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外文翻译--一维数学模型.doc

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外文翻译--一维数学模型.doc

英文原文3.1OneDimensionalMathematicalModel51TheConservationofInternalEnergyθωω.ddvpQhmhmdQduoutoutinin(3.1)whereθisangleofrotationofthemainrotor,hhθisspecificenthalpy,m˙m˙θismassflowrateppθ,fluidpressureintheworkingchambercontrolvolume,˙Q˙Qθ,heattransferbetweenthefluidandthecompressorsurrounding,˙V˙Vθlocalvolumeofthecompressorworkingchamber.Intheaboveequationthesubscriptsinandoutdenotethefluidinflowandoutflow.Thefluidtotalenthalpyinflowconsistsofthefollowingcomponentsoiloilglglsucsuvininhmhmhmh.,,...m3.2wheresubscriptsl,gdenoteleakagegainsuc,suctionconditions,andoildenotesoil.Thefluidtotaloutflowenthalpyconsistsoflllldisdisououthmhmhm,,..3.3whereindicesl,ldenoteleakagelossanddisdenotesthedischargeconditionswithm˙disdenotingthedischargemassflowrateofthegascontaminatedwiththeoilorotherliquidinjected.TherighthandsideoftheenergyequationconsistsofthefollowingtermswhicharemodelTheheatexchangebetweenthefluidandthecompressorscrewrotorsandcasingandthroughthemtothesurrounding,duetothedifferenceintemperaturesofgasandthecasingandrotorsurfacesisaccountedforbytheheattransfercoefficientevaluatedfromtheexpressionNu0.023Re0.8.ForthecharacteristiclengthintheReynoldsandNusseltnumberthedifferencebetweentheouterandinnerdiametersofthemainrotorwasadopted.Thismaynotbethemostappropriatedimensionforthispurpose,butthecharacteristiclengthappearsintheexpressionfortheheattransfercoefficientwiththeexponentof0.2andthereforehaslittleinfluenceaslongasitremainswithinthesameorderofmagnitudeasothercharacteristicdimensionsofthemachineandaslongasitcharacterizesthecompressorsize.ThecharacteristicvelocityfortheRenumberiscomputedfromthelocalmassflowandthecrosssectionalarea.Herethesurfaceoverwhichtheheatisexchanged,aswellasthewalltemperature,dependontherotationangleθofthemainrotor.Theenergygainduetothegasinflowintotheworkingvolumeisrepresentedbytheproductofthemassintakeanditsaveragedenthalpy.Assuch,theenergyinflowvarieswiththerotationalangle.Duringthesuctionperiod,gasenterstheworkingvolumebringingtheaveragedgasenthalpy,523CalculationofScrewCompressorPerformancewhichdominatesinthesuctionchamber.However,duringthetimewhenthesuctionportisclosed,acertainamountofthecompressedgasleaksintothecompressorworkingchamberthroughtheclearances.Themassofthisgas,aswellasitsenthalpyaredeterminedonthebasisofthegasleakageequations.Theworkingvolumeisfilledwithgasduetoleakageonlywhenthegaspressureinthespacearoundtheworkingvolumeishigher,otherwisethereisnoleakage,oritisintheoppositedirection,i.e.fromtheworkingchambertowardsotherplenums.Thetotalinflowenthalpyisfurthercorrectedbytheamountofenthalpybroughtintotheworkingchamberbytheinjectedoil.Theenergylossduetothegasoutflowfromtheworkingvolumeisdefinedbytheproductofthemassoutflowanditsaveragedgasenthalpy.Duringdelivery,thisisthecompressedgasenteringthedischargeplenum,while,inthecaseofexpansionduetoinappropriatedischargepressure,thisisthegaswhichleaksthroughtheclearancesfromtheworkingvolumeintotheneighbouringspaceatalowerpressure.Ifthepressureintheworkingchamberislowerthanthatinthedischargechamberandifthedischargeportisopen,theflowwillbeinthereversedirection,i.e.fromthedischargeplenumintotheworkingchamber.Thechangeofmasshasanegativesignanditsassumedenthalpyisequaltotheaveragedgasenthalpyinthepressurechamber.ThethermodynamicworksuppliedtothegasduringthecompressionprocessisrepresentedbythetermpdVdθ.Thistermisevaluatedfromthelocalpressureandlocalvolumechangerate.Thelatterisobtainedfromtherelationshipsdefiningthescrewkinematicswhichyieldtheinstantaneousworkingvolumeanditschangewithrotationangle.InfactthetermdV/dϕcanbeidentifiedwiththeinstantaneousinterlobearea,correctedforthecapturedandoverlappingareas.Ifoilorotherfluidisinjectedintotheworkingchamberofthecompressor,theoilmassinflowanditsenthalpyshouldbeincludedintheinflowterms.Inspiteofthefactthattheoilmassfractioninthemixtureissignificant,itseffectuponthevolumeflowrateisonlymarginalbecausetheoilvolumefractionisusuallyverysmall.Thetotalfluidmassoutflowalsoincludestheinjectedoil,thegreaterpartofwhichremainsmixedwiththeworkingfluid.Heattransferbetweenthegasandoildropletsisdescribedbyafirstorderdifferentialequation.TheMassContinuityEquationoutoutininhmhmdmd...