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Latin American Applied Research 32:195-204 (2002)
195
MASS AND ENERGY BALANCES AS STATE-SPACE MODELS FOR AEROBICBATCH FERMENTATIONS
R. Dondo and D. MarquésGrupo de Control de Procesos
Instituto de Desarrollo Tecnológico para la Industria Química (INTEC)Guemes 3450 (3000) Santa Fe – Republica Argentina
e-mail: [email protected]
Abstract:The main aim of this work is the development of mathematical models of aerobicbatch fermentations for its use in estimation and control algorithms. Most batchfermentation models are empirical and simple, and do not provideinterrelationships between state variables and measurements. In this work suchinterrelationships are obtained from mass and energy balances of the fermentationcomponents. Since aerobic fermentations with formation of a single metaboliteexhibit three degrees of freedom, three independent kinetic equations arenecessary to build the state space model. Test results on the batch fermentation ofxanthan gum are presented.
1. Introduction
Many publications have applied advanced control tofermentative processes carried out in continuous or fed-batch bioreactors [Wu et al. (1985); Takamatsu et al.(1985); Lim et al. (1986); San and Stephanopoulos(1986); Williams et al. (1986); Modak and Lim (1987);Agrawal et al. (1989); San and Stephanopoulos (1989);Shi et al. (1989); Harmon et al. (1989); Diener andGoldschmidt (1994); reviews by Shioya (1992) and byShimizu (1993)]. The main obstacle for applyingadvanced process control to batch fermentations is thepoor quality of the processes models, the relative lownumber of measurements, and the scarcely-knowninterrelationships between states and measuredvariables.In batch fermentations, the main system variables canvary widely along the process. Thus, there is anopportunity for driving the manipulated variables inoptimal fashion. Some publications examined theproblem of determining optimal control trajectories infermentors, considering different objective functionsand control variables. Constantinides et al. (1970)presented probably the first paper that proposed the useof optimal control in batch bioreactors. Reuss (1986)presented a review on the use of optimal control infermentative process, and only few of the reviewedworks considered batch operations. In recent years,Asenjo et al. (1995) and Lee et al. (1999) presentedarticles on the optimal control of batch reactors,including experimental validations. In spite of the factthat trajectory optimization is a well-known technique,it has not been widely applied to fermentative processes.The reason is that optimal control results are highlydependent on the process model, and many fermentationmodels do not accurately represent the dynamicbehavior.Erikson et al. (1978) is one of the first publicationswhere mass and energy balances have been used formodeling fermentations. The balances were used toderive optimal operating conditions in continuous
single-cell production reactors. Roels (1980)generalized the concepts presented by Erikson et al.(1978) to other fermentations (aerobic with or withoutproduct formation and anaerobic fermentations); andoutlined a scheme for building models on the basis ofthe available kinetic information. The relevance ofmacroscopic principles for modeling bioengineeringsystems was also discussed. Heijnen and Roels (1981)developed a slightly more complex scheme, specific toaerobic fermentations. Simple models were proposedfor estimating yield coefficients on substrates withdifferent degrees of reduction. The authors analyzed theeffect of the temperature on yields and on themaintenance coefficients. Minkevich (1983) modeledthe fermentation as a set of partial metabolisms, andused mass and energy balances for interrelating thepartial metabolism kinetics. The influence ofintracellular characteristics on the rate of physiologicalprocesses and on the culture productivity was discussed.Andrews (1989, 1993) presented a similar scheme, butwith the aim of estimating macroscopic yields. Andrews(1989) discussed the limitation of yield values for thedifferent product types.The mentioned works show the relation between thekinetics of the intracellular processes with differentlevels of complexity but assuming that the intracellularreaction rates and the mass transfer mechanismsbetween the cells and its environment are in the steadystate. Batch fermentations are time-varying processes,and we did not find any publication on the use ofbalances onto batch processes.In this work, we derive a time-varying state-spacemodel that is applicable to aerobic batch fermentations.This work is organized as follows. In section 2, state-space equations for batch fermentations are derivedfrom a model that involves five partial metabolisms.Microscopic balances are used to calculate the relationsbetween partial metabolisms and the net consumption ofthe main components. Also, macroscopic balances areused to calculate the relations between the partialmetabolism rates, the variation of main component
Latin American Applied Research 32:195-204 (2002)
196
concentrations and the net flow of these components tothe reactor. In section 3, these balances are combinedwith a simple kinetic model to derive the state spaceequations. In section 4, the procedure is applied topredict the evolution (consumption or production) of thegaseous components of a xanthan gum batchfermentation.
