Control Valve Sizing

Control Valve Sizing

Control Valve Sizing: Flow Coefficient, Choked Flow, Expansion Factor and Characteristics

Scope and sources

A control valve is sized by finding the flow coefficient that passes the required flow at the available pressure drop, and then choosing a valve whose rated coefficient leaves it at a sensible travel over the whole range of operating cases. The tool uses the sizing equations of ISA-75.01.01 and IEC 60534-2-1 for incompressible and compressible fluids. Those standards are copyrighted and are cited, not reproduced: the equations are written here as mathematics, every numerical constant that is a unit conversion is computed from the definition of the flow coefficient, and the standards' tables of typical valve factors are replaced by entered inputs. The public documents that print the pieces used are the U.S. Nuclear Regulatory Commission's Cavitation Guide for Control Valves (NUREG/CR-6031: the flow coefficient and its conversions, the cavitation index, choking and the pressure recovery factor F_L) and the U.S. Army Corps of Engineers Technical Instructions TI 810-11, reissued as UFC 3-410-02A (the liquid C_v and K_v equations, the simplified steam equations, and worked examples). Steam properties come from the IAPWS-IF97 formulation through the site's shared fluid-property library.

The flow coefficient

C_v is the number of US gallons per minute of water at 60 °F that pass the valve at a pressure drop of 1 lb/in², so that for any liquid Q = C_v √(ΔP/G) with Q in gpm, ΔP in lb/in² and G the relative density. In consistent units this is Q = C_v k_q √(ΔP/ρ) with k_q = (1 gpm) √(ρ₀/(1 lb/in²)) = 2.4016 × 10⁻⁵ m^7/2 kg^−1/2 and ρ₀ = 999.10 kg/m³, the density of water at 15 °C to which G is referred. The metric coefficient K_v is the flow in m³/h at 1 bar, K_v = 0.8650 C_v. (TI 810-11 warns that K_v is also quoted in L/s at 1 kPa, which is 1/36 of the value used here.)

All the "N" constants of the standards' equations that only carry units follow from k_q: N₁ = 1 (gpm, lb/in²), 0.865 (m³/h, bar), 0.0865 (m³/h, kPa); N₂ = π²/(8 k_q²) = 890 with the valve size in inches (0.00214 in mm), the constant of NUREG/CR-6031 equations 3.9 to 3.11, which relates C_v to the velocity-head loss coefficient, K = N₂ d⁴/C_v²; N₆ = 63.3 (lb/h, lb/in² absolute, lb/ft³), 27.3 (kg/h, bar, kg/m³), 2.73 (kg/h, kPa, kg/m³); N₈ = 19.3 (lb/h, lb/in² absolute, °R), 94.8 (kg/h, bar, K); N₉ = 7,320 (standard ft³/h at 60 °F and 14.73 lb/in², lb/in² absolute, °R), 21.2 or 22.4 (m³/h at 0 °C or 15 °C, kPa, K); N₄ = 2/√(π k_q/√2) = 17,300 (gpm, centistokes) or 0.0760 (m³/h, m²/s). The test script compares each computed value with the published rounded one. Two constants are the standards' own and are used as cited single facts: N₅ = 1,000 (inches; 0.00241 in mm) in the fitting correction of x_T, and N₃₂ = 127 (C_v, mm) in the reduced-trim Reynolds factor.

Reducers: the piping geometry factor

A valve smaller than its line sits between a reducer and an expander. Their effect is expressed by velocity-head coefficients on the valve bore d: the inlet reducer K₁ = 0.5 [1 − (d/D₁)²]², the outlet expander K₂ = [1 − (d/D₂)²]², and the Bernoulli coefficients K_B1 = 1 − (d/D₁)⁴ and K_B2 = 1 − (d/D₂)⁴ for the change of velocity head between the pipe and the valve. With ΣK = K₁ + K₂ + K_B1 − K_B2 the valve and its fittings behave as one device of coefficient F_P C_v, where F_P = 1/√(1 + ΣK C_v²/(N₂ d⁴)), which is simply the series addition of the loss coefficients K = N₂ d⁴/C_v². Because F_P contains the coefficient being sought, the tool iterates to convergence; for unchoked liquid flow the result has the closed form C_v = C₀/√(1 − ΣK C₀²/(N₂ d⁴)) with C₀ the coefficient without fittings, which the test script checks. If the reducers alone would take the whole pressure drop the tool says so. For a line-size valve every coefficient is zero and F_P = 1.

