流體力學(xué)（清華大學(xué)張兆順54講） PPT課件 116-6-1流體力學(xué)第六章

VI.VISCOUSINTERNALFLOW符松清華大學(xué)航天航空學(xué)院2023/4/61清華大學(xué)工程力學(xué)系IV.DIFFERENTIALRELATIONSFORAFLUIDPARTICLETodate,wehaveconsideredonlyproblemswheretheviscouseffectswereeither: a.known: i.e.-knownFDor hf

b.negligible: i.e.-inviscidflowThischapterpresentsmethodologiesforpredictingviscouseffectsandviscousflowlossesforinternalflowsinpipes,ducts,andconduits.Typically,thefirststepindeterminingviscouseffectsistodeterminetheflowregimeatthespecifiedcondition.Thetwopossibilitiesare: a.Laminarflow

b.Turbulentflow2023/4/62清華大學(xué)工程力學(xué)系IV.DIFFERENTIALRELATIONSFORAFLUIDPARTICLEForsteadyflowataknownflowrate,theseregionsexhibitthefollowing:Laminarflow: Alocalvelocityconstantwithtime,butwhichvariesspatiallyduetoviscousshearandgeometry.Turbulentflow:Alocalvelocitywhichhasaconstantmeanvaluebutalsohasastatisticallyrandomfluctuatingcomponentduetoturbulenceintheflow.2023/4/63清華大學(xué)工程力學(xué)系IV.DIFFERENTIALRELATIONSFORAFLUIDPARTICLETypicalplotsofvelocitytimehistoriesforlaminarflow,turbulentflow,andtheregionoftransitionbetweenthetwoareshownbelow.Fig.6.1(a)Laminar,(b)transition,and(c)turbulentflowvelocitytimehistories.2023/4/64清華大學(xué)工程力學(xué)系IV.DIFFERENTIALRELATIONSFORAFLUIDPARTICLE2023/4/65清華大學(xué)工程力學(xué)系IV.DIFFERENTIALRELATIONSFORAFLUIDPARTICLEWecannowdefinethe

Recr

ocriticalortransitionReynoldsnumber

Recr

oReynoldsnumberbelowwhichtheflowislaminar,abovewhichtheflowisturbulentWhiletransitioncanoccuroverarangeofRe,wewillusethefollowingforinternalpipeorductflow:2023/4/66清華大學(xué)工程力學(xué)系InternalViscousFlowAsecondclassificationconcernswhethertheflowhassignificantentranceregioneffectsorisfullydeveloped.Thefollowingfigureindicatesthecharacteristicsoftheentranceregionforinternalflows.Notethattheslopeofthestreamwisepressuredistributionisgreaterintheentranceregionthaninthefullydevelopedregion(duetofrictionaleffectsplustheaccelerationofthecoreflowastheboundarylayerdevelops).2023/4/67清華大學(xué)工程力學(xué)系InternalViscousFlowTypicalcriteriaforthelengthoftheentranceregionaregivenasfollows: Laminar:

Turbulent

where:Le=lengthoftheentranceregion2023/4/68清華大學(xué)工程力學(xué)系2023/4/69清華大學(xué)工程力學(xué)系InternalViscousFlowNote:Takecareinneglectingentranceregioneffects.

Intheentranceregion,frictionalpressuredrop/length>thepressuredrop/lengthforthefullydevelopedregion.Therefore,iftheeffectsoftheentranceregionareneglected,theoverallpredictedpressuredropwillbelow.Thiscanbesignificantinasystemwithshorttubelengths,e.g.someheatexchangers.

