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I'm working on temperature prediction and therefore also cooling. I stumbled upon Newton's Law of Cooling (NLC) and I do like its simplicity, but I'm not so happy about the condition that the surrounding temperature must be constant.

Original formula I'm working with $$T(t) = (T_0 - T_A) * e^{^-kt} + T_A$$

$T_0$ is temperature at $t = 0$

$T_A$ is ambient temperature

$k$ is the inverse time constant

$t$ is time.

The problem I have with this formula is that it assumes instant cooling from $T_0$ and that $T_A$ is constant. The resultant cooling is something like below:

enter image description here

Imagine an oven at 150$^oC$ with a sausage that is 70$^oC$. If the oven shuts off suddenly, the ambient temperature will start to drop. Therefore $T_A$ cannot be constant. Furthermore, $T(t)$ will not instantly drop when the oven drops, but will increase until $T_{oven} = T(t)$. So what if the formula was modified so that $T_A$ also is dependent on time?

Well that is exactly what I did. I played around a bit with the formula and ended up using the formula recursively, so that $T_A$ is also determined by NLC.

This gives me the formula:

$$T(t) = (T_0 - ((T_{A0} - T_{AA}) * e^{^-k_At} + T_{AA})) * e^{^-kt} + (T_{A0} - T_{AA}) * e^{^-k_At} + T_{AA}$$

I know, bear with me, please.

In out sausage example, $T_{A0}$ is now the oven temperature, and $T_{AA}$ is the ambient temperature outside the oven. The neat thing about this is that it is possible with acceptable accuracy to assume that the outside temperature of $T_{AA}$ will not change with time as it is a large room. $k_A$ is the time inverse constant for the interaction between oven and outside ambient temperature.

Now the problem arises when we want to know what $k$ and $k_A$ is, as the formula now is much more complex. I've managed to sidestep this problem with assumptions and some iterative coding to estimate the $k$ and assuming that $k_A$ can be derived from $$\frac{dT}{dt}=-k_A (T_{A0}-T_{AA})$$ This is all beyond this question though.

To conclude my question:

The results are quite nice giving me a much more realistic cooling curve as seen below:

Modified Cooling

Unfortunately, as you might have weaned from this question, I'm not math savvy and so I have no idea whether I've broken like a 100 fundamental rules of mathematics or not.

If the results are good, is this modification (gross infraction) acceptable?

Is there anything in the pure mathematics that says I cannot use this to predict/estimate cooling times of objects in given circumstances such as the oven example?

Comments, pointers, raging scolds and so on are all welcome

Cheers

Full disclosure: I asked this question in math.Stackexchange and was referred to this forum.

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  • $\begingroup$ infraction of what sort of law/principle are you referring too? $\endgroup$ – Buraian Oct 2 '20 at 9:13
  • $\begingroup$ I guess i'm worried if i'm breaking some basic rule within mathematics or physics. What i'm looking for is any information about this type of modification to Newtons law of physics, because for me it seemed obvious to do this, but then again i'm a philistine when it comes to mathematics. $\endgroup$ – brendbech Oct 2 '20 at 9:26
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    $\begingroup$ I edited the picture into your post. $\endgroup$ – Buraian Oct 2 '20 at 10:00
  • $\begingroup$ I mean, if you can find a better model for the situation, then why not? $\endgroup$ – Buraian Oct 2 '20 at 10:00
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Newton's law of cooling is a differential equation which states (in one form);

$\displaystyle \frac {dT}{dt} = -k(T-T_A)$

and you have given the solution to this equation when $T_A$ is constant and $T(0) = T_0$. But the differential equation still applies if $T_A$ is not constant - it just has a different solution.

In your extended scenario we have two linked differential equations, one for the temperature of the sausage $T(t)$:

$\displaystyle \frac {dT}{dt} = -k_1(T-T_{A})$

and the other for the temperature of the oven $T_{A}(t)$:

$\displaystyle \frac {dT_{A}}{dt} = k_2(T-T_{A}) - k_3(T_{A} - T_{AA})$

where $k_1, k_2, k_3$ are positive constants and $T_{AA}$ is the constant ambient temperature outside of the oven. You now need to solve these linked differential equations with initial conditions $T(0)=T_0, T_A(0) = T_{A0}$. Since they are linear ODEs, there should be an analytic solution. I doubt your ad-hoc solution is correct, but I haven't checked it.

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