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2013

Modelling of biological wastewater treatmentRubayyi Turki AlqahtaniUniversity of Wollongong

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Recommended CitationAlqahtani, Rubayyi Turki, Modelling of biological wastewater treatment, Doctor of Philosophy thesis, School of Mathematics andApplied Statistics, University of Wollongong, 2013. http://ro.uow.edu.au/theses/3951

Modelling of BiologicalWastewater Treatment

A thesis submitted in fullment of the

requirements for the award of the degree

Doctor of Philosophy

from

University of Wollongong

by

Rubayyi Turki Alqahtani

B.Sc., M.Sc. Mathematics

School of Mathematics and Applied Statistics

July, 2013

Dedicated to

My parents, wife and children

Certication

I, Rubayyi Turki Alqahtani, declare that this thesis, submitted in fullment of re-

quirements for the award of Doctor of Philosophy, in the School of Mathematics

and Applied Statistics, University of Wollongong, is wholly my own work unless

otherwise referenced or acknowledged. The document has not been submitted for

qualications at any other academic institution.

Rubayyi Turki Alqahtani

iii

Acknowledgements

In the Name of Allah, the Most Gracious, the Most Merciful.

First of all, I thank God for helping me and sending people who have been inuence

in my life.

I express my deep sense of gratitude to my supervisor Associate Professor Mark

Neslon, for his eort and guidance in helping me throughout my research. As a su-

pervisor, he gave me the opportunity to explore challenging mathematical research

problems and he has been a constant source of guidance.

I am deeply thankful to my co-supervisor, Associate Professor Annette Worthy for

her guidance and encouragement through my Journey.

I would like to express my deepest thanks and gratitude to my father Turki (Oh

God, forgive him and Rahma and insert it into the Prog bliss God), mother Sara,

my wife; Fatimah and my daughters; Sara, Yara and Dana as the gift of unbounded

love and support has no equal.

I would also like to extend my thanks to committee in school of mathematics and

applied statistics in UOW in particular Prof.Timothy Marchant, Dean of research

and Prof.Jacqui Ramagge, the head of the school.

I would also like to extend my thanks to my brothers and my sisters.

iv

v

Abstract

Many models have been proposed in the literature for the specic growth rate of

biomass, including those of Tessier, Monod, Moser and Contois kinetics. Of these

models the Monod model has become the default model used in bioreactor engineer-

ing. However, there is growing experimental evidence showing that in a number of

industrial processes, in particular wastewater treatment, the specic growth rate is

more accurately described by Contois kinetics rather than other models. Thus in

this thesis, we construct and analyze models for wastewater treatment where the

specic growth rate of biomass on biodegradable organic matter is assumed to be

given by the Contois growth rate.

This thesis is organized into four parts. Part 1 consists of three chapters (1, 2 and 3)

which contain general information on the wastewaster treatment and the pertinent

modeling studies.

For remaining parts, specic issues related to wastewater treatment modelling are

investigated. In part 2, which contain three chapters (4, 5 and 6), we develop a

model for wastewater treatment. In chapter (4), the analysis of wastewater treat-

ment in a single reactor with recycle is presented. In chapter (5), we consider a

n-cascade reactor with recycle around each reactor. We investigate how to tune

vi

vii

the parameters of a settling unit to minimize the euent concentration leaving the

reactor cascade. If only one settling unit is to be used, we ask the question, "Where

should it be placed in order to optimized the performance of the reactor cascade?"

The chapter (6) deals with the scenario in which the settling unit is placed after the

nal cascade reactor and the euent stream from the settling unit is recycled back

into the rst reactor. We show that there is a critical value of the total residence

time. If the total residence time is below the critical value then the settling unit

improves the performance of the reactor cascade whereas if the residence time is

above the critical value the performance of the cascade is reduced compared to that

of a cascade without a settling unit. We conclude by noting that the conguration of

the cascade where recycle occurs around each reactor (chapter (5)), outperforms the

conguration of the cascade where recycle is present around the whole cascade at

high total residence time. This is noteworthy as the latter is often used in industry.

A further three chapters (7, 8 and 9) are contained in part 3. In this part, we

extend the standard Contois expression to include both substrate inhibition and a

variable yield coecient. These extensions are important in both the theoretical

and practical application of wastewater treatment. We investigate parameter re-

gions in which either natural oscillations or bistable behaviour can occur. In this

rst of these chapters, we consider the case when the yield coecient is variable

with decay coecient whilst in the second of these chapter, the substrate inhibition

is analyzed. In the nal chapter of this part, the combination of substrate inhibition

and a variable yield coecient is investigated.

The nal part of this thesis investigates the use of a sludge disintegration unit to

viii

minimize the sludge production inside the reactor, represented by chapters (10 and

11). This is important to reduce the operation cost associated with the wastew-

ater process. In chapter (10), we investigate Yoon's model with innite reaction

rate by replacing Monod kinetic with Contois Kinetic. The operation condition for

two congurations of reactor, continuous ow reactor and membrane reactor under

which a sludge disintegration unit is required are investigated. In chapter (11), we

extend Yoon's model to include nite reaction rates and Contois kinetics and estab-

lish that the innite reaction rate assumption of Yoon's model is only correct for

specic value of the sludge solubilization eciency but a viable approximation in

only certain practical cases. The eect of the nite reaction rate and the relative

volume of the sludge disintegration unit (n) upon the production of activate sludge

inside the reactor are investigated. Finally, chapter (12) will summarize the thesis

ndings and give future research directions.

List of Publications

The following publications have been submitted by the author during his study.

Journal article

[1] R.T. Alqahtani , M.I. Nelson and A. Worthy. A fundamental analysis of

continuous ow bioreactor models governed by Contois kinetics. IV. Recycle around

the whole reactor cascade. Chemical Engineering Journal, 218, 99-107, 2013.

[2] R.T. Alqahtani. M.I. Nelson and A.L. Worthy. A fundamental analysis

of continuous ow bioreactor models with recycle around each reactor governed

by Contois kinetics. III. Two and three reactor cascades. Chemical Engineering

Journal, 183, 422-432, 2012.

[3] R.T. Alqahtani p, M.I. Nelson and A.L. Worthy. Analysis of a chemostat

model with variable yield coecient: Contois kinetics. In M. Nelson, M. Coupland,

H. Sidhu, T. Hamilton and A.J. Roberts, editors, Proceedings of the 10th Biennial

Engineering Mathematics and Applications Conference, EMAC 2011, ANZIAM J,

53, pages C155-C171. 2012.

[4] R.T. Alqahtani. M.I. Nelson and A.L. Worthy. Analysis of a chemostat model

with variable yield coecient and substrate inhibition: Contois growth kinetics.

chemical engineering communications, accepted.

ix

xConference paper

[1] R.T. Alqahtani , M.I. Nelson and A.L. Worthy. A mathematical analysis

of continuous ow bioreactor models governed by contois kinetics: A two reac-

tor cascade. In Proceedings of the Australasian Chemical Engineering Conference,

CHEMECA 2011, pages 1{11. Engineers Australia, 2011. On CDROM. ISBN 978

085 825 9225.

[2] R.T. Alqahtani , M.I. Nelson and A.L. Worthy. A mathematical model

for the biological treatment of industrial wastewaters in a cascade of four reac-

tors. In F. Chan, D. Marinova, and R.S. Anderssen, editors, 19th International

Congress on Modelling and Simulation, MODSIM 2011, pages 256-262. Modelling

and Simulation Society of Australia and New Zealand, 2011. On CDROM. ISBN

978-0-9872143-1-7.

