Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 11.4.1 - A phase-field model for multiphase electro-hydrodynamic flows
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 Material Information
Title: 11.4.1 - A phase-field model for multiphase electro-hydrodynamic flows Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Lin, Y.
Skjeten, P.
Carlson, A.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: droplets
two-phase flow
phase field
electrohydrodynamics
alternating/direct current
 Notes
Abstract: The principles of electro-hydrodynamics have been well known for more than a century. Extensive experimental studies have been carried out and its principles applied to industrial flows, e.g. fluid mixing and de-mixing. From a computational point of view analytic solution of the electric forces acting on a single droplet and interaction between droplets in an electric field is only possible for a limited number of cases. Thus, numerical modelling and simulation can provide significant insights. Previous numerical studies have been confined to steady direct current (DC) electric fields. In this paper we suggest a model for studying the electro-hydrodynamic forces acting on a droplet in an unsteady, alternating current (AC), electric field based on the phase field method. This AC model can be viewed as a generalized form of the DC model. The model formulation was validated under DC conditions. Excellent agreement is found with previous studies. The model was then used to study the effect of frequency on single droplet deformation, and how electro coalescence is influenced by the dielectric properties of the two phases, the viscosity ratio and surface tension between the two phases. Finally for a given set of physical parameters we investigate how AC frequency influences electro coalescence.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00284
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 1141-Lin-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Phase-field model for electro-hydrodynamic flow


Y. Lin*t, P. Skjetent and A. Carlson1

Comsol AB, Tegnbrgatan 23, Stockholm, 11140 Sweden
t Department of Process Technology, SINTEF Materials and Chemistry, Postboks 4760, 7465 Trondheim, Norway,
SDepartment of Mechanicics, Linne Flow center, The Royal Institute of Technology, Stockholm, Sweden
yuan@mech.kth.se, Paal.Skjetne@sintef.no and andreaca@mech.kth.se.
Keywords: Droplets, two-phase flow, Phase Field, electro-hydrodynamics, alternating/direct current.

March 24, 2010



Abstract

The principles of electro-hydrodynamics have been well known for more than a century. Extensive experimental
studies have been carried out and its principles applied to industrial flows, e.g. fluid mixing and de-mixing. From a
computational point of view analytic solution of the electric forces acting on a single droplet and interaction between
droplets in an electric field is only possible for a limited number of cases. Thus, numerical modelling and simulation
can provide, iii gi lk.iIi insights. Previous numerical studies have been confined to steady direct current (DC) electric
fields. In this paper we suggest a model for studying the electro-hydrodynamic forces acting on a droplet in an
unsteady, alternating current (AC), electric field based on the phase field method. This AC model can be viewed
as a generalized form of the DC model. The model formulation was validated under DC conditions. Excellent
agreement is found with previous studies. The model was then used to study the effect of frequency on single droplet
deformation, and how electro coalescence is influenced by the dielectric properties of the two phases, the viscosity
ratio and surface tension between the two phases. Finally for a given set of physical parameters we investigate how
AC frequency influences electro coalescence.


Introduction


Electro-hydrodynamics has attracted extensive research
interest in the last decade due to its industrial relevance
Jones (1995); Morgan & Green (2003); Eow & Ghadiri
(2003b). Electro-coalescence of water droplets in oil
continuous systems by electrophoresis and dielec-
trophoresis mechanisms, utilizing the large difference
in their electric properties, constitutes an important
separation method in the oil and gas industry. It has
been proven to be not only a highly efficient method to
enhance the separation of water from crude oil, but also
much more environment-friendly than the traditionally
used chemical demulsifies Ewo & Ghadiri (2002).The
application of electro-coalescence in petroleum-related
industries was first reported in 1911, and it is typically
applied for dehydration processing of crude oil Eow &
Ghadiri (2003a). Previous works on this subject sug-
gest that there exist optimal frequencies, voltages and
geometries for different systems of electro-coalescence
Eow & Ghadiri (2003b). Due to its wide adoption in


industries, it is of great practical interest to improve the
theoretical understanding of the basic mechanisms of
electro-coalescence.Analytical methods have obvious
limitations for such complex systems, thus numerical
methods are vital tools for investigating such phenom-
ena. In order to resolve the relevant physics in the
system it is important to have a front tracking method
with sufficient resolution of the interfacial physics.
Previous studies have relied on level set and VOF/FVM
methods or have represented the droplets as rigid
entities. Extensive experimental investigations have
been performed on binary droplet collisions, identifying
a wide range of physical phenomena.
In a study by Chabert, M. et al. (2010), two opposite-
charged droplets are observed to approach each other
first, and then they moves apart. The repulsive force be-
tween the droplets is believed to originate from a charge
transfer through the liquid bridge, formed between the
droplets as they come together. This results in a strong
repulsive force, if the field strength is sufficiently high.
Ristenpart et al. (2009) found that the coalescence or







