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Mathematical Problems in Engineering
Volume 2013, Article ID 125128, 11 pages
Research Article

Weighted Essentially Nonoscillatory Method for Two-Dimensional Population Balance Equations in Crystallization

1School of Materials Science and Engineering, Zhengzhou University, Zhengzhou 450002, China
2School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China
3Vehicle & Motive Power Engineering School, Henan University of Science and Technology, Luoyang 471023, China

Received 8 April 2013; Revised 15 July 2013; Accepted 10 August 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Chunlei Ruan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Population balance equations (PBEs) are the main governing equations to model the processes of crystallization. Two-dimensional PBEs refer to the crystals that grow anisotropically with the change of two internal coordinates. Since the PBEs are hyperbolic equations, it is necessary to build up high resolution schemes to avoid numerical diffusion and numerical dispersion in order to obtain the accurate crystal size distribution (CSD). In this work, a 5th order weighted essentially nonoscillatory (WENO) method is introduced to compute the two-dimensional PBEs. Several numerical benchmark examples from literatures are carried out; it is found that WENO method has higher resolution than HR method which is well established. Therefore, WENO method is recommended in crystallization simulation when the crystal size distributions are sharp and higher accuracy is needed.