Electrohydrodynamics of Dispersed Drops of Conducting ... · role is the electrical treatment of water-in-oil emulsions. On the crude oil extraction sites, the use of choke valves

I. INTRODUCTION

The study of the influence of an electric field on a system of two immiscible fluids with an initially well defined interface is an important part of Electrohydro-dynamics (EHD). The deformation and stability of free drops, bubbles, menisci, pendent drops, etc. have been investigated by numerous authors, particularly in the simplest case where the interface separates a conducting liquid from an insulating fluid. The first basic study in this EHD domain dates back to Rayleigh who analyzed the stability of an isolated perfectly conducting drop [1]. Another reference theoretical work is due to Taylor who determined the steady finite deformation and instability of an electrically neutral drop of conducting liquid subjected to a uniform electric field [2]. Shortly after, Latham and Roxburgh [3] extended the ellipsoidal approximation for deformed drop used by Taylor [2] to examine the problem of two conducting drops which interact in a uniform field; most often, the drops attract each other and merge, a process called electro-coalescence. This problem is also a basic one and it plays a fundamental role in several situations. A first situation is the behaviour of water droplets in electrified clouds: the interaction between close water drops is suspected to trigger drop disintegration and to contribute to the electrification of clouds whose summits are warmer than 0°C [3]. A second domain of application where the phenomenon of drops merging under the action of an electric field might become increasingly used is the lab-on-a-chip domain devoted to various

chemical or biological analyses on very small amounts of products; for example an appropriate electric field can be applied to handle and to merge two droplets of reactants, thus inducing a chemical reaction or a biological test [4]. A domain of very important practical interest where the interaction of close conducting drops plays a crucial role is the electrical treatment of water-in-oil emulsions. On the crude oil extraction sites, the use of choke valves on the wells generates very fine emulsions from the mixture of crude oil and salted water. Purifying the crude oil of sediments, water and corrosion inducing salts is the first step in oil treatment; elimination of catalyst poisons and salts is achieved by further introducing and dispersing fresh water and then removing the salted water. The problem with these usually stabilized emulsions is to separate water and oil. The natural settling out by gravity does not work with small water droplets ( <~ 50 µm) because their terminal velocity in oil is far too small. In practice, electrocoalescence has long been used [5] to break the water-in-oil emulsions by dramatically increasing their mean droplets size and thus substantially reducing the time for water settling out. The compact electrocoalescers presently in operation take advantage of the increased rate of electrocoalescence due to the more or less turbulent flow of the emulsions [6]. Despite numerous experimental and theoretical investigations, it is not so easy to determine the optimum operating conditions of compact electrocoalescers that, furthermore, do not work on all crude oils. The main reason for this is that crude oils contain numerous products like surfactants, waxes, asphaltenes which can modify the interface properties and, even, build an interfacial film whose rheological properties are far from being clearly characterized [7]. Now, even in the case of

Electrohydrodynamics of Dispersed Drops of Conducting Liquid: From Drops Deformation and Interaction to Emulsion Evolution

P. Atten

Grenoble Electrical Engineering laboratory (G2Elab), CNRS, Grenoble INP & Joseph Fourier University, France

Abstract—The interaction and behavior of droplets of conducting liquid suspended into an insulating medium and

subjected to an electric field are considered, in particular with relevance to electrical treatment of water-in-oil emulsions. After recap of the influence of an electric field on a single conducting drop, the action of a field on two close drops is examined. In the static case of anchored drops which attract each other, results are recalled concerning the interfaces deformation and disruption under the field action. In the dynamical case of free drops, their spacing decreases with time until an interface instability creates a bridge; in the asymptotic case of very close small droplets, an approximate model is briefly presented, which predicts the time for draining of the oil film between the droplets. The formation of a bridge between the drops does not necessarily lead to their merging. Examples of partial coalescence and bouncing with charge exchange are presented and qualitative explanations are proposed. The two-drop behavior under field leads to some insight on the evolution of a water-in-oil emulsion subjected to an AC electric field. In a stagnant emulsion, electrocoalescence results first in a fast increase of the droplets mean size and, finally, in a quasi steady arrangement of nearly equally spaced big drops in rows aligned with the field. In a flowing emulsion, the shear and turbulence play the major role in promoting quasi collisions of droplets, the electric field leading to coalescence if the film draining time is lower than the time of close proximity of drops.

