Capillary-Driven Ejection of a Droplet from a Micropore into a Channel: A Theoretical Model and a Computational Fluid Dynamics Verification

In this work, the problem of re-ejection of a permeating droplet through a membrane pore back to the feed channel when the transmembrane pressure (TMP) becomes zero is investigated. This problem is important in the context of oily water filtration using membranes. In particular, in the novel periodic feed pressure technique (PFPT), which has been proposed to combat membrane fouling, the TMP alternates between the operating value and zero in a periodic manner. During the period in which TMP is high, filtration occurs, and when it is zero, cleaning commences. We are particularly interested in what happens to a droplet, initially undergoing permeation, when the TMP becomes zero. It is evident that when the TMP is zero the meniscus inside the pore reverses its motion toward the feed channel rather than toward the permeate side by the action of interfacial tension force. A theoretical model is built to determine the rate at which the meniscus inside the pore advances when the TMP is zero. The conservation of momentum equation is used to establish a one-dimensional model that updates the location of the meniscus with time. The derived model considers both quasi-static and dynamic scenarios. In addition, the model accounts for both the viscosity contrast between the two fluids, as well as the gravity. A computational fluid dynamics (CFD) simulation has been built to provide a framework for model verification and validation. The model, based on quasi-static conditions, provides an overall similar trend to that obtained via CFD analysis. Nevertheless, the quasi-static model predicts a more rapid meniscus advancement inside the pore than the CFD simulation. When the dynamic contact angle is incorporated, very good matching is observed.