Dear anindya,

I’m not sure I can follow your doubts.
I’m not blaming you for this… it’s just my incapacity to resolve your doubts vs. my "knowledge".

In API 520 "2.4.1 DETERMINING REACTION FORCES IN AN OPEN DISCHARGE SYSTEM" first paragraph is saying "The following formula is based on a condition of critical steady-state flow of a compressible fluid that discharges to the atmosphere through an elbow and a vertical discharge pipe. The reaction force ( F ) includes the effects of both momentum and static pressure; thus, for any gas, vapor, steam,….etc"

So it’s clear that the intention is to consider a steady-state flow counted as critical where the jet exists in the atmosphere. In brackets…the formula is not looking to what is happening in PSV. Why? In my opinion because is nothing serious to give a reaction force there, at least in steady state flow. It’s funny sometimes Vendor is giving a reaction force in PSV but saying it’s counted as PSV is free discharging at outlet flange. In my opinion, in this case the real effect is "free jet" at outlet flange, not specifically the PSV presence…

In Fluid Mechanics the terminology "critical" is referring to Mach=1 in that section and the critical velocity is counted as sqrt(kRT*) with R=Ru/M.
To obtain T* , the theory is counting an isentropic evolution between stagnation conditions in vessel (T0 @ Mach=0) and critical condition (T* @ Mach=1) in exit section. The Fluid Mechanics rule for that is
T*=2*T0/(k+1).
Now you can get the API result, and you can calculate c*= the critical velocity sqrt(kRT*) as
sqrt(2Ru)* sqrt[kT0/(k+1)/M].
In SI, Ru=8314.47 J/kgmol/K so numerically sqrt(2Ru)=129.
So the first term of API's SI formula is
Wxc*=129*W*sqrt[kT0/(k+1)/M] with W "flow of any gas or vapor, in kilograms per second" as API says.

API is not entering in details for the second part of formula- effect of static. They are just saying is A = area of the outlet at the point of discharge, multiply by P = static pressure within the outlet at the point of discharge.

The formula has two parts and must include the effects of both momentum and static pressure.
Nothing is double counted- in my opinion. Both expressions must be evaluated- API says.

Now I’m not sure what’s your doubt.
Could be the assumption that the gas/vapor expands isentropically from protected vessel/ pipeline to the exit? True, it seems that the PSV existence is not specifically counted… And true again, that pipe will choke at more than one location along its length.
But this assumption (isentropic evolution on all path) just provides a convenient mean to determine the maximum capacity / reaction force, in my opinion.
And- again in my opinion- "choke" is not synonym with "free jet"...the real troubles are where we have a free jet giving a reaction force, so exactly where API is looking for...

You said "But , my question is, why this expression should be a substitute for (Pexit-Patm)* Exit area?"
No, shouldn't replace the second term. Just the fact the second term is not so easy to be counted and is moderate as value...