Heat Transfer Handbook part 60 pdf

BOOKCOMP, Inc. — John Wiley & Sons / Page 583 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 583 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [583], (11) Lines: 480 to 515 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [583], (11) the enclosure. Thus, in the case of opaque walls it is customary to speak of absorption by and emission from a “surface,” although a thin surface layer is implied. If radiation impinging on a solid or liquid layer is considered, a fraction of the energy will be reflected (reflectance ρ, often also referred to as reflectivity), another fraction will be absorbed (absorptance α, often also referred to as absorptivity), and if the layer is thin enough, a fraction may be transmitted (transmittance τ, often also referred to as transmissivity). Because all radiation must be either reflected, absorbed, or transmitted, ρ + α + τ = 1 (8.24) If the medium is sufficiently thick to be opaque, then τ = 0 and ρ + α = 1 (8.25) All surfaces also emit thermal radiation (or, rather, radiative energy is emitted within the medium, some of which escapes from the surface). The emittance  is defined as the ratio of energy emitted by a surface as compared to that of a black surface at the same temperature (the theoretical maximum). All of these four properties may vary in magnitudebetweenthevalues 0 and 1; for a black surface, which absorbs all incoming radiation and emits the maximum possible, α =  = 1 and ρ = τ = 0. They may also be functions of temperature as well as wavelength and direction (incoming and/or outgoing). One distinguishes between spectral and total properties (an average value over the spectrum) and also between directional and hemispherical properties (an average value over all directions). It may be shown (through another variation of Kirchhoff’s law) that, at least on a spectral, directional basis, α(T,λ, ˆ s) = (T,λ, ˆ s) (8.26) This is also true for hemispherical values if either the directional emittance or the incoming radiation are diffuse (they do not depend on direction). It is also true for total values if either the spectral emittance does not depend on wavelength or if the spectral behavior of the incoming radiation is similar to blackbody radiation at the same temperature. Typical directional behavior is shown in Fig. 8.4a (for nonmetals) and b (metals). In these figures the total, directional emittance, a value averaged over all wavelengths, is shown. For nonmetals the directional emittance varies little over a large range of polar angles but decreases rapidly at grazing angles until a value of zero is reached at θ = π/2. Similar trends hold for metals, except that at grazing angles, the emittance first increases sharply before dropping back to zero (not shown). Note that emittance levels are considerably higher for nonmetals. A surface whose emittance is the same for all directions is called a diffuse emitter, or a Lambert surface. No real surface can be a diffuse emitter because electromag- netic wave theory predicts a zero emittance at θ = π/2 for all materials. However, BOOKCOMP, Inc. — John Wiley & Sons / Page 584 / 2nd Proofs / Heat Transfer Handbook / Bejan 584 THERMAL RADIATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [584], (12) Lines: 515 to 529 ——— 0.70102pt PgVar ——— Normal Page * PgEnds: Eject [584], (12) a b Figure 8.4 Directional variation of surface emittances: (a) for several nonmetals; (b) for several metals. (From Schmidt and Eckert, 1935.) little energy is emitted into grazing directions, as seen from eq. (8.16), so that the assumption of diffuse emission is often a good one. Typical spectral behavior of surface emittances is shown in Fig. 8.5 for a few materials, as collected by White (1984). Shown are values for directional emittances in the direction normal to the surface. However, the spectral behavior is the same for hemispherical emittances. In general, nonmetals have relatively high emittances, which may vary erratically across the spectrum, and metals behave similarly for short wavelengths but tend to have lower emittances with more regular spectral dependence in the infrared. Mathematically, the spectral, hemispherical emittance is defined in terms of emis- sive power as  λ (T,λ) ≡ E λ (T,λ) E bλ (T,λ) (8.27) BOOKCOMP, Inc. — John Wiley & Sons / Page 585 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 585 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [585], (13) Lines: 529 to 564 ——— 2.31412pt PgVar ——— Normal Page PgEnds: T E X [585], (13) 1.0 0.8 0.6 0.4 0.2 0 012345678 Copper Carbon Gold Aluminum Tungsten Nickel Aluminum oxide Magnesium oxide White enamel Wavelength ( m)␭␮ Aluminum Silicon carbide Normal emissivity,  nλ Figure 8.5 Normal, spectral