7'" International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Experimental Qualification of an Eulerian Interfacial Area Transport Equation for
Modelling Dispersed Phase Behaviour in liquidLiquid Extraction devices
T. Randriamanantena: B. Cariteaut J.P. Magnaudi J. Duhamet: A. MonavonS
CEA,DEN,DTEC/SGCS/Laboratoire de Genie Chimique et d'Instrumentation,
BagnolssurCbze, Cedex 30207, France
t CEA,DEN,DM2S/SFME/Laboratoire d'Etude Experimentale des Fluides, GifsurYvette, 91191, France
SCEA,DEN,DM2S/SFME/Laboratoire d'Etude des Transferts et de Mecanique des Fluides,
GifsurYvette, 91191, France
5 UMR7190, Institut Jean le Rond d'Alembert, Universite Pierre et Marie CURIE,
4, Place Jussieu, PARIS Cedex 05, 75252, France
tojo.randriamanentena@cea.fr and benjamin.cariteau.fr
Keywords: Eulerian simulations, emulsion model, interfacial area density, drop fragmentation
Abstract
In the field of liquidliquid extraction, one of the most important parameters affecting extraction performance in
industrialscale devices is the interfacial area. An Eulerian emulsion model directly providing this interfacial area is
being developed on the basis of ensemble averaging, Lhuillier (2004).
The model validation involves two main steps. The first based on published experimental results, seeks to verify the
reproduction of single drop behaviour such in the case of low anisotropy or drop breakup. The second step validates
the behaviour and the gradual change of an entire drop population passing through a pulsed column. This second step
requires the acquisition of experimental data still rare in the literature. In accordance with its Eulerian philosophy, the
guideline is to acquire mean data on the emulsion at chosen locations in the device and at selected moments in the
pulsation cycle. Flow and emulsion characterization methods are used to obtain the data in a device specially designed
for this purpose. The flow is characterized using a particle imaging velocimetry method also possible in twophase
flow thanks to an optical index matching operation, Budwig R. (1994). In addition to velocity measurements for both
phases, this technique can also be used to qualify the influence of drops on the continuous phase flow. Drop populations
are characterized using image processing methods. Depending on the location in the device, images are obtained with
backlighting or by laser induced fluorescence. A Matlab program based on the HOUGH transform algorithm is used
to perform operations ranging from image binarization to plotting drop populations histograms. Histograms are used
to determine the mean interfacial area and, after solving an inverse problem, the model coefficients for the breakup
and coalescence terms.
In this model, the interfacial area is considered as a scalar quantity transported by the flow and evolving in accordance
with it. For each phase, a continuity equation is solved with an incompressible condition. Moreover, we solve
a momentum conservation equation of the emulsion associated with a droplet drift velocity model. Interfacial
variables characterizing emulsions appear in interfacial balance equations such as the interfacial area evolution and
the interfacial anisotropy equation.
Nomenclature Py,. fragmentation efficiency
Vo, coalescence velocity
Roman spubols
B anisotropy tensor
g gravitational constant (ms ) A interfacial area density(Inl)
Uk phase k velocity (m 1l) 0ssr(N 2
Pcon coalescence efficiency
7'" International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
According to the eulerian philosophy of the model,
mainly mean values are required. Contrary to a la
grangian approach describing each drop class with a
set of equations, an eulerian model emphasizes on the
behavior of an entire drop population. Basicly, the
dispersed phase is considered as a "pseudocontinuous
phase" whose characteristics depend on its volume frac
tion. In this sense we only focus on the influence of each
elementary phenomenon on the final level of interfacial
area density. Among those phenomena, in our operating
conditions, drop breakup plays a crucial role in addition
to coalescence. Both phenomena produce an irreversible
change in the interfacial area density. The adopted strat
egy aims at quantifying the effect of a given flow on the
change of volumic interfacial area of a population be
forehand characterized. To reach this aim an apparatus
based on the geometry of a pulsed column was espe
cially conceived making it possible to use optical tech
niques to characterize the emulsion.
