simulation of swirling gas–particle flows using an improved second-order moment two-phase...

Ž .Powder Technology 116 2001 178–189www.elsevier.comrlocaterpowtec

Simulation of swirling gas–particle flows using an improvedsecond-order moment two-phase turbulence modelq

L.X. Zhou a,), Y. Xu a, L.S. Fan b, Y. Li b

a Department of Engineering Mechanics, Tsinghua UniÕersity, Beijing 100084, People’s Republic of Chinab Department of Chemical Engineering, The Ohio State UniÕersity, Columbus, OH 43210-1180, USA

Received 1 March 2000; received in revised form 12 May 2000; accepted 12 May 2000

Abstract

Swirling gas–particle flows in a co-axial sudden-expansion chamber with different swirl numbers are simulated using an improvedsecond-order-moment two-phase turbulence model. The particle Reynolds stress equations and the two-phase fluctuation velocitycorrelation terms are closed based on a Lagrangian analysis, accounting for the crossing-trajectory effect, inertial effect and continuityeffect. Predictions give the gas and particle axial and tangential averaged and fluctuation velocities for swirl numbers of ss0, ss0.47and ss0.94. The swirl number is defined as the ratio of tangential momentum to axial momentum. Prediction results are in goodagreement with the PDPA measurement results for both mean and fluctuation velocities of single-phase swirling flows and two-phasemean velocities of swirling gas–particle flows. However, the normal components of Reynolds stresses or the fluctuation velocities of twophases for swirling gas–particle flows are still underpredicted. The results show that increasing swirl number changes the shape and sizesof recirculation zones, the size of the solid-body rotation zone, reduces the turbulent fluctuation of two phases in the upstream region andenhances it in the downstream region. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Swirling gas–particle flows; Two-phase turbulence; Second-order moment model

1. Introduction

Swirling gas–particle flows are encountered in cycloneseparators, hydrocyclones, cyclone combustors, tangen-tially fired furnaces and swirl burners. During coal com-bustion and NO formation in swirl burners, it is likely tox

use the swirl to increase the gas recirculation for flamestabilization. However, it is unclear whether increasingswirl number will enhance or reduce turbulence, increaseor reduce NO formation, since high fuel concentration inx

the recirculation zone is preferred for increasing flamestabilization and reducing NO formation, and the fuel–airx

mixing is dominated by turbulence. Most of the previous

q Research results of the Project, supported by the Special Funds forthe Major State Basic Research, PRC.

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

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

studies of swirling flows are related to single-phase flows.Ž .Recently, the phase Doppler particle anemometer PDPA

was used to study swirling gas–particle flows with swirlw xnumbers of 0.47, 2.08, 1.5 and 1.0 1 . The swirl number is

defined as:

D3r2 22 rwur d rH0Ss

D3r2 2D ru rd rH40

where D is the inlet diameter and D is the chamber3 4

diameter, u is the axial velocity and w is the tangentialvelocity. Some comparison has been made among theresults of these studies, but the effect of swirl number onthe mean flow field and turbulence properties is still notwell understood. Hence, it is suggested to clarify the effectof swirl number on swirling gas–particle flows using theCFD modeling approach. In regard to numerical modeling

w xof swirling gas–particle flows, Boysan et al. 2,3 used the

0032-5910r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.Ž .PII: S0032-5910 00 00396-X

( )L.X. Zhou et al.rPowder Technology 116 2001 178–189 179

Eulerian–Lagrangian approach–algebraic stress gas turbu-lence model and stochastic particle trajectory model tosimulate strongly swirling gas–particle flows in cycloneseparators and cyclone combustors. The predicted globalperformance—pressure drop and collection efficiency—isin good agreement with experimental results, but no com-parison is made between the predicted and measured time-averaged and fluctuation velocity fields. An improvedalgebraic stress model together with a stochastic particletrajectory model is used in simulating strongly swirlinggas–particle flows in a vortex combustor, and the gas flow

w xfield is well predicted 4 , but no verification of particleflow field was made. As for Eulerian–Eulerian or two-fluidmodeling of swirling gas–particle flows, a k–´–A two-p

Žphase turbulence model A denotes the algebraic particlep

turbulence model, based on the particle-tracking-fluid con-cept, according to which the ratio of particle turbulentkinetic energy k to the gas turbulent kinetic energy k isp

.always less than unity was used to simulate stronglyw xswirling gas–particle flows in a cyclone combustor 5 .

