Explicit versus implicit method for radiative heat transfer in gray and diffuse enclosures

The paper presents an explicit formulation of the radiative problem in gray enclosures. The method is based on a series expansion that derives the mutual radiation factors by means of summation of the contributions to the radiative thermal exchange given by the multi-reflection process. A mathematical proof that the infinite expansion converges to the well known solution, given by the implicit formulation and based on radiosities balance, is provided. Moreover a parametric analysis of the approximation induced by the explicit algorithm, as a result of the truncation order in the calculation of the series expansion, is carried out. The analysis shows how the precision of every approximation order is mainly related to the surface averaged infrared reflectance and is weakly dependent on the reflectance distribution. For high emissive and low reflecting surfaces, very few summations are required to meet high precision; the same level of precision is achievable also with medium and high reflecting surfaces, by increasing the truncation order of the series. More generally, the explicit formulation provides an alternative approach to problems involving multi-reflecting cavities in different engineering applications, from heat transfer to optics.