[Paper Review] Theoretical Analysis of Radiative Cooling for Mobile and Embedded Systems

Karel De Vogeleer, Gérard Memmi+2 more|arXiv (Cornell University)|Oct 1, 2014

Thermal Radiation and Cooling Technologies3 citations

TL;DR

This paper presents a theoretical analysis of radiative cooling in passively cooled mobile and embedded systems, showing that for cooling surfaces ≥1 dm², radiative cooling becomes non-negligible and deviates significantly from exponential cooling laws. It proposes a more accurate passive cooling model and demonstrates that exponential approximations remain valid only for small systems with low surface area and minimal thermal overhead.

ABSTRACT

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.

Motivation & Objective

• To develop an accurate analytical model for passive cooling in embedded systems, accounting for both radiative and convective heat transfer.
• To determine under what conditions the traditional exponential cooling law remains valid for passively cooled systems.
• To quantify the significance of radiative cooling relative to convective cooling in systems with varying surface areas.
• To evaluate the trade-off between model accuracy and computational overhead in thermal management for mobile and embedded systems.
• To assess the validity of exponential cooling approximations in the absence of precise temperature measurements.

Proposed method

• Derives a full analytical solution for transient heat dissipation in a passively cooled body with internal heat generation, considering both radiative and convective cooling.
• Uses the Stefan-Boltzmann law for radiative heat transfer and Newton’s law of cooling for convection, combined with energy balance equations.
• Applies binomial expansion to the fourth-power temperature term in radiative cooling to enable approximation techniques.
• Introduces first- and second-order O’Sullivan approximations to linearize the non-linear cooling equation for tractable solutions.
• Derives exact solutions for the quadratic approximation case using roots of a characteristic equation, with equilibrium temperature defined by the positive root.
• Compares the exact passive cooling model with the conventional exponential cooling law to assess accuracy across different system sizes.

Experimental results

Research questions

• RQ1 Under what conditions does radiative cooling become non-negligible in passively cooled embedded systems?
• RQ2 How does the size of the cooling surface affect the validity of exponential cooling approximations?
• RQ3 To what extent does the inclusion of radiative cooling alter the transient thermal behavior compared to standard exponential models?
• RQ4 In what scenarios is the exponential cooling approximation still accurate despite the presence of non-linear radiative effects?
• RQ5 How can thermal models be optimized for low-complexity embedded systems where precise temperature measurements are unavailable?

Key findings

• For cooling surfaces of 1 dm² or larger, radiative cooling becomes comparable to or even dominant over convective cooling, making it non-negligible.
• For systems with cooling surfaces below 1 dm², the difference between the exact passive cooling solution and the exponential approximation is negligible.
• The exponential cooling law remains a valid and accurate approximation for small-sized SoCs and systems requiring low processing overhead.
• The equilibrium temperature in the passive cooling model is determined by the positive root of a quadratic equation derived from the heat balance.
• The first-order O’Sullivan approximation yields an exponential decay solution, while the second-order approximation results in a more complex, non-exponential form that better captures non-linear behavior.
• The coefficients in the approximation model depend on system parameters such as surface area, emissivity, heat transfer coefficient, and internal power dissipation, and can be dynamically adjusted for better accuracy.