7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Phasefield model for electrohydrodynamic 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, twophase flow, Phase Field, electrohydrodynamics, alternating/direct current.
March 24, 2010
Abstract
The principles of electrohydrodynamics 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 demixing. 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 electrohydrodynamic 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
Electrohydrodynamics has attracted extensive research
interest in the last decade due to its industrial relevance
Jones (1995); Morgan & Green (2003); Eow & Ghadiri
(2003b). Electrocoalescence 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 environmentfriendly than the traditionally
used chemical demulsifies Ewo & Ghadiri (2002).The
application of electrocoalescence in petroleumrelated
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 electrocoalescence
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
electrocoalescence.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
noncoalescence of water droplets in oil mainly was
determined by the coneangle formed, as the droplets
approached each other. A distinct transition between
coalescence and noncoalescence 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
headon 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)
ReMa 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
nondimensional form of the governing equations,
1 Governing equation
In the current work, we adopt a phase field met
based on a CahnHilliard free energy formulation c
pled with NavierStokes 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
reinitialization 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 nondimensional governing equations:
1f
1 (V'((C )
Pe
Cn2V2C)
V (aE) + PiwV (cE) = 0
P1 Re
2v2LU
Pe = , Cn
3y
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 NavierStokes 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 CahnHilliard 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 electrocoalescence in an AC electric field
discussed below, we will discuss how the frequency in
fluences the efficiency of electrocoalescence.
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: dasheddot 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
Electrocoalescence 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 electrocoalescence. 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: Electrocoalescence 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. (ac)
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 electrocoalescence 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 electrocoalescence, 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 electrocoalescence 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 electrocoalescence.
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, CDAdapco, 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 DoQuang 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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