Liquids: choked flow, cavitation and flashing

The liquid flow rises with √ΔP only until the pressure at the vena contracta falls to the vapour pressure. Beyond that point more pressure drop passes no more flow: the valve is choked. NUREG/CR-6031 defines the pressure recovery factor as the ratio of the choked to the unchoked coefficient (its equation 4.8), which gives the choked pressure drop of the valve alone as ΔP_choked = F_L² (P₁ − F_F P_v). F_F = 0.96 − 0.28 √(P_v/P_c) is the liquid critical pressure ratio factor, which allows for the vena-contracta pressure falling somewhat below the vapour pressure before vapour forms; the guide notes that it can be taken as 1 for water below 200 °F. With reducers the recovery factor of the combination is F_LP = F_L/√(1 + F_L² (K₁ + K_B1) C_v²/(N₂ d⁴)) and the choked drop is (F_LP/F_P)² (P₁ − F_F P_v). The required coefficient is C_v = Q/(N₁ F_P) √(G/ΔP_sizing) with ΔP_sizing the smaller of the actual and the choked drop.

Cavitation begins long before choking. The guide measures its intensity with the cavitation index σ = (P₁ − P_v)/ΔP (its equation 2.1) and names a sequence of limits: incipient, critical, incipient damage, incipient choking and choking, the last being σ_ch = 1/F_L² when F_F = 1 (equation 4.10). The tool prints σ and its reciprocal x_F = ΔP/(P₁ − P_v) and compares x_F with two entered ratios: x_Fz, where cavitation begins, and optionally K_c, where the flow first departs from the square-root law (the guide's incipient choking, its equation 2.7). The default globe values rest on the measured 3 in globe valve of the guide's Figure 6.9: near 75 percent open σ is about 2.5 at incipient cavitation and 1.25 at choking, that is x_Fz = 0.40 and F_L = 0.89. The guide stresses that the early limits change with valve size and pressure (scale effects) and must come from tests of the valve type. If the outlet pressure is at or below the vapour pressure the liquid flashes: the flow is choked and part of it leaves as vapour, which the single-phase equations do not describe beyond the choked flow rate.

TI 810-11 uses a simpler rule for hot-water valves: the maximum pressure drop without cavitation is k_m (p_e − p_v) with k_m = 0.45, which is the choked drop of a valve with F_L = 0.67 and F_F = 1. Its example (200 °F water at 20 lb/in² gauge: 10.43 lb/in²; 50 gpm at 6 lb/in²: C_v = 20.41, K_v = 17.64) is reproduced by the test script.

Viscous flow: the Reynolds number factor

The valve Reynolds number is Re_v = N₄ F_d Q/(ν √(C_v F_L)) [F_L² C_v²/(N₂ D⁴) + 1]^1/4. Its length scale is the hydraulic diameter of one flow passage: the vena-contracta area of the valve is C_v F_L k_q/√2, the diameter of a circle of that area times the valve style modifier F_d is the hydraulic diameter, and the bracket adds the velocity of approach. Above 10,000 the flow is turbulent and F_R = 1. Below it the required coefficient is C_v/F_R, with F_R the smaller of the transitional form 1 + (0.33 √F_L/n^1/4) log₁₀(Re_v/10,000) and the laminar form (0.026/F_L) √(n Re_v), where n = N₂ d⁴/C_v² for a full-size trim and n = 1 + N₃₂ (C_v/d²)^2/3 for a reduced trim (rated C_v/d² below 0.016 with d in mm). These forms and their constants are the standards'; they are stated here as recalled and were checked against the open-source implementation in the fluids library, not against a copy of the standard. The standards size non-turbulent flow without the piping geometry factor, and so does the tool. Because F_R depends on the coefficient, the tool finds the smallest coefficient that satisfies C_v F_R(C_v) = C₀; for a full-size trim the product stops rising at large C_v, and when no coefficient satisfies the equation the tool reports that the body is too small for so viscous a flow.

Gas, vapour and steam: the expansion factor

For a compressible fluid the same coefficient is used with the inlet density and an expansion factor: W = N₆ F_P C_v Y √(x P₁ ρ₁), where x = ΔP/P₁ and ρ₁ = P₁ M/(Z R T₁) for a gas. Substituting the density gives the mass form with N₈ and the standard-volume form with N₉; the tool evaluates one consistent equation and converts a standard volume flow with the gas density at 60 °F and 14.696 lb/in² absolute. The expansion factor Y = 1 − x/(3 F_k x_T) accounts for the fall of density through the valve. It is linear in x, equals 1 at no pressure drop and 2/3 at x = F_k x_T, where the product Y √x reaches its maximum: the flow is choked and stays constant for any larger drop. x_T is the pressure differential ratio factor of the valve measured with air, and F_k = k/1.4 transfers it to a gas of another specific heat ratio. With reducers x_T is replaced by x_TP = (x_T/F_P²)/(1 + x_T (K₁ + K_B1) C_v²/(N₅ d⁴)). As a consistency check, the test script compares the linear Y with the expansion factor of an ideal isentropic nozzle that chokes at the same ratio: they differ by no more than 0.051 between zero drop and choking for k from 1.2 to 1.67.

