Short-Cut Method for the Estimation of Isothermal Effectiveness Factors

Short-Cut Method for the Estimation of Isothermal Effectiveness Factors

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Short-Cut Method for the Estimation of Isothermal Effectiveness Factors J. Alberto Ochoa-Tapia,* Francisco J. Valde ´ s-Parada, and Jose ´ de J. A Ä lvarez-Ramı ´ rez ² Divisio ´ n de Ciencias Ba ´ sicas e Ingenierı ´ a, Universidad Auto ´ noma Metropolitana-Iztapalapa, Apartado Postal 55 - 534, Iztapalapa, D.F., P.C. 09340 Mexico The computation of effectiveness factors (EFs) is commonly used for simplifying heterogeneous models of catalytic processes. To do this, the reaction rate in a catalyst pellet is expressed by its rate under surface/bulk conditions multiplied by a functional factor, namely, the EF. When many EF evaluations are required, it is desirable to dispose of short-cut methods to alleviate the computational burden. Existing short-cut methods are based on the availability of analytical solutions for a related linear, but nonhomogeneous, approximation to the nonlinear kinetics. However, the nonhomogeneous term can lead to spurious solutions, such as the presence of negative concentration values. This paper proposes a derivation of the EF short-cut method as an iteration procedure in the average concentration. On the other hand, aimed at avoiding negative concentration values, the proposed linear boundary-value problem is equipped with a nonactive region. In this way, the short-cut method is composed of two nonlinear equations on the average concentration and the position of the boundary between the active and nonactive regions. Numerical results show that such a modification increases the prediction capacity of EF short-cut methods for a practically acceptable Thiele modulus region. 1. Introduction The computation of effectiveness factors (EFs) is commonly used for simplifying heterogeneous models of catalytic processes. The underlying idea is to express the reaction rate in a catalyst pellet by its rate under surface/bulk conditions multiplied by a functional factor, namely, the EF. Although highly accurate numerical methods for the solution of the corresponding boundary- value problem have been reported elsewhere (see, for instance, work by Davis 1 ), in some cases their use can be computationally expensive. Such is the case of optimization-based process design where the EF has to be evaluated many times for different sets of param- eters. In other cases, one would like to have available simple-to-use methods for a preliminary evaluation of the process performance. In this way, there is an incentive to derive short-cut methods for the evaluation of EF for nonlinear kinetics. In a first approach, Wedel and Luss 2 used perturbation series to derive a rational expression for the EF as a function of the Thiele modulus. Haynes 3 proposed a modified Thiele modulus, obtained by either differentiation or integration of kinetic expressions, to obtain an approximate EF based on the expression for first-order kinetics. On the basis of this approximation idea, Marroquı´n de la Rosa et al . 4 obtained a short-cut method to estimate the EF from a modified Thiele modulus at the pellet surface conditions.
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Short-Cut Method for the Estimation of Isothermal Effectiveness Factors

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