simulation of swirling gas–particle flows using usm and k–ε–kp two-phase turbulence models

Ž .Powder Technology 114 2001 1–11www.elsevier.comrlocaterpowtec

Simulation of swirling gas–particle flows using USM and k–´–kptwo-phase turbulence models

L.X. Zhou), T. ChenDepartment of Engineering Mechanics, Tsinghua UniÕersity, Beijing 100084, People’s Republic of China

Received 1 February 1999; received in revised form 1 December 1999; accepted 17 March 2000

Abstract

Ž .The turbulent swirling gas–particle flows with swirl numbers 0.47 and 1.5 are simulated using a unified second-order moment USMŽ .two-phase Reynolds stress equations and a k–´–kp two-phase turbulence models. The results are compared with experiments. Both twomodels can well predict the axial time-averaged two-phase velocities in case of ss0.47, but the USM model is better than the k–´–kp

Ž .model in predicting the tangential time-averaged two-phase velocities of strongly swirling flows Ss1.5 . The anisotropic two-phaseturbulence can well be described only using the USM model. The results give the difference in flow behavior between weakly swirlingand strongly swirling gas–particle flows. q 2001 Elsevier Science S.A. All rights reserved.

Keywords: Swirling flows; Gas–particle flows; Two-phase turbulence model

1. Introduction

Weakly swirling and strongly swirling turbulent gas–particle flows are encountered in swirl coal burners, cy-clone coal combustors and cyclone separators, which weredeveloped and studied in the Laboratory of Two-phaseFlow and Combustion, Department of Engineering Me-chanics, Tsinghua University in recent years. A betterunderstanding of their flow behavior is of great importancefor increasing the flame stabilization and reducing the NOx

formation, or increasing the collection efficiency and re-ducing the pressure drop. Recently, PDPA measurementshave been used to study swirling gas–particle flows with

w x w xdifferent swirl numbers: ss0.47 1 and ss1.5 2 . Theswirl number is defined as the ratio of tangential momen-tum to the axial momentum. The results indicate thatdifferent swirl number will strongly affect the behavior ofswirling gas–particle flows. Therefore, it is necessary touse numerical simulation for studying the effect of swirlnumbers. In numerical simulation of swirling gas–particleflows, it has been found that the isotropic k–´–kp modelcan reasonably predict weakly swirling gas–particle flowsw x3 . For strongly swirling gas–particle flows, a unified

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

Ž .E-mail address: [email protected] L.X. Zhou .

Ž .second-order moment USM two-phase turbulence modelw xwas proposed 4 . In this paper, simulation of swirling

gas–particle flows with swirl numbers 0.47 and 1.5 iscarried out using the USM model and the k–´–kp model,and the results are compared with experiments to see thedifference in flow behavior between weakly swirling andstrongly swirling gas–particle flows.

2. USM and k–´–kp two-phase turbulence models andnumerical procedure

Starting from the instantaneous gas-phase and particle-phase momentum equations, after taking Reynolds averag-ing and making the modeling approximation, similar tothose used in single-phase flows, the closed form of gas-phase and particle-phase Reynolds stress equations can beobtained as:

E ErÕ Õ q rV Õ Õ sD qP qG qP y´Ž . Ž .i j k i j i j i j p i j i ji jEt Exk

1Ž .

E EN Õ Õ q N V Õ Õ sD qP q´Ž . Ž .p p i p j p pk p i p j p , i j p , i j p , i j

Et Exk

2Ž .

0032-5910r01r$ - see front matter q 2001 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 00 00254-0

( )L.X. Zhou, T. ChenrPowder Technology 114 2001 1–112

where, D , P , P , ´ are diffusion, production, pres-i j i j i j i j

sure-strain and dissipation rate terms in gas-phase Reynoldsstress equations, having the same meanings as those insingle-phase Reynolds stress equations, thus we have

E k EÕ Õi jD s c Õ Õ ,i j s k lž /Ex ´ Exk l

EV EVj iP syr Õ Õ qÕ Õi j i k j kž /Ex Exk k

´ 2P sP qP , P syc r Õ Õ y d ki j i j ,1 i j ,2 i j ,1 1 i j i jž /k 3

2 2P syc P y d P , ´ s d ´ ,i j ,2 2 i j i j i j i jž /3 3

EViPsyrÕ Õi k

Exk

and a new source term in case of two-phase flows

rpG s Õ Õ qÕ Õ y2Õ ÕŽ .Ýp , i j p i j p j i i j

trpp

is the gas Reynolds stress productionrdestruction due toparticles drag force. The transport equation of dissipationrate of gas turbulent kinetic energy is:

