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TRANSCRIPT
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NATIONALADVISORYCO’MMI’ITEE“g~FORAERONAUTICS
TECHNICALNOTE3451
ANALYSIS OF FULLY DEVELOPED TURBULENT HEAT TRANSFER
AND FLOW INAN ANNULUS WITH VARIOUS ECCENTRICITIES
By RobertG. DeisslerandMaynardF. Taylor
LewisFlightPropulsionLaboratoryCleveland,Ohio
Washington
May1955
z=.----
https://ntrs.nasa.gov/search.jsp?R=19930084129 2020-05-25T19:07:47+00:00Z
TECHLIBRARYKAFB,NM
Illlllllllllul[lllllifllllllllNATIONALADVISORYCOMMITTEE10.RAeronautic 00bb5atl
TECHNICALNOTIZ3451
ANALYSISOFXULLYDEWELOPEDTURBULENTHEATTRANSFERANDFLOW
INANANNULUSWITHVARIOUSECCENTRICITIES
By RobertG.DeisslerandMaynardF. Taylor
A previouswasgeneralizedExpressionsfor
SUMMARY
analysisforturbulentheattransferandflowintubesandappliedtoan annuluswithvariouseccentricities.eddydiffusivitywhichwereverifiedforflowandheat
tr-asferintubesw&e assmedto applyingeneralalonglinesnormaltoa wall. Velocitydistributions,wallshear-stressdistributions,andtiictionfactors,aswellaswallheat-transferdistributions,wall
q temperaturedistributions,andaverageheat-transfercoefficients,wereg~ calculatedforan annulushatinga radiusratioof3.5at various
eccentricities.+
INTRODUCTION
M&t oftheexistinganalysesforturbulentflowandheattransferinpassageshavebeenconfinedto circulartubesorparallelplates(refs.lto 5). Thesepassageshavebeenanalyzedextensivelybecauseoftheirimportanceintechnicalapplicationsandbecausetheirsimplic-itymakesthemamenableto analysis.
b recentyearstheproblemsassociatedwiththeuseof odd-shapedpassagesinheatexchangershavebecomeimportant.Inreference6,tem-peraturedistributionsforrectangularandtriangularductswerecalcu-latedby usinge~erimentalvelocitydistributionsandaverageheat-transfercoefficients.No attaptwasmadeto calculatethevelocitydistributionsorheat-trausfercoefficients.Saneworkonthecalcula-tionofthevelocityandshear-stressdistributionsin cornersisre-portedinreference7.
As a partofa generalinvestigationofheattransferandflowinpassagesofvariousshapesbeingconductedattheN/KMLewislaboratory,theeccentricsmnuluswasanalyzed.Themethodsusedinreferences4and5 forcalculatingtheheattransferandtiictionintubeswereex-
J tendedandappliedtoannulihavingvariouseccentricities.Themethodsusedhereinshouldbe generalenoughtobe appliedtopassageshavingvariousothercrosssections.
d
2 NACATN 3451
EA.SICEQUATIONSANDASSUMPTIONS
Thedifferentialequationsforshearstressandheattransfercanbe writtenInthefollowingform:
‘=(W4*
where e and ~h aretheeddydiffusivitiesformomentumandheattransfer,respectively,thevaluesforwhicharedependentupontheamountandkindofturbulentmixingat a point.Intheseequationsy istakenas theperpendicular.d.istancefromthewall. Equations
(1)
(2)
(1)and(2)canbe consideredas definitionscanbe writtenindimensionlessformas
Of E and ~h. Tliey
du+
3(3)
%)=wherethesubscriptOfinedinappendixB.
(kl P%?a”c
)
dt+.—k.Pro+ PoCpo ‘* g
(4)
referstovaluesata wall. Allsymbolsarede-
Expressionsforeddydiffusivity.- In_~rderto useequations(3)and(4),theeddydiffusivitye mustbe evaluatedforeachportionoftheflow. It isassumed,as inreferences4 and5,thatintheregionat a distancefromthewallthemechanismforturbulenttransferisde-pendentonlyonthevelocitiesinthevicinityofthepointmeasuredrelativeto thevelocityat thepointor onthespacederivativesofthevelocity.Intheregionclosetothewalltheturbulenceisassuneddependenton quantitiesmeasuredrelativetothewall.,thatis,u and
wheretheconstantn hasthe(ref.4).
gives,forthe
E = n%y
experimentally
regionclosetothewall
(5)
determinedvalue0.109
lJ.twasfoundinreference8 thatintheregionveryclosetothewall ~ appearstobe a functionalsoofl&mnaticviscosity,buttheeffectof thatfactorbecomesimportantonlyat Prandtlnumbersappre-ciablygreaterthan1.0.
.
b
NACAm 3451
Intheregionata distancefronthewall(y:> 26),e isassumedtobe dependentontherelativevelocitiesintheneighborhoodofthepoint. IYoma Taylor’sseriesexpansionfor u as a functionof yand.z,
where y and z aremeasuredinnormaldirectionsinthecrosssec-tionofthepassage.Inthecaseof flowina tubeorbetweenparallelplates,thederimtivesin thez-directionarezerobecausethevelocity-gradientlinesarestraightlinesnormalto thewalL. (Avelocity-gradientlineisa linewhichat eachpointisnormalto a constant-velocityline.)Inan eccent+icsmnulusthevelocity-gradientlinesnearthewallalsoarenormalto thesurface,buttheyareusuallycurvedinthecenterportionofthepassage.Inasmuchasthegreatestchangesofvelocitywithrespectto distancetakeplaceinlayersnearthewall,theeffectofthederivativeswithrespectto z willbe neg-lected.It sea reasonableto expectthatnearthecenteroftheflowpassagetheeffectofthederivativeswithresyectto z wouldbe toincreasetheturbtienceandflattentheprofileinthatregion.However,thenormalturbulentprofile(derivativeswithrespectto z absent)isalreadyveryflatinthatregion,sothattheincreasedturbulenceshouldnotproducesignificantchangesinthevaluesofthevelocities.There-fore,theexpressionfor e for @ > 26,obtainedby usingdimensionalanalysis,andwithonlythefirsttwoderivativeswithrespectto y con-sidered,is
6 .X2 J!MZ.L(d%@y2)2
(6)’
wherex hastheexperimentalvalue0.36(ref.4). ThisisvonK&r&a’sexpression(ref.1).
