énoncé

Ecoulements multiphasiques
TD8: Ébullition
UMPC. NSF16. 2009-2010
Jérôme Hoepffner & Arnaud Antkowiak
FROID
CHAUD
Ex2: Équation de Rayleigh-Plesset
On cherche à décrire la croissance (ou même les
oscillations) d’une bulle sphérique de rayon R(t). On suppose
que les mouvements du liquide environnant sont purement
radiaux, et on suppose que la vitesse du liquide au voisinage
de l’interface peut être assimilée à celle de l’interface (on
n’interdit pas pour autant les transferts de masse).
On rappelle les équations de conservation de la masse et de
la quantité de mouvement dans le liquide :
ρ�
�
∂u�
∂u�
+ u�
∂t
∂r
�
1 ∂ � 2 �
r u� = 0
r2 ∂r
=−
∂p�
+µ
∂r
�
∂ 2 u�
2 ∂u�
u�
+
−2 2
∂r2
r ∂r
r
�
où la vitesse u_l décroît vers 0 à l’infini, et la pression p_l
tend vers p_∞ dans cette même limite.
1/ Donner la vitesse du liquide en r=R en fonction de R, ainsi
que l’écart de pression p_v-p_l entre la vapeur et le liquide.
On négligera ici le phénomène de recul de vapeur. La valeur
de la contrainte visqueuse radiale est :
τrr = 2µ
∂u�
∂r
2/ Donner la structure du champ de vitesse, à une constante
multiplicative A(t) près. Que vaut A(t) en fonction de R ?
Ex1: Thermocapillarité.
Expliquer pourquoi une goutte (une bulle) se déplace
toujours vers les régions chaudes.
3/ Injecter cette expression dans l’équation de conservation
de la quantité de mouvement.
4/ En déduire la valeur de la pression p_l en un point r du
liquide, en fonction de p_∞, rho_l, r et R.
5/ Montrer que la pression dans la bulle est donnée par
l’équation suivante, dite de Rayleigh-Plesset :
�
�
Ṙ
3
γ
pv (R) − p∞ = ρ� RR̈ + Ṙ2 + 2 + 4µ
2
R
R
the shape of a Leidenfrost drop, we show that its size cannot
escribe the characteristics of the vapor layer on which it floats.
op size, and how both vary with time, as evaporation takes place.
r the lifetime of these drops. © 2003 American Institute of
1$
Ex4: Dégazage catastrophique : un volcan à la maison
II. DROPS SHAPES AND STABILITY
n a hot solid, of
of the liquid, the
olid temperature
drop is not anys above its own
of the film, the
oplet of water on
oat for more than
contact between
on of bubbles, so
evaporates. Such
after the name of
he phenomenon
These levitating drops can be considered as nonwetting.
We call contact the region where the drop interface is parallel to the solid surface. If the drop radius R is smaller than
the capillary length a (a! !& / ' g, denoting the liquid surface tension and density as & and '", the drop is nearly
spherical, except at the bottom where it is flattened. In this
limit, Mahadevan and Pomeau showed that the size ( of the
contact is given by a balance between gravity and surface
tension.9 Denoting ) as the lowering of the center of mass,
this balance can dimensionally be written: & ) * ' gR 3 . Together with the geometric Hertz relation (* !) R, this
yields9
(*R 2 /a.
e lifetime % of a
on a duralumin
T. Below 100 °C,
°C. At this point,
e surface. When
the droplet lifeactor 500, which
insulating vapor
mum defines the
ratures, % slowly
40 s at 350 °C.
f the Leidenfrost
–4
It depends on
quid6 !which can
n on the way the
er aspects of the
pe of the drops,
stics of the vapor
Lorsqu’on plonge une poignée de Mentos dans une bouteille de CocaCola, on observe une éruption impressionnante (faire la manip chez soi ou chez un ami- ou la regarder sur youtube).
