Relief sizing

With the scenarios for V-410 listed, Theo turns to sizing. A relief device must pass the required flow at the allowed relieving pressure; if it is too small, the pressure rises beyond what the vessel can hold. If it is much too large, it may chatter and damage itself at lower flows, so oversizing is not free either. This lesson presents the sizing equations of API Standard 520 for vapour and liquid relief, the API 521 method for estimating the heat input from a fire, and an introduction to two-phase relief from runaway reactions. The equations are given in SI units, as used in the company's calculation sheets; API 520 also gives them in US customary units with different constants.

Vapour relief

For gas or vapour relief at critical (choked) flow through the nozzle, which applies at almost all relieving pressures used in practice, API 520 gives the required effective discharge area as

with \(A\) in mm², \(W\) the required flow in kg/h, \(P_1\) the upstream relieving pressure in kPa absolute (set pressure plus allowed overpressure plus atmospheric pressure), \(T\) the relieving temperature in kelvin, \(Z\) the compressibility factor and \(M\) the molar mass. The coefficient \(C\) depends on the ratio of specific heats \(k\):

\(K_d\) is the effective coefficient of discharge, 0.975 for a relief valve in preliminary sizing; \(K_b\) corrects for back pressure on conventional or bellows valves; and \(K_c\) is 0.9 when a rupture disc is installed upstream of the valve and 1.0 otherwise. The result is compared with the standard effective orifice areas of API Standard 526, identified by letters from D to T, and the next larger orifice is chosen.

# Relief valve sizing in the style of API 520 (SI units) and the API 521 fire heat input
import math
# Vapour relief: A (mm2) = W / (C Kd P1 Kb Kc) * sqrt(T Z / M), W kg/h, P1 kPa abs, T K
def Cfac(k): return 0.03948 * math.sqrt(k * (2 / (k + 1)) ** ((k + 1) / (k - 1)))
W, M, k, Z, T = 9400.0, 88.1, 1.10, 0.95, 363.0       # fire case vapour from lesson 8 (illustrative)
Pset = 350.0; P1 = Pset * 1.21 + 101.3
Kd, Kb, Kc = 0.975, 1.0, 1.0
A = W / (Cfac(k) * Kd * P1 * Kb * Kc) * math.sqrt(T * Z / M)
print(f"vapour: C = {Cfac(k):.4f}, P1 = {P1:.0f} kPa abs, required area {A:.0f} mm2")
orifices = (("J", 830), ("K", 1186), ("L", 1841), ("M", 2323), ("N", 2800), ("P", 4116))   # API letter areas, mm2
pick = next((n, a) for n, a in orifices if a >= A)
print(f"select orifice {pick[0]} ({pick[1]} mm2)")
# Liquid relief: A (mm2) = 11.78 Q / (Kd Kw Kc Kv) sqrt(G / dP), Q L/min, dP kPa
Q, G = 18_000 / 60 / 0.90, 0.90      # 18 t/h of ethyl acetate -> L/min
dP = 350.0 * 1.10 - 0.0
Al = 11.78 * Q / (0.65 * 1 * 1 * 1) * math.sqrt(G / dP)
print(f"liquid: {Q:.0f} L/min, required area {Al:.0f} mm2")
# Fire heat input to a wetted surface: Q = 43,200 F A^0.82 W (adequate drainage and firefighting)
for Aw in (40.0, 80.0, 120.0):
    q = 43_200 * 1.0 * Aw ** 0.82
    print(f"wetted area {Aw:4.0f} m2: heat input {q / 1000:.0f} kW -> vapour {q / 1000 / 366 * 3.6:.1f} t/h (latent heat 366 kJ/kg)")

vapour: C = 0.0248, P1 = 525 kPa abs, required area 1465 mm2
select orifice L (1841 mm2)
liquid: 333 L/min, required area 292 mm2
wetted area   40 m2: heat input 890 kW -> vapour 8.7 t/h (latent heat 366 kJ/kg)
wetted area   80 m2: heat input 1570 kW -> vapour 15.4 t/h (latent heat 366 kJ/kg)
wetted area  120 m2: heat input 2190 kW -> vapour 21.5 t/h (latent heat 366 kJ/kg)

For the fire case on V-410, the vapour is mainly ethyl acetate at the relieving pressure of 424 kPa gauge, or 525 kPa absolute. With \(k = 1.10\), \(C\) is 0.0248, and a relief rate of 9,400 kg/h needs about 1,465 mm². The next larger standard orifice is L, with 1,841 mm². The liquid blocked-outlet case, 18 t/h at the single-valve relieving pressure, needs only about 290 mm². So the fire case governs despite its smaller mass rate, as lesson 8 anticipated. The existing data sheet for V-410 showed a K orifice, 1,186 mm², copied from an older drum. It is about 20 % too small for the fire case.

Liquid relief

For liquid relief through a spring valve certified for liquid service, API 520 gives

with \(A\) in mm², \(Q\) the flow in L/min, \(G\) the specific gravity, \(p_1 - p_2\) the differential pressure across the valve in kPa, and correction factors \(K_w\) for back pressure on bellows valves and \(K_v\) for viscosity. \(K_d\) is 0.65 for preliminary sizing of liquid valves. Because the flow scales with the square root of the pressure difference, liquids pass through small orifices easily. Liquid relief valves should be certified for liquid service; vapour-certified valves can behave poorly on liquid.

