Gas spreading on a heated wall wetted by liquid

0.88 MB·

by  Y Garrabos

Gas spreads on a heated wall wetted by liquid
Y. Garrabos and C. Lecoutre-Chabot
CNRS-ESEME, Institut de Chimie de la Mati`ere Condens´ee de Bordeaux,
Universit´e de Bordeaux I, Avenue du Dr. Schweitzer, F-33608 Pessac Cedex, France
J. Hegseth
Department of Physics, University of New Orleans, New Orleans, LA 70148
∗
V. S. Nikolayev and D. Beysens
6 ESEME, Service des Basses Temp´eratures, CEA-Grenoble, France†
1
0
2 J.-P. Delville
Centre de Physique Mol´eculaire, Optique et Hertzienne,
n
CNRS, Universit´e de Bordeaux I, Cours de la Lib´eration, 33405 Talence Cedex, France
a
(Dated: January 27, 2016)
J

6 This studydealswith asimplepurefluidwhosetemperatureisslightly belowitscritical temper-
2 atureanditsdensityisnearlycritical,sothatthegasandliquidphasesco-exist. Underequilibrium
conditions, such a liquid completely wets the container wall and the gas phase is always separated

] fromthesolidbyawettingfilm. Wereportastrikingchangeintheshapeofthegas-liquidinterface
n
influenced by heating under weightlessness where the gas phase spreads over a hot solid surface
y
showing an apparent contact angle larger than 90◦. We show that the two-phase fluid is very sen-
d
sitive to the differential vapor recoil force and give an explanation that uses this non-equilibrium
-
u effect. We also show how these experiments help to understand the boiling crisis, an important
l technological problem in high-power boiling heat exchange.
f
.
s
c
I. INTRODUCTION II. EXPERIMENTAL SETUP
i
s
y
h We report results that were obtained and repeated
p Singularpropertiesofasimplefluid[1,2]appearwhen using several samples of SF (T = 318.717 K, ρ =
6 c c
[ itisnearitscriticaltemperature,Tc,anditscriticalden- 742 kg/m3). These samples were heated at various rates

1 sity,ρc. Whenthefluid’stemperature,T,isslightlylower in cylindrical cells of various aspect ratios on several
v than Tc and the averagefluid density ρ is close to ρc the French/Russian and French/American missions on the
9 fluid exhibits perfect wetting (i.e., zero contact angle) of Mir space station using the Alice-II instrument [8]. This
6 practically any solid by the liquid phase in equilibrium. instrument is specially designed to obtain high precision
9 In this article we study a system that is slightly out of temperature control (stability of ≈15µK over 50 hours,
6 equilibrium. Our experiments, performed in weightless- repeatability of≈50µKover7 days). To place the sam-
0
ness [3–7] showed that when the system’s temperature ples near the critical point, constant mass cells are pre-
.
1 T is being increased to Tc, the apparent contact angle paredwithahighprecisiondensity,to0.02%,byobserv-
0 (see Fig. 7 below for definition) becomes very large (up ing the volume fraction change of the cells as a function
6 to 110◦), and the gas appears to spread over the solid of temperature on the ground [9].
1
surface. InsectionII wedescribeourexperimentalsetup
: Afluidlayerwassandwichedbetweentwoparallelsap-
v that allows the spreading gas to be observed. The gas-
phirewindowsandsurroundedbyacopperalloyhousing
i liquidinterfaceshapeatequilibrium,whichisconsidered
X in the cylindrical optical cell, the axial section of which
in section III, plays a crucial role as an initial condition
is shown in Figure 1. We consider here three cells of
r
a for the gas spreading phenomenon. The sections IV and the same diameter D =12 mm, the other parameters of
Vdealwiththeobservationsofthespreadinggas. Athe-
which are shown in Table I. The liquid-gas interface was
oreticalmodelthatallowsthisunusualphenomenontobe
visualized throughlight transmissionnormalto the win-
explained is proposed in section VI. In the section VII,
dows. Since the windows were glued to the copper alloy
we discuss the boiling crisis, a phenomenon that plays
wall,someoftheglueissqueezedinsidethecellasshown
an important role in industrial applications, and how it
in Fig. 1. This glue forms a ring that blocks the light
is relevant to the spreading gas.
transmission in a thin layer of the fluid adjacent to the
copper wall making it inaccessible for observations. Be-
causeofthisgluelayer,thewindowsmayalsobe slightly
tilted with respect to each other as discussed in sec. III.
∗email:[email protected]
A 10 mm diameter ring was engraved on one of the
†Mailing address:CEA-ESEME, Institut de Chimie de la Mati`ere
windows of each cell in order to calibrate the size of the
Condens´ee de Bordeaux, 87, Avenue du Dr. Schweitzer, 33608
PessacCedex, France visible area of the cell images as can be seen in each im-
2
the windows (Fig. 1) due to the large aspect ratio D/H
wire grid
of the cell.
