Batch Fermentation with Product Inhibition
The variables in batch fermentation processes are the biological dry mass or cell concentration , the substrate concentration , and the product concentration . The contents within the reactor volume are assumed well mixed and growth is assumed to follow kinetics described by the Levenspiel equation, based on one limiting substrate and product concentration. Substrate consumption is related to cell growth by a constant yield factor . Product formation is the result of growth associated with a rate of production. Lag and decline phases are not included in this model.
The plots , and versus (time) show that growth stops when the substrate is depleted. This can be confirmed by the specific growth rate plot. As the product concentration increases, the specific growth rate decreases due to an inhibitory effect. After that, the product concentration no longer increases.
The Levenspiel equation can be written as:
where is the maximum specific growth rate, is a saturation constant, is the inhibition effect constant related to the toxic power of the inhibitor, and is the critical concentration of the inhibitor (product concentration in this case) above which reaction cannot proceed. If , there are no inhibitory effects.
 I. J. Dunn, E. Heinzle, J. Ingham and J. E. Prvenosil, Biological Reaction Engineering, Weinheim, Germany: VCH Verlagsgesellschaft mbH, 2003.
 K. Han and O. Levenspiel, "Extended Monod Kinetics for Substrate, Product, and Cell Inhibition," Biotechnology and Bioengineering, 32(4), 1988 pp. 430–447. doi:10.1002/bit.260320404.