Heat Transfer Handbook part 64 ppt

BOOKCOMP, Inc. — John Wiley & Sons / Page 623 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA 623 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 [623], (51) Lines: 1627 to 1702 ——— 6.31003pt PgVar ——— Normal Page PgEnds: T E X [623], (51) where  λ (s) is the spectral emissivity of a homogeneous column (sothermal, and with constant concentrations of absorbing/emitting material) and τ λ (s) is its spectral transmissivity. For a homogeneous medium, on a spectral basis,  λ (s) = α λ (s) = 1 − τ λ (s) = 1 − e −κ λ s (8.96) where α λ (s) is the spectral absorptivity of the medium. 8.5.1 Mean Beam Length Method Relatively accurate yet simple heat transfer calculations can be carried out if an isothermal, absorbing–emitting, but not scattering medium is contained in an isother- mal, black-walled enclosure. While these conditions are, of course, very restrictive, they are met to some degree by conditions inside furnaces. For such cases the local heat flux on a point of the surface may be calculated by putting eq. (8.94) into eq. (8.20), which leads to q = [1 − α(L m )]E bw − (L m )E bg (8.97) where E bw and E bg are blackbody emissive powers for the walls and medium (gas and/or particulates), respectively, and α(L m ) and (L m ) are the total absorptivity and emissivity of the medium for a path length L m through the medium. The length L m , known as the mean beam length, is a directional average of the thickness of the medium as seen from the point on the surface. On a spectral basis, equation (8.97) is exact, provided that the foregoing conditions are met and that an accurate value of the (spectral) mean beam length is known. It has been shown that spectral depen- dence of the mean beam length is weak (generally less than ±5% from the mean). Consequently, total radiative heat flux at the surface may be calculated very accu- rately from eq. (8.97), provided that the emissivity and absorptivity of the medium are also known accurately. The mean beam lengths for many important geometries have been calculated and are collected in Table 8.5. In this table L 0 is known as the geometric mean beam length, which is the mean beam length for the optically thin limit (κ λ → 0), and L m is a spectral average of the mean beam length. For geometries not listed in Table 8.5, the mean beam length may be estimated from L 0  4 V A L m  0.9L 0  3.6 V A (8.98) where V is the volume of the participating medium and A is its entire bounding surface area. 8.5.2 Diffusion Approximation A medium through which a photon can travel only a short distance without being absorbed is known as optically thick. Mathematically, this implies that κ λ L  1 for a BOOKCOMP, Inc. — John Wiley & Sons / Page 624 / 2nd Proofs / Heat Transfer Handbook / Bejan 624 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 [624], (52) Lines: 1702 to 1702 ——— 7.74605pt PgVar ——— Normal Page PgEnds: T E X [624], (52) TABLE 8.5 Mean Beam Lengths for Radiation from a Gas Volume to a Surface on Its Boundary Geometric Average Mean Mean Characterizing Beam Beam Geometry of Dimension, Length, Length, Gas Volume LL 0 /L L m /L L m /L 0 Sphere radiating to its surface Diameter, L = D 0.67 0.65 0.97 Infinite circular cylinder to bounding surface Diameter, L = D 1.00 0.94 0.94 Semi-infinite circular cylinder to: Diameter, L = D Element at center of base 1.00 0.90 0.90 Entire base 0.81 0.65 0.80 Circular cylinder (height/diameter = 1) to: Diameter, L = D Element at center of base 0.76 0.71 0.92 Entire surface 0.67 0.60 0.90 Circular cylinder (height/diameter = 2) to: Diameter, L = D Plane base 0.73 0.60 0.82 Concave surface 0.82 0.76 0.93 Entire surface 0.80 0.73 0.91 Circular cylinder (height/diameter = 0.5) to: Diameter, L = D Plane base 0.48 0.43 0.90 Concave surface 0.53 0.46 0.88 Entire surface 0.50 0.45 0.90 Infinite semicircular cylinder to center of plane rectangular face Radius, L = R 1.26 Infinite slab to its surface Slab thickness, L 2.00 1.76 0.88 Cube to a face Edge, L 0.67 0.6 0.90 Rectangular 1 × 1 × 4 parallelepipeds: Shortest edge, L To 1 × 4 face 0.90 0.82 0.91 To 1 × 1 face 0.86 0.71 0.83 To all faces 0.89 0.81 0.91 BOOKCOMP, Inc. — John Wiley & Sons / Page 625 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA 625 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 [625], (53) Lines: 1702 to 1732 ——— 