The predicted particle velocity and concentration fieldsdiffer remarkably from those measured. The k–´–k two-p

Žphase turbulence model where k denotes the transportp.equation of particle turbulent kinetic energy was success-

fully used for simulating swirling gas–particle flows withswirl number of ss0.47 in a coaxial sudden-expansion

w xchamber 6 , the predicted axial and tangential two-phasevelocity fields and particle mass flux are in good agree-ment with the PDPA measurement results. The unified

Ž .second-order-moment USM two-phase turbulence modelwas adopted to simulate swirling gas–particle flows of

w xboth ss0.47 and ss1.5 7 . For the case of ss0.47, theUSM gives results similar to those obtained using thek–´–k model and for the case of ss1.5, the USM givesp

better results. Both of these two models underpredict two-phase turbulent fluctuations. In this paper, an improvedsecond-order-moment model is proposed, in which animproved closure of two-phase fluctuation velocity correla-tion is used. It is based on a Lagrangian analysis of gasturbulence seen by particles, accounting for the crossing-trajectory effect, inertial effect and continuity effect. Theproposed model is used to simulate swirling gas–particleflows, in order to understand the effect of swirl number onthe flow field and turbulence properties.

2. The two-fluid model and two-phase Reynolds stressequations and their original closure model

For a two-fluid model, the time-averaged equations ofisothermal turbulent gas–particle flows are obtained bytaking the decomposition of the instantaneous equations oftwo phases and then Reynolds averaging. When account-

ing for only the drag as the phase interaction force, theycan be written as:

Er Eq r Õ s0 1Ž . Ž .j

Et Ex j

E Er Õ q r Õ ÕŽ . Ž .i j i

Et Ex j

Ep EÕ EÕj isy qm qž /Ex Ex Exi i j

m Epqr g q N Õ yÕ y rÕ Õ 2Ž . Ž .Ž .Ýi p p i i j i

t Exr jp

EN E Epq N Õ sy n Õ 3Ž .Ž . Ž .p p j p p j

Et Ex Exj j

E EN Õ q N Õ ÕŽ . Ž .p p i p p j p i

Et Ex j

1sN g q N Õ yÕ yn ÕŽ .p i p i p i p p i

tr

Ey N Õ Õ qÕ n Õ qÕ n Õ 4Ž .Ž .p p j p i p j p p i p i p p j

Ex j

Ž . Ž .where Eqs. 1 – 4 are gas continuity, gas momentum,particle number density and particle momentum equations,respectively. The fourth term on the right-hand-side of Eq.Ž . Ž .2 and the second term on the right-hand-side of Eq. 4express the phase momentum interaction due to drag force.The Magnus force due to particle rotation and Saffmanforce due to velocity gradient are usually not importantcompared with the drag force in most regions of the flowfield except the near-wall region. In these equations, thegas and particle Reynolds stresses and the particle diffu-sion mass fluxes

Õ Õ ,Õ Õ ,n Õ ,n Õi j p i p j p p i p p j

Ž .are unknown correlations of fluctuating quantities , theirclosure should be made using a two-phase turbulence

Ž .model. In the unified second-order moment USM two-phase turbulence model, the two-phase Reynolds stressequations are obtained from the instantaneous gas-phaseand particle-phase momentum equations. After Reynoldsaveraging and making the modeling approximations, simi-lar to those used in single-phase flows, the closed form of

( )L.X. Zhou et al.rPowder Technology 116 2001 178–189180

gas-phase and particle-phase Reynolds stress equations canbe obtained 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 j

i jEt Exk

5Ž .

