Spring Identification for Portable Safety Relief Valves
Short Description
Spring Identification for Portable Safety Relief Valves describes both theory and deployment of flat end springs in mach...
Description
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. PROBLEM BACKGROUND Operational facilities should keep a design, selection, and service log for all Safety Relief Valves (SRV, RV) used in the facility. One of the most important items included in the RV record is the spring number identification for each RV. This number is required in all RV Authorization & Deployment Forms to verify the RV safety and compliance. Except for the small springs, each spring has an identification number punched on it. For springs with no identification, manufacturers suggest replacement by a newly purchased spring. This means that every time a valve is disassembled for repair or testing, the spring should be replaced. This arises from the fact that small springs have no ID number stamping. Thus the spring cannot be identified by a shop techinician from a simple visual inspection. Spring replacement at every service and test will be a costly exercise. A concern of this matter lead to the development of a method for RV spring identification. This method establishes a procedure which requires maintenance shops to take some additional RV's measurements. These measurements are submitted to an engineer. He then must do a set of calculations to identify the spring number of the deployed spring. Many facilities have hundreds of portable RV's in service. A portable RV is a small size valve. They generally have a threaded inlet and outlet. These RV's are commonly used for low relieving capacities with relatively high pressures applications. Like all the RV types, each valve type has about 20 different springs that can be fitted to it. The set pressure required for a specific valve is what fixes the required spring number. For each spring can only service a small pressure range. Except for springs of portable valves, spring numbers are usually punched on the top coils of the spring. Portable RV's springs are so small and weak that punching their numbers on them could damage them. When these springs are ordered, RV manufacturers send them in boxes with a label of their number. At any facility, each individual spring is supposed to be tagged with its number. It can happen that the stock is mixed or the tags are lost. A more complicated problem appears when the RV is disassembled and repaired. When the technician come to fill a RV Authorization Form he will not be able to truely identify the spring number. As a result, Inspection should put the RV on HOLD until the spring validity can be positively confirmed. This matter was discussed with Consolidated & Crosby, RV manufacturers. The discussion lead to one solution which is to purchase a replacement spring for every unidentified spring. Such solution would raise the cost of RV 's maintenance drastically. A cost estimate for one plant showed an annual material cost in the range of $17,000 to $30,000. Even though, if a new spring was installed the same problem will be encountered again at the next service. RV’s should be routinely serviced and also at any pop off between scheduled service. Here is a method of evaluating RV springs. The method determines the spring number based on measurement of spring and nozzle dimensions. Also, an air pressure pop off series of test data must be gathered and used in this method. The next section will explain the details of the measurements and procedure of the method. Flow test runs are required to validate the Drag Coefficient value used to determine a minimum set pressure of a RV for installation with a given spring. The following is a detailed description and analysis of the method.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. PROCEDURE Following is the procedure of spring verification of RV's. The procedure consists of two parts. The fist part should be carried before the reassembly of the RV while the second part should be done next to the repair and reassembly of the RV. Part One: While the RV is disassembled, Valve nozzle and spring dimensions need to be taken. These measurements include the spring free length (Xf), outer diameter (Do), inner diameter (Di), wire diameter (d) and total number of turns (N). Also, the nozzle bore diameter (Db) and outer diameter (Ds) should be measured. Readings to 3 decimal points of an inch is expected. The attached figures show the location of each measurement mentioned. Part Two: This part can not be done till the RV is prepared for pressure testing. This means that valve was insured for no leak as done in every regular repair procedure. Then, RV is to be tested on the Pressure Test Stand available. Test is started by screwing the compression screw till it touches the spring then the lock nut is tightened. Next, RV is popped and the popping pressure is recorded as well as the "compression screw" position (X). Compression screw position is measured from the top of the screw to the top surface of the lock nut. If the compression screw was just tightened to touch the spring, the popping pressure should not exceed 25 psig. If so, compression screw measurement will be designated as "Initial Position" (Xo). Then, screw is tightened for about 0.05 inches more and popped again. The same step is repeated for at least 4 times and the readings are recorded. Attached is a "RV Spring Verification Data Sheet", see appendix A. Measurements of both parts of the method described above are to be filled in this sheet. Later, this sheet will be the basis of the calculations and analysis in the next section.
