Radiative Heat Transfer PDF
Preview Radiative Heat Transfer
CHAPTER 1 FUNDAMENTALS OF THERMAL RADIATION 1.1 INTRODUCTION Thetermsradiativeheattransferandthermalradiationarecommonlyusedtodescribethescience oftheheattransfercausedbyelectromagneticwaves. Obviouseverydayexamplesofthermal radiationincludetheheatingeffectofsunshineonaclearday,thefactthat—whenoneisstanding infrontofafire—thesideofthebodyfacingthefirefeelsmuchhotterthantheback,andsoon. More subtle examples of thermal radiation are that the clear sky is blue, that sunsets are red, andthat, duringaclearwinternight, wefeelmorecomfortableinaroomwhosecurtainsare drawnthaninaroom(heatedtothesametemperature)withopencurtains. Allmaterialscontinuouslyemitandabsorbelectromagneticwaves,orphotons,bylowering orraisingtheirmolecularenergylevels. Thestrengthandwavelengthsofemissiondependon thetemperatureoftheemittingmaterial. Asweshallsee,forheattransferapplicationswave- lengthsbetween10−7mand10−3m(ultraviolet,visible,andinfrared)areofgreatestimportance andare,therefore,theonlyonesconsideredhere. Beforeembarkingontheanalysisofthermalradiationwewantbrieflytocomparethenature of this mode of heat transfer with the other two possible mechanisms of transferring energy, conductionandconvection. Inthecaseofconductioninasolid,energyiscarriedthroughthe atomiclatticebyfreeelectronsorbyphonon–phononinteractions(i.e.,excitationofvibrational energylevelsforinteratomicbonds). Ingasesandliquids,energyistransferredfrommolecule to molecule through collisions (i.e., the faster molecule loses some of its kinetic energy to the slower one). Heat transfer by convection is similar, but many of the molecules with raised kineticenergyarecarriedawaybytheflowandarereplacedbycolderfluid(low-kinetic-energy molecules),resultinginincreasedenergytransferrates. Thus,bothconductionandconvection require the presence of a medium for the transfer of energy. Thermal radiation, on the other hand,istransferredbyelectromagneticwaves,orphotons,whichmaytraveloveralongdistance withoutinteractingwithamedium. Thefactthatthermalradiationdoesnotrequireamedium foritstransfermakesitofgreatimportanceinvacuumandspaceapplications. Thisso-called “actionatadistance”alsomanifestsitselfinanumberofeverydaythermodynamicapplications. Forexample,onacoldwinterdayinaheatedroomwefeelmorecomfortablewhenthecurtains areclosed: ourbodiesexchangeheatbyconvectionwiththewarmairsurroundingus,butalso byradiationwithwalls(includingcoldwindowpanesiftheyarewithoutcurtains);wefeelthe heatfromafireadistanceawayfromus,andsoon. 1 2 1FUNDAMENTALSOFTHERMALRADIATION Another distinguishing feature between conduction and convection on the one hand and thermalradiationontheotheristhedifferenceintheirtemperaturedependencies. Forthevast majorityofconductionapplicationsheattransferratesarewelldescribedbyFourier’slawas ∂T q =−k , (1.1) x ∂x where q is conducted heat flux1 in the x-direction, T is temperature, and k is the thermal x conductivityofthemedium. Similarly,convectiveheatfluxmayusuallybecalculatedfroma correlationsuchas q=h(T−T∞), (1.2) wherehisknownastheconvectiveheattransfercoefficient,andT∞isareferencetemperature. Whilekandhmaydependontemperature,thisdependenceisusuallynotverystrong. Thus, formostapplications,conductiveandconvectiveheattransferratesarelinearlyproportionalto temperaturedifferences. Asweshallsee,radiativeheattransferratesaregenerallyproportional todifferencesintemperaturetothefourth(orhigher)power,i.e., q∝T4−T∞4. (1.3) Therefore,radiativeheattransferbecomesmoreimportantwithrisingtemperaturelevelsand