Atmospheric air enters the heated section of a circular tube at a flow rate of 0.005 kg/s and a temperature of 20°C. The tube is of diameter D = 50 mm, and full developed conditions with h = 25 W/m² ∙ K exist over the entire length of L = 3 m.

(a) For the case of uniform surface heat flux at q"_s = 1000 W/m², determine the total heat transfer rate q and the mean temperature of the air leaving the tube T_m,o. What is the value of the surface temperature at the tube inlet T_s,i and outlet Ts.o? Sketch the axial variation of T, and Tm. On the same figure also sketch (qualitatively) the axial variation of T_s and Tm for the more realistic case in which the local convection coefficient varies with x.

(b) If the surface heat flux varies linearly with x, such that q"_s (W/m²) = 500x (m), what are the values of q, T_m,o, T_s,i, and Ts.o? Sketch the axial variation of T, and Tm. On the same figure, also sketch (qualitatively) the axial variation of T_s and T_m for the more realistic case in which the local convection coefficient varies with x.

(c) For the two heating conditions of parts (a) and (b), plot the mean fluid and surface temperatures, T_m(x) and T_s(x), respectively, as functions of distance along the tube what effect will a fourfold increase in the convection coefficient have on the temperature distributions?

(d) For each type of heating process, what heat fluxes are required to achieve an air outlet temperature of 125°C? Plot the temperature distributions.