Let us first consider an ideally cylindrical cell as op-
posed to the real cell. At equilibrium, the windows and
glue engraved ring
CuBeCo the copper wall are wetted by the liquid phase. Because
the van der Waals forces from the walls act to make the
window
wetting film as thick as possible, the weightless bubble
liquid
H gas bubble
shouldbelocatedinthecell’scenter. Becausethebubble
window isflattenedandoccupiesone-halfoftheavailablevolume,
D
thedistanceofsuchacenteredbubbletothecopperwall
is large (Fig. 1). The lateral centering forces are then
incident light much weaker than the centering forces in the direction
of the cell axis. Any small externalinfluences in the real
cell can displace the bubble laterally from the cell’s cen-
FIG. 1: Sketch of a cross-section of the cylindrical sample ter. This displacement is illustrated in Fig. 2 that shows
cell (with parallel windows). The fluid volume is contained cell 10 at room temperature. We note that there are
between two sapphire windows that are glued to a CuBeCo twokindsofexternalinfluencesthatareeasilyidentified:
alloy ring. The dimensions H (see Table I) and D(=12 mm) residualaccelerationsin the spacecraftandcellasymme-
of the cell are indicated. Some glue is squeezed into the cell. try.
The thicknessof theglue layer is exaggerated for illustration Bubble images for cells 8 and10 were recordedin four
purposes. Inweightlessness,thegasbubbleshouldbelocated
Mir missionsbetween1996and2000. Severalimagesare
inthemiddleofsuchan’ideal’cell,seesec.IIIforthediscus-
reported in [3] (Cassiopeia mission, 1996) and in Fig. 6
sion.
below(GMSF2mission,1999)forcell10. Itisextremely
likelythatthespacestationchangeditspositionwithre-
Cell number Cell thicknessH (mm) (ρ−ρc)/ρc (%) spect to the residual gravity vector between these runs.
8 3.016 0.85 Thebubblepositionwithrespecttothecell,however,al-
10 1.664 0.25 waysremainedthesame. Thebubblelocationalsovaries
fromcellto cellwithoutanydependence onthe station’s
11 4.340 0.87
orientation. Therefore, we have no reason to attribute
theoff-centerpositionofthe bubbletothe residualgrav-
TABLEI:Physicalparametersoftheexperimentalcells. Cell ity.
11 hasamovablepiston tochange thecell volume. However, Although the cells were manufactured with high pre-
thevolume was kept constant duringthese experiments.
cision, the cell windows could not be exactly parallelbe-
cause of the glue layer as shown in Fig. 3. In the rest of
this section we will discuss the influence of the windows’
age. An out-of-focus wire grid, designed to visualize [10]
tilt on the position and on the shape of the bubble.
fluidinhomogeneitiesand/orafluidflowthroughlightre-
When the bubble’s surface is curved, there is a con-
fraction,wasalsoused. Thegridwasoccasionallymoved
stant excess pressure ∆p inside the bubble defined by
out of the light path, so that it is not always present in
the Laplace formula
all images.
ThesamplecellisplacedinsideofacopperSampleCell
∆p=σK, (1)
Unit(SCU)that,inturn,isplacedinsideofathermostat.
Heat is pumped into and out of the SCU using Peltier where σ is a surface tension and K is the surface curva-
elementsandheaters. Thetemperatureis sampledevery ture. This excesspressureactsonallpartsofthe bubble
second and is resolved to 1µK. interface. Inparticular,itactsonthepartA (wherethe
sg
Similar ground based experiments were done before index s standsfor “sapphire”andg for “gas”)ofthe flat
these experiments using a copy of the same instrument. window surface that contacts the gas directly (or, more
The gravity forces push the denser liquid phase to the accurately, through a wetting film that we assume to be
bottom of the cell and completely different behavior is of homogeneous thickness). This pressure creates reac-
seen, see [11]. tions forces F~(1) and F~(2) at each window, that act on
s s
the bubble. Each of these forces is perpendicular to the
corresponding window. The absolute values of F~(1) and
III. BUBBLE POSITION AT EQUILIBRIUM s
UNDER WEIGHTLESSNESS F~s(2) are equalto Asg∆p. When the windowsare exactly
parallel, F~(1)+F~(2) = 0 and the bubble remains at the
s s
The gas volume fraction φ (volume of the gas divided cell’s center. When the windows are tilted with respect
bythetotalcellvolume)isdefinedbyρandthedensities to each other, the non-zero force F~(1)+F~(2) pushes the
s s
ofgasandliquidforthegiventemperature. Inourexper- bubble in the direction of the increasing cell thickness.
iments, φ ≈ 0.5 and the gas bubble is flattened between Thismotioncontinuesuntilthe bubble touches(through
3
a b
FIG. 2: The experimental image of cell 10 at room temperature (a) and the equilibrium bubble shape simulated for the tilt
angle of 0.46◦ (b). When superposed, the images (a) and (b) give almost perfect match. The outer white circle in (a) (black
in (b)) shows the actual location of the cell wall. The inner black circles in (a) and (b) correspond to the engraved ring that
allows thesuperposition to bemade. Thedark space between two thesecircles in theimage (a) ismade bythering of glueas
shown in Fig. 1. The image (b) is a frontal projection of the bubbleshown in Fig. 4.
glue definedbytheknowngasvolumefractionandthecellvol-
gas bubble
tilt angle A ume. Onecanalsominimize thegas-liquidinterfacearea
A cg
sg with a bubble volume constraint. In both cases bound-
aryconditionsmustbesatisfied(zerocontactangleinour
case). The resulting bubble shape obviously depends on
the cell geometry. It is also nearly independent of tem-
→
Fs(1) F→ perature as can be seen from Eq. (2), because all three
→ c terms of this equation are proportional to the surface
F(2)
s tension σ, so that this force balance remains valid even
nearT ,whereσ disappears. Thereare,however,several
c
sources of weak temperature dependence of the bubble
shape. First,thereisweakdependenceofthegasvolume
fractionφ ontemperature atconstantaveragedensity ρ.
FIG. 3: Sketch of a cross-section of the sample cell with the This small deviationis smallest at the criticaldensity ρ
c
tilted windows at equilibrium in weightlessness. The wetting andslightlygreaterinthese experimentsdue to the very
film is not shown. The window tilt is possible due to the ex- small deviation (see Table I) of ρ from ρ . Second, the
c
istenceof aspace between thewindow’s edgeand thecopper
curvatureK dependsonthethicknessofthewettingfilm
wall, whichisfilledbyglue. Thisspaceandthetilt areexag-
that increases near T . The wetting film remains small,
geratedforillustrationpurposes. Basedonthemanufacturing c
however, in comparison with the cell thickness. Both of
process a maximum tilt angle of ≈ 1◦ is possible. The glue
these effects are very weak.
squeezed into the cell is not shown. The reaction forces that
act on the gas bubble are shown with arrows. The contact The force F~ , which is directed horizontally in Fig. 3,
c
areas of the gas bubblewith thesolid are indicated. causes a distortion of the bubble. This distortion results
in an ovalimage in Fig. 2 instead of a circle. The degree
ofdistortionincreaseswiththetiltanglebecausesodoes
a wetting film) the copper wall of the cell, thus forming F~ . This distortion can thus be used to estimate the tilt
a contact spot of the area A , where the index c stands c
cg angle.
for“copper”. Thisdirectcontactwiththesolidresultsin
anotherreactionforceF~ withtheabsolutevalueA ∆p, For these constant volume gas bubbles, the degree of
c cg
distortion should decrease with increasing cell thickness
such that
H forthesamewindowtilt. AlargervalueofH resultsin
F~(1)+F~(2)+F~ =0 (2) a less compressed (more sphere-like) bubble shape with
s s c
less area in contact with the wall. This smaller bubble
in equilibrium, see Fig. 3. curvature results in a smaller value for ∆p according to
Therearetwoequivalentwaystofindthebubbleshape Eq. (1). Consequently, the force F~ , the area A of the
c cg
at equilibrium. One can solve Eq. (1) that reduces to contact with the copper wall, and the bubble distortion
K =const, where the constant is obtained from the con- are smaller. This window tilt hypothesis is consistent
ditionofthe givenbubble volume. The bubble volumeis withobservations: wewerenotabletodetectanydistor-
4
tion of the gas bubble in cell 8 (see Fig. 10a below that impossible to verify.
corresponds to the nearly equilibrium shape) that is ap-
proximately twice as thick (Table I) as cell 10 shown in
Fig. 2. There is, however, some tilt in cell 8 because the IV. CONTINUOUS HEATING EXPERIMENTS
bubbletouchesthewall. Weexpectthatthetiltanglesin
allofthecellsareofthesameorderofmagnitudebecause In the continuous heating experiments, the cells 8, 10
they were all manufactured using the same method. and 11 were heated nearly linearly in time t. The evo-
To verify the window tilt hypothesis, we performed lution of the non-dimensional temperature τ for each of
a 3D numerical simulation of the bubble surface by us- theseexperimentsisshowninFig.5. Theparameterτ is
ing the Surface Evolverfinite element software [12]. The
result of this calculation is shown in Fig. 4 for cell 0.01
10. The experimentally observed bubble deformation
0
c
-0.01 b
τ a h
-0.02 0.002
g
0.001
-0.03 f
0
e
-0.001
-0.04 d
-0.002
1:22:00 1:24:00 1:26:00
-0.05
1:00:00 1:10:00 1:20:00 1:30:00
t (h:mm:ss)
FIG. 5: Reduced temperature evolution for the image se-
FIG. 4: The result of a 3D finite element calculation of the
quencesshown in Fig. 6 (solid line), Fig. 8 (dottedline), and
equilibrium gas-liquid interface for cell 10 with a window tilt
angleof0.46◦. Theverticesofthepolygonallinesindicatethe Fig.9(dashedline). Thetemperaturevaluesthatcorrespond
to each of the images (a-h) shown in these figures are indi-
location of thecylindrical copperwall andthey are shown to
catedbyarrowsandthecorrespondingletters. Thedefinition
guide the eye. A shape of the circular cylinder was input
of τ is discussed in the text. The temperature is measured
to the simulation. The contact angle is zero. A part of the
in the body of the SCU. The vicinity of the critical point is
image marked by the square is enlarged to show the contact
enlarged in theinsert.
