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One Dimensional Models for Pressure Drop, Empirical Equations for Void Fraction and Frictional Pressure Drop and Pressure Drop and other Effects in Fittings

  • Barry Azzopardi
  • John Hills
Part of the International Centre for Mechanical Sciences book series (CISM, volume 450)

Abstract

This chapter considers overall methods for pressure drop in gas liquid two-phase flow in pipes. The various elements of pressure drop are identified. Empirical methods are presented. The performance of these equations against available data are reported. Pressure drop in two-phase flow on the shell side of tube bundles is then considered. Geometries other than pipes are then considered. Firstly, pressure drop across fittings such and expansions and contractions are examined. Secondly, other aspects of two-phase flow at fittings are reviewed. Finally, the division of phase split at T-junctions is studied.

Keywords

Pressure Drop Void Fraction Liquid Flow Rate Annular Flow Orifice Plate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Andeen, G.B., and Griffiths, P. (1968) Momentum flux in two-phase flow. Journal of Heat Transfer 90: 211–222.Google Scholar
  2. Anderson, G.H., and Hills, P.D. (1974) Two-phase annular flow in tube bends. Symposium on. Multiphase Flow Systems, University of Strathclyde, paper J1, published in Instution of Chemical Engineers Symposium Series No. 38.Google Scholar
  3. Archer, W.H. (1913) Experimental determination of loss of head due to sudden enlargement in circular pipes. Transactions of the Americaan Society of Civil Engineers 76: 999–1026.Google Scholar
  4. Armand, A.A. (1946) The resistance during the movement of a two-phase system in horizontal pipes Izv. Vsesoyuznogo Tepl. Inst. 1:16–23.Google Scholar
  5. Armand, A.A., and Treschev, G.G. (1947) Investigation of resistance during the movement of steam-water mixtures in heated boiler pipes at high pressures. Izv. Vsesoyuznogo Tepl. Inst 4: 1–5.Google Scholar
  6. Arosio, S., Guglielmini, G., Lorenzi, A., Muzzio, A., and Sotgia, G. (1990) Two-phase pressure drop through sudden area contractions in horizontal flow. Heat Transfer 1990 (Proceedings of the 9th International Heat Transfer Conference, Jerusalem, 19–24 Aug. 1990), Hemisphere Publishing Corperation 6: 5964.Google Scholar
  7. Attou, A., and Bolle, L. (1997a) Integral formulation of balance equations for two-phase flow through a sudden enlargement–part 1: basic approach. Proceedings of the Institution of Mechanical Engineers 211C: 387–397.Google Scholar
  8. Attou, A., and Bolle, L. (1997b) Integral formulation of balance equations for two-phase flow through a sudden enlargement–part 2: a new interlocked volumes semi-empirical model. Proceedings of the Institution of Mechanical Engineers 211C: 399–408.Google Scholar
  9. Attou, A., Giot, M., and Seynhaeve, J.M. (1997) Modelling of steady-state two-phase bubbly flow though a sudden enlargement. International Journal of Heat and Mass Transfer 40: 3375–3385.MATHGoogle Scholar
  10. Azzopardi, B.J. (1984) The effect of side arm diameter on two phase flow split at a T junction., International Journal of Multiphase Flow 10:509–512.Google Scholar
  11. Azzopardi, B.J. (1985) Drop sizes in annular two-phase flow. Experiments in Fluids 3: 53–59.Google Scholar
  12. Azzopardi, B.J. (1988) Measurements and observations of the split of annular flow at a vertical T junction. International Journal of Multiphase Flow 14: 701–710.Google Scholar
  13. Azzopardi, B.J. (1989) The split of annular-mist flows at vertical and horizontal Ts. Proceedings of the Eighth International Conference on Offshore Mechanics and Arctic Engineering, The Hague, Netherlands, 19–23 March, ASMEGoogle Scholar
