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If U is internal energy, or the
energy if the constituents if the system, can we still think of it as being
conserved? It may or may not be constant, but energy that enters or leaves
the system can be accounted for. This internal energy would not be
constant, but it would be conserved.
Some folks try to simplify the equation by introducing the
“internal energy” U. However, to me that seems like one step
forward and two steps back. For one thing, the definition of
E is reasonably well known, whereas it’s hard to learn and
hard to remember what’s supposed to be included in U.
Secondly, in situations where the parcel moves significantly
in the dh direction, if you just ignore h then energy is not
conserved. You have to keep track of m g dh one way or another
(either as part of the dE of the parcel, or as part of the
energy flowing across the boundary). So the only remaining
question is whether you will need one new variable or two
(h alone versus h and U both).