BOOKCOMP, Inc. — John Wiley & Sons / Page 573 / 2nd Proofs / Heat Transfer Handbook / Bejan 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 [First Page] [573], (1) Lines: 0 to 90 ——— 11.3931pt PgVar ——— Normal Page * PgEnds: PageBreak [573], (1) CHAPTER 8 Thermal Radiation MICHAEL F. MODEST College of Engineering Pennsylvania State University University Park, Pennsylvania 8.1 Fundamentals 8.1.1 Emissive power 8.1.2 Solid angles 8.1.3 Radiative intensity 8.1.4 Radiative heat flux 8.2 Radiative properties of solids and liquids 8.2.1 Radiative properties of metals Wavelength dependence Directional dependence Hemispherical properties Total properties Surface temperature effects 8.2.2 Radiative properties of nonconductors Wavelength dependence Directional dependence Temperature dependence 8.2.3 Effects of surface conditions Surface roughness Surface layers and oxide films 8.2.4 Semitransparent sheets 8.2.5 Summary 8.3 Radiative exchange between surfaces 8.3.1 View factors Direct integration Special methods View factor algebra Crossed-strings method 8.3.2 Radiative exchange between black surfaces 8.3.3 Radiative exchange between diffuse gray surfaces Convex surface exposed to large isothermal enclosure 8.3.4 Radiation shields 8.3.5 Radiative exchange between diffuse nongray surfaces Semigray approximation method Band approximation method 573 BOOKCOMP, Inc. — John Wiley & Sons / Page 574 / 2nd Proofs / Heat Transfer Handbook / Bejan 574 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 [574], (2) Lines: 90 to 142 ——— -1.32596pt PgVar ——— Normal Page * PgEnds: Eject [574], (2) 8.4 Radiative properties of participating media 8.4.1 Molecular gases 8.4.2 Particle clouds Soot Pulverized coal and fly ash dispersions Mixtures of molecular gases and particulates 8.5 Radiative exchange within participating media 8.5.1 Mean beam length method 8.5.2 Diffusion approximation 8.5.3 P-1 approximation 8.5.4 Other RTE solution methods 8.5.5 Weighted sum of gray gases 8.5.6 Other spectral models Nomenclature References 8.1 FUNDAMENTALS Radiative heat transfer or thermal radiation is the science of transferring energy in the form of electromagnetic waves. Unlike heat conduction, electromagnetic waves do not require a medium for their propagation. Therefore, because of their ability to travel across vacuum, thermal radiation becomes the dominant mode of heat trans- fer in low pressure (vacuum) and outer-space applications. Another distinguishing characteristic between conduction (and convection, if aided by flow) and thermal ra- diation is their temperature dependence. While conductive and convective fluxes are more or less linearly dependent on temperature differences, radiative heat fluxes tend to be proportional to differences in the fourth power of temperature (or even higher). For this reason, radiation tends to become the dominant mode of heat transfer in high-temperature applications, such as combustion (fires, furnaces, rocket nozzles), nuclear reactions (solar emission, nuclear weapons), and others. All materials continuously emit and absorb electromagnetic waves, or photons, by changing their internal energy on a molecular level. Strength of emission and absorp- tion of radiative energy depend on the temperature of the material, as well as on the wavelength λ, frequency ν, or wavenumber η, that characterizes the electromagnetic waves, λ = c ν = 1 η (8.1) where wavelength is usually measured in µm(= 10 −6 m), while frequency is mea- sured in hertz = cycles/s), and wavenumbers are given in cm −1 . Electromagnetic