KS and KI. The parameters used are as follows: For all curves Pmax = 1.0 h-'. Curve A: KS = 1 and KI = 10, Curve B: KS = 0.1 and KI= 10; Curve C: KS = 1 and KI= 20; Curve D: KS = 0.1 and KI = 20. The units of KS and S are g ~ n -KI~has , units g2 m".

Thus a wide range of shapes can be achieved by varying the three parameters, but a maximum value of p is always obtained at some intermediate value of S. The substrate inhibition kinetic curve has the following characteristics, which result from the kinetic equation: When S = K s

59


When S = KI Pmax

p = 2+Ks/K1 The maximum for p occurs at S = (Ks K I ) ~ for . ~ which

1.9.2.6

Additional Forms of Inhibition

Inhibition occurs when a substance, inhibitor (I), reduces the rate of a biological reaction. Three simple types of reversible inhibition kinetics are given in Table 1.2. Here it is assumed that the biomass is essentially constant, and therefore that Vmax = pmaxX/Yx/S.

Table 1.2. Forms of substrate inhibition kinetics. Rate equation rS =

Vmaxs

Ks (1 + YKI) + S

Competitive

Vmax

S

Non-competitive

Vmax

S

Uncompetitive

rs = (1 + YKI) (Ks + S) rS = Ks

Inhibition

+ S (1 + YKI)

Here I is the concentration of the inhibitor.

1.9.2.7

Other Expressions for Specific Growth Rate

The Modified Monod Form

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

60


The Teisser Equation

P=

Pmax (1 -

relates p to S exponentially.

The Contois Equation

expresses the effective saturation constant as being proportional to the biomass concentration X. At high X, p is inversely proportional to X. This is sometimes used to represent a diffusion limitation in flocculating or immobilised biomass.

The Logistic Equation

p = (a-bX) encompasses exponential growth and the levelling off to zero growth rate at high X. For a batch fermentation the biomass balance is, dX dt = a x - b X 2

-

Thus when X is small, growth is exponential and given by dX dt = a X

~

When X is large,

At steady state or zero growth rate this gives X = ah.

The Moser Equation Pmax

P = 1 + Ks Sh introduces the parameter h to relate the growth rate to the limiting substrate concentration.


1.9.2.8

61

Multiple-Substrate Kinetics

In ecological systems, multiple substrate utilisation is the rule rather than the exception owing to the low nutrient levels found in natural waters. Multiplesubstrate Monod kinetics can often be used to describe the influence of many substrates (see PCPDEG). For two substrates, commonly oxygen and a carbon source. this takes the Double Monod form

With this expression either substrate may be limiting when the other is in excess. Note that the multiplicative effect gives for S1 = K1 and S2 = K2, the result p = pmax/4. This form is used in many examples, e.g., OXIBAT and AMMONOX. Double-Monod kinetics can also be written for two substrates (often two alternative sugars) as parallel reactions,

according to kl s1 k2 s2 1 p = p m a x ( m + m ) ( k l + k 2 ) This expression describes an additive, fractional contribution for each substrate. Thus for S1 = K1 and S2 = K2, the result is p = pmax/2. For the case S1 = 0 and S2 large, then p = pmax k2/(kl+k2). Thus each substrate alone allows a different maximal growth rate. If both S1 and S2 are large then p = pmax. Note that the flexibility of this kinetic form requires twice as many kinetic parameters as the simpler double Monod kinetics.

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

This form allows the consumption of substrate S2 to be repressed for suitably low values of KI until S1 is exhausted and is used in the simulation example PCPDEG.

62

1.9.2.9


Structured Kinetic Models

In many cases, the characterisation of biological activity by using simply the total biomass concentration is insufficient for an adequate model representation. A variation in the biomass activity per unit biomass concentration may be caused by many reasons, including a change of the population in a mixed culture, a change of enzyme content of the cells, an accumulation of storage materials and also a morphological change. Such variations in biomass activity and composition require a more complex description of the cellular metabolism and a more detailed approach to the modelling of cell kinetics. Structured models are usually based on a compartmental description of the cell mass as shown in Fig. 1.35.

Figure 1.35. Comparison of structured and unstructured models. 11, 12 are the cell compartment masses, e.g., protein, RNA, storage material. R is the remaining biomass, such that X=R+I1+12.

In general it is very difficult experimentally to obtain sufficient mechanistic knowledge about the cell metabolism for the development of a "realistic" structured model. Often the verification of even simple unstructured models is not possible owing to experimental difficulties. This problem becomes much more significant with increasing complexity of the model. For this reason, structured models are seldom used for design or control. Structured models may be useful to model transient behaviour of an experimental biological system, however.

1.9.2.10 Interacting Micro-Organisms The kinetics of a single microbial population were considered. In most environmental bioprocesses, however, there are multiple microbial species present. In some cases it is sufficient to treat all the species as one microbial population, but often it is necessary to consider the interactions between the individual species. In the following analysis the interactions between two

63


species are considered, and may be extended to multiple interactions using a similar method of analysis. The different types of interaction can be described using a graphical means of representation. In the following descriptions, the growth path kinetics are designated by solid arrows connecting the substrates to the products, and the organisms involved are given above the arrow. A solid arrow from one substrate symbol to the same symbol in another growth path indicates that the product from one organism acts as a substrate for another. Substrate or product inhibition is given by a dashed arrow connecting the component to the inhibited organism. The symbols +, - or 0 at the right side of the diagram indicate the type of interaction the organism has experienced. These can be a positive beneficial effect (+), a negative detrimental effect (-), or a neutral effect (0). This is shown in the following examples:

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

Simulation examples ALGAE, PCYCLE, COMPETE and MIXPOP demonstrate this type of system.

Commensalism Organism A uses substrate S2 to produce product P; organism B uses substrate S1 to produce product S2, which benefits organism A since product S2 acts as its substrate. A

b

P

+

The following processes (compound S2 in brackets) and simulation examples exhibit this form of commensalism: - nitrification (N02-), AMMONOX - anaerobic digestion (organic acids), MOLASSES - methanogenation (H2, C02), WHEY

64


Commensalism with Product Removed Organism A utilises a substrate S2, which inhibits the growth of B on substrate S1.

.m

B

S1

p1

+

This effect may be found in the removal of toxic wastes in mixed cultures with multiple carbon sources. An example is found in WHEY, in which the hydrogen substrate of the methanogens inhibits the acetogenic organisms.

Mutualism with Product Removed Organism A utilises substrates S2 to produce product P. Organism B utilises substrate S1 to produce S2, which inhibits organism B. A s2 S1

b

+

P

bb +

us2

An example of this would be found in anaerobic digestion for hydrogen gas production (see WHEY).

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

=: A

%+

s2+ pB

PA

+

pB

Other examples of interacting micro-organism effects are given in the simulation examples ACTNITR (neutralism) MIXPOP and COMPETE (predator-prey population dynamics) and TWOONE (competition between organisms). Other complex multi-organism examples are CURDS and PCPDEG.

65

1.10 Modelling of Bioreactor Systems

1.10

Modelling of Bioreactor Systems

1.10.1 Stirred Tank Reactors In what follows, material balance equations are derived for the total mass, the mass of substrate and the mass of cells assuming a generalised stirred-tank bioreactor system as shown in Fig. 1.36.

Figure 1.36. The variables for a tank fermenter.

In this generalised case, feed enters the reactor at a volumetric flow rate Qo, with cell concentration, Xo, and substrate concentration, So. The vessel contents, which are well-mixed, are defined by volume V, substrate concentration S1 and cell concentration X I . These concentrations are identical to those of the outlet stream, which has a volumetric flow rate Q1.

General Balance Form As shown previously, the general balance form can be derived by setting: (Rate of accumulation) = (Input rate)

-

(Output rate)

and be applied to the whole volume of the tank contents.

+

(Production rate)

66


Expressing the balance in equation form, gives: Total mass balance: = P (Qo-Qi)

Substrate balance: = QoSo-Q1S1 + r s V

Cell balance: d(\F1)

= Q ~ x ~ x1 - Q + r x~v

where the dimensions are: V (m3), p (kg m-3), Q (m3 s-l), S (kg m-3), X (kg m -3) with rS and rx (kg m-3 s -l). The rate expressions may be simply

with

and for a constant yield coefficient,

but other forms of rate equation may equally apply. As discussed in the simulation example CONTANK, these equations predict that the steady state for a continuous reactor with sterile feed is given by the condition p = D, where D, the dilution rate is the ratio of QN. Organisms will washout from the reactor if D > p, so that pmax represents an upper limit for dilution rate. The above generalised forms of equations can be applied to batch, semibatch and continuous bioreactor operation, as shown in the simulation examples BATREACT, SEMIBAT, CONTANK and ASCSTR.

67

1.10 Modelling of Bioreactor Systems

1.10.2 Modelling Tubular Plug-Flow Reactor Behaviour 1.10.2.1 Steady-State Balancing Tubular flow reactor behaviour can be modelled for steady-state conditions by considering the flow of a series of fluid elements or disks of liquid, each of which behaves as a batch reactor during its time of passage through the reactor. This can be understood by considering a pulse of unreacting tracer in Fig. 1.37, which passes from entrance to exit without mixing in the axial direction.

Tracer pulse input

Tracer output

Figure 1.37. Plug-flow idealisation of the tubular reactor with no axial mixing.

A reaction will cause a steady-state axial concentration profile, as shown in Fig. 1.38. Thus at steady state, the concentrations vary with distance in a manner which is analogous to the time-varying concentrations that occur in a batch reactor. I

Distance Figure 1.38. Axial profiles of steady-state concentrations in a tubular reactor.

This means that steady-state tubular reactor behaviour can be modelled by direct analogy to that of a simple batch reactor. Thus using the batch reactor substrate balance (p = constant),

.

68


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

where v = Q/A and Q is the volumetric flow rate through the tube with crosssectional area A. Thus substituting for dt gives, -dS- ! 3 dZ- v This is the steady-state tubular reactor design equation. With a kinetics model, rs = f(S), the equation can be integrated from the inlet, at position Z=O, to the outlet, at Z=L, to obtain the steady-state concentration profile of S. Additional component balances would be required for more complex kinetics.

1.10.2.2

Unsteady-State Balancing

If dynamic information is needed for tubular or column systems, then changes with respect to both length and time must be considered. In order to achieve this, the reactor can be considered by dividing the volume of the reactor into N finite-differenced axial segments (Fig. 1.39), and treating each segment effectively as a separate stirred tank.

Figure 1.39. Finite-differencing the tubular reactor.

As shown in Fig. 1.40 the substrate balance for S is formulated over a single segment n of volume AV = A AZ:

Figure 1.40. Balancing the difference segment n for the tubular reactor.

Accumulation rate o f s

(

rate of S

rate of S

Production rate of S by growth

69

I . 10 Modelling of Bioreactor Systems

The balances have the same form. Thus for segment n,

The constant density condition gives Qn-l = Qn = Q. Dividing by AV,

Setting AV = Adz and AS = dS gives the partial differential equation, which describes changes in both time and distance, as,

When the volumetric flow, Q, is constant,

At steady state, dS 0 = - v - dZ

+ rs

To model the dynamics by simulation methods, the partial differential equation must be written in difference form as,

or

where AVIQ = 2 , the residence time of the liquid in a single segment. Remembering the model of a continuous stirred tank the above dynamic simulation model for a tubular reactor is thus equivalent to the model for a series of stirred tanks, each with volume AV.

70


1.11

Mass Transfer Theory

1.11.1

Phase Equilibria

Knowledge of the phase equilibrium is essential for any mass transfer process. At phase equilibrium conditions, the driving force for mass transfer is zero and therefore further concentration changes via a mass transfer mechanism become impossible. The equilibrium, is therefore also important in determining the maximum extent of the concentration change possible by mass transfer. Equilibrium data correlations can be extremely complex. The approach in this text is based, however, on the basic concepts of ideal behaviour, as expressed by Henry's Law for gas absorption, This has the advantage that it enables a direct method of solution and avoids a cumbersome iterative type of procedure, which would otherwise be required.

1.11.2

Interphase Mass Transfer

Actual concentration profiles in the very near vicinity of a mass transfer interface are complex, since they result from an interaction between the mass transfer process and the local hydrodynamic conditions. As shown in Fig. 1.4, in each phase on both sides of the interface a relatively stagnant film exists, and this changes to more turbulent flow in the outer bulk phases. Concentration gradients of the transferred component develop on either side of the interface.

Figure 1.41. Concentration gradients at a gas-liquid interface.

71

1.11 Mass Transfer Theory

Mass transfer rate equations are usually based on overall coefficients of mass transfer, KG and KL, and overall concentration driving forces, where Q = & A (CG - CG*) = KL A (CL*- CL)

mole m mole - - m2 h - h m3 ~

and CG* and CL* are the respective equilibrium concentrations, corresponding to the bulk phase concentrations, CL and CG, respectively. The equilibrium concentrations are given by Henry's Law, which is a linear relation between the equilibrium solubility concentration in one phase to the bulk concentration in the other phase. Using Henry's Law, the corresponding concentrations can be established as

For components of low solubility, such as oxygen in water, the gas-side gradient will be small and can be neglected. In this case, the gradient would be zero on the gas side, and the concentration of the gas at the interface would be then the same as that in the bulk gas. Henry's Law can then be used to calculate the interfacial concentration in the liquid phase, as shown in Fig. 1.42.

cL

cL* Liquid phase conc.

Figure 1.42. The bulk phase concentrations determine the equilibrium concentrations.

In this case, there exists a relatively low resistance to mass transfer on the gas side of the interface, as compared to the much greater resistance to mass transfer on the liquid side. This condition is known as liquid film control and corresponds to the situation where the overall resistance to mass transfer resides almost entirely within the liquid phase. The overall mass transfer capacity coefficient is KLa, where a is the specific area, usually referred to the liquid volume, Hence the overall mass transfer rate equation used for slightly soluble

72


gases in terms of the specific area is KLa (CL* - CL) VL, with the dimensions of mass or moles per unit time.

1.1 1.2.1 Case A. Steady-State Tubular and Column Modelling Figure 1.43 represents a countercurrent-flow, packed gas absorption column, in which the absorption of a single solute occurs. The concentrations are low, and absorption does not change the flow rates appreciably. The mass transfer rate equation is expressed in terms of concentration units.

Figure 1.43. Steady-state gas absorption column.

Steady-State Design In the steady-state design application, the volumetric rates L and G, and concentrations CGjn, CLin, CGout and CLout will either be specified or established by an overall, steady-state solute balance, where

The problem then consists of determining the height of packing required to obtain the above separation or of determining the separation to be obtained for a given column length.

13

I . 1 1 Mass Transfer Theory

Component Mass Balance Equations For a small element of column volume dV, component balances can be written for each phase, where:

Taking distance Z to increase in the direction from top to bottom of the column, 0 = G ((CG+ dCG) - CG) - KLa ( CL* - CL) (1-E) A, dZ

0 = L (CL - (CL + dCL))

+

KLa ( CL* - CL) E A, dZ

where, KLa (sl ) is the overall mass transfer coefficient for the liquid phase based on the total volume, CL* is the equilibrium liquid concentration, and dV = A, dZ is the volume element in terms of the empty cross-sectional area of the column. The available area for flow of each phase is corrected for the fractional holdup of gas, I-&, neglecting the solid-phase volume. Hence dCG dZ

-

1 --KL~(CL*-CL) VGZ

and

where VGZ = G/( ~ - E ) A ,and VLZ = L/(E A,). Following the coordinate scheme in Fig. 1.43, the concentrations will increase in the direction of increasing Z. The above differential model equations linked by the equilibrium relationship provide sufficient detail for a solution of the problem by digital simulation (Ingham et al., 1994). The examples BIOFILT1 and BIOFILT2 apply the above modelling technique to the special case of a biofilter.

1.11.3 Case Studies 1.11.3.1 Case A. Aeration of a Tank of Water In many wastewater treatment processes air is sparged into a tank to provide oxygen for the aerobic growth of micro-organisms. It is therefore necessary to know the minimum rate at which air needs to be supplied.

74


As shown in Fig. 1.44, air is sparged into a tank, whose dissolved oxygen, as measured by an oxygen electrode, is initially very low. The oxygen balance in the liquid phase states that the rate of accumulation of oxygen in the liquid phase is equal to the rate of supply of oxygen to the liquid by mass transfer from the gas.

(Rate of accumulation) = (Rate of supply)

where CL is the oxygen concentration in the liquid phase.

Figure 1.44. Aeration of water in tank.

The equilibrium dissolved oxygen concentration, CL*,can be calculated from Henry's Law, where the Henry's Law constant H, now relates dissolved oxygen concentration CL*to gas partial pressure po2,

Integrating both sides of the oxygen balance gives KLa, assuming that initially the dissolved oxygen concentration is zero.

CL* In CL* - CL = KLat Thus a plot of In (CL*/(CL* - CL))against time has a slope of KLa (Fig. 1.45).

75

1.1 1 Mass Transfer Theory

Time Figure 1.45. Determination of KLa from a logarithmic plot.

Thus in principle, the mass transfer coefficient can be determined from a batch experiment and a logarithmic plot. In practice, a slow response of the oxygen electrode can, however, lead to inaccurate results, which may require more sophisticated modelling. A more complex situation involving gas phase and measurement dynamics is given by Dunn et al. (1992).

1 .1 1 .3 .2 Case B. Biological Oxidation in an Aerated Tank A batch reactor liquid is aerated with a continuous flow of air to support a biological reaction, as shown in Fig. 1.46.

Figure 1.46. A batch bioreactor with continuous aeration.

The biological reaction in the liquid phase is assumed to follow first-order kinetics with respect to oxygen concentration. Since oxygen is relatively insoluble (approximately 8 g m-3 saturation for air-water) a high transfer rate is important to maintain a high dissolved oxygen concentration CL.The oxygen balance for the batch liquid phase is then:

76


Rate of Transfer rate of (....myl?tiOn 0 2 in liquid Of) = (02 into the liquid) -

VL

rate of 0 2 ( Uptake by the cells

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

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

0 = KLa (CL*- CL)

-

k CL

Using this equation, the reaction rate constant, k, can be determined if CL is measured and KLa is known or measured. The equilibrium value, CL*, can be calculated from the gas phase concentration, and, assuming there is little oxygen depletion, can be calculated from the inlet gas conditions, where

and KLa CL* cL = KLa + k

Solving for CL gives, 1

The reaction rate, given by r = k CL, is then

Now it is possible to distinguish between two different extreme regimes for this system, mass transfer control and reaction kinetics control: 1. Reaction rate control applies for low values of MKLa, when r approaches k CL*and CL approaches CL*.

2. Transfer control applies for high values of MKLa, when r approaches KLa CL* and CL approaches 0. For intermediate values both rate effects will influence the overall apparent rate.

77

I . 1 1 Mass Transfer Theory

1.1 1 . 3 . 3 Case C. Determination of Biological Oxygen Uptake Rates by a Dynamic Method Low oxygen uptake rates, such as exist in slowly growing systems (plant and animal cell cultures, aerobic sewage treatment processes, etc.), cannot easily be measured by a gas balance method, since the measured difference between the inlet and outlet oxygen gas phase concentrations is usually very small. Owing to the low solubility of oxygen in the liquid media, quite small oxygen uptake rates will cause measurably large changes in the dissolved oxygen concentration. Thus it is possible to measure the oxygen uptake rate (OUR), here equivalent to specific uptake rate, qo2, multiplied by the biomass concentration, X, either by taking a sample for experiments in a oxygenelectrode respirometer or by using the reactor itself. Turning off the air supply causes a dynamic response in the dissolved oxygen concentration (Fig. 1.47), according to the liquid balance equation

When the time required for an appreciable decrease in dissolved oxygen is large, as compared to the electrode time constant, the method is quite accurate and no correction for the electrode measurement dynamics is required. See the simulation example RESPMET. Air off

Time Figure 1.47. Dynamic measurement of biological oxygen uptake rate.

It is possible to establish an automatic and quasi-continuous measurement system based on this method (Mona et al., 1979).

78

1.11.4


Gas-Liquid Phase Transfer Across a Free Surface

In many environmental processes, the transfer of a gas across a free surface, such as the surface of a lake or river, is an important factor. The rate of gas transfer across a free surface will still be proportional to the difference between the equilibrium dissolved gas concentration and the actual dissolved gas concentration as described in Sec. 1.1 1.2. However, the dissolved gas concentration in the liquid is not constant but varies as a function of position. Only the concentration at the liquid surface is determined by the oxygen transfer rate and not the concentration below the surface. However, in a turbulent stream, circulating eddies transfer elements of water from the liquid surface to the bulk liquid and back again (Fig. 1.48). By this process, oxygen is transferred from the surface to the bulk liquid, and oxygen depleted water is returned to the liquid surface. The size of the eddies determines the depth to which oxygenated water can penetrate. Oxygen can also diffuse through the water, but this process is very slow compared to the mechanism of eddy transport. The circulation rate of the eddies determines the time it takes for an element of fluid to travel from the liquid surface to the bulk liquid and back again, and hence determines the rate of mass transfer.

Figure 1.48. Transfer of oxygenated water in a turbulent natural stream.

O'Connor and Dobbins (1966) derived the following expression for mass transfer to natural streams:

79

1.1 1 Mass Transfer Theory

where Do2 is the diffusivity of oxygen in water, v is the mean river velocity, and H is the depth. The specific interfacial area of a stream (i.e., area of gas-liquid interface per unit volume of stream) is given approximately by:

1a = W L - ~W L V -WLH-H ~

-

where L is the eddy length and W is the width. Thus the overall mass transfer rate decreases with the depth of the stream as one might expect. The above analysis is based on the assumption that the flow is turbulent,. However, in some important processes the flow is often laminar or the water may be quiescent, i.e., the flow of wastewater through a trickling filter or a deep lake on a still day. In these cases the diffusion of the gas through the liquid becomes much more significant. The presence of waves increases the gasliquid interface area and therefore improves mass transfer. The examples DISCHARGE, OXSAG, RIVER and STREAM make use of surface aeration coefficients.

1.12

Diffusion and Biological Reaction in Solid Phase Biosystems

Diffusion is the transport process that occurs on the molecular level, owing to random kinetic motion. Diffusion in gases is generally faster than that in liquids which in turn is faster than that in solids. An increase in temperature increases the mean velocity of the molecules and so increases the rate of diffusion. Diffusion is an important process in many environmental processes, for example: - The diffusion of substrates and oxygen into flocs or biofilms and the diffusion of product carbon dioxide away. - The mixing of contaminated water with groundwater. - The transport of material from sediments into lakes and rivers.

Fick's Law: This law expresses the process of diffusion in gases, liquids and porous solids in mathematical form. The rate at which a quantity of dissolved substance diffuses across a unit area is proportional to the concentration gradient. The constant of proportionality, Ds, is known as the diffusion coefficient or diffusivity (L2 T-l). In mathematical terms: dS js = - D s z

mass time area

80


The minus sign represents the fact that the direction of diffusion is towards lower concentrations. Table 1.3 lists typical values for diffusivities.

Table 1.3. Typical values for diffusion coefficients (Bailey and Ollis, 1986). Molecule Oxygen Oxygen Hydrogen sulphide

Medium Water Microbial films Water

Diffusivity

Temperature

cm2s-1 x 10

OC

2.25 0.04 - 1.5 1.77

20 20 16

The diffusion coefficient of oxygen in biofilms and flocs is reported to be between 8 and 95% of the value in pure water (Christensen and Characklis, 1990). Reactions involving biofilms or flocs (e.g., trickle filters and activated sludge processes for wastewater treatment) are subject to the influence of mass transfer. In order to reach a reaction site, substrate S must first be transported by convection from the bulk liquid to the exterior stagnant diffusion film, and then by external mass transfer through the diffusion film to the surface of the biofilm, where surface reaction can take place. Within a biofilm or floc, an additional internal mass transfer step carries the substrate by diffusion to further reaction sites within the matrix. Similarly product P, formed within the matrix, must diffuse out of the matrix towards the surface, and then away from the surface via the external mass transfer laminar film to the bulk liquid. Two types of limitation within the reaction matrix can be identified for internal transfer: 1. Kinetic limitation when low substrate concentration causes the reaction rate to slow. 2. Penetration limitation when the substrate is completely exhausted with the biofilm or floc. Referring to Fig. 1.49, the concentration gradients in region B to D are caused by the joint effects of diffusion and reaction, which occur simultaneously throughout the reacting biofilm or floc matrix. Diffusion from both directions results in a symmetrical concentration profile. The gradients in regions A to B and D to E are due solely to diffusion in the external liquid diffusion film. Note that the concentration at the very centre of the matrix has fallen to zero, and the reaction must therefore cease due to penetration limitation.

1.12 Diffusion and Biological Reaction in Solid Phase Biosystems

81

Figure 1.49. Example of a concentration profile of substrate through a biofilm or floc.

In the case of a biofilm attached to an impermeable solid support, the substrate can enter from only one surface, as shown in Fig. 1.50.

I

I

A

6

I

Distance

C

Figure 1.50. Example of gradients of substrate and product in a biofilm attached to a carrier.

82


In the case shown, the substrate penetrates completely, and the reaction therefore proceeds throughout the matrix. Note that at the solid wall (C) a zero flux condition must exist, and therefore the concentration gradient is zero here.

1.12.1

External Mass Transfer

The rate of supply of substrate through the diffusion film to the surface is defined by mass transfer considerations, such that the mass flux to the catalyst surface is given by js = ~ S (SA L - SB) where, js is the mass flux (mole m-2 s-l), kSL is the mass transfer coefficient (m s - l ) and SA and SB are the substrate concentrations for the bulk and surface conditions (mole m-3), respectively. The steady-state balance can be written for the combined transport-reaction process, (Rate of supply by transfer) = (Surface reaction rate) = (Apparent overall rate)

In the following treatment, the surface reaction is assumed to be first-order, such as found for a biocatalytic reaction with Michaelis-Menten or Monod kinetics and SB cc KM. The apparent reaction rate per unit surface area, raPP (mole m-2 s-l), is equal to the rate of both processes. Solving the above equation, for the surface concentration, SB, gives

and hence

Two extreme conditions can be identified: 1. For ks/ksL >> 1, SB approaches zero, and the reaction is completely mass transfer controlled, with rapp = ksL SA.

2. For ks/ksL 1) and high diffusion rate ( 2Cr3++ 4H20 +30 > 6 k 3 + + 2Cr3++7H20

Certain inorganic compounds interfere with the test but these can usually be eliminated. For instance, chloride ions can react with dichromate, as shown by: Chloride interference:

Mercuric sulphate (20%w/v) is usually added (1 in 30 dilution) to suppress the above chloride interference. The COD test can be used for toxic wastes that are not oxidised biologically. A further advantage is that the results can be obtained in a few hours compared to the five days required for the BOD test. COD values are typically higher than BOD values since more constituents can be chemically oxidised than are oxidised biologically. The BODKOD ratio is also a good indicator for the presence of refractory pollutants. For typical untreated domestic wastes the ratio is usually 0.4 to 0.8. The ratio varies according to the degree of treatment and from one effluent to another, but once the relationship has been determined then the COD test can be used and the corresponding BOD value deduced from the COD.

2.1.3.3

Total Organic Carbon

The Total Organic Carbon (TOC) test is another method for determining the organic content of water, and is particularly suitable for low strength wastes. The sample is placed in a furnace at 9OOOC and the organic carbon is oxidised catalytically to carbon dioxide. The carbon dioxide produced is measured by infrared analysis. One problem with the test is that the sample size is only 30p1 and the presence of suspended solids may therefore affect the results. The greatest advantage is that the test can be performed rapidly, and for this reason the test is gaining in popularity. Certain organic compounds are not oxidised in this test and therefore the measured TOC values are generally lower than the actual amount in the sample. Typical values for domestic waste are between 80 to 290 mg L-l (see Table 2.2).

2.1 Wastewater Treatment Processes

2.1.3.4

111

Nitrogen Analyses

In addition to the removal of organic compounds, the wastewater treatment process is required to remove nitrogen-containing compounds. These compounds can be oxidised, by nitrifying bacteria found in soil and natural waters, to nitrates and nitrites and eventually to gaseous nitrogen. The discharge into rivers and lakes of wastes high in nitrogen compounds can therefore lead to a reduction in dissolved oxygen concentration and subsequent loss of aquatic life. Increased levels of nitrogen-containing compounds in lakes and reservoirs (an example of eutrophication) can lead to algal blooms. The simulation examples EUTROPH, NCYCLE and ALGAE are concerned with this problem. Nitrification is the process where nitrifying bacteria convert ammonium ions to nitrate. First Nitrosomas species oxidise ammonium ions to nitrite, and then Nitrobacter species oxidise nitrite to nitrate ions. The first process utilises 3.2 mg oxygen per mg of ammonium N oxidised and the second 1.1 mg oxygen giving a total of 4.3 mg of oxygen utilised for the complete oxidation of 1 mg of ammonium N to nitrate ion. Nitrification can occur in combination with carbon oxidation or can be accomplished in a separate system following carbon oxidation, with little difference in overall performance. For complete elimination of nitrogen, nitrification is often combined with an anoxic denitrification process, which is the reduction of nitrite and nitrate to release N2, The nitrogen containing compounds in wastewaters typically consist of organic nitrogen, ammonia, nitrite and nitrate. The simulation examples ACTNITR, AMMONOX and AMMONFED deal with the problem of ammonia removal.

2.1.3.5

Kjeldahl Total Nitrogen Test

The Kjeldahl method is used to determine the organic nitrogen content and the 'total Kjeldahl nitrogen' content (NH3, NH4+ and trinegative N in organic compounds) of wastewater. Not determined by this method is the nitrogen present in the form of nitrite, nitrate, azide, oxime, and nitro-, or nitroso-compounds. First the sample is boiled to remove any ammonia, then the sample is digested by boiling in sulphuric acid containing K2SO4 and HgS04 as a catalyst. The organic nitrogen is converted to ammonia which can be measured titrimetrically, colourimetrically or by ammonia sensitive electrode. To avoid interferences and to achieve higher accuracy, the solution is strongly alkalised, and the free ammonium is distilled into boric acid, where it is then determined. The total Kjeldahl nitrogen content is determined in the same manner except that the sample is not boiled first. Electrodes are available for the determination of nitrate and nitrite ion concentrations. Standard solutions (NH4Cl for ammonia, KN03 for nitrate) should always be used to check the calibration of the electrodes. Electrodes are used very conveniently in the laboratory, but there may sometimes be problems of

112

2 Environmental Bioprocess Descriptions

interference when using electrodes in the field. Automated colourimetric tests are also sometimes used. Detailed descriptions of the analytical methods for nitrogen are available in standard textbooks.

2.1.3.6

Phosphorus

Phosphorus-containing compounds are important in the development of algal blooms (see ALGAE, EUTROPH and PCYCLE), and their discharge therefore needs to be controlled. The major hosphate-containing compounds of interest are the orthophosphates (H3P04, PO$ and HP04*-) because these are available for biological metabolism without further breakdown. The polyphosphates (compounds with two or more phosphorus atoms) undergo slow hydrolysis to orthophosphates in aqueous solution. Organically bound phosphates are of minor importance in domestic wastes but can be important in some industrial wastes. The wide use of washing powders containing orthophosphates has lead to the increased discharge of phosphates into rivers and lakes. In recent years, however, the use of phosphate-free or low phosphate washing powders has dramatically increased and hence reduced the magnitude of this problem. Typical domestic wastewater contains around 10 mg litre-1 total phosphorus and the recommended maximum discharge limit for rivers is 0.5 mg litre-I. Orthophosphates in wastewater are usually determined by a colourimetric technique. A compound such as ammonium molybdate is added to the solution and a colour complex is produced and measured colourimetrically . Polyphosphates and organic phosphates can be converted to orthophosphates by acid digestion and determined in the same way.

2.1.4

Biological Characteristics and Analyses of Wastewater

It is important to know the micro-organisms present in treated and untreated waters. Several classes of organisms are of prime importance. These are respectively: I I1 I11

Pathogenic micro-organisms Pollution indicator organisms Micro-organisms responsible for biological treatment

2.1 Wastewater Treatment Processes

2.1.4.1

113

Pathogenic Micro-Organisms

Pathogenic organisms found in wastewaters may be discharged by humans or animals having a disease or being carriers of a disease, The organisms may be bacteria (e.g., Vibrio cholerae), viruses (e.g., hepatitis A), protozoa (e.g., Crypotosporidium) or helminths (e.g., tapeworms).

2.1.4.2

Pollution Indicator Organisms

The number of potential pathogenic organisms is very high, and the actual concentrations present in polluted waters need only be very low for there to be a health risk. Some of these organisms are difficult to isolate and identify, and therefore the use of indicator organisms has been widely applied. E. coli is the most widely used indicator organism. This is present in large number in faecal matter but does not occur in other natural sources. E. coli is easily counted and has a greater resistance to most pathogens. Total coliforms, including E. coli, is sometimes used as an indication of pollution. However, some coliforms are also found naturally in soil so their presence is not necessarily an indication of pollution. The following criteria have been used in the UK to determine adequate water quality (Forster 1986): Coliforms and E. coli should be absent, but may be present in five percent of samples provided that: no sample contains greater than 10 coliforms per 100 ml no sample contains greater than 2 E. coli per 100 ml no sample has 1-2 E. coli and greater than 3 coliform per 100 ml no coliform or E. coli occurs within consecutive samples.

2.1.4.3

Micro-Organisms Responsible for Biological Treatment

A wide range of micro-organisms are responsible for the biological treatment of wastewaters and these will be discussed individually with regard to each process. Bacteria, protozoa, rotifers, fungi and algae are the organisms of importance in most bioprocesses. The composition of the microbial population is often a good indicator of the state of the bioprocess and can easily be investigated by microscopic observation. Several staining techniques are used to aid in the process of species identification:

114

2 EnvironmentalBioprocess Descriptions

Gram Stain: Differentiates gram positive bacteria (e.g., Staphylococcus, Streptococcus) which take the stain, from gram-negative bacteria (e.g., Gonococcus, typhoid bacteria) which do not. Neisser Stain: A stain used in identifying filamentous organisms which are important in the formation of flocs. Alcian Blue Stain: A heat fixed slide smear is stained with Alcian Blue (5 g L-l) for two minutes, then washed with water. It is then counterstained with carbol Fuchsin (10 g L-l) for a further two minutes. Any bacteria present will stain red and polysaccharide, important in the formation of flocs, will appear blue.

2.2

Primary Treatment Processes

The purpose of primary treatment processes is to remove a large part of the solids. This has the effect of reducing the load on the subsequent biological oxidation processes, of minimising variations in the wastewater loading and of reducing the effects toxic shocks on the treatment process. Several procedures have been developed to achieve these aims.

2.2.1

Equalisation

Wastewater when it enters a treatment plant usually flows first into a large agitated tank or basin. The flow rate from this basin is then kept as constant as possible to minimise flow surges that could cause the washout of the organisms and also to maintain an adequate nutrient supply in the secondary treatment processes. The volume of liquid in the basin therefore varies with changing input flow rate. This also has the effect of damping variations in the organic content of the waste, dilutes toxic materials and minimises variations in pH. Mixing is achieved either by mechanical stirrers or by aeration. If the basin is aerated a limited reduction in BOD can be achieved so reducing the load on the subsequent processes (see BASIN and LAGOON).

2.2 Primary Treatment Processes

2.2.2

115

Neutralisation

The biological processes in the secondary treatment of wastewater have an optimum performance at values of pH values lying between 6.5 and 8.5. Industrial wastes sometimes have pH values far from this range and therefore some form of primary treatment of pH is required. The simplest method is the controlled mixing of an acidic and an alkaline waste stream, however this method is not usually feasible. Acid wastes can be neutralised by passage through limestone beds (both upflow and downflow) or by the addition of quick lime, NaOH, NH40H or Na2C03. Alkaline wastes can be neutralised by adding a strong acid such as H 2 S O 4 or HCl (usually too expensive), or by bubbling acidic waste gases containing carbon dioxide through the solution.

2.2.3

Sedimentation

The primary purpose of sedimentation is the removal of suspended solids such as sand, grit and organic material, hence reducing the load on the subsequent oxidation units (EPA, 1975). The mechanism of the settling of suspensions has been classified according to the following four categories : Class 1: Class 2: Class 3: Class 4:

Discrete Settling; unhindered settlement of discrete particles. Flocculent Settling; unhindered settlement of flocculent particles. Zone Settling; hindered settlement of Class 1 and 2 particles. Compression Settling; compressive settlement of particles at the bottom of the tank.

The simulation example SEDIMENT models a sedimentation unit.

2.2.3.1

Discrete Settling

A discrete particle settling in an infinitely deep liquid will accelerate until the gravitational force equals the frictional drag force and the particle will then continue to travel at constant velocity. The constant settling velocity for spherical particles can be calculated from a simple force balance leading to the following relationship, as used in the simulation example SEDIMENT:

116


where v,

Settling velocity Density of the particle Density of liquid g Acceleration due to gravity d Particle diameter CD Drag coefficient

pp pL

For fully turbulent conditions (Re > 1000) the value of the drag coefficient is given by C D = 0.4. For creep flow (Re 500 mg litre-l), the floc particles adhere together and settle as a blanket. This phenomenon is known as zone settling, as a clear interface can be seen between the flocs and the supernatant. Zone settling is characteristic of activated sludge behaviour. The initial rate of settling is constant and proportional to the solids concentration. As the density and viscosity of the liquid surrounding the particles increases, so the settling velocity decreases. The mode of settling is then termed compression settling. These phenomena are described in the example SETTLER.

2.2.3.4

Coagulating Agents

Chemicals may be added to wastewaters to improve their sedimentation characteristics. Alum (A12 (SO&. 14H20), ferric and ferrous sulphate (Fe(SO&, FeS04), ferric chloride (FeCl,) and lime (Ca(OH)2) have all been used as coagulating agents. Lime may be used alone, but this is usually used in conjunction with one of the other coagulating agents.

2.2.4

Flotation

Flotation is used as a primary treatment to remove suspended solids, oil and grease from wastewaters (see SETTLER). It is also used in the thickening of activated sludge. Small bubbles are introduced into the liquid and as they rise they take with them suspended solids and oil globules. The air-solid mixture is skimmed off at the liquid surface and the clarified liquid is withdrawn from the bottom. Various methods have been used to introduce air bubbles to the liquid as follows: 1. Frothing agents, usually organic polymers, may be added to assist the flotation. Usually this method is too expensive for wastewater treatment. 2. In vacuum flotation, the liquid is saturated with air at atmospheric pressure and then a vacuum is applied causing bubbles to form in the liquid.

3. In pressure flotation, the liquid is saturated with air under pressure (350 to 500 KPa), and the pressure is then reduced to atmospheric, causing small bubbles to be formed in the liquid. Units both with and without recycle have been used. Flotation is useful for removing very small or light particles that do not readily settle and also for very fragile flocs that are easily broken.

118


2.3

Secondary Treatment Processes

The aim of the secondary treatment processes is to reduce the suspended solids and organic concentration of the primary treated effluent to such levels that it may be treated by tertiary treatment processes. Typically this means a suspended solids concentration of about 30 g m-3 and a BOD of about 20 g m-3. The degree of nitrification, denitrification and the removal of bacteria depends on the particular process considered. A wide range of different processes have been used in the secondary treatment of wastewater. These can be divided into aerobic and anaerobic processes, fixed film and suspended culture processes (see Table 2.9, Sec. 2.3.4).

2.3.1

The Activated Sludge Process

The activated sludge process was developed by Arden and Lockett (1914) and is still used widely today (Eckenfelder,1989). Organic waste is introduced into a reactor in which an aerobic bacterial culture is maintained in suspension. The bacteria utilise the organic compounds in the waste to produce more cells, carbon dioxide and ammonia. The reactor is maintained in the aerobic state, by diffused or mechanical aeration. The mixture of cells and treated water is passed into a settling tank where the cells are separated from the water (see Sec. 2.2.3) Part of the settled sludge is recycled to the reactor and the rest is taken as waste for further processing and disposal A representation of the activated sludge process is shown in Fig. 2.1, and Sec. 3.3 contains a number of relevant simulation examples on this process Entering effluent

Treated water

Y

I Sedlrnentation

Reactor

4

Sludge recycle

Figure 2.1. Flow diagram for the activated sludge process.

b

2.3 Secondary Treatment Processes

2.3.1.1

119

Biology of the Activated Sludge Process

Bacteria are the major micro-organisms responsible for the decomposition of the organic material of the wastewater. The most important genera are Pseudomonas, Zoogloea, Acgromobacter, Flavobacterium, Nocardia, Mycobacterium and Bdellovibrio. The bacteria aggregate to form flocs (see Sec. 2.3.3.3, FLOCl and FLOC2), which can settle out and be separated from the liquid by zone settling. In addition nitrifying bacteria, i.e., Nitrosomonas and Nitrobacter are generally present. Filamentous bacteria are also sometimes found (see ACTNITR). Protozoa that are present consume the non-flocculated dispersed bacteria and rotifers consume the non-settled small biological flocs (see ANDREWS, CURDS and MIXPOP).

2.3.1.2

Process Analysis

An activated sludge process can often be modelled approximately as a well-mixed reactor with sludge recycle (see ASCSTR). Mass balances provide the basis of the model and the approximate kinetics are usually taken to be either of Monod form or simplified to first order. The recycle ratio, R, is defined as the ratio of the recycle flow to the influent flow rates. This is an important parameter for maintaining high biomass concentrations. The reactor residence time is defined as the volume of the reactor divided by the influent flow rate, and this is important for allowing sufficient time for growth. The settler or sedimentation tank is important for concentrating the sludge for recycle (see SETTLER). The concentration factor is defined as the ratio of recycle biomass to the effluent biomass concentrations. The combination of mass balance relationships for both biomass and substrate (usually expressed in terms of its BOD) in the reactor yields a very simplified but very useful model for the activated sludge process, as can be seen in detail in the simulation example ASCSTR. Several operating and design parameters are widely quoted in the literature:

Process-LoadingFactor: This substrate to micro-organisms ratio is defined as kg substrate consumed per day per kg of biomass in the reactor. It is also sometimes defined as kg substrate fed per day per kg biomass in reactor. and is called the food to biomass ratio. Typical values are 0.2 to 0.6 kg BOD kg-1 biomass. Sludge-ResidenceTime: The mean solid (active biomass and other solids) residence time in the reactor, € I , , is given by:

120


Solids in system

em = (Rate of biomass synthesis) + (Solids input to reactor)

Under steady-state conditions, the rate at which biomass is gained must equal the rate at which it is lost from the system and the denominator can be related to the wastage rate and the biomass death rate. In practice, the death rate is not known so the use of sludge residence as a design parameter is not popular. The alternative use of sludge age as a design parameter has therefore been adopted instead.

Sludge Age: The sludge age, ec,is defined as ec

Biomass in the reactor = Rate of biomass generation

Under steady-state conditions, the rate of biomass generation is equal to the rate at which the biomass flows out of the system, which can be measured. The value of the sludge age for a conventional activated sludge process usually varies from between 2.5 to 14 days.

2.3.1.3

Modifications of the Activated Sludge Process

Several modifications to the basic activated sludge process have been developed. In order to create a more uniform oxygen demand along the length of the reactor, stepwise addition of the untreated effluent has sometimes been used (see the simulation example STEPFEED and Sorensen, 1980). Multiple units with various flow patterns and a wide variety of aeration and agitation systems have also been used. Contact stabilisation is a modification of the basic process that involves the aeration of the recycled settled sludge in an extra tank before it is mixed with the raw wastewater in the aeration tank. This reaeration of the solids improves the performance of the reactor due to the adsorption of organics onto the flocs. One interesting reactor design that has been used for activated sludge processes is the Deep Shaft Reactor (Hines, 1975; Hemming, 1977). This reactor was developed by ICI Ltd. in England and is essentially an airlift reactor 50 to 150m deep and with a diameter of 0.5 to 10m that is installed in the ground, as shown in Fig. 2.2. The shaft is divided into a central core region in which the liquid flows downwards and an outer annulus region in which the liquid flows upwards. Air is injected initially into the riser and causes the liquid to circulate (1 to 2 m s-l) owing to the difference in density between the aerated and unaerated regions of the reactor. Once the liquid has begun to circulate, aeration is gradually transferred to the downcomer, where the bubbles travel downwards and then enter the riser. Owing to the high pressure at the base of the reactor, very high oxygen transfer rates (10 kg 0 2 m-3 h-l) can be obtained. The reactor is generally operated in


121

continuous mode, with the wastewater and recycled sludge entering at the top of the reactor, and having a residence time of about 80 minutes.

Figure 2.2. The Deep Shaft Reactor with its liquid and gas flow patterns.

2.3.2

Aerobic Fixed Film Processes

When given the proper conditions, micro-organisms will often attach themselves to solid surfaces and grow as a biofilm. Attachment is thought to occur by many different mechanisms, the most common of which is likely to be the formation of a polysaccharide slime; a mechanism which is not surface specific. The natural ability of organisms to grow on surfaces can be exploited to provide biomass retention for the design of biological reactors. Experience with aerobic, anaerobic, autotrophic and heterotrophic organisms has shown that after an undetermined adaptation period attached growth can be expected. Aerobic fixed film processes can be considered either as stationary processes, in which the water passes over the biomass (e.g., trickle filter) or as moving systems in which the biomass moves through the water (e.g., rotating biological contactors and fluidised beds).

122

2.3.2.1


Trickling Filters

The trickling filter (see TRICKLE) is a packed bed reactor with an upwards flow of air and a downwards countercurrent flow of liquid waste (Fig. 2.3). The packing is traditionally stone but recently plastic media have been employed successfully. The ecology of trickling filters is complex and can involve bacteria, protozoa, fungi, rotifers, nematodes and fly larvae. The wastewater runs in a thin sheet over the packing material supporting the growing microbial film. These microorganisms utilise the soluble organic matter in the waste but little or no degradation of suspended organic matter is achieved. Trickling filters have been widely used to treat domestic sewage in small to medium scale installations owing to their low operating costs.

Figure 2.3. Schematic diagram of a trickling filter.

2.3.2.2

Fluidised Sand Beds

Fluidised beds are an accepted technology for the contacting of a continuous fluid phase with a discontinuous particulate solid phase. Operating with solids denser than water, the particles are suspended in a column by an upflow stream of liquid having a linear flow velocity equal to the particle settling velocity. Biomass will naturally adhere to the surface and form a biofilm. As compared with a trickle filter or packed bed, a fluidised bed will generally give improved mass transfer and prevent channelling or stoppage. Sand is a convenient and inexpensive material, and a particle size of 0.5 mm or less has proved to be suitable. Good liquid-solid contact in a flowing system may be achieved with a fluidised bed, in which, for example, sand particles are


123

suspended in the upwardly flowing liquid. The flow velocity will depend on the particle size and density. The density of a biomass-covered sand particle will decrease as the biomass in the reactor accumulates, causing the bed to expand. Operating flexibility is achieved by using a column which has a larger crosssection at the top, to allow for bed expansion and hence to enhance retention of the smallest and lightest particles (see DENITRIF). With biomass adhering to a solid particulate carrier, the reactor can be operated at flow rates which are independent of the maximum specific growth rate. Thus the D < krnrestriction, which is applicable to suspended culture, has no significance for biofilm reactors, because the residence time of the liquid phase is uncoupled from that of the biomass. The activity of the reactor per unit volume will depend largely on the surface area per unit volume that can be provided by the solid carrier and on any mass transfer limitations that the biofilm may present. Particles which are covered by biofilm are less dense than clean particles and are therefore suspended at lower flow velocities. Fluidised beds thus exhibit a natural stratification with the less dense particles collecting at the top of the bed and the denser particles concentrating at the bottom. This stratification of the particles is thus very convenient for removing biomass by removing covered particles from the top of the column and thus also for controlling the biomass film thickness. The retention of biomass in high concentration (10 - 30 g L-I) creates a high activity per unit volume. If the reaction is aerobic, then an adequate oxygen supply to cells must be ensured. A fluidised bed reactor system can be operated as a twophase liquid-solid system, as shown in Fig. 2.4, or as a three-phase system. Aeration or oxygenation of a two-phase fluidised bed may be provided by an external oxygenator located in the liquid recycle loop.

Figure 2.4 Biofilm fluidised bed reactor with external oxygenation.

A three-phase reactor is the simplest configuration, but this involves additional problems concerning oxygen transfer. Very fine solid particles tend to promote bubble coalescence. Thus, in a fluidised bed of 0.5 mm sand, very large gas bubbles of several cm diameter will rise rapidly in the bed. This creates high

124


turbulence and shear conditions, to which most aerobic biofilms can adapt (see DCMDEG).

2.3.2.3

Rotating Biological Contactors

The rotating biological disc contactor comprises a series of discs (2-3 m diameter) mounted on a horizontal shaft, which is driven so that the discs rotate at right angles to the flow of wastewater (Fig. 2.5). The shaft is typically positioned above the liquid level so that the discs are only partially submerged. A biofilm grows on the discs and this is exposed alternatively both to the atmosphere, where oxygen is absorbed, and to the liquid phase where soluble organic matter is utilised. The discs are arranged in groups and several reactors may be used in series.

Figure 2.5. Rotating biological disc reactor.

