Chapter 8, Problem 8.109P

Consider the microchannel cooling arrangement ofProblem 8.107. However, instead of assuming theentire chip and cap to be at a uniform temperature.adopt a more conservative (and realistic) approach thatprescribes a temperature of $T_s=350$ K at the base ofthe channels ( $x=0$ ) and allows for a decrease in temperature with increasing x along the side walls of eachchannel.

(a) For the operating conditions prescribed in Problem 8.107 and a chip thermal conductivity of $k_{ch}=140\text{W/m}\cdot \text{K}$ , determine the water outlet temperature and the chip power dissipation. Heat transfer from the sides of the chip to the surroundings and from the side walls of a channel to the cap may be neglected. Note that the spacing between channels. $\delta =S-W$ , is twice the spacing between the side wall of an outer channel and the outer surface of the chip. The channel pitch is $\text{S }=\text{ L}/\text{N}$ , where $L=\text{ 1}0\text{ mm}$ is the chip width and $N=\text{ 5}0$ is the number of channels

(b) The channel geometry prescribed in Problem 8.107 and considered in part (a) is not optimized, and larger heat rates may be dissipated by adjusting related dimensions. Consider the effect of reducing the pitch to a value of $S=100\mu \text{m}$ . while retaining a width of $W=50\mu \text{m}$ and a flow rate per channel of $m_1={10}^{-4}$ kg/s.