Model development
PROGRAM 4.3 Multiplicities in Reactor Temperature and Concentration
5.4 CONTINUOUS STIRRED TANK BIOREACTOR (CSTB) .1 Process Description.1 Process Description
In a simple way, the biochemical reactor can be defined as a tank in which several biological reactions occur simultaneously in a liquid medium. In the bioreactor, the fermentation process is commonly performed with the substrate, in presence of other nutrients in the media, by the action of microorganisms, under optimum biological conditions. The fermentations give rise to a variety of products (Rao, 2005), such as: primary metabolites (e.g., alcohol, citric acid, etc.), biomass [e.g., baker’s yeast, single cell protein (SCP), etc.], transformed substrates (e.g., steroids), and purified solvents as in the case of water treatment.
In a simple biochemical reactor case study, we have considered two components, biomass and substrate. The substrate is the feed source for the cells. Consider the schematic of a biochemical reactor in Figure 5.3, where x is the biomass concentration (= mass of cells/volume) and S the substrate concentration (= mass of substrate/volume). F represents the volumetric flow rate of the feed stream, V the volume of the bioreactor, and xf and Sf the biomass and substrate concentrations respectively in the feed stream.
In the sample bioreactor, the feed is supplied continuously. To keep the cells in suspension, gentle stirring is allowed in the reactor. The agitator speed is chosen to provide sufficient mixing while avoiding excessive shear forces that may damage the cells (Lee, 1992). A stream is removed continuously from the bioreactor. The effluent stream contains unreacted substrate and biomass. The growth of microbial cells in a suitable medium results in the consumption of substrate and the formation of products. Here, the desired products are the cells themselves.
FIGURE 5.3 Schematic representation of the example CSTB.
This system is chosen because, despite this process being the simplest one, its dynamic behaviour is complex (Agrawal, Lee, Lim, and Ramkrishna, 1982). Moreover, several important industrial processes belong to this class (e.g., wastewater treatment process).
The bioreactor model will be constructed in the following way based on the conservation principle and biochemical reaction kinetics.
5.4.2 Mathematical Model
Assumptions
The following assumptions have been made to develop the mathematical model for the bioreactor.
The reactor contents are perfectly mixed.
The reactor is operating at a constant temperature (i.e., it is isothermal).
The feed is sterile (i.e., no biomass in the feed stream).
The feed stream and reactor contents have equal and constant density ().
The feed and product streams have the same flow rate (F).
The microbial culture involves a single biomass growing on a single substrate.
Model development Total Continuity Equation
Making overall mass balance, it is easy to get the following equation:
...(5.1)
Now it is obvious that the reactor volume (V) is constant since . Biomass Continuity Equation
We know,
Flow rate of biomass into the bioreactor = Fxf, Flow rate of biomass out of the bioreactor = Fx, Rate of generation of biomass by reaction = Vr1, and
Rate of accumulation of biomass within the bioreactor .
Substituting all these terms (mass/time) in Equation (3.2), we have ...(5.2)
where r1 is the rate of cell generation [= (mass of cells generated)/(volume) (time)]. Dividing both sides of the above equation by V, one obtains
...(5.3)
In the chemical reaction engineering, F/V is called space velocity (time–1) and V/F the residence time (time). But in biochemical engineering, F/V is referred to as the dilution rate4, probably due to the dilution of the biomass in the reactor with the addition of fresh feed. Accordingly, Equation (5.3) yields
4
It is defined as the number of tank liquid volumes (reactor volumes) that pass through the tank per unit time....(5.4)
or
...(5.5) where D denotes the dilution rate.
Substrate Continuity Equation For the substrate balance,
Flow rate of substrate into the bioreactor = FSf, Flow rate of substrate out of the bioreactor = FS, Rate of generation of substrate by reaction = –Vr2, and
Rate of accumulation of substrate within the bioreactor . Equation (3.2) gives
...(5.6)
Rearranging the above equation, we get
...(5.7)
where r2 is the rate of substrate consumption [= (mass of substrate consumed)/(volume) (time)].
