A finite element algorithm for particle/droplet trajectory tracking, tested in a liquid-liquid system in the presence of an external electric field
In this paper we are concerned with modeling a dispersion of electrically charged droplets in motion in a second immiscible liquid phase continuum in the presence of an external electric field. The system is highly relevant to solvent extraction processes and to liquid-liquid reactions such as phase transfer catalysis. Electrostatic enhancement of liquid-liquid contacting processes is well known and relies upon intensified drop breakup and drop acceleration due to the presence of significant electrical forces. The enhanced drop motion results in greatly increased rates of mass transfer. The scope of the current work is the mathematical modeling and experimental validation of the trajectories of multiple charged drops in three-dimensional motion in a liquid-liquid system in the presence of an externally applied field. A new particle tracking algorithm using a tetrahedral finite element mesh to solve the relevant system of differential equations is described in detail. Such elements are shown to be especially convenient for real word simulation because of their simplicity, their high stability with strictly linear interpolation, and their flexibility for dealing with complex computational domains. The calculations of trajectory tracking for several hundred particles taking in a mesh consisting of 50,000 tetrahedral elements was shown only to require a few seconds of computational time using a PC class computer. Using a double step stepping algorithm, with intermediate velocity interpolation, second-order convergence was proven in a rigorous benchmark test. Comparison of simulations of individual drop motion with experimental measurements proved to be very accurate. The full three-dimensional simulation, having greater requirements and involving more degrees of freedom than the earlier two-dimensional case was successfully demonstrated, with greater accuracy and economic use of computational time. Preliminary simulation of swarming motion of droplets using a cloud model is also presented. This shows significant promise for the evaluation of the mass distribution of very small droplets (similar to 1 to 100 mu m) in a real contactor for optimal shape design. The model, when coupled with the models for mass transfer kinetics and interfacial reaction kinetics, should provide a very valuable tool for liquid-liquid reactor design. (c) 2006 Elsevier Ltd. All rights reserved.
liquid-liquid system charged drops finite element analysis trajectory prediction swarming drops