Pseudo-hemispherical properties applied to an absorbing medium involving specular directional surfaces

Abstract

An analytical model is presented to treat the problem of radiative transfer in an absorbing-emitting medium bounded by specular surfaces with directionally dependent radiation properties. In this model, a set of a priori known pseudo-hemispherical properties is introduced to account for the directional radiation properties of the surfaces. It is shown that the equation of transfer for the case involving specular surfaces with directional surface properties can be written in terms of exponential integrals in a form which is similar, though not identical to the diffuse wall case. The specific example of radiative equilibrium is solved to obtain the exact solution as well as an approximate solution based on the present model. Comparisons are made among the exact solution, the present model and an existing approximation. LIST OF SYMBOLS a i pseudo-hemispherical properties of surface 1 b i pseudo-hemispherical properties of surface 2 E n nth exponential integral I intensity q R radiative flux T Temperature x optical thickness ε angular emittance θ angle from normal to surface θ 2 T 2/ T 1 μ cos θ ϱ angular reflectance τ optical thickness Φ constant defined in equation (4) φ normalized temperature 1 specular surface 1 2 specular surface 2