Diffusive mass transport in the fluid–porous medium inter-region: Closure problem solution for the one-domain approach

A challenging problem for diffusive mass transport is to describe and model the phenomena concerning the fluid-porous medium inter-region. Volume averaging techniques that provide a framework for rigorously addressing the issue of obtaining macroscopic models from pointwise models at fluid and porous medium scales have been used to attend the problem. The efforts have resulted in two modeling approaches. The first one, known as the one-domain approach (ODA), considers the system as a continuum where the geometrical (e.g., porosity) and transport parameters (e.g., diffusivity) display rapid spatial changes in the inter-region. The second one, known as the two-domain approach (TDA), uses different models for the fluid and the porous medium scales, and matches them via the development of corresponding jump conditions at the dividing surface. Recent results [Wood, B.D, Quintard, M., Whitaker, S. (2000). Jump condition at non-uniform boundaries: the catalytic surface. Chemical Engineering Science 55, 5231–5245; Valdés-Parada, F.J., Goyeau, B., Ochoa-Tapia, J.A. (2006). Diffusive mass transfer between a microporous medium and an homogeneous fluid: jump boundary conditions. Chemical Engineering Science 61, 1692–1704] have shown that the coefficients involved in the jump conditions can be computed by solving the associated closure problems. However, in the development of the jump conditions some complications arise due to the difficulty of modeling some of the “surface excess” transport mechanisms that take place in the inter-region. To address this problem, an implicit formulation based on the ODA and TDA is proposed. Although the ODA seems to be more suitable for modeling, it requires the knowledge of the spatial variations of the transport parameters. Heuristic interpolations between the fluid and the porous medium parameters have been commonly used; however, there is no guarantee that such models can provide an accurate description of the mass flux. Within a ODA framework, the aims of this paper are: (i) to show that the effective diffusivity coefficient for the case of passive diffusion in a fluid–porous medium inter-region can be posed as a closure problem derived from volume averaging techniques and (ii) to use a simple one-dimensional model to show that a complete knowledge of spatial variations of diffusivity and porosity are necessary for an accurate description of the mass transport phenomena in the entire fluid–porous medium system. The analysis has allowed us to identify a new contribution to the jump at the dividing surface. This contribution consists of the accumulation that occurs at the dividing surface even when there is no chemical reaction or adsorption taking place.