Transcript
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ViscousFlowinDuctsWewant tostudy theviscous flow inductswithvariousvelocities, fluidsandductshapes. Thebasic
problem is this:Given thepipegeometryand itsadded components (e.g. fittings,valves,bends,and
diffusers)plus thedesired flow rateand fluidproperties,whatpressuredrop isneeded todrive the
flow?
ReynoldsnumberregimesThemostimportantparameterinfluidmechanicsistheReynoldsnumber:
whereisthefluiddensity,Uistheflowvelocity,Listhecharacteristicslengthscalerelatedtotheflowgeometry, and is the fluid viscosity. The Reynolds number can be interpreted as the ratio ofmomentum(orinertia)toviscousforces.
Aprofound
change
in
fluid
behavior
occurs
at
moderate
Reynolds
number.
The
flow
ceases
being
smoothandsteady(laminar)andbecomesfluctuating(turbulent).Thechangeoveriscalledtransition.
Transitiondependsonmanyeffectssuchaswallroughnessorthefluctuations inthe inletstream,but
theprimaryparameter istheReynoldsnumber.Turbulencecanbemeasuredbysensitive instruments
suchasahotwireanemometerorpiezoelectricpressuretransducer.
Fig.1:Thethreeregimesofviscousflow:a)laminar(lowRe),b)transitionatintermediateRe,andc)
turbulentflowinhighRe.
Thefluctuations,typicallyrangingfrom1to20%oftheaveragevelocity,arenotstrictlyperiodicbutare
randomandencompassacontinuousrangeoffrequencies.InatypicalwindtunnelflowathighRe,the
turbulentfrequencyrangesfrom1to10,000Hz.
Thehigher
Re
turbulent
flow
is
unsteady
and
irregular
but,
when
averaged
over
time,
is
steady
and
predictable.ForafluidflowwithaveragevelocityUwhichitsshear layerhasthecharacteristicslength
scaleL, ,thefollowingapproximaterangesoccur:0
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103
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Fig.4:Reynoldsssketchesofpipetransition.
InternalandexternalviscousflowBothlaminarandturbulentflowmaybeinternalorexternal.
Internal flow:An internal flow iscontained (orbounded)bywallsand theviscous flowwillgrowandmeet and permeate the entire flow. As a result, there is an entrance region where nearly inviscid
upstreamflowconvergesandenterstheduct.
Fig.5:Developingvelocityprofilesandpressurechangesintheentranceofaduct.
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Beyond the entrance region,which is a finitedistance from the entrancex = Le, the velocity profilebecomesconstant,i.e.itnolongerchangeswithxandissaidtobefullydeveloped, .Forlaminarflow,theentranceregioncanbefoundformthefollowingempiricalcorrelation:
0.06
AssumingthemaximumReforthelaminarflowinaductisRed,crit=2300thelongestlaminardevelopingregionbecomes:Le=138d,i.e.138timesofthetubediameter.Turbulentflowboundarylayersgrowfaster,andtheentranceregion(Le)isrelativelyshorter: 4.4/ AssumingRe=10
6forturbulentflowinaduct,atypicalturbulentdevelopingregionbecomes:Le=44d.
Note: the entrance region length can vary from 40d to 100d for turbulent and laminar regimes,
respectively.However,fortypicalpipeflowapplication,thepipelengthis1000ofitsdiameterinwhich
casetheentranceeffectmaybeneglected.Thereforeasimpleanalysisforfullydevelopedregioncanbe
usedfortheentirepipelength.
Externalflow:Hasnorestrainingwallsandisfreetoexpandnomatterhowthicktheviscouslayersontheimmersedbodymaybecome.Asaresult,farfromthebodytheflowisnearlyinviscidandthereisno
externalequivalentoffullydevelopedinternalflow.
Flowheadloss(thefrictionfactor)Considerincompressiblesteadyflowbetweensections1and2oftheinclinedconstantareapipeshown
inFig.6.
Fig.6:Steadyfullydevelopedflowinaninclinedpipe.
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The1Dcontinuityequation:
Foraconstantareapipe,thesteadyenergyequationbecomes:
2 2 SinceV1=V2,thefrictionheadlostis:
Applyingthemomentumequation,thesummationontheforcesalongthexdirection,wefind:
2 0Rearrangingthis,wefindthattheheadlossisrelatedtowallshearstress:
2 4 where .Note that regardlessofwhether thepipe ishorizontalor tilted, thehead loss isproportionaltothewallshearstress.AGermanprofessor,JuliusWeisbach1850,arguedthatthefriction
factorisproportionaltoL/dandV2(observedexperimentallyinturbulentregime).Hethenproposedto
representthefrictionlesshead losswithadimensionlessparameterf(calledtheDarcyfrictionfactor),
thatisdefinedas:
2wheref=f(Red,ductroughness,ductshape).Byequatingtheaboveequations,wefind: 8LaminarfullydevelopedpipeflowConsiderfullydevelopedPoiseuilleflow inaroundductofdiameterd.Theanalyticalvelocitysolution,
fromthepreviouschapter,is:
1
4
2 8 8 4 2
32 128
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Note:thepressuredropininverselyproportionaltothepipediametertothepower4.So,ifthesizeof
thepipeisdoubled,thepressuredropwilldecreasebyafactorof16foragivenQ.
These formulas are valid whenever the Re
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Thethreecorrelationterms , , arecalledturbulentstressesbecausetheyhavethesame dimensions and occur right alongside the newtonian (laminar) stress terms.Actually, they are
convective acceleration terms (that is why the density appears), not stresses but they have the
mathematicaleffectofstress.
TurbulentpipeflowThereare three regions in turbulent flow, inner layer (orviscous sublayer)whereviscouseffectsare
dominant (near thewall),overlap layer (transition to turbulentoccurs)andouter region (the flow is
completelyturbulent)asshowninFig.8.
