Chapter 3, Problem 3.121P

A probe of overall length $L=200\text{mm}$ and diameter $D=12.5\text{mm}$ is inserted through a duct wall such that a portion of its length, referred to as the immersion length $L_i,$ is in contact with the water stream whose temperature, $T_{\infty ,i},$ is to be determined. The convection coefficients over the immersion and ambient-exposed lengths are $h_i=1100{\text{W/m}}^{\text{2}}\cdot \text{K}$ and $h_o=10{\text{W/m}}^{\text{2}}\cdot \text{K,}$ respectively. The probe has a thermal conductivity of $177\text{W/m}\cdot \text{K}$ and is in poor thermal contact with duct wall.

1. Derive an expression for evaluating the measurement error, $\Delta T_{err}=T_{tip}-T_{\infty ,i},$ which is the difference between the tip temperature, $T_{tip},$ and the water temperature, $T_{\infty ,i}.$Hint: Define a coordinate system with the origin at the duct wall and treat the probe as two tins extending inward and out-ward from the duct. but having the same base temperature. Use Case A results from Table 3.4.
2. With the water and ambient air temperatures at 80 and 20°C, respectively, calculate the measurement error, $\Delta T_{err},$ as a function of immersion length for the conditions $L_i/L=0.225,0.425,$ and 0.625.
3. Compute and plot the effects of probe thermal conductivity and water velocity $\left(h_i\right)$ on the measurement error.