Extension of the EMMS Model to Gas-Liquid Systems

  • Jinghai Li
  • Wei Ge
  • Wei Wang
  • Ning Yang
  • Xinhua Liu
  • Limin Wang
  • Xianfeng He
  • Xiaowei Wang
  • Junwu Wang
  • Mooson Kwauk


The Dual-Bubble-Size (DBS) model is an extension of the energy minimization multiscale (EMMS) approach for gas-liquid systems. The system is resolved into a liquid phase, small bubbles and large bubbles, and is jointly dominated by two movement tendencies; i.e., those of the small and large bubbles. A stability condition is formulated to reflect the compromise between these dominant mechanisms, offering another constraint in addition to mass and momentum conservation equations. The DBS model can theoretically predict the regime transition in bubble columns and physically explain the macro-scale evolution of flow structures through the jump change in the global minimum of the micro-scale energy dissipation changing from one point to another within the model space of the structure parameters. The DBS model is found to be an intrinsic model for gas-liquid systems in contrast to the models for single, triple, and multiple classes of bubble. A new model for the ratio of drag coefficient to bubble diameter, that is, the EMMS drag, is then integrated into the Eulerian-Eulerian computational fluid dynamics (CFD) models. The resulting improved prediction demonstrates the ability of the DBS model to reveal the multiscale nature and complexity of gas-liquid systems.


Bubble column Computational fluid dynamics Mesoscale Meso-scale Multiphase flow Multiscale Multi-scale Stability condition 



Drag coefficient for a bubble in a swarm, dimensionless


Drag coefficient for a bubble in a quiescent liquid, dimensionless


Drag coefficient for a particle in multi-particle systems, dimensionless


Coefficient of surface area increase, \( c_{\text{f}} = f_{\rm BV}^{2/3} + (1 - f_{\rm BV} )^{2/3} - 1 \), dimensionless


Bubble diameter, m


Bubble diameter of large bubbles, m


Bubble diameter of small bubbles, m


Eötvos number, dimensionless


Volume fraction of gas phase, dimensionless


Volume fraction of large bubbles, dimensionless


Volume fraction of small bubbles, dimensionless


Breakup ratio of daughter bubble to its mother bubble, dimensionless


Gravitational acceleration, m/s2


Rate of energy consumption due to bubble breakage and coalescence per unit mass, m2/s3


Rate of energy dissipation due to bubble oscillation per unit mass, m2/s3


Rate of energy dissipation in turbulent liquid phase per unit mass, m2/s3


Rate of energy dissipation for suspending and transporting particles per unit mass, m2/s3


