Supersonic discharge velocity

We may assume that the nozzle will be designed to produce a supersonic discharge velocity, and that the design pressure ratio will coincide with the lower critical discharge pressure ratio for the design inlet conditions, i.e. (pi/por)lo = (Pimi2/Por)lo-The nozzle will be choked at this point, and this implies that the pressure at the throat will be at its critical value. The associated throat critical pressure ratio is given by equation (14.55) ... 

If Pi is increased, the acoustic velocity may be shown to remain unaltered, but, since the density of the gas is increased, the rate of disharge is greater. The orifice and the venturi tube are alike insofar as discharge capacity is concerned. The only difference is that with the venturi tube, or a converging/diverging nozzle, a supersonic velocity may be attained at discharge from the device, while with the orifice the acoustic velocity is the maximum value possible at any point. 

When flow is subsonic, < 1 all terms on the right in these equations are then positive, and dP/dx < 0 and du/dx > 0. Pressure therefore decreases and velocity increases in the direction of flow. The velocity increase is, however, limited, because these inequahties would reverse if the velocity were to become supersonic. This is not possible in a pipe of constant cross-sectional area, and the maximum fluid velocity obtainable is the speed of sound, reached at the exit of the pipe. Here, dS/dx reaches its limiting value of zero. For a discharge pressure low enough, the flow becomes sonic and lengthening the pipe does not alter this result the mass rate of flow decreases so that the sonic velocity is still obtained at the outlet of the lengthened pipe. 

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