The Impact of Surface Size on the Radiative Thermal Behavior of Embedded Systems

This paper introduces a new global analytical model of the heat dissipation process that occurs in passively-cooled embedded systems, and explicits under what circumstances the folklore assumption that exponential cooling laws apply in such context is valid. Since the power consumption and reliability of microprocessors are highly dependent on temperature, both designers and, later on, run-time temperature management units must be able rely upon accurate heating and cooling models to handle heat generation and peak temperature. If exponential cooling models are justified for actively-cooled microprocessors, e.g., by forced air or water cooling, for passively cooled processors however, as frequently found in embedded systems such as mobile phones, an exponential law may not be theoretically justified. Here, we analyzed the tractability of the exact cooling law for a passively cooled body, subject to radiative cooling and a modest level of heat loss via convection. Focusing then on embedded microprocessors, we compare the performance difference between our new passive cooling law and the conventionally-used exponential one. We show that the differences between the exact solution and the exponential cooling law are not significant, and even negligible, for small surfaces of the order 10 cm. However, for larger surface sizes, the radiative cooling component may become comparable to the convective cooling component. Our results thus suggest that, in the absence of accurate temperature measurements, an exponential cooling law is accurate enough for small-sized SoC systems that require low processing overhead.