1、uI1.1IFORAERONAUTICS LTECHNICAL NOTE 2629ANALYTICAL AND EXPERIMENTAL INVESTIGATIONOF FULLYDEVELOPED TURBULENT FLOW OF AIRINA SMOOTH TUBEWITH HEAT TRANSFER WITH VARLABLE FLUID PROPERTIESBy R.G.DeisslerandC.S.EianLewisFlightPropulsionLaboratoryCleveland,Ohio.WashingtonFebruary1952. . .,.-. - .- . . .
2、. . . . . . ._ -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-(* _ . _ TECHLIBRARYK/lFB,NM”-IlllllunllllllllllllllllDClb5744NATIONALADVISORYCOMMITTEEFORAERONAIJTIUS- -ITECHNZCKLI?Ol?E2629ANALYTICALANDExl?ImmmTALmsTIGAmoN OFFULLYDEVELOPEDTWEWXWIFLOW
3、OFA3RINASIMIOTHTUBZWITHHEATTRANSFERWlX!13VARIAEGEFLUIDPROPIiZWTXSByR.G.DeisslerandC.S.EianSUMMARYA previousanalysisofturbulentflowandheattransferinsmoothtubeswithvariablefluidpropertiedwasmodifiedinordertob26),wheren snd K areexperhentalconstantshayingthevalues0.109and0.36,respectively.(3Thevariatio
4、nsacrossthetubeoftheshearstress% andtheheattrausferperunitarea q haveanegligibleeffectonthevelocityandtemperaturedistributions.Theeffectofthesevariationswi beinvestigatedinappendixB andinthesection“PredictedEffectofVar-iationofShesrStressandHeatTransferAcrossTube.”(4)Themoleculsxshearstressd heat-tr
5、ansfertermsinequa-tions(1and(2canbeneglectedintheregionatadistancefromthewall. TheeffectofthesetermswilJbeinvestigatedinappendixC andinthesection“PredictedEffectofMolecularShearStressandHeatTransferinRegionataDistancefromWall.”(5)Thestaticpressurecaubeconsideredconstantacrossthetube. .(6)ThePrandtln
6、umrsmdspecificheatcanbeconsideredconstant”withtemperaturevariation.Thevariationswithtemperatureofthespe-cificheatsndthePrandtlnumberofgasessreofa lowerjorderofmag-nitudethanthevariationsofviscosity,thermalconductivity,anddensity.,. . . . . . .-. .= . ._ . . . . . . - - -Provided by IHSNot for Resale
7、No reproduction or networking permitted without license from IHS-,-,-4 NACATIT2629cFlowclosetoWSIU-Forobtaininnthevelocity=d temperaturedistributionsclosetoa smoothwall.”theeressionfor andfromtheassumptionofconstantstaticpressure(assumption(5)andtheperfectgaslaw,p/p. canbereplacedby tO/t.Fromthedefh
8、itionsof and t+t - pt+%=1Equations(5)and(6thenbecome1= (1-pt+p+1du+(1-;t+%+Y+andI(7)(8)(9). . . - -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-t.IU.CATN2629,5Equations(8)and(9canbewritteninintegralformasI a71andJY+ -+u= w+ 2+o (l-pt+d+;t: ., (10
9、(n)Equations(10)and(U.)canbesolvedsimultaneouslybyiteration,thatis,assumedvaluesfor u+,y+,and t+ aresubstitutedintotherightsidesoftheequationsandnewvaluesof u+ and t+ arecalcu-latedbynumericalinteation.Thesenewvaluesarethensubstitutedintotherightsidesoftheequationsandtheprocessisrepeateduntilthevalu
10、esof u+and t+ correspontoeachvalueof y+ donotchangeappreciably.Equations(10and(U.)givetherelationsbetweenu+,t+,and y+ forvariousvaluesoftheheat-transferpsrameterforflowclosetoawall.Flowatadistancefromwall.- _ we ofammnptions(1),(2,73),(4),and(6),equations(13and(2canbewrittenindhen-sionlessformforflo
11、wata distancefromthewallas f)X2 (du+/dy+)3 du+Po (d2u+/dy+2)2andDividingequation(12)byequationJt+dt+%+(13andintegratingresultin.P+(12)(13). -_ _ . ._ _ _ _ _ _ _ _ - _. ._Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 CA,TN2629ort+-tl+=u+-Ul+ (14)
12、fwheretl+ and Ul+ arethevaluesof t+ ad u+ at Y1+9ch sthelowe;tvalueof y+.or,fromassumption(5)forwhichequations(12and(13);pply.Thent= 1- (U+-ul+tl+)p (15)oandtheperfectgaslaw,. (16)=l_ (U+-:l+tl+)pOnsubstitutingequation(16)inequation(12)thereresults+41- (U+-ul+tl+)p=xz(du+/w1(d%+/d.y+2)2Oneintegratio