E E E k E´r´ q rV ´ s c Õ ÕŽ . Ž .k ´ k lž /Et Ex Ex ´ Exk 1

´q c GqG yc r 3Ž .Ž .´1 p ´ 2 ´k

rp Ž .where the new source term is G sÝ Õ Õ yÕ Õp p p i i i itrp

D , P , ´ are the diffusion, production terms ofp, i j p, i j p, i j

particle Reynolds stress equations and the productionrde-struction term due to gas turbulence, respectively, and theycan be given as follows:

E k EpsD s N c Õ Õ Õ ÕŽ .p , i j p p pk p l p i p jEx ´ Exk p l

EVp iP sy V n Õ qN Õ ÕŽ .p , i j pk p p j p pk p j

Exk

EVp jy V n Õ qN Õ Õ qn Õ g qn Õ gŽ .pk p p i p pk p i p p j i p p i j

Exk

1´ s N Õ Õ qÕ Õ y2Õ ÕŽ .p , i j p p i j p j i p i p j

trp

q V yV n Õ q V yV n ÕŽ . Ž .i p i p p j j p j p p i

Ž . Ž . Ž .For a closed system, beside Eqs. 1 , 2 and 3 , thetransport equations of n Õ , n Õ , n n , Õ Õ , Õ Õ arep p i p p j p p p i j p j i

also used, see Appendix A.The transport equation of particle turbulent kinetic en-

ergy is:

E EN k q N V kŽ . Ž .p p p pk p

Et Exk

E k Ekp pss N c Õ Õ qP yN ´ 4Ž .p p pk p l p p pž /Ex ´ Exk p l

where

EVp iP sy N Õ Õ qV n ÕŽ .p p pk p i pk p p i

Exk

1 1´ sy Õ Õ yÕ Õ q V yV n ÕŽ .p p i i p i p i i p i p p i

t Nrp p

In the case of nearly isotropic turbulent flows, the USMmodel is reduced to a k–´–kp model, which consists ofthe following expressions and equations

2 EV EVi jÕ Õ s kd yn q 5Ž .i j i j t ž /3 Ex Exj i

2 EV EVp i p jÕ Õ s k d yn q 6Ž .p i p j p i j p ž /3 Ex Exj i

n EN n ENp p p pn Õ sy , n Õ sy 7Ž .p p i p p j

s Ex s Exp i p j

E E E m Ekerk q rV k s qPqG yr´Ž . Ž .j pž /Et Ex Ex s Exj j k j

8Ž .

E Er´ q rV ´Ž . Ž .j

Et Ex j

E m E´ ´es q c PqG yc r´ 9Ž .Ž .´1 p ´ 2ž /Ex s Ex kj ´ j

E EN k q N V kŽ . Ž .p p p p j p

Et Ex j

E N n Ekp p ps qP qP yN ´ 10Ž .p g p pž /Ex s Exj p j

( )L.X. Zhou, T. ChenrPowder Technology 114 2001 1–11 3

Fig. 1. Grid arrangement.

where

EV EV EVi j iPsm q ,t ž /Ex Ex Exj i j

2m Np p kG s c kk yk(Ý ž /p p ptrpp

EV EV EVp i p j p iP sN n q ,p p p ž /Ex Ex Exj i j

2m Np p kP s c kk yk(ž /g p p ptrp

1 1k´ sy 2 c k k yk q n Õ V yVŽ .(ž /p p p p p p i i p i

t Nrp p

k 2

m smqm , n sc , m srn ,e t t m t t´

k 2p

n scp m p < <´p

For boundary conditions: the inlet velocity of two phasesand the normal Reynolds stress components are given byexperiment. The shear Reynolds stress components aregiven by the eddy-viscosity assumption. The fully devel-oped flow condition for the two phases is adopted as outletcondition. At the walls, no-slip condition is used forgas-phase velocity, and the gas Reynolds stresses aredetermined via production term including the effect of wallfunction for near-wall grid nodes. Zero normal velocity,zero mass flux, zero gradients of longitudinal and tangen-tial velocities are used for particle phase. Moreover, thenormal gradients of particle Reynolds stresses are set to bezero. At the axis, symmetric conditions are adopted forboth the two phases. The differential equations are inte-

Ž .Fig. 2. Computational domain case 1 .

grated in the computational cells to obtain the finite differ-ence equations using a hybrid scheme. The FDEs aresolved using the extended version of SIMPLEC algorithmfor two-phase flows. The non-uniform grid nodes areshown in Fig. 1.

