Optimization of a Secondary Metabolite Fermentation Process: Effect of Cost Factor on the Optimal Feed Rate Control

Abstract

Optimisation of a fed-batch fermentation process is usually done by using the calculus of variations to determine an optimal feed rate profile. The obtained optimal feed rate profile consists of sequences of maximum, minimum and singular feed rates. In this paper, the optimal feed rate control of a secondary metabolite process was investigated under the different cost factors. A material balance model was used to describe the secondary metabolite production process, where the specific growth rate and the specific product formation rate were assumed to follow the substrate inhibition kinetic. This optimisation problem was formulated as a free final time problem in the optimal control literature where the control objective was to maximise the product at the end of the batch. It was shown mathematically that without a cost factor, the process was operated at the condition where the ratio of the specific product formation rate and the specific growth rate was maximum. With the presence of the cost factor, the operating condition was bounded by two extreme situations. One was at the maximum specific growth rate, where the cost factor is high and the operating time is at utmost important. The other was at the maximum ratio of the specific product formation rate and the specific growth rate, where the operating time was less important and the maximum product was required at the end of the batch. With the moderate cost factor, the process started with growth and gradually changed to product formation.