Abstract
This article reviews the most critical issues in the simulation of turbulent polydisperse gas-liquid systems. Here the discussion is limited to bubbly flows, where the gas appears in the form of separate individual bubbles. First, the governing equations are presented with particular focus on the generalized population balance equation (GPBE). Then, the mesoscale models defining the evolution of the gas-liquid system (e.g., interface forces, mass transfer, coalescence, and breakup) are introduced and critically discussed. Particular attention is devoted to the choice of the drag model to properly simulate dense gas-liquid systems in the presence of microscale turbulence. Finally, the different solution methods, namely, Lagrangian and Eulerian, are presented and discussed. The link between mixture, two- and multi-fluid models, and the GPBE is also analyzed. Eventually, the different methodologies to account for polydispersity, with focus on Lagrangian or direct simulation Monte Carlo methods and Eulerian quadrature-based moment methods, are also presented. A number of practical examples are discussed and the review is concluded by presenting the advantages and disadvantages of the different methods and the corresponding computational costs.
About the authors
Antonio Buffo received his master and PhD in chemical engineering from Politecnico di Torino (Italy) in 2012. Currently, he is a postdoctoral researcher at Politecnico di Torino. His main research interests are in the field of multiphase and reactive flows, transport phenomena, kinetic theory, mathematical modeling, numerical methods, and computer science.
Daniele L. Marchisio received his master and PhD in chemical engineering from Politecnico di Torino (Italy) in 2001. He then became postdoctoral researcher at Iowa State University (USA) and at Eidgenössische Technische Hochschule Zürich (Switzerland). In 2004 he was appointed assistant professor at Politecnico di Torino and he is currently associate professor in the same instituition. He held visiting positions at University College London (UK), Aalborg University (Denmark), and University of Valladolid (Spain). He authored many papers published in international journals on multiphase systems and reactor modeling and simulations. His monograph (coauthored with Rodney O. Fox) “Computational Models for Polydisperse Particulate and Multiphase Systems” offers an authoritative treatment of the field.
This work is the result of the research carried out at DISAT, Politecnico di Torino on gas-liquid systems over the past decade. This research was financially supported by ENI (Italy) and BASF (Germany). The important contributions of Miriam Petitti (DISAT), Marco Vanni (DISAT), Fabrizio Podenzani (ENI), Peter Renze (BASF), Rodney O. Fox (Iowa State University), Matteo Icardi (KAUST), and Djamel Lakehal and Chidu Narayanan (ASCOMP) are gratefully acknowledged.
References
Ali BA, Pushpavanam S. Analysis of unsteady gas-liquid flows in a rectangular tank: comparison of Euler-Eulerian and Euler-Lagrangian simulations. Int J Multiphase Flow 2011; 37: 268–277.10.1016/j.ijmultiphaseflow.2010.10.002Search in Google Scholar
Alopaeus V, Koskinen J, Keskinen KI, Majander J. Simulation of the population balances for liquid-liquid systems in a non ideal stirred tank. II. Parameter fitting and the use of the multiblock model for dense dispersions. Chem Eng Sci 2002; 57: 1815–1825.10.1016/S0009-2509(02)00067-2Search in Google Scholar
Andersson R, Andersson B. On the breakup of fluid particles in turbulent flows. AIChE J 2006; 52: 2020–2030.10.1002/aic.10831Search in Google Scholar
Bakker A, van den Akker HEA. A computational model for the gas-liquid flow in stirred reactors. Chem Eng Res Des 1994; 72: 594–606.Search in Google Scholar
Baldyga J, Orciuch W. Barium sulphate precipitation in a pipe – an experimental study and CFD modelling. Chem Eng Sci 2001; 56: 2435–2444.10.1016/S0009-2509(00)00449-8Search in Google Scholar
Balachandar S. A scaling analysis for point-particle approaches to turbulent multiphase flows. Int J Multiphase Flow 2009; 35: 801–810.10.1016/j.ijmultiphaseflow.2009.02.013Search in Google Scholar
Balachandar S, Eaton JK. Turbulent dispersed multiphase flow. Annu Rev Fluid Mech 2010; 42: 111–133.10.1146/annurev.fluid.010908.165243Search in Google Scholar
Batterham RJ, Hall JS, Barton G. Pelletizing kinetics and simulation of full scale balling circuits. In: 3rd International Symposium on Agglomeration. Nuremberg, West Germany, 1981: A136.Search in Google Scholar
Becker S, Sokolichin A, Eigenberger G. Gas-liquid flow in bubble columns and loop reactors: Part II. comparison of detailed experiments and flow simulations. Chem Eng Sci 1994; 49: 5747–5762.10.1016/0009-2509(94)00290-8Search in Google Scholar
