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Chapter1IntroductiontoHydraulicandPneumaticTransmission1.1StudyonHydraulicandPneumaticTransmission1.2OperatingPrinciplesofHydraulicsTransmission

1.3CompositionofHydraulicTransmissionSystem

1.4FeaturesofHydraulicandPneumaticTransmission1.5TheDevelopmentHistoryandApplicationofHydraulicandPneumaticTechnologyChapterlistmedium:

Fluidmediumenergyofcompressivefluid(pressureoilorcompressiveair)mediumcontrolmeansthecontents1.1StudyonHydraulicandPneumaticTransmissioncontrolmeans

Hydraulicandpneumatictransmissionsaresimilarinoperatingprincipleandcontrolmeans.Specialsub-circuitsbuiltbyvarioushydraulic(orpneumatic)componentsareusedtobuildupparticularhydraulic/pneumatictransmissionsystemstorealizeenergytransferandcontrol.contents:

(1)Physicalperformancesandstatic/dynamiccharactersoffluid.(2)Operatingprincipleandconstructseveralhydraulicelementsandsub-circuit(3)Hydraulic/pneumaticsystemdesignsarethebestwaytostudypower(hydraulic/pneumatic)transmissionandcontroltechnology.1.2OperatingPrinciplesofHydraulicsTransmissionPascal’slaw:pressureexertedonaconfinedliquidistransmittedundiminishedinalldirectionsandactswithequalforceonallequalareas.Fig.1-1Operatingprincipleofhydraulicjack

1-Smallactuator2-lever3-Heavyload4-Bigactuator5-Checkvalve6-Reservoir7,8-Non-returnvalve

(1-2)(1-1)1.

Powertransmission

Thusthemovingpistoncandoworkonanoutsideload.Wecangetthefirstimportantlawherethattheworkingpressuresinthehydraulicsystemdependedontheoutsideload.thehydraulicpressureintheactuator:thethrustonthesmallpistonofpumpF1:(1-4)Wecangetthesecondimportantlawherethatthemotionspeedofpistoninactuatordependsontheflow-inrate,andindependentoftheoutsideload.

Hydraulicoilpressureandtheflowratearethetwomainparametersinthehydraulicsystem.2.MotiontransferThevolume

ofdisplacementisequaltothevolumedrawnintoit,soweget(1-3)Dividingbythemovingtimetonthetwosidesintheaboveformula,wehave1.3ComposingofHydraulicTransmissionSystem

Fig.1-2Principleofatypicalhydraulicsystem

1-Hydraulicpump2-Adjustablethrottlevalve3-Closed-centerdirectionalvalve4-Hydrauliccylinder5-Workingload6-Reliefvalve7-Filter8-ReservoirFromanoperationalstandpoint,anyhydraulicsystemscanbedividedintofivelogicalsegments.(1)Powerinputsegment(energyportion):mechanicalforce→hydraulicpower.itusuallyconsistsofpumps.(2)Thepoweroutputportion(actuatorportion):

hydraulicpower→mechanicalforce.(3)Controlelements:

itcanbeusedinahydraulicsystemtorestrictthepressure,regulatethevolumeofoildirectedtoorfromtheactuator.(4)Assistantelements:

canbeusedastomaintainsystemworking(5)Workingoil:hydraulicoil.

2.Theadvantagesandshortages

ofpneumatictransmission

1.4FeaturesofHydraulicandPneumaticTransmission1.Theadvantagesanddisadrantages

ofhydraulictransmission

Advantages:1)Thehydraulicequipmentsystemhassmallervolume,light,highpowerconsistencyandcompactconfigurationatagivenpower.2)hasagoodworkingstability.3)canreachawiderangeofspeedregulation4)Thehydraulictransmissioncaneasilyrealizeautomation.5)canprotectfromover-loadeasily.6)thehydraulicsystemiseasierindesign,fabricationandapplication.7)Thehydraulicsystemiseasierthanmachineequipmentindoinglinemotion.Disadrantages:1)OilLeaksareinevitable.2)Workingtemperature.3)Thecostishigh.4)Itisdifficulttofindthereasonsoffault.Advantages:1)Theaircanbeobtainedandexpelledfromtheatmosphere.

2)Itisoflowviscosityandlowerpressurelossinpipes.3)Itisoflowworkingpressure.4)Thepneumatictransmissionhasasimpleservicing.5)SafetyDisadrantages:1)theworkingstabilitiesarepoorerthanthoseofhydraulictransmissionsystem.2)Thepushforceofpneumatictransmissionisusuallyverylower..3)Lowertransmissionefficiency.Hydraulictransmissionhasbeenexperiencingtheprocessasbelow.

