求解多维粒数衡算方程的高阶紧致差分方法

doi:10.11949/0438-1157.20240527

摘要/Abstract

摘要：

多维粒数衡算方程（MPBE）描述颗粒过程中两个或两个以上内在变量的粒度分布情况。由于大多数PBE缺乏解析解，通常需要采用计算昂贵的高阶或高分辨率（HR）方法来获得精确的数值解。如何高效、准确的获取数值解是其面临的挑战。针对以上问题，提出一种结合维度分裂的改进的高阶紧致差分（HOCD）方法，该方法能够使数值解具有四阶精度。采用维度分裂方法将多维问题分裂为几个一维问题，将分裂后的一维方程在时间和空间上进行离散，得到HOCD的三对角格式，并通过Tomas算法进行求解。在某些情况下，还进行了变量代换。除此之外，利用von Neumann稳定性分析方法证明了其稳定性。与HR方法相比，HOCD具有更高的计算精度和效率，且无严重的数值扩散。多个数值模拟证明了该方法的有效性。

关键词: 粒数衡算方程, 高阶紧致差分, 粒度分布, 维度分裂, 稳定性分析, 数值模拟

Abstract:

Multidimensional population balance equations (MPBE) describe the size distribution of a granular process over two or more intrinsic variables. Since most PBEs lack analytical solutions, computationally expensive high-order or high-resolution (HR) methods are often used to obtain accurate numerical solutions. The difficult part of the problem is finding an accurate and efficient numerical solution. To address the above problems, an improved higher-order compact difference (HOCD) method combining dimensional splitting is proposed, which enables the numerical solution with fourth-order accuracy. Dimensional splitting methodsare used to split the multidimensional problem into several one-dimensional problems. The split one-dimensional equations are then discretized in space and time to produce the tridiagonal format of the HOCD, which may be solved using the Tomas algorithm. Variable substitution is also carried out in some cases. Furthermore, the stability was demonstrated using the von Neumann stability analysis method. Compared with HR methods, the HOCD has higher computational accuracy and computational efficiency without numerical diffusion. The effectiveness of this method is demonstrated by multiple numerical simulations.

Key words: population balance equation, high-order compact difference, size distribution, dimensional splitting, stability analysis, numerical simulation

中图分类号:

TP391.9

李川, 洪振取, 单宝明, 徐啟蕾, 张方坤. 求解多维粒数衡算方程的高阶紧致差分方法[J]. 化工学报, DOI: 10.11949/0438-1157.20240527.

Chuan LI, Zhenqu HONG, Baoming SHAN, Qilei XU, Fangkun ZHANG. High-order compact difference method for solving the multidimensional population balance equation[J]. CIESC Journal, DOI: 10.11949/0438-1157.20240527.

图/表 6

表1 案例1不同网格尺寸的最大绝对误差及收敛阶

Table.1 Maximum absolute error and convergence rate with different grid sizes for Case 1

	HR		HOCD
h	MAE	Rate	MAE	Rate
0.04	25.408	-	4.395	-
0.02	11.809	1.105	0.282	3.964
0.01	4.246	1.476	1.42E-2	4.307
0.005	0.844	2.330	3.30E-4	5.428

图1 案例1当N=100，τ=2 s的仿真结果

Fig.1 Simulation results when N=100, τ=2 s for Case 1

图2 不同方法的RMSE及计算时间的比较

Fig.2 Comparison of the RMSE and calculation time of different methods

表2 案例2不同网格尺寸的最大绝对误差及收敛阶

Table 2 Maximum absolute error and convergence rate with different grid sizes for Case 2

	HR		HOCD
h	MAE	Rate	MAE	Rate
0.04	18.309	-	2.598	-
0.02	8.004	1.194	0.150	4.114
0.01	2.934	1.448	8.354E-3	4.167
0.005	0.921	1.672	6.791E-4	3.621

