Computers & Fluids 105 (2014) 91–100

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Study on CFD–PBM turbulence closures based on k–e and Reynoldsstress models for heterogeneous bubble column flows

http://dx.doi.org/10.1016/j.compfluid.2014.09.0230045-7930/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +49 (89) 289 13232; fax: +49 (89) 289 13513.E-mail address: olaf.hinrichsen@ch.tum.de (O. Hinrichsen).

Yefei Liu, Olaf Hinrichsen ⇑Catalysis Research Center and Chemistry Department, Technische Universität München, Garching b. München D-85748, Germany

a r t i c l e i n f o

Article history:Received 10 February 2014Received in revised form 6 September 2014Accepted 8 September 2014Available online 16 September 2014

Keywords:Bubble columnk–e modelRSMBubble-induced turbulenceCFD–PBM

a b s t r a c t

Gas–liquid heterogeneous flows in two cylindrical bubble columns are simulated using the computa-tional fluid dynamics–population balance model (CFD–PBM) implemented in the open source CFD pack-age OpenFOAM. The liquid phase turbulence is described by the k–e model and the Reynolds stress model(RSM). Simulation results are compared with experimental data from the literature. For the bubble col-umn operated at 0.10 m/s, minor difference is found in the predicted profiles when using 10 and 20 bub-ble classes. With the Rampure drag coefficient, Tomiyama lift coefficient and bubble-induced turbulence(BIT), the gas holdup is well predicted by both k–e model and RSM. For the bubble column operated at0.12 m/s, good agreement with experimental data is obtained when the k–e BIT model works with theTsuchiya drag coefficient and Tomiyama lift coefficient. The RSM with BIT also gives reasonable predic-tion when using the combination of Tsuchiya drag coefficient and Tomiyama lift coefficient.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Bubble columns are of considerable industrial importance dueto their wide applications in chemical, biochemical and petro-chemical industries. Numerous advantages of bubble column reac-tors are recognized such as excellent heat/mass transfercharacteristics, simple construction, no moving parts and lowoperating cost [1]. Most of bubble columns are operated under tur-bulent flow conditions. Turbulent fluid dynamics is physicallyrelated to gas dispersion, bubble breakup/coalescence and inter-phase transfer phenomena. The deep knowledge of turbulence inbubble columns is crucial to successful reactor design and scale-up.

In recent years computational fluid dynamics (CFD) hasemerged as an important tool to resolve the multiphase physicsin bubbly turbulent flows. The multiphase turbulence could bedescribed by the fully-resolved direct numerical simulation(DNS) methods [2]. Due to massive computational demand, theapplication of DNS-based methods is only restricted to very fewbubbles in gas–liquid systems. The Euler–Lagrange method alsohas limited applications in simulating the gas–liquid flows. Alter-natively, the Euler–Euler two-fluid model is widely employed tosimulate the gas–liquid turbulent flows with high gas fractions.As a result of the averaging procedure, the Reynolds stress termsin the two-fluid model should be closed by the appropriate

turbulence model. The gas–liquid two-fluid modeling approachstill remains some open questions due to the uncertainty regardingthe phase interaction terms, turbulence closure schemes, andmultiple bubble sizes [3].

Most of the two-fluid simulations were carried out using singlemean bubble size in bubble columns [4–7]. This assumption is usu-ally reasonable in the homogeneous flow regime. However, in thehighly turbulent heterogeneous flows, the knowledge of localbubble size distribution is very essential since a wide spectrumof bubble size is formed due to bubble breakup and coalescence.Many attempts have been made by coupling computational fluiddynamics with population balance model (CFD–PBM) to simulatethe gas–liquid flows [8–12]. In the CFD–PBM method the turbu-lence closure scheme not only determines the Reynolds stressterms but also governs the solution of the population balanceequations. Thus, it is of prime importance to examine theturbulence models used in the CFD–PBM method.

The two equation k–e turbulence models are widely applied insimulating bubbly turbulent flows [13–18]. The k–e models havemathematical simplicity and need low computational demand.However, they have the shortcoming from the isotropic eddy-viscosity assumption. In order to handle the anisotropic turbulentflows, the Reynolds stress model (RSM) can be coupled with theEuler–Euler multiphase algorithm [19]. The directional effect ofthe Reynolds stress field is represented with the transport equationof each Reynolds stress component. The Euler–Euler large eddysimulation (LES) method is often employed to track more

92 Y. Liu, O. Hinrichsen / Computers & Fluids 105 (2014) 91–100

turbulence details in the gas–liquid flows [20–22]. Comparing withthe RSM, the two-phase LES method requires much larger compu-tational effort because the finer grids should be used for resolvingthe large eddy structures. As a result, this LES form of turbulencemodel has not been widely applied for simulating large-scale flowreactors.

Comparative studies of turbulence models were performed forbubble column flows by various researchers. Zhang et al. [4]compared the different model constants of the sub-grid scale LESmodel, and three bubble-induced k–e models were also investi-gated. Tabib et al. [6] studied three different turbulence models(standard k–e, RSM and LES) used for the liquid phase. Ekambaraand Dhotre [7] assessed the performance and applicability of thestandard k–e, RNG k–e, k–x, RSM and LES models. Laborde-Boutetet al. [23] investigated three formulations of the k–e model(standard, RNG, realizable) combined with three different modali-ties to account for gas-phase effects. However, the bubble sizedistribution was not considered in the above work. The roles of dif-ferent turbulence models in the CFD–PBM method are still not clar-ified, especially for the heterogeneous flow regime in the bubblecolumns. Furthermore, the combined effects of interfacial forcemodels and bubble-induced turbulence models should be furtherinvestigated.

