群体平衡方程在搅拌反应器模拟中的应用

doi:10.3969/j.issn.0438-1157.2014.05.008

摘要/Abstract

摘要： 群体平衡方程（population balance equation，PBE）是描述多相流体系中分散相大小与分布随时空变化的通用方程。搅拌反应器内多为多相流体系，考虑到颗粒聚并、破碎等微观机制对颗粒大小、分布、粒数密度等宏观参量的影响，采用PBE对搅拌槽内多相流体系进行数值模拟，可以较准确预测搅拌槽内流场和颗粒的大小与分布。对群体平衡方程在搅拌反应器数值模拟中的应用进行了综述，在简要介绍PBE的基本形式后，讨论了PBE的主要数值求解方法，然后着重介绍近年来采用PBE对搅拌槽内液固沉淀过程、气液及液液体系进行数值模拟的情况，并对今后的研究方向进行了展望。

关键词: 群体平衡方程, 搅拌反应器, 数值模拟, 粒度分布

Abstract: The population balance equation (PBE) is generally used to describe the particle size distribution of the dispersed phase in multiphase systems. Considering the influence of particle events, such as aggregation and breakage, on particle size and number density, the PBE is a powerful tool for simulation of stirred reactors. The application of the PBE in stirred tanks is reviewed in this paper. After briefing the construction of the general form of the PBE, different numerical methods for solving the PBE are discussed. Specific applications of the PBE in simulation of precipitation processes, gas-liquid and liquid-liquid systems in stirred tanks are also presented.

Key words: population balance equation, stirred reactor, numerical simulation, particle size distribution

中图分类号:

TQ019

李倩, 程景才, 杨超, 毛在砂. 群体平衡方程在搅拌反应器模拟中的应用[J]. 化工学报, DOI: 10.3969/j.issn.0438-1157.2014.05.008.

LI Qian, CHENG Jingcai, YANG Chao, MAO Zaisha. Application of population balance equation in numerical simulation of multiphase stirred tanks[J]. CIESC Journal, DOI: 10.3969/j.issn.0438-1157.2014.05.008.

