Weber Number Reynolds

Ohnesorge

ε 0 Permittivity of vacuum: ε 0 = 8.85*10 -12 F/m. L Characteristic length (often refers to the diameter of droplet)

ρ o (kg/m 3 ) Oil density.

Mr Francis McCLUSKEY

Professeur des universités, Université Grenoble Alpesr e .

Mr Philippe MOLINIE

Professeur assistant, SUPELEC -Membre

Mr Jean-Luc REBOUD

Professeur des universités, Université Grenoble Alpes -Membre.

Preface

This work was funded by the project "Fundamental understanding of electrocoalescence in heavy crude oils"; co-ordinated by SINTEF Energy Research. The project was supported by The Research Council of Norway, under the contract no: 206976/E30, and by the following industrial partners:

Wärtsilä Oil & Gas Systems AS, Petrobras and Statoil ASA.

First I want to express my thanks to SINTEF (the research center of Norway) for enabling such an interesting joint work. I thank Mr Lars Lundgaard for his warm welcome at meeting, and Asmund

Ervik for his interesting discussion at G2ELab, Grenoble. I express my gratitude to my supervisor Pr Jean-Luc Reboud for his teaching and patience. He does not only fill me with his scientific knowledge and skills, but also gives me the urge to surpass myself. I also thank Dr Pierre Atten for his guidance through my PhD studies.

Thanks to the jury members of my PhD defense, Pr Laurent Berquez, Pr Christophe Louste, Dr Nelly Bonifaci, Pr Francis McCluskey and Dr Philippe Molinie for their thorough review of my work and for all their encouraging comments and insightful remarks. Especially I am grateful to Pr Francis

McCluskey for his help in English corrections of my manuscript.

In addition I thank to all the members of my equip MDE with whom I spent four years for my graduate study. Their kindness and helps accompany me to go through these years, and make me to have more confidence on myself.

Nomenclature

R cap (mm) Inner radius of used needle tip.

a (mm/s 2 ) Drop approach acceleration.

Introduction

Though the use of crude oil dates back over a millennium, it has nearly become ubiquitous over the last century. In the petroleum industry, refined products of crude oil are used in many domains, such as the chemical industry, plastics, synthetic fibers, fuel, lubricant… The formation of crude oil varies over a very long time while the consumption rate is fast. Thus worldwide depletion of crude oil comes into question. For an oil field, extraction of crude oil under the earth's crust will reduce the underground pressure to a point that the latter can no longer drive oil through wells. The method of oil extraction by injecting water and gas into oil reservoirs was commonly applied. When the crude oil and co-produced water and gas pass from the wellhead to the manifold, there is generation of oilin-water or water-in-oil emulsions. Water-cut can sometimes be higher than 80%, greatly increasing the crude oil volume and generating extra transportation cost. Moreover inorganic salts insoluble in oil are removed by washing or desalting the crude oil with relatively fresh water. Even at lower water-cut, water droplets in oil enhance the corrosion of instruments and should be removed before the refinery processes.

A fast rate of separation and a low value of residual water in the crude oil (typically few tenths of percent in weight) are desirable. Since the density of water is larger than that of oil, sedimentation allows the separation of large water drops. However, at low water-cut the emulsified small water droplets are particularly difficult to remove because the settling velocity in viscous oil decreases dramatically. Emulsion separation into oil and water can then be improved by adding chemical demulsifiers, by increasing the temperature of the emulsion to reduce the viscosity, or by increasing the drops size by merging them. This last is enhanced by applying electrostatic fields that promote coalescence and appears as the best way to speed up the demulsification process.

To optimize electrocoalescence with variable crude oil qualities, improvement of the understanding of the dynamics of water-oil interfaces and validation of numerical models is desirable. Consequently, it is necessary to provide detailed experimental data on the electrocoalescence of very small droplets.

Therefore, the aim of the present PhD is to perform experiments on small droplet pairs falling in stagnant model oil under DC and AC electric fields.

The first chapter of the thesis gives an overview of crude oil demulsification process using electrocoalescence. Former drop coalescence studies in laboratory conditions are presented, as drop collision in air in meteorology or coalescence under electric field of different liquid-in-liquid cases.

Attention is paid to different parameters influencing the drop coalescence process in gas or liquid medium. Existing numerical studies and experimental investigations are presented.

The second chapter presents actual existing drop injection methods with their advantages and disadvantages. Experimental set-ups for drop injection, including test cell design and measurement devices are also presented. Image processing of visualizations and measurement techniques (with related uncertainties) of physical properties such as temperature, interfacial tension, viscosity and density are discussed. Finally model oil properties are summarized.

The third chapter focuses on the well-controlled generation of very small droplets in the model oils.

As proposed by [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF], droplet-on-demand generation by the electrohydrodynamic (EHD) method is obtained by applying high voltage electric pulses to a pendant water meniscus. This EHD injection method using Multi-Stage Pulses (MSP) is improved for drop injection into new oils.

Influencing parameters are taken into account to perform droplet injections in the range of 20-300 micrometers in diameter. It is observed that most of the droplets are not electrically neutral and improvements to the Multi-Stage Pulses are proposed to reduce the droplet electric charge. Use of drop falling velocities is also proposed to characterize the drop size and electric charge, with the related uncertainties.

The fourth chapter illustrates a dual drop injection method from a single needle, for generating axisymmetric configurations of droplet pairs aligned with the electric field and gravity. Influence of hydrodynamic and electrostatic interactions between droplets are taken into account to characterize with accuracy the droplet pair size and charge. For that, the different forces acting on the droplets are presented and simplified models are discussed.

The fifth chapter presents the experimental cases of electrocoalescence of droplet pairs under DC and AC electric fields and summarizes a first data set of 70 different cases. Examples of the analysis of the transient evolution of drop pair spacing and a discussion of some observed phenomena are proposed. Finally improvements are suggested for further experiments.

Chapter 1. Previous studies on droplet coalescence

Introduction

Droplets coalescence phenomenon appears in liquidgas or liquidliquid two phase media, from nature as raindrop generation and falling, to industry use (sprays, emulsions, chemical or biological processes…). The major topic of our work will concern an immiscible liquidliquid two phase medium, specifically the coalescence of water droplets in oil under electric field. For droplet coalescence to take place, application of forces is involved, both external and internal. Droplet collision or attraction can bring into contact droplet interfaces, which play main roles in merging or bouncing processes. For drop merging processes, the coalescence will appear spontaneously driven by capillary forces of the droplets, under condition that separation forces remain smaller than capillary forces. If any extra force terms disturb this coalescence process, the coalescence will sometimes not be complete and partial coalescence or even no coalescence would appear. In this way, the coalescence efficiency comes onto stage, which can be expressed as the combination of a probability of contact and a ratio of total to partial coalescence.

This chapter presents first the context of crude oil extraction and refinery, where water drop coalescence techniques are used to break down water-in crude oil emulsions. Then an overview of research on liquid droplet coalescence is proposed, from early works on liquid droplets in gas, involved in meteorological studies, to later investigations of droplet coalescence in liquid-liquid medium...

Water in oil 1.2.1 Crude oil presence in history

Crude oil, also referred to as petroleum, is a brown to black liquid composed mainly of hydrocarbons, located under the Earth's crust. It is originally formed by organic matter which sediments and accumulates through biodegrading transformation, and is then compressed and heated over millions of years [START_REF] Tissot | Petroleum Formation and Occurrence[END_REF]. The use of crude oil has a long history which dates back to about five to six thousand years before Christ [START_REF] Simanzhenkov | Crude Oil Chemistry[END_REF]. In Greece, the crude oil was used as a kind of medicine to heal diseases in fourth to fifth century. Also the well-known 'Greek Fire' which made use of petroleum as a combustible, served as a military weapon in many battles in the seventh century. The Greeks put the mixture of asphalt (another physical form of known petroleum) with other chemical compounds on the end of sticks, and then threw them when lit. According to manuscript of (Herodotus 457AD), the ancient Greek historian, construction of Babylon walls used asphalt which was thrown up by the Euphrates river. In India, a huge pool constructed about five thousand years ago in Mohengo -Daro used a thin Asphalt layer on the pool wall as water resistant material [START_REF] Simanzhenkov | Crude Oil Chemistry[END_REF].

With advancement of history, more and more products have been derived from petroleum. Up to now there are many petroleum products such as plastics, ink, motor oils, gasoline and so on. With this increase of petroleum consumption, depletions of crude oil fields come into question. For oil productions, when the crude oil is extracted from an oil field, it often contains different proportions of water and other chemical or solid species. The water content is brine because it is a solution of salt, and this water content is always highly undesirable. Moreover, during maturation of the oil field the underground pressure is no longer sufficient to drive crude oil flow along producing well, thus gas and water are injected into the oil layer through a separate well, to increase the pressure inside. This process is called oil field recovery and the water injection can increase drastically water content inside crude oil, even to 80 % or 90 % [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF]. Finally, the high salinity inside crude oil is the major cause of corrosion in equipment. To reduce the salinity dilution water is introduced into oil body to dissolve brine water droplets. This process is called desalting during crude oil refinery [START_REF] Speight | The Chemistry and Technology of Petroleum, Fifth Edition[END_REF]). For such water contents inside crude oil, large water drops can be separated from the oil body by gravity effect. When the water cut is lowered to about 17 wt%, the average drop diameters is from 0.06 to 0.4 mm [START_REF] Borges | Use of near infrared for evaluation of droplet size distribution and water content in water-in-crude oil emulsions in pressurized pipeline[END_REF]. Thus other techniques other than the use of gravity are needed to treat the water emulsions.

When crude oil was first pumped from a well, apart from the hydrocarbon components it contained a quantity of water, chemical compounds such as nitrogen, oxygen, sulfur, and metallic species [START_REF] Speight | Handbook of Petroleum Product Analysis[END_REF]. Thus refinery process is needed to treat the crude oil. The first petroleum industry was founded in 1859 [START_REF] Speight | The Chemistry and Technology of Petroleum, Fifth Edition[END_REF]. Since then vast research has been carried out on crude oil refinery, among which one part concerns water emulsion breakup. Chemical method of dehydration in crude oil consists of adding chemical surface active agents which can remove already existing emulsifying agents inside crude oil, neutralize repulsive electrical charge on droplets, prevent small particles such as clays, drilling muds and ion sulfides from staying on the droplet surfaces in order to facilitate small droplets merging into large drops, so as to separate them by gravitational effect [START_REF] Balson | Chemistry in the Oil Industry VII: Performance in a Challenging Environment[END_REF].

The first attempt to separate water -oil emulsion with the help of electric fields was performed by [START_REF] Cottrell | Separating and collecting particles of one liquid suspended in another liquid[END_REF]. They found that when two electrodes with sufficient applied voltage were immerged in oil medium, the water droplets could form a chain between them. Coalescence between the adjacent drops in the chain could occur and form larger drops which then fell out of the chain under gravity effect, leading to phase separation [START_REF] Cottrell | Separating and collecting particles of one liquid suspended in another liquid[END_REF]. They also invented a separator with electric applications [START_REF] Cotrell | Process for separating and collecting particles of one liquid suspended in another liquid[END_REF]. The method of emulsion separation using electric fields is different than that by chemical agents because the former only uses electric forces to promote coalescence between drops, while the latter focus on the chemical compositions of the oilwater surroundings. [START_REF] Pearce | The mechanism of the resolution of water-in-oil emulsions by electrical treatment[END_REF] observed the chain formation of droplets in crude oil emulsion under electric fields as illustrated by Fig 1 .2 below. In his research the water cut of the oil is 11 % and the diameters of the coalesced drops can be as small as 0.05 mm. Based on this previous research, different influences of DC voltage [START_REF] Charles | The mechanism of partial coalescence of liquid drops at liquid/liquid interfaces[END_REF] and pulsed DC voltage [START_REF] Bailles | An experimental investigation into the use of high voltage D.C. fields for liquid phase separation[END_REF]) [START_REF] Bailes | Column liquid contacting with vigorous agitation balanced by electrostatic coalescence. I: Intensified coalescence-redispersion[END_REF] were investigated. The use of AC electric fields to promote the drop coalescence in crude oil emulsion was studied by Taylor and Spenser in 1988 [START_REF] Taylor | Investigations into the electrical and coalescence behaviour of waterin-crude oil emulsions in high voltage gradients[END_REF]. They observed conditions where chains of drops form, and other conditions without chain formation, with which drop coalescence appears more efficient.

Chemical additives were also studied to break stable emulsions in crude oil [START_REF] Jones | Water-In-Crude Oil Emulsion Stability And Emulsion Destabilization By Chemical Demulsifiers[END_REF], which lead to research into the combination of electric fields and chemical additives [START_REF] Dezhi | Demulsification of water-in-oil emulsion by wetting coalescence materials in stirred-and packed-columns[END_REF]) [START_REF] Eow | Electrostatic and hydrodynamic separation of aqueous drops in a flowing viscous oil[END_REF]. Up to now, available treatments on water emulsion breakdown in crude oil consist of mechanical, thermal, electrostatic, chemical method (Mohammed et al. 1993a;Mohammed et al. 1993b;[START_REF] Mohammed | Dewatering of crude oil emulsions 3. Emulsion resolution by chemical means[END_REF]Chen et al. 1994) or the combinations of the above mentioned methods [START_REF] Less | The electrocoalescers' technology: Advances, strengths and limitations for crude oil separation[END_REF].

Water emulsion breakup

In the modern petroleum industry, pumped crude oil from an oil field should first go through refinery process to reduce impurities contained inside. These impurities, generally consisting of water, salt, acid gas and other solid contents can cause corrosion, deposition inside the tubes and catalyst poison effects [START_REF] Fahim | Fundamentals of Petroleum Refining[END_REF]. The quantity of impurities may also cause extra transportation cost due to their extra volume when the water-cut in oil reaches more than 50 % [START_REF] Mhatre | Electrostatic phase separation: A review[END_REF]) [START_REF] Silset | Emulsions (w/o and o/w) of heavy Crude Oil. Characterization, Stabilization, Destabilization and Produced Water Quality[END_REF]. When crude oil is extracted from below sea water, foam formation can increase the viscosity of the liquid and cause pollutants [START_REF] Schramm | Petroleum Emulsions Basic Principles[END_REF]. Considering all the problems, only a small watercut of 0.001 % is allowed at the final stage of dehydration [START_REF] Silset | Emulsions (w/o and o/w) of heavy Crude Oil. Characterization, Stabilization, Destabilization and Produced Water Quality[END_REF].

The first stage of oil refinery is separation which aims at separating gas, liquid and solid phases inside crude oil body. The separation process is shown on figure below. achieved with the help of gravitational effect. After this preliminary separation phase, the next step is dehydration which means to remove water droplets emulsion contained in crude oil. This emulsion is formed at different steps: intense emulsification of water drops in crude oil can be produced both by shear forces which breakup large water drops into smaller ones during oil flow inside tubes [START_REF] Thompson | Emulsification and demulsification related to crude oil production[END_REF], and through pressure release valves from the well to the production manifold [START_REF] Less | Dehydration Efficiency of Water-in-Crude Oil Emulsions in Alternating Current Electrical Fields[END_REF]; also during crude oil refinery, desalting process requires fresh water injection to dissolve salt content. [START_REF] Manning | Oilfield Processing of Petroleum: Crude oil[END_REF]. To remove this stable water in oil emulsion, the dehydration method is very important. Fine water droplets formed stable emulsion inside crude oil is very difficult to be broken down, and the typical size of the droplets is less than 50 µm in diameter [START_REF] Atten | Electrocoalescence of water droplets in an insulating liquid[END_REF][START_REF] Less | The electrocoalescers' technology: Advances, strengths and limitations for crude oil separation[END_REF]. The gravitational effect for wateroil phase separation is no longer efficient because, according to Stokes law, the sedimentation speed of water droplets is proportional to the square of the drop diameter, and the finest droplets determine the residence time of the emulsion in the separator [START_REF] Less | The electrocoalescers' technology: Advances, strengths and limitations for crude oil separation[END_REF][START_REF] Hellesø | Experimental study of electrocoalescence of water drops in crude oil using near-infrared camera[END_REF]. Thus increasing droplet sizes by merging them are priority to accelerate drop sedimentation speed. As oil is a non-conductive medium, the conducting water drops subjected to an electric field are polarized, and the polarity effect of neighboring droplets will drive them to merge together. This merging process under electric fields is called electrocoalescence. As drops merge together, their mass increases and gravitational effect can easily cause them to sediment down. process. In petroleum industry, high electric fields are applied onto the flowing crude oil emulsion to make the dispersed water drops coalesce in order to separate them from bulk phase [START_REF] Eow | Electrostatic enhancement of coalescence of water droplets in oil: a review of the current understanding[END_REF].

Interaction of two drops in electric field

For most of the research on drop coalescence, the studies have been focused on the interaction of two drops because in chain formation under electric fields in crude oil, it was found that the coalescence occurred between drop pairs instead of several drops [START_REF] Galvin | Design principles for electrical coalescers[END_REF]). The smaller the distance between the drops was, the faster the coalescence took place. To simulate real crude oil environment, drops sizes should be chosen in the micrometer to tens of micrometer ranges in model oil. The choices of the latter are either oil with good stability in the face of oxidations, or crude oils.

Oils exhibit a low conductivity with compared to water. When an outer electric field is applied, the electric force will favor and accelerate coalescence process. When two electrically neutral drops in close vicinity are under electric fields, an induced charge will appear on the surface of the two drops with different polarities [START_REF] Davis | Two Charged Spherical Conductors in a Uniform Electric Field: Forces and Field Strength[END_REF]. For this case, more descriptive model is given in [START_REF] Pedersen | Forces and movement of water droplets in oil caused by applied electric field[END_REF]). Drop coalescence studies are not recent, especially in clouds physics. The coalescence is often referred to as merging of two colliding drops into a larger one. The first water drop coalescence phenomena were observed in air. The fact that a liquid body can either coalesce or bounce on another liquid body was observed by Rayleigh in using two fine liquid jets in 1880. He concluded that the scattering of fine jets was caused by the capillary forces which acted on each droplets, and that the force was so high that it made droplets bounce off each other when they collided [START_REF] Rayleigh | Further Observations upon Liquid Jets[END_REF].

For drop colliding model in air medium it is the competition between the surface tension which tends to conserve the drop shape, and the kinetic energy which can merge or even split the drop after coalescence. Drop colliding model was first widely used in interest of meteorological studies and droplets were set to collide in atmosphere air condition. Early research on drop coalescence was simulated by numerical method and it was found that collisions between small droplets inside clouds could give a wider range of droplet productions than that issued only from condensation procedures [START_REF] Mordy | Differences in Coalescence Tendencies in Computed Condensation Cloud Droplet Spectra[END_REF]. Some years later raindrop size distributions after coalescences were simulated by confining drop concentrations with their initial sizes in a discrete volume [START_REF] Srivastava | On the Role of Coalescence between Raindrops in Shaping Their Size Distribution1[END_REF]. Early experiment of [START_REF] Adam | The Collision, Coalescence, and Disruption of Water Droplets[END_REF]) used droplets of 60 μm -500 μm in sizes with 10 -15 C-10 -12 C variable electric charges. Some of these droplets were driven by electric field from their trajectories to collide in a normal gravitational zone. By varying velocities of the droplets, different collision behaviors are observed. They reported that when the velocity was less than 2.2m/s, permanent coalescence would occur, when velocity surpassed this value, drop splitting after coalescence would happen constantly. During their experiments, charges on droplets within this impact parameter range did not affect either collision or coalescence process.

Later followed experiments of [START_REF] Brazier-Smith | The Interaction of Falling Water Drops: Coalescence[END_REF] in which uneven drop pairs of r 1 > r 2 were used with radius varied from 150 μm to 750 μm, to collide in air. They aimed at studying the critical condition under which the coalescence drop broke up into fragments. x was defined as the perpendicular distance from the center of one drop to the undeflected trajectory of another drop. On Fig 1 .6 the parameter x is showed between the two drops, and U is the relative velocity of the small drop with radii r 2 to the big drop with radii r 1 . Then x is in the range of 0 to ±(r 1 +r 2 ). x c is defined as the critical value of x below which the two drops undergo permanent coalescence while above which the coalesced drop will further split. The coalescence efficiency is defined.

(1.1)

In simulations done by [START_REF] Martin | A Numerical Study of the Effect of Electric Charges on the Efficiency with which Planar Ice Crystals Collect Supercooled Cloud Drops[END_REF], it was proved that electric charges on colliding particles could enhance collision efficiencies. The model depicted critical trajectories of charged water drops of 0.02 mm -0.17 mm in diameters which collided with charged ice crystal plates of 0.1 mm -1.3 mm in sizes. In 1982 another paper came out with more detailed experiments to explain raindrop formation [START_REF] Low | Collision, Coalescence and Breakup of Raindrops. Part I: Experimentally Established Coalescence Efficiencies and Fragment Size Distributions in Breakup[END_REF]. In their experiments, drop sizes were uneven and the smaller drops were blown to the path of the larger one by compressed air, to perform drop collisions. They had analyzed in detail drop geometric deformations when drops collided in their free fall paths. It was concluded that for a high energy collision, a disk shape deformation of the colliding drop could favor coalescence because this deformation dissipated most of the impact energy of the system. If the colliding energy was much higher, splitting of the coalesced drop could occur. When a low energy collision took place, only an induced wave could travel from bottom to top. The above phenomenon demonstrated the influence of interfacial dynamics with a scale smaller than the whole drop diameter.

Fig1.7. Scheme of droplet collisions. Left: high energy collision; right: low energy collision. The diameter of the large drop D L varies from 0.18 cm to 0.46 cm while that of the small drop D S varies from 0.0395 cm to 0.18 cm.

On this figure D L >> D S [START_REF] Low | Collision, Coalescence and Breakup of Raindrops. Part I: Experimentally Established Coalescence Efficiencies and Fragment Size Distributions in Breakup[END_REF]. Drop-on-collision in gas is not only used for meteorological study, it is also expanded in other domains. Later another study in hydrocarbon droplet collision appeared with head-on and off-center cases [START_REF] Jiang | An experimental investigation on the collision behaviour of hydrocarbon droplets[END_REF]. They observed that the phenomenon was different to water droplets collisions. With increasing energy, they had successively permanent coalescence, bouncing, permanent coalescence during which a dimpled disc shape of drop appeared, and droplet splitting, because high collision energy exceeded surface energy which served to keep the drop in its spherical shape. Also they discussed the phenomenon as a function of Weber number (We) defined below [START_REF] Jiang | An experimental investigation on the collision behaviour of hydrocarbon droplets[END_REF].

(1.2)

Here ρ is the liquid density, r 1 and r 2 are the radii of the two colliding drops, U is the relative velocity, σ is the surface tension. In low Weber regimes, permanent coalescence occurred.

Generally in gas for drop coalescence, partial coalescence, bouncing and disintegration, it is the inertia and Weber number which determine the drop behaviors. For drop splitting, viscous dissipation plays a role and the behavior can be related to Ohnesorge number comparing the viscosity and surface tension effects. A further improvement in the drop collision model was done by [START_REF] Klimin | Collision efficiency calculation model as a software tool for microphysics of electrified clouds[END_REF]) and, in their simulations collision efficiencies were calculated for charged drops in an external electric field. Some year's later further investigation was provided on this study. [START_REF] Qian | Regimes of coalescence and separation in droplet collision[END_REF]) enlarged the collision conditions in experiments for both water and hydrocarbon droplets in environments of different gases such as air, nitrogen, helium and ethylene. By varying gas densities via pressure and molecular weight, droplet collisions exhibited different behaviors. It was found that high gas pressure favored drop bouncing, similar to gas viscosity. This was because the air layer trapped between the facing two drops was difficult to be broken up to let contact of two drop surfaces.

