chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102

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Chemical Engineering Research and Design

journa l h om epage: www.elsev ier .com/ locate /cherd

CFD-PBM approach with modified drag model forthe gas–liquid flow in a bubble column

Xiao-Fei Liang, Hui Pan, Yuan-Hai Su, Zheng-Hong Luo ∗

Department of Chemical Engineering, School of Chemistry and Chemical Engineering, Shanghai Jiao Tong University,Shanghai 200240, PR China

a r t i c l e i n f o

Article history:

Received 29 March 2016

Received in revised form 25 May

2016

Accepted 8 June 2016

Available online 18 June 2016

Keywords:

Bubble column

Euler–Euler approach

PBM

Drag model

Correction factor

a b s t r a c t

In this work, numerical simulations of cylindrical bubble column are performed using the

Euler–Euler approach incorporated with a population balance model (PBM). First, three drag

models and their corresponding modified models with the wake acceleration are incorpo-

rated into the coupled approach in order to evaluate the effectiveness of these drag models.

The simulated time-averaged local gas holdups and normalized axial liquid velocities using

different drag equations are compared with the experimental data, showing that only the

PBM-customized drag model with the wake acceleration (cf., the application of a correction

factor) can reproduce the measured flow field data. Subsequently, the applicability of the

coupled approach with the effective drag model is further evaluated at various superficial gas

velocities and gas distributors. The simulated results accord well with the experimental data

at high gas velocities. However, the model greatly underestimates the radial local gas holdup

and the total gas holdup at low gas flow rates. Additionally, the simulated results demon-

strate that the opening area and orifice geometry play a significant role in total aeration and

the triple-ring gas distributor produces more uniform radial profiles of local gas holdup and

normalized liquid velocity than the multi-orifice one, thus leading to poor mixing efficiency

in the bubble column.© 2016 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

imental results in the heterogeneous regime (Ishii, 1975). A

1. Introduction

Bubble columns serve as multiphase contactors and reactorsin the chemical, petrochemical, biochemical, and metallurgi-cal industries, primarily because of their simple construct withlow price, simple operation, ability to handle solids, excellentheat and mass transfer characteristics, etc. In spite of theirsimplicity in mechanical design, fundamental properties ofthe two-phase hydrodynamics associated with the operationof bubble column reactors essential for scale-up and processoptimization, are still not fully understood because of thecomplex nature of multiphase flow (Chen et al., 2005; Tabibet al., 2008).

A great deal of research on gas–liquid flows has beencarried out with computational fluid dynamics (CFD) simula-

tions for engineering purposes. Two approaches including the

∗ Corresponding author. Tel.: +86 21 54745602; fax: +86 21 54745602.E-mail address: luozh@sjtu.edu.cn (Z.-H. Luo).

http://dx.doi.org/10.1016/j.cherd.2016.06.0140263-8762/© 2016 Institution of Chemical Engineers. Published by Elsev

Euler–Lagrange (E-L) approach (Delnoij et al., 1997; Sokolichinet al., 1997) and the Euler–Euler (E-E) approach are primar-ily used (Drew, 1983; Ishii, 1975; Krishna et al., 1999; Lehret al., 2002). The E-L approach shows its advantages on a clearphysical description but its disadvantages on high computa-tional cost and the difficulties to involve the forces on thedeformable bubbles and bubble breakup and coalescence. Forthe E-E approach, the model equations for each phase have thesame form, showing obvious advantages in view of the com-puter memory and simulation time limitations. However, theE-E two-fluid model with the assumption of a constant bubblediameter gives reasonable predictions only for the homoge-nous regime in which the bubble size distribution is narrow.There are obvious deviations between simulated and exper-

profound progress in the CFD simulation of bubble columns

ier B.V. All rights reserved.

chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102 89

Nomenclature

ai,j bubble aggregation rate, s−1

b(v) bubble breakup rate, s−1

CD drag coefficientd bubble diameter, mdc critical size of bubbles having wake effect for

bubble coalescence, mEo Eötvös number, g(�l − �g)d2

b/�

flarge fraction of large bubblesg gravity, m2 s−1

G the production rate of turbulent energy, W m−3

k turbulent kinetic energy, m2 s−2

kb,large correction factor accounting for bubble wakeeffect

ki,j the covariance of the velocities of phase i andphase j

Mo Morton number, dimensionlessn number of particles per unit volume, m−3

p pressure, PaQ gas flow rate, m3 s−1

Re bubble Reynolds number, dimensionlesst time, stdrainage film drainage time, stcontact bubble contact time, sui,j the relative velocity between the liquid and dis-

persed phasesU velocity vector, m2 s−2

Ug superficial gas velocity, m2 s−2

v bubble volume, m3

xi pivot bubble size of the bubble interval (vi, vi+1),m3

Greek Letters˛ phase holdupˇ(v, xk) daughter bubble size distribution functionε turbulent dissipation rate, m2 s−3

� viscosity, kg m−1 s−1

�1(v, xi), �2(v, xi) bubble redistribution coefficientωi,jk transfer coefficient between bubble groups aris-

ing from bubble coalescence i,k transfer coefficient between bubble groups aris-

ing from bubble breakup˘k,i bubble-induced turbulence term, W m−3

˘ε,i bubble-induced turbulence term, W s−1 m−3

� density, kg m−3

� surface tension, N m−1

Subscriptsb bubble indexg gas indexl liquid indexi,j phase indexlarge large bubbles index

iC2aasa

s the coupling of the population balance model (PBM) intoFD models, namely the CFD-PBM coupled model (Duan et al.,011; Wang and Wang, 2007; Wang et al., 2006). The prominentdvantages of the coupled model could be expressed brieflys follows: (1) it combines the PBM into the CFD framework

o that bubble breakup and coalescence can be taken intoccount, beneficial for describing the bubble size distribution

and gas holdup in different flow regimes, and (2) it consid-ers the influence of bubble size on the interphase interaction,which allows it to well predict the local gas–liquid interfacialarea and the flow behavior in diverse flow regimes (Xing et al.,2013).

