Derivation of Cv (flow coefficient) 4

[olynyk]

Greetings everyone. I've just started at a small valve manufacturing company. I'm building a test rig to do the classic "flow rate vs. pressure drop" test on our products (injection quills and pressure relief valves). I'll then be preparing brochures/data sheets for each product. (Small place, Engineering gets mixed with Marketing, heh). I've been given some time to do some research on how to best collect and present the data to customers, so naturally I've been reading a lot about the Masoneilan flow coefficient Cv, introduced in 1944.

Here's my issue: every text or website I've read has given the well-known equation to relate the Cv value of a valve to pressure drops and flow rates other than 1 USGPM and 1 PSI:
Q = Cv * sqrt(dp/G)

My question is: how accurate is this relation? Nowhere in the Masoneilan valve sizing handbook, the Fisher Controls "Control Valve Handbook", my undergraduate fluid mechanics text, or any website or Eng-tips thread that I can find does it mention the derivation. I don't think it comes from Bernoulli's equation, since there is obviously a huge energy loss across a control valve.

I know the equation is accurate enough for general use, since it's been around so long, but since I'm doing R&D work I was hoping to get a little more background. Does anyone have a reference for the 1944 paper where the concept was introduced?

One last question: I see in the Fisher controls manual the following: "To aid in establishing uniform measurement of liquid flow capacity coefficients (Cv) among valve manufacturers, the Fluid Controls Institute (FCI) has developed a standard test piping arrangement, as shown in Figure...". Does anyone know the paper in which this standard test piping arrangement was introduced? I have searched the FCI, ASME, ANSI, ISO, and ISA websites to find a reference. ANSI/ISA S75.01 and S75.02 are on order, but I'm just trying to cover all my bases.

Sorry for all the questions! Thanks everyone.

 

[JNieuwsma]

Another resource you might find useful: Crane's Technical Paper 410 "Flow of Fluids", available at

http://www.cranevalves.com.

Derivation of Cv and the flow formulas from Darcy's equation is shown.

Jonathan Nieuwsam

 

[RJVan57]

The equation Q=Cv sqrt(dp/G) is, in fact derived from the bernoulli eqn. My old freshman physics book shows the derivation of dp=1/2 rho (v2^2-v1^2) and goes on to modify it for negligible v1. Regarding it's accuracy, remember this is for incompressible, inviscid flow. The term Cv is an empirical factor for the huge energy loss you referred to.

I believe the ANSI/ISA 75 papers you ordered are about the best resources I've seen to account for viscosity, choking and compressibiliy. They are very practical, but don't give a lot of theoretical justification to the equations. Also, they relate well to the Fischer Valve litersture.

Looks to me like you're on the right track. Good luck with your project. I personally enjoy the mixture of Marketing and Engineering. I believe it's a valuable skill to effectively explain technical concepts to non-technical audiences.

 

[olynyk]

Thanks JNieuwsma and RJVan57 for your helpful responses. I'm probably going to order Crane TP-410 the next time an order goes out for some companion flanges or something.

A few issues: RJVan I had that same equation you mention: dp = (1/2)*rho*(v2^2 - v1^2) which is a rearrangement of the Bernoulli equation, neglecting the potential-energy term. But I don't see why v1 can be taken to be negligible - the speed of the fluid coming into and out of the valve is equal due to mass continuity (same size pipe into and out of the valve). Because of this, the pressure loss cannot be due to velocity - it is entirely due to non-recoverable friction losses through the valve, and so I don't think the Bernoulli equation applies. That assumption you make (negligible v1) seems to be more applicable to something like a converging nozzle, where a fluid flows at (relatively) low speed through a large pipe and then exits a small orifice at a much higher speed. In this case there isn't the tortuous flow path like there is through a valve, so the friction losses will be low compared to the Bernoulli "losses" (in which pressure is converted into velocity with no energy loss).

I guess with regards to accuracy, I was looking for an order-of-magnitude percent accuracy of the incompressible Cv equation based on others' experience. When this rig is finally complete I'll be able to know for myself, but I was just wondering, for example, are real flows through valves usually within 2% of that predicted by the equation? 20%? 80%? The derivation of the equation is not so important to my actual work, it was more for my own interest, since the equation is quoted so widely and so often, I had wondered the original justification for it.

