on the modelling of bubbly flow in vertical pipes

Nuclear Engineering and Design 235 (2005) 597–611

On the modelling of bubbly flow in vertical pipes

Eckhard Krepper∗, Dirk Lucas, Horst-Michael Prasser∗

Institute of Safety Research, Forschungszentrum Rossendorf e.V., P.O. Box 510 119, 01314 Dresden, Germany

Received 27 April 2004; received in revised form 20 August 2004; accepted 30 September 2004

Abstract

To qualify CFD codes for two-phase flows, they have to be equipped with constitutive models standing for the interactionbetween the gaseous and the liquid phases. In case of bubbly flow this particularly concerns the forces acting on the bubbles andbubble coalescence and break-up. Applying a two fluid approach, besides the drag forces describing the momentum exchangein flow direction, the non-drag forces acting perpendicular to the flow direction play an important role for the development ofthe flow structure. Gas–liquid flow in vertical pipes is a very good object for studying the corresponding phenomena. Here, thebubbles move under clear boundary conditions, resulting in a shear field of nearly constant structure where the bubbles rise for acomparatively long time. The evolution of the flow within the pipe depends on a very complex interaction between bubble forcesand bubble coalescence and break-up, e.g. the lift-force, which strongly influences the radial distribution of the bubbles, changesits sign depending on the bubble diameter. The consequence is the radial separation of small and large bubbles. Neglecting thisphenomenon, models are not able to describe the correct flow structure. Extensive experiments measuring the radial gas volumef rmined ford of bubblefl . Using as . They givea is shown.©

1

i

DH

s likereseula-nsferr aofof

n em-

0

raction distribution, the bubble size distribution and the radial residence of bubbles dependent on their size were deteifferent distances from the gas injection. Basing on these experiments the applicability and the limits for the simulationow with current CFD-codes are demonstrated, using the simulation of vertical pipe flow with CFX-4 as an exampleimplified model focusing particularly on the radial phenomena described above, parametric studies were conductedn indication for necessary improvements of the codes. Finally a possible way for the improvement of the CFD-codes2004 Elsevier B.V. All rights reserved.

. Introduction

Two-phase flow phenomena play an important rolen many scenarios concerning safety analysis of nuclear

∗ Corresponding authors.E-mail addresses:[email protected] (E. Krepper),

[email protected] (D. Lucas),

[email protected] (H.-M. Prasser).

reactor systems. At present one-dimensional codeRELAP, TRAC, CATHARE, ATHLET are used foplant design, optimization and safety analysis. All thcodes imply empirical correlations, e.g. for the calction of the pressure drop or the heat and mass trarates. Most of these correlations are valid only fogiven flow regime. By this reason, the knowledgethe flow pattern is very important for the applicationsuch codes. Most of the existing codes are based o

029-5493/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.nucengdes.2004.09.006

598 E. Krepper et al. / Nuclear Engineering and Design 235 (2005) 597–611

Nomenclature

C constantdb bubble diameter (m)D diameter of the pipe (m)Eo Eotvos numberF force per volume (N/m3)g acceleration due to gravity (m/s2)J superficial velocity (m/s)K turbulent kinetic energy (m2/s3)L length of the pipe (m)R radius (m)w velocity (m/s)y distance (m)

Greek lettersα gas volume fraction (%)ρ density (kg/m3)σ surface tension (kg/s2)

SubscriptsG gasW water

pirical or theoretical flow pattern maps. These steadystate flow maps predict the flow regime for given vol-ume flow rates of liquid and gas or any equivalent pa-rameters (e.g.Taitel et al., 1980). That means, they arenot able to predict the development of the flow alongthe flow path in case of constant gas and liquid volumeflow rates. In particular they are not able to predict theflow pattern in case of transient flows.

Recently, attempts were made to solve this prob-lem by the introduction of additional equations for thebubble density or corresponding parameters like bub-ble diameter, bubble volume or interfacial area (e.g.Hibiki and Ishii, 2000; Milles and Mewes, 1996). Bub-ble coalescence and break-up rates, which form thesource terms in these equations, are determined by lo-cal events. That means, they depend on local parame-ters of turbulence as well as on the local bubble sizedistribution.

Three-dimensional CFD-codes are successfullyused for single phase flow simulations for many techni-cal systems, but the practical application of CFD-codes

for two-phase flow phenomena is still limited to specialcases. In principle CFD codes are able to predict thelocal flow structure instead of using flow maps. How-ever, for practical applications it is not possible to re-solve all the details of the interfacial area, i.e. to resolveevery bubble or drop within the flow. For this reasonan ensemble averaging is done and some assumptionsconcerning the interaction between the phases are nec-essary.

In case of bubble flow these assumptions especiallyconcern the forces acting on the bubbles and bubblecoalescence and break-up. From studying these micro-scopic phenomena relations have to be derived for themass and momentum balance equations representingthe ensemble averaged phase properties. Experimentaland numerical investigations show, that there are verycomplex dependencies of the microscopic processes onthe local structure of the flow itself, e.g. the forces act-ing on a bubble depend on the local gradient of theliquid velocity as well as on the bubble size.

Experimental investigations (e.g.Tomiyama et al.,1995) as well as numerical investigations using DNS(e.g. Erwin and Tryggvason, 1997) supply informa-tion on the behavior of single bubbles in case of ideal-ized flow situations. Constitutive models for the forcesacting on a bubble and for bubble coalescence andbreak-up were derived from such investigations. Atthe Forschungszentrum Rossendorf measurement tech-niques were developed, which allow a very detailed in-vestigation of complex flows with high gas fractions.B thed lida-t

thed ub-b k-up,m sary.I ovec persep row-i ncep im-p tionf Allg aluea thee truc-t om-

ecause of the high resolution in space and timeata are suitable for the improvement and the va

ion of CFD-codes.Modern CFD-codes include relationships for

rag-, lift-, wall- and dispersion-force acting on a ble. To consider bubble coalescence and breaodelling of different bubble size classes is neces

n principle the two fluid approach described aban be extended to simulate several gaseous dishases. In practice the procedure is limited by the g

ng computational effort and by increasing convergeroblems. In CFX-4 a multiple size group model islemented, which solves only one momentum equa

or all gas phases to limit the computational effort.as bubble velocities are related to the average vlgebraically. This makes it possible to reproducexperimental observed data concerning the flow sure for some special cases with limited additional c

E. Krepper et al. / Nuclear Engineering and Design 235 (2005) 597–611 599

putational effort. In other applications the simulationfails. This work shows the limits of the simulations forbubble flow according to the models implemented atpresent and discusses reasons for these limitations. Fi-nally possibilities for the improvement of the modelsare shown.

The investigation of bubble flows with high gasloads vertical pipe flow is of special interest becauseof two reasons. The first is, that pipes occur in nearlyall technical installations where two-phase flows mayappear. This is also true in case of nuclear reactors. Thesecond reason is, that the flow in vertical pipes is a rel-atively simple case because of the symmetry. Becauseof the clear boundary conditions the single phase flowfield within a pipe is well known. Therefore, for verti-cal pipe flow a lot of investigations were done and a lotof information is available.

A radial 1D model, which allows parametric studiesfor bubble and slug flow in vertical pipes was developed(Lucas et al., 2001a, 2001b). It is used to check the influ-ence of the constitutive models for the non-drag forces

imenta

acting on a bubble perpendicular to the flow directionand bubble coalescence and break-up. Requirementsfor the extensions of 1D system codes and 3D CFD-codes are investigated using this model.

2. Experimental observations

2.1. Experimental setup and measuring technique

The evolution of the two-phase flow was studiedin a vertical cylindrical pipe with an inner diameterof 51.2 mm. The pipe is part of a test loop shown inFig. 1. Water with a constant temperature of 30◦C iscirculated in this loop. The superficial velocity of theupwards flow in the vertical test section can be variedfrom 0 to 4 m/s. Air is injected by an injection deviceat the lower end of the test section. Different injectiondevices were used to vary the initial bubble sizes andthe radial position of the injected bubbles.Fig. 1showsthe scheme of an injection device, which consists of

Fig. 1. Scheme of the exper
l setup and the wire-mesh sensor.


