Dimensional analysis and scaling in two-phase gas–liquid stratified pipe flow–Methodology evaluation

ContentslistsavailableatScienceDirect

International Journal of Multiphase Flow

journalhomepage:www.elsevier.com/locate/ijmulflow

Dimensional analysis and scaling in two-phase gas–liquid stratified pipe flow–Methodology evaluation

Raheleh Farokhpoor

^a,∗

, Lan Liu

^a

, Morten Langsholt

^a

, Karin Hald

^a

, Joar Amundsen

^a

, Chris Lawrence

^b

aIFE, Institute for Energy Technology, Norway

bSchlumberger Software Technology, Oslo Technology Centre, Norway

a rt i c l e i n f o

Article history:

Received 10 July 2019 Revised 27 September 2019 Accepted 8 October 2019 Available online 11 October 2019 Keywords:

Multiphase flow Scaling principles Dimensionless analysis Experimental study

a b s t r a c t

Multiphaseflowmodelsarevalidatedbycomparisonwitharelativelygoodsupplyofhigh-qualitylabo- ratorydata,andarelativelysparsesupplyoffielddata,whichtendstohavepoorerquality.Oneofthe principalchallengesformultiphaseflowmodels,intermsofuncertainty,isthedifferenceinscale and someofthefluidpropertiesbetweenfieldandlaboratoryconditions.Therefore,themodelsmaybecome unreliablewhentheyareappliedtoconditionsthatareverydifferentfromthoseinthelaboratory.

IFE(InstituteforEnergyTechnology)hasrecentlydevelopedanddemonstratedscale-uprulesforthe mostbasicmultiphasepipeflows.Theobjectiveoftheworkpresented inthispaperwastoselect ap- propriatedatafromourexistingdatabaseanddesignnew,scaledlaboratoryexperiments,well-suitedto demonstrate(ortest)thescalingrulesbycomparingtheresults.Thedataincludefluidproperties,pipe configurationsandflowrates.Besidestheobservedflowpattern,liquidholdupandpressuregradientare thetwomainparametersforcomparison.

IFE’sCO2FlowLoopwithatestsectioninnerdiameter(ID)of44mm,operatesfortwo-phaseflows overalargerangeofpressuresandtemperaturesontheequilibriumlineofpureCO₂.Inordertover- ifyscale-upprinciples,seriesofexperimentswereconductedaccordingtothescalingrulestosimulate similarconditions.Theexperimentswereperformedwithgas–liquidtwo-phaseCO2forfully-developed, steady-stateflow,inahorizontalornear-horizontalpipe.Theflowregimesincludestratifiedandannular flows.Theexperimentalresultsshowedthatmeasurementsofliquidholdup,andpressuregradientinthe CO2FlowLoopareinexcellentagreementwithappropriatelyscaleddatafromthelarger-scalefacilities.

Theresults alsoconfirmthat thegas-to-liquiddensity ratioplaysan importantrole.The experiments providevaluabledatasetsforverifyingscalinglaws,whicharelackingintheliterature.

© 2019TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Therehasalwaysbeenadifficultyincomparinglaboratoryand field data formultiphase flow. The obviousreasonforthis isthe difference in scale. Typical field data come from pipelines with up to 1.2m in diameter and 120kg/m³ and more in gas den- sity whereas typical laboratory pipe diameters are 10cmor less, withjusta few testfacilities havinglarger diameters(0.2–0.3m).

Models and correlations are usually developed, and partly also validated, usinglaboratory data. The majority of laboratoriesuse low-density gases (e.g. air at atmospheric pressure); only a few havethe possibilityof usingdenser gases(e.g.SF₆ at8bara and

∼50kg/m³atIFE(InstituteforEnergyTechnology))orhigherpres-

∗ Corresponding author.

E-mail address: raheleh.farokhpoor@gmail.com (R. Farokhpoor).

sures(e.g.nitrogenat90baraand∼100kg/m³atSINTEF).Itisim- portanttovalidatetheapplicabilityofthemodelswithexperimen- talresultsobtainedforconditionssimilar tothoseexperienced in fieldsituations.

Ina recentstudyby(AlSarkhi,etal., 2016), amodelwaspro- posedtoscaleupordownthepressuredropandtheliquidholdup basedon dimensional analysis.In this study,the pressure coeffi- cient (so-called Euler Number) and Reynolds number of the gas phasewereusedtopredictthepressuregradientsathigh-pressure conditions.The modelwasvalidatedandshowedgoodagreement withnewexperiments fromtheTUFFP (Tulsa FluidFlow Project) high pressure facility for annular and stratified flow regimes. In anotherstudyby thesameauthors,(AlSarkhi,etal.,2016),anew dimensionlessnumber (the SlippageNumber) forgas–liquidflow inpipes,beingafunctionoftheFroudenumberwasproposed.Ac- cordingto thisstudy,the liquidholdup data fora wide rangeof https://doi.org/10.1016/j.ijmultiphaseflow.2019.103139

0301-9322/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

Nomenclature

ρ

G,

ρ

L,r_ρ gasandliquiddensity,densityratiokg/m³,[-]

μ_G,μ_L gasandliquidviscositymPa·s U_SG,U_SL gasandliquidsuperficialvelocitym/s

σ

GL gas–liquidinterfacialtensionmN/m D,

ε

^{pipediameter(inner)androughnessmm}

θ

^pipeinclinationangledegree,°

Fr_G,Fr_L gasandliquidFroudenumberdimensionless Re_SG,Re_SL gas and Liquid superficial Reynolds number di-

mensionless

|dp/dx| totalpressuregradientPa/m

h,H_L chordal“holdup” andcross-sectionalholdupfrac- tion

fluidsand flow conditions can be correlated witha single curve usingtheSlippageNumber.

Statistical analysis done by (Hasan et al., 2007) showed that incorrect predictionof the flow patternat highpressure will re- sultin erroneous prediction of pressure drop and liquid holdup.

