Chemical Engineering Science 60 (2005) 647–653

www.elsevier.com/locate/ces

A second-ordermoment three-phase turbulencemodel forsimulating gas–liquid–solid flows

L.X. Zhoua,∗, M.Yanga, L.S. FanbaDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, China

bDepartment of Chemical Engineering, The Ohio State University, Columbus, OH 43210, USA

Received 18 May 2004; received in revised form 19 August 2004; accepted 29 August 2004

Abstract

To simulate the bubble, liquid and particle turbulence properties and their interactions in three-phase flows, a second-order moment three-phase turbulence model for gas–liquid–solid flows is proposed. The bubble, liquid and particle Reynolds stress equations, bubble–liquid andliquid–solid two-phase correlation equations are derived using the mass-weighed and time averaging and the closure models of diffusion,dissipation and pressure-strain terms similar to those used in single-phase flows. The two-phase correlation equations are closed with atwo-time-scale dissipation term. The proposed model is applied to simulate gas–liquid flows and gas–liquid–solid flows in a channel. Theprediction results for two-phase flows are in good agreement with the PIV measurement results. The prediction results for three-phaseflows give the gas, liquid and solid velocities, volume fractions and Reynolds stresses, showing that in the case studied the turbulentfluctuation of 5mm bubbles is stronger than that of liquid, while the turbulent fluctuation of 0.5mm particles is weaker than that of liquid.Bubbles enhance liquid turbulence, while particles reduce liquid turbulence.� 2004 Elsevier Ltd. All rights reserved.

Keywords:Multiphase flow; Gas–liquid–solid flows; Second-order moment model; Turbulence; Simulation; Mathematical modeling

1. Introduction

The gas–liquid–solid (bubble–liquid–particle) three-phase turbulent flows are widely encountered in fluidizedbeds, oil–gas–sand transport, liquid metal laden with bub-bles and solid-inclusion in iron and steel making, spraysulfur removing of flue gas using calcic powders and cav-itation water flows with sand in rivers. Presently, eitherthe Eulerian–Lagrangian models or the Eulerian–Eulerianmodels for the continuous and dispersed phases are takenin the CFD modeling of three-phase turbulent flows. Manyresearchers (e.g.Mitra-Majumdar et al., 1997; Grevskottet al., 1996; Bahary et al., 1993) took only the liquid tur-bulence into account, the bubble turbulence is neglectedand frequently it is assumed that the there is no velocity

∗ Corresponding author. Tel.: +861062782231; fax: +861062781824.E-mail address:zhoulx@mail.tsinghua.edu.cn(L.X. Zhou).

0009-2509/$ - see front matter� 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.08.034

slip between the liquid and the particles. In fact, bubbles,liquid and particles have their own turbulent fluctuationsand there are strong interactions among them. The veloc-ity slip exists not only between the bubbles and liquid butalso between the liquid and the particles. The dispersionof bubbles and particles, heat and mass transfer betweenphases and reaction rates are determined significantly bytheir turbulent fluctuation. Recently, the direct numericalsimulation (e.g.Goz et al., 2000) and VOF simulation ofbubble–liquid flows (e.g.Li et al., 1999) indicate strongbubble fluctuation. Recently,Bourloutski and Sommerfeld(2003)use transient Euler/Lagrange calculation to simulatedense gas–liquid–solid three-phase flows, in which both bub-ble and particle motions are solved in Lagrangian coordi-nates. The results show instantaneous bubble and particle be-havior in bubble columns, and the predicted liquid velocity iscompared with the experimental results, but no comparisonis made for the bubble and particle velocities and Reynoldsstresses.

