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Safety Valve Calculations

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

[(Gnn)2 - (Goo)2] /2gc = - ∫ dP/ρ

Go ≈ 0 because Ao is usually very large compared to An

(Gnn)2 = - 2gc ∫ dP/ρ

(Gnn) = ( - 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 

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