maybetotallydominantoverconductionandconvectionatveryhightemperatures. Thus,ther- malradiationisimportantincombustionapplications(fires,furnaces,rocketnozzles,engines, etc.),innuclearreactions(suchasinthesun,inafusionreactor,orinnuclearbombs),during atmosphericreentryofspacevehicles,etc. Asmoderntechnologystrivesforhigherefficiencies, thiswillrequirehigherandhighertemperatures,makingthermalradiationevermoreimpor- tant. Otherapplicationsthatareincreasinginimportanceincludesolarenergycollectionand thegreenhouseeffect(bothduetoemissionfromourhigh-temperaturesun). And,finally,one ofthemostpressingissuesformankindtodayaretheeffectsofglobalwarming, causedbythe absorptionofsolarenergybyman-madecarbondioxidereleasedintotheEarth’satmosphere. Thesamereasonsthatmakethermalradiationimportantinvacuumandhigh-temperature applicationsalsomakeitsanalysismoredifficult,oratleastquitedifferentfrom“conventional” analyses. Under normal conditions, conduction and convection are short-range phenomena: Theaveragedistancebetweenmolecularcollisions(meanfreepathforcollision)isgenerallyvery small,maybearound10−10m. Ifittakes,say,10collisionsuntilahigh-kinetic-energymolecule hasakineticenergysimilartothatofthesurroundingmolecules, thenanyexternalinfluence is not directly felt over a distance larger than 10−9m. Thus we are able to perform an energy balanceonan“infinitesimalvolume,”i.e.,avolumenegligiblysmallincomparisonwithoverall dimensions,butverylargeincomparisonwiththemeanfreepathforcollision. Theprinciple of conservation of energy then leads to a partial differential equation to describe the temperature fieldandheatfluxesforbothconductionandconvection. Thisequationmayhaveuptofour independent variables (three space coordinates and time) and is linear in temperature for the case of constant properties. Thermal radiation, on the other hand, is generally a long-range phenomenon. Themeanfreepathforaphoton(i.e.,theaveragedistanceaphotontravelsbefore interactingwithamolecule)maybeasshortas10−10m(e.g.,absorptioninametal),butcanalso beaslongas10+10morlarger(e.g.,thesun’srayshittingEarth). Thus,conservationofenergy cannot be applied over an infinitesimal volume, but must be applied over the entire volume under consideration. This leads to an integral equation in up to seven independent variables (thefrequencyofradiation,threespacecoordinates,twocoordinatesdescribingthedirectionof travelofphotons,andtime). The analysis of thermal radiation is further complicated by the behavior of the radiative propertiesofmaterials. Propertiesrelevanttoconductionandconvection(thermalconductivity, 1Inthisbookweshallusethetermheatfluxtodenotetheflowofenergyperunittimeandperunitareaandthe termheatratefortheflowofenergyperunittime(i.e.,notperunitarea). 1.2THENATUREOFTHERMALRADIATION 3 kinematic viscosity, density, etc.) are fairly easily measured and are generally well behaved (isotropic throughout the medium, perhaps with weak temperature dependence). Radiative propertiesareusuallydifficulttomeasureandoftendisplayerraticbehavior. Forliquidsand solidsthepropertiesnormallydependonlyonaverythinsurfacelayer,whichmayvarystrongly withsurfacepreparationandoftenevenfromdaytoday. Allradiativeproperties(inparticular for gases) may vary strongly with wavelength, adding another dimension to the governing equation. Rarely,ifever,maythisequationbeassumedtobelinear. Becauseofthesedifficultiesinherentintheanalysisofthermalradiation,agoodportionof thisbookhasbeensetasidetodiscussradiativepropertiesanddifferentapproximatemethods tosolvethegoverningenergyequationforradiativetransport. 