Chapter 9, Problem 9.47P

Integrated circuit (IC) boards are stacked within a duct and dissipate a total of 500 W. The duct has a square cross section with $w=H=150\text{mm}$ and a length of 0.5 m. Air flows into the duct at 25°C and $1.2m^3/\min$ , and the convection coefficient between the air and the inner surfaces of the duct is ${\bar{h}}_i=50W/m^2$ . The entire outer surface of the duct, which is anodized with an emissivity of 0.5, is exposed to ambient air and large surroundings at 25°C.

Your assignment is to develop a model to estimate the outlet temperature of the air, $T_{m.o}$ and the average surface temperature of the duct, ${\bar{T}}_s$ .
(a) Assuming a surface temperature of 37°C, estimate the average free convection coefficient, $h_o$ , for the outer surface of the duct.
(b) Assuming a surface temperature of 37°C, estimate the average linearized radiation coefficient,
   ${\bar{h}}_{\text{rad}}$ , for the outer surface of the duct.
(c) Perform an energy balance on the duct by considering the dissipation of electrical power in the ICs, the rate of change in the energy of air flowing through the duct, and the rate of heat transfer from the air in the duct to the surroundings. Express the last process in terms of thermal resistances between the mean temperature, ${\bar{T}}_m$ , of the air in the duct and the temperature of the ambient air and the surroundings.
(d) Substitute numerical values into the expression of part (c) and calculate the air outlet temperature, $T_{m,o}$ . Estimate the corresponding value of ${\bar{T}}_s$ .Comment on your results and the assumptions inherent in your model.