19 capillaries with an inner diameter of 0.8 mm. Thecapillaries are equally distributed over the cross sectionof the pipe. In case of low gas volume flow rates someof the capillaries can be switched off to guarantee asymmetric injection of the bubbles. The air superficialvelocities can vary up to 12 m/s.

An electrode wire-mesh sensor is placed at a cer-tain distance from the injection device, which can var-ied between the tests. Because of the limited lengthof the test section, the maximal distance was aboutL/D= 60. The function of the sensor is based on themeasurement of the local instantaneous conductivity ofthe two-phase mixture (Prasser et al., 1998). It consistsof two planar electrode grids with 24 electrode wireseach (pitch of the electrodes 2 mm, diameter of thewires 120�m), placed at an axial distance of 1.5 mmbehind each other (Fig. 1). This results in 500 measur-ing positions inside the circular cross section. Duringthe signal acquisition, one plane of electrode wires isused as transmitter, the other as receiver plane. Thetransmitter electrodes are activated by a multiplex cir-cuit in a successive order. The currents arriving at thereceiver wires are digitalized by analogue/digital con-verters and stored in a data acquisition computer. Thisprocedure is repeated for all transmitter electrodes. Thetime resolution was 2500 frames per second. The wire-mesh sensor delivers a sequence of two-dimensionaldistributions of the local instantaneous conductivity,measured in each mesh formed by two crossing wiresi and j. Local instantaneous gas fractions are calcu-l frac-t nald a-n nce.A 2)a de-t ub-b dis-t lesw bec

2

pesis eri-m re-

fer to the different combinations of air and water vol-ume flow rates. Two columns always belong to onetest case. The left one shows a virtual side projec-tion obtained from the wire-mesh sensor by a simpli-fied ray-tracing algorithm described inPrasser et al.(2003). The right column shows a virtual side viewto the flow on a vertical plane cut along the diameterthrough the pipe. The vertical time-axis can be trans-formed into a virtualz-axis, when it is scaled accord-ing to the velocity of the gaseous phase (Prasser et al.,2003).

The case marked with 121 (JW = 4 m/s andJG = 0.22 m/s) represents a finely dispersed bubbly flowas defined, e.g. inTaitel et al. (1980). The next fourpoints show examples of bubble flow with an maxi-mum of the gas volume fraction near the wall (039,JW = 0.4 m/s andJG = 0.01 m/s), with a transition froma wall peak of the gas volume fraction to a corepeak (083,JW = 0.4 m/s andJG = 0.06 m/s), with a corepeak of the gas volume fraction (118,JW = 1 m/s andJG = 0.22 m/s) and with a core peak of the gas volumefraction associated with a bimodal bubble size distribu-tion (129,JW = 1 m/s andJG = 0.34 m/s). For point 140(JW = 1 m/s andJG = 0.53 m/s) slug flow is observedand point 215 (JW = 0.4 m/s andJG = 12 m/s) is an ex-ample for annular flow.

The classification to flow pattern is subjective, butobjective criteria can be defined by help of the mea-sured bubble size distributions (Fig. 3) and radialprofiles of the gas volume fraction (Fig. 4), e.g. tod ub-b them e di-a ass -s di-r les.F ernm

2

es-s forba n-v tt bub-

ated assuming a linear dependence between gasion and conductivity. The result is a three-dimensioata arrayi,j,k wherek is the number of the instanteous gas fraction distribution in the time sequespecial procedure, described inPrasser et al. (200

llows the identification of single bubbles and theermination of their volume and the equivalent ble diameter. Using this procedure bubble size

ributions as well as gas fraction profiles for bubbithin a predefined interval of bubble sizes canalculated.

