Chapter 8, Problem 8.55P

A common procedure for cooling a high-performance computer chip involves joining the chip to a heat sink within which circular microchannels are machined. During operation, the chip produces a uniform heat flux $q_c^n$ at its interface with the heat sink, while a liquid coolant (water) ¡s routed through the channels. Consider a square chip and heat sink, each L on a side, with microchannels of diameter D and pitch $S=C_{1}D$ , where the constant $C_1$ is greater than unity. Water is supplied at an inlet temperature $T_{m,i}$ and a total mass flow rate in (for the entire heat sink).

(a) Assuming that q is dispersed in the heal sink such that a uniform heat flux q:’ is maintained at the surface of each channel, obtain expressions for the longitudinal distributions 01’ the mean fluid. $T_m(x)$ , and surface, $T_s(x)$ , temperatures in each channel. Assume laminar, fully developed flow throughout each channel, and express your results in terms of $\text{m},C_1,2,q_c^n$ ,D, and/or L, as well as appropriate thermophysical properties.

(b) For $L=12\text{mm},D=1\text{mm},C_1=2,q_c^n=20{\text{W/cm}}^2$ , $m=0.010kg/s$ , and $T_{m,i}=290\text{K}$ , compute and plot the temperature distributions $T_m=(x)$ and $T_s=(x)$ .

(c)A common objective in designing such heal sinks is to maximize $q_c^n$ while maintaining the heat sink at an acceptable temperature. Subject to prescribed values of $\text{L}=\text{12 mm}$ and $T_{m,i}=290\text{K}$ , and the constraint that $T_{s,\max }\le 50°\text{C}$ , explore the effect on $q_c^n$ of variations in heat sink design and operating conditions.