Chemical Engineering Journal · Ankur Gupta, Shantanu Roy⇑ Department of Chemical Engineering,...

Chemical Engineering Journal 225 (2013) 818–836

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Chemical Engineering Journal

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Euler–Euler simulation of bubbly flow in a rectangular bubble column:Experimental validation with Radioactive Particle Tracking

Ankur Gupta, Shantanu Roy ⇑Department of Chemical Engineering, Indian Institute of Technology – Delhi, Hauz Khas, New Delhi 110 016, India

h i g h l i g h t s

" Exhaustive Euler–Euler CFD coupled with population balance, in air–water 2D-bubble column." Equivalence of four different population balance methods shown." Simulations show excellent agreement with Radioactive Particle Tracking (RPT) experiments." Systematic investigation of drag and lift forces.

a r t i c l e i n f o

Article history:Available online 10 November 2012

Keywords:Bubble columnRadioactive Particle TrackingEuler–Euler CFDDrag forceLift forceVirtual mass effectTurbulence modelsPopulation balance

1385-8947/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.cej.2012.11.012

⇑ Corresponding author. Tel.: +91 11 2659 6021; faE-mail address: [email protected] (S. Roy).

a b s t r a c t

This work presents an Euler–Euler (E–E) Computational Fluid Dynamics (CFD) model developed for flowinside a rectangular air–water bubble column, in which we have also implemented population balancemethod (PBM) based evolution of the various bubble size classes using the Homogeneous Discretemethod, Inhomogeneous Discrete method, Quadrature Method of Moments (QMOM) and Direct Quadra-ture Method of Moments (DQMOM) technique. The results are validated with liquid velocity field mea-surements obtained from the Radioactive Particle Tracking (RPT) technique. The study presents acomparative analysis of the effect of incorporating various interfacial closures like drag force, lift forceand virtual mass effect in the E–E CFD coupled with PBM framework. The importance of the lift forcein the transient three-dimensional simulations for predicting the time-averaged velocity profiles ishighlighted.

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1. Introduction

Bubble column reactors are industrially important reactors withapplications ranging from fermentation broths to slurry reactors[1]. Since reaction rates inside these columns are often limitedby heat and mass transfer, the knowledge of flow patterns insidethese columns becomes crucial in their sizing, troubleshootingand scale-up. To develop an insight into this arguably complexfluid flow phenomena and in particular the liquid velocity field,various experimental techniques like Particle Image Velocimetry(PIV), Laser Doppler Anemometry (LDA) and Radioactive ParticleTracking have evolved in recent decades [2–6]. Apart from experi-mental studies, numerous CFD studies are available which have at-tempted to model the flow by using various drag and lift models[7–16] within the Euler–Euler framework.

Bubble size distribution (BSD) inside conventional bubble col-umns is often highly dynamic and wide (especially in the so-called

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x: +91 11 2658 1120.

churn turbulent regime where a typical bubble size can vary from5 mm to 5 cm), an effect which is not captured in the conventionalEuler–Euler (E–E) description of flow because of its single bubblesize assumption. Further, in conventional E–E modelling, the dis-persed phase (gas) is considered to be a pseudo-continuum henceunless explicitly accounted for, would represent a single bubblesize in the closures for various forces such as drag, and would alsorepresent a single advection phase velocity. These issues can bepartially addressed by coupling E–E CFD with Population BalanceModelling (PBM) wherein an extra conservation equation (popula-tion balance equation or PBE) to capture the evolving bubble sizedistribution (BSD) is solved. PBE is an integro-differential equation[17] and therefore to solve it, special techniques like DiscreteHomogenous Method [17–19], Discrete Inhomogeneous Method[20], Quadrature Method of Moments (QMOM) [21] and DirectQuadrature Method of Moments (DQMOM) [22,23] have beendeveloped. A summary of the various works involving couplingof E–E CFD with PBM in bubble column flow modelling has beenshown in Table 1. Even though common in their philosophical ori-gin, the different PBM approaches are quite distinct from the point-

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Nomenclature

a(L1, L2) aggregation Kernel w.r.t. bubble sizes L1 and L2 (s�1)a(v1, v2) aggregation Kernel w.r.t. bubble volumes v1 and

v2 (s�1)Ba birth term due to aggregation (number of particles/

time)Ba,i birth term due to aggregation for the bin of index ‘i’

(number of particles/time)Bb birth term due to breakage (number of particles/time)Bb,i birth term due to breakage for the bin of index ‘i’ (num-

ber of particles/time)Ba;k integrated birth term due to aggregation after taking kth

momentBb;k integrated birth term due to breakage after taking kth

momentCf constant in breakage modelC1e constant in turbulence modelC2e constant in turbulence modelCl constant in turbulence modelCij convection term in turbulence modelCd drag coefficient (–)CL lift coefficient (–), constant in Luo modeld, dB diameter of the phase considered (m)di, dj diameter of the bubble represented by an index ‘i’ or ‘j’

(m)d3,2 Sauter mean diameter of the distribution (m)Da death term due to aggregation (number of particles/

time)Da,i death term due to aggregation for the bin of index ‘i’

(number of particles/time)Db death term due to breakage (number of particles/time)Db,i death term due to breakage for the bin of index ‘i’ (num-

ber of particles/time)Da;k Integrated death term due to aggregation after taking

kth momentDb;k integrated death term due to breakage after taking kth

momentDT,ij turbulent diffusion term in turbulence model (kg/ms3)DL,ij molecular diffusion term in turbulence model (kg/ms3)Eo Eotvos number (–)~Fk external force per unit volume (N/m3)fv fraction of the diameter into which the parent droplet is

breaking (–)~Flift lift force per unit volume (N/m3)~Fvm virtual mass force per unit volume (N/m3)g(L) breakage Kernel w.r.t bubble size (s�1)g(v) breakage Kernel w.r.t bubble volume (s�1)H height of column (free surface((m)k Turbulent Kinetic Energy (m2/s2), summation constant_mpk Mass transfer from phase ‘p’ to phase ‘k’ (kg/m3)

PC collision efficiency (–)I identity matrixKpk momentum exchange coefficient between phases p and

q (N s/m4)L bubble size (m)Li ith abscissa (m)

mk kth moment of bubble size distribution (mk)n(v) number size distribution for a given volume of bubble

size, location and timep pressure field, shared between all the phases (Pa)Pij stress production term in turbulence model~Rpq drag force per unit volume between phases p and q (N/

m3)Re Reynolds number (–)t time of flow (s)~uk velocity vector for kth phase(m/s)_ui bubble rise velocity for the bubble representative by in-

dex ‘i’ (m/s)�u0mi fluctuation in the mixture phase velocity in ‘i’ coordi-

nate direction (m/s)Ug gas superficial velocity (m/s)v volume of bubble~vm velocity vector for mixture phase (m/s)xk length variable ‘x’ in coordinate direction ‘k’n constant in breakage modeln(v) number size distribution for a given volume of bubble

size, location and timeW width of column (m)wi ith weight (–)Weij Weber number for coalescence of bubbles with diame-

ter di and dj

z Height above the distributor

Greek lettersak volume fraction for kth phase (–)ag volume fraction for gas phase (–)e Turbulent Kinetic Energy dissipation rate (m2/s3)qk density for kth phase (kg/m3)ql density for liquid phase (kg/m3)qg density for gas phase (kg/m3)qm density for mixture phase (kg/m3)sk stress tensor for kth phase (Pa)lk bulk viscosity for kth phase(kg/ms)lt,m turbulent viscosity for mixture (kg/ms)kk shear viscosity for kth phase (kg/ms)Ck turbulent intensity for kth phase (s�1)b(L|L0) probability distribution function of generating a daugh-

ter droplet of diameter L0 from parent droplet of diame-ter L

# a functional form which represents the number ofdaughter droplets generated upon breakage of parentdroplet

r surface tension between the two phases (N/m)rk constant in turbulence model (Eq. (8))re constant in turbulence model (Eq. (9))xc coalescence frequency (s�1)/ij pressure strain term in turbulence model (kg/ms3)eij dissipation term in turbulence model (kg/ms3)

A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) 818–836 819

of-view of their implementation, hence to perform ‘‘equivalent’’PBM implementations is a non-trivial task.