Most of previous bubble column simulations were conductedwith different CFD codes and the test cases were also of greatdifference. As a result, the distinct choices of solution algorithms,discretization schemes and grid arrangements make it difficult toclarify the intrinsic difference among various turbulence models.Jakobsen et al. [24] suggested a solution to this issue through a uni-fied code available for all research groups. Nowadays the opensource CFD package OpenFOAM (Open Field Operation And Manip-ulation) gains some success in the bubble column simulations[10,18,25]. The OpenFOAM package offers the possibility to haveinsight into the source codes and hence it is of great convenienceto implement new physical models.

In this work the population balance equation (PBE) is imple-mented into OpenFOAM and coupled with a two-fluid modelsolver. The k–e model and Reynolds stress model with bubble-induced turbulence models are also implemented to account forthe liquid phase turbulence. Two bubble columns operated at highgas inlet velocities are simulated. Numerical results are obtainedfor gas holdup, axial liquid velocity, bubble mean diameter andturbulence fields. Simulations of the k–e and Reynolds stress mod-els are compared with the experimental data in the literature.

2. Two-fluid model equations

Numerical simulations are performed with the Euler–Euler two-fluid model. For the bubble population balance, the bubble size dis-tribution is divided into a number of bubble classes. All the bubblesare assumed to travel at the same velocity Ug. This simplificationhas been employed with success by some researchers [9,10,12].Theoretically, the multi-fluid model is more accurate and it allowsdifferent bubble classes to move at different velocities. However,the momentum equation for each bubble class should be solved.Since the multi-fluid model is computationally expensive, thetwo-fluid model is used in this work. The continuity and momen-tum equations of the gas and liquid phases are written as

@ðakqkÞ@t

þr�ðakqkUkÞ¼0 ð1Þ

@ðakqkUkÞ@t

þr�ðakqkUkUkÞ¼�akrpþr�ðakskÞþakqkgþMk ð2Þ

where k refers to the phase (l for liquid and g for gas), U representsthe phase velocity, a denotes the volume fraction of each phase, s is

the effective stress tensor, Mk is the interfacial momentum transferterm due to various interphase forces.

In this study the drag and lift forces are considered, while theother forces are neglected since they have little effect on the flows[4]. The drag force resists the bubble motion in the surroundingliquid. The interfacial transfer term due to the drag force is definedas

MD;g ¼ �MD;l ¼34agalql

CD

dBjUg � UljðUl � UgÞ ð3Þ

where dB is the mean bubble size and CD is the drag coefficientwhich depends strongly on the flow regime and the liquid property.The drag coefficient for the single bubble proposed by Tsuchiyaet al. [26] is used in this work:

CD ¼max24Re

1þ 0:15Re0:687� �

;83

EoEoþ 4

� �ð4Þ

where the bubble Reynolds number Re and Eötvös number Eo aredefined as

Re ¼ qldBjUg � Uljll

; Eo ¼gðql � qgÞd

2B

r

In the heterogeneous flow, the bubble swarms are formed and theyexperience different drag forces from the single bubble. The dragcoefficient for bubble swarm is also used following the work ofRampure et al. [27] by correcting the drag coefficient of the singlebubble as

CD ¼ ð1� agÞ2 max24Re

1þ 0:15Re0:687� �

;83

EoEoþ 4

� �ð5Þ

In addition to the drag force, the bubble experiences a lift forceperpendicular to its relative motion. The lift force plays an impor-tant role in the radial gas holdup distribution. The momentumtransfer due to the lift force is calculated as

ML;g ¼ �ML;l ¼ agCLqlðUl � UgÞ � ðr � UlÞ ð6Þ

where CL is the lift coefficient. For the single bubble, the lift coeffi-cient varies with bubble size and shape. Small bubbles tend to movetowards the wall and large bubbles move towards the centre of thebubble column. Tomiyama [28] proposed a lift coefficient to capturethis phenomenon:

CL ¼min½0:288 tanhð0:121ReÞ; f ðEodÞ� Eod < 4f ðEodÞ 4 6 Eod 6 10:7�0:29 10:7 < Eod

8><>: ð7Þ

with

f ðEodÞ ¼ 0:00105Eo3d � 0:0159Eo2

d � 0:0204Eod þ 0:474

Eod ¼gðql � qgÞd

2H

rdH ¼ dBð1þ 0:163Eo0:757Þ1=3

The above lift coefficient is justified for the single bubble, but forbubble swarm the uncertainty still remains. Very few studies arefound on the lift force of the bubble swarms. Behzadi et al. [17] pro-posed a lift coefficient as a function of gas holdup:

CL ¼ 6:51� 10�4a�1:2g ð8Þ

However, the applicability of this lift coefficient should be fur-ther confirmed.