For axi-symmetric turbulent swirling gas–particle flows,the gas-phase and particle-phase basic equations in cylin-drical coordinates based on the USM two-phase turbulencemodel can be expressed in the generalized forms as

E 1 ErUF q r rVFŽ . Ž .

Ex r Et

E EF 1 E EFs G q G r qS 11Ž .F F Fž / ž /Ex Ex r Er Er

E 1 En U F q n rV FŽ . Ž .p p p p p p

Ex r Er

E EF 1 E EFp ps G q G r qS 12Ž .F F Fp p pž / ž /Ex Ex r Er Er

where the meanings of F , G , S , F , G , S areF F p F p F p

shown in Tables B.1 and B.2. The adopted empiricalconstants are given in Table B.3.

3. Prediction results and discussion

Swirling gas–particle flows of two cases with ss0.47Ž . Ž .weakly swirling and ss1.5 strongly swirling weresimulated. The first case imitates the weakly swirlingflows in swirl burners for coal-fired furnace and the sec-ond case imitates the strongly swirling flows in cyclonecombustors. The geometrical configuration and sizes of the

w xchamber in case 1 are shown in Fig. 2 1 . The grid nodesŽ .are 32=20 Fig. 1 . The particle phase is glass beads with

the size of 30 mm and density of 2.5=103 kgrm3. The

Ž .Fig. 3. Computational domain case 2 .


vŽ . Ž .Fig. 4. Particle axial velocity mrs case 1 —USM, ) k-´-kp, Exp.

vŽ . Ž .Fig. 5. Particle tangential velocity mrs case 1 —USM, ) k-´-kp, Exp.

vŽ . Ž .Fig. 6. Particle axial velocity mrs case 2 —USM, ) k-´-kp, Exp.

vŽ . Ž .Fig. 7. Particle tangential velocity mrs case 2 —USM, ) k-´-kp, Exp.


Ž . Ž . vFig. 8. Two-phase axial velocities mrs USM model, case 1 — Gas, — — Particle.

Ž . Ž . vFig. 9. Two-phase tangential velocities mrs USM model, case 1 — Gas, — — Particle.

inlet flow parameters are: central flow rate 9.9 grs, annu-lar flow rate 38.5 grs, and particle loading 0.034. Thegeometrical configuration and sizes of the chamber in case

w x2 2 are shown in Fig. 3. The grid nodes are 51=20, theparticle phase is the same as that for case 1, but with thesize of 50 mm. The axial inlet velocity is 5 mrs withparticle loading 0.01, and the tangential inlet velocity is 10mrs.

The convergence criteria for gas and particle phases aremass sources 3.0=10y3 and 10y2 , respectively. The USMcode is written in FORTRAN-77, consisting of 10 000statements.

Figs. 4 and 5 give the predicted particle axial andtangential velocities using the USM and k–´–kp modelsand their comparison with measurements for case 1.

Clearly, for weakly swirling flows, the predicted time-averaged velocities by both the two models are in goodagreement with the measurements. There is only a slightdifference between these two model predictions. For case1, the axial velocity has w-shaped profiles with an annularreverse flow zone. This reverse-flow zone will play impor-tant role in flame stabilization and NO reduction for ax

swirl burner. The tangential velocity profiles have a typicalRankine-vortex structure–solid-body rotation plus free vor-tex. Figs. 6 and 7 show the predicted particle axial andtangential velocities for case 2. It can be seen that for thiscase also, both the two models give the axial velocitiesnear to each other. However, the USM model is better inpredicting the typical Rankine-vortex structure of tangen-tial velocity profiles in strongly swirling flows. The USM




Ž . Ž . vFig. 12. Particle axial–tangential RMS velocities mrs USM, case 1 — — Axial, — Tangenital.

