# Physics behind carryover cooking

gnurf
I'm looking for a simple mathematical model that describes how the internal temperature of a piece of meat continues to rise beyond the point at which the external heat source is removed. The model's accuracy needs to be no better than to demonstrate that carryover cooking is a real effect. The simple model I had in mind is three regions: meat center (A), meat outer layer (B), and ambient surroundings (C). Initial conditions at t=0 is (arbitrary) TA=50C, TB=100C, and TC=25C. From there I was thinking about using Newton's Cooling Law from B to A and B to C at as many iterative discrete steps as necessary and ultimately plot the temperature overshoot in the meat's core vs time.

Does this approach make sense?

Staff Emeritus
Actually, you also need to know the thermal conductivity of the meat. This is because the temperature gradient between the outer layer and the center will cause further thermal energy to migrate into the center even when the meat is taken out of the oven or away from the heat source.

And yes, this is quite real, and depending on the size of the roast, the carry-over cooking can cause the center to rise 5-10 F. I have no found a "universal" quantitative model yet to be able to accurately describe this.

Zz.

_PJ_
I understand "carryover cooking" is not specifically a physical process (although of course, ultimately driven by physical relations) - Instead, it is more a combination of chemical reaction and the (okay, more physical) flow of liquid throughout the meat.
The chemical process is that of breakdown of large, complex, organic molecular structure and some chemical bonds which release that binding energy - which, can be modeled, but in an overly complicated description of quantum states of electrons - or at a far simpler chemical level. Further, liquids that have a greater heat capacity can traverse through the 'gaps' made as fibrous structures break down, allowing for the transfer of the hotter liquid to cooling regions of meat. This would necessitate (again, way overcomplicated) fluvial descriptions that are perhaps excessive when it may be that just to identify that heat energies can be retained and redistributed may be sufficient.

Good luck with a physical model. Sounds like quite a challenge.

Staff Emeritus
I understand "carryover cooking" is not specifically a physical process (although of course, ultimately driven by physical relations) - Instead, it is more a combination of chemical reaction and the (okay, more physical) flow of liquid throughout the meat.
The chemical process is that of breakdown of large, complex, organic molecular structure and some chemical bonds which release that binding energy - which, can be modeled, but in an overly complicated description of quantum states of electrons - or at a far simpler chemical level. Further, liquids that have a greater heat capacity can traverse through the 'gaps' made as fibrous structures break down, allowing for the transfer of the hotter liquid to cooling regions of meat. This would necessitate (again, way overcomplicated) fluvial descriptions that are perhaps excessive when it may be that just to identify that heat energies can be retained and redistributed may be sufficient.

It has nothing to do with that. If you have a large volume of an object with some non-zero thermal conductivity, and you heat the outside, you will have the same effect as "carryover" cooking at the center of that object. I've seen this effect in a "solid" cylinder of copper with an embedded thermocouple in the middle.

So it is a "physical process".

Zz.

Tom.G, nasu and russ_watters
Staff Emeritus
I actually found a fun site from MIT where you can test the cooking time for a particular thickness of meat.

http://up.csail.mit.edu/science-of-cooking/

It includes carryover cooking if in the final step, you set the temperatures of both side of the meat to room temp. So they obviously have a model that they are using for this. I haven't had the time to look around yet to see what model they are using.

Zz.

jim mcnamara
gnurf
Actually, you also need to know the thermal conductivity of the meat.[...]
Actually, my plan was to let the thermal conductivity be a variable and then show that for all but the two extreme cases of a perfect conductor (no overshoot because TA=TB always) and perfect insulator (TB is static) there will always be an overshoot as the B region transfers heat to the cooler regions A and C. This would, to my mind, be enough to show (crudely) that the concept is viable.

Mentor
It doesn't feel to me like this should be difficult to model, with a few not too impactful simplifications.

