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Cheung, C, Yeoh, G and Tu, J 2009, 'A review of population balance modelling forisothermal bubbly flows', Journal of Computational Multiphase Flows, vol. 1, no. 2,pp. 161-199.

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A Review of Population BalanceModelling for Isothermal Bubbly Flows

Sherman C.P. Cheung1, Guan Heng Yeoh2,3 and Jiyuan Tu1

1School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University,Victoria 3083, Australia

e-mail: [email protected] Nuclear Science and Technology Organisation (ANSTO), PMB 1, Menai,

NSW 2234, Australia3School of Mechanical and Manufacturing Engineering, University of New South Wales,

Sydney 2052, Australia

AbstractIn this article, we present a review of the state-of-the-art population balance modellingtechniques that have been adopted to describe the phenomenological nature ofisothermal bubbly flows. The main focus of the review can be broadly classified intothree categories: (i) Numerical approaches or solution algorithms of the PBE; (ii)Applications of the PBE in practical gas-liquid multiphase problems and (iii) Possibleaspects of the future development in population balance modelling. For the firstcategory, details of solution algorithms based on both method of moment (MOM) anddiscrete class method (CM) that have been proposed in the literature are provided.Advantages and drawbacks of both approaches are also discussed from the theoreticaland practical viewpoints. For the second category, applications of existing populationbalance models in practical multiphase problems that have been proposed in theliterature are summarized. Selected existing mathematical closures for modelling the“birth” and “death” rate of bubbles in gas-liquid bubbly flows are introduced.Particular attention is devoted to assess the capability of some selected models inpredicting bubbly flow conditions through detail validation studies againstexperimental data. These studies demonstrate that good agreement can be achieved bythe present model by comparing the predicted results against measured data withregards to the radial distribution of void fraction and sauter mean bubble diameter.Finally, weaknesses and limitations of the existing models are revealed are suggestionsfor further development are discussed. Emerging topics for future population balancestudies are provided as to complete the aspect of population balance modelling.

Keywords: Population balance; Computational Fluid Dynamics; Bubbly flow

1. INTRODUCTIONGas-liquid flows are featured in a wide diversity of industrial systems. One significant industrialapplication is venting of mixture vapors to liquid pools in chemical reactors. Here, bubble columnreactors are particularly used in many biochemical and petrochemical industries. Such reactors areknown as excellent systems for processes which require large interfacial area for gas-liquid masstransfer and efficient mixing for reacting species due to a host of gas-liquid reactions (oxidations,hydrogenations, halogenations, aerobic fermentations, etc.). In bubble column reactors, the size of gasbubbles is an important parameter influencing their performance. It determines the bubble risingvelocity and the gas residence time, which in turn governs the gas hold-up, the interfacial area, andsubsequently the gas-liquid mass transfer rate. More significantly, the prevalence of bubble-bubblephenomena such as bubble coalescence and break-up can profoundly influence the overallperformance by altering the interfacial area that is available for mass transfer between the phases. Theunderstanding of the bubble mechanistic behavior represents a crucial aspect in the rational design ofbubble column reactors. Such mounting industrial interests have certainly stimulated numerousscientific and engineering studies attempting to synthesize the behaviour of the population of particlesand its dynamical evolution subject to the system environments in which has resulted in a widelyadopted concept known as Population Balance.

161

In essence, population balance has been applied in numerous multiphase flow systems. Forexample: predicting the soot formation rate from incomplete combustion processes (Frenklach andWang, 1994; Zucca et al., 2007); solving the turbulent reacting flow in fluidized bed (Lakatos etal., 2008; Khan et al., 2007; Fan et al., 2004) and interfacial area in bubble column (Sha et al., 2006;Bordel et al., 2006; Jia et al., 2007). The population balance of any system is a record for the num-ber of particles, which may be solid particles, liquid nuclei, bubbles or, variables (in mathematicalterms) whose presence or occurrence governs the overall behaviour of the system under consider-ation. In most of the systems under concern, the record of these particles is dynamically dependedon the “birth” and “death” processes that terminate existing particles and create new particles with-in a finite or defined space. Mathematically, dependent variables of these particles may exist in twodifferent coordinates: namely “internal” and “external” coordinates (Ramkrishna and Mahoney,2002). The external coordinates refer to the spatial location of each particle which is governed byits motion due to convection and diffusion flow behaviour. On the other hand, internal coordinatesconcerns the internal properties of particles such as: size, surface area, composition and so on.Figure 1 shows an example of the internal and external coordinates involve in the population bal-ance for gas-liquid bubbly flows.

From a modelling perspective such as demonstrated in Figure 1, enormous challenges remain infully resolving the associated nucleation, growth and agglomeration processes of particles withinthe internal coordinates and flow motions of external coordinates which are subjected to interfacialmomentum transfer and turbulence modulation between gas and solid phases. Owing to the signif-icant advancement of computer hardware and increasing computing power over the past decades,Direct Numerical Simulations (DNS), which attempt to resolve the whole spectrum of possible tur-bulent length scales in the flow, provide the propensity of describing the complex flow structureswithin the external coordinates (Biswas et al., 2005; Lu et al., 2006). Nevertheless, practical mul-tiphase flows that are encountered in natural and technological systems generally contain millionsof particles that are simultaneously varying along the internal coordinates. Hence, the feasibility ofDNS in resolving such flows is still far beyond the capacity of existing computer resources. Thepopulation balance approach, which records the number of particles as an averaged function, hasshown to be a more promising way in handling the flow complexity because of its comparativelylower computational requirements. It is envisaged that the next stages of multiphase flow model-ling in research and in practice would most probably concentrate on the development of moredefinitive efficient algorithms for solving the population balance equation (PBE).

Figure 1. An exmple of the internal and external coordinates of population balance forgas-liquid bubbly flows

External Coordinates

Gas-liquid flow

Fractional volume occupy by gas bubbles at (x1,y1,z1,t1)

Internal Coordinates

Changes of internal properties caused by the ìBirthî or ìDeathî processes

Fractional volume occupy by gas bubbles at (x2,y2,z2,t2)

Changes of external variables resulted from the Fluid Motions

Number

Size

Number

Size

162 A Review of Population Balance Modelling for Isothermal Bubbly Flows

Journal of Computational Multiphase Flows

In the light of the emerging technique of the population balance modelling, this paper aims tofurther exploit the methodology of population balance modelling for isothermal bubbly flows. Atthe beginning, phenomenological behaviour of isothermal bubbly flow is firstly discussed whichfollows by a review of the solution algorithms of PBE that have been proposed within literatures.The second part will focus on the framework of the computational fluid dynamics which solves theexternal variables (e.g. field information) for the PBE. Details of the two-fluid model and the asso-ciated interfacial momentum transfer issues will be presented. In the third part of the paper, numer-ical studies are presented to demonstrate the performance of some selected population balancemodels in resolving the gas-liquid bubbly flows. Finally, fundamental weakness and limitations ofcurrent model development are addressed and possible directions for further development are alsodiscussed.

2. CHARACTERISTICS OF ISOTHERMAL BUBBLY FLOWSThrough the above discussion, one should now realize the broad application of population balancemodelling in practical engineering problems. Before going into details of existing algorithms forsolving the PBE, phenomenological discussion and background of bubbly flows is firstly providedallowing readers to develop the linkage between physical phenomena and mathematical models.

Figure 2 shows five most common flow regimes that could be encountered in the co-current gas-liquid flows within a vertical pipe. At low gas volume fractions, the flow is an amalgam of indi-vidual ascending gas bubbles co-flowing with the liquid. This flow regime refers to as the bubblyflow regime can be further subdivided into two sub-regimes – bubble flow at low liquid flow ratesand dispersed bubble flow at high liquid flow rates. As the volume fraction increases, a pattern isexhibited whereby slugs of highly aerated liquid move upwards along the pipe. These so-calledTaylor bubbles have characteristics of spherical cap nose and are somewhat abruptly terminated atthe bottom edge. The elongated gas bubbles are separated by liquid slugs which may have smallerbubbles near the skirt. Size of the slug units, Taylor bubble and liquid slugs may vary considerably.Subsequently, large unsteady occurrence of gas volumes accumulate within these mixing motionsand produce the flow regime known as churn-turbulent flow with increasing volume fractions. Atvery high gas velocities, an annular pattern is observed whereby parts of the liquid flows along thepipe and other parts as droplets entrained in the gas flow. At even higher gas velocities, a dispersepattern exists. There is now a considerable amount of liquid in the gas core.

Figure 2. Flow regimes for air-water flow in a vertical pipe

Flow

Direction

Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu 163

Volume 1 · Number 2 · 2009

In most situations, the topological distribution of different flow regimes is governed predomi-nantly by the bubble mechanistic behaviours such as bubble coalescence and bubble breakage.Figure 3 illustrates effects of bubble coalescence and breakage on the transition from bubbly-to-slug regime. Owing to the positive lift force created by the lateral velocity gradient, small bubbles(under 5.5 mm for air-water flows at 25oC) being injected at the bottom have a tendency to migratetowards the channel walls thereby increasing the bubble number density near the wall region. Thenet effect of bubble coalescence encourages the formation of larger bubbles downstream. Largerbubbles (above 5.5mm), driven by the negative lift force, will move towards to the centre core ofthe channel of which they will further coalesce with other bubbles to yield cap/taylor bubbles.

Figure 3. Schematic of the physical phenomenon embedded in isothermal bubbly flows

3. POPULATION BALANCE APPROACHES3.1. Population balance equationThe development of population balance model has a long standing history. Back to the end of 18th

century, the Boltzmann equation, devised by Ludwig Boltzmann, could be regarded as the first pop-ulation balance equation which can be expressed in terms of statistical distribution of molecules orparticles in a state space. Nonetheless, the derivation of a generic population balance concept wasactually initiated from the middle of 19th century. In 1960s, Hulburt and Katz (1964) and Pandolphand Larson (1964), based on the statistical mechanics and continuum mechanical frameworkrespectively, presented the population balance concept to solve particle size variation due to nucle-ation, growth and agglomeration processes. A series of research development were thereafter pre-sented by Fredrickson et al. (1967), Ramkrishna and Borwanker (1973) and Ramkrishna (1979,1985) where the treatment of population balance equations were successfully generalized with var-ious internal coordinates. A number of textbooks mainly concerning the population balance of aero-colloidal systems have also been published (Hidy and Brock, 1970; Pandis and Seinfeld, 1998;Friedlander, 2000). The flexibility and capability of population balance in solving practical engi-

Adiabatic walls

Fully developed liquid flows

Coalescence

Bubble injection

Breakage

Small bubble move towards the wall

Distorted/Cap bubble move towards the

core

Distorted/Cap bubbles



neering problems has not been fully exposed, until recently, where Ramkrishna (2000) wrote a text-book focusing on the generic issues of population balance for various applications.

