hydrodynamics of gas--liquid slug flow …paginas.fe.up.pt/ceft/pdfs/r37.pdf · hydrodynamics of...

HYDRODYNAMICS OF GAS--LIQUID SLUGFLOW ALONG VERTICAL PIPES INTURBULENT REGIMEA Simulation Study

T. S. Mayor, A. M. F. R. Pinto and J. B. L. M. Campos�

Centro de Estudos de Fenomenos de Transporte, Departamento de Engenharia Quımica, Faculdade

de Engenharia da Universidade do Porto, Porto, Portugal.

Abstract: A thorough study on the simulation of gas–liquid vertical slug flow, based onair–water co-current experimental data (Mayor et al., 2006a), is reported. The flow pattern in thenear-wake bubble region and in the main liquid is turbulent. The slug flow simulator includescomputation of gas phase expansion along the column and the introduction, at the column inlet,of distributed gas flow rates and liquid slug lengths. A slug flow entrance-length of 50–70 D wasfound for the ranges of superficial gas and liquid velocities studied (0.1–0.5 m s21). Generalexpressions are proposed to predict the modes and standard deviations of bubble velocity, bubblelength, and liquid slug length, as a function of several parameters (column vertical coordinate,superficial gas and liquid velocities and column diameter). Gas phase expansion was found toplay a major role in the evolution of the velocity and length of bubbles along the column. Theliquid slug length is shown to depend mostly on the coalescence effect.

Keywords: multiphase flow simulation; bubble columns; slug flow; gas expansion; lengthdistributions; bubble interactions.

INTRODUCTION

Several attempts have been made to simulategas–liquid slug flow in vertical co-current col-umns operating in the turbulent regime. Themain purpose is to gather information on theevolution of bubble and liquid slug length distri-butions along the column. Their average andmaximal values (in particular for slug length)as well as several other two-phase flow par-ameters are indispensable for many engineer-ing calculations. Barnea and Taitel (1993)simulated gas–liquid slug flow in vertical col-umns operating in the turbulent regime. Themodel assumes a liquid slug length distributionat the column inlet (random or uniform distri-bution) and predicts its evolution along thecolumn. The bubble velocity as a function ofthe liquid slug length ahead of the bubble (aswell as the minimal stable liquid slug length)must be introduced as an input relation. Theauthors adopted the correlation format of Mois-sis and Griffith (1962), which was establishedfor a short range of operation conditions (interms of gas and liquid flow rates). Hasaneinet al. (1996) implemented a similar strategyusing, however, a different correlation betweenthe bubble velocity and the length of the liquidslug ahead of it (based on air–kerosene

experimental data). More recently, Van Houtet al. (2001, 2003) reported a study on gas–liquid slug flow along vertical and inclinedpipes, respectively, in which some simulationresults are reported (only for slug length par-ameter). However, these slug flow simulationswere achieved without an exact implemen-tation of the gas phase expansion during theupward movement of the bubbles, and consid-ering constant volumetric gas flow rate at thecolumn inlet (an unreal scenario due to thechanging gas hold-up in the column). More-over, the effects of the superficial gas andliquid velocities and column diameter over theflow parameters were not yet thoroughly ana-lysed. Furthermore, despite the publishedmodels/simulators and experimental data,some doubts still exist concerning the prevail-ing mechanism in the development of theslug flow pattern: do the inlet gas and liquid dis-tributions determine the development of theslug flow pattern or, alternatively, does theovertaking mechanism by which bubblescoalesce along the column overcome the influ-ence of the inlet distributions and determinethe output of slug flow experiments? Thisissue, stressed by Fabre and Line (1992)remains an open question and, therefore,requires some attention.

1497 Vol 85 (A11) 1497–1513

�Correspondence to:Professor J.B.L.M. Campos,Centro de Estudos deFenomenos de Transporte,Departamento deEngenharia Quımica,Faculdade de Engenharia daUniversidade do Porto, RuaDr. Roberto Frias, 4200-465Porto, Portugal.E-mail: [email protected]

DOI: 10.1205/cherd06245

0263–8762/07/$30.00þ 0.00

Chemical EngineeringResearch and Design

Trans IChemE,Part A, November 2007

# 2007 Institutionof Chemical Engineers

The main goal of this work is to formulate a robust predic-tive model and, through simulation, to produce information ondeveloping and developed slug flow patterns for a largerange of operation conditions. The expansion of the gasphase during the rise of the bubbles and the gas flow rate dis-tribution at the column inlet are implemented. The empiricalcorrelation relating the velocity of a bubble and the lengthof the liquid slug ahead of it is given as input. This correlationwas determined for a large range of operating conditionsusing an non-intrusive technique (see Mayor et al., 2006afor details). The output of the simulation is compared notonly with the mentioned experimental results but also withthe findings of others researchers.

EXPERIMENTAL WORK

A series of air–water co-current slug flow experimentswere performed in 6.5 m long acrylic vertical columns withinternal diameters of 0.032 m and 0.052 m. These exper-iments are thoroughly described in Mayor et al. (2006a) buta brief summary of the experimental approach is givenhere. The experimental data was collected using an imageanalysis technique (Mayor et al., 2006b) at two vertical coor-dinates (3.25 m and 5.40 m from the base of the column) andfor several superficial gas and liquid velocities (UG and UL upto 0.26 m s21 and 0.20 m s21, respectively). The operatingconditions were designed to have turbulent regime in themain liquid and in the near-wake bubble region. The exper-imental study served mainly to gather information on theflow pattern characteristics (i.e., bubble length, bubble vel-ocity, liquid slug length) at a given vertical coordinate, andto establish the bubble-to-bubble interaction curve governingthe approach and coalescence of consecutive bubbles. Thefirst type of data is crucial for the validation of the simulationapproach (described later) whereas the second is required toimplement the relative motion of bubbles inside the column,i.e., to simulate the development of the slug flow pattern.The following chapter describes the assumptions andapproaches of the slug flow simulation.

SIMULATOR CHARACTERISTICS

Two adjacent domains are considered in a slug flowcolumn: the bubble formation domain, prior to the columnitself, and the simulation domain, where slug flow pattern isexpected. Bubble-to-bubble interaction and the expansionof the gas phase are considered only inside the simulationdomain. This domain is defined by the input of the columninternal diameter (ID) and height. A tank with a large crosssectional area is located at the top of the column (wheregas–liquid separation occurs) with a lateral exit (assuringan almost constant level of aerated liquid). The origin of thevertical coordinate matches the boundary between bubbleformation and simulation domains.

Model Assumptions and Inputs

Bubble shape, liquid slug shape and surroundingliquid filmThe bubble shape is taken cylindrical. The thickness of the

liquid flowing around the bubbles, d, is calculated assuming afree-falling film. Following the approach of Brown (1965) for

laminar regime in the film, one can use the following equationto compute the liquid film thickness, for continuous co-currentgas–liquid flow:

d ¼ {(3yRc)=½2 g(1� d=Rc)�

� ½(1� d=Rc)2Uexp

B � (UG þ UL)�}1=3 (1)

where y is the kinematic viscosity, Rc is the column internalradius, g is the acceleration of gravity, UB

exp is the experimen-tal upward bubble velocity in undisturbed conditions andUG and UL are the superficial gas and liquid velocities,respectively.The liquid slugs are considered non-aerated.

