Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 102
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The method assumes that radiative intensity at any point varies smoothlyBOOKCOMP, Inc. — John Wiley & Sons / Page 625 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1702 to 1732———8.72815pt PgVar———Normal Page* PgEnds: PageBreak[625], (53)626123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONwith direction. Thus, it is particularly suited for optically thick situations (indeed, thediffusion approximation is simply an extreme limit of the P-1 approximation) and forsituations 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) (limited to isotropic scattering) can be integrated over all directions, leading to∇ · q = κ(4Eb − G),∇G = −3βq(8.102)(8.103)where the incident radiation G = 4π I dΩ is intensity integrated over all solid angles.
These equations are subject to Marshak’s boundary conditions at the boundingwalls:2qw =w(4Ebw − G)2 − w[626], (54)(8.104)Lines: 1732 to 1781where w is the emittance of the wall, qw the net flux going into the medium, and Ebw———is emissive power evaluated at the wall temperature (as opposed to the temperature7.5251pt PgVar———of the medium next to the wall, which may be different in the absence of conductionShort Pageand convection). Note that for optically thick situations (κ large), G → 4Eb , and eq.(8.103) reduces to the diffusion approximation, eq. (8.100).* PgEnds: EjectFor multidimensional calculations it tends to be advantageous to eliminate thevector q from eqs.
(8.102)–(8.104), leading to an elliptic equation, which is readily[626], (54)incorporated into an overall heat transfer code:1∇·(8.105)∇G − 3κG = −12κEbβsubject to the boundary condition22 ∂G−−1+ G = 4Ebww3β ∂n(8.106)where ∂G/∂n is the spatial derivative of G, taken along the surface normal pointinginto the medium.Equation (8.105) and its boundary condition, eq. (8.106), can be solved for suitableaveraged 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 sourcefrom eq. (8.102). Alternatively, these equations are evaluated on a spectral basis,followed by spectral integration, ∞ ∞qλ dλ∇ ·q=κλ (4Ebλ − Gλ ) dλ(8.107)q=0BOOKCOMP, Inc. — John Wiley & Sons / Page 626 / 2nd Proofs / Heat Transfer Handbook / Bejan0RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA1234567891011121314151617181920212223242526272829303132333435363738394041424344458.5.4627Other RTE Solution MethodsThe diffusion approximation and P -1 approximation are powerful, yet simple methods that can give accurate solutions in many engineering applications (and are implemented in all important commercial heat transfer codes).
However, they cannotbe carried to levels of higher accuracy. This can be achieved by a number of finitevolume methods, notably by the discrete ordinates method (Modest, 2003; Raithbyand 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 MonteCarlo methods (Modest, 2003). The discrete ordinate method is probably the mostpopular higher-order method today and is also implemented in most commercial heattransfer codes.
In this method the RTE is solved for a set of discrete directions (ordinates) spanning the total solid angle of 4π. The resulting first-order differentialequations are solved along the various directions by breaking up the physical domaininto a number of finite volumes. In the presence of nonblack walls and/or scattering, because of the interdependence of different directions, the system of equationsmust be solved iteratively.
Integrals over solid angle are approximated by numerical quadrature (to evaluate the radiative flux and the radiative source). In the zonalmethod the enclosure is also divided into a finite number of isothermal volume andsurface area zones. An energy balance is then performed for the radiative exchangebetween any two zones, employing precalculated “exchange areas” and “exchangevolumes.” This process leads to a set of simultaneous equations for the unknowntemperatures or heat fluxes.
Once used widely, the popularity of the zonal methodhas waned recently, and it does not appear to have been implemented in any commercial solver. Monte Carlo or statistical methods are powerful tools to solve even themost challenging problems. However, they demand enormous amounts of computertime, and because of their statistical nature, they are difficult to incorporate into finite volume/finite element heat transfer solvers and are best used for benchmarking(Modest, 2003).8.5.5 Weighted Sum of Gray GasesThe weighted sum of gray gases (WSGG) is a simple, yet accurate method that hasbecome very popular to address the nongrayness of participating media, in particularfor molecular gas mixtures.
In this method the nongray gas is replaced by a numberof gray ones, for which the heat transfer rates are calculated separately, based onweighted emissive power. The total heat flux and/or radiation source are then foundby adding the contributions of the gray gases. The gray gases are determined by acurve fit from the total emissivity and absorptivity of a gas column, such as given byeqs.
