hydrodynamics of gas liquid slug flow along vertical pipes...

Hydrodynamics of Gas-Liquid Slug Flow along Vertical Pipes in the LaminarRegimesExperimental and Simulation Study

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

Centro de Estudos de Feno´menos de Transporte, Departamento de Engenharia Quı´mica, Faculdade deEngenharia da UniVersidade do Porto, Rua Dr. Roberto Frias 4200-465 Porto, Portugal

An experimental and simulation study on free-bubbling vertical slug flow in the laminar regime was developed.A non-intrusive image analysis technique and a developed simulation code (SFS, slug flow simulator) wereused for the purpose. A single correlation was obtained for the prediction of the bubble velocity as a functionof the length of the liquid slug ahead of the bubble. Strong bubble interaction was found for liquid slugsshorter than 2D, with weak and decreasing interaction persisting for longer liquid slugs. Coalescence, thoughsparse, was found to occur along the whole column length (6.5 m). These observations differ from the findingsregarding turbulent regime (bubble interaction for liquid slugs shorter than 8-10D and coalescence mainlyin the lower part of the column). A slug flow entrance length of 70-100D was obtained for inlet slug lengthdistributions centered on 2-4D, for the ranges of superficial gas and liquid velocities studied (0.05-0.20m/s). Different inlet slug length distributions (for instance centered on 2D and 5D) were found not to evolveto a single flow pattern within the 6.5 m length of the column. General expressions were proposed to predictthe evolution of the mode and standard deviation of bubble velocity, bubble length, and liquid slug lengthdistributions as a function of the vertical column coordinate and superficial gas and liquid velocities. Bubblecoalescence was found to govern the evolution of the liquid slug length along the column. Gas expansion andbubble coalescence were found to play important roles in the evolution of bubble length and velocity.

1. Introduction

When gas and liquid flow in a pipe, they assume variousconfigurations/flow patterns. These depend on fluid properties,pipe diameters, geometry, and superficial gas and liquidvelocities. Bubbly, slug, churn, wispy, and annular flow areexamples of patterns which occur for increasing ratios of gas/liquid flow rates.

Slug flow is a complex and intermittent two-phase flowpattern which can be found in several industrial applications1

such as air-lifts, nuclear and chemical reactors, geothermalpower plants, membrane and crystallization processes, hydro-carbon production, and transportation, and also in naturalvolcanic phenomena (such as at Stromboli volcano2).

In slug flow regime, bullet-shaped bubbles (known as Taylorbubbles) occupy most of the pipe cross sectional area and flowseparated by more or less aerated liquid plugs (termed slugs).The liquid flows around the Taylor bubble in a thin annularfilm, whose thickness stabilizes when the shear and gravitationalforces reach equilibrium (originating a free-falling film). Theannular film expansion at the rear of the bubbles creates arelatively confined flow (the bubble wake), with forms varyingfrom a closed well-defined region to an open random-likerecirculation (laminar and turbulent wakes, respectively). Thenature of the bubble wake determines the column length belowthe bubble (in a moving reference frame) required for fullrecovery of the undisturbed velocity profile in the liquid, afterbubble passage. The interaction of consecutive bubbles, aconsequence of the evolving velocity profiles in the near-wakeliquid region, controls the eventual merging of bubbles (coa-lescence).

Several studies on the motion of Taylor bubbles in stagnantor moving liquids are reported in the literature (Collins et al.,3

Davies and Taylor,4 Dumitresco,5 Fernandes,6 Mao and Dukler,7

Nicklin et al.,8 White and Beardmore,9 and Wallis10 are someearly examples). They set the basis for the early understandingand modeling of slug flow pattern. From an isolated bubble toa train of bubbles, several studies have since been reportedcovering different aspects of the flow pattern. For instance,studies based on particle image velocimetry (PIV) have provideddetailed descriptions of the flow characteristics in the vicinityof Taylor bubbles. Van Hout et al.11 studied the velocity fieldinduced by a Taylor bubble rising in stagnant water; Bugg andSadd12 reported results for a stagnant viscous fluid (0.084 Pas); Nogueira et al.13,14 mention studies on the flow aroundTaylor bubbles for a wide range of liquid viscosities (0.001-1.5 Pa s) and for stagnant and flowing conditions; Sousa etal.15,16report studies on non-Newtonian liquids, and in a differentdiameter scale, Thulasidas et al.17 studied the flow patterns inthe liquid slugs inside capillaries. Other interesting contributionscan be mentioned while addressing slug flow in small-scalediameters: Thulasidas et al.,18 Vandu et al.,19 van Baten andKrishna,20 and Kreutzer et al.21,22

The unsteadiness and complexity of slug flow pattern makesthe development of simulation methodologies for the predictionof characteristics difficult. However, these are essential for thedesign, optimization and operation of applications incorporatingsuch flow pattern.

Slug flow in turbulent regime has received much attentionfrom the research community. Several experimental and simula-tion studies on continuous co-current turbulent slug flow arereported (for instance, Barnea and Taitel,23 Pinto et al.,24 Mayoret al.,25,26 Van Hout et al.27). Considering mostly air-watermixtures, probably because of easy availability and handling,they provided valuable insight into the principles of continuousturbulent slug flow and have enabled the development of robustflow simulation codes.

Many industrial applications encompass slug flow in viscousor non-Newtonian fluids (e.g., air-lift reactors, hydrocarbon

* To whom correspondence should be addressed. Tel.:+351225081692. Fax:+351 225081449. E-mail: [email protected].

3794 Ind. Eng. Chem. Res.2007,46, 3794-3809

10.1021/ie0609923 CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 04/28/2007

production and transportation, depolarization in membraneseparation processes), and yet no information is found in theliterature regarding continuous co-current slug flow in laminarregime. Since this is a frequent scenario in these fluids/applications, there is a clear need for information on thistopic. This investigation proposes to extend experimental andsimulation studies to continuous laminar flow conditions. Theresulting comparison of laminar and turbulent experimental/simulation data is a further incentive for the work reportedhere.

2. Experimental Setup

The experimental apparatus is shown schematically inFigure 1. Experiments were performed in an acrylic verticalpipe 6.5 m long with an internal diameter of 0.032 m. An 85%(w/w) aqueous glycerol solution (µ ≈ 0.114 Pa s andF ≈ 1165kg/m3) was used as flowing medium at superficial velocitiesup to 0.21 m/s. The liquid flow rate was measured at the outletof the tank before and after each experiment. The liquidtemperature was monitored continuously during the experi-ments by thermocouples placed inside the tank and at thecolumn inlet. Temperature differences between the top and thebase of the column were constantly smaller than 0.5°C. Theviscosity of the glycerol solution was measured at the experi-ments temperature in a Brookfield rotating viscometer. Air froma pressure line was introduced laterally at the base of the columnthrough a 0.003 m internal diameter injector. The air flow ratewas measured by calibrated rotameters at superficial velocitiesup to 0.38 m/s (at 1 bar and 20°C).