  14. Azzopardi, B.J. (1999) Phase split at T junctions. Multiphase Science and Technology 11:223–329. Azzopardi, B. J. and Whalley, P.B. (1982) The effect of flow pattern on two phase flow in a T junction. International Journal of Multiphase Flow 8: 481–507.Google Scholar
  15. Azzopardi, B.J., Purvis, A., Govan, A.H. (1987) Annular two-phase flow split at an impacting T. International Journal of Multiphase Flow 13: 605–614.Google Scholar
  16. Azzopardi, B.J., Wagstaff, D., Patrick, L., Memory, S.B., and Dowling, J. (1988) The split of two-phase flow at a horizontal T - annular and stratified flow. UKAEA Report AERE R13059.Google Scholar
  17. Azzopardi B J and Memory, S.B. (1989) The split of two-phase flow at a horizontal T–annular and stratified flow. 4th International Conference on Multi phase Flow, Nice, France, 19–21 June (Pub. BHRA )Google Scholar
  18. Azzopardi, B.J., and Smith, P.A. (1992) Flow split at a T junction: effect of side arm orientation and downstream geometry. International Journal of Multiphase Flow 18: 861–875.MATHGoogle Scholar
  19. Azzopardi, B.J., and Sudlow, C.A. (1993) The effect of pipe fittings on the structure of two-phase flow. XIth UIT National Heat Transfer Conference, Milan, Italy, 24–26 June.Google Scholar
  20. Azzopardi, B.J., and Hervieu, E. (1994) Phase separation at junctions. Multiphase Science and Technology 8: 645–714.Google Scholar
  21. Azzopardi, B.J., and Teixeira, J.C.F. (1994) Detailed measurements of vertical annular two phase flow–Part I: drop velocities and sizes. Journal of Fluids Engineering 116: 792–795.Google Scholar
  22. Azzopardi, B.J., Colman, D.A., and Nicholson, D. (2002) Plant application of a T-junction as a partial phase separator. Transactions of the Institution of Chemical Engineers 80A: 87–96.Google Scholar
  23. Ballyk, J.D., and Shoukri, M. (1990) On the development of a model for predicting phase separation phenomena in dividing two-phase flow. Nuclear Engineering and Design 123: 67–75.Google Scholar
  24. Banerjee, S., Rhodes, E., and Scott, D.S. (1967) Film inversion of co-current two-phase flow in helical coils. American Institute of Chemical Engineers 13: 189–191.Google Scholar
  25. Bankoff, S.G. (1960) A variable density single-fluid model for two-phase flow with particular reference to steam-water flow. Journal of Heat Transfer 82: 265–272.Google Scholar
  26. Bao, Z.Y., Bosnich, M.G., and Haynes, B.S. (1994) Estimation of void fraction and pressure drop for twophase flow in fine passages. Transactions of the Institution of Chemical Engineers 72A: 625–632.Google Scholar
  27. Baroczy, C.J. (1963) Correlation of liquid fraction in two-phase flow with application to liquid metals. 6 th National Heat Transfer Conference American Institute of Chemical Engineers, Preprint No 26.Google Scholar
  28. Baroczy, C.J. (1966) A systematic correlation for two-phase pressure drop. Chemical Engineering Progress, Symposium Series, 62: 232–249.Google Scholar
  29. Beattie, D.R.H., and Whalley, P.B. (1982) A simple two-phase frictional pressure drop calculation method. International Journal of Multiphase Flow 8: 83–87Google Scholar
  30. Beggs, H.D., and Brill, J.P. (1973) A study of two-phase flow in inclined pipes. Journal of Petroleum Technology,25:607–617.Google Scholar
  31. Benedict, R.P. (1980) Fundamentals of pipeflow. Wiley-Interscience, New York.Bhaga,D. and Weber, M.E., (1972) Holdup in vertical two-and three-phase flow. Canadian Journal of Chemical Engineering 50: 323–328.Google Scholar
  32. Burkholz, A. (1989) Droplet Separation. VCH, Weinheim, Germany.Google Scholar
  33. Cemak, J.O., Jicha, J.J., and Lightner, R.G. (1963) Two-phase pressure drop across vertically mounted thick plate restrictions. ASME paper 63-HT-11.Google Scholar
  34. Chakbratai, P. (1976) Some aspects of annular two-phase flow in a horizontal tube PhD Thesis, Imperial College, London.Google Scholar
