Water enters a tube at 27°C with a flow rate of 450 kg/h. The heat transfer from the tube wall to the fluid is given as q'_s (W/m) = ax, where the coefficient a 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 T_m(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, T_m(x) and the tube wall temperature, T_s(x), as a function of distance along the tube for fully developed and developing flow conditions.

(d) What value of a uniform wall heat flux, q'_s (instead of q: = ax), would provide the same fluid outlet temperature as that determined in part (b)? For this type of heating, sketch the temperature distributions requested in part (c).