Table of Contents

Certication iii

Acknowledgements iv

Abstract vi

List of Publications ix

Table of Contents xi

1 Introduction 1

2 Literature review 82.1 Biological Wastewater Treatment . . . . . . . . . . . . . . . . . . . . 82.2 Anaerobic and aerobic wastewater treatment . . . . . . . . . . . . . . 102.3 Microbial growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Substrate inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Models with a variable yield coecient . . . . . . . . . . . . . . . . . 232.6 Endogenous processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7 Cascade reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Literature review of Contois growth kinetics 313.1 Contois growth kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Contois growth kinetic with inhibition . . . . . . . . . . . . . . . . . 333.3 Interpretations of the Contois kinetics function in microbiology . . . . 333.4 Modeling studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Analysis of single reactor with recycle 464.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Model equations and assumptions . . . . . . . . . . . . . . . . . . . . 474.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xi

CONTENTS xii

5 Analysis of continuous ow bioreactor models with recycle aroundeach reactor 725.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Model equations and assumptions . . . . . . . . . . . . . . . . . . . . 745.3 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 775.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Analysis of continuous ow bioreactor models with recycle aroundthe whole reactor cascade 1226.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7 Analysis of a chemostat model with variable yield coecient 1577.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8 Analysis of a chemostat model with substrate inhibition 1718.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9 Analysis of a chemostat model with variable yield coecient andsubstrate inhibition 1869.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1869.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1899.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10 Analysis of a model for the treatment of wastewater by the acti-vated sludge process 21010.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21010.2 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21210.3 Biochemical processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 21310.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21710.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

11 A nite rate of sludge disintegration unit model 24211.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24211.2 Yoon model (Innite rate SDA model) . . . . . . . . . . . . . . . . . 24311.3 Revised Yoon model (nite rate SDA model) . . . . . . . . . . . . . . 24411.4 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24911.5 Stability of the steady-state solutions . . . . . . . . . . . . . . . . . . 25011.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

CONTENTS xiii

12 Conclusion 26312.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

A The analysis for Chapter 6 268A.1 Parameter value for Monod kinetics . . . . . . . . . . . . . . . . . . . 269A.2 Stability of the washout solution . . . . . . . . . . . . . . . . . . . . . 272

B The analysis for Chapter 9 276B.1 Analysis of the quadratic equation . . . . . . . . . . . . . . . . . . . . 276B.2 Stability analysis for the case a=0. . . . . . . . . . . . . . . . . . . . 277B.3 Analysis of + b

p. . . . . . . . . . . . . . . . . . . . . . . . . . . 278

C The analysis for Chapter 8 280C.1 Claim 1. The steady state solution is not realist when 2 . . . . 280C.2 Claim 2. BR 1 , For any value of e on [a, 1K

d

Kd). . . . . . . . . . 281

C.3 Analysis of + bp. . . . . . . . . . . . . . . . . . . . . . . . . . . 281

D List of Symbols 283

Bibliography 287

Chapter 1

Introduction

Biochemical and bioprocess engineering is swiftly becoming important areas of ap-

plied mathematics as the use of living cells to produce marketable chemical products

becomes increasingly important. There have been many powerful advances that are

transforming the eld, including applications to genetic sequencing, the production

of pharmaceuticals, biologics, commodities and medical applications such as tissue

engineering and gene therapy. It is a broad eld drawing its fundamentals from

biochemistry, microbiology, molecular biology and mathematical modeling.

The mathematical modeling of the bioreactor dynamic, specially the treatment

plant model includes clarication, aeration, settling (gravity thickening) and aer-

obic/anaerobic digestion, has attracted a lot of attention since the last decades.

The study of the biological treatment of wastewater stands at the crossroads of

several elds including biochemistry, microbiology, molecular biology, ecology, en-

gineering, applied mathematics. The method of biological treatment of wastewater

has been extensively employed to reduce slurries and wastewaters pollutants that

include high concentrations of biodegradable organic materials before discharged in

1

Chapter 1 2

order to protect the environment. The bioreactor is a vessel in which wastewater

process is carried out which contain biomass grows through consumption of the

wastewater pollutants.

There are various dierent classications of bioreactors including batch, fed batch

or continuous, which generally refer to when and how much substrate is fed into the

bioreactor over time. The bioreactor's environmental conditions, such as gas (i.e. air,

oxygen, nitrogen, carbon dioxide), ow rates, temperature, pH and dissolved oxygen

levels, and agitation speed/circulation rate, need to be closely monitored and con-

trolled to ensure that the process continues as desired. Attracted by these tempting

mathematical challenges and encouraged by new tools developed in dynamical sys-

tems such as Lyapunov functions for stability, control, robust control [1], optimal

control, stochastic and software tools as Matlab and Maple, mathematicians have

studied the dynamic behavior, the steady state, the stability [1{8] and the control of

wastewater models [1,9{11]. However, in order to understand the evolution of math-

ematic modeling in wastewater treatment, we need to go back in history. Various

models have been used to model the wastewater processes. The most notable models

in this area are based upon the Monod specic growth rate expression [12]. These

models have been popular due to their ability to explain empirical data, and have

found wide application in many industries. However, extensive experimental work

has shown that the anaerobic and the aerobic degradation of wastewater originating

from industrial processes is often better described by the Contois specic growth

rate [13]. Interestingly, there has been only a limited number of modelling studies

employing the Contois model. Hence, this thesis begins by a presenting a case for

Chapter 1 3

Contois growth model's utility followed by developing and discussing mathematical

models where the degradation of a biodegradable organic material is represented by

the Contois expression.

This thesis contain four parts each of which is divided into several chapters. In part

1 we introduce a background and explain the principles behind wastewater treat-

ment and its mathematical modeling in chapter (2 followed by a survey of earlier

studies which have employed Contois model for specic growth rate in 3). In part

2, three dierent reactor conguration for wastewater treatment are studied across

three chapters: chapter (4), chapter (5) and chapter (6). In chapter (4), we ana-

lyze the steady-state operation of a generalized reactor model that includes both

a continuous-ow and a membrane reactor as limiting cases. We analyze the per-

formance of the reactor by nding the steady-state solutions and determining their

stability as a function of the residence time. We show that periodic solution do

not occur. As an application of our results, we show that the recycle improves the

performance of the reactor at moderate values of the residence time. However, when

the value of the dimensionless residence time is suciently large, the performance

of the reactor is independent of the recycle ratio.

In chapter (5), the steady-state treatment of industrial wastewaters in a cascade

reactor with recycle is analyzed. A number of cascades with alternative arrange-

ments of the settling units are considered. Specically, we consider the case when

the recycle stream leaving a settling unit placed around a reactor goes back into

the feed stream for that reactor. The steady-states for the model are found and

their stability determined as a function of the total residence time in the cascade.

Chapter 1 4

Asymptotic solutions in the limit of large total residence time are obtained for the

euent concentration leaving a cascade. This analysis is used to determine the reac-

tor conguration that minimizes the euent concentration leaving the nal reactor.

It is found that, the optimised reactor cascade is obtained by using perfect recycle

around the nal reactor and imperfect recycle around the preceding reactors. When

only one settling unit is used, the performance of the reactor cascade is optimized

at short residence times by placing it around the rst reactor whilst at large total

residence times the performance is optimized by placing it around the nal reactor.

However, at suciently large total residence times there is a little benet gained by

using any settling units.

A common reactor cascade conguration employs a settling unit to recycle biomass

from the nal cascade reactor back to the rst. In chapter (6), we use steady-state

analyse to examine the process eciency of such a reactor conguration. It is found

that there is a critical value of the total residence time which identies a turning

point in the performance of the reactor cascade. If the total residence time is be-

low the critical value then the settling unit improves the performance of cascade,

whereas, if the residence time is above the critical value then the performance of

the reactor cascade with the settling unit is inferior to that of a cascade without

one. It is shown that the critical values of residence time depends upon the values

of the recycle ratio and the concentration factor. We compare the performance of

a reactor conguration employing recycle around the whole cascade with that of a

cascade in which the settling unit recycles the euent stream leaving the ith reactor

into the feed stream for the ith reactor.

Chapter 1 5

In part 3, we extend the standard Contois growth-rate model allowing for either a

variable yield coecient and/or substrate inhibition which containing three chap-

ters, i.e. chapter (7), chapter (8) and chapter (9). In chapter (7), we investigate the

eect of a variable yield coecient including the death rate of the microorganisms.

Analysis shows that the system has natural oscillations for some range of the param-

eters. We also investigate the eects of the death rate parameter on the region of

periodic behaviour. In chapter (8), we investigate eects of the substrate inhibition

on the performance of the reactor. In chapter (9), the biochemical reaction kinet-

ics is governed by Contois growth model subject to both noncompetitive substrate

inhibition and a variable substrate yield coecient. The steady-state performance

of the reactor is predicted and stability of the steady-state solutions as a function

of dimensionless residence time reported. Our results identify two feature of prac-

tical interest. The rst feature corresponds to the case where the no-washout and

washout solutions are bistable. The second feature identies the parameter region in

which periodic solutions can occur when the yield coecient is not constant. These

features are often undesirable in practical applications and must be avoided. Scaling

of the model equations reveals that both the secondly bifurcation parameters are

functions of the inuent concentration. Our results show how the reactor behaviour

varies as a function of inuent concentration and identify the range of inuent con-

centration where the reactor displays neither periodic nor bistable behaviour.