  35. Charron, Y., and Whalley, P.B. (1995) Gas-liquid annular flow at a vertical tee junction–part I. Flow separation. International Journal of Multiphase Flow 21: 569–589.MATHGoogle Scholar
  36. Chisholm, D. (1967) A theoretical basis for the Lockhart-Martinelli correlation for two-phase flow. International Journal of Heat and Mass Transfer 10:1767–1778.Google Scholar
  37. Chisholm, D. (1972) An equation for velocity ratio in two-phase flow. N.E.L., Report No 535.Google Scholar
  38. Chisholm, D. (1983) Two-phase flow in pipelines and heat exchangers Pitman Press Ltd., Bath, England. Chisholm, D. (1985) Two-phase flow in heat exchangers and pipelines. Heat Transfer Engineering 6: 48–57.Google Scholar
  39. Cicchitti, A., Lombardi, C., Silvestri, M., Soldani, G., and Zavatarelli, R. (1960) Two-phase cooling experiments–pressure drop, heat transfer and burnout experiments. Energia Nucleare 7: 407–425.Google Scholar
  40. Clark, N.N., and Flemmer, R.L. (1985) Predicting the holdup in two-phase bubble upflow and downflow using the Zuber and Findlay drift-flux model. American Institute of Chemical Engineers Journal 31: 500–503.Google Scholar
  41. Crane (1983) Flow offluids through valves, fittings and pipe. Crane Ltd., London.Google Scholar
  42. Delhaye, J.M. (1981) Singular pressure drops. In Two-phase flow and heat transfer in the power and process industries. A.E. Bergles (Ed), Hemisphere Pub. Corp.Google Scholar
  43. Dowlati, R., Kawaji, M., and Chan A.M.C. (1988) Void fraction and friction pressure drop in two-phase flow across a horizontal tube bundle. American Institute of Chemical Engineers Symposium Series 84: 126–132.Google Scholar
  44. Dowlati, R., Kawaji, M., and Chan A.M.C. (1990) Pitch-to diameter effect on two-phase flow across an in-line tube bundle. American Institute of Chemical Engineers Journal 36: 765–772Google Scholar
  45. Dukler, A.E., Moye Wicks III, and Cleveland, R.G. (1965) Frictional pressure drop in two-phase flow–B: an approach through similarity analysis. American Institute of Chemical Engineers Journal 10: 44–51.Google Scholar
  46. El-Shaboury, A.M.F., Soliman, H,M., and Sims, G.E. (2001) Current state of knowledge on two-phase flow in horizontal impacting tee junctions. Multiphase Science and Technology 13:139–178.Google Scholar
  47. Engineering Science Data Unit (ESDU) (1989) Two-phase flow pressure losses in pipeline fittings. ESDU Item No. 89012.Google Scholar
  48. Fairhurst, C.P. (1983) Component pressure loss during two-phase flow. International Conference on the Physical Modelling of Multiphase Flow, Coventry, England, Paper Al, 1–24.Google Scholar
  49. Ferrell, J.K., and McGee, J.W. (1964) Two-phase flow through abrupt expansions and contractions. TID 23394.Google Scholar
  50. Fitzsimmons, D.E. (1964) Two-phase pressure drop in piping components. HW 80970 Rev. 1Google Scholar
  51. Friedel, L. (1979) Improved friction pressure drop calculations for horizontal and vertical two-phase pipe flow. European Two-phase Flow Group Meeting.Google Scholar
  52. Friedel, L., and Kissner, H.M. (1985) Pressure loss in safety valves during two-phase gas/vapour-liquid flow. Proceedings of the 2nd International Conference on Multiphase Flow, London, June (BHRA), 39–66.Google Scholar
  53. Fruendt, J., Steiff, A., and Weinspach, P.-M. (1997) Pressure relief with highly viscous fluids. Process Safety Progress 16: 57–59.Google Scholar
  54. Fryer, P.J., and Whalley, P.B. (1982) The effect of swirl on the liquid distribution in annular two-phase flow. International Journal of Multiphase Flow 8: 285–289.Google Scholar
  55. Fukano, T., and Ousaka, A. (1989) Prediction of the circumferential distribution of film thickness in horizontal and near-horizontal gas-liquid annular flow. International Journal of Multiphase Flow 15: 403419.Google Scholar
