Asymptotic expansion of the porous medium, effective diffusion coefficient in the Knudsen number

A variational principle based on the mean free path kinetic theory of gases is used to derive an expansion of the effective diffusion coefficient in the Knudsen number for a bed of overlapping, solid spheres. The effects of tortuosity are rigorously and properly considered by explicitly including the flux of diffusing material around obstructions in its path in the trial function. A formula for the ratio of Knudsen to bulk tortuosity in terms of the void fraction is derived. For small void fraction the reciprocal additivity of Knudsen and bulk effective diffusivities is obtained in analytical form.