The light system
Depending on the kind of measurement needed, either
a laser sheet or a backlight is used to illuminate the sys
tem. Hold up measurements, for instance, require the
drops to be illuminated by the laser sheet. Once illumi
nated with this kind of light, the drops fluoresce in their
section crossed by the laser becoming at the same time
the only visible phase in the images. On the contrary,
in the case of interfacial area density measurements, we
only use a diffuse backlight to see the totality of the vol
ume. In that case the fluorescence phenomena is not
used, drops are detected thanks to their color. As can
be seen on Fig. 1 the droplets are illuminated with an
UV laser sheet crossing the column. The laser sheet is
created with a semicylindrical lens placed at the exit of
the laser beam.
Description of the column
Taking into account the needs of the strategy and the
model's requirements, the apparatus was designed with
some specificities:
to reduce optical image distorsions, the main cyhin
der is enclosed in a parallelepiped envelope filled
up with the same fluid as the continuous phase. Ef
fects can be seen on Fig. 3. This correction is the
inevitable condition to get clean shots for a proper
use of image processing methods.
the internal plates are removable in order to make it
possible to create various kind of flows, dispersive
or not.
the drop population is produced with a set of hy
drophobic capillaries of millimetric diameters in
order to reduce the dispersion of the population.
Capillaries hydrophobia is needed to obtain lower
mean drop diameter.
Greek symbols
Phase k volume fraction
oT interfacial tension(Nm)
E emulsion stress tensor
Rotation rate tensor
pli phase k density (KgR.m )
ppphase k dynamic viscosity (Pa.s)
Subscripts
d dispersed phase
ccontinuous
Superscipts
0 intrinsic property
Introduction
Environmental concerns are playing an increasing role
in our time. Reprocessing of spent fuel is an early re
sponse to these many concerns. Among those reprocess
ing methods liquidliquid extraction is a nonnegligible
part. Liquidliquid extraction involves two immiscible
fluids. One phase is dispersed in the state of droplets in
the other phase. To predict the efficiency of the devices
implementing liquidliquid extraction several kinds of
models were created. Two main categories are distin
guished. The first one, first appeared in chemical engi
neering and called langragian models, initiated by Va
lentas K.J. (1966), focus on the behaviour of each drop.
Sets of equations are written for each drop size. The
second category of models called eulerian models and
steadily progressing since the advent of CFD, considers
each fluid as a pseudocontinuous phase. A more macro
scopic view is adopted. No more focus is made on drop
classes and mean values wheighted by volume fractions
are used. The model being developed here is part of the
second category. The innovation is based on the imple
mentation of interfacial area density transport equation
and anisotropy transport equation, Lhuillier (2004).
Calibrating such a model required the design of a spe
clal pulsed column and the development of new image
processing methods to access, by a non intrusive way,
mean quantities in the column. The following section
of this paper will focus on experimental materials and
methods used to collect the data required for the calibra
tion of the model. Then a presentation of the model's
equations is done and the assumptions used to simplify
it recalled. Before conclusion, experimental and numer
ical results are presented.
Experimental setup
The experimental strategy
Figure 3: Double envelope effects: as we can see here,
the filled part of the double envelope reduces the dis
torsions linked to the inner cylinder's curvature.
Table 1: Fluids physical properties.
Component Density Viscosity
TPH (Cl2H26) 0.76 1.29 103Pa.s
Drops 1.2 1.2 12Pa.S
method, first implemented by Budwig R. (1994), was
applied to reduce this refraction.
