First and second-order models for the vortex length in cylinder-on-cone cyclones based on large-eddy simulations

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Research article

First and second-order models for the vortex length in cylinder-on-cone cyclones based on large-eddy simulations

Ellinor Arguilla Svensen, Alex C. Hoffmann

DepartmentofPhysicsandTechnology,UniversityofBergen,Allegaten555007Bergen, Norway

A R T I C L E I NF O A B S T R A C T

Keywords:

Chemicalengineering Mechanicalengineering Mechanics

Appliedcomputing Computer-aidedengineering Cycloneseparators Naturalvortexlength Large-eddysimulations Multidimensionalregression Model

Thecommondesignofcycloneseparatorsisthecylinder–on–conedesign,andtheconicalshapehasastrong effectonthebehaviorofthevortexcorelowinthecyclone.The“vortexlength”isthedistancebetweenthe lipofthegasoutlettubeandthepositionatwhichthecoreofthevortexattachestothewallofthecyclone separationspace.Thisoccursspontaneouslyatanaxialpositionthat,atpresent,cannotbepredicted,although ithasaprofoundeffectonthecycloneoperation,since,ifthevortexistooshort,itcanleadtopluggingand wear.InthispapernumericalCFDsimulations,usingadvancedturbulencemodeling(LES),arethebasisfor theformulationofmodelsforthevortexlengthtakingintoaccountthegeometricalandoperationalvariables influencingit.Theworkleadstousefulmodelsforthevortexlengthandrevealsimportantinformationabout whichvariablesdetermineitandthenatureoftheireffects.

1. Introduction

A keyaspectfor cyclonedesigners andvendorsis tocontrolthe vortexendinthecyclonemakingsurethatthevortexpenetratestothe bottomofthecycloneseparationspacegivingtheoptimalseparation efficiency andavoidingproblemswith wearof thewallor plugging in thebottomof thecyclone. This paperwill extendtheknowledge in thisareaandinvestigate theeffectsofthedesignandoperational variablesonthebehaviorofthevortexinthisrespect.Earlierworkhas demonstratedthatusinganadvancedturbulencemodel,namelyLES, CFD can reflect thebehavior of thevortex in this respect correctly, evenquantitatively,anditwasthereforedecidedtousethistechnique tosimulatethevortexbehaviorincylinder–on–conecyclonesforcases wherethevortexendedintheseparationspaceitselftoelucidatethe phenomenonofthe“endofthevortex”andtheeffectsofthedesignand operationalvariablesonit.Fig.1showsthegeometryofacylinder-on- conecycloneandthegeometricalnotationusedinthispaper.

Alexander[1] pioneeredresearchonthevortexlengthincyclones.

Hebasedhisworkonexperimentsonfairlysmall,cylindricalcyclones withdiametersintherange30–50mmindiameter,limitingtherange ofapplicabilityofhisrelationforthevortexlength.Hispaperincludes important,moregeneral,researchintotheflowincyclones,usedasa

*

E-mailaddresses:alex.hoff[email protected],alex.hoff[email protected](A.C. Hoffmann).

basisformanystudiesafterhim.Amongotherthings,hedetermined thatasmallervortexfinderdiametergivesrisetoahigherseparation efficiency.Alexanderdefinedthenaturalvortexlength,𝐿_𝑛[m],asthe distancefromthebottomofthevortex findertothepointwherethe vortexturns,adefinitionthathasbeenusedinmoststudiessince.He alsostatedthatmeasurementsofthevortexturningpointwerecompli- catedduetothefactthatthevortexendmovesupanddownoverrange ofapproximatelyonequarterofthecyclonediameter.Heobservedthat alargeinletandsmalloutletgiverisetoashorteningofthenatural vortexlength.TherelationofAlexanderis:

𝐿_𝑛 𝐷 = 2.23

(𝐷_𝑥 𝐷

) (𝐷² 𝑎𝑏

)¹₃

(1) Bryantetal. [2] determinedthelengthofthevortexvisually.They fittedanempiricalexpressiontotheirexperimentalresults.Oneoftheir observations was thatthe position of theend of thevortex is inde- pendentoftheinletvelocityundernormaloperationconditions.They comparedthetheoreticallength(ascalculatedfromAlexandersequa- tion)withthelengthmeasuredbyvisualinspectionoferosioninthe conicalsectionandparticledepositsonthewall.Itwasobservedthat areductionofthevortexfinderdiametergaverisetoalongernatural length(contrarytowhatwasobservedbyAlexander).Thecyclonesthat

https://doi.org/10.1016/j.heliyon.2020.e03294

Received11August2019;Receivedinrevisedform17January2020;Accepted21January2020

2405-8440/©^{2020TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense}(http://creativecommons.org/licenses/by- nc-nd/4.0/).

Fig. 1.Figureshowingthegeometricalnomenclatureusedinthispaper,the SIunitsofallthedimensionsare[m],theReynoldsnumberisdefinedasRe≡

𝜌𝑣𝑖𝑛𝐷𝑒𝑞

𝜇 with𝑣_𝑖𝑛theinletvelocityand𝐷_𝑒𝑞thehydraulicequivalentdiameterof theinlet,inthegeometryusedinthispaperequalto4𝑏= 2𝑎,𝜇[kg/ms]isthe viscosityand𝜌[kg/m³]isthedensityofthefluid.

wereusedhadalongcylindricalsection,andwereclosertoaswirltube thananormalcylinder-on-conedesign.TherelationofBryantetal.is giveninadifferentform,butcanberewrittento:

𝐿_𝑛 𝐷 = 2.26

(𝐷_𝑥 𝐷

)−1( 𝐷² 𝑎𝑏

)−0.5

(2) Zhongli et al. [3] carried out experiments on longplexiglass cy- clones.Theyproposedanequationsuggestingthatthenaturalvortex length was longerthanpredicted bythe equation of Alexander.Ex- perimentswerecarriedoutwithbothcylindricalandconicalcyclone designs.Thetangentialvelocityandstaticpressureweremeasuredwith afive-holepitotprobe,andthetangentialvelocitywasseentodecline withaxialdepthinthecyclone.Bychargingdusttothecyclone,the vortexwasmadevisible,andastabledustringwasseenindicatingthe naturalturningpoint.Itwasfoundthatthenaturallengthwasslightly increasedwithanincreaseintheinletvelocity.Largerinletareasalso increasedthevortexlength.Theyalsoobservedthataconicalcyclone hadalongernaturallengththanacylindricalonewithotherwisethe samedimensions.Anexperimentalseparationperformancestudywas alsoconductedwithaverylowparticleconcentration[4].Intheirstudy theyfoundthatthecollectionefficiencyincreasedwithparticleconcen- tration,andtheeffectoftheinletvelocityontheefficiencybecameless pronounced.TherelationgivenbyZhonglietal.is:

𝐿_𝑛 𝐷 = 2.4

(𝐷_𝑥 𝐷

)−2.25( 𝐷²

𝑎𝑏 )−0.361

(3) Equations(2) and(3) bothinvolvethesamedimensionlessvariables asAlexander’sEquation(1),buttheeffectsofbothofthevariableson thedimensionlessvortexlengthareoppositetothosefoundbyAlexan- der.

Hoffmannetal.[5] performedexperimentsusingglasscyclonesvi- sualizing the vortex using smoke. They used a geometry called the Stairmand HE, withadiameter of 0.2m. Differentvortex finderdi- ameters anddifferent lengthsofthevortex finderweretested tosee theinfluenceonthevortexlength.Atubesectionwasplacedunder- neaththeconicalsectiontoturnthevortex,preventingthevortexcore fromreachingthesurfaceofthecollecteddust,reentrainingitinthe vortex.A stablevortexturning positionwas oftenfoundin thetube section.Smokewasinjectedtovisualizetheflowpattern.Thiswasob- servedtodisturbtheflowinitially,butthepositiontowherethesmoke extendedwasseenfairlyclearlyafterasteady-stateflowpatternwas

reestablished,thesmokeremaininginthesectionunderthevortexturn- ingpositiongivingtheinterestinginformationthatthereisverylittle exchangeofgasbetweenthesectionsaboveandbelowtheturningpo- sition.Thesmokeleftaliquidparaffinlayer,onwhichtheflowpattern atthewallcouldbeseenasanalternativevisualizationmethod.Acir- cumferentialringformedatastablepositionindicatingthelengthofthe vortex.However,theringseemedtoappearafewcentimeters abovethe topofthepersistingsmokeinthebottomsectionbelowthevortexend.

Itwasobservedthatthering movedlowerastheliquidonthewall dried,whichwasattributedtotheapparentroughness,thatarosebe- causeoftheliquidlayer,diminishing.Inthiswayitwasfoundthatthe vortexendwasdependentonwallroughness.Thenaturallengthwas foundtobelongerwithlargervortexfinderdiameterandinletveloc- ities.Alexandersequation predictedashortervortex thanwas found experimentally.Usingdustforvisualizationinsteadofsmoke,thesame phenomenawereobserved.Anamount ofdust wasdepositedon the wallfromthedustcollectionvesseluptoatcertainpositioninthetube sectionunderthecyclonedustexit.Withahighdustloading,theposi- tionofthevortexendwasfurtherupinthecyclonethanwithoutdust present,confirmingthatthehigherapparentwallroughnessduetothe dustonthewallgivesrisetoashortervortex.

Pengetal. [6,7] usedastroboscopetovisualizethevortexcorein cylinder–on–conecyclonesandswirltubes,andpressuretransducersto measurethepressureonthewall.Whenusingthestroboscopeasmall ringintheplaneofthewallcouldbeseenindustonthewall,which wasthecoreofthevortex(“eyeofthehurricane”),itsfastprecessing motion,normallymakingitinvisibletothenakedeye,wasfrozenby thestrobe-lightwhenthefrequencywassetexactlyright.Theyused pressuretransducerstodetecttherotationfrequencyoftheprecessing vortexcore. Inthemiddleof thecorethereis lowpressure andthe rotationcouldbemeasuredasasuddendropinwallpressureeachtime thevortexcorepassedtheangularpositionofthepressuretapping.By usingmultipletransducersplacedverticallyalongthecyclonebodyan axialprofileforthewallpressurewasobtainedfromwhichtheaxial positionofthevortexturningpoint(vortex“end”)couldbefound.

Qianetal. [8] usedresponsesurfacedesigntoanalyzetheresults fromtheirCFDsimulationstoobtainanewmodelforthevortexlength.

AReynoldsstresstransport(RSM)turbulencemodelwasused.Thevari- ableschosenwere:thevortexfinderdiameter(𝐷_𝑥),theinletlengthand width(𝑎and𝑏),theverticallengthofthecylindricalsectionofthecy- clonebodybeneaththevortexfinder(𝐻−𝐻_𝑐−𝑆)andtheReynolds number(Re).They foundthatthenaturallengthvaries slightlywith theinletvelocity,andthatwithincreasingvortexfinderdiameterthe lengthincreasedtoacertainpointandafterthisitbecomesshorter.For theinletareathefindingswereconsistentwiththerelationofAlexan- der,smallerinletareasresults inanincreaseofthevortexlength.In allthesimulationsthevortexendedinthetubesectionattachedtothe dustexitunderthecyclonebodylikethedesignusedbyPengetal.

Anexperimentalstudyfocusing ontheeffectsof theinletsection angleontheseparationperformancewasalsoperformed[9].Inmost cyclonedesignstheinletsectionis perpendiculartothecycloneaxis butinthisstudyotherangles,wherebytheinletwaspointingslightly downwardstowardsthedustoutlet,weretested.Theyfoundthatthe tangentialvelocity decreasedin some regions withincreasingangle.

