Quadrature-based moment methods for the population balance equation: An algorithm review

doi:10.1016/j.cjche.2018.11.028

Abstract

Abstract: The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.

Key words: Population balance equation, Quadrature-based moment method, Multiphase flow, Computational fluid dynamics

摘要： The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.

关键词: Population balance equation, Quadrature-based moment method, Multiphase flow, Computational fluid dynamics

Dongyue Li, Zhipeng Li, Zhengming Gao. Quadrature-based moment methods for the population balance equation: An algorithm review[J]. Chinese Journal of Chemical Engineering, 2019, 27(3): 483-500.

Dongyue Li, Zhipeng Li, Zhengming Gao. Quadrature-based moment methods for the population balance equation: An algorithm review[J]. 中国化学工程学报, 2019, 27(3): 483-500.

References 99

[1]	V. Buwa, V. Ranade, Dynamics of gas-liquid flow in a rectangular bubble column:experiments and single/multi-group CFD simulations, Chem. Eng. Sci. 57(2002) 4715-4736.
[2]	S. Wang, J. Wen, Y. Li, H. Yang, Y. Li, J. Tu, Numerical prediction for subcooled boiling flow of liquid nitrogen in a vertical tube with MUSIG model, Chin. J. Chem. Eng. 21(2013) 1195-1205.
[3]	J. Cheng, Q. Li, C. Yang, Y. Zhang, Z. Mao, CFD-PBE simulation of a bubble column in OpenFOAM, Chin. J. Chem. Eng. 26(2018) 1773-1784.
[4]	M. Bhole, J. Joshi, D. Ramkrishna, CFD simulation of bubble columns incorporating population balance modeling, Chem. Eng. Sci. 63(2008) 2267-2282.
[5]	A. Buffo, D. Marchisio, M. Vanni, P. Renze, Simulation of coalescence, break up and mass transfer in gas-liquid systems by using Monte Carlo and quadrature-based moment methods, Ninth International Conference on CFD in the Minerals and Process Industries, Melbourne, 2012.
[6]	A. Buffo, M. Vanni, D. Marchisio, R. Fox, Multivariate quadraturebased moments methods for turbulent polydisperse gas-liquid systems, Int. J. Multiphase Flow 50(2013) 41-57.
[7]	Y. Liao, D. Lucas, E. Krepper, Application of new closure models for bubble coalescence and breakup to steam-water vertical pipe flow, Nucl. Eng. Des. 279(2014) 126-136.
[8]	Y. Liao, D. Lucas, Poly-disperse simulation of condensing steam-water flow inside a large vertical pipe, Int. J. Therm. Sci. 104(2016) 194-207.
[9]	Y. Bao, J. Yang, B. Wang, Z. Gao, Influence of impeller diameter on local gas dispersion properties in a sparged multi-impeller stirred tank, Chin. J. Chem. Eng. 23(2015) 615-622.
[10]	X. Liang, H. Pan, Y. Su, Z. Luo, CFD-PBM approach with modified drag model for the gas-liquid flow in a bubble column, Chem. Eng. Res. Des. 112(2016) 88-102.
[11]	H. Pan, X. Chen, X. Liang, L. Zhu, Z. Luo, CFD simulations of gas-liquid-solid flow in fluidized bed reactors-a review, Powder Technol. 299(2016) 235-258.
[12]	D. Cheng, S. Wang, C. Yang, Z. Mao, Numerical simulation of turbulent flow and mixing in gas-liquid-liquid stirred tanks, Ind. Eng. Chem. Res. 56(2017) 13050-13063.
[13]	K. Guo, T. Wang, Y. Liu, J. Wang, CFD-PBM simulations of a bubble column with different liquid properties, Chem. Eng. J. 329(2017) 116-127.
[14]	G. Yang, K. Guo, T. Wang, Numerical simulation of the bubble column at elevated pressure with a CFD-PBM coupled model, Chem. Eng. Sci. 170(2017) 251-262.
[15]	M. Jaradat, M. Attarakih, H. Bart, Effect of phase dispersion and mass transfer direction on steady state RDC performance using population balance modelling, Chem. Eng. J. 165(2010) 379-387.
[16]	V. Alopaeus, Analysis of concentration polydispersity in mixed liquid-liquid systems, Chem. Eng. Res. Des. 92(2014) 612-618.
[17]	J. Mitre, P. Lage, M. Souza, E. Silva, L. Barca, A. Moraes, R. Coutinho, E. Fonseca, Droplet breakage and coalescence models for the flow of waterin-oil emulsions through a valve-like element, Chem. Eng. Res. Des. 92(2014) 2493-2508.
