An analytical treatment to spatially inhomogeneous population balance model

In modern liquid–liquid contact components, there is an increasing use of droplet population balance models. These components include differential and completely mixed contractors. These models aim to explain the complex hydrodynamic processes occurring in the dispersion phase. The hydrodynamics of these interacting dispersions include droplet breaking, coalescence, axial dispersion, and both entry and exit events. The resulting equations for population balance are represented as integro-partial differential equations, which rarely have analytical solutions, especially when spatial dependency is apparent. Consequently, the pursuit predominantly lies in seeking numerical solutions to resolve these complex equations. In this study, we have devised analytical solutions for inhomogeneous breakage and coagulation by employing the population balance equation (PBEs) applicable to both batch and continuous flow systems. The innovative approaches for solving PBEs in these systems leverage the Adomian decomposition method (ADM) and the homotopy analysis method (HAM). These semi-analytical methodologies effectively tackle the significant challenges related to numerical discretization and stability, which have often plagued previous solutions of the homogeneous PBEs. Our findings across all test examples demonstrate that the approximated particle size distributions utilizing these two methods converge to the analytical solutions continuously.