图3 案例2当N=100，τ=2 s的仿真结果

Fig.3 Simulation results when N=100, τ=2 s for Case 2

图4 不同方法的RMSE及计算时间的比较

Fig.4 Comparison of the RMSE and calculation time of different methods

参考文献 32

9	Patil D P, Andrews J R G. An analytical solution to continuous population balance model describing floc coalescence and breakage—a special case[J]. Chemical Engineering Science, 1998, 53(3): 599-601.
10	McCoy B J, Madras G. Analytical solution for a population balance equation with aggregation and fragmentation[J]. Chemical Engineering Science, 2003, 58(13): 3049-3051.
11	Muneer A, Schikarski T, Pflug L. Exact Method of Moments for multi-dimensional population balance equations[J]. ArXiv e-Prints, 2023: arXiv: .
12	Sun F R, Liu T, Nagy Z K, et al. Extended sectional quadrature method of moments for crystal growth and nucleation with application to seeded cooling crystallization[J]. Chemical Engineering Science, 2022, 254: 117625.
13	Inguva P K, Braatz R D. Efficient numerical schemes for multidimensional population balance models[J]. Computers & Chemical Engineering, 2023, 170: 108095.
14	Inguva P K, Schickel K C, Braatz R D. Efficient numerical schemes for population balance models[J]. Computers & Chemical Engineering, 2022, 162: 107808.
15	O'Sullivan D, Rigopoulos S. A conservative finite volume method for the population balance equation with aggregation, fragmentation, nucleation and growth[J]. Chemical Engineering Science, 2022, 263: 117925.
16	侯庆志, 沈嘉渊, 魏建国. 反应流模拟的有限体积法的比较[J]. 计算机工程与应用, 2017, 53(15): 63-7.
	Hou Q Z, Shen J Y, Wei J G. Comparison of finite volume methods for numerical simulation of reacting flow[J]. Computer Engineering and Applications, 2017, 53(15): 63-67.
17	Bhoi S, Sarkar D. Hybrid finite volume and Monte Carlo method for solving multi-dimensional population balance equations in crystallization processes[J]. Chemical Engineering Science, 2020, 217: 115511.
18	Ma D L, Tafti D K, Braatz R D. High-resolution simulation of multidimensional crystal growth[J]. Industrial & Engineering Chemistry Research, 2002, 41(25): 6217-6223.
19	Qamar S, Ashfaq A, Warnecke G, et al. Adaptive high-resolution schemes for multidimensional population balances in crystallization processes[J]. Computers & Chemical Engineering, 2007, 31(10): 1296-1311.
20	Szilágyi B, Nagy Z K. Graphical processing unit (GPU) acceleration for numerical solution of population balance models using high resolution finite volume algorithm[J]. Computers & Chemical Engineering, 2016, 91: 167-181.
1	Zhang F K, Shan B M, Wang Y L, et al. Progress and opportunities for utilizing seeding techniques in crystallization processes[J]. Organic Process Research & Development, 2021, 25(7): 1496-1511.
2	Maharana A, Sehrawat P, Das A, et al. Multi-dimensional population balance modeling of sonocrystallization of pyrazinamide with systematic estimation of kinetic parameters based on uncertainty and sensitivity analyses[J]. Chemical Engineering Research and Design, 2023, 200: 356-373.
3	Barrasso D, El Hagrasy A, Litster J D, et al. Multi-dimensional population balance model development and validation for a twin screw granulation process[J]. Powder Technology, 2015, 270: 612-621.
4	Qamar S, Rehman S M U. High resolution finite volume schemes for solving multivariable biological cell population balance models[J]. Industrial & Engineering Chemistry Research, 2013, 52(11): 4323-4341.
5	Kang Y S, Ward J D, Nagy Z K. A new framework and a hybrid method for one-dimensional population balance modeling of batch thermocycling crystallization[J]. Computers & Chemical Engineering, 2022, 157: 107588.
6	武首香, 王学魁, 沙作良, 等. 工业结晶过程的多相流与粒数衡算的CFD耦合求解[J]. 化工学报, 2009, 60(3): 593-600.
	Wu S X, Wang X K, Sha Z L, et al. Solution of population balance in multiphase flow field for industrial crystallization process[J]. Journal of the Chemical Industry and Engineering Society of China, 2009, 60(3): 593-600.
7	Omar H M, Rohani S. Crystal population balance formulation and solution methods: a review[J]. Crystal Growth & Design, 2017, 17(7): 4028-4041.
8	李倩, 程景才, 杨超, 等. 群体平衡方程在搅拌反应器模拟中的应用[J]. 化工学报, 2014, 65(5): 1607-1615.
	Li Q, Cheng J, Yang C, et al. Application of population balance equation in numerical simulation of multiphase stirred tanks[J]. CIESC Journal, 2014, 65(5): 1607-1615.
21	侯波, 葛永斌. 求解一维对流方程的高精度紧致差分格式[J]. 应用数学, 2019, 32(3): 635-642.
	Hou B, Ge Y B. A high-order compact difference scheme for solving the 1D convection equation[J]. Mathematica Applicata, 2019, 32(3): 635-642.
22	魏剑英, 葛永斌. 一种求解三维非稳态对流扩散反应方程的高精度有限差分格式[J]. 应用数学和力学, 2022, 43(2): 187-197.
	Wei J Y, Ge Y B. A high-order finite difference scheme for 3D unsteady convection diffusion reaction equations[J]. Applied Mathematics and Mechanics, 2022, 43(2): 187-197.
23	Zhao F X, Ji X, Shyy W, et al. Direct modeling for computational fluid dynamics and the construction of high-order compact scheme for compressible flow simulations[J]. Journal of Computational Physics, 2023, 477: 111921.
24	Wei J Y, Ge Y B, Wang Y. High-order compact difference method for solving two- and three-dimensional unsteady convection diffusion reaction equations[J]. Axioms, 2022, 11(3): 111.
25	Niu Y X, Liu Y, Li H, et al. Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media[J]. Mathematics and Computers in Simulation, 2023, 203: 387-407.
26	LeVeque R J. Finite volume methods for hyperbolic problems[M]. Cambridge: Cambridge University Press, 2002.
27	李青, 王能超. 解循环三对角线性方程组的追赶法[J]. 小型微型计算机系统, 2002, 23(11): 1393-5.
	Li Q, Wang N C. An algorithm for solving circulant tridiagonal systems[J]. Mini-micro Systems, 2002, 23(11): 1393-1395.
28	Gunawan R, Fusman I, Braatz R D. High resolution algorithms for multidimensional population balance equations[J]. AIChE Journal, 2004, 50(11): 2738-2749.
29	Kalita J C, Dalal D C, Dass A K. A class of higher order compact schemes for the unsteady two-dimensional convection–diffusion equation with variable convection coefficients[J]. International Journal for Numerical Methods in Fluids, 2002, 38(12): 1111-1131.
30	张莉, 罗春林, 张瀚月. 一类求解非线性对流扩散方程的线性多步法[J]. 西北师范大学学报(自然科学版), 2024, 60(2): 29-36.
	Zhang L, Luo C L, Zhang H Y. A second-order linear multi-step method for solving nonlinear convection-diffusion equations[J]. Journal of Northwest Normal University (Natural Science), 2024, 60(02): 29-36.
31	Yang X J, Ge Y B, Zhang L. A class of high-order compact difference schemes for solving the Burgers' equations[J]. Applied Mathematics and Computation, 2019, 358: 394-417.
32	Gunawan R, Fusman I, Braatz R D. Parallel high-resolution finite volume simulation of particulate processes[J]. AIChE Journal, 2008, 54(6): 1449-1458.