CALCULATIONS AND ANALYSIS Springs used in RV's are usually "Helical Compression Springs". The characteristics of these springs are measured by what is known as "Spring Rate" or "Spring Constant" (Ks). This constant determines the force (F) required to displace the spring a distance (dX). Appendix B shows the detailed description and derivations done to set the method of spring identification. Method described was tested and proved to a good accuracy for Consolidated RV types 1970, 1975C and 1990C. However, to generalize the method to verify springs for any valve type, flow tests are required to measure Drag coefficient (CD) and Effective Coefficient of Discharge (Kd). These constants are functions of the valve shape and testing the valve flow should give an accurate measure of them. By doing that it will be possible to generate a synthetic spring table for any RV type. This means that for any spring installed in any RV type, performing a set of calculations will reveal the maximum and the minimum set pressure that the spring can hold.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
APPENDIX A
MEASUREMENT AND CALCULATIONS SHEETS
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
SPRING NUMBER IDENTIFICATION STEPS
Using RV spring measurement data filled in "RV Spring Verification data Sheet", perform the following calculations to fill "RV Spring Verification Calculations Sheets": 1) Determine valve critical dimensions (Bore Area, Nozzle Outer Area and Lift) 2) Using RV pop test data, fill in the table in page 1. 3) Determine Spring Constant Ks using equation B1 and B2 page 2. Ks calculated using B1 is more accurate, but both results should be close. 4) Calculate the maximum compression allowed using equation B3. Use Ks calculated by B1. 5) Calculate Maximum Spring Set Pressure using equation B4 and B5. 6) If possible use Pmax calculated by equation B5 to fill the comparison of table page 3, otherwise go to step 7. A quick guess of the spring number can be made by observing the differences in the table. The spring number corresponding to the smaller difference is the most probable spring number. 7) Calculate Minimum Spring Set Pressure using equation B6 or B7. 8) If the spring number was not identified in step 6, it can be estimated by comparing the spring range calculated by equations B4 and B6 to the ranges available in the Manufacturer's Spring Charts. Select the spring number with the closest range. Also, from the Manufacturer's Charts find the required spring number according to the RV type and its set pressure. Use these results to fill the table in page 4. 9) Draw your conclusions based on the last table. The optimum result will be identifying of a spring number identical to the required, but a different spring number can be acceptable if close. If the spring numbers did not match, compare the spring range calculated to the set pressure required. If the set pressure falls roughly midway between the maximum and the minimum pressures, this spring number will be a safe spring to be used for this set pressure.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
RV SPRING VERIFICATION DATA SHEET RV B NO. RV TYPE
SPRING DIMENSIONS Spring free lenght (Xf) Spring outer dia. (Do) Spring inner dia. (Di) Spring wire dia. (d) Spring no. of turns(N): N1 N2 N3 N4 Average N
RV NOZZLE MEASUREMENT Nozzle bore dia. (Db) Nozzle outer dia. (Ds)
RV TEST DATA Po = #
Pressure (psi)
Xo = Compression (inches)
1 2 3 4 5 5
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
RV SPRING VERIFICATION CALCULATION SHEET RV B NO. RV TYPE
Calculation of RV Dimensions As =
Ds2/4 in2
As = Ab =
Db2/4 in2
Ab = dXL = Db2/4Ds
dXL =
in
Calculation of Spring Rate (Ks) Calculate Fs, dXc, Fs*dXc and dXc2 for every set of data measured. Fill the table below with the results. Fs = P*As dXc = Xo - Compression
# 1 2 3 4 5 6
P Po=
Compression Xo=
dXc
Fs
0
SUM () m=
total number of test points.
m=
6
FsdXc
dXc2
0
0
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. RV B NO. (
mFsdXc - FsdXc Ks = mdXc2 - (dXc)2
Ks =
Equation B1 Substitute next,
)(
)-(
)(
)
Ks = ( )( )-( )2
lb/in
Spring Rate Estimation (Ks) G(d)4 Ks = 8N(Do-d)3
( *106)( )4 Ks = )3 8( )(
Equation B2 Substitute next,
MATERIAL Ks =
G (psi)
lb/in Alloy & Carbon Steel Stainless Steel Brass Aluminum
11.5*106 10.6*106 5.0*106 4.0*106
Calculation of Maximum Pressure (Pmax) 1)Calculation of Maximum Pressure per ASME VIII: Maximum Compression Allowed: dXc(max) = 0.8(Xf - N*d) - dXL
dXc(max) =
Equation B3
dXc(max) = 0.8(
-
*
inch
Maximum Pressure Allowed: Pmax = Ks*dXc(max)/As.
Pmax =
Equation B4
Pmax =
psig
7
*
/
)-
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. RV B NO. 2)
Calculation of Maximum Pressure by Known RV Type Constants(C2):
Maximum Pressure Allowed: Pmax = Ks*C2
Equation B5
Pmax =
*
If your RV is available in the table below, use the previous Equation to calculate Pmax.
Pmax =
C2 1.0 1.28 1.73 2.2
RV TYPE 1970C (1") 1975C 1990C 1990xls
psig
By refering to Mnufacturer's Spring Tables you can compare Pmax you calculated to the closest two maximum pressures it falls between. Difference = Pmax(calculated) - Pmax(from the table) Pmax (from table)
Difference
Corresponding Spring Number
Upper Pmax Lower Pmax Spring number is the one with the small difference.
Calculation of Minimum Pressure (Pmin) CD: is the coefficient of drag. Its value varies according to the flow conditions. An average value of 3.55 is often used. Minimum Pressure of Liquids: KsdXLAD Pmin = K1CDAb2 - AsAD To simplify the calculations, assume AD=As. KsdXLAs Pmin = K1CDAb2 - As2
Equation B6
( )( )( ) Pmin = ( )( )( )2 - ( )2
K1=0.464(Kd/0.65)2 for Certified Liquid Flow. K1=0.479(Kd/0.62)2 for Non-Certified Liquid Flow.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. RV B NO. For cases which involve Consolidated RV's, API has allowed the RV manufacturer to use Kd specific to their RV, and these are listed below. Kd for Consolidated RV CASE 1990 and ASME VIII for Certified Liquid Flow API Certified Liquid Flow 1990 and API for Non-Certified Liquid Flow
Pmin =
Kd 0.669 0.744 0.62
psig
Minimum Pressure of Gases: KsdXLAD-8.6CDAb2 Pmin = 0.64CDAb2 - AsAD To simplify the calculations, assume AD=As. KsdXLAD-8.6CDAb2 Pmin = 0.64CDAb2 - As2
Pmin =
Equation B7
( )( )( )-8.6( )( )2 Pmin = ( )( )( )2 - ( )2
psig Identified Spring Number
Required Spring Number
Spring No. Spring Range
CONCLUSION: ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
APPENDIX B
DETAILED CALCULATIONS OF SPRING RANGE & SPRING NO.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
I. DETERMINATION OF SPRING MAXIMUM PRESSURE 1) Maximum Pressure vs. Spring Rate Relation: FIGURE 1 SPRING COORDINATES
dXc Xf-Xk
Xf
Xc Xk
POSITION A FREE SPRING
Ff Fs Ks P As Ds Xf XK Xs dXC dXL dXc N d A C1 C2
POSITION B FULL COMPRESSION
dXc Xs dX
POSITION C SET SPRING NO LIFT
POSITION D SET SPRING WITH FULL LIFT
Force of fluid pressure exerted on the disk. Force of spring exerted on the disk. (lbf) Spring constant. (lbf / inch) RV set pressure. (psig) Disk area affected by the set pressure, see figure 4. (inch2) Outer diameter of nozzle. (inch) Free spring length. (inch) Fully compressed spring length (XK=Nd). (inch) Spring length when valve relieves at full lift. (inch) Change in spring length to achieve set pressure. (inch) Change in spring length to achieve full disk lift. (inch) Difference in fully compressed spring length and the spring length at full disk lift relieving condition. (inch) Number of spring turns. Wire diameter. (inch) Constant relating maximum travel to the available travel. Fraction required by ASME VIII to be between 0.8 & 1.0.(1) Constant relating maximum set pressure of a spring to its constant.
By inspecting the above conditions and spring coordinates, it may be written for position "D", Figure 1 as: Xf = Xs + dXc + dXL Define dXs = (Xs - XK)
then
Xs = XK + dXs
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. Xf = XK + dXs + dXc + dXL (Xf - XK) = dXs + dXc + dXL To maintain spring linearity with force; F=KsX, it is required that dXs = fraction of (Xf -XK) or A(Xf - XK). (Xf - XK) = A(Xf - XK) + dXc +XL (1 - A)(Xf-XK) = dXc + dXL
call (1 - A) = C1 & C11
C1(Xf - XK) = dXc + dXL
(I)
ASME VIII requires that "the spring shall be designed so that the full lift spring compression ,(dXc + dXL), shall be no greater than 80% of the nominal spring deflection"(1). This means that C1 = 0.8 dXc + dXL 0.8(Xf - XK) F = Ks*dXc = P*As
(II) dXc = PAs/Ks
substituting in (II)
PAs/Ks + dXL 0.8(Xf - XK) PAs/Ks 0.8(Xf - XK) - dXL PAs/[0.8(Xf - XK) - dXL] Ks for a particular spring and RV type, dXL, As, Xf & XK will remain constant so we can say
C2 = As/[0.8(Xf - XK) - dXL]
For future reference C2 will be called the "RV Type Constant". So,
Pmax Ks/C2
(III)
More discussion on the RV type constant C2 is available in appendix C, section 1. The next section will discussed the determination of spring rate Ks.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
2) Calculations of Spring Rate (Ks): The free body diagram of the static forces on a closed nozzle gives the following: FIGURE 2 STATIC DISK FREE-BODY DIAGRAM force Fspring=KsdXc Fb=PbackA
F2 F1
C3
friction Ws+d
dX1 dX2 spring compression
Ff=PsetAs
STATIC FORCES: The static force exerted on the disk in a closed position by the fluid inside the nozzle is Ff = Pset*As = PsetDs2/4 Fup - Fdown = 0 Fdown = Fspring + Weight of spring disk + Friction of disk lip + Net forces of back pressure. Assuming no back pressure Fdown = KsdXc + Ws+D +Dso
:Coefficient of friction
Fup = Pset*Anormal = Ppopping Ds2/4 Rearranging the equation Fup - Fdown will give = Ppopping Ds2/4 - KdXc = Ws+D + Dso = constant = C3 By popping the valve at different set pressures. The constant forces of spring weight and friction may be eliminated to find the spring constant. (= Ppopping Ds2/4 - KsdXc)at test 1 = C3 = (= Ppopping Ds2/4 - KsdXc)at test 2 (Ppop at test 2 - Ppop at test 1) Ds2/4 = Ks(dXs at test 2 - dXs at test 1) (IV)
Ks = (P2-P1)Ds2/(4(dXs2-dXs1)) Note: subscript 1 & 2 represent first and second tests respectively.
