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1. Cv and Kv definition

2. Calculation of Cv and flow
through a valve - Case of liquids - SI units

3. Calculation of Cv and flow through a valve - Case of gases - SI unitsb

4. Control valves sizing

3. Calculation of Cv and flow through a valve - Case of gases - SI unitsb

4. Control valves sizing

5. Corrections of Cv for special
cases

6. Calculation of flow through a
valve - US units

The flow through a valve can be
calculated thanks to the pressure drop through the valve and **a
coefficient Kv which represents the number of m3/h of water that
goes through the valve at 15c and with a pressure drop of 1 bar.**

Manufacturers may specified the
coefficient Cv instead of Kv for their valve. **Cv is the number of
US gallons that goes through the valve at 60F and with a pressure
drop of 1 psi.** It is actually possible to convert Kv and Cv and
have the expression in the right unit for the calculation of interest.
**Cv=1.156Kv**

In order to calculate the flow of liquid through a valve thanks to
the Cv, the Engineer must 1^{st} determine if the flow is
subcritical or critical (cavitation / flashing). To do so, it is
necessary to compare the pressure drop to some limit values.

**If ΔP < F**_{L}^{2}(ΔP_{s}) : the flow is subcritical**If ΔP > F**_{L}^{2}(ΔP_{s}) : the flow is critical

With

ΔP=Pressure drop accross the valve (bar abs)

F_{L} = liquid pressure recovery factor = [(P_{1}-P_{2})/(P_{1}-P_{VC})]

P_{1} = upstream pressure (bar abs)

P_{2} = downstream pressure (bar abs)

P_{VC} = pressure at the vena contracta of the valve (bar
abs)

ΔP_{s} = critical pressure drop = P_{1} -
(0.96-0.28*√(P_{v}/P_{c}))*P_{v}

P_{v} = vapor pressure of the liquid at the temperature of
the flow (bar abs)

P_{c} = pressure at thermodynamic critical point (bar abs)

**The liquid pressure recovery factor is given by the valve manufacturer****ΔP**_{s}can be approximated as ΔP_{s}=P_{1}-P_{v}if P_{v}< 0.5*P_{1}

The relations given below are valid only for a **Newtonian liquid
in turbulent flow.**

**The volumetric flow of liquid** through a valve can be
calculated by Qv = Kv.√(ΔP/d) or Qv=Cv/1.156.√(ΔP/d) for subcritical
flow.

With

Q_{v}=Flow rate (m3/h)

C_{v}=valve flow coefficient (GPM)

K_{v}=valve flow coefficient (m3/h)

ΔP=Pressure drop across the valve (bar)

d_{15}^{t}=density of the liquid referred to water at
15 degrees (-) - as the density of water at this temperature is 999.13
kg/m3, d_{15}^{t} can be approximated as ρ/1000

ρ = density of the liquid at the temperature of flow (kg/m^{3})

C

K

ΔP=Pressure drop across the valve (bar)

d

ρ = density of the liquid at the temperature of flow (kg/m

**The mass flow rate** through the valve can be calculated
thanks to the following formula Cv=1.156*m/√(ΔP*d) ⇒
m=(Cv/1.156)*√(ΔP*d) :

With

m=Flow rate (t/h)

C_{v}=valve flow coefficient (GPM)

ΔP=Pressure drop accross the valve (bar)

d_{15}^{t}=density of the liquid referred to water
at 15 degrees (-) - as the density of water at this temperature is
999.13 kg/m3, d_{15}^{t} can be approximated as
ρ/1000

ρ = density of the liquid at the temperature of flow (kg/m^{3})

For critical flow, the following formula can be used to calculate **
the volumetric flow** through a valve of coefficient Cv : Qv=F_{L}*Cv/1.156.√(ΔP_{s}/d)

Q_{v}=Flow rate (m3/h)

F_{L} = critical flow factor

C_{v}=valve flow coefficient (GPM)

ΔP_{s}=as defined above at paragraph 2.1

d_{15}^{t}=density of the liquid referred to water
at 15 degrees (-) - as the density of water at this temperature is
999.13 kg/m3, d_{15}^{t} can be approximated as
ρ/1000

ρ = density of the liquid at the temperature of flow (kg/m^{3})

For calculating **the mass flow**, the following formula can be
used :

