Simple analytic formula for the light-tube optical efficiency under overcast sky conditions

In a previous study we developed an analytic tool for express optical efficiency analysis of cylindrical light-tubes under various standardized sky conditions. In case of the clear sky luminance pattern dominated by direct sunlight, which was of main interest, the tool generally provides better efficiency predictions than some other analytic formulae used for this purpose. However, this trend is not so clearly fulfilled in the case of the standard overcast sky, because some approximations made for diffuse light were not accurate enough due to the effort to apply them on all the sky types. The case of overcast sky is often regarded by lighting engineers as the reference one. Therefore, this contribution offers a new improved and moreover simplified analytic formula for the pipe's optical efficiency calculation, the results of which are in good coincidence with precise ray-tracing numerical computations and also with some experiments. In general, they appear to be better than those calculated according to the transmission efficiency formula proposed by CIE.