Most of these earlier studies have been done in modelling andvalidation in cylindrical bubble columns operating under churn-turbulent conditions. This poses some basic limitations vis-à-viscomparisons with CFD models. First, comparisons are done withradial profiles of velocity (and holdup), which are obtained by firsttime-averaging and then azimuthally averaging experimental data

(in case azimuthally resolved data is obtained; otherwise the datais usually obtained only at various radial location). This completelyaverages out any azimuthal dependence in the flow, which isclearly visible in any visual observation of a transparent cylindricalbubble column, wherein one sees spiral and helical structures ofgas rising around the central axis. Second, at high velocities(churn-turbulent conditions), there are multiple bubble plumesrising from the distributor which all merge and create a rather

Table 1Summary of studies done on bubble columns flow modelling using PBM methods to incorporate coalescence/breakage.

Reference Geometry Interfacial closurea PBMb Remarks

Wanget al.[24]

Cylindrical D + L + VM + WL + TD H-D (i) Various coalescence and breakage closures were compared

(ii) Good agreement of BSD was obtained for both heterogeneous and homogeneous regimes(iii) Holdup was also compared with the experimental data

Franket al.[25]

Cylindrical D + L + TD I-D (i) Velocity and hold up data was compared with experimental data, good agreement wasobtained

(ii) More detailed investigation into coalescence and breakage closures was suggested

Zhu et al.[26]

Cylindrical D + VM LSM (i) 2-D axisymmetric simulation, velocity and holdup data compared, good agreement obtained

Cheunget al.[27]

Cylindrical D + L + WL + TD ABND (i) Upward bubbly flow was studied

(ii) Void fraction, Sauter mean diameter, gas velocity, liquid velocity and interfacial areaconcentration were compared, good agreement was obtained

Bholeet al.[28]

Cylindrical D + L I-D (i) Axial liquid velocity, Radial Sauter Mean Diameter and Turbulent K.E. were compared, goodagreement was obtained

(ii) The results improved upon addition of PBM at the operated velocitiesBannari

et al.[29]

Rectangular D + L + VM + TD H-D (i) Velocity and hold up data was compared at a single level for different H/W ratios for twointerfacial closures

(ii) Results suggested that PBM marginally improves the results

Selmaet al.[30]

Rectangular,agitated reactor

D + L + VM DQMOMH-D

(i) DQMOM and H-D was compared for rectangular bubble column, liquid velocity and hold updata was compared

(ii) The model was then extended to a reactor

Chen et al.[31]

Cylindrical D H-D (i) 2-D axisymmetric simulations were performed, many aggregation breakage closures werecompared through holdup, velocity profile, KE and total interfacial area

Chen et al.[32]

Cylindrical D H-D (i) 2-D axisymmetric simulations were performed, the results showed some improvement in holdup profiles because of PBM

Chen et al.[33]

Cylindrical D H-D (i) 3-D simulations were performed, follow up study of [32]

Cheunget al.[34]

Cylindrical D + L + WL + TD ABND, H-D

(i) ABND and H-D approaches were compared by studying many parameters as done in [27]

(ii) Regime transition prediction could not be predicted numerically, reason given as assumptionof spherical bubbles

Olmoset al.[35]

Cylindrical D H-D (i) PBM was implemented and results were compared for liquid velocity, overall hold up, localhold up were compared with experimental data, data was taken for two different sparger designs

Olmoset al.[36]

Cylindrical D + L H-D (i) Regime transitions were predicted for two different sparger designs, initial bubble sizedistribution was carefully calculated which was essential for good agreement

Hansen[37]

Square D + L + VM + TD IACE (i) Systematic analysis was done for various closures, drag + lift force combination was predictedto be very good for flow modelling, virtual mass had little or no effect(ii) Not so good agreement with bubble size was obtained using IACE

Diaz et al.[38]

Rectangular D + L + VM H-D (i) Sauter mean diameter, Plume oscillation period and gas holdup was compared at high gasvelocities, good agreement obtained(ii) VM was shown not to have any effect(iii) Lift force led to large errors in results

Sanyalet al.[39]

Cylindrical D H-D,QMOM

(i) Axisymmetric simulations were done and H-D and QMOM methods were compared, merits ofQMOM were shown

a D – Drag, L – Lift, VM – Virtual Mass, TD – Turbulent Dispersion, WL – Wall Lubrication.b H-D – Homogeneous Discrete, I-D – Inhomogeneous Discrete, QMOM – Quadrature Method of Moments, DQMOM – Direct Quadrature Method of Moments, LSM – Least

Square Method, ABND – Average Bubble Number Density, IACE – Interfacial Area Concentration Equation.

820 A. Gupta, S. Roy / Chemical Engineering Journal 225 (2013) 818–836

complex cocktail of bubble coalescence-and-breakup, bubble-gen-erated turbulence, and multi-scale circulation cells in the liquid, sothat the role of individual forces and effects cannot be discerned bysimply examining time-averaged velocity profiles. Time-averagedvelocity profiles would be instructive only if we were to isolate asingle kind of phenomena and keep the flow simple, so that the ef-fect of various forces and effects on the isolated phenomena may

be discerned. Finally, the use of distributors like sintered platesand perforated plates (with perforation all across the cross-sec-tion), while realistic (vis-à-vis industrial bubble columns), contrib-ute little in isolating a certain kind of governing phenomena in thebubble column (as discussed above): rather they make the flow lotmore complex by contributing a large number of interacting gas-bubble plumes to the bulk of the bubble column.

Fig. 1. Geometry modelled in present study, taken from Upadhyay [27], (a) front, top and side view of the geometry. (b) Enlarged view of the distributor with 8 apertures ofdiameter 0.8 mm in a square pitch of 6 mm. (c) Modelling the ‘‘equivalent distributor’’ as a rectangular patch of 24 mm � 12 mm.


In light of the above discussion, what may be more suitable forcomparison with CFD codes are more sanitized flow conditions inbubble columns: namely low velocity flow, well-defined gas distrib-utor which should release a single plume of bubbles, and perhaps iso-lated rise of these bubbles (free from interactions with other plumes),so that the effect of various governing forces may be discerned.