In summary, Equation (IV) the slope of popping pressure to spring compression times orifice area can be used to find the spring rate. 13
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
3) Calculations of Disk Lift (dXL): dXL is the minimum lift required to assure valve does not operate with restricted flow. Restricted flow means a valve does not reach the relieving flow capacity required by the specifying engineer of a RV at a stated over pressure, accumulation. FIGURE 3 NOZZLE AND DISK COORDINATES SPRING WASHERS
SPRING DISK
Ds Db
dX A
Fluid Flow
NOZZLE
Ab POSITION A OPEN NOZZLE DISK SEATED
POSITION B CLOSE NOZZLE WITH DISK LIFTED
Flow Area = Ab = Db2/4
POSITION C NOZZLE WITH OPEN
See Figure 3, Position A.
To prevent restricted lift AL Ab AL = DsdXL
See Figure 3, Position C.
DsdXL Db2/4
cancels away.
dXL Db2/(4Ds) From which dXL may be calculated as: dXL = Aflow/(Ds) = Db2/(4Ds) = Db2/(4Ds) Since Ds Db.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. So, minimum lift require to supply a non restricted flow out of the valve can be approximated to: dXL = Db/4
(V)
A summary of dXL minimum is given in Table 1 for various Consolidated valves.
TABLE 1 Summary of Valve Major Dimensions for Portable RV RV MODEL
Ab (in2)
Ds (in)
As (in2)
1970 (3/4") 1970 (1") 1970 (1 1/2",2") 1975 (1/2",3/4") 1990/1995 HP 1993/1996 1997 1998 3999 1994 H/HP 1996 H
0.126 0.226 0.522 0.110 0.110 0.292 0.442 0.754 0.019 0.126 0.226
0.576 0.576 0.854 0.389 0.392 0.684 0.854 1.000 0.178 0.430 0.556
0.260 0.260 0.572 0.119 0.121 0.367 0.573 0.785 0.025 0.145 0.243
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LIFT = Ab/(Ds) (in) 0.070 0.125 0.195 0.090 0.089 0.136 0.165 0.240 0.034 0.093 0.129
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
II. DETERMINATION OF SPRING MINIMUM PRESSURE CALCULATION OF MINIMUM SET PRESSURE FOR FLUID FLOW: The dynamic forces acting on the RV disk are considered next to determine the minimum set pressure of a given spring in a given RV. FIGURE 4 DYNAMIC SPRING CONDITIONS
Fspring=Ks(dXc+dXL) D
As
fluid flow
Ws+d
Fup= Drag force of Relieving fluid
Ws+D Weight of the spring and disk assembly m Flow rate of fluid through nozzle d Fluid density at the disk holder FD Force of drag (lb) which is the force component parallel to the relative approach velocity, exerted on the body by the moving fluid. CD Coefficient of drag which is the momentum force imparted on a normal projected area for a velocity head of 1. AD Projected area on a plane normal to the flow. (in2) DD Diameter of the projected area on the plane normal to the flow. Fluid velocity. g Constant of gravity = 32.2 SG Specific Gravity of the fluid Ab Nozzle bore diameter. Acc Accumulation in percentage. As Disk area affected by the set pressure.
16
(lb) (lb/sec) (lb/ft3)
(in) (ft/sec) (lbf/lbm)*(ft/sec2) (SG= f / 62.4) (in) (in2)
Ab
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. At Steady State Flow condition of the allowable accumulation pressure: Fup - Fdown = 0 FD - [Ks(dXc + dXL) + Ws+D] = 0
(VI)
If Fup FDown then full lift will be exceeded and under no circumstance will lift be a limiting factor. However, as previously mentioned, lift plus spring compression is limited by ASME requirement i.e. The spring maximum set pressure. The minimum spring pressure will be determined so as to achieve full lift under flowing condition of the fluid at the RV nozzle. Drag Force of Fluid (FD) = CDADd2/2g
(VII) See Reference(2)
= 4m/(dDD2) = m/(dAD2)
then
2 = m2/(d2AD2)
(VIII)
Substituting (VIII) into (VII) FD = [m2/(d2AD2)](CDdAD)(1/2g) FD = m2CD/(dAD2g) = m2CD/(2gd)(4/DD2)/(144in2/ft2) Collecting constants gives:
144*4/(32.2*d) = 5.69
FD = 5.69CDm2/(2dDD2)
(IX)
Refer to Section 2 of Appendix A for more discussion on CD. Substituting (VIII) into (IX) 5.69CDm2/(2dDD2) - [Ks(dXc + dXL) + Ws+D] = 0
assuming Ws+D 0
5.69CDm2/(2dDD2) - Ks(dXc + dXL) = 0
5.69CDm2/(2dDD2) - Ks(dXc + dXL) = 0
(X)
What is the minimum set pressure required to achieve full lift ? 5.69CDm2/(2dDD2) - KsdXc - KsdXL = 0
but KsdXc = Pmin*As 17
So,
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. 5.69CDm2/(2dDD2) - KsdXL = Pmin*As Pmin = (1/As)[5.69CDm2/(2dDD2) - KsdXL] liquid density can be expressed in term of the specific gravity using the following relation = 62.4SG Using this relation Pmin will be
(XI)
Pmin = (1/As)[5.69CDm2/(2*62.4SGDD2) - KsdXL]
The case of Liquids and Gases will be considered next. API equation will be used to relate the flow rate to the set (popping) pressure, required to deliver full lift.
