The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 13
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The spectral distribution of the emissive power of a black surface is given by Planck’slaw.Ebl =C1[l5 e C2lT]-1,C1 = 3.7419 ´ 10 -16 Wm 2 ,C2 = 14, 388 mmK(4.3.2)where C1 and C2 are sometimes called Planck function constants. The total emissive power of a blackbodyis given by*From Modest, M., Radiative Heat Transfer, 1993; reproduced with permission of MacGraw-Hill, Inc.© 1999 by CRC Press LLC4-57Heat and Mass Transfer¥Eb =òE0bldl = sT 4 ,s = 5.670 ´ 10 -8 W m 2 K 4(4.3.3)with s known as the Stefan–Boltzmann constant.
Figure 4.3.1 shows the spectral solar irradiation thatimpinges on Earth, which closely resembles the spectrum of a blackbody at 5762 K. The general behaviorof Planck’s law is depicted in Figure 4.3.2, together with the fractional emissive power, f(lT), defined asf (lT ) =1EblòE0bl( l , T ) dlFIGURE 4.3.1 Solar irradiation onto Earth.FIGURE 4.3.2 Normalized blackbody emissive power spectrum.© 1999 by CRC Press LLC(4.3.4)4-58Section 4Note that 90% of all blackbody emission takes place at wavelengths of lT > 2200 mmK and at allwavelengths lT < 9400 mmK.
This implies that — for typical high-temperature heat transfer applicationsin the range between 1000 and 2000 K — infrared wavelengths in the range 1 mm < l < 10 mm governthe heat transfer rates. For solar applications shorter wavelengths, down to l @ 0.4 mm are also important.Also shown in Figure 4.3.2 is Wien’s law:Ebl =C1 - C2el5lT(4.3.5)which approximates Planck’s law accurately over the part of the spectrum that is important to heattransfer, and that is easier to manipulate mathematically.Example 4.3.1What fraction of total solar emission falls into the visible spectrum (0.4 to 0.7 mm)?Solution: With a solar temperature of 5762 K it follows that forl 1 = 0.4 mm,l 1Tsun = 0.4 ´ 5762 = 2304 mmKl 2 = 0.7 mm,l 2 Tsun = 0.7 ´ 5762 = 4033 mmKand forFrom Figure 4.3.2 we can estimate f(l1Tsun) @ 12% and f(l2Tsun) @ 48%.
Thus, the visible fraction ofsunlight is 48 – 12 @ 36%: with a bandwidth of only 0.3 mm the human eye responds to approximately36% of all emitted sunlight!Radiative Exchange between Opaque SurfacesRadiative Properties of SurfacesStrictly speaking, the surface of an enclosure wall can only reflect radiative energy and allow a part ofit to penetrate into the substrate. A surface cannot absorb or emit photons: attenuation takes place insidethe solid, as does emission of radiative energy (with some of the emitted energy escaping through thesurface into the enclosure).
In practical systems, the thickness of the surface layer over which absorptionof irradiation from inside the enclosure occurs is very small compared with the overall dimension ofan enclosure — usually a few nanometers for metals and a few micrometers for most nonmetals. Thesame may be said about emission from within the walls that escapes into the enclosure.
Thus, in thecase of opaque walls it is customary to speak of absorption by and emission from a “surface,” althougha thin surface layer is implied. Four fundamental radiative properties are defined:Reflectivity,rºreflected part of incoming radiationtotal incoming radiation( 4.3.6a )Absorptivity,rºabsorbed part of incoming radiationtotal incoming radiation( 4.3.6 b)Transmissivity,tºtransmitted part of incoming radiationtotal incoming radiation( 4.3.6c)© 1999 by CRC Press LLC4-59Heat and Mass TransferEmissivity,eºenergy emitted from a surfaceenergy emitted by a black surface at same temperature( 4.3.6d )Since all incoming radiation must be reflected, absorbed, or transmitted, it follows thatr+a+t =1(4.37)In most practical applications surface layers are thick enough to be opaque (t = 0, leading to r + a =1.