Since I've gotten into the reverse sear, I always use a thermometer now, and it wouldn't be hard to record the internal temperature every few minutes. And maybe with an infrared thermometer the surface temperature too.

gnurf
I actually found a fun site from MIT where you can test the cooking time for a particular thickness of meat.

http://up.csail.mit.edu/science-of-cooking/

It includes carryover cooking if in the final step, you set the temperatures of both side of the meat to room temp. So they obviously have a model that they are using for this. I haven't had the time to look around yet to see what model they are using.
Thanks for that link. From the About page on that link you can read that the simulator is based on the Crank-Nicolson Method [1] and I found the git repo at [2]. While the simulator does seem to back up what most people who cook recognize as carry over cooking, I'm looking for the simplest possible way to convince someone who does not believe that the interior can increase in temperature when the meat has left the oven. I find that notion quite illogical and a refutation does not, I think, need to involve heat diffusion through meat or any advanced math. Rather, I'm looking for a way to show, mathematically, that heat transfer from warm to cold is just as applicable from the outer part of the meat to the interior as it is from the outer part to the ambient surroundings.

[1] https://en.wikipedia.org/wiki/Crank–Nicolson_method
[2] https://github.com/laurabreiman/science-of-cooking.

Staff Emeritus
While the simulator does seem to back up what most people who cook recognize as carry over cooking, I'm looking for the simplest possible way to convince someone who does not believe that the interior can increase in temperature when the meat has left the oven.

Wait, if you want to find the "... simplest possible way..", then why are you going into the route of finding a mathematical model? Shouldn't something conceptual be enough?

The center of the meat does not know it has been taken out of the oven. All it knows is that the neighboring region has a higher temperature than it. And due to heat transfer, it will continue to be heated due to such a temperature gradient until the neighboring temperature is lower than its own temperature.

Is this not the "simplest" explanation one can give?

Zz.

nasu
Gold Member
Do not risk experimenting with expensive meat. The steaks could be too high!

Gold Member
The setup could be simplified by eliminating half the meat.

If you want to know how the internal temp changes in a piece of meat that is, say, four inches thick, it is tantamount to using a two inch "layer" of insulation made out of steak.

Apply heat to one side, place the probe on the other side of the meat. You'll get a fairly uncomplicated measure of the thermal conductivity of meat - independent of a lot of confounding factors.

Staff Emeritus
The setup could be simplified by eliminating half the meat.

If you want to know how the internal temp changes in a piece of meat that is, say, four inches thick, it is tantamount to using a two inch "layer" of insulation made out of steak.

Apply heat to one side, place the probe on the other side of the meat. You'll get a fairly uncomplicated measure of the thermal conductivity of meat - independent of a lot of confounding factors.

It doesn’t have to be that complicated. There are meat thermometers. Stick it in the center and put the roast into the oven. Then, after it has cooked long enough, take the roast out, and monitor the temperature. Both the thermometer probe and the meat surrounding it are at thermal equilibrium (as opposed to sticking the thermometer into the roast after it has been heated).

The increase in temperature is usually quite easy to observe during the first 5 minutes.

Zz.

rbelli1
Gold Member
It doesn’t have to be that complicated. There are meat thermometers. Stick it in the center and put the roast into the oven. Then, after it has cooked long enough, take the roast out, and monitor the temperature. Both the thermometer probe and the meat surrounding it are at thermal equilibrium (as opposed to sticking the thermometer into the roast after it has been heated).

The increase in temperature is usually quite easy to observe during the first 5 minutes.

Zz.
The OP is looking for a mathematical solution.

I am eliminating confounding factors - such as the shape of the meat - to get at the individual factors. The way you're doing it, you'll get different results for a round steak than a flat steak for example.

You'd want to be able to plug in numbers such as width and height thicknesses separately.

So: one independent factor is simply the linear thermal conductivity of meat. You'd have a generalized formula, to which to could apply various other factors as-needed, such as changing the shape of the meat in a known, controlled way.

Gold Member
The setup could be simplified by eliminating half the meat.

