where rQl is assumed to be constant and might be treated as a yield coefficient . During the product synthesis phase, when the rate of biomass formation is rather low, there is still significant heat generation associated with metabolic maintenance activities. Therefore, we have included the second term on the right hand side of Eq. 2.55 to account for the heat production during maintenance. Because the heat generation and C02 evolution show similar profiles, their production rate due to growth (dX/dt) and biomass (X) should have the same ratio as a first approximation. Based on this observation, rq2 is calculated and tabulated in Table 2.4. The energy balance is written based on a coiled type heat exchanger which is suitable for a laboratory scale fermentor :
The introduction of variables which are easy to measure yet important in terms of their information content has been very helpful in predicting other important process variables. One such variable is CO2 from which biomass may be predicted with high accuracy. In this work, CO2 evolution is assumed to be due to growth, penicillin biosynthesis and maintenance requirements as suggested by . The CO2 evolution is:
Here, the values of ai, ol-i and a.3 are chosen to give CO2 profiles similar to the predictions of .
The extended model developed consists of differential equations 2.11, 2.49, and 2.51-2.57 that are solved simultaneously.
Simulations have been carried out to check the performance of the simulator. In all runs, a batch operation has been followed by a fed-batch operation upon near complete depletion of the carbon source (glucose). This has been done by assigning a threshold value to glucose concentration which was chosen to be 0.3 g/L. The system switches to the fed-batch mode of operation when the level of glucose concentration reaches this threshold value. The predictions of the model under different conditions are compared with experimental data of Pirt and Righelato  and the simulation results of Bajpai and Reuss , Note that most of the parameters are functions of the strain, nature of the substrate and the environmental conditions like pH and temperature. The additional terms that were introduced increased the stiffness of the ordinary differential equations. For that reason, some of the parameter values are readjusted. These readjusted parameters are listed in Table 2.4.
Figures 2.2, 2.3, 2.4, 2.5, and 2.6 show the simulation results under normal operating conditions with the pH and temperature being controlled at 5.0 and 25°C, respectively. The model successfully predicted the concentration profiles of biomass (Figure 2.2), glucose (Figure 2.3), penicillin (Figure 2.4), dissolved oxygen (Figure 2.5), and carbon dioxide (Figure 2.6). Typical experimental data are also shown in Figure 1.4 for comparison. In the batch operation, glucose and oxygen are mainly used for biomass growth. In Figures 2.2, 2.3, 2.4, 2.5, and 2.6, phase I represents the lag phase where no biomass production is observed. Phase II represents the exponential growth phase where the specific growth rate is maximum and so is the substrate utilization rate. Phase III is the late exponential or early stationary phase where the operation is switched to fed-batch mode and penicillin production starts. At this stage, glucose and oxygen are used for both biomass growth and penicillin production. Phase IV is the stationary phase where biomass production is essentially negligible and penicillin production is high. When the concentration of penicillin reaches its high value and levels off, it is common practice to stop the operation. All phases are simulated successfully via the unstructured model.
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