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


non-coalescence of water droplets in oil mainly was
determined by the cone-angle formed, as the droplets
approached each other. A distinct transition between
coalescence and non-coalescence was found based
on field strength and the cone angle. It was reported
that to induce droplet coalescence, the electric field
strength should be within certain values (Eow & Ghadiri
(2003b); Ristenpart et al. II "i'. The frequency
of the AC current have also been identified as a key
parameter to trigger coalescence or droplet bouncing
see Chabert, M. et al. (2010). General flow maps for
head-on binary collision have been developed, Ashgriz
& Poo (1990); Qian & LAW (1997), which contains
three main regions: high Weber number coalescence,
intermediate Weber number bouncing and low Weber
number coalescence. The Weber number is defined as
a dimensionless relative approaching speed of droplets.
Recent experiments show that coalescence mainly takes
place as a result of competition between film drainage
and rupture dynamics caused by molecular interactions
of the approaching droplets surface Dupuy et al. (2009).

Furthermore, the governing equations are solved
with a finite element method and a projection scheme
has been applied that can handle two phases with high
viscosity and density ratios. It is feasible to simulate
a mixture of oil and water, with a viscosity ratio to be
more than one thousand. Such high viscosity ratios can
be encountered in actual industrial separation processes.


i+ Vu
at


1 2 2
Vp + -V2
Re


1 1(2
+ RE (- M 2VclE12 + V (cE)
Re-Ma 2


cov
Re -Ca- Cn
E)
(3)


V u 0 (4)
where we have added a new model to also account
for the electric field and the electric force. In the model
derivation Lin et al. (2010) the relaxation time of the
electric field is assumed to be small compared to the ob-
servation time (e.g. 1/24 of a second). Using this as-
sumption the DC and AC formulations become the same
except that the electric field vector (E) is replaced by its
root mean square (RMS) average, and the permittivity
becomes frequency dependent:

V (aE) + wV (eE) 0, (5)

f /(VeE| + V (E))dQ. (6)

Where E E0 exp(iwt) which reduces to E E0 in
the DC case, and E ERMS Eo cos(7r/4) for the
time averaged case.

The different phenomena are dictated by the rela-
tive importance of the forces in the flow, illustrated
by the seven dimensionless groups appearing in the
non-dimensional form of the governing equations,


1 Governing equation

In the current work, we adopt a phase field met
based on a Cahn-Hilliard free energy formulation c
pled with Navier-Stokes equations to simulate de
mation and coalescence of droplets in different elec
fields. The advantage of this approach also lies in its
trinsic mass conservation and that it avoids any exp
re-initialization of the phase function Anderson et
(1998) or numerical reconstructions of interface wh
are typically needed by Level Set methods and V
respectively. We adopt the Phase Field equations (
lanueva et al. (2007); Carlson, A. et al. (2010)) wh
the forces from an electric field is also included, gi
the non-dimensional governing equations:


1f
1 (V'((C )
Pe


Cn2V2C)


V (aE) + PiwV (cE) = 0


P1 Re

2v2LU
Pe = --, Cn
3--y

S o06mRV2
WeE -
YL2


hod
;ou-
for-
ctric
in-


pmUL
,Ca
t


2v2oU
3y


E PmUL
LMa 2,
SenOm V


licit The parameter P1 describes the effective ratio between
al. the electric permittivity and electric conductivity. The
which Reynolds (Re) number describes the relative importance
OF, between the inertia and viscous force, the Capillary (Ca)
Vil- number describes the ration between the viscous and
here the surface tension force. The Peclet (Pe) number de-
ing scribes the ratio between the convective and diffusive
mass transport and the Cahn (Ca) number defines the ra-
tio between the thickness of the interface and the droplet
radius. The Mason (Ma) number describes the ratio be-
tween the viscous and the electric force, and the electric
(1) Weber number (WeE) describes the ratio between the
electric and surface tension force. Here R is the radius
of the droplet, which is in the same order of magnitude
of characteristic length L; eo is the free space permit-
(2) tivity. Notice that the Mason number can be expressed


OC
S+ VC
at











as Ma = () and that we use the WeE
number in the analysis of the numerical result with the
theoretical prediction by Taylor (1966).