Thetangentialvelocityintheoutervortexincreasedandthatin the innervortexdecreased.Theyalsofoundthatthepressuredropwasre- ducedcomparedtoanormalcyclone,andthisdifferenceincreasedwith largerangles.Theseparationefficiencyalsoincreasedsignificantlywith increasinginletangle.UsingCFDandaresponsesurfacemethodology theinletsectionanglecouldbe studiedmorecomprehensively[10], theyfoundthataconventionalcyclonehadastrongershort-cutflow rate(see[11])thancycloneswithangledinlets.Thevortexlengthwas notstudiedinthiswork.AlsoBernadoetal.[12] studiedtheeffectof theinletangleontheworkingofcyclonesusingCFD.

Earlierworkinthegroupofthepresentauthors[13,14] hasshown that,whilebothLESandRSMturbulencemodelscansimulatetheend

ofthevortexphenomenoncorrectly,LESgivesresultswhicharequanti- tativelybetterinlinewithexperimentthanRSM,whichisnotsurprising inlightofthefactthatLES,whilebeingmuchmoredemandinginterms ofcomputerpower,ismoreconsistentwiththephysicsoftheflowfield, requiringonlyasimpleturbulencemodelfortheeffectsofthesubgrid- scale turbulence,which is almostisotropic. This is supportedin the comprehensive comparisonof CFDmethodsforcycloneflow simula- tionbyGronaldandDerksen[15].Sinceexperimentshadshownthat thewallroughnesshasastrongeffectonthevortexlength,CFDsimula- tionswerealsoperformedtocastlightonthiseffect.Simulationswith wallshavingabsoluteroughnessesof0.1and0.2mmwerecarriedout forvaryinginletvelocities[13,16],thesimulationresultsconfirming theexperimentalresultthatincreasingthewallroughnessshortensthe naturallengthofthevortex.

Responsesurfacemethodsrepresentarobustandhighlyempirical methodforidentifyingthesalientvariablesgoverningalmostanypro- cessparameterofinterestandconstructingpredictivemodels[17,18, 19].Thesemethodshavebecomeverypopularintheresearchliterature recently.

Chandrasekharanetal.[20] usedresponsesurfacemethodstostudy andoptimizethe performanceofcoalfired thermalpower plantsas functionsoftheoperatingvariables.Theycombinedtheiroverallmodel withmodelsfortheindividualprocessingunitsfordesignpurposes.

Sonsirietal.[21] studiedaspectsoftheworkingofarotarydryer, suchasthemoisturecontentoftheproductandtheelectricalpower used, asfunctions ofthe operationof associatedprocessequipment, namelyacycloneblowerandabiogasblower.

Suchstudiesillustratethepowerandthebroadrangeofapplicability oftheresponsesurfacemethod,andexplainthegrowingpopularityof themethodology.

SomeofthemostrelevantworkswhereinCFDsimulationsareused tostudyinstabilitiesin confinedflowsandtheflowincyclone sepa- ratorswillnowbediscussedbrieflyemphasizingthereliabilityofLES simulations[22].

Griffithsetal. [23] performedastudyusingCFDtostudycyclone performanceinsmallsamplingcyclones.Theymention,asisalsowell known[24],thatthestandardk-𝜖 turbulencemodelcausesexcessive levelsofturbulentviscosityandanunrealistictangentialvelocitydistri- butionandcanthereforenotbeused.SinceReynoldsStressmodelsat thetimewereverycomputationallyexpensive,anRNG-basedk-𝜖-model wasused.

Hoekstraetal. performedanexperimentalstudywhichverifiedCFD simulations[25].Theyusedcyclonesof0.29mindiameterwithascroll typeofinlet(see[11]).Thecycloneshadalongcylindricalsectionwith a shortconicalsection. Experiments were carriedout at a Reynolds numberof2.5× 10⁴.Tangentialandaxialvelocitycomponentsalongthe radialdirectionweremeasuredusinglaser-Doppleranemometry(LDA).

CFDsimulationswerealsocarriedoutusingdifferentturbulencemod- els,namelythek-𝜖-model,theRNG-k-𝜖-modelandtheReynoldsStress TransportModel(RSM).ComparingtheexperimentswiththeCFDsim- ulationscarriedoutwithRSM,theyfoundgoodagreement.Theother turbulencemodelsdidnotreflecttheaxialandtangentialvelocitiesac- curatelyandwerethereforenotfoundsuitable.

InhisPhDthesis[26] Hoekstraperformedanextensive studyon cycloneefficiency,bothexperimentallyandbyCFD.Thegasflowfield wasstudiedexperimentallyusingLDAandhefoundalow-frequency instability,relatedtoaphenomenoncalledtheprecessingvortexcore (PVC).ThisPVC-phenomenonwasfurtherstudiedusingLDA.

CFD simulationswereperformedusing different turbulencemod- elstocomparewiththeexperimentaldata,goodagreementwasfound whenusingtheReynoldsStresstransportturbulencemodel.TheLES- modelwasalsofoundtobeabletomodeltheprecessingmotionofthe vortexcore.Fromthisaregressionmodelwasformulatedtodescribe thecollectionefficiencyofthecyclone.Aresponsesurfacemodel(to testtheeffects ofdifferent geometricvariablesandoperating condi-

tions)wasalsopresented,whichcouldpredictthepressuredropand thecut-sizeoftheStairmandcyclone.

Derksenetal. [27] usedLEStosimulateturbulentflowinareverse- flowcyclone.Thecyclonewasoftheswirltubetype,butwithatangen- tial(scroll-type)inletinsteadofswirlvanes(see[11]).TheReynolds numberoftheflowwas1.4× 10⁴.Athree-dimensionalsimulationwith highresolutionwas used, and itwas foundthatan axi-symmetrical modelwouldnotproperlymodeltheswirlingflow.Goodagreement withtheLDA-experimentsexecutedbyHoekstra[25] wasfound.The studyalsoincludedsimulationswithparticlesintheflow,anditwas foundthatparticleslarger10 μmdidnotneedanyspecialmodeling.

However, smallerparticles would influence the subgrid-scale turbu- lenceof theLESsimulationsandthis hadtobe takenintoaccount.

