Average view factors for extended surfaces with fractal perforations

Abstract

Extended surfaces are often used for passive thermal management of electronic devices. By perforating extended surfaces in accordance with the Sierpinski carpet fractal pattern, an increase in surface area and a decrease in mass can be achieved. Intersurface thermal radiation, within the perforations, can account for a significant percentage of the total radiative heat transfer rate. As the perforations are of a non-uniform size, a correlation for the average fin view factor as a function of fractal iteration and width-to-thickness ratio was developed. For example, while a fin inspired by the fourth iteration of the Sierpinski carpet fractal pattern has 23.30% more surface area than a solid rectangular fin of equal width-to-thickness ratio the same fin only exchanges 67.37% of the radiation emitted with the surroundings due to intersurface thermal radiation. Regardless of width-to-thickness ratio, there was found to be a finite number of fractal iterations after which the average view factor of the extended surface approached zero. Similarly, the percentage of the total surface area that the perforations account for approaches an asymptote at approximately 100% by 10 iterations, regardless of width-to-thickness ratio. Finally, while there is an exponential increase in surface area with iteration, the increase in area did not offset the decrease in average view factor.