Immersion d’une boule
de cuivre surchauffée
dans de l’eau.
1) Proposer une explication du phénomène.
2) L’expérience marche beaucoup mieux avec les Mentos parfumés à la
menthe qu’avec n’importe quel autre parfum. Pourquoi ?
!1"
This relation was checked experimentally with nonwetting
liquid marbles.10 Drops larger than the capillary length form
Phys. Fluids, Vol. 15, No. 6, June 2003
FIG. 1. Lifetime % of a millimetric water droplet of radius R!1 mm, as a
function of the temperature T of the Duralumin plate on which it is deposited.
FIG. 2. Large water droplet deposited on a silicon surface at 200 °C.
puddles flattened by gravity, as it can be observed in Fig. 2,
and the contact becomes of the order of the puddle radius
(!"R).
distribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
The thickness h of this puddle is given by balancing the
surface tension #2$, per unit length, taking into account the
1) Décrire en un paragraphe l’expérience du film.
upper and the lower interface% with the hydrostatic force
( & gh 2 /2, also written per unit length%. This yields
1632
Leidenfrost drops
1633
FIG. 4. Largest possible radius R c of a Leidenfrost puddle without bubbling,
as a function of its height h. The data are obtained with different fluids *#!%
liquid nitrogen, #!% acetone, #&% ethanol, #"% water–ethanol mixtures of
various compositions, ##% water+ deposited on a duralumin plate at T
!300 °C. For drops of radius R larger than R c , bubbles such as photographed in Fig. 3 are observed.
© 2003 American Institute of Physics
Ex3: L’effet Leidenfrost.
2) Dans une seconde expérience, on dépose une grosse goutte d’eau sur une plaque
h!2a.
#2%
chauffée (fig. 2) et on reporte le ‘temps de vie’ de la goutte en fonction de la température
The temperature inside the water drop was measured and
(fig. 1). Commenter.
found to be constant and equal to 99"1 °C. This implies a
3) Ces deux phénomènes sont-ils liés ?
density &!960 kg/m3 and a surface tension $!59 mN/m,
and
thusinterprétation.
a capillary length a!2.5 mm. For Leidenfrost
Proposer une
puddles such as in Fig. 2 or larger, we measured
h!5.1 mm, in good agreement with Eq. #2%.
Up to now, the shape of these static drops was found to
be characteristic of a situation of nonwetting, close to what
can be obtained on superhydrophobic solids. But as an original property, it is observed that the radius R #and thus the
volume 2 ' R 2 a) of a Leidenfrost drop is bounded, by a
value of the order of 1 cm #corresponding to about 1 cm3 for
the volume%. If it is larger, a bubble of vapor #or possibly
several ones, for very large puddles% rises at the center of the
drop and bursts when reaching the upper interface, as reported in Fig. 3.
z! ( (1#cos kr), with ( k$1 and r the radial coordinate. The
smallest cost in surface energy being achieved for a single
bump centered in r!0, we choose a wave vector k! ' /R.
Considering capillary and gravitational effects, the difference
of pressure ) P between the center and the edge of the drop
for two points A and B at the same level is ) P! P A % P B
!2 & g ( * 1%3(ak) 2 /2+ . The perturbation increases for positive values of ) P and is stabilized for negative ones. The
threshold of the instability is thus for ) P!0, which leads to
Éruption de Coca-Cola
a critical radius R c !3.84a. Using Eq. #2%, we can express
this quantity as a function of the puddle height:
R c !1.92h.
#3%
Figure 4 shows the largest radius R c observed without
bubbles as a function of the puddle height, also measured.
Different liquids were used in order to vary the capillary
length, and thus the height. In particular, the thinner puddles
were obtained with liquid nitrogen and oxygen. The variation
is indeed linear, and the slope found to be 2, in close agreement with Eq. #3%.
III. THE VAPOR LAYER: STATIONARY STATES
Influence du soda sur
l’éruption