Critical slowing down and fading away of the piston effect in porous media

0.15 MB·English

by  Bernard Zappoli

Criti
al slowing down and fading away of the piston e(cid:27)e
t in porous media
Bernard Zappoli *
* CNRS-PMMH, ESPCI, 10 rue Vauquelin 75005 Paris, Fran
e
6
Raphaël Cherrier** and Didier Lasseux**
0
** CNRS-TREFLE, ENSAM, Esplanade des Arts et Métiers, 33405 Talen
e Fran
e
0
2
Jalil Ouazzani*** and Yves Garrabos***
n
***ESEME - ICMCB - CNRS - Université Bordeaux I - 87,
a
J avenue du Do
teur S
hweitzer, F 33608 Pessa
Cedex Fran
e.
(Dated: 6th February 2008)
0
1 We investigatethe
riti
al speeding upof heatequilibration bythepistone(cid:27)e
t (PE)ina nearly

super
riti
alvanderWaals(vdW)(cid:29)uid
on(cid:28)nedinahomogeneousporousmedium. Weperforman
]
h asymptoti
analysis of the averaged linearized mass, momentum and energy equations to des
ribe
c the response of the medium to a boundary heat (cid:29)ux. While nearing the
riti
al point (CP), we
e (cid:28)nd two universal
rossovers depending on porosity, intrinsi
permeability and vis
osity. Closer to
m the CP than the (cid:28)rst
rossover, a pressure gradient appears in the bulk due to vis
ous e(cid:27)e
ts, the
PE
hara
teristi
times
ale stops de
reasing and tendsto a
onstant. Inin(cid:28)nitlylong samples the
-
t temperature penetration depth is larger than the di(cid:27)usion one indi
ating that the PE in porous
a
mediais nota (cid:28)nitesize e(cid:27)e
tas itis in pure(cid:29)uids. Closer to theCP, a se
ond
ross over appears
t
s whi
h is
hara
terized by a pressure gradient in the thermal boundary layer (BL). Beyond this
t. se
ond
rossover, the PE time remains
onstant, the expansion of the (cid:29)uid in the BL drops down
a and thePE ultimately fades away.
m
- PACSnumbers: 65.40.De,47.10.ad,47.15.G-,65.20.+w
d
n
o
Near
riti
al pure (cid:29)uids are Newtonian, vis
ous and in in(cid:28)nite media but ultimately vanishes. The
on(cid:28)g-
c
heat
ondu
ting (cid:29)uids. Their diverging
ompressibility uration under study is that of two parallel and in(cid:28)nite
[ L
allows a fourth energy transport me
hanism,
alled the planesdistant by limiting a homogeneousrigidporous

1
piston e(cid:27)e
t (PE), to be
ome prominent [1, 2, 3℄. The mediumsaturatedbyanear
riti
al(cid:29)uidinitiallyatrest,
v ρc
(cid:29)uid
ontained in thermal boundary layers (BL) formed at thermodynami
equilibrium, at its
riti
al density
6 τ 1
9 along heated walls expands, generates a
ousti

ompres- and set just above its
riti
al temperaturet′s>o t0hat hh .
1 sions waves whi
h (cid:29)ash ba
k and forth in the (cid:28)nite size This de(cid:28)nes the referen
e
ondition′s. At , the left
φ(t)
1 sample(cid:29)uid,andindu
esanadiabati
heatingofthebulk boundary is heated with a (cid:29)ux while the right one
0
mu
hfasterthanitwouldbebyheatdi(cid:27)usion. The
har- remains thermally insulated. In the rest of this paper,
6 c 0
a
teristi
time of heat equilibration tends to zero to 0 as subs
ripts and are used to denote quantities in the
0 τ3/2 τ
/ for a vdW (cid:29)uid, where is the dime(TnsionTles)s/dTis- (cid:29)uid phase at the CP and in the referen
e
onditions re-
t C C
a Ttan
etothe
riti
alpoint(CP)de(cid:28)ned as T− , spe
tively.
m being the temperature of the (cid:29)uid and C its
riti- As super
riti
al (cid:29)uids
an be des
ribed as
ompress-
-
al temperature. This remarkable (cid:28)nite size me
hanism ible, vis
ous, heat
ondu
ting and newtonian van der
d
ontrasts with the
riti
al slowing down of heat di(cid:27)u- Waals (cid:29)uids, we
onsider the averaged transport equa-
n
o sion. The homogeneous
hara
ter of both temperature tionsoveranelementaryvolume
ontainingalargenum-
c andpressure(ex
eptinthethermalBLfortemperature) ber of pores to be identi
al to that obtained for an ideal
: aretypi
al featuresof the PE. Vis
ous e(cid:27)e
ts arealways gas in a porous medium. The masss
onservation equa-
v
i negligible in bulk super
riti
al (cid:29)uid although a

