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Control Valve Sizing Procedure

1. Introduction

The control valve is a flow controlling device. The orifice plate [just to name one] is a flow measuring

device. They both operate on the Torricelli principle.The orifice plate is sized to produce ΔP in relation

to the flow of fluid to be measured. The orifice sizing is iterated on itself and converges rapidly. Unlike

the orifice plate which is a fixed geometry, the control valve is a variable geometry and sizing does not

iterate on itself, it iterates on catalog pages of the designer trying to select the best compromise for the

flow range of the process and the service conditions. The material in this work sheet is largely based

on Fisher Controls and the ISA Handbook of Valve Sizing. More useful stuff is added for augmenting

the comprehension and best design of a project. The procedure is automated in the sense that it is based

on functions rather than nomographs.

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The purpose of two = valves in series is to drop by ½ the pressure across each valve.

Cv of two same valves in series Cv 12 sqrt / ≡

The speed of water in circular pipes is repertoried in Handbooks.

For pipe ID [40 mm ... 2000 mm] ... range [0.1 m/s ... 2.5 m/s].

PONT-A-MOUSON p: 214 ... 233

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pipe Reynolds Q D ρ μ Nr 0.353678 106^* Q * ρ D * μ * / :

Fluid velocity [m/s] Q D ρ V 0.353678 Q * ρ D 2^* / :

Demo => valve Reynolds Fr

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x log x log10 :

Rev f 1.54 5.9 Rev log / - 14 Rev log 2^/ + :

x Fr_70 170 x ≤ (x 3000 ≤ (& cases :

Rev Fr 150 Rev 0.7^/ Rev 70 ≤ 1 Rev 3000 ≥ Rev f cases / :

Re_min 70 : Re_max 700 : 21 sys

Re_min Fr Re_max Fr 21 mat 0.391 0.836 21 mat

x Fr Re_min Re_min Fr 1 2 mat Re_max Re_max Fr 1 2 mat 3 1 sys

Fr mid-laminar (70≤x)&(x≤3000)

x f 1 - ^ x Fr_70 * 70 1.5 70 0.5 - 2 2 mat 3000 1.5 3000 0.5 - 2 2 mat 3 1 sys

mid-laminar (70 ≤ x)&(x ≤ 3000)

The critical pressure factor:

A supplementary factor is needed to determine the Δ pressure differential producer.

This factor depends solely upon the vapor pressure at the valve inlet temperature and the

critical pressure of the liquid. For water, we enter only the vapor pressure Pv @ the

corresponding valve inlet temperature. For other liquids, there is a general nomograph and

formulation, we enter the vapor pressure Pv & the critical pressure Pc.

For convenience, we provide the two necessary approximations [water, other liquids].

]]> plot(data,#,size,#)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

H2O 0 0.96 170 0.905 500 0.838 1000 0.795 1500 0.775 2000 0.755 2500 0.725 3000 0.683 82 mat :

Water H2O o 5 black plot :

pv Fw 2.2403 9466.4 pv 12788 - 1.9766 107^* pv 1884.2 + 1.7743 106^* pv 996.8 + / + / + / + :

x w 10 x ≤ (x 3300 ≤ (& cases :

Other 0 0.96 0.025 0.895 0.1 0.853 0.2 0.825 0.3 0.804 0.4 0.788 0.5 0.777 0.6 0.764 0.7 0.753 0.8 0.738 0.9 0.718 1 0.672 122 mat :

Other_liquids Other o 5 black plot :

r Fo 0.7159 0.08051 r 0.15796 + 0.28034 r 0.90057 - 5.7899 103 -^* r 7.9245 103 -^* + / - / - / + :

x o 10 x ≤ (x 1 ≤ (& cases :

χ 30221 / : 0.1357

χ Fo 0.8413

χ absolute pressure ratio ≡

pv 750 :

pv Fw 0.8109

Water critical pressure factor

Water x Fw x w * pv pv Fw + 20 blue 1 5 mat 3 1 sys

Other liquids critical pressure factor

Other_liquids x Fo x o * χ χ Fo + 20 blue 1 5 mat 3 1 sys

F.F pv Fw ≡

Demo => Liquid pressure recovery factor Fy

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P1 398.7 : P2 114.7 : pv 200 : FL 0.84 : F.F 0.90 : 51 sys