Keywords—Electrocoalescence, electrical drops bouncing, water-in-oil emulsions

Corresponding author: Pierre Atten e-mail address: [email protected] Presented at the 2012 International Symposium on Electrohydrodynamics (ISEHD 2012), in September 2012

2 International Journal of Plasma Environmental Science & Technology, Vol.7, No.1, MARCH 2013

pure oils, the emulsion evolution under an applied field is not yet totally characterized and described. An overview is given here of the present knowledge concerning conducting drops suspended in insulating pure oil and subjected to a uniform field. Based on recent investigations, we focus on the system of two close drops, examining the basic phenomena involved in the deformation and approach of the drops and then, in their coalescence or bouncing; finally water-in-oil emulsions are considered and briefly discussed in the two cases of stagnant and flowing emulsions.

II. FIELD ACTION ON A SINGLE CONDUCTING DROP

Inside a highly conducting liquid, the electric field is

null so that the liquid is equipotential. The surface charge density s at the interface between conducting liquid and insulating fluid is related with the field E at interface : s = E where is the permittivity of the insulating fluid. The field exerts a force on the surface which results in the so-called electrostatic pressure pes = (1/2)E2 directed outside the conducting liquid.

The first study of the influence of electrostatic pressure on the stability of a liquid drop was performed by Rayleigh who considered a uniformly charged drop and showed that the maximum total charge Qmax that a drop can sustain is [1]:

3 8 RTQmax (1)

where R is the drop radius, the permittivity of the suspending fluid and T the interfacial tension. This maximum charge Qmax corresponds to the field Emax at the drop surface:

R

TEmax 2

22

(2)

By introducing the electrical Bond number Be defined as the ratio of electrostatic pes and capillary pcap pressures : Be = pes/pcap and taking the expressions (pes)crit = (1/2) (Emax)

2 and pcap = 2T/R, we obtain the critical value Becrit = 1 which means that instability occurs when the electrostatic pressure compensates for the capillary pressure. Another classical problem concerns a single conducting drop subjected to a uniform field E0. By using the approximation of a prolate spheroid, Taylor [2] derived the critical field inducing the drop instability:

R

TE crit0 2

.640

(3)

Clearly the interface disruption at the drop poles occurs when the electrostatic pressure overtakes the extra capillary pressure due to curvature, which would give Becrit = 0.653 if we would use the field and curvature

values at the pole; in practice, by retaining the applied uniform field E0 and the radius R of the undistorted drop, we obtain:

.0520 /2

)( 21

20 RT

EBe crit

crit (4)

There are several modes of ejection of liquid from the poles of the drop [8] which are analogous to some of the modes observed in the atomization process from a meniscus, a domain which has been and is still extensively investigated [9, 10].

A very interesting situation is the one known as electrowetting and sketched on Fig. 1. A drop of conducting liquid lies over a sheet of insulating material coating a planar counter-electrode. By increasing the applied voltage, the drop spreads out because the gain in electrostatic energy of the system compensates for the increase of energy associated with the generation of extra surface (see the review by Mugele and Baret [11]). This spreading process is limited by an instability of the interface due to the high field near the edge of the drop; then charged droplets are ejected (Fig. 2) that modify the field and result in a saturation value of the radius of the spread drop. An unexpected phenomenon observed with AC voltage of high enough frequency is a mean steady state between creation of new droplets and merging of previously ejected droplets into the big drop (electro-coalescence process).

metal

insulatingsheet

V

conductingdrop

metal

insulatingsheet

V

metalmetal

insulatingsheet

insulatingsheet

V

conductingdrop

conductingdrop

Fig. 1. Basic electrowetting set-up; partially wetting water droplet at

zero voltage (left) spreads out when subjected to a voltage (right); (after Mugele [12]).

Fig. 2. Image taken from a video clip of a water droplet on a silanized glass surface (after Mugele [12]).

Atten 3

III. FIELD INDUCED INTERACTION OF TWO DROPS

There exists a vast variety of behaviors of two conducting drops subjected to a field, depending on the size and initial spacing of the drops, on the DC or AC field strength, on the interfacial tension and, of course, on the drops charge. Keeping in mind the electrically induced separation of water-in-oil emulsions, only the case of uncharged drops is considered here. But, even globally uncharged, two close drops subjected to an electric field interact and deform. The interaction force can be approximated by the expressions valid for conducting spheres. A. Interaction Force between Uncharged Conducting Spheres

