Three-dimensional radiative heat-transfer solutions by the discrete-ordinates method

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Radiative heat transfer in a three-dimensional participating medium was predicted using the discrete-ordinates method. The discrete-ordinates equations are formulated for an absorbing, anisotropically scattering, and re-emitting medium enclosed by gray walls. The solution strategy is discussed and the conditions for computational stability are presented. Several test enclosures are modeled. Results have been obtained for the S2, S4, S6, and S8 approximation s that correspond to 8, 24, 48, and 80 fluxes, respectively, and are compared with the exact-zone solution and the P3 differential approximation. Solutions are found for conditions that simulate absorbing media and isotropically and anisotropically scattering media. Solution accuracy and convergence are discussed for the various flux approximations. The S4, S6, and S8 solutions compare favorably with the other methods and can be used to predict radiant intensity, incident energy, and surface heat flux. A an bn B C E G / L n q r S x y z a /U,,£,TJ p a a © V Nomenclature = north-south areas, m2 = coefficients of a Legendre series = coefficients of a modified Legendre series = east-west areas, m2 = front-back areas, m2 = emissive power ( = aT4), W/m2 = incident energy, /4w/d6, W/m2 = radiant intensity, W/(m2 • Sr) = enclosure dimension, m = unit normal = heat flux, W/m2 = position vector, m = source term, W/m3 = volume of pth control volume, m3 = weight function in a direction - m (fractional area of a unit sphere) = coordinate, m = coordinate, m = coordinate, m — finite-difference weighting factor = extinction coefficient, a -f K,m~l = surface emittance = absorption coefficient, m1 = ordinates p = cos0, £ = sin0 sin , TJ = sinG cos = outgoing direction of radiation = phase function = surface reflectance = scattering coefficient, m1 = Boltzmann's constant, 5.669 X 1(T 8 W/(m2