area Acg of thegas with thecopper wall (a small white rect-
anglecrossed bytwosymmetrylines). ThecontactareasAsg
withthewindowshavetheovalshape. Theprojection ofthis definedas(T−T )/T ,whereT isthetemperature
coex c coex
bubbleshape to thecell window is shown in Fig. 2b. of the coexistence curve that corresponds to the fluid’s
averagedensity shown in Table I. Note that T differs
coex
matches the calculation performed for a tilt angle of from T only by 1−50µK because the density is very
c
◦
0.46 ,seeFig.2. Thesimulationresultedintheinterface close to ρ for all cells. A 40 min temperature equili-
c
curvature K = 1.389 mm−1 and in A = 0.150 mm2 bration at τ ≈ −0.033 preceded the heating. The mean
cg
calculated for the bubble volume Vφ = 26.675 mm3. value of dT/dt at T was ≈7.2 mK/s.
c
From this data, it is easy to calculate the effective ac- Figure 6 shows the time sequence of the images of
celeration g that would create the equivalent buoy- the cell 10. The interface appears dark because the
eff
ancy force F = (ρ −ρ )Vφg = KσA . It turns liquid-gas meniscus refracts the normally incident light
c L V eff cg
out that g = 1.55·10−3g for T = 290 K, where g is awayfromthecellaxis. Afterthetemperaturerampwas
eff
thegravityaccelerationonEarth. Thisg acceleration started but still far from the critical temperature, the
eff
is much larger than the residual steady accelerations in bubble shape changed. The contact area A of the gas
cg
the Mir space station (∼10−6g) and this shows that the with the copper wall appears to increase. In other sys-
observed bubble deformation is not caused by residual tems the wetting film under a growing vapor bubble is
accelerations. We conclude that the window tilt hypoth- observed to evaporate [14]. In near-critical fluids, how-
esis about the origin of the bubble deformation and its ever, the heat transfer processes are more complex [15].
off-center position is correct. In this system we believe that there may be a similar
A similar off-center bubble position was observed un- drying process, i.e., at some time the thin wetting film
der weightlessness in a cell similar to ours by Ikier et. that separates the gas from the copper wall evaporates.
al. [13] and was attributed to a residual acceleration. In fact, we have observed low contrast lines that appear
However, they report only one run in a single cell mak- within the A area when the heating begins. An exam-
sg
ing the actual cause of the bubble off-centered position ple of such a line is indicated in Figure 6a by the white
5
a b c d
e f g h
FIG.6: Timesequenceofimagesofthecell10duringthecontinuousheatingthroughthecriticalpoint. Thetemperaturevalues
that correspond to each of the images (a-h) shown in these Figures are indicated in Fig. 5 by arrows and the corresponding
letters. This run is a repeat of the run shown in Fig. 2 of [3]. The gradual increase of the apparent contact angle as the gas
spreads with increasing temperature is clearly seen. The time corresponding to each image is shown to the left of the cell in
themiddle. The magnified upperregions close to the contact line from theimages (e-g) are shown in Fig. 7.
glue 6e 6f 6g
boundary
liquid-gas
meniscus
8e 8f 8g 8h
FIG.7: ThemagnifiedupperregionsclosetothecontactlinefromtheimagesFig.6(e-g)andfromtheimagesFig.8(e-h). The
apparent contact angle can be “measured” as the angle between the tangents to the black glue boundary and the liquid-gas
meniscus. Thelattercorrespondstotheboundarybetween thewidedarkand narrowbrightstripes ontheimages. Theliquid
domain is to theleft from the meniscus. Onecan see that this apparent contact angle exceeds90◦ in the images 6g and 8f.
arrows. The out-of-focus gridshows that these lines cor- rapidlyevolving. AtT ≈T ,thesurfacetensionvanishes,
c
respond to a sharp change in the wetting film thickness. thebubble’srelaxationfromsurfacetensionisnegligible,
These lines are most likely triple contact lines and we so that the interface shape is defined by the variation of
have actually seen them pinned by an imperfection on the localevaporationratealong the interface. The evap-
the windows as they advance andretreatin other exper- orationis strongerat the parts of the interface closestto
iments. Since the heat conductivity of copper is larger the copper heating wall. This effect leads to the waved
than that of sapphire, the heat is supplied to the cell interface shape shown in Fig. 6g. Diffusion causes the
mainly through the hotter copper wall. Therefore the disappearance of the interface at T > T as shown in
c
filmshouldevaporateonthecopperwallevenearlierthan Fig. 6h.
on the sapphire. A more refined analysis of the contact
Figure 8 shows the time sequence of the images of the
line motion will be discussed elsewhere.
cell 8, which is approximately twice as thick as cell 10.
