Valve Sizing Equations

By relationship you mean the ISA flow equation?

I don't have that equation handy but there should be a correction in the equation for choked flow.

Once the flow through a valve is chocked, flow depends directly on upstream density, eg. pressure. For example, PSVs usually operate in chocked flow and their flow relationship to pressure changes is linear (barring changes to any other parameters like Z). Fisher's valve equation, once you go to choked flow, also has flow being proportional to inlet pressure, not the traditional square root dependency.

Volumetric flow rate is dependent upon the upstresam pressure times the square root of x. X is the differential pressure divided by the upstream pressure. So the end results is not directly linear nor does it directly follow a square root curve. The results are proportional to a square root function.

bcd

If the upstream pressure is 65 and downs tream is 15 the valve is almost certainly choked.

For choked flow W=6.4 * 0.67*Cv *SQRT (Pi*Xt/sv,i)
but sv,i = kT/P (approx) so W= k2*Pi, or flow is proportional to inlet pressure. If you drop the inlet pressure by 1 psi, the new flow is 65/65 ths of the old flow.

See if the Chemical Engineering article from Fisher helps,


John

You don’t indicate what units of flow you desire to pop out at the end of your calculation but mass flow or volumetric there is a square root relationship to deal with.
Lets say your out flow is in pounds per hour.
You will need to assess if the change in pressure drop still provides choked flow. XT is valve specific so that will remain the same and as such Fk times XT will remain the same. X is the only variable in this case so, is X less than or larger than Fk times XT. The value of P1 has fallen, this is inversely proportional to X but both X and P1 have a square root relationship with the mass flow. The specific weight will have slightly changed due to the reduction in pressure this too has a square root relationship with the mass flow.

In other units of out flow the compressibility factor Z together with the relationship of inlet temperature and the pressure drop ratio will account for the square root relationship of pressure to flow.

ISA S75.01, Valtek, Fisher et al plus the European equivalent EN 60534 sizing equations make good reading plus a book entitled "Control Valve Primer" by Hans Baumann published by the ISA make good reading. Regarding the latter try to obtain the third edition there are a few (minor) errors in the first two that may cause some confusion.

Good luck with the exams.

Fog Jones

TD2K is correct.

In the case of choked flow, the flow is always approximately proportional to the inlet pressure. In the case of less than choked flow , the square root relationship governs , ie flow proportional to the square root of the pressure drop. The ISA valve sizing equations for choked flow can be converted to show this also as follows:
W=63.4*Cv*0.667*SQRT ( Pi*Xt/sv,i)
but approx Pi=ZRTi/sv,i, so

W=63.4*Cv*0.667*SQRT(Pi*Pi*ZRTi*Xt)
so the SQRT(Pi*Pi) is linear in Pi.

jsummerfield (John),

I believe equation (4) in Fisher's article (How to Size Liquid Control Valves) is inaccurate.

For an inlet reducer, K1 = 0.5[1-(d^2)/(D^2)] should be correct, i.e. without squaring the term in the outer bracket.

Same inaccuracy was found in ISA S75.01. I have informed the Manager, Publications and Standards, ISA Technical Publications.

I would appreciate if I could obtain the formal definitions of Kv and Av, [see equation (15) of the same ariticle]. I am not familiar with the terms used outside of North America.

Thank you.

4carats.