# How do I build a DE to model temperature inside something?

This is really a small cluster of questions on the same thing.

I was working with this problem to do with the temperature inside a cooler. The idea was that the rate of change of temperature inside the cooler was proportional to the difference in the inside and outside temperatures. So

$x'(t) = k(z(t))$

where z(t) is the difference in temps at time t.

But then I learned that this was only good for approximations around t=0, since it's a linearisation of

$x'(t) = f(z(t))$

where $f$ is some function on the temperature difference. If I wanted something more than an approximation and didn't want to linearise, what's a good choice of function for $f$?

Also, I could have built a function around the ratio of the inside to outside temps. What's the criteria for choosing between the two choices: the ratio and the difference?

Also, if I didn't know about logistic functions, and I'm building the DE from scratch, where do I begin so that a logistic function 'emerges'; what's the difference in the assumptions I make between the sum model and the ratio model that leads to a logistic equation?

If this question is too messy, I'll happily delete it.

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You could always take the Taylor expansion out to two terms! – Emily Nov 7 '12 at 18:23

## 1 Answer

Newton's law of cooling can be expressed by the following ODE:

$$\frac{dQ}{dt} = -hA(T_{a} - T)$$

Where $h$ is the heat transfer coefficient, $A$ is the surface area of the object being cooled, and $T_{a}$ is the ambient temperature.

The solution is:

$$\Delta T(t) = \Delta T(0) e^{-t/t_{0}}.$$

This will give you the difference in temperature at time $t$ in terms of the (presumably known value) of initial difference in temperature at time $t = 0$.

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