Evaluation of the geometric mean transmittance and total absorptance for two-dimensional systems

It has been found that the evaluation of the geometric mean transmittance and total absorptance between an infinitesimal area and an arbitrary finite area with an intervening absorbing and emitting medium is greatly simplified by making use of a mathematical identity which has never been recognized before. Solutions for three specific geometries were generated and identified as 'fundamental solutions'. While the obtained solutions are of great value for three-dimensional applications, they are difficult to use for systems with two-dimensional geometry. The present investigation has the objective to develop the corresponding two-dimensional fundamental solutions for the considered problem in radiative heat transfer. The application of the derived relations is illustrated with the aid of an example, involving the evaluation of the geometric mean beam length in the three limits for a circular cylinder of infinite length.