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.
It still remains to determine how the amount of water changes over time. I looked into a couple ways to tackle this, and I'm not sure how accurate my technique is, but it made the most sense to me: The particles of a substance at a certain temperature have a statistical distribution of energies. For ideal gases, there's the Maxwell-Boltzmann distribution, and even though that's not a great model for liquid water we'll go for it. It takes a certain amount of energy for water to evaporate, called the heat of vaporization. If we assume that all the molecules with this much energy do leave the solution, we can keep track of the current number of water molecules still in the cake. Taking out those high-energy particles will also cool the cake, so we can figure out the temperature change with the heat capacity.
I updated the previous code to include these new modifications. Once again, we start off by measuring the temperature after a given amount of time using the wider pan:
In contrast to the previous results, the center temperature goes well above water's boiling point! We can also look at how the amount of water changes over time:
I was surprised to see that the center of the cake experiences almost no loss in water. Even the edges don't lose much – The color scale varies from 50 to 60%. To check what effect the loss of water has, I reran the test with the same α calculated above, but left the amount of water constant. The results are almost identical:
As in the previous post, we can use that central temperature as a target for the narrower pan, and find the cook time:
As before, we get about 70 minutes, suggesting the water content is not as significant as I imagined. A big part of physics is figuring out what you can approximate, and what you can't – which results are "good enough". It seems cakes can be quite forgiving when it comes to baking precision!
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