Gas Critical Flow Across Orifice 5

[chemengbr]

I have been working on finding steam flow across an orifice, as this flow will dictate the relief requirements for a relief valve down stream of the orifice. Therefore, I decided to create my own excel calculator to be very detailed, so that future engineers can see how I arrived at my numbers as well as see my sources/citations.

Here is what is bothering me, for my calculations I've been using the critical flow equation from Perry's 8th edition (eqn 10-31, pg 10-19).

Flow = C * A_orifice * sqrt[g_c*k*(p1/v1)*(2/(k+1))^((k+1)/(k-1))]

where C is disch coeff, g_c is gravitational constant for English units, k is specific heat ratios, p1 is upstream pressure, and v1 is upstream specific volume. Besides my own calc, I have three other sources that are matching each other pretty closely, while my calc is about 15% lower. Two resources are excel sheets one of each I explain below, and the other resource is a website.

This second sheet I am looking at has Flow = C * A_orif * sqrt[g_c*k*density_1*r_c*(2/(k+1))] ... r_c is the critical pressure ratio defined as (2/(k+1))^(k/(k-1))
This equations is almost identical to mine since 1/v1 = density_1. Only difference is that last term in Perry's is (2/(k+1))^((k+1)/(k-1)), while this second sheet is r_c*(2/(k+1)). This sheet cites Mink, Chem Eng Aug 25, 1980 and Daugherty & Franzini, Fluid Mechanics. Unfortunately, I have not been able to locate these sources online.

Lastly, the site below also calculates the steam flow, but they don't explain how their equations were derived. Anyhow, I am thinking I might doing something slightly wrong since the other three resources pretty much match, while mine is 15% lower. Thoughts on what might be the issue? I did use the same disch coeff for all four calcs.

https://www.tlv.com/global/US/calculator/steam-flow-rate-through-orifice.html?advanced=on

 

[casflo]

The Crane´s TP 410 sugests in the graph of Y that the pressure ratio in the thin plate (downstream preesure/upstream pressure) may be less tan 0.4, the mínimum value of the graph, without choke.
Unless for saturated steam, my own calculations of Y,indicate that the thin plates don´t choke if d/D is equal or less than 0.5, but for d/D > 0.5, the thin orifice plates may choke to pressure ratios of 0.2 and great. NOTE that d is the orifice diameter and D the internal diameter of the pipe.
In relation with the thick plate orifices, the classification between thin and thick is not unanimous among the authors. Some of them consider also that the plates are thin with t/d < 2.
In any case, for values of t/d > 5 usually the thick plates may be named as long thick orifices. These type of orifices are installed in some system of the nuclear power plants and when the fluid is a liquid, some manufactures of high pressure pumps install them in the mínimum flow recirculation lines.