While the closure models are widely applied in quantita-tive and qualitative description of the heat and mass transferin reactors. In these models, bubble induce turbulence of liq-uid phase, interaction forces between gas and liquid phase,and local bubble size distribution are emphasized and highlycoupled with each other (Laborde-Boutet et al., 2009). It isgenerally believed that the drag is the predominant force inmodeling the gas–liquid flows of bubble columns (Chen, 2004)and the magnitude of the drag force was found to be over 100times than that of the other forces such as lift force, addedmass force, and turbulent dispersion force (Laborde-Boutetet al., 2009). Although there have been many studies on the liftforce (Díaz et al., 2009; Kulkarni, 2008; Lucas and Tomiyama,2011; Van Nierop et al., 2007), an appropriate model for thelift coefficient is not available and the role of lift force orturbulent dispersion force in two-fluid modeling is not clear.In some of these studies, the lift force was only consideredin order to adjust CFD simulations, and the turbulent dis-persion force was involved to compensate the destabilizingaction induced by a negative lift coefficient (Lucas et al., 2005).Many drag models such as Schiller and Naumann (1935), Ishiiand Zuber (1979), Tomiyama (1998), Zhang and VanderHeyden(2002) and Grace et al. (1976) models have been used to calcu-late the drag force in bubble columns (Gupta and Roy, 2013;Laborde-Boutet et al., 2009; Olmos et al., 2001; Silva et al.,2012; Zhang et al., 2006). For small spherical bubbles, theSchiller–Naumann model is frequently applied (Schiller andNaumann, 1935). While for larger bubble size, a model wasproposed by Ishii and Zuber (1979) which is suitable for vari-ous shapes of bubbles such as sphere, ellipse and cap shape(Ishii and Zuber, 1979; Xu et al., 2013). Recently, Yang et al.(2011) suggested a dual-bubble size (DBS) model (Chen et al.,2009; Yang et al., 2007, 2010) where the force balance equa-tions of two bubble classes are closed via a stability condition.Afterwards, they developed a stability-constrained multi-fluid(SCMF) model for the CFD simulation of bubble columns. Inthis SCMF model, the structure parameters are calculated viathe DBS model, and then the ratio of drag coefficient to bub-ble diameter can be obtained and fed into the two-fluid ormulti-fluid model frameworks. The simulation with a constantbubble diameter (5 mm) provides better predictions on thelocal and total gas holdup as well as the normalized axial liq-uid velocity, compared with a modified Tomiyama drag modelconsidering the bubble swarm effect (Tomiyama, 1998).

To evaluate and calculate the contribution of each bub-ble size group in PBM to the drag force per unit volume, aspecific formula is employed when PBM is solved by the dis-crete method (Bhole et al., 2008; Wang et al., 2006), with thehelp of the Tomiyama drag correlations (Tomiyama, 1998) oncalculating the drag coefficient for every bubble size group.Unlike the common formulas with adopting the Sauter meandiameter in three-dimensional (3-D) transient simulations,the PBM-customized model deals with the interphase dragforce directly based on the bubble size distribution for eachcomputational cell. The latter is considered to be closer toactual conditions since it takes the effect of each bubble sizegroup on the calculating drag force into account.

In this work, we present 3-D gas–liquid flow simulations

in a cylindrical bubble column operated in the heterogeneous

90 chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102

regime via a CFD-PBM model. The aim of this work is toillustrate the effects of different drag models, including theTomiyama, PBM-customized and DBS-based drag models, onthe multiphase flow behaviors in bubble columns. Meanwhile,the influence of a correction factor proposed by Wang et al.(2006) that accounts for the wake acceleration effect of largebubbles is thoroughly evaluated. Herein, we point out that theapplication of the DBS-based drag correlation is tentativelyextended to a non-uniform bubble size distribution condition.Subsequently, an effective drag model is chosen on the basisof its consistency with experimental data of Hills (1974), andits applicability at different superficial gas velocities and twotypes of gas distributors is investigated. A particular atten-tion should be paid that this study does not intend to describeexactly the complex coalescence and break-up phenomenaoccurring in bubble columns but rather shows the combinedeffects of the application of PBM and different drag forcemodels on hydrodynamics. In addition, the RNG k-� modelrecommended by Laborde-Boutet et al. (2009) is applied for abetter estimation of break-up rates during the implementa-tion of bubble-population balance, and only the drag force isconsidered in this work.

2. Model descriptions

2.1. Main governing equations of the Euler–Eulertwo-fluid model

For the gas–liquid flow system, the liquid is treated as a con-tinuous phase and the gas as a dispersed secondary phase.The main continuity and momentum conservation equationsfor the liquid and gas phases are given as follows:

∇ · (�i˛iUi) = 0 (1)

∂(˛i�iUi)∂t

+ ∇ · (˛i�iUiUi) = −˛i · ∇p + ∇(˛ii) + Fi,j + ˛i�ig (2)

where ˛i is the volume fraction of the ith phase (i = g for thegas phase and l for the liquid phase). The sum of the volumefractions of both phases is 1. The terms i and Fi,j repre-sent viscous stress tensor and interface momentum exchangeterm, respectively. The RNG turbulence model recommendedby Laborde-Boutet et al. (2009) is applied, and the k and ε foreach phase are computed by the following equations:

∂(˛i�iki)∂t

+ ∇ · (˛i�iUiki) = ∇ ·(˛i�i�k

∇ki)

+ ˛iGk,i

− ˛i�iεi + ˛i�i˘k,i (3)

∂(˛i�iεi)∂t

+ ∇ · (˛i�iUiεi) = ∇ ·(˛i�i�ε

∇εi)

+ ˛iεiki

(C1Gk,i − C2�iεi)

+ ˛i�i˘ε,i (4)

where k,i and ε,I are defined as follows:

˘k,i = 34˛jCd|ui,j| (ki,j − 2ki + ui,j · udr) (5)

˘ε,i = C3εiki˘k,i (6)

where �k, C1, C2, and C3 are constants.