TheincreaseoftheAcg areaisaccompaniedbyanevi- The images in Fig. 8 were taken exactly for the same
dentincreaseintheapparentcontactangle,seeFig.6d–f values of the non-dimensional temperature τ (shown in
and the corresponding magnified images in Fig. 7. Near Fig. 5) as the corresponding images in Fig. 6. The force
the critical temperature the apparent contact angle be- F~ pushes the bubble against the cell wall as in the case
comeslarger than90◦! We willanalyzetheseeffects the- c
of cell 10. As discussed above, this force is weaker than
oretically in section VI.
forthecell10becausethebubbleappearsalmostcircular
While crossing the critical point, the vapor bubble is at equilibrium (see Fig. 6a and Fig. 8a). By comparing
6
a b c d
e f g h
FIG.8: Timesequenceoftheimagesfrom cell8duringcontinuousheatingthroughthecriticalpoint. Images(a-h)weretaken
exactly for the same values of temperature (shown in Fig. 5) as corresponding images in Fig. 6. The magnified upper regions
close to thecontact line from theimages (e-h) are shown in Fig. 7.
images(e)ofbothsequences,wecanalsoseethattheva- images(a), (d) and (f) were takenfor the same values of
porspreadsslowerincell8. Theincreaseoftheapparent non-dimensionaltemperatureascorrespondingimagesin
contact angle is also slower. The waved shape interface Figs. 6 and 8. This cell contained three wetted thermis-
appears earlier in Fig. 8f, i.e. farther from T than for tors(Fig.9i)thatconstrainthe bubble surface(Fig.9a).
c
the cell 10. The interface is still quite sharp in Fig. 8h, The bubble is only slightly squeezed by the windows so
whileithasalreadydiffusedinthecaseofthethinnercell that the reaction forces that act on the bubble at equi-
(Fig. 6h). This difference can be explained by the differ- libriumareweak. As a result,the bubble doesnottouch
ence in the liquid-gas interface area, which is roughly the copper heating wallat all. Although this is notclear
proportional to the cell thickness. The surface tension intheimage(a),becauseofthegluenearthecopperwall,
force that tends to maintain the convex shape is not as itisclearinimage(f) wheresmallnewlyformedbubbles
strongforthethickercellwherealargerfluidvolumehas separate the initial bubble from the wall. These bubbles
to be moved during the same time. The diffusion time form from the local overheating of the fluid between the
is larger for cell 8 because the size of the inhomogeneity large bubble and the copper wall. There is enough fluid
(i.e. interface) is larger. between the large bubble and the wall so that a small
Figure 9 shows the time sequence of the images from bubble may grow in it. These small bubbles push the
cell11,whichisthickerthanboththecells8and10. The large bubble away from the wall before any coalescence
can take place.
The comparison of these three experiments clearly
a d shows that in order to obtain the bubble spreading, the
bubble needs to have a direct contact with the heating
wall,i.e. to be pushed to the heating wallbysome force.
Notethatnoneoftheimagesshowanyevidenceofsteady
fluidmotionthatwouldbenecessarytomaintainthedis-
torted bubble shapes in Figs. 6 and 8. We conclude that
f i thisdistortionofthebubbleequilibriumshapecannotbe
caused by fluid motion.
Similar continuous heating experiments are reported
by Ikier et. al. in [13]. However, a smaller heating rate
(1.7 mK/s) and erratic accelerations of the cell did not
allow gas spreading to be observed.
FIG.9: Timesequenceofimagesofthecell11duringthecon-
V. QUENCHING EXPERIMENTS
tinuousheatingthroughthecriticalpoint. Nobubblespread-
ing is seen. The bubbledoes not touch the copper wall. The
images(a,d,f)weretakenexactlyforthesamevaluesoftem- Figure10showsthetimesequenceoftheimagesofcell
perature (shown in Fig. 5) as corresponding images in Fig. 6 8 when it was heated by 100 mK quenches as shown in
and Fig. 8. The cell 11 contains three thermistors shown in Fig.11. Whiletheheatingrateisquitelargeduringeach
image (i) byarrows. This image was taken after the temper- quench, the time average of the heating rate 1.4 mK/s
ature equilibration above Tc. is smaller than that during the continuous heating due
7
a b c d
e f g h
FIG.10: Timesequenceofimagesofcell8duringtwo100mKquenches. Thegasspreadsduringeachquenchthatlastsabout
12 s. The equilibrium position of the vapor bubble with respect to the cell is shown by the white circle in each image for
comparison.
0 served during the heating of CO cells in other experi-
g h 2
f ments (Pegasus BV4705, Post-Perseus F14) carried out
e by our group in the Mir station. However, these experi-
-0.0005
ments were not designed to study the spreading gas and
we do not discuss them here.
d
-0.001
τ
-0.0015 VI. INTERFACE EVOLUTION DURING THE
HEATING
b c
-0.002
The above experimental data showed that the spread-
a ing gas and the associated interface deformation are
-0.0025 caused by an out-of-equilibrium phenomenon. This is
2:30:00 2:35:00 2:40:00 2:45:00
especially demonstrated by the analysis of the interface
t (h:mm:ss)
shape at equilibrium (sec. III) and by the return to the
equilibrium shape after each quench in sec. V. In this
section we analyze possible causes of the spreading gas.
FIG.11: Temperatureevolutionduringtheseriesofquenches.
Twocausesareconsidered: Marangoniconvectiondueto
ThepointsthatcorrespondtoeachoftheimagesinFig.10(a-
h) are indicated by arrows and corresponding letters. The thetemperaturechangeδTi alongthegas-liquidinterface
temperature is measured in thebody of the SCU. and the differential vapor recoil.
to the waiting time of ≈ 60 s after each quench. Dur- A. Marangoni convection
ing this waiting time a partial equilibration takes place.
