Finding Cv with K factors - Equivalent Resistance for Hydraulic System 5

[domc83]

Hello,

I am trying to find some resistances (Cv) for a hydraulic system for which I have the K-factors (K, or (f*L/D)^.5). I found an old copy of Crane's TP-410, which supplies the following:

Cv = Q*(rho/(dp*62.4))^.5 = 29.9*d^2/(f*L/D)^.5 = 29.9*d^2/K^.5

where

rho = density in (lb/ft3)
dp = pressure drop (lbf/in2)
d = internal diameter (in)
L/D = equivalent length
f = friction factor
K = resistance coefficient

However, no units are provided for those constants. If they are non-dimensional, the equation is false. Does anyone have any idea how the various steps are derived and what the units of those constants might be?

I need the Cv to use in an electrical analog for the Resistances (v = i*R, p^.5 = q*1/Cv)

Thanks in advance

 

[AndersE]

I don't remember the units in the Crane formula, but I have derived the same type of equation myself using SI units and the Fisher Control Valve Handbook formula for Cv.

Basically it can be done by replacing delta-P in the formula for Cv with K*rho*V^2/2. The velocity V is of course a function of volumetric flow rate and pipe inner diameter D.

It takes a few steps, but you will end up with a formula where K is a function of Cv, D and some constants. I have the formula written down somewhere, I'll come back when I find the paper.

 

[katmar]

The constants are dimensional, and are correct for the units you have listed. In the metric version of Cranes TP-410 there is the statement "there is not yet an agreed definition for a flow coefficient in terms of SI units" and the same equation that you have supplied (with these non-metric units) is given.

If all you want to do is convert between Cv and K then the only parameter to convert is d, the pipe ID. You could either convert your diameter to inches, or adjust the 29.9 constant to take your units for d into account.

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[AndersE]

OK, here is the formula:

K = D^4/(466.134*Cv^2)

where D is the inner diameter of the pipe in mm.

I don't remember the unit of the constant, so if you need them you should try and derive the formula yourself. It's not very difficult and it gives you a better understanding of how it works.

As I mentioned, I derived the formula from the Fisher Control Valve Handbook, using the equation for Cv with SI units (deltaP in bar, flow in kg/h and density in kg/m3).

I don't think the statement from Crane "there is not yet an agreed definition for a flow coefficient in terms of SI units" really matters that much. There are formulas for Cv with SI units published by Fisher and ISA and they are probably accurate enough in most cases.

 

[katmar]

Anders, sure you can have a formula for CV in terms of SI Units, but there is no definition of Cv in SI Units.

The Cv used in the USA is defined as Cv=1.0 for a water flow of 1.0 USgpm and a pressure drop of 1.0 psi. The English Cv is similar, but defined in Imperial gallons. The European equivalent is Kv which is defined as 1.0 for a water flow of 1.0 m[sup]3[/sup]/h and a pressure drop of 1.0 bar. Crane was simply saying that there is no definition of this type in SI Units.

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[AndersE]

Katmar,

You're right, but I just wanted to point out that no significant accuracy will be lost when using the formula for Cv in terms of SI units instead of the original US units.

 

[sheiko]

Taking the risk to deviate from the OP, i would like to point out that the relationship between K and Cv depends on the proper definition of d in the equation. This is an "effective" diameter for determining the friction loss, and should probably correspond to the minimum flow diameter within the valve (at the seat), but the exact definition isn't clear. It most probably is NOT the port diameter of the valve...

"We don't believe things because they are true, things are true because we believe them."

 

[Latexman]

A Cv value is usually specific to a certain control valve with a certain "trim". Since it is so specific, i.e. diameter and other factors are fixed, Cv does not have a size variable.

A Cv value that is derived from a K value is specific to all the assumptions that went into arriving at that K value, including the inside diameter that was the basis of the K.

Good luck,
Latexman

 

[katmar]

Once you have converted the Cv to a K value, you calculate the pressure drop by multiplying the K by V[sup]2[/sup]. The K value must therefore be based on the same D which is used to calculate V, and this is of course the inside diameter of the pipe.