Furtherassumptions.- Inorderto integrateequations(3)and{4},thefollowingasswptionsaremadeinadditionto thoseconcerningtheeddydiffusivtty(eqs.(5)and(6)):
(1)Thefluidpropertiescanbe consideredconstantandthePrandtlnumbercloseto 1.0(0.73}.Theheat-transferresultsarethereforeap-plicableto gaseswithmoderatetemperaturedifferences.Theanalysiscouldbe carriedoutforvariableproperties,butthecomplexitywouldbe increased.
(2)Theeddydiffusivitiesform=entum e andheattransfer~areeqyal,or a . 1. Previous analysesforflowintubesbasedonthisassumptionyieldedheat-transfercoefficientsthatagreewithexperiment(refs.2 and5}. At lowReynoldsorPecletnumbers(Pe= Re I&],a
4
appearstobe a functionofnumbersabove15,000,as inforgases.
Pecletnumber(ref.9),butforReynoldsthepresentanalysis,a isnearlyconstant
(3)Alongthelinesnormaltothewall,thevariationsof’theshearstressT andheattransferpertit area q havea negligibleeffecton thevelocityandtemperaturettistributions.Itis shownin figure11ofreference5 thattheassumptionofa linearvariationof shearstre~sandheattransferacrossa tube(T or q = O at tubecenter)givesverynearlythesamevelocityandtemperatureprofilesasthoseobtainedby assuminguniformshearstressandheattransferacrossthetube.
(4)Themolecularshear-stressandheat-transfertermsinequtions(3)em.d(4)canbe neglectedintheregionat a distance&om thewall.(y+> 26)(ref.5, fig.12).
Generalizedvelocityandtemperaturedistributions.- Eqwtions(3)and(4)havebeenintegratedusingtheforegoingassumptionsinrefer-ences4 and5,wheretheeqwtionsarealsoverifiedexperimentallyfarflowandheattransferintubes.Theresultsarereproducedinfiguresland 2. Thesecurvesgivetherelationsbetweenu+,y+,and t+whichwillbe usedinthefollowingcalculationsforeccentricannul-i.
CALCULATION0FW3XK!ITYDIS!EUJXJTION3l?CIRECCZMRICANNULI
Forapplyingtherelationbetweenu+’and y+ in figure1 tothecalculationofvelocitydistributionsinan eccentricannulus,an iter-ativeproceduremustbe used,inasmuchasthelinesofvelocitygradient(linesnormalto constant-velocitylines)areunknownattheoutset.Aneccentricannul-us,withseveralassumedvelocity-gradientlines,isshownin sketch(a).
Velcdty-gm.clientLines
%-
LineOf
veloeitie
il-
‘r
Kof
velocities(a)
&
d
NAcf.TN3451
By usingtheseassumedvelocity-gradientitycanbe calculated,aswillbe shown.
5
lines,linesof constantveloc-A newandmoreaccurateset
ofvelocity-gradientlinescanthenbe drawn,inasmuchas theymustbenormalto theconstant-velocitylines.Witha littlepractice,thevelocity-gradientIinsscanbe estimatedqyiteaccurate-thefirsttimesothatitisnotusuallynecess=yto gothroughtheproceduresmorethanonceortwice.
Calculationoflineofmaximumvelocities.- Afterthevelocity-gradientlineshavebeenestimated,thenextstepisto calculatethelineofmaximumvelocitiesshownon sketch(a}. Thevelocitieson ei-thersideofthislinearelowerthanthevelocitiesontheline. TwoadJacentvelocity-gradientEnes arealsoshowninthesketch.ThelengthsAZ1 and AZ2 areon theinnerandoutercylinders,respec-tively,andthelinedividingAAl and zW2 isthelineofmaMmumvelocities.Thelocatiofisof AZl and AZ2 mustbe:chosensothatstraightlinesdrawnnormalto themat theirmidpoints(showndashedinthesketch)meetattheMne ofmaxhumvelocities.Itwillbenecessarytomatchthevelocitiesat thatTointas calculated&amtherelationbetweenu+ and y+ alongthenormals&cm AZ1 andAlz. b ordertowriteforcebalancesontheelementsAAl and AA2,theshearforcesactingon AZ1 and AZ2 andthepressureforcesactingonthefacesoftheelementsmustbe considered.Therearenoshearforcesactingonthevelocity-gradientlines,becausethenormalvelocityderivativesarezexoalongthose&es. Writingforcebal-anceson AAl and AA2 resultsin
(7)
wherethepressuregradientdp/dx isuniformovertheannulusbecausetheflowis ftdlydeveloped.Iliviit@geqpation(8)by equation(7)gives
T2 AZlAA2—=— —T1 AX2AAl
“d
i
FTomthecurveof u+ against@ (fig.1),
u+=j7(y+)
6
Applyingthisequationto AAl and AAz gives
and
NACATN3451
(10) -
(n)
Whentheseequationsareevaluatedisdividedby equation(n], there
at q = U2 . ~, andeq,,ti.on(10)results
Eliminationof T1 andto (9)andconversionto
,
(12)
r2 fromequation(U?)byuseofequati.ons(7)dimensionlessformgive
w
AA#Ul * ‘~
‘1 ‘1 ~‘(’d %
F
=— =—Y~2AJ2 *_ ()
F Ha u&— ‘1 rlrl
(13)
G“-rldp/dx
where rl+,definedby r~= P rl,isa typeofReynoldsnum-WP
berwhichisassignedan arbitraryvalue.Theqyantityy~/rl,thatis,a pointonthelineofmaximumvelocities,canbe calculatedfrom
~ equation(13)by trial.Refemingto sketch(a),thevaluefor yw/rlis firstassumed.Valuesfor AL#(A22rl),dA1/(AZ1rl),and Yti/r~canthenbe calcuktedbymeasuringtheareasandlengthsinthesketch.Fora givenvalueof r~, then,y& and y% (termsinparentheses,eq.(13))cambe calculatedand u&Ju~ obtainedfromtheplotoftherelationbetweenu+ and y’+(fig.1). Ifthisvalueof u&/u& doesnotagreewiththetermontheextremeleftsideof equation(13),an.othervaluefor yj#l mustbe tried.InthiswaythevalueforyM/rljandthusa pointonthelineofE=*W ‘laitiesJ c- be found.
