Rarefied Pure Gas Transport in Non-isothermal Porous Media: Effective Transport Properties from Homogenization of the Kinetic Equation

Abstract

Abstract. Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper develops a model based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described e. g. by a digitized 3D image. The homogenization procedureneglecting the effect of gas density gradients on heat transfer through the solid -leads to macroscopic transfer relations, and to closure problems in R 6 for the obtention of effective properties. Coherence of the approach with previous literature on the subject is discussed. The asymptotic limits of the model (rarefied and continuum regimes) are also studied. One of the main results is that the effect of the geometry on thermal transpiration has to be described by a tensor which is distinct from the permeability and Knudsen diffusion tensors.

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Validation and Tests of the Model

“…Based on a homogenization model presented in a companion paper (Vignoles et al, 2008), numerical schemes have been produced for the computation of effective transport coefficients (i.e. Knudsen transport, viscous transport, and thermal transpiration coefficient) in whatever 3D image of a porous medium.…”

“…The associated energy transport is also treated. The performed change of scale provides a set of macroscopic variables and equations, as well as a set of closure problems that are to be solved at microscopic scale (Vignoles et al, 2008).…”

“…The full derivation of the model is discussed in (Vignoles et al, 2008) and (Charrier and Dubroca, 2003). Here the main steps of the procedure will be recalled.…”

“…Such a choice is conveniently guided by the analysis of the continuum and pure Knudsen regimes. In the continuum regime, remembering that the zero-order expansion of β is described by (Vignoles et al, 2008) :…”

“…In the free-molecule limit, where the collision operator vanishes, the result of the Knudsen limit analysis (Vignoles et al, 2008) yields a dependance of β to other functions, which are not polynomials of c, but are based on the normal vector associated to c, namely n i = c i |c| . So, it is chosen to add the following moments to the preceding ones :…”