BOOKCOMP, Inc. — John Wiley & Sons / Page 593 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 593 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 [593], (21) Lines: 679 to 687 ——— -5.903pt PgVar ——— Normal Page PgEnds: T E X [593], (21) Black Glass: = 1.517, = 0.546 mn ⊥ Ј Electromagnetic theory Experiment Reflectance, Ј Angle of incidence (degrees) Figure 8.12 Spectral, directional reflectance of blackened glass at room temperature. (From Brandenberg, 1963.) other hand, the emittance of amorphous solids (solids without a crystal lattice) tends to be independent of temperature. 8.2.3 Effects of Surface Conditions Up to this point, the present discussion of radiative properties has assumed that the material is pure and homogeneous, and that its surface is isotropic and optically smooth. Very few real material surfaces come close to this idealization. In usually hostile industrial environments, even an initially ideal material will have its surface composition and quality altered: Heating of the material may be accompanied by strong oxidation or other chemical reaction, producing an opaque surface layer of a material quite different from the substrate. Similar statements can be made about materials exposed to corrosive atmospheres for extended periods of time. In addition, very few surfaces have an optically smooth finish when new; exposing them to heat and/or corrosive atmospheres is generally accompanied by further roughening of the surface finish. Surface Roughness A surface is optically smooth if the average length scale of surface roughness is much less than the wavelength of the electromagnetic wave. Therefore, a surface that appears rough in visible light (λ 0.5 µm) may well be optically smooth in the intermediate infrared (λ 50 µm). This difference is the primary reason why results from electromagnetic wave theory cease to be valid for very short wavelengths. BOOKCOMP, Inc. — John Wiley & Sons / Page 594 / 2nd Proofs / Heat Transfer Handbook / Bejan 594 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 [594], (22) Lines: 687 to 698 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [594], (22) The character of roughness may be very different from surface to surface, depend- ing on the material, method of manufacture, and surface preparation, and classifica- tion of this character is difficult. A common measure of surface roughness is given by the root-mean-square (rms) roughness, σ m . The rms roughness can be measured readily with a profilometer (a sharp stylus that traverses the surface, recording the height fluctuations). Unfortunately, σ m alone is woefully inadequate to describe the roughness of a surface. Surfaces of identical σ m may have vastly different frequen- cies of roughness peaks, as well as different peak-to-valley lengths; in addition, σ m gives no information on second-order (or higher) roughness superimposed onto the fundamental roughness. In general terms it may be stated that surfaces will become less reflective, and the behavior of reflection will become less specular and more diffuse as surface rough- ness increases. This behavior may be explained through geometric optics by realizing that, for a rough surface, incoming radiation hitting the surface may undergo two or more reflections off local peaks and valleys (resulting in increased absorption), af- ter which it leaves the surface into an off-specular direction. Simple models predict sharp reflection peaks in the specular direction, and lesser reflection into other direc- tions, with the strength of the peak depending on the surface roughness. This was also found to be true experimentally for most cases as long as the incidence angle was not too large. For large off-normal angles of incidence, experiment has shown that the reflectance has its peak at polar angles greater than the specular direction, and for larger incidence angles, rough surfaces tend to display off-specular peaks, apparently due to shadowing of parts of the surface by adjacent peaks. Surface Layers and Oxide Films Even optically smooth surfaces have a sur- face structure that is different from the bulk material, due to either surface damage or the presence of thin layers of foreign materials. Surface damage is usually caused by the machining process, particularly for metals and semiconductors, which distorts or damages the crystal lattice near the surface. Thin foreign coats may be formed by chemical reaction (mostly oxidation), absorption (coats of grease or water), or