Proposed-correlation-of-data-for-isothermal-two-ph

PROPOSEDCORRELATIONOFDATAFORISOTHER~IALTWO-PHASE,TWO·COMPONENTFLOWINPIPES*R.W.LOCKHARTtandR.C.MARTINELLI.UiversityofCalifornia,Berkeley,California,,'Dataarepresentedforthesimultaneousflowofairandliquidsincludingbenzene,kerosene,waterand.variousoilsinpipesvaryingindiameterfrom.0586in.to1.011in.Fourtypesofisothermaltwo-phase,two­componentflowareshowntoexistdependinguponwhethereachphaseisflowingviscouslyorturbulently.ThepressuredropresultingfromthesevariousflowmechanismsiscorrelatedbymeansofparameterXequaltothesquarerootoftheratioofthepressuredropinthepipeiftheliquidflowedalonetothepressuredropifthegasflowedalone.Tentativecriteriaareproposedforthetransitionoftheflowfromonetypetoanother.ThepercentofpipefilledwithliquidunderanyflowconditionsisalsoshowntobecorrelatedforallfourflowtypesbymeansoftheparameterX.INtwopreviouspapers(4,5)onthesnbjectoftwo-phase,two­componentflow,.itwasdemonstratedthatfourtypesofflowmechanismscanexistduringthesimultaneousflowofaliquidandagasorvapor.Theseflowmechanismsare:1.FlowofboththeIiquidandthegasmaybeturbulent(turbulent-turbu­lentflow).2.Flowoftheliquidmaybeviscousandflowofgasmaybeturbulent(viscous­.turbulentflow)3.Flowoftheliquidmar.bcturbulentandflowofthegasVlSCOUS(turbu­lent-viscousflow)4.Flowofboth.theliquidandthegas.maybeviscous(viseous-viscoasHow)Inthepapersjustreferredto.thepressuredropperunitlengthduringtwo-phaseflow,.(-~~-L*AppendixBofthisarticle.2Spagesofexperimentaldata.areonfile(Docu­ment2533)withthe-AmericanDocumenta­!ionInstitute1719NStreet,N.W.,Wash­mgton,D.C.Canbeobtainedbyremitting$;50formicrofilmor$2.80forphotocopies.tPresentaddress:GeneralElectricCo..Shenctady,N.Y.1:Inthispaperanewparameter.1isin­troduced.Thetwo-phasepressuredrop,(~)=..~Gn,,where(~),isthepr~8suredropperunitlengththatwouldexistiftheliquidphaseisassumedto-flowalone.Vol.45,NQ.1is'expressedasisthepressuredropperunitlengthwhichwouldexistifthegaseousphaseisassumedtoflowalone.andq,gisaparameter.twhichisafunc­tionof.adimensionless'variable,X.ThevariableX.isa.functionoftheratioofweightratesoftheliquidandgas,theratioofdensitiesoftheliquidandgas,theratioofviscositiesoftheliquidandgas,andtubediameter.Inthesepreviouspublications.eachtypeofflowmechanismwasassociatedwithadifferentformofX.TableIsummarizesthesevariousparametersaspresentedin(4,5).Further,in(5)itwassuggestedthatthefractionofpipefilledwithliquid,Rwasalsoafunctionofthesameparameter.X.nataforR,vs.Xfor_viscous-viscousflowwere-pre­sentedin($).Thomsen(9)andTaylor(8)intheirresearchmeas­tiredthequantityRwhich,.whenplottedagainsttheparameterXforeachtypeofflow.indicatedreason-ablygoodcorrelation,.Inspectionof-thecurves'oftPtIvs.X,andR,vs.X.forthevariousflow.mechanisms,revealedasimilarityofCHEMICAl.ENGINEERINGPROGRESScurveshapeand-numericalmagnitudewhichseemedtoindicatethatamoregeneralizedtypeofcorrelationshouldexist,whichwouldallowthedataforallfourmechanismsofflowtobecompareddirectly.Itisthepurposeofthispapertodevelopsuchagen­eralizedcorrelation..._~_Themainvariablestobec~ted'aretheexperimentalvariablesq,g,andR1;inaddition,anumberofderivedvariableswillbediscussed.Allthesevariableswillbecorrelatedbj111ea,ofa.newparameter,similartothevariableX,butofamoregeneralizedformThe.newparameterwillbedesignatedbyXtodistingnishitfromtheoldparameterX..Aswas'discussedin(4)•thebasicpostulatesuponwhichisbasedtheanalysisofthepressuredrop'result­ingfromthesimultaneous.flowofa.liquidandagasinatube,are:1.StaticpressuredropfortheliquidPhase,mustequalthestaticpressuredroptorthegaseousphaseregardlessoftheflowpattern,aslongasanap­preciableradialstaticpressurediffer­encedoesnotexist2.Thevolumeoccupiedbytheliquidplusthevolumeoccupiedbythegasatanyinstantmustequal'thetotalvol­umeofthepipeThese-postulatesinlerthattheflowpatterndoesnotchangealongthetubelength;andthusslugflow,inwhichalternateslugsofliquidand-gasmovedownthetube,iseliminated-from~ide.ration:'---;-';'..----.Aswillbeseenbelow,thealgebraicformulationofthesetwopostulatesinvolvesanumberofstepswhichmaybeconsideredsomewhatoversimpli­fied.Theanalysis,however,leadstodimensionlessgroupswithdefinitephysicalsignificance,whichareloundtocorrelateexperimentaldatainwhichallphysicalvariableswerevariedover.widelimits,Assuch.theanalysisisfeittobeofmuchmoreworthandsignificancethanananaly­sis.utilizingthattoolofdubiousPage'99-I(7)(r;)(3)c,(4----n;',)(X):;aDl/l~i,=r&m=(:fJ~:J(9)WhenEquations(6),(9)aresubsti­tutedintoEquations(1)and(2),andEquations(1)and(2)equated,theresultisthatDIG-A=C,WJ,2-,ulpg/l2-M()-.111D,l-mC1-:-Vg2mp../,mpra2-ft4V.=(.-fJTDp)P..Thevelocities,VEandVPIinEqua­nons(1)and(2l'involvetherelativevelocitiesbetweenthefluids,andthussubstitutionoftheIuagnitudesofthevelocitiesfromEquations(6)and(7)intoEquations(1)and(2)isnotexact.Thisdifficultyisnotsoser!ous'asitfirstappears,fortheratiosaandf1areevaluatedIromex­perimentaldataand'thusprobablyincludetheeffectofrelativemotionaswellasthefluidgeometry.Thefrictionfactorsf.andf~maybeexpressedinthegeneralizedBlasiusform:C,f!=.R(:Li,(10)January,194'applies.Foramorecomplexer055­sectionalarea,sucha