Optimal temperature policy for reversible reactions with deactivation: Applied to enzyme reactors

Abstract

The optimal temperature policy which will maximize the h a 1 catalyst activity that gives a fixed conversion of reactants in a specified time for batch operation was determined by the formulation of a calculus of variations problem following the technique of Szepk and Levenspiel (1968). The method was applied to the general case of first-order reversible reactions which occur in the presence of catalysts deactivating by an irreversible firstorder mechanism. To reduce trial and error estimations and circumvent numerical instabilities, the two-point boundary value variational problem was reformulated in terms of an initial value problem with a parameter which includes the initial value of temperature. This initial value problem was solved by a regression technique.These techniques were applied to the industrially important enzymatic reaction of the isomerization of D-glucose to D-fructose catalyzed by glucose isomerase in solution. Kinetic and deactivation data are available for this endothermic reaction which obeys first-order reversible kinetics and for the isomerase denaturation which appears to be first order. The optimal temperature operational policy as stated above maximized final enzyme activity such that 1% less denaturation of glucose isomerase occurred when compared to final isomerase activity yielding the same conversion for the same reaction time when the reactor is operated at the optimal isothermal temperature. SCOPEThe problem of the optimal operation policy for chemical reactors is important when it is desired to improve the operational economics and product yield. Accordingly these problems have received attention in the recent literature where investigators have employed calculus of variations methods to the analysis of batch and tubular plug flow reactors. Irreversible nth order reactions with concentration independent and recently concentration dependent deactivation models were examined. Numerical methods and theoretical developments treated in the format of Pontryagin's Maximum Principle were applied to the problem of reversible reactions with concentration independent decaying catalysts in a tubular reactor. Biochemical reactions catalyzed by denaturable enzymes which may be in solution or immobilized on inert supports are of emerging interest in recent literature. The determination of the optimal policies of reactor operation for these biochemical reaction systems is as important as in previous needs because of the high costs of enzyme catalysts and the temperature sensitivity of enzyme denaturation.The work described herein applies the calculus of variations formulation to the problem of determining the optimal temperature policy which will maximize the final Correspondence concerning this paper should be addressed to L. T. catalyst activity that gives a fixed conversion of reactants in a specified time for a batchwise operated reactor. The technique can also be applied to the mathematically equivalent problem of maximizing conversion with a specified final catalyst activity in a given rea...

Optimal temperature policy for immobilized enzyme packed bed reactor performing reversible Michaelis–Menten kinetics using the disjoint policy

“…Under these assumed conditions the packed bed design equation is given by (Abu-Reesh and Faqir, 1996;Hass et al, 1974) …”

“…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 either a rising temperature policy with time to maintain a constant outlet conversion or a constant temperature pro®le that maximizes the average reactor productivity.…”

Fixed bed with catalyst deactivation: On‐line estimation of deactivation, control, and design

“…An immediate question is how we should control the operating conditions of a reactor designed in the usual way. This question has been the subject of many studies (Szepe and Levenspiel, 1968;Chou et al, 1967;Ogunye and Ray, 1971;Haas et al, 1974). The control policies resulting from these studies either lead to an open-loop control or require detailed knowledge of catalyst deactivation.…”

Optimal control policy for substrate inhibited kinetics with enzyme deactivation in an isothermal CSTR

“…Thus, it is possible to cease operation when the enzyme activity falls below a low level or to add fresh enzyme to make up for the activity loss (Verhoff and Schlager, 1981), or, one can manipulate the temperature or the pH of the reactor to achieve an overall optimal operation (Chou et al, 1967;Szepe and Levenspiel, 1968;Ogunge and Ray, 1971;Haas et al, 1974;Sadana, 1979;Park et a]., 1981). Although they produced definite improvements over the uncontrolled case, the above control schemes have various disadvantages.…”

“…From a practical standpoint there are four different control modes that can be applied to improve the operation of an enzymatic CSTR: A shut+ff operation for enzyme replacement when the conversion reaches a predetermined low level; continuous addition of enzyme to make up for the loss of activity; optimal manipulation of the temperature or the pH of the system; and varying the flow rate of the continuous system in a manner resulting in optimal overall operation. Previous studies have examined the first three of the above possibilities, (Verhoff and Schlager, 1981; Haas et al, 1974;Sadana, 1979; Part et al, 1981), and produced control schemes which resulted in improved reactor operation. However, there are several problems associated with the application of these control modes such as the requirement of a detailed description of the kinetics of both the main reaction and the deactivation process, the very narrow temperature and pH range within which enzymes remain active, and the fact that a rather complicated scheme is needed for the implementation of the controls.…”