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Chemical Engineering Science 56 (2001) 1049}1056

Phase distribution phenomena in two-phase bubble column reactorsHugo A. Jakobsen

Department of Chemical Engineering, The Norwegian University of Science and Technology, NTNU Sem Saelands vei 4, N-7491 Trondheim, Norway

Abstract

In a recent paper, Jakobsen, Sannvs, Grevskott and Svendsen (Ind. Eng. Chem. Res. 36(10) (1997) 4052}4074) presented a review ofthe present status on #uid dynamic modeling of vertical bubble-driven #ows. Special emphasis was placed on two-phase #ows inbubble column reactors. For these multiphase reactors, the averaged Eulerian multi#uid models have been found to representa trade-o! between accuracy and computational e!orts for practical applications. Unfortunately, in such multi#uid models

constitutive relations are needed to describe the phase interaction processes. It was concluded that the general picture from theliterature is that time-averaged liquid velocity "elds are reasonably well predicted both with steady-state and dynamic models of thistype. The prediction of phase distribution phenomena, on the other hand, is still a problem, in particular at high gas #ow rates. Thepresent paper gives an overview of the pertinent constitutive relations presented in the literature aiming at a "rm mechanisticprediction of the phase distribution phenomena. This includes transversal forces, steady drag forces, surface tension e!ects, andhydrodynamic bubble}bubble and bubble}wall interactions. Several interaction mechanisms in the turbulence "elds like the so-calledturbulent mass di!usion, turbulent migration, turbulent drift velocities, anisotropic turbulent drag forces, as well as the interactionsbetween these mechanisms, and the impacts of variations in bubble size and shape distributions are discussed. Various aspects of these

relations have been questioned. It is therefore the aim of this paper to compare the capabilities of the existing Eulerian multi #uidmodeling concepts and parameterizations. The various approaches are evaluated using an in-house 2D Euler/Euler steady-state code.

There are several reasons why we choose a steady 2D model for evaluation. First, the model has the advantage of being relatively

simple, thus the computational e!ort required for practical applications involving chemical reactions, and interfacial heat and masstransfer is feasible. Second, the dynamic axisymmetric 2D models do not give much improvement as the #ow phenomena possiblymissing are believed to be 3D. That is, they do not resolve the swirling motion of bubble swarms. Third, most parameterizations

presented in the literature are based on, and have so far only been applied to, 2D models. The results obtained indicate that thevarious phase interaction parameterizations available in the literature predict very di!erent phase distributions in bubble columns.For operating reactors these deviations will signi"cantly in#uence the predicted process performance. The results presented here thuscon"rm the demand for improved modeling including more accurate and stable numerical solution algorithms. Low-accuracyalgorithms may totally destroy the physics re#ected by the models implemented. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Multi#uid model; Forces; Turbulence closure; Bubble size and shape; Numerical methods

1. Introduction

There are several ways to formulate averaged Eulerian

#uid dynamic models intending to describe the bubblecolumn reactor performance. The traditional concept is"rst to formulate integral balances for mass, momentumand energy for a "xed control volume containing allphases in question. These balances must all be satis"ed atany time and point in space, and thus reduces into two

types of local equations, one being the local instan-

taneous equations for each phase and the other an ex-

pression of the local instantaneous jump conditions. The

next step in this fairly general procedure is to form the

E-mail address: [email protected] (H. A. Jakobsen).

average of the local instantaneous transport equations

and the corresponding jump conditions. For this pur-

pose, many di!erent averaging procedures have been

presented in the literature. These are the volume-aver-aging, time-averaging, and the ensemble-averaging

procedures, and combinations of these basic single-

averaging operators. Unfortunately, the resulting

averaged equations cannot be solved directly, as they

contain averages of products of the dependent variables.