2. A state-space model for an aerobic batchfermentation
An aerobic fermentation can be seen as a set of parallel“reactions” denoted partial metabolisms. Thesereactions are not simple chemical reactions. The partialmetabolisms either produce or consume carbon dioxide,water, oxygen, main substrate, and nitrogen source. Thehydrogen involved in oxidation/reduction reactions (H+
+ e-) is bound to electron carriers like NADH2
(nicotinamide adenine dinucleotides). Also, energycarriers like ATP (adenosine 5-triphosphate) transportthe produced or consumed free energy.Since fermentation models should be accurate and of theleast possible complexity, the aim is to find theminimum set of differential and algebraic equations thatadequately describes the process dynamics. In state-space formulation, a nonlinear model is represented by:
)(
)(),,( 00
xgy
txxpuxfx
=
==• (1.a)
(1.b)
where x is the state-vector, y is the vector ofmeasurements, u is the vector of manipulated variablesand p is a vector of model parameters.In a homogeneous and constant-volume stirred-tankreactor, the balance equations are expressed as follows[Roels (1980)]:
Φ+=•
CrC (2)
where C is a vector that represents the concentration ofthe main components, rC is the net conversion rate ofthese components, and Φ is a vector that represents thenet transport rate of the system components. To be usedas a state-space model, the Eq. (2) must be expressed inthe form of Eq. (1). When applied to a specificfermentor, Eq. (2) provides the interrelationshipsbetween reaction rates, concentration variations, and netinlet flows. Eq. (2) represents a macroscopic balancewhere the microbial metabolism determines theconversion rates of the different components (rC)together with the relations between these rates. Sincethe microbial metabolism must obey the conservationprinciples, microscopic balances can be used to derive aminimum number of dynamic equations that describethe process.In principle, microscopic balances should be applied toevery fermentation component and to every element.However, considering that hundreds of componentscould participate in the microbial metabolism, it is
necessary to limit the analysis to the so-called maincomponents and main elements. The four main elementsare C, H, O, and N, since these elements comprise about95% of the biological mass [Roels, 1980]. The maincomponents depend on the fermentation type, and theywill be identified later for an aerobic batch fermentationwith production of a single metabolite.
2.1 Microscopic balances
Figure 1 presents a model for the aerobic growth ofbiomass (X) using a single component as carbon andenergy source (S), and an independent nitrogen sourcethat can also contain carbon (SN). The generatedmetabolic component is denoted P, and CO2, H2O andO2 are components exchanged between the cells andtheir media. The growth is assumed to be approximatelybalanced, in the sense that microorganisms are able toproduce exact replicas of themselves. The fermentationis modeled by five partial-metabolisms or pseudo-reactions. Each pseudo-reaction is described by astoichiometry Ei and a kinetic rate ri. This simplifieddescription can be only explained from a stoichiometricpoint of view, since many biochemical reactionssimultaneously participate in different partialmetabolisms. The flow of main components can besubdivided according to such pseudo-reactions; andeach reaction can be kinetically modeled like an integralunit [Minkevich (1983)]. This description of an aerobicfermentation, although simple, is more complete thanmost of the empirical models normally found in controland estimation algorithms [Constantinides et al. (1970);Wu et al. (1985); Shimizu et al.(1989)].The concentration of intracellular components, ATP andNADH2, are assumed to be in the steady state [Roels(1983)]. Therefore, there is not accumulation term intheir balances.Since the element balances must be always satisfied,they represent the constraints to be met by eachstoichiometry EI. The vector of main components isdefined as follows:
CT = [X, S, P, SN, O2, CO2, H2O]
The composition of components X, S, P, and SN isexpressed by their atomic formulae CHb1Oc1Nd1,Ca2Hb2Oc2Nd2,, Ca3Hb3Oc3 and Ca4Hb4Oc4Nd4 respectively(the metabolite is assumed to be a nitrogen-freecomponent). The coefficients of the stoichiometricequations on Table 1 are expressed as the inverse of theyields (YI/J) of product I on each of the J components.The coefficients YX/ATP and YP/ATP indicate the moles ofenergy carriers (ATP) consumed in the anabolism and inthe product metabolism respectively; and YS/ATP
indicates the moles of ATP generated in the mainsubstrate catabolism.The production of biomass is a process with a quantifiednet consumption of ATP (YX/ATP = 10.5 g X/mol ATP[Andrews (1993)]). The formation of products caneither generate energy (e.g. by partial oxidation ofsubstrates in the alcoholic fermentations) or consume
R. DONDO, D. MARQUÉS
197
energy (e.g. by generating macromolecules). The netenergy demand of each reaction can be known bystudying the metabolic pathways that describe theproduct generation [Andrews (1993)]. Apart from theseprocesses, other numerous degradation and dissipation
reactions that imply ATP consumption are also present[Minkevich (1983)]. These so-called “cellularmaintenance reactions” imply a non measurable demandof ATP.
Biomassproduction
EX rX
Metaboliteproduction
EP rP
Cellularmaintenance
--- rMATP
Main substratecatabolism:
ESE rS
E
Oxidativephosphorilation:EPO rPO
CO H2O O2
S P SN
Biomass
Fig. 1: The aerobic fermentation model as a network of partial metabolisms. (The grey boxes denote partialmetabolisms that consume ATP while the white boxes denote partial metabolisms that produce ATP. The dashedarrows indicate the intracellular NADH2 flow. The continuous arrows indicate the intracellular ATP flow. The boldarrows denote exchange of main components between the biomass and its environment).
