Advances in numerical methods for the solution of population balance equations for disperse phase systems

Abstract

Accurate prediction of the evolution of particle size distribution is critical to determining the dynamic flow structure of a disperse phase system. A population balance equation (PBE), a non-linear hyperbolic equation of the number density function, is usually employed to describe the micro-behavior (aggregation, breakage, growth, etc.) of a disperse phase and its effect on particle size distribution. Numerical solution is the only choice in most cases. In this paper, three different numerical methods (direct discretization methods, Monte Carlo methods, and moment methods) for the solution of a PBE are evaluated with regard to their ease of implementation, computational load and numerical accuracy. Special attention is paid to the relatively new and superior moment methods including quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), modified quadrature method of moments (M-QMOM), adaptive direct quadrature method of moments (ADQMOM), fixed pivot quadrature method of moments (FPQMOM), moving particle ensemble method (MPEM) and local fixed pivot quadrature method of moments (LFPQMOM). The prospects of these methods are discussed in the final section, based on their individual merits and current state of development of the field. population balance equation, direct discretization method, Monte Carlo method, moment methods, disperse phase system

Parameter optimisation and validation for droplet population balances

“…Nevertheless, it has some disadvantages as it needs a long computational time and only guarantees the conservation of mass. [20] Long computational time is due to the defined number of classes and pivots used to refine the solution, which reflects on the number of equations to be solved simultaneously, that are considerably large and increasing in industrial cases. Attarakih et al [21] extended the fixed-pivot technique, which was first proposed by Kumar and Ramkrishna [22] to liquid extraction columns.…”

“…CM is a straightforward method, which gives information about the full particle size distribution with a high accuracy. Nevertheless, it has some disadvantages as it needs a long computational time and only guarantees the conservation of mass . Long computational time is due to the defined number of classes and pivots used to refine the solution, which reflects on the number of equations to be solved simultaneously, that are considerably large and increasing in industrial cases.…”

“…The MOM is widely used because of its advantages, such as efficiency, high accuracy and low computational time, but the main disadvantage is that it loses the shape of the distribution . Alternatively, Attarakih et al proposed a new technique that combines the advantages of the CM and MOM for localisation of the Quadrature Method Of Moments (QMOM) to solve the PBE, which is called the Sectional Quadrature Method Of Moments (SQMOM).…”

Coupling Eulerian‐Lagrangian method of air‐particle two‐phase flow with population balance equations to simulate the evolution of vehicle exhaust plume

“…The QMOM has a high computational efficiency and has emerged as a promising tool for solving PBE over the past several years . The method of moments tracks not only the numerical density function but its quadrature moments and is described by Equation …”

“…In this study, a coupling scheme of Eulerian‐Lagrangian method of two‐phase flow and PBE is developed. This involves an MP‐PIC method to accelerate the Lagrangian tracking and the use of quadrature method of moments (QMOMs) to solve PBE with high precision and few calculations . This coupling scheme can simulate the evolution of a multiphase polydisperse system in time, space, and property.…”