A common procedure for cooling a high-performance computer chip involves joining the chip to a heat sink within which circular micro channels are machined. During operation, the chip produces a uniform heat flux q"c at its interface with the heat sink, while a liquid coolant (water) is routed through the channels. Consider a square chip and heat sink, each L x L on a side, with micro channels of diameter D and pitch S = C, D, where the constant C, is greater than unity. Water is supplied at an inlet temperature Tm,i and a total mass flow rate m (for the entire heat sink).

(a) Assuming that q"c is dispersed in the heat sink such that a uniform heat flux q is maintained at the surface of each channel obtain expressions for the longitudinal distributions of the mean fluid, Tm(x) , and surface, Ts(x) , temperatures in each channel. Assume laminar, fully developed flow throughout each channel and express your results in terms of ,m, q:, C 1 , D, and/or L, as well as appropriate thermo physical properties.

(b) For L = 12 mm, D = 1 mm, C1 = 2, q"c = 20 W/cm2, In = 0.010 kg/s ∙ and Tm,i; = 290 K, compute and plot the temperature distributions Tm(x) and Ts(x).

Chip Solder joint 0000000O O Microchannel Water pi, Tm. heat sink fs=C, D-