I
ANDREAS ACRIVOS and PAUL
L.
CHAMBRi
University of California, Berkeley, Calif.
laminar Boundary layer Flows with Surface Reactions
AN and often used chemical engineering operation is that IMPORTANT
in which a fluid, containing several chemical substances, flows past a solid surface with which it reacts. In such cases, the solid itself may enter into the chemical reaction-for example, in the oxidation of metals or the burning of carbon-or it may act only as a catalyst. In all these problems, calculation of the rate of surface reaction is of prime interest. This rate is determined not only by the kinetic l a h that govern the chemical reaction, but also by the hydrodynamically induced transport mechanisms near the surface. This complicating factor exists because, before the chemical substance can react, they must be transported to the surface by the fluid, or phenomenon which is governed by the hydrodynamics of the system. These transport mechanisms then affect appreciably the rate of the surface reaction, especially where the chemical reaction is rapid, so that the process becomes diffusion-controlled. Therefore, to predict the rate of the surface reaction correctly, not only the kinetic laws but also the rate of mass transfer of the reacting species from the main part of the fluid to the surface must be considered. The problem of determining the effect of the transport mechanisms on the rate of the surface reaction is naturally complex. The authors ( 2 ) have recently shown, however, that when the flow past a catalytic surface is of the two-
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dimensional laminar boundary layer type and the surface reaction is firstorder and isothermal, then for certain surface geometrics it is possible to solve the equations, which describe the phenomenon, with accuracy and in terms of a convergent infinite series. As a contiquation of that work this article shows that such a formulation can be applied to surface reactions with kinetic laws of arbitrary complexity. A comparison between the accurate solution of the problem and an approximate method currently used by chemical engineers shows that the latter must be used with caution. This discussion covers simple conditions-laminar boundary layer flows with isothermal catalytic surface reactions. The more involved problems connected with laminar and turbulent flows of nonisothermal character, which can occur in either external or internal flow systems, have been the subject of a general analysis by ChambrC (7).
Basic Equations and Mathematical Formulation of the Problem
Problem. An inert fluid, containing a single reactant, A, flows past a solid surface which reacts with A . Assumptions. (1) Irreversible reaction with rate depending only on the local concentration of A next to the surface. (2) Concentration of A not so large that the properties of the fluid are affected by its composition.
In this important type of chemical engineering operation Problems involving catalytic surface reactions of arbitrary complexity can be solved accurately by a system of integral equations Chemical engineers should use with caution the current approximate method
The diffusion equation becomes, in the boundary layer approximation :
where c = concentration of reactant
D
A
= diffusion coefficient of A
x = distance along surface from leading
edge y = distance from surface u = velocity component in x direction u = velocity component in y direction The boundary conditions are : c = a for x
