Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 101
Текст из файла (страница 101)
Here κλ is the medium’s absorptioncoefficient, σsλ its scattering coefficient, βλ = κλ + σsλ is known at the extinctioncoefficient, and Φλ is the scattering phase function.As demonstrated in Fig. 8.25, this equation states that spectral radiative intensityIλ along a path s in the direction of ŝ is augmented by emission along the path,BOOKCOMP, Inc.
— John Wiley & Sons / Page 621 / 2nd Proofs / Heat Transfer Handbook / Bejan[621], (49)Lines: 1552 to 1605———-3.40993pt PgVar———Normal PagePgEnds: TEX[621], (49)622123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATION[622], (50)Figure 8.25 Coordinates for the formal solution to the radiative transfer equation.Lines: 1605 to 1627———diminished by extinction or absorption and outscattering (scattering of radiation awayfrom ŝ), and augmented by in-scattering (scattering from all other directions ŝi intodirection ŝ); κλ gives a measure of how much radiation is absorbed and/or emitted,σsλ gives a measure of how much is scattered, and Φλ is the probability that radiationis scattered from direction ŝi into direction ŝ.Finding a solution to eq.
(8.92), which is an integrodifferential equation in fiveindependent variables (three space coordinates and two direction coordinates), is atruly 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 literatureabounds with solutions to very simplistic scenarios as well as with approximatesolution methods. A few of these simplified cases and methods are outlined in thissection.In most engineering applications scattering can be neglected, and eq. (8.92) canbe formally integrated along a straight path from s = 0 at a bounding wall to a points = s inside the medium, to yield s−κλ s+Ibλ (s )e−κλ (s−s ) κλ ds (8.93)Iλ (s) = Iλ (0)e0where it was also assumed that κλ is constant along the path. If the medium isisothermal along the path, eq. (8.93) can be reduced further toIλ (s) = Iλ (0)e−κλ s + Ibλ (1 − e−κλ s )(8.94)Iλ (s) = Iλ (0)τλ (s) + Ibλ λ (s)(8.95)orBOOKCOMP, Inc.
— John Wiley & Sons / Page 622 / 2nd Proofs / Heat Transfer Handbook / Bejan0.06602pt PgVar———Normal PagePgEnds: TEX[622], (50)RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA123456789101112131415161718192021222324252627282930313233343536373839404142434445623where λ (s) is the spectral emissivity of a homogeneous column (sothermal, andwith constant concentrations of absorbing/emitting material) and τλ (s) is its spectraltransmissivity.
For a homogeneous medium, on a spectral basis,λ (s) = αλ (s) = 1 − τλ (s) = 1 − e−κλ s(8.96)where αλ (s) is the spectral absorptivity of the medium.8.5.1Mean Beam Length MethodRelatively accurate yet simple heat transfer calculations can be carried out if anisothermal, absorbing–emitting, but not scattering medium is contained in an isothermal, 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 localheat flux on a point of the surface may be calculated by putting eq. (8.94) into eq.(8.20), which leads toq = [1 − α(Lm )]Ebw − (Lm )Ebg(8.97)where Ebw and Ebg are blackbody emissive powers for the walls and medium (gasand/or particulates), respectively, and α(Lm ) and (Lm ) are the total absorptivity andemissivity of the medium for a path length Lm through the medium. The length Lm ,known as the mean beam length, is a directional average of the thickness of themedium 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 valueof the (spectral) mean beam length is known. It has been shown that spectral dependence 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 accurately from eq. (8.97), provided that the emissivity and absorptivity of the mediumare also known accurately. The mean beam lengths for many important geometrieshave been calculated and are collected in Table 8.5.
