[Programming Note: I've noticed looking through older posts that the equations have picked up extra + signs, due to my impolite hotlinking. I plan to go through and fix those when I have a chance, but until then, don't take them at face-value.]

Another great question from a reader, this time my own mother, Sally:

**I’m making a carrot cheesecake for Steve’s birthday. The cheesecake layer is supposed to go into an 9” springform pan but I only have 8”[...] I assumed I’d just make the cheesecake layer thicker, but how to determine baking time? The volume is the same but the height is 25% greater. Thicker tends to mean longer baking. But how do i determine that? What role does oven temp play? What role does moisture level of cheesecake play?**
I've talked about heat transfer before, in

this post about making granita, but I'd like to take a different approach this time, using the more general heat equation:

This says that the rate of temperature change at a point in the cake is proportional to the variation in temperature nearby, and the

thermal diffusivity, α. This quantity depends on the substance we're interested in, but I don't think it's been tabulated for cheesecake batter, so we'll assume it's about the same as water, 0.143 × 10

^{−6} m

^{2}/s [this is the role the moisture level plays].

Technically, this is a 3-dimensional problem, but thanks to the cylindrical symmetry of the cake, we can just consider a 2d cross-section through the center. We can assume the outer surface of the cake is fixed at the oven temperature, 325°F. Then we can use a numerical solver to find the temperature throughout the cake over time. As it happens, I wrote a solver for this equation a few years ago for a Computational Physics class. After a few adjustments, it was ready to go:

bake.py
First, we need to find the internal temperature that the cake reaches after the prescribed 45 minutes of baking.

The final temperature after 45 minutes is 206°F. This is a little close to the boiling point of water, 212°F, where I expect things to get a bit off from the approximations I'm making, but we'll go with it. Then we can start again with a narrower cake of the same volume, and find how long it takes to get to that temperature.

The final time is about 70 minutes, which isn't completely unreasonable, but I take no responsibility for any charred cakes this calculation results in. Thanks for a great question, Sally!

[Edit: One first posting, I mistakenly used 8/9" as radii, rather than diameter. It doesn't actually make a difference in the final results, since the thickness dominates.]