simulation of swirling gas–particle flows using a dsm–pdf two-phase turbulence model

Ž .Powder Technology 113 2000 70–79www.elsevier.comrlocaterpowtec

Simulation of swirling gas–particle flows using a DSM–PDF two-phaseturbulence model

L.X. Zhou), Y. Li 1

Department of Engineering Mechanics, Tsinghua UniÕersity, Beijing 100084, People’s Republic of China

Received 1 November 1998; received in revised form 1 August 1999; accepted 7 December 1999

Abstract

Ž .A DSM–PDF two-phase turbulence model, namely a Reynolds stress equation DSM model of gas turbulence combined with aŽ .probability density distribution function PDF equation model of particle turbulence, is proposed to simulate the swirling sudden-expan-

sion gas–particle flows with swirl number of 0.47. The predicted axial and tangential gas and particle time-averaged velocities and RMSfluctuation velocities using the DSM–PDF and k–´–kp models are compared with measurements reported in references. The results showthat for weakly swirling flows both models can reasonably predict the mean-flow behavior, but the DSM–PDF model can better predictthe anisotropy of two-phase turbulence and turbulence interaction between two phases, and hence may have the potential superiority inpredicting strongly swirling flows. q 2000 Elsevier Science S.A. All rights reserved.

Keywords: Two-phase turbulence model; Reynolds stress equation; PDF equation; Gas–particle flow

1. Introduction

Swirling gas–particle flows are studied for developingswirl coal burners, cyclone furnaces, cyclone separators,and oil–water hydrocyclones at Tsinghua University. Thetwo-phase flow characteristics in these facilities will affectthe pressure drop, collection efficiency, flame stabilization,combustion efficiency, and pollutant emission. Most of theprevious studies on turbulent swirling flows are relevant to

w xsingle-phase flows 1 . Recently, the phase Doppler parti-Ž .cle anemometer PDPA has been used to study weakly

Ž . w x w xswirling swirl number 0.47 2 and strongly swirling 3Ž .gas–particle flows swirl number greater than 2 to give

axial and tangential gas and particle time-averaged veloci-Žties, RMS fluctuation velocities or Reynolds stress com-

.ponents , and particle mass flux. Regarding numericalmodels, although the k–´–kp two-phase turbulence modelw x4 is good enough in predicting time-averaged velocitiesand particle mass flux for nonswirling and swirling two-

) Corresponding author. Tel.: q86-10-6278-2231; fax: q86-10-6278-5569.

Ž .E-mail address: [email protected] L.X. Zhou .1 Department of Chemical Engineering, The Ohio State University,

USA.

phase flows with swirl numbers less than 0.5, it givesunsatisfactory results about the Reynolds stress compo-nents of two-phase flows. For strongly swirling flows, allversions of k–´ models and algebraic stress models, pro-posed by Rodi, fail to reasonably predict the mean-flowbehavior, such as the size and location of the centralreverse-flow zone in axial velocity profiles and the Rank-ine-vortex structure of tangential velocity profiles. In orderto better simulate anisotropic turbulent gas–particle flows,including swirling gas–particle flows, a unified second-

Ž .order-moment USM model, namely a two-phase Reynoldsw xstress turbulence model, was proposed by the author 5

and used to simulate swirling sudden-expansion gas–par-w xticle flows 6 . However, the closure assumptions made in

the particle Reynolds stress equations, modeling the diffu-sion term and particle–gas fluctuation velocity correlationterm, need to be verified. To do this, a statistical probabil-

Ž .ity density distribution function PDF model of turbulentw xgas–particle flows was also proposed 7 . Therefore, a

k–´–PDF model, namely a combination of the k–´ gas-phase turbulence model with the PDF particle-phase turbu-lence model is proposed for simulating nonswirling sud-

w xden-expansion two-phase flows 7 . The results indicatethat the PDF model of particle phase can simulate theanisotropic features of particle turbulence, even when the

0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 99 00295-8

( )L.X. Zhou, Y. LirPowder Technology 113 2000 70–79 71

gas turbulence is nearly isotropic. Since the k–´ modelcannot simulate swirling flows, therefore, in this paper aDSM–PDF model, namely a combination of a Reynolds

Ž .stress equation DSM model of gas turbulence in two-w xphase flows 6 with a PDF model of particle turbulence

w x7 for simulating swirling gas–particle flows is proposed.The purpose of this study is to improve the simulation ofthe anisotropic two-phase turbulence and to compare thesemodeling results with those obtained using the k–´–kpmodel, so as to enhance the USM model.