waves or photons (which include what is perceived as “light”) travel at the speed of light, c. The speed of light depends on the medium through which the wave travels and is related to that in vacuum, c 0 , through the relation BOOKCOMP, Inc. — John Wiley & Sons / Page 575 / 2nd Proofs / Heat Transfer Handbook / Bejan FUNDAMENTALS 575 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 [575], (3) Lines: 142 to 184 ——— -1.13882pt PgVar ——— Normal Page PgEnds: T E X [575], (3) c = c 0 n c 0 = 2.998 × 10 8 m/s (8.2) 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, and the c in eq. (8.1) can be replaced by c 0 . Each wave or photon carries with it an amount of energy determined from quantum mechanics as = hν h = 6.626 × 10 −34 J · s (8.3) where h is known as Planck’s constant. The frequency of light does not change when light penetrates from one medium to another because the energy of the photon must be conserved. On the other hand, the wavelength does change, depending on the values of the refractive index for the two media. When an electromagnetic wave strikes an interface between two media, the wave is either reflected or transmitted. Most solid and liquid media absorb all incoming radiation over a very thin surface layer. Such materials are called opaque or opaque surfaces (even though absorption takes place over a thin layer). An opaque material that does not reflect any radiation at its surface is called a perfect absorber, black surface, or blackbody, because such a surface appears black to the human eye, which recognizes objects by visible radiation reflected off their surfaces. 8.1.1 Emissive Power Every medium continuously emits electromagnetic radiation randomly into all direc- tions at a rate depending on the local temperature and the properties of the material. The radiative heat flux emitted from a surface is called the emissive power E, and there is a distinction between total and spectral emissive power (heat flux emitted over the entire spectrum or at a given frequency per unit frequency interval), so that the spectral emissive power E ν is the emitted energy/time/surface area/frequency, while the total emissive power E is emitted energy/time/surface area. Spectral and total emissive powers are related by E(T ) = ∞ 0 E λ (T,λ)dλ = ∞ 0 E ν (T,ν)dν (8.4) It is easy to show that a black surface is not only a perfect absorber, but it is also a perfect emitter, that is, the emission from such a surface exceeds that of any other surface at the same temperature (known as Kirchhoff’s law). The emissive power leaving an opaque black surface, commonly called blackbody emissive power, can be determined from quantum statistics as E bλ (T,λ) = 2πhc 2 0 n 2 λ 5 (e hc 0 /nλkT − 1) (n = const) (8.5) where it is assumed that the black surface is adjacent to a nonabsorbing medium of constant refractive index n. The constant k = 1.3806 × 10 −23 J/K is known as BOOKCOMP, Inc. — John Wiley & Sons / Page 576 / 2nd Proofs / Heat Transfer Handbook / Bejan 576 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 [576], (4) Lines: 184 to 211 ——— -2.8009pt PgVar ——— Normal Page PgEnds: T E X [576], (4) 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Wavelength λ , mµ Blackbody emissive power (W/m . m)E b 2 µ T = 5 7 6 2 K 5 0 0 0 K 3 0 0 0 K 2 0 0 0 K 1 0 0 0 K 5 0 0 K Visible part of spectrum ETC b 3 (= /) Figure 8.1 Blackbody emissive power spectrum. Boltzmann’s constant. The spectral dependence of the blackbody emissive power into vacuum (n = 1) is shown for a number of emitter