BOOKCOMP, Inc. — John Wiley & Sons / Page 613 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE EXCHANGE BETWEEN SURFACES 613 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 [613], (41) Lines: 1329 to 1352 ——— 2.94913pt PgVar ——— Normal Page PgEnds: T E X [613], (41) Figure 8.21 Arrangement of parallel or concentric radiation shields. where R j−1,j = 1 j−1 A j−1 + 1 j − 1 1 A j (8.72) The analysis of radiation shields is one of the few applications where analysis of spec- ularly reflecting surfaces is relatively simple and may lead to substantially different answers for concentric shields with strongly varying radii. For a specularly reflecting shield A j (with A j−1 being specular or diffuse), the radiative resistance becomes R j−1,j = 1 j−1 + 1 j − 1 1 A j−1 (A j specular) (8.73) Note that it is desirable to make shields highly reflective (low ), and this tends to make them specularly reflecting (also desirable, because it also increases the resistance). Further simplifications arise if all shields are of identical material ( 2 = 3 = ··· = N−1 ); on the other hand, eqs. (8.71) through (8.73) remain valid for shields with different emittances on both of its sides (different values for j in R j−1,j and R j,j+1 ). While the network analogy can (and has been) applied to configurations with more than two surfaces seeing each other, this leads to very complicated circuits (because there is only one resistance between any two surfaces). For such problems the network analogy is not recommended, and the net radiation method, eq. (8.68), should be employed. BOOKCOMP, Inc. — John Wiley & Sons / Page 614 / 2nd Proofs / Heat Transfer Handbook / Bejan 614 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 [614], (42) Lines: 1352 to 1374 ——— -4.44827pt PgVar ——— Normal Page PgEnds: T E X [614], (42) 8.3.5 Radiative Exchange between Diffuse Nongray Surfaces In a number of important engineering problems the assumption of gray surface prop- erties may not provide adequate accuracy (when properties exhibit strong spectral variations across the important range of the spectrum). To deal with such effects, two simple models, known as the semigray approximation and the band approximation, will be described. Semigray Approximation Method This method employs the principle of su- perposition: The radiative flux at any given point is the sum of the contributions from the various emitters in the enclosure, each one acting independently. In some applica- tions there is a natural division of the radiative energy within an enclosure into two or more distinct spectral regions. For example, in a solar collector the incoming energy comes from a high-temperature source with most of its energy below 3 µm, whereas radiation losses for typical collector temperatures are at wavelengths above 3 µm. In the case of laser heating and processing, the incoming energy is monochromatic (at the laser wavelength); reradiation takes place over the entire near- to midinfrared (depending on the workpiece temperature). In such a situation, eq. (8.68) may be split into two sets of N equations each, one set for each spectral range, and with different radiative properties for each set. For example, consider an enclosure subject to external irradiation, which is confined to a certain spectral range (1). The surfaces in the enclosure, owing to their temperature, emit over spectral range (2). * Then from eq. (8.68), 1 (1) i Q (1) i A j − N j=1 1 (1) j − 1 F i−j Q (1) i A j + H oi = 0 (8.74a) 1 (2) i Q (2) i A j − N j=1 1 (2) j − 1 F i−j Q (2) j A j = E bi − N j=1 F i−j E bj (8.74b) Q i A i = Q (1) i A i + Q (2) i A i i = 1, 2, ,N (8.74c) where (1) j is the average emittance for surface j over spectral interval (1), and so on. The semigray approximation is not limited to two distinct spectral regions. Each surface of the enclosure may be given a set of absorptances and reflectances, one value for each different emission temperature (with its different emission spectra). How- ever, while simple and straightforward, the method can never become exact no matter how many different values of absorptance and reflectance are chosen for each surface. Band Approximation Method Another commonly used method to deal with nongray surfaces is the band approximation method. This method employs the fact * Note that spectral ranges (1) and (2) do not need