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1Review------------Continuityequation
連續(xù)性方程2Forincompressiblefluid,densityisaconstant.Atsteadystate3IntegratedformofcontinuityequationTheflowinchannelsofcircularcrosssectionIfdensityistakenasconstant,thus4------------Navier-StokesequationsTheequationsofmotion5Eulerequation:constantdensityandzeroviscosity64.3MACROSCOPICMOMENTUMBALANCES1.Momentumoftotalstream;momentumcorrectionfactor2.Layerflowwithfreesurface3.Angular-momentumequation7Thusequation(4.42)maybewrittenThemomentumcorrectionfactor(4.41)8Forcesactingonthefluidinthedirectionofthevelocitycomponentintheequationinclude:pressurechangeinthedirectionofflow.shearstressattheboundarybetweenthefluidstreamandtheconduitor(iftheconduititselfisconsideredtobepartofthesystem)externalforcesactingonthesolidwall.theappropriatecomponentoftheforceofgravity.
9One-dimensionalflowinthexdirection,whereFw=netforceofwallofchannelonfluidFg=componentofforceofgravity(writtenforflowinupwarddirection)
(4.43)10(4.43)Macroscopicmomentumbalanceequation112.Layerflowwithfreesurface12anewtonianliquid,steadyflow,constantrate&thickness,laminarflow
ControlvolumnL,b,rfilm-wisecondensation13where
=gravityforce=shearstressonlowersurfaceofcontrolvolumeA=areaoflowersurfaceofcontrolvolume(4.44)theshearontheuppersurfaceoftheelementisneglected.
14A=bL
Rearrangingeq.(4.44)andintegratingbetweenlimitsgive
Theflowislaminar
(4.47)(4.44)velocitydistributionisparabolic15Thetotalmassflowrateofthefluidthenis
(4.48)(4.47)16(4.49)whereΓiscalledtheliquidloading(凝液負(fù)荷).TheunitsofΓarekg/m.sorlb/ft.s.biscalledperimeterofchannelincontactwithliquid潤(rùn)濕周邊17Discussionoftheassumptions:1.Noshearstressactingontheinterfaceofliquidandgas(air)Shearforcewillincreaseifthevapor(orgas)shearexist.2.LaminarflowAcriticalRenumberof2100hasoftenbeenusedforlayerflow,butfilmthicknessmeasurementsindicatedatransitionatRearound1200.18DeviationofpredictedvaluesfromthatofmeasurementsForRe≈1000,
predictedvalueisapproximatelycorrectForRe<1000,predictedvalueislargerForRe>1000,predictedvalueissmallerAverticalsurface:19AlternateformofReynoldsnumberDe
:equivalentdiameter當(dāng)量直徑20hydraulicradius水力半徑:rHwhereS=crosssectionareaofchannel流通截面
Lp=perimeterofchannelincontactwithliquid
潤(rùn)濕周邊equivalentdiameter當(dāng)量直徑:De21Forcircularpipe
Forlayerflowwithfreesurfaceofacylinderoraflatplate
Cylinder:Flatplate:
22TheReynoldsnumberforflowdownaflatplateisdefinedbytheequation.
(4.51)233.Angular-momentumequation(角動(dòng)量方程)
Analysisoftheperformanceofrotatingfluid-handlingmachinerysuchaspumps,turbines,andagitatorsisfacilitatedbytheuseofforcemomentsandangularmomentum.angularmomentum角動(dòng)量forcemoment力矩24Impeller(葉輪)ofacentrifugalpump(離心泵)orturbine2526Equation(4.53)istheangularmomentumequationforsteadytwo-dimensionalflow.
ApplicationsofEq.(4.53)aregiveninChaps.8and9.
(4.53)
274.4MECHANICALENERGYEQUATION**
1.Energyequationforpotentialflow;Bernoulliequationwithoutfriction
2.Bernoulliequation(柏努利方程):correctionforeffectsofsolidboundaries
3.Kineticenergyofstream
4.CorrectionofBernoulliequationforfluidfriction5.PumpworkinBernoulliequation281.Energyequationforpotentialflow;Bernoulliequationwithoutfriction
Potentialflow:densityisconstantandviscosityiszero.ThereforeEulerequationisuseful29ThexcomponentoftheEulerequation[Eq.(4.31)]is
(4.31)(4.45)30uaubDatumsurface31Forthissteadyunidirectionalflowprocess,Eq.(4.45)(4.56)turnto:(4.45)32(4.57)Equation(4.57)isthepointformoftheBernoulliequationwithoutfriction.
