The energy equation that I found for steady, compressible fluids comes from source:
"Cengel, Y. A. & Cimbala, J. M., 2018. Chapter 8: Internal Flow. In: Fluid Mechanics Fundamentals and Applications. New York: McGraw-Hill Education, pp. 351-424."
and it looks as follows:
(qin-qout)+(win-wout)=cp(T02-T01)+g(z2-z1)
The equation for incompressible fluids considers the head experienced by the pump, turbine and minor and major head losses. Are these effects negligible in compressible fluids?
In principle, you can not say nothing only by considering compressible flows.
I suggest to go in deep in the proper non-dimensionalization of the system of equations.
Be careful in chosing the proper variables and have a look to the non-dimensional numbers such as Re, Pe, Ma, Nu, etc .
Make a proper analysis based on the correct choice of the characteristic scales.
Remember also when compressible flows are modelled by Euler or NS equations.
In general, when in physics we qualify any quantity as "negligible", we must specify "negligible compared to which other quantity". Otherwise, "negligible" in absolute terms is meaningless. Now, whatever the fluid, the pressure drop, for it to be negligible, must be very low vis-à-vis the effects of gravity and inertia. With regard to the incompressible fluid (gas in this case) the effects of inertia, gravity and pressure drops depend on the density and therefore on the pressure. If we place ourselves in the case of low pressures (around atmospheric pressure for example) and we propose to compare the pressure drops of two equivalent flows of liquid and gas, we can as a first approximation say: ( i) if we consider two flows of gas or liquid with the same volume flux, the gas flow pressure drops will be much lower than those of the equivalent liquid flow; if, on the contrary, two flows of gas or liquid are considered with the same mass flux, the pressure drops in liquid flow will be much lower than those of the equivalent gas flow. It suffices to calculate their ratios according to the ratios of the density and viscosity by considering the conservation of the mass.
The answer is sometimes they are negligible compared with say friction, and sometimes they dominate: for example a water turbine converts head to energy and the head must be maximised and flow losses minimised; the Archimedes screw is a cse of pusing water up-hill and overcoming the head difference.
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