1.2 THENATUREOFTHERMAL RADIATION Thermal radiative energy may be viewed as consisting of electromagnetic waves (as predicted by electromagnetic wave theory) or as consisting of massless energy parcels, called photons (as predicted by quantum mechanics). Neither point of view is able to describe completely all radiativephenomenathathavebeenobserved. Itis,therefore,customarytousebothconcepts interchangeably. Ingeneral,radiativepropertiesofliquidsandsolids(includingtinyparticles), andofinterfaces(surfaces)aremoreeasilypredictedusingelectromagneticwavetheory,while radiativepropertiesofgasesaremoreconvenientlyobtainedfromquantummechanics. Allelectromagneticwaves,orphotons,areknowntopropagatethroughanymediumata highvelocity. Sincelightisapartoftheelectromagneticwavespectrum,thisvelocityisknown asthespeedoflight,c. Thespeedoflightdependsonthemediumthroughwhichittravels,and mayberelatedtothespeedoflightinvacuum,c ,bytheformula 0 c c= 0, c =2.998×108m/s, (1.4) n 0 where n is known as the refractive index of the medium. By definition, the refractive index of vacuum is n ≡ 1. For most gases the refractive index is very close to unity, for example, air at room temperature has n = 1.00029 over the visible spectrum. Therefore, light propagates through gases nearly as fast as through vacuum. Electromagnetic waves travel considerably slowerthroughdielectrics(electricnonconductors),whichhaverefractiveindicesbetweenap- proximately1.4and4,andtheyhardlypenetrateatallintoelectricalconductors(metals). Each wavemaybeidentifiedeitherbyits frequency,ν (measuredincycles/s=s−1=Hz); wavelength,λ (measuredinµm=10−6mornm=10−9m); wavenumber,η (measuredincm−1);or angularfrequency,ω (measuredinradians/s=s−1). Allfourquantitiesarerelatedtooneanotherthroughtheformulae ω c ν= = =cη. (1.5) 2π λ Eachwaveorphotoncarrieswithitanamountofenergy,(cid:15),determinedfromquantummechan- icsas (cid:15)=hν, h=6.626×10−34Js, (1.6) where h is known as Planck’s constant. The frequency of light does not change when light penetratesfromonemediumtoanothersincetheenergyofthephotonmustbeconserved. On the other hand, wavelength and wavenumber do, depending on the values of the refractive 4 1FUNDAMENTALSOFTHERMALRADIATION Visible w Violet Blue Green Yello Red Infrared X rays Ultraviolet Microwave Gamma rays Thermal radiation 0.40 0.70 10–5 10–4 10–3 10–2 10–1 1 10 102 103 104 Wavelengthλ, µm 109 108 107 106 105 104 103 102 10 1 Wavenumberη, c m –1 1019 1018 1017 1016 1015 1014 1013 1012 1011 Frequencyν , Hz FIGURE1-1 Electromagneticwavespectrum(forradiationtravelingthroughvacuum,n=1). index for the two media. Sometimes electromagnetic waves are characterized in terms of the energythataphotoncarries,hν,usingtheenergyunitelectronvolt(1eV=1.6022×10−19J). Thus, lightwithaphotonenergy(or“frequency”)ofaeVhasawavelength(invacuum)of hc 6.626×10−34Js×2.998×108m/s 1.240 λ= = = µm. (1.7) hν a1.6022×10−19J a Sinceelectromagneticwavesofvastlydifferentwavelengthscarryvastlydifferentamounts of energy, their behavior is often quite different. Depending on their behavior or occurrence, electromagnetic waves have been grouped into a number of different categories, as shown in Fig. 1-1. Thermal radiation may be defined to be those electromagnetic waves which are emittedbyamediumduesolelytoitstemperature[1]. Asindicatedearlier,thisdefinitionlimits the range of wavelengths of importance for heat transfer considerations to between 0.1µm (ultraviolet)and100µm(midinfrared). 