Practical Work 2 - Part B :
Control of a sequential bioreactor

Preliminaries : This Practical Work uses version 3.0 of the scientific software Scilab , developed by INRIA and ENPC. All the installation files can be downloaded freely from the web page http://www.scilab.org.

The PW needs a Scilab command file simuseq.sce that one should store in his working directory.

In this part, we study several control laws for bioreactors working in sequential mode. It consists in a succession of cycles, each of them being constituted of 4 stages :

Stage 1 : the reactor contains a very small (but non null) volume of a solution of biomass (micro-organisms, bacteria, ...) and substrate.

Stage 2 : the reactor is progressively filled with a solution of high concentration $s_{in}$ of substrate. There is no output.

Stage 3 : when the reactor is filled at its maximum volume, the input rate if stopped and one waits (if it is necessary) until the concentration of the solution in the reactor (supposed to be homogeneous) has reached a minimal given threshold $s_{\min}$.

Stage 4 : after decantation, the liquid part (that can be now considered as depolluted) is emptied. It remains in the bottom of the reactor a small volume containing all the biomass.

Differently to continuous stirred reactors, for which input and output flows are identical, the volume of the solution inside the reactor is variable. The state of the reactor at a given time is given not only by the concentrations of biomass and substrate but also by the current filled volume :

1. The state of the reactor is represented by a vector in dimension $3$ :

$$x=\left(\begin{array}{c} s b V \end{array}\right)$$

$s$ and $b$ are the concentrations of the solution, respectively, in substrate and biomass. $V$ is the volume of the solution inside the reactor.

2. The manipulated variable is the input flow :

$$q \in [0,q_{\max}].$$

In the space $(s,b,V)$, the target to be reached (at stage 4) is the following set :

$${\cal T}= \{ (s,b,V) \; \vert \; s\leq s_{\min} \mbox{ et } V=V_{\max} \}$$

The dynamical equations of the system for the first three stages are :

$$({\cal S}): \left\{\begin{array}{lll} \dot{s} & = & \display... ...(s)b - \frac{q}{V}b [4mm] \dot{V} & = & q \end{array}\right.$$

where $c$ is the rate conversion factor ($c \leq 1$) of the reaction :

$$b + s \longrightarrow b$$

and $\mu()$ is the growth function, that can take two possible shapes :

• Monotonic growth : for instance Monod law :
  
  $$\mu_{M}(s)=\mu_{\max}\frac{s}{K+s}$$
  
  

• Inhibited growth : for instance Haldane law :
  
  $$\mu_{H}(s)=\mu_{\max}\frac{s}{k+s+s^{2}/K_{I}}$$
  
  

In order to simulate under Scilab the trajectories of the system $({\cal S})$, define first the function $\mu()$ :

deff('[m]=mu(t,s)','m=monod(t,s)');if the Monod law is chosen,

deff('[m]=mu(t,s)','m=haldane(t,s)');if the Haldane one is chosen,

For the control law, define the function $u()$, for instance for the maximal flow :
deff('[q]=u(t,x)','V=x(3);..
if V<Vmax then q=qmax;..
else q=0;..
end;');

For the parameters, one can choose the following values :

$$\begin{array}{c} \mu_{\max}=0.072 h^{-1}, \; K=2 mgl^{... ...ax} = 50 m^{3}h^{-1}, \; s_{\min}=10 mgl^{-1} \end{array}$$

and for the initial condition :

$$s(0)=50 mgl^{-1}, \; b(0)=1.3 10^{4} mgl^{-1}, \; V(0)=5 m^{3}$$

For this, define in Scilab the following variables :

mumax=0.072; K=2; Ki=50; c=0.5; Sin=200; Vmax=50; qmax=50; Smin=10;
x0=[50;13000;5];

Define also a length of the integration interval, for instance 3 hours :

tmax=3;

The simulations are then launched with the instruction :

exec("simuseq.sce");

A window then appears :

The history of the three state variables is represented on the three first graphs. The last graph gives a visual representation of the state of the reactor at different times :
- the height of the bars is proportional to the volume.
- the higher is the substrate concentration, the more red are the bars.

Question 1 : For the two kinds of growth functions, launch simulations and determine the time necessary for reaching the target ${\cal T}$. What comments can be made ?

Question 2 : For the Haldane law, determine the value $s^{*}$ maximizing the growth function $\mu$ :

Write a control law leading the substrate concentration as fast as possible to the value $s=s^{*}$, and then maintaining the concentration constant until the maximal volume is reached. Simulate and determine the time necessary for reaching the target ${\cal T}$. Compare this time with the one required by the previous control law. Choose the new initial condition $s(0)=5 mgl^{-1}$, and compare again. What can be concluded ? Explain.

END

Practical Work sheet prepared by Jérôme Harmand and Alain Rapaport.