(Abduvayt et al., 2003) performed experimental and modelling studies at high pressure conditions (20bar) for nitrogen–water two-phaseflow ina pipe withinnerdiameterof 106cm.The re- sultsshowedthatthestratifiedflowregionextendedtohigherliq- uidflow ratesthan atlower pressures. Itwassuggested that the mechanistic modeldeveloped forlow pressures should be modi- fiedforhigh-pressureconditionstopredictbettertheexperimental data.

Extendingdimensionalanalysistomultiphaseflowhashadlim- itedsuccessbecausethenumberofdimensionlessgroupsislarge, and parameters can be combined in unlimited ways to produce equallyvaliddimensionlessgroups.Scale-upusingmultiphaseflow modelshasbeendubiousduetopoor,orunknown, extrapolation propertiesofcorrelation-basedmodels.Thismaychangeinthefu- turewiththeimplementationofmoremechanisticmodels.

There are manymethodsofdimensional analysis, andpopular methodsofdimensionalanalysisincludeRayleigh’smethod,Buck- ingham’sPi theorem,thematrixmethod,andthemethodofsyn- thesis.Allofthesemethodsaredescribedin(Sharp,1981).In1914, (Buckingham, 1914) established thePi theorem fordescribing di- mensionlessparameters. Thetheorempostulatesthatifaphysical processsatisfiestheprincipleofdimensionalhomogeneityandin- volvesnrelevantvariablesandmindependentdimensions,thenit canbereducedto arelationshipbetweennandm dimensionless parameters. It is common to distinguish between three levels of similarity:

1.Geometrical similarityis satisfiedifall body dimensionsin all threecoordinates in the model andprototype havethe samelength-scaleratio.

2.Kinematic similarityrequiresthat themodel andprototype have the same length-scale ratio andthe same time-scale ratio (Langhaar,1951), i.e.thatthe velocitiesare scaled ac- cordingly.

3.Dynamicsimilarityexistswhenthemodelandtheprototype havethesamelength-scaleratio,time-scaleratio,andforce- scale(ormass-scale)ratio.

The paperfocuseson hydrodynamiceffectsintwo-phase flow, soourdimensionalanalysisdoesnottakeintoaccountheattrans- fereffects.In thelong-distancetransport ofoil andgas,theheat transferdoesnotstronglyinfluence thehydrodynamics.However, the temperature and pressure do influence the fluid properties (density,viscosityandsurfacetension),whichareincludedinour analysis.

Forfully-developed, steady-state,gas–liquid, two-phase, strati- fied or stratified wavy flow in an inclinedpipe, the 11 relevant variablesthatweconsiderare

1. Pipediameter,inclination,androughness(3variables):D,

θ

^,

ε

2. Densityandviscosityofliquidandgas(4variables):

ρ

G,

ρ

L, μ_G,μ_L

3. Superficialvelocityofgasandliquid(2variables):U_SG,U_SL 4. Gravityandinterfacialtension(2variables):g,

σ

GL

Insomeflowconditions,othervariablesmaybeimportant,in- cluding wall wetting and other surface chemical properties, but thesearenotconsideredhere.

According to Buckingham’sPi theorem, the numberof nondi- mensionalparametersisequaltothenumberofrelevantvariables minus the numberofindependent dimensions(time, length,and mass),giving11−3=8dimensionlessparametersforscalinganal- ysis. Fora perfect scaling,all 8dimensionless parameters should beidenticalforthetwoflowsbeingcompared.However,matching evenhalfofthese8dimensionlessparameterscanbedifficultdue tooperationallimitationsoftheflowloops.Thechallengeistode- terminewhichparametersaresignificantandwhich,ifany,canbe safelyneglected.

2. Scalingprinciplesandprocedure

IFE and its project partners have recently made significant progress inbuilding mechanistic models formultiphase flow. Al- though these models are still not exact, their accuracy against available, relevant data is greatly improved. The physical basis of the models means that they possess inherent scaling proper- ties, which have been demonstrated through comparison with a wide range of data from different laboratories and field sources (Lawrenceetal.,2012;Haldetal.,2013).

Toobtainwell-scaledinputparametersfortheexperiments,the followingdimensionlessparametersareconsidered:

1. Theinclination angle

θ

^{isan importantparameterwithre-}

specttogeometricalsimilitude.

2. Thedensityratior_ρ=^ρ_ρ^G_L.

3. The most important parameter to preserve dynamic sim- ilarity in hydraulic modelling of multiphase flows influ- enced by gravity is the Froude number. In this work, we haveassumed similaritythrough thesquared Froudenum- berFr²_SG=_(ρ^ρGU2SG

L−ρG)gD= ₍₁^r_−r^ρU_ρ^2SG_)gD todeterminethe targetvalue ofU_SG.

4. Theliquid-to-gassuperficialvelocityratioU_SL/U_SG isusedto obtainkinematicsimilarity.Togetherwith3,thisdetermines thetargetvalueofU_SL.

5. The Reynolds number is always relevant, with or without multiplephases.TheinverseReynoldsnumbersRe⁻¹_SL=_ρL^μU_SL^LD

andRe⁻_SG¹=_ρ^μG

GU_SGD canbeusedtodeterminethetargetvalues oftheviscosities

μ

Gand

μ

L.

6. MatchingtheinverseWeber numberWe⁻¹=_ρ^σGL

GU²

SGD isused todeterminethetargetvalueofthesurfacetension

σ

GL.

7. Thefinalparameteristheroughnesstodiameterratio,

ε

^/D.

Inthis study,the densityratiois prioritisedformatching.For thispurpose,two-phase gas–liquidCO₂ atdifferentpressuresand temperaturesis used.Withthe experimentallimitations,it isex- tremely difficultto find a fluid systemwhich meets all the scal- ingrequirements.Withjudgementderivedfrommodellingexperi- ence, andpreliminary studies,thelast fourdimensionlessparam- eters,items5,6and7inthenumberedlistabove,areconsidered tobelessimportant,andarenotmatched.Thetwodimensionless outputparametersforscalingcomparisonare:

Table 1a

Experimental conditions in earlier work at Tiller and IFE flow loops.

