Q$1.$ In a material processing experiment, a coated metal sphere of $5-$mm diameter is removed from a furnace at $900C$ and cooled to a temperature of $200C.$ Although properties of the metal vary over this temperature range, constant values may be assumed to a reasonable approximation, with $=9000k\frac{g}{m^3},c=300\frac{J}{k}g*K,$ and $k=60\frac{W}{m}.K$

a$)$ If cooling is implemented in a large, evacuated chamber whose walls are at $25C,$ determine the time required to reach the final temperature if the coating is polished and has an emissivity of $=0\mathrm{.}1.$ How long would it take if the coating is oxidized and $=0\mathrm{.}5?(3$ points assumption and sketch $($if any$)+2$ points calculation$)$

b$)$ To reduce the time required for cooling, consideration is given to immersion of the sphere in an inert gas stream for which $T_{}=25C$ and $h=200\frac{W}{m^2}.K.$ Neglecting radiation, what is the time required for cooling? $(2$ points calculation$)$

c$)$ Considering the effect of both radiation and convection, what is the time required for cooling if $h=200\frac{W}{m^2}.K$ and $=0\mathrm{.}1.$ Explore the effect on the cooling time of independently varying h and $.(3$ points calculation$)$