Becker S, De Bie H, Sweeney J. Dynamic flow behaviour in bubble columns. Chem Eng Sci 1999; 54: 4929–4935.10.1016/S0009-2509(99)00214-6Search in Google Scholar
Bird RB, Stewart WE, Lightfoot EN. Transport Phenomena. John Wiley and Sons, New York, United States of America, 1960.Search in Google Scholar
Borchers O, Busch C, Sokolichin A, Eigenberger G. Applicability of the standard k-turbulence model to the dynamic simulation of bubble columns. Part II. Comparison of detailed experiments and flow simulations. Chem Eng Sci 1999; 54: 5927–5935.10.1016/S0009-2509(99)00181-5Search in Google Scholar
Bove S. Computational fluid dynamics of gas-liquid flows including bubble population balances. PhD Thesis, Aalborg University, Esbjerg, Denmark, 2005.Search in Google Scholar
Bridge A, Lapidus L, Elgin J. The mechanics of vertical gas-liquid fluidized system. Part I. Countercurrent flow. AIChE J 1964; 10: 819–826.10.1002/aic.690100610Search in Google Scholar
Buffo A. Multivariate population balance for turbulent gas-liquid flows. Ph.D. Thesis. Politecnico di Torino, Torino, Italy, 2013.Search in Google Scholar
Buffo A, Marchisio DL, Vanni M. Multidimensional population balance model for the simulation of turbulent gas-liquid systems in stirred tank reactors. Chem Eng Sci 2012a; 74: 31–44.10.1016/j.ces.2011.04.042Search in Google Scholar
Buffo A, Marchisio D, Vanni M, Renze P. Simulation of mass transfer in gas-liquid systems by using monte carlo and quadrature-based moment methods. In: Ninth International Conference on CFD in the Minerals and Process Industries, 10–12 December 2012. Melbourne, Australia: CSIRO, 2012b.Search in Google Scholar
Buffo A, Marchisio D, Vanni M, Renze P. Simulation of polydisperse multiphase systems using population balances and example application to bubbly flows. Chem Eng Res Des 2013a; 91: 1859–1875.10.1016/j.cherd.2013.06.021Search in Google Scholar
Buffo A, Vanni M, Marchisio DL, Fox RO. Multivariate quadrature-based moments methods for turbulent polydisperse gas-liquid systems. Int J Multiphase Flow 2013b; 50: 41–57.10.1016/j.ijmultiphaseflow.2012.09.005Search in Google Scholar
Busciglio A, Grisafi F, Scargiali F, Brucato A. On the measurement of bubble size distribution in gas-liquid contactors via light sheet and image analysis. Chem Eng Sci 2010a; 65: 2558–2568.10.1016/j.ces.2009.12.031Search in Google Scholar
Busciglio A, Grisafi F, Scargiali F, Brucato A. On the measurement of local gas hold-up and interfacial area in gas-liquid contactors via light sheet and image analysis. Chem Eng Sci 2010b; 65: 3699–3708.10.1016/j.ces.2010.03.004Search in Google Scholar
Buwa VV, Deo DS, Ranade VV. Eulerian-Lagrangian simulations of unsteady gas-liquid flows in bubble columns. Int J Multiphase Flow 2006; 32: 864–885.10.1016/j.ijmultiphaseflow.2006.02.017Search in Google Scholar
Cachaza Gianzo EM. Hydrodynamics and mass transfer effects in bubble columns. PhD thesis, Universidad de Salamanca, Salamanca, Spain, 2011.Search in Google Scholar
Calderbank P. Physical rate processes in industrial fermentation. Part I. The interfacial area in gas-liquid contacting with mechanical agitation. Trans Inst Chem Eng 1958; 36: 443–467.Search in Google Scholar
Celata G, Cumo M, D’Annibale F, Di Marco P, Tomiyama A, Zovini C. Effect of gas injection mode and purity of liquid on bubble rising in two-component systems. Exp Therm Fluid Sci 2006; 31: 37–53.10.1016/j.expthermflusci.2005.08.006Search in Google Scholar
Chakraborty J, Kumar S. A new framework for solution of multidimensional population balance equations. Chem Eng Sci 2007; 62: 4112–4125.10.1016/j.ces.2007.04.049Search in Google Scholar
Chen C, Fan L-S. Discrete simulation of gas-liquid bubble columns and gas-liquid-solid fluidized beds. AIChE J 2004; 50: 288–301.10.1002/aic.10027Search in Google Scholar
Chen J, Kemoun A, Al-Dahhan MH, Duduković MP, Lee DJ, Fan L-S. Comparative hydrodynamics study in a bubble column using computer-automated radioactive particle tracking (CARPT)/computed tomography (CT) and particle image velocimetry (PIV). Chem Eng Sci 1999; 54: 2199–2207.10.1016/S0009-2509(98)00349-2Search in Google Scholar
Chen P, Duduković M, Sanyal J. Three-dimensional simulation of bubble column flows with bubble coalescence and breakup. AIChE J 2005a; 51: 696–712.10.1002/aic.10381Search in Google Scholar
Chen P, Sanyal J, Duduković M. Numerical simulation of bubble columns flows: effect of different breakup and coalescence closures. Chem Eng Sci 2005b; 60: 1085–1101.10.1016/j.ces.2004.09.070Search in Google Scholar
Cheung S, Yeoh G, Tu J. On the numerical study of isothermal vertical bubbly flow using two population balance approaches. Chem Eng Sci 2007; 62: 4659–4674.10.1016/j.ces.2007.05.030Search in Google Scholar
