Chapter 8, Problem 8.82P

A cold plate is an active cooling device that is attached to a heat-generating system in order to dissipate the heat while maintaining the system at an acceptable temperature. It is typically fabricated from a material of high thermal conductivity, $K_{cp}$ within which channels are machined and a coolant is passed. Consider a copper cold plate of height H and width W on a side, within which water passes through square channels of width w = h. The transverse spacing between channels $\delta$ is twice the spacing between the sidewall of au outer channel and the sidewall of the cold plate.

Consider conditions for which equivalent heat-generatingsystems are attached to the top and bottom of the coldplate, maintaining the corresponding surfaces at the sametemperature $T_s$ . The mean velocity and inlet temperatureof the coolant are $u_m$ and $T_{m,i}$ , respectively.

(a) Assuming fully developed turbulent flow throughout each channel, obtain a system of equations that maybe used to evaluate the total rate of heat transfer tothe cold plate. q. and the outlet temperature of thewater, $T_{m,o}$ , in terms of the specified parameters.

(b) Consider a cold plate of width W = 100 mm andheight H = 10 mm, with 10 square channels of width $\delta =4\text{mm}$ and a spacing of ô 4 nun betweenchannels. Water enters the channels at a temperature of $T_{m,i}=300\text{K}$ a velocity of $u_m=2\text{m/s}$ .If the top and bottom cold plate surfaces are at $T_s=360\text{K}$ , what is the outlet water temperature andthe total rate of heat transfer to the cold plate? Thethermal conductivity of the copper is $400\text{W/m}\cdot \text{K}$ ,while average properties of the water may be taken tobe $\rho =984\text{Kg}/{\text{m}}^3,c_p=4184\text{J/Kg}\cdot \text{K},\mu =489\times {10}^{-6}\text{N}\cdot {\text{s/m}}^2,k=0.65\text{W/m}\cdot \text{K}$ K, and Pr = 3.15. Isthis a good cold plate design? I-low could its performance be improved?