electrostatics (dust particles). All of these effects may have a severe impact on the radiation properties of metals and may cause considerable changes in the properties of semiconductors. Because metals have large absorptive indices k and thus high re- flectances, a thin, nonmetallic layer with small k can significantly decrease the com- posite’s reflectance (and raise its absorptance). Dielectric materials, on the other hand, have small k’s, and their relatively strong emission and absorption take place over a very thick surface layer. The addition of a thin, different dielectric layer cannot signif- icantly alter their radiative properties (Bennett et al., 1963; Dunkle and Gier, 1953). Figure 8.13 shows the spectral, normal emittance (or absorptance) of aluminum for a surface prepared by the ultrahigh vacuum method and for several other aluminum surface finishes. While ultrahigh vacuum aluminum follows the Drude theory for λ > 1 µm, polished aluminum (clean and optically smooth for large wavelengths) has a much higher absorptance over the entire spectrum. Still, the overall level of ab- sorptance remains very low, and the reflectance remains rather specular. As Fig. 8.13 shows, the absorptance is much larger still when off-the-shelf commercial aluminum BOOKCOMP, Inc. — John Wiley & Sons / Page 595 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 595 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 [595], (23) Lines: 698 to 704 ——— 12.721pt PgVar ——— Normal Page PgEnds: T E X [595], (23) Figure 8.13 Spectral, normal emittance for aluminum with different surface finishes. (From Bennett et al., 1963; Dunkle and Gier, 1953.) is tested, probably due to a combination of roughness, contamination, and slight at- mospheric oxidation. Deposition of a thin oxide layer on aluminum (up to 100 Å) appreciably increases the emittance only for wavelengths less than 1.5 µm. This clearly is not true for thick oxide layers, as evidenced by Fig. 8.13: Anodized alu- minum (electrolytically oxidized material with a thick layer of alumina, Al 2 O 3 )no longer displays the typical trends of a metal, but rather, shows the behavior of the dielectric alumina. The effects of thin and thick oxide layers have been measured for many metals, with similar results. As a rule of thumb, clean metal exposed to air at room temperature grows oxide films so thin that infrared emittances are not affected appreciably. On the other hand, metal surfaces exposed to high-temperature oxidizing environments (furnaces, laser heating) generally have radiative properties similar to those of their oxide layer. Although most severe for metallic surfaces, the problem of surface modification is not unknown for nonmetals. For example, it is well known that when exposed to air at high temperature, silicon carbide (SiC) forms a layer of silica (SiO 2 ) on its surface, resulting in a reflection band around 9 µm. Nonoxidizing chemical reactions can also significantly change the radiative properties of dielectrics. For example, the strong ultraviolet radiation in outer space (from the sun) as well as gamma rays (from inside Earth’s van Allen belt) can damage the surface of spacecraft-protective coatings such as white acrylic paint or titanium dioxide epoxy coating, and similar results can be expected for ultraviolet laser irradiation. BOOKCOMP, Inc. — John Wiley & Sons / Page 596 / 2nd Proofs / Heat Transfer Handbook / Bejan 596 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 [596], (24) Lines: 704 to 832 ——— 4.06808pt PgVar ——— Normal Page PgEnds: T E X [596], (24) 8.2.4 Semitransparent Sheets For an optically smooth semitransparent sheet of thickness L substantially larger than the laser wavelength, L λ, radiative properties are readily determined through geometric optics and raytracing. Accounting for multiple reflections, the absorptance A slab , reflectance R slab , and transmittance T slab of an absorbing layer with complex index of refraction m = n − ık(ı = √ −1) are given by R slab = ρ 1 + (1 − ρ) 2 τ 2 1 − ρ 2 τ 2 (8.38) T slab = (1 − ρ) 2 τ 1 − ρ 2 τ 2 (8.39) A slab = (1 − ρ)(1 − τ) 1 − ρτ (8.40) and A slab + R slab + T slab = 1 (8.41) In these relations ρ is the reflectance of both sheet–air interfaces and τ is the trans- mittance of the sheet, as given for a nonscattering material (one without defects, in- clusions, bubbles) by τ = e −κL (8.42) where κ = 4πk/λ is the absorption coefficient of the material, which is related to the absorptive index k as shown. If the thickness of the semitransparent sheet is on the order of the wavelength of the irradiation (thin film), interference effects need to be accounted for because phase differences between first- and second-surface reflected light make film reflectance a strongly oscillating function of wavelength, with near-zero reflectance at some wave- lengths and very substantial reflectances in between. This phenomenon is commonly exploited by putting antireflection coatings onto optical components, optimized to minimize the reflectance of the optical elements at desired wavelengths. 