Mass and energy balances as state-space models for aerobic batch fermentations
R. Dondo and D. Marqués
Grupo de Control de Procesos
Instituto de Desarrollo Tecnológico para la Industria Química (INTEC)
Guemes 3450 (3000) Santa Fe Republica Argentina
e-mail: rdondo@ceride.gov.ar
Abstract The main aim of this work is the development of mathematical models of aerobic batch fermentations for its use in estimation and control algorithms. Most batch fermentation models are empirical and simple, and do not provide interrelationships between state variables and measurements. In this work such interrelationships are obtained from mass and energy balances of the fermentation components. Since aerobic fermentations with formation of a single metabolite exhibit three degrees of freedom, three independent kinetic equations are necessary to build the state space model. Test results on the batch fermentation of xanthan gum are presented.
1. Introduction
Many publications have applied advanced control to
fermentative processes carried out in continuous or fedbatch
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 and
Goldschmidt (1994); reviews by Shioya (1992) and by
Shimizu (1993)]. The main obstacle for applying
advanced process control to batch fermentations is the
poor quality of the processes models, the relative low
number of measurements, and the scarcely-known
interrelationships between states and measured
variables.
In batch fermentations, the main system variables can
vary widely along the process. Thus, there is an
opportunity for driving the manipulated variables in
optimal fashion. Some publications examined the
problem of determining optimal control trajectories in
fermentors, considering different objective functions
and control variables. Constantinides et al. (1970)
presented probably the first paper that proposed the use
of optimal control in batch bioreactors. Reuss (1986)
presented a review on the use of optimal control in
fermentative process, and only few of the reviewed
works considered batch operations. In recent years,
Asenjo et al. (1995) and Lee et al. (1999) presented
articles on the optimal control of batch reactors,
including experimental validations. In spite of the fact
that trajectory optimization is a well-known technique,
it has not been widely applied to fermentative processes.
The reason is that optimal control results are highly
dependent on the process model, and many fermentation
models do not accurately represent the dynamic
behavior.
Erikson et al. (1978) is one of the first publications
where mass and energy balances have been used for
modeling fermentations. The balances were used to
derive 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 without
product formation and anaerobic fermentations); and
outlined a scheme for building models on the basis of
the available kinetic information. The relevance of
macroscopic principles for modeling bioengineering
systems was also discussed. Heijnen and Roels (1981)
developed a slightly more complex scheme, specific to
aerobic fermentations. Simple models were proposed
for estimating yield coefficients on substrates with
different degrees of reduction. The authors analyzed the
effect of the temperature on yields and on the
maintenance coefficients. Minkevich (1983) modeled
the fermentation as a set of partial metabolisms, and
used mass and energy balances for interrelating the
partial metabolism kinetics. The influence of
intracellular characteristics on the rate of physiological
processes and on the culture productivity was discussed.
Andrews (1989, 1993) presented a similar scheme, but
with the aim of estimating macroscopic yields. Andrews
(1989) discussed the limitation of yield values for the
different product types.
The mentioned works show the relation between the
kinetics of the intracellular processes with different
levels of complexity but assuming that the intracellular
reaction rates and the mass transfer mechanisms
between the cells and its environment are in the steady
state. Batch fermentations are time-varying processes,
and we did not find any publication on the use of
balances onto batch processes.
In this work, we derive a time-varying state-space
model that is applicable to aerobic batch fermentations.
This work is organized as follows. In section 2, statespace
equations for batch fermentations are derived
from a model that involves five partial metabolisms.
Microscopic balances are used to calculate the relations
between partial metabolisms and the net consumption of
the main components. Also, macroscopic balances are
used to calculate the relations between the partial
metabolism rates, the variation of main component concentrations and the net flow of these components to
the reactor. In section 3, these balances are combined
with a simple kinetic model to derive the state space
equations. In section 4, the procedure is applied to
predict the evolution (consumption or production) of the
gaseous components of a xanthan gum batch
fermentation.
2. A state-space model for an aerobic batch fermentation
An aerobic fermentation can be seen as a set of parallel "reactions" denoted partial metabolisms. These
reactions are not simple chemical reactions. The partial
metabolisms either produce or consume carbon dioxide,
water, oxygen, main substrate, and nitrogen source. The
hydrogen involved in oxidation/reduction reactions (H+ + e-) is bound to electron carriers like NADH2 (nicotinamide adenine dinucleotides). Also, energy
carriers like ATP (adenosine 5-triphosphate) transport
the produced or consumed free energy.
Since fermentation models should be accurate and of the
least possible complexity, the aim is to find the
minimum set of differential and algebraic equations that
adequately describes the process dynamics. In statespace
formulation, a nonlinear model is represented by:
![]() |
(1.a) |
(1.b) |
where x is the state-vector, y is the vector of
measurements, u is the vector of manipulated variables
and p is a vector of model parameters.
In a homogeneous and constant-volume stirred-tank
reactor, the balance equations are expressed as follows
[Roels (1980)]:
![]() |
(2) |
where C is a vector that represents the concentration of
the main components, rC is the net conversion rate of
these components, and Φ is a vector that represents the
net transport rate of the system components. To be used
as a state-space model, the Eq. (2) must be expressed in
the form of Eq. (1). When applied to a specific
fermentor, Eq. (2) provides the interrelationships
between reaction rates, concentration variations, and net
inlet flows. Eq. (2) represents a macroscopic balance
where the microbial metabolism determines the
conversion rates of the different components (rC)
together with the relations between these rates. Since
the microbial metabolism must obey the conservation
principles, microscopic balances can be used to derive a
minimum number of dynamic equations that describe
the process.
In principle, microscopic balances should be applied to
every fermentation component and to every element.
However, considering that hundreds of components
could participate in the microbial metabolism, it is
necessary to limit the analysis to the so-called main
components and main elements. The four main elements
are C, H, O, and N, since these elements comprise about
95% of the biological mass [Roels, 1980]. The main
components depend on the fermentation type, and they
will be identified later for an aerobic batch fermentation
with production of a single metabolite.
2.1 Microscopic balances
Figure 1 presents a model for the aerobic growth of biomass (X) using a single component as carbon and energy source (S), and an independent nitrogen source that can also contain carbon (SN). The generated metabolic component is denoted P, and CO2, H2O and O2 are components exchanged between the cells and their media. The growth is assumed to be approximately balanced, in the sense that microorganisms are able to produce exact replicas of themselves. The fermentation is modeled by five partial-metabolisms or pseudoreactions. Each pseudo-reaction is described by a stoichiometry Ei and a kinetic rate ri. This simplified description can be only explained from a stoichiometric point of view, since many biochemical reactions simultaneously participate in different partial metabolisms. The flow of main components can be subdivided according to such pseudo-reactions; and each reaction can be kinetically modeled like an integral unit [Minkevich (1983)]. This description of an aerobic fermentation, although simple, is more complete than most of the empirical models normally found in control and estimation algorithms [Constantinides et al. (1970); Wu et al. (1985); Shimizu et al.(1989)].
Fig. 1: The aerobic fermentation model as a network of partial metabolisms. (The grey boxes denote partial metabolisms that consume ATP while the white boxes denote partial metabolisms that produce ATP. The dashed arrows indicate the intracellular NADH2 flow. The continuous arrows indicate the intracellular ATP flow. The bold arrows denote exchange of main components between the biomass and its environment).
The concentration of intracellular components, ATP and
NADH2, are assumed to be in the steady state [Roels
(1983)]. Therefore, there is not accumulation term in
their balances.
Since the element balances must be always satisfied,
they represent the constraints to be met by each
stoichiometry EI. The vector of main components is
defined as follows:
The composition of components X, S, P, and SN is expressed by their atomic formulae CHb1Oc1Nd1, Ca2Hb2Oc2Nd2, Ca3Hb3Oc3 and Ca4Hb4Oc4Nd4 respectively (the metabolite is assumed to be a nitrogen-free component). The coefficients of the stoichiometric equations on Table 1 are expressed as the inverse of the yields (YI/J) of product I on each of the J components. The coefficients YX/ATP and YP/ATP indicate the moles of energy carriers (ATP) consumed in the anabolism and in the product metabolism respectively; and YS/ATP indicates the moles of ATP generated in the main substrate catabolism.
Table 1: Stoichiometric relations of the partial metabolisms (Ei)
The production of biomass is a process with a quantified
net consumption of ATP (YX/ATP = 10.5 g X/mol ATP
[Andrews (1993)]). The formation of products can
either generate energy (e.g. by partial oxidation of
substrates in the alcoholic fermentations) or consume energy (e.g. by generating macromolecules). The net
energy demand of each reaction can be known by
studying the metabolic pathways that describe the
product generation [Andrews (1993)]. Apart from these
processes, other numerous degradation and dissipation reactions that imply ATP consumption are also present
[Minkevich (1983)]. These so-called "cellular
maintenance reactions" imply a non measurable demand
of ATP.
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 the
level of oxidative phosphorylation (P/O). This function
depends on the specific metabolic oxidation pathway.
Thus, the amount of oxidized main substrate must be
calculated from the specific pathway function f(P/O) and from the ATP balance. The use of stoichiometric
relations and balances of intracellular components (ATP
and NADH2) allows to write the following relationships.
ATP balance:
![]() |
(3) |
NADH2 balance:
![]() |
(4) |
The consumption (or production) of main components exchanged between the cells and their media is calculated writing a mass balance for each component, as follows:
Nitrogen source consumption:
![]() |
(5) |
Main substrate consumption:
![]() |
(6) |
Oxygen consumption:
![]() |
(7) |
with YS/O2 = 2YS/NADH2; and YP/O2 = 2YP/NADH2.
Carbon dioxide production:
![]() |
(8) |
The intracellular balances of ATP and NADH2 provide
two interrelationships among the five intracellular
processes. Thus, an aerobic fermentation with formation
of a single metabolic product has three degrees of
freedom, and the unknown rates may be obtained from
the knowledge of three kinetic equations with the
appropriate stoichiometric yields YI/J.
The presented model provides a fairly accurate
description of an aerobic fermentation with formation of
a single metabolite, and is applicable to any reactor
type. However, the intracellular model parameters (P/O,
rM ATP) are generally unknown and their values could
change due to manipulated or non-manipulated changes
in the environment.
From a macroscopic point of view and without
considering intracellular components, a stoichiometry
for the main substrate oxidation is obtained by adding
the main substrate catabolism stoichiometry (ESE) and
the oxidative phosphorilation stoichiometry for the
NADH2 produced by the catabolism of the main
substrate (EPO/YS/NADH2). This oxidation occurs at a rate
rSE. Besides, the cellular maintenance 'reaction' consumes ATP but not main components. Thus, this
reaction can be ignored in the macroscopic mass
balances. For both reasons, the three independent
processes that are necessary for describing the
fermentation are:
• biomass growth (rX);
• metabolite production (rP); and
• main substrate oxidation (rSE).
In kinetic models of main substrate oxidation, the
unknown parameters (P/O, rMATP) are usually lumped
into a 'maintenance' coefficient KE that models the main
substrate oxidation as a first order reaction in the
biomass concentration. This assumption is common in
empirical models. However, a better model for the main
substrate oxidation can be obtained by reordering the
ATP balance (Eq. 3), as follows:
![]() |
(9) |
Since in batch reactors reaction rates rX and rP are
usually time-varying, Eq. (9) shows that the main
substrate oxidation is not strictly first order in the
biomass concentration. Therefore, the KE parameter
from a first order oxidation rate could show strong
variations. In effect, when the KE coefficients of several
experimental fermentations are compared [Heijnen and
Roels (1981)], it is observed that their values change for
similar fermentation conditions.
The assumption of a first order oxidation rate of the
main substrate is only valid if the reaction rates and the
mass transfer mechanisms between media and cells are
in the steady state. In this case, since rX, rP, and rATPM are in the steady state and the yields are constant, then
rSE is also in the steady state. However, in batch
fermentations, reaction rates and mass transfer
processes reach the steady state only when the
metabolic activity drops to zero (i.e., at the end of the
fermentation).
2.2 Macroscopic balances
The previous balance equations are independent of the
kinetics and provide relationships between the rates of
the different partial metabolisms and between these
rates and the net consumption (or production) of main
components. These relationships are also independent of
the reactor type. Consider now the derivation of
macroscopic relations between reaction rates,
concentration of main components, and net component
flows to the broth.
In aerated batch fermentors, X, P, S and SN are not
transferred through the surface boundary of the vessel.
Thus, the corresponding elements of the transport rate
vector Φ in Eq. (2) are all zero.
![]() |
(10) |
The oxygen concentration in the broth is considered to be in a quasi steady state because its variations are much faster than the oxygen flow transferred between the gas phase and the broth. Similarly, if the pH of the broth is constant, then the carbon dioxide concentration is also in a quasi steady state. Therefore:
![]() |
(11) |
Thus, in aerated batch bioreactors, Eq. (2) may be written as follows:
![]() |
(12) |
quation (12) indicates that the oxygen consumption rate (-rO2) and the carbon dioxide production rate (rCO2) can be obtained by measuring the oxygen transferred into the broth (-ΦO2) and the carbon dioxide transfer from the broth (ΦCO2). Except for the water, the concentration variation of the remaining main components are obtained from their formation or consumption rates.
