Chemical Engineering Science 58 (2003) 1479–1496www.elsevier.com/locate/ces
Exploration and exploitation of bifurcation/chaotic behavior of acontinuous fermentor for the production of ethanol
Parag Garhyan
a
;
∗
, S. S. E. H. Elnashaie
a
, S. M. AlHaddad
b
, G. Ibrahim
c
, S. S. Elshishini
d
a
Chemical Engineering Department, Auburn University, Auburn, AL 36849, USA
b
Hyprotech, 11, Rd #315, New Maadi, Cairo, Egypt
c
Faculty of Engineering, Menoa University, ShebeenElkom, Egypt
d
Chemical Engineering Department, Cairo University, Cairo, Egypt
Received 22 April 2002; received in revised form 29 July 2002; accepted 3 December 2002
Abstract
A fourdimensional model for the anaerobic fermentation process, developed and used earlier to simulate the oscillatory behavior of an experimental continuous stirred tank fermentor is utilized in the present investigation to explore the static/dynamic bifurcation andchaotic behavior of this fermentor, which is shown to be quite rich. The present investigation is a prelude to the experimental explorationof bifurcation and chaos in a membrane fermentor.Dynamic bifurcation (periodic attractors) as well as period doubling sequences leading to dierent types of periodic and chaotic attractorshave been uncovered. It is fundamentally and practically important to discover the fact that in some cases, periodic and chaotic attractorshave higher ethanol yield and production rate than the corresponding steady states.
?
2003 Elsevier Science Ltd. All rights reserved.
Keywords:
Dynamic simulation; Fermentation; Modeling; Nonlinear dynamics; Bifurcation; Chaos
1. Introduction
Multiplicity of steady states in chemically reactive systems was rst observed by Liljernoth (1919). However, it
wasnotuntilthe1950sthatgreatinterestinthisphenomenonin chemical reactors started, inspired by Minnesota schoolof Amundson, Aris and others (1958) and their students(e.g. Balakotaiah & Luss, 1981, 1983a, b). The Prague
school also made notable contribution to the eld (e.g.Hlavacek & Rompay, 1981). This phenomenon is treatedin the mathematical literature in more general and abstractterms under the title of “bifurcation theory” (Golubitsky &Schaeer, 1985). Uppal, Ray, and Poore (1974, 1976),
Ipsen and Schreiber (2000) and Melo, Sampaio, Biscaisa,and Pinto (2001) enriched the basic knowledge and understanding of both static and dynamic bifurcation of thesesystems. The bifurcation behavior of nonisothermal bub bling uidized bed catalytic reactors has been extensively
∗
Corresponding author. Tel.: +13348442033;fax: +13348442063.
Email address:
paragg@eng.auburn.edu (P. Garhyan).
investigated by Elnashaie (1977), Elnashaie and Yates
(1973) and Elnashaie and ElBialy (1980). Excellent re
views for the bifurcation behavior of chemically reactiveand biochemical systems were published by Ray (1977),Bailey and Ollis (1977), Bailey (1998), Gray and Scott
(1994), Elnashaie and Elshishini (1996) and Epstein and
Pojman (1998). The stability and consistency of bio processes has become more important with the development of new biological products and the tighter regulations on product quality. This in turn requires aquantitative knowledge of the bioculture stability anddynamics to understand, control and optimize a process (Davey et al., 1996; Wolf, Sohn, Heinrich, &
Kuriyama, 2001).
2. Fundamental and practical importance
In fermentation processes, many investigators havereported the presence of sustained oscillations in experimental fermentors and they have developed suitable mathematical relations to model these fermentors (Jarzebski, 1992;Ghommidh, Vaija, Bolarinwa, & Navarro, 1989; Daugulis,
00092509/03/$see front matter
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2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S00092509(02)006814
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P. Garhyan et al./Chemical Engineering Science 58 (2003) 1479–1496
McLellan, & Li, 1997; McLellan, Daugulis, & Li, 1999;
Jobses, Egberts, Ballen, & Roels, 1985; Jobses, Egberts,Luyben, & Roels, 1986b; Jobses, 1986a). Xiu, Zeng, and
Deckwer (1998), Zamamiri, Birol, and Hjortso (2001)and Zhang and Henson (2001) have carried out a detailedmultiplicity and stability analysis of microorganisms incontinuous cultures.This paper shows the rich static/dynamic bifurcation behavior of this system. It also shows that these oscillationscan be complex leading to chaotic behavior and that these periodicandchaoticattractorscanbeuseful.Usinganexperimentally veried mathematical model, it is shown that theaverage conversion of sugar and average yield/productivityof ethanol is sometimes higher for periodic and chaotic attractors than for the corresponding steady states despite of the fact that during oscillations, the values of the state variables fall below the average value of the oscillations for some time (Borzani, 2001).It is important to point out that carrying out bifurcationanalysis, rather than simply producing dynamic simulationsof the model equations for dierent parameter values andconditions has the following advantages:(i) For a slow process like fermentation, dynamic simulation may be inecient, inconclusive and may not beable to locate the model characteristics that are responsible for certain rich dynamic behavior such as bifurcation and chaos.(ii) Some dynamic characteristics may be completelymissed or neglected as only a limited number of dynamic simulation runs can be performed.
