Numerical evaluation of the porous medium effective diffusivity between the Knudsen and continuum limits

A variational principle based on the mean free path kinetic theory of gases is used to calculate the effective diffusion coefficient for gaseous diffusion through a bed of overlapping solid spheres for arbitrary Knudsen number and porosity φ. Effects of tortuosity of the diffusion paths are considered by including the flux around the solid spherical obstructions explicitly in the trial functions. Computer evaluations of the variational integrals are performed for Knudsen numbers from 0.1 to 100 and porosities from 0.01 to 0.99. The simple Bosanquet formula for series addition of the effective bulk and Knudsen diffusivities is compared with the variational calculations of the effective diffusivity. The Bosanquet formula is found to be exact for low porosities, and a suprisingly good approximation even at higher porosities. From the slopes at large Knudsen numbers, we propose a new variational expression for the Knudsen tortuosity τKn for a bed of overlapping spheres, τKn=9/8−(1/2) lnφ+(13/9−9/8)φ2/5.