Conservative mathematical model and numerical simulation of batch gravity settling with coalescence of liquid-liquid dispersions

Population balance equations (PBE) provide a suitable a framework for dealing with drop breakage and coalescence. In addition, modeling of the solid particle sedimentation process based on Kynch's theory has been successfully used and validated in mineral processing and wastewater treatment. In this work, we present a model that merges the coalescence process with hindered polydisperse sedimentation. The PBE model is projected onto a partial differential equation (PDE) system by discretizing the droplet volume. Because there is loss of mass in the system when the daughter droplets are greater than the larger species considered for the numerical solution, two terms that produce the conservation of the mass or total volume of the dispersed phase are incorporated into the PDE system. The resulting PDE system is split into two systems: homogeneous PDEs and ordinary differential equations (ODEs). The homogeneous PDEs and the ODEs are discretized using a first-order central differencing scheme and the second-order, two-stage Runge-Kutta method, respectively. The model predicts the vertical variation of the composition of the dispersed phase layer that forms at the top or bottom of the gravity settler. The proposed model was calibrated and validated through an experiment with an oil and water system. In particular, simulations illustrate the effects of: the continuity of the dispersion (oil-in-water and water-in-oil) and the standard deviation of the initial droplet volumes on phase separation quality, as well as the influence of the coalescence frequency on the average droplet volume. (C) 2019 Elsevier Ltd. All rights reserved.