2.3.3

Anaerobic Treatment Processes

Anaerobic processes have several advantages over aerobic processes for wastewater treatment, with especially high strength wastewaters. These are: a low sludge production (one-tenth that of aerobic treatment), zero aeration energy requirement, reduction of odours in a closed system, and associated methane energy production. The important disadvantages are that methanogenic organisms grow slowly, that the stability of anaerobic processes can be upset either by toxic


125

substrates or by overloading and that the process is not so well understood. The disadvantage of slow growth, however, can be largely overcome by advanced reactor designs employing biomass retention, and overloading can be avoided by control of the feed rate.

2.3.3.1

Reactions and Stoichiometry in Anaerobic Digestion

Anaerobic degradation is a very complex multi-substrate, multi-organism process. Although there are still many unknowns, most mechanistic models are based on a consideration of the four overall steps shown in Fig. 2.6. In a first step, polymer materials (carbohydrates, proteins or lipids) are hydrolysed to yield the monomer compounds (amino acids, sugars and fatty acids). In a second acidogenic step, these compounds are fermented to organic acids (acetic, propionic and butyric), carbon dioxide and hydrogen. In the third acetogenic step, organic acids with more than three carbon atoms are converted to acetic acid and hydrogen. The last methanogenic steps convert acetic acid to methane and reduce carbon dioxide with hydrogen to methane. With regard to reactor operation, there are two important conditions that apply: 1) If solid matter has to be degraded, then the hydrolysis step may be rate limiting and overloading will not be important. 2) Overloading of the reactor may occur if easily degradable material is to be treated. This can result in a pH decrease or a hydrogen gas increase. The influence of hydrogen is discussed in the simulation example WHEY,

2.3.3.2

Modelling Anaerobic Reactors

As discussed by Heinzle et al. (1993), many models have been used in an attempt to describe anaerobic systems, reflecting the need to simplify a complex system which is not completely understood. The following model is based on the work of Denac et al., 1988 and Ryhiner et al. (1993), as reviewed by Heinzle et al. (1993). The experimental system involved both the single and two-stage anaerobic treatment of whey in biofilm fluidised bed reactors. The six unknown stoichiometric coefficients of the acidification reaction, R1 in Table 2.7, were determined by measuring acid production and COD reduction in the first reactor of the two-stage system, which produced very little methane. Reactions R2 and R3 were assumed not to proceed under these conditions. The values found were for the stoichiometric coefficients of reaction RI: VBu = 0.5; Vpr = 0.12; VAc = 0.15; VCO2 = 0.23; v H 2 = 0.24; V H 2 0 = 0.095.

126


Figure 2.6. Reaction scheme for anaerobic degradation. The symbols are as follows: Poly - polymeric material (proteins, fats, hydrocarbons, etc.); XHYBiomass hydrolysing Poly; Mono - monomeric products of hydrolysis; XAG - acid forming biomass; HPr - propionic acid; Pr- - propionate; Xpr - biomass growing on propionate; HBu - butyric acid; Bu- butyrate; X B -~biomass growing on butyrate; HAc - acetic acid; Ac- - acetate; X A -~ biomass growing on acetate; XH - biomass growing on hydrogen and carbon dioxide.

The non-volatile substrates in the waste treatment system can be balanced as


127

where the possibility that each substrate may be consumed by several reactions is taken into account by the summation. Only the balances for the gaseous components (H2, C02 and CH4) include the mass transfer term.

Table 2.7. Reactions in the anaerobic degradation of whey. Hydrolysis and acidification of whey

Acetification

+ 2 H2 + Ht f CH3COO- + C02 + 3 H2

CH3(CH2)2COO- + 2 H20 f 2 CH3COOCH3CH2COO-

+ 2 H20

Methanogenesis CH3COO- + H+ f CH4 + C02 C02 + 4 H2 f CH4 + 2 H20

In a fluidised bed reactor, which retains biomass, the biomass balance is the same as for a batch reactor. In order to account for biomass losses an additional empirical term was added, giving Rate of Rate of Rate of loss in Rate of ( accumulation ) = ( production) - ( death ) - ( effluent

where MXLis the rate of biomass loss from the reactor by overflow. The growth rates were modelled in Monod fashion

where subscript i indicates the various substrates for biomass formation (whey, butyric, propionic and acetic acid, hydrogen).

128


The specific growth rates for whey, butyric acid, propionic acid and hydrogen have the form

Acetic acid has, however, an additional substrate inhibition term PAC

=

PAC

max CAC

c A c + KAC + c A c 2 / ~ I A c

Constant yield coefficients were used to relate biomass growth to substrate uptake:

Monod kinetics was assumed for the rate formation of a product j from substrate i as -Pi Xi Y P ~ / s ~ ‘Pj

-

Yxi/si

Henry’s Law

balances

I

Is

t

Transfer rate I

‘xi Yields

I

q-tq

-

ball ces

YCH4 yc02

balances

Q-

Controller

-

YCH4

Figure 2.7. Information flow diagram for the anaerobic degradation model.

I

129


The concentrations of the undissociated organic acid substrates are pH dependent, and can be calculated with the help of an ion balance. Also the thermodynamic inhibition by hydrogen can be described empirically. These calculations are discussed both by Heinzle et a.1. (1993) and in the example WHEY. The entire model, including the gas phase, can be summarised by means of an information flow diagram, as seen in Fig. 2.7 and which applies also for the case of control on the feed rate, Q.

2.3.3.3

Response Dynamics of Anaerobic Reactors

Dynamic responses of the uncontrolled reactors to step changes in feed concentration (+42%)were investigated both experimentally and by simulation. The responses for pH and gas production rate of the single stage reactor to a step increase in feed concentration are shown in Fig. 2.8. The corresponding response in the gas composition is given in Fig. 2.9.

8

0 0

1

2

3 Time [h]

4

5

Figure 2.8. Response of pH and gas rate to a feed concentration step change increase in single stage reactor (experiment and simulation)

130

2 Environmental Bioprocess Descriptions 68

1

140

66

38

36

E

N

34

8

60 32

58 56! 0

-

-

.

I

1

.

-

.

I

2

. .

-

I

3 Time [h]

.

- .

I

4

. . .

5

30

Figure 2.9. Response of gas composition to a feed concentration step change increase in single stage reactor (experiment and simulation).

2.3.3.4

Control of Anaerobic Reactors

The control of anaerobic reactors has been attempted on the laboratory and pilot scale, as well as on simulated systems, but little experience has been obtained on full scale equipment. A detailed review by Heinzle et al. (1993) is available. To emphasise the potential for modelling and simulation in this area, a control study from the work of Ryhiner et al. (1993) is presented below.

Proportional-Integral-DifferentialControl of p H PID digital control of pH by feed rate manipulation was investigated to counteract a step change in the feed concentration. Comparisons of the experimental and simulated PI control responses for step changes in the flow rate are given in Fig. 2.10 for the single stage reactor. Essentially identical controller settings for both simulated and experimental processes were applied. The differential part of the PID control equation was found to be ineffective for this system.

131


- 8

- 2

5.6

1

!

0

’

I

Time [h]

I

I-

2

3

0

Figure 2.10. Response of the pH controller to a step change doubling of the feed concentration to the single-stage reactor (Experimentand simulation). The manipulated feed rates from the controller are shown (Flowrate exp. and Flowrate sim.)

Control with Measurement of Organic Acids The organic acid response in anaerobic reactors is rapid (here with 15 min sampling times) but requires rather expensive analytical methods. Shown in Figs. 2.1 1 and 2.12 are the simulated and experimental results from PID control based on acetic acid measurement. The controller constants used in the experiment were similar to those used in the simulation. The gain was set somewhat higher in the experiment and caused oscillations. Similar results were obtained with propionic acid control.

132

2 Environmental Bioprocess Descriptions 25

Flowrate 20

=.E

'"1 0

5

1

2

3

4 Time [h]

5

6

Figure 2.11.

Acetic acid PID-control of the two stage system, simulated.

Figure 2..12.

Acetic acid PID-control of the two stage system, experimental.

7

133


Summary of Anaerobic Reactor Control Although adjusting the controller parameters requires a certain amount of trial and error, a simulation model is invaluable for determining an approximate controller setting. The integral and differential settings agreed well both in the experiment and the simulation, whereas the proportional controller constants needed readjustment. Table 2.8 summarises the experience with the feedback control systems in the above cited work .

Table 2.8. Evaluation of PID control schemes for anaerobic wastewater reactors. Controlled variable

Advantages

Disadvantages

PH

Fast response in systems with low buffer capacity. Simple measurement by electrode.

Slow response in buffered systems. Long term instability of measurement.

Propionic and acetic acid

Fast response.

Difficult and expensive analysis involving liquid phase sampling and GC analysis.

Product gas flow rate.

Simple and reliable measurement.

Not suitable for simple PID control because of influence of pH on C02 equilibrium.

Methane and C02 content in biogas.

Reliable and rather inexpensive measurement (IR or GC).

Slow response. Not stable with simple PID control.

Dissolved hydrogen concentration.

Fast response.

Expensive measurement (mass spectrometer). Hydrogen electrodes poisoned by H2S during long term application (Denac et al., 1988)

134

2.3.3.5


Anaerobic Reactor Design

Owing to the low growth rates and low specific rates rates of anaerobic organisms, it is important for the design of efficient treatment processes to achieve high biomass concentrations and to retain the active biomass within the reactor. In the following section, various reactor designs are discussed, based largely on the review of Schink (1988).

Anaerobic Contact Process One of the simplest ways to retain biomass is to settle the reactor effluent and recycle the sludge, as in the aerobic activated sludge process. The equivalent anaerobic system is referred to as the anaerobic contact process. Its efficiency is limited by the difficulty in achieving high sludge concentrations in the sedimentation tank, owing to the nature of the anaerobic sludge.

Anaerobic Filtration Anaerobic filters, both upflow and downflow have been used to treat liquid wastes (Chian and DeWalle, 1977). They are similar to trickle filters in that a microbial film grows on an inert solid support, either as a film or trapped within the pores of the solid packing. The advantage of this system is that it produces high concentrations of active biomass without the use of a settler. The disadvantage however, is that plugging may develop with high load conditions or if suspended solids are present

Upflow Anaerobic Sludge Blanket Reactor The anaerobic sludge blanket (UASB) reactor, developed and commercialised by Lettinga et al. (1979), is essentially an open column through which liquid waste is passed with a very low upflow velocity, sufficient to retain the flocs and granules of anaerobic biomass which develop and form a blanket at the bottom. Inclined settling screens at the top of the column provide separation of the gas and liquid phases from the solid material, which settles again to the bottom. Proper operation requires the formation of flocculating biomass, and startup may require inoculation of large amounts of sludge from another operating system.

Expanded and Fluidised Bed Reactors As discussed in Sec. 2.3.2, under flow conditions both aerobic and anaerobic micro-organisms can attach to surfaces to form films. This is the basis for anaerobic fluidised and expanded bed reactors, which employ small solid particles,

135


such as sand, as a biofilm carrier. The reactor is operated with an upflow fluid velocity sufficient to completely fluidised the bed (200% expansion) or to slightly expand the particle bed. This expansion has the advantage that channelling is reduced, liquid mixing is improved and blockage by solids is eliminated. Practical problems include providing the initial flow for expansion at startup, the time required for startup and the stability of the anaerobic biofilm during overloads or toxic shock conditions. These reactors can be operated at extremely high conversion rates.

2.3.4

Biofilms and Flocs

In many important environmental processes the cells are aggregated either into flocs or biofilms. This can be beneficial or detrimental according to the circumstances. Biofilms and flocs can be responsible for:

- Energy losses brought about by a reduction in heat transfer rate and increased fluid flow resistance in pipes and fittings; - Material deterioration due to corrosion; - Pathogenic organisms and diseases, for example, legionella in cooling towers and diseases of the lungs, intestine and urinary tract; - Reduced water quality caused by biofilm formation in drinking water pipes; - Dental plague and caries; - Reduced industrial product quality, for example in the pulp and paper industries; - Reduced effectiveness of ion exchangers. Biofilms and flocs are exploited in both the aerobic and anaerobic treatment of wastewater, as listed in Table 2.9 below.

Table 2.9. Uses of flocs and biofilms in wastewater treatment processes. Anaerobic

Aerobic Biofilm

Floc

Biofilm

Biological fluidised Activated sludge Anaerobic upflow filter bed Anaerobic Trickling filter Deep shaft downflow filter ExpandedIFluidised Rotating reactor biocontactor

Floc Upflow sludge blanket Plug-flow digestor Batch sludge system

136


The advantage of biofilms and flocs are that they enable biomass to be retained in a reactor at flow rates greater than that for washout. They often consist of several microbial populations which perform complementary processes. The rapid transfer of substrates between the populations facilitates rapid wastewater treatment. Another advantage is that the diffusional barrier renders the biomass less susceptible to irreversible damage due to shock loads and toxic shocks.

2.3.4.1

Formation of Biofilms

Biofilms can form on a variety of different surfaces, e.g.:

- Stones, clinker (e.g., trickling filter) - Sand, pumice, activated charcoal, kieselguhr (fluidised beds) - Plastic packing (trickling filters) - Plastic sheets, foams (downflow filter) - Metals (filters, fluidised beds) Flocs form by the cells adhering to other cells and to cell debris. There are several stages in the development of a biofilm, including:

- Formation of a conditioning film on the inert surface - Attachment of bacteria and extracellular polymer production (reversible) - Further attachment and growth (irreversible) - Attachment, growth and detachment

The Inert Surface Several factors influence the adsorption of bacteria to a surface:

Charge: Bacteria usually have a net negative charge and so will be attracted to surfaces having a positive charge but are repelled from surfaces having a negative charge (e.g., most plastics). The formation of a conditioning film can dramatically alter the surface properties. The pH of the liquid phase also influences the charge. Surface roughness: A rough surface provides an increased surface area for microbial attachment. Larger pores in the surface can provide shelter for attached bacteria from the turbulent forces occurring in the fluid at high flow rates. Surface free energy: The Gibbs free energy (G) is a measure of the potential for bonding at a surface due to free bonds. The adhesion of bacteria, molecules and ions reduces the free energy. High energy surfaces (e.g., glass) have a much


137

higher potential therefore for biofilm formation than low energy surfaces (e.g.,

PTFE). Wettability: The wettability of a surface is important because this influences the interaction of water with the surface and its susceptibility to displacement. Water participates in van der Waal's, polar, electrostatic and hydrogen bonding.

Cell Surface Properties The nature of the surface of a cell influences its attachment to an inert surface. The cell surface generally has a net negative charge but can often have regions of both positive and negative charges. The cell surface often has regions of hydrophobicity and hydrophilicity which influence the orientation of cells in a biofilm. In addition, the presence of excreted polymers, flagella and pilli influence the attachment process.

Liquid Phase Properties The pH of the liquid phase determines the dissociation of amino and carboxyl groups on the cell surface and therefore influences the electrical charge on the cell surface. The presence of surfactants and oil, and the ionic strength and nutrient concentration of the medium can also influence the attachment process.

Cell Attachment and Detachment Cells travelling in the bulk liquid are transported to a solid phase via momentum effects and by diffusion. In laminar flow there is no momentum component perpendicular to the surface so diffusion is the main process. The diffusion process is a combination of Brownian motion and in the case of motile cells the random motion of the cells. Once a cell is in contact with a surface, it can either be transported away again or it can be adsorbed to the surface. The probability of adsorption is express-ed in terms of a sticking efficiency. Physical adsorption is reversible and may be caused by van der Waal's forces, hydrogen bonding, coordination bonding and protonation. Chemical adsorption (or adhesion) is an irreversible process and involves the formation of an ionic or covalent bond. Cells can be removed from the biofilm by shear forces caused by the liquid flow. The rougher the surface the better the cells are protected. The physiological state of the cells also influences cell detachment.

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Modelling Biofilms

2.3.4.2

Adopting the methodology of Sec. 1.12.2, a mass balance for the viable cells in a biofilm can be derived, where:

[

Accumulation of cells in biofilm

Transport (rate of cells

to biofilm

)(zLzy) + (sib) (

)

- cell Rate Of death

-

(cell

Rate of detach-) ment

The growth of cells in a biofilm is often modelled by Monod type kinetics. If the biofilm is more than several micro-organisms thick, the transport of substrates through the biofilm may become rate limiting. A partial differential mass balance equation may be derived relating the variation of substrate concentration with both time and position. Rectangular geometry can be employed (see simulation example BIOHLM), or in the case of a floc, spherical geometry may be used to describe the geometry of the floc or film. In this case, the floc is divided into a number of concentric spherical shells (see simulation examples FLOC 1 and FLOC2). One problem with this approach is that as the biomass grows the thickness of the biofilm increases. This problem can be treated in several ways (Gujer and Wanner, 1990): a) The number of layers in the model increases or decreases depending on the thickness of the biofilm. All the layers are of constant thickness except the outer layer. When the outer layer grows to be thicker than the other layers, it is divided into two, parts, one having the same thickness as all the others and the remaining part being thinner. If the thickness of the outer layer is reduced to zero (e.g., excessive cell detachment) then the number of layers is reduced by one. b) The number of layers in the model remains constant but their thickness is variable. After each computational step the thickness of the layer is calculated by dividing the thickness of the biomass by the number of layers. c) It is possible to transform the coordinates with respect to the biofilm thickness so that there is no longer a moving boundary problem. In the above analysis several simplifying assumptions are made, including:

- Single species of biomass - Laminarflow - One dimensional geometry - Constant film density

139


It is also assumed that the biofilm is homogenous with respect to biomass, but experimental results have shown this to be the exception rather than the rule. Multispecies population models have been developed by Wanner and Gujer (1986) and by Kissel et al. (1984). Mixed populations of suspended micro-organisms compete for nutrients but in biofilms they also have to compete for space. The kinetics of different species vary and this can lead to some species growing near the liquid surface and others growing deep in the biofilm in the vicinity of the solid surface where the conditions can be considerably different. Wanner and Gujer used a mathematic model to study the coexistance of a heterotrophic and an autotrophic micro-organism within a biofilm. At first the heterotrophic species outgrows the autotrophic species but as the carbon supply in the depth of the biofilm decreases, conditions become more favourable for the autotrophic species, thus leading to both spatial and temporal variations in the differing microbial species within a biofilm. It was assumed that the flow by the biofilm was laminar, however, in many natural processes the flow is turbulent. The transport of cells to the walls in turbulent flow is primarily by eddy diffusion and molecular diffusion becomes insignificant. The processes whereby cells are detached from the biofilm are also different. In turbulent flows, drag and lift forces become more significant. The one-dimensional analysis presented above can be extended to two or three dimensions if sufficient experimental data has been obtained. Transport of material parallel to the solid surface is usually very small compared to transport perpendicular to the surface, and so may be neglected

2.3.4.3.

Bioflocs

Many organisms form flocs or pellets over a wide range of sizes as shown in the Table 2.10 below:

Table 2.10. Biofloc sizes for various cultures (Atkinson and Daoud, 1976). Organism type Agaricus blazei (mushroom) Aspergillus niger Mixed bacterial cultures Brewers' Yeast

Floc size range (mm) 2.8-25 0.2-0.5 0.025-5.0 c13

The size distribution of activated sludge flocs in a stirred tank reactor is shown in Fig. 2.13 below:

140


Figure 2.13. Size distribution of activated sludge flocs. (Taken from Parker, D.S., Kaufmann, W.J. and Jenkins, D., 1971).

2.4

Tertiary Treatment Processes

The tertiary treatment of wastewater is the further treatment of biologically treated sewage to improve its quality before the treated water is discharged into a river. Conventional biological treatment produces an effluent containing typically about 30 g m-3 suspended solids and 20 g m-3 BOD. Tertiary treatment processes aim to reduce both these values to around 2 g m-3 and also to reduce the total nitrogen to 5 g m-3 and the total phosphorus to 0.5 g m-3. The soluble organic matter, heavy metals and pathogenic micro-organisms are also reduced in concentration. Various processes are used in the tertiary treatment of wastewaters.

2.4.1

Grass Plots

Small sewage works traditionally have discharged secondary treated wastewater onto gently sloping (1 / 60) grass fields. The soil is impervious so that the liquid

2.4 Tertiary Treatment Processes

141

flows in a thin layer towards the receiving body of water. Purification of the water is achieved by the action of soil micro-organisms, by the uptake of nutrients by plants and by the filtering action of the soil. UV radiation from the sun reduces pathogenic organisms almost completely. Approximately 75% of suspended solids, 55% BOD and 50% ammoniacal nitrogen are removed by the use of grass plots. The grass plots however must be renewed every four years and they also need cutting and cleaning. Not all sewage works have suitable land for this type of treatment and so alternative methods are employed. In addition to grass plots, reed beds and aquatic plants have been used to treat wastewaters. The local climatic conditions, the nature of the wastewater, the soil characteristics and the area of land available determine the best method.

2.4.2

Lagoons

Shallow lagoons with a depth of l m and retention times of between 4 and 21 days and deep lagoons with a depth of 2 to 4m and retention times greater than 17 days have been used. Algal growth is often a problem and the death and decay of large quantities of algae may even increase the BOD concentrations. The difference between deep and shallow lagoons is that deep lagoons become anaerobic towards the bottom. Denitrification also occurs in deep lagoons. See the simulation example LAGOON.

2.4.3

Filters

A variety of designs of filter have been used to treat wastewater. The slow sand filter consists of two layers; a top layer of sand (0.5 mm) 40 to 70 cm thick, above a layer of clinker (38 mm) 20 to 40 cm thick. 60 percent of suspended solids, 40 percent of BOD and 35 percent of bacteria may be removed by this type of filtration (see FILTER). The top layer of sand needs to be replaced periodically and the filtered sludge removed from the top. The performance of the filter is affected by snow and ice. The rapid gravity sand filter is similar to the slow sand filter except that the sand layer is thicker (50 to 300 cm) and the sand coarser (1 mm). The layer of clinker is replaced by a layer of gravel which is finer (6 to 13 mm) and slightly deeper (50 to 60 cm). Slightly improved performance is obtained, much higher flow rates can be used and the filter can be operated for a longer period of time. Cleaning can be achieved by back washing. In the upward flow sand filter the water flows upwards, first through a layer of pebbles (25 mm), then through a layer of coarse gravel (6 to 13 mm), then fine gravel (2 to 5 mm) and finally through sand (1 mm). The filter can operate for a

142


much longer period of time and is washed by backflushing. Similar degrees of purification are obtained as with the rapid gravity sand filter, but the removal of bacteria is only about 5%.

2.4.4

Microstrainers

The water is fed continuously into a rotating (5 rpm) drum made of stainless steel or reinforced plastic with 20 to 35 mm diameter pores. High intensity UV lamps are used to reduce microbial growth. Blades or jets of water are used to remove solids that build up on the inside of the drum

2.4.5

Membrane Technology

Membranes have been defined as "an intervening phase separating two phases and/or acting as an active or passive barrier to the transport of matter between phases adjacent to it" (Gekas, 1986). Materials with membrane function can be solid, liquid or gaseous and have been widely used in separation processes. Membrane technology is fast developing and due to its separation efficiency coupled with low energy requirements it is becoming increasingly attractive in wastewater treatment. Membrane processes (ultrafiltration, etc.) are principally different from conventional filtration, in that:

1) Cross flow, a large tangential flow across the membrane surface, effectively prevents the formation of a filter cake. 2) Ultrafiltration performance is critically dependent on membrane characteristics. 3) Membrane geometry in the actual equipment, i.e., membrane modules, is of eminent importance. Membrane processes can be categorised based on the size of the retained particles as depicted in Fig. 2.14.


143

Figure 2.14. Example of membrane process classification based on the rejection characteristics (Cheryan and Nichols, 1992).

Membranes can also be categorised on the basis of whether they are pressure driven (ultrafiltration, microfiltration), electrically driven (electromembrane, electrodialysis) or driven by differences in partial vapour pressure (pervaporation, membrane distillation) or differences in chemical potential (reverse osmosis). Membrane processes are relatively expensive compared to traditional methods of wastewater treatment and are prone to fouling which necessitates periodic cleaning. Membrane life-time is generally limited. It is unlikely therefore that membrane processes alone will be used to treat wastewaters, but they have great potential to be used in conjunction with other treatment processes, or for the treatment of hazardous wastes and the wastes containing valuable substances which are to be recovered.

Examples of the Use of Membranes in Water Treatment Advances in membrane technology coupled with increasingly stringent emission standards have made the applications of reverse osmosis (RO) and ultrafiltration (UF) membranes to wastewater treatment attractive. Membranes have been used in primary, secondary and tertiary treatment processes usually with the aim of removing a specific product, either so it can be recycled or because it is toxic and would inhibit subsequent biological treatment processes.

144


Many waste streams contain valuable components, albeit at very low concentrations, and membranes have been used to concentrate these components which can then be reused in the process thereby increasing the efficiency of the process and reducing the concentration of the wastewater. An example of this is the recovery of E-caprolactam, a monomer used in the production of nylon, by PEC-1000 RO membranes (Nakagawa et al., 1985). UF membranes have been used to treat wastes from the metal processing industries (Rautenbach and Albrecht , 1989). Porter (1990) discuss the treatment of wastes from the metal plating industry, in particular nickel plating, using reverse osmosis membranes. Table 2.1 1 below gives typical rejection values that can be achieved.

Table 2.11. Rejection of ions by reverse osmosis membranes (Porter, 1990). ~~

Ion

Rejection range (%)

Nickel Copper Cadmium Chromate Cyanide Zinc

98-99 98-99 96-98 90-98 90-95 98-99

The Taio Pulp Company in Japan (Okamoto et al., 1985) used a tubular membrane reactor to reduce the effluent load to the plant's activated sludge process. The membrane had a molecular weight cut-off of 8000 and allowed small organic molecules to permeate, whilst retaining larger slowly digested molecules, so improving the efficiency of the activated sludge process. The ultrafiltration membrane removed 83% of the COD and the activated sludge process an additional 13%. The activated sludge process alone could only remove 51% of the COD. Porter (1990) also reports on the coupling of a ultrafiltration membrane to an activated sludge reactor. The membrane retains the biomass, so a clarifier is unnecessary, and the effluent contains < 10 g m-3 BOD and no suspended solids. Very high biomass concentrations can be achieved, allowing a smaller reactor volume than with a conventional activated sludge process. Membrane reactors have found applications for treating waste streams containing toxic components. A good example of this is the use of a flat plate membrane reactor to treat cyanide-containing wastewaters (Basheer et al., 1993). The cyanide can permeate through the membrane where it comes in contact with immobilised cyanidase. The products of the enzymatic degradation are ammonia and formate and can diffuse back across the membrane. High molecular weight compounds and the immobilised enzyme can not pass through the membrane so


145

never come into contact. The cyanide can be completely removed and the wastewater can then be subsequently treated by a conventional process. Membrane technology has also been applied to the treatment of industrial laundry wastewater (Porter, 1990), vegetable oil processing effluents (Kosseoglu and Engelgau, 1990), textile dyehouse effluents (Buckley et al., 1985) and oil containing wastes from metal processing industries (Rautenbach et al., 1984). Other applications include the removal of pesticides such as lindane (Skrinde et al., '1969),the removal of algae after tertiary treatment in lagoons (Oakey, 1972) and the production of drinking water. Recently, reverse osmosis membranes have been successfully applied for treatment of leachates from hazardous and domestic waste sites. As an example, the novel Rochem Disc Tube module system is shown in Fig. 2.15 (Gorler and Stevens, 1993).

Figure 2.15. Rochem Disc Tube reverse osmosis module. The membrane elements are in series. R=Raw wastewater, P=Permeate, C=Concentrate.

146


2.5

Water Pollution Modelling

2.5.1

Eutrophication of Lakes and Reservoirs

Eutrophication has been defined as the biological reaction of aquatic ecosystems to nutrient enrichment, and can lead to algal blooms, oxygen deficiency, fish death and other ecological changes. Eutrophication is most frequently caused by the fertilisation of water with nutrients from agricultural run-off contaminated by fertilisers and by sewage containing detergents, animal and human wastes. Lakes can be classified as being oligotrophic (from the Greek meaning 'little food') or eutrophic (Greek for 'well fed'). The intermediate state is referred to as mesotrophic and the terms ultra-oligotrophic and hypertrophic have also been used to describe the extreme lack or abundance of nutrients in a lake. The Organisation for Economic Cooperation and Development (OECD) have defined boundary conditions for the classification of lakes and reservoirs, as shown in Table 2.12.

Table 2.12. OECD boundary values for lake classification (OECD, 1982). Trophic Level

Total phosphorus (pg litre-')

Ultra-oligotrophic Oligotrophic Mesotrophic Eutrophic Hypertrophic

c4.0

Mean Maximum Mean chlorophyll a chlorophyll a Secchi (pg litre-l) (pg litre-l) depth (m)

10- 35 35 - 100

c1.0 c2.5 2.5 - 8 8 - 25

>lo0

>25

6.0

43.0 8 - 25

>6.0 6- 3

>3.0

25 - 75 >75

3 - 1.5 4.5

3 - 1.5 1.5 - 0.7 ~0.7

Phosphorus, especially when in the form of ionic orthophosphate (P043-) is widely accepted as being the critical nutrient in determining the degree of lake eutrophy, but in some cases inorganic nitrogen (N&+ or NO3-) can be the limiting nutrient, and in the seas and oceans it is thought that nitrates are the primary cause of the algal blooms that have been observed in recent years. Toxic algal blooms have been reported in the Baltic, the Irish Sea, and the Mediterranean. Chlorophyll a is a measure of the total phytoplankton content and is an important parameter for lake classification and characterisation. The Secchi depth is a measure of the degree of transparency of the water and is a function of the algal concentration and the concentration of dissolved components. Eutrophication generally reduces the transparency of the lake, which has a large influence on the absorption of light.

2.5 Water Pollution Modelling

2.5.1.1

147

Factors Influencing Lake Productivity

The productivity of a lake, i.e., the generation of new organic material, is dependent on many factors. Some of these are discussed below.

Light Sunlight provides the energy for the most important biological and physical processes occurring in a lake or a reservoir. Sunlight determines the rate of photosynthesis which determines the rate of production of plants and which in turn determines the rate of production of animals which feed on the plants. In the absence of light, plants respire and hence take up oxygen and thus reduce the dissolved oxygen concentration of the lake. Both the intensity and duration of sunlight are subject to regular seasonal variations and are usually modelled as sine functions. The degree of cloud and snow cover also influence the light energy available for photosynthesis under water. The light intensity at any given depth of the lake depends on the light intensity at the water surface, the depth beneath the surface and the transparency of the water. The transparency is dependent on the presence of dissolved and suspended matter (especially algae) in the water. The Secchi disc visibility method is a widely used technique for determining the transparency of a water body. In this the disc is lowered in the water until it is just no longer visible from the surface. The depth of water so obtained is a measure of the water transparency. This technique, however, is not very accurate and makes no distinction between light of different wavelengths but its very simplicity means it can be performed by nonexperts (e.g., fishermen) who use the lake regularly so large amounts of data can be obtained over a long period of time. Photoelectric cells have also been used to determine the transparency of lake water and can be fitted with filters so that the transmission of light of different wavelengths can be compared. It has been shown that blue-green light penetrates much deeper into clear water than light of other wavelengths. Respiratory processes predominate at depths below those for which insufficient light is available for photosynthesis to occur, and these can lead to the development of an anaerobic layer, within the lake.

Temperature and Stratification The uppermost layer of water in a lake absorbs long wavelength radiation as heat energy and this is transferred by convection to the rest of the water. The higher the temperature of the water the greater is the productivity of the lake. The solubility of oxygen in water however decreases with increasing temperature so increasing the risk of oxygen depletion. The water temperature is influenced by the

148


geographical position of the lake, the time of year and by the discharge of heated effluents into the lake. The temperature also influences the stratification of lakes. The density of water is temperature-dependent and is at a maximum at 4OC. Water that is either warmer or colder will therefore float on top of the water at this temperature leading to lake stratification. As seen in Fig. 2.16, often two layers of water of different temperature exist; a lower layer of cool water near its maximum density called the hypolimnion and an upper layer of water subject to seasonal variations in temperature called the epilimnion. The two are separated by a layer of water where the temperature rapidly changes and which is referred to as the thermocline. In temperate climates the water in lakes usually circulates during spring and autumn but in summer and winter the water stagnates. In winter the epilimnion is colder than the hypoliminion but in summer the converse is true. The summer stagnation period, which may last several months, is important because there is then no oxygen input to the hypolimnion which may become anaerobic.

\\

Summer / Winter

-

Wind

L Wind

/

/ Thermocline

Hypollmnion

."' -

Spring / Autumn Wind

Figure 2.16. Stratificationof lakes in winter and summer.

The mixing of the water in the lake is dependent on the temperature gradients, the wind velocity and direction and the surface area and depth of the lake (Fig. 2.16). It is often assumed for modelling purposes that the epilimnion is completely mixed due to the action of the wind.


149

Oxygen The dissolved oxygen concentration in a lake is a balance between the inputs via surface aeration and photosynthesis and the losses due to biological and chemical oxidation. During summer stagnation in eutrophic lakes, the hypolimnion can become anaerobic due to the absence of mixing from the epilimnion and the uptake of oxygen by respiring organisms. The oxygen production by photosynthesis is minimal owing to the low light intensity at this depth. The anaerobic degradation of organic compounds in the hypolimnion leads also to the production of hydrogen sulphide. In autumn the temperature of the epilimnion decreases, and it becomes denser than the hypolimnion, causing a circulation of the water in the lake. The oxidation of the hydrogen sulphide in the mixed water leads to a reduction in the dissolved oxygen content, and this can lead to the death of the fish in the lake. In winter if the lake freezes the surface aeration to the lake is reduced considerably. The short days and low light intensity mean that photosynthesis is at a minimum, and the oxygen content is therefore low. If snow covers the frozen lake then photosynthesis is reduced to practically zero, owing to the absence of light, and the oxygen level can fall even further and so result in the death of fish.

Phosphorus Phosphorus is an essential plant nutrient but, as it is not very soluble in water, it is usually found in relatively small concentrations in natural waters and often constitutes the growth limiting factor for many species (see ALGAE, EUTROPH and PCYCLE). Recently, human activity has accounted for a huge phosphorus discharge into surface waters. Plants utilise phosphorus mainly in the form of orthophosphate. The highest phosphorus levels are found at the end of winter when algal growth is at a minimum owing to the low temperatures, short days and low light intensity. In spring, algal activity increases and the phosphate is utilised until it is present only in trace amounts within the epilimnion layer. The algae containing phosphate fall slowly into the hypolimnion and eventually reach the sediment. The importance of phosphorus in lake management is evident from the fact that 1 g of phosphorus gives rise to at least 100 g of algal biomass which in turn requires about 150 g of oxygen to be decomposed aerobically. Detergents have been pinpointed as a major contributor (one third to one half) to the phosphorus in wastewaters entering natural waters. Detergents used to contain up to 65% by weight of complex phosphates, especially sodium tripolyphosphate. These phosphates were needed to chelate ionic Ca, Mg, Fe, and Mn in order to prevent them complexing with the chemical surfactants responsible for the cleaning process. In the USA the average per capita discharge to water was about 1.9 kg of phosphorus per year between 1965 and 1970. It has been estimated that in the case of Lake Erie over 87% of the phosphorus input to the lake has been caused by human activity.

150


Nitrogen After phosphorus, nitrogen is generally regarded to be the second most important nutrient and can sometimes be the rate limiting factor. Nitrogen exists in natural waters as dissolved nitrogen, ammonium ions, nitrates, nitrites and as combined nitrogen in organic compounds (see NCYCLE). As seen in Fig. 2.17, atmospheric nitrogen can be fixed by microbes in the water and in the sediments, and nitrogen can be released to the atmosphere by the action of denitrifying bacteria. The increased use of artificial fertilisers has lead to an increase in the quantity of nitrate entering natural waters. Nitrates have high solubility and low sorption characteristics so that they can be transported easily in the groundwater far from their site of application.

N-fixation Organic nitrogen

4

Atmospheric nitrogen

Denitrification

industrial N-fixation

Nitrite N02-

Nitrification

Figure 2.17. The various processes of the nitrogen cycle.

As phosphate is usually the rate-limiting nutrient, not all the nitrate can be utilised by the algae and thus nitrate contents in lakes have been increasing. High nitrate levels in drinking water have been implicated in the cause of infantile methaemoglobinaemia(blue baby syndrome) and nitrosamine induced cancers.

Carbon Carbon is present in both organic, inorganic, dissolved and suspended forms in natural waters. The largest quantity of carbon is present as carbonate, bicarbonate and carbon dioxide in relative quantites represented by the equilibrium situation:

151

2.5 Water Pollution Modelling Ca(HC03)z w CaC03 J,

+ H20 + C02

The presence of the lime-carbonate system buffers the pH of the lake to a value of around pH 7. The photosynthetic uptake of carbon dioxide by algae leads to calcium carbonate precipitation which can coat plants around the lake shoreline. The increase in colloidal and particulate calcium carbonate in the water reduces the transparency of the lake and hence reduces algal growth. The calcium carbonate can adsorb and co-precipitate phosphates and organic material and so reduce the nutrient supply for the algae. In lakes with severe weed growth and algal blooms the bicarbonate can be completely consumed. This leads to the hydrolisation of the calcium carbonate to form calcium hydrogen carbonate and calcium hydroxide, which can cause the pH to rise to as high as 11. This kills all the fish in the lake and many of the plants and renders the lake unsuitable for recreational purposes. Organic carbon is present in natural waters as detritus, biomass and dissolved forms. Organic carbon can enter the lake in effluent discharges and via dead leaves and other plant material etc. The carbon is utilised by both phyto- and zooplankton and by heterotrophic bacteria. Particulate organic carbon can sediment out and accumulate in the sediment where it can undergo methanogenesis (releasing methane), denitrification (releasing nitrogen), desulphurisation (releasing sulphur) and anaerobic degradation (releasing hydrogen). Methane released rises through the water and can be used by methane-utilising bacteria in the aerobic region of the lake.

2.5.1.2

Consequences of Eutrophication

Eutrophication has many consequences for the ecology of a lake or reservoir.

Ecological The follow ecological consequences have been identified: I

The biodegradation of the algal biomass utilises large amounts of dissolved oxygen and can lead to oxygen depletion and fish death.

I1

Sludge build-up and silting up of the body of water by decayed plant matter.

111

Useful organisms in the food chain are superseded by non-utilisable forms, in particular blue-green algae.

IV

The development of anaerobic zones and the subsequent production of hydrogen sulphide can lead to the death of the fish population.

152


V

Increased bacterial populations.

VI

Death of reeds along the shore of the lake. This is due to the stems becoming colonised by algae and breaking.

VII

Disappearance of nesting wildfowl species owing to the reduction in the fish population and reed habitat.

Lakes and reservoirs are used as sources of drinking water and for industrial and recreational use. The eutrophication of a lake has important consequences for all these activities.

Drinking Water Sources The development of algal blooms is undesirable if the lake is to be used as a source of drinking water for several reasons. The consequences for drinking water are as follows:

I

The presence of dissolved organic compounds can give rise to a pronounced odour and taste.

I1

The formation of toxic algal strains.

I11

The liberation of coloured organic material and vegetable pigments can discolour the drinking water.

IV

The formation of large quantities of particulate organic matter in the form of biomass places an extra demand on the treatment process.

V

Changes in the dominant algal species which water treatment plants have not been designed to cope with.

VI

Presence of chelating and complex-forming substances.

Ind ustria1 The industrial consequences are as follows: I

The reduction in dissolved oxygen and possible fish death renders eutrophic lakes unsuitable for commercial fishing.

I1

The taste and odour imparted to eutrophic water by the presence of algal blooms means the water can not be used in the production of beverages or alcoholic drinks.


153

I11

The massive growth of weeds can impede the flow of water to drainage and irrigation channels and so is detrimental to local agriculture.

IV

The growth of weeds and algal blooms can be an obstruction to navigation.

V

The presence of algae in the water can cause problems if the water is used for hydroelectric power generation or for cooling purposes.

VI

The high carbon dioxide levels can be corrosive to underwater pipelines.

Recreational The recreational consequences are as follows:

I

The presence of algal blooms makes a lake unsuitable for bathing.

I1

The release of toxins by the algae can lead to allergic responses in some people.

I11

Recreational fishing is not possible in a severely eutrophic lake.

IV

The change in colour of the lake and the release of unpleasant odours make the lake less attractive for tourists.

2.5.1.3

Eutrophication Models

Eutrophication models can be considered in terms of three differing categories: (a) those that describe the nutrient loading from the watershed, (b) those that describe the chemical and biochemical processes in the lake, and (c) models for lake management and the control of eutrophication. The three types of model are usually developed independently but may be used together to give an overall picture of the major processes affecting water quality and to allow an optimisation with the resources available (Somlyody and van Straten,1986).

a ) Watershed Models The aim of watershed models is to predict the nutrient loading reaching a lake or a reservoir. The models either predict the average annual nutrient loading or seasonal variations in the loading. They can also distinguish between nutrients that are biologically available or those that are biologically non-available. The source of the nutrients can be classified as a point source or as a non-point source. A typical point source could be, for example, the effluent from a wastewater treatment plant. In this case it is relatively easy to obtain data on the volume and

154


concentration of the important nutrients in the wastewater. Non-point sources can be considered as either urban or rural based sources. Rural non-point sources include the run-off and base flow from agricultural and forested areas. Runoff is the rapid drainage of an area of land after rainfall or snow melt. Base flow is the flow of water in the saturated layer of the soil which occurs regardless of weather conditions. The concentration of nutrients in rural non-point sources depends on the character of the soil, the overlying vegetation and fertiliser addition. Urban non-point sources due to run off are dependent on the extent of impervious surface in the drainage area, the degree of street cleaning and the time elapsed since the last storm. Accurate watershed modelling therefore requires a knowledge of the area of the watershed and how the land is used, details of the local weather conditions, and the siting and capacity of any wastewater treatment plants, etc.

b) Waterbody Models The output from a watershed model can often provide an input to a waterbody model. Waterbody models differ enormously in their degree of complexity, but several distinct sections can be identified.

Hydrology The input and output flows to and from the lake need to be identified and quantified. The volume of the lake and its dimensions also need to be determined. If the lake is small and shallow, it may be considered to be well-mixed and treated as a continuous stirred tank reactor. If the lake is long and thin, a plug-flow model may be more appropriate. Sometimes a lake is divided naturally into two or more basins. Each individual basin can then be considered to be well-mixed and the whole lake treated as several tanks-in-series. The degree of stratification, if any, also needs to be considered. If the lake is stratified then the epilimnion and hypolimnion layers are usually modelled as two separate systems with some exchange of material between them. In some cases, a model required to simulate the eutrophication of a lake during the course of a whole year would require information concerning both spring and autumn turnover events.

Ecology The aim of most eutrophication models is to predict the time and the extent to which algal blooms occur. To achieve this the kinetics of algal growth need to be known. In many cases (see ALGAE and EUTROPH), Monod or modified Monod kinetic relationships are assumed with phosphorus as limiting substrate, although other kinetic expressions have also been used, e.g., Langmuir-Hinshelwood. In some models several species of algae are included whereas in others all algae are lumped together as one group. Predation by zooplankton and higher trophic levels may also be important and may need to be included in the model.


155

Chemistry The chemical components can be described in detail. For example the model may include individual levels of orthophosphate, condensed phosphate, particulate organic phosphate and dissolved organic phosphate and their interactions. Alternatively, the chemistry can be simplified, e.g., accounting only for biologically available phosphate and phytoplankton phosphate. The influence of other compounds, e.g., nitrates, pesticides, carbonates, etc. has also been included in models. Sediment The exchange of nutrients between the waterbody and the sediment is a very complex process and is often neglected or oversimplified in models of eutrophication. Adsorption and chemosorption of nutrients on calcium and iron compounds in the sediment are both thought to be important processes that occur. These processes are both pH and oxygen concentration dependent but are usually modelled empirically. Climate A model of lake eutrophication requires data on the intensity and duration of the sunlight, the rainfall, temperature and possibly snowfall. Usually daily average values suffice, but in some cases shorter time scales are used if night and day need to be differentiated. Wind direction and intensity data may be required if the degree of wind-driven mixing needs to be determined.

c) Management Models Management models are used to identify the optimal eutrophication control strategy, where this may be defined on the basis of cost, the time needed to achieve the required water quality or a combination of both factors. Differing control strategies can be investigated and compared and the results used to formulate a water quality control policy.

2.5.1.4

Prevention and Reversal of Eutrophication

Eutrophication can be prevented by reducing the input of nutrients, and especially that of phosphorus into fresh water bodies. This has been achieved in several ways (Klapper, 1991; Ryding and Rast, 1989):

156


Phosphate Reduction In the long term the best method for preventing eutrophication or reversing eutrophication (or oligotrophication) is to reduce the nutrient load to the lake. In recent years public awareness and governmental legislation have lead to a decrease in the phosphate loading of many lakes. Detergents are now available that contain substitute chelating agents and little or no phosphorus and indeed in many parts of the world there are regulations concerning the maximum phosphate concentration permissible. This has lead to a large reduction in the typical phosphate concentration in domestic wastewaters in many countries. The controlled use of fertilisers has also been successful in reducing the nutrient input to natural waters.

Wastewater Treatment Phosphates in wastewaters can be reduced by improved wastewater treatment. Phosphates can be removed by flocculation or by the precipitation of phosphate compounds by aluminium, calcium, iron or by biological uptake by algae or macrophytes.

Desludging In severe cases of eutrophication where there are problems with the lake silting up, desludging may be employed. This is a very expensive alternative, but the cost may be reduced by selling the sludge as a soil conditioner or fertiliser. Destratification The stratification of lakes can be prevented by artificially induced circulation of the liquid. This helps to prevent by the formation of anaerobic regions at the bottom of the lake and thus the associated problems. Destratification can be achieved by the introduction of compressed air at the base of the lake so inducing circulation of the water. An alternative technique is to pump warm oxygen-rich water to the bottom of the lake in order to induce the formation of density-driven convection currents. Surface Aeration The surface aeration of lakes can be used to increase the dissolved oxygen content and hence to prevent the formation of anaerobic zones. Compressed air systems and venturi aerators have both been used, for fisheries applications in particular. Fountains are an attractive but expensive way of reaerating lakes. Centrifugal aerators have also been widely employed.


157

Deep-Level Aeration The hypolimnion is where the problems of lack of oxygen are likely to occur so it seems sensible to aerate this zone and not the whole lake. Another advantage is that in summer the hypolimnion is at a cooler temperature that the epilimnion and so the saturation dissolved oxygen concentration is higher. Deep-level aeration has been employed in reservoirs used for drinking water. Chemical Addition Sodium nitrate has been added to natural waters to combat the hydrogen sulphide produced by anaerobic bacteria. As the hydrogen sulphide is oxidised, the nitrate is reduced and the nitrogen escapes in gaseous form. This form of treatment is expensive but is often used after accidental spills of biodegradable substances. If the size of the spill is known it is possible to add exactly the right quantity of nitrate to oxidise the hydrogen sulphide. It is desirable to maintain the pH of lakes at or near neutrality, especially if they are to be used for recreational purposes. Aluminium sulphate has been added to lakes to reduce the pH and lime and soda to increase the pH.

2.5.2

Discharge of Pollutants into Rivers and Streams

Wastewaters are often discharged continuously into rivers and streams (see DISCHARG, RIVER and STREAM). The organic compounds in the water are utilised by aquatic micro-organisms and consequently reduce the dissolved oxygen concentration in the river. Simultaneously, oxygen is transferred from the atmosphere to the river and this replenishes the dissolved oxygen at a rate which is proportional to the oxygen deficit. At a certain distance downstream from the discharge point the reaeration rate equals the rate of microbial oxygen uptake and the dissolved oxygen level reaches a minimum. Further downstream the dissolved oxygen increases until it eventually reaches the saturation value again. Streeter and Phelps (1925) modelled this process using a simple differential equation that is the basis of most river models used today. The graph of dissolved oxygen concentration against distance from point of discharge is known as the sag curve, an example of which is shown in Fig. 2.18 (see also example OXSAG in Sec 3.1.2 and DISCHARG in Sec. 3.14.3). If the dissolved oxygen concentration falls too low then the plant and animal life in the river will die. The Streeter Phelps equation is useful in determining the minimum dissolved oxygen concentration caused by the discharge of an organic-containing wastewater into a river and also the distance downstream at which the minimum occurs.

158


I

Distance downstream Figure 2.18. Dissolved oxygen profile of a river downstream of a discharge.

2.5.2.1

Modifications to the Streeter-Phelps Theory

The Streeter-Phelps theory is based on the assumption that only two processes contribute to the dissolved oxygen concentration of a stream, namely the biological oxidation of organic matter and surface reaeration. In fact, as shown in Fig. 2.19 and as listed below, many other factors are also thought to be important (Krenkel and Novotny, 1979). These include the following: The removal of BOD by sedimentation or absorption. - The addition of BOD by diffusion from the benthal layer, also by scouring of the bottom of the stream bed. - The addition of BOD along the stream by run-off from local fields. - The addition of oxygen by the photosynthetic action of plankton and weeds. - The removal of oxygen by the respiration of plankton and plants. - Nitrification and denitrification. - The removal of oxygen by the purging action of gases rising from the benthal layer. - The presence of waterfalls, weirs, fountains, rapids, etc. - The continuous redistribution of both BOD and dissolved oxygen by longitudinal dispersion. - The oxygen demand of biological slimes etc. - Temperature effects. - Tributaries. -

159


Rainfall

1

1

Runoff

i

Surface aeration

Figure 2.19. Some of the processes important in determining the oxygen levels in rivers.