The foundation development of the PBE stems from the consideration of the Boltzman equation.Such equation is generally expressed in an integrodifferential form describing the particle size dis-tribution (PSD) as follow:

(1)

where f(ξ,r,t) is the particle size distribution dependent on the internal space vector ξ, whose com-ponents could be characteristics dimensions, surface area, volume and so on. r and t are the exter-nal variables representing the spatial position vector and physical time in external coordinaterespectively.u(ξ,r,t) is velocity vector in external space. On the RHS, the first and second termsdenote birth and death rate of particle of space vector ξ due to merging processes, such as: coales-cence or agglomeration processes; the third and fourth terms account for the birth and death ratecaused by the breakage processes respectively. a(ξ ξ´) is the coalescence or agglomeration ratebetween particles of size ξ and ξ´ . Conversely, b(ξ) is the breakage rate of particles of size ξ´. γ(ξ´)is the number of fragments/daughter particles generated from the breakage of a particle of size ξ´and p(ξ ξ´) represents the probability density function for a particle of size ξ to be generated bybreakage of a particle of sizeξ´.

Owing to the complex phenomenological nature of particle dynamics, analytical solutions onlyexist in very few cases of which coalescence and breakage kernels are substantially simplified(Scott, 1968; McCoy and Madras, 2003). Driven by practical interest, numerical approaches havebeen developed to solve the PBEs. The most common methods are Monte Carlo methods, Methodof Moments and Class Methods. Theoretical speaking, Monte Carlo methods, which solve the PBEbased on statistical ensemble approach (Domilovskii et al., 1979; Liffman, 1992; Debry et al.,2003; Maisels et al., 2004), are attractive in contrast to other methods. The main advantage of themethod is the flexibility and accuracy to track particle changes in multidimensional systems.Nonetheless, as the accuracy of the Monte Carlo method is directly proportion to number of simu-lations particles, extensive computational time is normally required. Furthermore, incorporating themethod into conventional CFD program is also not straightforward which greatly degraded itsapplicability for industrial problems. Because of their relevance in CFD applications, numericalapproaches developed for Method of Moments and Class Methods are then discussed.

3.2. Method of Moments (MOM) ApproachThe method of moments (MOM), first introduced by Hulburt and Katz (1964), has been consideredas one of the many promising approaches in viably attaining practical solutions to the PBE. Thebasic idea behind MOM centres in the transformation of the problem into lower-order of momentsof the size distribution. The moments of the particle size distribution are defined as:

(2)

From the above equation, the first few moments give important statistical descriptions on the pop-ulation which can be related directly to some physical quantities. In the case space vector ξ repre-sents the volume of particle, the zero order moment (k = 0) represents the total number density of

m t f t dk k( ) ( ) ( , )=∞

∫ ξ ξ ξ0

−b f t( ) ( , )ξ ξ

+ ′ ′ ′ ′∞

∫ γ ξ ξ ξ ξ ξξ

( ) ( ) ( / ) ( , )b p f t ds

− − ′ ′ ′ ′∞

∫f t a f t d( , ) ( , ) ( , )ξ ξ ξ ξ ξ ξ0

∂ ( )∂

+ ∇ ⋅( ) = − ′ ′f r t

tu r t f r t a f

ξξ ξ ξ ξ ξ

, ,( , , ) ( , , ) ( , ) (

1

2ξξ ξ ξ ξ

ξ− ′ ′ ′∫ , ) ( , )t f t d

0



population and the fraction moment, k = 1/3 and k = 2/3 gives information on the mean diameterand mean surface area respectively.

The primary advantage of MOM is its numerical economy which condenses the problem sub-stantially by tracking the evolution of limited number of moments (Frenklach, 2002). This becomesa critical value in modeling complex industrial systems when particle dynamics is coupled withalready time-consuming calculations of turbulence multiphase flows. Another significance of theMOM is that it does not suffer from truncation errors in the PSD approximation. Mathematically,the transformation from the PSD space to the space of moments is rigorous. Unfortunately,throughout the transformation process, fraction moments, representing mean diameter or surfacearea, are normally involved posing serious closure problem (Frenklach and Harris, 1987). In orderto overcome the closure problem, in the early development of MOM, Frencklach and his co-work-ers (Frenklach and Wang, 1991; Markatou et al., 1993; Frenklach and Wang, 1994) proposed aninterpolative scheme to determine the fraction moment from integer moments – namely Method ofmoments with interpolative closure (MOMIC).

3.2.1. Quadrature method of momentsAnother different approach for computing the moment is to approximate the integrals in Eq. (1)using numerical quadrature scheme – the quadrature method of moment (QMOM) as suggested byMcGraw (1997). In the QMOM, instead of space transformation, Gaussian quadrature closure isadopted to approximate the PSD by a finite set of Dirac’s delta functions as follow:

(3)

where Ni represents the number density or weight of the ith class consists of all particles per unitvolume with a pivot size or abscissa, χi. A graphical representation of the QMOM in approximat-ing the PSD is depicted in Figure 4.

Figure 4. Graphical presentations of the Quadrature Method of Moments (QMOM)

Although the numerical quadrature approach suffers from truncation errors, it successfully elim-inates the problem of fraction moment which special closure is usually required. The closure of themethod is then brought down to solving 2M unknowns, χi and Ni. A number of approaches in thespecific evaluation of the quadrature abscissas and weights have been proposed. McGraw (1997)first introduced the product-difference (PD) algorithm formulated by Gordon (1968) for solving

Particle Size Distribution (PSD)

1ξξ

N( iξ )

2ξ 3ξ 4ξ

1N 2N3N

4N

f t N xii

M

i( , ) ( )ξ δ ξ≈ −=∑

1



monovariate problem. Nonetheless, as point out by Dorao et al. (2006, 2008), the PD algorithm isa numerical ill-conditioned method for computing the Gauss quadrature rule (Lambin and Gaspard,1982). Comprehensive derivation of the PD algorithm can be found in Bove (2005). In general, thecomputation of the quadrature rule is unstable and sensitive to small errors, especially if large num-ber of moments is used. Later, McGraw and Wright (2003) derived the Jacobian MatrixTransformation (JMT) for multi-component population which avoids the instability induced by thePD algorithm. Very recently, Grosch et al. (2007) proposed a generalized framework for variousQMOM approaches and evaluated different QMOM formulations in terms of numerical quadratureand dynamics simulation. Several studies have also been carried out validating the method againstdifferent gas-solid particle problems (Barrett and Webb, 1998; Marchisio et al., 2003a,b,c).Encouraging results obtained thus far clearly demonstrated its usefulness in solving monovariateproblems and its potential fusing within Computational Fluid Dynamics (CFD) simulations. One ofthe main limitations of the QMOM is that moments are adopted to represent the PSD, each momentis “convected” in the same phase velocity which is apparently non-physical, especially for gas-liq-uid flow where bubble could be deformed and travel in different trajectory.

3.2.2. Direct quadrature method of moments (DQMOM)With the aim to solve multi-dimensional problems, Marchisio and Fox (2005) extended the methodby developing the direct quadrature method of moment (DQMOM) where the quadrature abscissasand weights are formulated as transport equations. The main idea of the method is to keep track theprimitive variables appearing in the quadrature approximation, instead of moments of the PSD. Asa result, the evaluation of the abscissas and weights are solved obtained using matrix operations.Substitute Eq. (3) into Eq. (1) and after some mathematical manipulations, transport equations forweights and abscissas are given by:

(4)

(5)

where ζi = Niχi is the weighted abscissas and the terms ai and bi are related to the “birth” and“death” rate of population which forms 2M linear equations of which unknowns can be evaluatedvia matrix inversion:

(6)

where the 2M × 2M coefficient matrix A= [A1 A2] is given by:

(7)

−

−

−

−=

−− 12

2

121

211

)1(2)1(2

0

1

0

1

A

MM

M

M xM

x

xM

x

O L

L

M

L

L

M

A dα =

∂ ( )∂

+ ∇ ⋅( ) =ζ

ζii i

r t

tu r t r t b

,( , ) ( , )

∂ ( )∂

+ ∇ ⋅( ) =N r t

tu r t N r t ai

i i

,( , ) ( , )



(8)

The 2M vector of unknowns α is defined by:

(9)

and the source on the RHS is:

(10)

The source term for the kth moment S–k is defined by:

(11)

One attractive feature of the DQMOM is that it permits weights and abscissas to be varied withinthe state space according to PSD evolution. Furthermore, different travelling velocity can be alsoincorporated into transport equations allowing the flexibility to solve poly-dispersed flows whereweights and abscissas travel indifferent flow fields (Ervin and Tryggvason, 1997; Bothe et al.,2006). In summary, the MOM represents a rather sound mathematical approach and an elegant toolof solving the PBE with limited computational burden. Such approach with no doubt is an emerg-ing technique for solving PBE, due to the considerably short development history, thorough vali-dation studies comparing model predictions against experimental data are however outstanding. Itcan be concluded that the DQMOM or other moment methods still require further assessments andvalidations for various multiphase flow problems.