The17thand18thcenturies—aproductiveperiodinthedevelopmentofhydraulictheory.The18thcentury—Thisprinciplewasfirstused.ThefirsthydraulicpressuremachinewasmanufacturedbyEnglandlateinthe18century.

The19centurytonow—HydraulicandPneumatictechnologyhaveagreatapplication……Intheworldwarone

……andworldwartwo…….p9.1.5TheDevelopmentHistoryandApplicationsofHydraulicandPneumaticTransmissionTab.1-1ExamplesofapplicationhydraulicandpneumatictransmissiontechnologyinindustriesFieldsExamplesEngineeringmachineGrab,loadingmachine,bulldozer,shovelmachine,etc.MinemachineCharge,digger,elevator,hydraulicsupportetc.ArchitecturemachinePiledriver,jack,flatmachine,etc.MetallurgymachineRollingmill,pressmachine,etc.fabricationToolmachine,numeric-controlmachiningcenter,automaticassemblyline,air-spanner,punch,model-forgemachine,etc.LightindustriesPacker,injection-plasticmachine,Foodpackager,vacuum-platingmachine,printinganddyeingmachine,etc.AutomobileindustryHighaltitudeoperatingcar,lift,redirector,etc.WaterprojectDam,strobe,shipmachine,ship-rudder,etc.Farmingindustryfertilizepackager,combineharvester,tractor,farmingsuspensionsystem,etc.yoursuggestions

arewelcome!TheEndFig.1-1Operatingprincipleofhydraulicjack

Fig.1-2PrincipleofatypicalhydraulicsystemChapter2FundamentalHydraulic

FluidMechanics2.1

PerformancesoftheHydraulicOil

2.2Hydrostatics

2.3Hydrodynamics

2.4CharacteristicsofFluidFlowinPipeline2.5FlowRateandPressureFeaturesofOrifice

2.6 HydraulicShockandCavitation

Chapterlist2.1PerformancesoftheHydraulicOil

2.1.1TheMainperformances2.1.2Therequestsandchoiceofhydraulicoil

1.Density(kg/m3)

2.Compressibility

2.1.1TheMainperformances

thecoefficientofcompressibility,isthebulkmodulusofelasticity(2-2)(2-1)isdefinedastheratioofthechangeinpressure()torelativechangeinvolume()whilethetemperatureremainsconstant.

3.Viscosity

TheexperimentshaveprovedthatfrictionforcebetweenthetwofluidmoleculescanbedescribedasWhereisviscositycoefficient,alsokinematicviscosity.Fig.2-1Thesketchofviscosity

ThesketchofviscosityisillustratedbyFig.2-1.

(2-3)Cohesionbetweentwomolecules……Therearethreemethodstodescribetheviscosity:absoluteviscosity,Kinematicviscosityandrelativeviscosity.(1)Dynamicviscosityorabsoluteviscosity

μ(Pa?s)or(N?s/m2)(2)Kinematicviscosityν(mm2/s)(2-4)(3)

Relativeviscosity(conditionalviscosity)Therelativeviscosity

whichusedinChinaistestedbytheviscometer,suchasFig.2-2.Fig.2-2Principleofviscometer

Takethenotedescribestheviscosity:Theconversionformulabetweentheandkinematicviscosityis

(m2/s)

(4)Viscosity-temperature:Fortheviscositylessthan15andthetemperature30℃~150℃,theviscosity-temperatureformulaisdescribeasfollowing(WecanalsolookupfromFig.2-3):(2-5)(2-6)(2-7)(5)Viscosity-pressure(2-8)(6)Othersperformances

:physicalandchemical,suchasanti-inflammability,anti-oxygenation,anti-concreting,anti-foamandanti-corrosionetc.Fig.2-3Theviscosity-temperatureofhomemadeoils

2.Choice

Thehydraulicoilinahydraulicsystematisrecommendedgenerally.Request

Theoilplaystworolesoftransmissionenergyandlubricationonthesurfacesofworkinginteraction.

