Heat transfer through fractal-like porous media: A tessellated continuum approach

Abstract

This paper tests the hypothesis that the analysis of heat transfer through a cellular structure represented by a fractal or a pre-fractal can be achieved through analysis on a tessellated continuum. A transport theory for fractals is introduced, which is coupled to a hole-fill mapping strategy, to facilitate the analysis of transport problems on a tessellated continuum. The hole-fill maps are constructed by means of an iterated function scheme similar to that applied in the fractal generation process. The method enables known analytical and numerical continuum solutions to be immediately transferred to the fractal medium. A feature of the approach is that complex fractal structures result in continuum transport equations with material properties that are inhomogeneous, anisotropic and discontinuous.It is demonstrated that a measurable temperature is possible on fractal structures, along with finite measures of heat flux and energy. Analytical steady-state thermal solutions are presented incorporating convective heat transfer.