3.3. Class Method (CM) ApproachInstead of inferring the PSD to derivative variables (i.e. moments), the class method (CM) whichdirectly simulate its main characteristic using primitive variable (i.e. particle number density) hasreceived greater attention due to its rather straightforward implementation within CFD softwarepackages. In the method of discrete classes, the continuous size range of particles is discretized intoa series number of discrete size classes. For each class, a scalar (number density of particles) equa-tion is solved to accommodate the population changes caused by intra/inter-group particle coales-cence and breakage. The particle size distribution is thereby approximated as follow:

(12)

The expression is exactly the same with QMOM in Eq. (3), however, the groups (or abscissas) ofclass methods are fixed and aligned continuously in the state space. A graphical representation ofthe CM in approximating the PSD is depicted in Figure 5.

f t N xii

M

i( , ) ( )ξ δ ξ≈ −=∑

1

S r t S r t dkk( , ) ( , , )=

∞

∫ ξ ξ ξ0

[ ]TMSSd 120= L

[ ]

==

b

abbaa T

MM LL 11α

−−

=

−− 22221

12

)12(

2

)2(

2

1

0

1

0

A

MM

M

M xM

x

xxM

x

M

L

O

L

M

L

L



Figure 5. Graphical presentations of Class Methods (CM)

3.3.1. Single Average Quantities ApproachThe simplest approach in class methods is adopted an averaged quantity to represent the overallchanges of the particle population. Kocamustafaogullari and Ishii (1995) first derived an interfacialarea concentration (IAC) transport equation for tracking the interfacial area between gas and liquidphase in bubbly flow problems. They concluded that the governing factor of the interfacial trans-fer mechanisms is strongly dominated by the interfacial area concentration. Modelling of the inter-facial area concentration is therefore essential. In addition, as the population of particle is repre-sented by a single average scalar, such average quantity approach requires very limited computa-tional time in solving the PBE, which provides an attractive feature for practical engineering prob-lems. Assuming the bubbles are in perfect spherical shape, the interfacial area concentration, aif,transport equation can then be written as (Ishii et al., 2002):

(13)

Following their study, Ishii and his co-workers preformed extended the capability of their (IAC)transport model to simulate different bubbly flow regime in different flow conditions (Wu et al,1998; Hibiki and Ishii, 2002; Fu et al., 2002a,b; Sun et al., 2004a,b). A series of experimental stud-ies covering a wide range of flow conditions have been carried out in order to provide a solid foun-dation for their model development and calibration. Recently, similar modelling approach has beenalso adopted by Yao and Morel (2004) with the attempt of better improving the bubble coalescenceand breakage kernels.

On the other hand, equivalent to the formulation of the interfacial area transport equation, anAverage Bubble Number Density (ABND) equation has been proposed very recently in our previ-ous studies (Yeoh and Tu, 2006; Cheung et al., 2007). The averaged bubble number density can beexpressed as:

(14)

where R represents the local volume-averaged source and sink rates, which need to be closedthrough constitutive relations. It can be observed that by solving the transport equation for the aver-aged bubble number density, the changes to the interfacial structure can be locally accommodated

∂∂

+ ∇ ⋅( ) =N

tN RdV

∂∂

+ ∇ ⋅( ) =

∂∂

+ ∇ ⋅( )

a

ta

a

tif d

if

if

d

dd dV V

2

3 αα α

+

1

3

2

ψα d

ifa

R



throughout the flow. The inclusion of the source and sink terms caused by the phenomenologicalmechanisms of coalescence and break-up as well as possible phase change processes allows thedescription of the temporal and spatial evolution of the geometrical structure of the disperse phase.

3.3.2. MUltiple SIze Group (MUSIG) modelBesides the proposed the average quantity approach, a more sophisticated model, namely homoge-neous MUltiple-SIze-Group (MUSIG) model which first introduced by Lo (1996) is becomingwidely adopted. Research studies based on the by Pochorecki et al. (2001), Olmos et al. (2001),Frank et al. (2004), Yeoh and Tu (2005) and Cheung et al. (2007) typified the application ofMUSIG model in bubbly flow simulations. In the MUSIG model, the continuous particle size dis-tribution (PSD) function is approximated by M number size fractions; the mass conversation ofeach size fractions are balanced by the inter-fraction mass transfer due to the mechanisms of parti-cle coalescence/agglomeration and breakage processes. The overall PSD evolution can then beexplicitly resolved via source terms within the transport equations.

Sanyal et al. (2005) examined and compared the CM and QMOM in a two-dimensional bubblycolumn simulation; both methods were found to yield very similar results. The CM solution hasbeen found to be independent of the resolution of the internal coordinate if sufficient number ofclasses were adopted. Computationally speaking, as the number of transport equations depends onthe number of group adopted, the MUSIG model requires more computational time and resourcesthan the MOM to achieve stable and accurate numerical predictions. For typical bubbly flow sim-ulations, the model requires around 10 groups to yield accurate results.

Nonetheless, unlike the QMOM, CM provides the feasibility of accounting different bubbleshapes and travelling gas velocities. The inhomogeneous MUSIG model developed by Krepper etal. (2005), which consisted of sub-dividing the dispersed phase into N number of velocity fields,demonstrated the practicability of such an extension. This flexibility represents a robust feature formultiphase flows modelling, especially for bubbly flow simulations where bubbles may deforminto different shapes. Figure 6 shows the concept of the inhomogeneous MUSIG in comparison tohomogeneous MUSIG. Useful information on the implementation and application of the inhomo-geneous MUSIG model can be found in Shi et al. (2004) and Krepper et al. (2007). In spite of thesacrifices being made to computational efficiency, the extra computational effort will rapidlydiminish due to foreseeable advancement of computer technology; the class method should there-fore suffice as the preferred approach in tackling more complex multiphase flows.

Figure 6. Schemetic diagram of the homogeneous and imhomogeneous MUSIG models

The formulation of the MUSIG model originates from the discretised PSE is given by:

(15)

To ensure overall mass conservation for all poly-dispersed vapour phases, the above bubble num-ber density equation for the inhomogeneous MUSIG model can be re-expressed in terms of the vol-

∂∂

+ ∇ ⋅( ) = ( )∑n

tu n Ri

i i i

⇀

ju

Bubble breakage Bubble coalescence

Homogeneous MUSIG

Inhomogeneous MUSIG

ju ju

d1 dm1 dm1+1 dm1+m2



ume fraction and size fraction of the bubble size class i, ∈ [1,Mj], of velocity group j,∈ [1,N]according to:

(16)

with additional relations and constraints:

;

; (17)

where mi is the mass fraction of the particular size group I and Rph is the mass transfer rate dueto phase changes which will be discussed in the coming sections.

On the right hand side of Eq. (16), the term represents thenet mass transfer rate of the bubble class i resulting from the source of PC, PB, DC and DB, whichare the production rates due to coalescence and breakage and the death rate due to coalescence andbreakage of particles respectively. They can be formulated as:

if

if

otherwise

with

(18)

with

and (19)Sjj

N

=∑ =

1

0S Sj j ii

M j

==∑ ,

1

i kik

N

==

∑1

D nB i i=

P v v nBj i

N

j i j= ( )= +∑

1

:

D n nC ij i jj

N

==

∑χ1

0

ν ν ν νi j k i< + < +1ν ν ν ν νi j k i i+ +− + −1 1( ) / ( )η jki =

ν ν ν νi j k i− < + <1( ) / ( )ν ν ν ν νj k i i i+ − −− −1 1

P n nC jki ij i jl

i

k

i

===∑∑1

2 11

η χ

S m R P P D Dj i i i C B C B, = ( ) = + − −( )∑

fii

M j

==∑ 1

1α αg l+ = 1

α αj ii

M j

==∑

1

α α αg jj

N

iji

M

j

N j

= == ==

∑ ∑∑1 11

∂∂

+ ∇ ⋅( ) =ρ α

ρ αj j ig j i j j i

f

tf u S ,

⇀



3.4. Bubble Interaction MechanismsThe interfacial area concentration represents the key parameter that links the interaction betweenthe phases. In gas-liquid flows, considerable attention has been concentrated towards describing thetemporal and spatial evolution of the two-phase geometrical structure caused by the effects of coa-lescence and break-up through the interactions among bubbles as well as between bubbles and tur-bulent eddies in turbulent flows. In view of this, the major phenomenological mechanisms havebeen identified and appropriate mechanistic models have subsequently been established. In bubblyflow conditions, they include:• Coalescence through random collision driven by turbulent eddies• Coalescence due to the acceleration of the following bubble in the wake of the preceding bub-

ble• Break-up due to the impact of turbulent eddies

The schematic illustrations of these mechanisms are shown in Figure 7. Final expressions for theparticle number source or sink terms due to such mechanisms for either the single average scalaror multiple bubble size approach are presented in the following sections.

Figure 7. Bubble interaction mechanisms in bubbly flow conditions

Bounce

Merge

Random Collision

Wake Entrainment

Turbulent Impact

Unaffected

Uneven breakage

due to fluid shearing

Even breakage

Coalescence Mechanism

Break-up Mechanism



3.4.1. Bubble Mechanism for Single Average Quantities ApproachAs shown in Eq. (13)(14), the term R in the right hand side of the equation must be specified bythe constitutive relations. The population balance of dispersed bubbles in bubbly flow conditions isgoverned by the three mechanisms of bubble coalescence and break-up of which the term can bewritten as:

(20)

where φ RCN , φ TI

N and φ WEN are the bubble number density changes due to random collision, tur-

bulent induced breakage and wake entrainment. With the assumption of spherical bubbles, thetransport equation of the averaged bubble number density is equivalent to the transport equation ofthe interfacial area concentration.Wu et al. (1998) Model

An empirical modeling of the bubble coalescence and bubble breakage that has been widelycited is the model developed by Wu et al. (1998). Considering the characteristic times for binarycollision and the mean traveling length between neighboring bubbles, they have modeled the ran-dom collision rate of bubble coalescence according to

(21)

where CRC = 0.021 and C = 3.0 are adjustable model constants representing the coalescence effi-ciency. The maximum allowable void fraction αmax = 0.8 takes the value of 0.8, which considersthe point of transition from slug to annular flow. Assuming a spherical bubble travelling with itsterminal velocity, the rate of collision caused by wake entrainment is however expressed as:

(22)

where CWE = 0.0073 is a model constant determining the effective wake length and the coales-cence efficiency. The terminal velocity of bubbles, Ur, is given by:

(23)

Turbulent induced break-up is derived from a simple momentum balance approach. In thismechanism, Wu et al. (1998) restricted only eddies with the same size as the bubbles responsiblefor breakage. The rate of bubble break-up is given by:

(24)

