cfd modelling of slug flowin vertical tubes

8/12/2019 CFD Modelling of Slug Flowin Vertical Tubes

1/12

Chemical Engineering Science 61 (2006) 676 687www.elsevier.com/locate/ces

CFD modelling of slug ow in vertical tubes

Taha Taha, Z.F. Cui Department of Engineering Science, Oxford University, Parks Road, Oxford OX1 3PJ, UK

Received 15 June 2004; received in revised form 4 July 2005; accepted 6 July 2005Available online 12 September 2005

Abstract

In this work we present a numerical study to investigate the motion of single Taylor bubbles in vertical tubes. A complete description

of the bubble propagation in both stagnant and owing liquids was obtained. The shape and velocity of the slug, the velocity distributionand the distribution of local wall shear stress were computed and compared favourably with the published experimental ndings. Thevolume of uid (VOF) method implemented in the commercial CFD package, Fluent is used for this numerical study.

2005 Elsevier Ltd. All rights reserved.

Ke ywords: Multiphase ow; Vertical slug ow; Numerical simulation; VOF

1. Introduction

When gas and liquid ow in a pipe they tend to distributethemselves in a variety of congurations. These character-istic distributions of the uiduid interface are called owpatterns or ow regimes. Much time and effort has beenexpended in determining these regimes for various pairsof uids, channel geometries, and inclinations ( Mandhaneet al., 1974; Taitel and Duckler, 1976; Taitel, 1986; Barnea,1987). For vertical co-current ow and at low gas ow rates,the ow pattern observed is bubbly. Here, the gas phase isdistributed as discrete bubbles within the liquid continuum.At higher gas ow rates, some of the bubbles have nearly thesame cross-sectional area as that of thechannel. These bullet-shaped bubblessometimes referred to as Taylor bubblesor slugsmove along and are separated by liquid plugs

that may or may not contain a dispersion of smaller gas bub-bles. An increase in the gas ow rate in a two-phase mixtureowing in slug ow will eventually result in a complete de-struction of slug ow integrity with consequential churningor oscillatory action. At very high gas ow rates, the owbecomes annular, in which, adjacent to the channel wall.

Corresponding author. Tel.: +44 1865273118; fax: +441865 283273. E-mail address: [email protected] (Z.F. Cui).

0009-2509/$- see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.07.022

There is a liquid continuum and the core of the channel is agas continuum ( Hewitt and Hall-Taylor, 1970 ).

Slug ow is the most important of the two-phase ow

regimes primarily because of the numerous industrial andpractical applications. Some of these include buoyancy-driven fermenters, production and transportation of hydro-carbons, boiling and condensation processes in thermalpower plants, and emergency cooling of nuclear reactors.Slug ow is characterised by its random intermittence andinherent unsteadiness. A xed observer would see a quasi-periodic occurrence of long, bullet-shaped Taylor bubblesfollowed by liquid plugs sometimes carrying dispersedbubbles and taking on the appearance of bubbly ow ina pipe. The bubble region may take on stratied or an-nular congurations depending upon the tube inclinationand ow conditions. In order to understand the complexfeatures of intermittent slug ow, mainly experimental re-search has been conducted to study the motion of isolatedTaylor bubbles in motionless and owing liquids for var-ious inclination angles. Here, vertical slug ow will beemphasised.

The shape and the velocity with which a single Taylorbubble ascending through a denser stagnant liquid is in-uenced by the forces acting on it, namely the viscous,inertial and interfacial forces. Dimensionless analysisbased on Pi-theorem leads to the following dimensionless
http://www.elsevier.com/locate/ceshttp://-/?-http://-/?-mailto:[email protected]:[email protected]://-/?-http://-/?-http://www.elsevier.com/locate/ces


2/12

T. Taha, Z.F. Cui / Chemical Engineering Science 61 (2006) 676687 677

groups:

U 2T B LgD t ( L G )

g( L G )D 2t ,g 4L ( L G )

2L

3 , L

G, L

G,

L T BD t

,

where D t is the diameter of the tube, U T B is Taylor bubblevelocity, L and G are the density of the liquid and gas, re-spectively, L and G are the viscosity of the liquid and gasrespectively, is the surface tension and L T B is the lengthof Taylor bubble. For cylindrical bubbles, the lm thicknessand the bubble rise velocity are independent of the bubblelength (Grifth and Wallis, 1961; Nicklin et al., 1962; Maoand Duckler, 1989; Polonsky et al., 1999 ). Under the as-sumption that inertial forces in the gas are far smaller thanthe inertial forces in the liquid ( L / G 1), L / G can beeliminated. And if the viscosity of the gas in the bubble isneglected the following set of three dimensionless groups issufcient to characterise the motion of a single bubble risingthrough a motionless liquid. These are the Etvs number Eo = g( L G )D 2t / , the Morton number dened asM =g 4L ( L G )/ 2L 3 and the Froude number dened asF r =U T B / gD t ( L G )/ L . The Etvs number rep-resents the relative signicance of surface tension and buoy-ancy. The Morton number is sometimes referred to as theproperty group. The Froude number represents the ratio of inertial to gravitational forces. Other dimensionless groupscould be used, e.g., Fabre and Line (1992) in their reviewpaper on the motion of Taylor bubbles used Froude numberas a unique function of Etvs number and a dimensionless

inverse viscosity number, N f , given by N f =(gD3t )