.2. Flow pattern in vertical pipes

The structure of the two-phase flow in vertical pis usually classified according to flow patterns.Fig. 2hows such flow pattern as received in our expents. The numbers at the bottom of the figure

istinguish bubble and slug flow the maximum ble size observed in the flow can be used. Ifaximum equivalent bubble diameter exceeds thmeter of the pipe the flow pattern is definedlug flow (Krussenberg et al., 1999). The radial poition maximum of the gas fraction can be seenectly from the measured radial gas fraction profiig. 5 shows an example of the resulting flow pattap.

.3. Microscopic effects

The migration of bubbles in radial direction isentially determined by the lift-force as introducedubble flow byZun (1980). DNS simulations (Erwinnd Tryggvason, 1997) as well as experimental iestigations (Tomiyama et al., 1995) showed, thahis force changes the sign depending on the


Fig. 2. Virtual side-views for different flow pattern.

Fig. 3. Examples of measured bubble-size distributions and classi-fication of the flow pattern.

Fig. 4. Examples of measured radial gas fraction profiles and clas-sification of the flow pattern.


Fig. 5. Measured flow pattern map forL/D= 60.

ble size. The classical lift-force was formulated forthe lateral migration of a spherical bubble within theliquid shear field, that means it is valid for smallbubbles. In co-current bubble flow the force actsperpendicular to the main flow in the direction tolower liquid velocity, i.e. in direction to the wall.For larger bubbles, which show a clear deformation,the lift-force acts towards higher liquid velocities,that means towards the center of the pipe.Tomiyama(1998) derived a correlation for the coefficient ofthe lift force, which depends on the bubble Eotvos-number

Eo = g(ρW − ρG)d2b

σ(1)

according to the results of experiments conductedwith single bubbles within a linear shear liquidfield. Using this correlation the lift-force chancesits sign in an air/water flow at ambient condi-tions at an equivalent bubble diameter of about5.5 mm.

By decomposition of the measured radial gas vol-ume fractions according to the bubble size (seeFig. 3)we were able to confirm the change of the sign of the liftforce in this region of bubble size by our experimentswith high gas volume fractions and a wide bubble sizedistribution (Prasser et al., 2002). This is demonstratedat Fig. 6. An example for such decomposed profiles isgiven inFig. 7, where the total gas fraction profiles is

and

3. Simulation of bubbly flow

3.1. Capabilities and limitations of presentCFD codes

Actually the most promising approach for CFD sim-ulation of a bubbly flow is the two fluid model. The twophases are regarded as interpenetrating. In each compu-tational cell the sum of volume fraction of both phasesis 1. For each phase the full set of Navier Stokes equa-tions is solved. The momentum exchange between bothphases is computed considering, that the gas phase hasthe form of dispersed bubbles of a constant diameterdb (drag forces). The consequences of this two fluidapproach are investigated by comparison of results ofsimulations with experimental observations. The CFDcode CFX-4 was used for the simulation. As an ex-tension of the two fluid model it includes a multiplebubble size group (MUSIG) approach. The advantagesas well as the remaining limitations are discussed be-low (Fig. 8).

3.1.1. Application of the two fluid model with amono-disperse approach

The k-epsilon turbulence model was applied forthe liquid phase. The influence of the gas bubbles onthe turbulence was considered according toSato andSekoguchi (1975). The task was solved in 2D cylindri-cal symmetry.

asFD

subdivided into fractions caused by bubbles largersmaller than 5.5 mm.

The radial volume fraction distribution of the gshould serve here as a criterion comparing the C


Fig. 6. Method of decomposition according to the bubble size.

Fig. 7. Radial gas fraction profiles decomposed according to thebubble size,JW = 0.64 m/s andJG = 0.09 m/s.

calculations with the experiments. The example forJW = 1.0 m/s andJG = 0.037 m/s shows, that the codeis able to reproduce the radial volume fraction profileand the gas velocity profile with good agreement to themeasurements.

