The temperature of flue gases flowing through the large stack of a boiler is measured by means of a thermocouple enclosed within a cylindrical tube as shown. The tube axis is oriented normal to the gas flow, and the thermocouple senses a temperature T, corresponding to that of the tube surface. The gas flow rate and temperature are designated as mg and Tg, respectively, and the gas flow may be assumed to be fully developed. The stack is fabricated from sheet metal that is at a uniform temperature Ts and is exposed to ambient air at Toc and large surroundings at Tsur. The convection coefficient associated with the outer surface of the duct is designated as ho, while those associated with the inner surface of the duct and the tube surface are designated as hi and hi respectively. The tube and duct surface emissivities are designated as e and es respectively.
(a) Neglecting conduction losses along the thermocouple tube develop an analysis that could be used to predict the error (Tg – Ts) in the temperature measurement.

(b) Assuming the flue gas to have the properties of atmospheric air, evaluate the error for Tl = 300oC,
Ds = 0.6 m, Dt = 10 mm, mg = 1 kg/s, T8 = Tsur = 27°C, e1 = e3 = 0.8, and ho = 25 W/m2· K.