Indeed, the above list of desirables is well-realised in a suitablydesigned rectangular bubble columns (with a thin gap for gas–liquidflow like shown in Fig. 1, the so-called ‘‘two-dimensional bubble col-umn’’, even though we refrain from using this term in this papersince the flow is really completely three-dimensional) and a local-ised inlet for gas. The bubble column depicted in Fig. 1 is inspiredby the work of Pfleger et al. [2], whose work, in turn was precededby the work of Becker et al. [63]. This kind of system provides an easybenchmarking option since bubble trajectories can be discernedclearly against a background of almost completely quiescent liquid.Several researchers have used rectangular bubble columns for vali-dating CFD codes and models ([2,8,10–12]). Thus, in this work, wehave restricted our attention for model validation in high resolutionvelocity data collected in flat, rectangular bubble columns. Further,in addition to a detailed treatment of various drag, lift and virtualmass force closures, in this work we have used different populationbalance implementations, through Homogenous Discrete Method,Inhomogeneous Discrete Methiod, QMOM and DQMOM methods,and validated against high-resolution Radioactive Particle Trackingexperiments (which has never been reported in a flat rectangularbubble column in quiescent conditions, thus far). The validation isthus attempted with a perspective for distinguishing the variouscontrolling effects in terms of their prediction of the time-averagedliquid velocity profile. Thus, we have the benefits of the rectangularbubble column as listed above, and we also have complete threedimensional liquid velocity profile information in that systems. Fur-ther, the velocity profiles have been compared at three different rep-resentative levels in the column, something which has not beendone before inside a rectangular bubble column (all prior compari-sons were at a single level or element within the vessel). This waspossible because of the rich velocity profile data that has been ob-tained by Upadhyay [40] recently, using Radioactive Particle Track-ing (RPT), the details of which have been described later in this

study. Rather than directly using all interfacial closures and bubblepolydispersity effects together, we have attempted to systemati-cally analyse the relative importance of interfacial closures (i.e. drag,lift and virtual mass) and the entire need and approach towards Pop-ulation Balance Modelling (PBM).

2. Mathematical model

Simulations in this study were performed using commercialCFD software package ANSYS Fluent� 14 as the base platform. Gov-erning equations of Euler–Euler multiphase model and PBM havebeen discussed in brief in this section.

2.1. Euler–Euler multiphase model

Eulerian description of fluid flow is based on the notion of pseu-do-continuum, i.e. the approach defines a point volume fraction foreach phase which represents the probability of a particular phaseto be present at that point in multiple realizations of flow [41].Same pressure field is shared between all the phases. The forceinteraction between phases is incorporated through various effec-tive ‘‘volumetric’’ force functions, such drag force, lift force and vir-tual mass effect (defined as net force between phases per unitvolume). The conservation equations for the Euler–Euler multi-phase model are as follows:

Continuity equation (solved for each phase):

@

@tðakqkÞ þ r � ðakqk uk

�!Þ ¼ Xn

p¼1;p–k

_mpk � _mkp ð1Þ

Momentum equation (solved for each phase):

@

@takqk uk

�!� �þr � ðakqk uk

�! uk�!Þ

¼ �akrpþr:sk þ akqk g!þ Fk!þ

Xn

p¼1;p–k

_mpkvpk�!� _mkpvkp

�!þ

Xn

p¼1;p–k

Rpk�!þ Flift

�!þ Fvm��! ð2Þ

Table 3Various models for lift coefficient in literature (Eq. (6)).

Reference Expression for lift coefficient

Auton [47] CL = 0.5Legendre and

Magnaudet [48]CL ¼

1þ16Re

1þ29Re

Tomiyama et al. [49] CL ¼minð0:288 tanhðReÞ; f ðEoÞf ðEoÞ ¼ 0:00105Eo3 � 0:0159Eo2 � 0:204Eoþ 0:474

Eo ¼ gðql�qg Þd2B

r


where

sk ¼ aklk r uk�!þr uk

�!T� �

þ ak kk �23lk

� �r � ðukIÞ ð3Þ

The term~Rpk in Eq. (2) represents the drag force between the phases‘p’ and ‘k’. The mathematical description of drag force is given asfollows:

Rpk�! ¼ Kpkð up

�!� uk�!Þ ð4Þ

where

Kpk ¼34akaplk

CdRe

d2p

ð5Þ

Kpk is the drag momentum exchange coefficient and Cd is the singleparticle (or single bubble or droplet) drag coefficient between thetwo phases. Essentially, Eq. (5) ‘‘corrects for’’ the fact that in factall particles/droplets ‘‘see’’ a modified drag from the continuousphase owing to the presence of other neighbouring dispersed phaseentities.

Some of the drag coefficient models used and compared in thisstudy have been summarized in Table 2. Note that these are all sin-gle particle (bubble or droplet) drag models and are all to be mod-ified in a similar fashion, as in Eq. (5). Note also that all models fordrag coefficient listed in Table 2, when modified for presence ofother particles of bubbles in the neighbourhood, leads to a diame-ter sensitivity that varies significantly. A detailed study regardingthe same can be looked up in Tabib et al. [46].

The term~Flift in Eq. (2) represents the lift force acting on second-ary phase (k) because of primary phase (p). The mathematicaldescription of lift force is [47]:

Flift�! ¼ CLqkakð uk

�!� up�!Þ � ðr� uk

�!Þ ð6Þ

CL is known the lift coefficient between the two phases. Since the liftforce expression involves the curl of the bubble velocity, in vectorproduct with the relative velocity between phases, its action is tomove the bubble or the dispersed phase in a direction that is trans-verse to the mean flow direction of the continuous phase. The liftcoefficient models used in this study have been summarized inTable 3.

The term ~Fvm in Eq. (2) represents the virtual mass force whichincreases the inertia of the secondary phase whenever the second-ary phase accelerated with respect to the primary phase. Themathematical description of virtual mass force is as follows [64]:

Fvm��! ¼ 0:5qkak

D uk�!Dt� D up

�!Dt

!ð7Þ

2.2. Turbulence model

The turbulence was modelled for the mixture i.e. for the mix-ture of primary and secondary phase(s). Turbulence model usedin the present study was RNG k–e model [50] for most of the sim-

Table 2Various models for drag coefficient in literature (Eq. (5)).

Drag law (Reference) Expression

Schiller–Naumann [42]Cd ¼

24Re ; Re 6 1000

0:44; Re P 1000Ishii–Zuber [43] Cd ¼ 2

3 Eo12

Tomiyama [44]

Cd ¼

83

Eoð1�E2ÞE

23Eoþ16ð1�E2ÞE

43

f ðEÞ�2

E ¼ 11þ0:163Eo0:757

sin�1ffiffiffiffiffiffiffiffiffi1�E2p

�Effiffiffiffiffiffiffiffiffi1�E2p

1�E2

Zhang and Vanderheyden [45] Cd ¼ 0:44þ 24Re þ 6

1þffiffiffiffiRep

ulations because of its merit for system where Reynolds number islow compared to the conventional k–e model [51]. RNG k–e modelhas a similar mathematical description as the k–e model [52], how-ever it is perhaps not as extensive because of its assumption ofisotropy of Reynolds stresses. Of course, this assumption can be re-laxed in the RSM model. Anyhow, the overall low Reynolds numberof the flow justifies the use of the RNG k–e model. To check the ef-fect of different turbulence closures, simulations with simple k–eand RSM models (with linear pressure strain closure, see [52])were also done. The governing equations for simple k–e modeland RSM have been tabulated in Table 4. Since, RNG k–e modelhas very similar equations as the k–e model, they are not describedin Table 4 and the details can be referred to in [65].

2.3. Population Balance Modelling (PBM)

In Eqs. (1)–(7), the value of secondary phase diameter is usedonly in the expression of Kpq (i.e., CD) and CL. Therefore, as men-tioned before, in conventional Euler–Euler multiphase descriptionof flow (in which the dispersed gas phase is treated as a pseudo-continuum), only a single value of diameter can be used, byassumption, which is held constant throughout the domain.Clearly, this may seem like a gross oversimplification, at least athigher gas velocities, since the whole dynamics and liquid flowin the bubble column is driven by the circulation induced by bub-bles in their sojourn from the inlet distributor to the exit of thebubble column. As they rise, bubbles coalesce, break, re-disperseand hence undergo size and shape change owing to their traverseinto varying pressure fields. All these factors contribute to bubblesize evolution, which in turn has an important role to play in thetype and velocity pattern of liquid circulation (note that the advec-tion velocity of each bubble size, in general, should also be size-dependent). The restriction of an assumed single bubble size canbe relaxed by using population balance where we solve indirectlyfor ‘secondary phase diameter’ at every cell and at every time step.