Y. Liu, O. Hinrichsen / Computers & Fluids 105 (2014) 91–100 93

3. Population balance model

Some approaches have been developed for solving the popula-tion balance equations, e.g. the method of classes [29], Monte Carlomethod [30], the parallel parent and daughter classes method [31],and the direct quadrature method of moments (DQMOM) [32]. TheDQMOM has proved an efficient technique for solving the popula-tion balance equation. Selma et al. [33] compared the DQMOMwith the class method. Both methods can give good agreementwith experimental data. In this work the class method is used,since the bubble size distribution can be directly defined. The bub-ble classes are represented through a finite number of pivotal gridxi, and the coalescence and breakup are transformed into the birthand death rates for each bubble class. The population balanceequation for the i-th class is written as

@ðagf iÞ@t

þr � ðagUgf iÞ ¼ Si ð9Þ

where fi represents the fraction of bubble class i occupied in the gasholdup, and Si is the source term accounting for bubble coalescenceand breakup:

Si ¼XjPk

j;kxi�16ðxjþxk Þ6xiþ1

1� 12

djk

� �gcðxj; xkÞ

agf j

xj

agf k

xkxi

� ag f i

XM

k¼1

cðxj; xkÞagf k

xkþXM

k¼i

ci;kbðxkÞagf k

xkxi � bðxiÞag f i ð10Þ

where

g ¼xiþ1�vxiþ1�xi

xi 6 v 6 xiþ1

v�xi�1xi�xi�1

xi�1 6 v 6 xi

8<:

ci;k ¼Z xiþ1

xi

xiþ1 � vxiþ1 � xi

bðv; xkÞdv þZ xi

xi�1

v � xi�1

xi � xi�1bðv; xkÞdv

where c(xj, xk) is the coalescence frequency, b(xi) is the breakup fre-quency of the bubble class i, and b(v, xk) is the daughter size distri-bution. The Sauter mean diameter d32 is calculated to represent thebubble size dB:

d32 ¼P

i f iPið f i=diÞ

ð11Þ

The coalescence frequency c(di, dj) is usually calculated as theproduct of the collision frequency -c(di, dj) and the coalescenceefficiency Pc(di, dj):

cðdi; djÞ ¼ -cðdi;djÞPcðdi; djÞ ð12Þ

where di and dj are the diameter of bubble group i and j, respec-tively. The bubble coalescence may occur due to a variety of mech-anisms in turbulent flows, e.g. turbulent fluctuation, global velocitygradient, eddy capture, buoyancy and wake effect [34]. Various coa-lescence models were proposed in the literature. Since we focus onassessing different turbulence models, the coalescence only due toturbulent fluctuation is considered in this work. The uncertaintyin other mechanisms would make the problem more complicated.The collision frequency resulting from turbulent fluctuation isexpressed as [35]

-cðdi; djÞ ¼p4

ffiffiffi2p

e1=3l ðdi þ djÞ2 d2=3

i þ d2=3j

� �1=2ð13Þ

Wu et al. [36] and Wang et al. [37] proposed some modifiedforms of the collision frequency. However, the uncertainty stillexists in their expressions. Accordingly, the coalescence efficiencyof the bubble group i and j is calculated as

Pcðdi;djÞ ¼ exp �C0:75 1þ n2

ij

� �1þ n3

ij

� �h i1=2

ðqg=ql þ 0:5Þð1þ nijÞ3We1=2

ij

8><>:

9>=>; ð14Þ

where C = 1.0, nij = di/dj and Weij is the Weber number.Many previous models considered the turbulent collision

between eddies and bubbles was the dominant reason for bubblebreakup. The breakup model by Luo and Svendsen [38] is used inthis work. It is based on the concept that the bubble breaks upwhen it collides with the turbulent eddy with sufficient energy.One advantage is that this model does not include empiricalparameters. Another advantage is that the daughter size distribu-tion is derived directly. This model has been widely used in theprevious work [8,10–12]. The breakup frequency of a bubble withvolume vi that breaks into two daughter bubbles with volume vj

and (vi � vj) is given as

Xðv j;v iÞ ¼ 0:923ð1� agÞel

d2i

!1=3

�Z 1

nmin

ð1þ nÞ2

n11=3 exp � 12cf r2qle

2=3l d5=3

i n11=3

!dn ð15Þ

where nmin = kmin/di, kmin = (11.4–31.4)g, g = (ll/ql)0.75/el0.25. cf is

calculated as

cf ¼v j

v i

� �2=3

þ 1� v j

v i

� �2=3

� 1 ð16Þ

The total breakup frequency of the bubble with volume vi is calcu-lated as

bðv iÞ ¼12

Z v i

0Xðv; v iÞdv ð17Þ

4. Turbulence closure models

The two-fluid model requires the closure relations for theReynolds stresses in the phase momentum equations. For theliquid phase, the effective stress tensor reads

sl ¼ ll;lam rUl þ ðrUlÞTh i

� 23ll;lamðr � UlÞIþ sl;turb ð18Þ

where ll,lam is the laminar viscosity and sl,turb is the Reynolds stresstensor which can be modeled either by solving the Reynolds stresstransport equation or by using the classical Boussinesq relation:

sl;turb ¼ �qlRl

¼ ll;turb rUl þ rUlð ÞTh i

� 23ll;turbðr � UlÞI�

23qlklI ð19Þ

where Rl is the Reynolds stress and ll,turb is the turbulent viscosity.For the gas phase, the effective stress tensor is calculated as

sg ¼ lg;lamþlg;turb

� �rUgþ rUg

Th i

�23ðlg;lamþlg;turbÞðr�UgÞI ð20Þ

where lg,turb is the turbulent viscosity of gas phase and calculatedfrom that of the liquid phase [39]: lg,turb = ll,turb qg/ql.

