Exact analytical expressions are derived for the optimal design (minimum overall reaction volume) of N perfectly mixed membrane reactors in series carrying out an enzyme catalysed reaction with Michaelis-Menten kinetics. These equations enable the direct calculation of the smallest total reactor volume (holding time) needed for a given overall conversion degree, as well as the individual reactor volume and conversion degrees. Results are compared with the ones obtained with a series of N CSTRs and with a plug flow reactor. The theoretical superiority of membrane reactors versus CSTRs is demonstrated.