  56. Gardel, A. (1957) Les pertes de charge dans les ecoulementes au travers de branchements en te. Bulletin Technique de la Suisse Romande 9:122–130 and 10: 143–148.Google Scholar
  57. Gardner, G.C., and Neller, P.H. (1969) Phase distributions in flow of an air-water mixture round bends and past obstructions at the wall of a 76 mm boil tube. Proceedings of the Institution of Mechanical Engineers 184: 36.Google Scholar
  58. Geiger, G.E., and Rohrer, W.M. (1966) Sudden contraction losses in two-phase flow. Journal of Heat Transfer:1–9.Google Scholar
  59. Guglielmini, G., Lorenzi, A., Muzzio, A., and Sotgia, G. (1986) Two-phase pressure drops across sudden area contractions - pressure and void fraction profiles. Heat Transfer 86, (Proceedings of the 8th International Heat Transfer Conference, San Francisco, 17–22 Aug., C.L. Tien et al. Ed. ), Hemisphere 5: 2361–2366.Google Scholar
  60. Guglielmini, G., Muzzio, A., and Sotgia, G. (1997) The structure of two-phase flow in ducts with sudden contractions and its effect on the pressure drop. Proceedings of the Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics:1023–1036..Google Scholar
  61. Haaland, S.E. (1983) Simple and explicit formulas for the friction factor in turbulent pipe flow. Journal of Fluids Engineering 105:89-Google Scholar
  62. Harmathy, T.Z. (1960) Velocity of large drops and bubbles in media of infinite or restricted extent. American Institute of Chemical Engineers Journal 6: 281–288.Google Scholar
  63. Harris, D.M. (1967) Calibration of a steam-quality-meter for channel power measurement in the prototype SGHW Reactor. European Two-Phase Flow Group Meeting, Bournemouth.Google Scholar
  64. Harris, D.M., and Shires, G.L. (1972) Two-phase pressure drop in a Venturi. National Engineering Laboratory UK, Report No. 549, pp. 18–33.Google Scholar
  65. Harshe, B., Hussain, A., and Weisman, J. (1976) Two-phase pressure drop across restrictions and other abrupt area changes. Cincinnati University Ohio, Report NUREG 0062.Google Scholar
  66. Hart, J., Hamersma, P.J., and Fortuin, J.M.H. (1989) Correlations predicting frictional pressure drop and liquid holdup during horizontal gas-liquid pipe flow with a small liquid holdup, International Journal of Multiphase Flow 15: 947–964.Google Scholar
  67. Hart, J., Hamersma, P.J., and Fortuin, J.M.H. (1991) A model for predicting liquid route preference during gas-liquid flow through horizontal branched pipelines. Chemical Engineering Science 46: 1609–1622.Google Scholar
  68. Hervieu, E. (1988) Ecoulement monophasique et diphasique a bulles dans un branchement en T: étude théorique et expérimentale. Thèse de Doctorat de l’Institut National Polytechnique de Grenoble, France.Google Scholar
  69. Hewitt, G.F., and Dell, F.R. (1968) Two-phase flow of air-water mixtures in horizontal coiled tubes. Private communication.Google Scholar
  70. Hewitt, G.F. (1978), Measurement of two phase flow parameters. Academic Press, LondonGoogle Scholar
  71. Hewitt, G.F. (1984) Two-phase flow through orifices, valves, bends and other singularities. 8th Lecture Series on Two-Phase Flow, Trondheim.Google Scholar
  72. Hewitt, G.F., Gill, L.E., Roberts, D.N., and Azzopardi, B.J. (1990) The split of low inlet quality gas/liquidGoogle Scholar
  73. flow at a vertical T - Experimental data. UKAEA Report AERE M3801. Report AERE M2459.Google Scholar
  74. Hewitt, G.F., and Govan, A.H. (1990) Phenomenological modelling of non-equilibrium flow with phaseGoogle Scholar
  75. change. International Journal of Heat and Mass Transfer 32:229–242.Google Scholar
  76. Hills, J.H., Azzopardi, B.J., and Barhey, A.S. (1996) Spatial unsteadiness–a way towards intensive gasliquid reactors Transactions of the Institution of Chemical Engineers 74A: 567–574.Google Scholar