648 L.X. Zhou et al. / Chemical Engineering Science 60 (2005) 647–653

On the other hand, in the framework of Eulerian–Eulerianmodeling, for turbulent gas–particle flows a series of two-phase turbulence models, including the unified second-ordermoment (USM) andk–�–kpmodels were developed and ap-plied to complex flows, such as swirling flows (Zhou andChen, 2001). For turbulent gas–liquid–solid flows, the tur-bulence interactions exist among three phases, the gas den-sity is much smaller than the liquid and solid densities, andthe density difference between the solid and liquid is muchsmaller than that between the gas and solid, so the turbu-lence phenomena is much more complex. We still do notunderstand whether bubble turbulence or particle turbulenceis stronger or weaker than the liquid turbulence and whetherbubbles or particles reduce or enhance the liquid turbulence.Up to now, in the framework of Eulerian–Eulerian models,no three-phase turbulence models accounting for the turbu-lent fluctuations of gas–liquid–particle three phases and theirinteractions are reported.In this paper, in the framework of multi-fluid models, the

mass-weighed and time-averaged continuity and momen-tum equations for each phase are derived. After that of thegas-phase, liquid-phase and particle-phase Reynolds stressequations with phase-interaction terms and the two-phasevelocity correlation equations are derived and closed. Theclosure for the diffusion, pressure-strain and dissipationterms in the Reynolds stress equations of each phase is likethe one used in single-phase flows. For the closure of dissi-pation term in two-phase velocity correlation equations, it isassumed that the dissipation is isotropic and proportional tothe summation of the normal components divided by a timescale, accounting for the crossing-trajectory effect (Zhouet al., 2001). Finally, the proposed model is used to simulategas–liquid and gas–liquid–solid flows in a channel.

2. The mass-weighed and time-averaged equations forturbulent gas–liquid–solid flows

The generalized volume-averaged instantaneous continu-ity and momentum equations in Eulerian coordinates foreach phase of gas–liquid–solid flows are

��m

�t+ �

�xk

(�mVmk) = 0 (m = b, l, p), (1)

��t

(�m�mVmi) + ��xk

(�m�mVmkVmi)

= �m0�mgi − �m1

[�m

�p

�xi

− ��xk

(�m�mki)

]

± �m2

�b�g

�lb

(Vbi − Vli)

± �m3

�p�p

�lp

(Vpi − Vli) (m = b, l, p), (2)

where�b0 = �b(1− �l/�g), �l0 = 1, �p0 = �p(1− �l/�p);�b1, �b3, �p1, �p2 are taken as 0;�b2, �l1, �l2, �l3, �p3 are

taken as 1;�lb, �lp are the relaxation times of bubble–liquidand particle–liquid interactions, determined by

�lp = �pd2p(1+Re2/3p /6)−1/18�1 �lb = 4�gd

2b /(3cd�lReb),

where

Rem = |−→Vl − −→Vm|dm/�l (m = b, p)

cd =max

[24

Reb

(1+0.15Re0.687b ), f8

3

Eo

Eo + 4

],

Eo = g�ld2b /�, f =

{1+ 17.67�9/7l

18.67�3/2l

}2.

Usually, the pure time averaging is used in single-phaseand gas–solid turbulent flows. For turbulent gas–liquid–solidflows, due to the remarkable change of volume fractionsof each phase, the pure time averaging may give a largenumber of equations and need more complex closures. Themass-weighed and time averaging can reduce the num-ber of unknowns and simplify the closure problem. Themass-weighed and time averaging or Favre averaging issuccessfully applied in single-phase compressible turbu-lent flows. Besides, the LDV/PDPA measurements give thenumber-weighed and time-averaged results closer to themass-weighed and time-averaged results.The mass-weighed and time averaging is defined by

Vi = Vi + V ′′i = V̄i + V ′

i Vi = �Vi/�.

For the pressure and volume fraction the pure time averagingis still used, that is

p = p + p′, � = �̄ + �′,

where one bar and one prime denote the averaged and fluc-tuation variables in the pure time averaging, and doublebars and double primes denote the averaged and fluctu-ation variables in the mass-weighed and time averaging.

Hence we have�v′′i = 0; v′

i = 0; Vi =(�Vi + �′v′

i

)/� =

Vi

(1+ �′v′

i/�Vi

).

