天大化工原理英文版課件Chapter42連續(xù)性方程

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(凝液負荷).TheunitsofΓarekg/m.sorlb/ft.s.biscalledperimeterofchannelincontactwithliquid潤濕周邊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

潤濕周邊equivalentdiameter當(dāng)量直徑:De21Forcircularpipe

Forlayerflowwithfreesurfaceofacylinderoraflatplate

Cylinder:Flatplate:

22TheReynoldsnumberforflowdownaflatplateisdefinedbytheequation.

（4.51）233.Angular-momentumequation(角動量方程)

Analysisoftheperformanceofrotatingfluid-handlingmachinerysuchaspumps,turbines,andagitatorsisfacilitatedbytheuseofforcemomentsandangularmomentum.angularmomentum角動量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勢能，位能kineticenergy動能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動能校正因子(4.61)5152534點反映動能和靜壓能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