A pressing problem associated with the operation of the activate sludge process is

the disposal of excess sludge. This can signicantly increase the operation cost as-

sociated with the process. A promising method to reduce the amount of the sludge

Chapter 1 6

is to destroy it in situ through the use of a sludge disintegration unit. In part

4, two models for sludge disintegration are considered across two chapters, namely

chapter (10), chapter (11).

In chapter (10), we analyze a model for the activated sludge process coupled to a

sludge disintegration unit. This model was proposed by Yoon [14], through in his

work the Monod specic growth rate law was used. Using steady state analysis we

nd a formula which identies a critical value of the inuent concentration. If the

inuent concentration is below criticality then the reactor is guaranteed to oper-

ate in a state of negative excess sludge production even in the absence of a sludge

disintegration unit. If the inuent concentration is higher than criticality then a

sludge disintegration unit must be used to ensure that the reactor is in a state of the

negative excess sludge production. For a continuous ow reactor the critical value

is relatively large and consequently a sludge disintegration unit is benecial for only

heavily contaminated industrial wastewaters. On other hand, for a membrane re-

actor the critical value is very low and in practice a sludge disintegration unit is

required to ensure that the reactor in state of the negative excess sludge production.

In chapter (11), we investigate a model for an activated sludge process consisting of a

membrane reactor coupled to sludge disintegration unit. This work extends a model

due to Yoon [14], which assumes that the processes inside the sludge disintegration

unit occur innitely quickly, to consider the processes in the sludge disintegration

unit occur at a nite rate. We investigate how the reaction rate in the sludge disin-

tegration unit aects the formation of sludge in the reactor. Steady state analysis

is used to study this system and compare it with Yoon's model. We consider two

Chapter 1 7

scenarios. In the rst scenario, we study the case in which the processes in the

sludge disintegration unit occur at an innite rate. As in chapter (10) there is a

critical value of the sludge disintegration factor above which the reactor system is

guaranteed to be in a state of negative excess sludge production. We show that, the

Yoon's model is only correct if the sludge solubilization eciency is equal to one but

in some cases suciently accurate for practical purposes.

For the second scenario, we investigate the system with a nite reaction rate. We

nd that there is a critical value of the reaction rate which depends upon the size

of the sludge disintegration unit. If the reaction rate is above the critical value

then the error in assuming an innite rate is less than 10% of the exact value using

nite rate. For such value for the reaction rate, the behaviour of the reactor can

be estimated, within experimental error, by assuming an innite rate. Finally, we

show that there is a second critical value of the reaction rate. If the reaction rate

in sludge disintegration unit is suciently small, then there is no longer a critical

value of the sludge disintegration factor above which the reactor operate in a state

of negative excess sludge production for all residence times. Instead negative excess

sludge production can be only achieved only when the residence time is suciently

large.

Chapter 2

Literature review

2.1 Biological Wastewater Treatment

Biological wastewater treatment is a natural process in which bacteria are used to

degrade waste organic matter to a mixture of carbon dioxide and methane. It is

widely used to minimize the volume of contaminated wastewaters that are produced

by industrial and municipal sources.

Biological wastewater treatment consists of several stages, including primary and

secondary treatments of the wastewater. In the primary treatment, particulate

solids are mostly removed by sedimentation. This is followed by secondary treat-

ment in which microorganisms, in the form of activated sludge, are used to remove

substrates (suspended organic solids and dissolved organic compounds) from the

wastewater. In the activated sludge process, microorganisms are suspended in the

wastewater, and this mixture is called 'mixed liquor'.

The secondary treatment process is generally conducted in two steps. In the rst

step, bacteria degrades organic pollutants in an aerated biological reactor as food

8

Chapter 2 9

to sustain their life processes, including energy production, reproduction, digestion,

and movement. These processes inevitably result in byproducts that include water

and carbon dioxide.

The second step involves a settling unit (or clarier). The purpose of this unit is to

allow the sludge to settle to the bottom. Then, the substances at the bottom of the

clarier, which include activated sludge and mixed liquor, are recycled into the reac-

tor, where they provide the nutrients required to keep the microorganisms alive and

active. The liquids that remain in the clarier are pumped to disinfection facilities

and discharged to receiving waters or sent for additional treatment (referred to as

"tertiary treatment"). A schematic of two stages of treatments of the wastewater is

shown in gure (2.1). The design of the activated sludge processes resolves around

Figure 2.1: Wastewater treatment stages

two main variables related to the treatment of sludge. These are: 1) The quantity

of suspended solids in the mixed liquor, often referred to as MLSS, and 2) the bio-

chemical oxygen demand (BOD) or the chemical oxygen demand (COD), which are

indicative of the extent to which the wastewater is loaded with organic materials.

The microbial growth that occurs when the microorganisms are utilizing the nutri-

ents in the wastewater can be used to dene the interaction between the concen-

Chapter 2 10

tration of microorganisms and the substrate. Microbial growth depends upon the

concentration levels of the substrate and the microorganisms in the bioreactor, these

levels in turn depend on the type of process and the operational mode. Thus, micro-

bial growth depends on various types of processes which can be simple or complex

to determine and control [15].

There are several types of bioreactors. They are classied generally as either open

or closed systems. For an open system, the ow through the system is continuous.

In a closed system, the rate of growth of the biomass decreases with time, eventu-

ally approaching zero, which is the inevitable result when insucient nutrients are

available to sustain growth [16].

In our research, we consider the secondary treatment of wastewater in a continuous

ow reactor. We assume that the substrate and microorganism concentrations in the

reactor are well mixed i.e. the substrate and microorganism concentrations leaving

the reactor are the same as those in the reactor.

2.2 Anaerobic and aerobic wastewater treatment

Organic matter is removed from wastewater primarily by two biological processes,

i.e., anaerobic and aerobic processes. They have been applied successfully for treat-

ing waste euents from various types of domestic and industrial wastewaters. The

anaerobic process does not require oxygen for bacteria to break down organic mat-

ter, whereas oxygen is required in the aerobic process. A major disadvantage of

the aerobic treatment process is the amount of energy that it requires [17]. Thus,

in most cases, anaerobic treatment processes are preferable for treating waste ef-

Chapter 2 11

uents that have high organic content [17{20]. The advantages of using anaerobic

treatment processes are [21],

1. Low cost. As oxygen is not required an anaerobic reactor is less expensive than

an aerobic reactor and it is simpler to operate and maintain.

2. Enhanced energy production. Anaerobic processes produce useful products in

the form of biogas that can be used to provide energy for other systems.

3. Very good stability. Anaerobic processes are stable under operating conditions

that would make other processes unstable.

4. Lower nutrient requirements. The requirement of nitrogen in the anaerobic

process is lower than that for an aerobic system.

5. Very high organic loading rate. The eciency of anaerobic systems for COD

removal is signicantly greater than that of aerobic systems.

6. Low area space.

7. No requirement for aeration.

The disadvantages of using anaerobic treatment processes,

1. The heavy metals and contaminants in sludge are accumulated.

2. The range of temperature control is narrowed .

3. Installing and managing an interrelated group of system to safely hand heating

of the tank.

4. Environmental Sensitivities

Chapter 2 12

The anaerobic process consists of three biological reaction stages in series. These are

the hydrolysis stage, the acidogenesis stage, and the methanogenesis stage. These

processes convert the organic matter to methane (CH4) and carbon dioxide (CO2)

[22,23]. Some researchers have proposed dierent numbers of stages, including two,

four, and nine stages [23,24].

In the rst stage, insoluble organic polymers are broken down by bacterial hydrolysis

to soluble substrate that the bacteria can utilize. In the second stage, acidogenic

bacteria convert the products from the rst stage into carbon, hydrogen and other

materials, i.e, CO2, H2, NH3, and organic acids such as acetates, propionates, and

butyrates. In the third stage, the products from the second stage are converted to

methane by methanogenic bacteria [25{27]. The processes in all three stages are

generally dependent on dierent bacterial species.

In the past few years there has been an increased interest in the kinetics associated

with the three stages of the anaerobic digestion process. As a result, variations in

the anaerobic digestion process have been described extensively by models that have

been developed to represent the process [24,28]. Mathematical models that are used

to describe anaerobic processes to predict the reactor behavior are normally divided

into four categories,

1. Simple basic models such as Tessier [29], Monod [12], Moser [30] and Contois

[13].

