Ind. Eng. Chem. Res. 1995,34,3514-3523
3514
Bulk Reaction Rate in a Heterogeneous Reaction System Jeffrey F. Morris? and John F. Brady* Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91 125
The steady reaction rate in a bed of spherical catalyst particles immersed in a flowing Newtonian fluid is studied. The dependence of the rate on the particle volume fraction = 4/3ma3,where n is the number density of spheres and a is the sphere radius, and the PBclet number Pe of the uniform bulk flow through the bed is determined analytically for q5 1,in which Nu has the form of A(#)Pe113, where A ( d ) is a n unknown function which is apparently O(1) for all q5. #J
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1. Introduction Heterogeneous reaction is commonplace in natural settings and engineering practice. Examples include dissolving minerals into groundwater flows, combustion of small fuel particles, and catalytic reactions in fixed or fluid beds. A description of the properties of such systems at a length scale much larger than the individual particle size or interparticle spacing is often desirable. Determination of these properties requires averaging the detailed description of the problem at the particle scale. We apply the method of averaged equations (Hinch, 1977)to determine the equation governing the average concentration field around a catalyst particle in a.fixed bed and from the solution of this equation determine the average flux of reactant to a particle or equivalently the bulk rate of reaction. Only rarely may one neglect the influence of transport processes upon heterogeneous reaction rates. In the absence of flow, the problem of determining the dependence of the diffusion-limited reaction rate on the volume fraction of particles, 4, has received abundant theoretical (Felderhof and Deutch, 1976; Muthukumar and Cukier, 1991) and simulation (Lee et al., 1989; Bonnecaze and Brady, 1991) attention. In many situations of interest, however, there will be fluid flow and thus an advective contribution to the transport of reactant. The advective contribution to the bulk reaction rate in fured beds has received less attention than the effect of particle fraction, in part because the theoretical developments necessary to determine the influence of particle fraction or advection strength alone, as well as the method of formulating the bulk problem, are nontrivial.
* Author to whom correspondence should be addressed. E-mail:
[email protected]. + Present address: Koninklijke/Shell-Laboratorium, Amsterdam (Shell Research BV); P.O. Box 3003, 1003 AA Amsterdam, The Netherlands. E-mail:
[email protected].
The physical processes of transport of mass, heat, and momentum have numerous aspects in common, a point well made by the authors Bird, Stewart, and Lightfoot (1960) whose textbook we celebrate in this issue. For similar flow conditions and model of a bed of particles, we expect the mass transport problem involved in determining the bulk reaction rate to be closely related to problems of heat and momentum transport in the bed. Indeed, the present work has much in common with the study of heat transfer to fluid flowing through a bed of spheres performed by Acrivos et al. (1980) and that of drag on the fluid flowing through the same model bed by Kaneda (1986). In both studies, 4 > 1,in which reactant transport is dominated by advection. Under this condition, diffusion is able to balance advection only in a narrow O(Pe-’”) boundary layer at the surface of a particle (the velocity satisfies a no-slip condition), and it is expected that Nu = A(4)Pe1l3,with A taking on O(1) values for all 4. For an isolated particle, Nu = O.625Pe1l3 (Levich, 1962; Batchelor, 1979), yielding A(0) = 0.625. Definitive analysis is not presented here. We instead present physical arguments for the expected form of the Nusselt number and discuss the considerations necessary to determine A($). The equation governingthe concentration disturbance about a reference particle is (15). First, we discard both PeV.(u’c’)l and R; and consider the problem of a single particle fixed in the large-Pe effective medium, in which case the problem governing C is again given by (351, with the conditions (16). The dominant balance near the particle involves only advection and diffision, and the reaction term is hereafter neglected. Although the 0 is correct, we observe far-field condition C(r -) that the actual concentration ( c ) ~goes from zero on the particle surface t o the fluid average (c)a = (c)d(l - 4) (rather than ( c ) ~ )in the boundary layer and only when r - 1= 0(1), Le. where a point may lie within a particle or in the fluid, does (c)1- ( c ) ~ .Thus, the concentration asymptotically approaches (c)fl immediately out of the boundary layer. This results in a stronger gradient a t the particle surface and larger flux to a particle than would be the case if the concentration went directly to the bulk average outside of the boundary layer. In dimensionless form, C undergoes a change from unity to -4/(l - 4) across the boundary layer: the gradient is 0[(1 4)Pe113/(l- 411. For all allowed 9 in a bed of equal-sized spheres, (1 4)/(1- 4) takes on O(1) values and thus the Nusselt number remains proportional to Pe1/3, with the form
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The detailed problem a i the particle surface must be analyzed to determine A. It is worth noting that the factor of (1 $)/(1 - 4) is associated with a definition of the dimensionless flux and bulk rate in terms of the bulk average concentration. If our formulation of the
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problem applied the fluid average concentration in these definitions, this factor would not appear. This does not alter any of the results reported for Pe
