Dispersion and reaction in two-dimensional model porous media

Abstract

Darcy-scale convective–diffusive–reactive phenomenological coefficients characterizing the transport of a reactive solute through the interstices of a two-dimensional, spatially periodic, model porous medium (on whose surfaces the solute undergoes a first-order, irreversible chemical reaction) are herein determined numerically as functions of the microscale Péclet (Pe), Damköhler (Da), and Reynolds (Re) numbers. The role of bed porosity ε and (circular) cylindrical array configuration are also studied, the latter encompassing square, staggered, and hexagonal arrays. Calculations are effected via generalized Taylor dispersion theory. The Darcy-scale reactivity coefficient K̄* characterizing the effective (first-order, irreversible) volumetric reaction rate is found, inter alia, to be (approximately) inversely proportional to Pe, a conclusion confirmed by analytical results for the limiting case of small Da. Configurational properties of the porous medium are observed to significantly influence K̄*, especially for small porosities and large Da. Moreover, it is found that the mean interstitial velocity vector Ū* of the reactive solute generally differs (often dramatically) from the comparable velocity vector (1/ε)Ū of the (inert) solvent as a consequence of the chemical reaction occurring at the surfaces of the cylindrical bed particles. These data reveal that the mean solute speed ‖Ū*‖ through the interstices may be larger or smaller than the comparable solvent speed (1/ε)‖Ū‖, depending upon the existence and nature of a diffusive boundary layer adhering to the cylindrical bed-particle surfaces. Finally, the longitudinal and lateral components of the solute’s (transversely isotropic) dispersivity dyadic D̄*, parallel and perpendicular, respectively, to the direction of mean flow, are generally observed to decrease with increasing Da. This behavior stems from the fact that, in the diffusion boundary-layer limit, an increasing proportion of the total depletion of solute (via microscale reaction at the cylinder surfaces) arises from those interstitial zones characterized by the existence of large velocity gradients.