Acrivos, A. and Goddard, J.D. (1965), Asymptotic expansions for laminar-forced convection heat and mass transfer, J. Fluid Mech., 23, 273-291.

Ahmedi, G. (1987), On the mechanics of incompressible multiphase suspensions, Adv. Water Res., 10, 32.

Allen, M.P. and Tildesley, D.J. (1990), "Computer Simulations of Liquids", Oxford Science Publications, Oxford.

Auton, T.R. (1983), The dynamic of bubbles, drops and particles in motion in liquids, PhD Thesis, University of Cambridge, UK.

Balzer, G., Boelle, A. and Simonin, O. (1995), Eulerian gas-solid flow modeling of dense fluidized bed, Fluidization VIII, Int. Symp. Of the Engineering Foundation, Tours, p. 1125.

Berlemont, A., Desjonqueres, P. and Gouesbet, G. (1990), Particle Lagrangian simulation in turbulent flows, Int. J. Multiphase Flow, 16, 19-34.

Berlemont, A., Simonin, O. and Sommerfeld, M. (1995), Validation of interparticle collision models based on large eddy simulations, AME/JSME Int. Conference of gas-solid flows, Hilton Head, SC 13-18 August, Proceedings FED, 228, 359-369.

Besnard, D.C. and Harlow, F.H. (1987), Turbulence in multiphase flow, Int. J. Multiphase Flow, 10, 32.

Besnard, D.C. and Harlow, F.H. (1988), Turbulence in multiphase flow, Int. J. Multiphase Flow, 14(6), 679.

Bolio, E.J., Yasuna, J.A. and Sinclair, J.L. (1995), Dilute turbulent gas-solid flow in riser with particle-particle interactions, AIChE J., 41, 1375.

Brackbill, J.U., Kothe, D.B. and Zemach, C. (1992), A continuum method for modelling surface tension, J. Computat. Phys., 100, 335-354.

Brenner, H. (1961), The slow motion of sphere through a viscous fluid towards a plane surface, Chem. Eng. Sci., 16, 242.

Brian, P.L.T. and Hales, H.B. (1969), AIChE J., 15, 419.

Cao, J. and Ahmadi, G. (1995), Gas-particle two phase turbulent flow in a vertical duct, Int. J. Multiphase Flow, 21, 1203.

Carrica, P.M., Drew, D.A., Bonetto, F. and Lahey, R.T. (1999), A polydisperse model for bubbly two phase flow round a surface ship, Int. J. Multiphase Flow, 25, 257.

Casey, M., Lang, E., Mack, R., Schlegel, R. and Wehrli, M. (1998), Applications of CFD for process engineering at Sulzer, Speedup J., 12, 43-51.

Chapman, S. and Cowling, T.G. (1970), "The Mathematical Theory of Non-uniform Gases", Cambridge Mathematical Library, Cambridge.

Chen, P.P. and Crowe, C.T. (1984), On the Monte Carlo method for modeling particle dispersion in turbulence, ASME Symp. Gas—Solid Flows, p. 37.

Clift, R., Grace, J.R. and Weber, M.E. (1978), "Bubbles, Drops and Particles", Academic Press, New York.

Csnady, G.T. (1963), Turbulent diffusion of heavy particles in the atmosphere, J. Atmos. Sci., 105, 329-334.

Dalla Ville, J.M. (1948), "Micromeritics", Pitman, London.

Delnoij, E. (1999), "Fluid dynamics of gas-liquid bubble columns", PhD thesis, University of Twente, The Netherlands.

Delnoij, E., Lammers, F.A., Kuipers, J.A.M. and van Swaaij, W.P.M. (1997), Dynamic simulation of dispersed gas-liquid two-phase flow using a discrete bubble model, Chem. Eng. Sci., 52, 1429.

Drew, D.A. (1983), Mathematical modeling of two phase flow, Ann. Rev. Fluid. Mech., 15, 261.

Drew, D.A. and Lahey, R.T. (1979), Application of general consecutive principles to the derivation of multidimensional two phase flow equations, Int. J. Multiphase Flow, 5, 243-264.

Elghobashi, S.E. (1991), Particle-laden turbulent flows: Direct simulation and closure models, Appl. Sci. Res., 48, 301-314.

Elghobashi, S.E. (1994), On predicting particle-laden turbulent flows, Appl. Sci. Res., 52, 309.

Enwald, H., Peirano, E. and Almstedt, A.E. (1996), Eulerian two phase flow theory applied to fluidization, Int. J. Multiphase Flow, 22, 21.

Ergun, S. (1952), Fluid flow through packed columns, Chem. Eng. Prog., 48(2), 89.

Foerster, S.F., Louge, M.Y., Chang, H. and Allia, K. (1994), Measurements of the collision properties of small spheres, Phys. Fluids, 6(3), 1108.

