Viscous Flow in Ducts

8/3/2019 Viscous Flow in Ducts

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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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M.Bahrami FluidMechanics(S09) ViscousFlowinDucts 2

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.

Viscous Flow in Ducts

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