From the above expression, CTI = 0.0945 and the critical Weber number Wecr = 2.0, which gov-erns the criterion of breakage, are adjustable parameters.Hibiki and Ishii (2002) Model

Some experimental observations (Serizawa and Kataoka, 1988) have argued that the coales-cence due to wake entrainment is only significant between pairs of large cap bubbles (slug flowregime) in fluid sufficiently viscous to maintain their wake laminar; whereas small spherical orellipsoidal bubbles tend to repel each other. In contrast to the model of Wu et al. (1998), Hibiki and

ϕα ε

nTI

TIg

S

cr crCD

We

We

We

We=

−

1 3

11 3

/

/1- exp

UD g

CrS

D l

=

3

1 2∆ρρ

/

ϕα

nWE

WE rg

S

C UD

= −2

4

ϕα ε

α α αnRC

RCg

S g

CD

= −−( ) −

2 1 3

11 3 1 3 1 3 1 31

/

/max

/max

/ /exxp −

−

C g

g

α αα α

max/ /

max/ /

1 3 1 3

1 3 1 3

R d dNRC

NTI

NWE= + +ρ α ϕ ϕ ϕ( )



Ishii (2002) have ignored the wake entrainment coalescence due to its insignificant event in bub-bly flow condition. By assuming that the bubble movement behaves analogously to ideal gas mol-ecules, the coalescence rate due to turbulent random collision can be determined as:

(25)

Instead of using the constant in the Wu et al. (1998) model, the coalescence efficiency is derivedfrom the liquid-film-thinning model (Oolman and Blanch, 1986a, 1986b) and dimensional consid-eration for turbulent flow (Levich, 1962) using the Coulaloglo and Tavlarides (1977) expression asthe main framework. The constants CRC = 0.03 and C = 1.29 are adjustable model constants thathave been calibrated through experiments.

Hibiki and Ishii (2000a) have also derived the breakage rate from kinetic theory. The breakagerate is correlated to the frequency for a given bubble colliding with the turbulent eddy as:

(26)

In the above expression, CTI = 0.03 and C = 1.37 are adjustable model constants that have beendetermined experimentally.Yao and Morel (2004) Model

Yao and Morel (2004) have pointed out that the aforementioned two models have been devel-oped based on two different considerations: the free travelling time or the interaction time. Theyargued that both characteristic times are identically important. Taking two considerations intoaccount, the bubble coalescence rate is derived as:

(27)

where the derived constants are CRC1 = 2.86, CRC2 = 1.017 and CRC3 = 1.922, respectivelySimilarly to the Hibiki and Ishii (2002) model, coalescence caused by wake entrainment is neg-lected.

For bubble break-up, they disputed that bubble breakage is mainly caused by the resonanceoscillation. Considering the natural frequency of the oscillating bubbles, the interaction time can beapproximated and the rate of bubble breakage is given by:

(28)

where the constants are: CTI1 = 1.6 and CTI1 = 0.42. The critical Weber number of 1.42 isemployed (Sevik and Park, 1973). Considering the transition point from the finely dispersed bub-bly flow to slug flow, the maximum allowable void fraction in Hibiki and Ishii (2002) and Yao andMorel (2004) models retains a value of 0.52.

3.4.2. Bubble Mechanism for Multiple Bubble Size ApproachFor the multiple bubble size approach, the Prince and Blanch (1990) coalescence and breakage ker-nels are widely applied for the bubbly flow simulations. For the merging of bubbles, the coales-cence of two bubbles is assumed to occur in three stages. The first stage involves the bubbles col-liding thereby trapping a small amount of liquid between them. This liquid film then drains until it

ϕα α α

nTI g

S

cr

D

We We=

+1 6

1 0 42

1 3

11 3.

.

/

/

(1- ) exp(- )

(1-g

αα g crWe We)

ϕα ε

αnRC

RCg

S

RC crCD

C We We= − 1

2 1 3

11 32

/

/

exp(- )

( max1/3 -- )/g maxα α α1 3

3/ + C We WeRC g cr

ϕα α ε

α ασ

ρ εnTI

TIg

S g l

CD

C= −−( ) −

2 1 3

11 3 2

(1- )expg

/

/max

// /3 5 3DS

ϕα εα α

ρ εnRC

RCg

S g

l SCD

CD

= −−( ) −

2 1 3

11 3

1 2 1 3/

/max

/ /

exp55 6

1 2

/

/σ



reaches a critical thickness. The last stage features the rupturing of the liquid film subsequentlycausing the bubbles to coalesce. The collisions between bubbles may be caused by turbulence,buoyancy and laminar shear. Only random collisions driven by turbulence are usually consideredfor bubbly flow conditions. The coalescence rate considering the turbulent collision taken fromPrince and Blanch (1990) can be expressed as:

(29)

where τij is the contact time for two bubbles given by (dij / 2)2/3 / ε1/3 and tij is the time requiredfor two bubbles to coalesce having diameter di and dj estimated to be [(dij /2)3 ρl / 16σ]0.5 ln(h0/ hf). The equivalent diameter dij is calculated as suggested by Chesters and Hoffman (1982): (dij= (2 / di + 2 / dj)

-1. According to Prince and Blanch (1990), experiments have determined the ini-tial film thickness ho = 1×10-4 m and critical film thickness hf = 1×10-8 m at which rupture for air-water systems. The turbulent velocity ut in the inertial sub-range of isotropic turbulence (Rotta,1972) which is given –by: ut = �–

2ε1/3 d 1/3. FC is the coalescence calibration factor.For the consideration of the breakage of bubbles, the model developed by Luo and Svendsen

(1996) is employed for the break-up of bubbles in turbulent dispersions. In this model, binarybreak-up of the bubbles is assumed and the model is based on surface energy criterion and isotrop-ic turbulence.v According to Luo and Svendsen (1996), bubble breakage rate of volume νj into vol-ume νi can be modelled based on the assumption of bubble binary breakage under isotropic turbu-lence situation. The daughter size distribution is accounted using a stochastic breakage volumefraction fBV. Denoting the increase coefficient of surface area as cf = [fBV

2/3 +(1-fBV)2/3-1], thebreakage rate can be obtained as:

(30)

where ξ = λ /dj is the size ratio between an eddy and a particle in the inertial sub-range and con-sequently ξmin = λmin /dj and C and β are determined from fundamental consideration of drops orbubbles breakage in turbulent dispersion systems to be 0.923 and 2.0. FB is the breakage calibra-tion factor.

With respect to the coalescence kernels, several other expressions have been proposed in addi-tion to the model by Prince and Blanch (1990). Instead of using water film thinning hypothesis,Chesters (1991) proposed an alternate expression for the coalescence efficiency purely based on thedimensionless Weber number which is given by:

(31)

Lehr et al. (2002) have however proposed different expressions not only to the coalescence effi-ciency but also for the characteristic velocity in the collision rate. In addition to the considerationof the turbulent velocity in the inertial sub-range of isotropic turbulence, the characteristic veloci-ty also accounts for the difference in rise velocities of the bubbles. It should be noted that the abovecoalescence efficiency models as well as the model by Prince and Blanch (1990) are based on phe-nomenological analysis and they provide probability functions that are employed to modify the col-lision frequency models in order to determine the appropriate coalescence rates. These are thus verystrongly influenced by the hydrodynamic conditions and the interfacial characteristics of the flowsystem.

P d d cWe

C i j, exp( ) = −

2

1 2

v v

nF C

dj i

g j

Bj

:( )−( ) =

+( )

1

12

1 3 2

11 3αε ξ

ξξ

/

/

minn

/ / /

1

2 3 5 3 11 3

12∫ × −

expc

ddf

l

σβρ ε ξ

ξ

χ πτij C i j ti tj

ij

ij

F d d u ut

= + +( ) −

4

2 2 2 0 5.exp



Concerning break-up kernels, other phenomenological models have also been proposed besidesthe model by Luo and Svendsen (1996). Lehr et al. (2002) have derived an expression in which cap-illary pressure constraint is incorporated and the interfacial and inertial forces are assumed to bal-ance each other. These additional constraints favor in the formation of larger break-up fractionsthus avoiding the evolution of infinitely small daughter bubbles. Another model for the bubblebreakup kernel proposed by Martínez-Bazán et al. (1999a) considered the balance between the tur-bulent stress and surface tension force. According to their mode, the expression is given by:

(32)

where the constant β = 8.2 has been ascertained from Batchelor (1956) and Kg = 0.25 was foundexperimentally by Martínez-Bazán et al. (1999a).

On the basis of the model by Luo and Svendsen (1996), it predicts the bubble undergoing break-age into a given combination of daughter bubble sizes that is defined by a U-shaped daughter bub-ble probability distribution function. On the other hand, the model by Martínez-Bazán et al.(1999b) predicts that the break-up process is described by an inverted U-shaped (I-shaped) daugh-ter bubble probability distribution function. The prediction of the daughter bubble probability dis-tribution function for lower breakage rates by the model of Lehr et al. (2002) yields maximumprobability for equivalent-size breakage but an M-shaped daughter bubble probability distributionfunction with two peaks close to zero breakage void fractions as the breakup rate increases. Sizedistribution of the daughter bubbles through this model is thus a function of the size ratio of givenbubbles to that of the maximum stable bubble size where larger bubbles are more likely to breakinto unequal daughter bubbles while small bubbles are more likely to break into equal size daugh-ter bubbles

Unfortunately, all these works were limited in solving isothermal bubbly turbulent co-flow prob-lems with low superficial gas velocity where there was no significant formation of cap/slug bub-bles. It should be noted that the adoption of the above kernels remain debatable for modelling thebubble dynamics beyond bubbly flow regime, especially for high superficial gas velocity.

4. MATHEMATICAL MODELS FOR EXTERNAL VARIABLESReferring back to the formulation of PBE, one should notice that the left-hand side of the equationdenotes the time and spatial variations of the PSD which depends on the external variables. Byincorporating the PBE within CFD solver, external variables can be obtained. In this section, gov-erning equations of the two fluid model and its associated model for handling interfacial momen-tum and mass transfer are introduced.

4.1. Two-Fluid modelThe three-dimensional two-fluid model solves the ensemble-averaged of mass, momentum andenergy transport equations governing each phase. Denoting the liquid as the continuum phase (αl)and the vapour (i.e. bubbles) as disperse phase (αg), these equations can be written as:Continuity equation of liquid phase

(33)

Continuity equation of vapour phase

(34)∂

∂+ ∇ ⋅( ) =

ρ αρ αg g i

g g i g i

f

tf u S

∂∂

+ ∇ ⋅( ) =ρ α ρ αl l

l l ltu 0

b M Kd d

dc

i g

ci

ci

i

εβ ε σ ρ

,( ) =( ) − ( )2 3

12



Momentum equation of liquid phase

(35)

Momentum equation of vapour phase

(36)

On the right-hand side of equation (34), Si represents the additional source terms due to coalescenceand breakage. For isothermal bubbly turbulent pipe flows, it should be noted that the mass transferrate Γ1g and Γg1 are essentially zero. The total interfacial force F1g appearing in equation (35) isformulated according to appropriate consideration of different sub-forces affecting the interfacebetween each phase. For the liquid phase, the total interfacial force is given by:

(37)

The sub-forces appearing on the right hand side of equation (37) are: drag force, lift force, walllubrication force and turbulent dispersion force which will be discussed below.

4.2. Inter-phase Momentum TransferInterfacial momentum transfer is rather crucial to the modeling of gas-liquid flows. Considered assources or sinks in the momentum equations, this interfacial force density generally contains theforce due to viscous drag as well as the effects of lateral lift, wall lubrication, virtual mass and tur-bulent dispersion, which are lumped together as non-drag forces. These interfacial force densitiesstrongly govern the distribution of the gas and liquid phases within the flow volume.

In the case of dispersed flows (bubbly, slug or churn-turbulent), the interfacial drag force is aresult of the shear and form drag of the fluid flow. It can be modelled according to

(38)

where the inter-phase drag term Bkl is expressed as:

(39)

where CD is the drag coefficient. The drag coefficients based on the correlations by Ishii and Zuber(1979) for different flow regimes are normally employed for gas-liquid flows. The function CD(Reb), known as the drag curve, can be correlated for individual bubbles across several distinctbubble Reynolds number regions; known as: stroke, viscous and turbulent regime.

On the other hand, non-drag forces have a profound influence on the flow characteristics espe-cially in dispersed flows. Bubbles rising in a liquid are subjected to a lateral lift force due to hori-zontal velocity gradient. This interfacial force density can normally be correlated to the slip veloc-ity and local vorticity of the continuous phase (curl of the velocity vector), which acts perpendicu-

B C akl D if

c l k= −1

8ρ U U

F U UDk drag

kll k

l

N

Bp

, ≡ −( )=∑

1

F F F F Flg = + + +lgdrag

lglift

lglubrication

lgdispersionn

+∇ ∇ + ∇ + − +[ ( ( ) )] ( )lgα µg ge

g gT

l g glu u u u FΓ Γgl⇀ ⇀ ⇀ ⇀

∂∂

+ ∇ ⋅( ) = − ∇ +ρ α

ρ α α α ρg g gg g g g g g g

u

tu u P g

⇀

⇀ ⇀ ⇀

+∇ ∇ + ∇ + − +[ ( ( ) )] ( )lg lgα µl le

l lT

g lu u u u FΓ Γgl⇀ ⇀ ⇀ ⇀

∂∂

+ ∇ ⋅( ) = − ∇ +ρ α ρ α α α ρl l l

l l l l l l l

u

tu u P g

⇀

⇀ ⇀ ⇀



lar to the direction of relative motion between two phases:

(40)

For the lift coefficient CL in above equation, Lopez de Bertodano (1992) and Takagi andMatsumoto (1998) suggested a value of CL = 0.1. Drew and Lahey (1979) proposed CL = 0.5 basedon objectivity arguments for an inviscid flow around a sphere. The constant of CL = 0.01 as sug-gested by Wang et al. (1987) has been found to be appropriate for viscous flows. Tomiyama (1998)however developed an Eotvos number dependent correlation that allows negative coefficients toemerge if the bubble diameter is larger than 5.5 mm for air-water system, which subsequentlyresults in a negative lateral lift force forcing large bubbles to be emigrated towards the centre of theflow channel. The lift coefficient can be expressed as:

(41)

where the modified Eotvos number Eod is defined by

(42)

in which DH is the maximum bubble horizontal dimension that can be evaluated through the empir-ical correlation of Welleck et al. (1966).

In contrast to the lateral lift force, wall lubrication force constitutes another lateral force due tosurface tension which is formed to prevent bubbles from attaching on the solid wall. This results ina low void fraction at the vicinity of the wall area. According to Antal et al. (1991), this force canbe modelled as:

(43)

where yw is the distance from the wall boundary and nw is the outward vector normal to the wall.The wall lubrication constants determined through numerical experimentation for a spheres areCw1 = –0.01 and Cw2 = 0.05. Following a recent proposal by Krepper et al. (2005), the model con-stants have been modified according to Cw1 = –0.0064 and Cw2 = 0.016. To avoid the emergenceof attraction force, the force is set to zero for large yw.

The virtual mass or added mass force arises because of acceleration of the gas bubble requiresacceleration of the fluid. It is generally taken to be proportional to the relative phase acceleration,which can be expressed as:

(44)F = FU

Dc

Dd d c

VM

d

CD

Dt, ,virtual mass virtual mass− = α ρ −−

D

Dt

cU

F = FU U U

Dc

Dd

d c d c

, ,lubrication lubrication− = −−( ) −α ρ dd c

w w

s

w ws

w

C

D

C CD

y

w

−( ) ⋅( )

× +

U n n2

1 2n

w

Eog D

d

c dH=

−( )ρ ρσ

2

C

f Eo

f EoL

b d

d=

( ) ( ) ( )

min . tanh . ,0 288 0 121Re Eo < 4

== 0.00105 0.0204 + 0.474 4Eo Eo Eod d d3 20 0159− − ≤. EEo ≤

−

10

0 29. Eo > 10

F = F U U UDc lift

Dd lift

Ld c d c cC, ,− ≡ −( ) × ∇ ×( )α ρ



where D / Dt is the material derivative. The virtual mass effect is significant when the dispersephase density is much smaller than the continuous phase density. For an inviscid flow around anisolated sphere, the constant CVM is taken to be equivalent to 0.5. Nevertheless, this particular con-stant is highly dependent on the shape and concentration and could be modified by further multi-plying a factor E´́ to CVM in order to account for the effect of surrounding bubbles, which is givenby Zuber (1964).

(45)

Considering turbulent assisted bubble dispersion, turbulence dispersion force taken as a functionof turbulent kinetic energy in the continuous phase and gradient of the volume fraction can beexpressed in the form according to Antal et al. (1991) as:

(46)

Values of constant CTD ranging from 0.1 to 0.5 have been employed successfully for bubbly flowwith diameters of the order of millimeters. In some situations, values up to 500 have been required(Lopez de Bertodano, 1998, Moraga et al., 2003). Burns et al. (2004) have however derived analternative model for the turbulence dispersion force based on the consistency of Favre-averaging,which is given by:

(47)

where CTD is normally set to a value of unity, µdT is the turbulent viscosity of the disperse phase

and Scb is the turbulent bubble Schmidt number with an adopted value of 0.9. In equation (6.32),the constant CD depicts the drag coefficient which essentially describes the interfacial drag force.This model therefore clearly depends on the details of the drag characteristics of the gas-liquid sys-tems. For situations where an appropriate value of CTD is not readily obtained through the turbu-lent dispersion force in equation (6.31), the Favre-averaged turbulent dispersion force formulatedin equation (6.32) is recommended.

4.3. Turbulence modelling for two-fluid modelIn handling bubble induced turbulent flow, unlike single phase fluid flow problem, no standard tur-bulence model is tailored for multiphase flow. For simplicity, the standard k-ε model has beenemployed with encouraging results in early studies (Schwarz and Turner, 1988; Davidson, 1990).Nonetheless, based on our previous study (Cheung et al, 2007a), the Menter’s (1994) k-ε basedShear Stress Transport (SST) model were found superior to the standard k-ε model. Similar obser-vations have been also reported by Frank et al. (2004). Based on their bubbly flow validation study,they discovered that standard k-ε model predicted an unrealistically high gas void fraction peakclose to wall. Interestingly, they also found that the two turbulence models behaved very similar byreducing the inlet gas void fraction to a negligible value. This could be attributed to a more realis-tic prediction of turbulent dissipation close to wall provided by the k-ε formulation. It revealed thatfurther development should be focused on multiphase flow turbulence modelling in order to betterunderstand or improve the existing models.

The SST model is a hybrid version of the k-ε and k-ε models with a specific blending function.Instead of using empirical wall function to bridge the wall and the far-away turbulent flow, it solvesthe two turbulence scalars (i.e. k and ω) explicitly down to the wall boundary. The ensemble-aver-aged transport equations of the SST model are given as:

F = FDc dispersion

Dd dispersion

TD DTd

db

d

C CSc

, ,− =∇µ

ρα

ααα

αd

c

c−

∇

F = FNDc dispersion

NDd dispersion

TDc c cC k, ,− = ∇ρ α

′′ =+−

Ed

d

1 2

1

αα



(48)

where σk3, σω3, λ3 and β3 are the model constants which are evaluated based on the blendingfunction F1. The shear induced turbulent viscosity µts,l is given by:

, (49)

The success of SST model hinges on the use of blending functions of F1 and F2 which govern thecrossover point between the k-w and k-e models. The blending functions are given by:

,

,(50)

Here, default values of model constants were adopted. More detail descriptions of these model con-stants can be found in Menter (1994). In addition, to account the effect of bubbles on liquid turbu-lence, the Sato’s bubble-induced turbulent viscosity model was also employed (Sato et al., 1981).The turbulent viscosity of liquid phase is therefore given by:

(51)

and the particle induced turbulence can be expressed as:

(52)

For the gas phase, dispersed phase zero equation model was adopted and the turbulent viscosi-ty of gas phase can be obtained as:

(53)

where σg is the turbulent Prandtl number of the gas phase

µρρ

µσt g

g

l

t l

g,

,=

µ ρ αµtd l p l g S g lC D U U, = −⇀ ⇀

µ µ µt l ts l td l, , ,= +

Φ2 20 09

500=

max

.,

k

d dl

l n

l

l l nωµ

ρ ωF2 2

2= tanh( )Φ

Φ1 20 09

500 4=

+min max

., ,

k

d d

k

Dl

l n

l

l l n

l l

ωµ

ρ ωρ

ωσσω 22dn

F1 14= tanh( )Φ

S S Sij ij= 2µ ρωts l

l

l

a k

a SF, max( , )= 1

1 2

− −∂∂

∂∂

+ −2 11

1 12

3ρ ασ ω

ω α γ ω ρ βω

ll

l

j

l

jl

l

lk l lF

k

x x kP( ) , 33

2ω l

∂∂

+ ∇ ⋅( ) = ∇ ⋅ + ∇

ρ α ω ρ α ω α µµσ

ωω

l l ll l l l l l

t llt

u ( ),

3 ⇀

∂∂

+ ∇ ⋅( ) = ∇ ⋅ + ∇

ρ α ρ α α µµσ

l l ll l l l l l

t l

kl

k

tu k k( ),

3 + − ′α ρ β ωl k l l l lP k,

⇀



Figure 8. Schematic drawing of the test section of Hibiki et al. (2001) experiment

5. NUMERICAL STUDIES ON GAS-LIQUID BUBBLY FLOWSSpecific modeling approaches and techniques in the context of computational fluid dynamics toresolve isothermal bubbly flows using the two-fluid formulation based on the interpenetratingmedia framework are discussed via numerical studies described below. Numerical details andexperimental setups adopted for validations are firstly presented.

5.1. Experimental detailsA brief discussion of the three experimental setups for the isothermal bubbly turbulent pipe flowsare provided below.

5.1.1. Experimental setup by Hibiki et al. (2001)This is an isothermal bubbly turbulent pipe flow experiments performed at the Thermal-Hydraulicsand Reactor Safety Laboratory in Purdue University (2001). The test section of the experimentcomprised of an acrylic round pipe with an inner diameter D = 50.8 mm and a length of 3061 mm.Temperature of the apparatus and circulation water was kept at a constant temperature (i.e. 20oC)within a deviation of ±0.2oC controlled by a heat exchanger installed in a water reservoir. Air bub-bles were introduced through a porous media with the pore size of 40µm by a compressor and pre-mixed with the purified water in a mixing chamber. Local flow measurements using the double sen-sor and hotfilm anemometer probes were performed at three axial (height) locations of z/D = 6.0,30.3 and 53.5 and 15 radial locations of r/R = 0 to 0.95. Experiments at a range of superficial liq-uid velocities jf and superficial gas velocities jg were performed covering most bubbly flow regionsincluding finely dispersed bubbly flow and bubbly-to-slug transition flow regions.

5.1.2. MTLOOP experiment by Lucas et al. (2005)Figure 9 depicts the schematic diagram of configuration details of the MTLOOP experiment byLucas et al. (2005). Two-phase flow was measured in a vertical cylindrical pipe with the height of3500mm and an inner diameter of 51.2mm. Water with a constant temperature of 30ºC was circu-

3061mm

50.8mm

z/D = 53.5

z/D = 30.3

z/D = 6.0

Bubble-water mixture introduced form mixing



lated from the bottom to the top of the pipe. Air was injected via an injection device which consistsof 19 capillaries with an inner diameter of 0.8mm. These capillaries were equally distributed overthe cross section area of the pipe. To measure the local instantaneous gas fraction void fraction aswell as bubble size distribution, an electrode wire-mesh sensor is placed above a certain distancefrom the injection device, which can be varied from 30mm to 3030mm (i.e. corresponding todimensionless axial location Z/D=0.6-60). Gas and liquid flow rates were closely controlled thecompressed air inlet and circulated pump with the maximum superficial velocities: jg=14m/s andjf=4m/s respectively.

Figure 9. Schematic drawing of the test section of MTLOOP experiment

5.1.3. TOPFLOW experiment by Prasser et al. (2007)In the TOPFLOW test facility, as depicted in Figure 10, large vertical cylindrical pipe with theheight 9000mm and inner diameter of 195.3mm inner diameter was adopted as test section (Prasseret al., 2007). Following the MTLOOP experiment, water was circulated from the bottom to the topwith a constant temperature of 30ºC maintain by a heat exchanged installed in the water reservoir.The maximum superficial velocities for gas and liquid phase were also identical to the MTLOOPexperiment (i.e. jg=14m/s and jf=4m/s). Nonetheless, entirely different gas injection device wasemployed. Different from MTLOOP, a variable gas injection system was constructed by equippingwith gas injection units at 18 different axial positions from Z/D=1.1 to Z/D=39.9. Three levels of

D=51.2mm

Z: The distance between measuring position and the gas injection capillaries

3500mm

Gas injection via capillaries and Water

Measuring Positions

Z/D=0.6

Z/D=59.2

Z/D=4.5

Z/D=29.9

Coalescence dom

inant

Large and wide range bubble size

Small and narrow range bubble size



air chambers were installed at each injection unit. The upper and the lower chambers have 72 annu-lar distributed orifices of 1mm diameter for small bubble injection; while the central chamber has32 annularly distributed orifices of 4mm diameter for large bubble injection. A fixed wire-meshsensor was installed at the top of the pipe where instantaneous information of gas volume fractionand bubble size distribution was measured.

Figure 10. Schematic drawing of the test section of TOPFLOW experiment

9000mm

D=195.3mm

Z: The distance between measuring position and the gas injection units

Gas injection unit locations

Z/D=1.7

Fixed wire-mesh measuring sensor

Z/D=13.0

Z/D=39.9

Z/D=1.1

Z/D=7.7

Water

Breakup dom

inant

Large and wide range bubble size

Small and narrow range bubble size



5.1.4. Effect of injection method on bubble size evolution from observationsExperimental observations attested that completely different trend of bubble coalescence andbreakup behaviour were exhibited in the three experiments due to the difference of the adoptedinjection methods. In the experiment of Hibiki et al. (2001), for the flow conditions within the bub-bly flow regime, experimental observations found that there is no significant development of thebubble size along axial direction. They concluded that such insignificant changes of bubble size arecaused by the breakage and coalescence rate among bubbles are nearly equilibrium throughout theexperiment.

In the TOPFLOW experiment, with the gas injection orifices located on the circumference of thepipe, high concentrated bubble were formed within a very close distance from the wall. This close-ly packed swarm of bubbles were then immediately merged with others forming larger bubble. Thisrapid bubble coalescence mechanism also explained why a bimodal bubble size distribution with awide range bubble size (i.e. from 0 to 60mm bubble diameter) was acquired by the sensor.Following the liquid flow direction, these bimodal distributed bubbles from the injection unit werethen gradually disintegrated into smaller bubbles and collapsed back to mono-peaked distributionat the top of the test section.

Unlike the TOPFLOW experiment where bubble breakup mechanism is observed to be domi-nant, bubble coalescence mechanism was found to be prevailing in MTLOOP experiment. With thegas being injected via 19 equally distributed capillaries, bubbles were uniformly distributed acrossthe pipe inlet and exhibited a mono-peaked distribution with considerably narrow size range (i.e.from 2 to 10mm bubble diameter). As the bubble travelled upward, larger bubbles were formed viacoalescence processes which eventually broaden the bubble size range at the outlet of the pipe.From the above observations, under the same flow condition but different gas injection methods,one should notices that a completely different trend of bubble coalescence and breakup processeswere found in the two experiments. This is also the initial motivation of selecting these two exper-iments for the present validation study.

5.2. Numerical detailsTwo population balance approaches based on the class method (i.e. MUSIG and ABND model) areselected for the present investigation. Predictions of both models are validated against the abovethree experimental data. For all isothermal gas-liquid bubbly flow conditions, the generic CFDcode ANSYS-CFX 11 (2006) was utilised to handle the two sets of equations governing conserva-tion of mass and momentum. The average bubble number density transport equation with appro-priate coalescence and break-up kernels by Yao and Morel (2004) was also implemented throughthe CFX Command Language (CCL). Radial symmetry has been assumed in all experimental con-ditions. Numerical simulations were performed on a 60o radial sector of the pipe with symmetryboundary conditions at both vertical sides. At the pipe outlet, a relative averaged static pressure ofzero was specified.

At the test section inlet of Hibiki et al. (2001), uniformly distributed superficial liquid and gasvelocities, void fraction and bubble size were specified. For the MTLOOP experiment, with refer-ence to its gas injection method, a uniform gas volume fraction was specified at the inlet boundary.On the other hand, for the wall injection method in TOPFLOW, 12 equally spaced point sources ofthe gas phase were placed at the circumference of the 60o radial sector. Gas injection rate of eachpoint source were assumed to be identical. For all flow conditions, reliable convergence criterionbased on the RMS (root mean square) residual of 1.0×10-4 were adopted for termination of simu-lation. Details of the boundary conditions for different flow conditions are summarized in Table 1.For the experiment of Hibiki et al. (2001), based on the equilibrium bubble interactions, the homo-geneous MUSIG model which assumed all bubbles travelling with the same velocity was applied.In contrast, subject to the more rigorous bubble interactions, inhomogeneous MUSIG model wasemployed for the MTLOOP and TOPFLOW experiments. To be consistent with the interfacial liftforce model where the lift force coefficient changes its sign if bubble large than 5.5mm (Tomiyama,1998), gas bubbles were grouped into two dispersed phases which travel with two individual flowfield. The first dispersed phase was assigned for bubbles range from 0-6mm while the second dis-persed phase was dedicated for larger bubbles.



Table 1. Bubbly flow conditions and its inlet boundary conditions employed in thepresent study

6. RESULTS AND DISCUSSION6.1. Prediction results for equilibrium breakage and coalescence conditions6.1.1. Local distributions of Void fractionFigure 11 compares the gas void fraction profiles obtained from the ABND and homogeneousMUSIG model with the experimental data measured at the location of z/D = 53.5 (i.e. close to theexit of channel) for three different bubbly turbulent pipe flow conditions. The high void fractionsclose to wall proximity typically characterised the “wall peak” behaviour. Its phenomenologicalestablishment can be best described by the balance between the positive lift force that acted toimpel the bubbles away from the central core of the flow channel and the opposite effect beingimposed by the lubrication force preventing the bubbles from being obliterated at the channel walls.