Outside a single sphere in a uniform field E0, the

sphere influence is equivalent to that of a dipole of moment: m = 4πR3E0 located at its centre. For two spheres with a large spacing s (see Fig. 3), to a good approximation the interaction force F between the spheres of radius R1 and R2 is given by the equivalent dipoles interaction; the force on sphere #2 then has the components [5, 13]:

1) cos (3 12 24

32

312

0 -d

RREFr (5a)

2sin 12 4

32

312

0 d

RREF (5b)

where is the angle between the field direction and the drops centres line (taken as r axis) and d is the centres distance (d = R1 + R2 + s). This force is attractive for

55 )3Arccos(1/ 1 and tends to align the

spheres centres with the applied field, but its strength is very limited for large spacing. When the spheres spacing s is small (lower than the smallest radius R2), the surface charge distributions are very different from the ones for a single sphere. The spheres being equipotential, the field at the facing poles is E V/s (V: potential difference between spheres) and it can take very high values for very close spheres. In that case we expect a strong attraction force between the facing surface charges. An asymptotic expression can be obtained by approximating the field by E V/r and replacing the spheres by paraboloids of same curvature at axis so that r = s + (2/2)(1/R1 + 1/R2) (see Fig. 3); for s << R2 the force on sphere #2 is [13]:

1

21

212

sRR

RRVF

(6)

The main difficulty here is to express V in terms of E0 (see a semi-empirical approach in [14]).

An exact treatment for two conducting spheres in a uniform field developed by using bispherical coordinates leads to the following expressions [15]:

)sin cos ( 4 22

21

22

20 FFREFr (7a)

)sin ( 4 23

22

20 FREF (7b)

where the coefficients F1, F2 and F3 are complicated series depending on the two ratios s/R2 and R2/R1. For very small spacing (s/R2 0), F1 diverges whereas F2 and F3 take saturation values [15]. An empirical expression of the form F1 ~ s- with β 0.8 has been proposed [13] and leads to the following expression [16] for the force between two spheres of same radius R: Fr 0.92 π E0

2 R2 (R/s)0.8. B. Static Case: Drops Deformation and Instability The electrically induced attraction force between two close free drops makes them moving towards each other, possibly up to contact. The applied field E0 also induces a deformation of the drops because of the non uniform distributions of field E and electrostatic pressure pes = (1/2)E2 at the interface. We can gain information on the drops deformation by considering the case of anchored drops (which are not free to move). A first configuration has been investigated which consists of a metallic sphere, immersed in oil, hanging above a layer of water (Fig. 4). This configuration can be viewed as the limit case of a droplet very much smaller than a neighbouring drop, so that the deformation of the droplet can be neglected due to the high level of its capillary pressure. By applying a potential difference V between the sphere and water, the water-oil interface rises (Fig. 4), the sum of capillary force due to its curvature and extra gravitational force balancing with pes.

Fig. 3. Two sphere configuration.

s0

R

Fig. 4. Sphere/water layer configuration with water/oil interface deformed by electrostatic pressure.

r

s

R1

R2

r


However a static solution for the interface deformation exists only when V < Vcrit [17, 18]. For V > Vcrit the interface disrupts and a column of water hits the sphere [19]. The critical spacing scrit is just a little higher than half the initial spacing s0 [17, 18]. The critical potential difference Vcrit is defined by the critical value Becrit which depends on the Bond number Bo proportional to the density difference m between water and oil [17, 18]. By taking, at axis, the electrostatic pressure pes characterising the undistorted interface and taking the sphere radius R as reference for the radius of curvature, the electrical Bond number Be was defined in [17, 18] by:

)(

2

20

2

T

R

s

ΔVBe

(8)

The critical value Becrit was determined analytically [17] in the asymptotic case of very small spacing (s0/R << 1) as a function of the Bond number Bo = m g s0 R/T (g: gravitational acceleration); the general solution (Fig. 5) was obtained by numerical simulation [16, 18] and the experimental results are in very good agreement with predictions [16, 18]. The critical value characterizing the effect of capillary forces alone (Bo = 0) is Becrit 0.05. The case of two facing anchored drops of same radius is sketched on Fig. 6. By assuming that the volume of the drops, outside the capillary tubes, is strictly constant, an elongation of the drops is induced by imposing a potential difference between them. Asymptotic solutions for the static deformation of the drops were also obtained by an analytical

