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Open Access Publications from the University of California

## Modeling and Optimization of Spatially Evolving Heat Sinks Using Volume Averaging Theory

Abstract

Today’s society has developed an ever-increasing dependence on electronic components, making it a critical challenge to match the rising demand for size reduction and performance reliability with efficient cooling strategies. Heat sinks are still the most common form of heat rejecting devices used in electronic cooling, and it has been shown in recent years that heat sinks with variable geometry can lead to efficiency improvement. The geometrical complexity and multi-scale nature of heat sinks make their modeling a challenging, and often time consuming, task. Volume Averaging Theory (VAT) has been shown to be a valid alternative to standard modeling techniques because of its ability to obtain accurate predictions of system performance parameters with a significant reduction in computational time. In this work, the theoretical fundamentals of VAT are examined in detail. Its mathematical fundamentals are discussed, and analogies with other averaging procedures are presented to illustrate the bases of the averaging process. The VAT transport equations are then derived and closed. The developed VAT model is applied to heat sinks with non-uniform geometry. Two numerical solution methods are applied to efficiently solve the resulting set of partial differential equations: a Galerkin method and a fractional step finite difference method. The fractional step method, based on Strang splitting, is used to address the coupling between the VAT solid and fluid energy conservation equations. In addition, a variable time-step approach is derived to accelerate the convergence to steady state. A new solution method, based on a spectral decomposition of the interface temperature and a quasi-Newton iteration method, is also proposed to address the coupling between the homogeneous base of the heat sink and the geometrically homogenized heat sink channel. Overall, the solution method provides a significant improvement in computational time over previously used methods. To determine the limits of applicability of the VAT model for systems with non-uniform geometry, a scaling procedure is applied to the governing equations. Through physical and mathematical arguments, it is determined that the momentum equation limits the applicability of the model, and it is shown that three non-dimensional parameters, M1, M2 and M3, can be used to provide estimates of these limits. For heat sinks with constant geometry, it is found that the solution is accurate when the boundaries of the system do not significantly affect the solution in the bulk, and the parameter M1 provides a quantitative estimate of these effects. For heat sinks with geometry variations in the cross-flow direction, it is determined that the accuracy of the solution is determined by the magnitude of the gradients induced by porosity variation, which are quantified through a parameter M2. Finally, for the case in which the geometry changes in the stream-wise direction, the VAT model is observed to be accurate when porosity variations do not affect local flow. This is quantified by a third parameter M3, which it is found to be Reynolds number dependent. In all three cases, it was shown that for low values of these parameters the VAT model is very accurate for a wide range of porosities, Reynolds numbers, geometries, and material combinations. The vast improvement in computational speed, along with the defined limits, is exploited to carry a series of optimization studies to determine the effects of the added geometric degrees of freedom of the system on its performance. A Genetic Algorithm is employed to determine optimal solutions for entropy generation and thermal resistance for three types of micro-channel heat sink geometries: straight, trapezoidal, and converging (or diverging). It is found that although straight channels provide an optimal combination of pumping power and thermal resistance, the limited geometric degrees of freedom do not allow for efficient heat transfer improvement. It is determined that straight channels present no efficient means to improve heat transfer and, in order to reduce the thermal resistance of a straight channel heat sink by 20%, a 200% increase in pumping power is required. It is also concluded that trapezoidal channels do not provide significant advantages over straight channels for either entropy generation or thermal resistance. On the contrary, an optimal converging channel configuration resulted in a 6% improvement in thermal resistance and a 23% decrease in pumping power, with respect to the thermally optimized straight channel. The results of the optimization studies are then combined to manually design a trapezoidal converging heat sink that features the same thermal performance of an optimal straight micro channel, but a 44% reduction in pumping power. Therefore, it is concluded that the added geometric degrees of freedom allow for a more efficient heat transfer improvement of the system.

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