A Numerical Framework for the Solution of Bivariate Population Balance Equation—Model Implementation and Verification

Abstract

In the bivariate population balance model, the dispersed phase particles are represented using two internal coordinates. The direct quadrature method of moments (DQMOM) is a computationally efficient and accurate numerical technique to solve monovariate and bivariate population balance equations (PBE) [4]. In the past, bivariate PBEs have been solved using DQMOM in a few commercial and non-commercial finite-volume computational fluid dynamics (CFD) packages. However, no code contains this solution technique as a standard implementation. Moreover, all previous implementations of DQMOM have been finite volume based. Recently, DQMOM was implemented for solving monovariate PBE in a highly parallelised finite element open-source framework—Fluidity [1]. The implementation was shown to be extremely efficient in the solution of monovariate PBEs and their integration to fluid flow problems. This is particularly true due to the anisotropic adaptive unstructured meshes utilised in the Fluidity code, which significantly reduces the computational cost maintaining the solution accuracy. In the present work, DQMOM is implemented in Fluidity for solving bivariate PBE. Test cases with homogeneous aggregation and homogeneous breakage were simulated and verified against the analytical solution, showing excellent agreement. In the extension to the implementation of DQMOM for solving bivariate PBE, the potential of this finite element framework to integrate the bivariate PBE with flow problems is maintained by default.