development [14] in the case of very small spacing (s0/R << 1). The results are qualitatively similar with those of the aforementioned configuration: as predicted by a very crude model [20], the variations of Be (defined by (8)) versus the amplitude of deformation exhibit a parabolic shape with maximum - defining the critical conditions - such that the critical spacing scrit is slightly higher than half the initial spacing s0 [14]. A numerical treatment of the problem confirmed the basic properties of the solutions [16, 18] and led to determination of the critical values Becrit (Fig. 7) depending on the initial relative spacing s0/R and on the angle m characterizing the initial shape of the drops (see Fig. 6). Experiments give values of the critical voltage of interface instability in fairly good agreement with the computed predictions [16, 18, 21]. C. Motion and Deformation of Two Free Drops For two free water drops in oil subjected to a uniform field, we are facing a dynamical problem because generally the drops move toward each other, tend to align their centres with the field and also deform. Their velocity of approach depends on the drainage force for which there is no simple expression when the drops are close. An accurate determination of the evolution of pairs of water drops can be obtained only through numerical simulations. It is nevertheless possible to have a qualitative picture of the behaviour that can be expected by considering the electrical Bond number Be. For two uncharged drops of same radius aligned with the field, the potential difference scales with 2RE0 [14] and the electrical Bond number relative to the poles is given by:

~

2

220

s

R

T

RE

p

pBe

cap

es (9)

Clearly the s-2 dependence of Be implies that Be(t) increases with time because the drops approach each other.

0,0

0,1

0,2

0,3

0,0 0,1 0,2 0,3

Bond number Bo

Be

crit

T = 25 mN/m Asymptotic case

Fig. 5. Rigid sphere/water layer configuration: critical value of the electrical Bond number Be as a function of Bond number Bo for a

polybutene oil (after [16, 18])

s0 R0

m

Fig. 6. Two facing water drops (of same radius) anchored at tube tips. The drops are characterized by their radius and the angle m.

0.01

0.1

1

0.001 0.01 0.1

= s 0/R0

Be c

rit m = π/3

m = π/2

m = 0.9 π

Fig. 7. Two anchored drops configuration: critical electrical Bond

number Becrit as a function of the relative drops spacing for 3 different shapes of the meniscus (after [22]).

Atten 5

During the drops approach, the interface deformation progressively grows; we guess that interface instability develops rapidly when Be ~ 1 which corresponds to spacing of the order of:

~ 00 BeR

Rss pole

inst (10)

where Rpole is the radius of curvature of the interface, at the axis, just before instability begins to grow and Be0 and R are the initial values for undistorted drops. This instability very shortly leads to contact between the drops, the interfaces contact being a necessary condition for coalescence. For big enough drops and/or strong electric field, when Be0 is not much smaller than 1, the drops deformation is noticeable and the interface instability starts for an instantaneous spacing sinst not so small compared with s0. This behaviour is illustrated in Fig. 8 showing the shape of drops at the moment where instability is triggered (result of numerical simulation of two drops evolution under field, performed using the commercial software COMSOL Multiphysics™ [18, 21]). For small water droplets or weak applied field, the electrostatic pressure is much smaller than the capillary pressure: Be0 << 1 and the droplets deformation can be neglected. In that case the spacing sinst at which instability triggers is much smaller than s0 and, to a first approximation, we can assume that the droplets move toward each other without being deformed. By further considering a small initial spacing (s0/R << 1), the electric field at facing poles of the spheres (with spacing s) takes the following approximate expression for 0.001 ≤ s/R ≤ 0.1 [16]:

1.87 0.85

0

s

REE (11)

For a viscosity of the oil oil much higher than the water viscosity, the thinning of the oil film between the droplets induces convection rings inside them; this fact together with the undistorted shape of the droplets and the pes distribution at the close interfaces results in the viscous force being proportional to the product oil R wdrop (like in Stokes formula!), wdrop being the droplets

velocity [16, 24]. In this asymptotic case, it is easy to determine the following time variation for spacing [16]:

1 )(

1.7

0

20

1.7

0t

s

REB

s

ts

oil

(12)

where B is a non dimensional constant. As can be seen on Fig. 9, this simple law (12) very well accounts for the spacing decrease obtained by numerical simulation. From (12) and (11) we deduce the following expressions for the time tdrainage of drainage of the oil until contact of (undistorted) drops [16, 24]:

1

0

1.70

20 es

oiloildrainage p

AR

s

EBt

(13)

where (pes)0 is the initial electrostatic pressure at drops facing poles and A is a non dimensional constant (related to B). Extensive numerical simulations give results which very well support relation (13) [16, 24] and confirm that, to a first approximation, the contact time versus (pes)0 is independent of the drop radius and of the initial (small) spacing (Fig. 10). Further simulations with drops of different radii and also by varying the oil viscosity and

Fig. 8. Numerical simulation of deformation and approach of two free water drops (R0 = 200 µm) in oil (initial distance s0 = 100 µm) under a field E0 = 7 kV/cm. Drops shape and oil velocity just before

interface disruption (after [23]).

s /s 0

(s /s 0)1.7

0

0,25

0,5

0,75

1

0 10 20 30 40 50Time t (µs)

Fig. 9. Time variation of spacing obtained by numerical simulation

of oil film thinning between two small and close water droplets; R = 20 µm, initial distance s0 = 2 µm, E0 = 4 kV/cm (after [23]).

Fig. 10. Numerically determined drainage time as a function of the maximum of initial electrostatic pressure for even droplets pairs, of

various radius and spacing, subjected to various field strengths (after [16]).

10

0.01

0.1

1

10 100 1000 10000

Initial maximum electrostatic pressure (pes)0 (Pa)

Dra

inag

e ti

me

(m

s)


the interfacial tension [16, 24] give results that are also consistent with (13). The drainage time has a decreasing trend a little stronger than 1/(pes)0 for high (pes)0 values (see Fig. 10); this presumably arises because the electrical Bond number Be, proportional to pes, is no more negligibly small and, therefore, the drops deformation is no more totally negligible and influences the droplets motion. A detailed examination of results showed that tdrainage begins to depart from the 1/(pes)0 law when Be0 takes values higher than a few 10-2 [16]; this observation appears to be consistent with the critical Be values for static instability (§III-B and Fig. 7).

IV. COALESCENCE AND PARTIAL COALESCENCE

The process of drops approaching and deforming until instability develops is reasonably well understood and modelled. Concerning the interface instability, from what is known of EHD meniscus disruption and atomization, at one of the drops facing poles the formation of a bump is expected that extremely rapidly elongates and hits the other drop, thus forming a bridge between the two water drops. What happens after appears to be trivial: the connected drops are expected to merge under the action of interfacial tension. However the electrical forces which are redistributed after the drops bridging can affect the coalescence process, particularly when the applied field is strong. Let us consider the case of two drops of very different size subjected to a field as shown in Fig. 11-a. Before the bridging of the globally uncharged drops, there exists a potential difference between the two drops and, therefore, a strong field in the thin oil film and strong electrostatic pressure at the facing interfaces; the resulting attraction force tends to expel the oil lying between the drops. Just after instability and bridging, due to the high conductivity of water, there is a quick charge exchange through the bridge that redistributes surface charges and pes; the resulting electric force (Fig. 1-b) then tends to draw the two initial drops apart. For a low enough applied field, this force partly counteracts the capillary forces which tend to smooth out the interface shape; consequently, as the applied field is increased, a slower and slower drops merging is expected. Experiments were performed in order to investigate the influence of an applied field on the coalescence of a small water drop ( 0.7 mm) and a bigger one, the two drops being immersed in a crude oil [25]. In these experiments the small drop was falling above the big one and the collision and coalescence could be observed using a fast near-infra-red (NIR) video camera [25, 26]. Although the surfactants and asphaltenes present in the crude oil may modify the properties of the water-oil interface, the observed phenomena give a qualitatively correct insight in the influence of electric forces. The photographs shown in Fig. 12 (crude oil at 40°C with

µoil = 62 mPa.s) illustrate the slowing down of the merging process due to the electric field action and show that the coalescence duration increases significantly with the applied field E0. Fig. 12 moreover reveals that the interface shape is sharper for E0 = 750 V/cm than for E0 = 250 V/cm due to pes acting mainly in the zone of maximum field. For a high enough applied field, the electric force on the top of the connected droplet can exceed the capillary force; then the upper part of the droplet tends to rise and does not merge into the big drop. The temporary water

Fig. 11. Schematic depiction of surface charge density and electrostatic pressure at the drops interfaces; a) water droplet close

above a water drop; b) just after bridging.

0 5 10 15 20 (ms)

t

Fig. 12. Images from 2 movies of drop coalescence in a crude oil at 40°C taken at the same instants after bridging. Top images: E0 = 250 V/cm, falling droplet diameter D = 710 µm. Bottom

images: E0 = 750 V/cm, D = 750 µm (after [25, 26]).