N.AC!ATN 3451 7
Calculationoflinesofconstautvelocity.- Forcalculatingveloc-itiesatvariouspointsinthesnnulus,itis convenientto definethefollowingvelocitypqmeter:
Thispsmmeterisusedinplaceof u+,becausetheshearstressinthede~tion of u+ varieswithposition.Eqpation(14)canbe writtenintermsof quantitiesalreadybown as
where
(15}
(16)
wherethefunctionF is obtainedfromtherelationbetween~+ -d@ in figure1. Equations(15) and(16)applytopointslyingbetweentheinnercylinderandthelineofmaximumvelocities.Forpointsbe-tweentheoutercylinderandthelineofmaximmvelocities,these.eqm-tionsarereplacedby
-L
where
(18)
/ and Y~/r~Thequantitiesy~, Y+&,YM rl, areatieadyknowntiomequation(13). Therelationbetween~+ ~ Y1/Yh or up and
Y2/Ya fora given r~ canthereforebe calculatedalonga givenlinenormaltotheinneror outercylinder.
8 NACATN 3451
By carryingoutthecalculationforvariouslinesnormaltotheinneroroutercylinder,linesof constantvelocity(constantu+) canbe obtained.As mentionedpreviously,newandmoreaccuratevelocity-gradientlinesarenetidrawnso asto intersecttheconstant-velocitylinesatrightangles.Thecalculationisthenrepeatedusingthenewvelocity-gradientlines.
Calculatedvelocitydistributions.- Severalvelocitydistributionsforeccentricannuli,as calculatedby themethoddescribed,areshownin figure3. Thecalculationswerecarriedoutforanannulushavinga
7diameterratioof3$ forvariouseccentricities.Theshapeoftheconstant-velocitylinesindicates,aswuul.dbe expected,thatthegreat-estvelocitiesoccuronthesidewheretheseparationofthecylindersisgreatest,andthatthevelocitieson thesidewheretheseparationisleastaremuchsmallerandgoto zerowhenthecylinderstouch.Althoughthevelocitydistributionsareplottedforan r~ of 200,theconstant-velocitylinesforan r~ of 4000havepracticallythessmeshape.Thevaluesof u/”b,avareslightlydifferentforthetwovaluesof r~j t
butthedifferenceisnotlarge.
InalJcasesthelineofmaximumvelocitiesliescloserto the Linnerthantotheoutercylinder,thatis,itliesclosertothesur-facehavingthesmallerarea.
Inmostcasesthecalculatedconstant-velocitylinesshownin fig-ure3 areverynearlynormaltothevelocity-gradientlines,asreqtired.However,in somecases,especiallyfortheMrge eccentricities,diffi-cultywasexperiencedinobtainingnormallinesin someregf.ons.Whetherthislackofnormalcyisduetothefactthattheiterationswerenotcarriedfarenoughortotheapproximatenatureof someoftheassump-tionsmadeintheanalysishasnotbeenestablished.However,thediffi-cultyoccursonlyinregionsinwhichthevelocitygradientsareverymail, so that theerrorinsmall.
l.?RIC!lZCIJ
thecomputedvalues
l?AC’IORANDREYNOLOS
ofvelocityshouldbe
mm
Oncethevelocitydistributionsfortheannul-ihavebeenobtained,trictionf%ctorsand~eynoldsnumberscanbe calculatedby integrating-thedistributionsto obtainbulkoraveragevelocities.Thebulkve-locitybetweentwoadjacentstraightlinesnormaltothewalJ.(dashedlinesinsketch(a))is firstobtained.Thisbulkvelocityvariestithpositionontheinnercylinderandisgivenby
NACATN 3451 9
&
f
UdlloUbz. All
wherehA isthetotalareabetweentwoadJacenttothewall.(onbothsidesofthelineofmaximum(19)canbe writtenindimensionlessformas
AA
J @~47= ‘AA
(19)
straightlinesnormalvelocities).Eqpation
(20)
Theaveragebulkvelocityforthewholeannuluscanbe writtenas
8 f%~av = ; %* w (21)
o
* whereA isthetotalareaoftheannulus.Thevariationof~*/~~av = ~/~,av +th me -d ecc~tricityiS shownin f-e ~.
Forlatercomparisonwithheattransfer,thelocalbasedontheshearstressontheinnercylinderandthelocityis introducedandisdefinedas
frictionfactoraveragebulkve-
or,intermsofknowndimensionlessquantities,canbe writtenas
Theaveragefriction
J
[
Y~
I
2fr=2H1 *(y-Jr~)‘b,av‘1
factorbasedontheshearstressis,then,
f 1=—‘l,av X
.!
f*de
o
(22)
{23]
(24)
where e istheazu?leatthecenteroftheinnercylindermeasuredfYomthelineofleastseparationofthetwocylinders.“Divisionofequation
● (22)by equation(24)gives
10
The ratio ~J71,avanda corresponding
NACAm 3451
frl ‘1—= —f ‘1,av
(25)‘1,av
couldhavebeenobtaineddirectlyfromeqwtion(7)equationforthewholeannulusas
sentedin figure5“.Thecurvesindica~ea decreasein shearstressonthesideoftheinnercylinderwhichisclosesttotheoutercylinder,anda shearstressof zerowherethecylinderstouch(e= 1.0).As inthecaseofthevelocitydistributions(fig.4)thevalueof r~ orReynoldsnumberhasa negligibleeffectontheshapeofthecurves.ThevaluesfortheReynoldsnumberswerecalculated&cm
which follows directlymomequivalentdiameterDe istersoftheouterandinner
(26)
thedefinitionofReynoldsnumber.Theeqpaltothe&l.fferencebetweenthediame-cylinders.A crossplotof figure5 forthe
u
decreases.