Table 1: Stoichiometric relations of the partial metabolisms (Ei)
Biomass production (EX):
2/
2
2/
2111
/4444
/2222
/
11
COXOHX
dcb
ATPX
dcba
NX
dcba
SX Y
CO
Y
OHNOCH
Y
ATPNOHC
YNOHC
Y++→++
Metabolite production (EP):
2/
2
2/
2
2/
2333
/2222
/
1
COPOHPNADHPcba
ATPPdcba
SP Y
CO
Y
OH
Y
NADHOHC
Y
ATPNOHC
Y+++→+
Main substrate catabolism (ESE):
2/
2
/2
2/2/
22222
1
NADHSATPSCOSOHSdcba Y
NADH
Y
ATPCO
YY
OHNOHC ++→+
Oxidative phosphorilation (EPO):
ATPYOHONADH NADHATP 2/222 2
1 +→+
To satisfy the ATP demand for the biomass production,the metabolite production, and the cellular maintenance,a certain amount of main substrate must be oxidized.But the generated ATP (YATP/NADH2) is a function of thelevel of oxidative phosphorylation (P/O). This functiondepends on the specific metabolic oxidation pathway.Thus, the amount of oxidized main substrate must becalculated from the specific pathway function f(P/O)and from the ATP balance. The use of stoichiometric
relations and balances of intracellular components (ATPand NADH2) allows to write the following relationships.
ATP balance:
)/(2/
2/2/2////
OPfY
YY
r
Y
r
Y
rr
Y
r
Y
r
NADHATP
NADHATPNADHP
P
NADHS
ES
ATPS
ESM
ATPATPP
P
ATPX
X
=
++=++
(3)
Latin American Applied Research 32:195-204 (2002)
198
NADH2 balance:
2/2/2/ NADHATP
PO
NADHP
P
NADHS
ES
Y
r
Y
r
Y
r=
+ (4)
The consumption (or production) of main componentsexchanged between the cells and their media iscalculated writing a mass balance for each component,as follows:
Nitrogen source consumption:
NX
XN Y
rr
/
−= (5)
Main substrate consumption:
++−= E
SSP
P
SX
XS r
Y
r
Y
rr
//(6)
Oxygen consumption:
2/2/2/2/2 2
1
OP
P
OS
ES
NADHP
P
NADHS
ES
O Y
r
Y
r
Y
r
Y
rr +=
+= (7)
with YS/O2 = 2YS/NADH2; and YP/O2 = 2YP/NADH2.
Carbon dioxide production:
2/2/2/2
COS
ES
COP
P
COX
XCO
Y
r
Y
r
Y
rr ++= (8)
The intracellular balances of ATP and NADH2 providetwo interrelationships among the five intracellularprocesses. Thus, an aerobic fermentation with formationof a single metabolic product has three degrees offreedom, and the unknown rates may be obtained fromthe knowledge of three kinetic equations with theappropriate stoichiometric yields YI/J.
The presented model provides a fairly accuratedescription of an aerobic fermentation with formation ofa single metabolite, and is applicable to any reactortype. However, the intracellular model parameters (P/O,rM
ATP) are generally unknown and their values couldchange due to manipulated or non-manipulated changesin the environment.From a macroscopic point of view and withoutconsidering intracellular components, a stoichiometryfor the main substrate oxidation is obtained by addingthe main substrate catabolism stoichiometry (ES
E) andthe oxidative phosphorilation stoichiometry for theNADH2 produced by the catabolism of the mainsubstrate (EPO/YS/NADH2). This oxidation occurs at a raterS
E. Besides, the cellular maintenance ‘reaction’consumes ATP but not main components. Thus, thisreaction can be ignored in the macroscopic massbalances. For both reasons, the three independentprocesses that are necessary for describing thefermentation are:• biomass growth (rX);• metabolite production (rP); and
• main substrate oxidation (rSE).
In kinetic models of main substrate oxidation, theunknown parameters (P/O, rM
ATP) are usually lumpedinto a ‘maintenance’ coefficient KE that models the mainsubstrate oxidation as a first order reaction in thebiomass concentration. This assumption is common inempirical models. However, a better model for the mainsubstrate oxidation can be obtained by reordering theATP balance (Eq. 3), as follows:
2/
2/
/
2/
2/
//
1
1
NADHP
NADHATP
ATPS
MATP
NADHP
NADHATP
ATPPP
ATPX
X
ES
Y
Y
Y
rY
Y
Yr
Y
r
r
+
+
−+
= (9)
Since in batch reactors reaction rates rX and rP areusually time-varying, Eq. (9) shows that the mainsubstrate oxidation is not strictly first order in thebiomass concentration. Therefore, the KE parameterfrom a first order oxidation rate could show strongvariations. In effect, when the KE coefficients of severalexperimental fermentations are compared [Heijnen andRoels (1981)], it is observed that their values change forsimilar fermentation conditions.The assumption of a first order oxidation rate of themain substrate is only valid if the reaction rates and themass transfer mechanisms between media and cells arein the steady state. In this case, since rX, rP, and rATP
M
are in the steady state and the yields are constant, thenrS
E is also in the steady state. However, in batchfermentations, reaction rates and mass transferprocesses reach the steady state only when themetabolic activity drops to zero (i.e., at the end of thefermentation).
2.2 Macroscopic balances
The previous balance equations are independent of thekinetics and provide relationships between the rates ofthe different partial metabolisms and between theserates and the net consumption (or production) of maincomponents. These relationships are also independent ofthe reactor type. Consider now the derivation ofmacroscopic relations between reaction rates,concentration of main components, and net componentflows to the broth.In aerated batch fermentors, X, P, S and SN are nottransferred through the surface boundary of the vessel.Thus, the corresponding elements of the transport ratevector Φ in Eq. (2) are all zero.