More recently, drop collisions in air medium of four drop behavior regimes were simulated and discussed: drop bouncing, coalescence, reflexive separation and stretching separation [START_REF] Pan | Numerical simulation of binary liquid droplet collision[END_REF]. For the stretching separation case more articles were focused on it. This splitting of merged drops after their collisions was further investigated by [START_REF] Ko | Numerical study on bouncing and separation collision between two droplets considering the collision-induced breakup[END_REF]) in simulations and [START_REF] Brenn | Satellite droplet formation by unstable binary drop collisions[END_REF] The same authors also studied the viscosity effect in drop collisions (Gotaas et al. 2007) and it was found that the borders of the coalescence area were shifted towards high Weber numbers for viscous drop collisions. It was concluded that this shift was due to the dissipation of kinetic energy inside the coalesced high viscous drop. When a portion of kinetic energy is dissipated inside merged drop, the available energy for drop separation is decreased. Thus the separation is expected for higher drop incoming velocities, which are at higher Weber number. Water drop collisions from 0.22 mm to 0.45 mm in diameters were studied by [START_REF] Rabe | Experimental investigation of water droplet binary collisions and description of outcomes with a symmetric Weber number[END_REF]) in a spray system. They show that the results obtained for unequal droplet sizes match on a single map, defining the regions of coalescence, reflection and stretching with respect to a new symmetric Weber number taking into account the sum of drop incoming kinetic energy and the sum of surface energy. Head-on collisions of unequal size drops were investigated for both hydrocarbon and water drops, and the relation between Weber number and size ratios were discussed [START_REF] Tang | Bouncing, coalescence, and separation in head-on collision of unequal-size droplets[END_REF].

In recent research in cloud physics, coalescence could occur between two colliding drops in air and the efficiency remained high for electrically charged drops (Mölders & Kramm 2014). With small impact energy of two drops in contact, coalescence will occur completely. However increasing impact energy can cause a flattening phenomenon of collided drops. To some extent this high impact energy can split the merged drop because the surface tension of the drop is no longer sufficient to hold the drop. In this way terms of partial coalescence and coalescence efficiency come into use.

Collision and coalescence of two electrically charged drops was simulated very recently [START_REF] Ghazian | Head-on collision of electrically charged droplets[END_REF]. The simulations consisted of two drops with diameters of 0.1 -0.5 mm colliding in air medium.

The boundaries were found by level-set method and the flow was simulated as axisymmetric, incompressible and laminar flow, coupled with electric fields and charge on the interfaces. Also the surface tension of drops was considered as an important parameter. The maximum electric charge that could be carried by a single drop was calculated via the Rayleigh limit, q Ray , under which the drop was not disintegrated:

(1.4)

Here the ε 0 was the vacuum permittivity, r was the drop radius and σ was the interfacial tension.

Results of simulations allow drawing maps of attraction or repulsion of two droplets carrying charges of the same polarity, with respect to the difference of their charge magnitude, or with respect to the head-on collision velocity. With fixed charges in drops, collisions of drops would lead to coalescence with increasing Weber number. Unequal drops could favor the coalescence process because of loss of symmetry in the system. Sometimes, with high kinematic energy, daughter droplets would be produced after coalescence. This model discussed in detail the influence of electric charge contained in pair colliding drops.

Drop interaction with a plane surface

For drop coalescence experiments in air, another configuration is of one drop falling onto a plane surface of the same liquid. This time the kinetic energy is not involved, instead that the Eötvös number which is the ratio of gravity over surface tension, plays a role. The first discovery of spreading liquid drops on the same surface was done by [START_REF] Mahajan | Liquid Drops on the Same Liquid Surface[END_REF]. Later experiments of liquid drop resting on a liquid surface were investigated by varying the two liquid body media [START_REF] Hazlehurst | Liquid Drops on Liquid Surfaces[END_REF]. It was said that there existed two kinds of drop formation on another liquid plane: one is of primary drop resting on a planar surface and the other is of splashing from the drop merged with the surface which produces a secondary drop. For the first case, it was found that it took more time for the same or alike liquid bodies to merge, than other body media of very different properties. For the second case some studies were investigated more in details recently by allowing one water drop to fall onto another liquid pool [START_REF] Manzello | An experimental study of a water droplet impinging on a liquid surface[END_REF].

In the experiments water kinetic energy was involved and the dynamics of splashing after waterplanar surface coalescence was studied. The same experiments were done later by adding different pool depth for drop splashing dynamics [START_REF] Fedorchenko | On some common features of drop impact on liquid surfaces[END_REF].

Recent studies involve the notion of thin film formation between two liquid bodies when they are supposed to merge. This thin film layer can either be an air layer or a liquid layer, which plays a crucial role in drop coalescence behavior. When a water drop was initially gently placed on another liquid surface, the air film was not penetrated immediately [START_REF] Vandewalle | Dancing droplets onto liquid surfaces[END_REF]. Then by vibrating the liquid surface the air film could be sustained and the drop would rest on the liquid surface for a long time. Another experiment was for droplet bouncing on a high viscosity thin layer inclined surface, it was discussed more in details on the air film surviving which prevented droplets to merge into liquid layer, and the incident normal Weber number [START_REF] Gilet | Droplets bouncing on a wet, inclined surface[END_REF]. It was said while normal Weber number We was between 2 and 15, droplets would partially immerge with the thin liquid layer and reproduce secondary droplets. When We was smaller than 2, air layer would not be broken and droplets would bounce. When We was larger than 15, the air layer would be broken and droplets would merge entirely with the liquid layer. Thus the air layer between the two liquid facing surfaces dominated the coalescence process.

Drops coalescence in liquid medium

Shear flow configuration

When two drops are in liquid medium, shear flow can bring them together and the stress can overcome the film drainage force and surface tension to make the drops coalesce. In some other cases the shear stress is so large that it can split the whole drop [START_REF] Erik Teigen | A diffuse-interface method for two-phase flows with soluble surfactants[END_REF]. The latter case is not the case that will be discussed in this thesis. [START_REF] Howarth | Coalescence of drops in a turbulent flow field[END_REF] proposed that in a shear flow, the drag force induced by the shear stress was a crucial factor for coalescence, more important than the attraction between the two colliding interfaces which was weak at molecular level. Following their theories, many experiments for droplet coalescence processes have been investigated and they all confirmed the theories above, and provided more details of this coalescence phenomenon. In shear flow the effects of confinement on drop coalescence were tested in experiments [START_REF] Chen | Effect of Confinement on Droplet Coalescence in Shear Flow[END_REF], and it was found that high confinement promoted drop coalescence. In their experiments two water drops of 0.1 -0.3 mm in diameters, were released into a channel with a certain distance of separation. This channel was filled up with polydimethylsiloxane oil, and between the two plates the shear rate was constant. The confinement was defined as the ratio of the drop diameter to the height of the flux channel. It was found that under low confinement of which 2R/H < 0.085, the walls had little or no influence. When confinement increased, it promoted drop coalescences. Simulations of the same configuration were first taken by [START_REF] Shardt | Simulations of Droplet Coalescence in Simple Shear Flow[END_REF], in which they used the lattice Boltzmann method to perform low speed droplet collisions. In their simulations, the fluid was simulated as incompressible flow and both chemical potential and capillary tensions on drop surfaces were considered. The phase field and the molecular collisions were taken into account. For the results, they reported cases of total coalescence and no coalescence of the drops. A critical capillary number was determined, depending on which drop coalescence is observed or not. The capillary number (Ca) is defined below, taking into account the shear stress, the viscous force and the interfacial force.

(1.5)

Here μ is the viscosity, R is the radius of the drop, σ is the interfacial tension and ϒ` is shear flow rate.

Thus if the capillary number of the system was lower than a critical capillary number, coalescence of the two drops would occur; if the capillary number was higher than a critical capillary number, no coalescence would take place. It should be added here that according to [START_REF] Shardt | Simulations of Droplet Coalescence in Simple Shear Flow[END_REF], another important parameter which influenced the capillary number was interface diffusivity which is related to the chemical potential of the droplet interface. If two contact droplets with a higher diffusivity of their interface meet, the separating film would rupture more easily, leading to interface contact and then coalescence of the drops.

Fig 1 .12. Simulations of drop contacting interfaces with different capillary number and approaching durations [START_REF] Shardt | The Critical Conditions for Coalescence in Phase Field Simulations of Colliding Droplets in Shear[END_REF].

Furthermore the same author studied critical conditions of drop coalescences in shear flow [START_REF] Shardt | The Critical Conditions for Coalescence in Phase Field Simulations of Colliding Droplets in Shear[END_REF]. In their simulations, the lattice Boltzmann method was adopted, and the chemical potential separated two phases. Simulated drops had the same density and viscosity as the outer fluid and Reynolds number was fixed to be 1. Under these simulation conditions, the figure above showed drop contacting interface behaviors during the drop coalescence process. The film-thinning time and locations varied with capillary number.

Drop interaction with a plane surface and partial coalescence

For drop coalescence with a liquid surface merging into another immiscible liquid medium, the initial kinetic energy of the falling drop is small and the affecting parameters are gravity, surface tension, viscosity. All the parameters mentioned above can be combined in forms of Eötvös number and Ohnesorge number. The film drainage process is always presented during the approach of the drop to the liquid plan. When external electric fields are applied, an extra parameter, the electric Bond number (B e ), should be added. First experiments were done by [START_REF] Charles | The mechanism of partial coalescence of liquid drops at liquid/liquid interfaces[END_REF], and in their experiments the notion of partial coalescence was investigated.

Fig1.13. Illustration of partial coalescence of liquid drop-face bulk in another liquid medium [START_REF] Charles | The mechanism of partial coalescence of liquid drops at liquid/liquid interfaces[END_REF].

When a liquid drop is resting on a plane surface of the same liquid and plunged in another immiscible liquid, a thin film layer is formed between the drop and the plane surface. This film layer is further thinned under the gravity effect of the resting drop, until a critical value above which the layer is drilled out and a liquid column is formed connecting the drop and the bulk phase. At the beginning of this column formation, a bulge may be formed and propagate. Under this effect the column may be broken, forming a satellite droplet. This phenomenon is called partial coalescence. Sometimes the bulge can give multiple satellites formations during the column breaking.

More recently [START_REF] Aryafar | Drop coalescence through planar surfaces[END_REF] studied the partial coalescence process as a function of

Ohnesorge number for the same experimental configuration. Oh=μ/(Rρσ 12 ) 0.5 , where R is the radius of the falling drop, μ is the viscosity, ρ is the density of the liquid, σ 12 is the interfacial tension between the two liquids. In addition, the drop ratio of partial coalescence decreased when Oh approached to 1 and a secondary drop disappeared when Oh was equal to or higher than 1. For one drop falling on another big drop inside crude oil under electric fields, experiments were then compared to numerical simulations to understand the effect of different oil viscosities [START_REF] Chiesa | Investigation of the role of viscosity on electrocoalescence of water droplets in oil[END_REF]. Zooming in on coalescence process, drop surface rupture mechanisms were discussed:

according to [START_REF] Priest | Controlled electrocoalescence in microfluidics: Targeting a single lamella[END_REF], the coalescence was caused by surface dynamic instabilities induced by electric field. The used drop diameters were of 0.2 mm and the placement of drops was very narrow.

Later it was again reported that the whole process of this mechanism was driven by the competition of electric strength and surface tension. If the electric strength was weaker than surface tension, drop coalescence could occur; otherwise the drops may retreat after their contact [START_REF] Jung | A novel actuation method of transporting droplets by using electrical charging of droplet in a dielectric fluid[END_REF]. In other experiments done by [START_REF] Aryafar | Electrocoalescence: Effects of DC Electric Fields on Coalescence of Drops at Planar Interfaces[END_REF] an external constant DC electric field was applied. It was reported that in the high Ohnesorge number region, partial coalescence would occur under DC electric fields while it should otherwise have been suppressed without electric fields. In addition, non-coalescence of an oppositely charged drop with the bulk phase was found above a critical electric fields [START_REF] Ristenpart | Non-coalescence of oppositely charged drops[END_REF]) and it was concluded that this non coalescence was caused by fast charge transfer between the drop and the bulk phase, which resulted in same electric potential for the two bodies. The same experiments were later repeated by [START_REF] Mousavichoubeh | Electro-coalescence of an aqueous droplet at an oil-water interface[END_REF]) in which higher electric fields were used. The drops used were of de-ionized water in diameter range of about 576 μm -1196 μm and the surrounding medium was sunflower oil. They said that the electric fields could increase the film rupture process and when a liquid column was formed between the two contacting drops, the electric fields would detach the draining drop while capillary surface tension tended to pull the drop into the bulk phase and enlarge the column. Thus, competition between capillary force and electric fields took place. If the capillary force exceeded the electric force, the drop would be pumped into the bulk phase. However, if the electric force was higher, the primary drop would not be drained completely and the column became narrower until its breakdown. Thus a secondary drop was formed, training satellites formation through its trail as the breakup of the neck occurred. The same phenomena were also found in other experiments even with oppositely charged drops (Hamlin et al. 2012). When the same experiments were done in crude oil environment, different coalescence behavior was exhibited. There were two series of experiments investigated, one at 40ºC and another at 60ºC [START_REF] Hellesø | Experimental study of electrocoalescence of water drops in crude oil using near-infrared camera[END_REF]. They found that without electric fields, no coalescence was observed.

Under low electric field intensity, coalescence of a falling drop with a stationary one was enhanced.

Increasing electric fields would then stop coalescence in experiments at 40ºC while it had no effect on the experimental series at 60ºC. Under rather high electric fields, a mist of fine charged droplets were injected from the stationary drop at the near contact point, instead of expected coalescence.

Further suppression of this secondary drop by pulsatile electric fields was studies by [START_REF] Mousavi | Electro-coalescence of water drops in oils under pulsatile electric fields[END_REF]. If drop coalescence processes are subjected to electric fields, drops can go from total coalescence to partial coalescence or even non coalescence according to electric field strengths [START_REF] Hellesø | Experimental study of electrocoalescence of water drops in crude oil using near-infrared camera[END_REF].

This partial coalescence or non-coalescence is driven by electric force, completely different from the theory of bouncing drops in an air medium. Under electric fields, when the interfaces break down between the two facing drops, a fast electric charges transfer can occur and thus the drop and the bulk body are of the same potential. At this moment a repulsive force tends to extract the drop from the bulk phase, then a secondary drop formed [START_REF] Hellesø | Experimental study of electrocoalescence of water drops in crude oil using near-infrared camera[END_REF]. The influence of water quality is discussed in [START_REF] Dong | Effects of Applied Electric Fields on Drop-Interface and Drop-Drop Coalescence[END_REF]. It is showed that addition of NaCl into water phase can decrease significantly the drops coalescence time, compared to identical configurations with non-ionic aqueous drops. The drop diameters varied from 1.6 mm to 1.9 mm.

Five series of experiments were done with different additives (HCl, NaCl, NaOH) which serve to form the drop and plan liquid bulk.

Drop pair coalescence

For drops pair coalescence in liquid medium experiments, in an axisymmetric system, the influencing parameters are gravity, viscosity and surface tension. If external electric fields are applied, an extra factor of electric force is added. For 3D system, where the falling drop direction is perpendicular to the electric field, the influencing parameters are the same as in the axisymmetric system, and one difference is that no gravity factor influences the electrocoalescence process. Film drainage phenomena are present in both cases. For two drop coalescence in an immiscible liquid medium, experiments were done first by [START_REF] Sartor | A laboratory investigation of collision efficiencies, coalescence and electrical charging simulated cloud droplets[END_REF]. In its experiments, two drops with sizes of 0.005 -0.05 mm diameter were first produced and then fell under a DC electric field. The test cell was made of Plexiglas, 47 cm in height and 30.5 cm 2 of base, and was fulfilled with mineral oil of 0.878 g/ml density. The drops were made of distilled water and the size of the produced drops was controlled by a 'stop cock' which began or stopped the formation of drops at the tip of a capillary tube. The consumed water volume could be read on the capillary tube. The drops pair collision occurred in axisymmetric configuration. It was found that drop coalescence was dominated by the strength of the applied electric field. Earliest experiments for interactions of phase bodies in another liquid medium took place around 1960, where a useful theory of film thinning came about [START_REF] Shinnar | Statistical Theories of Turbulence in Predicting Particle Size in Agitated Dispersions[END_REF]. It was said that when two bubbles came together, a liquid film would prevent the contact of their facing surfaces. Thus the coalescence of the two bubbles took place only if the film layer between them could drain out. Studies on two charged conducting spheres in a uniform electric field were investigated and electrostatic forces acting on the two close facing spheres were calculated accurately [START_REF] Davis | Two Charged Spherical Conductors in a Uniform Electric Field: Forces and Field Strength[END_REF]. Later, collision efficiencies between a drop pair falling under gravity in a viscous and incompressible flow field (described using the complete nonlinear and time dependent Navier -Stokes equations), were studied by [START_REF] Shafrir | A Numerical Study of Collision Efficiencies and Coalescence Parameters for Droplet Paris with Radii up to 300 Microns[END_REF].

For drop coalescence without electric fields, approaching time of two close drops was calculated and presented in [START_REF] Chesters | The modelling of coalescence processes in fluid-liquid dispersions : a review of current understanding[END_REF]) [START_REF] Abid | The drainage and rupture of partially-mobile films between colliding drops at constant approach velocity[END_REF]) [START_REF] Chesters | The applicability of dynamic-similarity criteria to isothermal, liquid-gas, twophase flows without mass transfer[END_REF]) for different cases. For drops of relatively small size (d < 1mm) with a large distance between them, different models were applied:

rigid spheres; deformable particles with immobile interface; deformable particles with partial mobile surfaces and with fully mobile surfaces.

This drop pair coalescence research is picked up in the 21 th century first by [START_REF] Lundgaard | Electrocoalescence of water drop pairs in oil[END_REF] for research into crude oil emulsions. Later studies on the effects of external electric fields and electrode geometry during electrocoalescence of a drops pair have been investigated [START_REF] Eow | Drop-drop coalescence in an electric field: the effects of applied electric field and electrode geometry[END_REF]. The viscosity effect of the surrounding medium on the coalescence process was discussed [START_REF] Chiesa | Investigation of the role of viscosity on electrocoalescence of water droplets in oil[END_REF][START_REF] Chiesa | Forces acting on water droplets falling in oil under the influence of an electric field: numerical predictions versus experimental observations[END_REF][START_REF] Pedersen | Forces Acting on Water Droplets in Electrically Energized Oil Emulsions[END_REF]). under electric fields. It was found that when the two drops were more than one radius apart, the system could be modelled using a dipole approximation. When the distance was less than one drop radius, the film thinning effect was added and the dipole approximation was no more valid.

Drop coalescence processes in liquid medium are different from that in air because of the influence of momentum and the dynamic viscosity of the surrounding fluid. When the two drops collide in air, they coalescence or bounce off depending mainly on the kinematic energies possessed by the drops, and the air layer does not strongly reduce the initial momentum of the drops. For drops approaching each other inside an immiscible liquid medium, the liquid layer can reduce the approach velocities due to viscosity, and in most cases drop coalescence occurs. deformations were more and more pronounced until interface broke up, leading to coalescence. Just after the coalescence, as shown in (e), the drop was not a spherical shape. Then capillary force of the interface pushed the two extremities towards the center and helped this coalesced drop to return a spherical shape. This simulation is consistent with experimental work done by [START_REF] Guo | Coalescence behaviour of two large water-drops in viscous oil under a DC electric field[END_REF]. In simulations done by [START_REF] Lin | A phase field model for multiphase electro-hydrodynamic flow[END_REF], two incompressible immiscible binary liquids were considered with Neumann boundary conditions, using the free energy electro-hydrodynamic model. The effects of viscosity compared to electric forces were investigated. The two drops were released from parallel micro syringes with a diameter of about 1.25 mm and electric field was perpendicular to their motion. They observed that when electric field was switched on, the two drops began to approach each other under a dipole-dipole system. After collision, the interfaces of two drops broke up and the two drops merged together immediately, forming a cylindrical liquid column. Then under push of capillary force, this cylinder came into spherical form through oscillations. More experiments were performed with a skew angle between the drops and it was found that more time was needed for coalescence process due to the skew angle. With large skew angles and moderate electric fields the drops would first aligned with electric fields and then coalesce. This experiment was compared with simulations done by the same author [START_REF] Mohammadi | Direct numerical simulation of water droplet coalescence in the oil[END_REF]. Moreover investigations of waveform effect on electrocoalescence process was added in their numerical research [START_REF] Mohammadi | Numerical prediction of the electrical waveform effect on electrocoalescence kinetic[END_REF]. In recent study, electric field intensities and drop deformations in coalescence processes were analyzed in details [START_REF] Guo | Coalescence behaviour of two large water-drops in viscous oil under a DC electric field[END_REF]. It was said that a low electric field would favor coalescence while a very high electric field made drops collide and repel from each other. showed that even when the beginning of coalescence has occurred, the electric force can separate again the drops pair. As relatively large droplets were used, of about 1.72mm to 1.80mm in diameter range, deformation of drop shape along electric field was also observed, and the duration of the bridge formed between the two colliding drops was discussed as it was longer with a moderate electric field. It should be noticed that here drops were distilled water mixed with alcohol in order to compare its density to that of surrounding white oil.

Influence of surfactants on moving droplets

For a water drop with free surfactants falling inside oil medium, fluid friction may cause inner fluid circulation within the drop, and thus the falling velocities of the drops increase. This phenomenon is depicted by Hadamard-Rybczynski equation, shown below.

(1.6) [START_REF] Milliken | The effect of surfactant on the transient motion of Newtonian drops[END_REF]. Then this gradient induces a Marangoni force which tends to redistribute the surfactants over the drop surface [START_REF] Teigen | Influence of surfactant on drop deformation in an electric field[END_REF]. When a stable state is reached, the Marangoni force is equal to friction force and the drops can be simulated as solid spheres whose motion can be described by the Stokes equation (Hamlin & Ristenpart 2012). In experiments, falling drops are of diameters D = 0.14 mm, the inertial force and interfacial tension variation are so small that they could not modify the spherical shape of drops [START_REF] Levan | Motion of a droplet containing surfactant[END_REF]. This assumption of spherical shape of drops is consistent with experimental observations. Surfactants can also influence on a micrometer scale the ''approaching process'' of a drop pair and modify the coalescence efficiencies ( [START_REF] Zhang | Discussion of the Drop Rest Phenomenon at Millimeter Scale and Coalescence of Droplets at Micrometer Scale[END_REF]. It is shown from Fig 1 .26 that while two drops were approaching each other, surfactants located between the interfaces would be swept to the fronts of two drops and thus formed a 'barrier ring' which would prevent the two interfaces from coming into contact with each other. In addition effects of surfactants on drop deformation subjected in an electric field was simulated by [START_REF] Teigen | Influence of surfactant on drop deformation in an electric field[END_REF]. Navier-Stokes equation was applied to solve both fluid equations and the interface was tracked using the level-set method. Leaky-dielectric model was adopted to simulate effects of electric fields. Surfactants were considered to be non-soluble and could move along drop interface. It

showed that surfactant effects depended on characteristics of the contacting two fluids and the inner drop circulation which could influence surfactant distributions on drop surface, could favor or decrease drop deformations. Later with the same method it was added that insoluble surfactant could give a solid cover over the falling drop in a liquid medium, while soluble surfactant could give a mobilize drop interface [START_REF] Erik Teigen | A diffuse-interface method for two-phase flows with soluble surfactants[END_REF]. Following the previous research, both experiments and simulations were investigated in studying a water drop covered by surfactant Span 80, falling inside model oil Marcol 52 [START_REF] Ervik | Experimental and computational studies of water drops falling through model oil with surfactant and subjected to an electric field[END_REF].

Conclusion

Crude oil contains a variable amount of water, and their separation is highly needed. Such separation concerns just as much high water-cut mixture, to reduce extra transportation cost, than low watercut mixture, prior to refinery processes. Water-in-crude-oil emulsions are difficult and long to breakup because of the very small size of water droplets, in the micrometers to tens of micrometers range.

To increase the sedimentation rate, thanks to the difference of density between water and oil, the coalescence of small droplets in larger ones is desirable. That can be done on water-in-crude oil flows in electocoalescers, through the assistance of applied AC or DC electric fields, to obtain droplets in the millimeter range.

