Darcy equation for compressible flow...eq. development question

[seasar]

Can anyone point me to a resource that would show how the version of the darcy formula found in crane tp410 was developed. The equation is:
w= 0.525*Y*d^2*(delP/(K*v))^.5

w=mass flow rate
Y = expansion factor (developed quantitavily)
d= pipe ID (inches)
delP = pressure drop across system of interest
K= res. coeff. of system
v= specific volume of fluid (cuft/lb)
Thanks!
SeaSAr

 

[Latexman]

That's not the Darcy equation. It's for compressible flow through a nozzle or orifice.

The Darcy equation is valid for liquid flow through pipes. With suitable restrictions, it can be used for gases and vapors in a pipe.

Good luck,
Latexman

 

[Latexman]

Start with q = YCA(2g(144)dP/rho)^0.5 and convert to mass flow rate and convert units.

Good luck,
Latexman

 

[katmar]

The Darcy equation is a practical application of Bernoulli's Theorem, but it is specifically developed for incompressible flow. In the same way that Darcy can be derived from Bernoulli (with a bit of empirical work for the friction factor) you can derive equations for compressible flow as well.

There is a density factor in the Bernoulli Equation. This is easy to handle when you integrate from beginning to end under incompressible conditions because it remains constant by definition. Under compressible conditions the fluid density changes along the flow path. In order to deal with the changing density you have to assume some relationship between the fluid density and the pressure. The usual assumptions are either isothermal or adiabatic. Now you have a relationship that you can plug into Bernoulli and integrate. This is where the Crane formula comes from.

Whenever you see the "Y" factor (or anything to do with the ratio of specific heats) appear it means that adiabatic conditions have been assumed. The calculations under isothermal conditions are easier to handle, but the results might be slightly conservative (i.e. higher pressure drop or lower flowrate) for short pipelines.

The best explanation of all of this, with the relevant derivations, that I have seen is from the series "Chemical Engineering" by Coulson and Richardson. See Chapters 2 and 3 from Volume 1.

Harvey


Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[vzeos]

seasar
The equation that you refer to is applicable to compressible flow through valves, fittings and pipe. The pipe friction factor, f, pipe length, L, and pipe diameter, D, are contained in the resistance factor, K, where K=fL/D. The net expansion factor, Y, compensates for the change in fluid properties during expansion, see Crane page 1-9. The Y factor is seldom considered in pipeline analyses in which pressure changes are not that great.

For a development of the compressible flow equation check out A Handbook of the Petroleum Industry by David Talbot Day, page 438:

http://books.google.com/books?id=Qq...stry",+"pipe-line+formula"&as_brr=1#PPA438,M1

Equation 10 corresponds to Crane equation 1-6. Equation 11 is pretty much the general pipe flow equation used today. Note that this book was published in 1922 and the Prandlt, von Karman and Colebrook equations for friction factor were not yet published. So for friction factor, use the more modern equations.

Also check out Engineering Thermodynamics by Charles Edward Lucke, page1111:

http://books.google.com/books?id=pp9KAAAAMAAJ&pg=PR6&dq="Thermodynamic",+"Lucke"#PPA1111,M1

 

[seasar]

Thanks for the help. The application is compressible flow through pipe..specifically venting of a liquid CO2 tank to atmosphere through a valve and pipe. The outlet is limited by sonic velocity, which is accounted for in that equation via "Y" and a modification to the maximum delP. I need to understand how that equation was derived although I am aware that a lot of it was developed experimentally.

 

[katmar]

See Milton Beychok's site at

http://www.air-dispersion.com/feature2.html

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[seasar]

Katmar,

I've seen that site before, unfortunately, this scenario involves a relatively minor release from a very large tank. A 100,000 lb tank of 240psig liquid CO2 is vented periodically to release non-condensables or relieve pressure. This occurs for a matter of a few seconds via 3/4" pipe. The very minor pressure drop in the tank isn't measurable with the installed equipment.

 

[vzeos]

Seasar,

Crane does not give an equation for Y for the flow through pipes. They only provide the charts on page A-22. I believe these charts are based on data determined by experimentation. However, some control valve manufacturers do publish something like the following equation which has the form of Crane’s charts:

Y = 1 – (x/(A*F_k*x_T))

x = dp/p₁
x_T = terminal value of x for choked flow.
F_k = k/1.40
k = gas specific heat ratio
A=constant, 3 for valve-pipe system.

Using Crane’s data, you should be able to determine constant(s) for the above equation such that it will correlate with Crane’s charts.

Also, see attached link for more details:

http://www.documentation.emersonprocess.com/groups/public/documents/book/cvh99.pdf