Theflrictionfactorfortheannulusbasedis definedby
De dp/tif~-
2~b,avz
or,intermsoflmowndimensionlessgroups,
f=1
2(rJDe)u~s,v2
point ofleastseparatism (TllTl,avagainsteccentricity)–isgiveninfigure6.
Thef?actthattheshearstressshoulddecreaseintheregionofleastseparationofthecylinderscanbe seendirectlyfromequation(7),whichindicatesthattheshearstressi.sproportionalto AAJZ31~.But Ml . AZlym,and,hencejtheshearstressisa~roximatelypro-portionalto Y,.,tiichdecreasesasthedistancebetweencylinders
on thepressuregradient
(27)
(28)
-.
aw-
*
NACATN 3451 l.1
*
.
Frictionfactorsbasedonthepressuregradientarepresentedin figure7 as a functionofReynoldsnumberandeccentricity.Comparisonofthesefrictionfactorswiththosefora circulartubeindicatesfairagreementforthecasewhenthecylindersareconcentric.Thevaluesofthefric-tionfactordecreaseastheeccentricityis”increased;and,for”thecasewherethecylinderstouch,thefrictionfactorssreapproximately70percentofthoseforconcentriccylinders.
HFAT-TRMH?ERDISTNE3UTIONO?i”llKERCJ!LINDER
WITEOUTERC!mumm 11’LWLATED
Withtheinformationonthevelocitydistributionsobtainedintheprecedingsectionandwithcertainassuqtions,thefuIlydevelopedheat-transferdistributionsontheinnercylinderwhentheinnercylind-er isheated(orcooled)andtheoutercylinderis insulatedcanbecalculated.Peripheralwalltarperaturedistributionswillbe calcu-latedfkomtheseheat-transferdistributionsinthenextsection.
Thehe&taddedbetweentwostraightlinesnormalto thewallperunitlengthofannulusis qlA21(seesketch(a)’).It isassumedhereinthatall this@at isusedinheatingthefluidelementbetweenthestraightlines(onbothsidesoftheline 03 maximLR velocities). That
is,thenetheattransferacrossthesidesof theelementis assumedsmallcomparedwiththattransferredthroughthewall. ThisassumptionisvalidwhenthevariationofwalltemperatureisnottoogreatandthePrandtlnumberiscloseto orgreaterthanunity.
Makinga heatbalanceontheelementof fluidbetweentwostraightlinesandusingtheforegoingassmptionresult
@l
Forthewholeagnul.us
‘l,avZ1=
in2
(29)
()‘tb,av‘%,avcp dx A(-) “ (30)
-J
.
‘Asan alternativeassmption,an elementboundedby velocity-gradientlinesratherthanby straightlineswasused. No netheattransferacrossthe~elocity-gradientlineswasassumed.Thisassump-tionwasfoundto giveessentiallythesamedistributionof ql as didtheassumptionusedinthetex%.
12 NACAm 3451
Itis shownina endixA that,forthefullydevelopedcase,dtb/dX= Tdtb,av dx whentheheattransferpertit areaata givenangle e doesnotvarywith x. Divisionofequation(29)by equa-tdon(30)andconversiontodimensionlessformgive -
It caneasilybe shownthatequation(31)alsogivestheNusseltnumbertoaverageNusseltnumberforthecaseofcumferentialwalltemperature,since,forthatcase,
—t~ - ‘b)av %, av‘-‘b,av
and
or
(31)
ratiooflocaluniformcir-
(32)
{33)
(34)
Equation(34)is strictlytrueonlyforthecaseofuniformcircumfer-ential~ &uperature,-because foranyothertemperaturedistributionthetemperaturedifferencesinequations(32)and(33)willnotcancel.
VtiuesOf q~/ql,av(h/havforuniformcircumferentialwalltem-perature)areshownas a functionofangleandofeccentricityin fig-ure8. As wasthecasewiththeshearstress(fig.5),theheattrans-ferislowonthesideofthecylinderwheretheseparationofthecyl-inderis leastandapproacheszerowhenthecylinderstouch.However,comparisonof figure$5 and8 indicatesthattheheattransferap-proacheszeromuchmorerapidlythandoestheshearstress.Thesametrendsareindicatedby comparisonoffigures6 and9,where ~l~~l,avti ‘l/ql,avareplottedagainsteccentricityforthepointofleast
separationofthecylinders.Comparisonofthesefiguresindicates,therefore,thattheassumptionof similarityoftheshear-stressandheat-transfer-coefficientdistributions,whichisusuall.ymadein anal-ysesforodd-shapedpassages,mightgiveheat-transfercoefficientsinthevicinityofa cornerthataretooM@. !Ihiswaspointedoutinrkference10. E2cperimentalwalltemperaturedistributionsinreference11 indicatedthatinsomecasesthepredicted”coefficientsweretoohigh.
+-
—
-—
%-
NACATN3451 13
.
.
Thereasonthattheheattransferdecreasesmorerapidlythantheshearstressasthelineofleastseparationofthecylindersisap-preachedcaneasilybeseenbycomparisonofequations(7)and(29).Bothecym.tionscontainAA/Allor AA1/A21,whichcausestheheattransferor shearstressto decreasenearthelineof leastseparation.However,theheat-transferequation(eq.(29)]contains,inaddition,thelocalbulkvelocityub,whichalsodecreases;therefore,theheattransferdecreasesmorerapidlythantheshearstressasthelineofleastseparationisapproached.