Therequestsforthehydraulicfluidsare:appropriateviscosity,thegoodinpropertyoffavorableviscosity-temperature,agoodlubricity,chemicallyandenvironmentallystabilities,compatiblewithothersystemmaterialsandsoon.2.1.2Therequestsandchoiceofhydraulicoil

Thehydraulicoilshouldbechoseninaccordingtotherequestofhydraulicpump.ThehydraulicoilviscosityadaptedfordifferenthydraulicpumpsislistedinTab.2-2.Tab.2-2TherangeofviscosityofhydraulicoiladaptedtopumpsTypesviscosities(10-6m2/s)TypesViscosities(10-6m2/s)5~40℃①40~80℃①5~40℃①40~80℃①VanePumpsP<7MPa30~5040~75Gearpumps30~7095~165P≥7MPa50~7050~90Radialpistonpumps30~5065~240Screwpumps30~5040~80Axialpistonpumps30~7070~150①5~40℃、40~80℃aredescribedthetemperaturesofhydraulicsystem.2.2

Hydrostatics

2.2.1CharacteristicsofHydrostatics2.2.2

Thebasicformulaofhydrostatics2.2.3TheprincipleofPascalapplication2.2.4Effectoffluidpressureoncurvedsurfaces1.ThehydrostaticsStaticpressure:theactionforceinnormalonaunitarea.Itisintituledpressureinphysicsandactionforceinengineeringusually.

2.Thecharacteristicsofhydrostatics

(1)Inanyhomogeneousfluidsystematrest,thepressureincreaseswiththedepthofthefluid.(2)Pressureatanypointinahomogeneousfluidsystematrestactsperpendicularlytosurfacesincontactwiththefluid.2.2.1CharacteristicsofHydrostatics

2.2.2Thebasicformulaofhydrostatics

Thebasicformulaofhydrostatics

Theactingpressuresonthefluidatrest,inacontainerincludetheweight,forceonthefluidsurface,showninFig.2-4a.

Fig.2-4ThedistributionofforcesinacontainerwithrestfluidThetotalbalanceforceformulais

Formula(2-9)divideby,then

(2-9)(2-10)Thepressureonarestfluidcontainedinvolvestwoparts:

Theformula(2-10)isthebasicequationforhydrostatic.Itstatesthatthedistributionstatusofhydrostaticsasfollowing:(2)Thepressureisincreasedwiththedepthh;(3)Isotonicpressuresurface,thatis,thepressuresareallequalatthesurfaceconsistedbyallpointsatgivendepthh,suchasatthelineofA-A;(4)Conservationofenergy

(2-11)(2-12)Here,theaspressureenergyatperunitmassfluid.2.Thedefinitionofpressure(1)AbsolutepressureRelativegaugepressure:Thepressuresmeasuredbyapressuregaugeareallrelativepressure(3)Vacuum(negativepressure)

1Pa=1N/m2;1bar=1×105Pa=1×105N/m2;1at=1kgf/cm2=9.8×104N/m2;1mH2O=9.8×103N/m2;1mmHg=1.33×102N/m2.

TherelationshipofthreepressuresisshowninFig.2-5.Theunitsofpressureandrelationsbetweendifferentpressures:Fig.2-5Absolute,relativeandvacuumpressureExample2-1:Theoilisfullinacontainer.Foragivencondition,thedensityofoil,theactionforceonthispistonsurfaceF=1000N,theareaofpistonA=1×10-3(m2),ifthemassofpistonisneglected,trytocalculatethestaticpressurepath=0.5m,asshowninFig.2-6.Fig.2-6Calculationoffluidstaticpressure2.2.3

TheprincipleofPascal

TheprincipleofPascal:pressureexertedonaconfinedliquidistransmittedundiminishedinalldirectionsandactswithequalforceonallequalareas.ItsapplicationisshowninFig.2-7.Fig.2-7TheexampleofPascalprinciple(1)Whenthewallisplane:F=PA(2)Whenwallisacurvedsurface:2.2.4Effectoffluidpressureoncurvedsurfaces

Example2-2.Fig.2-8showsacylindricalmemberofinsideradiiroflength.Calculation:theeffectforceFx

ontherightsegmentofthecylinderatxdirection.Fig.2-8Effectforceontheinnersurfaceofthecylinder

2.3

Hydrodynamics

2.3.1Equationofcontinuity—conservationofmass2.3.2BernoulliEquation—conservationofenergy2.3.3Equationofmomentum—conservationofmomentum

Theequationsofcontinuity,Bernoulliandmomentumarebasicmotionequationsthatdescribethedynamicslawsinflowingfluid2.3.1

Theequationofcontinuity—conservationofmass

Fig.2-9sketchofconservationmassaccordingtotheconservationofmass,Forincompressibleflow,,OrconstantFormula(2-16)istheequationofflowcontinuity.