13、nofequation(17)ves(17)wherethenegativesignwasselectedintakingthesquarerootinordertomake x positive.Integrationofequation(18)gives2xCll+ - 4,l-p(u+-ul+tl+)y+=_ e? “1l.p(u+-ul+tl+)-t1 +c2X2(19)Asthewallisapproached,thevelocitygradientbecomesverylargecom-paredwiththatatadistancefkomthewallsothat dy+/du
14、+approacheszeroas y+ approachestion(18)intoequationand u+ equalzero,Cwhen y+= yl+. Thenzero. If dy+/du+issubstitutedfromequa-(19)and dy+/du+issetequaltozerowhen y+equalszero.TodetermineCl,set U+=,%+ .- - - .- Provided by IHSNot for ResaleNo reproduction or networking permitted without license from I
15、HS-,-,-NACATN2629 7and(20Equation(20)givestherelationbetweenu+ and y+ forvariouEvaluesof forflowfarfromtbewall.Theparametert+ s.ndthusthetem-peraturedistributioncanbecslcatedfromequation(14).ForaPrandtlnumber-of1,tl+.U1+3andequation(20)reducestotheequationfOraPrandtlnuderof1 giv b referce2. For . ()
16、eqtition(20)isindeterminate.Forthiscase 13issetequaltozeroinequation(17)beforeintegrating,andthewell-knownlogarithmicequa-tionisobtained:+ +ogey+cNusseltnumber,Reynoldsnuniber,andfrictionfactor.-Itcaneasilybeshownfromthedefinitionofthequantitiesinvolved(seereference2)thattheNusseltnumber,Reynoldsnum
17、iber,andfrictionfac-torwiththefluidpropertiesevaluatedatgivenby thewalltemperaturearewhere2rolPrN=%Reo= tib+ro+(21)(22)(23). . - ._ _ - _ . -_ . _ . _ . _ ._-_z Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8and%+=-1- 2IW!ATN2629rro+ t+u+(ro-Y+)Jo
18、l+ m+Po+Ju+(ro+-y+)l-pt+ v+oJ%“-To u+(ro+-y+)dy+(24)%a(25)Therelationsbetweenu+,y+,and t+ me calculatedfromtheequa-tionsgiveninthetwoprecedingsections.TheI?usseltnumber,Reynoldsnuniber,andfrictionfactorwiththefluidpropertiesevaluatedatsometempeatureinthefluidotherthanthewallltemperaturecanbefoundbyu
19、singthefollowingF0+.J -Y+) +u+(roor0+U+(ro+-fil.-+equationfor(26)Boththethermal-conductivityratioandtheviscosityratioareequaltothetemperatureratiorai;edtotheexponentd inasmuchasthespecificheatandPrandtlnuniberareassumedconstant(assumption(6).Althoughtheprecedinganalysiswascarriedoutforthecaseerethec
20、ompressibilityeffectsduetohighvelocitiescanbeneglected,itisshowninreference2thatthesamecorrelationsapply,ingeneral,toflowskhighsubsonicvelocities.Forthelattercase(exceptforverysmalltemperaturedifferences)itisnecesssryonlytoreplacethestaticbulktemperaturebythetotslbulktemperatureinthedefinitionoftheh
21、eat-transfercoefficienth. . . _. _ -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,.N40ATN2629 9APPARATUSA schematicdiagramofthetestsectionandassociatedequipmentisshowninfigwe1. Compressedairfirstpassesthroughafilterandthenintosnalumina-typeairdry
22、er.Fromthedryertheairflowsthroughoneoffourorificesselectedrenmtelybyorificeselectionvalvesandthenthroughvalveswhichautomaticallycontroltheflowrate.Itromthecontrolvalvestheairflowsintoa surgetank,thentothe“inletmixingtank,throughthetestsection,andintotheoutletminingtankfrornwhichitisdischargedtotheat
23、mospherethroughapressureregulat-ingvalve.Electricpowerissuppliedtotheheatertubefroma 208-volt60-cyclesupplythroughanautotransformermd a step-downtransformerwithmxdmmmsecondaryvoltagesofeither15or25volts.Thelow-voltageleadsofthestep-downtramfomerareconnectedtotheflsngesontheendsofthetestsectionbycopp
24、erbusbsrs.Thecapacityoftheelec-tricequipmentis100kilovolt-amperes.Additionalheattocompensateforendlossesissuppliedtoeachendoftheheaterttiewhereguardheatercoilsofnichromewirearewrappedaroundthecone-shapedheater-tubeflanges.Individualcontrolofthe110-volt60-cyclesupplytoeachendofthetubeisobtainedbyvari