to cover the entire spectrum, and indeed, they may overlap. BOOKCOMP, Inc. — John Wiley & Sons / Page 615 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF PARTICIPATING MEDIA 615 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 [615], (43) Lines: 1374 to 1403 ——— -0.01006pt PgVar ——— Normal Page PgEnds: T E X [615], (43) that even for nongray materials, eq. (8.68) remains valid on a spectral basis (replacing emissive power E b by spectral emissive power E bλ , related to the total emissive power by E b = ∞ 0 E bλ dλ). In this method the spectrum is broken up into M bands, over which the radiative properties of all surfaces in the enclosure are constant. Therefore, 1 (m) i Q (m) i A i − N j=1 1 (m) i − 1 F i−j Q (m) j A j + H (m) oi = E (m) bi − N j=1 F i−j E (m) bj i = 1, 2, ,N, m = 1, 2, ,M (8.75a) E bj = M m=1 E (m) bj Q j A j = M m=1 Q (m) j A j H oi = M m=1 H (m) oi (8.75b) Here, E (m) b is the fractional emissive power contained in band m and so on. Equations (8.75) are, of course, nothing but a simple numerical integration of thespectral version of eq. (8.68), using the trapezoidal rule with varying steps. This method has the advantage that the widths of the bands can be tailored to the spectral variation of properties, resulting in good accuracy with relatively few bands. For very few bands the accuracy of this method is similar to that of the semigray approximation but is a little more cumbersome to apply. On the other hand, the band approximation method can achieve any desired accuracy by using many bands. 8.4 RADIATIVE PROPERTIES OF PARTICIPATING MEDIA In many high-temperature applications, when radiative heat transfer is important, the medium between surfaces is not transparent but is “participating”; that is, it absorbs, emits, and (possibly) scatters radiation. In a typical combustion process this interaction results in (1) continuum radiation due to tiny, burning soot particles (of dimension < 1 µm) and also due to larger suspended particles, such as coal particles, oil droplets, and fly ash; (2) banded radiation in the infrared due to emission and absorption by molecular gaseous combustion products, mostly water vapor and carbon dioxide; and (3) chemiluminescence due to the combustion reaction itself. While chemiluminescence may normally be neglected, particulates as well as gas radiation generally must be accounted for. 8.4.1 Molecular Gases When a photon (or an electromagnetic wave) interacts with a gas molecule, it may be absorbed, raising the energy level of the molecule. Conversely, a gas molecule may spontaneously lower its energy level by the emission of an appropriate photon. This leads to large numbers of narrow spectral lines, which partially overlap and together form vibration–rotation bands. As such, gases tend to be transparent over most of the spectrum but may be almost opaque over the spectral range of a band. The absorption coefficient κ λ is defined as a measure of how strongly radiation is BOOKCOMP, Inc. — John Wiley & Sons / Page 616 / 2nd Proofs / Heat Transfer Handbook / Bejan 616 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 [616], (44) Lines: 1403 to 1417 ——— 0.98203pt PgVar ——— Long Page PgEnds: T E X [616], (44) 2300 2325 2350 2375 (cm ) Ϫ1 0 100 200 300 400 Ϫ (cm /bar) 1 Tpp= 300 K, = 1 bar, = 0 bar CO2 Figure 8.22 Absorption coefficient spectrum for the CO 2 4.3-µm band. absorbed or emitted along a path in a participating medium. Figure 8.22 shows the absorption coefficient of the important 4.3-µm vibration–rotation band of CO 2 (per partial pressure of CO 2 ), for small amounts of CO 2 contained in nitrogen, with a temperature of 300 K and a mixture pressure of 1 bar, generated from the HITRAN database (Rothman et al., 1998). The figure shows that the band consists of a large number of strong spectral lines, and a number of weak lines can also be observed. In reality, there are many more spectral lines than appear in the figure. However, at the relatively high total pressure of 1 bar, the lines strongly overlap, giving a relatively smooth appearance. Lowering the pressure would decrease line overlap, and more and more of the ≈ 12,500 lines contained in the HITRAN database for this band would become distinguishable. Similarly, with increasing temperature, lines become narrower (less overlap), and many additional “hot lines” must