Don Foreman writes:

What waste? Heat is still energy. If the downstream air is colder
than ambient then heat is not lost by conduction or radiation to
ambient, so that energy is still in the downstream gas.

The point of the dispute was whether overcompressing and then regulating
back down to application pressure was more/less/equally efficient
compared to just compressing to application pressure (or just above).
In short, are regulators inherently "lossy".

Yes, air regulators are inherently lossy. The answer is, all other
things being equal, overcompressing/regulating-down is always *less*
efficient, in a degree increasing with the degree of overcompression and
down-regulating.

This is not to say that practical considerations won't make the
inefficiency necessary. For example, you may have only one supply pipe
across a building going to a variety of tools, and you therefore need to
transmit the pressure for the highest-pressure tool. Or your pipes may
be too small to transmit enough air at lower pressure. Or you may need
very close regulation of the application pressure and widely varying
delivery volumes.

If I may risk another analogy, compressing and transmitting air at
excess pressure (above what is needed for the application) is *not* an
efficiency measure like boosting voltage up and down for long-distance
electric power transmission.

Maybe we're getting closer to understanding, though. We've agreed
that a certain amount (mass) of air is compressed, and some heat is
lost in that process. Compression to higher pressure results in more
work being converted to heat (usually lost) at the compressor or in
the reservoir as that air cools by conduction to ambient -- and
perhaps a little radiation as well. But what about after that, as
in from just before the regulator to point of exhaust? I think that
was the original question.

Yes, that was the original question. But the answer is, regulators are
lossy, even if you assume the compressor is 100 percent efficient and
lossless.