thick ice

It has been below, or around, freezing for a while now, and there’s a lot of ice around.  Today I noticed that the ice in one of our garden water tubs has expanded about 1cm above the rim of the tub.

Looking at it, I wondered how thick the ice was in total, to give that amount of expansion.  And then, could I calculate the thickness from just this observation?  And could I calculate it without knowing the height of the tub?

So I drew a sketch:

On the right is the liquid situation: water to the brim of the tub (which it was before the freeze).  On the left is the current situation, the top x units of water frozen to make x+d units of ice.  That does indeed look independent of the depth of the tub (unless maybe it is all frozen, which I doubt).

Let’s assume both the ice and water are at 0°C.  Wikipedia tell us that ice has a 9% increase in volume over water, which means that d = 0.09 x.

Since the observed d = 1cm, this implies that x = 11cm (much less than the depth of the tub, so no need to worry about it being frozen through), and so the whole ice layer is about 12cm (nearly 5 inches) thick.

That’s a thick layer of ice!