25、abletransfofirwithcapacitiesof1kilovolt-ampere.TestSectionThetestsection,shownindetailinfigure2,isa commercially-smoothInconeltubehavings.ninsidedismeterof0.87inchandanout-sidedismeterof1.25incheswitha sharpright-angleentrance.Steelflangeswithcone-shaped-centersweldedtothetubeateachendprovideelectri
26、calcontactwiththetrsmformerleadsfromthepowersupply.Thetube,whichisthermallyinsulatedbya5-inchthicknessofhightemperaturequartvzwool,actsasaheatingelementwithaneffectiveheat-transferlengthof87inches.Staticpressuresaremeasuredthrough0.03-inchholesdrilledinthetubeatthepositionsshowninfigure2. Outsidetub
27、e-wall.teratures=e measuredlychromel-alumelthermocouples(twothermocoupleslocated180apartateachof21sta-tionsandoneneareachendofthetubeanda self-balancingindicating-typepotentiometer.AirtotaltemperaturessremeasuredbythermocoupleslocatedinthecentralpassagesoftheiziLetandoutletmLdngtanofthetestsectionas
28、shownh figure1. Theinletn=n tankconsistsoftwoconcentricpassagessoarrs.ngedthattheairmakestwopassesthroughthetankbeforeenteringthetestsection,whiletheoutlet,t*is comprisedoffourconctritank-s.Thepassageofthe throughthecentral-chambersndconcentricpassagespreventsradia-tionerrorsinthetemperaturemeasurem
29、ents.Bafflesareprovidedinthecentralpassagestoinmrethoroughmtingoftheairbeforeitstemperatureismeasured. - - .- .- - _ . . .-. .-. - - . . .- - -.- -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10Total-pressureMeasurementsW TN2629Anopeningfortaking
30、total-pressuremeasurementsacrossthetubeislocated6inchesfromtheetitofthetubeasshowninfigure2. Aholehav3ngaO.15-inchdiameterthroughwhichatotal-pressureprobeentersthetubewasdrilledinthetubewallatrightanglestothestatic-pressuretaps.A probeactuatortonmvetheprobeandmeasureitsdistanceintothetubew fittedtoa
31、 6-inchlengthoftubingattheopening.Thelocationofthetotal-pressureprobewithrespecttotheopeningisshowninfigure2(b).TheprobetiphasaflatopeningwithaheightofO.W inchandaO.002-inchwallandismadesothatthetipjustclearstheedgeofthe0.15-inchholeinthetestsection.Thetotalprojectedareaoftheprobeinthedirectionofflo
32、wwiththetipatthecenterofthetubeisaboutlPercmtoftheareaofthetube,buttheeffectiveblockingareaatthetipisbecausethemainportionoftheprobeisdownstreamTotal-TemperatureMeasurementsconsiderablylessofthetip.Thesameopeninginthe%ubeusedfortakingtotal-pressuremeasure-mentsisusedfortakingtotsl-temperaturemeasure
33、ments.Figure2(c)showsthelocationofthetotal-temperatureprobewithrespecttothe .,openinginthetube.Oneofthetwoprongsoftheprobeismadeofchromelwhiletheotherismadeofalumel.ThethermocoupleoftheprobeconsistsofO.001-inchchromelandalumelwirebutt-weldedbetweenthepwngswhichwere0.07-inchapart.Temperaturesasprobew
34、erereadona self-balancirindicating-typeMETHODSTestProceduremeasuredbythepotentiometer.Preliminaryrunsheatlossthroughthefortheendheaters.withnoairflowwerefirstmadetodeterminebothtest-sectioninsulationandrequiredpowersettingsThetubewasheatedtovszioustemperaturesbyadjustingtheelectricalpowerinputtothet
35、ube,andthepowersettingsfortheendheaterswereadjustedtogiveauniformwalltemperatureforeachrun.Foreachtestruntheflowcontrolwasadjustedtogivethedesiredflowrateandtheelectric-powercontrolforthetubewassettogivethedesiredtube-walltemperature.Thepowersettingstobeusedfortheendheatersateachtemperatureweredeter
36、minedfromtheprelnaryruns.Fortherunsmadethetube-walltemperaturenesrtheexitvsriedbetween300and1500l?,andtheReyuoldsnumberwithfluidpropertiesbasedonthebulktemperaturevariedbetween8000and500,000.Themax-imumMachnumberwasabout0.6. . -Provided by IHSNot for ResaleNo reproduction or networking permitted wit