Cheung SCP, Yeoh GH, Tu J. A review of population balance modelling for isothermal bubbly flows. J Comput Multiphase Flows 2009; 1: 161–199.10.1260/175748209789563928Search in Google Scholar
Clift R, Grace JR, Weber ME. Bubbles, crops, and particles. New York: Academic Press, 1978.Search in Google Scholar
Coulaloglou CA, Tavlarides LL. Description of interaction processes in agitated liquid-liquid dispersions. Chem Eng Sci 1977; 32: 1289–1297.10.1016/0009-2509(77)85023-9Search in Google Scholar
Crowe CT. Multiphase flow handbook. Boca Raton, FL: CRC Press, 2006.Search in Google Scholar
Crowe C, Stock D, Sharma M. The particle-source-in cell (PSI-CELL) model for gas-droplet flows. J Fluids Eng 1977; 99: 325–332.10.1115/1.3448756Search in Google Scholar
Cussler EL. Diffusion: mass transfer in fluid systems, 2nd ed., Cambridge Series in Chemical Engineering. Cambridge, UK: Cambridge University Press, 1997.Search in Google Scholar
Danckwerts PV. Significance of liquid-film coefficients in gas absorption. Ind Eng Chem Res 1951; 43: 1460–1467.10.1021/ie50498a055Search in Google Scholar
Davidson J, Harrison D. The behavior of a continuously bubbling fluidized bed. Chem Eng Sci 1966; 21: 731–738.10.1016/0009-2509(66)87001-XSearch in Google Scholar
Deen N, Solberg T, Hjertager B. Large eddy simulation of the gas-liquid flow in a square cross-sectioned bubble column. Chem Eng Sci 2001; 56: 6341–6349.10.1016/S0009-2509(01)00249-4Search in Google Scholar
Deen N, Solberg T, Hjertager B. Flow generated by an aerated rushton impeller: two-phase PIV experiments and numerical simulations. Can J Chem Eng 2002; 80: 1–15.10.1002/cjce.5450800406Search in Google Scholar
Derksen J, Van den Akker HEA. Large eddy simulations on the flow driven by a rushton turbine. AIChE J 1999; 45: 209–221.10.1002/aic.690450202Search in Google Scholar
Desjardins O, Fox RO, Villedieu P. A quadrature-based moment method for dilute fluid-particle flows. J Comput Phys 2008; 227: 2514–2539.10.1016/j.jcp.2007.10.026Search in Google Scholar
Dhotre M, Niceno B, Smith B. Large eddy simulation of a bubble column using dynamic sub-grid scale model. Chem Eng J 2008; 136: 337–348.10.1016/j.cej.2007.04.016Search in Google Scholar
Dhotre M, Niceno B, Smith B, Simiano M. Large-eddy simulation (LES) of the large scale bubble plume. Chem Eng Sci 2009; 64: 2692–2704.10.1016/j.ces.2009.02.040Search in Google Scholar
Dhotre M, Deen N, Niceno B, Khan Z, Joshi J. Large eddy simulation for dispersed bubbly flows: a review. Int J Chem Eng 2013; 2013: 22. Article ID 343276. http://dx.doi.org/10.1155/2013/343276.10.1155/2013/343276Search in Google Scholar
Diaz ME, Iranzo A, Cuadra D, Barbero R, Montes FJ, Galan MA. Numerical simulation of the gas-liquid flow in a laboratory scale bubble column influence of bubble size distribution and non-drag forces. Chem Eng J 2008a; 139: 363–379.10.1016/j.cej.2007.08.015Search in Google Scholar
Diaz ME, Montes FJ, Galan MA. Experimental study of the transition between unsteady flow regimes in a partially aerated two-dimensional bubble column. Chem Eng Process 2008b; 47: 1867–1876.10.1016/j.cep.2007.10.012Search in Google Scholar
Diemer RB, Olson JH. A moment methodology for coagulation and breakage problems: Part 2 – moment models and distribution reconstruction. Chem Eng Sci 2002; 57: 2211–2228.10.1016/S0009-2509(02)00112-4Search in Google Scholar
Drahos J, Zahradnk J, Punochr M, Fialov M, Bradka F. Effect of operating conditions on the characteristics of pressure fluctuations in a bubble column. Chem Eng Process 1991; 29: 107–115.10.1016/0255-2701(91)87019-YSearch in Google Scholar
Drew D, Lahey R Jr. The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Int J Multiphase Flow 1987; 13: 113–121.10.1016/0301-9322(87)90011-5Search in Google Scholar
Drew DA, Passman SL. Theory of multicomponent fluids. New York: Springer-Verlag, 1999.10.1007/b97678Search in Google Scholar
Duan X, Cheung S, Yeoh G, Tu J, Krepper E, Lucas D. Gas-liquid flows in medium and large vertical pipes. Chem Eng Sci 2011; 66: 872–883.10.1016/j.ces.2010.11.031Search in Google Scholar
Ekambara K, Nandakumar K, Joshi J. CFD simulation of bubble column reactor using population balance. Ind Eng Chem Res 2008; 47: 8505–8516.10.1021/ie071393eSearch in Google Scholar
Fan R, Marchisio D, Fox R. Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds. Powder Technol 2004; 139: 7–20.10.1016/j.powtec.2003.10.005Search in Google Scholar
Farzpourmachiani A, Shams M, Shadaram A, Azidehak F. Eulerian-Lagrangian 3-D simulations of unsteady two-phase gas-liquid flow in a rectangular column by considering bubble interactions. Int J Nonlinear Mech 2011; 46: 1049–1056.10.1016/j.ijnonlinmec.2011.04.024Search in Google Scholar