(2-14)(2-15)

(2-16)Theassumptions:noenergyloss(meansin-viscidandincompressible),accordingtheequationofBernoulli—Conservationofenergy.OrFormulas(2-17)isthewell-knowBernoulliequation.Itstatesthatidealfluidincludepressureenergy,potentialenergy,andkineticenergy.Thesethreeenergiescanbetransferredbetweeneachother,butthetotalenergyisalwaysinvariable.

2.3.2BernoulliEquation—conservationofenergyFig.2-10SketchofBernoulliequation

(2-17)1.IdealequationofBernoulli2.RealequationofBernoulliInmanyhydraulicsystems,theenergiescanbelost(thetotallossisdescribedashw),ontheotherhand,therealvelocityisanon-uniformdistributionandsetakineticcorrectionfactortooffsetthislost,andthecoefficientdefinedby:Hereα=1.1whenitisturbulentflow,andα=2whenlaminarflow,butusuallyinpracticesettheα=1.

Afterintroducingtheenergylossandkineticcorrectionfactor,theequation(2-17)willbechangeto(2-18)(2-19)Notes:seep27,(1)across-sectionarea1and2shouldbeselectedalongthestreamlinedirectionoffluidflow……3.ApplicationexampleoftheequationofBernoulli

Example2-3TheVenturimetershownreducesthepipediameterfrom0.1mtoaminimumof0.05masshowninFig.2-11.Calculatetheflowrateandthemassfluxassumingidealconditions.Fig.2-11Venturemeter

Example2-4.TrytoanalysetheconditionofapumpdrawingintooilfromareservoirbytheequationofBernoulli(Fig.2-12).Setthepressureat2-2across-sectionisp2,thepressureat1-1across-sectionisp1,andp1=pa.andthedistancefrompumporificetohydraulicoilsurfaceish.Fig.2-12Setupofhydraulicpump2.3.3Equationofmomentum-conservationofmomentumFig.2-13SketchofoilflowthroughapipelinewithapressurevesselFig.2-14Sketchofoilflowthroughapipeline

Fig.2-15SketchofoilthroughcurvedpassagesInanysystemofabove,therateofchangeofmomentuminthesystemequalsthenetappliedexternalforce.

Theequationlooksthesameastherelationship(2-20)(2-21)

Assumeafrictionless,incompressibleliquidinacylindricalpassageasshowninFig.2-14.

Theforcebalanceis,fromequation(2-20):

Because

q=Av,so

(2-22)(2-23)(2-24)Fig.2-15,isachangeinmomentumasdefinedinequation2-20.TheforcescanberesolvedintoacomponentFxwhichisaxialtotheinletdirectionandacomponentFywhichisnormaltotheinletdirection.

(2-25)Example2-5.Fig.2-16showsasketchofaspoolvalve.Whenoilfluidflowthroughthevalve,calculate:theaxialeffectforceofoilfluidonthespoolsurface.Fig.2-16Hydraulicdynamiconthespoolvalve

Example2-6.Fig.2-17showsasketchofapoppetvalve,wherethepoppetcoreis2.Whenfluidrateflowqthroughthevalveunderthepressureandthefluidflowdirectionat

bothstatusesofout-flowingFig.2-17

aandin-flowingFig.2-17

b,calculate:actionforcemagnitudeanddirectiononthispoppetcore.Fig.2-17Hydraulicdynamiconthepoppetvalve

FortwocasesabovethefluidactionpressuresonthepoppetareallequaltoF.TheactiondirectionsareshowninFig.2-17aandFig.2-17brespectively.

FortheFig.2-17athefluiddynamicpressuremakesthepoppetorificestendtobeclosed,andfortheFig.2-17btendtobeopened.Soweshouldbeconsideredaccordingtothedetailstatusandcouldnotconsideralltendspoolorificetobeclosedinanyconditions.2.4CharacteristicsofFluidFlowinPipeline

2.4.1StatesoffluidflowandReynoldsnumber

2.4.2Lossesalongcircleparallelpipe2.4.3MinorlossesinpipesystemWhenacontinuityviscousfluidflowsthroughvariablesection,fluidwilllosepartsofenergy.Thiscanbepresentedbythepressurelosshwandkineticcorrectionfactor

,i.e.,intheabovementionedrealfluidBernoulli’sequation

herehwincludestwoparts:pressurelossesalongparallelpipesandminor(orlocal)losses.

