Resistance coefficient K
Δ p  : Pressure drop of the component 
K 
: Dimensionless resistance coefficient (friction factor) of the component.
K is assumed to be constant. 
ρ  : Density 
v  : Average speed in the characteristic cross section 
 : Volume flow 
A  : Characteristic cross section 
d  : Inner diameter of the characteristic cross section. For valves as a general rule the
characteristic diameter equals the nominal diameter DN. 
Equation (1) is derived from the equation of DarcyWeisbach
.
The resistance coefficient K is assumed to be constant. This assumption
is correct for completely turbulent flow. In many other cases the inaccuracy is acceptable. With K=const. equation (1) can be written as follows:
The valve manufacturer MasonNeilan (MA, USA) introduced in the 1940s
the flow coefficient c_{v} . In the 1950s c_{v}
was introduced as k_{v}
value (k_{vs}value for rated travel) in the metric system [Früh 1957].
The flow coefficients c_{v} and k_{v} describe a valve´s reference point of operation
_{}. The reference fluid
is water (ρ_{o} = density of water
at 15,6 ºC) [VDI/VDE 21732007 page 8].
k_{v}  : _{o} = k_{v} [m³/h]  @ Δp_{o} = 1 [bar] 
 
c_{v,us}  : _{o} = c_{v,us} [USgal/min]  @ Δp_{o} = 1 [psi] 
 ≡  
_{o}[m³/h] = 0.2271 c_{v,us} [USgal/min]  @ Δp_{o} = 0.0689 [bar] 
c_{v,uk}  : _{o} = c_{v,uk} [UKgal/min]  @ Δp_{o} = 1 [psi] 
 ≡  
_{o}[m³/h] = 0.2728 c_{v,uk} [UKgal/min]  @ Δp_{o} = 0.0689 [bar] 
With equation (4) we can describe the correlation between
k
_{v} and c
_{v}
Putting equation (1) to K and with A = π/4 d
^{2} the correlation between
resistance coefficient K and flow coefficient k
_{v} becomes equation (5):
By lowering the pressure behind a valve the volume flow can´t rise without limits. Pressures
below the vapour pressure lead to cavitation. The noise level rises and at maximum cavitation
we face choked flow. According to
[VDI/VDE 21732007 page 10] we can note for
control valves:


p_{1}: Absolute pressure at the inlet
p_{v}: Vapour pressure
p_{c}: Critical pressure (Water: p_{c}=221,2 bar)


Ball valve  : F_{L}= 0.50.7 
Butterfly valve (aperture angle 6070º)  : F_{L}= 0.550.75 
Lownoise control valve  : F_{L}= 0.880.98 

More detailed information (including not turbulent flow) can be found in
[DIN EN 60534 2012].
Literature

[DIN EN 60534 2012] DIN EN 6053421: Stellventile für die
Prozessregelung, BeuthVerlag Berlin, Januar 2012 
[Früh 1957] K.F.Früh: Berechnung des Durchflusses in
Regelventilen mit Hilfe des k_{v}Koeffizienten, Regelungstechnik Heft 9;
5. Jahrgang 1957 
[VDI/VDE 21732007] VDI/VDE 2173: Stömungstechnische Kenngrössen
von Stellventilen und deren Bestimmung, BeuthVerlag Berlin, September 2007 