Optimal Temperature Profile for Penicillin Fermentation

The two-point boundary-value problem of Eq. (5.161) can now be expressed in terms of the orthogonal collocation method as 

169) represents a set of (n + 2) simultaneous nonlinear equations, two of which correspond to the boundary conditions (the first and the last equation) and should be neglected when solving the set. The solution of the remaining n nonlinear equations can be obtained using Newton s method for nonlinear equations. 

The orthogonal collocation method is more accurate than either the finite difference method or the collocation method. The choice of collocation points at the roots of the orthogonal polynomials reduces the error considerably. In fact, instead of the user choosing the collocation points, the method locates them automatically so that tlie best accuracy is achieved. 

Example 5.5 Solution of the Optimal Temperature Profile for Penicillin Fermentation. Apply the orthogonal collocation method to solve the two-point boundary-value problem arising from the application of the maximum principle ofPontryagin to a batch penicillin fermentation. Obtain the solution of this problem, and show the profdes of the state variables, the adjoint variables, and the optimal temperature. The equations that describe the state of the system in a batch penicillin fermentation, developed by Constantinides et al.(6], are  

These parameter-temperature functions ai-e inverted parabolas that reach their peak at 30 °C for bf and 20°C for by. The values of the parameters decrease by a factor of 2 over a 10°C change in temperature on either side of the peak. The inequality, / , 0, restricts the values of the parameters to the positive regime. These functions have shapes typical of those encountered in microbial or enzyme-catalyzed reactions. 

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Example 5.5 Optimal Temperature Profile for Penicillin Fermentation... 

Figs. E5.5a to E5.5d show the profiles of the system variables and the optimal control variable (temperature). For this particular formulation of the penicillin fermentation, the maximum principle indicates that the optimal temperature profile varies from 30 to 20°C in the pattern shown in Fig. E5.5c/. 

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