Fundamental Equations for Two-Phase Flow in Tubes

  • Masahiro KawajiEmail author
Reference work entry


Two-phase flow of gas and liquid is often encountered in the design and operation of heat exchangers, oil/gas transport lines, chemical and bioreactors, and mass transfer equipment. The two-phase pressure drop governs the pumping requirement in forced-circulation systems, while the pressure drop dictates the circulation rate and, hence, various system parameters in natural-circulation systems. All three components of pressure drop (gravitational, frictional, and accelerational) are dependent on void fraction or quality, so the design of energy systems and their performance are highly dependent on accurate predictions of both the two-phase pressure drop and void fraction. In this chapter, basic parameters are defined first, followed by descriptions of two-phase flow patterns, flow pattern maps and transition criteria, the conservation equations used in two-phase flow analyses, and the correlations and models available for predicting void fraction and pressure drop in simple flow channel geometries such as circular and noncircular tubes. In particular, advanced two-phase flow models including multidimensional two-fluid models and the constitutive relations for interfacial transfer terms are presented. Examples of two-dimensional and one-dimensional two-fluid models applied to predict radial void fraction distributions in bubbly flow and interfacial wave characteristics in inverted annular flow, respectively, are also described.


Two-phase flow Void fraction Flow pattern Pressure drop Gas-liquid flow Conservation equations Two-fluid model Constitutive relations 