Images of the flow pattern 3.25 m above the base of thecolumn were recorded using a Canon digital video camcorder(model XM1) operating at a frequency of 25 Hz (exposure timesvarying from 1/4000 to 1/8000 s). Image distortion wasminimized by the use of a rectangular transparent acrylic box(filled with the liquid medium) surrounding the column test-section. Images of up to 0.47 m of column were captured inthe camera field of view.

Uniform illumination over the whole test section was achievedby means of the illumination system illustrated in Figure 2a.Two fluorescent lamps, equipped with an electronic ballast toavoid light scintillation problems (boosting scintillation frequen-cies to kHz range), were mounted inside an opaque box with adiffusive surface in front of the lamps for greater lightuniformity. The illumination kit was placed in contact with thetransparent acrylic box as illustrated in Figure 2b.

3. Video Processing

Each video frame was scrutinized by a set of custom-madeimage analysis routines for the purpose of bubble tracking. Theseroutines (built in MATLAB28) implement image loading,conversion, enhancement, filtering, and erosion steps. A thresh-old length of 0.5D is considered in order to distinguish betweenTaylor bubbles and small bubbles in the liquid slugs. Thethresholding procedure allows to spot the position of the Taylorbubble nose and to obtain a rough estimate of the positionof the Taylor bubble rear (the oscillations of the bubble wakeand the aeration level of liquid slugs makes bubble reartracking difficult). The estimate of the position of the bubblerear is further improved by tracking the lowest pixel of thecentral area of each bubble (the lighter region). This strategyassures a more accurate estimation of the bubble rear boundaryand thus provides more reliable bubble length and slug lengthdata.

The image processing comprises two different types ofapproaches: moving-point data analysis, which aims at thedefinition of the bubble-to-bubble interaction correlation (veloc-ity-wise), and fixed-point data analysis, focusing on the flowpattern at a fixed vertical coordinate. The former allows theestablishment of an empirical bubble-to-bubble interaction curvefor velocity, governing the approach and coalescence ofconsecutive bubbles, and the latter enables the gathering ofinformation on bubble characteristics (length, velocity, distance,and bubble frequency) at a given vertical coordinate. These twotypes of data are crucial inputs for the simulation of slug flowpattern. The following section describes in detail the dataobtained.

4. Experimental Data

Several experimental conditions were studied. Superficial gasand liquid velocities (UG andUL, respectively) were chosen tofulfill the requirement of laminar regimes in the main liquidbetween bubbles, in the near wake bubble region and in theannular film around bubbles. All operating conditions exceptcondition f comply with this requirement (according to thecorresponding Reynolds numbers in Table 1). The Reynoldsnumber based onVS (liquid velocity as seen by the bubble) forcondition f indicates transitional regime in the near-wake bubbleregion. More details on the calculation of these Reynoldsnumbers and corresponding critical values can be found in theappendix.

Moving-point and fixed-point data analysis25 were performedon all video recordings of the flow in order to draw as much

Figure 1. Schematic representation of the experimental setup for thestudy of co-current continuous slug flow (test column and adjacentmodules).

Figure 2. Schematic representation of the illumination system: (a)fluorescent lamps placed inside an opaque box, with a translucent diffusivesurface; (b) system positioning during operation.

Ind. Eng. Chem. Res., Vol. 46, No. 11, 20073795

information as possible from slug flow pattern in laminar regime.The data resulting from these approaches are described in detailbelow.

4.1. Moving-Point Data Analysis.Several thousand frames(up to 4000) containing more than one Taylor bubble wereanalyzed with moving-point data analysis. Focus was put onthe variation of the trailing bubble velocity according to thelength of the liquid slug flowing ahead. In Figure 3a, thevelocities of the leading and trailing bubbles are plotted againstthe liquid slug length for all operating conditions shown in Table1. Notice that the bubble velocities are normalized by theexperimental average upward bubble velocity in undisturbed

conditions,U Bexp (whose computation is described later), and

the liquid slug length is normalized by the column internaldiameter.

An analysis of Figure 3a shows that the acceleration of thetrailing bubble toward the leading one occurs for all operatingconditions, mainly for liquid slugs shorter than 2D. A verystrong acceleration is observed particularly for liquid slugsshorter than 1D, where the trailing bubble travels up to twiceas fast as the leading bubble. The strong increase of the ratioUi

trail/UBexp for short liquid slugs indicates (approximately) the

length of the bubble wake. From the shape of the bubble-to-bubble interaction curves, one can conclude that the length ofthe bubble wake is approximately 1D for all experimentalconditions studied (a figure of 0.4D is reported for stagnantconditions in a PIV study by Nogueira et al.;13 also using PIV forstagnant conditions, Bugg and Saad12 report that, at 0.77D belowthe bubble, the liquid velocity at the column axis is about 10%of the bubble velocity, a figure that provides a rough estimateof the length of the bubble wake). This parameter seems to beindependent of the superficial gas and liquid velocities (in theranges studied). However, there is still some interaction betweenconsecutive bubbles for liquid slugs longer than 3D, since thetrailing bubble travels slightly faster than the leading bubble(about 5%, 3.5%, and 1% faster forhs ≈ 3D, 5D, and 10D,respectively). This indicates that the full recovery of theundisturbed laminar profile in the liquid occurs very slowly(requiring up to 10-12D, in a moving reference frame).

By averaging the normalized velocities shown in Figure 3a,for each slug length class (0.3D wide), the smoother bubble-to-bubble interaction curve shown in Figure 3b is obtained. Errorbands corresponding to a 95% confidence level are representedfor the trailing bubble velocity. The shorter the liquid slug thelonger the error bands, behavior that is related to the higherstandard deviation of the velocity samples as bubbles approachcoalescence. The increasing trailing bubble velocity for decreas-ing liquid slug length has been fitted by an empirical exponentialequation of the form

where Uitrail refers to the trailing bubblei flowing behind a

liquid slug with lengthhs,i-1 (in column diameters; bubbles andslugs numbered from top to bottom). The first exponential termfits the strong acceleration of the trailing bubble in the vicinityof the leading bubble (near its wake), whereas the secondexponential term fits the slow acceleration of the trailing bubblefor longer liquid slugs (where the velocity profile in the liquidgradually evolves from the laminar profile). Note however that eq

Figure 3. (a) Bubble-to-bubble interaction curves for operating conditionsa-h of Table 1; (b) average bubble-to-bubble interaction curve with 95%confidence intervals; (c) bubble-to-bubble interaction curve for turbulentregime (water, after Mayor et al.25) and laminar regime (glycerol aqueoussolution, present data).

Table 1. Superficial Liquid and Gas Velocities and ReynoldsNumbers in Main Liquid, in the near Wake Region and in theLiquid Film, for Several Experiments

Reynolds number

condition UL (m/s)(Tamb, Pamb)

UG (m/s)(Tamb, Pacq)UG (m/s)

liquidReUM

wakeReVS

filmReUδ

a 0.020 0.060 0.046 22 75 11b 0.035 0.060 0.046 26 78 11c 0.104 0.060 0.046 49 94 11d 0.205 0.117 0.089 96 144 13e 0.205 0.187 0.141 113 156 12f 0.205 0.379 0.288 161 202 13g 0.211 0.110 0.083 96 144 13h 0.211 0.187 0.142 115 162 13

Uitrail

UBexp

) -3.85+ 0.82e-((hs,i-1-0.3)/0.48)+

4.91e-((hs,i-1-0.3)/1002.48) (1)

3796 Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007

1 is used to predict the bubble relative displacement, only forliquid slugs up to 12D. For longer liquid slugs, no bubble-to-bubble interaction is observed and, thus, the ratioUi

trail/UBexp is

set to unity.Mayor et al.25 reported a study on co-current slug flow in

turbulent regime (using water as flowing medium). The presentbubble-to-bubble experimental data (for laminar regime) areplotted together with the homonymous data of Mayor et al. (forturbulent regime) in Figure 3c.

The bubble-to-bubble interaction curves for laminar andturbulent regimes differ considerably. The acceleration of thetrailing bubble for turbulent regime occurs for liquid slugsshorter than 8-10D. Moreover, no evidence of bubble interac-tion has been found for longer liquid slugs. These two differentinteraction curves originate, consequently, different degrees ofcoalescence. Indeed, according to these curves there is a lotmore frequent coalescence (mostly) in the lower part of thecolumn in turbulent regime than in laminar regime. This happensbecause bubbles flowing separated by a givenhs require aconsiderably smaller portion of column to coalesce whenflowing in turbulent regime, than they do when flowing inlaminar regime (trailing bubbles catch up more slowly in laminarregime, mainly if flowing more than 2D apart).

4.2. Fixed-Point Data Analysis.Fixed-point data analysiscovered between 1000 and 2300 bubbles. The minimum numberof bubbles to have converged statistics (1000) was determinedby comparing smaller subsets of data. The superficial gas andliquid velocities (in Table 1) were chosen to allow direct analysisof the influence of those parameters over the characteristics ofthe slug flow pattern. Thus, conditions a-c feature increasingUL for constantUG while conditions d-f feature increasingUG

for constantUL. Snapshots of the flow patterns for thoseoperating conditions are shown in Figure 4. Different rangesof superficial velocities are discerned in the operating conditions.Conditions a-c, regarding low superficial mixture velocities(0.08 m/s< UM < 0.16 m/s), show bubble rear axisymmetryand nonaerated liquid slugs. On the contrary, conditions d-f,concerning higher superficial mixture velocities (0.32 m/s<UM < 0.58 m/s), show bubble rear asymmetry and oscillation,as well as increasingly aerated liquid slugs. Recall that conditionf (in Table 1) concerns already transitional regime in the near-wake bubble region (ReVS ) 202; critical values according toPinto et al.29). The influence that several parameters have onthe flow pattern was analyzed quantitatively as shown below.

4.2.1. Superficial Liquid Velocity (UL). In this section, threeexperiments with increasing superficial liquid velocity are

Figure 4. Snapshots of flow pattern for different superficial liquid and gas velocities (for operating conditions, see Table 1).


compared (for constant superficial gas velocity). The frequencydistribution curves for the main flow parameters are plotted inFigure 5. The average, mode, and standard deviation of the log-normal fits are plotted against the superficial liquid velocity inFigure 6. The log-normal fitting procedure aims at simplifyingthe comparison between different frequency distribution curves.Similar approaches are followed for all distribution curvesshown in this document.

The most probable (mode) and average values of thedistribution of bubble velocities increase with superficial liquidvelocity (Figure 5a-c and Figure 6a). Both the modes and theaverages of the distributions of bubble length (hb) and of liquidslug length (hs) decrease with increasingUL (Figure 5d-f,g-iand Figure 6a).

The variations in the bubble length and liquid slug lengthwith UL are the result of the competition between two effects:the inlet slope effect and the coalescence effect. These arediscussed in detail below. On the one hand, asUL increases(for constantUG) longer slugs and shorter bubbles are formedat the inlet of the column (an experimental observation explainedby flow continuity). Thus, if the chart of Figure 6a was focusingon the data at the column inlet, one would observe an increasein hs (longer slugs) and a decrease inhb (shorter bubbles), forincreasingUL. Said differently, one would have an inlet positive-slope trend forhs and an inlet negative-slope trend forhb,respectively. On the other hand, the fact that for increasingUL,the bubbles enter the column at higher distances (the inletpositive-slope trend forhs), implies that there is a decrease inthe number of coalescences along the column. This leads, in

Figure 5. Frequency distribution curves and log-normal fit for experiments withUL ≈ 0.020, 0.035, and 0.10 m/s;UG ≈ 0.060 m/s; vertical coordinate:3.25 m.

Figure 6. Log-normal fit parameters: (a) average and mode and (b)standard deviation for experiments withUL ≈ 0.020, 0.035, and 0.10 m/s;UG ≈ 0.060 m/s; vertical coordinate: 3.25 m.


turn, to a decrease in bothhb andhs. There is thus a competitionbetween the inlet slope effect and the coalescence effect. Thecoalescence effect is dominant in the variation ofhb andhs withUL. Indeed, the smaller number of coalescences strengthens theinlet (negative-slope) trend ofhb and inverts the inlet (positive-slope) trend ofhs. As a result, at 3.25 m from the base of thecolumn both variables decrease slightly with increasing super-ficial liquid velocity.

The standard deviation of all mentioned parameters decreaseswith increasing superficial liquid velocity (Figure 5a-c,5d-f,g-i and Figure 6b). A slight deviation from this trend existsfor the operating condition b (regarding the standard deviationof bubble velocity, Figure 6b). However, the relative contiguityof UL values for operating conditions a and b (0.020 and 0.035m/s; see Table 1) and the very low standard deviation of bubblevelocity when compared to the corresponding average (<4%)diminish the relevance of the aforementioned deviation. Thevariation of the standard deviation ofhb andhs indicates a morestabilized flow pattern for increasingUL. Similar findings arereported for studies in turbulent regime.25

4.2.2. Superficial Gas Velocity (UG). Data from threeexperiments with increasing superficial gas velocity (andconstant superficial liquid velocity) are compared in this section.The frequency distribution curves for the main flow parametersare plotted in Figure 7. The average, mode and standarddeviation of the log-normal fits are plotted against the superficialgas velocity in Figure 8.

The average and the most probable values of the distributionsof all parameters (U, hb, and hs) increase with increasingsuperficial gas velocity (Figure 7a-c,d-f,g-i and Figure 8a).

Figure 7. Frequency distribution curves and log-normal fit for experiments withUL ≈ 0.21 m/s andUG ≈ 0.12, 0.19 and 0.38 m/s; vertical coordinate: 3.25m.

Figure 8. Log-normal fit parameters: (a) average and mode and (b)standard deviation for experiments withUL ≈ 0.21 m/s andUG ≈ 0.12,0.19, and 0.38 m/s; vertical coordinate: 3.25 m.


The variation of the bubble length withUG is the result ofthe strengthening of the inlet positive-slope trend by thecoalescence effect along the column. Recall that increasingUG

while maintainingUL induces longer bubbles and shorter liquidslugs, which in turn favors coalescence (and therefore botheffects point toward the growth of bubble length withUG). Tothe contrary, the variation of the liquid slug length withUG

results from the competition of the aforementioned effects (inlettrend and coalescence). As before, the coalescence effect isdominant sincehs increases withUG (the inlet negative-slopetrend is overcome by the more frequent coalescence events,increasinghb andhs).

