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STUDY OF THE OPERABILITY OF NONIDEAL CONTINUOUS
BIOREACTORS
A. Ajbara
a Department of Chemical Engineering, King Saud University, Riyadh, Saudi Arabia
Online publication date: 25 November 2010
To cite this Article Ajbar, A.(2011) 'STUDY OF THE OPERABILITY OF NONIDEAL CONTINUOUS BIOREACTORS',
Chemical Engineering Communications, 198: 3, 385 — 415
To link to this Article: DOI: 10.1080/00986445.2010.512544
URL: http://dx.doi.org/10.1080/00986445.2010.512544
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Study of the Operability of Nonideal
Continuous Bioreactors
A. AJBAR
Department of Chemical Engineering, King Saud University, Riyadh,
Saudi Arabia
The article investigates the operability of a nonideal unstructured kinetic model of
continuous bioreactors. The study of operability (interactions between design and
control) can help in identifying control problems that could be overcome during
the design stage. The issue of operability has been studied extensively in the litera-
ture for chemically reactive systems but has not received the same attention for bio-
reactive systems. The bioreactor model investigated here allows for general
dependence of growth rate and yield coefficient on the substrate. The nonideality
of the bioreactor is described by the fraction of reactor total volume that is perfectly
mixed and by the fraction of the feed entering the perfectly mixed zone. The static
and dynamic analyses of the model enable the delineation of regions of input and
output multiplicity, as well as the conditions for the existence of oscillatory behavior
in the bioreactor. This allows the determination of boundaries between safe and
washout regions and the delineation of regions of undesired oscillations. The general
analysis is illustrated through a specific example for substrate-inhibited growth rate
and linear dependence of the yield on the substrate. The analysis illustrates how
modeling decisions and ultimately the design and operating parameters influence
the operating characteristics of the bioreactor.
Keywords Bioreactors; Multiplicity; Singularity; Operability; Oscillations
Introduction
The continuous stirred tank bioreactor (CSTBR) is at the heart of many biological
processes ranging from wastewater treatment to the production of useful biopro-
ducts. As is the case for all reactive systems, the control of this unit is essential for
the optimization of the operation of the biological process. Continuous bioreactors
are, however, known to present operational problems that render their control a
challenging task (Aguilar et al., 2001). These problems manifest themselves essen-
tially in the form of input and output multiplicities and undesired oscillatory beha-
vior. The occurrence of these nonlinear phenomena in bioreactors has been
investigated in the literature since the sixties. In this regard, the existence of multiple
steady states in the chemostat with a single substrate-inhibited growth was studied
by both Andrews (1968) and Chi et al. (1974). The experimental evidence for the
existence of multiple stable steady states and the transition from one state to another
were presented, respectively, by Pawlowski et al. (1973) for phenol oxidation and
Address correspondence to A. Ajbar, Department of Chemical Engineering, King Saud
University, P.O. Box 800, Riyadh 11421, Saudi Arabia. E-mail: aajbar@ksu.edu.sa
Chem. Eng. Comm., 198:385–415, 2011
Copyright #Taylor & Francis Group, LLC
ISSN: 0098-6445 print=1563-5201 online
DOI: 10.1080/00986445.2010.512544
385
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DiBiasio et al. (1981) for methanol bioxidation. A theoretical investigation of
dynamic behavior of the chemostat was carried out by Agrawal et al. (1982) and
Mihail and Straja (1988). The stability of the activated sludge process was, studied
by Rozich and Gaudy (1985) for a single species system growing on a toxic waste,
and by Bertucco et al. (1990) for substrate inhibition kinetics and solids recycle.
Sheintuch (1993) and Sheintuch et al. (1995) also carried out studies on the stability
of activated sludge reactors as well as on the multiplicity in a continuous nitrification
process. Recently, Nelson and Sidhu (2009) carried out a detailed stability analysis
of a structured model of the activated sludge reactor. Smith and Waltman (1995)
compiled in their book their extensive work on the dynamics of the chemostat. They
included, in particular, the general chemostat, the chemostat with an inhibition, and
the variable yield model. Zhang and Henson (2001) showed the usefulness of the
bifurcation theory in studying the multiplicity and the oscillatory behavior observed
in continuous cultures of some organisms. Other studies on bioreactor stability
include the static multiplicity of a temperature-controlled chemostat (Kurtanjek,
1987), the steady-state analysis of production of polysaccharide by Methylomonas
mucosa (Lin and Lim, 1990), the effect of medium viscosity on the multiplicity
patterns of a chemostat (Edissonov, 1996), the bifurcation analysis of an aerated
continuous flow bioreactor (Pinheiro et al., 2004), and the stability of the chemostat
for microbial reduction of sulfur dioxide (Dutta et al., 2007).
The occurrence of nonlinear phenomena in the bioreactor is known to affect the
closed-loop control performance. Input multiplicities occur when different values of
the input variable produce the same value of the output variable. The occurrence
of such behavior is known to affect the closed-loop performance regardless of the
control scheme used (Russo and Bequette, 1995). Output multiplicities are also
known to occur in bioreactors. This type of multiplicity arises when the same value
of an input variable produces different values of the output variable. The hysteresis
phenomenon is the most common form of output multiplicity and is associated with
the existence of a region of open-loop unstable behavior. Output multiplicities are
also known to adversely affect control performance (Radhakrishnan et al., 1999).
Moreover, oscillatory behavior has long been known to occur in continuous cultures
of some microorganisms such as Saccharomyces cerevisiae (Parulekar et al., 1986)
and Zymomonas mobilis (Jo
¨bses et al., 1985).
An early detection of difficult operating regions in bioreactors would be a useful
piece of information. This would allow the removal or at least the reduction of these
operational problems in the early stage of process design and would ultimately
improve the operability of the bioreactor. The detection of operational problems
in bioreactors is best carried out through the study of the open-loop process. The
dynamics of the open-loop system may provide useful information on possible diffi-
culties and may yield guidelines for control structure design (Szederke
´enyi et al.,
2002). The classification of the parameter space into regions of different nonlinear
behavior is an important objective. This classification is, however, a challenging task
since the number of parameters in any bioreactor model can be quite large. Various
nonlinear dynamics tools have been used in the literature for the analysis of reac-
tive systems. Among them the singularity theory (Balakotaiah and Luss, 1982;
Golubitsky and Schaeffer, 1985) is recognized as being a useful tool for nonlinear
analysis since it can provide a general framework for classifying branching phenom-
ena in which different kinds of multiplicity in the nonlinear model can be expected.