Fox RO. Optimal moment sets for multivariate direct quadrature method of moments. Ind Eng Chem Res 2009; 48: 9686–9696.10.1021/ie801316dSearch in Google Scholar
Fox RO. Large-eddy-simulation tools for multiphase flows. Annu Rev Fluid Mech 2012; 44: 47–76.10.1146/annurev-fluid-120710-101118Search in Google Scholar
Garnier C, Lance M, Mariè J. Measurement of local flow characteristics in buoyancy-driven bubbly flow at high void fraction. Exp Therm Fluid Sci 2002; 26: 811–815.10.1016/S0894-1777(02)00198-XSearch in Google Scholar
Gautschi W. Orthogonal polynomials: computation and approximation. Oxford, UK: Oxford University Press, 2004.10.1093/oso/9780198506720.001.0001Search in Google Scholar
Geary NW, Rice RG. Bubble size prediction for rigid and flexible spargers. AIChE J 1991; 37: 161–168.10.1002/aic.690370202Search in Google Scholar
Gillis PA, Hommersom G, Schäfer M. A comparison of several CFD approaches for predicting gas-liquid contacting in a cylindrical tank agitated with a single rushton turbine. American Society of Mechanical Engineers, PVP 2002; 448: 23–36.10.1115/PVP2002-1571Search in Google Scholar
Gimbun J, Rielly CD, Nagy ZK. Modelling of mass transfer in gas-liquid stirred tanks agitated by Rushton turbine and CD-6 impeller: A scale-up study. Chem Eng Res Des 2009; 87: 437–451.10.1016/j.cherd.2008.12.017Search in Google Scholar
Griffith P, Wallis G. Two phase slug flow. J Heat Transfer 1961; 83: 307–320.10.1115/1.3682268Search in Google Scholar
Haberman WL, Morton RK. An experimental study of bubbles moving in liquids. Trans Am Soc Civ Eng 1956; 121: 227–252.10.1061/TACEAT.0007317Search in Google Scholar
Hartmann H, Derksen J, Montavon C, Pearson J, Hamill I, van den Akker H. Assessment of large eddy and RANS stirred tank simulations by means of LDA. Chem Eng Sci 2004; 59: 2419–2432.10.1016/j.ces.2004.01.065Search in Google Scholar
Hibiki T, Ishii M. Lift force in bubbly flow systems. Chem Eng Sci 2007; 62: 6457–6474.10.1016/j.ces.2007.07.034Search in Google Scholar
Higbie R. The rate of absorption of a pure gas into a still liquid during short periods of exposure. Trans Am Inst Chem Eng 1935; 31: 364–389.Search in Google Scholar
Hirt C, Nichols B. Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 1981; 39: 201–225.10.1016/0021-9991(81)90145-5Search in Google Scholar
Hjertager BH. Multi-fluid CFD analysis of chemical reactors. In: Marchisio DL, Fox RO, editors. Multiphase reacting flows: modelling and simulation. CISM International Centre for Mechanical Sciences, No. 492. Udine: Springer-Verlag, 2007: 125–179.Search in Google Scholar
Ho M, Yeoh G, Tu J. Population balance models for subcooled boiling flows. Int J Numer Methods Heat Fluid Flow 2008; 18: 160–172.10.1108/09615530810846310Search in Google Scholar
Hu G, Celik I. Eulerian-Lagrangian based large-eddy simulation of a partially aerated flat bubble column. Chem Eng Sci 2008; 63: 253–271.10.1016/j.ces.2007.09.015Search in Google Scholar
Icardi M, Marchisio D, Labois M. Efficient simulation of a two-phase vertical pipe flow with population balance method. In: Ninth International Conference on CFD in the Minerals and Process Industries, 10–12 December 2012. Melbourne, Australia: CSIRO, 2012.Search in Google Scholar
Icardi M, Marchisio D, Narayanan C, Fox R. Equilibrium-Eulerian LES model for turbulent poly-dispersed particle-laden flow. Int J Nonlinear Sci Numer Simul 2013; 14: 139–158.10.1515/ijnsns-2012-0086Search in Google Scholar
Ishii M, Mishima K. Two-fluid model and hydrodynamic constitutive relations. Nucl Eng Des 1984; 82: 107–126.10.1016/0029-5493(84)90207-3Search in Google Scholar
Ishii M, Zuber N. Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AIChE J 1979; 25: 843–855.10.1002/aic.690250513Search in Google Scholar
Jakobsen HA. Chemical reactor modeling: multiphase reactive flows. Berlin, Germany: Springer-Verlag, 2008.Search in Google Scholar
Jia X, Wen J, Feng W, Yuan Q. Local hydrodynamics modeling of a gas-liquid-solid three-phase airlift loop reactor. Ind Eng Chem Res 2007a; 46: 5210–5220.10.1021/ie061697lSearch in Google Scholar
Jia X, Wen J, Zhou H, Feng W, Yuan Q. Local hydrodynamics modeling of a gas-liquid-solid three-phase bubble column. AIChE J 2007b; 53: 2221–2231.10.1002/aic.11254Search in Google Scholar
Jia X, Wen J, Wang X, Feng W, Jiang Y. General research CFD modeling of immobilized phenol biodegradation in three-phase airlift loop reactor. Ind Eng Chem Res 2009; 48: 4514–4529.10.1021/ie800816dSearch in Google Scholar
Jia X, Yuan Q, Wen J, Feng W. Fluid flow modeling of a gas-induced pulsating flow bubble column. Chem Biochem Eng Q 2011; 25: 27–36.Search in Google Scholar
Kataoka I, Serizawa A. Basic equations of turbulence in gas-liquid two-phase flow. Int J Multiphase Flow 1989; 15: 843–855.10.1016/0301-9322(89)90045-1Search in Google Scholar
