Application of Population Balance Model in the Simulation of SlurryBubble ColumnLijia Xu, Zihong Xia, Xiaofeng Guo, and Caixia Chen*

Key Laboratory of Coal Gasification and Energy Chemical Engineering of Ministry of Education, East China University of Science andTechnology, Shanghai, 200237, China

ABSTRACT: Numerical simulations of a slurry bubble column with particle loadings up to 40 vol % and superficial gasvelocities up to 0.26 m/s have been performed using the Euler−Euler approach with the renormalized group (RNG) k−εturbulence model. Special attention was paid to the bubble size distributions, interphase closure models, and liquid−solid dragforces. A modified Luo−Lehr population balance model was used to simulate the changes of mean bubble size and the overall gasholdup as functions of particle loadings. A pseudo gas−slurry closure model was proposed to overcome the drawbacks of existinginterphase momentum exchange closure models. The newly constructed model was different from the traditional gas−slurryclosure model in that the hydrodynamics of three phases are solved by three sets of momentum equations describing gas, liquid,and solid phases, respectively. Three different drag force models, i.e., the Schiller−Naumann model, the Wen−Yu model, andthe energy-minimization multiscale (EMMS) model, were used for the computation of the liquid−solid interaction force and thesimulated particle settlings were compared with experiments. A series of numerical simulations were performed. The followingconclusions were drawn from the simulation results: (1) The Luo−Lehr population balance model could simulate the meanbubble size changes as a function of particle loadings. The gas holdup and its variation trend predicted by the modified Luo−Lehrpopulation balance model were in agreement with the experimental data by setting a coalescence coefficient C0 = 0.4. (2) Apseudo gas−slurry closure model that took the influence of solid concentration into account was recommended for gas−liquid−solid three-phase simulation when the particle size was much smaller than the bubble size. The simulated average gas holdupsshowed good agreement with the experimental data. (3) The axial distribution of solid concentration simulated by the EMMSdrag model was much closer to experimental results than other drag models for the simulation of slurry bubble columns.

1. INTRODUCTIONSlurry bubble column reactors are widely used in petrochemicaland coal liquefaction processes due to their advantages of highcapacity, easy operation, and good heat and mass transfer char-acteristics. Traditionally, the design and scale-up of thesereactors rely on a large number of empirical correlations ob-tained from different scale experiments. Those correlationsinclude estimations of the gas holdup, the interphase heattransfer coefficient, the axial solid concentration, and otherhydrodynamic parameters.1 However, applications of theempirical correlations are limited by the experimental con-ditions, such as the superficial gas velocities, liquid properties,particle sizes and concentrations, among others. Computationalfluid dynamics (CFD) has provided the state-of-the-art methodfor the description of fluid dynamics in gas−liquid flows (e.g.,Chen and Fan2 and Buwa et al.3). Nevertheless, the applicationof the CFD approach to the predictions of gas−liquid−solidfluid dynamics is still immature due to the inherent com-plexities including the description of bubble size distributions,the uncertainty of the interphase closure models, and the choiceof liquid−solid drag model.Over the years, the Eulerian multifluid model incorporated

with the population balance model (PBM) has been appliedin the simulations of air−water bubble columns operated inthe churn-turbulence regime, and the predicted distributions ofthe axial liquid velocity and the gas holdup agree with the ex-periments.4,5 The application of the PBM technique to thesimulation of a gas−liquid−solid three-phase system needsconsiderable elaboration of modeling the influence of solids on

hydrodynamics, including the breakup and coalescence ofbubbles, and the interaction forces between liquid−solids andgas−solids. Chen and Fan6 suggested that the effect of bubble−particle collisions on the bubble breakup could be neglectedwhen the particle size is much smaller than the bubbles.Hooshyar et al.7 studied the dynamics of a single rising bubblein liquid−solid suspensions, and found that the small particles(78 and 587 μm) remained on the liquid flow stream aroundthe bubble. Their experimental findings demonstrated that theliquid−particle mixture behaved more like a single pseudoclearliquid than two discrete media. In this regard, the gas−liquid−solid three-phase problem can be simplified as a gas−slurrytwo-phase problem, and the influence of small solid particles onthe bubble breakup and coalescence are taken into accountthrough a change of physical properties of the slurry instead ofdirect bubble−particle collisions. The viscosity of slurry wasreported to increase with particle concentrations, and a higherparticle concentration would usually result in an increase ofbubble size and a decrease of gas holdup.8 Hence, the increasesof bubble size can be interpreted as a result of increases in theviscosity of slurry in a gas−liquid−solid bubble column. Follow-ing this scenario, the gas−liquid−solid three-phase problemcould be simplified as a gas−slurry two-phase problem, as long