- Multiple point sources of pollution. - Variations in the flow rate, depth and velocity of the stream. The basic Streeter-Phelps equation can be modified to include expressions for each of these factors. In the following sections, some of the factors are examined in more detail, but to allow for only a few of the more important factors, model equations for the variations in BOD concentration and nitrate concentration obviously need to be included. If temperature effects are also to be included, then an additional energy balance relationship is required.

2.5.2.2

Surface Reaeration

Mass transfer theory predicts that the rate of oxygen transfer from the atmosphere to the river will be proportional to the difference between the saturation value of the dissolved oxygen concentration and the actual dissolved oxygen concentration in the river, according to

The constant of proportionality, K,, is known as the atmospheric reaeration coefficient and is proportional to the water surface area to volume ratio and to the

160


turbulent intensity of the flow at the water surface. Empirical relations are available for its dependency on river velocity, depth and temperature, as used in the examples RIVER, STREAM and DISCHARGE. A relation for the dissolved oxygen concentration dependency on temperature is also available in these examples. Other relationships for reaeration coefficients that take into account the mean vertical eddy diffusivity, ED, (m2 s-l) (Krenkel and Novotny, 1979), and the channel slope have been used (O'Connor and Dobbins, 1958 and Foree, 1976).

2.5.2.3

River Parameters

The flow rate of a river varies during the course of the year depending on the climatic conditions. This variation can be modelled by mean of an empirical polynomial equation if sufficient experimental data exist. Alternatively, models may be based on the minimum annual flow in order to simulate the worst possible scenario. The effect of temperature can also be modelled in a similar way, with the highest temperature being used. The velocity, VR, of the river can be determined by injecting fluorescent dyes into the river and detecting the dyes at a known time and distance downstream. In addition the depth, DR, and width, W, of the river need to be measured at several different volumetric flow rates, Q. Correlations relating the depth, width and velocity of the river as a function of flow rate can be derived from experimental data. These correlations often take the following form: DR = a Q b

W = eQf where a, b, c, d, e and fare constants. The value of b is generally between 0 and 0.6 and that of d between 0.4 to 1.O. For a rectangular cross-section river channel

which serves as a useful approximation if no other data is available. Simulation examples are RIVER, STREAM and DISCHARG.

161


2.5.2.4

Photosynthesis and Respiration of Green Plants and Algae

Green plants and algae obtain energy from sunlight by photosynthesis and release oxygen. They also, regardless of the light conditions, respire and take up oxygen and release carbon dioxide. In daylight the rate of photosynthesis is usually about four times greater than the rate of respiration resulting in a net oxygen production. At night there is no photosynthesis so there is a net uptake of oxygen. Plant density is an important factor in determining the oxygen balance. Initially oxygen production increases with increasing plant density but then reaches a maximum value from which it then decreases. This is probably due to self shading of the plants reducing the photosynthetic activity and can lead to extremely low values of dissolved oxygen at night-time conditions. The rate of oxygen production by photosynthesis is independent of the dissolved oxygen concentration and can thus lead to oxygen supersaturation. On bright, sunny days photosynthetic oxygen production rates of up to 50 g 0 2 m-2 day-' have been measured. In general the difference between the photosynthetic production rate, Rphot,, and respiratory demand, Rresp, is usually modelled by sine curve functions, i.e.:

where C1 and C2 are constants and t is the time in hours.

2.5.3

Groundwater Pollution

The rocks of the Earth's surface contain many cavities, which vary from large caverns to minute pores, many of which are filled with water. The contained water is known as ground water and it has been estimated that there is 50 million cubic kilometres of groundwater in the Earth of which 4 million cubic kilometres are fresh water (Table 2.13). Groundwater is a very good source of drinking water owing to the purification properties of the soil and is the only source of water for many people. Groundwater is also used for irrigation, spraying and industrial use.

Table 2.13. Distribution of water on the Earth (New Scientist, 16th Feb 1991). Water source

Estimated volume ( lo6 * km3)

Sea Groundwater Ice Rivers and Lakes

1370 50 30 0.2

162


Layers of rock that are sufficiently porous and permeable to store water and to allow water to flow through them easily are called aquifers. Far below the Earth's surface the rocks are so compressed that the rocks are non porous and water is unable to flow through. The depth of this barrier to water flow is usually about 1 km but can be as much as 10 km. Above this barrier is a layer of rock that is saturated with water. The top of this layer is known as the water table. Groundwater is the water contained in the rocks beneath the water table. Above the water table is the unsaturated or vadose layer. The depth of the water table varies throughout the year depending on the amount of rainfall. In some places the aquifer lies underneath a layer of impermeable rock (a confined aquifer) and in other places is covered by permeable rocks or soil (an unconfined aquifer). Water enters the aquifer in the unconfined part as shown in Fig. 2.20. Groundwater is under hydrostatic pressure that is greater than atmospheric pressure and can therefore flow upwards as well as in the downwards direction. If a bore hole is drilled into a confined aquifer, water will rise owing to the hydrostatic pressure of the aquifier. The level to which the water rises is called the potentiometric surface. It is possible for this surface to be above ground level, in which case water will flow out of a bore hole drilled into the aquifer. Such bore holes are called artesian wells. For unconfined aquifers the water table and the potentiometric surface coincide.

Figure 2.20. Groundwater, water table and aquifers. (New Scientist, 16th Feb 1991).

163


2.5.3.1

Sources of Groundwater Pollution

Several different causes of groundwater pollution have been identified. These include: 1) industrial pollution arising possibly from industrial accidents (spills and leaks), rain infiltrating through waste disposal sites, disposal of chemically, biologically or radioactively contaminated wastewaters; 2) domestic pollution due to rain infiltration through sanitary landfills, accidents (e.g., septic tanks);

3) agricultural pollution, owing to possible leaching of fertilisers, herbicides, pesticides etc.; 4) environmental pollution, for example infiltration of seawater. Although groundwater is more protected than surface water, it is still subject to pollution and once this occurs, restoration is usually difficult, lengthy and costly.

A Little Hydrology It is important to be able to model and predict the movement of groundwater in order that the effects of accidents can be assessed and the pollution of the groundwater minimised. When a borehole is drilled into an aquifer, water will rise to a certain height. This height is referred to as the hydraulic potential or head. If the hydraulic potential at some other point in the same aquifer is different then water will flow from the region of high potential to that of low potential. The Bernoulli energy equation can be applied to two separate points in the aquifer: Pi

+ p g h i + 0.5 p v i 2

= P2 + p g h2 + 0.5 p vz2 + AH

where AH represents the energy losses due to heat dissipation in the flow. Use of the above relationship thus enables the velocity of flow of the water in an aquifer to be estimated.

Hydraulic Gradient The hydraulic gradient between point 1 and point 2 is defined as the difference in hydraulic head between the two points divided by the distance between:

.

1=-

h2-hl L

164


The hydraulic gradient is taken in the direction of flow, and is therefore negative by definition since water always flows from a region of high potential to one of low potential.

Velocities The apparent or macroscopic velocity (VM)is the velocity of the bulk flow through a porous medium. The real or microscopic velocity (Vm) is the velocity of the water as it follows a tortuous path through the interconnecting pores of the rock. It follows that the real velocity is always greater than the apparent velocity because the actual path length is always greater, as shown in Fig. 2.21.

Figure 2.21. Comparison of apparent and actual paths through a porous medium.

Darcy's Law Darcy's Law states that the flow velocity of the groundwater is directly proportional to the hydraulic gradient:

The negative sign indicates that flow is always in the direction of decreasing hydraulic gradient. KM is a constant dependent on the porous medium. Extending the flow equation to three dimensions gives:

VM = - K ~ g r a d h Such problems however are the proper subject of a more specialised text.

165


2.5.3.2

Modelling Groundwater Pollution

The following processes are important in modelling the transport of pollutants in groundwater:

Adsorption Pollutants can be adsorbed onto the surface of particles in the aquifer. The attraction to the solid surface may be caused by electrical attraction, van der Waal's forces (intermolecular attraction) and chemisorption (chemical interaction between the solid and the adsorbed substance). The main factors affecting adsorption are the physical and chemical characteristics of the soil and the nature of the pollutant. For instance the surface of clay particles are negatively charged and will attract positively charged ammonium, potassium and calcium ions but not negatively charged nitrate (NO3-) ions which will be free to move. In contrast, tropical soils are often positively charged, so the opposite case then applies. Adsorption will also be influenced by the temperature and pH of the liquid phase. Various forms of adsorption isotherms have been used to model the adsorption characteristics of pollutants. Two basic classes of isotherm have been used; i.e., equilibrium and non-equilibrium isotherm relationships. Equilibrium isotherms are based on the assumption that any change in concentration of either the adsorbed or free component produces an instantaneous change in the concentration of the other component. Non-equilibrium isotherms assume that the equilibrium is approached at a rate depending upon concentration. The following are examples of isotherm relationships that have been used to relate the adsorbed concentration, Ca, to the concentration in solution, C. a) Freundlich isotherm C, = b C m where b and m are constants. b) Linear equilibrium isotherm Ca = KdC This is based on the Freundlich isotherm with the special condition that the value of m is equal to 1. The constant K d is known as the distribution coefficient or the partitioning coefficient. This relationship assumes the adsorption is linear, instantaneous and reversible. It also assumes that the solid phase never becomes saturated, and is therefore valid only at low concentration. c) Langmuir isotherm aC Ca = l + b C ~

166


where a and b are constants. Compare this non-linear equilibrium isotherm with the forms of relationship describing Michaelis-Menten enzyme kinetics and Monod microbial culture kinetics. d) Simple non-equilibrium, irreversible isotherm

e) Non-equilibrium Freundlich isotherm dCa dt

-=

a ( b C m - F)

f) Exponential equilibrium isotherm

The choice of an appropriate isotherm should be based on experimental studies based on the particular soil under consideration.

Degradation and Decay If the pollutant is microbial, the concentration may be reduced by natural death. This is often modelled as a first order process, where:

where kd is a constant, determined experimentally, with units of time-l. If the pollutant is biodegradable it will decrease in concentration owing to the action of the natural soil micro-organisms. The rate of degradation is usually expressed as a Monod-type relationship (see SOIL): dC = -c, x P dt pmaxKp’+Cp If the pollutant is radioactive, the radioactive decay relationship applies:

where Tl,2 is the half-life of the radioactive species.

167


Aquifer The aquifer needs to be characterised. This encompasses the need to describe 1 ) the geometry of the boundaries; 2) the material comprising the aquifer, including such factors as the aquifier isotropy, homogeneity, tortuosity, permeability, mean pore size and pore size distribution; 3) the mode of flow; 4) the flow regime (laminar/turbulent); 5) the presence of any water sources and water sinks.

Transport of Pollutants Pollutants are transported in groundwater primarily by the processes of advection, diffusion and dispersion. A simplified one-dimensional modelling approach is shown here, but this can be easily extended to the modelling of two or three dimensional flow problems. Consider a small volume of groundwater that is contaminated with a high concentration of pollutant. It is assumed that the density difference between the polluted water and the unpolluted water is very small so that density-driven flows are negligible. The advective flux (ML-2T-1) is the flux of polluted water carried by the ground water at the latter's average velocity as determined by Darcy's Law. Hence: CVM = -cKMi According to Darcy's Law the volume and concentration of the polluted water will be unchanged, but experimental evidence shows this not to be the case. In addition to advective transport, diffusion and dispersion processes cause a spreading of the polluted water and mixing with unpolluted water. Molecular diffusion is the random movement of molecules in a fluid, and this results in the transfer of solute molecules from high to low concentration regions. Molecular diffusion is a time-dependent process and therefore becomes more important the lower the groundwater velocity. Molecular diffusion is usually modelled by Fick's Law (see Sec. 1.8 and 1.12): J =

-D- dC dx

where D is the coefficient of diffusion and x the linear displacement. In a porous medium the coefficient of diffusion needs to be modified to take into account the tortuosity, z, of the medium: D* = D T

168


where D is the diffusion coefficient in a continuous fluid and D* in a porous medium. The tortuosity is a function of the ratio of water to solid in the medium. Dispersion is a result of the polluted water flowing through pores. If it is assumed that the velocity at the pore surface is zero, then there will be a velocity profile across the pore. The maximum velocity is at the centre of the pore and this is dependent on the pore diameter. There is a variety of pore shapes and sizes available to the liquid flow, so causing the polluted water to spread, both in the direction of flow and (to a lesser extent) perpendicular to the main flow. This is usually modelled by a Fick-type relationship: dC dx

Js = - D -

where D, is the coefficient of dispersion. The dispersion coefficient has been shown to be a function of the Peclet number, the pore size, the medium permeability and tortuosity. The total flux of pollutant due to advection, diffusion and dispersion can therefore be modelled by: dC dC Q = CVM-D*-dx - D S dx The diffusion and dispersion terms are sometimes combined to give a coefficient of hydrodynamic dispersion, Dhyd: hence

In practice, two or three-dimensional models are used and the techniques of finite element and finite difference analysis can be applied to the problem as shown in Sec. 1.12.2. If the polluted water has a density significantly different from that of the unpolluted water, as is the case with the intrusion of sea water into an aquifer, then density-driven flows also need to be included in the model.

2.5.3.3

Nitrates in Groundwater

Nitrates from fertilisers have found their way into groundwater and subsequently into drinking water. In 1975, the European Union set a legal limit of 50 mg nitrate per litre of drinking water (Dudley, 1990). Plants require nitrogen to produce proteins and their growth is stunted if there is insufficient nitrogen in the soil. Plants are unable to utilise nitrogen from the air,but can assimilate ammonium or nitrate ions from the soil. Farmers increase the nitrogen content of the soil by


169

adding manure, artificial fertiliser or by periodically growing nitrogen-fixing crops such as clover (crop-rotation). Nitrates are water soluble so that when it rains they are transported with the water through the vadose layer to the groundwater, by water leaching. The rate of leaching is dependent on the rainfall and the soil type. Sandy soils are much more prone to leaching than clay soils, owing to their higher porosity. Leaching is also influenced by the crops that are grown in the field. For example, potatoes have short roots and relatively low nitrogen requirements and therefore require little fertiliser. Wheat, however, has long well-developed roots that can take up large amounts of nitrogen. The time of fertiliser addition is also important. Plants take up nitrogen when they are growing rapidly, i.e.,in spring and summer, but take up very little nitrogen when the temperatures are low. However, if the soil is frozen, leaching is prevented. A heavy downpour of rain can wash away large quantities of nitrates, so it makes more sense to fertilise often with small amounts. If the soil becomes waterlogged, however, the soil becomes anaerobic and denitrifying bacteria can convert nitrates to nitrogen gas. Excess nitrate in drinking water has been associated with blue baby syndrome (infantile methaemoglobinaemia). Bacteria in the baby's gut convert nitrate to nitrite which is taken up by haemoglobin in the blood so causing a reduction in the transport of oxygen and causing respiratory problems for the baby. Nitrates have also been implicated in cancers of the stomach and windpipe although the data are conflicting.

2.6

Solid Waste Treatment and Disposal

2.6.1

Sources of Solid Wastes

Solid waste consists of both municipal waste and waste sludge from wastewater treatment processes. These are usually treated separately but in some cases they are mixed to aid processing.

2.6.1.1

Municipal Solid Waste

Modern society generates large volumes of solid waste which must be disposed of cheaply and safely. For example, the United Kingdom produces 30 million tonnes of solid refuse per year, over 90% of which is disposed of in over 5000 landfill sites. As environmental controls become more stringent and land becomes scarcer, this means of solid waste disposal is becoming more expensive. One possibility of making the process more attractive is to extract and utilise the gas that is formed

170


naturally. This can be done either in-situ, in which case the gas produced is known as landfill gas, or it can be done in specially designed digestors in which case the gas produced is called biogas. Biogas and landfill gas both consist of mixtures of methane and carbon dioxide, but biogas contains a higher methane content, thus making it a better fuel.

Table 2.14. Analysis of municipal solid waste in the UK (Dept. of Trade and Industry). Classification Paper Plastic film Dense plastic Textile Misc. combustible Misc. non-combustible Glass Ferrous metal Non-ferrous metal Putrescible Unsorted fines (c10 mm) Wood Garden waste

2.6.1.2

Waste fraction (wt%)

29.2 4.2 2.8 3.0 5.8 4.0 8.4 8.0 1 .o 19.0 8.6 2.2 3.8

Waste Sludge

The sludge both generated by wastewater treatment processes needs to be disposed of both safely and cheaply. A city of one million people produces approximately 3000 m3 of sludge per annum. In the UK 35 million tonnes of wet sludge need to be disposed of every year and it was estimated that in 1982 40%of the operational costs of wastewater treatment could be attributed to sludge treatment. Wastewater treatment plants generate a variety of different solid wastes which varying solids concentration, as shown in Table 2.15.

171

2.6 Solid Waste Treatment and Disposal

Table 2.15. Sources and concentrations of waste sludge. Typical solids concentration (%)

Treatment process

5.O 0.8 1.5 5.O 4.0 7.0 1.2

Primary settling tank Secondary settling tank Trickling filter humus sludge Scum Waste activated sludge Anaerobic digestor Aerobic digestor

2.6.2

Sludge Processing and Disposal

2.6.2.1

Summary of Disposal Methods

Many different methods of sludge disposal have been employed. The following Table 2.16.summarises the methods used in various EU countries.

Table 2.16. Sludge disposal methods in the European Union. Current sludge production Sludge treatment ( 103 tonnes /year) (%I Country ~

ion

sation

Disinfection

(%)

@)

80 95 60 75

1 29 3 17

40 75 most 65 60

NA 1 NA

~~

Belgium Denmark France Germany Greece Ireland Italy* Lux .bourg Holland UK

*

70 130 840 2200 3 20 1200 11 230 1500

7 1

20% of sludge disposal unspecified.

Disposal routes

(%I Agricul- Landfills Incin- tural eration land

15 45 30 39 0 4 20 90 53 41

83 45 50 49 100 5 55 10

32 26

2 10

20 8 0 0 5 0 3 4

Dump- Pipeline ing at sea

0 0 0 0 0 45 0 0 0 27

0 0 0 0 0 45 0 0 13

2

172

2.6.2.2


Sludge Thickening and Dewatering

The first priority of waste sludge treatment is to reduce the volume of the sludge by reducing the water content. This provides the opportunity for significant reductions in sludge volumes to be obtained (see Table 2.17) and therefore in reducing the subsequent transport and storage costs.

Table 2.17. Reduction of sludge volume as it is dewatered (5 kg solids m-3). Moisture content (%)

Volume (m3)

99.5 99 97 90 70

1.o 0.5 0.167 0.050 0.0 167

Various methods have been used to reduce the water content of sludges. These involve: sedimentation and flotation, air drying beds, filter presses, belt filters, vacuum filters, centrifuges, sludge concentrators. The method employed in any given application will depend on the sludge characteristics, which in turn depend on the nature of the secondary treatment processes.

2.6.2.3

Use of Sludge on Agricultural Land

Both stabilised and raw sludge can be applied to agricultural land. In most developed contries, regulations exist concerning the frequency, timing and content for the addition of both stabilised and raw sludge to be applied to agricultural land. In addition the subsequent grazing and human consumption of crops is controlled with the metal content, pH and toxic elements content needing to be determined.

2.6.2.4

Dumping at Sea and Pipeline Discharges

One of the easiest and cheapest methods for disposing of waste sludge is either to dump it at sea or to discharge it into the sea or other large bodies of water via pipelines. This method of disposal is particular attractive to countries having a long shoreline relative to the land area, i.e., the United Kingdom and the Republic


173

of Ireland. This method of disposal is, however, limited by legislation and especially in relation to the use of nearby beaches for recreation.

2.6.3

Composting

Composting has been defined (Haug, 1980) as "the biological decomposition and stabilisation of organic substrates under conditions which allow development of thermophilic temperatures as a result of biologically produced heat, with a final product sufficiently stable for storage and application to land without adverse environmental effects". Composting is usually applied to solid or semi-solid materials and can be carried out under either aerobic or anaerobic conditions. The products of aerobic composting consist of carbon dioxide, water, ammonia and heat, whereas methane, carbon dioxide and volatile acids are produced by anaerobic composting. Anaerobic composting releases significantly less energy per unit weight of decomposed organics than aerobic composting. Problems of odour control may be encountered with anaerobic composting, owing to the nature of the intermediate metabolites. For this reason, almost all engineered compost systems are aerobic, but mass transfer limitations may produce anaerobic zones. A schematic of the composting process is shown in Fig. 2.22. Most composts are too low in nutrients to be used as fertilisers, but because the nutrients are bound organically, they are much less prone to leaching than soluble fertilisers and are therefore used often as a soil conditioner. At the high temperatures (60-70°C) generated by the microbial action the rate of degradation is 5 to 7 times greater than at the average ambient temperature (12OC). In addition, prolonged exposure to these high temperatures is an effective way of killing pathogenic organisms, weeds, seeds and diseased plant material. Composting leads to an approximately 50% reduction in weight and an even greater reduction in volume.

2.6.3.1

Composting Processes

Many different composting processes have been used over the years. Composting systems can be classified as reactor or non-reactor processes (see COMPOST). Reactor processes can be either vertical (tower) , horizontal or inclined flow. Examples of horizontal flow reactor compost systems include the rotating drum reactor and the agitated solids bed reactor. Non-reactor systems can be subdivided into static beds and agitated solids beds, such as the windrow system (see WINDROW). Aeration can be forced or may be caused only by natural ventilation. The process may be continuous, semi-continuous or batch and the bed may be continuously agitated, turned periodically or left undisturbed.

174


Figure 2.22. Schematic of the major processes in composting. (Taken from Biddlestone et al., 1989).

2.6.3.2

Compost Ecology

A bewildering variety of bacterial organisms can be isolated from composting material. Mesophilic bacteria dominate in the early stages of composting but thermophilic types take over as the temperature increases to 40-5OoC. If the compost is left undisturbed then filamentous colonies of Streptomyces and Actinomycetes are observed towards the end of the composting process, whereas in agitated systems no such colonies occur. Bacteria are thought to be responsible for about 90% of the microbial activity during composting and over two thousand different strains of bacteria have been isolated from compost sources. Poincelot

175


(1977) analysed the microbial population during aerobic composting and his results are summarised in Table 2.18.

Table 2.18. Number of organisms per wet gram of garden compost (Poincelot, 1977). Microbe Bacteria Mesophilic Thermophilic Actinomycetes Thermophilic Fungi Mesophilic Thermophilic

Initial temp . p and simulation allows the startup, steady state and washout phenomena to be investigated. Under conditions of washout (D > pm) it can be seen that the cell concentration X declines to zero and the effluent substrate concentration rises to attain the feed inlet concentration SF.

215

3.2 Basic Biological Reactor Examples

Figure 3.10. Chemostat with model variables.

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

dX z = -QX+rx

or dX

dt

= -DX+rx

For the substrate

dS V z = Q(SF-S)

+

rsV

dS dt = D ( S F - S )

+

rs

or -

where D is the dilution rate, and SF is the concentration of the limiting substrate in the feed. Note that when D=O the model becomes equivalent to the batch reactor case, BATREACT. The same simple Monod type kinetics as in BATREACT are assumed to apply.

216

3 Simulation Examples of Environmental Bioprocesses

Program :File CONTANK : CHEMOSTAT STARTUP AND OPERATION : Starts as batch with D=O.

CONSTANT UM=O.3,KS=O.l,Y=0.8 CONSTANT SF=lO,SO=lO,XO=l CONSTANT CINT=l,TFIN=40 CONSTANT D=O.O : DILUTION RATE(=F V 1 SIM; INTERACT; RESET; GOT0 1 INITIAL

x=xo s=so

DYNAMIC : Biomass balance equation XI=-D*X+RX S = D * ( S F - S )+ R S : Substrate balance equation RX=U*X : Kinetic equations U=UM*S/(KS+S) RS=-RX/Y IF (s.LT.O.O) S=O : Limits to zero IF (X.LT.O.0) X=o PLOT T,X, 0, TFIN, 0,10 OUTPUT T,X,S PREPARE T ,X, S ,U, RS ,RX

Nomenclature

r S V X Y

Dilution rate Saturation constant Specific growth rate coefficient Volumetric feed rate Reaction rate Substrate concentration Volume Biomass concentration Yield coefficient

Subscripts F m S X 0

Refers to feed Refers to maximum Refers to substrate Refers to biomass Refers to the initial reactor conditions

D

KS c1 (U>

Q

h-I kg m-3 h-I m3 h-I kg m-3h-1 kg m-3 m3 kg m-3 kg kg-1

217


Exercises 1. The program starts as a batch system with D initially zero. From this starting point, increase D interactively to obtain washout. Note that it may be necessary to increase the value of TFIN interactively using the VAL command. 2. Note the resultant steady-state values of X and S obtained for each value of D; calculate Y from these. 3. Change SF. Does this alter the value of S at steady state? Why?

4.Calculate the value of S at steady state for some given value of D. Verify by simulation. 5. Change the program to account for biomass in the feed.

6. Try to find the optimal switchover time from batch operation so as to reduce the time needed to reach steady-state operation. 7. Rapidly changing dynamic fermentations do not follow instantaneous Monod kinetics. Modify the model and the program to show a dynamic lag effect on F, such that p = ( P M o n o d - p)/Z. Compare the response to step changes in D for suitable values of the growth time lag constant 7.

8. Partial retention of biomass can be modelled simply by letting the outlet biomass concentration be some fraction of its concentration in the reactor. Alter the program to describe this situation and investigate the consequences.

Results m.m

Ism

- s 1

1,W

-

-2.88

R

A

I

-3.me:e

Figure 3.11. Startup for three dilution rates, D=O.O, 0.1 and 0.3 (Curves A, B and C).

Ism

RRS

m'.0

!

26. e

3.e

de

Figure 3.12. Variation of substrate uptake rate during the same runs.

218

3.2.3


SEMIBAT - Semi-Batch Reactor with Batch Startup

System In fed-batch bioreactors (Fig. 3.13), the continuous feeding of substrate in the absence of outflow from the reactor will cause an increase in volume (total accumulation of mass) in the reactor. Simulation methods can be used to study the characteristics of these reactors and to compare their performance to that of batch and continuous flow processes.

Figure 3.13. Fed-batch reactor with model variables.

Model For fed-batch operation, the component balance equations become: Total balance dV dt

- = Q

Cell balance

Substrate balance

~d(V 'I dt

- QSF

+ rsv

219


where Q is the volumetric feed rate, SF is the feed concentration and V is the volume of the fermenter contents at time t. The kinetics are the same as in BATREACT and CONTANK.

Program Note that the mass quantities, [VX] and [VS] are, at each time interval, first calculated and then divided by the total volume obtained, V, to obtain the concentrations X and S. The parameter STARTFED is the time at which continuous operation begins, and the parameter Qfeed is the volumetric flow rate. By changing the values of STARTFED and Qfeed, different operating strategies can be investigated using the VAL command. :

FILE

SEMIBAT

: :

SEMI-BATCH BIOREACTOR WITH BATCH Flow rate is initially zero.

START

UP

CONSTANT UM=0.3,KS=0.1fY=0.8 CONSTANT S F = 1 0 CONSTANT Qfeed=l CONSTANT STARTFED=20 CONSTANT C I N T = l ,T F I N = 5 0 1 SIM; INTERACT; RESET; GOT0 1 INITIAL Q=O x=o.o1;s=1o;v=1 vx=v*x vs=v*s DYNAMIC V'=Q : MASS BALANCES VX'=RX*V VS'=Q*SF+RS*V

x=vx/v s=vs/v RX=U*X : KINETIC EQUATIONS U=UM*S/(KS+S) R S = -RX / Y IF (S.lt.O.0) S=O : Limits IF (X.LT.O.0) X=O IF (T.EQ.STARTFED) Q=Qfeed PLOT T,S,O,TFIN,O,10 PREPARE T ,X S V V X VS OUTPUT T ,X ,S ,V

Nomenclature D KS

Dilution rate Saturation constant

hkg m-3

220


CL (U)

Flow rate Reaction rate Substrate concentration Reactor volume Biomass concentration Yield coefficient Specific growth rate coefficient

Subscripts F S X

Refers to feed Refers to substrate Refers to biomass

Q r S

v X Y

m3 h-1 kg m-3 h-I kg m-3 m3 kg m-3 kg kg- I h-1

Exercises 1. Operate the fed batch with high values of SF and low values of Q. Note that a quasi-steady state can be reached, where p = D and X remains constant at low levels of S , even though the effective value of D decreases, owing to the increasing volume of the tank contents. 2. Operate at high SF and note the slope of the X and VX versus time curves under linear growth conditions. Relate this rate of growth to the feed rate. 3. Vary Q interactively to maintain S well above Ks, and plot the growth rates and feed rates versus time. Compare the growth rates with those of Exercise 2.

4. Start the reactor under batch conditions, and then switch to fed batch operation. Try to minimise the startup time for maximum substrate uptake.

Results 16.9 A X

16.0

R S 12.e

-

Ism

12.9-

8.8 4.8

e. e

e

Figure 3.14. Feeding starts at T=20 (A) and (B) and T=30 (C). Values of QFEED: 1 (A), 2 (B) and for curve C: QFEED=2 with SF=15.

Figure 3.15. The variation in biomass concentration for the same runs.


3.2.4

22 1

UPTAKE - Substrate Uptake with Monod Kinetics

System For some environmental processes, the rate of growth of the micro-organisms is very slow compared to the rate of substrate utilisation. Often this occurs because the level of nutrients may be very low and the temperature and pH may be far from the optimum. In other processes, the biomass may be maintained rather constant by sedimentation and recycle or by some means of cell retention, e.g., in a biofilm reactor. In these cases, a simple model of substrate uptake can be applied. In this model, Monod kinetics are assumed for the uptake of a substrate, but the biomass is assumed to be constant and there is no flow into or out of the system.

Model The model consists of a balance for the single substrate. For substrate

where the kinetics are simply

Program :FILE UPTAKE : SUBSTRATE UPTAKE WITH MONOD KINETICS : NO GROWTH CONSTANT VM=0.3,KS=1 CONSTANT CINT=O.S,TFIN=60 1 sim; interact; reset; gotol INITIAL s=10 DYNAMIC S ' = RS : SUBSTRATE BALANCE IF (S.LT.O.O)S=O RS = - VM*S/(KS+S) : MONOD RATE EQUATION PLOT T,S,0,60,0,10 OUTPUT T,S PREPARE T, S, RS

222


Nomenclature

Vm

Saturation constant Reaction rate Substrate concentration Maximum velocity

Subscripts m S

Refers to maximum Refers to substrate

KS

r S

kg kg kg kg

m-3 m-3 h-I m-3 m-3 h-l

Exercises 1. Vary the values of the kinetic parameters VM and Ks using the VAL command and see how they influence the substrate uptake kinetics. 2. By using a DO loop, as in example BATREACT, generate a series of curves for different values of either VM or Ks.

3. Rewrite the model to allow for different substrate uptake kinetics and compare the shape of the curves to those for the original Monod kinetics (see Secs. 1.9.2.5 and 1.9.2.6).

Results

Figure 3.16. Degradation of substrate for various parameter values: Curve A: Parameters as in program, Curve B: v ~ = 0 . 5and Ks=2, Curve C: Ks=2, Curve D: Ks=O. 1.

Figure 3.17. Values of the negative reaction rate for the same runs.


3.2.5

223

OXIBAT - Oxidation of Substrate in an Aerated Tank

System In the previous examples, it was assumed that the growth of the microorganisms was limited by only one substrate. Here the general case of a batch oxidation bioreactor is considered, in which substrate is degraded by oxidation. It is assumed that the reaction kinetics are limited by both oxygen and by soluble substrate concentrations (see Sec. 1.9.2.8). This situation may be encountered in aerobic wastewater treatment processes where air is sparged into the reactor and mixing is provided by an impeller (Fig. 3.18).

Figure 3.18. Schematic of the batch biological oxidation reactor.

Model The reaction kinetics are described by a double Monod relation, which allows for either substrate or oxygen to become limiting.

where Ks and KCL are the respective saturation constants for substrate, S, and dissolved oxygen concentration, CL.

224


The batch mass balances lead to: For substrate

For dissolved oxygen

where KLa is the overall mass transfer capacity coefficient for the transfer of oxygen from the air to the liquid. The value of KLa may be taken to vary with stirring speed (N) and aeration rate (G) according to: KLa = k N 3 G0.5 where k = 4.78 x

with N in h-l, G in m3 h-l and KLa in h-l.

Program :File : :

OXIBAT

OXIDATION OF SUBSTRATE IN AERATED TANK KLA VARIES WITH STIRRING RATE N AND GASSING RATE G

K=4.78E-13 CONSTANT N=30000, G = 6 0 , CONSTANT VM=20, KS=5, KI=5O,YOS=5 CONSTANT CLS=8, KCL=O.5,SO=1OO,CLO=8 CONSTANT CINTPO.1, TFIN=5 1 SIM; INTERACT; RESET; GOT0 1 INITIAL S=SO :IN G/M3 CL=CLO :IN G/M3 DYNAMIC :INFLUENCE OF N(rev/h) AND G(m3/h ON KLA(l/h) KLA=K*(N**3)*(G**0.5) RS=-(VM*S/(KS+S+(S**2/KI)))*(CL/(KCL+CL)) :KINETICS S ’ =RS :MASS BALANCE FOR S CL‘=KLA*(CLS-CL)+RS*YOS :MASS BALANCE FOR DISS. 0 2 IF (CL.lt.0) CL=O IF (S.LT.0) S=O PLOT T, CL, 0, TFIN, 0,8 PREPARE T,S,CL,RS OUTPUT T ,S ,CL, KLA, RS

225


Nomenclature

Y

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

Subscripts 0 S

Refers to oxygen Refers to substrate

CL CLS, CL* G KCL

KLa KS

k N r S Vm

g m-; g mm3 h-I kg m-3 h-I kg m-3 complex hg m-3 h-I g m-3 g m-3 h-I kg kg-1

Exercises 1. Reduce the aeration rate G until the reaction becomes oxygen limited. This can be done interactively. Set G at this level and reduce the biomass concentration by 1/2 (reduce Vm by one half). Does the initial oxygen uptake rate decrease by 1/2? Is the reaction still oxygen-limited?

2. Operate the reactor oxygen-limited and observe the increase of CL at the end of the reaction. 3. Increase N by 25% and notice the change of KLa and CL.

4.Vary N and G under non-oxygen-limiting conditions. What is the effect on the reaction rate?

226


Results 11.98

lW.0 A S

AQ 6.98

s.0 W.0

l.0

Figure 3.19. Reduction of substrate for various parameter values: curve A as in program; curve B n=15000, C n=10000, D p~ =lo, E KI =lo.

3.2.6

Figure 3.20. Dissdved oxygen concentration for the same runs.

RESPMET - Oxygen Uptake Experiment with Respirometer

System In analysing environmental processes it is often important to be able to measure changes in the dissolved oxygen concentration. A respirometer is an instrument that is capable of providing useful information about the oxygen uptake rate and therefore the biological activity in a system. The dissolved oxygen has to diffuse across a membrane from the liquid phase to the electrode, and, consequently, there is a delay between the change in the actual dissolved oxygen concentration and that recorded by the respirometer. This effect can be modelled by introducing first-order lag equations to represent the electrode response. In addition to the lag caused by the membrane there may also be a lag caused by the boundary layer on the membrane surface.

Model The oxygen uptake rate OUR is assumed to be proportional to the biomass concentration which in this example is assumed to be constant. Therefore

OUR=qX where q is the specific oxygen uptake rate, g oxygen 1 g biomass h.


227

The specific oxygen uptake rate can be modelled by a double substrate Monod relationship (see Sec. 1.9.2.8) with organic substrate inhibition (see Sec. 1.9.2.5), as given by

The mass balance for oxygen gives

and the electrode dynamics can be modelled by simple first-order lag equations of the form: For the boundary layer film

For the measurement electrode

where ZF and ZE are the first order lag constants for the boundary layer film and for the electrode, respectively.

Program :

File

RESPMET

: : :

OXYGEN UPTAKE EXPERIMENT WITH RESPIROMETER Includes electrode dynamics Carbon substrate assumed constant

QM=0.5,KO=1.O,X=1.0,S=10 CONSTANT CONSTANT KLA=O.O,CLS=8,KS=1,KI=lO CONSTANT TF=2, TE=10 CONSTANT TFIN=BO,CINT=1.0 1 SIM; INTERACT; RESET; GOT0 1 IN ITIAL CL=8 CE=8 CF=8 DYNAMIC

228


:KINETICS OUR=Q*X :MG/L SEC, SPEC.RATE*BIOMASS CONC. Q=QM*(CL/(KO+CL))*(S/(KS+S+(S**2/KI))) :MASS BALANCE FOR 02 CL'=KLA*(CLS-CL)-OUR :ELECTRODE DYNAMICS CF'=(CL-CF)/TF CE'=(CF-CE)/TE IF (CL.LT.0) CL=O IF (CE.LT.0) CE=O IF (CF.LT.0) CF=O PLOT T, CL, 0, TFIN, 0, CLS OUTPUT T,CL,CE,OUR PREPARE T, CL, CE, OUR

Nomenclature Dissolved oxygen concentration Saturation oxygen concentration Saturation constants Substrate inhibition coefficient Oxygen transfer coefficient Oxygen uptake rate Specific oxygen uptake Biomass concentration Maximum degradation rate coefficient Time lag constants

Subscripts E F m 0 S

g m-; g m-3 g mg*l m-6 S-

g m-3 s-1 S-'

g m-3 g m-3 s-1 S

Refers to electrode Refers to liquid film Maximum Refers to oxygen Refers to substrate

Exercises 1. Vary the values of the time lag constants ZF and ZE, and see how they and their relative magnitudes influence the difference between the actual dissolved oxygen concentration and the electrode reading under dynamic conditions. Note there is no effect on the resultant steady state. 2. Investigate the kinetics for the influence of substrate concentration on the rate by changing the initial concentration in each run, and hence use the results of the simulation to find the values of Ks and KI.

229


3. The sensitivity of the culture to oxygen can be investigated by noting the slope of the dissolved oxygen curve, i.e., dCL/dt, over the range of oxygen values. Use this to find the value of KO.

Results a.88 AGL

ECE

6.00

CCI

4.88

2.88

0.88

Figure 3.21. Response of the electrode, film and liquid phases, according to the program's parameter values.

3.2.7

REPEAT

-

Figure 3.22. Curves A and B are the electrode and film responses for the same run; curves C and D result from an increase in TF to 4s and in TE to 15 s.

Repeated Fed-Batch Culture

System An alternative operating strategy to the normal constant volume processes for the treatment of wastewaters is the fed-batch or semi-continuous process, also called sequential batch processing. The tank is initially charged with a small volume of micro-organism-containing effluent, to which an additional supply of wastewater is fed until the tank becomes full. After mixing and aeration the biomass is allowed to settle. The tank is then partially drained, leaving a residual volume behind. Effluent is then again fed to the reactor, and the whole process repeated over many cycles. In this mode of operation the volume of liquid in the tank is constantly changing such that the concentrations of biomass and substrates can be controlled by regulation of the inlet flow rate, thus conferring a high degree of flexibility to the system.

Model The volume of the reactor is no longer constant but now increases at a rate equal to the volumetric flow rate into the reactor:

230


dV

x=Q A mass balance on the substrate in the reactor must take into account the change in reactor volume, and hence has the following form:

d(VS) dt = Q So + rS V Likewise for the biomass:

The growth of the biomass is assumed to follow Monod kinetics with substrate inhibition: rx = p X

The rate of substrate uptake is related to the biomass growth by a constant yield factor:

The concentrations of biomass and substrate at any time are calculated by dividing the total mass of each component by the current volume of the reactor.

In this type of reactor the dilution rate is not constant but decreases as the volume of liquid in the reactor increases:

23 1


At time TFIN, the volume of liquid in reactor is reduced from VF to Vo, where F = VdVF. The total amount of biomass and substrate in the reactor therefore is reduced in direct proportion, i.e.,

[VXO] = [VX] VO VF ~

Program This model does not include the concentration effects of biomass by settling, but this can be included by increasing the initial conditions for biomass at the start of each repeated run above that calculated by direct proportion. :REPEAT :VARIABLE VOLUME REPEATED FED BATCH :AT TFIN THE VOLUME IS REDUCED FROM VF TO Vo :VX AND VS ARE REDUCED BY A FRACTION F=VO/VF :SIMULATION IS REPEATED WITH THESE NEW CONDITIONS CONSTANT FO=0.5, S0=5 CONSTANT UM=0.8, KS=O.l,KI=lO,Y=0.5 CONSTANT CINT=0.5, TFIN=10, NOCI=2 CONSTANT VS=O.1 CONSTANT VF=O.1,VSF=O,VXF=O.O1 CONSTANT V O = 1 1 SIM; INTERACT;TFIN=TFIN+lo; GOT01 INITIAL

v=vo F=VO/VF V X = F*VXF VS=F*VSF DYNAMIC V'=FO VS'=FO*SO+RS*V VX' =RX*V RX=U*X RS= - RX/Y U=UM*S/(KS+S+(S**2/KI))

x=vx/v s=vs/v D=FO/V VF=V VXF=VX VSF=VS PLOT T,VX,0,40,0,25

232


PREPARE T,VX,VS,X, S,U,V OUTPUT T,VX,S,X,V

In this program the line containing the statement 1

SIM;

INTERACT;

TFIN=TFIN+10;

GOT01

allows the results of several fed-batch operations to be plotted consecutively. Note that the range of the x axis is set to 40 not to TFIN as usual. As the value of TFIN is 10, this allows four consecutive batch operations to be plotted on the graph. By varying the values of TFIN and the range of the x axis any number of consecutive operations can be plotted.

Nomenclature

V [VSI [VXI X Y

Dilution rate Ratio of initial to final volumes Substrate inhibition constant Volumetric flow rate Reaction rate Substrate concentration Growth rate Volume of liquid in tank Total substrate in reactor Total biomass in reactor Biomass concentration Yield coefficient

Subscripts 0 S X F

Refers to feed Refers to substrate Refers to biomass Refers to final conditions

D

F KI

Q r S

v

hm3 m-3 g2 m-6 m3 h-l g m-3 h-I g 7-3 hm3 g g g m-3 -

Exercises 1. Run the program and observe the number of cycles required to reach a pseudo-steady-state operating condition, at which each cycle is identical to the previous one.

2. Alter the starting conditions and see how this effects the number of cycles required to reach the pseudo-steady state.

233


3. Compare the operation of a fed-batch process with both a batch and a continuous process, using the same kinetic constants. What are the relative advantages and disadvantages of each process?

4. Increase the initial biomass conditions for the repeated fed-batch runs by an arbitrary factor. This will represent biomass settling to give higher concentrations, as would be obtained with a realistic biomass settling period.

Results xsm

nm

Figure 3.23. Three runs were made with parameters as in the program. At T=30 h, the inlet concentration was changed to 10 g m-3.

3.2.8

CONTI Uptake

Figure 3.24. The specific growth rate during the same four repeated fed batches.

- Continuous Reactor with

Substrate

System This example differs from BATREACT in that there is flow into and out of the reactor. It is assumed that the growth of the micro-organisms is negligible compared to the rate of substrate uptake, so that effectively a constant biomass in the reactor can be assumed. A similar situation exists in activated sludge systems, in which the settler largely determines the biomass concentration.

Model A mass balance for the substrate in the reactor gives: dS = dt

-

Q v

(SF-S)+rS

234


The rate of substrate uptake is assumed to follow Monod kinetics rS = -vM

S

Program :File CONTI : CONTINUOUS REACTOR WITH SUBSTRATE UPTAKE : Model assumes constant biomass concentration. CONSTANT VM=0.3,KS=O.1 CONSTANT SF=lO,V=2,SO=lO CONSTANT CINT=l,TFIN=40 CONSTANT Q= 1.0 :FLOW RATE 1 SIM; INTERACT; RESET; GOT0 1 INITIAL

s=so DYNAMIC S'=(Q/V)*(SF-S)+RS :SUBSTRATE RS=-VM*S/(KS+S) :KINETICS IF (S.LT.0) S=O PLOT T,S,O,TFIN,O,lO OUTPUT T,S PREPARE T, S ,RS

BALANCE

EQUATION

Nomenclature

V

Saturation constant Volumetric flow rate Reaction rate Substrate concentration Volume of reactor Specific reaction rate

Subscripts F M S

Refers to feed Refers to maximum Refers to substrate

KS

Q r S V

kg m-3 m3 h-l kg m-3 h-1

kg m-3 m3

h-

Exercises 1. Vary the volumetric flow rate Q, using the VAL command and follow the substrate concentration in the reactor.

235


2. From a series of runs, use the resulting steady-state concentrations to calculate the substrate reaction rate for each run, and use these values to establish the kinetic model and parameter values used in the model. 3. Repeat Exercise 2 but apply different forms of kinetic relationship, such as zero and first order, in the program. 4. Investigate the behaviour of the reactor for substrate inhibition kinetics.

Results xsm

Figure 3.25. Variation of inlet flow rate from 1.0 to 0.2 to 0.0 (Curves A, B and C).

3.2.9

Figure 3.26. For the same runs, the reaction rate is constant (zero order), except for the batch case when it drops to zero.

FEEDINH - Control of Inhibitory Substrate Feed Rate to a Continuous Reactor

System Considered here is a continuous reactor for treating an inhibitory substrate. The degradation rate is influenced both by the concentration of substrate and by the dissolved oxygen concentration, which is measured with an electrode. The biomass concentration can be assumed constant, as maintained by a sedimentation tank. Liquid phase balances for both the substrate and for dissolved oxygenare therefore used to model the process. This is important since the reactor is operated to avoid high concentrations of substrate, which give rise to inhibitory behaviour. This can be achieved by feeding the substrate at controlled flow rates, such that the substrate concentration does not become too high. A schematic representation of the controlled bioreactor is shown in Fig. 3.27.

236


Figure 3.27. Schematic representation of the continuous bioreactor with control of feed rate.

Model The liquid phase mass balances are as follows: For substrate, dS V L d t = F(SF-S)-rSVL For oxygen, VL

% = KLa (CL*

- CL) VL - ro VL

The kinetics are as follows: rS =

vm S CL Ks + S + (S2/K~) KO + CL

where ro = rs

yo/s

and ro is the oxygen uptake rate per unit volume, equal to OURNL. Proportional control of the feed rate can be either based on the substrate concentration, or can be implemented by means of the following controller equation, based on measured OUR values.

here Kp is an effective proportional controller gain constant.

237


Program :

File

: : :

FEED CONTROL BASED ON OXYGEN UPTAKE RATE CONTINUOUS DEGRADATION OF INHIBITING SUBSTRATE ACTIVATED SLUDGE REACTOR ASSUMING CONSTANT BIOMASS

CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT CONSTANT

FEEDINH

VM=50 :[g/m3h] Maximum Uptake Rate KS=lO :[g/m31 Substr. Saturation Const. KI=100 :[g/m31 Substr. Inhibition Const. KO=0.5 :[g/m31 Oxygen Saturation Const. YOS=50.0 : [-] Oxygen Yield Coeff. KLA=lSO:[l/h] Oxygen Transfer Coeff. FO=10 : [m3/hl Feed Rate KP=O.O :[(m3/h)/(g/hm3)1 Control Constant OURSET=150 : [g/m3h] Set point : [m31 Vol. Liquid VL=lOO : [g/m31 Saturation for oxygen CLS-8.0 SF=500 : [g/m3] Substr. Conc. in feed SI=40, CLI=8 :[g/m31 Initial concentrations ALGO=3, CINTe0.01, TFIN=50

1 SIM; INTERACT; RESET; GOT0 1 INITIAL S=SI :[g/m3] Substr. Conc. in Reactor CL=CLI :[g/m31 Oxygen Conc. in Reactor DYNAMIC : Kinetics RS=VM*S/(KS+S+S*S/KI)*CL/(KO+CL):Uptake RO=RS*YOS OUR=RO

rate

:Mass Balances ( F / V L )* ( S F - S )- R S IF(S.LT.O)S=O CL'=-RO+KLA*(CLS-CL) IF (CL.LT.O)CL=O :Control F=FO+KP*(OURSET-OUR) :Control equation :Prevents negative values IF (F.LT. 0 )F=O S' =

PLOT T, CL, 0, TFIN, 0 , 8 PREPARE T,S,F,CL,OUR,RS, SF OUTPUT T, S ,CL, F ,SF, OUR

Nomenclature CL

CLS

Dissolved oxygen concentration Saturation oxygen concentration

g m-; g m-

of

F

23 8

F FO KI

Subscripts I


Flow rate Flow rate base for controller Inhibition constant Transfer coefficient Oxygen saturation constant Proportional control constant Saturation constant Oxygen uptake rate Reaction rate for oxygen Reaction rate for substrate Substrate concentration Substrate concentration in feed Reactor liquid volume Maximum reaction velocity Yield coefficient, oxygen to substrate

m3 h-l m3 h-' g 7-3 hg m-3 m6 g;' g mg m-3 h-' g m-3 h-l g m-3 h-l g m-; g mm3 g m-3 h-1

-

Refers to initial

Exercises 1. Observe the inhibition effect of high substrate concentration by following the variation of OUR, CL and S with time, for differing starting values of S. 2. Set K, = 0 to turn off the automatic control, and then change F via Fo interactively to obtain maximum substrate utilisation rates, while at the same time avoiding conditions of oxygen limitation and substrate inhibition. This can be done by following the changing CL values. Repeat this manual, interactive control by following the OUR values.