3. Model development and discussion
In microbial fermentation processes, biomass acts as thecatalyst for substrate conversion and is also produced bythe process. This is a biochemical example of autocatalysis(Gray & Scott, 1984; Lynch, 1993; Bailey, 1998).
Mathematical models can be classied according to their description of or assumptions of biomass (Roels, 1983).Segregated(orcorpuscular)modelsregardbiomassasapopulation of individual cells. Consequently, the applied mathematics is based on statistical equations. These models arevaluable for describing the variations in a given populations,like for instance the age distribution amongst the cells. Thistype of model is also useful for the description of stochasticevents, in which case probability statistics are applied.If there is no need for the description of variations in the biomass population or stochastic events, and provided thatthe number of cells under consideration is large, continuum(or unsegregated) models are more convenient. Continuummodels regard biomass as a chemical complex in solution,or as a multiphase system without concentration gradientswithin the separate phases. A more detailed model for ocsof microorganism was introduced by Elnashaie and Ibrahimfor an industrial fermentor (Elnashaie & Ibrahim, 1991).In fact, the description of nonstochastic interactions betweenthebiomassanditsenvironmentbycontinuummodelsequals the description by segregated models in the case of largepopulationswithanormalvariancedistribution(Roels,1983).Continuum models can be subdivided into unstructuredand structured models. Unstructured models regard biomassas one compound which does not vary in composition uponenvironmental changes. Structured models regard biomassas consisting of at least two dierent compounds, and describe the interactions between the constituting compoundsand between the biomass and the environment.It is interesting to study the occurrence of oscillations inanaerobic cultures experimentally and theoretically. The occurrence of oscillations (which in some cases can hardly beavoided) is a perfect tool for indepth studies of the microorganism physiology (Kaeppeli, Arreguin, & Rieger, 1985; Neubauer, Haeggstroem, & Enfors, 1995; Andersen et al.,2001).Several models have been proposed to account for the oscillatory behavior of
Zymomonas mobilis
as mentioned earlier; some of these are discussed briey here.Jarzebski (1992) proposed a tricompartment model including substrate limitation and product inhibition (withviable, nonviable and dead cells), which was an extensionof the work of Ghommidh et al. (1989) with the view thatthe sustained oscillations were present only at high substrate concentration. It was concluded that the mechanismof decrease in product inhibition with decrease in substrateconcentration was a feasible explanation for the sustainedor slowly damped oscillations at substrate saturation andstrongly damped oscillations leading to stable steady stateat substrate shortage. With appropriate mass balance equations, the model was able to represent the data obtained by their group during the oscillatory behavior of the
Zymomonas mobilis
culture. Some experimental support for the presence of viable and nonviable or dead cells was provided by plate count and slide culture estimations.Daugulis et al. (1997) and McLellan et al. (1999), pro
posed a model which considers inhibitory culture conditionsin the recent past aecting subsequent cell behavior. They proposed the concept of “dynamic specic growth rate” toexplain the inhibition of the instantaneous specic growthrate due to “ethanol concentration change rate history”. Thischange in “ethanol concentration change rate history” is aresult of change in the physiological state of the culture.This model is more phenomenological in that macroscopicvariables such as ethanol, substrate and biomass concentrations are the only experimental quantities which need to bemeasured as opposed to other structured models.Another model (Jobses et al., 1985, 1986b) is an
unsegregatedstructured twocompartment representation,it considered biomass as being divided into compartments(Kcompartment and Gcompartment) containing specic groupings of macromolecules (e.g. Kcompartment is
P. Garhyan et al./Chemical Engineering Science 58 (2003) 1479–1496
1481
identied with RNA, carbohydrates and monomers of macromolecules while as the Gcompartment is identiedwith protein, DNA and lipids). Jobses et al. (1985, 1986b)
