The nozzle extension $($skirt$)$ on the second stage of the Delta launch vehicle is made of columbium and is radiation cooled. The skirt attaches to an ablatively cooled chamber at an expansion ratio of $5$:$1.$ The engine operates at a chamber pressure of $300$ psi and has a throat diameter of $7\mathrm{.}48$ inches. The fuel is a $60/40$ blend of hydrazine and unsymmetrical dimethyl hydrazine $($UDMH$)$ and the oxidizer is nitrogen tetroxide. These propellants are combined at a mixture ratio of $2\mathrm{.}1.$

a$.$ What is the highest temperature in the skirt? Use the McAdams correlation for Nu$.$

b$.$ Assuming the engine starts instantaneously $($pc @ t $=0=$ pc $=300$ psi$)$ and that radiation heat transfer can be neglected during heat up of the wall, how long will it take the $200$ mil thick skirt to reach $95\%$ of the calculated temperature calculated in part $($a$)?$ Assume an ambient temperature of $300$ K prior to ignition.

Properties for columbium are rho $=8750$ kg$/$m $3$ and cp $=300$ J$/$kg$-$K