Chapter 8, Problem 8.75P

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 thermo couple senses a temperature T, corresponding to that of the tube surface. The gas flow rate and temperature are designated as the $m_g$ and $T_g$ . 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 $T_{\infty }$ and is exposed to ambient air at $T_s$ , and large surroundings at $T_{sur}$ . The convection coefficient associated with the outer surface of the duct is designated as $h_0$ , while those associated with the inner surface of the duct and the tube surface are designated as $h_i$ and $h_t$ , respectively. The tube and duct surface emissivities are designated as ${\epsilon }_t$ and ${\epsilon }_s$ respectively.

(a) Neglecting conduction losses along the thermocouple tube, develop an analysis that could be used topredict the error $\left(T_g-T_t\right)$ in the temperature measurement.

(b) Assuming the flue gas (o have the properties ofatmospheric air, evaluate the error for $T_t=300°\text{C},D_s==0.6\text{m},D_t=10\text{mm,mg}=1\text{kg/s},T_{\infty }=T_{sur}=27°\text{C},{\epsilon }_t={\epsilon }_s=0.8,$ and $h_0=25{\text{W/m}}^2\cdot \text{K}$ .