The images (a–c) show a slight bubble spreading that IfatemperaturechangeδT exists,itwillcreateasur-
i
appears during a quench that is farther from the criti- face tension change δσ = (dσ/dT)δT that will drive a
i
cal point than the quench shown in images (d-h). After thermo-capillary (Marangoni) flow in the bulk of both
each quench as soon as the heating stops, the bubble in- fluids[16–18]. Theimagesobtainedinourexperimentare
terface begins to return to its initial form (Fig. 10c,d). capable of visualizing convective flows from the shadow-
This shows that the spreading vapor is caused by a non- grapheffect. Wehavenotseenanyevidenceofthesteady
equilibrium effect. The second quench that precedes the convectionthatisrequiredtocreateandmaintaintheob-
crossing of the critical point (Fig. 10d–h) shows very servedbubbleshapecontinuouslyduringtheheating. We
rapid interface motion accompanied by fluid flows. conclude that the Marangoni convection is absent.
While the interface returns to its initial state during This conclusion is an apparent contradiction with
the waiting time of the first quench (Fig. 10c), it does many works that study the Marangoni effect caused by
not return in the second quench (Fig. 10h). This occurs evaporation (see e.g. [19]). The main difference between
because the characteristic equilibration time grows dra- these works and ours is in the conditions of evaporation.
matically near Tc. Theseworksconsiderthe evaporationintoanopen space
The same phenomenon of spreading gas was also ob- where the vapor pressure is very small. The interface
8
temperature thus follows the temperature in the bulk of According to the quasi-staticargument[23], the inter-
the liquid and a very large evaporation rate is possible, face shape canbe determined from the modified Laplace
limitedonlybytheaveragevelocityofthefluidmolecules. equation
In our case, the gas phase is almost at saturation pres-
σK =∆p+P (~x). (4)
sure. This means that the total evaporation (over the r
whole gas liquid interface) is small and limited by the
The 3D curvature K is equal to the sum of the 2D cur-
amountofthesuppliedheatconsumedbythelatentheat.
vature c in the image plane and the 2D curvature in the
Therefore, any variation δT is rapidly dampened by the
i perpendicular plane shown in Fig. 1. For the small cell
correspondingchangeintheevaporationrate,stabilizing
thickness H, this latter curvature can be accurately ap-
the interface against Marangoni convection, see [15] for
proximated by the constant value 2/H. This is possible
an extended discussion. This conclusion is confirmed by
because the relatively small heat flow through the less
theexperiments[20],inwhichMarangoniconvectionwas
conductivesapphirewindowsimpliesasmallP nearthe
carefullystudiedinaclosed cellwithverycleanwaterin r
contact line on the windows, as compared to the large
contactwithitsvapor. No surface-tension-drivenconvec-
value of ∆p at this small H. The interface shape can
tionwasregisteredinspite ofa largeMarangoninumber
thus be obtained from the 2D equation
that was much greaterthan its critical value obtained in
the classical Marangoni-Benard experiments with non- σc=∆p′+P (l), (5)
r
volatile liquids [16]. It was argued in [16] that the con-
′
vection was absent due to a hypothetical interface con- where∆p isaconstanttobedeterminedfromtheknown
tamination present in spite of many careful preventive bubblevolumeandlisacoordinatethatvariesalongthe
measures. According to our reasoning, a variation δT bubble contour in the image plane.
i
would have been strongly dampened in [16] because of In order to find the distribution n(~x) at the interface
the saturationconditionsinthe sealedcell. We alsonote it is necessary to solve the entire heat transfer problem.
that even in evaporative driven Marangoni convection Thisproblemiscomplicatedbyseveralimportantfactors.
far from saturation, the convection cells may also tend First,wedealwithaproblemthatcontainsafreebound-
tostabilizetheinterfaceresultinginintermittentcellular ary (gas-liquid interface) the position of which should
formation as was observed in [17]. It was also observed be determined. Second, this interface contains lines of
in [17] that the velocity of convection and frequency of singularities (gas-liquid-solid contact lines) where vari-
intermittent cell formation decreases as the external gas ous divergences are possible. Third, the adiabatic heat
becomes more saturated. transfer[15,21,22](“the pistoneffect”) shouldbe taken
into account for near-critical fluids. The first two com-
plications were addressed in [23–25] for plane geometry,
B. Differential vapor recoil i.e. for the gas bubble growing on a plane. We have
shown that n(~x) can exhibit a divergence at the contact
line and that it decreases exponentially far away from
We now analyze another possible source of bubble de-
it. Because the bulk temperature varies sharply in the
formingstressthatdoesnotrequireatemperaturegradi-
boundary layer adjacent to the walls of the cell [21] and
ent alongthe interface. The bubble may be deformedby
theinterfacetemperatureisconstant,thelargestportion
the normal stress exerted on the interface by the recoil
of mass transfer acrossthe interface takes place near the
from departing vapor [19]. Let n(~x) be the evaporating
triple contact line. Thus n(~x) is large in the vicinity
mass per unit time per unit interface area atthe point~x
of the contact line. In this work, we present first the
onthe interface. The evaporatinggasmovesnormallyto
scaling arguments and then an approximate calculation
theinterface,andexertsaforceperunitarea(a“thrust”)
of the bubble shape to illustrate our explanation of the
on the liquid of
spreading gas in the cylindrical geometry.