Population balance modelling involves writing an extra conser-vation equation for the bubble size distribution that varies in timeor space. In the present context, implementing population balancewould means writing an extra conservation equation for bubblesize distribution (BSD). Population balance equation (PBE) relevantfor the present system has been described below, essentially basedon the development presented by Ramkrishna [17]:

@

@tðnðvÞÞ þ r � ð~ugnðvÞÞ ¼ Ba � Da þ Bb � Db ð8Þ

Ba ¼12

Z v

0aðv � v 0;v 0Þnðv � v 0Þnðv 0Þdv 00 ð9Þ

Da ¼Z 1

0aðv; v 0ÞnðvÞnðv 0Þdv 00 ð10Þ

Bb ¼Z 1

vtgðv 0Þbðvjv 0Þnðv 0Þdv 0 ð11Þ

Db ¼ gðvÞnðvÞ ð12Þ

Eq. (7) is the number conservation equation with right hand sidebeing the source term. The source term is obtained through combi-

Table 4Description of turbulence models.

Simple k–e model RSM model

@@t ðqmkÞ þ r � ðqm um

�!kÞ ¼ r � ðlt;mrkrkÞ þ Gk;m � qme @

@t ðqmu0m;iu0m;jÞ þ Cij ¼ �DT;ij þ DL;ij � Pij þuij � eij � Fij

@@t ðqmeÞ þr � ðqm um

�!eÞ ¼ r � ðlt;mrkreÞ þ e

k ðC1eGk;m � C2eqmeÞ Cij ¼ @@xkðqmum;ku0m;ium;jÞ

lt;m ¼ qmClk2

eDT;ij ¼ @

@xkðqmu0m;iu

0m;ju

0m;k þ pðdkju0m;i þ dkju0m;jÞ

Gk;m ¼ lt;mðrv ;m�!þ vT

;m

�!Þ : rv ;m

�!Pij ¼ qðu0m;iu0m;k

@um;j��!@xkþ u0m;ju

0m;k

@um;i��!@xk

qm ¼Pn

i¼1aiqi uij ¼ pð@u0m;i

@xjþ

@u0m;j

@xiÞ

vm�! ¼Pn

i¼1aiqi ui!

qmeij ¼ 2l

@u0m;i

@xk

@u0m;j

@xkÞ


nation of four terms (Eqs. (9)–(12)) representing birth and deathdue to coalescence (aggregation) and breakage, respectively. Sincethe general population balance equation is an integro-differentialequation, analytical solutions exist for very few special cases. There-fore, numerical methods are used to solve this equation.

The closures used for PBM in the following study is the closurefor aggregation and breakage by Luo and Svendsen [54,55] model.The coalescence model for Luo and Svendsen [53] considers onlythe coalescence induced by turbulence. Other works in literature,for instance the work of Wang et al. [56], describes alternative coa-lescence closures that may be used. The mathematical descriptionof the closures (for Eqs. (9) and (10)) is as follows:

aðdi;djÞ ¼ xcðdi;djÞPcðdi;djÞ ð13Þ

xc ¼p4

ffiffiffi2p

e13ðdi þ djÞ2ðd

23i þ d

23j Þ

12 ð14Þ

PC ¼ exp �CL

0:75ð1þ 12ijÞð1þ 13

ijÞ12

h i12

ðqg

qlþ 0:5Þð1þ 1ijÞ

3

8><>:

9>=>;We

12ij ð15Þ

where 1ij ¼didj; uij ¼ ð _ui þ _ujÞ2.

The breakage model of Luo [54] considers breakage only due toisotropic turbulence. This model has been very widely used be-cause of its ability to give the daughter distribution function aswell the breakage function simultaneously [39,56]. The mathemat-ical description of the closures (for Eqs. (11) and (12)) is as follows:

bðfv ;dÞ ¼ 0:923ð1� agÞne

d2

� �13Z 1

nmin

ð1þ nÞ2

n113

exp � 12cf rbqln

118 e2

3d53

!dn ð16Þ

gðvÞ ¼Z 0:5

0bðfv jdÞdfv ð17Þ

bðfv jdÞ ¼2bðfv ; dÞR 1

0 bðfv jdÞdfvð18Þ

There are various methods that are available for solving PBE such asthe Homogeneous Discrete method, Inhomogeneous Discrete meth-od, Quadrature Method of Moments (QMOM) and Direct Quadra-ture Method of Moments (DQMOM). Generally speaking, thesemethods can be classified into two types: ones in which we directlydiscretize the governing equations and solve directly for the num-ber density function, and the other category being in which we inte-grate out the internal coordinate and solve for the moments usingsome numerical approximation. Naturally, the computational ex-pense for the methods based on moments will be less as comparedto direct solving of the distribution because of the less number ofthe variables involved, since we solve transport equation for mo-ments only and not the entire distributions. The idea for findingmoment based methods stems from the fact that for all practicalpurposes, we do not need the entire distribution and are generallymore focussed on obtaining information about valuable variableslike Sauter-mean diameter, which can be obtained from the lowerorder moments.

It is important to note that Homogeneous Discrete method andQuadrature Method of Moments assume that the secondary phasemoves with the same velocity even for different bubble sizes,whereas this crucial assumption is relaxed in Inhomogeneous Dis-crete method and Direct Quadrature Method of Moments.

A pertinent question that arises is whether these models areequivalent, and if they are, does having a difference advectionvelocity for different bubble sizes make any significant difference?In principle, they should be equivalent but numerical implementa-tion and approximations may render them quite different in imple-mentation. While there have been few attempts to comparedifferent PBM approaches coupled with Euler–Euler CFD, thus farno published work has compared all the different approaches to-gether, and also benchmarked the results against high-resolutionexperimental data. The next section briefly describes these fourdifferent population balance modelling methods.

2.3.1. Discrete Homogenous and Discrete Inhomogeneous MethodsDiscrete method is one of the few numerical methods which

were developed to solve this equation [17–19]. In this method,the PBE is solved for specific bubble diameters (also referred toas bins, or classes) and it is assumed that bubble diameters canonly take up the values given to different bins or classes. Thismethod is generally less preferred because it is computationallyexpensive [39] and requires a prior knowledge of expected bubblesize distribution (BSD) for proper bin size selection. However, it hasan advantage that it directly gives us the number density function,which has to be otherwise built from the moments (in other meth-ods like QMOM and DQMOM).

The homogenous discrete method assumes the same velocityfor all the bins and only one dispersed phase is assumed. Thismethod can be described by the following equations, details ofwhich are presented in [17]:

@ni

@tþr � ð ug

�!niÞ ¼ Si ð19Þ

Si ¼ Ba;i � Da;i þ Bb;i � Db;i ð20Þ

Ba;i ¼XjPk

v i�16v jþvk6v i

1� 12

djk

� �hðv j þ vkÞaðv j;vkÞnjnk ð21Þ

Da;i ¼XM

j¼1

aðv i;v jÞninj ð22Þ

Bb;i ¼XM

j¼1

thijgðv iÞni ð23Þ

Db;i ¼ gðv iÞni ð24Þ

h ¼v iþ1�ðv jþvkÞ

v iþ1�v i;v i6ðv j þ vkÞ 6 v iþ1

ðv jþvkÞ�v i�1v i�v i�1

;v i�16ðv j þ vkÞ 6 v i

ð25Þ

hij ¼Z v iþ1

v i

v iþ1 � vv iþ1 � v i

bðv ;v jÞdv þZ v iþ1

v i

v � v i�1

v i � v i�1bðv ;v jÞdv ð26Þ

Table 5Description of various grids generated.

Grid name Description

Grid-1 Full domain – 4 � 20 � 75, distributor – 1 � 2Grid-2 Full domain – 5 � 20 � 120, distributor – 1 � 2Grid-3 Full domain – 9 � 36 � 218, distributor – 4 � 2Grid-4 Full domain – 20 � 13 � 60, distributor – 3 � 4


The Inhomogeneous Discrete method [20] relaxes the assumption(to some extent) of all the bins moving with same velocity byassuming multiple phases for the dispersed phase, allowing eachphase to have its own velocity. For instance, if we divide a dispersedphase into M phases and each phase has N classes, then every phase(and its N classes) will move with one velocity whereas each phasecan have its own velocity. Hence, for homogenous discrete model,M = 1.

For this method, continuity equation becomes slightly more in-volved because of the presence of mass transfer between the mul-tiple phases (within a single dispersed phase). Also, numberdensity equation will have a different dispersed phase velocitydepending upon the phase under which the bin is assigned. Sincethe equations are very similar to the homogenous method, theyare not described here (the reader is advised to refer to [20] fordetails).

2.3.2. Quadrature Method of Moments (QMOMs) and DirectQuadrature Method of Moments (DQMOMs)

QMOM solves the population balance equations by solving con-servation equations for moments (and not number density, henceno notion of bins and classes) by using the quadrature pointapproximation [21]. Generally three quadrature points (or six mo-ments) are sufficient for most flow problems of interest [21]. Sinceonly six moments are used, this method is computationally lessexpensive than the discrete method. Also it does not require a priorknowledge of expected bubble size distribution. However, becauseQMOM solves directly for moments, number distribution, ifneeded, has to be extracted from moments and is not readily avail-able as in discrete method. Also, QMOM assumed that all bubblesizes move with same velocity and hence is not able to capture seg-regation [23].

mk ¼XN

i¼1

wiLki ð27Þ

@

@tðmkÞ þ r � ð u!mkÞ ¼ Sk ð28Þ

Sk ¼ Ba;k � Da;k þ Bb;k � Db;k ð29Þ

Ba;k ¼12

XN

i¼1

wi

XN

j¼1

wjðL3i þ L3

j Þk3aðLi; LjÞ ð30Þ

Da;k ¼XN

i¼1

Lki wi

XN

j¼1

wjaðLi; LjÞ ð31Þ

Bb;k ¼XN

i¼1

wi

Z 1

oLkgðLiÞbðLjLiÞdL ð32Þ

Db;k ¼XN

i¼1

wiLki gðLiÞ ð33Þ

d3;2 ¼m3

m2ð34Þ

In Direct Quadrature Method of Moments (DQMOM), rather thansolving for moments (mk),transport equation of abscissas (Li) andweights (wi) are directly solved which allows us to specify a differ-ent velocity at each quadrature point, unlike QMOM [23]. Hence,DQMOM is able to capture the effect that different bubble sizesmove with different velocities without compromising the advanta-ges mentioned earlier for QMOM. The governing equations forDQMOM are presented below. Detailed derivation of DQMOM ispresented in [23].

@wi

@tþr � ðui

!wiÞ ¼ ai ð35Þ

@wiLi

@tþr � ðui

!wiLiÞ ¼ bi ð36Þ

where ai and bi can be solved numerically by applying Gauss-Siedelmethod to the system of equations described below:

XN

i¼1

ð1� kÞLki ai þ kLk�1

i bi ¼ Sk ð37Þ

where Sk is same as described in Eqs. (29)–(34). For tracking six mo-ments in QMOM and DQMOM, N = 3 or k varies from 0 to 5 and ivaries from 1 to 3.

3. Experimental

The experimental data for the present study has been takenfrom Upadhyay [40] where velocity and kinetic energy profileswere obtained for an air–water rectangular bubble column usingRadioactive Particle Tracking (RPT). In the Radioactive ParticleTracking technique [4,57,58] as applied to the present system, asingle, small-sized radioactive tracer was designed to be neutrallybuoyant with respect to the liquid water phase, and its motion wasscanned with an arrangement of (eight in number in this case) ofNaI/Tl scintillation detectors positioned around the rectangularbubble column (Fig. 1). The system was operated for a long time(over 8 h or so), so that over multiple realizations of flow, the tracerparticle visited every location in the column many times. The posi-tion time series was reconstructed using an established algorithm[40,57] for this purpose, followed by data processing to get theLagrangian velocity time series and from that the Eulerian velocityfield. Upadhyay et al. [59] have shown that the RPT results ob-tained in this geometry agree very well with the LDA results ob-tained in the same geometry by Pfleger et al. [2], but as a bonus,RPT results could be obtained with greater detail and at conditionsat which LDA would not provide any meaningfully accurate results.Thus, Upadhyay’s work [40] on the rectangular bubble column pro-vides a rich database of information for CFD validation, and hasbeen correspondingly used as the main database for the presentsimulation work.

4. Results and discussion

The geometry considered in present study is an air–water rect-angular bubble column of dimensions 120 � 20 � 5 cm (Fig. 1a).The gas distributor of the column consists of eight holes of0.8 mm diameter each, arranged in two centrally positioned rows,and with a pitch of 6 mm (Fig. 1b). Similar experimental setup hasalso been used in Pfleger et al. [2], Buwa and Ranade [10,11], andDiaz et al. [38]. In the grids generated to model the present geom-etry, gas distributor (inlet) was modelled as a rectangular patch ofsize 24 mm � 12 mm (Fig. 1c).

For the numerical simulations, four grid densities were gener-ated to first check for grid independence. The details of variousgrids have been described in Table 5. Numerical diffusion waschecked by using three different discretization schemes, i.e. QUICK,Second order upwind, and Third Order MUSCL. Grid independenceand numerical diffusion were tested using only drag as the interfa-cial force, with constant bubble size of 5 mm for a gas superficialvelocity of 1.33 mm/s. The results of time-averaged axial liquidvelocity profile (at a height of 13 cm from distributor) for different


grids have been compiled in Fig. 2. From Fig. 2a, it can be clearlyseen that all the grids are practically converging to the same solu-tion (in the rest of the paper, Grid-4 was used). It should be notedthat different grids have different cell aspect ratios as well, andhence the results are independent of this parameter. FromFig. 2b, it is clear that different discretization schemes are also con-verging to the same solution and hence it can be concluded thatnumerical diffusion is not playing a prominent role in these simu-lations. In the rest of the paper, QUICK scheme was used as the dis-cretization scheme because it converges fastest among the threeschemes. It may be noted that in several papers available in theopen literature related to bubble column flow simulations, the gridconvergence of the experimental results has not been rigorouslyimplemented, so that several of the trends observed have beenerroneously ascribed to ‘‘physical effects’’, when in reality theywere merely artefacts of numerical diffusion. Therefore, an a priorielimination of mesh refinement issues was essential to continuewith exploring the effects of various closures for forces and thepopulation balance.

Initial value of bubble diameter specified was 5 mm, which isthe value estimated by Buwa and Ranade [10] at the distributorthrough experiments (in conventional E–E CFD, 5 mm value has

Fig. 2. (a) Test simulations for grid independence (Table 5). (b) Test simulations for variodrag, bubble diameter = 5 mm, RNG k–e turbulence model, no PBM.)

been specified throughout the column). Also, single bubble forma-tion models like the widely used Vogelphohl–Gaddis model [60]also predicts nearly a value of 5 mm in these conditions. Theboundary conditions and important properties used for different li-quid velocity vectors at the central plane have been shown usingonly drag as the interfacial simulation cases used have been de-scribed in Table 6. Table 7 reports the conditions and parametersof a base case simulation. As the various forces and parameterswere varied in the many simulations whose results are reportedbelow, all other parameter values and models were kept fixed atthe conditions mentioned in this table.