8.72815pt PgVar ——— Normal Page * PgEnds: PageBreak [625], (53) characteristic dimension L, across which the temperature does not vary substantially. For such an optically thick, nonscattering medium, the spectral radiative flux may be calculated from q λ =− 4 3κ λ ∇E bλ (8.99) similar to Fourier’s diffusion law for heat conduction. Note that a medium may be optically thick at some wavelengths but thin (κ λ L  1) at others (such as inmolecular gases). For a medium that is optically thick for all wavelengths, eq. (8.99) may be integrated over the spectrum, yielding the total radiative flux q =− 4 3κ R ∇E b =− 4 3κ R ∇(σT 4 ) =− 16σT 3 3κ R ∇T (8.100) where κ R is a suitably averaged absorption coefficient, termed the Rosseland mean absorption coefficient. For a cloud of soot particles, κ R  κ m from eq. (8.76) is a reasonable approximation. Equation (8.100) may be rewritten by defining a radiative conductivity k R , q =−k R ∇Tk R = 16σT 3 3κ R (8.101) This form shows that the diffusion approximation is mathematically equivalent to conductive heat transfer with a (strongly) temperature-dependent conductivity. The diffusion approximation can be expected to give accurate results for gas- particulate suspensions with substantial amounts of particulates (such as for very sooty flames and in fluidized beds), and for semitransparent solids and liquids (such as glass or ice/water at low to moderate temperatures, that is, where most of the emissive power lies in the infrared, λ > 2.5 µm, and where these materials exhibit large absorption coefficients). The method is not suitable for pure molecular gases (such as non- or mildly luminescent flames), because molecular gases are always optically thin across much of the spectrum. Indeed, more accurate calculations show that in the absence of other modes of heat transfer (conduction, convection), there is generally a temperature discontinuity near the boundaries (T surface = T adjacent medium ), and unless boundary conditions that allow such temperature discontinuities are chosen, the diffusion approximation will do very poorly in the vicinity of bounding surfaces. 8.5.3 P-1 Approximation For the vast majority of engineering applications, very accurate (spectral) values for radiative fluxes q (and internal radiative sources ∇·q) can be obtained using the P -1 approximation, also known as the spherical harmonics method and differential ap- proximation. The method assumes that radiative intensity at any point varies smoothly BOOKCOMP, Inc. — John Wiley & Sons / Page 626 / 2nd Proofs / Heat Transfer Handbook / Bejan 626 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 [626], (54) Lines: 1732 to 1781 ——— 7.5251pt PgVar ——— Short Page * PgEnds: Eject [626], (54) with direction. Thus, it is particularly suited for optically thick situations (indeed, the diffusion approximation is simply an extreme limit of the P-1 approximation) and for situations in which radiation is emitted isotropically from a hot participating medium (as in combustion applications). Under the smooth intensity assumption, the radiative transfer equation (8.92) (lim- ited to isotropic scattering) can be integrated over all directions, leading to ∇·q = κ(4E b − G), (8.102) ∇G =−3βq (8.103) where the incident radiation G =  4π IdΩ is intensity integrated over all solid an- gles. These equations are subject to Marshak’s boundary conditions at the bounding walls: 2q w =  w 2 −  w (4E bw − G) (8.104) where  w is the emittance of the wall, q w the net flux going into the medium, and E bw is emissive power evaluated at the wall temperature (as opposed to the temperature of the medium next to the wall, which may be different in the absence of conduction and convection). Note that for optically thick situations (κ large), G → 4E b , and eq. (8.103) reduces to the diffusion approximation, eq. (8.100). For multidimensional calculations it tends to be advantageous to eliminate the vector q from eqs. (8.102)–(8.104), leading to an elliptic equation, which is readily incorporated into an overall heat transfer code: ∇·  1 β ∇G  − 3κG =−12κE b (8.105) subject to the boundary condition −  2  w − 1  2 3β ∂G ∂n + G = 4E bw (8.106) where ∂G/∂n is the spatial derivative of G, taken along the surface normal pointing into the medium. Equation (8.105) and its boundary condition, eq. (8.106), can be solved for suitable