Chapter 4, Problem 4.25P

A small device is used to measure the surface temperature of an object. A thermocouple bead of diameter $D=12\mu \text{m}$ is positioned a distance $z=100\mu \text{m}$ from the surface of interest. The two thermocouple wires, each of diameter $d=25\mu \text{m}$ and length $L=300\mu \text{m,}$ are held by a large manipulator that is at a temperature of $T_m=23°\text{C}\text{.}$

If the thermocouple registers a temperature of $T_{tc}=29°\text{C,}$ what is the surface temperature? The thermal conductivities of the chromel and alumel thermocouple wires are $k_{\text{Ch}}=19\text{W/m}\cdot \text{K}$ and $k_{\text{Al}}=29\text{W/m}\cdot \text{K,}$ respectively. You may neglect radiation and convection effects.