Briefly, four different modes of transport existing in porous media are incorporated within the model, namely, Knudsen flow
, bulk/viscous flow, ordinary diffusion, and surface flow and we consider a similar schematic diagram for the counter-current flow membrane used previously by Shindo et al.
Steckelmacher, "Knudsen flow
75 years on: the current state of the art for flow of rarefied gases in tubes and systems," Reports on Progress in Physics, vol.
It is a general expression to capture continuum, transition, and Knudsen flow for the apparent gas permeability of tight porous media.
It is noted that the Knudsen flow relies only on the Knudsen number and the intrinsic permeability of the porous medium.
At very low pressures, Equation (8) approaches the expression for the diffusion coefficient in the Knudsen flow regime (Loeb, 1961):
In the Knudsen flow regime the diffusion coefficient is independent of pressure and the governing PDE simplifies to Fick's second law:
However, at p = 0.13 Pa, the resulting D represents the limiting value for the Knudsen flow regime.
This can be: i) transport through a dense (nonporous) layer, ii) Knudsen flow in narrow pores, iii) viscous flow in wide pores or iv) surface diffusion along the pore wall (12).
When the pore size is reduced to sizes between 0.002 and 0.05 [micro]m (13), the main transport mechanism is Knudsen flow. This pore size, found in ultrafiltration membranes, is so small that the molecules interact primarily with the pore wall.