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 modelsto 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 systemssuch 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 10cm^2 . 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.
A new global analytical model of the heat dissipation process that occurs in passively-cooled embedded systems is introduced, and we explicit under what circumstances the traditional assumption that exponential cooling laws apply in such context is valid. Since the power consumption and reliability of microprocessors are highly dependent on temperature, management units need accurate thermal models. Exponential cooling models are justified for actively-cooled systems . Here, we analyze the tractability of the cooling law for a passively cooled body, subject to radiative and convective cooling, including internal heat generation . Focusing then on embedded system-like objects , we compare the performance difference between our new passive cooling law and the conventionally-used exponential one. We show that , for quasi isothermal cooling surfaces of the order of 1\,dm^2 or greater, the radiative cooling effect may become comparable to the convective cooling one. In other words, radiation becomes non-negligible for systems with a cooling surface larger than about 1\,dm^2. Otherwise for surfaces below 1\,dm^2, we show that the differences between the exact solution and the exponential cooling law becomes negligible. In the absence of accurate temperature measurements, an exponential cooling model is shown to be accurate enough for systems, such as small-sized SoCs, that require low processing overhead.
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