Russo and Bequette (1995, 1996), for instance, applied the singularity theory to
386 A. Ajbar
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study the operability of a jacketed exothermic continuous stirred tank reactor. The
authors studied the impact of process design on the multiplicity behavior of the
reactor. The same authors studied the operability of a styrene polymerization appli-
cation (Russo and Bequette, 1997). Gamboa-Torres and Flores-Tlacuahuac (2000),
on the other hand, carried out a study of the effect of process modeling on the non-
linear behavior of consecutive reactions in a CSTR, while Elnashaie and Mohamed
(2004) studied the implications of bifurcation on the design, operation, and control
of industrial FCC units.
The operability of bioreactors, on the other hand, did not receive the same
attention in the literature (Davison and Scott, 1988; Montague et al., 1992; Ajbar,
2001b). Recently, Alrabiah and Ajbar (2008) addressed the issue of bioreactor oper-
ability when applied to a continuous pre-fermentation of cheese culture. Using the
concepts of the singularity theory applied to a validated model, the authors analyzed
the potential operating problems that may be inherent in the continuous cheese
production.
In this article we analyze the operability of a large class of unstructured models
of bioreactors with cell recycle. There are a number of aspects that make this analysis
quite general. First, a nonideal bioreactor is considered. The nonideality follows the
Lo-Chollette model (Lo and Cholette, 1983; Liou and Chien, 1995), which was used
to study nonideality in reactive systems. In the Lo-Cholette model, the nonideal
behavior is described by the fraction of the total volume that is perfectly mixed
and by the fraction of the feed entering the perfectly mixed zone. This model was
used to study the effect of nonideal mixing on input and output multiplicity of a
PI-controlled CSTR (Pellegrini and Possio, 1996) and was also used to investigate
the complex dynamics of a bioreactor model under a PI controller (Ibarra-Junquera
and Rosu, 2007). The second aspect of the generality of the analysis carried out in
this study concerns the selection of growth kinetics. The analysis is carried out for
arbitrary substrate-dependent growth rate. The third generality aspect concerns
the variations of the yield coefficient. Both cases of constant and substrate depen-
dent yield coefficient are considered. The variability of the yield coefficient was
shown in the literature to be the case for a number of microbial populations (Tang
and Wolkowicz, 1992). A linear model for the dependence of the yield coefficient on
the substrate was proposed by Ivanitskaya et al. (1989) to describe the continuous
growth of Saccharomyces cerevisiae. The yield for this organism is known to depend
on the substrate concentration. This is attributed mainly to the Crabtree effect
(Porro et al., 1988) as the yield for fully oxidative and fully fermentative regimes
are known to be significantly different (Parulekar et al., 1986). The variability of
the yield coefficient depends therefore on the coexistence of these mechanisms.
The relative impact of these pathways on the microbial metabolism is, on the other
hand, affected by the inhibitive action that glucose has on the oxidative pathway at
high substrate concentration.
The organization of this article includes the description of the process model
followed by the study of static and dynamic singularities. A specific example is
provided to illustrate the analysis and is followed by concluding remarks.
Process Model
We consider the continuous bioprocess shown in Figure 1. The process consists of a
bioreactor and a settler. Nonideal behavior is assumed to prevail in the bioreactor.
Study of Operability of Bioreactors 387
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The nonideality follows the Lo-Cholette model (Lo and Cholette, 1983). The para-
meter mrepresents the fraction of the reactor volume that is perfectly mixed. The
parameter n, represents the fraction of the reactant feed that enters the zone of per-
fect mixing. The substrate concentration in the mixed part of the bioreactor is
denoted by S
1
. When the values of mand nare equal to 1, the bioreactor is ideally
mixed. The feed conditions consist in the feed flow rate Qand the substrate and bio-
mass feed concentrations S
f
and X
f
. The recycle conditions are described by the
recycle ratio Rand the recycle biomass concentration X
R
. The settling unit is
described by the fraction wof the sludge wasted after passing through the settling
unit. The growth rate of biomass is denoted by r, which depends arbitrarily on the
substrate S
1
, i.e., r¼r(S
1
). The yield coefficient is denoted by Yand is an arbitrary
positive function of the substrate, i.e., Y¼Y(S
1
)>0. The mass balances of both sub-
strate and biomass are described by the following equations:
Substrate mass balance:
nðQSfþRQSÞrðS1ÞXmV
YðS1Þ¼nQð1þRÞS1þmV dS1
dt ð1Þ
Biomass mass balance:
nðQX fþRQXRÞþrðS1ÞXmV ¼nQð1þRÞXþmV dX
dt ð2Þ
Figure 1. (a) Schematic diagram of the bioreactor-settler system; (b) schematic diagram of the
bioreactor with nonideality conditions.
