Supercritical Fluids PSV Sizing
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PROCESS/PLANT OPTIMIZATION
SPECIALREPORT
Fine-tune relief calculations for supercritical fluids Improved process simulation assists with relief load and valve sizing P. NEZAMI, Jacobs Engineering, Houston, Texas; and J. PRICE, Jacobs Consultancy, Houston, Texas
37
48 47
36 46 35
45 44
34 43 33
42 41
Mass method 1 Mass method 2 Volume method 1 Volume method 2
40 39 505 FIG. 1
510
515
520 525 Temperature, °F
Volume relief rate, 100 ft3/hr
where: VR = Volumetric relief rate Q = Heat input h1 = Initial specific enthalpy h2 = Final specific enthalpy 1 = Initial density 2 = Final density The mass relief rate can be determined using the average of the initial and final densities for each interval. ⎛ ρ + ρ2 ⎞⎟ M R = ⎜⎜⎜ 1 ⎟×V (2) ⎝ 2 ⎟⎠ R where: MR = Mass relief rate
Both the volumetric and mass relief rates will change during the course of a relief as the specific volume and enthalpy of the fluid change. To estimate the relief rates at different intervals, one can generate a property table in a process simulator to calculate the densities and specific enthalpies of the fluid at a constant relief pressure over a given temperature range. The volumetric and mass relief rates for each interval can be calculated using Eq. 1 and Eq. 2, respectively. In this study, a series of calculations were conducted for randomly selected n-paraffins, i-paraffins and aromatic compounds from C1 to C16 , using the Peng-Robinson equation of state (EOS). The results indicate that the maximum mass relief rate occurs at lower temperature than the maximum volumetric relief. Both temperatures where the maximum relief rates occur are greater than the critical temperature. Improving the calculation precision by reducing the temperature increments does not affect the temperatures at which the mass and the volume relief rates peak. (Smaller temperature increments result in a smaller enthalpy change, Δh, which translates to a smaller time span.) In fact, it is possible to mathematically prove that the two peaks occur at two different temperatures for real gas. This is where this article differs from the one presented at the API meeting.3
Mass relief rate, 1,000 lb/hr
I
n the past 40 years, several different methods have been suggested for relief load and pressure relief valve (PRV) orifice sizing calculations for a supercritical fluid exposed to an external heat source. The following sources include some of these methods: • API 521 suggests the use of a latent heat of 50 Btu/lb for hydrocarbons near the critical point. In the absence of a better method, this led to the use of 50 Btu/lb for even supercritical fluids. • “A Calculation of Relieving Requirements in the Critical Region”1 • “Rigorously Size Relief Valves for Supercritical Fluids”2 • “Calculation of Relief Rate Due to Fluid Expansion and External Heat.”3 The most recent method, “Calculation of Relief Rate Due to Fluid Expansion and External Heat,” was presented at the API 2010 Summer Meeting. As the title suggests, the relief load is calculated based on the expansion of the fluid due to absorbed heat. This method can be used for any fluid, including vapor and liquid, as long as no phase change occurs. To maintain a constant pressure at a fixed volume, the relief rate at any interval must be equal to the additional volume created by the change in specific volume from heat input to the fluid. However, some assumptions must be made and some basis must be set to make this method viable: • Other than the relieving stream, no fluid enters or leaves the vessel during the course of relief • There is no change of phase during the course of relief. A simple equation can be set to calculate the relief rate at each interval: ⎛1 Q 1⎞ VR = ×⎜⎜ − ⎟⎟⎟ (1) ⎜ h2 − h1 ⎝ ρ2 ρ1 ⎟⎠
32
530
535
31 540
Volumetric and mass relief rates (10 data points).
