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SOURCEMODELS
Source Models 1
Mr.RogerDeo
Source Models 2
Source Models 3
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Source
Model
Dispersion
Model
Release
Incident
Source Models 4
Fire and
Explosion
Model
Effect
Model
Mitigation
Risk
Background
Mosthazardousincidentsstartwithsomeformofdischargeof
flammableortoxicmaterialoutofitsnormalcontainment.
Dischargescanbeofliquid,gasesortwophasedischarge.
Modelsareavailablethatprovideanestimationofthetotal
quantityreleased,thedischargerate andthetimetakenfor
discharge.
Thisinformationisusedasessentialinputsintoothermodels.
Source Models5
Dispersion
Model
Discharge
Rate Model
Dispersion
Model
Source Models 6
Flash and
EvaporationFlash and
Evaporation
Discharge
Rate Model
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Description Thefirststageistheidentificationofconceivable
dischargescenarios withtheresultingeffectsonthematerialreleasephase.
Severalimportantissueshastobeconsideredare
theholesize,leakduration,releasephase,end
point,thermodynamicpath etc.
Source Models 7
Topics
Dischargeofaliquidoutofavessel/pipe
Dischargeofagasoutofavessel/pipe
Dischargeoccurringin2phases
Dr. H. FarabiSource Models 8
Source Models 9
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Scenarios
Liquiddischarge
Holeinatmosphericstoragetankat
atmosphericpressure fromvesselorpipe
underliquidhead
Holeinvesselorpipecontaining
pressurizedliquidbelowitsboilingpoint
Source Models 10
Scenarios
Gasdischarges
Holeinequipmentpipe/vesselcontaining
gasunderpressure
Reliefvalvedischargeofvaporonly
o ngo evapora ono qu poo
Reliefvalvedischargefromtopofpressurizedstoragetank
Generationoftoxiccombustionproducts
asaresultoffire
Source Models 11
Scenarios
Twophasedischarges
Holeinpressurizedstoragetank/pipe
containingliquidaboveitsnormalboilingpoint
Reliefvalvedischarge(e.g.,dueto
foamingliquidorrunawayreaction)
Source Models 12
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Source Models 13
Thermodynamicpathandendpoint Thespecificationofathermodynamicpathand
endpointisimportanttothedevelopmentofthe
sourcemodel.
T especi icationo w et ert epat is
isenthalpic,isentropic,isothermalandadiabatic
areimportantintheselectionofanappropriate
modelandthecalculationofthatmodel.
Source Models14
Thermodynamicpath Forisentropicreleasestheequilibriumflashmodel
canbeusedtodeterminethefinaltemperature,
composition andphasesplitsatambient
temperature.Ifaphasechangeisencounteredthen
.
willflashatitsboilingpointandamixturewillflash
continuously.
Forreleasesofgaseseitheradiabaticorisothermal
modelsareavailable.
Source Models 15
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LeakdurationandholesizesThereisnogeneralconsensusonthesize
andlengthsofeitherofthese.Studies has
expressedaleakdurationof3to10minutes
andaholesizeofbetween 2inchesandthe
sizeofthepipe(fullbreakage)
Source Models 16
FundamentalequationsFundamentalequationdischargescanbemodeledbythemechanicalenergyequation
Where
P ressure(force/area)
( ) ( ) 0m
Wevv
g2
1zz
g
gdP sf
21
22
c
12
c
P
P
1
1
=++++ &
(Equation 1)
density(mass/volume)g accelerationduetogravity(length/time2)
gc gravitationalconstant(force/massaccelaeration)
z verticalheightfromsomedatum(length)
v finalvelocity(length/time)
Ws istheshaftwork(energy/time)m massflowrate(mass/time)Source Models 17
EquationsThefrictionallosstermisgivenas
=
c
2
ffg2
vKe
fL4
Equation 2
Wheref fanningfrictionfactor
L length
d diameter
Kf istheexcessheadlossduetopipeandfittings
Source Models 18
df = Equat on 3
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LiquiddischargesThenormaldrivingforcefordischargeisnormally
pressure,withthepressureenergybeingconvertedto
kineticenergyduringthedischarge.Sincethedensity
remainsconstantduringthedischargethenthe
pressureintegralinthemechanicalenergybalancecan
beintegratedtogivethefollowingsimplifiedformula
(equation10)
Source Models 22
( ) ( ) 0m
Wevv
g2
1zz
g
gPP sf
21
22
c
12
c
12 =++++
(Equation 10)
Liquiddischargefromapipe
P1 P2
Source Models 23
A
v1
PipeflowThemassfluxthroughthepipeisconstantand,for
pipesofconstantcrosssectionalarea,theliquid
velocityisconstantalongthepipe.