Bubble velocity as a function of the liquid slug lengthahead of it—an input relationConsider a train of Taylor bubbles flowing upwards in a ver-

tical column, in slug flow pattern (as a set of consecutive slugunit cells, i.e., bubble plus the liquid slug below it). The vel-ocity of a trailing bubble flowing in the column depends onthe length of the liquid slug ahead of it. In the experimentalstudy mentioned previously, Mayor et al. (2006a) found asingle curve for this dependency, for the two diameters andcolumn vertical coordinates tested (0.032 m and 0.052 m;3.25 m and 5.40 m, respectively). The obtained averagebubble-to-bubble interaction curve is shown in Figure 1(a)together with the best fit equation. The form and parametersof this equation are shown below:

Utraili =Uexp

B ¼ 1þ 2:4e�0:8(hs, i�1=D)0:9

(2)

where Uitrail refers to a trailing bubble i flowing behind a liquid

slug with length hs,i21 (bubbles and slug numbered from topto bottom), and D stands for the column diameter. The trailingbubble velocity is normalized by the experimental upwardbubble velocity in undisturbed conditions (UB

exp). This par-ameter was obtained focussing on bubbles flowing morethan 10 D apart from the previous bubbles [as shown inFigure 1(a), such slug length assures negligible bubble-to-bubble interaction].The estimates of UB

expobtained in the experimental studyare plotted against the superficial mixture velocity (UM ¼

ULþUG) in Figure 1(b), together with the predictions fromNicklin et al. (1962) for co-current flow in turbulent regime.The Nicklin correlation is of the form

UB ¼ U1 þ CUM (3)

where U1, the drift velocity, is the bubble rising velocitythrough a stagnant liquid and C an empirical parameterdepending on the flow regime in the liquid. The drift velocitycan be computed by 0.35

p(gD) for inertial controlled

regime (White and Beardmore, 1962) whereas parameterC is generally taken equal to 1.2 for turbulent regime in theliquid (Nicklin et al., 1962; Collins et al., 1978; Bendiksen,1984).For the narrower column reported in Mayor et al. (2006a)

(0.032 m), a good agreement was obtained between the par-ameter UB

exp and the estimates of UB [equation (3)] based onvalues of C and U1 as in the literature (C ¼ 1.2 andU1 ¼ 0.196 m s21). For the larger column (0.052 m), thevalues of UB

exp were found higher than the estimates of UB

Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A11): 1497–1513

1498 MAYOR et al.

computed in the same way [Equation (3) with C and U1 as inthe literature]. This discrepancy was ascribed to two factors.First, the aeration of the liquid slugs (particularly for highsuperficial gas velocities) can result in an increase in theexperimental drift velocity (Van Hout et al., 2002). Secondly,the level of turbulence in the liquid between bubbles (higherfor the larger column), generates instantaneous velocity pro-files considerably different from the average ones. Consider-ing that the bubble nose always follows the higherinstantaneous liquid velocity ahead of it, there is a continuousacceleration and deceleration of the bubbles rising in thecolumn even when no Taylor bubbles are ahead. This maylead, ultimately, to the increase of parameter C (and thereforeUB

exp). The simulations for the larger column are thus basedon estimates of UB

exp computed using the obtained best fitvalues of C and U1 (1.628 and 0.314 m s21, respectively).

Slug Flow Conditions at Inlet

The simulation algorithm considers two independent vari-able distributions at the column inlet: slug length and gasflow rate distributions. Several types of slug length distri-butions were implemented (normal, uniform, constant anduser-defined type). A normal distribution was implementedfor the gas flow rate. Normal distributions were preparedusing the Box Muller algorithm. The following sections pro-vide information about the relations governing the inlet slugflow for constant and distributed gas flow rates.

Bubble length as a function of the sluglength—constant gas flow rateConsider Figure 2(a), representing a train of Taylor bubbles

flowing in the bubble formation domain (prior to the columninlet). In order to assure constant gas and liquid superficial vel-ocities, a relationship must exist (at the referred coordinate)between the length of each bubble and the length of theliquid slug, in each unit cell. Notice in Figure 2(a) the presenceof long liquid slugs flowing immediately after long bubbles.Assuming a cylindrical bubble shape, one can write

UGScDti ¼ Sbhb, i (4)

where Dti refers to the time interval required for the entrance

of a slug unit cell (bubbleþ slug), hb,i is the length of bubble i,and Sc and Sb stand for column and bubble cross sectionalarea, respectively. Considering that in the bubble formationdomain no bubble-to-bubble interaction is acknowledged,bubbles move upwards in the column at their undisturbedvelocity (given by UB

exp). Therefore

UexpB Dti ¼ hb, i þ hs, i (5)

By combining the previous equations and rearranging, itfollows:

hb, i ¼ hs, i=½(Sb=Sc)(UexpB =UG)� 1� (6)

Equation (6), valid at the column inlet, relates the length ofeach bubble to the length of the liquid slug flowing bellow, forgiven superficial gas and liquid velocities [UB

exp encloses theinfluence of UL as in equation (3)]. Thus, having prepared aslug length distribution and defined the superficial gas andliquid velocities, one can prepare the corresponding bubblelength distribution by using equation (6) for all unit cells.

Figure 2. Representation of gas liquid distributions, in the bubbleformation domain, for (a)–(b) constant gas flow rate and (c)–(d)variable gas flow rate.

Figure 1. (a) Average bubble-to-bubble interaction curve with 95% confidence intervals; (b) Experimental average upward bubble velocityplotted against UM after correction for vertical coordinate: 5.4 m; internal diameter: 0.032 m and 0.052 m; data after Mayor et al. (2006a).


SIMULATION OF TURBULENT SLUG FLOW 1499

Notice that the slug length distribution is the independentdistribution, whereas the bubble length distribution is thedependent one.

Bubble length as a function of the slug length—variablegas flow rateThe turbulence at the column base, caused by the injection

of the gas followed by bubble formation (intense bubblecoalescence and break-up), and the constant alteration ofthe hydrostatic pressure in this region due to the changinggas hold-up in the column, advise acknowledgement of avariable gas flow rate at the column inlet. For this purposea distributed gas flow rate (with a certain average and stan-dard deviation) was implemented in the simulation code.Figure 2(c) depicts this issue. Notice in Figure 2(d) thateach unit cell has a different gas superficial velocity in thisnew scenario.At the column inlet, the relationship between the length of

the bubbles and the length of the liquid slugs for each unit cellis now given by

hb,i ¼ hs,i= Sb=Scð Þ UexpB =UG,i

� �� 1

� �(7)

where UG,i is the superficial gas velocity associated to eachunit cell [Figure 2(d)]. Having defined a slug length distri-bution and a gas flow rate distribution (and defined UL) onecan thus prepare the corresponding bubble length distri-bution, by using equation (7), for all unit cells. In this new situ-ation two independent distributions exist: slug lengthdistribution and gas flow rate distribution.The assumption of a variable gas flow rate at the column

inlet is important for relatively short columns. However,such assumption is of little importance when consideringvery long columns such as in petroleum production wells.

