5.3 Transition Regime – Constant and Non-uniform Pressure

Diffusion in the transition regime is distinguished by the fact that resistance to diffusion offered by both molecule-molecule and molecule-particle collisions must be considered. That is, the second term on the right of Equation 14 must be retained. For the isobaric condition, the relation between the component fluxes is given by Graham’s law, Equation 21. Upon introduction of Equation 21 into Equation 22, followed by some algebraic manipulation, the equation for the flux of species A under isobaric conditions is derived to be Equation 27.

\displaystyle N_{A}^D=-\frac{DC\ dx_{A}/dl}{\left ( 1+D/D_{A}^{K} \right )-\left ( 1-M_{AB}^{0.5} \right )x_{A}} (27)

The corresponding result for the non-isobaric condition is obtained by using Equation 20 in Equation 22 and adding advection via viscous flow to obtain Equation 28.

\displaystyle N_{A}=-\frac{DC\ dx_{A}/dl+\left ( D+D_{B}^{K} \right )x_{A}dC/dl}{\left ( 1+D/D_{A}^{K} \right )-\left ( 1-M_{AB}^{0.5} \right )x_{A}} \displaystyle -x_{A}\frac{k_{g}p}{\mu }\frac{dC}{dl} (28)

As usual, the corresponding equation for the flux of species B is obtained by interchanging the subscripts.

We appealed to the conditions D/D_{i}^{K} < < 1, i = A, B and D_{i}^{K}\mu /k_{g}p < < 1, i = A, B to justify simplifications leading to Equation 26, applicable in the molecular regime. Neither of these conditions generally applies in the transition regime now under consideration. However, important to groundwater scientists and engineers is the circumstance in which the species of interest, say species A, is present only in trace concentrations. Equation 28 can then be simplified to Equation 29.

\displaystyle N_{A}=-\frac{DC\ dx_{A}/dl}{(1+D/D_{A}^{K})} \displaystyle -\left ( \frac{D+D_{B}^{K}}{1+D/D_{A}^{K}}+\frac{k_{g}p}{\mu } \right )\left ( x_{A} \frac{dC}{dl}\right ) (29)

This result is of the same mathematical form as the widely used advection-diffusion model. Webb and Pruess (2003) calculated the transport of trace species using a flux equation that can be derived from Equation 29 written on a mass flux basis. Click on this exercise link to view an example problem Exercise 5.


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