4.1. Two equation k–e model

In the majority of publications on simulations of turbulent bub-bly flows, the k–e model has been employed to calculate the turbu-lent viscosity. The transport equations of the kl and el are written as

@ðalqlklÞ@t

þr�ðalqlUlklÞ¼r� al ll;lamþll;turb

rk

� �rkl

� �þalðGk;l�qlelÞþSk;l ð21Þ

@ðalqlelÞ@t

þr�ðalqlUlelÞ¼r� al ll;lamþll;turb

re

� �rel

� �þal

el

klðCe1Gk;l�Ce2qlelÞþSe;l ð22Þ

0

20

40

60

80

100

Spec

ific

bre

akup

fre

quen

cy, 1

/s

Bubble diameter, m

εl = 0.25 m2/s3

εl = 0.50 m2/s3

εl = 1.00 m2/s3

(a)

0.000 0.005 0.010 0.015 0.020

0.000 0.005 0.010 0.015 0.0200

10

20

30

40

50

εl=0.5 m2/s3, λ

min=11.4η

εl=0.5 m2/s3, λ

min=31.4ηSp

ecif

ic b

reak

up f

requ

ency

, 1/s

Bubble diameter, m

εl=1.0 m2/s3, λ

min=11.4η

εl=1.0 m2/s3, λ

min=31.4η

(c)

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

dB = 0.006 m

Daughtersizedistribution

Breakup volume fraction

l = 0.25 m2/s3

l = 0.50 m2/s3

l = 1.00 m2/s3

(b)

Fig. 1. (a) Specific breakup frequency; (b) daughter bubble size distribution; (c)effect of the minimum eddy size.

Initialization

Solve gas phase

continuity equation

Solve population balance

equations

Calculate the Sauter mean

diameter d32

Update drag and lift

coefficients

Construct the velocity

equations

PISO correction loop

Solve turbulence model

equations

Post processing

Nex

t tim

e st

ep

Fig. 2. Solution algorithm implemented in the CFD–PBM coupled solver.

Table 1Numerical schemes used in the test cases.

Term Discretization scheme

ow/ot Euler

rw cellMDLimited Gauss linear 1

rp Gauss linear

r � (UkUk) Gauss limitedLinearV 1

r � (Ukak) Gauss limitedLinear01 1

r � (Ulkl) Gauss limitedLinear 1

r � (Ulel) Gauss limitedLinear 1

r � (UlRl) Gauss limitedLinear 1

r � sk Gauss linear

r2w Gauss linear uncorrected

r\w uncorrected

(w)f linear

Note: w is a dependent variable, r2 is the Laplacian term, r\ is the surface normalgradient term, (w)f is the face interpolation operator.

94 Y. Liu, O. Hinrichsen / Computers & Fluids 105 (2014) 91–100

where Gk,l is the production term of turbulent kinetic energy andcalculated as

Gk;l ¼ rUl : sl;turb ð23Þ

The shear-induced turbulent viscosity ll,SI is calculated as

ll;SI ¼ Clqlk2

l

elð24Þ

In this work the standard k–e model constants are used asCl = 0.09, Ce1 = 1.44, Ce2 = 1.92, rk = 1.0, re = 1.3.

The source terms Sk,l and Se,l represent the influence of thebubble phase on the liquid phase turbulence, i.e. bubble-inducedturbulence. In this work, these source terms are set to zero. The

bubble-induced turbulence is considered by adding an extrabubble-induced contribution to the shear-induced turbulentviscosity [40]:

ll;turb ¼ ll;SI þ ll;BI ð25Þ

The bubble-induced turbulent viscosity ll,BI is calculated as

ll;BI ¼ qlCl;BdBag jUr j ð26Þ

where Ur = Ug–Ul and the model constant Cl,B is set to 0.6.

0.0

0.1

0.2

0.3

0.4

0.5

Gas

hol

dup

Radial position, m

Exp. (Rampure et al. 2007) 10 bubble classes 20 bubble classes

(a)

k-ε ; CD: Rampure; CL: Tomiyama; BIT: C.

0.00 0.05 0.10 0.15 0.20

0.00 0.05 0.10 0.15 0.20-0.8

-0.4

0.0

0.4

0.8

1.2

k-ε ; CD: Rampure; CL: Tomiyama; BIT: C.

Axi

al li

quid

vel

ocity

, m/s

Radial position, m

Exp. (Sanyal et al. 1999) 10 bubble classes 20 bubble classes

(b)

Fig. 3. Effect of the number of bubble classes on the gas holdup and axial liquidvelocity predicted by the k–e model. BIT: bubble-induced turbulence; C.:considered.

Y. Liu, O. Hinrichsen / Computers & Fluids 105 (2014) 91–100 95

4.2. Reynolds stress model

The transport equation of the Reynolds stress Rl in the liquidphase is written as

@ðalqlRlÞ@t

þr � alqlUlRlð Þ ¼ r � al qlCskl

elRl

� �rRl

� �þ alqlPl

þ alqlUl �23alqlelIþ SR;l ð27Þ

The production term is calculated as

Pl ¼ �Rl � rUl þ ðrUlÞTh i

ð28Þ

The pressure–strain term is modeled as

Ul ¼ �C1el

ktRl �

23

ktI� �

� C2 Pl �23

trðPlÞI� �

ð29Þ

The source term SR,l accounts for the bubble-induced turbu-lence, which is also set to zero in this work. To consider the bub-ble-induced turbulence, the bubble-induced turbulent viscosityll,BI calculated in Eq. (26) is added to the laminar viscosity. Thetotal turbulent kinetic energy is calculated from the trace of thetotal Reynolds stress:

kt ¼12

trðRl þ RBIÞ ð30Þ

The bubble-induced Reynolds stress is calculated following Arnoldet al. [41]:

RBI ¼ agCvm a Ur � Urð Þ þ bðUr � UrÞI½ � ð31Þ

where Cvm = 1.2, a = 0.1 and b = 0.3 in this work.To close the Reynolds stress model, the transport equation of

the dissipation rate el should be solved and it is expressed as

@ðalqlelÞ@t

þr � ðalqlUlelÞ ¼ r � al qlCekl

elRl

� �rel

� �

þ alel

klðCe1Gk;l � Ce2qlelÞ þ Se;l ð32Þ

where the source term Se,l is set to zero in this work. The model con-stants are listed as C1 = 1.8, C2 = 0.6, Cs = 0.22, Ce = 0.15, Ce1 = 1.44,Ce2 = 1.92.

5. Numerical solution

5.1. Model implementation

OpenFOAM is employed as the basic framework which is a flex-ible and efficient C++ library for manipulating scalar, vector andtensor fields [42]. In our developed solver, the discretized popula-tion balance equations are constructed using the C++ template:PrtList<fvScalarMatrix>. The partial differential equationsare discretized with the operator fvc (finite volume calculus)and fvm (finite volume method). The fvc functions calculate theexplicit terms, while the fvm functions are used to discretize the

Table 2The diameters of the 10 and 20 bubble classes.

Index 1 2 3 4

(a) 10 bubble classesDiameter (mm) 3.1 3.9 4.9 6.1

(b) 20 bubble classesIndex 1 2 3 4Diameter (mm) 2.3 2.6 3.0 3.4Index 11 12 13 14Diameter (mm) 8.8 10.1 11.6 13.2

implicit derivatives. To calculate the triple integration in thebreakup frequency term, the incomplete Gamma function is imple-mented following the work of Alopaeus et al. [43]. Fig. 1 shows thecalculated results of breakup frequency and daughter bubble sizedistribution. The dissipation rate of kinetic energy has obviousinfluence on breakup frequency and daughter size distribution.Meanwhile, the breakup frequency is not sensitive to the differentvalues of the minimum eddy size.

Since the liquid phase is not present in the whole domain of thebubble column, the discretized turbulence model equations resultin the singular system of linear algebraic equations. Following thework of Oliveira and Issa [44], the phase-intensive forms of the tur-bulence model equations are employed by dividing the originaltransport equations by the liquid volume fraction. Oliveira and Issafound the phase-intensive formulation gives stable solutions andthe predicted results have quite small difference from those pre-dicted by the original equations.

5 6 7 8 9 10

7.7 9.7 12.2 15.4 19.4 24.5

5 6 7 8 9 103.9 4.5 5.1 5.9 6.7 7.715 16 17 18 19 2015.2 17.4 19.9 22.7 26.0 29.8

0.0

0.1

0.2

0.3

0.4

0.5

RSM; CD: Rampure; CL: Tomiyama; BIT: C.

Gas

hol

dup

Radial position, m

Exp. (Rampure et al. 2007) 10 bubble classes 20 bubble classes

(a)

0.00 0.05 0.10 0.15 0.20

0.00 0.05 0.10 0.15 0.20-0.8

-0.4

0.0

0.4

0.8

1.2

RSM; CD : Rampure; CL: Tomiyama; BIT: C.

Axi

al li

quid

vel

ocity

, m/s

Radial position, m

Exp. (Sanyal et al. 1999) 10 bubble classes 20 bubble classes

(b)

Fig. 4. Effect of the number of bubble classes on the gas holdup and axial liquidvelocity predicted by the RSM.

0.0

0.1

0.2

0.3

0.4

0.5

k-ε ; CD: Rampure; BIT: C.; 10 bubble classes

Gas

hol

dup

Radial position, m

Exp. (Rampure et al. 2007)CL:TomiyamaCL: Behzadi

(a)

0.00 0.05 0.10 0.15 0.20

0.00 0.05 0.10 0.15 0.20-0.8

-0.4

0.0

0.4

0.8

1.2

k-ε ; CD: Rampure; BIT: C.; 10 bubble classes

Axi

al li

quid

vel

ocity

, m/s

Radial position, m

Exp. (Sanyal et al. 1999)CL :TomiyamaCL : Behzadi

(b)

Fig. 5. Effect of the lift coefficient on the gas holdup and axial liquid velocitypredicted by the k–e model.

96 Y. Liu, O. Hinrichsen / Computers & Fluids 105 (2014) 91–100

The linear equation systems resulting from the discretizationprocedure are solved in a segregated fashion. The pressure–veloc-ity coupling is handled using the PISO solution algorithm [45]. Theinterphase coupling terms in the momentum equations are treatedusing the semi-implicit method [46]. The pressure equation issolved and the predicted phase velocities are corrected by the pres-sure change. The solution procedure is schematized in Fig. 2.Table 1 gives the details of the discretization schemes used forthe different terms in the governing equations.

5.2. Test case descriptions

The bubble column (2.0 m height and 0.2 m i.d.) built by Ram-pure et al. [27] is simulated using the CFD–PBM method. The waterin the column has unexpanded height of 1.0 m. Air was suppliedthrough the column bottom at 0.10 m/s. The heterogeneous flowregime was formed with high gas holdup. The simulations are alsoperformed to predict the hydrodynamics in a bubble column of2.0 m height and 0.19 m diameter. The experimental data of gasholdup, liquid velocity and turbulent kinetic energy were mea-sured in this column [47–49]. The static water height was0.95 m. The heterogeneous flow regime was achieved by operatingthe column at a superficial gas velocity of 0.12 m/s.

