化工原理课程(全英文)教学课件-5

©2015YanweiWangBasicPrinciplesofChemicalEngineeringProcessesLecture5FallSemester,2014YanweiWang(王衍伟)Email:[e-mail]©2015YanweiWang2PreviouslecturePages33-43oftextbookBasicequationsoffluidflowMassbalanceinfluidflowEnergybalanceandBernoulli’sequationFluidflow©2015YanweiWang3Homeworkfrompreviouslecture#1,#2,#3#1:Measuresofflow#2:TankDrainingThroughaPipe#3:PumpflowsystemDuedate:March16,2015at15:10pm.©2015YanweiWang4Today’stopicPages43-47&67-69oftextbookBernoulliequationanditsapplicationsTankDrainingThroughaPipe(Example1.4)SiphonandcavitationDeterminetheflowdirectioninapipe(Problem1.5)Pumpflowsystem(Example1.5)Pitottube(pp.67-69oftextbook)FeedbackonHW-3-4(Couetteflow)MacroscopicmomentumbalancesDate:March16,2015©2015YanweiWang5Review:BernoulliequationOverallmechanicalenergybalanceforanincompressiblefluidatsteadystate:𝑝𝑎𝜌+𝑔𝑍𝑎+12𝑉𝑎2+𝑊𝑒=𝑝𝑏𝜌+𝑔𝑍𝑏+12𝑉𝑏2+ℎ𝑓AlltermshavetheunitofJ/kg.Theequationaboveiscalledthe“energy”formoftheEngineeringBernoulliEquation.©2015YanweiWang6TankDrainingThroughaPipe(1/6)©2015YanweiWang7TankDrainingThroughaPipe(2/6)Torricelli’sLaw𝑉𝑏=2𝑔𝐻©2015YanweiWang8TankDrainingThroughaPipe(3/6)Howlongdoesittakefortheliquidleveltodropbye.g.1m???Toanswerthisquestion,weneedtoknowtheareaofthelargeopentank.Supposeitsdiameteris1m.©2015YanweiWang9TankDrainingThroughaPipe(4/6)Continuity:−𝑑𝑧𝑑𝑡14𝜋𝐷2=𝑉𝑏14𝜋𝑑2𝑑𝑡=−𝐷2𝑑2𝑑𝑧𝑉𝑏Quasi-steadystateassumption(准稳态假设):Assumesteadystateduringasmalltimeinterval[𝑡,𝑡+𝑑𝑡],andapplytheBernoulliEquation.©2015YanweiWang10TankDrainingThroughaPipe(5/6)Continuity:𝑑𝑡=−𝐷2𝑑2𝑑𝑧𝑉𝑏Underthequasi-steadystateassumption,BernoulliEq.gives𝑔𝑧=𝑉𝑏22+ℎ𝑓Furtherassumeℎ𝑓isnegligible,wehave𝑉𝑏=2𝑔𝑧𝑑𝑡=−𝐷2𝑑2𝑑𝑧2𝑔𝑧𝑡=0𝑡𝑑𝑡=𝑧=5m𝑧=4m−𝐷2𝑑2𝑑𝑧2𝑔𝑧©2015YanweiWang11TankDrainingThroughaPipe(6/6)𝑡=0𝑡𝑑𝑡=𝑧=5m𝑧=4m−𝐷2𝑑2𝑑𝑧2𝑔𝑧𝑧:Heightofliquidlevel,unitm.𝑡:Time,units.𝐷:Diameteroftank,𝐷=1m𝑑:Diameterofthedrainingpipe,d=50mm=0.05m𝑔:standardgravity,𝑔=9.80665ms2.𝑡=0𝑡𝑑𝑡=𝑧=5m𝑧=4m−𝐷2𝑑2𝑑𝑧2𝑔𝑧=120.05212𝑔𝑧=4𝑧=5𝑑𝑧𝑧=400×0.2258×(2𝑧)𝑧=4𝑧=5s=400×0.2258×(25−4)𝑧=4𝑧=5s=42.6s.©2015YanweiWang12Siphon(1/5)SiphonAsiphon(虹吸管)isadeviceforremovingliquidfromacontainerusingapipethatrisesabovetheliquidlevelinthecontainer.Asketchofatypicalsiphonisshowntotheleft.Aswecanseetherearenopumpsorturbineshere,andifweneglectthesmalllossesinvolved,wecanapplytheBernoulliEq.toobtainanestimateofthevelocityoutofthepipe,𝑉3.Allthatmattersistheheight𝐻ofthefreesurfaceoftheliquidinthecontainerabovethelocationofthesiphonoutlet.Wehavethereforeselectedthelocation3asdatum,andlocation1asthefreesurfaceoftheliquidinthecontainer.