1.3 BASICLAWSOFTHERMAL RADIATION When an electromagnetic wave traveling through a medium (or vacuum) strikes the surface of another medium (solid or liquid surface, particle or bubble), the wave may be reflected (either partially or totally), and any nonreflected part will penetrate into the medium. While passingthroughthemediumthewavemaybecomecontinuouslyattenuated. Ifattenuationis complete so that no penetrating radiation reemerges, it is known as opaque. If a wave passes throughamediumwithoutanyattenuation,itistermedtransparent,whileabodywithpartial attenuation is called semitransparent.2 Whether a medium is transparent, semitransparent or opaquedependsonthematerialaswellasonitsthickness(i.e.,thedistancetheelectromagnetic wave must travel through the medium). Metals are nearly always opaque, although it is a 2Amediumthatallowsafractionoflighttopassthrough,whilescatteringthetransmittedlightintomanydifferent directions,forexample,milkyglass,iscalledtranslucent. 1.4EMISSIVEPOWER 5 T = const. T FIGURE1-2 Kirchhoff’slaw. common high school physics experiment to show that light can penetrate through extremely thinlayersofgold. Nonmetalsgenerallyrequiremuchlargerthicknessesbeforetheybecome opaque,andsomearequitetransparentoverpartofthespectrum(forexample,windowglass inthevisiblepartofthespectrum). An opaque surface that does not reflect any radiation is called a perfect absorber or a black surface: Whenwe“see”anobject,oureyesabsorbelectromagneticwavesfromthevisiblepart ofthespectrum,whichhavebeenemittedbythesun(orartificiallight)andhavebeenreflected by the object toward our eyes. We cannot see a surface that does not reflect radiation, and it appears “black” to our eyes.3 Since black surfaces absorb the maximum possible amount of radiativeenergy,theyserveasastandardfortheclassificationofallothersurfaces. It is easy to show that a black surface also emits a maximum amount of radiative energy, i.e., more than any other body at the same temperature. To show this, we use one of the manyvariationsofKirchhoff’slaw:∗ Considertwoidenticalblack-walledenclosures,thermally insulatedontheoutside,witheachcontainingasmallobject—oneblackandtheotheronenot— asshowninFig.1-2. Afteralongtime,inaccordancewiththeSecondLawofThermodynamics, both entire enclosures and the objects within them will be at a single uniform temperature. Thischaracteristicimpliesthateverypartofthesurface(oftheenclosureaswellastheobjects) emits precisely as much energy as it absorbs. Both objects in the different enclosures receive exactly the same amount of radiative energy. But since the black object absorbs more energy (i.e.,themaximumpossible),itmustalsoemitmoreenergythanthenonblackobject(i.e.,also themaximumpossible). Bythesamereasoningitiseasytoshowthatablacksurfaceisaperfectabsorberandemitter at every wavelength and for any direction (of incoming or outgoing electromagnetic waves), andthattheradiationfieldwithinanisothermalblackenclosureisisotropic(i.e., theradiative energydensityisthesameatanypointandinanydirectionwithintheenclosure). 1.4 EMISSIVEPOWER Everymediumcontinuouslyemitselectromagneticradiationrandomlyintoalldirectionsata rate depending on the local temperature and on the properties of the material. This is some- timesreferredtoasPre´vost’slaw(afterPierrePre´vost,anearly19thcenturySwissphilosopher and physicist). The radiative heat flux emitted from a surface is called the emissive power, E. We distinguish between total and spectral emissive power (i.e., heat flux emitted over the entire 3Notethatasurfaceappearingblacktooureyesisbynomeansaperfectabsorberatnonvisiblewavelengthsand viceversa;indeed,manywhitepaintsareactuallyquite“black”atlongerwavelengths. ∗GustavRobertKirchhoff(1824–1887) Germanphysicist. AfterstudyinginBerlin,Kirchhoffservedasprofessorofphysicsatthe UniversityofHeidelbergfor21yearsbeforereturningtoBerlinasprofessorofmathematical physics. TogetherwiththechemistRobertBunsen,hewasthefirsttoestablishthetheory ofspectrumanalysis. 