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

6Ž .

where, D , P , P , ´ are terms having the samei j i j i j i j

meanings as those in single-phase Reynolds stress equa-tions. If we use the Daly–Harlow model of diffusion term,

Ž .IPCM Isotropization of Production and Convection modelw xof pressure–strain term and isotropic dissipation model 8 ,

we have

E k EÕ Õ EV EVi j j iD s c Õ Õ P syr Õ Õ qÕ Õi j s k l i j i k j kž /ž /Ex ´ Ex Ex Exk l k k

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

1P syc P yc y d Gyc ;Ž .Ž .i j ,2 2 i j i j i j k k3

2 EVi´ s d ´ , GsyrÕ Õi j i j i k3 Exk

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

rpG s Õ Õ qÕ Õy2Õ 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 Er´ q rV ´Ž . Ž .k

Et Exk

E k E´ ´s c Õ Õ q c GqG yc r´Ž .´ k l ´1 p ´ 2ž /Ex ´ Ex kk l

7Ž .

rp Ž .where the new source term is G sÝ Õ Õ yÕ Õ .p 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 and the productionrdestructionterm due to gas turbulence, respectively. The productionand phase interaction terms keep their exact form. We use

Table 1Empirical constants

c c c c c c cs 1 2 ´ ´1 ´ 2 p s

0.24 3.0 0.3 0.24 1.44 1.92 0.24

the Daly–Harlow model for the diffusion term, as that forthe gas phase, hence they can 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 Õy2Õ Õ q V yV n ÕŽ .Ž .p , i j p p i j p j i p i p j i p i p p j

trp

q V yV n ÕŽ .j p j p p i

Ž . Ž . Ž .For a closed system, beside Eqs. 5 , 6 and 7 , the termsn Õ ,n Õ ,n n ,Õ Õ ,Õ Õ need to be modeled. For exam-p p i p p j p p p i j p j i

ple, the transport equations of n Õ and n n are:p p i p p

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 nÕ

i i iEt Exk

8aŽ .

where

E k EpÕD s n c Õ Õ n ÕŽ .nÕ p n pk p l p p i

i 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 i

i trp

E En n n q n V n nŽ . Ž .p p p p pk p p

Et Exk

EV Enpk psD y2n n n y2n n Õ 8bŽ .nn p p p p p pk

Ex Exk k

E k Epnw Ž .xwhere D s n c Õ Õ n n .nn p n pk p l p pEx ´ Exk p l


Fig. 1. The swirl chamber for single-phase flows.

The two-phase fluctuation velocity correlation Õ Õ canp i j

be closed using the transport equation

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 d 8cŽ .pk j k p i p i i i jž /Ex Ex kk k

In case if the convection term and diffusion term, i.e. theterms on the left-hand-side and the first term on the

Ž .right-hand-side of Eq. 8c are neglected, as that is made inthe algebraic stress model of single-phase flows, then analgebraic expression can be obtained as:

rt EV EV 1rp j p iÕ Õ sy Õ Õ qÕ Õp i j p i k j pkž /rqr Ex Ex rqrp k k p

=rt ´rp

rÕ Õ qr Õ Õ y Õ Õ d 9Ž .Ž .i j p p i p j p i i i jrqr kp

where

2 2 < <n snqn , n sc k r´ , n sc k r ´e T T m p m p p p

Ž .c s0.09 and c s0.06. Eq. 9 is the original closurem mpw xmodel used in Ref. 7 . The transport equation of particle

turbulent kinetic energy 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 ´ 10Ž .p p pk p l p p pž /Ex ´ Exk p l

where

EVp iP sy N Õ Õ qV n Õ ,Ž .p p pk p i pk p p i

Exk

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

t nrp p

The empirical constants for the USM two-phase turbu-lence model are given in Table 1.

Simulation of swirling gas-particle flows with differentswirl numbers using the USM model with the closure

Ž . w xmodel of Eq. 9 7 shows that the two-phase mean flowfield is well predicted, but the predicted Reynolds stressesare not in good agreement with those measured. Since theclosure models of all terms other than the phase interactionterms G , ´ are based on the method used in single-p, i j p, i j

phase second-order moment model and they are widelyadopted and tested in different problems, so the onlyimportant closure model in the two-phase Reynolds stressequations is the closure of G , ´ , or the closure ofp, i j p, i j

Õ Õ ,Õ Õ . The role of these terms is like that of thep j i p i j

pressure-strain term in the single-phase Reynolds stressequations. Hence, the closure model of two-phase fluctua-tion velocity correlation for complex flows remains to be

Ž .studied. One approach is using the transport Eq. 8c but it

Ž .Fig. 2. Streamlines ss0.53, IPCM version .