Experiment D Fluids Pipe angle Pressure U SL U SG

mm [–] ° bara m/s m/s

Tiller LS1

289 N 2–Naphtha 5.0 45 0.05, 0.15 3.0–5.0

5.0 90 0.05, 0.15 3.0–5.2

IFE WFL1 99 SF 6–Exxsol D80 5.0 7 0.03, 0.095 2.0–5.0 Tiller LS2 189 N 2–Naphtha 1.0 21 0.01, 0.06, 0.2 0.5–12.0 IFE WFL2 99 SF 6–Exxsol D80 1.0 4 0.007, 0.04, 0.15 0.04–9.0

Table 1b

Fluid properties used in Experiments at Tiller and IFE flow loops.

Experiment Density ratio Original Fluid properties in each Lab. Scaled values at IFE WFL ( D = 99 mm)

ρG/ ρL μL μG σGL μL μG σGL

[–] mPa ·s mPa ·s mN/m mPa ·s mPa ·s mN/m Tiller

LS1 0.078 0.256 0.018 13.3 0.064 0.004 2.0

0.155 0.258 0.02 10.9 0.068 0.003 1.8

IFE WFL1 0.057 1.8 0.015 21

Tiller LS2 0.035 0.312 0.10 14.4 0.14 0.005 4.8

IFE WFL2 0.035 1.8 0.015 22.6

1. Total pressure gradient per unit length in the form of ρGDU²_SG(^dp_dx+

ρ

Ggsin

θ

)^;

2. LiquidholdupH_L

Inthe contextofourdimensional analysis, theflowpatternis alsoanoutputparameter.

Theobjectiveofthisstudywastoselectappropriatedatafrom our existing data sets anddesign scaled-down laboratory experi- mentsfortheCO₂flowloopatIFE,toreproducethesame(scaled) flowconditions.Thisdemonstratesaprocedurethatcouldbeused todesignlaboratoryexperimentsrelevanttofullscalefieldcondi- tions.

Itisstraightforwardtoevaluatethescale-uprulesbycomparing results.Theaimistogeneratedataforcomparisonandverification ofthe scale-up principlesinthesimplestcases. Forthispurpose, thestepsdescribedherehavebeenundertaken:

1. High-quality laboratorydata frommedium- andlarge-scale pipesfortwo-phase,gas–liquidstratifiedflowarechosenfor scalingcomparisons,listedinTable2.

2. Byassumingequaldimensionlessnumbers(items1–4inthe dimensionless parameterlist above),the fluid flow proper- ties (flow rates, pipe inclination and density ratios) were scaledtodesignexperimentswithCO2astheworkingfluid inapipewithID=44mm.

3. Finally,measurementsofliquidholdupandadimensionless formof thepressure gradient were compared,aswere the observedflowpatterns.

3. Earlierstudy

In2012,asmallexperimentalcampaignwascarriedoutinIFE’s WellFlow Loop,(Lawrence etal.,2012),toverifythescalingrules with respect to experiments from SINTEF’s Large Scale Loop at Tiller.Theobjectiveofthisexperimentalcampaignwastogenerate some dataforcomparisonandverificationofthescale-up princi- ples inthe simplestcases.Thetest matricesforthetwo datasets aregiveninTable1a:Tillerdatafrom1993to1995(Hedne,1996; Heggum,1993)namedasTillerLS1withpipediameterof289mm and5.0°upward inclination andTillerLS2 withpipediameterof 189mmand1.0°upwardinclination.Thefluidswerenaphthaand nitrogenatnominalpressuresof21,45and90bara,withtheden- sityratiosof0.035,0.078and0.15respectively.Subsetsofthedata withU_SLvaluesintherangefrom0.01m/sto0.2m/s,instratified wavyflow,wereidentified.

Table 2

Overview of original datasets from Tiller.

Dataset Dataset 1 Dataset 2

Year 1986 1995

Pipe D, mm 194 289

Pressure, bara 45, 65 and 90 45 and 90

Fluids, two phase gas–liquid N 2–Naphtha N 2–Naphtha, N 2–Diesel Inclination, ° 1.0, 0.0, −1.0 5.0

Original U SL, m/s 0.06–1.3 0.1–0.3 Original U SG, m/s 0.5–12.0 1.0–5.0

Correspondingly, we have the IFE Well Flow Loop (Lawrence et al., 2012) experiments with 99-mm pipe diame- ter andthe same inclinations. InIFE WFL1 experiments,SF₆ and Exxsol D80 at 7 bara with density ratio of 0.057 were used to simulateTiller LS1 experiments with densityratios of 0.078 and 0.15. Due to experimental limitations, the density ratio was not matched with Tiller LS1 experiments; these experiments were intended to assess the importance of the density ratio. In IFE WFL2,with asimilar fluid systematnominal pressureof4 bara, the density ratio of 0.035 was closely matched with Tiller LS2.

The superficial velocity values, U_SL and U_SG, used in the IFE experimentsgiveninTable 1a,were designedto matchtheTiller experimentsbasedon thescaling rules describedintheprevious section.

Fluid properties including the gas and liquid viscosities and gas–liquidinterfacialtension valuesare givenin Table 1b.In the lastthreecolumns,theviscosity(andsurfacetension)valuesgiven fortheTillerflow loopare hypotheticalvaluesthat wouldbe re- quiredto achieve a perfectscaling.These hypotheticalvaluesare obtainedfromtheactualfluidpropertiesoftheTillerexperiments byscalingdowntotheIFEWellFlowLoopscale(ID=99mm)us- ingReynoldsnumberandWebernumber.

InFig.1,measuredtotalpressuregradient(magnitude)andliq- uidholdupforTillerandIFEexperimentsare compared.The aim wastoreplicatetheTillerexperiments.Here,theTillerresultshave beenscaleddowntoIFEWFLscale(withpipediameterof99mm).

Theerror linesinthe two panelshavea differentappearance. In theupperpanel,theerrorinthepressuredropmeasurementisan absoluteerrorforsmaller measurements, anda relative errorfor largermeasurements,sotheerrorlinesdivergeforlargervalue.In thelower panel,theerrorintheholdupmeasurementisan abso- luteerror,sotheerrorlinesareparallel.