Total rate of energy dissipation


Bubble breakup probability, dimensionless


Superficial gas velocity, m/s


Superficial gas velocity for large bubbles, m/s


Superficial gas velocity for small bubbles, m/s


Superficial liquid velocity, m/s


Relative velocity between gas and liquid, m/s

Greek Letters


Volume fraction of liquid, dimensionless


Character size of eddy, m


Viscosity, Pa·s


Density, kg/m3


Surface tension, N/m


Collision frequency, 1/s









Large bubble




Small bubble


  1. Bi HT, Grace JR (1996) Regime transitions: analogy between gas-liquid co-current upward flow and gas-solids upward transport. Int J Multiph Flow 22:1–19MATHCrossRefGoogle Scholar
  2. Camarasa E, Vial C, Poncin S, Wild G, Midoux N, Bouillard J (1999) Influence of coalescence behaviour of the liquid and of gas sparging on hydrodynamics and bubble characteristics in a bubble column. Chem Eng Process 38(4–6):329–344Google Scholar
  3. Chen J, Yang N, Ge W, Li J (2009a) Computational fluid dynamics simulation of regime transition in bubble columns incorporating the dual-bubble-size model. Ind Eng Chem Res 48(17):8172–8179CrossRefGoogle Scholar
  4. Chen J, Yang N, Ge W, Li J (2009b) Modeling of regime transition in bubble columns with stability condition. Ind Eng Chem Res 48(1):290–301CrossRefGoogle Scholar
  5. Chen J, Yang N, Ge W, Li J (2012) Stability-driven structure evolution: exploring the intrinsic similarity between gas–solid and gas-liquid systems. Chin J Chem Eng 20(1):167–177CrossRefGoogle Scholar
  6. Clift R, Grace JR, Weber ME (1978) Bubbles, drops, and particles, vol 3. Academic, New YorkGoogle Scholar
  7. De Swart JWA, van Vliet RE, Krishna R (1996) Size, structure and dynamics of “large” bubbles in a two-dimensional slurry bubble column. Chem Eng Sci 51(20):4619–4629CrossRefGoogle Scholar
  8. Eissa SH, Schugerl K (1975) Holdup and backmixing investigations in cocurrent and countercurrent bubble columns. Chem Eng Sci 30(10):1251–1256CrossRefGoogle Scholar
  9. Ellenberger J, Krishna R (1994) A unified approach to the scale-up of gas-solid fluidized bed and gas-liquid bubble column reactors. Chem Eng Sci 49(24B):5391–5411Google Scholar
  10. Fan LS, Tsuchiya K (1990) Bubble wake dynamics in liquids and liquid–solid suspensions. Butterworth–Heinemann, StonehamGoogle Scholar
  11. Ge W, Li JH (2002) Physical mapping of fluidization regimes—the EMMS approach. Chem Eng Sci 57(18):3993–4004CrossRefGoogle Scholar
  12. Ge W, Chen F, Gao J, Gao S, Huang J, Liu X, Ren Y, Sun Q, Wang L, Wang W, Yang N, Zhang J, Zhao H, Zhou G, Li J (2007) Analytical multi-scale method for multi-phase complex systems in process engineering—Bridging reductionism and holism. Chem Eng Sci 62(13):3346–3377CrossRefGoogle Scholar
  13. Grace JR, Wairegi T, Nguyen TH (1976) Shapes and velocities of single drops and bubbles moving freely through immiscible liquids. Trans Inst Chem Eng 54(3):167–173Google Scholar
  14. Harteveld WK (2005b) Bubble columns: structure or stability? Ph.D. Thesis, Delft University of Technology, The NetherlandsGoogle Scholar
  15. Harteveld WK, Mudde RF, Van den Akker HEA (2005) Estimation of turbulence power spectra for bubbly flows from laser Doppler Anemometry signals. Chem Eng Sci 60(22):6160–6168CrossRefGoogle Scholar
  16. Hills J (1974) Radial non-uniformity of velocity and voidage in a bubble column. Trans Inst Chem Eng 52(1):9Google Scholar
  17. Joshi JB, Deshpande NS, Dinkar M, Phanikumar DV (2001) Hydrodynamic stability of multiphase reactors. Adv Chem Eng 26:1–130CrossRefGoogle Scholar
  18. Kostoglou M, Karabelas AJ (2005) Toward a unified framework for the derivation of breakage functions based on the statistical theory of turbulence. Chem Eng Sci 60(23):6584–6595CrossRefGoogle Scholar
  19. Krishna R, Urseanu MI, van Baten JM, Ellenberger J (1999) Influence of scale on the hydrodynamics of bubble columns operating in the churn-turbulent regime: experiments vs. Eulerian simulations. Chem Eng Sci 54(21):4903–4911CrossRefGoogle Scholar
  20. Letzel HM, Schouten JC, Krishna R, van den Bleek CM (1997) Characterization of regimes and regime transitions in bubble columns by chaos analysis of pressure signals. Chem Eng Sci 52(24):4447–4459CrossRefGoogle Scholar
  21. Li J, Kwauk M (1994) Particle-fluid two-phase flow: the energy-minimization multi-scale method. Metallurgical Industry Press, BeijingGoogle Scholar
  22. Li J, Kwauk M (2003) Exploring complex systems in chemical engineering—the multi-scale methodology. Chem Eng Sci 58(3–6):521–535Google Scholar
  23. Li J, Cheng C, Zhang Z, Yuan J, Nemet A, Fett FN (1999) The EMMS model—its application, development and updated concepts. Chem Eng Sci 54(22):5409–5425CrossRefGoogle Scholar
  24. Li J, Ge W, Zhang J, Kwauk M (2005) Multi-scale compromise and multi-level correlation in complex systems. Chem Eng Res Des 83(A6):574–582CrossRefGoogle Scholar
  25. Li J, Ge W, Wang W, Yang N (2010) Focusing on the meso-scales of multi-scale phenomena—In search for a new paradigm in chemical engineering. Particuology 8(6):634–639CrossRefGoogle Scholar
  26. Lucas D, Prasser HM, Manera A (2005) Influence of the lift force on the stability of a bubble column. Chem Eng Sci 60(13):3609–3619CrossRefGoogle Scholar