Ž . Ž . vFig. 13. Particle axial–tangential RMS velocities mrs exp., case 1 — — Axial, — Tangenital.

Ž . Ž . vFig. 14. Particle axial–tangential RMS velocities mrs USM, case 2 — — Axial, — Tangenital.


Ž . Ž . vFig. 15. Particle axial–tangential RMS velocities mrs exp., case 2 — — Axial, — Tangenital.

model can clearly predict the free-vortex zone, whereas thek–´–kp model tends to eliminate the measured free-vortexregion. Comparing case 1 with case 2 shows that in case 2,the axial velocity has a more or less uniform profile withmaximum value at the axis and no annular reverse-flowzone and no w-shaped profiles. This flow behavior isunfavorable to flame stabilization and NO reduction in ax

cyclone combustor. However, if we add a contraction partto the exit of the combustor, the reverse-flow zone can beinduced. This method was really used in the practicalcyclone coal combustor developed in our laboratory. Forcase 2, the tangential velocity profile also has a typicalRankine-vortex structure, but the solid-body rotation zoneis much larger than that of case 1. In general, we like tokeep the tangential velocity profile more or less uniform, ifwe try to reduce the turbulence level for reduce NOx

formation. Therefore, we should find some measure tomodify the tangential velocity profiles. Many years ago,when we develop an innovative cyclone separator, we puta central body to change the tangential velocity distribu-tion, making it more or less uniform, hence, reduce thepressure drop and increase the collection efficiency. Now,we should modify the flow field in order to reduce theNO formation. As for the relation between two-phasex

velocities, for case 1, the particle axial velocity obviouslyŽ .exceeds the gas one Fig. 8 , but the particle tangential

Ž .velocity lags behind the gas one Fig. 9 . For case 2, theŽ .axial velocity slip decreases to near zero Fig. 10 , whereas

Ž .the tangential velocity slip increases Fig. 11 .Figs. 12–15 show USM predicted and PDPA measured

particle axial and tangential fluctuation velocities. Bothpredictions and measurements indicate that in case 1, theaxial fluctuation velocity distribution has two or three

Žpeaks, where the axial velocity gradient is large Figs. 12.and 13 . The tangential fluctuation velocity distribution is

more or less uniform, particularly in the near-axis regionwhere the tangential velocity gradient is constant. In orderto reduce the particle turbulence, we need to reduce bothaxial and tangential velocity gradients. For case 2, bothaxial and tangential fluctuation velocity distributions be-come more or less uniform with only small peaks near the

Ž .wall 14, 15 . Clearly, the particle turbulence is anisotropic,

i.e. the axial fluctuation velocity is larger than the tangen-tial one for both the two cases. In the downstream region,the anisotropy decreases, and in case 2, the anisotropy ingeneral is smaller than that for case 1. The USM modelunderpredicts the axial fluctuation. However, this modelgives properly the above-stated behavior of anisotropicturbulence, which cannot be predicted by the k–´–kpmodel.

4. Conclusions

Ž .1 The numerical simulation points out the differencein flow behavior between weakly swirling flows in a swirlburner and strongly swirling flows in a cyclone combustor.

Ž .2 For weakly swirling flows, there is an annularreverse-flow zone, which is favorable to flame stabilizationand NO reduction, but the problem is how to reduce thex

turbulent mixing in order to keep high particle concentra-tion in this zone.

Ž .3 For strongly swirling flows, there is no reverse flowzone, so we need to add a contraction part to induce areverse-flow zone.

Ž .4 Both weakly and strongly swirling flows have aRankine-vortex structure of tangential velocity profiles–solid-body rotation plus free vortex, but the size of thesolid-body rotation zone for strongly swirling flows ismuch larger than that for weakly swirling flows. Theproblem is to create more uniform tangential velocityprofiles in order to reduce turbulence.

Ž .5 For weakly swirling flows, the particle axial veloc-ity exceeds the gas one, while for strongly swirling flows,the former lags behind the latter.

Ž .6 The particle tangential velocity lags behind the gasone for both weakly and strongly swirling flows.

Ž .7 The particle axial fluctuation velocity has a two-peakand three-peak distribution, whereas the particle tangentialfluctuation velocity distribution is more uniform. To re-duce particle turbulence, we need to reduce both axial andtangential velocity gradients.

