A thermocouple bead is largely solder, $1mm$ in diameter. It is initially at room temperature and is suddenly placed in a $200C$ gas flow. The heat transfer coefficient $h$ is $250\frac{W}{m^2}K,$ and the effective values of other physical properties are as follows

$k=45\frac{W}{m}K$

rho $=9300K\frac{g}{m^3}$

$C_p=0\mathrm{.}18k\frac{J}{K}g*K$

$($a$)$ Derive model equations for the system and

$($b$)$ Evaluate the response of thermocouple with the passage of time.