Itwasmentionedthatflowsathigher(andmorecommoninindustrial cyclones)Reynoldsnumbers,wasnotstraight-forwardtosimulateus- ingauniformly spacedgridbecauseof thelackofresolution inthe boundarylayer.Locallyrefinedgridwasrecommendedasasolution.A moreextensivestudywithsimulationsusingLESwassubsequentlydone [28].Itismentionedin thisworkthatusingLESforswirlingflowis betterthanusingRMSbecauseflowcontainingparticlesismorerealis- ticallymodeledwithLES,alsoswirlingflowsoftenexperiencecoherent, quasi-periodicfluctuations(suchastheprecessingofthevortexcore, PVC)whichLESsimulatesbetterthanRMS.Threedifferent subgrid- scale(SGS)turbulencemodelswhereused, using thesamedamping function.Theyalsoobservedthephenomenonofvortexbreakdownin thesimulations.Whentheeffectofthediameterofthevortexfinder diameterwastested,theyfoundthattheaxialvelocitywasstronglyde- pendentonthisandalsoontheaxialpositionofthevortextube.The Mixed-scale-model(MSM)wasthebestoftheSGS-modelstested.They alsofoundthatLESrevealedmoreofthephysicsinturbulentflowthan othermodels,butatacomputationalcost[15,29].

This presentwork aimsto make use of the advancing computer powertoformulateanimprovedmodelforthenaturalvortex length incylinder-on-conecyclonesbasedonCFDsimulationsusingstate-of- the-artturbulencemodeling,namelyLES.Aresponsesurfacemethod similartothatinref. [8] willbeusedtoformulatebothfirst-orderand second-ordermodelsforthevortexlengthandtoadvancephysicalin- sightintothefactorsaffectingthevortexlengthandthenatureoftheir effects.Inaddition, modelsofdifferent functionalformsfittedtothe numericalresultswillbeexploredwiththegoaloffindinganoptimal modelforuseinpractice.

2.Numericalmethods

TheequationstobesolvednumericallyaretheNavier-Stokesequa- tions:

𝛁⋅𝐯= 0 𝜌 𝐷𝐯

𝐷𝑡=𝜌𝐠−𝛁𝑝+𝜇𝛁²𝐯 (4)

Anumericalmeshisappliedtocoverthedomainofinterest,anddis- creterepresentationsoftheequationsaresolvedonthemesh.Themesh hastobefineenoughtoobtainanaccuratesolution,andascoarseas possibletolimitthecomputationaleffort.

2.1. Meshing,timediscretizationandboundarytreatment

ThecommercialCFDpackageStar-CCM+wasusedforthenumeri- calsimulations.Thispackageoffersmeshcellsofavarietyofshapes,for examplehexahedra,tetrahedraorblocks.Whichofthesetypesisopti- maldependsonthegeometryofthedomain.Alsounstructuredgrids areoffered.Themeshingmodelsusedinthisworkare“surfaceremesh- er”,“trimmer”and“prismlayermesher”.Thetrimmerofferstheoption ofusingaregulargrid,whichisadvantageous,atthesametimeasen- suringthatthecellsdonotextendbeyondthecomputationaldomain.

Theprismlayermeshermakesitpossibletohavesmaller,prismshaped,

Fig. 2.Figureshowingthemeshontheboundaryofthecomputationaldomain.

cellsnearthewall.Aprismlayerisorthogonalprismaticcellscloseto thewall,forwhichadifferentsizethanthecorecellscanbeset.The surfacemeshisshowninFig.2.

ThemethodforwalltreatmentintheLESturbulencemodelisde- scribedinmoredetailelsewhere[13].Briefly,theshearvelocity,𝑢_𝜏,is estimatedfrominvertingathird-orderSpaldingwallfunction:

𝑦⁺=𝑢⁺+ 1 𝐸

(

exp(𝑘𝑢⁺) − 1 −𝑘𝑢⁺−(𝑘𝑢⁺)² 2! −(𝑘𝑢⁺)³

3!

)

(5) where𝐸isanempiricalfactorand𝑘isthevonKarmanconstant.The momentumfluxfromthewallisthencalculatedfromtheestimated𝑢_𝜏. Comparisonwithexperimentdemonstratedthatitisimportantthatthe wallvelocityisevaluatedattheappropriate𝑦⁺valuewhendetermining thevelocitygradientatthewall.

Griddependencytestswerecarriedout,anditwasfoundthatthe largest cellsizegivinggrid-independentresults fortheend-of-vortex phenomenonwas4.5mmcells,correspondingtoatotalof500000cells.

Animplicitschemewasused,andthetime-stepsetsothattheCourant- numberdidnotexceed1.5.

2.2. Turbulencemodeling

Asmentionedabove,earliersimulationsandexperiments[13] have shownthat, althoughunsteadysimulationsusingReynoldsstresstur- bulencemodeling(RSM)cancorrectlyreflecttheend-of-vortex(EoV) phenomenon,namelythevortexcorebendingspontaneouslytothecy- clonewallandprecessingaroundthewallatamoreorlessfixedaxial position,quantitative agreementbetween simulationandexperiment forthepositionoftheEoVisbetterwhenusingLarge-eddysimulations (LES).

Forlargeeddysimulationsoftheflowofincompressiblefluidsaspa- tialfilterisappliedtotheflowvariables.Foranarbitraryflowvariable, 𝜙(𝐱),whichcouldbeacomponentofthefluidvelocityorthepressure, thespatialfilteris:

̄𝜙=

∫ 𝐺(𝐱−𝐱^′,Δ)𝜙(𝐱, 𝑡)𝑑𝐱^′ (6) where𝐺(𝐱)issomesuitablefunctionofthespatialcoordinates,related tothesizeofonecomputationalcell.Δisthewidthofthefilter,similar inmagnitudetothesizeofthecomputationalcells.Byapplyingthisfil- terturbulenteddiessmallerthanthegridsizearefilteredout,andtheir effect onthe flowis modeled witha turbulencemodel.This model, however,canbequitesimple,sincethisfine-scaleturbulenceisalmost isotropic. Inthisworkthewell-knownandverysimpleSmagorinsky

Table 1

Listofthesimulationscarriedoutduringthisworkandtheresultsforthedi- mensionlessvortexlength.