Fig. 1. Experimental results for the inclination from earlier works at Tiller and IFE (pressure gradient in upper graph and liquid holdup in lower graph).

As can beseen inFig.1,forexperiments withthe densityra- tio of 0.35 and pipe with 1.0° upward inclination, there is very goodagreementinbothpressuregradientandliquidholdup.InIFE WFL1experiments,wherethedensityratiois1.5–3 timessmaller thanintheTillerexperiments andpipeinclination ishigher,5.0° upward, the pressure gradients are overestimated, butthere is a good matchin the liquid holdup. We believe that the large dif- ferencein thedensityratioisthe mainreasonforthe deviations foundinthepressuregradients.

4. Availabledatasets

As wementioned earlier,theaimisto generatesome datafor comparisonandverificationofthescale-up principlesinthe sim- plestcases.Forthispurpose,wehaveselected38subsets,atotalof 145experiments,whicharecategorizedasDataset1andDataset2 inTable2.29subsetsarefromDataset 1(Hedne,1988;Lingaand Hedne,1987;LingaandØstvang,D.,1985),wherethepipediame- teris194mm,thefluidsarenitrogenandnaphthaatnominalpres- suresof45,65,and90bara.Thepipeinclinationsincludehorizon- tal,1.0°upward,and1.0°downwardinclinations.Liquidsuperficial velocity ranges from 0.06m/s to 1.3m/s. For a constant U_SL, gas superficialvelocityvariesfrom0.5m/stoamaximumof12m/sin eachofthesubsets.

Dataset 2 consists of nine subsets of experiments with pipe diameterof 289mm and 5.0° upward inclination. Twofluid sys- tems were used; naphtha and nitrogen at nominal pressures of 45baraand90bara,anddieselandnitrogenatnominalpressure of45bara.Foreachfluidsystem,atacertainpressure,threesub- setsofthedatawithU_SL valuescloseto0.1,0.15and0.3m/sand USG ranging from1.0to 5.0m/swere identified.The flow regime changesfromstratifiedwavyatlow U_SG todispersedflowathigh U_SG.

5. TheCO₂ flowloop

A CO₂ test righas been constructedat IFE, referred to asthe CO₂ Flow Loop. The test section isa stainlesssteel pipewithdi- ameterof44mmand5.0

μ

^{mwallroughness.Thispipe,13-mlong}

(∼300 diameters) is mounted on an inclinable rigid steel beam.

The beamcan be tilted toroughly ±10°inclination. Thetest sec- tionhasthefollowinginstrumentsandequipment(Fig.2):

• ThreeFujidP-transducers(TS.dP1,TS.dP2andTS.dP3)

• Three absolute pressure transducers (TS.P1–P3) and three temperature transducers (TS.T3–T5) and four temperature sensors at the inlet andoutlet of the test section (two at eachend).

• Oneclamp-onnarrowbeamgammadensitometer(TS.

γ

⁾

• Onevisualizationsectionwithtwosightglasses(TS.SG)

• Heat exchanger made from longitudinally attached copper tubingforcontrollingthetemperatureofthetestsection

Tokeep thefluidintheentiretestrigascloseaspossibletoa uniformtemperature,allthepartsofthecoolingsystemrunasan integrated systemwith one set-point. This works very smoothly, andtemperatures in the range of −10°C–+40°C can normally be obtained.ThedataacquisitionsystemfortheCO₂-loopisbasedon NationalInstrument’s (NI) Compact FieldPoint data loggingmod- ules (PLCs) andLabVIEW software.The gasand liquidmass flow rates are measured using two Rheonik RHM 20 Coriolis meters, which are specified for measuring gas and liquidphase CO₂, re- spectively

A narrow-beam gamma densitometer is used to measure the chordal liquid fraction, h/D, across a diameter of the pipe in a verticalplane.Thegammadensitometerincludesan11GBq²⁴¹Am gammasource,a sourceholder,adetector,digiBASE-E multichan- nelanalyser fromORTEC, anda collimator.APC withaLabVIEW applicationcommunicateswiththebase,andthe dataarelogged usingin-housesoftware.

The holdup measurements compared inthe paper comefrom two different types ofinstrument. The broad beamgamma den- sitometer on the IFEWell flow loop gives a direct measurement of holdup (within a certain margin of error). The narrow beam gammadensitometersinourexperiments(IFE’sCO₂flowloop)and atSINTEFgiveameasurementoftheliquidheightonly.Inorderto compare thesemeasurements, a conversionis necessary, andthe formulabasedona flatinterface isthesimplestandmostrobust waytodothis.Theuseofthisconversionhasbeenassessedmany times atSINTEF by comparing with direct holdupmeasurements obtainedusingquickclosingvalves.The conversionintroducesan additionaluncertaintywhichisincludedinthestateduncertainty rangeofthemeasurements.Forthenarrowbeaminstruments,the pipe cross-sectional holdup H_L is estimated assuming a flat gas–

liquidinterface:

HL= AL

A = 1

2

π ⁽

²

δ

−sin2

δ )

, h/D= 1

2

(

¹−cos

δ )

⁽¹⁾

where A is the pipe cross sectional area, A_L is the cross section ofthepipefilledwithliquidand

δ

^isthewettedhalf-angle,illus- tratedinFig.3.

The CO₂ flow loop can operate in two-phase flow over the rangeof−10°C–+30°C(thecriticalpoint),correspondingtopres- suresfrom26.5to73.7bar,whichallowsaccesstogas–liquidden- sity ratios in the range0.07–1.0. It is difficult to control experi- ments near the criticalpoint, but theoperating temperature can beincreasedabovethecriticalpointandupto+40°CwhereCO₂ isinsupercriticalphase.Thegas–liquidphaseboundaryforCO2is showninFig.4, whichalsoshowsthegas–liquiddensityratioat thephaseboundaryasafunctionofthetemperature.

Fig. 2. Top: Schematic diagram of the test section of the CO 2flow loop. Bottom: part of test section with and without insulating materials.

Fig. 3. A schematic diagram of the cross-section geometry in two-phase stratified flow.

Fig. 4. The phase diagram for CO 2for the CO 2flow loop operating conditions and the gas–liquid density ratio as a function of temperature.

6. Experimentalprocedure

1. Thepipeinclinationwasadjustedandfixed.

2. Priorto theexperiments,thetestsection wasfloodedwith gaseousCO₂andthesight-glasswasusedtomakesurethat therewasnoliquidleftinthetestsection.

3. Theimpulselinesforthedp-transducerswere bledoff and thesignal/output was zeroed.The zeropoint of the trans- ducerswassetatstaticconditionswithsingle-phasegasin the pipe andwith the test section at the desired inclina- tion.Therefore,toconvertthemeasureddifferentialpressure (magnitude)totheactualtotalpressuregradient,thefollow- ingexpressionshouldbeused:

dp/dx=dp/dx_measured+

ρ

^Ggsin

θ

⁽²⁾

where

θ

^isthepipeinclination.

4. Thegammadensitometer wascalibrateddailyorwhen the fluid temperature changed significantly (by say, 2–3 °C).

Countrateswere measured forboth purevapour andpure liquidphasesandgivenasinputtotheLabVIEWprogramme, asthisisrequiredinputtotheholdupalgorithm.

5. ForafixedU_SL,U_SG-sweepswereperformedbystartingwith a high value of U_SG and then step-wise reducing U_SG ac- cordingtothe plannedtest matrix.(In thismanner, reach- ing steady-state flow is faster than the other way around, i.e.stepping up inU_SG). ForeachU_SG,holdupandpressure gradientsweremeasured.

6. To match the density ratio, we used pure CO₂ at differ- ent temperatures (and thus pressures) in the range from

−12°C–+10°C,dependingontheexperiment.Thetempera- tureattheinletofthetestsectionwasusedtocalculatethe thermodynamicpropertiesofliquidandgasCO₂.

7. Measurementuncertainty

Allmeasurementsare subjecttodegreesofuncertainty, which for the CO2 Flow Loop have been estimatedby combining Type Bstandarduncertainties(instrumentaccuracy,repeatability,linear- ity,ambientconditions,drift,offset,etc.)andtheTypeAstandard

Table 3

Estimated measurement uncertainties in input and measured data.

Parameter Uncertainty in CO 2loop Uncertainty in Tiller loop

Pipe inclination ±0.05 ° ±0.1 °

Pipe diameter ±0.4 mm ±1%

Absolute pressure ±0.1 bara Not available

Temperature ±1.0 °C Not available

Liquid holdup ±0.02–0.03 (absolute) ±0.01–0.02

Liquid superficial velocity ±2–7% ±3%

Gas superficial velocity ±2–5% ±1%

Pressure gradient ( > 100 Pa/m) ±5–10% ±7.5%

Pressure gradient ( < 100 Pa/m) ±10 Pa/m ±10 Pa/m

Liquid density ±0.5–1.0% ±0.5%

Gas density ±2.5–3.5% ±5%

uncertainty, which is based on statistical treatment of repeated measurements.Thebestestimatedtotaluncertaintiesaregivenin Table3.

Theuncertaintyintheholdupmeasurementsisgenerallyofthe order±0.02–0.03(absoluteerror).Theuncertaintyinthepressure gradientisgenerallyintherange±5%–10%ofreading(relativeer- ror).Insomeexperimentswherethepipehas1°or5°downward inclination,thepressuregradientreachestozeroatlowU_SGsothe errorinisassumedtohavealowerlimitwhichisthevaluespeci- fiedbytheinstrumentmanufacturer,10Pa/m(absoluteerror).Be- causethefluidmassflowratesaremeasured,ratherthanthevol- umetricflow rates,theuncertaintiesinthepipediameterandgas andliquiddensitiescontributetotheresultinguncertaintiesinthe gas andliquid superficial velocities. Gas and liquid densities are takenfromNationalInstitute ofStandards andTechnology (NIST) (USDepartmentofCommerce,n.d.),andtheuncertaintyindensi- tiesgiveninTable3ismostlyduetovariationsinthetemperature.

For the output data (flow regimes, H_L, |dp/dx|) in this study, minorchangesinthepipewallroughness,liquidandgasviscosity, andinterfacialtensionarenotexpectedtohaveastronginfluence ontheresults.We,therefore,anticipatethat themeasured values arerelativelyinsensitivetouncertaintiesintheseparameters.The effect of the uncertainty in the pipe inclination (±0.05°) on the measurementsisalsoverysmall.

8. Experimentalresultsfordataset1

Inthissection,theresultsfromtheCO₂FlowLoopexperiments andsubsequent dataanalysisare presented.Firstly, wecompared TillerDataset1experimentswithourexperimentsintheCO₂Flow Loop.Exp1–Exp29correspond tothe29subsetsfromDataset 1, withthe experimental conditions listed in Table 4. As explained

earlierin thispaper, theaim wasto simulateTillerexperiments, withmatchingdensityratiobeingprioritized.Therefore,inExp1–

Exp29,twophaseCO2 atdifferenttemperatures,−7,0and10°C, wasusedtomatchthedensityratioincorrespondingexperiments Tiller 1–Tiller 29 (see Table 4). The detailed results are given in Table6intheAppendix.

Gasandliquidsuperficialvelocities inTillerexperiments were scaled down to the CO₂ flow loop according to the scaling pro- cedure explained earlier. In Table 4, the U_SL values used in the TillerexperimentsandthescaledvaluesusedintheCO₂flowloop aregiven.Theliquidviscosityandgas–liquidinterfacialtensionare listed forpureCO₂ atthe giventemperature. Thevaluesofthese propertiesfortheTillerexperimentsarescaleddowntoCO₂ flow loop scale (with pipe diameterof 44mm), using liquid Reynolds number andWeber number (asin Table 1b), andthe target val- uesare giveninTable4.In all29experiments, foreach USL,USG

isslowlydecreasedinquitesmallstepsfromapproximately5m/s–

0.5m/s.

Exp1–Exp13wereperformedwithahorizontalpipeforthree differentfluidsystemsrepresentingdensityratiosof0.15,0.11and 0.08.Tomatchthegas-to-liquiddensityratioofthefluidsusedin the Tiller experiments, two-phase CO₂ at temperatures of 10° C, 0.0̊Cand−7°Cwasused.Theliquidviscosityandgas–liquidsur- face tension in the CO₂ Flow Loop are significantly higher than the scaled valuesfromthe corresponding Tillerexperiments. The comparisonofresults(below)demonstratesthatthisdifferenceis notveryimportant.Exp14–Exp21correspondtotheeightsubsets fromDataset 1fora pipewith1.0°upward inclination.Tillerex- periments werecarriedout withnitrogenandnaphthaat45and 90baraand20°C.ToreachsimilardensityratiosintheCO₂ Flow Loop,two-phase,CO₂at−7°Cand10°Cwasused.Lastly,Exp22–

Exp29includeeight subsetsofexperimentsthat wereconducted

Table 4

Experimental conditions and fluid properties used in CO 2Flow Loop and Dataset 1.

Experiment D Pipe angle Temp. Press. ρG/ ρL Original U SL Scaled down to CO 2Flow Loop

mm ° °C bara [–] m/s μLmPa ·s σGLmN/m

Exp 1–Exp 4 44 0.0 10 45 0.15 0.05, 0.1, 0.5, 0.8 0.083 ∼3

Tiller 1–Tiller 4 194 0.0 20 90 0.15 0.06, 0.1, 0.2, 1.0 0.045 0.72

Exp 5–Exp 9 44 0.0 0.0 35 0.11 0.1, 0.2, 0.3, 0.4, 0.6 0.10 ∼4.5

Tiller 5–Tiller 9 194 0.0 20 65 0.11 0.2, 0.4, 0.6, 0.8, 1.3 0.033 0.88

Exp 10–Exp 13 44 0.0 −7 28.8 0.081 0.03, 0.05, 0.1, 0.5 0.11 ∼6.0

Tiller 10–Tiller 13 194 0.0 20 45 0.082 0.06, 0.1, 0.2, 1.0 0.05 0.97

Exp 14–Exp 17 44 1.0 10 45 0.15 0.03, 0.05, 0.1, 0.5 0.083 ∼3

Tiller 14–Tiller 17 194 1.0 20 90 0.15 0.06, 0.1, 0.2, 1.0 0.042 0.7

Exp 18–Exp 21 44 1.0 –7 28.8 0.081 0.03, 0.05, 0.1, 0.5 0.112 ∼6.0

Tiller 18–Tiller 21 194 1.0 20 45 0.076 0.06, 0.1, 0.2, 1.0 0.047 0.8

Exp 22–Exp 25 44 −1.0 10 45 0.15 0.03, 0.05, 0.1, 0.5 0.08 ∼3

Tiller 22–Tiller 25 194 −1.0 20 90 0.15 0.06, 0.1, 0.2, 1.0 0.04 0.7

Exp 26–Exp 29 44 −1.0 –7 28.8 0.081 0.03, 0.05, 0.1, 0.5 0.112 ∼6.0

Tiller 26–Tiller 29 194 −1.0 20 45 0.076 0.06, 0.1, 0.2, 1.0 0.046 1.0

Fig. 5. Total pressure gradient in CO 2flow loop and corresponding Tiller experiments.

usingtwo-phaseCO2 at10and−7°Crespectively, ina pipewith inclinationof1.0°downward.

The pressure gradient measurements obtained from the CO₂ Flow Loop experiments are compared with scaled values of the pressure gradient measurements from the Tiller experi- ments. The dimensionless pressure parameter in the form of ρGDU²_SG(^dp_dx+

ρ

G gsin

θ

)^{isusedtodeterminetheexpectedtotalpres-} sure gradient in the CO₂ Flow Loop. In Fig. 5, the scaled val- uesofthepressuregradient(magnitude)arecomparedwithmea- sured values from the CO₂ Flow Loop for three different pipe inclinations.

As can be seen in Fig. 5, there is excellent agreement in the pressure gradients between the CO₂ Flow Loop and the Tiller experiments for all density ratios and all three pipe inclina- tions. The totalpressure gradientsat highgassuperficial velocity (U_SG >4m/s),weremeasured slightlyhigherthanthescaledval- uesfromtheTillerexperiments.Oneexplanationforthisdeviation can be that athigher U_SG, thereis moredroplet entrainment,so that effects of interfacialtensionand liquidviscosity are not en- tirelynegligible.

The uncertainty lines are the combination of uncertaintiesin data from the CO₂ flow loop and the Tiller loop, calculated by

Eq.(3):

Totalerrorindp/dx=

Errorin dp/dxCO2FlowLoop2

+Errorin dp/dxTillerLoop2

¹/2 (3) Therefore,

1. Fordp/dx>100Pa/m;uncertaintyis12.5%ofreading(rela- tiveerror).

2. Fordp/dx < 100Pa/m; uncertaintyis 14Pa/m (absoluteer- ror).

For1.0°upward pipe inclination,both theCO₂ Flow Loopand the Tiller experiments show a similar minimum in the pressure gradients, associated with the change from gravity-dominatedto friction-dominatedflow.Likewise,for1.0°downwardpipeinclina- tion,asU_SG decreasesstepbystep,andforU_SG<1m/s, theaver- agepressuregradientapproacheszeroandthentheflowbecomes gravity dominated. For better understanding, the total pressure gradientforthreeexperimentsintheCO₂ flowloopwithhorizon- talpipe,andfor1.0°upward anddownwardpipe inclinationsare showninFig.6.Allexperimentsarefortwo-phaseCO₂atthetem- peratureof10°C(densityratioof0.15)withU_SL =0.1m/s. Scaled

Fig. 6. Total pressure gradient measured directly in CO 2flow loop and the scaled values from Tiller loop for three different inclinations with density ratio = 0.15 and U SL= 0.095 m/s.

Fig. 7. Liquid holdup in CO 2flow loop and corresponding Tiller experiments.

values from corresponding Tiller experiments are also shown in thefigure.

Sinceliquidholdupisalreadydimensionless,themeasuredliq- uidholdups fromthe CO₂ Flow Loopare compareddirectly with measuredvaluesfromtheTillerexperiments.AsexhibitedinFig.7, the liquid holdup values match very closely with the equivalent experimentsatTiller.Thecombineduncertaintyinliquidholdup iscalculatedinthesamewayastheone forpressuregradientby Eq.(3)andis0.036absoluteerror.Largerelativedeviationsinthe measuredliquidholdupcanbe seenwhenit fallsbelow10%. But thesedifferencesareofsimilarmagnitudetothecombineduncer- taintyintheholdupmeasurements.

The flow pattern determination in the Tiller experiments is basedonvideorecordings.Forhorizontalandnearhorizontalflow, gas–liquiddistributionisclassifiedintofourmajorpatterns;annu- larflow,stratifiedwavyflow,slugflowanddispersed-bubbleflow.

Inourexperiments,weusevideorecordings,timeseriesofliquid holdupandtheprobability distribution functionofthe holdupto determinetheflowpattern.

Therearesystematictrendsintheflowpattern.Intheseexper- iments,for all three densityratios (0.15, 0.11 and 0.078) andall threeinclinations(−1.0,0.0and1.0°),forU_SL >0.38m/s,theflow regimeis largewave flow, withtheinterface becoming smoother asU_SG reduces. Forlow U_SL (0.03, 0.05 and0.095m/s),the flow ismainly stratified-annular flow athigh U_SG and stratified wavy flowatlower U_SG,exceptforthepipewithinclinationof1.0°up- wardanddensityratioof0.078,forwhichlargewave flowisob- servedforUSG < 1m/s.Inall 29Tillerexperiments(named Tiller

1–Tiller 29), for all density ratios and pipe inclinations, and for U_SL < 0.1m/s, the flow is described as stratified wavy flow for the entire range of U_SG. For U_SL < 0.2m/s and U_SG > 3.5m/s, the flow pattern is identified as annular flow. The flow pattern mapsforboth CO₂ flow loopexperiments andTillerexperiments (for all three inclinations and density ratios) are presented in Fig. 8 with scaled velocities. The lines in this figure are indica- tions or the approximate boundaries between the different flow patterns.

The flow patternmap for CO₂ Flow Loop look differentfrom Tillerloop.Wethinkthatthedifferencesbetweentheflowpattern observationsin theleft andrightpanelsarecaused bytwo main factors:

1. FortheTillerexperiments,conductedin1986,theinforma- tion available todetermine the flow patternwasvery lim- ited.Thismadeitvery difficulttoidentifytheflowpattern withanydegreeofconfidence,soitisquitepossiblethatthe reportedflowpatternisnotalwayscorrect.

2. Eveninthebestcircumstances,theidentificationoftheflow pattern represents the subjective opinion of the observer.

Theexperimentswereconductedbydifferentresearchersat differentlaboratorieswithmanyyearsinbetween, andthis mayexplain thesystematicdifferencesinthereportedflow patterns.

Nevertheless,thethree-flowpatterns,stratified-wavy,stratified- annular and large waves are rather similar with no sharp distinctions. They could all be regarded as variations of one flow pattern, e.g. wavy-stratified annular, so that the appar- ent large difference in observed flow patterns may not be very significant.

9. Experimentalresultsfordataset2

Tiller Dataset 2 from 1995was obtained using a pipe with a diameter of 289mm and5.0° upward inclination. Two fluid sys- tems were used; naphtha and nitrogen at nominal pressures of 45 bara and 90 bara, and diesel and nitrogen at nominal pres- sureof45bara.Foreach fluidsystemandspecificpressure,three subsets of the data with U_SL values of 0.1, 0.15 and 0.3m/s in stratified-wavyflowwereidentified.InTable5,theCO₂ FlowLoop experiments andcorresponding Tillerexperiments are listed. The detailedresultsaregiveninTable7intheAppendix.Asmentioned earlier,tomatchthe Tillerexperiments’densityratios,two-phase

Fig. 8. Flow pattern map for CO 2flow loop experiments (to the left) corresponding to Tiller dataset 1 (to the right).

Table 5

Experimental conditions and fluid properties used in Exp 30–Exp 38 and corresponding Tiller experiments for the pipe with inclination of 5.0 °upward.

Experiment ID Liquid Temp. Press. ρG/ ρL

Original U SL

Scaled down μL σGL

mm [–] °C bara [–] m/s mPa ·s mN/m

Exp 30–Exp 32 44 CO 2 10 45 0.15 0.04, 0.06, 0.12 0.083 ∼3 Tiller 30–Tiller 32 289 Naphtha 20 90 0.15 0.1, 0.15, 0.3 0.021 0.33 Exp 33–Exp 35 44 CO 2 −8 28 0.078 0.04, 0.06, 0.12 0.114 ∼6.0 Tiller 33–Tiller 35 289 Naphtha 20 45 0.078 0.1, 0.15, 0.3 0.022 0.45 Exp 36–Exp 38 44 CO 2 −12 25 0.067 0.04, 0.06, 0.12 0.122 ∼6.5 Tiller 36–Tiller 38 289 Diesel 20 45 0.06 0.1, 0.15, 0.3 0.156 0.65

Fig. 9. Total pressure gradient in CO 2flow loop and scaled values from the corresponding Tiller experiments for the pipe with inclination of 5.0 °upward and for different density ratios.

gas–liquid CO₂ experiments atthree different temperatures were used. The same procedure as explained for Dataset 1 is applied here.

Exp 30–Exp 32 are designedto replicate the three subsetsof experiments from Tiller that were performed with nitrogen and naphthaat90bara and20°Ctemperature. At thiscondition,the nitrogentonaphthadensityratiois0.15andtomatchthisvalue, two-phaseCO₂ at10 °C wasused.For Exp33–Exp 35, two-phase gas–liquid CO₂ at −8 °C gives the density ratio of 0.078. These experiments are representative of Tillerexperiments fornitrogen andnaphtha at45 baraand20 °Ctemperature, as mentionedin Table 5. In Exp 33–Exp 35, CO₂ liquid viscosity and gas–liquid interfacial tension are significantly higher than the scaled values fromtheTillerexperiments.Again,thecomparisonofresults(be- low)demonstrates that thisdifferenceisnot very important.Exp 36–Exp38correspondtoTillerexperimentsfordieselandnitrogen at45bara.Atthispressure,thenitrogentodieseldensityratiois 0.06andwiththe temperaturelimitationsinthe CO₂ Flow Loop, with two-phase CO₂ at −12 °C, a densityratio of 0.067 wasthe lowest that could be achieved.Intheseexperiments,for thefirst time,CO₂ liquidviscosity(0.122mPa.s) isveryclosetothescaled value (0.156mPa.s),butthegas–liquidsurfacetensionistentime higher.

The measurements oftotal pressure gradient(magnitude) and liquidholduparecomparedinFig.9andFig.10,withsymbolsin- dicatingthedensityratio.Pressuregradientsareinexcellentagree- ment with the scaled values from the Tiller experiments, with slightdeviationathighervalues,butstill withinthe rangeofex-

perimentaluncertainty.Forexperimentswithdensityratiosof0.15 and0.078,theliquidholdupmatchesverywellwiththeTillerex- periments.Forexperimentswithlowestdensityratio(0.06), there aresome discrepanciesintheliquidholdups,with10–20%higher valuesmeasured intheCO₂ FlowLoopcomparedtotheTillerex- periments.

In these experiments, for all three density ratios (0.15, 0.078 and 0.06) and U_SL (0.04, 0.06 and 0.12m/s), the flow pattern is quite similar. Stratified-annular flow was identified at high U_SG, changingtostratified-wavyflowand,finally,tolargewavesatU_SG values where the pressure gradient reaches its minimum. With lower density ratio,the minimum inpressure gradient occurs at slightly higher U_SG, which means that large waves flow pattern happensathigherU_SG.IntheTillerexperiments,theflowpattern forexperimentswithdensityratiosof0.15and0.078 wasidenti- fiedasstratified-wavy,exceptforthelowestU_SG=0.45m/s,where slugflowwasreported.For thelowest densityratioof0.06,slug flow was reportedfor a larger rangeof U_SG < 1.2m/s. The flow pattern maps forthe CO₂ Flow Loop and the Tiller experiments arecomparedinFig.11.Bothexperimentspresentstratified-wavy flowregime withsimilarboundaries. For U_SG < 1.5m/s, the flow patternswereobserved differentlyintwo loops,butwe thinkfor thesame reasonaswe mentioned inprevious section, itis diffi- culttodistinguishbetweenthesetwo flowregimes,andthey can bevery similar. Aswe explained intheprevious section, thedif- ferencesmaybeduetothelimitedinformationavailableaboutthe Tillerexperiments,coupledwithsubjectivedifferencesinflowpat- ternidentification.

Fig. 10. Liquid holdup in the CO 2flow loop and the corresponding Tiller experiments for the pipe with inclination of 5.0 °upward and for different density ratios.

Fig. 11. Flow pattern map for CO 2flow loop experiments (to the left) corresponding to Tiller dataset 2 (to the right).

10. Conclusion

In earlierstudies atIFE, the concepts ofgeometric, kinematic, anddynamicsimilaritieswereusedtoselectasetofeightdimen- sionless parameters to characterizetwo-phase pipe flow. The ex- periments that were designed to test the scale-up rules, gave a muchbettermatchwiththecorrespondingexperimentswhenthe densityratio was matched. The work presented in this paper is carriedoutusingtheIFECO₂FlowLoopwhichallowsmatchingof thedensityratiotobeimprovedsignificantly.

Thisworkhasfocusedondimensionalanalysisandscalingrules for multiphase flow in fully-developed, steady-state, two-phase, gas–liquid,stratifiedandstratified-wavyflows.Atotalof38exper- imentalsubsets wereconducted withpure CO₂ atdifferentpres- sure/temperature combinations, all on the equilibrium line, and witha few differentpipe inclinations. It is challengingto finda fluidsystemthat matchesallthesimilaritycriteria,andweprior- itized a system that has the correctdensity ratio. Totalpressure gradients, liquidholdups andflow patternswere compared with scaledvaluesfromcorrespondingexperimentsatlargerscales.The discrepancybetweenthenewexperimentalresultsandthescaled datafrompreviousexperimentsisgenerallyverysmallandwithin theuncertaintyofthedata.Theconclusionisthatthescale-upap- proachworksverywell. Theseexperimentsconfirmthat thegas–

liquid densityratio is a very important parameter of multiphase pipe flow. One should therefore be very cautious to use results fromlabexperimentsdirectlytofieldapplications,unlessthegas–

liquiddensityratiosaresimilar.

Moreover,wehaveobserveddiscrepanciesinthepressuregra- dientsforhighsuperficialgasvelocities.WehaveusedtheFroude numberto preserve dynamicsimilarity in hydraulic modellingof multiphaseflowsinfluencedbygravity.Forhighgassuperficialve- locity, there may be a lot of droplet entrainment and assuming similarity through the squared Froude number may not be fully adequate.

Inclosing,wehavedemonstratedthatmatchingthegas–liquid density ratio, the pipe inclination, the Froude number, and the liquid-to-gas velocity ratio are sufficient to give very good scal- ing behaviour of experiments in two-phase stratified and strati- fied wavy flow, with excellent agreement even when scaling the diameterwithafactorof6.5.Valuesofliquidviscositiesandgas–

liquidinterfacialtensionsscaleddown fromthe Tillerloop tothe CO₂ loop deviate by factors of 0.8–4 and 5–10, respectively. De- spite thesignificant differencesin thesetwo parameters, our re- sultsshowedanexcellentmatchwiththeTillerdatasetswiththe same density ratios, indicating that the surface tension and vis- cosityhavelittleinfluenceontheoverallflowbehaviourforthese conditions.