  1. Antal SP, Lahey RT Jr, Flaherty JE (1991) Analysis of phase distribution in fully developed laminar bubbly two-phase flow. Int J Multiphase Flow 17(5):635–652CrossRefGoogle Scholar
  2. Armand AA (1946) Resistance to two-phase flow in horizontal tubes. Izv VTI 15(1):16–23Google Scholar
  3. Baker O (1954) Simultaneous flow of oil and gas. Oil Gas J 53:185–195Google Scholar
  4. Banerjee S, Chan AMC (1980) Separated flow models – I analysis of the averaged and local instantaneous formulations. Int J Multiphase Flow 6:1–24CrossRefGoogle Scholar
  5. Bankoff SG (1960) A variable density single-fluid model for two-phase flow with particular reference to steam-water flow. J Heat Transf 82:265–272CrossRefGoogle Scholar
  6. Baroczy CJ (1965) A systematic correlation of for two-phase pressure drop. Chem Eng Prog Symp Ser 62(44):232–249Google Scholar
  7. Basset AB(1888) On the motion of a sphere in a viscous liquid. Philos Trans Royal Soc London, Ser A Math Phys Sci 179:43–63; also A treatise on hydrodynamics, 1961, Dover, New York, Chap. 22Google Scholar
  8. Beattie DRH, Whalley PB (1982) A simple two-phase frictional pressure drop calculation method. Int J Multiphase Flow 8:83–87CrossRefGoogle Scholar
  9. Bergles AE, Roos JP, Bourne JG (1968) Investigation of boiling flow regimes and critical heat flux. NYO-3304-13Google Scholar
  10. Cheng L, Ribatski G, Thome JR (2008) Two-phase flow patterns and flow-pattern maps: fundamentals and applications. Appl Mech Rev 61(5):050802-050802-28. Scholar
  11. Chichitti A, Lombardi C, Silvestri M, Soldaini G, Zavattarelli R (1960) Two-phase cooling experiments – pressure drop, heat transfer and burnout measurement. Energ Nucl 7(6):407–425Google Scholar
  12. Chisholm D (1973) Pressure gradients due to friction during the flow of evaporating two-phase mixtures in smooth tubes and channels. Int J Heat Mass Transf 16:347–358CrossRefGoogle Scholar
  13. Chisholm D, Laird ADK (1958) Two-phase flow in rough tubes. Trans ASME 80(2):276–286Google Scholar
  14. Coddington P, Macian R (2002) A study of the performance of void fraction correlations used in the context of drift-flux two-phase flow models. Nucl Eng Design 215:199–216CrossRefGoogle Scholar
  15. Collier JG (1972) Convective boiling and condensation. McGraw Hill, LondonGoogle Scholar
  16. Collier JG, Thome JR (1994) Convective boiling and condensation. Oxford University Press, New YorkGoogle Scholar
  17. De Jarlais G (1983) An experimental study of inverted annular flow hydrodynamics utilizing an adiabatic simulation. NUREG/CR-3339, ANL-83-44Google Scholar
  18. Drew DA, Lahey RT Jr (1987) The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Int J Multiphase Flow 13:113–121CrossRefGoogle Scholar
  19. Dukler AE, Taitel Y (1977) Flow regime transitions for vertical upward gas liquid flow: a preliminary approach through physical modeling. Progress Report No. 1, NUREG-0162Google Scholar
  20. Dukler AE, Wicks M, Cleveland RG (1964) Frictional pressure drop in two-phase flow: an approach through similarity analysis. AICHE J 10:44–51CrossRefGoogle Scholar
  21. Faghri A, Zhang Y (2006) Transport phenomena in multiphase systems. Elsevier, BurlingtonGoogle Scholar
  22. Franca F, Lahey RT (1992) The use of drift-flux techniques for the analysis of horizontal two-phase flows. Int J Multiphase Flow 18(6):787–801CrossRefGoogle Scholar
  23. Friedel L (1977) Momentum exchange and pressure drop. In: Whalley PB (ed) Two-phase flows and heat transfer. Oxford University Press, OxfordGoogle Scholar
  24. Friedel L (1979) Improved friction drop correlations for horizontal and vertical two-phase pipe flow. Paper E2 presented at the European Two-phase Flow Group Meeting, IspraGoogle Scholar
  25. Friedel L, Diener R (1998) Reproductive accuracy of selected void fraction correlations for horizontal and vertical up flow. Forsch im Ingenieurwes 64:87–97CrossRefGoogle Scholar
  26. Godbole PV, Tang CC, Ghajar AJ (2011) Comparison of void fraction correlations for different flow patterns in upward vertical two-phase flow. Heat Transf Eng 32(10):843–860CrossRefGoogle Scholar
  27. Govier GW, Aziz K (1972) The flow of complex mixtures in pipes. Van Nostrand Reinhold, New YorkGoogle Scholar
  28. Hasan AR, Kabir CS (1992) Two-phase flow in vertical and inclined annuli. Int J Multiphase Flow 18(2):279–293CrossRefGoogle Scholar
  29. Hewitt GF (1982) Flow regimes. “Pressure drop” and “void fraction”, sections 2.1–2.3. In: Hetsroni G (ed) Handbook of multiphase systems. McGraw-Hill, New YorkGoogle Scholar
  30. Hewitt GF, Roberts DN (1969) Studies of two-phase flow patterns by simultaneous X-ray and flash photography. UKAEA Report AERE-M2159Google Scholar