Received: October 14, 2013Revised: February 22, 2014Accepted: February 27, 2014Published: February 27, 2014

Article

pubs.acs.org/IECR

© 2014 American Chemical Society 4922 dx.doi.org/10.1021/ie403453h | Ind. Eng. Chem. Res. 2014, 53, 4922−4930

as the particles mix with the liquid perfectly or the particle sizeis small.9 The gas−slurry two-phase model is referred to as“closure A” in the present paper, and is depicted in Figure 1a.Such a simplification avoided the explicit closures of the gas−solid and liquid−solid interphase momentum exchanges andprovided reasonable results in certain cases.9,10

However, when the particle settling is obvious in some gas−liquid−solid flows, the liquid and solids need to be treated intwo distinct phases, and closure models which describe theinteraphase momentum exchanges between gas−liquid−solidthree phases are required. In most published literature, only theliquid−gas and liquid−solid drag forces were considered,11,12

and the gas−solid interaction force was neglected, referred to as“closure B” as shown in Figure 1b. In recent years, an increasingnumber of researchers suggested that the gas−solid interac-tion did exist in the gas−liquid−solid flows, based on the factthat the particles in the vicinity of a bubble tend to follow thebubble. Padial et al.,13 Schallenberg et al.,14 Panneerselvamet al.,15 and others directly used a general form of the drag forcemodel (such as the Wen−Yu and Schiller−Naumann dragmodels) to calculate the gas−solid interaction forces, andtreated the gas phase as a continuous phase and the particlesas a dispersed one, respectively. This treatment made a closedgas−liquid−solid−gas loop of interface exchanges, and isreferred to as “closure C” as shown in Figure 1c. Here, theclosure C model involves three major uncertainties: first, inthe majority of slurry bubble column reactors, the gas phase ispresented as bubble foam and the particles are more likely tobe presented in the liquid. That is, the particles can barelypenetrate the gas−liquid interface and dance around inside agas bubble. A direct, explicit gas−solid interfacial drag forcewould overpredict the gas−solid interactions. Second, whenbubbles rise in a slurry, a negative pressure generated in thebubble wake will induce the liquid to move, and the movementof the liquid drives the particles to rise faster than those that are

not in the wake. Rather than a direct action, movements of gasbubbles influence the solids motion in an indirect way. Third,the liquid is simultaneously susceptible to the upward forcefrom bubbles and the downward force from solids, and thesolids are driven by the upward forces from bubbles and liquid,simultaneously. This effect may cause a higher velocity ofparticles than the liquid phase, or a decrease of the relativevelocity between liquid and solids. Therefore, the closure Cmodel may underestimate the particle settling.For the simulations of particle settling, a reasonable closure

model of liquid−solid interaction forces plays a critical role.The Wen−Yu and Gidaspow drag models were recommendedfor dilute and dense gas−solid or liquid−solid flow systems,respectively. Visuri et al.16 simulated the liquid−solid (particlesize 1.5 mm) flows with different drag models, and found thatthe bed height obtained with the Wen−Yu model was higherthan that obtained in the experiments. On the other hand, theGibilaro model was reported to be appropriate for flows withhigh particle Reynolds numbers, and it predicted the particleconcentrations in good agreement with experiments.16 Yanget al.17 simulated a gas−solid flow in a circulating fluidized bed,and found that the Wen−Yu/Ergun drag model produced amore homogeneous structure. In addition, they found that theWen−Yu and Gidaspow drag models overestimated the gas−solid drag force in a dilute gas−solid system.In the present paper, the gas−liquid−solid flows were simu-

lated by the Euler−Euler approach implemented in ANSYSFluent software. The experimental setup of Gandhi et al.18 wasused as a benchmark case. The PBM equations were solved incombination with a mixture manifold of the renormalizedgroup (RNG) k−ε model. A pseudo gas−slurry closure modelreferred to as the “closure D” model as indicated in Figure 1dwas proposed. The newly constructed model was different fromthe existing closure A model in that the hydrodynamics of threephases were solved by three sets of momentum equations

Figure 1. Closure law for the gas−liquid−solid interphase momentum exchange.

