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热镀锌Jet Wiping in Hot-Dip Galvanization JFE 2007

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JetWipinginHot-DipGalvanization

e-mail:gosset@vki.ac.be

AnneGosset1

Jean-MarieBuchlin

e-mail:buchlin@vki.ac.be

vonKarmanInstituteforFluidDynamics(VKI),

ChausséedeWaterloo72,10Rhode-St-Genèse,Belgium

Thispaperpresentsananalysisofthegas-jetwipingprocessinhot-dipgalvanization.Thistechniqueconsistsofreducingtheliquidﬁlmthicknessonamovingsubstratebyapplyinggasslotjets.Atheoreticaldevelopmentallowsthecomputationoftheﬁlmthicknessevolutioninthewipingzone.Itisfurthersimpliﬁedtoanengineeringmodelwhichpredictsdirectlytheﬁnalcoatingthickness,ingoodagreementwithwipingexperi-ments.Thelimitofapplicabilityofjetwipingisduetotheoccurenceofaviolentﬁlminstability,calledsplashing,whichtakestheformofaliquiddropletemissionjustup-streamthenozzle.Anexperimentalinvestigationofthisphenomenonisconductedonawater-modelfacility.Twonozzledesignsaretested.Theeffectofprocessparameterssuchasthestripspeed,thenozzlepressure,thestandoffdistance,andthetiltangleofthenozzleonsplashingisemphasized.Adimensionlesscorrelationisestablishedtopredicttheoperatingconditionsleadingtosplashingoccurence.Itissuccessfullyconfrontedtoobservationsmadeongalvanizationlines.͓DOI:10.1115/1.2436585͔Keywords:coatingﬂows,thinﬁlms,gas-jetwiping,dropletformation,spray

1Introduction

Thedepositionofaverythinliquidﬁlmonasolidsurfaceisthebasisofnumerouscoatingtechniquesusedinindustrialpro-cessessuchaspaperandphotographicﬁlmmanufacturing,wirecoating,andintheironandsteelindustries.Inthecaseofhot-dipgalvanization,movingsteelstripsarecoatedwithathinlayerofzincinordertoresisttooxidization.Theprotectivemetalisap-pliedinitsliquidstate,bydippingthesubstrateintoabathofcoating.However,atcommonstripspeeds,theliquidlayerdraggedbythemovingwebisfartoothickandunevenfortheapplicationsconsidered.Theﬁnalcoatingthicknessisthereforecontrolledbytheapplicationoftwo-dimensionalhigh-speedgasjetsimpingingontheliquidlayer.Theyleadtotheformationofarunbackﬂowdowntothebath,leavingauniformandconstantcoatingthicknessdowsntreamonthesubstrate.Thisprocessisreferredtoasthejetwipingtechnique,orair-knifecoating,theprincipleofwhichisillustratedbytheschematicshowninFig.1.Theﬁlmthicknessafterwiping,hf,dependsonthesubstrateve-locityU,thenozzlepressurePn,thenozzletosubstratestandoffdistanceZ,thenozzleslotwidthd,andthenozzletiltangle␣,aswellastheliquidproperties.

Asthestripspeedincreases,thewipingtechniqueishoweverlimitedbytheoccurenceofaratherviolentﬁlminstabilitycalledsplashing,whichoccursupstreamthejetnozzle.Typicalvisual-izationofincipientsplashingonagalvanizationlineisshowninFig.2͑a͒,whileFigs.2͑b͒and2͑c͒demonstratethatthesplashingphenomenonisqualitativelywellreproducedonawatermodelfacility.Thismechanismischaracterizedbytheejectionofliquiddropletsfromtherunbackﬁlmﬂow,followedbyitscompleteexplosioninfullydevelopedstate.Splashingaffectstheﬁnalcoat-ingqualitybecausetherunbackﬂowseparatesfromthesubstratewhenitdevelops.Suchabehaviordegradessigniﬁcantlythewip-ingconditions͑pressuregradientandshearstressatjetimpinge-ment͒,andleadstounstableconditionsandtotheformationofzincdroplets.Theresultisaverypoorwipingefﬁciency,i.e.,theﬁlmthicknessislessefﬁcientlyreducedforgivenoperatingcon-ditions,andapoorcoatingqualitydownstream.Moreover,theejecteddropletsmayreachthenozzleslit,andaftersolidiﬁcation,

Correspondingauthor.

ContributedbytheFluidsEngineeringDivisionofASMEforpublicationintheJOURNALOFFLUIDSENGINEERING.ManuscriptreceivedApril24,2006;ﬁnalmanu-scriptreceivedDecember27,2006.Assoc.EditorTheodoreHeindel.

blockit.Inthecaseofgalvanization,thereisalsoasafetyconcernbecausethemoltenzincdropletsat460°Cmayreachtheworkersinchargeofremovingregularlythelayerofoxidizedzincatthefreesurfaceofthebath.

Itisthereforeofinteresttohaveatdisposalapredictivemodelforsplashingoccurrenceongalvanizationlines.Meanwhile,ama-jorpracticalconcernisthemaintenanceofhighproductionratesofconstantcoatingthicknessproducts.Therefore,thepredictionofthewipingefﬁciencyisnecessarytodeﬁnecompletelyanop-timumprocesswindow.

Thispaperdescribesasimpleengineeringmodelfortheﬁnalcoatingthicknessafterwiping.Itisvalidatedwithexperimentaldatainaprocesswindowinwhichsurfacetensioncanbene-glected.Aphenomenologicalapproachforthepredictionofsplashingoccurrenceisthenproposed.Adatabasegatheringsplashingconditionsforawiderangeofparametersonawatertestfacilityisusedforthederivationofadimensionlessempiricalmodelforsplashing.Theanalysisemphasizestheeffectofnozzletilting.Optimumprocesswindowscanthusbedeﬁnedfromthecombinationofthewipingandsplashingmodels.Finally,theen-gineeringtoolsareappliedtotypicalgalvanizationconditions.

2JetWiping

Ananalyticalmodelisdevelopedforthepredictionoftheﬁnalﬁlmthickness.Theeffectofnozzletiltingisincludedinthismodel.Itisvalidatedwithexperimentaldata.

