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Boiling in porous media: Model and simulations
J. Bénard, Robert Eymard, Xavier Nicolas, C. Chavant
To cite this version:
J. Bénard, Robert Eymard, Xavier Nicolas, C. Chavant. Boiling in porous media: Model and simula- tions. Transport in Porous Media, Springer Verlag, 2005, 60 (1), pp.1-31. �10.1007/s11242-004-2594-9�.
�hal-00694587�
Johann Benard,Robert Eymardand XavierNiolas
Laboratoire d'
EtudedesTransferts d'
Energieetde Matiere
UniversitedeMarnelaVallee-B^at.Lavoisier-77454ChampssurMarne-Frane
ClementChavant
EDFR&D,Departement Analyses MeaniquesetAoustiques
1, avduGeneraldeGaulle-92141ClamartCedex-Frane
April2,2004
Abstrat. Wepresentamodelizationoftheheatandmasstransferswithinaporous
medium,whihtakesintoaountphasetransitions.Classialequationsarederived
for the mass onservation equation, whereas the equation of energy relies on an
entropybalaneadaptedtotheaseofarigidporousmedium.Theapproximationof
thesolutionisobtainedusinganitevolumeshemeoupledwiththemanagement
ofphasetransitions. Thismodelis showntoapplyinthease ofanexperimentof
heatgenerationinaporousmedium.Thevaporphaseappearaneiswellreprodued
by thesimulations, and the size of the two-phase regionis orretly predited.A
resultofthisstudyistheevideneofthedisrepanybetweentheair-waterapillary
andrelativepermeabilityurvesandthewater-watervaporones.
Keywords:two-phaseowinporousmedia,phasetransition,nitevolumemethod,
water-watervaporapillaryurve.
Nomenlature:
A
ij
Measureofinterfaebetweengridbloksiandj,m 2
B Equationandinequality assoiatedwiththethermodynamistate
Cp Massheatapaityofphasep,J:kg 1
:K 1
D Disretesystemofequations
dij Distanebetweenentersofontrol volumesiandj,m
E Internalenergypervolumeunit,J:m 3
F
w
Massuxofwater,kg:s 1
F
h
Heatux,W
gp Gibbspotentialofphaseppermassunit,J:kg 1
h
p
Enthalpyofphaseppermassunit,J:kg 1
K Absolutepermeability,m 2
krp Relativepermeabilityofphasep
mp Massofphasepperporousvolumeunit,kg:m 3
Mw Molarweightofwater,kg:mol e 1
N
v
Numberofontrolvolumes
P
p
Pressureofphasep,Pa
P Capillarypressure,Pa
q Condutivethermalux,W:m 2
Q Heatsoureterm,W:m 3
Q
i
Heatsoureterm,W
S Liquidsaturation
T Temperature,K
t Time,s
U Disreteunknowns
Vi Measureofgridbloksi,m 3
Vp Volumiuxofphasep,m 3
:s 1
v
p
Speiuxofphasep,m:s 1
Greeksymbols
Entropypermassunit,J:K 1
:kg 1
~
Entropypervolumeunit,J:K 1
:m 3
Thermalondutivity,W:m 1
:K 1
wet Thermal ondutivity of the saturated porous
medium,W:m 1
:K 1
dry
Thermalondutivityofthedryporousmedium,
W:m 1
:K 1
p Dynamivisosityofphasep,Pa:s
p Bulkdensityofphasep,kg:m 3
w
l!v
Massrateofwatertransferfromphaseltophase
vpervolumeunit,kg:m 3
:s 1
Porosity
' Dissipations
Indiatorofthermodynamistateofgridblok i
(=1;2;3)
Subsripts andsupersripts
Capillarity
h Heat
l Liquid
p Phase
s Solid
v Vapor
w Water
w a Relativetotheouplewaterandair
w v Relativetotheouplewaterandwatervapor
1. Introdution
Heattransferanduidowwithliquid-vaporphasetransitioninporous
media arise in a number of sienti and engineering disiplines. Im-
portant tehnologial appliations an be found in various domain.
The mehanial behavior of drying porous materials must be known
in ivilengineering appliations(Whitaker, 1998; Coussyet al, 1998).
Inpetroleumengineering,multipleowingphasesarepresentinnatural
oilreservoirsandvariousenhanedmulti-phaseexploitationtehniques,
suh as water and vapor ooding, are employed (Woods, 1999). For
the purpose of the nulear reator safety analysis, the understanding
of ow and transport mehanism of vapor through the onrete en-
losure is essential (Medhekar et al, 1991). The study of the storage
or disposal of nulear waste strongly involves the predition of the
long-term heating of porous media due to the residual radioativity.
