国外文献[4] Design_optimization_of_a_low-speed_fan_blade_with_s(1)

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Designoptimizationofalow-speedfanbladewithsweepandlean

S-JSeo,S-MChoi,andK-YKim*

DepartmentofMechanicalEngineering,InhaUniversity,Nam-Gu,Incheon,RepublicofKorea

Themanuscriptwasreceivedon4January2007andwasacceptedafterrevisionforpublicationon22October2007.DOI:10.1243/09576509JPE410

Abstract:Thepresentworkperformsnumericaloptimizationfordesignofbladestackinglineforanaxial?owfanwithresponsesurfacemethodusingthree-dimensionalNavier–Stokesanalysis,andevaluatestheeffectsofsweepandleanontheperformanceofthefanblade.Rey-nolds-averagedNavier–Stokesequationsarediscretizedwith?nite-volumeapproximationsusingunstructuredgrids.Fourgeometricvariablesconcerningspanwisedistributionsofsweepandleanofbladestackinglinearechosenasdesignvariablesto?ndmaximumef?-ciency.Thecomputationalresultsshowgoodagreementswithexperimentaldata.Thetotalef?-ciencyissuccessfullyincreasedincomparisonwiththereferencefanbyoptimizingthree-dimensionalstackinglinewithsweepandlean.Couplingofsweepandleanalsoimprovesoff-designperformanceofthebladeremarkably.Keywords:designoptimization,fan,blade,sweep,lean

1INTRODUCTION

Thebladeshapes,whicharebeingusedinturboma-chinerysofar,hasbeendesignedmostlybyapplyingtwo-dimensionalstackingline,andtheperformancehasbeenimprovedsteadilybyminimizingthelosses.Theenergylossesduetobladetipleakage,?owseparation,andsecondary?owsarethemajorfactors,whichdeterioratetheperformance.Amongthenumerousattemptstoimprovetheperformancebyminimizingtheselosses,thestudies[1–6]oneffectsofthree-dimensionalstackinglineemployingsweepandleanonbladeperformancearenoticeable.Itwasreportedthatsweepandleanimproveef?-ciencyandalsostabilizetheperformancebycontrol-lingthelossesinturbomachinery.Forexample,Gallimoreetal.[3]introducedthree-dimensionalbladedesignsusingsweepandleaninanaxial?owcompressorrotor.Theyshowedthatthepositiveleanreducedahubcornerandtipclearancelossesexceptingnearthemid-spanregion.DentonandXu

*Correspondingauthor:DepartmentofMechanicalEngineering,InhaUniversity,253Yonghyun-Dong,Nam-Gu,Incheon402-751,RepublicofKorea.email:kykim@inha.ac.kr

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[4]investigatedtheeffectsofsweepandleanontheperformanceofatransonicfan,andshowedthatthestallmarginwassigni?cantlyimprovedwiththeforwardsweptbladealthoughaverylittlechangeinthepeakef?ciencywasproducedbythebladesweeporlean.However,inspiteoftheseinterestsinthree-dimensionalbladedesign,thecouplingeffectsofsweepandleanontheperformanceoftur-bomachinerybladesarenotstillclear.

Recently,designoptimizationusingnumericaloptimizationtechniquecoupledwithNavier–Stokesanalysisisacceptedasanewtrendinturbomachinerydesign.Amongavarietyofoptimizationmethods,responsesurfacemethod(RSM)[7],asaglobaloptim-izationmethod,hasmanyadvantagesovergradient-basedmethods[7,8].Someworksonnumericaloptimizationofstackinglineofbladebasedonthree-dimensionalRANS(Reynolds-averagedNavier–Stokesequations)analysishavebeenperformed.AhnandKim[9]improvedperformanceofanaxial?owcom-pressorbyoptimizingskewedstackinglineofrotorbladeusingthree-dimensionalthin-layerNavier–StokesanalysisandRSM[7].

Inthecurrentstudy,theRSMusingthree-dimensionalRANSanalysisisappliedtothe

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88S-JSeo,S-MChoi,andK-YKim

aerodynamicdesignofanaxial?owfanusingastack-inglinethatintroducessweepandleantomaximizethe?owef?ciency.Theeffectsofsweepandleanontheperformanceofthefanarealsodiscussed.

2FLOWANALYSIS

ThecommercialsoftwareCFX5.7[10]isusedforthe?owanalysisinanaxial?owfan.Three-dimensionalsteadyincompressibleRANSequationsaresolved.Governingequationsarediscretizedusing?nite-volumeapproximations.Standardk–1turbulencemodelisusedasaturbulenceclosure.Andtheimplementationofwallboundaryconditionsintur-bulent?owiscompletedbytheuseofempiricalwallfunction.

