I. INTRODUCTION

Compact electrostatic Coalescers (CECs) are growingly used in the petroleum industry to increase the size of water droplets suspended in crude oils by merging and, therefore, to reduce the time required to separate water and oil phases under the gravity effect [1]. The control and increase of electrocoalescers efficiency are very difficult as the numerous phenomena involved in electrocoalescence are far from being fully understood. That justifies our investigations on the deformation and coalescence of two closely spaced drops of conducting liquid suspended in an insulating fluid under the action of an electric field (often referred to as the second stage of electrocoalescence) [2]. The disintegration of pairs of water drops in an electric field was examined theoretically more than 40 years ago [3], extending the initial work of Taylor on single drops [4], but without proposing very clear conclusions and laws concerning the coalescence conditions for close small drops subjected to a uniform electric field. The observed large deformations of the interfaces involve the strong coupling of hydrodynamics and electrostatics and numerical models should be used to improve the analyses [5, 6]. The development of efficient numerical methods promotes now multi-physics models in EHD [7, 8] and allows improving electrical or chemical behavior modeling [9, 10].

The problem of deformation and disruption of a water-oil interface has been studied recently numerically and experimentally in the particular configuration of a metallic sphere hanging above a horizontal layer of water (asymptotic case of interaction between a very small and a big droplet). The major fact is that a static shape of the

electrically stressed interface exists only for a limited deformation: the sphere-interface distance cannot decrease below about half of its initial value (i.e. without field action) [11]. General considerations and an approximate treatment extended the basic properties of instability conditions for two identical drops [12]. However, these results are questionable as two free drops attracting each other move and the conditions of interface disruption are different from the ones characterising immobile drops. The paper presents: - 2D axisymmetric simulations, taking into account electrostatic pressure and interfacial tension effects and using ALE method for mesh deformation handling. Computations are performed with the commercial software COMSOL MULTIPHYSICS™. - an experimental study of water drops anchored at the tip of capillary tubes and subjected to a potential difference in stagnant oil. The critical conditions for existence of a stationary solution are determined from visualizations and represented by the critical electric Bond number drawn as a function of the relative initial spacing between the facing drop surfaces, for different anchoring angles. First comparisons between numerical and experimental results are performed. - first simulations of the behavior of free droplets suspended in stagnant oil under the influence of a uniform applied electric field. Differences with the anchored drops configuration are discussed.

II. ANCHORED DROPS: NUMERICAL MODEL

We consider two water drops of equal radius R0

formed at the tip of capillary tubes and immersed in oil (insulating fluid) with an initial spacing S0 (Fig. 1). The angle m characterizes the initial shape of the drop at the end of the capillary tubes: m = 90° for hemispherical drops and m is larger than 90° when the drops diameter is larger than the diameter of the capillary tube as illustrated in Fig. 1.

Electrocoalescence of Two Water Drops in Oil: Experiment and Modeling

J. Raisin, J.-L. Reboud, and P. Atten

Grenoble Electrical Engineering laboratory, CNRS, France

Abstract—We investigate the deformation and coalescence of two closely spaced drops of conducting liquid suspended

in an insulating fluid under the action of an electric field. First, electrically induced deformations of water-oil interfaces and interaction of two droplets are simulated, taking into account electrostatic pressure and interfacial tension effects and using ALE method for mesh deformation handling. Secondly, an experimental study of water drops anchored at the tip of capillary tubes and subjected to a potential difference in stagnant oil is performed. Critical conditions for existence of a stationary solution are determined from visualizations and represented by the critical electric Bond number drawn as a function of the relative initial spacing between the facing drop surfaces, for different anchoring angles. First comparisons between numerical and experimental results are presented. Finally, the behavior of free droplets suspended in stagnant oil and under the influence of a uniform applied electric field is simulated and differences with the anchored drops configuration are discussed.

Keywords—Electro hydrodynamics, fluid-fluid interfaces, drop deformation, coalescence, numerical simulations

Corresponding author: J. -L. Reboud e-mail address: jean-luc.reboud@grenoble.cnrs.fr Presented at the International Symposium on Electro-hydro-dynamics, in March 2009

Raisin et al. 127

The application of a potential difference V between the drops induces an electric field E and an electrostatic pressure at the interfaces. The problem is to determine the subsequent deformation of the drops and, in particular, of the facing zones of the interfaces. We aim at determining the critical conditions of electrocoalescence of the two conducting drops, i.e. the critical potential difference Vcrit above which no static drops deformation exist.

