Static and Dynamic Analysis of a Continuous Bioreactor Model for the Production of Biofuel from Refinery Wastewater Using Rhodococcus opacus

Abstract

This paper studied the bifurcation phenomena that can occur in a continuous bioreactor for lipid-rich biomass production using Rhodococcus opacus that feeds on refinery wastewater. An unstructured model of the bioreactor was developed based on experimentally validated studies reported in the literature. The analysis of the unsteady-state model was carried out both analytically and through numerical simulations. It was shown that the inhibition effect of the biomass growth rate can lead to the occurrence of static bistability in the model. Periodic behavior was also found for some range of model parameters. The effect of model parameters on the productivity of the bioreactor was also investigated. The analysis carried out in this paper allowed for the detection of unsafe operating regions in the bioreactor which, would help in the attenuation of these operational problems in the early stage of process design.

1. Introduction

Some microorganisms are known to be natural oil producers in their cellular compartments. Microorganisms that accumulate more than 20% w/w of lipids on a cell dry weight basis are considered as oleaginous microorganisms [1,2]. Research on the use of such microorganisms to produce biodiesel has been quite active in the past few years [2]. This approach is poised to be more eco-friendly and sustainable compared to biofuels production using lignocellulosic biomass. The latter strategy utilizes materials that compete for human food, whereas raw materials that are considered for the production of single cell oil-derived biodiesel can utilize a variety of industrial wastes as substrate for lipid-rich biomass production. The use of oleaginous microorganisms in wastewater treatment is also known to be inexpensive and environmentally friendly and has a short fermentation period [1].

R. opacus is an aerobic oleaginous bacterium with great potential for triacylglyceride accumulation and efficient utilization of various substrates [3]. This microorganism has recently attracted growing attention in the literature because of its capability to produce lipid-rich biomass through the use of recalcitrant organics in wastewater as carbon substrate. A number of studies on R. opacus for lipid production have been performed using substrates such as sugar beet molasses, sucrose, dairy, and refinery wastewater as the sole carbon source [4,5,6,7,8,9,10]. In this regard, Paul et al. [6] investigated a batch system for the production of lipid-rich biomass by R. opacus utilizing refinery wastewater as the sole substrate. The authors also carried out kinetics modeling by fitting the data to a number of substrate utilization kinetic models available in the literature.

In this paper, we complement the batch studies carried out in [6] on the use of R. opacus in refinery wastewater, and we carry out a study of the potential production on a continuous scale. We developed a simple model for the bioreactor using experimentally validated kinetics [6] and investigated the bioreactor static and dynamic behavior using standard bifurcation techniques [11].

Continuous bioreactors are known to be highly nonlinear and therefore can present operational problems that can manifest themselves in the form of steady-state multiplicity and/or undesired oscillations [12,13]. Such nonlinear phenomena are generally detrimental to the operation of the bioreactor. The objective of this paper is to examine the operational problems, if any, that may be encountered during the continuous operation of the bioreactor. Another objective is to examine the productivity of the bioreactor. Again, we will delineate the operating regions that may be safe and/or optimal.

The rest of the paper is structured as follows. In Section 2, the model is presented, followed in Section 3 by an analysis of its positivity and boundedness of its solutions. In Section 4, the stability of the model is discussed, and Section 5 presents numerical simulations to support and explain the theoretical analysis.

2. Process Model

We consider a continuous stirred tank bioreactor where the substrate with feed concentration $S_f$ is degraded to biomass $\left(X\right)$ and products by R. Opacus. We assume clean feed conditions. The unsteady-state mass balances of substrate and biomass are given by the following equations:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}S\left(t\right)}{\mathrm{d}t}& =D(S_f-S)-q\left(S\right)X\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}X\left(t\right)}{\mathrm{d}t}& =-DX+\mu \left(S\right)X\hfill \end{array}$$

(2)

where t is time in hours $\left[h\right]$, D [h${}^{-1}$] is the bioreactor dilution rate, $S\left(t\right)$ [g/L] is the substrate concentration, $X\left(t\right)$ [g/L] is the biomass concentration, $q\left(S\right)$ [h${}^{-1}$] is the specific substrate utilization rate, and $\mu \left(S\right)$ [h${}^{-1}$] is the biomass specific growth rate.