Thefrictionlossesoccurduetotheliquidflow.If
shaftwork:
( ) ( ) 0vvg2
1zz
g
gPP 21
22
c
12
c
12 =++
Source Models 24
(Equation 11)
(Bernoulli equation)
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Dischargeofpurenonflashingliquidthroughasharporifice
Themodelisdevelopedfromthemechanicalenergy
balancebyassumingthatthefrictionallossisrepresentedbyadischargecoefficientCD
(Equation12)
Where
Aisthearea(length2)
P1 istheupstreampressure
P2 isthedownstreampressure
Source Models 25
21cD PPg2ACm =&
ThefollowingaresuggestedforthedischargecoefficientCD:
ForsharpedgedorificesandReynoldsnumberin
excessof30,000CD approaches0.61.
ForawellroundedorificetheCD approachesunity
notgreaterthan3,theCDapproaches0.81
Wherethedischargecoefficientisunknownor
uncertainaCD=1shouldbeusedtomaximize
computedflowstoobtainaconservativevalue
Source Models 26
DischargecoefficientFromthe2KMethodthedischargecoefficientcan
becalculatedasfollows
Equation13
Source Models 27
+=
f
D
K1
1C
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Generalprocedurefordeterminingthemassdischargefromapipingsystem
1.Given:length,diameter,typeofpipe,elevations changesandpressures;work
inputoroutputtothefluidduetopumps
turbinesetc.Numberoffittingsandthe
propertiesofthefluid,includingdensity
andviscosity.Source Models 28
2. Specifytheinitialandfinalpoint,this
mustbedonecarefullysincethe
individualtermsintheequation(10)are
highlydependentonthisspecification
3. Determinethepressuresandelevation
attheinitialpointanddeterminethe
initialfluidvelocityatpoint1Source Models 29
4. Guessavalueforthevelocityatpoint2.Ifafully
developedturbulentflowisexpectedthenthis
stepisnotrequired
5. Determinethefrictionfactorofthepipeusing
eitherequationsorthechart.
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6. Determinetheexcessheadlossinthepipeusing
equation(3)andthefittings,usingequation4.Sumthe
headlosstermsandcomputethenetfrictionlossterm
usingequation(2). Usethevelocityatpoint2
=c
2
ffg2
vKe Equation 2
7. Computethevaluesofallthetermsinequation(10).If
itisnotequaltozerothengobacktothe4th step.
8. DeterminethemassflowratevAm =&
Source Models 31
d
fL4K f = Equation 3
Dischargeofliquidfromatank
Initial level V1
Dr. H. FarabiSource Models 32
hL
AV2
LiquidoutflowoutofatankForholesintanksthedischargeofmaterial
throughaholeresultsinthelossofliquidand
theloweringoftheliquidlevel.Forthiscase,
equation(12)iscoupledwithamassbalanceontheliquidinthetanktoobtainthe
followingexpressionforthetankdrainage
time.
