Problem 3. Mars's atmosphere is primarily made of CO₂ (more than 95%), and has a pressure of only 6300 Pa. Dust particles on Mars surface are typically about 3µm in average diameter, but can be as large as about 20 µμm. A volumetrically-heated metallic block that is 15 cm wide, 15 cm tall, and 10 mm in thickness is covered all around by insulating layers, each 4 mm in thickness. The block is to be cooled by Martian atmospheric gas through parallel microchannels. The gas flow is caused by a fan which causes the pressure at inlet to the microchannels to be at most 100 Pa larger than the pressure at the exhaust end of the channels. The channels exhaust into a chamber with Martian atmospheric pressure. The velocity and temperature at inlet are 100 m/s and 150 K, respectively. Based on design considerations the porosity of the plate is not to exceed 25%. The plate is made of a high thermal conductivity material, and its temperature should not exceed 200 K anywhere. Furthermore, Martian atmospheric gas may contain entrained regolith particles and therefore the flow passages should not be smaller than 50 um in diameter. By considering uniform-size parallel cylindrical micorchannels with diameters in the 50 µm to 4 mm, calculate the maximum thermal load that can be disposed by the cooling gas. Based on these calculations, determine the optimum coolant channel diameter. For simplicity neglect heat loss through the insulation layers, and wherever entrance-dominated predictive methods are not available you may use thermally developed correlations. Also, note that the flow passages must pass through the insulation layers as well.

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