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describing gas, liquid, and solid phases, respectively. Using theclosure D model, the particle settlings with various particleloadings were simulated using different liquid−solid dragmodels. The simulated results were compared with exper-imental data of literature.

2. DESCRIPTION OF COMPUTATIONAL METHODS2.1. Governing Equations of the Pseudo Gas−Slurry

Closure Model. For the Eulerian multifluid model, bubblesand particles are treated as dispersed phases. The continuityand momentum of gas, liquid, and solids are solved by using thephase-coupled SIMPLE algorithm (PC-SIMPLE). The conser-vation equations of each phase are presented as follows:

ραα

∂∂

+ ∇· =t

u( ) 0i ii i (1)

α ρα ρ

α α τ α ρ

∂∂

+ ∇·

= − ·∇ − ∇ + +

u

tu u

p F g

( )( )

( )

i i ii i i i

i i i i j i i, (2)

where αi is the volume fraction of the ith phase. The sumof the volume fractions of all phases equals 1. ρi, τi, ui, and Fi,jrepresent the density, viscous stress tensor, velocity, and inter-face momentum exchange term, respectively.The liquid−gas (Fl,g) and liquid−solid (Fl,s) drag models

used in the pseudo gas−slurry closure model (closure D) areexpressed as

ρα α

= | − | −F Cd

u u u u34

( )l,g slu,g slul g

gl g l g

(3)

= −F K u u( )l,s l,s l s (4)

In the closure D model, the density and viscosity of theliquid−solid slurry are used, which poses the major differencecompared to the closure B and C models, where the pureliquid properties are used in the computations. The ex-change coefficients Kl,s of liquid−solid drag forces are presentedin eqs 11−15.Note that the closure D model is different from closure A

in that the hydrodynamics of three phases are solved by threesets of momentum equations describing gas, liquid, and solidphases, respectively. The liquid and solids are treated as a singleslurry phase in the closure A model, and the viscosity anddensity of the slurry phase are functions of solid concentrations;the surface tension of slurry remains unchanged when the solidweight loading increases.19

The viscosity of the liquid−solid slurry is calculated using theThomas semitheoretical correlation,20 and the density is thevolume-weighted average density:

μ μ α α= + + + α(1 2.5 10.05 0.00273e )slu l s s2 16.6 s (5)

ρ ρ α ρα= +slu l l s s (6)

When the population balance model is incorporated in thesimulation, the Sauter mean diameters of bubbles change in awide range. The Ishii−Zuber model21 (nonspherical bubbledrag model) is used to calculate the liquid−gas drag coefficient:

= +CRe

Re24

(1 0.1 )d,spherical0.75

(7)

ρ ρ

σ

α

=− +

=

α

α

α

⎛⎝⎜⎜

⎞⎠⎟⎟C d

g f

f

f

23

( ) 1 17.67

18.67,d,distorted

slu g6/7 2

slu1.5

(8)

α=C83d,cap slu

2

(9)

>

=

=

C C C

C C

C C

If ,

; else

min( , )

d,spherical d,distorted d

d,spherical d

d,distorted d,cap (10)

The liquid−solid drag force used in closures B, C, and D arecomputed using the Schiller−Naumann correlation22 (eqs 11and 12), which is also used to calculate the gas−solid interac-tion force in closure C.

α α ρ=

| − |−K C

u ud

34sl,Schiller Naumann d

l s l l s

s (11)

=+ ≤

>

⎧⎨⎪⎩⎪

C ReRe Re

Re

24(1 0.15 ) 1000

0.44 1000d s

s0.687

s

s (12)

For comparison, the Wen−Yu23 and the EMMS17 liquid−soliddrag models are also used in the simulations of the closure Dmodel.

