TechnicalReportonPressure-reliefSystemCalculations

APITECHNICALREPORT522

FIRSTEDITION,APRIL2022

AP

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iii

Contents

Page

1Scope 1

2NormativeReferences 1

3TermsandDefinitions 1

4ForceBalanceAssessment-VaporExample 1

4.1General 1

4.2ExampleCalculation 1

5ForceBalanceAssessment-LiquidExample 6

5.1General 6

5.2ExampleCalculation 6

6CriticalLineLength-VaporExample 12

6.1General 12

6.2Example 12

Bibliography 14

Figures

1IsothermalBulkModulusofElasticityat100F 9

V

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TechnicalReportonPressure-reliefSystemCalculations

1Scope

Thistechnicalreportisnotadesigncode.Itonlyprovidesequationsandexamplesforperformingreliefsystemcalculations.Usersareresponsibleforperformingtheirowncalculationsandusingappropriatereferencesforequations.ThisreportcontainsavarietyofcalculationexamplesforequationsandmethodsfoundinAPIStandard520,Sizing,Selection,andInstallationofPressure-relievingDevices,PartI/—Installation.

2NormativeReferences

Thefollowingreferenceddocumentsareindispensablefortheapplicationofthisdocument.Fordatedreferences,onlytheeditioncitedapplies.Forundatedreferences,thelatesteditionofthereferenceddocument(includinganyamendments)applies.

APIStandard520,Sizing,Selection,andInstallationofPressure-relievingDevices,PartII-Installation,7Edition

3TermsandDefinitions

3.1

physicalacousticlinelength

TheaxiallineardistancefromthePRVinletflangetothefirstsignificantacousticreflectionpoint.SeeAPI520PartII,AnnexC.

3.2

speedofsound

Thedistancetraveledperunittimebyasoundwaveasitpropagatesthroughanelasticmedium.

3.3

springconstant

Acharacteristicofaspringthatistheratiooftheforceaffectingthespringtothedisplacementcausedbyit.

4ForceBalanceAssessment—VaporExample

4.1General

ThisisanexampleofaforcebalanceassessmentfromAPI520,PartI,Section7.3.6.dforsimpleinstallationinvaporserviceusingreference[1]and[4].

4.2ExampleCalculation

1)Obtainthevalveandinstallationinformation.

—Valve:1/·F·2bellows,vaporcertified.

—RelieffluidphaseatinletofPRV:Vapor.

—Certifiedorificearea(A)=0.3568in2.

—Certifiedcoefficientofdischargevalue(K)=0.855unitless.

—Discbackpressurearea(A)=0.4638in2.

—Lit(xm)=0.182in.

1

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-Setpressure(P)=50psig—VesselMAWP=50psig.

—Blowdown(PBD)=7%ofsetpressure=3.5psi

—Overpressure(Por)=10%ofsetpressure=5.0psi.

—PRVmountedtoa2-in.Schedule40[innerdiameter(ID)of2.067in.andcross-sectionalarea(A)of3.354in2]inletlinepipethatisrouteddirectlytoapressurevesselwithnobends.PRVisnotmountedonaprocessline.

—Physicalinletpipelengthfromvesseltoequipment(L)=15ft.

—Physicalacousticlinelength(L)=15ft.

—Non-recoverablefrictionalinletpressureloss(4P)=6%ofsetpressure=3.0psi.

—Ratedcapacityofreliefvalve(basedonhexaneandcertifiedvalues)=2860Ib/hr.

—Thetotalbackpressure(P)=2.0psig.Thisexampleassumesnoeffectsonsetpressurefromsuperimposedbackpressure,sothebuilt-upbackpressureisequaltothetotalbackpressure.Thebuilt-upbackpressureisequalto4%ofthesetpressure.

—TotalmassofthePRV(MpRv)=451bm.

NOTEThisisahypotheticalreliefvalveforwhichthespringconstant,massinmotion,anddampedopeningtimewillbeestimated.

2)Determinethefluidproperties.

