Atmospheric air enters the heated section of a circular tube ata flow rate of .005 kg/s and a temperature of 20 degrees Celsius.The tube is of diameter D = 50 mm, and fully developed conditionswith h = 25 W/m^2K exist over the entire length of L = 3m.

a) For the case of the uniform surface heat flux atq''_s = 1000 W/m² , determine the total heattransfer rate q and the mean temperature of the air leaving thetube T_m,o. What is the value of the surface temperatureat the tube inlet T_s,i and the outlet T_s,o?Sketch the axial variation of T_s and T_m. Onthe same figure, also sketch (qualitatively) the axial variation ofT_s and T_m for the more realistic case inwhich the local convection coefficient varies with x.

b) If the surface heat flux varies linearly with x, such thatq''_s(W/m²) = 500x(m), what are the values ofq, T_m,o, T_s,i, and T_s,o? Sketchthe axial variation of T_s and T_m. On the samefigure, also sketch (qualitatively) the axial variation ofT_s and T_m for the more realistic case inwhich the local convection coefficient varies with x.

c) For the two heating conditions of parts (a) and (b), plot themean fluid and surface temperatures, T_m(x) andT_s(x), respectively, as functions of distance along thetuble. What effect will a fourfold increase in the convectioncoefficient have on the temperature distributions?

d) For each type of heating process, what heat fluxes arerequired to achieve an air outlet temperature of 125 degreesCelsius? Plot the temperature distributions.