{"title":"求解间歇结晶中种群平衡方程的高阶紧致差分法","authors":"Fangkun Zhang, Zhenqu Hong, Chuan Li, Zimu Diao, Bin Lian, Baoming Shan, Yinglong Wang, Qilei Xu","doi":"10.1021/acs.iecr.4c02816","DOIUrl":null,"url":null,"abstract":"As there is typically no analytical solution to most population balance equations (PBEs) of interest, computationally expensive high-order or high-resolution methods are typically used to obtain accurate numerical solutions. In this study, a new high-order compact difference (HOCD) method is proposed to solve the PBE with fourth-order accuracy in both space and time. This method provides high computational accuracy with a computationally efficient solution for one-dimensional population balance modeling in batch cooling crystallization processes. A compact difference scheme is proposed based on a two-layer format, with three grid points involved at each time level. Tridiagonal linear equations are solved directly using Tomas’ algorithm. Stability is demonstrated through von Neumann stability analysis. Compared to the Upwind, Lax–Wendroff, and high-resolution finite volume (HR-FVM) methods, the HOCD method offers higher computational accuracy and efficiency, without numerical diffusion or dispersion. The effectiveness of this method is demonstrated through multiple case studies.","PeriodicalId":39,"journal":{"name":"Industrial & Engineering Chemistry Research","volume":"22 1","pages":""},"PeriodicalIF":3.9000,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"High-Order Compact Difference Method for Solving Population Balance Equations in Batch Crystallization\",\"authors\":\"Fangkun Zhang, Zhenqu Hong, Chuan Li, Zimu Diao, Bin Lian, Baoming Shan, Yinglong Wang, Qilei Xu\",\"doi\":\"10.1021/acs.iecr.4c02816\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"As there is typically no analytical solution to most population balance equations (PBEs) of interest, computationally expensive high-order or high-resolution methods are typically used to obtain accurate numerical solutions. In this study, a new high-order compact difference (HOCD) method is proposed to solve the PBE with fourth-order accuracy in both space and time. This method provides high computational accuracy with a computationally efficient solution for one-dimensional population balance modeling in batch cooling crystallization processes. A compact difference scheme is proposed based on a two-layer format, with three grid points involved at each time level. Tridiagonal linear equations are solved directly using Tomas’ algorithm. Stability is demonstrated through von Neumann stability analysis. Compared to the Upwind, Lax–Wendroff, and high-resolution finite volume (HR-FVM) methods, the HOCD method offers higher computational accuracy and efficiency, without numerical diffusion or dispersion. The effectiveness of this method is demonstrated through multiple case studies.\",\"PeriodicalId\":39,\"journal\":{\"name\":\"Industrial & Engineering Chemistry Research\",\"volume\":\"22 1\",\"pages\":\"\"},\"PeriodicalIF\":3.9000,\"publicationDate\":\"2025-02-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Industrial & Engineering Chemistry Research\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://doi.org/10.1021/acs.iecr.4c02816\",\"RegionNum\":3,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"ENGINEERING, CHEMICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Industrial & Engineering Chemistry Research","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1021/acs.iecr.4c02816","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ENGINEERING, CHEMICAL","Score":null,"Total":0}
High-Order Compact Difference Method for Solving Population Balance Equations in Batch Crystallization
As there is typically no analytical solution to most population balance equations (PBEs) of interest, computationally expensive high-order or high-resolution methods are typically used to obtain accurate numerical solutions. In this study, a new high-order compact difference (HOCD) method is proposed to solve the PBE with fourth-order accuracy in both space and time. This method provides high computational accuracy with a computationally efficient solution for one-dimensional population balance modeling in batch cooling crystallization processes. A compact difference scheme is proposed based on a two-layer format, with three grid points involved at each time level. Tridiagonal linear equations are solved directly using Tomas’ algorithm. Stability is demonstrated through von Neumann stability analysis. Compared to the Upwind, Lax–Wendroff, and high-resolution finite volume (HR-FVM) methods, the HOCD method offers higher computational accuracy and efficiency, without numerical diffusion or dispersion. The effectiveness of this method is demonstrated through multiple case studies.
期刊介绍:
ndustrial & Engineering Chemistry, with variations in title and format, has been published since 1909 by the American Chemical Society. Industrial & Engineering Chemistry Research is a weekly publication that reports industrial and academic research in the broad fields of applied chemistry and chemical engineering with special focus on fundamentals, processes, and products.