All four properties may be functions of wavelength, temperature, incoming direction (except emissivity), and outgoing direction (except absorptivity).Directional Behavior. For heat transfer applications, the dependence on incoming direction for absorptivity (as well as r and t) and outgoing direction for emissivity is generally weak and is commonlyneglected; i.e., it is assumed that the surface absorbs and emits diffusely. Then, for an opaque surface,for any given wavelengthe l = a l = 1 - rl(4.38)Published values of emissivities are generally either “normal emissivities” (the directional value of elin the direction perpendicular to the surface) or “hemispherical emissivities” (an average value over alloutgoing directions).
The difference between these two values is often smaller than experimental accuracyand/or repeatability.Reflected energy (due to a single, distinct incoming direction) may leave the surface into a singledirection (“specular” reflection, similar to reflection from a mirror for visible light), or the reflectionmay spread out over all possible outgoing directions. In the extreme case of equal amounts going intoall directions, we talk about “diffuse” reflection.
Smooth surfaces (as compared with the wavelength ofradiation) tend to be specular reflectors, while rough surfaces tend to be more or less diffusely reflecting.Analysis is vastly simplified if diffuse reflections are assumed. Research has shown that — except forsome extreme geometries and irradiation conditions susceptible to beam channeling (irradiated opencavities, channels with large aspect ratios) — radiative heat transfer rates are only weakly affected bythe directional distribution of reflections.
Therefore, it is common practice to carry out radiative heattransfer calculations assuming only diffuse reflections.Spectral Dependence. The emissivity of a surface generally varies strongly and in complex ways withwavelength, depending on the material, surface layer composition, and surface structure (roughness).Therefore, unlike bulk material properties (such as thermal conductivity) the surface emissivity maydisplay significant differences between two ostensibly identical samples, and even for one and the samesample measured at different times (due to surface roughness and contamination).
Despite these difficulties, surfaces may be loosely grouped into two categories — metals and nonconductors (dielectrics),and some generalizations can be made.Polished Metals. Smooth, purely metallic surfaces (i.e., without any nonmetallic surface contamination,such as metal oxides) tend to have very low emissivities in the infrared. For many clean metals el < 0.1for l > 2 mm, and spectral as well as temperature dependence are generally well approximated by theproportionality el µ T / l in the infrared. However, for shorter wavelengths (l < 1 mm), emissivityvalues may become quite substantial, and temperature dependence is usually reversed (decreasing, ratherthan increasing, with temperature). Typical room temperature behavior of several metals is shown inFigure 4.3.3. Caution needs to be exercised when choosing an emissivity value for a metal surface:unless extraordinary care is taken to keep a polished metal clean (i.e., free from oxidation and/or surfacecontamination), its emissivity may soon become several times the value of the original, polished specimen(for example, consider the formation of aluminum oxide on top of aluminum, Figure 4.3.3).© 1999 by CRC Press LLC4-60Section 4FIGURE 4.3.3 Normal, spectral emissivities for selected materials.Ceramics and Refractories.
Smooth ceramics tend to have fairly constant and intermediate emissivityover the near- to mid-infrared, followed by a sharp increase somewhere between 4 and 10 mm. At shortwavelengths these materials display strong decreases in emissivity, so that a number of them may appearwhite to the human eye even though they are fairly black in the infrared.
The temperature dependenceof the emissivity of ceramics is rather weak; generally a slight increase with temperature is observed inthe infrared. The spectral emissivity of a few ceramics is also shown in Figure 4.3.3.Other Nonconductors. The behavior of most electrically nonconducting materials is governed by surfacestructure, nonhomogeneity, dopants, porosity, flaws, surface films, etc. The emissivity may vary irregularly across the spectrum because of various emission bands, influence of flaws, etc., making anygeneralization impossible. This irregularity may be exploited to obtain surfaces of desired spectralbehavior, so-called selective surfaces.
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