A combined lossy capacitor population balance model (LCPBM) for calculating the influence of frequency on electric field enhanced coalescence in a static-mixer settler setup

In this work a combined lossy capacitor population balance model (LCPBM) is developed to predict the effect of a square wave frequency on electric-field coalescence (drop size) in a static-mixer settler setup for the caprolactam-toluene-water system. The static-mixer is used to mix the organic and aqueous phases. The electric field is applied by placing insulated electrodes at the end of the static-mixer. The electrical-circuit model of the system is based on the lossy capacitor model where the insulation and the dispersion are considered as leaky dielectrics. The charge and the electric field are determined from the electrical-circuit model and coupled in the hydrodynamic equation to calculate the velocities of drops. The velocities are in turn used in the population balance model (PBM) to determine the time evolution of the drop size. Three approaches are used to calculate the electrical force and the charge on the drops from the circuit model: (1) the Bailes′ charge hypothesis where the drops pick their charge from the free dispersion-insulation interfacial charge, and the electrical force calculated based on electrophoretic contribution, (2) the electrical force determined from the Taylor-Melcher leaky dielectric model considering the electrical force due to the free charge and the force due to the difference in dielectric permittivity's of the dispersion and insulation, and (3) by accounting for the free charge convection by the fluid motion which results in equal redistribution of the polarization charge between the dispersion and the insulation. The (LCPBM) model was validated by measuring drop sizes using a square wave of 0.4 kV/cm and frequencies between 3 and 100 Hz for two flowrates. With all approaches, the mean sauter diameters were calculated within 10% relative error at lower frequencies for both flowrates. At higher frequencies of 50 Hz and 100 Hz, a 20% relative error was obtained for the first approach. A better prediction within 10% was found for the second and third approaches with the later approach giving the best prediction.