2.1AnalyticalModel.Theﬁlminterfaceshapeinjetwipingisduetothepressuregradientandshearstressdistributionspro-ducedbytheimpingingjet͓1–3͔.TypicalproﬁlesofthewipingactuatorstogetherwiththeinterfaceshapeareshowninFig.3.Variousmodelsfortheﬁlmthicknessproﬁleinthewipingregionareproposedinliterature.ThepioneerstudyinthisﬁeldistheonebyThorntonandGraff͓1͔,whoassumethattheinterfacedefor-mationisdueonlytothepressuregradientcreatedbytheimping-ingjetontheﬁlm.Itisconsideredasabodyforcesupplementinggravity.Tuck͓2͔adoptsasimilarapproach,andchecksthestabil-ityofthesolutionsforlongwavelengthperturbations.EllenandTu͓3͔proposeamodelwhichfortheﬁrsttimetakestheshearstressintoaccount.Theyshowthatitparticipatesfor20–40%tothewipingaction.Tucketal.͓4͔quantifytheinhibitingeffectofsurfacetensiononjetwiping,whereasitisneglectedinpreviousstudies.Followingthis,Yoneda͓5,6͔presentsacompletenumeri-TransactionsoftheASME

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Fig.3Typicalﬁlmshapeinjetwipingwithpressuregradientandshearstressproﬁlesduetothejet

Fig.1Schematicofthejetwipingoperationappliedtothegalvanizationprocess

calsolutionfortheﬁlminterface,includingbothsurfacetensioneffectsandshearstress.Moreover,hepredictsthetransitionfrom“weak”air-knifeaction͑norunbackﬂow͒to“strong”action͑ef-fectivejetwiping͒.Finally,theoriginalityofthedevelopmentbyBuchlin͓7͔liesinthefactthatheproposesdifferentlevelsofsolutionofthemodel,accordingtotheassumptionsmade.For

instance,whensurfacetensionistakenintoaccount,hepresentsanumericalsolutionforthethicknessdistributioninthewipingregion.Whensurfacetensionisrelaxed,heshowsthataone-dimensionalanalyticalsolutionexists.Assumingﬁnallythatthemainwipingactuators͑ﬁrstmaximumpressuregradientandshearstressmetbythedraggedﬁlm͒actatthesamelocation,hepro-posestheideaofazero-dimensionalmodelpredictingdirectlytheﬁnalﬁlmthickness.

Thetheoreticaldescriptionofgas-jetwipingisusuallybasedonthelubricationapproach,whichassumesnegligibleinertiawithrespecttoviscous,gravityandpressureterms.If͑Ox͒istheaxisalongthesubstrate,and͑Oy͒theoneacrossthesubstrate,itturnsoutthatinjetwiping,variationsalongthexdirectionareveryslowcomparedtovariationsalongyandtotheﬁlmthickness.Thescalingfactor⑀=2␲h/␭Ӷ1canthusbeintroduced,wherehisthemeanﬁlmthickness,and␭isthecharacteristicwavelengthofaxﬂuctuation.Makingthechangeofvariablesx*=␧xandy*=y,theinertiatermsintheNavier–Stokesequationsarefoundtobeoforder⑀RefӶ␧.SincetheﬁlmReynoldsnumberRefbasedontheﬁlmvelocityinthesubstratereferentialandonthemeanthicknessisverysmall,inertiacanbeneglected,andtheﬂowcanbeas-sumedparallel.Theresulting͑Ox͒momentumequationoftheﬁlmstatesthattheshearstressbalancestheweightandpressure

ץ2udPl␮l2=␳lg+ץydxq=

͑1͒

͵h͑x͒

u͑x,y͒dy͑2͒

whereu͑x,y͒isthelocalﬁlmvelocity;h͑x͒isthelocalﬁlmthick-ness;␮land␳lare,respectively,theliquiddynamicviscosityand

density;gistheaccelerationofgravity;Plisthepressureintheliquid;andqistheﬁlmvolumetricﬂowrate.TheﬁlmisthinenoughtoassumethattheliquidpressurePlisequaltothegas-jetpressureatimpingementPgcorrectedbyasurfacetensiontermduetothemeniscusformation

dhxxdPldPg

=−␴l

3/2

dx͑1+h2dxdxx͒

ͫͬ͑3͒

Fig.2„a…Splashingphenomenononagalvanizationline;„b…

fullydevelopedsplashingwithwater;therunbackﬂowiscom-pletelyseparatedalongthestripwidth;and„c…sideviewvisu-alizationofsplashingwithwater

where␴listheliquidsurfacetension;andsubscriptxreferstoderivationwithrespecttox.Subscriptlreferstotheliquidphase,andsubscriptgtothegasphase.Thetermcontaining␴lrepresentsthecontributionofsurfacetensiontothepressuregradientjumpthroughthegas–liquidinterface.Themeanﬁlmcurvaturehxx/͑1

3/2+h2canbelinearizedtogethxxxonthegroundsthatthefreex͒

surfaceslopeissmall.TheboundaryconditionstobeassociatedwithEq.͑1͒expresstheno-slipconditionoftheﬁlmonthesub-strate,andthecontinuityofshearstressattheinterface

u=U

y=0

͑4͒

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␮l

ץu

=␶͑x͒ץy

aty=h͑x͒͑5͒

where␶istheshearstressduetotheimpingingjetontheﬁlmfreesurface.Equation͑1͒togetherwithEqs.͑4͒and͑5͒canbereadilyintegratedtogivetheﬁlmvelocityproﬁleu͑x,y͒fromwhichtheﬂowrateqisdeterminedthroughEq.͑2͒.Theresultingﬂowrateequationhasthefollowingnondimensionalform

⌫

ˆˆˆ2ˆ͑X͒d3hˆ͑X͒+2Q−3h−1.5Th=1+ٌPg

dX3ˆ3h

͑6͒

ˆThenormalizedvariablesaredeﬁnedasfollows:X=x/d,h

=h/h0withh0=ͱ␮lU/␳lg;⌫=␴lh0/␳lgd3;Q=q/q0withq0

ˆ=␶/␶with␶=ͱ␮U␳g.Sub-ˆ=ٌP/␳g;andT=2/3Uh0;ٌPggl00ll

script0referstothedraggedﬁlmﬂowwithoutwiping.⌫isthedimensionlessparameterwhichmeasurestheinﬂuenceofsurfacetension͓4͔.ThoughthepressurePistheonlyfunctionofx,wechoosedeliberatelytouseٌPg͑x͒insteadofdPg/dxforconve-nienceinwriting.