Frenkel, D. and Smith, B. (1996), "Understanding Molecular Simulations, from Algorithm to Applications", Academic Press, San Diego, CA.

Garside, J. and Al-Dibouni, M.R. (1977), Velocity voidage relationship for fluidization and sedimentation, I&EC Proc. Des. Dev., 16, 206.

Gidaspaw, D. (1994), "Multiphase Flow and Fluidization", Academic Press, Boston.

Gidaspaw, D., Huilin, L. and Magner, E. (1996), Kinetic theory of multiphase flow and fluidization validation and extension to binary mixture, 19th Int. Congress on Theoretical and Applied Mech., Kyoto, Japan, pp. 25-31.

Gore, R.A. and Crowe, C.T. (1989), Effect of particle size on modulating turbulent intensity, Int. J. Multiphase Flow, 15(2), 279.

Gouesbet, G., Berlemont, A. and Picart, A. (1984), Dispersion of discrete particles by continuous turbulent motions. Extensive discussion of the Tchen's theory, using a two-parameter family of lagrangian correlation functions, Phys. Fluids, 27(4), 827.

Gouesbet, G. and Berlemont, A. (1999), Eulerian and Lagrangian approaches for predicting the behavior of discrete particles in turbulent flows, Prog. Energy Combustion Sci., 25, 133-159.

Gunn, D.J. (1978), Transfer of heat or mass to particle in fixed and fluidized beds, Int. J. Heat Mass Transf., 21, 467-476.

Harlow, F.H. and Amsden, A.A. (1975), Numerical calculation of multiphase flow, J. Computat. Physics, 17, 19.

Hoomans, B.P.B. (2000), Granular dynamics of gas solid two phase flows, PhD thesis, University of Twente, Enschede, The Netherlands.

Hoomans, B.P.B., Kuipers, J.A.M., Briels, W.J. and van Swaaij, W.P.M. (1998), Comment on the paper 'Numerical simulation of the gas solid flows in fluidized bed combining discrete particles method with computational fluid dynamics' by Xu, B.H. and Yu, A.B., Chem. Eng. Sci., 53, 2645.

Ishii, M. (1975), "Thermo-Fluid Dynamic theory of Two Phase Flow", Eyrolles.

Jenkins, J.T. and Richman, M.W. (1985), Grad's 13 moment system for a dense gas of inelastic spheres, Arch. Ratio. Mech. Anal., 87, 355-377.

Jenkins, J.T. and Mancini, F. (1987), Balance laws and constitutive relations for plane flows of a dense binary mixture of smooth, nearly elastic circular disks, J. Appl. Mech., 54, 27.

Johansen, S.T. and Boysen, F. (1988), Metal. Trans., 19B, 755.

Johnson, P.C. and Jackson, R. (1987), Frictional collisional constitutive relations for granular materials with application to plain shearing, J. Fluid Mech., 176, 67.

Kataoka, I. and Serizawa, A. (1989), Basic equation of turbulence in gas-liquid two phase flow, Int. J. Multiphase Flow, 15(5), 843.

Kohnen, G., Ruger, M. and Sommerfeld, M. (1994), Convergence behavior for numerical calculations by the Eular/ Lagrange method for strongly coupled phases. ASME Symp. On Numerical Methods in Multiphase Flows FED, 185, 191.

Krishna, R. and van Baton, J.M. (1999), Simulating the motion of gas bubbles in a liquid, Nature, 398, 208.

Kuipers, J.A.M. and van Swaaij, W.P.M. (1997), Application of computational fluid dynamics to chemical reaction engineering, Rev. Chem. Eng., 13, 1-118.

Kunni, D. and Levenspiel, O. (1991), "Fluidization Engineering", John Wiley and Sons, New York.

Kuo, K.K.Y. (1986), "Principles of Combustion", John Wiley and Sons, New York.

Kuo, J.T. and Wallis, G.B. (1988), Flow of bubbles through nozzles, Int. J. Multiphase Flow, 14, 547.

Lahey, R.T. (1987), Turbulence and phase distribution phenomena in two-phase flow. ICHMT International Seminar on Transient Phenomena in Multiphase Flow, Dubrovnik, Yugoslavia, May 24-30.

Lahey, R.T. and Drew, D.A. (1989), The three dimensional time and volume averaged conservation equations for two-phase flows, Adv. Nucl. Sci. Technol., 20, 1.

Lane, G.L., Schwarz, M.P. and Evans, G.M. (1999), CFD simulation of gas liquid flow in a stirred tank, 3rd Int. Symp. on Mixing in Industrial Processes, Japan.

Levenspiel, O. (1972), "Chemical Reaction Engineering", 2nd edition, John Wiley and Sons, New York.