The third step is thus to obtain a solvable set of equations

by relating the averages of products to expressions con-

taining products of averaged variables only. This has

commonly been done either by employing procedures

similar to conventional single-phase Reynolds decompo-

sition, or procedures similar to single-phase Favre

0009-2509/01/$- see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 09 - 2 5 0 9 ( 0 0 ) 0 0 3 2 1 - 3

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decomposition before averaging. Anyway, these proced-

ures give rise to extra terms in the transport equations,

containing covariances of the #uctuating components,analogous to the Reynolds stress terms obtained in the

case of single-phase turbulence modeling. The "nal stepin this procedure is therefore to formulate proper closure

laws and parameterizations.

Similar models have also been formulated based onalternative concepts like kinetic and probabilistic the-

ories. It is beyond the scope of this paper to consider all

possible model formulations available. Fortunately, the

model formulations most frequently used describing

bubble column reactors can roughly be divided into two

groups:

One group of bubble column models is based on the

traditional concept described above including a double-

averaging operator, the time- (or ensemble-) after volume-

averaged models (see Jakobsen et al., 1997, and references

therein). The covariance terms obtained from the volume

averaging part of the procedure are usually neglected. Inthe time- or ensemble-averaging part, the Reynolds or

Favre decomposition and averaging rules are normally

applied, in agreement with single-phase turbulence

modeling procedures, in order to separate the #uctuatingcomponents of the variables from the time- (or ensemble-)

averaged variables. In case of applying Favre averaging

only the velocity variables are treated by the Favre ap-

proach, the remaining variables are averaged in accord-

ance with the standard Reynolds procedure.

The second group of bubble column models is based

on the same modeling concept but includes a single-

ensemble averaging operator only (e.g. Kumar et al.,

1995; Friberg & Hjertager, 1998; Mudde & Simonin,1999). In contrast to the "rst group of bubble columnmodels, the main purpose of applying the Reynolds or

Favre decomposition and averaging procedures is now

to separate the averages of products into products of

averages. In most cases, applying the Favre procedure

several variables have been weighted, not only the velo-

city as in the other group of models. The resulting trans-

port equations are thus similar, but not identical to the

equations obtained in the "rst group of bubble columnmodels. In other words, the closure laws found in the

literature are thus valid for the model approach deter-

mining the basis for their derivation only, and are notalways generally valid for any model formulation.

2. Closure

Several alternative parameterizations are available in-

tending to describe the pertinent phase distribution

mechanisms. As the distinction between the di!erentconstitutive relationships are not always obvious, care

has to be taken to ensure that the resulting model formu-

lation chosen is consistent. In the following, the various

phase interaction parameterizations commonly used in

bubble column modeling are very brie#y described. Forfurther details, the interested readers are referred to the

original papers.

2.1. Interfacial momentum transfer

The investigation of the fundamental behavior of dis-persed gas}liquid #ow systems in bubble columns areperformed by modeling the generalized or total drag

force as a linear combination of the various drag compo-

nents. The forces assumed to be important for a funda-

mental description of the dispersed #ow in bubblecolumns are: the steady-interfacial drag, the added mass

force, the transversal lift forces, the Basset force and the

interfacial mass momentum transfer rate. The approach

chosen is to separate the various forces as far as possible

in order to verify the signi"cance of the di!erent e!ects asthey are developed and validated. In this paper, focus will

be put on the steady interfacial drag and the transversallift forces as they are considered the important ones

determining the phase distribution in these reactors. The

standard interfacial steady-drag force given by Ishii and

Mishima (1984) is usually applied. In this formulation,

the ratio of the Sauter mean diameter to the drag dia-

meter appears as a shape factor. In this work, the shape

factor is set equal to 1.

