Figure 7.2. Profiles of St (dashed curve ABCDE) and overall product yields for repeated batch operation without recycle (lower solid curve) and repeated fed-batch operation (upper solid curve). Singular control in each cycle of a repeated fed-batch operation terminates at a locally stable limit point [Si lying on the upper branch (portion ABC) of the limit point curve ABCDE], The non-labeled dashed curve represents the upper (open) bound on the profiles of the overall product yield for repeated batch operations with recycle [447].

curve are unstable. Parulekar [447] has established that fed-batch operations terminating at an unstable limit point are not feasible. The profiles of overall product yield in Figure 7.2 illustrate the superiority of cyclic fed-batch operation with singular control terminating at the stable limit point over cyclic batch operations. These profiles also reveal the substantial improvement in yield that can be obtained with recycle in a cyclic batch operation.

The optimizations based on the highly lumped models such as the ones considered in Eqs. 7.66 and 7.67 may be sensitive to variations in the kinetic parameters in these, some of which have significant uncertainty. For the kinetic parameters considered in Eq. 7.69, the predicted maximum product yield exceeded the theoretical maximum yield beyond certain Sf-This indicates that these parameter values are not accurate enough to be used for fed-batch optimization. Sensitivity of the objective function (max imum product-to substrate yield at the termination of a cycle in a cyclic fed-batch operation, Pf/Sp) to variations in the kinetic parameters was therefore examined. The effects of ±25% variations in each of the kinetic parameters on the maximum product yield are illustrated in Table 7.1. The kinetic parameters, with the exception of the parameter being varied, were assigned the values listed in Eq. 7.69. In each case, a repeated fed-batch operation provided the maximum Pj for a given Sf- For both examples of kinetics, the maximum product yield increased with increases in K\, em and biomass-to-substrate yield {YX/s — 1 /p) and decreases in K2 and fim. The maximum product yield increased with increased cell growth inhibition by the desired product (signified by a decrease in or an increase in c*i) and with reduced inhibition/repression of product synthesis by the desired product (signified by an increase in or a decrease in Bm). The maximum product yield is very sensitive to fim, em, p, K3 (example 1), and am (example 2), moderately sensitive to K4 (example 1) and 3m (example 2), and less sensitive to K\ and K2. The results in Table 7.1 clearly demonstrate the need for accurate estimation of the maximum specific cell growth and product formation rates, biomass-to-substrate yield, and product-inhibition coefficients and the necessity of frequent updating/retuning of kinetic parameters via on-line estimation when lumped kinetic models are employed for bioprocess optimization.

Case 3. The three specific rates are functions of S and P but have no linear relations among them.

The objective function in Eq. 7.3 is considered to be independent of Xj and tj, both of which are not specified, and A5 is trivial as a result. Further, g in Eq. 7.3 is considered to be trivial. Termination of bioreactor operation in a particular cycle must occur in singular control or batch mode, the final reactor volume being Vm in either case. It has been shown that A2 is trivial during singular control. Triviality of ho and dhi/dt during singular control then provides the following necessary and sufficient condition for admissibility of singular control and description of singular arc

The feeding policy during singular control (obtained from d?h\/dt2 = 0) is described in Eq. 7.58 with a, (3 and 7 being defined as

The projections of the batch bioreactor trajectories on an S — P plane will in general be nonlinear due to the nonlinear nature of a and e and may

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