3. Set KLa to a value sufficiently high to avoid oxygen limitation, and vary F to cause S to move through a range of values from 1.0 to 100. Plot the values of S versus OUR and observe the inhibition kinetics. 4. See the discussion on control in Sec. 1.13. and operate the reactor with

control of the feed rate but employing differing values of the controller constant K,. Note that the simple proportional control strategy is not adequate to properly control the reactor for all values of S.

5. Modify the program to operate the reactor with sinusoidal changes in the feed concentration SF. Try to vary F interactively to maintain a constant value of s.


239

6. Change the program to base the controller action on measured values of CL or S , rather than on OUR, and investigate the subsequent control. 7. Modify the control equation to include integral and differential control, as discussed in Sec. 1.13.

Results

Figure 3.28. Response of dissolved oxygen to interactive changes in FO with the controller off (KP=O).

Figure 3.29. Variation of OUR and S for the same run.

3.2.10 CONINHIB - Continuous Bioreactor with Inhibitory Substrate System Inhibitory substrates at high concentrations reduce the specific growth rate below that predicted by the simple Monod equation. The inhibition function may often be expressed empirically as

where KI is the inhibition constant (kg m-3). If substrate concentrations are low, the inhibition function reduces to the Monod equation. In batch cultures the term S 2 / K ~may be significant during the early stages of growth, even for higher values of KI, owing to the initially high values of S. As shown in Sec. 1.9.2.5 the value of p passes through a maximum at Smax= (Ks KI)O.~.

240


Model A continuous bioreactor with its variables is represented in Fig. 3.30.

Figure 3.30. Model variables for a continuous bioreactor.

Biomass balance, dX V -dt = p V X - F X

where D is the dilution rate (= FN). Substrate mass balance,

dS

dt

= D ( S 0 - S ) - CLX y

where Y is the yield factor. Thus steady-state behaviour, where dX/dt = 0, is represented by the conditions that p = D. Also since dS/dt = 0, the steady-state cell concentration, X = Y (So - S). A continuous inhibition culture will often lead to two possible steady states, as defined by the steady-state condition p = D, as shown in Fig. 3.31.

24 1


S

Figure 3.31. Possible steady states for a chemostat with inhibition kinetics.

One of these steady states (A) can be shown to be stable and the other (B) to be unstable. Thus only state A and the washout state (S = So) are possible. Using simulation techniques the behaviour of the dynamic approach to these states can be investigated.

Program When the system equations are solved dynamically, one of two distinct steadystate solutions is obtained, i.e., the reactor passes through an initial transient but then ends up under steady-state conditions either at the stable operating condition, represented by A, or the washout condition, for which X=O. The initial concentrations for the reactor will influence the final steady state obtained. :

File

CONINHIB

: :

CONTINUOUS CULTURE WITH SUBSTRATE INHIBITION KINETICS

CONSTANT UM=0.53, KS-0.12, KI=2.2 CONSTANT Y=O.5, S0=5, D=0.25, XI=1, SI=o CONSTANT CINT=2.0, TFIN=40 :INITIAL S IS INCREASED FOR EACH RUN 1 SIM; INTERACT; RESET; SI=SI+l.O; GOT0 1 INITIAL x-XI S=SI DYNAMIC X'=(U-D)*X S'=D*(SO-S)-U*X/Y U=UM*S/(KS+S+(S*S/KI)) PLOT T, S , 0, TFIN, 0, SO PREPARE T,X,S,U OUTPUT T,X,S,U

242


Nomenclature D F

P (U)

Dilution rate Flow rate Inhibition constant Saturation constant Substrate concentration Volume Biomass concentration Yield coefficient Specific growth rate coefficient

Subscripts 0 I m

Refers to inlet Refers to initial value Refers to maximum

KI

KS S V X

Y

h-1 m3 h-' kg m-3 kg m-3 kg m-3 m3

kg m-3 kg kg-' h-

Exercises 1. Vary Ks and KI,and observe the changes in the inhibition curve by plotting p versus S. Note especially the position and values of p and S for the maxima. 2. Start the reactor with the initial concentration conditions of S = 0, X = 2 and D c p m , and observe the approach to steady state. Did washout occur? 3. Start the reactor with S > Smax and X = 0.1. What is the steady-state result?

4. Run a simulation under the conditions of Exercise 3, and stop the feed (D = 0) interactively just before washout. Allow S to return to near zero. Plot p versus S. 5. Rerun Exercise 2 until near steady state. Increase S to just below Smax by setting S o very high for a short period, return to the original value, and let the system reach steady state. Repeat this, but let S become much larger than Smax. Compare the results. 6. Rerun Exercise 2 with S o = 20. Stop the flow when S > Smax, and allow the system to return to S c Smax before starting again.

7. Change the model to include feedback control of the feed flow rate, assuming the substrate concentration can be measured (see Sec. 1.13).

24 3


8. Make suitable changes in the initial conditions of X and S, and plot the phase plane diagram X versus S. By making many runs at a range of initial conditions, the washout region can be identified. 9. Use the following DO loop statements to investigate the influence of initial conditions: DO S I = 0,5,1 DO X I = 1,5,1 SI M T=O END END

Results Ism 6-88 4-88 2.88

e,me:m

9.k

X

1:s

2.h

3.b

Figure 3.32. Results for various vaiues of initial conditions (SI varied with XI=^).

9.m

X

Figure 3.33. In these runs XI =2 and S1 was varied.

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

244


3.3

Activated Sludge Wastewater Treatment Processes

3.3.1

ANDREWS - Model of a Batch Activated Sludge Process

System Andrews and colleagues have developed structured dynamic models for the activated sludge process, which are intended mainly for control purposes. The model here considers a batch activated sludge process, in which the total suspended solids consist of three main components. These components, stored mass, active mass and inert mass, are all derived from substrate and are related according to the scheme shown in Fig. 3.34. Attachment

- -

Synthesis Stored mass

f

Respiration

Residue

Active mass

Inert mass

f

Respiration

Figure 3.34. Scheme of Andrews' activated sludge model.

The inlet substrate is rapidly transferred from the liquid phase and into the form of flocs by the processes of adsorption, absorption and physical entrapment (attachment). This stored mass is then assimilated by the biomass and converted by synthesis to produce active mass. Cell death leads to a conversion of active mass to inactive mass (residue).

Model The total solids concentration (X) is composed of the sum of the stored mass concentration (Xs), the active mass concentration ( X a ) and the inert mass concentration (Xi):

The rate of the attachment step (rl) is taken as:

245

3.3 Activated Sludge Wastewater Treatment Processes

where ks is a mass transfer coefficient and K, is a saturation coefficient. The factor f, is the maximum fraction of the substrate (both suspended and soluble) that can become attached to the flocs. The uptake of substrate into the biomass follows Monod kinetics, where:

Cell death is assumed to follow first-order kinetics:

where Kd is the specific death rate. Mass balances on the attached, active and inactive mass fractions lead to the following equations:

In the above equations, Ya is a yield coefficient relating the biomass produced to the attached substrate consumed, and Yj is a yield coefficient relating the inactive biomass produced to the active biomass.

Nomenclature

kS

Ks2

r S

X

Storable substrate fraction Specific death rate coefficient Mass transfer coefficient Saturation coefficient Reaction rate Substrate concentration Solids concentration

hg y-3 h-

246


Y

h-

Yield coefficient Maximum growth rate

Pmax

Subscripts a i

Active biomass Inert biomass Stored mass Total

S

T

Exercises 1. Test the model by carrying out repeated simulations to see whether the system is sensitive to the starting conditions.

2. Extend the model to simulate a continuous process with recycle (see example ASCSTR) Assume that equal fractions of the biomass components are recycled and that there is no biomass in the feed.

Results 600:

B'0 400.-c

U

0 4 . 0

'

.

'

.

9

20 30 Time (hours)

10

.

I 40

Figure 3.35. Variations in substrate, total solids and active, inert and stored mass concentrations, for the conditions in the program.

m 4 0 a

1 0

20 30 Time (hours)

10

40

Figure 3.36. Substrate utilisation for different initial values of stored mass of 10, 50, 100, and 200 g ~ n - Note ~ . the rapid initial rate of substrate uptake to stored mass at low concentrations of initial stored mass and its absence at higher values.

Reference Busby, J. B. and Andrews, J. F. (1975) Dynamic Modeling and Control Strategies for the Activated Sludge Process, J. Water Pollut. Control Fed. 47, 5 , 1055-1080.


3.3.2

247

ASCSTR - Continuous Stirred Tank Reactor Model of Activated Sludge

System The activated sludge process, depicted in Fig. 3.37, involves basically the aeration and agitation of an effluent in the presence of a flocculated suspension of micro-organisms which are supported on particulate organic matter. After a predetermined residence time (usually several hours) the effluent is passed to a sedimentation tank where the flocculated solids are separated from the treated liquid. A reduction of BOD from 250-350 mg litre-l to a final value of 20 mg litre-' is achieved under typical operating conditions. Part of the settled sludge is usually recyyled to the aeration tank in order to maintain biological activity. In this model the aeration tank is modelled as a continuous stirred tank reactor.

Figure 3.37. Continuous stirred tank reactor and settler with sludge recycle.

Model The total volumetric flow rate entering and leaving the reactor, Q, is the sum of the influent and recycle flow rates:

Q

=

Qo+QR

The recycle ratio is defined as

R = QR Q ~ Combining the above equations gives:

248


The reactor residence time is defined by the volume of the reactor divided by the volumetric influent flow rate: V

e=Qo

The concentration factor p is defined as the ratio of the recycle biomass to effluent biomass concentrations

The growth of biomass in the reactor is assumed to follow Monod kinetics with a first-order death rate. A mass balance on the biomass in the reactor yields the following differential equation (assuming that no biomass enters the reactor in the feed):

Kdx Similarly, assuming there is only negligible substrate in the recycle, a mass balance on the substrate (BOD) in the reactor gives:

-dS dt

-

(SO-S)

e

PmXS

- Y(K,+s)

As discussed in Sec. 2.3.1 various operating parameters can be calculated as follows: (i) Substrate to micro-organism ratio or process load factor, Load =

Substrate consumed per day Biomass in reactor

(ii) Mean solids residence time in the reactor, ,8 Solids in reactor 8, = (Rate of biomass synthesis) + (Solids input to reactor)

249


(iii) The sludge age, €I,, Biomass in the reactor 8, = Rate of biomass generation

Nomenclature

Kd KS

Load

Q R S V X Y

P Pm 8 9, 9 , Subscripts 0 e R W

Death rate constant Monod constant Loading factor Volumetric flow rate Recycle ratio Substrate concentration Volume Biomass concentration Yield biomass/substrate Sludge concentration factor Maximum growth rate Reactor residence time Sludge age Mean solids residence time

hg f” hm3 h-l g m-3 m3 g m-3 hh h h

Refers to inlet Refers to effluent Refers to recycle Refers to waste

Exercises 1. Vary the recycle ratio and see how this influences the process performance. 2. Change the model so that the reactor is modelled as two or more tanks in series and compare the results with those obtained for the single tank model.

3. Simulate a shock loading by changing the values of either the flow rate or the effluent concentration interactively.

250


4. Add a calculation of the operating parameters substrate consumption rate to micro-organisms ratio (Load), solids residence time (0,) and sludge age (0,) to the program. Experiment with variations in the values of these parameters under various conditions of flow and feed concentration to test their value as process control parameters.

Results

Biomass

I

200

Q)-

100

-e

200

E O

100

bstrate

0

v)

0

10

20 30 Time (hours)

40

Figure 3.38. Startup of the continuous activated sludge reactor with recycle ratio R=0.3.

0

10 20 30 Time (hours)

40

Figure 3.39. Substrate concentration profiles for different values of the recycle ratio.

Reference Grady, C. P. L. and Lim, H. C. (1980) Biological Wastewater Treatment: Theory and Applications. Pollution Engineering and Technology 12, Chapter 16.

3.3.3

ASPLUG - Plug-Flow Model of an Activated Sludge Process

System Activated sludge processes using plug-flow types of contactors are still widely used for wastewater treatment. Wastewater and recycled sludge are fed into one end of a long narrow basin, typically 6-10 m wide, 30-100 m long and 4-5 m deep. The basin is equipped with aeration devices to supply oxygen along the length of the basin. In all other respects, the system is the same as that described in the previous example ASCSTR. Fig. 3.40 illustrates the modelling of such a system.

25 1


Profiles of substrate and biomass concentrations along the length of the reactor can be generated by simulation of the model equations. The concentrations of biomass and substrate in the recycle stream depend on both the performance of the reactor and that of the settling tank, which in turn depend on the influent stream.

Figure 3.40. Modelling of plug-flow configuration for activated sludge treatment.

Model The volumetric flow rate, Q, entering the reactor is the sum of the influent and recycle flow rates, Qo and QR: Q = Qo+QR The recycle ratio, R, is defined as the ratio of the recycle flow rate to that of the influent flow rates.

R = QR Q ~ Combining these equations gives:

The concentration factor p is defined as the ratio of the recycle biomass to effluent biomass concentrations

The inlet substrate concentration to the reactor is therefore obtained by a steady-state balance around the inlet mixing point to give SI =

QO SO + QR

Q

SR

252


Similarly the inlet biomass concentration is given by XI = Qo XO+ QR XR

Q

The net rate growth of biomass in the reactor is assumed to follow Monod growth kinetics combined with a first order death rate. The plug flow equations can be expressed in terms of the liquid velocity, U, to give

The of substrate utilisation is related to the biomass growth rate by a constant yield factor Y: 1 pxs -dSdZ - - U Y (Ks + S ) where the velocity is

u = QA Program An initial guess of the recycle biomass and substrate concentrations is made so that the inlet concentrations can be calculated. The model is run with these conditions, and the actual values of the recycle concentrations are compared with the estimated values. If the values are not close enough, then the model is rerun with a new estimate of the recycle concentrations. Using a DO loop, this is repeated until satisfactory agreement is achieved or a predetermined number of iterations have been performed (in this case 20). In the program the equations are integrated from inlet (Z=O) to outlet (Z=L). A first estimate of the initial conditions is used, and revised values of SI and XI are calculated from the inlet mass balances. After 10 or so iterations the solution converges.

Nomenclature A

Kd KS

Q R S U

Reactor cross-sectional area Death rate constant Monod constant Volumetric flow rate Recycle ratio Substrate concentration Liquid velocity

m2 h-I g m-3 m3 h-l g m-3 m h-'

253


z

Biomass concentration Yield biomadsubstrate Distance along the reactor Sludge concentration factor Maximum growth rate Reactor residence time

Subscripts 0 I R

Influent Initial estimated value Recycle

X

Y Z

P Pmax

g m-3 -

m hh-

Exercises 1. Vary the recycle ratio R between 0.1 and 0.3 and see how this influences the performance of the process. 2. Vary the sludge concentration factor (P) between 1 and 5 and see how this influences the process. Vary both R and P together and determine the region of stable operation for the contactor, i.e., the region where no washout occurs.

3. Keep the overall volume of the reactor constant but vary the dimensions of the reactor. How do the dimensions influence the reactor performance? Are the washout conditions affected by reactor geometry? 4. Simulate a shock load entering the reactor by interacting with the program and changing the value of either the flow rate (Q) and/or the substrate concentration (S) for a short time. How does the system cope with such shocks? Compare this performance to that the ASCSTR model.

5. This model assumes ideal plug flow which is unlikely to be achieved in practice. Axial dispersion can be included in the model by defining an eddy diffusivity constant for the reactor. For further details, see Ingham et al., (1994). 6. Compare the simulation results obtained with a similar reactor but modelled by two or more tanks in series.

254


Figure 3.41. The results of the iterations to calculate the axial substrate profile

Figure 3.42. Results from the same run, giving the iterations for biomass.

References Grady, C. P. L. and Lim, H. C. (1980) Biological Wastewater Treatment: Theory and Applications, Pollution Engineering and Technology 12, Chapter 16. Ingham, J., Dunn, I. J., Heinzle, E., and Prenosil, J. E. (1994) Chemical Engineering Dynamics, VCH.

3.3.4

CURDS

-

Curds' Model of Sludge Ecology

System In many models used to describe activated sludge processes, the microbial population is considered as a whole, and no distinctions are made between the different micro-organisms that exist in activated sludge. Curds (1973) developed a model that considers the ecology of the activated sludge process, in which the major interactions between the bacteria (sewage and sludge) and the protozoa (attached and free bacteria-consuming and carnivorous protozoa) are summarised in Fig. 3.43.

Model The activated sludge reactor in this simulation is similar to that of example ASCSTR. The recycle ratio is R and the concentration factor is given by p.


255

Figure 3.43. Microbial interactions in an activated sludge process.

It is assumed that the growth of the bacteria and flagellated protozoa follow Monod kinetics:

where the index i designates the organism type, as listed in the Nomenclature.

1. Sewage Bacteria It is assumed that the sewage contains bacteria which contribute to the removal of substrate. These bacteria do not flocculate and hence do not settle in the sedimentation tank. The sewage bacteria are preyed upon by the ciliated protozoa. A mass balance for the sewage bacteria in the reactor gives:

Note that the mass balance equations are written in terms of dilution rate (volume of system divided by volumetric flow rate), recycle and concentration.

256


2. Sludge Bacteria The sludge bacteria flocculate and are assumed to settle out completely in the sedimentation tank. These bacteria are not found in the feed and are not consumed by the protozoa. A mass balance on the sludge bacteria in the reactor gives:

3. Flagellated Protozoa The flagellated protozoa consume substrate but not bacteria and are found in activated sludge but do not enter the activated sludge process in the feed. A mass balance on these organisms in the reactor gives:

4. Ciliated Protozoa 4.1 Free-Swimming Ciliates These organisms are not present in the sewage and do not flocculate. They can feed on sewage bacteria but not on flocculated sludge bacteria and are themselves consumed by carnivorous ciliates. A mass balance on the free-swimming ciliated protozoa in the reactor gives:

In the above balance relationship, the terms 0 1 and 0 2 represent the predation rates on the free swimming ciliates by both free-swimming and attached carnivorous protozoa, respectively. The predation terms can be modelled by Monod-type equations as 01 =

max5 1 x41 x5 1 KS5l + x41

0 2 = pmax52 x41 x 5 2 KS52 + x41

4.2 Attached Ciliates Attached ciliates are not present in the feed but are found in the activated sludge. They feed on free-swimming bacteria but are themselves consumed by


25 7

carnivorous protozoa. The attached ciliates are closely associated with the sludge flocs and therefore settle out on sedimentation.

where the protozoa predation rates are 0 3 =

max51 x 4 2 x 5 1 KS5 1 + x 4 2

04 =

Prnax52 x 4 2 x 5 2 Ks52 x 4 2

4.3 Crawling Ciliates Crawling ciliates are not so closely associated with the sludge flocs as the attached ciliates and therefore do not settle out so well on sedimentation. This can be represented in the model by a lower value of p. Otherwise they are modelled in exactly the same way as the attached ciliates.

with 0 5 and 0

6

again representing the protozoa predation rates given by 05 =

06 =

PrnaxS 1 x 4 3 x 5 1 KS5 1 + x 4 3

'

max52 x 4 3 x 5 2 KS52 + x 4 3

5. Carnivorous Ciliates 5.1 Free-Swimming Carnivorous Ciliates Free-swimming carnivorous ciliated protozoa, such as Litonutus, Hemiophyrs, Truchelophyllurn and Amphileptus, are found in activated sludge but generally not in sewage. They consume ciliated protozoa but are not concentrated in the sedimentation tank.

5.2 Attached Carnivorous Ciliates Attached carnivorous ciliated protozoa, such as Acinetu and Podorphryu, are also found in activated sludge but not in sewage. They consume both free-

258


swimming and attached ciliated protozoa and are concentrated in the sedimentation tank to the same degree as the attached ciliated protozoa and the sludge bacteria.

It is assumed that there is only one limiting substrate in the sewage and that this is consumed by the bacteria and flagellated protozoa but not by the ciliated protozoa. The mass balance for the substrate is therefore: dS (P1 x1+ P2 x2 + P3 X3) Y dt = D S o + D R S - ( l + R ) D S -

-

The volumetric flow rate is not constant but shows a diurnal rhythm with a peak at about midday and a minimum at about midnight. The maximum flow rate will thus vary with the installation. Here the maximum flow rate is taken as three times the minimum. The concentration of the sewage follows a similar trend with the most concentrated sewage occurring at midday and varying between 140 and 260 g m-3 in concentration. This effect can be modelled by applying a sine wave function to both the inlet flow rate and substrate concentration:

So = 200+60sin ( 2 x t)

Fo = 100 + 50 sin (2 x t)

The sewage bacteria concentration is similarly assumed to show diurnal fluctuations, represented by

Xo = 30 + 15 sin (2 71: t)

Nomenclature D KS

R S X Y

P

0

P

Dilution rate Monod saturation constant Recycle ratio

hg m-3 -

Substrate concentration Biomass concentration Yield Concentration factor Predation rate Specific growth rate coefficient

g m-’3 g mg m-3 h-l h-

259


Subscripts 0 1 2 3 41 42 43 51 52

Inlet concentration Sewage bacteria Sludge bacteria Flagellated protozoa Free-swimming ciliated protozoa Attached ciliated protozoa Crawling ciliated protozoa Free-swimming carnivorous ciliated protozoa Attached carnivorous ciliated protozoa

Exercises 1. Run the model with the different predator prey interactions (i) to (iv), as shown in Fig. 3.43. Which combinations lead to stable populations and which lead to wash out of one or more of the populations? 2. Investigate the influence of diurnal variations in the sewage flow rate and concentration on the bacterial and protozoal populations. Are the microbial fluctuations in phase with the sewage fluctuations? 3. Plot one organism concentration versus that of another. Are the oscillations sustained?

Results

0

200 300 Time (hours)

100

400

Figure 3.44. Simulation of sludge ecology with predator-prey interaction i only. Conditions as in the program. The curves are as follows: Substrate (a), Sludge bacteria (b), Sewage bacteria (c), Flagellated protozoa (d).

0

100

200 300 Time (hours)

400

Figure 3.45. Simulation of sludge ecology with predator-prey interactions i and ii. Conditions as in the program. The curves are as follows: Sewage bacteria (a), Flagellates (b), Carnivorous ciliates, Bacteria-consuming ciliates (c).

260


Reference Curds, C. R. (1973) A Theoretical Study of Factors Influencing the Microbial Population Dynamics of the Activated Sludge Process, Water Res. 7, 12691284.

3.3.5

FLOCl and FLOC2 in a Sludge Floc

- Diffusion

and Reaction

System Diffusion and reaction (see Sec. 1.12) take place within a spherical floc of activated sludge (see Sec. 2.3.3). It is interesting to find the penetration distance of oxygen for differing specific activities and diameters of the floc (Fig. 3.46). In the first example, (FLOCl), oxygen is assumed to be the only limiting substrate. A second example, (FLOC2) considers both oxygen and organic substrate, which is degraded both aerobically and anaerobically at low oxygen concentrations.

n+l

1 R

0

r

Figure 3.46. The finite differencing of the spherical sludge floc.

The system is modelled by taking small spherical shell increments of volumes 4n/3 (rn3- rn-l3),where r represents the floc radius. The outside area of the nth increment is 4n rn2 and the inside surface area is 4n rn-i2. The diffusion fluxes for the nth spherical shell are considered as shown in Fig. 3.47.

26 1


Figure 3.47. The diffusion fluxes entering and leaving the spherical nth shell.

Model Mass balances for oxygen and organic substrate can be written for the spherical shell as

][

Accumulation Diffusion rate = into element rate within volume AV element AV

][ -

Diffusion rate out of element volume AV

The oxygen balance for the shell n becomes

where the diffusion fluxes in and out of the shell are

Substitution gives

The mass balance for the inner spherical element 1 becomes

because there is no diffusional flux away from the centre.

Reaction rate within element volume AV

][ +

]

262


Since rl = Ar, this becomes

The uptake rate of oxygen is expressed as

In FLOC2 the organic substrate @)uptake rates are described by equations of the same form, and oxygen (0)uptake rates are related by a constant yield coefficient.

Program A different increment numbering is used in both programs, with the inner segment taken as element number 6. Also the outer segment has a radius rl =

R. The sequential, unsorted operation of ISIM allows separating the substrate balances in two parts as given in the program.

Nomenclature The nomenclature for FLOCl is as described below. In FLOC2 the symbols are much the same and are given in the program with the units mg, cm, s. Ar D

t

Increment length for r, R/6 Diffusion coefficient Saturation constant Maximum specific oxygen uptake rate Reaction rate Radius of sphere Radial length variable Oxygen concentration Biomass concentration in floc Time

Subscripts 1 2 6 n P S

Refers to outer segment 1 Refers to segment 2 Refers to inner spherical segment Refers to segment n Refers to particle Refers to substrate

KS OURmax rS R r S X

m m2 h-I g m-3 g kg-lh-l g m-3 h-I m m g m-3 kg m-3 h


263

Exercises 1. Investigate the response of the system to changes in the outside surface concentration of oxygen. Can you relate the time to reach steady state to the magnitude of DR2?

2. Show that the ratio of maximum diffusion rate to maximum reaction rate is proportional to the dimensionless group [OUR X R2/D S o ] . Vary the parameters of this group keeping its value 1.0 and compare the results. Show that small values of this group lead to little diffusion influence and that large values cause important oxygen gradients. 3. Note that the position of zero oxygen concentration within the floc depends upon the values of Do2 and R. 4. Write the model in dimensionless form. What are the governing parameters

applying for first and zero order kinetics? Verify the equations by simulation.

5. Establish a model and program with rectangular coordinates (see BIOFILM), taking the thickness as VsphereIAsphere. Compare the results of the spherical model with those for rectangular geometry. 6. In FLOC2, show by simulation that the ratio of external concentrations will determine whether organic substrate or oxygen will be limiting and that this is related to the oxygen to substrate yield constant.

Results

Figure 3.48. FLOCl oxygen concentrationtime profiles at six different positions for a 1 mm floc. The external oxygen concentration was set to zero at t=0.042 h.

Figure 3.49. Organic substrate-time profiles for FLOC2. At t=O the external oxygen was set to zero, causing oxygen depletion throughout the 0.1 mm floc.

264

3.3.6


ASTEMP - Temperature Gains and Losses in an Activated Sludge Process

System Poor performance of some wastewater treatment plants during periods of cold weather has been reported. High temperature operation can also cause problems such as excessive carry-over of solids. It is necessary, therefore, to be able to accurately predict and control the temperature of wastewater treatment processes in order to obtain optimum performance. In this example a heat transfer model for a continuous activated sludge process is developed (Brown and Enzminger, 1991).

Figure 3.50. Major sources of heat gain and loss from an activated sludge reactor.


265

Model The design and operation of the activated sludge process is exactly the same as that employed in the simulation example ASCSTR. Here the reactor is rectangular and has walls 60 cm thick (Fig. 3.50).

Heat Gains Heat is generated by the biological degradation of the wastewater, the action of the stirrer and by sunlight. 1. Biological Heat Generation The heat generated by the biological degradation of the wastewater can be estimated by use of the expression:

where V is the volume of the tank, YQS is the amount of heat released per unit mass of substrate removed. The exact value will depend on the nature of the wastewater. In this example a value of 1.359 x 104 kJ kg-1 (the value for acetic acid) is assumed. Yxs is the yield of biomass per unit mass of substrate removed.

2. Mechanical Heat Generation The heat generated by the mechanical agitation depends on the motor power (P) and efficiency (Me) and the mechanical efficiency (re) of the mixing:

3. Solar Heat Generation The solar heat gain can be estimated as follows: Q, = A, E (sin a) 6 e - ams where As is the exposed liquid surface, E is the solar flux (J m-2 hour-l), ams the dimensionless molecular scattering coefficient, n the dimensionless air turbidity factor, a the solar angle in radians and 6 the dimensionless solar angle correction factor. a is equal to the latitude of the treatment plant minus n/2 radians. m and ams can be derived from a in the following way: m = cosec a ams = (0.128 - 0.054 log m) The total heat gained by the system is therefore given by:

266

3 Simulation Examples of Environmental Bioprocesses Qgain = Qb + Qs + Qm

Heat Losses Heat is lost from the system by radiation, conduction, evaporation and in the air stream. 1. Radiant Heat Loss The radiant heat loss can be estimated by the Stefan-Bolzmann Law (see Glossary):

where Awall and A, are the tank wall and the exposed liquid surface areas respectively. &wall is the emissivity of the tank walls, E~ the emissivity of water, and Tamb the ambient temperature. The temperature of the outer walls of the reactor, Twall, can be estimated as

where d is the wall thickness, Km the thermal conductivity of the tank and ho the air film heat transfer coefficient. Here it is assumed that negligible heat transfer resistance by the liquid film on the inner tank walls occurs.

2. Evaporative Heat Losses Heat losses due to evaporation can be estimated as follows:

(& = - [ 19974 (1-H)+ 11%7 (Tw- Tamb) ] w &0*95 e 0.0604Tam,

-

n3)

where H is the humidity (0 to l), W is the wind speed (m s-l), As is the exposed liquid surface area. 3. Conductive Heat Losses Conductive heat is lost through the tank walls, through the liquid surface and through the base of the reactor. Heat losses through the tank wall are estimated by the empirical relation:

where UA is the overall wall heat transfer coefficient, and 3600 converts seconds to hours. Heat losses through the water surface can be estimated similarly as:

267


Conductive heat losses through the base of the reactor to the ground can be estimated by the following equation: Qcg = - 3600 Ug A (Tw - Tg) where Tg is the ground temperature, Ug is the overall heat transfer coefficient and A is the area of the tank base. The overall heat lost by conduction is therefore given by: Qc

=

Qct+Qc~+Qcg

4. Heat Losses in the Aeration Stream The heat lost in the air stream depends on the humidity of the air, H, and the temperature difference between the wastewater and the surroundings.

+ 4.401 e 0.0604 (Tarnb - 273) (1 - H) + + 2.638 e 0.0604 (Tarnb - 273)(Tw- Ta)] The total heat lost to the surroundings is therefore: Qloss

= Qr + Qc + Qe + Qa

Heat Balance The temperature in the reactor can be derived by writing a heat balance for the reactor assuming the contents are well mixed:

-dt- - F (Tin -TI + (Qgain + Qloss) dT v c, P

v

Program Note the expression for Qa has been split into 2 lines.

Nomenclature A ams b

Exposed surface area Molecular scattering coefficient Decay coefficient

m2 -

268

CP E F Fair h0

Km

m Me n P

Q R P

re

S T UA V

w

X YQS

yxs a

P

6 E

P d

Subscripts amb b C

cg ct

cw e m r S

Wall


Specific heat capacity Solar flux Volumetric wastewater flow rate Volumetric air flow rate Air film heat transfer coefficient Thermal conductivity of the tank Geometrical factor for solar heat gain Motor efficiency Air turbidity factor Motor power Generation or loss rate of heat energy Recycle ratio Liquid density Mechanical efficiency Substrate concentration Temperature Overall wall heat transfer coefficient Volume of reactor Wind speed Biomass concentration Heat released per unit mass of substrate Biomass per unit mass of substrate Solar angle Concentration factor Solar angle correction factor Emissivity Specific growth rate Wall thickness Ambient conditions Biological heat Conduction Conduction ground Conduction tank Conduction wall Evaporation Mechanical heat Radiation Surface Tank wall

J kg-1 K-l J m-2 h-l m3 h-' m3 h-l J m-2 s-1 K-I J m-1 5 1 K-1 -

J h-1 J h-1 kg m-3 kg m-3 K J m-2 ,-1 K-1 m3 m s-1 kg m-3 J kg-1 kg kg-1 radians hm


269

Exercises 1. See how the temperature of the reactor changes during startup. Which of the various heat transfer processes are the more important? 2. Consider what happens to the reactor during the night when there is zero solar energy and the ambient temperature is lower. Model the diurnal changes in the ambient temperature by using a sine wave function and find a suitable function for the changes in solar flux. 3. Investigate the effect of insulating the walls of the reactor to prevent extremes of temperature developing in the reactor. Compare the performance of an above ground reactor with a below ground reactor.

4. In the model it was assumed that the yield coefficient Yxs was constant. However, YXS has been shown to be influenced by both temperature and solids retention time. The following empirical relationship has been proposed to model this effect:

yxso yxs = (1 + b 8,)

where Tw is the temperature of the wastewater (K) and 8 and is the temperature activity coefficient and has a value of 1.02. 8, is the solids residence time and has a value of around 23 days. bo is the decay coefficient at 29313 and has a value of 0.07 days-'. Yxso is the yield coefficient at 293 K. Include these effects in the model. Do they have any significant effect on the results?

5. See how the temperature in the tank varies with changes in the wastewater, i.e., flow rate, temperature and substrate concentration. Include relationships in the problem specification and in the ISIM program for a controller to maintain the temperature in the tank between defined limits.

270


Results

Figure 3.51. The response of the water temperature to changes in Tamb from 293 to 320 is shown.

Figure 3.52. The same ambient temperature changes influenced the fractional heat losses as shown.

Reference Brown, E. V. and Enzminger, J. D. (1991) Temperature Profile and Heat Transfer Model for a Chemical Wastewater Treatment Plant, Environ. Progress 10, 3, 159-168.

3.3.7

STEPFEED - Step Feed Activated Sludge Process with Structured Kinetics

System A modification to the standard activated sludge process described in the preceding examples is the step feed activated sludge process (Fig. 3.53). In this, the influent waste flow can be divided into various streams which enter the aeration basin at differing locations. This mode of operation has the following advantages:

1. Better control of the hydraulic and sludge residence times. 2. Prevention of hydraulic overload and sludge bulking. 3. Better control of effluent quality. 4.Optimisation of dissolved oxygen utilisation and sludge production.

27 1


f0

SO

*1

I

f01

I

f02

f03

fl$-l-rl-lI-. f

v2

xs1

f

s2 xs2

v3

'3 s3 xs3

L

fE s4

v4

.

1

f5 s4 xs4 xA4 s43 xS49 xA4

Figure 3.53. Flow diagram for the step feed activated sludge process.

As shown the system consists of three tanks in series, followed by a settler. If the loading is low, all the wastewater can be directed to tank three. Under these conditions the sludge in tanks 1 and 2 is not being used and is being stabilised. If the loading increases, then tank 2 and also, if needed, tank 1 can be brought into operation. At very high loadings all the effluent is directed to tank 1 so that all the effluent passes through the three tanks.

Model Kinetics The kinetic equations are those derived by Andrews (see the example ANDREWS) based on substrate BOD, dissolved oxygen and a biomass consisting of active, inert and stored constituents. Rate of substrate consumption (BOD removal): rs = I-1 Rate of formation of active mass:

S

(Ks+s)

XT

212


Rate of formation of stored mass (reserves):

Some of the substrate is stored and some is used up for the synthesis of active biomass. Rate of formation of inert mass:

XI = Pi Y2 XA Oxygen uptake rate: r0X =

1-Y (*)

rXA -k

1-Y (*)

rX1

Mass Balances The components substrate (S), stored biomass (Xs), active biomass (XA) and dissolved oxygen (C) can be balanced for the three tanks and the settler. Component balances tank 1:

Component balances for tanks 2, 3 are similar to those for tank 1, but allowing for the respective stream quantities and noting that fR = 0. Component balances for the settler tank: (assuming no biodegradation)

273


x T 4 = xS4 + x A 4 + XI4

Flow Balances f0 = fOl -k f02 + f03

fR =

a fo

f l = fol -k fR

f2 = fo2

+ fl

f3 = fo3

+ f2

Nomenclature Dissolved oxygen concentration Concentration factor Flow rate Saturation constant Oxygen transfer coefficient Michaelis constant Reaction rate Substrate concentration Volume of tank Active biomass concentration Inert biomass concentration Stored biomass concentration

g m-3 -

m3 h-' g 5-1

hg m-3 g m-3 h-l g m-3 m3 g m-; g g m-

274 XT

Y1 y2

a

P


Total biomass Active biomass/stored biomass Inert biomasdactive biomass Recycle ratio Growth rate constant

g m-; g Kl g gh-

Subscripts 1,293 4 5 A E I

ox R S T W

Tank number Settler Recycle stream leaving settler Active Effluent Inert Oxygen uptake Recycle Stored Total WaSte

Exercises 1. Compare the operation of the process under the following conditions: a) The feed all goes into tank 3, b) The feed is half fed to tank 2 and half fed to tank 3. c) Feeding is distributed equally between all three tanks. How does the concentration of the feed stream influence the process strategy? 2. Plot the dissolved oxygen concentrations, C4 and C3, versus time and explain the shape of the curves.


275

Results

Figure 3.54. At 45h the feeding was changed from the third to the first and second stages (FR1=0.5, FR2=0.5, FR3=0).

Figure 3.55. The dissolved oxygen concentrations were influenced by the feeding changes.

References Sorensen, P. E. (1980) Evaluation of Operational Benefits to the Activated Sludge Process Using Feed Control Strategies, Progress in Water Technology, 12. 109-125. Sorensen, P. E. (1982) Modelling of Effluent Quality Control for Activated Sludge plants, Water Sci. Technol. 14, 133-146. This example was developed at the ETH by L. Rincon.

3.3.8

SETTLER - Solids-Liquid Separation in a Continuous Settler

System The separation of activated sludge from treated water is an important process in the overall performance of an activated sludge plant. Failure of the settler to clarify the secondary effluent adequately will lead to a reduction in effluent quality. Inadequate sludge thickening is also detrimental to the aeration stage of the activated sludge process.

276


Model As shown in Fig. 3.56, the settler is modelled as n horizontal layers, each of which has an equal depth z. It is assumed that the concentration of suspended solids is constant within each layer. The feed is introduced into the centre of the settler at a depth defined by layer m (lcmcn). Clarified water is removed from the top (layer 1) of the settler and thickened sludge is removed from the bottom (layer n). Qf = Qf + Qo

QO

Qn Figure 3.56. Schematic diagram of the settler showing the numbering of the layers.

The gravitational settling flux for the sludge particles is a function both of the gravitational settling velocity and the suspended solids concentration where:

Gs = vSX The gravitational settling velocity may be estimated by means of the empirical correlation of Vesilind (1979):

Values of

Vi

= 2 m h-' and b = 3

L g-' are used in this model.


277

The bulk downward flux G, through the settler is determined by the underflow withdrawal rate (Qu):

where A is the cross-sectional area of the settler. Similarly, the upward bulk flux through the settler is determined by the overflow rate (Qo)

The mass flux of liquid into any given layer cannot exceed the maximum mass flux that the element is capable of passing, nor can it exceed the maximum mass flux relating to the layer immediately below. If the solids concentration is greater than some critical value, Xt, then the settling flux in that layer will affect the rate of settling in the adjacent layers, i.e., hindered settling behaviour will occur. These conditions are represented by

where i = 1 to n. Mass balances for the suspended solids within each of the various layers of the settler lead to the following set of defining model equations: In the top layer (i=l)

Above the feed zone layer elements (i=2 to m-1)

Feed zone (i=m)

Below feed zone (i=m+l to n-1)

278


The effective concentration factor of the settler can be calculated by dividing the quantity of the suspended solids in the underflow by the suspended solids in the feed.

Program In the program the settler is divided into 9 sections with the feed entering at layer 5.

Nomenclature A b G

Q V

Vi

X Z

Subscripts f m n 0

r S U

Cross section of the settler Empirical constant Mass flux Volumetric flow rate Velocities Empirical constant Solids concentration Thickness of one layer

m2 L g-1 g m h-I L-1 m3 h-I m h-1 m h-I g L-1 m

Feed Layer feed is introduced into Total number of layers Overflow Critical sludge concentration Settling Underflow

Exercises 1. Run the model at different effluent loadings and observe the level of the sludge in the settler. Find the limits of satisfactory operation.


279

2. Investigate how the volume and geometry of the settler influences its performance.

3. Investigate the behaviour of the settler with fluctuating flow rates and the ability of the settler to cope with shock loads.

Results

5UJ

20

UJ

m

.-m5

10 0

1

2

3

4

5

6

7

8

9

Layer number Figure 3.57. Biomass profile in the settler. Layer 1 is at the top and layer 9 at the bottom. Feed is introduced at layer 5.

Figure 3.58. Response of three layers to changes in inlet flow rate and biomass concentration at T=l 1 h.

References Vesilind, P. A. (1979) Treatment and Disposal of Wastewater Sludges, Ann Arbor Science Publishers. Vitasovic, Z. (1989) Continuous Settler Operation: A Dynamic Model, Dynamic Modelling and Expert Systems in Wastewater Engineering (eds.. Patry, G.G. and Chapman, D.), Lewis Publishers, USA.

280


3.4

Fixed Film Reactors for Wastewater Treatment

3.4.1

ROTATE - Model of a Rotating Biological Disc Reactor

System As described in Sec. 2.3.2.3 and shown in Fig. 3.59, three rotating biological disc reactors in series are used for the aerobic treatment of primary effluent. Each reactor consists of N discs with each disc having a diameter of D metres. Each disc is submerged to a depth R metres.

Figure 3.59. A series of rotating biological disc reactors.

Primary effluent enters the reactor and is degraded by both suspended biomass and by the biofilm on the discs. The degradation is assumed to follow Monod kinetics with one limiting substrate. It is assumed that the rotation of the discs provides sufficient oxygen for this process so that the oxygen supply rate is non-limiting and also that the rotation keeps the contents of each reactor well mixed. The geometry of the discs is shown in Fig. 3.60.

3.4 Fixed Film Reactors for Wastewater Treatment Processes

28 1

Figure 3.60. Construction of the rotating biological disc reactor.

Model The total area available for biomass growth is given by the geometry as A = 2 N x (Ro2 - Ru2)

The factor of 2 is introduced because biomass can grow on both sides of the discs. A mass balance on biodegradable substrate on the ith (i=1,2,3) reactor gives:

where Yfand Ys are the yield coefficients for the fixed and suspended biomass, V is the volume of the reactor and F is the volumetric flow rate of effluent. The specific growth rates pf and ps are given by: Pf = Pmax,f

S S

Ps = Pmax,s K~~ +

s

Suspended biomass is assumed to attach to the attached biomass on the disc at a rate proportional to the suspended biomass concentration. Biomass is also sheared off from the disc at a rate which is proportional to the thickness of the biofilm.

282


Mass balances on the fixed and suspended biomass in tank i give therefore:

In the above relationships, Kd is the first order death rate constant. Vf is the volume of the fixed biomass and dj is the thickness of the biomass in reactor i. The biomass thickness can be calculated from

where p is the density of biomass in the biofilm. The volume of biofilm is then given by:

Nomenclature A

D d F

Kd Kfl KS

Ksh

N R S V X Y

P P

Surface area of biofilm Diameter of disc Thickness of biofilm Volumetric flow rate Biomass death rate constant Flocculation or attachment rate Monod saturation coefficient Shear rate constant Number of discs per reactor Depth of disc submergence Substrate concentration Reactor volume Biomass concentration Yield coefficient Specific growth rate Biofilm cell density per unit volume

m2 m m m3 h-' hm h-' g y-3 h-

*

m g m-3 m3 g m-3 hg m-6


Subscripts f fl i S

sh

283

fixed biomass flocculation reactor number suspended biomass shear

Exercises 1 . Investigate the response of the system to fluctuations in flow rate and concentration. 2. Modify the program to investigate different operating strategies for example parallel and series operation, number and volume of reactors, recycling etc.

3. Extend the model to include nitrification by Nitrosomonas and Nitrobacter species (see ACTNITR example).

4.The biofilm in the last reactor sees only low concentrations of substrate and growth is consequently slow. The level of biomass that is maintained may therefore periodically not be able to adequately cope with shock loads. One possibility of overcoming this problem is to reverse the flow direction so that all reactors can maintain a viable biofilm. Investigate the effect on the system of reversing the direction of flow and varying the frequency of flow reversal.

Results Ism

n

i

,/

e m R.R Figure 3.61. Substrate transients in tanks 1 to 3. At t = 12h the inlet concentration was increased from 150 to 300 mg L-'.

Figure 3.62. Variation in biofilm thickness in tanks 1 to 3 for the same run.

284


References Gujer, W. and Boller, M. (1990) A Mathematical Model for Rotating Biological Contactors, Water Sci. Tech. 22, 1/2, 53-73., Hansford, G. S. (1979) Mathematical Models for the Rotating Biological Disc Reactor, Mathematical modelling in biology and ecology, Proceedings, Pretoria, South Africa (ed. Getz, W. M.), Springer.

3.4.2

- Model

TRICKLE

of a Trickle Filter

System The trickling filter is made up of a packed bed reactor with an upwards countercurrent flow of air and a downwards flow of liquid waste as shown in Fig. 3.63. Wastewater 1

1

1

1

1

7 1 1

1

1

1

1

1

1

3

A

6 7

1Alrflow

I

b l Effluent

Figure 3.63. Schematic diagram of a trickle filter.

The packing is traditionally stone, but recently plastic media have also been successfully employed. The ecology of trickling filters is complex and can involve bacteria, protozoa, fungi, rotifers, nematodes and fly larvae. The wastewater runs in a thin sheet over the packing, upon which a microbial film forms and grows. The micro-organisms in the film utilise the soluble organic matter in the waste but little or no degradation of suspended organic matter is


285

achieved. Trickling filters have been widely used to treat domestic sewage in small to medium scale installations owing to their low operating costs

Model The trickle filter is considered in terms of n horizontal layers each of thickness z. It is assumed that the concentration of substrate and the biofilm thickness within each layer is uniform. If the total volume of the trickle filter is V then the volume of each layer is V/n.

Substrate

-w

Microbial film

Products

Liquid layer

Figure 3.64. Transport processes in the biofilm.

It is assumed that the biomass grows according to Monod kinetics in the absence of any limitation due to oxygen. The transport processes in the biofilm are shown schematically in Fig. 3.64. Considering any layer n in the reactor, the biomass growth rate is given by

The thickness of the biofilm in each layer can be calculated from the biomass concentration and assuming each of N particles to be spherical with radius r:

286


The substrate in the effluent passes down through the bed and is utilised by the bacteria in the biofilm. Thus for any layer n, the substrate balance equation is obtained as

The free space between the packing for the flow of wastewater through the reactor bed is defined by Vn =

(1-&)V n

where E is the volume fraction of packing, and 1-E is the free volume of gas plus liquid. The volume occupied by the gas phase is neglected here. As the biofilm grows the free flow volume is reduced and can be replaced by

As a result, the flow velocity of the liquid through the pores will increase, and the performance of the reactor will decrease. It is assumed that at some critical biofilm thickness, the excess biofilm will slough off so the filter never blocks completely.

Program In the program the trickle filter is divided into 9 layers. For a quicker, but less accurate simulation, reduce the number of layers.

Nomenclature d Ks

n N

Q

r S V X

Biofilm thickness Monod constant Number of layers Number of packing particles Volumetric flow rate Radius of packing Substrate concentration Volume of trickle filter Biomass concentration

m kg m-3 m3 h-' m kg m-3 m3 kg m-3

287


P

Yield biomass/substrate Layer thickness Volume fraction of packing Maximum growth rate Cell density in biomass

Subscripts n

Layer number

Y z E

CLmax

m h-1 kg m-3

Exercises 1. Vary the model so that the packing size is varied through the bed and different layers have different sizes of packing. See how this influences the performance. Is it better to have smaller or larger particles at the top of the filter?

2. See how the trickle filter behaves with fluctuating flow rates and also how it copes with shock loads. 3. Extend the model to include diffusional effects in the biofilm. See FLOCl and FLOC2 as examples of this type of model.

Results 0.24

ri

:

d0~E 0

sa g

n

.

1

6

E

:"m

g g 0.08

0.08

3

v)

0.00

0.00 0

10 20 30 40 Time (hours)

50

Figure 3.65. Substrate concentration-time profiles for the different layers in the trickle filter.

1

2

3

4 5 6 7 8 9 Layer number

Figure 3.66. Substrate concentration profile down the trickle filter at times T=18, 30 and 50 hours.

Reference Grady, C. P. L. and Lim, H. C. (1980) Biological Wastewater Treatment: Theory and Applications, Pollution Engineering -and Technology 12. Chapter 18.

288

3.4.3


BIOFILM

- Biofilm Tank Reactor

System Active biomass can be retained effectively within continuous reactors by providing a carrier surface upon which a biofilm can develop. Examples are trickling filters (see TRICKLE) and fluidised bed reactors (see NITBED, FBR, and DENITRIF). In this example, biomass is retained within an aerated continuous tank reactor as a biofilm (Fig 3.67). Wastewater entering the reactor is mixed throughout the tank and the pollutant diffuses into the biofilm, along with oxygen, where it is degraded. The overall reaction rate in such a case will depend on the magnitude of the concentration gradients within the film and the concentration sensitivity of the reaction. The biofilm diffusionreaction process can be described by a diffusion model (see Section 1.12). As shown in Fig. 3.68, the biofilm is divided into segments for simulation purposes. Mass balances are written in terms of diffusion fluxes for each segment, and the diffusion fluxes at each position are expressed in terms of the corresponding concentration driving forces. The local reaction rate is expressed in terms of an appropriate kinetic model and the local concentration within the segment. The bulk phase can be coupled to the biofilm by means of an additional relationship representing the diffusion flux at the liquid-biofilm interface.

