1. UNIVERSITÀ DEGLI STUDI DI SALERNO
Bachelor Degree in Chemical Engineering
Course: Process Instrumentation and Control
(Strumentazione e Controllo dei Processi Chimici)
CONTROL VALVE SIZING
FLOW CURVE
Rev. 1.9 – March 20, 2019

2. Process Instrumentation and Control - Prof M. Miccio
SMOOTH CONTRACTION
LIQUID
SEC. AREA. S1
DIAM D1
PRESS P1
TEMP T1
VELOCITY v1
SEC. AREA S2 < S1
DIAM D2 < D1
PRESS P2 < P1
TEMP T2 = T1
VELOCITY v2 > v1
HYPOTHESES
CIRCULAR PIPE
HORIZONTAL POSITION
ρ = CONSTANT
v # f (r )
ΔPDISTR = ΔPLOC = 0
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

3. FLOW EQUATION across a CONTRACTION
We can write the continuity equation between sec. 1 and 2 considering the previous hypotheses:
(§ pag. 9of Magnani, Ferretti e Rocco, 2007):
ρS1v1=ρS2v2  S1v1 =S2v2 (S1# S2) (1)
Mechanical energy balance (Bernoulli’s principle)
(§ pag. 15 of Magnani, Ferretti e Rocco, 2007)
(2)
The difference between z1 and z2 is zero because of the horizontal position of the valve.
We evaluate v1 from (1) and insert it in (2):
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

4. FLOW EQUATION across a CONTRACTION (cont.ed)
= FLOW COEFFICIENT OF A CONTRACTION
D2/D1 = CONTRACTION RATIO
Multiplying for the sectional area S2:
v2S2=v1S1=
GENERAL FLOW EQUATION ACROSS A CONTRACTION FOR AN INCOMPRESSIBLE FLUID
Multiplying eq. (3) for ρ we obtain the mass flowrate :
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

5. FLOW EQUATION across a CONTRACTION (for liquid)
1st HYPOTHESIS: contraction with the same pressure drop (P1 – P2) and the same sectional area S for:
Generic liquid Water
Dividing the respective members of eqs. (1) and (2):
2nd HYPOTHESIS: The control valve in “nominal” condition is considered as an ideal contraction:
Cvn [=] US gpm(H2O)/psi1/2
We have:
Replacing eq. (5) in (3):
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

6. FLOW EQUATION across a CONTRACTION (for liquid)
For NON nominal conditions eq. (6) becomes:
eq. (5.7)
where:
Cv(h) [=] US gal(H2O) / (min psi ½)
(Ρ1 – P2) [=] psi
Gf = specific gravity [=] -
Volumetric flow rate [=] US gal/min
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

7. FLOW EQUATION across a CONTRACTION (for liquid)
The previous Eq. (7) can be expressed in SI units for mass flow rate:
eq. (5.7)
where:
N1= [(kg/s)/(gpm(Pa/psi)1/2)]
Cv(h) [=] US gal(H2O) / (min psi ½)
(Ρ1 – P2) [=] Pa
Gf = specific gravity [ =] -
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

8. GLOBE CONTROL VALVE ΔP = P1 – P2 Φ=1= (Cvn/Cvmax) ΔPn Cvmax
MINIMUM TRAVEL
h= Φ=Φ0
Φ=Φ0=0
Cvmin = 0
The valve has also
an isolating function
Φ=Φ0 #0 e >0
ΔPmin
Cvmin # 0
MAXIMUM TRAVEL
(nominal condition) h= Φ=1
Φ=1= (Cvn/Cvmax)
ΔPn
Cvmax
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

9. Process Instrumentation and Control - Prof M. Miccio
FLOW curve
Now we analyze the effect of the pressure drop on the valve flowrate. A qualitative representation is showed in figure. The continuous line represents the common trend of the mass flowrate of a liquid across a valve vs. the square root of the pressure drop P
HYPOTHESES:
ρ=constant
P1=constant
Newtonian fluid
Re > 2100
Slope ≈ Cv (h)
h = constant
P2 decreasing ↓
From the diagram in figure, we can distinguish three operating zones:
Normal flow, where the flow rate is proportional to P1/2;
Semi-critical flow, where the flow rate increases less than proportional to P;
Choked flow, for which flow rate does not depend on P and is equal to wmax.
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

10. CAVITATION, FLASHING and NOISE
VENA CONTRACTA
HYPOTHESES:
LIQUID
valve represented as a contraction (convergent followed by divergent)
Horizontal flow
Equal inlet and outlet cross sectional area
PRESSURE DROP ΔP = P1 - P2
The inlet pressure of the valve P1, for liquid, at the beginning of the convergent, is always higher than the outlet pressure P2 (pressure drop).
The qualitative trends of pressure and velocity of fluid inside the valve are reported in figure with a continuous and a plotted line, respectively.
Along the converging section, following the Bernoulli’s equation, the velocity increases and the pressure decreases as the cross sectional area decrease. The opposite occurs in the diverging section.
The minimum cross−sectional area of the flow stream occurs just downstream of the actual physical restriction at a point called the vena contracta, because the fluid vein continues to contract, as shown in figure.
In this point, the pressure (Pvc, vena contracta pressure) is minimum while the fluid velocity is maximum.
If the vena contracta pressure Pvc becomes lower than the vapor pressure Pv, vapor bubbles form inside the valve and cavitation or flashing occur depending on the value of the outlet pressure P2.
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

