7 Summary and Conclusions

The molecules of gases exist in a state of continuous random, chaotic motion. In multi-component gases with non-uniform composition, individual components spontaneously diffuse from locations of high to low concentrations because the component energy per unit volume is greater at high concentration than at low concentration. Diffusing molecules in systems free of solid particles experience a resistance to their collective motion due to collisions with molecules of other components. The rate at which there occurs a net transfer of component molecules from one location to another is represented by Fick’s law and is characterized by the molecular diffusion coefficient.

Gases diffusing through porous media experience an additional resistance to motion caused by the collision of molecules with solid particles embedded in the gas. The effect of molecule-solid collisions on the rate of diffusion is characterized by the effective Knudsen diffusion coefficient. When the diffusion problem of interest occurs in the transition regime (porous solids of small to moderate permeability), the applicable flux equations involve both the molecular and Knudsen diffusion coefficients. This is because neither inter-molecular nor molecule-solid collisions dominate in the transition regime. These flux equations will appear unfamiliar to those who are used to thinking about diffusion in spaces free of obstruction where Fick’s law applies.

The molecular diffusion regime prevails when the pores are so large that molecule-molecule collisions account for essentially all resistance to diffusion. This regime occurs in porous media with moderate to high permeability. Even in this case, diffusion fluxes are not independent of molecule-solid collisions and the component flux equations are generally not the familiar Fick’s law expressions. Molecule-solid collisions during isobaric diffusion in the molecular diffusion regime are responsible for the fact that component fluxes are not of equal magnitude. The difference between the magnitudes of the fluxes is a diffusion-generated bulk flow that is not a viscous flow (i.e., there is no pressure gradient associated with this non-equimolar flow). This case becomes a familiar Fick’s law problem if the molecular weights of the diffusing species are equal or if the species of interest is present only in trace quantities (i.e., mole fraction of one component everywhere is much less that unity). The latter of these conditions is often encountered in environmental applications.

Isobaric diffusion can be established in the laboratory where the pressure on the system boundaries can be externally controlled, but diffusion-generated viscous flow and associated pressure gradients are to be expected in virtually all field settings. We have provided an example of how this occurs in a semi-open system at steady state. Even in a completely open system, transient pressure gradients develop during unsteady flow. However, the practical significance of diffusion-generated pressure gradients is a separate question, one for which an answer can be provided only on a case-by-case basis.


Flux Equations for Gas Diffusion in Porous Media Copyright © 2021 by David B. McWhorter. All Rights Reserved.