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 air flow over the sleeve.

(a) Under conditions for which a uniform surface temperature T, is maintained around the circumference of the heater and the temperature Too and convection coefficient h of the air flow are known obtain an expression for the rate of heat transfer per unit length to the air. Evaluate the heat rate for T1 = 300°C, D = 20 mm, an aluminum sleeve (k = 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 k h = 400 W/m ∙ K, what is the required volumetric heat generation within the heater and its corresponding centerline temperature? (c) 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.

Transcribed Image Text:

Fins, N ПЛР Sleeve, k, Air flow D- T, h Heater (à, k) ЛЛЛИ