averaged values (across the spectrum) of the absorption and extinction coefficient, followed by the evaluation of wall fluxes from eq. (8.104) and/or the radiative source from eq. (8.102). Alternatively, these equations are evaluated on a spectral basis, followed by spectral integration, q =  ∞ 0 q λ dλ ∇·q =  ∞ 0 κ λ (4E bλ − G λ )dλ (8.107) BOOKCOMP, Inc. — John Wiley & Sons / Page 627 / 2nd Proofs / Heat Transfer Handbook / Bejan RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA 627 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 [627], (55) Lines: 1781 to 1802 ——— 7.34795pt PgVar ——— Short Page PgEnds: T E X [627], (55) 8.5.4 Other RTE Solution Methods The diffusion approximation and P -1 approximation are powerful, yet simple meth- ods that can give accurate solutions in many engineering applications (and are im- plemented in all important commercial heat transfer codes). However, they cannot be carried to levels of higher accuracy. This can be achieved by a number of finite volume methods, notably by the discrete ordinates method (Modest, 2003; Raithby and Chui, 1990; Chai et al., 1994; Fiveland and Jessee, 1994) and the zonal method (Modest, 1991; Hottel and Sarofim, 1967), and by statistical methods, called Monte Carlo methods (Modest, 2003). The discrete ordinate method is probably the most popular higher-order method today and is also implemented in most commercial heat transfer codes. In this method the RTE is solved for a set of discrete directions (or- dinates) spanning the total solid angle of 4π. The resulting first-order differential equations are solved along the various directions by breaking up the physical domain into a number of finite volumes. In the presence of nonblack walls and/or scatter- ing, because of the interdependence of different directions, the system of equations must be solved iteratively. Integrals over solid angle are approximated by numeri- cal quadrature (to evaluate the radiative flux and the radiative source). In the zonal method the enclosure is also divided into a finite number of isothermal volume and surface area zones. An energy balance is then performed for the radiative exchange between any two zones, employing precalculated “exchange areas” and “exchange volumes.” This process leads to a set of simultaneous equations for the unknown temperatures or heat fluxes. Once used widely, the popularity of the zonal method has waned recently, and it does not appear to have been implemented in any commer- cial solver. Monte Carlo or statistical methods are powerful tools to solve even the most challenging problems. However, they demand enormous amounts of computer time, and because of their statistical nature, they are difficult to incorporate into fi- nite volume/finite element heat transfer solvers and are best used for benchmarking (Modest, 2003). 8.5.5 Weighted Sum of Gray Gases The weighted sum of gray gases (WSGG) is a simple, yet accurate method that has become very popular to address the nongrayness of participating media, in particular for molecular gas mixtures. In this method the nongray gas is replaced by a number of gray ones, for which the heat transfer rates are calculated separately, based on weighted emissive power. The total heat flux and/or radiation source are then found by adding the contributions of the gray gases. The gray gases are determined by a curve fit from the total emissivity and absorptivity of a gas column, such as given by eqs. (8.77) and (8.81), (T g ,p a ,s)  K  k=0 a k (T g ,p a )(1 − e −κ k s ) (8.108a) BOOKCOMP, Inc. — John Wiley & Sons / Page 628 / 2nd Proofs / Heat Transfer Handbook / Bejan 628 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 [628], (56) Lines: 1802 to 1850 ——— 3.9239pt PgVar ——— Short Page PgEnds: T E X [628], (56) α(T g ,T w ,p a ,s)  K  k=0 a ∗ k (T g ,T w ,p a )(1 − e −κ k s ) (8.108b) For mathematical simplicity the gray gas absorption coefficients κ k are chosen to be constants, while the weight factors a k may be functions of temperature. Neither a k nor κ k are allowed to depend on path length s. Depending on the material, the quality of the fit, and the accuracy desired, a K value of 2 or 3 usually gives results of satisfactory accuracy (Hottel and Sarofim, 1967). Because, for an infinitely thick medium, the absorptivity approaches unity, K  k=0 a k (T ) = 1 (8.109) Still, for a