The physical phenomena giving rise to lift on single

bubbles in liquids can roughly be divided into three

groups. The Magnus lift force is due to initial bubble

rotation causing an asymmetric pressure distribution

around the bubble. It was early observed that the lift

force on a sphere could act in the opposite direction ofthe standard Magnus force at high Reynolds numbers

due to the transition from laminar to turbulent boundary

layers on the two sides of the sphere. A second lift force,

the Sa!man force, acts on non-rotating bodies due toshear in the continuous phase #ow pattern. It has beenfound that at low Reynolds numbers, the lateral force due

to shear should be an order of magnitude larger than

that due to particle rotation. At intermediate to high

Reynolds numbers, as in bubble columns, only very

idealized theories on these forces based on potential

theory have been presented. Auton (1983, 1987), and

Drew and Lahey (1987) developed equal formulations forthe lift force on a sphere in an inviscid #uid. The surfacepressure variations which depend on the #uid viscosity inthe immediate vicinity of the bubble were estimated by

prescribing the boundary velocity pro"le.The tendency for bubbles to deform under various #ow

conditions allows for yet another lift force. Kariyasaki

(1987) observed a lift force on deformable particles of

opposite sign to that on rigid spheres under linear shear

#ow. In a later paper by Tomiyama, Miyoshi, Tamai,Zun and Sakaguchi (1998) this force was expressed as

a transversal force caused by a slanted wake behind

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a distorted bubble in a shear "eld (i.e. a turbulent wakeforce).

2.2. Turbulent mechanisms

It seems to be generally accepted that a turbulent mass

di!usion *or dispersion force should occur in the aver-

aged multi#uid model equations. In the double-aver-aging operator procedure, these mechanisms occur as

a phasic di!usion term in the continuity equations ap-plying the Reynolds-like averaging procedure, model 1.

Alternatively, these physical mechanisms occur as a dis-

persion force in the momentum equations if a Favre-like

procedure is applied (e.g. Laux, 1998), model 2.

In contrast, if a single-ensemble averaging procedure is

applied, the phase fraction is de"ned as an ensembleaverage and no dispersion forces occur explicitly in the

continuity equation in this model formulation. Conse-

quently, to account for these physical mechanisms,

Lahey, Lopez de Bertodano and Jones (1993) formulateda semi-empirical constitutive parameterization which do

not originate directly from the single-averaging proced-

ure applied. However, the resulting relation (model 2c) is

here considered to be an alternative to the two models

mentioned above.

Particle dispersion due to transport by #uid turbulencehas also been modeled applying the drift velocity concept

of Simonin and co-workers (e.g. Mudde & Simonin,

1999), model 3. This term occurs in the single-average

model equations due to correlations between the distri-

bution of the particles and the turbulent #uid motion.The parameterization of these correlations which is

based on semi-empirical analysis, is another alternative(i.e. based on Favre-averaged equations) to the di!usiveterms in the mass balance (i.e. which originates from the

Reynolds-averaged equations, model 1).

Simonin and co-workers also developed several more

sophisticated versions of this model including additional

transport equations based on kinetic *and probabilistictheories. To the author's knowledge, these models haveso far only been applied to gas-particle #uidization, andwill thus not be described in more detail.

If the turbulence is anisotropic, the dispersion coe$c-ient must be substituted with a tensor. Based on the

work by Reeks (1992a), Jakobsen (1993) (in co-operationwith Johansen, 1993) developed a turbulent drag force to

explicitly account for this anisotropic e!ect in theReynolds-averaged model. This term was found neces-

sary to enable a proper description of the phase distribu-

tion phenomena as an isotropic k} model was applieddescribing turbulence.

The spatial distributions of the turbulent normal stres-

ses will also act as forces on the dispersed phase. This

mechanism is based on the theory of bulk liquid turbu-

lent structure control (Subbotin, Ibraginor, Bobkov

& Tychinskii, 1971). If the liquid-phase turbulence struc-

ture alone controls the void distribution, the void peak-

ing would occur at the region of the largest turbulent

kinetic energy of the liquid phase. This model has been

used by Lahey and co-workers (e.g. Lahey, 1987, 1988,

1990; Lopez de Bertodano, Lee, Lahey & Drew 1990).

Applying the gradient hypothesis in modeling the

Reynolds-like stresses, the turbulent migration term is

embedded in the model through the turbulent kineticenergy term.