No. 𝐷_𝑥∕𝐷 (𝐻−𝐻_𝑐)∕𝐷 𝐻_𝑐∕𝐷 (𝑎𝑏)^0.5∕𝐷 Re 𝐿_𝑛∕𝐷

1 0.3 1.0 2.0 0.2 30000 2.5

2 0.5 1.0 2.0 0.2 30000 2.5

3 0.3 3.0 2.0 0.2 30000 4.5

4 0.5 3.0 2.0 0.2 30000 4.5

5 0.3 1.0 5.0 0.2 30000 2.02

6 0.5 1.0 5.0 0.2 30000 2.48

7 0.3 3.0 5.0 0.2 30000 4.17

8 0.5 3.0 5.0 0.2 30000 4.53

9 0.3 1.0 2.0 0.55 30000 1.17

10 0.5 1.0 2.0 0.55 30000 1.32

11 0.3 3.0 2.0 0.55 30000 1.62

12 0.5 3.0 2.0 0.55 30000 1.58

13 0.3 1.0 5.0 0.55 30000 1.47

14 0.5 1.0 5.0 0.55 30000 1.28

15 0.3 3.0 5.0 0.55 30000 1.88

16 0.5 3.0 5.0 0.55 30000 1.75

17 0.3 1.0 2.0 0.2 3000000 2.50

18 0.5 1.0 2.0 0.2 3000000 2.50

19 0.3 3.0 2.0 0.2 3000000 4.50

20 0.5 3.0 2.0 0.2 3000000 2.50

21 0.3 1.0 5.0 0.2 3000000 2.06

22 0.5 1.0 5.0 0.2 3000000 2.86

23 0.3 3.0 5.0 0.2 3000000 3.05

24 0.5 3.0 5.0 0.2 3000000 3.45

25 0.3 1.0 2.0 0.55 3000000 2.50

26 0.5 1.0 2.0 0.55 3000000 2.50

27 0.3 3.0 2.0 0.55 3000000 2.08

28 0.5 3.0 2.0 0.55 3000000 1.92

29 0.3 1.0 5.0 0.55 3000000 3.10

30 0.5 1.0 5.0 0.55 3000000 2.11

31 0.3 3.0 5.0 0.55 3000000 2.18

32 0.5 3.0 5.0 0.55 3000000 1.68

33 0.4 2.0 3.5 0.35 1515000 2.11

34 0.3 2.0 3.5 0.35 1515000 2.88

35 0.5 2.0 3.5 0.35 1515000 2.00

36 0.4 1.0 3.5 0.35 1515000 3.05

37 0.4 3.0 3.5 0.35 1515000 2.26

38 0.4 2.0 2.0 0.35 1515000 3.00

39 0.4 2.0 5.0 0.35 1515000 2.11

40 0.4 2.0 3.5 0.2 1515000 5.00

41 0.4 2.0 3.5 0.55 1515000 3.28

42 0.4 2.0 3.5 0.35 30000 1.85

43 0.4 2.0 3.5 0.35 3000000 5.00

subgridturbulencemodelwasused.ThelargereddiesareinLESsimu- lateddirectly.

3. Simulations

Atotalof43simulationswerecarriedouttoconformwitha“central composite”experimentaldesign,whichwasseenasthesimplestdesign togiveusefulinformationabouttheeffectsofallthesalientvariables andisaschemesimilartothatusedin[8].Thegeometricalandopera- tionalvariableswerechosensoastocovertherangeofgeometriesused inpracticeasgivenin[11].Thesimulationsandtheresultsarelistedin Table1.Thenomenclatureis,asmentioned,giveninFig.1. Ifthevor- texendisnotseeninthecyclonethevortexissaidtobe“centralized”, althoughinourexperienceitisthennormallyattachedtothedustout- letlip,exceptincaseswherethedustcollectionvesselisveryshallow allowingthevortextopenetrateitandendonthebottomofthedust hopperoronthesurfaceofthecollecteddustinit(whichisnotgood).

4. Results

AnexamplesimulationisshowninFig.3,thevortexcoreisclearly visibleinthiscontourplotforthestaticpressureandisseentobend fromtheaxisofthecycloneandattachtotheconicalpartofthewall.

TheresultsofthesimulationsarelistedinTable1.

Fig. 3.Examplesimulationsnapshot,thelow-pressurevortexcoreisclearly visibleandisseentobendfromthecycloneaxisandattachtotheconicalpart ofthewall.Thelengthofthevortex,fromthelipofthevortexfindertothe pointofattachmentisindicated.Thesimulationisforaninletvelocity,𝑣𝑖𝑛,of 20m/s.

Duringthesimulationsthefollowingobservationsweremade:

• InthesimulationswithlowRethevortextooksometimetoform insidethecyclonebody.

• Forthe simulationswithhigh Rethevortex formed almostim- mediatelybothwhenthevortexcorewascentralizedandwhenit attachedtothewalltoformtheEoV.

• Alltheshorter cyclonesexperiencedacentralizedvortex,in the mostofthelongercyclonestheEoVphenomenonwasseen.

• Alongerconicalsectionseemedtodestabilizethevortexandgave risetoashorternaturalvortexlength.

5. Discussionandformulationofmodels

Inthissectionaseriesofmodelequationswillbefittedtothenumer- icalresultsforthevortexlength,attemptswillbemadetofindrelatively simplemodelequations thatneverthelessareusefulin practice. The methodwillbetofitthemodelstothesimulationresultsusingthero- bustzeroth-ordermultidimensionalsearchtechniquedevisedbyRosen- brock[30], forthisthenaturallogarithmwas takenoftheReynolds numbertoavoidround-off-errorsarisingfromorder-of-magnitudedis- crepanciesintheindependentvariables.Ananalysis-of-variancetable generatedforthesecondordermodelbythesoftwareSPSSwillalsobe used.