State dynamics.
As mentioned before, rX, rP, and rSE are the three independent processes present in an aerobic fermentation with formation of a single metabolite. The dynamics of biomass growth (rX) and of metabolite production (rP) are characterized by the evolution of the biomass concentration and the metabolic product concentration. Thus, it is possible to use the following empirical equations:
![]() |
(13) |
The main substrate oxidation is represented by:
![]() |
(14) |
Note that the second equality of Eq. (14) is the Eq. (6) reordered. The state associated to this reaction is the amount of oxidized main substrate SE. Empirical expressions for rX, rP, and rS. with Eq. (14) provides the kinetics of rX, rP, and rSE and therefore complete the fermentation dynamics. Alternatively, other less usual kinetic equations together with the yields YI/J also can be used to complete the state space model.
State-measurements relations.
The theoretical relation between states and measured variables is provided by the stoichiometric coefficients. The microscopic mass balances indicate that the total consumption rate of main substrate and oxygen and the total production rate of carbon dioxide are the sum of their respective evolutions in rX, rP, and rSE (Eqs. 6, 7 and 8). The measurement of ΦO2 and ΦCO2 provides more information on the fermentation rates than on the states. Thus, it is preferable to use the cumulative oxygen consumption (∆O2(t)) and the cumulative carbon dioxide production (∆CO2(t)) at time t as measurement variables:
![]() |
(15.a) |
![]() |
(15.b) |
In these equations, the yield coefficients are extracted
from the stoichiometric relations presented in Table 1.
The yields of biomass on the other components can be
exactly known only in pure cultures of a perfectly
identified biomass. Therefore, their values are in general
approximate, and can be estimated using some
fermentation regularities, element balances and the
knowledge of the broth composition [Erikson et al.
(1978)].
If the main substrate and the nitrogen source
concentrations are measurable, then the following
equations provide relationships between the states and
the two mentioned main components:
![]() |
(16.a) |
![]() |
(16.b) |
In these expressions, X(t)-X0, P(t), and SE(t) are
respectively the amounts of produced biomass, of
produced metabolite, and of oxidized main substrate at
time t.
Note that when the measured variables belong to the set
of main components, then the stoichiometric yields
allow to find linear relations between measured
variables and state-variables.
2.3 Effect of the control variables and modeling of uncertainties
Batch fermentation models present the following
peculiarities:
• The influence of intensive variables (pH,
temperature) on the fermentation dynamics is
usually expressed as empirical functions π(u) that
show the dependence of the kinetic parameters π on
such variables. If the intensive variables can be
manipulated, then they can be used as control
variables, but in this case important deviations from
the empirical functions π(u) are to be expected.
• Large changes in the control variables may cause
unpredictable changes in the kinetic parameters and
even a variation in the structure of the dynamic
model. This limits the use of empirical models that
cannot quantify the physiological effects.
• Variations in the model parameters from one
fermentation to another are to be expected, even
when the same control police is applied.
Thus, the control variables affect the system via the
kinetic parameters. Calling p(u,t) the function of the
kinetic parameters with respect to the control variables
u, then the batch reactor dynamics can be described
through:
![]() |
(17) |
Usually only a crude empirical function π(u) which is an approximation of the real p(u,t) is known. If an accurate knowledge of the parameter values is required, then an estimation algorithm can be used, expressing p(u,t) as follows:
![]() |
(18) |
where ∆ π(u,t) is an unknown deviation between p(u,t) and π(u). To identify this deviation, the control variable must vary along the whole range of variation. In contrast, when the control variable is kept in a narrow interval during the whole fermentation time, then is not possible to identify ∆ π(u,t). In this case, p(u,t) can be written as follows:
![]() |
(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 to use the logistic equation for the biomass kinetics together with the Luedeking-Piret equation for the product and main substrate kinetics [Weiss and Ollis (1980)]:
|
(20.a) |
![]() |
(20.b) |
![]() |
(20.c) |
Consider the development of a state-space model that uses this set of kinetic equations. As explained in section (2.1), the biomass growth, the product formation and the main substrate oxidation describe an aerobic fermentation with production of a single metabolite. For the first two processes, Eqs. (20.a) and (20.b) can be used as dynamic models. Equation (20.c) corresponds to the total main substrate consumption (the sum of the substrate consumption in each of the three independent processes). Thus, a model for the main substrate oxidation may be obtained by subtracting Eqs. (20.a) and (20.b) divided by their respective stoichiometric yields YX/S and YP/S from Eq. (20.c). Alternatively, it can be assumed that this process is first order in the biomass concentration:
![]() |
(21) |
When empirical equations that express the kinetic parameters ( µ, XS, a, b and KE in this case) as functions of the control variables are available, these equations together with Eqs. (20.a), (20.b) and (21) constitute the dynamic model f(x, π(u)). But as explained in section 2.3, it is convenient to add a deviation term or'disturbance-parameter' to each parameter. These disturbances include the errors of the functions π (u). Thus, the process dynamics can be expressed as:
![]() |
(22) |
The uncertainty in the 'disturbance-parameters' dynamics suggests to model them as constants with a real-time state and parameter estimator estimating their values.