6 1FUNDAMENTALSOFTHERMALRADIATION (cid:50)(cid:53)(cid:48)(cid:48) (cid:66)(cid:108)(cid:97)(cid:99)(cid:107)(cid:98)(cid:111)(cid:100)(cid:121)(cid:32)(cid:101)(cid:109)(cid:105)(cid:115)(cid:115)(cid:105)(cid:118)(cid:101)(cid:32)(cid:112)(cid:111)(cid:119)(cid:101)(cid:114)(cid:32)(cid:97)(cid:116)(cid:13) (cid:53)(cid:55)(cid:55)(cid:55)(cid:32)(cid:75)(cid:44)(cid:32)(cid:110)(cid:111)(cid:114)(cid:109)(cid:97)(cid:108)(cid:105)(cid:122)(cid:101)(cid:100)(cid:32)(cid:116)(cid:111)(cid:32)(cid:49)(cid:51)(cid:54)(cid:54)(cid:32)(cid:87)(cid:47)(cid:109)(cid:50) (cid:109) µ (cid:50)(cid:48)(cid:48)(cid:48) (cid:79) (cid:69)(cid:120)(cid:116)(cid:114)(cid:97)(cid:116)(cid:101)(cid:114)(cid:114)(cid:101)(cid:115)(cid:116)(cid:114)(cid:105)(cid:97)(cid:108)(cid:32)(cid:115)(cid:111)(cid:108)(cid:97)(cid:114)(cid:32)(cid:115)(cid:112)(cid:101)(cid:99)(cid:116)(cid:114)(cid:117)(cid:109)(cid:44)(cid:32)(cid:49)(cid:51)(cid:54)(cid:54)(cid:32)(cid:87)(cid:47)(cid:109)(cid:50) (cid:50) (cid:51) (cid:0)(cid:0) (cid:109) (cid:34)(cid:65)(cid:105)(cid:114)(cid:32)(cid:109)(cid:97)(cid:115)(cid:115)(cid:32)(cid:111)(cid:110)(cid:101)(cid:34)(cid:32)(cid:115)(cid:111)(cid:108)(cid:97)(cid:114)(cid:32)(cid:115)(cid:112)(cid:101)(cid:99)(cid:116)(cid:114)(cid:117)(cid:109) (cid:87)(cid:47) (cid:110)(cid:44)(cid:32) (cid:49)(cid:53)(cid:48)(cid:48) (cid:79)(cid:50) (cid:100)(cid:105)(cid:97)(cid:116)(cid:105)(cid:111) (cid:72)(cid:79)(cid:50)(cid:79)(cid:50) (cid:97) (cid:111)(cid:108)(cid:97)(cid:114)(cid:32)(cid:105)(cid:114)(cid:114) (cid:49)(cid:48)(cid:48)(cid:48) (cid:0)(cid:0)(cid:0)(cid:0)(cid:72)(cid:50)(cid:79)(cid:72)(cid:50)(cid:79) (cid:0)(cid:0) (cid:0) (cid:115) (cid:99)(cid:116)(cid:114)(cid:97)(cid:108)(cid:32) (cid:53)(cid:48)(cid:48) (cid:72)(cid:50)(cid:79) (cid:83)(cid:112)(cid:101) (cid:79)(cid:85)(cid:51)(cid:86) (cid:86)(cid:105)(cid:115)(cid:105)(cid:98)(cid:108)(cid:101) (cid:73)(cid:110)(cid:102)(cid:114)(cid:97)(cid:114)(cid:101)(cid:100) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:72)(cid:50)(cid:79) (cid:72)(cid:0)(cid:0)(cid:50)(cid:79) (cid:72)(cid:50)(cid:79)(cid:67)(cid:79)(cid:50) (cid:0) (cid:48) (cid:48)(cid:46)(cid:50) (cid:48)(cid:46)(cid:52) (cid:48)(cid:46)(cid:54) (cid:48)(cid:46)(cid:56) (cid:49)(cid:46)(cid:48) (cid:49)(cid:46)(cid:50) (cid:49)(cid:46)(cid:52) (cid:49)(cid:46)(cid:54) (cid:49)(cid:46)(cid:56) (cid:50)(cid:46)(cid:48) (cid:50)(cid:46)(cid:50) (cid:50)(cid:46)(cid:52) (cid:50)(cid:46)(cid:54) (cid:50)(cid:46)(cid:56) (cid:51)(cid:46)(cid:48) (cid:51)(cid:46)(cid:50) (cid:87)(cid:97)(cid:118)(cid:101)(cid:108)(cid:101)(cid:110)(cid:103)(cid:116)(cid:104)λ, µ(cid:109) FIGURE1-3 (cid:0)(cid:0) (cid:0) SolarirradiationontoEarth. spectrum,oratagivenfrequencyperunitfrequencyinterval),sothat spectralemissivepower,Eν≡emittedenergy/time/surfacearea/frequency, totalemissivepower, E≡emittedenergy/time/surfa(cid:0)cearea. Here and elsewhere we use the subscripts ν,λ, or η (depending on the choice of spectral variable)toexpressaspectralquantitywhenevernecessaryforclarification. Thermalradiation ofasinglefrequencyorwavelengthissometimesalsocalledmonochromaticradiation(since,over the visible