WuL Tmmwm DISTRIBUT~ON
Theinnercyldnderoftheannul.usisassumed,inthissection,tobe theoutsideswfaceofa thin-walledtube.Heatistransferreduni-formlytotheinsidesurfaceofthetube. Becauseoftangentialconduc-tionaroundthetube,theheattransferthroughtheoutersurfacetothefluidwillnotbe uniformandwillhavethedistributionobtainedintheprecedingsection.Sincethetubeis thin-wall.edandtherearenoheatsourcesinthetube,theheattransferperunitm?eathroughtheinnersurfaceis equalto ql,avjtheaverageheattransferperunitareathroughtheoutersurfaceto thefluid.As int’heprevioussection,theoutercylinderoftheannulusis insulated.,
Inorderto obtainthetemperaturedistributionaroundthetube,aheatbalanceis firstmuleonan elementof tubeof circumferentiallengthd2. Thisheatbalancegives
(35)
where b isthethiclmessconductionforunitarea.formas
ofthetubeand ~ isthetangentialheatEqyation(35)canbe writteninintegral
where Z1/rl~ e and ~ is zerofor 21= O,thepointofleastsepa-rationofthecylinders,becausethe~ temperaturedistributionissymmetricaboutthatyoiqt(thetemperaturegradientis zero).But
14 NACATN3451
(37)
Substitutingequation(37)inequation(36)andintegratingagainresultin
where t~,= isthewalltemperatureat thepointofleastseparationof thecylinders.By usingthevaluesof ql/ql,avobtainedintheprecedingsection,thedifferencebetweenthem&imumwall.temperatureandthewalltemperatureatanyanglefora givenheatfluxcanbe calc-ulatedfromequation(38).A dimensionlessparsmetercontainingthedifferencebetweenthemaximumandtheaveragewalltemperaturescanbeobtainedby integratingequation(38),thatis
Therelationfor
tractingequation
‘l,avrlLxJ %,avrl’
o
bkt(tl- tl,avv(!ll,av+Canbeob*~edbY sub-(38)fromequation(39).
(39)
Theresultsforthewalltemperaturedistributionsarepresentedin figure10,where bk& - ‘l,av)/(ql,a#12)isplottedagainstan-gleforvariouseccentricities,andinfigure1.1,whereb’t(tl,~x- ‘l)av)/(ql,a#12)iSplottedagainsteccentricity.It iSof interestthat tl,mx - *l,avJtiichis theqpantityofgreatestpracticalinterest,canbe obtainedfroma singlecurve.Thatis,fora givenannulus,onlytwodimensionlessparametersareinvolved.Fora giveneccentricity,.1,- - tl,av isdirectlyproportionaltotheuniformheatfluxthroughtheinsidesurfaceofthetube.
Walltemperaturedistributionsforvariousothercases,suchasthecasewherethetubewallisnotthinorwheretiternal.heatsourcesarepresentin thetubewall,canalsobe readilycalculatedby usingtheheat-transferdistributionsin figure8. b ordertousethesedistributionsitis,of course,necessarythattheoutercylinderbeinsulated.
.
*-
I?ACA‘IN3451
NUSSELTNJ31BER
Forobtainingalltheresultsthusfar
15
(wallheat-transferdistri-butions,walltem&raturedistributions,etc.),itwasnotnecessarytousethegeneralizedtemperaturedistributionin figure2. However,itisnecessarytousethatdistributionforobtainingthedifferencebe-tweenwallandbulkt@nperatureswhichcorrespondstoa givenheatflow,thatis,forobtainingtheheat-transfercoefficient.
Averageheat-transfercoefficient.- Theaverageheat-transferco-efficientfortheinnerwalloftheannulusisdefinedby
‘1,av%3TT= tq,av- %,av
where
Ltu dA
tb= ou#
and
%,av =Jo
%,av%
TheaverageNusseltnmber correspondingto theaverage
(40)
(42)
(43)
heat-transfercoefficient(eq.(40))canbe calculatedfromthefollowingequation,whichcanbe-veri&ed-bysubstitutingthedefinitionsofthevariousquantities:
.
-<’
.
16 NAcl’1TN3451
.
1 ‘l/De~=
f~H(ub/ub,~~)d(A/Ao)-
+%ro
rl krl r“k.p(tl- %,av——‘e%b %,avr12 ‘(*)’(*)
where
and
(45)
(46)
Equation(46)appliestotheregionbetweentheinnercyld.nderandthelineofmaximumvel~cities.Fortheregionbetweentheoutercylinderandthelineofmaximumvelocities,YM and y& arereplacedby y~and y~, respectively.As inthecaseofvelocitydistributions,thevaluesof y+ aremeasuredalongnormalstotheinneroroutercylinderwhichextendtothelineofmaximumvelocities.Thevaluesof t+ areobtainedfromfigure2. Inthepresentcase,wheretheouterwallisinsulated,figure2 indicatesthatthetem.pe?mtureisuniformalongthenomalsdrawnfra theoutercylinderto.thelineofmaximumvelocities.Since q#o,t2 - t mustalsobe zero,inasmuchas t+ isfinite.Intheactualcase t wouldvarysomewhatalongthelinesnormaltotheoutercylinder.However,a constantt isconsistentwiththeassump-tionsmadeinthepresentanalysis,andtheerrorproducedissmallbe-causetheactualtemperaturegradientsaresmallintheregionbetweentheoutercylinderandthelineofmaximumvelocities.
AverageNusseltnmnberscalculatedby eqyation(44)areplottedagainsteccentricityfora rangeofReynoldsnumbersandforvaluesoftheparameter@~/~b of zeroand0.01infigure12. Thewalltemperaturedistributionswereobtainedfh?omfigure10. Forthecasewherethecylindersareconcentric,theNusseltnumbersareingood
.
u
(44)
i!=
“
NACATN 3451 17
agreementwiththoseobtainedby theconventionalNusselteqyationfortubesorthepredictedrelationfortubesinreference5. TheaverageNusseltnumibersdecreasewithincreasingeccentricity(fig.12)as didthefrictionfactors(fig.6). Forthemaximumeccentricitytheaver-ageNusseltnumbersdecreaseto aboutone-fifththeirvaluesforzeroeccentricity.Thecurvesalsoindicatea substantialeffectoftemperaturedistributionkrl/~b ontheNusseltnumbers.Thenumbersdecreaseas lcrl/~bincreasesorastubeconductivitythicknessdecreases.Theeffectofperipheralwalltemperaturelmtionon averageNusseltnumberisnotgenerallyconsideredin
Localheat-transfercoefficient.- Thelocalheat-transfercientata givenangle“6 isdefinedas
ThelocalNusseltnumbercorres~ondingto h canbe calculated
NU .
‘ml-lNusseltor .distri-analyses
coeffi-
(47)
(48)
Itcanbe seen.fromequation(48)that,foruniformcircumferentialwalltemperature,Nu/Nuav=h/~v = ql/ql,avjasmentionedpre~o~~(fig.8).
