Lie group analysis and exact solutions for a class of population balance equations

doi:10.3969/j.issn.1000-5641.201911008

Abstract

Abstract: In this paper, exact solutions for a class of population balance equations were investigated. The population balance equation was first transformed to a partial differential equation; symmetries of the partial differential equation were then obtained by use of the classical Lie group analysis method. In addition, the paper presents symmetries, optimal system of subalgebras, reduced ordinary differential integral equations, and group invariant solutions of the population balance equation. Exact solutions of the reduced ordinary differential integral equations were subsequently found using trial functions. Finally, exact solutions for a class of population balance equations are derived.

Key words: population balance equation, Lie group analysis method, method of trial function, group invariant solution, exact solution

CLC Number:

LIN Fubiao, ZHANG Qianhong. Lie group analysis and exact solutions for a class of population balance equations[J]. Journal of East China Normal University(Natural Science), 2020, 2020(2): 15-22.

References

[1] RAMKRISHNA D. Population Balances:Theory and Applications to Particulate Systems in Engineering[M]. San Diego:Academic Press, 2000.
[2] HULBURT H M, KATZ S. Some problems in particle technology:A statistical mechanical formulation[J]. Chemical Engineering Science, 1964, 19(8):555-574. DOI:10.1016/0009-2509(64)85047-8.
[3] RANDOLPH A D. A population balance for countable entities[J]. Canadian Journal of Chemical Engineering, 1964, 42(6):280-281. DOI:10.1002/cjce.5450420612.
[4] RANDOLPH A D, LARSON M A. Theory of Particulate Processes:Analysis and Techniques of Continuous Crystallization[M]. 2nd ed. San Diego:Academic Press, 1988.
[5] MULLIN J W. Crystallization[M]. 4th ed. Oxford:Butterworth-Heinemann, 2001.
[6] CAMERON I T, WANG F Y, IMMANUEL C D, et al. Process systems modelling and applications in granulation:A review[J]. Chemical Engineering Science, 2005, 60(14):3723-3750. DOI:10.1016/j.ces.2005.02.004.
[7] FILBET F, LAURENÇOT P. Numerical simulation of the Smoluchowski coagulation equation[J]. Society for Industrial & Applied Mathematics, 2003, 25(6):2004-2028.
[8] ZHANG N, XIA T C. A new negative discrete hierarchy and its N-fold Darboux transformation[J]. Communications in Theoretical Physics, 2017, 68:687-692. DOI:10.1088/0253-6102/68/6/687.
[9] LI Q, XIA T C, YUE C. Algebro-geometric solutions for the generalized nonlinear Schrödinger hierarchy[J]. Journal of Nonlinear Science & Applications, 2016, 9:661-676.
[10] SCHUMANN T E W. Theoretical aspects of the size distribution of fog particles[J]. Quarterly Journal of the Royal Meteorological Society, 1940, 66(285):195-208.
[11] 田畴. 李群及其在微分方程中的应用[M]. 北京:科学出版社, 2001.
[12] BLUMAN G W, KUMEI S. Symmetries and Differential Equations[M]. Berlin:Springer, 1989.
[13] OLVER J P. Applications of Lie Group to Differential Equation[M]. 2nd ed. New York:Springer, 1993.
[14] OVSIANNIKOV L V. Group Analysis of Differential Equations[M]. New York:Academic Press, 1982.
[15] IBRAGIMOV N H. Elementary Lie Group Analysis and Ordinary Differential Equations[M]. Chichester:John Wiley & Sons, 1999.
[16] IBRAGIMOV N H. Transformation Groups and Lie Algebra[M]. Beijing:Higher Education Press, 2013.
[17] MELESHKO S V. Methods for Constructing Exact Solutions of Partial Differential Equations:Mathematical and Analytical Techniques with Applications to Engineering[M]. New York:Springer, 2005.
[18] GRIGORIEV Y N, IBRAGIMOV N H, KOVALEV V F, et al. Symmetries of Integro-differential Equations:with Applications in Mechanics and Plasma Physics[M]. New York:Springer, 2010.
[19] ZHOU L Q, MELESHKO S V. Group analysis of integro-differential equations describing stress relaxation behavior of one-dimensional viscoelastic materials[J]. International Journal of Non-Linear Mechanics, 2015, 77:223-231. DOI:10.1016/j.ijnonlinmec.2015.08.008.
[20] ZHOU L Q, MELESHKO S V. Invariant and partially invariant solutions of integro-differential equations for linear thermoviscoelastic aging materials with memory[J]. Continuum Mechanics & Thermodynamics, 2017, 29(1):207-224.
[21] ZHOU L Q, MELESHKO S V. Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory[J]. Journal of Applied Mechanics & Technical Physics, 2017, 58(4):587-609.
[22] SURIYAWICHITSERANEE A, GRIGORIEV Y N, MELESHKO S V. Group analysis of the Fourier transform of the spatially homogeneous and isotropic Boltzmann equation with a source term[J]. Communications in Nonlinear Science & Numerical Simulation, 2015, 20(3):719-730.
[23] GRIGOREV Y N, MELESHKO S V, SURIYAWICHITSERANEE A. Exact solutions of the Boltzmann equations with a source[J]. Journal of Applied Mechanics & Technical Physics, 2018, 59(2):189-196.
[24] MKHIZE T G, GOVINDER K, MOYO S, et al. Linearization criteria for systems of two second-order stochastic ordinary differential equations[J]. Applied Mathematics & Computation, 2017, 301:25-35.
[25] LONG F S, KARNBANJONG A, SURIYAWICHITSERANEE A, et al. Application of a Lie group admitted by a homogeneous equation for group classification of a corresponding inhomogeneous equation[J]. Communications in Nonlinear Science & Numerical Simulation, 2017, 48:350-360.
[26] BARENBLATT G I. Scaling, Self-similarity, and Intermediate Asymptotics[M]. Cambridge:Cambridge University Press, 1996.
[27] LANGHAAR H L. Dimensional Analysis and Theory of Models[M]. New York:John Wiley and Sons, 1951.
[28] LIN F B, FLOOD A E, MELESHKO S V. Exact solutions of population balance equation[J]. Communications in Nonlinear Science & Numerical Simulation, 2016, 36:378-390.
[29] LIN F B, MELESHKO S V, FLOOD A E. Symmetries of population balance equations for aggregation, breakage and growth processes[J]. Applied Mathematics and Computation, 2017, 307:193-203. DOI:10.1016/j.amc.2017.02.048.
[30] LIN F B, MELESHKO S V, FLOOD A E. Exact solutions of the population balance equation including particle transport, using group analysis[J]. Communications in Nonlinear Science & Numerical Simulation, 2018, 59:255-271.