Biochemical Reaction Kinetics
It is well-known that for the chemical reaction,
we can write
(–rA) = k(CA)n...(5.8) or
(rA) = –k(CA)n...(5.9) where (–rA) = rate of disappearance of reactant A
(rA) = rate of formation of A k = reaction rate constant
CA = concentration of reactant A
n = order of the reaction with respect to component A For a first-order reaction, n = 1 and accordingly,
(–rA) = kCA...[from Equation (4.7)]
The reaction kinetics involved in biochemical operations is comparatively complicated than the chemical reaction kinetics. The interested reader may consult the book by Bailey and Ollis (1986) for details about the biochemical reaction kinetics. In general, a biochemical reaction proceeds with the intervention of living organisms (or cells) in the presence of nutrients of the medium under optimum conditions of temperature, pH etc. The cells grow by consuming the substrate and essential nutrients from the medium of fermentation. Remember that the growth pattern is not same for all types of the cells. The growth processes of interest to us have two different manifestations. The unicellular organisms (organisms that composed of one cell only), which eventually divide as they grow, increase in the number of cells (population growth) or increase the biomass, whereas the moulds [fungi that grow as a mesh of fine branched filaments (mycelium)] increase in size and hence density of the cells but not necessarily in numbers. Notice that two interacting systems are involved in the biochemical operation—the biological phase consisting of a cell population and the environmental phase or growth medium. Indeed, various parameters of the broth (a fluid culture medium) have been changed with the cell growth.
In the sample bioreactor, (i) the cell population is treated as one component solute with cell-to-cell homogeneity, (ii) there are no intracellular reactions occurring within the cell, and (iii) the growth phenomena are described based on a single limiting substrate. The mathematical model of such a biochemical reactor is commonly referred to as unstructured model. This model describes a condition called balanced growth. This is a quasi-steady state assumption, which requires that the environment of the biomass changes sufficiently slowly so that the biomass can adjust its internal composition to adapt to the changes. For the example CSTB, this is often a fairly good assumption because the cell population can adjust to a steady environment and achieve or closely approximate a state of balanced growth.
In the following, the cell population kinetics has been discussed for the unstructured models where balanced growth condition is assumed. In the component mass balance Equations (5.5) and (5.7), r1 and r2 represent the rate of cell growth and rate of substrate consumption respectively. The following form of equation normally represents the net rate of cell mass growth:
r1 = x...(5.10)
where is known as the specific growth rate or specific growth rate coefficient (time–1). We may think that is similar to the first-order reaction rate constant k [Equation (4.7)]. However, is originally not a constant. It is true that the microorganisms cannot grow without the supply of food. In the subsequent discussion, we will see that the specific growth rate is a function of the concentration of some
nutrient(s).
Now we wish to define the term yield. It is generally defined as the ratio of mass or moles of product formed to the mass or moles of reactants consumed. It is also called yield factor or yield coefficient or yield production coefficient. The yield Y of product P with respect to the reactant A is expressed by:
...(5.11)
For the case of bioreactor,
...(5.12)
It gives
...(5.13)
or
...(5.14)
Inserting Equation (5.10) into Equation (5.14), one obtains:
...(5.15)
Actually, the yield coefficient varies linearly with the substrate concentration (Ramaswamy, Cutright and Qammar, 2005) as:
Y(S) = a + bS...(5.16)
where a and b are positive constants. In the present case study, we have assumed that Y is a constant parameter.
Final Form of Modelling Equations
Substituting Equation (5.10) into Equation (5.5) and Equation (5.15) into Equation (5.7), the following equations are obtained respectively as:
...(5.17)
...(5.18)
We have assumed that the feed stream does not contain any biomass, i.e., xf = 0. Therefore, the bioreactor modelling equations finally get the following forms:
...(5.19)
...[from Equation (5.18)]
Specific Growth Rate
As previously mentioned, the specific growth rate is not a constant parameter; it is generally a function of the concentration of some nutrient(s). In the following, we will know a little more about it.
The general goal of a medium in the biochemical operations is to support handsome growth and/or high rates of product synthesis. It does not necessarily mean that all medium constituents or nutrients should be supplied to the reactor in great excess. Notice that excessive concentration of a nutrient may inhibit or even poison cell growth. Moreover, if the cells grow too extensively, their accumulated metabolic end products will often disrupt the normal biochemical processes of the cells. Therefore, it is common practice to control total cell growth by limiting the amount of one nutrient in the medium. Often, a single substrate exerts a dominant influence on rate of growth and this component is known as the growth limiting substrate, or more simply, the limiting substrate.