HYDROCARBON PROCESSING JUNE 2012
I 77
SPECIALREPORT
PROCESS/PLANT OPTIMIZATION
The subject was examined using two different approaches to calculate maximum relief rates (volumetric and mass) for n-hexane at 660-psia relief pressure with 5 million Btu/hr absorbed heat and a one-hour duration. In the first approach, the relief rates were calculated by setting up property tables and using Eqs. 1 and 2 for three different temperature increments. The second approach was based on stepwise simulation models with three different time spans. The initial and final temperatures were made the same to apply the same bases for all calculations. Results are plotted in Figs. 1, 2 and 3. The time spans in these plots are six minutes for Fig. 1, three minutes for Fig. 2, and two minutes for Fig. 3. It is clear that the impact of reducing time span on the temperatures at which the relief rates peak is insignificant. It is also obvious that the two methods yield almost the exact same results for the volumetric relief rates and very similar results for the mass relief rates. The small difference in mass relief rate is due to the fact that, in the first approach, at each interval the average of the initial and the final densities are used to convert volumetric relief rate to mass 37
48 47
35
44 34 43 42
33
41
Mass method 1 Mass method 2 Volume method 1 Volume method 2
40 39 500 FIG. 2
505
510
515 520 525 Temperature, °F
32
530
535
PRV orifice calculation. The API 520 equation for compress-
ible gas, which is derived from an ideal gas along an isentropic path, is not a suitable method for supercritical fluids, since supercritical fluids are far from ideal gas. Instead, an isentropic mass flux expression should be used for sizing relief valves in supercritical service: P ⎡ ⎤ ⎢ −2 v×dP ⎥ ⎢ ∫ ⎥ ⎢ ⎥ (3) P1 2 ⎢ ⎥ G = 2 ⎢ ⎥ vt ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎦ MAX where: ⎣ G = Mass flux v = Fluid-specific volume P = Fluid pressure vt = Specific volume at throat conditions P1 = Fluid pressure at the inlet of the nozzle 120 100 Maximum value, %
45
Volume relief rate, 100 ft3/hr
Mass relief rate, 1,000 lb/hr
36 46
relief rate. In the second approach, only the final density is used to convert volumetric relief rate to mass relief rate. The main objective of this exercise (and the next step in the relief valve calculation) is to size the PRV orifice area. The PRV orifice area is a function of relief valve set pressure, relief load, density and some other properties of the relieving fluid. In a scenario where a vessel or container is exposed to external heat, the fluid properties (and the relief load) vary during the course of a relief. The goal is to find the maximum required orifice area, as outlined below.
31 540
80 60 40
0 1.05 37
48
Volume Mass Orifice area
20
Volumetric and mass relief rates (20 data points).
FIG. 4
1.15
1.25
1.35 1.45 1.55 Reduced temperature
1.65
1.75
Methane relief at 1,346 psia.
47
44 34 43 33
42 Mass method 1 Mass method 2 Volume method 1 Volume method 2
41 40 39 500
FIG. 3
78
505
510
515 520 525 Temperature, °F
32
Volume Mass Orifice area
100 80 60 40 20
530
535
Volumetric and mass relief rates (30 data points).
I JUNE 2012 HydrocarbonProcessing.com
120
Maximum value, %
35
45
Volume relief rate, 100 ft3/hr
Mass relief rate, 1,000 lb/hr
36 46
31 540
0 1.05 FIG. 5
1.10
1.15
1.20 1.25 1.30 Reduced temperature
Iso-octane relief at 745 psia.
1.35
1.40
SPECIALREPORT
PROCESS/PLANT OPTIMIZATION
Eq. 3 is the result of a volumetric energy balance for an isentropic nozzle, and it is valid for any homogeneous fluid regardless of the non-ideality or compressibility of the fluid. Derivation details of the equation and the numerical examples for mass flux calculation are presented in Appendix B of API 520. Eq. 3 can be solved with a numerical integration technique. With the use of a process simulator, a property table can be generated along the isentropic line to find specific volumes at various pressures, beginning at relief pressure and moving down to the relief valve back pressure. Solving Eq. 3 for each downstream pressure will result in a series of mass fluxes, which will peak when the flow is choked in the nozzle. The required orifice area for the relief valve may be simply calculated by dividing the mass flux by the mass relief rate and the discharge coefficient: (4) MR A= G Kd where: A = Required orifice area Kd = Relief valve discharge coefficient It is surprising that the maximum required orifice area is not in line with either the maximum mass relief rates or the maximum volumetric relief rates. Figs. 4–7 illustrate the relationship
between the maximum relief rates (mass and volumetric) and the maximum required orifice area for the relief valve for some of the hydrocarbons used in this study. Fig. 8 shows the relationship between the maximum mass relief rate, the maximum volumetric relief rate, and the maximum required orifice area for n-pentane at various relief pressures. The maximum required orifice area appears at a temperature between the corresponding temperatures of the maximum volumetric and maximum mass relief rates for relief pressures from PR = 1 to PR = 7. Similar patterns were observed for other pure hydrocarbons used in the study. Numerical example. The following example illustrates