[18]	M. Attarakih, S. Alzyod, M. Hlawitschke, H. Bart, OPOSSIM:a population balance-SIMULINK module for modelling coupled hydrodynamics and mass transfer in liquid extraction equipment, Comput-Aided, Chem. Eng. 37(2015) 257-262.
[19]	J. Favero, L. Silva, P. Lage, Modeling and simulation of mixing in water-in-oil emulsion flow through a valve-like element using a population balance model, Comput. Chem. Eng. 75(2015) 155-170.
[20]	A. Buffo, V. Alopaeus, Solution of bivariate population balance equations with highorder moment-conserving method of classes, Comput. Chem. Eng. 87(2016) 111-124.
[21]	Z. Gao, D. Li, A. Buffo, W. Podgorska, D. Marchisio, Simulation of droplet breakage in turbulent liquid-liquid dispersions with CFD-PBM:Comparison of breakage kernels, Chem. Eng. Sci. 142(2016) 277-288.
[22]	A. Bourdillon, P. Verdin, C. Thompson, Numerical simulations of drop size evolution in a horizontal pipeline, Int. J. Multiphase Flow 78(2016) 44-58.
[23]	D. Li, A. Buffo, W. Podgorska, Z. Gao, D. Marchisio, Droplet breakage and coalescence in liquid-liquid dispersions:comparison of different kernels with EQMOM and QMOM, AICHE J. 63(2017) 2293-2311.
[24]	D. Li, A. Buffo, W. Podgorska, D. Marchisio, Z. Gao, Investigation of droplet breakup in liquid-liquid dispersions by CFD-PBM simulations:The influence of the surfactant type, Chin. J. Chem. Eng. 25(2017) 1369-1380.
[25]	S. Alzyod, M. Attarakih, A. Hasseine, H. Bart, Steady state modeling of Kuhni liquid extraction column using the Spatially Mixed Sectional Quadrature Method of Moments (SM-SQMOM), Chem. Eng. Res. Des. 117(2017) 549-556.
[26]	A. Misra, L. De Souza, M. Illner, L. Hohl, M. Kraume, J. Repke, D. Thevenin, Simulating separation of a multiphase liquid-liquid system in a horizontal settler by CFD, Chem. Eng. Sci. 167(2017) 242-250.
[27]	C. Qin, C. Chen, Q. Xiao, N. Yang, C. Yuan, C. Kunkelmann, M. Cetinkaya, K. Mulheims, CFD-PBM simulation of droplets size distribution in rotor-stator mixing devices, Chem. Eng. Sci. 155(2016) 16-26.
[28]	L. Xie, Q. Liu, Z. Luo, A multiscale CFD-PBM coupled model for the kinetics and liquid-liquid dispersion behavior in a suspension polymerization stirred tank, Chem. Eng. Res. Des. 130(2018) 1-17.
[29]	R. Fox, F. Laurent, M. Massot, Numerical simulation of spray coalescence in an Eulerian framework:direct quadrature method of moments and multi-fluid method, J. Comput. Phys. 227(2008) 3058-3088.
[30]	R. Fox, A quadrature-based third-order moment method for dilute gasparticle flows, J. Comput. Phys. 227(2008) 6313-6350.
[31]	O. Desjardins, R. Fox, P. Villedieu, A quadrature-based moment method for dilute fluid-particle flows, J. Comput. Phys. 227(2008) 2514-2539.
[32]	A. Passalacqua, R. Fox, Implementation of an iterative solution procedure for multifluid gas-particle flow models on unstructured grids, Powder Technol. 213(2011) 174-187.
[33]	W. Yan, J. Li, Z. Luo, A CFD-PBM coupled model with polymerization kinetics for multizone circulating polymerization reactors, Powder Technol. 231(2012) 77-87.
[34]	M. Hussain, M. Peglow, E. Tsotsas, J. Kumar, Modeling of aggregation kernel using Monte Carlo simulations of spray fluidized bed agglomeration, AIChE J. 60(2014) 855-868.
[35]	W. Yan, Z. Luo, Y. Lu, X. Chen, A CFD-PBM-PMLM integrated model for the gas-solid flow fields in fluidized bed polymerization reactors, AIChE J. 58(2012) 1717-1732.
[36]	M. Hussain, J. Kumar, E. Tsotsas, Micro-macro transition of population balances in fluidized bed granulation, Procedia. Eng. 102(2015) 1399-1407.
[37]	Y. Yao, J. Su, Z. Luo, CFD-PBM modeling polydisperse polymerization FBRs with simultaneous particle growth and aggregation:the effect of the method of moments, Powder Technol. 272(2015) 142-152.
[38]	T. Nguyen, F. Laurent, R. Fox, M. Massot, Solution of population balance equations in applications with fine particles:mathematical modeling and numerical schemes, J. Comput. Phys. 325(2016) 129-156.