2. PDF transport equations

The Eulerian PDF of any instantaneous quantity at timew xt and location x is defined as 7 :j

p 'p V ,V ; x ,t 1Ž .Ž .s s i pi j

where p is the joint PDF of instantaneous quantities ins

two-phase velocity space and V , V are the gas andi pi

particle velocity coordinates respectively. One possiblerealization is:

pX V ,V ; x ,tŽ .s i pi j

3 3

s d U x ,t yV d U x ,t yV 2Ž .Ž . Ž .Ž . Ž .Ł Łi j i pi j pijs1 js1

and the PDF is the ensemble-averaged value of one real-ization in the phase space:

² X :p V ,V ; x ,t s p V ,V ; x ,t 3Ž .Ž . Ž .s i pi j s i pi j

² :where expresses the statistical average or mathematicalexpectation. Similarly, the PDF for any fluctuating quanti-ties can be defined as:

² X :p Õ ,Õ ; x ,t s p Õ ,Õ ; x ,t 4Ž .Ž . Ž .f i pi j f i pi j

where Õ , Õ are the gas and particle fluctuation velocityi pi

coordinates, and

pXÕ ,Õ ; x ,tŽ .f i pi j

3 3

s d u x ,t yÕ d u x ,t yÕ 5Ž . Ž .Ž . Ž .Ž .Ł Łj j i pi i pijs1 js1

The statistically averaged value of any variable in thegeometrical space should be:

² :Q V ,V s p V ,V Q V ,V dV dV 6Ž .Ž . Ž . Ž .HHi pi s i pi i pi i pi

² :q Õ ,Õ s p Õ ,Õ q Õ ,Õ dÕ dÕ 7Ž . Ž . Ž . Ž .HHi pi f i pi i pi i pi

For dilute gas–particle flows, starting from the instanta-neous continuity and momentum equations of two phasesand the definition of PDF and neglecting all the forces

other than the drag force between two phases, the jointPDF transport equations for p and p can be obtaineds fŽ w x.see Ref. 7 as:

Ep Epsi siq U qUŽ .j pj

Et Ex j

E PXEP pX

Etsi si ijs y d ij¦ ; ¦ ;EV r Ex r Exi i j

r EU Epp i si² : ² :y g qÝ U y U qUŽ .i pi i pjrt Ex EVrp j i

1 m EU Ep˙ p pi si² : ² :y g q q U y U qUŽ .i pi i jž /t m Ex EVrp p j pi

E EX X² :y p Ý f y p f 8² : Ž .Ž .si rpi si ri

EV EVi i

Ep Ep Epfi fi fi² : ² :q U q U q u quŽ .Ž .j pj j pjEt Ex Exj j

E pXEPX pX

EtX

fi fi ijs y d ij¦ ; ¦ ;EÕ r Ex r Exi i j

² : ² :Ep E U Ep E Ufi i fi piq u q uj pjž / ž /EÕ Ex EÕ Exi j pi j

rX

Eu Epi fiX ² :y p g y U quŽ .fi i pj pj¦ ;ž /r Ex EÕj i

Eu Ep Epi fi X² :y U qu y p Ý f² :Ž .Ž .j j fi rpiEx EÕ EÕj pi i

EX² :y p Ý f 9Ž . Ž .fi ri

EÕpi

Ž .Ž . ŽŽ .where f s r rrt u y u , f s 1rt qrpi p rp pi i ri rpŽ ..Ž .m rm u yu express the fluctuating drag forces˙ p p i pi

acting on the gas and particle phases respectively, and U ,jU are gas and particle instantaneous velocities respec-pj

² : ² :tively, U s U qu , U s U qu , u and u arej j j pj pj pj j pj

gas and particle fluctuation velocities respectively.

