Dear Anindya,

A "k+1" in denominator formula is naturally coming-up from Laval nozzle theory. See, for example, Roberson, Crowe‘s "Engineering Fluid Mechanics", paragraph "mass-flow rate through a Laval nozzle".

API formula is correct against the assumptions made: a perfect gas evolving isentropic from stagnation state (M=0) to critical state (M=1). You can correct this formula for a specific real gas. But staying under API calculation it means to follow the formula.

Now few ideas not related to your comment but to Laval theory.

Although isentropic flow is an idealization, it often is a good approximation for the actual behavior of nozzles. Since a nozzle is a device that accelerates a flow, the internal pressure gradient is favorable. This trends to keep the wall boundary layers thin and to minimize the effects of friction, Fox- McDonald "Introduction to fluid mechanics"

In my opinion that’s why we cannot extrapolate directly the Laval theory as valid for a PSV model (I refer here only to the PSV). A lot of friction is generated in the PSV's body (and we want to have that friction!) and it’s hard to think that after the PSV throat the fluid accelerates to supersonic flow.

It is one reason I don’t agree with Brandmaier and Knebel’s article…They said "The flow leaving the valve orifice can be characterized as an underexpanded jet exhausting into a larger diameter cylindrical pipe. The flow expands across a series of expansion waves to supersonic velocities until its static pressure equals the pressure surrounding the jet…."
IMO, there are few IFs here: IF the wall boundary layer is thin; IF the friction effects are kept to minimum, IF the internal pressure gradient is favorable… Nevertheless, with such turbulence and friction developed in PSV body, for me it’s hard to believe such behavior.

I reattach this article; I remember you have posted it some time ago. Thank you for your generosity!

My kind regards