For small enough droplets the gravitational force is negligible. The time dependent Navier-Stokes equations are solved in the water (NS1) and oil (NS2) media (Fig. 2), ensuring, in particular, the volume conservation of the droplets. These fluid dynamic equations are coupled with the successive solutions of the Laplace equation in the oil medium, water being considered as perfectly conductive and thus at constant electric potential. The deformation of the water-oil interface is implemented by using an Arbitrary Lagrangian-Eulerian formulation with deforming meshes. This approach allows us to take into account the forces that act on the interface, such as interfacial tension and electrostatic pressure, and to track the interface deformation in a very accurate way. Transient calculations have been performed with a very slow voltage rise: V/t = 0.05 V/s (the inertial and viscous terms remain in practice negligibly small, except when the interface instability occurs). All the test cases were computed considering axial symmetry, and, in this particular case of two droplets of same radius, using the plane of symmetry between the drops (Fig. 2). The main modification of the standard software concerns the computation of the interfacial tension forces performed through a weak formulation of the fluid dynamics equation [13, 14] adapted to the axial symmetry conditions. At each time step, the electric field E is computed in the oil domain, taking into account the actual interface deformation of the water droplet. Boundary conditions imposed for Navier Stokes equations in the two liquid domains are summarized on Fig. 2: i) Constant static pressure at the upper boundary of the

oil domain. The pressure p2 computed in oil (NS2), modified by electrostatic pressure En

2 / 2, is applied as boundary condition at the interface for the water domain (interfacial tension is considered indirectly through the weak formulation). Normal and tangential components of the viscous stress τ on water and oil are balanced at the interface.

ii) The velocity field u1 computed within the water droplet (NS1), is applied as boundary condition at the

interface for the oil domain. Liquid velocity at the water-oil interface is applied in a Lagrangian way to move it. The water-oil interface is closed, above the anchorage point, by a vertical impervious fixed boundary. An example of the evolution of the spacing S between the drops is drawn on Fig. 3 as a function of the slowly increasing potential difference V. It can be seen that the appearance of the interfacial instability corresponds to a very sharp drop of the curve that provides an accurate evaluation of Vcrit. The critical spacing Scrit appears to be slightly larger than 0.6 S0. Varying the initial spacing for different anchoring angles, allows drawing the numerical stability limit (critical value of the electric Bond number Becrit) as a function of the relative initial spacing = S0/R0, in the range = 0.01 – 0.5. The asymptotic approach presented in [12, 15] for << 1 is compared to the numerical results showing a good agreement up to = 0.1 and validating the numerical model (Fig. 4). The general numerical method gives critical values slightly lower than the asymptotic ones because the real electric field on

s0

m

R0

Fig. 1. Drops anchored at the end of capillary tubes.

Fig. 2. Boundary conditions of the EHD coupled problems.

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 ΔV (V)

S/So

Fig. 3. Quasi-steady relative spacing S/S0 vs increasing voltage V. m= 90°, Ro= 0.28 mm, = S0/R0 = 0.7, = 25 mN/m, = 21.2

pF/m ; the result here gives Becrit = (/2) V2 / S02 (R0 / ) = 0.512.

NS2 (oil): u2, p2

Laplace eq. : ε, E

Symmetry: 0.5 x applied electric potential

Grounded

u2 = u1

p1 - τ1n = p2 - τ2n - pelec + PInt. tension (n)

p2 = 0

Symmetry: n·u2 = 0

τ1t = τ2t (t)

NS1 (water):u1, p1

n

t

128 International Journal of Plasma Environmental Science and Technology Vol.3, No.2, SEPTEMBER 2009

the interfaces is slightly higher than the simplified field retained in the asymptotic approach, thus requiring a slightly lower applied voltage.

III. ANCHORED DROPS: EXPERIMENTS The experimental set-up (Fig. 5) consists in a cell

built of Plexiglas for visualization purposes filled with synthetic polybutene oil.

Drops volume is controlled using 10 l syringes. For small enough droplets, gravity effect can be neglected, and we observe a horizontal axis of symmetry. The potential difference is created by applying an AC sine voltage of frequency (f = 900 Hz) to one of the drops, the

other being grounded. The signal is produced by an arbitrary wave generator and magnified using a TREK high voltage amplifier. The quasi static interfaces deformation is captured using a CCD camera (25 frames per seconds) and the key geometrical parameters of each experiment are determined through image analysis (Fig. 6).

Experimental results of Becrit as a function of ψ are drawn on Fig. 7 for three different values of the anchorage angle m. Horizontal error bars correspond to an indeterminacy of 2 pixels on the value of the initial separation distance S0 between the drops. Vertical error bars account for the influence of this error on S0

cumulated with 0.002 N/m uncertainty on the measured

0.01

0.1

1

0.001 0.01 0.1 1

So / Ro

Becrit

Analytical 60° Numerical 60° Analytical 90° Numerical 90° Analytical 172° Numerical 172°

Fig. 4. Critical value of the electric Bond number Becrit as a function of the relative initial spacing ψ for three values of the angle m. Asymptotic

(analytical) and numerical approaches.

XYZ Translation table

Plexiglas walls

Grounded Needle

10 µl syringes

HV Needle

Fig. 5. Anchored droplets test cell.

Raisin et al. 129

interfacial tension (here = 0.014 N/m). Experimental values are lower but very close to those determined by the numerical simulations. The same trends are observed: (Becrit)110° > (Becrit)120° > (Becrit)130° and the numerical curve lies within the domain defined by the error bars of the corresponding experimental measurements. Moreover, the limited accuracy (± 5 degrees) on the measurement of the anchorage angles m, used to plot Fig. 7, can explain the dispersion of the experimental results.