1) Calculations of Minimum Set Pressure for Liquid Flow: A) CERTIFIED LIQUID FLOW: Based on API, Equation for certified liquid flow(3); with 10% accumulation _ A = (Q/38KdKwKv) [SG/P1-P2] A Required effective discharge area = Ab. Q Flow rate in gallon per minute = GPM. Kd Effective coefficient of discharge. For a preliminary sizing estimation, a discharge 0.65 can be used. Kw Correction factor due to back pressure. If the back pressure = atm., Kw = 1. Kv Correction factor due to viscosity. P1 Over pressure = (1+Accumulation)Pset. P2 Back pressure. Kv = Kw = 1
and
Kd = 0.65
Collecting the constants gives:
38KdKwKv = 38*0.65*1*1 = 24.7
If any correction in the value of Kd is later needed, 24.7 could be replaced by 24.7(Kd/0.65) _ Ab = (GPM/24.7) [SG/((1+Acc/100)Pset - Pb)] _ GPM = 24.7Ab [((1+Acc/100)Pset - Pb)/SG] Accumulation = 10 % 18
coefficient of
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. _ _ GPM = 24.7Ab [(1.1Pset - Pb)]/ SG
500SG gpm = lb/hr
Assuming no back pressure; Pb = 0 _ _ lb/hr = 24.7(500)SGAb [1.1Pset]/ SG _ _ lb/sec = 24.7(500)Ab SG [1.1Pset]/3600
lb/hr (hr/3600sec) = lb/sec
_ Collecting the constants gives: (24.7*500* 1.1)/3600 = 3.6 Let lb/sec = m mass flow rate. _
m = 3.6Ab [SGPset] _
m2 = {3.6Ab [SGPset]}2 = 12.96Ab2SGPset Substituting in (XI),
Note=12.96=12.96(Kd/0.65)2
Pmin= (1/As)[5.69CDm2/(2*62.4SGDD2) - KsdXL] gives,
Pmin = (1/As)[5.69CD{12.96Ab2SGPset}/(2*62.4SGDD2) - KsdXL] Note that Pset = Pmin Collecting constants gives: (5.69*12.96)/(2*62.4)=0.591
or
Pmin = (1/As)[0.591CDAb2SGPmin]/(SGDD2) - KsdXL]
DD2 = 4AD/
SG
0.591(Kd/0.65)2
cancels away
Pmin = (1/As)[0.591CDAb2Pmin]/(4AD/) - KsdXL] Collecting the constants gives:
0.591*/4 = 0.464
Pmin = (1/As)[0.464CDAb2Pmin/AD - KsdXL] AsPmin = [0.464CDAb2Pmin/AD - KsdXL]
or
0.464(Kd/0.65)2
Multiply both sides by As Collecting Pmin from both sides
Pmin[0.464CDAb2/AD - As] = KsdXL
Pmin = KsdXL/[0.464CDAb2/AD - As]
(XII) Certified Liquid Flow i.e. Acc=10% Note 0.464 0.464(Kd/0.65)2
This equation gives the minimum set pressure required for liquids to achieve full lift. Therefore, the minimum set pressure is based on the valve dimensions, spring rate and liquid viscosity.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. B) NON-CERTIFIED LIQUID FLOW: Based on API Equation for non-certified liquid flow (4); with 25% accumulation: _ A = (Q/38KdKwKvKp) [SG/1.25P-Pb] A Q Kd Kw Kv Kp P Pb
Required effective discharge area = Ab Flow rate in gallon per minute = GPM Effective coefficient of discharge. For a preliminary Sizing estimation, a discharge coefficient of 0.62 can be used Correction factor due to back pressure. If the back pressure = atm. , Kw = 1. Correction factor due to viscosity. Correction factor due to over pressure. At 25% Accumulation, Kp = 1. Set pressure = Pmin = Pset. Back pressure.