Ž .Fig. 3. Streamlines ss0.53, GL version .

is rather complex. So, in this paper an improved but stillsimple closure is used.

3. An improved closure model based on Lagrangiananalysis

An alternative closure method can be used. The startingpoint is the expression derived by Zaichik based on aLagrangian PDF transport equation and its simple solution

w xfor uniform and isotropic turbulence 8

EVj² : ² : ² :Õ Õ sa Õ Õ yb Õ Õi p j i j i jEx j

For complex flows, an improvement should be made indetermining a , b as functions of the particle relaxationtime and the Lagrangian time scale. The key point is thatthe Lagrangian time scale is taken not as that for thesingle-phase gas flows, but as the gas turbulence scale seenby particles. In doing so, the crossing-trajectory effect,inertia effect, continuity effect and anisotropic turbulencebased on Lagrangian analysis are taken into account. Thecrossing-trajectory effect is simulated by setting the parti-cle–eddy interaction time as the minimum value from theeddy lifetime and the crossing time:

leTsmin t ,ež /< <Vrel

where V is the relative velocity between the particle andrel

the fluid, t is the eddy lifetime or the fluid integral timee

scale seen by the particle, l is the eddy length or thee

Eulerian spatial integral space scale seen by the particle, asparticle moves through the fluid. The fluid integral timescale seen by the particle, t , is only important when thee

drift velocity is small. Very small particles, that have noŽ .drift velocity or inertia fluid points , see the fluid La-

grangian integral time scale T . Very large particles withL

no drift velocity staying at the same location, on the otherhand, see the Eulerian scale of fluid moving T . WangmE

w xand Stock 9 found an empirical relationship betweenthese two scales for isotropic and homogeneous turbulencewith turbulence structure parameter of 1:

1yT rTL mEt s2T 1ye mE 0.4ž /Ž .1qSt 1q0.01StŽ .

where Stst rT , T rT s0.356 I. The fluid integralp mE L mE

time scale seen by the particle t is changing with thee

particle inertia, and represents the inertia effect. Huang andw xStock 10 correlate the spatial length scale l with thee

longitudinal spatial length scale L , using the followingf

expression

Lf 2l s 1qcos uŽ .e 2

where u is the angle between the fluid velocity and theaverage velocity of the particle. It is expected to simulateboth crossing-trajectory effect and continuity effect as achanging Eulerian spatial scale with the angle between theparticle velocity vector and the fluid velocity vector. Gra-

w xham 11 suggested that in the eddy interaction model, the

Ž .Fig. 4. Axial velocity ss0.53, mrs .


Ž .Fig. 5. Tangential velocity ss0.53, mrs .

interaction time in the lateral direction should be twice ofthe interaction time in the longitudinal direction for consid-ering the continuity effect. The above-described modelsare given for the isotropic turbulence. The validity of thesemodels is questionable for complex flows. Mehrotra et al.w x12 argued that the continuity effect is not isotropic. Hesuggested a three-eddy interaction model as each eddy hasits own integral length scale and time scale. The particleinteracts with different eddies in different directions. So itnaturally represents the effect of anisotropic turbulence onthe particle dispersion. The anisotropic effect is included inthe fluid velocity correlation along the particle trajectoryas

t le i ipR t sexp y , T smin t ,Ž .f i i i i e i iž / ž /< <T Vii rel

Correspondingly, the coefficients a ,b in the above statedequations should be changed from scalars to vectors asa ,b . The time scale T is determined by either the fluidi i i i i i

integral time scale seen by the particle in that direction orthe crossing time of the eddy in that direction caused bythe drift velocity. The drift velocity causes the decrease ofthe crossing time in each direction. For the fluid integral

time scale seen by the particle t , we can still takee i i

Wang’s expression:

1yT rTL i i mE i it s2T 1ye i i mE i i 0.4ž /1qSt 1q0.01StŽ .Ž . i ii i

The spatial correlation depends on the velocity and thedirection of separation. The Eulerian spatial integral scale,when the direction of separation and the direction of thevelocity are the same, is twice of that as the directions ofvelocity and of separation are perpendicular to each other.To avoid establishing the new coordinate, the spatial scaleis set as:

Lf i i 2l s 1qcos uŽ .e i i 2

The Lagrangian integral time scale and the Eulerian lengthscale were estimated using the following correlation pro-

w xposed by Lu et al. 13

Õ Õi i 2(T s0.235 I L s2.5T ÕL i i f i i L i i´

2 2 2 2Ž .where Õ s1r3 Õ qÕ qÕ and ´ is the turbulent ki-1 2 3

netic energy dissipation rate. Therefore, to make the model

Ž .Fig. 6. Axial fluctuation velocity ss0.53, mrs .