Coalescence of droplet was studied for long time in air: water drop in meteorological field or more recently hydrocarbons droplets in sprays. The studies of colliding drop pairs enlighten the influence of surface tension and the associated Weber number as the main parameter influencing the coalescence or separation behaviors. The influence of external electric field has been taken into account in some papers, but through the electrophoresis effect on electrically charged droplets. For colliding droplets, good agreements are generally observed between experiments and models when inertia and surface tension effects are dominant. However difficulties remain in the description of the surface effects more locally, such as the one involved in bouncing.

Droplets behavior is different in another liquid, mainly because there is a strong influence of the viscosity of the bulk phase, and of gravity. Different studies have been published for configurations of drop falling on larger one or on a plane surface under electric field. Phenomena as partial coalescence or bouncing have been discussed, involving the viscosity of the liquid bulk and the electrostatic pressure applying on the interfaces. Drop pair coalescence was studied experimentally only in very few papers since year 2000, notably in very recent ones, published after the beginning of the PHD.

Existing experiments have some common characteristics: quite large droplets are generated and observed (not far under the millimeter scale), only few cases of interactions of droplets pair are reported, always in configuration of side-by-side fall in transverse electric fields (that is perpendicular to the gravity). Thus, the observed phenomena are fully 3D. On the other side, numerical models have been developed but rarely compared systematically to experimental data.

In this context, G2Elab works in collaboration with SINTEF energy research on ElectroCoalescence (EC) projects for more than ten years. The present PHD is a part of ECIII project "Fundamental understanding of electrocoalescence in heavy crude oils". Among the conclusions of the previous project (ECII), it was stated that experimental data on the electrocoalescence of small droplets will be useful to improve the understanding of the dynamics of water-oil interfaces and to validate numerical models. The aim of the research project, proposed in Grenoble, concerned initially experiments on droplets pair in oil flows. As a first step the present PHD will focus on pairs of small droplets falling in stagnant fluid, under electric field aligned with the gravity and the droplets pair axis. The choice of this configuration aims at providing simple experimental data to compare with models, thanks to the axial symmetry. The difficulty remains on generating and controlling the injection of very small droplets, up to one order of magnitude smaller than in existing experimental data.

Chapter 2. Experimental set-up and measurements

Introduction

The goal of the thesis is to study coalescence of two drops under electric fields. The drop pair is of micrometer scale and aligned with electric field lines while falling inside model oil. For this study a dual drop injection system combined with high tension application system is needed. The background light is chosen as conventional light, and the test cell filled with model mineral oil is transparent. Camera visualizations are needed for different angles for the drop coalescence process.

A waveform generator is used for dual drop injection process and a high tension connecting electrode is needed for introducing electric fields. The physical measurements consist of liquid temperature, oil-water interfacial tension, densities, viscosities and related uncertainties. This chapter is divided into four parts, and they are respectively drop injection methods, experimental set-up, physical measurements with uncertainties and oil properties.

Existing drop injection methods

In drop injection experiments, different techniques have been developed such as piezoelectric injection [START_REF] Harris | A low-cost, precise piezoelectric droplet-on-demand generator[END_REF], ultrasonic injection [START_REF] Lévin | Micro-Injection System Using Ultrasonic Vibrations for Drop on Demand Ejection[END_REF], thermal bubble injection [START_REF] Chen | Bubble growth and ink ejection process of a thermal ink jet printhead[END_REF], electrohydrodynamic (EHD) injection [START_REF] Chen | A new method for significantly reducing drop radius without reducing nozzle radius in drop-on-demand drop production[END_REF]. These techniques will be presented briefly hereafter. However it can first be mentioned that in most cases the obtained droplet diameter depends on the nozzle size and the most often only droplet diameters of millimeter scale can be obtained. Electrohydrodynamic drop injection technique appears to be the only one allowing us to obtain droplet sizes independent of the nozzle size. Moreover the injected droplets are much smaller than the nozzle tip diameter [START_REF] Raisin | A novel actuation technique for the on-demand injection of charge-free conducting droplets in a viscous dielectric liquid[END_REF]. For the desired experiments on drop electrocoalescence in model oil, the drop diameters are required to be of the order of tens of micrometer. Such drops scale can only be achieved by using EHD injection method. For drop injection into air, up to now, there have been many techniques for injection processes such as thermal method, pneumatic method, acoustic method, ultrasonic method and finally electrohydrodynamic method, often named as EHD method. In thermal drop-on-demand method, heat is transferred to a thin film resistor at the bottom of ink reservoir and this leads to an augmentation in temperature in this reservoir. This would by consequence cause the growth of a vapor bubble, the growth of which pushes a small amount of ink to be ejected from the nozzle of the reservoir [START_REF] Chen | Bubble growth and ink ejection process of a thermal ink jet printhead[END_REF]). This method is often known as ''thermal bubble'' and this is a train ejection, not a single droplet extraction. Thus the diameter of a single drop cannot be exact. Another drop actuation method is called spark bubble and this depends mainly on bubble dynamics and during the growth of which a liquid column is ejected from nozzle. This method is also used in ink jet printing technology, and the advantage of spark bubble is that it favors a large droplet with smaller satellites. Control of drop size can be achieved (Dadvand et al. 2008a). However, both thermal and spark bubble method have the possibility to alter chemical compositions of the fluid in its reservoir due to heating [START_REF] Lee | Microdrop Generation[END_REF].

Drop injection in air

For the pneumatic method, a drop ejected from the nozzle depends on the variation of gas pressure which is introduced via a pneumatic generator to push the liquid meniscus in an ''in-out'' sequence.

The gas pressure pulse is sinusoidal and it is out of phase with the meniscus oscillation. When a jet from the meniscus is pushed out of the nozzle, the jet could be broken when it retracted back to the meniscus, leaving a formed drop. Drops with diameters much smaller than the nozzle size can be produced, but the pneumatic generator requires adjusting the liquid viscosity range to the nozzle size, to inject droplets of a given size (Amirzadeh & Chandra 2010a). For this pneumatic generator, applications are varying from deposition of molten metal droplets [START_REF] Cheng | Producing molten metal droplets with a pneumatic droplet-ondemand generator[END_REF] to recent research into the evaluation of pneumatic methods in biochemical use, which aims at producing drops of pesticide aqueous solution for crop plants and weeds [START_REF] Basi | Evaluation of a pneumatic dropon-demand generator for application of agrochemical solutions[END_REF]. Recently piezoelectric droplet-on-demand generators have come onto the market and are stable with good repeatability of drop production. This method is similar to the pneumatic method because it uses pressure to push out a liquid meniscus. With the application of an electric pulse, a pressure is induced onto the piezoelectric buzzer which then presses the fluid chamber to push out a precise quantity of liquid through the nozzle [START_REF] Harris | A low-cost, precise piezoelectric droplet-on-demand generator[END_REF]. This generator is reported to be able to give good production of water droplets with sizes of 0.5-1.4 mm in diameter. Further decreasing the droplet sizes to around 0.15 mm would be difficult with this technique. Thus other experiment set-ups should be considered.

The acoustic drop-on-demand method, which was first discovered by [START_REF] Meacham | Droplet formation and ejection from a micromachined ultrasonic droplet generator: Visualization and scaling[END_REF] for oil drop injection into air, was developed in many research domains both scientific and industrial, such as in picoliter drug transfer [START_REF] Ellson | Picoliter: enabling precise transfer of nanoliter and picoliter volumes[END_REF] and ink jet printing [START_REF] Krause | Focusing Ink Jet Head[END_REF]. The acoustic energy is focused near the liquid surface to achieve drop injections without any physical contact. The drop volume can be controlled by the applied acoustic energy, frequency and the used liquid property (Arrondo & Alonso 2007). During acoustic drop injection process, propagation and reflection of acoustic waves plays a crucial role [START_REF] Bogy | Experimental and Theoretical Study of Wave Propagation Phenomena in Drop-on-Demand Ink Jet Devices[END_REF].

For the ultrasonic drop-on-demand technique, an ultrasonic pressure wave is generated by a transducer and focused through lenses onto the liquid-air interface at the nozzle tip [START_REF] Lee | Microdrop Generation[END_REF]) [START_REF] Lee | Microdrop Generation[END_REF]. Then this focused ultrasonic wave will interact with capillary wave of liquid-air interface to give a full drop injection process. With this ultrasonic injection method, both individual drop injection and a continuous jet could be achieved. Experiments were done by [START_REF] Meacham | Droplet formation and ejection from a micromachined ultrasonic droplet generator: Visualization and scaling[END_REF] who used nozzle diameters from 5 µm to 10 µm to study water drop ejections into air dynamics and the ejected droplets were of a diameter of 7.2 µm with a nozzle size of 5.93 µm. The drawback of this ultrasonic method is that it can cause nearby equipment to vibrate while ejecting droplets [START_REF] Lévin | Micro-Injection System Using Ultrasonic Vibrations for Drop on Demand Ejection[END_REF]).

Finally we come to the electrohydrodynamic (EHD) method which consists in applying electrostatic pressure to the liquid meniscus, in order to deform and extract a droplet from it. This technique is often used in inkjet printing because it is easy to reduce and control droplet sizes. In the experiments of [START_REF] Rahman | Study of drop-on-demand printing through multi-step pulse voltage[END_REF], they had a minimum droplet size of 40 µm in drop train production.

Droplet injection in liquids

Injection in microchannels

Drop injection into liquids has been the subject of much research in different domains. In microfluidics, small drop injection into other immiscible liquids with successful control of drop diameters over a wide range were done by [START_REF] Nisisako | Droplet formation in a microchannel network[END_REF]. Pushing one liquid medium flowing through a channel forth to meet another liquid which goes inside a perpendicular channel at a variable given space, can cut the former liquid body into drops under drag force. Using this method, droplets of 100 -380 µm which are of a size smaller than the given channel, have been produced.

Some people have done experiments on drop coalescence in microfluidic channels [START_REF] Jose | Droplet arrangement and coalescence in diverging/converging microchannels[END_REF]) [START_REF] Tan | Droplet coalescence by geometrically mediated flow in microfluidic channels[END_REF], however the boundary conditions can influence drop shapes and motion during coalescence process.

Injection of single droplet

In other cases the drop injection techniques are mostly discussed in electrocoalescence experiments.

The investigation of this research is mainly focused on petroleum use and improvement of actual techniques. The drop injection method is to hold a glass cannula with tip size of 0.03 -3.1 mm in diameter, immersed into model oil. The water in the glass cannula is distilled with 3.5 wt% NaCl and the formed water drop at the tip of the cannula is 0.2 -0.5 mm in diameter. To detach the water drop, the glass cannula is pulled upwards and the water drop falls off at the oil-air interface [START_REF] Pedersen | Forces Acting on Water Droplets in Electrically Energized Oil Emulsions[END_REF]). Later this method of injecting water drops into model oils is used for further experiments on drop coalescence under electric fields. Globally there are two types of experiments: one injected water drop falling onto a stationary bigger one and two drops falling side-by-side. In the former case, the research has been led by [START_REF] Ingebrigtsen | The Electrocoalescence of two water drops in oil[END_REF]) in the early years with a test cell filled with crude oil and images visualized by infrared camera. With the same experimental configurations, other research has been performed to complete drop coalescence parameters in more details. [START_REF] Mousavichoubeh | Electro-coalescence of an aqueous droplet at an oil-water interface[END_REF]) did experiments with falling drop size of about 0.98 mm in diameter. [START_REF] Chiesa | Investigation of the role of viscosity on electrocoalescence of water droplets in oil[END_REF]) did experiments with drop sizes 0.17 It is noticed that all the experiments above have large drop diameters for electrocoalescence research, and the coalesced drops are uneven.

Injection of drop pair

To simulate falling drops inside crude oil environment, two water drops falling side-by-side were released and submitted to a transverse electric field. Two falling droplets were first examined by [START_REF] Mohammadi | Electrocoalescence of binary water droplets falling in oil: Experimental study[END_REF]) and the two electrodes were oriented leftright. As with previous experiments, two micro-syringes were used for producing electrically uncharged water droplets with 3.0 wt% NaCl added while transparent transformer oil filled the test cell. When the two water drops fell into the electrode zone, an electric field was applied. The goal of this research was to study droplet moving times with their different initial positions when injected into model oil. It should be noticed here that average size of injected water droplets is 1.25mm. As it was mentioned before, the water droplets in crude oil have a diameter range smaller than 0.05 mm. Thus a further reduction in drop diameters should be considered. Moreover, this system of two drops side-by-side was not axially symmetric, which could complicate the calculations of drop coalescence process. Consequently these drop injection methods were not suitable for experiments of small drop electrocoalescence in axisymmetric system.

Electrohydrodynamic (EHD) injection method

To inject liquid drops into another immiscible liquid without constraint on the size of the injecting nozzle, up to now there are not so many existing methods. A leading research of injecting drops independent of nozzle sizes was done by [START_REF] Chen | A new method for significantly reducing drop radius without reducing nozzle radius in drop-on-demand drop production[END_REF] using piezoelectric transducers.

Further improvement of this injection technique was done by [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF] in which water drops were injected into polybutene oil with the help of multi-step electric pulses. This injection is called Electrohydrodynamic (EHD) because the whole drop formation process is based on water meniscus The application of high voltage on the bottom electrode can induce the upper water meniscus to deform and then detach water drops from the hanging needle. The advantage of this technique is that the injected drop diameters were independent of the needle inner diameter, and could vary over a large range. Thus this EHD injection method is chosen for further experiments of two falling drops in axisymmetric configuration.

Experimental set-up

Experimental apparatuses

For the EHD drop injection set-up, the connection of the different devices is presented on

EHD injection set-up

The first test cell set-up, hereafter called EHD injection set-up, was used for first sets of single drop injection. This first test cell is cubic, with 0.5 mm thick Plexiglas walls and of 9 cm inner length on each side. A grounded needle is placed approximately 1.5 cm above the bottom electrode. The inner diameter of the needle is 0.46 mm.

Needle form and water recipient connections

The forms of needles used in both the EHD injection and the drop coalescence set-ups have the same shape. The tip of the needle is blunt to make sure the water meniscus can anchor on its inner diameter without bordering out. The only difference between the injection needles is the diameter value. For the EHD injection set-up the needle has an inner diameter of D = 0.46 mm, while the needle used for drop coalescence was D = 0.72 mm inner diameter. The outer diameter of the needle used in this latter set-up is D ext = 1.057 mm which is often used for calibrations purposes.

Test cell set-up

Test When zooming in on the test cell, the electrode connected to the high voltage is placed between the needle and the bottom grounded metallic plate. The electrode is 0.5 cm in thickness, and the distance between the needle tip and the upper surface of the electrode was 0.8 mm. The distance between the lower surface and the bottom metallic plate surface was 1.5 cm. The hanging needle was 1.057 mm outer diameter, 0.72 mm inner diameter. Generally the outer diameter was used for calibration of the cameras. This set-up will be called hereafter the ''drop coalescence set-up''. Further zooming in around the boundary of the drop (Fig 2.16) shows that the gray levels of the neighboring pixels are not smoothed (Gray level is uniform on one pixel, whose length is measured as '1' by the software). During all the measurements of length using ImageJ, the starting point and the end point of the yellow line segment should be fixed manually. The uncertainty of the diameter measurement is estimated to be ∆n = ± 1 pixel, with n the measured diameter of the drop in pixels.

For the uncertainties of the calibration, the length of the inner needle diameter was also estimated to within ± 1 pixel.

At the moment of calibration, the length in pixels of the needle inner diameter x was set as reference and was measured manually. Thus this involved also one pixel of uncertainty. The real needle inner diameter is 2R cap which is a fixed value in millimeters, n is the number of pixels across the drop and D

is the real drop diameter in millimeters.

(2.1)

The combined standard uncertainty is computed by summation in quadrature [START_REF] Bell | A Beginner's Guide to Uncertainty of Measurement[END_REF], assuming that the origins of the different individual uncertainties are independent. The summation in quadrature is applied on the uncertainty values when the combination is a sum or a difference and on the fractional uncertainties when the combination is a product or division. The combined uncertainty on drop diameter D calculated using equation (2.1) is:

(2.2)

The droplet diameter D is 3 to 15 time smaller the needle diameter. Then inside the square root the second term is much smaller than the first one and can be neglected. Thus the uncertainties can only take account of the uncertainties of the drop diameters measured in pixels.

(2.3)

For a drop with 134 pixels diameter, the relative uncertainty of drop diameter is ∆D/D = 1/134 = 0.75 %. The relative uncertainty is small and thus, in single drop injection experiments done in the EHD injection setup, drop diameters were estimated using this method.

For experiments done in the drop coalescence setup, the Dalsa Cameras are placed lower and the needle tip is largely out of the camera vision field. It should be noted that the cameras are turned at 90° to get the largest recording length in the direction of the drop falling path. The vision field of the Dalsa camera is 1392*1024 pixels. Thus the two drops are aligned horizontally on all photos. The calibrations are done by lowering the needle into the camera field of vision. Two photos were taken through the needle descending process and the distance between the two can be read on the mechanical micro-screw holding the needle. This calibration will then be verified by measuring the outer diameter of the needle. The real value of the needle outer diameter is D ext = 1.057 mm. This calibration uncertainty can also be neglected compared to the measured drop diameter uncertainties (in pixels). The image is taken with a stronger zoom-in, so that the drop diameter D is only about 20 pixels on the image. A line is drawn across the drop as a probe of measurement of gray levels. On the right of

AOS S-MOTION 1683 Camera visualization

The AOS camera has a fast motion which helps to visualize meniscus dynamics through drop injection process and drop coalescence moments when two falling drops approach each other. The AOS

Uncertainty on transient high applied voltage

Waveform Generator

The electric devices used in experiments have calibration uncertainties. The most important electric device is the waveform generator which determines the transient value of voltage during the pulse trains, and the return to zero after the pulse. During the drop injection process, the water bridge between formed drop and the meniscus should be cut after the shutdown of voltage to ensure electric neutrality of injected droplets [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF]. Thus a possible offset zero is very important to ensure that when the pulse is finished, there is no tension applied on the electrode which might potentially influence the water meniscus on the needle tip. The functions of the Wavetek 395 are not sure to work well and it had a small deviation of 0.1V for the offset value. This deviation is amplified, 2000 times and at the exit of the amplifier, the deviation could reach to 200V. One conclusion was to change it for a newly fabricated waveform generator for a better accuracy.

The offset value helps to modify a waveform location compared to the grounded value 0. For the generator Agilent 33500B Series, the 0 offset has a small derivation to the grounded value. Checked on the oscilloscope, this value is of +140 µV above the ground. This value is not due to the accuracy of the oscilloscope because the latter has an uncertainty much smaller than that value, of about 3.8

µV. Thus for this +140 µV amplified by the amplifier 20/20C, it gives 0.28 V at the exit of the amplifier.

Trek amplifier

The high voltage amplifier Trek 20/20C connects directly to the electrode in the test cell. According to the commercial information, the offset of the amplifier is less than ± 2V. This value is good given that the electrode well returned to zero after application of high voltage. The slew rate of the amplifier is defined as the slope of the transient voltage: dV/dt. According to the manufacturer, the amplifier allows a maximum slew rate greater than 450 V/µs (Anon n.d.). This amplifier can generally give accurate values of electric fields following the created electric signals in waveform generator. Fig

2.
22 is one photo taken from the screen of the oscilloscope. The dark blue curve is the wave form created on the waveform generator, and the green curve is the output signal of the amplifier already divided by a factor of 2000. In this case, the other two curves of the light blue one and the purple one are the outputs of the two Dalsa cameras. When this photo was taken, the two cameras were not triggered up. In other experiments one of the light blue curve or the purple curve follows the signals sent from the computer to see whether triggers were generated correctly. On the oscilloscope, the dark blue line is of 2.00 V scale while the green curve is of 1.00 V. Clearly the two curves are of the same shapes, and the output of the amplifier follows the input signal well.

However limitations of the amplifier dynamics have been reached when trying to generate pulses with high amplitude and fast changes of polarity. F. This wire cannot be reduced to help approach the amplifier to the test cell because of spatial constrains. The electric wire acts as a capacitor and it slows down the changes of voltage. To put it in calculations, two points are taken marked with quadrilaterals and pentagons on the figure below.

The peak to peak voltage of the blue curve is identical to the green curve, of about 12 kV. The total duration of the MSP10 pulse is 1.52 ms and according to the pulse shape, the horizontal distance of the blue curve (pentagon mark to next pentagon mark on the figure) is 24.1 µs. That of the green curve revealed from the figure (quadrilateral to next quadrilateral) is 100 µs. For the blue curve, the peak to peak voltage divided by the time used is 12 kV / 24.1 µs = 498 V/µs. This value of slew rate should be allowed by the amplifier theoretically. However in reality the amplifier follows the green curve. For the green curve, the slope is 12 kV/ 100 µs = 120 V/µs, much smaller than the maximum slew rate indicated by the manufacturer. The limiting current of Trek 20/20C is 20 mA. Using this value, the equation of current with an addition of capacitor can be written as follows.

(2.4)

Here C is the capacity, i is the limit current, and V is the voltage. dV/dt is the slew rate. Taking the slope of the green curve, dV/dt = 120 V/µs, with i = 20 mA, then the calculated capacitor is C = 1.67*10 -10 F. The capacitor has the same order of magnitude as the electric wire between the amplifier and the electrode. It is confirmed that this deformation of the pulse shape may be due to the electric wire which connects the amplifier to the electrode in the test cell.

Uncertainty on mechanical position

The position of the needle with respect to the HV electrode has an influence on the electric field. For the EHD injection setup, the height of the needle is measured by a ruler with 1 mm uncertainty on 15 mm of total height. The relative uncertainty is 6.7 %. In the drop coalescence setup, the needle position is surveyed by a CDD camera and the distance can be measured on the screen with a magnification factor of about 30. The exact scale is given by measuring on the screen the outer diameter b' of the needle (b' = 3 cm), and comparing with the real needle diameter b = 1.057 mm.

Then the height H of the needle over the HV electrode can be calculated for the distance H' on screen:

(2.5)

(2.6) Equation (2.6) gives relative uncertainty of δH/H.

(2.7)

The second term under the square root is much smaller than the first one, because of the small needle diameter and the calculated uncertainty is: [START_REF] Benmekhbi | Dynamic and Rheological Properties of Span 80 at Liquid-Liquid Interfaces[END_REF]. Marcol is a mixture of saturated hydrocarbons and its advantage compared to other model oils lies in its components stability to oxidation. Such properties assure that water-oil interfacial tension change is not due to any possible molecular chemical structure change caused by oxidation since this oil content generally stayed in test cell for weeks once filled up. Primol 352 exhibits the same properties as Marcol 52, the difference is that Primol 352 is more stable while with a higher viscosity (Anon n.d.). During the PHD Primol was only used in EHD injection experiments.

δH/H » δb'/b' » 3 %.

Oil properties

Model oils and measured characteristics

Interfacial tension measurements

Interfacial tension is the force on the interface between two immiscible phases which holds a liquid medium in its proper environment. This interfacial tension is often referred to as capillary tension which is generally discussed in the domain of fluid mechanics as it plays an important role in fluid movements, especially in liquid column breakups and droplet formations. The movements of a liquid drop in its surrounding space or the deformation of its own shape depend on many parameters, including its interfacial tension with respect to the surrounding medium. This interfacial tension between different liquid mediums, including liquidair interface, can be measured in different experiments. The most used are Pendant drop formation and the Du Noϋy Ring method. For our experiments, this interfacial tension is measured for oils such as Marcol 52, Primol 352 or Marcol 52 mixed with Span 80.

The Du Noϋy Ring method and Pendent Drop method were used to measure water-oil interfacial tension.

