Florence 1993

8/12/2019 Florence 1993

1/11

Engineering Turbulence Modelling and Experiments 2W. Rodi and F. Martelli (Editors)( ) 1993 Elsevier Science Publishers B.V. Ali rights reserved. 947

Modelling of Turbulent Dispersion in Two Phase Flow JetsR I Issaa and P J Oliveirabar>epartment of Mineral Reoources EngineeringImperial College of Science, Technology and Medicine, London, UKbDepanamento de Electromecnica, Universidade da Beira Interior, 6200 Covilh, PortugalAbstract

The paper describes the application of a turbulence model especially developed for two-phase flows to the pred:iction of dispersion of partic1esin co-axial confined air jets.In this model, the transport equations describing dispersed two-phase flow in an Eulerianframe are ensemble averaged using phase-fraction weighted quantities. A number of terrosarise from the averaging process in two-phase flow in addition to those which result from asimilar averaging process in single-phase flow. The effect of these additional terms on theprediction of partic1e dispersion is evaluated by comparing calculations with existing data forthe case of two co-axial jets, one of which is particle laden.The results show that two terms in the averaged equations are mainly responsible indetermining the computed rate of dispersion. They also show that the assumed partic1eresponse to eddy fluctuations has a marked influence on these predictions.1. INTRODUCTION

There are a number of two-phase flow phenomena in engineering for which reasonablepredictions cannot be obtained by application of single-phase turbulence models; such modelsdo not account for the dispersion of one phase into the other by the action of the eddies.Two-phase flow models have been developed in the recent past and the present work isconcerned with the development and assessment of one such mode .The base model was proposed by Gosman et al (1989) and developed and implementedby Politis (1989). It is based on the two-fluid concept (Ishii 1975) in which transportequations are formulated for each of the phases. These equations are then ensemble averagedusing the phase-fraction as a weighting quantity for the averaging of alI other fluctuatingquantities; this is akin to Favre averaging in variable density flow. Such an averagingprocess possesses the distinct advantage of leading to equations with far less terms than whatis obtained from straight ensemble averaging (e.g. Elghobashi Abou-Arab 1983). Themodel employs the eddy viscosity concept and equations for k and e are derived aloo usingphase-fraction weighting.Correlations appear in the averaged equations which involve fluctuations of both thecontinuous and dispersed phase velocities. Such terros are modelled with the aid of theassumption that the dispersed velocity fluctuation is directly proportional to that of thecontinuous phase, the proportionality factor (Ct) being obtained from a consideration of theresponse of a single partic1e traversing an eddy.The objective of this paper is to assess and develop the two-phase turbulence modelmentioned above. The equations involved are solved by a numerical finite-volume

8/12/2019 Florence 1993

2/11

948

methodology which is detailed by Oliveira (1992) and need not be described here. Themethodwas applied to the problemof a two-phaseparticulatejet formedby an inner air jetladen with solid particles, confined by an externallow-velocity unladen air stream. Thenumerical results obtained after the systematic inclusion of each term of the extendedturbulence model were compared with experimental measurements of Hishida Maeda(1991)therebyenablingan assessmentof themoreimportantfactorsin the turbulencemodel.This assessmentshowsthat someof the extra terms in the turbulencemodel are requiredin order to predict dispersion of the particle phase; however, additional refinement of themodel is still neededforpredictingthisdispersionaccurately.2. EQUATIONS AND TURBULENCE MODELLING

The extension of the single-phase k-e turbulence model to two-phase flows is nowpresented. This is done by introducing into the equations of motion, turbulence kineticenergy (k) and dissipation (e), the additional terms resulting from correlations of volumefractionand velocities. This followsthe workofGosman et al (1989)andPolitis (1989).2.1. The a-weighted, ensemble-average equations of motionThese are based on the Eulerian treatment of both phases, following the two-fluid model(Ishii 1975). The continuity and momentum equation obtained after applying a doubleaveraging procedure (volume-average followed by ensemble-average) to the usual single-phase equations can be written as (Oliveira 1992):

(1)

p~ i ~k~k + V.~k~k~k)=-~kVp+~kV.~k+V ~k~~+ Pk~kg+Fl\ . (2)In these equations the subscript k denotes the phases (c for the continuous and d for thedispersed) and p, p, u and t are the density, volume-fraction, pressure, velocity vectorand stress tensor. The turbulent stresses are denoted with a superscript t. The interphasemomentum exchange is represented by FD resulting from the action at the interface of the~ressure forces (Fp) and viscous stresses (F t) The sum of F t with part of F p can beidentified as the usual drag force, whereas the remaining part of Fpcontributes to the virtualmass and inviscid lift forces, which will not be considered here. The reason for neglectingthese terms is based on an order-of-magnitude analysis for the case of particle laden air jetswhere piPc - HP. In equations (1) and (2) the overbar is used, as usual, to denoteensemble-averaging. The symbol - is used to denote a-weighted averaging, which is similarto Favre averaging used in variable density single-phase flows. The definition of a-weightedaveraging is:

111= llll (3)where 111epresents any phase averaged quantity (i.e. one obtained after the first volume-average operation) and which may be split into a mean plus a fluctuating value, as:

8/12/2019 Florence 1993

3/11

949111 = 111+ 111 = j; + 111The expression for the turbulent stress in the momentum equation (2) is:

(4)

2.2. Main modelling assumptionsTo solve the momentum equation for each phase it is necessary to define the correlationsappearing in those equations. The turbulent stress (eqn. 4) for each phase is modelledfollowing the Boussinesq approximation:(5)

where is the identity tensor. For the continuous phase, the turbulence kinetic energy kc willbe obtained from its own transport equation and the turbulent viscosity ~cl is given by the k-Emodel, as explained in section 2.5. On the other hand, the dispersed phase turbulentviscosity and kinetic energy need to be specified as functions of the respective continuousphase values. To do this, and to develop alI the correlations, two main model assumptionsare required.The first is the gradient diffusion for the transport of volume fraction by velocityfluctuations:

In these equations TIis the turbulent diffusivity of which will be obtained from Tlc=vl/(Jawith the Schmidt number here taken as unity, (Ja = 1.

The second main model assumption (introduced by Gosman et aI 1989) links theinstantaneous velocity fluctuations of one phase to the velocity fluctuations of the other. Thisis a key point in the modelling and it is expressed as:

(8)where the turbulence correlation function C1is given by:c = l-exp(-t/t).I E P (9)Expression (9) is derived from the integration of the Lagrangian equation of motion of aparticle:

u = -TI , (6)c c c- - dU = - Tld d . (7)

8/12/2019 Florence 1993

4/11

8/12/2019 Florence 1993

5/11

951

however is the same. Now, sinee ultimately the veloeities solved for are the a-weightedones it becomesneeessaryto transforro intou.This ean be donebyusing the usual Favre-averagerelations seeeqns. 3) 6)and 7, toobtain:16)

where it has been assumed that the two diffusivities Tldand Tlcare equal. It ean be seen thatthe drag is eomposed of the usual mean drag plus a eontribution proportional to the void-fraction gradient whieh arises from turbulent fluetuations of a and u. The eontribution of theturbulent drag ean be quite important, mainly in the radial direction where volume-fraetiongradients are high and the mean drag is small.2.4. Modelling the dispersed phase turbulent stress and kinetic energyThe expression for the turbulenee kinetic energy of the dispersed phase is:

17)akwhieh is somewhatsimilar to the expressionfor tdt eqn.4). The simplifiedexpressionstobe usedfor kdand ti turn out to be:

18)

Pd -tC - tk Pc cwhere Ck =C? The turbulent kinematic viseosity of the dispersed phase is obtained byeomparing eqn. 19) with the Boussinesq stress model eqn. 5) to obtain:

t tVd =Cvv c

19)

20)where again Cy=C?-2 5 The k and E equations and the modelling of the additional termsThe a-weighted equationsfor the transponof turbuleneekinetieenergy k)and its rate ofdissipation E),for theeontinuousphase, arewritten as:

{ )ta - - - - - - Jl - - - kP :.. a k + V a u k = V. a ~ V k) + a G -P E ) + SdU c c. c c c c c:r c c c ck 21)

{ a - - - - -) - Jlt - - E -P :.. a E + V a u E = V a ~VE ) + a ~C IG - C2P E ) + SEd .U cc ccc Cc:r c c- ccE kc

22)

8/12/2019 Florence 1993

6/11

952

These equations are a standard generalisation of the single-phase k-e model (Jones andLaunder 1972) applied to the continuousphase, except for the additional tenns Sdwhichaccount for the interaction between dispersed and continuous phase turbulence. Theturbulentor eddyviscosityandthegenerationofk arecomputedfrom:?- C -=-- 11 - ,eetVe (23)t - - ..:rG = Jl V u . (V u + V u ),e. e e

and the constants used in the present work are the standard ones (C1 = 1.44,C2= 1.92,CI1= 0.09,