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Balance around the bioreactor:
nQð1þRÞS1þð1nÞðQSfþRQSÞ¼Qð1þRÞSð3Þ
Balance around the settler:
Qð1þRÞX¼RQX RþwQX Rð4Þ
Equation (3) yields the following relation between the concentrations S
1
and S:
S1¼Sð1þRnÞþSfðn1Þ
nð1þRÞð5Þ
Taking the derivatives yields
dS1
dt ¼ð1þRnÞ
nð1þRÞ
dS
dt ð6Þ
Substituting for the expression of S
1
and dS1
dt (Equations (5) and (6)) in the
substrate mass balance Equation (1) and rearranging yields
QSfQS rðS1ÞXmV
YðS1Þ¼mVð1þRnÞ
nð1þRÞ
dS
dt ð7Þ
The settler balance Equation (4) yields
XR¼ð1þRÞX
ðRþwÞð8Þ
Substituting this equation in Equation (2) and rearranging yields
nQX fnwð1þRÞ
RþwQX þrðS1ÞmVX ¼mV dX
dt ð9Þ
Introducing the dilution rate D¼Q
V, the mass balances Equations (7) and (9) are
equivalent to
DðSfSÞrðS1ÞXm
YðS1Þ¼mð1þnRÞ
nð1þRÞ
dS
dt ð10Þ
and
nDX fnð1þRÞw
RþwDX þrðS1ÞmX ¼mdX
dt ð11Þ
For the asymptotic case of n¼m¼1, Equation (5) yields S
1
¼S, and we recover
the model for the ideal bioreactor with cell recycle:
DðSfSÞ rðSÞ
YðSÞX¼dS
dt ð12Þ
DX fð1þRÞw
RþwDX þrðSÞX¼dX
dt ð13Þ
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Static Singularities
The nonlinear model Equations (10) and (11) can exhibit a number of steady-state
multiplicities that can influence the control of the process. The study of these multi-
plicities and the associated process control ramifications can be suitably carried out
within the framework of the singularity theory. The starting point of the analysis is
to see whether the steady-state equations of the model can be reduced to a single
variable algebraic equation. This is fortunately the case in the studied model. The
model Equations (10) and (11) are written in the following form:
DðSfSÞrðS1ÞXm
YðS1Þ¼kdS
dt ð14Þ
nDX fnaDX þrðS1ÞmX ¼mdX
dt ð15Þ
with
k¼mð1þnRÞ
nð1þRÞand a¼ð1þRÞw
RþwÞð16Þ
Combining the steady state forms of Equations (14) and (15) yields the following
equation for X:
YðS1ÞDðSfSÞþnDX fnXDa¼0ð17Þ
or equivalently
X¼YðS1ÞðSfSÞþnXf
nað18Þ
Substituting Equation (18) into Equation (14), at the steady state, yields the
following single variable equation in S:
DðSfSÞmrðS1Þ
YðS1Þ
ðYðS1ÞðSfSÞþnX fÞ
na¼0ð19Þ
or by recasting the value of a(Equation (16)),
DðSfSÞ rðS1Þ
YðS1Þ
mðRþwÞ
nwð1þRÞðYðS1ÞðSfSÞþnX fÞ¼0ð20Þ
Denote by F(S,u) the left-hand side of Equation (20):
FðS;uÞ:¼DðSfSÞ rðS1Þ
YðS1Þ
mðRþwÞ
nwð1þRÞðYðS1ÞðSfSÞþnXfÞð21Þ
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where Sis the output (controlled variable) and uis the selected input or manipulated
variable, which is a distinguished parameter in the singularity theory. It can be seen
that the single algebraic Equation (21) includes a number of potentially variable
parameters. These include the dilution rate D, the substrate feed concentration S
f
,
the biomass feed concentration X
f
, the recycle ratio R, and the fraction wof sludge
wasted. The parameters mand ncan be considered as ‘‘design’’ parameters since they
depend on the mixing conditions.
The selection of the controlled variable for the bioreactor is obviously an impor-
tant issue. In a number of applications, pH or temperature are used as regulation
variables for optimizing the microbial growth Vicente et al. (1998). These variables
are easier to measure and control and have negligible perturbations. However, vari-
ables subject to large fluctuations such as substrate concentration, selected in this
work, can be just as important for growth optimization (Barron and Aguilar,
1998; Dondo and Marques, 2001). Large levels of substrate can be toxic for the
microbial growth while too little can force an early stationary or decay phase. For
these reasons the control of bioreactors based on substrate concentration has
become an important issue in many applications such as the production of biopro-
ducts of high added value, alcoholic fermentation, and wastewater bio-treatment
where strict environmental regulations require limits on organic matter released in
effluents (Marcos et al., 2004; Ramaswamy et al., 2005). The use of substrate as a
controlled variable is not, however, without practical implementation problems.
The lack of reliable on-line sensors for substrate may require the use of observers
to estimate the unmeasured states (Marcos et al., 2004).
In the following section we examine which of the mentioned model parameters,
if chosen as the manipulated variable, can lead to multiplicity in the nonlinear
model. From a control point of view, both input and output multiplicities should
be considered. The necessary conditions for the existence of input multiplicity for
the variable uare that
F¼@F
@u¼0ð22Þ
while the necessary conditions for output multiplicity are that
F¼@F
@S¼0ð23Þ
Input or output multiplicities, when they occur, manifest themselves in the
form of specific behavior, as the selected manipulated variable is varied. These types
of behavior are called static singularities. Figure 2 shows the basic singularities a
nonlinear model can predict. The simplest static bifurcation is the saddle-node bifur-
cation (Figure 2(a)). It can be seen from the figure that saddle-node bifurcation is
one form of output multiplicity. The necessary conditions for the existence of this
singularity are given by Equation (23) in addition to:
@F
@u6¼ 0and @2F
@S26¼ 0ð24Þ
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Should either one or both these conditions be violated, then this gives rise to
higher order singularities. The second possible change that can occur in the steady-
state locus is the formation of an isola and the development of isola into
Figure 2. Various static singularities: (a) saddle-node; (b) isola; (c) mushroom; (d) hysteresis;
(e) pitchfork; (f) singularity that may result from perturbations in the pitchfork.
392 A. Ajbar
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a mushroom, as shown in Figure 2(b) and (c). These singularities can involve both
input and output multiplicities. They arise when the following conditions are
satisfied:
F¼FS¼Fu¼0ð25Þ
FSu 6¼ 0;FSS 6¼ 0;Fuu 6¼ 0ð26Þ
F
x
and F
xx
designate respectively the first- and second-order partial derivatives
of Fwith respect to the variable x. The third qualitative change in the behavior of the
model is the appearance of Sor inverse S-shaped, as shown in Figure 2(d). This is the
famous hysteresis singularity that produces output multiplicity. The S-shaped
input-output curve is generally associated with a jump from low to high operating
conditions such as low and high substrate conversion. The conditions for the hyster-
esis singularity are that
F¼FS¼FSS ¼0ð27Þ
Fu6¼ 0;FSu 6¼ 0;FSSS 6¼ 0ð28Þ
The last singularity to be considered is the pitchfork singularity (Figure 2(e)).
