Tham khảo tài liệu 'heat transfer mathematical modelling numerical methods and information technology part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Radiative Heat Transfer and Effective Transport Coefficients 109 matter. In the sequel we will discuss a few practically relevant closure methods. We will then argue that the preferred closure is given by an entropy production principle. For clarity we will consider the two-moment example generalization to an arbitrary number of moments is straight-forward. The appropriate number of moments is influenced by the geometry and the optical density of the matter. For symmetric geometries like plane cylindrical or spherical symmetry less moments are needed than for complex arrangements with shadowing corners slits and the same. For optically dense matter the photons behave diffusive which can be modelled well by a low number of moments as will be discussed below. For transparent media beams or even several beams that might cross and interpenetrate may occur which makes higher order or multipole moments necessary. Two-moment example The unknowns are Pe Pf and n which may be functions of the two moments E and F. For convenience we will write Pe fM - E 17 Pf -4eff F 18 where we introduced the effective absorption coefficients Kgeff and Kpeff that are generally functions of E and F. Because the second rank tensor n depends only on the scalar E and the vector F by symmetry reason it can be written in the form n E c 1 X z 3x 1 FnFm A 19 nnm E I 2 Unm 2 p2 19 where the variable Eddington factor VEF X is a function of E and F and where ỏkl 0 if k l and ỗfci 1 if k l is the Kronecker delta. Assuming that the underlying matter is isotropic j 1 eff d b dfti f E d Ke Kp and X can ue expressed as functions oi E and v E 20 with F F . Obviously it holds 0 v 1 with v 1 corresponding to a fully directed radiation beam free streaming limit . According to Pomraning 1982 the additional E-dependence of suggested or derived VEFs often appears via an effective E-dependent single scattering albedo which equals . for gray matter KE eq aE k a E. The task of a closure is to determine the .