  77. Holt, A.J., Azzopardi, B.J., and Biddulph, M.W. (1995) The effect of density ratio on two-phase frictional pressure drop. International Symposium on Two-Phase Flow Modelling and Experimentation, Rome, 9–11 October.Google Scholar
  78. Holt, A.J., Azzopardi, B.J., and Biddulph, M.W. (1997) Two-phase pressure drop and void fraction in narrow channels. 5th U.K. National Heat Transfer Conference, London.Google Scholar
  79. Holt, A.J., Azzopardi, B.J., and Biddulph, M.W. (1999) Calculation of two-phase pressure drop for vertical upflow in narrow passages by means of a flow pattern specific model. Transactions of the Institution of Chemical Engineers 77A: 7–15.Google Scholar
  80. Honan, T.J., and Lahey, R.T. (1981) Measurement of phase separation in wyes and tees. Nuclear Engineering and Design 64: 93–102.Google Scholar
  81. Hoogerndoorn, C.J., and Welling, W.A. (1965) Experimental studies on the character of annular-mist flow in horizontal pipes. Symposium on Two-phase Flow, Exeter, 21–23 June, Paper C3.Google Scholar
  82. Hughmark, G.A. (1962) Hold-up in gas-liquid flow. Chemical Engineering Progress 58: 62–65.Google Scholar
  83. Isbin, H.S., Moen, R.H., Wickey, R.O., Mosher, D.R., and Larson, H.C. (1958) Two-phase steam-water pressure drops. Nuclear Science and Engineering Conference, Chicago.Google Scholar
  84. Issa, R.I., and Oliveira, P.J. (1994) Numerical prediction of phase separation in two-phase flow through T-junctions. Computers Fluids 23: 347–372.MATHGoogle Scholar
  85. Issa, R.I, and Adechy, D. (2002) Modelling of annular flow through pipes and t-junctions. Submitted.Google Scholar
  86. James, P.W. Azzopardi, B.J. Graham, D.I., and Sudlow, C.A. (2000) The effect of a bend on droplet size distribution in two-phase flow. 7 th International Conference on Multiphase Flow in Industrial Plants, Bologna, 13–15 September.Google Scholar
  87. Janssen, E., and Kervinen, J.A. (1964) Two-phase pressure losses - final report. US Atomic Energy Comm., Report No. GEAP 4634.Google Scholar
  88. Kirillov, P.R., Smogalev, I.P., Doroshenko, V.A. (1982) A graphical method of predicting the losses of pressure due to friction with a rising steam-water flow in round tubes. Thermal Engineering 29: 171172Google Scholar
  89. Kooijman, J.M., and Lacey, P.M.C. (1968) Unpublished experiments. University of Exeter.Google Scholar
  90. Lahey, R.T., and Moody, F.J. (1977) The Thermal Hydraulics of a Boiling Water Nuclear Reactor, American Nuclear Society.Google Scholar
  91. Lahey, R.T., Azzopardi, B.J., and Cox, M. (1985) Modelling two-phase flow division at T-junctions. 2nd International Conference on Multiphase Flows, London, 19–21 June (ed. BHRA ).Google Scholar
  92. Lahey, R.T. (1990) The analysis of phase separation and phase distribution phenomena using two-fluid models. Nuclear Engineering and Design 122: 17–40.Google Scholar
  93. Lemonnier, H., and Hervieu, E. (1991) Theoretical modelling and experimental investigation of singlephase and two-phase division at a tee junction. Nuclear Engineering and Design 125: 201–213.Google Scholar
  94. Levy, S. (1960) Steam-slip–theoretical prediction from momentum model. Journal of Heat Transfer 82: 113–124.Google Scholar
  95. Lin, Z.H. (1985) Two-phase flow measurements with orifices. In Encyclopedia of Fluid Mechanics (Ed. N. Cherenmisinoff) Gulf Publishing Co. Houston.Google Scholar
  96. Lockhart, R.W., and Martinelli, R.C. (1949) Proposed correlation of data for isothermal, two-phase, two-component flow in pipes. Chemical Engineering Progress 45: 39–48.Google Scholar
  97. McNown, J.S. (1954) Mechanics of manifold flow. ASCE Transactions 119: 1103–1142.Google Scholar
  98. McQuillan, K.W., and Whalley, P.B. (1984) The effect of orifices on the liquid distribution in annular two-phase flow. International Journal of Multiphase Flow 10: 721–73.Google Scholar