Levich, V.G. (1962), "Physicochemical Hydrodynamics", Prentice-Hall, Englewood Cliffs, NJ.

Logtenberg, S.A., Nijemeisland, M. and Dixon, A.G. (1999), Computational fluid dynamics simulations of fluid flow and heat transfer at the wall particle contact points in a fixed bed reactor, Chem. Eng. Sci., 54, 2433-2440.

Lun, C.K.K. and Savage, S.B. (1986), The effect of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials, Acta Mechanica, 63, 15.

Luo, H. and Svendsen, H.F. (1996), Theoretical model for drop and bubble breakup in turbulent flow, AIChE J., 42(5), 1225.

Ma, D. and Ahmadi, G.J. (1986), An equation of state for dense rigid sphere gases, J. Chem. Phys., 84(6), 3449.

Ma, D. and Ahmadi, G.J. (1989), A thermodynamical formulation for dispersed turbulence flows, Int. J. Multiphase Flow, 16, 323.

Magner, E. (1996), Modeling and Simulation of gas/solid flow in curvilinear coordinates. PhD thesis, Telemark Institute of Technology, Norway.

Mathiesen, V. (1997), An experimental and computational study of multiphase flow behavior in circulating fluidized beds, PhD thesis, Norwegian University of Science and Technology, Norway.

Mathiesen, V., Solberg, T. and Hjertager, B.H. (2000), An experimental and computational study of multiphase flow behavior in a circulating fluidized bed, Int. J. Multiphase Flow, 26, 387.

Maxey, M.R. and Riley, J.J. (1983), Equation of motion for a small rigid sphere in a non uniform flow, Phys. Fluids, 26(4), 883.

Mei, R. (1993), History force on a sphere due to a step change in the free stream velocity, Int. J. Multiphase Flow, 19, 509-525.

Mikami, T. (1998), Agglomeration fluidization of liquid/solid bridging particles and its control, PhD thesis, Tokyo University of Agriculture and Technology, Department of Chemical Engineering, Japan.

Molerus, O. (1980), A coherent representation of pressure drop in fixed beds and of bed expansion for particulate fluidized beds, Chem. Eng. Sci., 35(6), 1331-1340.

Morsi, S.A. and Alexander, A.J. (1972), An investigation of particle trajectories in two phase flow system, J. FluidMech., 55(2), 193-208.

Nelson, P.A. and Galloway, T.R. (1975), Particle to fluid heat and mass transfer in dense systems of fine particles, Chem. Eng. Sci., 30, 1-6.

Nieuwland, J.J., van Sint Annaland, M., Kuipers, J.A.M. and van Swaaij, W.P.M. (1996), Hydrodynamic modeling of gas/particle flows in riser reactors, AIChE J., 42(6), 1569.

Oesterle, B. (1994), Une etude de l'influence des forces transversales agissant sur les particles dans les ecoulements gaz-solide, Powder Technol., 79, 81.

Oesterle, B. and Petitjean, A. (1993), Simulation of particle-particle interaction in gas-solid flows, Int. J. Multiphase Flows, 19, 199-211.

Ogawa, S., Umemura, A. and Oshina, N. (1980), On the equations of fully fluidized granular materials, J. Angew. Math. Phys., 31, 483.

Patel, M.K., Pericleous, K. and Cross, M. (1993), Computat. FluidDyn., 1, 161-176.

Peirano, E. (1998), Modeling and simulation of turbulent gas-solid flows applied to fluidization. PhD thesis, Chalmers University of Technology, Goteborg, Sweden.

Peirano, E. and Leckner, B. (1998), Fundamentals of turbulent gas solid flow applied to circulating fluidized bed combustion, Proc. Energy Combustion Sci., 24, 259-296.

Peirano, E. and Leckner, B. (2000), A mean diameter for numerical computations of polydispersed gassolid suspensions in fluidization, Chem. Eng. Sci., 55(6), 1189-1192.

Picart, A., Berlemont, A. and Gouesbet, G. (1982), De l'infulence du terme de Basset sur la diepersion de particules discretes dans le cadre de la theorie de Tchen, C.R. Acad. Sci., Paris, Series II, T295, 305.

Prince, N.J. and Blanch, H.W. (1990), Bubble coalescence and breakup in air sparged bubble columns, AIChE J., 36, 1485.

Ranade, V.V. (1995), Computational fluid dynamics for reactor engineering, Rev. Chem. Eng., 11(3), 225-289.

Ranade, V.V. (1998), Multiphase reaction engineering, Speedup J., 12, 26-33.

Ranz, W.E. and Marshall, W.R. (1952), Evaporation from drops, Chem. Eng. Prog., 48, 141-146.