2.2. PBM

Since the bubble breakup and coalescence exist in bubblecolumns within the heterogeneous regime, these phenomenashould be considered in the model. The usual approach isto use population balance models, which describe the vari-ation in a given population property over space and time ina velocity field. In bubble column modeling, the applicationof population balance models is to determine the bubble sizedistribution over space and time, and how this distributiondevelops due to the breakup and coalescence processes. Thegeneral form of the PBM equation for the gas–liquid bubblyflow can be expressed as follows:

∂n(v, t)∂t︸ ︷︷ ︸I

+ ∇ · [Ubn(v, t)]︸ ︷︷ ︸II

= 12

∫ v

0

n(v − v′, t)n(v′, t)a(v − v′, v′)dv′

︸ ︷︷ ︸III

−∫ ∞

0

n(v, t)n(v′, t)a(v, v′)dv′

︸ ︷︷ ︸IV

+

∫ ∞

v

ˇ(v, v′)b(v′)n(v′, t)dv′

︸ ︷︷ ︸V

− b(v)n(v, t)︸ ︷︷ ︸VI

(7)

n(v, t) =M∑k=1

Nk(t)ı(v − xk) (8)

Ni(t) =∫ vi+1

vi

n(v, t)dv (9)

where the bracketed terms represent time variation (I), con-vection (II), birth due to coalescence (III), death due toaggregation (IV), birth due to breakage (V), and death due tobreakage (VI), respectively. The population balance equation(PBE) can be solved by different methods, such as the discretemethod, the standard method of moments (SMM), the quadra-ture method of moments (QMOM), etc. The discrete methoddeveloped by Ramkrishna (2000) is applied in this work. Itis based on the continuous particle size distribution with aset of discrete size classes and each class is represented bya pivot size xi, showing the outstanding characteristics onrobust numerics and directly giving the particle size distribu-tion (PSD). Eq. (7) is integrated over each size interval [vi, vi+1],resulting in the following expression:

∂Ni(t)∂t︸ ︷︷ ︸I’

+ ∇ · [UbNi(t)]︸ ︷︷ ︸II’

= 12

∫ vi+1

vi

dv

∫ v

0

n(v − v′, t)n(v′, t)a(v − v′, v′)dv′

︸ ︷︷ ︸III’

−∫ vi+1

vi

n(v, t)dv

∫ ∞

0

n(v′, t)a(v, v′)dv′

︸ ︷︷ ︸IV’

+

∫ vi+1

vi

dv

∫ ∞

v

ˇ(v, v′)b(v′)n(v′, t)dv′ −∫ vi+1

vi

b(v)n(v, t)dv

︸ ︷︷ ︸V’

︸ ︷︷ ︸VI’

(10)

chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102 91

obo(tf

�

�

s

︸

ω

H(r

a

wi

geir1

2

Ia

Jiang et al. (2015). Then, the drag correlation can be expressedas:

The population in a representative volume xi has a fractionf bubbles born in the size range (xi, xi+1) or (xi−1, xi). For bub-les born in the size range (xi, xi+1), bubbles with a percentagef �1(v, xi) are assigned to xi, and for those born in the range

xi−1, xi), bubbles with a percentage of �2(v, xi+1) are assignedo xi+1. The values of �1(v, xi) and �2(v, xi) are given by theollowing equations.

1(v, xi)xi + �2(v, xi+1)xi+1 = v (11)

1(v, xi) + �2(v, xi+1) = 1 (12)

The final discrete PBM after all terms in Eq. (10) are recon-tructed is expressed as:

dNi(t)dt

︷︷ ︸I”

+ ∇ · [UbNi(t)]︸ ︷︷ ︸II”

=j≥k∑

gi−1≤(gj+gk)≤gi+1

(1 − 1

2ıj,k

)ωi,jkaj,kNj(t)Nk(t)

︸ ︷︷ ︸III”

− Ni(t)

M∑k=1

aj,kNk(t)︸ ︷︷ ︸IV”

+M∑k=i

i,kb(gk)Nk(t)︸ ︷︷ ︸V”

− b(gi)Ni(t)︸ ︷︷ ︸VI”

(13)

i,jk ={

(xi+1 − v)/(xi+1 − xi), xi < v ≤ xi+1

(v − xi−1)/(xi − xi−1), xi−1 < v ≤ xi

v = vj + vk (14)

i,k =∫ xi+1

xi

xi+1 − v

xi+1 − xiˇ(v, xk)dv +

∫ xi

xi−1

v − xi−1

xi − xi−1ˇ(v, xk)dv (15)

ere, the breakup rate b(v) proposed by Luo and Svendsen1996) and the aggregation rate ai,j of Luo (1993) are used,espectively. The formulas are described briefly as follows:

b(v) = 0.9238(1 − ˛g)n(ε

d2

)1/3

∫ 1

�min

(1 + �)2

�11/3exp

{−12[f 2/3 + (1 − f )2/3 − 1]��−1ε−2/3d−5/3�−11/3

}d�

(16)

i,j = 1.43�4

(di2 + dj

2)(di3/2 + dj

3/2)1/2ε exp{−tdrainage/tcontact}

(17)

here f is the volume fraction of one daughter bubble, and �

s the ratio of eddy size to parent bubble.The bubbles with different sizes are classified into 20

roups as recommended by Sanyal et al. (2005) and Wangt al. (2006) which is enough to capture significant featuresnduced by the discrete method, when using the geomet-ic method (vi+1 = vir) with the smallest bubble volume (vi =.0 × 10−10 m3) and a factor (r = 1.7).