Figure 3.67. Continuous tank reactor with biofilm.

289


Figure 3.68. Finite differencing of the concentration profiles within the biofilm into segments 1 to N.

Model Multicomponent reaction within a biofilm, can be described by diffusionreaction equations. A component mass balance is written for each segment and for each component, respectively, where: (Accu;:ation

Diffusion

Diffusion

Production rate by reaction

) = ( rate in ) - ( rate out ) ( +

dS A M a dt = jn-l A - j , A + r S n A A Z Using Fick's Law (Sec. 1.12), the diffusional flux can be described by

giving

Thus N dynamic equations are obtained for each component at each position, within each segment. The equations for the first and last segment must be written according to the boundary conditions. The boundary conditions for this case correspond to the bulk tank concentration, S = So at the external surface of the biofilm, where Z = 0, and a zero flux at the far end of the biofilm corresponding to the wall of solid surface, on which the biofilm is immobilised, dS/dZ = 0 at Z = L.

290


The kinetics used here consider carbon-substrate inhibition and oxygen limitation. Thus, S 0 rS = vm K s + S + ( S 2 / K ~ ) KO+O A constant yield coefficient Yos describes the oxygen uptake rate

ro = yos rs

For the well-mixed continuous-flow liquid phase shown in Fig. 3.67, the balance equations for oxygen and substrate must account for the supply of each component both by convective flow and by gas-liquid transfer, as well as the diffusion rate into the biofilm. For substrate with continuous inflow and outflow from the reactor:

For oxygen transferred from the gas phase __-

do' dt

- KLa (0s - 00) - a Do 0 0A-z0 1

In the above equations, symbol a represents the area of biofilm per unit volume of bulk liquid. The diffusion rate into the biofilm is driven by concentration differences between the bulk liquid (0) and the outer biofilm segment (1).

Nomenclature Specific area perpendicular to the flux Area perpendicular to diffusion flux Diffusion coefficients Volumetric flow rate Diffusion flux Saturation constants Inhibition constant Oxygen transfer coefficient Biofilm thickness Dissolved oxygen concentration Saturation concentration for oxygen Reaction rate Substrate conc. of carbon source Volume of tank Maximum reaction rate

m-1 m2 m2 h-' m3 h-l g m-* h-l g m-; g hm g m-; g mg m-3 h-l g m-3 m3 g m-3 h-1

y-

29 1


Z

Yield for oxygen uptake Length of element

Subscripts 0 1 - 10 I 0 S n Feed

Refers to bulk liquid Refer to sections 1 - 10 Refers to inhibition Refers to oxygen Refers to carbon source Refers to section n Refers to feed

yos

m

Exercises 1. Observe the response time of the reactor for changes in operating parameters, SFeed, F and KLa. Is the biofilm always the slowest to respond? Relate the response to the diffusion time constant L2/D. 2. Experiment by varying the magnitude of the parameter KI. 3. Vary the ratio of substrate and oxygen in the bulk phase, Oo/So, above and below the value of Yos for a series of runs. Note that the steady-state penetration limitation of one or the other substrates depends on whether the bulk ratio is greater or equal to the stoichiometric requirement, Yos.

4.Vary the amount of biomass by changing the numerical value of a and note the influence on the reaction rate. 5 . Use the model to perform steady-state experiments to determine the apparent kinetics. Calculate the rates by sampling the substrate in the feed and effluent. Avoid the inhibition effect by setting K ~ v e r yhigh, while keeping So high enough to maintain zero order throughout the biofilm. By varying KLa run the reactor at various bulk dissolved oxygen concentrations. Plot the results as substrate uptake rate versus 00to determine the apparent Ks value for oxygen.

6. Repeat the Exercise 5 for substrate kinetics, while operating at constant 0 0 .

7. Use the model to simulate a respirometer to measure the oxygen uptake kinetics. Guidance can be obtained from the example RESPMET. Make the dynamic measurements for a series of dissolved oxygen concentrations. Set the diffusion coefficients very high to simulate the suspended biomass situation and compare the apparent Ks value obtained for oxygen.

292


8. Alter the model to include external film transfer. Show by simulation that its influence will increase as the value of the diffusion coefficients increase.

Results &a1

Figure 3.69. Response of oxygen profiles in the tank and at four internal film positions (1, 5, 8 and 10) to a change in feed flow rate from 0.014 to 0.06 at T=0.37.

Ipm

Figure 3.70. For the same run, substrate profiles for S at the same positions.

3.5

Nitrification of Wastewater

3.5.1

ACTNITR - Nitrification in a Single-Stage Activated Sludge System

System Nitrification takes place in a single-stage activated sludge system, as shown in Fig. 3.71 and is the process of ammonia oxidation caused by specialised organisms, called nitrifiers. Their growth rate is much slower than that of the organisms which oxidise organic carbon, and the nitrifying organisms can easily be washed out of the reactors by the sludge wastage stream (F5).In an activated sludge system when the organic load (F S O N ) is high, high biomass growth rates consequently require high wastage rates. Nitrification is not possible under such conditions since the concentration of the nitrifying organisms becomes very low.

3.5 Nitrification of Wastewater

293

Figure 3.71. Configuration and streams for the activated sludge system.

Model Dynamic balance equations can be written for all components based on the total system, the reactor and the settler, considering these as separate units. The settler is simplified to act as a well-mixed system with the effluent streams reflecting the cell separation.

Dynamic Balance Equations Organic substrate balance in the reactor:

Ammonia substrate balance in the reactor:

Reactor balance for the organic stabilising organisms:

Reactor balance for the nitrifying organisms:

Organic substrate balance in the settler:

294


v2is2 -

~-

F1 S1- F3 S2 - F4 S2

Ammonia substrate balance in the settler:

Balance for the organic stabilising organisms in the settler:

Balance for the nitrifying organisms in the settler:

Flow Rate Equations Recycle flow rate F2 = Fo R

(R = recycle factor)

Reactor outlet flow F1 = F2+Fo = F o R + F o Flow of settled sludge where C is the concentration factor. Flow of exit substrate F4 = F1 -F3 Flow of exit waste F5 = F3 - F2

Organisms Growth Rate Equations

295


Nomenclature A C F FO-5

K1 K2 N 0 R R1 R2 S V Y

PI P2

Ammonia substrate concentration Concentrating factor for settler Flow rate Flow rates, referring to the figure Saturation constant of organic stabilising organisms Saturation constant of nitrifying organisms Concentration of nitrifiers Concentration of organic stabilising organisms Recycle factor Growth rate of organic stabilising organisms Growth rate of nitrifying organisms Organic substrate concentration Volumes Yield coefficients Specific growth rate of organic stabilising organisms Specific growth rate of nitrifying organisms

kg m-3 -

m3 h-l m3 h-l kg m-3 kg m-3 kg m-3 kg m-3 -

kg m-3 h-l kg m-3 h-I kg m-3 m3 kg kg-1 hh-

Subscripts The flow and concentration indices are as follows: 0 1 2 3 4 5 max

Refers to feed and initial values Refers to reactor and organic oxidation Refers to settler and ammonia oxidation Refers to recycle Refers to settler effluent Refers to sludge wastage Refers to maximum

Exercises 1. Investigate the response of substrates organic carbon (S2) and ammonia (A2) to a change in feed rate (Fo) and feed concentrations (So, Ao). 2. Noting that pmaxfor the nitrifiers is 0.04 h-1, set the feed rate to wash the nitrifying organisms out, but not the organic stabilising organisms, for which pmaxequals 0.5 h-1.

296


3. Alter the program to allow the wastage flow to maintain the total biomass constant. What operating variables influence the rate of wastage? Explain how the wastage rate influences the nitrification. 4. How might the influence of low dissolved oxygen be brought into consideration into the modelling of this situation? Write the necessary balances and kinetic expressions. How many new parameters does this change introduce?

Results

0

Time (hours) Figure 3.72.

Ammonia (A) and substrate (S) concentrations in the reactor (1) and settler (2). Flow rate = 20 m3 h-l.

3.5.2

5 10 Time (hours)

15

Figure

3.73. Effect of varying the volumetric flow rate ( 5 to 40 m3 h-l) on the ammonia concentration in the reactor.

AMMONOX - Continuous Nitrification with Immobilised Biomass

System The consecutive oxidation of ammonium ions to nitrite and then to nitrate ions is an important environmental process. The organisms grow very slowly, but continuous nitrification can be carried out effectively using immobilised biomass, as in trickling filters (see TRICKLE), rotating biological contactors (see ROTATE) and packed (see MOLASSES) or fluidised beds (see NITBED, and FBR). In this example, it is assumed that the biomass within the system remains essentially constant and that the reactor behaves as a well-mixed continuous tank (Fig. 3.74).


297

Figure 3.74. Continuous tank reactor for nitrification.

The sequential oxidation of NH4+ to N 0 2 - and NO3- proceeds according to the following reaction sequence: 3 NH4+ + 5 0 2

+ 1

N02-+ H20 +2H+

NO2-+502

-+

NO3-

The overall reaction is thus NH4+ + 202

+

NO3-

+ H20 + 2H+

Model The apparent kinetics of this process can be approximated by relatively simple homogeneous reaction kinetics which follow the form of a double MichaelisMenten or Monod kinetics:

The component mass balances are represented by: For NH4+ (S 1)

298


For N02- (S2) dS2 dt

-=

Q v (S2,o - S2) + rl - r2

For NO3- (S3)

For oxygen (CL)

The stoichiometry for the first reaction step is:

and for the second reaction step:

Nomenclature CL CLSand CL* K KLa

Y

Dissolved oxygen concentration Saturation oxygen concentration Saturation constants Oxygen transfer coefficient Volumetric flow rate Reaction rates Concentration of NH4+ - N Concentration of NO2- - N Concentration of NO3- - N Volume of tank Maximum oxidation rates Yield coefficients

Subscripts 0 1,293 0 S

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

Q

r S1 s2 s3

V

Vm


299

Exercises 1. Run the simulation at differing values of the volumetric flow rate and determine whether oxygen or substrate subsequently becomes rate limiting. 2. Assume maximum rates of reaction and calculate the KLa value required to maintain the dissolved oxygen above the sensitive Ks level. Verify this with by simulation. 3. Investigate whether oxygen limitation has any influence on the nitrite level of the effluent. Explain the results. 4. Operate with only nitrite in the feed, and determine the value of the kinetic parameters Ks2 and vm2 for this substance by analysing the data from several differing steady-state runs.

5. Determine the required operating conditions to maintain the nitrite concentration below 2 mg-N L-' with an entering ammonium ion concentration of 50 mg-N L-l. 6. Stop the flow rates and observe how the oxygen uptake rate and dissolved oxygen can be used to detect the onset of complete consumption of substrate.

Results

26.0

0.0

Figure 3.75. Startup and approach to steady-state operation for the N components.

Figure 3.76. Dissolved oxygen profile during the same run.

3 00

3.5.3


AMMONFED - Fed-Batch Nitrification with Immobilised Biomass

System In this case, the model equations describe the continuous feeding of a wastewater stream containing ammonia to a semi-continuous or fed-batch tank reactor (Sec. 1.10.1.2). The increase in volume (total accumulation of mass) in the reactor is shown schematically in Fig. 3.77.

Figure 3.77. Semi-batch reactor for nitrification with model variables.

Model The same nitrification kinetics are used as in the previous example AMMONOX. For semi-batch operation, the mass balance equations become: Total balance dV - = Q dt For the nitrogen components i

where i = 1 to 3. The term C r i V represents the net sum of the individual reaction rates, as already shown in example AMMONOX.


30 1

For oxygen the mass balance is d(V CL) = KLa (CL*-CL) V -Y1 r1 V - Y2r2 V dt here the relatively small quantity of dissolved oxygen entering in the feed is neglected, further the value of KLa is assumed to be independent of volume. Thus the mass quantities [VSi] and [VCL] are calculated and divided by V at each time instant to obtain the concentration of each component.

Nomenclature Dissolved oxygen concentration Saturation oxygen concentration Saturation constants Oxygen transfer coefficient Volumetric flow rate Reaction rates Concentration of NH4+ - N Concentration of N 0 2 - - N Concentration of NO3- - N Volume of liquid Maximum degradation rates Yield coefficients

Subscripts 0 3 0 1 9 2 ,

S

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

Exercises 1. Vary the magnitude of KLa, and hence vary the oxygenator supply capacity of the system, and note the influence of oxygen limitation on both uptake rate and on the N02- levels. 2. Calculate the value of KLa required to maintain CL above the value of Kol . Try this value in a simulation.

3. Vary the Kol/K02 ratio and notice the effect on N02- levels at low oxygen conditions.

302


4. Vary F interactively to maintain S well above Ks,and plot the reaction rates

and feed rates versus time. 5. Startup in batch mode and then switch to fed-batch operation. Try to minimise the time required to reach steady state.

6. Alter the program so that the initial conditions can be changed to simulate a repeated fed-batch operation. Here the values of [VX], [VP] and [VS] need to be reset using the ratio factor V(initial)N(total). Change the output statement to obtain these values. See the example REPEAT.

Results

9.0

Figure 3.78. Profiles of ammonia and nit-

Figure 3.79. Profiles of volume and dis-

rite during fed-batch operation.

solved oxygen during the same run.

3.5.4

NITBED Reactor

- Nitrification in

a Fluidised Bed

System As explained in the example AMMONOX, nitrification is a process that benefits greatly from biomass retention, owing to the relatively slow growth rates of the nitrifiers. In this case, a fluidised biofilm sand bed reactor for nitrification, as investigated by Tanaka et al. (1981), is modelled as three tanks-in-series with a recycle loop (Fig. 3.80). With continuous operation, ammonium ion is fed to the reactor, and the products nitrite and nitrate exit in the effluent. The bed expands in volume because of the constant circulation flow of liquid upwards through the bed. Oxygen is supplied external to the bed in a well-mixed gasliquid absorber.


303

Figure 3.80. Biofilm fluidised-bed recycle loop reactor for nitrification.

The nitrification kinetics involve a sequence of two oxidation steps, as discussed in AMMONOX. Both steps are influenced by dissolved oxygen and the corresponding substrate concentration.

Model The model balance equations are developed by considering both the individual tank stages and the absorber. Component balances are required for all components in each section of the reactor column and in the absorber, where the feed and effluent streams are located. Although the reaction actually proceeds in the biofilm phase, a homogeneous model apparent kinetics model is employed, which is justified by its simplicity. In the absorber, oxygen is transferred from the air to the liquid phase. The nitrogen compounds are referred to as S1, S2, and S3, respectively. Dissolved oxygen is referred to as 0. Additional subscripts, as seen in Fig. 3.80, identify the feed (F), recycle (R) and the flows to and from the tanks 1, 2 and 3, each with volume V, and the absorption tank with volume VA. The fluidised bed reactor is modelled by considering the component balances for the three nitrogen components (i) and also for dissolved oxygen. For each stage n, the component balance equations have the form

3 04


Similarly for the absorption tank, the balance for the nitrogen containing components must include the input and output of the additional feed and effluent streams, giving dt The oxygen balance in the absorption tank must account for mass transfer from the air, but neglects oxygen kpply and removal by the feed and effluent streams, which have relatively low rates. This gives

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

The oxygen uptake rate is related to the above reaction rates by means the constant yield coefficients, Y1 and Y, according to

The reaction stoichiometry provides the yield coefficient for the first step

and for the second step Y2 = 1.1 mg 02/(mg "02)

305


Nomenclature F

FR KLa K K1 K2

0 0 s and O* OUR r S V VA Vm

Y

Subscripts 1,2, 3 1,293 A F ij m 0 1 and 0 2 Sl, s2 S and *

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

L h-1 L h-' h-l mg ~ - 1 mg L-1 mg L-' mg L-1 mg L-1 mg h-1 mg L-' h-' mg L-1 L L mg L-l h-' mg mg-1

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

Exercises 1. Assuming maximum reaction rates, choose a value of F and SlF, and estimate the outlet substrate concentration at steady state. What oxygen supply rate would be required? Verify by simulation. 2. Choose values of KLa and VA to give sufficient oxygen transfer capacity; run the simulation and compare the results obtained with your prediction. 3. Vary the operating parameters and interpret the results in terms of the axial oxygen gradients in the reactor column.

4. Vary the rates of substrate and oxygen supply, and note the interaction of both these factors on the outlet values of S13 and 0 3 .

306


5. Create a differential reactor by either increasing FR, decreasing V or decreasing Vm. Show that these conditions allow estimating the reaction rate by the average of the inlet and outlet concentrations. Conduct simulated experiments for varying average concentration levels. 6. Choose the conditions to separate the concentration effects of the oxygen and ammonia substrates. Verify the kinetics of the reaction from the simulated steady-state data by running the reactor at various ammonia and dissolved oxygen levels. Make a linear plot of the rate versus concentration data from each run to verify the assumed Monod kinetics. Obtain the kinetic constants from these plots. 7. Operate the reactor to measure the influence of the nitrite concentration on the kinetics. 8. Operate the reactor under oxygen-limiting conditions and investigate the influence of differing values for Kol and K02 on the residual nitrite concentration. Study the influence of KLa on the N02- levels.

Results

3. en 1.0

Figure 3.81. After startup, the dissolved oxygen became limiting. KLa was changed from 40 to 200 h-l at 8 h. The feed was decreased from 0.5 to 0.1 L h-l at 20 h.

I F

Figure 3.82. The response of the reactor effluent to the changes in the same run.

Reference Tanaka, H., Uzman, S. and Dunn, I. J. (1981). Kinetics of Nitrification Using a Fluidized Sand Bed Bioreactor with Attached Growth, Biotechnol. Bioeng., 23, 1683.

307

3.6 Primary Treatment of Wastewater

3.6

Primary Treatment of Wastewater

3.6.1

SEDIMENT - Removal of Solids in a Sedimentation Tank

System Settling tanks are used to remove grit, sand and other suspended-organic materials from the wastewaters in order to reduce the load on the subsequent oxidation processes. The rate of settling depends on the size and the density of the particles and also on the degree of flocculation (see Section 2.2.3), as shown in Fig. 3.83. Small particle carried over

-

\ \

\

h

Flow out

\ \

\

Flocculation \

\

\

Large particle settles

Figure 3.83. Particles settling in a sedimentation tank.

Model A particle falling in an infinitely deep liquid will accelerate until the gravitational force equals the frictional drag force, and it will then travel at constant velocity. For settling at constant velocity:

Gravitational force = Frictional force

308


The gravitational force depends on the density difference between the solid, ps, and that of the liquid, p1, phases and the volume of the particle (Vp), where

The frictional force depends on the drag coefficient, CD, the cross-sectional area, A, and the velocity, v of the particle, where

Assuming that all particles are spherical

The value of the drag coefficient depends on the Reynolds number of the particle, where CD = 0.4 for Re > lo3

c D = 24R e + G CD =

+ 0.3 D 24

for l hcrit. A rearrangement of the above equations gives:

where A is the tank surface area. In other words, the settling velocity must be greater than the tank overflow rate and is independent of the tank depth. In this example the particles are of uniform density but have variable diameters which are modelled by a random number generator using the special ISIM function "RAND". D = ( 1.0 + 2( RAND(0 to 1 )

- 0.5 )

)O.OOl

This gives a random variation in particle diameter of 0 to 2 mm. The probability of particle flocculation may be given any value between 0 (no flocculation) and 1 (complete flocculation). If a particle meets another particle, its diameter and weight are then assumed to double so that the particle settling velocity increases. As a particle can flocculate more than once, the settling velocity can therefore increase even further. The particles are assumed to be randomly distributed in the feed which enters in the top ten percent of the tank volume. This effect is again modelled using a random number generator:

zo = h (0.9 + RAND(0 to 0.1))

Program The random number generator requires an initial seed value. The program asks the user to provide a value for the seed prior to the start of the simulation. This is done by typing VAL SEED=x, where x is any real number. The user must then type GO (or G) to start the simulation. A DO loop causes the program to be rerun with different initial conditions. The variable "number" determines the number of times the program is rerun. For each run is calculated the percentage of particles that are removed by sedimentation.

3 10


Nomenclature A

CD D F g h L

Q

Re t V

V VP

W Z

P P Subscripts 0 f g 1 S

Particle cross-sectional area Drag coefficient Particle diameter Force acting on particle Gravitational acceleration Height of tank Length of tank Volumetric flow rate Reynolds number Time Particle velocity Volume of tank Volume of particle Tank width Vertical position of particle Liquid viscosity Density

m2 m

mN m s-2 m m m3 s-1 S

m s-1 m3 m3 m m m s g-1 g m-3

Initial conditions Friction Gravitation Liquid phase Solid phase

Exercises 1. Vary the degree of flocculation and see how this influences the sedimentation of tlhe particles. 2. Vary the tank dimensions, keeping the total volume constant, and see how the tank geometry influences the performance.

3. For a constant tank geometry vary the volumetric flow rate. What is the optimum tank geometry?

311


Results Muen

1.m

Y

Y

7.58

7,58

5.m

5.W

2. a

2.a

8. en

E

0

Figure 3.84. Trajectories of particles in a sedimentation tank, showing some particles that are carried over, some that settle, and some that flocculate so enhancing their settling.

Figure 3.85. Trajectories of particles as in the previous run but with the probability of flocculation PROB increased to 0.05.

Reference Eckenfelder, W. W. (1989) Industrial Water Pollution Control, Chapter 3, McGraw-Hill Series in Water Resources and Environmental Engineering.

3.6.2

LAGOON - Aerated Lagoon for the Treatment of Industrial Wastewaters

System Aerated lagoons consist of basins 2.5 to 5 m deep with a surface area of more than 10,000 m2 (Fig. 3.86). Oxygenation is accomplished by mechanical or diffused aeration and via induced surface aeration. The dissolved oxygen and suspended solids concentration are maintained uniformly throughout the basin. The major objectives of aerated lagoons are to convert soluble organic matter into insoluble microbial cells and thus to achieve the stabilisation of the organic material. The biomass can subsequently be separated in settling basins.

312


0

A

Aerator Figure 3.86. Aerated lagoon.

Model In this example, various industrial wastewaters are held in an aerated lagoon and are degraded biologically according to Monod kinetics: dS -

Pmax

dt = - Y (K,

XS + S)

It is assumed that the contents of the lagoon are well mixed and that there is no oxygen limitation.

Table 3.1. Kinetic parameters for the biological treatment of wastewaters.

Domestic wastewater

0.5

100

0.0027

0.6

Shellfish processing

0.43

96

0.058

0.58

Yeast processing

0.038

680

0.0033

0.88

Plastics processing

0.83

167

0.0033

0.3

Skimmed milk

0.1

100

0.002 1

0.48

313


The kinetic parameters listed in Table 3.1 have been found for various industrial wastewaters (Bailey and Ollis, 1986).

Program The value of pmaxfor skim milk in the program was set arbitrarily to 0.05 h-'.

Nomenclature

Kd Ks

S T X Y Pmax

Death rate constant Monod constant Substrate concentration Temperature Biomass concentration Yield biomasshbstrate Maximum growth rate

hmg L-1 mg L-1 OC

mg L-1 h-

Exercises 1 . Run the model using the differing values of the kinetic parameters, as shown in Table 3.1 by varying the parameter WASTE, which identifies the type of wastewater, with the VAL command. 2. Vary the holding time of the lagoon (TFIN) and investigate how this effects the rate of treatment of the wastewater. 3. Modify the program to model the continuous operation of an aerated lagoon. Compare the performance of a single large tank to two or more lagoons in series with the same total volume as the single tank.

4. Aerated lagoons are subject to temperature variations, which influence the kinetics of the biomass growth rate and the rate of substrate utilisation. P m a x values, in particular, have been found to be temperature sensitive. This temperature dependence has been modelled by a modified van't HoffArrhenius equation: Pmax = ~ m a x ( 2 0"C) 1.08(T-20) Investigate the performance of the lagoon as the temperature varies.

3 14


Results

E, 158

--

lime (hours) Figure 3.87. Operation of the lagoon for a variety of wastes as follows: a - Plastics processing, b - Domestic waste, c - Shellfish processing, d - Skimmed milk, e - Yeast processing.

Figure 3.88. Shellfish waste lagoon operation for differing initial waste concentrations.

References Gellman, I. and Berger, H. F. (1968) Advances in Water Quality Improvement, Vol 1, University of Texas Press, Austin, Texas, USA. Grady, C. P. L. and Lim, H. C. (1980) Biological Wastewater Treatment: Theory and Applications, Chapter 15, Pollution Engineering and Technology 12, Marcel Dekker. Bailey, J. E., and Ollis, D. F. (1977) Biochemical Engineering Fundamentals, McGraw-Hill.

3.7 Tertiary Water Treatment Processes

3.7

Tertiary Water Treatment Processes

3.7.1

FILTER - Tertiary Water Treatment by Filtration

3 15

System Rapid filtration is frequently used as a tertiary treatment following biological wastewater treatment. In this, the water is passed through a packed bed of particles and suspended solids are removed primarily by straining but also possibly by adsorption. Sand (0.35 to 0.6 mm) and coal (1.0 to 1.7 mm) are the major filter media particles, but garnet and metal oxides have also been used. Filters may employ only one media type or may consist of multiple layers with the coarser material at the top of the filter, as shown in Fig. 3.89. Filtration is terminated either when the loss of pressure head through the bed exceeds or equals the total pressure driving force or when the effluent concentration exceeds some permitted value.

Figure 3.89. Schematic diagram of a rapid filter.

316


Model The filter is composed of particles having a uniform diameter d and has a total depth L with a cross-sectional area A. The volumetric flow rate of influent is Q, hence the superficial velocity through the filter is given by

The rate of accumulation of suspended particles in the filter will vary both with vertical position and with time. In order to model the filter, it is considered in terms of a number of elements (N) each of uniform cross-sectional area (A) and height (r=L/N). In each element, conditions are assumed to be completely uniform. The greater the number of elements, the greater is the accuracy of the model but the slower the run time. In this model only five elements are taken so the simulation takes an acceptable time to run. A mass balance for suspended solids (C) in element n gives:

where k is the filtration coefficient. The efficiency of the filter changes with respect to time owing to the increasing accumulation of deposits in the pores of the filter. The following expression, based on the change of velocity and geometry of the pores, is employed to predict the change in k (Ives, 1978):

k = ko(1

+ 11.8 dO.2 07 )1.5 (1 -7) 0.75 (I -$)O (T

P

where

p

0.95 = d0.61 v0.24

and (TO, the saturation value of deposit is given by: (T"

=

0.44 (1 + v p 7 5

The rate of increase in the concentration of material suspended in the filter must equal the rate of loss of material from the effluent:

317


The initial filter coefficient, ko, is given by:

ko

5.9 x 10-3 = (d1.35 ,0.25)

A clean filter shows a loss of pressure head which can be estimated by the Kozeny-Carman equation: 2

Ho = 5 p L & ( F )

2

($)

The build up of deposit in the pores gives rise to an increase in the head loss, that has been found experimentally (Mohanka, 1969) to be:

The deposit in each section is calculated from: kCv dt =

An average value of

(5

p

can be used in the estimation of the head loss, where N

The available driving force is estimated by: Hlim = 0.8 (L + W)

where L is the depth of the filter and W is the depth of the standing water on top of the filter. The fraction 0.8 is introduced to account for head losses in underdrains and piping. H increases until it reaches a value of Hlim, after which no further filtration occurs.

Nomenclature A C

d

Cross-sectional area of filter Concentration of suspended solids in influent Diameter of filter particles

cm2 g cm-3 cm

318 f g

H k L n N

Q r

V

w P

P P 0


Filter porosity Acceleration due to gravity Pressure head Filter coefficient Depth of filter Element number (1 is at top of filter) Number of elements Volumetric flow rate of effluent Height of one element Liquid velocity through filter Depth of standing water on filter Exponent for filtration coefficient Viscosity Particle density Specific deposit (DP in program)

cm min-2 cm cm-1 cm

cm3 min-1 cm cm min-1 cm

-

g cm-1 min-1 g cm-3

Subscripts

*

0 lim

Saturation value Initial value Limiting value

Exercises 1. Vary the height and cross-sectional area of the filter and see how it influences the filter performance. 2. Study the response to changes in effluent concentration and flow rate.

3. Investigate the possibility of using different size particles in different sections of the filter. In this case the head loss in each element of the filter will need to be calculated separately and then summed to give the total head loss.

319


Results

Time (hours) Figure 3.90. Head loss and clarification with time. Parameters are as in the program.

Time (hours) Figure 3.91. Concentration of suspended solids in filter sections 1 to 5.

References Ives, K. J. (1978) Optimisation Model for Tertiary Treatment Rapid Filtration, in Mathematical Models in Water Pollution Control (ed. James, A.), Wiley. Mohanka, S. S. (1969) Theory of Multilayer Filtration, J. San. Eng. Div. Proc. Amer. SOC.Civ. Engrs. 95, (SA6), 1079.

3.8

Sludge Disposal and Processing

3.8.1

COMPOST - Microbial Kinetics in a Continuous Compost Reactor

System Composting is the decomposition of moist organic material by a mixed microbial population and occurring at elevated temperature. High temperature is caused by the heat generated from the aerobic and exothermic microbial reactions. Material is fed into the composting reactor on a continuous basis, where the contents are assumed to be completely mixed. The reactor is cylindrical in shape. Final material is withdrawn at a constant rate from the

320


mixed volume. The feed is activated sludge with wood chips as a bulking agent.

Figure 3.92. Continuous mixed compost reactor.

Model The model here is taken from the work of Haug (1980).

Feed Composition The feed to the reactor consists of a mixture of activated sludge waste and wood chips, which act as bulking agent. Both materials contain a fraction of solids (S) and water (1-S). Part of the solids is organic or volatile (V), and the rest is inert ash (1-V). Of the volatile fraction, a fraction is biodegradable (k), and the rest is non-biodegradable (1-k). Thus the feed can be expressed in terms of the total wet density of the sludge, X, and the wood chips, BULK, and can be given in terms of the concentrations of its respective components, as shown in Table 3.2. The fractional values for the sludge ( S , V and k) and for the bulking agent (SB, VB and kB) are all in the range of 0 to 1. The ash components and the non-volatile solids are not consumed during the composting process and are found unchanged in the material leaving the reactor. Using Table 3.2, the total nonbiodegradable volatile solids (NVsolids) content in the feed (sludge and the bulking agent) becomes NVsolids = (1- k) V s X

+ (I-

kB) VB SB BULK

3.8 Sludge Disposal and Processing

32 1

The solid mass fraction of the mixture, SM, is calculated as follows: SM =

S X + SB BULK (X + BULK)

Table 3.2. Expressions for the concentrations in the compost feed. Quantity

(kr: m-3) Total wet density Biodegradable volatile solids Non-biodegradable volatile solids Ash content Water content

Expressions for sludge

Relations for wood chips

X

kV SX

BULK kB VB SB BULK

(1 - k) V S X

(1 - kB) VB SB BULK

(1 - V ) s x (1 - S) x

(1

- VB) SB BULK

(1 - SB) BULK

Biodegradation It is assumed that the oxidation of the biodegradable volatile solids of the sludge (BVS) and the bulking agent (VBULK) follows first order degradation kinetics as rd = & BVS + 0.2 Kd VBULK with rate constant Kd (day-l), The value of Kd for wood chips has been estimated to be 0.2 times the value for the sludge. The magnitude of the rate constant is affected by the temperature, moisture content and oxygen concentration of the reactor. The rate of degradation increases with temperature, reaching a maximum at around 70°C and then falling rapidly to zero by about 80°C as all the microbes are killed off. This can be modelled by the following expression: Kd

= 0.0126

( (1.066)T-20 - (1.21)T-60)

A correction factor F1 is introduced into the degradation rate equation to account for the effect of moisture content, 1-sM. The drier the compost the slower is the rate of reaction. An empirical fit to experimental data gives the following expressions:

322


The further correction factor F02 is introduced to account for the effect of oxygen concentration on the degradation rate. Low oxygen concentrations give rise to reduced reaction rates. Experimental data was best fitted by a Monod type relationship with a half maximum rate constant of 1% oxygen by volume: PO2 F02 = 1.0+po2 The volume of the free air space is thought to influence the reaction rate if it is too low, but in this case it is assumed not to be a limiting factor, owing to the effect of the bulking agent, which provides additional free air space. Combining the above parameters, the following expression for the rate constant is obtained: Kd = F1 F02 0.0126 ( (1.066)T-20 - (1.21)T-60)

Stoichiometry The sludge is assumed to be oxidised according to the following reaction stoichiometry C10H1903N + 12.5 0 2 + 10 CO;! + 8 H20 + NH3 The bulking agent is assumed to be primarily cellulose and is therefore assumed to be oxidised according to the following equation:

Allowing for the appropriate molar mass of each component, the mass uptake rate of oxygen (kg m-3 day-1) can then be related to the rate of degradation by 0 2 u p = &BVS

400 201 + 0.2 & VBulk 1192 80

The production rates of water, carbon dioxide and ammonia (kg m-3 day-') can also be calculated as


323

Mass Balances for the Volatile Solids The rate of change in the volatile solids contents, BVS and VBULK, in the wellmixed reactor can be described by mass balance equations. For the sludge volatiles: dBVS V O ~7 = F (BVSin - BVS) - Kd BVS V O ~ and for the biodegradable volatiles in the wood chips, '01

dVBULK = F (VBULKin - VBULK) - 0.2 €&VBULK VOI dt

Water Balance Water enters the system in the feed and in the bulking agent as well as in the form of water vapour in the incoming air. The mass flow rate of water vapour in the incoming air, AFRi, (kg day-'), is related to the temperature of the air (TAir) and the vapor pressure of the water, PV (mm Hg) by the following relationship: PV WVi, = 0.622 AFRin (760 - pv) where the partial pressure of the water, PV is calculated via the humidity of the air, @ (kg waterkg dry air), by

and by the Antoine vapour pressure relationship

--

Pvsat

108.896 - 2238/(Tair + 273)

The mass rates of water entering in the feed and produced by reaction can be found by the previously obtained equations. Water leaves the system as wet solid and as water vapour in the air and is also lost by evaporation. The mass rate of water leaving in the air stream (kg day-') can be estimated by the following expression: wvOUt

= o*622AFROUt

PV0"t (760 - pVout)

where PVout is calculated by

In the above equation, PVSatis calculated based on the reactor temperature and F1 is calculated as shown previously. The use of the factor F1 in the above

324


equation is entirely empirical, but has been found give a reasonable fit to experimental data. The rate of loss of water due to evaporation (kg day-') is given by EVAP = A r e v p where A is the surface area of the reactor, rev is taken to be a uniform volumetric rate of evaporation per unit surface area, and p is the density of water. Rainwater is assumed not to be important. A mass balance equation can therefore be written for the water in the reactor:

Gas Balances It is assumed that the incoming air contains by volume 21% oxygen, 0.05% carbon dioxide and that the remainder is nitrogen. Oxygen is consumed by the micro-organisms and carbon dioxide and ammonia are produced according to reaction stoichiometry. The volumetric flow rates of the individual gases are calculated from their mass flow rates, Mi, and molecular weights, MWti, and applying the ideal gas law, assuming the gases are at atmospheric pressure and are at the temperature of the compost. Hence: Mi R T P MWti

Q = -

Heat Balances Heat enters the system in the gas and in the solid feeds and is produced by the exothermic heat of reaction. Heat leaves the system in the outlet gas and in the solids, via conduction to the surroundings and is also lost during periodic turning. A dynamic heat balance can be employed to predict the temperature of the reactor:

where: Hin = Hmix

b i n + HH20in Hbio i-

Hout = HSout + HH2Oout + HGASout + HWVout + Hvap + Hevap + Hcond The enthalpies or heat energy rates, Hi (kcal day-I), for the various components in the solids feed were all estimated, based on a reference temperature of O O C , and using the relationship:

325


where Mi is the mass flow rate of each component and Cpj the respective specific heat capacity. The values for Cpi are given in Table 3.3

Table 3.3. Heat capacities for composting. Substance

Heat capacity kcal kg-1 OC-'

Water Solids Dry gases Water vapour

1 .oo 0.25 0.24 0.444

The above relations are applied to calculate the individual energy inputs as:

The energy outputs in the solids and gas streams are calculated assuming the gases and compost have achieved the same temperature as the compost. The energy output terms are: HSout = F BVS Cp T

The heat produced by the biological degradation of the organic material is estimated as

where Z, the heat of combustion is assumed to be 6766 kcal kg-1 for sludge and 3739 kcal kg-I for the bulking agent.

326


The heat generated by mixing (kcal day-') can be estimated as

Here a is the fraction of the day that the motor is in operation, and W is the power of the motor. The quantity of water vaporised can be calculated as the difference between the input and output of water vapours from the system. The energy required to vaporise this quantity of water is given by Hvap = L (WVout - WVin)

kcal day-'

The latent heat of vaporisation (kcal kg-I) for water, L, is temperature dependent and is given by L = 597.0 - 0.581 T where T is the temperature in the reactor. The heat lost from the reactor by transfer through the walls can be estimated by: Hcond = u A (T - Tair) where U is the overall heat transfer coefficient and is taken as, as 36 kcal m-2 day-' OC-l. The area for heat loss, by conduction A, is assumed to comprise the base and walls of the reactor which, for a cylindrical reactor is equivalent to an area A = 2

Vol

+ 2 (K h VOI)'.~

Reactor Performance The dry weight of material leaving the reactor is given by: DWTout = VBULKOut + BVSout+ ASH + NVsolids and the performance can be represented by DWTout + WATERout DWT;, + WATER;, 100

Nomenclature A (Aland A2)

AFR VAFR

Area Air mass flow rate Volumetric air flow rate

m2 kg day-' m3 day-'

327


ASH BULK BVS CPi D DWT EVAP F F1 F02 H h H20 k kB

Non-biodegradable content Bulking agent density Biodeg. volatile solids in sludge Specific heat capacity of component i Diameter of reactor Total dry weight (ash +volatiles) Rate of loss of water by evaporation Volumetric flow rate of solids Correction factor for moisture content Correction factor for oxygen levels Heat energy rates Height of reactor Water content Biodegradability coefficient for sludge Biodegrad. coeff. for bulking agent Degradation rate constant Kd L Latent heat of evaporation M Mass flow rate of gas MIXtime Fraction of day reactor is mixed Mwt Molecular weight NVsolids Non biodegradable volatile solids P Pressure Partial pressure 0 2 Po2 PV Vapour pressure Volumetric flow rate of gas Q R Universal gas constant (for air = 287) rev (EVAPrate) Rate of evaporation Rate of BVS + BULK degradation rd Solids mass fraction of feed S Solids mass fraction of bulking agent SB Solids fraction SM Temperature T (TEMP) t Time Tau Residence time U Heat transfer coefficient V Volatile solids fraction of sludge Volatile solids fraction, bulking agent VB Volumetric prod. and uptake rates Vi VBULK Biodeg. volatile solids in bulking agent Volume of reactor Vol W (POWER) Power of the mixer WATER Water content wv Water vapour mass flow rate Wet density of the solids X Z Heat of combustion

kg m-3 kg m-3 kg m-3 kcal kg-I OC-' m kg m-3 kg day-' m3 day-' kcal day-' m kg m-3 day-' kcal kgkg day-' Da kg m-3 Pa mm Hg mm Hg m3 day-' J kg-' K-' m3 m-2 day-' kg m-3 day-' -

OC days days kcal m-2 day-' "C-1 m3 m-3 day-' kg m-3 m3 kcal day-' kg m-3 kg day-' kg m-3 kcal kg-

328

a P

@

Subscripts bio B cond evap H20 in M mix prod out S

sat sol UP

VaP


Motor operation fraction Density of water vapour Humidity

kg m-3 kg waterkg dry air

Biological degradation Bulking agent Conductivity Evaporation Water Inlet values Mixture Mixing Produced Outlet values Solids Saturation Solids Uptake Vaporisation

Exercises 1. Vary the air flow rate of air. How does this influence the temperature and rate of degradation of the sludge.

2. Modify the model to include a variable air flow rate such that this can be regulated to maintain constant temperature or a temperature close to optimum. 3. Vary the sludge residence time to see the effects on reactor performance.

4.Vary the proportion of bulking agent to sludge to see the influence. Instead of using a bulking agent, modify the model to include a recycle of compost. 5 . Investigate the influence of the reactor geometry and material of

construction (i.e., heat transfer coefficient, U) on the performance.

6 . Does covering the reactor to prevent water losses by evaporation (and gains from rainfall). Do surface heat losses have an effect on the performance?

329


Results

-

A

20

08 c -m o p Q '5 D 0

Carbon dioxide

- 'a

?!

2

.= 10.

p f

0

E

Ammonia

I

0

10 Time (days)

60 40

Oxygen

0

80

I-

20

Figure 3.93. Percentage composition of the exit gas during startup and at steady state. Remainder of the gas not accounted for is assumed to be nitrogen.

20

O0.0

1.o Time (days)

2.0

Figure 3.94. Temperature profiles for different air flow rates (m3 day-'). If the air flow rate is too low, then the microbial degradation is very slow and the temperature rise is slow. If the air flow rate is too high, then the rate of cooling is very high and the compost does not heat up.

Reference Haug, R. T. (1980) Compost Engineering: Principles and Practice, Ann Arbor Science, USA.

3.8.2

WINDROW Process

- Batch

Windrow Compost

System The windrow system is an agitated solid bed composting system. Mixed compost material is placed in rows and is turned periodically by mechanical equipment. Conventional windrows rely on natural aeration and gas exchange during turning, but forced aeration processes have also been widely used.

Model In this simulation model (Haug, 1980) the batch composting of dewatered activated sludge (with recycled product as an amendment) is considered in a windrow of approximately triangular cross-section. The same kinetic and

330


physical model as in the example COMPOST is used. The composition of the starting material is adjusted so that it always contains 40% solids.

Figure 3.95. Dimensions and characteristics of the windrow system.

The surface area of the windrow is given by:

AL = 2SL+hw where WZ

S2 = h 2 + q In open systems water can enter in the form of rain, where the rate from this source is given by RAIN = A I p Here A is the area open to the rain, I is the rain intensity and p the density of water. This term must be added to the water balance of COMPOST. If rain occurs during the composting process, the resulting rate of heat input is calculated as: Hrain = RAIN CPH20 Tair Here it is assumed that the temperature of the air is the same as the ambient temperature. This term must be added to the heat balance of COMPOST. Heat will be lost every time the compost is turned. The rate of heat lost will depend on the number of turnings per day, 8, and also on the temperature difference between that of the compost and the surroundings, according to

33 1


After mechanically turning the windrow, the surface of the compost will lose heat until it reaches the ambient temperature. In order to estimate this, it is assumed that a linear temperature gradient is established between the surface and a depth of 0.2m. The average rate of heat loss, then, due to turning can be estimated as:

Nomenclature The nomenclature for this example is the same as that for the example COMPOST but with the following additions: A A1 D

I (INTENSE) L RAIN S W

P

e

Subscripts rain surf turn

Surface area of windrow exposed to rain Total surface area of windrow Depth of compost influenced by surface cooling Rain intensity Length of windrow Water contribution by rain Length of slope Width of windrow Density of rainwater Turning frequency, TURNUM in program

m2

m2 m m day-' m kg day-' m m kg m-3 day-

Rain influence Surface of the windrow Turning of the compost

Exercises 1. Compare the performance of the WINDROW system with the COMPOST system. 2. See how climatic conditions, i.e., air temperature, rainfall and humidity influence the process. The effect of a shower can be simulated by stopping the simulation by hitting any key, giving a value of between 0.05 and 0.25 to the parameter INTENSE and then typing GO to continue the simulation. A second interaction can be used to set INTENSE back to zero to simulate the end of the shower.

332


3. Investigate the sensitivity of the process to the turning frequency of the heap. Vary the turning frequency with the temperature i.e., only allow turning if the temperature is above the optimum.

Results

A

i5 20.-.-5 , CI

Ammonia

0 u

0

-

-

1

*

.

m

2

-

'

.

'

3 4 Time (days)

.

.

5

Figure 3.96. Volumetric exit gas composition during startup of a windrow.

0

io

Time (days)

20

Figure 3.97. Temperature-time profile for a windrow. Parameters as in the program.

Reference Haug, R. T. (1980) Compost Engineering: Principles and Practice, Ann Arbor Science, USA.

3.9

Biodegradation Processes

3.9.1

PCPDEG and PCPDEGCF Batch and Continuous Biodegradation of Pentachlorophenol by Mixed Cultures

System The biological treatment of pentachlorophenol (PCP) has been the subject of numerous investigations. Klecka and Maier ( 1988) investigated the influence of the addition of substrate analogues to the batch degradation of PCP by bacterial cultures. They found that the addition of some analogues (e.g., 2,4,5

333

3.9 Biodegradation Processes

trichlorophenol) decreased the initial rate of PCP biodegradation but increased the overall rate due to the increase in biomass caused by the simultaneous growth on both substrates. They used a modified Monod equation to model this process with the aim of defining the optimum conditions for the biological treatment of complex mixtures of phenolic compounds, according to the scheme shown below in Fig. 3. 98. Biomass

Kd

b Cell death

W Figure 3.98. Schematic representation of the uptake of PCP and TCP by mixed microbial cultures.

Model The cells utilise PCP (Sl) and TCP (S2) simultaneously for growth. The overall growth rate on the mixed substrate in this case is assumed to be the sum of the growth rates on each individual substrate. In addition a first order cell death rate is assumed to apply with a rate constant G. For the batch case, the mass balances are as follows:

Yield constants Y1 and Y2 relate the rate of increase in biomass to the rate of utilisation of the particular substrate, i.e., PCP and TCP respectively:

There is an inhibitory effect of TCP on the rate of PCP utilisation and similarly an inhibitory effect of PCP on TCP utilisation. Interaction coefficients 11 and I2 are used to represent this behaviour in the kinetic relationships shown

334


below. In addition, high concentrations of both TCP and PCP are also inhibitory. A substrate inhibition term, Kj, in the denominator of the kinetic equation is used here to model high concentration inhibition:

Program The programs PCPDEG, for the batch mode, and PCPDEGCF for the continuous bioreactor degradation are on the diskette.

Nomenclature I

Pmax

Interaction coefficient Decay coefficient Inhibition coefficient Monod constant Substrate concentration Time Biomass concentration Yield coefficient Growth rate Maximum growth rate

Subscripts 1 2

PCP (primary substrate) TCP (secondary substrate)

Kd Ki KS

S

t X Y P

h-1 mg mg mg h mg

L-1 L-1 L-1

L-1

-

hh-1

Exercises 1. Run the model PCPDEG first with no TCP and then with increasing concentrations of TCP and see how this effects both the rate of biomass production and the rate of PCP degradation. What is the optimum quantity of TCP to add? Is the ratio of the two compounds a sufficient criteria, or do the absolute amounts have to be taken into consideration?

335


2. Using the program PCPDEGCF run until a steady state is reached and investigate the influence of flow rate on the degradation rate, on the fractional degradation and on the possible washout of biomass.

Results 0.61

1

I

g5 € 4 Y

c

0.4'

.e 3 c. E 2

c

1 1

g

0

0.04 . ". 0 10

' . y . ' 20 30 40 Time (hours)

- I

50

Figure 3.99. Batch biodegradation of PCP and TCP (0.5mg L-l).

8

a0 40 80 Time (hours)

120

Figure 3.100. Influence of adding TCP (0 to 5 mg L-l) to the batch biodegradation of PCP.

Reference Klecka, G. M. and Maier, W. J. (1988) Kinetics of Microbial Growth on Mixtures of PCP and Chlorinated Aromatic Compounds, Biotechnol. Bioeng. 31, 328-335.

3.9.2

BIOFILTl and BIOFILT2 - Biofiltration Column for Removing Ketones from Air

System Biofiltration is a process for treating contaminated air streams. Moist air is passed through a packed column in which the pollutants in the contaminated air are adsorbed onto the moist packing and there biologically degraded by the resident population of organisms. Such columns can be run with a liquid phase flow (bio-trickling filter) or just with moist packing (biofilter). The work of Deshusses et al. (1994, 1995), inyestigated the removal of two ketones, methyl isobutyl ketone (MIBK) and methyl ethyl ketone (MEK) in such a biofilter The kinetics are especially interesting since both substances exhibit mutual inhibitory effects on their rates of degradation. The original dynamic model is

336


greatly simplified in this example to calculate the steady-state concentration profiles in the column.

dZ

I

C+dC

S+dS

I

Cl Figure 3.101. Steady-state biofiltration column with cocurrent flow.

Model The kinetics can be described approximately according to the following competitive inhibition relationships For MEK For MIBK

The steady-state mass balances for cocurrent flow can ,e formulated by considering changes in the component mass rates over a differential length of column dZ (see Sec. 1.11.2.1). In this case the length is measured from the top of the column (Z=O) to the bottom, as shown in Fig. 3.101. The column is assumed to consist of a moist solid or liquid phase with volume Vs, and a gas phase of volume VG. The actual support volume is included in Vs.