temperatures in Fig. 8.1. It is seen that emission is zero at both extreme ends of the spectrum with a maximum at some intermediate wavelength. The general level of emission rises with temperature, and the important part of the spectrum (the part containing most of the emitted energy) shifts toward shorter wavelengths. Because emission from the sun (“solar spectrum”) is well approximated by blackbody emission at an effective solar temperature of T sun = 5762 K, this temperature level is also included in the figure. Heat transfer problems generally involve temperature levels between 300 and, say, 2000 K (plus, perhaps, solar radiation). Therefore, the spectral ranges of interest in heat transfer applications include the ultraviolet (0.1 to 0.4 µm), visible radiation (0.4 to 0.7 µm, as indicated in Figure 8.1 by shading), and the near- and mid-infrared (0.7 to 20 µm). For quick evaluation, a scaled emissive power can be written as E bλ n 3 T 5 = C 1 (nλT) 5 (e C 2 /nλT − 1) (n = const) (8.6) where C 1 = 2πhc 2 0 = 3.7419 × 10 −16 W · m 2 C 2 = hc 0 k = 14,388 µm · K Equation (8.6) has its maximum at (nλT) max = C 3 = 2898 µm · K (8.7) BOOKCOMP, Inc. — John Wiley & Sons / Page 577 / 2nd Proofs / Heat Transfer Handbook / Bejan FUNDAMENTALS 577 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 [577], (5) Lines: 211 to 250 ——— 3.77525pt PgVar ——— Normal Page * PgEnds: Eject [577], (5) which is known as Wien’s displacement law. The constants C 1 ,C 2 , and C 3 are known as the first, second, and third radiation constants, respectively. The total blackbody emissive is found by integrating eq. (8.6) over the entire spectrum, resulting in E b (T ) = n 2 σT 4 (8.8) where σ = 5.670 × 10 −8 W/m 2 · K 4 is the Stefan–Boltzmann constant. It is often desirable to calculate fractional emissive powers, that is, the emissive power con- tained over a finite wavelength range, say between wavelengths λ 1 and λ 2 .Itisnot possible to integrate eq. (8.6) between these limits in closed form; instead, one resorts to tabulations of the fractional emissive power, contained between 0 and nλT , f(nλT)= λ 0 E bλ dλ ∞ 0 E bλ bλ = nλT 0 E bλ n 3 σT 5 d(nλT) (8.9) so that λ 2 λ 1 E bλ dλ = [f(nλ 2 T)− f(nλ 1 T)]n 2 σT 4 (8.10) An extensive listing of f(nλT), as well as of the scaled emissive power, eq. (8.6), is given in Table 8.1. Both functions are also shown in Fig. 8.2, together with Wien’s distribution, which is the short-wavelength limit of eq. (8.5), E bλ 2πhc 2 0 n 2 λ 5 e −hc 0 /nλkT = C 1 n 2 λ 5 e −C 2 /nλT hc 0 nλkT 1 (8.11) As seen from the figure, Wien’s distribution is actually rather accurate over the entire spectrum, predicting a total emissive power approximately 8% lower than the one given by eq. (8.8). Because Wien’s distribution can be integrated analytically over parts of the spectrum, it is sometimes used in heat transfer applications. 8.1.2 Solid Angles Radiation is a directional phenomenon; that is, the radiative flux passing through a point generally varies with direction, such as the sun shining onto Earth from essentially a single direction. Consider an opaque surface element dA i , as shown in Fig. 8.3. It is customary to describe the direction unit vector ˆ s in terms of polar angle θ (measured from the surface normal ˆ n) and azimuthal angle ψ (measured in the plane of the surface, between an arbitrary axis and the projection of ˆ s); for a hemisphere 0 ≤ θ ≤ π/2 and 0 ≤ ψ ≤ 2π. The solid angle with which a surface A j is seen from a certain point P (or dA i in Fig. 8.3) is defined as the projection