Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 94
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Metals are generally excellent electrical conductors because of an abundance of free electrons. For materials with large electrical conductivity, both√the real and imaginaryparts of the complex index of refraction, m = n − ık(ı = −1), become large andapproximately equal for long wavelengths, say λ > 1 µm, leading to an approximaterelation for the normal, spectral emittance of the metal, known as the Hagen–Rubensrelation (Modest, 2003),nλ 22√= 1 − ρnλn0.003λσdcλ in µm,σdc in Ω−1 · cm−1(8.32)Lines: 564 to 599where σdc is the dc conductivity of the material.
Equation (8.32) indicates that forclean, polished metallic surfaces the normal emittance can be expected to be small,√and the reflectance large (using typical values for conductivity, σdc ), with a 1/ λwavelength dependence. Comparison with experiment has shown that for sufficientlylong wavelengths, the Hagen–Rubens relationship describes the radiative propertiesof polished (not entirely smooth) metals rather well, in contrast to the older, moresophisticated Drude theory (Modest, 2003). However, for optically smooth metallicsurfaces (such as vapor-deposited layers on glass), radiative properties closely obeyelectromagnetic wave theory, and it is the Drude theory that gives excellent results.Directional Dependence The spectral, directional reflectance for an opticallysmooth interface is given by Fresnel’s relations (Modest, 2003).
As noted before,in the infrared, n and k are generally fairly large for metals, and Fresnel’s relationssimplify toρ =(n cos θ − 1)2 + (k cos θ)2(n cos θ + 1)2 + (k cos θ)2(8.33a)ρ⊥ =(n − cos θ)2 + k 2(n + cos θ)2 + k 2(8.33b)Here ρ is the spectral reflectance for parallel-polarized radiation, which refers toelectromagnetic waves whose oscillations take place in a plane formed by the surfacenormal and the direction of incidence.
Similarly, ρ⊥ is the spectral reflectance forperpendicular-polarized radiation, which refers to waves oscillating in a plane normalto the direction of incidence. In all engineering applications (except lasers), radiationconsists of many randomly oriented waves (randomly polarized or unpolarized), andthe spectral, directional emittance and reflectance can be evaluated fromBOOKCOMP, Inc. — John Wiley & Sons / Page 586 / 2nd Proofs / Heat Transfer Handbook / Bejan[586], (14)———-2.06184pt PgVar———Normal PagePgEnds: TEX[586], (14)RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445587[587], (15)Lines: 599 to 610———0.62703pt PgVarFigure 8.6 Spectral, directional reflectance of platinum at λ = 2µm.λ = 1 − ρλ = 1 −1ρ + ρ⊥2(8.34)The directional behavior for the reflectance of polished platinum at λ = 2 µmis shown in Fig.
8.6 and is also compared with experiment (Brandenberg, 1963;Brandenberg and Clausen, 1965; Price, 1947). As already seen from Fig. 8.4, reflectance is large for near-normal incidence, and the unpolarized reflectance remainsfairly constant with increasing θ. However, near grazing angles of θ 80–85°, theparallel-polarized component undergoes a sharp dip before going to ρ = ρ⊥ = 1at θ = 90°. This behavior is responsible for the lobe of strong emittance near grazing angles commonly observed for metals.
Fortunately, these near-grazing angles arefairly unimportant in the evaluation of radiative fluxes, due to the cos θ in eq. (8.21);that is, even metals can usually be treated as “diffuse emitters” with good accuracy. Itneeds to be emphasized that the foregoing discussion is valid only for relatively longwavelengths (infrared). For shorter wavelengths, particularly the visible, the assumption of large values for n and k generally breaks down, and the directional behaviorof metals resembles that of nonconductors (discussed in the next section).Hemispherical Properties Equation (8.33) may be integrated analytically overall directions to obtain the spectral, hemispherical emittance.
Figure 8.7 is a plot of theratio of the hemispherical and normal emittances, λ /nλ . For the case of k/n = 1 thedashed line represents results from integrating equation (8.33), while the solid lineswere obtained by numerically integrating the exact form of Fresnel’s relations. Formost metals k > n > 3, so that, as shown in Fig. 8.7, the hemispherical emittance isBOOKCOMP, Inc.