Sincethepressuregradientandshearstressdistributionsduetothejetareusuallyassumedtobeknown,Eq.͑6͒hasbasicallytwo

ˆandﬂowrateQ.unknowns:thedimensionlessﬁlmthicknessh

Becauseitwasshownthatthepresenceoftheliquidﬁlmalmostdoesnotaffectthejetproﬁles͓8͔,thepressuregradientandshearstressproﬁlesaregivenbyexperimentalornumericalsimulationofanimpingingjetonaﬂatplate͓9,10͔.Assuminganegligibleeffectofsurfacetension,whichwillbejustiﬁedlater,Eq.͑6͒canbefurthersimpliﬁedto

ˆ͒h−1.5Tˆh−3hˆ+2Q=0͑1+ٌP

ˆ2

ˆ3

͑7͒

Fig.4Comparisonofthedimensionlessvolumetricﬁlmﬂow

ratepredictedbytheknifemodelandvariousmodelsfromlit-eratureasafunctionofthedimensionlessjetpressuregradientforCa=0.01

C␶=−10−5Rej+0.129

͑13͒

wherePisnowtheliquidpressure,whichisequaltothegaspressure.ThesolutionofEq.͑7͒isobtainedbysolvinglocallythecubicequation,usingthepressuregradientandshearstressdistri-ˆ͑X͒andTˆ͑X͒ofanimpingingjetonaﬂatsurface.ItisbutionsٌP

generallyagreedthatthepressuredistributionatthewalliswelldescribedbyaGaussianlaw͓9,10͔

P͑␰͒=Pse−0.693␰2

Otherwise,C␶=0.067.

Inthewall-jetregion,theshearstressdistributionisbetterap-proximatedby͓9͔

␶͑␰͒=0.26

dPnRe0.2j

ͩͪͩͪ1

͉␰͉+4

1bP

͑14͒

͑8͒

wherePsisthemaximumpressureatthestagnationpoint;and␰=x/bP,wherebPisthedistancebetweenthejetaxisandthelocationofPs/2.Thepressuregradientproﬁleisthereforegivenbythefollowingexpression

ٌP͑␰͒=−1.386

Ps−0.693␰2−1.386␰e=␰P͑␰͒bPbP

͑9͒

PsandbParebothfunctionsofZandd.Whenthepotentialcore

impingesontotheplate͑Z/dഛ5͒,thestagnationpressureremainsconstantandequaltothejetpressurePn.ForZ/dϾ5,Psde-creaseswhenZ/dincreases

dPs

=Cp

ZPn

͑10͒

Insteadyconditions,continuityimpliesthattheliquidﬂowrate

Qisconstantthroughoutthewipingregion.Ontheotherhand,theﬁlmthicknesshvariesalongxwiththeٌP͑x͒and␶͑x͒distribu-tions.ThecubicEq.͑7͒,whichhastobesolvedforallxvalues,hastwo,ornopositiveroots.ThereexistsonlyonevalueofQforwhichtheﬁlmthicknessevolutionisphysicallyacceptable.Itcor-respondstotheconditionwhenthederivativeoftheﬁlmﬂowrate

ˆ=0͒,thatistosaywithrespecttotheﬁlmthicknessisnull͑dQ/dh

whentheﬂowrateisoptimum.Azero-dimensionalmodel͑here-afterreferredtoastheknifemodel͒canbederivedfromEq.͑7͒,relyingontheobservationthatthepositionoftheupstreammaxi-mumpressuregradientandshearstresscorrespondsapproxi-matelytothelocationwheretheﬁlmsurfacevelocityisequaltozero͑calledlocationofﬁlmsurfacebifurcation͓͒10͔.Therefore,weconsideronlythispositionwherethenetﬂowrateis“opti-mum”andnameitXopt.Equation͑7͒iswrittenatthatpositionto

ˆgiveEq.͑15͒,andthelocalﬁlmthicknessisnamedhopt

ˆ͒hˆ3ˆˆ2ˆ͑1+ٌPmaxopt−1.5Tmaxhopt−3hopt+2Q=0

͑15͒

Then,inordertosolveEq.͑15͒,asecondequationisfoundby

ˆ=0.usingthefactthatthenetﬂowrateisoptimum,i.e.,dQ/dh

ˆcanthusbefoundthroughEq.͑16͒Thevalueofhopt

ˆˆ2ˆTmax+Tmax+4͑1+ٌPmax͒ˆhopt=

ˆ͒2͑1+ٌPmax

withCpӍ6.5.

Intheimpingementregion,theshearstressproﬁleﬁtscloselythedistributiongivenby͓9͔

␶͑␰͒=␶max͓erf͑0.833␰͒−0.2␰e−0.693␰͔

͑11͒

whereerfistheerrorfunction.Foradevelopedimpingingjet͑Z/dϾ8͒,themaximumshearstressisgivenby͓9,10͔

͑16͒

␶max=C␶͑Rej͒

PnZ/d

͑12͒

whereC␶isavaluedependingonthejetReynoldsnumberRejbasedonthenozzleslotwidthdandthejetexitvelocityV0j,whenitislowerthanabout6000468/Vol.129,APRIL2007

ThisvalueisreplacedinEq.͑15͒,andQisfound.Fardown-streamthewipingregion,theﬁlmthicknessbecomesconstantandequaltoitsﬁnalvaluehf.UsingthevalueofQandEq.͑2͒,hfiscomputed͑thevelocityproﬁleacrosstheﬁlmisalmostuniformandequaltothesubstratevelocityUatthatpositions͒.

ThedifferentaforementionedmodelsarecomparedinFig.4,

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wheretheevolutionofthedimensionlessﬁlmﬂowrateQisplot-ˆtedasafunctionofthedimensionlesspressuregradientٌPmaxfor

acapillarynumberCa=␮lU/␴lequalto0.01.Ifpurewaterisconsideredastheworkingﬂuid,itcorrespondstoasubstrateve-ˆˆlocityU=0.67m/s.ThevaluesofٌPmaxandTmaxusedforthe

computationsinFig.4arebasedontheexperimentalcorrelationsEqs.͑9͒–͑14͒fromBeltaosetal.͓9͔.Thecomparisonofthemod-elsofThorntonetal.͓1͔andTuck͓2͔revealsthatthesurfacetensioneffectbecomesnegligiblebeyondacertainvalueofthepressuregradient;thejetactionthenovercomesthecapillaryforceinthewipingmeniscus.ThesameconclusionisreachedwhencomparingtheresultsofYoneda͓5,6͔withtheonesoftheknifemodel,whichbothtakeshearstressintoaccount,incontrasttothepreviouslycitedmodels.Asnotedpreviously,themodelproposedbyYoneda͓5,6͔predictsthejetthresholdbeyondwhichtheﬁlmthicknessisreduced.ItappearsasthestraightlineportioninFig.4.