of the surface onto a plane normal to the direction vector, divided by the distance squared, as also shown in Fig. 8.3 for an infinitesimal BOOKCOMP, Inc. — John Wiley & Sons / Page 578 / 2nd Proofs / Heat Transfer Handbook / Bejan 578 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 [578], (6) Lines: 250 to 309 ——— -1.53975pt PgVar ——— Normal Page PgEnds: T E X [578], (6) TABLE 8.1 Blackbody Emissive Power E bλ /n 3 T 5 E bλ /n 3 T 5 nλT(W/m 2 · µm ·K 5 ) nλT(W/m 2 · µm ·K 5 ) (µm · K) ×10 −11 f(nλT) (µm · K) ×10 −11 f(nλT) 1,000 0.02110 0.00032 5,600 0.56332 0.70101 1,100 0.04846 0.00091 5,700 0.54146 0.71076 1,200 0.09329 0.00213 5,800 0.52046 0.72012 1,300 0.15724 0.00432 5,900 0.50030 0.72913 1,400 0.23932 0.00779 6,000 0.48096 0.73778 1,500 0.33631 0.01285 6,100 0.46242 0.74610 1,600 0.44359 0.01972 6,200 0.44464 0.75410 1,700 0.55603 0.02853 6,300 0.42760 0.76180 1,800 0.66872 0.03934 6,400 0.41128 0.76920 1,900 0.77736 0.05210 6,500 0.39564 0.77631 2,000 0.87858 0.06672 6,600 0.38066 0.78316 2,100 0.96994 0.08305 6,700 0.36631 0.78975 2,200 1.04990 0.10088 6,800 0.35256 0.79609 2,300 1.11768 0.12002 6,900 0.33940 0.80219 2,400 1.17314 0.14025 7,000 0.32679 0.80807 2,500 1.21659 0.16135 7,100 0.31471 0.81373 2,600 1.24868 0.18311 7,200 0.30315 0.81918 2,700 1.27029 0.20535 7,300 0.29207 0.82443 2,800 1.28242 0.22788 7,400 0.28146 0.82949 2,900 1.28612 0.25055 7,500 0.27129 0.83436 3,000 1.28245 0.27322 7,600 0.26155 0.83906 3,100 1.27242 0.29576 7,700 0.25221 0.84359 3,200 1.25702 0.31809 7,800 0.24326 0.84796 3,300 1.23711 0.34009 7,900 0.23468 0.85218 3,400 1.21352 0.36172 8,000 0.22646 0.85625 3,500 1.18695 0.38290 8,200 0.21101 0.86396 3,600 1.15806 0.40359 8,400 0.19679 0.87115 3,700 1.12739 0.42375 8,600 0.18370 0.87786 3,800 1.09544 0.44336 8,800 0.17164 0.88413 3,900 1.06261 0.46240 9,000 0.16051 0.88999 4,000 1.02927 0.48085 9,200 0.15024 0.89547 4,100 0.99571 0.49872 9,400 0.14075 0.90060 4,200 0.96220 0.51599 9,600 0.13197 0.90541 4,300 0.92892 0.53267 9,800 0.12384 0.90992 4,400 0.89607 0.54877 10,000 0.11632 0.91415 4,500 0.86376 0.56429 10,200 0.10934 0.91813 4,600 0.83212 0.57925 10,400 0.10287 0.92188 4,700 0.80124 0.59366 10,600 0.09685 0.92540 4,800 0.77117 0.60753 10,800 0.09126 0.92872 4,900 0.74197 0.62088 11,000 0.08606 0.93184 5,000 0.71366 0.63372 11,200 0.08121 0.93479 5,100 0.68628 0.64606 11,400 0.07670 0.93758 5,200 0.65983 0.65794 11,600 0.07249 0.94021 5,300 0.63432 0.66935 11,800 0.06856 0.94270 5,400 0.60974 0.68033 12,000 0.06488 0.94505 5,500 0.58608 0.69087 (continued) BOOKCOMP, Inc. — John Wiley & Sons / Page 579 / 2nd Proofs / Heat Transfer Handbook / Bejan FUNDAMENTALS 579 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 [579], (7) Lines: 309 to 369 ——— 0.47249pt PgVar ——— Normal Page * PgEnds: Eject [579], (7) TABLE 8.1 Blackbody Emissive Power (Continued) E bλ /n 3 T 5 E bλ /n 3 T 5 nλT(W/m 2 · µm ·K 5 ) nλT(W/m 2 · µm ·K 5 ) (µm · K) ×10 −11 f(nλT) (µm · K) ×10 −11 f(nλT) 12,200 0.06145 0.94728 19,200 0.01285 0.98387 12,400 0.05823 0.94939 19,400 0.01238 0.98431 12,600 0.05522 0.95139 19,600 0.01193 0.98474 12,800 0.05240 0.95329 19,800 0.01151 0.98515 13,000 0.04976 0.95509 20,000 0.01110 0.98555 13,200 0.04728 0.95680 21,000 0.00931 0.98735 13,400 0.04494 0.95843 22,000 0.00786 0.98886 13,600 0.04275 0.95998 23,000 0.00669 0.99014 13,800 0.04069 0.96145 24,000 0.00572 0.99123 14,000 0.03875 0.96285 25,000 0.00492 0.99217 14,200 0.03693 0.96418 26,000 0.00426 0.99297 14,400 0.03520 0.96546 27,000 0.00370 0.99367 14,600 0.03358 0.96667 28,000 