0=
ΦΦΦΦ
SN
S
P
X
(10)
The oxygen concentration in the broth is considered tobe in a quasi steady state because its variations are muchfaster than the oxygen flow transferred between the gas
R. DONDO, D. MARQUÉS
199
phase and the broth. Similarly, if the pH of the broth isconstant, then the carbon dioxide concentration is alsoin a quasi steady state. Therefore:
02
2 =
CO
O
dt
d (11)
Thus, in aerated batch bioreactors, Eq. (2) may bewritten as follows:
ΦΦΦ
+
=
OH
CO
O
OH
CO
O
N
S
P
X
N
r
r
r
r
r
r
r
OH
S
S
P
X
dt
d
2
2
2
2
2
2
2
0
0
0
0
0
0
(12)
quation (12) indicates that the oxygen consumption rate(-rO2) and the carbon dioxide production rate (rCO2) canbe obtained by measuring the oxygen transferred intothe broth (-ΦO2) and the carbon dioxide transfer fromthe broth (ΦCO2). Except for the water, the concentrationvariation of the remaining main components areobtained from their formation or consumption rates.
State dynamics.
As mentioned before, rX, rP, and rSE are the three
independent processes present in an aerobicfermentation with formation of a single metabolite. Thedynamics of biomass growth (rX) and of metaboliteproduction (rP) are characterized by the evolution of thebiomass concentration and the metabolic productconcentration. Thus, it is possible to use the followingempirical equations:
=
P
X
r
r
P
X
dt
d (13)
The main substrate oxidation is represented by:
SP
P
SX
XS
ES
E
Y
r
Y
rrr
dt
dS
//
−−−=−= (14)
Note that the second equality of Eq. (14) is the Eq. (6)reordered. The state associated to this reaction is theamount of oxidized main substrate SE. Empiricalexpressions for rX, rP, and rS. with Eq. (14) provides thekinetics of rX, rP, and rS
E and therefore complete thefermentation dynamics. Alternatively, other less usualkinetic equations together with the yields YI/J also can beused to complete the state space model.
State-measurements relations.
The theoretical relation between states and measuredvariables is provided by the stoichiometric coefficients.The microscopic mass balances indicate that the total
consumption rate of main substrate and oxygen and thetotal production rate of carbon dioxide are the sum oftheir respective evolutions in rX, rP, and rS
E (Eqs. 6, 7and 8). The measurement of ΦO2 and ΦCO2 providesmore information on the fermentation rates than on thestates. Thus, it is preferable to use the cumulativeoxygen consumption (∆O2(t)) and the cumulativecarbon dioxide production (∆CO2(t)) at time t asmeasurement variables:
2OS
E
2OP2OX
0
2OS
t
0
E
S
2OP
t
0P
2OX
t
0Xt
02O
t
02O2
Y
tS
Y
tP
Y
XtXY
dtr
Y
dtr
Y
dtrdtrdttO
///
///
)()()(
)(
++−
=
++=−=Φ=∆∫∫∫
∫∫(15.a)
2COS
E
2COP2COX
0
2COS
t
0
E
S
2COP
t
0P
2COX
t
0Xt
02CO
t
02CO2
Y
tS
Y
tP
Y
XtXY
dtr
Y
dtr
Y
dtrdtrdttCO
///
///
)()()(
)(
++−=
++=−=Φ=∆∫∫∫
∫∫ (15.b)
In these equations, the yield coefficients are extractedfrom the stoichiometric relations presented in Table 1.The yields of biomass on the other components can beexactly known only in pure cultures of a perfectlyidentified biomass. Therefore, their values are in generalapproximate, and can be estimated using somefermentation regularities, element balances and theknowledge of the broth composition [Erikson et al.(1978)].If the main substrate and the nitrogen sourceconcentrations are measurable, then the followingequations provide relationships between the states andthe two mentioned main components:
)()()(
)(
//
0
0/
0
/
0
00
tSY
tP
Y
XtX
dtrY
dtr
Y
dtr
dtrdtdt
dStS
E
SPSX
tE
SSP
t
P
SX
t
Xt
S
t
++−=
++−===∆ ∫∫∫
∫∫
NX
0
NX
t
0Xt
0S
t
0
NN Y
XtX
Y
dtrdtrdt
dt
dStS
N
//
)()(
−=−===∆∫
∫∫
(16.a)
(16.b)
In these expressions, X(t)-X0, P(t), and SE(t) arerespectively the amounts of produced biomass, ofproduced metabolite, and of oxidized main substrate attime t.Note that when the measured variables belong to the setof main components, then the stoichiometric yieldsallow to find linear relations between measuredvariables and state-variables.
Latin American Applied Research 32:195-204 (2002)
200
2.3 Effect of the control variables and modeling ofuncertainties
Batch fermentation models present the followingpeculiarities:• The influence of intensive variables (pH,
temperature) on the fermentation dynamics isusually expressed as empirical functions π (u) thatshow the dependence of the kinetic parameters π onsuch variables. If the intensive variables can bemanipulated, then they can be used as controlvariables, but in this case important deviations fromthe empirical functions π (u) are to be expected.
