Bulk Reaction Rate in a Heterogeneous Reaction System

Abstract

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 φ = 4/3 πn a^3, where n is the number density of spheres and a is the sphere radius, and the Péclet number Pe of the uniform bulk flow through the bed is determined analytically for φ << 1 and Pe << 1. The Péclet number is the ratio of transport by advection to diffusion; Pe = Ua/D, where U is the bulk velocity through the bed and D is the molecular diffusivity of the reactant in the fluid. In the absence of a bulk flow, it is shown that a bulk reaction in the bed results in a screening of the concentration disturbance beyond a distance of O(aφ^(-1/2)) from a particle. In this case, the nondimensional bulk reaction rate or Nusselt number is Nu ~ 1 + (3φ)^(1/2). For an isolated particle, it is well-known that weak uniform advection balances diffusion at a length scale of O(aPe^(-1)) and Nu ~ 1 + Pe/2. When there is weak flow through a bed of catalytic particles, the influence of flow strength and particle fraction upon the reaction rate depends on the ratio of the two length scales, Pe/(φ^2), characteristic of these two limiting situations. For Pe/φ^(1/2) = O(1), there is a balance of advection, diffusion, and reaction at a distance of O(aφ^(-1/2)) from a particle. The first correction to Nu for all Pe/φ^(1/2) is shown to be (3φ + (Pe^2)/4)^(1/2), in which the dependence of Nu on Pe and φ is not, in general, separable, although it becomes separable in the limits Pe/φ^(1/2) → 0 and e/φ^(1/2) → ∞. We also discuss the case of Pe >> 1, in which Nu has the form of A(φ)Pe^(1/3), where A(φ) is an unknown function which is apparently O(1) for all φ.