The Growth Rate of Ice on a Pond
Ice will start to form on a pond when the temperature of the air falls below 0 ° C. Suppose the air temperature is θ° C and the water temperature is 0 ° C. Take the thickness of the ice to be
and suppose that in a time dt the thickness increases by
The heat released as the ice freezes has to escape through the layer of ice that has already formed.
The heat released by the water as a mass
turns to ice is
where
is the specific latent heat of fusion of water. If the density of ice is
then
where 
is the volume of ice formed and
is the area of the ice, so the energy given up is
If this freezing happens in a time
the amount of heat that escapes through the ice is
Equating the two expressions gives
Cancellingand rearranging gives
We can separate variables and integrate. Take
when
Sincewhen
we have
hence
