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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