Simulation Start-Up

The simulation start-up begins by defining the superficialgas velocity, UG

inlet, at the inlet hydrostatic pressure. This

parameter relates the volume of gas passing at the inlet coor-dinate with the time required for the entrance of n unit cells,and it can be determined by

UinletG ¼ Sb

Xni¼1

hb,i

!,Sb

Xni¼1

Dti

!(8)

Notice that the Box Muller algorithm (Campos Guimaraesand A. Sarsfield Cabral, 1997), used to prepare the gasflow rate distribution, requires the input of the arithmeticaverage of the distribution to be created, UG

inletjm, as in the

following equation:

UinletG jm ¼

Xni¼1

UG,i

.n ¼ Sb

Xni¼1

hb,iDti

!.(Scn) (9)

Bearing in mind that equations (8) and (9) define two differ-ent quantities, an iterative procedure had to be implementedto initiate the simulation. Figure 3 illustrates the strategypursued.The iterative cycle converges when the parameter Uinlet

G[evaluated at the inlet through equation (8)] reaches the pro-posed value, introduced as input. Once the convergence isachieved the simulator holds two distributions [slug lengthand bubble length distributions, related by equation (7)]which assure the required average superficial liquid andgas velocities at the column inlet (and at the inlet pressure).However, in many practical situations (e.g., when comparingsimulation and experimental data) it is more convenient touse as reference the ambient pressure rather than the inletpressure. And yet, due to the random length of the slugsand bubbles entering the column, the average superficialgas velocity at ambient pressure is only obtainable as oneof the simulation outputs [computed as in equation (8) butfocusing on the outlet coordinate]. Thus, an additional itera-tive procedure was implemented in the simulator in order toprepare it to receive as input the average superficial gas vel-ocity at ambient pressure. This external iterative cyclerequires several slug flow simulations with differing values

Figure 3. Representation of the iterative approach for the simulation start-up.


1500 MAYOR et al.

of UinletG to run iteratively, until the required average superficial

gas velocity is achieved at the top of the column. Bisections’method was used within the mentioned iterative cycles.

The Displacement of the BubblesAlong the Column

The displacement of bubbles along the column isimplemented as the incremental movement of their bound-aries (bubble nose and rear) between consecutive instants(tj and tjþ1). The position of the rear of bubble i at instanttjþ1, z rear,i

tjþ1 , is computed by updating its position at tj, zrear,itj ,

according to its velocity (Ui).

ztjþ1rear,i ¼ z

tjrear,i þ Ui (tjþ1 � tj) (10)

Bubble velocity Ui has two contributions: one related to thelength of the slug ahead of the bubble [given by equation (2)],and another related to the expansion of the gas bubbles flow-ing upstream (below) the bubble. This latter contribution isdescribed in the following section. The update of the positionof the bubble rear is achieved assuming a constant bubblevelocity between consecutive time instants, an assumptionwhose accuracy increases for decreasing time increments.The position of the bubble nose is updated according to

ztjþ1nose,i ¼ z

tjþ1rear,i þ h

tjþ1b,i (11)

Taking the boundaries of two consecutive bubbles, thelength of the liquid slug flowing in-between them is given by

htjþ1s,i ¼ z

tjþ1rear,i � z

tjþ1nose,iþ1 (12)

This strategy extends to all the bubbles flowing in the column.

Expansion of the Gas Phase Along the Column

As gas bubbles rise along a vertical column there is adecrease in the pressure acting on each bubble. Accordingto the ideal gas law this decrease produces the expansionof the gas phase, i.e., an increase in the volume of eachbubble. Discarding the pressure drops in the liquid phase

(at the wall and at the wake of the bubbles) the pressurealong the column can be predicted taking only the hydrostaticpressure gradient.It is reasonable to assume that, with an open tank at

the column top, the bubble expansion occurs as a rise ofthe bubble nose region (reference frame attached to thebubble), since there is no volume, upstream (below) ofeach bubble, to accommodate the extra volume resultingfrom the bubble expansion. Thus, the bubble expansioninduces the upward displacement of everything ahead ofthe bubble (liquid and gas).In order to calculate the volume (or length) of bubbles at a

given column position, the hydrostatic pressure acting oneach bubble and the number of moles of air in each bubblemust be known. This latter parameter can be assessed, forinstance, at the entrance of the column (inlet position)where the hydrostatic pressure can easily be computed.

Evaluation of the amount of air in a bubble, at thecolumn inletAs bubbles enter the simulation domain (where expansion

phenomena are acknowledged), different hydrostatic press-ures act on each bubble depending on the gas hold-up inthe column. Figure 4(a) depicts one of these instants. Thehydrostatic liquid height above a bubble i, at the instant itenters the column, can be calculated taking only theamount of liquid inside the column at that instant:

Hhyd,i ¼ zliq: � Sb=Sc

Xi�1

k¼1

(hb,k ak) (13)

where Hhyd,i is the hydrostatic liquid height above the bubble iand ak a parameter informing on the positioning of thebubbles relative to the tank base (zT). This parameter isdefined as follows:

bubbles totally inside the column ) ak ¼ 1

bubbles crossing the tank base ) ak ¼ (zT�zrear,k)=hb,k

bubbles totally inside the tank ) ak ¼ 0

8>>>><>>>>:

(14)

Figure 4. Representation of the upward movement of a Taylor bubble (a) at inlet and (b) inside the column.



The last parcel of the right hand side of equation (13)accounts for the decrease in Hhyd,i (relative to the scenarioof a column full of liquid), due to the presence of bubblesinside the column. The use of parameter ak is related to thefollowing. The tank cross sectional area is considerablyhigher than the column’s and, thus, it is reasonable toassume that the pressure at the base of the tank dependsonly on the height of liquid above that position (zliq–zT),regardless of the presence of a bubble entering the tank (oreven totally inside the tank). However, the portion of abubble still inside the column (i.e., below zT) should not beneglected when computing the hydrostatic liquid heightabove the bubble at the column inlet (Hhyd,i). The volume ofliquid inside the column is reduced by the presence of abubble crossing the tank base (or totally inside the column)and, consequently, the pressure acting on the bubble at thecolumn inlet is also reduced. Notice, for instance, that thepresence of a bubble totally inside the tank does not alterthe pressure acting at the column inlet and, in agreementwith this, the summation in the previous equation does notdepend on the length of such a bubble (since ak ¼ 0).Once defined Hhyd,i, the hydrostatic pressure acting on

bubble i at the inlet coordinate can then be computed by

Phyd,i ¼ r gHhyd,i (15)

where r is the density of the liquid and g is the acceleration ofgravity. An algebraic transformation of the ideal gas law withfurther substitution of the pressure according to equation (15)gives origin to an expression that computes the number ofmoles of air in a bubble i, at the inlet coordinate:

ni ¼ hb,i Sb½Patm þ rgHhyd,i�=RT (16)

where Patm stands for ambient pressure, T refers to the temp-erature, R is the ideal gas constant and ni is the number of

moles of air in bubble i. As before, this strategy extends toall the bubbles entering the column. The computation of theprevious parameter is a requirement for the implementationof the bubble expansion along the column. Note that bubbleshaving the same length might contain different number ofmoles of air, provided that the hydrostatic pressure actingon the bubbles is different by the time they enter the simu-lation domain. This fact is related to the requirement of adetermined volumetric gas flow rate (distributed but nonethe-less determined) at the inlet coordinate, at the expense of achanging mass flow rate.

Gas expansion—effect over the length of the bubbleFigure 4(b) illustrates an instant in the upward movement

of bubbles inside the simulation domain. The hydrostaticpressure acting on bubble i is given by an expression similarto equation (13), with a correction to account for the position-ing of the bubble (znose,i).

Hhyd,i ¼ zliq: � znose,i � Sb=Sc

Xi�1

k¼1

(hb,kak) (17)

If equation (16) is transformed to isolate hb,i and Hhyd,i isfurther substituted according to equation (17), one obtains

hb,i ¼ ni RT=Sb½Patm þ r gHhyd,i� (18)

Knowing the positioning of a bubble i, equation (18) allowsthe computation of the length of a bubble i as a function of thehydrostatic pressure acting on it. This is not, however, asequential calculation. The length of a bubble is a functionof the vertical coordinate of the bubble nose [see equation(18)] whose computation, in turn, requires an estimate ofthe length of the bubble [see equation (11)]. This requiresthus an iterative approach. Figure 5 illustrates this procedure.

Figure 5. Iterative procedure for the implementation of the upward bubble movement and consequent expansion.


1502 MAYOR et al.

Gas expansion — effect over the velocityof the bubbleConsider two consecutive instants (tj and tjþ1) in the move-

ment of a train of bubbles inside the column. From instant tj totjþ1, all bubbles had their positions updated according to theirupward velocity. The hydrostatic height above each bubbledecreased from instant tj to tjþ1 and, therefore, all bubblesexpanded accordingly. Consider a bubble i inside thecolumn and the liquid flowing ahead of it. The expansion ofthe bubbles under bubble i induces a raise in the liquid andgas ahead of them, proportional to the sum of the individualexpansions undergone by each bubble (Dh1, . . . , Dhn), andgiven by

Dzahead iexpans ¼ Sb=Sc

Xnk¼iþ1

Dhk (19)

This ‘extra’ upward displacement of liquid and gas can beseen as an increase in the liquid and gas superficial vel-ocities. This increase can be calculated dividing Dzahead i

expans: bythe time increment between the two consecutive instantsunder focus (tjþ1– tj), as in the following equation:

DUahead iexpans: ¼ Dzahead i

expans:=(tjþ1 � tj)

¼ Sb= (tjþ1 � tj)Sc

� � Xnk¼iþ1

Dhk(20)

where DUahead iexpans: is the increase in the flow velocity ahead of

bubble i, due to the expansion of all bubbles flowing belowit. The upward velocity of gas bubbles flowing in a co-cur-rent liquid flow depends on the velocity profile of the liquidphase ahead. This dependence is usually introduced byparameter C (equal to the ratio between the maximumand average liquid velocity), whose value depends on theflow regime (and velocity profile) in the liquid. Thus, theoverall velocity of a trailing bubble flowing in co-currentflow is the result of two contributions: one related to thelength of the liquid slug ahead of it [equation (2)], andanother related to the extra ‘upward’ displacement of theliquid and gas due to the gas phase expansion [equation(20)]. The following equation allows the computation of theoverall velocity of a trailing bubble i, in a train of bubblesflowing upwards:

Utraili ¼Uexp

B 1þ 2:4e�0:8(hs, i�1D)0:9

h i

þ CSb=½(tjþ1 � tj)Sc�Xnk¼iþ1

Dhk(21)

where UBexp must be computed by equation (3) after substi-

tution of UG by UinletG . This substitution allows for the esti-

mation of the undisturbed upward bubble velocitydiscarding the effect of the gas phase expansion alongthe column. Recall that this effect is computed by the lastparcel of the right hand side of equation (21). In addition,the values of C and U1 required for the computation ofUtrail

i are set as discussed earlier. Finally, the estimates ofUtrail

i given by equation (21) are used in equation (10) toupdate the positioning of bubble boundaries (implementingtherefore the upward movement of bubbles).

RESULTS AND DISCUSSION

Grid Tests

Several simulations were performed to determine the timeincrement and the initial number of bubbles needed to assureadequate representativity of the simulation results. Theseanalyses were based on the bubble-to-bubble interactioncurve and on the frequency distribution curves for bubble vel-ocity, bubble length and liquid slug length. The simulationresults were found to be adequately represented when simu-lations were based on minimum of 2500 bubbles with a timeincrement of 0.005 s.

Simulation Results Versus Experimental Data

In the following sections a comparison between experimen-tal data and simulation results for three column diameters(0.052, 0.032 and 0.024 m) is shown. The experimentaldata for the larger columns (0.052 m and 0.032; 6.5 m long)are thoroughly described in Mayor et al. (2006a) whereasthe data for the narrower column (0.024 m; 10 m long) aretaken from Van Hout et al. (2001) and Van Hout et al.(2003). Normal distributions of slug length (hs) and superficialgas velocity (UG) are introduced at the inlet, for the three col-umns reported (distribution of hs: m � 5D and s � 2D; distri-bution of UG: s/m � 10%). The inlet distributions, however,are shown later not to determine the outlet results.

0.052 m column diameterTwo flow conditions are compared in this section: one with

superficial liquid and gas velocities equal to 0.074 m s21 and0.10 m s21 [Figures 6(a)–(c) and 7(a)–(b)] and another with0.10 m s21 and 0.21 m s21, respectively [Figures 6(d)–(f)and 7(c)–(d)]; ambient pressure is used as the referencethroughout the whole document, unless told otherwise.Experimental values of C and drift velocity [C ¼ 1.628 andU1 ¼ 0.314 m s21, from Mayor et al. (2006a)] are used toestimate the experimental upward bubble velocity (UB

exp).The focus is on results at 5.4 m from the base of thecolumn (vertical column coordinate at which the experimentaldata were acquired).The distributions of bubble velocity, resulting from the simu-

lation of both flow conditions, have lower standard deviationsthan the corresponding experimental distributions [Figures6(a) and (d), in terms of the frequency distribution curves,or Figures 7(b) and (d) directly from the standard deviationcharts]. The high standard deviations of the experimental dis-tributions are related to the continuous acceleration and slow-ing down of bubbles rising in the column, in part due to thelevel of turbulence of the flow and to the aeration level ofthe slugs. More details on this issue can be found in Mayoret al. (2006a) (recall that the simulation approach does notconsider slug aeration). Despite the discrepancy in the stan-dard deviations, good agreement exists in terms of the aver-age and mode of the velocity distributions [as can be seen inthe charts of Figures 7(a) and (c)].Excellent agreement is found between experimental data

and simulation results regarding the distribution of bubblelengths, for both flow conditions compared [Figures 6(b)and (e)]. Quite reasonable agreement is obtained for thedistribution of slug lengths (Figures 6(c) and (f)]. The charts



Figure 6. Frequency distribution curves: (a) bubble velocity, (b) bubble length and (c) slug length, for an experiment/simulation withUL � 0.074 m s21 and UG � 0.10 m s21; (d) bubble velocity, (e) bubble length and (f) slug length, for an experiment/simulation withUL � 0.10 m s21 and UG � 0.21 m s21; 0.052 m ID; vertical coordinate: 5.4 m.

Figure 7. Log-normal fit parameters: (a) average, mode and (b) standard deviation, for an experiment/simulation with UL � 0.074 m s21 andUG � 0.10 m s21; (c) average, mode and (d) standard deviation, for an experiment/simulation with UL � 0.10 m s21 and UG � 0.21 m s21;0.052 m ID; vertical coordinate: 5.4 m.


1504 MAYOR et al.

of Figures 7(a) and (c), showing the corresponding log-normal fit parameters, corroborate these conclusions.The experimental slug length frequency distribution curves

show wider tails than the corresponding simulation curves[Figures 6(c) and (f)]. This discrepancy may be partiallyrelated to the experimental methodology used. In fact, inorder to calculate the length of liquid slugs, constant velocitywas assumed for the preceding bubbles between themoment those bubbles cross a certain reference line andthe moment the following bubbles do. Considering the highstandard deviation of the experimental distributions ofbubble velocitiy [Figures 7(b) and (d)], this assumption,although unavoidable, may lead to a marginal increase inthe occurrence of extreme slug length values (either shortor long).

0.032 m column diameterFollowing an approach similar to that used for the larger

column (0.052 m), two different flow conditions are com-pared: one with a lower gas flow rate [UG � 0.088 m s21;UL � 0.10 m s21; Figures 6(a)–(c)] and another with ahigher gas flow rate [UG � 0.26 m s21; UL � 0.10 m s21;Figures 6(d)–(f)]. Experimental values of C and drift velocity,for the 0.032 m internal diameter column, are estimatedaccording to the literature (C ¼ 1.2 and U1 ¼ 0.196 m s21.There is a very good agreement between experimental

data and simulation results regarding the average and

mode of the bubble velocity distributions [Figures 9(a) and(c)], for both flow conditions. As for the larger column, how-ever, different standard deviations are obtained [Figures9(b) and (d)]. The bubble length distributions from the simu-lations of both flow rate conditions represent the correspond-ing experimental data [Figures 8(b) and (e)] reasonably well.There is a slight underestimation, though, for the lower flowrate condition. The slug length distributions from the simu-lations of both flow conditions are in good agreement withthe experimental data. Slightly wider tails in the experimentalfrequency distribution curves occur, however, particularly forthe higher flow rate condition [Figure 8(f)]. Similar behaviourhas been observed for the larger column. Nevertheless, verysimilar modes are obtained [Figures 9(a) and (c)].

0.024 m column diameterIn order to extend the validity ranges of the proposed simu-

lator, the experimental results by Van Hout et al. (2001)regarding a 0.024 m internal diameter column were com-pared to simulation results. The flow in a 10 m long columnwas simulated considering two reported conditions: onewith UL � 0.01 m s21 and UG � 0.41 m s21 [Figures 10(a)and (b)] and another with UL � 0.10 m s21 andUG � 0.63 m s21 [Figure 10(c) and (d)]. Experimentalvalues of C and drift velocity are considered well-predictedby expressions in the literature (C ¼ 1.2 and U1 ¼

0.17 m s21, following White and Beardmore, 1962), an

Figure 8. Frequency distribution curves: (a) bubble velocity, (b) bubble length and (c) slug length, for an experiment/simulation withUL � 0.10 m s21 and UG � 0.088 m s21; (d) bubble velocity, (e) bubble length and (f) slug length, for an experiment/simulation withUL � 0.10 m s21 and UG � 0.26 m s21; 0.032 m ID; vertical coordinate: 5.4 m.



Figure 9. Log-normal fit parameters: (a) average, mode and (b) standard deviation, for an experiment/simulation with UL � 0.10 m s21 andUG � 0.088 m s21; (c) average, mode and (d) standard deviation, for an experiment/simulation with UL � 0.10 m s21 and UG � 0.26 m s21;0.032 m ID; vertical coordinate: 5.4 m.

Figure 10. Frequency distribution curves of (a) bubble length and (b) slug length, for an experiment/simulation with UL � 0.01 m s21 andUG � 0.41 m s21; (c) bubble length and (d) slug length, for an experiment/simulation with UL � 0.10 m s21, UG � 0.63 m s21; (a)–(d)0.024 m ID, vertical coordinate: 6.88 m.


1506 MAYOR et al.

assumption corroborated by the author’s findings (transla-tional bubble velocities obtained approaching the upwardbubble velocity as defined by Nicklin, equation (3), near thetop of the column). Focus is put on the data acquired at6.88 m from the base of the column.A very reasonable agreement between experimental data

and simulation results is obtained, for both flow rate con-ditions, regarding the frequency distribution curves forbubble length [Figures 10(a) and (c)]. Moreover, a verygood agreement is obtained for the slug length variable[Figures 10(b) and (d)]. Indeed, very similar modes areobtained from the distribution curves for this variable, forboth flow rate conditions. In Figures 10(b) and (d), the simu-lated frequency distribution curves from Van Hout et al.(2001, 2003) for the slug length variable are also shown(curves drawn directly from the charts of the mentioned pub-lications). No simulation curves are given, unfortunately,regarding the frequency distribution for bubble length. Never-theless, from the analysis of the frequency distribution curvesfor slug length, it can be concluded that a better represen-tation of the reported experimental data is obtained byusing the simulator described in the present work.

On the Influence of the Inlet Slug LengthDistribution

Three simulations with different inlet slug length distributionsare compared in order to assess the extent of the influence ofthis parameter over the outlet results. Normal distributions with

increasing inlet average lengths (2 D, 5 D and 8 D) and similarstandard deviations (1 D, 2 D and 2 D, respectively) are used.All average slug lengths are shorter than 10 D in order toassure that most bubbles entering the column are within theinteraction range (up to 8–10 D) regarding the precedingbubbles [see Figure 1(a) or Mayor et al., 2006a]. A lower stan-dard deviation is used for the slug distribution centred on 2 D inorder to avoid the unreal scenario of negative slug lengthvalues. Even so, for a normal distribution, only 95% of thevalues are within an interval of two standard deviations fromthe average. Therefore, the 2.5% of the values of the distri-bution that theoretically fall on the negative side of the axisare transformed into the corresponding symmetric values.No major difference exists between the distribution obtainedand the theoretical normal distribution. All inlet parametersother than slug length and bubble length are similar for thethree simulations being compared [slug and bubble lengthare related by equation (7) at inlet]. Long columns (6.5 m)with an internal diameter of 0.032 m are considered. Figures11(a) and (c) show the frequency distribution curves for theinlet liquid slug length and bubble length, respectively,whereas Figures 11(b) and (d) show the correspondingcurves at 5.4 m from the base of the column.From an analysis of the charts it is clear that, despite the

inlet differences, similar frequency distribution curves areobtained at 5.4 m from the base of the column, for bothliquid slug and bubble length [Figures 11(b) and (d)]. Inorder to further analyse the influence of the inlet distributionsalong the column, focus was put on the evolution of those

Figure 11. Frequency distribution curves of (a) slug length at inlet, (b) slug length at outlet, (c) bubble length at inlet and (d) bubble lengthat outlet, for simulations with different inlet average slug length (2 D, 5 D and 8 D); UL � 0.10 m s21, UG � 0.26 m s21; 0.032 m ID, verticalcoordinate: 5.4 m.



distributions as the bubbles move upwards. For this purpose,the above simulations were compared once again, at severalobservation points along the column (a point every 0.6 m).Log-normal curves were afterwards fitted to the slug andbubble length frequency distribution curves obtained ateach observation point. In Figure 12, the maximal relativedifferences of the average and mode of the log-normal localfits are plotted against the vertical coordinate of the column.It can be seen in chart (a) that the maximal relative differencebetween the log-normal fit parameters for the three simu-lations decreases along the column. Indeed, despite thedifferences in the inlet distributions, similar frequency distri-bution curves are obtained for vertical column positionsabove 65 D, when UL � 0.10 m s21 and UG � 0.26 m s21