• Large changes in the control variables may causeunpredictable changes in the kinetic parameters andeven a variation in the structure of the dynamicmodel. This limits the use of empirical models thatcannot quantify the physiological effects.
• Variations in the model parameters from onefermentation to another are to be expected, evenwhen the same control police is applied.
Thus, the control variables affect the system via thekinetic parameters. Calling p(u,t) the function of thekinetic parameters with respect to the control variablesu, then the batch reactor dynamics can be describedthrough:
)),(,( tupxfx =• (17)
Usually only a crude empirical function π (u) which isan approximation of the real p(u,t) is known. If anaccurate knowledge of the parameter values is required,then an estimation algorithm can be used, expressingp(u,t) as follows:
),()(),( tuutup ππ ∆+= (18)
where ∆π (u,t) is an unknown deviation between p(u,t)and π (u). To identify this deviation, the control variablemust vary along the whole range of variation. Incontrast, when the control variable is kept in a narrowinterval during the whole fermentation time, then is notpossible to identify ∆π(u,t). In this case, p(u,t) can bewritten as follows:
)()(),( tputup ∆+≅ π (19)
The sum of π (u) and the estimated value of ∆p(t)provides a better representation of p(u,t) than π (u).
3. A batch kinetic model
A common practice in batch fermentations models is touse the logistic equation for the biomass kineticstogether with the Luedeking-Piret equation for theproduct and main substrate kinetics [Weiss and Ollis(1980)]:
XX
Xr
S
X
−= 1µ (20.a)
bXdt
dXarP +=
+−= X
dt
dXrS βα
(20.b)
(20.c)
Consider the development of a state-space model thatuses this set of kinetic equations. As explained insection (2.1), the biomass growth, the product formationand the main substrate oxidation describe an aerobicfermentation with production of a single metabolite. Forthe first two processes, Eqs. (20.a) and (20.b) can beused as dynamic models. Equation (20.c) corresponds tothe total main substrate consumption (the sum of thesubstrate consumption in each of the three independentprocesses). Thus, a model for the main substrateoxidation may be obtained by subtracting Eqs. (20.a)and (20.b) divided by their respective stoichiometricyields YX/S and YP/S from Eq. (20.c). Alternatively, it canbe assumed that this process is first order in the biomassconcentration:
XKdt
dS EE
=(21)
When empirical equations that express the kineticparameters (µ, XS, a, b and KE in this case) as functionsof the control variables are available, these equationstogether with Eqs. (20.a), (20.b) and (21) constitute thedynamic model f(x,π(u)). But as explained in section2.3, it is convenient to add a deviation term or‘disturbance-parameter’ to each parameter. Thesedisturbances include the errors of the functions π (u).Thus, the process dynamics can be expressed as:
( )
( ) ( )( )
∆+∆++∆+
∆+
−∆+
=
∆∆∆
∆∆
•
0
0
0
0
0
)(
)()(
)(1)(
XKuK
XbubXaua
XXuX
Xu
K
b
a
X
S
P
X
dt
dEE
SS
R
S
R
µµ
µ (22)
The uncertainty in the ‘disturbance-parameters’dynamics suggests to model them as constants with areal-time state and parameter estimator estimating theirvalues.
4. Application to the batch production of xanthangum. Results and discussion.
Xanthan gum is an extracellular polysaccharideproduced by Xanthomonas campestris. This gum hasnumerous applications and is produced in largequantities. The kinetic model of Eqs. (20) has been usedby Weiss and Ollis (1980), Pinches and Pallent (1986)and Shu and Yang (1991) to describe the production ofxanthan gum in batch fermentors. Since these authorsdo not provide a complete information on the main
R. DONDO, D. MARQUÉS
201
component dynamics, the model of section 2 is used tocomplete the fermentation description. Pons et al.(1989) developed a structured model for thisfermentation and estimated the stoichiometric relationsfor the gum production, the catabolism of the mainsubstrate (glucose), and the oxidative phosphorilation.The chemical formula of a “mol” of xanthan gumproposed by Pons et al. (1989) is C32.34H48.58O27.36Na1.38.The stoichiometry of xanthan gum production providesthe values YP/S = 0.917 g P/g S, YP/CO2 = 34.326 g P/gCO2 and YP/O2 = 15.820 g P/g O2. Also, YS/O2 = 0.937 g S/g O2 and YS/CO2 = 0.682 g P/g CO2 are derived fromthe stoichiometry of the glucose oxidation. Thestoichiometry of biomass production was not reportedbut can be estimated from some regularities of aerobicfermentations, element balances and the knowledge ofthe broth composition [Erikson et al. (1978)]. Then, theyields YX/S, YX/O2 and YX/CO2 can be calculated from theexperimental value of YX/N.The cellular maintenance coefficient of Eq. (21) isestimated using Eq. (14). By modelling rP and rS
through Luedeking-Piret type equations, then KE results:
SP
E
Y
bK
/
−= β (23)
Kinetic expressions for rO2 and rCO2 can be calculatedreplacing Eqs. (20.a), (20.b), and (21) into Eqs. (7) and(8). Since rP is modeled by the Luedeking-Piretequation, then the ‘calculated rates’ rO2 and rCO2 willalso have the same structure.In Table 2, the resulting values of some kineticparameters are compared with experimental values byPinches and Pallent (1986); Pons et al. (1989); andPeters et al. (1992). As the rates rO2 and rCO2 depend onthe biomass concentration, the evolution of thisconcentration must be simulated for driving these rates.The fermentation conditions of the cited papers and thekinetic equations used for simulating the evolution ofthe biomass concentration are also summarized in Table2. Computer simulations are presented in Figs. 2 to 5. Inthese figures the evolution of the biomass concentrationwas simulated; the model-predicted evolution of theoxygen consumption (computed as explained in theprevious paragraph) is compared with their simulatedevolution using the experimental kinetic parameters(Eq. (15) with experimental parameters of Table 2). Thepredicted carbon dioxide production is also presented.Unfortunately, there is not experimental information onthe evolution of this main component.For the fermentation presented in Pinches and Pallent(1986) with sodium glutamate as nitrogen source, theexperimental and theoretical evolution of consumedoxygen are quite similar. Interestingly, the theoreticalvalue of c is smaller than the experimental value, whilethe opposite occurs with the parameter d, indicating acompensation of values. This can be attributed to errorsin the graphical method used by Pinches and Pallent(1986) to identify the experimental parameters. For thefermentation of Peters et al. (1992), the agreement
between experimental and theoretical parameters isquite good, as it can be seen in the simulation of Fig. 5.