Fig.8:turbulentflowregimes.
Sincetheinnerwall layerformstypicallyonly2%ofthefluidflowprofile,wecanneglectit.Therefore,
weonlyconsidertheoverlaplayervelocityprofilefortheentirepipeflow:
1 ln where
0.4 5.0,
/isthekinematicviscosityofthefluid,and
/ iscalledthefrictionalvelocity,because ithasvelocitydimensions,although it isnotactuallyaflowvelocity.
Thefrictionfactorfortheturbulentpipeflowcanbecalculatedfromthefollowingcorrelations:
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0.316/ 4000 101.8 6.9
where
.
Forahorizontalpipe,wehave:
0.158////Orintermsof 4 ,wefindanalternativeform:
0.24//..Note:Thepressuredrop for the turbulent flowdecreaseswithdiameterevenmore sharply than the
laminarflow.Doublingthepipesizedecreasesthepressuredropbyafactorof27foragivenQ.
Note:As
depicted
in
Fig.
9,
aturbulent
velocity
profile
is
very
flat
in
the
center
and
drops
off
sharply
to
zeroatthewall.
Fig.9:Comparisonofa)laminarandb)turbulentpipeflowvelocity.
EffectofroughwallsThe surface roughness has an effect on friction resistance; for laminar flow, however, this effect is
negligible.Theturbulentflow isstronglyaffectedbyroughness.Nikuradse(1933)simulatedroughness
bygluinguniformsandgrainsontothe innerwallsofthepipes.Hethenmeasuredthepressuredrops
andflowratesandcorrelatedfrictionfactorversusReynoldsnumber,asshowninFig.10.
Note:
Thelaminarfrictionisunaffected, Theturbulent friction,afteranonsetpoint, increasesmonotonicallywiththeroughness ratio,/, Thefrictionfactorbecomesconstant(fullyrough)athighReynoldsnumbers.
Threeregionscanbedetectedinthechart:
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Fig.10:Effectofwallroughnessonturbulentpipeflow.
5:Hydraulicallysmoothwalls,noeffectofroughnessonfriction5 70:Transitionalroughness,moderateReynoldsnumbereffect
70:
Fully
rough
flow,
sub
layer
totally
broken
up
and
friction
independent
of
Reynolds
number.
Thelogarithmlawmodifiedforroughnessbecomes:
1 8.5Integratingthisequation,wecanfindtheaveragevelocityinthepipe:
2.44 3.2For
fully
rough
flow,
we
have:
1/ 2.0 /3.7 NoticethatthereisnoReynoldsnumbereffect;hencetheheadlossvariesexactlyasthesquareofthe
velocityinthiscase.
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TheMoodychartColebrook(1939)combinedthesmoothwallandfullyroughrelationsintoaclevercombined
interpolationformula:
1 2.0 /3.7 2.51
Itisplottedin1944byMoody(1944)intowhatiscalledtheMoodychartforpipefriction,Fig.11.
Moodychartisoneofthemostfamousandusefulfigureinfluidmechanics.Itisaccurateto15%and
canbeusedforcircularandnoncircularductsandopenchannels.
Fig.11:TheMoodychartforpipefrictionwithsmoothandroughwalls.
TheColebrookequationiscumbersometoevaluatef,Haaland(1983)proposedanothercorrelation:
1 1.8 6.9 /3.7 .HydraulicdiameterIftheductisnoncircular,theanalysisoffullydevelopedflowfollowsthatofthecircularpipebutismore
complicatedalgebraically.Forturbulentflow,thelogarithmlawvelocityprofilecanbeused.Ingeneral,
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forcomplexgeometries,itisverychallengingtoperformtheanalysisdirectly.Therefore,theconceptof
thehydraulicdiameterisintroducedforconvenience.
Fig.12:Flowinnoncircularducts.
For noncircular ducts, the above analyses are correct, however, the crosssectional area A does not
equaltoandthecrosssectionalperimeterwettedbytheshearstress,,isnotequalto2. Forthisreasonanoncircularductissaidtohaveahydraulicdiameter:
4 4
Weusethehydraulicdiameter,asalengthscalefornoncircularducts,andusetheanalysesderivedfor
thecircularpipes.
MinorlossesinpipesystemsForanypipesystems,inadditiontotheMoodytypefrictionloss,computedforthelengthofpipe,there
areadditionalminor losses, including:pipeentranceorexit, suddenexpansionorcontraction,bends,
elbows,tees,otherfittings,valves(openorpartiallyclosed),andgradualexpansionsorcontractions.
Theflowpatternsinmostoftheabovementionedgeometriesarequitecomplex,seeFig.13,andthus
thelosses
commonly
measured
experimentally
and
correlated
with
the
pipe
flow
parameters.
Themeasuredminor loss isusuallygivenasaratioofthehead lossthroughthedevicetothevelocity
head/2oftheassociatedpipingsystem: /2 12
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Fig.13:Typicalcommercialvalvegeometriesa)gatevalve,b)globevalve,c)anglevalve,d)swingcheck
valve,e)disktypegatevalve.
Asinglepipesystemmayhavemanyminor losses.Sinceallarecorrelatedwith/2,theycanbesummedintoasingletotalsystemlossifthepipehasconstantdiameter:
2
Fig.14:Kfactorfor90degreeelbows.
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K isanondimensional factor, likef,and isoftencorrelatedwith the raw sizeof thepipe, tablesand
figuresareavailable inthetextbook forvariousminorchanges,seeFig.14forexample.Notethatwe
must sum the losses separately if thepipe size changes so thatV2 changes.The length L is the total
lengthofthepipeaxis.
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