8.2.5 Summary Reflectance and absorptance for many materials have been compiled in a number of books and other publications, notably the handbooks by Touloukian et al. (Touloukian and DeWitt, 1970, 1972; Touloukian et al., 1973). These tabulations show large amounts of scatter, and radiative properties for opaque surfaces, when obtained from tabulations and figures in the literature, should be taken with a grain of salt. Unless detailed descriptions of surface purity, preparation, and treatment are available, the data may not give any more than an order-of-magnitude estimate. One should also keep in mind that the properties of a surface may change during a process or overnight BOOKCOMP, Inc. — John Wiley & Sons / Page 597 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 597 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 [597], (25) Lines: 832 to 832 ——— 0.19804pt PgVar ——— Normal Page PgEnds: T E X [597], (25) TABLE 8.2 Total Emittance and Solar Absorptance of Selected Surfaces Temperature Total, Normal Solar (°C) Emittance Absorptance Alumina, flame-sprayed −25 0.80 0.28 Aluminum foil As received 20 0.04 Bright dipped 20 0.025 0.10 Aluminum, vacuum-deposited 20 0.025 0.10 Hard-anodized −25 0.84 0.92 Highly polished plate, 98.3% pure 225–575 0.039–0.057 Commercial sheet 100 0.09 Rough polish 100 0.18 Rough plate 40 0.055–0.07 Oxidized at 600°C 200–600 0.11–0.19 Heavily oxidized 95–500 0.20–0.31 Antimony, polished 35–260 0.28–0.31 Asbestos 35–370 0.93–0.94 Beryllium 150 0.18 0.77 370 0.21 600 0.30 Beryllium, anodized 150 0.90 370 0.88 600 0.82 Bismuth, bright 75 0.34 Black paint Parson’s optical black −25 0.95 0.975 Black silicone −25 to 750 0.93 0.94 Black epoxy paint −25 0.89 0.95 Black enamel paint 95–425 0.81–0.80 Brass, polished 40–315 0.10 Rolled plate, natural surface 22 0.06 Dull plate 50–350 0.22 Oxidized by heating at 600°C 200–600 0.61–0.59 Carbon, graphitized 100–320 0.76–0.75 320–500 0.75–0.71 Candle soot 95–270 0.952 Graphite, pressed, filed surface 250–510 0.98 Chromium, polished 40–1100 0.08–0.36 Copper, electroplated 20 0.03 0.47 Carefully polished electrolytic copper 80 0.018 Polished 115 0.023 Plate heated at 600°C 200–600 0.57 Cuprous oxide 800–1100 0.66–0.54 Molten copper 1075–1275 0.16–0.13 Glass, pyrex, lead, and soda 260–540 0.95–0.85 Gold, pure, highly polished 225–625 0.018–0.035 (continued) BOOKCOMP, Inc. — John Wiley & Sons / Page 598 / 2nd Proofs / Heat Transfer Handbook / Bejan 598 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 [598], (26) Lines: 832 to 837 ——— -10.08395pt PgVar ——— Normal Page PgEnds: T E X [598], (26) TABLE 8.2 Total Emittance and Solar Absorptance of Selected Surfaces (Continued) Temperature Total, Normal Solar (°C) Emittance Absorptance Gypsum 20 0.903 Inconel X, oxidized −25 0.71 0.90 Lead, pure (99.96%), unoxidized 125–225 0.057–0.075 Gray oxidized 25 0.28 Oxidized at 150°C 200 0.63 Magnesium, polished 35–260 0.07–0.13 Magnesium oxide 275–825 0.55–0.20 900–1705 0.20 Mercury 0–100 0.09–0.12 Molybdenum, polished 35–260 0.05–0.08 540–1370 0.10–0.18 2750 0.29 Nickel, electroplated 20 0.03 0.22 Polished 100 0.072 Platinum, pure, polished 225–625 0.054–0.104 Silica, sintered, powdered, fused silica 35 0.84 0.08 Silicon carbide 150–650 0.83–0.96 Silver, polished, pure 40–625 0.020–0.032 Stainless steel Type 312, heated 300 h at 260°C 95–425 0.27–0.32 Type 301 with Armco black oxide −25 0.75 0.89 Type 410, heated to 700°C in air 35 0.13 0.76 Type 303, sandblasted 95 0.42 0.68 Titanium, 75A 95–425 0.10–0.19 75A, oxidized 300 h at 450°C 35–425 0.21–0.25 0.80 Anodized −25 0.73 0.51 Tungsten, filament, aged 27–3300 0.032–0.35 Zinc, pure, polished 225–325 0.045–0.053 Galvanized sheet 100 0.21 (by oxidation and/or contamination). A representative list of total normal emittances and total normal absorptances (= 1− reflectance) for solar irradiation is given in Table 8.2 for a number of metals and nonmetals, which may be enlisted for a gray analysis. All values are for stated temperature ranges and stated surface conditions: As explained earlier, these values may change significantly with temperature, surface roughness, and oxidation. 