2.4.1StatesoffluidflowandReynoldsnumber

therearethreemainstatesofflow,suchaslaminar,transitionandturbulentinapipe.NowtakeFig.2-18forexample.Fig.2-18.SetupofReynoldstestTheexperimentprovedthat,Reynoldsnumber,isconsistedofthreeparameters.TheReynoldsnumberwasobservedtobearatiooftheinertialforcetotheviscousforce.(2-26)1-Overflowpipe2-Supplypipe3,6-Reservoir4,8-Checkvale5-Smallpipe7-Largepipeisacriticalvaluebetweenlaminarandturbulenceusuallydeterminedbyexperimentaldata.(showinTab.2-3)pipesRecrpipesRecrsmoothmetalpipe2320Smoothpipewitheccentricannularitygap1000hosepipe1600-2000Columnvalveorifice260smoothpipewithconcentricannularitygap1100Poppetvalveorifice20-100Tab.2-3FamiliarcriticalReynoldsnumberbasedondifferentpipematerialForflowinnoncircularducts

(2-27)HereRishydraulicradius,definedby:(2-28)2.4.2

Lossesalongcircleparallelpipe

Thelossesduetoviscosityinequaldiameterpipeisreferredaslossesinparallelpipe,whichwillchangewiththedifferentflowingstates.Lossesinparallelpipeatlaminarflow

(1)Velocityprofileinalaminarpipeflow

Fig.2-19Laminarflowinacirclepipe(2-29)

Integrateitandundertheboundaryofu=0atr=R,weobtain

Itsaysthatvelocityprofileinalaminarpipeflowalongradiidirectionisaparabolaprofileandthemaximumvelocityisattheaxiscenterr=0andAsshowinFig.2-19,aforcebalanceinthex-directionyields,thusSetthen

(2-30)(2)Theflowrateinpipe

Formula(2-32)saysthattheaveragevelocityis1/2ofthemaximumvelocity.(2-32)(2-31)Integrateitweobtain(3)Averagevelocityinpipe

Accordingtothedefinitionofaveragevelocity,Fromformula(2-30)(4)LossesalongcircleparallelpipeFromformula(2-32),thelossis

Dosomechange,Theformula(2-33)canbewrittenas(2-33)(2-34)

Whereistheresistancecoefficientalongacirclepipe.Intheory,,butinapracticalcase,forametalpipe,forahosepipebecauseinfluenceoftemperatureneedtobeconsidered.Whenturbulenceflowhashappened,Theexperimenthasshownthatresistancecoefficientis

Here?isrelatedwithmaterialofpipe,suchassteeltube0.04mm,copperpipe0.0015~0.01mm,aluminum0.0015~0.06mmandhosepipe0.03mm.2.Lossesinparallelpipeatturbulenceflow

Theresistancecoefficientcanbecalculatedbyexperimentalformulaasfollowsforwater-powerslipperypipe,

(2-35)(2-36)

Thevelocityiswelldistributionatturbulenceflow,themaximumvelocityas2.4.3Minorlossesinpipesystem

Usuallytheminorlossescanbecalculatedby

Thereasonsofminorlosses:(2-37)

Thenwecancalculatetheflowrateexcepttheratingratebypressurelossformula,(2-38)

Thetotalenergylossesinawholehydraulicsystemcanbesummedaftercalculatingoutseveralsection’slossesby

(2-39)

2.5.1Thinwallorifice

2.5.2Stubbyorificeorslotorifice

2.5.3Plateclearance

2.5.4Cylinderannularclearance

2.5

FlowRateandPressureFeaturesofOrifice

2.5.1Thinwallorifice

ThinwallorificedefinedastheradioofflowlengthLtodiameteroforificedislessthan0.5asshowninFig.2-20,usuallytheorificeissharpedged.Fig.2-20Fluidflowthroughorifice

Fortheorificebeforeandaftersection1-1and2-2,TheBernoulliequationis

Thenwecanobtain

Hereisthespeedcoefficient.