Putting the above-stated relationships into the instanta-neous equations, taking the time averaging, neglecting thefluctuation of viscous force and the pressure–volume frac-tion correlation, assuming that the phase interaction times�lb, �lp do not fluctuate, and expressing each variable interms of volume-fraction-weighed averaged value, the gen-eralized form of mass-weighed and time-averaged continu-ity and momentum equations of each phase in turbulentgas–liquid–solid three-phase flows can be obtained as

��m

�t+ �

�xk

(�mVmk) = 0 (m = b, l, p), (3)

L.X. Zhou et al. / Chemical Engineering Science 60 (2005) 647–653 649

��t

(�m�mVmi

)+ �

�xk

(�m�mVmkVmi

)

= �m0�mgi − �m1

[�m

�p

�xi

− �m

��xk

(�mki)

]

− ��xk

(�m�mv′′

mkv′′mi

)

± �m2

�b�g

�lb

(Vbi − Vli − v′′

li,b

)

± �m3

�p�p

�lp

(Vpi −Vli − v′′

li,p

)(m = b, l, p). (4)

In Eq. (4), there are phase Reynolds stressesv′′mkv

′′mi and

the averaged liquid velocities seen by bubbles and particles

v′′li,b, v

′′li,p. These terms need to be closed.

3. A second-order moment three-phase turbulencemodel

For simulating complex turbulent gas–liquid–solid flows,a second-order moment three-phase turbulence model isproposed. The derivation procedure is similar to that usedfor single-phase flows. The generalized form of bubble,liquid and particle Reynolds stress transport equations forgas–liquid–solid three-phase flows can be obtained as

��t

(�m�mv′′

miv′′mj

)+ �

�xk

(�m�m Vmk v′′

miv′′mj

)= Dmij + Pmij + mij + Smij + �mij ,

(m = b, l, p), (5)

where

Dmij = − ��xk

�m�mv′′

mkv′′miv

′′mj + �mp′v′′

miik

+ �mp′v′′mjjk − ��m

�v′′miv

′′mj

�xk

,

Pmij = −�m�m

(v′′mkv

′′mi

�Vmj

�xk

+ v′′mkv

′′mj

�Vmi

�xk

)

lij = �lp′(

�v′′li

�xj

+ �v′′lj

�xi

), bij = 0, pij = 0

Slij = �b�g

�lb

(Vbi v′′

lj ,b + v′′biv

′′lj + Vbj v′′

li,b + v′′bj v

′′li

−Vli v′′lj ,b − Vlj v′′

li,b − 2v′′liv

′′lj

)+ �p�p

�lp

(Vpi v′′

lj ,p + v′′piv

′′lj + Vpj v′′

li,p

+ v′′pj v

′′li − Vli v′′

lj ,p − Vlj v′′li,p − 2v′′

liv′′lj

),

Sbij = �b�g

�lb

(v′′biv

′′lj + v′′

bj v′′li − 2v′′

biv′′bj

),

Spij = �p�p

�lp

(v′′piv

′′lj + v′′

pj v′′li − 2v′′

piv′′pj

),

�lij = −2��l

�v′′li

�xk

�v′′lj

�xk

�bij = 0 �pij = 0.

The convection, production and phase-interaction termsin Eq. (5) are exact terms, need not be closed. The dif-fusion, dissipation terms and pressure-strain terms in thisequation are closed using the approaches similar to thoseused in single-phase flows, i.e., the Daly–Harlow’s gradientmodeling, isotropic dissipation and Launder–Rotta’s return-to-isotropy model (IP model). Hence we have

Dmij = ��xk

Csm�m�m

km

�mv′′mkv

′′ml

�v′′miv

′′mj

�xl

(m = b, l, p)

�lij = − 2

3�1�1�1ij

lij = lij ,1 + lij ,2

= − Cl1�lkl

�1�1

(v′′liv

′′lj − 2

3klij

)

− Cl2

(Plij − 2

3Plij

).