0 10 20 30 40 500
2
4
6
Biomass concentration∆O
2(experimental)
∆O2
(theoretical)
∆CO2
(theoretical)
(g./l
t.)
Time (h.)
Fig. 2: Simulated evolution of the gaseous componentsfor the fermentation of Pinches and Pallent (1986).
Nitrogen-source: sodium glutamate
0 10 20 30 400
2
4
(g./l
t.)
Biomass concentration∆O
2(experimental)
∆O2
(theoretical)
∆CO2
(theoretical)
Time (h.)
Fig. 3: Simulated evolution of the gaseous componentsfor the fermentation of Pinches and Pallent (1986).
Nitrogen-source: peptone
0 10 20 300
2
4
6
8
10
12
(g./l
t.)
Time (h.)
Biomass concentration∆O
2(experimental)
∆O2
(theoretical)
∆CO2
(theoretical)
Fig. 4: Simulated evolution of the gaseous componentsfor the fermentation of Pons et al. (1989). Nitrogen-
source: peptone + yeast.
Latin American Applied Research 32:195-204 (2002)
202
Table 2: Experimental and calculated values of yields and kinetic parameters for four xanthan gum fermentationscarried out in batch reactors (The reported experimental values are given in normal font. The calculatedvalues are given in Italics, and below the experimental values.)
Pinches and Pallent (1986) Pons et al. (1989) Peters et al. (1992)
Fermentation N-source Sodium glutamate Peptone Peptone + Yeast NH4ClConditions
Temperature 30 °C 30 °C 29 °C 28 °C
pH 7 6.9 7 7
rX 0.106X (X ≤ 1.77)0.0 (X ≥ 1.77) X
X
−
145.21285.0
0.084X (X ≤ 4.12)0.0 (X ≥ 4.12)
XX
−
816.11448.0
X(0) 0.250 0.140 0.063 0.075
Stoichiometric YX/N 12.02 10.03 6.35 8.0Yields
YX/S 2.69 ∞ ∞ 0.850.782
YX/O2 ∞ 25.45 1.23 ∞YX/CO2 9.47 9.66 0.878 8.74
Kinetic a 1.314 0.474 N/A 0.97Parameters
α 2.932.025
1.2430.517
N/A---
2.302.23
c 0.3860.083
---0.069
N/A---
0.590.64
γ N/A0.144
N/A0.117
N/A---
N/A0.143
b 0.107 0.148 0.250 0.107
KR 0.052 0.045 0.158 0.069
β 0.169 0.206 0.431 0.193
d 0.05540.0622
0.07190.0574
0.1280.184
0.0780.080
δ N/A0.0794
N/A0.0703
N/A0.239
N/A0.104
N/A: not available.
0 10 20 30 40 50 60 700
2
4
6
8
10
12
14
16
18
Biomass concentration∆O
2(experimental)
∆O2
(theoretical)
∆CO2
(theoretical)
(g./l
t.)
Time (h.)
Fig. 5: Simulated evolution of the gaseous componentsfor the fermentation of Peters et al. (1992). Nitrogen-
source: NH4Cl
For the fermentations of Pinches and Pallent (1986)with peptone as nitrogen source and Pons et al. (1989),the results are not too good. Nevertheless, thedifferences can be explained by the possible generationof by-products from a complex nitrogen source. Someexperimental information by Flores Candia (1994)supports this statement.In summary, the agreement between the calculated andexperimental parameters is acceptable, considering thelarge uncertainties in this process and the poormeasurements of some of the key variables. The use of‘disturbance-parameters’ seems appropriate in this case.
