Question 9.31: In order to measure the temperature of a gas flowing through......
In order to measure the temperature of a gas flowing through a copper pipe, a thermometer pocket is fitted perpendicularly through the pipe wall, the open end making very good contact with the pipe wall. The pocket is made of copper tube, 10 mm o.d. and 0.9 mm wall, and it projects 75 mm into the pipe. A thermocouple is welded to the bottom of the tube and this gives a reading of 475 K when the wall temperature is at 365 K. If the coefficient of heat transfer between the gas and the copper tube is 140 W/m² K, calculate the gas temperature. The thermal conductivity of copper may be taken as 350 W/m K. This arrangement is shown in Figure 9.88.
If θ is the temperature difference (T – T_{G}), then:
\theta=\theta_{1}{\frac{\cosh m(L-x)}{\cosh m L}}At x = L: \theta={\frac{\theta_{1}}{\cosh m L}}
m² = \frac{h b}{kA}
where the perimeter: b = π x 0.010 m, tube i.d. = 8.2 mm or 0.0082 m
cross-sectional area of metal: A={\frac{\pi}{4}}(10.0^{2}-8.2^{2})=8.19\ \pi\ \mathrm{mm}^{2} or 8.19 π × 10^{-6} m²
∴ m^{2}={\frac{(140\times0.010\pi)}{(350\times8.19\pi\times10^{-6})}}=488\ \mathrm{m}^{-2}
and: m=22.1{\mathrm{~m}}^{-1}
\theta _{1} = T_{G} – 365, \theta _{2} = T_{G} – 475
{\frac{\theta_{1}}{\theta_{2}}}=\cosh m L∴ \frac{T_{G}-365}{T_{G}-475}=\cosh(22.1\times0.075)=2.72
and: \underline{\underline{T_{G}=539\ K}}