ounting tion is given in [5℄. The momentum balan
e is expressed
X fora strong divergen
eof the bulk vis
osity [4℄ may lead by the unsteady form of the Dar
y equation [6℄ whi
h
r to small s
ale thermal boundary layer gradients. needs to be valid that the pressure gradient at the pore
a
Inthisletter,weaddressthe1Dproblemofheattrans- s
aleissmall
omparedto thatat the ma
ros
opi
s
ale.
portinanear
riti
al(cid:29)uid
ontainedinaporousmedium The energy balan
e whi
h a
ounts for the work of pres-
. We
onsiderthegravityve
tororientedperpendi
ularly sure for
es is similar to that of pure (cid:29)uids, sin
e the one
totheplanarboundariesofthethinsample(1-Dapprox- temperature model as derived in [7℄ leads to a solid
on-
imation)orthatgravityisequibratedbyinertiafor
esas tribution whi
h only modi(cid:28)es the heat
apa
ity and the
in spa
e laboratories. We basi
ally (cid:28)nd that pressure thermal
on′du
tivity of the pure (cid:29)uid.′
t = 0 x = 0
gradientswhi
h build up progressivelywhile nearing the At time , heat deposited at di(cid:27)uses into
CP((cid:28)rstinthebulkandtheninthethermalBL)
onsid- themediumandwederivetheasymptoti
analysisofthe
erably weaken and slow down the PE whi
h still exists
oupled equations. To this end, we divide the domain
2
δ ε/K
iδnto1two sub systems: a thin di(cid:27)usion BL of thi
hness , The ratio
an v1a0ry−3over1s0e4v)eral ordeεr2s/Kof mag-
hh when referred to the sample length, and the bulk. nitude (typi
ally from to while is al-
The
hara
teristi
heat (cid:29)ux at the wall is su
h that the ways mu
h smaller than unity. Equation (3) is the in-
averaged density, velo
ity, pressure ηa˜n,dη˜t,eηm˜ peratuη˜re of ternal energy bala(n
e(3e/q2u)aδt−io1nη˜ wu˜h)i
h shows the work
ρ u P T u z
respe
tive (cid:28)rst orders of magnitude and in of pressure for
es − and heat di(cid:27)usion
(ε γ0 δ−2η˜ τ−.05T˜ )
the BL undergo small perturbationsθw=itςhtreςsp1e
tθto the Pr0(γ0−1) T zz terms. The near
riti
al
freerfreerden
toe
aontdimiteions
sa.leIfwwhei
dhenisotςe−1 time,s lhohng,eritshraen- (cid:29)u(cid:16)id thermal(cid:17)
ondu
tΛivFity=, ΛΛF′F, is=deΛs
′bri+bedτ−b0y.5th=emτ−e0a.n5
thesamplea
ousti