4. Application to the batch production of xanthan gum. Results and discussion.
Xanthan gum is an extracellular polysaccharide
produced by Xanthomonas campestris. This gum has
numerous applications and is produced in large
quantities. The kinetic model of Eqs. (20) has been used
by Weiss and Ollis (1980), Pinches and Pallent (1986)
and Shu and Yang (1991) to describe the production of
xanthan gum in batch fermentors. Since these authors
do not provide a complete information on the main component dynamics, the model of section 2 is used to
complete the fermentation description. Pons et al. (1989) developed a structured model for this
fermentation and estimated the stoichiometric relations
for the gum production, the catabolism of the main
substrate (glucose), and the oxidative phosphorilation.
The chemical formula of a "mol" of xanthan gum
proposed by Pons et al. (1989) is C32.34H48.58O27.36Na1.38.
The stoichiometry of xanthan gum production provides
the values YP/S = 0.917 g P/g S, YP/CO2 = 34.326 g P/g CO2 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 from
the stoichiometry of the glucose oxidation. The
stoichiometry of biomass production was not reported
but can be estimated from some regularities of aerobic
fermentations, element balances and the knowledge of
the broth composition [Erikson et al. (1978)]. Then, the
yields YX/S, YX/O2 and YX/CO2 can be calculated from the
experimental value of YX/N.
The cellular maintenance coefficient of Eq. (21) is
estimated using Eq. (14). By modelling rP and rS through Luedeking-Piret type equations, then KE results:
![]() |
(23) |
Kinetic expressions for rO2 and rCO2 can be calculated
replacing Eqs. (20.a), (20.b), and (21) into Eqs. (7) and
(8). Since rP is modeled by the Luedeking-Piret
equation, then the 'calculated rates' rO2 and rCO2 will
also have the same structure.
In Table 2, the resulting values of some kinetic
parameters are compared with experimental values by
Pinches and Pallent (1986); Pons et al. (1989); and
Peters et al. (1992). As the rates rO2 and rCO2 depend on
the biomass concentration, the evolution of this
concentration must be simulated for driving these rates.
The fermentation conditions of the cited papers and the
kinetic equations used for simulating the evolution of
the biomass concentration are also summarized in Table
2. Computer simulations are presented in Figs. 2 to 5. In
these figures the evolution of the biomass concentration
was simulated; the model-predicted evolution of the oxygen consumption (computed as explained in the
previous paragraph) is compared with their simulated
evolution using the experimental kinetic parameters
(Eq. (15) with experimental parameters of Table 2). The
predicted carbon dioxide
Table 2: Experimental and calculated values of yields and kinetic parameters for four xanthan gum fermentations carried out in batch reactors (The reported experimental values are given in normal font. The calculated values are given in Italics, and below the experimental values.)
production is also presented.
Unfortunately, there is not experimental information on
the evolution of this main component.
For the fermentation presented in Pinches and Pallent
(1986) with sodium glutamate as nitrogen source, the
experimental and theoretical evolution of consumed
oxygen are quite similar. Interestingly, the theoretical
value of c is smaller than the experimental value, while
the opposite occurs with the parameter d, indicating a
compensation of values. This can be attributed to errors
in the graphical method used by Pinches and Pallent
(1986) to identify the experimental parameters. For the
fermentation of Peters et al. (1992), the agreement 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, the
differences can be explained by the possible generation
of by-products from a complex nitrogen source. Some
experimental information by Flores Candia (1994)
supports this statement.
In summary, the agreement between the calculated and
experimental parameters is acceptable, considering the
large uncertainties in this process and the poor
measurements of some of the key variables. The use of'disturbance-parameters' seems appropriate in this case.
