Flow through an "annular orifice". 2

[Goosen]

Greetings Mechanikals,

I need some help in determining the flow through what I will call an “annular orifice”.

1. I have a smooth pipe with an inside diameter of 24 in.

2. Centered in that pipe is a thin plate with an outside diameter that is 0.08 in. smaller than the inside diameter of the pipe.

3. I know the pressure drop across the plate to be 0.5 psid.

4. The medium is water at 80 deg F.

What is the formula for calculating the flow through this annulus?

Thanks in advance,

Goosen

 

[ve7brz]

Try:

http://www.efunda.com/formulae/fluids/calc_orifice_flowmeter.cfm

 

[Goosen]

Thanks for the link but this refers to an orifice in a pipe. My problem is a bit different since it involves an annulus. My understanding is that simply determining the area of the annulus and converting that area to an orifice diameter is not an appropriate approach.

Goosen

 

[BobPE]

Goosen:

No, you will have to consider it as an annulus, you are right. Unfortunately, I don't have the time to look it in my Fluids book, but any advanced fluid mechanics text will have that, it's fundumental to training engineers since it's such an interesting flow problem.

BobPE

 

[ve7brz]

Sorry, I didn't wasn't thinking in terms of an annulus. The same web site gives calculations for venturiis, which might give a better approximation to your annulus.

 

[Goosen]

Thanks for confirming my understanding of the problem, Bob.

I only have my undergraduate Fluids text and it does not address this problem. That is why I am here seeking the expertise of this gathered assembly.

Goosen

 

[zdas04]

I think you'll have to wait for BobPE to find the time to look in his books. I dug through all my MS fluids books and can't find the thread that unravels this problem.

You might turn the annular area into an equivilent diameter [((ID+OD)^2)*(ID-OD)^3]^(1/5)] and use orifice calcs, but I wouldn't have a lot of faith in the answer.

 

[vanstoja]

Flow through an annulus is more complicated than flow through an orifice so empiraical data seems to be needed. This is discussed in Lenkei,A. (1965), &quot;Close Clearance Orifices&quot;, Product Engineering, 4/26/65. pp. 57-61. Empirical data from one or more of 5 cited references is plotted to determine a K-factor that multiplies velocity. The K-factor must be determined by trial and error solution using the ratio of radial clearance to length and gap Reynolds Number. K values range from 0.1 to 0.7 for Re's from 5 to 20,000 when c/l > 0.8. For c/l < 0.8, K values drop asymptotically toward zero for Re of 200 or less while higher Re's peak around c/l =0.2 and then drop off as c/l gets lower. I set the equations up in an Excel spreadsheet but can't yet match a sample problem results and need to do more debugging before I can input your values and use the graphs to determine flowrates. There is another classic ASME paper on annular orifice flows from the early 1960's by Tao and Donovan which I haven't seen for awhile and can't cite exactly. One of the references cited by Lenkei is Bell,K.G. & Bergelin,O.P. (1956), ASME Transactions titled &quot;Flow Through Annular Orifices&quot;

 

[abeltio]

Goosen...
The annular flow problem simplified for industrial applications can be found in the following classic reference:
Process Heat Transfer by Donald Q. Kern
Chapter 6, Counterflow: Double Pipe Heat Exchangers

The whole chapter deals with Heat Transfer and Pressure drop in annular flow...

In your case, being the OD of the internal plate only 0.08in smaller than the ID of the 24in pipe you may have laminar flow... which is a whole different story.
HTH
Saludos.
a.

 

[hacksaw]

Simple problem. You estimate the free area, assume a discharge coefficient, and calculate the flow. It is only an estimate, but it is actually pretty good.

Most flow or mechanical engineering handbooks have all the information you'll need.

 

[Goosen]

Vanstoja,

Thank you for the references. I will investigate them. I also appreciate your time spent iterating with the spreadsheet. Please let me know if you can confirm the sample problem results.

Hacksaw,

I tried the approach that you recommend early on and I have difficulty accepting the results. That is, the flow seems higher than I would expect. That is not to say that it is not approximately correct, only that the result caused me to pause and wonder if this problem, with the very small annulus width-to-diameter ratio did not call for some other approach. I found other formulations with regard to annular flow, of the inner and outer pipe (very unlike an orifice) and 2. the annulus width-to-diameter ratios were not nearly as small. The results from these equations were even more unrealistic. This is truly an &quot;annular orifice&quot; problem.

Goosen

 

[Goosen]

Oops,

I left out some text in my response. It should have said,

****************

Hacksaw,

I tried the approach that you recommend early on and I have difficulty accepting the results. That is, the flow seems higher than I would expect. That is not to say that it is not approximately correct, only that the result caused me to pause and wonder if this problem, with the very small annulus width-to-diameter ratio did not call for some other approach. I found other formulations with regard to annular flow, *where 1. there was relative significant length* of the inner and outer pipe (very unlike an orifice) and 2. the annulus width-to-diameter ratios were not nearly as small. The results from these equations were even more unrealistic. This is truly an &quot;annular orifice&quot; problem.

Goosen

 

[vanstoja]

I found the reference for Tao & Donovan (1955), &quot;Through-Flow in Concentric and Eccentric Annuli of Fine Clearance with and without Relative Motion of the Boundaries&quot;, Trans. ASME, Vol. 77, Nov., pp. 1291-1301. Unfortunately, I haven't located the paper yet which, as I recall, has a multiple-graphic solution scheme. Haven't had time to further debug Lenkei's equations but will try again soon. There are many solution schemes for labyrinth seal flows in the literature I've surveyed but they won't help much for a plain annulus geometry. Many of the more recent impeller seal annular gap studies focus on rotordynamics properties rather than leakage flows so they're not much help either.

 

[vanstoja]

I think I got Lenkei's equations debugged but &quot;buried&quot; the article somewhere and still have to check it to be sure. Ran his example problem with a 6 in. OD, radial clearance of 1 mil, gap length of 10 mils at 140F and got a flow of 0.313554 lbs/sec(about 2.25 GPM) with a K factor of 0.71 at a Reynold's No. of 1240. I need to know your plate thickness to input gap length. If you're not sure what it is, I can try several lengths and/or any other parameter changes you want to try.

 

[vanstoja]

PS The gap delta P was 20 psi in the example.

 

[Goosen]

The gap length is 0.060 in.

Thanks again for your efforts, vanstoja, I really appreciate the help.

Goosen

 

[vanstoja]

Found the &quot;lost&quot; article and the example problem checked out OK. I input your 60 mil gaplength with appropriate viscosity and density values for 80F and got for your 0.667 c/l ratio a K factor of 0.71 with a flowrate of 7.985 lbs/sec (approximately 57.449 GPM).
At your c/l of 0.667, the curve shows all Reynolds numbers from 500 to 20,000 bunched around K=0.71. Your Re is 4518.296 which would be turbulent.
Good news or bad news?

 

[Goosen]

vanstoja,

A colleague did a simple CFD analysis and your result comports well with what he found.

It is somewhat &quot;bad news&quot; in that we were hoping the leakage would be much less than this, but &quot;the facts is the facts&quot;, as they say, and THAT is why I like practicing engineering.

Thanks again for your time and efforts.

Goosen

 

[sloquick]

Goosen

I'm interested in knowing of the references you found that would help calculate/estimate capacity loss of gas flow through a pipe into which a smaller pipe (d/D < .25) is inserted. In other words annular flow outside of innerpipe. Thanks!

 

[hacksaw]

Goosen,

The calculation is commonly used in estimating the leakage flows for large (swing-thru) dampers and butterfly valving, etc.

If the gap is greater than the thickness of the plate, then you can use the thin restriction discharge coefficient, if the thickness is comparable to or greater than the gap, you have to assume what is known as a thick plate restriction.

The real problem is not your fluids calculation, but your ability to confirm the annular gap.With 0.5 psid on a thin plate you will actually bend the plate unless it is re-inforced, and the leakage will be much greater.

Differential thermal expansion is also an issue. Even a 10 deg difference between shop and service conditions will cause a considerable difference in the calc. flow.

Back to the flow calc. All flow calc. are based on the bernoulli equation (ultimately). You only have to sort out the discharge coeff. based on hydraulic similarity, and the hydraulic radius of the flow channel.

good luck,


Valve