Vtiuesof Nu/N~v= h~l, fora valueof lml/~b of 0.01areplottedas functionsofangle e andof eccentricityin figure13.Thesecurvesarestronglyaffectedby changesinReynoldsnmber,incontrasttomostoftheprece~”results,becausethesec6ndteimirithedenominatorof eqpatio~(48)variesconsiderabJ.yw5.’dReynoldsnumber,whereasthefirsttermisnearlyconstant.
Itmightbementionedthatlocalheat-transfercoefficientsarenotessentialto givea completedescriptionoftheheattransferin eccen-tricannuli.Mostqutities of interestcouldbe obtainedfromthe”plotsofwallheat-transferdi&tribution,walltemperaturedistribution,andaverageNusseltnuniber.Thelocalheat-transfercoefficientsaregivenas a matterof interest,inasmuchasmostpersonsconcernedwithheattransferusuallyconsider10cA. h,es.t-trmsfer coefficientsratherthanlocalheat-transferrates.
18 NACATN3451
SUMMARYOFRESULTS
Thefollowingresultswereobtainedfromtheanalyticalinvestiga-tionof fullydevelopedturbulentheattransferandflowineccentricannlili:
1.Themaximumvelocitiesintheannulusoccurredclosertotheinnerthanto theoutercylinder.Thelocationofthelineofmaxi-mumvelocitieswasessentiallyindependentofReynoldsnumber.
2.Whenthecylinderswerenotconcentric,thevelocitiesandshearstressesin theregionwherethese~ation ofthecylinderswasleastwerelowerthantheaveragevaluesandapproachedzerowhenthecylin-derstouched.
3. Whentheiimercylinderwasheatedandthecylinderswerenotconcentric,theheattransfertothefluidintheregionwheretheseparatimofthecylinderswasleastwaslowerthantheaveragevalueandapproachedzerowhenthecylinderstouched.Forthecasewherethecylinderstouched,theheattransferapproachedzeromuchmorerapidlythandidtheshearstressas thelineofcontactwasa~roached.
4.Thefrictionfactorsfortheannuluswiththecylindersconcen-tricwereveryslightlyhigherthanthosefora circulartube.As theeccentricityincreased,thefrictionfactorsdecreased.
5. Whentheoutsidesurfaceofa thin-walledtubewasassumedtoformtheinnercylinderoftheannulusandheatwasa~lieduniformlyto theinsidesurfaceofthetube,thedifferencebetweenthemaximumandaveragewalltemperaturesfora giveneccentricitywasdirectlyproportionaltotheheatflux.
6.TheaverageNusseltnumbersfortheannuluswiththecylindersconcentricwereveryslightlyhigherthanthosefora tube.As theec-centricityincreased,theNusseltnumbersdecreased.Theaveragelhsseltnumberwasalso-foundtobe a functionofperipheralwalltem-peraturedistribution.
w
LewisFlightPropulsionLaboratoryNationalAdtisoryCommitteeforAeronautics
Cleveland,Ohio,March16,1955
L-
.
NACATN 3451 19
APPENDIXA
CONDITIONFCIRl!WZLYDEWELOPEDHEATTRANSFER
Forfullydevelopedheattransferandconstantfluid
q
t~ - ta ‘%
properties,
(Al)
where ~, ansrbitraryheat-transfercoefficient,is independentof x.If ~ werenotindependentof x at a greatdistancetromtheentrance(cyclicvariationsof ~ excluded),theabsolutevalueof ~ would.becomearbitrarilylargeas x increasedsothatforfinitetemperaturedifferencestheabsolutevalueof ql wouldbecomearbitrarilylarge.Thefollowingeq~tionsarespecialcases
%.=t~ - ‘b)av
ofequation(Al):
h
ql .h
t~-tb 1
Forthecasewherethewallheattransferperentof x (butnotof e),eq=tions{AZ)and(A3)to givethefollowingresults:
dtl ‘% ,avF= dx
dtl dtbZF-‘Z
or
d% ‘% ,av=“ dx
Theqwntity d~/dx isthereforeindependentoftransferis independentof x.
(A2)
(M)
unitareais independ-canbe differentiated
(A4)
6’whentheheat
20 NACATN 3451
APPENDIXB—.
.
%
Al
A2
b
%
D
De
g
h
k
%
2
P
SYMBOLS
Thefollowtngsymbolsareusedinthisreport:
totalareabetweeninnercylinderandoutercyl-irider,sqft
totalareabetweeninnercylinderandlineOYmax-imumvelocities,scfft
totalareabetweenoutercylinderandlineofmax-tiumvelocities,sqf%
wallthicknessof’innertube,ft
specificheatof fluidat constantm’essure,‘Btu/(lb)(°F)
diameter,f%
equivalent
conversion
diameterofannulus,
factor,32.2ft~secz
,
D2 - Dl,f%
localheat-tramsfer
Btusec)(sqft)(oF)
coefficient,t1 - ‘b,av$
‘l)avaverageheat-transfercoefficient,‘l)av- %,av’
Btusec)(sqf%){oF)
thermalconductivityoffluid,Btu/(sec)(sqftj(OF/ft).
thermalconductivityofmaterialofinnercylinder,Btu/(see)(sqft}(°F/ft)
peripheraldistancealongwall,ft
staticpressure,lb/sqftabs
.
