Scaling Cake Baking Times

[erobz]

I have a recipe that was a scratch cake of my grandmothers, but it is made for a $9 ~\text{in} \times 13 ~\text{in}$ baking pan. I wish to bake three $9 ~\text{in}$ rounds from the recipe. I can look up baking times online for the size, but times are dependent somewhat on the recipe.

Just for fun I wanted to see what a model would predict.

I was thinking very basically to neglect thermal gradients (lumped capacitance - likely bad approximation for a cake, but still...), and assume convection. Then I suspect there is some critical temperature that is reached, and then bubbling of gases and structural change would act like a latent heat. Such that a prescribed bake time at 350 F would be comprised of a time to reach some critical temperature (via sensible heat), then the remaining bake time would be in effect a phase transformation; formation of gas bubbles, evaporation, cake structure, etc...

Any thoughts?

 

[DaveC426913]

What is the depth of your off-recipe cakes versus the depth of the recipe cake?

Presumably, a cake's geographical centre will cook from top and bottom, not so much from edges in. Effectively, I'm looking at the cake as an infinite sheet, with no confounding edge effects to worry about. So, as long as the depth is similar, you should get similar results.

 

[erobz]

What is the depth of your off-recipe cakes versus the depth of the recipe cake?

Presumably, a cake's geographical centre will cook from top and bottom, not so much from edges in. Effectively, I'm looking at the cake as an infinite sheet, with no confounding edge effects to worry about. So, as long as the depth is similar, you should get similar results.

I would just generalize that if in the rectangular pan we have $V = abc$, ($c$ being the height dimension)then in each cylindrical pan we have $\frac{1}{3} abc = \frac{\pi D^2 c'}{4}$, so $c'$ ( the batter depth in each round pan) is fixed. The surface areas can be expressed in terms of $a,b,c,D$ (all known - measurable).

That part isn't really an issue. I'm more curious to see what the baking chemistry physical model is generally understood to be? It for sure seems complex, but it seems to me like there is effectively some kind of latent energy at some critical temperature as the cake structure forms. Is it a gross underestimation of reality?

 

[Vanadium 50]

Any thoughts?

a) I don't want to eat the first one.
b) I would see if your model correctly predicts cupcake baking time.

 

[erobz]

a) I don't want to eat the first one.
b) I would see if your model correctly predicts cupcake baking time.

A) Unfortunately the unsuspecting party goers will.
B)The cupcake idea is good!

Anyone know the surface area of a cupcake shell off hand?

 

[Vanadium 50]

The cupcake idea is good!

I do this for a living, after all. Just not with cake. But if the answer is "cook for 150 years in a 2 degree oven" for cupcakes, I would definitely be more concerned than if the model got it right.

 

[erobz]

I do this for a living, after all. Just not with cake. But if the answer is "cook for 150 years in a 2 degree oven" for cupcakes, I would definitely be more concerned than if the model got it right.

That being said...would you let me suffer through experimentation with no real hope...I don't know. However, I'm going take this as "it's not the most half-baked idea" you've pondered.

 

[Vanadium 50]

"it's not the most half-baked idea" you've pondered.

That remains to be seen, doesn't it?

 

[erobz]

That remains to be seen, doesn't it?

I need to petition the government for funds...so it might be a while.

 

[erobz]

So, I gather "experimentally" I'm going to put a few digital temperature probes into a round cake and look for the signature of a latent heat (a zero temp change for some period of time). What do I need to know about the digital thermometers?

I'm thinking this one a ThermoPro TP20 off amazon with the following relevant specs:

• 2 Probes Monitor: Wireless meat thermometer for grilling and smoking features food-grade stainless steel probes with step-down tip design allow users to get readings more precisely and quickly, acurrate to 1.8°F; Probe wires can withstand up to 716°F
• displays both food and oven temps simultaneously with built-in backlight, say goodbye to the tedious setup steps

I'm guessing one probe measures oven temp, the other the food, but I should be able to just verify oven temp and use both in cake simultaneously. It won't be a waste because I don't have one, and probably should!

Questions/Concerns: I'm going to have to make some kind of a bracket to hold the probes so they are suspended in the mixture. Perhaps this is a destructive test, as the probes are probably significantly longer than the cake depth. Does having significant portion of probe exposed to ambient oven effect the output, or is the measurement done in the tip?

 

[Chestermiller]

Write down the equations that govern the heat transfer behavior at the surface as well as within the cake, and then reduce the equations to dimensionless form. This will give you your answer.

 

[erobz]

Write down the equations that govern the heat transfer behavior at the surface as well as within the cake, and then reduce the equations to dimensionless form. This will give you your answer.

It initially feels out of my league. I'll try to think it over.