ItcanbethusinferredthattheknifemodelisperfectlyreliableforagivenCawhenthejetactionis“strong”͑comparedtocap-illarityinthemeniscus͒.Itshouldbenotedthattheﬁnalﬁlmthicknesshfwhichisobtainedveryquicklywiththezero-dimensionalknifemodeldiffersbygenerallylessthan1%fromthevaluefoundwhencomputingthewholethicknessdistributionwiththeone-dimensional͑1D͒modelEq.͑7͒.Acompletethick-nessproﬁlethusobtainedwasvalidatedwithnumericaldatafromvolumeofﬂuid-largeEddysimulation͑VOF-LES͒numericalsimulationsbyLacanetteetal.͓8͔forZ/d=8andajetpressurePn=1450Pa.Thepressuregradientandshearstressdistributionsusedforthe1Dmodelpredictionshavebeenmeasuredexperi-mentally͓8͔,andtheyﬁtcloselytheempiricalcorrelationsEqs.͑9͒–͑10͒and͑11͒–͑14͒fromtheliterature͓9͔.Theﬁlminterfaceshapesobtainedanalyticallyandwithtwo-phasesimulationsareingoodagreementbothinthemeniscusregionanddownstreamthejet,provingthatsurfacetensioncanbeneglectedinsuchcon-ditions.Theﬁlmthicknessafterwipinghfisunderestimatedofabout15%bythewipingmodelwithrespecttotheinterfaceob-tainednumerically.Upstreamtheimpingingzone,therunbackﬂowfromthe1Dmodelissigniﬁcantlythickerthanthesimulatedinterface.Thereasonforthatmaybethehighaspectratioofthemeshcellsinthatregion,inducingapoorspatialresolutionfortheinterfacedetectioninVOF-LESsimulations,orthefactthatthelubricationassumptionmadeintheanalyticalmodelisnotbevalidintherunbackﬂowregion.

2.2EffectofNozzleTilting.Whenthenozzleistilted,theimpingementangleofthejetinvolvesanasymmetryofthepres-sureandshearstressproﬁles,andthusamodiﬁcationofthewip-ingactuators.Forthepurposeofsplashingdelaystudy,wewillconsideronlydownwardnozzletilting.Inthatcase,itisexpectedthattheusptreammaximumpressuregradientisreducedduetotheﬂatteningofthepressuredistribution.ExpressionsforٌPmaxand␶maxcanbederivedfromtheempiricalrelationsprovidedbyBeltaosforobliqueimpingingjets͓11͔.Itfollowsthat

ٌPmax=

5.712d

Pncos␣ZЈbP

͑17͒

Fig.5Schematicofthewaterjetwipingfacility

remainslowerthan30deg͓11͔.Beltaos͓11͔proposesavalueof

0.06fordevelopedimpingingjets͑Z/dϾ8͒.Themaximumshearstressforatiltedjetthereforevariesincos␣.

ThepressuregradientbeingthemainactuatorfortheZ/dval-uesconsideredinthepresentstudy,itmaybeexpectedthattheoverallwipingefﬁciencyislowerwhenthejetistilteddownward.2.3ValidationWithExperimentalData.Theknifemodelhastheadvantageofbeingverysimpleandfasttoimplement.Itsuseforthepresentinvestigationandinindustrialconditionsre-quiresitsvalidationwithexperimentaldata,inverywellcon-trolledconditions.

2.3.1ExperimentalSetupandTechniques.Therefore,wipingexperimentsareperformedonadedicatedwatermodelfacility,whichsimulatesagalvanizationline.Figure5showsaschematicofthesetup,whichincludesaverticalrubberstrip5mlongand0.5mwide,stretchedbetweentworolls.Thestripissetintomo-tionbytheupperroll,whichisentrainedbyanelectricmotor.Thestripvelocitycanbeadjustedintherange0.5–4.5m/s,anditismeasuredusingatachometer.Thesubstratedipsinabathofwa-ter,towhichasurfactantwasaddedtoensureagoodwettability.Thewatersurfacetensionthusdropsto0.03N/m±0.001N/m.Normally,surfactantmaterialstendtoremainatthefreesurfaceofaliquid,andsurfacetensiongradientsmayoccur.However,thecapillarynumbersremainsufﬁcientlylowheresothatsuchaneffectisnotfelt.Moreover,thewipingprocessensuresacontinu-ousmixingoftheworkingﬂuidandsurfactant.Theslotnozzleispositionedat0.8mabovethefreesurfaceofthebath.Itstiltangleisadjustable,andthestandoffdistanceZbetweenthenozzleandthestripistunedthankstoshimswithin±0.1mm.Twonozzlegeometriesaretested.TheyaresketchedinFig.6,togetherwiththeirmaingeometricalcharacteristics.NozzleT1isfullysymmet-ric,whileT2exhibitsabottomclearingandamoreconﬁnedge-ometry.Theyareboth0.6mwidetoavoidedgeeffects.The

ZЈisthenozzletowalldistancealongthejetaxis,suchthatZ͑␣=0deg͒=ZЈcos␣.ThequantitybP/Zisgivenbyanempiri-calrelation͓11͔,andisalmostinsensitiveto␣foranglessmallerthan30deg.Themaximumpressuregradienthasthereforeanevolutionincos2␣.

Theshearstress␶maxcanbeapproximatedbythefollowingexpression͓11͔

␶max=C*͑␣͒

PndZЈ

͑18͒

Fig.6Nozzledesigns

whereC*isacoefﬁcientwhichdependsonthenozzlegeometry,andisshowntobealmostinsensitivetothejetangle␣whenitJournalofFluidsEngineering

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nozzleisfedwithcompressedairupto10kPa.Theslotwidthdcanbeadjustedbetween0.7mmand2.1mmwithin±0.005mm.Inalltheexperimentspresentedhere,d=1.4mm.Forsakeofsimplicity,thewipingmechanismisstudiedonlyononeside.Toensureagoodstabilityofthestripintheimpactzone,therearfaceisslidingonanaluminumplatelubricatedbytheentrainedwater.Themeanliquidﬁlmthicknessinspaceandtimeisdeterminedfromthemeasuredmassﬂowrateafterwipingq

hf=

q␳lLU

͑19͒

whereListhestripwidth.Theassumptionofauniformvelocityproﬁleacrosstheliquidﬁlmintroducesanerroroflessthan0.1%onhf,withrespecttothevaluecomputedwiththerealﬁlmve-locityproﬁlecomputedfromEq.͑1͒.Theliquidﬁlmiswithdrawnatthetopofthebandbytheactionofarubberscrapercombinedtoasuckingapparatusbasedonairejectors.Twolateralsmalljetsdrawtheliquidﬁlmtowardssuctionports,whichareconnectedtoacycloneseparator.Abalancemeasurestheamountofcollectedwater,andastopwatchtheweightingtime.Dependingontheﬂowrate,themeasuringtimerangesbetween1and6min.Eachex-perimentisrepeatedﬁvetimes.Theuncertaintyonthemeanvalue

