Re: exponential cooling

[palmer@sfu.ca (Leigh Palmer)]

> So are we concluding that there is no exponential factor in the solution or
> just that it is not simply exponential?

I conclude that it can't be solved in closed form. If you mean "exponential"
in sort of a vulgar way, then yes, the cooling of the brick will be faster
at first, then slow down, and it will approach equilibrium asymptotically.
I don't think we should ever tell students that, since an exponential has
that behaviour, any behaviour of that sort is "exponential".

For the benefit of thoes of you who would like to try, here's my Maple
result:

Given a hot brick of constant heat capacity C, area A, emissivity epsilon,
placed in a vacuum surrounded by walls at temperature To: the temperature
of the brick, T, varies according to the relation

                 dT                        4     4
              C ---- = - epsilon sigma A (T  - To )
                 dt

sigma is the Stefan radiation constant. Integrating this one obtains

 1   ln(T-To)     1   ln(T+To)     1   arctan(T/To)
--- ---------- - --- ---------- - --- -------------- - K = - epsilon sigma A t
 4        3       4        3       2          3
        To               To                 To

where the constant of integration K is given in terms of the initial
temperature Ti:

        1   ln(Ti-To)    1   ln(Ti+To)    1   arctan(Ti/To)
   K = --- ---------- - --- ---------- - --- --------------
        4        3       4        3       2          3
               To               To                 To

I don't particularly cotton to the task of inverting that function to
find T(t). I leave that as an exercise to the industrious reader. I will
point out that while it is difficult to do in closed form, the curve can
easily be plotted as t vs. T. Lying on one's side would then make the
curve look like T vs. -t. Maybe you can put it on a transparency and
look at it lying on your side from the back side. Oh, you might plot it
initially as T vs. t as well! How about that?

I think that making a numerical plot of this solution is a valuable
exercise, easily accomplished nowadays by students with a computer or
even a sophisticated graphing calculator. As I pointed out in my "Big
Crunch" problem earlier, such exercises are a good way to get students
rid of the idea that they are impotent in the face of an equation
they've not been explicitly taught to solve.

All of this is probably quite irrelevant to the application of the
original questioner, but it is certainly appropriate to the introductory
college physics course. Students should finish such a course with a
sense of increased ability to handle real world problems using physics.

Leigh