2. Un-structured non-segregated model [31].

3. Un-structured segregated models [32].

Chapter 2 13

4. Structured kinetic models such as IWA models [33{38].

The last three groups of models provide more comprehensive descriptions of the

anaerobic processes and are normally complicated than the simple basic models.

However, the simple basic models are often sucient for technical purposes and

are an accepted method to predict the behavior of biological wastewater treat-

ment [39{42]. The study [39] is of interest because all the numerical simulation

reported in this thesis use the kinetics value from this study.

Several accurate descriptive structured models have been internationally accepted

to simulate activated sludge processes such as the ASM-1, ASM-2, ASM-3 [37, 43].

The ASM-1 model includes 13 dierential equations and involves eight chemical

processes including the growth and decay of heterotrophic and autotrophic biomass,

ammonication of soluble organic nitrogen and hydrolysis. The model can predict

nitrogen, chemical oxygen demand, nitrication and denitrication processes in-

cluding aerobic, anaerobic processes. Henze et al. [37] developed the ASM-1 model

by taking into account the biological nutrient removal in activated sludge systems,

leading to the ASM-2 model. This was achieved by incorporating the biological

uptake of phosphorus into the basic ASM1 framework. Biological nutrient removal

is handled by the addition of a module to model enhanced biological phosphorus

removal, and also by including chemical removal of phosphorus via precipitation.

Enhanced biological phosphorus removal refers to the biological uptake and removal

of phosphorus by the activated sludge system in excess of the amount removed by

normal aerobic activated sludge system.

Henze et al. [43] further extended ASM-1 by allowing the storage of organic sub-

Chapter 2 14

strates as a new process and improving the modeling of lysis (decay) of endogenous

respiration process. These models are analyzed by direct integration of the gov-

erning equations [43]. Nelson and Sidhu [38] investigated the steady states of the

ASM-1 model without recycle considering how the performance of the reactor change

as function of control parameters including the oxygen transfer coecient and the

residence time. The result of this study showed that there is a critical bifurcation

point in residence time which determines the performance of reactor.

Mosey [24] studied four groups of bacteria during the degradation of organic matter

to carbon dioxide and methane. These groups are acid-forming, acetogenic, aceto-

clastic methane, and hydrogen-utilising methane bacteria. A mathematical model

was developed to describe each of the steps of the kinetics processes where the Monod

model is used to simulate the reaction. Kalyuzhnyi [44] developed a mathematical

model of the batch anaerobic digestion of glucose, which involved ve dierent types

of bacteria, i.e., acidogenic, ethanol-degrading acetogenic, butyrate-degrading ace-

togenic, acetoclastic methanogenic, and hydrogenotrophic methanogenic bacteria.

This model also considered several factors and potential inhibitors that are depen-

dent on bacterial action. A kinetic model was developed earlier for use in simulating

the various steps that occur when given substrate materials are degraded in an anaer-

obic sequencing batch reactor (ASBR) [45].

Munch et al. [46] developed a volatile acid production model and validated it against

the experimental data reported in the literature. Knobel and Lewis [47] developed a

mathematical model that can be used to assess the process of anaerobic digestion in

wastewater in which sulfate species exist in high concentrations. Their model incor-

Chapter 2 15

porated the Debye-Huckle theory, and it can be used to calculate activity coecients

and several other process parameters includes sulphate reduction, hydrolysis, acido-

genesis, beta oxidation of long chain fatty acids, acetogenesis, and methanogenesis.

In this thesis, we represent the biological processes by considering a single substrate

concentration consumed by a single microorganism species in which the reaction rate

is described by the Contois model (to be described later, see chapter (3)). Thus, the

three stages at lined above are replaced by a single rate-determining step.

2.3 Microbial growth

Microbial growth increases the number of cells and the mass of the bacteria as

a consequence of cell division. Several factors can aect microbial growth, and

these are classied as intracellular and extracellular factors. Intracellular factors

include the internal structure, metabolic mechanisms, and genetic material of the

cell. Extracellular factors are external environmental conditions that aect the cells,

including pH, temperature, oxygen concentration, water, and food. The growth of

biomass in a culture requires a suitable environment in which the microorganisms

can live and grow. This environment must satisfy several conditions, including viable

inoculums, an energy source, nutrients, absence of inhibitors which prevent growth,

and suitable physicochemical conditions.

In the investigation of the kinetic behaviour of such a system, it is important to

understand how the concentrations of representative components of the system, i.e.,

cells, substrate, products, and byproduct, change with time. The study of microbial

growth in terms of various variables (various growth parameters) is useful for many

Chapter 2 16

purposes and is required to predict and control the behavior of the system. These

growth parameters are dened to describe the growth of both simple and complex

cultures. These parameters include specic growth rate, growth yield, metabolic

quotients for substrate utilization and product formation, and maximum biomass.

2.3.1 Specic growth rate

If all the requirements for growth are satised, then the biomass concentration

increases exponentially with time, since the overall rate of change of biomass is

proportional to the mass of biomass:

dX

dt= X: (2.3.1)

The dierential coecient (dXdt

) is the population growth rate. X is the cell concen-

tration (kg cell=m3) and the parameter , which represents the rate of growth per

unit of biomass, is termed the specic growth rate.

2.3.2 Growth yield

Growth yield is a biological variable that allows us to assess the rates of production

and consumption of energy and mass in a biological system. From the standpoint of

modeling and describing such systems, the measured growth rate is quite benecial.

The growth yield is dened by,

Yx=s =rxrs; (2.3.2)

where rx is the amount of biomass produced and rs is the amount of substrate

consumed. When the reaction rates are equal to accumulation rates, which occurs

Chapter 2 17

in batch systems, the growth yield becomes,

Yx=s = dXdtdSdt

= dXdS

; (2.3.3)

For a continuous system, the growth rate is given by,

Yx=s = XS

= X X0S S0 ; (2.3.4)

where X0 and S0 are the initial biomass and substrate concentrations, respectively,

and X and S are the corresponding concentrations during the growth of the culture.

The growth rate of X and S have opposite signs, so the use of the negative sign

is required to reect that reality. For a growth-limiting substrate when the cul-

ture reaches its maximum biomass (Xm) and is approximately 0, the growth yield

becomes,

Yx=s =Xm X0

S0: (2.3.5)

Growth yield is an important variable because it is a quantitative expression of

the nutrient requirements of an organisms. As early as 1869, Raulin expressed the

nutrient requirements of fungus in term of growth yield. In some bacterial cultures,

the growth yield is constant when the conditions are maintained constant [48]. In

other studies, it is considered to be variable [49].

2.3.3 Metabolic quotient

The rate at which the substrate is utilized by organisms is called the rate of con-

sumption of the substrate. This rate is given by

dS

dt= qX; (2.3.6)

Chapter 2 18

where X is the biomass, and the coecient q is known as the metabolic quotient or

specic metabolic rate. If the biomass concentration is constant and the environ-

mental factors are constant, then q must be constant. In terms of growth, the rate

of consumption of a substrate is given by:

dS

dt=

X

Yx=s; (2.3.7)

2.3.4 Eect of substrate concentration on growth rate

The microorganism growth rate (and therefore the microorganism concentration) is

related directly to the concentration of the substrate that is normally consumed by

the microorganisms. In some bacterial cultures, however, the growth rate is virtually

unaected by substrate concentration, i.e., zero order kinetics exists. When the

substrate consumption follows enzyme kinetics, the metabolic quotient is given by:

q = qmax

S

Ks + S

; (2.3.8)

where Ks is the saturation constant, which is equivalent to the Michaelis-Menten

constant [50], and qmax is the maximum value of q. If we make the substitutions,

q = Yand qmax =

maxY

, then it follows that:

= max

S

Ks + S

: (2.3.9)

This equation is known as the Monod equation.

2.3.5 Kinetic models of microbial growth

Two variables that may determine the rate at which microbial growth occurs are

the microbial population's specic growth rate () and the concentration of the sub-

Chapter 2 19

strate (S) such as in equation (2.3.9). This relationship is used to great benet in

many dierent specialties, including biotechnology, ecology, genetics, microbiology,

and physiology. During the last two centuries, extensive studies of microbial cul-

tivation were conducted, and, as early as the 1830s, Cagniard de Latour, Kutzing,

and Schwann revealed that the growth of yeasts and other protists is responsible

for fermentation. An overview of the historical development of knowledge concern-

ing microbial growth is presented in [16]. The understanding of microbial growth

has been improved by the understanding of principle of metabolic uxes and by

a number of mathematical models that have been proposed [51, 52]. In the 19th

century, various classical models were used to characterize the growth of microbial

populations, such as the Verhulst and Gompertz function [53{55].

In 1912, the rst kinetic approach that was associated with the growth of microbes

was posited by Penfold and Norris [56], who proposed a "saturation" type of curve

to express the relationship between and S. This curve indicated that the maximum

rate of growth of the organisms (max ) is independent of high levels of substrate

concentrations [56]. Monod's model is consistent with the Penfold hypothesis, but

some have been critical of it due to the fact that it includes the determination of

at a wide range of substrate concentrations, ranging from high to low [57,58]. Later,

Monod introduced a model that incorporated the concept that the substrate could

limit the rate of growth of the microbes,

= max

S

Ks + S

; (2.3.10)

where = specic growth rate, max = maximum specic growth rate, S = sub-

strate concentration, and Ks = substrate saturation constant. Monod used several

Chapter 2 20

parameters to dene the relation between growth rate and the utilization of the

substrate [48], i.e, Ks, max and the yield coecient, Yx=s [58],

Yx=s =dX

dS(2.3.11)

=Yx=sX

:dS

dt Yx=sq:

Many researchers have attempted to improve Monod's representation for the growth

kinetics of cells by using three approaches. These are 1) investigating the eect of

physicochemical factors on the Monod growth parameters [59{63] ; 2) adding a

new parameter in Monod's model to account for the inhibition of the substrate or

product, the diusion of the substrate, maintenance, or the eects of cell density

on max [13,16,28,57,63{66] ; and 3) the development of innovative kinetic theories

that support the mechanistic and empirical models [65,67{72]. Contois adapted the

Monod model after discovering evidence that showed the specic microbial growth

rate may also depend on the microorganism concentration. The Contois expression

is given by

(S;X) = max

S

KsX + S

; (2.3.12)

where = specic growth rate, max = maximum specic growth rate, X= microor-

ganism concentration, S = substrate concentration, and Ks = substrate saturation

constant. In this thesis, we use the Contois model to describe the growth rate.

The choice of this model is motivated by the increasing number of the experimental

studies showing the Contois model giving excellent t with experimental data. For

more detail see chapter 3.5.

Chapter 2 21

2.4 Substrate inhibition

Monod proposed the dependence of growth rate on the concentration of the sub-

strate, and the concept was explored further by additional researchers in the eld.

These models t experimental data for biological processes in wastewater treatment

quite well, but Powell's [73] showed that they are not valid at low concentrations of

some substrates or when high concentrations inhibit the growth of organisms.

Since high concentrations of a nutrient inhibit the growth of microorganisms, it

could be concluded that the Monod models only represent a special case rather

than a general relationship. High concentrations that inhibit growth may occur

during start-up or transient conditions. Models that describe the inhibitory eect

of substrate concentrations are very useful for modeling the biological treatment of

waste euents from industrial operations. For example, such euents may contain

ammonia, nitrates, phenols, thiocyanates, and volatile acids, all of which are known

to inhibit the growth of microbes [28].

Several empirical models that describe the inhibitory eect of substrates on mi-

crobial growth have been proposed. Haldane [74] proposed a popular homologous

equation that represents the uncompetitive inhibition of enzymes. Andrews [28]

later applied Haldane's equation to substrate-inhibited microbial growth. Other re-

searchers used Haldane's equation to describe the kinetics involved in the inhibition

of microbial growth by various substrates [75,76]. The equation describes some ex-

perimental data quite well, but it has been shown to be less than adequate for other

sets of data [77,78]. Andrew's equation is the most widely used substrate inhibition

Chapter 2 22

model [79{81].The Andrew/or Haldane's equation is given by,

= m

S

KM + S +S2

KI

!: (2.4.13)

The rate increases to its maximum value and then decreases as S increases, with

the shape of the curve depending on the values of KM and KI . At low substrate

concentrations, we have:

KM S + S2

KIthen = m

S

KM

: (2.4.14)

In this case the rate increases linearly as substrate concentration increases. At high

substrate concentrations we have :

KM + S S2

KIthen = m

KIS

: (2.4.15)

In this case the rate decreases as the substrate concentration increases. There is a

maximum value of that is achieved at an intermediate value of S . This maximum

value of that can be obtained from [15] is given by,

S =pKMKI ; (2.4.16)

=m

2(q

KsKI) + 1

: (2.4.17)

Han and Levenspiel [82] proposed a more general expression that has been used

to describe substrate inhibition,

= m

(1 S

Sm)n

Ks + S (1 SSm )m!; (2.4.18)

where Sm is critical inhibitor concentration, and n and m are constants.

Most of the current information concerning the role of the substrate in inhibiting

biodegradation has been derived from studies using phenol. High concentrations

Chapter 2 23

of phenol inhibit the growth of microorganisms, and the kinetics of inhibition have

been described by several models [78, 83{85]. Among these models, most of them

are empirical models, but they are quite eective for modeling at high phenol con-

centrations. Rozich et al. [86] reported that, among ve models that they evalu-

ated, Andrew's equation described the experimental data most accurately. However,

there is disagreement among various researchers concerning the viability of Andrew's

equation for the intended purpose. Pawlowsky and Howell [84] observed that the

dierences between ve inhibition models were statistically insignicant. Yang and

Humphrey [78] made a similar observation for three models for describing the degra-

dation of phenol by Pseudomonas putida and Trichosporon cutaneum. Tan et al. [87]

generalized a model for substrate inhibition by describing the inhibition of micro-

bial growth associated with the substrate using statistical thermodynamics. The

selection of an appropriate model for representing the inhibition of the substrate

based on theoretical considerations was discussed by Goudar et al. [88] who found

Andrew's equation to be suitable for describing the biodegradation of phenol.

2.5 Models with a variable yield coecient

In many models of bioreactors it is assumed that the yield coecient is a constant

during the course of the reaction. However, some experimental evidence has shown

that the yield coecient can be a function of substrate concentration. In this section,

we provide a brief overview of models using a variable substrate yield by extending

the literature review appearing in [89]. An example of Contois model with variable

Chapter 2 24

yield coecient can be presented as follows, see [49]

VdS

dt= F (S0 S) V X(S;X)

Y (S); (2.5.19)

VdX

dt= FX(R(C 1) 1) + V X(S;X)KdV X: (2.5.20)

where Y (S) = + S, with ; > 0.

Note that the study based upon Contois kinetics is related to chapter (3).

Ajbar [2] analysed a continuous bioreactor using a general dependence for the growth

rate and a variable yield coecient upon the substrate concentration. A non-ideal

bioreactor is considered in which a bioreactor is divided into two regions, a well-

mixed region and an unreacted region. This study showed that the variability of

the substrate yield is a necessary condition for the existence of periodic behavior.

This extends earlier nding of Crooke et al [90], who showed that periodic solutions

cannot occur for systems with a constant yield, from an ideal reactor to a non-ideal

reactor.

Zhu et al. [91] investigated a bioreactor in which the growth rate was assumed to

depend only on the substrate concentration. They claimed that limit cycles can

occur in a chemostat with two competitors for a single nutrient when there is a con-

stant substrate yield. Sari [92] re-investigated the reactor model of Zhu et al [91]

and showed that there was an error in the computation of the eigenvalues in [91]

and that consequently the result in [91] is false. In this study the growth rate was

assumed to depend only on the substrate concentration.

A number of authors have studied the behavior of a chemostat with two microorgan-

isms and a single substrate with variable yields [93{95]. Song and Li [93] assumed

that the growth rate follows Monod kinetics with a variable substrate yield. Huang

Chapter 2 25

and Zhu [94] extended the study [93] by considering quadratic substrate yield. Huang

and Zhu [95] furthermore generalised the two linear yield functions to nth and mth

order-polynomials and replaced the standard Monod specic growth rate by a gen-

eral non deceasing function. Sari [96] studied a chemostat model in which n species

compete for a single growth-limiting substrate by extending the results of [97] to

the case when yields are variable. The growth rate includes both monotone and

non-monotone response functions (which also includes Monod kinetics).

2.6 Endogenous processes

The endogenous processes, acting to reduce the amount of biomass, incorporates a

number of mechanisms including endogenous respiration, predation, cell death and

lysis [36]. The most important of theses is usually cell death. Decay has a signicant

inuence on the performance of biological reactors. Such eects include reducing the

sludge yield, an increase in electron acceptor utilization and a deterioration of the

performance of the reactor [98{100]. Several investigators [38, 101] have combined

these endogenous processes into a single parameter as a rst-order processes due to

the dicultly of separation between these processes experimentally. Therefore, the

endogenous processes are included in all-but-one of the modeling chapter.

2.7 Cascade reactors

In this section we provide a brief historical discussion of the literature relating to the

mathematical analysis of cascade reactors. Cascade reactors using Contois kinetic

Chapter 2 26

are discussed separately in chapter (3).

The optimal design of a reactor cascade has been receiving attention since the early

1960s. A reactor cascade is optimized by adjusting the physical properties of the

reactor cascade, such as volume, interconnection structure of dierent tanks, recircu-

lation rates, tanks shapes, with respect to the design objective such as the residence

time.

Harmand et al. [10] analyzed the optimal design of a two-reactor cascade including

two feed streams (multi-stream ow) and/or a recirculation loop in which a single

reaction occurs. The model assumed that the microorganism does not decay. The

objective of this study is to minimize the total volume of the cascade given a spec-

ied conversion. The steady-state design problem was analyzed for a generalized

growth rate law that included Monod kinetics subject to substrate inhibition.

The obtained result was that the appropriate allocation of the inuent into the re-

actor cascade in order to minimize the total volume depend on the ratio rate. The

ratio rate depends upon the euent concentration leaving the cascade. They showed

that the distribution of the inuent and introducing a recycle into the reactor cas-

cade is only required to minimize the total volume when the ratio rate is less than

one. Harmand et al. [9] revisited the optimal design of a two-reactor cascade whilst

investigating enzymatic biological reactions. Latter Rapaport et al. [102] extended

the study on optimal design of two-reactor cascade to includes several microbial

species, instead of a single specie. The important nding of their study was that

the optimal design does not allow the coexistence of several species.

Powell and Lowe [103] studied the behaviour of a cascade of N reactors of equal vol-

Chapter 2 27

ume with recycle and with no death rate. Recycle was considered to occur between

the last reactor of the cascade and the rst reactor. As the number of the reactors

increased, they found that the behaviour of the system approached that of an ideal

tubular fermenter with plug ow.

Erickson and Fan [104] studied the optimum hydraulic regime for several activated

sludge systems composed of N reactors (N = 2 or 3) with recycle and a non-zero

death rate. They considered two points of view: the minimum total reactor volume

of the cascade required to produce the desired euent concentration; and an esti-

mate of the cost of the organic waste being discharged and the total cost for the

volume of the total reactor cascade. They compared the results from the reactor

cascades with these obtained using a single tank.

Erickson et al. [105] examined the optimisation of a multi-stream cascade with N

reactor (N = 2, 3, 4, and 5), recycle, and a zero death rate. They considered two

stages in each reactor, a mixing stage and an aeration stage. The inuent entered

the reactor cascade in the mixing stage, where no reactions occurred. The objective

function was to minimize the total volume of the reactor cascade by determining

the appropriate allocation of the inuent into the mixing tanks and the distribution

of reactor volumes for specic concentrations of the organisms and substrate in the

recycle stream. The parameter value of the concentration factor (C) was 2 which is

also commonly used in practice.

The optimal cascade designed can be divided to two groups which are static [106]

or dynamical [107]. Several investigators have sought to determine the optimum

design of two-reactor cascade in series [107{110]. Yang and Su [110] conducted their

Chapter 2 28

experiment in each reactor whilst residence times in each were equivalent. In their

ndings, they observed a superior performance for the two reactor cascade as com-

pared to a single reactor of the same dilution rate. Basing their arguments on the

later observation, they inferred that enormous output will be experienced when two

reactor cascade are connected in series.

Chen et al. [109] examined the possibility of increasing performance in a single re-

actor and two-reactor cascade in series with a variable yield coecient as a result of

natural oscillations generation. For a single reactor using Monod, Tessier and Moser

growth models, they found out that the improvement of the reactor performance can

be achieved by operating at the optimal steady-state residence time which always

above the oscillatory region. Basing their study on the later observations, they com-

pared the cascade reactor which has equal residence time against the single reactor

that has substrate inhibition in its growth kinetics. While analyzing two more com-

plex microbial systems, Balakrishnan and Yang [108] visited again Monod-growth

model. In general, the output of a single chemostat system is poor as compared to

a two-chemostat-in-series system with the same total residence time.

Nelson and Sidhu [107] re-investigated the biological system of a two-reactor cascade

in series with no death rate that has been originally considered by Balakrishnan and

Yang [108], Chen et al. [109] and Yang and Su [110] in which the growth rate is

given by a Monod growth kinetics with a variable yield coecient. The criteria they

used to compare between the two congurations were dierent from that proposed

in [108{110] wherein the performance of optimal single reactor compared with a

cascade performance when the residence time of former is the same, or smaller than

Chapter 2 29

that of the latter.

They found that the improvement of performance is obtained by using two-reactor

cascade than a single reactor for the choice of specic parameter. The nding of the

comparison with respect of the reactor productivity, is the single reactor surprisely

is superior than the cascade. It also showed that an insignicant improvement of

the cell mass eciency of the cascade which is not as the rst nding that reported

in [108] and [110].

Grady and Lim [80] studied biological wastewater treatment using reactor cascades

with and without recycle. Their study included the use of a single feed stream and

multiple feed streams. They investigated the optimisation of the reactor design from

two points of view, i.e., 1) the minimum euent concentration that can be obtained

for a specied total volume and 2) the minimum reactor volume that is required to

deliver a specied euent concentration. They found that the recycle ratio had no

signicant eect on either the substrate concentration or the microorganism con-

centration in each reactor. They concluded that recycle is not a signicant tool to

reduce the substrate concentration.

Using several and dierentiated equipments, Mantzaris, et al. [111] explained how

the growth process in multiple bioreactors can be modelled and also addressed the

following issues: (1) the chemical environment is likely to vary in diverse reactors

and (2) the biomass arises as distinct cells which are in reality separate and remain

separated from each other and thus a constituent of the biomass that go into the

reactors will not mix up with the biomass that was previously there. They devel-

oped dierent numerical algorithms for dierent models to solve the steady-state

Chapter 2 30

and transient problems and concluded that the heterogeneity in the biomass can be

ignored for the purposes of calculating the concentrations of biomass, nutrients, and

products in the abiotic environment if the structured model is linear in the state

vector.

Chapter 3

Literature review of Contois

growth kinetics

3.1 Contois growth kinetics

In many biological processes, the rate at which a population of microorganisms (X)

grows and the rate at which the substrate (S) is consumed are essential part of the

kinetic models. Often researchers have assumed that the specic growth rate is not

dependent on the concentration of the microorganisms [30,48,112{115]. However, it

has been found experimentally that in some systems, the specic growth rate model

should include the concentrations of both the substrate and the microorganisms.

Such a growth kinetics model was introduced by Contois [13], who presented evidence

to show that the specic growth rate can also depend upon the microorganism

concentration from studies of batch cultures growing under conditions of nutrient

31

Chapter 3 32

limitation. The Contois growth model is given by,

(S;X) = max

S

KsX + S

; (3.1.1)

where the specic growth rate (S;X) is a function of microorganism concentra-

tion (X) and the concentration of the substrate (S). The parameters max and Ks

are the maximum specic growth rate and the saturation constant respectively.

These are constants under dened conditions.

