Fluid Streams in Two-Phase Systems

  • Massoud Kaviany
Part of the Mechanical Engineering Series book series (MES)


In this chapter we examine the representation (i.e., modeling) of fluid streams in two-phase systems using an effective, single medium. In this effective medium, the two phases are assumed to be in local thermal equilibrium. The flow and heat transfer in this effective medium is described by models which can be derived from the local volume (and time) averaging. Depending on the complexity of the phase distributions and the velocity fields, various assumptions and simplifications are made to arrive at models which can be used with a reasonable effort.


Heat Transfer Porous Medium Energy Equation Heat Mass Transfer Representative Elementary Volume 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, K.G., and Jackson, R., 1992, “A Comparison of the Solutions of Some Proposed Equations of Motion of Granular Materials for Fully Developed Flow Down Inclined Planes,” J. Fluid Mech., 241, 145–168.ADSCrossRefGoogle Scholar
  2. Aris, R., 1956, “On the Dispersion of a Solute in a Fluid Flowing Through a Tube,” Proc. Roy. Soc. (London), A235, 67–77.ADSCrossRefGoogle Scholar
  3. Bear, J., and Bachmat, Y., 1991, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht.CrossRefzbMATHGoogle Scholar
  4. Bennen, W.D., and Incropera, F.P., 1987, “A Continuum Model for Momentum, Heat and Species Transport in Binary Solid-Liquid Phase Change Systems-I., and -II.,” Int. J. Heat Mass Transfer, 30, 2161–2187.CrossRefGoogle Scholar
  5. Brenner, H., 1980, “Dispersion Resulting from Flow Through Spatially Periodic Porous Media,” Phil Trans. Roy. Soc. (London), 297, 81–133.ADSCrossRefzbMATHGoogle Scholar
  6. Brewster, M.Q., 1992, Thermal Radiative Transfer and Properties, John Wiley and Sons, New York.Google Scholar
  7. Brewster, M.Q., and Tien C.-L., 1982, “Examination of the Two-Flux Model for Radiative Transfer in Particulate Systems,” Int. J. Heat Mass Transfer, 25, 1905–1907.ADSCrossRefzbMATHGoogle Scholar
  8. Carboneil, R.G., and Whitaker, S., 1983, “Dispersion in Pulsed Systems—II. Theoretical Development for Passive Dispersion in Porous Media,” Chem. Eng. Sci., 38, 1795–1802.CrossRefGoogle Scholar
  9. Couderc, J.-P., 1985, “Incipient Fluidization and Particulate Systems” in Fluidization, Second Edition, Davidson, J.F., et al., Editors, Academic Press, London.Google Scholar
  10. Crowe, C.T., and Smoot, L.D., 1979, “Multicomponent Conservation Equations,” in Pulverized-Coal Combustion and Gasifications, Smoot, L.D., and Pratt, D.T., Editors, Plenum Press, New York.Google Scholar
  11. Davidson, J.F., Clift, R., and Harrison, D, 1985, Fluidization, Second Edition, Academic Press, London.Google Scholar
  12. Drolen, B.L., and Tien, C.-L., 1987, “Independent and Dependent Scattering in Packed Spheres Systems,” J. Thermophys. Heat Transfer, 1, 63–68.ADSCrossRefGoogle Scholar
  13. Dybbs, A., and Edwards, R.V., 1984, “A New Look at Porous Media Fluid Mechanics: Darcy to Turbulent,” in Fundamentals of Transport Phenomena in Porous Media, Bear, J., and Corapcigoln, M.Y., Editors, Martinus Nijhoff Publishers, Dordrecht.Google Scholar
  14. Elghobashi, S.E., and Abou-Arab, T.W., 1983, “A Two-Equation Turbulence Model for Two-Phase Flows,” Phys. Fluids, 26, 931–938.ADSCrossRefzbMATHGoogle Scholar
  15. Elghobashi, S.E., and Truesdell, G.C., 1993, “On the Two-Way Interaction Between Homogeneous Turbulence and Dispersed Solid Particles,” Phys. Fluids, A5, 1790–1801.ADSCrossRefzbMATHGoogle Scholar
  16. Faeth, G.M., 1983, “Evaporation and Combustion of Sprays,” Prog. Energy Combust. Sci., 9, 1–16.ADSCrossRefGoogle Scholar
  17. Fatehi, M., and Kaviany, M., 1994, “Adiabatic Reverse Combustion in a Packed Bed,” Combust. Flame, 99, 1–17.CrossRefGoogle Scholar
  18. Ganesan, S., and Poirier, D.R., 1990, “Conservation of Mass and Momentum for the Flow of Interdendritic Liquid During Solidification,” Metall. Trans., 21B, 173–181.CrossRefGoogle Scholar
  19. Gates, B.C., 1992, Catalytic Chemistry, John Wiley and Sons, New York.Google Scholar
  20. Gosman, A.D., Lekakou, C., Politis, S., Issa, R.I., and Looney, M.K., 1992, “Multidimensional Modeling of Turbulent Two-Phase Flows in Stirred Vessels,” AIChE J., 38, 1946–1956.CrossRefGoogle Scholar
  21. Gray, W.G., Leijnse, A., Kolar, R.L., and Blain, C.A., 1993, Mathematical Tools for Changing Spatial Scales in the Analysis of Physical Systems, CRC Press, Boca Raton, FL.zbMATHGoogle Scholar
  22. Gunn, D.J., 1968, “Mixing in Packed and Fluidized Beds,” Chem. Eng. (London), CE153–CE172.Google Scholar
  23. Hills, R.N., Loper, D.E., and Roberts, P.H., 1992, “On Continuum Models for Momentum, Heat and Species Transport in Solid-Liquid Phase Change Systems,” Int. Commun. Heat Mass Transfer, 19, 585–594.CrossRefGoogle Scholar
  24. Kaviany, M., 1999, Principles of Heat Transfer in Porous Media, Corrected Second Edition, Springer-Verlag, New York.Google Scholar
  25. Kaviany, M., 2001, Principles of Heat Transfer, John Wiley and Sons, New York, in press.Google Scholar
  26. Kim, C.-J., and Kaviany, M., 1992, “A Fully Implicit Method for Diffusion-Controlled Solidification of Binary Alloys,” Int. J. Heat Mass Transfer, 35, 1143–1154.CrossRefzbMATHGoogle Scholar
  27. Lahey, R.T., Jr., 1991, “Void Wave Propagation Phenomena in Two-Phase Flow (Kern Award Lecture),” AIChE J., 37, 123–135.CrossRefGoogle Scholar
  28. Lahey, R.T., Jr., and Drew, D.A., 1989, “The Three-Dimensional Time- and Volume-Averaged Conservation Equations for Two-Phase Flows,” Advan. Nuclear Sci. Tech., 20, 1–69.CrossRefGoogle Scholar
  29. Lahey, R.T., Jr., and Drew, D.A., 1990, “The Current State-of-the-Art in the Modeling of Vapor/Liquid Two-Phase Flows,” ASME paper no. 90-WA/HT-13, American Society of Mechanical Engineers, New York.Google Scholar
  30. Louge, M., Yusof, J.M., and Jenkins, J.T., 1993, “Heat Transfer in the Pneumatic Transport of Massive Particles,” Int. J. Heat Mass Transfer, 36, 265–275.ADSCrossRefzbMATHGoogle Scholar
  31. Mao, C.-P., Szekely, G.A., Jr., and Faeth, G.M., 1980, “Evaluation of Locally Homogeneous Flow Model of Spray Combustion,” J. Energy, 4, 78–87.CrossRefGoogle Scholar
  32. Martynenko, O.G., and Korovkin, V.N., 1992, “Concerning the Calculations of Plane Turbulent Jets on the Basis of the κ-ϵ Model of Turbulence,” Int. J. Heat Mass Transfer, 35, 3389–3395.CrossRefzbMATHGoogle Scholar
  33. Meyer, R.E., 1983, Theory of Dispersed Multiphase Flow, Academic Press, New York.zbMATHGoogle Scholar
  34. Michaelides, E.E., 1986, “Heat Transfer in Particulate Flows,” Int. J. Heat Mass Transfer, 29, 265–273.CrossRefGoogle Scholar
  35. Michiyoshi, I., and Serizawa, A., 1986, “Turbulence in Two-Phase Bubbly Flow,” Nuc. Eng. Design, 95, 253–267.CrossRefGoogle Scholar
  36. Nunge, R.J., and Gill, W.N., 1969, “Mechanisms Affecting Dispersion and Miscible Displacement,” in Flow Through Porous Media, American Chemical Society, Washington, 180–196.Google Scholar
  37. Ochoa, J.A., Stroeve, P., and Whitaker, S., 1986, “Diffusion and Reaction in Cellular Media,” Chem. Eng. Sci., 41, 2999–3013.CrossRefGoogle Scholar
  38. Poirier, D.R., Nandapurkar, P.J., and Ganesan, S., 1991, “The Energy and Solute Conservation Equations for Dendritic Solidification,” Metall. Trans., 22B, 889–900.CrossRefGoogle Scholar
  39. Rappaz, M., and Voller, V.R., 1990, “Modeling of Micro-Macrosegregation in Solidification Processes,” Metall. Trans., 21A, 749–753.CrossRefGoogle Scholar
  40. Revankar, S.T., and Ishii, M., 1992, “Local Interfacial Area Measurement in Bubbly Flows,” Int. J. Heat Mass Transfer, 35, 913–925.CrossRefGoogle Scholar
  41. Rutland, C.J., and Trouvé, A., 1993, “Direct Simulations of Premixed Turbulent Flames with Nonunity Lewis Number,” Combust. Flame, 94, 41–47.CrossRefGoogle Scholar
  42. Sahraoui, M., and Kaviany, M., 1993, “Slip and No-Slip Temperature Boundary Conditions at Interface of Porous, Plain Media: Conduction,” Int. J. Heat Mass Transfer, 36, 1019–1033.CrossRefGoogle Scholar
  43. Sahraoui, M., and Kaviany, M., 1994, “Slip and No-Slip Temperature Boundary Conditions at Interface of Porous, Plain Media: Convection,” Int. J. Heat Mass Transfer, 37, 1029–1044.CrossRefzbMATHGoogle Scholar
  44. Sangani, A.S., and Didwania, A.K., 1993, “Dispersed-Phase Stress Tensor in Flows of Bubbly Liquids at Large Reynolds Numbers,” J. Fluid Mech., 248, 27–54.ADSCrossRefzbMATHGoogle Scholar
  45. Scheidegger, A.E., 1974, The Physics of Flow Through Porous Media, Third Edition, University of Toronto Press, Toronto.Google Scholar
  46. Seshadri, K., Berlad, A.L., and Tangirala, V., 1992, “The Structure of Premixed Particle-Clouds Flames,” Combust. Flame, 89, 333–342.CrossRefGoogle Scholar
  47. Silverman, I., Greenberg, J.B., and Tambour, Y., 1993, “Stoichiometry and Poly-disperse Effects in Premixed Spray Flame,” Combust. Flame, 93, 97–118.CrossRefGoogle Scholar
  48. Singh, B.P., and Kaviany, M., 1991, “Independent Theory Versus Direct Simulation of Radiation Heat Transfer in Packed Beds,” Int. J. Heat Mass Transfer, 34, 2869–2882.CrossRefzbMATHGoogle Scholar
  49. Singh, B.P., and Kaviany, M., 1992, “Modeling Radiative Heat Transfer in Packed Beds,” Int. J. Heat Mass Transfer, 35, 1397–1405.CrossRefGoogle Scholar
  50. Singh, B.P., and Kaviany, M., 1994, “Effect of Particle Conductivity on Radiative Heat Transfer in Packed Beds,” Int. J. Heat Mass Transfer, 37, 2579–2583.CrossRefGoogle Scholar