We assume that n(~x) has the following form:
Pr(~x)=n2(~x)(1/ρG−1/ρL), (3) n(~x)=g(~x)(Tc−T)a (6)
where ρ denotes mass density and the subscripts L and as T → T , i.e., it has the same local behavior with re-
c
G refer to liquid and gas respectively. spect to temperature as the critical temperature is ap-
Theinterfaceshapecanbeobtainedfromaquasi-static proached. The integral rate of change of mass M of the
argumentwhentheexperimentallyobservedinterfaceve- gas bubble is defined as
locity v is smallerthanthe characteristichydrodynamic
i
velocity σ/η, where η is the shear viscosity. A numerical dM/dt= n(~x)d~x∼(T −T)a, (7)
Z c
estimateshowsthatthequasi-staticapproximationholds
for the images (a-f) in Fig. 6, in which the spreading is where the integration is performed over the total gas-
observed. The quasi-static approximation does not ap- liquid interface area. On the other hand,
peartoholdforthe quenchexperiments (Fig.10), where
the interface moves rapidly. dM/dt=d/dt(Vφρ ), (8)
G
9
whereV isthecellvolume,andφ=0.5isassumed. Near Eq.(10)canbedeterminedfromEq.(11), wherethe up-
the critical point, the co-existence curve has the form per integration limit can be replaced by infinity without
ρ =ρ −∆ρ/2,where∆ρ∼(T −T)β withtheuniversal any loss of accuracy. Although the expression Eq. (10)
G c c
exponentβ =0.325,so thatdM/dt∼(T −T)β−1dT/dt for the vapor recoil pressure is not rigorous, it contains
c
as T → T according to Eq. (8). Thus Eq. (7) results themainphysicalfeaturesofthesolutionoftheheatcon-
c
in a =β−1 and the curvature change due to the vapor duction problem: a weak divergence at the contact line
recoil scales as and a rapid decay away from it. It is shown in [25] that
the rigorous numerical solutions obtained far from the
Pr/σ ∼(Tc−T)3β−2−2ν, (9) critical point follow this behavior.
TheresultofthiscalculationisshowninFig.12. Since
whereEq.(3)andthescalingrelationshipσ ∼(T −T)2ν
c
(ν =0.63)wereemployed. Becausethiscriticalexponent N=0.01 N=0.5
(3β −2−2ν ≈ −2.3) is very large, it should manifest
itself even far from the critical point in agreement with
the experiments. In summary, as T → T , the vapor
c
mass growthfollows the growthof its density (the vapor
volume remains constant), so that the diverging vapor
productionnearthecriticalpointdrivesadivergingrecoil
¹¹¹``‰˚(cid:28)fiÛ´````¿L_`,˜``````,`(`Û+&```^•T`z•T`à•T`¹¹ñ•T`¹¹¹¹¹¹¹¹``````````(`»`Û´òj_`¿L_`````Wˆ(cid:31)˜`¿````````````````˚`´`˚`````l•T`````Ł```````````````````````````````````X`h`Û´òj_`¿L_`````````````−`L`Û´òj_`¿L_`````````````Ÿ`–`−`Û+Ł`````````````````
force.
This curvature change has a striking effect on the N=1.0 N=1.5
bubble shape because it is not homogeneously dis-
tributed along the bubble interface. Since the evapora-
tion is strongest near the copper heating wall where the
strongest temperature gradients form, both P and c in-
r
creasestronglynearthis wall,i.e. nearthe triple contact
line. Note that c is proportionalto the secondderivative
of the bubble shape function, i.e. to the first derivative
of the bubble slope. If c is large, then the slope of the
FIG. 12: Calculated bubble shape for different values of
bubble contour changes sharply when moving along the
the non-dimensional strength of vapor recoil N that goes to
bubble contour towards the contact line, see [5, 23, 24]
infinity at the critical point. Note that the actual contact
for more details. Because the interface slope changes so
angle is zero for all thecurves.
abruptly near the contact line, the apparent contact an-
gle should be much larger than its actual value.