Kazakis N, Mouza A, Paras S. Experimental study of bubble formation at metal porous spargers: effect of liquid properties and sparger characteristics on the initial bubble size distribution. Chem Eng J 2008; 137: 265–281.10.1016/j.cej.2007.04.040Search in Google Scholar
Kerdouss F, Bannari A, Proulx P, Bannari R, Skrga M, Labrecque Y. Two-phase mass transfer coefficient prediction in stirred vessel with a CFD model. Comput Chem Eng 2008; 32: 1943–1955.10.1016/j.compchemeng.2007.10.010Search in Google Scholar
Khopkar AR, Rammohan AR, Ranade VV, Dudukovic MP. Gas-liquid flow generated by a Rushton turbine in stirred vessel: CARPT/CT measurements and CFD simulations. Chem Eng Sci 2005; 60: 2215–2229.10.1016/j.ces.2004.11.044Search in Google Scholar
Krepper, E., Frank, T., Lucas, D., Prasser, H-M., Zwart, PhJ Inhomogeneous musig model – A population balance approach for polydispersed bubbly flows. Proceedings – 12th International Topical Meeting on Nuclear Reactor Thermal Hydraulics, NURETH-12, 2007.Search in Google Scholar
Krepper E, Morel C, Niceno B, Ruyer P. CFD modeling of adiabatic bubbly flow. Multiphase Sci Technol 2011; 23: 129–164.10.1615/MultScienTechn.v23.i2-4.30Search in Google Scholar
Kumar S, Ramkrishna D. On the solution of population balance equations by discretization – I. A fixed pivot technique. Chem Eng Sci 1996a; 51: 1311–1332.10.1016/0009-2509(96)88489-2Search in Google Scholar
Kumar S, Ramkrishna D. On the solution of population balance equations by discretization – II. A moving pivot technique. Chem Eng Sci 1996b; 51: 1333–1342.10.1016/0009-2509(95)00355-XSearch in Google Scholar
Kumar J, Peglow M, Warnecke G, Heinrich S, Morl J. Improved accuracy and convergence of discretized population balance for aggregation: the cell average technique. Chem Eng Sci 2006; 61: 3327–3342.10.1016/j.ces.2005.12.014Search in Google Scholar
Kumar J, Peglow M, Warnecke G, Heinrich S. The cell average technique for solving multi-dimensional aggregation population balance equations. Comput Chem Eng 2008a; 32: 1810–1830.10.1016/j.compchemeng.2007.10.001Search in Google Scholar
Kumar J, Peglow M, Warnecke G, Heinrich S. An efficient numerical technique for solving population balance equation involving aggregation, breakage, growth and nucleation. Powder Technol 2008b; 182: 81–104.10.1016/j.powtec.2007.05.028Search in Google Scholar
Kumar J, Warnecke G, Peglow M, Heinrich S. Comparison of numerical methods for solving population balance equations incorporating aggregation and breakage. Powder Technol 2009; 189: 218–229.10.1016/j.powtec.2008.04.014Search in Google Scholar
Laakkonen M, Alopaeus V, Aittamaa J. Validation of bubble breakage, coalescence and mass transfer models for gas-liquid dispersion in agitated vessel. Chem Eng Sci 2006; 61: 218–228.10.1016/j.ces.2004.11.066Search in Google Scholar
Laakkonen M, Moilanen P, Alopaeus V, Aittamaa J. Modelling local bubble size distributions in agitated vessels. Chem Eng Sci 2007; 62: 721–740.10.1016/j.ces.2006.10.006Search in Google Scholar
Lahey RT Jr, Lopez de Bertodano M, Jones OC Jr. Phase distribution in complex geometry conduits. Nucl Eng Des 1993; 141: 177–201.10.1016/0029-5493(93)90101-ESearch in Google Scholar
Lane G, Schwarz M, Evans G. Numerical modelling of gas-liquid flow in stirred tanks. Chem Eng Sci 2005; 60: 2203–2214.10.1016/j.ces.2004.11.046Search in Google Scholar
Lee CH, Erickson LE, Glasgow LA. Bubble breakup and coalescence in turbulent gas-liquid dispersions. Chem Eng Commun 1987; 59: 65–84.10.1080/00986448708911986Search in Google Scholar
Lehr F, Mewes D. A transport equation for the interfacial area density applied to bubble columns. Chem Eng Sci 2001; 56: 1159–1166.10.1016/S0009-2509(00)00335-3Search in Google Scholar
Lehr F, Millies M, Mewes D. Bubble-size distributions and flow fields in bubble columns. AIChE J 2002; 48: 2426–2443.10.1002/aic.690481103Search in Google Scholar
Letzel H, Schouten J, Krishna R, Van den Bleek C. Characterization of regimes and regime transitions in bubble columns by chaos analysis of pressure signals. Chem Eng Sci 1997; 52: 4447–4459.10.1016/S0009-2509(97)00290-XSearch in Google Scholar
Li Y, Zhang J, Fan L-S. Numerical simulation of gas-liquid-solid fluidization systems using a combined CFD-VOF-DPM method: bubble wake behavior. Chem Eng Sci 1999; 54: 5101–5107.10.1016/S0009-2509(99)00263-8Search in Google Scholar
Liao Y, Lucas D, Krepper E, Schmidtke M. Development of a generalized coalescence and breakup closure for the inhomogeneous musig model. Nucl Eng Des 2011; 241: 1024–1033.10.1016/j.nucengdes.2010.04.025Search in Google Scholar
Lin T-J, Reese J, Hong T, Fan L-S. Quantitative analysis and computation of two-dimensional bubble columns. AIChE J 1996; 42: 301–318.10.1002/aic.690420202Search in Google Scholar
