One of the ways I have learned a technology is to get down into the basics and derive key working equations that are found in the references.
It is common practice to model a safety valve on a pressure vessel as a flow nozzle, and NOT as an orifice. The theoretical model from the pressure vessel to the throat of the flow nozzle is an isentropic converging flow nozzle:
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Flow [→] ----- - ----- Z = 0, adiabatic, frictionless
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Derivation
P/ρ + V^{2}/2gc + gZ/gc = constant Bernoulli’s equation
Z = 0
P/ρ + V^{2}/2gc = constant
Differentiate
dP/ρ + d [ V^{2}/2gc ] = 0
G = w/A = ρV Continuity equation
V = G/ρ and, therefore, V^{2} = (G/ρ)^{2}
Substitute and rearrange
d [ (G/ρ)^{2} /2gc] = - dP/ρ
Integrate
∫ d [ (G/ρ) ^{2} /2gc ] = - ∫ dP/ρ Equation 1
Integrate LHS of Equation 1 from Go to Gn
[(Gn/ρn)^{2} - (Go/ρo)^{2}] /2gc = - ∫ dP/ρ
Go ≈ 0 because Ao is usually very large compared to An
(Gn/ρn)^{2} = - 2gc ∫ dP/ρ
(Gn/ρn) = ( - 2gc ∫ dP/ρ )^{1/2}
Gn = ρn ( - 2gc ∫ dP/ρ )^{1/2}
Evaluate ∫ dP/ρ (the RHS of Equation 1) numerically from Po to Pn until Gn reaches a maximum (sonic flow) OR Pn = Pbp (subsonic flow).
The beauty of this method is . . . . no restrictive assumptions were made!
Notes
- The method was derived with the nozzle oriented horizontally. Most safety valve nozzles are oriented vertically. However, gas pressure changes very little with elevation changes due to the small density of a gas. It is common practice to ignore this effect on gas flow evaluations. This is especially true for the small elevation change from the pressure vessel to the safety valve nozzle on most safety valve installations.
- We did not assume an ideal gas. Any PVT relationship can be used to calculate the temperature and density at each pressure increment in the numerical integration.
- Pressure increments should be chosen sufficiently small for accuracy and sufficiently large for calculation speed. A dP = 1% of the safety valve set pressure is a good starting point. For most problems, a dP = 1 psi works quite well.
- The method is extremely easy to implement in a spreadsheet. I created a spreadsheet which uses the ideal gas law as the PVT relationship. A copy of the input and output is included further below.
Nomenclature
P = pressure, lbf/ft^{2}
ρ = density, lbm/ft^{3}
V = velocity, ft/sec
g = gravitational acceleration, 32.174 ft/sec^{2}
gc = gravitational constant, 32.174 lbm.ft/lbf/sec^{2}
Z = elevation, ft
G = mass velocity, lbm/ft^{2}/sec
w = mass flow rate, lbm/sec
A = area, ft^{2}
Subscripts
o = in the vessels head space.
n = in the throat of the nozzle.
back pressure = the pressure of the surroundings where the gas exits the nozzle. In a safety valve, this is the back pressure created by the tailpipe attached to the outlet connection.