This distribution is mainly influenced by the non-drag forces, which act perpendicular to the flow di-rection. A lift-force, a wall-force and a turbulentdispersion-force are considered here.

For the lift force, the formulation ofZun (1980)wasapplied:

Flift = −Clift ρW( �wG − �wW) × rot �wW (2)

In a cylindrical symmetry Eq.(2) reduces to

Flift = −Clift ρW(wG − wW)dwW

dR(3)


Fig. 8. (a) Measured and calculated radial distributions of the gasvolume fraction and for the phase velocity forJW = 1.0 m/s andJG = 0.037 m/s.

Fig. 9. Radial profiles of the water velocity and of the velocity gra-dient.

In the calculationsClift was assumed to be 0.06.Fig. 9shows, that for a typical flow in a vertical tube the ra-dial gradient of the fluid velocity is always negative.That means, with a positiveClift the lift force is alwaysdirected towards the wall.

The turbulent dispersion-force was calculated ac-cording toLahey et al. (1993)with the coefficientCdispwas set to 0.1:

Fdisp = CdispρWKW ∇α (4)

For the wall force, the formulation ofAntal et al. (1991)was considered:

Fwall = αGρW|wG − wW|2 max

(0,

CW1

db+ CW2

ywall

)�n

(5)

CW1 was assumed to be−0.0064 andCW2 0.016, sothat the wall force is acting only near the wall.

The resulting profiles for the non-drag forces show(seeFig. 10), that the gas volume fraction profile ismainly determined by the lift and the wall force.Fig. 11shows the change of the volume fraction profile, in-creasing the gas superficial velocity. The former near-wall maximum is shifted to a centre maximum.

Applying the described modelling approach with thesame coefficients to this series of shifted boundary con-ditions, results in the profiles presented inFig. 12.

The two fluid approach assuming a mono-dispersebubble size and applying the described model conceptsi ow-i oww ow,

s not able, to describe the transition from a flow shng a near wall gas volume fraction maximum to a flith core maximum. The experimental results sh

Fig. 10. Radial profiles of the non-drag forces.


Fig. 11. Measured radial profiles of the gas volume fraction withincreasing gas superficial velocityJG (JW = 1.0 m/s).

that indeed tests with a higher gas superficial veloc-ity show a bubble size distribution which contain alsolarger bubbles (seeFig. 3).

3.1.2. Application of a multiple bubble size groupapproach

In the CFD code CFX a multiple bubble size groupapproach (Lo, 1996) is implemented (MUSIG model).The model considers bubble coalescence and break-up between different bubble phases having differentbubble size diametres. To reduce the additional com-putational effort and limit the connected convergenceproblems of the solver, the model assumes that all thebubble velocities can be related to the average value al-

F bbles

gebraically so that it is only necessary to solve one setof momentum equations for all the bubble phases. Themultiphase approach is reduced to a two fluid approachwith one velocity field for the continuous phase andone for the dispersed phase. The continuity equationsof the particle size groups are retained, however, andsolved to represent the size distribution. This approachwas successfully applied to model chemical reactors orbubble columns, where the exact determination of theinterfacial area and the correct calculation of the dragforce dependent on the bubble size is necessary.

The application on the actual task yield indeed aplausible bubble size distribution (seeFig. 13, left side).However, since the non-drag forces were implementedin the same way as in the previous section, this approachalso yields the wrong gas volume fraction profiles (seeFig. 13, right side).

3.1.3. Consideration of the dependency of the liftcoefficient on the bubble size

As shown in the previous sections, all attempts todescribe radial volume fraction profiles for higher gassuperficial velocities failed, which do not consider thedependency of the lift coefficientClift on the bub-ble size. In Section2.2 microscopic effects were al-ready described. Different authors reported this de-pendency in the literature. Also in the described testsat the FZ-Rossendorf MT-loop this dependency wasobserved.

Applying the classical implementation of theM allb epa-r signo e isn s oft ec-t n-c e ise eldsr thep ow-e de-t mallb

tain-ip zesw ribu-

ig. 12. Calculated radial void profiles for mono-disperse buizedb = 4 mm.

