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.)
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 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 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.
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