  99. Maddock, C., Lacey, P.M.C., and Patrick, M.A. (1974) The structure of two-phase flow in a curved pipe. Symposium on. Multiphase Flow Systems, University of Strathclyde, paper J2, published in Instution of Chemical Engineers Symposium Series No. 38.Google Scholar
  100. Marchaterre, J.F., and Hoglund, B.M. (1962) Correlation for two-phase flow. Nucleonics 8:142-Google Scholar
  101. Martinelli, R.C., and Nelson, D.B. (1948) Prediction of pressure drop during forced circulation boiling of water“, Transaction of the American Society of Mechanical Engineers 70: 695–702.Google Scholar
  102. Massena, W.A. (1960) Steam-water pressure drop. Hanford report H.W. 65706.Google Scholar
  103. Mishima, K., And Hibiki, T. (1996) Some characteristics of air-water two-phase flow in small diameter vertical tubes. International Journal of Multiphase Flow 22: 703–712.MATHGoogle Scholar
  104. Morris, S.D. (1984) A simple model for estimating two-phase momentum flux. Institution of Chemical Engineers Symposium Series No. 86, 2: 773–784.Google Scholar
  105. Morris, S.D (1985) Two-phase pressure drop across valves and orifice plates. European Two-phase Flow Group Meeting, Marchwood Engineering Laboratory, Southampton, England.Google Scholar
  106. Muller-Steinhagen, H., and Heck, K. (1986), A Simple Friction Pressure Drop Correlation for Two-Phase Flow in Pipes, Chemical Engineering Process 20: 297–308.Google Scholar
  107. Norstebo, A. (1985) Pressure drop in bends and valves in two-phase refrigerant flows. 2nd International Conference on Multiphase Flows, London, 19–21 June (ed. BHRA ).Google Scholar
  108. Oshinowo, T., and Charles, M.E. (1974) Vertical two-phase flow. Canadian Journal of Chemical Engineering 52: 438–448.Google Scholar
  109. Owens, W.L. (1961) Two-phase pressure gradient. International Heat Transfer Conference, Boulder, Colorado.Google Scholar
  110. Patrick, M., and Swanson, B.S. (1959) Expansion and contraction of an air-water mixture in vertical flow. American Institute of Chemical Engineers 5: 440–445.Google Scholar
  111. Popp, M., and Sallet, D.W. (1983) Experimental investigation of one and two-phase flow through a tee-junction. Paper B3, International Conference on the Physical Modelling of Multiphase Flows, Coventry, England, April 19–21.Google Scholar
  112. Premoli, A., Francesco, D., and Prina, A. (1970) An empirical correlation for evaluating two-phase mixture density under adiabatic conditions. European Two-Phase Flow Group Meeting.Google Scholar
  113. Ribeiro, A.M., Bott, T.R., and Jepson, D.M. (2001) The influence of a bend on drop sizes in horizontal annular two-phase flow. International Journal of Multiphase Flow 27: 721–728.MATHGoogle Scholar
  114. Richardson, B.E. (1959) Some problems in horizontal two-phase two component flows. ANL 5949.Google Scholar
  115. Romie, F. (1958) Unpublished information.Google Scholar
  116. Rooney, D.H. (1968) Void fraction prediction under saturated conditions. N.E.L. Report No 386.Google Scholar
  117. Rose, S. (1964) Some hydrodynamic Characteristics of bubbly mixtures flowing vertically upwards in tubes. ScD Thesis, Massachusetts Institute of Technology.Google Scholar
  118. Saba, N., and Lahey, R.T. (1984) The analysis of phase separation phenomena in branched conduits. International Journal of Multiphase Flow 10:1–20.Google Scholar
  119. Salcudean, M., Chun, C.H., and Groenveld, D.C. (1983) Effect of flow obstructions on void distribution in horizontal air-water flow. International Journal of Multiphase Flow 9: 91–96.Google Scholar
  120. Schadel, S.A., Leman, G.W., Binder, J.L., and Hanratty, T.J. (1990) Rates of atomisation in vertical annular flow. International Journal of Multiphase Flow 16: 363–374.MATHGoogle Scholar