2 Numerical methodology


The finite element toolbox FemLego developed by Am-
berg et al. (1999) is used to perform the numerical sim-
ulations. It is a symbolic tool where the ordinary or
partial differential equations, boundary conditions, ini-
tial condition and numerical solvers are all specified in
a single Maple worksheet. All equations are approxi-
mated with piecewise linear functions and an adaptive
mesh refinement method is applied in order keep a high
spatial resolution at the moving interface. The multi-
fluid Navier-Stokes equations are solved by a fractional-
step projection method proposed by Guermond & Quar-
tapelle (1998). The discretized linear system for the
momentum equations are solved by the generalized min-
imal residual method. The pressure and electric field
equation are solved by an Incomplete Cholesky Conju-
gate Gradient (ICCG) method. Due to the stiff nature of
the Cahn-Hilliard equation with fourth order derivatives,
special care needs to be taken in the solution procedure.
It has been solved with a modified ICCG solver in ac-
cordance with Villanueva et al. (2007).


3 Results and Discussion

Droplet deformation in DC and AC electric fields
For a leaky dielectric fluid we find excellent agree-
ment with Taylor's theoretical predictions Taylor (1966).
The simulations were performed for two different Cahn
numbers Cn=0.05 and Cn=0.02, to verify that the inter-
face is sufficiently resolved and identify the influence of
a diffuse interface, see Figure 1. In Figure 1 the con-
ductivity ratio has been fixed and the permittivity ratio
varies. Similar tests were done by keeping the permit-
tivity ratio fixed and letting the conductivity ratio vary.
For small deformations both cases give excellent com-
parison with Taylor's theory. Taylor's theory predicts
that the droplet deformation mode depends on the ra-
tio of the electrical permittivity of the dispersed phase
to that of the continuous phase, with a transition from
a oblate deformation relative to the electrical field, to
a prolate deformation as the permittivity ratio increases
past a value of approximately 10. This is illustrated in
Figure 2. Notice how both the internal and external cir-
culation changes in this transition.
Similarly for a perfect dielectric fluid Sherwood Sher-
wood (1998) developed a theory which confirmed exper-
imental observations that droplets in such systems al-
ways deform into prolate shapes. Our simulations are


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010




0.1 ....
~------------ ---A
0

-0.1
o '0
0.2 '
'\
-0.3 *
Taylor's theory
Simulation when Cn=0.05
-0.4 Simulation when Cn=0.02

0.5
10' 100 101 102
Ed/Em (when d/c -5)

Figure 1: Comparison of the numerical results and the
theoretical prediction of small deformations,
for WEE = 0.2 and od/Ur, 5. The dashed
line part indicates that the theoretical predic-
tion is not expected to be valid when deforma-
tion is large.



again in excellent with the theoretical analysis and is re-
ported in detail in Lin et al. (2010).
The relationship between the droplet deformation and
the frequency in the AC current is plotted in figure ??.
A striking observation is that both prolate and oblate
droplet shapes are obtained by just changing the fre-
quency. For frequencies below 500 Hz we observe that
the deformation is negative, and the droplet takes an
oblate shape. For frequencies above 500 Hz the droplet
takes a prolate shape. This is understandable because
in the limit of small and large frequencies the droplet
behavior would converge to the approximation of leaky
dielectric droplet and a perfect dielectric droplet in a DC
field, respectively. Figure ?? shows the interfacial shape
the droplets take for some different frequencies. In the
section on electro-coalescence in an AC electric field
discussed below, we will discuss how the frequency in-
fluences the efficiency of electro-coalescence.
Droplet coalescence in DC and AC electric fields
For leaky droplets under DC conditions, the affore men-
tioned oblate and prolate deformation modes have a
strong influence on the interaction between the droplets.
Depending on e.g. the permittivity ratio, the prolate
or very weakly deformed droplets are seen to coalesce
whereas oblate droplets seem to repel each other. Fur-
thermore, the influence of the viscosity ratio between
the droplets and the suspending medium was investi-
gated. Again the viscosity ratio incluences the defor-
mation mode of the pair of droplets. E.g. for a viscos-
ity ratio of 0.01 (droplets more viscous than suspend-







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


0

0.05

Q 0.1

0.15

0.2

0.25






1



0.5


(b)

Figure 2: A leaky droplet in a DC electric field for
We 0.2 and Cd/oT, 5; (a) Ed/m,
10, (b) d/e, = 60.


Figure 3: The deformation of a single droplet ina a
AC electric field as a function of frequency.
(a) When WeE 1, the deformation ratio
as function of the frequency, w. (b) When
WeE 10, steady state of a droplet with dif-
ferent frequencies: dashed-dot line: w = 200
Hz; solid line: w = 500 Hz; dashed line:
w = 1000 Hz.