Source Models 33
=1
2
V
VD h
)h(dV
g2AC
1t (Equation 14)
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Where tisthetimetodrainthetankfrom
=1
2
V
VD h
)h(dV
g2AC
1t (Equation 14)
volumeV2 tovolumeV1 (time)
Vistheliquidvolumeinthetankabove
theleak(length3)
histheheightoftheliquidabovetheleak
(length)
Source Models 34
LiquidoutflowofatankThemassdischargerateisdeterminedbythe
followingequation.Thefrictionisrepresentedbythe
dischargecoefficientCDandaccountsforthe
pressureduetotheliquidheadabovethehole
Source Models 35
+
== L
gc
D ghPg
2ACvAm&
(Equation 15)
Where
visthefluidvelocity(length/time)
Aistheareaofthehole(length2)
CD
isthemassdischargecoefficient(dimensionless)
+
== L
gc
D hgPg
2ACvAm& (Equation 15)
gc isthegravitationalconstant(force/massacceleration)
Pg isthegaugepressure
istheliquiddensity(mass/volume)
gistheaccelerationduetogravity(length/time2)
hL istheheightofliquidabovethehole
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GasdischargesGasdischargeoccursbyavarietyofmeans
fromaholein avesseltoleaksonpipelines
andtheopeningofreliefvalves.Thereare
differentprocedurestodealwithallofthese.
Themajorityofcasesthedischargeisinitially
eithersonicorchoked.
Source Models 37
Gasdischarge Asthepressuredropsthegasexpands.The
pressureintegralinthemechanicalenergybalance
(equation1)wouldrequireanequationofstate
an a ermo ynam cpa spec ca on o
completetheintegration.
Source Models 38
( ) ( ) 0m
Wevv
g2
1zz
g
gdP sf
21
22
c
12
c
P
P
1
1
=++++ &
(Equation 1)
ForgasdischargethroughholesEquation(1)is
integratedalonganisentropicpath
Theequationassumesthatthegasis
ideal,thereisnoshaftworkandthatthere
isnoheattransfer.
Thedischargerateisgivenbelow
Source Models 39
( )
=
k
1k
1
2k
2
1
2
1g
c1D
P
P
P
P
1k
k
TR
Mg2APCm&
(Equation 16)
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GasdischargeWhere
m - mass flow rate of gas through the hole (mass/time)
CD - mass discharge coefficient (dimensionless)
A - area of the hole (length2)
P1 - pressure upstream of the hole (force/area)
Source Models 40
gc - grav tat ona constant ( orce mass-acce erat on)
M - molecular weight of the gas (mass/mole)
k - heat capacity ratio, Cp/Cv (unitless)
Rg - ideal gas constant (pressure-volume/mol-deg)
T1 initial upstream temperature of the gas (deg)
P2 downstream pressure (force/area)
GasdischargeTheequationrepresentingthesonicorchockedflowis
( )( )1k
1k
1g
c1D
1k
2
TR
MkgAPCm
+
+
=& (Equation 17)
Source Models 41
The condition for a choked flow
(i.e. the pressure ratio required to achieve choking)
)1k(k
1
choked
1k
2
P
P
+
= (Equation 18)
GasDischargeForgasreleasedthroughpipestheissueofwhether
thereleaseisadiabaticorisothermalisimportant.
Forbothcasesthevelocitywillincreaseduetothe
dropinpressureofthegas.Foradiabaticflowsthe
temperaturemayincreaseordecreasedependingon
thetemperatureandthefrictionalforcesactingon
it.
Source Models 42
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Foradiabaticflowsthechokingpressureis
lessthantheisothermalchokingpressure
Forrealflows,theactualflowratewillbeless
thantheadiabatic redictionbut reater
thantheisothermalprediction.Thereisvery
littledifferencebetweentheadiabaticand
isothermalflows
Source Models 43
Whencompressiblefluidsdischargefromtheendofareasonablyshortpipe ofuniformcrosssectionareaintoanareaoflargercrosssectiontheflowisusuallyconsideredtobeadiabatic.Thisstatementissupportedby
130and220pipediametersdischargingairintotheatmosphere.Therefore,theadiabaticflowmodel isusedforcompressiblegasdischargedthroughpipes.