ofameasurandXisgivenbyU=ͱB2+͑tS¯X͒,whereBisthebias

¯error,S¯XistheprecisionindexofthemeanX,andtistheStudent

valuewhichisafunctionofthedegreesoffreedomusedincal-culatingS¯X.Forthepresentmeasurements,tisequalto2.776foraconﬁdenceintervalof95%.Theﬂowratestandarddeviationoverrepeatedmeasurementsisanincreasingfunctionofqf͑higherﬂowratesaremoredifﬁculttomeasure͒anditisoftheorderof±0.5%,whichgivesaprecisionindexof±1.4%with95%conﬁdence.Thebiaserrorontheﬁlmﬂowrateisestimatedto±5.10−5kg/s,fromwhichweﬁnallygetqf±5.2%fortheoveralluncertainty.Asforthestripvelocity,itsuncertaintyis±0.6%at95%conﬁdencelevel,innominalwipingconditions.Thepreci-sionindexonhfistherefore±0.5%andthebiaserrorabout±5%,whichgivesaﬁnaluncertaintyof±5.2%at95%conﬁdence.Theuncertaintyonthenormalizedﬁlmthicknesshf/h0andﬁlmRey-noldsnumberReisalsoabout±5.2%.Theuncertaintyonthejetpressureisabout±6Pa,whiletheoneonZ/disestimatedto±4%.

2.3.2Validation.TheimportanceofcapillaryeffectsinthepredictionoftheﬁnalﬁlmthicknessisillustratedinFig.7,forawaterﬁlmﬂow.ThestandoffdistancebetweenthenozzleandthesubstrateisZ/d=10,andthestripvelocityis1.5m/s,which

ˆ,givesacapillarynumberof0.055.AtthelowestvaluesofٌPmax

theknifemodelunderestimatestheﬁlmthickness͑normalizedbytheﬁlmthicknesswithoutwipingh0͒ofalmost35%withrespecttotheexperimentalvalues,whilethecompletemodelofYoneda͓5,6͔succeedsinprovidingreasonablevalues͑overestimationof9%atmost͒.Soitcanbeassertedwithsomeconﬁdencethatthediscrepancybetweentheknifemodelandthemeasurementsismainlyduetotheomissionofsurfacetension.Asthejetpressuregradientincreases,theknifemodelpredictionbecomesmoreac-curate,untilitalmostmatchesthemeasurementsstartingfromˆٌPmax=40approximately.Itshouldbepointedoutthatweareinterestedprimarilyinthewipingefﬁciencyintheneighborhoodofsplashingconditions,wherethecapillarynumberrangesfrom0.07to0.14,thuswheresurfacetensionhasalowereffect.More-over,closetosplashing,thejetpressuregradientactionover-comeslargelythecapillaryeffects,andtheknifemodelisex-pectedtogiveaccuratepredictionsforhf.

TypicalresultsfornozzleT1inconditionswheresurfaceten-sionhaslittleeffectareshowninFig.8,wherethenozzledy-namicpressurePniskeptconstant͑Pn=1450Pa,whichcorre-spondstoajetReynoldsnumberRejof4500͒,whilethestandoffdistanceZ/dvariesbetween2and14.Theknifemodelpredic-tionsandthemeasurementsareingoodagreement͑within5–9%͒,470/Vol.129,APRIL2007

Fig.7EffectofsurfacetensioninthepredictionofﬁnalﬁlmthicknessatZ/d=10andCa=0.055

exceptforlargeZ/dwhereitcanbeanticipatedthatthepressuregradientstartstobetooweaktoneglectcapillarity͑16%under-estimationofhf/h0͒.ThewipingplateauuntilZ/dӍ8iswellrecoveredbytheanalyticalmodel.Itisduetothespeciﬁcbehav-iorofthewipingactuatorsٌPmaxand␶maxintheregionofthepotentialcore͓10,12͔.ThemoreimportantunderestimationofhfatZ/d=12illustratestherisingeffectofsurfacetensionwhichbecomesnon-negligibleduetothelargestandoffdistance.TheuncertaintyofthethicknesspredictionhereliesmainlyintheuncertaintyoftheexperimentalornumericalestimationofٌPmaxand␶max.Inthepresentstudy,weusetheresultsofapriorchar-acterizationofnozzleT1onaﬂatsurface,performedduringaparallelexperimentalstudy͓12͔.Thefacilityusedforthisinves-tigationconsistsofaninstrumentedplateonwhichthejetim-pingesnormally.BydisplacingthisplateequippedwithastaticpressureholeandStantonprobes,thepressureandshearstressdistributionsatimpingementaremeasured.Theuncertaintyonthe

Fig.8Comparisonofﬁlmthicknessafterwipingobtainedex-perimentally,andpredictedbytheknifemodelforRej=4500andU=1.5m/s

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Fig.9Experimentalwipingcurvesforthetwonozzlegeom-etriesT1andT2atZ/d=10andU=1.5m/s

Fig.10EvolutionofnormalizedﬁnalﬁlmthicknesswiththenozzletiltingangleatconstantRej=5100fordifferentZ/dvalues

wipingactuatorsisestimatedto±4%forthepressuregradient,andpresumablymorethan±10%fortheshearstress,duetothefavorablepressuregradientwhichstronglyaffectstheStantonprobemeasurements͓8͔.Asensitivitystudyoftheknifemodelshowsthatthesevaluesleadtoaﬁnal±6%uncertaintyonthethicknesshf.ItwasalsoshownbyLacanetteetal.͓8͔thattheresultsofVOF-LESnumericalsimulationstakingfullyintoac-countthesurfacetensioneffectsareinreasonableagreementwiththemeasurementsandmodelpredictions,exceptforsmallZ/ddistances͑Z/dഛ4͒,wheretheimportantunderestimationofhfisprobablyduetoanoverestimationofٌPmaxand␶max.Thelatteristypicalofanartiﬁcialconﬁnementproducedbythedomainboundariesinimpingingjetsnumericalsimulations͓13͔.

TypicalexperimentalwipingcurvesobtainedbyvaryingthenozzledynamicpressurePnatconstantZ/d=10areshowninFig.9.ItappearsthatnozzleT2isabout15%moreefﬁcientinwipingthanT1,thatistosayforaconstantnozzlepressurePn,theﬁnalﬁlmthicknessis15%lowerwithT2thanwithT1.Thenozzleexternalandinternalgeometriesseemthereforetohaveanon-negligibleinﬂuenceonthewipingactuatorsٌPmaxand␶max.Thiscouldbequantiﬁedbycharacterizingtheimpingingjetproducedbyvariousnozzlegeometriesonaﬂatplate.Sincewipingefﬁ-ciencyiscontrolledbytheupstreammaximumpressuregradientandshearstress,wecanspeculatethatthesewipingactuatorsaremodiﬁedbythesquarecornershapeofT2,whichisfavorabletoﬂowrecirculations.

