7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Axisymetric film evaporation : an investigation of the transfers at the triple line trough numerical
modeling and experimental measurement
LoTc TACHON, Stephan GUIGNARD and Lounes TADRIST
AIX MARSEILLE UNIVERSITY (U1,U2)POLYTECH MARSEILLELaboratoire IUSTI CNRS
UMR 6595
guignard@polytech.univmrs.fr
Keywords: Film evaporation, phase change, Numerical modelling, Triple line, Laser sounding
Abstract
The aim of the present study is to understand the physical phenomenon rose in the vicinity of the triple line thanks to the
comparison between numerical simulation and experimental investigation. However, this paper presents only the experimental
method. As no thermal boundary condition of the experiment is available, the author did not find relevant to present the
numerical model and its results.
The experimental method's purpose is to compute the shape of the interface in the vicinity of a contact line after a film breakup.
The principle of the experiment consists in a controlled volume of evaporating liquid (HFE7100) deposited in a millimetric
cylindrical vertical well. During the evaporation process, the liquidgas interface takes a toroidal shape delimiting an
axisymmetric film with the well bottom and side walls. At the beginning, the evaporation process occurs without triple line on
the bottom. Then a circular triple line appears on the bottom. The meniscus interface shape and the triple line location are
instantaneously measured by laser sheet sounding from underneath and numerical inversion: This technique analyses the
variation of the light intensity along a laser sheet due to its refraction trough consecutive interfaces (solidliquidvapor).The
laser sheet impacts intensity on a perpendicular screen is inverted. The inversion result is the shape of the interface and the
position of the triple line during the evaporation process.
INTRODUCTION
1.1 General ideas
Heat transfer and fluid flow mechanisms in the vicinity
of the triple line during the evaporation and boiling process
is a recurrent research subject since the early seventies.
A better understanding of the transfer in the vicinity of the
triple line allows to increase dramatically the heat and mass
fluxes densities and thus to decrease drastically heat
exchanger size.
Many studies exhibit the influence of the triple line velocity,
contact angle and film thickness on the heat and mass
transfer at the triple line.
Many measurement techniques of the dynamic interface
shape exist for drop and meniscus on flat surface. Recently,
Chau (2i" 1') has made a review of the existing techniques
for drop shape. He summarizes the measurement techniques
on flat plates in some categories.
The drop profile technique consists in a direct measurement
of the angle by viewing the drop profile. Firstly developed
by Bigelow et al. (1946) who do it with a telescope
goniometer this technique has been considerably improved
with the CCD camera. Hunter (2001) showed that for
contact angle greater than 200 can be determinate to an
accuracy of 20.
The drop dimensions methods consists of measuring
curvature radius by a photograph of the drop that determine
the entire profile of an axially symmetrical drop starting
with the Laplace equation describing the shape of fluid
interface. Fisher (1979) developed a relationship between
contact angle, radius, and volume in order to measure small
contact angles. Limits of this method are sawn when the
shape of the drop is strongly influenced by gravity.
An improvement of this method is the axisymmetric drop
shape analysisprofile method (ADSAP). Firstly developed
by Rotenberg (1983) and then improved by Spelt (1987)
and Cheng (1990) this technique consist to find the
theoritical profile based on the Laplace equation that best
matches the drop profile extracted from an image of the real
drop, from which the surface tension, contact angle, drop
volume, and surface area can be computed. Del Rio (1998)
reported that ADSAP is a more reliable and less sensitive
to surface heterogeneity and /or roughness and therefore
more suitable for mineral surfaces. RodriguezValverde
(2' '2) reported that the captivebubble method in
conjunction with the ADSAP technique allows
comfortable, automatic and reproducible measurements of
contact angle on porous stones.
To sum up, this kind of method provides good results for
contact angle greater than 200 but a side view on the
interface is necessary.
Considering widly spreaded small volume of liquids (thin
films, drops or meniscus), a good accuracy is obtained by
image analyzing interferometry (liquid wedge fringes)
Wayner ( 2002) et Panchamgam (2" 1',), but this method
only holds with small contact angles.
Hegseth J (2005) has developed an indirect optical
technique that consist of observing the deformation of a
projected grid due to the presence of a gasliquide curvated
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
interface. This technique compute the interface shape of the
film and the apparent contact angle.
Other indirect methods have been developed using laser
reflection and refraction. Allain (1985) proposed a method
that consists in illuminating the drop at a normal incident to
the solid with a large laser beam. The beam is reflected by
the drop into a light cone, whose angle is a function of the
mean contact angle around the perimeter. Since, several
method using laser exist to probe interface shape and
contact angle.