AtPRVinletpipeentrance:

—Fluidcomposition=100%n-hexanevapor—Relieftemperature(T)=300F.

—Reliefpressure(P)=(50psig*1.1+14.7)=69.7psia.

-Vapordensity@55psig=0.8323Ib/ft3.

—Compressibility(Z)=0.8852.

—Isentropicexpansioncoefficient(K)=0.963.

3)DeterminethespeedofsoundinthevaporatthePRVinletpipe.

Assumethatn-hexanebehavesasarealgasandignoretheelasticityofthepipe.ThespeedofsoundforarealgasisgivenbyEquation(1):

USCunits:

(1)

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Slunits:

Where:

Cspeedofsound(USCunits:fts1,Slunits:m.s?1)

Kisentropicexpansioncoefficient(dimensionless)

Zcompressibilityfactor(dimensionless)

Rgasconst.(USCunits:10.731ff3psiR-1/b-mol',SIunits:8314.46m3PaK-'kmol)Tabsolutetemperature(USCunits:R,Slunits:K)

Mmolecularweight(USCunits:/b../bmol',Slunits:kg.kmol")

4)EstimatethespringconstantusingGrolmes'equations[1]

Where:

ksValvespringconstant(USCunits:Ibjn1,SIunits:N.m-1)

PuPressurewhichratedcapacitywascalculated(USCunits:psig,Slunits:Pa(g))

TSetpressureofthereliefvalve(USCunits:psig,Slunits:Pa(g))

AMinimumflowarea(USCunits:in2,Slunits:m2)

AopDiscbackpressurearea(USCunits:in2,Slunits:m)

xmarMaximumvalvelift,orrestrictedlift(USCunits:in,Slunits:m)

5)EstimatethemassinmotionfromtheGrolmes'equations[1].

Where:

momassinmotion(USCunits:/bm)

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MPRvtotalmassofPRV(USCunits:/bm)

6)EstimatethePRVopeningtime.USCunits:

Slunits:

Where:

T

undampednaturalperiod(s)

@

undampednaturalfrequency(radians.s')

f

naturalfrequencyofthevalve(s?1orHz)

K?

springconstant(USCunits:/bjn1,Slunits:N.m)

m,

massinmotion(USCunits:/b.Slunits:kg)

tpnundampedvalveopeningtime(s)

tpenddampedvalveopeningtime(s)

5dampingcoefficient(dimensionless)=0.5basedonbestfit[1]

TECHNICALREPORTONPRESSURE-RELEFSYSTEMCALCULATIONS5

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7)Calculatetheinletlinepressuredrop[1,4](4P).

△Pm=△Pme+4Pve

Where:

4Pm△P

△PeUSCunits:

inletlinepressuredrop(USCunits:psi,SIunits:Pa)

wavepressuredrop(USCunits:psiorSlunits:Pa)

frictionalwavepressuredrop(USCunits:psiorSlunits:Pa)

Slunits:

Mp

A,C%PoT

massflowrateofthereliefvalveat10%overpressure,duringPRVopening(USCunits:/bms?1,Slunits:kg.s?1)

inletlineinsidepipearea(USCunits:in2,Slunits:m)

speedofsoundofthefluidatthepressuresource(USCunits:ft.s-1,Slunits:m.s?)densityofthefluidatthepressuresource(USCunits:/bmt3,Slunits:kg.m3

ratioofwavetraveltime(ta)todampedvalveopeningtime(opend

0≤t≤1

t=2L/c?

Lacousticinletlinelength(USCunits:ft,Slunits:m)

C%speedofsoundinrelievingfluid(USCunits:ft.s?1,Slunits:m.s1)tpmddampedopeningtime(s)

t=2(15f)/611fts-10.0491sec

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4P(psi)=4.65psi

△Ppe=t2AP,

4Pe=12(3psi)=3psi

△P=3psi+4.65psi=7.65psi

8)Calculatetheforcebalance.