3. Statistically averaged equations

Neglecting the fluctuation of gas density and masssource, multiplying the instantaneous equations by p ors

p , and integrating them in the velocity space, we canf

( )L.X. Zhou, Y. LirPowder Technology 113 2000 70–7972

Ž .Fig. 1. Sudden-expansion chamber. 1 Primary air flows with particles,Ž .2 secondary air flows.

obtain the statistically averaged equations of gas and parti-cle phases for nonreacting gas–particle flows as follows.

Gas phase continuity equation:

Er E² :q r U s0 10Ž .Ž .j

Et Ex j

Particle phase continuity equation:

Er Ep ² :q r U s0 11Ž .Ž .p pjEt Ex j

Gas phase momentum equation:

² : ² : ² :E U E U E u ui i i j² :q U qjEt Ex Exj j

1 EP 1 Et ijsy q qgi¦ ; ¦ ;r Ex r Exi j

rp ² : ² :qÝ U y U q Ý f 12² : Ž .Ž .pi i rpirtrp

Particle phase momentum equation:

² : ² : ² :E U E U E u upi pi pi pj² :q U qpjEt Ex Exj j

1 EP rp ² : ² : ² :sy qg qÝ U y U q fŽ .i pi pi ri¦ ;r Ex rti rp

13Ž .Gas phase Reynolds stress transport equations:

² : ² : ² :E u u E u u E u u uk l k l j k l² :q U qjEt Ex Exj j

² : ² : 2E U E U El k² : ² : ² :sy u u y u u qm u uj k j l k l2Ex Ex Exj j j

Eu Eu Ek l X² :y2m y y r u d qu dŽ .k il l ik¦ ;Ex Ex Exj j i

Eu Eu rX

l kXq P q q g u d qu dŽ .i k il l ik¦ ;¦ ;ž /Ex Ex rk 1

q Ý f u d qu d 14² :Ž . Ž .Ž .rpi k il l ik

Particle phase Reynolds stress transport equations:

² : ² : ² :E u u E u u E u u upk pl pk pl pj pk pl² :q U qpjEt Ex Exj j

² : ² :E U E Upl pk² : ² :sy u u y u upj pk pj plEx Exj j

² :q f u d qu d 15Ž .Ž .ri pk il pl ik

Gas phase turbulent kinetic energy equation:

² 2:Ek Ek E u uj i² :q UjEt Ex Exj j

22² :E U E Euk k² :sy u u qm kymj k 2 ¦ ;ž /Ex ExExj jj

E Eu rX

kX X² :y P u q P q g uk i k¦ ;¦ ;Ex Ex ri l

q Ý f u 16² : Ž .Ž .rpi k

Particle phase turbulent kinetic energy equation:

² 2 :Ek Ek E u up p pj pi² :q U qpjEt Ex Exj j

² :E Upk² : ² :sy u u q f u 17Ž .pj pk ri kEx j

Obviously, the statistically averaged equations have a formsimilar to that of the equations obtained by Reynolds

w xaveraging 8 ; however, the former is equivalent to themass-averaged equations.

4. The DSM–PDF model

Since the modeling of the particle phase is the moreimportant component in the simulation of turbulent gas–particle flows, and solving PDF equations is more expen-sive, the PDF transport equation model can be applied onlyto particle phase, and the k–´ model or DSM model canbe still used for gas phase. Hence, k–´–PDF and DSM–PDF two-phase turbulence models are proposed. Integrat-

Ž .ing Eq. 4 in the gas fluctuating velocity space, and usingthe gradient modeling for the term of fluctuating drag

Fig. 2. Grid node arrangement.