37、hout license from IHS-,-,-NK!ATN2629 U.Thefollowingquantitiesweremeasuredforeachrun: airflow,staticpressuresatthewall,outsidetube-wsIltemperatures,inletandoutlettotalbulk-airtemperature,andelectrical-powerinputtothetube.Total-pressureandtemperaturesurveysata section6inchesfromtheexitofthetubewhereth
38、eflowwaspracticallyfuJ3.ydevelopedweretakenduringsomeofthetests.Measurementsweretakenatpointsbetweenthewd.1.inwhichtheprobeopeningwaslocatedandthecenterofthetube.Mesmrementsmadeneartheoppositewallwereinaccuratebecauseofdisturbanceduetotheprobe.Total-pressuresurveysweretakenfortube-walltemperaturesup
39、to1300F,whereastotal-temperaturesurveysweremadeonlyatlowwalltemperatures(300F)becauseofthedelicacyofthetemperatureprobe.ReductionofLocal.heat-trsnsferrate.-areafromthetubetothesiratlatedfromExperimentalDataThelocalheat-transferrateperunita cross-sectionofthetubewascalcu-(27)wherethechangeofenthalpyA
40、H wasobtainedfromthebulktotsltem-peraturesoftheairattheinletandoutletofthetube.Theheatgen-erationpersquarefootoftubeareaG wasobtainedbymeasurementofelectriccurrentandtuberesistanceandtheheatloss L wasdeter-minedbyheatingthetubetovarioustemperatureswithnoairflowing.-ThequantitiesG,L,and S areallfunct
41、ionsoftubetemperdej thesubscript10C meansthatthequantitiesweredeterminedatthelocaltubetemperature,whereasthesubscriptave meansthattheyweredeterminedattheaveragetubetegrperature.Thequantityinthefirst parenthesisistheaverageheattransferperunitarea;thequantityinthesecondparenthesisisa correctionfactorf
42、orlocalheattransfer.The1OCSJ-heat-trsmferrate wasdeterminedfromequation(27)ratherthanfrom oc-L1oc becausethemeasurentsofinletandout-letairtemperaturewere,ingeneral,moreaccuratethanthoseoftuberesistanceandheatloss.Theuseofthisequationalsoreduceserrorsduetopossiblenonequilibriumconditions.Thequantityi
43、nthesecondparenthesisintheequationdidnotvsrygreatlyfrom1formostofthetests.Insidetube-wallandlocalbulktemperatures.-Theinsidetube-walltemperatureto wasobtainedfromthemeasuredoutsidetube-walltem-peratureta bythefollowlngequationderivedinreference4:. - .- . -.- . - - - - - - _ - -_ - _. _ - . -Provided
44、 by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12 NACATN2629.(28)Inequation(28theassumptionsaremadethatheatisgenerateduniformlyacrossthetubewallthicl.messandthattheheatflowateverypointinthetubeisradiaJlyinward.Forthepresentteststheradialtempera-turedropthro
45、ughthetubewallwasverysmallcomparedwiththediffer-encebetweentheinsidewallandtheairbulktemperature.Thelocaltotalbu3Js-airtemperatureataointwasobtainedbysubtractingfromthemeasuredbulktemperatureatthetubet thetem-peraturechangeproducedbytheheataddedtotheairintheportionofthetubebetweentheexitandthepointi
46、nquestionPhysicalpropertiesofair.-Inordertocompsretheexperimentalresultswiththeanalysisgiveninreference2,theviscosiiyandther-malconductivitywerebothassumedtobeproportional.to t0.68.mespecificheatandPrandtlnumberwereassumednottovarywithtempera-ture.Substantiallydifferentresultswouldbeobtainedifadiffe
47、rentvsriationofthermalconductivitywithtemperaturewereused.A conduc-tivityproportionalto toa7185 issometimesvtbenmmentwn-pressuregradientswerecalcu-latedfrom utipW2. %amm=2A2g2b (30).Equation(30)isexactonlywhenthevelocityprofileisuniform;theeffectofnonuniformvelocityprofileoncslculatedshearstresswascheckedforturbulentflowandwasfoundtobenegligibleforthepres-enttests.Inequation(30) wasfoundfromtheperfectgaslawPPb (31) . . . . - .- .-. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,