Pitchfork singularity is an example of both input and output multiplicities. This
can be seen in Figure 2(f) showing an example of diagrams that can arise when slight
perturbations of parameters are made around the basic pitchfork diagram of
Figure 2(e). The conditions of the existence of pitchfork are
F¼FS¼Fu¼FSS ¼0ð29Þ
and
FSu 6¼ 0;FSSS 6¼ 0ð30Þ
By characterizing the type of static singularity the model can predict, one can
understand how to minimize or even eliminate the associated multiplicity. In parti-
cular, if the multiplicity between the output variable Sand any of the operating para-
meters is eliminated then the bioreactor will generally be easier to control and safer
to operate, since the process gain would remain with the same sign over the operating
region of interest. However, this does not guarantee that the bioreactor is asympto-
tically stable, since limit cycles (periodic behavior) can also occur. Limit cycles are
examined in a later section. The derivatives of F(Equation (21)) with respect to
the different parameters are given by:
@F
@D¼SfS;@F
@R¼mrðS1Þðw1ÞðYðS1ÞðSfSÞþnXfÞ
nð1þRÞ2wYðS1Þð31Þ
@F
@w¼mrðS1ÞRðYðS1ÞðSfSÞþnX fÞ
nð1þRÞw2YðS1Þ;@F
@Sf
¼DmrðS1ÞðRþwÞ
nð1þRÞwð32Þ
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@F
@Xf
¼mrðS1ÞðRþwÞ
ð1þRÞwYðS1Þ;@F
@m¼rðS1ÞðRþwÞðSfSÞþnX fÞ
nð1þRÞwY ðS1Þð33Þ
@F
@n¼mrðS1ÞðSSfÞðRþwÞ
n2ð1þRÞwð34Þ
Using the steady-state equation Equation (21) and substituting in Equations
(31)–(34) yields
@F
@R¼DðSfSÞðw1Þ
ð1þRÞðRþwÞ;@F
@w¼DRðSfSÞ
wðRþwÞð35Þ
@F
@Sf
¼nDX f
YðS1ÞðSfSÞþnX f
;@F
@Xf
¼nDðSfSÞ
YðS1ÞðSfSÞþnXf
ð36Þ
@F
@m¼DðSfSÞ
m;@F
@n¼DðSfSÞ2YðS1Þ
nðYðS1ÞðSfSÞþnXfÞð37Þ
Examining these equations we can conclude that the following conditions should
be satisfied for the system to exhibit input multiplicities (F
u
¼0) for the selected
variables:
.For u¼D, input multiplicity implies necessarily that S¼S
f
.
.For u¼R, this implies that D¼0, w¼1, or S¼S
f
.
.For u¼w, this implies that D¼0, R¼0, or S¼S
f
.
.For u¼S
f
, this implies that D¼0, n¼0, or X
f
¼0.
.For u¼X
f
, this implies that D¼0, n¼0, or S¼S
f
.
.For u¼m, this implies that D¼0orS¼S
f
.
.For u¼n, this implies that D¼0orS¼S
f
.
Barring the trivial cases of D¼0 and n¼0, i.e., all the feed is bypassed, S¼S
f
,
i.e., no conversion, w¼1 and R¼0, i.e., no recycle, it can be seen that input
multiplicity may be possible only for X
f
¼0, i.e., sterile feed. If input multiplicity
is ruled out then we can also rule out the existence of isola and mushroom singula-
rities as well as the occurrence of pitchfork singularities, since all these singularities
require that F
u
¼0. The case of sterile feed (X
f
¼0) is examined in more detail in a
later section.
Having examined the input multiplicities we turn our attention to the study of
possible output multiplicities. The existence of the hysteresis singularity is defined
by the conditions of (Equations (27)–(28)). It can be noted that since input multi-
plicity F
u
¼0 is not satisfied except for the cases w¼1, R¼0, i.e., no recycle and
X
f
¼0, i.e., sterile feed, we therefore anticipate that output multiplicity may exist
for other parameters. However, the condition F
Su
6¼ 0 is still to be satisfied. Taking
the derivatives F
S
and F
SS
of Equation (21) yields:
FS¼DmðRþwÞ
nwð1þRÞrS1ðSfSÞrðS1ÞþnX f
r
Y
S
ð38Þ
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FSS ¼mðRþwÞ
nwð1þRÞrS1S1ðSfSÞ2rSþnX f
r
Y
SS
ð39Þ
Taking the derivative F
uS
(with ubeing one of the model parameters) yields:
FDS ¼1;FSfS¼mðRþwÞ
nwð1þRÞrSð40Þ
FXfS¼mðRþwÞ
wð1þRÞðr
YÞS;FmS ¼ ðRþwÞ
nwð1þRÞrSðSfSÞrðS1ÞþnXf
r
Y
S
ð41Þ
FnS ¼mðRþwÞ
n2wð1þRÞrSðSfSÞrðS1ÞþnX f
r
Y
S
ð42Þ
FRS ¼ mð1wÞ
nwð1þRÞ2rSðSfSÞrðS1ÞþnX f
r
Y
S
ð43Þ
FwS ¼mR
nð1þRÞw2rSðSfSÞrðS1ÞþnX f
r
Y
S
ð44Þ
Using the condition F
S
¼0, Equation (38), these derivatives are reduced to
FmS ¼D
m;FnS ¼Dm
nð45Þ
FRS ¼Dð1wÞ
ð1þRÞðRþwÞ;FwS ¼DR
wðRþwÞð46Þ
It can be seen therefore that barring the trivial cases of D¼0, m¼0, w¼1, and
R¼0, the conditions F
mS
6¼ 0, F
nS
6¼ 0, F
RS
6¼ 0, F
wS
6¼ 0 are always satisfied. The
conditions F
X
f
S
and F
S
f
S
are, on the other hand, no-nil when r
S
1
and ðr
YÞS1are no-nil.
We conclude therefore that output multiplicity in the form of hysteresis is possible in
the bioreactor for nonsterile feed conditions.
