The build-up of contaminants at the wall of cross-flow membrane filtration systems can be detrimental to the operation of such systems because of, amongst other things, the osmotic backflow it may induce. In this paper, we propose a strategy to avoid the negative effects of backflow due to osmosis by using 2D channels bounded by walls with a combination of permeable and impermeable segments. We show that preventing flow through the final portion of the channel can increase the efficiency of filtration and we determine the optimal fraction occupied by the permeable wall that maximizes efficiency. Our analysis uses a combination of numerical techniques and asymptotic analysis in the limit of low wall permeabilities. Finally, we consider how the energy cost of filtration depends on the Péclet number and show that the energy cost per unit of filtered water may be minimized by appropriately choosing both the Péclet number and the permeable-region fraction.
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