Chapter 7, Problem 7.69P

A thermocouple is inserted into a hot air duct to measure the air temperature. -me thermocouple $\left(T_1\right)$ is soldered to the tip of a steel thermocouple well of length $L=0.15\text{m}$ and inner and outer diameters of $D_i=5\text{mm}$ and $D_o=10\text{mm}\text{.}$ A second thermocouple $\left(T_2\right)$ is used to measure the duct wall temperature.

Consider conditions for which the air velocity in the duct is $V=3\text{m/s}$ and the two thermocouples register temperatures of $T_1=450\text{K}$ and $T_2=375\text{K}\text{.}$ Neglecting radiation, determine the air temperature $T_{\infty }.$ Assume that, for steel, $k=35\text{W/m}\cdot \text{K,}$ and, for air, $\rho =0.774{\text{kg/m}}^{\text{3}},\mu =251\times {10}^{-7}\text{N}\cdot {\text{s/m}}^{\text{2}},k=0.0373\text{W/m}\cdot \text{K,}$ and $\mathrm{Pr}=\mathrm{0.686.}$