FromAPI520,PartII,Section7.3.6:

Thetotalinletpressureloss+0.1*thebuilt-upbackpressure≤overpressure+blowdown.Theadjustmenttothebuilt-upbackpressuretermrecognizesthatthebellowsareaisolatesalargepercentageofthediskareafromthebackpressure.

Inequationform:

△P+0.1Pbk≤Pop+Poforbellowsvalves

Where:

4Pmtotalinletpressureloss,includingboththewavecomponentandthefrictional(non-

recoverable)componentofpressureloss(USCunits:psi;Slunits:Pa)

Popoverpressure(USCunits:psi;SIunits:Pa)

PBDblowdown(USCunits:psi;Slunits:Pa)

Pbockbuilt-upbackpressureatthereliefdevicedischarge(USCunits:psi,SIunits:Pa)

7.65psi+0.1·2psi≤5psi+3.5psi

7.85psi≤8.5psiistrue;therefore,thePRVpassestheforcebalanceassessment.

Notethattheforcebalancemayonlybeonecriterionofmanyinanengineeringanalysisfortheacceptabilityofelevatedinletpressuredrops.RefertoAPIStandard520,PartII,Section7.3.6formoreinformation.

5ForceBalanceAssessment—LiquidExample

5.1General

ThisisanexampleofaforcebalanceassessmentfromAPI520,PartI,Section7.3.6.dforasimpleinstallationinliquidserviceusingreference[1].

5.2ExampleCalculation

1)Obtainthevalveandinstallationinformation.

—Valve:1·D·2bellows,liquidcertified.

—RelieffluidphaseatinletofPRV:Liquid.

TECHNICALREPDRTONPRESsuRE-BELEESYSTEMCALCULATIONS7

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—Certifiedorificearea(A)=0.128in2.

—Certifiedcoefficientofdischargevalue(K)=0.67unitless—Discbackpressurearea(A)=0.1664in2.

—Lift(xm)=0.095in.

-Setpressure(P)=50psig.

-VesselMAWP=50psig.

—Blowdown(PB)=10%ofsetpressure=5.0psi.

-Overpressure(Pop)=10%ofsetpressure=5.0psi.

一PRVmountedtoa2-inSchedule40[innerdiameter(ID)of2.067in.andcross-sectionalarea(A)of

3.354in2]inletlinepipethatisrouteddirectlytoapressurevesselwithnobends.PRVisnotmountedonaprocessline.

—Physicalinletpipelengthfromvesseltoequipment(L)=15ft.

—Physicalacousticlinelength(L)=15ft.

—Non-recoverablefrictionalinletpressureloss(4P)=8%ofsetpressure=4.0psi.

—Ratedcapacityofreliefvalve(basedonliquidhexaneandcertifiedvalues)=29USGPM=9287b/hr.

—Thetotalbackpressure(Pbc)=4.0psig.Thisexampleassumesnoeffectsonsetpressurefromsuperimposedbackpressuresothebuilt-upbackpressureisequaltothetotalbackpressure.Thebuilt-upbackpressureisequalto8%ofthesetpressure.

—TotalmassofthePRV(MpRv)=40lbm

NOTEThisisahypotheticalreliefvalveforwhichthespringconstant,massinmotion,anddampedopeningtimewillbeestimated.

2)Determinethefluidproperties.

AtPRVinletpipeentrance:

—Fluidcomposition=100%n-Hexaneliquid.

—Relieftemperature(T)=100F.

—Reliefpressure=(50psig*1.1+14.7)=69.7psia—Liquiddensity@55psig(p)=40.3Ib/ft3

—Specificgravity(60F/60F)(S)=0.647

—Realheatcapacityratio(C/C)=1.31

3)DeterminethespeedofsoundoftheliquidatthePRVinletpipe.

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lgnoretheelasticityofthepipe.

Thespeedofsoundforaliquidisgivenbythefollowingequation(API520,PartII,C.4).

