## An Unstructured Model for Penicillin Production

In this case study, the mechanistic model of Bajpai and Reuss [36] was used as starting point for model development. The original model has been extended by including additional input variables such as agitation power, and aeration rate. Functional relationships among the process variables are summarized in Table 2.3 and all inputs and outputs are listed in Figure 2.1. A variety of mathematical representations have been suggested for describing certain biological behaviors by researchers referenced earlier in the text and others. We used the representations by Bajpai and Reuss [36] but readers are cautioned that several other representations may also be used to describe the penicillin fermentation process as we discussed earlier.

Unstructured Models

Mass balance equations can be summarized as follows.

Input Variables

Glucose Feed Temperature Glucose Feed Flow Rate Aeration Rate Agitator Power Input Coolant Flow Rate Acid/Base Flow Rate

Output Variables

Output Variables

Culture Volume Fermenter Temperature Generated Heat PH

Concentrations of Glucose Biomass Penicillin

Dissolved Oxygen Carbon Dioxide

Figure 2.1. Input/output structure of the process.

Table 2.3. Functional relationship among the process variables

Model Structure

X = f (X, S, Cl, h, T) S = f (X, S, CL, h, T) CL = f (X, S, CL, H, T) P = f(X, S, CL, H, T, P) C02 = f (X, h, T) h = f (X, h, T)

Biomass: The dependence of specific growth rate on carbon and oxygen substrates was assumed to follow Contois kinetics [36] to consider the biomass inhibition. The biomass growth has been described by Eq. 2.11 with Cx = X and finet = ¡x and the specific growth rate ¡1 being

in the original model [36]. The variables and parameters used are defined in Table 2.3 and 2.4.

In order to include the effects of environmental variables such as pH and temperature, biomass formation can be related to these variables by introducing their effects in the specific growth rate expression [61] to give:

This would in turn affect the utilization of substrate and the production of penicillin. Direct effects of pH and temperature on penicillin production

 Time: t (h)