α α ρα=

| − |−

−K Cu ud

34sl,Wen Yu d

l s l l s

sl

2.65

(13)

α α ρα=

| − | −K Cu ud

H34sl,EMMS d

l s l l s

sl

2.65D

(14)

= +H a Re b( )cD s (15)

Here, the definitions of a, b, and c are referred to Hong’swork.24 It should be noted that the existing EMMS modelbased on superficial gas velocities and solid flux is only used tosimulate a gas−solid fluidized bed. It is not easy to build aEMMS model for liquid−solid interaction in a slurry bubblecolumn. In the present paper, the Hong-EMMS model isdirectly used to calculate the liquid−solid interaction force.The drag force coefficient Cd of the Wen−Yu (eq 13) and

EMMS (eq 14) models can be calculated as

αα

=+ <

>

⎧⎨⎪⎩⎪

C ReRe Re

Re

24(1 0.15( ) ) 1000

0.44 1000d l s

l s0.687

s

s (16)

In our previous research work,4 it was found that reasonablegas holdup distributions could be obtained only by taking thelift force into account, and the Tomiyama lift force25 was used:

ρ α= − − × ∇ ×C u u uF ( ) ( )lift lift slu g l g l (17)

where

=<

< <

− <

⎧⎨⎪

⎩⎪C

Re f Eo Eo

f Eo Eo

Eo

min[0.288 tanh(0.121 ), ( )] 4

( ) 4 10

0.27 10

lift

(18)

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= − − +f Eo Eo Eo Eo( ) 0.00105 0.0159 0.0204 0.4743 2 (19)

where Eo is the Eotvos number.2.2. Numerical Scheme and Benchmark Cases. The

simulations were performed based on the experimental setup ofGandhi et al.18 The slurry bubble column was operated withparticle loadings up to 40 vol % and gas velocities up to 0.26 m/s(ds = 35 μm, ρs = 2452 kg/m3). The diameter and height of theslurry bubble column were 0.15 m and 2.50 m, respectively. Thestatic liquid height of the column was 1.50 m. Hamidipour et al.26

performed three-phase fluidized bed simulations with differentsolid wall boundary conditions (no slip, partial slip, and free slip),and discovered the simulation results in excellent agreement withexperimental results when a near-zero value of the specularitycoefficient was used. The same free-slip boundary condition wasused by Schallenberg et al.14 and Panneerselvam et al.15 There-fore, in all simulations of the present paper, the free-slip wallboundary condition is used for the bubbles and particles.For bubbly flow, 3-D time dependent simulation is recom-

mended.27 The mixture RNG k−ε turbulence model is used for themodeling of bubble induced turbulence. The QUICK discretizationscheme is used for momentum and volume fraction equations, andthe first order upwind scheme is employed to discretize otherequations. More information about simulation numerical schemeand result processing can refer to our previous work.4

3. RESULTS AND DISCUSSION3.1. Selection of Population Balance Model. The

bubble number density equation of population balance modelis expressed as

where CB, CD, BB, and BD represent the birth rate and death ratedue to coalescence and the birth rate and death rate due tobreakup, respectively.By far, the population balance model has been mainly used in

the simulation of air−water systems. In the present work, anattempt has been made to test the existing population balancemodels in the simulation of gas−liquid−solid flows with dif-ferent particle loadings. As the first step of the effort, the effectsof particle concentrations on the breakup rates calculatedby different bubble breakup models (Table 1) have beeninvestigated and are compared in Figure 2. The breakup rates of

Martinez-Bazan, Luo and Svendsen, and Zhao and Ge increasewith the increase of solid concentrations; only that of the Lehr et al.model decreases with the increase of solid concentrations. There-fore, the Lehr breakup kernel is supposed to be able to model theeffect of solid concentration on the overall gas holdup.Subsequently, the Lehr breakup model and the Luo coa-

lescence model are used to simulate the gas−slurry flows with

Table 1. Models for Bubble Breakup Kernelsa

authors breakage rate kernel

Martinez-Bazan et al.28ε

Ω =− σ

ρd

d

d( ) 0.25

8.2( ) 12i

i d

i

2/3ic

Luo and Svendsen29

∫Ω = Ωd d d f( ) ( , ) di i j0

0.5

bv

∫α ε ξξ

σρ ε

ξ ξΩ = +ξ

−⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟d d

dcd

( , ) 0.9238(1 )