Eq.(4.56)turnto33Infpsunits
(4.58a)(4.58b)Equation(4.58)isknownastheBernoulliequationwithoutfriction.Itisaparticularformofamechanicalenergybalance
IntegratingEq.(4.57)overthesystemshowninFig.4.8gives
34DiscussionaboutBernoulliEquationStaticenergy靜壓能壓力能potentialenergy勢(shì)能,位能kineticenergy動(dòng)能J/kgtotalmechanicalenergyperunitmass,J/kg(a)EachterminEq.(4.58)isascalarandhasthedimensionsofenergyperunitmass,
35(b)Totalmechanicalenergyisconstant.Mechanicalenergycanchangefromoneformtootherform.363,4高度反映靜壓能37(c)forastationaryfluid,densityisconstantEq.(4.58a)become
(2.4)EquationofhydrostaticequilibriumistheparticularformofBernoulliequation38(d)toidentifythestreamlineorstreamtube,tochoosedefiniteupstreamanddownstreamstations,andtochoosedatumsurface.39EXAMPLE4.4.Brine,specificgravity60°F/60°F=1.15,isdrainingfromthebottomofalargeopentankthrougha50-mmpipe.Thedrainpipeendsatapoint5mbelowthesurfaceofthebrineinthetank.Consideringastreamlinestartingatthesurfaceofthebrineinthetankandpassingthroughthecenterofthedrainlinetothepointofdischarge,andassumingthatfrictionalongthestreamlineisnegligible,calculatethevelocityofflowalongthestreamlineatthepointofdischargefromthepipe.
40aa’bb’5mSolution.choosestationaasupstreamstationatthebrinesurfacestationbasdownstreamstationattheendofthestreamlineatthepointofdischarge.Thedatumformeasurementofheightscanbetakenthroughstationb41(4.58a)pa=pb=atmosphericpressureuaisnegligible,Zb=0andZa=5m.
aa’bb’5m42Fluidisflowinginasiphon.Frictionalongthestreamlineisnegligible.Calculatethepressureat2-2’,3-3’,4-4’,5-5’.43Take1-1’asupstreamstationand6-6’asdownstreamstation.6-6’asdatumface44UsingBernoulliequationfrom1-1’to2-2’.2-2’istakenasdatumlevelZ1=3m,p1=101330Pa,u2=4.43m/sGetp2=120990PaSamemethod,wecangetp3=91560Pa,p4=86660Pa,p5=91560Pa45p2=120990Pap3=91560Pa,p4=86660Pa,p5=91560Pa46threemodification:Kineticenergyofstream
FluidfrictionPumpwork
2.Bernoulliequation:correctionforeffectsofsolidboundaries
473.KineticenergyofstreamConsideranelementofcross-sectionalareadS.Themassflowratethroughthisis48KineticenergyflowratethroughareadSisThetotalrateofflowofkineticenergythroughtheentirecrosssectionSis,assumingconstantdensitywithintheareaS,
(4.59)49(4.60)Defineα50αis2.0forlaminarflowandisabout1.05forhighlyturbulentflow.
Kineticenergycorrectionfactor動(dòng)能校正因子(4.61)5152534點(diǎn)反映動(dòng)能和靜壓能544.CorrectionofBernoulliequationforfluidfriction
Forincompressiblefluids,theBernoulliequationbecomes(4.62)55hfFriction,frictionloss,energyloss阻力損失Unit:J/kgenergyperunitmass
hf,isalwayspositive.Itiszeroinpotentialflow
hfrepresentsthelossofmechanicalenergyatallpointsbetweenstationsaandb.hfisnotinterconvertiblewiththemechanicalenergyquantities.
56TwokindsoffrictionlossFrictiongeneratedinunseparatedboundarylayersiscalledskinfriction(表面摩擦阻力).
Whenboundarylayersseparateandformwakes,additionalenergydissipationappearswithin
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