Abstract
The process of transport and trapping of arsenic ions in porous water filters is treated as a classic mass transport problem which, at the pore scale, is modeled using the traditional convection-diffusion equation, representing the migration of species present in very small (tracer) amounts in water. The upscaling, conducted using the volume averaging method, reveals the presence of two possible forms of the macroscopic equations for predicting arsenic concentrations in the filters. One is the classic convection-dispersion equation with the total dispersion tensor as its main transport coefficient, and which is obtained from a closure formulation similar to that of the passive diffusion problem. The other equation form includes an additional transport coefficient, hitherto ignored in the literature and identified here as the adsorption-induced vector. These two coefficients in the latter form are determined from a system of two closure problems that include the effects of both the passive diffusion as well as the adsorption of arsenic by the solid phase of the filter. This theoretical effort represents the first serious effort to introduce a detailed micro–macro coupling while modeling the transport of arsenic species in water filters representing homogeneous porous media.
Highlights
We will employ the volume averaging method to upscale the phenomenon of solute transport accompanied with adsorption in homogeneous porous media
Plumb and Whitaker [34] presented the upscaling theory corresponding to diffusion, adsorption and advection in porous media composed of porous particles in Section 5 of [34]
Note that for single-phase flow of the β phase through our porous medium, e β will be equal to the porosity of the porous medium, since the latter is defined as the ratio of the total pore volume within representative elementary volume (REV) to the total REV volume
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. We will employ the volume averaging method to upscale the phenomenon of solute transport (which include both diffusion and advection) accompanied with adsorption in homogeneous porous media. Such media are found in commercial water filters where the cartridges created by packing particles or beads that can be assumed to be of mono-modal size distribution and create single-scale porous media. Plumb and Whitaker [34] presented the upscaling theory corresponding to diffusion, adsorption and advection in porous media composed of porous particles in Section 5 of [34] This one-equation model approach was a multi-scale treatment that involved lower-scale averaging inside what will be our solid phase here. A researcher experienced in the volume average method may find several portions of the manuscript repetitions of what is available in the literature; the authors feel that all the main derivations should be presented in the paper here in order to improve its readability and bring diverse aspects into a single presentation
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