Analytical heat diffusion models for heat sinks with circular micro-channels

This study explores the prediction of temperature distribution in a heat sink containing an array of circular micro-channels, which is found mostly in electronic cooling applications. The analytical heat diffusion models for most common micro-channel shapes are based on one-dimensional fin models with varying degrees of complexity. Because of a singularity in the governing one-dimensional heat diffusion equation for a fin with circular profile, no exact solution is possible for the circular heat sink geometry. In this paper, an alternative analytical power series solution technique is presented in which the differential equation is recast in polynomial form. Predictions of the power series solution are validated for different channel diameters and spacings and both one-sided and two-sided heating conditions using one-dimensional and two-dimensional numerical simulations. Overall, maximum percent differences in temperature and heat transfer rate between the analytical and two-dimensional numerical results of 0.23% and 1.33%, respectively, prove that the present analytical models are very accurate and effective tools for the design and thermal resistance prediction of micro-channel heat sinks found in electronic cooling applications.