Water enters a tube at 27°C with a flow rate of 450 kg/h. The rate of heat transfer from the tube wall to the fluid is given as qs ′ = x, where the coefficient is 20 W/m2 and x(m) is the axial distance from the tube entrance.

(a) Beginning with a properly defined differential control volume in the tube, derive an expression for the temperature distribution Tm(x) of the water.
(b) What is the outlet temperature of the water for a heated section 30 m long?
(c) Sketch the mean fluid temperature, Tm(x), and the tube wall temperature, Ts(x), as a function of distance along the tube for fully developed and developing flow conditions. In other words, draw the plot we drew in our notes, but now for this different case where the heat flux varies with x.
(d) What value of a uniform wall heat flux, qs ′′ (instead of qs ′ = x), would provide the same fluid outlet temperature as that determined in part b? For this type of heating, sketch the same temperature distributions requested in part c.