exp12

di ji i

slu 2

1/31 2

11/3f

slu2/3 5/3

11/3

min

Lehr et al.30 ∫Ω = Ωd d d f( ) ( , ) di i j0

0.5

bv

∫σρ ε

ξξ

σρ ε

ξ ξΩ = +ξ

−⎛⎝⎜⎜

⎞⎠⎟⎟d d

d fWed f

( , )1.19 (1 )

exp2

di ji islu

1/3 7/3bv

1/3

1 2

13/3crit

slu2/3 5/3

bv1/3

2/3

min

∫Ω = Ωd d d f( ) ( , ) di i j0

0.5

bv

Zhao and Ge31 ∫φ ε ξξ

λλ

ξΩ = − + −ξ

⎛⎝⎜

⎞⎠⎟d

de d

e( ) 0.923(1 )

(1 )( )

exp( , )

( )di

i

i

i

1/31 2

5/3 11/3c

min

λπ σ πσλ=

−

⎛⎝⎜⎜

⎞⎠⎟⎟e d

c dC d f f

( , ) max ,3 min[( ), 1 ( )]i

i

ic

f2

ed bv bv1/3

aThe volume fraction of daughter bubble f bv = (dj/di)3, the ratio of eddy size to the parent bubble diameter ξ = λ/di, the mean eddy energy ei(λ) =

πβLρslurryε2/3λ11/3/12, and the coefficient of surface increment during bubble breakup cf = f bv

2/3 + (1 − f bv)2/3 −1.

Figure 2. Comparison of different bubble breakup kernels with thechange of solid concentration.

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different particle concentrations (0, 10, 20, 30 vol %) and a gasvelocity of 0.25 m/s. The coalescence rate cij is defined as

In eq 21, tdrainage and tcontact represent film drainage time andbubble contact time, respectively.It is worth noting that C0 is an adjustable parameter and

takes very different values in the existing collision frequencymodels. For example, in the models of Lee et al.,32 Prince andBlanch,33 and Luo,34 the values of C0 are 1.4142π/4, 0.28, and1.43π/4, respectively. Theoretically, the bubble collisionfrequency is contributed by three different mechanisms:turbulence, buoyancy-driven, and laminar shear stress collisions.However, in the Luo coalescence model, only the turbulencecollision is implemented and the other two mechanisms areignored; thus a much larger value of C0 (∼1.1) was used. Inour previous work,4 we found this value overestimated thebubble collision frequency, and 0.5 was used for C0 to better fitthe experimental data. For small particles, the influence ofparticle concentration on bubble breakup and coalescence is theproperty change of the slurry instead of the particle−bubblecollision. The effect of liquid viscosity on the bubble coa-lescence rates can be modeled by introducing a film drainagetime; see, e.g., Chesters35 and Martin et al.36 Since the Luocoalescence model does not take into account the influence ofliquid viscosity on the bubble coalescence, a slightly smaller C0of 0.4 is used to simulate a lower bubble coalescence rate due toan increase in slurry viscosity. Figure 3 shows the simulated gas

holdups using original and modified Luo−Lehr models and theexperimental data as well. The Sauter mean bubble sizes arealso simulated and compared with an empirical correlation asshown in Figure 4. The simulated bubble sizes as a function ofparticle loading by using the modified Luo−Lehr model agreewith the Wilkinson model, which reads37

σ μ ρ ρ= − − − −d g U3g,Wilkinson0.44 0.34

l0.22

l0.45

g0.11 0.02

(22)

The gas holdup and liquid streamline of gas−slurry flow with30 vol % particle concentration are shown in Figure 5. A highvelocity of liquid is located in the center of the column, andcomplicated fine structures of reverse flow are formed in thenear-wall zone. The radial distributions of bubble size and axialliquid velocity profiles with different particle loadings aresimulated by the modified Luo−Lehr model, and the results are

shown in Figure 6. An increase of slurry viscosity leads toincreases of bubble size and axial liquid velocities with theincreases of the particle loading. Bubble size distribution ofgas−liquid−solid flows will be simulated by the above modifiedLuo−Lehr model in the following simulations.

