(350aa) Implementation and verification of numerical solution of the bivariate population balance equation in a finite-element open-source framework

Polydisperse multiphase flows appear in a large number of processes in the chemical industry. The population balance equation (PBE) is frequently used for the modelling of the evolution of such dispersed-phase systems. Monovariate PBE models, characterized by one specific property (or one internal coordinate), can be effectively managed by a variety of numerical techniques, while the solution of bivariate (or, in general, multivariate) PBEs is still considered a challenging task. While in the past much research has been conducted on the coupling of the solution of the PBE to computational fluid dynamics (CFD) packages, numerical methods have typically been computational expensive. An implementation of an accurate and computationally affordable numerical method for solving multivariate PBEs is still required. Notably, to date, an implementation of DQMOM for solving the bivariate PBE is not available in any open-source CFD package as a standard feature. The present work focuses on the implementation of the direct quadrature method of moments (DQMOM), a favourable numerical method for the PBE-CFD coupling and computationally economical for solving the bivariate PBE in a finite element (FE) open-source code – Fluidity. This FE framework is highly-parallelised with adaptive-mesh refinement on fully-unstructured meshes. In order to investigate the accuracy of the bivariate PBE solution in the proposed framework, transient homogeneous bivariate aggregation and bivariate breakage simulations were performed and verified against analytical solutions, showing excellent agreement. The results show that the proposed finite element framework produces precise numerical solutions for the bivariate PBEs and is a robust tool that can be exploited for the modelling of complex polydisperse multiphase flows.