8/12/2019 Florence 1993

7/11

953

Those niOOwere amongst the models found in the literature that are listOObelow:

]1 2C~ = 1 + 0.45 u:{ ~ k) / C~ (Mostafa Mongia 1987)Furthermore, many workers use values for

8/12/2019 Florence 1993

8/11

954

standard model of Gosman et al (with Cy=Ct2 as defined in eqn. (20 ean be clearly seen tounderprediet the dispersion effect exhibited by the data. An attempt to enhance the dispersivemechanism by taking Cy=1gives someimprovementbut not quitewhatis sought Thiseanbe explained by examining the quantities responsible for the particle dispersion: themagnitudes of the fluetuations of the dispersed phase. Fig. 3 shows the plOfiles of the rmsfluctuations of the radial and axial velocities respectively at the second measuring station.The first thing to note is the anisotropy of the turbulenee whieh eould not be eaptured by thepresent k and Emodel. However, this is on1ypart of the reason why the predietions with thestandard model (with C; =Ct12as defined in eqn. (18 give too low a spread rate. FlOmFig. 3 it is apparent that the fluetuations whieh are responsible for the dispersive effect aregrossly underpredieted. It takes the use of Ct2 and C... for C; (see section 2.6) to increasethe eomputed fluetuations up to the measured levels of vd and udIn Fig. 4, the plOfiles for the normalised particle flux at the two axial stations aredisplayed for different formulations of C;. Comparison with the data shows that like Ctl assuggested by Gosman et al, C~ also gives too low a dispersion rate. On the other hand C...(see section 2.6) yields too high a value. The sensitivity of the predietions to the value of C;.ean also be gleaned from the predietions when C; is ehosen to be 0.5 Ct4; the dispersion rateis now much lower than what is measured suggesting that further refinement of the model forealeulating Ct is essential in arder to prediet the real behaviour bener.

4. CONCLUSIONSThe paper presents developments and applieation of a turbulenee model for two-phaseflows. The model is applied to the predietion of a particle-laden jet flow whieh is aeonvenient plOblem for model assessment sinee the dispersion of the partieles is mainly dueto turbulence effects. This modelleads to additional terms in the equations as compared with

the standard k-e model applied to the eontinuous phase. Sueh terms were introdueedsystematieally in the equations and the resulting predietions were analysed and comparedwith data. The main eonclusions from this study are:1. The standard model of Gosman et aI (1989) prediets dispersion of the particle-phase;

however the dispersion is under-predieted, as revealed by a eomparison with measuredparticle-flux profiles.2. The main terms plOmoting dispersion are the turbulent drag and the -2/3VC;ak term in

the dispersed-phase radial momentum equation; in this last term, Ck:should be higherthan the function Ct12as mentioned earlier and smaller than 1.

3. The dispersed-phase eddy-viscosityobtained by sening vdt=Ct/vct appears to be toosmall; with sueh small vdt(=O) the results become very sensitive to the imposed nidialvelocity for both phases at inlet. With higher v (either vi =vct or v =Ct4 Vct), theresults are not sensitive to the given inlet radial-velocities.

4. If the C(funetion used in the term -2I3VCk:ak is too high (e.g. C; = Ct4) an overshootof the distribution of along the axis oceurs close to inlet (x/D ~ 5). This overshoot is

8/12/2019 Florence 1993

9/11

955

more aceentuated if vi is small. However, even for medium or high Vdt the overshoot ispresent, although less aeeentuated. Use of C;=Ctl or Ck=Ct2 (related to inertia effectsonly) under-prediet the dispersion; use of C;=Ct4 (related with erossing-trajeetorieseffeet) yields over-predietions. This suggests use of a CCfunetion for Ck whieh takesinto account both effects (inertia and crossing-trajectories).5. The predietions of particle-dispersion are mueh improved with the modifieations to theCcfunetions mentioned above; agreement with the particle-flux data (and also with a) isstill not perfeet but it is similar to other authors (in Sommerfeld Wennerberg 1991).Further improvement eould be obtained by using a Sehmidt number (Ja smaller than 1inthe turbulenee drag term (as Simonin 1991).