Light absorption techniques have been used by Goodwin
(1991). Johnson (1997) presented a fluorescent imaging
system for global measurement of liquid film thickness and
dynamic contact angle in free surface flows. In an another
method applied by Buguin (1999) to axisymmetric
dcl'. eiilin, a laser beam is deflected by the interface and its
deviation is followed versus time while the liquid dewets. A
similar technique using a laser sheet has been developed by
Rio ('" 14). Its exploit the deflection of the sheet after its
refraction across liquidvapor interface to probe the contact
angle at the triple line of a dry patch.
The presented measurement method is also based on the
light refraction across the liquidvapor interface. We show
in this paper the capability of our method to probe
instantanously the contact angle value and the interface
shape of an evaporative axisymmetric film.
1.2 Axisymmetric film evaporation problem
To illustrate our method, we here apply it to an
axisymmetric film problem. This problem consists to study
the evaporation that occurs in a cylindrical well. A
controlled volume of evaporating liquid (HFE 7100) is
deposited in a millimetric cylindrical vertical well. During
the evaporation process, the liquidgas interface takes a
toroidal shape delimiting an axisymetric film with the well
bottom and side wall. At the beginning, the evaporation
process occurs without any triple line on the bottom, the
interface is dug by the center. Then a circular triple line
appears in the middle on the bottom, delimiting a dry patch.
e lWell t
Interface i
t+d
i t t+2dt
Figure 1Axisymetric film evaporation problem
MATERIAL AND METHODS
2.1 Experimental setup
A 7 mm inner diameter cylindrical well is constituted by a
1mm thick slice (Teflon or inox) deposited on a 1 mm thick
glass plate. This well is filled with a controlled volume of
low boiling temperature fluid. A laser diode (X=635 nm) is
used to generate triangular laser sheets with a 30 opening
angle. Laser sheets illuminate the plate from underneath and
after refraction across the interface, they impact a screen
located at roughly 15 cm above the glass plate. A high
resolution (2848x4288) CCD camera is
screen picture.
used to catch the
Camera
Screen
Refracted laser
sheets
Evaporating well
Laser diode
Figure 2 Experimental setup illustration
2.2 Screen visualization and interpretation
Figure 3 shows rays produced by three laser sheets
impacts on the screen.
During a first step a dilatation along the "x" axis and a
curvature of the lateral rays is observed, that is due to the
interface increasing concavity. In this case, the fluid volume
is a plano convex lens.
When the triple line appears, leaving a dry patch in the
middle on the well, non distorted rays appear in the center
of the figure. They are the dry patch imprint on the screen.
When looking into detail on the screen, the boundary
between the distorted and non distorted zone is a narrow
circular shadow area ("C" on Figure 3). Two interpretations
of this observation are detailed in 2.4 section. In both cases,
this circle is a discontinuity imprint: the triple line image on
the screen.
When the liquid has evaporated, we obtain straight
undistorted rays on the screen.
x No triple
line
Triple
2 line
07 line
I
& 4 \ No fluid
Figure 3 Different configuration of Laser sheets on the
screen
The screen intensity is then evaluated with a calibrated
photographic camera. The summation of the light intensity
on the central laser sheet is done to evaluate de variation of
the light signal. This leads to exploitable values that allow
us to compute triple line velocity, contact angle value and
interface shape or film thickness.
SIntensity value
;4 ; NNo triple
line
e ~ Triple line
Ii position
^ ^^^   V\\^).\ '
Triple
line
n Triple
s\\\ A\ line
P sition on the screen No fluid
Figure 4 Different light intensity profile of the central laser
sheet on the screen
As we can see on the Figure 4, the triple line position
and the width "AB" can be automatically detected on the
screen. So after some geometrical transformation we can
compute the real position of the triple line inside the well
and the value of the contact angle. To compute the shape of
the interface, a polynomial interpolation of this signal is
done and inverted by the methodology presented below.
2.3 Triple line position computation
With the light intensity profile the triple line image on
the screen is detected. A simple geometrical description and
the Snell Descartes lead us to a simple formulation of the
real triple line position inside the well.
Position on the
Screen where
L intensity collapsing
c / RLT:Real radius of
triple line position
S e: Glass blade thickness
b !
Laser light source
Figure 5 Direction of the closest beam of the triple line
Considering the notation of the Figure 5, the formulation
of the triple line position is given by the following
expression.
L b+
RLT= L s (1)
e
b+ +ce
glass
2.4 Interface shape reconstruction method
The principle of our method for interface shape
reconstruction is based on a direct inversion of the light
signal of the central ray observed on the Figure 3. The key
point is to understand how this intensity variation is
correlated to the interface geometrical characteristics.
The first step (describe in section 2.4.1) for evaluating
this correlation is to obtain a function of this signal. To do it,
we make a polynomial interpolation of the signal
summation on the width of the central laser sheet of the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
matrix CCD response. In fact it consists in interpolating the
signal drawn on Figure 4.
In a second step (section 2.4.2) a first numerical integration
of this interpolation signal is done by considering the light
energy conservation under geometrical optics assumption
Figure 7. This leads to the value of the angle of each beam
that composes the laser sheet with the vertical axe along the
laser sheet image on the screen.
In the third step (section 2.4.3) we use the Snell Descartes
law with some geometrical formulations on this angle to
compute the tangent of the interface along the diameter of
the well. After the computation of the tangent a second
numerical integration of it along the diameter gives the film
thickness and thus the interface shape.
As we can see on the Figure 4 at the triple line, the light
intensity collapses. This is due to the presence of the triple
line. To explain the suddenly decrease and increase of the
light intensity signal we will explore two different
configurations.
The first will consider that the contact angle is null (slope
continuity) and the decrease and increase of the intensity is
due to strong variation of the curvature in the micro region
in the vicinity of the triple line. So in this case, the part of
the signal (labeled "AB" on Figure 4) is a part of the signal
that has to be integrated for the interface shape
reconstruction.
In the second configuration, the contact angle is assumed
not to be null and thus, a slow discontinuity occurs at the
triple line implying a strong divergence of the light rays
propagating on different sides of the contact line (Figure 6).
This divergence produces the shadow circle on the screen
(C on Figure 3) In this case, the part of the signal "AB" is
not taken into account for the interface reconstruction but is
the distance value AB is used to compute the contact angle
(used for the computation of one of the integration constant
of the interface reconstruction).
2.4.1 Interpolation of the light intensity signal
 Polynomial Interpolation
 Experimental
N I )
No shadow area => Part between A and B interpolated
sheet
Microscopic contact angle NULL
Glass
Laser
sheet
Microscopic contact not NULL
Figure 6 Interpolation of light intensity signal
The problem is axisymetric so the study of the intensity
signal drawn in Figure 4 is concentrated of an half of it.
This step consists firstly to detect points labeled A and B on
the Figure 6. After, the type of interpolation will depend of
the theory about the contact angle.
If we consider a null contact angle the part of the
signal between the point A and B is interpolated by a high
order polynomial function (order 7) and the part from the
point B and is interpolated by a second order polynomial
function. So the collapse of the intensity is interpolated.
If we consider that the contact angle is not null, the
part of the signal between the point A and B is not
interpolated. This is the image on the screen of the shadow
area represented on the Figure 6. So the interpolation of the
intensity profile start from the point B and is done by a
second order polynomial function.
2.4.2 I,,i. ii,.,, i of the light intensity signal
The aim of this step is to compute the incident angle
with the vertical axe of each beam that constituted the laser
sheet when it impacts the screen.
1(x)
0 (X)
[ rX
dX Screen
0, +
x
I(x)=I,
Figure 7 Electromagnetic light energy conservation
To calculate this angle labeled Oscreen on the Figure 7
the energy conservation law is used. So the following
formulation is correct if all the electromagnetic light energy
that comes from the laser diode impact the screen. It follow
then that we consider that the transmittivity of the fluid is
close to 1. The energy loss due to multi reflection on fluid
interfaces is also considered negligible. As it is represented
on the Figure 7 the energy conservation law is defined by
the following formulation:
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Screen. The relations are the following. The name of the
variable is labeled on the Figure 7:
x = X H tan ....en
dX = H (tan (0 ....n + dO, ...J) tan (0 .... ))+ dx
The injection of the equation ( 2 ) in the ( 4 ) give:
d(tanOcrn) 1 I (X
SdO, I I dX
dO H H Io
screen
The left hand side can be analytically integrated. The part
on the right is numerically integrated between the triple line
position on the screen labeled "Xo" and the end of the
intensity signal on the screen. So finally the equation ( 5 ) is:
tan O. tan 0 = 1o )du
H x, o
After integration of the left hand side a constant tanOo
appears. This is the angle between the vertical and the first
beam of the laser sheet that see the fluid in presence of triple
line after the refraction of itself through the interface. It
follows then that the value of this constant will depend of
the theory about the contact angle
2.4.3 Interface shape computation
Considering the equation ( 3 ) and ( 5 ), and
according to the Figure 8 we can express the value of the
tangent of the interface by the following formulation:
tan 0ront =nHFE7100 i sin screen (7
cos n_ cos O,
COS screen HFE7100 COS laser
By integration of this formulation we obtain the
formulation of the shape of the interface:
z(x)= J ta Font,,du
RLT
In the situation of the contact angle is considered different
of zero, the value of it is given by:
contact ( 100 1
H (nHFE7100 1)
Idx = I(X)dX
It means that all the electromagnetic light energy that cross
the interface through the element labeled "dx" on the Figure
7 impact the screen through the element labeled "dX" on the
Figure 7.