be considered, which are negligible at room temperature. The new HITEMP database (Rothman et al., 2000), which is designed for temperatures up to 1000 K, includes ≈ 185,000 lines for the 4.3-µmCO 2 band alone! Fortunately, for many engineering problems, for simple heat transfer calculations, it is sufficient to determine the total emissivity for an isothermal, homogeneous path of length L, = 1 E b ∞ 0 (1 − e −κ λ L )E bλ (T g )dλ (8.76) For a mixture of gases the total emissivity is a function of path length L, gas tem- perature T g , partial pressure(s) of the absorbing gas(es) p a , and total pressure p. BOOKCOMP, Inc. — John Wiley & Sons / Page 617 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF PARTICIPATING MEDIA 617 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 [617], (45) Lines: 1417 to 1466 ——— * 28.55127pt PgVar ——— Long Page * PgEnds: PageBreak [617], (45) Especially important in combustion application, the total emissivity in mixtures of nitrogen with water vapor and/or carbon dioxide may be calculated from Leckner (1972). First, the individual emissivities for water vapor and carbon dioxide, respec- tively, are calculated separately from (p a L,p,T g ) = 0 (p a L,T g ) 0 (p a L,p,T g ) (8.77a) 0 (p a L,p,T g ) = 1 − (a −1)(1 − P E ) a +b − 1 + P e exp −c log 10 (p a L) m p a L 2 (8.77b) 0 (p a L,T g ) = exp M i=0 N j=0 c ji T g T 0 j log 10 p a L (p a L) 0 i (8.77c) Here 0 is the total emissivity at a reference state, which is p = 1 bar total pressure and p a → 0(butp a L>0). The correlation constants a, b, c, c ji ,P E ,(p a L) 0 ,(p a L) m , and T 0 are given in Table 8.4 for water vapor and carbon dioxide (for convenience, plots of 0 are given in Fig. 8.23 for CO 2 and Fig. 8.24 for H 2 O). The total emissivity of a mixture of nitrogen with both water vapor and carbon dioxide is calculated from CO 2 +H 2 O = CO 2 + H 2 O − ∆ (8.78) ∆ = ζ 10.7 + 101ζ − 0.0089ζ 10.4 log 10 (p H 2 O + p CO 2 )L (p a L) 0 2.76 (8.79) TABLE 8.4 Correlation Constants for the Determination of the Total Emissivity for Water Vapor and Carbon Dioxide H 2 OCO 2 M,N 2,2 2,3 c 00 ···c N0 −2.2118 −1.1987 −0.035596 −3.9893 −2.7669 −2.1081 −0.39163 . . . . . . . . . −0.85667 −0.93048 −0.14391 −1.2710 −1.1090 −1.0195 −0.21897 c 0M ···C NM −0.10838 −0.17156 −0.045915 −0.23678 −0.19731 −0.19544 −0.044644 P E (p + 2.56p a / √ t)/p 0 (p + 0.28p a )/p 0 (p a L) m /(p a L) 0 13.2t 2 0.054/t 2 ,t <0.7 0.225t 2 ,t>0.7 a 2.144,t<0.75 1 + 0.1/t 1.45 1.888 − 2.053 log 10 t,t > 0.75 b 1.10/t 1.4 0.23 c 0.5 1.47 T 0 = 1000 K,p 0 = 1 bar,t = T/T 0 ,(p a L) 0 = 1 bar · cm Source: Leckner (1972). BOOKCOMP, Inc. — John Wiley & Sons / Page 618 / 2nd Proofs / Heat Transfer Handbook / Bejan 618 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 [618], (46) Lines: 1466 to 1478 ——— 5.854pt PgVar ——— Normal Page * PgEnds: Eject [618], (46) Figure 8.23 Total emissivity of carbon dioxide at a total pressure of 1 bar and zero partial pressure. (From Leckner, 1972.) Figure 8.24 Total emissivity of water vapor at a total pressure of 1 bar and zero partial pressure. (From Leckner, 1972.) BOOKCOMP, Inc. — John Wiley & Sons / Page 619 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE PROPERTIES OF PARTICIPATING MEDIA 619 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 [619], (47) Lines: 1478 to 1521 ——— 0.88713pt PgVar ——— Normal Page PgEnds: T E X [619], (47) where ζ = p H 2 O p H 2 O + p CO 2 and where the ∆ compensates for overlap effects between H 2 O and CO 2 bands, and the CO 2 and H 2 O are calculated from eq. (8.77). If radiation emitted externally to the gas (e.g., by emission from an adjacent wall at temperature T w ) travels through the gas, the total amount absorbed by the gas is of interest. This leads to the absorptivity of a gas path at T g with a source at T w : α(p a L, p, T g ,T w ) = 1 E b (T w ) ∞ 0 [1 − e −κ λ (T ) g L ]E bλ (T w )dλ (8.80) which for water vapor or carbon dioxide may be estimated from α(p a L, p, T g ,T w ) = T g T w 1/2 p a L T w T g ,p,T w (8.81) where is the emissivity calculated from eq. (8.77) evaluated at the temperature of the surface, T w , and using an adjusted pressure path length, p a LT w /T g . For mixtures of water vapor and carbon dioxide, band overlap is again accounted for by taking α CO 2 +H 2 O = α CO 2 + α H 2 O − ∆ (8.82) with ∆ evaluated for a pressure path length of p a LT w /T g . 