The case of X
f
¼0 warrants more discussion, since it arises in the conditions per-
tinent to input multiplicity. The steady-state value of Xis given by Equation (18):
X¼YðS1ÞðSfSÞ
nað47Þ
Substituting this equation in Equation (21) yields the following algebraic
equation that depends solely on S:
FðS;uÞ:¼ðSfSÞDmðRþwÞ
nwð1þRÞrðS1Þ
¼0ð48Þ
The washout solution S¼S
f
always exists for the sterile feed case. The other
solution satisfies the following condition:
FðS;uÞ¼DmðRþwÞ
nwð1þRÞrðS1Þ¼0ð49Þ
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For this function, the hysteresis conditions yield
F¼0;rS¼0;and rSS ¼0ð50Þ
Let us examine the no-nil conditions
Fu6¼ 0;FSu 6¼ 0andFSSS 6¼ 0ð51Þ
When u¼D, it can be seen that F
D
¼1 therefore F
SD
¼0 and no hysteresis can
be found for D. This also holds for all other variables, n,w, and R. We conclude
therefore that hysteresis singularity can not exist for the sterile feed case. With
F
DD
¼0 we also rule out the existence of isola and mushroom singularity for D,
and the same holds for the other parameters. As for pitchfork singularity, the
requirements are that
FS¼Fu¼0 and FuS 6¼ 0ð52Þ
When u¼D, we have that F
D
¼16¼ 0. Therefore, pitchfork singularity cannot
exist. For the other parameters the condition F
S
¼0 requires that r
S
¼0, but as
was shown previously, this violates the condition F
uS
6¼ 0. Therefore, the only singu-
larity that may exist for the clean feed conditions is the simple saddle-node bifurca-
tion defined by
F¼0 and FS¼0ð53Þ
The interesting feature of the sterile feed case is that total washout (S¼S
f
)is
always a solution. Therefore, the crossing of the washout line with the nontrivial
steady state of Equation (49) may yield bistability. Substituting S¼S
f
, i.e., S
1
¼S
f
in Equation (49) yields the critical value for the dilution rate, D
w
, at the crossing
of the washout line with the non-trivial steady state:
Dw¼mðRþwÞ
nwð1þRÞrðS1¼SfÞð54Þ
Dynamic Singularities
In this section we investigate the existence of periodic behavior, i.e., Hopf points in
the model and the associated Hopf singularities. The two-dimensional system exhi-
bits a periodic solution, i.e., Hopf bifurcation point if the Jacobian matrix has pure
imaginary eigenvalues. The Jacobian matrix for this model is
J¼f1Sf1X
f2Sf2X
ð55Þ
where f
1
and f
2
denote the mass balances in Equations (10) and (11). The eigenvalues
kof the Jacobian matrix are the solutions of the characteristic equation
k2kðf1Sþf2XÞþf1Sf2Xf1Xf2S¼0ð56Þ
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The conditions for this equation to have pure nonzero eigenvalues are that
f1Sþf2X¼0ð57Þ
f1Sf2Xf1Xf2S>0ð58Þ
Substituting the expressions of the gradients yields
f1S¼ðDmX ðr
YÞSÞ
k;f1X¼mr
YðSÞkð59Þ
f2S¼XrS;f2X¼ðnaDþmrÞ
mð60Þ
The condition f
1S
þf
2X
¼0 is equivalent to
D
kmX
k
r
Y
SnaD
mþr¼0ð61Þ
while the condition f
1S
f
2X
f
1X
f
2S
>0 is equivalent to
naD2mrD þnaDmX r
Y
Srm2Xr
Y
Sþrm2X
YrS>0ð62Þ
These two conditions are the Hopf conditions. Before the analysis is carried out
any further we examine the important case of constant yield coefficient. For this case
the second Hopf condition, Equation (62) is equivalent to
naD2mrD þnaDmX rS
Y>0ð63Þ
which is equivalent to
naDmr þnamX rS
Y>0ð64Þ
The first Hopf condition, Equation (61), on the other hand, is equivalent to
mXrS
kY¼D
knaD
mþrð65Þ
Substituting Equation (65) into Equation (64) yields
naDmr þnak rD
knaD
m
>0ð66Þ
which is equivalent to
rðnak mÞ>n2a2kD
mð67Þ
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Recasting the expression of aand kfrom Equation (16), it can be shown that
ðnak mÞ¼ðwn1ÞR
Rþw, which is always negative (since the product wn is smaller than
1). The condition of Equation (67) is therefore never satisfied. We reach therefore
the important conclusion that the unstructured model, even with nonideality cannot
exhibit oscillatory behavior for constant yield, regardless of the expression of the
substrate dependent growth rate. This result joins the conclusion reached before by
Ajbar (2001a), who showed that neither the presence of a maintenance term nor
the assumption of spatial inhomogeneity due to the adhesion of microorganisms
to the reactor wall can cause the appearance of limit cycles in the model. It seems,
therefore, that a dependence of the yield coefficient on the substrate is a necessary
condition for the existence of limit cycles in the two-dimensional model, a result
that was proved before for the case of the basic model of the ideal chemostat by
Ajbar (2001a) and Pilyugin and Waltman (2003).
For the general case where the yield is variable, a Hopf point is not ruled out.
There are two Hopf degeneracies that can be studied. The first one is the so-called
F
1
degeneracy and is associated with the interaction of static limit point and Hopf
point. The F
1
degeneracy is obtained by solving the steady-state equation F¼0 and
the two Hopf conditions (Equations (61) and (62)) with the equality replacing the
inequality. The H
01
singularity, on the other hand, corresponds to the appearance
or the coalescence of two Hopf points. The conditions for this singularity are that:
H¼HS¼0;HSS 6¼ 0ð68Þ
where
H:¼f1Sþf2Xð69Þ
with
f1Sf2Xf1Xf2S>0ð70Þ
This condition is equivalent to the simplest type of static bifurcation, i.e., the
turning points of the steady-state curve in the continuity diagram. H
01
represents,
therefore, the turning points of Hopf points curve. In the following section we pro-
vide an explicit example of the results that were derived in the previous section.
Application to Specific Case
In this section we consider a model with Haldane growth rate (Rozich and Gaudy,
1985) and linear dependence of the yield on the substrate. The choice of a linear
model to describe the variations of the yield is essentially due to its simplicity. The
coefficients of the linear relationship have a meaning of a regression of the experi-
mental values. Moreover, the assumption of linear dependency was shown to be suf-
ficient to predict the existence of oscillatory behavior in continuous cultures (Crooke
et al., 1980; Ivanitskaya et al., 1989; Ajbar, 2001a). The specific growth rate and the
yield coefficient are given by:
rðSÞ¼ lmS
KSþSþS2=KI
ð71Þ
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YðSÞ¼aþbS ð72Þ
The original model (Equations (10) and (11)) with growth rate (Equation (71))
and yield expression (Equation (72)) is rendered dimensionless using the following
variables:
S¼S
Sf
;S1¼S1
Sf
;X¼X
aSf
;D¼D
lm
;t¼tlmð73Þ
b¼Ks
Sf
;c¼Sf
KI
;d¼bSf
að74Þ
The model (Equatins (10) and (11)) in dimensionless form is given by
Dð1SÞrðS1ÞXm ¼kdS
dtð75Þ
nDX fnaDX þrXm ¼mdX
dtð76Þ
with
rðS1Þ¼ S1
bþ1þcS2
1
ð77Þ
S1¼Sð1þRnÞþðn1Þ
nð1þRÞð78Þ
k¼mð1þnRÞ
nð1þRÞ;a¼ð1þRÞw
RþwÞð79Þ
For this specific model, the algebraic steady-state equation ðFðSÞ¼0Þ
(Equation (21)) consists of a polynomial of fourth order, and its coefficients are
Table I. Nominal values of the parameters used in the
simulations, unless specified otherwise
Parameter Value
b0.1
c2
R 1.5
w 0.1
n 0.1
m 0.1
d5
Xf0.1
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given in the appendix. In the following, the simulations are carried out using the
nominal values of the parameters shown in Table I, unless specified otherwise.
