Chapter 4, Problem 4.36P

The elemental unit of an air heater consists of a longcircular rod of diameter D, which is encapsulated by afinned sleeve and in which thermal energy is generatedby ohmic heating. The N fins of thickness t and length L are integrally fabricated with the square sleeve of widthw. Under steady-state operating conditions, the rate ofthermal energy generation corresponds to the rate of heattransfer to airflow over the sleeve.

1. Under conditions for which a uniform surfacetemperature $T_s$ is maintained around the circumferenceof the heater and the temperature $T_{\infty }$ and convection coefficient h of the airflow areknown, obtain an expression for the rate of heattransfer per unit length to the air. Evaluate theheat rate for $T_s=300°\text{C,}D=20\text{mm,}$ an aluminium 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 $h=500{\text{W/m}}^2\cdot \text{K}\text{.}$ For the foregoing heat rate and a copper heater ofthermal conductivity $k_h=400\text{W/m}\cdot \text{K,}$ what is therequired volumetric heat generation within the heaterand its corresponding centerline temperature?
2. With all other quantities unchanged, explore theeffect of variations in the fin parameters (N, L, t ) onthe heat rate, subject to the constraint that the finthickness and the spacing between fins cannot beless than 2 mm.