Advanced method for the solution of multi-dimensional radiative heat transfer problems is developed. The basic idea of the method lies in construction of the special subdivision of the total solid angle surrounding an arbitrary point into a set of smaller solid angles (cells) and in representation of radiation intensity in each solid angle by P₁ -approximation. Similar approach was suggested for the first time thirty years ago in Ref. [1] as applied to neutron transport problem and then later independently in Refs. [2-3] for solution of radiant transport equation and has shown excellent results in solving different 1-D radiative heat transfer problems. The close approach has been used in Ref. [4] for mitigation of ray-effect in neutron transport problems. The essence of the proposed method can be formulated by the following way.

Let us divide the total solid angle surrounding an arbitrary point into M smaller solid angles in such a manner that where cells , are symmetric to each other relative to the center of a unit sphere, while the radiant intensity in each solid angle is given by an expression analogous to P₁ -approximation (for simplicity the grey radiation is considered):

(1)

where are the direction cosines and . Radiant transport equation is satisfied in the mean over each elementary cell and the system of the second order partial differential equations with respect to the local zeroth- or first moments of radiation intensity are derived in rectangular and cylindrical coordinates. Subdivision of a unit sphere [1] can be carried out by different ways. The simplest one consists in the arrangement of in such a manner that its boundaries coincide with the coordinate lines of spherical coordinate system. On the other hand, in our approach the elementary solid angles can depend on spatial coordinates, that is, can be varied from point to point. The latter is one of the main advantages of the proposed method because permits to take into account the specific features of the problem under consideration and thus to get required accuracy using quite coarse angular partitions. Note that both approaches [1,4] do not offer such possibility. It is shown that if the subdivision does not depend on spatial coordinates, the coefficients A_j^m are determined from the following system of equations

(2)

where , while R>_mn and H_il^nq are expressed through the zeroth and second moments of scattering phase function, while the zeroth moments I₀^m are connected with A_jⁿ in the following manner

The boundary conditions to Eq. (2) in the case of the diffuse reflection take the form

where

,

On the other hand, in the case of axisymmetric problems the spatially variable subdivisions have to be used. In particularly, introducing at the surface of a unit sphere the set of lines and , () , where is normalized radial coordinate and are the polar and azimuth angles of a spherical coordinate system, the following set of equations relative to I₀^rq are obtained (here for simplicity the scattering is absent)

(3)

where I_0,i^rq is normalized to s_rq and signs "+" and "-" correspond to different values of i. In this case the coefficients s_rq, p^rq and p_z^rq turn out to be dependent on radial coordinate .

As examples, the radiative transfer in a plane layer of absorbing and linear scattering media and radiative heat transfer in a circular cylinder with specularly reflective side surface have been considered. In latter case we have used the spatially dependent subdivision with respect to both angle and in order to take into account the phenomenon of the total inner reflection. This gave possibility at the first time to calculate the radiative heat transfer in the optically dense circular cylinder of finite length with transparent specular side surface. Some attention was given also to the application of the proposed method to the calculation of radiative transfer in a non-participating media.

REFERENCES

1. Lebedev V.I., P_NI- Equations, J. Numer. Math. and Math. Physics, Vol. 7, No. 4, pp 813-824, 1967 (in Russian).
2. Yuferev V.S., Generalization of Differential Approximation for Calculation of Radiative-Conductive Heat Transfer in a Rectangular Enclosure with Transparent Specularly Reflective boundaries, High Temperature, Vol. 33, pp 961-967, 1995 (in Russian).
3. Yuferev V.S. and Vasil'ev M.G., New Approach to Solution of Radiant Transport Equation, submitted to publication in JQSRT.
4. Briggs L.L., Miller W.F., and Lewis E.E., Ray-Effect Mitigation in Discrete Ordinate-Like Angular Finite Element Approximation in Neutron Transport, Nuclear Sci. and Eng., Vol. 57, pp. 205-217, 1975.