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

 

[hacksaw]

That is normal as you are assuming a "thin plate restriction," in you formulas.

 

[chemengbr]

Latexman, thanks for the article, and like you said not sure if it helped me or confused me more.

The article seems to argue against the existence of a maximum flow caused by critical conditions,but unfortunately I am still a bit confused on how to take that and translate into a reliable mass flow calculation for pressure ratios below the critical ratio. Are you aware if there are any widely accepted equations for max flow assuming critical flow is real? Likewise, is there a standard equation that assumes critical flow to not limit total flow?

Perry's seems to be a reliable source and that's the equation I am currently using. Being honest I've only found Perry's equation so far for flow at critical or below critical pressure. It just really bothers me Perry's and this other sheet that I have are almost identical equations, but just different enough to make a difference. Then the TLV website equation seems completely random to me, but flow matches almost identically to the flow on this other sheet that I can't seem to verify the equation. Have you ever seen this sheet's equation anywhere by any chance? I've attached a picture of the perry's eqn and one for the other sheet.

 

[chemengbr]

Last post only attached the sheet formula. So here is the one from Perry's.

 

[Latexman]

chemengbr,

On May 2 I was having a senior moment and could not remember the name of the guy that brought Cunningham's article to my attention, but I knew where I had the article stashed. Today, I remember! It was Dennis Kirk! He is/was a member here at ET. His post that introduced the subject to me in 2003 was thread798-51260 . Using Search on "Dennis Kirk" I found the 2006 thread378-164311 where I found his website

http://www.ozemail.com.au/~denniskb

and this PDF that tries to tie it all together.

Again, while not an answer to your post, it is highly pertinent. I hope it helps. Remember, Search (under thread title and between Forum and FAQs) is a powerful tool here at ET.

I will reply to your May 5 question above shortly; I'm not in a position to do so right now.

Good Luck,
Latexman
Pats' Pub's Proprietor

 

[chemengbr]

Had some time between meetings today, and cracked Perry's open one more time to read it a bit more closely. Here is what I found.

"For the case of critical flow through a square- or sharp-edged concentric circular orifice (where r ≤ rc, as discussed earlier in this subsection), use Eqs. (10-31), (10-32), and (10-33) as given for critical-flow nozzles. However, 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. See Grace and Lapple, loc. cit.; and Benedict, J. Basic Eng.,
93, 99–120 (1971)".

So looks like I have the right equation from Perry's and I also found the exact same equation for choked flow in the NCEES PE Chemical Reference Handbook. In fact, for an orifice application mass flow continues to increase at critical conditions due to a change in disch coeff as described above. For my application I was gonna use a discharge coefficient of 1 as a conservative number, since this is for a safety application. However, it would nice to refine my spreadsheet so that disch coeff could be estimated. My plan is to continue towards refining my calculator/sheet, but for my current application I think I will use Perry's equation with a discharge coefficient of 1.

 

[hacksaw]

The experimental discharge coefficients for square edge orifices for plate thickness comparable to the bore size is 0.84, for thin plate devices, 0.72, you only approach 0.99 for critical flow devices (venturia or nozzles). There are plenty of flow tests in the literature.

 

[Latexman]

It appears one equation can be converted to the other if you focus on the differences and you substitute the definition of the critical pressure ratio:

r_c = (2/(k+1))^(k/(k-1))

I've seen both.

Good Luck,
Latexman
Pats' Pub's Proprietor

 

[casflo]

The initial post says that the fluid through the restriction orifice is steam. In the design and operation of the power plants, the Crane Technical Paper No. 410 is the more known source used by the engineers for this type of calculations. Besides the flow coefficient C of the orifice plate, it takes into account the expansión factor Y.
See the thread378-466996 where Latexman supplied the graphs of Y.
According to my experience in the design and calculation of the steam systems of nuclear and thermal power plants, the flow of steam through restriction orifice sharp edged thin plates, have not critical conditions if do/D is equal or less than 0.5 and the pressure before the plate, less than 2500 psia.

 

[chemengbr]

Thanks for all the replies and resources. I'll try to review tonight and tomorrow the documents and links posted and will return afterwards.

Latexman, at first I also thought that one equation could turn into the other, but that is not the case here. I tried to work the algebra out, but did not obtain the same equation. Now, admitting that algebra is not my greatest strength, I also did a number test to see if the number came out to be the same. Focusing only on the differences between the equations and using k=1.4 I get the following.

Perry's: [2/(k+1)]^[(k+1)/(k-1)] = (2/2.4)^(2.4/0.4) = 0.335
Other Sheet: r_c*2/(k+1) = {[2/(k+1)]^[k/(k+1)]}*[2/(k+1)] = [(2/2.4)^(1.4/2.4)]*(2/2.4) = 0.749

So the equations are almost identical, but not quite the same.

 

[Latexman]

Yes, I see what you mean now.

Wikipedia's formula matches Perrys.

For the second sheet equation to equal Perry's it would be:
Flow = C * A_orif * sqrt[g_c*k*density_1*r_c*(2/(k+1)^(1/(k-1)))]

Good Luck,
Latexman
Pats' Pub's Proprietor

 

[chemengbr]

Wow, there is a lot more nuance to this matter than I thought at first. It is almost a bit overwhelming, here is what I've learned and will do next.

