Steady, Frictionless, Adiabatic, One-Dimensional Flow of a Perfect Gas

Many of the most interesting features of high-velocity gas flow can be seen in the simplest of all cases, the steady, frictionless, adiabatic, one-dimensional flow of a perfect gas. We study this type of flow in detail other types are treated more briefly, because they have so much in common with this one. 

The flow iS assumed to be in some kind of a duct or pipe or closed channel of varying cross-sectional area A. As long as this cross-sectional area changes slowly with distance down the duct, the velocities in the direction perpendicular to the main flow will be small enough to neglect and we can treat the flow as one-dimensional. The gas is assumed to be perfect and to have a constant heat capacity Cp. 

The open-system energy balance (Chap. 4) between any two points R and 1 in such a duct, for steady flow without heat transfer or turbines or compressors, is 

It can be readily shown (Prob. 8.8) that the potential-energy changes Ag2 are negligible for most high-velocity gas flows, so we drop the gz terms from this equiation. Next we assume that state R is some upstream reservoir, where the cross-sectional area perpendicular to the flow is very large therefore, Vff is negligible. This condition is referred to in various texts as the reservoir, stagnation, or total condition. We call it the reservoir condition and use the subscript R. 

If we now divide both sides of this equation by RkTIM, we find 

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