Despite of similar trends predicted by both model, the predicted void fraction peaks, on closerexamination, appeared to be leaning more towards the channel wall in contrast to the actual bubbledistributions observed during experiments. Against the assessment on other wall lubrication mod-els by Frank et al. (2004) and Tomiyama (1998), a much lower than expected wall force determinedvia Antal et al. (1991) model purported to be the most probable cause of the discrepancy. As afore-mentioned, the two-phase turbulence modelling that could affect the magnitude of the turbulenceinduced dispersion force could also contribute to the additional modelling uncertainty. Overall, all

MTLOOP Experiment

Case M107 Case M118

[0/ =DZlj ] (m/s) 1.017 1.017

[0/ =DZlj ] (m/s) 0.140 0.219

[0/ =DZgα ] (%) [12.1] [17.72]

[00.D/zSD

=] (mm) [5.14] [6.38]

TOPFLOW Experiment

Case T107 Case T118

[0/ =DZlj ] (m/s) 1.017 1.017

[0/ =DZlj ] (m/s) 0.140 0.2194

[0/ =DZgα ] (%) [12.1] [17.72]

[00.D/zSD

=] (mm) [20.18] [23.28]

Hibiki et al. (2001) Experiment

Case H01 Case H02 Case H03

[0/ =DZlj ] (m/s) 0.491 0.986 0.986

[0/ =DZlj ] (m/s) 0.0556 0.0473 0.113

[0/ =DZgα ] (%) [10.0] [5.0] [10.0]

[00.D/zSD

=] (mm) [2.5] [2.5] [2.5]



the model predictions of the void fraction profile at the two measuring locations were in satisfac-tory agreement with measurements.

Figure 11. Predicted radial void fraction distribution and experimental data of Hibiki et al.(2001) at measuring station of z/D =53.5

6.1.2. Local distributions of Sauter Mean bubble diameter Figure 12 illustrates the predicted and measured Sauter mean bubble diameter radial profiles at thelocation of z/D = 53.5. As the bubble breakage and coalescence rate are in equilibrium, the Sautermean bubble diameter profiles remained roughly unchanged throughout the whole channel.Overall, bubble size changes were found mainly due to the bubble expansion caused by the staticpressure variation along the axial direction. As demonstrated by the good agreement between thepredicted and the measured bubble diameters at the channel core, the ensemble bubble expansioneffect was adequately captured by the two-fluid approach via the ideal gas assumption. The slight-ly larger bubbles were formed near the wall might be due to the tendency of small bubbles migrat-ing towards the wall creating higher concentration of bubbles thereby increasing the likelihood ofpossible bubble coalescence. In general, predictions from all models agreed reasonably well withthe measurements. For all the flow cases and locations especially for the MUSIG, predictions ofthe model were in remarkable agreement with the measurement and were also obviously superiorto the ABND model.

<jf>=0.491 m/s <jg>=0.0556 m/s

<jf>=0.986 m/s <jg>=0.0473 m/s (a)

(b)

<jf>=0.986 m/s <jg>=0.113 m/s

(c)

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Gas

Vol

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tion

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Mesurement ABND MUSIG

Z/D=53.5

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Z/D=53.5



Figure 12. Predicted Sauter mean bubble diameter distribution and experimental data ofHibiki et al. (2001) at measuring station of z/D =53.5

The above results clearly that population balance of bubbles was accurately captured by usingthe “Multiple Size Groups” approach. Compared to the single average parameter of ABND mod-els, higher resolution of multiple size groups were found to be better resolved through the dynam-ical changes of bubbles size distribution. Since the bubble Sauter diameter is generally closely cou-pled with the interfacial momentum forces, as demonstrated in our previous study (Cheung et al.,2007b), better predictions of the bubble Sauter diameter could improve the prediction of liquid andgas velocities. Unfortunately, as extra transport equations were required in the numerical calcula-tions, additional computational effort is required to solve these equations.

6.1.3. The evolution of bubble size distributionOn the other hand, the advantage of the MUSIG model lies on its flexibility in tracking the evolu-tion of bubble size distribution throughout the whole computational domain. Figure 13 shows thedevelopment of the size fraction of each bubble classes along the radial direction at the location ofz/D = 53.5. Since the turbulence intensity is relatively low at the channel core (i.e. r/R = 0.05), bub-ble sizes remained unaffected owning to the insignificant bubble coalescence and bubble breakagerates. With increasing number of bubbles driven by the lift force and the rising turbulence intensi-ty within the boundary layer towards the channel walls (i.e. r/R=0.8 and 0.95), bubble coalescenceand bubble breakage became increasingly noticeable forming larger bubbles and re-distributed theBSD to higher bubble classes. Such behaviour was amplified especially for cases with relativelyhigher void fraction (e.g. Figure 11a,c).

<jf>=0.491 m/s <jg>=0.0556 m/s

<jf>=0.986 m/s <jg>=0.0473 m/s

(a) (b)

<jf>=0.986 m/s <jg>=0.113 m/s

(c)

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

SDi [

mm

]



Z/D=53.5

0.0 0.2 0.4 0.6 0.8 1.00

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SDi [

mm

]



Z/D=53.5

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

SD

i [m

m]



Z/D=53.5



Figure 13. Predicted size fraction of each bubble classes and its evolution along radialdirection at the measuring station of z/D =53.5

6.2. Prediction results for Coalescence or Breakage dominant conditions6.2.1. Local distributions of Void fractionBased on the above encouraging results, discussion is now focusing on more rigorous bubble inter-actions. Figure 14 shows the predicted volume fraction distributions obtained from the ABND andinhomogeneous MUSIG model in comparison with the MTLOOP measurements at dimensionlessaxial position (i.e. Z/D=4.5 and 60) for the flow conditions of case M107 and M118. In general, itis observed that both population balance models are capable to capture the transition process from“wall peak” to “core peak” of the gas volume fraction distribution. This clear attested that theadopted interfacial force models (especially for the lift force model) successfully predicted thetrend of the gas phase lateral motions and its sequential de-mixing behaviour of bubbles. However,as depicted in Figure 14c-d, both models under-predicted the core peaking gas volume fraction atthe measuring station Z/D=60.

One the other hand, Figure 15 shows the comparison of the predicted volume fraction distribu-tions of both models and the TOPFLOW experimental data at dimensionless axial positionsZ/D=1.7 and 40. Align with the above observation, the dynamical changes of wall peaking and corepeaking behaviours were been successfully captured by the ABND and inhomogeneous MUSIGmodel. For the Case T118, as shown in Figure 15d, the predicted gas volume fraction profiles fromboth models agreed well with the measurements. Overall, for all flow conditions of the two exper-iments, predictions of the ABND model are comparable to the inhomogeneous MUSIG model.Nonetheless, similar to the MTLOOP experiment, both models under-predicted the core peak valuein Case T107 (see Figure 15c).

One plausible reason for the under-estimation of the core peak could be attributed to the neigh-bouring effects of swarm of bubbles. Within the swarm of bubbles, interfacial forces acting on bub-bles are affected by the closely packed neighbouring bubbles. Based on the findings from experi-ments, Simonnet et al. (2007) concluded that the drag force acting on bubbles decreases signifi-cantly if the gas volume fraction exceeded the critical value around 15%. Back to our present study,gas volume fractions at the core peak ranged from 18% to 32% suggesting that the neighbouringeffect could become influential at centre of the pipe. Nonetheless, in our calculation, interfacialforce models which predominantly developed based on isolated bubbles were adopted. Withoutconsidering the influence of neighbouring bubbles, these models may introduce errors to the inter-facial momentum transfer calculation causing the resultant under-estimation of the core peak val-ues

12

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i [-

]

Bubble classRadial lo

cation

<jf>=0.986 m/s <jg>=0.113 m/s z/D=53.5



Figure 14. Comparison of the predicted radial gas volume fraction distributions of theABND and Inhomogeneous MUSIG model with the MTLOOP measurments: (a) CaseM107 — Z/D =4.5; (b) M118 — Z/D =4.5; (c) M107 — Z/D =60 and (d) M118 — Z/D=60

6.2.2. Local distributions of Sauter Mean bubble diameter The dynamical changes of bubble size distribution dictates fundamental interfacial area betweengas and liquid phase; which is closely coupled with the interfacial momentum transfer and the gasvolume fraction profile, a close investigation of both population balance approaches in predictingthe dynamical changes of bubble size is essential. Table 2 summarizes the measured and predictedcross-sectional averaged sauter mean bubble diameter of both models corresponding to flow con-ditions and locations in Figure 14 and Figure 15. In general, predictions from the ABND modelwere comparable with the measurements and numerical results of MUSIG model. Although a rel-atively simple mathematical expression is adopted, encouraging results clearly demonstrated thatthe main trends of the bubble size evolution were well captured by the ABND model. By trackingthe bubble size distribution explicitly with discrete bubble class, it is no surprise that predictions ofthe inhomogeneous MUSIG appear marginally superior to those from ABND model. Nonetheless,one should be also noticed the MUSIG requires to solve extra transport equations posing signifi-cant burden to both computational time and resources

0.0 0.2 0.4 0.6 0.8 1.00.00

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0.40

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(a) (c)

(b) (d)



Figure 15. Comparison of the predicted radial gas volume fraction distributions of theABND and Inhomogeneous MUSIG model with the TOPFLOW measurments: (a) CaseT107 — Z/D =1.7; (b) T118 — Z/D =1.7; (c) T107 — Z/D =39.9 and (d) T118 — Z/D=39.9

6.3. Limitations and shortcomings of existing modelsThe above encouraging results clearly demonstrated the capability of existing population balancemodel and its viable applications in resolving isothermal bubbly turbulent pipe. Nevertheless, theflow cases that have been investigated from above are predominately within the bubbly-flowregime. As discussed in previous sections, most of the existing bubble mechanism kernels weredeveloped for bubbly flow regime. Directly adopt these kernels to other flow regimes couldnonetheless introduce significant error with the calculations. This could be exemplified in the fol-lowing investigation where transitional bubbly-to-slug flow condition with liquid superficial veloc-ity <jf>=0.986m/s and gas superficial velocity <jg>=0.242m/s as measured by Hibiki et al. (2001)were considered.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.06