1/ 2/ .Wallis (1969) used a set of Fr , N f and Archimedes number

Ar ( Ar = 3/ 2 L / 2L g 1/ 2( L G )1/ 2). Other and thesewidely used dimensionless groups can all be derived bymanipulating and/or combining two or more of groupsadopted in this work, e.g., N f =(E 3o /M) 1/ 4, Ar =(1/M) 1/ 2and the capillary number, Ca =Fr (M Eo)1/ 4 which is theratio of viscous to surface tension forces. White andBeardmore (1962) described a wide spectrum of experimen-tal results on Taylor bubbles drifting through motionlessliquids in vertical tubes. The authors presented a cross plot(Fig. 6 in their paper) showing the regions in which vari-

ous retarding forces may be neglected. The density of airwas neglected and instead of Fr , Fr is plotted to have areasonable spread of data ( White and Beardmore, 1962 ). Inthe region where surface tension dominates the bubble doesnot move at all where the hydrostatic forces are completelybalanced by surface tension forces. This occurs at Eo < 3.37(Hattori, 1935; Bretherton, 1961 ). For inertia-controlledregion when viscosity and surface tension can be neglected( Eo > 100 , N f > 300, 2L gD t /

2L > 3 105), the bubblerise velocity is given solely in terms of Fr (Nicklin et al.,

1962; White and Beardmore, 1962; Zukosk i, 1966; Maoand Duckler, 1990 ). In the centre of the graph, the relativemagnitude of all retarding forces, namely the viscous, iner-

tial, and interfacial forces are signicant. In this region therelationship between Fr , Eo , and M for vertical tubes hasbeen presented by White and Beardmore (1962) as a graph-ical map which plots lines of constant M on FrEo axis.Similar maps have been produced for non-vertical tubes byWallis (1969) and Weber et al. (1986) .

The description of the motion of single Taylor bubblesin owing liquids dates back to the pioneering work of Nicklins et al. (1962) who placed the corner stone of slugow modelling by recognising the fact that the bubble ve-locity is a superimposition of two components:

U T B =C1U m +U 0 . (1)The second term represents the drift due to buoyancy (thebubble velocity in a stagnant liquid) and the rst term refersto the transport by the mean ow, U m (U m =U SL +U SG ). C1is a dimensionless coefcient that depends on the velocityprole ahead of the bubble, and can be seen as the ratio of the maximum to the mean velocity in the prole. Hence forturbulent ows, C1 1.2 while for laminar pipe ow, C1 2(Nicklin et al., 1962; Collins et al., 1978; Grace and Clift,1979; Bendiksen, 1985; Polonsky et al., 1999 ). The veloc-ity eld around the bubble has been measured by many re-searchers primarily because of its importance regarding theinteraction between two Taylor bubbles. Several techniqueshave been adopted, e.g.: the photochromic dye activation(PDA) method ( DeJesus et al., 1995; Kawaji et al., 1997 ;Ahmad et al., 1998 ), the particle image velocimetry (PIV)technique ( Polonsky et al., 1999 ; Nogueria et al., 2000 ) andthe laser doppler velocimetry (LDV) technique ( Kvernvoldet al., 1984). Measurement of wall shear stress in stagnant

(Mao and Duckler, 1989 ) and owing (Nakoryakov et al.,1989) water showed clearly the reversal of the ow in theliquid lm. Experimental work on the propagation of Tay-lor bubbles in downward ow has been limited to a fewstudies on ascending bubbles in vertically downward ow(Martin, 1976 ; Polonsky et al., 1999 ) and one single study (toour knowledge) on inclined downward ow by Bendiksen(1984) . Theoretical treatments of the problem for predict-ing the rise bubble velocity have been successful for thespecial case of vertical tubes. This is, however, restrictedto the case where both viscous and surface tension effectsare negligible ( Dumitrescu, 1943 ; Davies and Taylor, 1950 ),

or the case where viscous effects dominate ( Goldsmith andMason, 1962 ). Later, Bendiksen (1985) extended the the-oretical approach of Dumitrescu (1943) and Davies andTaylor (1950) to account for the surface tension. Mao andDuckler (1990, 1991) and Clarke and Issa (1997) performednumerical simulations in order to calculate the velocity of the bubble and the velocity eld in the liquid lm. Propa-gation of Taylor bubbles in non-vertical tubes has been lesswell studied experimentally, and even less well theoretically,for the obvious reason of asymmetry.

Thus, studying the motion of a single Taylor bubble instagnant and in moving liquid is essential in order to un-derstand the intrinsically complicated nature of slug ow.
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


3/12

678 T. Taha, Z.F. Cui / Chemical Engineering Science 61 (2006) 676 687

The rise of a single bubble in both stagnant and owing liq-uid inside vertical tubes has been extensively studied by nu-merous researchers, both experimentally and theoretically.All the published numerical methods to model slug oware restricted only to vertical tubes and they assume ei-ther the shape of the bubble or a functional form for the

shape (Dumitrescu, 1943; Davies and Taylor, 1950 ). Theseassumptions constrain the nature of the solution, while theapproach adopted here (the volume of uid method, VOF)lays no such a priori foundations.The solution domain in thepresent model not only includes the eld around the bubble,as in the study of Mao and Duckler (1990) , but also extendsbehind the bubble, allowing eld information to be obtainedin the wake region. The difculty of obtaining local data iseven more complicated in horizontal and inclined slug owby the fact that the ow is asymmetric, and consequently,very few detailed data have been published in the open liter-ature. Previous models available in literature also fail to pro-vide a fully satisfactory mechanistic model of slug ow inhorizontal conguration ( Duckler and Hubbard, 1975 ; Taiteland Barnea, 1990 ). Thus, it is evident that more insight intoslug ow is needed to attain a thorough understanding of theinternal structure of the slug ow pattern. In this work, anattempt is made to calculate the shape and rising velocity of a single Taylor bubble in stagnant and in moving liquid invertical tubes. The velocity eld in a slug unit (Taylor bub-ble + liquid plug) and wall shear stress are also calculated.The intention of future work is to characterise the hydrody-namics of slug ow when the tube is tilted away from thevertical and also to investigate the effect of the leading bub-ble on the shape and velocity of the trailing one.