molecular gas with its spectral windows, it would take very large path lengths indeed for the absorptivity to be close to unity. For this reason, eq. (8.108) starts with k = 0 (with an implied κ 0 = 0), to allow for spectral windows. Substituting this into eqs. (8.102) through (8.104) leads to ∇·q k = κ k (4a k E b − G k ) (8.110) ∇G k =−3κ k q k (8.111) and for the bounding walls, 2q w,k =  w 2 −  w  4a ∗ k E bw − G k  (8.112) Total wall flux and internal source are then found from q w = K  k=0 q w,k ∇·q = K  k=1 ∇·q k (8.113) Note that for κ 0 = 0 (spectral windows), the enclosure is without a participating medium, and q w can (and should) be evaluated from eq. (8.68), while ∇·q 0 = 0. Mathematically, the weighted-sum-of-gray-gases method is equivalent to the “step- wise gray” assumption, that is, asystemwhere the absorption coefficient is considered a step function in wavelength, with a gray value κ k over the fraction a k (based on emissive power) of the spectrum. Some weighted-sum-of-gray-gases absorptivity fits for important gases have been reported in the literature (Modest, 1991; Smith et al., 1982; Farag and Allam, 1981). Very recent work has shown that the weighted-sum- of-gray-gases method is a crude implementation of the also simple full-spectrum correlated k distribution method (FSCK) (Modest and Zhang, 2002), which produces almost exact results. BOOKCOMP, Inc. — John Wiley & Sons / Page 629 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 629 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 [629], (57) Lines: 1850 to 1917 ——— 5.5036pt PgVar ——— Short Page PgEnds: T E X [629], (57) 8.5.6 Other Spectral Models More sophisticated spectral modeling than the weighted-sum-of-gray-gases method is rarely warranted, in particular, because accurate values for the spectral radia- tive properties are seldom known: in particle-laden media spectral behavior depends strongly on particle material, shape, and size distributions, which are rarely known to a high degree of accuracy; in gases the temperature and pressure behavior is still not perfectly understood, although the new HITEMP database (Rothman et al., 2000) is nearing that goal. For more detailed descriptions of particle models, the reader should consult textbooks (Modest, 2003; Bohren and Huffman, 1983), while the state of the art in gas modeling is described in Goody and Yung (1989) and Taine and Soufiani (1999). NOMENCLATURE Roman Letter Symbols A matrix of view factors, dimensionless A area, m 2 absorptance of a slab, dimensionless a radius of sphere, m ¯a average particle size, m a k ,a ∗ k gray gas weight factors, dimensionless b column vector of heat fluxes, W/m 2 C 0 constant for particulate absorption coefficient, dimensionless C 1 ,C 2 ,C 3 radiation constants, dimensions vary c speed of light, m/s c 0 speed of light in vacuum, 2.998 × 10 8 m/s E emissive power, W/m 2 e b column vector of emissive powers, W/m 2 F i−j view factor, dimensionless f fractional Planck function, dimensionless f A projected area of particles per unit volume, m −1 f v soot volume fraction, dimensionless G incident radiation, W/m 2 H irradiation, W/m 2 H 0 irradiation from external source, W/m 2 h Planck’s constant, 6.626 × 10 −34 J · s I radiative intensity, W/m 2 · sr i index or counter, dimensionless ˆ ı unit vector, dimensionless J surface radiosity, W/m 2 ˆ j unit vector, dimensionless j index or counter, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 630 / 2nd Proofs / Heat Transfer Handbook / Bejan 630 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 [630], (58) Lines: 1917 to 1945 ——— 0.20847pt PgVar ——— Normal Page PgEnds: T E X [630], (58) ˆ k unit vector, dimensionless k Boltzmann’s constant, 1.3806 × 10 −23 J/K imaginary part of complex index of refraction, dimensionless k R radiative thermal conductivity, W/m · K L path length, m thickness of sheet, m L m spectrally averaged mean beam length, m L 0 geometric mean beam length, m l direction cosine in x-coordinate direction, dimensionless m complex index of refraction, dimensionless direction cosine in y-coordinate direction, dimensionless N number of surfaces in enclosure, dimensionless ˆ n normal unit vector, dimensionless n refractive index, dimensionless direction cosine in z-coordinate direction, dimensionless P point, dimensionless p pressure, Pa Q radiative heat flow, W q radiative