流体力学往届课件08-9chapter

华南理工大学化工学院，伍钦,,1.3.6MacroscopicMomentumBalances2.LayerflowwithfreesurfaceInoneformoflayerflowtheliquidlayerhasafreesurfaceandflowsundertheforceofgravityoveraninclinedorverticalsurface.华南理工大学化工学院，伍钦,,Ifsuchflowisinsteadystate,withfullydevelopedvelocitygradients,thethicknessofthelayerisconstant.Thereissolittledragatthefreeliquidsurfacethattheshearstresstherecanbeignored.Iftheflowislaminarandtheliquidsurfaceisflatandfreefromripples,thefluidmotioncanbeanalyzedmathematically.华南理工大学化工学院，伍钦,,Alayerofliquidflowinginsteadyflowatconstantrateandthicknessoveraflatplate.Theplateisinclinedatanangleφwiththevertical.Thebreadthofthelayerinthedirectionperpendiculartotheplaneofthefigureisb,andthethicknessofthelayerinthedirectionperpendiculartotheplateisδ华南理工大学化工学院，伍钦,,Theuppersurfaceofthecontrolvolumeisincontactwiththeatmosphere,thetwoendsareplanesperpendiculartotheplateatadistanceL

apart,andthelowersurfaceistheplaneparallelwiththewallatadistancerfromtheuppersurfaceofthelayer.华南理工大学化工学院，伍钦,,Thepossibleforcesactingonthecontrolvolumeinadirectionparalleltotheflowarethepressureforcesontheends,theshearforcesontheupperandlowerfaces,andthecomponentoftheforceofgravityinthedirectionofflow.华南理工大学化工学院，伍钦,,Fromthisequation,notingthatA=bLand华南理工大学化工学院，伍钦,,SincetheflowislaminarRearrangingandintegratingbetweenlimitsgive华南理工大学化工学院，伍钦,,Equation(1.3-40)showsthatinlaminarflowonaplatethevelocitydistributionisparabolic.(1.3-40)华南理工大学化工学院，伍钦,,Industrialprocessesnecessarilyrequiretheflowoffluidsthroughpipes,tubes,andchannelswithanoncircularcrosssection.Thissectiondealswiththesteadyflowofpressiblefluidsthroughclosedpipesandchannels1.4pressibleFlowinPipesandChannels华南理工大学化工学院，伍钦,,FigureFluidelementinsteadyflowthroughpipe.RFlowpττdL-(p+dp)rConsiderthesteadyflowoffluidofconstantdensityinfullydevelopedflowthroughahorizontalpipe.Visualizeadisk-shapedelementoffluid,concentricwiththeaxisofthetube,ofradiusrandlengthdL,asshowninFig.华南理工大学化工学院，伍钦,,

forcesactingontheelement:Forcefromthepressureontheupstreamfacesofthedisk:Abpb=πr2(p+dP)Forcefromthepressureonthedownstreamfacesofthedisk:Aapa=πr2p华南理工大学化工学院，伍钦,,Shearforceactingontherimoftheelement:(2πrdL)τSothetotalforceactingontheelementisSimplifyingthisequationanddividingbyπr2dL(1.4-1)华南理工大学化工学院，伍钦,,Insteadyflow,thepressureatanygivencrosssectionofastreamtubeisconstant,sothatdp/dLisindependentofr,dp/dL=Δp/ΔL,Eq.(1.4-1)thenes1.4-1华南理工大学化工学院，伍钦,,Thismeansthatthemomentumfluxvarieslinearlywiththeradius,asshowninFig.,andthemaximumvalueoccursatr=Ratthewall.华南理工大学化工学院，伍钦,,SubstitutingNewton'slawofviscosity,intoEq.weobtainthefollowingdifferentialequationforthevelocity:华南理工大学化工学院，伍钦,,Integratingusingtheboundaryconditionthatatthewall,u=0atr=R,weobtaintheequationforthevelocitydistribution1.4-15

Thisresultshowsthatthevelocitydistributionisparabolic,asshowninFig华南理工大学化工学院，伍钦,,TheaveragevelocityVforacrosssectionisfoundbysummingupallthevelocitiesoverthecrosssectionanddividingbythecross-sectionalareasubstitutinguintoequationandintegrating华南理工大学化工学院，伍钦,,EquationcalledHagen–Poiseuilleequation,relatesthepressuredropandaveragevelocityforlaminarflowinahorizontalpipe.华南理工大学化工学院，伍钦,,ThemaximumvelocityforapipeisfoundfromEq.(1.4-15)andoccursatr=0:华南理工大学化工学院，伍钦,,wefindthat1.4-16Also,1.4-17

