气体动力学第三讲 控制体分析2

CoursefortheGE3program

Fundamentalsof

GasDynamics

Lecture3Prof.YanchengYou(尤延铖)yancheng.you@Lecture3:ControlVolumeAnalysis2ControlVolumeAnalysis(continue)ConservationofEnergyCommentsonEntropyPressure-EnergyEquationTheStagnationConceptStagnationPressure-EnergyEquationConservationLaws:EnergyThefirstlawofthermodynamicsisastatementofconservationofenergyQ=thenetheattransferredintothesystemW=thenetworkdonebythesystemE=thechangeintotalenergyofthesystemAlsovalidatanyinstantoftimeLecture3:ControlVolumeAnalysis2ConservationLaws:EnergyδQ/dtandδW/dtrepresentinstantaneousratesofheatandworktransferbetweenthesystemanditssurroundings(notmaterialderivatives)dE/dtisamaterialderivativesinceenergyisapropertyofthesystem.Lecture3:ControlVolumeAnalysis2ConservationLaws:EnergyForone-dimensionalflow,e,ρ,andVareconstantAssumingthatthevelocityVisperpendiculartothesurfaceAPositivewherefluidleavesthecontrolvolumeandnegativewherefluidentersthecontrolvolumeChoosethecontrolsurfacecarefullyLecture3:ControlVolumeAnalysis2ConservationLaws:EnergyChoosethecontrolsurfacecarefullyExceptionwherefluidentersandleavesthesystem,orwhereamechanicaldevicesuchasashaftcrossestheboundariesofthesystem.Lecture3:ControlVolumeAnalysis2ConservationLaws:EnergyδWs/dtisisaccomplishedbyshearstressesbetweenthedeviceandthefluid(subscriptsforshearstressesorshaftwork)Theotherworkquantitiesconsideredarewherefluidentersandleavesthesystem.(pressureforces)Lecture3:ControlVolumeAnalysis2ConservationLaws:EnergyTheshadedareaattheinletrepresentsthefluidthatentersthecontrolvolumeduringtimedt.Lecture3:ControlVolumeAnalysis2ConservationLaws:EnergyThustherateofdoingflowworkisThetotalworkRewritetheenergyequationinamoreusefulformLecture3:ControlVolumeAnalysis2SteadyFlow??ConservationLaws:EnergyIfthereisonlyonesectionwherefluidleavesandonesectionwherefluidenters(flowcontinuity)WedefineLecture3:ControlVolumeAnalysis2HeatfluxrateWorkfluxrateConservationLaws:EnergysubstitutionforeandusethedefinitionofenthalpyLecture3:ControlVolumeAnalysis2DifferentialFormofEnergyEquationFluidleavesthecontrolvolumewithpropertiesthathavechangedslightlyasindicatedbyρ+dρ,u+du,andsoon.Lecture3:ControlVolumeAnalysis2DifferentialFormofEnergyEquationCancelliketermandthehighorderterm(HOT)NotingthatWeobtainLecture3:ControlVolumeAnalysis2DifferentialFormofEnergyEquationSinceWehaveThiscanbeintegrateddirectlytoproduceequationLecture3:ControlVolumeAnalysis2ExamplesLecture3:ControlVolumeAnalysis2Page40ExamplesLecture3:ControlVolumeAnalysis2Page41Lecture3:ControlVolumeAnalysis2ExamplesSteps:1.Sketchtheflowsystemandidentifythecontrolvolume.2.Labelsectionswherefluidentersandleavesthecontrolvolume.3.Notewhereenergy(QandWs)crossesthecontrolsurface.4.Recordallknownquantitieswiththeirunits.5.Makeanylogicalassumptionsregardingunknowninformation.6.Solvefortheunknownsbyasystematicapplicationofthebasicequations.State:asimpledensityrelationsuchasp=ρRTorρ=constantContinuity:derivedfromconservationofmassEnergy:derivedfromconservationofenergyLecture3:ControlVolumeAnalysis2CommentsonEntropyDefinitionofEntropyRmeansfictitiousreversibleprocess,itmaynotrepresentthetotalentropychange.Actualheattransferbetweenthesystemandits(external)surroundings.dSecanbeeitherpositiveornegative.ItdependsonδQObviously,dSe=0foranadiabaticprocess.Lecture3:ControlVolumeAnalysis2CommentsonEntropydSirepresentsthatportionofentropychangecausedbyirreversibleeffects,whichareinternalinnature.AllirreversibilitiesgenerateentropydSi=0onlyforareversibleprocess.Isentropic,dS=0TakingthecyclicintegralofequationLecture3:ControlVolumeAnalysis2CommentsonEntropyEntropy(S)isapropertyIrreversibleeffectsalwaysgenerateentropyThus,InequalityofClausiusLecture3:ControlVolumeAnalysis2Pressure-energyequationStartingwiththethermodynamicpropertyrelationWeintroduceds=dse+dsiandv=1/ρ,RecallingtheconservationofenergyLecture3:ControlVolumeAnalysis2Pressure-energyequationPressure–energyequationForinstance,ifnoshaftworkcrossestheboundary(δws=0)andiftherearenolosses(dsi=0),InvisicdEulerequationLecture3:ControlVolumeAnalysis2TheStagnationConceptStagnationstate:thisisareferencestatedefinedasthatthermodynamicstatewhichwouldexistifthefluidwerebroughttozerovelocityandzeropotential.Thestagnationstatemustbereached(1)withoutanyenergyexchange(Q=W=0)(2)withoutlosses.dse=0dsi=0Lecture3:ControlVolumeAnalysis2TheStagnationConceptThestaticpropertiesshownas(a).Atlocation(b)thefluidhasbeenbroughttozerovelocityandzeropotentialundertheforegoingrestrictions.Lecture3:ControlVolumeAnalysis2TheStagnationConceptThestaticpropertiesshownas(a).Atlocation(b)thefluidhasbeenbroughttozerovelocityandzeropotentialundertheforegoingrestrictions.Lecture3:ControlVolumeAnalysis2TheStagnationConceptCondition(b)representsthestagnationstatecorrespondingtothestaticstate(a).Wecallhbthestagnationortotalenthalpycorrespondingtostate(a)anddesignateitashta.Orforanystate,wehaveingeneral,Lecture3:ControlVolumeAnalysis2TheStagnationConceptWhendealingwithgases,potentialchangesareusuallyneglected,andwewriteLecture3:ControlVolumeAnalysis2TheStagnationConceptIntroductionofthestagnation(ortotal)enthalpymakesitpossibletowriteequationsinamorecompactformbecomesLecture3:ControlVolumeAnalysis2TheStagnationConceptLecture3:ControlVolumeAnalysis2TheStagnationConceptInanyadiabaticno-worksteadyone-dimensionalflowsystem,thestagnationenthalpyremainsconstant,irrespectiveofthelosses.Youshouldnotethatthestagnationstateisareferencestatethatmayormaynotactuallyexistintheflowsystem.Lecture3:ControlVolumeAnalysis2TheStagnationConceptAlso,onemustrealizethatwhentheframeofreferenceischanged,stagnationconditionschange,althoughthestaticconditionsremainthesame.Lecture3:ControlVolumeAnalysis2Stagnationpressure-energyequationIfweconsiderainfinitesimalprocess,Eventhoughthestagnationstatesdonotactuallyexist,theyrepresentlegitimatethermodynamicstateLecture3:ControlVolumeAnalysis2TheStagnationConceptRecalltheenergyequationwrittenintheformBysubstitutingdhtRecallthatthestagnationpressure–energyequation:Lecture3:ControlVolumeAnalysis2TheStagnationConcep