2. CFD model development

The CFD software FLUENT (Release 5.4.8, 1998) is usedto simulate the motion of a single Taylor bubble rising ina motionless or owing liquid through a vertical tube. InFLUENT, the control volume methodsometimes referredto as the nite volume methodis used to discretise thetransport equations. The movement of the gasliquid inter-face is tracked based on the distribution of G , the volumefraction of gas in a computational cell, where G =0 in theliquid phase and G

=1 in the gas phase (Hirt and Nichols,

1981). Therefore, the gasliquid interface exists in the cellwhere G lies between 0 and 1.

2.1. Model geometry

For axisymmetric simulations, a 2D coordinate system as-suming axial symmetry about the centreline of the pipe isused. The length of the domain is 11 D t , where D t is thetube diameter. The grids used to generate the numerical re-sults throughout this work are either uniform grids contain-ing quadrilateral control elements/volumes or uniform gridswith extra renement near the walls. Condence of grid in-

dependence results is gained by selecting simulations thatwere run with the grid cells number doubled (i.e., choosingresults from a 52 560 grid rather than a 26 280). Priorto simulations simple mass balance was performed to testwhether the lm is turbulent. This is discussed below. Foraxisymmetric simulations, the grid was uniform containing

52 560 elements when the liquid lm is laminar. If theliquid lm is turbulent, the last row of cells near the walls issub-divided three times. The result is a 59 560 elementsin the domain. This renement method, ensuring lm regiongrid independence, does, however, not always guarantee fullgrid independence in regions where the airliquid interfaceis highly curved.

In Fig. 1 the boundary conditions and the initial bubbleshape used in the simulation are displayed. The initial bub-ble shape consists of one hemisphere connected to a cylinderof the same radius. If other shapes were used (e.g. only acylinder), the nal shape of the bubble is found to be similarexcept the convergence is slower in the latter case. Thus, forthe simulations the former initial shape is adopted. The ini-tial guess for the lm thickness and the bubble rise velocityare calculated using simple mass balance and Eq. (1), re-spectively. The no-slip wall condition is applied to the walls.The uid mass ux at the inlet is specied using a prole fora fully developed ow through a pipe. The governing equa-tions are solved for a domain surrounding a Taylor bubblein a frame of reference attached to the rising Taylor bubble.With these coordinates, the bubble becomes stationary andthe pipe wall moves with a velocity U wall , equal to that of the Taylor bubble rise velocity, U T B . The liquid is fed atthe inlet with an average velocity U inlet , which is equal to

U T B U SL . A fully developed velocity prole is imposedat the inlet and the relative movement between the liquidand the wall generates a velocity prole shown in Fig. 1.The value of U T B is adjusted after the initial guess untilthe nose of the bubble ceased to move in the axial direc-tion. Trial simulations were conducted to examine the effectof using a xed frame of reference; they run longer withthe same nal result as with that of a moving frame of ref-erence. Throughout this work moving wall simulations arepresented. The frame of reference, initial and the boundaryconditions for 3D simulations are similar to those adoptedin axisymmetric simulations. The geometric reconstruction

scheme that is based on the piece linear interface calcula-tion (PLIC) method of Youngs (1982) is applied to recon-struct the bubble free surface. The surface tension is approx-imated by the continuum surface force model of Brackbillet al. (1992) . In all calculations, Courant number was set toa value of 0.25 and time step was set to 10 3 s.

2.2. Turbulence model and grid renement

Prior to simulations a simplemomentum andmass balance(Taitel and Barnea, 1990 ) was conducted to test whether theliquid lm is turbulent. The RNG k-epsilon model was the
http://-/?-http://-/?-


4/12


Axis-SymmetryBoundary

Out flow

Inlet FlowU inlet = U TB - U SL

Bubble

Moving WallU wall = U TB

Axis

Fig. 1. Initial and boundary conditions for a Taylor bubble rising in avertical pipe in a moving coordinate moving with the bubble.

turbulence model used throughout this work.A mass balanceon the liquid lm, relative to a coordinate system movingwith the bubble, yields the following differential equation:

dU f dx =

g( 1 L / G )(U T B U f ) +

(2/D t )f w(U T B U SL )

U f |U f | , (2)

where f w = Cf (D t F f U f L / L )m , F f being the liquidholdup in lm which can be written as a function of liquidlm thickness, F f =2h/R h 2/R 2 where h is the liquidlm thickness and R is the tube radius. For laminar owC f =16 and m = 1, while for turbulent ow C f =0.046and m = 0.2. The above differential equation was solvedusing a 4th RungeKutta with initial condition: U f =U SLat x =0 where x is axial distance from the bubble nose. Thiswill give us the local liquid lm thickness, h(x) and the lo-cal lm velocity, U f (x) . When the ow is turbulent in thelm ( Ref > 2000), a turbulence model is introduced ( Taiteland Barnea, 1990 ).