It should be noted that a batch model of the same system under study was experimentally investigated before in [6]. In this regard, the authors in [6] investigated the utilization of refinery wastewater containing complex hydrocarbons as the sole substrate for lipid-rich biomass production by R. opacus in the batch system. They concluded that their experimental results suggested that refinery wastewater with a high content of recalcitrant organics can be used as the sole substrate for lipid-rich biomass production by R. opacus. This, of course, adds significant value to the wastewater treatment process. Moreover, among the various possible expressions of $q\left(S\right)$ and $\mu \left(S\right)$, the authors [6] found that the following expressions best fitted their experimental data:

$$\begin{array}{c}\hfill q\left(S\right)=\frac{q_{m}S}{q_s+S+\frac{S^2}{q_i}}\end{array}$$

(3)

and

$$\begin{array}{c}\hfill \mu \left(S\right)=\frac{{\mu }_{m}S}{k_s+S+\frac{S^2}{k_i}(1+\frac{S}{k_s})}.\end{array}$$

(4)

Here, $q\left(S\right)$ follows the well known Haldane substrate inhibition expression, and $\mu \left(S\right)$ is the Edward expression that also includes a substrate inhibition term [6]. The expression $q_m$ [h${}^{-1}$] is the maximum specific substrate utilization rate, $q_s$ [g/L] is the half-saturation constant, and $q_i$ [g/L] is the inhibition constant. For $\mu \left(S\right)$, the term ${\mu }_m$ [h${}^{-1}$] is the maximum biomass specific growth rate, $k_s$ [g/L] is the half-saturation constant, and $k_i$ [g/L] is the inhibitory concentration of the substrate.

3. Positivity and Boundendness

For the proposed model to be biologically reasonable, it is necessary to show that all the state variables are positive and bounded.

Theorem 1.

Let initial conditions ($S\left(0\right)$ and $X\left(0\right)$) be non-negative. Then the solutions of the model coupled Equations (1) and (2) are positively invariant, i.e., belong to the region $(S,X)\in {\mathbb{R}}_{+}^2$, and they are uniformly bounded for all $t\ge 0$.

Proof. 

First, we prove that $X\left(t\right)>0$ for all $t>0$ as follows: From Equation (2), we have

$$\begin{array}{cc}\hfill \frac{dX}{X}& =(-D+\mu )dt,\mathrm{with}X\ne 0.\hfill \end{array}$$

(5)

Taking the integral on both sides leads to

$$\begin{array}{c}\hfill \int \frac{dX}{X}=\int (-D+\mu )dt,\end{array}$$

(6)

Thus

$$\begin{array}{c}\hfill X=X\left(0\right)exp\left({\int }_0^t\left((-D+\mu )d\tau \right)\right)>0\mathrm{for}\mathrm{each}t\ge 0\end{array}$$

(7)

Consider $S=0,X\ge 0$, then $\frac{dS\left(t\right)}{dt}=D{S}_f>0$, so the vector field of S is pointed inside ${\mathbb{R}}_{+}^2$. We conclude then that all trajectories starting in ${\mathbb{R}}_{+}^2$ remains in it.

Boundedness: Following the procedure used in [14], multiplying Equation (1) by $\mu \left(S\right)$ and Equation (2) by $q\left(S\right)$ and combining the two equations, we obtain

$$\begin{array}{c}\hfill \mu \frac{dS}{dt}+q\frac{dX}{dt}=D(\mu S_f-(\mu S+qX)).\end{array}$$

(8)

Let $Z\left(t\right)=\mu S+qX$, we have

$$\begin{array}{c}\hfill \frac{dZ\left(t\right)}{dt}=D(\mu S_f-Z\left(t\right)),\end{array}$$

(9)

which can be rewritten as

$$\begin{array}{c}\hfill \frac{dZ\left(t\right)}{dt}+DZ\left(t\right)=D\mu S_f.\end{array}$$

(10)

Equation (10) is a linear differential equation of the first order. Using the integrating factor method, we have $v\left(t\right)=exp(\int Ddt)=exp\left(Dt\right)$. Thus, multiplying Equation (10) by the integrating factor $v\left(t\right)$, we obtain

$$\begin{array}{c}\hfill exp\left(Dt\right)·[\frac{dZ\left(t\right)}{dt}+DZ\left(t\right)]=exp\left(Dt\right)·D\mu S_f.\end{array}$$