Ž .Fig. 7. Tangential fluctuation velocity ss0.53, mrs .

more reasonable, the fluid Lagrangian time scale seen bythe particle T should be used instead of T in thei i L

above-described equations for taking above-stated threeeffects and the anisotropic turbulence into consideration.

w xThe final result gives 14

le i ia sT r t qT b sa T , T smin t ,Ž .i i L p L i i i i L L e i iž /< <Vrel

1yT rTL i i mE i it s2T 1y ,e i i mE i i 0.4ž /1qSt 1q0.01StŽ .Ž . i ii i

tpSts , T rT s0.356.L mETmE

L Õ Õf i i i i2l s 1qcos u , T s0.235 ,Ž .e i i L i i2 ´

12 2 2 2 2(L s2.5T Õ , Õ s Õ qÕ qÕŽ .f i i L i i 1 2 33

where t is the particle relaxation time.p

4. Numerical procedure

The differential equations are integrated in the controlvolume to obtain finite-difference equations using a hybrid

Fig. 8. The swirl chamber for two-phase flows.

scheme with staggered grid nodes. The normal stresses arestored at the pressure nodes and the shear stresses arestored at the velocity nodes. The FDEs are solved using

Žthe SIMPLEC Semi-Implicit Pressure Linked Equations-. w xCorrected algorithm 15 —p–Õ corrections with TDMA

Ž .tri-diagonal marching algorithm line-by-line iterationsand under-relaxation. For boundary conditions, at the inletthe two phase velocities are given by experiments, theturbulent kinetic energy and its dissipation rate are takenby empirical expressions, as

k rU 2s0.03, ´ sk 3r2rl , l rDs0.05in in in in in in

The normal stresses are assumed to have an isotropic inletdistribution and the shear stresses are determined byeddy–viscosity expressions. At the exit, the fully devel-oped flow condition is adopted. The axi-symmetric condi-tion is taken at the axis. At the walls, no-slip condition andzero normal mass flux or zero gradient conditions aretaken for the gas and particle phases, respectively. At thenear-wall grid nodes, the wall-function approximation is

Ž .used. for detailed boundary conditions see Appendix A .20=43 non-uniform staggered grid nodes are adopted.However, fine grid nodes of 40=80 have also been used,and it is found that these two grid systems give the sameresults. The under-relaxation factors for the gas phase,

Table 2Inlet flow parameters

Primary air mass flow rate 9.9 grsSecondary air mass flow rate 38.3 grsInlet Reynolds number 53256Swirl numbers 0, 0.47, 0.94Mean particle size 45.5 mm

3Particle material density 2500 kgrmParticle mass flow rate 0.34 grsParticle mass loading 0.034


Ž .Fig. 9. Gas streamlines ss0 .

particle number density and particle velocities are 0.3, 0.05and 0.15, respectively. The convergence criteria are 10y7,10y4 , 4=10y3 and 10y5 for gas mass source, particlemass source, and the dissipation rates of gas turbulentkinetic energy and other variables, respectively.

5. Simulation results and discussion

To verify the gas-phase Reynolds stress equation model,simulation was made first for single-phase swirling flows

w xmeasured in Ref. 16 . The geometrical configuration andsizes of the swirl chamber to be simulated are shown inFig. 1. The predicted streamlines, time-averaged axial andtangential velocities, axial and tangential fluctuation veloc-ities are shown in Figs. 2–7, where the swirl number isstill defined as the ratio of tangential momentum to theaxial momentum. The predictions were made using differ-ent versions of the Reynolds stress model with differentclosure models of the pressure–strain term—IPCM and

ŽGL models, where the IPCM Isotropization of Production.and Convection Model is the basic closure model, based

w xon the assumption of return to isotropy 17 , and GLw xdenotes Gibson–Launder’s closure model 18 . It can be

Ž .seen from the predicted streamlines Figs. 2 and 3 that theIPCM version gives more reasonable results—corner and

Ž .Fig. 10. Gas axial velocity ss0, mrs .

near-axis recirculation zones. For the axial time-averagedŽ .velocity Fig. 4 , the IPCM version also gives better result.