— John Wiley & Sons / Page 587 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal PagePgEnds: TEX[587], (15)5881.40Hemispherical emittance ⑀,⑀nNormal emittance123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONk/n = 41.3021.2011.1001.00[588], (16)0.90110Refractive index, nLines: 610 to 636100———0.623pt PgVarFigure 8.7 Ratio of hemispherical and normal spectral emittance for electrical conductors asa function of n and k. (From Dunkle, 1965.)larger than the normal value, due to the strong emission lobe at near grazing angles.Again, this statement holds only for relatively long wavelengths.Total Properties Equation (8.32) may be integrated over the spectrum, using eq.(8.29), and applying the correction given in Fig. 8.7 to convert normal emittanceto hemispherical emittance. This leads to an approximate expression for the total,hemispherical emittance of a metal,(T ) = 0.766Tσdc1/2TT− 0.309 − 0.0889 lnσdc σdc(8.35)where T is in K and σdc is in Ω−1 · cm−1 .Because eq.
(8.35) is based on the Hagen–Rubens relation, this expression is validonly for relatively low temperatures (where most of the blackbody emissive powerlies in the long wavelengths; see Fig. 8.1). Figure 8.8 shows that eq. (8.35) does anexcellent job predicting the total hemispherical emittances of polished metals as compared with experiment (Parker and Abbott, 1965), and that emittance is essentiallylinearly proportional to (T /σdc )1/2 .Surface Temperature Effects The Hagen–Rubens relation, eq. (8.32), predicts√that the spectral, normal emittance of a metal should be proportional to 1/ σdc .Because the electrical conductivity of metals is approximately inversely proportionalBOOKCOMP, Inc.
— John Wiley & Sons / Page 588 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal PagePgEnds: TEX[588], (16)589RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS0.3Total hemispherical emittance ⑀1234567891011121314151617181920212223242526272829303132333435363738394041424344450.2Theoretical (Hagen-Rubens)TungstenTantalumNiobiumMolybdenumPlatinumGoldSilverCopperZincTinLead0.100.00.10.20.3公T/dc (⍀ . cm . K)0.40.5[589], (17)0.61/2Figure 8.8 Total, hemispherical emittance of various polished metals as a function of temperature. (From Parker and Abbott, 1965.)to temperature, the spectral emittance should therefore be proportional to the squareroot of absolute temperature for long enough wavelengths.
This trend should also holdfor the spectral, hemispherical emittance. Experiments have shown that this is indeedtrue for many metals. A typical example is given in Fig. 8.9, which shows the spectraldependence of the hemispherical emittance for tungsten for a number of temperatures.Note that the emittance for tungsten tends to increase with temperature beyond acrossover wavelength of approximately 1.3 µm, while the temperature dependenceis reversed for shorter wavelengths.
Similar trends of a single crossover wavelengthhave been observed for many metals. Because the crossover wavelength is fairly shortfor many metals, the Hagen–Ruben temperature relation often holds for surprisinglyhigh temperatures.8.2.2Radiative Properties of NonconductorsElectrical nonconductors have few free electrons and thus do not display high reflectance/opacity behavior across the infrared as do metals.Wavelength Dependence Reflection of light by insulators and semiconductorstends to be a strong, sometimes erratic function of wavelength.
Crystalline solidsgenerally have strong absorption–reflection bands (large k) in the infrared commonlyknown as Reststrahlen bands, which are due to transitions of intermolecular vibrations. These materials also have strong bands at short wavelengths (visible to ultraviolet), due to electronic energy transitions. In between these two spectral regions thereBOOKCOMP, Inc. — John Wiley & Sons / Page 589 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 636 to 647———-5.903pt PgVar———Normal PagePgEnds: TEX[589], (17)5900.5Hemispherical emittance, ⑀123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATION0.4Tungsten0.30.2T = 1600 KT = 2000 KT = 2400 KT = 2800 K0.10.00.00.51.01.5Wavelength (m)2.02.53.0Figure 8.9 Temperature dependence of the spectral, hemispherical emittance of tungsten.(From Weast, 1988.)[590], (18)Lines: 647 to 655———0.79701pt PgVargenerally is a region of fairly high transparency (and low reflectance), where absorption is dominated by impurities and imperfections in the crystal lattice.
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