A geometrical approach gives us the relation between
the position on the screen and the position on the radius
inside the well and between the element "dx", "dX" and
HFE 7100
Glass
Figure 8 Refraction of a beam at the interface
2.5 Validation of the method
To check our optical method, we have applied it on
planoconcave lens with a known geometry and refractive
index. This corresponds of a solid thin film. We apply the
presented method to compute the shape of the interface of
the lens with the inversion of the central laser beam sheet
figure refraction. We compare values of the theoretical
shape and the computed shape. We also compare value of
the theoretical curvature with the computed curvature.
Diameter (mm) 12
Radius of curvature (mm) 14.12
Refraction indice 1.785
Tableau 1 Lens characteristics
relative error of the computed
curvature radius along radius
015
0.1
1 0 0
o.05
0 1 2 3
relative error of the computed
s 10 haped along the radius
410
2 2\
22
x(m) x10 x(m) x 10
Figure 9 Relative error of the computed curvature radius
the interface shape and the computed shape
The relative error on the curvature radius is less than 15
percent and it implies and errors on the computed shape
less than 1 percent.
RESULTS AND DISCUSSION
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
p kg.m3 2530 7850 2200
E = effisivity 1480 9500 740
Tableau 2 Physicals characteristics of experimented well
Triple line position
0 20 40 60
time (s)
Figure 10 Triple line positions versus the time
Triple line velocity on the radius
0,0003
S0,00025
E 0,0002
Z, 0,00015
Z o,ooais
0,0001 Inox
o COC"s Teflon
0 0,001 0,002 0,003 0,004
Distance from de center (m)
Figure 11 Triple line velocities on the radius
Figure 10 shows that the evaporating process is slightly
faster in the iron case. The Figure 11 confirms this. The
velocity of the triple line along the radius is more important
in case of iron than in case of Teflon.
3.2 Interface shape computation diff'ivreii between the
both models of contact angle
With ours triangular laser sheets, this shape computation
method does not give us the shape in the vicinity of the
S second triple, the triple line on the well. To solve this
of problem parallel laser sheets can be used. In the presented
results, to probe the shape until the wall of the well we
extend the computed shape by a circle with the same radius
of the end of the computed shape. This part of the shape is
in this case equal to 1mm of the end of the radius. The
Laplace equation can also be used to probe this part of the
shape. Anyway we don't have the contact angle of the
contact line on the well.
3.1 Triple line position and velocity results
We have applied this method to track the triple line
during the time in two different kind of well, one in iron and
on other in Teflon. Both well have the same dimensions so
the same volume. The experiment is repeatable so we were
in the same condition for both experiments. During the
evaporating process we take one picture every second of the
refracted laser sheet picture.
Glass Iron Teflon
C J.K'.kg' 720 450 1000
" W.m K 1.2 26 0.25
Interface shape with NULL
contact angle at different time step
5x 10 Circular shape
251 Circular shape
A
.
K
Interface shape with a NOT NULL
contact angle at different time step
25[ 10 Circular shape
2 ,
1.5
x:
2 3 4 10
(m) x 10'
3 4
x 10s
Figure 12 Interface shape at different time step
On the Figure 12, we represent the interface shape for
different time step for both theories with or without a null
Inox
Teflon
contact angle. A 4 seconds time step separates each curve of
the interface shape.
The first conclusion is that the volume of liquid is not the
same. The computed interface shape lead to a bigger
volume of fluid.
CONCLUSION AND PERSPECTIVES
In this paper we present an original method to compute
the instantaneous shape of the interface in the vicinity of a
contact line. The validation of this technique to probe
interface shape on optical lens with well known geometry
has given very encouraging results with a very low error.
This method also computes the apparent contact angle with
a good precision even for contact angle lower than 20.
To discriminate one of the hypothesis about the contact
angle, we will test it on a well known geometry film with a
not null contact angle.
In addition to this study, the authors have developed a
numerical model dedicated to the numerical simulation of
two phase flows with phase change. This model is based on
a finite element axisymmetric formulation of the energy,
momentum and mass conservation equations and a sharp
interface method: At each step of the computation, the
interface is a grid line moving through the computation grid
that is iteratively adapted to the interface shape. The
conservation problem being solved successively, the heat
flux discontinuity at the interface gives, according to the
massenergy conservation law, the velocity field
discontinuity at the interface which is one of the boundary
condition of the momentum conservation equation solver.
A further work at this time consist in performing the
experiment of the present study with I.R thermal analysis
in order to use the temperature results as inputs for the
numerical model. Then only, a comparison between
numerical and experimental results seems interesting.
ACKNOWLEDGMENTS
The authors gratefully appreciate the support of the
National Agency for Research (ANR) with project
ANR06BLAN011901 "INTENSIFILM".
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ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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