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Taylor series expansion scheme applied for solving population balance equation

  • Mingzhou Yu

    Mingzhou Yu is now a professor at China Jiliang University and a guest professor at Key Laboratory of Aerosol Chemistry and Physics, Chinese Academy of Sciences. He was an Alexander von Humboldt research fellow at Karlsruhe Institute of Technology. He has published more than 40 SCI-indexed journal papers. Currently, he is an editorial board member of Journal of Hydrodynamics. His research interests include fine particle process modeling, nanoparticle-laden multiphase flow, and particle technology.

    and Jianzhong Lin

    Jianzhong Lin is a professor at Zhejiang University. He was the associate editor of International Journal of Multiphase Flow and is currently serving as the associate editor of Applied Mathematics and Mechanics and as an editorial board member of over ten journals. He has published more than 600 papers (including 218 SCI-indexed journal papers), nine books and one technical manual, and has six patents. His scientific interests are multiphase fluid flows, dynamics of fiber suspensions, micro fluid dynamics, and turbulence, and fluid machinery.

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Published/Copyright: May 11, 2017
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Abstract

Population balance equations (PBE) are widely applied to describe many physicochemical processes such as nanoparticle synthesis, chemical processes for particulates, colloid gel, aerosol dynamics, and disease progression. The numerical study for solving the PBE, i.e. population balance modeling, is undergoing rapid development. In this review, the application of the Taylor series expansion scheme in solving the PBE was discussed. The theories, implement criteria, and applications are presented here in a universal form for ease of use. The aforementioned method is mathematically economical and applicable to the combination of fine-particle physicochemical processes and can be used to numerically and pseudo-analytically describe the time evolution of statistical parameters governed by the PBE. This article summarizes the principal details of the method and discusses its application to engineering problems. Four key issues relevant to this method, namely, the optimization of type of moment sequence, selection of Taylor series expansion point, optimization of an order of Taylor series expansion, and selection of terms for Taylor series expansion, are emphasized. The possible direction for the development of this method and its advantages and shortcomings are also discussed.

About the authors

Mingzhou Yu

Mingzhou Yu is now a professor at China Jiliang University and a guest professor at Key Laboratory of Aerosol Chemistry and Physics, Chinese Academy of Sciences. He was an Alexander von Humboldt research fellow at Karlsruhe Institute of Technology. He has published more than 40 SCI-indexed journal papers. Currently, he is an editorial board member of Journal of Hydrodynamics. His research interests include fine particle process modeling, nanoparticle-laden multiphase flow, and particle technology.

Jianzhong Lin

Jianzhong Lin is a professor at Zhejiang University. He was the associate editor of International Journal of Multiphase Flow and is currently serving as the associate editor of Applied Mathematics and Mechanics and as an editorial board member of over ten journals. He has published more than 600 papers (including 218 SCI-indexed journal papers), nine books and one technical manual, and has six patents. His scientific interests are multiphase fluid flows, dynamics of fiber suspensions, micro fluid dynamics, and turbulence, and fluid machinery.

Nomenclature
A

Constant

r

Particle radius

v

Particle volume

N

Particle number concentration density

xi

Position coordinate

t

Time

u

Particle velocity

DB

Brownian diffusion coefficient

Gr

Particle surface growth rate

J

Nucleation rate

v*

Critical monomer volume

a, b

Parameters in breakage rate model

mk

k-th moment of particle size distribution

B1

Collision coefficient for the free molecular regime

B2

Collision coefficient for the continuum-slip regime

u0

Taylor series expansion point of the classic TEMOM

q0

Taylor series expansion point of the GTEMOM

C

Cunningham correction factor

kb

Boltzmann constant, J·K

Kn

Particle Knudsen number

g

=m0m2/m12

t

Time, s

T

Temperature, K

vg

Geometric mean particle volume, m3

Df

Mass fractal dimension

f

1/Df

Greek letters
ν

Kinematic viscosity, m2·s−1

β

Particle collision kernel

μ

Gas viscosity kg·m−1·s−1

λ

Mean free path of the gas, m

σg

Geometric mean deviation of size distribution

Acknowledgments

M.Z. thanks the Alexander von Humboldt Foundation (grant no. 1136169) and the joint venture Forschungsförderungseinrichtung der Deutschen Forschungsgemeinschaft (DFG)/National Natural Science Foundation of China (NSFC) (grant no. GZ 971). The authors thank for the joint support from the National Natural Science Foundation of China (grant nos. 11372299, 11632016).

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Received: 2016-12-07
Accepted: 2017-04-05
Published Online: 2017-05-11
Published in Print: 2018-07-26

©2018 Walter de Gruyter GmbH, Berlin/Boston

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