The Du Noϋy Ring method

The Du Noϋy Ring method uses a platinum ring initially immersed below the interface separating the two immiscible liquids (here in water, the heaviest fluid). Slowly lifting this ring from the interface into the upper liquid permits the measurement of the net fluid force coming from attached liquid meniscus on the ring. This maximum lifting force just before the rupture of the liquid meniscus is directly related to the interfacial tension. This Noϋy Ring method is used for measurements in pure

Marcol 52, and for pure Primol 352 with distilled water, and the tensiometer used is KRUSS School Tensiometer K6. This tensiometer has a correction coefficient only under conditions of measured interfacial tension smaller than 25 mN/m and with a density difference between the two liquids of more than 1 g/ml. Thus for our experiments, no correction term needed to be considered. For pure Marcol 52 with distilled water, the measurements were performed from five minutes (the time for

The Pendant Drop method

The Pendent Drop method was used to measure the interfacial tension of Marcol 52 with different additives. Two kinds of measurements were done: one of 3.5 wt% NaCl contained tap water with pure Marcol 52, another of the same salt water with 0.001 wt% Span80 addition in Marcol 52. The tensiometer used for measurements was KRÜSS Drop Shape Analyzer -DSA30s model. During By dosing the syringe slowly, water content exits from the needle tip and forms a drop. Combined with gravity, this spherical drop slowly deforms into a pear shape, and the analysis of the interfacial tension is based on the water drop shape viewed by a camera placed behind ( http://www.kruss.de/services/education-theory/glossary/pendant-drop/). The calibration of the image depends on the outer diameter of the needle used and the shape parameter calculated by the software should reach more than 0.5 to begin measurements. Generally the measurements end upon detachment of the hanging drop. Due to the accumulation of surfactants on the drop surface the interfacial tension decreased further until a point where it could no longer sustain the drop hanging on the needle tip under gravity effect.

Sum up of all the results with Marcol 52

The liquid [START_REF] Von Szyszkowski | Experimentelle Studien über kapillare Eigenschaften der wasserigen Losungen von Fettsauren[END_REF]. The equation between the surfactant concentration on the liquid interface and interfacial tension is shown below [START_REF] Ervik | Multiscale modelling using molecular dynamics and interface-capturing methods for two-phase flow simulation of droplets covered with surfactants or asphaltenes, and applications to electrocoalescence[END_REF].

(2.8)

Here β = R gas TΓ ∞ /σ 0 , R gas is the constant of ideal gas, T temperature in Kelvin and Γ ∞ the maximum surface concentration of surfactant on drop interface. In equation (2.8) σ 0 is the interfacial tension without surfactant and Γ should be the equilibrium interfacial concentration for a given bulk concentration.. We use this equation although Γ is in fact in our case a transient value that increases with the hanging time of the droplet. The equation (2.8) can be rewritten as follows.

(2.9)

For the calculations, T = 295 K and σ 0 is taken as the first value for all interfacial tension measurement series. The value Γ ∞ = 4.18*10 -6 mol/m 2 is taken from [START_REF] Benmekhbi | Dynamic and Rheological Properties of Span 80 at Liquid-Liquid Interfaces[END_REF] As the interfacial tension decreases logarithmically, the uncertainties of the interfacial tension over a certain time interval can be calculated by using trend line equations. The trend lines have following forms.

(2.10)

Here A and B are constants, x is a variable and y describes the interfacial tension. Then the uncertainty of y can be calculated as:

(2.11)

The relative uncertainty of y is written as: The interfacial tension (after 2 mn) is σ = 42 mN/m ± 6.7 %.

Interfacial tension with Primol 352

The Du Noϋy ring method is used for interfacial tension measurements of Primol 352 -distilled water.

Primol 352 exhibits a more stable interfacial tension trend compared to Marcol 52. Primol 352 has interfacial tension decreases from 43.2 mN/m at the beginning, to about 36.2 mN/m.

Temperature measurements

The temperatures of the test oils are measured by EXTECH instruments 3 Channel Temperature Datalogger, Model SD200. The measured range is -50.0 -1300.0 °C. The resolution of the apparatus is 0.1 °C.

Viscosity & density measurements

The densities and viscosities of the Marcol 52 with 0.001 wt% Span 80 in the test cell were measured with increasing temperatures. The viscosity of the oil was measured with a HAAKE Falling Ball Viscometer C. This viscometer serves to measure viscosities of transparent Newtonian liquids, and it comprises a temperature control bath filled with an appropriate liquid, and two cylindrical tubes, one inside the other. A nickel iron alloy ball of density ρ R = 8.157 g/cm 3 is placed inside the inner cylinder which should be later filled with the liquid to be measured. Between the inner and the outer cylinder wall the thermal bath liquid circulates to fix the temperature of the inner liquid. The viscosity value relates directly to the time of the ball falling a defined distance marked on the cylindrical tube:

(2.13)

The µ is the dynamic viscosity, k is the ball constant given by the manufacturers with k = 0.0685, ρ R is the density of the metallic ball given as 8.157 g/cm 3 , ρ o is the oil density at the bath temperature, and t is the ball drop time measured in seconds. Before each measurement it should take at least 15 minutes after the test temperature was achieved, to wait for the stabilization of the thermal transfer between the inner cylinder tube and the outer circulating fluid. The reproducibility of the machine is 0.5 % according to the manufacturers.

The oil density values are measured by Fischer hydrometer of CAT.No.11-555-D with measurement range of 0.820 -0.890 g/ml, and increment of 0.0005 g/ml. This is a glass tube which floats inside the test liquid if the density of the latter is in the range of hydrometer measurements. The graduations are marked on the glass tube and the oil surface height read on the glass tube corresponds to the value of the oil density. This hydrometer floats in the oil medium in a large recipient and the entire recipient is immersed into the temperature controlled bath.

Conductivity and relaxation time

The (2.15)

The calculated time is τ = 1000-5000 s. Thus the wandering free charge can stay a very long time inside the oil medium and influence the falling drops.

Summation of results

The liquids mentioned above all have their own properties such as capillary

Derived characteristics

Capillary time tcap

Equation of t cap

During the drop injection process, the dynamics of the liquid-liquid interface are affected by electric force, surface tension and surrounding liquid viscosity. With respect to inertia, surface tension has its characteristic time, which is called surface energy time scale. This time characterizes the dynamics of the liquid body under the effect of surface tension [START_REF] Hsiao | The critical Weber number for vortex and jet formation for drops impinging on a liquid pool[END_REF]). Moreover when a fluid is flowing as a cylindrical column, disturbances propagating on its surface, called capillary waves, tend to break it up and the breakup time is of the order of the capillary time. The capillary time is calculated as follows [START_REF] Vicent | Dynamique de ligaments fortement étirés[END_REF]):

(2.16)

In the case of drop injection experiments, the t cap is capillary time, ρ is viscosity of the water, R cap is the inner radius of the needle and σ is interfacial tension. This capillary time gives an idea of the life time of the water column generated by stretching the initial water meniscus, and this may give a rough estimation of the starting value of the electric pulse duration range.

Uncertainty calculations of t cap

The relative uncertainty of t cap can be expressed as a function of relative uncertainties of water density, inner radius of the needle and the interfacial tension respectively.

(2.17)

By comparing the relative uncertainties of the water density, the inner radius R cap and the interface tension, it is found that the relative uncertainty of t cap depends mainly on the last:

(2.18)

Here the interfacial tension varies with different model oils used in experiments.

Viscous time tµ

Equation of t µ

All fluids have a different viscosity and this parameter influences fluid dynamic behaviors under external forces. Thus for liquid thread breakup the viscosity also plays an important role [START_REF] Mckinley | Visco-Elasto-Capillary Thinning and Break-Up of Complex Fluids[END_REF]. With high viscosity fluids , a liquid column or liquid bridge may increase the life time before break-up [START_REF] Furbank | Pendant drop thread dynamics of particle-laden liquids[END_REF]. The viscous time is an important factor used (as is the capillary time) in most cases of liquid thread breakup studies. In liquid-liquid configurations it may involve the viscosities of the two fluids:

(2.19)

For water drop injection in oil, ρ w is water density, R cap is the inner radius of needle, μ o is the viscosity of oil. Generally this viscous time is the upper limit of pulse duration.

Uncertainty calculations of t µ

As used previously, the relative uncertainties on the water density and the needle inner radius are considered to be much smaller than the relative uncertainty on the viscosities, due to the possible variations of temperature. Moreover, the viscosity of water µ w and its associated uncertainty is much smaller than the corresponding values for oil. Then the relative uncertainty on the viscous time can by simplified to be:

(2.20)

Here the viscosities vary with different oils used in experiments.

Reynolds number Re

The Reynolds number is a dimensionless parameter which compares the importance between inertial and viscous forces in fluid mechanics. It is represented as the ratio of momentum force to viscous force.

(2.21)

In the case of our experiments, here ρo is the oil density, U is a characteristic velocity of the falling drop, L is the characteristic length which corresponds to diameters of droplets [START_REF] Mckinley | Wolfgang von Ohnesorge[END_REF], and µ o is the oil viscosity. In experiments of falling drops inside a stagnant oil medium, one example will be taken as a water drop of diameter D = 0.14 mm falling inside Marcol 52 with its terminal velocity U = 0.28 mm/s. Then the calculated Reynolds number is Re = 0.003. This number is very small and it means that the experiments are completely within the classical Stokes regimes [START_REF] Brennen | Fundamentals of Multiphase Flow[END_REF].

Weber number We

The Weber number is also a dimensionless parameter often used in fluid mechanics, and it compares the fluid disruptive hydrodynamic forces to surface tension forces [START_REF] Pilch | Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop[END_REF]) (Anon n.d.).

(2.22)

Here, the ρo is the oil density, U is the velocity of falling drops, L is the characteristic length and σ is the surface tension.

Ohnesorge number Oh

The Ohnesorge number is a dimensionless parameter which relates the internal viscosity dissipation to surface energy. It can also be written as the square root of Weber number divided by the Reynolds number [START_REF] Li | Ohnesorge Number[END_REF]. This can also be referred to as the viscous number because it characterizes viscous effects on drop dynamics [START_REF] Pilch | Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop[END_REF]. Oh number is widely used in describing drop ejection mechanisms, and this number is independent of liquid motion [START_REF] Lefebvre | Atomization and Sprays[END_REF]).

(2.23)

Here, μ o is the viscosity of the fluid, ρ o is the density, σ is the surface tension and L is the characteristic length [START_REF] Mundo | Droplet-wall collisions: Experimental studies of the deformation and breakup process[END_REF]. A low Oh number means that the dynamics of droplet deformation is mainly controlled by the surface tension, while a high Oh number denotes the dominant effect of viscous dissipation [START_REF] Li | Ohnesorge Number[END_REF].

Eötvös number

For a drop falling inside another immiscible liquid, the drop shape should be determined because its shape can possibly be modified due to the resultant effects of body force, viscosity force and capillary force. This determination involves Eötvös number calculations, which is a dimensionless number and shows the relative importance of the surface tension force and gravity force on the moving drop. If the Eötvös number is relatively low, the surface tension force is stronger than the gravity force and the deformation of the drop is small. With a relative high Eötvös number, the drop can have a more important deformation of its shape, even breakup in liquid surroundings [START_REF] Fakhari | Simulation of falling droplet by the lattice Boltzmann method[END_REF][START_REF] Clift | Bubbles, Drops, and Particles[END_REF]. The expression for the Eötvös number is written as:

(2.24)

The estimation of the Eötvös number values is done by assuming one given falling salt water droplet inside 0.001 wt% Span80 contained in Marcol 52 at 23 °C. The density difference Δρ = ρ w -ρ o = 196 kg/m 3 , acceleration g = 9.8 m/s 2 , drop radius r = 0.07 mm (D = 0.14 mm for drop diameter) and surface tension σ = 42 mN/m. These numerical numbers are just shown as an example not far from values taken from experiments. The result is Eo ≈ 10 -4 . To be more precise, this value should be combined with previously calculated Reynolds number Re = 10 -3 on Table 2.3 to confirm that the free falling drops are well in spherical shape.

Electric capillary number

When the droplet is subjected to an electric field stress, the electrocapillary number should be introduced to determine whether the drop is subsequently deformed. The electrocapillary number is the balance between electric pressure and capillary pressure [START_REF] Bird | Critical Angle for Electrically Driven Coalescence of Two Conical Droplets[END_REF]).

(2.25)

Here ε c is the electrocapillary number, ε r is the relative permittivity of the medium and ε 0 is the permittivity of the void. E is the electric fields, r is the drop radius and σ is the surface tension of the drop. The transitions of the drop shape occur at ε c ≈ 1 [START_REF] Bird | Critical Angle for Electrically Driven Coalescence of Two Conical Droplets[END_REF].

For an estimation of falling drop shapes under electric fields in experiments, we assume a drop with diameter D = 0.14 mm (with D = 2r) falling in 0.001 wt% Span80 in Marcol 52 subjected to a field strength of E = 60 kV/m. Combining Table 2.2 for relative parameter values, the calculated electrocapillary number is ε c ≈ 10 -4 F/m. Thus the falling drops remain almost spherical under electric fields.

Summation of results

To sum up all the characteristics calculated above, a Ohnesorge number calculations. For cases using Marcol 52 the Oh number is small, which means for water phase inside the oil it is the surface tension that dominates the water dynamics. On the contrary, the Oh number for Primol 352 is high, thus the viscous force is dominant with respect to surface tension on the dynamics of water body.

Conclusion

This chapter is dedicated to describe the experimental set-ups, to discuss related uncertainties and to present oil properties. This can give readers a global view of the whole experimental design and process, and let them become familiar with the model oils.

The electric devices ensure that relatively small uncertainties are related to the choices of apparatuses. On the other hand, main sources of uncertainty results on image processing of the visualizations of very small droplets with limited spatial resolution, on the effect of temperature on oil characteristics (mainly the viscosity) and on large range transient variations of interfacial tension between the model oils and water.

The summation of different oils properties in tables and the presentation of characteristic times and non-dimensional parameters aim to simplify the comparisons of different cases. With sufficient amplitude of the applied voltage, the stretched drop detaches from the initial meniscus. This break off from the water body is generally named necking breakdown, or water ligament cut off. It should be noticed that the shutdown of the high voltage pulse should be earlier than the water necking breakdown to ensure the electrical neutrality of the injected drop. The surface charge induced by the electric field on the meniscus should have enough time to evacuate through the water necking to the grounded needle after the high voltage cut off. Thus it is considered that the method can allow the injection of electrically neutral droplets into oil. However, despite this precaution, we will see hereafter that small droplets generally remain slightly charged. technique to characterize the droplet size distributions inside crude oil and they found distributions varying between 19 µm and 59 µm. Another experiment confirmed that the droplet sizes were in microscale range, the drop sizes being observed in the very narrow range 13 -15 µm [START_REF] Rodionova | Preparation and characterization of reference fluid mimicking behavior of North Sea heavy crude oil[END_REF]. However with so small diameters the falling velocities in oil would be very slow.

Injected drop size requirements

We can take the example of one droplet of diameter D = 0.05 mm (50 microns) to calculate the falling velocity. The drop is supposed to be falling in Marcol 52 oil with 0.001 wt% Span80, and the oil characteristics are given in Table 2.2 with assumed temperature of 23 °C. Under these experimental conditions, the Reynolds number based on the droplet diameter, falling velocity and oil characteristics will be very low (to be verified later), the surfactants deposited on drop surface can block the surface velocity and the drop is considered to be a rigid sphere. Thus the falling drop is in the Stokes regime. When the drop reaches steady state, its terminal velocity can be deduced from the force balance:

(3.1)

The ∆m is the difference between the water mass and the oil mass, ∆m = m w -m o . The falling drop is considered to be a sphere of radius r, thus the mass can be replaced by other terms and:

(3.2)

Then the final drop falling velocity is U ≈ 0.03 mm/s. This velocity is too small and can be easily influenced by other parameters inside oil bulk such as convection fluxes. Through simulations, if the test wall is only 0.1 °C warmer than the bulk oil content, (for example due to heating by LED light source in experiments), oil velocities of U = 0.03 mm/s can be observed under free convection effects. If the diameter of the drop is changed to D = 0.14 mm, the calculated velocity becomes U ≈ 0.2 mm/s. This range of drop diameter seems then more convenient to perform further experiments of drop coalescence in oil medium, because the falling velocity will not be so widely influenced by convection flux inside the test cell.

Previous experiments on EHD injection

Drop injection in polybutene oil

The Drop-on-demand technique of EHD injection was previously proposed and tested by [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF] in polybutene oil. The liquid used to form drops was tap water, and the interfacial tension was Part 1 deforms the meniscus into a blunt-tip shape, which is due to the fast shrinking of the sides compared to the tip displacement [START_REF] Wright | Self-consistent modeling of the electrohydrodynamics of a conductive meniscus[END_REF]. Part 2 forces the meniscus, especially the locality of highest curvature which is away from the axis, to retract itself due to inertial effects as the voltage amplitude decreases. At this stage a cone will be formed at the tip of the meniscus due to the ''push'' force of the mass located away from the axis during the step before. Then Part 3 is applied which will enforce the highest curvature at the cone tip and drag this quantity of mass downwards.

At the end of the pulse, a quantity of mass is connected to the meniscus via a thin necking. Due to the inertial and capillary forces, the necking will break off and thus the quantity of mass forms a drop and falls. A good drop injection needs a suitable pulse shape and any excessive pulse deformation can give charged jets. For example, the prolongation of Part 1 will give lateral jets while a strong increasing of Part 3 will favor axial jets [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF]. A series of good drop injections is illustrated on -First, as the electric voltage was returned to zero before the breakup of the water thread which connected the tip head to the rest of the initial meniscus, the drop formed and ejected was supposed to be electrically neutral because the induced charge in the tip head had enough time to evacuate to the grounded needle. All through the experiments the tap water which has a high conductivity was used, and the relaxation time of charge in the drop through the necking was estimated around 0.15 µs [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF].

-Second, the efficiency of the Multi-stage pulse seemed to rely on the ratio between the second and third part of the pulse shape.

-Third, with a higher value of h/R cap , the drops injected were of bigger diameters with the same voltage amplitude.

Active parameters

For water drop injection by MSP j into polybutene oil, the theoretically required lowest voltage needed to be applied on the electrode was calculated [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF]. Since the Eötvös number is small (Eo=( /σ < 0.005), the gravitational effects can be neglected and the meniscus can be considered as a hemispherical shape. To deform the meniscus, the electrostatic pressure must overcome the capillary pressure [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF].

=> (3.3)

Here, p E , p cap , R cap and σ are respectively electrostatic pressure, capillary pressure, radius of the metallic needle and the interfacial tension. An approximation of threshold field intensity allowing deforming the meniscus at its axial tip, the highest electric field region, is calculated as E Max [START_REF] Coelho | Properties of the tip-plane configuration[END_REF]):

(3.4)

In this calculation, the form of the meniscus is supposed to be hyperboloid and have the same radius of curvature as the metallic needle. Then the electric potential necessary for quasi-static deformation and injection of a charged drop is written as:

(3.5)
This estimation is a rough calculation because the real applied voltage would be higher than V low due to the short duration of the pulse. That means that a large quantity of energy is given to the meniscus for the injection in a limited time [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF]. The numerical calculation results are summed up and shown in the next section.

Uncertainty on V low

For uncertainty calculations, the R cap , ε r and ε 0 are assumed to be constant. Then V low depends only on σ and H.

(3.6)

The uncertainty on V low is not calculated for polybutene oil here. It is used for the calculations of the new oils in our experiments.

Pulse energy parameter

A number representative of the pulse energy is defined as the area of the non-dimensional pulse shape multiplied by the applied voltage squared over the pulse duration [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF]. It takes into account the pulse shape when comparing different cases.

(3.7)

Here V is the voltage applied for drop injection and t max is the pulse maximum duration. This allowed us to plot diameters of injected drops as a function of applied pulse energies and helped, in comparing different pulse shapes.

Improvement of EHD injection parameters

for new model oils

Electric field in EHD injection set-up

For experiments done in EHD injection setups, the electric fields inside the test cell are calculated using COMSOL Multiphysics software. The conditions are chosen as the application of 1 kV voltage on the electrode, while the distance between the needle and the electrode is H = 15 mm. The axis z(m) takes positive direction upwards from the upper surface of the electrode, and beyond the distance of 0.012 m the electric field intensity increased abruptly. Equation 3.4 gives a maximum electric field at the needle tip of 1.1 kV/mm, but the numerical simulation is very sensitive to the oil level above the needle tip (here 2.2 cm) and the condition applied at the free surface (here grounded, as the needle).

Definition of new multistage pulses

Marcol 52 and Primol 352 have interfacial tension higher than polybutene oil, and the direct application of MSP j did not give any water drop injection in these new oils. Thus new pulse shapes must be defined to have successful drop injection. To overcome the high interfacial tension, elongation of the first plateau and increase of the second plateau based on MSP j should be considered. New pulse shapes were tested and compared and, as an example, one of them (MSP6) mainly used hereafter, is drawn in comparison with MSP j on the graph of Fig 3 .8. In experiments the duration of the pulse is of an order of three times the t cap and the amplitude is of order of V low . experiments. Through experiments it is observed that, in addition to former conclusions, the relative duration of Part 1, on the total duration, also plays a crucial role in drop injection. An insufficient duration of Part 1, with higher voltage, will lead to long necking under inertia effect, which often results in big satellite formations [START_REF] Tjahjadi | Satellite and subsatellite formation in capillary breakup[END_REF].

Drop injection in Marcol 52

Previous EHD drop injection was performed in polybutene oil which was a type of unsaturated organic medium. This oil may be oxidized, which can modify its interfacial tension with water. In the present study, polybutene oil has been changed for Marcol 52 and Primol 352, both saturated hydrocarbon mixtures, to ensure that no oxidation could take place in these organic liquids. However, compared to polybutene oil, surface tension with water is significantly higher. Thus droplet injection conditions should be again put in question. The Marcol 52 oil is a transparent mineral oil, and its characteristics have been presented in Table 2.2 and 2.3. The calculation of the Ohnesorge number of Marcol 52 showed that the capillary force dominates the meniscus dynamics compared to the viscosity effect. The calculations of capillary time and viscous time gave a global idea about the pulse durations for EHD drop injection. However this range is too large and more precision is needed to better define a convenient pulse shape. Thus experiments are needed to test pulse durations in order to find a best suitable pulse shape.

Summation of characteristics of drop injection in Marcol 52

To sum up the characteristics of drop injection in Marcol 52 with two types of oil corresponding to

Pulse duration tests

To fix pulse duration, experiments were done in EHD injection set-ups into model oil (Marcol 52) with H = 15 mm. The needle was the same as used for polybutene oil and the inner radius was of R cap = 0.23 mm. Experiments were done with a fixed pulse shape named MSP6 by varying the pulse total durations by 1.25 ms, 1.30 ms, 1.37 ms and 1.5 ms. The diameter values were determined by visualizations with the help of ImageJ. On Fig 3 .10 the curves are grouped together for small drop diameters but for larger droplets it seems that the efficiency is better for intermediate duration of 1.30 ms. For drop injection, the pulse duration is fixed at 1.30 ms which gives the largest drop diameter range on the figure. This value is located between the t cap and t µ calculated for Marcol 52 on Table 2.3.

Effects of oil interfacial tension on drop injection

The oil interfacial tension changes may affect drop injection processes and drop diameters. Studying these two curves, the trend of the 'old oil' is not as smooth as the 'new oil', and it starts at a lower voltage value. This is because the surfactants or impurities present in the 'old oil' lowered the surface tension on the water meniscus, which could favor drop injection. For the 'new oil', an obvious gap occurred in the middle of the curve (no injection of single droplet observed between 4 and 4.1 kV), which is not as evident for the 'old oil'. This gap in the curve is caused by the nonbreakup of the water necking, the drop beginning to form is completely absorbed by the meniscus.

The reason of such a meniscus dynamic relates to the oil-water interfacial tension. Above this voltage the two curves combine. This phenomenon may be due to the meniscus refreshment by successive drop injection which drains parts of total surfactants on the interface until the drained meniscus reached the same state as the meniscus of 'new oil'. To conclude, a low interfacial tension favors water necking breakup which leads to drop injection as seen in 'old oil' series. This change of interfacial tension seems not to affect the injected drop diameter range. This lowing of interfacial tension is supposed to come from the particle pollution such as falling dusts or impurities in oil. drawn with horizontal axis V/V low . Error bars are drawn from results of table 3.3: For new oil it gives 2.6%. For old oil surface tension is varying and V low is taken as 3.7 kV +/-∆V low = 0.6 kV. "Error bars" are drawn using as an uncertainty ∆V low /V low = 16.2 %.

On Fig 3 .12 it is shown that if the variation of surface tension during the successive injections for old oil is seen as an uncertainty on the parameter and is then used to draw large error bars, the data obtained for old oil and new oil are located within those errors bars.

Complete results for drop injection in Marcol 52

Many experiments were carried out by adjusting pulse shape adaptation to the Marcol 52 model oil, in the EHD injection set-up for a needle of R cap = 0.23 mm. Some selected results are presented below to show the global trend of the curves and the problems generally encountered during drop injection.

Improvements of pulse shapes for drop injection

Improvements of drop injection pulse shapes

During the experiments of droplet injection in Marcol 52 presented above, gaps in the injected drop diameter curves are observed which is due to meniscus dynamics. Moreover, for large drop diameters, the pulse shapes tested are not so convenient to use because of jets and satellite production. Thus to better control meniscus dynamics in drop injection, improvement of pulse shapes is necessary. The needle was changed for R cap = 0.36 mm to facilitate larger drop injection.

The improvements of pulse shapes are focused on two points: first, to obtain larger drops with better smooth curves without any gaps inside, without producing jets; second, reduce or even eliminate satellite production after main drop injection. This leads to a new Multi-stage pulse shape with five distinct parts. That is, a new plateau is added at the end of a normal MSP pulse shape. This pulse shape is named MSP4.

The advantage of the pulse shape shown on Fig 3 .18 compared to normal pulses is that it removes the gaps present on other curves. This effect is mainly due to adding Part 5 which serves to reapply a little force onto meniscus apex while the latter is dragged under the force of Part 3. This helps to prevent a long necking which would result in large satellite formation. This new pulse shape can give a larger range of drop diameter injection. The important thing in this pulse shape is that Part 5 should not be higher than Part 3. Otherwise, this would give axial jets during injection. This pulse shape especially favors droplet injection in viscous liquid medium to get small size droplets, which would otherwise not be possible due to large necking formation. As will be discussed hereafter, among other attempts to suppress the generation of electrically charged droplets, this pulse shape (MSP4) was modified to get MSP inv . For this inversed pulse, the voltage of Part 1 is kept positive, while the voltage of later parts is negative. This inversion of voltage polarity aims at neutralizing charge injection.

EHD drop injection with improved pulse shape

The drop injection was performed in EHD injection setups, with a straight needle of R cap = 0.36 mm hanging above the electrode. The distance between the needle and the electrode is H = 18 mm. To drain the electric charges possibly injected into oil during successive drop injections, a grounded metallic plate was added just below the oil free surface. The drop injections were done with different additives to compare their effects. One series was performed by injecting tap water drops into

Marcol 52, and this is named 'tap water'. The second curve was done with the same drop injection, but adding 3.5 wt% NaCl to the tap water. The third injection series involved injecting 3.5 wt% NaCl solution into Marcol 52 with 0.001 wt% Span 80. These three curves are drawn on the same graph below. green curve may be due to some pollution of the oil medium by falling dust, which decreased the w/o interfacial tension.

Calculation of drop diameter based on falling velocity

Regime determination

For electrocoalescence experiments, cameras need to be placed at the potential drop coalescence region, far from the needle tip. Drop sizes can be revealed by two methods: either directly from the camera vision field or through calculations based on the drop falling velocities. Generally the second method is adopted because later electric charge of drops could be calculated using the same method.

When one water drop is falling inside the oil medium, the first thing is to determine the regime of the 2.3, the Reynolds number is much smaller than 1, which denotes laminar flow in the Stokes regime.

Surface mobility discussions

When a liquid drop is falling inside an immiscible liquid, surfactants on the drop surface will be swept toward the rear point on the drop by viscous stress and lead to an interfacial tension gradient [START_REF] Pawar | Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants[END_REF] [START_REF] Milliken | The effect of surfactant on the transient motion of Newtonian drops[END_REF]). As a result Maragoni stress grows in opposition to the viscous stress to the point where the fluid velocity vanishes along the drop surface, fixing the surfactant distribution [START_REF] Teigen | Influence of surfactant on drop deformation in an electric field[END_REF]. In experiments, falling drops are of diameter D = 0.14 mm, the inertial force and interfacial tension variations are so small that they will not modify the spherical shape of drops [START_REF] Levan | Motion of a droplet containing surfactant[END_REF]. This assumption of spherical shape of drops is consistent with experimental observations on the videos.

Falling drop velocities are accessible by camera records, and with given parameters of test oil, this velocity is in accordance with the Stokes Law rather than the Hadamard -Rybczynski model. The latter model depicts a moving drop surface when it is falling inside oil medium, while the former describes a solid sphere behaviour (Hamlin & Ristenpart 2012). From this phenomenon it is concluded that, as shown by [START_REF] Levan | Motion of a droplet with a newtonian interface[END_REF], the interfacial tension gradient on the drop surface could form a cap over the water drop surface and block the circulations of the liquid inside the water drop.

Calculation of drop diameter from measured falling velocity

When a water drop reached its terminal falling velocity inside oil medium, the total forces acting on the water drop equals zero. It combines gravity force, buoyancy force and Stokes' drag force, as The Stokes drag force is written f = -6πµ o rU 0 , where r is the radius of the drop and U 0 the terminal falling velocity of the drop without electric field. The balance equation can be written as follows:

(3.8) ∆m is the difference between the mass of the water drop and the mass of the equivalent oil drop.

(3.9)

Here ∆ρ is the difference in densities of water and oil, with ∆ρ = ρ w -ρ o . Combining the equations leads to the following expression for D (=2r).

(3.10) This calculated drop diameter will replace the visual revealed diameter. For the calculations of the drop diameters, there are also uncertainties involved in the equation. This should be discussed to determine the relative uncertainty of the drop diameter value.

Uncertainty on drop diameter

For the uncertainty calculation of drop diameter based on equation (3.10), the variables are respectively µ o , U 0 , and ∆ρ. The uncertainty equations can be written as below [START_REF] Bell | A Beginner's Guide to Uncertainty of Measurement[END_REF]).

(3.11) The difference of densities is then respectively 195.8 and 196.7 kg/m 3 . Finally, by linearizing, the variation of density difference due to a temperature uncertainty of 1°C is 0.15 kg/m 3 and the total relative uncertainty is very small and can often be neglected facing to other terms.

(3.12)

The uncertainties of drop falling velocities rely on image quality. During experiments Dalsa cameras survey drop falling journeys at 100 fps. The calibration of images gave 0.0033 mm/pixels. For a drop of 0.15 mm in diameter this gives 50 pixels in length. The uncertainty is of ± 1/50 = ± 2 %. There are two ways to track the falling drop using Spotlight software: center tracking and threshold tracking.

The center tracking follows the drop center through image steps. The threshold tracking locates drop boundary in downwards direction. No matter which tracking method is used, the uncertainty of drop location is 1 pixel, which corresponds to 0.0033 mm after calibration. The time interval between two successive photos is 0.1 s, and the relative uncertainty is the division of drop location uncertainty over the time interval between two successive photos.

(3.13)

Here δx is the uncertainty on the drop position δt is the time interval between two successive photos. For example at 21 °C ± 1°C, δµ o /µ o = 3.8 %, drop falling velocity over 2 seconds, then the uncertainties of the drop diameter can be written as:

(3.14)

For a falling velocity of 0.1 mm/s the relative uncertainty on the velocity is 1.65 10 -6 /10 -4 = 0.0165 and it gives a relative uncertainly on the diameter of 2 %. This uncertainty is smaller than those obtained by direct visualization for small droplets, and is mainly due to the oil density variation. With a shorter averaging time on the velocity, the second term in equation (3.14) increases, as the total uncertainty: for example with an averaging on 1 second instead of 2, the relative uncertainty on the velocity becomes 3.3 10 -6 /10 -4 = 0.033 and the relative uncertainly on the diameter increases to 2.5 %.

At 23 °C and 25°C the main changes concern the viscosity relative uncertainty with respectively δµ o /µ o = 3.5 % and 3.2 %, with a small influence on the total relative uncertainty. However the relative uncertainty is smaller for larger drop diameter, through the increasing term of falling velocity.

Section 3.5.4 will show the calculated drop diameters with error bars.

Comparison of drop sizes obtained from visualization and calculation

When drops are injected and falling in model oil, the sizes of the drops should be determined. One example is illustrated by 3.5 wt% NaCl contained water drop injected in Marcol 52 with 0.001 wt% Span80, with -MSP4 pulse shape in the drop coalescence setup. Since it was shown before that error bars of drop diameters were not larger than the markers for visualization data, only the error bars associated with the calculated drop diameter uncertainties are presented on the figure above. For small drops no calculations of drop diameter are available because it was not possible to detect these drops with the Dalsa camera. For larger drops they fell well into the camera detection range and motion was recorded. Both diameters through visualizations and calculations coincide well. The error bars are no larger than markers of calculated drop diameters, thus for following figures the error bars of drop diameters will not be presented.

Drop diameters of -MSP4 revealed by visualizations and calculations

drop diameter visualizations drop diameter calculations

Electric charge of generated droplets

Electric charge: first observations

The electric charge of water drops was first observed by applying a DC field on the electrode while injected water drops were falling inside oil medium. At the instant of DC application, the motion of the water drops changed immediately. Moreover, these changes of motion vary with the polarity of the DC field. One may conclude that the injected water droplets are electrically charged. However according to previous experiments described in section 3.3.1, the water droplets were assumed to be electrically neutral after their injection in the oil, because the voltage was always shut down sufficiently before the complete separation from the grounded meniscus. Moreover during drop injection no jets were produced, as verified by AOS camera at 5000 fps with 4 µs shutter speed. A possible reason is that wandering electric space charge was trapped in the oil medium. The electric relaxation time of the model oil was measured in section 2.5.1.4 in the range τ = 1000-5000s, thus the trapped space charge could not be evacuated efficiently. Consequently the changes remain inside oil medium and affect injected water drops.

Different methods were tested to evacuate the electric space charge from the model oil, and the objective of all them was to increase the surface contact of the model oil with grounded metallic parts. A grounded metal mesh was deposited along the wall of the test cell or a grounded metal plate covered the free surface in direct contact with the oil. Even in association with the application of a small DC voltage, no solution was found sufficiently efficient in suppressing the phenomenon. Thus it was decided to improve the understanding and control of the electric charge of the injected droplets.

All the experiments reported hereafter have been carried out with the drop coalescence set up. The test cell includes an intermediate electrode, at which the high voltage is applied, while the needle and the bottom electrode are grounded. A hole was drilled in the intermediate electrode. The idea was to separate the generation of droplets to the upper part of the cell, and their analysis to the lower part, in a more uniform electric field. In fact, because of the low falling velocity of the droplets, it was quite difficult to observe with a good repeatability the interaction of droplet pairs after their passage through the hole and it was decided to carry out all analyses in the upper side. The main consequence is that the electric field is not absolutely uniform, because of the needle-electrode configuration. The distance between the needle tip and the electrode is H = 8 mm. The height z = 0 corresponds to the upper surface of the bottom metallic plate, and the intermediate electrode is located between z = 0.015 m and z = 0.02m. The result of a numerical simulation is presented in figures 3.24 and 3.25 for a DC voltage of 1 kV applied on the electrode.

Droplet charge measurement

Principles of calculation

To determine the charge contained in water drops, a DC field was applied on the electrode during free fall of the injected drops. Since the drops are charged, this electric field will change the forces acting on the drops and then modify their motion. A Dalsa camera was placed along the path of the falling drops and when the drop entered the camera field of vision, the DC field was turned on. The velocities were revealed with the help of Spotlight software. A droplet without applied electric field, is subjected to only gravity force, buoyancy force and viscous force. However, when an electric field is applied, the supplementary electric force will influence this droplet to modify its falling motion.

Neglecting the possible dielectrophoresis effect, this can be illustrated as: For the calculation of the forces, the upwards direction of the vertical axis is taken as positive direction. The forces of gravity and buoyancy combine to give:

(3.15)

Here the ∆m is the difference between the mass of the water drop and the equivalent oil mass. The viscous force can be calculated in Stokes regime as:

(3.16)

Where μ o is the dynamic viscosity of the oil, r is the radius of the droplet and U is the stabilized falling velocity of the drop under electric field.

The force balance is:

(3.17)

With F e = qE and F dep is the dielectrophoresis force which depends on electric field gradient.

F s F w F e (3.18)
The dielectrophoresis force is not strong, and it can be calculated as following [START_REF] Benselama | Calcul de la force diélectrophorétique dans les microsystèmes biologiques: comparaison du modèle dipolaire avec le modèle du tenseur de Maxwell[END_REF]).

(3.19)

(3.20)
Here ε 1 is the permittivity of the water drop and ε 2 is the permittivity of the model oil. For the values of the permittivity refer to table 2.2. K is the polarizability factor of the drop. The form of the drop is assumed to be spherical. For the calculations, the factor K is taken equal to 1.

For an experiment in the drop coalescence set up for a drop of diameter D = 7.82*10 -5 m with an applied DC voltage V = +1000V on the electrode, the calculated force is: F dep + F e = -3.28 * 10 -10 N, with F dep = 1.25 * 10 -11 N. The percentage of the dielectrophoresis force over total force is 4 %. Thus the dielectrophoretic force can be neglected to first order.

(3.21)

Combined with equation (3.9), the equation becomes as follows:

(3.22)

The electric field and drop velocities are aligned vectors and the sign of the calculated result gives the polarity of the charge.

Drop charge uncertainties

Starting from equation (3.22), with r = D/2, the relative uncertainty of r is δr/r = δD/D. D is the diameter of the drops.

(3.23)

The uncertainties of U 0 and U depend on image quality, and δU 0 = δU = 1.65*10 -6 m/s. The parameter δD/D was calculated previously and the expression depends on temperature. In the expression for δD/D, the used drop velocity is the value without electric field, and it is changed for U 0 in calculations of drop charge uncertainty. The drop charge uncertainties can be simplified and written (for example at 21 °C) as:

(3.24)

(3.25)

For equation (3.25), the term δU 0 /(U-U 0 ) can diverge if the drop falling velocity under electric fields does not change much comparing to its free fall velocity. This case often happens for droplets with very small charges. The term of δE/E depends on the voltage applied to the electrode. Noting y as the distance between the drop (or the middle point of a drop pair) and the electrode a curve fitting on the electric field obtained by numerical simulation and drawn on Figure 3.27a gives the following expression for a voltage V applied at the electrode:

(3.26)

The term δE/E depends on the vertical distance y between the droplet position and the bottom electrode, and can be expressed as follows:

(3.27)

As mentioned previously the calibration is 0.0033 mm/pixels. The uncertainty of the position of the droplet is 1 pixel which corresponds to 0.0033 mm but the bottom of the electrode is not so clear and its position can vary by +/-5 pixels. The corresponding uncertainty on y is close to = +/-0.2 mm. Using the equation (3.25) and (3.26) the relative uncertainty δE/E is found equal to 0.5%.

Experimental methods for revealing electric charge of injected drops

The Dalsa registered drop falling velocities are drawn on the figure below and the changing velocities under DC field is obvious. As already described chapter 3.5.2, the duration of record without and with electric field allows averaging the velocity on a duration of 2s and the uncertainty on resulting velocity is 1.65 10 -6 m/s.

The drop falling velocity without DC field is estimated to be 0.081 mm/s and during DC field it is estimated to be 0.132 mm/s. This method for drop falling velocity calculations is used for all experiments hereafter. At 23 °C ± 1 °C, δD/D = 2 %, and δq/q = 6.4 %.

Results of experiments

Drop injection was performed in the drop coalescence setup, using water containing 3.5 wt% NaCl and Marcol 52with 0.001 wt% Span80. Multistage pulses applied are MSP4 and MSP inv with negative polarity of DC voltage (noted hereafter -MSP4 and -MSP inv ). The drop diameters and contained charge are drawn on the figure below. Once the drops had been injected, a DC = 1 kV was applied on the electrode to change drop falling velocities, and reveal drop charge. On the figure all the series of drop diameters coincide well. The charge of droplets generated with -MSP4 are all negative. That is coherent with the idea that positive space charge is injected around the needle tip during the negative pulse, and, remaining after the voltage shutdown, can induce the opposite polarity on the meniscus and detached droplet. Thus the polarity of the injected drop is in accordance with that of applied voltage on the electrode.

For the drop falling velocities, in some cases the drop did not change its motion during applications of DC. One example is shown below for -MSP inv injection at -5.40 kV with DC = +1 kV. In experiments when the drop fell into Dalsa camera vision field, three seconds later a DC field was applied on the electrode with a finite duration of three seconds. Generally it is expected to see drop The different characteristics of the drop injection presented above will be summed up and showed in a table. The experiments were -MSP4, -MSP inv and MSP10. The injections were performed in Marcol 52 model oil with 0.001 wt% Span80 and water containing 3.5 wt% NaCl solution. On Table 3.5 the maximum voltages for -MSP4 and -MSP inv are in the same range, and only MSP10 can reach a much higher voltage. The pulse shape of MSP10 is special and the injected drop charge is mostly neutralized. Thus until a high voltage the injected drops are not obviously charged.

Different phenomena observed during drop injection

Influence of meniscus height

During drop injection there are many parameters which could influence the injection process. One important factor is meniscus height h/R cap which influences directly injected drop diameters. The following graph represents the relation between the meniscus heights h/R cap and the diameter of injected drops. Two series of experiments were done by MSP4 at 5400V and MSP10 at -6350V with different meniscus height. All the experiments were performed in the drop coalescence set-up.

Jets formed during meniscus stretching

During drop injection, jets can be formed on the meniscus at different instants of meniscus dynamics

Failure of neck breaking

As said in chapter 3.4.3, a gap in the droplet diameter versus applied voltage curves is sometimes observed. This could be related to the fact that the dynamics of stretched meniscus fails sometimes in detaching a droplet. This failure seems due to very small differences in the sequence, as illustrated

Conclusion

In this chapter, EHD single drop injections are investigated in details. For such injections, oil viscosity, interfacial tension, meniscus height and pulse shape were found influencing the meniscus dynamics and the resulting droplets. Starting from the results obtained by [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF] However our experiments on drop coalescence require drop diameters of smaller scale to simulate real crude oil water emulsion conditions. For this reason the EHD drop injection technique was adopted. Dual drop injection from two parallel needles was tested by [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF]. In this chapter, dual drop injection from a single needle is proposed. The analysis of the drops pair falling velocity should take into account strong hydrodynamic coupling effect obtained at very low Reynolds number.

The electric charge of the two drops can then be deduced from the falling velocity.

Previous dual drop injection from parallel needles

The EHD drop injection technique allows injecting droplets with variable diameter smaller than nozzle size. This drop injection method was first used by [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF] for drop electrocoalescence experiments. In his studies two parallel needles were mounted with identical heights and meniscus shapes. By applying a multistage electric pulse, two equal drops were produced successfully.

However, the motions of the two drops are off-axis as illustrated on Uneven droplets, generation of large satellites or failure in detaching one of the two droplets are often observed, and appear very sensitive to initial conditions. Thus a new drop injection method should be thought of for more stable and repeatable drop production to perform drop coalescence experiments.

Dual drop injection from single needle

The new dual drop injection method aims at injecting a pair of drops from the same needle, hanging straight above the bottom electrode. The injected drop pair then falls along the vertical axis in model oil and remain aligned with the electric field. This system is axisymmetric and this drop injection method shows a better repeatability than that using two parallel needles. The falling drops approach each other before they reach the bottom of the test cell, to allow coalescence. The experiments of dual drop injection were performed both in EHD injection setup and in drop coalescence setup. The oil used was Marcol 52 with 0.001 wt% Span80, and injected drops were made of tap water with 3.5 wt% NaCl.

Parameters influence on dual drop injection

For dual drop injection, the first important factor influencing the injection process is the distance between the first injected drop and the needle tip during the second drop injection. As it is known that a single drop exposed to high voltage fields may disintegrate [START_REF] Taylor | Disintegration of Water Drops in an Electric Field[END_REF], it is important to ensure a sufficient distance between the first drop and the meniscus apex to give no split of the former and good injection of the second droplet. To give an idea about this distance, a simulation was done using COMSOL software. Here σ is the interfacial tension, ε 0 is the permittivity of free space and ε r the relative permittivity of the medium. D is the water droplet diameter. The electric field intensities were obtained by numerical simulation for both the first drop and the meniscus apex prior to the second drop injection.

For the calculations, the first injected drop diameter is assumed to be D = 0.15 mm, and the constant applied voltage on the electrode is 7.0 kV. The distance between the needle tip and the electrode is H = 18 mm. The permittivity of used oil is given in Table 2.2. The distance is between the meniscus apex and the drop surface, and it begins at 0.075 mm which corresponds to one half of the first drop radius. When the drop is very close to the meniscus, the electric field intensity on the drop surface is stronger than that on the meniscus. On the contrary when the distance is quite large (about 10 times the diameter) the electric field on the droplet surface is smaller than Taylor limit. It should be noticed that the voltage applied on the electrode is fixed to extract one droplet and not to make the meniscus fully disintegrate. On the other hand the Taylor critical electric field does not take into account the dynamics of the surface deformation for short duration pulses. The intersection of the two curves indicates only an order of magnitude of the minimum distance that should be ensured between the first injected drop and the needle tip, and it corresponds to a little less than twice the droplet diameter. The series of frames taken from the high speed video illustrated the splitting process of the first injected drop during the second pulse and the influence on the meniscus cancelling the second drop injection. To make the two droplets meet during their descent down in cell, the best method consists in generating a second droplet a little larger than the first one. The processes were recorded using the Dalsa camera at 100 fps frame rate. This phenomenon of non-coalescence is due to the separation effect of thin oil film which locates between the drops pair facing poles.

Diameter range of droplet pairs

On the optimal conditions for dual drop injection were found, a series of injections were performed to test the widest range of drop diameters. The experiments were done in the EHD injection setup containing Marcol 52 with 0.001 wt% Span80. The water contained 3.5 wt% NaCl and the distance between the needle tip and the electrode was H = 18 mm. The needle inner radius was R cap = 0.36 mm. For this injection series a metallic cover was placed on the top of the test cell. The applied electric pulse for drop injection was of MSP4 at identical voltage for the two drop injections.

Between the two successive pulses, a time delay of 600 ms was applied so that the first injected drop was far enough to have a sufficiently small influence on the second injection. As it was shown previously the meniscus heights could influence injected drop sizes. Our analyses are that this difference in drop size is due to the difference in meniscus heights for the two drop injection.

When the first injected drop is of a small size, the meniscus height is not modified significantly. When the first injected drop enlarges, the meniscus height is then reduced because of the extraction of the

Determinations of diameter and charge of the two falling droplets

For small droplet size and falling velocity, the Reynolds number is very low, of the order of 10 -3 , and the interaction between the two drops is fully symmetric. The strong coupling of the two drops is called reflection, as described by [START_REF] Batchelor | Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory[END_REF].

Drop diameter corrections

For a drops pair falling inside oil medium, the diameters and forces acting on the two drops can be calculated through their falling velocities. However these parameters do not connect to each other through a simple law because of the strong coupling effect between the two drops. Thus more developed calculations were adopted based on principles described in [START_REF] Batchelor | Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory[END_REF] and (Jeffrey & Onishi 1984). For drop diameter calculations, in experiments on falling drops pairs only translations of the two drops should be considered as there is no rotational effect on these two drops. Thus the velocity of the two drops can be depicted as following.

(4.2)

(4.3)
With U m the velocity of the drops m (m = 1, 2) (bold characters denotes a vector), M mn the mobility tensors [START_REF] Batchelor | Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory[END_REF] and F n the exterior forces exerted on the drop n (n=1, 2).

The mobility tensors can be decomposed along axial (with respect to the axis joining the two drop centers) and perpendicular directions by involving the mobility coefficients X mn and Y mn respectively which gives the mobility tensor M mn another form.

(4.4)

In the above formula, I is the identity matrix; r is the radius of the drops and the distance between the drops center to center (bold character denotes the vector form and is then a tensor). Thus U 1 and U 2 can be rewritten in the more developed forms.

(4.5) (4.6)

In the experiments of dual drop injection, the two injected drops fall one after another in an axisymmetric fashion. This implies that firstly there is no force acting perpendicular to the drop motion, and thus the parameter Y mn = 0. Secondly, according to symmetrical properties of the system, these mobility coefficients have symmetries X mn = X nm and X mm = X nn . Thirdly, as the system is axisymmetric, there is only one component (z axis) and the vectors U m and F n are projected along this z axis. Thus they are described by scalars U m and F n . Summing up all the above conditions, the equations of drop velocities can be much simplified.

