Pressure-relief valves are reliable and effective pressure-relief devices that protect personnel from the dangers of over-pressurizing equipment, prevent damage to equipment, and minimize release of hazardous materials. Sizing relief valves involves determination of the rate of material release through the relief valve during the identified worst-case contingency. Relief valves are designed to relieve liquids, vapors or two phases from protected pressure vessels before excessive pressures are developed. A mistake in the relief-valve sizing can result in catastrophic failures because relief valves are usually the last defense to the process equipment against instrument failures, process upsets and operator errors.
This article focuses on sizing the pressure-relief valves for critical flow of gases or vapors. Two-phase sizing methods1,2 are not well established, but it is generally understood that sizing methods for vapor or gas are well established and the results are relatively accurate. Relief-system designers favor simple sizing equations, and use the conventional vapor-sizing equation in American Petroleum Institute (API) Standard 520, using the ideal gas specific heat ratio.3 Although rigorous calculations using isentropic flash calculations give the most accurate results, the simple API relief-valve sizing equation is still preferred because of its simplicity.
However, sometimes the real gas specific heat ratio is more readily available from a process simulator than the ideal gas specific heat ratio. Thus, a simple sizing equation using the real gas specific heat ratio was developed. Emerging from that development is an improved sizing equation using the real gas specific heat ratio. The results are compared with the conventional API sizing equation. Furthermore, a rigorous critical-flow equation with the isentropic flash is introduced as a recommended estimation tool for gas or vapor critical flow when the gas or vapor is known to deviate significantly from ideal conditions.
Conventional API flow equation.
Eq. 1 is the conventional API relief-valve sizing equation for critical vapor flow. The conventional API sizing equation requires five fluid property data: absolute pressure, P; kPa, absolute temperature, T; K, molecular weight, M, compressibility factor, Z, and ideal gas specific heat ratio, k*, at inlet conditions. The equation provides satisfactory sizing results over a wide range of process conditions. However, the sizing equation is only valid for 0.8 < Z < 1.1.3 This means that the sizing results may not be satisfactory for very high pressure conditions or critical-point regions.
A = Relief valve orifice area, mm2
W = Mass flow rate, kg/h
Ck* = A function of the ratio of ideal gas specific heat at inlet conditions
Kd = The coefficient of discharge
Kb = The capacity correction factor due to backpressure
Kc = The combination correction factor for installation with a rupture disk upstream
1 = Fluid conditions at the inlet of the relief valve, where velocity is equal to zero
choke = Choked (critical) conditions.
Real gas specific heat ratio.
The new equation uses the real gas specific heat ratio in the sizing equation instead of the ideal gas specific heat ratio. In order to figure out what assumptions are needed to use the gas specific heat ratio as an isentropic expansion coefficient, it is required to check with Eq. 3. This equation is one of the widely used methods for calculating the isentropic expansion coefficient where rigorous relief-valve sizing is deemed necessary. The equation is based on the assumption that the isentropic expansion coefficient is constant. Although the isentropic expansion coefficient is actually not constant during the expansion process, the sizing results with the isentropic expansion coefficient are relatively good. The derivatives in Eq. 3 for the Redlich-Kwong and Peng-Robinson equations are readily available in the literature.4,5
The Peng-Robinson equation of state appears to be the most favorable with the SRK equation. Eq. 4 shows the derivative of pressure with respect to specific volume, v; m3/kg, at constant temperature and constant compressibility factor. The gas specific heat ratio becomes the isentropic expansion coefficient (n) when the compressibility factor is constant. Of course, the real gas specific heat ratio, k, will be the ideal gas specific heat ratio, k*, if the compressibility factor is 1. In conclusion, it is required to assume the constant compressibility factor when developing a sizing equation with the real gas specific heat ratio.
Pv = ZRT
where R is a universal gas constant, 8.314 kPa-m3/kg-mole-K.
Eq. 5 is easily found in textbooks.6 When ideal gases are expanded, they follow Eq. 5. However, Eq. 6 is for real gases provided that the compressibility factor is constant. Eq. 6 explains why the inlet compressibility factor is not to be included in Eq. 7. The real gas specific heat ratio already accounts for the value of the compressibility factor and non-ideality at high pressure conditions. Therefore, when the real gas specific heat ratio is used in the sizing equation, the compressibility factor is not necessary. If one uses the real gas specific heat ratio in Eq. 1, the compressibility factor will be accounted for twice. This may result in inadequate relief valves, as addressed in API-520.7 A statistical analysis shows that about 7% of the equipment in the oil, gas and chemical industries had pressure-relief devices undersized.8
Pvk* = constant (5)
The authors developed Eq. 7 for relief-valve sizing for critical vapor flow with the real gas specific heat ratio. The simple equation follows fundamental thermodynamic rules.
where Ck is a function of the ratio of real gas specific heat at inlet conditions.
Although Eq. 7 may be satisfactory for critical-flow relief-valve sizing, the critical pressure prediction is not sufficiently accurate. In order to predict the accurate critical pressure, Eq. 9 is derived based on Eq. 7 being equal to Eq. 10. Instead of solving Eq. 9 for the isentropic expansion coefficient with the real gas specific heat ratio and compressibility factor, Eq. 12 fits well the correlation among the compressibility factor, real gas specific heat ratio and isentropic expansion coefficient over the range of 0 < n < 2.5.
Finally, Eq. 10 can be obtained as a critical-flow sizing equation with the compressibility factor and isentropic expansion coefficient. Eq. 7 and Eq. 10 are identical, and the two equations give the same sizing results. However, Eq. 11 predicts better choked pressures than Eq. 8.
a = 4.8422E-5 b = 1.98366 c = 1.73684 d = 0.174274 e = 1.48802
where Cn is a function of the isentropic expansion coefficient at inlet conditions.