0.00324 0.99429 14,800 0.03205 0.96783 29,000 0.00284 0.99482 15,000 0.03060 0.96893 30,000 0.00250 0.99529 15,200 0.02923 0.96999 31,000 0.00221 0.99571 15,400 0.02794 0.97100 32,000 0.00196 0.99607 15,600 0.02672 0.97196 33,000 0.00175 0.99640 15,800 0.02556 0.97288 34,000 0.00156 0.99669 16,000 0.02447 0.97377 35,000 0.00140 0.99695 16,200 0.02343 0.97461 36,000 0.00126 0.99719 16,400 0.02245 0.97542 37,000 0.00113 0.99740 16,600 0.02152 0.97620 38,000 0.00103 0.99759 16,800 0.02063 0.97694 39,000 0.00093 0.99776 17,000 0.01979 0.97765 40,000 0.00084 0.99792 17,200 0.01899 0.97834 41,000 0.00077 0.99806 17,400 0.01823 0.97899 42,000 0.00070 0.99819 17,600 0.01751 0.97962 43,000 0.00064 0.99831 17,800 0.01682 0.98023 44,000 0.00059 0.99842 18,000 0.01617 0.98081 45,000 0.00054 0.99851 18,200 0.01555 0.98137 46,000 0.00049 0.99861 18,400 0.01496 0.98191 47,000 0.00046 0.99869 18,600 0.01439 0.98243 48,000 0.00042 0.99877 18,800 0.01385 0.98293 49,000 0.00039 0.99884 19,000 0.01334 0.98340 50,000 0.00036 0.99890 element dA j . If the surface is projected onto a unit sphere above point P , the solid angle becomes equal to the projected area, or Ω = A jp dA jp S 2 = A j cos θ 0 dA j S 2 = A jp (8.12) where S is the distance between P and dA j . Thus, an infinitesimal solid angle is simply an infinitesimal area on a unit sphere, or BOOKCOMP, Inc. — John Wiley & Sons / Page 580 / 2nd Proofs / Heat Transfer Handbook / Bejan 580 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 [580], (8) Lines: 369 to 395 ——— 0.84001pt PgVar ——— Long Page * PgEnds: Eject [580], (8) dΩ = dA jp = (1 × sin θ dψ)(1 × dθ) = sin θ dθ dψ (8.13) Integrating over all possible directions yields 2π ψ=0 π/2 θ=0 sin θ dθ dψ = 2π (8.14) Figure 8.2 Normalized blackbody emissive power spectrum. n dAi s 0 sin d d dAjp dAj n 0 1 dA j Љ P d Figure 8.3 Definitions of direction vectors and solid angles. BOOKCOMP, Inc. — John Wiley & Sons / Page 581 / 2nd Proofs / Heat Transfer Handbook / Bejan FUNDAMENTALS 581 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 [581], (9) Lines: 395 to 443 ——— 0.30309pt PgVar ——— Long Page PgEnds: T E X [581], (9) for the total solid angle above the surface. If a point inside a medium removed from the surface is considered, radiation passing through that point can strike any point of an imaginary unit sphere surrounding it; that is, the total solid angle here is 4π, with 0 ≤ θ ≤ π, 0 ≤ ψ ≤ 2π. Similarly, at a surface one can talk about an upper hemisphere (outgoing directions, 0 ≤ θ < π/2), and a lower hemisphere (incoming directions, π/2 < θ ≤ π). 8.1.3 Radiative Intensity The directional behavior of radiative energy traveling through a medium is charac- terized by the radiative intensity I , which is defined as I ≡ radiative energy flow/time/area normal to the rays/solid angle Like emissive power, intensity is defined on both spectral and total bases, related by I( ˆ s) = ∞ 0 I λ ( ˆ s,λ)dλ (8.15) However, unlike emissive power, which depends only on position (and wavelength), the radiative intensity depends, in addition, on the direction vector ˆ s. Emissive power can be related to emitted intensity by integrating this intensity over the 2π solid angles above a surface, and then realizing that the projection of dA normal to the rays is dA cos θ. Thus, E(r) = 2π 0 π/2 0 I(r, θ, ψ) cos θ sin θ dθ dψ = 2π I(r, ˆ s) ˆ n · ˆ s dΩ (8.16) which is, of course, also valid on a spectral basis. For a black surface it is readily shown, through a variation of Kirchhoff’s law, that I bλ is independent of direction, or I bλ = I bλ (T,λ) (8.17) Using this relation in eq. (8.16), it is observed that the intensity leaving a blackbody (or any surface whose outgoing intensity is independent of direction, or diffuse) may be evaluated from the blackbody emissive power (or outgoing heat flux) as I bλ (r,λ) = E bλ (r,λ) π (8.18) In the literature the spectral blackbody intensity is sometimes referred to as the Planck function. 