θω(3.4)Themassinflowrateconsistsofoilglsucininmmmhm.,..(3.5)3.1OneDimensionalMathematicalModel53Themassoutflowrateconsistsof.,..lldisoutmmm3.6EachofthemassflowratesatisfiesthecontinuityequationAm.3.7wherewm/sdenotesfluidvelocity,ρ–fluiddensityandA–theflowcrosssectionarea.TheinstantaneousdensityρρθisobtainedfromtheinstantaneousmassmtrappedinthecontrolvolumeandthesizeofthecorrespondinginstantaneousvolumeV,asρm/V.3.1.2SuctionandDischargePortsThecrosssectionareaAisobtainedfromthecompressorgeometryanditmaybeconsideredasaperiodicfunctionoftheangleofrotationθ.ThesuctionportareaisdefinedbysucosucAAsin,suc3.8wheresucmeansthestartingvalueofθatthemomentofthesuctionportopening,andAsuc,0denotesthemaximumvalueofthesuctionportcrosssectionarea.Thereferencevalueoftherotationangleθisassumedatthesuctionportclosingsothatsuctionendsatθ0,ifnotspecifieddifferently.ThedischargeportareaislikewisedefinedbysecodisAAsin,dis3.9wheresubscriptedenotestheendofdischarge,cdenotestheendofcompressionandAdis,0standsforthemaximumvalueofthedischargeportcrosssectionalarea.SuctionandDischargePortFluidVelocities212hh3.10whereμisthesuction/dischargeorificeflowcoefficient,whilesubscripts1and2denotetheconditionsdownstreamandupstreamoftheconsideredport.Theprovisionsuppliedinthecomputercodewillcalculateforareverseflowifh2h1.543CalculationofScrewCompressorPerformance3.1.3GasLeakagesLeakagesinascrewmachineamounttoasubstantialpartofthetotalflowrateandthereforeplayanimportantrolebecausetheyinfluencetheprocessbothbyaffectingthecompressormassflowrateorcompressordelivery,i.e.volumetricefficiencyandthethermodynamicefficiencyofthecompressionwork.Forpracticalcomputationoftheeffectsofleakageuponthecompressorprocess,itisconvenienttodistinguishtwotypesofleakages,accordingtotheirdirectionwithregardtotheworkingchambergainandlossleakages.Thegainleakagescomefromthedischargeplenumandfromtheneighbouringworkingchamberwhichhasahigherpressure.Thelossleakagesleavethechambertowardsthesuctionplenumandtotheneighbouringchamberwithalowerpressure.Computationoftheleakagevelocityfollowsfromconsiderationofthefluidflowthroughtheclearance.TheprocessisessentiallyadiabaticFannoflow.Inordertosimplifythecomputation,theflowisissometimesassumedtobeatconstanttemperatureratherthanatconstantenthalpy.Thisdeparturefromtheprevailingadiabaticconditionshasonlyamarginalinfluenceiftheanalysisiscarriedoutindifferentialform,i.e.forthesmallchangesoftherotationalangle,asfollowedinthepresentmodel.Thepresentmodeltreatsonlygasleakage.Noattemptismadetoaccountforleakageofagasliquidmixture,whiletheeffectoftheoilfilmcanbeincorporatedbyanappropriatereductionoftheclearancegaps.AnidealizedclearancegapisassumedtohavearectangularshapeandthemassflowofleakingfluidisexpressedbythecontinuityequationglllAm.3.11whererandwaredensityandvelocityoftheleakinggas,Aglgδgtheclearancegapcrosssectionalarea,lgleakageclearancelength,sealingline,δgleakageclearancewidthorgap,μμRe,Matheleakageflowdischargecoefficient.Fourdifferentsealinglinesaredistinguishedinascrewcompressortheleadingtipsealinglineformedbetweenthemainandgaterotorforwardtipandcasing,thetrailingtipsealinglineformedbetweenthemainandgatereversetipandcasing,thefrontsealinglinebetweenthedischargerotorfrontandthehousingandtheinterlobesealinglinebetweentherotors.Allsealinglineshaveclearancegapswhichformleakageareas.Additionally,thetipleakageareasareaccompaniedbyblowholeareas.Accordingtothetypeandpositionofleakageclearances,fivedifferentleakagescanbeidentified,namelylossesthroughthetrailingtipsealingandfrontsealingandgainsthroughtheleadingandfrontsealing.Thefifth,throughleakagedoesnotdirectlyaffecttheprocessintheworkingchamber,butitpassesthroughitfromthedischargeplenumtowardsthesuctionport.Theleakinggasvelocityisderivedfromthemomentumequation,whichaccountsforthefluidwallfriction3.1OneDimensionalMathematicalModel5502211Dgdxfdpdl3.12wherefRe,MaisthefrictioncoefficientwhichisdependentontheReynoldsandMachnumbers,Dgistheeffectivediameteroftheclearancegap,Dg≈2δganddxisthelengthincrement.FromthecontinuityequationandassumingthatT≈consttoeliminategasdensityintermsofpressure,theequationcanbeintegratedintermsofpressurefromthehighpressuresideat

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