Thanksgiving put baking on my mind, and I started thinking about a post I wrote several years ago about baking a cake in different sized pans. I had been concerned in that post that I didn't take into account the evaporation of water from the cake, which would change the way it heated. This week though, I thought of a way to account for that, and decided to follow up on my previous post (and continue the French Revolution theme).
To recap, my mother Sally was baking a cake, but her pan was smaller than the recipe called for. That meant the cake would be thicker, and she wondered how that would change the cooking time. The key to figuring out how temperature changes spread through an object is the heat equation:
This says that the change in temperature at a point is proportional to the variation in temperature at the surrounding points. I mentioned last time that it was the proportionality constant, α, that depended on the moisture content of the cake batter. I realized that we could account for a changing amount of water by varying α according to
where m is the mass of water and of other ingredients in the cake, and α the individual diffusivities. I was surprised to find that a group of food engineers (!) had actually measured the thermal diffusivity of flour. Taking a sheet cake recipe as an example, we can approximate the amount of liquid (~water) and the amount of flour to get the effective α.