Now we will discuss the Monod model. This model is used to express growth based on a single limiting substrate S. The Monod model is commonly employed in the biochemical reactor modelling and is the basis of almost all growth models.
Monod model
If the concentration of the limiting substrate is varied, experimental study shows that the specific growth rate typically changes in a hyperbolic fashion, as Figure 5.4 shows. Obviously, the growth rate passes through various phases, such as high growth phase, low growth phase and finally cessation. The variation of specific growth rate with the concentration of the growth limiting substrate is well explained by an empirical equation proposed by Monod in 1942. This equation has the following form:
...(5.20)
where m is the maximum achievable specific growth rate and Km represents the limiting substrate concentration when the specific growth rate is equal to half of the maximum specific growth rate, as illustrated in Figure 5.4. These parameters are obtained experimentally and generally they do not have a direct physical interpretation (Bailey and Ollis, 1986).
Substitution of = (m/2) in Equation (5.20) and after simplification, we get Km = S. Marison (1988) has reported some typical values of Km. Note that the Monod equation is valid only for balanced growth and should not be applied when growth conditions are changing rapidly.
Monod Equation (5.20) has the same form as the Langmuir adsorption isotherm and the standard rate equation for enzyme-catalyzed reactions with a single substrate (Michaelis–Menten kinetics). If S is very large, Equation (5.20) yields:
...(5.21)
FIGURE 5.4 Dependence of on the concentration of the limiting substrate.
Inserting m in Equation (5.10), we have r1 mx...(5.22)
It reveals that the Monod description is similar to a first-order reaction when S is very high.
Similarly, if S is very small, the Monod equation gives:
...(5.23)
and the rate of cell growth is represented by the following form:
...(5.24)
Therefore, when S is very low, the Monod growth kinetics is similar to a second-order (bimolecular) reaction kinetics.
From some studies of the cell’s biochemistry, it is apparent that the Monod equation is a great over-simplification. However, there are variants to the Monod model that have also been used, such as the models by Tessier, Moser, Contois, etc., reported in Table 5.1. These models differ in their substrate dependence and some include terms to account for saturation due to high substrate concentration and inhibition due to product or a competing inhibitor. But these models do not differ significantly from the Monod model in the fact that they are empirical and represent all of the cellular processes with just a single equation for the specific growth rate.
Table 5.1 Kinetic structures of fermentation models
Monod (1942) Tessier (1942) Moser (1958) Contois (1959) Aiba et al. (1965)
Powell (1967)
Edwards (1970)
Peringer et al. (1972)
Jackson and Edwards (1975)
Olsson (1976)
Dourado and Calvet (1983) Williams et al. (1984)
S = substrate concentration, x = cell mass concentration, P = product concentration, Pf = inhibition constant, A = dissolved oxygen concentration, H+
= hydrogen ion concentration, K(any subscript) = constant, = specific growth rate, m = maximum specific growth rate.
5.4.3 Dynamic Simulation
In order to know the dynamic behaviour of the sample biochemical reactor, we have to simulate the developed model. The model structure of the CSTB is comprised of the following set of equations:
...(5.25)
For solving the above differential–algebraic system, the required data are given in Table 5.2. In the preceding chapters, we have used the fourth-order Runge–Kutta method (Chapter 3) and the second-order Runge–Kutta method (Chapter 4) to solve the ordinary differential equations. Here, the Euler method (detailed in Chapter 2) has been used to simulate the model. The bioreactor simulator is developed using Fortran (90) programming language and is given in Program 5.1. Considering different initial guesses of x and S, we obtain two steady state solutions as:
Equilibrium 1:...xs = 0.0...Ss = 4.0...(guess values: x = 0.0; S = 1.0) Equilibrium 2:...xs = 1.53735...Ss = 0.15653...(guess values: x = 1.0; S = 1.0) Here the subscript s indicates steady state. The unit of both x and S is g/litre.
Table 5.2 Steady state data for the bioreactor
m 0.53 h–1
Km 0.12 g/litre
D 0.3 h–1
Sf 4.0 g/litre
Y 0.4
Integration time interval = 0.005 h