NACATN 3451 “ 21
‘l)av
r
t
ta
%
.‘o
‘1
u
#
%
rateofheattransferfromtil ofinnercyfinderperunitarea,Btu/(sec)(sqf%)
averagerateofheattransferfrcnnwallofinnercyltiderperunitarea,Btu/(sec)(sqi%)
radius,ft
t~eratureof fluidat a point,*F
arbitrarytemperature
localbulktqrature of fluidat givenangleatcrosssectionofannulus,‘F
walltauperature,‘F
localwdl temperatureof innercylinderortube,OF
maximum wall temperatureof innercylinderortube,OF
the-averagedvelocityft/sec
localbullsvelocityatofannulus,ft/sec
velocityat a paintonft[sec
paralleltowallat a point,
givenangleat crosssection
lineofmaximumvelocities,
axialdistancealongannulus,l%normaldist=cetianwall,N
valueoffyl at u-~,ft
valueof y2 at u = ~, ft
coefficientofeddydiffusitityformomentum,sqft/sec
coefficientof eddydiffusivityforheat,sqft~sec
zunglemadewithcenterofinnertube,radians
22 NACATN3451
v absoluteviscosityof fluid,(lb)(sec)/sqf%
P massdensity,(lb)(sec2)/ft4
T shearstressin fluid,lb/sqf%
Dimensionlessquantities:
b%(tl,M,X- ‘l,av~
ql,E#I2
bqtl - ‘l)av1
r++1
‘1,av’1
e
f
fT1
Nu
%v
n
Pr
Re
wall-temperature-distributionparsmeterwhereau-—gl.eis zero
wall-temperature-distributionparsmeter
eccentricityparsmeter,distancebetweenof cylindersdividedby r2 - rl
.De~frictionfactor,—
2@%,av
2T~frictionfactor,-
centers
*
pub,av
Nusseltnumberbasedon
Nuseltnumberbasedon
* ,– c1
melocalheattransfer,~
averageheattransfer,‘apek
constant,0.109
Prandtlnumber,
Reynoldsnumber,
Cpvdkpu@e
.
b
P
7/-rldp/dxinner-tuberadiusparameter, PK/P
rl-.
.
mm TN34s1 23
t+
-H-ub
(t. - *}~owotemperatureparameter,
%4=
bulk-temperatureparameter,ft@4k)
temperatureparameter,d!l~ “
velocityp~ameter,—
@ o Po
‘uvelocityparameter,
F
-rldp/dx
P
bulk-velocity
bulk-velocity
AArudA
doU1
velocityparameter,~w1P
‘%velocityparameter,.~
$ evaluatedatlineof’maxinnmvelocities>
24 NACATN3451
%n
x
Subscripts:
a
o
1
2
1.
2.
3.
4.
5.
~ evaluated
wall-distance
wall-distance
Y; evaluated
Y; evaluated
ratioof eddy
at lineof
psmmeter,
parameter,
atlineof
atMne of
maximumvelocities
mad.mumvelocities
maximumvelocities
diffusivityforheattransferto
.
.
eddydiffusi.vityformomentumtransfer,~h/e
K&m& constant,0.36
arbitrary
pertainingto
pertainingto
pertainingto
awau
innercylinder
outercylinder
REFERENCES
vonK&m&n,Th.: TurbulenceandSkinFriction.Jour.Aero.Sci.,vol.1,no.1,Jan.1934,pp.1-20.
vonK&mu!n,%.: TheAnalogyBetweenFluidErictionandHeatTrans-fer. Trans.A.S.M.E.,vol.61,no.8,Nov.1939,pp.705-710.
B&’tindlij R. C.: HeatTransfertoMoltenMetals.Trans.A.S.M.E.,vol.69,no.8,Nov.1947,pp.947-959.
Deissler,RobertG.: *ical and~ertientalI&estigationofAdiabaticTurbulentFlowinSmoothTubes.NACATN 2138,1950.
Deissler,R. G.: HeatTransferandFluid”l&ictionforFullyDevel-opedTurbulentFlowofAirandSupercriticalWaterwithVariableFluidProperties.Trans.A.S.M.E.,vol.76,no.1,Jan.1954,pp.73-85.
b
,
NACATN 3451
6.Eckert,E.
25
R. G.,md Low,GeorgeM.: T~eratureDistributioninfiternallyHeatedWallsofHeatExchangersCcmposedofNoncircularFlowPassages.NACARep.1022,1951. (SupersedesNACATN 2257.)
7.Elrod,HaroldG.,Jr.: TurbulentHeatTransferinPolygonalFlowSectTons.NDA-10-7,NuclearDev.Associatesj~c. (NewYork},Sept.22,1952. (PurchaseOrder267031.)
8.Deissler,RobertG.: AnalysisofTurbulentTransfer,andFrictionin ~ooth TubesatSchmidtNmbers. NACATN 3145,1954.
HeatTransfer,MassHighPradtl &d
9.Deissler,RobertG.: AnalysisofFullyDevelopedTurbulentHeatTransferatLowPecletI?tmbersinSmoothTubeswithApplicationtoLiquidMetal.s.NACARME52F05,1952.
10.Eckert,E.R. G.: TemperatureDistributionin theWallsofEeatEx-_ers withNoncircularFlowPassages. HeatTransferandFluidMech.Inst.,Univ.Calif.,1952,Pp.175-186.
11.LowdermilX,WarrenH.,Weiland,WalterF.,Jr.,andLivingood,JohnN.B.: MeasurementofHeat-!lM3.nsferandfrictionCoefficientsforFlowofAirinNoncircularDuctsatH@ SurfaceTemperatures.NACARME53J07,1954.
I 20 40 100 200 400 6C
Reynoldsnumber,Re
Wall-distanceparameter,y+
Figure 1. - Generalizedvelocity distributionfor ?W1-lYdevelopedturbulentflow insmooth tubes (ref. 10).
I .1 , ,
30
2s
++ 20
5
0
~sL?&Y
1 2 46 e 10 20 40 60 60 103 ‘KM 403 6008001003 2030 4CQ0 6CO0
Wall-distance p=wneter, p
Figure 2. - Rxdicted generalized temperature distribution for flow of air. lhndtl number, 0.73.
~
NACATN 3451
%~av
——
(a)Eccentricityparameter,O.
.
.
(b)Ecce.ntricityparameter,0.6.
Figure3.- Predictedvelocitydistributionsforan annulus.Reynoldsnumber,20,000}r.+lr 3.5.
-s
m
29
u
ub,av.
(c)Eccentricityp.rameterj0.8.
%Yav
(d)Eccentricitypammeter,1-0.
Figure3. - Copcluded.Pcedictedvelocitydistributionsforan anuulus.Reynoldsnumber,20,m0Jr2/rl~3.5.
..
1.25
Eccentricity —
paremeter,e
o1.00
///
/
//
.75 /
5
<’ > ~
/
3~
.50 /
/ /a 1
rz.E1 h
e
.25 /
/ ‘1.0 /
o 20 40 60 80 NM 120 140 16JI 120e, deg
(a) Reynolds number, 20,CM.
