Optimal Temperature Policy for Immobilized Enzyme Packed Bed Reactor Performing Reversible Michaelis±Menten Kinetics Using the Disjoint Policy Naim M. Faqir, 1 Menwer M. Attarakih 2 1 Chemical Engineering Department, The University of Jordan, Amman 11942, Jordan: e-mail: faqir@fet.ju.edu.jo 2 Chemical Engineering Department, Amman-College for Engineering Technology, Al-Balqa Applied University, Amman, Jordan Received 9 September 2000; accepted 1 September 2001 Abstract: The optimal temperature policy that maximizes the time-averaged productivity of a continuous immobi- lized enzyme packed bed reactor is determined. This optimization study takes into consideration the enzyme thermal deactivation with substrate protection during the reactor operation. The general case of reversible Micha- elis±Menten kinetics under constant reactor feed ¯ow rate is assumed. The corresponding nonlinear optimiza- tion problem is solved using the calculus of variations by applying the disjoint policy. This policy reduces the op- timization problem into a differential-algebraic system, DAE. This DAE system de®nes completely the optimal temperature±time pro®les. These pro®les depend on the kinetic parameters, feed substrate concentration, oper- ating period, and the residence time and are character- ized by increasing form with time. Also, general analytical expressions for the slopes of the temperature and residual enzyme activity pro®les are derived. An ef- ®cient solution algorithm is developed to solve the DAE system, which results into a one-dimensional optimiza- tion problem with simple bounds on the initial feed temperature. The enzymatic isomerization of glucose into fructose is selected as a case study. The computed productivities are very close to that obtained by numer- ical nonlinear optimization with simpler problem to solve. Moreover, the computed conversion pro®les are almost constant over 90% of the operating periods, thus producing a homogeneous product. ã 2002 John Wiley & Sons, Inc. Biotechnol Bioeng 77: 163±173, 2002. Keywords: optimal temperature policy; Michaelis±Men- ten kinetics; disjoint policy INTRODUCTION The problem of optimal operating policy for immobi- lized enzyme packed bed reactors is a subject of vast operational research due to the economical potential of characterizing such optimal operations. This optimal operating policy can be practically achieved in mainly two control modes of reactor operation. The ®rst one is the control of feed rate to the reactor in such a way that it decreases as function of time in order to compensate for the loss of enzyme activity. The second mode is the temperature and/or pH pro®le that is set according to a well-de®ned time pro®le (Faqir, 1998; Haas et al., 1974; Kimetal.,1982;Parketal.,1981;Straatsmaetal.,1983; Vos and Luyben, 1993). Temperature is one of the most important variables due to its eect on the kinetic pa- rameters, enzyme activity, and substrate protection factor. However, one limitation of this control variable is the narrow range within which enzymes remain active (Lee, 1992). In spite of this, many researchers have used two types of temperature operating policies utilizing the enzymatic isomerization of glucose to fructose as a model system (Abu-Reesh and Faqir, 1996; Faqir, 1998; Hass et al., 1974; Kim et al., 1982; Park et al., 1981; Straatsma et al., 1983). These operating policies are ei- ther a rising temperature policy with time to maintain a constant outlet conversion or a constant temperature pro®le that maximizes the average reactor productivity. Parketal.,(1981)studiedthetemperaturecontrolpolicy for the enzymatic isomerization of glucose to fructose using immobilized glucose isomerase in a continuous packed bed reactor. They found that the average pro- ductivity is improved by approximately 8% upon using the optimal increasing temperature policy of operation as compared to the optimum isothermal operation. Most of the published works on determining the opti- mumtemperatureoperatingpolicyforcontinuouspacked bed reactors performing reversible Michaelis±Menten kinetics use either Pontryagin's maximum principle or nonlinearprogrammingmethods(Faqir,1998;Haasetal., 1974;Kimetal.,1982;Parketal.,1981;Straatsmaetal., 1983;VosandLuyben,1993).Themaindisadvantagesof these optimization methods are the tedious formulation and complex solution algorithms used to obtain the de- sired solution. Moreover, they lack general applicability whenappliedtothesameclassofproblems. The present work deals with the optimization of con- tinuous immobilized enzyme packed bed reactors under constant feed ¯ow rate. For such reactors, the optimal Correspondence to: Naim M. Faqir ã 2002 John Wiley & Sons, Inc. DOI 10.1002/bit.10118