relief load and orifice-sizing calculations for a vessel containing n-hexane and absorbing 5 million Btu/hr of heat with a relieving pressure of 660 psia (PR = 1.5). Relief load calculation. A spreadsheet is used to calculate the relief rates at various stages of a relief incident. Utilizing a process simulator, a property table was created to calculate densities, along with specific enthalpies and entropies of the fluid at various temperatures. Using Eqs. 1 and 2, the volumetric and mass relief rates are calculated at different temperatures. The relief rates will peak if
TABLE 1. Volumetric and mass relief rates at different temperatures Temperature, °F
Reduced temperature
Density, lb/ft³
Enthalpy, Btu/lb
Entropy, Btu/lbmol–°F
504.2
1.054
14.93
–679.84
52.46
3,037
45,903
506.4
1.057
14.55
–677.12
52.70
3,143
46,331
508.7
1.059
14.19
–674.41
52.94
3,242
46,596
510.9
1.062
13.84
–671.73
53.18
3,331
46,693
513.2
1.064
13.50
–669.07
53.42
3,410
46,621
515.4
1.067
13.18
–666.44
53.65
3,478
46,390
517.7
1.069
12.87
–663.86
53.88
3,534
46,014
519.9
1.072
12.57
–661.31
54.10
3,578
45,511
522.2
1.074
12.29
–658.81
54.32
3,612
44,901
524.4
1.076
12.03
–656.34
54.54
3,635
44,205
526.7
1.079
11.78
–653.92
54.75
3,650
43,443
528.9
1.081
11.54
–651.55
54.96
3,657
42,633
531.2
1.084
11.32
–649.21
55.16
3,657
41,791
533.4
1.086
11.11
–646.90
55.36
3,651
40,929
535.7
1.089
10.91
–644.64
55.56
3,640
40,060
100
100
80 60 40 Volume Mass Orifice area
20 0 1.01 FIG. 6
80
Maximum value, %
120
Maximum value, %
120
1.03
1.05
1.07 1.09 1.11 Reduced temperature
Hexadecane relief at 412 psia.
I JUNE 2012 HydrocarbonProcessing.com
1.13
Vol. relief rate, ft³/hr Mass relief rate, lb/hr
80 60 40 Volume Mass Orifice area
20 1.15
0 1.05 1.075 1.1 1.125 1.15 1.175 1.2 1.225 1.25 1.275 1.3 Reduced temperature FIG. 7
Benzene relief at 1,428 psia.
PROCESS/PLANT OPTIMIZATION
Reduced pressure
the temperature range is wide enough to cover the temperatures at which the peaks occur. Table 1 is a sample calculation for n-hexane at PR = 1.5. As shown in Table 1, the maximum mass relief rate occurs when the temperature in the vessel reaches 510.9°F (TR = 1.062) and the maximum volumetric relief rate is 528.9°F (TR = 1.081). Relief valve orifice calculation. In the process simulator, a constant entropy table has been developed for each entropy between the maximum mass and the maximum volumetric relief rates in Table 1. The property tables include the specific volume of the fluid at different pressures, from relief pressure to PRV back pressure. Using a spreadsheet, the mass flux is calculated by numerically integrating “v ΔP ” along the range of pressures, from relief pressure to the PRV back pressure. The maximum mass flux represents the choked conditions in the nozzle. Tables 2–4 show sample calculations for three different entropies. Now the final table can be generated to calculate the maximum required orifice area throughout the relief event. Each row of the table will include throat pressure, specific entropy, mass relief rate, maximum mass flux, and the required orifice area, which is calculated from the mass relief rate and the mass flux using Eq. 4. The orifice area calculation is presented in Table
SPECIALREPORT
5. For a relief valve with a 0.95 discharge coefficient, the actual required orifice area would be 0.564/0.95 = 0.594 in2. Takeaway. As process simulator capability increases, the ability of engineers to utilize this software allows for a significantly more precise calculation process. The possibility to generate additional data points for this calculation by decreasing the step change in enthalpy will help increase the precision of the calculation. However, it is shown that, at extremely small step changes, the temperatures at which the maximum mass rate and maximum volume rate are generated do not approach each other. Sizing a relief device in this fashion will ensure that the orifice is adequately sized without the application of an overly conservative factor. HP
TABLE 3. Mass flux calculation for s = 54.10 Btu/lbmol–°F Pressure, psia
Specific volume, ft³/lb
∫–2vdP, ft²/s²
Mass flux, lb/sec.–ft2
660.0
0.07954
–
–
610.4
0.08749
38,414
2,240.2
8
560.7
0.09839
81,162
2,895.6
7
511.1
0.11339
129,865
3,178.3
6
461.4
0.13367
186,683
3,232.3
411.8
0.16047
254,328
3,142.8
362.2
0.19550
336,193
2,965.9
312.5
0.24187
436,777
2,732.5
262.9
0.30539
562,633
2,456.2
213.3
0.39756
724,295
2,140.7
163.6
0.54390
940,810
1,783.3
114.0
0.81382
1,253,057
1,375.5
64.3
1.48679
1,782,147
897.9
14.7
6.46163
3,610,108
294.0
5 4 3 Max. mass Max. volume Max. orifice
2
1 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 Reduced temperature FIG. 8
N-pentane supercritical relief.
