Numerical solution of bivariate population balance equation using an open-source finite element CFD framework (MS)

Abstract

The population balance equation (PBE) is an established mathematical framework to explain the evolution of polydisperse multiphase systems. The employment of a bivariate or multivariate PBE model, characterized by two or more internal coordinates, is required to analyse a variety of systems. The direct quadrature method of moments (DQMOM), a computationally economical and favourable numerical method for the PBE–computational fluid dynamics (CFD) coupling, can be conveniently applied for solving bivariate or multivariate PBEs. In the past, DQMOM has been implemented in a few commercial finite volume CFD packages to solve bivariate PBEs. However, to date, no open-source CFD package contains this numerical technique for solving the bivariate PBEs as a standard implementation. In this work, for the numerical solution of the bivariate PBE the DQMOM is for the first time implemented in an open-source CFD code—Fluidity. This efficient numerical framework is a highly-parallelised finite element (FE) CFD code that allows for the use of mesh adaptivity on fully-unstructured meshes. To evaluate the accuracy of the bivariate PBE solution using DQMOM in the present FE framework, various test cases to solve spatially homogeneous bivariate PBEs with aggregation, breakage, growth and dispersion (diffusion in phase space) were simulated and verified against analytical solutions, resulting in excellent agreement. Benchmarking, by comparison with the Monte-Carlo method solutions from the literature, with realistic kernels in a gas-liquid system for simultaneous bivariate aggregation and breakage was also performed to show the feasibility of this implementation for realistic applications. This open-source framework demonstrates its impressive potential in the case of bivariate PBE and can be exploited for the simulation of the complex polydisperse multiphase system.