流体力学往届课件-08-9chapter华南理工化工学院

1.31.3BasicEquationsofFluidand/orprocessInfluiddynamicsfluidsareinmotion.Theyaremovedfromplacetoplacebymeansand/orprocess华南理工大学化工学1.3.2MassBalanceinaFlowingFluid;ContinuityTheprinciplesofphysicsmostusefulintheapplicationsofthe fluidmechanicsaremass-balance,orcontinuity;thelinear-andangular-momentum-balanceequation,andthemechanical-energybalance.华南理工大学化工学院One-dimensionalOne-dimensionalIndiscussingfluidflowitishelpfultovisualize,inthefluidstream,fluidpathscalledstreamlines.Astreamlineisanimaginarypathinamassofflowingfluidsodrawnthatateverypointthevectorofthenetvelocityalongthestreamlineuistangenttothestreamline.华南理工大学化工学院SteadyFlowalongastreamlineisthereforeone-dimensional,andasingletermforvelocityisallthatisneeded.Astreamtubeisatubeoflargeorsmallcrosssectionandofanyconvenientcross-sectionshapethatisentirelyboundedby华南理工大学化工学院SimpleMassConsidertheflowthroughaconduitofcross-sectionalareaSaattheentranceandareaSbatthe‘exit.TheaveragevelocityanddensityattheentranceareVaandρa;attheexittheyareVbandρbaab华南理工大学化工学院Atsteadystatethemassflowinequalsthemassflowout,

(1.3-6Theequationisalsocalledtheequationof华南理工大学化工学院For pressibleabIfthefluidflowsthroughachannelsofcircularcrosssection,thenthevolumetricflowrateisQVS

d4华南理工大学化工学院 ssesthroughtwostationaandb,therelationshipbetweentwovelocitiesVaandd dQ

d dfromV

2 b

da华南理工大学化工学daanddbarediametersofthechannelattheupstreamanddownstreamstations,华南理工大学化工学院1.3.3OverallEnergyBalanceforSteady-stateFlow华南理工大学化工学院Theprincipleoftheconservationofenergytoacontrolvolumeismuchthesamemannerastheprincipleofconservationof华南理工大学化工学院Theenergy-conservationequationwillthenbecombinedwiththe lawofthermodynamicstoobtainthefinaloverallenergy-balanceequation.华南理工大学化工学DerivationofOverallEnergy-BalanceSincemasscarrieswithitassociatedenergyduetoitsposition,motion,orphysicalstate,wewillfindthateachofthesetypesofenergywillappearintheenergybalance.Inaddition,wec sotransportenergyacrosstheboundaryofthesystemwithouttransferringmass华南理工大学化工学lawofthermodynamicsWelawofthermodynamicsE

QW

whereEisthetotalenergyperunitmassoffluid,Qistheheatexchangedbetweensystemandenvironmentperunitmassoffluid,andWistheworkofallkindsdoneperunitmassoffluid,canbedividedintopurelymechanicalshaftworkandthepressure–volumework.华南理工大学化工学TheenergyEinsystemcanbePotentialenergyzgofaunitmassoffluidistheenergyduetothepositionofthemassinagravitationalfieldg,wherezistherelativeheightfromareferenceplane.Kineticenergyu2/2ofaunitmassoffluidistheenergypresentbecauseofmotionofthemass.InternalenergyUofaunitmassofafluidisalloftheotherenergypresent,suchasrotationalandvibrationalenergyinchemicalbonds.华南理工大学化工学院ThetotalenergyofthefluidperunitmassisE

u22

Toobtaintheoverallenergybalance,we Eq.(1.3-9)intotheentitybalanceEq.(1.3- u2 U 2

QW华南理工大学化工学院WisbedividedintopurelymechanicalshaftWsandthepressure–volumeworkWWs EnthalpyHisdefinedHU 华南理工大学化工学院substitutingtheEqs.(2)forWand(3)forintoequation(1),andrearrangingH