5. Conclusions
A procedure was outlined for building dynamicalmodels by combining macro and microscopic balanceswith available kinetic information. A model of themicrobial metabolism that is based on elemental andintracellular component balances was presented. Themodel divides the metabolism into five partialmetabolisms or pseudo-reactions. The balances allow to
R. DONDO, D. MARQUÉS
203
calculate how errors in the kinetics parameters or in thestoichiometry of a partial metabolism influence theremaining process, and also allow to calculate some ofthe unknown rates. This is useful in a batch processwhere the biomass grows in a changing environmentthat can generate strong kinetic disturbances. Kineticinformation on fermentations is generally scarce,particularly in the case of the gaseous componentsinterchanged between the broth and the gas phase (O2
and CO2). A procedure was proposed to deriveinterrelationships between these measured variables andthe state variables.As it was seen in section 2.1, in batch fermentations themain substrate oxidation rate can show considerablevariations. Contrary to what is usually assumed, it wasshown here that this rate is not first order in the biomassconcentration.Some modern control strategies require of dynamicalmodels. In batch fermentations the kinetic rates aregenerally unreliable, and for this reason the concept ofdisturbance-parameters estimated via an estimationalgorithm were introduced to improve the quality of thekinetic information.The proposed procedure was applied to xanthan gumbatch fermentations, and the gum production wasmodeled trough the Luedeking-Piret equation. Theequation parameters and some stoichiometriccoefficients were used to predict the oxygenconsumption and the carbon dioxide production rates. Afairly good agreement with experimental data wasobserved.
Acknowledgments
We would like to thank Gregorio Meira for helpful anddetailed comments on a draft of this paper. This workwas supported by the Consejo Nacional deInvestigaciones Científicas y Técnicas (CONICET) andthe Universidad Nacional del Litoral (UNL).
Nomenclature
µ
α
β
γ
δ
∆p∆CO2
∆S∆SN
∆O2
ΦI
a
specific biomass growth on the logisticequation (h-1)growth associated to the specific mainsubstrate consumptio (g substrate/g biomass).steady specific main substrate consumption(g substrate/g biomass h)growth associated to the specific carbondioxide production (g CO2 /g biomass)steady specific carbon dioxide production (gCO2 /g biomass h)
vector of disturbance-parameterscumulative carbon dioxide productioncumulative main substrate consumptioncumulative nitrogen source consumptioncumulative oxygen consumptionnet flow rate of component I into the reactor(g I/L h)growth associated specific metaboliteproduction (g metabolite/g biomass)
ATPb
c
Cd
EP
EPO
ESE
EX
f(•)g(•)KE
NADH2
P
pP/O
rATPM
rC
rCO2
rN
rO2
rP
rPO
rS
rX
S
SN
uXxXS
YYI/J
adenosine 5-triphosphate (energy carriers)steady specific metabolite production (gmetabolite/g biomass h)growth associated specific oxygenconsumption (g O2 /g biomass)vector of main components concentrationsteady specific oxygen consumption (g O2 /gbiomass h)stoichiometry of the metabolite productionstoichiometry of the oxidativephosphorylationstoichiometry of the main substratecatabolismstoichiometry of the biomass productionvector of dynamic functions for the statesvector of state -measurements relationsmacroscopic specific coefficient ofmaintenance (g main substrate/g biomass.h)reduced equivalents (electron carriers)concentration of metabolic product (gmetabolite/L)vector of model parameterslevel of oxidative phosphorylation (ratiobetween ATP formed and oxygen consumedon EPO )ATP consumption for maintenance (molesATP/g biomass h)vector of conversion rates for the maincomponentsproduction rate of carbon dioxide (g CO2/L h)consumption rate of the nitrogen source (gnitrogen source/L.h)consumption rate of oxygen (g O2/L h)production rate of metabolite (gMetabolite/L h)oxidative phosphorylation rate (molesATP/L.h)consumption rate of main substrate (g mainsubstrate/L h)production rate of biomass (g biomass/L h)main substrate concentration (g mainsubstrate/L)concentration of nitrogen source (g Nitrogensource/L)vector of control variablesbiomass concentration (g biomass/L)vector of state variablesstationary biomass concentration (gbiomass/L)vector of measurementsstoichiometric yield of component I oncomponent J (g I/g J)
References:
Agrawal P., Koshy G. and Ramseier M. “An Algorithmfor Operating a Fed-Batch Fermentor at OptimumSpecific Growth Rate,” Biotechnol. Bioeng. Vol. 33,115-125 (1989).
Andrews G. “Estimating Cell and Product Yields,”Biotechnol. Bioeng. Vol 33, 256-265 (1989).
Latin American Applied Research 32:195-204 (2002)
204
Andrews G. “The Yield Equations in the Modeling and
Control of Bioprocesses,” Biotechnol. Bioeng. Vol42, 549-556 (1993).
Asenjo J., W. Sun and J. Spencer. “Optimal Control ofBatch Processes Involving Simultaneous Enzimaticand Microbial Reactions,” Bioproc. Eng. 323-329(1995)
Constantinides A., J. Spencer and E. Gaden. “IIOptimun Temperature Profiles for Batch PenicilinFermentation,” Biotechnol. Bioeng. Vol 12, 1081-1098 (1970).
Diener A. Goldschmidt B. “Suboptimal Control of Fed-Batch Bioprocesses Using Phase Properties” Journalof Biotechnology. Vol. 33, 71-85 (1994)
Erikson L., I. Minkevich and V. Eroshin. “Applicationof Mass and Energy Balance Regularities inFermentation,” Biotechnol. Bioeng. Vol 20, 1595-1521 (1978).
