Pressure Relief Valve and Rupture Disk Sizing: Method

Relief Valve and Rupture Disk Sizing: the Nozzle Equation and Its Companions

Scope

A pressure relief valve or a rupture disk is sized by finding the smallest flow area through which the required relieving rate can pass at the relieving pressure, and then choosing the standard orifice that provides at least that area. The tool does this for the six relieving cases met in practice: a gas or vapour, steam, a liquid, a flashing or frozen two-phase mixture, the vapour generated by an external fire, and the normal breathing of an atmospheric tank. Every result is a closed form or a property evaluation: the gas equation is the isentropic-nozzle flow of a perfect gas, whose US-unit form is the API 520 / ASME VIII equation with its coefficient C; the steam flux comes from the IAPWS-IF97 formulation through the shared fluid-property library; the liquid equation is Bernoulli's orifice with the API viscosity correction; the two-phase flux is Leung's omega method; the fire-case rates are those printed in 29 CFR 1910.106 and 1910.110 or the API 521 form of the wetted-area heat input. The standards themselves (API 520, 521, 526, 2000, ASME VIII and I, CGA S-1.3) are cited and not reproduced: their tables of correction factors are replaced by the computation they summarise or by an entered value with its source named.

Relieving pressure and back pressure

The relieving pressure P₁ is the set pressure plus the allowable overpressure, converted to an absolute pressure: P₁ = P_set(1 + overpressure/100) + P_atm. ASME VIII allows 10 percent (or 3 psi) above the maximum allowable working pressure for a single device, 16 percent for multiple devices and 21 percent for the fire case; ASME I limits a boiler to 6 percent above the highest set pressure, as TM 5-650 paragraph 2-13b repeats; 29 CFR 1910.110 rates LP-gas container valves at 120 percent of the start-to-discharge pressure. The back pressure P₂ is the total pressure at the valve outlet during relief, superimposed plus built-up, absolute. The ratio P₂/P₁ against the critical ratio decides whether the nozzle is choked.

The gas equation derived

Treat the valve as an ideal nozzle fed from a reservoir at P₁, T (stagnation conditions) with a perfect gas of molecular weight M, specific-heat ratio k and compressibility Z, so that ρ₁ = P₁M/(ZRT). Isentropic expansion to a throat pressure p gives the density ρ = ρ₁(p/P₁)^1/k and, from the energy equation u²/2 = ∫_p^P₁ dp/ρ, the velocity. The mass flux G = ρu is therefore G² = 2P₁ρ₁ (k/(k−1)) [r^2/k − r^(k+1)/k] with r = p/P₁. This expression rises from zero at r = 1 to a maximum at the critical ratio r_c = [2/(k+1)]^k/(k−1), where the throat velocity equals the local speed of sound; the nozzle cannot pass more, however low the downstream pressure, so for P₂/P₁ ≤ r_c the flow is choked and

G = P₁ √(kM/(ZRT)) [2/(k+1)]^{(k+1)/(2(k−1))}.

The tool checks this closed form against a numerical maximisation of the integral at five gas states (agreement to 1 part in 10¹²). The required effective area is A = W/(K_dK_bK_cG) with W the relieving rate, K_d the coefficient of discharge of the device, K_b the back-pressure factor and K_c the combination factor. Writing W in lb/h, A in in², P₁ in psia, T in °R and M in lb/lbmol, the conversion factors collect into

W = C K_d P₁ K_b K_c A √(M/(TZ)), C = 519.45 √{k [2/(k+1)]^{(k+1)/(k−1)}},

with 519.45 = (3600/lb)(in²)(psi)√(10⁻³ × 1.8/R_u) for R_u = 8.314462618 J/(mol K); API 520 and ASME VIII round the constant to 520, and their tabulated C (315 at k = 1.001, 356 at k = 1.4, 400 at k = 2.0) is this function. In SI working units (kg/h, mm², kPa, K, kg/kmol) the constant is 0.039481, printed as 0.03948. As k approaches 1 the bracket tends to e⁻¹, so C tends to 519.45 e^−1/2 = 315.1, the conservative value used when k is unknown.