1)I reviewed Dennis Kirk technical paper that Latexman recommended and it is impressive the amount of work that Dennis put into it. Looks like he compared his model to the Perry's eqn that I am using. It also seems that for Perry's equation he used the disch coefficients from the Grace-Lapple article cited in Perry's and the results were very similar. I was able to get a hold of that article and plan on reading it next. As Dennis mentioned the error for Perry's eqn increases a lot for diameter ratios above 0.6, which is expected since Perry's equation is derived by assuming diameter ratio of 0.2 or less I believe.

2)While looking through one of the threads linked above. I also saw a mention of the following article: Ward-Smith "Critical Flowmetering: The Characteristics of cylindrical nozzles with Sharp Upstream edges" Int J Heat Fluid Fl vol 1 No 3 pp 123-132 1979. This article explains that choke flow occurs for thicker orifice plates, and finds the choked coeff of disch for different orifice thickness to diameter (t/d) ratios. This is very interesting to me as a lot of the "orifices" being used around me are actually what I would call an orifice block where t/d is ~ 6.

3) Ended up purchasing Crane 410 2018 edition this afternoon to see what's there. Not sure if it is possible or not, but ideally I am trying to create a calculator that takes into account orifice t/d, and also the impact of different diameter ratios (beta). This way it can be used for orifice blocks and plates, and for a wide range of beta. Hoping crane might be able to help me with that a little bit. Am I being to ambitious here?

 

[Latexman]

Crane doesn’t address t/d IIRC.

Good Luck,
Latexman
Pats' Pub's Proprietor

 

[chemengbr]

hacksaw, do you have any literature recommendation regarding the relationship between t/d and discharge coefficient? So far, I’ve only skimmed through the Ward-Smith article above, which seem to indicate the following.

sharp edge, t/d= 0, Cd = 1.0
thin plate (0<t/d<1)Cd varies from 1 to 0.81 as function of t/d.
thick plate ( 1<t/d<7) Cd = 0.81 constant
very thick plate (t/d > 7) Cd less than 0.81 per Fanno friction

 

[hacksaw]

NASA Technical Note, TN D-5467 might be useful with a number of what appear to be useful references.

 

[Latexman]

Great idea, I forget about NASA and the wealth of knowledge they have on their servers.

 

[hacksaw]

For your orginal query, just search 'ASME Benedict critical pressure ratio', a couple of papers are listed

 

[chemengbr]

Latexman, hacksaw,and others that have posted, thanks for all your help. I think I just need to sit down for a few hours now and digest all the information and resources provided, as well as organize my thoughts. Later in the weekend or early next week, I'll come back and try to summarize my learnings and new understanding of this topic. At that point, I'd love your feedback (hopefully for a final time), to just make sure I did not misunderstand something.

Thanks

 

[chemengbr]

Ok, so here is my current understanding of this matter.

Thin plate orifices do not experience choked flow. As shown by Cunningham 1951, and many others, mass flow rate across the orifice continues to increase with decreasing downstream pressure. This occurs because the vena contracta is not contained by the orifice, so as pressure downstream is reduced the vena contracta cross-section is increased.

Critical Flow Rate Calculation for Thin Plate Orifices: Perry's Handbook (8th Edition) recommends the following, "For the case of critical flow through a square- or sharp-edged concentric circular orifice (where r ≤ r_c), use Eqs. (10-31), (10-32), and (10-33) as given for critical-flow nozzles. However, 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 r_c. 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 = r_c, C is about 0.75, while at r ≅ 0, C has increased to about 0.84. See Grace and Lapple, loc. cit.; and Benedict, J. Basic Eng.,93, 99–120 (1971)".

Critical Flow Rate for Thick Plate Orifices: In reviewing Ward-Smith "Critical Flowmetering: The Characteristics of Cylindrical Nozzles with Sharp Upstream Edges," which is also cited in the RW Miller Handbook, shows thick plate orifices do obtain choke flow. Here there is a bit of variation, but choked disch coeff for thick plates seem to be around 0.84. The definition of a thick plate is a bit tricky though. Literature reviewed by this article indicates that this disch coeff would be applicable to orifices with t/d ratios (thickness/diameter) on the low end starting at 0.5-1 t/d and going up to t/d around 7-10. After that c_d decreases as t/d increased due to fanno choking. My personal inclination is to assume a thick plate orifice with c_d = 0.84 to be orifices with t/d ratios of 0.5-10, and then after that estimated the discharge coeff change based on the data provided by Ward-Smith.

Couple Questions: 1) Any flaws on my analysis above? As well as on how I plan on treating thick plates in the future? 2)I am still waiting on my Crane copy, but how do people utilize crane for pressure ratios below 0.4? Do you just extrapolate the line to the desired pressure ratio?

 

[ImRB]

Hello Chemengbr,

my suggestion is to go through the Crane's TP 410 for the same.

it has some solved examples for the type of calculation which you are doing. i am sure it will certainly help you.

Take care,

regards,

ImRB