Contois's model shows that as the microorganism concentration increases, the growth

rate ((S;X)) decreases. The growth rate is given by,

G = max

S

KsX + S

X = max

S=X

Ks + S=X

X: (3.1.2)

There are two extreme cases for the Contois model as the population density of

biomass increases. In these cases, the Contois model reduces to rst-order kinetics

for either biomass or substrate concentration. These approximations are [116],

1. First-order kinetics for biomass growth.

S

X Ks =) G = maxX: (3.1.3)

2. First-order kinetics for substrate consumption.

S

X Ks =) G = max

S

Ks

: (3.1.4)

The Contois growth rate has been used as a surface limiting model to explain the

mass transfer limitations due to the limited surface area when the biomass con-

centration is high [46, 117{120]. In this case, the specic growth rate is expressed

as,

= max

S=X

Ks + S=X

: (3.1.5)

Chapter 3 33

Thus, if the population density of biomass increases, leading to increased obstruction

to substrate uptake and growth of any particular microbe, then the Contois rate law

reduces to,

G maxS

Ks

: (3.1.6)

3.2 Contois growth kinetic with inhibition

Substrate inhibition is a vital component (see chapter (2)) in microbial degradation

processes. Monod model does not include substrate inhibition. Andrew extended the

Monod model to include substrate inhibition and his extended form has been applied

to many biochemical processes. In the following sections, we establish that there

are many experimental situations where the Contois model is the most appropriate.

Hence we believe that extending the Contois model to include substrate inhibition,

could improve its applicability greatly over many domains. This is achieved by

modifying the growth rate model as follows,

= m

S

KMX + S +S2

KI

!: (3.2.7)

3.3 Interpretations of the Contois kinetics func-

tion in microbiology

The Contois model is sometimes found to t experimental data better than mod-

els that are independent of the biomass concentration (such as the Monod model).

Researchers in several scientic elds have attempted to determine why this phe-

nomenon occurs. Both Fujimoto [121] and Characklis [122] reported that the use of

Chapter 3 34

the Contois model can be derived by mechanistic means on the basis of saturation

kinetics when mass transfer is growth-limiting or when enzyme kinetics is growth-

limiting. Furthermore, the availability of a local medium that is heterogeneous and

that surrounds the cell, producing restricted access to the substrate by the biomass,

is one of the most important features in the use of the Contois model [123]. Arditi

and Ginzburg, [124] evaluated the role of predator-prey in the procedure that de-

termines growth rate. They found that describing population growth-rate processes

using both predator-prey ratios was more appropriate than considering only the prey

in the growth process. Other evidence that supports this hypothesis were provided

by Bail [125], Greenleaf [126] and Pearl and Parker [127]. They specically stud-

ied spatial heterogeneity and its role the growth process. Bail studied cell space,

and Greenleaf evaluated the reproduction rate of infusoria and how cell proximity

aected it.

3.4 Modeling studies

In this section we summarize the literature regarding reactor models that use Con-

tois kinetics. Although the Monod expression has been the most popular model

used to describe microbial growth rate, several researchers have used Contois ki-

netics to simulate the behavior of the growth rate in wastewater treatment sys-

tems [3, 49,101,128{131].

Nelson et al. [101] were the rst to use the Contois model to simulate growth rate

in a single reactor with recycle. They showed that the euent concentration can

be decreased to any desired level by operating the reactor at a suciently large

Chapter 3 35

residence time. This is not true for processes controlled by Monod kinetics in which

the euent concentration has a limiting value [132].

Nelson and Holder [128] studied the behavior of a cascade of N reactors without

recycle and reported the steady state solutions with associated stability analysis.

Their objective was to minimize the euent concentration as a function of residence

time and as the number of tanks in series. For the majority of cases considered,

a signicant decrease in the euent concentration was achieved by increasing the

number of reactors in the reactor cascade. They found that the value of the sub-

strate concentration leaving a N reactor cascade (Sn) reduces by1

n as the number

of the reactor increase where t is the total residence time.

Several authors have investigated the behavior of reactors in which the specic

growth rate follows the Contois model with a linear yield coecient [3,49,129{131].

Nelson and Sidhu [49] analyzed a well-mixed continuously stirred reactor for both

a single reactor and a double reactor cascade. Assuming that the death rate was

zero, they considered two cases. In the rst case they analyzed the model for a

single reactor by nding the steady state solutions and identifying the condition for

washout. They also found a parameter region in which self -sustained oscillation was

generated. After a Hopf bifurcation, the time-averaged euent concentration is an

increasing function of the residence time so they concluded that periodic behaviour

is undesirable.

In the second case, they examined how the performance of a two-reactor cascade

changes as the residence time in the rst reactor is varied by assuming that the total

residence time in the cascade is xed. Operating at the same residence time, the

Chapter 3 36

performance of the cascade can be inferior to that of the single reactor. However,

the optimal performance of the cascade was always better than that of the single

reactor. Nelson et al. [129] further investigated a membrane reactor with a non-zero

death rate of the microorganism. They studied the model using the same analysis

approach as in [49] and investigated how the death rate aected the performance of

a single reactor and a double reactor cascade as well as the Hopf bifurcation region.

Ajbar et al. [3] extended Nelson et al's study [49] to take into account oxygen transfer

limitations. The steady state solutions and their stability were determined and the

performance of the reactor was carried out by analyzing the process eciency and

the productivity. It was found that a decrease in the inuent concentration resulted

in an increase in the eciency and a decrease in the productivity of the reactor.

The eciency of the reactor was found to decrease as the oxygen transfer coecient

decreased. It was found that a variable yield coecient is necessary to produce a

periodic solution but it has a minor eect on the performance of the reactor.

Ajbar [131] considered a chemostat model in which two microbial populations com-

peted for the same substrate with two dierent variable yield coecient. In this

study, the specic growth rates of both species are assumed to follow the Contois

model. Analysis of this model showed that the assumption of non-constant yield

may lead to complex behavior such as chaotic behavior.

3.5 Experimental studies

In this section we provide an overview of experimental research reported in the

literature which have employed the Contois growth model. We provide specic (and

Chapter 3 37

many) instances where the Contois model has proved to be the best choice for tting

experimental data when compared with alternate models like Monod. Specically,

in section 3.5.1, applications of this model in the processing of industrial wastewater

are discussed. Other types of applications are discussed in section 3.5.2.

3.5.1 Industrial wastewater

The Contois growth model has been applied to both aerobic and anaerobic indus-

trial wastewater treatment processes in recent years. It has been shown that the

Contois model is suitable for tting various experimental data from a broad range

of organic materials [133{135].

Beltran et al. [136] presented a study of the oxidation treatment of black olive

wastewater which was performed by an aerobic biological degradation process and

biological ozonation. In the aerobic biological treatment, the Contois kinetic was

used to describe the specic decomposition rate. It was found that the Contois

model ts the experimental data very well which conrmed that the Contois model

is suitable for the present experimental system.

Bhattacharya and Pham [137] studied how Monod and Contois kinetics t experi-

mental data from the anaerobic treatment of cow dung digesters. The result of the

comparison indicated that the Contois kinetic model is more suitable for tting the

experimental data of cow dung digesters than the Monod kinetic model.

Hu et al. [39] investigated the process kinetics of the anaerobic digestion of ice-cream

wastewater using two kinetic models, Monod and Contois. It was found that the root

mean square for the Contois kinetic is much greater than that of the Monod kinetic.

Chapter 3 38

Thus, the comparisons of experimental data and predicted values obtained from

Monod and Contois models suggested that the Contois model is more appropriate

than the Monod model for modeling the process kinetics of the anaerobic digestion

of ice-cream wastewater, i.e., predicting the performance of the anaerobic digester

reactor, with the highest correlation coecient of 0.918. Both the performance of

the anaerobic digester reactor and microbial growth were predicted better by the

Contois model than by the Monod model. This was because the Contois kinetic

model considered the eect of inuent substrate in making its prediction [39].

Hu et al. [42] also studied the process kinetics for the anaerobic digestion of sulphate-

rich wastewater using models based on Monod and Contois kinetics at continuously

stirred tank reactor. The results of the kinetic studies indicated that the Contois

model predicted the kinetic reactions of the process very well, showing good agree-

ment with the experimental data and having a greater correlation coecient of 0.989

than Monod model. Both the performance of the anaerobic digester reactor and the

microbial growth were predicted better (very well) by the Contois kinetic model

than by the Monod kinetic model. Hu et al. [42] suggested that the reason for the

good prediction of Contois kinetic model was due to the attachment of biomass at

the walls of the reactor which can provide a source of inoculum.

Isik and and Sponza [41] used several models to study the process kinetics of the

anaerobic treatment of textile wastewater in a lab-scale upow anaerobic sludge

blanket reactor, including Monod, Contois, Grau second order, modied Stover-

Kincannon, and rst order kinetic. The results of their kinetic studies showed that

the Contois kinetic model, with a correlation coecient of 0.967, is more appropri-

Chapter 3 39

ate than the Monod and rst order kinetic models for describing microbial kinetics.

