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
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whereλ(s) is the spectral emissivity of a homogeneous column (sothermal, and
with constant concentrations of absorbing/emitting material) andτλ(s) is its spectral
transmissivity Fora homogeneous medium, on a spectral basis,
λ(s) = αλ(s) = 1 − τλ(s) = 1 − e−κλs (8.96)
whereαλ(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 − (Lm )E bg (8.97)
whereE bw andE bgare blackbody emissive powers for the walls and medium (gas
and/or particulates), respectively, andα(Lm ) and (L m ) are the total absorptivity and
emissivity of the medium fora path lengthL mthrough the medium The lengthL 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 tableL0is known as the geometric mean beam length, which is the mean beam length for the optically thin limit (κλ → 0), and L mis a spectral average of the mean beam length For geometries not listed in Table 8.5, the mean beam length may be estimated from
L0 4V
A L m 0.9L0 3.6 V
whereV 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 thatκλL 1 fora
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TABLE 8.5 Mean Beam Lengths for Radiation from a Gas Volume to a Surface
on Its Boundary
Geometric Average
Characterizing Beam Beam Geometry of Dimension, Length, Length,
Sphere radiating to its surface Diameter,
L = D
Infinite circularcylinderto bounding surface
Diameter,
L = D
Semi-infinite circularcylinderto: Diameter,
L = D
Circular cylinder (height/diameter= 1) to:
Diameter,
L = D
Circular cylinder (height/diameter= 2) to:
Diameter,
L = D
Circular cylinder (height/diameter= 0.5) to:
Diameter,
L = D
Infinite semicircularcylinderto centerof plane rectangular face
Radius,
L = R
1.26
Infinite slab to its surface Slab thickness,
L
Rectangular1× 1 × 4 parallelepipeds:
Shortest edge,
L
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characteristic dimensionL, across which the temperature does not vary substantially.
For such an optically thick, nonscattering medium, the spectral radiative flux may be calculated from
qλ = − 4
similar to Fourier’s diffusion law for heat conduction Note that a medium may be optically thick at some wavelengths but thin (κλL 1) at others (such as in molecular
gases) Fora medium that is optically thick forall wavelengths, eq (8.99) may be integrated over the spectrum, yielding the total radiative flux
q= − 4
3κR∇E b= −
4
3κR∇(σT4) = −
16σT3
whereκR is a suitably averaged absorption coefficient, termed the Rosseland mean absorption coefficient Fora cloud of soot particles,κR κm from eq (8.76) is a reasonable approximation Equation (8.100) may be rewritten by defining a radiative conductivityk R,
q= −k R ∇T k R =16σT3
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 gas-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- ormildly luminescent flames), because moleculargases 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 (Tsurface= Tadjacent 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
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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
where the incident radiation G =4πI dΩ is intensity integrated over all solid
an-gles These equations are subject to Marshak’s boundary conditions at the bounding
walls:
2q w= w
wherewis the emittance of the wall,q wthe net flux going into the medium, andE 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 foroptically thick situations (κ large), G → 4E b, and eq
(8.103) reduces to the diffusion approximation, eq (8.100)
Formultidimensional 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
subject to the boundary condition
−
2
w − 1
2
3β
∂G
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
qλdλ ∇ · q =
∞ 0
κλ(4E bλ − Gλ) dλ (8.107)
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8.5.4 Other RTE Solution Methods
The diffusion approximation andP -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
popularhigher-ordermethod 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 numberof finite volumes In the presence of nonblack walls and/orscatter-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/fifi-nite 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),
(Tg ,p a ,s)
K
k=0
a k (T g ,p a )(1 − e−κk s ) (8.108a)
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α(Tg ,T w ,p a ,s)
K
k=0
a∗
k (T g ,T w ,p a )(1 − e−κk s ) (8.108b)
For mathematical simplicity the gray gas absorption coefficients κk are chosen to
be constants, while the weight factorsa k may be functions of temperature Neither
a k norκk are allowed to depend on path lengths Depending on the material, the
quality of the fit, and the accuracy desired, aK value of 2 or3 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, fora moleculargas with its spectral windows, it would take very large path lengths indeed forthe absorptivity to be close to unity Forthis reason, eq (8.108) starts withk = 0 (with an implied κ0 = 0), to allow forspectral windows
Substituting this into eqs (8.102) through (8.104) leads to
and forthe bounding walls,
