An analytical solution to the fixed pivot fragmentation population balance equation

Abstract

Drop breakup in high-energy emulsification is often modelled using a population balance equation (PBE) framework. For cases with high emulsifier to disperse phase concentration, the PBE is dominated by the fragmentation term. Since there are no analytical solutions to the continuous PBE for physically reasonable fragmentation rate expressions, a discretization is often needed to evaluate the PBE, turning it into a set of ordinary differential equations that can be solved numerically. The fixed pivot technique (Kumar and Ramkrishna, 1996) is one of the most often applied class discretization methods. This contribution suggests an analytical solution to the fixed pivot technique fragmentation equation, that can be used instead of the traditional numerical approach provided that the fragmentation rate is constant over time. The proposed solution compares favorably to two special cases where analytical solutions for the continuous PBE are available, and to two more realistic PBE emulsification problems (with varying fragmentation intensity and varying number of fragments formed per breakup), while offering a substantial reduction in computational time compared to the traditional approach of solving the discretized equations numerically.