[39]	E. Ghadirian, J. Abbasian, H. Arastoopour, CFD simulation of particle size change during the coal char gasification process using the population balance model with FCMOM, Powder Technol. 323(2017) 128-138.
[40]	B. Kong, R. Fox, A solution algorithm for fluid-particle flows across all flow regimes, J. Comput. Phys. 344(2017) 575-594.
[41]	B. Kong, R. Fox, H. Feng, J. Capecelatro, R. Patel, O. Desjardins, Euler-euler anisotropic gaussian mesoscale simulation of homogeneous cluster-induced gas-particle turbulence, AIChE J. 63(2017) 2630-2643.
[42]	S. Shekar, A. Smith, W. Menz, M. Sander, M. Kraft, A multidimensional population balance model to describe the aerosol synthesis of silica nanoparticles, J. Aerosol Sci. 44(2012) 83-98.
[43]	R. Guichard, A. Taniere, E. Belut, N. Rimbert, Simulation of nanoparticle coagulation under Brownian motion and turbulence in a differential-algebraic framework:Developments and applications, Int. J. Multiphase Flow 64(2014) 73-84.
[44]	M. Yu, J. Lin, Hybrid method of moments with interpolation closure-Taylor-series expansion method of moments scheme for solving the Smoluchowski coagulation equation, Appl. Math. Model. 52(2017) 94-106.
[45]	M. Yu, Y. Liu, A. Koivisto, An efficient algorithm scheme for implementing the TEMOM for resolving aerosol dynamics, Aerosol Sci. Eng. 1(2017) 119-137.
[46]	P. Marchal, R. David, J. Klein, J. Villermaux, Crystallization and precipitation engineering-I. An efficient method for solving population balance in crystallization with agglomeration, Chem. Eng. Sci. 43(1988) 59-67.
[47]	F. Puel, G. Fevotte, J. Klein, Simulation and analysis of industrial crystallization processes through multidimensional population balance equations. Part 1:a resolution algorithm based on the method of classes, Chem. Eng. Sci. 58(2003) 3715-3727.
[48]	S. Qamar, G. Warnecke, Numerical solution of population balance equations for nucleation, growth and aggregation processes, Comput. Chem. Eng. 31(2007) 1576-1589.
[49]	J. Liu, C. Ma, X. Wang, Imaging protein crystal growth behaviour in batch cooling crystallization, Chin. J. Chem. Eng. 24(2016) 101-108.
[50]	A. Majumder, V. Kariwala, S. Ansumali, A. Rajendran, Lattice boltzmann method for population balance equations with simultaneous growth, nucleation, aggregation and breakage, Chem. Eng. Sci. 69(2012) 316-328.
[51]	S. Cheung, L. Deju, G. Yeoh, J. Tu, Modeling of bubble size distribution in isothermal gas-liquid flows:Numerical assessment of population balance approaches, Nucl. Eng. Des. 265(2013) 120-136.
[52]	T. Vetter, M. Iggland, D. Ochsenbein, F. Hanseler, M. Mazzotti, Modeling nucleation, growth, and ostwald ripening in crystallization processes:a comparison between population balance and kinetic rate equation, Cryst. Growth Des. 13(2013) 4890-4905.
[53]	A. Hasseine, H. Bart, A domian decomposition method solution of population balance equations for aggregation, nucleation, growth and breakup processes, Appl. Math. Model. 39(2015) 1975-1984.
[54]	J. Cheng, C. Yang, M. Jiang, Q. Li, Z. Mao, Simulation of antisolvent crystallization in impinging jets with coupled multiphase flow-micromixing-PBE, Chem. Eng. Sci. 171(2017) 500-512.
[55]	S. Rigopoulos, Population balance modelling of poly-dispersed particles in reactive flows, Prog. Energy Combust. 36(2010) 412-443.
[56]	E. Abbasi, H. Arastoopour, Numerical simulation of CO2 removal process using solid sorbent in a fluidized bed:A CFD-PBE model, 2011 AIChE Annual Meeting, Minneapolis, USA, 2011.
[57]	J. Akroyd, A. Smith, R. Shirley, L. McGlashan, M. Kraft, A coupled CFD-population balance approach for nanoparticle synthesis in turbulent reacting flows, Chem. Eng. Sci. 66(2011) 3792-3805.