Initially, the static water exists in the bubble columns and thegas holdup is set to be zero within the static water. The gas distrib-utor is treated as a uniform inlet with the gas volume fraction of1.0. The liquid inlet velocity is set to zero for all test cases becauseof no water supply into the bubble columns. The pressure at theinlet is specified using the zero gradient boundary condition. Atthe outlet, the pressure is specified as atmospheric pressure. The

no-slip boundary condition is applied at the wall for the velocitiesof gas and liquid phases. The wall function proposed by Launderand Spalding [50] is used to specify the turbulent quantities. Thelaw of the wall for mean velocity gives

U� ¼ 1j

lnðEy�Þ ð33Þ

with

y� ¼qlC

1=4l k1=2

P yP

llð34Þ

where j is the von Karman constant, j = 0.42, E is an empirical con-stant, E = 9. In OpenFOAM, the log-law equation is employed wheny⁄ > 11.6.

The turbulent kinetic energy is solved in the whole domainincluding the wall-adjacent cells. At the wall, the zero gradientboundary condition is used for turbulent kinetic energy. The pro-duction term of turbulent kinetic energy and its dissipation rateat the wall-adjacent cells are computed as

GP ¼llUPC1=4

l k1=2P

qljyPð35Þ

eP ¼C3=4

l k3=2P

jyPð36Þ

where yP is the distance from point P to the wall, UP is the meanvelocity at the point P, and kP is the turbulent kinetic energy atthe point P.

0.0

0.1

0.2

0.3

0.4

0.5

RSM; CD: Rampure; BIT: C.; 10 bubble classes

Gas

hol

dup

Radial position, m

Exp. (Rampure et al. 2007)CL:TomiyamaCL: Behzadi

(a)

0.00 0.05 0.10 0.15 0.20

0.00 0.05 0.10 0.15 0.20-0.8

-0.4

0.0

0.4

0.8

1.2

RSM; CD : Rampure; BIT: C.; 10 bubble classes

Axi

al li

quid

vel

ocity

, m/s

Radial position, m

Exp. (Sanyal et al. 1999)CL :TomiyamaCL : Behzadi

(b)

Fig. 6. Effect of the lift coefficient on the gas holdup and axial liquid velocitypredicted by the RSM.

0.00

0.05

0.10

0.15

0.20

0.25

CD: Rampure; CL: Tomiyama; BIT: C.

Tur

bule

nt k

inet

ic e

nerg

y, m

2 /s2

Radial position, m

CFD-PBM-k-ε model CFD-PBM-RSM

(a)

0.00 0.05 0.10 0.15 0.20

0.00 0.05 0.10 0.15 0.200.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CD : Rampure; CL : Tomiyama; BIT: C.Kin

etic

ene

rgy

diss

ipat

ion

rate

, m2 /s

3

Radial position, m

CFD-PBM- k-ε model CFD-PBM-RSM

(b)

Fig. 7. (a) Turbulent kinetic energy and (b) dissipation rate predicted by k–e modeland RSM.

Y. Liu, O. Hinrichsen / Computers & Fluids 105 (2014) 91–100 97

Simulations are conducted using two-dimensional computa-tional meshes with 10 mm cell size. The diameters of 10 and 20bubble classes are listed in Table 2. The simulations are carriedout for 60 s real time and the time-averaged results are obtainedin the last 55 s. The governing equations are solved in a transientway using adaptive time step method to improve the stability.The time step is adapted by the Courant number. The Courantnumber is defined as

Co ¼ DtjUrjDx

ð37Þ

where Dt is time step, Ur is the relative velocity through the cell, Dxis the cell size in the direction of relative velocity. The maximumCourant number is set to 0.1.

6. Results and discussion

6.1. Test case I: superficial gas velocity of 0.10 m/s

Sensitivity study on the number of bubble classes is performedthrough the 10 and 20 bubble classes for the k–e model and RSM.The bubble-induced turbulence is considered in the two turbu-lence models. Fig. 3 shows the effect of the number of bubble clas-ses on the gas holdup and axial liquid velocity when using the k–emodel. Clearly, increasing the number of bubble classes does notsignificantly increase the agreement with the experimental data.However, much more computational time is needed when increas-ing the number of bubble classes to 20. Fig. 4 shows the effect ofthe number of bubble classes on the gas holdup and axial liquid

velocity predicted by the RSM. Also, the simulation results of theRSM are not sensitive to the number of bubble classes. From Figs. 3and 4, the 10 bubble classes are sufficient to resolve the bubble sizedistribution. The profile of gas holdup is well predicted when usingthe 10 and 20 bubble classes. However, small underestimation onaxial liquid velocity is observed for the k–e model and RSM.

Fig. 5 displays the effect of the lift coefficients of the single bub-ble and the bubble swarm on the simulated results of the k–emodel. The Rampure model is adopted to account for the swarmeffect on the drag coefficient. In Fig. 5a the Tomiyama model givesbetter prediction on gas holdup than the Behzadi model, althoughit was formulated based on the data of single bubbles. The Tomiy-ama model has the lift coefficient in the range 0 < CL 6 0.288 for thesmall bubbles with diameter less than 6 mm, whereas the negativevalues in the range �0.29 < CL 6 0 are used for the large bubbleswith diameter larger than 6 mm. The negative lift coefficientmakes the bubbles move towards the center of the bubble column.However, only the positive lift coefficients are predicted by theBehzadi model. The positive coefficient leads to the bubble migra-tion towards the walls. Thus, the flat profile of gas holdup is pre-dicted by the Behzadi model. In Fig. 5b both lift coefficientmodels give the underestimation on the axial liquid velocity.

Fig. 6 shows the effect of the lift coefficient on the gas holdupand axial liquid velocity predicted by the RSM. The poor predictionon gas holdup and axial liquid velocity is also observed for theBehzadi lift coefficient. Although the Behzadi lift coefficient wasoriginally proposed for the pipe flows, its applicability is not gen-eral in simulating the bubble column flows. Since the strong liquidcirculation exists in the bubble columns, the bubble column flow isquite different from the concurrent flow in the pipe.

0.0

0.1

0.2

0.3

0.4

0.5

CFD-PBM-k-ε model

Exp. (Kumar 1994)CD:Tsuchiya; CL:Tomiyama; BIT: N.C.CD:Tsuchiya; CL:Tomiyama; BIT: C.CD:Rampure; CL:Tomiyama; BIT: C.CD:Tsuchiya; CL:Behzadi; BIT: C.

Gas

hol

dup

Dimensionless radius

(a)

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

CFD-PBM-k-ε model

Exp. (Degaleesan 1997)CD :Tsuchiya; CL:Tomiyama; BIT: N.C.CD :Tsuchiya; CL:Tomiyama; BIT: C.CD :Rampure; CL:Tomiyama; BIT: C.CD :Tsuchiya; CL:Behzadi; BIT: C.

Axi

al li

quid

vel

ocity

, m/s

Dimensionless radius

(b)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.00.00

0.04

0.08

0.12

0.16

0.20

CFD-PBM-k-ε model

Exp. (Degaleesan, 1997)CD:Tsuchiya; CL :Tomiyama; BIT: N.C.CD:Tsuchiya; CL :Tomiyama; BIT: C.CD:Rampure; CL :Tomiyama; BIT: C.CD:Tsuchiya; CL :Behzadi; BIT: C.

Tur

bule

nt k

inet

ic e

nerg

y, m

2 /s2

Dimensionless radius

(c)

Fig. 8. Effect of the different closure models with the k–e model. NC: notconsidered.

0.0

0.1

0.2

0.3

0.4

0.5

CFD-PBM-RSM

Exp. (Kumar 1994)CD :Tsuchiya; CL:Tomiyama; BIT: N.C.CD :Tsuchiya; CL:Tomiyama; BIT: C.CD :Rampure; CL:Tomiyama; BIT: C.CD :Tsuchiya; CL:Behzadi; BIT: C.

Gas

hol

dup

Dimensionless radius

(a)

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Exp. (Degaleesan 1997)CD :Tsuchiya; CL :Tomiyama; BIT: N.C.CD :Tsuchiya; CL :Tomiyama; BIT: C.CD :Rampure; CL :Tomiyama; BIT: C.CD :Tsuchiya; CL :Behzadi; BIT: C.

Axi

al li

quid

vel

ocity

, m/s

Dimensionless radius

(b) CFD-PBM-RSM

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

CFD-PBM-RSM

Exp. (Degaleesan 1997)CD :Tsuchiya; CL :Tomiyama; BIT: N.C.CD :Tsuchiya; CL :Tomiyama; BIT: C.CD :Rampure; CL :Tomiyama; BIT: C.CD :Tsuchiya; CL :Behzadi; BIT: C.

Tur

bule

nt k

inet

ic e

nerg

y, m

2 /s2

Dimensionless radius

(c)

Fig. 9. Effect of the different closure models with the RSM.

98 Y. Liu, O. Hinrichsen / Computers & Fluids 105 (2014) 91–100

In Fig. 7 the predicted profiles of turbulent kinetic energy andits dissipation rate are compared between the k–e model and theRSM. For the k–e model, the radial profile of turbulent kineticenergy exhibits the two maximums at the two sides near the walls.However, the parabolic profile is predicted by the RSM. In the RSM,the bubble-induced kinetic energy is added to the shear-inducedkinetic energy. Therefore, the large kinetic energy is predicted bythe RSM in the column center. In Fig. 7b the simulated dissipationrate increases from the center to the wall for the k–e model and theRSM. Near the walls, the smaller values are predicted by the RSMand the large difference is observed. It should be mentioned that,due to the lack of experimental data of the turbulence quantitiesin this bubble column, the applicability of the k–e model and theRSM cannot be judged in predicting the turbulence parameters.

Hence, the turbulence models should be further investigated bysimulating other test cases.

6.2. Test case II: superficial gas velocity of 0.12 m/s

To further assess the k–e and Reynolds stress models in theCFD–PBM method, the bubble column operated at 0.12 m/s is sim-ulated. Fig. 8 shows the experimental and simulated results of gasholdup, axial liquid velocity and turbulent kinetic energy. The dif-ferent models for drag and lift coefficients are used with the k–emodel. The bubble size distribution is represented by 10 bubbleclasses. For the gas holdup profile in Fig. 8a, the best predictionis obtained by the combination of the Tsuchiya drag coefficient,Tomiyama lift coefficient and bubble-induced turbulence model.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

CD:Tsuchiya; CL :Tomiyama; BIT: C.

Kin

etic

ene

rgy

diss

ipat

ion

rate

, m2 /s

3

Radial position, m

CFD-PBM- k-ε model CFD-PBM-RSM

(a)

0.00 0.05 0.10 0.15 0.20

0.00 0.05 0.10 0.15 0.200.000

0.002

0.004

0.006

0.008

0.010

CD :Tsuchiya; CL:Tomiyama; BIT: C.

Saut

er m

ean

diam

eter

, m

Radial position, m

CFD-PBM-k-ε model CFD-PBM-RSM

(b)

Fig. 10. Comparison of the dissipation rate and Sauter mean diameter predicted bythe k–e model and RSM.

Y. Liu, O. Hinrichsen / Computers & Fluids 105 (2014) 91–100 99

However, the gas holdup profile is poorly predicted by the othermodel combinations. The Behzadi lift coefficient predicts the flatprofile when it works with the Tsuchiya drag coefficient. Further-more, it is found that the bubble-induced turbulence should betaken into account to obtain good agreement with the experimen-tal data. In Fig. 8b the good prediction on axial liquid velocity isalso obtained using the Tsuchiya drag coefficient, Tomiyama liftcoefficient and bubble-induced turbulence model. The under-pre-diction is given by the Rampure drag coefficient. This is becausethe Rampure drag coefficient is smaller than the Tsuchiya dragcoefficient. In Fig. 8c the best prediction on turbulent kineticenergy is obtained when the k–e model is used with the Tsuchiyadrag coefficient, Tomiyama lift coefficient and bubble-induced tur-bulence model.

Fig. 9 gives the profiles of gas holdup, axial liquid velocity andturbulent kinetic energy predicted by the RSM with different clo-sure models. It is seen from Fig. 9a that good agreement is obtainedwhen the RSM is used with the Tsuchiya drag coefficient, Tomiyamalift coefficient and the bubble-induced turbulence model. Similar tothe k–e model, the RSM with the Behzadi lift coefficient predicts theflat profile of gas holdup in the column center. Although the Ram-pure drag coefficient was proposed to account for the swarm effect,it is still poor in predicting the gas holdup in the bubble column. Theswarm effect in the heterogeneous flow regime needs to be mod-eled based on more sound physics. In Fig. 9b and c the predictionon axial liquid velocity and turbulent kinetic energy is greatlyimproved when the RSM works with the bubble-induced turbu-lence, Tsuchiya drag coefficient and Tomiyama lift coefficient.Without the bubble-induced turbulence model, the predicted pro-file of turbulent kinetic energy is much lower than the experimental

data. This discrepancy is due to the lack of the bubble-inducedkinetic energy. For the Rampure drag coefficient, although the bub-ble-induced turbulence is considered, the large over-prediction onturbulent kinetic energy is observed in the column center. Similarto the k–e model, the RSM with the Behzadi lift coefficient givesunder-prediction on the turbulent kinetic energy.

From Figs. 8 and 9, the closure models of the drag and lift coeffi-cients and the bubble-induced turbulence play very important rolesin evaluating the performances of the k–e and the RSM. The simu-lated results are determined by the combined effect of various mod-els. In principle, the RSM is more physically sound than the k–emodelin handling the anisotropic flows. However, the uncertainty in theinterfacial force terms and the bubble-induced turbulence modelwould ruin the validity of the RSM. Some effort should be made toestablish more accurate models of the interfacial force and bubble-induced turbulence. Furthermore, when the population balance isconsidered, the bubble coalescence and breakup should be accu-rately modeled. The multi-fluid model framework would be morereasonable to calculate the velocities of different bubble classes.

Finally, the comparison of the profiles of the dissipation rateand Sauter mean diameter is made for the two turbulence models.In Fig. 10a, the larger values of the dissipation rate are predicted bythe k–e model near the walls. Since the bubble size distribution isinfluenced by the dissipation rate, the large difference in the pro-files of Sauter mean diameter is also found near the walls. FromFig. 1a, the increase in the dissipation rate promotes the bubblebreakup. Thus, the smaller values of the bubble diameter are pre-dicted by the k–e model, as shown in Fig. 10b. The profiles of thedissipation rate and Sauter mean diameter should be validated inthe future study.

7. Conclusions

The heterogeneous bubble column flows are simulated for theevaluation on the k–e and Reynolds stress models. The coupledCFD–PBM method is implemented into OpenFOAM. Simulationsare compared with the experimental data in the literature. Forthe bubble column operated at 0.10 m/s, minor difference is foundin the predicted profiles of gas holdup and axial liquid velocitywhen using 10 and 20 bubble classes. The Behzadi lift coefficientgives poor prediction when it is used with the k–e model and theRSM. With the Rampure drag coefficient, Tomiyama lift coefficientand bubble-induced turbulence, the k–e model and the RSM givesgood prediction on gas holdup.

For the bubble column operated at 0.12 m/s, the k–e modelshould work with the Tsuchiya drag coefficient and Tomiyama liftcoefficient to get good prediction. The bubble-induced turbulenceshould also be considered in the k–e model. Similarly, good agree-ment with experimental data can be obtained when the RSM isused with the Tsuchiya drag coefficient, Tomiyama lift coefficientand bubble-induced turbulence model. For the k–e model, thesmaller bubble diameters are predicted, because the larger valuesof the dissipation rate are obtained by this model.

Acknowledgements

Yefei Liu acknowledges the financial support from ChinaScholarship Council and TUM Graduate School. The authors thankthe people involved in the development of the OpenFOAM package.

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