3.2. Comparison of Interphase Closure Models. Asdescribed in section 1, closure A is a gas−slurry two-phasemodel, and closures B, C, and D are three-phase models; i.e.,bubbles, liquid, and solid particle three-phase momentum equa-tions are solved, separately. While the particle settling cannotbe simulated in the closure A model, the other three models dotreat the liquid and solids as distinct phases, and the particlesettling can be modeled. The newly constructed closure Dmodel is similar to the closure A model in treating the liquid−solid as a slurry while computing the gas−liquid drag force; the

Figure 3. Comparison of gas holdups with particle loadings up to30 vol % using different population balance models.

Figure 4. Comparison of Sauter mean diameters obtained by Wilkinsoncorrelation and simulation method with different population balancemodels.

Figure 5. Gas holdup and liquid streamline of gas−slurry flow with30 vol % particle concentration.

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closure D model treats the solids as a distinct phase and modelsthe liquid−solid drag force in a similar way to the closure B andC models. The simulation results of the four closure schemesare compared and discussed as follows.The distributions of gas volume fractions and axial liquid

velocities of a slurry bubble column operated at a superficialgas velocity of 0.25 m/s and a slurry concentration of 0.2 weresimulated using the closure A, B, C, and D models, respec-tively. The simulated radial distributions of axial liquidvelocities and gas holdups are shown in Figure 7. As can befound in Figure 7, since closure B underestimates the interac-tion between bubbles and other phases, the simulated liquidvelocity of closure B is relatively low, and the gas holdupis underestimated. Taking the gas−solid interaction intoaccount, the bubble lifting resistance increases, and the gasholdup is overestimated in closure C. The axial liquid velocityincreases with the increase of gas holdup. The overall gasholdups simulated by the four closure models are compared inTable 2. The gas holdups of closure A and closure D are ingood agreement with the experimental data. The closure Bmodel underestimates and the closure C model overestimatesthe gas holdup.Figure 8 compares the axial velocities of liquid and solid

simulated by closure C. It is interesting to find that the solidvelocity is higher than the liquid’s. Therefore, closure C is in-capable of predicting the particle settling phenomenon whenthe particle size in the slurry reactor is much smaller than thebubble size. The simulated gas volume fraction and axialliquid velocity of closure D are similar to those of closure Adue to the contributions of solids to the increase of gas−slurryinteractions.3.3. Comparison of Drag Force Models. The axial

distribution of the solid phase is largely dependent on the

superficial gas velocity and the properties of the liquid andparticles. Smith and Ruether38 studied the particle dynamicsin a slurry bubble column and proposed a one-dimensionalsedimentation dispersion model. The model is represented asfollows:

= ⎜ ⎟⎛⎝

⎞⎠C C B

zH

exps s0 (23)

= −C C B B(exp( ) 1)s0 s (24)

ψ= −B

u H

Dl p

s (25)

Figure 6. Time-averaged radial distributions of Sauter diameters andaxial liquid velocities with different particle loadings.

Figure 7. Time-averaged radial distributions of gas holdup and axialliquid velocities simulated with different interphase closure models.

Table 2. Comparison of the Whole Column Gas HoldupsSimulated by Four Closure Models with Experimental Value

expt closure A closure B closure C closure D

gas holdup 0.185 0.173 0.154 0.279 0.169

Figure 8. Comparison of axial velocities of liquid and solid simulatedby closure C.

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where up and Ds represent the particle hindered settling velocityand axial dispersion coefficient which are calculated by the fol-lowing equations, respectively:

ψ=u u u1.10p g0.026

t0.80

l3.5

(26)

= +u D

DFr Re Re9.6( / ) 0.019

g T

sg

6g

0.1114p

1.1

(27)

In eqs 26 and 27, Frg, ut, Reg, and Rep represent the Froudenumber, particle terminal settling velocity, gas Reynolds num-ber, and particle Reynolds number, respectively.For a cylindrical slurry bubble column, an accurate solid

concentration distribution can be obtained using the Smith−Ruether empirical correlation (Figure 9a). When the geometryshape is complex or internals are presented in a column, theempirical model is no longer applicable. In this situation, CFDprovides an alternative approach. The Schiller−Naumannmodel, Wen−Yu model, and EMMS model are used tosimulate the axial solid concentrations in slurry bubble columnswith different particle concentrations.The axial solid concentration distributions simulated by the