(11)

This leads to

$$\begin{array}{c}\hfill \frac{d}{dt}[exp\left(Dt\right)·Z\left(t\right)]=exp\left(Dt\right)·D\mu S_f.\end{array}$$

(12)

Integrating both sides of Equation (12) with respect to t leads to

$$\begin{array}{c}\hfill Z\left(t\right)=\mu S_f+(Z\left(0\right)-\mu S_f)exp(-Dt)>0\mathrm{for}\mathrm{all}t\ge 0.\end{array}$$

(13)

Therefore

$$\begin{array}{c}\hfill max\{Z\left(0\right),\mu S_f\}\ge Z\left(t\right)\ge min\{Z\left(0\right),\mu S_f\}>0\mathrm{for}\mathrm{all}t\ge 0\end{array}$$

Thus, we conclude that the model solutions $\left(X\right(t),S(t\left)\right)$ are uniformly bounded. This completes the proof of Theorem 1. □

Model Equilibria

In this section, we investigate the existence of the model’s equilibrium points as a function of the dilution rate $\left(D\right)$ selected, for convenience, as the main bifurcation parameter. The equilibrium points of Equations (1) and (2) are solutions of the following system of algebraic equations:

$$\begin{array}{cc}\hfill & D(S_f-S)-qX=0\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & -DX+\mu X=0\hfill \end{array}$$

(15)

Obviously, the washout condition $E_0:(S=S_f,X=0)$ is always an equilibrium point of the given model for all $D>0$. For the nontrivial solution (i.e., $X\ne 0$), Equation (15) yields necessarily that $\mu =D$. Substituting in Equation (14) yields

$$\begin{array}{cc}\hfill & X=\frac{\mu }{q}(S_f-S)\hfill \end{array}$$

(16)

It is clear that $X>0$ if and only if $S<S_f$ holds. Substituting for the expression of $\mu$ (Equation (4)), we obtain the S-component of the equilibrium point, which satisfies the following polynomial:

$$\begin{array}{c}\hfill D{\mathit{S}}^3+D{k}_s{\mathit{S}}^2+(D-{\mu }_m)k_s{k}_i\mathit{S}+D{k}_i{k}_s^2=0\end{array}$$

(17)

In order to check the existence of real and positive roots of Equation (17), we apply Descartes rule of signs [15], which stipulates that the number of positive real zeros in a polynomial function is the same or less than by an even number as the number of changes in the sign of the coefficients. It can be seen that when $D>{\mu }_m$, there are no sign changes in Equation (17) and, therefore, there is no positive solution. For $D<{\mu }_m$, there are two sign changes and, therefore, the possible number of real and positive solutions is either 2 or 0.

Vieta’s formulas [15] can also be useful for a further analysis of Equation (17). Let $S_1$, $S_2$, and $S_3$ be the roots of Equation (17). Then we have:

$$\begin{array}{c}\hfill S_1+S_2+S_3=-k_s<0\\ \hfill S_1.S_2.S_3=-k_i{k}_s^2<0\end{array}$$

There are two possible cases here:

(1)

Equation (17) has (exactly) two complex-conjugate roots of the form $S_{1/2}=a\pm ib$, where $i=\sqrt{-1}$ and $b\ne 0.$ Then:

$$\begin{array}{c}\hfill S_1.S_2=a^2+b^2>0,\end{array}$$

and the second Vieta’s formula implies that $S_3<0$. Thus, in this case, Equation (17) has no positive roots.

(2)

Equation (17) has (only) real roots. If there are exactly two positive roots $(S_1,S_2>0)$, then by applying Vieta’s formulas, we have $(S_3<0)$:

$$\begin{array}{c}\hfill 0<S_1+S_2=-k_s-S_3\\ \hfill 0<S_1.S_2=-\frac{k_i{k}_s^2}{S_3}.\end{array}$$

Thus, $S_3<-k_s$, i.e., $|S_3|>|k_s|$, and $S_1$, $S_2$ are the solutions of the (quadratic) equation:

$$\begin{array}{c}\hfill S^2+(k_s+S_3)S-\frac{k_i{k}_s^2}{S_3}=0.\end{array}$$

Although the roots of Equation (17) can be obtained analytically, they are complicated and not easily amenable to analytical manipulations. For this reason, the numerical solution of this cubic equation is be carried out in a later section using the efficient AUTO-based continuation techniques.