Ž .8 The particle axial fluctuation velocity is larger thanthe tangential one for both weakly and strongly swirlingflows, which can be predicted only by the USM model.


Ž .9 The USM model can better predict time-averagedtwo-phases flow field of strongly swirling flows, and bothUSM and k–´–kp models can well predict the time-aver-aged two-phase flow field of weakly swirling flows.

Nomenclaturec Empirical constantsD Diffusion term in Reynolds stress equationsG Source term due to gas–particle interactiong Gravitational accelerationk Turbulent kinetic energym Particle massN Particle number densityn Particle number density fluctuationP Production term of Reynolds stresses or turbulent

kinetic energyp Pressuret TimeV Time-averaged velocityÕ Fluctuation velocityx Coordinate in geometrical space

Greek alphabetsd Unit tensor´ Turbulent kinetic energy dissipation rateP Pressure–strain termr Densityt Relaxation timem, n Dynamic and kinematic viscosities

Subscriptsi, j, k, l Coordinates directionsp Particler Relaxation

Acknowledgements

This study was sponsored by the National Natural Sci-ence Foundation, P.R. China.

Appendix A. Additional transport equations

The transport equation of two-phase fluctuation velocitycorrelation

E EÕ Õ q V qV Õ ÕŽ .Ž . Ž .p i j k pk p i j

Et Exk

E Es n qn Õ ÕŽ . Ž .e p p i j

Ex Exk k

1q rÕ Õ qrÕ Õ y rqr Õ ÕŽ .p i p j i j p p i j

rtrp

EV EV ´p i jy Õ Õ qÕ Õ y Õ Õ dpk j k p i p i i i jž /Ex Ex kk k

The transport equation of particle diffusion mass flux

E EN n Õ q N V n Õ sD qP q´Ž . Ž .p p p i p pk p p i nÕ nÕ nÕi i iEt Exk

where

E k EpÕD s N c Õ Õ n ÕŽ .nÕ p n pk p l p p ii Ex ´ Exk p l

ENpP sy N Õ Õ qV n Õ q2V n ÕŽ .nÕ p pk p i pk p p i p i p pki Exk

EVp iy N n Õ qV n nŽ .p p pk pk p p

Exk

EVpky2 N n Õ qV n nŽ .p p p i p i p p

Exk

1´ s V yV n n qN n ÕŽ .nÕ i p i p p p p p ii trp

The transport equation of particle number–density fluc-tuation

E EN n n q N V n nŽ . Ž .p p p p pk p p

Et Exk

EV Enpk psD y2 N n n y2 N n Õnn p p p p p pk

Ex Exk k

where

E k EpnD s N c Õ Õ n nŽ .nn p n pk p l p pEx ´ Exk p l

Appendix B. Tables

Table B.1 shows a list of the gas-phase equations where

1ks uuqÕÕqww ,Ž .

2

k 2 EU EUm smqc r G sy2 r uu quÕ ,e m uu ž /´ Ex Er

EV EV WG sy2 r uÕ qÕÕ y Õw ,Õ Õ ž /Ex Er r

EW EW VG sy2 r uw qÕw q ww ,w w ž /Ex Er r


Table B.1Gas-phase equations

Equation F G SF F

Gas continuity 1 0 0Ep E EU 1 E EV E 1 E 1

Ž . Ž . Ž .Gas axial momentum U m y q m q mr y ruu y r ruÕ q r U yUp pž / ž /Ex Ex Ex r Er Ex Ex r Er trp

Ep E EU 1 E EV E 1 EŽ . Ž .Gas radial momentum V m y q m q mr y ruÕ y r rÕÕž / ž /Er Ex Er r Er Er Ex r Er

2mV rW rww 1Ž .y2 q q q r V yVp p2 r r tr rp

W E E 1 E rÕw 1 rVWŽ . Ž . Ž . Ž .Gas tangential momentum W m y rm y ruw y r rÕw y q W yW yp2 Er Ex r Er r t rr rp