hatra
teristi
timewhi
hde(cid:28)nesthe (cid:28)eld thΛe′ory [8℄ i.e. Λ′0F Λ′0FΛ′F ∼
non dimensional time as indi
ated below. The non- where b isits ba
kgroundvalue and 0 its
riti
alam-
dimensional,res
aled,governingequationsintheBLare: plitude.
ςη˜ ρ˜ +δ−1η˜ u˜ =0 In (3), we have negle
ted the solid heat
ondu
tivity
ρ θ u z
ςη˜ u˜ +δ−1γ−1η˜ P˜ = (4/3)(ε/K)η˜ u˜ (1)
ompared to the diverging one of the near-
riti
al (cid:29)uid
u θ 0 P z − u (2) and the Maxwell approximation [9℄ was used to derive
the equivalent heat
ondu
tivity of the porous medium
2εp ΛF = 2εp τ−0.5 Pr
to give 3−εp 3−εp . 0 is an equivalent
(ρCv)∗ςη˜TT˜θ = +−23(cid:16)δP−r10η˜(γγu00u˜−z1)(cid:17)εδ−2η˜Tτ−0.5T˜zz (3) Prerfaenredntc
l(cid:16)′entuh=merbm(cid:17)eγr0aldrdei(cid:28)(cid:27)nu(cid:16)esdivbityy(cid:17)Pder(cid:28)0n=edεDbpTyν00DwTh0e=re3D2−εεTpp0ρΛcisc′0F′p0a
where p0 γ0−1 is the spe
i(cid:28)
heat at
onstant pres-
η˜ P˜ = 3 η˜ T˜+ 9τη˜ ρ˜ sure of the ideal gas.
P 2 T 4 ρ (4)
−δ−ρ˜1=τ−u˜(cid:0)0.=5(cid:1)η˜TT˜T˜=z =P˜ϕ=W0(θ) at zθ ==00 (5) τ−O1ndeivsehrgoeunld
enooftethtehhaetatth
eapmae
aitny(cid:28)aetld
otnhsetaonryt pyτri0ee.sl5dsusrae
and thus a heat di(cid:27)usivity whi
h goest∗o zero as . In
at (6) (ρC )
v
u˜(z,θ)=0 z =0 thelefthandsideof(3),theterm istheequivalent
at (7)
Intheseequations,dimensionlesstemperature,density, bnyon(-ρdCimv)e∗ns=ionγa01−l h1e+at1
−εapεppaρ
sictsy,awth
eornestcavn0t=vocl′vru0m=e gγi0v1−en1
pressureand velo
ityarede(cid:28)ned fromthe
orresponding is the dimensionless heat
apa
ity at
onstant volume of
dimensional quantities denoted with a prime as follows: the near-
riti
al (cid:29)uid
oτn−s0i.d11ered as
onstant be
ause of
• T(x,t)= T′(TxC′,t′) =1+τ +T˜(z,θ)η˜T; iρtsscsρwetchaek=ddiiρmv′secer′sngseino
neleasssheat
awpah
eintynoefartihnegstohliedCdPe(cid:28)annedd
ρ(x,t)= ρ′(x′,t′) =1+ρ˜(z,θ)η˜ by s s ρcr .
• ρC ρ;
Equation (4) is the linearized vdW equation of state.
• P(x,t)= P′γ(0xc′20,t′) =P0+η˜PP˜(z,θ) E(zqu=at0i)onwh(5e)reis0t6heWbo(uθ)nd6ar1yi
sotnhdeittioimneamt tohdeullaetftiownaolfl
where r, γ0, c0 are the ideal gas
onstant per unit mass, the (cid:29)ux deϕposited at this boundary. The dimensionless
t
hoerrerasptioonodfinsgpeid
ie(cid:28)a
l ghaesa;tsua(xn,dt)th=eus′o(uxc0′n,td′)v=eloη˜
uiu˜ty(zi,nθ)t.he pisatri
amheeatetr(cid:29)uxindeb(cid:28)onuenddaarsyϕ
o=ndWiTtcm′iΛoa′0nxFL(5w)hiesrtehWe
mhaaxrais
ttehre-
maximum value over time of the deposited (cid:29)ux at the
Thze=dimxe′nsiδo−n1les=s bxoδ−un1dary layxer
oordinate is given boundary. Equation (6)is the initial
onditions.
by: L where is the dimensionless
(cid:16) (cid:17) Inthebulkregρi¯o(nx,,tθh)eu¯go(vxe,rθn)inPg¯e(xqu,θa)tionsfT¯or(xth,θe)per-
spa
e variable at the sample length s
ale while the time turbedvariables , , and are
t x δ
varita=blec0t′isεreferred to the a
ousti
time and is de(cid:28)ned similar to (1-4) ex
ept the spa
e variable is , is re-
by L . is the ratio, far fromL the
riti
al tempera- pη¯la,
η¯ed,η¯by,η¯1, the orders of magnitude are respe
tively,
ture,ofthesamplea
ousti
time c0 tothesamplevis
ous ρ u P T andtheboundaryandinitial
onditionsare:
di(cid:27)usiontime Lν02,ν0 being thekinemati
vis
osityofthe T¯x =0 x=1
at (8)
(cid:29)uid in the referen
e
onditions. ρ¯=u¯=T¯ =P¯ =0 θ =0
ε at (9)
10F−o8rusual
riti
al(cid:29)uids isasmall parameteroforder u¯(x,θ)=0 x=1
; largervalues
orrespondtomorevis
oussuper
rit- at (10)
K
i
al (cid:29)uids or to thinner sample′s. The Dar
y number
K Mat
hing
onditions between the bulk and the BL re-
is the intrinsi
permeability divided by the squared
gions
lose the equations, namely
ε
hpa:raK
t=eriKsti′
leεnpgLt2h.ofKt′hiessparmoppolertaionndalbtyotthheepsqoruoasrietdy zl→im∞η˜FF˜(z,θ)=xli→m0η¯FF¯(x,θ) , (11)