and Jobses (1986a) studied the oscillatory behavior utilizingthis model in which the synthesis of a cellular component“
e
” (which is essential for both growth and product formation) had a nonlinear dependence on ethanol concentration.Hence the inhibition by ethanol did not directly inuencethe specic growth rate of the culture, but its eect wasindirect. This twocompartment model is the one used inthe present investigation.One of the most widely used models to model fermentation processes is the maintenance model (Pirt, 1965), inwhich substrate consumption is expressed in the form:
r
S
=
1
Y
SX
r
X
+
m
S
C
X
:
(1)The rst term accounts for growth rate, and the secondterm accounts for the maintenance. The growth term and themaintenance factor have their classical denition (Bailey &Ollis, 1977).The rate of growth of biomass is usually given by
r
X
=
C
X
:
(2)Jobses et al. (1985, 1986b) and Jobses (1986a), suggested
a relatively simple unsegregatedstructured model based onintroducing an internal key compound (
e
) of the biomass.The activity of this compound is expressed in terms of concentrations of substrate, product and the compound (
e
) of the biomass itself. So the rate of formation of the key com pound (
e
) is given by
r
e
=
f
(
C
S
)
f
(
C
P
)
C
e
;
(3)where the substrate dependence function
f
(
C
S
) is given byMonod type relation,
f
(
C
S
) =
C
S
K
S
+
C
S
:
(4)The experimental data of Jobses et al. (1985, 1986b) and
Jobses (1986a) show that the relation between alcohol concentration
C
P
and alcohol dependence function
f
(
C
P
) is asecondorder polynomial in
C
P
having the following form,
f
(
C
P
) =
k
1
−
k
2
C
P
+
k
3
C
2
P
:
(5)The model developed by Jobses et al. (1985, 1986b) and
Jobses(1986a)isafourdimensionalmodelwiththeconcentrations of the following components: substrate (
S
), product (
P
), microorganism or biomass (
X
) and the internal keycomponent (
e
).Based on the above, we have modied the dynamic modelrepresenting the concentrations of three components:
X; S
and
P
, together with a mass ratio of components
e
and
X
.Here we dene
E
= (
C
e
=C
X
)
≡
(
kge=kgX
) (thus
E
is thefraction of biomass that is component
e
). The factor
p
used by Jobses et al. (1985, 1986b) is the maximum possible
specic growth rate (
max
) that would be obtained if
E
=1, i.e. the whole biomass was active. Thus we replace thefactor
p
used by Jobses et al. (1985, 1986b) by
max
, thuswith this change the specic growth rate can be written as
=
C
S
E
max
=K
S
+
C
S
and the dynamic model is given bythe following set of ordinary dierential Eqs. (6)–(9).
d
E
d
t
=
k
1
max
−
k
2
max
C
P
+
k
3
max
C
2
P
−
E;
(6)d
C
X
d
t
=
C
X
+
D
(
C
X
0
−
C
X
)
;
(7)d
C
S
d
t
=
−
1
Y
SX
+
m
S
C
X
+
D
(
C
S
0
−
C
S
)
;
(8)d
C
P
d
t
=
1
Y
PX
+
m
P
C
X
+
D
(
C
P
0
−
C
P
)
:
(9)It is interesting to point out that the balance Eq. (6) for the mass ratio of component
e
and
X
(denoted by
E
) isindependent of the type of reactor used. It states that the rateofformationof
E
(representedby[(
k
1
=
max
)
−
(
k
2
=
max
)
C
P
+(
k
3
=
max
)
C
2
P
)
]) must be atleast the same as the dilutionrate of
E
(represented by
E
). In Eqs. (6)–(9), the value of
max
is taken to be equal to 1 h
−
1
(Jobses et al., 1986b). If
needed, Eq. (6) can be replaced by a dierential equationfor component
e
concentration:d
C
e
d
t
= (
k
1
−
k
2
C
P
+
k
3
C
2
P
)
C
S
C
e
K
S
+
C
S
+
D
(
C
e
0
−
C
e
)to get the same results. It should also be noted that
D
=
q=V
(dilution rate), where
q
is the constant ow rate into thefermentor and
V
is the active volume of fermentor.For steady state solutions, the set of four dierential equations (6) –(9) reduces to a set of four coupled nonlinear al
gebraic equations which can only be solved simultaneously.Jobses et al. (1985, 1986b) and Jobses (1986a) used suc
cessfully the above fourdimensional model to simulate theoscillatory behavior of an experimental continuous fermentor in the high feed sugar concentration region.In the present investigation, the abovedescribed model isusedtoexplorethedierentpossiblecomplexstatic/dynamic bifurcation behavior of this system in the twodimensional(
D
−
C
S
0
) parameter space (Fig. 1), and to study the im plications of these phenomena on substrate conversion andethanol yield and productivity.The system parameters for one of the experimental runs of Jobsesetal.(1986b)andJobses(1986a)showingoscillatory
behavior are used as the base set of parameters in the presentinvestigation and are given in Table 1.