Copy of Safety Valve with Ideal Gas.xls:
P_{o} = 100 psia dP = 1 psia
T_{o} = 25 C
MW = 29 lb/lb.mole
k = 1.4
d_{nozzle} = 1 inch
P_{n} = 53 psia
T_{n} = -24.5 C
ρ_{n} = 0.320 lbm/ft3
Σ(dP/ρ_{ave}) = -115.293 lbf.ft3/(in2.lbm)
G_{n} = 330.746779 lbm/(ft2.sec)
w = 6494 lbm/hr
P_{n} | T_{n} | ρ_{n} | Σ(dP/ρ_{ave}) | G | w |
---|---|---|---|---|---|
psia | ^{o}C | lbm/ft^{3} | lbf.ft^{3}/(in^{2}.lbm) | lbm/(ft^{2}.sec) | lbm/hr |
100 | 25.0 | 0.504 | |||
99 | 24.1 | 0.500 | -1.993 | 67.944540 | 1334 |
98 | 23.3 | 0.496 | -4.000 | 95.566345 | 1876 |
97 | 22.4 | 0.493 | -6.022 | 116.401889 | 2286 |
96 | 21.5 | 0.489 | -8.059 | 133.663328 | 2624 |
95 | 20.7 | 0.485 | -10.111 | 148.601443 | 2918 |
94 | 19.8 | 0.482 | -12.178 | 161.860800 | 3178 |
93 | 18.9 | 0.478 | -14.262 | 173.826018 | 3413 |
92 | 18.0 | 0.474 | -16.361 | 184.748745 | 3628 |
91 | 17.1 | 0.471 | -18.477 | 194.804455 | 3825 |
90 | 16.2 | 0.467 | -20.609 | 204.121365 | 4008 |
89 | 15.2 | 0.463 | -22.759 | 212.796582 | 4178 |
88 | 14.3 | 0.460 | -24.926 | 220.905767 | 4337 |
87 | 13.4 | 0.456 | -27.110 | 228.509242 | 4487 |
86 | 12.4 | 0.452 | -29.312 | 235.656017 | 4627 |
85 | 11.5 | 0.448 | -31.533 | 242.386553 | 4759 |
84 | 10.5 | 0.445 | -33.773 | 248.734707 | 4884 |
83 | 9.5 | 0.441 | -36.031 | 254.729138 | 5002 |
82 | 8.6 | 0.437 | -38.310 | 260.394347 | 5113 |
81 | 7.6 | 0.433 | -40.608 | 265.751468 | 5218 |
80 | 6.6 | 0.429 | -42.926 | 270.818868 | 5318 |
79 | 5.6 | 0.426 | -45.265 | 275.612616 | 5412 |
78 | 4.6 | 0.422 | -47.626 | 280.146852 | 5501 |
77 | 3.5 | 0.418 | -50.008 | 284.434085 | 5585 |
76 | 2.5 | 0.414 | -52.412 | 288.485432 | 5664 |
75 | 1.5 | 0.410 | -54.840 | 292.310811 | 5740 |
74 | 0.4 | 0.406 | -57.290 | 295.919101 | 5810 |
73 | -0.6 | 0.402 | -59.764 | 299.318276 | 5877 |
72 | -1.7 | 0.398 | -62.263 | 302.515514 | 5940 |
71 | -2.8 | 0.394 | -64.786 | 305.517290 | 5999 |
70 | -3.9 | 0.390 | -67.335 | 308.329458 | 6054 |
69 | -5.0 | 0.386 | -69.910 | 310.957313 | 6106 |
68 | -6.1 | 0.382 | -72.512 | 313.405650 | 6154 |
67 | -7.2 | 0.378 | -75.141 | 315.678815 | 6198 |
66 | -8.4 | 0.374 | -77.798 | 317.780744 | 6240 |
65 | -9.5 | 0.370 | -80.485 | 319.715001 | 6278 |
64 | -10.7 | 0.366 | -83.201 | 321.484810 | 6312 |
63 | -11.9 | 0.362 | -85.947 | 323.093080 | 6344 |
62 | -13.1 | 0.358 | -88.725 | 324.542432 | 6372 |
61 | -14.3 | 0.354 | -91.535 | 325.835216 | 6398 |
60 | -15.5 | 0.350 | -94.378 | 326.973534 | 6420 |
59 | -16.7 | 0.345 | -97.256 | 327.959251 | 6439 |
58 | -18.0 | 0.341 | -100.168 | 328.794009 | 6456 |
57 | -19.2 | 0.337 | -103.116 | 329.479243 | 6469 |
56 | -20.5 | 0.333 | -106.102 | 330.016186 | 6480 |
55 | -21.8 | 0.329 | -109.126 | 330.405881 | 6488 |
54 | -23.1 | 0.324 | -112.189 | 330.649185 | 6492 |
53 | -24.5 | 0.320 | -115.293 | 330.746779 | 6494 |
52 | -25.8 | 0.316 | -118.439 | 330.699170 | 6493 |