$(2011$ Mid$-$term Exam #$6)$ The roll$-$to$-$roll process is a very popular printing technology to manufacture flexible, paper$-$like displays, circuits, and solar cells. A typical substrate material for these applications is polyimide which has a good thermal and mechanical stability together with a thickness of $2$ mm $.$ Right after the roll$-$to$-$roll printing process, the substrate temperature is $80C$ and the substrate is cooled down in a room whose temperature is always controlled at $20C$ by an air conditioning system. The convective heat transfer coefficient of the air forced by the air conditioner is $200\frac{W}{m^2}*K.$ What will be the temperature of the polyimide substrate at the $1$ mm depth from its surface if the substrate is exposed to the air for $10$ sec $?$ Provide proper assumptions, a model, and calculations. Physical properties of the polyimide is $k=0\mathrm{.}13\frac{W}{m}*K,=1430k\frac{g}{m^3},$ and $c=1\mathrm{.}15k\frac{J}{k}g*K.$