  31. Hubbard MG, Dukler AE (1966) The characterization of flow regimes for horizontal two-phase flow. In: Saad MA, Miller JA (eds) Proceedings of the 1966 heat transfer and fluid mechanics institute, Stanford University Press, Palo Alto, pp 100–121Google Scholar
  32. Idzinga W, Todreas N, Bowring R (1977) An assessment of two-phase pressure drop correlations for steam-water systems. Int J Multiphase Flow 3:401–413CrossRefGoogle Scholar
  33. Ishii M (1975) Thermo-fluid dynamic theory of two-phase flow. Eyrolles, PariszbMATHGoogle Scholar
  34. Ishii M (1977) One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes. ANL Report ANL-77-47Google Scholar
  35. Ishii M, Chawla TC (1979) Local drag laws in dispersed two-phase flow. ANL-79-105, NUREG/CR-1230Google Scholar
  36. Ishii M, De Jarlais G (1986) Flow regime transition and interfacial characteristics of inverted annular flow. Nucl Eng Des 95:171–184CrossRefGoogle Scholar
  37. Ishii M, Hibiki T (2006) Thermo-fluid dynamics of two-phase flow. Springer US. 10.1007/978–0–387-29187-1.
  38. Ishii M, Mishima K (1980) Study of two-fluid model and interfacial area. Argonne National Laboratory Report, ANL-80-111, NUREG/CR-1873Google Scholar
  39. Ishii M, Mishima K (1984) Two-fluid model and hydrodynamic constitutive relations. Nucl Eng Des 82:107–126CrossRefGoogle Scholar
  40. Ishii M, Zuber N (1979) Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AICHE J 25:843–855CrossRefGoogle Scholar
  41. Ishii M, Kim S, Uhle J (2002) Interfacial area transport equation: model development and benchmark experiments. Int J Heat Mass Transf 45(15):3111–3123CrossRefGoogle Scholar
  42. Ishii M, Kim S, Kelly J (2005) Development of interfacial area transport equation. Nucl Eng Technol 37(6):525–536Google Scholar
  43. Jones OC, Zuber N (1975) The interrelation between void fraction fluctuations and flow patterns in two-phase flow. Int J Multiphase Flow 2:273–306CrossRefGoogle Scholar
  44. Kawaji M, Banerjee S (1987) Application of a multifield model to reflooding of a hot vertical tube, part 1. Model structure and interfacial phenomena. J Heat Transf 109(1):204–211CrossRefGoogle Scholar
  45. Kawaji M, Anoda Y, Nakamura H, Tasaka T (1987) Phase and velocity distributions and holdup in high-pressure steam/water stratified flow in a large diameter horizontal pipe. Int J Multiphase Flow 13(2):145–159CrossRefGoogle Scholar
  46. Kim S, Ishii M, Sun X, Beus SG (2002) Interfacial area transport and evaluation of source terms for confined air water bubbly flow. Nucl Eng Des 219(1):61–65CrossRefGoogle Scholar
  47. Kocamustafaogullari G, Ishii M (1995) Foundation of the interfacial area transport equation and its closure relation. Int J Heat Mass Transf 38(3):481–493CrossRefGoogle Scholar
  48. Koizumi Y, Yamamoto N, Tasaka K (1990) Air/water two-phase flow in a horizontal large-diameter pipe (1st Report, Flow regime). Trans. JSME 56(532, B):3745–3749CrossRefGoogle Scholar
  49. Lahey RT Jr, Lopez de Bertodano M, Jones OC Jr (1993) Phase distribution incomplex geometry conduits. Nucl Eng Des 141:117–201Google Scholar
  50. Lamb H (1932) Hydrodynamics, 6th edn. Cambridge University Press, Cambridge, UKzbMATHGoogle Scholar
  51. Liu TJ, Bankoff SG (1993) Structure of air-water bubbly flow in a vertical pipe – II. Void fraction, bubble velocity and bubble size distribution. Int J Heat Mass Transf 36:1061–1072CrossRefGoogle Scholar
  52. Lockhart RW, Martinelli RC (1949) Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chem Eng Prog 45:39–48Google Scholar
  53. Mandhane JM, Gregory GA, Aziz K (1974) Critical evaluation of holdup prediction methods for gas–liquid flow in horizontal pipes. J Pet Technol 27:1017–1026CrossRefGoogle Scholar
  54. Martinelli RC, Nelson DB (1948) Prediction of pressure drop during forced-circulation boiling of water. Trans ASME 70:695–702Google Scholar
  55. McAdams WH, Wood WK, Bryan RL (1942) Vaporization inside horizontal tubes: II, benzene-oil mixtures. Trans ASME 64:193–200Google Scholar
  56. Mei R, Adrian RJ, Hanratty J (1991) Particle dispersion in isotropic turbulence under stokes drag and Basset force with gravitational settling. J Fluid Mech 225:481–495CrossRefGoogle Scholar
  57. Michaelides EE (1997) Review-the transient equation of motion for particles, bubbles and droplets. J Fluids Eng 119:233–247CrossRefGoogle Scholar
  58. Mishima K, Ishii M (1984) Flow regime transition criteria for upward two-phase flow in vertical tubes. Int J Heat Mass Transf 27(5):723–737CrossRefGoogle Scholar
