m ~ _ o_ ~ -.* I%- NATIONAL ADVISORY CO’MMI’ITEE “g ~ FORAERONAUTICS TECHNICAL NOTE3451 ANALYSIS OF FULLY DEVELOPED TURBULENT HEAT TRANSFER AND FLOW INAN ANNULUS WITH VARIOUS ECCENTRICITIES By RobertG.DeisslerandMaynardF.Taylor LewisFlightPropulsionLaboratory Cleveland, Ohio Washington May1955 z =. ---- https://ntrs.nasa.gov/search.jsp?R=19930084129 2020-05-25T19:07:47+00:00Z
43
Embed
NATIONALADVISORYCO’MMI’ITEE“g~ z=. FORAERONAUTICS€¦ · NATIONALADVISORYCO’MMI’ITEE“g~ FORAERONAUTICS TECHNICALNOTE3451 ANALYSISOF FULLY DEVELOPED TURBULENT HEAT TRANSFER
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
m~_o_~
-.* I%-
NATIONALADVISORYCO’MMI’ITEE“g~FORAERONAUTICS
TECHNICALNOTE3451
ANALYSIS OF FULLY DEVELOPED TURBULENT HEAT TRANSFER
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-
where e and ~h aretheeddydiffusivitiesformomentumandheattransfer,respectively,thevaluesforwhicharedependentupontheamountandkindofturbulentmixingat a point.Intheseequationsy istakenas theperpendicular.d.istancefromthewall. Equations
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
(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.
(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).
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.
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.
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.
where Z1/rl~ e and ~ is zerofor 21= O,thepointofleastsepa-rationofthecylinders,becausethe~ temperaturedistributionissymmetricaboutthatyoiqt(thetemperaturegradientis zero).But
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.
6.TheaverageNusseltnumbersfortheannuluswiththecylindersconcentricwereveryslightlyhigherthanthosefora tube.As theec-centricityincreased,theNusseltnumbersdecreased.Theaveragelhsseltnumberwasalso-foundtobe a functionofperipheralwalltem-peraturedistribution.
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: