## Contois Equation of Growth

### Biochemical Engineering

Problem Statement: The following Contois equation of growth describes high-density cell fermentation where cell growth is suppressed when the fermentor becomes too crowded.

```    u=um*s/(K*x+s)
```
In the above equation, x is the cell concentration, s is the limiting substrate concentration, and u is the specific growth rate. The kinetics of formation of the desired product is mixed (i.e., partially cell growth related and partially nongrowth related). A cell-free feed stream continuously enters the fermentor, and the fermentor content is also continuously withdrawn. Assume that the limiting substrate is utilized only for cell growth but not for product formation.
1. Write the dynamic equations for the bioreactor. Identify the operating parameters. List the equations that need to be solved in order to find the operating conditions that maximize cell productivity.

Solution:

2. Analytically find the steady-state conditions in the bioreactor.

Solution:
Set each of the above dynamic equations to 0: dx/dt=ds/dt=dp/dt=0.

3. List all the equations that need to be solved in order to find the operating condition(s) that maximize product productivity.

Solution:
In addition to dx/dt=ds/dt=dp/dt=0 from the last part, add d(D*p)/dD=d(D*P)/dsf=0. In practical cases, both D and f are constrained between 0 and an upper limit.

4. Someone proposed that a repeated fed-batch mode of operation might give a higher productivity. Do you agree? Why?

Solution:
No. Product productivity is proportional to cell concentration and cell growth rate. A distinct advantage of a continuous bioreactor is that it is more productive. Remember that a fed-batch fermentor utilizes only a fraction of the available volume, especially at the beginning. You can also convince yourself through dynamic simulation.

5. Now, let's install a cell recycle loop. Write the dynamic equations for this configuration.

Solution:
Note that normally only the cells are concentrated (with centrifugation, filtration, or sedimentation) and returned back to the bioreactor. Soluble substrate and product are not concentrated, and their dynamic equations remain unchanged from Part (a.

6. Describe the effect of cell recycle on steady-state cell, substrate, and product concentrations.

Solution:

```  Cell concentration:      up    (See the Mathcad worksheet below to be exact.)
Substrate concentration: down
Product concentration:   up
```

7. Since cell growth is suppressed at high cell concentration, as mentioned earlier in the problem, do you expect product productivity to increase with the installation of a cell recycle loop? Why?

Solution:
Cell recycle increases cell concentration in the bioreactor and pushes the washout dilution rate to a higher level. Remember that the product is synthesized by cells; thus, more cells, more product (both concentration and productivity) -- whether or not cell growth is suppressed at high cell concentration.

Contois Equation of Growth