  27. Luo H, Svendsen HF (1996) Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J 42(5):1225–1233CrossRefGoogle Scholar
  28. Magnaudet J, Eames I (2000) The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu Rev Fluid Mech 32(1):659–708MathSciNetCrossRefGoogle Scholar
  29. Monahan SM, Fox RO (2007) Linear stability analysis of a two-fluid model for air-water bubble columns. Chem Eng Sci 62(12):3159–3177CrossRefGoogle Scholar
  30. Monahan SM, Vitankar VS, Fox RO (2005) CFD predictions for flow-regime transitions in bubble columns. AIChE J 51(7):1897–1923CrossRefGoogle Scholar
  31. Mudde RF (2005) Gravity-driven bubbly flows. In: Annual review of fluid mechanics, vol 37. Annual review of fluid mechanics, pp 393–423Google Scholar
  32. Mudde RF, Harteveld WK, van den Akker HEA (2009) Uniform flow in bubble columns. Ind Eng Chem Res 48(1):148–158CrossRefGoogle Scholar
  33. Olmos E, Gentric C, Midoux N (2003) Numerical description of flow regime transitions in bubble column reactors by a multiple gas phase model. Chem Eng Sci 58(10):2113–2121CrossRefGoogle Scholar
  34. Ruthiya KC, Chilekar VP, Warnier MJF, van der Schaaf J, Kuster BFM, Schouten JC, van Ommen JR (2005) Detecting regime transitions in slurry bubble columns using pressure time series. AIChE J 51(7):1951–1965CrossRefGoogle Scholar
  35. Ruzicka MC, Zahradnik J, Drahos J, Thomas NH (2001) Homogeneous–heterogeneous regime transition in bubble columns. Chem Eng Sci 56(15):4609–4626CrossRefGoogle Scholar
  36. Ruzicka MC, Drahos J, Mena PC, Teixeira JA (2003) Effect of viscosity on homogeneous-heterogeneous flow regime transition in bubble columns. Chem Eng J 96(1–3):15–22CrossRefGoogle Scholar
  37. Ruzicka MC, Vecer MM, Orvalho S, Drahos J (2008) Effect of surfactant on homogeneous regime stability in bubble column. Chem Eng Sci 63(4):951–967CrossRefGoogle Scholar
  38. Shaikh A, Al-Dahhan MH (2007) A review on flow regime transition in bubble columns. Int J Chem Reactor Eng 57:1--68 Google Scholar
  39. Taitel Y, Bornea D, Dukler AE (1980) Modeling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AIChE J 26(3):345–354CrossRefGoogle Scholar
  40. Tomiyama A (1998) Struggle with computational bubble dynamics. Multiph Sci Technol 10(4):369–405Google Scholar
  41. Wang TF, Wang JF, Jin Y (2003) A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow. Chem Eng Sci 58(20):4629–4637CrossRefGoogle Scholar
  42. Wang TF, Wang JF, Jin Y (2005) Theoretical prediction of flow regime transition in bubble columns by the population balance model. Chem Eng Sci 60(22):6199–6209CrossRefGoogle Scholar
  43. Wang Y, Xiao Q, Yang N, Li J (2012) In-depth exploration of the dual-bubble-size model for bubble columns. Ind Eng Chem Res 51(4):2077–2083CrossRefGoogle Scholar
  44. White FM (1974) Viscous fluid flow. McGraw-Hill, New YorkMATHGoogle Scholar
  45. Wu Z, Yang N, Li J (2010) Eulerian simulation incorporating a dual-bubble-size drag model for a bubble column. Chemeca 2010: engineering at the edge. Hilton Adelaide, p 2649, 26–29 Sept 2010Google Scholar
  46. Xiao Q, Yang N, Li JH (2013) Stability-constrained multi-fluid CFD models for gas-liquid flow in bubble columns. Chem Eng Sci. doi: 10.1016/j.ces.2013.02.027
  47. Yang N, Chen JH, Zhao H, Ge W, Li JH (2007) Explorations on the multi-scale flow structure and stability condition in bubble columns. Chem Eng Sci 62(24):6978–6991CrossRefGoogle Scholar
  48. Yang N, Chen JH, Ge W, Li JH (2010) A conceptual model for analyzing the stability condition and regime transition in bubble columns. Chem Eng Sci 65(1):517–526CrossRefGoogle Scholar
  49. Yang N, Wu ZY, Chen JH, Wang YH, Li JH (2011) Multi-scale analysis of gas-liquid interaction and CFD simulation of gas-liquid flow in bubble columns. Chem Eng Sci 66(14):3212–3222CrossRefGoogle Scholar
  50. Zahradnik J, Fialova M (1996) The effect of bubbling regime on gas and liquid phase mixing in bubble column reactors. Chem Eng Sci 51(10):2491–2500CrossRefGoogle Scholar
  51. Zahradnik J, Fialova M, Ruzicka M, Drahos J, Kastanek F, Thomas NH (1997) Duality of the gas-liquid flow regimes in bubble column reactors. Chem Eng Sci 52(21–22):3811–3826Google Scholar
  52. Zakrzewski W, Lippert J, Lubbert A, Schugerl K (1981) Investigation of the structure of 2-phase flows in bubble column bioreactors. 6. Turbulence structures. Eur J Appl Microbiol Biotechnol 12(3):150–156CrossRefGoogle Scholar
  53. Zhang JP, Grace J, Epstein N, Lim K (1997) Flow regime identification in gas-liquid flow and three-phase fluidized beds. Chem Eng Sci 52(21):3979–3992CrossRefGoogle Scholar
  54. Zhang D, Deen NG, Kuipers JAM (2009) Euler–Euler modeling of flow, mass transfer, and chemical reaction in a bubble column. Ind Eng Chem Res 48(1):47–57CrossRefGoogle Scholar
  55. Zhao H (2006) Multi-scale modeling of gas-liquid (slurry) reactors. Unpublished doctoral dissertation. Chinese Academy of SciencesGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jinghai Li
    • 1
  • Wei Ge
    • 1
  • Wei Wang
    • 1
  • Ning Yang
    • 1
  • Xinhua Liu
    • 1
  • Limin Wang
    • 1
  • Xianfeng He
    • 1
  • Xiaowei Wang
    • 1
  • Junwu Wang
    • 1
  • Mooson Kwauk
    • 1
  1. 1.Institute of Process EngineeringChinese Academy of SciencesBeijingPeople’s Republic of China

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