Numerical simulation and convergence analysis of a finite volume scheme for solving general breakage population balance equations

Abstract

This paper presents a finite volume scheme (FVS) for solving general breakage population balance equations (BPBEs). In particular, the number density based BPBE is transformed to the form of a mass conservation law. Then it becomes easy to apply the well known FVSs that have an important property of mass conservation. Following Kumar and Warnecke for fixed pivot (FP) method [16] and cell average technique (CAT) [15], the stability and the convergence analysis of the semi-discretized FVS are studied. Unlike the CAT and the FP method, the FVS is second order consistent independently of the type of meshes. We also observe that FVS gives second order convergence rate on four different types of uniform and non-uniform meshes with non-decreasing behavior of mesh width. Nevertheless, one order better accuracy than the FP method is achieved on locally uniform meshes. It is also noticed that on non-uniform random meshes the FVS shows one order and two orders higher accuracy than the CAT and the FP method, respectively. The mathematical results of convergence analysis are validated numerically by taking three test problems. The numerical simulations are also compared with the results obtained by the CAT and the FP method.