Effect of Area Ratio on Thermal Spreading Resistance of a Cubic Heat Spreader

Abstract

A unique non-dimensional scheme has been employed to analytically determine the steady-state temperature field within a cubic heat spreader of unity aspect-ratio undergoing centralized, uniform heat flux with square footprint. This ‘square-on-square’ boundary value problem was solved using method of Fourier expansion. Plots of the heat spreader non-dimensional maximum temperature and non-dimensional thermal spreading resistance are provided for various Biot numbers, heat spreader thicknesses and a newly-defined ‘area ratio’ — the ratio of heater cross-sectional area to heat spreader cross-sectional area (i.e. footprint). The proposed solution is advantageous for determining optimal heat spreading configurations with low Biot numbers — typical of many electronics packaging applications and heat spreaders of very high effective thermal conductivity. The presented results indicate that the non-dimensional, maximum temperature increases as the area ratio decreases and that a limiting, non-dimensional thermal spreading resistance exists for relatively thick heat spreaders regardless of area ratio or Biot number. A critical, non-dimensional thickness was also found in which the non-dimensional, maximum temperature becomes near unity regardless of Biot number or area ratio.