2.3. Axial wall force

According to Lahey et al. (1993), there is an axial wall

force acting on the bubbles near an axial wall due to the

"nite size of the bubbles and the steep velocity gradientnear the wall. It is supposed that there is a region of

interaction between the bubble and the wall although the

bubbles do not wet the wall. Between the wall and bub-

bles, located in the immediate vicinity of the wall there is

a thin liquid "lm of thickness of the order of the laminarsub-layer thickness. By introducing rough assumptions

estimating the gas wall shear, the shear area, the friction

coe$cient, and the laminar sub-layer thickness, theyderived an approximate wall shear force.

2.4. Hydrodynamic interaction forces

Antal et al. (1991) proposed a model of a lubrication

wall force that pushes bubbles located in the immediate

vicinity of the wall towards the column center.

Tomiyama et al. (1995, 1998) noted that this model pos-

sesses the defect that a bubble located far from the wall isattracted towards the wall. They modi"ed the originalmodel formulation in accordance with experimental data

for bubble trajectories for single bubbles released from

a near-wall region in a stagnant liquid.

Delnoij (1999) considered the relative importance of

the hydrodynamic bubble}bubble interaction force com-pared to the other components of the generalized drag

acting on a non-deformable spherical bubble. In the

study of Delnoij (1999) an Euler}Lagrangian model wasused to describe a bubble column operating in the homo-

geneous regime at very low void fractions ((5%). The

only indirect hydrodynamic interaction e!ect taken intoaccount was the one due to the disturbance of the localvelocity "eld around a speci"c bubble induced by theneighboring bubbles. This induced velocity will contrib-

ute to the lift force, the steady drag and the added mass

force acting on the bubble under consideration. Delnoij

(1999) neglected the hydrodynamic interaction contribu-

tions to both the lift and the steady drag forces. It was

then shown that the hydrodynamic interaction force

exerted by the bubbles on each other, being a modi"edadded mass e!ect only, did not signi"cantly in#uence themacroscopic #ow pattern observed.

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3. Model validation

The basic model used in this work originates from

more than 10 years research on bubble column modeling

at our department (e.g. Torvik & Svendsen, 1990; Torvik,

1990; Svendsen, Jakobsen & Torvik, 1992; Jakobsen,

1993; Jakobsen, Svendsen & HjarbDo, 1993; Jakobsen,

Sannvs, Grevskott & Svendsen, 1997). The originalmodel belong to the "rst group of bubble column modelsdescribed above, derived using a double time after vol-

ume-averaging operator approach. In the present work,

three versions of the original program code have been

developed.

Model 1 can be looked upon as an updated version of

the previous model (Jakobsen, 1993). In the time-aver-

aging part of this model version a Reynolds-averaging-

like procedure was adopted. For model evaluation and

sensitivity studies, the previous model has been updated

to include the interaction terms suggested in the litera-

ture described above. The reference model 1 was de"nedcontaining the following sub-models: the steady dragcoe$cient relation by Tomiyama, Sou, Zun, Kanami andSakaguchi (1995), Tomiyama et al., (1998), the added

mass, the turbulent mass di!usion in the continuity equa-tions, the turbulent migration due to the turbulent nor-

mal stresses, and the anisotropic drag force by Jakobsen

(1993), and the transversal forces by Tomiyama et al.

(1998). Note that the anisotropic drag force coe$cientapplied here has been tuned to experimental data. For

sensitivity analyses, the results predicted by the model

formulations de"ned below are compared with the corre-sponding data obtained using this reference model (i.e.

model 1). Model 1 variations: Model 1a, Reynolds-aver-aged basis model with no wake force (Tomiyama et al.,

1998). Model 1b, Reynolds-averaged basis model with no

anisotropic drag force. Model 1c, Reynolds-averaged

basis model with no turbulent migration force. Model 1d,

Reynolds-averaged basis model with lift force parameter

set to 0.35.