In this table L0 is known as thegeometric mean beam length, which is the mean beam length for the optically thinlimit (κλ → 0), and Lm is a spectral average of the mean beam length. For geometriesnot listed in Table 8.5, the mean beam length may be estimated fromL0 4VALm 0.9L0 3.6VA(8.98)where V is the volume of the participating medium and A is its entire boundingsurface area.8.5.2Diffusion ApproximationA medium through which a photon can travel only a short distance without beingabsorbed is known as optically thick. Mathematically, this implies that κλ L 1 for aBOOKCOMP, Inc. — John Wiley & Sons / Page 623 / 2nd Proofs / Heat Transfer Handbook / Bejan[623], (51)Lines: 1627 to 1702———6.31003pt PgVar———Normal PagePgEnds: TEX[623], (51)624123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONTABLE 8.5 Mean Beam Lengths for Radiation from a Gas Volume to a Surfaceon Its BoundaryGeometry ofGas VolumeCharacterizingDimension,LGeometricMeanBeamLength,L0 /LAverageMeanBeamLength,Lm /LLm /L0Sphere radiating to its surfaceDiameter,L=D0.670.650.97Infinite circular cylinder tobounding surfaceDiameter,L=D1.000.940.94Semi-infinite circular cylinder to:Diameter,L=DElement at center of baseEntire base[624], (52)1.000.810.900.650.900.80Circular cylinder(height/diameter = 1) to:Element at center of baseEntire surfaceDiameter,L=DCircular cylinder(height/diameter = 2) to:Plane baseConcave surfaceEntire surfaceDiameter,L=DCircular cylinder(height/diameter = 0.5) to:Plane baseConcave surfaceEntire surfaceDiameter,L=DInfinite semicircular cylinder tocenter of plane rectangularfaceRadius,L=RInfinite slab to its surfaceSlab thickness,L2.001.760.88Cube to a faceEdge, L0.670.60.90Rectangular 1 × 1 × 4parallelepipeds:To 1 × 4 faceTo 1 × 1 faceTo all facesShortest edge,L0.900.860.890.820.710.810.910.830.91BOOKCOMP, Inc.
— John Wiley & Sons / Page 624 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1702 to 1702———0.760.670.710.600.920.900.730.820.800.600.760.730.820.930.910.480.530.500.430.460.450.900.880.901.267.74605pt PgVar———Normal PagePgEnds: TEX[624], (52)RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA123456789101112131415161718192021222324252627282930313233343536373839404142434445625characteristic dimension L, across which the temperature does not vary substantially.For such an optically thick, nonscattering medium, the spectral radiative flux may becalculated fromqλ = −4∇Ebλ3κλ(8.99)similar to Fourier’s diffusion law for heat conduction. Note that a medium may beoptically thick at some wavelengths but thin (κλ L 1) at others (such as in moleculargases).
For a medium that is optically thick for all wavelengths, eq. (8.99) may beintegrated over the spectrum, yielding the total radiative fluxq=−4416σT 3∇Eb = −∇(σT 4 ) = −∇T3κR3κR3κR(8.100)[625], (53)where κR is a suitably averaged absorption coefficient, termed the Rosseland meanabsorption coefficient. For a cloud of soot particles, κR κm from eq. (8.76) is areasonable approximation. Equation (8.100) may be rewritten by defining a radiativeconductivity kR ,q = −kR ∇TkR =16σT 33κR(8.101)This form shows that the diffusion approximation is mathematically equivalent toconductive heat transfer with a (strongly) temperature-dependent conductivity.The diffusion approximation can be expected to give accurate results for gasparticulate suspensions with substantial amounts of particulates (such as for verysooty flames and in fluidized beds), and for semitransparent solids and liquids (such asglass or ice/water at low to moderate temperatures, that is, where most of the emissivepower lies in the infrared, λ > 2.5 µm, and where these materials exhibit largeabsorption coefficients).
The method is not suitable for pure molecular gases (suchas non- or mildly luminescent flames), because molecular gases are always opticallythin across much of the spectrum.Indeed, more accurate calculations show that in the absence of other modes of heattransfer (conduction, convection), there is generally a temperature discontinuity nearthe boundaries (T surface = T adjacent medium), and unless boundary conditions that allowsuch temperature discontinuities are chosen, the diffusion approximation will do verypoorly in the vicinity of bounding surfaces.8.5.3P-1 ApproximationFor the vast majority of engineering applications, very accurate (spectral) values forradiative fluxes q (and internal radiative sources ∇· q) can be obtained using the P -1approximation, also known as the spherical harmonics method and differential approximation.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
