Chapter 1, Problem 1.78P

A thin electrical heating element provides a uniform heat flux $q_o^{"}$ to the outer surface of a duct through which airflows. The duct wall has a thickness of 10 mm and a thermal conductivity of $20\text{W/m}\cdot \text{K}\text{.}$

1. At a particular location, the air temperature is $30°\text{C}$ and the convection heat transfer coefficient between the air and inner surface of the duct is $100{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ What heat flux $q_o^{"}$ is required to maintain the inner surface of the duct at $T_i=85°\text{C?}$ For the conditions of part (a), what is the temperature $\left(T_o\right)$ of the duct surface next to the heater?
2. With $T_i=85°\text{C,}$ compute and plot $q_o^{"}$ and $T_o$ as a function of the air-side convection coefficient h for the range $10\le h\le 200{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ Briefly discuss your results.