Model 2 is also based on the previous model (Jakob-

sen, 1993). However, in the time-averaging part of this

model formulation, a Favre-averaging like procedure

was adopted instead of the previous Reynolds procedure.

For model evaluation and sensitivity studies the same

updated interaction terms as used in model 1 have beenimplemented. The turbulent dispersion force of Laux

(1998) is used instead of the turbulent mass di!usionterms in the continuity equations. Model 2 variations:

Model 2a, Favre-averaged basis model with the lubri-

cation force by Tomiyama (1998). Model 2b, Favre-

averaged basis model with the axial wall force by Lopez

de Bertodano (1992) and Lahey et al. (1993). Model 2c,

Favre-averaged basis model with the turbulent disper-

sion force by Lopez de Bertodano (1992) and Lahey et al.

(1993). Model 2d, Favre-averaged basis model with the

drift velocity model by Mudde and Simonin (1999).

Model 3, Favre-averaged basis model with the drift

velocity model by Mudde and Simonin (1999), without

the phase dispersion force by Laux (1998).

For model validation, experimental data for an ordi-

nary bubble column have been used. The #ow is isother-mal and no interfacial mass transfer occur. Air/water

system data: J"1000 kg/m,

J"1.0;10\ Ns/m,

J"0.070 N/m, and E"1.2 kg/m

at the top of thereactor. Inner column diameter, D"0.288 m, column

height, H"4.25 m, temperature 253C, pressure 1 bar,gas and liquid super"cial velocities at inlet 0.08 and0.01 m/s, respectively. The void fraction data were mea-

sured at axial level 2 m above column inlet, whereas the

velocity variables are measured at level 1.6 m above inlet.

4. Results and discussion

The in#uence of drag on the phase distribution has

been studied. For high gas void fraction #ows, dragcorrelations based on empirical single bubble data havebeen found unable to predict gas velocity pro"les withreasonable accuracy (i.e. compared to experimental data

given by Jakobsen, 1993 and Yu and Kim, 1991), as can

be seen from Fig. 1(g). This may be due to the fact that

none of these drag relations do explicitly take into ac-

count the hydrodynamic bubble}bubble interaction ef-fects. Bubbles #owing in line or in wakes rise faster thansingle bubbles, but models intending to describe this

e!ect by including the e!ect of the bubble swarm's voidfraction into the expression for C

"remain unsatisfactory,

as observed by Jakobsen (1993). In particular, the void

fraction variable used for a bubble cluster does not ident-ify whether it is composed of many small bubbles or a few

larger ones. Delnoij (1999) did only consider very low

void fraction #ows ((5%) when concluding that thesemechanisms did not in#uence the main #ow pattern. Inbubble columns operating in the heterogeneous #owregion, having local void fractions up to about 30}40%,these interaction mechanisms are expected to be impor-

tant. Delnoij (1999) also suggested that the contributions

to the steady drag and lift forces can be implemented by

introducing steady drag and lift coe$cients that dependon the local gas fraction. For the Eulerian}Eulerian mod-

els usually applied for higher void fraction #ows, Jakob-sen (1993) found that such a procedure was not su$cientwhen using a single bubble size model. Bubble size and

shape distribution models are needed to enable a good

implementation of the hydrodynamic interaction e!ects.In line with this conclusion, Krishna, Urseanu, van Baten

and Ellenberger (1999) reported that the large bubble

swarm velocity was found to be 3}6 times higher thanthat of single isolated bubbles. It was stated that the

limited success in modeling bubble column #uid dynam-ics in the past is due to the lack of reliable procedures for

estimating the large bubble sizes and the corresponding


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Fig. 1. Results from the 2D steady Euler/Euler model versus measured data. The graphs in the "gures indicate: (a) Gas void fraction, predicted bymodels 1 and 2. (b) Gas void fraction, predicted by model 2. (c) Gas void fraction, predicted by models 2 and 3. (d) Axial liquid velocity pro"les,predicted by models 1 and 2. (e) Axial liquid velocity pro"les, predicted by models 2 and 3. (f) Turbulent intensity, predicted by models 1 and 2. (g) Axialgas velocity, predicted by models 1 and 2. (h) Bubble size distribution, predicted by models 1 and 2.

drag coe$cients. The development of such a procedurewas the main object of their paper. However, any "rmvalidation of the resulting drag relations has not been

reported yet. There are thus several reasons for further

studies on the hydrodynamic interaction forces.