USIG model considering only one flow field forubble size classes, the simulation of the radial sation of bubbles by consideration of a change off the lift coefficient depending on the bubble sizot possible. In flow situations, where large portion

he flow field are determined by a defined flow dirion (e.g. chemical airlift reactors with riser and dowomer), the exact determination of the drag forcssential. Here the classical MUSIG approach yiesults with good agreement to experiments. Inresent case of a bubbly flow in a vertical tube hver, this approach fails since the flow is mainlyermined by the radial separation of large and subbles.

Therefore, calculations with two gas phases conng small (Clift > 0) and large bubbles (Clift < 0) wereerformed. In this preliminary model the bubble siere calculated from measured bubble size dist


Fig. 13. Calculated bubble size distribution and radial volume fraction profile applying the multiple size group approach for tests with increasinggas superficial velocityJG (JW = 1.0 m/s).

tions by averaging all found bubbles withdb < 6 mm(small bubbles) anddb > 6 mm (large bubbles). The liftforces for these two bubble phases are considered withan opposite sign. Wall and turbulent dispersion-forcewere calculated in the same way described in Section3.1.1 for both gaseous phases.Figs. 14–16show theresults for the three test cases forJG = 0.140, 0.219 and0.342 m/s.

The comparison of the distribution of the overall gasvolume fractions on the right side ofFigs. 14–16withthe measurements inFig. 11shows, that this approachis very promising. In a developed model the initial set ofthe bubble diameters could be replaced by appropriatemodelling of bubble coalescence and break-up, whichis here not yet included.

lated g

3.1.4. Concept for improvement of the multiplesize group approach

The number of possible bubble size classes will belimited for numerical reasons. The idea consist in acombined approach (seeFig. 17). Instead of solving themomentum equation only for one gas phase, several (atleast two) phases each with its own momentum equa-tion should be implemented. This enable modelling thelift force in different directions depending on the bub-ble size. Doing this, the separation of small and largebubbles on the flow path can be simulated. In addi-tion each gas phase shall contain several scalar bubblesize groups. The modelling of bubble coalescence andbreak-up shall comprise all scalar groups. The appro-priate number of gas phases and scalar groups will be

Fig. 14. Measured bubble size distribution (left) and calcu
as volume fraction profile (right)JG = 0.140 m/s andJW = 1.0 m/s.


Fig. 15. JG = 0.219 m/s andJW = 1.0 m/s.

Fig. 16. JG = 0.342 m/s andJW = 1.0 m/s.

the result of the model development. A correspondingdevelopment of the model, actually is performed in FZ-Rossendorf (seeShi et al., 2003). The implementationin a CFD code is intended.

Fig. 17. Concept for improvement of the standard MUSIG model.

3.2. Radial 1D model

To investigate the details of the proposed conceptand prove its applicability, a radial 1D model was used,


Fig. 18. Variations of the lubrication- and dispersion-force(JW = 1 m/s andJG = 0.14 m/s).

to calculate radial gas fraction profiles from a givenbubble size distribution. As shown in the previous sec-tion several bubble size classes have to be consideredfor an appropriate modeling of the radial distributionof the bubbles. The model resolves the variables in ra-dial direction. It bases on the assumption of a fully de-veloped flow showing an equilibrium of the non-dragforces. A detailed description of the model can be foundin Lucas et al., 2001a, 2001b. The model allows to con-sider a large number of bubble classes. The measuredbubble size distribution is used to define the volumefractions of each bubble class. The radial volume frac-tion profile is calculated by solving of the balance offorces separately for each bubble class. Because of thedependency of the forces on the local gradient of theliquid velocity and the local turbulent kinetic energythese properties have to be calculated. For the calcu-lation of the profile of the liquid velocity the modelof Sato et al. (1981)was used. This model considersbesides the turbulence caused by the wall also the tur-bulence caused by the bubbles. The turbulent viscosityis also calculated by the model from Sato. It can beused together with the balance equation for the turbu-lent kinetic energy and the eddy viscosity hypothesisto calculate the radial profile of the turbulent energy.

