Calculation of the radiant characteristics of a plane diffuse surface covered by a specular hemisphere

A mathematical formulation for calculating precisely the effective emissivity ( epsilon a ) of a plane diffuse surface covered by a specular hemisphere is presented. Exact expressions are developed for the exchange factors governing radiation exchange between points on the plane surface after (multiple) reflection from the hemisphere, with allowance for an aperture at the top and a gap between surface and hemisphere. The resulting integral equation for epsilon a is solved numerically. The results show that epsilon a varies non-uniformly along a radius as different numbers of specular reflections come into play. The top aperture only slightly decreases values of epsilon a below those for a complete hemisphere; the gap produces substantial and non-uniform changes. A prior closed-form expression for epsilon a is shown to be a good approximation within one-third of the radius from the centre. The effects of temperature gradients across the surface are shown to affect epsilon a considerably.