1. The addition of rectangular brass (k = 60 Wm-1 K-1 ) fins has been suggested as a way of increasing the rate of heat transfer from water to air through a 2.0-mm thick brass wall. The fins would be 0.8 mm thick, 2.50 cm long, and spaced 1.25 cm apart. The water-side and air-side convective heat transfer coefficients are 170 Wm-2 K-1 and 17 Wm-2 K-1 , respectively. Determine the increase in heat transfer rate achieved by adding fins to the water side or the air side. Where should the fins be added?
2. A cylindrical brass (k = 60 Wm-1 K-1 ) rod of 2.5 cm diameter is welded to the outside of a combustion furnace wall that is covered by a 20 cm-thick layer of insulation. The rod protrudes normally outside the insulation layer and is used as a hanger for instrument cables. The steadystate furnace wall temperature is 200°C, and the section of the rod outside the insulation layer is exposed to 25°C air with a convective heat transfer coefficient of 15 Wm-2 K-1. The portion of the rod inside the insulation layer may be considered perfectly insulated on its surface so that heat transfer within the insulated section of the rod can be considered to be one-dimensional.
a. To avoid damaging the instrument cables hanging on the rod, the temperature of the exposed section of the rod (outside the insulation layer) must be kept below 100°C everywhere. Does a 40 cm-long rod meet this specification?
b. If not, what design parameters (rod material, rod length, or insulation thickness) would you change? Justify your answer.
c. (Extra credit) If a tube with the same outside diameter and wall thickness of 3 mm were to be used instead of the rod, would it meet the specified operating limit? Conduction inside the air within the tube can be neglected.
3. When an integrated circuit (chip) is activated, it dissipates power through uniform volumetric heating. To increase the rate of heat removal from a square-shaped chip, a 4 X 4 array of copper (k = 400 Wm-1 K-1 ) pin fins is attached to the top surface of the chip. The chip is 12.7 mm on a side and the cylindrical pin fins are 1.5 mm in diameter and 1.5 cm in length. The finned surface of the chip is cooled by 20°C air with convective heat transfer coefficient of 250 Wm-2 K-1 , while all other chip surfaces are perfectly insulated. The chip is sufficiently thin to have uniform temperature.
a. If the maximum allowable chip operating temperature is 75°C, what is the maximum power that can be dissipated by the chip?
b. What is the effectiveness of each fin and the overall enhancement in heat transfer rate provided by the array of fins?
c. Examine the effect of changes in pin fin geometry (number of fins in the square array, fin diameter, fin length) on the maximum heat dissipation rate. Consider arrangements of 16, 25, and 36 pin fins, and plot the effects of fin diameter and length on the heat dissipation rate. Recommend a design for enhanced chip cooling.