On the mixture model for multiphase flow - vtt.fi

1 VTT PUBLICATIONS 288On the mixture model for multiphase flowMikko Manninen & Veikko TaivassaloVTT EnergySirpa Kallio bo AkademiVALTION TEKNILLINEN TUTKIMUSKESKUSESPOO 1996 ISBN 951 38 4946 5 ISSN 1235 0621 Copyright Valtion teknillinen tutkimuskeskus (VTT) 1996 JULKAISIJA UTGIVARE PUBLISHERV altion teknillinen tutkimuskeskus (VTT), Vuorimiehentie 5, PL 2000, 02044 VTTpuh. vaihde (90) 4561, telekopio 456 4374 Statens tekniska forskningscentral (VTT), Bergsmansv gen 5, PB 2000, 02044 VTTtel. v xel (90) 4561, telefax 456 4374 Technical Research Centre of Finland (VTT), Vuorimiehentie 5, 2000, FIN 02044 VTT, Finlandphone internat. + 358 0 4561, telefax + 358 0 456 4374 VTT Energia, Ydinenergia, Tekniikantie 4 C, PL 1604, 02044 VTTpuh. vaihde (09) 4561, faksi (09) 456 5000 VTT Energi, K rnkraft, Teknikv gen 4 C, PB 1604, 02044 VTTtel. v xel (09) 4561, fax (09) 456 5000 VTT Energy, Nuclear Energy, Tekniikantie 4 C, 1604, FIN 02044 VTT, Finlandphone internat.

2 + 358 9 4561, fax + 358 9 456 5000 Technical editing Leena UkskoskiVTT OFFSETPAINO, ESPOO 19963 Manninen, Mikko, Taivassalo, Veikko & Kallio, Sirpa. On the mixture model for multiphaseflow. Espoo 1996, Technical Research Centre of Finland, VTT Publications 288. 67 :51 7 wordsmultiphase flow, mixtures, models, flow, flow control, simulation, dispersions,mathematical models, equations, computersABSTRACTN umerical flow simulation utilising a full multiphase model is impracticalfor a suspension possessing wide distributions in the particle size or approximations are usually made to simplify the computationaltask. In the simplest approach, the suspension is represented by ahomogeneous single-phase system and the influence of the particles is takeninto account in the values of the physical properties. The multiphase natureof the flow cannot, however, be avoided when the concentration gradientsare large and the dispersed phases alter the hydrodynamic behaviour of themixture or when the distributions of the particles are studied.

3 In manypractical applications of multiphase flow, the mixture model is a sufficientlyaccurate approximation, with only a moderate increase in the computationaleffort compared to a single-phase study concentrates on the derivation and closing of the modelequations. The validity of the mixture model is also carefully from the continuity and momentum equations written for eachphase in a multiphase system, the field equations for the mixture arederived. The mixture equations largely resemble those for a single-phaseflow but are represented in terms of the mixture density and , an additional term in the mixture momentum equation arises fromthe slip of the dispersed phases relative to the continuous phase. Thevolume fraction for each dispersed phase is solved from a phase approaches applied in closing the mixture model equations arereviewed.

4 An algebraic equation is derived for the velocity of a dispersedphase relative to the continuous phase. Simplifications made in calculatingthe relative velocity restrict the applicability of the mixture model to casesin which the particles reach the terminal velocity in a short time periodcompared to the characteristic time scale of the flow of the mixture . Theterms for the viscous and turbulent stresses in the mixture momentumequation are usually combined to a generalised mixture model applications reported in the literature are brieflysummarised. The areas of application include gravity settling, rotationalflows and turbulent flows. The multiphase models in three commercialcodes, PHOENICS, FLUENT and CFX 4, are reviewed. The mixture modelapproach, in a simplified form, is implemented only in interest of applying computational fluid dynamics in industrialmultiphase processes has increased during the last few years.

5 Fluidisedbeds, polymerisation processes, settling tanks, chemical reactors, gasdispersion in liquids and air-lift reactors are typical examples in processindustry. Modelling of multiphase flows is, however, very complicated. Fullmultiphase modelling requires a large computing power, especially ifseveral secondary phases need to be this study, we investigate the mixture model , which is a simplification ofthe full models. This approach is a considerable alternative in simulatingdilute suspensions of solid particles or small bubbles in work is part of the project dynamics of Industrial multiphase Flows(MonDy) within the Finnish National CFD Technology Programme fundedand managed by Technology Development Centre of Finland (TEKES). TheMonDy project is carried out jointly by the University of Jyv skyl ,Tampere University of Technology, VTT Energy and bo authors are grateful to the members of the theory group of the MonDyproject for useful discussions on the various topics in multiphase flows.

6 Inparticular, we wish to thank Mr. Hannu Karema for providing valuableinformation on the general theory of multiphase flows and bringing to ourattention many important INTRODUCTION92 MODELLING OF multiphase MODELLING BASIC EQUATIONS123 MATHEMATICAL FORMULATION OF THE mixture FIELD Continuity equation for the Momentum equation for the Continuity equation for a THE RELATIVE Drag Force balance VALIDITY OF THE mixture CONSTITUTIVE model mixture model Implementation484 mixture model ONE-DIMENSIONAL APPLICATIONS WITH GRAVITATIONAL ANDCENTRIFUGAL TURBULENT FLOWS525 multiphase MODELS IN COMMERCIAL COMPUTER CODES CFX FLUENT576 SUMMARY AND DISCUSSION59 REFERENCES6267 NOTATIONSL atin letterscmass fraction [-]erradial unit vector [-]ggravitational acceleration [m/s2]jvolumetric flux [m/s]kkinetic energy of turbulence [m2/s2]mmass [kg]

7 Nnumber of phases [-]ppressure [N/m2]rradius [m]ttime [s]u,uvelocity [m/s]uIklocal instant velocity of phase k [m/s]uuuFkIkk= fluctuating component of the velocity of phase k [m/s]uuuCkkc= velocity of phase k relative to the continuous phase [m/s]umvelocity of the mixture mass centreuuuMkkm= diffusion velocity - velocity of phase k relative to themixture mass centre [m/s]uujVkkm= drift velocity - velocity of phase k relative to the mixture volume centre [m/s]Aarea [m2]CDdrag coefficient [-]Ddiffusion coefficient [m2/s]Dkiinterfacial extra deformation tensorFdrag force [N]Mmomentum source [N/m3]ReReynolds number [-]Urterminal velocity correctionVvolume [m3]Greek letters volume fraction [-] curvature of the interface [m-1] dynamic viscosity [kg/m s] material density [kg/m3] surface tension [N/m] , stress tensor [N/m2] angular frequency [s-1] second (bulk) viscosity [kg/m s] rate of the mass transfer [kg/m3s]8 Subscriptsccontinuous phaseeeffectiveicoordinate indexkphase indexmmixturepdispersed phase, particlerradialssolidtterminalx,y,zrecta ngular coordinatesr, ,zcylindrical coordinatesCrelative to the continuous phaseDdiffusionFfluctuating componentIlocal instant valueMrelative to the mass centreTturbulentVrelative to the volume centreOther symbols and operatorsabdyadic product of two vectorsaaverage of aATtransposed tensor difference gradient operator91 INTRODUCTIONA multiphase system is defined as a mixture of the phases of solid, liquidand gas.

8 Common examples are water droplets falling in air, gas bubblesrising in a liquid and solid particles transported by a fluid . multiphase flowsare often classified according to the nature of the system (Ishii 1975):dispersed flows (particles or droplets in liquid or gas, bubbles in liquid),separated flows (annular flows in vertical pipes, stratified flows inhorizontal pipes) and transitional flows, which are combinations of theother two classes. Free-surface flows can be described as stratified two-phase many cases in which the flow phenomena are dominated by one phaseand the amounts of the other, unimportant phases are small (like dusty gasflows, small gas bubbles in a liquid), multiphase flow is in practicedescribed as single phase flow and all effects of the secondary phases areneglected.

9 In this report we focus on multiphase flows where the secondaryphases cannot be ignored due to their influence on the fluid dynamicbehaviour of the mixture and partly also due to their importance for theprocess studied. Depending on the strength of the coupling between thephases, different modelling approaches are suggested. They can beclassified into homogeneous flow models, mixture models and multiphasemodels. Combinations of these are possible, too. In most models, eachphase is treated as an interpenetrating continuum with a volume fractionparameter, which is analogous to the porosity assigned to a fluid phase inflow through a porous simplest, most common formulations of the hydrodynamics of amixture refer to the motion of the centre of mass of the system. The motionsof individual components are treated in terms of diffusion through themixture.

10 This homogeneous flow model is applicable in drag dominatedflows in which the phases are strongly coupled and their velocities equaliseover short spatial length scales. All phases are assumed to move at the samevelocity. The velocity of the mixture is solved for from a single momentumequation. For each phase, an individual continuity equation is solved for toobtain its volume multiphase mixtures, gravity and centrifugal forces tend to cause velocitydifferences which have to be accounted for. A group of models has beendeveloped on the basis of an assumption of a local equilibrium. Dependingon the exact formulation of the equations used to determine the velocitydifferences (and on the personal preference of the author), this model iscalled the drift-flux model (Zuber & Findlay 1965), the mixture model (Ishii1975), the algebraic-slip model (Pericleous & Drake 1986), the suspensionmodel/approach (Verloop 1995), the diffusion model (Ungarish 1993, Ishii1975) or the local-equilibrium model (Johansen et al.)