The standard deviations of bubble velocity and bubblelength escalate for increasingUG (Figure 8b). The standarddeviation of the slug length does not have a clear dependence,however (Figure 8b). The increasing standard deviation impliesa less stabilized flow pattern for increasing superficial gasvelocity. This conclusion is corroborated by the snapshots ofthe flow pattern shown in Figure 4 (in particular for the operatingcondition f). Similar findings are reported for turbulentregime.25

4.2.3. Experimental Values ofC and Drift Velocity. Theexperimental average bubble velocity in undisturbed conditions(UB

exp) is also assessed through fixed-point data analysis. Forthe purpose of assuring undisturbed bubble velocity only bubblesflowing behind liquid slugs longer than 5D are considered.However, a slight bubble-to-bubble interaction still exists inthese conditions (see section 4.1). The establishment of 5D asthe minimum liquid slug length for the computation ofUB

exp

results from a balance between representativity and accuracy.Indeed, computingUB

exp values based only on bubbles flowingbehind longer liquid slugs would jeopardize the representativityof the estimates as they would be based on a relatively smallsample of bubbles. Besides, according to the bubble-to-bubbleinteraction fit (eq 1), a given trailing bubble flows less than3.5% faster than its leading bubble, provided that the liquid slugbetween them is longer than 5D. Thus, although the values ofUB

exp obtained in these conditions are slightly overestimated,the increasing representativity of the estimates (based on 500-1200 bubbles) clearly makes up for the eventual drawback inaccuracy.

In order to estimate parameterC and the drift velocity (U∞)for the experiments reported in the previous sections, theexperimental average bubble velocity (UB

exp) is plotted againstthe average superficial velocity of the mixture (UM ) UL +UG; see Figure 9). The values ofUM were corrected to thepressure at coordinate 3.25 m (see Table 1). The linear fit of

the experimental data and correlation-based predictions ofUB

are also shown in this figure. The predictions ofUB arecomputed following Nicklin:8

with C ) 2 andU∞ ) 0.181 m/s. The value ofC for laminarregime was obtained theoretically by Collins et al.3 andexperimentally for instance by Nicklin.8 The value of U∞used in the computation ofUB was estimated following Whiteand Beardmore9 considering column diameter and liquidproperties.

A very good agreement was obtained between predictionsand experimental results. The experimental value ofC obtained(1.93) differs less than 3.5% from the theoretical value (2) andis within the range reported by Nicklin8 (1.80-1.95). Theexperimental drift velocity (0.16) is 12% lower than thatfollowing White and Beardmore9 (0.181 m/s). Notice thatalthough the aerated level of the liquid slugs for conditionf isfairly considerable (Figure 4f), the observed value ofUB

exp isstill consistent with the correlation-based predictions.

The agreement between experimental results and correlation-based predictions indicates that the latter can be used as inputfor simulations of this flow pattern. This is of significantimportance as it widens the range of applicability of thesimulation code (in terms of column diameter, liquid properties,etc.). Moreover, the aforementioned agreement further confirmsthe irrelevance of the accuracy concession in the computationof UB

exp.

5. Slug Flow Simulation

A slug flow simulator (SFS) was used in the present study.The following section addresses the main assumptions of thesimulator.

5.1. Simulator Assumptions, Approaches and Output.Inthe simulator, a given number of randomly distributed liquidslugs (and Taylor bubbles) is assumed to enter the column atits base. The bubble length distribution is created usingtwo independent normal distributions (prepared using BoxMuller algorithm30): the liquid slug length distribution and thesuperficial gas velocity distribution. The liquid slug lengthdistribution allows to “introduce” in the simulation theinfluence of the gas injection system (in terms of the lengthof the gas bubbles and liquid slugs formed). The super-ficial gas velocity distribution is an attempt to include in thesimulation the effect of the changing hydrostatic pressure atthe column inlet, due to the variable gas hold-up inside thecolumn.

The elements of the bubble length distribution (hb,i) arecalculated based on the mentioned distributions of the liquid

Figure 9. Undisturbed average upward bubble velocity plotted againstUM;vertical coordinate: 3.25 m.

Figure 10. (a) Representation of a train of slug unit cells (bubble+ slug)and (b) superficial gas and liquid velocities for each slug unit cell.

UB ) U∞ + C(UL + UG) (2)


slug length and superficial gas velocity, using the followingequation, for each slug unit cell (i.e., slug+ bubble):

wherei refers to the index of the slug unit cell, andSb andSc

stand for the bubble and the column cross sectional area,respectively. The estimate ofSb is computed based on the liquidfilm thickness determined following Brown31 for free-fallingconditions. The variableshs,i andUG.i refer to theith elementsof the slug length and the superficial gas velocity distributions,respectively. Equation (3) was obtained by combining the

following equations, at the inlet coordinate:UG,iSc∆ti ) Sbhb,i

andUBexp∆ti ) hb,i + hs,i, where∆ti is the time interval required

for the entrance in the column of a slug unit cell. Figure 10represents a set of slug unit cells, prepared as described above(note the distributed bubble lengths, slug lengths and superficialgas velocities and the constant superficial liquid velocity). Asshown in Figure 10, the bubble shape is considered cylindricalin the simulation. Although this assumption does not alter theway slug flow pattern evolves along the column, a correctionhas been implemented over the length of the bubbles (consider-ing spherical bubble noses, under the condition of equal bubblevolume) in other to determine more realistic estimates of bubblelength. The estimate ofUB

exp in eq 3 is retrieved using eq 2.The relation (3) serves to ensure a given averageUL andUG atthe column inlet (inputs to the simulator). With the distributionsof bubble length and liquid slug length at the column inlet, it isthen possible to simulate the process of evolution of thesedistributions along the column.

The displacement of bubbles along the column is implementedas the incremental movement of their boundaries (nose and rear),between consecutive instants, according to their velocities. Thevelocity of a given bubblei is computed, at each instant, as theresult of the bubble-to-bubble interaction (given by eq 1) andof the expansion of all bubbles flowing below. Bubble expansion(taken as a rise of the nose boundary) is implemented iterativelyfor each bubble, after updating its position, according to thehydrostatic pressure gradient along the column. The hydrostaticpressure inside the column, at a given vertical coordinate, iscomputed taking only the liquid inside the column (i.e., withoutthe Taylor bubbles). This is a reasonable approach whendiscarding pressure losses in the liquid phase. A simulation timeincrement of 0.005 s is used based on grid testing (comparisonof similar simulations with decreasing time increments).

The main outputs of the simulator are the coalescence ratealong the column and the distributions of bubble velocity, bubblelength and liquid slug length at any vertical column position.Furthermore, it is possible to determine the local/global gas hold-up values and the positioning of every bubble and liquid sluginside the column, for any time instant during the simulation.

5.2. Validation of the Simulation.Two issues are dealt within this section: the applicability of the correlation-basedestimates of parametersC andU∞ for simulation purposes, andthe comparison between experimental results and simulationdata.