Sine, in this last tehnial area, it is partiularlydiÆult to manage
aurate experimentsforlong-term storage, it hasbeenundertaken to
establish the mainphysialand hemialmehanismsthat govern the
behaviorof waste pakagesdisposals(Toulhoat, 2002). The predition
of thresholds for aeptable heating, in nominal oraidental operat-
ing onditions,dependsonmodelsvalidated undertheatualdisposal
onditions (Castelier,2001).
Here,westudysomefeaturesof thisproblem,fousingonthemod-
elization and the numerial simulation of heat and mass transfers in
initiallysaturated porous media, taking into aount phase transition
phenomena.
In this paper, our approah for treating this problem onsists in
onsidering the various liquid phases as distint uids with individ-
ual thermodynami and transportproperties and with dierent phase
speiuxes. Thetransportphenomenaare thenmathematiallyde-
sribed by the basi balane equations for eah phase separately. In
this diretion,(Ramseh et al, 1993) propose a model where interfaes
separatingsinglefromtwophaseregionsaretraked,whereas(Daurelle
et al, 1998) proposea model wherethe liquidand thegaseous phases
oexist in any point. The model that we present here inludes the
determination of the thermodynami equilibrium state at eah point
of the porous medium, and the appearane or the disappearane of
phases.Thismodel,whihisanextensionof(Wangetal,1993;Ghar,
2000;Najjarietal,2002)toaseswheretheenergyequationannotbe
writtenasanenthalpybalane, anbedesribedbyasetofequations,
oupledwith inequalities.
The outline of the artile is as follows. In Setion 2, we desribe
the ontinuous equations of the model and the management of the
phasetransition.Then,inSetion3,weoutlinethenumerialmethods
usedto ndan approximatesolutiontothesystemofequations(inan
appendix, a omparison between numerial and analytial solutions,
in some simplied ases, provides a validation of these methods). In
Setion4,we proeedtoomparisonbetweenexperimentalresultsand
omputationalones.Amethodofdeterminingthevaporapillarypres-
sure and a parametri study of the water vapor relative permeability
is detailed in Setion 5. Some onlusions and future works are then
drawninSetion6.
2. Desription of the model
2.1. Mass onservation and energyequations
The porous medium is treated as a unique ontinuous medium re-
sulting from the super-impositionof the skeleton and uid ontinua.
The skeleton ontinuum isrigid and the uidontinuum is omposed
by two uids (liquid water and water vapor), assuming that there is
no dry air. We assume the existene of a representative elementary
volume whih is relevant at the marosopi sale for all the physi-
al phenomenainvolved intheintendedappliation.Moreover, at any
pointoftheontinuousmedium,thethreephasesareloallyatthermal
equilibrium (T
s
= T
l
= T
v
= T). Under these onditions, we use the
followinglassialequationsto modeltheows intheporous medium.
The liquidphase onservation as wellasthevaporphase onservation
are expressedby:
8
>
>
>
>
>
<
>
>
>
>
>
: m
l
t
+ r(
l v
l
) =
w
l !v
m
v
t
+ r(
v v
v
) =+ w
l !v
(1)
where
m
l
=S
l
and m
v
=(1 S)
v
: (2)
In the above equations, is the porosity of porous mediumsupposed
to be a onstant, S is the liquid saturation,
p (P
p
;T) is the density
of phase p = l;v, state funtion of the pressure P
p
of phase p and
of thetemperature T. We denote byv
p
the spei uxof thephase
p = lorv. The term
l !v
represents the mass rate per volume unit
transferedfrom theliquidto thevaporphase.
