Re: cooling

Henry Allen wrote:
> 
> So,
> This maybe be a little crazy but I guess that never stopped anyone
> from posting to usenet...
> 
> It is well known that the rate of cooling of a volume depends largely
> on the surface area of that volume.  I am curious what happens in the
> limit...
> 
> Suprisingly, It is possible to constuct a curve that has an infinite
> perimeter while only containing a finite area (e.g. snowflake curve).
> And, if you rotate this curve in space, you wind up with a volume that
> has an infinite surface area while only containing a finite volume.
> So, direct application of most rate of cooling equations would imply
> an infinite rate of cooling. It's entirely possible, then, to
> construct a theoretical object you can hold in your hand that would
> drop the earth to zero kelvin if it came into contact with it.;)  Idle
> speculation on the properties of such an object is amusing, I find...
> 
> Granted as atoms have a finite size and all this isn't terribly
> possible, but it does indicate that it's possible to construct
> surfaces with a huge surface to volume ratio- how do the conventional
> heat flow equations break down when you have really enormously large
> surface areas?
> Thanks!
> henry

OK, take a Sierpinski (Menger) sponge with infinite perimeter and zero
area,

<http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=3524&objectType=FILE>
http://www.h33.dk/sierpinski_index.en.html
http://world.std.com/~j9/sponge/index.html

Most of everything will radiate into itself as you asymptotically
approach nothing/volume.  You'll get a transparent body (no stuff, no
optical absorption, no emittance) with the apparent extrneal surface
area of the polyhedron.  A good physical model for this is very low
density silica aerogels

http://eande.lbl.gov/ECS/aerogels/saprops.htm
http://stardust.jpl.nasa.gov/spacecraft/aerogel.html

with densities as low as 0.003 g/cm^3.  You have a nearly invisible
lump (fair approximation to nothing/volume) that does not transmit
heat or sound.  One supposes that an *ordered* lattice constituting a
photonic crystal would exhibit narrow intense transitions.

--
Uncle Al
http://www.mazepath.com/uncleal/qz.pdf
http://www.mazepath.com/uncleal/eotvos.htm
 (Do something naughty to physics)