Fire heat input

A fire around a vessel heats the part of its wall that is wetted by liquid, and the heat boils the liquid. API 521 gives an empirical heat input, based on tests, for vessels exposed to an open pool fire:

in watts, where \(A_{ws}\) is the wetted surface area in m² within a height of about 7.6 m above the fire's base, and \(F\) is an environment factor: 1.0 for a bare vessel, and less for insulated vessels or those with water spray or fireproofing that meet specified standards. That form applies where there is adequate drainage and prompt firefighting; without them, the constant becomes 70,900. In US customary units the corresponding constants are 21,000 and 34,500 with the heat input in Btu/h and area in ft². The relief rate is the heat input divided by the latent heat of vaporization at relieving conditions. For a wetted area of 40 m² and a latent heat of 366 kJ/kg, the heat input is about 890 kW and the vapour rate about 8.7 t/h; V-410's wetted area of about 44 m² gives the 9.4 t/h used above.

Because the exponent is less than one, doubling the wetted area does not double the relief rate. Insulation that is fire-rated and protected by weather jacketing can reduce \(F\) substantially, which is one reason to insulate vessels containing volatile flammable liquids. Drainage matters too: a sloped surface that carries burning liquid away from the vessel reduces the fire exposure and justifies the lower constant.

Fire sizing also asks what else happens to the vessel in a fire. Above the liquid level, the wall is not cooled, and its temperature can rise until the steel loses strength, even while the relief valve holds the pressure within limits. For vessels containing mainly vapour, or for long fire exposures, API 521 recommends considering depressuring systems that lower the pressure quickly, reducing the stress in the hot wall, and fire-resistant insulation or water spray that keeps the steel cool. For V-410, a liquid-filled drum in a well-drained area, the relief valve with the plant's fire water monitors is judged adequate, but the team records the reasoning in the relief data sheet so that it can be revisited if the drum's service changes.

The relieving temperature must be estimated with care too. In a fire the liquid boils at the relieving pressure, which is higher than its normal boiling point, so the vapour is hotter than in normal operation and the latent heat is somewhat lower. Using the normal boiling point and its latent heat usually gives a slightly larger relief rate, which is conservative. For mixtures, the light components boil off first, and the composition of the vapour changes as the fire continues; a careful study checks the relief rate at several points during the boil-off.

Two-phase relief and runaway reactions

Some scenarios produce a mixture of liquid and vapour at the relief device. Runaway reactions are the clearest case: the reacting liquid boils, and the vapour bubbles swell the liquid until a foam fills the vessel and enters the relief line. Two-phase flow through a relief device carries much less volume per unit area than vapour alone, so a device sized for vapour can be far too small. The work of the Design Institute for Emergency Relief Systems, DIERS, sponsored by the American Institute of Chemical Engineers in the 1970s and 1980s, provided methods for these cases, including the omega method for estimating two-phase mass flux and the use of adiabatic calorimeters to measure how fast pressure and temperature rise in a runaway. For the pre-reactor of CHE-345, the team's relief design for the runaway scenario was based on calorimeter data and a DIERS method, with a rupture disc discharging to a catch tank rather than a spring valve.

Two-phase sizing is a specialist area. Its results are sensitive to the composition, the reaction kinetics and the vessel fill level, and it is common to find that the governing scenario for a reactor needs a relief area many times larger than a simple vapour calculation would suggest. In the course capstone, Theo's role is to make sure the scenario has been identified and handed to the specialists, not to perform the DIERS calculation himself.

Back pressure and installation effects

The sizing equations assume that the valve sees the full pressure difference between the vessel and its discharge. In practice, built-up back pressure from flow in the outlet piping and superimposed back pressure from other valves discharging into the same header reduce the capacity of conventional valves and can change their set pressure. API 520 gives back pressure correction factors for balanced bellows and pilot valves, and practice limits built-up back pressure on conventional valves to about 10 % of the set pressure. The inlet piping is just as important: a long or restricted inlet causes pressure loss that makes the valve chatter. For V-410, the existing inlet nozzle is large enough for the L orifice, but the outlet piping must be enlarged from 80 to 100 mm to keep the built-up back pressure within the limit at the higher flow.

Relief valves also need clean and dry service. Ester and ethanol are clean, but polymerizing or solids-laden services can plug inlets and seats, and a plugged relief valve gives no warning until it is needed. Inspection records for each valve show the as-found condition when it is removed for testing; a valve that repeatedly fails its as-found test is evidence that the test interval is too long or the service is unsuitable for that valve type. Theo adds V-410's new valve to the mechanical integrity database with an initial test interval of three years, to be adjusted on the basis of its as-found results.

What Theo recommends

Theo's calculation note for V-410 reaches three conclusions. The fire case governs and needs at least 1,465 mm². The installed K orifice is undersized, and an L orifice should be installed with its inlet and outlet piping checked for the larger flow, including the 3 % inlet loss rule. And the drum's relief data sheet must be corrected, with a management of change record explaining the error. Nadia reviews the calculation and adds one point: the header and flare that receive V-410's discharge must also be checked, because a larger valve may now be able to deliver more flow than the header was designed to carry. The capstone in lesson 12 includes these actions.