©2015YanweiWang13Siphon(2/5)SiphonBecauselocations1and3arebothopentotheatmosphere,thegagepressureiszeroatbothlocations.Thevelocityatlocation1canbesetequaltozerobecauseofthelargecross-sectionalareainthecontainer.Bychoiceofdatum,𝑧1=𝐻and𝑧3=0.SubstitutingallofthisinformationintotheBernoulliEquationyields𝑉3=2𝑔𝐻,sameasthedrainingvelocityinthetankdrainingthroughapipeproblem.TheBernoulliEquationis𝑉3=2𝑔𝐻©2015YanweiWang14Siphon(3/5)SiphonBecausethepipehasthesamediameterthroughout,thevelocity𝑉2=𝑉3.Substitutingalltheknowninformation,weobtain：Anegativegagepressuremeansthatthepressureatlocation2islessthanatmosphericpressure????Now,letusfindthegagepressureatlocation2.WecanwritetheBernoulliEquationbetweenlocations2and3.©2015YanweiWang15Siphon(4/5)Clearlythereisalimittohowlowwecangobelowatmosphericpressure.Atfirstglance,youmightthinkthatwecanincreasetheheightdifferencebetweenthelowestandhighestpointsinthesiphonpipeuptothevaluewheretheabsolutepressureatlocation2willbezero.Thiswouldbeanincorrectconclusion.Actually,whenthepressureisgraduallyloweredinaliquid,itwillfirstreachavaluewhereitequalsthevaporpressure(蒸汽压)atroomtemperature.Whenitgoesslightlybelowthisvalue,vaporbubbleswillbegintoform,typicallyatlocationsonthepipewallthatcontaincreviceswithtrappedair,aprocessknownasheterogeneousnucleation（异相成核）.©2015YanweiWang16Siphon(5/5):CavitationWhenvaporbubblesarenucleated,wesaythat“cavitation”(气蚀现象)isoccurringintheliquid.Thepresenceofsuchvaporbubbleswillcauseproblemswithoperatingthesiphon,interferingwiththeflow.Cavitationcanbeaseriousprobleminmachineryinwhichthepressuredropsbelowatmosphericpressure,perhapsbecauseofahighvelocitybeingreachedatcertainlocations.Thevaporbubblesthatformwillcollapsewhentheliquidmovestoadifferentlocationwherethepressureishigher.Suchcollapsingofvaporbubblescanbeviolentandcancausepittinganderosionofmaterialontheblades(叶片)ofcentrifugalpumps,andonpropeller（螺旋桨）bladesusedonships.©2015YanweiWang17Determinethedirectionofflow(1/4)Thethreereservoirproblem(tobestudied)©2015YanweiWang18Determinethedirectionofflow(2/4)Problem1.5ofTextbook,pp.77-78Waterdischargesfromthereservoirthroughthedrainpipe,whichhasathroatofdiameter𝑑.Theratioof𝐷𝑑equals1.25.TheverticaldistanceℎbetweenthetankAandaxisofthedrainpipeis2m.Whatheight𝐻fromthecenterlineofthedrainpipetothewaterlevelinthereservoirisrequiredfordrawingwaterfromtankAtothethroatofthe