5.2.1. ParametersC and U∞sExperimental versus Cor-relation-Based Estimates.Two similar simulations of theoperating conditiond (Table 1) are compared in this section.All parameters of the simulations exceptC andU∞ are similar.One simulation is based on the experimental values of these

Figure 11. Frequency distribution curves of (a) bubble velocity, (b) bubble length, and (c) slug length, for simulations with experimental and predictedvalues ofC andU∞ (1.932, 0.161, and 2, 0.181, respectively);UL ≈ 0.20 m/s,UG ≈ 0.12 m/s.

Figure 12. Log-normal fit parameters (a) average, mode and (b) standarddeviation, for simulations with experimental and predicted values ofC andU∞ (1.932, 0.161 and 2, 0.181, respectively);UL ≈ 0.20 m/s,UG ≈ 0.12m/s.

hb,i )hs,i

SbUBexp

ScUG,i- 1

(3)


parameters (1.932 and 0.161 m/s, respectively) while anotheris based on the correlation predictions (2 and 0.181 m/s,respectively; see section 4.2.3). The frequency distributioncurves of bubble velocity, bubble length and liquid slug length,obtained at 3.25 m from the base of the column, are shown inFigure 11. The average, mode and standard deviation of thecorresponding log-normal fits, are shown in Figure 12.

The analysis of the charts shows that the output of bothsimulations is very similar. Both the frequency distributioncurves (Figure 11) and the log-normal fit parameters (Figure12) of bubble length and liquid slug length show considerableresemblance. Indeed, the main difference in the outputs concernsthe frequency distribution curves of bubble velocity (and,accordingly, the corresponding log-normal fit parameters). The

Figure 13. Frequency distribution curves: (a) bubble velocity, (b) bubble length, and (c) slug length, for an experiment/simulation withUL ≈ 0.020 m/s,UG ≈ 0.060 m/s; (d) bubble velocity, (e) bubble length, and (f) slug length, for an experiment/simulation withUL ≈ 0.20 m/s,UG ≈ 0.19 m/s; verticalcoordinate: 3.25 m.

Figure 14. Log-normal fit parameters: (a) average, mode, and (b) standard deviation, for an experiment/simulation withUL ≈ 0.020 m/s andUG ≈ 0.060m/s; (c) average, mode, and (d) standard deviation, for an experiment/simulation withUL ≈ 0.20 m/s andUG ≈ 0.19 m/s; vertical coordinate: 3.25 m.


simulation based on the correlation predictions ofC and U∞shows a bubble distribution whose velocities slightly exceedthe ones observed for the simulation based on the experimentalparameters (the modes of the corresponding log-normal fit differby about 6%). However, both frequency distribution curves aresimilar in shape (similar standard deviation). Therefore, norelevant bias is introduced in the simulations by using thecorrelation predictions ofC and U∞ instead of the observedexperimental estimates. As already mentioned, this fact widensthe range of applicability of the proposed simulator.

5.2.2. Experimental Data versus Simulation Results.Acomparison between experimental data and simulation results,for operating conditions a and e, is shown in this section. The

comparison only focuses on two operating conditions (out ofeight possible) for the sake of simplicity of the analysis. Similarconclusions can be drawn for the other operating conditions.Normal distributions of slug length are introduced at the inlet(average and standard deviation equal to 3D and 1D, respec-tively). Distributed superficial gas velocity is also considered(with standard deviation equal to 10% of the correspondingaverage). The focus is on the results at 3.25 m from the base ofthe column (vertical coordinate of acquisition). Notice that thecharacteristics of the inlet slug length distribution are shownlater not to affect the results at the mentioned coordinate (seesection 5.4).

The frequency distribution curves resulting from the simula-tion of both flow conditions indicate that simulation slightlyoverestimates the bubble velocity (Figure 13a,d; the modes ofthe corresponding log-normal fits differ about 10%). This minordiscrepancy is partially related to the use of the correlation-based estimates ofC andU∞ instead of the experimental valuesobtained. The simulation results for both flow conditions showsome degree of underestimation for bubble length parameter(Figure 13b,e). This discrepancy is small for operating conditiona (≈10%) and moderate for operating condition e (≈30%).Nevertheless, these deviations should be understood in a contextof small absolute values. A very good agreement exists,however, between experimental data and simulation resultsfor slug length variable (Figures 13c and 13f). Accordingly,the log-normal fits of the corresponding frequency distri-bution curves have very similar modes and standard deviations(Figure 14).

5.3. The Coalescence Events along the Column.Thecoalescence occurring along a column, for simulations basedon different bubble-to-bubble interaction correlations, is dis-cussed in this section. The correlation (1) and that proposed by

Figure 15. Coalescence events along the column (steps of 0.1 m) forsimulations for turbulent regime (water, after Mayor et al.26) and laminarregime (glycerol aqueous solution); inlet slug length distributions centeredon 3D; UL andUG equal to 0.1 m/s.

Figure 16. Maximum relative difference of the mode and average of log-normal fits, along the column, for simulations with increasing average inletslug lengths: (a) 2D, 3D, 4D, and 5D; (b) 2D, 3D, and 4D; UL ) 0.20 m/sandUG ) 0.20 m/s.

Figure 17. Entrance length of slug flow for simulations withUL andUG

equal to 0.05, 0.1, 0.15, and 0.20 m/s; average inlet slug lengths equal to2D, 3D, and 4D; maximum relative difference of (a) 10% and (b) 12%.


Mayor et al.25 for turbulent regime (both shown in Figure 3c)are used for simulations withUL and UG equal to 0.10 m/s.Similar inlet slug length distributions are used for both simula-tions (centered on 3D). The dispersion of coalescence eventsalong the column for both simulations is shown in Figure 15Coalescence occurs mostly in the lower part of the column forturbulent regime. As is evident in the corresponding cumulativecurve, about 75% of all coalescence events occur in the first30D of the column. To the contrary, the simulation based oncorrelation 1 shows a wider coalescence curve, covering thewhole column length. Indeed, for the corresponding simulation,the first 30D of the column encompasses only 30% of allcoalescence events. These results show that the coalescence ofbubbles in laminar slug flow must be considered as omnipresentalong the whole length of the column. The evolution of the slugflow pattern is, thus, always affected by this phenomenon. Theseconclusions refer to the length of column used in this study.

5.4. The Entrance Length of Slug Flow.The gas injectionsystem (nozzle) influences the way gas bubbles are formed inthe liquid medium and affects, consequently, the length of thebubbles and liquid slugs. Thus, it determines the characteristicsof the distributions of bubble length and liquid slug lengthobtained at the inlet of the column (changing, for instance, themode, the median or the standard deviation of the distribution).However, the differences in the inlet distributions tend todissipate along the column due to coalescence. Moreover, thelength of column along which the influence of the different inletdistributions (the entrance effects) can be observed is usuallyknown as the entrance length of slug flow. This concept isgrounded on the idea that the slug flow evolves to a single slug

flow pattern, regardless of the initial characteristics of the flow(in terms of bubble and slug distributions, for instance).However, although this idea is very compelling there arepractical limitations to its applicability, namely, in terms of thecolumn length. Indeed, two very contrasting bubble length (andslug length) distributions, evolving in a column in the slug flowregime, might never reach the “final” slug flow pattern due tothe finite nature of the column. Thus, it is more useful toestimate a range of inlet distributions (for instance in terms ofslug length) that do converge to a single slug flow pattern fora given finite column length. This is a very interesting approachfor practical applications.

