Flux Equations for Gas Diffusion in Porous Media
David B. McWhorter
The molecules in a gas are in constant, chaotic thermal motion. Because the molecules are free to move about, constituent molecules intermingle, collide with one another, and impinge upon particles that may be embedded in the gas. If the concentration of a particular constituent is non-uniform, there occurs an aggregate motion of that constituent from locations of high concentration toward locations of low concentration. This aggregate motion is diffusion and can be attributed to differences in partial pressure or chemical potential, as well as concentration.
Diffusion of a particular constituent in systems free of any solid surfaces (e. g. sand grains or walls) is resisted only by collision with molecules of different mass. The resistance to diffusion arising from these molecule-molecule collisions at the molecular scale manifests at the macroscopic scale as the molecular diffusion coefficient that appears in Fick’s law. Diffusion in a gas occupying a porous medium experiences additional resistance that is attributable to molecule-particle collisions. The resistance to diffusion arising from the complicated and unresolved molecule-particle collisions on the molecular scale manifests at the macroscopic scale as the Knudsen diffusion coefficient.
The above concepts underpin the equations in this book for the one-dimensional flux of constituents of an ideal, isothermal, binary gas in porous media. Explanations are provided for Graham’s law, diffusion engendered non-viscous bulk gas flow, and why diffusion-created pressure gradients are to be expected in most field settings. The equation presented herein for constituent diffusion affected by a phase pressure gradient (so-called pressure diffusion) includes both the molecular and Knudsen diffusion coefficients, even when Knudsen diffusion is unimportant. These and other results are unique to diffusion in porous solids and do not follow from the usual Fick’s law treatment of diffusion.