Lumped Capacitance: Your company has a process that exhausts a hot gas at a very high temperature $(600C).$ The EPA tends to frown upon shotting this into the atmosphere $($although before the EPA weighed in$,$ it did a wonderful job of keeping Canadian Geese away from the facility!$)$ so you are tasked with finding something useful to do with the gas before cooling it$.$ You decide to see if you can store some of the energy from this stream into a solid slab of aluminum $($because you have a lot of them lying around$).$ You take a slab of aluminum that is $0\mathrm{.}05m$ in width and is initially at a temperature of $25C.$ The hot gas is passed over both sides of the aluminum slab. The gas has a temperature of $T_{}=600C$ with a film heat transfer coefficient of $h=100\frac{W}{m^2*K}.($Note that a characteristic length for a slab of this type can be W$/2)$

How long will it take to achieve $75\%$ of the maximum possible energy storage? What is the temperature of the aluminum at this time?

Properties of aluminum are:

$k=231\frac{W}{(m)*K}$

$=2702\frac{kg}{m^3}$

$C_p=1033\frac{J}{kg*K}$