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22.06
Engineering
of
Nuclear
Systems
MIT
Department
of
Nuclear
Science
and
Engineering
NOTES
ON
TWOPHASEFLOW,BOILINGHEATTRANSFER,ANDBOILINGCRISES
IN
PWRs
AND
BWRs
Jacopo
Buongiorno
AssociateProfessorofNuclearScienceandEngineering
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Definitionofthebasictwophaseflowparameters
In
this
document
the
subscripts
v
and
indicate
the
vapor
and
liquid
phase,
respectively.
Thesubscriptsgandfindicatethevaporandliquidphaseatsaturation,respectively.
AvVoidfraction: Av A
M Av v vStatic
quality: xst
Mv M Avv A
m
v A
v v v vFlow
quality:
x
mv m vvAvv vAvSlip
ratio: S
v
v
Mixture
density: m v (1)
Flowquality,voidfractionandslipratioaregenerallyrelatedasfollows:
1
1xv1 S
x
Figures1and2showthevoidfractionvsflowqualityforvariousvaluesoftheslipratioandpressure,
respectively,
for
steam/water
mixtures.
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In22.06weassumehomogeneousflow,i.e.vv=vorS=1. Thisassumptionmakesitrelativelysimpleto
treattwophasemixtureseffectivelyassinglephasefluidswithvariableproperties. IfS=1,itfollows
1
x
1
ximmediatelythatxst=xand,aftersometrivialalgebra, .
m v
Notethattheassumptionofhomogeneousflowisveryrestrictive,andinfactaccurateonlyunder
limited
conditions
(e.g.
dispersed
bubbly
flow,
mist
flow).
In
general,
a
significant
slip
between
the
two
phases
is
present,
which
requires
the
use
of
more
realistic
models
in
which
vvv. Suchmodelswillbediscussedin22.312and22.313.
Twophaseflowregimes
With
reference
to
upflow
in
vertical
channel,
one
can
(loosely)
identify
several
flow
regimes,
or
patterns,
whoseoccurrence,foragivenfluid,pressureandchannelgeometry,dependsontheflowqualityand
flowrate. ThemainflowregimesarereportedinTable1andshowninFigure3. Notethatwhatvalues
offlowqualityandflowratearelow,intermediateorhighdependonthefluidandpressure. In
horizontal
flow,
in
addition
to
the
above
flow
regimes,
there
can
also
be
stratified
flow,
typical
of
low
flowratesatwhichthetwophasesseparateundertheeffectofgravity.
Table1. Qualitativeclassificationoftwophaseflowregimes.
Flowquality Flowrate Flowregime
Low Lowandintermediate Bubbly
High
Dispersed
bubbly
Intermediate Lowandintermediate Plug/slug
High Churn
High High Annular
High
(postdryout)
Mist
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Morequantitatively,onecandeterminewhichflowregimeispresentinaparticularsituationofinterest
resortingtoanempiricalflowmap(seeFigure4). Notethatsuchmapsdependonthefluid,pressure
andchannelgeometry,i.e.,thereisnouniversalmapfortwophaseflowregimes. However,there
existmethodstogenerateaflowmapforaparticularfluid,pressureandgeometry. Thesemethodswill
be covered in 22.312 and 22.313.
Typical configuration of (a) bubbly flow, (b) dispersed bubbly (i.e. fine bubbles
dispersed in the continuous liquid phase), (c) plug/slug flow, (d) churn flow,
(e) annular flow, (f) mist flow (i.e. fine droplets dispersed in the continuous vapor
phase) and (g) stratified flow. Note: mist flow is possible only in a heated channel;stratified flow is possible only in a horizontal channel.
a b c
g
d e f
Image by MIT OpenCourseWare.
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L 1 G 2
Pfric
0 f D 2
dz
(3)
e m
L
Pgrav
m gcosdz (4)0
Notethattheaboveequationsareformallyidenticaltothesinglephasecasewiththemixturedensity,
m=v+(1),usedinsteadofthesinglephasedensity. ThefrictionfactorinEq.3canbecalculated
fromasinglephasecorrelationandusingtheliquidonlyReynoldsnumber,Reo=GDe/.
For
each
abrupt
flow
area
change
in
the
channel
(again
including
the
inlet
and
outlet),
the
associated
pressurelossisthesumofanaccelerationtermandanirreversiblelossterm:
Pform Pform,acc Pform,irr 1
(G22 G1
2 )K G 2
(5)2 2m m
wherethesubscripts1and2refertothelocationimmediatelyupstreamanddownstreamofthe
abrupt
area
change,
respectively.
Boilingheattransfer
Boilingisthetransitionfromliquidtovaporviaformation(ornucleation)ofbubbles. Ittypically
requiresheataddition. Whentheboilingprocessoccursatconstantpressure(e.g.intheBWRfuel
assemblies,
PWR
steam
generators,
and
practically
all
other
heat
exchangers
in
industrial
applications),
theheatrequiredtovaporizeaunitmassofliquidishfghghf,whichcanbefoundinthesteamtables.