Subcritical flow and the back-pressure factor

When P₂/P₁ exceeds r_c the throat pressure equals the back pressure and the same integral, evaluated at r = P₂/P₁, gives G = F₂ √(2ρ₁(P₁ − P₂)) with

F₂ = √{ (k/(k−1)) r^2/k (1 − r^(k−1)/k) / (1 − r) },

which API 520 writes as A = W/(735 F₂ K_d K_c) √(ZT/(M P₁(P₁ − P₂))); 735 = 519.45 √2. At r = r_c the two forms give the same flux, which the tool checks. For a conventional spring-loaded valve this is the whole back-pressure effect and K_b = 1 while the flow is choked; the second figure on the results page draws the subcritical ratio G(r)/G_choked against the gauge back pressure in percent of the set pressure. Balanced-bellows and pilot-operated valves behave differently: their capacity falls with back pressure through the lift of the disk, not through the nozzle, and the makers (and API 520 Figure 30 for bellows valves at 10 percent overpressure) give K_b curves that are applied to the choked form. The tool takes such a factor as an entered value or interpolates it from entered points of the maker's curve.

Coefficients

K_d is the ratio of the actual to the ideal-nozzle flow. For a certified relief valve sized preliminarily API 520 uses 0.975 for gas, vapour and steam and 0.65 for liquid; the certified value of the chosen valve replaces it in the final check. A rupture disk alone is sized as an orifice with K_d = 0.62 on its minimum net flow area (ASME VIII UG-127, API 520). A valve with a rupture disk at its inlet keeps the valve's K_d and takes the combination capacity factor K_c = 0.9 unless a certified combination factor exists (ASME VIII UG-127(a)(3)(b)). The Device dropdown fills these values and they stay editable.

Steam

Steam is not a perfect gas near saturation, so the practice uses Napier's empirical form W = 51.5 A P₁ (lb/h, in², psia; Napier 1866) for dry saturated steam, corrected for superheat by K_SH and, above 1,500 psia, by K_N = (0.1906 P₁ − 1,000)/(0.2292 P₁ − 1,061). API tabulates K_SH against pressure and temperature; the tool computes it instead as the ratio of the isentropic-nozzle mass flux at the entered temperature to that at saturation, both from IF97: along the isentrope from the inlet state the flux G = ρ √(2(h₀ − h)) is maximised over the throat pressure. With condensation in equilibrium (the homogeneous-equilibrium path, which the steam enters at a throat ratio near 0.58) the saturated flux agrees with Napier's 51.5 P₁ within 3 percent up to 1,500 psia; on the metastable dry-vapour path of a short nozzle (IF97's equation 18, below 10 MPa) it is 1 to 6 percent higher. At 100 psia and 500 °F the computed K_SH is 0.935, the value the API table prints as 0.93. The page gives the area by Napier's form with the computed K_SH and K_N, and by the IF97 equilibrium flux directly, and takes the larger. For a boiler the required rate is the maximum designed steaming capacity; for a high-temperature hot-water boiler or an economizer TM 5-650 paragraph 2-13b sets it at the Btu/h output divided by 1,000.

Liquid

For an incompressible liquid Q = K_dK_wK_cK_v A √(2ΔP/ρ) with ΔP = P₁ − P₂. With Q in gpm, A in in², ΔP in psi and the specific gravity G referred to water at 1,000 kg/m³ this is A = Q/(38 K_dK_wK_cK_v) √(G/ΔP); the constant follows from the unit conversions as 37.97, and 11.78 in L/min, mm², kPa. K_w is the back-pressure factor of a balanced-bellows valve in liquid service (the maker's curve, API 520 Figure 31); for a conventional valve it is 1 because the differential pressure already contains the back pressure. The viscosity correction is the API relation K_v = (0.9935 + 2.878/Re^0.5 + 342.75/Re^1.5)⁻¹ with the Reynolds number at the orifice Re = 2ρQ/(√π μ√A), which API writes as Q(2,800 G)/(μ√A) in gpm, cP and in²; because Re contains the area, the tool iterates A = A₀/K_v to convergence.

