A Simplified Model for the Evolution of the Drop Size Distribution in an Emulsion by Breakup: A New Breakup Kernel

A theoretical study has been developed for the evolution of a drop size distribution in an emulsion under conditions where drop breakup dominates drop coalescence. Previous experiments and numerical simulations demonstrate that the size of daughter drops produced by breakup in a particular fluctuation scale with the critical size drop for the fluctuation; the volume of the parent drop only determines the number of daughter drops produced by the breakup event. A simplified population balance model predicated on this observation will be presented. The essential simplification involves the replacement of the usual two-variable daughter-drop distribution function by a single-variable distribution that describes the volume of daughter drop relative to the critical size drop. The resulting model equations are analyzed and a long-time similarity formulation is derived. Analytical formulas are derived for certain limiting cases and a complete set of numerical results are presented for a wide range of parameter values. Experiments in the literature (Lam, Sathyagal, Kumar and Ramkrishna, 1996) are shown to strongly support our new drop breakup model.