emittances for selected materials. (From White, 1984.) This property may be extracted from the spectral, directional emittance   λ by inte- grating over all directions,  λ (T,λ) = 1 π    λ (T,λ,θ,ψ) cos θ dΩ = 1 π  2π 0  π/2 0   λ (T,λ,θ,ψ) cos θ sin θ dθ dψ (8.28) and finally, the total, hemispherical emittance may be related to the spectral hemi- spherical emittance through (T ) = E(T ) E b (T ) =  ∞ 0 E λ (T,λ)dλ E b (T ) = 1 n 2 σT 4  ∞ 0  λ (T,λ)E bλ (T,λ)dλ (8.29) Here a prime and subscript λ have been added temporarily to distinguish directional from hemispherical properties, and spectral from total (spectrally averaged) values. If the spectral emittance is the same for all wavelengths, eq. (8.29) reduces to (T ) =  λ (T ) (8.30) Such surfaces are termed gray, and for the very special case of a gray, diffuse surface, this implies that (T ) =  λ =   =   λ (8.31) BOOKCOMP, Inc. — John Wiley & Sons / Page 586 / 2nd Proofs / Heat Transfer Handbook / Bejan 586 THERMAL RADIATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [586], (14) Lines: 564 to 599 ——— -2.06184pt PgVar ——— Normal Page PgEnds: T E X [586], (14) Although no real surface is truly gray, it often happens that  λ is relatively constant over that part of the spectrum where E bλ is substantial, making the simplifying assumption of a gray surface warranted. 8.2.1 Radiative Properties of Metals Wavelength Dependence Electromagnetic theory states that the radiative prop- erties of interfaces are strong functions of the material’s electrical conductivity. Met- als are generally excellent electrical conductors because of an abundance of free elec- trons. For materials with large electrical conductivity, both the real and imaginary parts of the complex index of refraction, m = n − ık(ı = √ −1), become large and approximately equal for long wavelengths, say λ > 1 µm, leading to an approximate relation for the normal, spectral emittance of the metal, known as the Hagen–Rubens relation (Modest, 2003),  nλ  2 n  2 √ 0.003λσ dc = 1 − ρ nλ λ in µm, σ dc in Ω −1 · cm −1 (8.32) where σ dc is the dc conductivity of the material. Equation (8.32) indicates that for clean, polished metallic surfaces the normal emittance can be expected to be small, and the reflectance large (using typical values for conductivity, σ dc ),witha1/ √ λ wavelength dependence. Comparison with experiment has shown that for sufficiently long wavelengths, the Hagen–Rubens relationship describes the radiative properties of polished (not entirely smooth) metals rather well, in contrast to the older, more sophisticated Drude theory (Modest, 2003). However, for optically smooth metallic surfaces (such as vapor-deposited layers on glass), radiative properties closely obey electromagnetic wave theory, and it is the Drude theory that gives excellent results. Directional Dependence The spectral, directional reflectance for an optically smooth 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 relations simplify to ρ  = (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 to electromagnetic waves whose oscillations take place in a plane formed by the surface normal and the direction of incidence. Similarly, ρ ⊥ is the spectral reflectance for perpendicular-polarized radiation, which refers to waves oscillating in a plane normal to the direction of incidence. In all engineering applications (except lasers), radiation consists of many randomly oriented waves (randomly polarized or unpolarized), and the spectral, directional emittance and reflectance can be evaluated from BOOKCOMP, Inc. — John Wiley & Sons / Page 587 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 587 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [587], (15) Lines: 599 to 610 ——— 0.62703pt PgVar ——— Normal Page PgEnds: T E X [587], (15) Figure 8.6 Spectral, directional reflectance of platinum at λ = 2µm.   λ = 1 − ρ  λ = 1 − 1 2  ρ  + ρ ⊥  (8.34) The directional behavior for the reflectance of polished platinum at λ = 2 µm is 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, re- flectance is large for near-normal incidence, and the unpolarized reflectance remains fairly constant with increasing θ. However, near grazing angles of θ  80–85°, the parallel-polarized component undergoes a sharp dip before going to ρ  = ρ ⊥ = 1 at θ = 90°. This behavior is responsible for the lobe of strong emittance near graz- ing angles commonly observed for metals. Fortunately, these near-grazing angles are fairly 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. It needs to be emphasized that the foregoing discussion is valid only for relatively long wavelengths (infrared). For shorter wavelengths, particularly the visible, the assump- tion of large values for n and k generally breaks down, and the