  51. Slattery, J.C., 1981, Momentum, Energy, and Mass Transfer in Continua, Robert E. Krieger Publishing Company, Huntington, NY.Google Scholar
  52. Smoot, L.D., and Pratt, D.T., Editors, 1979, Pulverized-Coal Cumbustion and Gasification, Plenum Press, New York.Google Scholar
  53. Sohn, C.W., and Chen, M.M., 1981, “Micro-Convective Thermal Conductivity in Disperse Two-Phase Mixtures as Observed in a Low-Velocity Convective Flow Experiment,” ASME J. Heat Transfer, 103, 47–51.CrossRefGoogle Scholar
  54. Somorjai, G.A., 1994, Introduction to Surface Chemistry and Catalysis, John Wiley and Sons, New YorkGoogle Scholar
  55. Song, J. H. and Ishii, M., 2000, “The Well-Posedness of an Incompressible One-Dimensional Two-Fluid Model,” Int. J. Heat Mass Transfer, 43, 2221–2231.CrossRefzbMATHGoogle Scholar
  56. Soo, S.-L., 1989, Particles and Continuum: Multiphase Fluid Mechanics, Hemisphere Publishing Corporation, New York.Google Scholar
  57. Szekely, J., Evans, J.W., and Sohn, H.Y., 1976, Gas-Solid Reactions, Academic Press, New York.Google Scholar
  58. Taylor, G.I., 1953, “Dispersion of Soluble Matter in Solvent Flowing Slowly Through a Tube,” Proc. Roy. Soc. (London), A219, 186–203.ADSCrossRefGoogle Scholar
  59. Vafai, K., and Tien, C.-L., 1981, “Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media,” Int. J. Heat Mass Transfer, 24, 195–203.CrossRefzbMATHGoogle Scholar
  60. Vodak, F., Cerny, R., and Prikryl, P., 1992, “A Model of Binary Alloy Solidification with Convection in the Melt,” Int. J. Heat Mass Transfer, 35, 1787–1793.CrossRefzbMATHGoogle Scholar
  61. Vortmeyer, D., 1978, “Radiation in Packed Solid,” Proc. 6th Int. Heat Transfer Conf., 6, 525–539, Hemisphere Publishing Company, Washington, DC.Google Scholar
  62. Warmeck, J.J., 1979, “Modeling Multidimensional Systems,” in Pluverized-Coal Combustion and Gasification, Smoot, L.D., and Pratt, D.T., Editors, Plenum Press, New York.Google Scholar
  63. Whitaker, S., 1986a, “Flow in Porous Media. I: A Theoretical Derivation of Darcy’s Law,” Transp. Porous Media, 1, 3–25.CrossRefGoogle Scholar
  64. Whitaker, S., 1986b, “Transient Diffusion, Adsorption and Reaction in Porous Catalysts: The Reaction-Controlled Quasi-Steady Catalytic Surface,” Chem. Eng. Sci., 41, 3015–3022.CrossRefGoogle Scholar
  65. Whitaker, S., 1991, “Constraints for the Principles of Local Thermal Equilibrium,” Ind. Eng. Chem. Res., 30, 983–997.CrossRefGoogle Scholar
  66. Worster, M.G., 1991, “Natural Convection in a Mushy Layer,” J. Fluid Mech., 224, 335–359.ADSCrossRefzbMATHGoogle Scholar
  67. Yuan, Z.-G., Somerton, W.H., and Udell, K.S., 1991, “Thermal Dispersion in Thick-Walled Tubes as a Model of Porous Media,” Int. J. Heat Mass Transfer, 34, 2715–2726.CrossRefzbMATHGoogle Scholar
  68. Zakharov, L.V., Ovchinikov, A.A., and Nikolayev, N.A., 1993, “Modelling of the Effect of Turbulent Two-Phase Flow Friction Decrease Under the Influence of Dispersed Phase Elements,” Int. J. Heat Mass Transfer, 36, 1981–1991.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Massoud Kaviany
    • 1
  1. 1.Department of Mechanical Engineering and Applied MechanicsUniversity of MichiganAnn ArborUSA

Personalised recommendations