Chapter 4, Problem 4.30P

A long constantan wire of 1-mm diameter is butt welded to the surface of a large copper block, forming a thermocouple junction. The wire behaves as a tin, permitting heat to flow from the surface, thereby depressing the sensing junction temperature $T_j$ below that of the block $T_o.$

1. If the wire is in air at 25°C with a convection coefficient of $10{\text{W/m}}^{\text{2}}\cdot \text{K,}$ estimate the measurement error $\left(T_j-T_o\right)$ for the thermocouple when the block is at 125°C.
2. For convection coefficients of 5, 10, and $25{\text{W/m}}^{\text{2}}\cdot \text{K,}$ plot the measurement error a function of the thermal conductivity of the block material over the range 15 to $400\text{W/m}\cdot \text{K}\text{.}$ Under what circumstances is it advantageous to use smaller diameter wire?