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P. Garhyan et al./Chemical Engineering Science 58 (2003) 1479–1496
Table 1The base set of parameters usedParameter Value
k
1
(h
−
1
) 16.0
k
2
(m
3
=
kg h
−
1
) 4
:
97
×
10
−
1
k
3
(m
6
=
kg
2
h
−
1
) 3
:
83
×
10
−
3
m
S
(kg
=
kg h
−
1
) 2.16
m
P
(kg
=
kg h
−
1
) 1.1
Y
SX
(kg
=
kg) 2
:
44498
×
10
−
2
Y
PX
(kg
=
kg) 5
:
26315
×
10
−
2
K
S
(kg
=
m
3
) 0.5
C
X
0
(kg
=
m
3
) 0
C
P
0
(kg
=
m
3
) 0
C
e
0
(kg
=
m
3
) 0
4. Discussion of the model used
The parameters of the maintenance and the twocompartment model oers a physiological more adequatedescription of growth and fermentation of
Z. mobilis
(Jobses et al., 1985). Furthermore, the twocompartmentmodel is consistent with the experimental results, in thatit predicts qualitatively the response of the steady stateRNA content of the biomass on elevated ethanol concentrations. The eect of elevated ethanol concentrationon the fermentation kinetics resembles the eect of elevating the temperature of fermentation broth (Fieschko& Humphrey, 1983). Also, elevated temperature enlargesthe inhibitory eect of ethanol (Lee, Skotnicki, Tribe, &Rogers, 1981).The oscillatory behavior of productinhibited culturescannot simply be described by a common inhibition term inthe equation of biomass growth (Kurano, Kotera, Okazaki,& Miura, 1984; Wolf et al., 2001). A better description
necessitates the inclusion of an indirect (or delayed) eectof the product on the growth rate as was experimentallydemonstrated (Kurano et al., 1984). Kurano et al. (1984)
introduced a decay rate of
max
caused by the accumulationof the inhibitory product pyruvic acid. Jobses (1986a) pro posed a more mechanistic, structured model, in which
max
is related to an internal keycompound (
e
). The inhibitoryaction of ethanol is realized by the inhibition of the formation of this keycompound (Jobses et al., 1985, 1986b;Jobses, 1986a).Mathematically these descriptions are equivalent, exceptthatthekeycompoundiswashedoutasapartofthebiomassin continuous cultures, and the rate constant
max
is not. The proposed indirect inhibition model provides qualitatively agood description of the experimental results as shown inFig. 2 (Jobses et al., 1985). The quantitative description
is however not optimal, as it was necessary to adapt some parameters values for the description of the oscillations atdierent dilution rates. A quantitative adequate model, must probably also account for inhibition of the total fermentation(including growth rate independent metabolism) and dyingo of the biomass at long contact times at high ethanolconcentrations.
5. Presentation techniques and numerical tools
The bifurcation diagrams are obtained using the software package AUTO97 (Doedel et al., 1997). This package is
able to perform both steady state and dynamic bifurcationanalysis, including the determination of entire periodic solution branches using the ecient continuation techniques(Kubaiecek & Marek, 1983).The DIVPAG subroutine available with IMSL Librariesfor FORTRAN with automatic step size to ensure accuracyfor sti dierential equations is used for numerical simulation of periodic as well as chaotic attractors. A FORTRAN program is written for plotting the Poincare plots.The classical time trace and phase plane for the dynamics are used. However, for high periodicity and chaoticattractors these techniques are not sucient. Therefore,other presentation techniques are used. These techniquesare based upon the plotting of discrete points of intersection(return points) between the trajectories and a hypersurface(Poincare surface) chosen at a constant value of the statevariable (
C
X
= 1
:
55 kg
=
m
3
in the present investigation).These discrete points of intersection are taken such that thetrajectories intersect the hyperplane transversally and crossit in the same direction.The return points are used to construct a number of im portant diagrams namely,1.
Poincar e one parameter bifurcation diagram
: A plot of one of the coordinates of the return points (e.g.
C
S
)versus a bifurcation parameter (e.g.
D
).2.
Return point histogram
: A plot of one of the coordinatesof the return points (e.g.
C
S
) versus time.