8.1.4 Radiative Heat Flux Emissive power is the total radiative energy streaming away from a surface due to emission. Therefore, it is a radiative flux, but not the net radiative flux at the surface, BOOKCOMP, Inc. — John Wiley & Sons / Page 582 / 2nd Proofs / Heat Transfer Handbook / Bejan 582 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 [582], (10) Lines: 443 to 480 ——— 4.75412pt PgVar ——— Normal Page PgEnds: T E X [582], (10) because it only accounts for emission and not for incoming radiation and reflected radiation. Extending the definition of eq. (8.16) gives (q λ ) out = cos θ>0 I λ ( ˆ s) cos θ dΩ ≥ 0 (8.19) where I λ (θ) is now outgoing intensity (due to emission plus reflection). Similarly, for incoming directions (π/2 < θ ≤ π), (q λ ) in = cos θ<0 I λ ( ˆ s) cos θ dΩ < 0 (8.20) Combining the incoming and outgoing contributions, the net radiative flux at a surface is (q λ ) net = q λ · ˆ n = (q λ ) in + (q λ ) out = 4π I λ ( ˆ s) cos θ dΩ (8.21) The total radiative flux, finally, is obtained by integrating eq. (8.21) over the entire spectrum, or q · ˆ n = ∞ 0 q λ · ˆ n dλ = ∞ 0 4π I λ ( ˆ s) ˆ n · ˆ s dΩ dλ (8.22) Of course, the surface described by the unit vector ˆ n may be an imaginary one (located somewhere inside a radiating medium). Thus, removing the ˆ n from eq. (8.22) gives the definition of the radiative heat flux vector inside a participating medium: q = ∞ 0 q λ dλ = ∞ 0 4π I λ ( ˆ s) ˆ s dΩ dλ (8.23) 8.2 RADIATIVE PROPERTIES OF SOLIDS AND LIQUIDS Because radiative energy arriving at a given point in space can originate from a point far away, without interacting with the medium in between, a conservation of energy balance must be performed on an enclosure bounded by opaque walls (i.e., a medium thick enough that no electromagnetic waves can penetrate through it). Strictly speak- ing, the surface of an enclosure wall can only reflect radiative energy or allow a part of it to penetrate into the substrate. A surface cannot absorb or emit photons: Atten- uation takes place inside the solid, as does emission of radiative energy (and some of the emitted energy escapes through the surface into the enclosure). In practical systems the thickness of the surface layer over which absorption of irradiation from inside the enclosure occurs is very small compared with the overall dimensions of an enclosure—usually, a few angstroms for metals and a few micrometers for most non- metals. The same may be said about emission from within the walls that escapes into . models Nomenclature References 8.1 FUNDAMENTALS Radiative heat transfer or thermal radiation is the science of transferring energy in the form of electromagnetic waves. Unlike heat conduction, electromagnetic waves do. the figure. Heat transfer problems generally involve temperature levels between 300 and, say, 2000 K (plus, perhaps, solar radiation). Therefore, the spectral ranges of interest in heat transfer applications. (2) Lines: 90 to 142 ——— -1.3 2596 pt PgVar ——— Normal Page * PgEnds: Eject [574], (2) 8.4 Radiative properties of participating media 8.4.1 Molecular gases 8.4.2 Particle clouds Soot Pulverized