2

Q

VThe

inequationaboveiskineticenergyofaunitmassallofwhichisflowinginthesamevelocityV.华南理工大学化工学院Kinetic-energycorrectionWhenthevelocityvariesacrossthestreamcrosssection,thekineticenergyisfoundinthefollowingmanner u2

WhereEktotalflowrateofkineticenergythroughtheentirecrosssection华南理工大学化工学院AssumingconstantdensitywithintheareaS

u3dS AThekineticenergyperunitmassofflowingfluid m

u3dS 2

u3dSS6华南理工大学化工学院ItisconvenienttoeliminatetheintegralbyafactoroperatingV2/2togivethecorrectvalueofthekineticenergyascalculatedfromequation6.thisfactorisdenotedbyαanddefinedbyV2

m

u3dS 2

u3dSSu3dS V3S

华南理工大学化工学院kinetic-energycanbecalculatedfromaveragevelocitybyusingαV2/2.α=2.0forlaminarflowandisabout1.05forhighlyturbulentItisusualtotakeαto1inthe华南理工大学化工学院1.3.4OverallMechanicalEnergyBalanceforSteady-stateFlowSystemThetotalenergybalance,Eq.(4)isnotoftenusedwhenappreciableenthalpychangesoccurorappreciableheatisadded(orsubtracted),sincethekinetic-andpotential-energytermsareusuallysmallandcanbe华南理工大学化工学院Asaresult,whenappreciableheatisaddedorsubtractedorlargeenthalpychangesoccur,themethodsfor ngheatbalancesdescribedaregenerallyused.华南理工大学化工学OverallMechanical-EnergyAmoreusefultypeofenergybalanceforflowingfluidsismechanicalenergy.Mechanicenergyincludestheworkterm,kineticenergy,potentialenergy,andtheflowworkpartoftheenthalpyterm,andexceptfortheheattermsandinternal华南理工大学化工学院toEnergyconvertedtoheatorinternaltoItisconvenienttowriteanenergybalanceintermsofthisloss,Σhf,whichisthesumofallfrictionallossesperunitmass.华南理工大学化工学Forthecaseofsteady-stateflow,whenaunitmassoffluidpassesfrominlettooutlet,theenthalpydifferenceisthesumofheatQexchangedbetweenthesystemandenvironment,frictionallossesΣhf,(convertedintoheat),andpressure-H

Qhf

华南理工大学化工学院Finally,wesubstituteEq.(7)into(4)and1/ρforv,toobtaintheoverallmechanical-energy-balanceequation:hf

p2

V2

(1.3-23Ifthefluidisan pressibleliquid,the es(p2-p1)/ρandEq.(1.3-23)华南理工大学化工学院V2Vhf

p2p1

2

g(z2

)Ws

1.3-25Whenthemechanicalenergyisactuallydeliveredtothefluidbythepump,thenWs<0,rearrangingEq.1.3-252222

hf

p22

华南理工大学化工学院1.3.5DiscussionontheOverallMechanicalEnergyBalanceEquationInthespecialcasewherenomechanicalenergyisadded(WS=0)andfornofriction(Σhf=0),thenEq.(1.3-25) estheBernoulliequation,Eq.(1.3-26). V V 1

