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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.

tryst with freedom
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brendbech
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1 Answers1

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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.

gandalf61
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