3. Results and discussion

3.1. Drift of Taylor bubbles in stagnant liquids

Consider a single Taylor bubble rising in stagnant liquidinside a vertical tube with velocity U T B . The bubble may bemade stationary by superimposing a downwardvelocity U T Bto the liquid and to the tube walls. The bubble has a round

nose and lls almost the cross sectional area of the tube(Fig. 2). The liquid ahead of the bubble moves around thebubble as a thin liquid lm moving downwards in the annularspace between the tube wall and the bubble surface. Along-side the bubble, the liquid lm accelerates until it reachesits terminal velocity under the condition of a long enoughbubble. At the rear of that bubble, the liquid lm plungesinto the liquid plug behind the bubble as a circular wall jetand produces a highly agitated mixing zone in the bubblewake. As shown clearly in Fig. 2, this recirculation zonesometimes contains small bubbles shed from the bubble taildue to the turbulent jet of the liquid lm.

Experimental data obtained by extensively varying theuid properties and pipe diameter, indicated that the termi-nal rising velocity and shape are signicantly affected byviscosity, surface tension, buoyancy and inertia. By takinginto account these effects, White and Beardmore (1962) con-ducted a dimensional analysis using a large matrix of ex-perimental data. They also proposed a graphical correlationof the terminal rise velocity of Taylor bubble inside verticaltubes. The three dimensionless numbers, Eo, M and Fr , arethe Etvs, Morton and Froude numbers, respectively, de-ned above. Fig. 3 presents the computed results for waterwhere the Froude number is plotted as a function of Etvsand M < 108. It can be seen that the results agree well


5/12


Fig. 2. Numerical simulation of a Taylor bubble rising through glycerinein a vertical tube.

with a wide range of experimental results and correlationsreported in literature. For low viscous liquids, all the datapoints reported follow the same course where there is a crit-ical Eo value ( Eo < 3.37) below which the bubble ceasedto move. Increasing Eo would increase the bubble veloc-ity as Etvs number represents the relative signicance of

surface tension and buoyancy. For more viscous liquids, thecalculated terminal rise velocity of a Taylor bubble in sev-eral liquids covering a wide range of Etvs and Mortonnumbers is presented in Fig. 4, where the results comparedfavourably with experimental data reported by White andBeardmore (1962) . In Fig. 4 the separation of the curves canbe attributed to the increasing importance of viscous forceson the terminal rise velocity.

Goldsmith and Mason (1962) observed in their experi-ments that in the case of high viscosity both ends of thebubble rising in stagnant liquid are spheroids; the top endis prolate and the bottom is oblate. They also indicated thatthe degree of prolateness of the nose and oblateness of thetail increase with surface tension. In addition, they reportedthat a wave disturbance would appear at the tail of the bub-ble when the viscosity is high. Also, the lm thickness de-creases as the surface tension increases. Fig. 5 shows theresults from the VOF method of the effects of Eo and M on the bubble shape. With decreasing Morton number undera constant Etvs number, the bluntness of the nose of thebubble increases and the bubble tail is attening, which re-sults in an increment of the liquid lm thickness around thebubble. The bluntness of the nose increases with increasing Eo . The wavelet disturbance shown in Fig. 5 appears when Eo is low. In the above calculations, it is found that the lm

thickness and the bubble rise velocity are independent of thebubble length, which is consistent with experiments by otherresearchers ( Grifth and Wallis, 1961; Nicklin et al., 1962 ).

The understanding of the hydrodynamic characteristic of wake behind the bubble is of great importance to the task of modelling transient slug ow (Moissis and Grifth, 1962;Duckler et al., 1985; Fabre and Line, 1992 ). The wake re-gion is believed to play a pivotal role in explaining the inter-action and coalescence between two successive Taylor bub-bles (Moissis and Grifth , 1962; Shemer and Barnea , 1987 ;Taitel and Barnea, 1990; Pinto and Campos, 1996; Talvyet al., 2000). For situations where heat and mass transfer is

augmented by introducing air slugs this understanding canbe critical.Inspired by the work of Maxworthy (1967) , Campos

and Guedes de Carvalho (1988) conducted a photographicstudy of the wake behind Taylor bubble. The liquids usedwere water, glycerol, and mixture of the two in differ-ent proportions, covering a wide range of viscosities. Theauthors adopted Fr , N f and Eo as the set of dimension-less group to characterise the wake pattern behind thebubbles. In their experiments the surface tension may beneglected ( Eo > 680) and for N f > 250 the Froude number,Fr =0.351. Thus the wake nature depends solely on the di-mensionless number, N f (Campos and Guedes de Carvalho,
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


6/12


Fig. 3. Taylor bubble rise velocity in stagnant water and dilute aqueous solutions contained in vertical tubes: ( ) Dumitrescu (1943) ; ( ) Laird andChisholm (1956) ; (

) Gibson (1913) ; ( ) Barr (1926); ( ) Hattori (1935) ; (

) Nicklin et al. (1962) ; (

) Grifth and Wallis (1961) ; (+) White and

Beardmore (1962) ; ( ) White Correlation (1962); ( ) Harmathy Correlation (Harmathy, 1960) ; ( ) Polonsky et al. (1999) ; ( ) CFD-This work.

Fig. 4. Taylor bubble rise velocity in stagnant viscous liquids contained in vertical tubes, experiment, White and Beardmore (1962) : ( ) M =4.7 105;( ) M =1.6 102 ; () M =0.33; () M =8.0; ( ) M 100; () CFD-This work.