[conditions reported in chart (a); accepting a maximal differ-ence of 10%]. This value defines the extent of the entrancelength of the slug flow (for the UL and UG mentioned). Byextending the aforementioned approach to a set of increasingsuperficial gas and liquid velocities (0.10, 0.23, 0.36 and0.50 m s21) the chart of Figure 12(b) is obtained, showingthe variation of the entrance length with these parameters.A linear surface (both in UL and in UG), was fitted to thedata and it is shown also in the chart. From this chart it canbe concluded that for the given UL and UG range the entrancelength ranges from 50 D to 70 D. Additionally, within thisrange this parameter increases slightly with superficial gasvelocity while it decreases with superficial liquid velocity.This behaviour is in agreement with the variation of the gashold-up in the column. Notice that the vertical column positionused extensively as reference in the previous sections (5.4 mequals 169 D for the narrower column and 104 D for thelarger one) is far above the mentioned entrance-lengthrange. A similar entrance-length range was obtained for the0.052 m internal diameter column. A figure of 60 D is givenby Van Hout et al. (2003) for the homonymous parameter(based on the analysis of the coalescence rate along thecolumn). Figure 13 gives information on the dependence ofthe coalescence curve on the inlet slug length distributions(for a given flow condition). The coalescence curve is givenas the percentage of coalescence along the column (com-pared to the total number of coalescences occurring in the6.5 m long column).

Most coalescence occurs in the entrance length of the slugflow (50–70 D), in particular for the simulations with the smal-ler inlet average slug lengths (about 90% and 80% of the totalcoalescences occur under 70 D, for inlet averages equal to2 D and 5 D, respectively). Moreover, the coalescencecurve broadens for increasing inlet average slug length[Figure 13(a)]. Additionally, the mode and median of thecurves (corresponding to the position of maximal and 50%of total coalescences) shift upwards as the inlet averageslug length increases [Figure 13(b)].The results introduced in this section show that the stabil-

ized slug flow pattern depends more on the overtaking mech-anism (which determines the bubble coalescence) than itdoes on the type of bubble injector/nozzle (which changesthe inlet distributions). Barnea and Taitel (1993) arrived atsimilar conclusions, although they were using a differentovertaking model and discarding the gas phase expansion.

Values of the Main Flow Parameters

Several simulations with increasing superficial gas andliquid velocities (0.10, 0.23, 0.36 and 0.50 m s21) and threecolumn diameters (0.024, 0.032 and 0.052 m) are comparedin order to study the influence of these parameters over thedistributions of bubble velocity, bubble length and liquidslug length, along the column (6.5 m long). Normal distri-butions of slug length (m ¼ 5 D, s ¼ 2 D) are acknowledgedat the inlet of the column. The column diameter is used fornormalisation purposes. Log-normal distributions were fittedto the curves obtained. Nonlinear estimation wasimplemented in order to fit the modes and standard devi-ations (for U, hb and hs) computed by the log-normal fits.The general form of the equation is:

z ¼ a(H)2 þ bH þ c(UL)2þ dUL þ e(UG)

2þ fUG

þ gD2 þ hDþ i HUL þ j HUG þ k HDþ

þ l ULUG þmULDþ nUGDþ o (22)

where quadratic, linear and crossed terms in Hi, the verticalcoordinate, UL, UG and D are acknowledged. Consideringthat not all parameters in the above equation are required to

Figure 12. (a) Maximal relative difference of the mode and average of log-normal fits, along the column, for simulations with increasing averageinlet slug lengths (2 D, 5 D and 8 D), UL � 0.10 m s21 and UG � 0.26 m s21; (b) entrance length of slug flow for simulations with UL and UG

equal to 0.10, 0.23, 0.36 and 0.50 m s21; 0.032 m ID.


1508 MAYOR et al.

adequately represent some of the results obtained, relevantcoefficients were determined (for a 95% confidence level) toeach case. Non-significant coefficients were excluded fromthe fits. Standard errors for each coefficient were also calcu-lated. The coefficient estimates and corresponding standarderrors obtained are shown in Tables 1 and 2.For an easier understanding of the results, 3-D represen-

tations of the data are shown for two particular cases: atthe column top (H � 5.3 m) and along the column for increas-ing UG.

Results at the column topThe mode, standard deviation and corresponding ratio

(from the log-normal fits) for bubble velocity, bubble lengthand liquid slug length are plotted against increasing valuesof UL and UG (Figure 14), for the 0.032 m ID column. Focus

is put on the frequency distribution curves obtained at5.3 m from the column base. The corresponding 3-D surfaces[computed by equation (22) with H � 5.3 m] are also shownin the charts.As expected, the most probable (mode) bubble velocity

increases with superficial gas and liquid velocities. Moreover,the variation referred to is linear on UL and almost linear onUG [in equation (22), c ¼ 0 and e is very small, as shown inTable 1]. An excellent agreement exists between the surfacefit and the simulation results for the three column diameters(r 2 ¼ 0.998). Additionally, the corresponding standard devi-ation increases with UG and slightly decreases with UL. More-over, this parameter reaches no more than 18% of thecorresponding modes [Figure 14(g)]. This low percentageconfirms that the bubble velocity values are low and reason-ably centred on the corresponding mode (as seenpreviously).

Table 1. Coefficients (estimate and standard error) and residuals (SSE: sum of squares of error; SSE/ndat: average sum of squares of error) ofthe surface fits in Figures 14 and 15, focusing modes; equation form:

z ¼ a (H)2 þ bH þ c (UL)2þ d UL þ e (UG)

2þ f UG þ gD2 þ hDþ i HUL þ j HUG þ k HDþ l ULUG þmULDþ nUGDþ o

Mode (U) Mode (hb) Mode (hs)

Estimate Stand. error Estimate Stand. error Estimate Stand. error

a 1.24 � 1023 2.98 � 1024 21.69 � 1023 2.70 � 1024 25.32 � 1023 1.54 � 1024

b 21.22 � 1022 2.46 � 1023 2.40 � 1022 1.93 � 1023 3.75 � 1022 1.12 � 1023

c . . . . . . 6.10 � 1021 3.09 � 1022 27.32 � 1022 1.75 � 1022

d 7.03 � 1021 1.74 � 1022 24.04 � 1021 2.44 � 1022 5.92 � 1022 1.25 � 1022

e 8.01 � 1022 3.41 � 1022 21.08 � 1021 3.09 � 1022 21.87 � 1022 3.44 � 1023

f 5.64 � 1021 2.69 � 1022 6.81 � 1021 2.44 � 1022 . . . . . .g 4.93 � 10þ2 8.32 � 10þ0 21.13 � 10þ2 7.54 � 10þ0 . . . . . .h 23.33 � 10þ1 6.71 � 1021 8.17 � 10þ0 6.01 � 1021 5.10 � 10þ0 7.88 � 1022

i 1.01 � 1022 2.68 � 1023 23.98 � 1022 2.43 � 1023 . . . . . .j 3.41 � 1022 2.68 � 1023 6.27 � 1022 2.43 � 1023 . . . . . .k 1.38 � 1021 3.39 � 1022 . . . . . . 7.51 � 1021 1.75 � 1022

l 28.10 � 1022 2.75 � 1022 28.72 � 1021 2.49 � 1022 . . . . . .m 1.73 � 10þ1 3.48 � 1021 3.42 � 10þ0 3.15 � 1021 24.85 � 1021 1.79 � 1021

n 1.26 � 10þ1 3.48 � 1021 22.76 � 10þ0 3.15 � 1021 . . . . . .o 7.29 � 1021 1.42 � 1022 21.31 � 1021 1.26 � 1022 2.11 � 1022 3.39 � 1023

SSE (m2 or m2 s22) 6.69 � 1022 5.50 � 1022 1.80 � 1022

SSE/ndat. (m2 or m2 s22) 1.55 � 1024 1.27 � 1024 4.16 � 1025

r 2 0.998 0.985 0.996

Figure 13. (a) Coalescence events along the column (intervals of 0.1 m) for slug length distributions with increasing average and (b) verticalposition of maximal and 50% of the total coalescences (mode and median of the curves, respectively); 0.032 m ID; UL � 0.10 m s21,UG � 0.26 m s21.