The synchronisation system
The synchronization system is used to localize shots
precisely on the pulsation cycle in order to calculate pha
sic averages. Synchronization is based on measurement
of differential pressure on either side of a stricture lo
cated in the submerged part of the pulsation system. The
differential pressure signal is then filtered and derived
aHRlOgically to make it possible to the trigger on ex
trema of the signal. The evolution of this differential
pressure signal will follow and describe precisely the hy
(a) (b)
Figure 4: (a) an example of two liquid phases with no
index matching, (b)both phases have the same refractive
index in this case. Contrary to previous example the in
terface between the two liquids can only be seen thanks
to a color difference.
7'" International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Fluorescing Droplet
Laser Sheet(UV)
Pulse o grc~ e Inlet
Figure 1: Laser illumination of the column.
,21 7
Figure 2: A geometrical description of the internals.
*the design also includes symetric shapes so as to
provide reversibility in operating conditions. It is
possible to operate with both organic and aqueous
continuous phases.
The phase system, an index matched emulsion
The continuous phase is Tetrapropylene Hydrogene
(TPH, Cl2H26). The dispersed one is a mixture of wa
ter, Glycerol(CHO3) and a fluorescent dyestuff called
fluoresceine (C20HloNaO2). The physical properties of
the components can be examined in the Table 1. The
presence of drops in the system is a potential source of
refraction phenomena at the interface between the TPH
and the dispersed phase. Some authors like GALINAT
S. (2005) mentioned possible apparition of droplets
shadows in the images. The refractive index matching
7'" International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
drodynamic behaviour of the pulsed column.
The pulse is pneumatic type, it can work at frequen
cies ranging from 0.5Hz to 2.5Hz at a pressure between
some mBar and 1Bar.
TLLTLIvT,,
Synchronisation Signal
Synchronisation
Box
Noisy Signal
Diffeenta
Pressure
Sensor
Pressure
Increase
beoethroat
Convergent I~vergent Device
Figure 5: The synchronisation system
The video camera, a PIKE F421B with a resolution
of 1280 x 1024 and a maximum frequency of 500fp~s,
was used to acquire 31 images per cycle for volume frac
tion and interfacial area density acquisitions. The in
ternal memory could store more than 100cycles which
provided a total of 500 images for average calculation
on each moment of the cycle.
Image processing for flow and drop population
characterisation
Mean Volume Fraction
The holdup reflects the probability of finding a phase
at a given point of space at a given time. We consider a
pixel as the smallest spatial dimension. Each pixel can
be affected either a 1 or a 0 depending on the phase in
volved, which is done by binarizing images.
Volume fraction is then calculated from the average
image obtained from a set of 500 images for each time
on the pulsation cycle. Once the average image calcu
lated, a color code is applied on it to highlight areas of
importance.
I I
Figure 6: Holdup calculation process
Mean Interfacial Area density
Interfacial Area density is the amount of exchange
SUTrfCO COntained in a unity of volume. To get this infor
mation from our shots we made the assumption of drops
sphericity and neglected the effects of perspective on di
ameter measurements. The hypothesis of sphericity is
only verfied upstream and downstream the internals, far
from nonzero deformation fields, which explains why
we only used this method in those areas.
Contrary to volume fraction, backlight is used here.
Fluorescence property of the drops is not exploited, the
color difference between phases is the only property
used to differentiate them. Despite numerous cases of
drop superpositions, the images we got were still work
able thanks to the use of HOUGH algorithm for circles
detection.
In the same way as with holdup measurements, pixels
are taken as the spatial unity. The amount of surface in
tercepted by each pixel is calculated taking into account
its position relative to the center of each drop contain
ing the studied pixel. It is then divided by the volume
of the column corresponding to the pixel studied. All
informations (drop radius, drop center's position, col
umn's axe position) necessary for these calculations are
provided by HOUGH algorithm and geometrical infor
mations collected on each image.
As can be seen on Fig.8 drop superpositions produce
7'" International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
the aqueous dispersed phase.
Intercepting Pixel
Overlapping droplets
detected wilth Hough algorithm
Figure 7: Case of a
droplets
1111
Intercepted surfaces
Droplet
pixel intercepting 3 overlapping
high level of interfacial area density. Though repre
sented on 2D support these results reflect a 3D infor
mation.
Figure 9: Image of diphasic flow with droplets loaded of
particles at their interface. As can be seen here the level
of particles concentration at the interface of the droplets
is not high enough to create shadows. However the con
tinuous phase is quite poor in particles since they pro
gressively migrate to the drops borders. The particles
hydrophilicity limited the duration of PIV.
Equations
Ensemble averaging
The equations used in the model were built on the ba
sis of ensemble averaging, Lhuillier (2004). Let V be
the volume were the average is to be calculated and V~
the volume actually occupied by the phase k. The phase
k presence function is denoted Xk (z, t). The volume of
the phase k is calculated with Vl =v f~ ukdV and th
volume fraction with kl (Xk). The ensemble average
of a physical quantity L*,' intrinsic to the phase k is given
by (\. <,':. In the same way we get pk (Xkp) and
PlVk. k (\i :, 
Continuity equation
With no production or destruction of mass at the in
terface, we get for the phase k :
Sk+ div( k k~) = 0 (1)
Which gives from summation of all phases :
div(Ek k Tk~) = divU = 0 (2)
Figure 8: (a) Volumic interfacial area distribution after
drops detection, (b) mean interfacial area density from
the averaging process of 300 (a)like images.
Particle image velocimetry(PlV)
For the time being, PIV Fincham A.M. (1997),Meu
nier P. (2003), is mainly used to provide boundary con
ditions for the simulations. The continuous phase was
seeded with 3MS60 particles whose density is very close
to the continuous phase one's (760KgR.m3). Their sizes
range from 20pum to 30pm.
Even in diphasic cases the continuous phase was the
only phase to be seeded. The previously described index
matching method allows the laser to pass through drops
without creating shadow zones as long as the hydrophilic
particles seeded into the TPH do not pollute too severely
7'" International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
D. Equation (4) is then replaced by the simple relation :
Momentum equation
In the same way as with the continuity equation,
emulsion's momentum equation is obtained from sum
mation of phasic balance equations.
B=kD
where k is a coefficient built in order to ensure the bal
ance between deformation process and return to isotropy
which traduces the reversibility of a deformation not
leading to breakup. Such a condition balances the two
first terms on the right hand side of equation (3).
Considering the successive steps involved in the evo
lution of an amount of interfacial area intended to in
crease by fragmentation after excessive deformation, it
is necessary to split equation (3) solving into two steps.
To model a possible fragmentation it is necessary to sep
arate deformation from return to isotropy. The first solv
ing step simulates fragmentation after an excess of de
formation by the use of equation (3) with only the first
and third terms of its right hand side activated. The sec
ond step, occurring half a time step after the first one,
reproduces the return to isotropy of the drops, canceling
the previous effects of deformation if no fragmentation
happened. Assuming an adequate dilution of the emul
sion we neglect the coalescence phenomena.
d
p
dt
PcPd
div(E)div (R, it) @ (RI it)
pc + pd
V is the emulsion mean velocity weighted by densities.
Interfacial balance equations
Interfacial area density A is defined by A (6,)
where 6, is the interface density function giving the
amount of interfacial surfaces contained in a volume V
from S6, (z, t)dV. The interfacial area density trans
port equation is then :
(1 +H(md) .. >$:3
(qH ( d)\ B:B
S20 C ~d j
COGoOAV P a2
tv,.P,.A
In addition to transport, the interfacial area density is
subjected to deformation due to the external flow. This
phenomenon is represented by the first term on the right
hand side. The following term, the return to isotropy
term, represents the natural ability of a drop to become
spherical again when not streched. Both streching and
return to isotropy lead to a reversible deformation. The
third term represents coalescence and the last one is
linked to fragmentation phenomena. Both irreversibly
affect the level of interfacial area density.