.3. Drag model

n multiphase flows the correlations of interfacial forcesre developed to study the interaction between different

phases. The interaction between the liquid and gas phasesis dominated by the drag force and it is determined by solv-ing the momentum equation (Krishna and Van Baten, 2003;Laborde-Boutet et al., 2009; Pfleger and Becker, 2001; VanBaten and Krishna, 2001). In many studies only the drag forcewas considered among various interfacial forces to simulatehydrodynamic properties in bubble columns (Chen et al., 2005;Gupta and Roy, 2013; Olmos et al., 2001, 2003; Zhang et al.,2006). The lift force is neglected due to the uncertainty inassigning values of the lift coefficients for the small and largebubbles (Olmos et al., 2003). The added mass force is alsoneglected based on the fact that the concept of added massis not applicable for the large bubbles without existing closedwakes (Krishna et al., 1999). Therefore, only the drag force istaken into account in this work.

2.3.1. The DBS model-based drag correlation.On the basis of previous work by Chen et al. (2009) and Yanget al. (2007, 2010, 2011) employing the Energy-MinimizationMulti-Scale (EMMS) method to understand the complex natureof gas–liquid two-phase flow, a simplified correlation is pro-posed to calculate the ratio of effective drag coefficient tobubble diameter for the gas phase CD/db as the followingequation:

CDdb

= f (Ug) ={

422.5 − 5335Ug + 21640.5Ug2, Ug ≤ 0.128

139.3 − 795Ug + 1500.3Ug2, Ug > 0.128(18)

FD = 34˛g�lf (Ug)(ug − ul)|ug − ul| (19)

As can be seen, the value of CD/db will be a constant actuallywhen Ug is determined, so FD in Eq. (20) can be termed as theaverage drag force.

2.3.2. The Tomiyama drag model concerning liquidfractionSimple but reliable constitutive equations for a drag coefficientCD of single bubbles with different diameters and the gravityacceleration in a wide range of fluid properties, were devel-oped by Tomiyama (1998) based on a balance of forces actingon a single bubble in infinite stagnant liquid and availableempirical correlations of terminal rising velocities of singlebubbles. The expression of CD is given by:

CD0 = max{

min[

16Re

(1 + 0.15Re0.687),48Re

],

83

Eo

Eo + 4

}(20)

where CD0 denotes the drag coefficient of a single bubble in aninfinite medium, and it needs to be modified when consideringthe effect of bubble swarms with a correction term:

CD = CD0(1 − ˛g)p (21)

where p is considered to be 1 in this work, as recommended by

FD = 3CD4d

˛g(1 − ˛g)�l(ug − ul)|ug − ul| (22)

92 chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102

Table 1 – Formulas of modified drag models.

Modified model Correlation of drag force

Model T-k

FD = 0.75 max(1.0, 50.0˛gfb,large) max{

min[

16Re

(1 + 0.15Re0.687)m,48Re

],

83(Eo/(Eo + 4))

}˛g(1 − ˛g)�l

(ug − ul)

d|ug − ul|

(28)

Model S-k

FD = 0.75 max(1.0, 50.0˛gfb,large)∑M

i=1fi˛g�l max

{min

[16Rei

(1 + 0.15Re0.687),48Rei

],

83

Eo

Eo + 4

}(ug − ul)

dbi|ug − ul|,

(29)

, 0˛gf

173,451, 278,488, 342,230 and 414,498) are applied to discretize

Model D-k FD = 0.75 max(1.0, 50

2.3.3. PBM-customized drag modelWhen PBM is solved by the discrete method, the drag forcecan be calculated using a specific formula proposed by Wangand Wang (2007), Wang et al. (2006) and Bhole et al. (2008) inwhich the contribution of different bubble size groups to thetotal drag coefficient in each computational cell is computed.The drag coefficient for the bubbles in the size group i can beexpressed as the following:

CDi = max{

min[

16Rei

(1 + 0.15Rei0.687),

48Rei

],

83

Eo

Eo + 4

}(23)

where the Tomiyama drag coefficient correlation (Tomiyama,1998) corresponding to pure systems developed by Wang andWang (2007), Wang et al. (2006) and Xing et al. (2013) is used forobtaining the drag coefficient CDi in contaminated systems.

After taking into account the effect of each discrete bubblesize on the interphase force, the drag force can be expressedas follows:

FD = 34

∑M

i=1fi˛g�l

CDidbi

(ug − ul)|ug − ul| (24)

where fi is the volume fraction of group i in the gas holdup.

2.3.4. Correction factorkb,large in Eq. (26) proposed by Wang et al. (2006) accounts for thewake acceleration effect of large bubbles, and it is correlatedto the fraction of large bubbles and the local gas holdup

kb,large = max(1.0, 50.0˛gfb,large) (25)where, fb,large is the fraction of large bubbles.

CD,correction = kb,largeCD (26)

The value of dc that distinguishes between small and largerbubbles is estimated by the correlation of dc = 4(�/g �)1/2

(Ishii and Zuber, 1979). For the air-water system, this criticalvalue is about 10 mm.

In the next discussion, in a simplified way we refer tothe DBS-based model as model D, Tomiyama single bubblemodel as model T, and PBM-customized drag model as modelS. When the correction factor kb,large is considered, modelD-k, model T-k, and Model S-k are specified, respectively.Moreover, formulas of three modified models are shown inTable 1.

2.4. Numerical detail

The bubble column of Hills (1974) simulated in this study is

0.138 m in inner diameter and 1.38 m in height. Initially, thecolumn is filled with water up to a static height of 0.9 m.

b,large)f (Ug)˛g�l(ug − ul)|ug − ul| (30)

Pressure-outlet boundary condition is applied for the topsurface of the column. Along the walls, no-slip boundary con-ditions are adopted. In particular, the multiple orifice spargerwith 61 holes of 2 mm diameter is modeled instead of theexperimental sparger with 61 holes of 0.4 mm diameter, inorder to avoid a huge number of meshes and to minimizemesh skewness. Correspondingly, due to the fact that theopening area increases from 0.0512% to 1.28%, the gas veloc-ity at holes is actually lower than that of real cases, whichmay lead to the underestimation of absolute axial liquid veloc-ity. The arrangement of holes in the simulation is exactlythe same with the experiments. Fig. 1 shows the geometryof the cylindrical bubble column and the meshes generatedby a commercial grid-generation tool, GAMBIT 2.3.16 (AnsysInc., US). Initial grid sensitivity studies are carried out. Hybrid(tetrahedral and hexahedral) grids with different sizes (i.e.

Fig. 1 – Arrangement of holes in perforated sparger andmeshes generated.

chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102 93

Fig. 2 – The evolution of instant gas holdup distribution simulated with six different drag force models: (a) model T; (b)model T-k; (c) model S; (d) model S-k; (e) model D; (f) model D-k.

ttoisugcbttImTPuel

he column geometry, and the grid number of 342,230 is foundo be sufficient for achieving a periodic 3D flow field. For casesf the triple-ring gas distributor, a special treatment of the

nlet condition brings the grid number to 235,621. Moreover,erious numerical problems may arise due to the large vol-me fraction gradient at the gas–liquid interface and the highas volume fraction in the freeboard. These problems can beircumvented by a variation of drag coefficient for the free-oard. It is set as 0.05 when the liquid volume fraction is lesshan 0.55 according to Zhang et al. (2009). The governing equa-ions are solved by the commercial CFD code FLUENT (Ansysnc., US) with double precision mode and the various drag

odels are implemented by user-defined functions (UDFs).he pressure-velocity coupling is handled through the SIM-LE procedure. The hybrid-upwind discretization scheme issed for the convective terms. The bubble diameter at the

xit of the sparging pipes was assumed based on the calcu-ation from Khurana and Kumar (1969) correlation for bubble

formation, VB = (2�)0.25(15�LQ/2�Lg)0.25. The time step is set tobe 0.0005 s in the beginning. When the physical time reaches10 s, the time step is changed to 0.001 s till the flow timereaches 60 s. Then the time step is fixed to be 0.005 s. Thepseudo-steady state is considered to achieve in 40 s, whichcan be concluded from the evolution of the instantaneous gasholdup distribution and the total gas holdup versus the flowtime with different drag laws, as illustrated in Figs. 2 and 3.The time-averaged radial gas holdup at the cross section ofH = 0.6 m are obtained at the flow time of 80 s. It should benoted that all the results presented in this study are obtainedat unsteady-state cases over a period of time long enoughso that all variables have a cyclic behavior. In general, everycase takes about 8 h of CPU (16 cores) time to compute a 1-sphysical process. Simulations of hydrodynamics in both thehomogeneous and heterogeneous regimes are performed and

the numerical data are compared with the experimental dataof Hills (1974).

94 chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102

Fig. 3 – The total gas holdup with different drag forcemodel versus flow time.

Fig. 4 – The effect of different drag models ontime-averaged redial gas holdup distribution andcomparison of simulation with the experiment of Hills

Fig. 6 – Gas holdup distribution at the axial cross section

(1974) (Ug = 0.095 m/s, OA: opening area).

3. Results and discussion

3.1. The effect of different drag models

Gas holdup profile is one of the most important propertiesin bubble column reactors. The spatial variation of gas holdupleads to the pressure variation, which results in liquid recircu-lation in the bubble columns. This liquid recirculation governsthe mixing rate, heat and mass transfer. As the feasibility topredict radial gas holdup profile in the bubble column reac-tors, liquid mixing, heat and mass transfer becomes higher,the scale-up and optimization design of bubble columns aremore reliable (Wu et al., 2001).

Figs. 4 and 5 compare the distribution of gas holdup alongradial direction and the total gas holdup from six different drag

Fig. 5 – Simulated total gas holdup with different dragmodels and the correction factor (Ug = 0.95 m/s).

with different models for t = 80 s (Ug = 0.95 m/s).

force models with experimental data (Hills, 1974) under thecondition of the superficial gas velocity Ug = 0.095 m/s. Fig. 6depicts the contour plot of gas holdup in the axial cross sec-tion of the whole bubble column at t = 80 s. It can be seen thatunder such a condition the flow patterns simulated insidethe bubble column are heterogeneous with all cases of var-ious models. As reported by Olmos et al. (2003), Wang et al.(2006) and Yang et al. (2011) the utilization of a correction fac-tor for the standard drag coefficient CD0 has great influenceon simulations. Meanwhile, the strong effect of the correc-tion factor on the total gas holdup is evident (cf. Fig. 5 andTable 2). All of three unmodified models underestimate boththe radial profile of local gas holdup and the overall gas holdupremarkably. Encouragingly, an obvious improvement can beobserved after the coupling of the correction factor (i.e. kb,large).

Compared with the simulated results of Wang et al. (2006),where the gas holdup decreases due to considering kb,large, our

Table 2 – Comparison of experimental data andcalculated total gas holdup obtained using differentmodels.