2q w,k = w
2− w
4a∗
Total wall flux and internal source are then found from
q w= K
k=0
q w,k ∇ · q = K
k=1
Note that for κ0 = 0 (spectral windows), the enclosure is without a participating
medium, andq wcan (and should) be evaluated from eq (8.68), while∇ · q0= 0
Mathematically, the weighted-sum-of-gray-gases method is equivalent to the “step-wise gray” assumption, that is, a system where the absorption coefficient is considered
a step function in wavelength, with a gray valueκk overthe fractiona 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
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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
absorptance of a slab, dimensionless
¯a average particle size, m
a k , a∗ gray gas weight factors, dimensionless
b column vectorof heat fluxes, W/m2
C0 constant for particulate absorption coefficient, dimensionless
C1, C2, C3 radiation constants, dimensions vary
c0 speed of light in vacuum, 2.998× 108m/s
e b column vectorof emissive powers, W/m2
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
H0 irradiation from external source, W/m2
h Planck’s constant, 6.626 × 10−34J· s
I radiative intensity, W/m2· sr
i index orcounter, dimensionless
ˆı unit vector, dimensionless
ˆj unit vector, dimensionless
j index orcounter, dimensionless
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ˆk unit vector, dimensionless
k Boltzmann’s constant, 1.3806 × 10−23J/K
imaginary part of complex index of refraction, dimensionless
k R radiative thermal conductivity, W/m· K
thickness of sheet, m
L m spectrally averaged mean beam length, m
L0 geometric mean beam length, m
l direction cosine inx-coordinate direction, dimensionless
m complex index of refraction, dimensionless
direction cosine iny-coordinate direction, dimensionless
N number of surfaces in enclosure, dimensionless
ˆn normal unit vector, dimensionless
n refractive index, dimensionless
direction cosine inz-coordinate direction, dimensionless
q radiative heat flux vector, W/m2
R radiation resistance, m−2
reflectance of a slab, dimensionless radius, m
ˆs direction unit vector, dimensionless
transmittance of a slab, dimensionless
V volume of participating medium, m3
x Cartesian length coordinate, m
y Cartesian length coordinate, m
z Cartesian length coordinate, m
Greek Letter Symbols
β extinction coefficient, m−1
surface emittance, dimensionless
gas column emissivity, dimensionless
κ absorption coefficient, m−1
κR Rosseland mean absorption coefficient, m−1
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σ Stefan–Boltzmann constant, 5.67 × 10−8W/m2· K4
σdc electrical conductivityΩ−1· m−1
σs scattering coefficient, m−1
τ transmittance, dimensionless
transmissivity of gas column, dimensionless
Φ scattering phase function, dimensionless
φ size parameter, dimensionless
Roman Letter Subscripts
slab slab (multiple reflections)
parallel polarized component
⊥ perpendicular polarized component
Greek Letter Subscripts
Superscripts
directional quantity
(m) spectral rangem, orband 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.
Trang 101 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)
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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) The Directional Spectral Emittance of
Sur-faces between 200 and 600 C, in Symposium on Thermal Radiation of Solids, S Katzoff, NASA-SP-55, pp 313–319.
Buckius, R O., and Hwang, D C (1980) Radiation Properties for Polydispersions:
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, R V (1965) Emissivity and Inter-reflection Relationships for Infinite Parallel
Specu-larSurfaces, in Symposium on Thermal Radiation of Solids, S Katzoff, ed., NASA-SP-55,
pp 39–44
Dunkle, R V., and Gier, J T (1953) Snow Characteristics Project Progress Report, Technical report, University of California–Berkeley, June
Farag, I H., and Allam, T A (1981) Gray-Gas Approximation of Carbon Dioxide Standard
Emissivity, J Heat Transfer, 103, 403–405.
Felske, J D., and Tien, C L (1977) The Use of the Milne–Eddington Absorption Coefficient
forRadiative Heat Transferin Combustion Systems, J Heat Transfer, 99(3), 458–465.
Fiveland, W A., and Jessee, J P (1994) Finite Element Formulation of the Discrete-Ordinates
Method forMultidimensional Geometries, J Thermophys Heat Transfer, 8(3), 426–433.
Goody, R M., and Yung, Y L (1989) Atmospheric Radiation: Theoretical Basis, 2nd ed.,
Oxford University Press, New York
Hottel, H C., and Sarofim, A F (1967) Radiative Transfer, McGraw-Hill, New York.
Leckner, B (1972) Spectral and Total Emissivity of WaterVaporand Carbon Dioxide, Com-bust Flame, 19, 33–48.
Modest, M F (1991) The Weighted-Sum-of-Gray-Gases Model for Arbitrary Solution
Meth-ods in Radiative Transfer, J Heat Transfer, 113(3), 650–656.
Modest, M F., and Zhang, H (2002) The Full-Spectrum Correlated-k Distribution For
Ther-mal Radiation from Molecular Gass–Particulate Mixtures, J Heat Transfer, 124(1), 30–38.
Modest, M F (2003) 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 forPredicting 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