Eq. (9) implies
Because c is proportional to the second derivative of
the bubble shape function, Eq. (5) is a differential equa- N ∼(T −T)−2.3 →∞ (12)
c
tion with the boundary condition given by the actual
as T → T , the N increase mimics the approach to
contact angle [23]. This actual contact angle defines the c
the criticalpoint andqualitatively explains the observed
first derivative (the slope) of the bubble shape function
shape of the vapor bubble (see Fig. 6). The increase of
at the solid wall. It is also specified by the interfacial
the apparent contact angle and of the gas-solid contact
tension balance andmust be zero near the criticalpoint.
area A can be seen in Fig. 12. Note that such a cal-
Thisconditionofthezerocontactanglegivesaboundary cg
culation is not able to predict the wavy interface shapes
conditionforEq.(5). In orderto illustrate a possibleso-
like those in Fig. 6g or Fig. 8f-h, because these images
lution of Eq. (5), we solved it using the same expression
correspondeither to T >T (images g and h of the both
for P (l) as in [23] c
r
figures) or to the close vicinity of T where σ < v η, see
c i
P (l)∝−N log(l/L) exp{(−[l/(0.1L)]2}, (10) the discussion of the validity of the quasi-static approxi-
r
mation earlier in this section.
where l ∈ [0,L], L being a length of the bubble half-
contour with l = 0 at the solid wall. We use a non-
VII. SPREADING GAS AND THE BOILING
dimensionalparameterN tomeasuretheinfluenceofthe
CRISIS
vapor recoil force relative to the surface tension. It is
defined as
Averysimilarbubblespreadingwasobservedfarfrom
L T during boiling at large heat flux [26, 27]. When the
1 c
N = P (l)dl. (11) heating to a surface is increased past a Critical Heat
σ Z r
Flux (CHF) there is a sudden transition to “film” boil-
0
ing,wherethe heaterbecomescoveredwithgasandmay
wheretheintegrationisperformedoverthedropcontour burnout [14]. This “boiling crisis” is an important prac-
intheimageplane. Thenumericalcoefficient(see[23])in tical problem in many industries where large heat fluxes
10
arefrequentlyused. Weinterpret[23,25]the boilingcri- apparent contact angle that appeared despite the zero
sis to be similar to the gas spreading shown here. The actual contact angle with the solid. The spreading gas
main difference is that the large value of N is made by is a phenomenon that can occur in a sealed heated fluid
a large vapor production that can be achieved during cell only when the bubble is pressed against the heating
strong overheating rather than by the critical effects. wall. The 3D numerical calculation of the equilibrium
It is well-documented from experiments [14] that the bubble shape showed that the slightly tilted windows of
CHF decreases rapidly when the fluid pressure p ap- theexperimentalcellpressedthebubbleagainstthecop-
proaches the critical pressure p , i.e., when T → T in per side-wall. Weightless conditions are needed in the
c c
our constant volume system. Previously, this tendency near-critical region in order to observe this phenomenon
hasnotbeenwellunderstood. Thedivergenceofthefac- whenthesurfacetensionissmallandabubble-likeshape
tor N, discussed above, helps to understand it. We first persists. Thesamephenomenoncanbeobservedfarfrom
note that the evaporation rate n scales as the applied thecriticalpointduringboilingathighheatfluxeswhere
heat flux q and N ∼ q2, where Eqs. (3) and (11) are itisknownasthe “boilingcrisis”. While the gasspreads
used. By assuming that the boiling crisis (q = q ) veryquicklyduringthe boilingcrisisfarfromthe critical
CHF
begins when N attains its critical value N ∼ 1 (see point, the near-critical region allows a very slow spread-
CHF
[23]), one finds that ing gas to be observed in great detail.
q ∼(T −T)1+ν−3β/2 ∼(T −T)1.1 (13) We explain this phenomenon as induced by the vapor
CHF c c recoil force that changes the shape of the vapor-liquid
interface near the triple contact line. Our preliminary
from Eq. (12). The same exponent is also valid for the
calculations of the gas-liquid interface shape are quali-
pressure scaling,
tatively consistent with the experimental images. The
q ∼(p −p)1.1. (14) scaling analysis gives the critical exponent for the criti-
CHF c
calheatfluxdecreasenearthecriticalpointandexplains
Eq. (14) explains the observed tendency q → 0 as the increase of the vapor recoil effect near the critical
CHF
p→p . point. Webelievethatthereismuchtobelearnedabout
c
Although the strict requirements on temperature sta- theboilingcrisisinthenear-criticalregionandhopethat
bility and the necessity of weightlessness lead to experi- these experiments inspire more investigations.
mental difficulties to study the boiling crisis in the near-
critical region, they also present some important advan-
tages. Only a very small heating rate (heat flux) is
needed to reach the boiling crisis because q is very
CHF
small. At such low heat fluxes, the bubble growth is ex-
Acknowledgments
tremelyslowdue tothe criticalslowing-down. Inourex-
periments wewereable toobservethe spreadinggas(i.e.
the dry-out that leads to the boiling crisis, see Fig. 6) This work was supported by CNES and NASA Grant
during 45min! Such experiments notonly permit anex- NAG3-1906. A part of this work was made during the
cellent time resolution, but also allow the complicating stayofV.N.atthe UNO andhe wouldliketo thank the
effects of rapid fluid motion to be avoided. Department of Physics of the UNO for their hospitality.
WethankalloftheAliceIIteamandeveryoneinvolvedin
theMirmissions. WeespeciallythankJ.F.Zwilling,and
VIII. CONCLUSIONS the french cosmonauts Claudie Andr´e-Deshays, L´eopold
Eyharts, and Jean-Pierre Haigner´e. We thank Kenneth
In our experiments we observed a gas bubble spread- Brakkefor creating the Surface Evolversoftware and for
◦
ing over a solid wall and a large value (> 90 ) of the making it available for the scientific community.
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