Figure 3 shows the bifurcation diagram for the parameter values shown
in Table I. The diagram was generated using the continuation software AUTO
(Doedel, 1985). This software is perhaps the most widely used numerical bifurcation
code. AUTO can trace out the entire steady-state branches, locate the static limit
points, and continue these points in two parameters, as well as locating Hopf bifur-
cation points. The capabilities of AUTO also include the continuation of Hopf
points and the determination of stable and unstable periodic branches as well as
any torus or period doubling bifurcations. Figure 3 is characterized by the existence
of two stable branches separated by an unstable region. There is also the appearance
of a Hopf point as result of dynamic bifurcation. Stable branches can be seen to ema-
nate from the Hopf point and terminate as they collide with the unstable static
branch. Therefore for dilution rates smaller than LP
2
, the operation of the bioreactor
leads to high conversion (low values of S). Between the Hopf point HB and LP
1
the
operation of the bioreactor may lead either to oscillatory behavior or to the low con-
version branch. Figure 4 shows an example of oscillatory behavior for D¼3:266
and for start-up conditions S¼0:81 and X¼20:68. Small variations in start-up
conditions such as S¼0:811 lead to the upper branch, as seen in Figure 4(b).
Two-parameter diagrams are useful to delineate the effect of kinetic and other
model parameters on the bireactor behavior. Figure 5 shows the effect of the inhi-
bition constant c. The dashed line of Hopf points exhibits a minimum (sign of H
01
Figure 3. Continuity diagrams showing periodic behavior emanating from Hopf point. Solid
line, stable branch; dashed line, unstable branch; LP
1
and LP
2
, static limit points; HB, Hopf
point.
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degeneracy) and terminates along another static branch. A number of behaviors can
be delineated in the diagram. We start with the point of minimum of Hopf points
(dashed line). It can be seen in the enlargement of Figure 5(a) (Figure 5(b)) that
the minimum A
1
of Hopf points is lower than that of the static branch A
2
. Therefore
for the region extending between A
1
and A
2
only Hopf points are expected. An
example of this behavior is shown in Figure 6 for c¼0.286. It can be seen that
two Hopf points characterize the bifurcation diagram. Therefore for dilution rates
between the two Hopf points, oscillatory behavior alone is to be expected. This is
Figure 4. Time trace dynamics for D¼3:266 of Figure 3 for different initial conditions: (a)
S¼0:81;X¼20:68; (b) S¼0:811;X¼20:68.
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an interesting result since it shows that oscillatory behavior alone can exist for some
range of dilution rates without the occurrence of any static multiplicity.
Going back to Figure 5(b) it can be seen that above point A
2
and until the first
crossing A
3
, the behavior of the system is characterized by the existence in this order
of HB, HB, LP, LP. An example of this behavior is shown in Figure 7 for c¼0.35.
Again oscillatory behavior alone is expected for dilution rates between the two Hopf
points. However, hysteresis is also expected between the two limit points. Going
Figure 5. Diagram showing the domain of hysteresis (solid) and Hopf points (dashed) in para-
meter spaces: (a) ðD;cÞ; (b) enlargement of (a).
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Figure 6. Continuity diagrams for the case of c¼0.286 in Figure 5(b) showing periodic beha-
vior emanating from the two Hopf points. Solid line, stable branch; dashed line, unstable
branch; HB
1
and HB
2
, Hopf points.
Figure 7. Continuity diagrams for the case of c¼0.35 in Figure 5(b). Solid line, stable branch;
dashed line, unstable branch; LP
1
and LP
2
, static limit points; HB
1
and HB
2
, Hopf points.
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back to Figure 5(a), another behavior is expected between the first crossing A
3
and
the second one, A
4
. Here the system is characterized by the occurrence in this order
of HB, LP, HB and LP. Figure 8 shows an example of this behavior for c¼0.46.
For this case and because of the relative location of Hopf and static limit points,
Figure 8. Continuity diagrams for the case of c¼0.46 in Figure 5(a). Solid line, stable branch;
dashed line, unstable branch; LP
1
and LP
2
, static limit points HB
1
and HB
2
, Hopf points.
Figure 9. Diagram showing the domain of hysteresis (solid) and Hopf points (dashed) in para-
meter spaces ðD;bÞ.
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it can be seen that between HB
1
and LP
2
only oscillatory behavior exists. For
dilution rates between LP
2
and HB
2
, oscillations coexist with the upper stable
branch, while for values between LP
2
and LP
1
simple hysteresis occurs. The effect
of the other kinetic parameter bis shown in Figure 9. It can be seen that the Hopf
point terminates along the static limit point branch. A Hopf point exists therefore
only for some range of values of b(in this case for b<0.32). For larger values, only
hysteresis can be expected.
Having examined the effect of the kinetics of biodegradation on the behavior of
the bioreactor, we turn our attention to the study of the effect of operating para-
meters of the unit. Figure 10 shows the effect of the recycle ratio and the purge frac-
tion. For each value of the recycle ratio, a Hopf point (dashed line) is expected
Figure 10. Diagram showing the domain of hysteresis (solid) and Hopf points (dashed) in
parameter spaces: (a) ðD;RÞ; (b) ðD;wÞ.
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between two static limit points, a situation similar to Figure 3. It can also be seen
that both the hysteresis and the Hopf regions increase with increasing values of
the recycle ratio. Instability of the bioreactor is therefore favored by the increase
in the recycle ratio. Figure 10(b) shows that the Hopf point line also exists for each
value of the purge fraction w, although the region of periodic behavior, in terms of
dilution rate, is quite narrow.