Finally,theeffectofnozzletiltingwhichisincludedintheknifemodelinSec.2.2isvalidatedbyexperimentsinFig.10,forajetReynoldsnumberof5100,attwodifferentsubstratespeedsU=1.5m/sand2m/s.Asitcouldbeexpected,thewipingefﬁ-ciencydropsmoreimportantlywiththetiltanglewhenthestand-offdistanceZ/dislarge.BecausethejetismoredevelopedatimpingementwhenZ/dislarger,thepressuregradientandshearstress͑andthusthewipingefﬁciency͒aremoresensitivetothetiltangle.Thepredictionsoftheknifemodelareinagreementwithin±7%withthemeasurements,whichisconsideredacceptableforthemodelvalidation.Itwillthusbeusedforthefuturewipingpredictionsatconstantcoatingthickness.

throughaphenomenologicalapproachoftheproblem.Anexperi-mentaldatabaseisusedfortheﬁndingofanempiricalmodeltopredictsplashing.

3.1PhenomenologicalApproach.Theejectionofdropletsfromliquidfreesurfacesexposedtogasﬂowshasbeenexten-sivelystudiedinbasicconﬁgurations:shearedliquidﬁlmsonpipewalls͓14͔,stillliquidfreesurfacesimpingedbyroundjets͓15͔.However,thereexistveryfewstudiesaboutsplashinginthecon-textofjetwiping.Thisﬁlmsprayingmechanismmaybeseenastheultimatedevelopmentofaﬁlminstability͓16͔,whoseampli-ﬁcationfactormaybehighenoughsothatatomizationoccurs.Yoneda͓5,17͔usesHinze’sdropbreakupmodelinturbulentgasﬂow͓18͔toderiveacriterionforsplashinginjetwiping.Infact,Yoneda͓5,17͔seestheliquid“bump”formedbythewipingme-niscusasadropletofdiameter͑2h0−hf͒,whichissubjectedtoaparallelincominggasﬂow.Tothisextent,theproblemcanberelatedtothebreakupofasingledropletinaturbulentﬂow,andYoneda͓5,17͔postulatesthatthecriticalconditionsfordropletdisintegrationintosmallerdropletsarethoseofsplashingoccur-rence.Asatypicallengthforthecapillarycontribution,hechoosesthequantity͑2h0−hf͒,forwhichhegetsacorrelationdependingontheﬁlmcapillarynumberCaandjetpressurePn.HeendsupwithasplashingcriterionintermsofCaandcriticaljetpressureP*n.Amoreempiricalapproachisattemptedheretoiden-tifythewipingconditionsthatmayallowretardingsplashing.Thepresentmodelingpostulatesthattheonsetofsplashingoc-curswhenthesheareffectproducedbythedownwardgaswalljetovercomesthestabilizingeffectofsurfacetensionmodeledhereby␴l/R,whereRisthemeniscusradiusofcurvature.Thelattercanbeapproximatedby͑2h0−hf͒,asdepictedinFig.3,butsincehfӶ2h0,wechooseRӍ2h0.Expressingthewallshearstressin

,where␳isthetermsofdynamicpressureofthewalljet͑0.5␳Vwj

gasdensity,andVwjisatypicalwall-jetvelocity͒,andevaluatingtheratioofthedominatingforcescontrollingthesplashingmechanismleadstotheeffectivejetWebernumberWe

h0/␴l.ThecriticalWebernumberWe*abovewhich=␳gVwj

splashingdevelopsiscorrelatedwiththeﬁlmReynoldsnumberbasedonthestripvelocity,U,andtheﬁnalcoatingthicknessRe=Uhf/␯l.Thewalljetvelocity,Vwj,ismodeledbythefollowingrelation,whichincludestheeffectofnozzletilting͓11͔

APRIL2007,Vol.129/471

3Splashing

Thesplashingphenomenonoccurringupstreamthewipingjetisnowinvestigated.TypicaldimensionlessparametersarederivedJournalofFluidsEngineering

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Fig.11DimensionlesssplashingcurvesforZ/d=10,nozzleT1

Fig.12DimensionlesssplashingcurvesforZ/d=10,nozzleT2

Vwj=

V0j

ͱ1+sin␣Z/d

͑20͒

Figure13pointsoutthatthenormalizedstandoffdistanceZ/ddoesnotaffectsubstantiallythesplashingcorrelation,whichinturncanbeformulatedasfollows

We*=e͑a␣+b͒Re−n

͑21͒

Thevalueofcoefﬁcientsaandbaswellastheexponentndependsonthenozzledesign.ForT1,a=0.043,b=5.5,andn=1.44,whileforT2,a=0.018,b=7.9,andn=1.91.ThedifferentcoefﬁcientsforT1andT2incorrelation͑21͒resultfromacom-binationofthedifferentperformancesofT1andT2inwipingandsplashing.TheimprovementofthecoatingwindowreachedbytiltingthenozzledownwardcanbequantiﬁedbythenetincreaseofthestripspeedU,whichisobtainedforagivenhfvalue.Toevaluatethisenhancement,anoptimumoperatingpointisdeﬁnedasthehighestproductivityconditionbelowsplashing.Itcorre-spondstotheintersectionbetweenthesplashingcriticalcurvegivenbyEq.͑21͒andthewipingcurveatconstantthickness.The

Itcanbeanticipatedthatdownwardtiltingofthenozzledelaystheoccurrenceofsplashing.Indeed,thetendencyofthejetstreamlinestoimpactnormallyontheﬁlminspiteofitsobliquepositionmightforcetheﬁlmtosticktothesubstratejustbelowthestagnationpoint.Thisintuitionwasconﬁrmedbypreliminarytests.

Ithastobeseennowwhetherthelossofwipingefﬁciencyinducedbyjetdownwardtiltingcompensatesforthesplashingdelayatconstantﬁlmthickness.

3.2EmpiricalModel.Thesplashingexperimentsareper-formedonthefacilitydescribedinSec.2.3.1.Theyconsistofthevisualdetectionofsplashingfordifferentwipingconditions.Atconstantjetpressure,thesubstratevelocityisincreaseduntilsplashingoccursinitsfullydevelopedstatealongthestrip.Anhysteresisphenomenonisobserved:thesubstratespeedatwhichsplashingdisappearsisonaverage7%lowerthantheoneatwhichitappears.Wewillonlyconsidersplashingappearancehere.Thesplashingtestsaresystematicallyrepeatedﬁvetimes.Atﬁrst,twoobserversreproducedindependentlythesameexperi-ments,andthedifferenceintheresultswasfoundnottoexceed

͑U*max−Umin͒overasetofﬁvetestsperformedbyasingleexperi-menter.ThestandarddeviationofthedetectedcriticalvelocityU*is±2%.ThebiaserroronUbeingof0.006m/s,theﬁnaluncer-taintyonU*isoftheorderof3%at95%conﬁdence.Thepropa-gationofaforementionederrorsleadtoaﬁnal±6.5%uncertaintyonReand±11%onWe*insplashingconditions.