Ž .w Ž . xTKE dissipation rate ´ m rs ´rk c GqG yc r´e ´ ´ 1 p ´ 2rp Ž .Gas axial normal stress uu uu m rs G qP q´ q2 u uyuue k uu uu uu ptrp

rm rÕw pe Ž . Ž .Gas radial normal stress ÕÕ m rs G qP q´ y2 ÕÕyww q2 Wq2 Õ ÕyÕÕe k Õ Õ Õ Õ Õ Õ p2 r ts r rpk

rm rÕw pe Ž . Ž .Gas tangential normal stress ww m rs G qP q´ y2 wwyÕÕ y2 Wq2 w wywwe k w w w w w w p2 r ts r rpk

rruw m uÕ peŽ . Ž .Shear stress x,r uÕ m rs G qP q´ q Wy q u ÕqÕ uy2uÕe k uÕ uÕ uÕ p p2r ts r rpk

rruÕ m uw peŽ . Ž .Shear stress x,u uw m rs G qP q´ y Wy q u wqw uy2uÕe k uÕ uw uw p p2r ts r rpk

rr ÕÕyww m ÕwŽ . peŽ . Ž .Shear stress r,0 Õw m rs G qP q´ y Wy4 q Õ wqw Õy2Õwe k Õw Õw Õw p p2r ts r rpk

EU EU EV EV WG syr uÕ qÕÕ quu quÕ y uw ,uÕ ž /Ex Er Ex Er r

EU EU EW EW VG syr uw qÕw quu quÕ q uwuw ž /Ex Er Ex Er r

EV EU EW EWsyr yuw qÕw quu quÕž /Er Er Ex Er

EV EV EW EWG syr uw qÕw quÕ qÕÕÕw ž Ex Er Ex Er

V Wq Õwy ww /r r

EV EU EW EW Wsyr uw yÕw quÕ qÕÕ y wwž /Ex Ex Ex Er r

EU EU EV EVGsyr uu quÕ quÕ qÕÕž Ex Er Ex Er

EW EW V Wquw qÕw q wwy Õw /Ex Er r r

rpG s u uqÕ Õqw wy2kŽ .p p p p

trp

´ 2P sP qP , P syc r Õ Õ y d ki j i j ,1 i j ,2 i j ,1 1 i j i jž /k 3

2 2P syc G y d G , ´ sy d ´ ,i j ,2 2 i j i j i j i jž /3 3i , jsu ,Õ ,w

Table B.2 shows a list of the particle-phase equationswhere

k 2 1pn sc , k s u u qÕ Õ qw wŽ .p m p p p p p p p p

´ 2p

1´ sy n u uqÕ Õqw wy2kŽ .p p p p p pn tp r p

q UyU n u q VyV n Õ q WyW n wŽ . Ž . Ž .p p p p p p p p p

EUpP sy2 n u u qU n uŽ .u u p p p p p pp p Ex

EUpy2 n u Õ qV n uŽ .p p p p p p

Er

EVpP sy2 n u Õ qU n Õ y2 n Õ ÕŽ . ŽÕ Õ p p p p p p p p pp p Ex

EV Wp pqV n Õ q2 n Õ w qW n Õ. Ž .p p p p p p p p p

Er r

EWpP sy2 n u w qU n w y2 n Õ wŽ . Žw w p p p p p p p p pp p Ex

EW Vp pqV n w y2 n w w qW n w. Ž .p p p p p p p p p

Er r


Table B.2Particle-phase equations

Equation F G Sp F Fp p

E 1 EŽ . Ž .Particle continuity 1 0 y n u y rn Õp p p p

Ex r Ern E 1p Ž . Ž .Particle axial momentum U 0 UyU y n u u q2n u y n up p p p p p p p pt Ex trp rp

1 EŽ .y n ru Õ qrU n Õ qrV n up p p p p p p p pr Er

n E 1 Ep Ž . Ž . Ž .Particle radial momentum V 0 VyV y n u Õ qU n Õ qV n u y n rÕ Õ q2 rV n Õp p p p p p p p p p p p p p p p pt Ex r Errp

1 1 1 22y n Õ q n W q n w w q W n wp p p p p p p p p p

t r r rrp

n Ep Ž . Ž .Particle tangential momentum W 0 WyW y n u w qU n w qW n up p p p p p p p p p pt Exrp