directional behavior of metals resembles that of nonconductors (discussed in the next section). Hemispherical Properties Equation (8.33) may be integrated analytically over all directions to obtain the spectral, hemispherical emittance. Figure 8.7 is a plot of the ratio of the hemispherical and normal emittances,  λ / nλ . For the case of k/n = 1 the dashed line represents results from integrating equation (8.33), while the solid lines were obtained by numerically integrating the exact form of Fresnel’s relations. For most metals k>n>3, so that, as shown in Fig. 8.7, the hemispherical emittance is BOOKCOMP, Inc. — John Wiley & Sons / Page 588 / 2nd Proofs / Heat Transfer Handbook / Bejan 588 THERMAL RADIATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [588], (16) Lines: 610 to 636 ——— 0.623pt PgVar ——— Normal Page PgEnds: T E X [588], (16) 10 100 1.40 1.30 1.20 1.10 1.00 0.90 Refractive index, n kn/=4 2 1 0 1 Hemispherical emittance , ⑀ ␭ Normal emittance ⑀ n␭ Figure 8.7 Ratio of hemispherical and normal spectral emittance for electrical conductors as a 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 emittance to hemispherical emittance. This leads to an approximate expression for the total, hemispherical emittance of a metal, (T ) = 0.766  T σ dc  1/2 −  0.309 − 0.0889 ln T σ dc  T σ dc (8.35) where T is in K and σ dc is in Ω −1 · cm −1 . Because eq. (8.35) is based on the Hagen–Rubens relation, this expression is valid only for relatively low temperatures (where most of the blackbody emissive power lies in the long wavelengths; see Fig. 8.1). Figure 8.8 shows that eq. (8.35) does an excellent job predicting the total hemispherical emittances of polished metals as com- pared with experiment (Parker and Abbott, 1965), and that emittance is essentially linearly proportional to (T /σ dc ) 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/ √ σ dc . Because the electrical conductivity of metals is approximately inversely proportional BOOKCOMP, Inc. — John Wiley & Sons / Page 589 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 589 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [589], (17) Lines: 636 to 647 ——— -5.903pt PgVar ——— Normal Page PgEnds: T E X [589], (17) 0.3 0.2 0.1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Theoretical (Hagen-Rubens) Tungsten Tantalum Niobium Molybdenum Platinum Gold Silver Copper Zinc Tin Lead Total hemispherical emittance ⑀ ͌␴ ⍀T/ dc 1/2 ( . cm . K) Figure 8.8 Total, hemispherical emittance of various polished metals as a function of tem- perature. (From Parker and Abbott, 1965.) to temperature, the spectral emittance should therefore be proportional to the square root of absolute temperature for long enough wavelengths. This trend should also hold for the spectral, hemispherical emittance. Experiments have shown that this is indeed true for many metals. A typical example is given in Fig. 8.9, which shows the spectral dependence of the hemispherical emittance for tungsten for a number of temperatures. Note that the emittance for tungsten tends to increase with temperature beyond a crossover wavelength of approximately 1.3 µm, while the temperature dependence is reversed for shorter wavelengths. Similar trends of a single crossover wavelength have been observed for many metals. Because the crossover wavelength is fairly short for many metals, the Hagen–Ruben temperature relation often holds for surprisingly high temperatures. 8.2.2 Radiative Properties of Nonconductors Electrical nonconductors have few free electrons and thus do not display high re- flectance/opacity behavior across the infrared as do metals. Wavelength Dependence Reflection of light by insulators and semiconductors tends to be a strong, sometimes erratic function of wavelength. Crystalline solids generally have strong absorption–reflection bands (large k) in the infrared commonly known as Reststrahlen bands, which are due to transitions of intermolecular vibra- tions. These materials also have strong bands at short wavelengths (visible to ultravi- olet), due to electronic energy transitions. In between these two spectral regions there BOOKCOMP, Inc. — John Wiley & Sons / Page 590 / 2nd Proofs / Heat Transfer Handbook / Bejan 590 THERMAL RADIATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [590], (18) Lines: 647 to 655 ——— 0.79701pt PgVar ——— Normal Page PgEnds: T E X [590], (18) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 Tungsten Wavelength T = 1600 K T = 2000 K T = 2400 K T = 2800 K Hemispherical emittance, ⑀ ␭ ␭␮(m) Figure 8.9 Temperature dependence of the spectral, hemispherical emittance of tungsten. (From Weast, 1988.) generally is a region of fairly high transparency (and low reflectance), where absorp- tion is dominated by impurities and