Coefficient Of Discharge For Depressurizing Orifices

When the pressure ratio is less than critical,
a)many literature sources indicate Cd continues to increase when r<rc, and is of the order of 0.9 or so when r approaches zero.
b)for a thin plate RO, flow continues to increase beyond what is indicated by choked flow limitations, even when the beta ratio is less than 0.2. Flow is choked when t/d ( t= plate thickness, d= RO inner dia) exceeds 7 or so. A thicker plate RO is thus required for plant emergency depressuring services to enable limiting flow to choked conditions when the pressure drop ratio is less critical ( ie r<rc).
c)I get the impression that flow through globe type control valves is limited to critical values when r is = or < than critical, as evidenced by control valve vendor sizing calc outputs.

I agree with georgeverghese, I copy the answer to your post on another forum :
when depressuring P, T (and W for state changes) are not constant, fluid properties and process change passing from critical to subcritical conditions, different procedures may include different parameters and you may contact Aspen for the details,

I have a different solution (based on Prode Properties) for piping I adopt the methods proposed by Leung and for orfices (or valves) several correlations (considering critical and subcritical conditions during depressurization), the procedure calculates the flows depending from fluid properties (process conditions) and dimensions of restriction orifice, I suppose Aspen has equivalent methods, in both cases you can specify or obtain the dimensions of restriction orifice,

I have not an extensive comparison but I do not expect large differences (from different procedures) except, perhaps, in some specific cases..

The discharge coefficient Cd of a thin plate restriction orifice is approximately 0.6 to 0.62, depending on the Reynolds number and the ratio do/D, being do the hole diameter and D the internal piping diameter. See the Crane Technical Paper Nº 410.
Other thing is the flow coefficient C = Cd/[1-(do/D)^{4]0.5 that according to the Crane depends of do/D and the Reynolds number in the pipe and varies between 0.6 and 0.76
The square-edge orífices of the thin plates have not critical conditions.
If the plate is thick, it may have critical conditions and in this case Cd is 0.84}

Sorry for english, I have rare practice.
Do not believe them. Here is my point of view:

1. Critical flow is not well researched. Sources have not common position or consensus.

2. Only labaratory tests were conducted. Only air was used as a medium.

3. Cd is a consequence of vena cotracta effect.

Shape of vena contracta depends on plate thikness, bore shape, fluid properties and upstream velocity profile. Fluid properties depend on fluid, pressure, temperature and velocity. It means that Cd can vary when upstream pressure/temperature varies during depressurization. It means that Cd values measured in labaratory with air are not suitable for process industry.

4. Shape of vena contracta haviely depends on shape of bore inlet port and upstream velocity profile.

Cd values mentioned above are suitable only:
a. for square edge orifice
b. when bore diameter is much less than diameter of upstream pipe
c. when swirls and local disurbances are avoided (stright run of upstream pipe)
d. when upstream liquid accumulation is avoided (no liquid carryover or vertical installation or drain holes)
Some corelations for velocity approach factor do exist (e.g. see equation 9.47 Miller Flow Measurement Handbook 3rd ed.) but they were obtained in labaratory with air (see statement 3).

5. Cd depents on t/d. "Cd vs t/d" curve has some flat part, but start and end points of this flat part depends on fluid properties and metal roughness (see statement 3).


6. Plate thickness depends not only on choise of Cd. Minimum plate thikness should be provided to prevent plate bending (e.g. see mim plate thikness in Miller Miller Flow Measurement Handbook).

7. No labaratory tests were conducted for fluids that partially condense at critical velocity or downstream pipe. No labaratory tests were conducted for two-phase fluids (saturated or liquid carryover). No labaratory tests were conducted for effect of wear of bore inlet port due to liquid drops abrasion. It means depressurization shall be carefully studied and liquid carryover/formation shall be avoided.

It is my opinion - all mentioned above means that Cd is site specific. Any Cd obtained from an academic literature should not be applied to process industry, even this Cd value has been obtained during reliable labaratory test.

Some links.

Thank you for all the replies and in particular to Shvet for the articles.
I cite from Perry:
“…, unlike nozzles, the flow through a sharp-edged orifice continues to increase as the downstream pressure drops below that corresponding to the critical pressure ratio rc. This is due to an increase in the cross section of the vena contracta as the downstream pressure is reduced, giving a corresponding increase in the coefficient of discharge. At r = rc, C is about 0.75, while at r ≅ 0, C has increased to about 0.84.”

The coefficient of discharge 0.62 appear to be the asymptote for high Reynolds number for liquid or gas flow in non-chocking conditions.
Now, what I don’t understand is why API 520 I specify 0.62 as a coefficient of discharge to be used for sizing rupture disks for critical gas flow. Any idea?

