A COMPARATIVE STUDY OF NUMERICAL APPROXIMATIONS FOR SOLVING THE SMOLUCHOWSKI COAGULATION EQUATION

M Singh,J Kumar,I Nopens,G Kaur,T De Beer

doi:10.1590/0104-6632.20180354s20170050

Abstract

ABSTRACT In this work, numerical approximations for solving the one dimensional Smoluchowski coagulation equation on non-uniform meshes has been analyzed. Among the various available numerical methods, finite volume and sectional methods have explicit advantage such as mass conservation and an accurate prediction of different order moments. Here, a recently developed efficient finite volume scheme (Singh et al., 2015) and the cell average technique (Kumar et al., 2006) are compared. The numerical comparison is established for both analytically tractable as well as physically relevant kernels. It is concluded that the finite volume scheme predicts both number density as well as different order moments with higher accuracy than the cell average technique. Moreover, the finite volume scheme is computationally less expensive than the cell average technique.

Highlights

Population balance equations (PBEs) generally describe the changes taking place in the particles due to certain mechanisms like coagulation, breakage, growth and nucleation
The results revealed that both methods predict the number distribution functions with comparable accuracy, but the various order moments are more accurately predicted by the cell average technique as compared to the finite volume scheme
The results reveal that the Finite Volume Scheme (FVS) predicts both order moments with more accuracy than the Cell Average Technique (CAT)

Summary

Introduction

Population balance equations (PBEs) generally describe the changes taking place in the particles due to certain mechanisms like coagulation (aggregation), breakage, growth and nucleation. Many other applications which involve these mechanisms are crystallization, granulation, fluidized bed etc., as illustrated in Marchal et al (1988), Ramkrishna (2000), Peglow et al (2005, 2006) and Kettner et al (2006). Population balance equations are classified in the class of integro-partial differential equations of the hyperbolic or parabolic type. Our main concern is to solve the one dimensional Smoluchowski coagulation population balance equation, which in the continuous form can be written as (Hulburt and Katz, 1964): ∂n (t,u) = ∂t 1 2 u ∫ o b ( v, u − v ) n

Objectives

Methods

Results

Conclusion