(2-40)(2-41)

Thefluidflowratethatflowsthroughthisorificeasbelow,Where:A0—theacross-sectionareaofthisorifice;

Cc—thesectioncontractioncoefficient,;Cd—flowratecoefficient,Cd=CvCc。(2-42)Inthecaseofcompletecontraction,,canbecalculated

InthecaseofRe>105,=0.60~0.61inthecaseofincompletecontraction,canbeselectedbyTab.2-4Tab.2-4Flowratecoefficientsinincompletecontraction0.10.20.30.40.50.60.7Cd0.6020.6150.6340.6610.6960.7420.804

Thisisthereasonoflowresistancelosseswhenfluidflowsalongthelengthofthepipeinthinorifice.Ithaslesssensitivitytotemperature,andthinorificeisthususuallyusedtothrottleadjustor.Poppetandspoolvalveorificesaresimilartothethinorifice,sobothareallusedtothehydrauliccomponentorifices.(2-43)Fig.2-21SketchofcylinderspoolorificeAisavalveseatBisaspoolcore

Theflowratethatflowthroughtheorificeiscalculatedbelowbyequationasfollow

Ifxv>>Cr,neglectCr,theflowrateas

TheflowratecoefficientcanbeobtainedbyFig.2-22,theReynoldsnumbercanbecalculatedbyfollowing,

(2-44)(2-45)(2-46)Forahydraulicvalvewhateverflowinginorout,istheanglebetweenstreamlineandspoollineandiscalledspeeddirectionangle,itisusually.Fig.2-22Flowcoefficientontheorificeofspoolvalve

ThepoppetvalveorificeisshowninFig.2-23,Whenpoppetmovesupadistanceof,theaveragediameterof,,thentheflowrateis

Fig.2-23OrificeshapeofpoppetvalveFig.2-24Flowcoefficientofpoppetvalveorifice(2-47)WheretheflowratecoefficientcanbeobtainedbyFig.2-242.5.2Stubbyorificeorslotorifice

Fig.2-25FlowratecoefficientsinStubbyorificeTheflowrateequationforslotorificeobeystheformula

(2-31),i.e.

Theflowrateequationforthestubbyorificeisthesameasformula(2-42),buttheflowratecoefficientcanbeobtainedfromthecurveinFig.2-25.Thestubbyorificeisdefinedas,slotorifice2.5.3

PlateclearanceFig.2-26FlowinparallelplainclearanceTheflowratefluidflowthroughtheplainplateclearanceis

(2-48)Theformula(2-48)hastwostatuses:1)Fluidflowatpressuredifferential:

(2-49)2)Fluidflowbyviscosityshear:(2-50)

ThefluidflowsunderpressuredifferentialandvelocityasshowninFig.2-26.2.5.4Cylinderannularclearance1.TheflowrateequationinaconcentricannularorificeFig.2-27showsasketchofconcentricclearanceflow

Fig.2-27SketchofconcentricclearanceflowLet’sconsiderannularclearanceexpandedalongthelengthdirectionisthesameasaplainplateclearance,sosubstitutingintoformula(2-48)

Ifthemotiondirectionofcylinderisthesameasthedirectionofpressuredifferential,thesymbolin(2-51)chooses“+”,otherwise“-”.theflowrateis(2-51)(2-52)

asshowninFig.2-28,wecanabtainForverysmallclearances,isverysmalland,thenBecauseofsmallclearance,,

canbeconsideredasPlatesclearanceflow,theincrementalflowiswhereTheflowrateequationineccentricannularorifice

Fig.2-28Eccentricannularorifice(2-53)(2-54)(2-55)Ife=h0,theflowisgreaterthanitwouldbeindicatedbytheuseofequation(2-51).Substitute(2-54)into(2-55)(2-56)(2-57)Integrating:Or

(2-58)3.TheflowratethroughaconicalannularclearanceBecauseofmachiningirregularities,suchaspistonorbore,valvecoreorseatcore,somedegreeofconicmustalwaysbeexpected,asshowninFig.2-29.Fig.2-29Fluidflowthroughaconicalannularclearancea)Converseconeb)Sequencecone

WhenitiscalledinversedegreeofconicasshowninFig.2-29a;

otherwisesequencedegreeofconicasshowninFig.2-29bForthestatusofFig.2-29a,substitutingintoformula(2-51),

Becauseh=h1+xtanθ,substitutingintoformula(2-59):Integratingandsubstitutinginto

Weobtaintheflowrateas

(2-59)(2-60)(2-61)(2-62)When,flowrateis

Integratingformula(2-61)thepressuredistributioninthisclearanceflowing,andsubstitutingtheboundaryconditionath=h1,p=p1,weobtainSubstitutingformula(2-62)andinto(2-64),

,Whenu0=0,wehave

(2-64)(2-63)(2-65)(2-66)

ForthestatusofFig.2-29b,thesequencedeg

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