E0 = 1.33 kV/cm

E0 = 1.66 kV/cm

Fig. 13. Selected images from 2 movies of partial coalescence in a crude oil at 60°C; droplet diameter: D = 710 µm and 730 µm for

top and bottom images respectively (after [25]).

E0

b)

E0

a)

Atten 7

column (between the droplet upper part and the bottom drop) pinches off, leaving a daughter droplet. This process, known as partial coalescence, was observed in the experiment with crude oil at 60°C (moderate viscosity oil = 25 mPa.s) [25] and two examples are shown in Fig. 13. Notice that increasing the field results in a daughter drop of increasing radius (Fig. 13). The threshold field Eth for partial coalescence can be estimated by balancing the electrical and capillary forces; in conditions prevailing for the images shown in Fig. 13 Eth ~ 1 kV/cm [26]. Otherwise, the very thin water ligament remaining at the end of partial coalescence process (at 13 ms in Fig. 13) eventually breaks up giving a string of very small droplets. This droplets string shortens but does not disappear and prevents the daughter droplet to reach the bottom drop [25]; presumably very intricate phenomena of coalescence, droplets ejections and charge exchanges, etc. are involved, depending on the field frequency.

V. DROPS BOUNCING UNDER STRONG FIELD

The rapid increase with E0 of the size of the daughter droplet might suggest that for E0 >> Eth the partial coalescence cannot be distinguished from bouncing. However, in crude oil at 40°C (high viscosity oil = 62 mPa.s), it was not necessary to apply a very strong field to observe a bouncing: as a matter of fact, there is an abrupt change from coalescence to non coalescence at a critical field also on the order of 1 kV/cm [25, 26]. As can be seen on Fig. 14, there is no partial coalescence and the droplet remains hanging above the tip of the conically deformed interface of the bottom drop. This steady state can be explained by the occurrence of an exchange of charge between the two drops at every alternation of the AC applied field (1 kHz bipolar square wave) and by the subsequent sequences of attraction and repulsion of the droplet. A fact to underline here is that the charge exchange is not accompanied by any significant mass transfer. If a DC field would be applied, after charge exchange, there would be a bouncing of the droplet, carrying a net charge, which would rise under an upward directed electrical force.

This phenomenon is not specific to the studied crude oil at moderate temperature and was also observed with different pure oils [27-30]. Fig. 15 exhibits a similar bouncing phenomenon under DC applied field [27]. Oppositely charged drops of similar size also bounce when the applied field is strong enough [27-29]. The bouncing mechanism is the following: the local instability of the interface(s) generates a short and thin bridge and the life time can be very short (<< 10-3 s) [27, 28, 30] or long (~ 1 s see [29]) depending on the various parameters (in particular the drops size and the oil viscosity). In practice, as can be seen on Fig. 16 (taken from [29]), in a way similar to the one depicted in Fig. 11, the charge exchange through the bridge promotes

a repelling force on the drops which drives the drops away and elongates the bridge that becomes thinner; under the action of the capillary force, the linking thread eventually breaks up (Fig. 16). Depending on the detailed mechanism of ligament break-up, a small mass exchange is possible between the drops but it would result in a vanishingly small percentage of change in the volume of drops. Increasing the applied field induces an increase of the spacing sinst at which the interface instability begins to develop and, therefore, an increase in the initial length of the bridge is expected. Moreover the repelling force

Fig. 14. Images from a movie with crude oil at 40°C (oil viscosity oil = 62 mPa.s) showing non coalescence; applied field E0 = 1250

V/cm, droplet diameter D = 730 µm (after [25]).

Fig. 15. Electrically driven bouncing of a salted water drop in

silicone oil (oil 900 mPa.s). Left: diagram of experimental set-up. Right: droplet coalescence (top, DC field E0 = 1.6 kV/cm) and bouncing (bottom, E0 = 3 kV/cm) (after Ristenpart et al. [27]).

Fig. 16. Formation and break-up of a bridge between two

oppositely charged water drops in silicone oil (oil 45 mPa.s). DC field E0 = 4 kV/cm, droplet diameter: D 1.6 mm (after Jung &

Kang [29]).


( E02) accelerates the elongation of the ligament so that

its break-up occurs for an increasing value of the length/diameter ratio. For high enough values of this ratio, the ligament break-up leaves tiny satellite droplets. Very likely such a mechanism of break-up of long enough ligament is responsible for the generation of extremely fine droplets forming a ring cloud at the bottom of the fallen droplet as can be seen in Fig. 17. In this case, presumably at least two fine droplets are created per period of applied voltage. The zone of mist was slowly swelling with time until it (after about 2 seconds) resulted in changes in electric field and forces distributions (Fig. 17).