hara
rterist(cid:14)i
(cid:0)pore(cid:1)size and the porosity is the ratio of
F
the total pore volume to the total volumeof the sample. where is a physi
al unknown.
3
Finding an approximate solution of the equations smaτl−le0r.5thantheheatdi(cid:27)usiontimewhi
hgoestoin(cid:28)nity
(cid:28)rst needs determining the orders of magnitude fun
- as . This
riti
al speeding up of heat equilibration
ttihoants.η˜TBδτou0.n5dϕar=y
η˜oTn.ditTiohne [
5o℄ufpolrintgemofpetrhaetutrheerimmpodliyes- I
nonottrhaestrswwoirtdhstahsefa
rritais
a
lohnedaittiodni(cid:27)τuisiiτocn1 s=lowεin2/gKdo23wnis.
nη˜Tami=
vη˜aPria=bleτsη˜ρthrough the equation of state [4℄) gives ful(cid:28)lled, the PE is not τa(cid:27)e
ted by the poro(cid:0)us ma(cid:1)trix.
gives ςη˜ρ = δ−1η˜uw.hMileortehoevemr,aasss t
hoensmeravta
thioinngebqeutawteioenn τWc1heanndnaeaprirnegssuthreegCraPd,ientbeb
uoimldessu(cid:28)prisnt tohfethbeulkorpdherasoef.
di(cid:27)usionandtheworkofpressurefor
esintheboundary Therefore, a (cid:28)rst
rossoverto porous e(cid:27)e
ts arises when
layer(di(cid:27)usionmat
heswηi˜thδt−h1e=biεgδe−st2τte−r0m.5η˜intheequa- the pressure gradient and the average vis
ous stress in
u T
tion) [10℄ is expressed by , the pre- (16) be
ome of the same order of magnitude, i.e. when
viousequations(cid:28)nallylinkspa
eandtimes
alesthrough
ςδ2 =ετ0.5 2
the relation and give the order of magnitude ε2 3
η˜ = εϕ τ τ =
u of the velo
ity in the BL The above orders of ≈ c1 K (17)
(cid:18) (cid:19)
magnitude allow writing the Dar
y equation as
ςu˜θ+ε−1τ0.5P˜z/γ0 = 4/3(ε/K)u¯ When written in dimensional parameters, this
rossover
− (12)
K/ε 1 value be
omes
hh (13) T′ T (ν′/c )4/3
− c = 0 0
If
ondition (13) is ful(cid:28)lled, whi
hτ wτill b=e suε2p/pKose2d Tc K′2/3 (18)
c2
valid throughout the paper, and if ii ,
τ
c1
whi
h will be further
ommented later in the t(cid:0)ext, th(cid:1)en it thus is a universal value, only depending on the
the following hierar
hy holds (cid:29)uid and solid matrix prνo′pe=rties0..66.10−7m2s−1 c′ =
ς 1 ε/K ε−1τ0.5 3.1F0o2rm
.asr−b1on dioxide 0 , 0
hh hh hh (14) K′/ε =10−,14anmd2 for reasonable values of the0.r0a9tKio
P˜ =0 P ,the
rossovervalueisatabout
z
and the Dar
y equation in the BL redu
es to .
awayfromthe
riti
altemperature. Oneshouldnotethat
The BL solution shows that all the variables go to a
formorevis
ous
riti
al(cid:29)uids,this
rossovervaluewould