FollowingCoussy(1995, 2004)for theformulation ofthe energyequa-
tionintheaseofanopenvolumeunitofarigidporousmedium,under
quasi statis evolutions assumptions, the rst law of thermodynamis
produes
E
t
= r X
p=l ;v h
p
p v
p
rq+g X
p=l ;v
p v
p +
Q; (3)
whereE,h
p (P
p
;T),q,g and
Qarerespetivelytheinternalenergyper
volumeunit,theenthalpypermassunitofphasep,theondutiveheat
ux,thegravityaelerationandtheheatsoureterms.Theseondlaw
of thermodynamisan be expressed by
~
t
r
0
X
p=l ;v
p
p v
p +
q
T 1
A
+
Q
T
; (4)
with
~
=S
l
l
+(1 S)
v
v
+(1 )
s
s :
In the above relations, we denote by
p (P
p
;T) the entropy per mass
unit of the liquid and vapor phases (p = landv) and we denote by
s
(T)=C
s
log (T=T
0
) theentropypermassunitof thesolidphase.We
suppose that
s
is onstant. The expression of rq+
Q, obtained
from(3),isintroduedin(4)multipliedbyT.Thankstotheexpression
g
p (P
p
;T)=h
p (P
p
;T) T
p (P
p
;T)oftheGibbspotentialpermassunit
of phase p=l;v,we get
'
int +'
ow +'
therm
0; (5)
where theterm'
int
,representingtheintrinsidissipation,is given by
'
int
=T ~
t E
t
X
p=l ;v g
p r(
p v
p );
the term '
ow
, representing the dissipation due to the mass transfer,
is given by
'
ow
= X
p=l ;v
p v
p (rg
p +
p
rT g )= X
p=l ;v
p v
p (
1
p rP
p g );
andtheterm'
therm
,representingthedissipationduetotheheattrans-
fer, isgiven by
'
therm
= q
T rT:
Sine the porous mediumis assumed to be rigid,theintrinsi dissipa-
tion '
int
is equalto zero. Thisgives, thanksto Equation (3)
T ~
t X
p=l ;v g
p r
p v
p
= r
0
X
p=l ;v h
p
p v
p +q
1
A
+g X
p=l ;v
p v
p +
Q (6)
Usingmass onservation Equations(1), we get
T
~
t +
X
p=l ;v g
p m
p
t +
w
l !v (g
l g
v
)= r X
p=l ;v h
p
p v
p +q
+g X
p=l ;v
p v
p +
Q:
Assuming thatthere isno dissipationdueto thephase transition,i.e.
w
l !v (g
l g
v
)=0; (7)
andnegletingthemehanialenergyduetothevolumiweightfores,
weget:
T d~
dt +
X
p=l ;v g
p dm
p
dt
= r
0
X
p=l ;v h
p
p v
p 1
A
rq+
Q: (8)
We satisfy the ondition '
therm
0, assuming that the ondu-
tive heat ux is given by Fourier's law, in whih we use an ee-
tive ondutivity taking into aount the water ontent of the porous
medium:
q= (S)rT: (9)
Dierent expressions of (S) are available in the literature (see for
example (De Vries, 1964; Kelly et al, 1983)). Note that the inuene
of a given law is essentially governed by the values (0) =
dry and
(1)=
wet
,sinefor0<S<1,thetemperatureisdeterminedbythe
equilibriumbetweenwaterandwatervapor,leadingtosmallgradientin
thetwo-phase zone.Nevertheless, following(Wang etal,1993), we use
alineareetivethermalondutivitylaw(S)=S
wet
+(1 S)
dry .
Finally,we satisfytheondition'
ow
0,assumingthatthevelo-
ityof phasep isgiven byDary'slaw:
v
p
= k
rp K
p
( rP
p +
p
g ); (10)
where K is theabsolute permeabilityof theporousmedium(assumed
here to be onstant), k
rp
(S), the relative permeability of phase p, is
a funtion of the liquid saturation (k
rl
(S) is an inreasing funtion
suh that k
rl
(0) = 0 and k
rv
(S) is a dereasing funtion suh that
k
rv
(1) = 0) and
p
(T) is the dynami visosity of phase p, assumed
to onlydependonthetemperature.UsingEquations(1), (2)and(10),
the massonservation equation writes:
t (S
l
+(1 S)
v )+r
2
6
4
l k
rl K
l
( rP
l +
l g )
+
v k
rv K
v
( rP
v +
v g )
3
7
5
=0: (11)
Using Equations (1), (2), (8), (9) and (10), the energy equation is
expressed by
2
6
6
6
6
4 T
t (S
l
l
+(1 S)
v
v )
+T
t
((1 )
s
s )+
g
l
t (S
l )+g
v
t
((1 S)
v )
3
7
7
7
7
5 +r
2
6
6
6
6
4 h
l
l k
rl K
l
( rP
l +
l g )
+h
v
v krvK
v
( rP
v +
v g )
(
wet S+
dry
(1 S))rT 3
7
7
7
7
5
=
Q:
(12)
The vaporpressureP
v
isrelatedto theliquidpressureusingtheapil-
lary pressure,whih isa dereasingfuntionof theliquidsaturation:
P
(S)=P
v P
l
: (13)
Equation (7) is not suÆient to lose system (11), (12), (13) with
respet to (P
l
;P
v
;S;T). We therefore give in the next setion suÆ-
ient onditions, whih ensure (7), and whih enableto alulate the
thermodynamistate at eah point.
