48
Ind. Eng. Chem. Fundam. 1981, 20, 48-54
Greek Letters defined in eq 9b 0, = defined in equation 9d A = aspect ratio of pellet (= R / L ) t = small parameter, defined in eq 4e tp = particle voidage 9 = intraphase effectiveness factor 7, = eigenvalue corresponding to eigenfunction T,, 8, = defined in eq 9a a, =
Xmn
= ( 9 2 + h2t;m2)1'2/+
t;, = eigenvalue corresponding to eigenfunction K ,
+
= Thiele modulus (= (R2k,(l -- tP)/tpa))1/2) $, = defined in eq 9c
= reactant in bulk fluid B = product Bo = product in bulk fluid A0
Literature Cited Ark, R. Chem. Eng. Sci. 1957, 6 , 262. Do, D. D.; Weiland, R. H. Ind. Eng. Cbem. Fundam. 1980a, preceding article in this issue. Do, D. D.; Weiland, R. H., submitted for publication in I&. Eng. Chem. Fundam. 1980b. Ho, T. C.; Hsiao, G. C. Chem. Eng. Sci. 1977, 32, 63. Lamba, H. S.; Dudukovic, M. P. Chem. Eng. J . 1978, 76, 117. Masamune, S.;Smith, J. M. AIChEJ. 1966, 72, 384.
Received for review October 18, 1979 Resubmitted July 11, 1980 Accepted September 26, 1980
Subscripts A = reactant
Deactivation of Single Catalyst Particles at Large Thiele Modulus. Travelling Wave Solutions Duong D. Do and Ralph H. Weiland" Department of Chemical Engineering, Univers.sity of Queensland, St. Luck, Queensland, Australia 4067
The method of matched asymptotic expansions is used to obtain simple analytical expressions for the timede-
pendent effectiveness factor of variously shaped porous catalyst particles undergoing slow fiist-order self-poisoning at large Thiele modulus. These asymptotic solutions are barely distinguishable from their exact numerical counterparts for Thiele moduli above 100. The position of the activity wave as a function of time is given by compact analytical expressions and the structure of the wave front and the way in which it is formed are described in detail.
Introduction Deactivation in catalytic systems has been the subject of reviews by Butt (19721, Carberry (1976), and Schlosser (1977). The behavior of a single porous particle of catalyst undergoing deactivation by the reactant (parallel poisoning) or by the product of the reaction (series poisoning) is well described by a nonlinear parabolic equation coupled to one of first order. The earliest attempt to solve these equations, and thereby obtain a description of the timedependent catalyst effectiveness factor, appears to be that of Masamune and Smith (1966). They reported both numerical solutions and an approximate result based on a shell model. This was followed by the work of Chu (19681, who considered the effect of a Langmuir isotherm. Khang and Levenspiel (1973) repeated the numerical work of Masamune and Smith but in the context of a lumped parameter approximation to the equations. Recently, Lamba and Dudukovic (1978) extended these analyses to catalysts of slab geometry. It is an unfortunate fact that there are no experimental data with which the predictions of any of these analyses can be compared. Catalyst poisoning appears to be one area of kinetics sadly neglected from an experimental viewpoint. One of the parameters characteristic of reacting systems with diffusion is the Thiele modulus. When the Thiele modulus is O(l), species concentrations and catalyst activity vary gradually within the particle. However, when the Thiele modulus is large the particle can be represented to a first approximation by a central core of high activity
* Clarkson
College of Technology, Potsdam, N.Y. 13676. 0 196-43 13/8 1/ 1020-0048$01.OO/O
surrounded by a shell of inactive (inert) catalyst. Large Thiele modulus systems have been reported by Carberry (1976) for conventional catalysis, and Tai and Greenfield (1978) have commented that liquid-phase reactions catalyzed by immobilized enzymes are generally large-Thiele modulus systems. The dead zone gradually moves inward from the catalyst surface at a rate which is a priori unknown. Such moving boundary problems are characteristically nonlinear and like similar problems in heat transfer with change-of-phase (e.g., freezing and melting), they present serious mathematical and numerical difficulties. In addition, the fact that the activity exhibits a sharp jump over a very narrow region in the vicinity of the moving boundary necessitates the use of very small step sizes in the numerical integration of the equations. The need to maintain temporal stability also dictates that very small steps in time must be used. In the current work we use the methods of singular perturbation theory to obtain asymptotic solutions to the problem for large Thiele modulus. The group of techniques collectively known as singular perturbation methods represents an extremely powerful class of analytical tools largely unknown to, and rarely used by, chemical engineers. Originally developed in the context of fluid dynamics, perturbation methods have found wide application in solid mechanics, vibration analysis, population dynamics, and so on. One of their great advantages is that to a large extent ad hoc modelling assumptions (such as the shell model in the present context) can be avoided. The solutions we obtain here are extremely simple and result in explicit formulas for the time required to deactivate plane, cylindrical, and spherical pellets to any extent. Simple 0 1981 American Chemical
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Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981
analytical expressions are obtained for the time dependence of the thickness of the dead shell, and the structure of the wave front or interface between dead and active zones is described in detail. In an effort to make the techniques more readily accessible to chemical engineering analysts, the mathematics will be demonstrated in some detail for the slab pellet. The leading order results found for the slab pellet are equivalent to those found from the shell model, but it must be emphasized that results for spherical and cylindrical pellets are not the same as those of Masamune and Smith (1966) and are distinctly different from shell model solutions even at leading order. Shell model analyses represent a first approximation which cannot be improved; by contrast, the results found here allow for improvement to any extent, provided one is willing to do the algebra. Formulation and Approach Consider a porous catalyst particle of slab geometry; a first-order chemical reaction takes place within the particle. At the same time the catalyst is being deactivated by the reactant (parallel poisoning) according to a first-order mechanism, and at a rate which is slow compared with the rate of the main reaction. The nondimensional reactant balance equation is
49
whereas catalyst deactivation occurs over a much longer time. This allows us to break time into two distinct periods in which separate asymptotic expansions are valid. Within each time period we separate the catalyst particle into two or more distinct regions in space, each characterized by different balances between the rates of diffusion, reaction, and deactivation. Within each region the equations possess separate asymptotic solutions which can be made continuous throughout space and time through the application of asymptotic matching principles. This will allow us to obtain simple analytical approximations for the velocity and position of the moving activity front. The method also allows US to improve the accuracy of the approximations to any degree desired and to obtain solutions uniformly valid throughout space and time-the cost is more tedious algebra but the ability to improve systematically is important in that simple answers can still be had, which, unlike shell models allow one to deal with poisoning rates not SO very slow. Initial Time Period In the initial period the rate of diffusion of reactant into the partcle must be balanced by the rates of accumulation and reaction. This balance suggests that distance should be stretched according to (2)
$' = X / P
Then eq l a becomes and the rate expression for loss of catalyst activity is
in which time has been scaled on the rate constant for the main reaction and A is the reactant concentration made nondimensional on its bulk value Co. Here y 2 = a)/kL2 is the inverse of the square of the Thiele modulus, 4, and t = k,Co/k is the small parameter ( E > 1 or y 2