heat flux vector, W/m 2 q radiative heat flux, W/m 2 R radiation resistance, m −2 reflectance of a slab, dimensionless radius, m r position vector, m S distance between points, m s distance vector, m ˆ s direction unit vector, dimensionless T temperature, K transmittance of a slab, dimensionless V volume of participating medium, m 3 x Cartesian length coordinate, m y Cartesian length coordinate, m z Cartesian length coordinate, m Greek Letter Symbols α absorptance, dimensionless β extinction coefficient, m −1 Γ contour, m  surface emittance, dimensionless gas column emissivity, dimensionless η wavenumber, cm −1 θ polar angle, rad κ absorption coefficient, m −1 κ R Rosseland mean absorption coefficient, m −1 λ wavelength, µm BOOKCOMP, Inc. — John Wiley & Sons / Page 631 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 631 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 [631], (59) Lines: 1945 to 1994 ——— 0.54597pt PgVar ——— Normal Page PgEnds: T E X [631], (59) ν frequency, Hz ρ reflectance, dimensionless σ Stefan–Boltzmann constant, 5.67 × 10 −8 W/m 2 · K 4 σ dc electrical conductivity Ω −1 · m −1 σ m rms roughness, m σ s scattering coefficient, m −1 τ transmittance, dimensionless transmissivity of gas column, dimensionless Φ scattering phase function, dimensionless φ size parameter, dimensionless ψ azimuthal angle, rad Ω solid angle, sr Roman Letter Subscripts A area b black g gas in incoming n normal direction out outgoing p associated with point P slab slab (multiple reflections) v volume w wall  parallel polarized component ⊥ perpendicular polarized component Greek Letter Subscripts λ spectral (wavelength) ν spectral (frequency) Superscripts  directional quantity (m) spectral range m, or band m REFERENCES American Institute of Physics (1972). American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York, Chap. 6. Bennett, H. E., Silver, M., and Ashley, E. J. (1963). Infrared Reflectance of Aluminum Evap- orated in Ultra-high Vacuum, J. Opt. Soc. Am., Vol. 53(9), 1089–1095. Bohren, C. F., and Huffman, D. R. (1983). Absorption and Scattering of Light by Small Particles, Wiley, New York. BOOKCOMP, Inc. — John Wiley & Sons / Page 632 / 2nd Proofs / Heat Transfer Handbook / Bejan 632 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 [632], (60) Lines: 1994 to 2034 ——— 6.0pt PgVar ——— Normal Page PgEnds: T E X [632], (60) Brandenberg, W. M. (1963). The Reflectivity of Solids at Grazing Angles, in Measurement of Thermal Radiation Properties of Solids, J. C. Richmond, NASA-SP-31, pp. 75–82. Brandenberg, W. M., and Clausen, O. W. (1965). 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Radiative Heat Transfer, 2nd ed., Academic Press, New York, 118–131. Oppenheim, A. K. (1956). Radiation Analysis by the Network Method, Trans. ASME, J. Heat Transfer, 78, 725–735. Parker, W. J., and Abbott, G. L. (1965). Theoretical and Experimental Studies of the Total Emittance of Metals, in Symposium on Thermal Radiation of Solids, S. Katzoff, ed., NASA- SP-55, pp. 11–28. Price, D. J. (1947). The Emissivity of Hot Metals in the Infrared, Proc. Phys. Soc., 59(331), 118–131. Raithby, G. D., and Chui, E. H. (1990). A Finite-Volume Method for Predicting Radiant Heat Transfer Enclosures with Participating Media, J. Heat Transfer, 112(2), 415–423. Rothman, L. S., Rinsland, C. P., Goldman, A., Massie, S. T., Edwards, D. P., Flaud, J. M., Perrin, A., Camy-Peyret, C., Dana, V., Mandin, J. Y., Schroeder, J., McCann, A., Gamache, R. R., Wattson, R. B., Yoshino, K., Chance, K. V., Jucks, K. W., Brown, L. R., Nemtchinov, V., and Varanasi, P. (1998). The HITRAN Molecular Spectroscopic Database and HAWKS (HITRAN Atmospheric Workstation): 1996 Edition, J. Quan. Spectrosc. Radiat. Transfer, 60, 665–710. . Applica- tion to Coal, Heat Transfer, 102, 99–103. Chai, J. C., Lee, H. S., and Patankar, S. V. (1994). Finite Volume Method for Radiation Heat Transfer, J. Thermophys. Heat Transfer, 8(3), 419–425. Dunkle,. Standard Emissivity, J. Heat Transfer, 103, 403–405. Felske, J. D., and Tien, C. L. (1977). The Use of the Milne–Eddington Absorption Coefficient for Radiative Heat Transfer in Combustion Systems, J. Heat Transfer, . Chui, E. H. (1990). A Finite-Volume Method for Predicting Radiant Heat Transfer Enclosures with Participating Media, J. Heat Transfer, 112(2), 415–423. Rothman, L. S., Rinsland, C. P., Goldman,