2023/4/610清華大學(xué)工程力學(xué)系FullyDevelopedPipeFlowTheanalysisforsteady,incompressible,fullydeveloped,laminarflowinacircularhorizontalpipeyieldsthefollowingequations:

and

Q=AVavg=pR2

Vavg2023/4/611清華大學(xué)工程力學(xué)系FullyDevelopedPipeFlowKeyPoints:Thusforlaminar,fullydevelopedpipeflow(notturbulent):Thevelocityprofileisparabolic.Themaximumlocalvelocityisatthecenterline(r=0).Theaveragevelocityisone-halfthecenterlinevelocity.Thelocalvelocityatanyradiusvariesonlywithradius,notonthestreamwise(x)location(duetotheflowbeingfullydeveloped).Note:AllsubsequentequationswillusethesymbolV(nosubscript)torepresenttheaverageflowvelocityintheflowcrosssection.2023/4/612清華大學(xué)工程力學(xué)系DarcyFrictionFactor:2023/4/613清華大學(xué)工程力學(xué)系DarcyFrictionFactor:KeyPoint:Itiscommoninindustrytodefineandusea“fanning”frictionfactorffinviscouspipeflowanalyses.ThefanningfrictionfactordiffersfromtheDarcyfrictionfactorbyafactorof4.Thus,careshouldbetakenwhenusingunfamiliarequationsordatasinceuseofffinequationsthatweredevelopedfortheDarcyfrictionfactorwillresultinsignificanterrors(afactorof4).2023/4/614清華大學(xué)工程力學(xué)系DarcyFrictionFactor:Youremployerwillnotbehappyifyouordera10hpmotorfora2.5hpapplication.Theequationsuitableforusewithffis fanningfrictionfactoronly: 2023/4/615清華大學(xué)工程力學(xué)系Laminarflow:ApplicationoftheresultsforthelaminarflowvelocityprofiletothedefinitionoftheDarcyfrictionfactoryieldsthefollowingexpression: laminarflowonly(Re<2300)Thus,withthevalueoftheReynoldsnumber,thefrictionfactorforlaminarflowiseasilydetermined.2023/4/616清華大學(xué)工程力學(xué)系Turbulentflow:Asimilaranalysisisnotreadilyavailableforturbulentflow.However,theColebrookequation,shownbelow,providesanexcellentrepresentationforthevariationoftheDarcyfrictionfactorintheturbulentflowregime.NotethattheequationdependsonboththepipeReynoldsnumberandtheroughnessratio,istranscendental,andcannotbeexpressedexplicitlyforf. turbulentflowonly(Re>2300)Wheree=nominalroughnessofpipeorductbeingused.(Note:Takecarewithunitsfore;e/Dmustbenon-dimensional)2023/4/617清華大學(xué)工程力學(xué)系Turbulentflow:AgoodapproximateequationfortheturbulentregionoftheMoodychartisgivenbyHaaland’sequation:Noteagaintheroughnessratioe/Dmustbenon-dimensionalinbothequations.Graphically,theresultsforbothlaminarandturbulentflowpipefrictionarerepresentedbytheMoodychartasshownbelow.2023/4/618清華大學(xué)工程力學(xué)系2023/4/619清華大學(xué)工程力學(xué)系Table6.1AverageroughnessvaluesofcommercialpipeHaaland’sequationisvalidforturbulentflow(Re>2300)andiseasilysetuponacomputer,spreadsheet,etc.2023/4/620清華大學(xué)工程力學(xué)系KeyfluidsystemdesignconsiderationsforlaminarandturbulentflowMostinternalflowproblemsofengineeringsignificanceareturbulent,notlaminar.Typically,averylowflowrateisrequiredforinternalpipeflowtobelaminar.Ifyouopenyourkitchenfaucetandtheoutletflowstreamislargerthanakitchenmatch,theflowisprobablyturbulent.Thus,checkyourworkcarefullyifyouranalysisindicateslaminarflow.Thefollowingcanbeeasilyshown: Laminarflow:

Turbulentflow:

2023/4/621清華大學(xué)工程力學(xué)系KeyfluidsystemdesignconsiderationsforlaminarandturbulentflowThesubscript‘f’oneachofthesetermsindicatesthattheserepresenttheeffectsduetofrictiononlyandarenotthetotalpressuredroporpowerrequirements.Thus,bothpressuredropandpumppowerareverydependentonflowrateandpipe/conduitdiameter.Smallchangesindiameterand/orflowratecansignificantlychangecircuitpressuredropandpowerrequirements.