4. Stability of Equilibria

Having determined the equilibrium points of the model, the next step is to check their stability and carry out a classification of these equilibrium points. This analysis has important implications on the operation of the bioreactor, as it allows us to establish the conditions under which the dangerous washout solution is stable. It also allows us to see how the equilibrium points are approached when small perturbations are introduced in the process. An important part of the stability of dynamical systems deals with asymptotic properties of solutions and the trajectories, i.e., the long-term behavior of the system. Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium. The asymptotic stability of equilibrium points of a non-linear system is often examined using the corresponding behavior of the Jacobian matrix (J). More precisely, if all eigenvalues of (J) are negative real numbers or complex numbers with negative real parts, then the equilibrium point is asymptotically stable. The analysis of eigenvalues of the Jacobian matrix is is also useful in classifying the nature of equilibrium points. These points can be nodes (attracting or repellent), saddles, foci, or centers [16].

For the model equations (Equations (1) and (2)), the Jacobian matrix is given by

$$\begin{array}{c}\hfill J=\left[\begin{array}{cc}{J}_{11}& J_{12}\\ J_{21}& J_{22}\end{array}\right]\end{array}$$

(18)

where

$$J_{11}=-D-q^{\prime }\left(S\right)X,J_{12}=-q,J_{21}={\mu }^{\prime }\left(S\right)X,J_{22}=-D+\mu$$

(19)

and $q^{\prime }\left(S\right)$ and ${\mu }^{\prime }\left(S\right)$ are, respectively, the first derivatives of q and $\mu$.

4.1. Stability of Washout Solution

For $E_0:(S=S_f,X=0)$, substituting in Equation (19), the eigenvalues of the Jacobian matrix can be found using the simple equation

$$\begin{array}{c}\hfill det(J-\lambda I)=det\left(\left[\begin{array}{cc}-D-\lambda & -q\\ 0& -D+\mu -\lambda \end{array}\right]\right)=0,\end{array}$$

(20)

which yields

$$(\lambda +D)(\lambda +D-\mu )=0$$

(21)

The eigenvalues of J are therefore ${\lambda }_1=-D$, ${\lambda }_2=-D+\mu$. Therefore, the washout solution is stable provided that

$$\begin{array}{c}\hfill D>\mu (S=S_f)=\frac{{\mu }_m{S}_f}{k_s+S_f+\frac{S_f^2}{k_i}(1+\frac{S_f}{k_s})}.\end{array}$$

(22)

It can also be noted that, because the two eigenvalues are real, then the washout solution is always a node.

4.2. Stability of the Non-Trivial Solution

For any non-trivial solution, the characteristic equation is defined by

$$\begin{array}{c}\hfill det(J-\lambda I)=det\left(\left[\begin{array}{cc}{J}_{11}-\lambda & J_{12}\\ J_{21}& J_{22}-\lambda \end{array}\right]\right)=0,\end{array}$$

(23)

which yields

$$\begin{array}{c}\hfill {\lambda }^2+C_1\lambda +C_2=0\end{array}$$

(24)

where $C_1=-(J_{11}+J_{22})$, and $C_2=J_{11}{J}_{22}-J_{21}{J}_{12}$. Following the Routh–Hurwitz criterion [16], it suffices to show that $C_1>0$ and $C_2>0$ to conclude that all the roots of Equation (24) have negative real parts. Because at the nontrivial solution $J_{22}=-D+\mu =0$, then the first condition $C_1=-(J_{11}+J_{22})>0$ is reduced to $J_{11}<0$ or, equivalently,

$$\begin{array}{c}\hfill -D-q^{\prime }\left(S\right)X<0\end{array}$$

(25)

Substituting for the expression of X (Equation (16)) into Equation (25) yields

$$-D-q^{\prime }\left(S\right)\frac{\mu }{q}(S_f-S)<0$$

(26)

Because $\mu =D$, this is reduced to

$$\mu (1+\frac{q^{\prime }\left(S\right)(S_f-S)}{q})>0$$

(27)

or, equivalently,

$$q+q^{\prime }\left(S\right)(S_f-S)>0$$

(28)