7. R F R N S1 Chen, c.P. and Wood, P.E. ~ A turbulenee closure model for dilute gas-partieleflows. CanoJ. Chem. Eng. 63, pp. 349-360.2 Csanady,G.T. ~ Turbulentdiffusionof heavy partic1esin the atmosphere. J. Arm.Sei. 20, pp. 201-208.3 EIghobashi, S.E. and Abou-Arab, T.W. .l2.81 A two-equation turbulenee model fortwo-phase flows. Phys. F/uids 26, pp. 931-938.4 Faeth, G.M. 12ll Mixing, transport and eombustion in sprays. Prog. EnergyCombust. Sei. 13, pp. 293-345.5 Gosman, A.D., Issa, R.I., Lekakou, C., Looney, M.K. Politis, S. l2..8..2Multidimensionalmodellingof turbulent two-phase flows in stirred vessels. In AiChEannua/meeting 5-10Nov.6 Hishida, K. and Maeda,M. .l22.1 Turbulenee eharaeteristies of gas-solids two-phaseeonfined jet (Effeet of partiele density). In Proc. 5th Workshop on Two Phase F/owPredictions Erlangen,March 19-22,1990,pp. 3-14.7 Ishii, M. .1lli Thenno F/uidDynamicTheoryqfTwo Phase F/ow. Eyrolles, Paris.8 Issa, R.I. and Oliveira,P.I. .l22.lMethodfor predietionof partieulatejets. In Proc. 5thWorkshopon Two Phase F/owPredictions Erlangen,March 19-22,1990,pp.39 41.9 Jones, W.P. and Launder, B.E. .l2li The predietion of laminarisation with a two-equationmodel of turbulenee. Int. J. Heat Mass Transf. 15, p. 301.10 MeTigue, D.F. 2ll Mixture theory of turbulent diffusion of heavy particles. InTheoryof Dispersed Mu/tiphaseF/ow Ed. R. Meyer,AeademiePress, pp. 227-250.11 Melville, W.K. and Bray, K.N.C. l212 A model of the two-phase turbulentjet. Int.J. Heat Mass Transf 22, pp. 647-656.12 Mostafa, A.A. and Mongia, H.C. 1m On the modelling of turbulent evaporatingsprays:EulerianversusLagrangianapproach.Int J HeatMass Transf30 pp 2583-2593.13 Oliveria, P.J. 1992 Computer modelling of multidimensional multiphase flow andapplieationto T-junetions. PhDThesis Apri192,Univ. ofLondon.14 Pieart, A., Berlemont,A. and Gouesbet,G. .lilllli Modellingand Predieting turbuleneefields and the dispersion of diserete particles transported by turbulent flows. Int. J.Mu/tiphaseF/ow 12, pp. 237-261. .15 Politis, S. .l2.8.2Predietion of two-phase solid-liquidturbulent flow in stirred vessels.PhD Thesis ImperialCollege,UniversityofLondon.16 Simonin,O. .l22.1 An Eulerian approaeh for turbulent two-phase flows loaded withdiserete partieles: eode deseription. In Proc. 5th Workshop on Two Phase F/owPredictions Erlangen,March 19-22,1990,p.40 andpp 156-166.17 Sommerfeld,M. and Wennerberg,D. (Eds.) 1.2.2.LProc.5th Workshopon Two PhaseF/ow Predictions. Erlangen, Mareh 19-22, 1990. Fersehungszentrum Jlieh Cmbh,KFA.

8/12/2019 Florence 1993

10/11

956

.g 0.6q :-~ 0.4

,--,

: u,,,, ,,I:I,, b i-,I .

Lf/2

1REGlON1 I

I

J

REGIONII

Fig.l Experimental FIow Configuration

1.0

.g 0.6q :-~ 0.4

X D=10.8 - C .=1 2--_n C .=C ~doto

0.20.0 o 2

Y/R3 4

Fig.2

1.00.8 - C .=1 2--n- C .=C~do to

0.2

0.0o 2

Y/R4

Radial Profiles of the Nonnalised ParticIe FIuxat Two Axial Stations with Different

8/12/2019 Florence 1993

11/11

0 5X D=20- pred. ~

pred. 0.4

. . ,(jJ 0.3C- 0.2 ., . -- ,... - ~-- ... ~... ,...- - ~0.1

0.0 o 2 3Y/R

4

(jJ 1 0C .2.0 X D=20- pred. k

_uuu pred. 1.5 ... .

~ 0.4~

0.0 o 2 3Y/RFig 3 Radial Profiles of the RMS FluctuatingDispersed Phase Velocities

1.0 X D=100.8 -c,=c..- - - - C,=Cnnm C,=0.5.dolo.6~ 0.4~0.0

o 2Y/R 3 4

Fig 4

1.0 X D=200.8 -C,=C..C,=C_Um- C,=0.5.dolo.6

0.2

0.0o 2Y/R 3 4

Radial Prof esof the Nonnalised Particle Fluxat Two Axial Stations with Different C c

957

4