337


Thus for MIBK in the gas phase:

and for MEK:

Thus in these equations for steady-state conditions to apply, the convective terms are balanced by the transfer terms, . Here the KLa value is based on the total volume VT and is assumed to be equal for each component. The linear velocities, expressed in terms of the volumetric flow rates and the empty tube cross-section, for the gas and liquid phases respectively, are:

VZL =

LF

Assuming the reaction to occur within the wet packing volume Vs, the liquid phase balances for MIBK and MEK are:

Note that the liquid phase balances, reaction, convective and transfer terms are all applicable. The gas-liquid equilibrium relationships for the two ketones are represented by SIS = MI CI and SES = ME CE Very low liquid flow rates correspond to the case of biofilter operation and higher rates to that of the trickling filter case.

Program The program BIOFILT1 solves the steady-state countercurrent biofiltration operation for the single compound MEK. The integration is started at the top of the column by assuming a desired concentration in the outlet air at the base of the column. The integration is stopped with a TERMINATE statement when

338


the inlet gas concentration exceeds the actual inlet value. Thus the required length of column is determined. In BIOFILT2, a column of given length with both MEK and MIBK is operated cocurrently. The program allows for recycling the liquid from the bottom of the column to the top and feeding fresh liquid. Fresh water at flow rate F enters the recycle stream at the top, and effluent at the same rate leaves the loop at the bottom. Since the entering liquid concentrations at steady state are not known, these are found iteratively by balancing around the feed point and in this way replacing the initial inlet values with the result from the balances, as seen from the last program statements. This program is run by entering GO for each iteration. The first run, before any iterations apply, simulates the countercurrent column without recycle. Very low liquid flow rates are appropriate for biofilter description.

Nomenclature

KS LF M r S VM

VG VS

Z &

Subscripts E G I L M S Z

Cross-sectional area of empty tube Concentration in gas phase Diameter Gas flow rate Inhibition coefficient Mass transfer coefficient based on total volume Monod coefficient Liquid flow rate Partition coefficient Reaction rate Concentration in moist or liquid phase Maximum reaction velocity Volume of gas phase Volume of solid or liquid phase Length or height Bed porosity Refers to MEK Refers to gas Refers to MIBK Refers to liquid Refers to maximum Refers to saturation Refers to axial direction

m2 kg m-3 m m3 s-1 kg m-3 S-'

kg m-3 m3 s-1 kg m-3 s-1 kg m-3 kg m-3 s-1 m3 m3

m


339

Exercises 1. Using BIOFILTl, investigate the influence of the entering gas flow rate and entering concentration on the column length required to obtain 90% removal of MEK. 2. Alter BIOFILTl to describe a cocurrent column. Compare the results with the countercurrent case for various extremes of liquid flow rate. Using the length found in Exercise 1, operate the column cocurrently to see whether the same degree of removal can be obtained. Show that the comparison depends on whether the column performance is reaction rate or transfer rate controlled. 3. Show that the amount of biomass and its activity are important on the performance of the column by varying the values of VME.

4. The mass transfer rate is largely dependent on the wetted surface area and the concentration driving force. Vary KLa and the equilibrium constant in separate simulation runs to study these influences. 5. Investigate the inhibition kinetics in BIOFILT2 by operating at medium (0.001 kg m-3) and constant concentrations of MIBK, while varying the operation to change the concentration of MEK. Plot the rates as a function of this concentration. Repeat this for different values of KiI and KiE.

6 . With BIOFILT2 investigate the influence of liquid feed rate and effluent flows. 7. Operate BIOFILT2 as a trickling filter by setting reasonably high values of liquid flow. Experiment with the recycle system by changing the feed flow F. Removing the INTERACT statement will cause repeated iterations to be made.

8. Develop a model and program this for the case of a countercurrent column removing both components. Use a dynamic model to avoid the difficulty of making iterations for steady state.

340


Results

Figure 3.102. With BIOFILTI using two gas flow rates (0.5E-4 and 0.5E-4 m3 s'l) gave column lengths of 0.74 and 1.3 m, respectively.

Figure 3.103. BIOFILT2 was used with a very low liquid flow rate to simulate a biofilter column. Curves A and B: MEK and MIBK liquid phase concentrations. Curves C and D: MEK and MIBK equilibrium liquid phase concentrations.

Figure 3.104. Biodegradation rates for the previous biofilter column run. Curve A is for the MEK degradation rates, and curve B for those of MIBK. Significant inhibition of MIBK degradation is noticeable at the top of the column (low Z).

Figure 3.105. BIOFILT2 was used to simulate a trickling filter (LF=2E-6) and gave these iterations for the liquid recycle concentrations, combined with feed. Curves A to D are for MEK, and E to H are for MIBK. Additional iterations are needed here for convergence.

References Deshusses, M. A (1994) ETH-Zurich, PhD Thesis, No. 10633. Deshusses, M. A, Hamer, G. and Dunn, I. J. (1995) Part I, Behavior of Biofilters for Waste Air Biotreatment: Part I, Dynamic Model Development and Part 11, Experimental Evaluation of a Dynamic Model, Environ. Sci. Technol.

34 1


3.9.3

DCM 1 and DCM 2 - Airlift Biofilm Sandbed for Dichloromethane-Waste Air Treatment

System Dichloromethane (DCM) is a common industrial solvent, of which 500,000 tons per year has been estimated to contaminate the atmosphere. A three-phase fluidised sandbed airlift reactor, shown in Fig. 3.106, was designed for the removal of DCM from an air stream (Niemann, 1993). The organic DCM is absorbed from the air stream into the liquid phase and is degraded there by an adapted mixed culture growing as a biofilm on sand particles. The degradation reaction is as follows: CH2C12 + 0 2

-+ C02 + 2HC1

In addition, a small part of the carbon goes into the biomass growth. The production of HCI by the reaction requires the control of pH by neutralisation. The regulation of salinity also requires the addition of a continuous liquid feed and the removal of liquid effluent.

Model The actual three phase reactor, shown in Fig. 3.106, can be represented for modelling purposes by the system shown in Fig. 3.107. Owing to the high liquid recycle rate through the system, the liquid phase can be taken to be equivalent to a single well-mixed phase, while the gas phase is best described by a series of well-mixed tanks. The reaction is assumed to be described by a homogeneous model in the liquid phase. No biomass growth is considered. The mass balances for each component i in the gas and liquid phases are summarised below. For DCM and oxygen in the gas phase, with mass transfer from the gas to the liquid phase and assuming the same value for KLa :

For DCM and oxygen in the liquid phase with reaction and mass transfer from the gas to the liquid:

342


Air NaOH $ 7 Feed

Effluent

3-Phase airlift reactor

- Air with DCM

Figure 3.106. Airlift biofilm sandbed reactor for air treatment.

For the chloride and hydrogen ions in solution the balances are of the form: dCiLn VLT = FL (CiLn-1- CiLn) - rin VL The equilibrium relationship for each component is represented by Henry's Law:

343

3.9 Biodegradation Processes G

4

R

3-Phase biofilm fluidised bed reactor

ciLF, 4

I 1

Figure 3.107. The two phase fluidised bed schematic model for the airlift.

The reaction kinetics are relatively complex in that they consider both DCM, and possible oxygen limitation, as well as inhibition by C1- according to the following type of relationship:

In addition, the reaction kinetics are influenced by pH and by temperature, with the temperature dependency or influence modelled as:

where the pH dependency, f(pH), is given by f(pH) = a pH2- b pH3 - C pH + d The above relationship represents an empirical correlation in which the parameters a, b, c and d are fitted to the pH-rate data. The oxygen uptake rate is related to the DCM uptake rate by a constant yield factor YO/DCM,where rOn = YO/DCM rDCMn

344


Additional relationships are needed for the liquid stream mixing points. The concentrations entering stage 1 are obtained by a simple balance as

The pH is calculated from the H+ concentration by

The percentage of DCM (OVERF) leaving in the liquid is 100.0 OVERF = F CDCML~O G CDCMGO

and the percentage (STRIP) leaving in the gas phase is 100.0 STRIP = G CDCML~O G CDCMGO The experimental process parameters needed to solve this model are the reaction kinetic parameters, the transfer coefficients for DCM and 0 2 , the Henry coefficients, and the temperature and pH rate dependencies for DCM and 0 2 .

Program The programs, DCMl and DCM2, represent simplifications of the above model. In DCM1, a single well-mixed liquid phase is used and the gas phase is modelled as five stages. In DCM2, single well-mixed gas and liquid phases are used, thus neglecting all axial gradients. Oxygen was found to be non-limiting in the experiments of Niemann (1993) and is neglected in the program.

Nomenclature Constants in pH-rate function Concentration Activation energy-temperature factor Flow rate of liquid Flow rate of gas Saturation coefficients Inhibition coefficient for chloride Mass transfer coefficient Percentage lost in outflow liquid

various mg L-' -

L h-1 L h-1 mg L-1 mg L-1 h%

345


R r RTMi STRIP T t V

Y

Recycle flow rate Reaction rate Distribution coefficient for substance i Percentage lost in exit gas Temperature Time Volume Yield coefficient

L h-1 mg L-l h-I 9%

K h L -

Subscripts

*

CL DCM G 1

L max n 0

Refers to saturation Refers to chloride Refers to dichloromethane Refers to gas Refers to component i Refers to liquid Refers to maximum Refers to stage n Refers to oxygen

Additional Nomenclature in Program

VR

Buffer factor Chloride concentration Activation energy Gas fraction Fraction liquid in solid-liquid system Hydrogen ion concentration Base reaction rate Gas constant Temperature Volume of 3-phase mixture

Subscripts S 0 1 1,293,475

Refers to DCM Refers to inlet of stage 1 Refers to single liquid stage Refers to gas stages

BUFF CL EA EG EPS H RO RGC TEMP

mg L-' J mole-l mole L-1 mg L-' h-' J mole-' K-l K L

Exercises 1. Run the programs DCMl and DCM2 and note the difference in the results. Compare especially the gas phase results and the overall removal at steady state.

346


Run simulations for differing values of the DCM concentration in the inlet air. How do the results obtained with DCMl and DCM2 compare?

2. Investigate the influence of pH and chloride concentration on the rate of removal. Vary the inlet liquid feed rate and note the influence. 3. Change the program to maintain the pH constant, corresponding to controlled pH operation. Note this does not require a controller equation and can be done simply by means of the CONSTANT statement.

4. Investigate the influence of temperature on the distribution coefficient and the overall operation of the column.

5. Find a set of values of inlet air flow rate and inlet DCM concentration that gives better than 95% removal.

Results

Figure 3.108. Response of reactor DCMl to a step change in inlet gas concentration from 5 to 10 to 1 m& at t=0.1 and 0.3 h.

Figure 3.109. For the same run, response of chloride concentration and the reaction rate.

References Niemann, D. (1993) ETH-Zurich, PhD Thesis 10025. Dunn, I. J. (1994) "High-Rate Biofilm Fluidized Bed Reactors for Specialized Wastewater Treatment" In Adv. in Bioprocess Eng., Kluwer.

347


3.9.4

FBR - Biofilm Fluidised Bed with External Oxygen Supply

System Biofilms are exploited for both the aerobic and anaerobic treatment of wastewater (see Sec. 2.3.3). The advantage of biofilms is that they enable biomass to be retained in a reactor at flow rates greater than the washout flow rate. They often consist of several microbial populations which perform complementary processes. The rapid transfer of substrates between the populations facilitates rapid wastewater treatment. Another advantage is that the diffusional barrier renders the biomass less susceptible to irreversible damage due to shock loads and toxic shocks.

Model A fluidised bed column reactor can be described as 3 tanks-in-series (Fig. 3.1 10). Substrate, at concentration So, enters the circulation loop at flow rate F.

The flow rate through the reactor due to circulation is FR. Oxygen is absorbed in a well-mixed tank of volume VT. The reaction rate for substrate (rs) depends on both S and dissolved oxygen (CL). The rate of oxygen uptake (ro) is related to S by a yield coefficient (Yos). The gas phase is not included in the model, except via the saturation concentration (CLS). The oxygen uptake rate of reactor can be determined by a mass balance based on the difference in CL inlet and outlet values. The model balance equations are developed by considering the individual tank stages and the absorber separately. The gas phase in the absorber is assumed to be air.

Substrate Balances: For the absorption tank:

45 dt

=

F FR (S3 - S) + - (So - S ) VT VT

For each stage n of the reactor:

348


Figure 3.110 Biofilm fluidised bed with external aeration.

Oxygen Balances: For the absorption tank:

Kinetics for Stage n: vm Sn rSn = Kn+S,

CLn &+CLn

Nomenclature CL CLS F FR

Dissolved oxygen concentration Saturation oxygen concentration Feed flow rate Recycle flow rate

349

3.9 Biodegradation Processes ~

KLa K

KO rOn rS n t S V VT Vm

Ym

Mass transfer coefficient Saturation constant Saturation constant for oxygen Reaction rate for oxygen Reaction rate for substrate Time Substrate concentration Reactor volume of one stage Volume of absorber tank Maximum velocity Yield coefficient, oxygen to substrate

hkg m-3 g m-3 g m-3 h-I kg m-3 h-l h kg m-3 m3 m3 kg m-3 h-1 g kg-'

Subscripts 0 1,2, 3, n m 0

S T

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

Exercises 1. Assuming maximum uptake rates, choose values of F and So and hand calculate the outlet substrate concentration. What oxygen supply rate will be required? Choose values of KLa and VT to give the desired results, run the simulation and compare the results. 2. Change the operating parameters and interpret the results in terms of the axial concentration gradient in the reactor column.

3. Vary the substrate and oxygen supplies, and note the interaction of both on outlet values of S3 and C L ~ .

4. Operate the reactor to obtain a very small conversion across the bed (differential reactor) by increasing FR or decreasing v,. Use the simulated data to verify the kinetics of the reaction by running the reactor at various substrate (S and CL) levels.

350


Results

.-g

10'

D

Time (hours)

Time (hours)

Figure 3.111. Dissolved oxygen concentrations in tanks 1 to 3 during startup.

Figure 3.112. Dissolved oxygen concentration in tank 3 (CL3) for three different values of recycle flow rate (2000, 3000 and 4000 m3 h-l).

3.10

Anaerobic Digestion Processes

3.10.1 ANAEROBE Digestion

- Andrew's

Model of Anaerobic

System Anaerobic digestion (see Sec. 2.7.5) is the process where suspended organic matter is stabilised by a mixed population of bacteria in the absence of oxygen. There are two major groups of bacteria in the culture, the first hydrolyses the solids to soluble short-chained fatty acids and a stable insoluble residue, and the second converts the fatty acids to methane gas. The process (Fig. 3.113) is slower than an activated sludge process and therefore requires larger reactor volumes. This disadvantage is often offset by the lower energy requirements and the associated production of methane, which can be used as an energy source. An additional advantage of an anaerobic digestor as compared to an activated sludge reactor is that much less sludge is produced by the anaerobic process. The most common reactor configuration is the continuous stirred tank reactor, sometimes with recycle or with two-stages, but plug-flow reactors have also been investigated.

35 1

3.10 Anaerobic Digestion Processes

Bacterial cells

Volatile acids C 0 2 + Hq

Other products Methane-producing bacteria

Bacteria cells Figure 3.113. Schematic diagram of the major processes in anaerobic digestion.

Model

Figure 3.114. Major processes in Andrew's model of anaerobic digestion.

Biological Kinetics In the Andrew's model of anaerobic digestion (Fig. 3.114) the growth of biomass is assumed to follow Monod kinetics with inhibition caused by high

352


substrate concentration and by the presence of toxin (TOX). The rate limiting substance is assumed to be the nonionised substrate:

A mass balance on the biomass in the reactor gives the following equation: dX - = dt

D(Xo-X)+pX-KtTOX

where D is the dilution rate:

F D = v and F the influent flow rate. A mass balance for substrate gives the following relationship where Yxs is the yield of biomass with respect to substrate uptake:

Gas Production Rates The rates of production for carbon dioxide (CPR) and methane (MPR) are related to the biomass growth rate by constant yield factors: CPR = Y c x X p

Liquid Phase The substrate can exist in both an ionised ( S ) and a non-ionised (HS) form. The concentration of non-ionised substrate can be calculated from the hydrogen ion concentration and the dissociation constant, Ka:

The pH in the liquid phase is calculated from the hydrogen ion concentration:

353


The hydrogen ion concentration is calculated from the equilibrium concentrations of dissolved carbon dioxide and bicarbonate ion (BC) and the equilibrium constant, K1: K1 (C02)d Hion = BC The carbon dioxide produced by the biomass can exist as dissolved C02, as bicarbonate ions and in the gaseous form. The rate of transfer of C02 (MTR) between the gaseous and the dissolved forms can be modelled by applying a mass transfer coefficient, KLa:

The equilibrium dissolved carbon dioxide concentration can be determined knowing the partial pressure of carbon dioxide in the gas phase and the Henry constant KH: (C02)dEQ = KH ppC02 The bicarbonate i o n concentration is needed to calculate the hydrogen ion concentration. Thih can be found by comparing the positive and negative ions in solution and balancing for electroneutrality: H+ + c = HC03-

+ S- + OH- + a + 2 C032-

where a is the anion and c the cation concentration. If it assumed that the concentrations of H+, C032- and OH- are negligible in the normal pH range of anaerobic digestors, then the equation can be simplified to:

z = S+BC where z is the difference between the cation and anion concentrations and BC is the bicarbonate ion concentration: z = c-a To a good approximation, assuming the sulphide ion concentration to be low, the value of z corresponds to that of the ammonium ion concentration, A mass balance for the ammonium ions, assuming negligible microbial utilisation. gives:

Differentiating the simplified ion balance gives: dBC _ dt

-

d(NH4)_ dt

dS -~ - dt

354


A mass balance on the bicarbonate ions in the reactor therefore gives: BCPR = D (BCo-BC)

dS + dt

-

d(NH4)

7

Carbon dioxide is produced by the biomass (CPR). Some comes out of solution and leaves the reactor in the gas phase (MTR) and some is ionised (BCPR). Hence a mass balance can be written for dissolved carbon dioxide as: = D ( (CO2)do - (C02)d) + CPR + BCPR + MTR d(Co2)d dt

A mass balance for the non-consumed toxic substance in the reactor gives:

-dToX -

- D (TOXo-TOX)

dt

Gas Phase

It is assumed that all the methane produced by reaction enters the gas phase. The carbon dioxide entering the gas phase has been discussed previously in terms of MTR. The volumetric gas flow rates can be as: Q C H ~=

V MPR p V MTR P

Qc02 = --

The negative sign designates a reversal in the transfer direction, where Qco2 is the volumetric flow rate from the liquid into the gas phase. The volumetric gas flow rate consists mainly of carbon dioxide and methane: Qtot

= Q C H ~+ Q C O ~

The partial pressure of carbon dioxide in the reactor head space is calculated from a mass balance between the rate of mass transfer of C02 from the liquid to the gas phase and the total mass flow rate of gas out of the reactor:

The percentage carbon dioxide in the exit gas is frequently used as a parameter for judging the state of the anaerobic digestion and can easily be calculated from the C02 and total gas flow rates:

355


co2pc = 100.0

Qc02 __

Qtot

Nomenclature a BC BCPR C

C02PC CPR D F H+ Hion HCO3HS K1 Ka KH Ki KLa KS

Kt MPR MTR NH4+ PP PPtot

Q S

t

TOX V

v, X

ycx

ymx yxs

z P c1 (MU)

Concentration of anions Bicarbonate concentration C02 production rate from bicarbonate Concentration of cations Carbon dioxide composition Carbon dioxide production rate Dilution rate Influent flow rate Hydrogen ion concentration H+ ion concentration Bicarbonate ion concentration Non-ionised substrate Bicarbonate dissociation constant Acetic acid dissociation constant Henry's constant (C02) Substrate inhibition constant Mass transfer coefficient Monod constant Toxic death constant Methane production rate C02 mass transfer rate Ammonium ion concentration Partial pressure Total atmospheric pressure Volumetric flow rate Substrate in non-ionised form Time Toxin concentration Volume of reactor Volume of gas phase Biomass Yield factor (C02hiomass) Yield factor (methanehiomass) Yield factor (biomass/substrate) Net cation concentration (c - a) Gas density Specific growth rate

mmole mmole mmole mmole

L-1 ~ - 1

L-l day-' L-1

%

mmole L-l day-' dayL day-' mmole L-1 mole L-1 mmole L-1 mmole L-' mmole L-I mmole L-l mmHg-l mmole L-1 daymmole ~ - 1 daymmole L-1day-' mmole L-' day-' mmole L-1 mmHg mmHg L day-1 mmole L-' day g L-1

L L g L-1 mmole g -1 mmole g -1 g mmole mmole L-' mmole L-I day-

356

Subscripts 0

co2 CH4 EQ max Pc tot d


Inlet conditions Carbon dioxide Methane Equilibrium value Maximum Percent composition Total Dissolved

Exercises 1. See how the systems responds to changes in the feed once steady-state conditions have been achieved. Is it possible to return to a steady state?

2. Vary the liquid flow rate and observe the partial pressure of carbon dioxide in the exit gas. Is there any correlation between the two parameters. If so what is the lag time between a change in the flow rate and a change in the ppCO2?

3. Vary the bicarbonate concentration in the feed and observe the change in the buffering effect.

Results

Figure 3.115. A change in dilution rate from D=0.1 to D=0.2 at T=8 d caused these responses for biomass and substrate.

Figure 3.116. The responses for CH4 and C 0 2 flow rates are shown for the same run.

References Andrews, J. F. (1989) Dynamics, Stability and Control of the Anaerobic Digestion Process, Dynamic Modelling and Expert Systems in Wastewater Engineering, (eds. Patry, G.G. and Chapman, D.) Lewis Publishers, USA.

357


Graef, S. P. and Andrews, J. F. (1 974) Mathematical Modelling and Control of Anaerobic Digestion, Water- 1973 (ed. Bennet), AIChemE Symposium Series 136, 70, 101-1 13, New York, USA.

3.10.2 WHEY - Model for the Anaerobic Degradation of Whey System Anaerobic processes can be described by multi-substrate, multi-organism kinetics (Dunn et a]., 1992). In the production of cheese, the proteinaceous solids (curds) are separated from the liquid (whey). The whey contains lactose and lactic acid and can be degraded anaerobically. This example follows the work of Ryhiner (1990), in developing a model for the batch anaerobic degradation of whey. The reaction stoichiometry for the process is represented in Table 3.4. The stoichiometry for Step 1, the hydrolysis and acidification of whey, was determined by comparing experimental data for the distribution of organic acids with simulation results. In Step 2, higher acids are converted into acetic acid by relatively well established reactions. In Step 3, methane is formed both from acetic acid and by the combination of H2 and C02. Overloading a continuous reactor causes an accumulation of hydrogen, and this thermodynamically inhibits the reactions of Step 2, which causes acid accumulation, a drop in pH and further inhibition.

Table 3.4. Stoichiometry of the anaerobic degradation of whey. Step 1: Hydrolysis and acidification of whey c6H11.105.12 + 0.57 H20

+ 0.75 CH3(CH2)2COOH + 0.24 CH3(CH2)2COOH + 0.5 CH3COOH + 1.38 C02 + 1.44 H2

(Whey1

Step 2: Organic acid degradation to produce acetic acid CH3(CH2)2COO- + 2 H20 CH3CH2COO- + 2 H20

2 2 CH3COO- + 2 H2 + H+ 2 CH3COO- + 3 H2 + C02

AGO = 71.7 kJ AGO = 48.3 kJ

Step 3: Methane production from acetic acid, hydrogen and carbon dioxide __

CH3COOH

+ CH4 + C02

358


Model A multi-organism model is used involving five different types of organisms. Each intermediate compound appearing in Table 3.4 is utilised as a substrate for one organism type and is also produced as a product by another type of organism. The rates of production and consumption of each substrate are related to the organism growth rates. Thus for the batch system, the substrate balances take the form

where rConsi and rproi represent the rates of consumption and production of substrate Si, respectively. The mass transfer rates for the volatile substrates (hydrogen, carbon dioxide and methane) to the gas phase are not considered here, but are assumed to obtain a liquid phase saturation value. The effect of mass transfer could be included, if required, by adding appropriate mass transfer terms to the balances for the volatile components. The respective reaction rates ri for the consumption of the substrates Si and for the formation of product Pi correspond to those from the reactions of Table 3.4 and where: - Pi Xi rconsi = ___ Yxi/si

Here rproj is calculated from rconsi using the corresponding growth of Xi. The specific growth rates for each organism type are assumed to take the Monod form, where

and are modified in the case of substrate inhibition for acetate-consuming bacteria as PmaxAc SAC PAC = SAC+ KAC + SAC’KIAC Note that in the case of organic acids, the substrate concentration S refers to the total acid (undissociated plus dissociated) concentration. The undissociated forms are assumed to serve as the growth substrates, and the dissociation equilibria to calculate these concentrations are included in the model. These equilibria are strongly influenced by pH.


359

Reaction Equilibrium According to the positive free energy changes for the acetogenic reactions of Step 2 in Table 3.4, excess hydrogen will shift the thermodynamic equilibrium to the left, thus reducing the conversion to acetic acid. This effect is incorporated in the model by making use of the equilibrium constants for the reactions, which can be calculated from the free energies of reaction given in Table 3.4. Thus for butyrate and propionate

The factor 2/3 arises because concentrations are expressed here in C-mole. When equilibrium is approached, the net forward reaction rate decreases and is equal to zero at equilibrium. This effect is accounted for by an empirical factor, which is related to the ratio of the equilibrium constant calculated from the momentary concentrations to the true equilibrium constant. This relation shown in Fig. 3.117 has the effect of progressively slowing the rate of reaction as the system approaches equilibrium. The ratios, K B ~ / K * Band ~ , KPr/K*Pr are used to determine the empirical factors FEQi, which vary from 0 to 1 as the ratios vary from 1 at equilibrium to 2 away from equilibrium. The corresponding growth rates are multiplied by these factors, causing both Step 2 1.2' 1.0.

0.8. FEQi

0.6. 0.4.

0.2.

-0.21 0

I

1

Ki/K'i

I

I

2

3

Figure 3.117. Empirical equilibrium factors (FEQ) to slow the growth rates near equilibrium.

3 60


reactions to stop when equilibrium is reached (FEQi =O when Ki/Ki* < l), and to proceed normally away from the equilibrium (FEQi =O when Ki/Ki*> 2).

Ion Charge Balance to Estimate p H As discussed in Dunn et al. (1992) and Heinzle et a1.(1993), to calculate the pH an ion charge balance can be written, which includes the organic acid dissociation equilibria. Considering the dissociation of organic acid HSi into anion Si- and cation H+, the dissociation constants are

Using the sum Csi

+ CH equal to Ctoti, the anion concentrations for Si- become

For electroneutrality, the sum of the charges of all the cations must be equal to the sum of the charges of all the anions, as given by an ion balance. This ion balance forms an algebraic loop within the dynamic model, and must be solved by iteration for each integration time step, such that the charge difference between the cations and anions, 6, is always zero, where

where CAi and Cci are the total anion and cation concentrations for the strong acids and bases, and Zi are their ionic charges. If no acids or bases are added to the reactor, as in the case without pH control, XCci Zi and CCAi Zi can be assumed constant and can be estimated initially by experiment, using this balance with a measured value of pH. The iteration to maintain 6 = 0 involves varying CH+ until 6 becomes essentially zero. A very robust algorithm is used in the program, which always h o l e m-3, CH+ is increased by a factor f n converges. Starting at CH+ = (usually 10) until 6 c 0. Then CH+ is decreased by a factor of l/fn, and fn+l is set to fi.The previous procedure is repeated until 6 < 0, and is stopped after reaching an error criterion. If a base is fed to the reactor to control pH, an additional dynamic mass balance must be used to calculate the additional accumulation rate, d(Cci Zi)/dt for strong bases and dCtoti/dt for weak bases, whose dissociation must additionally be included in the ion balance.


36 1

Nomenclature The nomenclature of the program is also defined within the program. The balances for organic acids are written in terms of the total concentrations (Csi + CH), designated by BU, PR, AC and CG. C Buffer Cons f (FACTH) Fi FEQi Ions KD KIAC Ki Ki* KS

Si Pro i r X YPJS

s'" c1

Subscripts Ac Bu Buf cg CH HY in

w

wh Pr Tot

Concentration Total buffer concentration Consumption rate Factor for the pH iteration Stoichiometric coefficients Equilibrium factor for substrate i Z (Cations - Anions) Dissociation constants Inhibition constant for acetic acid Reaction equilibrium constant Momentary equilibrium constant Monod saturation constant Substrate concentration Production rate Reaction rate Biomass concentration Yield, product from substrate Yield, biomass from substrate Conc. difference (cations - anions) Specific rate of biomass synthesis

C-mole m-3 mole m-3 C-mole m-3 h-' -

mole m-3 mole m-3 C-mole m-3 c-mole4 m-12 c-mole4 m-12 C-mole m-3 C-mole m-3 C-mole m-3 h-' C-mole m-3 h-1 C-mole m-3 C-mole C-mole-1 C-mole C-mole-1 mole m-3 h-

Acetic acid Butyric acid Buffer Bicarbonate plus carbon dioxide Methane Hydrogen gas Initial Water Whey substrate Propionic acid Total

Exercises 1. In the model as programmed, the pH is allowed to vary. Modify the model to keep pH constant by adjusting the feeding rate of a strong base. In this case,

362


the molar flow rate of base anions would then be equal to the total production rate of dissociated acid protons. The concentration of these can easily be estimated knowing pH, dissociation constants and total concentrations, and no iteration will be necessary. An additional balance to calculate the total ions will be necessary. 2. Modify the model to consider the continuous feeding of whey at flow rate Q to the reactor of volume V. Assuming immobilised biomass, allow a small fraction of the biomass to leave the reactor with the liquid.

Results In Fig. 3.1 18 are shown the dynamic profiles of substrate whey (Wh), with acetate (Ac), propionate (Pr) and butyrate (Bu). In Fig. 3.119 th pH profile is shown. In this run the whey concentration was increased at t=0.025 h. The ion balance iteration causes the simulation to run rather slowly.

8.3881

Ism

6.231

1 Figure 3.118. Dynamic profiles of substrate whey (Wh), with acetate (Ac), propionate (Pr) and butyrate (Bu). An increase in the whey concentration was made at t=0.025 h.

Figure 3.119. The pH profile for the same run, showing the influence of adding additional whey to the reactor.

References Dunn, I. J., Heinzle, E., Ingham, J., and Prenosil, J. E. (1992) Biological Reaction Engineering: Principles, Applications and Modelling with PC Simulation, VCH, Weinheim. Rhyiner, G. B, (1990) Ph.D. thesis, ETH-Zurich, No. 9207. Rhyiner, G. B., Heinzle, E., and Dunn, I. J. (1993) Modelling and Simulation of Anaerobic Wastewater Treatment and its Application to Control Design: Case Whey, Biotechnology Progress, 9, 332-343.


363

Heinzle, E., Dunn, I. J. and Ryhiner, G. B. (1993), "Modelling and Control for Anaerobic Wastewater Treatment", in Adv. in Biochem. Eng., 48, 79- 1 14.

3.11

Anaerobic Fixed Film Processes

3.11.1 DENITRIF - Denitrification of Drinking Water in a Fluidised Bed Reactor System Many sources of drinking water, especially in areas of intensive agriculture, contain intolerably high levels of nitrate ion, above the safe level of 50 mg litre*. Certain autotrophic organisms are known (e. g. Micrococcus denitrif-icans ) which will reduce nitrate ions to nitrite ions and eventually to nitrogen gas using hydrogen as the electon donor reaction partner.

An obvious advantage of this autotrophic process is that it is necessary to add only hydrogen to the process.

Model Reactor In the system shown in Fig. 3.120, denitrification of drinking water is performed in a conical tapered fluidised bed reactor containing sand particles. Hydrogen gas absorption occurs in a separate tank, and the fluidised bed reactor and the absorption tank are linked by a pumped recycle loop, which supplies the hydrogen to the bed. This allows the possibility of monitoring the difference in dissolved hydrogen concentration across the bed, which is directly related to the reaction rate. In addition the recycle loop make it easy to adjust the pH and temperature of the liquid recycle and hence the reactor.

3 64


I

Fluidised reactor

t

FR

It

Figure 3.120. Flow diagram for the anaerobic fluidised bed reactor.

Kinetics The kinetics for each reaction step can be expressed by using a double Michaelis-Menten relationship with hydrogen and nitrite or nitrate, respectively, as limiting substrates. Thus:

where Vmax is the maximum rate for each reaction.

Balancing Both the total reactor system (fluidised bed reactor and absorption tank) and the absorption tank itself can be assumed to act as well-mixed stages. Owing to the very low change in concentration occurring across the fluidised bed reactor,, at high rate recycle flow rates the reactor can also be considered to be well mixed with respect to the nitrogen components. Mass balances for nitrate and nitrite for the combined system can be given as

3.1 1 Anaerobic Fixed Film Processes

365

where VA and VR are the volumes of the adsorption tank and reactor respectively. F is the volumetric flow rate of water into the system A hydrogen balance around the fluidised bed can also be applied, giving

Nomenclature

VR

Dissolved hydrogen concentration Nitrite concentration Nitrates concentration Feed flow rate Circulation flow rate K, value for H2 Ks value for nitrate K, value for nitrite Rate of nitrate to nitrite conversion Stoichiometric parameter Stoichiometric parameter Volume of absorption tank Maximum reaction rate Volume of fluidised bed reactor

Subscripts in 1 2

Feed concentration nitrate to nitrite conversion nitrite to nitrogen conversion

cH2

CN02 CN03 F FR KH2 KN02 KN03 r S1 s2

VA Vmax

mg Lmg Lmg LL h-1 L h-l mg Lmg L-1 mg L-1 mg L-1 h-1 mg H2 (mg N03)-' mg H2 (mg N02I-l L mg L-1 h-I

Exercises 1. Vary the feed flow rate through the system and see how the nitrate removal efficiency is influenced.

2. Modify the model to include the effects of hydrogen mass transfer in the absorber. Experiment with the influence of the gas-liquid transfer coefficient to determine the value required to avoid hydrogen limitation. Study the influence of hydrogen on the kinetics if the Ks values in each reaction step were not the same.

3. For the bed to be fluidised but no particles to be carried over, the circulation flow rate must be between 10 and 100 litres hour1. Vary the circulation flow rate within this range.

366


4.Design a large scale process for the denitrification of drinking water. What assumptions made in the above model are then no longer valid?

Results 30-

e

CI

8 '

C

m

Time (hours) Figure 3.121. Concentration-time profiles for nitrate, nitrite and hydrogen. Parameters values as in the program.

Figure 3.122. Response of the reactor to a change in flow rate from 0.6 to 3 at T=6.

Reference Kurt, M., Dunn, I. J., and Bourne, J. R. (1987) Biological Denitrification of Drinking Water Using Autotrophic Organisms with H2 in a Fluidized-Bed Biofilm Reactor, Biotechnol. Bioeng. 29, 493-501.

3.1 1.2 MOLASSES - Anaerobic Degradation of Molasses in a Packed Bed Reactor System Molasses consists of a variety of sugars, mainly sucrose, which can be converted anaerobically to methane by a sequence of four reaction steps comprising hydrolysis, acidogenesis, acetogenesis and methanogenesis. Following the hydrolysis of complex sugars to simple sugars, the further degradation is known to proceed via rapidly growing and pH insensitive bacteria to form organic acids (butyric, propionic and acetic acids) carbon dioxide and hydrogen. In the next step, slowly growing, pH-sensitive bacteria further oxidise the higher acids to acetic acid, carbon dioxide and hydrogen. Methanogenesis involves the reduction of carbon dioxide with hydrogen to methane by relatively fast growing, pH-sensitive autotrophic bacteria. Highly

367


pH-sensitive methanogens also catalyse the reduction of acetic acid to methane. The process is shown schematically in Fig. 3.123. Degradation in this case is effected by the use of a small laboratory packed bed reactor with a clay spherical packing and operated with liquid recycle and control of both pH and temperature (Denac et al, 1988).

Higher sugars Anaerobic hydrolysis

-

Acidogenic fermentation

I

t

1

I

Propionic acid Butyric acid

cid

genesis

I

Methanogenesis

Figure 3.123. Schematic of anaerobic degradation pathways.

Model For stoichiometric reasons, glucose is assumed to be the starting reactant. The reactions considered in the model are listed in Table 3.5. The problem is formulated in terms of chemical oxygen demand, COD, (see Sec. 2.3.1.2). For this reaction system, there are five reactions and seven components, such that the stoichiometric coefficient matrix in terms of COD can be written as follows:

Glu -1 0 0 0 0

HBt 0.028 -1 0 0 0

HPr 0.053 0 -1 0 0

HAc 0.614 0.8 0.57 1 -1 0

,

H2

CH4

0.302 0.2 0.429 0 -1

0 0 0 1 1

rl r2 r3 r4 r5

368


Table 3.5. Stoichiometry of the anaerobic reactions. Step 1 rl

Acidogenic fermentation of glucose C6H 1 2 0 6 -+ CH3(CH2)2COOH+2C02+2H2 C6H12O6 C6H12O6

Step 2 r2 r3

Step 3 r4 r5

+ 2H2 +

+ 2H20 +

2CH3CH2COOH + 2H20 2CH3COOH + 4H2 + 2H20

Acetogenic oxidation reactions CH3(CH2)2COOH + 2H20 CH3CH2COOH + 2H20

+ +

2CH3COOH + 2H2 CH3COOH + C02 + 3H2

Methanogenic reactions CH3COOH 4 H 2 + C0 2

+ +

CH4+C02 CH4+H20

The above coefficients correspond to the chemical oxygen demand (COD) yields (mass COD of a component produced or consumed per unit mass of COD consumed from the limiting reactant in reaction ri). The procedure is very similar to a carbon balance and is essentially equivalent to an electron balance for the reactions. It provides a check on the stoichiometry since each row must add to zero. Generally, the amount of oxygen required in a reaction (zero for anaerobic reactions) is equal to the change in the oxygen demand of the substrate minus the oxygen demand of the material formed. The COD values are given in Table 3.6. So, for example, the r2 line in the matrix means that one gram HBt-COD is converted to 0.8 g HAc-COD and 0.2 g H2-COD by reaction r2. COD values for specific compounds can be calculated by working out the oxygen requirement for complete combustion and these values are given below.

Table 3.6. Chemical oxygen demand (COD) values of the reactants. Reactant

By weight

Glu HAc HPr HBt H2 CH4

(g COD/g) 1.07 1.07 1.51 1.82 8.0 4.0 0.0

co2

By mole (g COD/mole) 192 64 112 160 16 64 0.0

369


In the work of Denac et. al. (1988) Step I , involving three individual reactions, was simplified to one reaction, The fractions of glucose converted to the indicated acids must be known, and these were determined by fitting experimental data of the measured product distribution by simulation. From this the above stoichiometric coefficient matrix for rl was determined. It can be seen that a very high relative production of HAc was assumed, corresponding to the observation that HAc was favoured over the other acids under the experimental conditions. The stoichiometric coefficients for reactions, r2 to r5 were calculated using Table 3.6 directly. The rates of biomass growth and substrate utilisation are assumed to follow Monod kinetics with constant yield factors and diffusional effects are assumed negligible.

where the subscript i represents the following: 1 Glucose 2 Butyric acid 3 Propionic acid 4 Acetic acid 5 Hydrogen The total biomass is given by X = CXi Assuming a well-mixed reactor, the following liquid phase balance can be written for each substrate component as

Considering the COD stoichiometry, the individual component balances are as follows: Glucose: Acetic acid:

~dSac dt

- 0.614 rl

+ 0.8 r2 + 0.571 r3 - r4

370


Propionic acid:

$f

Butyric acid:

= 0.053 rl - r3

di:t

_ _ -- 0.028 rl - r2

Hydrogen:

-d z-2

- 0.305 rl

+ 0.2 r2 + 0.429 r3 - r5 - N

H ~

Methane: Carbon dioxide: _ _ _-- 0.01 r l dSco2

dt

+ 0.009 1-3 + 0.016 1-4 - 0.016 r5 - N c 0 2

Owing to its low solubility, the methane is assumed to be transferred immediately to the gas phase. The transfer rates for C02 and H2 from the liquid phase to the gas are described by:

where Si* is the saturation gas concentration, as determined by Henry's law:

The partial pressure of each component, pi, in the gas phase is calculated by means of a component balance around the gas phase giving:

where Cpj = P, i.e., the sum of the partial pressures equals the total pressure which in this case is assumed to be atmospheric. A total gas phase balance at steady state gives the volumetric flow rate of gas

Q:

Q=

(NH2 + NC02 + NCH4) VR M

As the biomass grows the thickness of the biofilm, Rf, is assumed to increase. Allowing for the spherical nature of the clay packing, a balance for the forming biomass film gives:

n ((Rp + Rf13 - Rp3) 24 3

=

x VR 1000

3.11 Anaerobic Fixed Film Processes

37 1

In the above equation, n is the total number of packing particles in the reactor, and p is the density of the biofilm. The factor of 1000 converts the reactor volume from Iitres to cm3.

Nomenclature F

H

Kd KLa KS M n N

P P

Q r

Rf RP

S t

VR VG X Xi

Y P

Pmax

Subscripts ac bt f 1

Pr

Volumetric flow rate of molasses Henry's constant First order cell decay constant Mass transfer coefficient Monod constant Molar volume Number of particles in reactor Gas production rates Partial pressure Head space pressure Volumetric gas flow rates Reaction rates Biofilm thickness Radius of particle Substrate concentration Time Volume of reactor Volume of head space Total biomass Biomass growing on substrate i Yield factors Density of film Maximum growth rate

L h-1 L atm mole-1 hh-1 g COD L-1 L mole-1 moles L-I h-* atm atm L h-l g L-I h-1 cm cm g COD L-l h

L L g L-1 g L-1 g L-1 h-1

Acetic acid Butyric acid Feed component Propionic acid

Exercises 1. Investigate the response of the system to step changes in the flow rate of glucose. 2. Does the volume of the head space influence the performance of the reactor?

372


3. Simulate the first 100 days of a reactor start-up with a molasses feed of 2.7 g COD L-' and a residence time VR/F = 3 h. Note that a large amount of HAc is formed initially. Identify a maximum in the H2 concentration and a C02 limitation period. When is a steady state attained? 4. It is assumed in this model that the reactions are not diffusion limited. Modify the model to include diffusional effects. To do this, divide the biofilm up into elements, and write mass balances for each element (see the example FLOCl and FLOC2). The following diffusion coefficients are needed from Table 3.7.

Table 3.7. Diffusion coefficients in water. Substance Glucose Butyric acid Propionic acid Acetic acid Hydrogen Carbon dioxide Methane

Diffusivity in water cm2 h-l 0.00 19 0.0021 0.0025 0.0027 0.042 0.0072 0.0042

Results nm

:.;\ 2.25

Carbon dioxide 0

15

30 45 Time (hours)

60

Figure 3.124. Exit gas composition during startup.

r Figure 3.125. Response to a change of flow rate from 0.08 to 0.2 L h-l at T = 23h.

Reference Denac, M., Miguel, A. and Dunn, I. J. (1988) Modeling Dynamic Experiments of the Anaerobic Degradation of Molasses Wastewater, Biotechnol. Bioeng. 31, 1-10.

3.12 Microbial Interaction Kinetics

3.12

373

Microbial Interaction Kinetics

3.12.1 MIXPOP - Predator-Prey Population Dynamics System The growth of a predator-prey mixed culture in a continuous biological flow process can be described via a reaction kinetics formulation. In this growth process, the dissolved substrate S is consumed by organism X1 (the mouse), while species X2 (the monster) preys on organism X I , as shown in Fig. 3.126.

Figure 3.126. Monster attacks mouse while it unsuspectingly feeds on substrate S.

Model The model involves the continuous flow balances for each species allowing for the corresponding kinetics. The variables are shown in Fig. 3.127, where the reactor is represented by a constant volume chemostat with sterile feed (X=O).

Figure 3.127. Chemostat predator-prey reactor.

374


Substrate balance, dS 1 = D(So-S1) --PI XI y1 dt

-

Species 1 (prey) balance,

dX _ _ I dt - P I X I - D X 1 Species 2 (predator) balance,

P2 x2

--

y2

where D is the dilution rate.

The kinetics of growth for both population species are given by Monod relations,

Nomenclature D F K S V

X

Y

P

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

Subscripts 0 1 2

m

Refers to feed stream Refers to prey Refers to predator Refers to maximum

h-I m3 h-l kg m-3 kg m-3 m3 kg m-3 kg kg-I h-

375


Exercises 1 . It is known that predator-prey systems exhibit stable oscillatory conditions. Investigate the influence of dilution rate and inlet substrate concentration on the behaviour of this system. Phase plane plots, in which the concentrations X I , X 2 or S1 are plotted against each other, are useful to exhibit the oscillatory behaviour.

2. The initial conditions may have an influence on the response curves obtained, as do, of course, the kinetic parameters. Investigate this feature of the model by simulation. 3. Set the value of D and simulate for the following: a) to cause washout of X I , b) to cause washout of X2, and c) batch behaviour. 4. Examine the influence of the relative growth rates by varying the p m l / p m 2

ratio in a range from 1 to 10.

5. Study the influence of substrate affinity by varying the values of the kinetic parameters K1 and K2. 6. Vary D between 0.1 and 1.0 interactively. 7. Increase So interactively to provide more substrate to the system.

Results m- 6 x2

5 5 r

4

.P 3 c Q

b 2 E $ 1 E O 0 0

100

200 300 400 Time (hour)

500

Figure 3.128. Oscillations in the microbial populations caused by setting p 1 = O . 5 , p2=0.49.

0'

0

-

'

1

-

'

2

-

x1

-3

.

'

4

Figure 3.129. Phase plane diagram of above graph showing unstable oscillations.

376


3.12.2 TWOONE - Competition Between Organisms System Consider organism A and organism B with their respective specific growth rates, p~ and pg, and which both grow independently on substrate S. Assume the kinetics S

pB

= pMB

S

K ~ ~ + s

Depending on the values of p~ and Ks. these two functions may occur in two different forms, as shown in Fig. 3.130.

< P

P

D1=

P

b

S1

S

P' pinter

L .

7 Sinter

s

Figure 3.130. Comparison of growth rate curves for the competitive chemostat growth.

It is clear that the curves B and A will cross each other if ~ M Pinter , A will grow the faster, causing B to be washed out.

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

In addition, the Monod relations, P A = f(S) and pg = f(S), are required. Solution of these equations will simulate the approach to steady state of A and B competing for a single substrate.

Nomenclature hm3 h-1 kg m-3 kg m-3 m3 kg m-3 kg kg-’ h-

P

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

Subscripts A B M 0 inter

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

D F KS

S V

x Y

378


Exercises 1. Vary ~ M and B MA and plot S versus p~ and p ~ Note . the point of intersection, Sinter. 2. Do the same as in Exercise 1, but varying KSB and KSA.

3. Vary D around the pinter point, and observe washout of either XA or XB.

4.Try to operate exactly on the Sinter conditions.

Results n

5

r

1.0 0.8 0.6

9! ! 0.4

c

3 E Q

0.2 0.0

0 1 2 3 4 5 6 Substrate concentration (kg/m3)

Figure 3.131 In the above graph ~ M B MA and KSB < KSA therefore the curves do not cross.

3.12.3 COMPETE - Lotka-Volterra Model of Competition System The Lotka-Volterra model for predator-prey interaction was developed in the late 1920s and has been applied to numerous different ecological interactions, e.g., mixed microbial cultures, foxes and rabbits, lynxes and hares, fish populations etc. The assumptions of this simple model are that the prey in the absence of predator grows in an unlimited manner, the rate of predation is proportional to the product of the two populations and that the prey is the only source of food for the predator which dies out in the absence of prey.

379


Model The Lotka-Volterra model can be expressed mathematically by the following differential equations.

where N1 and N2 are the prey and predator populations respectively. y is a coefficient related to the efficiency of the predator killing and consuming the prey at each encounter. The factor E is the amount by which the predator population increases per kill of prey. a is the rate of growth of prey in the absence of predator and b is the death rate of the predator in the absence of Prey * The steady-state population levels are found by setting the left-hand side of equations 1 and 2 to zero:

Nomenclature Growth rate of prey Death rate of predator Increase in predators per kill Population fraction of prey Population fraction of predators Efficiency of predation Number of prey Number of predators Initial numbers of prey Initial numbers of predators

T-1 T-1 -

Exercises 1. Plot the phase plane graph (N1 against N2) and note the resulting trajectories. 2. Vary the starting conditions and see how these influence the population fluctuations.

380


Results

Figure 3.133. Oscillation in populations for the prey and the predator.

Figure 3.134. Phase plane plot of the same run for the oscillations of the populations.

References Lotka, A. J. (1920) Undamped Oscillations Derived from the Law of Mass Action. J. Am. Chem. SOC.42. 1595. May, R. M. (1973) Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA.

3.13

Ecological Population Studies

3.13.1 BLOW FLY - Cycling Populations of the Austra ian Blowfly System The Australian sheep blowfly (Lucila cuprina) is an important pest in Australian sheep farming. A regular periodic oscillation in the numbers of blowfly can be observed, the period being 35 to 40 days. May (1975) modelled the phenomenon by means of a simple delay model. In this, the logistic growth model is extended by assuming that the regulatory effect is proportional to the population of flies existing at an earlier date and not to the current population level. The length of the delay is assumed to be the time required for the larva to mature to adult.

3.13 Ecological Population Studies

381

Model The following differential delay model is used to model the change in the numbers of the blowfly population:

_ _ _- -r N(t) (1 - N(t K- T)) dN(t) dt where N(t) is the current population and N(t-T) is the population at some prior time. T is the time required for the larva to mature into adult form, r represents the maximum linear growth rate and K is a constant which is related to the available food levels.

Program A fixed step integration routine (ALGO=2) is used so that each time the DYNAMIC section is executed a defined time interval elapses and which is equivalent to the integration step length. With a second-order numerical integration the DYNAMIC region is executed every CINT/2 units of time. The value of the time delay is thus set by the value of CINT.

Nomenclature K N(t), NO N(t-T), Ndelay r T

Food availability constant Number of blowflies at time t Number of blowflies at time t-T Maximum linear growth The delay in the regulatory effect

Exercises 1. Plot the phase plane of the actual population No versus the delayed population Ndelay and note the limit cycle behaviour. 2. Vary the food availability parameter K in the range of 1000 to 10000 and see how this effects the period and amplitude of the oscillations. 3. Vary the time delay and determine the value needed for oscillations to occur. Delays longer than 12 days cause the model to fail because of the integration step length. This problem can be overcome by increasing the number of previous data points saved (currently 10) and by reducing the integration step length. An alternate time delay procedure is given in Ingham et al. (1994).

382


4.Vary the initial conditions to see if there is any influence on the subsequent oscillations.

Results

.

15000

“0

. 100 . 200 . 300 . SAO Time (days)

Figure 3.135. Cyclic behaviour of the blowfly population, NO, with time.

0

I

5000

NO

10000 150( I

Figure 3.136. Phase plane diagram of Ndelay against the number of blowfly, No, from the same run showing limit cycle behaviour.

References May, R. M. (1975) Stability and Complexity in Model Ecosystems, 2nd Edition, Princeton University Press, USA. Nisbet, R. M. and Gurney, W. S. C. (1982) Modelling Fluctuating Populations, Wiley, New York, USA.

3.13.2 BUDWORM - Dynamics of the Growth of the Spruce Budworm in Canada System The spruce budworm can cause major problems in Canada by defoliating the balsam fir. An outbreak of budworm, during which balsam fir trees are denuded of foliage, generally lasts about four years. The trees then die and birch trees take over. Over the next 50 to 100 years the fir trees make a comeback and drive out the birch trees and the cycle is repeated. Note that the

383


Figure 3.137. The spruce budworm feeds on the balsam fir.

model here does not take into account the tree dynamics and considers only the dynamics of the budworm (Fig. 3.137). From a pest control point of view, the numbers of budworm need to be kept low in order to prevent an outbreak that would destroy many trees. However, the aim is not to wipeout the budworm completely, as the budworm also serves as a food source for some birds and other animals. A delicate balance in the budworm population dynamics is therefore required.

Model The growth rate of the spruce budworm, rg, can be described by the following equation: rg = rb n ( 1 --)

n

KB

where rb is the linear birth rate, n is the size of the population at time t, and KB is the carrying capacity which is related to the density of foliage available on the trees. Spruce budworms are prey to birds. At low densities of budworm the birds tend to seek food elsewhere and the level of predation is low. At high population densities, the rate of predation reaches a saturation value as there are not enough birds to eat all the budworms. This behaviour is best described by an equation of the form:

384


where A and B are constants. The rate of increase in the population of budworms is given by the difference between the rate of growth and the rate of predation, as in the following number balance dn n = rg-p = rbn( 1-K)B dt

-

B n2 A +n2

The equation can be expressed in dimensionless terms by the following substitutions: t which give rise to the dimensionless balance equation: dN N N2 n=RN(l--) ___ K -1+N2 Note that the dimensionless form of this equation only has two parameters, R and K, as compared to four parameters for the dimensional version of the equation. At steady state (i.e., dN/dt = 0), the following relationship must be satisfied: N2 N R N (1 - -) - ___ K -1+N2

Eliminating the solution N=O (i.e., extinction of the budworm) and rearranging gives: K N3-KN2+N(~+l)-K = 0 The cubic form of the equation means that (in addition to the trivial solution N=O) there will be one, two or three steady-state values of N depending on the values of K and R. The parameters from field observations confirm the existence of three naturally occurring steady states for the population. The lower steady state represents the refuge equilibrium, whereas the higher steady state is equivalent to the outbreak equilibrium. The middle steady state is unstable, as any slight deviation from this position will tend to give rise to a change to one or other of the other two steady states.

385


Program Note the nested DO loop around the SIM and RESET commands, which allows the simulation to be run several times using different values of K and R and with different initial values of N. The inner two loops are inactivated by the comment signs (:), which can be removed to activate all the loops.

Nomenclature A B KB

K n N P rb rg

R

t

T

Related to predation threshold Related to predation rate Carrying capacity Dimensionless constant Number of budworms Dimensionless number of budworms Predation rate Linear birth rate Budworm growth rate Dimensionless constant Time Dimensionless time

Budworm Budworm Budworm Budworm Budworm Time-' Budworm Time -

number number Time-I number number number Timenumber Time-l

Exercises 1. Vary the values of the dimensionless parameters K and R and note their effect on the resulting steady-state values for the budworm population. Change the starting conditions (i.e., No) and see how this effects the steady-state values.

2. Find values of K and R for which the model gives three steady states. Investigate different pest control strategies for maintaining the numbers of budworm at the lower steady-state value. For instance, (1) the introduction of more birds to increase the predation rate, (2) the spraying of foliage to reduce the carrying capacity, or (3) reduction of the birth rate. From the definition of the dimensionless numbers K and R, attempt to see how changes in field parameters causes changes in their values. 3. In the model described above, the carrying capacity is a constant, whereas in practice it will decrease as the number of budworms increase and consume the tree foliage. Include this effect in the model and see how it affects the steadystate values obtained.

386


Program For this model N=2 is an exact solution for the unstable steady-state. An initial value of N less than 2 will give the refuge equilibrium steady state (N=0.68), and an initial value greater than 2 gives the outbreak equilibrium (N=7.32). Hence a very small change in budworm number can lead to an outbreak.

Results

Outbreak equilibrium

n = 4

'

Z

0

20

40 60 80 Time (days)

100

Figure 3.138. Graph showing the three possible equilibrium states for the values K=10, R=0.5 for different initial values of N from 0.5 to 4.5.

Figure 3.139. Results obtained using the nested DO loops for two values of R and a range of K values: R=0.1 (K=2,4,....20) and R=0.2 (K=2,4, .... 16).

Reference Ludwig, D., Jones, D. D., Holling, C. S. (1978) Qualitative Analysis of Insect Outbreak Systems: the Spruce Budworm and Forest, J. Animal Ecol. 47, 315332.

3.14 River and Stream Modelling

3.14

387

River and Stream Modelling

3.14.1 RIVER - Dissolved Oxygen and BOD Dynamic Profiles along a River System The aquatic life of a river is dependent upon the maintenance of an adequate dissolved oxygen concentration along its whole length. The dissolved oxygen concentration in the river is influenced by several factors such as the inputs due to the discharge of water from sewage works or from industrial processes, agricultural run-off waters, the presence of tributaries, and aeration due to waterfalls and weirs. Models of dissolved oxygen profiles are useful in predicting the possible consequences of additional river discharges or aeration strategies on the river.

Figure 3.140. Flows into and out of a typical river.

Model There is generally little variation in the dissolved oxygen concentration across the width of the river, and if the river is not too deep the vertical differences are also unlikely to be very significant. It is therefore acceptable to model a river

388


in terms of variations with respect to the distance along the river only. In this, the river is divided into twelve sections or reaches. It is assumed that each reach is well-mixed and that the dissolved oxygen and BOD concentrations within each reach is therefore constant. In this development, the substrate is actually a mixture of many substances, and the biomass represents the many organisms involved in the aerobic degradation. A mass balance for dissolved oxygen and substrate can be written for the water in each reach. Additional flows from tributaries, entering a particular reach, are accounted for by a total mass balance and by additional terms in the component balances as follows: Qn = Qtribn + Qn-1

don dt

-

dS, dt

-

Qn-1

Qn-1

On - 1 + Qtribn Otribn - Qn Vn

on

Sn-1 + Qtribn Stribn- Qn Sn Vn

-

+ Ra,n -

Rbi0.n y

Rbio,n + Rrun,n

where Qn-l, Qtribn and Qn are the flows in and out of reach n, respectively. Ra is the aeration rate which depends on the river gradient (g) and velocity (v), according to Ra = Ka(O,at-O) where Ka is given by the empirical expression as Ka = 0.142 g v Osat is the saturated dissolved oxygen concentration, and this is a function of the river temperature, according to OSat = 14.652 - 0.41Temp + 0.008 Temp2 The velocity of the river,v, is related to the volumetric flow rate, Q, by

where Wn and dn are the width and depth of the river at each location n, respectively. It is assumed that the biological rate of substrate uptake depends on the substrate and oxygen concentrations according to overall second-order kinetics


389

The oxygen uptake rate is calculated with a constant yield as RbidY RrUn represents the BOD input to the river due to agricultural runoff, and is given by: - 2 L V n Cn Rrun,n - voln where L is the length of the reach, VOln is its volume and Cn is the specific agricultural loading. The factor 2 applies because BOD can enter from both banks of the river.

Program This example demonstrates the use of subroutines in ISIM. Each reach involves mass balances for substrate and oxygen and makes use of two subroutines to complete the model for each reach section. Thus for reach 6, midway along the river, the program is as follows: :Reach 6 ~6=Q6/(W6*D6) reaerate (Aer6,06,G6, V6, temp6 1 Call S6l=Q6*(SS-S6)/VO16-MU*S6 06i=Q6*(05-06)/Vo16-MU*06/Y+Aer6 call inflow(Q6,0.2,Q7,06,8,07,S7,Temg6,15,Temp7)

The subroutines are programmed as follows: INFLOW(Qr,Qtrib,Qd,O,Otrib,Od,S,Strib,Sd,Temp,TempTr ib,Temgd) Qd=Qtrib+Qr O=((Otrib*Qtrib)+(O*Qr))/Qd sd=((Strib*Qtrib)+(S*Qr))/Qd Tempd=((TempTrib*Qtrib)+(Temp*Qr))/Qd Return REAERATE(Aer,O,grad,velocity,temp) Osat=14.652-(0.41*Temg)+(0.008*Temg*Temp*T~mp~

Ka=O.l42*grad*velocity Aer=Ka*(Osat-0) Return

Note that the sequence of arguments in the call statement must correspond to those in the subroutine. The first two lines of INFLOW are actually one line in the program.

390


Nomenclature Aer C d g Ka L 0

Y Pmax (MU)

Aeration rate Specific agricultural loading Depth of river River gradient Aeration transfer coefficient Length of one reach Dissolved oxygen concentration Volumetric flow rate Rate of river aeration Rate of biological oxygen uptake Rate of BOD runoff inflow to the river Rate of runoff to reach n Substrate concentration River temperature Velocity of the river Volume of one reach Width of the river Yield factor (S/02) Maximum specific growth rate

Subscripts 0 n sat trib

Initial value Reach or section number Saturation Tributary

Q Ra Rbio Rrun

Runoff, S Temp V

Vol W

g m-3- s-1 g m-2 m m m-1 S- 1

m g m-3 m3 s-1 g m-3 sg m-3 sg m-3 sg m-3 sg m-3 OC

m s-1 m3 m S- 1

Exercises 1. Run the model and see which are the most important factors in determining the dissolved oxygen concentrations of the river. Include a weir in the river to improve reaeration, by changing the factor g as seen in the program for the 10th reach.

2. Vary the position of the industrial discharges into the river and see how this influences the water quality of the river. What other factors (social, political, geographical etc.) need to be considered when siting industrial outflows?

Results In Fig. 3.141, the initial decrease in oxygen concentration and increase in BOD is due to agricultural run-off. The increase in dissolved oxygen in reach 6 is

39 1


due to a less polluted tributary joining the main river, and in reach 10 the decrease is caused by a weir which improves aeration. I 200

n

"12

3 4 5 6 7 8 9 101112 Reach number

Figure 3.141. Dissolved oxygen (solid line) and BOD concentrations (dashed line) along the course of a river.

Ism

.-

.

Figure 3.142. Dynamics of the dissolved oxygen in reaches 2 to 1 1 of the river.

References Bach, H. K., Brink, H., Oleson, K. W. and Havno, K. (1989) Application of PCBased Models in River Water Quality Modelling, Hydraulic and Environmental Modelling of Coastal, Estuarine and River Waters (eds. Falconer, R. A., Goodwin, P, and Mattew, R. G. S . ) , Gower Technical, England. Beck, M. B. (1978) Modelling of Dissolved Oxygen in a Non-Tidal Stream, in Mathematical Models in Water Pollution Control (ed. James, A.), Wiley.

3.14.2 STREAM - One-Dimensional Steady-State Model of Aeration and Degradation in a Stream System In this model, steady-state BOD (organic carbon, ammonia and nitrite) and dissolved oxygen profiles along a river are simulated. The gradient and velocity are assumed to vary with position, so that the varies from a fast-flowing stream near the source to a slow-moving river towards its end. In the previous example RIVER, a river was divided up into a number of reaches and the dynamics in each reach were calculated using a tanks-in-series approach. In the present example, an alternative method is used which assumes the river to be at steady state. Thus the calculation gives the concentration-

392


distance profiles along the river, essentially like a plug-flow tubular-reactor. This type of model can be used when there are continuous and constant flows, but can not be used when there are time-variable flows or transient discharges into the river, for which the dynamic behaviour is important.

Model The simulation starts at position Z=O and continues downstream until Z=L. The profile of the river dictates the linear velocity at any downstream position, which is most simply modelled using IF statements to represent new inputs and outputs to the river. Since the velocity of the river is not constant, the following expression is used to define the relationship between time (s) and position (km): dZ dt

-

-~v - 1000

The mass balances are formulated below as batch dynamic equations, which are integrated with respect to time and then transformed into distance relationships using the above equation distance-time relationship. The organic carbon substrate in the river is assumed to be degraded with first order kinetics, according to:

As a result of this biodegradation, oxygen is consumed with a yield constant Y. In addition, oxygen is consumed due to nitrification, and the river is reaerated from the atmosphere. The steady-state oxygen balance is therefore given by:

As in RIVER, the rate of reaeration is given by:

where Ka is the reaeration coefficient, which is a function of the velocity (v) and gradient (8) of the river, according to the empirical relation: Ka = 0.142 g v The saturation oxygen concentration, Osat, is temperature dependent where: Osat = 14.652 - 0.41T + 0.008T2

393


Nitrification also consumes oxygen and the kinetics are assumed to be first order for the first and second steps, according to: dNH3 dt = - Kn NH3 dN02 dt = Kn NH3 - Ky NO2 dNO3 dt = Ky NO2 The nitrification rate constant depends on the temperature and velocity of the river where: 9.2783 - 0.6272 + 0.0789 T + 7

Additionally, if the oxygen concentration drops below 2 mg/L, then the rate of nitrification is reduced in proportion to the oxygen concentration. From the reaction stoichiometry the rate of oxygen uptake as a consequence of nitrification can be determined as:

Whenever a tributary or other liquid flow joins the river, the new flow rate, temperature, dissolved oxygen concentration BOD and ammonia concentrations must be recalculated by steady-state, mixing-point mass balances: Qnew = Q + Qtrib

0 Q + Otrib Qtrib

Onew = Snew = Tnew =

Qnew

S Q + Strib Qtrib Qnew

T Q + Ttrib Qtrib Qnew

3 94


Program Subroutines are used to calculate the new values of the river parameters every time a new stream (either a tributary, industrial, outflow or wastewater treatment plant effluent) enters the river. Also, a subroutine is used to recalculate the reaeration coefficient every time the temperature, velocity or gradient of the river changes. Note the use of the IFIX parameter to give the integer value of z to ensure that the subroutine is only ever called once when a new stream enters the river. The TERMINATE command is used in this model to stop the simulation when Z=Length, i.e., the end of the river.

Nomenclature Aer g Ka Kn KY Length NH3

NO3 NH3

0

Q

Aeration rate Gradient of river Reaeration coefficient Nitrification constant Nitrification constant Length of river Concentration of ammonia Concentration of nitrate Concentration of nitrite Dissolved oxygen concentration River volumetric flow rate

r

Reaction rate

rnit (Nit) S T (Temp)

P M (MU)

Oxygen uptake rate by nitrification BOD Temperature Velocity of river Yield coefficient Distance downstream Maximum specific growth rate constant

Subscripts 0 aer d new nit r trib

Initial Aeration from the atmosphere Downstream New values of parameters below a tributary stream Nitrification River Tributary

V

Y Z

395


Exercises 1. Compare this model with the simulation example RIVER. What are the advantages and disadvantages of each type of model? Under what circumstances would this kind of model be most useful?

2. Devise a mathematical expression that to give more realistic changes in the river velocity and gradient.

Results

5,m

i

Figure 3.143. Steady-state profiles of BOD and dissolved oxygen in the stream.

2,m i

Figure 3.144. From the same run, steadystate profiles of ammonia, nitrite and nitrate in the stream.

References Bach, H. K., Brink, H., Oleson, K. W. and Havno, K. (1989) Application of PCBased Models in River Water Quality Modelling, Hydraulic and Environmental Modelling of Coastal, Estuarine and River Waters (eds. Falconer, R. A., Goodwin, P, and Mattew, R. G. S.), Gower Technical, England. Beck, M. B. (1978) Modelling of Dissolved Oxygen in a Non-Tidal Stream, in Mathematical Models in Water Pollution Control (ed. James, A,), Wiley.

396


3.14.3 DISCHARG - Dissolved Oxygen and BOD Steady-State Profiles along a River System Wastewater treatment plants and most factories discharge water into a river system. The wastewater will differ from the river water in temperature, BOD and dissolved oxygen concentration and nitrogen content. If the wastewater is discharged at an approximately constant rate then it may be assumed that steady state will be attained at some point along the length of the river with respect to temperature, BOD and dissolved oxygen. The minimum dissolved oxygen concentration likely to be encountered in the river and its position downstream from the point of the discharge can therefore be predicted.

Figure 3.145. Discharge of a treatment plant effluent into a river.

Model Wastewater flows into a river and is immediately mixed with the river water. The flow rate of the river is increased as a result, where

397


Similarly, the dissolved oxygen concentration (DO), BOD ( S ) , nitrates (N) and temperature of the river will also be effected according to DO = DOr Qr + DOw Qw

Q

S = Sr Qr + Sw Qw

Q

Temp = Temp, Qr + Temp, Qw

Q

N = Nr Qr + Nw Qw

Q

The dissolved oxygen deficit can be defined as the difference between the actual dissolved oxygen concentration and the saturation concentration for oxygen in the river at the same conditions.

The saturated dissolved oxygen concentration can be related to the temperature: Dosat

=

14.652 - 0.41 Temp

+ 0.008 Temp2

Oxygen is consumed by micro-organisms as they utilise the substrate and also as a result of nitrification, but oxygen is gained by the river by processes of surface aeration, so that:

The rate of nitrification is assumed to be first order with respect to nitrate concentration, and the value of the rate constant, Kn, is a function of temperature.

The rate of pollutant degradation is also first order and temperature dependent:

K, = Ks20 1.OSTemP- 2o

398


The rate of reaeration is dependent on the velocity, v, of the river and its depth D, where 2.26 v Ka20 =

D0.667

The reaeration rate is also temperature dependent: Ka20 e 0.024(Temp - 20) K, = 24

Assuming steady-state conditions, the distance downstream (km) can be related to the time of passage along the river (hours) by:

z = 3.6 v t The velocity of the river is related to the volumetric flow rate and the river depth: v = 3600QD W where this assumes approximately rectangular cross-sectional area of the river, W being the width.

Nomenclature E

a D DO Dosat Ka

Kn KS

N

Q S t

Temp

Oxygen deficit Stoichiometric parameter Depth of river Dissolved oxygen concentration Saturated dissolved oxygen concentration Reaeration coefficient Nitrification rate coefficient Biodegradation rate coefficient Nitrate concentration Volumetric flow rate BOD substrate concentration Time Temperature

mg L-1 m mg ~ - 1 mg h-' h-I h-' mg m3 mg h OC

L-'

L-1 h-1 L-1

399


V

W z

Subscripts r W

20

Velocity of river Width of river Distance downstream River Wastewater Reference temperature

m s-1 m km

(OC)

Exercises 1. Compare the results obtained with those for the models OXSAG and RIVER. What assumptions are appropriate to each particular model, and how valid are these likely to be in reality? 2. Investigate the discharge of a pollutant into rivers of different sizes, flow rates and velocities. What is the best and the worst type of river for the discharge of industrial wastewaters (from the point of view of minimising environmental damage)?

3. Model seasonal variations in temperature, flow rate and river depth. At what time of year is the river most at risk?

4. Is there any difference between the discharge of a small volume of highly concentrated waste and a large volume of more dilute waste if the overall amount of pollutant is the same? 5 . Is it more worthwhile to reduce the temperature of the wastewater by 10% or the concentration by lo%? Is this always the case or does it depend on the

absolute values?

400


Results 12’

c 7.0 a6.0

rE O

Dissolved

55.0

l?g

2 6 4.0

8.

2 P

8 3.0 2.0

0

60 120 180 240 300 360 Distance downstream (m)

Figure 3.146. Dissolved oxygen profile for the discharge of three different wastewaters (15, 25 and 35OC). Note that the higher temperature leads to a lower minimum dissolved oxygen concentration that occurs nearer to the discharge point, but also recovers more quickly.

0

60 120 180 240 300 360 Distance downstream (m)

Figure 3.147. Dissolved oxygen, BOD and nitrate concentration-time profiles (25OC wastewater).

Reference Jorgensen, S. E. (1986) Fundamentals of Ecological Modelling, Elsevier.

3.15

Lake and Reservoir Modelling

3.15.1 NCYCLE - Nitrogen Cycles in a Reservoir in Slovakia System Nitrogen cycles in the Slnava Reservoir in western Slovakia have been studied intensively for many years as part of a collaboration between the Slovak Academy of Science and the International Institute for Applied Systems Analysis in Austria (Toth and Tomasovicova, 1989). The reservoir has a volume of 12 million cubic metres and has a catchment area of 4,000 ha of intensively used arable land. The amounts of nitrogen fertilisers and pesticides

40 1

3.15 Lake and Reservoir Modelling

applied to the land each year in this region are 500,000 kg and 20,000 kg, respectively. It is estimated that between 10 and 30% of these amounts reach the reservoir, where they accumulate in the sediments. Some of these pesticides persist for more than five years. The River Vhh flows into the Slnava Reservoir, and this is heavily polluted by chemical and paper factories upstream. The water within the reservoir is used for hydroelectricity generation and for industrial uses. It can not support fish nor can it used for drinking water.

Model

Dissolved

D4

4 % -

D3

D2 Figure 3.148. Nitrogen cycle in the aquatic environment.

Overview Nitrogen-containing compounds in the reservoir are interconverted according to the scheme depicted in Fig. 3.148. Nitrogen-containing compounds are taken up by bacteria and plankton, converted and then excreted. The biomass also dies and contributes to the detritus (particulate) fraction of the total nitrogen. Some of the processes are temperature dependent (see exercises). The variables N1 to N9 represent nitrogen concentrations contained in the organic and inorganic substances and living organisms specified as follows: Micro-organisms 1 Heterotrophic bacteria 2 Nitrosomonas 3 Nitrobacter 4 Phytoplankton 5 Detritus (particulates)

402


Dissolved material 6 Dissolved organic nitrogen (DON) 7 Ammonium N ,8 Nitrite N 9 Nitrate N

Reservoir Water flows into the reservoir at a volumetric flow rate Q and thus has a mean residence time of V/Q. It is assumed that the contents of the reservoir are well mixed. Substrate Uptake The different micro-organisms have different substrate preferences, i.e., Heterotrophic bacteria Nitrosomonas Ni trobacter Phytoplankton

Dissolved organic nitrogen Ammonia Nitrite Ammonia + Nitrate + DON

The specific rate constants for substrate uptake (g substrate/g biomass day) are based on Monod kinetics: Nit5 Si = vmax;i Ksi + Ni+5

for i = 1, 2, 3

where Vmax,i is the maximum uptake rate coefficient for organism i and Ksi is the substrate saturation constant, respectively. With plankton the situation is a little more complicated because plankton are capable of taking up DON, ammonia and nitrate. It is assumed that these processes occur simultaneously and are independent. The nitrogen-containing compounds that can be utilised by the plankton are lumped together into a total nitrogen pool (P) with coefficients (U), which are used to describe the preferential uptake of some nitrogen compounds over others.

The total specific uptake rate of the nitrogen-containing compounds by phytoplankton is therefore given by:

The individual nitrogen uptake rates by phytoplankton for DON ammonia and nitrate, respectively, are:

403


u6 N6 P(j = Vrnax:4 Ks4+ P

U7 N7 P7 = vmax;4 Ks~+ P

Excretion The excretion rate constants for ammonia by heterotrophic bacteria, nitrite by nitrosomonas and nitrate by nitrobacter are assumed to be given by the following expression:

where Kei is a saturation constant, and Emaxi is the maximum rate of excretion per unit biomass. Similarly the excretion rate constant of DON by the phytoplankton is given by:

Mortality The mortality rates for all the micro-organisms are assumed to be of the form:

where Ci and KD,i are constants and Ei is the excretion rate constant. Thus there is a constant rate of mortality when the concentrations of nutrients are low. When the nutrient concentration increases, the organism activity also intensifies and the specific rates of mortality increase.

Micro-Organisms The following groups of organisms are considered in the model: i.e., heterotrophic bacteria, nitrosomonas and nitrobacter species, and phytoplankton. Heterotrophic Bacteria The heterotrophic bacteria utilise dissolved organic nitrogen as substrate (S I ) , excrete ammonium ions (El) and exhibit a first order mortality rate (D1).They flow into and out of the reservoir at a dilution rate of QN. A mass balance for heterotrophic bacteria gives:

404


Nitrosomonas Nitrosomonas species utilise ammonium ions as substrate (S2), excrete nitrite ions (E2) and exhibit a first order mortality rate (D2). A mass balance similar to that for the heterotrophic bacteria can be derived, giving:

Nitrobacter Similarly, nitrobacter utilise nitrite and excrete nitrate. A mass balance on nitrobacter in the reservoir has the following form:

Phytoplankton Phytoplankton take up dissolved organic nitrogen, ammonium and nitrate ions and excrete dissolved organic nitrogen as described above, such that.

Nitrogen Compounds Dissolved organic nitrogen, ammonium, nitrite and nitrate are the major nitrogen-containing compounds of this model. Stepwise transformation of these compounds in natural waters depends on bacterial activity. The kinetics of these transformations is assumed to be first order. Dissolved Organic Nitrogen (DON) DON is produced by the decomposition of detritus (K5)and is excreted by phytoplankton. It can sediment out (K6)and be taken up by heterotrophic bacteria and phytoplankton and flows in and out of the reservoir. A mass balance for DON in the reservoir gives:

Ammonium (NH4) Ammonium ions are excreted by heterotrophic bacteria and taken up by nitrosomonas species and by phytoplankton. A mass balance for the NH4 in the reservoir gives:

405


Nitrite Nitrite ions are excreted by nitrosomonas species and taken up by nitrobacter. A mass balance gives:

~dN8 dt -

E2 N2 - s3 N3 -

Q v (N8 - N8in)

Nitrate Nitrate ions are excreted by nitrobacter and taken up by phytoplankton. Hence:

d2

v

_ _-- E3 N3 - P9 N4 - Q (N9 - Ngin)

Detritus Nitrogenous detritus is produced from dead micro-organisms at a rate proportional to the biomass. The detritus decomposes to give DON, and it is assumed that this decomposition follows first-order kinetics. A mass balance for detritus in the reservoir gives:

Dissolved Oxygen The dissolved oxygen concentration is determined by the processes of atmospheric reaeration, the biochemical oxidation of nitrogenous compounds, plankton respiration and photosynthesis. The mass balance for oxygen in the liquid phase of the reservoir is then :

The rate of atmospheric reaeration is assumed to be proportional to the difference between the saturation dissolved oxygen concentration and the actual dissolved oxygen concentration:

The uptake of oxygen by respiration is related to the organism activity and is modelled by the following equation:

where q is a stoichiometric parameter.

406


Nomenclature Ci Di DO Dosat Ei Emaxi K5 K6 KD,i

Kei KLa Ksi Ni

P Pi

Q 9i Rres

Rae,

Si t Ui V Vmax,i

Mortality rate constant (i=l to 4) Mortality rate constant (i=l to 4) Dissolved oxygen concentration Saturated dissolved oxygen concentration Excretion rate constant(i=l to 4) Maximum specific excretion rate Detritus decomposition rate constant Sedimentation coefficient Constants for cell mortality (i=l,4) Excretion rate constant (i=l to 4) Reaeration coefficient Constants for limiting substrate uptake kinetics (i=l to 4) Concentration of nitrogen in different fractions (i=l to 9) Pooled nitrogen fraction Phytoplankton substrate uptake rate constants (i=6,7,9) Volumetric flow rate of water into reservoir Stoichiometric coefficients (i=l to 4) Rate of 02 uptake by respiration Rate of surface aeration Substrate uptake rate constants Time Phytoplankton preferences for DON, NH4 and nitrate (i=6,7,9) Volume of reservoir Maximum substrate uptake rates (i=1,4)

day-' dayg m-3 g m-3

dayday- 1 day- 1 dayg m-3 dayg m-3

g m-; g m-

daym3 day-' g m-3 day-' g m-3 day-' daydays

m3 day-

Subscripts in 1

Value in the water flowing into the reservoir Nitrogen fractions 1 to 9

Exercises 1. Run the model as it stands and observe the accumulation of nitrogencontaining compounds.

407


2. The diurnal variation in temperature T can be approximated by a sine function: T = Tav + Tamp sin 2n t where Tav is the average temperature and Tamp is the amplitude. The reaeration coefficient KLa is temperature dependent: KLa = K ~a 2 01.05 T - 2 0

0

where K~a20is the reaeration coefficient at 2OoC and T is the temperature. The saturated dissolved oxygen concentration is a function of temperature (T) and can be estimated from the following polynomial function: Dosat = 14.61996 - 0.4042 T

+ 0.00842 T2 - 0.00009 T3

Change the simulation program to describe these temperature effects.

3. Pesticide enters the reservoir from the surrounding land. There undergoes first-order biodegradation and inhibits bacterial activity. A mass balance gives:

where Kd is the first-order degradation constant. Vary the concentration of the pesticide, and see how this upsets the nitrogen cycle. Investigate how long it takes to return to normal after several years of high pesticide concentrations.

Results

ISM

Figure 3.149. Response of the compounds nitrite (N8) and nitrate (N9).

Figure 3.150. For the same run, response of the microbial biomass in the reservoir: N1 - Heterotrophic bacteria, N2 - Nitrosomonas. N3 - Nitrobacter.

408


Reference Toth, D. and Tomasovicova, D. (1989) Microbial interactions with Chemical Water Pollution, Chapter 6, Ellis Horwood Series in Wastewater Technology.

3.15.2 PCYCLE

- Phosphorus Cycles in a

Lake

System Phosphorus is generally regarded to be the critical nutrient in lake eutrophication. Consequently attempts are being made to reduce the phosphorus levels in wastewaters which are discharged into natural waters (see Sec 2.6.1.4). In the lake phosphorus can be taken up by micro-organisms and can also accumulate in the sediment. The cycling of phosphorus in natural waters is also dependent on the phytoplankton population which in turn is strongly influenced by the climatic conditions, in particular temperature and the hours of sunlight per day.

Model

Figure 3. 151. Phosphorus cycling in natural waters.

409


This simulation example is intended to investigate the influences on the phosphorus cycles in a shallow lake or in the epilimnic upper layer of a deep lake and is based on the work of Kmet and Straskraba (1989).

Climatic Variations The water temperature varies throughout the year with a range from 2 to 22OC. TEMP = 1 2 + 10 sin

2 n (t + 220) 365

where the condition in January 1st applies to t = 0. The number of daylight hours per day varies from 8 to 16:

2n t PHOTOP = 12 - 4 cos 365

Light Intensity The average light intensity at the surface of the lake also varies throughout the year: 2n (t + 220) INT = 1200 + 800 sin 365 The total light energy per day (J cm-2) is given by the product of the intensity and the photoperiod:

I* = INT

PHOTOP 24

The average light intensity over the water depth depends on the light extinction factor E and the depth of the fully mixed lake region Zmix where:

Photosynthetic Growth Phytoplankton photosynthesise at a rate which is dependent on the temperature, light intensity, phosphorus concentration and biomass concentration. The temperature dependence is modelled by: PTmax

=

0.3665 e 0.09 TEMP

where PTmaxis the maximum rate of photosynthesis at the temperature of the water.

410


The light dependency is modelled by:

where IK is the photosynthesis light sensitivity parameter, and P1,ax is therefore the maximum rate of photosynthesis at a given temperature and light intensity. The rate of photosynthesis is also influenced by the concentration of phosphate, F. It is assumed that the kinetics are of Monod form, with constant Ks. Thus: PSmax = Plmax

F

The maximum rate of photosynthesis at a given temperature, light intensity and phosphate concentration is therefore Psmax. The rate of photosynthesis decreases with increasing biomass density, primarily due to self-shading, i.e., the density of biomass blocks out the sunlight so the light intensity is much lower than in clear water. The overall rate of photosynthesis can therefore be estimated from:

Sedimentation To simulate the annual changes of stratification in a deep water body with respect to the effect on photoplankton sedimentation, a periodic function STRAT with maxima in spring and autumn and a minimum in mid-summer is used: 2lr t STRAT = 0.8 + 0.25 cos 365 - 0.12 cos (0.0349 t) The rate of phytoplankton sedimentation is then calculated from the following expression: dS dt = S , CHP where S , = SINK STRAT P and CHP is the conversion factor relating chlorophyll to phosphorus.

Phytoplankton Phytoplankton take up phosphorus during photosynthesis and release phosphorus during respiration. They die by natural mortality and by predation. Zooplankton graze on the phytoplankon and assimilate their

41 1


phosphate content. Phytoplankton can also settle out and accumulate in the sediment. A mass balance on phytoplankton chlorophyll (g phosphate m-3 day - l ) in the lake gives: dP dt = PH P - R, - S , - Prey + D (Pi" - P) where D is the lake dilution rate and Rp is the rate of respiration. temperature dependent and is modelled by:

Rp is

Rp = RESP TEMP P where RESP is the specific rate of respiration. The rate of sedimentation S, has been derived previously. The rate at which zooplankton graze on the plankton is proportional to the phytoplankton and zooplankton concentrations and is given by: Prey = Graze P Z where GRAZE is the zooplankton grazing rate.

Zooplankton The rate of predation of zooplankton on the phytoplankton is assumed to follow Monod-type kinetics: P Pred = Graze k8 Z and k8 is the kinetic constant for phytoplankton uptake. A mass balance for the zooplankton phosphate (g phosphate m-3 day - l ) in the lake gives: dZ dt = - D, - RZ + CHP Eff Pred + (Zin - Z) D where D, is the first order zooplankton mortality, CHP is a factor converting g chlorophyll to g phosphate and Eff is the efficiency of the zooplankton in assimilating phosphate from the phytoplankton. R, is the respiration rate which is assumed to be independent of temperature and is modelled by: R,=RESP Z where RESP is the specific rate of respiration. Similarly: D, = MORTZZ

412


and MORTZ is the specific mortality rate.

Phosphate Phosphate enters the lake water via the following mechanisms: 1. Respiration of the phyto- and zooplankton (3rd term), 2. The inefficiency of the grazing of the zooplankton on the phytoplankton (4th term), 3. The death of zooplankton (2nd term) and 4. In the water flowing into the lake (5th term). Phosphate is taken up by the phytoplankton during photosynthetic growth (1 st term) and leaves in the water flowing out of the lake (5th term). A mass balance on phosphate in the lake thus gives the following expression:

dF

dt = -

CHP PH P + Dz+ CHP (R,

+ Rp) + (1-Eff)

Red + (Fin - F) D

Nomenclature CHP

D

DZ E Eff F GRAZE I

IK INT

KS MORTZ P PTmax Plmax

Conversion factor chlorophyll to phosphate Lake dilution rate Mortality rate loss of phosphate Light extinction factor Efficiency of zooplankton grazing Phosphate concentration Rate of zooplankton grazing Average daily light intensity over depth of lake Photosynthetic light sensitivity parameter Average daily light intensity at lake surface Monod constant for zooplankton predation Specific mortality rate Phytoplankton chlorophyll concentration Maximum rate of photosynthesis with respect to temperature Maximum rate of photosynthesis with respect to light

day-' g m-3 day-] m-1 g m-3 day-'

J cm-2 J cm-2 J cm-2 day-' g chlorophyll m-3 dayg m-3 dayday-'

413


PSmax PH PHOTOP Pred Prey

Q

R ESP S SP

SINK STRAT t TEMP V Z Zmix

gubscripts in f P Z

Maximum rate of photosynthesis with respect to substrate Overall rate of photosynthesis Number of hours sunlight per day Rate of chlorophyll predation of zooplankton on phytoplankton Rate of chlorophyll loss of phytoplankton by zooplankton predation Volumetric flow rate of water into lake Rate of chlorophyll respiration Specific rate of respiration Concentration of phosphate in sediment Sedimentation rate of phytoplankton chlorophyll Sedimentation coefficient Periodic sedimentation function Time Temperature Volume of lake Zooplankton concentration Depth of fully mixed layer

daydayhours g m-3 day-' g m-3 day-' m3 day-' g m-3 day-' dayg m-3 g m-3 day-' daydays

OC m3 g phosphate m-3 m

Input values Phosphate Phytoplankton Zooplankton

Exercises 1. See how the volume and depth of the lake influence the water quality. Is a deep lake more at risk than a shallow lake of equal volume? 2. Simulate the addition to the lake of a fish species that preys on the zooplankton. How will this influence the water quality of the lake?

414


Results m n 5

E 5 4

40 6

5 4 n 0

loo

200

300

Time (days) Figure 3.152. Dynamics of the phytoplankton and phosphorus concentrations.

Figure 3.153. Algal blooms caused by different phosphate loadings (50 to 250 glrn3).

Reference Kmet, T. and Straskraba, M. (1989) Global Behavior of a Generalized Aquatic Ecosystem Model, Ecological Modelling 45, 95-1 10.

3.15.3 ALGAE - Algal Growth in a Deep Lake in Canada System Kootenay Lake in British Columbia, Canada has a surface area of nearly 500 km2 and a mean depth of over 100m. The lake is long and narrow and about 100 km along its north-south axis. The Kootenay River enters the lake from the south and water leaves through a long shallow arm extending westwards from a point midway along the lake. Two algal species are common in the lake: one which is prevalent in August and September, consists largely of Phacotid and the second, prevalent in April and May is dominated by Cryptomonas and Stephanodiscus. The zooplankton can be loosely divided into cladocerans (Daphnia, Bomina, Diaphanosoma) and copepods (Diaptomus, Cyclops). Parker (1973) developed the following model, used in this example.


415

Model

Figure 3.154. Schematic of major processes in a deep lake.

River Kootenay The flow rate of the Kootenay River, Q, is not constant but shows seasonal variations, which are modelled by a sine function:

where Qav and Qamp are the average and amplitude of the variation in the river volumetric flow rate respectively. The dilution rate D (day-l) is given by:

Climatic Conditions The number of hours of daylight, DayLight, and the average daily temperature also show seasonal variations: 2nt n DayLight = DayLightav + DayLightampsin (365-

z)

416


27Ct It sin (365- 2)

TEMP = TEMPaV+ TEMP,,,

Photosynthesis The average light intensity over the depth of water considered depends on the self-shading by the algae (A1 + A2) where. EF = EFo + 0.1 (A1 + A2) EFo is the light extinction of clear water and EF is the light extinction factor in water with algae. The light energy available to the algae for photosynthesis is dependent on the light extinction factor, the number of hours of daylight per day, the depth of water, and the light intensity at the liquid surface:

The rate of photosynthesis of the algae under defined conditions of illumination is dependent on the light energy actually used, IA , given by

IA =

I S

I

exp(1 - -)

IS

where Is is the optimum illumination intensity. The rate of photosynthesis for each algal species is also influenced by the temperature:

Note that the temperature dependencies for the two groups of algae are different, and this therefore accounts for their appearance at different times of the year. The rate of photosynthesis is also influenced by the nutrient concentration, where phosphor and nitrogen compounds (ammonia and nitrate) can both be rate limiting. No distinction is made between the uptake of nitrate (N2) and that of ammonia (N3), and both are lumped together in a Monod style relationship: N1 N I A = 0.01 + N 1


417

Growth Rates The overall algal growth rates have maximum values which are modified by temperature, light intensity, and nutrient concentrations, as described by

GA2 = 8 IA2 TA2 NlA N2A The growth rates for the zooplankton species depend on the temperature and on the abundance of the algae on which they feed, where:

Gcl = 0.47 TEMP B (0.7 A1 + 0.1 A2) Gc2 = 0.47 TEMP B (0.3 A1 + 0.15 A2) and B is a factor depending on the total algal concentration:

Mortality The rate of mortality includes the rate of sedimentation, and the balance between respiration and the rate death from natural causes. These rates are assumed to be temperature dependent and take the combined general form of M = &TEMP

Predation The zooplankton prey on the algae at a rate which is temperature dependent and which is also dependent on the concentrations of the zooplankton. Both species of zooplankton feed on each species of algae but with different efficiencies. The efficiency of the predation increases as the algal concentration increases, and the rates of predation differ for each of the two algal classes: PA1 = TEMP B (0.7 C1 + 0.3 C2)

PA^

= TEMP B (0.1 C1 + 0.15 C2)

The zooplankton are themselves preyed upon by fish with a predation rate which follows first order temperature dependent kinetics:

Pc2 = 0.5 TEMP C2

418


Upwelling The upwelling of phosphates, nitrates and ammonia occurs at a constant rate which is proportional to the difference in concentration between the upper wellmixed layer and that of the deeper water. The concentration of the deeper water is assumed to be equal to that of the upper water at the time of the autumn overturn (about week 46) and then remains constant for the rest of the year.

where i is 1, 2 or 3.

Biomass Balances Balances for the two algal groups in the lake give

Similarly for the two classes of zooplankton:

Phosphate Balances Phosphate is taken up by the algae and the phytoplankton during growth. Phosphate is also released due to the less than 100% efficiency of predation, and also enters due to upwelling and in the input flow to the lake from the Kootenay River. A mass balance for phosphate gives _ dN1 _-- PredN1- GrowthN1 + D (Nlin - N1)

dt

where: and

+ UPWELLN~


419

Nitrogen Balance The nitrogen balances for the nitrates and ammonia have a similar form as that for phosphate. The rate of nitrates uptake is assumed to be proportional to the fraction of the total nitrogen containing compounds, i.e., N2/(N2+N3), where:

Similarly for ammonia:

Nomenclature A1 A2 B C1 c2

D Day Light EFO EF G GROWTH H I IA I0 IS

K

Kd

M N1 N2 N3 P Pred

Q

Phacotid algae Cryptomonas and Stephanodiscus algae Algal predation efficiency Cladocera Copepoda Dilution rate of lake Number of hours of daylight per day Extinction factor for clear water Extinction factor for light in lake Growth rates of algae Uptake rate of components by growth Depth of well-mixed layer Light energy available for photosynthesis Fraction of light actually used Light intensity at lake surface Optimum light intensity Zooplankton predation efficiency factor First-order mortality constant Mortality rate Phosphate concentration Nitrate concentration Ammonium concentration Predation rate for algae Release of components by predation Flow rate of Kootenay River

Zooplankton Zooplankton dayh m-1 m-1 dayg rw3day-' m cal cm-2 day-' cal cm-2 day-' cal cm-2 day-' m3 g-1 day-' OC-l dayg m-; g m-3 g mday-' g m-3day-1 m3 day-'

420 t

TEMP T Ts

UPWELL V

Subscripts av amp A1 A2 c1 c2 Nl

N2 N3 0 in


Time Temperature Temperature function for growth Temperature constant for algal growth Rate of upwelling of components Volume of lake

day OC

-

OC g m-3 day-'

m3

Average Amplitude Phacotid algae Cryptomonas and Stephanodiscus algae Cladocera Copepoda Phosphate Nitrate Ammonium Upwell value Inlet

Exercises 1. Run the model with variable daylight and river flow rates. What climatic conditions give the most severe algal blooms? 2. Vary the nutrient flows into the lake and see if this affects the appearance of algal blooms. What reduction in nutrient loading is required to prevent this problem? How many years would it take for the lake to be returned to normal?

Results

Figure 3.155. Variation in algae and copepoda concentrations during the year.

Figure 3.156. Same run: phase plane plot of the phosphate and phacotid algae species


42 1

Reference Parker, R. A. (1973) Some Problems Associated with Computer Simulation of an Ecological System, In: Mathematical Theory of the Dynamics of Biological Populations (ed. M. S. Bartlett and R. W. Hiorns) Academic Press.

3.15.4 EUTROPH - Eutrophication in a Shallow Lake in Hungary System Eutrophication has been defined as the biological reaction of aquatic ecosystems to nutrient enrichment, and this can lead to algal blooms, oxygen deficiency, fish death and other ecological changes (see Sec. 2.6.1). Phosphorus is widely accepted as the critical nutrient in determining the degree of lake eutrophy. Dissolved inorganic phosphorus and organic phosphorus (detritus) concentrations in Lake Balaton in Hungary are considered in this example, based on Somlyody and van Straten (1986).

Figure 3.157. Lake Balaton showing only the main tributary and the four basins.

Phosphorus enters the lake via the flow from the River Zala and from other rivers, as well as from wastewater treatment plants. Two classes of algae are modelled; blue-green algae, which are the major species in summer, and diatoms, which predominate in spring.

Hydrology Lake Balaton consists of four basins each with dimensions as given in Table 3.8. The River Zala flows into basin 1 with an average volumetric flow rate of 9 m3 s-l. Water passes sequentially from basin 1 to basins 2, 3 and 4 before leaving the lake. Smaller rivers and wastewater treatment plants also discharge into the lake adding to the eutrophication problem. Each basin is assumed to be well mixed and the lake is modelled as four tanks-in-series. The lake is

422


Model

Figure 3.158. Phosphorus cycle in a lake.

shallow (average depth 3.2 m) and the region windy (2 to 5 m s-l average wind speed) which helps prevent thermal stratification, and also ensures that the lake remains aerobic, even in the upper layers of the sediment.

Table 3.8. Lake Balaton's geometric parameters. Basin

Volume (106m3)

1 2 3 4 Totals

82 413 600 802 1907

4.3 21.8 31.6 42.3 100

Depth (m) 2.3 2.9 3.2 3.7 3.2

Surface (km2) 38 144 186 228 596

area

(%I 6.4 24.4 31.1 38.1 100

Climatic Conditions The average daily water temperature varies from O°C to 25OC and is modelled by a sine function:

423


271 t T = 12.5 + 12.5 sin ( 365

-

71

2)

where t is the time in days and T is temperature. The minimum temperature is therefore at the beginning of the year and the maximum temperature in mid summer. The average daily sunlight intensity varies from 70 to 490 cal cm-2 day-', and this is also modelled by a sine function:

R = 280 + 210 sin

271t 471 (m+ 7)

The minimum sunlight intensity is therefore in January whereas the maximum occurs in July.

Algal Dynamics The growth rates for both blue-green algae and for diatoms are based on Monod kinetics with dissolved inorganic phosphorus, P, as the limiting nutrient:

is the maximum growth rate and k, the Monod constant. The values of these parameters for both blue green algae and diatoms are shown in Table 3.9. The light available to the algae varies with depth, time of day and selfshading by other algae. The light intensity correction factor, ki, is based on integrals of the intensity over the depth for one day. kT is the temperature dependency correction factor. The extinction factor for water, EO, is modified by the presence of algae:

pmax

where a is the self shading coefficient. The light intensity at the liquid surface is given by:

h is the daylight fraction of the day, and Is is the optimum light intensity for algal growth, which also depends on temperature, according to Is = I,,

+ Ise T

where I,, is the temperature correction factor for Is, and Ism is the base optimum light intensity.

424


At a depth H the light intensity is given by:

The factor ki can now be calculated from:

The temperature dependence factor KT is calculated using the following expression:

where Tca, Tca, Toa, Tod are the critical and optimum temperatures for the blue green algae and the diatoms, respectively and are given in Table 3.9. The mortality of the algae is modelled as a first-order temperature dependent process: RDTH = b O T - 2 0 X The mortality rate constants, kd, are also given in Table 3.9. Cell death is assumed to be temperature dependent with 0 equal to 1.14. It is assumed that zooplankton grazing is not a major cause of algal death and therefore this effect is not included in the model.