Substituting for expression of $q\left(S\right)$ (Equation (3)) and its derivative $q^{\prime }\left(S\right)$ yields

$$2S^3+(q_i-S_f)S^2+q_i{q}_s{S}_f>0$$

(29)

The second condition $C_2=J_{11}{J}_{22}-J_{21}{J}_{12}>0$ is reduced to $J_{21}{J}_{12}<0$. Since $J_{12}=-q$, $J_{21}={\mu }^{\prime }\left(S\right)X$ and substituting for the expression of X (Equation (16)), then the condition $J_{21}{J}_{12}<0$ is reduced to

$$\mu \left(S\right){\mu }^{\prime }\left(S\right)(S_f-S)>0$$

(30)

Substituting for the derivative of $\mu$ (Equation (4)) yields

$$2S^4+(k_s-2S_f)S^3-k_s{S}_f{S}^2-k_i{k}_s^{2}S+k_i{k}_s^2{S}_f>0$$

(31)

The two Equations (29) and (31) provide the conditions for the stability of the nontrivial steady state.

4.3. Hopf Bifurcation Analysis

In this section, we study the conditions under which the model (Equations (1) and (2)) exhibits Hopf bifurcation. A nonlinear model can exhibit periodic or non-periodic oscillations [11,12,13]. Solutions that form a closed curve for some model parameter values are called limit cycle (or periodic orbit). This is associated with bifurcation from equilibria to limit cycles. This bifurcation is commonly called a Hopf bifurcation [13].

The conditions for the two dimensionless system to have a Hopf bifurcation are that:

$$\begin{array}{c}\hfill J_{11}+J_{22}=0,J_{11}{J}_{22}-J_{21}{J}_{12}>0\end{array}$$

(32)

These conditions evaluated along the washout solution $(S=S_f,X=0)$ yield

$$\begin{array}{c}\hfill J_{11}+J_{22}=-2D+\mu =0,J_{11}{J}_{22}-J_{21}{J}_{12}=-D(-D+\mu )>0\end{array}$$

(33)

These equations are incompatible, as they lead to

$$\begin{array}{c}\hfill \mu =2D,\mathrm{and}\mu <D\end{array}$$

(34)

We conclude, therefore, that a Hopf point can not occur on the washout solution branch.

Evaluating the Hopf conditions on the nontrivial steady state, the conditions are equivalent to Equations (29) and (31) with the change of the inequality sign:

$$2S^3+(q_i-S_f)S^2+q_i{q}_s{S}_f=0$$

(35)

$$2S^4+(k_s-2S_f)S^3-k_s{S}_f{S}^2-k_i{k}_s^{2}S+k_i{k}_s^2{S}_f>0$$

(36)

These two conditions form the Hopf conditions. A simple analysis of Equation (35) using Descartes rule of sign reveals that whenever $S_f<q_i$, Equation (35) does not exhibit a change of sign, and therefore no positive solutions (i.e., no Hopf point) can exist, which is quite an interesting result. In contrast, if $q_i<S_f$, there are two changes in the sign of Equation (35), and therefore up to two Hopf points can be found. We can carry out the analysis further and compute the discriminate of the cubic equation (Equation (35)):

$$\mathrm{\Delta }=-{(q_i-S_f)}^3{q}_i{q}_s{S}_f-27{\left(q_i{q}_s{S}_f\right)}^2$$

(37)

The roots of a cubic equation with real coefficients are real and distinct if the discriminant is positive.

$$\mathrm{\Delta }>0\to {(q_i-S_f)}^3+27q_i{q}_s{S}_f<0,$$

(38)

which yields

$$q_s<-\frac{{(q_i-S_f)}^3}{27q_i{S}_f}$$

(39)

This inequality defines a convenient region that, together with (Equation (36)), delineate the domain where two Hopf points exist, as illustrated numerically in the next section.

5. Numerical Simulations

In the following section, we provide explicit examples of the results that were derived in the previous section using the nominal values of model parameters shown in

Table 1. Nominal values of model parameters [6].

${\mathit{k}}_{\mathit{i}}$	3.479 g/L
$k_s$	3.373 g/L
$q_m$	0.087 h${}^{-1}$
$q_i$	2.038 g/L
$q_s$	2.104 g/L
$S_f$	3 g/L
${\mu }_m$	0.046 h${}^{-1}$

. These values were taken from the experimental work [6].