Two-phase flow: the omega method

A flashing liquid, a liquid with dissolved or entrained gas, or wet steam leaves the nozzle as a two-phase mixture whose density falls steeply with pressure. Leung (1986) showed that the homogeneous-equilibrium expansion of such a mixture is described to within a few percent by the single parameter ω in the equation of state v/v₀ = ω(P₀/P − 1) + 1, where v₀ and P₀ are the inlet (stagnation) specific volume and pressure. For a flashing one-component fluid ω = x₀v_fg/v₀ + c_plT₀P₀/v₀ (v_fg/h_fg)², the first term the compressibility of the vapour already present and the second the vapour generated by flashing; for a frozen gas-liquid mixture ω = α₀/k. Substituting the equation of state into the nozzle integral G = √(2∫v dP)/v and maximising gives the critical pressure ratio η_c as the root of η_c² + (ω² − 2ω)(1 − η_c)² + 2ω² ln η_c + 2ω²(1 − η_c) = 0, the choked flux G_c = η_c √(P₀/(ω v₀)) and, below the critical ratio, G = √(P₀/v₀) √(−2[ω ln η + (ω − 1)(1 − η)])/(ω(1/η − 1) + 1). The tool checks these forms against a numerical integration of the same equation of state (seven values of ω, 1 part in 10⁹) and, for water, prints beside them the flux obtained by maximising the IF97 homogeneous-equilibrium expansion directly; the omega approximation sits 1 to 6 percent low for the cases tested. The coefficient of discharge for two-phase flow is entered (API 520 Annex C suggests 0.85 when no certified value exists). The method assumes a saturated or two-phase inlet; a subcooled inlet, which flashes only after the pressure has fallen to saturation, needs Leung's extension and is not on the page.

Fire exposure

Three published rates are offered. For a pressure vessel the heat absorbed from a pool fire is Q = C F A^0.82 (Btu/h, A the wetted area in ft² up to 25 ft above grade), C = 21,000 where drainage and fire fighting are adequate and 34,500 where they are not, F the environment factor (1.0 bare, lower with insulation or water spray); API 521 tabulates these, and the note to 29 CFR 1910.106(b)(2)(v)(d) prints Q = 21,000 A^0.82. The vapour generated is W = Q/λ with the latent heat at the relieving pressure, which the tool takes from IF97 for water (checked against the latent-heat column of TM 5-650 Table 1-9) or as an entered value. For an aboveground flammable-liquid storage tank 29 CFR 1910.106(b)(2)(v) prints Table H-10 of required venting in cubic feet of free air per hour against the wetted area (55 percent of a sphere, 75 percent of a horizontal tank, the first 30 ft of a vertical shell), on a hexane basis, with CFH = 1,107 A^0.82 above 2,800 ft²; the rate for another stable liquid is the table value times 1,337/(L√M), and the protection factors of paragraph (f) apply. The factor 1,337 is the hexane value of L√M (144 Btu/lb × √86.18 = 1,337) and the free-air equivalent of a vapour flow W of molecular weight M is V_air = (W/M) V_m √(M/M_air) with V_m = 379.5 ft³/lbmol at 60 °F and 14.7 psia, so that 21,000 A^0.82 × 70.5/1,337 reproduces the rule's 1,107 A^0.82. For an LP-gas container 29 CFR 1910.110(b)(10)(ii) prints the minimum rate of discharge in cubic feet per minute of air at 120 percent of the start-to-discharge pressure against the total outside surface area, with 53.632 A^0.82 above 2,000 ft². In each case the device is then sized for the vapour: water through the steam route, another liquid as a vapour with its molecular weight at the entered gas state, LP-gas as air.

Orifice letters and rated capacity

API 526 designates the effective orifice areas of flanged steel relief valves by letters D (0.110 in²) to T (26.0 in²); the tool lists them (as reproduced in Bhowmik and Sabharwall 2024, Table 6, in cm²), picks the smallest that covers the required area and prints the rated capacity of that orifice at the relieving conditions, the flow a valve of that orifice passes when it opens. The piping checks and the reaction force are made at that rated capacity, as API 520 Part II requires, not at the required rate. The first figure draws the rated capacity of the selected orifice against the set pressure with the required rate as a horizontal line.