IV. SIMULATION OF TWO FREE DROPS IN A STAGNANT FLUID

Simulation of a second configuration is performed.

Water drops are initially suspended in oil, close to each other and between two plane electrodes. Axial symmetry is retained, but the two drops are meshed separately to enable the computation of different radii and variable position of the drop pair with respect to the electrodes. The two drops are now completely free to move from their initial position. Their electric potential is computed

by assuming that the total charge of each drop is zero and thus depends on their shape and on their position between the electrodes. From the hydrodynamics, three different Navier-Stokes problems are solved: one for the oil and the two others separately for the two water drops. Remeshings of the computational domain are performed throughout the simulations to ensure the accuracy of the resolution especially in the oil film between the drops. We present preliminary results obtained with a pair of two identical drops of radius R0 = 4 mm initially spaced by 1 mm, at equal distance of two electrodes separated by 100 mm and subjected to a potential difference increasing, first, up to 3 kV and then up to 30 kV in 0.1 µs (Fig. 8). The latter corresponding to the order of magnitude of applied electric field E0 intensity in up to date CECs [1]. Fig. 8 shows the deformation and relative displacement of the two drops and the associated oil film thinning. Highest oil flow velocities are located between the facing drop surfaces and are of about 0.2 m/s at E0 = 0.3 kV/cm and 0.7 m/s at E0 = 3 kV/cm. In this case of quite large drops, two coalescence

(a) (b)

Fig. 6. Experimental pictures of the electrically induced deformation of two anchored meniscus. (a) represents the initial configuration with no

electric field while (b) shows the critical configuration (last stable deformation before coalescence).

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

ψ = So / Ro

Be crit

Numerical 120°

Numerical 172°

Experiments 110°

Experiments 120°

Experiments 130°

Fig. 7. Critical value of the electric Bond number Becrit as a function of the relative initial spacing ψ for different values of the angle m. Numerical and experimental approaches.

130 International Journal of Plasma Environmental Science and Technology Vol.3, No.2, SEPTEMBER 2009

schemes can be observed depending on the applied electric field intensity. At a low E0, the drops are first slightly deformed, with an elongation of their axis of symmetry of less than about 1 percent at t ≈ 0.1 seconds (Fig. 9). Then, the drops start to move toward each other (as illustrated by the velocity field on Fig. 8a), under the effect of electrostatic attraction, until coalescence (between t ≈ 0.1 s and t ≈ 0.45 s). The progressive film thinning (Fig. 10) induces a progressive increase in the electric field intensity in the oil film between the drops and with respect to the previous cases of anchored droplets, no steady state can be obtained here. Another major difference is that, due to the shear stresses exerted by the oil flow at the drops interface, very low separation distance can be achieved before coalescence occurs. At relatively high E0 and thus higher electrostatic pressure, the oil film thinning is mainly promoted by the deformation of the droplets (axis elongation rising up to 6 percent) as the displacement of the drops center of mass is negligible (Fig. 8b). According to Fig. 10 drops surface instability (and thus coalescence) seem to appear at a higher separation distance S probably because of the cone-like shape of the facing interfaces. In the case of small droplets (R0 smaller than 50 µm), which are of primary interest in the understanding of failed electrocoalescence phenomena, the effect of interfacial tension is greater and deformations of the droplets will remain very small. Coalescence is believed to occur mainly at very low separation distances as in the former case E0 = 0.3 kV/cm).

V. CONCLUSION

An analysis of the deformation and coalescence of

two closely spaced drops of conducting liquid suspended in an insulated fluid under the action of an electric field has been proposed. Numerical simulations combined with experimental measurements have confirmed and

extended results of analytical treatments on the determination of the quasi-steady limit of stability of two facing anchored droplets subjected to a potential difference. The electrocoalescence appears in this configuration once the distance between droplets has approximately been reduced to one half of its initial value. Some preliminary simulation results on the dynamics of electrocoalescence, obtained in the case of two freely suspended drops have revealed completely different behaviours. The coalescence time of small droplets should correspond directly to the time required to remove all the oil contained between the facing drop surfaces. Further works will thus focus on varying geometry, fluid and electric related parameters and study their influence on the film thinning process.

ACKNOWLEDGMENT

This work was developed in the framework of a collaborative research between the French CNRS and the Consortium working on the project ”Electrocoalescence – Criteria for an efficient process in real crude oil systems”; co-ordinated by SINTEF Energy Research (contact person is L.E. Lundgaard). The project is supported by The Research Council of Norway, under the contract no: 169466/S30, and by the following industrial partners: Aibel AS, Aker Solutions AS,

(a) (b)

Fig. 8. Radial velocity (colour) and velocity field (arrows) in oil at time t = 0.405 s with E0 = 0.3 kV/cm (left) and at time t = 0.00453 s with E0 = 3 kV/cm (right). Initial drops position and shape in black

lines. Colour scale unit is m/s and arrows have been normalized.

Fig.9: Relative elongation of drops axis vs time.

Fig.10: Minimum oil film thickness vs time.

Raisin et al. 131

StatoilHydro ASA, BP Exploration Operating Company Ltd, Shell Technology Norway AS, Petrobras, Saudi Aramco.

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