Eq. 14, which requires a few iterations in an isentropic flash routine, is the most accurate sizing equation. The rigorous method uses the best predictions of the actual fluid properties since the calculated isentropic expansion coefficient is constant between two data points. The first trial of isentropic flash can start with an initial estimate of choked pressure at 55% of the inlet pressure. The choked condition is usually attained when the downstream pressure is about 45% to 65% of the inlet pressure. Repeat the isentropic flash with the new estimate until it stops changing. In case of liquid formation during the isentropic flash, the overall specific volume of the fluid has to be used in Eq. 15. Otherwise, it is better to use the temperature and compressibility factor in Eq. 15 to indicate that there is no condensation during expansion. The calculation details will be illustrated in the example calculations section. However, the equivalent results can be obtained using numerical integration with numerous flash calculations. The intensive numerical integration method is presented in API Standard 520.
where Cr is a function of the rigorous isentropic expansion coefficient, and Subscript 2 refers to fluid conditions at the outlet of the relief valve (at the nozzle throat).
The predictions of the simple flow equations have been compared with the most accurate estimates for the following two cases at six different pressures. The two cases include high pressures and critical-point regions to evaluate the limitations of the new simple method. The Peng-Robinson equation of state was used for the estimation of necessary fluid properties.
Case 1: Air at 250 K. The first case considers the discharge of air at 250 K that is relieving at six different pressure levels of the inlet reduced pressure from 0.5 to 8. The inlet compressibility factors are in the range of 0.9 < Z < 1.1. Fig. 1 shows the mass flux (W/A) deviations of Eqs. 1 and 7 from Eq. 14. Here, the compressibility factor decreases during expansion. Therefore, both of the simple sizing equations appear to oversize the relief valves. The difference between the compressibility factors increases with increasing the inlet reduced pressure. Fig. 1 shows that both simple equations give conservative estimates as expected. The results of both equations are satisfactory at low pressures. However, if the vapor or gas is at high pressure and low temperature, one should use them with caution. Generally, the vapor or gas tends to behave ideally at high temperatures.
| Fig. 1. Calculation results for high-pressure air |
at 250 K (Case 1).
Case 2. Saturated n-hexane vapor. The second case considers the discharge of a pure component-saturated vapor (n-hexane) that is relieving at six different pressure levels of the inlet reduced pressure from 0.15 to 0.9. The inlet compressibility factors are in the range of 0.4 < Z < 0.9. Here, the compressibility factor increases during expansion. The difference between the compressibility factors increases with increasing the inlet reduced pressure. Therefore, the simple sizing equations appear to undersize the relief valves. Fig. 2 shows that the conventional simple API equation gives profoundly unconservative estimates as expected. Unlike the conventional simple equation, the results of Eq. 7 or Eq. 10 are satisfactory up to the inlet reduced pressure of 0.75. However, if the inlet reduced pressure is greater than approximately 0.7, one should use it with caution.
| Fig. 2. Calculation results for saturated |
n-hexane vapor under critical conditions (Case 2).
A relief valve should release 10,000 kg/h of saturated n-hexane vapor (M = 86.18) at a relief pressure of 1,807.38 kPa (inlet reduced pressure of 0.6) and a relief temperature of 474 K. The compressibility factor at the conditions is 0.6279. Calculate the required actual orifice area based on Kd = 0.877, Kb = 1 and Kc = 1. The authors used a process simulator with a selection of the Peng-Robinson equation of state to obtain the necessary fluid properties.
Calculations with Eq. 14. The results of isentropic flashes for Eq. 14 that were obtained from a process simulator are summarized in Table 1. Iterations were stopped after four trials, as the new choked pressure was close enough to the old one. The best estimate of required orifice area is 556 mm2.
Calculations with Eq. 1. Using the ideal gas specific heat ratio along with the inlet compressibility factor gives the required orifice area of 482 mm2, which is much smaller than the best estimate of 556 mm2.
Calculations with Eq. 7. Using the real gas specific heat ratio gives the required orifice area of 567 mm2.
Calculations with Eq. 10. Eq. 10 is technically identical to Eq. 7. This method will give a better estimate of the choked pressure than the Eq. 7 method.
All calculation results are summarized in Table 2 for comparison. The required orifice area estimated by the API sizing equation is not satisfactory as expected, and this may result in an under-sized relief valve. On the other hand, the new method by Eq. 7 or Eq. 10 produces satisfactory results.
Proper sizing of a relief valve requires not only using an accurate critical-flow equation, but also using accurate fluid properties. The simple sizing equation for critical vapor flow using the real gas specific heat ratio has been tested on two cases: air at higher pressures and n-hexane under critical conditions. Although in both cases the conventional API equation is difficult to properly size pressure relief devices, the new approach results in a significant improvement. It is also important to note that the compressibility factor should be removed in the simple sizing equation where the real gas specific heat ratio is used as an isentropic expansion coefficient. However, relief-system designers should be careful when sizing the pressure relief devices for vapor or gas at critical regions or high pressures where the gas or vapor deviates significantly from the ideal conditions.
Any simple equation involves assumptions that generally introduce errors. As demonstrated in the example calculations, the rigorous flow equation with isentropic flash is not so complicated and difficult to use. The rigorous flow equation with no assumptions uses the most accurate fluid properties. Therefore, using the rigorous flow equation with isentropic flash is recommended when the gas or vapor is known to behave significantly non-ideally. HP
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