[58]	E. Yapp, D. Chen, J. Akroyd, S. Mosbach, M. Kraft, J. Camacho, H. Wang, Numerical simulation and parametric sensitivity study of particle size distributions in a burner-stabilised stagnation flame, Combust. Flame 162(2015) 2569-2581.
[59]	E. Abbasi, J. Abbasian, H. Arastoopour, CFD-PBE numerical simulation of CO2 capture using MgO-based sorbent, Powder Technol. 286(2015) 616-628.
[60]	E. Yapp, C. Wells, J. Akroyd, S. Mosbach, R. Xu, M. Kraft, Modelling PAH curvature in laminar premixed flames using a detailed population balance model, Combust. Flame 176(2017) 172-180.
[61]	A. Boje, J. Akroyd, S. Sutcliffe, J. Edwards, M. Kraft, Detailed population balance modelling of Tio2 synthesis in an industrial reactor, Chem. Eng. Sci. 164(2017) 219-231.
[62]	S. Mosbach, W. Menz, M. Kraft, Outlier analysis for a silicon nanoparticle population balance model, Combust. Flame 177(2017) 89-97.
[63]	A. Zucca, D. Marchisio, A. Barresi, R. Fox, Implementation of the population balance equation in CFD codes for modelling soot formation in turbulent flames, Chem. Eng. Sci. 61(2006) 87-95.
[64]	D. Chen, Z. Zainuddin, E. Yapp, J. Akroyd, S. Mosbach, M. Kraft, A fully coupled simulation of PAH and soot growth with a population balance model, Proc. Combust. Inst. 34(2013) 1827-1835.
[65]	B. Wang, S. Mosbach, S. Schmutzhard, S. Shuai, Y. Huang, M. Kraft, Modelling soot formation from wall films in a gasoline direct injection engine using a detailed population balance model, Appl. Energy 163(2016) 154-166.
[66]	G. Song, Y. Li, W. Wang, K. Jiang, Z. Shi and S. Yao, Hydrate agglomeration modelling and pipeline hydrate slurry flow behavior simulation, Chin. J. Chem. Eng., doi:10.1016/j.cjche.2018.04.004.
[67]	O. Emre, R. Fox, M. Massot, S. De Chaisemartin, S. Jay, F. Laurent, Towards Eulerian modeling of a polydisperse evaporating spray under realistic internal-combustionengine conditions, Flow Turbul. Combust. 93(2014) 689-722.
[68]	M. Hussain, J. Kumar, E. Tsotsas, A new framework for population balance modeling of spray fluidized bed agglomeration, Particuology 19(2015) 141-154.
[69]	D. Kah, O. Emre, Q. Tran, S. De Chaisemartin, S. Jay, F. Laurent, M. Massot, High order moment method for polydisperse evaporating sprays with mesh movement:application to internal combustion engines, Int. J. Multiphase Flow 71(2015) 38-65.
[70]	M. Essadki, S. De Chaisemartin, M. Massot, F. Laurent, A. Larat, S. Jay, A new high order moment method for polydisperse evaporating sprays dedicated to the coupling with separated two-phase flows in automotive engine, 9th International Conference on Multiphase Flow (ICMF), Italy, 2016.
[71]	M. Essadki, S. De Chaisemartin, F. Laurent, M. Massot, High order moment model for polydisperse evaporating sprays towards interfacial geometry, SIAM J. Appl. Math. 78(2018) 2003-2027.
[72]	A. Sibra, J. Dupays, A. Murrone, F. Laurent, M. Massot, Simulation of reactive polydisperse sprays strongly coupled to unsteady flows in solid rocket motors:Efficient strategy using Eulerian Multi-Fluid methods, J. Comput. Phys. 339(2017) 210-246.
[73]	D. Marchisio, R. Fox, Computational models for polydisperse particulate and multiphase systems, Cambridge University Press, 2013.
[74]	D. Ramkrishna, M. Singh, Population balance modeling:current status and future prospects, Annu. Rev. Chem. Biomol. 5(2014) 123-146.
[75]	J. Solsvik, H. Jakobsen, The foundation of the population balance equation:a review, J. Dispers. Sci. Technol. 36(2015) 510-520.
[76]	H. Arastoopour, D. Gidaspow, E. Abbasi, Computational Transport Phenomena of Fluid-Particle Systems, Springer, 2017.
[77]	D. Ramkrishna, Population balances:Theory and applications to particulate systems in engineering, Academic Press, 2000.
[78]	P. Lage, Comments on the "An analytical solution to the population balance equation with coalescence and breakagethe special case with constant number of particle" by D.P. Patil. and J.R.G. Andrews.[Chem. Eng. Sci. 53(3) 599-601], Chem. Eng. Sci. 57(2002) 4253-4254.
