Quadrature-based moment methods for the population balance equation: An algorithm review

Dongyue Li; Zhipeng Li; Zhengming Gao

doi:10.1016/j.cjche.2018.11.028

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Quadrature-based moment methods for the population balance equation: An algorithm review

Review | Updated：2026-01-06

- Quadrature-based moment methods for the population balance equation: An algorithm review
- Quadrature-based moment methods for the population balance equation: An algorithm review
- 中国化学工程学报（英文版） 2019年27卷第3期页码：483-500
- Affiliations：
  
  2 State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing,Beijing,China,100083
- Author bio：
- Funds：
- DOI：10.1016/j.cjche.2018.11.028
  中图分类号：
- 纸质出版：2019
- Accepted：
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Dongyue Li, Zhipeng Li, Zhengming Gao. Quadrature-based moment methods for the population balance equation: An algorithm review[J]. 中国化学工程学报（英文版）, 2019,27(3):483-500.

Dongyue Li, Zhipeng Li, Zhengming Gao. Quadrature-based moment methods for the population balance equation: An algorithm review[J]. Chinese Journal of Chemical Engineering, 2019, 27(3): 483-500.
Dongyue Li, Zhipeng Li, Zhengming Gao. Quadrature-based moment methods for the population balance equation: An algorithm review[J]. 中国化学工程学报（英文版）, 2019,27(3):483-500. DOI： 10.1016/j.cjche.2018.11.028.

Dongyue Li, Zhipeng Li, Zhengming Gao. Quadrature-based moment methods for the population balance equation: An algorithm review[J]. Chinese Journal of Chemical Engineering, 2019, 27(3): 483-500. DOI： 10.1016/j.cjche.2018.11.028.

摘要

Abstract

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport

loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years

many QBMMs have been developed for different problems

such as the quadrature method of moments (QMOM)

direct quadrature method of moments (DQMOM)

extended quadrature method of moments (EQMOM)

conditional quadrature method of moments (CQMOM)

extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper

we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows

in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.

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