Fig. 3. Gas axial velocity.

force, the closed form of particle-phase PDF transportequation of p can be obtained as:fp

Ep Epfp fp² :q U quŽ .pj pjEt Ex j

² :E U Eppi fps upjž /Ex EÕj pi

1 m 2ck kyk E2 p˙ Ž .p p p fpy q 18Ž .r 2ž /t m EÕrp p pi1q

rp

Ž .If p is obtained by solving Eq. 18 , then the particlefp

Reynolds stress and turbulent kinetic energy can be di-rectly obtained without solving their transport equations,that is:

² :u u s p u dÕ p u dÕ 19Ž .H Hpi pj fpi pi pi fpj pj pjž / ž /21 1

2² :k s u s p u dÕ 20Ž .Ý Ý Hp pi fpi pi piž /2 2i i

Ž . Ž .Eqs. 18 – 20 constitute the PDF model of particle phase.For swirling gas–particle flows, the DSM–PDF two-phaseturbulence model is preferred, that is, a combination of thesecond-order-moment model for gas-phase turbulence intwo-phase flows with the PDF model for particle phase.

The gas-phase Reynolds stress transport equations can bew xwritten as 8 :

E E² : ² :² :r u u q r U u uŽ . Ž .i j k i j

Et Ex k

sD qP qP y´ qG 21Ž .ij ij ij ij p ,ij

where D , P , P , ´ , G are diffusion, production,ij ij ij ij p,ij

pressure-strain, dissipation terms of gas stresses, and thesource term due to gas–particle interaction. The meaningsof the first four terms are the same as that for the single-phase flows. For the last term, the dimensional analysis

w xgives 8 :

2 rp kG sG Pd s C kk yk Pd(Ý ž /p ,ij p ij p p ijtpp

where k and kp are the turbulent kinetic energy of gas andparticle phases. Moreover, the transport equation of gasturbulent kinetic energy dissipation rate for two-phase

w xflows is 8 :

E E² :r´ q r U ´Ž . Ž .k

Et Ex k

2E C r E´ ´m ks q C G qG yC r´Ž .´1 k p ´ 2ž /Ex s ´ Ex kk ´ k

22Ž .

Fig. 4. Particle axial velocity.


Fig. 5. Gas tangential velocity.

Ž . Ž .Eqs. 18 – 22 constitute the DSM–PDF two-phase turbu-lence model.

5. Solution procedure

The gas-phase and particle-phase differential equationsin their generalized form are integrated in the computa-tional cell to obtain the finite difference equations using ahybrid scheme. The empirical constants taken in this model

w xare the same as those taken in Refs. 4,6 . A nonuniformstaggered grid system is adopted. The gas-phase Reynolds

² : ² : ² : ² :stresses uu , ÕÕ , ww , gas tangential velocity W ,and PDF functions p , p , p are stored in grid nodesfpu fpv fpw

p. The SIMPLEC algorithm with p–Õ corrections is usedto solve the gas-phase equations and the FVG-FD methodw x7 is used to solve the PDF equation of particle phase. Theconvergence criterion for gas phase is mass source lessthan 0.005, and for particle phase is mass source less than

Ž0.01. The computer code DPTF DSM–PDF model for.two-phase flows is developed with nearly 15 000 state-

ments in FORTRAN-77 language for solving 11 gas-phaseŽequations three momentum equations, six stress equations,

.one pressure correction equation and one equation and sixŽparticle-phase equations three momentum equations and

.three PDF equations .

6. Simulation of swirling gas–particle flows

The geometrical configuration and sizes of the sudden-expansion chamber with coaxial swirling gas–particle flows

w xto be predicted are shown in Fig. 1, taken from Ref. 2 .The non-swirling primary air with particles is supplied intothe central tube, and the swirling secondary pure air issupplied through the annular space. The vertically placedchamber is 950 mm long with diameters D s32 mm,1

D s70 mm and D s194 mm. The PDPA measurements2 3w xmade in Ref. 2 give axial, radial and tangential time-aver-

aged velocities and RMS fluctuation velocities of gasphase and particle phases with sizes of 30, 45 and 60 mm

Ž .for seven cross sections from the inlet xs3 mm to theŽ .exit xs950 mm . The predictions were made for parti-

cles of only 45.5-mm size. The predictions using theDSM–PDF model are compared with the measurements

w xmade in Ref. 2 and those using the k–´–kp modelw x4 .The inlet-flow parameters are: inlet-flow velocity 9.9mrs; annular-flow velocity 38.3 mrs; swirl number 0.47;mean particle size 45.5 mm; material density 2500 kgrm3;

Žmass loading particle mass flux over gas mass flux at the.inlet 0.034; particle mass flow rate 0.34 grs. The bound-

ary conditions for gas phase are: measured inlet velocitiesand Reynolds stresses; fully developed flow exit condi-tions; axi-symmetric conditions at the axis, no-slip condi-

Fig. 6. Particle tangential velocity.


Fig. 7. Gas axial fluctuation velocity.

tions at the walls and wall function approximations fornear-wall grid nodes. Particle-phase boundary conditionsare: measured inlet velocities and stresses and assumedthree-peak inlet PDF as

0.5 Õ s0pi< ²p sinfpi < < < <0.2 Õ s"0.1 Upi pin

fully developed flow exit conditions; symmetrical condi-tions at the axis and zero normal velocities and zero fluxesof other quantities at the walls. Nonuniform 32=20 grid

Ž .nodes are adopted for the computed domain Fig. 2 . Theunder-relaxation factors for all variables are taken as 0.1.Running a case with one particle size in a 386-33 PCcomputer needs about 60 h for the DSM–PDF model and14 h for the k–´–kp model.

7. Results and discussion

Figs. 3–6 show predicted axial and tangential velocitiesŽ .of gas and particle phases using the DP DSM–PDF and

Ž .KP k–´–kp models and the comparison of predictionsw xwith the measurements made in Ref. 2 . It can be seen that

both DP and KP models are in agreement with the mea-

surement results for predicting the mean flow behavior.The difference between these two models is not obvious inthis aspect. There are w-shaped axial velocity profiles withan annular reverse-flow zone, which is important for flamestabilization in a swirl burner. The gas axial-velocity lagsbehind the particle one, since the particles are fed into theprimary air flow and so particles have more inertia than

Ž .the gas in the axial direction Figs. 3 and 4 ; The so-calledŽRankine-vortex a solid-body rotation core plus free vortex

.in the near-wall region structure of tangential velocityŽprofiles is given by predictions using two models Figs. 5

.and 6 . The particle tangential velocity lags behind the gasŽ .one Figs. 5 and 6 , since particles have no initial tangen-

tial velocity and their tangential velocity is caused purelyby swirling secondary-air flows. Particle inertia makes itsvelocity lag in tangential direction. Quantitatively, there isstill obvious difference between DP model predictions andmeasurements, which may be caused by the uncertainty inspecifying boundary conditions and numerous empiricalconstants in the Reynolds stress model, and also measure-ment errors. However, in comparison with the KP model,the DP model can better predict the general features ofvelocity profiles, for example, more obvious three peaks inaxial velocity profiles and two-zone vortex structure of

Fig. 8. Particle axial fluctuation velocity.


Fig. 9. Gas tangential RMS velocity.

Fig. 10. Particle tangential RMS velocity.

Ž .Fig. 11. Gas RMS velocity experimental .

Ž .Fig. 12. Gas RMS velocity prediction .


Ž .Fig. 13. Particle RMS velocity experimental .

Ž .Fig. 14. Particle RMS velocity prediction .

Ž .Fig. 15. Two-phase tangential RMS velocity experimental .

Ž .Fig. 16. Two-phase tangential RMS velocity DP .


