Investigation-of-Particle-Trajectories-in-2-Phase-

J.FluidMe&.(1972),vol.55,paTt2,pp.193-208Printedin@eatBritain193Aninvestigationofparticletrajectoriesintwo-phaseflowsystemsByS.A.MORS1ANDA.J.ALEXANDERDepartmentofMechanicalEngineering,UniversityofSurreyLoughboroughUniversityofTechnology(Received12November1971)Thispaperdescribesatheoreticalinvestigationinto(i)theresponseofasphericalparticletoaone-dimensionalfluidflow,(ii)themotionofasphericalparticleinauniformtwo-dimensionalfluidflowaboutacircularcylinderand(iii)themotionofaparticleaboutaliftingaerofoilsection.Inallthreecasesthedragoftheparticleisallowedtovarywith(instantaneous)Reynoldsnumberbyusingananalyticalapproximationtothestandardexperimentaldrag-Reynolds-numberrelationshipforsphericalparticles.1.IntroductionThepredictionofparticlevelocitiesandtrajectoriesinfluidflowisofacon-siderableimportanceinmanyfields.Instudyingtheerosiondamagesustainedbynozzlesorbladesinsteamorgasturbines,wherethesepartsaresubjectedtotheflowofamixtureofgasandsolidsorgasandwaterdrops,Martlew(1960)andNeilson&Gilchrist(1968)havefoundthattheerosiondependsonthewallmaterial,theparticlevelocityandtheangleofattack.Similarly,inthedesignofinertiaandimpingementfilters,thepredictionofparticletrajectorieswillbeofgreathelpinassessingtheefficiencyofthesefiltersincapturingsolidparticlesfromthecarrierfluid.Inthecaseofflowthroughaturbineitisdesirableforparticlesnottocontactthebladeswhereasinflowthroughafiltertheoppositeeffectisneeded.Ineithercaseitisnecessarytoknowwhatpathaparticlewilltakeandalthoughcalcula-tionshavebeenmadebyLangmiur&Blodgett(1946)andMichael&Norey(1969),itisusualtoassumethattheparticleReynoldsnumberissmallandthattheStokeslinearapproximationtothedragcoeficient(C,=24/RN)willbevalid.ThisisnotalwaysthecaseandatsufficientlyhighReynoldsnumbersverylargeerrorswillresultfromtheuseofthisformula.Inthispaperanexpressionforthedragcoefficientwhichcloselyapproximatestothestandardexperimentaldrag-Reynolds-numberrelationshipisusedtogiveananalyticalsolutioninthecaseofone-dimensionalflowandtosavecomputationtimeinthecaseoftwo-dimen-sionalflow.13194S.A.MorsiandA.J.Alexander1OZI04RNFIGURE1.Dragcoefficientforsphericalparticlesvs.Reynoldsnumber.2.Responseofasphericalparticletoaone-dimensionalfluidflowWhenaparticle,oracloudofparticlesoflowconcentration,isintroducedintoanairstreamtheresponseoftheparticledependsontherelativevelocityoftheparticleandthefluid.Thisrelativevelocitydeterminesthedrag,whichisthesoleforcedeterminingthemotionoftheparticleifitisassumedthatthereisnoparticleinteractionandfurtherthatthepresenceoftheparticlesdoesnotchangethebasicflowpattern.Theequationofmotionofasingleparticleism,dUp/dt=cdx+pg(Ug-Up)*A,,(1)wherethesubscriptpreferstotheparticleandthesubscriptgtothegas;A,isthesurfaceareaoftheparticle.Themassofasphericalparticleisgivenbymp=&~Dzp,.(2)ThedragcoefficientcdforasphericalparticleisplottedagainsttheReynoldsnumberRNinfigure1.TheReynoldsnumberforasphericalparticleisgivenbyRN=Pg(Ug-qPpIP-(3)AtverysmallReynoldsnumbers(RN-0-I),theflowisknownasStokesflowandundertheseconditionsC,=24/RN.SolutionsobtainedusingthisconditionaregivenbyZenz&Othmer(1960).AtveryhighReynoldsnumbers(RNN103)thevalueofC,becomesapproximatelyconstantatabout0.4,butintheinter-mediaterangeofRN,whichistherangeofpracticalinterest,cdvarieswithRA,inacomplicatedmanner.Manyempiricalformulaehavebeensuggestedbutthesearesuitableonlyincertainregion.InthepresentpaperC,iscalculatedatthecorrectReynoldsnumberandisalwayswithin1-2%oftheexperimentalvalue.Particletrajectoriesintwo-phaseflowsystems195RN0.10.20.30-50.71.02.03.05.07.010.020.030.050.070.0Cdcaca(experimental)(calculated)RN(experimental)240.0120.080.049.036.526.514-410.46.95.44.12.552.001.501.27240-01119.5980-4549.50136.33526-50314.83310.516.95-314.12.652-031-501-257100.0200.0300.0500.0700.01000~02000~03000.05000.07000*010000.020000.030000.050000.01.070.770.650.550.500.460.420.400-3850,3900.4050.450.470.49TABLE1.DragcoefficientforsphericaIparticlesCd(calculated)1.070.7710.66130-550.49910.460.420.40160.3850.3910.4100.4520.46970.488InordertoobtainthisaccuracytheexperimentaldragcurveisdividedintoanumberofregionsandthecurveinthatregionisapproximatedtobyanKlK’2Cd=-+-+K3.equationoftheformRNR%(4)Thiscurvecanbemadetofittheexperimentalcurveatthreepoints,andthewidthoftheregionchosenisadjustedsothatthediscrepancybetweenthetwocurvesisnegligible,seetable1.TherangeofReynoldsnumberstakenis0.1-5xlo4.Intheappendixthevaluesoftheconstantsandthedragcoefficientequationsaregiven.Theadvantageofusing(4)isthat(1)canbesolvedanalyticallyandtheerrorsassociatedwithnumericaltechniquesavoided.Equation(1)maybewrittenasdU,/dt=$1-$2Up+A1Ui,(6)whereA1=3K3PgI4DpPp.(5)Equation(5)isaformoftheRiccatiequationwithconstantcoefficientsifU,isconstant.Underthesecircumstancesananalyticsolutioncanbeobtained.ForU,variabletheequationcanbereducedtotheAbelform,inwhichcaseananalyticsolutioncanbeobtainedonlyforcertaintypesofvelocityfunction.ForconstantvelocityU,(5)becomesdU,I(U,-rlJ(Up-72)=AIdt,71.2=$2/24rt[($2/242-41/41**(9)(10)whererlandy2aretherootsoftheright-handsideof(5)anda