Total gas holdup Relative error

Experimental 0.19 –Model T 0.09182 51.57Model T-k 0.1391 26.79Model S 0.1058 44.32Model S-k 0.1839 3.21Model D 0.1255 33.95Model D-k 0.2083 9.63

Relative error =∣∣ Expt value−Predicted value

Expt value

∣∣ × 100.

chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102 95

Fig. 7 – The effect of correction factor on the volume-averaged bubble size distribution and comparison of distributions withd

prgmatebfhtwobswier

ifferent models.

redicted data seem to be on the contrary. In fact, the cor-ection factor not only helps to increase the local and totalas holdups, but also produces larger bubbles (cf. Fig. 7). Itay result from two aspects: (1) 3-D transient simulations

re somewhat different with two-dimensional (2-D) computa-ions (Ekambara et al., 2005; Mudde and Van Den Akker, 2001),specially the 2-D axisymmetric steady-state approach usedy Wang et al. (2006), (2) the breakup and coalescence kernelunctions used by Wang et al. (2003, 2006) are more compre-ensive than those in this work. A more systematic study onhe effect of kb,large on the hydrodynamics in 3-D simulationsith other breakup and coalescence models will be done inur group in the next work. In general, model S-k has muchetter simulation performance than other models in spite oflight deviation in the column center and the regions near thealls. A large disagreement can be found between the numer-

cal results obtained from model T and the experimental data

ven if kb,large is considered. There is a joint effect of the cor-ection factor and the bubble size groups when calculating CD,

which is of importance and necessity for satisfactory predic-tions. The simulated radial gas holdup profiles from modelD and D-k are more uniform than that from other models.This can be attributed to the fact that CD/db in these two mod-els remains constant in the entire flow field except for thefreeboard, even though the lift force effect is neglected in thecomputations. It should be pointed out that the discrepancybetween the simulated results from model D or D-k and theexperimental data is still distinct.

In a given system, the bubble coalescence and breakuprates are mainly affected by the local gas holdup, the inter-phase force and the turbulent energy dissipation rate. Asdiscussed above, a prominent advantage of the CFD-PBM cou-pled model is to display the bubble size distribution directly.The comparison of bubble size distribution between the orig-inal drag models and the corrected models is illustrated inFig. 7. The fraction of bubble class with large sizes (db > 10 mm)

remarkably increases and the bubble size distribution has anobvious tail after kb,large is applied. Large bubbles have strong

96 chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102

Fig. 8 – Comparison of the normalized axial liquid velocity

Fig. 9 – Gas holdup distribution at the axial cross sectionwith different Ug.

using different drag model.

wake effects, and swarms of large bubbles result in an increasein the drag force, which has been indicated in most pub-lished studies (Rampure et al., 2007; Shen and Finch, 1996;Sokolichin et al., 2004). As a result, the turbulent energy dis-sipation rate (ε) will increase, which influences the balance ofbubble coalescence and breakup dramatically (Laborde-Boutetet al., 2009). It is worth noting that the distribution shifts fromdisorder to “bimodal” or “trimodal” size distribution when tak-ing the correction factor into account, corresponding to thealteration of total gas holdup. In addition, we can observe thatthe Tomiyama single bubble model always generates a nar-rower bubble size range and smaller bubble sizes no matterwhether kb,large is integrated into the drag force correlation ornot (i.e. model T and model T-k). Besides, both model S-k andmodel D-k produce a larger percentage of large bubbles thatare helpful to maintain high overall gas holdup. It is of inter-est to note that the large bubbles have higher rise velocitiesthan small bubbles. Thus the residence time of large bubblesdecrease, leading to the decrease in the rate of increasing gasholdup.

Considering the obvious difference in the inlet porositybetween the generated grids structure and the experimen-tal geometry, we argue that only qualitative comparison ofthe normalized axial liquid velocity obtained from differentmodels is performed. The radial distribution of normalizedaxial liquid velocity magnitude is presented in Fig. 8, in whichthe distribution generated by three unmodified models arenot shown due to large deviations from experimental datain terms of the gas holdup. Obviously, all these three mod-els integrating the correction factor enable to match the maincharacteristics of experimental flow field (i.e. the parabolicdistribution). In particular, model S-k provides better simu-lation performance than the other ones in a specific region(r/R = 0.2–0.5). In other words, the good agreement betweenthe experimental data and the simulation results from modelS-k exists not only in the gas holdup but also in the normal-ized axial liquid velocity. This implies that it is reasonable tochoose model S-k as the optimal model for predicting hydro-dynamic characteristics of the bubble column under differentconditions in the following study.

3.2. The applicability of model S-k at differentsuperficial gas velocities

The superficial gas velocity (Ug) is one of the key operating

variables that affect the hydrodynamics in bubble columns. Asmentioned above, the gas holdup profile varies significantly

with the change of gas velocity. The steeper the gas holdupprofile is, the faster the liquid recirculation rate is. Accordingly,this will improve the liquid mixing, heat and mass transferrate (Wu et al., 2001). To investigate the applicability of modelS-k at various superficial gas velocities, two more set of sim-ulations were performed in which Ug is 0.038 and 0.127 m/s,respectively. The simulation is carried out at these three vari-ous superficial gas velocities, covering both the homogeneousand heterogeneous regimes according to the experiments car-ried out by Yang et al. (2011).

