Diffusion and surface reaction in random fibrous media

Abstract

In the classic “trapping” problem, the reactant is uniformly generated at a constant rate across a matrix phase, and diffuses to the interface of a dispersed cylinder phase, where it instantly reacts. Prager’s upper bound on the effective reaction rate constant k (a “best” bound based on void-point nearest-neighbor-surface statistics) is derived for randomly placed, freely overlapping, infinitely long cylinders of radius a, where each cylinder has an arbitrary orientation with respect to the others. To compare with simulation data for an isotropic bed of overlapping spheres of radius a, the variational upper bound is considered for the case of the three-dimensional, isotropic overlapping cylinder bed with totally random mutual cylinder orientations. Once a correction factor of 3/2 for the surface area ratio of the overlapping sphere to overlapping cylinder beds is applied, the analytical variational cylinder bed bound is nearly coincident with the sphere dispersion simulation curve. The small differences observed at lower dispersion densities are consistent with the expected diffusion flux differences in the sphere and cylinder geometry. By comparing the variational upper bound with other simulation results for randomly overlapping, aligned, elongated, prolate spheroids, a maximum range for the k change due to mutual rotation between neighboring cylindrical reaction sites (29% and 32%, respectively, at solid volume fractions of 0.30 and 0.50) is obtained. The necessary distribution moments for the corresponding “relaxation time” lower bound are also given. The Doi lower bound on the effective reaction rate (a “best” bound based on two-point void–void Fvv, void–surface Fvs, and surface–surface Fss correlations) is addressed. The two- point correlations for an isotropic bed of overlapping cylinders with random mutual orientation in three dimensions are presented. Their behavior is discussed, and a logarithmic singularity in Fss at the two-point distance of 2a is pointed out, that renders the Doi bound indeterminate.