Re: exponential cooling

[palmer@sfu.ca (Leigh Palmer)]

> > Does the exponential law of cooling apply to the radiational mechanism,
> > for example, to a very hot brick suspended in very good vacuum?

Joseph said

> Since in radaition the latter depends on
> the 4th power of the temperature, I wouldn't expect it to be exponential.

and Dewey said

> I recall this mechanism goes as the difference between T^4 of the brick and
> T^4 of 'its surroundings'.  Wouldn't everything that goes as some
> difference be exponential?  It seems to me like the time constant might be
> affected by the power of T involved, but the cooling would be exponential.

Joseph is correct; the cooling rate would not be exponential. I've been
trying to integrate this intermittently while answering email (and through
a power failure). I finally resorted to Maple, and I'm afraid that a three
term expression with two logarithms and an arctangent are the result. The
solution is certainly not exponential, but while I can display it
numerically, I certainly can't express it in closed form.

The "exponential law of cooling" (or "Newton's law of cooling", as I have
heard it called) is not a law of Nature. It is the mathematical result of
a simple model in which a system with a temperature independent heat
capacity (and no phase changes) is weakly coupled to surroundings to which
it loses heat at a hypothetical rate proportional to the difference in
their temperatures. There is no physical process which approximates this
model (usually we work the other way 'round) closely, but the qualitative
result, that the temperature difference decreases rapidly at first, then
more slowly as equilibrium is approached, is true for many systems. For
engineering purposes it is usually necessary to delve into the messy
empirical business of hydrodynamics &c. to get useful answers to *a priori*
problems. Unforced convection in air answers pretty nearly to the hypotheses
of the problem, however. Radiative cooling in a vacuum does not.

Leigh