1 EŽ .y n rÕ w qV rn w qW rn Õp p p p p p p p pr Er

n Õ w V n w W n Õ n w1 p p p p p p p p p p py n V W y y y yp p pr r r r trp

2w Ž . Ž . xParticle axial normal stress u u n n rs P q n u uyu u q UyU n up p p p p u u p p p p p p pp p trp

n n n W 2p p p pŽ . w Ž .Particle radial normal stress Õ Õ n n rs P y2 Õ Õ yw w q2 Õ w q n Õ ÕyÕ Õp p p p p Õ Õ p p p p p p p p p p2p p r ts r rpp

Ž . xq VyV n Õp p p

n n n W 2p p p pŽ . w Ž .Particle tangential w w n n rs P y2 w w yÕ Õ y2 Õ w q n w wyw wp p p p p w w p p p p p p p p p p2p p r ts r rpp

Ž . xnormal stress q WyW n wp p p

n n n W 1p p p pŽ . w Ž .Shear stress x,r u Õ n n rs P y u Õ q u w q n u ÕqÕ uy2u Õp p p p p u Õ p p p p p p p p p2p p r ts r rpp

Ž . Ž . xq UyU n Õ q VyV n up p p p p p

n n n W 1p p p pŽ . w Ž .Shear stress x,u u w n n rs P y u w y u Õ q n u wqw uy2u wp p p p p u w p p p p p p p p p2p p r ts r rpp

Ž . Ž . xq UyU n w q WyW n up p p p p p

n n n W 1p p p pŽ . Ž . w Ž .Shear stress r,u Õ w n n rs P y4 Õ w y Õ Õ yw w q n Õ wqw Õy2Õ wp p p p p Õ w p p p p p p p p p p p2p p r ts r rpp

Ž . Ž . xq VyV n w q WyW n Õp p p p p p

EUpP sy n u Õ qU n Õ y n Õ ÕŽ . Žu Õ p p p p p p p p pp p Ex

EU EVp pqV n Õ y n u u qU n u. Ž .p p p p p p p p p

Er ExEVp

y n u Õ qV n uŽ .p p p p p pErWp

q n u w qW n uŽ .p p p p p p rEUp

P sy n u w qU n wŽ .u w p p p p p pp p ExEUp

y n Õ w qV n wŽ .p p p p p pErEWp

y n u u qU n uŽ .p p p p p pEx

EWpy n u Õ qV n uŽ .p p p p p p

ErVp

y n u w qW n uŽ .p p p p p p rEVp

P sy n u w qU n wŽ .Õ w p p p p p pp p ExEVp

y n Õ w qV n wŽ .p p p p p pEr

EWpy n u Õ qU n ÕŽ .p p p p p p

Ex

Table B.3Empirical constants

kc c c s s c c c c s cm ´ 1 ´ 2 k ´ 1 2 m p g p p p

0.09 1.45 1.92 1.0 1.22 3.0 0.3 0.0064 1.0 0.7 0.75


EWpy n Õ Õ qV n ÕŽ .p p p p p p

Er

Wpq n w w qW n wŽ .p p p p p p r

Vpy n Õ w qW n ÕŽ .p p p p p p r

1P s P qP qPŽ .p u u Õ Õ w wp p p p p p2

n En n Enp p p pn u sy , n Õ sy , n w s0p p p p p p

s Ex s Erp p

Õ Õ sc r Õ Õ qrÕ Õ r rqrŽ .Ž .p i j g p p p i p j i j p

Table B.3 shows a list of the empirical constants used.

References

w x1 M. Sommerfeld, H.H. Qiu, Detailed measurement in a swirlingparticulate two-phase flow by a phase-Doppler anemometer, Int. J.

Ž .Heat Fluid Flow 12 1991 15–32.w x2 L.X. Zhou, T. Chen, Experimental studies on strongly swirling turbu-

lent gas–particle flows in a cyclone combustor, First Asia-PacificConference on Combustion, Osaka, Japan, 1997, pp. 74–77.

w x3 L.X. Zhou, W.Y. Lin, K.M. Sun, Numerical simulation of sudden-ex-pansion swirling gas–particle flows using a k – ´ – kp model, J. Eng.

Ž . Ž .Thermophys. China 16 1995 481–485.w x4 L.X. Zhou, T. Chen, A unified second-order moment two-phase

turbulence model for simulating gas–particle flows, Numer. MethodsŽ .Multi-Phase Flows, ASME FED 185 1994 307–313.

simulation of swirling gas–particle flows using usm and k–ε–kp two-phase turbulence models

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