Chapter 3, Problem 3.150P

As a means of enhancing heat transfer from high-performance logic chips, it is common to attach a heat sink to the chip surface in order to increase the surface area available for convection heat transfer. Because of the ease with which it may be manufactured (by taking orthogonal sawcuts in a block of material), an attractive option is to use a heat sink consisting of an array of square fins of width w on a side. The spacing between adjoining fins would be determined by the width of the sawblade, with the sum of this spacing and the fin width designated as the fin pitch S. The method by which the heat sink is joined to the chip would determine the interfacial contact resistance, $R_{t,c}^{"}.$

Consider a square chip of width $W_c=16\text{mm}$ and conditions for which cooling is provided by a dielectric liquid with $T_{\infty }=25°\text{C}$ and $h=1500{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ The heat sink is fabricated from copper $\left(k=400\text{W/m}\cdot \text{K}\right),$ and its characteristic dimensions are $w=0.25mm,S=0.50mm,L_f=6\text{mm,}$ and $L_b=3\text{mm}\text{.}$ The prescribed values of w and S represent minima imposed by manufacturing constraints and the need to maintain adequate flow in the passages between fins.

1. If a metallurgical joint provides a contact resistance of $R_{t,c}^{"}=5\times {10}^{-6}{\text{m}}^{\text{2}}\cdot \text{K/W}$ and the maximum allowable chip temperature is $\text{85°C,}$ what is the maximum allowable chip power dissipation $q_c?$Assume all of the heat to be transferred through the heat sink.
2. It may be possible to increase the heat dissipation by increasing w, subject to the constraint that $\left(S-w\right)\ge 0.25mm,$ and/or increasing $L_f$ (subject to manufacturing constraints that $L_f\ge 10\text{mm)}\text{.}$ Assess the effect of such changes.