Figs. 1(a), (d), (f)}(h) show that the predicted void, bubblesize, velocity and turbulent kinetic energy pro"les are notsigni"cantly altered by the lift forces (i.e. comparing mod-els 1 and 1d). The "gures also show that the importance ofthe turbulent wake force is negligible too (i.e. comparing

models 1 and 1a). This latter result is somewhat surprising

as Tomiyama et al. (1995, 1998) found that the transversal

forces caused by bubble deformation determine the phase

distribution in laminar bubble columns. Several reason-

able explanations exist. First, there may be other mecha-

nisms determining the phase distribution in bubble-driven

turbulent #ows. Second, the numerical solution algo-

rithms applied to multiphase #ow simulations are oftenchosen among the very stable ones, and less focus has been

put on accuracy. It may be that numerical di!usion isa!ecting the solution to a large extend. Third, the lift andwake force formulations used are developed for single-

bubble motions and may be ampli"ed incorporating thee!ects of bubble interactions, coalescence and breakup.Further experimental analyses are therefore needed to

validate these models. Fourth, applying the bubble-size

model given by Jakobsen et al. (1993), the wake force

coe$cient given by Tomiyama et al. (1995, 1998) is zero inmost cases because Eo(4. However, the bubble-size

model predicts an averaged bubble-size distribution in

reasonable agreement with the experimental data, as can

be seen from Fig. 1(h). An averaged bubble-size model is

thus not su$cient for modeling these phenomena.A bubble size and shape distribution model is needed.


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The anisotropic drag force has a signi"cant impact onthe solution, as can be seen from Fig. 1(a), (d), (f)}(h),comparing models 1 and 1b. This should be expected as

the model parameter has been tuned to experimental

data. However, without this force both the velocity and

the void pro"les become almost #at in contrast to theexperimentally obtained data which show a marked

parabolic pro"le. From Figs. 1(a), (d) and (g), comparingmodels 1 and 1c, it can be seen that the turbulent migra-

tion terms also have a signi"cant in#uence on the pre-dicted pro"les. In fact, the anisotropic drag force and theturbulent migration force dominate the phase distribu-

tion in these model simulations (i.e. for the various model

1 formulations). Zun (1990) stated that the principal

weakness of this theory is that it ignores bubble size as

one of the key variables. The model also relies on the

accuracy re#ected by the turbulence models predictingthe normal Reynolds stresses. As has been discussed in

a number of papers (e.g. Speziale, 1987; Martinuzzi

& Pollard, 1989, Pollard & Martinuzzi, 1989; Launder,1991; Hrenya, Bolio, Chakrabarti & Sinclair 1995), the

popular k} model has severe limitations even for single-phase #ows. One of the undesired consequences of this isthat the model predicts spuriously high generation rates

of turbulent energy and thus too high levels of turbulent

viscosity. Fig. 1(f) shows the turbulent intensity pro"le. Itcan be seen that the turbulent normal stresses are not

accurately predicted with the k} model. Furthermore,the simulated results, Figs. 1(f) and (h), show that the

largest discrepancies occur in the wall region where the

basic turbulence model is strictly not valid and single-

phase wall functions are normally applied. As discussed

by Bradshaw and Huang (1995), various aspects of thelogarithmic wall law for velocity have been questioned.

