A scheme for concurrently heating separate water and air streams involves passing them through and over an array of tubes, respectively, while the tube wall is heated electrically. To enhance gas-side heat transfer, annular fins of rectangular profile are attached to the outer tube surface. Attachment is facilitated with a dielectric adhesive that electrically isolates the fins from the current-carrying tube wall.

 

(a) Assuming uniform volumetric heat generation within the tube wall, obtain expressions for the heat rate per unit tube length (W/m) at the inner (r_i) and outer (r_o) surfaces of the wall. Express your results in terms of the tube inner and outer surface temperatures, T_s,i and T_s,o, and other pertinent parameters. (b) Obtain expressions that could be used to determine T_s,i and T_s,o in terms of parameters associated with the water- and air-side conditions.

(c) Consider conditions for which the water and air are at T_∞,i - T_∞,o = 300 K, with corresponding convection coefficients of h_i = 2000 W/m² ∙ K and h_o = 100 W/m² ∙ K. Heat is uniformly dissipated in a stainless steel tube (kw = 15 W/m ∙ K), having inner and outer radii of r_i = 25 mm and r_o = 30 mm, and aluminum fins  mm) are attached to the outer surface, with  10^-4 m² ∙ K/W. Determine the heat rates and temperatures at the inner and outer surfaces as a function of the rate of volumetric heating . The upper limit to  will be determined by the constraints that Ts,i not exceed the boiling point of water (100°C) and T_s,o not exceed the decomposition temperature of the adhesive (250 C).