The catalytic effectiveness factor under nonisothermal conditions

Abstract

The effectiveness factor for a poroussolid-catalyzed reaction is a measure of the extent to which species transport by Knudsen diffusion within the particle retards the over-all rate process. The literature reveals correlations based upon analytical solution of the appropriate equations when isothermal conditions prevail throughout the catalyst particle. Following Thiele's original work ( 5 ) , Wheeler (8) and Weisz (7) have made significant contributions to the discussion of the isothermal effectiveness factor.The problem of reaction heat release or abstraction within a catalyst particle has been recognized as a potentially serious one with respect to the influence of the resulting temperature gradient upon the effectiveness factor for both reactant and product. However analytical solution of the governing differential equations presents marked difficulties. Wheeler (8) and more recently Prater (2) have shown that for a simple firstorder irreversible reaction the maximum temperature difference between the external and interim surface of the pellet is given by Valuable as the above relation is for purposes of estimating the maximum possible temperature gradient, the precise influence of the temperature gradient upon effectiveness remained, until recently, undefined.Schilson and Amundson ( 3 ) were the first to undertake the task of solving the nonisothermal problem analytically. The heat-generation functions for a single reaction and for two simultaneous reactions were approximated by single linear functions and also by two linear relationships. Analytical solutions were then found by Schilson and Amundson in the form of an integral equation expressing temperature as a function of pellet radius. Specific cases were solved in terms of the effectiveness factor, the details of which can be found in their forthcoming publication ( 4 ) .Tinkler and Pigford (6) applied perturbation techniques to obtain an approximate analytical expression for the nonisothermal effectiveness factor in terms of the heat-generation function and the Nusselt number for heat transfer at the external surface. Their attention was fociised upon a first-order irreversible reaction catalyzed by a spherical pellet.The work reported in this communication embraces both first-and secondorder irreversible catalytic reactions. Pellet geometry is riot specified, since the model is written for a single pore. Aris (1 ) has shown that a proper definition of particle size ( L = l/a) renders the isothermal effectiveness factor virtually independent of particle shape, while Prater's derivation ( 2 ) proves that AT is also independent of geometry. Thus it is assumed here that the single-pore treatment (equivalent to flat-plate geometry) is of general applicability.The following dimensionless equations are assumed to apply for kinetics of order n, where, in the case of secondorder reaction, equality of species diffusivities is assumed. For mass J where C , AHD T o K p = I -__-. , k = k, exp [ -E / R T , (tl)] (4) J and The boundary conditions, in terms of the ...

Principles of Heterogeneous Catalysis

“…The onset of diffusion limited regimes can be altered by changing the reactor design, the catalyst pore structure, the catalyst particle size, and the distribution of the active sites in the catalyst particles. Values of various dimensionless groups can be calculated to estimate the extents to which transport phenomena may control catalytic performance for specific operating conditions [23][24][25][26][27][28][29][30][31]; however, these calculations are most reliable for cases where the intrinsic reaction kinetics are known. In these cases, it is possible to make catalysts with structures designed to provide adequate rates of diffusion and yet offering high surfaces areas, leading to high rates of reaction per reactor volume, such as the design of specific pore size distributions (e.g., bimodal distributions containing large pores leading to high accessibility of the active sites within the interior of the catalytic pellet, and small pores that branch from the larger pores leading to high surface areas), the formulation of unique pellet shapes (that lead to high accessibility of the active sites but do not cause large pressure drops through the catalytic reactor), and the synthesis of catalyst pellets containing a spatial distribution of the active material within the catalyst pellet [32].…”

“…In some cases, transport effects can be used to improve the selectivity of a catalyst, such as in the case of shape-selective catalysis in zeolites [33][34][35][36]. In the following sections, we focus on various factors controlling the intrinsic reaction kinetics of catalysts, and we refer the reader to other articles for further discussion on transport effects in heterogeneous catalysis [23][24][25][26][27][28][29][30][31].…”

Diffusion and reaction in a porous catalyst slab: Perturbation solutions

“…The problem was subsequently approached numerically by, among others, Carberry,8 who examined first and second order kinetics in a catalyst of slab geometry, and Weisz and Hicks, 9 who considered first order kinetics in a spherical catalyst particle. Both these studies presented results for the effectiveness factor as a function of the Thiele modulus.…”

A method of representing the nonisothermal effectiveness factor for fixed bed calculations

“…Carberry (6) has shown that this is not a bad approximation if 6 < 6, and Liu (12) restricts himself to this range. However, this excises a large part to the range of nonuniqueness which is under consideration here.…”