(4.7) (4.8) U 1 and U 2 are velocities of the two droplets that can be taken directly from the Dalsa cameras in experiments. r 1 and r 2 are drop diameters which are firstly calculated as single drops falling freely inside oil medium, by using Stokes equation (3.2). Then the forces F 1 and F 2 can be calculated as:

(4.9) (4.10)

Introducing two non-dimensional parameters λ and s, with λ = r 2 /r 1 and s = 2 /(r 1 +r 2 ), with the drop center to center distance, the mobility coefficients are given by [START_REF] Batchelor | Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory[END_REF] as:

(4.11) (4.12)

For a simplified model when the two drops are sufficiently far from each other (typically s > 4), the terms of the equations (4.11) and (4.12) can be limited to O (s -4 ). If the two drops are much closer, all the terms in the equations will be considered. For very close droplets (typically s < 3) the terms in equations (4.11), (4.12) are decreasing only very slowly with the increasing exponent of (1/s) and the accuracy of the relation between velocities and forces is degraded.

When the forces acting on the drops are calculated through equation (4.9) and (4.10), the values of drop radii r 1 and r 2 can be obtained using a technique of successive approximation. For free fall drops inside oil medium, the forces are balanced. (4.17) (4.18)

These values of drop radius can then be substituted into equations (4.9) and (4.10) to calculate the corrected forces acting on the two drops. In this method of successive approximations, the initial radii can be given by the method without interaction (equation 3.4). It was observed that generally one or two iterations are sufficient to get a pretty good convergence of the results (radii, velocities and forces). Moreover at the second iteration, the convergence can be improved by applying Newton's method.

Drop charge calculations

To determine the electric charge of water drops, an application of DC field on the electrode is programmed during the free fall journey of the drops. The drop descents are recorded by the Dalsa camera. When the falling drops pair is subjected to the electric field, the motion is modified due to the contained electric charge. The drops change their velocities within 0.1 seconds and they readjusted immediately to an equilibrium state. For this system the positive direction is upwards along the vertical axis z and the applied electric fields lines are parallel to the drop center-center lines.

The forces acting on the two falling drops can be calculated using the complete method suggested by [START_REF] Davis | Two Charged Spherical Conductors in a Uniform Electric Field: Forces and Field Strength[END_REF].

(4.19) (4.20)
Here ε is the permittivity of the medium with ε = ε 0 ε r , q 1 and q 2 are the electric charges carried by the two drops, E the electric field intensity. F 1 to F 7 in equation (4.20) are coefficients given in [START_REF] Davis | Two Charged Spherical Conductors in a Uniform Electric Field: Forces and Field Strength[END_REF] which can be noted as F j . It should be noticed that the coefficients F j ' in equation (4.19) and F j in equation (4.20) are not the same because the system described is not symmetric, and the coefficients F j depend on the drop ratio 1/λ = r 1 /r 2 and the ratio of drop ''inter-distance'' on drop radius ( -r 1 -r 2 )/r i with r i the drop radius of either the first drop or the second one according to calculations of F j . The calculations of F j involve a series of summations at different drop ''interdistances'', thus a simplified model is used.

The third terms in equation (4.19) and (4.20), F z3 (1) = and F z3 (2) = , represent the electrostatic interaction between the two charged droplets. For a simplified form the charges are considered as concentrated at the drop centers, and the resulting force, opposing for to the two drops, can be written as:

(4.21)

For the first drop the attraction force is positive and the repulsion is negative. It is the contrary for the second drop. The dipole-dipole interaction force corresponds to the first term in equation (4.19) and (4.20), F z1 (1) = and F z1 (2) = and the equations are taken from [START_REF] Atten | Electrocoalescence of water droplets in an insulating liquid[END_REF]. In experiments to reveal drop charge, the distance between the two drops is larger than the smallest drop radius, and no coalescence process occurs in these cases. Thus the equation is taken for large separations without deviation angle between drop center-center lines and the electric field: The last term is the drop weight minus buoyancy force, which is presented as the difference between the drop weight and the replaced quantity of oil weight. With equation (4.34) and (4.35), the drop charge can be calculated as well as the relative uncertainties on drop charge using the same method as described in section 3.5.2. It should be noticed again that the calculations are fully valid for relatively large drop distance (non-dimensional parameter s > 4) and less accurate when the distance decreases (s < 3). The differences between the forces given in [START_REF] Davis | Two Charged Spherical Conductors in a Uniform Electric Field: Forces and Field Strength[END_REF] and that of the simplified model is calculated and drawn on Fig 4.10 for the example of a drops pair at variable distance. The condition is taken for r 1 = r 2 . The electric field

applied is E = -0.125kV/mm (i.e. -1kVDC for H = 8mm). The charge on the drops are assumed to be q 1 = 0.01 pC and q 2 = -0.02 pC. drops becomes smaller than 2 to 3 times the drop radius, the differences between the model given in [START_REF] Davis | Two Charged Spherical Conductors in a Uniform Electric Field: Forces and Field Strength[END_REF]) and the simplified models of forces increase. Fig 4 .11 quantifies the error made by using simplified models: a little more than 5% when the distance equals two times the radius (s = 4) , less than 2% when it exceeds 4 times the radius. The calculations of velocities can be explained from the graph drawn on Fig 4 .12. The positions of the drops are obtained using the Spotlight software applied to a set of successive images, and drop velocities are calculated from those position values.

On

Experimental results

Experiments

(4.36)

Here x n is the drop position at time t n . The time step is 0.1 seconds and the change of drop velocities under DC fields began at 1.8 s and ended at 2.7 s, which corresponds more or less to the DC duration of 1 second. After 9 seconds the acquisitions stopped in that experiment. From the graph it can be seen that there are many quantification noises on the velocity curves. This is due to the uncertainties of the drop positions revealed by visual imaging. For drop velocity trends, one can observe three successive phases. In the first phase which starts from the beginning to 1.7 s, the drop velocity is steady, which means all the forces acting on it are in equilibrium. This is the drop free falling velocity in oil medium without DC application. The second phase began from 1.8 s to 2.6 s when the falling drop is subjected to an electric field generated with the DC voltage +1000V. The velocities changed within 0.1 s and droplets appear again falling in steady state. The shutdown of this DC field began at 2.7 second and the drop velocities changed again. The positions of the first drop and the second drop are also drawn on the graph and the yellow curve is the distance between the two drops. With the application of the DC field, the first drop decelerated while the second drop accelerated, which gives the sign of the electric charge of the two droplets: there the electric field is negative from electrode (+1 kVDC voltage) to the needle (grounded): a positive charge tends to oppose the Coulomb force to the motion (falling from needle to the bottom) and to slow down the droplet. The estimated drop diameters are D 1 = 0.137 mm and D 2 = 0.157 mm. The electric charges of the two drops are respectively q 1 = 0.033 pC and q 2 = -0.055 pC. the two drops were in their free fall. The drops were injected by MSP10 at -6280V and -6200V. The drop diameters are D 1 = 0.14 mm and D 2 = 0.16 mm, measured directly by visualisation on Spotlight because the initial distance between the two drops is very small, (less than one radius, giving a nondimensional parameter s < 3). The drop mass or charge calculations are not accurate because the hydrodynamic interaction model used for force calculations is only valid when the distance between the two drops is large enough (s > 4).

Conclusion

EHD injection of drops pair from the same needle was performed successfully. At very low Reynolds number, due to the very small size of the droplets and their associated small falling velocity, the hydrodynamic interaction between the two droplets is very strong. Under electric field, the combination of hydrodynamic and electrostatic drops interactions further complicates the analysis.

A method of identifying the drops diameter and charge is proposed, based on the coupled analysis of the falling velocity of the two droplets, with and without application of external DC electric field.

Validity of the hydrodynamic and electrostatic used models and possible simplifications are discussed Different examples of application to the characterization of injected drops pairs are presented.

Chapter 5. Drop electrocoalescence

Introduction

Water-oil emulsions, drop electrocoalescence occurs under application of an electric field when two drops come in close contact and merge into a bigger one. In our experiments of drop electrocoalescence, the drops pair are injected in oil by the EHD drop injection method. All of the experiments in this chapter are performed in the drop coalescence set-up and the model oil used is Marcol 52 with 0.001 wt% Span80. The drops are of tap water with 3.5 wt% NaCl. When the injected drops pair is falling inside oil medium in the axisymmetric system, a DC or AC field is applied at the electrode. Under influence of the electric field the two drops coalesce, and the process is recorded by two Dalsa cameras and an AOS camera. In this chapter the first section shows experimental results of drop coalescence visualized by the Dalsa cameras. The drop charges are calculated using the method described in Chapter 4. The second section presents AOS fast camera recordings in which we zoom in on the drop coalescence process. The third section summarizes the results obtained and problems encountered during experiments and suggestions for future improvement.

Global review of electrocoalescence experimental results

In petroleum industry research, electrocoalescence is thought to be the best way to break up water in crude oil emulsions because the electric fields can promote an efficient, clean and fast phase separation [START_REF] Eow | Electrostatic enhancement of coalescence of water droplets in oil: a review of the current understanding[END_REF]. For experiments with crude oil, [START_REF] Hellesø | Experimental study of electrocoalescence of water drops in crude oil using near-infrared camera[END_REF]) used an infrared camera to visualize the drop coalescence process. Crude oil is transparent to near infrared rays while water is opaque to them. In their experiments, they used droplets, of about one half to one millimeter diameter, falling on a big stationary drop posed at the bottom of the test cell. It has been found that with increasing voltage on the electrode, the total coalescence is first fastened, but evolves to partial coalescence, or even no coalescence. Since experiments in crude oil require a high For all experiments carried out to date, only large drops have been used. The number of tests is limited, and among them most experiments are of one drop falling onto a bigger stationary one. For drops falling side by side, the flux inside the test cell is not in the same direction as the applied electric fields. Such cases are three dimensional non-axisymmetric systems. Thus it will be difficult to perform simulations corresponding to such a configuration. In our experiments, up to 70 cases have been done in the axisymmetric system, and these will be compared to simulations done in Norway.

Electrocoalescence in axisymmetric system

Electrocoalescence experiments are performed in the drop coalescence set-up with Marcol 52 containing 0.001 wt% Span80 as model oil and tap water with 3.5 wt% NaCl droplets. The system is theoretically axisymmetric as described in Chapter 4. Two cameras register drops falling and coalescence processes in two plans rotated by 45 or 90 degrees around the vertical axis, allowing the verification of the alignment of the drops center axis with the electric field and the gravity. The cameras used are one or two Dalsa cameras to record the falling velocities of the drops pair over a large distance, and one AOS high speed camera to zoom in on drop coalescence to analyze the coalescence processes at 8000 fps. Because of the strong zoom and the variations on droplet diameter and charge and on their initial distance, the location of the coalescence phenomenon is not always in the field of the AOS camera, and only a few cases have been possible to study. Under sufficiently high electric field, the two drops attract each other and coalesce. In experiments, the distance between the two drops should be adjusted to ensure on the one hand the minimum distance during drop injection, on the other hand the sufficiently short distance to let two drops approach and coalesce with each other under electric fields before crossing the total inter electrode distance. This needs controls over drop sizes through injection processes. Gravity is from left to right. (5.1)

Electrocoalescence under DC/AC electric fields

Tracking of falling droplets

After coalescence, the two AOI overlap on the same drop, and it is the same for the thresholds. Thus the locations of the drop are not well determined and the inter drop distance may be larger than reality. (5.2)

Here D 3 is the diameter of the coalesced drop. The first AOI is threshold tracking and the second is center tracking. It is seen that the position of the latter is not well placed due to the overlap of the two AOIs and associated image processing. The center positioned by Spotlight on the coalesced drop is shifted with respect to the real center. Thus the calculated inter distance ∆e is not exactly equal to zero.

Using the distance between the two droplets, the non-dimensional parameter s = 2 /(r 1 +r 2 ) (see equations 4.11 and 4.12) can be calculated as s = /(D 1 +D 2 ). The initial non dimensional distance between the two drops s 0 , is taken at the time when the DC or AC voltage is turned on.

Electrocoalescence under DC fields

The procedure for dealing with the falling drops pair falling and coalescence in model oil is similar to that described in Chapter 4. The difference is the duration of the applied DC field. For = 0.005 pC, q 2 = -0.026 pC; Δm/m = 1.9 %, Δq/q = 0.8 %, DC=1.2kV (24/03/2015, 21 °C ± 1 °C, case No. 6). 'distance' which represents the inter drop distance ∆e shows a small peak at time close to 6s. For the first part of the curve, preceding the peak, the calculation is based on equation (5.1). The peak corresponds to sudden switch from equation (5.1) to equation (5.2) which serves for the rest part of the curve 'distance'. As it was explained in 5.3.2.1, the locations of the two tracking points may then not be exact, thus a slight derivation can be found from coalesced drop radius. This leads to the nonzero value of the second part of the curve 'distance'. Moreover, the switch moment is obtained when ∆e < 0, the equation used is immediately switched from (5.1) to (5.2). However, it is possible that in some cases the two drops are not perfectly aligned and a slight overlap occurs when they are in close vicinity. At this moment the equation is switched to (5.2) a little too early, which can explain the more or less pronounced peak on the curve. Increasing the duration of the applied DC electric field leads to the coalescence of the droplets, and the resulting single droplet continues its falling trajectory, first under the same DC electric field, and later without. Thus, the analysis of the single drop velocity during the second part of the record allows us to deduce its mass and associated diameter and its electric charge by comparing the velocities with and without electric field.

The results deduced from the analysis of the four stages of the record (falling drops pair without and with DC electric field, then falling coalesced droplet with and without DC electric field) can be used to verify the mass and electric charge balance between initial drops pair and final coalesced droplet. We propose to define the two following non-dimensional imbalance parameters:

-Mass imbalance:

(5.3)

Here m 1 , m 2 and m 3 are the masses for the two initial droplets and the final coalesced one taken from calculations from drop different falling velocities. The values are calculated from the different velocities deduced from the analysis of the initial and final stages of the record, without DC electric field. For large droplets the coalesced drop fell out of the camera vision field so fast that the last stage cannot be registered. The mass of the coalesced drop cannot be calculated from drop falling velocity. Then the mass of the coalesced droplet is supposed to be equal to those of the pair of droplets, and the mass imbalance is 0.0%. That artifact allows us to calculate the charge imbalance as if the mass of the coalesced droplet was deduced from its last velocity.

-Charge imbalance:

(5.4)

Here q 1 , q 2 and q 3 are the charge of the two initial droplets and of the final coalesced one. Because q 1 and q 2 are often of opposite sign, |q 3 | can be much smaller than |q 1 | or |q 2 |, or even equal to zero.

Thus the |q 3 | is not chosen as the denominator in equation expression (5.4).

The imbalances of mass and charge corresponding to The injections are performed by MSP4 -MSP inv , with drop diameters of D 1 = 0.159 mm, D 2 = 0.171 mm, and charges of q 1 = 0.026 pC, q 2 = 0.006 pC. The imbalances of mass and charges are respectively Δm/m = 0.6 % and Δq/q = 22.4 %. q 1 = 0.030 pC, q 2 = 0.009 pC; Δm/m = -7.6 %, Δq/q = 6.9%, DC=1.8kV (15/04/2015, 23 °C ± 1 °C, case No. 27).

It can be seen that under DC field, the first drop went upwards because the strong electrophoresis is larger than the other forces. Under DC field, the average falling velocity of the two droplets and the falling velocity of the coalesced droplet are very small and there is no visible effect of the decreasing distance before coalescence. The injections are performed by MSP4 -MSP inv , with drop diameters of D 1 = 0.130 mm, D 2 = 0.142 mm, and charges of q 1 = 0.030 pC, q 2 = 0.009 pC. The imbalances of mass and charges are respectively Δm/m = -7.6 % and Δq/q = 6.9 %.

Electrocoalescence under AC fields

Experiments on drop electrocoalescence under AC fields is an improvement of that under DC fields because the former can eliminate the electrophoresis effect of the charged drops [START_REF] Chabert | Droplet fusion by alternating current (AC) field electrocoalescence in microchannels[END_REF]). Thus it is expected that electrocoalescence will be more similar to that of uncharged droplets in oil medium. In experiments, peak voltage is fixed to 1414 V which corresponds to 1000 V in DC fields. During the first experiments the AC frequencies did not seem to influence the drop coalescence process. Thereafter, the frequency was set to the fixed value f = 1000 Hz. To reveal drop charges a short DC field of one second is applied at the moment when the two drops fall inside the Dalsa camera vision field. After the application of the DC field the drops pair enters into the free fall regime for a duration of two seconds, and then the AC field is applied. of AC field is that as the drop coalescence processes occur during the AC application, the charge on the coalesced drops cannot be revealed. Moreover, in the previous cases electrophoresis under DC field tended to increase the approach velocity of the two droplets. The drop approaching process takes more time under AC fields and the initial distances between the two drops should be reduced in order to allow them coalesce before reaching the bottom of the test cell. Some results of this electrocoalescence are shown below.

AC frequency testing without drop charge calculation

The following two figures present the frequency influences of the applied AC field on electrocoalescence. However in these first experiments the short DC voltage was not applied on the electrode, thus no drop charges could be calculated from drop falling velocities. , the drop coalescence looks quite identical, thus it can be concluded that the AC field frequency does not affect drop coalescence process. The drop diameters used in these cases are of visual diameters because the initial distance between the drops pair is very small (s < 4), thus the calculated diameters are not exact.

Acceleration and approach velocity

Under electric field, the distance between the two droplets decreases with time up to contact and When the droplets get closer, the approach velocity is negative. The consequence is that a negative value of acceleration a means that the approaching droplets are slowed down by oil film drainage:

the hydrodynamic force being stronger than the electrocoalescence force. A positive value of acceleration means that the electrocoalescence force increases faster than the hydrodynamic interaction. The curve 'distance' is fitted with a polynomial equation, and a =2*(-4.98*10 -3 ) = -0.010 mm/s 2 ∆U is calculated from application of AC field until drop contact time, and the result at drop contact is ∆U = -0.045 mm/s.

Sum up table for AC fields

All experimental results carried out for different AC field frequencies are shown in

Drop electrocoalescence under AC fields with detection of charge

The frequency of the AC field is chosen to be 1000 Hz and electrocoalescence of drops pair with small electric charges and large electric charges may be compared. The drop diameters are revealed by visualization and image processing using Spotlight. The initial distance between the two drops is very small and s < 3. Thus the calculations on drop diameters and charges may be not very accurate. From calculations, the drop diameters are D 1 = 0.138 mm, D 2 = 0.151 mm while the measured diameters are D 1 = 0.15 mm, D 2 = 0.15 mm. The charges are found to be q 1 = -0.001 pC and q 2 = -0.004 pC. This jump of distance at t ≈ 6s is due to the switch of equations through distance calculations, identical to the 'jumps' of drop distance previously presented for electrocoalescence under DC fields.

Table of results

The 70 different cases studied are summed up in the two following tables:

-Table 5.2 gives the initial state of the generated droplet pair, with the two drop diameters and charges, and the conditions of the injection pulses.

-Table 5.3 gives the conditions of electrocoalescence, applied electric field (DC or AC), coalesced droplet obtained, mass and charge imbalance and acceleration and final approach velocity. All the results presented are cases where electrocoalescence is achieved. Their parameters are then not randomly distributed, but affected by the fact that effect of the difference in weight and charge of the two droplets should mainly favor their approach, and in any case does not prevent possible contact. The second droplet is generally larger than the first one, and there are combinations that are better than others for droplet charge and polarity of the DC electric field applied.

case No. 4.12) are limited to O(s -10 ) and O(s -9 ). For the one point where the imbalance is larger than 20 % it was verified that the diameter of the coalesced drop was not well calculated because the duration of the drop falling path was too short to obtain a stable falling velocity. When the initial non dimensional distance s 0 is larger than 5, ∆m/m is always smaller than 6 %. This value is good because the average time duration for revealing drop falling velocity is often only 1 second. According to equation (3.4) the uncertainty on the estimate of the droplet diameters is then 2.3%. The uncertainty on the drop mass is close to 7%.

Pulse shape

Date MSP Maximum Voltage (V) Drop 1 Drop 2 V 1 V 2 D 1 (mm) q 1 (pC) D 2 (mm) q 2 (pC) DC 1 MSP4-MSP4 3/20/

Droplet electric charge and charge conservation

The calculation of drop charge follow application of the DC field, at the beginning of which the inter drop distance parameter is calculated as s 0 . The calculation of drop charge is presented in section 4.4.2 and the model adopted is good for non-dimensional distance s > 4.

-10,0% -5,0% The calculation of drop charge is after application of the DC field, at the beginning of which the inter drop distance parameter is calculated as s 0 . The calculation of drop charge is presented in section 4.4.3 and the model adopted is considered good for s 0 > 4. On Fig 5 .14 the charge imbalances ∆q/q decrease with s 0 and the imbalances remain smaller than 30 % for s 0 > 5. Only the cases where the mass imbalance has been verified have been drawn on the figure: when the diameter of a coalesced drop has not been calculated from the videos but from application of mass conservation the charge imbalances ∆q/q sometimes exceeds 200 % (Table 5.3). The three points remaining largely out of the 140,0% 0,0 5,0 10,0 15,0 20,0 25,0 ∆q/q s 0 ∆q/q with values of s 0 points cluster, with ∆q/q around 100%, are due to the limited duration of coalesced drop falling path.

Charge conservation is not as good as mass conservation. Estimation of charge uncertainty using equation (3.16) is not sufficient to explain the few cases where the charge imbalance is much larger than 30 %.

Droplet approach velocity

As proposed in section 5.3.1.4, the distance between the two droplets, decreasing with time up to contact, can be fitted by a 2 nd order polynomial function, from which acceleration and approach velocity can be deduced.

Image processing of AOS high speed videos

The time interval between the two successive photos is 1/8000 = 0. 125 ms and the uncertainties in The drops sometimes appear slightly deformed when they are in close proximity. Thus the diameter estimate could be different along the two axes of the drop, as illustrated on Fig 5 .21b, where the shape of the drops is a little flattened as if pushed by the trapped oil film. Contrarily it can sometimes be observed that facing poles deformed in a cone-like shape. This observation confirmed well the assumptions of [START_REF] Ghazian | Head-on collision of electrically charged droplets[END_REF] for the redistribution of electric charge.

Analyses of AOS camera data

The drop approach velocity and acceleration can be deduced from the data of the AOS camera, and can be compared to the results obtained previously by image processing of the Dalsa camera records.

The decrease with time of the inter droplet distance ∆e is drawn on Fig 5 .22. Because of uncertainty sources discussed previously, the value ∆e can take negative values when approaching contact.

Because there is no exact time matching between Dalsa and AOS records, the correspondence of the two sets of points is adjusted manually. (2 nd order).

On Table 5.4, the values of approach velocity and acceleration deduced from 4 th order polynomial fitting on the Dalsa data are clearly closer to the values determined from the AOS camera for case No.

65. The same trend is observed for the case No. 57, but the differences remain very strong. The case

No. 67 shows a different result: the time step chosen to extract points from the AOS camera data set is almost as large as the time step of the Dalsa camera records and the two curves present a very similar behavior. Thus the fit of the two data sets by parabolic curves directly gives quite a good agreement, and the 4 th order polynomial fitting applied on the Dalsa data diminishes this good agreements.

Discussion

Estimation of possible charge injection in the oil from the needle tip

It is confirmed that the injected droplets contain electric charge. Consequently the nature and provenance of the charge should be discussed.

First of all, it should be reminded that the shut-off of the multistage pulses, used to extract a droplet from the water meniscus, takes place before the neck breakdown and that the conductivity of water is in theory sufficient to relax all the surface charges. No offset of the high voltage supply was observed with the oscilloscope, thus the voltage seems really returned to zero. During the successive stages of water drop injection, no jets either lateral or axial were observed. Thus the questionable point is whether the application of the high voltage on the electrode can induce jets of electric charge at the needle tip since it has the smallest curvature of the whole system. For the calculations of injected charge by the needle tip curvature, one example is taken in drop coalescence setup, and the applied voltage on the electrode is 5 kV for H = 8 mm. Under this condition, if positive or negative ions are injected, then the mobility of one ion can be calculated.