When the wall law fails the current turbulence models are

likely to fail too. The accuracy of the predicted pro"les ofturbulent kinetic energy are thus often very low for multi-

phase #ows, however, as for single-phase #ows, theReynolds shear stress terms have been reasonably well

predicted in many #ow situations. The low accuracyre#ected by the turbulent energy pro"les predicted bythis type of turbulence model, limits the possibility of

predicting the phase distribution phenomena in these

reactors. Lopez de Bertodano et al. (1990) and Jakobsen

(1993) extended their models to account for the e!ect ofanisotropy by adopting a full Reynolds stress and anAlgebraic Reynolds Stress model, respectively. However,

in both cases, severe problems were encountered due to

the lack of numerical convergence and unphysical solu-

tions. In a later paper, Lopez de Bertodano, Lahey and

Jones (1994) instead proposed a modi"ed form of thestandard k} model valid for very low void fractions andincluding two-time scales and an anisotropic matrix to

account for the bubble induced turbulence. It was also

suggested that the single-phase law of the wall might not

be valid for high void fraction multiphase #ows, due to

the existence of a second length scale imposed by the

bubble size. This is in accordance with the conclusions by

Jakobsen (1993). Further work is needed to enable "rmpredictions of the turbulence quantities in high gas frac-

tion bubble column #ows. This may also indicate thatlarge eddy simulations (LES) are the natural next step

towards a proper description of the phase distribution

phenomena. However, the high resolution required bythis type of models and the inherent system disability of

de"ning proper separation of scales, limit the applicationof LES since the bubble-size distribution normally en-

countered covers the range from less than 0.5mm to

about 2}3 cm. A suitable LES model has to resolve thisdispersed phase, making the computational task prohibi-

tively large.

Considering the Reynolds-averaged model version, no

solution was obtained without the turbulent mass di!u-sion terms in the continuity equations. However, as

discussed by Gray (1975) and Jakobsen (1993), these

dispersion terms have several undesirable propertieswhen considering reactive #ows and should thus beavoided. A Favre-like procedure is recommended. How-

ever, for non-reactive #ows, both formulations shouldproduce the same #ow pattern. The simulated resultsshow that this is the case when comparing the basic

models (i.e. models 1 and 2, Figs. (a),(d), (f)}(h)). Thepredicted pro"les for all variables shown are hardly dis-tinguishable applying the basic Reynolds-averaged equa-

tions containing a turbulent dispersion term in the

continuity equations and the alternative dispersion force

suggested by Laux (1998). When implementing the sec-

ond alternative dispersion force, suggested by Lahey et

al. (1993), the model did not converge properly. However,by adding this force on top of the basic model 2 formula-

tion (model 2c, Fig. 1b), this force did not have much

in#uence on the predicted gas void fraction pro"le, butproduces results that are less consistent with the basic

model 1.

The drift velocity model of Mudde and Simonin (1999)

is very similar to the model of Laux (1998). In fact, both

models are based on semi-empirical analyses having

a similar basis. Reeks (1992b) discussed the turbulent

particle dispersion e!ect and argued that the approach ofSimonin and co-workers is a more appropriate model

than the alternative gradient hypothesis. The model ofLaux (1998) represents a simpli"ed formulation com-pared to the drift velocity model, as only the void-velo-

city covariances are considered. The length of the relative

velocity vector occurring in the drag formulation is as-

sumed not to be a!ected by turbulence. The anisotropicdrag force on the other hand was developed accounting

for these mechanisms. The model of Laux (1998) is also

based on the assumption that the turbulent viscosity

predicted by the model will be the same applying both

Reynolds- and Favre-averaged models. In the present

simulations, this seems to be a reasonable assumption,


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but it is not veri"ed that this assumption holds for any#ow situation. The drift velocity model does not rely onthese assumptions. Instead, it is based on a semi-empiri-

cal probalistic analysis of an undisturbed turbulent #ow"eld of the continuous phase. As can be seen from Figs. 1(b), (c) and (e), the predicted pro"les are signi"cantlyaltered by the drift velocity. In model 3 the dispersion