The bubble length mode and corresponding standard devi-ation increase with UG whereas they decrease with UL

[Figures 14(b) and (e)]. This behaviour, a direct consequenceof flow continuity and coalescence along the column, is

analysed in detail in Mayor et al. (2006a). This somewhat pro-portional variation results in a quasi-constant ratio betweenthe mode and standard deviation of bubble length variable[30–40%; see Figure 14(h)].

Figure 14. Mode (a)–(c), standard deviation (d)–(f) and corresponding ratio (g)–(i) of log-normal fits; (a), (d) and (g) bubble velocity; (b), (e) and(h) bubble length; (c), (f) and (i) liquid slug length; simulations with UL and UG equal to 0.10, 0.23, 0.36 and 0.50 m s21; 0.032 m ID; verticalcoordinate: 5.3 m.

Table 2. Coefficients (estimate and standard error) and residuals (SSE: sum of squares of error; SSE/ndat: average sum of squares of error) ofthe surface fits in Figures 14 and 15, focusing s; equation form:

z ¼ a (H)2 þ bH þ c (UL)2þ d UL þ e (UG)

2þ f UG þ gD2 þ hDþ i HUL þ j HUG þ k HDþ l ULUG þmULDþ nUGDþ o

s (U) s (hb) s (hs)

Estimate Stand. error Estimate Stand. error Estimate Stand. error

a 2.26 � 1023 1.15 � 1024 . . . . . . 22.51 � 1023 1.67 � 1024

b 21.40 � 1022 9.03 � 1024 5.99 � 1023 6.52 � 1024 1.43 � 1022 1.22 � 1023

c . . . . . . 2.24 � 1021 1.30 � 1022 . . . . . .d . . . . . . 21.64 � 1021 1.02 � 1022 . . . . . .e 6.62 � 1022 7.90 � 1023 . . . . . . . . . . . .f . . . . . . 2.37 � 1021 6.64 � 1023 . . . . . .g 2.98 � 10þ1 3.21 � 10þ0 24.10 � 10þ1 3.17 � 10þ0 . . . . . .h 22.11 � 10þ0 2.55 � 1021 3.07 � 10þ0 2.56 � 1021 2.38 � 10þ0 6.73 � 1022

i 25.74 � 1023 4.76 � 1024 21.17 � 1022 1.02 � 1023 . . . . . .j 3.61 � 1022 9.85 � 1024 1.85 � 1022 1.02 � 1023 . . . . . .k 24.86 � 1022 1.31 � 1022 23.52 � 1022 1.29 � 1022 3.36 � 1021 1.90 � 1022

l . . . . . . 23.04 � 1021 1.05 � 1022 2.57 � 1022 6.64 � 1023

m . . . . . . 1.44 � 10þ0 1.33 � 1021 . . . . . .n 2.12 � 10þ0 1.17 � 1021 21.26 � 10þ0 1.33 � 1021 24.81 � 1021 7.83 � 1022

o 5.91 � 1022 4.80 � 1023 24.36 � 1022 5.37 � 1023 8.08 � 1023 2.68 � 1023

SSE [m2 or m2 s22] 1.01 � 1022 9.75 � 1023 2.14 � 1022

SSE/ndat. [m2 or m2 s22] 2.33 � 1025 2.26 � 1025 4.95 � 1025

r 2 0.983 0.978 0.974


1510 MAYOR et al.

Similar slug length modes (10–13 D) are obtained for theranges of superficial gas and liquid velocities studied(0.10–0.50 m s21). These results indicate that this parameteris almost independent of UL and UG. However, a lineardependence on column diameter exists (corresponding coef-ficient equal to 5.10, in Table 1). As for the bubble velocityparameter, there is very good agreement between the sur-face fit and the modes of slug length, for the three columndiameters (r 2 ¼ 0.996). The standard deviation/mode ratiofor the slug length parameter is approximately 35–45% forthe ranges of UL and UG studied.Finally, both mode and standard deviation surfaces

become flatter for increasing column diameters. This beha-viour is related to the normalisation procedure. Additionally,similar s/mode ratios are obtained for the three column diam-eters studied.

Results along the columnThe mode and standard deviation of the main flow par-

ameters are plotted against H and UG (Figure 15), for UL ¼

0.23 m s21 and D ¼ 0.032 m. The corresponding 3-D sur-faces [computed by equation (22)] are also shown in thecharts.The most probable (mode) bubble velocity value slightly

increases along the column [Figure 15(a)]. Although one

would expect lower velocity values for higher column verticalcoordinates (due to less frequent coalescences), the gasphase expansion camouflages and overtakes the effect ofthe decreasing coalescence (see Figure 16, showing the vari-ation of the most probable bubble velocity along the column,with and without gas phase expansion). As confirmation ofthis, the increase in bubble velocity along the column isslightly more pronounced for higher superficial gas velocities.There is excellent agreement between the surface fits andthe simulation results for this parameter for the threecolumn diameters. The standard deviation values reach nomore than 18% of the corresponding modes.The bubble length mode and corresponding standard devi-

ation increase along the column and with increasing UG. Thevariation of the most probable value is the result of thecoalescence along the column (recall that coalescenceincreases the bubble length), of the flow continuity effectsby which an increasing superficial gas velocity favours theformation of longer bubbles, and of the gas phase expansion(see Figure 16). The corresponding standard deviationreaches 30–45% of the corresponding mode, for theranges of UG and H studied.The most probable slug length increases along the column

due to the coalescence phenomena. No change is perceivedin this parameter for increasing UG [surface of Figure 15(c)approximately parallel to UG axis]. There is a weak

Figure 15. Mode (a)–(c), standard deviation (d)–(f) and corresponding ratio (g)–(i) of log-normal fits along the column; (a), (d) and (g) bubblevelocity; (b), (e) and (h) bubble length; (c), (f) and (i) liquid slug length; simulations with UL � 0.23 m s21 and UG � 0.10, 0.23, 0.36 and0.50 m s21; 0.032 m ID.



dependence on column diameter, however (mode of hsslightly decreases for increasing D). Nevertheless, the mostprobable slug length varies in the range 10–13 D atH � 5.3 m (6–8 D, at H � 0.6 m), for the three column diam-eters studied. Similar behaviour is found for the standarddeviation of the slug length parameter leading, therefore, toa reasonably steady s/mode ratio (35–50%).