2.2. Conditions for the phase transition
The equilibriumthermodynamistate of the water (one phase liquid,
one phasevapor,ortwo-phase)an bedetermined forgiven liquidand
gaseous pressures and temperature onditions, usingthe omputation
of the Gibbs potential for eah phase. When Gibbs potentials are
equal, both phases are in equilibrium;otherwise, the phase with the
maximumGibbs potentialdisappears to thebenet ofthe phase with
the minimum Gibbs potential. Therefore, three equilibrium states are
possible:
State1 :no vaporphase;S=1andg
l (P
l
;T)<g
v (P
v
;T)
State2 :liquid-vaporequilibrium;g
l (P
l
;T)=g
v (P
v
;T)
and0<S<1
State3 :no liquidphase;S =0andg
l (P
l
;T)>g
v (P
v
;T)
(14)
Note that,inState1,we getm
v
=0 andv
v
=0,whihdelivers,using
(1), w
l !v
= 0. In State 3, we then have m
l
= 0 and v
l
= 0, and the
same onlusion holds.Therefore, equation(7) issatised.
System (11), (13), (8) and (14) is now losed, with respet to the
fourunknownsP
l ,P
v
,S and T.
2.3. State funtions for the liquid and vapor water phases
In this model, we need the expressions of the density, the dynami
visosity, the enthalpy and the entropy of eah water phase p = l;v
as expliit state funtions of the pressure of the phase and of the
ommon temperature. We assume that for p = l;v, the mass heat
apaity h
p
T (P
p
;T)doesnotdependon thepressureP
p
,andtherefore
veries hp
T (P
p
;T) = C
p
(T). By integration, introduing a referene
state (speiedbelow)denedbythepressureP
0
andthetemperature
hp
Pp (P
p
;T) suh that
h
p (P
p
;T)=h
p0 +
Z
T
T0 C
p
()d+ Z
Pp
P0
()d: (15)
Sine we have
dh
p (P
p
;
p
)=Td
p +
1
p dP
p
;
weget that
d
p (P
p
;T)= C
p (T)
T dT
1
T 1
p (P
p )
!
dP
p :
This impliesusingMaxwellrelations that:
T 1
p (P
p )
T
=0;
whihgivestheexistene of afuntion(P
p
) suh that
1
p (P
p )
T
=(P
p ):
We thusdeduethefollowingexpressionforthedensityof thephase:
p (P
p
;T)=
1
T(P
p
)+(P
p )
: (16)
We thenobtainthat
d
p (P
p
;T)= C
p (T)
T
dT (P
p )dP;
and therefore,there exists anintegration onstant
p0
suh that
p (P
p
;T)=
p0 +
Z
T
T
0 C
p ()
d
Z
Pp
P
0
()d: (17)
Relations(15-17)thenprovideonsistentthermodynamifuntionsfor
the lass of materials whose mass heat apaity only depends on the
temperature. Forthesake ofsimpliity,weonsideraonstant density
for the liquid water. This orresponds to the hoie
l (P
l
) = 0 and
l (P
l
) = 1=
l 0
, with
l 0
= 957:9 kg:m 3
, and we set C
l
(T) = C
l 0
=
4196 J:kg 1
:K 1
.Note thatit ispossibleto inreasetheaurateness
of these funtions, setting
l (P
l
) = a=
l 0 and
l (P
l
) = (1 b(P
l
P
0 ) aT
0 )=
l 0
. It then suÆes to selet a and b with respet to the
ompressibility and the dilatability of liquid water in the onsidered
range of temperatureand pressure.
Assuming the water vapor to be an ideal gas, we write
v (P
v ) =
R
MwPv
, with R = 8:315 J:K 1
:mole 1
and M
w
= 18 10 3
kg:mole 1
,
v (P
v
)=0,and weset C
v
(T)=C
v0
=1870 J:kg 1
:K 1
.
The four onstants h
l 0 ,
l 0 , h
v0
and
v0
annot be hosen inde-
pendently. Indeed, onsidering the referene equilibrium state at the
atmospheri pressure P
0
= 1:01325 10 5
Pa, T
0
= 373 K, we must
ensure g
v (P
0
;T
0 ) = g
l (P
0
;T
0
) and
v (P
0
;T
0
)
l (P
0
;T
0 ) = L
0
=T
0 ,
where the latent heat L
0
at this referene state is equal to L
0
=
2257 10 3
J:kg 1
. We an therefore take h
l 0
= 0,
l 0
= 0, h
v0
= L
0 ,
v0
=L
0
=T
0
.Gatheringthepreviousexpressions,weobtainthetableI.
UsingtheexpressionsgivenbytableI, itan be veriedthattheequi-
librium pressure funtion P(T) suh that g
v
(P(T);T) = g
l
(P(T);T)
and theequilibriumlatent heat L(T) =T(
v
(P(T);T)
l
(P(T);T))
areloseto thatwhih anbefoundintheliterature(Rohsenow etal,
1998) intheonernedrangeoftemperatures andpressures(see gure
1). We remark that the above expressions of the phase densities are
suÆientto ensureKelvin'slaw, that is,fortwoequilibriumstates P,
