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
(a) Under conditions for which a uniform surface temperature Ts is maintained around the circumference of the heater and the temperature T∞ and convection coefficient h of the airflow are known, obtain an expression for the rate of heat transfer per unit length to the air. Evaluate the heat rate for Ts = 300°C, D = 20 mm, an aluminum sleeve (ks = 240 W/m∙ K), w 40 mm, N = 16, t = 4 mm, L = 20 mm, T∞ = 50°C, and h 500 W/m2 ∙ K.
(b) For the foregoing heat rate and a copper heater of thermal conductivity kh = 400 W/m∙ K, r heater of thermal conductivity kh = 400 W/m∙ K, what is the required volumetric heat generation within the heater and its corresponding centerline temperature?
With all other quantities unchanged, explore the effect of variations in the fin parameters (N, L, t) on the heat rate, subject to the constraint that the fin thickness and the spacing between fins cannot be less than 2 mm.
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