华南理工大学化工学院，伍钦,,ShellMomentumBalanceforFallingFilmWenowuseanapproachsimilartothatusedforlaminarflowinsideapipeforthecaseofflowofafluidasafilminlaminarflowdownundertheforceofgravityoveraverticalsurface.ThecontrolvolumeforthefallingfilmisshowninFig.2.9-3a.华南理工大学化工学院，伍钦,,FluidthicknessisΔxandhasalengthofLintheverticalzdirection.Thisregionissufficientlyfarfromtheentranceandexitregionssothattheflowisnotaffectedbytheseregions.Thismeansthevelocityu(x)doesnotdependonpositionz.华南理工大学化工学院，伍钦,,Theshearstresscanbeignoredatthefreeliquidsurface.Iftheflowislaminarandtheliquidsurfaceisflatandfreefromripples.Thefluidmotioncanbeanalyzedmathematically华南理工大学化工学院，伍钦,,Thepossibleforcesactingonthecontrolvolumeinadirectionparalleltothefloware

thepressureforcesattheends

theshearforcesonthefaces

thecomponentoftheforceofgravityinthedirectionofflow.华南理工大学化工学院，伍钦,,Sincethepressureontheoutersurfaceisatmospheric,thepressuresonthecontrolvolumeattheendsofthevolumeareequaland‘oppositelydirected.Also,byassumption,theshearonthefreeliquidsurfaceoftheelementisneglected.华南理工大学化工学院，伍钦,,Thetwoforcesremainingarethereforetheshearforceontherightsurfaceofthecontrolvolumeandthecomponentofthegravityinthedirectionofflow.Fromthisequation,notingthatA=WLandFg=gρLWx华南理工大学化工学院，伍钦,,SoSincetheflowislaminar,τ=-µdu/dxandRearrangingandintegratingbetweenlimitsgives(theboundaryconditionthatu=0atx=δ)华南理工大学化工学院，伍钦,,andWhereδisthetotalthicknessoftheliquidlayer2.9-25

华南理工大学化工学院，伍钦,,Thismeansthevelocityprofileisparabolic,asshowninFig.2.9-3b.Themaximumvelocityoccursatx=0inEq.(2.9-25)andis2.9-26

华南理工大学化工学院，伍钦,,Theaveragevelocitycanbefound:2.9-27

SubstitutingEq.(2.9-25)into(2.9-27)andintegrating2.9-28

华南理工大学化工学院，伍钦,,

VelocityProfilesinPipesFigureisaplotoftherelativedistancefromthecenterofthepipeversustheratiooflocalvelocitytomaximumvelocityu/umax.Forlaminarflow,thevelocityprofileisatrueparabola.Thevelocityatthewalliszero.华南理工大学化工学院，伍钦,,InmanyengineeringapplicationstheaveragevelocityVinapipeisthemostuseful.Insomecasesonlytheumaxatthecenterpointofthetubeismeasured.TherelationshipbetweenumaxandVcanbeusedtodetermineV.experimentallymeasuredvaluesofV/umaxareplottedasafunctionoftheReynoldsnumbers华南理工大学化工学院，伍钦,,Theaveragevelocityisprecisely0.5timesthemaximumvelocityatthecenterforlaminarflow.Ontheotherhand,forturbulentflow,thecurveissomewhatflattenedinthecenterandtheaveragevelocityisabout0.8timesthemaximum.Thisvalueof0.8variesslightly,dependingupontheReynoldsnumber。华南理工大学化工学院，伍钦,,Theliquidlayerhasafreesurfaceandflowsinlaminarflowundertheforceofgravityoveraverticalsurface,thevelocityprofileis()Averagevelocityis()Problem1华南理工大学化工学院，伍钦,,Theaveragevelocityis()timesthemaximumvelocityatthecenteroftubeforlaminarflow.Theaveragevelocityis()timesthemaximumvelocityatcenterforturbulentflow华南理工大学化工学院，伍钦,,Themaximumvalueofthemomentumfluxoccurs()foralaminarflow.Themaximumvalueofthevelocityoccurs()foralaminarflowinpipe.Inlaminarflowthevelocitydistributionwithr