A spherical thermocouple whose surface is diffuse and gray with an emissivity of $0\mathrm{.}5$ indicates a temperature of $180C$ when used to measure the temperature of a gas flowing through a large duct whose walls have a uniform temperature of $450C.$

a$.$ If the convection coefficient between the thermocouple and the gas stream $h=125\frac{W}{m^2}-K,$ determine the actual temperature of the gas under steady state conditions. Neglect conduction losses from the thermocouple leads and assume the duct is much larger than the thermocouple.

b$.$ Assume an actual gas temperature of $125C.$ Generate a plot $($actual plot, not a sketch$)$ of the thermocouple measurement error $($i$.$e$.,$ the difference between what the thermocouple measures and the actual gas temperature$)$ for convection coefficients ranging from $10\frac{W}{m^2}-K$ to $1000\frac{W}{m^2}-K.$