8.3 RADIATIVE EXCHANGE BETWEEN SURFACES In many engineering applications the exchange of radiative energy between surfaces is virtually unaffected by the medium that separates them. Such (radiatively) non- participating media include vacuum as well as monatomic and most diatomic gases BOOKCOMP, Inc. — John Wiley & Sons / Page 599 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE EXCHANGE BETWEEN SURFACES 599 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 [599], (27) Lines: 837 to 846 ——— 3.026pt PgVar ——— Normal Page PgEnds: T E X [599], (27) (including air) at low to moderate temperature levels (i.e., in the absence of ionization and dissociation). Examples include spacecraft heat rejection systems, solar collector systems, radiative space heaters, illumination problems, and so on. This implies that photons will travel unimpeded from surface to surface, possibly over large distances. To account for all the radiative energy arriving at a point in space from all directions, the analysis has to be carried out over a complete enclosure of opaque surfaces. One or more of these surfaces may not be material, such as windows, in which case they are assigned equivalent properties and equivalent temperatures to account for the ra- diative energy entering or leaving the enclosure through them. In the following sections the analysis of radiative heat transfer in the absence of a participating medium will be considered for different levels of complexity. To make the analysis tractable it is common practice to make the assumption of an idealized enclosure and/or ideal surface properties. An enclosure may be idealized in two ways, as indicated in Fig. 8.14: by replacing a complex geometrical shape with a few simple surfaces, and by assuming surfaces to be isothermal with constant (or average) heat flux values through them. Obviously, the idealized enclosure approaches the real enclosure for sufficiently small isothermal subsurfaces. Surface properties may be idealized in a number of ways. The greatest simpli- fication arises if all surfaces are assumed black: For such a situation no reflected radiation needs to be accounted for and all emitted radiation is diffuse. The next level of difficulty arises if surfaces are assumed to be diffuse gray emitters (and, thus, ab- sorbers) as well as diffuse gray reflectors. This level of idealization tends to give results of acceptable accuracy for the vast majority of engineering problems. If the directional reflection behavior of a surface deviates strongly from a diffuse reflector (such a polished metal, which reflects almost like a mirror), one may often approxi- mate the reflectance to consist of a purely diffuse and a purely specular component. However, this greatly complicates the analysis, in particular, if the enclosure includes curved surfaces. Luckily, the effects of specular (or, indeed, any type of nondiffuse) reflections tend to be very small in most engineering enclosures. Exceptions include light concentrators and collimators (in solar energy applications), long channels with specular sidewalls (optical fibers), and others. For the treatment of specular reflections the reader is referred to more detailed textbooks, such as the one by Modest (2003). Figure 8.14 Real and ideal enclosures for radiative transfer calculations. BOOKCOMP, Inc. — John Wiley & Sons / Page 600 / 2nd Proofs / Heat Transfer Handbook / Bejan 600 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 [600], (28) Lines: 846 to 1030 ——— 3.66231pt PgVar ——— Normal Page PgEnds: T E X [600], (28) Of greater engineering importance is the case when the assumption of a gray surface is not acceptable. Simple methods to treat nongray behavior are outlined briefly at the end of this section. 