The effect of nonideality of the bioreactor is shown in Figure 11. We recall that
the nonideality is described by the fraction mof the reactor volume that is perfectly
mixed and the fraction nof the reactant feed that enters the zone of perfect mixing. It
can be seen that for each value of m(Figure 11(a)) or that of n(Figure 11(b)), a Hopf
Figure 11. Diagram showing the domain of hysteresis (solid) and Hopf points (dashed) in
parameter spaces: (a) ðD;mÞ; (b) ðD;nÞ.
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point is expected between two static limit points. It can also be noted that increasing
the mixing in the reactor (by increasing the value of m) widens the region of hyster-
esis (the two solid lines) and also widens the region between the Hopf point and the
limit point. This shows that increasing the mixing widens (in term of the dilution
rates) the region of instability (either hysteresis or periodic behavior). The effect
of the other parameter nis shown in Figure 11(b). It can be seen that for all values
of nthe region of periodic behavior is narrow (in terms of dilution rates). Moreover,
for nnear the asymptotic values of n¼1 (perfect case) or n¼0 (imperfect case), the
hysteresis region is narrow. The domain of instability (i.e., hysteresis) is larger for
values of nin the middle (in this case for values between 0.1 and 0.4).
As was mentioned in the analysis carried out in previous sections, the sterile feed
case can give rise to saddle-node bifurcation. Figure 12 shows the continuity diagram
for this case for values shown in Table I. The diagram is characterized by the pres-
ence of single Hopf and limit points. The point BR is the bifurcation point resulting
from the crossing of the total washout line with the nontrivial steady state. For
dilution rates smaller than BR, the lower stable branch is the only solution. For
dilution rates between BR and HB, bistability is expected, as stable periodic
branches coexist with the washout line. For dilution rates larger than LP, washout
occurs for any initial conditions.
The effect of kinetic parameter bis shown in Figure 13(a). The diagram shows
the locus of three critical points: LP, HB, and BR. It can be seen that the Hopf line
terminates along the static branch. Therefore, the Hopf point is expected only for
values of bsmaller than the crossing. For larger values, no Hopf point is expected,
and the behavior of the system is similar to that of Figure 12 but with no periodic
behavior. The effect of inhibition coefficient cis shown in Figure 13(b). For this
Figure 12. Continuity diagrams for the case of sterile feed. Solid line, stable branch; dashed
line, unstable branch; LP, static limit point; HB Hopf point; BR, bifurcation point (crossing
of washout with nontrivial steady state).
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case the Hopf line forms a minimum (degeneracy) and collides with the static limit
point (another degeneracy). For values of csmaller than the minimum point A
1
,
the bifurcation diagram exhibits no Hopf point. For values of cbetween A
1
and
A
2
(the crossing with the washout line), the bifurcation diagram is characterized
by the occurrence in this order of HB, HB, BR, and LP. An example of this behavior
is shown in Figure 14 for c¼0.3. For dilution rates between the two Hopf points,
stable oscillatory behavior is inevitable. For dilution rates between HB
2
and BR,
the lower static branch is the only outcome of the process. Between BR and LP, there
is bistability between the static branch and the total washout line, while for dilution
rates larger than LP, total washout occurs.
Figure 13. Diagram showing the domain of limit point (solid), Hopf point (dashed) and wash-
out critical point (dash-dot) of Figure 12. (a) ðD;bÞ; (b) ðD;cÞ.
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Going back to Figure 13(b), it can be seen that for values of cbetween A
2
and
A
3
, the bifurcation diagram is characterized by the occurrence of HB, BR (washout),
HB, and LP. An example of this behavior is shown in Figure 15 for c¼0.35. It can
Figure 14. Continuity diagrams for c¼0.3 of Figure 13(b). Solid line, stable branch; dashed
line, unstable branch; LP, static limit point; HB
1
and HB
2
, Hopf points; BR, bifurcation point
(crossing of washout with nontrivial steady state).
Figure 15. Continuity diagrams for c¼0.35 of Figure 13(b). Solid line, stable branch; dashed
line, unstable branch; LP, static limit point; HB
1
and HB
2
, Hopf points; BR, bifurcation point
(crossing of washout with nontrivial steady state).
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be seen that for dilution rates between the HB and BR points, stable oscillatory
behavior is inevitable. For dilution rates between BR and HB
2
, the oscillatory beha-
vior coexists with the upper total washout line. Between HB
2
and LP, the lower static
branch coexists with the total washout line, while for values larger than LP total
washout occurs.
The effect of some bioreactor parameters are shown in Figure 16. In the same
diagram are plotted the locus of the limit point LP, the Hopf point HB, and the
bifurcation point BR for the washout line. The plots for ðD;mÞand ðD;nÞshow that
for each value of the parameters mor nthe bifurcation diagram is characterized by
the occurrence of BR, HB, and LP points. Similarly to the case of nonsterile feed, it
Figure 16. Diagram showing the domain of limit point (solid), Hopf point (dashed), and criti-
cal washout point (dash-dot) of Figure 12. (a) ðD;mÞ; (b) ðD;nÞ.
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can be seen that an increase in mixing (Figure 16(a)) substantially widens the region
of bistability between the limit point and the BR (washout line). The region between
LP and HB also increases, but modestly, with the increase in m. The effect of the
other nonideality parameter nis shown in Figure 16(b). It can be seen that close
to the asymptotic cases of n¼1orn¼0 the domain of bistability decreases, but
the unsafe range of operation is larger for values of nin the middle.
Conclusions
This article has studied in some detail the operability of a large class of unstructured
kinetic models of a bioreactor-recycle system. A number of features has made the
analysis quite general. The model of the bioreactor assumes nonideal behavior by
dividing the bioreactor into a well-mixed region and an unreacted region. The analy-
sis was made for a general growth rate expression and for general dependence of
yield coefficient on the substrate. The analysis of the model static and dynamic sin-
gularities has shed some light on the operability of the bioreactor. The analysis has
shown that input multiplicity cannot occur in the studied model. However, output
multiplicity in the form of hysteresis in the case of nonsterile feed and saddle-node
in the case of sterile feed are possible. The study has also shown that when the yield
coefficient is independent of substrate, the model even with nonideality cannot pre-
dict periodic behavior regardless of the expression of the growth rate. When, on the
other hand, the yield is variable the model can predict oscillatory behavior, and a
number of Hopf degeneracies were studied. These general results were illustrated
for the case of substrate inhibition growth rate with linear dependence of the yield
on the substrate, for cases of both sterile and nonsterile feed conditions. The study
of this example has allowed the delineation of the different interactions between the
static and Hopf points. It also allowed the construction of practical diagrams that
showed the effect of the different model parameters on the occurrence of the differ-
ent behavior (point attractor, static multiplicity, and periodic behavior).
For the case of nonsterile feed conditions, it was found that an increase in the
substrate inhibition constant increases the instability of the process in the form of hys-
teresis and=or the appearance of stable limit cycles. However, depending on the values
of this kinetic parameter a variety of behaviors can be found. For cultures with weak
substrate inhibition effects, it is interesting to note that the model predicts that the
only operating problem in the bioreactor is the occurrence of spontaneous stable
periodic behavior for some range of dilution rates with no appearance of any static
multiplicity. If the substrate inhibition effect is stronger, then the model predicts, in
addition to stable oscillations, the existence of a different region of static multiplicity
at higher dilution rates. For even stronger inhibition effects, the model predicts the
existence of distinct regions of dilution rate where stable periodic behavior and hys-
teresis are expected. But the model also predicts a region of dilution rates where oscil-
lations can exist or die out, depending on the start-up and=or feed conditions. For
cultures subject to very strong substrate inhibition effects, the operating problems
in the chemostat manifest themselves in the presence of hysteresis for some range
of dilution rates while periodic behavior is also expected in a smaller range of dilution
rates. However, all the oscillations are dependent on start-up and=or feed conditions.
For the practical case of sterile feed conditions, the same patterns are observed, except
that the static multiplicity consists in a saddle-node instead of a hysteresis and the
oscillations for some cases coexist with the unsafe washout region.
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For the effect of operating parameters, it was found that the region of unsafe
behavior (bistability or periodic behavior) increases with the increase in the recycle
ratio but decreases with the increase in the purge fraction. The increase, in the mixing
widens, in terms of the dilution rates, the region of unsafe behavior. The increase, on
the other hand, in the fraction of the reactant feed that enters the zone of perfect
mixing tends to decrease the unsafe behavior but only past certain values. The
domain of instability is larger for values of this parameter in the range of 0.1 and 0.4.
Acknowledgment
This work was made possible by a generous grant from the Research Center of the
College of Engineering at King Saud University.
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Appendix
Coefficients of the polynomial
FðS;uÞ:¼a4S4þa3S3þa2S2þa1Sþa0¼0
a4¼ðDdcwÞ3DdcnRw 3Ddcn2R2wDdcn3R3w
a3¼dmR þ2dmnR2þdmn2R3þ4Ddcwþdmw Ddnw Dcnw 3Ddcnw
DdnRw DcnRw þ9DdcnRw þ2dmnRw 2Ddn2Rw 2Dcn2Rw
6Ddcn2Rw 2Ddn2R2w2Dcn2R2wþ6Ddcn2R2wþdmn2R2wDdn3R2w
Dcn3R2w3Ddcn3R2wDdn3R3wDcn3R3wþDdcn3R3w
a2¼3dmR þmnR þ2dmnR þmnR24dmnR2þmn2R2þ2dmn2R2þmn2R3
dmn2R36Ddcw3dmw þ3Ddnw þ3Dcnw þ9Ddcnw þmnw þ2dmnw
Dn2w2Ddn2wbDdn2w2Dcn2w3Ddcn2wþ3DdnRw þ3DcnRw
9DdcnRw þmnRw 4dmnRw 2Dn2Rw þ2Ddn2Rw 2bDdn2Rw
þ2Dcn2Rw þ12Ddcn2Rw þmn2Rw þ2dmn2Rw Dn3Rw 2Ddn3Rw
bDdn3Rw 2Dcn3Rw 3Ddcn3Rw Dn2R2wþ4Ddn2R2wbDdn2R2w
þ4Dcn2R2w3Ddcn2R2wþmn2R2wdmn2R2w2Dn3R2wDdn3R2w
2bDdn3R2wDcn3R2wþ3Ddcn3R2wDn3R3wþDdn3R3w
bDdn3R3wþDcn3R3w
414 A. Ajbar
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a1¼3dmR 2mnR 4dmnR þmn2Rþdmn2R2mnR2þ2dmnR22dmn2R2
mn2R3þ4Ddcwþ3dmw 3Ddnw 3Dcnw 9Ddcnw 2mnw 4dmnw
þ2Dn2wþ4Ddn2wþ2bDdn2wþ4Dcn2wþ6Ddcn2wþmn2wþdmn2w
Dn3wbDn3wDdn3wbDdn3wDcn3wDdcn3w3DdnRw
3DcnRw þ3DdcnRw 2mnRw þ2dmnRw þ4Dn2Rw þ2Ddn2Rw
þ4bDdn2Rw þ2Dcn2Rw 6Ddcn2Rw 2dmn2Rw Dn3Rw 3bDn3Rw
þDdn3Rw bDdn3Rw þDcn3Rw þ3Ddcn3Rw þ2Dn2R2w2Ddn2R2w
þ2bDdn2R2w2Dcn2R2wmn2R2wþDn3R2w3bDn3R2wþ2Ddn3R2w
þbDdn3R2wþ2Dcn3R2wþDn3R3wbDn3R3wþbDdn3R3wmn2RX f
mn2R2Xfmn3R2Xfmn3R3Xfmn2wX fmn2RwX f
mn3RwX fmn3R2wX f
a0¼ðdmRÞþmnR þ2dmnR mn2Rdmn2RþmnR2mn2R2Ddcwdmw
þDdnw þDcnw þ3Ddcnw þmnw þ2dmnw Dn2w2Ddn2wbDdn2w
2Dcn2w3Ddcn2wmn2wdmn2wþDn3wþbDn3wþDdn3wþbDdn3w
þDcn3wþDdcn3wþDdnRw þDcnRw þmnRw 2Dn2Rw 2Ddn2Rw
2bDdn2Rw 2Dcn2Rw mn2Rw þ2Dn3Rw þ3bDn3RwDdn3Rw
þ2bDdn3Rw þDcn3Rw Dn2R2wbDdn2R2wþDn3R2wþ3bDn3R2w
þbDdn3R2wþbDn3R3wþmn2RX fmn3RX fþmn2R2Xfmn3R2Xf
þmn2wXfmn3wX fþmn2RwX fmn3RwX f
Study of Operability of Bioreactors 415
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