On the Solution of the Smoluchowski Coagulation Equation Using a Conservative Discretization Approach (CDA)

• Published in: Computer Aided Chemical Engineering Vol. 46; pp. 691 - 696
• Main Authors: Attarakih, Menwer, Bart, Hans-Jörg
• Format: Book Chapter
• Language: English
• Published: 2019
• Subjects: CDA; Coagulation; Population Balances; Smoluchowski; Smoluchowski; CDA; Population Balances; Coagulation
• ISBN: 9780128186343; 0128186348
• ISSN: 1570-7946
• DOI: 10.1016/B978-0-12-818634-3.50116-8

The continuous Smoluchowski coagulation equation, which is known as the population balance equation (PBE) for particle coagulation, is a nonlinear integro-partial differential equation with no general analytical solution. In this work, we are concerned with extending our discrete formulation of the PBE for particle breakage using a Conservative Discretization Approach (CDA) (Attarakih et al., 2004) to solve the Smoluchowski coagulation equation coupled with particle growth. The method is based on introducing auxiliary functions to modify the discrete loss and formation terms in the discrete PBE. These are then uniquely determined by exactly reproducing two arbitrary chosen integral quantities from the continuous PBE. The CDA is validated using many test cases with known analytical solutions including coupled particle coagulation and growth dynamics as a simplified model for a batch crystallizer. The discrete approximate solutions for the number concentration function is found to converge with an order O(1/M) where M is the number of grid points.