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ChemicalEngineeringandProcessing442005447–451AbstractseparationradialgradetheK1.lutionandricate,harshmostutilizedoftribtions.ofories.etraincollectionries.particles,galDevelopmentofanewmethodforevaluatingcycloneefficiencyBingtaoZhao∗DepartmentofEnvironmentalEngineering,DonghuaUniversity,No.1882WestYananRoad,Shanghai200051,PRChinaReceived29March2004receivedinrevisedform9June2004accepted15June2004Availableonline20August2004Anewtheoreticalmethodforevaluatingcycloneefficiencyisdevelopedbasedontheinvestigationofflowpattern,thecriticalparticlesizetheoryandtheboundarylayerseparationtheory.Theradialparticleconcentrationgradient,insteadoftheusuallyassumeduniformparticleconcentrationwithinthecyclone,isconsideredinthismathematicalmodel.Thelocalparticleconcentrationandthecycloneefficiencycanbecalculatedonthebaseofatimeofflightmodelintermsoftheparticlesizedistributionofthefeed.Theavailabilityofmethodisverifiedbycomparisonofthecalculatedgradeefficiencywithexperimentaldataandtheoreticalcounterpartsintheliterature.eywordsCycloneCollectionefficiencyMathematicalmodelIntroductionCycloneseparatorsarewidelyusedinthefieldofairpolanyparticlesizeisdeterminedfromtheratioofitssettlingvelocitytotheterminalsettlingvelocityofthestaticparticle.DirgoandLeith3modifiedtheBarththeoryandfoundcontrolandgassolidseparationforaerosolsamplingindustrialapplications.Duetorelativesimplicitytofablowcosttooperate,andwelladaptabilitytoextremelyconditions,cycloneseparatorshavebecomeoneoftheimportantparticleremovaldeviceswhicharepreferablyinbothengineeringandprocessoperation.Inordertodescribethecycloneperformance,thetheoriescycloneparticlecollectionaredevelopedbymanyconutorsusingdifferentmethodswithsimplifyingassumpRepresentativeresearchesmainlyincludeThetheoryLapple1isthemostwidelyusedexampleofthecuttheLappleassumedthatparticlesenteringthecyclonearevenlydistributedacrosstheinletopening.Theparticlethatvelsfromtheinlethalfwidthtothewallduringthetimethecycloneiscollectedwith50efficiency.Barths2theoryisanotherrepresentationofcutsizetheoBarthcalculatedtheterminalsettlingvelocityforstaticbasedontheexactbalancebetweenthecentrifuforceandthedragforce.Thecollectionefficiencyfor∗Tel.862162373718fax862162373482.Emailaddresszhaobingtaomail.dhu.edu.cnB.Zhao.aciencLappledescribetriescadjustmentLichtdesigntrationalloparticlesunfortunatelyfdirectiontheciencandoccurs,manDietzresimpleexpressionforBarthsplotofthecollectioneffiyversustheratio.IoziaandLeith4,basedonthesandBarthstheory,proposedalogisticfunctiontothefractionalefficiencyforcycloneswithgeomewhichwerevariationsoftheStairmandhighefficiencyyclone.Thisequationhasanempiricalparameterthatallowsofthesharpnessofthecutofcyclone.Leithand5developedanotherpopularapproachforcycloneassumingthatturbulencekeepstheparticleconcenatanyheightinthecyclonewellmixed.Thetheorywsadirectcalculationofthecollectionefficiencyforofanysizeandcyclonesofarbitrarydesign.But,thereisexperimentalevidencetosupporttheactthatthereis,indeed,aconcentrationgradientintheradialofcyclones.Cliftetal.6modifiedtheestimateofmeangasresidencetimeandrederivedthegradeeffiyequationbasedontheoriginalassumptionsofLeithLichtstheoryandadifferentdependenceonparticlesizeapproachingthetypicalSshapecurvesobtainedbyyresearchers.Ahybridcollectiontheorydevelopedby7dividesthecycloneintothreeregionsTheentrancegion,thedownfloworannularregion,andtheupflow448andProrclesassumedtrationThisLentgionefMoreawpresentcdifdifognizingintheKimclonesoryandlentthegionfromAnother11,12theoryroughness,sizeandmethod13dictingtigationtheorysidersofticlesby2.inminingareparticleticle,samene2.1.Gasflowpatternandparticlemotion2.1.1.GasflowpatternThecircumferentialgasvelocityprofileforeffectiveseparationregionRe≤r≤RwwasobtainedbyMothesandL¨offler8,withsimplifyingthecyclonetotherightcylinderandconsideringthewallroughnessasfollowsvθvθwr/Rw1P1−r/Rw1vθwvdξh∗parenleftBiggradicalBigg14ξh∗v∗θwvd−12parenrightBigg2v∗vπR2w3B.Zhao/ChemicalEngineeringcoreregion.Dietzproposedtheinterchangeofpartibetweentheannularandcoreregions.ButDietzalsothatturbulenceproducesauniformradialconcenprofileforuncollectedparticleswithineachregion.assumptionaswell,shouldbejustified.Mothesand¨offler8haveextendedtheconceptsofDietzofdifferflowregionswithinthecyclonetoincludeafourthreclosetothedustexitatthecyclonebottom,wherethefectofdustreentrainmentcanbeincludedinprinciple.important,however,seemstobetheconsiderationoffiniteturbulentdiffusivityinboththedownwardandupardflowregions.ThisapproachavoidsthediscontinuitiesintheDietzmodelandthetransportofparticlesinycloneseparatorsisthenviewedasasuperimpositionofafusivemotionwithadeterministicmeanmotion.OtherferenttheoriesincludeLiandWang9althoughrectheroleofafiniteturbulentparticlediffusivityproducingradialconcentrationgradients,havedroppedhypothesisofdifferentflowregionswithinacyclone.andLee10proposeatheoryforhighefficiencycybasedontheboundarylayercharacteristics.Thisthedividesthecycloneintotworegions,theturbulentregionthenearwallregion.Particlestrajectoriesintheturburegionwerecalculatedfromthemeanfluidmotionandcollectionprobabilityofparticlesinthenearwallrewascalculatedbythedepositionvelocitywhichresultsbothturbulentthediffusionandthecentrifugalforce.typicaltheorydevelopedbyMuschelknautzetal.foralongtime.MuschelknautzimprovedBarthsbyconsideringtheeffectsoftheparticleload,thewallthesecondaryflow,andthechangeinparticledistributionwithinthebodyonthecollectionefficiencythepressureloss.Thistheorymaybethemostpracticalformodelingcycloneseparatorsatthepresenttime.Inthepresentpaper,anewmathematicalmethodforprecycloneefficiencyisdevelopedbasedontheinvesofflowpattern,thecriticalparticlesizeseparationandtheboundarylayerseparationtheory,whichconavariedradialparticleconcentrationgradientinsteadauniformradialconcentrationprofileforuncollectedparwithinthecyclone.Thecollectionefficiencycalculatedthismodeliscomparedwithexperimentaldata.TheoryTheconventionalcycloneseparatorgeometryisshownFig.1.Todevelopthenewcalculationmethodfordeterthecollectionefficiency,thefollowingassumptionsmadetheparticleissphericalinshape,themotionofaisnotinfluencedbythepresenceofneighboringparthetangentialvelocitycomponentoftheparticleistheasthatofthegasstream,andtheradialgasvelocityisglected.vhPQξAlthoughagreestheofthesometimes2.1.2.thedirectionocessing442005447–451Fig.1.Schematicdiagramofcyclonegeometry.θwdab−0.204b/Rw0.889dQπR2w4∗aRwbracketleftbigg2π−arccosb/Rw−12π−1bracketrightbigghRw5vθwvdparenleftbiggξξsinεparenrightbigg6abvin70.0065∼0.00758theexpressionforvθlookscomplicated,theresultverywellwiththetypicalvelocityprofilebasedonpowerlawcorrelationofAlexander14.Theadvantagethisexpressionisthatitpresentsaquantitativevalueoftangentialvelocityattheedgeofthecore,v∗θw,whichhasbeenassumedequaltotheinletvelocity.ParticlemotionUndertheinitialassumptions,theforcebalanceactingonparticlethedragforceobeysStokeslawintheradialgivesd2rdt218µgρpd2pCcdrdt−v2θpr09andPrEq.obtainedisinAccordingCombiningFromAccordingityInteθofθ2.2.necessarysize.log–normalfAccordingcleofdAssumingspecifiedtheCW2.3.SeparationefficiencyFig.2showsschematicallythegasparticleseparationprocessinahorizontalcrosssectionofthecyclone.ThenumberofparticlesperunitvolumeCisafunctionofrandθ.Supposethatduringthetimedtallparticles,whosedistancetothecyclonewallisdrorless,willbecollected,andthatmeanwhiletheparticleswilltraveladistancerdθinthetangentialanddzintheverticaldirection.B.Zhao/ChemicalEngineering9isnotreadilysolvable.Anapproximatesolutioncanbebyneglectingthesecondderivatived2r/dt2,whichtheconsequenceoftheassumptionthattheparticlemotiontheradialdirectionisstationary.Eq.9thenbecomesdrdtρpd2pCc18µgv2θpr10totheEquationofparticletrack,itisdrdtvrp11dθdt1rvθp12Eqs.11and12drdθrvrpvθp13Eqs.10,11and13drdθparenleftBiggρpd2pCc18µgparenrightBiggvθp14totheassumptionofequalgasandparticleveloc,Eqs.1and14canbecombineddrdθparenleftBiggρpd2pCc18µgparenrightBiggvθwr/Rw1P1−r/Rw15gratingEq.15andconsideringtheboundarycondition00,r0ReyieldstheformulaforcalculatingthetrackaparticleinanypositionparenleftBigg18µgρpd2pCcparenrightBiggparenleftbigg1vθwparenrightbiggbracketleftBiggparenleftBigg1PRw12