range, the human eye perceives electromagnetic waves to have the colors of the rainbow). Itisclearfromtheirdefinitionsthatthetotalandspectralemissivepowersarerelated by (cid:90) ∞ E(T)= Eν(T,ν)dν. (1.8) 0 BlackbodyEmissivePowerSpectrum Scientistshadtriedformanyyearstotheoreticallypredictthesun’semissionspectrum,which weknowtodaytobehaveverynearlylikeablackbodyatapproximately5777K[2]. Thespectral solarfluxfallingontoEarth,orsolarirradiation,isshowninFig.1-3forextraterrestrialconditions (asmeasuredbyhigh-flyingballoonsandsatellites)andforunityairmass(airmassisdefined asthevalueof1/cosθ ,wherethezenithangleθ istheanglebetweenthelocalverticalanda S S vectorpointingtowardthesun)[3,4]. Solarradiationisattenuatedsignificantlyasitpenetrates throughtheatmospherebyphenomenathatwillbediscussedinSections1.12and1.14. Lord ∗ † Rayleigh (1900) [5] and Sir James Jeans (1905) [6] independently applied the principles of ∗JohnWilliamStrutt,LordRayleigh(1842–1919) Englishphysicalscientist.RayleighobtainedamathematicsdegreefromCambridge,where helaterservedasprofessorofexperimentalphysicsforfiveyears.Hethenbecamesecretary, andlaterpresident,oftheRoyalSociety. Hisworkresultedinanumberofdiscoveriesin thefieldsofacousticsandoptics,andhewasthefirsttoexplainthebluecolorofthesky (cf.theRayleighscatteringlawsinChapter12). Rayleighreceivedthe1904NobelPrizein Physicsfortheisolationofargon. †SirJamesHopwoodJeans(1877–1946) English physicist and mathematician, whose work was primarily in the area of astro- physics. He applied mathematics to several problems in thermodynamics and electro- magneticradiation. 1.4EMISSIVEPOWER 7 classical statistics with its equipartition of energy to predict the spectrum of the sun, with ‡ dismalresults. WilhelmWien(1896)[7] usedsomethermodynamicargumentstogetherwith experimentaldatatoproposeaspectraldistributionofblackbodyemissivepowerthatwasvery § accurateoverlargepartsofthespectrum. Finally,in1901MaxPlanck[8] publishedhisworkon quantumstatistics: Assumingthatamoleculecanemitphotonsonlyatdistinctenergylevels, he found the spectral blackbody emissive power distribution, now commonly known as Planck’s law,forablacksurfaceboundedbyatransparentmediumwithrefractiveindexn,as 2πhν3n2 Ebν(T,ν)= c2(cid:2)ehν/kT−1(cid:3), (1.9) 0 where k = 1.3807×10−23J/K is known as Boltzmann’s constant.4 While frequency ν appears to be the most logical spectral variable (since it does not change when light travels from one mediumintoanother),thespectralvariableswavelengthλ(primarilyforsurfaceemissionand absorption) and wavenumber η (primarily for radiation in gases) are also frequently (if not more often) employed. Equation (1.9) may be readily expressed in terms of wavelength and wavenumberthroughtherelationships c c c (cid:34) λdn(cid:35) c (cid:34) ηdn(cid:35) ν= 0 = 0η, dν=− 0 1+ dλ= 0 1− dη, (1.10) nλ n nλ2 ndλ n ndη and (cid:90) ∞ (cid:90) ∞ (cid:90) ∞ Eb(T)= Ebνdν= Ebλdλ= Ebηdη, (1.11) 0 0 0 or Ebνdν=−Ebλdλ=Ebηdη. (1.12) Hereλandηarewavelengthandwavenumberfortheelectromagneticwaveswithinthemedium of refractive index n (while λ = nλ and η = η/n would be wavelength and wavenumber of 0 0 the same wave traveling through vacuum). Equation (1.10) shows that equation (1.9) gives convenientrelationsforEbλandEbηonlyiftherefractiveindexisindependentoffrequency(or wavelength,orwavenumber). Thisiscertainlythecaseforvacuum(n=1)andordinarygases (n (cid:39) 1), and may be of acceptable accuracy for some semitransparent media over large parts of the spectrum (for example, for quartz 1.52 < n < 1.68 between the wavelengths