A Simple Numerical Method for Gas/Vapor Flow in a Safety Valve

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:
[Pre]

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Flow [→] ----- - ----- Z = 0, adiabatic, frictionless
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[↑] [↑] [↑]
P[sub]o[/sub], G[sub]o[/sub] P[sub]n[/sub], G[sub]n[/sub] P[sub]back pressure[/sub]
[/Pre]
Derivation

[Pre]P/ρ + V[sup]2[/sup]/2g[sub]c[/sub] + gZ/g[sub]c[/sub] = constant Bernoulli's equation[/Pre]

Z = 0

P/[ρ] + V[sup]2[/sup]/2g[sub]c[/sub] = constant

Differentiate

dP/[ρ] + d [ V[sup]2[/sup]/2g[sub]c[/sub] ] = 0

[pre]G = w/A = ρV Continuity equation[/pre]

V = G/[ρ] and, therefore, V[sup]2[/sup] = (G/[ρ])[sup]2[/sup]

Substitute and rearrange

d [ (G/[ρ])[sup]2[/sup] /2g[sub]c[/sub]] = - dP/[ρ]

Integrate

[pre]∫ d [ (G/&rho[sup]2[/sup] /2g[sub]c[/sub] ] = - ∫ dP/ρ Equation 1[/pre]

Integrate LHS of Equation 1 from G[sub]o[/sub] to G[sub]n[/sub]

[(G[sub]n[/sub]/[ρ][sub]n[/sub])[sup]2[/sup] - (G[sub]o[/sub]/[ρ][sub]o[/sub])[sup]2[/sup]] /2g[sub]c[/sub] = - [∫] dP/[ρ]

G[sub]o[/sub] [≈] 0 because A[sub]o[/sub] is usually very large compared to A[sub]n[/sub]

(G[sub]n[/sub]/[ρ][sub]n[/sub])[sup]2[/sup] = - 2g[sub]c[/sub] [∫] dP/[ρ]

(G[sub]n[/sub]/[ρ][sub]n[/sub]) = ( - 2g[sub]c[/sub] [∫] dP/[ρ] )[sup]1/2[/sup]

G[sub]n[/sub] = [ρ][sub]n[/sub] ( - 2g[sub]c[/sub] [∫] dP/[ρ] )[sup]1/2[/sup]

Evaluate [∫] dP/[ρ] (the RHS of Equation 1) numerically from P[sub]o[/sub] to P[sub]n[/sub] until G[sub]n[/sub] reaches a maximum (sonic flow) OR P[sub]n[/sub] = P[sub]BP[/sub] (subsonic flow).

The beauty of this method is . . . . no restrictive assumptions were made!

Notes

1. 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.
2. 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.
3. 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.
4. 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, lb[sub]f[/sub]/ft[sup]2[/sup]
[ρ] = density, lb[sub]m[/sub]/ft[sup]3[/sup]
V = velocity, ft/sec
g = gravitational acceleration, 32.174 ft/sec[sup]2[/sup]
g[sub]c[/sub] = gravitational constant, 32.174 lb[sub]m[/sub].ft/lb[sub]f[/sub]/sec[sup]2[/sup]
Z = elevation, ft
G = mass velocity, lb[sub]m[/sub]/ft[sup]2[/sup]/sec
w = mass flow rate, lb[sub]m[/sub]/sec
A = area, ft[sup]2[/sup]

Subscripts

o = in the vesselÆs 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[sub]o[/sub] = 100 psia dP = 1 psia
T[sub]o[/sub] = 25 C
MW = 29 lb/lb.mole
k = 1.4
d[sub]nozzle[/sub] = 1 inch

P[sub]n[/sub] = 53 psia
T[sub]n[/sub] = -24.5[sup]o[/sup] C
[ρ][sub]n[/sub] = 0.320 lb[sub]m[/sub]/ft[sup]3[/sup]
[Σ](dP/[ρ][sub]ave[/sub]) = -115.293 lb[sub]f[/sub].ft[sup]3[/sup]/(in[sup]2[/sup].lb[sub]m[/sub])
G[sub]n[/sub] = 330.746779 lb[sub]m[/sub]/(ft[sup]2[/sup].sec)
w = 6494 lb[sub]m[/sub]/hr

[Pre]
P[sub]n[/sub] T[sub]n[/sub] ρ[sub]n[/sub] Σ(dP/ρ[sub]ave[/sub]) G w
psia [sup]o[/sup]C lb[sub]m[/sub]/ft[sup]3[/sup] lb[sub]f[/sub].ft[sup]3[/sup]/(in[sup]2[/sup].lb[sub]m[/sub]) lb[sub]m[/sub]/(ft[sup]2[/sup].sec) lb[sub]m[/sub]/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[/Pre]

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