Dynamic behavior of pipes conveying gas–liquid two-phase flow

ContentslistsavailableatScienceDirect

Nuclear

Engineering

and

Design

j o ur na l h o me p a g e:w w w . e l s e v i e r . c o m / l o c a t e / n u c e n g d e s

Dynamic

behavior

of

pipes

conveying

gas–liquid

two-phase

flow

Chen

An

a

_,

_Jian

_Su

b,∗

a_{OffshoreOil/GasResearchCenter,ChinaUniversityofPetroleum-Beijing,Beijing102249,China}

b_{NuclearEngineeringProgram,COPPE,UniversidadeFederaldoRiodeJaneiro,CP68509,RiodeJaneiro21941-972,Brazil}

h

i

g

h

l

i

g

h

t

s

•Dynamicbehaviorofpipesconveyinggas–liquidtwo-phaseflowwasanalyzed. •Thegeneralizedintegraltransformtechnique(GITT)wasapplied.

•Excellentconvergencebehaviorandlong-timestabilitywereshown.

•Effectsofvolumetricqualityandvolumetricflowrateondynamicbehaviorwerestudied. •Normalizedvolumetric-flow-ratestabilityenvelopeofdynamicsystemwasdetermined.

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received13October2014

Receivedinrevisedform23June2015

Accepted25June2015

a

b

s

t

r

a

c

t

Inthispaper,thedynamicbehaviorofpipesconveyinggas–liquidtwo-phaseflowwasanalyticallyand numericallyinvestigatedonthebasisofthegeneralizedintegraltransformtechnique(GITT).Theuseof theGITTapproachintheanalysisofthetransversevibrationequationleadtoacoupledsystemofsecond orderdifferentialequationsinthedimensionlesstemporalvariable.TheMathematica’sbuilt-infunction, NDSolve,wasemployedtonumericallysolvetheresultingtransformedODEsystem.Thecharacteristics ofgas–liquidtwo-phaseflowwererepresentedbyaslip-ratiofactormodelthatwasdevisedandusedfor similarproblems.Goodconvergencebehavioroftheproposedeigenfunctionexpansionsisdemonstrated forcalculatingthetransversedisplacementatvariouspointsofpipesconveyingair–watertwo-phase flow.Parametricstudieswereperformedtoanalyzetheeffectsofthevolumetricgasfractionandthe volumetricflowrateonthedynamicbehaviorofpipesconveyingair–watertwo-phaseflow.Besides,the normalizedvolumetric-flow-ratestabilityenvelopeforthedynamicsystemwasobtained.

©2015ElsevierB.V.Allrightsreserved.

1. Introduction

Internalflow-inducedstructuralvibrationhasbeenexperienced innumerousfields,includingheatexchangertubes,chemicalplant pipingsystems,nuclearreactorcomponentsandsubsea produc-tionpipelines.Thesystematicandextensiveinvestigationshave beencarriedoutinthepastdecadestounderstandthedynamic behavior of pipes conveying single-phase flow (Païdoussis and Li, 1993; Païdoussis, 1998, 2008; Kuiper and Metrikine, 2005). Recently,manyachievementshavebeenmadeinunderstanding thedynamiccharacteristicsofpipesconveyingfluid.Kang(2000) investigatedtheeffectsofrotaryinertiaofconcentratedmasseson thefreevibrationsandsysteminstabilityoftheclamped-supported pipeconveyingfluid, andconcluded thatintroductionof rotary

∗ Correspondingauthor.Tel.:+552139388448;fax:+552139388444.

E-mailaddresses:anchen@cup.edu.cn(C.An),sujian@lasme.coppe.ufrj.br(J.Su).

inertia can influence the higher natural frequencies and mode shapesoffluid-conveyingpipes.Sinhaetal.(2005)utilizeda non-linearoptimizationmethodtopredicttheflow-inducedexcitation forcesandthestructuralresponsesofthepipingsystem,whose validationwaspresentedthroughasimulatedexperimentonalong straightpipeconveyingfluid.Basedontheprincipleofeliminated element-Galerkinmethod,Huangetal.(2010)reducedthebinary partialdifferentialequationtoanordinary differentialequation usingtheseparationofvariables, thenutilizedGalerkinmethod toobtainthenatural frequencyequationsofpipelineconveying fluidwithdifferentboundaryconditions,andfoundthattheeffect of Coriolis forceonnatural frequency is inappreciable. Liet al. (2011)presentedaTimoshenkobeamtheory-basedmodelforthe dynamicsofpipesconveyingfluid,andproposedthedynamic stiff-nessmethodtoanalyzethefreevibrationofmulti-spanpipe.Zhai etal.(2011)determinedthenaturalfrequenciesoffluid-conveying Timoshenkopipesviathecomplexmodalanalysis,andcalculated the dynamic response of pipeline under random excitation by

http://dx.doi.org/10.1016/j.nucengdes.2015.06.012

thepseudoexcitationalgorithm.ZhangandChen(2012)analyzed theinternalresonanceofpipesconveyingfluidinthe supercrit-icalregime,inwhichthestraightpipeequilibriumconfiguration becomesunstableandbifurcatesintotwopossiblecurved equi-librium configurations. As the extension of the previous work, ZhangandChen(2013)andChenetal.(2014)studiedthe nonlin-earforcedvibrationofaviscoelasticpipeconveyingfluidaround thecurvedequilibriumconfigurationresultingfromthe supercrit-icalflowspeed,wherethefrequencyandamplituderelationships ofstead-stateresponsesinexternalandinternalresonanceswere derived.

However,theliteratureonthedynamicbehaviorofthepipes subjected to two-phase internal flow, which commonly occurs inpipingelements, isstillsparse.Afewstudieshavebeen con-ductedtoinvestigatethevibration behaviorof pipesconveying gas–liquid two-phase flow. Hara (1977) performed the experi-mentaland theoreticalanalysesof theexcitation mechanismof air–watertwo-phaseflowinduced vibrationsof astraight hori-zontalpipe,andfoundthatvibrationsweredominantlyexcitedby parametricexcitationandbynaturalvibration resonancetothe externalforcethroughtheinstabilityanalysisofasystemof Math-ieuequations.Fluidelasticinstabilityoccursatacriticalvelocityfor whichtheenergy fromthefluidisnolongerdissipated entirely bythedampingofthepipingsystem(RiverinandPettigrew,2007), whichisthemostimportantofseveralflow-inducedvibration exci-tationmechanismsthatcancauselargeoscillationsandearlypipe failure.Monetteand Pettigrew(2004) reportedtheresultsof a seriesofexperimentstostudythefluidelasticinstabilitybehavior ofcantileveredflexibletubessubjectedtotwo-phaseinternalflow, whereamodifiedtwo-phasemodelwasproposedtoformulatethe characteristicsof two-phaseflows.Pettigrewand Taylor(1994) outlinedcylinders experienced fluidelasticinstabilitiesin liquid flowintheformofbucklingoroscillations,andfluidelastic insta-bilitiescanbeaffectedbyseveralparameterssuchasflowvelocity, voidfraction,flexuralrigidity,endconditionsandannular confine-ment.Pettigrewetal.(1998)analyzedtherelativeimportanceof flow-inducedvibrationexcitationmechanisms(fluidelastic insta-bility,periodicwakeshedding,turbulence-inducedexcitationand acousticresonance)forliquid,gasandtwo-phaseaxialflow.Riverin etal.(2006)measuredthetime-dependentforcesresultingfroma two-phaseair–watermixtureflowinginanelbowandatee,and exploredarelationbetweenthefluctuatingforcescausedbythe two-phaseflow andthecharacteristicsof theflow.Riverin and Pettigrew(2007)conductedanexperimentalstudytoinvestigate thegoverningvibrationexcitationmechanismgoverningin-plane vibrationsobservedonU-shapedpipingelements.ZhangandXu (2010)carriedoutanexperimentalstudyofpipevibrations sub-jectedtointernalbubblyflow.Zhangetal.(2010)examinedthe characteristicsofchannelwallvibrationinducedbyinternalbubbly flow,anddevelopedamathematicalmodelthatcanwellpredicted thespectralfrequenciesofthewallvibrationsandpressure fluctu-ations,thecorrespondingattenuationcoefficientsandpropagation phasespeeds.Recently,flow-inducedvibrationofpipelines con-veyinghigh-pressureoilandgashasreceivedattentionsince it causesdamagetosupportingstructuresorpipelineruptures lead-ingtocostlyshutdownandsevereenvironmentalproblemsGuo etal.(2014).PatelandSeyed(1989)examinedthecontributionof timevaryinginternalslugflowtothedynamicexcitationforces appliedtoaflexibleriser.SeyedandPatel(1992)deducedan ana-lyticalmodelforcalculatingthepressureandinternalslug flow inducedforcesonflexible risers.Ortegaetal.(2012)reporteda computationalmethodtoanalyzetheinteractionbetweenan inter-nalslugflowandthedynamicresponseofflexiblerisers.Extending previouswork,Ortegaetal.(2013)studiedtheinteractionbetween aninternalslugflowplusanexternalregularwaveandthedynamic responseofaflexibleriser.

In this study, the dynamic behavior of pipes conveying gas–liquidtwo-phaseflowisanalyticallyandnumerically investi-gatedonthebasisofthegeneralizedintegraltransformtechnique (GITT),whichhasbeensuccessfullydeveloped inheatandfluid flowapplications(Cotta,1993,1998;CottaandMikhailov,1997) and further appliedin solving thedynamic response of axially moving beams (An and Su, 2011), axially moving Timoshenko beams(AnandSu,2014b),axiallymovingorthotropicplates(An and Su, 2014a), damaged Euler–Bernoulli beams (Matt, 2013a), cantileverbeamswithaneccentrictipmass(Matt,2013b), fluid-conveyingpipes(Guetal.,2013),thewind-inducedvibrationof overheadconductors(Matt,2009)andthevortex-induced vibra-tionoflongflexiblecylinders(Guetal.,2012).Themostinteresting feature in this technique is the automatic and straightforward globalerrorcontrolprocedure,whichmakesitparticularly suit-ablefor benchmarkingpurposes, and theonlymild increase in overallcomputationaleffortwithincreasingnumberof indepen-dent variables. The main contribution is the application of the effective tool, the generalized integral transform technique, to developing an analytical-numerical approach to determine the dynamiccharacteristicsofpipesconveyinggas–liquidtwo-phase flow,aimingatprovidingmorereferencedatainthistopicforfuture comparison.

Therestofpaperisorganizedasfollows:inthenextsection, themathematical formulationof thetransversevibration prob-lemofpipesconveyinggas–liquid two-phaseflowispresented. InSection3,thehybridnumerical–analyticalsolutionisobtained bycarryingoutintegraltransform.Numericalresultsofproposed methodincludingtransversedisplacementsandtheir correspond-ingconvergencebehaviorarepresentedinSection4.Aparametric studyisthenperformedtoinvestigatetheeffectsofthe volumet-ricqualitiesonnatural frequenciesand vibration amplitudesof pipesconveyingair–watertwo-phaseflow,respectively.Besides, thevolumetric-flow-ratestabilityenvelopeforthesystemisalso presented.Finally,thepaperendsinSection 5withconclusions andperspectives.

2. Mathematicalformulation

Weconsideraverticalpipewithclamped–clampedboundary conditionssubjectedtogas–liquidtwo-phaseinternalflow,as illus-tratedinFig.1,wheretheinternalflowofthepipeisconstituted bythediscretegaseousbubblesinacontinuousliquid.Ifinternal damping,externalimposedtensionandpressurizationeffectsare eitherabsentorneglected,theequationofmotionofthepipecanbe derivedfollowingtheNewtonianderivationbymeansof decom-posinganinfinitesimalpipe-fluidelementintothepipeelement andthefluidelementundertheassumptionsoftheEuler–Bernoulli beam theory, according to the procedures given by Païdoussis (1998)andMonetteandPettigrew(2004):

EI

∂

4 w

∂

x4 +



k MkUk2

∂

2 w

∂

x2 +2



k MkUk

∂

2 w

∂

x

∂

t +





k Mk+m



∂

2 w

∂

t2 +





k Mk+m



g



(x−L)

∂

2 w

∂

x2 +

∂

w

∂

x



=0, (1a)

subjectedtotheclamped–clampedboundaryconditions w(0,t)=0,

∂

w(0,t)

∂

x =0, w(L,t)=0,

∂

w(L,t)

∂

x =0, (1b–e)

where w(x,t) is the transversedisplacement, EI is the flexural rigidityofthepipewhich dependsupon bothYoung’smodulus Eandtheinertialmomentofcross-sectionareaI,MkandUkare

Fig.1.Illustrationofaclamped–clampedpipeconveyinggas–liquidtwo-phase

flow,whereU1andU2expressthevelocitiesofthegasandtheliquidphases.

respectivelythemassesperunitlengthandthesteadyflow veloc-itiesofthegasandtheliquidphases(ktakesthevaluesof1forgas (G)and2forliquid(L)),misthemassofthepipeperunitlength, Listhepipelength,andgisthegravitationalacceleration.Note thatthetermsofEq.(1a)representsrespectively:flexuralforce,

centrifugalforce,Coriolisforce,inertiaforceand gravity,where centrifugalandCoriolisforcesaregeneratedbybothgasandliquid phases.Thefollowingdimensionlessvariablesareintroduced x∗₌x L, w∗= w L, = t L2



EI



kMk+m , (2a–c) k=UkL



Mk EI , ˇk= Mk



kMk+m , =gL3



kMk+m EI . (2d–f) SubstitutingEq.(2)intoEq.(1)givesthedimensionlessequation (droppingthesuperposedasterisksforsimplicity)

∂

4 w

∂

x4 +



k 2 k

∂

2 w

∂

x2 +2



k kˇ_k1/2

∂

2 w

∂

x

∂

 +

∂

2 w

∂

2 +



(x−1)

∂

2 w

∂

x2 +

∂

w

∂

x



=0, (3a)

togetherwiththeboundaryconditions w(0,)=0,

∂

w(0,)

∂

x =0, w(1,)=0,

∂

w(1,)

∂

x =0, (3b–e) Theinitialconditionsaredefinedasfollows:

w(x,0)=0, w(x,˙ 0)=

v

0sin(x), (4a,b)

wherethesign‘._{’denotesthetimederivativeofthedimensionless}

transversedisplacement.

3. Integraltransformsolution

Accordingtotheprincipleofthegeneralizedintegraltransform technique,theauxiliaryeigenvalueproblemneedstobechosenfor thedimensionlessgoverningequation(3a)withthehomogenous boundaryconditions(3b–e).Thespatialcoordinate‘x’iseliminated throughintegraltransformation,andtherelatedeigenvalue prob-lemisadoptedforthetransversedisplacementrepresentationas follows:

d4Xi(x)

dx4 = 4

iXi(x), 0<x<1, (5a)

withthefollowingboundaryconditions Xi(0)=0, dXi(0)

dx =0, (5b,c)

Xi(1)=0, dXi(1)

dx =0, (5d,e)

whereXi(x)andiare,respectively,theeigenfunctionsand

eigen-values of problem(5).The eigenfunctions satisfythe following orthogonalityproperty

1 0

Xi(x)Xj(x)dx=ıijNi, (6)

withıij=0fori /= j,andıij=1fori=j.Thenorm,ornormalization

integral,iswrittenas Ni=

1 0 X2 i(x)dx. (7)

Table1

Convergencebehaviorofthedimensionlesstransversedisplacementw(x,)ofapipeconveyingair–watertwo-phaseflow.

x NW=4 NW=8 NW=12 NW=16 NW=20 NW=24 =5 0.1 0.00127 −0.00676 −0.00498 −0.00468 −0.00462 −0.00460 0.3 0.0121 −0.0479 −0.0360 −0.0340 −0.0336 −0.0334 0.5 0.0233 −0.102 −0.0770 −0.0729 −0.0719 −0.0716 0.7 0.0226 −0.112 −0.0873 −0.0832 −0.0822 −0.0819 0.9 0.00560 −0.0258 −0.0213 −0.0206 −0.0204 −0.0204 =20 0.1 0.00516 0.00756 0.0105 0.0103 0.0103 0.0102 0.3 0.0298 0.0504 0.0768 0.0773 0.0772 0.0772 0.5 0.0703 0.0855 0.157 0.161 0.161 0.162 0.7 0.0892 0.0552 0.136 0.142 0.143 0.143 0.9 0.0228 0.00682 0.0216 0.0227 0.0230 0.0231 =50 0.1 0.00673 −0.000719 −0.00826 −0.00673 −0.00623 −0.00604 0.3 0.0569 0.00513 −0.0705 −0.0613 −0.0583 −0.0573 0.5 0.134 0.0396 −0.157 −0.146 −0.141 −0.140 0.7 0.121 0.0685 −0.147 −0.143 −0.139 −0.138 0.9 0.0215 0.0170 −0.0257 −0.0259 −0.0256 −0.0254

Problem(5)isreadilysolvedanalyticallytoyield

Xi(x)=

⎧

⎪

⎨

⎪

⎩

cos[i(x−1/2)] cos(i/2) − cosh[i(x−1/2)]

cosh(i/2) , foriodd,

sin[i(x−1/2)] sin(i/2) − sinh[i(x−1/2)] sinh(i/2) , forieven, (8a,b)

wheretheeigenvaluesareobtainedfromthetranscendental equa-tions:

tanh(i/2)=



−tan(i/2), foriodd,

tan(i/2), forieven,

(9a,b)

andthenormalizationintegralisevaluatedas

Ni=1, i=1,2,3,... (10)

Therefore, thenormalized eigenfunction coincides, in this case, withtheoriginaleigenfunctionitself,i.e.

˜

Xi(x)=Xi(x)

N1/2 i

. (11)

Thesolutionmethodologyproceedstowardsthepropositionof theintegral transformpairforthepotentials,theintegral trans-formation itself and the inversion formula. For the transverse displacement: ¯ wi()=

1 0 ˜ Xi(x)w(x,)dx, transform, (12a) w(x,)= ∞



i=1 ˜ Xi(x) ¯wi(), inverse. (12b)

Theintegraltransformationprocessisnowemployedthrough operationof(3a)with



₀1X˜i(x)dx,tofindthetransformedtransverse

displacementsystem:

(a)

(b)

Fig.2. GITTsolutionswithdifferenttruncationordersNWforthetransversedisplacementprofiles(a)w(x,)|=30and(b)w(x,)|=50ofapipeconveyingair–watertwo-phase

(a)

(b)

(c)

Fig.3.Timehistories(a)w(0.5,)|25≤≤30and(b)w(0.5,)|45≤≤50ofthetransversedisplacementatthecentralpointofapipeconveyingair–watertwo-phaseflow;and(c)

amplitudespectrumofthestructuralresponsew(0.5,)|0≤≤50.

d2w¯i() d2 +2



k kˇ1/2k ∞



j=1 Aij d ¯wj() d + 4 iw¯i() +



k 2 k ∞



j=1 Bijw¯j()+ ∞



j=1 Cijw¯j() + ∞



j=1 Aijw¯j()=0, i=1,2,3,..., (13a)

wherethecoefficientsareanalyticallydeterminedfromthe follow-ingintegrals Aij=

1 0 ˜ Xi(x) ˜Xj(x)dx, (13b) Bij=

1 0 ˜ Xi(x) ˜Xj(x)dx, (13c) Cij=

1 0 ˜ Xi(x)(x−1) ˜Xj(x)dx. (13d)

Inthesimilarmanner,initialconditionsarealsointegral trans-formedtoeliminatethespatialcoordinate,yielding

¯ wi(0)=0, d ¯wi(0) dt =

v

0

1 0 ˜ Xi(x)sin(x)dx, i=1,2,3,... (14a,b) Forcomputationalpurposes,theexpansionforthetransverse deflectionistruncatedtofiniteordersNW.Eqs.(13)and(14)in thetruncatedseriesaresubsequentlycalculatedbytheNDSolve routine of MathematicaWolfram (2003). Once thetransformed

Fig.4.Variationofthedimensionlessfrequency(ω1)withthevolumetricgasfraction(εG).

potential, ¯wi() has been numerically evaluated, the inversion

formulasEq.(12b)isthenappliedtoyieldexplicitanalytical expres-sionforthedimensionlesstransversedeflectionw(x,).

4. Resultsanddiscussion

4.1. Two-phaseflowmodel

Threefundamentalparametersofgas–liquidtwo-phaseflow, thevolumetricgasfractionεG,thevoidfraction˛,andtheslipfactor

K,aredefinedasfollows: εG= QG QG+QL, (15) ˛= _A AG G+AL, (16) and K=U_UG L, (17)

whereQGandQLarethevolumetricflowratesofthegasandliquid

phases,respectively,AGandALaretheareaoccupiedbythegasand

theliquidintheinnercrosssectionofthepipe,respectively.The flowvelocitiesofthegasandtheliquidaregivenby

UG=QG

AG and UL=

QL

AL. (18)

Thevoid fraction˛is related tothevolumetric gasfractionεG

throughtheslipfactorK: 1−εG εG =



_1−˛ ˛



₁ K. (19)

Inthiswork,theslipfactorKiscalculatedfromthemodelproposed byMonetteandPettigrew(2004):

K=



₁₋εG_ε

G



1/2

. (20)

AspointedoutbyMonetteand Pettigrew(2004),theslipfactor KcalculatedfromEq.(20)isconsistentwithobservedflow pat-ternsandisnotverydifferentfromChisholm’sslipratio(Chisholm, 1983).

4.2. Convergencebehaviorofthesolution

Wenowpresenttheconvergencebehaviorofnumericalresults for the transverse displacement w(x,) of a pipe conveying air–water two-phase flow calculated using the GITT approach. Tomake theanalysiscomputationally tractable,theparameters ofthepipeandthetwo-phaseflow withintheparameterrange givenbyMonetteandPettigrew(2004)aretakeninEq.(1).The inner andouterdiametersof pipecrosssectionared=12.7mm andD=15.9mm,respectively.Thepipeisconsideredtobea flex-ible tube(Tygon®R-3606)withthedensityof 1180kg/m3_{, the}

Young’smodulusof3.6MPaandthelengthof1m.Fortheflow condition,the volumetric air and water flow rates are known: QG=0.0005m3/sandQL=0.0002m3/sandwiththedensitiesofthe

Fig.5. Variationofthedimensionlessvibrationamplitude(wmax|x=0.5)withthevolumetricgasfraction(εG).

ofeachphaseperunitlengthcanbeobtainedbyMG=AGG and

ML=ALL.Thedimensionlessvariables canbeobtainedthrough

Eq.(2).Inaddition,

v

0=1.0isemployedintheinitialconditions

(2).Thesolutionofthesystem,Eqs.(13)and(14),isobtainedwith NW_≤20toanalyzetheconvergencebehavior.

Thedimensionless transversedisplacementw(x,)at differ-entpositions,x=0.1,0.3,0.5,0.7and 0.9,ofthepipeconveying air–watertwo-phaseflowarepresentedinTable1.The conver-gencebehavioroftheintegraltransformsolutionisexaminedfor increasingtruncationtermsNW=4,8,12,16,20and24at=5,20 and50,respectively.NotethattheoptionMaxStepsforNDSolve, whichspecifiesthemaximumnumberofstepsthatNDSolvewill evertakeinattemptingtofindasolution,issettobe106_.Itcanbe

observedthatalmostallthesolutionsconvergetothevalueswith twosignificantdigits,whilesomeofthemconvergetothevalues withthreesignificantdigitswitha truncationorderofNW=20. Theresultsat=50indicatethattheconvergencebehaviorofthe integraltransformsolutiondoesnotchangewithtime,verifying thegoodlong-timenumericalstabilityofthescheme.Forthesame cases,theprofilesofthetransversedisplacementat=30and=50 areillustratedinFig.2withdifferenttruncationorders.Inaddition, thetimehistoriesfor∈[25,30]and∈[45,50]ofthetransverse displacementatthecentralpointofthepipeareshowninFig.3, whichalsodemonstratesthattheconvergenceisachievedata trun-cationorderofNW=20.Inordertoidentifythefrequencycontent ofthestructuralresponse,theFastFourierTransform(FFT) ampli-tudespectrumofthetimehistoryfor∈[0,50]ofthetransverse displacementatx=0.5isshowninFig.3(c),fromwhichitcanbe clearlyseenthatthethreepeaksintheamplitudespectrumappear atthefrequenciesof0.520(thefundamentalfrequency),8.34and 18.2,respectively.

4.3. Parametricstudy

In this section, transverse displacement of pipes conveying air–watertwo-phaseflowwithclamped–clampedboundary con-ditionsisanalyzedtoillustratetheapplicabilityoftheproposed approach.Thevariationsofthefundamentalfrequencyω1andthe

vibrationamplitudewiththevolumetricgasfractionεGandthe

liquidflowrateQLarestudied.Inthefollowinganalysis,weusea

relativehightruncationorder,NW=20,forasufficientaccuracy. 4.3.1. Variationofthefundamentalfrequency

Toobtainthefundamentalcircularfrequencyforthetransverse vibrationofthesystem,thecoupledODEs,Eq.(13a),canbe repre-sentedinthematrixformasfollows:

M ¨w(t)+C ˙w(t)+Kw(t)=F(t). (21)

ConsiderthegeneralizedeigenvalueproblemofEq.(21),andthe fundamentalcircularfrequencycanbeobtainedbyusingstandard eigenvalueroutineforacomplexgeneralmatrix.Thedimensionless fundamentalnaturalfrequenciesofthepipeconveyingair–water two-phaseflowfordifferentvolumetricgasfractions0≤εG≤1and

liquidflowratesQL=0.0001,0.0002,0.0003m3/sarecalculated,as

showninFig.4,wheretheothergeometricalandphysical parame-tersaresameasinSection4.2.Itcanbeseenthatthefundamental frequencydecreaseswiththevolumetricgasfractionandthe liq-uidflow rate.WhenεG=0, ω1 equal to43.9, 36.6and 22.7for

QL=0.0001,0.0002and0.0003m3/s,respectively,whichrepresent

thefundamentalfrequenciesofthepipeconveyingliquid.When the volumetric gas fraction reaches a critical value (εG=0.974,

Fig.6. Normalizedvolumetric-flow-ratestabilityenvelopeforthepipeconveyingair–watertwo-phaseflow,anditscomparisonwiththatcalculatedusingthemethod

proposedbyMonetteandPettigrew(2004).

QL=0.0001,0.0002and0.0003m3/s,respectively,whichmeansthe

dynamicsystemlosesitsstabilitybydivergence. 4.3.2. Variationofthevibrationamplitude

Thedimensionlessvibrationamplitudesatthecentralpointof thepipeconveyingair–watertwo-phaseflowfordifferent volu-metric gasfractions0≤εG≤1and liquid flowrates QL=0.0001,

0.0002,0.0003m3_{/sarecalculated,asshowninFig.5,wherethe}

othergeometrical andphysical parameters aresame asin Sec-tion4.2.Theamplitudeisthemaximumabsolutevaluefoundfrom thecalculatedtime-historyresponseofpipeconveyinggas–liquid two-phaseflow for∈[0,50].It canbeseen thatthevibration amplitudesincreasewiththevolumetricgasfractionandthewater flow rate.WhenεG=0,wmax|x=0.5 equal to0.0237,0.0258 and

0.0327forQL=0.0001,0.0002and0.0003m3/s,respectively,which

representthevibrationamplitudesatthecentralpointofthepipe conveyingliquid.Foreachcase,thevibrationamplitudestendsto infinity(viz.,instabilityofthedynamicsystemoccurs)whenthe liquidflowratereachesacriticalvalue,whichisthesamevalue giveninFig.4,provingthecoherenceoftheanalysis.

4.4. Volumetric-flow-ratestabilityenvelope

Whenthewaterflowrateisspecified,thecriticalvalueofthe volumetricgasfractionfortheonsetofinstabilityofthedynamic systemcanbecalculatedasmentionedinSection4.3.1.Following thesameprocess,thenormalized volumetric-flow-ratestability envelopefor the pipeconveyingair–water two-phaseflow can beobtained,asdemonstratedinFig.6.QG,max=0.00823m3/sand

QL,max=0.000372m3/s represent thecritical volumetric gas and

liquidflowrates,respectively,whenconsideringthesingle-phase internalflow.Itshouldbenotedthatallpointslocatedoutsidethe enveloperepresentatwo-phaseflowwiththespecifiedvolumetric gasandliquidflowratesthatwillcausethedynamicsystemtolose itsstability.

ToverifyheresultsgiveninFig.6,themethodtocalculatethe dimensionlesscriticalvelocityofeachphaseandthedimensionless frequencyproposedbyMonetteandPettigrew(2004)(Section2. FluidelasticinstabilitytheoryinMonetteandPettigrew(2004))is employedinthiswork.Theclassicalmodesummationresponse formulationcanbeexpressedas

w(x,)=

∞



i=1

aiX˜i(x)eiω. (22)

Thenthesystemofequations,Eq.(13a),isexpressedasa summa-tionofmodes ∞



j=1



(4 j−ω2)ıij+2



k kˇ_k1/2Aijωi+



k 2 kBij+Cij+Aij



aj=0. (23)

Tofindthecriticalfrequencyofthetubeatinstability,the imagi-narypartofthefrequencytermissettozero.Thesolutionofthe systemofequationsisobtainedbyallowingthedeterminantofthe coefficientsofajtobeequaltozero.Inotherwords,therealandthe

imaginarypartofthedeterminantareequaltozero,respectively, whichyieldstwoequations.Bycombiningthemodelforslipfactor K(Eq.(20)),thethreeunknownsincludingthedimensionless veloc-ityofeachphaseandthedimensionlessfrequencycanbesolved.

TheresultscalculatedusingthemethodproposedbyMonetteand Pettigrew(2004)arealsoshowninFig.6.Theexcellentagreement betweentheresultsobtainedandtheonesgivenbyGITTindicates thevalidityoftheproposedmethodinthisstudy.

5. Conclusions

Thegeneralizedintegraltransformtechnique(GITT)hasproved inthispapertobea goodapproachfortheanalysisofdynamic behaviorofpipesconveyinggas–liquidtwo-phaseflow,providing anaccuratenumerical–analyticalsolutionforthenatural frequen-ciesand transverse displacements. The solutions confirm good convergence and long-time numerical stability of the scheme. Theparametricstudiesindicatethatthefundamental frequency decreasewiththevolumetricgasfractionandthewaterflowrate. Whenthewater flowratereachesacritical value,thedynamic systemlosesitsstabilitybydivergence.Thevibrationamplitudes increasewiththevolumetricgasfractionandthewaterflowrate. Inaddition,thenormalizedvolumetric-flow-ratestabilityenvelope forthepipeconveyingair–watertwo-phaseflowisobtained.The proposedapproachcanbeemployedtopredictthedynamic behav-iorofpipesconveyinggas–liquidtwo-phaseflowinassociatedwith othermoretwo-phasemodelsforfutureinvestigation.

Acknowledgements

The work was supported by Science Foundation of China University of Petroleum, Beijing(No. 2462013YJRC003and No. 2462015YQ0403),andbyCNPq(GrantNo.306618/2010-9),CAPES andFAPERJ(GrantNo.E-26/102.871/2012)ofBrazil.

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