The tensor B quantifies the level of anisotropy of the
emulsion. Applied on a deformed drop it would quantify
its difference from a spherical shaped one. Its transport
equation is given by :
dB 8 *
~+ BR + OB= AD
dt 15
A H1(4d) c Df (4)
15 P 1
8 TH'(4d) *
15 2pUc #d j
Simplified model
Initially a simplified version of the model built on the
basis of physical hypothesis is tested. The first hypothe
sis is to assume the absence of drift between both phases,
The second, aiming at reducing the number of equations
to solve assumes, on the basis of deformation character
istic times, the proportionality between anisotropy ten
sor B and the continuous phase deformation rate tensor
Results and Discussion
Experimental results, volume fraction
As described previously mean volume fraction is ob
tained from the average of 500 binary images. A de
scription of the pulsation cycle in the first level of the
column's internals is given below. The pulsation cycle
begins with an ascending motion preventing drops from
falling. As a result, a drops accumulation can be seen
on the first plate and on the edge of the first ring. This
accumulation leads to an increase in the local amount of
mean volume fraction of the dispersed phase up to values
exceeding 11 I' . Once the half of the cycle is reached, a
descending motion begins and leads to the draining of
the previously accumulated drops. Increase in volume
fraction here are mainly due to accumulation processes.
Preferential paths are highlighted, drops never seem to
cross some areas of the pulsed column internals.
Experimental results, interfacial area density
The interfacial area density fields are 2D representa
tions of 3D results. Their use for simulations require first
a deconvolution process. Moreover, due to limitations
of the algorithm used for this image processing method,
results are only available upstream and dowstream the
internals of the column, it should give good informa
tions on the model's potential at least for global breakup
quantification. Drops accumulation on the first plate at
t 0.5s leads to a slight increase of interfacial area den
sity. The main source of growth of interfacial area den
sity remains the drop beakup events as can be seen at
7'" International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Figure 11: Mean interfacial area density fields upstream
and downstream the internals of the column. Due to drop
breakup the global amount of interfacial area density
has increased.
reversible drop deformation is rather well calculated by
the simplified model. Overestimation of the amount of
deformation could be explained by both the experimen
tal errors and the negligence of anisotropy transport phe
nomena.
Though interesting drop fragmentation phenomena
were studied in Godbille F.D. (1998) no quantitative re
sults were found. The author focused on qualitative de
scriptions of drop breakup mechanisms he observed in
his experiments. Windhab E.J. (2005) studies of emul
sification processes in various devices is a good start
ing point for a quantitative drop fragmentation study. To
quantify the relative contributions of elongation and pure
shear in fragmentation mechanisms Windhab E.J. (2005)
used the parameter a defined by :
a 
Where if the deformation rate tensor of the continuous
phase is
Figure 10: Description of the mean volume fraction's
cycle. The first mean volume fraction field on the top
left image corresponds to t = 0.13s. The last field in the
bottom night image represents t = 0.90s from a cycle of
t = 1s downstream the internals where the level of in
terfacial area density is clearly higher than the upstream
level despite the lack of drops accumulation. Those re
sults will b used as a basis for the validation of fragmen
tation.
Preliminary numerical calibration from literature re
sults
All the calculations mentioned in this paper were
done using the finite elements code CASTEM from
French Atomic Comission (CEA). Before making sim
ulations with a complex and time consuming geometry
and due to some experimental methods limitations, we
firt test and calibrate our model on a simpler geome
try. To this end we reproduced numerically a part of
the work of Godbille ED. (1998). On the one hand the
aim is to validate the reversible increase of interfacial
area density when drops are deformed without breaking
with some experimental results of Godbille F.D. (1998).
On the other hand we want to calibrate the increase of
interfacial area density due to fragmentation phenomena
with our experimental results. Tests are all conducted
with the assumption of sufficient level of dillution of the
emulsion so as to neglect coalescence. As can be seen
on Fig.12 the increase of interfacial area density due to
an
ay
]
Dix D12)
D21 D22/
& = D~i~ D 2
\ ="
D = 1 ( a Z s
S\2( "y ''
7' International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
S1.
