New Geometric Theorems Derived from Integral Equations Applied to Radiative Transfer in Spherical Sectors and Circular Segments

Semicircles and circular sectors are both ubiquitous in the natural realm. However, mathematically speaking they have represented an enigma since antiquity. In recent years, the author has worked in integral equations with sections of spheres as related to radiative heat transfer and their associated form factors, to the point of defining new postulates. The main theorems thus far enunciated refer to the radiative exchange between circles and half disks, but recently the possibility to treat circular sectors has arrived, thanks to the research already conducted. As is known, to find the exact expression of the configuration factor by integration is complex. In the above mentioned problem of the circular sectors, the author reached the first two steps of the basic formulation for radiant exchange. Subsequently, the novelty of the procedure lies in introducing a finite differences approach for the third and fourth integrals which still remain unsolved, once we have been able to find the preliminary integrals. This possibility had not been identified by former research and the output provides us with an ample variety of unexpected scenarios. As a consequence, we are able to analyze with more precision the spatial transference of radiant heat for figures composed of circular sectors. We already know that spherical shapes cannot be discretized with any accuracy. Therefore, we would be able to reduce a considerable amount of hindrance in the progress of thermal radiation science. Important sequels will be derived for radiation in the entrance to tunnels, aircraft design and lighting as well.