In Fig. 3, time-averaged axial liquid velocity results obtainedafter incorporating various PBM with E–E CFD model has beenshown. The results have been compared at three different repre-sentative heights of 13 cm (z/H = 0.29), 25 cm (z/H = 0.56) and37 cm (z/H = 0.82) for gas superficial velocity of 1.33 mm/s (eventhough RPT data is available at other heights as well, in fact every-where in the column). Schiller–Naumann drag model was used asthe interfacial force and lift force was turned off for the present setof simulations. It can be seen from Fig. 3 that though velocity pro-file are slightly improved because of the addition of PBM in themodel, results at all levels have not changed significantly for all

us discretization schemes. (Gas superficial velocity = 1.33 mm/s, Schiller–Naumann

Table 6Summary of the boundary conditions and simulations setup.

Boundary conditionsInlet Velocity specified

Population balance variables specified for mono-dispersed BSD for gasTurbulent Kinetic Energy specified as unityTurbulent Kinetic Energy dissipation rate specified as unity

Outlet Pressure specified at atmospheric pressureGradient of population balance variables specified zeroTurbulent Kinetic Energy specified as unityTurbulent Kinetic Energy dissipation rate specified as zero

Others Stationary walls

Initial conditionsInitial height of water 45 cm or aspect ratio = 2.25

Time step and under relaxation factorsTime step 0.01 s, implicit scheme usedUnder relaxation factors Pressure – 0.6

Momentum – 0.4Volume fraction – 0.2Turbulence variables – 1.0Population balance variable – 0.5

Setup for homogenous discrete modelNumber of bins 9Geometric ratio exponent 1.4 (ratio of two consecutive bubble diameters)Minimum bubble diameter 1 mm

Setup for Inhomogeneous Discrete modelNumber of secondary phases 3Number of bins/phase 3Value of bin diameters Same of homogenous but first three bins in one phase,

next three bins in second phase and last three bins in third phase

Setup for QMOMNumber of moments 6

Setup for DQMOMNumber of secondary phases 3

Table 7Properties and settings of base case simulation.

Drag model Schiller–Naumann [42]

Lift model On (unless mentioned),CL = 0.5 (unless mentioned)

Virtual mass effect Off (unless mentioned)Initial diameter (for PBM cases) 5 mm (using approximation from [60])Diameter (without PBM cases) 5 mm (using approximation from [60])Turbulence model RNG k–e model [50]H/W 2.25Gas superficial velocity 1.33 mm/s


the PBM models and at all levels. Also, it can be clearly seen thatresults are not in particularly good agreement with experimentaldata. An important observation in this regard is that all these pro-files at lower levels (z = 13 cm (z/H = 0.29), z = 25 cm (z/H = 0.56))deviate from the experimental RPT data largely in the central re-gion (which, by the way, is the region of the rising bubble plume)than the off-centre region. This might be because of incorrect orincomplete description of the coupling of two phases because devi-ation is more in central region where gas-holdup is relatively high-er. In other words, solution seems to agree well with data awayfrom central region where there is practically no hold up and hencewe are solving a single phase flow equation.

Since it was observed that the deviation of the model predic-tions seem from experimental RPT data to be higher in the regionof larger holdup, the next step was to improve the coupling be-tween the two phases. A small analysis was done where lift forcewas turned on with constant lift coefficient of value +0.5 and vir-tual mass effect was also included. Then, PBM was incorporatedinto the model. Fig. 4 depicts the time-averaged profiles for allthe simulations after incorporation of the lift force and virtual masseffect. Observing the results of Drag and Drag + Virtual Massshown in Fig. 4, one can clearly see that virtual mass effect margin-ally improves the answers near the centre whereas not much effectis seen near the wall. However, only slight improvement in seenwhen Drag + Lift is compared to Drag + Lift + Virtual Mass. Hence,though virtual mass does improve the results very marginally, itsincorporation slows down the convergence; hence it will not beconsidered in the remaining simulations reported in this paper.

One also observes in Fig. 4 that after the addition of lift force tothe system, velocity profiles agree much better with the experi-mental data (both in the Drag + Lift comparison, as well as theDrag + Lift + Virtual Mass). In fact, agreement of the simulated pro-files with experimental RPT data is remarkably for z = 13 cm (z/

H = 0.29) and z = 25 cm (z/H = 0.56) whereas a minor deviation isseen at z = 37 cm (z/H = 0.82). However, even this deviation is notlarge and might also be removed if we try to model the flow moreextensively, suggestions for which have been explained later in thepaper. Be that as it may, to understand more deeply effect of liftforce, we tracked the axial liquid velocity vectors at the centralplane with respect to time and few key snapshots have been shownin Fig. 5.

In Fig. 5a, instantaneous and time-averaged snapshots of liquidvelocities have been shown for the D and D + L case of Fig. 4. Theseplots clearly suggest that when drag is used as the interfacial force,the central region is the region of action. In other words, the gassimply zips through the centre without any lateral movement. Thishowever is not the case in reality where plume oscillations are ob-served. It may also be noted that velocity vectors remain the prac-tically the same when population balance is also added to systemof equations (figure not shown) and hence insignificant effect ofaddition of population balance was observed. This is perhaps tobe expected, since the gas velocities involved are very small andhence the actual rates of bubble coalescence are not significant

Fig. 3. Simulated liquid velocity profile with different PBM methods at gas superficial velocity of 1.33 mm/; Schiller–Naumann drag, 5 mm initial bubble diameter, RNG k–eturbulence model: (a) z = 13 cm (z/H = 0.29), (b) z = 25 cm (z/H = 0.56) and (c) z = 37 cm (z/H = 0.82).


for the effect of bubble polydispersity to affect the time-averagedliquid velocity profile. However, as soon as we add lift to the sys-tem, as shown in Fig. 5b, the bubble plumes start to oscillate aboutthe centre. This phenomenon is observed because lift force acts inthe direction perpendicular to the flow and therefore takes gasaway from the centre. Since now the velocity vectors also have a

lateral component, the axial liquid velocity in the centre dropssignificantly.

Though it was observed that the simulation results are more orless insensitive to use of any of the PBM techniques (as discussedabove, owing arguably to the low gas superficial velocity), whenDrag was the only interfacial force (Fig. 3), a set of simulations

Fig. 4. Simulated liquid velocity profile with combination of drag (Schiller–Naumann) and lift (constant Auton lift) and virtual mass, but no PBM (at gas superficial velocity of1.33 mm/s, 5 mm bubble diameter, RNG k–e turbulence model): (a) z = 13 cm (z/H = 0.29), (b) z = 25 cm (z/H = 0.56) and (c) z = 37 cm (z/H = 0.82).


were performed to reflect whether there is some effect of PBMwhen it is used with Drag + Lift being the interfacial forces. The re-sults have been compiled in Fig. 6. The results clearly suggest thatas in Fig. 3, all the PBM methods are not adding significant differ-ence to the velocity profiles, though slight improvements are pres-ent. This is not very unexpected because we are in deep bubblyflow where the bubble size distribution is narrow and hence the

impact of population balance will not be very high. Therefore,the importance of these methods at higher velocities needs to berigorously tested both in cylindrical and rectangular geometry.