ŽHowever, for the tangential time-averaged velocity Fig..5 , the GL version gives better result. For axial and

Ž .tangential fluctuation velocities Figs. 6 and 7 , differentversions give the similar results, near to those measuredwith minor differences among each other. In general, theReynolds stress model predicts single-phase swirling flowspretty well.

The next step is to simulate swirling gas–particle flows.The geometrical configuration and sizes of the swirl cham-

w xber measured by Sommerfeld and Qiu 19 are shown inFig. 8. In these cases, the swirl numbers are ranging fromss0 to ss0.94 by changing the tangential velocity,keeping the total inlet flow rate and the ratio of primary airto secondary air unchanged. The inlet flow parameters areshown in Table 2. The length of the swirl chamber is 950mm. The particle size is 45 mm. 32=20 grid nodes areadopted.

Figs. 9–17 show predicted results for the case of ss0.Unlike the cases of swirling flows, in this case there is no

Ž .central reverse flow Fig. 9 . The length of the cornerrecirculation zone is near to nine times of sudden-expan-sion step height. The predicted gas-phase axial velocity

Ždistribution is in good agreement with that measured Fig..10 . There is a shear zone between the primary-air flow

Ž .and secondary-air flow Fig. 10 . Figs. 11, 12 and 13 give

Ž .Fig. 11. Gas axial fluctuation velocity ss0, mrs .


Ž .Fig. 12. Gas radial fluctuation velocity ss0, mrs .

Ž .Fig. 13. Gas tangential fluctuation velocity ss0, mrs .

the predicted gas-phase axial, radial and tangential fluctua-tion velocities near to those measured, and quantitativelythe predicted values are somewhat lower than those mea-sured. The axial fluctuation is stronger than the radial and

Ž .Fig. 14. Particle axial velocity ss0, mrs .

Ž .Fig. 15. Particle axial fluctuation velocity ss0, mrs .

Ž .Fig. 16. Particle radial fluctuation velocity ss0, mrs .

Ž .Fig. 17. Particle tangential fluctuation velocity, ss0, mrs .

tangential ones. Figs. 14, 15, 16 and 17 show the predictedparticle-phase axial velocity, axial, radial and tangential

Ž .Fig. 18. Gas axial velocity ss0.47, mrs .

Ž .Fig. 19. Gas tangential velocity ss0.47, mrs .


Ž .Fig. 20. Particle axial velocity ss0.47, 45 mm, mrs .

fluctuation velocities in good agreement with those mea-sured. It can be seen that the particle axial fluctuationvelocity is near to that of gas phase, but the particletangential and radial fluctuation velocities are about half ofthe axial one, which indicates that the anisotropy of parti-cle turbulence is stronger than the gas phase.

Figs. 18–25 give the prediction results for the gas-phaseflow field of ss0.47. Figs. 18–21 give predicted gas andparticle axial and tangential mean velocities. In general,predictions are in good agreement with the measurements,and the IPCM version gives better results, but the particleaxial velocity is predicted not so well as other velocities.Figs. 22–25 show gas and particle axial and tangentialfluctuation velocities. There are two peaks of fluctuationsin the upstream region. At the downstream region, the

Ž .Fig. 21. Particle tangential velocity ss0.47, 45 mm, mrs .

Ž .Fig. 22. Gas axial fluctuation velocity ss0.47, mrs .

Ž .Fig. 23. Gas tangential fluctuation velocity ss0.47, mrs .

Ž .Fig. 24. Particle axial fluctuation velocity ss0.47, 45 mm, mrs .

profiles become more or less uniform. The improved clo-sure model gives better results than other models, but the

Ž .Fig. 25. Particle tangential fluctuation velocity ss0.47, 45 mm, mrs .

Ž .Fig. 26. Gas axial fluctuation velocity mrs .


