A cold plate is an active cooling device that is attached to a heat-generating system is order to dissipate the heat while maintaining the system at an acceptable temperature. It is typically fabricated from a material of high thermal conductivity, kcp, 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 δ is twice the spacing between the sidewall of an outer channel and the sidewall of the cold plate. Consider conditions for which equivalent heat-generating systems are attached to the top and bottom of the cold plate, maintaining the corresponding surfaces at the same temperature Ts. The mean velocity and inlet temperature of the coolant are um and Tm, I, respectively.
Assuming fully developed turbulent flow throughout each channel, obtain a system of equations that may be used to evaluate the total rate of heat transfer to the cold plate, q, and the outlet temperature of the water, Tm, o, in terms of the specified parameters.