Schiller−Naumann, Wen−Yu, and EMMS liquid−solid dragmodels are shown in Figure 9, parts b, c, and d, respectively.Figure 9 shows very little difference between the axial solidconcentration distributions of the slurry bubble columns withdifferent particle loadings simulated with the Schiller−Naumann model, which does not consider the effect of thevolume fraction of the continuous phase as indicated in eqs 11and 12. The simulated results obtained with the Wen−Yu andEMMS models reveal that the particle settling tendencybecomes weaker with the decrease of solid holdup. Thesimulation results are in good agreement with the experimentalresults. The distributions of solid holdup computed with theWen−Yu model are more homogeneous than that of theEMMS model. The reason for a large variation obtained withthe EMMS model is that the model takes the local hetero-geneous structure into account for the drag force computation.Similar results can be found in the simulations of gas−solidflow of Yang et al.17 and Chalermsinsuwan et al.39

4. CONCLUSIONA slurry bubble column was simulated using the Euler−Eulerapproach implemented with a population balance model. Apseudo gas−slurry closure model was proposed and applied inthe predictions of gas holdup and axial liquid velocity, and theresults were compared with the existing closure models. Threedifferent drag force models, i.e., the Schiller−Naumann model,the Wen−Yu model, and the EMMS model, were used for thecomputation of the liquid−solid interaction force, and thesimulated particle settlings were compared with experiments.The following conclusions were drawn from the simulationresults: (1) The Luo−Lehr population balance model couldsimulate the mean bubble size changes as a function of particleloading. The gas holdup and its variation trend predicted by themodified Luo−Lehr population balance model were inagreement with the experimental data by setting the Luo coa-lescence coefficient C0 = 0.4. (2) A pseudo gas−slurry closuremodel which took the influence of solid concentration intoaccount was recommended for gas−liquid−solid three-phasesimulation when the particle size was much smaller than thebubble size. The simulated average gas holdups showed goodagreement with the experimental data. (3) The axial distribu-tion of solid concentration simulated by using the EMMSdrag model was much closer to the experiments than thoseof other drag models for the simulation of slurry bubblecolumns.

Figure 9. Comparison of radial distributions of solid concentrationcalculated with (a) one-dimensional sedimentation dispersion modeland CFD method using (b) Schiller−Naumann, (c) Wen−Yu, and (d)EMMS drag models.

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■ AUTHOR INFORMATIONCorresponding Author*E-mail: cxchen@ecust.edu.cn.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work is supported by China’s National ScienceFoundation (NSFC) under Grants 21276085 and 20876049.

■ NOTATIONB = dimensionless parameter defined in eq 23BB = production of bubble by breakup, m−3·kg−1·s−1

BD = destruction of bubble by breakup, m−3·kg−1·s−1

c = coalescence rate, m−3·s−1

CB = production of bubble by coalescence, m−3·kg−1·s−1

CD = destruction of bubble by coalescence, m−3·kg−1·s−1

Cd = drag coefficient, dimensionlessClift = lift coefficient, dimensionlessCs = particle concentration, kg·m−3

d = bubble or solid particle diameter, mDs = axial dispersion coefficient, dimensionlessEo = Eotvos number, dimensionlessf = volume fraction, Vi/VjF = interface momentum exchange termFlift = lift force, N·m−3

Fr = Froude number, dimensionlessg = gravity acceleration, m·s−2

g(V′) = breakup frequency, s−1

H = liquid height, mn = number of bubbles per unit volume, m−3

p = pressure, PaRe = Reynolds number, dimensionlesst = time, stcontact = bubble contact time, stdrainage = film drainage time, sT = temperature, Ku = velocity, m·s−1

up = particle hindered settling velocity, m·s−1

ut = particle terminal settling velocity, m·s−1

V = bubble volume, m3

z = axial position from bottom of column, m

Greek Symbolsα = void fraction, dimensionlessβ(V,V′) = probability density function (pdf), m−3·kg−1

ξ = eddy size divided by parent bubble sizeρ = density, kg·m−3

μ = viscosity, kg·m−1·s−1

σ = surface tension, N·m−1

ψ = volume fraction in slurryε = turbulent dissipation rate, m2·s−3

ΩB(vi,vj) = breakup frequency, m−3·s−1

Subscriptsg = gas phasel = liquid phases = solid phaseslu = slurry

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