On the estimation of the leaked volume of an oil droplet undergoing breakup in crossflow filtration: CFD investigation, scaling, and a macroscopic model

Abstract

The fate of an oil droplet at the surface of a membrane in crossflow filtration has been identified as either of four destinies; namely, pinning, permeation, rejection or breakup. These fates are determined according to the operating conditions (transmembrane pressure, TMP, and crossflow velocity, CFV) in relation to the critical conditions (critical entry pressure,pcrit, and critical velocity of dislodgment, vcrit). These conditions have enabled the establishment of fate maps for every combination of pore and droplet sizes. Such maps have been the bases for the newly developed multicontinuum modeling approach, which describes the permeation process not only of the continuous phase but also of the inevitably associated dispersed phase. In the multicontinuum approach, both the droplet and pore size distributions are used to establish multitude of continua that interact according to the previously mentioned maps. Therefore, every oil continuum is checked against all membrane continua to determine how the droplets behave. According to this procedure, the permeation flux of the oil as well as the rejection capacity of the membrane can be estimated. At the moment of breakup, the portion of the droplet that has advanced inside the pore depends on the ratios TMP/pcrit andCFV/vcrit. In this work, we investigate this topic thoroughly by conducting computational fluid dynamics (CFD) analysis of various combination of operating and critical conditions. A macroscopic relationship is developed to allow for the estimation of the volume of the droplet inside the pore opening upon breakup under different conditions. This formula allows the accurate estimation of the permeation flux of those oil droplets that have experienced breakup conditions, which allow its use in the context of multicontinuum approach. From this study it is found that the volume of the droplet inside the pore is large the larger the ratio of the transmembrane pressure with respect to the critical entry pressure. It is also found that the larger the feed velocity compared with the critical velocity of dislodgment the smaller the size of that part of the droplet inside the pore. The latter is a manifestation of the increase in hydrodynamic drag as a result of the increased velocity. A scaling analysis has also been conducted to show how the breakup time is correlated with the ratio of the feed stream velocity and the critical velocity of dislodgment.