imperfections in the crystal lattice. As such, these spectral regions often show irregular and erratic behavior. Defects and impurities may vary appreciably from specimen to specimen and even between different points on the same sample. As an example, the spectral, normal reflectance of silicon at room temperature is shown in Fig. 8.10. The strong influence of different types and levels of impurities is clearly evident. Therefore, looking up properties for a given material in published tables is problematical unless a detailed description of surface and material preparation is given. In spectral regions outside Reststrahlen and electronic transition bands the absorp- tive index of a nonconductor is very small; typically, k<10 −6 for a pure substance. While impurities and lattice defects can increase the value of k, one is very unlikely to find values of k>10 −2 for a nonconductor outside the Reststrahlen bands. This im- plies that Fresnel’s relations can be simplified significantly, and the spectral, normal reflectance may be evaluated as ρ nλ =  n − 1 n + 1  2 (8.36) Therefore, for optically smooth nonconductors the radiative properties may be cal- culated from refractive index data. Refractive indices for a number of semitransparent materials at room temperature are displayed in Fig. 8.11 as a function of wavelength. All of these crystalline materials show similar spectral behavior: the refractive index drops rapidly in the visible region, then is nearly constant (declining very gradually) until the midinfrared, where n again starts to drop rapidly. This behavior is explained BOOKCOMP, Inc. — John Wiley & Sons / Page 591 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 591 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [591], (19) Lines: 655 to 679 ——— 7.54106pt PgVar ——— Normal Page PgEnds: T E X [591], (19) Figure 8.10 Spectral, normal reflectance of silicon at room temperature. (Redrawn from the data of Touloukian and DeWitt, 1972.) by the fact that crystalline solids tend to have an absorption band due to electronic transitions near the visible and a Reststrahlen band in the infrared: The first drop in n is due to the tail end of the electronic band; the second drop in the midinfrared is due to the beginning of a Reststrahlen band. Directional Dependence For optically smooth nonconductors, for the spectral region between absorption–reflection bands, experiment has been found to closely follow Fresnel’s equations of electromagnetic wave theory. Figure 8.12 shows a comparison between theory and experiment for the directional reflectance of glass (blackened on one side to avoid multiple reflections) for polarized, monochromatic irradiation. Because k 2  n 2 , the absorptive index may be eliminated from Fresnel’s relations, and the relations for a perfect dielectric become valid. For unpolarized light incident from vacuum (or a gas), this leads to   λ = 1 − 1 2  ρ  + ρ ⊥  = 1 − 1 2    n 2 cos θ −  n 2 − sin 2 θ n 2 cos θ +  n 2 − sin 2 θ  2 +  cos θ −  n 2 − sin 2 θ cos θ +  n 2 − sin 2 θ  2   (8.37) Comparison with experiment agrees well with elecromagnetic wave theory for a large number of nonconductors. BOOKCOMP, Inc. — John Wiley & Sons / Page 592 / 2nd Proofs / Heat Transfer Handbook / Bejan 592 THERMAL RADIATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [592], (20) Lines: 679 to 679 ——— -0.96301pt PgVar ——— Normal Page PgEnds: T E X [592], (20) Figure 8.11 Refractive indices for various semitransparent materials. (From American Insti- tute of Physics, 1972.) Temperature Dependence The temperature dependence of the radiative prop- erties of nonconductors is considerably more difficult to quantify than for metals. Infrared absorption bands in ionic solids due to excitation of lattice vibrations (Rest- strahlen bands) generally increase in width and decrease in strength with tempera- ture, and the wavelength of peak reflection–absorption shifts toward higher values. The reflectance for shorter wavelengths depends largely on the material’s impurities. Often, the behavior is similar to that of metals, that is, the emittance increases with temperature for the near infrared while it decreases with shorter wavelengths. On the . BOOKCOMP, Inc. — John Wiley & Sons / Page 583 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 583 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [583],. all materials. However, BOOKCOMP, Inc. — John Wiley & Sons / Page 584 / 2nd Proofs / Heat Transfer Handbook / Bejan 584 THERMAL RADIATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [584],. ≡ E λ (T,λ) E bλ (T,λ) (8.27) BOOKCOMP, Inc. — John Wiley & Sons / Page 585 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 585 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [585],