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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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