onstant at the outer edge of the BL. Considering the
lim η˜ u˜(z,θ)= be mu
h farther from the
riti
al temperature.
mat
hing
ondition(11)forthevelo
ity,z→∞ u
limη¯ u¯(x,θ) The solution of the system of equations when nearing
u
x→0 , and looking for a temperature homo- the CP is obtained by performing a
rossovertype solu-
τ
geneizationtimes
ale(bothtemperatureinthebulkand tion, e.g. by rτ˜ep=eaτti/ngε2t/hKe a2b/o3ve analysis when tends
intheBLareofthesameorderofmagnitude)thusgives: to zero while is kept
onstant. Doing
so leads to the followi(cid:0)ng ord(cid:1)ers of magnitude:
η˜T =η¯T =δϕτ0.5 and η˜u =η¯u =εϕ,. (15) η˜T =η˜p =η¯T =η¯P =η¯ρ =τc31/2ϕ
,
and (19)
By
arryingthelatterintothebulkenergybalan
eand η˜u =η¯u =εϕ
mat
hing the transient term with the term representing
ς =K/ε
the work of pressure for
es (di(cid:27)usion in the bulk being
δ = τ as well as to the following time s
ale parameter 2
negligible), we obtain the BςL=tεhτi−
k3n/2ess and thus and thermal boundary layer thi
kness δ = ε2/K 3.
the value of the time s
ale .
As beyond the
rossoverwe have per de(cid:28)(cid:0)nition(cid:1)
These orders of magnitude allow writing the Dar
y's
Equation in the bulk as: τ3/2 ε2/K
hh
τε32/2 u¯θ+τ3/2 Pγ¯0x =−34(cid:0)ε2/K(cid:1) u¯ (16) whi
h implies θ = ε2ε/tKhh εt/τ3/2 . (20)
2
If
onditionτiiετc1 =K ε2ε/Kτ33/2isveri(cid:28)ed,thefollowing As θPE = εt/τ3/2 is the time variable
orresponding
hierar
hy holdsτ3/2hhε(cid:0)hhKhh(cid:1)ε aP¯nd=Da0r
y's equation tothePE in thepure(cid:29)uid, we
an
on
ludethatthePE
x
in the bulk phase also redu
es to . Under these time s
ale in porous media is longer than the one in the