Fy 1 FL P1 F.F pv * - P1 P2 - / sqrt * 1 ≥ FL P1 F.F pv * - P1 P2 - / sqrt * cases :

ratio P1 P2 - P1 F.F pv * - / :

Fy ratio 21 mat 0.7371 1.2986 21 mat

multi-plot Fy vs 'x' = ratio x λ F.y λ 1 x / sqrt * :

x 0.4 F.y x 0.5 F.y x 0.6 F.y x 0.7 F.y x 0.8 F.y x 0.9 F.y x 1 F.y ratio Fy 1 2 mat ratio 1 ratio 0 2 2 mat Fy 10 1 sys

Typical Fp calculations

The burden of the installed Fp is greatly simplified by

the functional relationship Fp(Cv,d,D) . . . =>

Note that the installed Fp is Cv dependent .

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d 75 : D 200 : Cv 121 : 31 sys

Cv d D Fp 1467 1.5 1 d D / (2^- (2^* (* Cv d 2^/ (2^* + (0.5 -^:

Fp Cv d D Fp : 0.8982

x d D Fp Cv 1 Cv 0 2 2 mat Cv d D Fp 3 1 sys

Valve installed w/o reducers ΔP.max FL 2^P1 Fp pv * - (* : 154.6

Valve c/w identical reducers ΔP.max FL Fp / (2^P1 Fp pv * - (* : 191.6

Remark: We have assumed the same FL for both valve size, i.e: full size and reduced size.

In reality, FL for both choices aren't exactly the same, but it shows that a valve installed

with reducers, if she can meet the demand, will support a higher ΔP

The choice of the reduced size is interesting in two counts:

1. Economics fron the original cost and durability.

2. Higher ΔP for reduced flashing/cavitation.

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Recovery factor F_L

At this point of our insight, the fluid vein has contracted and expanded within the Bernoulli law

and some energy has been consumed by the work of contraction-expansion. The pressure at the

valve outlet has decreased. If the vein had passed through a flow nozzle or Venturi, the recovery

would be 99% or so.Valves don't recover as well and by far. For globe valves [the best

streamlined configuration], recovery is 80 .. 90 % and constant for any opening.

For ball valves, butterfly, gate, plug, pinch ... recovery varies with the % opening.

The recovery factor is F_L for the valve alone, i.e: not associated with reducers.

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Explain: Under choked flow DP < P1-P2 cavitation or flashing will develop. Once it is known that

choked flow will develop within specified valve design and in operation, a further distinction can be

made to determine whether the choked flow is caused by flashing or cavitation.

If the outlet pressure of the valve P2 is > Pv of the fluid => cavitation.

If the outlet pressure of the valve P2 is < Pv of the fluid => flashing.

Flashing is a temporary state of the fluid. Cavitation is a permanent state of the fluid. In both cases

bubbles form and hit [collapse] the nearest target at very high impact pressure of several thousands

atmos, thus damaging the surrounding material. Flashing is "milder cavitation". On extreme case

of cavitation from wrong valve specification & design installation, the valve plug may simply

vaporize in no time (so to speak). Cavitation occurs on gases as well.

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Liquid Critical Pressure Ratio Factor F_F

... multiplied by the vapor pressure, predicts the theoretical vena contracta

pressure at the maximum effective [choked flow] pressure drop across the valve.

Liquid Pressure Recovery Factor F_L

... predicts the amount of pressure recovery that will occur between the vena contracta and the valve outlet.

F_L is an experimentally determined coefficient that accounts for the influence of the valve's internal geometry

at the maximum capacity of the valve. F_L also varies according to the valve type. High recovery valves such

as butterfly and ball valves have significantly lower pressure at the vena cobntracta and hence recover much

farther for the same pressure drop than a globe valve. Thus, they tend to choke [cavitate] at smaller pressure

drop than globe valves.

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Example 1. Sizing a globe valve [F_L is constant]

Full port, linear plug, nomial 3" [12, 1-77]

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Port diameter [mm]
Pipe ID [mm]
Recovery factor d 87.3 : D 202.72 : F.L 0.82 : 31 sys

C.v20% 32.9 : C.v80% 133 : 21 sys

Constants N2 467 : N4 76000 : N6 0.0274 : 31 sys

USER U 0.2273 : T_hr 1 : psia 1 : Fr 1 : 41 sys

Cv C.v20% C.v80% 21 mat : 32.9 13321 mat

Cvmin C.v20% : Cvmax C.v80% : 21 sys

Service: hot water

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Hot water [°C]
Critical pressure [bar]
Vapor pressure [psia]
Vapor pressure [bar] T 200 ≡ Pc 220.64 ≡ Pv 225.75 ≡ bar 15.55 ≡ 41 sys

kg/m³[IAPWS]
upstream pressure [psia]
back pressure [psia]
Vapor pressure [bar] ρ 864.4 : P1 398.7 : P2 274.7 : Pv 14.5 : 41 sys

114.7

274.7

Σk 1.5 1 d D / (2^- (2^* : 0.9952

i 12 range F.p i el 1 N2 Σk * Cv i el d 2^/ (2^* + (0.5 -^: for

F.p 0.9957 0.936 21 mat

F.p F.p :

Otherwise Fp ... sanity check

sanity check Cv d D Fp 1467 1.5 1 d D / (2^- (2^* (* Cv d 2^/ (2^* + (0.5 -^≡

i 12 range Fp i el 1467 1.5 1 d D / (2^- (2^* (* Cv i el d 2^/ (2^* + (0.5 -^: for

Fp 0.9957 0.936 21 mat

Cv d D Fp 1467 1.5 1 d D / (2^- (2^* (* Cv d 2^/ (2^* + (0.5 -^:

Cv 01501 range :

Fp.installed Cv Cv d D Fp vectorize augment :

Fp.installed Cvmin 1 Cvmin 0.9 2 2 mat Cvmax 1 Cvmax 0.9 2 2 mat Fp 1 el Fp 2 el 5 1 sys

Fp.installed Cvmin 1 Cvmin 0.9 22 mat Cvmax 1 Cvmax 0.9 22 mat Fp 1 el Fp 2 el 51 sys

critical pressure factor F.F Pv Fw : 0.9546

Fy F.L P1 F.F Pv * - P1 P2 - / sqrt * : 1.4446

Ki K1 B1 + ≡ (0.5 1 d D1 / (2^- (2^* 1 d D1 / (4^- (+ ≡

ratio P1 P2 - P1 F.F Pv * - / sqrt : 0.568

Ki 0.5 1 d D / (2^- (2^* 1 d D / (4^- (+ : 1.2973

Cv C.v20% C.v80% 21 mat : 32.9 13321 mat

i 12 range F.Lp i el 1 F.L 2^/ N2 Ki * Cv i el d 2^/ (2^* + (0.5 -^: for

F.L 0.82

F.Lp 0.817 0.773 21 mat

F.Lp F.Lp :

valve w/o reducers DP.allow F.L 2^P1 F.F Pv * - (* ≡

valve c/w reducers DP.allow F.Lp F.p / (2^P1 F.F Pv * - (* ≡

i 12 range DP.allow i el F.Lp i el F.p i el / (2^P1 F.F Pv * - (* : for

<= @ min Cv =>
<= @ max Cv => DP.allow 259.1 262.8 21 mat

Fp 0.9957 0.936 21 mat

DP P1 P2 - : 124

Conclude valve sizing

At this point that the actual P1-P2 does

not exceed the smallest of DP.allow,

we can resume/tabulate the project.

This procedure serves two purposes:

1. size/select the valve from the given

serive conditions.

2. sanity check an installation.

Further, design conditions may vary.

principaly wrt P1, P2 . . . Some branched

users may be shut upstream/downstream

the valve. Thus affecting P1, P2. Under

such variations and if P1-P2 are already

border line DPallow, choked flow will

develop and flashing/cavitation result.

A valve that has supported cavitation

may have been sufficiently destroyed to

lose controllability, must be replaced.

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DP.psia P1 P2 - :

Q U Fr * Fp Cv * DP.psia sqrt * (vectorize * ρ 1000 / sqrt * : 7729321 mat

V 353700 Q * D 2^ρ * / : 0.77 2.92 21 mat

Header m/s Fp Cv DP psia Q T/hr 15 mat : Sizing V Fp Cv DP.psia Q 15 mat : Sizing Header Sizing stack : 31 line

Sizing m/s Fp Cv DP psia Q T/hr 0.77 2.92 21 mat 1 0.94 21 mat 32.9 13321 mat 124 77.09 292.94 21 mat 25 mat

The rated valve rangeability Cv 2 el Cv 1 el / 4

The installed valve rangeability Q 2 el Q 1 el / 3.8

Example 2. Sizing a U- valve [F_L varies with % opening]

Flow ring, approx = %, nomial 3" [12, 1-155]

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1. Catalogue data

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C.v20% 5.94 : C.v80% 123 : 21 sys

F.L20% 0.94 : F.L80% 0.7 : 21 sys

Port diameter [mm]
Pipe ID [mm] d 76.2 : D 202.72 : 21 sys

USER U 0.2273 : T_hr 1 : psia 1 : Fr 1 : 41 sys

2. Assign to procedure

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Cv C.v20% C.v80% 21 mat : F.L F.L20% F.L80% 21 mat : 12 mat

Cvmin C.v20% : Cvmax C.v80% : 21 sys

Constants N2 467 : N4 76000 : N6 0.0274 : 31 sys

Service: hot water

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Hot water [°C]
Critical pressure [bar]
Vapor pressure [psia]
Vapor pressure [bar] T 200 ≡ Pc 220.64 ≡ Pv 225.75 ≡ bar 15.55 ≡ 41 sys

kg/m³[IAPWS]
upstream pressure [psia]
back pressure [psia]
Vapor pressure [bar] ρ 864.4 : P1 398.7 : P2 114.7 : Pv 14.5 : 41 sys

114.7

274.7

Σk 1.5 1 d D / (2^- (2^* : 1.1061

i 12 range F.p i el 1 N2 Σk * Cv i el d 2^/ (2^* + (0.5 -^: for

F.p 1 0.901 21 mat

F.p F.p :

X_Y  plot Fp F.p :

Cv d D Fp 1467 1.5 1 d D / (2^- (2^* (* Cv d 2^/ (2^* + (0.5 -^:

Cv 01501 range :

Fp.installed Cv Cv d D Fp vectorize augment :

Fp.installed Cvmin 1 Cvmin 0.85 2 2 mat Cvmax 1 Cvmax 0.85 2 2 mat Fp 1 el Fp 2 el 5 1 sys

Cvmin d D Fp Cvmax d D Fp 21 mat 1 0.901 21 mat

Fp.installed Cvmin 1 Cvmin 0.85 22 mat Cvmax 1 Cvmax 0.85 22 mat Fp 1 el Fp 2 el 51 sys

critical pressure factor F.F Pv Fw : 0.9546

Fy F.L P1 F.F Pv * - P1 P2 - / sqrt * : 1.0943 0.8149 21 mat

Ki K1 B1 + ≡ (0.5 1 d D1 / (2^- (2^* 1 d D1 / (4^- (+ ≡

ratio P1 P2 - P1 F.F Pv * - / sqrt : 0.859

Ki 0.5 1 d D / (2^- (2^* 1 d D / (4^- (+ : 1.3487

Cv C.v20% C.v80% 21 mat : 5.94 12321 mat

i 12 range F.Lp i el 1 F.L i el 2^/ N2 Ki * Cv i el d 2^/ (2^* + (0.5 -^: for

F.L 0.94 0.7 21 mat

F.Lp 0.94 0.656 21 mat

F.Lp F.Lp :

valve w/o reducers DP.allow F.L 2^P1 F.F Pv * - (* ≡

valve c/w reducers DP.allow F.Lp F.p / (2^P1 F.F Pv * - (* ≡

i 12 range DP.allow i el F.Lp i el F.p i el / (2^P1 F.F Pv * - (* : for

<= @ min Cv =>
<= @ max Cv => DP.allow 34020421 mat

Fp 0.9997 0.901 21 mat

DP P1 P2 - : 284

Type Normal flow DP.allow 1 el DP.allow 2 el | (DP ≥ Choked flow cases : Choked flow