USCunits:

Slunits:

Theisentropicbulkmodulusofelasticitymaybecalculatedas:

Where:

K,isentropicbulkmodulusofelasticity(USCunits:psi,Slunits:Pa)

C,/Crealheatcapacityratio(dimensionless)

Kisothermalbulkmodulusofelasticity(USCunits:psi,Slunits:Pa)

βisothermalcompressibility(USCunits:psi1,SIunits:Pa-1)

Sspecificgravityofthefluid

Theisothermalcompressibilitycanbeobtainedfromaprocesssimulatordirectlyorbycalculatingtheslopeofthedensitytopressureatconstanttemperature,thendividingbythedensity.Arecommendedrangeis±20%ofthesetpressureattherelieftemperature.

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Figure1-IsothermalBulkModulusofElasticityat100F

4)EstimatethespringconstantusingGrolmes'equations[1].

Where:

ksvalvespringconstant(USCunits:/bjn1,Slunits:N.m1)

PfouPressurewhichactualcapacitywascalculated(USCunits:psig,Slunits:Pa(g))

PSetpressureofthereliefvalve(USCunits:psig,Slunits:Pa(g))

AMinimumflowarea(USCunits:in2,SIunits:m2)

AopDiscbackpressurearea(USCunits:in2,SIunits:m)

xmrMaximumvalvelift,orrestrictedlift(USCunits:in,Slunits:m)

5)CalculatethemassinmotionfromtheGrolmesequation[1].

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Where:

m,massinmotion(USCunits:/bm)MPRVtotalmassofPRV(USCunits:/bm)

6)EstimatethePRVopeningtime.USCunits:

Where:

@f

ksm,

undampednaturalperiod(s)

undampednaturalfrequency(radians.s-?1)

naturalfrequencyofthevalve(s?1orHz)

springconstant(USCunits:/b,in?1,Slunits:N.m)massinmotion(USCunits:/b.Slunits:kg)

tpundampedvalveopeningtime(s)

tpnddampedvalveopeningtime(s)

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ζdampingcoefficient(dimensionless)=0.5basedonbestfit

7)Calculatetheinletlinepressuredrop(4Pm)[1].

△Pm=△P+4PveWhere:

△Pe

△P

△PeUSCunits:

inletlinepressuredrop(USCunits:psi,SIunits:Pa)

wavepressuredrop(USCunits:psiorSlunits:Pa)

frictionalwavepressuredrop(USCunits:psiorSlunits:Pa)

Slunits:

Where:

Mp

A,

C%

P?

T

massflowrateofthereliefvalveat10%overpressure,duringPRVopening(USCunits:/bms?1,Slunits:kg.s?1)

inletlineinsidepipearea(USCunits:in2,Slunits:m)

speedofsoundofthefluidatthepressuresource(USCunits:ft.s?1,SIunits:m.s?)densityofthefluidatthepressuresource(USCunits:/bmt3,Slunits:kg.m-)

ratioofwavetraveltime(tv)todampedvalveopeningtime(ton)

te=2L,/c。

Where:

L。

C?

tpend

Acousticinletlinelength(USCunits:ft,Slunits:m)

speedofsoundinrelievingfluid(USCunits:ft.s?1,Slunits:m.s?1)dampedopeningtime(s)

te=2(15f)/3482fts-1=0.0086sec

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4P(psi)=36.64psi

4Pe=t2AP,

4Pe=0.442(4psi)=0.78psi

4P=36.64psi+0.78psi=37.4psi

8)Calculatetheforcebalance.

FromAPI520,PartII,Section7.3.6:

Thetotalinletpressureloss+0.1*thebuilt-upbackpressure≤overpressure+blowdown.Theadjustmenttothebuilt-upbackpressuretermrecognizesthatthebellowsareaisolatesalargepercentageofthediskareafromthebackpressure.

Inequationform:

△Pm+0.1Pk≤Pop+PD

Where:

△Ptotalinletpressureloss,includingboththewavecomponentandthefrictional(non-

recoverable)componentofpressureloss(USCunits:psi;Slunits:Pa)

Popoverpressure(USCunits:psi;Slunits:Pa)

PBDblowdown(USCunits:psi;Slunits:Pa)

Pbockbuilt-upbackpress