Richardson, J.F., Davidson, F. and Harrison, D. (eds) (1971), "Fluidisation", Academic Press, London and New York.

Richardson, J.F. and Zaki, W.N. (1954), Sedimentation and fluidization: part I, Trans. Inst. Chem. Eng., 32, 35.

Rider, W.J. and Kothe, D.B. (1995), Stretching and rearing interface tracking methods, Los Alamos National Laboratory (

Rietema, K. and Van den Akker, H.E.A. (1983), On momentum equations in dispersed two phase system, Int. J. Multiphase Flow, 9, 21.

Rimmer, P.L. (1968), Heat transfer from a sphere in a stream of small Reynolds number, J. Fluid Mech., 32, 1-7.

Rudman, M. (1997), Volume tracking methods for interfacial flow calculations, Int. J. Num. Methods Fluids, 24, 671.

Schaeffer, D.G. (1987), Instability in the evolution equations describing incompressible granular flow, J. Diff. Eq., 66, 19-50.

Schiller, L. and Naumann, Z. (1935), Z. Ver. Deutsch. Ing., 77, 318.

Seibert, K.D. and Burns, M.A. (1998), Simulation of structural phenomenon in mixed-particle fluidized beds, AIChE J., 44, 528.

Simonin, O. (1995), Summerschool on 'Numerical modeling and prediction of dispersed phase flows', IMVU, Meserburg, Germany.

Sinclair, J.L. and Jackson, R. (1989), Gas-particle flow in a vertical pipe with particle-particle interactions, AIChE J., 35, 1473.

Soo, S.L. (1990), "Multiphase Fluid Dynamics", Science Press, Gower Technical, New York.

Sommerfeld, M. (1990), Numerical simulation of the particle dispersion in turbulent flow: the importance of particle lift forces and particle/wall collision models, in Numerical Methods for Multiphase Flows, Vol. 91, ASME, New York.

Sommerfeld, M. (1993), Reviews in numerical modeling of dispersed two phase flows, Proceedings of 5th Int. Symp. on Refined Flow Modeling and Turbulence Measurements, Paris.

Sommerfeld, M. (1995), The importance of inter-particle collisions in horizontal gas-solid channel flows, in "Gas-solid Flows", ASME FED 228.

Spalding, D.B. (1983), Chemical reactions in turbulent fluids, Physico Chem. Hydrodynamics, 4, 323-336.

Squire, K.D. and Eaton, J.K. (1994), Effect of selective modification of turbulence on two equation models for particle laden turbulent flows, ASME J., Fluids Eng., 116, 778.

Stokes, G.G. (1851), On the effect of the internal friction of fluids on the motion of pendulums, Trans. Cambridge Phil. Soc., 9, 8.

Stover, R.L., Tobías, C.W. and Denn, M.M. (1997), Bubble coalescence dynamics, AIChE J., 43, 23852392.

Tchen, C.M. (1947), Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid, PhD thesis, Delft University of Technology, The Netherlands.

Theologos, K.N., Nikou, I.D., Lygeros, A.I. and Markatos, N.C. (1997), Simulation and design of FCC riser-type reactors, AIChE J., 43, 486-494.

Thomson, D.J. (1987), Criteria for the selection of stochastic models of particle trajectories in turbulent flows, J. Fluid Mech., 180, 529-556.

Tsuji, Y., Morikawa, Y. and Shiomi, H. (1984), LDV measurement of two phase air solid flow in vertical pipe: J. Fluid Mech., 139, 417-434.

Wehrli, M.B., Borth, J., Drtina, P. and Lang, E. (1997), Industrial applications of CFD for mass transfer processes, Speedup J., 10, 19-26.

Westerterp, K.R., van Swaaij, W.P.M. and Beenackers, A.A.C.M. (1984), "Chemical Reactor Design and Operation", 2nd edition, John Wiley and Sons, New York.

Wijngaarden, van L. (1976), Hydrodynamic interaction between gas bubbles in liquid, J. Fluid Mech., 77, 27-44.

Xu, B.H. and Yu, A.B. (1997), Numerical simulation of gas solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics, Chem. Eng. Sci., 52, 2785.


Eddy lifetime or the integral time scale of turbulence can be expressed in the framework of the k-e model as


1 e1

Subscript 1 indicates continuous phase and 2 indicates dispersed phase. CD is a parameter of the standard k-e model (0.09), k\ is turbulent kinetic energy and ei is turbulent energy dissipation rate. The eddy lifetime seen by dispersed phase particles will in general be different from that for continuous phase fluid particles due to the so-called crossing-trajectory effect (Csnady, 1963). This can be expressed in the form:

where Ur is relative velocity and Cp is a constant which depends on the type of flow. The particle relaxation time, which represents entrainment of particles by the continuous phase, is defined by

4dp p2

0 0

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