Sonic flow for a polytropic expansion

It is shown in specialized texts on fluid dynamics that a convergent-diveigent nozzle is needed to accelerate a gas from subsonic to supersonic conditions, since gas acceleration in the subsonic regime requires the flow area to diminish with speed, while gas acceleration from sonic to supersonic speeds requires the flow area to expand with speed. The subsonic, convergent part of the nozzle is linked to the supersonic, divergent part of the nozzle by a duct of constant flow area, known as the throat, which is kept very short in practice in order to avoid frictional losses. The throat is the only section of the nozzle in which sonic flow can occur, and it is impossible for the throat to support any speed greater than sonic. The above remarks apply to all polytropic 

It follows from the previous paragraph that a convergent-only nozzle cannot produce supersonic velocities. Such nozzles are most often used to produce subsonic outlet velocities, but it is possible for them to yield a sonic velocity at the outlet, since the outlet in a convergent-only nozzle is equivalent to the throat of a convergent-divergent nozzle, and we will often refer to it as the throat . 

For either type of nozzle, the speed of the gas in the nozzle throat will increase as the pressure ratio decreases, until the speed of sound sets a limit. The speed of sound in the throat can be shown to be a function of the local thermodynamic state  

To give an idea of the pressure ratios involved, consider a diatomic gas such as nitrogen, where the specific-heat ratio, y = 1.4. For an isentropic expansion, the polytropic index is equal to the specific-heat ratio, and substituting n = y = 1.4 into equation 

Equation (5.51) is indeterminate at = 1.0, but nevertheless we can gain an insight into the behaviour by setting n = 1.01, when pdP = 0.4978. (A more complete treatment of the theoretical case of sonic velocity in an isothermal expansion is given in Section 5.4.3.) 

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