Thereferencefan,whichisabouttobeoptimizedinthecurrentwork,isthattestedbyJangetal.[11].ThisfanhasNACA65bladesection,anditsmajorspe-ci?cationsarelistedinTable1.Oneoftheninebladesisselectedfornumericalanalysisthatusesperiodicconditions.Inthepresentedwork,tetrahedralunstructuredgridsystemisemployedforthegrids,andoptimumgridsystemhasbeenselectedaftergriddependencytestforthereferencefan.The?nesthybridmeshusedinthetestconsistsof6124648tetrahedronsalongwithfourlayersofin?ationcomprising405614prismsonthehub,shroud,andbladewallsforanoverallmeshsizeof6530262elements(1270824nodes).Severalothermesheswerealsotestedtoevaluatemeshindepen-dence:anoverallmeshsizeof1378211elements(1157801tetrahedrons/220410prisms)and1924461elements(1703733tetrahedrons/220728prisms)withfourlayersofin?ation,1266464elements(1156259tetrahedrons/110205prisms)and1763313(1652937tetrahedrons/110376prisms)withtwolayersofin?ation,and1248422elementsand886316elementswithtetrahedronsonly.Fromthistest,incomparisonwiththe?nestmeshwith6530262elements,meshindependenceisalmostachievedbythemeshwith1924461elements(417867nodes)withfourlayerofin?ation,andthereforethesimilarmeshsizesareusedforallothercalculations.Figure1showsanexampleofthegridsystemonthesurfacesofbladeandhub.Sincek–1turbulencemodelemployswallfunction

Table1

Speci?cationsofreferencefan

0.410.3

1000r/min287.5mm0.5268.8863.88

basedontheempiricallog-lawnearthewall,thegridpointsadjacenttothewallarearrangedtobelocatedintheregion,yþ¼50–150.Here,yþiswallcoordinatede?nedbyy(tw/r)1/2/n,wherey,tw,r,andnaredistancefromthewall,wallshearstress,?uiddensity,andkinematicviscosity,respectively.Uniformpro?lesareassumedattheinlet,andcon-stantpressuresareappliedattheexitboundary.Theworking?uidis208Cair.Toobtainacompletelyconvergedsolutionforthepresentanalysis,theCPUtimewasapproximately12hwithaPentium-IV,3.0GHzprocessor.33.1

NUMERICALOPTIMIZATIONResponsesurfacemethod

RSM[7]isaseriesofstatisticalandmathematicaltechniques;generationofdatabynumericalcompu-tationsorexperiments,constructionofresponsesur-facebyinterpolatingthedata,andoptimizationoftheobjectivefunctiononthesurface.Inthecurrentwork,theresponsesurfaceisapproximatedbyasecond-orderpolynomial.Todeterminethecoef?-cientsofthisresponsemodel,standardleast-squaresregressionisused.Toestimatethesigni?canceofeachindividualinthequadraticpolynomialcoef?-cient,analysisofvariance(ANOVA)[7]andregressionanalysisyieldameasureoftheuncertaintyinthecoef?cientstoincreasetheef?ciencyoftheresponsesurface.Inordertoreducethenumberofdataneededforconstructingresponsesurfaceandtoimprovetherepresentationofthedesignspace,D-optimaldesign[12]asthedesignofexperimentisusedforselectingdesignpoints.3.2

Objectivefunctionanddesignvariables

Theobjectivefunctionforthepresentoptimizationproblemisanef?ciencyde?nedasfollows

h¼

ðpt;outÀpt;inÞÁQ

tÁv

ð1Þ

Flowcoef?cient

Totalpressurecoef?cientRotorrotationfrequencyTipradiusHub–tipratio

InletangleatrotortipOutletangleatrotortip

whereptisthetotalpressure,andthesubscripts,inandex,respectively,indicateinletandexitofthefan.Qisthe?owrate,andtandvaretorqueandangularvelocity,respectively.

Sweepandleaninthestackinglineofthereferencefan[11]aretheshapevariablestobeoptimizedinthiswork.Sweep(g)andlean(d)anglesareshowninFig.2,andthede?nitionsfortheseanglesaresameasthoseusedbyDentonandXu[4].Fourvariablesofgt,gm,dtanddmareselectedasdesignvariables,whichindicatedegreesofsweepandleanattipandmiddleoftheblade,respectively.Forresponse

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Fig.1Anexampleofcomputational

grids

surface-basedoptimization,settinguptheexper-imentalrangesofdesignvariablesisveryimportant.Therefore,therangesofdesignvariablesweredeter-minedbypre-calculationssothatoptimumpointcanbelocatedintheexperimentaldomain.TherangesofdesignvariablesareshowninTable2.

