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Safety Relief Valve engineering (PSV) FAQ

Safety Valve Calculations

 A Simple Numerical Method for Gas/Vapor Flow in a Safety Valve by Latexman faq1203-1293 Posted: 1 Jan 07 (Edited 29 Sep 15) 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:  ****** * * * ****** Flow [→] ----- - ----- Z = 0, adiabatic, frictionless ****** * * * ****** [↑] [↑] [↑] Po, Go Pn, Gn Pback pressure  Derivation P/ρ + V2/2gc + gZ/gc = constant Bernoulli's equation  Z = 0 P/ρ + V2/2gc = constant Differentiate dP/ρ + d [ V2/2gc ] = 0 G = w/A = ρV Continuity equation  V = G/ρ and, therefore, V2 = (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 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, lbf/ft2 ρ = density, lbm/ft3 V = velocity, ft/sec g = gravitational acceleration, 32.174 ft/sec2 gc = gravitational constant, 32.174 lbm.ft/lbf/sec2 Z = elevation, ft G = mass velocity, lbm/ft2/sec w = mass flow rate, lbm/sec A = area, ft2 Subscripts o = in the vesselÆs head space. n = in the throat of the nozzle. ôback pressureö = 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: Po = 100 psia dP = 1 psia To = 25 C MW = 29 lb/lb.mole k = 1.4 dnozzle = 1 inch Pn = 52 psia Tn = -25.8o C ρn = 0.316 lbm/ft3 Σ(dP/ρave) = -118.439 lbf.ft3/(in2.lbm) Gn = 330.699 lbm/(ft2.sec) w = 6493 lbm/hr  Pn Tn ρn Σ(dP/ρave) G w psia oC lbm/ft3 lbf.ft3/(in2.lbm) lbm/(ft2.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  Back to Safety Relief Valve engineering (PSV) FAQ Index Back to Safety Relief Valve engineering (PSV) Forum

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