11. CAVITATION, FLASHING and NOISE
HYPOTHESIS : Pvc ≤ Pv
if P2 > Pv, cavitation occurs: vapor bubbles collapses when the fluid pressure reaches values higher than Pv. Collapsing of the vapor bubbles releases energy and produces a noise similar to what one would expect if gravel were flowing through the valve.
Se P2 Pv, flashing occurs: vapor bubbles remain in the outlet fluid, where there may be a mixture of liquid and vapor or just vapor. Also flashing produces loud noise.
Both phenomena tear and damage the metallic surface in contact with the fluid. Special valves are commercially available with anti-cavitation trim able to reduce noise and damaging effects of cavitation and flashing.
For the preservation of the valve, cavitation has to be avoided. On the other hand, flashing can not always be avoided since it depends of the inlet fluid conditions as vapor pressure (Pv) and outlet pressure.
The negative effects of flashing (noise and damage) increase as the velocity increases.
In order to keep down it, harder trim are used and a maximum fluid velocity about 3.5 m/s is considered sizing the valve properly.
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

12. Process Instrumentation and Control - Prof M. Miccio
FLOW CURVE
We can now explain the flow behavior of a valve in the three operating zones:
Normal flow occurs when Pvc> Pv. No cavitation occurs and the flowrate can be expressed by the Bernoulli’s equation; Pc defines the value of P for which Pvc = Pv. In the normal flow zone we have 0 P< Pc
Semi-critical flow occurs when Pvc  Pv and P2 > Pv. Cavitation occurs, vapor bubbles form inside the valve, with consequent increase in velocity and pressure drop and the slope of the flowrate decreases as the pressure drop increases. Pf indicates the value of the P for which P2 = Pv; in the semi-critical flow zone we have Pc  P < Pf . The flowrate becomes maximum when there is vapor in the entire section of the vena contracta, which has reached the velocity of sound in the fluid itself. At this point, Pvc is no longer tied to P2, but depends only on Pv, (see below, regarding the FF coefficient).
Choked flow occurs when Pvc < Pv and P2  Pv. The flowrate remains constant as the pressure drop increases, because in the vena contracta the conditions of maximum evaporation were reached. The average fluid velocity corresponds to the sound velocity. The outlet fluid is a liquid-vapor mixture with the vapor fraction increases as P2 decreases. The word flashing is commonly used to indicate the vapor and the vapor-liquid mixture formation. In the choked flow zone we have P  Pf.
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

13. Process Instrumentation and Control - Prof M. Miccio
FLOW CURVE
 The flow curve can be calculated and drawn for a valve at nominal condition as well as at any prefixed h value
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

14. CHARACTERISTIC POINTS of the FLOW CURVE
They can be evaluated from specific experimental coefficients provided by valve manufacturers:
Kc is the incipient cavitation index
FL is the recovery factor
FF is the liquid critical pressure ratio factor.
NOTE:
Kc and FL only depends on the internal geometry of the valve and NOT from the fluid properties.
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

15. CHARACTERISTIC POINTS of the FLOW CURVE
Saturation Flowrate (Point S)
where:
FL [ - ] is the recovery factor
 dependent only on the internal geometry of the valve and NOT on the properties of the fluid
FF [ - ] is the liquid critical pressure ratio factor.
 dependent only on the properties of the fluid
in nominal condition:
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

16. RECOVERY FACTOR Definition of recovery factor
High recovery valve: FL< 0.8
Low recovery valve: FL> 0.8
from “Perry’s Chemical Engineering Handbook”
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

17. LIQUID CRITICAL PRESSURE RATIO FACTOR FF
The FF factor is used to calculate ΔPmax in flow curve.
The liquid vaporizes in vena contracta when Pvc is equal to Pv. The flowrate assumes the maximum values when in vena contracta the fluid is completely vaporized and has the sound velocity. In this condition Pvc is related to Pv through the FF factor (Pvc=FFPv) and does not depend on P2. Therefore:
FF = Pvc/Pv
with Pvc is evaluated in choked flow condition.
Considering the fluid in saturation condition as a liquid finely dispersed in its own vapor in thermodynamic equilibrium and where the liquid and vapor velocities are equal, the following equation is commonly us for calculating FF:
with:
Pc= critical pressure of the fluid
Usually: FF  0.956
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

18. CHARACTERISTIC POINTS of the FLOW CURVE
Incipient Cavitation (Point C)
where:
Kc [ - ] is the incipient cavitation coefficient
 dependent only on the internal geometry of the valve and NOT on the properties of the fluid
in nominal condition:
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

19. INCIPIENT CAVITATION INDEX Kc
The incipient cavitation index allows to evaluate the value of the pressure drop at which the cavitation starts inside the valve
It is defined as:
It can be calculated by:
ΔP* is the value of pressure drop with constant P1 when water is the inlet fluid and the measured flowrate at minimum is lower of 2% of the predicted value.
With this definition, ΔP* does not correspond to ΔPc .
However, the difference between these two values are considered negligible ΔP*ΔPc
When Kc is unknown and is not provider by manufacturer, it can be estimated by:
GLOBE VALVE Kc = 0.8 FL2
ROTARY VALVE Kc= 0.6FL2 ÷ 0.8FL2
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

20. CHARACTERISTIC POINTS of the FLOW CURVE
Incipient Flashing (Point F)
in nominal condition:
11/10/2019
Process Instrumentation and Control - Prof M. Miccio

21. CAVITATION and the IEC 60534 NORM
The semi-critical flow zone is difficult to determine. For this reason, the Norm IEC neglects the semi-critical flow zone in the relation for the valve sizing.
The continuation of the normal and choked flow lines are considering to approximate the semi-critical flow zone, as in figure. The lines intercept in point S ( , wmax).
The flow equations in nominal conditions are:
Normal flow zone ( 0 ≤ ΔP < ΔPmax):
Choked flow zone (ΔP > ΔPmax)
(from point S)
11/10/2019
Process Instrumentation and Control - Prof M. Miccio