of 0.2 and 2.4µm). Thus,withtheassumptionofconstantrefractiveindex, 2πhc2 Ebλ(T,λ)= n2λ5(cid:2)ehc0/nλ0kT−1(cid:3), (n=const), (1.13) ‡WilhelmWien(1864–1928) Germanphysicist,whoservedasprofessorofphysicsattheUniversityofGiessenandlater at the University of Munich. Besides his research in the area of electromagnetic waves, hisinterestsincludedotherrays,suchaselectronbeams,X-rays,andα-particles. Forthe discoveryofhisdisplacementlawhewasawardedtheNobelPrizeinPhysicsin1911. §MaxPlanck(1858–1947) Germanphysicist.PlanckstudiedinBerlinwithH.L.F.vonHelmholtzandG.R.Kirchhoff, butobtainedhisdoctorateattheUniversityofMunichbeforereturningtoBerlinasprofessor intheoreticalphysics. HelaterbecameheadoftheKaiserWilhelmSociety(todaytheMax PlanckInstitute). ForhisdevelopmentofthequantumtheoryhewasawardedtheNobel PrizeinPhysicsin1918. 4Equation(1.9)isvalidforemissionintoamediumwhoseabsorptiveindex(tobeintroducedinChapter2)ismuch lessthantherefractiveindex. Thisincludessemitransparentmediasuchaswater,glass,quartz,etc.,butnotopaque materials.Emissionintosuchbodiesisimmediatelyabsorbedandisofnointerest. 8 1FUNDAMENTALSOFTHERMALRADIATION 108 m Visiblepartofspectrum µ 2m 107 W/ ,λ Eb106 er w o E emissivep 110045 T=5777K 5000K 0K bλ(T=C3/λ) dy 00 K Blackbo 103 3 2000 1000K K 500 102 101 100 101 - Wavelengthλ,µm FIGURE1-4 Blackbodyemissivepowerspectrum. 2πhc2η3 Ebη(T,η)= n2(cid:2)ehc0η/n0kT−1(cid:3), (n=const). (1.14) Figure 1-4 is a graphical representation of equation (1.13) for a number of blackbody temper- atures. Asonecansee,theoveralllevelofemissionriseswithrisingtemperature(asdictated by the Second Law of Thermodynamics), while the wavelength of maximum emission shifts toward shorter wavelengths. The blackbody emissive power is also plotted in Fig. 1-3 for an effective solar temperature of 5777K. This plot is in good agreement with extraterrestrial solar irradiationdata. Itiscustomarytointroducetheabbreviations C =2πhc2 =3.7418×10−16Wm2, 1 0 C =hc /k=14,388µmK=1.4388cmK, 2 0 sothatequation(1.13)mayberecastas Ebλ = C1 , (n=const), (1.15) n3T5 (nλT)5[eC2/(nλT)−1] whichisseentobeafunctionof(nλT)only. Thus,itispossibletoplotthisnormalizedemissive power as a single line vs. the product of wavelength in vacuum (nλ) and temperature (T), as showninFig.1-5,andadetailedtabulationisgiveninAppendixC. Themaximumofthiscurve maybedeterminedbydifferentiatingequation(1.15), (cid:18) (cid:19) d Ebλ =0, d(nλT) n3T5 leadingtoatranscendentalequationthatmaybesolvednumericallyas (nλT) =C =2898µmK. (1.16) max 3 Equation (1.16) is known as Wien’s displacement law since it was developed independently by WilhelmWien[9]in1891(i.e.,wellbeforethepublicationofPlanck’semissivepowerlaw). Example1.1. Atwhatwavelengthhasthesunitsmaximumemissivepower? Atwhatwavelength Earth? 1.4EMISSIVEPOWER 9 15.0 1.00 12.5 wer 0.75T)λ epo5mK 10.0 Planck’slaw fn( emissiv2W/mµ 7.5 RWaiyelne’isghd-isJteraibnustdioisntribution 0.50epower kbody1210 1% missiv aledblac35E/nT,bλ 5.0 10% 90% 0.25ctionale Sc 2.5 98% Fra 0.0 0.00 0 5 10 15 20 Wavelength×(cid:13)temperature nλT,103µmK FIGURE1-5 Normalizedblackbodyemissivepowerspectrum. Solution Fromequation(1.16),withthesun’ssurfaceatT (cid:39)5777Kandboundedbyvacuum(n=1),itfollows sun that C 2898µmK λmax,sun= T 3 = 5777K =0.50µm, sun whichisnearthecenterofthevisibleregion. Apparently, evolutionhascausedoureyestobemost sensitiveinthatsectionoftheelectromagneticspectrumwherethemaximumdaylightisavailable. In contrast,Earth’saveragesurfacetemperaturemaybeinthevicinityofT =290K,or Earth 2898µmK λmax,Earth(cid:39) 