CONCLUSIONS

A detailed study on the simulation of gas–liquid verticalslug flow is reported. The slug flow simulation is groundedon the bubble-to-bubble empirical correlation drawn fromthe experimental data reported in Mayor et al. (2006a). Theflow pattern in the near wake bubble region and in the mainliquid is turbulent for the ranges of parameters studied.An algorithm for implementation of the gas phase expan-

sion along the column (considering effect over bubblelength and velocity) is proposed. Distributed gas flow ratesand liquid slug lengths are acknowledged at the columninlet. Simulation data are validated for three column diam-eters (0.052, 0.032 and 0.024 m).The inlet slug length distribution was shown not to influ-

ence the development of the slug flow pattern for distancesabove 50–70 D. This defines the extent of the entrancelength of slug flow for the ranges of superficial gas andliquid velocities studied (0.10–0.50 m s21).General expressions are proposed (with a set of fit coeffi-

cients) to compute the mode and standard deviation (of log-normal fits) for bubble velocity, bubble length, and liquidslug length, as a function of H, the vertical coordinate, UL,UG and D. These expressions are shown to adequately rep-resent the simulation data for the ranges of parametersstudied.Bubble velocity is shown to increase with UL, UG and H.

The evolution of U along the column confirms the dominantinfluence of the gas phase expansion over the overall vari-ation of this parameter. Bubble length is shown to increasewith UG and H, a natural outcome of the coalescence effectand of the gas expansion. The liquid slug length increases

along the column, and is almost independent of UL and UG.Liquid slug length modes between 10 D and 13 D areobtained at the column top (H � 5.3 m), for the ranges ofUL, UG and D studied. This parameter is shown to mostlydepend on the coalescence effect along the column.Reasonably similar s/mode ratios are obtained for the

three column diameters studied.

NOMENCLATUREC empirical coefficientD column internal diameter, mg acceleration of gravity, ms22

hb,i length of gas bubble i, mHhyd,i hydrostatic liquid height above bubble i, mH vertical coordinate along the column, mhs,i length of liquid slug i, mLslug flow entrance length of slug flow, mn number of slug unit cellsndat. number of data used in the non linear estimationni number of moles of air in bubble i, molPatm ambient pressure, kg m21s22

Phyd,i hydrostatic pressure acting on bubble i, kg m21s22

r 2 coefficient of determination of fits (¼[SST-SSE]/SST)

R universal gas constant, JK21mol21

Rc column internal radius, mSb bubble cross section area, m2

Sc column cross section area, m2

SSE sum of squares of error (sum of squares ofresiduals), m2 or m2 s22

SSE/ndat. average sum of squares of error, m2 or m2 s22

SST total sum of squares (sum of squares about themean), m2 or m2 s22

t time, sT temperature, Ktj, tjþ1 consecutive time instants, sU1 upward bubble velocity in a stagnant liquid (drift

velocity), m s21

UB upward bubble velocity (according to Nicklin’sequation), m s21

UexpB experimental upward bubble velocity, m s21

UG superficial gas velocity, m s21

UG,i superficial gas velocity of slug unit i, m s21

UinletG superficial gas velocity, at column inlet, m s21

UinletG jm arithmetic average of superficial gas velocity, at

column inlet, m s21

U upward velocity of bubble (Utraili and Ulead

i for the ithtrailing and leading bubbles, m s21

UL superficial liquid velocity, m s21

z parameter to be fit by nonlinear estimationzliq. liquid free-surface coordinate, mznose,i vertical coordinate of bubble nose (bubble i), mzrear,i vertical coordinate of bubble rear (bubble i), mzT vertical coordinate of the tank base, m

Greek symbolsak parameter informing on the bubble positioning

relative to the tank base (zT)d film thickness, mDhi increase of bubble length due to expansion (bubble

i), mDti time interval for complete entrance of slug unit i, sDU ahead i

expans increase in flow velocity, ahead bubble i, due toexpansion of bubbles below, m s21

DZahead iexpans raise in liquid and gas, ahead bubble i, due to

expansion of bubbles below, mm average (mean)r density of liquid, kg m3

s standard deviationy liquid kinematic viscosity, m2 s21

Figure 16. Mode of log-normal fits for bubble velocity and bubblelength along the column, for simulations with and without gasphase expansion; UL � 0.23 m s21 and UG � 0.36 m s21(at inlet);0.032 m ID.


1512 MAYOR et al.

REFERENCESBarnea, D. and Taitel, Y., 1993, A model for slug length distribution ingas-liquid slug flow, Int J Multiphase Flow, 19(5): 829–838.

Bendiksen, K.H., 1984, An experimental investigation on the motionof long bubbles in inclined tubes, Int J Multiphase Flow, 10(4):467–483.

Brown, R.A.S., 1965, The mechanics of large gas bubbles in tubes.I—Bubble velocities in stagnant liquids, 43: 217–223.

Campos Guimaraes, R. and A. Sarsfield Cabral, J., 1997, Estatıstica(McGraw-Hill de Portugal Limitada).

Collins, R., De Moraes, F.F., Davidson, J.F. and Harrison, D., 1978,The motion of large gas bubble rising through liquid flowing in atube, J Fluid Mech, 28: 97–112.

Fabre, J. and Line, A., 1992, Modeling of two-phase slug flows. AnnRev Fluid Mech, 24: 21–46.

Hasanein, H.A., Tudose, G.T., Wong, S., Malik, M., Esaki, S. andKawaji, M., 1996, Slug flow experiments and computer simulationof slug length distribution in vertical pipes, AIChE SymposiumSeries, 92(310): 211–219.

Mayor, T.S., Ferreira, V., Pinto, A.M.F.R. and Campos, J.B.L.M.,2006a, Hydrodynamics of gas-liquid slug flow along verticalpipes in turbulent regime. An experimental study, InternationalJournal of Heat and Fluid Flow (in revision).

Mayor, T.S., Pinto, A.M.F.R. and Campos, J.B.L.M., 2007, An imageanalysis technique for the study of gas-liquid slug flow along verti-cal pipes—Associated uncertainty, Flow Measurement and Instru-mentation, 18: 139–147.

Moissis, R. and Griffith, P., 1962, Entrance effects in a two-phase slugflow, J Heat Transfer, 84: 29–39.

Nicklin, D.J., Wilkes, J.O. and Davidson, J.F., 1962, Two-phase flowin vertical tubes, Transactions of the Institution of Chemical Engin-eers, 40: 61–68.

Van Hout, R., Barnea, D. and Shemer, L., 2001, Evolution of statisti-cal parameters of gas-liquid slug flow along vertical pipes, Int JMultiphas Flow, 27(9): 1579–1602.

Van Hout, R., Barnea, D. and Shemer, L., 2002, Translational vel-ocities of elongated bubbles in continuous slug flow, Int J MultiphasFlow, 28(8): 1333–1350.

Van Hout, R., Shemer, L. and Barnea, D., 2003, Evolution of hydro-dynamic and statistical parameters of gas-liquid slug flow alonginclined pipes, Chem Eng Sci, 58(1): 115–133.

White, E.T. and Beardmore, R.H., 1962, The velocity of single cylind-rical air bubbles through liquids contained in vertical tubes, ChemEng Sci, 17: 351–361.

ACKNOWLEDGEMENTSThe authors gratefully acknowledge the financial support of Funda-

cao para Ciencia e a Tecnologia through project POCTI/EQU/33761/1999 and scholarship SFRH/BD/11105/2002. POCTI(FEDER) also support this work via CEFT.

The manuscript was received 19 December 2006 and accepted forpublication after revision 3 July 2007.



hydrodynamics of gas--liquid slug flow …paginas.fe.up.pt/ceft/pdfs/r37.pdf · hydrodynamics of...

Documents