8.4.2 Particle Clouds Nearly all flames are visible to the human eye and are therefore called luminous (sending out light). Apparently, there is some radiative emission from within the flame at wavelengths where there are no vibration–rotation bands for any combustion gases. This luminous emission is today known to come from tiny char (almost pure carbon) particles, called soot, which are generated during the combustion process. The dirtier the flame, the higher the soot content and the more luminous the flame. Soot Soot particles are produced in fuel-rich flames, or fuel-rich parts of flames, as a result of incomplete combustion of hydrocarbon fuels. As shown by electron microscopy, soot particles are generally small and spherical, ranging in size between approximately 5 and 80 nm and up to about 300 nm in extreme cases. Although mostly spherical in shape, soot particles may also appear in agglomerated chunks and even as long agglomerated filaments. It has been determined experimentally in typical diffusion flames of hydrocarbon fuels that the volume percentage of soot generally lies in the range 10 −4 to 10 −6 %. Because soot particles are very small, they are generally at the same temperature as the flame and therefore strongly emit thermal radiation in a continuous spectrum over BOOKCOMP, Inc. — John Wiley & Sons / Page 620 / 2nd Proofs / Heat Transfer Handbook / Bejan 620 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 [620], (48) Lines: 1521 to 1552 ——— 4.96216pt PgVar ——— Normal Page * PgEnds: Eject [620], (48) the infrared region. Experiments have shown that soot emission often is considerably stronger than the emission from the combustion gases. For a simplified heat transfer analysis it is desirable to use suitably defined mean absorption coefficients and emissivities. If the soot volume fraction f v is known as well as an appropriate spectral average of the complex index of refraction of the soot, m = n − ık(ı = √ −1), one may approximate the spectral absorption coefficient from Felske and Tien (1977) as κ λ = C 0 f v λ C 0 = 36πnk (n 2 − k 2 + 2) 2 + 4n 2 k 2 (8.83) and a total or spectral-average value may be taken as κ m = 3.72f v C 0 T C 2 (8.84) where C 2 = 1.4388 cm · K is the second Planck function constant. Substituting eq. (8.84) into eq. (8.76) gives a total soot cloud emissivity of (f v TL) = 1 − e −κ m L = 1 − e −3.72C 0 f v TL/C 2 (8.85) Pulverized Coal and Fly Ash Dispersions To calculate the radiative proper- ties of arbitrary size distributions of coal and ash particles, one must have knowledge of their complex index of refraction as a function of wavelength and temperature. Data for carbon and different types of coal indicate that its real part, n, varies little over the infrared and is relatively insensitive to the type of coal (anthracite, lignite, bituminous), while the absorptive index, k, may vary strongly across the spectrum and from coal to coal. If the number and sizes of particles are known and if a suitable average value for thecomplex index of refraction can be found,thespectral absorption coefficient of the dispersion may be estimated by a correlation given by Buckius and Hwang (1980). They observed spectral behavior to be weak (similar to that of small soot particles) and that spectrally averaged properties do not depend appreciably on the optical properties of the coal. However, due to their larger size, coal particles tend to scatter radiation as well as absorb and emit radiation, leading to the definition of the scattering coefficient σ s and extinction coefficient β = κ + σ s . Interpolating the data of Buckius and Hwang, crude approximations for spectrally averaged absorption and extinction coefficients may be determined from κ m f A = 0.0032 1 + φ 425 1.8 −6/5 + 10.99 φ 0.02 −6/5 −5/6 (8.86) β m f A = 0.0032 1 + φ 650 2.0 −5/4 + 13.75 φ 0.13 −5/4 −4/5 (8.87) BOOKCOMP, Inc. — John Wiley & Sons / Page 621 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA 621 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 [621], (49) Lines: 1552 to 1605 ——— -3.40993pt PgVar ——— Normal Page PgEnds: T E X [621], (49) where f A is the total projected area of particles per unit volume (e.g., f A = πa 2 N for uniform spheres of radius a and a particle density of N particles/unit volume), and φ is a size parameter defined as φ =¯aT ¯a = 3f v 4f A a in µm, T in K (8.88) where ¯a is an average particle size. This leads to total coal cloud emissivity and absorptivity: α(φ) = (φ) = 1 − e −κ m L (8.89) On the other hand, if one is interested in transmitted radiation (i.e., radiation not absorbed or scattered away), the cloud transmissivity becomes τ(φ) = e −β m L (8.90) If both soot as well as larger particles are present in the dispersion, the absorption coefficients of all constituents must be added before applying eqs. (8.89) and (8.90). Mixtures of Molecular Gases and Particulates To determine the total emis- sivity of a mixture, it is generally necessary to find the spectral absorption coefficient κ λ of the mixture (the sum of the absorption coefficient of all contributors), followed by numerical integration of eqs. (8.89) and (8.90). However, because molecular gases tend to absorb only over a small part of the spectrum, to some degree of accuracy mix gas + particulates (8.91) Equation (8.91) gives an upper estimate because overlap effects result in lower emis- sivity [compare eq. (8.78) for gas mixtures]. 8.5 RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA To calculate the radiative heat transfer rates within—and to the bounding wall of—a participating medium, it is necessary to solve the radiative transfer equation (RTE), dI λ ds = ˆ s ·∇I λ = κ λ I bλ − β λ I λ + σ sλ 4π 4π I λ ( ˆ s i )Φ λ ( ˆ s i , ˆ s)dΩ i (8.92) to some degree of accuracy, followed by integration over all directions and all wave- lengths, to obtain the radiative heat flux desired. Here κ λ is the medium’s absorption coefficient, σ sλ its scattering coefficient, β λ = κ λ + σ sλ is known at the extinction coefficient, and Φ λ is the scattering phase function. As demonstrated in Fig. 8.25, this equation states that spectral radiative intensity I λ along a path s in the direction of ˆ s is augmented by emission along the path, BOOKCOMP, Inc. — John Wiley & Sons / Page 622 / 2nd Proofs / Heat Transfer Handbook / Bejan 622 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 [622], (50) Lines: 1605 to 1627 ——— 0.06602pt PgVar ——— Normal Page PgEnds: T E X [622], (50) Figure 8.25 Coordinates for the formal solution to the radiative transfer equation. diminished by extinction or absorption and outscattering (scattering of radiation away from ˆ s), and augmented by in-scattering (scattering from all other directions ˆ s i into direction ˆ s); κ λ gives a measure of how much radiation is absorbed and/or emitted, σ sλ gives a measure of how much is scattered, and Φ λ is the probability that radiation is scattered from direction ˆ s i into direction ˆ s. Finding a solution to eq. (8.92), which is an integrodifferential equation in five independent variables (three space coordinates and two direction coordinates), is a truly daunting task for all but the most trivial situations, even at the spectral level. Integration over all wavelengths, due to the complicated nature of radiative properties, tends to add another dimension to the level of difficulty. Consequently, the literature abounds with solutions to very simplistic scenarios as well as with approximate solution methods. A few of these simplified cases and methods are outlined in this section. In most engineering applications scattering can be neglected, and eq. (8.92) can be formally integrated along a straight path from s = 0 at a bounding wall to a point s = s inside the medium, to yield I λ (s) = I λ (0)e −κ λ s + s 0 I bλ (s )e −κ λ (s−s ) κ λ ds (8.93) where it was also assumed that κ λ is constant along the path. If the medium is isothermal along the path, eq. (8.93) can be reduced further to I λ (s) = I λ (0)e −κ λ s + I bλ (1 − e −κ λ s ) (8.94) or I λ (s) = I λ (0)τ λ (s) + I bλ λ (s) (8.95) . EXCHANGE WITHIN PARTICIPATING MEDIA To calculate the radiative heat transfer rates within—and to the bounding wall of—a participating medium, it is necessary to solve the radiative transfer equation. RADIATIVE PROPERTIES OF PARTICIPATING MEDIA In many high-temperature applications, when radiative heat transfer is important, the medium between surfaces is not transparent but is “participating”; that. tem- perature T g , partial pressure(s) of the absorbing gas(es) p a , and total pressure p. BOOKCOMP, Inc. — John Wiley & Sons / Page 617 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE