General overview of compressible pipe-flow

The entrance to the pipe acts as a nozzle, accelerating fluid that is originally at rest in the upstream vessel from a velocity C = 0 to a velocity cy. The entrance to the pipe is shown as a well-rounded, convergent nozzle, assumed frictionless, with the pipe 

Chapter 5 presented a mathematical model for adiabatic flow through a frictionless nozzle, and we shall use some of that chapter s results. It will be helpful for our present purpose if we derive the Mach number at station 2 , where the Mach number, M, is defined as the ratio of the velocity to the local speed of sound. Recalling equation (5.47), the Mach number at station 2 will be  

Very little energy is lost to friction in the convergent part of any nozzle, and so we may write down the velocity at station 2 by using equation (5.22), setting n = y for frictionless, adiabatic flow  

The isentropic expansion through the nozzle will, of course, obey the relation 

Combining equations (6.1) and (6.5) gives the Mach number at station 2 as  

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