Fig. 9 indicates that the streamlines in the gas–liquidtwo-phase flow become more tortuous with the increase ofsuperficial gas velocity, implying the important role of gasbubbles on the flow field. At the lowest Ug studied, noticeablebubbles only exists inside the central region of the column.The gas bubbles gather together in the zone near the sparger,while they spread in the zone close to the top liquid surface.It is clear that the partial aeration occurs in the whole bub-ble column under this low superficial gas velocity condition.As Ug increases up to 0.095 m/s, small bubbles are trapped bythe moving liquid vortices at the top part of the bubble col-umn and then they move downwards along the sidewalls. Ascan be seen, the total aeration occurs at the top part of thebubble column. At the maximum Ug applied, the downwardmovement of the liquid draws bubbles to a region close to thegas inlet, resulting in the total aeration in a large part of thebubble column.

Fig. 10 shows the radial distribution of gas holdup at thesethree different superficial gas velocities. Fig. 11 compares thesimulated total gas holdup with the experimental data. Thesimulation accuracy is acceptable merely for relatively highgas velocities (Ug = 0.095 and 0.127 m/s), yet the dampeningtendency of increasing gas holdup with the increase of super-ficial gas velocity can be successfully captured, showing aplateau in the curve, which signifies the transition to heteroge-neous regime. Additionally, the relative deviation of the totalgas holdup between the simulated and experimental results

is almost 40% at the lowest Ug studied, which is much largercompared with 10% at another two velocities. As one can

chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102 97

Fig. 10 – Influence of the superficial gas velocity on the radial gas holdup distribution.

oaelitm

Uvwbbwfft2

Fw

btain from Figs. 4–7, the correction factor (kb,large) modifiesnd enhances effects of large bubbles on gas holdup. How-ver, the fraction of large bubbles in homogeneous regime isimited, so model S-k should be further adjusted when appliedn such a regime. Meanwhile, for low superficial gas veloci-ies one may need to consider other forces for the interphase

omentum transfer in addition to the drag force.Bubble size distribution in the entire flow field at different

g values is displayed in Fig. 12. At the lowest superficial gaselocity studied, the bubble size distribution is fairly uniformith small sizes. As the superficial gas velocity increases, theubble size distribution becomes less uniform, and smallerubbles exist in this wider bubble size distribution comparedith the case of the lowest superficial gas velocity, resulting

rom the coalescence and breakup of bubbles. This is beneficialor the liquid mixing and the mass transfer process due to

he more chaotic and irregular oscillation (Buwa and Ranade,002).

ig. 11 – Comparison of the simulated total gas holdupith experiment of Hills (1974).

Fig. 13 shows the radial distribution of normalized axial liq-uid velocity at three different superficial gas velocities. Thenormalized axial liquid velocity is underestimated at mostpositions along the radial direction for the low superficialgas velocity (Ug = 0.038 m/s) but it is well predicted for themedium velocity (Ug = 0.095 m/s). Since the experimental dataof axial liquid velocity for higher superficial gas velocitiesare not available, only the simulation results are plotted inthis case (e.g. Ug = 0.127 m/s). Considering both the simula-tion results of gas holdup (Figs. 9–11) and liquid axial velocity(Fig. 13), it seems that the modified drag force can satisfythe numerical simulations in bubble columns for mediumand high superficial gas velocities. However, for low super-ficial gas velocities one may need to consider other forcesfor the interphase momentum transfer except for the dragforce.

A good agreement is obtained between the experimen-tal and simulation data performed at only higher superficialvelocities, which confirms the correct choice of the numer-ical model. At lower values of Ug, the agreement betweenthe experimental and simulation data becomes poorer, whichprobably results from the inaccurate breakup and coalescencemodels, drag force laws or the range of initial bubble size dis-tribution used in these simulations. It is worth noting thatthe movement of gas bubbles in the liquid phase with avery low Morton number (Mo = 2.5 × 10−11) is rather difficultto simulate, and most simulations cannot be found for thelow-Reynolds/low-Eötvös regime (i.e. spherical and ellipsoidalbubbles at low values of Ug) (Roghair et al., 2011). Further-more, considering industrial bubble columns are frequentlyoperated in the turbulent flow regime (at high Ug values),we suggest that model S-k can be employed to generate rea-

sonable predictions in this regime, at least for superficial gasvelocity around 0.1 m/s.

98 chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102

Fig. 12 – Comparison of the simulated bubble size distribution with variable U .

3.3. Effect of the gas distributor configuration

After understanding the sensitivity of superficial gas velocity,the effect of gas distributor geometry on the hydrodynamic

characteristics of the bubble column is investigated. Two gas

Fig. 13 – The effect of Ug on simulated

g

distributors with different geometries are employed, as shownin Fig. 14. One is the multi-orifice gas distributor used above(61 holes of � 2 mm uniformly spaced, opening area: 1.28%)and the other is a triple-ring gas distributor (3 rings of 2 mm

wide, opening area: 11.0%) proposed based on the previous

normalized axial liquid velocity.

chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102 99

Fig. 14 – Configurations of gas distributors used in theb

wgot

cgagttidvgtoitstmtF

Fd

Fig. 16 – Comparison of the time-averaged radial gasholdup profiles with the experimental data with differentgas distributors.

Fig. 17 – Comparison of the total gas holdup using differentgas distributors with experimental data.

ubble column (a: gas distributor A; b: gas distributor B).

ork of Ranade and Tayalia (2001). Herein, the multi-orificeas distributor is referred as gas distributor A and the otherne as gas distributor B. The simulation is carried out underhe condition of the superficial gas velocity Ug = 0.095 m/s.

