Modelling of multi-component gas flows in capillaries and porous solids

Abstract

The paper presents a derivation of the governing equations for multi-component convective-diffusive flow in capillaries and porous solids starting from a well-defined model and clear assumptions. The solution for the continuum regime is discussed in detail including a derivation of the diffusion slip boundary condition based on an improved momentum transfer theory. The Stefan–Maxwell species momentum equations are also re-examined and important distinctions made between the local and tube-averaged equations. An equation for the pressure gradient is derived and some examples of binary flows in capillaries are discussed. The theory for free-molecule flow is standard but the equations are recast into a form identical to the continuum equations which suggests an obvious method of interpolation for flow at arbitrary Knudsen number. There are no problems concerning viscous terms which have marred other derivations. The extension to flow in porous bodies is achieved by introducing a porosity–tortuosity factor but, unlike other treatments, this parameter is not absorbed into the gas diffusivities and flow permeability. It can then be eliminated from all but one of the equations and, with appropriate boundary conditions, the flux ratios can be obtained in terms of a mean pore radius only. The porosity–tortuosity parameter simply controls the absolute flux level and is best interpreted as a length scale-factor. The theory is applied with success to the prediction of some experimental data for helium–argon counter-diffusion and it is shown that, contrary to common belief, the mean pore radius is well-defined by flux ratio measurements if these are made with non-zero pressure differences.