This study addresses the issue of oil removal from water using hydrophilic porous membranes. The effective separation of oil-in-water dispersions involves high flux of water through the membrane and, at the same time, high rejection rate of the oil phase. The effects of transmembrane pressure and crossflow velocity on rejection of oil droplets and thin oil films by pores of different cross-section are investigated numerically by solving the Navier-Stokes equation. We found that in the absence of crossflow, the critical transmembrane pressure, which is required for the oil droplet entry into a circular pore of a given surface hydrophilicity, agrees well with analytical predictions based on the Young-Laplace equation. An analytical expression for the critical pressure in terms of geometric parameters of the pore cross-section is validated via numerical simulations for a continuous oil film on elliptical and rectangular pores. With increasing crossflow velocity, the shape of the oil droplet is strongly deformed near the pore entrance and the critical pressure of permeation increases. We determined numerically the phase diagram for the droplet rejection, permeation, and breakup depending on the transmembrane pressure and shear rate. The critical pressure of permeation is identified as the line separating permeation and rejection regions. Using a novel method for computing the critical pressure, we investigated the effect of various physical and geometrical parameters on the critical pressure of permeation and breakup of droplets under shear flow. It is demonstrated numerically that the critical pressure of permeation increases with shear rate, viscosity ratio, surface tension coefficient, contact angle, and droplet size. On the other hand, droplet breakup at the pore entrance is facilitated at lower values of the surface tension coefficient, higher oil-to-water viscosity ratio, and larger droplet size. Using simple force and torque balance arguments, an estimate for the increase in critical pressure due to crossflow and the breakup capillary number is obtained and validated for different viscosity ratios, surface tension coefficients, contact angles, and drop-to-pore size ratios.
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