Table 3.9. Algal growth kinetics parameters. Algae-type Diatoms Blue-green algae

Pmax (day-') 2.0 6.0

ks (mg L-') 10.2 10.2

kd (day-l) 0.13 0.13

TO("c>TC("c) 12 25 25 30

Phosphorus Cycling The total phosphorus load can be considered as shown in Fig. 3.159. The particulate organic phosphorus fraction is approximately 50 percent incorporated into the phytoplankton. Two fractions are considered as follows: 1) organic phosphorus or detritus (D) 2) inorganic phosphorus (P). The fraction of non-sedimenting, i.e., soluble, detritus is denoted by y.

425


phosphate

organic P

organic P

inorganic P

Figure 3.159. Division of phosphorus-containing compounds in the model.

Sedimentation Insoluble detritus, blue green algae and diatoms settle in the lake, and dissolved inorganic phosphorus is released from the sediment, at rates which may expressed by

vs is the sedimentation velocity (assumed the same for all fractions), y is the fraction of the detritus that is soluble and krel is the rate of release of inorganic dissolved phosphorus per unit area of sediment. The release of phosphorus from the sediment is assumed to be temperature dependent with a temperature sensitivity parameter, 0, with value 1.04 Mineralisation Heterocyclic bacteria in lakes mineralise dissolved organic P into the form of dissolved inorganic P which can then be utilised by the algae. It is assumed that the bacterial processes are fast compared to the algal processes and therefore the dynamics of the bacteria population are not modelled explicitly. The rate of mineralisation is temperature dependent (with 0 = 1.18), and this reflects the bacterial temperature dependent growth.

426


Biogenic Lime Precipitation Calcium carbonate is an important constituent of the suspended solids fraction of Lake Balaton water and during the year considerable Ca precipitation occurs in the phytoplankton growing season, mostly as biogenic lime precipitation. The rate of biogenic lime precipitation is given by:

where klime is the biogenic lime coprecipitation coefficient.

Bulk Sediment Exchange Phosphorus adsorption is also seen to occur outside the growing season, and for this it is assumed that there is a continuous adsorption-desorption process, probably involving the sediment. This process is modelled by a dynamic relationship which assumes an equilibrium concentration of phosphorus, Peq. The rate of adsorption exchange is given by:

where kex is the transport coefficient for sorption exchange and Peq is the effective equilibrium P concentration.

Nutrient Loading The loadings for dissolved inorganic phosphorus (P) and detritus (D) in each of the basins is summarised in Table 3.10. It is assumed that rainwater and sewage discharges contain only dissolved inorganic phosphorus. Table 3.10. Average phosphorus loads (kg day-') into Lake Balaton. Load (kg day-') River Zala Tributaries Sewage Runoff Atmospheric Total

Basin 1 P D 115 129 130 100 3 0 6 17 1 0 255 246

Basin 3 P D

Basin 4 P D

95 58 26 36 6 0 8 0 7 17 39 21 5 1 5 0 7 0 123 97 62 87

4 6 8 0 26 63 7 0 115 69

Basin 2 P D

Total P 115 255 9 5 70 20 555

lake D 129 200 0 170 0 499


427

Mass Balances The following phosphorous mass balances can be written in terms of detritus, inorganic phosphorus, blue-green algae and diatoms and for all four basins (n=1,2,3 and 4):

Nomenclature All rates and concentrations are in terms of mg phosphorous. The symbols used in the program are given in brackets. Detritus phosphorous Volumetric flow Rate Lake depth Light intensity at surface Mortality rate constant Sorption exchange coefficient Light intensity correction factor Lime coprecipitation coefficient Lime coprecipitation coefficient Monod constant Temp correction factor Nutrient loading Light intensity at surface and depth H Dissolved inorganic phosphorus Average light intensity Sorption exchange rate with sediment Rates Lime precipitation rate Mineralisation rate Release rate from sediment Settling or sedimentation rate Algal death rate

mg m-3 m3 day-' m cal m-2 day-1 mg-1 m3 mg-1 m3 mg m-3 mg day-' cal m-2 mg m-3 cal cm-2 day-' mg m-3 day-' mg m-3 day-' mg m-3 day-' mg m-3 day-' mg m-3 day-' mg m-3 day-' mg m-3 day-'

428 RGRi,n t T (TEMP) Ti (TEMPi)

vn

X

a E

Y

h Clrnax

0 Subscripts 0 1,29394 a C

d eq ex

H lime min n 0

re1 S

se set sm


Algal growth rate Time Water temperature Critical and optimal temperature Volume of basin n Biomass conc. as phosphorous Self shading coefficient Light extinction factor Soluble fraction of detritus Daylight fraction of day Maximum growth rate Temperature sensitivity parameter

mg m-3 day-' days K

K m3 mg m-3 mg m-2 m-1 day-

Surface Basins in lake Blue-green algae Critical temperature Diatoms Equilibrium Sorption exchange Refers to depth Biogenic lime precipitation Mineralisation (organic P converted to inorganic P) Basin number (1 to 4) Refers to depthoptimum temperature Release of inorganic P from the sediment Optimum light intensity Correction factor for light intensity Sedimentation Base optimum light intensity

Exercises 1. In summer Lake Balaton is a popular tourist location. As a result, the volume of sewage during the summer months almost doubles. Include this factor into the model and see how this influences the algal blooms.

2. When the wind is in the right direction and sufficiently strong, there is a transfer of water in the reverse direction to the overall flow (i.e. from basin 4 to 3, 2, 1). Include in the model the effects of a return flow of the order of 0.25 lo6 m3 day-1 and see whether this has any effect on the eutrophication.


429

Results

0

0

50 100 150 200 250 300 350 Time (days)

Figure 3.160. Total algal concentrations in basins 1 to 4. Why is the problem most severe in basin l ?

Time (days) Figure 3.161. Variation of the total phosphorus concentration in basins 1 to 4.

References Somlyody, L. and van Straten, G. (1986) Modelling and Managing Shallow Lake Eutrophication with Application to Lake Balaton, Springer-Verlag, Germany. van Huet, H. (1991) Modelling Water Transport and Phosphorus Eutrophication in an Interconnected Lake System: A Scenario Study, Dissertation, University of Wageningen, Netherlands.

3.15.5 METAL - Transport of Heavy Metals in Water Column and Sediments System Heavy metals find their way into the water cycle via the natural processes of erosion, weathering and volcanic activity. In addition the human activities of mining, smelting and burning fossil fuels contribute significantly to the metals found in natural waters. These metals can be accumulated by some microorganisms and thus enter into the food chain, so presenting a possible toxic risk to wildlife and humans. Rivers passing through industrial and mining areas transport metals, partly as metal in solution and partly as metal adsorbed to suspended material. This suspended material sediments out in lakes and estuaries and there accumulates in the sediment. The model presented here, based on van de Vrie (1987), aims at predicting the concentration of zinc,

430


cadmium and lead in the sediment and water of a lake in the southwest of the Netherlands.

Model

Figure 3.162. Balance regions for heavy metal transport for a lake in the Netherlands.

The whole lake is divided into a number of vertical elements, each comprising a column of water and the top layer of sediment. In this simulation only one such element is considered in order to improve the speed of execution. The general principle however can be extended to many elements. It is assumed that the metals are distributed homogeneously in the water column as well as in the upper sediment layer. In both compartments, the metals can exist in either a dissolved form or as adsorbed to solid particles.

The dissolved fraction, fd, is given by the distribution coefficient, KD:

43 1


where m is the total concentration of the suspended particles. The suspended particles settle with a velocity v,, and enter the top sediment layer, at a rate given by Js = Vs A Ca = Vs A (1 - fd) CT The particles in the top layer of sediment also settle until they reach a certain depth at which they are no longer influenced by the water column. There is a diffusion exchange between the two compartments owing to differences in the dissolved metal concentration in the water column and in the top layer of the sediment. The rate of this exchange is proportional to the concentration difference, and the area of the sediment-water interface and is defined by proportionality constant KL.

Water flows into the water column at a volumetric flow rate Q containing a total metal concentration Cin. It is assumed that water flows out from the column at the same rate and at the same concentration as in the water column. Mass balances on the water column and top sediment layer give:

where V1 and V2 are the volumes of the water column and top layer of sediment, respectively, and H1 and H2 are the corresponding heights. Substituting and rearrangement gives & d dt

Nomenclature A C fd

H1

H2 J

Area of water-sediment interface in each element Concentration of metal Fraction of metal that is dissolved Height of water column Depth of top sediment layer Fluxes of metals

m2 g m-3 m m g m-2 day-1

432 KD

KL m

Q t V1 v2 VS


Distribution coefficient Diffusion exchange coefficient Concentration of suspended particles Volumetric flow rate of water into water column Time Volume of water column Volume of top sediment layer Sedimentation velocity

g-

m day-' g m-3

m3 day-1 days m3 m3 m day-1

Subscripts 1

2 a d S

T

Water column Top sediment layer Adsorbed Dissolved settling Total

Exercise Run the model with the following parameters and compare the relative rates of accumulation for the different metals.

Table 3.11. Parameters for METAL. Parameter Thickness of top sediment layer (m) Distribution coefficient water column (g-1) Distribution coefficient sediment (g-9 Diffusion exchange coefficient (m day-')

Zinc 0.39

Cadmium 0.37

Lead 0.26

114

81

309

18.8

22.0

38.0

1.O

8.1

0.88

433


Results

3.OE-4 m -

E

'a, 2.0~-4 v

ti

1.OE-4

0

O.OE+O,

-

100 200 Time (days)

0

Figure 3.163. Variation in adsorbed (A) and dissolved (D) metal both in the water column (1) and in the sediment (2).

Figure 3.164. Response of the dissolved and total metals concentration to a change in flowrate from 1.0 to 0.5 m3 d-* at T=140 days.

References van de Vrie, E. M. (1987) Modelling and Estimating Transport and Fate of Heavy Metals in Water Column and Sediment Layer in Some Enclosed Branches of the Sea in the SW Netherlands, In: Dynamical Systems and Environmental Models (eds. Bothe, H. G., Ebeling, W., Kurzhanski, A. B. and Peschel, M.), Akademie-Verlag Berlin. Salomons, W. and Forstner, U. (1984) Metals in the Hydrocycle, Springer Verlag, Berlin.

3.16

Land Pollution Modelling

3.16.1 LANDFILL - One-Dimensional Transport of Pollutant from a Landfill Site System The pollution of groundwater can occur in various ways. Thus, it may leach from polluted surface water, from leaking wastewater pipes, ponds or cesspools. It may be leached from the top soil by rain and be carried down into the saturated zone by seepage (e.g., pesticides, nitrates). Similarly, seepage through

434


landfills and waste deposits is of major current interest. The transport of pollutant in the unsaturated (vadose) zone is essentially vertical, whereas that in the ,saturated zone is mainly horizontal. The long range transport of pollutants mainly occurs in the saturated zone. Models for the transport of pollutants within the aquifer are important in predicting the ultimate fate of pollutants particularly with respect to sources of drinking water (see Sec. 2.6.3.2). Heydarpour and Slotta (1989) have developed software for PCs that models the one and two-dimensional transport through porous media. The model presented here is a simplified version that demonstrates the basic ideas. Atmosphere

Figure 3.165. Leaching of a pollutant from a landfill site.

Model A pollutant is released from a landfill site with the result that its concentration in

the aquifer varies with time according to the following equation: C o = A e - a t - Be-BT The pollutant is transported horizontally in the confined aquifer with a dispersion mechanism which occurs in the Z direction only, and there it undergoes a first-order decay reaction with rate constant KD. 6C 62C (l+R)==D--

622

6C

V B -

KDC

435

3.16 Land Pollution Modelling

In order to solve the above model expression by digital simulation, the aquifer is divided into n zones of equal volume (see Sec. 1.12.2). In each zone it is assumed that the pollutant concentration is uniform. By applying a mass balance over each zone the following equations can be derived:

It is assumed that the aquifer thickness and the groundwater velocity is constant, the groundwater flowing only in the Z direction. At time zero the pollutant is uniformly mixed throughout the height of the aquifer. R is a retention coefficient which is a function of the soil characteristics and is estimated by: ES R = Eo ( sFsKs + PcFcKc + POFIJKIJ) where Es and Eo are the actual and effective porosities, respectively. The soil is assumed to be a mixture of sand (>0.005 mm), clay (~0.005mm) and organics. The fraction composition F, density p and linear equilibrium sorption coefficient K are needed to characterise the soil for each pollutant.

Program In this model, the release of the pollutant is modelled with A = B = 50 and a = 0.04 and p = 0.1, The aquifer is divided into 10 zones.

Nomenclature

ES F A B C D

Kd K R t V

Pollutant release constant Pollutant release constant Soil density Actual sorption porosity Effective sorption porosity Fraction soil cornposition Pollutant release constant Pollutant release constant Pollutant concentration Dispersion coefficient Decay constant of pollutant Equilibrium sorption coefficient Retention coefficient Time Average pore water velocity

day-' daykg m-3

kg m-3 kg m-3 kg m-3 m2 day-' daycm3 g-1 day m day-'

436


rn rn

Length of one zone Axial distance

AZ Z

Subscripts Clay Number of zones in the model Organics Sand Sorption

C

n 0 S

S

Exercises 1. Vary the release of pollutant (i.e., change values A, B, a and p) and see how the change influences the transport. 2. Vary the rate of biodegradation, &, and see how it influences the transport of the pollutant.

Results h

cr,

2

0

7

E

9

0

7

1

3 -

W

W

c

g E

;m 2

c

g E

10

c

c

c

c

a 0

8

c

0 ('I n - v

0

20

40

60

80

100

Time (days) Figure 3.166. The concentration time profile for different positions, n, in the aquifer. Note that the peak becomes broader due to dispersion as travels along the aquifer with increasing n.

10-

0

20

40

60

80

100

Time (days) Figure 3.167. Concentration time profiles at position 10 for different values of Kd (0.0, 0.025, 0.05, 0.075, 0.1).

Reference Heydarpour, J. and Slotta, L. S. (1989) 1-D and 2-D Personal Computer Programs for Solute Transport Through Porous Media, Environmental Software 4, 2, 62-69.


437

3.16.2 LEACH - One-Dimensional Transport of Solute Through Soil System Groundwater is a major source of drinking, industrial and agricultural water. Its quality is a primary environmental concern, not only for health reasons, but also because of the decrease in crop productivity caused by pollutants. The ability to model the migration of pollutants through the vadose soil layer plays an important part in the combat against the degradation of groundwater.

Model In this model from Corwin et al. (1991), the transport of a solute such as a pesticide, through the soil is simulated from ground level to the water table. As shown in Fig. 3.168, the soil column is divided into N elements each of thickness z, with element 1 at the surface and element N directly above the water table. Hence:

The water fraction for any element n is expressed as the ratio of the volume of water to the total volume of the element:

The water fraction of each element of soil can not exceed a certain maximum value, known as the field capacity. If the water content is less than the field capacity, then no water is able to leave the element. If the water content is equal to or greater than the field capacity, then water is able to leave the element at the same rate as water enters. The water is also taken up by the roots'of plants. Here it is assumed that the uptake rate of water is proportional to the length of root. If the water content falls below a critical value, the plants can no longer take up water, and they wilt and eventually die. A mass balance based on the volume of water in each element leads to the following equation: _ dWn_ dt - Qn-1 - Qn - ETn where Qn-l and Q n are the volumetric flow rates of water in and out of element n, and ET, is the rate of assimilation by the plant or evapotranspiration rate.

438


Figure 3.168. Solute transport through the vadose layer.

The limiting behaviour on the rate of evapotranspiration is expressed by the logical condition, that with ET, = 0

when Fn F C

439


The roots are assumed to grow at a constant rate (GR), so that the length of the root at any time t is given by Roottot = Rook + GR t where Roottot is the total root length at time t, and Rooto is the root length at time t=O. The growth stops if the water content is too low, given by the condition

If the water content is less than the minimum then growth stops. Thus the root length in any element n is given by:

Thus the root occupies the total length of the element for those elements less than the total root length. When the root penetrates only part way into the soil element,, as shown in Fig. 3.168, Root, = Roottot - (n - 1)z

for (n - 1) z c Roottot< n z

and Root, = 0

for Roottot c (n - 1) z

The input of water to the top element is by rain or via human irrigation. A water balance for the top element 1 therefore leads to the mass balance

~dW1 dt - A (Rain

+ Irrig) - Q1 - ET1

where Rain and Irrig are specific rates given in cm day-', and the crosssectional area, A, of the soil column is related to the height, z, and the volume, V, of each element by V

If a pesticide, or any other solute, is sprayed onto the field it will travel in the water through the soil, but may also be adsorbed onto the soil. It is assumed that the process of adsorption occurs almost instantaneously, as compared to the slower rate of transport through the soil. Only solute dissolved in the mobile water column is transported from one element to the next. A mass balance on the total solute concentration, C, including both adsorbed and mobile solute for element n yields the following relationship:

440


where Cs, the concentration in the soil water, is related to the total and adsorbed solute concentrations by the water fraction, Fn, i.e.,

An assumed instantaneous distribution between the adsorbed and free solute is modelled by a Langmuir-type adsorption isotherm, where cads = Cmax

cs

Cmax is the maximum concentration of solute that can be adsorbed by the soil, and K is the Langmuir adsorption coefficient. Combining the above two equations, a quadratic equation can be derived in terms of Cs, which on solving for the positive root gives, - b + (b2 - 4a s)o.5 cs = 2a where a =F b = F K - C + Cmax (1 -F) s =-CK Hence the corresponding values for C, Cs and Cads can be determined for each element. It is assumed that the plants do not take up solute and that the solute is not biodegraded. As water is taken up by the plants at a greater rate than the rate of supply, the value of F will decrease, and the solute will become more concentrated.

Program Five elements are used to model the soil column.

Nomenclature a A b C Cn cads

cs

Cmax

Constant in quadratic equation Cross-sectional area of soil column Constant in quadratic equation Solute concentration in imgation water Total solute concentration in element Concentration of adsorbed solute Concentration of solute in liquid phase Maximum adsorbed solute conc.

cm2 g g g g g

cm-3 cm-3 cm-3 cm-3 cm-3

44 1


ETn Fn FC Fmin

GR Irrig K N

Q"

Rain Root, Roottot S

t V Wn Wtable 2

Evapotranspiration rate in element n Water fraction in element n Field capacity fraction Minimum moisture content for plant growth Plant root growth rate Specific rate of human irrigation Langmuir adsorption coefficient Number of elements in the soil model Volumetric flow rate from element n Specific rainfall rate Length of root in element n Total length of root Constant in quadratic equation Time Volume of one element Water content in element n Depth of water table Height of one soil element

cm3 day-' cm3 cm-3 -

cm day-' cm day-' g cm-3 cm3 day-' cm day-' cm cm -

day cm3 cm3 cm cm

Exercises 1. Run the simulation under the following conditions: (a) Dry conditions time 0 to day 2, (b) spraying day 2 to day 5 and (c) rain day 8 to day 13. See how the concentrations of adsorbed and unadsorbed solute vary with time and depth. Vary the time of spraying to before, during and after rainfall.

2. Change the chemical equilibrium parameters (K and Cmax) to see how they effect the distribution and transport of the solute. See Lyman et al. (1982) for a comprehensive set of data. Experiment by using a different model for the adsorption of the solute e.g., Freundlich linear adsorption. 3. In practise some water will always be transported through the soil even if the local water content is less than the field capacity owing to preferential flow through cracks and large pores in the soil structure. This can be modelled simply by defining a bypass parameter to account for the fraction of water that can pass through each element. Modify the model to include bypass and see how this influence the solute profiles. See Corwin et al. (1991) for further details. 4. Different soil types have differing field capacities and minimum water contents.

442


Typical values are: Soil type

Field capacity (cm3 cm-3)

Minimum water content (cm3 cm-3)

Sandy soil Clay soil

0.08 0.35

0.05 0.16

Vary the parameters within the above ranges. How does the soil type affect the transport of solutes? Modify the model so that the soil characteristics vary with depth, e.g., a sandy soil near the surface becoming clay deeper below the surface.

5. The assumption of linear water uptake along the length of the plant roots is not usually true in practise. Often the uptake is divided 40:30:20:10 i.e. 40% of water is taken up by the top quarter of the roots, 30% by the second quarter and so on. Modify the model to take this into account. Try other models of water uptake (e.g., 44:31:19:6). 6. Modify the model for a solute that is capable of being biodegraded or being assimilated by the plants. An appropriate kinetic relationship must be assumed.

Results

Figure 3.169. Time profiles for the overall concentration of solutes at five different depths. Irrigation between days 2 and 5 . Rain on days 8 to 13.

Figure 3.170. Water fractions for the same run. The influence of irrigation and rain is clearly seen near the surface at depth 1.

References Corwin, D. l., Waggoner, B. L. and Rhoades, J. D. (1991) A Functional Model of Solute Transport that Accounts for Bypass, J. Environ. Qual. 20, 647-658.

443


Lyman, W. J., Reehl, W. F. and Rosenblatt, D. H. (1982) Handbook of Chemical Property Estimation Methods, Chapter 4,Adsorption Coefficients for Soils and Sediments, McGraw-Hill.

3.16.3 SOIL

- Bioremediation

of Soil Particles

System Soil that has been contaminated with an organic compound can be cleaned by naturally occurring soil bacteria. These bacteria are present as suspended forms existing within the liquid in the pores of the soil particles as well as existing as microcolonies attached to the solid surface. The micro-organisms utilise the organic compounds as a carbon source but also require oxygen. Oxygen and the contaminant are transported by diffusion within the liquid phase of the soil particles. The time required for the contaminated soil to be cleaned, or the bioremediation time, is an important parameter in influencing the course of action to minimise the environmental impact of the contamination.

Figure 3.171. Leaching of an organic compound into the soil.

Model The soil aggregates are assumed to be spherical in form and to have constant temperature and to contain initially uniform distributions of substrate (contaminant) and biomass. The external concentrations of biomass and substrate are assumed to be zero and the external oxygen concentration is

444


constant. Substrate is adsorbed onto determined by an equilibrium partition Thus for the substrate Ssol = and for the biomass: Xsoi =

the solid phase to an extent which is coefficient. Sliq Kps Xliq Kpx

Substrate is present both in both the liquid and soil phases of the soil particle. A factor Fs can be defined such that all the substrate can be considered to be found in the total volume V, where

This can be simplified to: Fs = Ea

+ (1 - Ea )- Ssol Sliq

which is Fs=Ea

+ (1 - E a ) K p ,

A similar expression can be derived for biomass, which gives Fx = Ea

+ (1 - Ea )Kpx

where Ea is the volumetric fraction of liquid, and Kps and K,? are the partition coefficients. The factors F, and Fx are introduced to avoid having to use separate differential biomass and substrate balances for both the solid and liquid phases. Oxygen is assumed to be present only in the liquid phase. The biomass is assumed to follow Monod growth kinetics, depending on both the oxygen and substrate concentrations in the liquid phase and to decline according to a first order decay term, where:

The rates of uptake of substrate and oxygen are related to the biomass growth rate by appropriate yield constants:

-RS- - v Fx

PmaxX

Y,

S 0 K,+S Ko+O

445


Oxygen, substrate and biomass are all transported by diffusion within the liquid phase contained in the aggregate. The modelling of this process is achieved via the use of a finite differencing technique, as discussed in Sec. 1.12.2. In this, the spherical aggregate is divided into a number of shells, as seen in Fig. 3.172.

Figure 3.172. Finite differencing of a soil particle.

The outer diameter of the nth shell is rn and its inner diameter is rn-l. Therefore the volume of shell n is given by

and the outside area of the shell by

Mass balances for element n give

dS Fs V n x = Js,n + Rs,n

446


where Jx, Js and J, are the respective nett rates of diffusion of biomass, substrate and oxygen into the nth element which are approximated by

Do* and Ds* are the diffusion coefficients for biomass, oxygen and substrate in the liquid phase of the aggregates. It is assumed that the rates of diffusion in the solid phase are negligible compared to those in the liquid phase. The biomass diffusion coefficient also incorporates the effect of cell motility. The rate of diffusion through the aggregate will depend on the volume fraction of liquid in the aggregates and the degree of tortuosity of the pore space. The diffusion constants in the aggregates can be related to the diffusion constants in free liquid by the following relationship:

Dx*,

where i is substrate, biomass or oxygen, E, is the liquid volume fraction in the aggregate and z is a measure of the degree of tortuosity of the pore space.

Program The model equations are unfortunately very stiff, requiring small values of CINT and consequent long simulation times. Large values of CINT lead to oscillations in the dissolved oxygen concentrations and result in errors in the substrate uptake rate.

Nomenclature A D DO Ea

Fx

FS

Outside area of shell Diffusion coefficient Dissolved oxygen concentration Liquid fraction in aggregate Biomass concentration correction factor Substrate concentration correction factor

cm2 cm* s-1 g cm-3 dimensionless dimensionless dimensionless

447


J

Kd KS KO KPS

KPX

r R S t V

X YO ys Pmax

z

Subscripts ex liq n 0

sol S X

in

Diffusion rate First order cell death constant Monod constant for substrate Monod constant for oxygen Partition coefficient substrate Partition coefficient biomass Radius of aggregate Reaction rates Substrate (contaminant) concentration Time Volume of aggregate Biomass concentration Yield oxygenhiomass Yield substratehiomass Maximum growth rate Tortuosity of the pore space

g s.1 S-

g cm-3 g cm-3 dimensionless dimensionless cm g s-1 g cm-3 S

cm3 g cm-3 dimensionless dimensionless S- 1

dimensionless

External Liquid phase element number (1 to 6) Oxygen Solid phase Substrate Biomass Initial

Exercises 1. Vary the radius of the soil aggregates in the range 0.1 to 10 cm, and see how this influences the bioremediation time. 2. Vary the substrate partition coefficient in the range 1 to 1500 and see how this also influences the bioremediation time.

3. Also vary the initial substrate concentration, i.e. the degree of contamination in the range 1.0 x 10-6 to 0.1, and see how this influences the bioremediation time.

4.Investigate the oscillations caused by a large CINT value (>50)on the results for 0 6 in the region of zero. Note the resultant change in all rates.

448


5 . Compare the bioremediation time for a small, highly contaminated particle to

that of a larger, less contaminated particle having an equal initial mass of contaminant.

Results rsm

Figure 3.173. Variation in oxygen concentration with time for three different elements. Note the integration errors in 0 1 near zero due to CINT.

Figure 3.174. Response of oxygen to an increase in the external oxygen concentration to 0.8E-5 at t=0.28E5.

Reference Dhawan, S., Fan, L. T., Erickson, E. and Tuitemwong, P. (1991) Modeling, Analysis and Simulation of Bioremediation of Soil Aggregates, Environmental Progress 10, 4,251-260.

3.17 Miscellaneous Examples

3.17

449

Miscellaneous Examples

3.17.1 DEADFISH - Distribution of an Insecticide in an Aquatic Ecosystem System The environmental impact of a new product needs to be assessed before it can be released for general use. Chemicals released into the environment can enter the food chain and be concentrated in plants and animals. Aquatic ecosystems are particularly sensitive, in this respect, since chemicals, when applied to agricultural land, can be transported in the ground water to rivers and then to the lakes, where they can accumulate in fish and plant life. The ecokinetic model presented here is based on a simple compartmental analysis and is based on laboratory ecosystem studies (Blau et al., 1975). The model is useful in simulating the results of events, such as the accidental spillage of an agrochemical into a pond, where it is not ethical to perform actual experimental studies. The distribution of the insecticide Dursban in a simple aquatic environment consisting of water, fish, soil and plants is simulated, as shown in Fig. 3.175. The kinetic constants were obtained from radio label experiments in a test aquarium.

Figure 3.175. Kinetic model of the water-fish-plant-soil interactions.

450


Model It is assumed that at time zero a defined quantity of Dursban enters a lake and is distributed subsequently between the fish, the soil and the plants. The soil and plants are lumped together as one compartment. The quantities in each compartment are expressed as a percentage of the initial contamination. Dursban in the pond water can be taken up by the soil and plants and also by the fish in the pond, and it is assumed that there is an equilibrium between the amount of insecticide in the plants and soil and in the pond water. Inside the fish, Dursban is metabolised to pyridinol and then excreted. Some insecticide is stored separately within the fish. The released pyridinol is taken up by the soil and plants but not again by the fish. In this model only the quantities of the components in the different compartments are considered, and therefore no assumptions have to be made about the sizes of the different compartments. Writing mass balances for compartments 1 to 6 and referring to Fig. 3.175 we obtain: For the Dursban in the water

For the Dursban in the plants and soil

For the Dursban in the fish

For the Pyridinol in the water

For the Pyridinol in the plants and soil

45 1


For the stored Dursban in the fish

~dX6 dt - k36

x3 - k63X6

Nomenclature kij t

Xi

Rate constant for reaction from compartment i to compartment j Time Percentage insecticide in compartment i

h-1 h %

Subscripts 1

2 3 4 5 6

Dursban in water Dursban in plants and soil Dursban in fish Pyridinol in water Pyridinol in plants and soil Stored Dursban in fish

Exercises 1. Compare the accumulation of the insecticide in the fish in systems with and without the plant and soil compartments. 2. Modify the model to simulate the slow release of insecticide (for instance via runoff) into the lake.

Results

Figure 3.176. Insecticide distribution (percent) between the various compartments as a function of time.

Figure 3.177. For the same run, insecticide plus metabolite distribution (percent) between the fish, plants and water.

452


References Blau, G. E., Neely, W. B. and Branson, D. R. (1975) Ecokinetics: A Study of the Fate and Distribution of Chemicals in Laboratory Ecosystems, AIChE J. 21, 5 , 854-861. Butte, W. (199 1) Mathematical Description of Uptake, Accumulation and Elimination of Xenobiotics in a F i s W a t e r System, In: Bioaccumulation in Aquatic Systems: Contributions to the Assessment (ed. Nagel, R. and Loskill, R.), VCH, Weinheim.

3.17.2 GAIA

- The

Parable of Daisyworld

System In 1972, James Lovelock proposed in the "Gaia Hypothesis", that the earth's atmosphere is actively maintained and regulated by life on the surface. To test this hypothesis, he developed a simple model for the temperature of a hypothetical planet called Daisyworld. Daisyworld is a cloudless planet where the only life consists of two species of daisy, one black the other white. The black daisy reflects less light than the bare ground, while the white daisy reflects more. Simulations show that the area of the planet covered by the two species of daisy can maintain the temperature of the surface of the planet within a remarkably narrow range.

Figure 3.178. Daisyworld.

453


Model The growth and death of the daisies follow first-order kinetics:

where D is the fraction of the total area of the planet covered by one species of daisy, X is the fraction of the planet not covered by daisies of either species, p is the growth rate and the death rate. The uncolonised area of ground is given by

where P is the proportion of the planet's area which is fertile. The growth rate of the daisies is a parabolic function of the local temperature, TI: p = 1 - 0.003265 (22.5 Hence the growth rate is zero at temperatures greater than 40°C and at temperatures less than 5 ° C but with a maximum at 22.5"C. By equating the rates at which radiation is absorbed and emitted, the effective temperature at which the planet radiates, Te, can be found:

o (Te + 273)4 = S L (1 - A ) where o is the Stefan-Boltzmann constant, S is a flux constant and A is the albedo of the planet. Ignoring the variations in absorption due to the spherical geometry, the albedo of the planet can be estimated by:

A = AwDw+ABDB+AGX where AG, AB, Aw are the albedos of the bare ground, and the ground covered by black and white daisies, respectively and X is the fractional area of bare ground. The local temperature is a function of the effective temperature and the local albedo. (TI + 273)4 = Q (A - A]) + (T, + 273)4 where Q is a positive constant which expresses the degree to which solar energy, having been absorbed by the planet, is then redistributed amongst the three types of surface. If Q = 0 the local temperature will always equal the effective temperature, and if Q = S L/o then there is perfect insulation between the different temperature

454


zones. A value of Q greater than S L/o implies a transfer of heat against the temperature gradient; so in this model the value of Q always lies between 0 and S L /o.The value of the parameter L is 1.0 for the earth, but in this model it is varied linearly with time between 0.65 and 1.65 to investigate the temperature modification of a planet that is heating up.

Nomenclature A Di

Kd

L

P

Q

Total S T1 X

P

o Subscripts B

G

w e

Albedo Fraction of area covered by daisy species i Death rate constant Solar radiation Fraction of fertile ground on earth Energy distribution constant Fraction of ground available Flux constant Local temperature Fraction of earth not covered by daisies Specific growth rate Stefan-Boltzmann constant

ergs cm-2 s-1

K S- 1

ergs cm-2 K-4

Black daisies Ground White daisies Effective

Exercises 1. Run the model with (a) no daisies, (b) only black daisies or (c) only white daisies, and (d) with both species present. Compare the results obtained for the temperature of the planet as the sun heats it up.

2. Vary the parameter Q between 0 and 1 and investigate the influence of heat transfer between the different temperature zones. 3. Investigate the ability of the system to recover from a plague that kills off one or both species of daisy. Run the model at constant value of L and when the steady state has been reached INTERACT via the keyboard and reset the values of White and/or Black using the VAL command. Is the new steady state the same as the old one?


455

4. Investigate having more species on the planet with varying growth rates and albedos. 5. Investigate the influence of the rate at which the planet heats up. If L is allowed to increase very slowly compared to the growth rate of the daisies, then the model becomes very stiff and runs very slowly. By using ALGO=3 some improvement can be achieved, but at the risk of the solution not converging. 6. Vary the parabolic growth rate temperature dependence to other relationships, e.g., rectangular, semi-circular.

Results

Solar radiation, L

Figure 3.179. Shown is the heating up of the planet without any daisies and the moderating effect of the daisies.

Figure 3.180. Increasing the death constant from 0.3 to 3.0 at T=250 causes the daisies to die off and the temperature to drastically increase.

References Watson, A. J. and Lovelock, J. E. (1983) Biological Homeostasis of the Global Environment: the Parable of Daisyworld, Tellus 35B, 284-289. Margulis, L. and Lovelock, J. E. (1974) Biological Modification of the Earth's Atmosphere, Icarus 21, 47 1-489.

456


3.17.3 CABBAGE - Structured Growth of the White Cabbage System In the literature there are numerous examples of various plant growth models. Classical plant growth models consider the growth of the whole plant, are generally non-linear and contain only a few parameters. Such models are applicable to a large number of growth processes but have the disadvantage that they can not simulate plant physiological behaviour, e.g., the response to fertiliser addition. At the other extreme there are models based on the plant biochemistry and physiology that can simulate physiological behaviour. These models, however, have the disadvantage that they contain a large number of parameters, for which it is often difficult to obtain accurate values. The model presented in this section is a compromise between these two approaches and attempts to simulate the growth of the white cabbage as a function of fertiliser addition. The overall rate of growth is the sum effect of the individual rates of growth for the roots, stem, leaves and the head of the cabbage.

Model Biomass is produced primarily in the leaves and therefore the growth of all other organs are closely related to the photosynthetic activity within the leaves. The rate of photosynthesis is proportional to leaf area so long as the leaves do not overlap. Growth of the leaves starts at the beginning of the vegetation period, and is triggered by temperature or daylength but stops according to other external environmental variables. A mass balance on leave biomass (L) gives: dL dt = R L L ~ ( ~ ) - M u L L

-

where RL is the specific growth rate, MULis a resistance to growth or leaf decay factor and f(t) is a switching function to account for the temporal changes in leaf growth due to environmental conditions. The total nitrogen available to the plant is the sum of the nitrogen in the soil plus the nitrogen in the fertiliser.

The growth rate of the leaves is dependent on the available nitrogen and is modelled by a Michaelis-Menten or Monod type relationship:

457


The switching function, f(t), is given by: a+ 1 f(t) = a + exp(rho t) The parameter a determines the leaf growth constant and the parameter rho the smoothness of the transition. Combining the above equations gives the following expression for the rate of growth of leaf biomass: dL(a+ l ) L dt - RL a + exp(rho t) - MULL The partitioning of assimilates between the head and stem follows a certain temporal pattern that can also be simulated using switching functions. The growth of the stem is proportional to the leaf area. The growth rate is also related to the ratio of the stem to leaf biomass and growth stops if the ratio attains a value of h.

The growth of the head only starts after a critical time tH. This effect can be modelled by a simple switch constant, expressed by the logical conditions that Switch(t) = 0

t < tH

Switch(t) = 1

t > tH

The value of tH can be linked to environmental parameters if sufficient data exist. A mass balance for head biomass is given by: dH dt = (RH L - MUHH) f(t>

-

where RH and MUHare growth and decay constants, respectively. Neglecting the roots, the total biomass is given by:

X = H+S+L

45 8


Nomenclature

FL FS H K L

h MUH MUL Nfert Nsoil

RH rho RLmax RS S Switch tH t

X

Leaf growth constant Switching function Fraction of biomass in head Fraction of biomass in leaves Fraction of biomass in stems Head biomass Monod constant Leaf biomass Critical stendleaves ratio Head decay constant Leaf decay constant Nitrogen added as fertiliser Nitrogen content of soil Head growth rate constant Leaf growth constant Maximum leaf growth rate Stem growth rate constant Stem biomass Switching constant Head growth begins Time Total biomass

kg ha-' kg ha-1 kg ha-1 daysdayskg ha-' kg ha-' daysdaysdayskg ha-' days days kg ha-l

Exercise Run the model with values of Nfert varying from 0 to 300 kg N ha-'. How does the level of fertilisation influence the total biomass and the distribution of biomass in the cabbage?

459


Results 3.m

-

2.m

-

1.1

-

6

1.w1

rsm

!

Figure 3.181. Growth of the white cabbage with 100 kg N ha-' added as fertiliser. A switch at T=60.7 days triggers the head growth.

Figure 3.182. For the same run, showing the fraction biomass as leaves (A), head (B) and stem (C).

Reference Richter, 0. and Sondgerath, D. (1990) Parameter Estimation in Ecology: The Link Between Data and Models, VCH, Weinheim.

Dynamics of EnvirOnmental Bioprocesses: Modelling and Simulation

Jonathan B. Snape, Irving J. D u n , John Ingham & Jiii E. pkenosil OVCH Verlagsgesellschaft mbH, 1995

Appendix: Instructions for Using ISIM

1

ISIM Installation Procedure

1. Computer requirements: a) DOS 2.00 or higher b) Hard disk Free RAM required: 0.25 Mbyte c) Supported graphics are VGA, CGA, and EGA d) 2. Before installation, check your computer configuration: Do you have a hard disk? If not, another ISIM version is available. a) Do you have a 8087 mathematical coprocessor ? b) What kind of graphics adapter do you have? c) Make a one-to-one copy of the original disk. d)

3. The installation disk contains the following files: INS-ISIM .BAT INSTALL .EXE ISIM86 .EXE ISIM87 .EXE ISIMREF .DOC UPDATE .DOC It also contains all of the simulation examples: NAMESIM

4. To install ISIM 2.06 and the simulation examg,cs, this version of SIh requires the following: a hard disk, with 0.25 Mbyte of free space; approximately 350 Kbyte memory; and a graphics adaptor of the types CGA, EGA or VGA. a) Place distribution disk in drive A: b) Type:

A :INSTALL

The installation program will prompt you for the following information: a) Directory name which is created for the ISIM program, and the library files (EX1.SIM etc), e.g., C:USIM.PRG.

462


b) Directory name for the ISIM start file, this can be the same as for the directory in (a), or you may specify any directory in the "path". Such a directory must exist! c) Type of screen; VGA, EGA, CGA. d) Whether there is a numerical co-processor present. Following installation, the directory for the ISIM start file must appear in the "path". If necessary edit the C:\AUTOEXEC. BAT file to include the directory specified in (b) above. It is important that only one path exists. To use ISIM type: ISIM

2

Programming with ISIM

2 . 1 Getting Started To enter ISIM, simply type ISIM from DOS and press ENTER. After a few seconds, a dollar sign prompt should appear. The user now has two options; to read an existing file from disk, or to create a new model. To exit ISIM and return to DOS type QUIT or the abbreviation Q.

2 . 2 Reading Files From Disk To read a file from the current directory into ISIM type READ (this can be abbreviated to just R) followed by the filename. A file type specifier of SIM is assumed and need not be entered. Files from directories other than the current one can be read in by specifying the full DOS filename. Examples:

READ SIMULATE READ A:MODELS\SIMULATE R SIMULATE.SIM

The file that is being read is displayed on the screen. The user can then decide whether to run the simulation as it stands or modify the model. Only one model can be operational at any one time.

2 . 3 Running Simulations Files that have been previously read into ISIM can be run by typing START after the $ prompt. A graph of one variable against time is plotted on the

Instructions for Using ISIM

463

screen, and numerical values of other variables may also be displayed. In Section 2.8 it is shown how to control this output.

2 . 4 Interacting with ISIM Simulations An ISIM simulation can be interrupted by hitting any key. The user then has a choice of one of four options. HOLD:

This option allows the value of variables to be modified or inspected using the VAL command. For example, VAL KLA, VAL OUR would give the current values of the variables KLA and OUR. The command VAL KLA=100 would change the value of the variable KLA to 100. The run can be continued by typing GO or can be restarted by typing START. The variable TFIN can also be changed in this way to allow longer or shorter runs. The command XVAL resets all variables to their original value. VAL will list the values of all variables and is useful for debugging.

QUIT:

Returns the user to DOS.

SKIP:

Current run is aborted; next run commences.

COMMAND:

Current run is aborted; control is passed to user allowing values to be modified or inspected as with HOLD.

Note: Only the first letter is required (i.e.. H, Q, C or S). Any other key causes the simulation to continue.

2 . 5 Editing ISIM Files ISIM files can be edited either by exiting to DOS, using any convenient editor, or word processor and then returning to ISIM and rereading the file or by using the ISIM line editor. Particularly convenient is a Windows installation in conjunction with WORD. After completion of an edit, the program should be returned to command mode by typing the $ sign followed by return. The important commands of the ISIM line editor are: APPEND:

The current line is added to the end of the program. This is also used to start writing a new program.

LIST:

Displays the whole of the program. By specifying a number, or range of numbers before the LIST command, the number of lines displayed can be modified e.g., 25,30 L.

464


INSERT:

The current line is inserted at the line number specified, e.g., 25INSERT or 251.

CHANGE:

Specify the line that needs to be changed, e.g., 12C. After the question mark prompt type in the new line.

CLEAR:

The screen is cleared.

SAVE:

The file is saved as the specified filename in the default directory, unless otherwise stated. For example: SAVE SIMULATE SAVE C:MODEL\SIMULATE Care must be taken when saving; a changed file will replace the old one if the same name is used. Saving is advisable before running a new example.

KILL:

The current file is removed and a new file can now be read in. It must be used when a new file is read.

QUIT:

Return to DOS.

2 . 6 Writing ISIM Models

General Comments Start the program writing with $APPEND, and the prompt will appear. When writing your own program, it is best to follow an existing program for guidance. ISIM programs usually consist of four sections, namely CONTROL, INITIAL, DYNAMIC and TERMINAL. In addition ISIM programs can contain blank lines, comment lines and end-of-line sections, which must start with a colon. Any line (except PLOT and OUTPUT statements) may be labelled by a number (1 to 8 digits) before the statement. More than one statement may be written on one line if each statement is separated by a semicolon. ISIM does not distinguish between upper and lower case characters. Derivatives with respect to time are indicated by an apostrophe, e. g., X', P', S'. 'I?"

a) Always save a new program, using a suitable name, before trying to run it; a failure, such as numerical overflow, could cause the computer to quit ISIM, and your program will be lost unless previously saved. b) Debugging is easiest when all the important variables are in the PREPARE statement. All the calculated values can be obtained at any time by VAL.


465

c) When changing an existing program, save it under a different or related name, otherwise the original program will be replaced. d) When changing an existing program list the program after every change to see if it is correct.

e) Referring to the original constant values is easiest with XVAL. f) Problems with integration are often caused by errors in the model equations or program. Reducing the value of CINT may sometimes solve problems of numerical overflow at the beginning of an integration. Changing from the default integration method using ALGO will not usually give a faster integration.

g) ISIM is best learned by studying and running the examples.

Program Structure CONTROL section This section must be started by a SIM statement. Commands in this section are executed after the simulation has been completed and include commands for printing messages and resetting values to their original values. INITIAL section In this section the initial values of variables can be defined or calculated by the INITIAL statement. DYNAMIC section All differential equations and all calculations to be performed at each time interval must be in this section. TERMINAL section End-of-run calculations and print instructions are placed here. This should be placed at the end of the program, following OUTPUT and PLOT.

Reserved Variables The variable names listed below are used as ISIM system variables and can not be used for any other purpose.

466 T TFIN ALGO

CINT

NOCI

RERR


The independent variable. The value of T at which the run is terminated. Selects the integration method. 0 variable step. 1 4th order fixed step. 2 2nd order fixed step. 3 implicit (stiff) 2nd order fixed-step. The calculation interval. This is the integration step length for fixed step methods. It also influences the first step of all methods, and if it is too large an error with crash may occur when running new programs - so save them first. Integration difficulties can often be solved by reducing CINT. The values of the variables displayed on the screen and defined in the OUTPUT statement are updated at a time interval of CINT * NOCI. NOCI must be a positive integer. The permitted per-unit relative error for variable-step integration.

2 . 7 ISIM Statements and Functions

Arithmetic Functions Y =ABS(=X) Y=SIN(=X) Y=cos(=X) Y=ATAN(=X) Y =EXP(=X) Y=INTG(=X,YO)

Y =RAND(=X) Y=LOG(=X) Y=SQRT(=X) .LT. .GT. .EQ. .LE. .GE.

*

I

**

Absolute value X in radians It 11

ex Integrates X with respect to time with Y=YO at time T=O. Must be in the dynamic section Returns a pseudo-random number in Y in the range 0 to 1. X is the seed for the random number generator Natural logarithm X must be positive less than greater than equal to less than or equal to greater than or equal to multiply divide exponent


467

ISIM Statements GOTO

Transfers control to the line indicated by the label, e.g., GOTO 7 ...code.. . 7 RESET

IF

Conditional execution of a statement, e.g., IF (OUR.LT.20) AFR=AFR+l

DO

Allows repeated execution of a group of statements, e.g., DO J=O. 1, 1.O, 0.2 ...code.. . END The above loop is executed 5 times with values of J= 0.1, 0.3, 0.5, 0.7 and 0.9

0 o N S l " T X=?

Variables defined as CONSTANT may be modified interactively via the VAL command

RESET

The values of all variables are reset to their original values

INTERACT

A simulation can be stopped at any point by the INTERACT command, for instance when a variable exceeds a certain value. The user can then change the values of any variables by the VAL command as described in Section 2.4

2 . 8 Output Output to the Screen The OUTPUT statement allows the values of the specified variables to be displayed on the screen at intervals of CINT * NOCI during the course of the simulation. For example, OUTPUT T,X,S,P The PLOT statement usually appears in the DYNAMIC part of the model and controls the output to the screen during a run after a START command. Only two variables may be specified. The format of this command is PLOT x variable, y variable, xmin, xmax, ymin, ymax

468


The GRAPH keyboard command, however, allows any number of variables to be plotted against one other variable. A GRAPH command can be given after a simulation has finished, or after an interrupt. The axes can be defined for only one pair of variables or will be automatically assigned. Only variables that have previously appeared in a PREPARE statement can be used in a GRAPH statement. If the simulation has been run several times, using a DO loop or RESET the user can select which runs are to be plotted by specifying the number of the run immediately after the GRAPH command. For example, GRAPH 1,5,7,T,X,P,S,0,100,0,25 will plot X, P and S against T for run numbers 1, 5 and 7 with x axis 0 to 100 and y axis 0 to 25. If no run numbers are specified in the GRAPH command all runs are plotted. For example, GRAPH T,X,S Graphing too many variables will often disguise the details of the smallest scale, due to the automatic scaling. Choosing the range of the scaling (for two variables only) is useful for revealing details. The AGRAPH command is similar to the GRAPH command except the output is produced on the alternative graphics device. The TGRAPH command is to be used to produce an alpha-numeric output if the user does not have a graphics terminal.

Printing ISIM Output The ISIM command MONITOR will cause all subsequent output to the screen to be printed. The command NORMAL terminates this printing procedure. Graphs can be printed on most systems simply by pressing the Print Screen Key.

Saving Data The data produced by running a simulation can be saved in a text file by using the OPEN command. For example, the command OPEN C:ENVIROUSIM.DAT will transfer all subsequent data to the file 1SIM.DAT in the directory ENVIRO until the command ENDFILE is entered or QUIT. The amount of data that is stored can be controlled by use of the NOCI command in exactly the same way as with the output statement. The data file can be read into other graphical software and the graphs replotted and printed. Cricketgraph was used in this way used to produce some of the plots in this book.


469

2 . 9 Useful Sequences of Statements 1 SIM INTERACT RESET GOT0 1 This series of statements in the program allows the user, after the completion of a run, to change the values of any of the variables (using the VAL command) and rerun the simulation (using the GO command). In this way, the influence of the value of one or more variables on the simulation can easily be seen. IF (S.LT.O.0) S=O In some stiff problems, this statement may be replaced advantageously by: IF (S.LT.1E-20) S=O These are a useful program statements to prevent variables, such as substrate concentrations, becoming negative. An alternative way of coding this is:

S=S *COMP(=S) where COMP(=S) equals 1 for S>1 and equals 0 for S

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