(5.3)

Here the k is mobility coefficient, e is one electron charge with e = 1.602*10 -19 C. x is the radius of the anion with x ≈ 10 -10 m. Calculation gives, k ≈ 10 -8 m 2 /Vs.

The electric field was assumed to be E ≈ V/L with V the voltage applied on the electrode, L the distance around the needle concerned with charge injection, seen here as a variable. It is assumed that the injected charge neutralized the electric fields around the needle tip, then the space charge density ρ verifies divE = ρ/ε, with ε = ε 0 ε r . With divE ≈ E/L, it can be written ρ ≈ εV/L 2 . The current density is j = kρE, and the electric current limited by the space charge I = j*πL 2 if the surface is supposed to be a disc of radius L. We deduce the expression:

(5.4)

Taking L = H the numerical applications gives I ≈ π*10 -8 *2.1*8.85*10 -12 *5000 2 /(8*10 -3 ) ≈ 10 -9 A. Then the total injected charge is Q = I*t with t the duration of the pulse, of about 1 ms. Then Q ≈ 2 10 -12 C.

In fact this supposes that the system has the time to stabilize itself: with this indicated mobility the spatial charge injected from the needle tip has moved (during 1 ms under 5 kV, and L = 8 mm) of a distance of k*U/L*t = 6 microns.

The above calculations only give an order of magnitude of injected charge. The charge of the injected drops can then be explained in two different ways. The first possibility relates to the fact that when the high voltage was applied on the bottom electrode, electric charges of the opposite polarity are injected around the needle tip. After the shut-down of the electrode voltage, for electroneutrality purposes, the needle should be of the opposite polarity to the surrounding charge and thus of the same polarity as the electrode. The water droplet has the same potential as the needle, thus it is charged with the same polarity as the initially applied voltage. The second possibility is that the injected charge around the needle tip neutralized the electric field around it, and this electric-field screening charge is absorbed by falling drops through its journey. This effect may cause the drops to have the opposite polarity to the electrode.

During experiments drops were observed to have the same polarity as the bottom electrode, which confirmed the first possibility. However, through series of injection or DC applications on the electrode, the charge on drops lost their concordance with the electrode polarity. This is because the accumulation of injected charge in oil medium cannot be evacuated easily due to low conductivity of the oil. Then the resultant electric charge of the accumulation is not as clearly linked to the applied voltage, and needs to be defined through drop falling velocities under DC field.

Comparison with numerical simulation of falling droplet pair under electric field

In sections 5.4 and 5.5, it was seen that, despite the large number of configurations studied, it remains difficult to extract some of the main features of electrocoalescence behavior. In analyses it is particularly difficult to distinguish the different influences: effects of weight difference and of electrophoresis due to the charged droplets, strong non-linear interactions between droplets. That is why a direct comparison of the records with numerical simulations would probably be the best way to improve the knowledge of the physical interactions and to develop prediction capabilities. As an example, we can compare the results of case n° 6, already illustrated in 5.31. Drop coalescence of MSP4 -MSP4 with drop diameters of D 1 = 0.105 mm, D 2 = 0.113 mm; charges of q 1 = 0.005 pC, q 2 = -0.026 pC; Δm/m = 1.9 %, Δq/q = 0.8 % (24/03/2015, 21 °C ± 1 °C, case No. 6). Radial velocity just before contact (drainage of the oil fime between the two facing water-oil interfaces).

Additional experiments

For drop coalescence under electric fields, most experiments were done in the region between the injection needle and the intermediate high voltage electrode. However, in this region the electric field is not perfectly uniform, its intensity decreasing from the needle tip to the electrode. To have a constant electric field environment, in the drop coalescence set-up test cell, some experiments were done between the high voltage electrode and the bottom grounded metallic plate. The free falling droplet pair traveled through the hole in the middle of the high voltage electrode and the DC voltage is applied on the electrode only when the two droplets have reached the other side. Thus the electric fields are generated between the two parallel metallic plates and are far more uniform as expected.

The distance between the two plates is 1.5 cm, and it increases the total time of observation with respect to previous experiments. However, it was far more difficult to catch good droplet pair configurations (good alignment with the vertical axis, good range of the inter-drop distance) in the vision fields of the cameras after a longer journey from the injection point. An example is presented in Fig 5 .32. between the drop center line and the vertical axis remains quite large (> 30°), and the falling velocity is high. When the DC field is applied on the electrode, from (c) to (h), the falling motion of the drops is slowed down while the drops began to rotate their center line to align with electric field, as predicted theoretically [START_REF] Lundgaard | Coalescence efficiency of water drop pairs in oil influenced by electric field[END_REF]. There, the initial distance between droplets is not sufficient to obtain a perfect alignment before their final coalescence.

Conclusion and recommendations for further experiments

In this chapter, 70 different cases of drops coalescence under DC or AC fields are presented. First analysis of data from Dalsa and AOS camera are proposed to describe the near contact drop approaching process. Discussions on the origin of the droplet charge, on the possible comparisons with numerical simulations, and on a better configuration for additional experiments, are also proposed.

For further research of electrocoalescence in the configuration depicted in this PhD, analysis of the results allows proposing different aspects to be improved:

-Reducing the uncertainty related to temperature effects on viscosity and density. The temperature should be registered separately for each case and adjustment of the viscosity and density should be taken into account for data analysis.

-Managing the variations of surface tension. First a special attention should be paid to the prevention of oils dust contamination. Secondly, a primary ejection of a large droplet can be applied to clean the meniscus interface from surface active component, after that the injection of the droplets pair can be done with a fixed delay. This procedure will allow the comparison with the time dependent interfacial tension measured with the pendant drop tensiometer. A better alternative would consist in verifying in situ the surface tension by transient analysis of droplets oscillations under application of square DC voltage. However, it will be difficult to visualize because of the droplets' small size and their associated high frequency oscillations.

-Working on longer time period of falling droplet pair and on more uniform electric field. It can be done by recording the electrocoalescence process between HV electrode and bottom of the cell as illustrated in chapter 5.6.3.

-Reducing the uncertainties on the droplet diameter and velocity estimations by improving the spatial resolution (mainly for high speed camera) or by developing improved image processing, for example by using correlation technique of the images with moving template circles.

-Improving the use of alternate pulses for drop-on-demand generation to minimize the droplet charge. That can be based on MSP10 pulse, but needs reducing the capacitive effect of the HV cable to enable the increase of the frequency.

-Applying AC electric fields, instead of DC, to minimize the effect of droplet electric charge.

Conclusion

When electric fields are applied in oil-water mixtures small water droplets are attracted to others and merge in larger drops. This electrocoalescence process makes more efficient the oil-water separation by sedimentation.

Experimental data on the electrocoalescence of very small droplets will be useful to improve the understanding of the dynamics of water-oil interface and to validate numerical models. The simple studied configuration consists in a small droplet pair falling in stagnant model oil, under electric field aligned with the symmetry axis of the droplet pair and the direction of gravity.

First part of the work consisted in the well-controlled generation of very small droplet pair (range 20-200 micrometers) aligned with electric field. Droplet-on-Demand generation by EHD method was improved for a better control of the diameter and electric charge of droplets injected from a single metallic needle. This was obtained by applying to a pendant water meniscus optimized multistage high voltage electric pulses.

Then, electrical and hydrodynamic characterization of the droplet pairs and their coalescence are mainly deduced from the analysis of falling velocities, with and without applied DC electric field. A complete data set of droplet position and velocity is deduced from video. A special attention was paid to the visualizations of very small droplet and small falling velocities, involving multiple angles of view, strong zooming and high speed video.

Modelling the different terms of hydrodynamic and electrostatic interactions between droplets allows deducing, from the recorded velocities, their respective mass and electric charge. When coalescence occurs, a record of the resulting single droplet velocity, with and without applied voltage, allows controlling the mass and charge conservations and validating the method.

A first data set of about 70 different cases, taking into account the variation of droplets pair diameter (with a limited range in order to maintain falling velocities between 0.1 and 0. This plateau showed no increase in main drop sizes, however the hidden information is that with increasing voltage, water thread became longer and longer, then thread breakup caused big satellites formations. Among the four curves, the curve of 3.4 ms does not have any 'hole' inside. Thus the pulse duration for drop injection into Primol 352 was fixed to be 3.4 ms. The four curves are placed two by two with a certain distance between, which correspond to different dates for experiments. The two series of 4.2 ms and 3.4 ms done two days earlier are placed before the other two curves 3.2 ms and 3.0 ms done later. This phenomenon may be due to changes of water-oil interfacial tension.

The refractive index was from 1.473 to 1.479 (Anon n.d.,p.35). Then for calculations of relative permittivity, the refractive index was chosen to be 1.476. For Primol 352 ε r ≈ n 2 = 2.1786. This relative permittivity value is used for V low calculation. The V low corresponding to the curves of 43 mN/m is 3.33 kV and to the curves of 41 mN/m is 3.25 kV.

Curves are drawn in function of pulse energies W to test curve horizontal positions with different given pulse energies. 2. The series with the largest pulse duration is placed to a higher W number while the series with the lowest pulse duration is placed in between the extremities. For the experiments on the same date, the series of 3.4 ms was done before that of 4.2 ms, and the series of 3.2 ms was done before that of 3.0 ms. This seemed not due to the accumulated charge inside bulk oil because the behaviors of series done in order are not the same, for the experiments on the two dates respectively. It is supposed that the dynamic influence which relates to the used pulse shape played the role to influence the injected drop sizes. L'électrocoalescence consiste à appliquer un champ électrique, continu ou alternatif, à l'écoulement de l'émulsion de pétrole brut. Les fines gouttelettes d'eau vont être attirées et fusionner sous l'effet des forces électrostatiques créées par ce champ électrique pour former des grosses gouttes. Puis sous l'effet de la gravité, les grosses gouttes sédimentent rapidement. Le champ électrique typique est de l'ordre de 2 à 10kV/cm avec réchauffement du pétrole pour diminuer sa viscosité. L'objectif du traitement de desémulsification est de rendre la quantité d'eau inférieure à 0.1-1%. inconvénients des expériences est que la taille des gouttes est très grande, ce qui n'est pas en accord avec la composition réelle des émulsions. En effets les tailles de gouttelettes d'eau observée dans les émulsions sont de l'ordre quelques microns à quelques dizaines de micron [START_REF] Aichele | Water in oil emulsion droplet size characterization using a pulsed field gradient with diffusion editing (PFG-DE) NMR technique[END_REF][START_REF] Less | Light beam reflectance measurement of droplets diameter distribution in crude oil emulsions[END_REF][START_REF] Rodionova | Preparation and characterization of reference fluid mimicking behavior of North Sea heavy crude oil[END_REF]. De plus pour la configuration de deux gouttes tombant parallèlement, le problème est complètement tridimensionnel parce que le champ électrique est perpendiculaire à leur direction de chute. Cela alourdit beaucoup les calculs qui devraient être fait pour développer des modèles de simulation. Dans cette thèse on cherche donc à diminuer la taille des gouttelettes étudiées expérimentalement, passant de diamètres de l'ordre du millimètre à quelques dizaine de microns. On va chercher à générer et suivre des paires de gouttelettes tombant dans une huile modèle stagnant dans une cuve, alignées avec le champ électrique et la gravité.

Durant la thèse on génèrera ainsi des bases de données expérimentales qui pourront ainsi être comparées avec des modèles numériques en géométrie axisymétrique. La figure B.9 à droite présente le résultat d'un calcul de champ électrostatique en appliquant applique la haute tension sur l'électrode. On peut voir les lignes de champs dans la cellule. On s'intéresse principalement à trois domaines : le domaine autour de l'aiguille où le champ électrique est plus fort, pour effectuer des injections électrohydrodynamiques de gouttelettes ; le domaine entre l'électrode et l'aiguille où le champ électrique est plus uniforme, pour étudier l'interaction et l'électrocoalescence de deux gouttelettes ; le domaine entre l'électrode et la plaque métallique en du fond sera évoqué dans la conclusion de ce travail.

B.2 Montage expérimental et mesures

On a mesuré les caractéristiques des liquides utilisés, résumés Table B.1. Les valeurs de l'eau salée sont tirées de documents, celles pour les huiles mesurées. On constate qu'avec l'augmentation de la température, la viscosité et la masse volumique de l'huile diminuent. La permittivité relative de l'huile et de 2.1 et le temps de relaxation électrique est compris entre 1000 et 5000 s. La tension interfaciale entre l'huile Marcol 52 avec 0.001wt% de surfactant Span80 et l'eau salée est de 42mN/m, avec une incertitude de 6.7%. La mesure de la tension interfaciale est effectuée par la méthode de la goutte pendante comme présenté figure B.10. On applique cette tension sur une durée finie de l'ordre de temps capillaire t cap qui désigne le temps caractéristique de la dynamique de l'interface entre les liquides.

(B.4) Avec l'application numérique, t cap =0.5ms. électriquement neutre [START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF]. En réalité on a observé que les gouttes étaient malgré tout légèrement chargées à cause de l'influence des charges injectées dans l'huile lors de l'applications des tensions élevées. Dans cette région pour tension de 1kV appliqué sur l'électrode, E(y) varie de 40V/m à 60V/m. Pour une goutte à l'équilibre sous champ électrique, on peut calculer sa charge. L'expression des forces exercées sur les deux gouttes est donnée par [START_REF] Batchelor | Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory[END_REF] :

(B.18) (B.19)
où X ij sont les coefficients de mobilité, dépendant des paramètre sans dimension λ = r 2 /r 1 et s = 2 /(r 1 +r 2 ). On peut en déduire les forces à partir des vitesses de chute U 1 et U 2 , avec une influence non linéaire des tailles de gouttes r 1 et r 2 .

Quand on applique le champ électrique, il y a aussi une interaction électrostatique. On peut utiliser pour la caractériser le modèle de [START_REF] Davis | Two Charged Spherical Conductors in a Uniform Electric Field: Forces and Field Strength[END_REF]. Avec les valeurs des rayons r et des vitesses associées U, on peut en déduire les charges q 1 et q 2 .

B.4.2 Electrocoalescence

On utilise le logiciel Spotlight pour suivre les trajectoires des deux gouttes, comme montré Figure

B.5 Conclusion

Dans cette thèse on a cherché à produire une base de données expérimentales sur l'électrocoalescence de paires de très petites gouttelettes tombant dans une cuve d'huile stagnante en configuration axisymétrique. On a pour cela dans un premier temps amélioré une méthode d'injection EHD pour obtenir des très petites gouttes (20-200 microns) dans des conditions de forte tension interfaciale eau/huile. On a appliqué cette méthode pour parvenir à l'injection d'une paire de gouttelettes depuis la même aiguille. Grace à la prise en compte des interactions hydrodynamiques et électrostatiques on a pu caractériser précisément les propriétés de la paire de goutte (taille/charge électrique). On a ainsi généré une première base de données de 70 cas pour comparaison à des simulations numérique.

Pour le futur, l'analyse des résultats obtenus nous permet de suggérer des pistes d'améliorations :

réduire l'incertitude liée à la température sur la viscosité et la masse volumique.

-mieux contrôler la tension interfaciale en fixant la durée entre deux éjections de goutte.

réduire l'incertitude sur le diamètre de goutte et la vitesse par un traitement d'images amélioré (sub-pixel).

-réaliser les expériences de coalescence dans la région située entre l'électrode haute tension et le fond de la cuve, où le champ électrique est plus uniforme.

-appliquer de préférence un champ alternatif et générer les gouttes en utilisant l'impulsion alternée (MSP10) pour réduire la charge des gouttelettes et les effets qui y sont associés.

Fig 1. 1 .

 1 Fig 1.1. Location of crude oil under Earth crust (http://www.green-planet-solar-energy.com/fossil-fuelformation.html).

Fig 1. 2 .

 2 Fig 1.2. Chain formation of water drops in crude oil emulsion between electrodes (Pearce 1954).

Fig 1. 3 .

 3 Fig 1.3. Separator of crude oil contents ( http://www.dowcorning.com/content/oilgas/foam-control.aspx?e=Solutions+by+Application).

Fig 1. 4 .

 4 Fig 1.4. Merging droplets increases the settling velocity in crude oil ( https://www.sintef.no/en/projects/compact-separation-by-electrocoalescence-/).

Fig 1. 4

 4 Fig 1.4 shows a global view of electrocoalescence occurring in crude oil medium during dehydration

Fig 1. 5 .

 5 Fig 1.5. Scheme of uneven pair of conductive drops in an insulating medium in a uniform electric field E (Pedersen et al. 2004).

Fig 1. 5

 5 Fig 1.5 shows the diagram of an uneven pair of water drops in an insulating oil medium under a uniform electric field E. The two drops can be simulated as a dipoledipole configuration and forces acting on the two drops are transversal and radial which are presented as F transverse and F radial on the figure respectively. The r 1 and r 2 are the radius of the two drops with given separated distance d for

Fig

  Fig 1.6. Parameters of falling drops pair (Brazier-Smith et al. 1972).

Fig 1. 7

 7 Fig 1.7 shows drop collisions with different values of the drop kinematic energy. The diagram on the left presented drop collisions with higher kinematic energy, and the out coming result was either coalescence of drops, or splitting of the drop. The diagram on the right described the collisions with lower energy, and generally coalescence took place.

Fig 1. 8 .

 8 Fig 1.8. From left column to right collision energy increases. The drop diameters are of about 0.35 mm (Jiang et al. 1992).

  in both simulations and experiments. The former simulation model was based on mass, momentum and energy conservations before and after drop collisions. In addition energy dissipation which consisted of stretching and reflexive separation regimes in drop bouncing model was added. The simulation model showed the correct number of satellite drops with respect to experimental data. The latter experiments combined with simulations used two streams of monodisperse drops of various liquids and the measurements were done on drop sizes, velocities, dimensions of ligaments and sizes of formed satellites. For the diagram of drop behaviors after collisions, coalescence, bouncing, stretching separation and reflexive separation were studied by (Gotaas, Havelka, Jakobsen & Svendsen 2007) in both experiments and numerical modelling. They defined parameter X as the drops relative position. (1.3) Here the x is defined the same as on Fig 1.6, and r 1 and r 2 are drop radii. The results of collisions were presented as a function of Weber number and initial relative drop positions.

Fig

  Fig 1.9. A typical collision map for droplet-droplet collisions with respect to Weber number and drops relative position X. (Gotaas et al 2007).

Fig 1 .

 1 Fig 1.10. Coalescence of two identical drops increased with Weber number with given charge (Ghazian et al. 2014).

Fig1. 11 .

 11 Fig1.11. Droplet coalescences in a shear flow influenced by capillary number[START_REF] Shardt | Simulations of Droplet Coalescence in Simple Shear Flow[END_REF]).

Fig 1 .

 1 Fig 1.14. Complete coalescence of distilled water drop falling onto a stationary one in sunflower oil medium under electric fields of 56V/mm (Mousavichoubeh et al. 2011).

Fig 1 .

 1 Fig 1.16. Diagram of partial drop coalescence in liquid medium (Hellesø et al. 2015).

  Fig 1.16 shows an example of partial coalescence occurring between a falling drop and a bulk phase in liquid medium. Electric charge distributions on the two bodies are drawn explicitly. The diagram (a) showed the distribution of electric charges before drop contact to each other. The diagram (b) showed that after a fast transfer of the electric charges, the whole body was in the same potential and at this time the electric force tended to separate the drop from the bulk phase.

Fig 1 .

 1 Fig 1.17. Significant decreasing coalescence time by ionization in aqueous drop phase (Dong et al. 2002).

Fig 1 .

 1 Fig 1.18. Diagram of drop coalescence apparatus (Sartor 1954).

Fig 1 .

 1 Fig 1.19. Two falling water drops inside synthetic oil under electric fields (Pedersen 2008).

Fig 1 .

 1 Fig 1.20. Simulation of droplet deformations and coalescence under electric fields (Bjørklund 2009).

Fig 1 .

 1 Fig 1.21. Coalescence process simulation of two drops under different viscosity medium (Lin et al. 2012).

Fig 1 .

 1 Fig 1.22. Two falling water drops in oil under electric field with strength of 2.8 kV/cm (Mohammadi et al. 2014).

Fig1. 23 .

 23 Fig1.23. Two water-ethanol drops in oil under electric field. The field strength for the left series is 193 kV/m and for the right series is 200 kV/m (Guo & He 2014).

Fig 1 .

 1 Fig 1.23 shows drop coalescence on the left photo and non-coalescence on the right photo. It is

Fig 1 .

 1 Fig 1.24 Water drop falling in a viscous liquid. Simulation using Comsol Multiphysics. Navier-Stokes equations are solved in the two liquids, with mobile interface. Results are drawn in a frame moving with the drop (drop radius R = 50µm, water viscosity µ w = 1 mPa.s outer fluid viscosity µ o = 10 mPa.s, surface tension s = 25mN/m). Velocities are non dimensionnalized by U cap = (s/ρ w /R) 1/2 . U stockes / U cap = 1.3 10 -4 and U HR /U cap = 1.9 10 -4.

Fig 1 .

 1 Fig 1.25. Surfactants are concentrated on the rear of the falling drop, and the Marangoni force equals to the friction force.

Fig 1 .

 1 Fig 1.26. Contact of two surfactant covered drops (Zhang et al. 2012).

Fig 2. 1 .

 1 Fig 2.1. Process of single drop formation by pneumatic method (Amirzadeh & Chandra 2010a).

Fig 2. 2 .

 2 Fig 2.2. Piezoelectric droplet-on-demand generator (Harris et al. 2015).

Fig 2. 3 .

 3 Fig 2.3. Experimental set-up for drop injection into model oil by (Pedersen 2008).

  -0.35 mm of the falling one onto a stationary big drop of 4 mm in diameter, and the viscosity effects of surrounding model oil on water drops coalescence were investigated. (Hellesø et al. 2015) performed drop coalescence with drop diameters of 0.7 mm of the falling one and 4 mm of the bottom stationary one. In their experiments on drop coalescence, the test cell was filled up with crude oil and visualization was done by using a near-infrared camera. The experiment configuration and set-up is shown on Fig 2.4.

Fig 2. 4 .

 4 Fig 2.4. Drop coalescence experiments in crude oil using near-infrared camera for visualization (Hellesø et al. 2015).

Fig 2. 5 .

 5 Fig 2.5. Drop coalescence experimental setup (syring injection) for droplet diameters of 1.25 mm (Mohammadi et al. 2014).

Fig 2. 6 .

 6 Fig 2.6. Drop coalescence experimental setup (syring injection) for alcohol contained water drops of diameter range from 1.72 mm to 1.80 mm (Guo & He 2014).

  dynamics. In experiments, one grounded needle is fixed at the top cover of the test cell with its tip immersed into model oil. The needle is filled with tap water and the water meniscus is anchored on the inner diameter of the needle. Below the needle a high voltage electrode is placed. The test cell configuration is shown below.

Fig 2. 7 .

 7 Fig 2.7. Principle of experimental design for drop injection.

  Fig 2.8. They include: a PC sending and receiving signals through standard I/O ports or a special card (video), a waveform generator (Agilent 33500B series, or Wavetek 395), a high voltage amplifier (Trek 20/20C), an oscilloscope (Tektronix DPO 4034), two Dalsa Falcon 1.4M100 XDR cameras and a fast camera (AOS S-MOTION 1683). Two supplementary Bosch Dinion CDD cameras are directly linked to video screens for live reading of water meniscus and injected droplet position in the cell.

Fig 2. 8 .

 8 Fig 2.8. Diagram of electric devices.

Fig 2. 9 .

 9 Fig 2.9. Scheme of different signals from the electric devices recorded by the oscilloscope.

Fig 2 .

 2 Fig 2.10. EHD injection set-up of test cell.

Fig 2 .