1988). They identied three different patterns in the wakedepending on N f . The authors observed a closed axisym-metric wake for N f < 500, closed unaxisymmetric wake for500 < N f < 1500 and opened wake with recirculatory owfor N f > 1500. Their experimental results are simulatedhere. The wake patterns for ve different N f values arepresented in Fig. 6. For lowest value of N f , the annular lmconforms to the body of the tail of the bubble. With increas-ing N f , the liquid jet starts to separate from the body of thebubble with the free streamlines rejoining together at somepoint downstream forming a closed region, i.e., the wake,containing closed vortices travelling steadily at the samevelocity of the bubble. The wake here is enclosed into theoblate tail of the bubble and is axisymmetric with respect of

the tube axis. No bubble shedding is observed. Campos andGuedes de Carvalho (1988) observed the same type of pat-tern and they referred to them as laminar wakes. IncreasingN f further causes the wake to stretch downstream. At highN f , the free streamlines may not rejoin downstream result-ing in an open wake thereby allowing for vortex shedding.Here, 3D simulation of the bubble is conducted to capturethe 3D nature of the ow. The unstable and transient natureof the ow surrounding the bubble from laminar to turbu-lent produces signicant modication to the bubble proleand stability. As one can see from Fig. 6 the bubble tailoscillates and small bubbles can be seen shed at the bubblerear. Fig. 7 depicts a sequence of pictures of a 3D shape of the bubble and the velocity eld around it obtained with a


7/12


Fig. 5. Bubble shape prole: ( )Eo =477, M =4.7105 ; () Eo =120, M =4.7105 ; ( ) Eo =1986, M =1.6102 ; ( ) Eo 1142 , M =1.6102 ;() Eo =642, M 1.6 102 ; bubble volume =1.03ml; xaxial distance from bubble nose.

Fig. 6. Wake ow pattern at different values of N f for bubbles ris-ing through stagnant glycerol solutions inside a 19mm vertical tube;the frame of reference is moving with the bubble: (a) =1223 kg / m3;=9.7105 m2 / s; Eo =0.066; Fr =0.30; N f =84, (b) =1206 kg / m3; = 4.67 105 m2/ s; Eo = 0.064; Fr = 0.341; N f = 176, (c)=1202kg / m3 ; =4.0105 m2 / s; Eo=0.064; Fr =0.351; N f =205, (d)=1190 kg / m3; =2.5105 m2/ s; Eo=0.063; Fr =0.351; N f =325, (e)

=1129kg / m3;

=547

106 m2 / s; Eo

=0.064; Fr

=0.341; N f

=1528.

Fig. 7. Progression of wake ow pattern behind a Taylor bubble ris-ing through a stagnant glycerol solution inside a 19mm vertical tube:

=1181kg / m3; =1.9105 m2/ s; Eo =0.062; Fr =0.351; N f =437;t =0.01s; the frame of reference is moving with the bubble.


8/12


Fig. 8. Dependence of length of bubble wake on N f : ( ) numericalresults; the liquids used are stagnant glycerol solutions similar to thosepresented in Fig. 5, D t =19mm; ( ) experiment, Campos and Duedesde Carvalho (1988); () correlation, Campos and Duedes de Carvalho(1988).

time interval of 0.01s between the consecutive images atmoderate N f . It can be seen that the bubble nose propagatessmoothly, while the tail is characterised by vigorous oscilla-tions accompanied by small bubbles being shed at the bubbletrailing end. This was observed experimentally by Polonskyet al. (1999) . The wake of the bubble is closed but lacks sym-metry and it tends to oscillate and was referred to as tran-sitional wake by Campos and Guedes de Carvalho (1988) .Fig. 7 also shows that a streaming tail can be seen trailingthe travelling transitional wake in accordance with the ex-perimental ndings by Lighthill (1968) . The dependence of wake length on N f for low and intermediate N f is depictedin Fig. 8. The length of the wake is dened as the distancebetween the bottom of the bubble at the central plane of thetube and the stagnation area behind the bubble ( Nogueria etal., 2000 ). The linear relationship between the wake length

Fig. 9. Wall shear stress distribution in a slug unit at different values of N f : D t =19mm; liquids used are stagnant glycerol solutions; 1- N f =84, Eo =0.066; 2- N f =176, Eo =0.065; 3- N f =437, Eo =0.062; bubble length =3D t ; xaxial distance from bubble nose; the frame of reference is asseen by a xed observer.

and N f suggested by the Campos and Guedes de Carvalho(1988) is plotted together with their experimental results andcompared favourably with the theoretical values.

Fig. 9 illustrates the calculated wall shear stress arounda slug unit (Taylor bubble + liquid plug) for a bubble ris-ing into stagnant glycerol solutions inside a 19mm diameter

tube at different values of N f . The frame of reference is asseen by a xed observer and not moving with the bubble.The shear stress is plotted against a dimensionless distancefrom the bubble nose. The character of the wall shear stressdistribution is similar for all cases. The wall shear stressrapidly increases attaining its maximum positive value, in-dicating downow in the lm, near the bubble tail and thenstarts to decrease to zero at the end of the bubble wake. Thepositive value of the shear stress just ahead of the bubble in-dicates the existence of upward ow upstream of the bubblenose. This bubble expansion-induced velocity is connedto approximately 0 .5D t ahead of the bubble in agreementwith Polonsky et al. (1999) and van Hout et al. (2002) . Thefalling liquid lm is accelerated under gravity along the Tay-lor bubble while the lm thickness is continually narrowed(Fig. 10) until it is stabilised under the action of the frictionforce at the wall provided that the bubble is long enough.In this case the lm attains its terminal lm thickness andthe corresponding terminal velocity. Figs. 9 and 10 suggestthat the distance from the bubble nose where the liquid lmreaches its terminal thickness and velocity increases with in-creasing N f . Increasing N f also results in an increase in thewall shear stress and a reduction in the liquid lm thickness.

3.2. Rise of Taylor bubbles in owing liquids

For a Taylor bubble rising in a moving liquid it is gener-ally agreed that the translational bubble velocity, U T B , canbe expressed as the superimposition of two components: the


9/12


Fig. 10. Bubble shape prole at different values of N f :D t =19 mm; liquids used are stagnant glycerol solutions; 1- N f =84, Eo =0.066; 2- N f =176, Eo =0.065; 3- N f =437, Eo =0.062; x axial distance from bubble nose.

Fig. 11. Parity plot of Taylor bubble velocity: D t = 20 mm; airwatersystem.

Fig. 12. Wall shear stress distribution in a slug unit: D t =20mm; airwater system; ( ) U SL =0.625, U T B =0.81, LT B =2.5D t ; ( ) U SL =0.714,U T B =0.90, L T B =2.7D t ; ( ) U SL =1.0, U T B =1.2, L T B =2.5D t ; () U SL =1.25, U T B =1.44, LT B =2.6D t ; () U SL =1.42, U T B =1.63,L T B

=2.7D t ; xaxial distance from bubble nose; the frame of reference is as seen by a xed observer.

drift due to buoyancy (the bubble velocity in a stagnant liq-uid) and the velocity due to the transport by the mean ow(Eq. (1)). It is generally assumed that the value of C1U mis equal to the maximum local supercial liquid velocity(Nicklin et al., 1962 ; Collins et al., 1978 ). Thus, for turbu-lent pipe ow, C 1 1.2, while for a fully developed lami-nar ow in a pipe the value of C1 approaches 2. The bubblepropagation velocity inside a 20 mm diameter tube was cal-culated and found to be U 0 =0.155m / s, corresponding to0.351 gD t , which is in agreement with Dumitrescu (1943)and Nicklin et al. (1962) . Fig. 11 shows a parity plot of thetranslational bubble velocity predicted by Eq. (1) and thecalculated values with reasonable agreement.

Fig. 12 illustrates the calculated wall shear stress around aslug unit for various supercial liquid velocities for a Taylor


10/12


bubble moving inside a 20 mm diameter tube. The frame of reference is as seen by a xed observer and not moving withthe bubble. The character of the wall shear stress distributionis similar for all cases. The wall shear stress sign changestwice in a slug unit. The rst change takes place near the noseof the Taylor bubble and the second near the top of the liquid

plug. The negative shear stress, indicating upow, exists overthe liquid plug ahead of the bubble and persists beyond thenose of the Taylor bubble, before becoming positive as thedownow is established in the liquid lm around the bubble.

As the liquid ow increases, the portion of downward owbecomes shorter and the sign does not change if the bub-ble is not long enough ( U T B =1.63). This is expected asthe velocity of the Taylor bubble increases with increasingliquid velocity, and to accelerate the falling liquid lm upto the velocity of the bubble (which corresponds to a zerowall shear stress in the bubble region) it is necessary that thebubble length to be long enough. Near the bubble tail uc-tuations of the wall shear stress can be seen correspondingto the stirring nature of the wake trailing the bubble. Thismixing zone caused by the annular lm impinging the liquidplug behind the bubble plays a pivotal role in mass and heattransfer augmentation. The length of the mixing zone canbe determined from the wall shear stress prole and foundto be approximately 2 D t conrming the experimental nd-ings of other researchers ( Nakoryakov et al., 1989; Mao andDuckler, 1989 ).

3.3. Effect of angle of inclination

Although upward vertical two-phase slug ow has re-ceived considerably more attention in the open literature thandownward and inclined ows for the obvious reason that thesymmetry with respect to the tube axis is lost once the tubeis tilted away from the vertical. The difculty in handlingthe 3D nature of the ow limits the existence of experimen-tal data. When the tube is titled away from the vertical, theaxisymmetry of slug ow is breached. The degree of this ec-centricity increases when the inclination increases from thevertical to the horizontal. Fig. 13 shows the shape and thevelocity eld around a Taylor bubble rising through motion-less water in a 20 mm diameter tube when the tube is tilted20 away from the vertical. This feasibility study of inclinedslug ow shows that the VOF method adopted in this work proved to be powerful in tackling 3D slug ows, in general,and should meet the challenges in downward and inclinedslug ow to provide a comprehensive picture of such com-plex ows.

4. Conclusions

In this paper we presented a numerical study to investi-gate the motion of single Taylor bubbles in vertical tubes.A complete description of the bubble propagation in bothstagnant and owing liquids was obtained. The bubble was

Fig. 13. Numerical simulation of a Taylor bubble rising through waterinside a 20mm diameter tube titled 20 away from the vertical.

found to have a cylindrical body with a spherical nose anda uctuating tail. Sometimes, small bubbles were seen to

be sheered off the tail due to the liquid jet coming downfrom the annular region that separates the bubble from thetube walls. As the bubble moves up, the liquid ahead of it is picked up, and at a certain distance from the bub-ble nose, it starts to accelerate downwards in the annu-lar region. This distance becomes longer for faster bub-bles. Under the condition of a long enough bubble, theliquid lm accelerates until it eventually reaches its ter-minal velocity. At the rear of the bubble, the liquid lmimpinges the liquid plug behind the bubble as a circularwall jet and produces a highly agitated mixing zone in thebubble wake. Depending upon the inverse viscosity dimen-sionless number, N f , the wake takes on different patterns.
http://-/?-


11/12


When N f < 500, the wake was found to be composed of two closed toroidal vortices which are mirror image of eachother. The wake is enveloped into the oblate bubble tail. At500 < N f < 1500, the bubble tail is nearly at and the wake,still closed, tends to lose symmetry around the tube axisand to show periodic undulation, the frequency of which in-

creases with increasing N f . At N f > 1500, the bubble wakeopens and turbulent eddies are shed from the main bubblewake. For the low and moderate range of N f , the lengthof the bubble wake was found to be linearly dependent onN f .

The bubble shape was found to be dependent uponliquid viscosity and surface tension but not on the bub-ble length. The degree of prolateness of the nose andoblateness of the tail increase with surface tension. Awavelet was seen at the bubble tail when viscosity ishigh. The lm thickness around the bubble decreasesas surface tension increases. The bluntness of the bub-ble nose increases with decreasing viscosity, which re-sults in an increment of the liquid thickness. The bub-ble velocity was calculated over a wide range of tubediameter and liquid properties and the values obtainedcompared favourably with experimental results reportedin literature.

Notation

Ar Archimedes number, dimensionlessC1 dimensionless coefcient, dimensionless

C f dimensionless coefcient, dimensionlessCa capillary number, dimensionlessD t diameter of the tube, m Eo Etvs number, dimensionlessf w friction factor, dimensionlessF f liquid holdup in the liquid lm, dimensionlessFr Froude number, dimensionlessg acceleration due to gravity, m / s2

h liquid lm thickness, mL LP length of liquid plug, mL T B length of Taylor bubble, mm dimensionless coefcient, dimensionless

M Morton number, dimensionlessN f inverse viscosity dimensionless number, dimen-sionless

R bubble radius, mU 0 Taylor bubble drift velocity, m/sU f liquid lm velocity, m/sU inlet inlet velocity, m/sU m mixture velocity, m/sU SG supercial gas velocity, m/sU SL supercial liquid velocity, m/sU T B Taylor bubble velocity, m/sU wall wall velocity, m/sx axial coordinate, m

Greek letters

G volume fraction of the gas phase in the compu-tational cell, dimensionless

G gas molecular viscosity, kg/m sL liquid molecular viscosity, kg/m s

kinematic viscosity, m2/ s

G gas density, kg / m3

L liquid density, kg / m3

surface tension, N/m

Acknowledgements

T. Taha is grateful to the Karim Rida Said Foundationfor nancial support and to Dr. David Kenning for helpfulsuggestions. This work is partially sponsored by EPSRC(GR/66438).

References

Ahmad, W., DeJesus, J.M., Kawaji, M., 1998. Falling lm hydrodynamicsin slug ow. Chemical Engineering Science 53, 123130.

Barnea, D., 1987. A unied model for predicting ow pattern transition inthe whole range of pipe inclination. International Journal of MultiphaseFlow 13, 112.

Bendiksen, K.H., 1984. An experimental investigation of the motion of the long bubble in inclined tubes. International Journal of MultiphaseFlow 10, 467483.

Bendiksen, K.H., 1985. On the motion of long bubbles in vertical tubes.International Journal of Multiphase Flow 11, 797812.

Brackbill, J.U., Kothe, D.B., Zemach, C., 1992. A continuum method formodeling surface tension. Journal of Computational Physics 100, 335354.

Bretherton, F.P., 1961. The motion of long bubbles in tubes. Journal of Fluid Mechanics 10, 166188.

Campos, J.B.L.M., Guedes de Carvalho, J.R.F., 1988. An experimentalstudy of the wake of gas slugs rising in liquids. Journal of FluidMechanics 196, 2737.

Clarke, A., Issa, R., 1997. A numerical model of slug ow in verticaltubes. Computers and Fluids 26, 395415.

Collins, R., De Moraes, F.F., Davidson, J.F., Harrison, D., 1978. Themotion of a large gas bubble rising through liquid owing in a tube.Journal of Fluid Mechanics 89, 497514.

Davies, R.M., Taylor, G., 1950. The mechanics of large bubble rising

through extended liquids and through liquids in tubes. Proceedings of the Royal Society of London, Series A 200, 375390.DeJesus, J.D., Ahmed, W., Kawaji, M., 1995. Experimental study of ow

structure in vertical slug ow. Proceedings of the Second InternationalConference Multiphase Flow, Kyoto, vol. 4. pp. 51559.

Duckler, A.E., Hubbard, M.G., 1975. A model for gas liquid slug owin horizontal and near horizontal tubes. Industrial and EngineeringChemistry Fundamentals 14, 337347.

Duckler, A.E., Maron, D.M., Brauner, N., 1985. A physical model forpredicting the minimum stable slug length. Chemical EngineeringScience 40, 13791386.

Dumitrescu, D.T., 1943. strmung an einer Luftblase im senkrechten Rohr.Zeitschrift fuer Angewandte Mathematik und Mechanik 23, 139149.

Fabre, J., Line, A., 1992. Modelling of two-phase slug ow. AnnualReview of Fluid Mechanics 24, 2146.


12/12


Gibson, A.H., 1913. On the motion of long air-bubbles in a vertical tube.Philosophical Magazine 26, 952.

Goldsmith, H.L., Mason, S.G., 1962. The movement of single largebubbles in closed vertical tubes. Journal of Fluid Mechanics 14, 4258.

Grace, J.R., Clift, R., 1979. Dependence of slug rise velocity on tubeReynolds number in vertical gasliquid ow. Chemical EngineeringScience 34, 13481350.