6. Results and discussion
The results of the bifurcation analysis are classied in twodierent sections:(A) Dilution rate (
D
) as the bifurcation parameter.(B) Feed sugar concentration (
C
S
0
) as the bifurcation parameter.The reason for choosing these two bifurcation parameters(
D
and
C
S
0
) is that they are the easiest to manipulate duringthe operation of a laboratory or fullscale fermentor.Fig. 1A is a twoparameter continuation diagram of
D
vs.
C
S
0
showing the loci of static limit points (SLPs) andHopf bifurcation (
HB
) points. One parameter bifurcationdiagrams are constructed by taking xed value of
C
S
0
andconstructing the
D
bifurcation diagrams, then taking xedvalues of
D
and constructing the
C
S
0
bifurcation diagrams.Fig. 1B is an enlargement of dotted box of Fig. 1A.
P. Garhyan et al./Chemical Engineering Science 58 (2003) 1479–1496
1483

1.01.03.05.07.0230110 190150
S
0
C
D
0.000.030.060.090.12160110 150120
S
0
C
D
140130
(A) (B)
Fig. 1. (A) Two parameter continuation diagram of
C
S
0
vs.
D
: (——) loci of HB points, () loci of SLP, (B) enlargement of box of gure A.
In order to evaluate the performance of the fermentor asan alcohol producer, we should calculate the conversion of substrate, the product (ethanol) yield and its productivityaccording to the simple relations incorporated into the FORTRAN programs,Substrate (sugar) conversion =
X
S
=
C
S
0
−
C
S
C
S
0
,Ethanol yield =
Y
P
=
C
P
−
C
P
0
C
S
0
.Ethanol productivity (production rate per unit volume,kg
=
m
3
h) of the fermentor is
P
P
=
C
P
D
.For the oscillatory and chaotic cases also, the averageconversion
X
S
, average yield
Y
P
and the average productionrate
P
P
as well as the average ethanol concentration
C
P
arecomputed. They are dened as,
X
S
=
0
X
S
d
t ;
Y
P
=
0
Y
P
d
t ;
P
P
=
0
P
P
d
t ;
C
P
=
0
C
P
d
t ;
where the
values in the periodic cases represent one periodof the oscillations, and in the chaotic cases, they are takenlongenoughtobereasonablerepresentationofthe“average” behavior of the chaotic attractor.
6.1. (A) Dilution rate
D
as the bifurcation parameterCase
(A.1):
C
S
0
= 140 kg
=
m
3
.Jobses et al. (1986b) and Jobses (1986a) used this value
of
C
S
0
in their experiments together with a dilution rateof
D
= 0
:
022 h
−
1
. An example of the comparison betweenthe dynamic modeling and experimental results obtained byJobses et al. (1986b) is shown in Fig. 2. Details of static
and dynamic bifurcation behavior for this case are shownin Figs. 3A–E, with the dilution rate
D
as the bifurcation parameter.Fig. 3A shows the bifurcation diagram for substrateconcentration (
C
S
) with clear demarcations between the
Fig. 2. Comparison of experimental and simulation results (Jobses et al.,1986b): (——) measured ethanol concentration, () simulatedethanol concentration.
dierent regions using dotted vertical lines. It is clear thatthe static bifurcation diagram is an incomplete
S
shape hysteresis type with a SLP at very low value of
D
=0
:
0035 h
−
1
.The dynamic bifurcation shows a
HB
at
D
HB
= 0
:
05 h
−
1
with a periodic branch emanating from it. The region in theneighborhood of the SLP is enlarged in Fig. 3B. The periodic branch emanating from
HB
terminates homoclinically(with innite period) when it touches the saddle point veryclose to the SLP at
D
HT
=0
:
0035 h
−
1
. Fig. 3C is the bifurcation diagram for the ethanol concentration (
C
P
). It is clear from Fig. 3C that the average ethanol concentrations for the periodic attractors are higher than those corresponding tothe unstable steady states. Figs. 3D and E show the bifurcation diagrams for ethanol yield (
Y
P
) and ethanol productionrate (
P
P
) where the average yield (
Y
P
) and production rate(
P
P
) for the periodic branch are shown as diamond shaped points. Fig. 4 shows the period of oscillations as the periodic branch approaches the homoclinical bifurcation point;the period tends to innity indicating homoclinical termination of the periodic attractor at
D
HT
= 0
:
0035 h
−
1
(Keener,1981).The region (Region 1) with three point attractors(
D¡D
HT
) is characterized by the fact that two of themare unstable and only the steady state with the highest conversion is stable. The highest conversion (almost complete