4RESULTSANDDISCUSSIONS

Computationalresultsarevalidatedincomparisonwiththeexperimentaldata[11]inFig.3.Figure3comparesthecomputedtotalandstaticpressureriseswithexperimentaldata.InFig.3,solidanddottedlinesdenotetotalandstaticpressurerises,respectively,obtainedbytheprediction[11]basedonapproximatedanalysis,andblacktriangleisthetotalpressureriseobtainedbytheexperimentatthedesign?owrate.And,thesymbols,hollowrec-tangleandcircleindicatetotalandstaticpressure

risescomputedinthecurrentwork,respectively.Inthis?gure,discrepanciesarefoundinthelow?ow-rateregion,butcomputationalresultsagreewellwiththepredictedpro?lesaswellastheexperimentaldatanearthedesignpoint.Sincethepredictionmethods[11]arebasedonapproximatedtheoryof?uiddynamics,thereliabilityoftheresultsisloweredintheregionoflow?owcoef?cientswhereviscouseffectsaredominant.And,accuracyofthepresentcalculationisalsodeterioratedinthisregion,becausestandardk–1turbulencemodelgenerallydoesnotgivegoodresultsforseparated?ows,whichmayoccurasthestallpointisapproached.Thesearethemainreasonsforthediscrepanciesbetweenpre-dictedandcalculatedresults.

Todeterminethecoef?cientsinthepolynomial,31pointsforresponseevaluationsareselectedusingD-optimaldesignamong81fullfactorialpoints.Thevaluesofunknowncoef?cientsareobtainedwiththedataattheselectedpointsusingthecommercialstatisticssoftware,SPSS(statisticalpackageforthesocialsciences),whichisusedtoperformANOVAandregressionanalysisalongwithotherstatisticalanalysesforawidevarietyof?elds.Thereliabilityoftheresponsesurfaceisimprovedbyt

-statistics

Table2

Variables

Rangesofdesignvariables

Lowerbounds20.0220.0320.0420.01

Upperbounds0.040.030.020.01

Fig.2De?nitionofsweepandlean

gtgmdtdm

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90S-JSeo,S-MChoi,andK-YKim

Table4

Designvariable

Optimalvaluesofdesignvariables

Optimumsweeponly0.022920.00940.00000.0000

Optimumleanonly0.00000.00000.02702

0.0003

Optimumsweepandlean0.038120.02200.032620.0030

gtgmdtdm

Fig.3

Comparisonofperformancecurvesbetweenexperimentandcomputationforthereference

fan

andadjustedR2method.AsaresultofANOVAandregressionanalysisforpresentsurface,thevalueofadjustR2is0.950,whichmeanstheresponsesurfaceisreliable.Theoptimumpointontheresponsesur-facehasbeenfoundbyalinearprogrammingmethod.

ResultsofoptimizationandoptimalvaluesofdesignvariablesaresummarizedinTables3and4,respectively.Asamainresultofoptimization,showninTable3,theef?ciencyissuccessfullyincreasedby1.75percentfortheoptimumshape.Ef?ciencyhasnotbeenimprovedremarkablyassuggestedbyDentonandXu[4],andtheincrementisofthesimilarlevelthatcanbeexpectedfromthepreviousresults[4–6].

Theadditionaloptimizationsforthecaseswithsweeponlyandleanonly,respectively,usingtheobtainedresponsesurface,havebeencarriedouttoexaminetheeffectsofsweepandleanseparatelyontheef?ciencyasshowninTables3and4.Letopt-1,opt-2,andopt-3indicatetheoptimaincaseswithbothsweepandlean,sweeponly,andleanonly,respectively.Table4showsthatoptimumgtanddthavepositivevalues,butoptimumgmanddmarenegative.Thisagreeswellwiththepreviousresults[1–3,6]thatthebladeswithforwardsweeporfor-wardleanshowbetterperformancethanthebladeswithbackwardsweeporbackwardlean.

PerformancesobtainedbythenumericalanalysesfortheoptimumandreferenceshapesareshowninFig.4.Asaresultofoptimization,totalef?ciency

Table3Resultsofoptimizations

Ef?ciency(%)

Reference

OptimumwithsweepandleanOptimumwithsweeponlyOptimumwithleanonly

85.1086.8586.3586.43

Increment(%)21.751.251.33

isincreasedinneighbourhoodofthedesignpoint(w¼0.41).Inthecaseofopt-3,whereonlyleaniscon-sidered,theef?ciencycurvedoesnotdiffermuchfromthatofreferenceshapeovertheentire?owraterange(Fig.4(a)).However,incasesofopt-1andopt-2,wheresweepisapplied,theperformanceinlow?ow-rateregionisimprovedremarkably.Thus,highef?ciencydrivingisavailableinthewide?owrateregion,andthebestef?ciencypointmovestothelow?owrateregion.InFig.4(b),itisobviousthatsweepmakeslocationofpeakpressuretoshifttothelow?ow-rateregion.Incaseofopt-1,thecouplingofsweepandleanlowersthestaticpressurelargelyinmostofthe?owraterange,butproducesthehighestpeakpressureatthelowest?owrate,whereasincaseofopt-2sweepimprovestheperformanceinthelow?owrateregionwithoutchangingthepressureinhigh?owrateregion.

