Chemical Engineering Scimce, Vol. 47. No. 15/16, pp. 3869-3884.1992. wa9-2509p2 55.00 + 0.00 Printed in zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Great Britain. 0 1992 PergPmon Press Ltd OPTIMAL MODE OF OPERATION OF BIOREACTOR FOR FERMENTATION PROCESSES zyxwvutsrqponmlkjihgfedcbaZYXWV JAYANT M. MODAK Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India and HENRY C. LIM Biochemical Engineering Program, University of California, Irvine, CA 92707, U.S.A. (Received 15 April 1991; accepted fir publication 2 Jonunry 1992) Abstract-A systematic analysis of the problem of maximizing metabolite yield and bioreactor productivity for fermentation processes is presented in this paper. The periodic operation of a variable-volume stirred tank bioreactor is examined in the framework of optimal control theory for multiple control variables. The feasible combinations of inlet and outlet flow rates are identified. The candidates for optimal mode of operation are shown to be. either steady-state continuous, repeated batch or repeated fed-batch operation. The optimal mode of operation among these three candidates is analytically determined for four types of fermentation processes. A numerical example is pravided to illustrate the analytical results. 1. INTRODUCTION Batch, continuous and fed-batch (semibatch) are the most commonly used modes of operation of a bio- reactor for production of a number of valuable metabolites by fermentation processes. In addition, re- peated batch and repeated fed-batch are two cyclic modes operation which are commonly employed in research as well as in fermentation industry. In recent years, a number of studies have been reported for determining optimal operating policies for batch, con- tinuous and fed-batch (semibatch) bioreactors. These include determining optimal temperature or pH pol- icies for batch processes (Constantinides and Rai, 1974), and the rate of addition of substrate for fed- batch processes (Modak et al., 1986; Park and Ramirez, 1988; San and Stephanopoulos, 1989). A few studies have also been reported in which optimal policies for more than one operating variable have been determined (Chu and Constantinides, 1988; Modak and Lim, 1989). However, in all of the above studies, the mode of operation of a bioreactor, namely, batch, continuous or fed-batch, is fixed zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH a priori and the optimal policies determined accord- ingly. An important and challenging problem which has received very little attention is that of determining the mode of operation of a bioreactor which is opti- mal for achieving the desired objective of either max- imizing the metabolite yield or the bioreactor pro- ductivity. The processes studied in this regard in the previous investigations have been cell mass produc- tion with either constant (Weigand, 1981; Matsubura et al., 1985) or variable (Weigand et al., 1979) cellular yield and metabolite production with Ludenking- Piret kinetics (Hasegawa et al., 1987). However, the studies cited above are restrictive in scope due to several assumptions regarding the fermentation pro- cesses. A comprehensive and systematic analysis of determining the optimal mode of operation for com- plex fermentation processes is lacking thus far in the literature. In the present work, we present such an analysis, utilizing variable-volume semibatch reactor frame- work proposed earlier for chemical reactions (Waghmare and Lim, 1981; Parulekar et aZ.. 1988). In this framework, the problem of determining the opti- mal mode of operation of a bioreactor is posed as an optimal bontrol problem with the inlet and outlet flow rates of the bioreactor as control variables. In the first part of this work, feasible modes of operation of a bioreactor are identified by examining the combina- tions of inlet and outlet flow rates within the frame- work of optimal control theory for multiple control variables (Goh, 1964, Gabasov and Kirillova, 1972). In the second part of the work, the optimal per- formance of each of the feasible modes of operation is compared and the optimal mode of operation is iden- tified for two different objectives, namely, maximizing the metabolite yield and the bioreactor productivity. 2. PROBLEM FORMULATION We consider a fermentation process in which a microorganism X produces a metabolite P while util- izing a substrate S. The process is occurring in a stirred tank bioreactor in which the inlet and outlet flow rates are not necessarily equal. The dynamic mass balance equations describing such a process in a variable-volume semibatch reactor can be written as follows: i =/4x -FIX/v B = F,(S, - s)/V - ax 6 = 7cx - FIPIV ti=FF,-FF2 (1) 3869