Fig. 15 depicts the contour plot of gas holdup in the axialross section of the bubble column at t = 80 s. It can be seen thatas distributor B produces a more dramatic local and total aer-tion leading to a higher total gas holdup as compared withas distributor A. Fig. 17 indicates a higher total gas holdup forhe case of the use of gad distributor B. This can be attributedo the fact that gas distributor B holds a 10-fold larger open-ng area than gas distributor A. Figs. 16 and 18 show the radialistribution of local gas holdup and normalized axial liquidelocity at H = 0.6 m, respectively. The obtained profiles whenas distributor B is applied are relatively flat compared withhese when gas distributor A is applied, which results from thebvious difference in opening area and the disappearance of

nterval between gas sparging pipes in the triple-ring gas dis-ributor. In addition, the gas distributor configuration shows aignificant impact on the axial liquid velocity since the nega-ive value of this parameter near wall region becomes positive

ostly even at H = 0.6 m, giving rise to poor mixing charac-eristics and particle suspension inside bubble column. Both

igs. 16 and 17 reveal that the application of gas distributor A

ig. 15 – Predicted instantaneous gas holdup contours withifferent gas distributor configurations at t = 80 s.

results in lower local and total gas holdup. This is due to thefact that the use of sparging pipes gas distributor gives riseto poor gas phase dispersion (cf. Fig. 15) and bubble plumeinduces large circulation vortices inside the bubble column.As illustrated by Li et al. (2009), the existence of such vorticescauses poor entrainment of the bubbles around the vortices.In the case of using gas distributor B, the gas phase is spargedthrough this gas distributor and then enters the bubble col-umn effectively, generating more uniform vortices with themedium sizes. Such vortices are beneficial for the interactionbetween the gas bubbles and the liquid phase. As a result, thegas phase can be highly dispersed in the liquid phase, whichis helpful to give rise to both local and total gas holdup.

The population balance model is employed in the present

study to account for the change in bubble diameter since the

Fig. 18 – Radial profiles of the simulated time-averaged andthe experimental normalized axial liquid velocity withdifferent gas distributor configurations.

100 chemical engineering research and design 1 1 2 ( 2 0 1 6 ) 88–102

Fig. 19 – Comparison of the bubble size distribution withdifferent gas distributors.

Chen, P., 2004. Modeling the Fluid Dynamics of Bubble Column

bubbles experience the break-up and coalescence once thegas phase is sparged to form dispersed bubbles in the bub-ble column. It can be expected that the configuration of gasdistributors also affects the bubble sizes. The bubble size dis-tribution in the bubble column for these two gas distributorsare shown in Fig. 19. It can be seen that the fraction of smallbubble class for gas distributor B decreases significantly andbubbles with the diameter larger than 40 mm almost disap-pear. Most of small bubbles derive from the interaction amongliquid, gas and spargers and they exist in the region near thegas distributor (Polli et al., 2002) whose configuration is of vitalimportance in the bubble formation. Thus, the reduction ofsmall bubble class fraction may be attributed to the differencebetween two applied gas distributors. On the other hand, witha given gas input, the larger opening area means a reduction inthe gas flow rate in each individual sparging pipe or ring. Con-sequently, this provides more sufficient time for small bubblesin the vicinity of the gas distributor to oscillate and coalesce.

As can be seen from profiles of gas holdup and liquidvelocity at a high H/D value (i.e. H = 0.6), the effect of the con-figuration gas distributors is not confined to the vicinity of thegas distributor. This observation is in contradiction with otherresearches (Li et al., 2009; Thorat et al., 1998). The deviationmay be attributed to two factors including the huge discrep-ancy in opening area and the great alternation in the spargergeometry (Li et al., 2009; Ranade and Tayalia, 2001).

4. Conclusions

The Euler–Euler two-fluid model coupled with the PBM wasemployed to investigate the gas–liquid flow in a cylindricalbubble column with 138 mm in diameter. The work focusedon simulating the effects of various drag models as well asa correction factor accounting for wake acceleration of large

bubbles. Main conclusions drawn from this work are as follow:

(1) Even though the deviation of simulated results reduceafter kb,large was implemented into drag models, thenumerical results show that only the modified PBM-customized drag model (i.e. model S-k) could producereasonable predictions with the experiment data in termsof radial distribution of local gas holdup and overall gasholdup. Both the Tomiyama and the DBS-based drag mod-els cannot accurately predict local gas hold-up profile andtotal gas holdup where they either overestimate or under-estimate the experimentally reported data. Meanwhile, itshould be mentioned that the parabolic distribution of nor-malized axial liquid velocity could be obtained by all threemodified drag model.

(2) Comprehensive comparisons among the simulationresults on gas holdup, liquid phase velocity, and bubblesize distribution indicate that model S-k was recom-mended to be applied in churn-turbulent instead of bubblyflow regime. Poor agreement is obtained between theexperimental data and the simulations performed at lowersuperficial gas velocity, indicating that the modified dragmodel need to be adjusted further, especially the correc-tion factor.

(3) Simulated results of two types of gas distributors areanalyzed briefly, where the triple-ring gas distributorgenerates much more obvious total aeration correspond-ing to the higher total gas holdup compared with themulti-orifice gas distributor. The radial profiles of localgas holdup and normalized axial liquid velocity becomeapparently flatter obtained by former gas distributor asa result of larger opening area and significant differencein configuration between multi-orifice and triple-ring,which deteriorate mixing and mass transfer rate in flowfield.

Though PBM-customized drag model and the correctionfactor in the CFD-PBM model still require further validation,the simulation in this study shows their potential and advan-tage in 3-D transient simulation of the flow field in bubblecolumn.

Acknowledgments

The authors thank the State Key Laboratory of HeavyOil Processing (China University of Petroleum) (No.SKLOP201403001), National Ministry of Science and Tech-nology of China (No. 2012CB21500402), the National NaturalScience Foundation of China (No. U1462101) and the Center forHigh Performance Computing, Shanghai Jiao Tong Universityfor supporting this work. The authors also thank Prof. WangT.F. from Tsinghua University for his revision and suggestion.

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