  121. Schmidt, J., and Friedel, L. (1997) Two-phase pressure drop across sudden contractions in duct area. International Journal of Multiphase Flow 23: 283–299.MATHGoogle Scholar
  122. Schrage, D.S., Hau, J.T., and Jensen, M.K. (1987) Void fractions and two-phase multipliers in a horizontal tube bundle. American Institute of Chemical Engineers Symposium Series No 257.Google Scholar
  123. Shoham, O., Brill, J.P., and Taitel,Y. (1987) Two-phase flow splitting in a Tee junction–experiment and modelling. Chemical Engineering Science 42: 2667–2676.Google Scholar
  124. Simpson, H.C., Rooney, D.H., and Gratton, E. (1983) Two-phase flow through gate valves and orifice plates. International Conference on the Physical Modelling of Multiphase Flow, Coventry, England, Paper A2,: 25–40.Google Scholar
  125. Simpson, H.C., Rooney, D.H., and Callender, T.M. (1985) Pressure loss through gate valves with liquid-vapour flows. 2nd International Conference on Multiphase Flows, London, 19–21 June (ed. BHRA):67–80.Google Scholar
  126. Smith, S.L. (1971) Void fraction in two-phase flow–a correlation based upon equal velocity head model. Heat and Fluid Flow 1: 22–39Google Scholar
  127. Taitel, Y., and Dukler, A.E. (1976) A model for predicting flow regime transitions in horizontal and near-horizontal gas-liquid flow. American Institute of Chemical Engineers Journal 22: 47–55.Google Scholar
  128. Thom, J.R.S. (1964) Prediction of pressure drop during forced circulation boiling of water. International Journal of Heat and Mass Transfer 7: 709–624.Google Scholar
  129. Ulbrich, R., and Mewes, D. (1995) Experiemtnal study of gas void fraction for two-phase flow across tube bundles. Proceedings of the International Symposium on Two-Phase Flow Modelling and Experimentation, Rome, 9–11 October, 1211–1218.Google Scholar
  130. Wallis, G.B. (1961) Flooding velocities for air and water in vertical tubes. UKAEA Report AEEW R123.Google Scholar
  131. Wallis, G.B. (1969) One-dimensional Two-phase Flow.,McGraw-Hill.Google Scholar
  132. Ward Smith, A.J. (1971) Pressure losses in ducted flows. Butterworth, London.MATHGoogle Scholar
  133. Weisman, J. (1974) Two-phase pressure drop studies. Second Water Reactor Information Meeting, Washington, October.Google Scholar
  134. Whalley, P.B. (1980) Air-water two-phase flow in a helically coiled tube. International Journal of Multiphase Flow 6: 345–356.Google Scholar
  135. Wiafe, F.K. (1970) Two-phase flow through rough tubes. PhD Thesis, University of Strathclyde.Google Scholar
  136. Williams, L.R., Dykhno, L.A., and Hanratty, T.J., (1996) Droplet flux distributions and entrainment in horizontal gas-liquid flows. International Journal of Multiphase Flow 22: 1–18.MATHGoogle Scholar
  137. Wooley, D.M., and Muller-Steinhagen, H. (1989) Prediction of frictional pressure drop for two phase flow in horizontal pipes. Proceedings of the Seventeenth Australian Chemical Engineering Conference, 184190.Google Scholar
  138. Zivi, S.M. (1964) Estimation of steady state steam void fraction by means of the principle of minimum entropy production. Journal of Heat Transfer 86: 247–252.Google Scholar
  139. Zuber, N., and Findlay, J.A. (1965) Average volumetric concentration in two-phase flow systems. Journal of Heat Transfer 87: 453–468.Google Scholar
  140. Zuber, N., and Hench, J. (1962) Steady state and transient void fraction of bubbling systems and their operating limits, Part I: Steady state operation. General Electric Report 62GL100.Google Scholar

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© Springer-Verlag Wien 2003

Authors and Affiliations

  • Barry Azzopardi
    • 1
  • John Hills
    • 1
  1. 1.School of Chemical, Environmental and Mining EngineeringUniversity of NottinghamUK

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