Figure 4. . Predicted variation of ~j~,av vith angle e for various values of eccentricityy parmneter.r21r1, 3.5.
L t
1.25
.Eccentricity
— —
o1.00 L
/ +
/
/ /1 1
.75/ //
5’
<’ /
# ---%-
. .-/
.50 / ““/
.8
//
ESziJ ~
,25—
1.0
~ ‘
o 20 40 60 ‘%3 100 120 140 m) 1000,deg
..-
(b) Re~olds number, 600,000.
Figure 4, - Concluded. predicted variation of Ub/~, ~v with angle .9 for various valuea of eccentricityparameter. r2~rl, 3.5.
1
2.8
2.4
Q
/ ‘“
r2rl / ‘
2.0 - /
/
{
1.6 / ~
./
1.2Eccentricityparmneterj
/
e
o
!.e
.4 !3 // ‘ /
A / /
.9, Ueg
F@ur@ 5. - Fxedlctedvariationof shear stree6cm Inner.@Under with angle .9 for variousva.1.w?mof eccen-tricityparameter. r2/rl,3.5j Reynoldsnumber,20>IXI0& m>~.
1
0!N
. 4 # a‘,1 9 ●
,, ,,
NACATN 3451
.
.
n
1..0. I
\
.0
.6 \
.4 i
.2
0 .2 .4 .6 .8 1.0Eccentricityparameter,e ,
Figure 6. - tied~ctedvaria.tlonwitheccentricityparameterofshearstressoninnercylinderatpointofleastseparation.r2/rl~3.5jReynoldsnumber,20,030and600,000.
‘m
.
%
1X10-2
.8
.6
Eccentricity
.4 -e
—Annulus (presentanalysis)———Circular tube (ref. 5) 1.0
.2
.1 .2 .4 .6 .8 1 z 4 6 8 10x
I?igure7. - Predictedvariation ofReynolds number and eccentricity
Reynolds number, Re
frictionfactor based on static-pressuregradientwithpaxsmeter. r#l, 3.5. F
s
Q
#PJ
, 4* ?
14Eccentricitymeter,
e
~a’
12
rz r~
10
8
6
4
2
0
— : ~/ /
~/ “
0’20 40 60 80 100 120 140 120 1.93.9,deg
Figure 8. - Fredlctedvariationof heat traneferfrom inner cylinderwith angle 8 for variouOvaluesof eccentricityparameter. r2/rl,3.5; Reynoldsnumber,20,000and W, CJY3. Outer cylinderinmlated. CA
VI
36 NACATN 3451
1.0
.8
.6
.4
.2
0
\
\
.2 .4 .6 .8 1.0Eccent&icityparameter,e
Figure9. - Predictedvariationof heattransferfrominnercylinderat pointof leastseparationwitheccentricityparameter.r2/rl;3.5;Reynoldsnumber,20,0CKIand600,000.Outercylinderinsulated.
●
☛
.
.
N/WATN 3451 37
.
.
2.0+
Eccmtrlcity~t=s
1.5 — e
\ 1.0
\ \
1.0
\
.5
00
\
-.5\
-1.0
\
-1.5 \
\
-2.0 \\
-2.50 20 40 m So 103 1.20 140 lm 180
e,deg
illFigure10. - Predictedvarhtionofvailtemperatureon innertubetithanglee forvariountiueBofeccentricityparameter.Con6tantheatfluxattilde vail of Innertube;cutir cylindero? amulmi~U.lStdj r~rl,3.5JReymld.snumber,KI,W and8X3,000.
1.6
1.4
F
o
1.2
1.0
u!-!
$?&. .8
.6
.4
.2
E
.2 .4 .6 .8 1.0Eccentricityparameter,e .-
Figure11. - Predictedvariationof maximumtemperaturedifferenceon insidetubewitheccentricityparameter.Constantheatfluxatinsidewallofinnert~bejoutercylinderof annulusinsulated;r2/rl,3.5;Reynoldsnumber.,20,000and 600,000.
t
.
?
39
.
.
10
8
6
4
2
1
.8
.6
.4
.2
.1 .2 .4 .6.8”1 2 4 6 8 10xReynoldsnumber,Re
~1(a) ~ = O (unifbrmperipheralwalltemperaturedistribution).“
t
:105
H.gure12.- PredictedvariationofaverageNusseltnumberwithReynoldsnumberandeccentricityparameter.Outercylinderinsulated;Prandtlnumber,0.73.
.
40
.10X102I I I I I I I III I I I I t I I I I8
6
.4
2
1
.8
.6
.4
.2
.1 .2 .4 .6 .8 1 2 4 6 8 10x
.
.
Reynoldsnumber,Re
(b)~ = 0.01 (non~formperipheralwalltemperaturedistribution).t
Figure12.- Concluded.-PredictedveriationofaverageNusseltnumberwithReynoldsnumberandeccentricityparameter.Outercylinderinsulated.;Prandtlnumber,0.73.
.
.
I IEccentricity
14 wrsm,eter,
(0
\
Iz
Q
%F1
10
8I .8
//
6 //
II
4 .6
A
2.=-
/~ : ~
//+ 0
— ~
0 20 40 WI WI MM 120 - 140 Ma MO0, deg
(a)Reynoldenumber,ZO,WO.
FigureM. - Predictedvuriationor ratioof 10cal hmt.-tmfer coefficient @ ~ver~e ~at-trmfer ~oeffjc~en~ ~j~~
angle 0 for v~oum value~ of eccentricity pm-meter. Outer rylinder hsu.lat.d) X2/rl, 3 .!5J lmlf~b, 0.01.
7!.%
28EccentrlciQ
24
20
16
a J1
n
g ~%E
12
8
4 I
1
0 20 40 53, 80 m 120 140 15) I& 58, deg 9
(b)Reynoldenumber,KX),C130. 2
Fi&wxeS3. - Concluded. Fredict.edYarlaticmof ratioof localheat-transfercoefficientta arerwe heat-refer COe&fici@XJtwith @I13k S for tiGUB @W CIfeOCent15Cityp31WIWkr. Outercylinder inatiteds .2/rl,3.5j!@+b, O.Cl. Y
P
I
al