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Is there a way to calculate how fast water will cool if you know the water's starting temperature and temperature of the environment that you put it in, such as a freezer?

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Does the container shape matter much to your solution? Cylinder, box? Resting on a surface? Embedded in a medium? in-ground (like a swimming pool?) –  New Alexandria Oct 4 '12 at 4:46
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up vote 5 down vote accepted

Check out Fourier's law. The rate of heat flux (transfer of thermal energy per unit time per unit surface area) is proportional to the difference in temperature:

Heat eqn

Since a change in temperature is directly proportional to the change in energy, you can form this into a fairly simple first-order differential equation. If we take one side (air in a room), at a temperature we'll call $T_{0}$ as a cold reservoir, we can say:

$$ \frac{d(cT_{1})}{dt}=c\frac{dT_{1}}{dt}=-kA\frac{T_{1}-T_{0}}{w} $$

Where $T_{1}$ is the temperature of the water, $c$ is the water's specific heat, $A$ is the surface area, $k$ is the thermal conductivity of the material between them (i.e. styrofoam cup, say it has a styrofoam lid too so it surrounds the water completely), and $w$ is the thickness of the container.

You will notice that this equation leads to an exponential decay in the temperature difference.

Simply plug in the appropriate constants and solve this equation for any given $\Delta T$ and you can find how long it takes to get that close (it will never quite reach zero).

I'm a bit rusty, so please forgive any notational errors

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