Inlet and outlet piping

A relief valve whose inlet line loses more than about 3 percent of the set pressure at the rated flow may chatter, because the pressure at the valve falls below the reseat pressure as soon as it opens; API 520 Part II sets the 3 percent limit on the non-recoverable loss. The tool computes it by Darcy, ΔP = (ρV²/2)(fL/D + ΣK), at the inlet density with the Colebrook friction factor of the shared pipe-flow library. The outlet line builds up a back pressure that must stay within what the valve tolerates (10 percent of the set pressure for a conventional valve at 10 percent overpressure, more for balanced valves per the maker). For a gas discharging to atmosphere the tool solves the isothermal compressible-flow relation P_o² − P_atm² = G²(RT/M)[fL/D + ΣK + 2 ln(P_o/P_atm)] for the pressure at the valve outlet, and flags an exit Mach number above the isothermal limit 1/√k, beyond which the exit chokes and the line pressure is higher than the isothermal solution; steam is treated with M = 18.02 and k = 1.3 in this check, a liquid by Darcy.

Reaction force

The discharge from an open vent pushes back on the piping with F = Wu_e + (P_e − P_atm)A_e, the momentum flux plus the pressure thrust at the exit plane. If the exit is choked, T_e = 2T₀/(k+1), u_e = √(kRT_e/M) and P_e = G√(RT_e/(kM)); the tool first evaluates this exit pressure and, when it is below atmospheric, treats the exit as subsonic at P_atm, finding the exit Mach number from the mass flux G = P_atmM_e√(kM/(RT₀))√(1 + (k−1)M_e²/2). API 520 Part II writes the choked form as F = (W/366)√(kT/((k+1)M)) + AP in lbf, lb/h, °R, in² and psig; the constant follows from 3,600 g_c/√(2R_ug_c) as 367.3, and the page prints the API form beside the momentum balance. The force acts along the vent axis away from the exit and must be taken by the supports; TM 5-650 Figure 2-23 shows the discharge pipe anchored to the structure with the drain and clearances the boiler installation needs.

Atmospheric tank breathing

An atmospheric or low-pressure tank breathes in as liquid is pumped out and as its vapour space cools, and breathes out as it is filled and warms. API 2000 (ISO 28300) gives the thermal terms as V_IT = C V^0.7 R_i and V_OT = Y V^0.9 R_i in normal m³/h of air with the tank volume in m³, the constants depending on latitude, the liquid's vapour pressure and insulation, and the movement terms as the pump-out rate (in) and the pump-in rate times a factor (out). The standard is copyrighted and its tables are not reproduced; the constants are entered, and the defaults are the builder's recollection, not verified against a copy, so the standard's values should be entered for a design.

Checks and limits

The library is checked by test-scripts/test_relief_valve_sizing.php against an independent numpy / scipy script (References/Relief-Valve/checks/): the C(k) constant and its SI counterpart, the choked flux against the numerical maximisation of the nozzle integral, the F₂ continuity at r_c, the liquid constants and the K_v iteration, the omega method against the integration of its equation of state, the fire-case arithmetic and the reproduction of the CFR constants 1,107 and 1,337, the reaction-force momentum balance, the isothermal outlet solution; the IF97 latent heats against TM 5-650 Table 1-9; the steam flux against Napier's constant; and the worked case of Bhowmik and Sabharwall (2024), Table 7 (93.17 cm² for 50,000 kg/h at 4.725 bar, k = 1.27), to the printed digits. Not on the page: the maker's certified K_d and lift curves, rupture-disk burst tolerances and manufacturing ranges, pilot-valve flow curves, subcooled two-phase inlets, non-condensable gases in flashing flow, dynamic (chatter) analysis of the inlet line, and the thermal-relief case of blocked-in liquid lines.