Productive DP DP.psia 28420421 mat :

Q U Fr * Fp Cv * DP.psia sqrt * (vectorize * ρ 1000 / sqrt * : 2133521 mat

V 353700 Q * D 2^ρ * / : 0.21 3.33 21 mat

Header m/s Fp Cv DP psia Q T/hr 15 mat : Sizing V Fp Cv DP.psia Q 15 mat : Sizing Header Sizing stack : 31 line

Sizing m/s Fp Cv DP psia Q T/hr 0.21 3.33 21 mat 1 0.9 21 mat 5.94 12321 mat 28420421 mat 21.15 334.51 21 mat 25 mat

The rated valve rangeability Cv 2 el Cv 1 el / 20.7

The installed valve rangeability Q 2 el Q 1 el / 15.817

% open
3" globe Cv
3" U-ball Cv valve 102030405060708090100 14.5 32.9 52.1 70.4 88.5 105118133142148 0.704 5.94 15 27.5 43.8 63.1 87.3 123174220310 mat transpose :

Cv installed factor r Q 2 el Q 1 el / Cv 1 el Cv 2 el / * : 0.7638

GlobeCv valve 1 col valve 2 col augment : U_ballCv valve 1 col valve 3 col augment : U_ballCvInstalled valve 1 col r valve 3 col (* augment : 31 line

GlobeCv U_ballCv U_ballCvInstalled 3 1 sys

Example 3. Sizing a globe valve [F_L is constant]

Full port, linear plug, nomial 3" [12, 1-77], 6" pipe

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1. Catalogue data

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C.v20% 32.9 : C.v80% 133 : 21 sys

XT.20% 0.699 : XT.80% 0.650 : 21 sys

Port diameter [mm]
Pipe ID [mm] d 87.3 : D 161.47 : 21 sys

USER U 0.2273 : T_hr 1 : psia 1 : Fr 1 : 41 sys

2. Assign to procedure

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Cv C.v20% C.v80% 21 mat : X.T XT.20% XT.80% 21 mat : 12 mat

Cvmin C.v20% : Cvmax C.v80% : 21 sys

Constants N2 467 : N4 76000 : N6 0.0274 : 31 sys

Service: saturated steam

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Saturated steam [°C]
Critical pressure [bar]
Vapor pressure [psia]
Vapor pressure [bar] T 200 ≡ Pc 220.64 ≡ Pv 225.75 ≡ bar 15.55 ≡ 41 sys

kg/m³[IAPWS]
upstream pressure [psia]
back pressure [psia]
isentropic expansion
isentropic factor
pressure drop ratio
pressure drop ρ 7.863 : P1 398.7 : P2 274.7 : k 1.296 : F.k k 1.4 / : X P1 P2 - P1 / : ΔP P1 P2 - : 71 sys

114.7

274.7

F.k 0.9257

Σk 1.5 1 d D / (2^- (2^* : 0.7512

Ki 0.5 1 d D / (2^- (2^* 1 d D / (4^- (+ : 1.165

i 12 range F.p i el 1 N2 Σk * Cv i el d 2^/ (2^* + (0.5 -^: for

F.p 0.9967 0.9505 21 mat

F.p F.p :

i 12 range X.TP i el X.T i el F.p i el 2^/ 1 N2 X.T i el * Ki * Cv i el d 2^/ (2^* + (1 -^* : for

X.TP 0.6986 0.6495 21 mat

Y 1 X 3 F.k * X.TP * / vectorize - : 0.8397 0.8276 21 mat

X.TP X.TP :

Cv d D Fp 1467 1.5 1 d D / (2^- (2^* (* Cv d 2^/ (2^* + (0.5 -^:

Cvmin d D Fp Cvmax d D Fp 21 mat 0.9967 0.9505 21 mat

Q U Fr * Y Fp * Cv * ΔP sqrt * (vectorize * ρ 1000 / sqrt * : 6.2 22.3 21 mat

V 353700 Q * D 2^ρ * / : 10.69 38.4 21 mat

Header m/s Fp Cv DP psia Q T/hr 15 mat : Sizing V Fp Cv ΔP Q 15 mat : Sizing Header Sizing stack : 31 line

Sizing m/s Fp Cv DP psia Q T/hr 10.69 38.4 21 mat 1 0.9 21 mat 32.9 13321 mat 124 6.2 22.26 21 mat 25 mat

%CvX.T 10 14.5 0.671 20 32.9 0.699 30 52.1 0.697 40 70.4 0.72 50 88.05 0.733 60105 0.718 70118 0.707 80133 0.65 90142 0.63 100148 0.62 103 mat :

x f 8.9 104 -^* x * 0.6125 + 4.314 x 78 - 801 x 75 - / + / - :

dataXT %CvX.T 1 col %CvX.T 3 col augment :

dataXT x f 2 1 sys

Note that Y cannot exceed 1 and that it cannot be less than 0.67 since the latter is exactly the density of the gas, i.e., 67% of the inlet density when the valve is choked (when it sees sonic velocity in the valve orifice).

Conclusion

This document aims at selecting control valves wrt their hi/lo massflow rate.

A pretty good guide line to size pipe, especially useful for expansive materias.

At this point, the procedure is incomplete. P1, P2 are just demonstrative of

the effect on the valve capacityto control the hi/lo operating massflow rate.

Valve sizing goes in pair with pipe sizing [pump curve, pressure sources ...]

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data 0.1 0.00017 0.2 0.00058 0.3 0.00118 0.4 0.00197 0.5 0.0029 0.6 0.00409 0.7 0.00541 0.8 0.00689 0.9 0.00855 1 0.01 1.1 0.012 1.2 0.014 1.3 0.017 1.4 0.019 1.5 0.022 1.6 0.025 1.7 0.028 1.8 0.031 1.9 0.034 2 0.037 2.1 0.041 2.2 0.045 2.3 0.049 2.4 0.053 2.5 0.057 252 mat :

Linear losses

Typical pressure drop[m/m], water 10 °C

pipe 100 mm, k = 0.03 mm rugosity

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x Linear 0.4672 x * 24.298 - 1267.96 x 52.184 + / + :

x Linear data 2 1 sys

Pulp stock correction factor Fc . . . sewage sludge, fibrous slurries . . .

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Fc 1 1.0 1.0 2 0.97 0.99 3 0.90 0.95 4 0.84 0.92 5 0.80 0.90 53 mat :

Chemical stock Chemical Fc 1 col Fc 2 col augment :

Mechanical stock [ground wood] Pulp Fc 1 col Fc 3 col augment :

x Chem 0.78783 0.41766 x 1.7812 + 21.715 x 5.5665 - / + / - :

x Mech 0.88832 0.12477 x 3.0059 + 28.654 x 6.5883 - / + / - :

Q U Fr * Fc * Fp Cv * DP.psia sqrt * (vectorize * ρ 1000 / sqrt * ≡

x Chem x Mech Chemical Pulp 4 1 sys

valve style modifier from mf'r catalog

Sizing for viscuous fluids :

No examples are given. Reader is recommended

to consult valve suppliers . . . manufacturers.

Further reading:

Baumann-ControlValvePrimer_Chapter5.pdf

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Rev N4 Fd * Q * μ F.Lp Cv * sqrt * / (1 N2 F.Lp Cv * D1 4^/ * - (0.25^* ≡

* Remember, there is no sonic velocity if flow is non-turbulent; therefore, critical flow equations don't apply.