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 (ri) and outer (ro) surfaces of the wall. Express your results in terms of the tube inner and outer surface temperatures, Ts,i and Ts.o, and other pertinent parameters.

(b) Obtain expressions that could be used to determine Ts,i and Ts.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 hi = 2000 W/m2 ∙ K and ho = 100 W/m2 ∙ K. Heat is uniformly dissipated in a stainless steel tube (kw = 15 W/m ∙ K), having inner and outer radii of rj = 25 mm and ro = 30 mm, and aluminum fins (t = S = 2 mm, r, = 55 mm) are attached to the outer surface, with Rt,c = 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 q. The upper limit to q will be determined by the constraints that Ts.i not exceed the boiling point of water (100°C) and Ts,o not exceed the decomposition temperature of the adhesive (250°C).

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Gas flow Liquid flow Air Tube, , k Adhesive, Re