The non-drag forces can be easily modified totest their influence on the lateral bubble distributions.Fig. 18gives an example for the influence of modifica-tions on the wall-force (Eq.(5)) and on the dispersion-force (Eq. (4)). At the standard calculation the lift-a )

Fig. 19. Comparison of calculated (lines) and measured (stars) totaland decomposed volume fraction profiles (JW = 1 m/s andJG =0.14 m/s).

with coefficients dependent on the bubble size and forthe dispersion the correlation fromLahey et al. (1993)(Eq.(4)) with a coefficient ofCdisp= 0.5. Starting fromthis calculation the influence of a bisection and a dou-bling of these forces on the radial gas fraction profile isshown. Comparing the profiles decomposed accordingto the bubble size with accordant experimental data,the reasons for discrepancies can be analysed.

For the standard calculation there is a very goodagreement between calculated and measured profiles.Fig. 19demonstrates this for the total gas fraction pro-file and the decomposed profiles for four classes ofbubble sizes. For the calculation equidistant (regardingthe bubble diameter) bubble size classes with a width

Fig. 20. Influence of the number of bubble size groups on the radialg
nd the wall-force were taken fromTomiyama (1998 as fraction profile (JW = 1 m/s andJG = 0.14 m/s).


of 0.25 mm where used. InFig. 19these bubble classeswere summed up to only four classes.

Fig. 20 demonstrates the influence of the numberof bubble size groups used for the calculation. 2 or 4bubble size groups are not sufficient to achieve a goodagreement between calculated and measured profiles.But the differences between the calculations with 8 and50 bubble size groups (width 0.25 mm each) are smallin this example. An detailed analyses have to be donefor a large number of combinations of volume flowrates to determine an appropriate number of bubble sizeclasses for CFD simulations as shown in Section3.1.

4. Simulation of the transition from bubblyflow to slug flow

The development of the flow along the pipe is de-termined by the migration of the bubble as discussedin the previous section as well as by bubble coales-cence and break-up. The frequency of bubble break-upincreases with the dissipation rate of turbulent energy(see e.g.Luo and Svendsen, 1996). Fig. 21shows theradial profile of the dissipation rate of turbulent en-ergy calculated by CFX-4 for the testJW = 1 m/s andJG = 0.037 m/s. There is a strong maximum near thewall.

Because of the completely different radial profilesfor the decomposed volume fractions for bubbles ofdifferent sizes and the pronounced profile of the dissi-p o bec k-up.

Fig. 22. Stable bubbly flow (left) and transition to slug flow (right)(from Lucas et al., 2003).

The development of the flow is determined by an inter-action of bubble migration, coalescence and break-up(Lucas et al., 2003). This is sketched atFig. 22.

In Fig. 22 an upward air–water flow is consid-ered. In both considered cases small bubbles (diam-eter < 5.5 mm) are injected. In the left side of the figurea low superficial gas velocity was assumed. The smallbubbles tend to move towards the wall. The local gasfraction in the wall region is larger than the averagedgas fraction, but it is still low. In this case bubble coa-lescence and break-up are in equilibrium and an stablebubble flow is established. If the gas superficial veloc-ity is increased (Fig. 22, right side), the equilibriumbetween bubble coalescence and break-up is shiftedtowards a larger bubble diameter, because the coales-cence rate increases with the square of the bubble den-sity, while the break-up rate is only proportional to thebubble density (Prince and Blanch, 1990). The bubblebreak-up rate strongly increases with the bubble diam-eter. By a further increase of the gas superficial veloc-ity, more and more large bubbles (diameter > 5.5 mm)are generated. They start to migrate towards the pipecentre. If enough large bubbles are generated by coa-lescence in the wall region, some of them can reach the

ation rate of turbulent energy local values have tonsidered for the bubble coalescence and brea

Fig. 21. Profile of the dissipation rate of turbulent energy.


core region without break-up. Because of the lower dis-sipation rate of turbulent energy they can then grow byfurther coalescence at much lower break-up rates, typ-ical for the low shear in the centre. This mechanism isthe key for the transition from bubble to slug flow. Thatmeans, for an appropriate modelling of the transition anumber of bubble classes as well as radial gas fractionprofiles for each bubble class have to be considered.