Slug flow in a 6.5 m long (0.032 m internal diameter) columnwas simulated. Several simulations with different inlet sluglength normal distributions were compared. Focus was put onthe evolution along the column (in steps of 0.6 m) of the averageand the most probable values of the bubble length and sluglength distributions. Log-normal distributions were fitted to theslug and bubble length frequency distribution curves obtainedalong the column, to allow for a straightforward comparison.As a first approach, the evolution of inlet slug length distribu-tions centered on 2D, 3D, 4D, and 5D was monitored (standarddeviation equal to 0.5D, 1D, 1D, and 1D, respectively; noattempt was made to address inlet slug length distributionscentered on values lower than 2D because the correspondingbubbles would be too short to be considered as Taylor bubbles,a scenario that would be outside the scope of the present work.Nevertheless, such small bubbles would undergo intensecoalescence due to the short liquid slugs between them (seethe interaction curve in Figure 3), which would result in slug

Table 2. Coefficients (Estimate and Standard Error) and Residuals (SSE, Sum of Squares of Error; SSE/ndat, Average Sum of Squares ofError) of the Surface Fits of Figures 18 and 19, Focusing Modesa

mode(U) mode (hb) mode (hs)

estimate stand. error estimate stand. error estimate stand. error

a 9.42× 10-4 7.80× 10-5 8.60× 10-4 5.30× 10-5 -3.77× 10-4 8.00× 10-5

b -7.01× 10-3 5.83× 10-4 9.87× 10-3 5.98× 10-4

c 8.16× 10-1 8.80× 10-2 -3.57× 10-1 8.13× 10-2

d 2.05× 10+0 3.54× 10-3 -1.12× 10-1 2.50× 10-2 8.05× 10-2 2.18× 10-2

ef 1.33× 10+0 7.66× 10-3 3.92× 10-1 1.18× 10-2

g -3.51× 10-2 2.12× 10-3 7.73× 10-3 2.14× 10-3

h 1.03× 10-1 2.09× 10-3 6.03× 10-2 2.12× 10-3

i -1.59× 10+0 7.04× 10-2

j 1.97× 10-1 1.29× 10-3 8.99× 10-3 1.86× 10-3 1.06× 10-1 1.60× 10-3

SSE(m2 or m2/s2) 9.83× 10-4 1.17× 10-3 1.02× 10-3

SSE/ndat (m2 or m2/s2) 6.15× 10-6 7.33× 10-6 6.36× 10-6

r2 1.000 0.992 0.970

a Equation form: z ) a(H)2 + bH + c(UL)2 + dUL + e(UG)2 + fUG + gHUL + hHUG + iULUG + j; z in Si units.

Table 3. Coefficients (Estimate and Standard Error) and Residuals (SSE, Sum of Squares of Error; SSE/ndat, Average Sum of Squares ofError) of the Surface Fits of Figures 18 and 19, Focusingσa

σ (U) σ (hb) σ (hs)

estimate stand. error estimate stand. error estimate stand. error

a 5.97× 10-4 7.30× 10-5 7.22× 10-4 5.00× 10-5 1.08× 10-3 1.66× 10-4

b -3.78× 10-3 5.03× 10-4 -2.59× 10-3 4.08× 10-4 1.03× 10-2 1.13× 10-3

c 3.59× 10-1 5.07× 10-2

d 8.46× 10-2 8.10× 10-3 -5.37× 10-2 1.45× 10-2

e 1.98× 10-1 6.84× 10-2

f 1.63× 10-1 1.50× 10-2 1.45× 10-1 7.04× 10-3

g -1.55× 10-2 1.87× 10-3 -1.26× 10-2 1.34× 10-3

h 4.20× 10-2 1.87× 10-3 2.91× 10-2 1.34× 10-3 -6.83× 10-3 2.05× 10-3

i -4.03× 10-1 5.47× 10-2 -6.34× 10-1 4.05× 10-2

j 4.56× 10-3 1.33× 10-3 3.77× 10-2 1.56× 10-3

SSE(m2 or m2/s2) 9.18× 10-4 3.88× 10-4 4.41× 10-3

SSE/ndat (m2 or m2/s2) 5.74× 10-6 2.42× 10-6 2.76× 10-5

r2 0.983 0.986 0.966

a Equation form: z ) a(H)2 + bH + c(UL)2 + dUL + e(UG)2 + fUG + gHUL + hHUG + iULUG + j; z in SI units.


length distributions similar to those considered above, at avertical coordinate not far from the column base). The resultingmaximum relative differences of the average and mode of thelog-normal local fits are plotted against the vertical coordinateof the column in Figure 16a. This chart shows that even at thelast observation point (185D or 5.9 m from the base of thecolumn) the maximum relative difference between the afore-mentioned simulations exceeds 10%. The column is not longenough to allow the full attenuation of the inlet differences.Or, alternatively, the inlet simulations are too contrasting so asto allow the definition of the entrance length of the slug flow,for a 6.5 m long column.

A second approach was attempted regarding simulations withinlet slug length distributions centered instead on 2D, 3D, and4D. The results of this comparison are shown in Figure 16b.Unlike the first scenario, strong attenuation of the inlet differ-ences occurs within the length of the column. It is thus possibleto define the entrance length of slug flow for such a columnand such inlet differences (and, obviously, such operatingconditions). A figure of 75D (2.4 m) is obtained forUG ) UL

) 0.20 m/s (accepting a maximal difference of 10%). Spanningthe aforementioned approach for a set of increasing superficialgas and liquid velocities (0.05, 0.10, 0.15, and 0.20 m/s), oneobtains the chart of Figure 17a, showing the variation of theentrance length with these parameters (for 10% maximum

relative difference). The entrance length ranges from 70D to100D, for UG andUL in the range referred to. However, if oneis less demanding in terms of the maximum relative difference(accepting for instance 12% instead of the 10%) much smallerentrance lengths are obtained (as shown in Figure 17b).Nevertheless, this approach shows that the inlet slug lengthdistributions must be centered on 2-4D to ensure reducedentrance effects in the simulation results, from the acquisitioncoordinate upward (3.25 m or 101.5D from the base of thecolumn). Notice that a study26 on turbulent slug flow revealedan entrance length ranging from 50D to 70D for convergenceof inlet slug length distributions centered on 2D, 5D and 8D(for similar column length). Thus, in turbulent slug flow, muchmore contrasting inlet slug length distributions converge withinthe same column length, due to the more frequent coalescence(as shown in section 4.1).

5.5. Simulation Study.An extensive simulation study wasdeveloped in order to expound the influence ofUG andUL onthe evolution of 3 variables along the column: bubble velocity(U), bubble length (hb), and liquid slug length (hs). For thatpurpose, several simulations with increasingUG andUL (0.05,0.10, 0.15 and 0.20 m/s) were compared systematically (16simulations). Normal distributions of slug length (average andstandard deviation equal to 3D and 1D, respectively) areacknowledged at the inlet of the column for all simulations.

Figure 18. Mode (a-c), standard deviation (d-f) and corresponding ratio (g-i) of log-normal fits along the column; (a, d, and g) bubble velocity; (b, e, andh) bubble length; (c, f, and i) liquid slug length; simulations withUL ≈ 0.10 m/s andUG ≈ 0.05, 0.10, 0.15, and 0.20 m/s.


Distributed superficial gas velocities are also considered (stan-dard deviations equal to 10% of the corresponding averages).The evolution of the resulting distributions ofU, hb andhs wasmonitored along the column, in steps of 0.6 m (10 verticalcoordinates). Log-normal distributions were fitted to eachfrequency distribution curve obtained in order to favor astraightforward comparison of different curves. This originatestwo parameters (mode and standard deviation) per curve. Thisprocedure finally results in a considerable amount of data (3variables× 2 statistical parameters× 16 simulations× 10vertical coordinates). However, data can be sorted into 6different groups: 3 regarding modes (ofU, hb and hs) and 3regarding the corresponding standard deviations. Nonlinearestimation was implemented within each group of data in orderto expound its variation withUL, UG andH (vertical coordinatealong the column). The general form of the fitting equation is

where quadratic, linear and crossed terms inH, UL, andUG areacknowledged. For each group of data, relevant fit coefficientswere determined (for a 95% confidence level). Non-significantcoefficients were excluded from the fits. Standard errors for

each coefficient were also calculated. The coefficient estimatesand corresponding standard errors obtained are shown in Tables2 and 3.

3-D representations of the variation ofU, hb andhs (modesand standard deviations) withH, UL andUG are shown in thefollowing sections. Focus is put on the variation along thecolumn for increasingUG and at column outlet.

5.5.1. Results along the Column.The mode and standarddeviation of the main flow parameters are plotted againstHand UG (Figure 18) for constantUL (equal to 0.10 m/s). Thecorresponding 3-D surfaces (computed by eq 4) are also shownin the charts.

The most probable value (mode) of liquid slug length has analmost steady increase along the column (Figure 18c) and showsno dependence onUG. The evolution of this parameter withHis mainly related to the coalescence of bubbles along the column(expansion effects have little influence over the evolution ofhs; the fact that hs shows almost no dependence onUG

corroborates this observation). Moreover, the steady variationrate of this parameter along the column confirms that thecoalescence of bubbles occurs along the whole column (asalready mentioned in section 5.3). As a consequence, both shortliquid slugs (between coalescing bubbles) and increasinglylonger slugs (sincehs increases withH) coexist in the column.To further confirm this, the standard deviation ofhs increases

Figure 19. 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) bubblelength; (c, f, and i) liquid slug length; simulations withUL andUG equal to 0.05, 0.10, 0.15, and 0.20 m/s; vertical coordinate: 5.9 m (Table 1). Superficialliquid and gas velocities and Reynolds numbers in main liquid, in the near wake region and in the liquid film, for several experiments.

z ) a(H)2 + bH + c(UL)2 + dUL + e(UG)2 + fUG +gHUL + hHUG + iULUG + j (4)


strongly along the column. Indeed, the corresponding standarddeviation/mode ratio increases from 35-40% near the base, to70-95%, at the top of the column (Figure 18i). A reasonablyconstant standard deviation/mode ratio (35-50%) was reportedby Mayor et al.,26 for a study on turbulent regime. Additionally,in that study26 the liquid slug length evolved almost asymptoti-cally along the column to about 11-12D at the top. No similarevolution of slug length parameter was found in the presentstudy (laminar regime). The liquid slug length does not stabilizebefore the column outlet and, therefore, no asymptotical valueis reached within the 6.5 m length of the column (Figure 18c).This is related to the fact that bubble coalescence occursanywhere along the whole vertical coordinate of the column.For turbulent regime in the liquid, the coalescence occurs mainlyin the lower part of the column.26

The most probable value and the standard deviation of bubblelength increase along the column and withUG (Figures 18b and18e). Coalescence and expansion effects can account for thevariation of the most probable value alongH, while flowcontinuity justifies its variation withUG. Recall that due to flowcontinuity, increasingUG for constantUL favors the formationof longer bubbles. The standard deviation of this parameterranges between 27% and 43% of the corresponding mode, forthe ranges ofUG andH studied (Figure 18h). A quite similarrange is reported for turbulent regime26 (30-45%).

Both the most probable value and the standard deviation ofbubble velocity parameter increase withH andUG (Figures 18aand 18d). Additionally, although not very evident in the charts,their variation rates withH are slightly more pronounced forhigher superficial gas velocity (i.e., for higher gas hold-up). Thisindicates that the gas-phase expansion plays an important rolein the evolution of these parameters along the column. Howeveras was already mentioned, the coalescence of bubbles also hasa relevant influence in the evolution of bubble velocity alongthe column. The standard deviation of this parameter reachesno more than 12% of the corresponding mode (Figure 18 g),for the ranges ofUG andUL studied. As for the bubble lengthvariable, this value is quite similar to that reported for turbulentregime26 (18%).

5.5.2. Results at Column Outlet.The mode and standarddeviation of the main flow parameters are plotted againstincreasing values ofUL and UG in Figure 19a-f. The chartsfocus on the frequency distribution curves obtained at 5.9 mfrom the base of the column.

The most probable bubble velocity increases linearly withboth superficial gas and liquid velocities (Figure 19a). Thecorresponding standard deviation escalates with increasingUG

and decreasingUL (Figure 19d). The standard deviation/moderatio is smaller than 14% for the ranges ofUL andUG studied(Figure 19g). Despite the wider ranges ofUL andUG studied(0.10-0.50 m/s) Mayor et al.26 reports a quite similar figure(18%), for turbulent regime.

The most probable bubble length and corresponding standarddeviation augment with increasingUG and decreasingUL (Figure19b,e). Flow continuity accounts for these contrasting variations.Approximately constant standard deviation/mode ratios areobtained (ca. 40%; Figure 19h), however, for the ranges ofUL

and UG studied. Similar ratios (30-40%) are reported forturbulent regime.26

Similar slug length modes and standard deviations areobtained at the column outlet, for the ranges of superficialvelocities studied (Figures 19c and 19f). Those parameters show,therefore, weak dependence onUL and UG. The standarddeviations are substantial, however, when compared to the

corresponding modes (ca. 70-95% as shown in Figure 19i).Lower standard deviation/mode ratios are reported for turbulentregime26 (35-45%). This is obviously related to the differentcoalescence curves in both regimes (very scarce coalescence inthe upper part of the column for turbulent regime, as opposedto coalescence along the whole column for laminar regime; seesection 5.3 for more details).

6. Conclusions

An experimental and simulation study on free-bubblingvertical slug flow is reported. A non-intrusive image analysistechnique and a slug flow simulator (SFS) were used.

A single correlation for the bubble-to-bubble interaction isproposed, relating the trailing bubble velocity to the length ofthe liquid slug ahead of the bubble. Strong bubble interactionwas found for liquid slugs shorter than 2D. However, someinteraction (though weak) was found for longer liquid slugs (upto 10-12D of column). The shape of the bubble-to-bubbleinteraction curve showed that bubbles flowing more than 2Dapart require a long portion of column to coalesce. This differsfrom the findings for turbulent regime. In turbulent conditions,bubble interaction occurs for liquid slugs shorter than 8-10Dand bubbles flowing within this range are bound to coalesce ina short portion of column.

Flow stability was shown to increase with increasing super-ficial liquid velocity and with decreasing superficial gas velocity.Good agreement was obtained between experimental undis-turbed bubble velocities and theoretical predictions for laminarregime.

Slug flow simulation was developed based on the bubble-to-bubble interaction correlation obtained. Simulation resultsshowed that bubble coalescence, although not very frequent,occurs at any point along the length of the column (in turbulentregime coalescence occurs mainly in the lower part of thecolumn). Inlet slug length distributions centered on 2-4D werefound to evolve to similar slug flow patterns within the 6.5 mof the column. An entrance length of 70-100D was found forthese inlet distributions, for the ranges of superficial gas andliquid velocities studied (0.05-0.2 m/s).

General expressions are proposed to compute the mode andstandard deviation (of log-normal fits) for bubble velocity,bubble length, and liquid slug length, as a function of the verticalcoordinate (H) and the superficial gas and liquid velocities (UG

andUL, respectively). These expressions adequately representthe simulation data for the ranges of parameters studied.

Bubble velocity is shown to increase withUL, UG andH. Itsevolution along the column is mainly the result of the gas-phaseexpansion (and coalescence). Bubble length is shown to increasewith UG andH, as a result of coalescence, gas expansion andflow continuity. The liquid slug length increases along thecolumn and is almost independent ofUG (and, therefore,UL).Coalescence is shown to be the main effect governing theevolution of this parameter. Unlike in turbulent slug flow,26 nostabilization ofhs occurs within the column length studied (6.5m).

The outlet standard deviation/mode ratios for bubble velocity,bubble length and liquid slug length are smaller than 14%, about40% and in the range 70-95%, respectively, for the ranges ofsuperficial gas and liquid velocities studied.

7. AppendixsCalculation of Reynolds Numbers (in theLiquid, in the Wake, and in the Annular Film)

All Reynolds numbers are computed after correction ofUG

for the experimental/ambient temperature and pressure at theacquisition coordinate (at 3.25 m from the column base).


The Reynolds number in liquid is based on the superficialmixture velocity (UM ) UL + UG) in a fixed reference frame.Thus, it can be obtained using the following equation:

It is generally assumed that laminar regime in the liquid isobtained forReUM smaller than 2100.

The Reynolds number in the near-wake bubble region is basedon the downward liquid velocity as seen by the bubble (VS )UB

exp - UL - UG), thus, in a moving reference frame. It can beobtained using

Laminar regime in the near wake bubble region is obtainedfor ReVS smaller than 175 (Pinto et al.29).

The Reynolds number in the annular film is based on thedownward liquid velocity in the film around the bubble, in afixed reference frame,uδ. The computation ofuδ requires someconsiderations. In a moving reference frame (inside the bubble),flow continuity requires consistency between the liquid flowrate in the column cross sectional area (ahead of a bubble), andthe homonymous parameter in the annular cross sectional areabetween the bubble interface and the pipe wall. This balancecan be expressed by the following equation:

whereVδ is computed by

andπRc2 andπ[Rc

2 - (Rc - δ)2] are the column and annularfilm cross sectional area, respectively. By solving eq 7 forVδone obtains the liquid velocity in the annular film around thebubble, in a moving reference frame (attached to the bubble):

Using eq 8 one can transform the previous expression tocompute the liquid velocity in the annular film, in a fixedreference frame:

The Reynolds number in the annular film can thus be obtainedusing an expression similar to eq 5 with the film thickness (δ)instead of the column diameter (D):

The end of laminar regime in the annular film occurs forReuδ

in the range 250-400 (Fulford32).

8. Notation

C ) empirical coefficientD ) column internal diameter (m)H ) vertical coordinate along the column (m)

hb ) length of gas bubble (m)

hs ) length of liquid slug (m)

Lslugflow ) entrance length of slug flow (m)

i ) index of bubble/slug

ndat. ) number of data used in the nonlinear estimation

r2 ) coefficient of determination of fits () (SST- SSE)/SST)

Rc ) column internal radius (m)

SSE) sum of squares of error (sum of squares of residuals)(m2 or m2 s-2)

SSE/ndat. ) average sum of squares of error (m2 or m2 s-2)

SST) total sum of squares (sum of squares about the mean)(m2 or m2 s-2)

U ) upward velocity of bubble (Uitrail and Ui

lead for the ithtrailing and leading bubbles, respectively) (m s-1)

U∞ ) upward bubble velocity in a stagnant liquid (drift velocity)(m s-1)

UB ) upward bubble velocity (according to Nicklin’s equation)(m s-1)

UBexp ) experimental average upward bubble velocity inundisturbed conditions (m s-1)

uδ ) liquid velocity in the annular film in a fixed referenceframe (m s-1)

UG ) superficial gas velocity (m s-1)

UL ) superficial liquid velocity (m s-1)

UM ) superficial mixture velocity () UL + UL) (m s-1)

Vδ ) liquid velocity in the annular film in a moving referenceframe (m s-1)

VS ) liquid velocity relative to the bubble () UBexp - UM) (m

s-1)

z ) parameter to be fit by nonlinear estimation (m or m s-1)

znose,i ) vertical coordinate of the nose of bubble i, within eachimage frame (pixel)

zrear,i ) vertical coordinate of the rear of bubble i, within eachimage frame (pixel)

Dimensionless Groups

ReUM Reynolds number based on the mixture velocity() FUMD/µ)

ReVS Reynolds number based on the liquid velocity relativeto the bubble () FVSD/µ)

Reuδ Reynolds number based on the liquid velocity in theannular film in a fixed reference frame () Fuδδ/µ)

Greek Symbols

δ liquid film thickness (m)

σ standard deviation (of bubble length or bubble velocity)(m) or (m s-1)

µ liquid viscosity (Pa s)

F liquid density (kg m-3)

ReUM)

FUMD

µ(5)

ReVS)

FVSD

µ(6)

πRc2VS ) π[Rc

2 - (Rc - δ)2]Vδ (7)

Vδ ) UBexp + uδ (8)

Vδ )Rc

2VS

δ(2Rc - δ)(9)

uδ )Rc

2[UBexp - (UL + UG)]

δ(2Rc - δ)- UB

exp (10)

Reuδ)

Fuδδµ

(11)


Acknowledgment

The authors gratefully acknowledge the financial support ofFundac¸ ao para Cieˆncia e Tecnologia through project POCTI/EQU/33761/1999 and scholarship SFRH/BD/11105/2002. POC-TI (FEDER) also supported this work via CEFT.

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ReceiVed for reView July 28, 2006ReVised manuscript receiVed February 26, 2007

AcceptedMarch 23, 2007

IE0609923


hydrodynamics of gas liquid slug flow along vertical pipes...

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