Figure 1 shows the bifurcation diagram using the dilution rate as the main bifurcation parameter. Figure 1a shows the variations of substrate concentration, and Figure 1b shows the profile of productivity $Pr=DX$. In addition to the horizontal washout branch, the diagram of Figure 1a exhibits a static limit point $LP$ at $D=0.0128$ [h${}^{-1}$] on the nontrivial static branch. Therefore, a region of bistability is expected between the static limit point and the crossing of the nontrivial branch with the washout branch. The value of dilution rate at this crossing is defined by Equation (22). Operating the bioreactor in the region of bistability may lead to washout conditions unless feed conditions are controlled. The profile of productivity (Figure 1b) shows that the productivity increases montonically and reaches a maximum value at dilution rate $D=0.0108$ [h${}^{-1}$] before decreasing. It can be seen that the maximum production occurs at a stable operating point.

In order to examine the Hopf bifurcation of the model, Figure 2 shows the plot corresponding to Equation (39) that delineates the region of Hopf points. Figure 2 divides the $(q_i,q_s)$ domain into a region where two Hopf points are expected and a region of no Hopf points. Figure 3a shows an example of the bifurcation diagram for the value of inhibition constant $q_i=0.10$ and the rest of parameters of Table 1. It can be seen that, in addition to the region of bistability around the static limit point, the diagram is characterized by the appearance of two Hopf points at dilution rates $D=0.00686$ [h${}^{-1}$] and $D=0.0109$ [h${}^{-1}$]. Stable periodic branches are expected in this region. An example of a limit cycle is shown in Figure 4 for $D=0.1$ [h${}^{-1}$]. Figure 3b shows the corresponding productivity profile. Between the two Hopf points, the productivity should be calculated as a time average of the periodic oscillations. The productivity still reaches a maximum before decreasing. As was noted in a number of studies [17,18], the average efficiency or productivity within the periodic regimes is never larger than that of the static branch. Therefore, the performance between the Hopf points is lower than that in the static branches, and therefore the operation of the bioreactor between these points should be avoided.

The effect of the different model parameters on the productivity are shown in Figure 5a–c. The unstable regions were omitted from the graphs. Figure 5a shows the effect of an increase in substrate feed concentration. An increase in $S_f$ from 3 to 4.5 [(g/L)] increases the productivity. This range of increase in $S_f$ was reported in the batch experiments of [6]. The dilution rate at the maximum value $D_{max}$ of productivity also increases. Figure 5b shows the effect of the maximum specific substrate utilization rate $q_m$. An increase in $q_m$ from 0.087 to 0.1 [h${}^{-1}$] decreases the productivity, whereas the maximum productivity occurs at larger dilution rate. The effect of an increase in the saturation constant from $q_s=2.104$ to 2.50 [g/L] increases the productivity and decreases $D_{max}$. Finally, Figure 5c shows that an increase in the inhibition constant from $q_i=2.038$ to 3 [g/L] decreases the productivity and decreases the value of $D_{max}$. It should be noted that, apart from changes in $S_f$, moderate changes (in the order of $\pm 20\%$) in the rest of the parameters yielded only marginal effects on the productivity.

6. Conclusions

This paper presented a study on the potential production on a continuous scale of lipid-rich biomass from waste refinery using R. opacus. We studied the stability of the continuous bioprocess based on an experimentally validated kinetic model. We analytically uncovered the main static and dynamic behavior of the model. In addition to washout conditions, the model predicted, for some range of dilution rates, the existence of bistability and a periodic behavior. Both bistability and oscillations are detrimental to the safe operation of the bioreactor and should be avoided. The productivity of the bioreactor was found to exhibit a maximum value as a function of dilution rate. Depending on the model parameters, the maximum productivity was found to be followed by a region of instability or was encompassed by two zones of instability. Hence, a special procedure must be deployed to reach it. That procedure could simply be batch cultivation to enter the stable domain of cell concentration $\left(X\right)$ before operating the bioreactor in continuous mode.

Sensitivity analysis was also carried out for the effect of model parameters on the productivity of the bioreactor.

The model examined in this paper is quite simple. It is an unstructured and two-dimensional model that simplified the analytical manipulations to a great extent. However, the major strength of the model is that it was experimentally validated, which provided the results of the analysis with a good deal of credibility.