2023/4/622清華大學(xué)工程力學(xué)系Example(Laminarflow):Water,20oCflowsthrougha0.6cmtube,30mlong,ataflowrateof0.34liters/min.Ifthepipedischargestotheatmosphere,determinethesupplypressureifthetubeisinclined10oabovethehorizontalintheflowdirection.2023/4/623清華大學(xué)工程力學(xué)系2023/4/624清華大學(xué)工程力學(xué)系2023/4/625清華大學(xué)工程力學(xué)系Example(turbulentflow):Oil,r=900kg/m3,n=1E-5m2/s,flowsat0.2m3/sthrougha500mlengthof200mmdiameter,castironpipe.Ifthepipeslopesdownward10ointheflowdirection,computehf,totalheadloss,pressuredrop,andpowerrequiredtoovercometheselosses.2023/4/626清華大學(xué)工程力學(xué)系2023/4/627清華大學(xué)工程力學(xué)系2023/4/628清華大學(xué)工程力學(xué)系Example(turbulentflow):

ans.ans.Notethatthisisnotnecessarilythepowerrequiredtodriveapump,asthepumpefficiencywilltypicallybelessthan100%.Theseproblemsareeasilysetupforsolutioninaspreadsheetasshownbelow.Makesurethatthecalculationforfrictionfactorincludesatestforlaminarorturbulentflow(i.e.Re>or<2300)withtheanalysisthenusingthecorrectequationforfrictionfactor,f.Alwaysverifyanycomputersolutionwithproblemshavingaknownsolution.2023/4/629清華大學(xué)工程力學(xué)系2023/4/630清華大學(xué)工程力學(xué)系SolutionSummary:Tosolvebasicpipeflowfrictionalheadlossproblem,usethefollowingprocedure:UseknownflowratetodetermineReynoldsnumber.Identifywhetherflowislaminarorturbulent.Usecorrectexpressiontodeterminefrictionfactor(withe/Difnecessary).Usedefinitionofhftodeterminefrictionheadloss.Usegeneralenergyequationtodeterminetotalpressuredrop.2023/4/631清華大學(xué)工程力學(xué)系UnknownFlowRateandDiameterProblemsProblemsinvolvingunknownflowrateanddiameteringeneralrequireiterative/trial&errorsolutionsduetothecomplexdependenceofRe,frictionfactor,andheadlossonvelocityandpipesize.UnknownFlowRate:Forthespecialcaseofknownfrictionloss,hf,aclosedformsolutioncanbeobtainedfortheproblemofunknownQ.2023/4/632清華大學(xué)工程力學(xué)系UnknownFlowRateandDiameterProblemsThesolutionproceedsasfollows:Given:KnownvaluesforD,L,hf,r,andm,calculateVorQ.(Note:Itisthefrictionheadloss,hf,notthetotalheadloss,ht,usedinthesolution.Definesolutionparameter:2023/4/633清華大學(xué)工程力學(xué)系UnknownFlowRateandDiameterProblemsNotethatthissolutiondoesnotcontainvelocityandtheparameterzcanbecalculatedfromknownvaluesforD,L,hf,r,andm.TheReynoldsnumberandsubsequentlythevelocitycanbedeterminedfromzandthefollowingequations:Turbulent:

Laminar:

andlaminartoturbulenttransitioncanbeassumedtooccurapproximatelyat

z=73,600(checkReatendofcalculationtoconfirm).2023/4/634清華大學(xué)工程力學(xué)系UnknownFlowRateandDiameterProblemsNotethatthisprocedureisnotvalid(exceptperhapsforinitialestimates)forproblemsinvolvingsignificantminorlosseswheretheheadlossdueonlytopipefrictionisnotknown.Fortheseproblemsatrialanderrorsolutionusingacomputerisbest.Forexample,usingthepreviousspreadsheet,assumevaluesforQuntilthecorrecthtisobtained.2023/4/635清華大學(xué)工程力學(xué)系Example6.9Oil,withr=950kg/m3andn=2E-5m2/s,flowsthrough100mofa30cmdiameterpipe.Thepipeisknowntohaveaheadlossof8mandaroughnessratioe/D=0.0002.Determinetheflowrateandoilvelocitypossiblefortheseconditions.2023/4/636清華大學(xué)工程力學(xué)系Example6.9Withoutanyinformationtothecontrary,wewillneglectminorlossesandKEheadchanges.Withtheseassumptions,wecanwrite:;turbulentchecks,turbulent

ans.

ans.

