A fundamental assessment of the concept of mean beam length for application for multi-dimensional non-gray radiative heat transfer

The traditional concept of mean beam length (MBL) and its recommended empirical expression is demonstrated to be inaccurate for general application for the evaluation of multi-dimensional non-gray radiative heat transfer. A new concept, namely Point Mean Beam Length (PMBL), is proposed and the formulation of PMBL is provided. The mathematical properties of PMBL for three common geometries including sphere, cylinder, and slab are presented. Results show that PMBL is more effective in generating an accurate evaluation of radiative heat transfer from a differential area to any finite area at the boundary of an enclosure with an isothermal absorbing/ emitting medium. The deficiency of the traditional mean beam length empirical expression is illustrated. A concept of "optimal" point mean beam length (OPMBL) is demonstrated to be a more accurate length scale for practical applications. In contrast to the traditional MBL, a single value of OPMBL is applicable for all gas absorption bands, independent of the strength and shape of the absorption bands. The proposed work provides a mathematically validated approach to efficiently and accurately evaluate the radiation heat transfer within an isothermal, non-gray, multi-dimensional medium.