In order to fully close the momentum and Reynolds stressequations, we should further derive and close the transport

equations forv′′li,b, v

′′li,p andv′′

miv′′lj (m=b, p). The first two

correlations are neglected and the generalized form of thetransport equation for the two-phase velocity correlation is

��t

(�̄mv′′miv

′′lj ) + �

�xk

(�̄mVmkv′′miv

′′lj )

= Dml,ij + Pml,ij + Sml,ij + ml,ij

− �ml,ij (m = b, p), (6)

where the terms on the right-hand side of Eq. (6) are diffu-sion, production, phase interaction, pressure-strain and dis-sipation terms. The production and phase interaction termsare exact terms, need not be modeled. The closure of dif-fusion and pressure-strain terms is similar to that used inReynolds stress equations. Thus, we have

Dml,ij = ��xk

cml

(�̄m

km

�mv′′mkv

′′mn

+ �̄1kl

�1v′′lkv

′′ln

) �v′′liv

′′lj

�xn

,

Pml,ij = −�̄m

(v′′mkv

′′lj

�Vmi

�xk

+ v′′lkv

′′mi

�Vlj

�xk

),

650 L.X. Zhou et al. / Chemical Engineering Science 60 (2005) 647–653

Sml,ij = �m

�l�l�lm

[�m�mv′′

miv′′mj + �1�1v

′′liv

′′lj

− (�m�m + �1�1

)v′′miv

′′lj

]ml,ij = ml,ij ,1 + ml,ij ,2,

ml,ij ,1 = − cml2

�ml

�̄m

(v′′miv

′′lj − 1

3ij v

′′miv

′′li

),

ml,ij ,2 = −cml3

(Pml,ij − 2

3ij

√PmPl

).

The key point is the closure of the dissipation term. As-suming that the dissipation is isotropic, it is proportional tothe summation of its normal components divided by a timescale. For this time scale we can take the liquid turbulencetime scalek/�, the bubble or particle relaxation time�lm intoaccount. According to the Lagrangian analysis (Zhou et al.,2001), we should take the phase interaction time, accountingfor the crossing-trajectory effect. So, we have

�ml,ij = cml,1�̄m

�e

v′′miv

′′liij , (7)

where�=min(�lm, k/�). Eqs. (3)–(7) constitute the second-order moment three-phase turbulence model.

4. Simulation of bubble–liquid and gas–liquid–solidflows

At first the bubble–liquid flows in a 2-D channel are sim-ulated using the proposed model. The 2-D channel is of1.1m high, 15.2 cm width and 1.27 cm thick (Fig.1). Thereare three injection holes of 1.6mm diameter opened at thebottom. The mean bubble size is 5mm. The gas superficialvelocity is 1.0 cm/s. The liquid is initially stagnant (in batchmode). The grid size is 5mm. The differential equations arediscretized into finite-difference equations (FDEs) using ahybrid scheme. The FDEs are solved using the SIMPLECalgorithm with line-by-line TDMA iterations and under-relaxations. The convergence criterion is the liquid-phasesummation of maximum residual mass source less than 10−3

and the gas maximum residual mass source less than 10−2.Running a case on the PentiumIII-450PC takes about 40min.Figs. 2–5give the predicted bubble and liquid velocities

and normal components of Reynolds stresses in the verticaldirection. The predictions are in good agreement with thePIV measurement results. It can be seen that although thetwo-phase velocities are rather small (Figs.2and3), but boththe bubbles and the liquid have strong turbulent fluctuation(Figs. 4 and5), and the bubble fluctuation is much largerthan the liquid fluctuation. Bubbles induce liquid turbulence.The liquid turbulent fluctuation is produced by both its ownvelocity gradient (shear) and the liquid–bubble interaction.As the next step, the turbulent gas–liquid–solid flows

were simulated. This is a case of adding solid particlesinto the above-stated 2-D channel with a volume fraction

1.1m

15.2cm

38mm

1.6mm

Fig. 1. The 2-D channel.

Fig. 2. The bubble velocity.

of 5% size of 0.5mm and material density of 1346 kg/m3.Figs. 6–9show the predicted bubble and particle volumefractions, bubble, liquid and particle vertical velocities, nor-mal Reynolds stresses. The bubbles and the particles havesimilar volume fraction profiles (Fig.6) with peak valuesat the centerline, and the particle volume fraction is smallerthan the bubble volume fraction at each horizontal loca-tion. Like the case of bubble–liquid flows, in the case ofgas–liquid–solid flows the bubble vertical velocity is still

L.X. Zhou et al. / Chemical Engineering Science 60 (2005) 647–653 651

Fig. 3. The liquid velocity.