5.1. First-ordermodel

Thefirstmodeltobefittedwillbeafirst-ordermodel,themodel isconstructedasaseriesofadditivefirst-ordertermswhereallthead- justableparametershavebeenfittedtobestmatchthesimulationresults forthelengthofthevortex,𝐿_𝑛:

Fig. 4.Parity plot for the fitted first order model, Equation (8).

𝐿_𝑛 𝐷 =𝑝1+

∑5

𝑖=1𝑝_𝑖+1×𝑥_𝑖 (7)

wherethe𝑝_𝑖aretheadjustableparametersandthe𝑥_𝑖aretheindepen- dentgeometricalandoperationalvariableslistedinTable1.

Thefittedfirst-ordermodelis:

𝐿_𝑛

𝐷 = 2.56 − 0.212 (𝐷_𝑥

𝐷 )

− 0.360

(𝐻−𝐻_𝑐 𝐷

)

−

0.0688 (𝐻_𝑐

𝐷 )

− 4.03 ((𝑎𝑏)^0.5

𝐷 )

+ 0.0938 (ln Re) (8) ThePearson’sris0.701,theresidualsumofsquaresis24.46anda parityplotisshowninFig.4.Thefitisseentobereasonablewithout havingaveryhighpredictivepowerforuseinpractice.

Thequalitativeeffects,however,onthenaturalvortexlengthofthe variablescanbegleanedfromthisrelationship,anincreasein𝐷_𝑥ap- pearstoweaklydecreasethelengthofthevortex,whileanincreasein thecylindrical-sectionheight,(𝐻−𝐻_𝑐)doesthesame.Anincreasein thedimensionlessinletareahasastrongereffectindecreasingthenat- uralvortexlength,whileanincreaseinReincreasesthenaturallength.

Theeffectsof𝐷_𝑥and𝑎𝑏arequalitativelyconsistentwiththerelation ofZhonglietal.and Bryantetal.,butnotwiththatofAlexanderal- thoughitdoesagreewiththetextof Alexanders’paperasquotedin theIntroduction,whilethatofReisconsistentwithliteratureingen- eral.

Itcanbeconcludedfromthisthatthevariableschoseninthisstudy doinfluencethelengthof thevortex,andthefirst-order modeldoes reflecttheireffects,butthatthismodelisnotsufficienttoreducethe residualvarianceintheresultstoalowvalue.Alsothefamousmodel ofAlexanderdoesnotseemtoapplytocylinder-on-conecyclones(see alsobelow).

5.2. Second-ordermodel

Thefirst-ordermodeldoesnotreducetheresidualvariancesomuch thatitgivesasufficientlygooddescriptionofthelengthofthevortex, andthiscanbethemotivationtoexploreasecond-ordermodelwhere alsothenon-lineareffects ofthevariablesandtheirinteractions are takenintoaccount.Thedraw-backofconstructingsuchamodelisthat itisverycomplexandinvolvesalargenumberofadjustableparameters.

Theformofthismodel,whichissimilartotheoneinvestigatedbyQian etal. [8] is:

𝐿_𝑛 𝐷 =𝑝1+

∑5 𝑖=1

∑5

𝑗=𝑖𝑝_𝑖𝑗×𝑥_𝑖𝑥_𝑗 (9)

This,inprinciple,involves21adjustableparameters,averylargenum- ber.Forthemultidimensionalsearchthechoiceismadeheretoreduce thisalittlebydecidinginadvance,basedonpreliminaryfitresults,that thetwo-factorinteractions:(𝐷_𝑥∕𝐷)((𝐻−𝐻_𝑐)∕𝐷)and(𝐷_𝑥∕𝐷)(𝐻_𝑐∕𝐷)are notimportant,reducingthenumberofparametersto19.

Twomethodswereusedtofitthissecond-ordermodeltothesimu- lationresults:amulti-dimensionalsearchasfortheothermodelsand usingthecommercialsoftwareSPSS.Themultidimensionalsearchre- quired in excess of 2million iterations, butis coded in a compiled language,namelyFORTRAN,whichexecutesfasterthananyotherlan- guage exceptC/C++,whichexecutesat aboutthesamespeed.The resultofthesearchwas:

𝐿𝑛

𝐷 = 12.6 + 54.5(𝐷𝑥

𝐷 )

+ 4.71(𝐻−𝐻𝑐

𝐷 )

+ 1.97(𝐻𝑐

𝐷 )

−

32.0 ( √𝑎𝑏

𝐷 )

− 3.71 ln(Re) − 61.8 (𝐷_𝑥

𝐷 )2

− 6.70 (𝐷_𝑥

𝐷 ) ( √𝑎𝑏

𝐷 )

−

0.216(𝐷𝑥

𝐷 )

ln(Re) − 0.483(𝐻−𝐻𝑐

𝐷 )2

− 0.050(𝐻−𝐻𝑐

𝐷

) (𝐻𝑐

𝐷 )

−

2.56

(𝐻−𝐻𝑐

𝐷

) ( √𝑎𝑏 𝐷

)

− 0.1017

(𝐻−𝐻𝑐

𝐷 )

ln(Re) − 0.258 (𝐻𝑐

𝐷 )2

−

0.502 (𝐻_𝑐

𝐷 ) ( √

𝑎𝑏 𝐷

)

− 0.0251 (𝐻_𝑐

𝐷 )

ln(Re) + 36.1 ( √𝑎𝑏

𝐷 )2

−

0.542 ( √𝑎𝑏

𝐷 )

ln(Re) + 0.158(ln(Re))²

(10) ThePearson’s ris 0.911,theresidual sumof squaresis 8.105anda parityplotisshowninFig.5.Thefitisseentobegood.

ThesecondmethodoffittingEquation(9) tothesimulationresults wasusingthecommercialsoftwareSPSS.Theresultsofthisisshownin Table2,theresidualsumofsquaresis8.110.

Fig. 5.Parity plot for the fitted second order model, Equation (10).

ComparingEquation(10) andtheunstandardizedcoefficientsinTa- ble2itisclearthatthecoefficientsaresimilarbetweenthetwo,the multidimensionalsearchhasachievedaslightlylowerresidualsumof squaresthanSPSSinspiteoftwointeractionshavingbeendisregarded.

AdirectcomparisonwiththefitofasimilarmodelbyQianetal. [8] is notpossible,sincethevariableschoseninthisstudyaredifferentfrom theirs.Theresults inTable 2makeitpossible toconcentrateonthe mostsignificanteffectsonthevortexlength,thiswillbeexploredlater inthispaper.

5.3. EquationofAlexander,Bryantetal.,andZhonglietal.

Thesethreemodelshavethesamefunctionalformanddependon thesamedimensionlessindependentvariables[1,2,3]:

𝐿_𝑛 𝐷 =𝑝1

(𝐷_𝑥 𝐷

)𝑝2( 𝐷² 𝑎𝑏

)𝑝3

(11) Table 2

ResultsoffittingthesecondordermodeltothesimulationresultsusingSPSS,thedegreesforfreedom forthe𝑡-valueis22.

Variable Unstandardized

Coefficients

Std. error Standardized Coefficients

𝑡 Significance

Level,p-value

Constant 12.05 20.37 .584 .565

(𝐷_𝑥∕𝐷)

59.55 31.82 4.998 1.872 .075

(𝐷_𝑥∕𝐷)2

-68.06 38.75 -4.577 -1.756 .093

((𝐻−𝐻_𝑐)∕𝐷)

4.693 1.748 3.938 2.685 .014

((𝐻−𝐻_𝑐)∕𝐷)2

-.468 .387 -1.582 -1.208 .240

(𝐻_𝑐∕𝐷)

1.905 1.318 2.398 1.445 .162

(𝐻_𝑐∕𝐷)2

-.252 .172 -2.236 -1.466 .157

(√

𝑎𝑏∕𝐷) -32.373 10.690 -4.765 -3.028 .006 (√

𝑎𝑏∕𝐷)² 36.61 12.93 4.091 2.832 .010

ln(Re) -3.747 3.638 -7.59 -1.030 .314

ln(Re)² .159 .144 8.090 1.104 .281

(𝐷_𝑥∕𝐷) (

(𝐻−𝐻_𝑐)∕𝐷)

-.095 1.073 -.037 -.089 .930

(𝐷_𝑥∕𝐷) ( 𝐻_𝑐∕𝐷)

.055 .716 .033 .077 .939

(𝐷_𝑥∕𝐷) (√

𝑎𝑏∕𝐷) -6.733 6.130 -.459 -1.098 .284 (𝐷_𝑥∕𝐷)

ln(Re) -.221 .460 -.298 -.479 .636

((𝐻−𝐻_𝑐)∕𝐷) ( 𝐻_𝑐∕𝐷)

-.049 .072 -.201 -.691 .496

((𝐻−𝐻_𝑐)∕𝐷) (√

𝑎𝑏∕𝐷) -2.563 .613 -1.161 -4.182 .000 ((𝐻−𝐻_𝑐)∕𝐷)

ln(Re) -.102 .046 -1.176 -2.211 .038

(𝐻_𝑐∕𝐷) (√

𝑎𝑏∕𝐷) .505 .409 .368 1.236 .230

(𝐻_𝑐∕𝐷)

ln(Re) -.025 .031 -.444 -.818 .422

(√

𝑎𝑏∕𝐷) ln(Re) .543 .263 1.104 2.066 .051

Fig. 6.Parity plotforthefittedequationoftheformofAlexander/Zhongliet al./Bryantetal.,Equation(12).

Whetheramodelofthisformcanpredictthelengthofthevortexin cylinder-on-conecyclonescanbeexploredbyfittingtheparametersto thesimulationresults.Theresultofdoingthisis:

𝐿_𝑛 𝐷 = 1.53

(𝐷_𝑥 𝐷

)0.0338( 𝐷² 𝑎𝑏

)0.260

(12) ThePearson’srforthisfitis0.5942,theresidualsumofsquares is31.16andtheparityplotisshowninFig.6.Theparityplothasthe clearcharacteristicsofafitwhereintheeffectsofsomeoftheimportant governingvariableshavenotbeentakenintoaccount,anditcanbecon- cludedthatthisfunctionalformisnotsufficienttopredictthelengthof thevortexincylinder-on-conecyclones.Theeffectsofthephysicalvari- ablesontheRHSagreequalitativelywiththerelationofAlexander.As mentioned,however,thisrelationhaslimitedvaluefortheprediction ofthenaturalvortexlengthincylinder-on-conecyclonesinpractice.

Thequestioncouldbeposedifitispossibletomodifythisstandard expressiontotakeintoaccounttheeffectofothervariables.Toexplore this furthera modelequation wherein theeffects of othervariables areaccountedforinanadditiveproduct-of-powersterm,implicitlyas- sumingthattheireffectscanbesuperposedontheeffectsof thetwo parametersfeaturinginthestandardexpression,wasformulatedand fittedtothesimulationresults.Thisequationisoftheform:

𝐿_𝑛 𝐷 =𝑝1

(𝐷_𝑥 𝐷

)𝑝2( 𝐷²

𝑎𝑏 )𝑝3

+𝑝4Π⁵_𝑖=3𝑥^𝑝_𝑖^𝑖+2 (13) featuringanadditiveproduct-of-powerstermtotakeintoaccountthe effects of theothervariables consideredin this study. Theresultof fittingEquation(13) tothesimulationresultsis:

𝐿_𝑛 𝐷 = 0.195

(𝐷_𝑥 𝐷

)0.182( 𝐷²

𝑎𝑏 )0.737

+ 0.743

(𝐻−𝐻_𝑐 𝐷

)0.380

× (𝐻_𝑐

𝐷 )−0.122

((ln(Re))^0.0548

(14)

resultinginafitwithaPearsonsrof0.697,aresidualsumofsquares of24.8andtheparityplotshowninFig.7.

Thisisclearlyanimprovementcomparedwiththeoriginalformof theequationofAlexander,Zhonglietal.andBryantetal.,aswouldbe expected.