parenleftBiggparenleftbiggrRwparenrightbigg2−parenleftbiggReRwparenrightbigg2parenrightBigg−PRw13parenleftBiggparenleftbiggrRwparenrightbigg3−parenleftbiggReRwparenrightbigg3parenrightBiggparenrightBiggbracketrightBigg16ConcentrationdistributionTodescribetheparticleconcentrationdistribution,itistodefinethespecifiedpointofcriticalparticleTheparticlesizedistributionissupposedtoobeythedistributionlndp1lnσ√2πexpbracketleftBigg−lndp−lndp5022lnσ2bracketrightBigg17totheequationofparticletrack,thecriticalpartisizeinanypositionshouldalsobeexpressedasafunctionrandθpcradicaltpradicalvertexradicalvertexradicalbtparenleftbigg18µgρpCcparenrightbiggparenleftbigg1vθwθparenrightbiggbracketleftBigg1PRw12parenleftBiggparenleftbiggrRwparenrightbigg2−parenleftbiggReRwparenrightbigg2parenrightBiggv−ThebeNAccordingtionobtained−ocessing442005447–451449Fig.2.Crosssectionofthecyclonewithschematicseparationprocess.thatparticleslargerthandpccanbeseparatedatthepointandparticleslessthandpccannotbeseparated,particleconcentrationatthispointshouldbeobtainedr,θC01−∞integraldisplaylndpcflndpdlndp.19ecallthisthecriticalseparationtheory.−PRw13parenleftBiggparenleftbiggrRwparenrightbigg3−parenleftbiggReRwparenrightbigg3parenrightBiggbracketrightBigg18ThenthenumberofcollectedparticleswithinthecontrololumewillbedNCRw,θRw−drdθdzdrCRw,θRwdθdzdr−CRw,θdθdzdr2≈CRw,θRwdθdzdr20totalnumberofparticleswithinthecontrolvolumewillRwintegraldisplayReCr,θrdθdzdr21totheboundarylayerseparationtheory,thefracofparticlesremovedwithinthecontrolvolumecanbebycombiningEqs.20and21dNNCRw,θRwdrRwintegraltextReCr,θrdr22450andPrNearCombining−ThenηHere,θLLH3.perimentalcwwith5.97inletecomparisonneMothescloserLichtwell4.lectionmodelload,dimensionstheB.Zhao/ChemicalEngineeringFig.3.Comparisonoftheoreticalandexperimentalresults.thewall,theequationoftrackisdrdθparenleftBiggρpd2pCc18µgparenrightBiggvθw23andintegratingtheaboveequationleadstodNNparenleftBiggρpd2pCc18µgparenrightBiggvθwCRw,θRwdθRwintegraltextReCr,θrdr24thegradeefficiencyisi1−NN01−exp−parenleftBiggρpd2pCc18µgparenrightBiggvθwθTintegraldisplay0CRw,θRwRwintegraltextReCr,θrdrdθ25θTisgivenbyLiandWang9T2πSLa262.3DeparenleftbiggD2abparenrightbigg1/3forL≤H−h27−hforLH−S28ResultsanddiscussionToverifytheusefulnessofthemathematicalmodel,anexstudyiscarriedoutonaStairmandhighefficiencyyclone15withadiameterof0.3m.Thetestedpowderastalcumpowderobeyingalog–normalsizedistributionaskeletaldensityof2700kg/m3,ameanparticlesizeofH9262mandageometricalstandarddeviationof2.08.Thevelocitywas20.18m/sandthedustloadwas5.0g/m3.Fig.3comparesthepresenttheoreticalmethodwiththexperimentaldataandseveralpreviousclassicaltheories.TheshowsthatthegradeefficiencycalculatedbytheasametoAabBCCCddDDhHLNNPQrRRRStvzzGrεηµθθθocessing442005447–451wmethodagreeswellwiththedataandthetheoriesofandL¨offlerandofIoziaandLeith.Otherwise,itistotheexperimentaldatathanthetheoryofLeithandforsmallerparticles,andthanthetheoriesofDietzasasofCliftetal.forlargerpaiticles.ConclusionsAtheoreticalmethodwasproposedtoestimatethecolefficiencyforcycloneseparators.Althoughthenewdoesnottakeintoaccounttheeffectsofinletparticleparticlereentrainmentandsomecyclonegeometricalincludingthoseofthevortextubeandthecone,workcanserveasastaringpointforthedevelopmentofnewapproachtocalculatethecycloneefficiency.Atthetime,furtherworkonthissubjectisrequiredinorderenhancetheadaptabilityofthemodelinpractice.ppendixA.Nomenclatureinletheightinletwidthparticleoutletdiameterparticleconcentration0inletparticleconcentrationcCunninghamcorrectioncoefficientparticlediameterp50meanparticlesizecyclonebodydiameteregasoutletdiametercyclonecylinderdiametercycloneheightnaturallengthofcycloneparticlesnumber0inletparticlesnumberparameterofmomentumexchangebetweengasandthewallvolumetricgasflowrate0initialradialpositionofparticleradialdimensionwD/2eDe/2gasoutlettubelengthtimegasvelocitycoordinatedirectionseekletterstheconeslopecollectionefficiencyggasdynamicviscosityangularcoordinatedirections0initialangularpositionofparticleTtotalturningangle
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