Lockett M, Kirkpatrick R. Ideal bubbly flow and actual flow in bubble columns. Trans Inst Chem Eng 1975; 53: 267–273.Search in Google Scholar
Lopes RJG, Perdigoto MLN, Quinta-Ferreira RM. Euler-Lagrange CFD simulation of a gas-liquid fluidized bed reactor for the mineralization of high-strength phenolic wastewaters. Ind Eng Chem Res 2012; 51: 8891–8902.10.1021/ie2021366Search in Google Scholar
Lopez de Bertodano MA, Saif AA. Modified k-ε model for two-phase turbulent jets. Nucl Eng Des 1997; 172: 187–196.10.1016/S0029-5493(97)00033-2Search in Google Scholar
Lopez de Bertodano M, Sun X, Ishii M, Ulke A. Phase distribution in the cap bubble regime in a duct. J Fluids Eng 2006; 128: 811–818.10.1115/1.2201626Search in Google Scholar
Luo H, Svendsen F. Theoretical model for drop and bubble break-up in turbulent dispersions. AIChE J 1996; 42: 1225–1233.10.1002/aic.690420505Search in Google Scholar
Manninen M, Taivassalo V. On the mixture model for multi-phase flow. Espoo: VTT Publications 288, 1996.Search in Google Scholar
Marchisio DL, Fox RO. Solution of population balance equations using the direct quadrature method of moments. J Aerosol Sci 2005; 36: 43–73.10.1016/j.jaerosci.2004.07.009Search in Google Scholar
Marchisio D, Fox R. Computational models for polydisperse particulate and multiphase systems. Cambridge, UK: Cambridge University Press, 2013.10.1017/CBO9781139016599Search in Google Scholar
Marrucci G. Rising velocity of a swarm of spherical bubbles. Ind Eng Chem Fundam 1965; 4: 224–225.10.1021/i160014a022Search in Google Scholar
Mazzei L, Marchisio D, Lettieri P. New quadrature-based moment method for the mixing of inert polydisperse fluidized powders in commercial cfd codes. AIChE J 2012; 58: 3054–3069.10.1002/aic.13714Search in Google Scholar
McGraw R. Correcting moment sequences for errors associated with advective transport, 2006. Retrieved from http://www.ecd.bnl.gov/pubs/momentcorrection_mcgraw2006.pdf. Accessed on 17 November, 2013.Search in Google Scholar
Migdal D, Agosta VD. A source flow model for continuum gas-particle flow. J Appl Mech 1967; 34: 860–865.10.1115/1.3607848Search in Google Scholar
Monahan SM, Fox RO. Effect of model formulation on flow-regime predictions for bubble columns. AIChE J 2007; 53: 9–18.10.1002/aic.11042Search in Google Scholar
Montante G, Paglianti A, Magelli F. Experimental analysis and computational modelling of gas-liquid stirred vessels. Chem Eng Res Des 2007; 85: 647–653.10.1205/cherd06141Search in Google Scholar
Montante G, Horn D, Paglianti A. Gas-liquid flow and bubble size distribution in stirred tanks. Chem Eng Sci 2008; 63: 2107–2118.10.1016/j.ces.2008.01.005Search in Google Scholar
Mudde R, Simonin O. Two- and three-dimensional simulations of a bubble plume using a two-fluid model. Chem Eng Sci 1999; 54: 5061–5069.10.1016/S0009-2509(99)00234-1Search in Google Scholar
Narsimhan G, Gupta JP, Ramkrishna D. A model for transitional breakage probability of droplets in agitated lean liquid-liquid dispersion. Chem Eng Sci 1979; 34: 257–265.10.1016/0009-2509(79)87013-XSearch in Google Scholar
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. Chem Eng Sci 2001; 56: 6359–6365.10.1016/S0009-2509(01)00204-4Search in Google Scholar
Osher S, Sethian JA. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 1988; 79: 12–49.10.1016/0021-9991(88)90002-2Search in Google Scholar
Peebles FN, Garber HJ. Studies on the motion of gas bubbles in liquid. Chem Eng Prog 1953; 42: 88–97.Search in Google Scholar
Peskin CS. Numerical analysis of blood flow in the heart. J Comput Phys 1977; 25: 220–252.10.1016/0021-9991(77)90100-0Search in Google Scholar
Petitti M, Caramellino M, Marchisio DL, Vanni M. Two-scale simulation of mass transfer in an agitated gas-liquid tank. In: 6th International Conference on Multiphase Flow, Leipzig, Germany, July 09–13, 2007: S6 Tue A 16.Search in Google Scholar
Petitti M, Nasuti A, Marchisio D, Vanni M, Baldi G, Mancini N, Podenzani F. CFD modelling coupled to population balance to describe bubble size distribution in agitated vessels and bubble columns. In: 6th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries, Trondheim, Norway, June 10–12, 2008.Search in Google Scholar
Petitti M, Marchisio DL, Vanni M, Baldi G, Mancini N, Podenzani F. Effect of drag modeling on the prediction of critical regime transitions in agitated gas-liquid reactors with bubble size distribution modeling. Multiphase Sci Technol 2009; 21: 95–106.10.1615/MultScienTechn.v21.i1-2.80Search in Google Scholar
Petitti M, Nasuti A, Marchisio DL, Vanni M, Baldi G, Mancini M, Podenzani F. Bubble size distribution modeling in stirred gas-liquid reactors with QMOM augmented by a new correction algorithm. AIChE J 2010; 56: 36–53.10.1002/aic.12003Search in Google Scholar
