An Improved Short-Cut Method for Effectiveness Factor Estimation

An Improved Short-Cut Method for Effectiveness Factor Estimation

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An Improved Short-Cut Method for Effectiveness Factor Estimation Francisco J. Valde ´ s-Parada,* Jose ´ A Ä lvarez-Ramı ´ rez, ² J. Obet Marroquı ´ n de la Rosa, and J. Alberto Ochoa-Tapia ² Departamento de Ingenierı ´ a de Procesos e Hidra ´ ulica, Di V isio ´ n de Ciencias Ba ´ sicas e Ingenierı ´ a, Uni V ersidad Auto ´ noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me ´ xico D.F., 09340 Me ´ xico, and Instituto Mexicano del Petro ´ leo, Eje central La ´ zaro Ca ´ rdenas 152, Me ´ xico D.F., 07730 Me ´ xico Two main short-cut methods for catalyst particle effectiveness factor (EF) estimation that use kinetics linearization about surface and average concentration conditions have been recently considered in the literature. The former produces simple computations with acceptable estimations for low Thiele modulus values, although negative particle concentrations can be generated. The latter is intended to improve the particle concentration profiles but at the expense of increased computations. The aim of this paper is to propose a hybrid method that combines the advantages of the two approaches to obtain EF and concentration profile estimations that are better than those of the individual ones. This is done by taking a kinetics linearization at an artificial concentration condition resulting from a type of Crank - Nicholson scheme. From a simple error analysis, it is shown that the resulting short-cut procedure displays enhanced convergence properties, which is corroborated by means of numerical computations. 1. Introduction Some approaches have been proposed for short-cut computa- tion of the catalyst particle effectiveness factor (EF). Wedel and Luss 1 used perturbation series to derive a rational expression for the EF as a function of the Thiele modulus. Haynes 2 proposed a modified Thiele modulus, obtained by either differentiation or integration of kinetic expressions, yielding an approximate EF based on a first-order kinetics expression. Marroquı´n de la Rosa et al. 3 proposed a short-cut method by departing from a kinetics first-order Taylor series expansion at surface conditions (SCC). By incorporating the linear driving force concept, 4 Szukiewicz and Petrus 5 improved the surface concentration-based EF estimation scheme and reported accept- able results for small and moderate Thiele modulus values. Although the use of the SCC approach 5 yields acceptable EF estimation for relatively small Thiele modulus, significant errors can be obtained for large Thiele modulus since the particle concentration profile becomes very sharp. In fact, such errors are caused by estimated concentration profiles with negative values, which lead to overestimated EF values. To address this problem, Ochoa-Tapia et al. 6
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This note was uploaded on 03/10/2011 for the course CBI 101 taught by Professor O.tapia during the Spring '11 term at UNAM MX.

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An Improved Short-Cut Method for Effectiveness Factor Estimation

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