How can heat exchange alone (no mixing) bring cold blue water at 1 degree up to 99 degrees using an equal...

physics/thermal.inversion New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

How can heat exchange alone (no mixing) bring cold blue water at 1
degree up to 99 degrees using an equal amount of hot red water at 99
degrees, so that the red ends at 1 degree?

[ Editor: problem statement reconstructed from the postings.  The
  original puzzle is not in the collected material; the reply speaks of
  red and blue water, of temperatures 1 and 99 degrees, and of "total
  thermal inversion", and the reconstruction may not match the puzzle as
  first posed. ]

Show solution

The problem is, every time you let heat flow between parts in thermal
contact, the entropy increases.  But the final state you want (total
thermal inversion) has the same entropy as you started with.  So we have
to let the heat flow adiabatically(!)  In our case this means the
temperature of any two pieces in contact must be infinitesimal.  This is
possible, for example, if we separate the blue water into infinitesimal
pieces, and each piece has a different temperature, ranging from 1` to
99`.  We get this situation by letting an infinitesimal piece of red
water contact each piece of blue water in turn.  Repeating for all the
red pieces, the final state has all the red pieces at 1` and all the
blue pieces at 99`.

Another way to look at it would be to let the water slowly flow in
opposite directions in two adjacent, ideal tubes.  All the water to the
right of the center point (in both tubes) will be 99`, and all the water
to the left will be 1`.  There will be a transition region in the middle
which will widen with time, but taking appropriate limits, and
idealizations, the region can be made as small as desired.

-- David Wayne Ring <Cdude@phys.tamu.edu>