Re: exponential cooling

[palmer@sfu.ca (Leigh Palmer)]

Ah, David Bowman, you're a physicist after me own heart. You say:

> I think it would be quite instructive to numerically solve it for a
> simple case (such as for a uniform spherical brick) and monitor the
> time dependence of the temp. field as a function of radius.

and many other appropriate comments on this problem. The transient heat
transfer solution is something I didn't consider in my case, assuming
as I did that the brick's temperature remained constant throughout.

Like me, you tend to leap to the spherical brick problem. I think
another approach is well worth considering, however, and that is the
purpose of writing this note. What do you think of this as a way to
approach the problem, and as a way to introduce such problems to
students?

It occurs to me that the problem can be attacked numerically for a
brick-shaped brick (for which even the surface temperature will be
nonuniform after startup) using a marvelous tool, a spreadsheet on a
powerful microcomputer (I have a personal Power Macintosh 7500/100,
soon to have its processor and clock replaced to make it a 604/120).
These machines are becoming common in schools, and I have a Power Mac
native version of Excel. The interior of the brick can be set up as a
relaxation calculation. I've solved many Laplace's equation problems in
just that way. The boundary can have a Stefan-Boltzmann condition
imposed upon it.

In this case the brick configuration is more natural, since
spreadsheets are fundamentally rectangular! Only 1/8 of the brick need
be represented if symmetry conditions are exploited appropriately. This
will involve special treatment at the three interior boundaries, but it
should speed up the computation by quite a bit.

I really like using spreadsheets for doing problems when they are
appropriate. This one is a natural.

This could become the standard of the heat transfer problems in the
modern era - the hot brick problem! Students will curse us long after
we are all dead. What do you think?

Leigh