Writingthe(YoungLaplace)equationforthemechanicalequilibriumofabubblesurroundedbyliquid,it
canbeshownthatthevaporpressurewithinthebubblemustbesomewhathigherthanthepressureof
thesurroundingliquid. Itfollowsthatthevapor(andliquid)temperaturemustbesomewhathigher
thanthesaturationtemperature,Tsat,correspondingtotheliquid(orsystem)pressure. Usingthe
ClausiusClapeyron
equation
to
relate
saturation
temperatures
and
pressures,
it
can
then
be
shown
that
themagnitudeofthesuperheat(TnucleationTsat)requiredtosustainthebubbleisinverselyproportionalto
theradiusofthebubbles,rbubble:
1T
nucleation Tsat (6)r
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etc.)isfullofmicrocavities,naturallypresentduetoroughness,corrosion,etc. Therefore,airorvapor
trappedinthosecavitiesserveasnucleationsitesforthebubblesandensureinitiationoftheboiling
process.
The
bottom
line
is
that
one
needs
to
raise
the
temperature
of
the
surface
a
little
bit
above
Tsatifboilingistobeinitiatedandsustained. ForarigorousderivationofEq.6andmoredetailsabout
bubblenucleation,refertoanyboilingheattransfertextbook(e.g.Boiling,CondensationandGasLiquid
FlowbyP.B.Whalley,OxfordSciencePublications,1987).
PoolBoiling
Considerasimpleexperimentinwhichwater(orotherfluid)boilsofftheuppersurfaceofaflatplate.
The
plate
is
connected
to
an
electric
power
supply,
and
thus
is
heated
via
Joule
effect.
This
situation
is
referredtoaspoolboiling(vsflowboiling)becausethepooloffluidabovetheheaterisstagnant. Inthis
experimentwecontroltheheatfluxq"(viathepowersupply)andwemeasurethewalltemperatureTw
(viaathermocouple). Thenaq"vsTwTsatcurvecanbeconstructedfromthedata. Thiscurveisknown
astheboilingcurveandisshowninFig5.
)/(
2mWq
505 1500
106C
O
A
B
Tw Tsat (C)
Figure5. Qualitativeboilingcurveforwateratatmosphericpressure(Tsat=100C)onaflatplate.
Asoneprogressivelyincreasestheheatflux,severalregions(orheattransferregimes)canbeidentified
from
this
curve.
A
depiction
of
the
physical
situation
associated
with
these
heat
transfer
regimes
is
shown
in
Fig
6.
OA
ThewalltemperatureisnotsufficientlybeyondTsattoinitiatebubblenucleation. Inthisregiontheheat
from the wall is transferred by single phase free convection
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Manybubblesareformedandgrowatthewall,anddetachfromthewallundertheeffectofbuoyancy.
Thisheattransferregimeiscallednucleateboiling. Thebubblescreatealotofagitationandmixingnear
the
wall,
which
enhances
heat
transfer.
As
a
result,
nucleate
boiling
is
a
much
more
effectivemechanismthanfreeconvection,assuggestedbythehigherslopeoftheboilingcurveinthisregion.
BC
Atacritical (high)valueoftheheat flux,therearesomanybubblescrowdingthewallthattheycan
mergeandformacontinuousstablevaporfilm. Thus,thereisasuddentransitionfromnucleateboiling
to
film
boiling.
This
sudden
transition
is
called
Departure
from
Nucleate
Boiling
(DNB),
or
Critical
Heat
Flux
(CHF),
or
burnout,
or
boiling
crisis,
and
is
associated
with
a
drastic
reduction
of
the
heat
transfer
coefficient
because
vapor
is
a
poor
conductor
of
heat.
Consequently,
the
wall
temperature
increasesabruptlyanddramatically(to>1000C),whichmayresultinphysicaldestructionoftheheater.
Candbeyond. Thisisthefilmboilingregion. Heattransferfromthewalloccursmostlybyconvection
withinthevaporfilmandradiationacrossthevaporfilm.
Intheabsenceofanexperiment,thevarioussegmentsof theboilingcurvecanalsobepredictedby
Newtonslawofcooling, q"h(Tw
Tsat
),withtheheattransfercoefficient,h,givenbythefollowing
empiricalcorrelations:
OA Si l h f ti f fl t l t b di t d b th Fi h d d S d
Film boiling (C and beyond)
The heat transfer regimes in pool boiling.
Onset of nucleate boiling (A)
Nucleate boiling (A B)
Free convection (O A)
Image by MIT OpenCourseWare.
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In this correlation (valid for 107
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Flowboiling
The
situation
of
interest
in
nuclear
reactors
is
flow
boiling.
Consider
a
vertical
channel
of
arbitrary
cross
sectional
shape
(i.e.
not
necessarily
round),
flow
area
A,
equivalent
diameter
De,uniformlyheated
(axiallyaswellascircumferentially)byaheatfluxq". LetTinbetheinlettemperature(Tin
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transfermostlyoccursthroughnucleationanddetachmentofbubblesatthewall,while intheforced
evaporationsubregiontherearetypicallynobubbles,andheattransferoccursmainlythruconvection
within
the
liquid
film
and
evaporation
at
the
liquid
film/vapor
core
interface.