Numerical resull
PCHIP fining


10 10
Frequency [Hz]


o200Hz
o=500Hz
-- ,= 1000Hz



1 -
.5 *- 0.5





*%,e







-0.5 0 0.5 1


(a)








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


ing medium) an oblate coalescence with the aforemen-
tioned cone tip formation being prominent, whereas for
a viscosity ratio of 100 a prolate coalescence takes place
where the "contacting" surface is dimpled at the cen-
ter, thus producing a multiple emulsion, i.e. a smaller
satellite droplet inside the newly formed droplet. The ef-
fect of surface tension on under DC conditions was also
investigated by varying the capillary number between 1
and 1000. From experiments it is known that the sur-
face tension is relates to an optimal voltage for coales-
cence, since a too strong electrical field could lead to tip
streaming of small satellite droplets or post coalescence
fragmentation. Simulations clearly show this same be-
haviour. Details of these resluts will be reported in Lin
et al. (2010).
As was seen for the single droplet deformation under
AC electric fields, varying the frequency of the field will
change the characteristics of the droplet deformation. A
prolate deformation mode is seen to give more rapid
coalescence than an oblate deformation. A crossover
between oblate and prolate deformation is again seen
around 500Hz for the selected parameters studied. The
reason the coalescence time is reduced for prolate defor-
mations is the fact that the distance between the inter-
faces is reduced and thus the CM translation needed for
the surfaces to come in close contact is smaller. In ad-
dition the circulation patterns and pressure field formed
between the droplets favor film drainage thus aiding co-
alescence. See Figure 4 ? for details.


4 Conclusions


Electro-coalescence has become an important separa-
tion technique for dehydration of crude oil. Extensive
experimental investigations have been performed, but
a recurring challenge is how to separate the effects of
different physical parameters. Thus there is a need
for theoretical formulations that can be used to study
such systems numerically in order to understand the
fundamental physics of the system and to aid in opti-
mized process design. The current work suggests a the
phase field method to simulate electro-coalescence. The
deformation of a single droplet in a DC electric field
by the current model is validated against previous theo-
retical and numerical studies, and excellent agreement
is achieved. The model is also capable of simulating
droplets in an AC electric field. Both the magnitude
as well as the direction of the droplet deformation was
found to depend on the electric field frequency. It should
be noted that these simulations were carried out under
the assumption that the frequency of the AC electric
field is rather high, so that the relaxation time is much
smaller than the observation time scale. The current
method can also be used for low frequency AC fields, by


0.5


-0.5


-0.5 0 0.5
r

---- t=
t=47
,." .". ...... t=150


-0.5 0 0.5
r


Figure 4: Electro-coalescence of two leaky droplets in
an AC electric field. d/em = 2, oad/cm =
100 when WeE = 2.4. The snapshots are for
times t = 0, 47 and 150, respectively. (a-c)
for frequencies 200, 500 and 1000 Hz, respec-
tively.


---- t=O
-t=47
.- ...... t=150


-0.5 0 0.5
r

S---- t=
t=47
7, s -...... t=150


N Oh











simply substituting (2) with (5). In this case, the solved
electric field is no longer the RMS value but the real
time value, thus being the time resolved computation
of the electric force exerted on the fluids. Furthermore,
we simulated electro-coalescence of two droplets in
uniform DC and AC electric fields respectively. In the
former case, we investigate the influence of permittivity
ratio for leaky droplets and confirm that when the
permittivity ratio is much less than the conductivity
ratio, the hydrodynamic force might overcome the
dielectrophoresis force and prevent the occurrence of
coalescence. To investigate the effects of viscosity
and surface tension on electro-coalescence, simulations
were performed with different viscosity ratios and Ca
numbers, respectively. The first category of simulations
suggest that when the viscosity ratio is different from
one, a higher electric field is needed to overcome the
hydrodynamic pressure or the dissipation of energy.
The second category of simulations suggest that for
emulsions having a lower surface tension, the applied
electric field should be lower. Finally, we investigate
how the frequency affects the electro-coalescence for
a certain system to demonstrate the feasibility of this
model to simulate electric coalescence in an AC electric
field.

The theoretical formulation and numerical model
is believed to capture physical flow phenomena that
are relevant to real applications in industries where
separation of oil and gas appear. Further studies should
include complex geometries that mimic the industrial
applications. Furthermore, systems employing multiple
superimposed electric fields would also be of industrial
as well as fundamental interest to study with the current
approach. Real systems are almost always contami-
nated by surfactants an additional phenomenon to be
considered in future studies of electro-coalescence.




5 Acknowledgements*


This work was funded by the Flow Assurance Center
(FACE) a center for research based innovation funded by
The Research Council of Norway, and by the following
industrial partners: Statoil ASA, Norske ConocoPhillips
AS, Vetco Gray Scandinavia AS, Scandpower Petroleum
Technology AS, FMC, CD-Adapco, ENI Norge AS, and
Shell Technology Norway AS. Part of the work has been
performed in collaboration with the Linne Flow Center.
The authors wish to thank Professor Gustav Amberg, Dr.
Minh Do-Quang and Professor Sanjoy Banerjee for in-
spiring discussions.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


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