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Anyone seen this equation
3

Anyone seen this equation

Anyone seen this equation

(OP)
I was reviewing some material for a course I may be teaching and came across the attached equation for gas flow through a hole in a pipe.  I've never seen this equation before (and I've been looking for something like this for years), so I plotted it up (also in the attachment).  

According to this equation, there is a significant discontinuity in flow as you go from Mach=1.0 to Mach=0.999.  Nothing I've ever seen supports this discontinuity.

Has anyone ever seen this equation before (and what is it called)?

David Simpson, PE
MuleShoe Engineering
www.muleshoe-eng.com
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RE: Anyone seen this equation

(OP)
BigInch,
That was an interesting article, and I'm going to spend some time comparing it to the models that I've been using, but both the models in the article are choked-flow models.

What I'm looking for is a non-choked flow model.  I'm not sure I'm ready to accept the one I posted, and I'm looking for someone who has used it and found it to adequately reflect field conditions (my choked flow model has always been within a couple of percent of field observations, so I'm pretty comfortable with it).

David

RE: Anyone seen this equation

Thought you were looking for an independent source for the same eq.

If not choaked flow, can't you use the general equation for gas flow through an orifice.

RE: Anyone seen this equation

Taken from equations for isentropic flow in the limit of vanishingly small hole diameter and subsonic flow conditions


 

RE: Anyone seen this equation

3
I have seen something similar (but not properly the same) to the equation you've posted for the non-choked compressible flow through an orifice.
Attached what I am referring to with the limitation of orifice to pipe diameter ratio lower than 0.25).

There is also something in the link below from page 15 on.

http://www.osti.gov/bridge/purl.cover.jsp?purl=/7318185-YThVku/
 

RE: Anyone seen this equation

(OP)
ione,
Thank you for that, while it doesn't have the exact equation I was asking about, it certainly points to the origins of that equation.  It also has great explainations of several concepts that I've struggled to explain for years (e.g., the incompressible flow equations have an upper limit of velocity where they can be applied).

David

RE: Anyone seen this equation

David,

Not that I consider this source to be one of the best amongst those available (sometimes free contribution without further validation leads to nothing but garbage)....anyway

Check the expression (6) in the attached link

http://en.wikipedia.org/wiki/Orifice_plate
 

RE: Anyone seen this equation

The equation in the document brought by ione is the root for all variations of incompressible flow equation. It is free of  numerical constants induced by specific unit system and therefore less prone to errors. It can be used with any system units as long as it is used consistently.  

RE: Anyone seen this equation

(OP)
ione,
If I could give you another star for that post I would.  I'm not a huge fan of Wikipedia, but that article is really useful.

Equation 6 has a 2/(k-1) inside the radical that the equation I was asking about is missing.  When I added that term into my equation, the discontinuity dropped from 75% to 34% and the shape of the curve above Mach=1.0 (where it isn't valid) had some similarity to the choked flow curve.  I expect that there is another equation to describe the flow between choked conditions and velocity of 0.6 Mach where most authors start beleiving that the incompressible flow calculations start to become valid with decreasing velocity.  I expect that in real flows, the time that that equation would be valid is really short (miliseconds) and it wouldn't be material.

Thanks again.

David

RE: Anyone seen this equation

David,

One final note to be taken with extreme care (please check this as here in Italy it's already Friday evening and I could have taken apples for pears).

Whilst equation (6) reported in the wikipedia link is correct from a dimensional point of view (mass flow rate results expressed in kg/s) it seems the equation you have presented in your first post is not consistent as well.
 

RE: Anyone seen this equation

What the heck.  I gave him one 4U, but not for the wiki.  I had already cached that one.

RE: Anyone seen this equation

(OP)
ione,
The one I posted becomes dimensionally correct with the "gc" term.  Whenever I see an equation with that in it I take GREAT care to make sure that it is right since so many people mess it up.  MathCAD takes 32.17 ft-lbf/(s^2-lbm) to equal 1.0 with all units cancelled.  For this equation, I put it in with units on all the variables and got a mass flow rate number.  Then I put it in with all the units removed (and including the 32.17) and got the same number.

The equation was diminsionally correct, just arithmetically wrong.

BigInch, thanks.

David

RE: Anyone seen this equation

"iwon" and and UOme12. smile

RE: Anyone seen this equation

David,

I have disregarded the subscript c and taken gc as it were gravity acceleration (ft/sec^2). Sorry to have asked to check this (I would never have to doubt that you had already done this dimensional control and probably double-checked it, before starting a new thread): it really was friday evening!
 

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