With :

m=Flow rate (t/h)

F_{L} = critical flow factor

C_{v}=valve flow coefficient (GPM)

ΔP_{s}=as defined above at paragraph 2.1

d_{15}^{t}=density of the liquid referred to water at
15 degrees (-) - as the density of water at this temperature is 999.13
kg/m3, d_{15}^{t} can be approximated as ρ/1000

ρ = density of the liquid at the temperature of flow (kg/m^{3})

F

C

ΔP

d

ρ = density of the liquid at the temperature of flow (kg/m

A control valve has a Cv of 5. The instrumentation on the line show that the pressure drop through the valve is 0.1 bar ? What is the flow through the valve. It is water at 50°c.

The density of water at 15°c is 999.13 kg/m^{3}

The density of water at 50°c is 988.07 kg/m^{3}

The density of water at 50°c compared to water at 15°c is : 988.07/999.13 = 0.9889

The flowrate is Qv=Cv/1.156.√(ΔP/d) = 5/1.156*√(0.5/0.9889) = 3.07
m^{3}/h

Note : the reverse calculation can be done when knowing a flow, a pressure and wanting to calculate the Cv.

**The volumetric flow rate of gas** through a valve can be
calculated from the valve Cv thanks to the following formula :

With

Q_{v}=Flow rate (m3/h) at 15c and 101325 Pa abs

F_{L} = critical flow factor

C_{v}=valve flow coefficient (GPM)

P_{1} = upstream pressure (bar abs)

d = gas specific gravity vs air (d_{air} = 1) = M/29

M = molar mass of the gas (g/mol)

T = temparature (K)

Z = compressibility factor (-)

y = (1.63/F_{L})*√(ΔP/P_{1})

- If y < 1.5, subcritical flow
- if y > 1.5, then y is capped at y = y
_{max}=1.5 for critical flow

**The mass flowrate** through the valve can be calculated thanks
to the following formula :

With

m=Flow rate (t/h)

F_{L} = critical flow factor

C_{v}=valve flow coefficient (GPM)

P_{1} = upstream pressure (bar abs)

d = gas specific gravity vs air (d_{air} = 1) = M/29

M = molar mass of the gas (g/mol)

T = temparature (K)

Z = compressibility factor (-)

y = (1.63/F_{L})*√(ΔP/P_{1})

- If y < 1.5, subcritical flow
- if y > 1.5, then y is capped at y = y
_{max}=1.5 for critical flow

The formula given above are valid when the valve diameter is equal
to the pipe diameter. However, it sometimes happens that the valve
is mounted in between pipe reducers which has as an effect to reduce
the capacity of the valve. To account for this effect, a coefficient
F_{p}, the piping geometry factor is calculated. The actual
Cv is then :

**Cv_corrected = Cv / Fp**

Fp = [(Cv^{2}*ΣK)/(0.00214*d^{4})+1]^{-0.5}

With

Cv_corrected = actual Cv of the valve in between pipe reducers

Cv = calculated Cv without pipe reducers

F_{p} = pipe geometry factor

d = valve diameter (mm)

ΣK = K_{1} + K_{2} + K_{B1} - K_{B2}

K_{1} = loss coefficient at inlet = 0.5*[1-(d/D_{1})^{2}]^{2}

K_{2} = loss coefficient at outlet = [1-(d/D_{2})^{2}]^{2}

K_{B1} = Bernoulli coefficient = 1-(d/D_{1})^{4}

K_{B2} = Bernoulli coefficient = 1-(d/D_{2})4

D_{1} = inside diameter of upstream pipe (mm)

D_{2} = inside diameter of downstream pipe (mm)

According to [Baumann], the correction of the valve Cv due to
viscosity is to be applied only if the fluid has a viscosity > 40
centistokes. The
correction procedure requires the calculation of a correction
factor F_{R} which will be used in a similar way as
the coefficient Fp for pipe geometry.

The
case of laminar flow is according to Baumann calculated the
same way as a correction of viscosity, which appears logic as a
higher viscosity will often lead to laminar flow.

The relations above, in US units, are expressed the following ways to calculate valves of Cv in liquid and gas applications

With P and ΔP in Psi abs, Q_{v}in GPM, C_{v}in
GPM, m in lb/h, d is specific gravity (water = 1 at 60 F), T flowing
temperature in R