VI. FIELD ACTION ON WATER-IN-OIL EMULSIONS

Up to now the interaction of two drops only, subjected to a uniform electric field, has been considered. In an emulsion there are a huge number of droplets with size varying on a large range and each droplet is subject to the influence of numerous neighbouring droplets. However the problem is not so hopelessly intricate because the interaction force between close enough drops varies greatly with the spacing s ( s-0.8) so that, most often, only the interaction with the nearest drop is decisive. Therefore the understanding of two electrically interacting drops plays a key role in the study of water-in-oil emulsions under the influence of electric field. The electrical treatment of these emulsions is generally based on the use of AC field of moderate frequency in order to limit the disturbing effects of the bare or coated electrodes [31]. In a stagnant emulsion subjected to a uniform AC field, pairs of close droplets tend to align with the field direction and the droplets draw near each other up to contact. In mineral or silicone oils, coalescence occurs for small enough droplets size because the threshold fields for partial coalescence or

bouncing scales with (T/R)1/2 [26] (like the critical field (3) for single drop instability). Electrocoalescence makes the mean size of droplets to increase with time and the total droplets number, of course, to decrease; this implies a progressive increase of their mean spacing and, therefore, of the mean time required for droplets pairs to get into contact. The emulsion evolution which is fast at the beginning thus progressively slows down [6, 32]. Fig. 18 shows that, after a few seconds of field application, the size distribution of the emulsion has dramatically changed and there are a very limited number of rather big droplets, the size of which has an upper bound corresponding to the instability condition (3). In practice, the interaction with other drops clearly makes the maximum size lower than this upper bound. In fact Fig. 18 highlights the tendency of the biggest drops to arrange in strings of approximately equidistant drops aligned with the field, behaviour reminiscent of the electrorheological (ER) effect in suspensions of solid particles. However, there is a difference with ER fluids because of the finite spacing between the drops; we conjecture that the quasi-stability of these strings of interspaced drops is due to a periodic (at doubled frequency) exchange of charge between the drops and the related subtle balance between their attractions and repulsions. Generally, in the electrocoalescence process, three stages can be distinguished. In the first stage, close drops are approaching one another, their relative motion leading them from typically a finite spacing (on the order of R2 for low enough water cut R2 being the radius of the smallest drop) to a very small one (small fraction of R2). In the second stage, there is drainage of the oil film between the close drops; this stage is of short duration in pure oils (see Fig. 10). In the last stage, the drops in contact merge. In practice, drops merging depends on several parameters like the applied field, the local shear rate, etc. and might also be influenced by interfacial films in crude oils. In a stagnant emulsion, the first stage can take a rather long time because, for a spacing s >~ R2, the attraction force is of very small amplitude (given by relation (5-a)). In flowing emulsions, that is the case in the commercial compact electrocoalescers, the rate of drops

Fig. 17. Images from a movie with crude oil at 40°C (oil = 62 mPa.s) showing non coalescence and generation of a mist of tiny droplets; applied field E0 = 1.5 kV/cm, frequency f = 1 kHz,

droplet diameter D = 800 µm (after [25, 26]).

E E

Fig. 18. Photographs of water droplets in a mineral oil.

Left: initial stagnant emulsion. Right: quasi steady state after field application, E0 = 4 kV/cm, f = 1 kHz, picture width 2 mm (after

[33]).

Atten 9

collision is mainly determined by the properties of the emulsion flow (and not by the field) which explains why the electrocoalescence rate can be enhanced by the turbulent flow [6, 32]. Fig. 19 illustrates the relative motion of two similar droplets in a shear flow, in the absence of field. The spacing between the drops passes through a minimum value smin which, most often, is definitely non-zero (very small coalescence rate induced by the flow). In a flowing emulsion there is presumably both a statistical distribution of smin on a rather narrow interval and a drops collision rate depending on the flow characteristics. Under the action of an electric field (for instance, vertical field in Fig. 19), the attraction force when drops are close enough (horizontal component of drop centres distance ≤ 2R) will bring the drops closer, possibly down to contact. Typically, there will be contact between the drops if the drainage time becomes lower than the time of close proximity. Considering the case depicted by Fig. 19, the axial component of the drops relative velocity is Rzzuu 2 ~ ) ( ~ 1212 where

denotes the shear rate; assuming that the drops are close

when Rxx 12 leads to the time of proximity t:

1

~

2 ~

12 uu

Rt

(14)

Coalescence of close droplets is presumably possible only when:

1 ~

1

1.7

20

tR

s

EBt minoildrainage

(15)

If the statistical mean value / Rsmin of the ratio smin/R

in the emulsion is known, we can estimate the critical field for electrocoalescence (in a shear flow):

1

~ 0.85

R

s

BE minoil

elcoal

(16)

which has a moderate dependence on the oil viscosity and on the shear rate. The problem of time evolution of an emulsion in turbulent flow has been addressed using crude approximations, in particular concerning the coalescence probability of colliding drops [32]. A more realistic modelling requires knowing the coalescence probability

as a function of the applied field, the size and relative positions of the drops and the flow characteristics (local shear rate, turbulence rate). Such information might be obtained through experiments on drop pairs in a well defined shear flow. The terminal statistically steady state of a flowing emulsion under electric field is presumably characterized by a balance between electrocoalescence (droplets merging) and droplets generation through dispersion of the biggest drops, due to their instability.

VII. CONCLUSION

For two free drops in a pure oil subjected to an

electric field, the process of approach until bridging is reasonably well understood and modelled and the electrical Bond number Be appears to be the main controlling parameter.

The subsequent fate of the bridged drops depends on the electric field magnitude and, of course, on the drops size and interfacial tension; for low enough field, drops merging is slowed down by the electrostatic pressure; above a threshold field, for drops of clearly different size, partial coalescence occurs in some cases; in other cases there is an abrupt transition from coalescence to bouncing. Further investigations are necessary to determine the system properties (oil viscosity, interfacial tension, drops size, etc…) which control further partial coalescence or bouncing.

The knowledge of two-drop behaviour under electric field plays a very important role in understanding the time evolution of a water-in-oil emulsion under electric field. This helps to understand the qualitative evolution of a stagnant emulsion. In flowing emulsions, the flow determines the rate of quasi collisions (pairs of drops temporary in close proximity); the field determines the rate of coalescence of the pairs of very close drops. Very simple considerations lead to an order of magnitude estimation of the field necessary to achieve a noticeable electrocoalescence rate. This is just a preliminary attempt and refined approaches are necessary to have a realistic description of the evolution of flowing emulsions.

For the practical applications with crude oil, the crucial question is to characterize the properties of the water-oil interface affected by numerous compounds as surfactants and asphaltenes. These properties may affect or even subvert the phenomena occurring with pure oils and this opens a new domain of investigations.

ACKNOWLEDGMENT

The author is indebted to J. Raisin and J. -L. Reboud in the lab and to G. Berg, S. M. Helleso and L. Lungaard from Sintef Energy Research (Trondheim, Norway) for collaborative work and discussions continued for the past five years. Work partly developed in the framework of

u x

z

Fig. 19. Relative positions of two nearly colliding close drops in a

shear flow at 4 successive instants (no electric field).


the project “Fundamental understanding of electrocoalescence in heavy crude oils”; coordinated by SINTEF Energy AS and funded by The Research Council of Norway, (contract no. 206976/E30) and by the following industrial partners: Hamworthy Technology & Products AS, Petrobras, Statoil ASA.

NOMENCLATURE

A, B : non dimensional coefficients Be : electric Bond number Bo : Bond number d : distance between two drops centres E and E : electric field and field strength E0 : strength of applied uniform field Eth : threshold field for partial coalescence F1, F2, F3 : non-dimensional coefficients Fr, F : force components in polar coordinates g : gravitational acceleration m : dipole moment pcap : capillary pressure (due to interfacial tension) pes : electrostatic pressure Qmax : maximum total charge of drop R : drop radius Rpole : radius of curvature at the pole of a deformed drop s : two drop spacing s0 : initial value of drops spacing sinst : spacing at which a dynamical instability starts T : interfacial tension tdrainage : time of thinning of oil film between two drops u : axial component of drop velocity wdrop : drop velocity r : interfaces separation at distance of symmetry axis t : time of proximity of two drops V : difference of electric potential between two drops m : density difference between water and oil : permittivity of suspending fluid oil : dynamic viscosity of the oil : angle between electric field and drops centres line m : anchoring angle of a drop at the tip of a capillary : distance to symmetry axis : shear rate

SUBSCRIPTS

crit : critical value (instability of static solution) max : maximum value min : minimum value

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Atten 11

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