To consider bubble coalescence and break-up themultiple size group model was introduced into CFX-4.It allows the introduction of a number of bubble sizeclasses, which belong to the gas phase. That meansit is not possible to combine the calculation with twogas phases, representing two bubble size groups (seeSection3.1) with the multiple size group model. Forthis reason the applicability of the CFX-4 code on thesimulation of the development of the flow along theflow path is limited to very special cases.

The simplified 1D model (see Section3.2) was com-bined with models for bubble coalescence and break-up to simulate the development of bubbly flow alongthe pipe including the transition from bubbly to slugflow. The extended model was introduced in (Lucaset al., 2001b). Since it is a radial 1D model it doesnot resolve the parameters over the height of the pipe.Instead a bubble rise velocity, which is equal for allbubble sizes and radial positions is assumed. This al-lows the approximate evaluation of the flow patternover the height of the pipe in case of stationary flowsby introducing a dependence on time. Because of the

F t theua

Fig. 24. Experimental and calculated bubble size distributions at theupper end of the test section (3030 mm from the gas inlet),JW = 1 m/sandJG = 0.53 m/s.

assumed uniform bubble velocity the time correspondsto a height position within the pipe. Starting from aninitial bubble size distribution for each time step radialprofiles and new bubble size distributions according tolocal bubble coalescence and break-up are calculated.

Figs. 23 and 24show comparisons of calculated andmeasured bubble size distributions at the upper endof the pipe (L/D= 60). The figures also comprise thebubble size distributions close to the injection device.These data were taken as an input for the model. Inthe case shown atFig. 23bubbly flow is observed atthe upper end of the pipe in the experimental data aswell as in the results of the simulation.Fig. 24givesan example for the transition to slug flow along thepipe. A bimodal bubble size distribution in which theequivalent diameter of the largest bubbles exceeds thediameter of the pipe occurs at the upper end of the pipe.

5. Conclusions

For the simulation of two-phase flow the structure ofthe flow has to be considered. Because it is not possibleto resolve the complete interface between the phases incase of flows relevant for technical systems, simpli-fying assumptions concerning the interaction betweenthe phases have to be made. Constitutive models arerequired. In case of bubble flow the forces acting onthe bubbles as well as bubble coalescence and break-up have to be modeled. This requires the subdivision

ig. 23. Experimental and calculated bubble size distributions apper end of the test section (3030 mm from the gas inlet),JW = 1 m/sndJG = 0.22 m/s.


of the gas phase into several bubble size groups. Be-cause of the dependence of the bubble forces on thebubble size the bubble migration may be very differentfor the single size groups. In case of vertical co-currentpipe flow small bubbles can be found preferred in thenear wall region, whereas large (deformed) bubble arelocated preferred in the core region of the pipe. The de-velopment of the flow is the result of a complicated in-teraction of bubble migration, bubble coalescence andbubble break up. An extensive database for vertical pipeflow was used to test the applicability and the limits ofthe present CFD codes using the CFX-4 code as an ex-ample. While some cases with a wall peak of the radialgas volume fraction profile can be modeled by a mono-disperse approach, it fails for cases with a core peak ofthe radial gas volume fraction profile. An improvementwas achieved by using two gas phases in the simulationrepresenting small and large bubbles.

A simplified model was used to test the constitu-tive models as well as the required number of bubblesize groups. It is also possible to simulate the transi-tion from bubbly flow to lug flow using this model.The mechanism for this transition is also determinedby a complex interaction of bubble migration, bubblecoalescence and bubble break up, as discussed in thepaper. A concept for the improvement of CFD codeson basis of these investigations is introduced.

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