(8.77) and (8.81),(Tg ,pa ,s) Kak (Tg ,pa )(1 − e−κk s )k=0BOOKCOMP, Inc. — John Wiley & Sons / Page 627 / 2nd Proofs / Heat Transfer Handbook / Bejan(8.108a)[627], (55)Lines: 1781 to 1802———7.34795pt PgVar———Short PagePgEnds: TEX[627], (55)628123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL RADIATIONα(Tg ,Tw ,pa ,s) Kak∗ (Tg ,Tw ,pa )(1 − e−κk s )(8.108b)k=0For mathematical simplicity the gray gas absorption coefficients κk are chosen tobe constants, while the weight factors ak may be functions of temperature.
Neitherak nor κk are allowed to depend on path length s. Depending on the material, thequality of the fit, and the accuracy desired, a K value of 2 or 3 usually gives resultsof satisfactory accuracy (Hottel and Sarofim, 1967). Because, for an infinitely thickmedium, the absorptivity approaches unity,Kak (T ) = 1(8.109)k=0[628], (56)Still, for a molecular gas with its spectral windows, it would take very large pathlengths indeed for the absorptivity to be close to unity.
For this reason, eq. (8.108)starts with k = 0 (with an implied κ0 = 0), to allow for spectral windows.Substituting this into eqs. (8.102) through (8.104) leads to∇ · qk = κk (4ak Eb − Gk )∇Gk = −3κk qk(8.110)(8.111)Lines: 1802 to 1850———3.9239pt PgVar———Short PagePgEnds: TEXand for the bounding walls,2qw,k =w ∗4ak Ebw − Gk2 − w[628], (56)(8.112)Total wall flux and internal source are then found fromqw =Kqw,kk=0∇ ·q=K∇ · qk(8.113)k=1Note that for κ0 = 0 (spectral windows), the enclosure is without a participatingmedium, and qw can (and should) be evaluated from eq.
(8.68), while ∇ · q0 = 0.Mathematically, the weighted-sum-of-gray-gases method is equivalent to the “stepwise gray” assumption, that is, a system where the absorption coefficient is considereda step function in wavelength, with a gray value κk over the fraction ak (based onemissive power) of the spectrum. Some weighted-sum-of-gray-gases absorptivity fitsfor 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-sumof-gray-gases method is a crude implementation of the also simple full-spectrumcorrelated k distribution method (FSCK) (Modest and Zhang, 2002), which producesalmost exact results.BOOKCOMP, Inc. — John Wiley & Sons / Page 628 / 2nd Proofs / Heat Transfer Handbook / BejanNOMENCLATURE1234567891011121314151617181920212223242526272829303132333435363738394041424344458.5.6629Other Spectral ModelsMore sophisticated spectral modeling than the weighted-sum-of-gray-gases methodis rarely warranted, in particular, because accurate values for the spectral radiative properties are seldom known: in particle-laden media spectral behavior dependsstrongly on particle material, shape, and size distributions, which are rarely known toa high degree of accuracy; in gases the temperature and pressure behavior is still notperfectly understood, although the new HITEMP database (Rothman et al., 2000)is nearing that goal.
For more detailed descriptions of particle models, the readershould consult textbooks (Modest, 2003; Bohren and Huffman, 1983), while the stateof the art in gas modeling is described in Goody and Yung (1989) and Taine andSoufiani (1999).[629], (57)NOMENCLATURERoman Letter SymbolsAmatrix of view factors, dimensionlessAarea, m2absorptance of a slab, dimensionlessaradius of sphere, māaverage particle size, mgray gas weight factors, dimensionlessak , ak∗bcolumn vector of heat fluxes, W/m2C0constant for particulate absorption coefficient, dimensionlessradiation constants, dimensions varyC1 , C2 , C3cspeed of light, m/sspeed of light in vacuum, 2.998 × 108 m/sc0Eemissive power, W/m2ebcolumn vector of emissive powers, W/m2Fi−jview factor, dimensionlessffractional Planck function, dimensionlessprojected area of particles per unit volume, m−1fAfvsoot volume fraction, dimensionlessGincident radiation, W/m2Hirradiation, W/m2H0irradiation from external source, W/m2hPlanck’s constant, 6.626 × 10−34 J · sIradiative intensity, W/m2 · sriindex or counter, dimensionlessı̂unit vector, dimensionlessJsurface radiosity, W/m2ĵunit vector, dimensionlessjindex or counter, dimensionlessBOOKCOMP, Inc.
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