 

[NGiLuzzu]

olynyk,
in my opinion, Cv is not derived from any equation in particular, but empirically defined (and measured) as "the flow rate which flows through a valve at a given travel referred to 1 psi of dp across the valve and water at room temperature"; then the definition must be taken "as is" and considered for its practical value, i.e. because it provides an empirical coefficient good to calculate the flow rate of a valve under given conditions.
So, to be precise, it is not applicable to discuss about the accuracy of the definition; one should consider instead the accuracy of the measurements (or of the predictive algorithms) adopted to determine the Cv values, and then the accuracy of the flow rate calculations in which Cv values are used.
Just to give a standard reference, I can mention IEC 60534, Part 2 (edition 2000; formerly known as IEC 534-2)...

Regarding Bernoulli's Equation, as far as I can remember, energy losses are usually expressed (by means of dimensionless coefficients) as a percentage of the kinetic term, and therefore in function of the speed (which for incompressible fluids, as you observed, is constant through a constant-section pipe because of the mass conservation); but they may be expressed in function of the pressure term as well...
In any case, unlike long pipes that represent dispersed, distributed losses (usually expressed per unit of length: try searching the web for "Moody diagram", "Colebrook-White", "Hazen-Williams", etc.), a valve represents a discrete, concentrated, localized (even if variable with obturator travel) loss, like a curve, an angle, a fitting and so on: this may be taken in account by means of a coefficient... but my feeling is that there is no direct correlation with the valve Cv.
Please note that the loss coefficient is a pure number, while the flow coefficient depends on the units of measure.

Hope this helps, 'NGL

 

[shahyar]

Anegri,
If we want to translate back your definition to formula, we come up with Q=Cv .(dp/G) and not Q=Cv sqrt(dp/G) .
I want to say origin of Cv formula is this: dp=K (Rho)v2/2 which is applicable for every element in pipe. If we rearrange that for A, we come up with :A=K1.Q/sqrt(dp/G), so we can say: Cv=Q/sqrt(dp/G) . so :
"Cv is an index of valve area for flow".

Just My Thought.

 

[olynyk]

Anegri, I didn't express myself very well. What I meant by accuracy was this:

Cv is defined as the volume flow rate of water at standard conditions that passes through the valve (in USGPM) when a pressure drop of 1 psi is across the valve. This I'm sure can be measured quite accurately, as you say, subject only to limits of the test equipment.

The equation I'm wondering about the accuracy of is the standard incompressible-flow-through-a-valve equation:
Q = Cv * sqrt(dp/G)
which is used to calculate flows through a valve when the pressure drop is something other than 1 psi. For example, a pressure drop of 4 psi across the valve would lead to a flow rate of twice the Cv, according to the above equation.

That equation assumes a square-root relationship between pressure drop and flow, and I'm not sure where that initial assumption (square root relationship) came from. Upon looking at the variables involved in the equation, I think it comes from one of two places:
1) Bernoulli's equation (as others have mentioned). This is because the equation above involves only density (or rather, specific gravity) and pressure differential, not any kind of viscosity term. These are the same variables involved in Bernoulli's equation. (It also means it has the same limitations: if I pump syrup of a high viscosity but a specific gravity of 1 through the valve, there will most certainly be a higher pressure drop than that equation predicts!)
2) Empirical data (from tests done by Masoneilan in the 40s? I don't know) supporting the idea that flow through a valve follows the square root of the pressure drop. I'll have my own empirical data in a few weeks, and I'll update this thread with my findings. It doesn't make sense that every valve manufacturer in the world would publish the equation above if it wasn't reasonably accurate, so I have a feeling I will indeed see a square-root relationship.

Thanks again everyone for all your ideas. Hopefully this thread can be useful to other valve R&D people in the future!

 

[olynyk]

I've found some confirmation of what I'm looking for:

http://www2.psu.ac.th/PresidentOffice/EduService/journal/27-2-pdf/13ball-valve.pdf

This person is doing tests on flow characteristics of ball valves (Trying to come up with a linear ball valve) but as part of the paper includes tests of Q vs dp. (on pages 358 - 360, which are page 6 - 8 of the PDF). He finds that ln (Q) = 0.5023 ln(dp) + const for a valve which is 70-degrees open. This means that for that valve, at that degree of opening (which is almost fully open), the flow rate Q varies as the (approximately) 0.5'th power of the pressure drop, exactly what the equation predicts.

Unfortunately the slope drops to ln (Q) = 0.4106 ln(dp) + const for when the valve is only 20-degrees open, which means that for a partly-open ball valve (which has a more tortuous flow path), the flow does not follow a square-root relationship. In conclusion: I'd better stop posting so much in this thread and get back to finishing the test rig, so I can collect my own data. I now have a feeling I won't see the perfect square-root relationship since the flow path through our in-line pressure-relief valves is quite a bit more convoluted than that of a ball valve.