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Critical entry pressure of a droplet pinning over multitude of pore openings

Porous membranes filter by the virtue of their pore sizes in relation to the sizes of dispersals. While this is essentially true for solid dispersals, it needs to be reframed when dispersals are droplets. That is, without the existence of other selectivity criterion (other than pore sizes), droplets are prone to permeation, irrespectively. Fortunately, this extra criterion exists via the use of interfacial phenomena. That is, if the materials of the membrane are cast such that they are nonwetting with respect to droplets, interfaces are formed at pore openings that prevent droplets from permeation if the operating pressure is kept smaller than the entry pressure. Therefore, it is important to estimate such critical entry pressure under the different wettability conditions and droplet to pore ratios. Previous works have looked at droplets pining over single pore openings. In this work, the case in which relatively larger size droplets pin over multiple pore openings is investigated theoretically and via the tools of computational fluid dynamics. An exact formula is derived that account for the volumes of that part of the droplet hanging at the pore openings. An approximate formula is also highlighted that ignores this volume and compares very well with the exact formula. This derivation is based on the assumption that the droplets maintain their spherical shape, which is typically the case for smaller size droplets in produced water applications. The study shows that a pining droplet permeates first through the largest size pore until its size matches the critical size associated with the next larger pore opening when it starts to permeate.