) )3
013
SI 0la
3.97E02
0 .19
0 17
0.3s
0.14
0 5 10 15 20 25 30 35 40
45 50
Distance from convergent entrance (mm)
Figure 12: Comparison of measured ans calculated in
terfacial area density on the centerline of the convergent
studied by Godbille F.D. (1998). Despite an overesti
mation of the expected result the model gives a good
approximation of the mean increase of interfacial area
density due to deformation.
a is close to 1 for flows dominated by pure elongation or
compression and close to 0 for shear dominated flows.
Drop breakup happens when external solicitations
outbalance the interfacial effects. The capillary number
measuring the balance between both previously described
quantities is often used as a criteria in breakup study. In
our case the capillary number Ca is defined with :
The mean radius r is evaluated from the local interfa
cial area density and holdup with A = ~. Critical
capillary numbers, functions of the flow type and phases
viscosity ratio, are used to determine if breakup happens
or not. Critical capillary curves can be found in litera
ture. For a first test we took WINDHAB's curve for a
viscosity ratio of 8 (A = 10 in our case). We expect an
overestimation of fragmntation phenomena due to this
temporary choice.
Fragmentation will happen at all points in the mesh
having a capillary number exceeding the critical value
shown in Fig.13. Calculations are underway to provide
a correct volume fraction level and spatial distribution in
order to evaluate precisely r mean values and capillary
values in the pulsed column. Experimental interfacial
area density results will be exploited to check the proper
reaction of the model for breakup events.
(a)
Figure 13: (a) isoa field calculated with the continuous
phase velocity field at t 1.10 3s by Castem. At this
moment of the cycle the flow is compressionelongation
dominated, which predict a good dispersive behaviour.
(b) critical capillary numbers field obtained from liter
ature's curveWindhab E.J. (2005). Breakup events are
expected to happen at low Cac points.
Conclusions
A new image processing method was developed on the
basis of HOUGH transform algorithm to access inter
facial area density in a nonintrusive way upstream and
downstream the internals of a pulsed column. Combined
with a mean holdup measurement this method offers the
possibility to calibrate and validate our eulerian emul
sion model. Though only at an embryonic state the first
validations are promising. More systematic validations
based on experimental results are still needed to com
plete the test of the model.
References
Budwig R., Refractive index matching methods for liq
uid flow investigations, Exp. Fluids, Vol. 17, pp. 350
355, 1994
Fincham A.M. and Spedding G.R., Low cost High res
olution DPIV for measurement of turbulent fluid, Exp.
Fluids, Vol. 23, pp. 449462, 1997
Galinat S., Etude Exp~rimentale de la rupture de gouttes
7'" International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
dans un Ccoulement turbulent, Thbse de Doctorat,
Toulouse, 2005
Godbille ED., Drop deformation and breakup in com
bined shear and extensional flow, Master Degree Thesis,
University of New Brunswick, 1998
Lhuillier D., Smallscale and coarsegrained dynam
ics of interfaces: the modelling of volumetric interfa
cial area in two phase flows, 3rd International Sympo
sium on TwoPhase flow Modeling and Experimenta
tion, Pisa,September 2004
Meunier P. and Leweke T., Analysis and treatment of
errors due to high velocity gradients in Particle Imaging
Velocimetry Exp. Fluids, Vol. 27, pp. 408421, 2003
Valentas K.J., Analysis of breakage in dispersed phase
systems, Ind. Eng. Chem. Fundamentals, Vol. 5, pp. 271
279, 1966
Windhab E.J. and Dressler M. and others, Emulsion
processingfrom single drop deformation to design of
complex process and products, Chem Eng Sci, Vol. 60,
pp. 21012113, 2005