[79]	B. McCoy, G. Madras, Analytical solution for a population balance equation with aggregation and fragmentation, Chem. Eng. Sci. 58(2003) 3049-3051.
[80]	G. Bhutani, P. Brito-Parada, Analytical solution for a three-dimensional nonhomogeneous bivariate population balance equation:a special case, Int. J. Multiphase Flow 89(2017) 413-416.
[81]	M. Attarakih, A. Hasseine, H. Bart, On the solution of the PBE by orthogonal expansion of the maximum entropy functional, Comput-Aided, Chem. Eng. 40(2017) 2053-2058.
[82]	S. Kumar, D. Ramkrishna, On the solution of population balance equations by discretization-Ⅲ. Nucleation, growth and aggregation of particles, Chem. Eng. Sci. 52(1997) 4659-4679.
[83]	D. Jo, S. Revankar, Investigation of bubble breakup and coalescence in a packed-bed reactor-Part 1:A comparative study of bubble breakup and coalescence models, Int. J. Multiphase Flow 37(2011) 995-1002.
[84]	J. Morchain, M. Pigou, N. Lebaz, A population balance model for bioreactors combining interdivision time distributions and micromixing concepts, Biochem. Eng. J. 126(2017) 135-145.
[85]	H. Hulburt, S. Katz, Some problems in particle technology:A statistical mechanical formulation, Chem. Eng. Sci. 19(1964) 555-574.
[86]	J. Brock, J. Oates, Moment simulation of aerosol evaporation, J. Aerosol Sci. 18(1987) 59-64.
[87]	X. Luo, Y. Cao, H. Xie, F. Qin, Moment method for unsteady flows with heterogeneous condensation, Comput. Fluids 146(2017) 51-58.
[88]	M. Nicmanis, M. Hounslow, Finite-element methods for steady-state population balance equations, AIChE J. 44(1998) 2258-2272.
[89]	J. Solsvik, H. Jakobsen, Evaluation of weighted residual methods for the solution of a population balance model describing bubbly flows:The least-squares, galerkin, tau, and orthogonal collocation methods, Ind. Eng. Chem. Res. 52(2013) 15988-16013.
[90]	J. Solsvik, P. Becker, N. Sheibat-Othman, H. Jakobsen, On the solution of the dynamic population balance model describing emulsification:Evaluation of weighted residual methods, Can. J. Chem. Eng. 92(2014) 250-265.
[91]	J. Solsvik, P. Becker, N. Sheibat-Othman, H. Jakobsen, Numerical solution of the drop population balance equation using weighted residual and finite volume methods, J. Dispers. Sci. Technol. 37(2016) 80-88.
[92]	M. Smith, T. Matsoukas, Constant-number Monte Carlo simulation of population balances, Chem. Eng. Sci. 53(1998) 1777-1786.
[93]	H. Zhao, A. Maisels, T. Matsoukas, C. Zheng, Analysis of four Monte Carlo methods for the solution of population balances in dispersed systems, Powder Technol. 173(2007) 38-50.
[94]	Y. He, H. Zhao, H. Wang, C. Zheng, Differentially weighted direct simulation Monte Carlo method for particle collision in gas-solid flows, Particuology 21(2015) 135-145.
[95]	Z. Xu, H. Zhao, C. Zheng, Fast Monte Carlo simulation for particle coagulation in population balance, J. Aerosol Sci. 74(2014) 11-25.
[96]	P. Fede, O. Simonin, P. Villedieu, Monte-Carlo simulation of colliding particles or coalescing droplets transported by a turbulent flow in the framework of a joint fluid-particle pdf approach, Int. J. Multiphase Flow 74(2015) 165-183.
[97]	Z. Xu, H. Zhao, C. Zheng, Accelerating population balance-Monte Carlo simulation for coagulation dynamics from the Markov jump model, stochastic algorithm and GPU parallel computing, J. Comput. Phys. 281(2015) 844-863.
[98]	W. Zhang, C. You, Numerical approach to predict particle breakage in dense flows by coupling multiphase particle-in-cell and Monte Carlo methods, Powder Technol. 283(2015) 128-136.
[99]	Z. Xu, H. Zhao, H. Zhao, CFD-population balance Monte Carlo simulation and numerical optimization for flame synthesis of TiO2 nanoparticles, Proc. Combust. Inst. 36(2017) 1099-1108.