Fig. 4. The bubble normal Reynolds stress in vertical direction.

Fig. 5. The liquid normal Reynolds stress in vertical direction.

Fig. 6. Volume fractions.

652 L.X. Zhou et al. / Chemical Engineering Science 60 (2005) 647–653

Fig. 7. Vertical velocities.

larger than the liquid vertical velocity (Fig.7). The parti-cle velocity is larger than the liquid velocity and smallerthan the bubble velocity near the wall, but is smaller thanboth liquid and bubble velocities at the centerline. This iscaused by the larger particle inertia than the liquid and gasinertia. The velocity slip between the liquid and the parti-cles may reach about 50% of liquid velocity, so the no-slipassumption taken by many investigators is obviously inap-propriate. The peak value of the vertical particle Reynoldsstress profiles is near the wall (Fig.8), like that of the liquidReynolds stress profiles, but unlike that of bubble Reynoldsstress profiles, which is located at the centerline. The par-ticle Reynolds stress is smaller than the liquid and bubbleReynolds stresses everywhere. Near the centerline, the bub-ble Reynolds stress is larger than the liquid Reynolds stress,and in turn the liquid Reynolds stress is larger than the parti-cle Reynolds stress. The horizontal components of gas, liq-uid and solid normal Reynolds stresses are smaller than theirvertical components (Fig.9). All of the three-phase turbu-lences are anisotropic; the anisotropy of the bubble turbu-lence is the largest. In general, the bubbles enhance liquidturbulence, but particles reduce liquid turbulence. The rea-son is that in the present situation, the initial turbulent ki-netic energy of injected bubbles is much higher than that ofinitially stagnant liquid, and large-size (5mm) bubbles havewake effect leading to the enhancement of liquid turbulence;while particles are initially suspended in stagnant liquid andsmall-size(500�m) particles with negligible wake effectcan only dissipate the liquid turbulent kinetic energy due

Fig. 8. Vertical component of normal Reynolds stresses.

Fig. 9. Horizontal component of normal Reynolds stresses.

to their drag effect. However, if the liquid has an initiallyhigher velocity and the bubble volume fraction is low, bub-bles may reduce liquid turbulence (Zhou and Xiao, 2004).

L.X. Zhou et al. / Chemical Engineering Science 60 (2005) 647–653 653

5. Conclusions

(1) A second-order moment (SOM) three-phase turbulencemodel for simulating turbulent gas–liquid–solid flowsis proposed. The anisotropic bubble, liquid and particleturbulences and their interaction are fully taken into ac-count.

(2) Simulation of bubble–liquid flows and its comparisonwith PIV measurements indicate that the proposedmodel is reasonable.

(3) The SOM model can well predict the bubble, liquidand particle volume fractions, velocities and Reynoldsstresses, gives a better understanding to the three-phaseturbulences and their interactions.

(4) In the cases studied the bubble turbulence and itsanisotropy are stronger than the liquid ones, while theparticle turbulence is weaker than the liquid one. Bub-bles enhance liquid turbulence, while particles reduceliquid turbulence.

Notation

c empirical constantscd drag coefficientD diffusion termd bubble sizeEo Eotvos numberf function defined in the textg gravitational accelerationk turbulent kinetic energym massP production termp pressureRe Reynolds numberS phase interaction termt timev velocity componentsx coordinate

Greek letters

� volume fraction� coefficients defined in the text unit tensor� dissipation rate of turbulent kinetic energy� empirical constant� dynamic viscosity� kinematic viscosity pressure strain term� density� Prandtl number; surface tension force�rb bubble relaxation time�rp particle relaxation time�ij molecular viscous stress

Subscripts

b bubblee effectiveg gasi, j, k, n coordinate directionsl liquidm bubble or liquid or particlep particleT turbulent

Superscripts

= mass-weighted averaged value′′ fluctuating component of mass-weighting

averaging- time-averaged value′ fluctuating component of time averaging

Acknowledgements

The study at Tsinghua University was supported by theSpecial Funds for Major State Basic Research, PRC, underthe Grant G1999-0222-08. The study at the Ohio State Uni-versity was supported by the US DOE.

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