Both in Figs. 3 and 6, the results are almost same for all the fourPBM model implementations. This is important as this establishedthe equivalence between the four models. To compare the equiva-lence of all the PB models even further, average Sauter-mean diam-

Fig. 5. (a) Instantaneous and time-averaged snapshots of liquid velocity vectors at central plane with only Schiller–Naumann drag as interfacial force and (b) instantaneousand time-averaged snapshots of liquid velocity vectors at central plane with Schiller–Naumann drag and constant lift coefficient (Auton) lift force as interfacial force.


eter was plotted for against the height of the column. The resultshave been shown in Fig. 7 for the cases shown in Fig. 3. It is clearfrom Fig. 7 that all the four models predict similar trend of bubblesizes and hence it is safe to say that all the four models are equiv-alent, at least in for the simulations presented here. ThoughDQMOM and Inhomogeneous discrete method are arguably supe-rior as compared to other methods because they can take into ac-count that the different bubble sizes can move with differentadvective velocities, in the present case, no significant differencehas been observed between all the methods. It is notable that ina previous reference (Selma et al. [30]), the homogenous discretemethod was compared with DQMOM wherein both methods werealso shown to have similar results. It should also be noted that thetrend of increasing average Sauter-mean diameter with increase inaxial coordinate is consistent with reality because coalescence rate

is higher than breakage rate as we move away from the distributor.It should also be noted that variation of the Sauter-mean diameteris not wide, ranging from 5 mm to 1 cm and hence velocity profilesare relatively insensitive to PBM (as pointed out above). This insen-sitivity might be due to the fact that upon implementing PBM (andas compared to the case of a single prescribed bubble size), size-wise gas fraction (and corresponding interfacial area) simply redis-tributes as a result of the PBM implementation itself but overall va-lue of gas volume fraction remains the same (assuming noconsumption or production of the gas phase). Hence, the netmomentum exchange between liquid and gas phase remains prac-tically the same even after implementation of PBM. In reality, how-ever, it is entirely possible that the average size of the bubblepopulation in the column may be quite different from the bubblesize at the distributor (and the frequency with which the bubbles

Fig. 6. Simulated liquid velocity profile with many PBM methods at gas superficial velocity of 1.33 mm/with Schiller–Naumann drag, constant lift coefficient force, 5 mminitial bubble diameter and RNG k–e turbulence model: (a) z = 13 cm (z/H = 0.29), (b) z = 25 cm (z/H = 0.56) and (c) z = 37 cm (z/H = 0.82).


are formed and released at the distributor). In that case we expectthe incorporation of the population balance model to make a sig-nificant impact. However, at this point such an argument is specu-lative, particularly in the context of CFD validation with asophisticated technique such as RPT, since at least with the currentset of results one cannot make any generic statements about the

role of PBM at higher velocities. In any case, the story is quite dif-ferent for the mass transfer problem, since in that case, many smallbubbles of high interfacial area would lead to much more efficientmass transfer than that from a few large bubbles. Thus, even withthe current results, we expect the PBM to play an important role inmodelling mass transfer in bubble columns.

Fig. 7. Average Sauter mean diameter profile with respect to height for differentPBM methods. Gas superficial velocity of 1.33 mm/ with Schiller Naumann drag,5 mm initial bubble diameter and RNG k�e turbulence model.


Further simulations were performed to search for a bettercombination of drag and lift model. All other parameter valueswere kept at those mentioned in Table 7. These set of results havebeen compiled in Fig. 8. In Fig. 8, few drag models were varied tosee the effect of change in various drag models available. It wasobserved that drag models Schiller–Naumann [42], Ishii–Zuber[43], Tomiyama [44], and Zhang and Vanderheyden [45] predictedsimilar profiles (see Table 2 for expressions). This is not unex-pected because in the present system we are in extremely bubblyflow where holdup is low and hence single bubble drag formula-tion like Schiller–Naumann can be safely used in all the simula-tions. Hence, Schiller–Naumann drag is a good representative ofthe drag force and was used in the rest of the simulations. Also,the effect of addition of few lift models (see Table 3 for expres-sions) on Schiller–Naumann drag has been shown. It can be seenthat good agreement comes from both constant lift coefficient (orAuton lift [47], CL = 0.5) and Magnaudet and Legendre [48] lift. Itwas seen that Tomiyama et al. [49] lift severely over predicts theprofiles especially in the central region. The problem of over pre-diction in the central region from Tomiyama lift [49] has alsobeen reported by Hansen [37] and Zhang et al. [9] in a squarebubble column. To understand this in greater detail, liquid veloc-ity vectors were observed and it was seen that Tomiyama lift isnot able to capture the movement of the bubble plume as thebubbles generally stay at centre (clear from profiles as well asthe velocity vectors). In Magnaduet and Legendre lift [48], this ef-fect is well captured and the profiles are similar to the ones ob-tained with constant lift [47].

Finally, effect of turbulence models was investigated by usingno turbulence, simple k–e model, RNG k–e model and RSM model.All other simulation parameters were held at the values in Table 7.The results have been compiled in Fig. 9. It can be seen from Fig. 9aand b, not a very significant difference is there between all themodels. This is not very unexpected because in the present system,slip velocities are low and hence there should not be much differ-ence because of turbulence models anyway. However, in Fig. 9c,there is a distinct difference in between the models. The agreementwith the model at the third level (z/H = 0.82) is not very good, andthis is perhaps owing to the inability of these ‘‘isotropic’’ models tocapture the dynamics as one gets closer to the free surface. It issubjective to choose the best between all these models but it canbe observed that adding turbulence makes the solution stable,and hence is a numerical convenience even in these bubbly flowsimulations.

In summary, RNG k–e turbulence model + Schiller–NaumannDrag + Constant CL or Legendre and Magnaduet CL [48] seems tobe and adequate model for modelling these kind of bubbly flows.To test the ability of the model proposed in the present communi-cation, simulation was done at a higher gas superficial velocity of2.38 mm/s. In Fig. 10, the results of Schiller–Naumann drag + Con-stant CL + DQMOM and Schiller–Naumann drag + CL have been re-ported for 2.38 mm/s gas superficial velocity (initial diameter of5 mm was specified was calculated through Vogelpohl–Gaddis[60] model). It can be seen that very good agreement has been ob-tained at all the levels though this model. However, even at this gassuperficial velocity the PBM is not having any significant role toplay in the prediction of velocity profiles. As mentioned before,and thought not reported here, adding the Virtual Mass effectseems to marginally improve the results.

To further test the effect of proposed model in a new geometry,we did a similar study on a square bubble column (size15 � 15 cm) where in the velocity profile data was obtained byDeen [3] at a gas superficial velocity of 4.9 mm/s and initial aspectratio of 3.0 (please refer to Deen [3] or Hansen [37] for detailsregarding the geometry and setup). The results have been com-piled in Fig. 11. Acceptable results were obtained with the modelproposed. One can clearly see that the role of PBM (throughQMOM) is not very significant in predicting the velocity profilesand that the role of lift is extremely important. However, the high-er gas superficial geometry in this case does show a strongerdependence of the results of PBM, compared to the bubbly flowcases from the data of Upadhyay [40].