If you want to know how the internal temp changes in a piece of meat that is, say, four inches thick, it is tantamount to using a two inch "layer" of insulation made out of steak.

Apply heat to one side, place the probe on the other side of the meat. You'll get a fairly uncomplicated measure of the thermal conductivity of meat - independent of a lot of confounding factors.
This is the equivalent to Lee's Disc experiment for measuring insulators. But is there not data on thermal conductivity of almost everything one can think of? You could maybe assume the conduction is via the water in the meat (not the convection of course.

russ_watters
Gold Member
Heat flow in a solid is governed by the diffusion equation, which describes a "slow" process. An element of solid that is somewhere between surface and center begins to cool when the temperature at the surface suddenly falls (roast taken out of oven). Its heat flows to cooler volumes, namely towards the surface but also towards the cool center. This is why the center continues to heat for a short time.

To understand it mathematically, realize that all driven diffusion problems (including eddy currents) display a phase shift compared to the driving function--what happens inside is time-lagged relative to the outside driver. This is covered in all treatments of diffusion. You can certainly simulate it with a numerical code, but analytic solutions are also available. Here, for example, are plots of temperature versus time at various depths in a semi-infinite slab subjected to square-wave temperature variation at its surface, taken from Carslaw & Jaeger, Conduction of Heat in Solids, 2nd ed., 1959, sec. 2.6:

Note the phase shift--at all depths the temperature continues to rise a bit after the surface temperature falls. What you call carryover heating is most noticeable at depth "2" in this plot. Although this is a slab, similar behavior is seen in all geometries.

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russ_watters and phinds
Gold Member
It doesn’t have to be that complicated. There are meat thermometers. Stick it in the center and put the roast into the oven. Then, after it has cooked long enough, take the roast out, and monitor the temperature.

This would, to my mind, be enough to show (crudely) that the concept is viable.

Why do we need to prove that it is viable? This thing happens in practice. Any reasoning as to why it can't happen is plainly wrong.

BoB

Homework Helper
Gold Member
I know little about this, but it rang a bell with me, something to do with the origin of the Fourier transform.
So maybe something like this and this is of interest? (I haven't looked at them myself, just the first results of searching on Fourier transform and heat transfer.)

Gold Member
, something to do with the origin of the Fourier transform.
Fourier's Law of heat transfer is not related to the Fourier transform afaiaa (apart from the same clever bloke being responsible).

votingmachine
FWIW, I cooked a rib-roast with the thermometer in it, following an internet recipe and the temperature rise was about as predicted. I took the meat out with the internal temperature at 110-degrees-F, covered it with aluminum fool tent, and the internal temperature was at 132-degrees-F after about 20 minutes.

https://therecipecritic.com/prime_rib_recipe/

It makes perfect sense that the process is simply heat transfer that averages the surface temperature and internal temperature. I assume that if I had taken the temperature for each mass layer, and averaged that, then the final average temperature was close to the final internal temperature.

My guess is that the temperature expectations are already available in different recipes.

rbelli1
Gold Member
My guess is that the temperature expectations are already available in different recipes.

Certainly. What size roast did you cook? The 2-6 pound range in the recipe will have a wide range of "perfect" cooking regimes depending on size.

BoB

votingmachine
Certainly. What size roast did you cook? The 2-6 pound range in the recipe will have a wide range of "perfect" cooking regimes depending on size.

BoB
I was right in the center of that range ... a little over 4 pounds.

I will propose a possible added factor - Cooking is typically the Denaturing of Proteins and breaking down of cells, this process is temp dependent ( Aging meat is similar and takes more time) - but as most reactions are moving to a lower energy state, I will guess and say it is somewhat exothermic.
(xx Consider the combustion of compost xx - see edit)

While I am totally speculating - I do have an expert to consult! And will return with their feedback.

E-> In writing up my query I realized/remembered compost heat is due to bacteria braking down the material and giving of heat ( respiration) - so not relatable here!

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