Kv = Kw = Kp = 1
and
Kd = 0.62
Collecting the constants gives:
38KdKwKvKp = 38*0.062*1*1*1 = 23.56
If any correction in the value of Kd is later needed, 23.56 could be replaced by 23.56(Kd/0.62) _ Ab = (GPM/23.56) [[SG/(1.25Pset - Pb)] _ _ GPM = 23.56Ab [1.25Pset - Pb)]/ SG and 500SG gpm = lb/hr Assuming no back pressure; Pb = 0 _ _ lb/hr = 23.56(500)SGAb [1.25Pset]/ SG lb/hr (1hr/3600sec) = lb/sec _ _ lb/sec = 23.56(500) SGAb [1.25Pset]/3600 _ Collecting the constants gives: (23.56*500* 1.25)/3600 = 3.66 Let lb/sec = m mass flow rate. _ m = 3.66Ab [SGPset] _ m2 = {3.66Ab [SGPset]}2 = 13.4Ab2SGPset Substituting in (XI),
Note=12.96=13.4(Kd/0.62)2
Pmin= (1/As)[5.69CDm2/(2*62.4SGDD2) - KsdXL] gives,
Pmin = (1/As)[5.69CD{13.38Ab2SGPset}/(2*62.4SGDD2) - KsdXL] Note that Pset = Pmin Collecting the constants gives: (5.69*13.38)/(2*62.4) = 0.61 Pmin = (1/As)[0.61CDAb2SGPmin]/(SGDD2) - KsdXL] SG
cancels away
and
Since DD2 = 4AD/
Pmin = (1/As)[0.61CDAb2Pmin]/(4AD/) - KsdXL] 20
or
0.61(Kd/0.62)2
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. 0.61*/4 = 0.479
Collecting the constants gives:
Pmin = (1/As)[0.479CDAb2Pmin/AD - KsdXL] AsPmin = [0.479CDAb2Pmin/AD - KsdXL]
or
0.469(Kd/0.62)2
Multiplying both sides by As gives Collecting Pmin from both sides.
Pmin[0.479CDAb2/AD - As] = KsdXL Pmin =
KsdXL/[0.479CDAb2/AD
- As]
(XII) Non-Certified Liquid Flow i.e. Acc=25% Note 0.469 0.479(Kd/0.62)2
2) Calculations of Minimum Set Pressure for Gas Flow: Based on API (RP 521) equation 4.3.2.1.2 for gas flow through nozzle under critical conditions(6): _ A = (W/CgP1KdKb)( [ZT/M] ) A W Cg
Required effective discharge area = Ab Required flow rate in lb/hr Coefficient determined from an expression of the ratio of the specific heats of the gas at standard conditions. Kd Effective coefficient of discharge. Kd = 0.975. Kb Capacity correction factor due to back pressure. P1 = Set Pressure + Over Pressure + Atmospheric Pressure = Pacc T Relieving temperature. Z Compressibility factor. _ W = AbCgKbKdPacc [M/(ZT)] Kb = 1, Kd = 0.975
and
Cg = 315
Collecting constants gives: Let W = lb/hr = m
Kb*Kd*Cg = 1*0.975*315 = 307
_
m = lb/hr = 307 Ab Pacc [M/(ZT)] Changing to lb/sec
m2 = [307 Ab Pacc/3600]2[M/(ZT)] m2 = (307/3600)2(Ab2Pacc2)M/(ZT) m2 = [(307/3600)2(Ab2Pacc2)M/(ZT)] Substituting into equation (X), {5.69CDm2/(2dDD2) - Ks(dXc + dXL) = 0} 5.69[(307/3600)2(Ab2Pacc2)M]CD/(2oZTDD2) - Ks(dXc + dXL) = 0
21
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. by definition of gas constant: gas = [PM/(10.73ZT)] (M/ZT)(1/o) = (M/ZT)(10.73ZT/PM) = 10.73/Pd 5.69(10.73)(307/3600)2CD(Ab2Pacc2)/(2PdDD2) - Ks(dXc + dXL) = 0 Since the valve relieves under critical conditions, a ratio between Pd and Pacc is defined. Figures 27 and 30 of API RP-520 show that for Kw to be 1, Pd must be less than 0.3*Pacc or Pd = 0.3*Pacc, psia(6). Collecting the constants gives:
5.69(10.73)(307/3600)2/(2*0.3) = 0.74
0.74CDAb2Pacc/DD2 - Ks(dXc + dXL) = 0
(XIIII)
Since Ks*dXc= Pset*As and Pacc = (Acc+1)/100*Pset + 14.7 Substituting into equation (VIIII) 0.74CDAb2[Pset(Acc+1)/100 + 14.7]/DD2 - Pset*As = KsdXL Let accumulation = 10%, then (Acc+1)/100 = 1.1 0.74CDAb2[1.1Pset + 14.7]/DD2 - Pset*As = KsdXL 0.74(1.1)CDAb2Pset/DD2 + (14.7)0.74CDAb2/DD2 - Pset*As = KsdXL Collecting constants gives:
0.74(1.1) = 0.81
and
0.81CDAb2Pset/DD2 + 10.9CDAb2/DD2 - Pset*As = KsdXL
(14.7)0.74 = 10.9 Collecting Pset from both sides gives:
Pset(0.81CDAb2/DD2- As) + 10.9CDAb2/DD2 = KsdXL Pset = [KsdXL - 10.9CDAb2/DD2]/(0.81CDAb2/DD2- As)
and
DD2 = AD*4/
Pset = [KsdXL - 10.9CDAb2/(AD*4/)]/(0.81CDAb2/(AD*4/)- As) Collecting the constants gives:
10.9/4=8.6 and
0.81/4=0.64
Pset = [KsdXL - 8.6CDAb2/AD]/(0.64CDAb2/AD- As) Rearranging the equation gives: Pset = (KsdXLAD - 8.6CDAb2)/(0.64CDAb2 - AsAD) Note that Pset = Pmin Pmin = (KsdXLAD - 8.6CDAb2)/(0.64CDAb2 - AsAD)
(XV)
This equation gives the spring rate for gases, accuracy may be improved if Cg for the specific gas is used. The minimum pressure for gases will always be less than that of liquids when all dimensional factors are the same. This due to the subtraction term atop the denominator
22
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. Estimation of Ks & dXs: A quick estimation of spring rate may be obtained by the Wahl equation for springs(7):
Ks = d4G/[8N(Do - d)3]
d Do G
(XVI)
Spring wire diameter Spring outer diameter Bulk Modulus of Rigidity = E/[2(1+)]. E: Modulus of Elasticity.