8.3.1 View Factors To make an energy balance on a surface, the incoming radiative flux, or irradiation, H must be evaluated. In a general enclosure the irradiation has contributions from all parts of the enclosure surface. Therefore, one needs to determine how much of the energy that leaves any surface of the enclosure travels toward the surface under consideration. The geometric relations governing this process for diffuse surfaces (which absorb and emit diffusely, and also reflect radiative energy diffusely) are known as view factors. Other names used in the literature are configuration factor, angle factor, and shape factor. The view factor between two surfaces A i and A j is defined as F i−j ≡ diffuse energy leaving A i directly toward and intercepted by A j total diffuse energy leaving A i (8.43) where the word directly is meant to imply “on a straight path, without intervening reflections.” A list of relationships for some common view factors is given in Table 8.3, and Figs. 8.15 through 8.17 give convenient graphical representations of the three most important view factors. Radiation view factors may be determined by a variety of methods, such as direct integration (analytical or numerical integration), statistical evaluation [through sta- tistical sampling using a Monte Carlo method (Modest, 2003), or through a variety of special methods, some of which are described briefly in what follows. Direct Integration Mathematically, view factors can be expressed in terms of a double surface integral, that is, F i−j = 1 A i A i A j cos θ i cos θ j πS 2 ij dA j dA i (8.44) where S ij is the distance between points on surfaces A i and A j , and θ i and θ j are the angles between S ij and the local surface normals, as shown in Fig. 8.18. Using Stokes’ theorem, eq. (8.44) can be converted into a double contour integral, A i F i−j = 1 2π Γ i Γ j ln S ij ds j · ds i (8.45) where Γ i is the contour of A i (as also indicated in Fig. 8.18) and s i is a vector to a point on contour Γ i . While the integration of eqs. (8.44) and (8.45) may be straightforward for some simple configurations, it is desirable to have a more generally applicable formula at BOOKCOMP, Inc. — John Wiley & Sons / Page 601 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE EXCHANGE BETWEEN SURFACES 601 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 [601], (29) Lines: 1030 to 1030 ——— 4.08975pt PgVar ——— Normal Page PgEnds: T E X [601], (29) TABLE 8.3 Important View Factors 1. Two infinitely long, directly opposed parallel plates of the same finite width: H = h w F 1−2 = F 2−1 = 1 + H 2 − H 2. Two infinitely long plates of unequal widths h and w, having one common edge, and at an angle of 90° to each other: H = h w F 1−2 = 1 2 1 + H − 1 + H 2 3. Two infinitely long plates of equal finite width w, having one common edge, forming a wedgelike groove with opening angle a: F 1−2 = F 2−1 = 1 − sin α 2 w w A 1 A 2 ␣ 4. Infinitely long parallel cylinders of the same diameter: X = 1 + s 2r F 1−2 = 1 π sin −1 1 X + X 2 − 1 − X A 1 A 2 rr s 5. Two infiniteparallel cylinders of different radius: R = r 2 r 1 S = s r 1 C = 1 + R + S F 1−2 = 1 2π π + C 2 − (R + 1) 2 − C 2 − (R − 1) 2 +(R − 1) cos −1 R − 1 C − (R + 1) cos −1 R + 1 C A 1 A 2 r 1 r 2 s (continued) BOOKCOMP, Inc. — John Wiley & Sons / Page 602 / 2nd Proofs / Heat Transfer Handbook / Bejan 602 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 [602], (30) Lines: 1030 to 1030 ——— 5.43091pt PgVar ——— Normal Page PgEnds: T E X [602], (30) TABLE 8.3 Important View Factors (Continued) 6. Exterior of infinitely long cylinder to unsymmetrically placed, infinitely long parallel rectangle; r ≤ a: B 1 = b 1 a B 2 = b 2 a F 1−2 = 1 2π (tan −1 B 1 − tan −1 B 2 ) 7. Identical, parallel, directly opposedrectangles: X = a c Y = b c F 1−2 = 2 πXY ln (1 + X 2 )(1 + Y 2 ) 1 + X 2 + Y 2 1/2 + X 1 + Y 2 tan −1 X √ 1 + Y 2 +Y 1 + X 2 tan −1 Y √ 1 + X 2 − X tan −1 X − Y tan −1 Y 8. Two finite rectangles of same length, having one common edge, and at an angle of 90° to each other: H = h l W = w l F 1−2 = 1 πW W tan −1 1 W + H tan −1 1 H − H 2 + W 2 tan −1 1 √ H 2 + W 2 + 1 4 ln (1 + W 2 )(1 + H 2 ) 1 + W 2 + H 2 W 2 (1 + W 2 + H 2 ) (1 + W 2 )(W 2 + H 2 ) W 2 H 2 (1 + H 2 + W 2 ) (1 + H 2 )(H 2 + W 2 ) H 2 9. Disk to parallel coaxial disk of unequal radius: R 1 = r 1 a R 2 = r 2 a X = 1 + 1 + R 2 2 R 2 1 F 1−2 = 1 2 X − X 2 − 4 R 2 R 1 2 . (radiatively) non- participating media include vacuum as well as monatomic and most diatomic gases BOOKCOMP, Inc. — John Wiley & Sons / Page 599 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE. 8.14 Real and ideal enclosures for radiative transfer calculations. BOOKCOMP, Inc. — John Wiley & Sons / Page 600 / 2nd Proofs / Heat Transfer Handbook / Bejan 600 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 [600],. off-the-shelf commercial aluminum BOOKCOMP, Inc. — John Wiley & Sons / Page 595 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS 595 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 [595],