Filters that act by adsorbing contaminant onto their pore walls willexperience a decrease in porosity over time, and may eventually block. Asadsorption will generally be larger towards the entrance of a filter, where theconcentration of contaminant particles is higher, these effects can also resultin a spatially varying porosity. We investigate this dynamic process using anextension of homogenization theory that accounts for a macroscale variation inmicrostructure. We formulate and homogenize the coupled problems of flowthrough a filter with a near-periodic time-dependent microstructure, solutetransport due to advection, diffusion, and filter adsorption, and filterstructure evolution due to the adsorption of contaminant. We use thehomogenized equations to investigate how the contaminant removal and filterlifespan depend on the initial porosity distribution for a unidirectional flow.We confirm a conjecture made in Dalwadi et al. (2015) that filters with aninitially negative porosity gradient have a longer lifespan and remove morecontaminant than filters with an initially constant porosity, or worse, aninitially positive porosity gradient. Additionally, we determine which initialporosity distributions result in a filter that will block everywhere at once byexploiting an asymptotic reduction of the homogenized equations. We show thatthese filters remove more contaminant than other filters with the same initialaverage porosity, but that filters which block everywhere at once are limitedby how large their initial average porosity can be.
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