Re: exponential cooling

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Leigh asked me:

> 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).
> ....

It sounds ok to me if you are motivated enough to do it.  I think the only
important reason why one would want to go to the bother of numerically solving
the 3-spatial dimension PDE for the brick-shaped problem is if there is some
overwhelming desire to see how differently the corners of the brick cool than
the sides.  If one just wants to see (and demonstrate to students) how the
temp. profile tends to behave with depth I think the much simpler 1-spatial
dimensional problem of the spherical brick should suffice.  OTOH I suppose
that doing the 3-d calculation may be instructive if one wants to make (and
show to students) graphs of the concentric isothermal surfaces and see how
they are "rounder" at depth and more "brickish" with sharper corners in shape
near the surface.

> ....                     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.

The diffusive parabolic equation may involve the Laplacian operator but it is
fundamentally different to solve than the elliptic PDE's such as Laplace's
equation and the Poisson equation.  One typically doesn't need to do a
relaxation calculation on a parabolic PDE; one starts with the initial Cauchy
data and just propagates the spatial state at each time slice forward in time
(after making sure that the chosen time step is small enough relative to the
spatial grid size so that numerical instabilities do not occur).

> 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.

If you want to speed up the 3-d calculation and still show corner effects,
how about using a cubic brick.  In this case you only need to explicitly
calculate the solution on 1/48 of the brick.  In this case you only need to
consider the simplex-shaped region 0<= x <= y <= z <= L for a cube of side L.

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

I've never liked spreadsheets that much.  (Maybe I'm just too old and set in
my ways.)  Whenever I want to numerically solve a Diff. Eq. I program it using
one of the standard algorithms for that purpose (such as are found in the
_Numerical_Recipes_ book).

> 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?

Just what I need!  The students curse me enough now already as it is.

David Bowman
dbowman@gtc.georgetown.ky.us