heating a slab of pie

Calculating the time it takes for a pie to cook – Fourier’s law

Cooking a pie in the oven is always a delicate balancing act. You don’t want the pie to dry out and become tough. But under cooking is probably worse, with raw liquid filling being left in the pie. Imagine how important this is for food manufacturers. Every pie has to come out right.

What if you could only calculate just how long that pie needed to go into the oven? Unfortunately, that’s a lot more complicated than it may sound. There are a lot of things going on simultaneously in that pie. Proteins set, moisture evaporates, temperature changes throughout, fat melts and the list goes on.

It is what makes food complicated, and at the same time so interesting. And it doesn’t mean that none of your traditional chemical engineering formulas such as Fourier’s law are useless. They are just more indicative that precisely predictive. So let’s have a look how far we get into calculating just exactly when that pie is cooked in the oven. (It is easier to calculate the cooling down of a well stirred pot of soup!)

Setting the stage: a pie in an oven

We’ll be having a look at what happens when baking that pie in the oven. So let’s have a what it is that we have here:

  • A round pie with a diameter of 10 cm (4 inch)
  • A pre-baked crust – 3 mm thick on the bottom and sides
  • A filling – 2 cm (0.8 inch) thick
  • In an oven of 180°C

We can assume that nothing happens with the pie as long as it is not in the oven. Once you put it in the oven though, a lot of things start happening. First of all, the pie enters this hot air. The pie and filling are static though so all the heat will have to move through the pie through conduction. The outside of the pie will be the temperature of the oven almost immediately, however, the center will take a lot longer to heat. The heat has to penetrate in slowly.

For now we assume that all we have to do is heat the complete filling to 80°C and that all that is needed for this is the heat from the oven to travel through our pie.

cherry chocolate pie
Pie!

Heating the pie

We will now have to make some assumptions and start simplifying our pie. We know that heat will surround the entire pie. We are interested in knowing when every part of the pie has reached the required temperature. As such, we have to look at the part that takes longest to heat up.

The sides will cook quickly since they are surrounded by heat on several sides. The middle section though, only gets heat from the top and bottom. As a result, that takes longest to cook.

So let’s have look at a slice in the middle of our pie. It gets heat from the top and from the bottom and that heat slowly seeps in. Initially, only the most outer layer of the pie is warm. The heat will diffuse in and start warming up the inside of the pie. The outer parts heat up more quickly than the inside, but there will be a gradual gradient of heat throughout the slice of pie.

The image below visualizes this for half of the pie. The top block represents the pie when it has just started to heat up. The center is still the original temperature, but you can see that the outside is already hot and the area in between is warming up. The middle block represents further heating of the pie and at the bottom you can see that the center is also warmer than the initial temperature! Of course, this process occurs at both the top and bottom simultaneously.

heating up a slice of pie

Fourier’s law

Those sketched grey curves drawn into the pie above can be described by a mathematical equation, called Fourier’s law.

Fourier’s law states that the heat transfer over a constant surface area is proportional to the temperature gradient divided by the length gradient:

dQ ∝ – k * A (dT/dx)

Where:

  • dQ = rate of heat flow
  • k = thermal conductivity, value depends on the properties of the materials you use
  • A = surface area
  • dT/dx = temperature gradient over the product (our pie)

You solve this formula by defining boundary conditions. In our case such a boundary condition is that the temperature on the outside of the pie always stays the same. It also says that the temperature of the pie starts at room temperature. These boundary conditions, combined with a few others, the fact that we treat the pie as a flat surface and some math will give you a formula for the temperature at each point within your pie:

d2T/dx2 = (1/α) * (dT/dt)

Where:

  • α = thermal diffusivity = k / ( ρ * cp)
  • dT/dT = temperature gradient over time
  • k = thermal conductivity (how easily the product conducts heat)
  • ρ = density
  • cp = specific heat capacity (how much energy you need to heat up a mass, in J/(kg*K)

As you can see in the formula, this example of heating a pie is actually a very complicated problem to solve mathematically. This is because it is not a steady state situation. The temperature of a pie of the pie depend both on the location in the pie as well as on the time. The temperature will be different when it just goes in versus when it comes out and the temperature at the center will be different than that at the end. This dependency on both time & temperature makes the baking of a simple pie quite complex!

If you have a computer and good software skills, this is a solvable problem though. There are even more complex problems out there, that are even harder if not almost impossible to solve for.

lemon custard pies

The impact of chemical reactions, evaporation, etc.

In reality, once the pie starts to heat up a lot of things will start to happen. First of all, solid fats (e.g. butterfat) will melt and proteins will start to denature. Moisture will be bound and some moisture will evaporate. All those processes require heat to occur and take energy to take place. This is what makes calculating the temperature of that pie so hard. It is tricky to exactly quantify how much energy is required.

Also have a look at the factor α in the formula. We assume in this formula that α remains constant throughout the process. However, you can see that α depends on the density. If you’ve ever baked a pie, you will have seen that the volume and thus the density of the pie can change during baking. In other words, a factor that you’re assuming to be constant, isn’t actually so, making it even more complicated.

Sources

Amir Faghri, Yuwen Zhang, John R. Howell, Advanced heat and mass transfer, 2010, Chapter 3.3, link ; this book discusses the more advanced mathematics involved with these heat transfer problems

Rathore, M.M., Kapuno, R. R., Engineering heat transfer, 2011, chapter 2, link

Rudromoorthy, R. Heat and mass transfer, 2010, p. 16, link

Soloviev, Vladimir, Transient conduction – analytical methods, link

Soloviev, Vladimir, Steady State Conduction in the Plane Wall with Uniform Heat Generation, link

Soloviev, Vladimir, Fourier’s law, link

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