Figure11showstheevolutionofcriticalconditionsintermsofRe−We*stabilitycurvesforthesymmetricnozzleT1.Splashingoccursabovethecurve,whilejetwipingisstablebelowthecurve.Asexpected,downwardnozzletiltingallowsdelayingsplashing.Splashingdelaymeansthatsplashingoccursathigherstripspeedsforaconstantnozzlepressure.ThecomparisonwithFig.12forT2revealsthatintermsofsplashing,thenozzlegeometrydoesnotmodifymuchthecriticalconditions,atleastforlowertiltingangles.For␣=30deg,itseemsthatT1isslightlymoreefﬁcient͑thesametrendisobservedforsplashingdisappearance͒.TheworseperformanceofT2isverylikelyduetothesquareshapeofthenozzleupstreamwiping,whichmakesitmoresentitivetothetiltangle.

472/Vol.129,APRIL2007

Fig.13DimensionlesssplashingcurvesforZ/dvalues,fortiltangles␣=0degand10deg

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Fig.14EffectofsurfacetensioninthepredictionofﬁnalzinccoatingthicknessongalvanizationlinesatZ/d=8andCa=0.0065„U=1.5m/s…

Fig.15Measuredandpredictedcoatingthicknessongalvani-zationlines

latterispredictedbytheknifemodel,inwhichcorrelations͑17͒and͑18͒foranobliquejetareused.Thisexercisewillbedoneforindustrialconditionsinthefollowingsection.

Therangecoveredbythedimensionlessparametersallowsavalidationoftheempiricalcorrelationswithgalvanizationlinedata.Indeed,itisfoundthat0.1ഛWeഛ1.8and50ഛReഛ250forthewatermodel,whileWeജ0.19–0.23andReജ120–150corre-spondtolineconditions.Thecapillarynumberrangeisgenerallylowerintheindustrialprocesswindow:0.004ഛCaഛ0.0125whileitisequalto0.07ഛCaഛ0.14onthelaboratoryfacility.

4ApplicationtoIndustrialConditions

Dependingonthedesiredﬁnalproduct,typicalwipingcondi-tionsingalvanizationincludeastripvelocitybetween50and170m/min,anozzlepressurebetween3and45kPa,toreachcoatingweightsMbetween50and140g/m2͑M=1000␳Znhf͒,thatistosayzincﬁlmthicknessesbetween5and20␮m.ThestandoffdistanceZislargeenoughtoavoidcontactbetweenthestripandthenozzle:8ഛZഛ14mm.Thenozzleslotistunedbe-tween0.8and1.2mm.Theliquidzincpropertiesare␳Zn=6540kg/m3,␮Zn=0.0035Pas,and␴Zn=0.7N/m.Typically,verythincoatingsarereservedtotheautomotiveindustry,whileheavycoatedproductsserveforfurnituresanddomesticappli-ances.

Theimportanceofsurfacetensionintypicalwipingconditionsongalvanizationlinesiscomparativelylowerthanonthewatermodelfacility.Althoughthecapillarynumbersarelower͑0.004ഛCaഛ0.012͒,thejetpressuregradientinnominalworkingcon-ditionsovercomeslargelythesurfacetensioneffect,asitcanbeseeninFig.14forCa=0.0065.Theknifemodelisthereforesuf-ﬁcienttoprovidereliablepredictionsofthecoatingthickness.ThecoatingweightsmeasuredongalvanizationlinesareaspresentedversusknifemodelpredictionsinFig.15.Thein-linedataarebasedonthestatistictreatmentofalargenumberofmeasurementsondrysamples͑solidiﬁedzinc͒withathicknessgauge.Thestandarddeviationislowerthan0.5g/m2.Theinter-polationofthisdataforconstantcoatingweightsconstitutesasimpliﬁedmodelwhichisusedinpracticeonthelineremotecontrolsystems.TheempiricalcoefﬁcientsforthecomputationofthewipingactuatorsٌPmaxand␶maxareadjustedintheknifemodeltoﬁtthemeasurementsinthehighercapillaryandWebernumberranges,thatistosaywhensurfacetensionhasthelowestJournalofFluidsEngineering

effect.Indeed,thereisnopossibilitytocharacterizetheindustrialnozzleslikeitwasdoneforlaboratorytests.Followingthisad-justment,theagreementbetweentheknifemodelandthemea-suredthicknessesremainsgood͑±4%͒overtheprocesswindowsforthedifferentcoatingweights.Aslightdeparture͑underestima-tionofupto15%͒fromthepredictionscanbedepictedinFig.15forhigherhf/h0values,forwhichsurfacetensionstartstobenon-negligible.Indeed,wipingconditionsforlargecoatingmassesgenerallyincludelownozzlepressures,andthereforelow-pressuregradientandshearstress.Insuchasituation,itcanbeanticipatedthatcapillaryeffectscomeupgradually.

Ongalvanizationlines,splashingistypicallyobservedforhighsubstratevelocities͑above150m/min͒andhighcoatingweights͑130–140g/m2͒atZ/dabout10.AlltheoperatingpointsofFig.15areplottedinFig.16togetherwiththedimensionlesssplashingcorrelationEq.͑21͒forthesymmetricnozzleT1.Mostoftheprocesspointswhichareinthenonsplashingregimearefoundtobebelowthecriticalcurve,whiletheonesforwhichsplashingis

Fig.16Typicalgalvanizationprocesspoints„stableandwithsplashing…withempiricalcorrelationforsplashingoccurrence

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Fig.17Parallelevolutionofsplashingandwipingat␣=0degand30degforaconstantcoatingthicknessof20␮m,ingalvanizationwithliquidzinc

Fig.18Evolutionoftheestimatedmaximumsubstratespeedbelowsplashingforthetwonozzlegeometries„Z/d؍10,hf=20␮m…ingalvanizationwithliquidzinc

observedarecorrectlypositionedabovethecurve.Theagreementisthereforegood,exceptforafewpointswhichareveryclosetocriticalconditions.Thisisnotsurprisingconsideringthatthelineprocessparametersareverydifﬁculttoassess.Thecurvecorre-spondingtothecriterionproposedbyYoneda͓5,17͔appearsalsoontheplot.HeﬁndsthatthecriticaljetpressureP*nisafastdecreasingfunctionofthecapillarynumber,whichisconsistentwithexperimentalobservationswhenUrisesforagivenliquid.Inthatconditionsindeed,Caincreases,andonegetsclosertosplash-ing.However,whenplottedintermsofRe−We*withwaterandzincasworkingﬂuidsandusingtheknifemodelforhf,itisfoundthatthecriticalWe*increasesslightlywithRe͑Fig.16͒,instrongdisagreementwiththepresentexperimentaldata.Itisnotverysurprisingactuallythatthedropbreakupmodeldoesnotcomparewellwithsplashingdata,sincetheassumptionsofYoneda͓5,17͔areratherdistantfromtherealwipingconﬁguration.