Ž .Fig. 17. Two-phase axial RMS velocity experimental .

tangential velocity profiles, while the KP model tends tosmooth these structures due to its isotropy assumptions.Figs. 7–10 give the predicted axial and tangential RMSfluctuation velocities of gas and particle phases using DPand KP models and the comparison between predictionsand measurements. Experiments give two peaks and threepeaks in gas axial fluctuation velocity and particle axial

Ž .fluctuation velocity profiles respectively Figs. 7 and 8 ,but predictions using two models show more uniformdistribution and predicted values are lower than the experi-mental ones. For gas and particle tangential fluctuation

Ž .velocities the situation is better Figs. 9 and 10 . Predic-tions using two models give the distribution similar to thatobtained in measurements. Initially there are two peaksnear the axis and afterwards there are peak values near thewall. The predicted gas and particle tangential fluctuationvelocities are also lower than those measured, but thedifference between predictions and measurements is smallerthan that in axial direction. In general for the normalcomponents of Reynolds stresses of two phases, the agree-ment of predictions using both models with measurementsis worse than that for the time-averaged velocities, inparticular, in the axial direction. This indicates that themodels still need to be improved. However, it is worth-while to point out that PDPA measurements may give

higher fluctuation velocities than the actual ones due to theeffect of particle size range. Figs. 11–14 give the compari-son between the axial and tangential RMS fluctuationvelocities of two phases obtained by predictions and mea-surements. The DP model can properly predict the anisot-ropy of both gas and particle turbulence — obviousstronger axial turbulent fluctuations than the tangential oneat the upstream cross sections, observed in measurements.This is due to the larger gradients of axial velocity thanthat of tangential velocity at the initial stage. Subsequently,in the downstream region, the two-phase turbulence tendsto become isotropic due to the Areturn-to-isotropy effectB.Moreover, it can also be seen from these figures that theanisotropy of particle turbulence is stronger than the gasturbulence. Clearly, the KP model fails to predict thisphenomena. Figs. 15–18 show the comparison betweengas-phase turbulent fluctuation and particle-phase fluctua-tion obtained by predictions and measurements. The DPmodel properly predicts that the particle tangential fluctua-tion is always smaller than the gas one, as observed in

Ž .experiments Figs. 15 and 16 , whereas the particle axialfluctuation is greater than the gas one in some regionsŽ . ŽFigs. 17 and 18 for predictions at xs52 mm, for

.measurements at xs112 to 195 mm . The KP model,once again, cannot give this result.

Ž .Fig. 18. Two-phase axial RMS velocity DP .


8. Conclusions

Ž .1 For gas and particle turbulence and their interac-tions, the DSM–PDF model is better than the k–´–kpmodel in predicting the anisotropy of normal Reynoldsstress components.

Ž .2 For predicting the time-averaged two-phase flowfield in moderately or weakly swirling flows the k–´–kpmodel is sufficiently good, and there is no need to usemore sophisticated DPS-PDF model. But for simulatingstrongly swirling flows the latter may have the potentialsuperiority due to the above stated reason.

Ž .3 For simulating turbulent swirling gas–particle flows,the DSM–PDF model still cannot give satisfactory results,and needs to be further improved, before it can be used asa basis for improving the unified second-order-momentmodel.

9. Nomenclature

c Empirical coefficientD Diffusion term in Reynolds stress equationsf Fluctuating drag forceG Source term due to gas–particle interaction;

production term of turbulent kinetic energyk Turbulent kinetic energyM Particle massP Pressure; production term of Reynolds stress

Ž .p Probability density distribution function PDFQ, q Generalized variablet TimeU Instantaneous velocityu Fluctuation velocityV Instantaneous velocity coordinateÕ Fluctuation velocity coordinatex Coordinate in geometrical space

Greek lettersd Kronic-delta function; unit tensor´ Turbulent kinetic energy dissipation rate

Ł Productr Densityt Viscous force; relaxation time

Subscriptsf Fluctuationi, j, k, l Coordinates directionsp Particles Instantaneous

Acknowledgements

This study is sponsored by the Special Science Founda-tion for Training Ph.D. Students of Higher Universities,and the National Natural Science Foundation, PRC.

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simulation of swirling gas–particle flows using a dsm–pdf two-phase turbulence model

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