[1]	卢飞, 鲁波娜, 许光文. 气固微型流化床反应分析仪的理想流型判据分析[J]. 化工学报, 2024, 75(6): 2201-2213.
[2]	黄斌, 丰生杰, 傅程, 张威. 液滴撞击单丝铺展特性的数值研究[J]. 化工学报, 2024, 75(6): 2233-2242.
[3]	李娟, 曹耀文, 朱章钰, 石雷, 李佳. 仿生正形尾鳍结构微通道流动与传热特性数值研究及结构优化[J]. 化工学报, 2024, 75(5): 1802-1815.
[4]	王金山, 王世学, 朱禹. 冷却表面温差对高温质子交换膜燃料电池性能的影响[J]. 化工学报, 2024, 75(5): 2026-2035.
[5]	师毓辉, 邢继远, 姜雪晗, 叶爽, 黄伟光. 基于PBM的离心式叶轮内气泡破碎合并数值模拟[J]. 化工学报, 2024, 75(5): 1816-1829.
[6]	李怡菲, 董新宇, 王为术, 刘璐, 赵一璠. 微肋板表面干冰升华喷雾冷却传热数值模拟[J]. 化工学报, 2024, 75(5): 1830-1842.
[7]	刘帆, 张芫通, 陶成, 胡成玉, 杨小平, 魏进家. 歧管式射流微通道液冷散热性能[J]. 化工学报, 2024, 75(5): 1777-1786.
[8]	李静, 张方芳, 王帅帅, 徐建华, 张朋远. 凹腔结构对正丁烷部分预混火焰可燃极限的影响[J]. 化工学报, 2024, 75(5): 2081-2090.
[9]	谢磊, 徐永生, 林梅. 不同截面肋柱-软尾结构单相流动传热比较[J]. 化工学报, 2024, 75(5): 1787-1801.
[10]	王文雅, 张玮, 楼小玲, 钟若菲, 陈冰冰, 贠军贤. 纳米纤维素嵌合型晶胶微球的多微管成形与模拟[J]. 化工学报, 2024, 75(5): 2060-2071.
[11]	吉笑盈, 郑园, 李晓鹏, 杨振, 张维, 邱诗蕊, 张倩颖, 罗沧海, 孙东鹏, 陈东, 李东亮. 微流控可控制备液滴、颗粒和胶囊及其应用[J]. 化工学报, 2024, 75(4): 1455-1468.
[12]	李昂, 赵振宇, 李洪, 高鑫. 微波诱导高分散Pd/FeP催化剂构筑及其电催化性能研究[J]. 化工学报, 2024, 75(4): 1594-1606.
[13]	陈好奇, 史博会, 彭琪, 康琦, 宋尚飞, 姚海元, 陈海宏, 吴海浩, 宫敬. 基于稳定性分析的含酸/醇烃水体系相平衡计算[J]. 化工学报, 2024, 75(3): 789-800.
[14]	谷世良, 谭博仁, 程全中, 姚玮洁, 董志鹏, 许峰, 王勇. 轴流泵式混合室内水力学特征的数值模拟[J]. 化工学报, 2024, 75(3): 815-822.
[15]	徐百平, 梁瑞凤, 喻慧文, 吴桂群, 肖书平. 双螺杆挤出机强化三角形转子作用下的腔内分布混合模拟[J]. 化工学报, 2024, 75(3): 858-866.