To manufacture an aluminum soda can, a thin sheet of aluminum of thickness 10-4m (100 micrometer) is placed in a die of a metal forming machine that punches the aluminum sheet into the shape of the can, making it 5.5 cm in diameter and 12.3 cm in height. During the metal drawing process, the temperature of the can rises to 50 °C. At this point the can has the side walls and the base, but does not have the top. This aluminum can is then placed in air at 20°C, where it cools down with a heat transfer coefficient of h = 10 W/m2K. As an engineer in this manufacturing plant, you have been asked to calculate the temperature history T(t) in this aluminum can as it cools down. You can ignore the surface area due to the edge on the top, and the surface area difference between the inner and outer surface due to the small aluminum thickness. Required:
a. What would be your choice of control volume?
b. Using the first law of thermodynamics for this control volume, derive the governing equation for the temperature history.
c. Solve the governing equation to find the temperature history.
d. How long will it take for the whole aluminum can to cool down below 30 °C?