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

(1.3-6Theequationisalsocalledtheequationof華南理工大學化工學院For pressibleabIfthefluidflowsthroughachannelsofcircularcrosssection,thenthevolumetricflowrateisQVS

d4華南理工大學化工學院 ssesthroughtwostationaandb,therelationshipbetweentwovelocitiesVaandd dQ

d dfromV

2 b

da華南理工大學化工學daanddbarediametersofthechannelattheupstreamanddownstreamstations,華南理工大學化工學院1.3.3OverallEnergyBalanceforSteady-stateFlow華南理工大學化工學院Theprincipleoftheconservationofenergytoacontrolvolumeismuchthesamemannerastheprincipleofconservationof華南理工大學化工學院Theenergy-conservationequationwillthenbecombinedwiththe lawofthermodynamicstoobtainthefinaloverallenergy-balanceequation.華南理工大學化工學DerivationofOverallEnergy-BalanceSincemasscarrieswithitassociatedenergyduetoitsposition,motion,orphysicalstate,wewillfindthateachofthesetypesofenergywillappearintheenergybalance.Inaddition,wec sotransportenergyacrosstheboundaryofthesystemwithouttransferringmass華南理工大學化工學lawofthermodynamicsWelawofthermodynamicsE

QW

whereEisthetotalenergyperunitmassoffluid,Qistheheatexchangedbetweensystemandenvironmentperunitmassoffluid,andWistheworkofallkindsdoneperunitmassoffluid,canbedividedintopurelymechanicalshaftworkandthepressure–volumework.華南理工大學化工學TheenergyEinsystemcanbePotentialenergyzgofaunitmassoffluidistheenergyduetothepositionofthemassinagravitationalfieldg,wherezistherelativeheightfromareferenceplane.Kineticenergyu2/2ofaunitmassoffluidistheenergypresentbecauseofmotionofthemass.InternalenergyUofaunitmassofafluidisalloftheotherenergypresent,suchasrotationalandvibrationalenergyinchemicalbonds.華南理工大學化工學院ThetotalenergyofthefluidperunitmassisE

u22

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

QW華南理工大學化工學院WisbedividedintopurelymechanicalshaftWsandthepressure–volumeworkWWs EnthalpyHisdefinedHU 華南理工大學化工學院substitutingtheEqs.(2)forWand(3)forintoequation(1),andrearrangingH

2

Q

VThe

inequationaboveiskineticenergyofaunitmassallofwhichisflowinginthesamevelocityV.華南理工大學化工學院Kinetic-energycorrectionWhenthevelocityvariesacrossthestreamcrosssection,thekineticenergyisfoundinthefollowingmanner u2

WhereEktotalflowrateofkineticenergythroughtheentirecrosssection華南理工大學化工學院AssumingconstantdensitywithintheareaS

u3dS AThekineticenergyperunitmassofflowingfluid m

u3dS 2

u3dSS6華南理工大學化工學院ItisconvenienttoeliminatetheintegralbyafactoroperatingV2/2togivethecorrectvalueofthekineticenergyascalculatedfromequation6.thisfactorisdenotedbyαanddefinedbyV2

m

u3dS 2

u3dSSu3dS V3S

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

Qhf

華南理工大學化工學院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)華南理工大學化工學院V2Vhf

p2p1

2

g(z2

)Ws

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

hf

p22

華南理工大學化工學院1.3.5DiscussionontheOverallMechanicalEnergyBalanceEquationInthespecialcasewherenomechanicalenergyisadded(WS=0)andfornofriction(Σhf=0),thenEq.(1.3-25) estheBernoulliequation,Eq.(1.3-26). V V 1

22

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

whereuisthevelocityat2IfthegaugesatAandBarereplacedbytubesfilledwithwaterandconnectedtoaU-tubecontainingmercuryofrelativedensity13.6,giveasketchshowinghowthelevelsinthetwolimbsoftheU-tubedifferandcalculatethevalueofthisdifferenceinmetres.[k=0.344,華南理工大學化工學院ProblemProblem華南理工大學化工學院Movingthewaterfromonecontainerintotheotherplacebygravityheadwithapipeofconstantcross-sectionalareaisshownasinfigure.find(1)ThevelocityatstationThepressureat ThemechanicalenergyinstationA,B,CAssumingthatthewaterflowsthroughthepipewithoutfrictionlosses.

B

D華南理工大學化工學院華南理工大學化工學院HereisanexampleofusingtheBernoulliequationtodeterminepressureandvelocityatwithinacontractingandexpandingpipe.Thetubeishorizontal,withz1=z2soBernoulligivesusthefollowingequationforpressureatsection2:Sowecannowcalculatethepressureatsection2華南理工大學化工學院Afluidofconstantdensity=960kg/m3isflowingsteadilythroughtheabovetube.Thediametersatthesectionsared1=100 andd2=80mm.Thegaugepr