While we took efforts to implement PBM in our CFD model andattempted to test it against the RPT experimental data, from thecurrent study one cannot claim validation of the PBM models perse. Rather it clearly seems to be a rigorous and arguably excellentvalidation of the drag versus lift debate, whose effects are clearlyvisible in the aforementioned results. Directionally it seems thatthe effect of PBM needs to be studied in greater detail in both rect-angular and cylindrical bubble columns where high velocities canoperate and where we can expect very wide BSD. In such a case,it will be interesting to see whether PBM affects the profiles andif it does not, is there a need to use PBM for flow modelling? Itshould be kept in mind that in tall cylindrical columns, holdupstarts to become constant with respect to height above a certainheight [31]. In that case, it will be interesting to see whether thereis any effect of PBM in that region. Earlier works like [31] haveinvestigated the effect of PBM, but in those cases it is never clearwhether the ‘‘good agreement’’ is owing to drag, lift, VM or becausethe PBM was incorporated. It is clear from our study that at leastunder bubbly flow conditions, acceptably good agreement is ob-tained even without incorporation of PBM (by any of the ap-proaches). Though there have been studies that have used PBMin various regimes, studies with data at multiple levels and moredetailed analysis can give us a better answer about the importanceof PBM for flow modelling.

Though significant improvements were obtained after additionof lift force, there is a definite need for understanding the lift forcein greater detail, especially in multiple bubble systems. Kulkarni[61] studied the lift force experimentally in a cylindrical bubblecolumn (multiple bubbles) and concluded that lift coefficient hasa radial variation in the system. It was shown that the values of liftcoefficient varied from �0.5 to +1, a range which is not captured inany of the single bubble models. Also, lift and drag forces havebeen generally studied as two separate forces whereas new re-search seems to be progressing along the lines to study both ofthem together in a coupled fashion [62]. More research into thetransient nature of these forces is required to understand theseforces in detail.

Fig. 8. Simulated liquid velocity profiles at gas superficial velocity of 2.38 mm/s with various combinations of drag and lift force, 5 mm initial bubble diameter, RNG k–eturbulence model, with and without PBM. (a) z = 13 cm (z/H = 0.29), (b) z = 25 cm (z/H = 0.56) and (c) z = 37 cm (z/H = 0.82).


5. Summary and conclusions

This paper is based on the argument that prior efforts at validat-ing population balance and closures for other forces in CFD model-

ling of bubble columns, is often clouded by the complexity of theflow itself, such as high levels of turbulence and interaction of oscil-lating bubble plumes. Such validation has been attempted withexperimental data (sometimes even collected with invasive tech-

Fig. 9. Simulated liquid velocity profile at gas superficial velocity of 1.33 mm/s with Schiller–Naumann drag and constant lift force, 5 mm initial bubble diameter, variousturbulence models and without PBM. (a) z = 13 cm (z/H = 0.29), (b) z = 25 cm (z/H = 0.56) (c) z = 37 cm (z/H = 0.82).


niques) that is reported as radial profiles of axial liquid velocity, andthis averages out and azimuthal variations. In this work, we haveconsidered a flat-rectangular bubble column (hence restricting flowvariations to the axial and the radial directions only), with a singlecentrally located inlet (hence isolating the dynamics of a single bub-ble plume), and considering only low velocities (hence bubbly flow).

These conditions have been investigated with the non-invasive RPTtechnique, which provides us detailed information about the flow aswell as statistically long-time averaged (by the nature of the exper-iment) velocity profiles of liquid.

In this communication, population balance was successfullyimplemented (as homogenous discrete model, Inhomogeneous

Fig. 10. Simulated liquid velocity profile at gas superficial velocity of 2.38 mm/s with Schiller–Naumann drag and constant lift force, 5 mm initial bubble diameter, RNG k–eturbulence model, with and without PBM.

Fig. 11. Simulated liquid velocity profile at gas superficial velocity of 4.9 mm/s with Schiller–Naumann drag and constant lift force, 4 mm initial bubble diameter, RNG k–eturbulence model, with and without PBM in a square bubble column [3].


Discrete model, QMOM and DQMOM) in a transient, three-dimen-sional Euler–Euler CFD model of rectangular bubble columns. Thepredicted results were used to predict time-averaged liquid veloc-ity profiles obtained from high-resolution RPT experiments atthree different levels. Monitoring the velocity profiles at three dif-ferent levels helps us to have more confidence on our model.

The first set of simulations was performed at the gas superficialvelocity of 1.33 mm/s. Our results indicate that incorporation ofthe population balance had relatively minor effect on the predictedvelocity profiles under those conditions. Our results are suggestiveof the fact that a good estimate of the initial bubble diameter couldactually obviate the need for rigorous population balance couplingwith CFD. This is particularly true in the context of low superficialgas velocities, which puts the column in bubbly flow regime.Essentially, what population balance does is to redistribute interfa-cial area amongst difference bubble classes (and hence the gas–li-quid momentum exchange), but that does not appreciably alter the

overall momentum exchange coupling (cell by cell in the computa-tional domain) between the two-interpenetrating continua thatmake up the Euler–Euler simulation (of course for a situation whenthere is no phase change in the gas–liquid dispersion). It was alsoobserved that all schemes give similar velocity profile results (andare therefore consistent with each other), even under those condi-tions where deviation from experimental data was observed.Equivalence of all population balance schemes was furtherstrengthened shown by comparing Sauter-mean diameter profiles(as a function of level or height in the column). It was seen that allmodels give agreeably close Sauter-mean diameter profiles.

Even though PBM does not seem to influence the velocity pro-file, when lift force and virtual mass effect were incorporated intothe model, the predicted results significantly improve and verygood agreement was obtained with the experimental data. It wasalso observed that though the effect of the virtual mass effectwas very marginal; incorporation of appropriate lift forces were


the major reason for improvement of agreement of the simulationand experimental profiles. This happened because addition of lifeforce forced the bubble plume to oscillate, an effect which wasnot captured by the drag force alone. This brings down the centralpeak of axial liquid velocity in the centre of the column because ofwhich a good agreement is then obtained between simulated andexperimental results. In a continued search for the holy grail ofthe ‘‘ideal parameters’’ for an successful two-fluid implementationof gas–liquid flow in bubble columns, drag models like Schiller–Naumann [42], Ishii–Zuber [43], Tomiyama [44], and Zhang–Vanderhyden [45] were implemented. It was observed that allthe models are practically giving same solutions and hence itwas concluded that Schiller–Naumann model is sufficient to modeldrag inside the column. Lift coefficients models like Auton lift [46],Tomiyama lift [47], and Magnaduet and Legendre Lift [48] wereincorporated in the model to improve the coupling between thetwo phases. It was observed that Tomiyama lift severely over pre-dicted the profiles at the centre of the column while Auton lift, andMagnaudet and Legendre lift both gave good agreement with theexperimental data. It was also concluded that RNG k–e model is agood description of turbulence and hence was used in this study.

Very good agreement was observed for velocity profiles at ahigher gas superficial velocity of 2.38 mm/s with Schiller Naumanndrag model with constant lift and RNG k–e model at all the threelevels. Similar conclusions were drawn from set of simulationsdone on a square bubble column with experimental data takenby Deen [3].

Future work from this study should try to understand the rela-tive importance of these forces at higher gas velocities in cylindri-cal bubble column and rectangular bubble column. More workshould also be done to understand the lift forces in greater detail.Another possible avenue for exploration maybe to understand bet-ter the bubbling mechanism at the distributor and incorporate abetter bubble formation model into the overall modelling suite. At-tempt at really validating the role of PBM in Euler–Euler CFD in asetup like this is in order, perhaps with a distributor which allowswidely different sized bimodal bubble injection (hence distinctlytailoring the drag and lift forces), and studying the validation ofvarious PBM implementations.

Acknowledgements

The authors would like to thank Dr. Jay Sanyal and Dr. Feng Liuof AnSys Inc., USA for helpful discussions and many clarifications.This work was conducted as part of the IIT Delhi’s High Impact Re-search Initiative for XTL Technology, for which SR is thankful to IITDelhi administration.

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