290K =10µm, thatis,Earth’smaximumemissionoccursintheintermediateinfrared,leadingtoinfraredcamerasand detectorsfornight“vision.” It is of interest to look at the asymptotic behavior of Planck’s law for small and large wavelengths. For very small values of hc /nλkT (large wavelength, or small frequency), the 0 exponentinequation(1.13)maybeapproximatedbyatwo-termTaylorseries,leadingto 2πc kT hc Ebλ = nλ04 , nλk0T (cid:28)1. (1.17) Thesameresultisobtainedifoneletsh→0,i.e.,ifoneallowsphotonsofarbitrarilysmallenergy contenttobeemitted,aspostulatedbyclassicalstatistics. Thus,equation(1.17)isidenticalto theonederivedbyRayleighandJeansandbearstheirnames. TheRayleigh–Jeansdistributionis alsoincludedinFig.1-5. Obviously,thisformulaisaccurateonlyforverylargevaluesof(nλT), where the energyof the emissive power spectrum isnegligible. Thus, this formula is of little significanceforengineeringpurposes. Forlargevaluesof(hc /nλkT)the−1inthedenominatorofequation(1.13)maybeneglected, 0 leadingtoWien’sdistribution(orWien’slaw), Ebλ (cid:39) 2nπ2hλc520e−hc0/nλkT = nC2λ15e−C2/nλT, nhλck0T (cid:29)1, (1.18) since it is identical to the formula first proposed by Wien, before the advent of quantum me- chanics. ExaminationofWien’sdistributioninFig.1-5showsthatitisveryaccurateovermost 10 1FUNDAMENTALSOFTHERMALRADIATION of the spectrum, with a total energy content of the entire spectrum approximately 8% lower thanforPlanck’slaw. Thus,Wien’sdistributionisfrequentlyutilizedintheoreticalanalysesin ordertofacilitateintegration. TotalBlackbodyEmissivePower Thetotalemissivepowerofablackbodymaybedeterminedfromequations(1.11)and(1.13)as (cid:90) ∞ (cid:90) ∞ d(nλT) Eb(T)= 0 Ebλ(T,λ)dλ=C1n2T4 0 (nλT)5(cid:2)eC2/(nλT)−1(cid:3) =CC14 (cid:90) ∞ eξξ3−dξ1n2T4, (n=const). (1.19) 2 0 The integral in this expression may be evaluated by complex integration, and is tabulated in manygoodintegraltables: E (T)=n2σT4, σ= π4C1 =5.670×10−8 W , (1.20) b 15C4 m2K4 2 where σ is known as the Stefan–Boltzmann constant.∗ If Wien’s distribution is to be used then the −1 is absent from the denominator of equation (1.19), and a corrected Stefan–Boltzmann constantshouldbeemployed,evaluatedas σ = 6C1 =5.239×10−8 W , (1.21) W C4 m2K4 2 indicatingthatWien’sdistributionunderpredictstotalemissivepowerbyabout7.5%. Histor- ically, the “T4 radiation law,” equation (1.20), predates Planck’s law and was found through thermodynamicarguments. Ashorthistorymaybefoundin[10]. It is often necessary to calculate the emissive power contained within a finite wavelength band,saybetweenλ andλ . Then 1 2 (cid:90) λ2 C (cid:90) C2/nλ1T ξ3dξ λ1 Ebλdλ= C421 C2/nλ2T eξ−1n2T4. (1.22) Itisnotpossibletoevaluatetheintegralinequation(1.22)insimpleanalyticalform. Therefore, it is customary to express equation (1.22) in terms of the fraction of blackbody emissive power containedbetween0andnλT, (cid:82) λ f(nλT)= (cid:82)00∞EEbbλλddλλ =(cid:90)0nλT(cid:18)n3EσbTλ5(cid:19)d(nλT)= π154(cid:90)C2∞/nλT eξξ3−dξ1, (1.23) ∗JosefStefan(1835–1893) Austrianphysicist. ServingasprofessorattheUniversityofVienna,Stefandeterminedin 1879that,basedonhisexperiments,blackbodyemissionwasproportionaltotemperature tothefourthpower. LudwigErhardBoltzmann(1844–1906) Austrian physicist. After receiving his doctorate from the University of Vienna he held professorshipsinVienna,Graz(bothinAustria),Munich,andLeipzig(inGermany). His greatestcontributionswereinthefieldofstatisticalmechanics(Boltzmannstatistics). He derivedthefourth-powerlawfromthermodynamicconsiderationsin1889.
The list of books you might like