Fig. 7.Parity plotforthefittedequationoftheformofEquation(13),anequa- tiontoimproveAlexandersequationtotakeintoaccountadditionalparameters incylinder-on-conecyclones.

5.4. Otherfunctionalforms

Thesimulationresultsallowustotryoutdifferentfunctionalforms inaquesttofindarelativelysimplepredictiverelationforthenatural vortexlengthincylinder-on-conecyclones.

5.4.1. Product-of-powers

A simple product-of-powers expression is often used in multi- dimensionalregression.Thefunctionalformis:

𝐿𝑛

𝐷 =𝑝1Π⁵_𝑖=1𝑥^𝑝_𝑖^𝑖+1 (15)

withatotalof6adjustableparameters.Thisformshouldofferquitea lotofflexibilityatthesametimeasnotinvolvingtoomanyadjustable parameters,infactonlysixforthepresentcase.Theresultoffittingthis relationtothesimulationresultswas:

𝐿_𝑛 𝐷 = 1.28

(𝐷_𝑥 𝐷

)0.0256(𝐻−𝐻_𝑐 𝐷

)0.309(𝐻_𝑐 𝐷

)−0.0948(√

𝑎𝑏 𝐷

)−0.565

ln(Re)^0.0225 (16) Pearson’srforthisfitis0.7371,theresidualsumofsquaresis:22.03, andtheparityplotisshowninFig.8A).Thisfitisbetterthantheother expressionsdiscussedhithertointhispaper,exceptforthesecond-order models.

5.4.2. Identifyingthemostimportantvariablesandinteractions

StudyingtheresultsofthestatisticalanalysisinTable2andidentify- ingthemostimportantvariablesinfluencingthelengthofthevortexthe followingvariablesandinteractionswereidentified,basedonthesig- nificancelevelsoftheireffects:(𝐷_𝑥∕𝐷),(𝐻−𝐻_𝑐)∕𝐷,√

𝑎𝑏∕𝐷,(𝐷_𝑥∕𝐷)², ((𝐻−𝐻𝑐)∕𝐷)²,(√

𝑎𝑏∕𝐷)²and((𝐻−𝐻𝑐)∕𝐷)(√

𝑎𝑏∕𝐷).Amodelequation involvingonlytheseeffectsandinteractionswasfittedtothesimulation results.Theresultwas:

𝐿𝑛

𝐷 = −11.5 + 82.3(𝐷𝑥

𝐷 )

+ 1.31(𝐻−𝐻𝑐

𝐷 )

− 16.3 (√𝑎𝑏

𝐷 )

− 99.9(𝐷𝑥

𝐷 )2

+26.7 (√𝑎𝑏

𝐷 )2

− 6.71 (𝐷_𝑥

𝐷 ) (√𝑎𝑏

𝐷 )

− 2.56 (𝐻−𝐻_𝑐

𝐷

) (√𝑎𝑏 𝐷

)

(17)

Fig. 8.Parity plotsforthefittedequationsoftheformsA):product-of-powers andB):selectedparametersandinteractions.

ThePearson’srforthisfitwas0.814andtheresidualsumofsquares was16.3.TheparityplotisshowninFig.8B).Althoughnotappearing thatgoodinthefigure,thisistheclosestfittothesimulationresults exceptforthatofthesecond-ordermodels.

6. Furtherdiscussion

Itisawell-establishedprocedureto“spottheresultthatisnotinthe familyofresults”andeliminatethisresultbeforetheregressionanalysis iscarriedout,basedontheargumentthatthiseliminatesanoddresult thatmightwellbewrongforavarietyofreasonsandthereforeshould notinfluencetheregressionparameters.Inthepresentcasethiswould be theresultof thevery last simulationin Table1, whichwas also chronologicallythelastsimulationtobecarriedout.Ifthisresultwas disregarded,thePearson’srvalueswouldbesignificantlyhigher,and theresidualsumofsquaresconsiderablylowerthanthevaluesquoted inthispaper,forexample,forthesecond-ordermodelinEquation(10) Pearson’srwouldbe0.959andtheresidualsumofsquareswouldbe

3.46.Thechoicehasbeenmadenottodothisinthispaper,butitmay bekeptinmindthatthefitswouldbebetterifthiswasdone.

7. Conclusions

Thisworkhasclearlydemonstratedthattheclassicalformofmodels topredictthelengthofthevortexincyclones,namelyEquation(11), isinsufficientforcylinder-on-conecyclones,thetypeofcyclonealmost universallyusedthesedays,andthatothervariables,bothgeometrical andoperational,needtobetakenintoaccountforthistypeofcyclone.

Thesecond-order model,Equation (10) gave avery goodfitand shouldpredictthedimensionlessvortexlengthincylinder-on-conecy- clonesquitewell,butitisunwieldy,involving19adjustableparame- ters.Noneofthesimplerfunctionalformstestedinthisworkgaveas goodafittothesimulationresults.Thebestofthesimplifiedmodels testedhasbeenshown,bythePearson’sr-valuesgiven,tobeEquation (17),involvingonlythevariablesthatwerefoundtobethemostsignif- icantindeterminingthelengthofthevortex.Thisequation,although notasgoodasthesecond-order-modelinEquation(10),maybeused inpracticeasaguideforthedesignofcylinder-on-conecyclones.

Declarations

Authorcontributionstatement

A.Hoffmann: Conceivedanddesignedtheexperiments;Analyzed andinterpretedthedata;Wrotethepaper.

E.A.Svensen:Performedtheexperiments;Analyzedandinterpreted thedata;Wrotethepaper.

Fundingstatement

Thisresearchdidnotreceiveanyspecificgrantfromfundingagen- ciesinthepublic,commercial,ornot-for-profitsectors.

Competingintereststatement

Theauthorsdeclarenoconflictofinterest.

Additionalinformation

Noadditionalinformationisavailableforthispaper.

Acknowledgements

TheauthorswishtoextendtheirgratitudetoDrGlebPisarevand TorillRødlandforhelpwithexperimentalworkinsupportofthesimu- lations.

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