For the Fick’s law or the Mixture-averaged diffusion models, the diffusion coefficient is corrected with the Wall diffusion coefficient in the following way

For the Maxwell–Stefan diffusion model (see Ref. 9), the following term is added to the species force balance (Equation 3-75)

The wall velocity uW, representing the velocity of the fluid in a representative region adjacent to the pore wall, is here defined as

It should be noted that uW can become nonzero also in the absence of pressure gradients. For a proper description of the fluid momentum balance, uW needs to be added either as a contributing velocity (Darcy’s law) or as a corresponding volume force F = μκ−1uW (Brinkman equations).

The Maxwell–Stefan model formulated in this way correctly captures the Klinkenberg effect, and has large similarities with other models in literature incorporating Knudsen diffusion such as the Dusty Gas Model and the (Modified) Binary Friction Model. In addition, for the limit of vanishing permeabilities (κ → 0), the formulation reduces to the Lightfoot model for membrane transport.