0.12

0.18

0.24

0.30

Gas

Vol

ume

Frac

tion

[-]



0.0 0.2 0.4 0.6 0.8 1.00.00

0.06

0.12

0.18

0.24

0.30


Gas

Vol

ume

Frac

tion

[-]


0.0 0.2 0.4 0.6 0.8 1.00.00

0.06

0.12

0.18

0.24

0.30

Gas

Vol

ume

Frac

tion

[-]



0.0 0.2 0.4 0.6 0.8 1.00.00

0.06

0.12

0.18

0.24

0.30


Gas

Vol

ume

Fra

ctio

n [-

] Measurement ABND Inhomo. MUSIG

(a) (c)

(b) (d)



Table 2. Comparison of bubbly size diameter predictions by inhomogeneous MUSIG andABND Method

Figure 16 shows the measured and predicted local radical void fraction, Sauter mean bubblediameter, IAC and gas velocity distribution at the measuring station of z/D=53.5. Comparing thepredicted Sauter mean diameters, the inhomogeneous MUSIG model was found to yield compara-tively better prediction when compared against the measured data. This could be attributed to themerit of splitting the bubble velocity with two independent fields which facilitated the model to re-capture the separation of small and big bubbles caused by different lift force actuation.Nevertheless, notable discrepancies were found when comparing against other variables (i.e. voidfraction, gas velocity and IAC) against the experimental measurements. As depicted in Figure 16b,void fractions of both models were obviously over-predicted at the channel core but under-predict-ed at the wall region, which resulted in unsatisfactory IAC predictions (see Figure 16c).Interestingly enough, the consideration of multiple velocity fields in the inhomogeneous MUSIGmodel did not contribute to the desired expected improvements when comparing the gas velocitypredictions against those of the homogeneous MUSIG model (see Figure 16d). In this transitionflow regime, coalescence due to wake entrainment and breakage of large bubbles caused by sur-face instability may prevail beyond the critical void fraction limit. The existing kernels which onlyfeatured coalescence due to random collision and breakage due to turbulent impact for sphericalbubbles have to be extended to account for additional bubble mechanistic behaviours for cap/slugbubbles. Several papers in the literature have attempted to deal with this problem by the develop-ment of a two-group interfacial area transport equation (Fu et al., 2002a,b; Sun et al., 2004a,b).Encouraging results attained thus far not only suggested the feasibility of the proposed approachbut more importantly the prevalence of large bubble mechanisms that are substantially differentfrom spherical bubble interactions in turbulent gas-liquid flows. Furthermore, as discussed above,

MTLOOP Experiment Case M107 Case M118

[.ExpSD ] (mm) [5.29] [6.54]

[Inh.MUSIGSD ] (mm) [5.45] [6.42] Z/D=4.5

[ABNDSD ] (mm) [4.98] [6.35]

[.ExpSD ] (mm) [5.79] [7.6]

[Inh.MUSIGSD ] (mm) [5.57] [5.92] Z/D=59.2

[ABNDSD ] (mm) [5.27] [8.57]

TOPFLOW Experiment Case T107 Case T118

[.ExpSD ] (mm) [19.43] [21.96]

[Inh.MUSIGSD ] (mm) [18.78] [18.20] Z/D=1.7

[ABNDSD ] (mm) [17.58] [17.00]

[.ExpSD ] (mm) [7.49] [7.50]

[Inh.MUSIGSD ] (mm) [8.82] [6.86] Z/D=39.9

[ABNDSD ] (mm) [8.86] [8.48]



the interfacial force models which have been developed principally for isolated bubbles rather thanon a swarm or cluster of bubbles.

Figure 16. Predicted local radical void fraction, Sauter mean bubble diameter, IAC andgas velocity distribution of transition bubbly-to-slug flow condition by homogenous and

inhomogeneous MUSIG model

7. CONCLUSIONS AND FURTHER DEVELOPMENTSIn this paper, the state-of-the-art methodologies of population balance modelling for isothermalbubbly flows are reviewed. Some widely adopted or representative solution algorithms for the PBEare briefly discussed. Theoretical concepts and advantages of each approach are also revealed. Todemonstrate the capability of existing population balance model in capturing the bubble coales-cence and bubble breakage mechanisms, a complete three-dimensional two-fluid model coupledwith two population balance models (i.e. MUSIG and ABND model) of the class method was pre-sented to handle the complex hydrodynamics processes of various bubbly turbulent pipe flow con-ditions. In conjunction with the single average quantity approach (i.e. represented by the ABNDmodel), the homogenous MUSIG model which assumed all bubbles travelling with the same veloc-ity was applied and validated against the measurements of Hibiki et al. (2001). Overall, both ABNDand MUSIG model yielded good agreement for the local radial distributions of void fraction andSauter mean bubble diameter against measurements.

With the aim to thoroughly assess the performance of existing coalescence and breakage ker-nels, numerical studies were then performed to validate predictions of both models against twoexperimental data of Lucas et al. (2005) and Prasser et al. (2007). With two different gas injectionmethods, bubble coalescences were found to be dominant in MTLOOP experiment while the com-pletely opposite bubble breakup behaviours were prevalent in TOPFLOW experiment. Albeit flow

Radial Position [-]

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ter

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<jf>=0.986 m/s <jg>=0.242 m/s z/D=53.5

<jf>=0.986 m/s <jg>=0.242 m/s z/D=53.5

(a) (b)

<jf>=0.986 m/s <jg>=0.242 m/s z/D=53.5

<jf>=0.986 m/s <jg>=0.242 m/s z/D=53.5

(c)

(d)



conditions with wide range of bubble size and rigorous bubble interaction were considered, pre-dictions of both models were in satisfactory agreement with the measurements. The transition of“wall peak” to “core peak” gas volume fraction profiles were successfully captured by both mod-els. Encouraging results clearly exemplified the capability of both population balance models incapturing the dynamical changes of bubble size due to coalescence and breakup processes.Compared with inhomogeneous MUSIG model, the ABND model produced marginally inferiorresults in terms of sauter mean bubble diameter. These less favourable results could be resultedfrom its drawback of representing the wide bubble size range with a single average quantity. On theother hand, even though perceptible advantages have been demonstrated by the inhomogeneousMUSIG model, it also suffered from the extra computation resources needed for calculating thetransportation of each discrete bubble class.

To reveal the limitation of existing model, numerical studies were also performed to investigatethe feasibility of application in handling transition bubbly-to-slug flow. It was observed that theinhomogeneous MUSIG gave better prediction of the Sauter mean bubble diameter distributionswhen compared to the homogeneous model. The more complicated inhomogeneous MUSIG modelprovided the premise of better capturing the different effects of the lift force acting on the small andlarge bubbles. Less encouraging results were however ascertained between the predicted and meas-ured void fraction, interfacial gas velocity and IAC profiles.

From the above numerical study, as an example of population balance modelling, some impor-tant numerical issues have been demonstrated. Firstly, it exposed the limitation of the existing inter-facial forces models. Most of the existing interfacial forces models were developed and calibratedfrom isolated single particle which may not be strictly applicable if particles are closely packed.This is due to the fact that particle motion may be influenced by neighbouring particles resultingdifferent momentum transfer. According to the recent experimental study by Simonnet et al. (2007),significant decrement of gas bubble drag coefficient was found if the gas void fraction exceeded15%. In fact, very recently, some research studies have been performed attempting to improve theexisting interfacial drag and lift force models (Marbrouk et al., 2007; Hibiki and Ishii, 2008; Liu etal, 2008)

Secondly, it unveiled the constraint of the current coalescence and breakage kernels. For bubblysimulations presented in this review, coalescence and breakage kernels were derived based on thespherical bubble assumption. This assumption limits the kernels to be only applicable to bubblyflow regime where bubbles are in spherical shape. This also explains why less encouraging resultswere obtained when bubbly-to-slug flow was considered. In bubbly-to-slug flows, subject to thebalance of the surface tension and surrounding fluid motion, large bubbles may be deformed intocap or taylor bubbles. Thus, coalescence due to wake entrainment may become significant whichunfortunately was not modelled in current adopted kernels. Unfortunately, fundamental knowledgeof bubble coalescence and breakage mechanisms still remain elusive forming a bottleneck for thedevelopment of more robust population balance kernels. In essence, not only for gas-liquid sys-tems, similar problems and challenges are also prevalent in other PBE applications; for example:soot formation prediction in combustion system (Frenklach and Wang, 1994). It is certainly thatsubstantial research works would be centred in kernels development in near future.

Finally, although encouraging results had been obtained from the MUSIG model based on CM,one should be also reminded that QMOM or other moment methods which represent a rather soundmathematical approach and an elegant tool of solving the PBE with limited computational burdencould be alternatively considered for modelling practical multiphase flow systems for future inves-tigative studies. Figure 17 shows the predicted void fraction and sauter mean bubble diameter pro-files of the DQMOM in comparison with MUSIG model predicted and measured results of Hibikiet al. (2001). Overall, DQMOM model predictions were in satisfactory agreement with measure-ments and comparable with the MUSIG model results. The bubble size distributions near the pipecenter and near the pipe wall at z/D = 53 are shown in Figure 18. As can be seen in the two figures,the dominant bubble size predicted by DQMOM was 2.5 mm near the pipe center and 5.0 mm nearthe pipe wall, which provided some insight into the shape of the underlying different bubble sizedistributions. The above encouraging results clearly exemplified the potential of the method ofmoment (MOM) in population balance modelling of gas-liquid bubbly flows.



Figure 17. Local predicted and measured void fraction and sauter mean bubble diameterprofiles at z/D = 53.5

Figure 18. Bubble Size Distributions Near the Pipe Center (top) and Near the Pipe Wall(bottom) at z/D = 53.5



ACKNOWLEDGMENTThe authors would like to acknowledge Dr. Lucas and Dr. Krepper on their valuable discussionsand assistances on the present work. The financial support provided by the Australian ResearchCouncil (ARC project ID DP0877734) is also gratefully acknowledged.

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