TABLE 4. Mass flux calculation for s = 54.96 Btu/lbmol–°F
TABLE 2. Mass flux calculation for s = 53.18 Btu/lbmol–°F ∫–2vdP, ft²/s²
Pressure, psia
–
–
660.0
0.08665
–
–
34,720
2,367.1
610.4
0.09576
41,951
2,138.8
3,075.2
560.7
0.10785
88,777
2,762.8
0.12383
142,057
3,043.8
Specific volume, ft³/lb
660.0
0.07225
610.4
0.07872
560.7
0.08788
73,034
Specific volume, ft³/lb
∫–2vdP, ft²/s²
Mass flux, lb/sec.–ft2
Pressure, psia
Mass flux, lb/sec.–ft2
511.1
0.10124
116,526
3,371.9
511.1
461.4
0.12049
167,519
3,396.8
461.4
0.14473
203,819
3,119.3
411.8
0.14697
229,028
3,256.3
411.8
0.17184
276,623
3,060.8
362.2
0.18199
304,681
3,033.0
362.2
0.20708
363,766
2,912.5
0.25379
469,756
2,700.6
312.5
0.22824
399,025
2,767.6
312.5
262.9
0.29126
518,499
2,472.2
262.9
0.31796
601,247
2,438.7
213.3
0.38228
673,399
2,146.6
213.3
0.41135
768,974
2,131.8
163.6
0.52631
882,356
1,784.8
163.6
0.55995
992,352
1,779.0
0.83444
1,313,031
1,373.2
114.0
0.79141
1,185,404
1,375.7
114.0
64.3
1.45149
1,701,223
898.6
64.3
1.51947
1,854,378
896.2
14.7
6.32557
3,489,778
295.3
14.7
6.58830
3,718,987
292.7
HYDROCARBON PROCESSING JUNE 2012
I 81
SPECIALREPORT
PROCESS/PLANT OPTIMIZATION
TABLE 5. Mass flux and PRV orifice area calculation Throat pressure, psia
Entropy, Btu/lbmole–°F
Vol. relief rate, lb/hr
Mass relief rate, lb/hr
Mass flux, lb/s–in.²
Orifice area, in.²
488.8
52.46
3,037
45,903
3,606
0.509
488.8
52.70
3,143
46,331
3,531
0.525
488.8
52.94
3,242
46,596
3,464
0.538
475.6
53.18
3,331
46,693
3,405
0.549
475.6
53.42
3,410
46,621
3,354
0.556
475.6
53.65
3,478
46,390
3,308
0.561
475.6
53.88
3,534
46,014
3,267
0.563
475.6
54.10
3,578
45,511
3,230
0.564
462.5
54.32
3,612
44,901
3,196
0.562
462.5
54.54
3,635
44,205
3,167
0.558
462.5
54.75
3,650
43,443
3,140
0.553
462.5
54.96
3,657
42,633
3,115
0.547
462.5
55.16
3,657
41,791
3,092
0.541
462.5
55.36
3,651
40,929
3,070
0.533
462.5
55.56
3,640
40,060
3,050
0.525
LITERATURE CITED
Piruz Latifi Nezami is a process engineering section manager with Jacobs
1
Francis, J. O. and W. E. Shackelton, “A Calculation of Relieving Requirements in the Critical Region,” API Proceedings—Refining Department, 50th MidYear Meeting, 1985.
Engineering in Houston, Texas. He holds a BS degree in chemical engineering from Sharif University of Technology in Tehran, Iran, and has more than 30 years of experience in the design and engineering of chemical, petrochemical and refining projects.
2
Ouderkirk, R., “Rigorously Size Relief Valves for Supercritical Fluids,” Chemical Engineering Progress, August 2002.
3
Freeman, S., and D. Huyen, “Calculation of Relief Rate Due to Fluid Expansion and External Heat,” API Summer Meeting, 2010.
Jerry Price is a refining and petrochemicals consultant for Jacobs Consultancy Inc. in Houston, Texas. Jacobs Consultancy provides expert consulting services to the global oil, refining and chemical industries. Mr. Price previously worked as a process engineer for Jacobs Engineering Group. He holds a BS degree in chemical engineering from Washington University in St. Louis, Missouri.
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