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[domc83]

Thank you all for your help.

I determined the units for the numerical constants provided in the equations. Now I can meaningfully use the equation that relates K, Q, and dP in both unit systems, which is what I was looking for.

I have attached a pdf with my work. If you would like to look it over and verify that my conclusions were sound, I would appreciate it.

Thanks again,

Dom

 

[sheiko]

Dear Katmar,

Is it correct to use the inside diameter of the pipe to convert Cv to K?

"We don't believe things because they are true, things are true because we believe them."

 

[katmar]

Sheiko, the Crane TP-410 formula (Equation 3-16 in my copy) certainly specifies d as the pipe internal diameter. For the reasons I gave in my post of 14 July 09 11:37 I agree with this. In my own workings I had derived this conversion before I found it in Crane (my constant was 29.83) and that was also based on the pipe ID. AndersE mentioned above that he had done the same thing, and describes the procedure.

This conversion has been discussed here before (see thread378-134079) and we all seemed to be in ageement there that the d was the pipe ID. To be honest, I cannot see any other likely candidate. What alternative is there?


Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com

 

[vzeos]

As katmar said, this conversion does come up every so often and it would probably be a good idea to elaborate on it a bit more. The equation is a means of converting C[sub]v[/sub] values obtained from experimentation to K values that can be used in the Darcy pipe equation. Consequently, the diameter used in the equation has nothing to do with the valve trim. As an illustration, if you were doing a flow calculation on a piping system and you had the K values of all the pipe and fittings you could use the C[sub]v[/sub] to K conversion to determine the K value of valves to be included in the calculation. The diameter that would be used in the C[sub]v[/sub] to K conversion for each valve would be the same diameter to be used in the flow calculation.

Below is a step by step development of the C[sub]v[/sub] to K conversion equation.
(see Crane TP-410 for nomenclature)
From continuity:
Q=Av
A=[π]r[sup]2[/sup]
Q=[π]/4 (d/12)[sup]2[/sup]v = [π]d[sup]2[/sup]v/(4*144)
v=4*144Q/ [π]d[sup]2[/sup]
v[sup]2[/sup]=33615.93Q[sup]2[/sup]/d[sup]4[/sup]

From the Darcy equation:
h[sub]L[/sub]=Kv[sup]2[/sup]/2g = (K/2g)*(33615.93Q[sup]2[/sup]/d[sup]4[/sup]) = 522.47326 (K/d[sup]4[/sup])*Q[sup]2[/sup]

gpm is required for C[sub]v[/sub] equation,
q(gal/min)=Q(ft[sup]3[/sup]/sec)*7.48052(gal/ft[sup]3[/sup])*60(sec/min) = Q*448.8312

q(gal/min)/448.8312=Q(ft[sup]3[/sup] /sec)

h[sub]L[/sub]=(522.47326/(448.8312[sup]2[/sup]))(K /d[sup]4[/sup])q[sup]2[/sup]

psi is required for C[sub]v[/sub] equation,
[Δ]P= ([ρ]/144)*0.0025935(K/d[sup]4[/sup])*q[sup]2[/sup]

[ρ]=62.367 lb/ft[sup]3[/sup] (at 60 F and 14.73 psia, per NIST)

[Δ]P=(62.367*0.0025935/144)*(K/d[sup]4[/sup])*q[sup]2[/sup]

q[sup]2[/sup]=([Δ]P/0.00112325565625)*(d[sup]4[/sup]/K)

q=(29.83738051725*d[sup]2[/sup][√][Δ]P)/[√]K

For [Δ]P=1 psi (@60 F), q [≡] C[sub]v[/sub]

C[sub]v[/sub]=(29.8373*d[sup]2[/sup])/[√]K

K= (890.2692*d[sup]4[/sup])/C[sub]v[/sub][sup]2[/sup]

 

[MechEng2005]

Thanks for the document Domc83!

 

[domc83]

MechEng2005 -

Just FYI that document was more of a question than anything else, so take it with a grain of salt. It'd be good to have others check the work to make sure it and my conclusions are correct.

Dom