Fig.4

Comparisonofperformanceandef?ciencycurvesbetweenoptimumandreferencebladeshapes:(a)totalef?ciencyand(b)staticpressure

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mass?owdistributioninspanwisedirection.Asiswellknown,leakage?owisgeneratedduetobladerotationinthetipclearanceregion.But,asshowninFig.6,theapplicationofleancausestheaxialvel-ocitytobecomepositiveinthisregion.Thisoffersmorestable?owcharacteristicsincomparisonwiththereferencecaseastheleakagevortexisreduced,andalsocontributestotheuniformspanwisedistri-butionof?ow.5

Fig.5

Comparisonoflocalef?ciencypro?lesbetweenoptimumandreferencebladeshapesattrailing

edge

CONCLUSIONS

Localef?ciency(ht,local)distributionsatthetrailingedgeofbladewithchangesindynamicpressureinspanwisedirection,areshowninFig.5.Pressureand?uxesaremassaveragedateachspanwiselocation.Therefore,localef?ciencyindicateslocaltotalef?ciencyateachspanwiseposition.AsshowninFig.5,ef?ciencyreducesat85percentspaninthecasesofopt-1andopt-3whereleanisapplied.Especially,thereductionatthisspanwiselocationisnoticableincaseofopt-1wherebothsweepandleanareoptimized.However,itshouldbenotedthatthecouplingofsweepandleanincreasestheef?-ciencyremarkablybelow60percentspan,whichcausestheincreaseinoverallef?ciency.Thus,theeffectofthecouplingobviouslyistomakethespan-wisedistributionoflocalef?ciencymoreuniform.Thepeaklocalef?ciencynear85percentspanincasesofreferenceandopt-2,isrelatedtothesuddenincreaseinaxialvelocityatthislocationasshowninFig.6.However,axialvelocitydistributionsincasesofopt-1andopt-3donotshowthispeakvel-ocity,whichindicatesthatleansmoothesoutthe

Indesignofanaxial?owfan,sweep,andleanofbladestackinglineareoptimizedusingthree-dimensional?owanalysis.Thecomputationalresultsarevalidatedbycomparingpressureandvelocitycomponentswithexperimentaldata.Thefanef?-ciencyisincreasedby1.75percentcomparedwithreferencefanbyusingresponsesurfaceoptimizationmethod.Theperformanceisimprovedbysweepinthelow?owrateregion;highef?ciencyregionisextended,andstallpointisshiftedtolow?owrateside.Couplingofsweepandleanenhancestheper-formance,butreducesthestaticpressureneardesignpoint.Thecouplingisalsoeffectivetomakethespanwisedistributionoflocalef?ciencymoreuniform.Thisisrelatedtothefactthatleansmoothesoutthemass?owdistributioninspanwisedirectionbysuppressingtheleakagevortex.ACKNOWLEDGEMENTS

TheauthorsthankDrChoon-ManJang,seniorresearcherofKoreaInstituteofConstructionTech-nologyforhisvaluableadvice.Thecurrentworkwassupportedbycentreforunderground?reandenvironmentresearch,andalsosupportedpartlybyKISTIunder‘TheEighthStrategicSupercomputingSupportProgramme’.

REFERENCES

1Sasaki,T.andBreugelmans,F.Comparisonofsweepanddihedraleffectsoncompressorcascadeperform-ance.ASMEJ.Turbomach.,1998,120,454–464.

2Wadia,A.R.,Szucs,P.N.,andCrall,D.W.Innerwork-ingsofaerodynamicssweep.ASMEJ.Turbomach.,1998,120,671–682.

3Gallimore,S.J.,Bolger,J.J.,Cumpsty,N.A.,Taylor,M.J.,Wright,P.I.,andPlace,J.M.M.Theuseofsweepanddihedralinmultistageaxial?owcompressorblad-ing–partI:universityresearchandmethodsdevelop-ment.ASMEJ.Turbomach.,2002,124,521–532.

4Denton,J.D.andXu,L.Theeffectsofleanandsweepontransonicfanperformance.InProceedings

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Fig.6

Comparisonofaxialvelocitydistributionsforoptimumandreferencebladeshapeattrailingedge

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