onditions, the governingequationsareexa
tlythe same pure (cid:29)uid. The e(cid:27)e
t of the solid matrix is thus to slow
as those for the pure super
riti
al (cid:29)uid [10, 11℄ and the downthetemperaturehomogenizationwhose
harateris-
ε/K
solution in the bulk displays homogeneoςu−s1temperature, ti
time tends to a
onstant equal to when referred
pressureanddensity. ThePEtimes
ale ,i.e. thetime to the a
ousti
time instead of being shorterand shorter
s
aleonwhi
htemperatureinthesampleishomogenized as in pure (cid:29)uid. We
all the phenomenon the porous
τ3/2
bythermo-a
ousti
heatinggoestozeroas . Itisthus saturation of the
riti
al speeding up.
4
τ 10−5K
c2
Combining(1-5)withtherelevantordersofmagnitude ( ≈ for
arbon dioxyde for L=10 mm) may
leadsto adi(cid:27)usion equationfordensity in the BLwhose suggestseverelimitationstotheNavierStokesapproa
h,
solution
an be written as follows anew
rossoveranalysisleadsto adi(cid:27)usion equationfor
θ density in the boundary layerwhose solution,
2
ρ˜(θ,z)=−3AZ KT (θ−u,z)W(u)du (21) ρ˜(θ,z)= −23A2(τ˜2)τ˜2−1/2
0 θ
K (θ u,z)W (u)du, (25)
T2
A(τ˜)= 4(γ /γ 1)Pr−1τ˜0.5 Aτ˜0.5 0 −
where 3 0 0− 0 ≡ is the dif- R
fusion
oe(cid:30)
ient and
1 z2 where the asso
iated di(cid:27)usion
oe(cid:30)
ient
KT (u)= πA(τ˜)uexp(cid:18)4−A(τ˜)u(cid:19) (22) A2(τ˜2)= 2176γ0τ˜2−3/2+A(τ˜2)−1 −1 (26)
p (cid:20) (cid:21)
is a normalized Gaussian di(cid:27)usion kernel. Going now
to the bulk equations we obtain a di(cid:27)usion equation for and di(cid:27)usion kernel
temperature whose solution is: 1 z2
K (u)= exp
T2
θ πA2(τ˜2)u! (cid:18)−4A2(τ˜2)u(cid:19) (27)
A
T¯(θ,x)= ∗ K¯T (θ u,x)W(u)du, p
(ρC ) − (23)
v Z indi
ates that both mole
ular di(cid:27)usion and thermo-
0
a
ousti
energy transfers o

ur at the BL s
ale. The
expresionfor the expansion velo
ity whi
h generates the
where
∞ heating a
outi
(cid:28)eld,
K¯ (u,x)=1+2 ( 1)n
T
n=1 − u∞(θ) = lim u(z,θ)=u¯(θ,x=0)
P ×ceoxsp[n−πn(12π2Bx)(]tildeτ)u (24) =z32→A∞2(τ˜2)τ˜2−1.5W (θ) , (28)
× (cid:0) − (cid:1)
B(τ˜) = τ˜1/2
is the kernel as∗so
iated with temperature and indi
atesthatthePEfadesawayas 2 whennearingthe
27/16γ (ρC ) τ˜3/2
0 v is an e(cid:27)e
tivedi(cid:27)usion
oe(cid:30)
ient.
riti
al point whi
h is howeverbeyond any experimental
(cid:0)The(cid:28)rst
orre
ti(cid:1)onbroughtbythepresen
eoftheporous
apability in normally vis
ous super
riti
al (cid:29)uids.
n=1
medium appearsτ˜in1theK¯ term in the series (24).
T
In the limit hh , be
omes the di(cid:27)usion ker-
K¯ (u) = 1 exp z2 u
nel T √πB(τ˜)u −4B(τ˜) whi
h indi-
(cid:18) (cid:19) (cid:16) (cid:17)
[1℄ A. Onuki, H. Hao, and R. A. Ferrell, Phys. Rev. A 41,

ates that there is an e(cid:27)e
tive energy penetration depth
B(τ˜) 2256 (1990)
inanin(cid:28)niteporousmediumsaturatedbyanear
[2℄ H. Boukari, J. N. Shaumeyer, M. E. Briggs, and R. W.

priti
al (cid:29)uid, larger than the heat di(cid:27)usion penetration Gammon, Phys.Rev. A 41 2260 (1990)
depth on the same time s
ale. An e(cid:27)e
tive energy trans- [3℄ B. Zappoli, D. Bailly, Y. Garrabos, B. le Neindre, P.
fer pro
ess in an in(cid:28)nite bulk thus exists whose penetra- Guenoun,andD.Beysens,Phys.Rev.A41,2264(1990)
B(τ˜)
tion depth, is longer than the di(cid:27)usion one on [4℄ P. Carlès, Phys.Fluids 10, 2164 (1998)
the same timpe s
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onve
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ousti
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(cid:29)uid [11℄.
τc1 [9℄ J.Cn.dMaxwellAtreatiseonele
tri
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When nearing the CPτ muτ
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c2
already-mentioned value ≈ is rea
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ond
rossoverfor usual super
riti
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