Source Models 44
GasdischargeForidealgasflowthemassflowforboth
sonicandsubsonicdischargeisgivenby
theDarcyequation
(Equation19)
Source Models 45
( )
=f
2111
K
PPg2YAm&
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Where
mmassflowrateofgasthroughthehole(mass/time)
Y gasexpansionfactor(unitless)
A areaofthedischarge(length2
)gc gravitationalconstant(force/massacceleration)
1 upstreamgasdensity
P1 pressureupstreamofthehole(force/area)
P2 downstreampressure(force/area)
kf Theexcessheadlossusing2Kmethod
Source Models 46
Thegasexpansionfactorisdependent
onlyontheheatcapacityratioofthegas,
k,andonthefrictionalelementsonthe
flow ath, k .Itisdeterminedusin onl
acompleteadiabaticflowmodel.
Thefollowingprocedureisusedto
determinethegasexpansionfactor:Source Models 47
TheupstreamMachnumber,Ma,isdetermined and
thensubstitutedintothelatterequationthe
procedureinvolvesthedeterminationoftheupstream
Manumber bytrialanderrorandifitmeetsthe
conditionsbelow.Useaspreadsheet.
(Equation20)
Source Models 48
( ) =+
+
+01
1
1
2ln
2
122
1
fKMaMak
Yk
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Thenextstepistodeterminethesonicpressure
ratio
(Equation21)
Iftheactualratiois reaterthanthisthentheflowis
1
2
11
1
21
+=
k
Y
MaP
PP
sonicorchokedandthepressuredroppredictedby
(21)isused.Ifless,thentheflowisnotsonic,and
theactualpressuredropratioisused
Source Models 49
Gasdischarge FinallytheexpressionYiscalculatedfrom
(Equation22)
Source Models 50
=21
1
2 PPMaY
f
Gas discharge
Source Models 51
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2Phasedischarge Whenanypressurizedliquidaboveitsnormal
boilingpointisreleasedtoatmosphericpressure
itwillbegintoflashandthe2phaseflowwill
occur
Themodelsthatwillbepresentedwillpredict
minimummassfluxesandmaximumpressure
drops
Source Models 55
2PhasedischargeThenonreactive(reactivecasenotconsideredyet)case
involvesflashingoftheliquidsastheyaredischargedfrom
containment.
Twos ecialconsiderationsaretakenintoaccount
1. Iftheliquidissubcooled,thedischargeflowwillchokeatits
saturationvaporpressureatambienttemperature
2. Iftheliquidisstoredunderitsownvaporpressure,amore
detailedanalysisisrequired
Source Models 56
Bothsituations areaccountedforbythefollowing
expression
(Equation23)
Where
misthetwophasemassdischargerate
N
GGAm
2
ERM2
sub +=&
Aistheareaofthedischarge(length2)
Gsub isthesubcooledmassflux(mass/areatime)
GERM istheequilibriummassflux(mass/areatime)
Nthenon equilibriumparameter(dimensionless)
Source Models 57
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Thesubcooledmassfluxisgivenby
(Equation24)
Where
[ ]satDsub PPgCG = 112
Psat isthesaturatedvaporpressure
Pisthestoragepressure
1 istheliquiddensity
CD isdischargeCoefficient
Source Models 58
Forsaturatedliquidsifthedischargelengthisgreaterthan0.1m(approximatelygreaterthan10diameters)theequilibriummassfluxisgivenby;
(Equation25)
Where P
c
fg
fg
ERMTC
g
v
hG =
Hfg istheenthalpychangeinvaporization
Vfg isthechangebetweenspecificvolumebetweenliquidandvapor
Tisthestoragetemperature
CpisthespecificheatcapacitySource Models 59
TherelationshipforNthenonequilibriumparameter
isgivenbelow
(Equation26)
Where
CPfgD
fg
L
L
TCvCP
hN +
=
22
1
2
2
Pisthetotalavailablepressuredrop
Lpipelengthtoopening
Lc isthedistancetoequilibriumconditions
(usually0.1m)
Source Models 60