2023/4/637清華大學(xué)工程力學(xué)系Example6.9Withoutanyinformationtothecontrary,wewillneglectminorlossesandKEheadchanges.Withtheseassumptions,wecanwrite:;turbulentchecks,turbulent

ans.

ans.

2023/4/638清華大學(xué)工程力學(xué)系Example6.9Thisisthemaximumflowrateandoilvelocitythatcouldbeobtainedthroughthegivenpipeandgivenconditions(hf=8m).Again,thisproblemcouldhavealsobeensolvedusingacomputerbasedtrialanderrorprocedureinwhichavalueisassumedforthefluidflowrateuntilaflowrateisfoundwhichresultsinthespecifiedheadloss.Notealsothatwithamoregeneralcomputerbasedprocedure,theproblembeingsolvedcanincludetheeffectsofminorlosses,KE,andPEchangeswithnoadditionaldifficulty.2023/4/639清華大學(xué)工程力學(xué)系UnknownPipeDiameter:Asimilardifficultyarisesforproblemsinvolvingunknownpipedifficulty,exceptaclosedform,analyticalsolutionisnotgenerallyavailable.Again,atrialanderrorsolutionisappropriateforusetoobtainthesolutionandtheproblemcanagainincludelossesduetoKE,PE,andpipingcomponentswithnoadditionaldifficulty.Thisprocedureisshowninthefollowingexampleusingthepreviousspreadsheetbasedsolution.Ex.6.11Afluidwiththeindicatedpropertiesisknowntoflowthrougha100mlongpipewithe=.06mmandaflowrateof0.342m3/sandafrictionalheadloss,hf=8m.Whatdiameterpipeisrequiredtoprovidetheseconditions?Again,assumevaluesofDuntilthevalueofhf=8misobtainedinthesolution.2023/4/640清華大學(xué)工程力學(xué)系2023/4/641清華大學(xué)工程力學(xué)系2023/4/642清華大學(xué)工程力學(xué)系MinorLossesInadditiontofrictionallossesforalengthLofpipe,wemustalsoconsiderlossesduetovariousfittings(valves,unions,elbows,tees,etc.).Theselossesaretypicallyexpressedaswhere hm=theequivalentheadlossacrossthefittingorflowcomponent V=averageflowvelocityforthepipesizeofthefitting

Ki=theminorlosscoefficientforgivenflowcomponent;valve,union,etc.i.e. LossCoefficient 2023/4/643清華大學(xué)工程力學(xué)系Table6.5Minorlosscoefficientforcommonvalvesandpipingcomponents2023/4/644清華大學(xué)工程力學(xué)系Figurebelow

showsminorlossKvaluesforseveraltypesofcommonvalves.

NotethattheKvalvesshownherearefortheindicatedfractionalopening.Also,thefullyopenvaluesmaynotbeconsistentwithvaluesindicatedinTable6.5forfullyopenvalvesorforthevalveofaparticularmanufacturer.Ingeneral,usespecificmanufacturer’sdatawhenavailable.Averagelosscoefficientsforpartiallyopenvalves2023/4/645清華大學(xué)工程力學(xué)系NotethatexitlossesareK@1forallsubmergedexits,e.g.fluiddischargedintoatankatalevelbelowthefluidsurface.Also,foranopenpipedischargetotheatmosphere,thereisnolosscoefficientwhentheenergyequationiswrittenonlytotheendofthepipe.Ingeneral,donottakepoint1forananalysistobeintheplaneofaninlethavinganinletloss.Youdonotknowwhatfractionoftheinletlosstoconsider.Fig.6.21Entranceandexitlosscoefficients(a)reentrantinlets;(b)roundedandbeveledinlets2023/4/646清華大學(xué)工程力學(xué)系NotethatthelossesshowninFig.6.22donotrepresentlossesassociatedwithpipeunionsorreducers.Thesemustbefoundinothersourcesintheliterature.Alsonotethatthelosscoefficientisalwaysbasedonthevelocityinthesmallerdiameter(d)ofthepipe,irrespectiveofthedirectionofflow.Assumethatthisisalsotrueforreducersandsimilarareachangefittings.Fig.6.22Suddencontractionandexpansionlosses2023/4/647清華大學(xué)工程力學(xué)系Theseandothersourcesofdatanowprovidetheabilitytodeterminefrictionallossesforboththepipeandotherpiping/ductflowcomponents.ThetotalfrictionallossnowbecomesorTheseequationswouldbeappropriateforasinglepipesize(withaveragevelocityV).Formultiplepipe/ductsizes,thistermmustberepeatedforeachpipesize.KeyPoint:

Theenergyequationmuststillbeusedtodeterminethetotal

headlossandpressuredropfromallpossiblecontributions.

2023/4/648清華大學(xué)工程力學(xué)系Example6.16Water,r=1.94slugs/ft3andn=1.1E-5ft2/s,ispumpedbetweentworeservoirsat0.2ft3/sthrough400ftof2–indiameterpipewithe/D=0.001havingtheindicatedminorlosses.Computethepumphorsepower(motorsize)required2023/4/649清華大學(xué)工程力學(xué)系Example6.16Writingtheenergyequationbetweenpoints1and2(thefreesurfacesofthetworeservoirs),weobtainForthisproblem,thepressure(P1=P2)andvelocity(V1=V2=0)headtermsarezeroandtheequationreducesto2023/4/650清華大學(xué)工程力學(xué)系Example6.16ForaflowrateQ=0.2ft3/sweobtainWithe/D=0.001andTheflowisturbulentandHaaland’sequationcanbeusedtodeterminethefrictionfactor:2023/4/651清華大學(xué)工程力學(xué)系Theminorlossesfortheproblemaresummarizedinthefollowingtable:Note:Thelossforapipebendisnotthesameasforanelbowfitting.Iftherewerenotankatthepipedischargeandpoint2wereatthepipeexit,therewouldbenoexitlosscoefficient.However,therewouldbeanexitK.E.term.2023/4/652清華大學(xué)工程力學(xué)系Theminorlossesfortheproblemaresummarizedinthefollowingtable:SubstitutingintheenergyequationweobtainNotethedistributionofthetotallossbetweenstatic,pipefriction,andminorlosses.Thepowerrequiredtobedeliveredtothefluidisgivenby2023/4/653清華大學(xué)工程力學(xué)系Theminorlossesfortheproblemaresummarizedinthefollowingtable:Ifthepumphasanefficiencyof70%,thepowerrequirementswouldbespecifiedby2023/4/654清華大學(xué)工程力學(xué)系SolutionSummary:Tosolveabasicpipeflowpressuredropproblem,usethefollowingprocedure:1. UseknownflowratetodetermineReynoldsnumber.2. Identifywhetherflowislaminarorturbulent.3. Useappropriateexpressiontofindfrictionfactor(withe/Difnecessary).4. Usedefinitionofhftodeterminefrictionheadloss.5. Tabulateandsumminorlosscoefficientsforpipingcomponents.6a.Usegeneralenergyequationtodeterminetotalpressuredrop,or6b.Determinepumpheadrequirementsasappropriate.7. Determinepumppowerandmotorsizeifrequired.2023/4/655清華大學(xué)工程力學(xué)系Multiple-PipeSystemsBasicconceptsofpipesystemanalysisapplyalsotomultiplepipesystems.However,thesolutionprocedureismoreinvolvedandcanbeiterative.Considerthefollowing:a. Multiplepipesinseriesb. Multiplepipesinparalle

2023/4/656清華大學(xué)工程力學(xué)系SeriesPipeSystemTheindicatedpipesystemhasasteadyflowrateQthroughthreepipeswithdiametersD1,D2,&D3.Twoimportantrulesapplytothisproblem.2023/4/657清華大學(xué)工程力學(xué)系SeriesPipeSystemTheflowrateisthesamethrougheachpipesection.Forincompressibleflow,thisisexpressedas Q1=Q2=Q3=QorD12V1=D22V2=D32V3Thetotalfrictionalheadlossisthesumoftheheadlossesthroughthevarioussections.