IIRC, ASME suggested 0.62 for rupture disks many, many years ago. I don't recall seeing any proof.

Good luck,
Latexman

To a ChE, the glass is always full - 1/2 air and 1/2 water.

that number may work for some orifice types under certain conditions,
but, as generic approach for blowdown models, IMHO : better to consider the different conditions and orifice types (see my previous post)...

These days I look for the "certified flow coefficient" from each vendor/model/phase.

Good luck,
Latexman

To a ChE, the glass is always full - 1/2 air and 1/2 water.

I would go with Latxman and the 0.62 for design. As far as i recall the Miller book example (it may be a different book by Miller) is about a orifice for flow measurement where the dP is much smaller than a orifice for flow control.

It's very surprising the sentence 7 of shvet that "No laboratory tests were conducted for two-phase fluids as saturated wáter". I recommend to see, as a sample the following papers: "The Flow of Saturated Water Through Throtting Orifices" by M. W. Benjamin and J. G. Miller, Tansactions of the ASME July, 1941 and "Low Pressure Differential Discharge Characteristics of Saturated Liquids Passing Through Orifices" by T. J. Rohloff and I. Catton, Journal of Fluids Engineering, Transactions of the ASME, September 1996.
I agree that there is no a complete consensus in the t/d values that define a restriction orifice plate as thin or thick and that the the design of the entrance to the hole, affects to Cd, but don't forget that the engineering calculations for fluid systems in the industrial plants have not the same exactness as the laboratory tests and usually use the more conservative data.

yes, the problem depends from different RO types, conditions etc.
and, for two phase flow, things are much more complex with boiling delay, vapor-liquid slip (in HNE models) etc.

We are talking about Cd. Cd is a difference beetwen calculated and measured mass flowrate through orifice.
Let's back to reality. There are so many gases and their mixtures. There are many thermodynamic models of critical flow. There are so many physical factors that can not be assessed. A scientist conducted a laboratory test and measured gas mass flowrate. A scientist implied assumptions, created model of discharge and calculated "ideal" mass flowrate. Then a scientist divided measured by "ideal" flowrate and obtained Cd. What does it mean?
1. Cd refers to particular model and particular set of assumptions. When one is talking about Cd the model and assumptions shall be defined.
2. Any model has some accuracy and limits. When one is talking about Cd accuracy and limits shall be defined.
3. Laboratory test was conducted using one (set of) gas(es). Measured Cd can be applied with precautions to the same gas(es) in the same conditions in process industry. Measured Cd can be applied with precautions to the similar gases and similar conditions using similarity parameters. When one is talking about Cd reference gas(es) or group of gases (e.g. one-atomic, di-atomic) shall be defined.
<<<It is my opinion - measured Cd shall not be applied to complex poly-atomic gases and even more to two phase mixtures bacause no one conducted laboratory tests with these.>>>
3. Cd is not a phisical factor or phenomena. Cd is a measured differenc beetwen a model and reality. It means that Cd includes effects from many factors and one group increases Cd while other decreases. Real facility and laboratory device differ each other. Real facility tests shall be conducted and real flowrate shall be measured. When one is talking about Cd scale factor shall be defined.
<<<I have not found any link to real facility tests for critical flow orifices. Is there any evidence these were conducted?>>>
4. Critical flow through orifice is not well researched. Critical flowrate through orifice depends on critical velocity and shape of vena contracta. Critical velocity depends on fluid parameters at vena contracta and fluid parameters at vena contracta depend on critical velocity. If fluid parameters (especially heat capacity ratio and isentropic index) vary during flowing through orifice critical velocity calculation is complex as requires integration of group of interrelated parameters. One-equation calculation of mass flux through orifice is simple but inaccurate.
<<<Two-phase critical flow model does not exist at all>>.

When one is asking on an anonym forum: "Hey, guys. Give a value for coefficient of discharge?" and immidiatelly takes a reply: "Be sure, 0.62 (0.83, 0.95 - no matter) is correct." I ask myself: "What the hell is going on here?". Which is t/d, shape of bore? Which model? Which gases? Can two phase mixture occur? Which conditions? What are the tolerable consequences of inaccuracy (flame out, excess flare radiation, piping vibration, liquid abrasive erosion, ex-cloud formation etc)?

Reader, ask yourself:
Engineering handbooks (e.g. Perry's) are created by qualified and competent scientists. There are so many handbooks. There are so many groups of authors. They are so smart, they have so many knowledge, they have so many authority. They are paid money and respect for their work. Coefficient of discharge in a handbook is one line on sheet with 5-10 words. Why they did not write coefficient(s) of discharge in a handbook? Why here (on an anonym forum) we are discussing only one book - ~25 years old Miller's Handbook? Why here we are discussing some laboratory test reports which are 30-80 (!) years old?