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Macroscopic
processes in the real world are almost without exception irreversible ...
We
calculate the change in entropy of a system by devising an imaginary
*reversible* thermodynamic path from the initial to the final state.
In any event, the heat we are
talking about here is *always* a quasistatic exchange of energy between
two systems that occurs specifically as a result of an infinitesimal
difference in temperature.
Furthermore, I think that the modern perspective of virtually all
textbooks is that the short and long term result in the Joule experiment
is increased internal energy, not heat.
I'd like to reserve the word heat to mean *essentially* this same thing in
all circumstances. I say "essentially" because I am willing to soften my
definition to include *nonquasistatic* exchanges between systems that
occur as a result of *finite* temperature differences. Otherwise, I'd
just as soon call everything else work to make a clear distinction between
work--which can be arbitrarily distinguished from heat for use in the
first law--and heat--which must conform to a far more rigid definition for
use in the second law.