Petitti M, Vanni M, Marchisio D, Buffo A, Podenzani F. Application of the conditional quadrature method of moments for the simulation of coalescence, breakup and mass transfer in gas-liquid stirred tanks. In: 14th European Conference on Mixing, 10–13 September 2012, Warsaw, Poland.Search in Google Scholar
Petitti M, Vanni M, Marchisio D, Buffo A, Podenzani F. Simulation of coalescence, breakup and mass transfer in a gas-liquid stirred tank with CQMOM. Chem Eng J 2013; 228: 1182–1194.10.1016/j.cej.2013.05.047Search in Google Scholar
Pfleger D, Becker S. Modelling and simulation of the dynamic flow behaviour in a bubble column. Chem Eng Sci 2001; 56: 1737–1747.10.1016/S0009-2509(00)00403-6Search in Google Scholar
Pfleger D, Gomes S, Gilbert N, Wagner HG. Hydrodynamic simulations of laboratory scale bubble columns fundamental studies of the Eulerian-Eulerian modelling approach. Chem Eng Sci 1999; 54: 5091–5099.10.1016/S0009-2509(99)00261-4Search in Google Scholar
Podowski M. On the consistency of mechanistic multidimensional modeling of gas/liquid two-phase flows. Nucl Eng Des 2009; 239: 933–940.10.1016/j.nucengdes.2008.10.022Search in Google Scholar
Pope S. Self-conditioned fields for large-eddy simulations of turbulent flows. J Fluid Mech 2010; 652: 139–169.10.1017/S0022112009994174Search in Google Scholar
Prince MJ, Blanch HW. Bubble coalescence and break-up in air-sparged bubble columns. AIChE J 1990; 36: 1485–1499.10.1002/aic.690361004Search in Google Scholar
Ramkrishna D. Population balances: theory and applications to particulate systems in engineering. San Diego, CA: Academic Press, 2000.Search in Google Scholar
Rafique M, Chen P, Duduković MP. Computational modeling of gas-liquid flow in bubble columns. Rev Chem Eng 2004; 20: 225–375.10.1515/REVCE.2004.20.3-4.225Search in Google Scholar
Ranganathan P, Sivaraman S. Investigations on hydrodynamics and mass transfer in gas-liquid stirred reactor using computational fluid dynamics. Chem Eng Sci 2011; 66: 3108–3124.10.1016/j.ces.2011.03.007Search in Google Scholar
Richardson J, Zaki W. Sedimentation and fluidisation: Part I. Trans Inst Chem Eng 1954; 32: 35–53.Search in Google Scholar
Rigopoulos S. Population balance modelling of polydispersed particles in reactive flows. Prog Energ Comb Sci 2010; 36: 412–443.10.1016/j.pecs.2009.12.001Search in Google Scholar
Saffman PG. The lift on a small sphere in a slow shear flow. J Fluid Mech 1965; 22: 385–400.10.1017/S0022112065000824Search in Google Scholar
Sajjadi B, Raman AAA, Ibrahim S, Shah RSSRE. Review on gas-liquid mixing analysis in multiscale stirred vessel using CFD. Rev Chem Eng 2012; 28: 171–189.10.1515/revce-2012-0003Search in Google Scholar
Sanyal J, Marchisio DL, Fox RO, Dhanasekharan K. On the comparison between population balance models for CFD simulation of bubble columns. Ind Eng Chem Res 2005; 44: 5063–5072.10.1021/ie049555jSearch in Google Scholar
Sato Y, Sekoguchi K. Liquid velocity distribution in two-phase bubble flow. Int J Multiphase Flow 1975; 2: 79–95.10.1016/0301-9322(75)90030-0Search in Google Scholar
Scaglia E. Simulazione di reattori agitati gas-liquido con metodi CFD e Monte Carlo. Master’s thesis, Politecnico di Torino, Torino, Italy, 2012.Search in Google Scholar
Scargiali F, D’Orazio A, Grisafi F, Brucato A. Modelling and simulation of gas-liquid hydrodynamics in mechanically stirred tanks. Chem Eng Res Des 2007; 85: 637–646.10.1205/cherd06243Search in Google Scholar
Schiller L, Naumann A. Uber die grundlegende berechnung bei der schwekraftaufbereitung. Ztg Ver Deutsch Ing 1933; 44: 318–320.Search in Google Scholar
Shohat JA, Tamarkin JD. The problem of moments. New York: American Mathematical Society, 1943.10.1090/surv/001Search in Google Scholar
Simonnet M, Gentric C, Olmos E, Midoux N. Experimental determination of the drag coefficient in swarms of bubbles. Chem Eng Sci 2007; 62: 858–866.10.1016/j.ces.2006.10.012Search in Google Scholar
Song Q, Jin J, Wu G, Lu J, Zhang K. Experiment and cfd simulation of bubble size distribution in 2D gas-liquid bubble column. Huagong Xuebao (Chin Ed) 2008; 59: 335–340.Search in Google Scholar
Sungkorn R, Derksen JJ, Khinast JG. Modeling of turbulent gas-liquid bubbly flows using stochastic Lagrangian model and lattice-Boltzmann scheme. Chem Eng Sci 2011; 66: 2745–2757.10.1016/j.ces.2011.03.032Search in Google Scholar
Szalinski L, Abdulkareem L, Da Silva M, Thiele S, Beyer M, Lucas D, Hernandez Perez V, Hampel U, Azzopardi B. Comparative study of gas-oil and gas-water two-phase flow in a vertical pipe. Chem Eng Sci 2010; 65: 3836–3848.10.1016/j.ces.2010.03.024Search in Google Scholar
Tomiyama A. Drag, lift and virtual mass forces acting on a single bubble. In: 3rd International Symposium on Two-Phase Flow: Modeling and Experiments, Pisa, Italy, September 22–25, 2004.Search in Google Scholar
