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).

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/ball vlave(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.

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