force of Laux (1998) is substituted by the drift velocityformulation. In model 2(d) the drift velocity is added on

top of the basic model 2. In these simulations the drift

velocity formulation predicts pro"les less in accordancewith the measured data compared to the other model

formulations. Model 2 seems to be consistent with model

1, whereas the magnitude of the dispersion force inherent

in model 3 is larger. This discrepancy may be due to

di!erent handling of the so-called &crossing trajectory'e!ect in the various model formulations. The fact that theparticles pass from one turbulent eddy to the next more

quickly than if the particles translated at the mean #uid

velocity, e!ectively reduces the time available for#uid}eddy interaction and, in turn, the particle di!usion.This phenomenon has been discussed in several papers

(see Elghobashi, 1994, and references therein). It may be

that the drift velocity concept overestimates the bubble

dispersion e!ect, which should be reduced by the &cross-ing trajectory' phenomena. However, no turbulent dis-persion force model preference can be given as none

of them are able to predict unknown #ow "elds withreasonable degree of accuracy.

The lubrication wall force parameterization developed

by Antal et al. (1991) and later modi"ed by Tomiyamaet al. (1995,1998), as well as the axial wall force of Lahey

et al. (1993) have no signi"cant impact on the phasedistribution, as can be seen from Fig. 1(b), models 2a and

b. The lubrication force represents a wall lift force due to

viscous phenomena close to the wall (i.e. in contrast to

the other lift forces used derived applying potential the-

ory). The physical origin of this force has been ques-

tioned, as all wall e!ects are supposed to be lumped intothe law of the wall. Therefore, the wall force may be

looked upon as a correction to the single-phase wall law

valid for a limited number of multiphase #ow situations.The generality of the wall forces is strictly not validated

yet, and detailed analysis of the multiphase wall phe-

nomena in turbulent bubbly #ow is highly recommended.A "rm model cannot be developed before the actualphysical mechanisms determining the #ow pattern in thewall region are su$ciently understood. Recall thatproper predictions of the signi"cant turbulence pro"lesin the wall region can hardly be accurately performed

even for the much simpler single-phase #ows.The computational time of these simulations having

a very coarse-grid resolution is several hours on a Cray

T3E supercomputer. The number of iterations is very

high, typically 100 000}500000. This is far beyond rea-sonable limits for practical applications in industry. In

addition, in most multiphase codes, "rst-order numericalschemes are used for the convective terms due to the

inherent stability. It is however well known that the low

accuracy obtained using these algorithms may totally

destroy the physics (which may be poor by itself) re#ectedby these models. Therefore, there is a severe demand for

better numerical solution algorithms, both in terms of

computational time, and in terms of numerical accuracyand stability for bubble-driven #ows.

5. Conclusion

Constitutive models and parameterizations suggested

in the literature for prediction of the phase distribution

phenomena in bubble column reactors have been evalu-

ated.

The phase distribution in bubble columns are believed

to be determined by complex interactions between the

bubble size and form distributions, transversal forces,and several turbulence mechanisms. Accurate modeling

of these phenomena is still limited, both by physical

understanding and numerical accuracy.

The present models are still on a level aiming at rea-

sonable solutions with several model parameters tuned

to known #ow "elds. For predictive purposes, these mod-els are hardly able to predict unknown #ow "elds withreasonable degree of accuracy. It appears that the CFD

evaluations of bubble columns by use of multidimen-

sional multi#uid models still have very limited inherentcapabilities to fully replace the empirically based analysis

in use today. This type of models are thus yet not very

useful for industrial applications.The results presented here con"rm the demand for

improved modeling including more accurate and stable

numerical solution algorithms.

Acknowledgements

This work has received support from The Research

Council of Norway (Programme for Supercomputing)

through a grant of computing time.

The author gratefully acknowledges contributions of

Mr. I. Bourg, Mr. J. "vstaas, Ms A. S. H. Flateb+ andMs I. Berg performing model runs.

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