The
heat
transfer
coefficientishighinbothsubregions. ThephysicalsituationisshowninFig8.
q"
q"
(a) (b)
Figure
8.
(a)
Nucleate
boiling
(typical
of
bubbly
and
plug
flow),
(b)
Forced
evaporation
(typical
of
annularflow).
Klimenkoscorrelationcomprisesthefollowingequations:
Nucleateboilingdominatedsubregion:
hLc
4.9103
Pem0.6
Prf0.33
PLc
0.54
(8)kf
Forced
evaporation
subregion:
0.2
hLc 0.6 1/ 6
g
k
0.087Rem Prf (9)
f f
Where
in
Eq.
8,
P
is
the
operating
pressure,
PrfistheliquidPrandtlnumberandthefollowingdefinitions
areusedforPem,RemandLc:
q"Lcfcp,fPe m
h fgg k f
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Klimenko recommends use of the following criterion to determine if nucleate boiling or forced
evaporation
dominates
at
any
given
location
in
the
channel:
IfNCB1.6104,heattransferisforcedevaporationdominateduseEq.9
h
1/ 3
whereN
G
q
fg
"
1x
g
f 1
g
f
CB
Note thatKlimenkoscorrelation isvalidup to thepointofdryout,butnotbeyond it. Dryout isan
undesirable
condition
(asocalledboilingcrisis)becausetheheattransfercoefficientdropsdrastically
andthusthewalltemperaturerisesabruptly,whichcanresultindamageofthewall. SincePWRsand
BWRsaredesignednottoreachthepointofdryout,Klimenkoscorrelationisagoodpredictivetoolto
estimatetheheattransfercoefficientinourapplications.
Forthesakeofsimplicity,theabovediscussionhasignoredthermalnonequilibriumeffects. Inreality,
boiling
starts
at
the
wall
for
zTsat)whiletheliquiddropletsareatsaturation. ThisregionisimportantinBWRfuel
assemblies ifthepointofdryout isexceeded(e.g.duringanaccident). Postdryoutwillbecoveredin
22.312and22.313.
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Boiling crises in LWRs
Jacopo BuongiornoAssociate Professor of Nuclear Science and Engineering
22.06: Engineering of Nuclear Systems
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the thermal limits associated with the
Ex lain how the thermal limits are verifiedin core design:
- DNBR for PWR
- CPR (for BWR)
2
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remove before it experiences a thermal crisis,i.e., a sudden and drastic deterioration of itsability to remove heat from the fuel.
,overheats and fails, thus releasing fission
.
To understand the thermal crisis we need tounderstand boiling.
3
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Thermal-hydraulics of PWR Core (2)
q
Hot channel
Temperature
Subcooled boiling in
Tsat
Tco
Tb
z
5
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Thermal-h draulics of PWR Core 4q (T ,G, P)Heat flux to cause DNB depends on Tb, G and P DNB b
Heat flux Heat flux
DNB
z
q
qDNB . .
q the US
7
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Thermal-h draulics of PWR Core 5Correlation (Tong 68) to calculate qDNB
qDNB
KTon
G0.4
0f
.
.6hfg
e
Where:
KTong
[1.767.433xe
12.222xe
2 ]
p,
sat
bx
< 0 in a PWRe
hfg
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Thermal-h draulics of BWR CoreFuel pin
CoolantChannel
16.2 mm
12.3 mm
geometryq
Average or hot
channel
Temperature
Saturated boiling inTco
Tsat
Tb
z 9
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Thermal-h draulics of BWR Core 2q
Dryout (thermal crisis)in hot channel
Dryout of the
liquid film in
annular flow
Clad tem erature
increases sharplywhen dryout occurs
z
10
Temperature
Tb
co
Tsat
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Thermal-h draulics of BWR Core 4
Correlation (CISE-4) to calculatexcr
Ph z
x
acrP zbw
Where:
cr
1/ 3
b0.199(Pcr
/ P1)0.4GDe
1.4
Note:
P is the thermodynamic critical pressure of water (22.1 MPa)
The units for G and De must be kg/m2-s and m, respectively
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Boiling demos
Jacopo BuongiornoAssociate Professor of Nuclear Science and Engineering
22.06: Engineering of Nuclear Systems
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Boilin a sim le ex erimentPool boiling of water at 1 atm. The current delivered to
.
Power su l
Electric resistance of the
wire
RIq 2
Water q
L
Wire LrwHeat flux at the
Wire radiuswire surface
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crisis N or
Boiling a simple experiment (2)Plot of wire temperature vs. heat flux (the boiling curve)
/ 2mWq Boilin crisis or DNB or CHF
Film Boiling106
Burnout
typically
occurs
Nucleate Boiling
ONB
)( CTT satw
Main message: Nucleate boiling = good, DNB = bad 16
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Lets run the demo!
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-Injection of air into column of water.
Air compressor
Injection valve
Watch how the flow pattern changes as air flow is increased21
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22.06 Engineering of Nuclear Systems
Fall2010
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