A. Specular and imperfectly diffuse surfaces

The concept of a perfectly diffuse surface was an artifice introduced to simplify formal mathematical analysis of radiant transfer. The concept is widely used in engineering design and analysis for its convenience, and, surprisingly, the answers so obtained are found in many instances to be remarkably close to the answers found with more realistic analytical models. There are situations — for example, in transmission through long passages with specular side walls — where the assumption of perfectly diffuse reflection will lead to serious error, however. Thus the designer or analyst needs to be able to carry out calculations when one or more surfaces are not perfectly diffuse.

Perfectly diffuse reflection is where the bidirectional reflectance is a constant independent of all four angles, the two angles of incidence and the two of emergence. The antithesis of perfectly diffuse reflection is specular reflection, where the bidirectional reflectance is identically zero for all directions of emergence except the specular angle where it has an integrable singularity. Imperfectly diffuse is the term usually used to denote that the bidirectional reflectance is nonzero but not constant with angles of emergence. Mixed specular diffuse reflection occurs when there is a specular component, for example, from the smooth surface of the binder of a glossy enamel paint, and a (perfectly or imperfectly) diffuse component, for example, from the underlying particles of pigment of such a glossy enamel.

B. The mirror-image concept

The mirror-image concept was introduced formally into thermal radiation transfer analysis by Eckert, Sparrow, and co-workers (Eckert and Sparrow,1961; Sparrow et al., 1964). The concept is useful primarily when the enclosure in question contains only a few plane specular surfaces arranged so that the number of multiple specular reflections is either limited or forms an easily summed chain. The concept is based on the fact that a ray coming from an element of diffuse surface i and reflected by mirror m to an element of diffuse surface j can be regarded as an uninterrupted straight line from i to the mirror image of j. Thus the shape factor of Equation 206.5 can be applied between surface i and the mirror image of j in calculating the transfer between i and j via m. The image of j as seen in m is denoted j(m). The mirror-image shape factor is then written F_i–j(m).