Is the integral of temperature over time useful?

Just a hypothetical question. Suppose I have an oven that heats food, and I can measure the temperature (over time) accurately at the surface of the food being heated (this may or may not be feasible). Is the integral of the temperature measurement (wrt time) useful in controlling the cooking time? (The actual heat transfer function is unknown, as is the heat capacity of the food, as is the nature of cooking. It is assumed that such issues are too complex to calculate. The oven may contain a heating element that turns on and off with some duty cycle.)

• Measuring the surface temperature is certainly feasible - you can just put a thermometer there - but I'm not sure I understand why you're discounting the possibility of knowing anything about the food or the cooking technique. Commented Oct 22, 2019 at 23:32
• It seems to me that, at a minimum, one needs to know the heat capacity of the food. Otherwise, it doesn't matter what the surface temperature is doing (for example, if the heat capacity is very large). Commented Oct 23, 2019 at 1:07
• Chet, I think what you say makes sense. The oven would have to be calibrated for the type of food. I conclude that my idea for better cooking is useless. It's good to eliminate bad ideas. Commented Oct 23, 2019 at 13:36
• Also, from a purely mathematical/thermodynamic point of view, adding temperatures (which is what you're doing when you integrate the temperature over time) is not mathematically meaningful and therefore not physically meaningful, even if you are using an absolute scale for the temperature. Commented Oct 23, 2019 at 15:58
• @march Integrating temperature over time does not mean adding temperatures. It means adding the product of temperature and time. It can correspond to heat transferred, given constant (or approximately constant) food surface temperature. It can correspond to other things in other situations. One analogy is Integrating speed over time means adding distances (the product of speed and time), and does not mean to add speed (which does not make sense here). Commented Jan 17, 2020 at 19:14

6 Answers

Yes, the integral is useful for control, for example in a PID controller. It is then the difference between the measured temperature and the setpoint. The power to the heater is primarily proportional to this difference. Adding a term proportional to the integral will make it reach the setpoint sooner.

• I don't think this answers the OP's question. Commented Jan 5, 2021 at 16:21
• @DavidWhite OP explicitly mentioned control.
– user137289
Commented Jan 5, 2021 at 16:26
• Well, the dynamic integral of temperature error with respect to temperature setpoint is not the same as the integral of temperature with respect to time. Perhaps the OP should rephrase his question, because I definitely read it differently than you did. Commented Jan 5, 2021 at 16:31
• I stand by my mistake. I also like what Pieter posted. Cheap toaster ovens are notoriously unreliable when doing almost any kind of cooking. I still think that adding a cheap optical infrared temperature measuring device, looking at the food surface, with suitable electronic control, could potentially result in superb cooking results. But proving this based on physics is quite a challenge. Commented Jan 6, 2021 at 22:51

I think we should answer this with a clear no. Your oven might expose food to 350 F for 40 minutes and bake a nice cake. Or it might put food in a 3500 F blast furnace for 4 minutes and make not such a nice cake. Or it might put food in a 4000 F blast furnace for 4 minutes, then try to correct things out a bit by refrigerating to -50 F for 40 minutes. Or you could try slow cook at 70 F for 800 minutes.

Normally temperature calculations are more sane in kelvin, but I think if you go through the above scenarios you can see that you could do all the same permutations. We could make this quantity slightly relevant if we start looking at heat transmission to an extremely cold reservoir that is proportional to temperature, but it would be at best contrived.

• The question was about process control.
– user137289
Commented Jan 5, 2021 at 15:33
• I think that perhaps Mike didn't read the question carefully. I could be wrong. Commented Jan 6, 2021 at 22:59
• I reread the question - it still seems consistent with this response. I listed several conditions where the integral of T(t)dt would be the same, but which would lead to very different outcomes. Commented Jan 7, 2021 at 14:59

Normally temperature probes measure the internal temperature of food being cooked. The surface of food can heat quickly, while the interior is still cool. Most food being cooked will have a known cooking time at a certain temperature setting (from previous experience).

• So true. Irrelevant to the question, though. I was trying to evaluate a new idea. Commented Oct 23, 2019 at 13:37
• sorry, keep the new ideas coming, Commented Oct 23, 2019 at 15:20

The chemical reactions important in cooking typically have a strong dependence on temperature. For this reason, it is the temperature reached that is more important, rather than the time spent at any given temperature.

Having said that, spending a longer time at a given temperature does lead to more cooking, so, yes, the integral you mentioned would be of some value. More valuable, arguably, would be to integrate a quantity proportional to the cooking rate. This would be a strong function of temperature, such as an exponential. The temperature would have to be measured inside the item, of course, not just at the surface.

• Cooking is obviously a very chaotic system. If all the oven can do is turn heating elements on and off, then there is no algorithm that can cook perfectly. There will likely be a good probability that there is no period of time that works: below a time t the food is not completely cooked, and above t the food is overcooked. I was hoping that adding an inexpensive thermistor (optically focused on the food surface) and logic could remove the chaos. Perhaps controlling a fixed duty cycle might work. Commented Feb 22, 2020 at 1:12

About the only use I see for integrating Kelvin temperature over time would be an attempt to squeeze an entropy measurement to the quantum limit. Entropy has units of Joules/Kelvin. In base SI units, that is kg⋅(m^2)⋅(s^−2)⋅(K^−1). The corresponding measurement involved in the Heisenberg uncertainty principle would require units of (s)⋅(K) in order to satisfy (delta entropy)⋅(delta integrated temperature with time) = (h_bar), which has units of angular momentum.

Our oven does not keep temp well, varies a lot around the setpoint. I've noticed that it always seems to take longer to cook than recipes indicate. So I'm implementing a monitor (on a small processor with a thermocouple in the oven) to try and predict when something will be done given that the temp varies. Something like "growing degree days" in horticulture (the integral of average daily temp - 10 c) Given that I know what the recipe says should be done time at a given temp, I would calculate the number of "cooking degree minutes" that represents assuming a perfectly constant temp. Then track the actual temp over time and calculate a running "degree minutes" to see when it should be done. No extreme temperatures, so the crust/interior problems are not relevant, they are determined by the original recipe. My question is, what should the base temp be, analogous to the 10C of growing degree day? About 100F? Thoughts? Mike