Finally,theparallelevolutionofsplashingandwipingatcon-stantﬁlmthicknessismonitored,inordertocomputetheopti-mumworkingpointintermsofsubstratespeedU.ThisexerciseisdoneinFig.17forgalvanizationconditionswhen␣variesfrom0degto30deg,andforacoatingthicknessof20␮m.ItappearsthatthemaximumlikelylinespeedU*beforeﬁlmsprayingcanbesubstantiallyincreasedbytiltingdownwardsthenozzle.ThistrendisdetailedinFig.18,wheretheevolutionofU*isplottedversusthejetangle.Velocitygainsofuptoabout40%canbereachedwithnozzleT1,whileonly8%isobtainedwithT2,prob-ablyasaresultofitsdifferentperformancesbothinwipingandsplashing.Inindustrialconditionshowever,thenozzlecanhardlybetiltedofmorethan10deg,whichlimitsthevelocitygaintoabout12%.

pointsarealsowellpredicted,providedthatanadjustmentoftheempiricalcoefﬁcientsdependingonthenozzlegeometryismade.Theknifemodelisthereforeanefﬁcienttooltobeusedintheremotecontrolsystemofindustriallines.

Anexperimentalinvestigationofsplashinginjetwipingisthenundertaken.Thisphenomenonisfeaturedbyastronginstabilityoftherunbackﬂowleadingtotheejectionofdropletsthatmayim-pingeontheslotnozzle.Suchaneventconstitutesaseverelimi-tationtothewipingtechnique.

AnempiricalsplashingcorrelationexpressedintermsofthecriticaljetWebernumberandﬁlmReynoldsnumberisproposed.Verygoodagreementisfoundbetweenthesplashingcriterionandtheobservationsonindustriallines.

Comparingthesplashingandwipingmodelsatconstantcoatingthicknessleadstothedeﬁnitionofanoptimumprocesswindow.Tiltingthenozzledownwarddelaystheoccurrenceofsplashing,andallowshigherlinespeed.Anetincreaseof40%ofthepro-ductivitycanbeexpectedwithatiltangleof30deg.Thenozzledesign,onwhichbothwipingandsplashingperformancesdepend,isshowntoinﬂuencestronglythesensitivityofthejettothetiltangle.

Acknowledgment

TheauthorsgratefullyacknowledgetheﬁnancialsupportofAr-celor.

Nomenclature

RomanLetters

aϭsplashingcorrelationcoefﬁcientbϭsplashingcorrelationcoefﬁcient

bPϭdistancebetweenthejetaxisandthelocation

ofPs/2͑m͒

Cϭempiricalcoefﬁcientforimpingingjetshear

stressestimation

Caϭﬁlmcapillarynumber,␮lU/␴lCpϭempiricalcoefﬁcientC␶ϭempiricalcoefﬁcientdϭnozzleslotwidth͑m͒

gϭaccelerationofgravity͑ms−2͒hϭlocalﬁlmthickness͑m͒

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5Conclusions

Anoverallanalysisofthejetwipingtechniqueforthegalvani-zationprocessispresented.Atheoreticaldevelopmentforthemeanﬂowcomputationisﬁrstdiscussed.Itisextendedtoazero-dimensionalmodelforthedirectpredictionoftheﬁlmthicknessafterwiping.Theimportanceofsurfacetensionisemphasized,anditisshownthatitcanbeneglectedinthetypicalwipingprocesswindowwhichisunderstudy.Insuchconditions,theknifemodelprovidesthicknesspredictionswhichcomparewellwiththeresultsofwipingexperiments.Typicalindustrialworking474/Vol.129,APRIL2007

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hfLMnPPlPgPnPsQqRReRefRejˆTUuV0jVwjWeX

ϭϭϭϭϭϭϭϭϭϭϭϭϭϭϭϭϭϭϭϭϭϭ

xϭyϭZϭZЈϭ

GreekLetters

␣ϭ⌫ϭ␧ϭ␭ϭ␮ϭ␯ϭ␰ϭ␳ϭ␴ϭ␶ϭ

ﬁlmthicknessafterwiping͑m͒stripwidth͑m͒

coatingweight,␳lhf͑kg/m2͒splashingcorrelationcoefﬁcient

staticpressureatjetimpingement͑Pa͒pressureintheliquid͑Pa͒pressureinthegas͑Pa͒

nozzleexitdynamicpressure͑Pa͒

maximumpressureatthestagnationpointforanimpingingjet͑Pa͒

dimensionlessvolumetricﬂowrateﬁlmvolumetricﬂowrate͑m3/s͒localﬁlmradiusofcurvature͑m͒ﬁlmReynoldsnumber,Uhf/␯l

ﬁlmReynoldsnumberbasedontheﬁlmveloc-ityinthereferentialofthesubstrateandthelocalthickness

jetReynoldsnumber,V0jd/␯lgdimensionlessshearstresssubstratevelocity͑m/s͒

liquidﬁlmlocalvelocity͑m/s͒jetexitvelocity͑m/s͒

characteristicwalljetvelocity͑m/s͒jetWebernumber,␳gV2wjh0/␴l

dimensionlessspatialcoordinateinthemovingsubstratedirection

spatialcoordinateinthemovingsubstratedi-rection͑m͒

spatialcoordinateinthedirectionnormaltothesubstrate͑m͒

nozzletosubstratedistance͑m͒

nozzletosubstratedistancealongthejetaxisforobliquejets͑m͒

nozzletiltangle͓deg͔

surfacetensionterm,␴h0/␳lgd3scalingfactor,2␲h/␭

characteristicwavelengthofxﬂuctuations͑m͒dynamicviscosity͑Pas͒kinematicviscosity͑m2/s͒

normalizedabsissaforjetimpingement,x/bPdensity͑kg/m3͒

surfacetension͑N/m͒

shearstressduetoimpingingjet͑Pa͒

lmaxopt0*ˆZnϭϭϭϭϭϭϭliquidmaximum

valuesatthelocationofthewipingpointwithoutwiping

criticalforsplashingdimensionlessquantitiesliquidzinc

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Subscripts/Superscripts

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JournalofFluidsEngineeringAPRIL2007,Vol.129/475

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