Steam is sized with the same equation. The inlet density and the isentropic exponent k = ρ w²/P (w the speed of sound) are taken from IAPWS-IF97 at the inlet pressure and temperature, dry saturated when no temperature is entered, and the outlet state for the velocity check follows from the constant-enthalpy drop. Beside it the tool prints the simplified steam equations of TI 810-11, C_v = W (1 + 0.0007 T_sh)/(2.11 √((p_e + p_o) Δp)) below the critical drop 0.45 p_e and C_v = W (1 + 0.0007 T_sh)/(1.74 p_e) at or above it (lb/h, lb/in² absolute, °F of superheat). They contain no valve-specific factor, so they agree with the equation above only for a valve whose F_k x_T is near 0.45; the manual's examples (250 lb/h from 5 to 2 lb/in² gauge: C_v = 11.34; 6,000 lb/h at the critical drop with 48 °F of superheat: K_v = 77.5) are reproduced by the test script. Two details of the second example differ: the inch-pound value printed, 89.1, corresponds to an inlet pressure rounded to 40 lb/in² absolute (39.7 gives 89.8), and IAPWS-IF97 gives 45.7 °F of superheat for the throttling from 125 to 25 lb/in² gauge.

Characteristic, travel, installed characteristic and authority

The inherent characteristic is the fraction f of the rated coefficient at the travel fraction h under constant pressure drop: linear f = h; equal percentage f = R^{h − 1}, in which equal steps of travel multiply the coefficient by equal ratios and R is the rangeability; quick opening, idealised here as f = √h. The travel of each operating case is the inverse of that function at f = C_v/C_v,rated.

In a system the pressure drop across the valve is not constant. The pump or supply and the rest of the circuit leave the valve a drop that falls roughly with the square of the flow, ΔP(F) = ΔP₀ − (ΔP₀ − ΔP_d)(F/F_d)², from the entered no-flow value to the drop of the design case. The installed characteristic is the flow at each travel for which the valve equation and this curve agree, found numerically for every fluid. For an unchoked liquid valve without reducers it has the closed form q = f/√(N + (1 − N) f²), where q is the flow as a fraction of the fully open flow and N, the valve authority, is the pressure drop of the fully open valve divided by the no-flow drop. A low authority bends an equal-percentage characteristic toward linear and a linear one toward quick opening. The installed gain dq/dh is tabulated; a loop tuned at one operating point keeps its margin where the gain changes little. TI 810-11 requires the valve to take at least half of the circuit pressure drop at full flow.

Velocity checks and what is left out

The outlet velocity is the volume flow at the outlet state through the valve bore; for gas and steam the Mach number refers to the speed of sound at that state. Both are compared with entered limits, since the acceptable values depend on the valve design and the service. Noise prediction (IEC 60534-8-3 for gases, 8-4 for liquids) and two-phase sizing are outside the tool. The factors F_L, x_T, F_d and x_Fz filled by the style list are typical values for a first estimate: only the globe F_L and x_Fz are tied to a public measurement (NUREG/CR-6031 Figure 6.9); the others are recalled typical values and unverified. The final sizing uses the certified factors of the selected valve at the travel in question.

Worked examples reproduced

NUREG/CR-6031: the 24 in butterfly valve of section 3.2 (21,000 gpm at 0.355 lb/in² gross and 0.227 net: C_v = 35,200 and 44,100); Example 6.1 (σ = 2.13); Example 6.4 (6 in butterfly valve, C_d = 0.60: C_v = 805; σ = 1.77 against σ_ch = 2.9: choking, ΔP_ch = 30.54 lb/in²). The guide's last line of that example prints Q = 805/(30.54)^0.5 = 145 gpm; the coefficient equation gives Q = 805 √30.54 = 4,449 gpm, which is what the tool returns. TI 810-11: the examples named above. The default case of the tool (water, 400 gpm at 60 lb/in², 3 in valve in a 4 in line, rated C_v 120) is illustrative and has no printed source: C_v = 51.93, K_v = 44.91, travel 78.6 percent.