Grifth, P., Wallis, G.B., 1961. Two-phase slug ow. Journal of HeatTransfer 83, 307320.Harmathy, T.Z., 1960. Velocity of large drops and bubbles in media of

innite or restricted extent. A.I.Ch.E. Journal 6, 281288.Hattori, S., 1935. Report, Aeronautical Research Institute, Tokyo Imperial

University, No. 115.Hewitt, G.F., Hall-Taylor, N.S., 1970. Annular Two-phase Flow. Pergamon

Press, London.Hirt, C.W., Nichols, B.D., 1981. Volume of uid (VOF) method for the

dynamics of free boundaries. Journal of Computational Physics 39,201225.

Kawaji, M., DeJesus, J.M., Tudsoe, G., 1997. Investigation of owstructures in vertical slug ow. Nuclear Engineering and Design 175,3748.

Kvernvold, O., Vindoy, V., Sontvedt, T., Saasen, A., Selmen-Oslen, S.,

1984. Velocity distribution in horizontal slug ow. International Journalof Multiphase Flow 10, 441457.Laird, A.D., Chisholm, D.A., 1956. Pressure and forces along cylindrical

bubbles in vertical tube. Journal of Industrial and EngineeringChemistry 48, 1361.

Lighthill, M.J., 1968. Pressure-forcing of tightly pellets along uid-lledelastic tubes. Journal of Fluid Mechanics 34, 251272.

Mandhane, J.M., Gegory, G.A., Aziz, K., 1974. A ow pattern map forgasliquid ow in horizontal pipes. International Journal of MultiphaseFlow 1, 537553.

Mao, Z.S., Duckler, A.E., 1989. An experimental study of gasliquid slugow. Experiments in Fluids 8, 169182.

Mao, Z.S., Duckler, A.E., 1990. The motion of Taylor bubbles invertical tubesI. A numerical simulation for the shape and the risevelocity of Taylor bubbles in stagnant and owing liquids. Journal of

Computational Physics 91, 132160.Mao, Z.S., Duckler, A.E., 1991. The motion of Taylor bubbles in vertical

tubes: II. Experimental data and simulations for laminar and turbulentow. Chemical Engineering Science 46, 20552064.

Martin, C.S., 1976. Vertically downward two-phase slug ow. Journal of Fluids Engineering 98, 715722.

Maxworthy, T., 1967. A note on the existence of wakes behind largerising bubbles. Journal of Fluid Mechanics 27, 367368.

Moissis, R., Grifth, P., 1962. Entrance effect in a two-phase slug ow.Journal of Heat Transfer 84, 2939.

Nakoryakov, V.E., Kashinsky, O.N., Petukhov, A.V., Gorelik, R.S., 1989.Study of local hydrodynamic characteristics of upward slug ow.Experiments in Fluids 7, 560566.

Nicklin, D.J., Wilkes, J.O., Davidson, J.F., 1962. Two-phase ow invertical tubes. Transactions of the Institution of Chemical Engineers40, 6168.

Nogueria, S., Dias, I., Pinto, A.M.F.R., Riethmuller, M.L., 2000. Liquid

PIV measurements around a single gas slug rising through stagnantliquid in vertical pipes. Tenth Symposium on Laser Techniques Appliedto Fluid Mechanics, Lisbon.

Pinto, A.M.F.R., Campos, J.B.L.M., 1996. Coalescence of two gas slugsin rising in a vertical column of liquid. Chemical Engineering Science51, 4554.

Polonsky, S., Shemer, L., Barnea, D., 1999. The relation between theTaylor bubble motion and the velocity eld ahead of it. InternationalJournal of Multiphase Flow 25, 957975.

Shemer, L., Barnea, D., 1987. Visualization of the instantaneous velocityproles in gas-liquid slug ow. PhysicoChemical Hydrodynamics 8,243253.

Taitel, Y., 1986. Stability of severe slugging. International Journal of Multiphase Flow 12, 203217.

Taitel, Y., Barnea, D., 1990. Two-phase slug ow. Advances in Heat

Transfer 20, 83132.Taitel, Y., Duckler, A.E., 1976. A model for predicting ow regimetransitions in horizontal and near horizontal gasliquid ow. A.I.Ch.E.Journal 22, 4755.

Talvy, C.A., Shemer, L., Barnea, D., 2000. On interaction between leadingand trailing elongated bubbles in a vertical pipe ow. InternationalJournal of Multiphase Flow 26, 19051923.

van Hout, R., Gulitski, A., Barnea, D., Shemer, L., 2002. Experimentalinvestigation of the velocity eld induced by a Taylor bubble rising instagnant water. International Journal of Multiphase Flow 28, 579.

Wallis, G.B., 1969. One-dimensional Two-phase Flow. McGraw Hill, NewYork.

Weber, M.E., Alarie, A., Ryan, M.E., 1986. Velocities of extended bubblesin inclined tubes. Chemical Engineering Science 41, 22352240.

White, E.T., Beardmore, R.H., 1962. The velocity of rise of single

cylindrical air bubbles through liquids contained in vertical tubes.Chemical Engineering Science 17, 351361.

Youngs, D.L., 1982. Time-dependent multi-material ow with large uiddistortion. In: Morton, K.W., Baibnes, M.J. (Eds.), Numerical Methodsfor Fluid Dynamics. Academic Press, New York, p. 273.

Zukoski, E.E., 1966. Inuence of viscosity, surface tension, and inclinationangle on motion on long bubbles in closed tubes. Journal of FluidMechanics 25, 821837.

cfd modelling of slug flowin vertical tubes

Documents