Tomiyama A, Kataoka I, Zun I, Sakaguchi T. Drag coefficient of single bubbles under normal and microgravity conditions. JSME Int J 1998; 41: 472–479.10.1299/jsmeb.41.472Search in Google Scholar
Tomiyama A, Tamai H, Zun I, Hosokawa S. Transverse migration of single bubbles in simple shear flows. Chem Eng Sci 2002; 57: 1849–1858.10.1016/S0009-2509(02)00085-4Search in Google Scholar
Unverdi SO, Tryggvason G. A front-tracking method for viscous, incompressible, multi-fluid flows. J Comput Phys 1992; 100: 25–37.10.1016/0021-9991(92)90307-KSearch in Google Scholar
Vale M, McKenna F. Solution of the population balance equation for two-component aggregation by an extended fixed pivot technique. Ind Eng Chem Res 2005; 44: 7885–7891.10.1021/ie050179sSearch in Google Scholar
van Sint Annaland M, Deen N, Kuipers J. Numerical simulation of gas-liquid-solid flows using a combined front tracking and discrete particle method. Chem Eng Sci 2005; 60: 6188–6198.10.1016/j.ces.2005.04.038Search in Google Scholar
van Wachem B, Almstedt A. Methods for multiphase computational fluid dynamics. Chem Eng J 2003; 96: 81–98.10.1016/j.cej.2003.08.025Search in Google Scholar
Verloop W. The inertial coupling force. Int J Multiphase Flow 1995; 21: 929–933.10.1016/0301-9322(95)00029-WSearch in Google Scholar
Walter JF, Blanch HW. Bubble break-up in gas-liquid bioreactors: break-up in turbulent flows. Chem Eng J 1986; 32: B7–B17.10.1016/0300-9467(86)85011-0Search in Google Scholar
Wang T. Simulation of bubble column reactors using CFD coupled with a population balance model. Front Chem Eng 2011; 5: 162–172.10.1007/s11705-009-0267-5Search in Google Scholar
Warmoeskerken M, Smith J. Flow regime maps for Rushton turbines. In: Proceedings of the 3rd World Congress of Chemical Engineering, W-624, The Society of Chemical Engineers, Japan, Tokyo, 1986.Search in Google Scholar
Wellek RM, Arawal AK, Skelland AHP. Shapes of liquid drops moving in liquid media. AIChE J 1966; 12: 854–862.10.1002/aic.690120506Search in Google Scholar
Wijngaarden LV, Jeffrey DJ. Hydrodynamic interaction between gas bubbles in liquid. J Fluid Mech 1976; 77: 27–44.10.1017/S0022112076001110Search in Google Scholar
Wright DL. Numerical advection of moments of the particle size distribution in Eulerian models. J Aerosol Sci 2007; 38: 352–369.10.1016/j.jaerosci.2006.11.011Search in Google Scholar
Wu H, Patterson G. Laser-doppler measurements of turbulent-flow parameters in a stirred mixer. Chem Eng Sci 1989; 44: 2207–2221.10.1016/0009-2509(89)85155-3Search in Google Scholar
Yang L. Numerical investigation on bubbly flow and heat transfer characters in a liquid-reactor. In: 2008 Proceedings of the ASME Summer Heat Transfer Conference, HT 2008, Vol. 2. p6. 525–530.10.1115/HT2008-56246Search in Google Scholar
Yeoh G, Tu J. Numerical modelling of bubbly flows with and without heat and mass transfer. Appl Math Modell 2006; 30: 1067–1095.10.1016/j.apm.2005.06.012Search in Google Scholar
Yeoh GH, Tu J. Computational techniques for multiphase flows. Oxford, UK: Butterworth-Heinemann, 2010.10.1016/B978-0-08-046733-7.00009-6Search in Google Scholar
Yeoh G, Cheung S, Tu J. On the prediction of bubble size distribution and void fraction in vertical gas-liquid flows. J Comput Multiphase Flows 2012; 4: 1–22.10.1260/1757-482X.4.1.1Search in Google Scholar
Yuan C, Fox RO. Conditional quadrature method of moments for kinetic equations. J Comput Phys 2011; 230: 8216–8246.10.1016/j.jcp.2011.07.020Search in Google Scholar
Yuan C, Laurent F, Fox R. An extended quadrature method of moments for population balance equations. J Aerosol Sci 2012; 51: 1–23.10.1016/j.jaerosci.2012.04.003Search in Google Scholar
Zhang D, Deen N, Kuipers J. Numerical simulation of the dynamic flow behavior in a bubble column: a study of closures for turbulence and interface forces. Chem Eng Sci 2006; 61: 7593–7608.10.1016/j.ces.2006.08.053Search in Google Scholar
Zhang Y, Yang C, Mao Z-S. Large eddy simulation of the gas-liquid flow in a stirred tank. AIChE J 2008; 54: 1963–1974.10.1002/aic.11516Search in Google Scholar
Zhang X, Li Z, Gao Z. Numerical simulation of gas-liquid flow in a dual cd-6 impeller stirred tank. Journal of Beijing University of Chemical Technology (Natural Science Edition) 2011; 38: 1–6.Search in Google Scholar
Zhang Q, Yang C, Mao Z-S, Mu J. Large eddy simulation of turbulent flow and mixing time in a gas-liquid stirred tank. Ind Eng Chem Res 2012; 51: 10124–10131.10.1021/ie202447nSearch in Google Scholar
Zun I. Transverse migration of bubbles influenced by walls in vertical bubbly flow. Int J Multiphase Flow 1980; 6: 583–588.10.1016/0301-9322(80)90053-1Search in Google Scholar
©2014 by Walter de Gruyter Berlin Boston