22

华南理工大学化工学院Equation(1.3-26)isknownastheBernoulliequationwithoutfriction.Itisaparticularformofamechanicalenergybalance.Eachtermintheequationisascalarandthedimensionsofenergyperunitmass.ThetermsgzandV2/2arethepotentialandkineticenergy,respectively,ofaunitmassoffluid.华南理工大学化工学院Bernoulli'sBernoulli'sequationhassomeFlowisDensityisconstant(whichalsomeansthefluidispressible)FrictionlossesareTheequationrelatesthestatesattwopointsalongasinglestreamline,(notconditionsontwodifferentstreamlines).华南理工大学化工学院TheunitforEq(1.3-26)canbepkg/m3N/m2NmTorepresentthemechanicalworkdonebyforces，externaltostream，onthefluidinpushingitintothetubeortheworkrecoveredfromthefluidleavingthetube。华南理工大学化工学院Equation(1.3-26)isdividedbyg,PP12Z112gP2222gu2ThedimensionpgJfInjouleperunit华南理工大学化工学院Inthesamewaytheequation(1.3-26)ismultipliedbyρg1Z1g P2Z2g u2u222m2NNmmm2Thetermismechanicalworkdonebyforcesinjouleperunitvolumetricfluid.华南理工大学化工学院DiscussionDiscussionofBernoulliEquation(1.3-26)isusefulindealingwiththeflowof pressiblefluids.Equation(1.3-26)showsthatintheabsenceoffriction,whenthevelocityisreduced,eithertheheightabovedatumZorthepressureorbothmustincrease.Whenthevelocityincreases,itdoessoonlyattheexpenseofZorp华南理工大学化工学院TheBernoulliequationhasagreaterrangeofvaliditythanitsderivationimplies.Theprincipleofconservationofenergypermitstheextensionoftheequationtopotentialflowtakingplaceincurvedstreamtubeofvariablecrosssection.Theequationcanbemodifiedforuseinboundarylayerflow.华南理工大学化工学院Itisessentialtochooseupstreamanddownstreamstation.Stationaandbarechosenonthebasisofconvenienceandareusuallytakenatlocationswherethemostinformationaboutpressure,velocity,andheightis华南理工大学化工学院Bernoulli'sequationhasfollowingrestrictionsThetotalenergybalanceisnotoftenusedwhen( changesoccuror( )isexchangedbetweenthesystemandtheenvironment.华南理工大学化工学院Mechanicenergyincludes ),( ),and( )oftheenthalpyterm.Whenthevelocityincreases,itdoessoonlyattheexpenseof( kinetic-energycanbecalculatedfromaveragevelocitybyusingαV2/2.α=(2.0)forlaminarflowandisabout(1.05)forhighlyturbulent华南理工大学化工学院华南理工大学化工学院WaterfromthereservoirflowsthroughthepipeofdiameterD,whichthethroatdiameterisd.theratioofDtodis2,theverticaldistancehbetweenthesurfaceofliquidin Aandtheaxisofthepipeis1m.HowmuchtheHneededifthewater Aisleftto throatoftheAssumingthat AflowisapotentialA华南理工大学化工学华南理工大学化工学院 ticalpipecarryingwater,pressuregaugesareinsertedatpointsAandBwherethepipediametersare0.15mand0.075mrespectively.ThepointBis2.5mbelowAandwhentheflowratedownthepipeis0.02m3/s,thepressureatBis14715N/m2greaterthanthatatA.华南理工大学化工学AssumingthelossesinthepipebetweenAanducanbeexpressedasfindthevalueof

whereuisthevelocityat2IfthegaugesatAandBarereplacedbytubesfilledwithwaterandconnectedtoaU-tubecontainingmercuryofrelativedensity13.6,giveasketchshowinghowthelevelsinthetwolimbsoftheU-tubedifferandcalculatethevalueofthisdifferenceinmetres.[k=0.344,华南理工大学化工学院ProblemProblem华南理工大学化工学院Movingthewaterfromonecontainerintotheotherplacebygravityheadwithapipeofconstantcross-sectionalareaisshownasinfigure.find(1)ThevelocityatstationThepressureat ThemechanicalenergyinstationA,B,CAssumingthatthewaterflowsthroughthepipewithoutfrictionlosses.

B

D华南理工大学化工学院华南理工大学化工学院HereisanexampleofusingtheBernoulliequationtodeterminepressureandvelocityatwithinacontractingandexpandingpipe.Thetubeishorizontal,withz1=z2soBernoulligivesusthefollowingequationforpressureatsection2:Sowecannowcalculatethepressureatsection2华南理工大学化工学院Afluidofconstantdensity=960kg/m3isflowingsteadilythroughtheabovetube.Thediametersatthesectionsared1=100 andd2=80mm.Thegaugepr