The problem is simple, I suppose…
I have a beverage at room temperature (let's say 23°C) in an aluminum can (it's a Barq's root beer.) I want to put it in a freezer (let's say it's at -17°C). How long do I have to wait for it to reach a chilled temperature (let's say 8°C).
I need to know for a couple reasons. One, I don't want the can to explode. Two, I don't want it to get too cold for my teeth (perhaps I shouldn't be drinking pop? 🙂 ) Three, I want don't my coworkers to steal my nice, cold beverage on the first hot day of the year.
If I could find the right equations, I would just plug in the numbers, but I can't seem to find the right ones.
Best Answer
I would probably use Newton's Law of Cooling;
$\frac{dT}{dt}=-k(T - T_0)$
where $T_0$ is the ambient temperature.
As far as what k is, you could just do a little experiment. Find the temperature of the beer, put it in for 5 minutes, take it out, take the temperature again.
If you aren't the experimental type, $k = A \cdot H$ where $A$ is the surface area (a nice cylinder) and $H$ is the heat transfer coefficient.
For water to air, through metal, you get about $11 \frac{W}{m^2K}$.
But I bet non-linear effects will cause enough issues that you will probably have a pretty wide error, (Specific type of metal of a can (aluminum), the shape of it, thickness, fact that beer is not just water, etc.). Probably an experimental approach of finding k will give you the most accurate way of going about it.