Flores Candia J. “Metabolic Flux Distribution,Modelling and Process Optimization of XanthanProduction,” Ph.D. Thesis. Technical University ofBraunschweig, Germany (1997).
Harmon J., Lyberatos G. and Svoronos S. “A NewMethod for Adaptive Determination of Optimun pHand Temperature,” Biotechnol. Bioeng. Vol. 33,1419-1424 (1989).
Heijnen J.J. and J.A. Roels. “A Macroscopic ModelDescribing Yield and Maintence Relationships inAerobic Fermentation Processes,” Biotechnol.Bioeng. Vol 23, 739-763 (1981).
Lee J., I. Lee and Y. Park. “Optimal pH Control ofBatch Processes for Production of Curdlan byAgrobacterium Species,” Journal of Ind. Microbiol.and Biotech. 23, 143-148 (1999).
Lim H., J. Tayeb, J. Modak and P. Bonte.“Computational Algorithms for Optimal Feed Ratesfor a Class of Fed-batch Fermentation: NumericalResults for Penicillin and Cell Mass Production,”Biotechnol. Bioeng. Vol. 28, 1408-1420 (1986).
Minkevich I.G. “Mass-Energy Balance for MicrobialSysntesis – Biochemical and Cultural Aspects,”Biotech. Bioeng. Vol 25, 1267-1293 (1983).
Modak J. and Lim H. “Feedback Optimization of Fed-batch Fermentation,” Biotechnol. Bioeng. Vol. 30,528-540 (1987).
Peters H., Suh I., Schumpe A. and W. Deckwer.2Modeling of Batchwise Xanthan Production,” Can.J. Chem. Eng. Vol 70, 742-750 (1992).
Pinches A. and L. Pallent. “Rate and YieldRelationships in the Production of Xanthan Gum byBatch Fermentations Using Complex andChemically Defined Growth Media,” Biotechnol.Bioeng. Vol 28, 1484-1496 (1986).
Pons A., C. Dussap and J. Bros. “ModellingXanthomonas Campestris Batch Fermentation in aBubble Column,” Biotechnol. Bioeng. Vol 33, 394-405 (1989).
Reuss M. Computer Control of Bioreactors. “PresentLimits and Challenges for the Future. ChemicalProcess Control,” 641-687. Elsevier Austin. (1986)
Roels J.A. “Application of Macroscopic Principles toMicrobial Metabolism” Biotechnol. Bioeng. Vol 22,2457-2514 (1980).
Roels J.A. “Energetics and Kinetics in Biotechnology,”Elsevier (1983).
San K. and Stephanopoulos G. “The Effect of GrowthRate Delays in Substrate Inhibited Kinetics on theOptimal Profile of Fed-Batch Reactors,” Biotechnol.Bioeng. Vol. 28, 356-361 (1986)
San K. and Stephanopoulos G. “Optimization of a Fed-batch Penicillin Fermentation. A Case of SingularOptimal Control with State Constraints,” Biotechnol.Bioeng. Vol. 34, 72-78 (1989).
Shi Z., Shimizu K., Watanabe N. and Kobayashi T.“Adaptive On-Line Optimizing Control ofBioreactor Systems,” Biotechnol. Bioeng. Vol.33,999-1009 (1989).
Shimizu H., Takamatsu T, Shioya S. and K.I. Suga. “AnAlgorithmic Approach to Constructing the On-LineEstimation System for the Specific Growth Rate,”Biotechnol. Bioeng. Vol.33, 354-364 (1989).
Shimizu K. “An Overview of the Control SystemDesign of Bioreactors,” Advances in BiochemicalEngineering. Vol 50, 65-84 (1993).
Shioya S. “Optimization and Control in Fed-BatchBioreactors,” Advances in Biochemical Engineering.Vol. 46, 112-142 (1992).
Shu Ch. and Yang Sh. “Effects of Temperature on CellGrowth and Xanthan Production in Batch Culturesof Xanthomonas Campestris,” Biotechnol. Bioeng.Vol. 35, 454-468 (1991).
Takamatsu T., Shioya S, Okada Y. and Kanda M.“Profile Control scheme in Baker’s Yeast Fed-batchCulture,” Biotechnol. Bioeng. Vol. 27, 1675-1686(1985).
Weiss R. and D. Ollis. “Extracellular MicrobialPolisaccharides. I Substrate , Biomass and ProductKinetic Equations for Batch Xanthan GumFermentation,” Biotechnol. Bioeng. Vol 22, 859-873 (1980).
Williams D., Yousefpour P. and E. Wellington. “On-line Adaptive Control of a Fed-batch Fermentationof Sacharomyces Cerevisiae,” Biotechnol. Bioeng.Vol. 28, 631-645 (1986).
Wu W. T., Chen K.Ch. and Chiou H.W. “On-lineOptimal Control for Fed-Batch Culture of Baker’sYeast Production,” Biotechnol. Bioeng. Vol. 27,756-760 (1985).
Received: January 15, 2001. Accepted for publication: November 21, 2001. Recommended by Subject Editor R. Sánchez Peña.