2023/4/658清華大學(xué)工程力學(xué)系SeriesPipeSystemNote:Becarefulhowyouevaluatethetransitionsfromonesectiontothenext.Ingeneral,losscoefficientsfortransitionsectionsarebasedonthevelocityofthesmallersection.2023/4/659清華大學(xué)工程力學(xué)系ExampleGivenapipesystemasshowninthepreviousfigure.ThetotalpressuredropisPa–Pb=150kPaandtheelevationchangeisZb–Za=–5m.Giventhefollowingdata,determinetheflowrateofwaterthroughthesection.2023/4/660清華大學(xué)工程力學(xué)系ExampleTheenergyequationiswrittenaswherehfisgivenbythesumofthetotalfrictionallossesforthethreepipesections.Withnopump;hpis0,Zb-Za=-5m,ht=15.3mforDP=150kPa,andhf(net)=20.08m

(includingKEeffects).SincetheflowrateQandvelocityaretheonlyremainingvariables,thesolutioniseasilyobtainedfromaspreadsheetbyassumingQuntilDP=150kPa.2023/4/661清華大學(xué)工程力學(xué)系2023/4/662清華大學(xué)工程力學(xué)系ExampleThusitisseenthataflowrateof10.17m3/hrproducestheindicatedheadlossthrougheachsectionandanettotalDP=150kPa.Asolutioncanalsobeobtainedbywritingalltermsexplicitlyintermsofasinglevelocity,however,thealgebraisquitecomplex(unlesstheflowislaminar),andaniterativesolutionisstillrequired.Allequationsusedtoobtainthesolutionarethesameasthosepresentedinprevioussections.2023/4/663清華大學(xué)工程力學(xué)系ParallelPipeSystemsAflowrateQTenterstheindicatedparallelpipesystem.Thetotalflowsplitsandflowsthrough3parallelpipesections,eachwithdifferentdiametersandlengths.Twobasicrulesapplytoparallelpipesystems:2023/4/664清華大學(xué)工程力學(xué)系ParallelPipeSystemsThetotalflowenteringtheparallelsectionisequaltothesumoftheflowratesthroughtheindividualsectionsThetotalpressuredropacrosstheparallelsectionisequaltothepressuredropacrosseachindividualparallelsegment.Notethatifacommonjunctionisusedforthestartandendoftheparallelsection,thevelocityandelevationchangeisalsothesameforeachsection.Thus,theflowratethrougheachsectionmustbesuchthatthefrictionallossisthesameforeachandthesumoftheflowratesequalsthetotalflow.

2023/4/665清華大學(xué)工程力學(xué)系ParallelPipeSystemsForthespecialcaseofnokineticorpotentialenergychangeacrossthesections,weobtain:ht=(hf

+hm)1=(hf

+hm)1=(hf

+hm)3

andQT=Q1+Q2+Q3Again,theequationusedforboththepipefrictionandminorlossesisthesameaspreviouslypresented.Theflowandpipedimensionsusedforthepreviousexamplearenowappliedtotheparallelcircuitshownabove.2023/4/666清華大學(xué)工程力學(xué)系ExampleAparallelpipesectionconsistsofthreeparallelpipesegmentswiththelengthsanddiametersshownabove.Thetotalpressuredropis150kPaandtheparallelsectionhasanelevationdropof5m.Neglectingminorlossesandkineticenergychanges,determinetheflowrateofwaterthrougheachpipesection.Thesolutionisiterativeandisagainpresentedinaspreadsheet.Thenetfrictionheadlossof20.3mnowoccursacrosseachofthethreeparallelsections2023/4/667清華大學(xué)工程力學(xué)系2023/4/668清華大學(xué)工程力學(xué)系ExampleThestrongeffectofdiametercanbeseenwiththesmallestdiameterhavingthelowestflowrate,eventhoughitalsohastheshortestlengthofpipe.Againthetotalflowisdistributedbetweenthethreeparallelsectionssuchthattheheadlossacrosseachsectionisthesame,inthiscase20.3m.2023/4/669清華大學(xué)工程力學(xué)系BernoulliObstructionTheoryAnobstr