The results further emphasized that Contois model is the best model and is more

suitable for predicting the performance of anaerobic digester reactors compared to

the other applied models with a correlation coecient of 0.97. Krzystek et al. [138]

showed that the Contois kinetic model had good agreement with experimental data

from the aerobic biodegradation of solid municipal organic waste with a correlation

coecient of 0.985.

Moosa et al. [139] investigated the kinetics of the anaerobic reduction of sulphate

using several kinetic expressions including the Monod, Chen & Hashimoto and Con-

tois models at continuous bioreactors. Among the models that were tested, the

Contois kinetic model provided the best agreement with the experimental data of

the anaerobic reduction of sulphate for describing the bacterial specic growth rate

on sulphate concentration with the highest accuracy. This process has several ap-

plications such as the cleaning of sulphate containing industrial euents and in the

cleaning of acid mine drainage. In their study, the kinetic coecients of the Contois

model were determined for dierent sulphate concentrations in the inuent. It was

found that the death rate coecient is not aected by variations in initial concen-

tration of sulphate, and it remained constant.However, as the initial concentration

of sulphate increased, the saturation constant (Ks) increased signicantly while the

change in the value of the maximum specic growth rate (max) was insignicant.

These values are stated in table (3.1).

Sun et al. [140] studied the biological treatment of pharmaceutical wastewater by a

functional strain Xhhh employing three ions, Mn+2, Cu+2 and Zn+2. Three dierent

Chapter 3 40

biodegradation kinetic models, Tessier, Monod and Contois models, were used to

simulate the process. Each of the models gave a good t to the experimental data.

However, the Contois kinetic model was found to be the most appropriate for de-

scribing the microbial growth of Xhhh growth with the highest correlation coecient

of 0:984. Similarly, using three kinetic models, i.e., the Monod, Contois and Chen

& Hashimoto kinetic models, Abdurahman et al. [141, 142] reviewed the treatment

of palm oil mill euent using membrane anaerobic system. In both studies, they

found that the Contois model provided an excellent t with experimental data with

correlation coecients of 0:962 and 0:997, respectively. In the latter case the Contois

model gave the highest value for correlation coecient. Poh et al. [143] cultivated

a thermophilic mixed culture for treating the palm oil mill euent at thermophilic

conditions using a continuous stirred tank reactor. In this study, the Contois model

provided a good t with the experimental data with R2 values ranging between 0.82

and 0.97 depending on the euent concentration.

Pinna et al. [144] investigated caeic aside biodegradation using a mixed culture in

a batch reactor. The Contois model, with an R-squared value of 0.851, was found to

give a minor improvement in tting experimental data over the Monod model, which

had an R-squared value of 0.848. Zhang et al. [145] developed a kinetic model to

describe simultaneous saccharication and co-fermentation of paper sludge using the

xylose-utilizing yeast Saccharomyces cerevisiae RWB222 and the commercial cellu-

lase preparation Spezyme CP. Their results showed that the Contois model gave the

best description of the specic rate of growth on xylose.

Paulo et al. [146] used a bioreactor to treat mining inuenced water that had been

Chapter 3 41

discharged from abandoned mining sites. The Contois model and the rst-order

model were used to simulate the decomposition process. They compared their re-

sults with the predictions based on the Contois and the rst-order models to conclude

that the Contois model stayed closer to the experimental data. Emerald et al. [147]

investigated the kinetics of the activated sludge process treating synthesized dairy

wastewater using four kinetic models, i.e. the Monod, Moser, Contois and Chen &

Hashimoto models, at an organic loading rate of 1200 mg L1. Among the tested

models, the Contois kinetic model, with a correlation coecient of 0.95, was found

to be an appropriate model and the best kinetic model for describing the kinetic

reactions of the activated sludge process due to tting the data very well.

Karim et al. [148] proposed a new model, including the Contois kinetic model and an

endogenous decay model to predict the methane production rate from the anaerobic

digestion of cow manure in bench-scale gas-lift digesters. The proposed model and

two other well-known kinetic models, the Chen & Hashimoto [149] and Hill [150]

models, were used to t the experimental data obtained for methane production

rate. It was found that the tting of the Chen & Hashimoto and Hill models with

the experimental data was poor with the values of the correlation coecient being

0.86 and 0.51 respectively whereas their new model featuring Contois kinetics t-

ted the experimental data of observed methane production rate with a correlation

coecient of 0.99. Thus, the comparisons of experimental values with the results

produced by the three models showed that the proposed model that included Con-

tois model is more suitable for modeling the methane production rate.

The rst-order kinetic model has been traditionally used to simulate hydrolysis re-

Chapter 3 42

action in anaerobic digestion which is independent of the hydrolytic microorganism

concentrations. However, some researchers have shown that the hydrolysis step de-

pends on the microorganism concentration and activity [151]. The rst-order kinetic

may not describe the hydrolysis steps accurately [135] due to the complexity of the

multi-step process involved in the hydrolysis of carbohydrase, proteins and lipids

hydrolysis while the Contois model was found to t the experimental data very

well [135]. This result has been supported by several studies in which the Contois

model was used to illustrate the hydrolysis of particulate wastes [116{118,152].

Ramirez et al. [116] developed anaerobic digestion model No.1 (ADM1) to describe

the thermophilic anaerobic digestion of thermally pre-treated waste activated sludge

using a batch reactor that involved three disintegration biochemical parameters, nine

hydrolytic biochemical parameters and four stoichiometric parameter values. The

Contois model was used to simulate the disintegration and hydrolysis processes. It

was found that the Contois model gave a better t to the experimental data than the

standard ADM1 model which uses a rst-order kinetic expression. Vavilin et al. [118]

discussed four kinetic models, including Monod, rst-order, two-phase and Contois

kinetics, for the hydrolysis of particulate organic material in anaerobic digestion.

The obtained results showed that using the Contois kinetic for the hydrolysis ki-

netics of swine waste, sewage sludge, cattle manure and cellulose provided excellent

ts to the experimental data. Gawande et al. [153] presented the development

of a generalized biochemical process model. The hydrolysis of particulate matter

was modeled using Contois kinetics which also described the microbiologically me-

diated reaction. Gawande et al. [152] used their model to simulate the anaerobic

Chapter 3 43

reduction of municipal solid waste. The use of Contois kinetics was found to give

an excellent prediction in the reduction of solid concentrations. Myint and Nir-

malakhandan [117] compared three models, i.e., the rst-order, the second-order

and the surface-limiting reaction models (also known as Contois kinetic model) for

simulating hydrolysis-acidogenesis in the digestion of cattle manure residues using a

batch reactor. They found that the Contois kinetic model t the experimental data

very well, with a correlation coecient of 0.914, which was better than the other

applied models. Therefore, the results discussed above provide convincing evidence

that the substrate-microorganism ratio (S=X) used in the Contois model could be

a better limiting factor in the hydrolysis of particulate substrate, rather than the

substrate concentration (S) as modeled by the rst-order reaction model.

Several researchers have conducted theoretical studies of a continuous ow bioreactor

model using Contois kinetics [39,41,118,137,139,154]. One should note that at high

feed substrate concentrations, the growth rate as a function of residual substrate

concentration has been predicted successfully by the Contois kinetic model [31].

3.5.2 Other applications of the Contois model

In this section we review other experimental applications of the Contois model.

Hidaka et al. [155] developed a model to describe the lactate fermentation char-

acteristics of B. Coagulans in a batch reactor. The model included the inhibitory

eects of the substrate, lactate (product), NaCl, and bacterial growth. The Contois

kinetic model was used to simulate the degradation of particulate carbohydrates.

Their results indicated that the Contois model is more suitable for simulating the

Chapter 3 44

hydrolysis of particulate kitchen garbage than rst order reaction. This result is in

agreement with other studies that used the Contois model to describe the anaerobic

hydrolysis of particulate organic wastes [116,156].

Several studies have indicated that the Contois equation can be used to describe

fungal growth kinetics [157]. Zhou et al. [158] studied the growth kinetics of Rhizo-

pus nigricans fungal on glucose using the Contois model. The results showed that

the Contois model is useful for simulating the kinetics of cell growth due to the high

value of the correlation coecient of 0:99.

Mazutti et al. [159] developed a phenomenological model containing 19 kinetic mod-

els for microbial growth to simulate inulinase production in a batch reactor by the

yeast Kluyveromyces marxianus NRRL Y 7571, employing a medium containing

agroindu