Finding Cv with K factors - Equivalent Resistance for Hydraulic System
Finding Cv with K factors - Equivalent Resistance for Hydraulic System
(OP)
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
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





RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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.
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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.
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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 m3/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
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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.
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
"We don't believe things because they are true, things are true because we believe them."
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
Katmar Software
Engineering & Risk Analysis Software
http://katmarsoftware.com
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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."
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
This conversion has been discussed here before (see thread378-134079: Cv conversion to other units) 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
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
Below is a step by step development of the Cv to K conversion equation.
(see Crane TP-410 for nomenclature)
From continuity:
Q=Av
A=πr2
Q=π/4 (d/12)2v = πd2v/(4*144)
v=4*144Q/ πd2
v2=33615.93Q2/d4
From the Darcy equation:
hL=Kv2/2g = (K/2g)*(33615.93Q2/d4) = 522.47326 (K/d4)*Q2
gpm is required for Cv equation,
q(gal/min)=Q(ft3/sec)*7.48052(gal/ft3)*60(sec/min) = Q*448.8312
q(gal/min)/448.8312=Q(ft3 /sec)
hL=(522.47326/(448.83122))(K /d4)q2
psi is required for Cv equation,
ΔP= (ρ/144)*0.0025935(K/d4)*q2
ρ=62.367 lb/ft3 (at 60 F and 14.73 psia, per NIST)
ΔP=(62.367*0.0025935/144)*(K/d4)*q2
q2=(ΔP/0.00112325565625)*(d4/K)
q=(29.83738051725*d2√ΔP)/√K
For ΔP=1 psi (@60 F), q ≡ Cv
Cv=(29.8373*d2)/√K
K= (890.2692*d4)/Cv2
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
RE: Finding Cv with K factors - Equivalent Resistance for Hydraulic System
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