# Polydispersity and the Population Balance Model

## Introduction to Polydisperse Systems and the Method of Moments Solution Technique

In fluid-particle flow systems, when the disperse-phase (e.g., particle phase) property distribution is wide or changing due to the particulate processes (such as changes in size distribution due to breakage, agglomeration, attrition, or growth or changes in disperse-phase density due to heterogeneous chemical reactions), use of average values for disperse-phase properties is no longer accurate. Successful computational fluid dynamics (CFD) simulations of polydisperse flows must include the distribution of particulate phase properties and its variation caused by the particulate processes.

One method to account for polydispersity in the system is using a multi-fluid model by dividing the disperse phase into different classes based on the desired property by assigning one fluid for each class. As discussed in Chap. 2, Iddir and Arastoopour (2005) extended the kinetic theory to granular mixtures of different mechanical properties (size, density, and/or restitution coefficient) for multi-type particle systems where each particle group was considered as a separate phase with different average velocity and granular energy. An alternative method to account for polydispersity of the particulate phase is based on the population balance approach. This approach is expected to be computationally more attractive and is also able to account for changes in the disperse-phase property distributions.

The population balance equation (PBE) is a balance equation based on the number density function and accounts for the spatial and temporal evolutions of the particulate phase internal variable distribution function in a single control volume. This equation is an integrodifferential equation and involves both integrals and derivatives of the distribution function.

Ramkrishna (2000) and Vanni (2000) provided a comprehensive review of the subject of PBE in terms of formulation of PBE, solution techniques, applications, © Springer International Publishing AG 2017

H. Arastoopour et al., *Computational Transport Phenomena of Fluid-Particle Systems, *Mechanical Engineering Series, DOI 10.1007/978-3-319-45490-0_4

and theoretical considerations. Solution methods of PBE include method of successive approximations, method of Laplace transform, method of moments (MOM), method of weighted residuals (MWR), discrete formulations for the solution, and the Monte Carlo method. Among them, MOM is widely accepted as a computationally attractive method and has gained significant attention.

MOM is based on solving the distribution function transport equation in terms of its lower-order moments. For fluid-particle flow systems, some of the variables in PBE need to be calculated from the CFD model, and, in turn, solution of PBE gives some of the phase properties needed in the CFD model. Therefore, PBE and CFD need to communicate via a two-way coupling. However, in its original form, this method is capable of modeling only very simple particulate processes due to some mathematical limitations (e.g., closure problem) (Strumendo and Arastoopour 2008). To overcome these limitations, different solution methods have been proposed by many researchers. The various forms of MOM can be expressed in four categories: