ChemicalEngineeringandProcessing44(2005)447451AbstractseparationradialgradetheK1.lutionandricate,harshmostutilizedoftribtions.ofories.etraincollectionries.particles,galDevelopmentofanewmethodforevaluatingcycloneefficiencyBingtaoZhaoDepartmentofEnvironmentalEngineering,DonghuaUniversity,No.1882WestYananRoad,Shanghai200051,PRChinaReceived29March2004；receivedinrevisedform9June2004；accepted15June2004Availableonline20August2004Anewtheoreticalmethodforevaluatingcycloneefficiencyisdevelopedbasedontheinvestigationofflowpattern,thecriticalparticlesizetheoryandtheboundarylayerseparationtheory.Theradialparticleconcentrationgradient,insteadoftheusuallyassumeduniformparticleconcentrationwithinthecyclone,isconsideredinthismathematicalmodel.Thelocalparticleconcentrationandthecycloneefficiencycanbecalculatedonthebaseofatime-of-flightmodelintermsoftheparticlesizedistributionofthefeed.Theavailabilityofmethodisverifiedbycomparisonofthecalculatedgradeefficiencywithexperimentaldataandtheoreticalcounterpartsintheliterature.eywords:Cyclone；Collectionefficiency；MathematicalmodelIntroductionCycloneseparatorsarewidelyusedinthefieldofairpol-anyparticlesizeisdeterminedfromtheratioofitssettlingvelocitytotheterminalsettlingvelocityofthestaticparti-cle.DirgoandLeith3modifiedtheBarththeoryandfoundcontrolandgas-solidseparationforaerosolsamplingindustrialapplications.Duetorelativesimplicitytofab-lowcosttooperate,andwelladaptabilitytoextremelyconditions,cycloneseparatorshavebecomeoneoftheimportantparticleremovaldeviceswhicharepreferablyinbothengineeringandprocessoperation.Inordertodescribethecycloneperformance,thetheoriescycloneparticlecollectionaredevelopedbymanycon-utorsusingdifferentmethodswithsimplifyingassump-Representativeresearchesmainlyinclude:ThetheoryLapple1isthemostwidelyusedexampleofthecutthe-Lappleassumedthatparticlesenteringthecyclonearevenlydistributedacrosstheinletopening.Theparticlethatvelsfromtheinlethalfwidthtothewallduringthetimethecycloneiscollectedwith50%efficiency.Barths2theoryisanotherrepresentationofcutsizetheo-Barthcalculatedtheterminalsettlingvelocityforstaticbasedontheexactbalancebetweenthecentrifu-forceandthedragforce.ThecollectionefficiencyforTel.:+862162373718；fax:+862162373482.E-mailaddress:(B.Zhao).aciencLappledescribetriescadjustmentLichtdesigntrationalloparticlesunfortunatelyfdirectiontheciencandoccurs,manDietzresimpleexpressionforBarthsplotofthecollectioneffi-yversustheratio.IoziaandLeith4,basedonthesandBarthstheory,proposedalogisticfunctiontothefractionalefficiencyforcycloneswithgeome-whichwerevariationsoftheStairmandhigh-efficiencyyclone.Thisequationhasanempiricalparameterthatallowsofthesharpnessofthecutofcyclone.Leithand5developedanotherpopularapproachforcycloneassumingthatturbulencekeepstheparticleconcen-atanyheightinthecyclonewellmixed.Thetheorywsadirectcalculationofthecollectionefficiencyforofanysizeandcyclonesofarbitrarydesign.But,thereisexperimentalevidencetosupporttheactthatthereis,indeed,aconcentrationgradientintheradialofcyclones.Cliftetal.6modifiedtheestimateofmeangasresidencetimeandrederivedthegradeeffi-yequationbasedontheoriginalassumptionsofLeithLichtstheoryandadifferentdependenceonparticlesizeapproachingthetypicalS-shapecurvesobtainedbyyresearchers.Ahybridcollectiontheorydevelopedby7dividesthecycloneintothreeregions:Theentrancegion,thedown-flow(orannular)region,andtheup-flow448andPr(orclesassumedtrationThisLentgionefMoreawpresentcdifdifognizingintheKimclonesoryandlentthegionfromAnother11,12theoryroughness,sizeandmethod13dictingtigationtheorysidersofticlesby2.inminingareparticleticle,samene2.1.Gasowpatternandparticlemotion2.1.1.GasowpatternThecircumferentialgasvelocityprofileforeffectivesep-arationregionRerRwwasobtainedbyMothesandLoffler8,withsimplifyingthecyclonetotherightcylinderandconsideringthewallroughnessasfollows:v=vw(r/Rw)1+P(1r/Rw)(1)vw=vdhparenleftBiggradicalBigg14+hvwvd12parenrightBigg(2)v=vR2w(3)B.Zhao/ChemicalEngineeringcore)region.Dietzproposedtheinterchangeofparti-betweentheannularandcoreregions.ButDietzalsothatturbulenceproducesauniformradialconcen-profileforuncollectedparticleswithineachregion.assumptionaswell,shouldbejustified.Mothesandoffler8haveextendedtheconceptsofDietzofdiffer-flowregionswithinthecyclonetoincludeafourthre-closetothedustexitatthecyclonebottom,wherethefectofdustre-entrainmentcanbeincludedinprinciple.important,however,seemstobetheconsiderationoffiniteturbulentdiffusivityinboththedownwardandup-ardflowregions.ThisapproachavoidsthediscontinuitiesintheDietzmodelandthetransportofparticlesinycloneseparatorsisthenviewedasasuperimpositionofafusivemotionwithadeterministicmeanmotion.Otherferenttheoriesinclude:LiandWang9althoughrec-theroleofafiniteturbulentparticlediffusivityproducingradialconcentrationgradients,havedroppedhypothesisofdifferentflowregionswithinacyclone.andLee10proposeatheoryforhighefficiencycy-basedontheboundarylayercharacteristics.Thisthe-dividesthecycloneintotworegions,theturbulentregionthenear-wallregion.Particlestrajectoriesintheturbu-regionwerecalculatedfromthemeanfluidmotionandcollectionprobabilityofparticlesinthenearwallre-wascalculatedbythedepositionvelocitywhichresultsbothturbulentthediffusionandthecentrifugalforce.typicaltheorydevelopedbyMuschelknautzetal.foralongtime.MuschelknautzimprovedBarthsbyconsideringtheeffectsoftheparticleload,thewallthesecondaryflow,andthechangeinparticledistributionwithinthebodyonthecollectionefficiencythepressureloss.Thistheorymaybethemostpracticalformodelingcycloneseparatorsatthepresenttime.Inthepresentpaper,anewmathematicalmethodforpre-cycloneefficiencyisdevelopedbasedontheinves-offlowpattern,thecriticalparticlesizeseparationandtheboundarylayerseparationtheory,whichcon-avariedradialparticleconcentrationgradientinsteadauniformradialconcentrationprofileforuncollectedpar-withinthecyclone.Thecollectionefficiencycalculatedthismodeliscomparedwithexperimentaldata.TheoryTheconventionalcycloneseparatorgeometryisshownFig.1.Todevelopthenewcalculationmethodfordeter-thecollectionefficiency,thefollowingassumptionsmade:theparticleissphericalinshape,themotionofaisnotinfluencedbythepresenceofneighboringpar-thetangentialvelocitycomponentoftheparticleistheasthatofthegasstream,andtheradialgasvelocityisglected.vhPQAlthoughagreestheofthesometimes2.1.2.thedirectionocessing44(2005)447451Fig.1.Schematicdiagramofcyclonegeometry.wdab0.204(b/Rw)+0.889d=QR2w(4)=aRwbracketleftbigg2arccos(b/Rw1)21bracketrightbigg+hRw(5)=vwvdparenleftbigg+sinparenrightbigg(6)=(ab)vin(7)=0.00650.0075(8)theexpressionforvlookscomplicated,theresultverywellwiththetypicalvelocityprofilebasedonpower-lawcorrelationofAlexander14.Theadvantagethisexpressionisthatitpresentsaquantitativevalueoftangentialvelocityattheedgeofthecore,vw,whichhasbeenassumedequaltotheinletvelocity.ParticlemotionUndertheinitialassumptions,theforcebalanceactingonparticle(thedragforceobeysStokeslaw)intheradialgives:d2rdt2+18gpd2pCcdrdtv2pr=0(9)andPrEq.obtainedisinAccordingCombiningFromAccordingityInteof2.2.necessarysize.lognormalfAccordingcleofdAssumingspecifiedtheCW2.3.SeparationefciencyFig.2showsschematicallythegas-particleseparationpro-cessinahorizontalcross-sectionofthecyclone.ThenumberofparticlesperunitvolumeCisafunctionofrand.Supposethatduringthetimedtallparticles,whosedistancetothecy-clonewallisdrorless,willbecollected,andthatmeanwhiletheparticleswilltraveladistancerdinthetangentialanddzintheverticaldirection.B.Zhao/ChemicalEngineering(9)isnotreadilysolvable.Anapproximatesolutioncanbebyneglectingthesecondderivatived2r/dt2,whichtheconsequenceoftheassumptionthattheparticlemotiontheradialdirectionisstationary.Eq.(9)thenbecomes:drdt=pd2pCc18gv2pr(10)totheEquationofparticletrack,itis:drdt=vrp(11)ddt=1rvp(12)Eqs.(11)and(12):drd=rvrpvp(13)Eqs.(10),(11)and(13):drd=parenleftBiggpd2pCc18gparenrightBiggvp(14)totheassumptionofequalgasandparticleveloc-,Eqs.(1)and(14)canbecombined:drd=parenleftBiggpd2pCc18gparenrightBiggvw(r/Rw)1+P(1r/Rw)(15)gratingEq.(15)andconsideringtheboundarycondition0=0,r0=Reyieldstheformulaforcalculatingthetrackaparticleinanyposition:=parenleftBigg18gpd2pCcparenrightBiggparenleftbigg1vwparenrightbiggbracketleftBiggparenleftBigg(1+P)Rw12parenleftBiggparenleftbiggrRwparenrightbigg2parenleftbiggReRwparenrightbigg2parenrightBiggPRw13parenleftBiggparenleftbiggrRwparenrightbigg3parenleftbiggReRwparenrightbigg3parenrightBiggparenrightBiggbracketrightBigg(16)ConcentrationdistributionTodescribetheparticleconcentrationdistribution,itistodefinethespecifiedpointofcriticalparticleTheparticlesizedistributionissupposedtoobeythedistribution:(lndp)=1ln2expbracketleftBigg(lndplndp50)22(ln)2bracketrightBigg(17)totheequationofparticletrack,thecriticalparti-sizeinanypositionshouldalsobeexpressedasafunctionrand:pc=radicaltpradicalvertexradicalvertexradicalbtparenleftbigg18gpCcparenrightbiggparenleftbigg1vwparenrightbiggbracketleftBigg(1+P)Rw12parenleftBiggparenleftbiggrRwparenrightbigg2parenleftbiggReRwparenrightbigg2parenrightBiggvThebe:NAccordingtionobtainedocessing44(2005)447451449Fig.2.Cross-sectionofthecyclonewithschematicseparationprocess.thatparticleslargerthandpccanbeseparatedatthepointandparticleslessthandpccannotbeseparated,particleconcentrationatthispointshouldbeobtained:(r,)=C01+integraldisplayln(dpc)f(lndp)d(lndp).(19)ecallthisthecriticalseparationtheory.PRw13parenleftBiggparenleftbiggrRwparenrightbigg3parenleftbiggReRwparenrightbigg3parenrightBiggbracketrightBigg(18)Thenthenumberofcollectedparticleswithinthecontrololumewillbe:dN=C(Rw,)(Rwdr)ddzdr=C(Rw,)RwddzdrC(Rw,)ddz(dr)2C(Rw,)Rwddzdr(20)totalnumberofparticleswithinthecontrolvolumewill=RwintegraldisplayReC(r,)rddzdr(21)totheboundarylayerseparationtheory,thefrac-ofparticlesremovedwithinthecontrolvolumecanbebycombiningEqs.(20)and(21):dNN=C(Rw,)RwdrRwintegraltextReC(r,)rdr(22)450andPrNearCombiningThenHere,LLH3.perimentalcwwith5.97inletecomparisonneMothescloserLichtwell4.lectionmodelload,dimensionstheB.Zhao/ChemicalEngineeringFig.3.Comparisonoftheoreticalandexperimentalresults.thewall,theequationoftrackis:drd=parenleftBiggpd2pCc18gparenrightBiggvw(23)andintegratingtheaboveequationleadsto:dNN=parenleftBiggpd2pCc18gparenrightBiggvwC(Rw,)RwdRwintegraltextReC(r,)rdr(24)thegradeefficiencyis:i=1NN0=1expparenleftBiggpd2pCc18gparenrightBiggvwTintegraldisplay0C(Rw,)RwRwintegraltextReC(r,)rdrd(25)TisgivenbyLiandWang9:T=2(S+L)a(26)=2.3DeparenleftbiggD2abparenrightbigg1/3forLHh(27)hforL=HS(28)ResultsanddiscussionToverifytheusefulnessofthemathematicalmodel,anex-studyiscarriedoutonaStairmandhigh-efficiencyyclone15withadiameterof0.3m.Thetestedpowderastalcumpowderobeyingalognormalsizedistributionaskeletaldensityof2700kg/m3,ameanparticlesizeofH9262mandageometricalstandarddeviationof2.08.Thevelocitywas20.18m/sandthedustloadwas5.0g/m3.Fig.3comparesthepresenttheoreticalmethodwiththexperimentaldataandseveralpreviousclassicaltheories.TheshowsthatthegradeefficiencycalculatedbytheasametoAabBCCCddDDhHLNNPQrRRRStvzzGrocessing44(2005)447451wmethodagreeswellwiththedataandthetheories