(inch) (inch) (psi) (psi)
: Poison's Ratio. Table 2 gives G values of different materials.
Table 2 Modulus of Rigidity MATERIAL Carbon Steel Alloy Steel Brass Aluminum
G (psi) 11 5*106 11 5*106 5 1*106 3 8*106
MATERIAL Inconel Monel Bronze Stainless Steel
G (psi) 11 0*106 9 5*106 5 5*106 10 6*106
For carbon and alloy spring material of RV's it is recommended to use a value of G = 11.5*106. The next equation estimates the amount of compression needed to set the RV to its set pressure.
dXset = 8N[PsetDs2/4](Do-d)3/(d4G)
(XVII)
Figure 5 compares the 2 calculations method of Ks determination. The less dependable method would be the Wahl Equation, Eq. XVI, because of the uncertainty introduced by estimating G, as given in Table 2. However, the Wahl Equation for Carbon Steel and Alloy Steel springs is generally accurate when compared to Ks as measured by the popping test outlined in section I.2, appendix B. Generally, the Wahl equation result is within 10-20% of the measured Ks.
23
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. FIGURE 5 VALIDITY OF Ks BY POPPING TEST & WAHL 800 CALC' D BY WAHL EQN
700
PARITY LINE
600 500 400 300 200 100 0 0
100
200
300
400
500
600
MEASURED Ks BY POPPING TEST, #/INCH
24
700
800
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
APPENDIX C
EXTRA DISCUSSION OF THEORIES
25
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
Discussion of Theories: SECTION ONE: Equation (III) shows that ASME limits to the maximum set pressure a particular spring can take. This maximum pressure is proportional to the spring constant Ks. It also shows that the maximum pressure of a spring installed in a certain RV is related to the spring constant by a simple constant "C2". C2 is a function of the valve orifice and the spring free and compressed lengths.
FIGURE 6 SPRING RATE FROM MAX SET PRESSURE Ks= C2* Pmax of spring charts 1600.0 1975c
1400.0
1990c
1200.0 1000.0 800.0 600.0 400.0 200.0 0.0 AC-14
AA-14
AA-15
AA-16
AA-17
AA-18
A-19
A-21
A-23
A-25
A-26
SPRING NUMBER
The valve dimensions can only allow springs with suitable dimensions to fit in. So, springs manufactured for a specific RV type will almost have the same dimensions. Hence, we can say that every RV type has a constant "C2" that relates the maximum pressure of a spring to the spring rate. This relationship has been proved for Consolidated 1975C & 1990C RV types, as shown by the above Graph. The data of this graph shows the calculated spring rate from the maximum set pressures of the spring as listed in the Consolidated spring tables (8). Since the spring rate for identical springs are identical, the use of separate constants should give identical spring rates, as shown by the graph. The C1 factor used for this graph was 1.38 for the 1975C and 1.73 for the 1990C valve. The AB25 spring did not have a listed pressure for use in the 1975C valve. The points where the deviation begins are for the larger springs which have spring identification markings and are not important to this discussion.. Additional investigation has shown that Consolidated does not use the full spring travel allowed by the ASME code. The amount of spring compression allowed in a given RV is determined by the length of the valves' spring compression screw. The travel of most set screws are limited to approximately 50% of the ASME allowed travel but this may vary depending upon a particular RV and spring. The effect of limiting this travel to 50% is that twice as many springs are required for a given RV body. One valve manufacturer replied that the limited travel is due to considerations of keeping the spring coils in a parallel position at the maximum set pressure. The graph below shows the results of a survey for 23 various springs evaluated in portable RV's.
