Optimum design of a series of CSTRs performing reversible Michaelis-Menten kinetics: a rigorous mathematical study N.M. Faqir, M.M. Attarakih Abstract The optimum design of a given number of CSTRs in series performing reversible Michaelis-Menten kinetics in the liquid phase assuming constant activity of the enzyme is studied. In this study, the presence of product in the feed stream to the first reactor, as well as the effect of the product intermediate concentrations in the downstream reactors on the reaction rate are investi- gated. For a given number of N CSTRs required to perform a certain degree of substrate conversion and under steady state operation and constant volumetric flow rate, the re- actor optimization problem is posed as a constrained nonlinear programming problem (NLP). The reactor op- timization is based on the minimum overall residence time (volume) of N reactors in series. When all the reactors in series operate isothermally, the constrained NLP is solved as an unconstrained NLP. And an analytical expression for the optimum overall residence time is obtained. Also, the necessary and sufficient conditions for the minimum overall residence time of N CSTRs are derived analytically. In the presence of product in the feed stream, the re- versible Michaelis-Menten kinetics shows competitive product inhibition. And this is, because of the increase in the apparent rate constant K 0 m that results in a reduction of the overall reaction rate. The optimum total residence time is found to increase as the ratio (w 0 ) of product to sub- strate concentrations in the feed stream increases. The isomerization of glucose to fructose, which follows a re- versible Michaelis-Menten kinetics, is chosen as a model for the numerical examples. List of symbols C E;0 [mg/l] initial concentration of active enzyme C p [mole/l] product concentration C p;e [mole/l] product concentration at equilibrium C p;0 [mole/l] product concentration at the inlet of the first reactor C s [mole/l] substrate concentration C s;e [mole/l] substrate concentration at equilibrium C s;0 [mole/l] substrate concentration at the inlet of the first reactor D [] determinant HU [] upper triangular hessain matrix K e [] equilibrium constant K m [mole/l] apparent Michaelis-Menten constant K 0 m [] dimensionless Michaelis-Menten constant K m =C s;0  K p [mole/l] Michaelis-Menten constant for product K s [mole/l] Michaelis-Menten constant for substrate K 1 ; K 2 [h 1 ] rate constants K 1 ; K 2 [l/mole  h] rate constants N [] number of reactors in series Q [l/h] volumetric flow rate RC s  [mole/l  h] reaction rate t [h ] time T [  C] temperature V [l] reactor volume V m [mole/l  h] maximum apparent reaction rate V 0 m [h )1 ] apparent reaction rate defined as V m =C s;0  V p [mole/l  h] maximum reaction rate for product V s [mole/l  h] maximum reaction rate for sub- strate X [] substrate conversion Greek symbols a [] dimensionless substrate concentra- tion C s =C s;0 a  [] dimensionless optimum substrate concentration s [h] residence time V =Q s  [h] optimum residence time w [] ratio of product to substrate con- centrations C p =C s Subscripts e equilibrium i refers to ith reactor j refers to jth reactor k order of submatrix p product s substrate 0 initial Bioprocess Engineering 20 (1999) 329 – 335 Ó Springer-Verlag 1999 329 Received: 28 April 1998 N.M. Faqir, M.M. Attarakih Chemical Engineering Department, The University of Jordan, Amman 11942, Jordan e-mail: faqir@fet.ju.edu.jo Correspondence to: N.M. Faqir