  59. Müller-Steinhagen H, Heck K (1986) A simple friction pressure correlation for two-phase flow in pipes. Chem Eng Process 20:297–308CrossRefGoogle Scholar
  60. Nakoryakov VE, Kashinskii ON, Koz’myenko BK, Goryelik RS (1986) Study of upward bubbly flow at low liquid velocities. Izv Sib otdel Akad nauk SSSR 16:15–20Google Scholar
  61. Nigmatulin RI (1979) Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int J Multiphase Flow 4:353–385CrossRefGoogle Scholar
  62. Noghrehkar GR, Kawaji M, Chan AMC (1999) Investigation of two-phase flow regimes in tube bundles under cross-flow conditions. Int J Multiphase Flow 25:857–874CrossRefGoogle Scholar
  63. Oshinowo T, Charles ME (1974) Vertical two-phase flow: part 11. Holdup and pressure drop. Can J Chem Eng 56:438–448CrossRefGoogle Scholar
  64. Owens WL (1961) Two-phase pressure gradient. ASME Int Develop Heat Transf Part II 363–368Google Scholar
  65. Rouhani SZ, Axelsson E (1970) Calculation of void volume fraction in the sub cooled and quality boiling regions. Int J Heat Mass Transf 13:383–393CrossRefGoogle Scholar
  66. Rouhani SZ, Sohal MS (1983) Two-phase flow patterns: a review of research results. Prog Nucl Energy 11(3):219–259CrossRefGoogle Scholar
  67. Saadatomi M, Sato Y, Saruwatari S (1982) Two-phase flow in vertical non-circular channels. Int J Multiphase Flow 8(6):641–655CrossRefGoogle Scholar
  68. Sadatomi M, Kawaji M, Lorencez CM, Chang T (1993) Prediction of liquid level distribution in horizontal gas-liquid stratified flows with interfacial level gradient. Int J Multiphase Flow 19(6):987–997CrossRefGoogle Scholar
  69. Sato Y, Sadatomi M (1986) Two-phase flow in vertical non-circular channels. In: Cheremisinoff NP (ed) Encyclopedia of fluid mechanics, vol 3. Gulf Publishing, Houston, pp 651–664Google Scholar
  70. Serizawa A, Kataoka I, Michiyoshi I (1975) Turbulence structure of air-water bubbly flow, part II: local properties. Int J Multiphase Flow 2:235–246CrossRefGoogle Scholar
  71. Stuhmiller JH (1977) The influence of interfacial pressure on the character of two-phase flow model equations. Int J Multiphase Flow 3:551–560CrossRefGoogle Scholar
  72. Taitel Y, Dukler AE (1976a) A model for predicting flow regime transition in horizontal and near horizontal gas-liquid flow. AICHE J 22:47–55CrossRefGoogle Scholar
  73. Taitel Y, Dukler AE (1976b) A theoretical approach to the Lockhart-Martinelli correlation for stratified flow. Int J Multiphase Flow 2:591–595CrossRefGoogle Scholar
  74. Taitel Y, Bornea D, Dukler AE (1980) Modelling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AICHE J 26(3):345–354CrossRefGoogle Scholar
  75. Thom JRS (1964) Prediction of pressure drop during forced circulation boiling of water. Int J Heat Mass Transf 7:709–724CrossRefGoogle Scholar
  76. Tomiyama A, Kataoka I, Zun I, Sakaguchi T (1998) Drag coefficients of single bubbles under normal and micro gravity conditions. JSME Int J, Ser B 41(2):472–479CrossRefGoogle Scholar
  77. Tomiyama A, Tamai H, Zun I, Hosokawa S (2002) Transverse migration of single bubbles in simple shear flows. Chem Eng Sci 57:1849–1858CrossRefGoogle Scholar
  78. Wallis GB (1969) One-dimensional two-phase flow. McGraw-Hill, New YorkGoogle Scholar
  79. Wang X, Sun X (2010) Three-dimensional simulations of air–water bubbly flows. Int J Multiphase Flow 36:882–890CrossRefGoogle Scholar
  80. Weisman J, Duncan D, Gibson J, Crawford T (1979) Effects of fluid properties and pipe diameter on two-phase flow patterns in horizontal lines. Int J Multiphase Flow 5:437–462CrossRefGoogle Scholar
  81. Woldesemayat MA, Ghajar AJ (2007) Comparison of void fraction correlations for different flow patterns in horizontal and upward inclined pipes. Int J Multiphase Flow 33:347–370CrossRefGoogle Scholar
  82. Wu Q, Kim S, Ishii M, Beus SG (1998) One-group interfacial area transport in vertical bubbly flow. Int J Heat Mass Transf 41(8–9):1103–1112CrossRefGoogle Scholar
  83. Zuber N (1964) On the dispersed flow in the laminar flow regime. Chem Eng Sci 19:897–917CrossRefGoogle Scholar
  84. Zuber N, Findlay JA (1965) Average volumetric concentration in two-phase flow systems. J Heat Transf 87:453–468CrossRefGoogle Scholar
  85. Zun I (1980) The transverse migration of bubbles influenced by walls in vertical bubbly flow. Int J Multiphase Flow 6:583–588CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.City College of New YorkNew YorkUSA
  2. 2.University of TorontoTorontoCanada

Section editors and affiliations

  • Vijay K. Dhir
    • 1
  1. 1.Mechanical and Aerospace EngineeringUniversity of California Los AngelesLos AngelesUSA

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