Newtonian Granita

Last night I made some Watermelon Granita, but the pan I used to freeze it was smaller than the recipe called for.  It got me thinking about how the dimensions of the pan affect the rate of freezing.

Assuming we have a rectangular pan, the volume is simply

and the surface area is

In comparing pans, we'll want the same volume, so we can combine these to eliminate a dimension:

According to Newton's Law of Cooling, the difference in temperature between the pan and the surrounding air in the freezer will be given by

where r is defined as

and h is the heat transfer coefficient, m is the mass of the substance being cooled, and cp is the specific heat capacity of the substance.

We're interested in how long it takes the granita to freeze, so solving Newton's equation for t gives

Notice that the only thing that will change with the container dimensions in this equation is A, so for any pair of containers, we can write

This is a surprisingly simple equation – a pan of half the area will take twice as long to freeze.  This isn't perfectly accurate, since Newton's Law of Cooling is an approximation, but I still expected something much more complicated.