参考文献 54

[1]	Hulburt H M, Katz S. Some problems in particle technology: a statistical mechanical formulation [J]. Chem. Eng. Sci., 1964, 19(8):555-574
[2]	Randolph A D, Larson M A. Theory of Particulate Processes [M]. 2nd ed.San Diego, CA: Academic Press,1988
[3]	Cheng Jingcai(程景才). Precipitation kinetics of nickel hydroxide and numerical simulation of continuous precipitation process in a stirred tank [D]. Beijing: Institute of Process Engineering, Chinese Academy of Science, 2009
[4]	Cheng J C, Yang C, Mao Z S. CFD-PBE simulation of premixed continuous precipitation incorporating nucleation, growth and aggregation in a stirred tank with multi-class method [J]. Chem. Eng. Sci., 2012, 68(1):469-480
[5]	Smith M, Matsouka T. Constant-number Monte Carlo simulation of population balances [J]. Chem. Eng. Sci., 1998, 53(9):1777-1786
[6]	McGraw R. Description of aerosol dynamics by the quadrature method of moments [J]. Aerosol Science and Technology, 1997, 27(2):255-265
[7]	Marchisio D L, Pikturna J T, Fox R O, et al. Quadrature method of moments for population-balance equations [J]. AIChE J., 2003, 49(5):1266-1276
[8]	Fan R, Marchisio D L, Fox R O. Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds [J]. Powder Technol., 2004, 139: 7- 20
[9]	Attarakih M M, Drumm C, Bart H J. Solution of the population balance equation using the sectional quadrature method of moments [J]. Chem. Eng. Sci., 2009, 64: 742-752
[10]	Rigopoulos S, Jones A G. Finite-element scheme for solution of the dynamic population balance equation [J]. AIChE J., 2003, 49: 1127-1139
[11]	Kumar S, Ramkrishna D. On the solution of population balance equations by discretization(1): A fixed pivot technique [J]. Chem. Eng. Sci., 1996, 51(8): 1311-1332
[12]	Kumar J, Peglow M, Warneke G, et al. Improved accuracy and convergence of discretized population balance for aggregation: the cell average technique [J]. Chem. Eng. Sci., 2006, 61(10): 3327-3342
[13]	Kostoglou M. Extended cell average technique for the solution of coagulation equation [J]. Journal of Colloid and Interface Science, 2007, 306(1): 72-81
[14]	Kumar S, Ramkrishna D. On the solution of population balance equations by discretization(2): A moving pivot technique [J]. Chem. Eng. Sci., 1996, 51(8):1333-1342
[15]	Kumar J, Peglow M, Warneke G, et al. An efficient numerical technique for solving population balance equation involving aggregation, breakage, growth and nucleation [J]. Powder Technology, 2008, 182(1): 81-104
[16]	Wang Tiefeng(王铁峰). Experimental study and numerical simulation on the hydrodynamics in gas-liquid (slurry) reactors [D]. Beijing: Tsinghua University, 2004
[17]	Wei H. Application of computational fluid dynamics techniques to the modelling of precipitation processes [D]. UK: UMIST,1997
[18]	Garside J, Wei H. Pumped stirred and maybe precipitated: simulation of precipitation process using CFD [J]. Acta Polytech. Scand., Chem. Technol. Metall. Ser., 1997, 244: 9-15
[19]	Jaworski Z, Nienow A W. CFD modelling of continuous precipitation of barium sulphate in a stirred tank [J]. Chem. Eng. J., 2003, 91: 167-174
[20]	Wang Z, Mao Z S, Yang C, Shen X Q. Computational fluid dynamics approach to the effect of mixing and draft tube on the precipitation of barium sulfate in a continuous stirred tank [J]. Chin. J. Chem. Eng., 2006, 14: 713-722
[21]	Vicum L, Mazzotti M. Multi-scale modeling of a mixing-precipitation process in a semibatch stirred tank [J]. Chem. Eng. Sci., 2007, 62: 3513-3527
[22]	Cheng J C, Yang C, Mao Z S, Zhao C J. CFD modeling of nucleation, growth, aggregation, and breakage in continuous precipitation of barium sulfate in a stirred tank [J]. Ind. Eng. Chem. Res., 2009, 48(15): 6992-7003
[23]	Ba?dyga J, Orciuch W. Closure problem for precipitation [J]. Chem. Eng. Res. Des., 1997, 75: 160-170
[24]	O'Hern H A, Rush F E. Effect of mixing conditions in barium sulfate precipitation [J]. I&EC Fundam., 1963, 2: 267-272
[25]	Pohorecki R, Baldyga J. The effects of micromixing and the manner of reactor feeding on precipitation in stirred tank reactors [J]. Chem. Eng. Sci., 1988,43(8): 1949-1954
[26]	Bakker A, van den Akker H E A. A computational model for the gas-liquid flow in stirred reactors [J]. Chem. Eng. Res. Des., 1994, 72: 594-606
[27]	Venneker B C H, Derksen J J, van den Akker H E A. Population balance modeling of aerated stirred vessels based on CFD [J]. AIChE J., 2002, 48(4): 673-685
[28]	Moilanen P, Laakkonen M, Visuri O, et al. Modelling mass transfer in an aerated 0.2 m3 vessel agitated by Rushton, Phasejet and Combijet impellers [J]. Chem. Eng. J., 2008, 142(1): 95-108
[29]	Montante G, Horn D, Paglianti A. Gas-liquid flow and bubble size distribution in stirred tanks [J]. Chem. Eng. Sci., 2008, 63(8): 2107-2118
