Chapter 4, Problem 4.36P

The elemental unit of an air heater consists of a long circular rod of diameter D, which is encapsulated by a finned sleeve and in which thermal energy is generated by ohmic heating. The N fins of thickness t and length L are integrally fabricated with the square sleeve of width w. Under steady-state operating conditions, the rate of thermal energy generation corresponds to the rate of heat transfer to airflow over the sleeve.

1. Under conditions for which a uniform surface temperature $T_s$ is maintained around thecircumference of the heater and the temperature $T_{\infty }$ and convection coefficient h of the airflow areknown, obtain an expression for the rate of heat transfer per unit length to the air. Evaluate the heat rate for $T_s=300°\text{C,}D=20\text{mm,}$ an aluminum sleeve $\left(k_s=240\text{W/m}\cdot \text{K}\right),w=40\text{mm,}N=16,t=4\text{mm,}L=20\text{mm},T_{\infty }=50°\text{C,}$ and $h=500{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$
2. For the foregoing heat rate and a copper heater of thermal conductivity $k_h=400\text{W/m}\cdot \text{K,}$ what is the required volumetric heat generation within the heater and its corresponding centerline temperature?
3. With all other quantities unchanged, explore the effect of variations in the tin parameters $\left(N,L,t\right)$ on the heat rate, subject to the constraint that the tin thickness and the spacing between fins cannot be less than 2 mm.