26
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. FIGURE 7 % OF MAXIMUM ASME TRAVEL AT MANUF. MAX SET PRESSURE, 23 SPRINGS/2 MANUF'S 10 9 8 7
SURVEY RESULTS
6 5 4 3 2 1 0 5%
15%
25%
35%
45%
55%
65%
75%
85%
95%
% of ASME TRAVEL AT MAX SPRING PRESSURE
The majority of springs surveyed limited the travel to about 50% of the ASME travel. However the limits were between 35% and 85%. The next section will determine the Minimum Pressure of a spring. SECTION TWO: Analysis of Impulse Turbine is the best illustration of why CD must have a maximum value, limited by conservation of momentum. If CD is greater than the value of the momentum a perpetual motion machine could be conceived. The power gain is limited only by the upper limit of CD. However, the 2nd law of thermodynamics does not allow such perpetual motion machine. The Impulse Turbine: FIGURE 8 FLUID FLOW THOUGH IMPULSE TURBINE NOZZLE
V2
27
SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. Force on the wheel = F = Q(V1-V2Cos)/g Note that the fluid jet contours remain constant for an incompressible fluid, hence the scalar value of V1=V2=V, allowing the following substitution: F = QV1(1-Cos)/g, the force term is greater. Noting that Q=VA, "A" being the area of the nozzle: F = AV1V1(1-Cos)/g and equating to the drag force allows for solution of CD: FD = AV2CD/2g = AV2(1-Cos)/g and elimination of like terms gives: CD = 2(1-Cos) For =180, Cos = -1 For =90, Cos = 0
CD(max) = 4 CD(max) = 2
In practice, the velocity used for such analysis is based on the jet area which set the velocity into the wheel rather than working with the force vectors determined by the normals in the X direction. Therefore, AD can be set to As. Another example of hydro equipment which operates in a manor similar to the RV nozzle and disk is the Rotoflow meter. The drag coefficient for the roto-meter has been extensively compiled allow accurate calculation of fluid flow rates (11, 12). For the Roto-flow meter the fluid passes between an annular area formed by a weighted disk and the containing conical wall. The force exerted by the fluid on the weighted disk moves the disk up or down so as to indicate flow rate. The angle of momentum transfer, , would be between 900 and 450. Also some allowance would need to be made for wall friction. The typical values of CD were found to range between 1.45 and 0.30 for the roto-flow meter. Another example of momentum transfer handled by the CD method is in liquid mixing applications11. For mixing calculations the charts show CD is independent of Reynolds number for Reynold numbers greater than 10,000. In the high Reynolds number range the value of CD is dependent on the shape factor, a type of pseudo . The validity of this type analysis is verified by checking the viscosity correction factor, Kv , for RV's. The API listed value of Kv becomes independent of Reynold number at Reynolds numbers greater about 10,000. Table 3 gives CD values for different flow conditions and body shapes in a free liquid streams.
TABLE 3 CD Values for Different Flow Conditions BODY SHAPE
CD
REYNOLD S NUMBER
Triangle Cyl. = 120o Triangle Cyl. = 30o Open Semi-Tube
2.0 1.8 2.3
104 105 4*104
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E. For the conditions of Table 3 one may deduce the fluid free stream angle of momentum transfer to be about 900. The above analysis is valid only for liquids. For gases the Mach number is used in place of the Reynolds number, and at Mach numbers greater than 1 the CD becomes more dependent on the shape factors. The minimum set pressure of a spring is based on liquid conditions, except where specific spring tables are developed for gases and steam. Most RV designs are set-up to give of 180 to achieve maximum lift. The lift is somewhat variable by use of adjustable blowdown collars. The design of these collars are to give maximum force reactions. However the design of Atwood-Morrel safety relief valves are such that no blowdown collars are used. Instead the gas impinges onto a flat disk, giving equal to 90. MEASUREMENT OF Ks The proposed method used for determination of the spring rate is a standard practice in the RV industry. Most major RV suppliers provide a device which allows the set pressure of a RV to be checked without bringing the pressure up to the full setting of the popping pressure. For Consolidated10 this device is called an On-site Testing Device, model 1556. The operation principle is identical to the method of spring rate determination outlined here-in. The results of this practice are considered accurate enough to guarantee the set pressures of ASME I boilers.
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SPRING IDENTIFICATION METHODFOR "PORTABLE RELIEF VALVES" by OP Armstrong P.E.
REFERENCES 1. ASME STANDARDS Excerpts from ASME code Section VIII - division 1 "Pressure Vessels", 1989 edition. paragraph UG 136 (a)(2), page 95.
2. V. L. Streeter. "Fluid Mechanics". 1971. McGraw Hill, page, 272. 3. API Recommended Practices 520 5th edition. "Sizing, Selection and Installation of
Pressure Devices in
Refineries". Equation 4.5.1.9 for certified liquid flow, page 36.
4. API Recommended Practices 520 5th edition. "Sizing, Selection and Installation of
Pressure Devices in
Refineries". Equation 4.6.12 for non-certified liquid flow, page 39.
5. API Recommended Practices 520 5th edition. "Sizing, Selection and Installation of
Pressure Devices in
Refineries". Equation 4.3.2.1.2 for critical gas flow, page 27.
6. API Recommended Practices 520 5th edition. "Sizing, Selection and Installation of
Pressure Devices in
Refineries". Figures 27 & 30, page 30 & 35.
7. A. M. Wahl. "Mechanical Springs". 1963. McGraw Hill NYC, page, 8. Consolidated safety Relief Valves. "Spring Selection Charts". August 1990, pages
R2-2 & R2-9,
Dresser Industries, Alexandria La.
9. Consolidated Operation Manual, Dresser Industries, Alexandria La. 10. Consolidated Maintenance and Installation Manual. Dresser Industries, Alexandria La. 11. L. M. Polent. "How to do it Ideas for", Hydrocarbon Processing, Houston 12. McCable & Smith, Unit Operations of Chemical Engineering, Mcgraw-Hill NYC pages, 234 - 239. 13. Armstrong, Otis P.E: Portable Safety Relief Valve Spring Identification, APOE/ICOPU-93-225, June/22/1993
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