Water at 20°C and a flow rate of 0.1 kg/s enters a heated, thin-walled tube with a diameter of 15 mm and length of 2 m. The wall heat flux provided by the heating elements depends on the wall temperature according to the relation q"s (x) = q"x,o [1 + a(Ts - T ref)] where q"s,o = 104 W/m2, a = 0.2 K-1, T ref = 20°C, and Ts is the wall temperature in dc. Assume fully developed flow and thermal conditions with a convection coefficient of 3000 W/m2 ∙ K.

(a) Beginning with a properly defined differential control volume in the tube, derive expressions for the variation of the water, Tm(x), and the wall, Ts(x), temperatures as a function of distance from the tube inlet.

(b) Using a numerical integration scheme, calculate and plot the temperature distributions, Tm(x) and Ts(x), on the same graph. Identify and comment on the main features of the distributions

(c) Calculate the total rate of heat transfer to the water.