5. Conclusions
A procedure was outlined for building dynamical
models by combining macro and microscopic balances
with available kinetic information. A model of the
microbial metabolism that is based on elemental and
intracellular component balances was presented. The
model divides the metabolism into five partial
metabolisms or pseudo-reactions. The balances allow to calculate how errors in the kinetics parameters or in the
stoichiometry of a partial metabolism influence the
remaining process, and also allow to calculate some of
the unknown rates. This is useful in a batch process
where the biomass grows in a changing environment
that can generate strong kinetic disturbances. Kinetic
information on fermentations is generally scarce,
particularly in the case of the gaseous components
interchanged between the broth and the gas phase (O2 and CO2). A procedure was proposed to derive
interrelationships between these measured variables and
the state variables.
As it was seen in section 2.1, in batch fermentations the
main substrate oxidation rate can show considerable
variations. Contrary to what is usually assumed, it was
shown here that this rate is not first order in the biomass
concentration.
Some modern control strategies require of dynamical models. In batch fermentations the kinetic rates are
generally unreliable, and for this reason the concept of disturbance-parameters estimated via an estimation
algorithm were introduced to improve the quality of the
kinetic information.
The proposed procedure was applied to xanthan gum
batch fermentations, and the gum production was
modeled trough the Luedeking-Piret equation. The
equation parameters and some stoichiometric
coefficients were used to predict the oxygen
consumption and the carbon dioxide production rates. A
fairly good agreement with experimental data was
observed.
Acknowledgments
We would like to thank Gregorio Meira for helpful and detailed comments on a draft of this paper. This work was supported by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and the Universidad Nacional del Litoral (UNL).
Nomenclature
µ | specific biomass growth on the logistic equation (h-1) |
α | growth associated to the specific main substrate consumptio (g substrate/g biomass). |
β | steady specific main substrate consumption (g substrate/g biomass h ) |
γ | growth associated to the specific carbon dioxide production (g CO2 /g biomass) |
δ | steady specific carbon dioxide production (g CO2 /g biomass h) |
∆p | vector of disturbance-parameters |
∆CO2 | cumulative carbon dioxide production |
∆S | cumulative main substrate consumption |
∆SN | cumulative nitrogen source consumption |
∆O2 | cumulative oxygen consumption |
ΦI | net flow rate of component I into the reactor (g I/L h) |
a | growth associated specific metabolite
production (g metabolite/g biomass) adenosine 5-triphosphate (energy carriers) |
ATP | steady specific metabolite production (g metabolite/g biomass h ) |
b | growth associated specific oxygen |
c | consumption (g O2 /g biomass) |
C | vector of main components concentration |
d | steady specific oxygen consumption (g O2 /g biomass h) |
EP | stoichiometry of the metabolite production stoichiometry of the oxidative |
EPO | phosphorylation |
ESE | stoichiometry of the main substrate catabolism |
EX | stoichiometry of the biomass production vector of dynamic functions for the states |
f(•) | vector of state -measurements relations |
g(•) | macroscopic specific coefficient of
maintenance (g main substrate/g biomass.h) |
KE | reduced equivalents (electron carriers) |
NADH2 | concentration of metabolic product (g metabolite/L) |
P | vector of model parameters |
p | level of oxidative phosphorylation (ratio |
P/O | between ATP formed and oxygen consumed on EPO ) |
rATPM | ATP consumption for maintenance (moles ATP/g biomass h) |
rC | vector of conversion rates for the main components production rate of carbon dioxide (g CO2/L h) |
rCO2 | consumption rate of the nitrogen source (g |
rN | nitrogen source/L.h) consumption rate of oxygen (g O2/L h) |
rO2 | production rate of metabolite (g |
rP | Metabolite/L h) oxidative phosphorylation rate (moles |
rPO | ATP/L.h) consumption rate of main substrate (g main |
rS | substrate/L h) production rate of biomass (g biomass/L h) |
rX | main substrate concentration (g main |
S | substrate/L) concentration of nitrogen source (g Nitrogen |
SN | source/L) vector of control variables |
u | biomass concentration (g biomass/L) |
X | vector of state variables |
x | stationary biomass concentration (g biomass/L) |
XS | vector of measurements |
Y | stoichiometric yield of component I on |
YI/J | component J (g I/g J) |
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Received: January 15, 2001.
Accepted for publication: November 21, 2001.
Recommended by Subject Editor R. Sánchez Peña.