[30]	Ranganathan P, Sivaraman S. Investigations on hydrodynamics and mass transfer in gas-liquid stirred reactor using computational fluid dynamics [J]. Chem. Eng. Sci., 2011, 66(14): 3108-3124
[31]	Kerdouss F, Bannari A, Proulx P, et al. Two-phase mass transfer coefficient prediction in stirred vessel with a CFD model [J]. Comput. Chem. Eng., 2008, 32(8): 1943-1955
[32]	Martin M, Montes F, Galan M A. Mass transfer rates from bubbles in stirred tanks operating with viscous fluids [J]. Chem. Eng. Sci., 2010, 65(12): 3814-3824
[33]	Yang J, Bao Y Y, Lin M L, Zhu S, Gao Z M. Experimental study and numerical simulation of local void fraction in cold-gassed and hot-sparged stirred reactors [J]. Chem. Eng. Sci., 2013, 100: 83-90
[34]	Gimbun J, Rielly C D, Nagy Z K. Modelling of mass transfer in gas-liquid stirred tanks agitated by Rushton turbine and CD-6 impeller: a scale-up study [J]. Chem. Eng. Res. Des., 2009, 87: 437-451
[35]	Petitti M, Nasuti A, Marchisio D L, et al. Bubble size distribution modeling in stirred gas-liquid reactors with QMOM augmented by a new correction algorithm [J]. AIChE J., 2010, 56(1): 36-53
[36]	Maggioris D, Goulas A, Alexopoulos A H, et al. Prediction of particle size distribution in suspension polymerization reactors: effect of turbulence nonhomogeneity [J]. Chem. Eng. Sci., 2000, 55(20): 4611-4627
[37]	Maaß S, Metz F, Rehm T, Kraume M. Prediction of drop sizes for liquid-liquid systems in stirred slim reactors(Ⅰ): Single stage impellers[J]. Chem. Eng. J., 2010, 162(2): 792-801
[38]	Maaß S, Rehm T, Kraume M. Prediction of drop sizes for liquid-liquid systems in stirred slim reactors(Ⅱ): Multi stage impellers [J]. Chem. Eng. J., 2011, 168(2): 827-838
[39]	Maaß S, Paul N, Kraume M. Influence of the dispersed phase fraction on experimental and predicted drop size distributions in breakage dominated stirred systems [J]. Chem. Eng. Sci., 2012, 76: 140-153
[40]	Roudsari S F, Turcotte G, Dhib R, Ein-Mozaffari F. CFD modeling of the mixing of water in oil emulsions [J]. Comput. Chem. Eng., 2012, 45: 124-136
[41]	Sowbna P R, Yadav G D, Ramkrishna D. Population balance modeling and simulation of liquid-liquid-liquid phase transfer catalyzed synthesis of mandelic acid from benzaldehyde [J]. AIChE J., 2012, 58(12): 3799-3809
[42]	Olmos E, Gentric C, Vial C, Wild G, Midoux N. Numerical simulation of multiphase flow in bubble column reactors. Influence of bubble coalescence and break-up [J]. Chem. Eng. Sci., 2001, 56(21/22): 6359-6365
[43]	Wang T F, Wang J F, Jin Y. Population balance model for gas-liquid flows: influence of bubble coalescence and breakup models [J]. Ind. Eng. Chem. Res., 2005, 44(19): 7540-7549
[44]	Wang T F, Wang J F, Jin Y. A CFD-PBM coupled model for gas-liquid flows [J]. AIChE J., 2006, 52(1): 125-140
[45]	Wang T F, Wang J F. Numerical simulations of gas-liquid mass transfer in bubble columns with a CFD-PBM coupled model [J]. Chem. Eng. Sci., 2007, 62(24): 7107-7118
[46]	Xing C T, Wang T F, Wang J F. Experimental study and numerical simulation with a coupled CFD-PBM model of the effect of liquid viscosity in a bubble column [J]. Chem. Eng. Sci., 2013, 95: 313-322
[47]	Dhanasekharan K M, Sanyal J, Jain A, Haidari A. A generalized approach to model oxygen transfer in bioreactors using population balances and computational fluid dynamics [J]. Chem. Eng. Sci., 2005, 60(1): 213-218
[48]	Silva M K, d'Avila M A, Mori M. CFD modelling of a bubble column with an external loop in the heterogeneous regime [J]. Can. J. Chem. Eng., 2011, 89(4): 671-681
[49]	Law D, Battaglia F. Numerical simulations for hydrodynamics of air-water external loop airlift reactor flows with bubble break-up and coalescence effects [J]. J. Fluids Eng.-Trans. ASME, 2013, 135(8): 0813021-0813029
[50]	Nandanwar M N, Kumar S. A new discretization of space for the solution of multi-dimensional population balance equations: simultaneous breakup and aggregation of particles [J]. Chem. Eng. Sci., 2008, 63: 3988-3997
[51]	Chauhan S S, Chakraborty J, Kumar S. On the solution and applicability of bivariate population balance equations for mixing in particle phase [J]. Chem. Eng. Sci., 2010, 65: 3914-3927
[52]	Kumar R, Kumar J, Warnecke G. Numerical methods for solving two-dimensional aggregation population balance equations [J]. Comput. Chem. Eng., 2011, 35(6): 999-1009
[53]	Chiney A, Kumar S. On the solution of bivariate population balance equations for aggregation: pivotwise expansion of solution domain [J]. Chem. Eng. Sci., 2012, 76: 14-25
[54]	Buffo A, Vanni M, Marchisio D L. Multidimensional population balance model for the simulation of turbulent gas-liquid systems in stirred tank reactors [J]. Chem. Eng. Sci., 2012, 70: 31-44