Ž .Fig. 27. Gas tangential fluctuation velocity mrs .

gas and particle fluctuation velocities are still underpre-dicted. Similar situation occurred with the original USM

w xmodel predictions 7 , which gives still worse results and isalso shown in these figures. However, in the above-statedsingle-phase swirling flow predictions the gas fluctuationis well predicted by the second-order moment model. Thediscrepancy may be caused by many reasons. First, the gasturbulence model for two-phase flows, in particular theclosure of dissipation rate needs to be reconsidered. Sec-ond, the empirical constants in diffusion terms of particleReynolds stress and turbulent kinetic energy equations still

Žremain to be optimized but computation shows that they.do not have significant effect . Furthermore, in PDPA

measurements of gas–particle flows, gas and particle ve-locities are determined by larger particles and small parti-cles taken as the tracer of the gas phase. Each size range ofparticles may cause the increased measured fluctuationvelocity not due to turbulence.

Figs. 26 and 27 show the gas axial and tangentialfluctuation velocities for ss0.94 and 0.47. These figurestell us that increasing swirl number will cause reduction ofturbulence in the near-entrance region and enhancement ofturbulence in the downstream region.

6. Conclusions

Ž .1 Modeling of single-phase swirling flows shows thatthe adopted IPCM and GL versions of gas Reynolds stressmodel give good results for both mean and fluctuationvelocities.

Ž .2 Modeling of non-swirling gas–particle flows usingthe improved second-order-moment model give good re-sults for both two-phase mean and fluctuation velocities.

Ž .3 The anisotropy of particle turbulence is muchstronger than the gas one.

Ž .4 The improved second-order-moment model can wellpredict two-phase mean velocities of swirling gas–particleflows, gives better results for fluctuation velocities thanthose given by the existing closure model, but still under-predicts the fluctuation velocities. The reason needs to befurther analyzed. Possibly, the gas turbulence model fortwo-phase flows, in particular the closure of dissipationrate should be improved and the empirical constants should

be optimized. Besides, PDPA measurements give higherfluctuation velocities than the actual ones caused by turbu-lence.

Ž .5 As the swirl number increases, the size of the cornerrecirculation zone decreases and the size of the centralrecirculation zone increases, finally forming an annularrecirculation zone.

Ž .6 As the swirl number increases, the solid-body rota-tion core extends to the near-wall region and the free-vortexzone gradually disappears.

Ž .7 Increasing swirl number leads to reduction of turbu-lence in the near-entrance region and enhancement ofturbulence in the downstream region.

Nomenclaturec Empirical coefficientg Gravitational accelerationG Production termk Turbulent kinetic energyL, l Length scaleN Particle number densityn Particle number density fluctuationp Pressurer Radial coordinateS Source terms Swirl numbert TimeT Time scaleU,V,W Time-averaged velocityu,Õ,w Fluctuation velocityx Axial coordinate

Greek alphabetsa , b Functions defined in the textG Generalized transport coefficient´ Dissipation rateF Generalized dependent variablem ViscosityP Pressure–strain termr Densitys Prandtl numbert Relaxation time

Subscriptse Effectivep Particler Relaxation

Acknowledgements

This study is the research results of the Project, sup-ported by the Special Funds for the Major State BasicResearch, PRC.


Appendix A. Boundary conditions

A.1. Gas Phase

Inlet: Given by experimentsExit: Fully developed flow condition

Efs0;fsu ,Õ ,w ,uu ,ÕÕ,ww,uÕ,uw,Õw

Ex

Wall: No-slip condition

fs0;fsu ,Õ ,w ,uu ,ÕÕ,ww

Axis: Symmetrical condition

Efs0;fsu ,Õ ,w ,uu ,ÕÕ,ww,uÕ,uw,Õw

Er

A.2. Particle Phase

Inlet: Given by experimentsExit: Fully developed flow condition

Efs0;fsu ,Õ ,w ,u u ,Õ Õ , w w ,u Õ ,u w ,Õ wp p p p p p p p p p p p p p p

Ex

Wall: Zero normal flux or gradient condition

Eu Ew Enp p pÕ s0, s s s0p

Er Er Er

Axis: Symmetrical condition

Efs0;fsu ,Õ ,w ,u u ,Õ Õ ,w w ,u Õ ,u w ,Õ wp p p p p p p p p p p p p p p

Er

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