 2 Fig 2.10 shows the test cell of EHD injection set-up. Here only two Dalsa cameras were mounted along two perpendicular symmetry axes to visualize drop injection dynamics. One camera serves to survey the meniscus shape and the other to take photographic images of drop injection procedure. The light sources were LED lights, and two lenses just behind each source were used to focus the lights. Due to the good reproducibility of the drop injection processes (Raisin 2011), the second Dalsa camera takes one photo for each injection series with a variable delay (fixed as a multiple of a 200 μs time step) and a shutter speed of 40 μs synchronized with a strobe light. All the images are then combined together to give a full series of drop injection dynamics. It is impossible to trigger the camera to take a full sequence of drop injection because the maximum frame rate of the camera is 100 fps. The whole drop injection process is approximately 1.3 ms, and the time interval between two frames of the Dalsa camera is too large to capture the injection dynamics. To survey the drop electrocoalescence process in model oil, the test cell is replaced by an octagonal test cell and a fast camera (AOS S-Motion) of up to 16000 fps is used. In experiments the configuration of the AOS camera uses a spatial resolution of 250*200 pixels, with 8000 fps.

Fig 2 .

 2 Fig 2.11. Diagram of the needle used for drop injection (the diagram to the left) and the connection of the test cell with the water recipient (the diagram to the right).

  cell set-ups are very important because they can influence drop injection. With different test cell configurations, the obtained results may differ from one to another. Thus it is important to specify the test cell used in each set-up in order to explicitly indicate experimental parameters. For drop coalescence experiments, the test cell was an octagonal Plexiglas cell measuring 4.8 cm on each inner side. This test cell design allows visualizations along four axes perpendicular to its sides, as illustrated on Fig 2.8 and Fig 2.12.

Fig 2 .

 2 Fig 2.12. Test cell configurations of drop coalescence set-up.

Fig 2 .

 2 Fig 2.13. Drop coalescence setup test cell viewed under LED illumination (left) and axisymmetric view of the cell drawn by COMSOL (right). Electric field norm is drawn in color when electrode is set to a high voltage while bottom and needle are grounded.

Fig 2 .

 2 Fig 2.13 shows the test cell configuration of the drop coalescence set-up as a real photo graph and for electric field simulations. The latter uses an equivalent axisymmetric configuration and 1kV voltage is applied on the electrode. The calculated electric field is drawn on one meridian half plane (axis of symmetry on the left). The electrode was held by Nylon screws on the bottom metallic plate, and it has a small hole at the middle of the disc. This hole is 5 mm in diameter and is drilled on the electrode of 80 mm diameter. This hole can be seen clearer on the right hand photo graph of Fig 2.13.

Fig 2 .

 2 Fig 2.15. Measurements of drop diameter by using ImageJ software, according to measurements at different angles (seen 'Length' on the window), the uncertainties done by experimenter is ± 1 pixel.

Fig 2 .

 2 Fig 2.17. Move of the needle in Dalsa Camera vision field for distance calibration (03/06/2015).

Fig 2 .

 2 Fig 2.18 is the measurement window with the gray value curve presented as 'line profile luminance', 'pixels' gives the number of pixels on the line and 'line length' gives the value after converting pixels into millimeters by the calibration process. On Fig 2.18 the boundaries of the drops appear very fuzzy. One straight line cut the drop into two parts, and the variations of gray values were measured along this line. The maximum gray value was located out of the drop and the minimum values were situated on the boundary of the drops. At the center of the drop, the gray value is high because the drop was a bit transparent to background light. The boundary location was determined at the average gray value between these maximum and minimum. For the measurement it is important to avoid the reflection light spot on the drop (white spot on the image).

Fig 2 .

 2 Fig 2.19. Measurements of drop diameters (26/05/2015, -MSP4 injection, -5250 V).

  Camera was also turned 90° to have the maximum number of pixels along the drop path. The vertical positions of the AOS camera are different depending on experiments. In single drop injection experiments, this camera was used to observe the meniscus dynamics. Thus the vision field was placed at the height of the needle tip, and the calibration of the images used the diameter of the needle. In drop coalescence experiments, the AOS camera only took photos in the zone where the drop coalescence would occur, with a strong zoom in. The calibration of AOS was done by taking first, one photo of the needle tip. The diameter of the needle was used for calibrations. For both cases the AOS Camera was speeded up of 8000 fps with a 4 µs shutter speed. This reduced the spatial resolution to 250*200 pixels. One example of single drop injection, identical to the case on Fig 2.18 and Fig 2.19 is shown below (Fig 2.20).

Fig 2 .

 2 Fig 2.20. Visualization of injected drops by using AOS camera (26/05/2015, -MSP4 injection, -5250 V).

Fig 2 .

 2 Fig 2.20 shows the photo taken by the AOS camera during drop injection process, and the drop contour was drawn using ImageJ. Despite this software having subpixel accuracy, it cannot define drop boundaries within pixel resolutions. To overcome this problem, newly developed softwareAMILab by[START_REF] Trujillo-Pino | Accurate subpixel edge location based on partial area effect[END_REF]) is used for measurements. Once the drop boundary was determined within pixel uncertainties and was drawn on the image, the diameter of the drop could be measured using ImageJ.

Fig 2 .

 2 Fig 2.21. Diameters are measured by ImageJ with the help of subpixel resolution given by AMILab, of photo taken by AOS Camera (26/05/2015, -MSP4 injection, -5250 V).

  Fig 2.23 shows the pulse shape MSP10 returned by the amplifier and compared to initial signal.

Fig 2 .

 2 Fig 2.22. Verification of sinusoidal signals by the oscilloscope Tektronix DPO 4034. The blue curve is the input signal generated by the house developed software in the computer and the green curve is the output signal from the amplifier Trek 20/20C, already divided by a factor of 2000.

Fig 2 .

 2 Fig 2.24. Screen image of MSP10 with two marked points shown by oscilloscope Tektronix DPO 4034 which survey the signals of all outputs.

  preparation of the liquids) to 240 hours (about ten days). During the waiting time before making the measure, the beaker containing Marcol 52 is covered by a reversed beaker as shown on Fig 2.25, in order to prevent dusts falling into the oil. It should be mentioned that the Marcol 52 oil is taken from the bottle, and is not directly the oil used in the test cell. On Fig 2.26, it shows that the values measured decrease from 43 mN/m, for immediate measurement, to 21.8 mN/m at the end of 240 hours. The interfacial tension of pure Marcol 52 and distilled water decreases with time to a quite stable value around 22 mN/m while at the beginning the interfacial tension decreased very fast. One point of σ=20.5 mN/m at the end of 72h24min is largely out of the tendency curve, probably due to some accidental pollution (dust particles) of the interface.

Fig 2 .

 2 Fig 2.25. Marcol 52 pure in beaker is covered by another reversed beaker to avoid contamination by dusts.

  tension of Marcol 52 & distilled water measurements, a glass cuvette is filled up with oil and a metallic needle with plate tip is plunged into the oil medium. This needle connects to a syringe filled with salt water.

Fig 2 .

 2 Fig 2.27. Photo of KRÜSS Drop Shape Analyzer --DSA30S model ( http://www.kruss.de/products/contactangle/dsa30/drop-shape-analyzer-dsa30/).

  Fig 2.28. (1) Pear form of drop shape hanging on the needle tip; (2) Hanging drop captured on the screen of analyze.

  results of interfacial tension (IFT) measurements are summed up below. The measurements are of Marcol 52 & tap water by the Du Noüy Ring method, Marcol 52 & 3.5 wt% NaCl solution, 0.001 wt% Span80 in Marcol 52 & 3.5 wt% NaCl solution, 0.0025 wt% Span80 in Marcol 52 & 3.5 wt% NaCl solution and 0.010 wt% Span80 in Marcol 52 & 3.5 wt% NaCl solution.

Fig 2 .

 2 Fig 2.29. Interfacial tension measurements of Marcol 52 with different additives.

  Fig 2.30. Surfactant concentration on liquid drop interface for Marcol 52 with different quantities of Span80.

  by σ, x is replaced by t, A and B are replaced by the coefficients on fig 2.31. During experiments of drop injection or drop coalescence, before experiments of the day were carried out, one large water drop was pushed through to refresh the meniscus which stayed inside oil medium overnight. Between each drop injection the interval time was estimated to be 1.5 minutes. During drop injection, extracted drops could only partly refresh the meniscus. Considering all above parameters, the estimated interfacial tension of the water-oil meniscus occurs at t = 3 minutes with uncertainties corresponding to ± δt = 2 minutes of time durations. The uncertainties can then be expressed as a function of time. Starting from equation (2.10), the uncertainty can be calculated in a simpler way. For 3.5 wt% NaCl solution & Marcol 52, A = 50.844, B = 2.872, t = 3 min and δt = 2 min. The interfacial tension (after 2 mn) is σ = 48 mN/m ± 3.4 %. For 3.5 wt% NaCl solution & 0.001 wt% Span80 in Marcol 52, A = 47.086, B = 5.207, t = 3 min and δt = 2 min.

Fig 2 .

 2 Fig 2.31. Interfacial tension measurements of Primol 352 with distilled water (DoD excel Primol 352).

Fig 2 .

 2 Fig 2.32. Graph of oil density ρ o with increasing temperature T.

  capacity and the conductivity of the oil are measured with a high sensibility resistivimeter IRLAB -LDTRP2 model. Both capacitance C and conductance G were measured in the test cell. The capacity value was measured with a very good repeatability C = 45.8 pF. With air the value was C 0 = 22.3 pF.That gives the relative permittivity, ε r = C/C 0 = 2.1, which is the classical value for white oils, and the relative permittivity of water is taken equal to the classical value of 80.1. The conductance value G was a more fluctuating value, measured between 10 -14 and 4.10 -14 Ω -1 . It corresponds to oil conductivity s = e/t between 0.4 and 1.5 10 -14 S/m. The discharge time of the oil can be calculated directly as:

  EHD drop injection is based on the electrostatic force applied across a liquid meniscus to cause its dynamic motion leading to break-up of the liquid thread into droplets. In a test cell, a grounded needle hanging from the top is filled with water with the meniscus anchored on its inner orifice. At the bottom of this test cell a metallic electrode is connected to high voltage. This test cell is filled up with model oil and the hanging needle is merged into the oil (Fig 3.1).

  Fig 3.1. EHD injection principle (Raisin 2011).

Fig 3 .

 3 Fig 3.2 shows two photos taken in experimental conditions (the cameras were rotated of 90 ° as mentioned before and gravity is from left to right).

Fig

  Fig 3.2. Photos taken as illustrations of convection flux inside test cell: particles exiting at the needle tip are displaced by oil movement (05/05/2015).

  measured, using the Du Noϋy Ring tensiometer as being close to σ = 0.021 mN/m. The distance between the needle tip and the electrode was H = 20 mm and the inner radius of the needle R cap = 0.23 mm. Several multi stage pulse (MSP) shapes were tested and the best one, named MSP 6 on Fig 3.3, was chosen so that the injected drop diameters obtained with this pulse shape vary linearly over a wide range of applied high voltage.

Fig 3. 3 .

 3 Fig 3.3. Injected drop diameters vs maximum applied voltage: results obtained with different pulse shapes in polybutene oil by (Raisin 2011).

Fig 3. 4 .

 4 Fig 3.4. Multistage pulse shape used by (Raisin 2011) to perform EHD water drop injection into polybutene oil.

Fig

  Fig 3.5.

Fig 3 .

 3 Fig 3.5 shows meniscus dynamics under electric pulse were taken by (Raisin 2011) at different instants to give an overview of the drop injection process. From 0 to 900 µs the processes correspond to blunt-tip formation of Part 1. From 900 µs to 1350 µs the retraction of the meniscus takes place over the small influence of Part 2. The rest, until 1750 µs, corresponds to Part 3, which gave a short drag force on the tip of the meniscus. The pulse was shut down at 1750 µs, thus the breakup of the water necking was driven only by the capillary force. Some conclusions were drawn from former experiments:

Fig

  Fig 3.6. Electric potential simulated by COMSOL inside HED injection setups with H = 15 mm. Electric potential varies from 0 (ground -dark blue) to 1kV (high voltage electrode-red color).

Fig 3. 8 .

 8 Fig 3.8. Comparison of MSP j with new defined pulse shape MSP6 (29/05/2015).

Fig

  Fig 3.11: the 'old oil' and the 'new oil', Table3.2 is drawn with all the parameters and calculated

Fig 3. 9 .

 9 Fig 3.9. Single drop injection performed in EHD injection set-ups by MSP6 with tested different pulse durations. The temperature is taken of 21 °C ± 1 °C and H = 15 mm.

  The injection experiments were done in the EHD injection setup. Two series of experiments were done and their results are compared on Fig 3.11. Since the error bars are no larger than the size of markers, on the following graphs the error bars will be omitted. Fig 3.11 shows the drop injection data using the MSP6 pulse of 1.3 ms duration. The series of droplet injection were carried out in Marcol 52 with different ageing. The 'old oil' refers to the Marcol 52 which rested in the test cell for five or six days while the 'new oil' means the Marcol 52 was just poured into the test cell from the container. For these two oils the only significant change seems to be the interfacial tension.

Fig 3 .

 3 Fig 3.11. Drop injection were performed by MSP6 of 1.3 ms into Marcol 52 with different ageing. The temperature is taken of 21 °C ± 1 °C, H = 15 mm and the interfacial tension is σ = 48 mN/m ± 3.4 % for the new oil, but can vary between 25 and 48 mN/m for the old oil.

(

  Figure 3.11 the surface tension is supposed to increase from the beginning (extraction of the smallest droplets) to the end (extraction of the largest droplets).

Fig 3 .

 3 Fig 3.13. Drop injections were performed by different MSP into Marcol 52. The temperature is taken of 21 °C ± 1 °C and H = 15 mm.

Fig 3 .

 3 Fig 3.14. Drop injection as function of V/V low for different MSP.

Fig 3 .

 3 Fig 3.16. Drop injection in Marcol 52 with non-dimensional pulse energy parameter.

Fig 3 .

 3 Fig 3.18. New MSP4 pulse shape for EHD drop injection.

Fig 3 .

 3 Fig 3.19. New MSPinv pulse shape for EHD drop injection.

Fig 3 .

 3 Fig 3.20. Drop injection are performed in EHD injection set-ups with different additives, and the oil free surface is covered by a grounded metallic plate (23 °C).

Fig 3 .

 3 Fig 3.21. Drop injection in Marcol 52 with different additives drawn on V/V low .

  Fig 3.22. Forces acting on falling water drop inside oil medium.

Fig

  Fig 3.23. Density of seawater (ρ w = (1000+σ)kg/m 3 ) from (http://hyperphysics.phyastr.gsu.edu/hbase/chemical/seawater.html.).

  Fig 3.24. Drop falling velocities and distance in drop coalescence set-up. The purple curve is the average drop falling velocity over 2 seconds (26/05/2015, -MSP4, -5300V).

Fig 3 .

 3 Fig 3.25. Single drop injection were performed by -MSP4 into Marcol 52 + 0.001 wt% Span 80 in drop coalescence set-up (26/05/2015).

Fig 3 .

 3 Fig 3.26. Norm of electric field and streamlines inside drop coalescence set up. Applied voltage is 1 kV DC on the intermediate electrode, the needle and bottom electrode are grounded. At the oil free surface, a no charge condition is applied there.

Fig 3 .

 3 Fig 3.28. Forces acting on falling water drop inside oil medium.

Fig 3 .

 3 Fig 3.29. The falling velocities were calculated of one drop injected by -MSP4 at -5300V, into Marcol 52 + 0.001 wt% Span80, DC = 1 kV (23 °C ± 1 °C, 26/05/2015).

Fig 3. 30 .

 30 Fig 3.30. Drop diameters injected into Marcol 52 + 0.001 wt% Span 80 by -MSP4, -MSP inv pulse shapes, with drop coalescence set-up (26/05/2015, 23°C ± 1 °C). The error bars on drop contained charge are taken for the calculated largest uncertainty of one point, 7%.

Fig 3 .

 3 Fig 3.30 shows two series of drop injection: -MSP4 and -MSP inv . The curves consist of visual drop diameters, calculated drop diameters and electric charge of drops. Since the uncertainties of calculated drop diameters are no larger than the size of the marks, the error bars are not drawn here.

  Fig 3.31. Drop falling velocities without and under DC fields for -MSP inv injection at -5.40 kV with DC = +1 kV (26/05/2015, 23°C ± 1 °C).

  motion changing as on Fig 3.29. However no obvious drop velocity changing was observed Fig 3.31. In this case the drop charge uncertainty is difficult to calculate because the third term in equation (3.25) diverges.

Fig

  Fig 3.32. Pulse shape of MSP10.

Fig 3 .

 3 Fig 3.33. Drop diameters injected into Marcol 52 + 0.001 wt% Span 80 by MSP10 pulse shape, with drop coalescence set-up (08/06/2015, 25 °C ± 1 °C).

Fig 3 .

 3 Fig 3.33 presents two series of drop injection by MSP10 performed the same day. The drop diameters are taken from calculations from the falling velocity. Since the visual diameters are in good agreement with calculated diameters, the former are not drawn on the figure. It is the same for diameter uncertainties because the error bars are no larger than markers. The two series of data are in good agreement, which shows the good repeatability of drop injection. The charges of the drops are small, thus the calculated charge uncertainty is large.

Fig 3 .

 3 Fig 3.34. Injected drop diameters vary in function of h/R cap for MSP4 5400V (26/05/2015) and MSP10 -6350V (12/06/2015).

  Fig 3.35. Jets off-axis produced from blunt-tip salted water meniscus for injection into Marcol 52 + 0.001 wt% Span 80 with MSP4, -5450V, h/R cap = 0.75 (26/05/2015).

  Fig 3.38. Injection performed with MSP10 6.8 kV (04/06/2015) case A.

  Fig 3.41. Used pulse shapes MSP4, MSP inv and MSP10 for following chapters.

  Fig 4.1.

Fig 4. 1 .

 1 Fig 4.1. Dual drops injection from parallel needles (Raisin 2011).

Fig 4. 2

 2 Fig 4.2 illustrates some examples of different types of failure in generating a pair of similar droplets.

Fig 4. 3 .

 3 Fig 4.3. Dual drop injection principle.

Fig 4. 3

 3 Fig 4.3 shows the principle of dual drop injection in EHD injection setup. The needle is grounded and the electrode below is connected to high voltage. All are immersed in model oil and the two drop extracted from the needle fall, their axis remaining aligned with the electric field. On their way down, before they reach the electrode, a DC field or AC field may be applied to induce drop electrocoalescence.

Fig 4

 4 Fig 4.4. 3D numerical simulation of electric fields faced by the first injected drop around the needle tip.

Fig 4. 4

 4 Fig 4.4 shows the electric field intensity and streamlines around the meniscus hanging at the needle tip and a first injected drop.With the presence of the drop, the electric fields at the needle tip will be modified. Also the first drop will be exposed to a very high electric field during the second drop injection if it remains in the close vicinity of the needle tip and in the reinforced electric field. The critical value of the uniform electric field has been calculated by[START_REF] Taylor | Disintegration of Water Drops in an Electric Field[END_REF] as:

Fig 4. 5 .

 5 Fig 4.5. Maximum electric field calculated at water meniscus and at the first drop apex during the second drop injection.

Fig 4 .

 4 Fig 4.6 illustrates a real behavior of the first droplet, when located too close to the meniscus apex when the second pulse is applied. . The photographs were taken during drop injection by MSP4 pulse with 1.5 ms duration and -8.5 kV voltage amplitude in EHD injection setup (H = 18 mm). The video

Fig 4. 6 .

 6 Fig 4.6. First drop was split during application of the second electric pulse (30/06/2014, 25 °C ± 1 °C).

  Fig 4.7. Falling two drops without DC fields (injected by MSP10 -MSP10 at +6060V and +6200V). (08/06/2015, 25 °C ± 1 °C).

Fig 4. 8 .

 8 Fig 4.8. Dual drop injection with identical MSP pulse shapes (02/07/2014, T = 25 °C ± 1 °C).

Fig 4. 8

 8 Fig 4.8 shows that the drop diameter range is from about 25 μm to 300 μm. For small drops the diameters obtained are identical. With increasing voltage, a discrepancy occurs between the sizes of the two drops injected which enlarges with voltage. At intermediate diameters, the second drop is a little larger than the first one, while for the higher voltage the second droplet is clearly smaller than the first one.

  water. As shown on Fig 3.34 the injected drop diameter is not a monotonic function of the meniscus height and it can explain that a small reduction of this height increases slightly the droplet size while a larger one decreases it. The curve stopped at 8 kV because jets were produced.

Fig

  Fig 4.9. Diagram of the coupling of the two falling drops.

  equations (4.13), (4.14), (4.15) and (4.16), the radius of the drops can be calculated again.

  (1) and F es (2) are the electrostatic forces acting on the first drop and the second drop respectively.

  dipole forces are attractive, thus the force on the first drop is positive and that on the second drop is negative. involving the external electric field E corresponds to the second term combined with the fourth term in equation (4.19) and (4.20), F z24 (1) = and F z24 (2) = and its simplified form when the charges are considered as concentrated at the drops center is: F dep described in equation (3.19) is considered always to be upwards because the up hanging needle has a much larger curvature than the electrode below. It is not taken into account in the Davis paper where the electric field E is uniform. The hydrodynamic coupling forces, dealing with viscous effects, F 1 and F 2 are calculated by equation (4.9) and (4.10).

  Fig 4.10 the red solid curve is the first term F z1 in equation (4.20) and the red dashed line is the value F dd of the dipole-dipole model of equation (4.27). The green solid line is the second term combined with the fourth term F z24 in equation (4.20) and the green dashed line is the value F c of the Coulomb force model of equation (4.29). The blue solid line is the third term F z3 in equation (4.20) and the blue dashed line is F es from the electrostatic force model of equation (4.23). The purple solid line is the sum of all the forces F z calculated in equation (4.20) and the dashed purple line is the sum of the forces in simplified models. It is shown on the graph that when the distance between the two

Fig 4 .

 4 Fig 4.10. Different part of electric forces on drop 2 are calculated (full lines) and compared with their simplified models (dotted lines).

  Fig 4.12. Drop falling positions a) and velocities b) with dual injection MSP4 -MSP4 at +5300V and -5300V respectively, DC=1kV(29/05/2015, 23 °C ± 1 °C).Duration of application of DC voltage is 1 second. Velocities and position are positives from top to bottom. The position 0 corresponds to the upper entry of the video screen.

Fig 4 .

 4 Fig 4.13 shows dual drop injection of MSP4 -MSP inverse at +5300V and -5300V respectively. The estimated drop diameters are D 1 = 0.115 mm and D 2 = 0.120 mm. The electric charge of the drops areq 1 = 0.019 pC and q 2 = 0.005 pC. Thus at the moment of DC application, both drops decelerated.

Fig 4 .

 4 Fig 4.14 shows the injection of MSP4 -MSP inverse both at -5300V. The estimated drop diameters are D 1 = 0.114 mm and D 2 = 0.112 mm.The electric charge of the drops are q 1 = -0.012 pC and q 2 = 0.010 pC. It can be seen that during DC application, the first drop speeded up while the second drop slowed down.

Fig 4 .

 4 Fig 4.15 shows the injection of MSP inverse -MSP inverse at +5300V and -5300V respectively. The estimated drop diameters are D 1 = 0.117 mm and D 2 = 0.133 mm. The electric charge of the drops are q 1 = 0.006 pC and q 2 = 0.011 pC.

  level of camera imaging due to oil opacity to visible light, more experiments have been done in transparent model oils to investigate the electrocoalescence process. In existing experiments two configurations were discussed: one drop falling onto a stationary one, as in[START_REF] Hellesø | Experimental study of electrocoalescence of water drops in crude oil using near-infrared camera[END_REF] experiment , and two drops falling side-by-side. The first arrangement was studied in detail over the last ten years, after the early experiments ofCharles and Masson (1960):[START_REF] Chiesa | Investigation of the role of viscosity on electrocoalescence of water droplets in oil[END_REF],[START_REF] Aryafar | Electrocoalescence: Effects of DC Electric Fields on Coalescence of Drops at Planar Interfaces[END_REF],[START_REF] Ristenpart | Non-coalescence of oppositely charged drops[END_REF],[START_REF] Mousavichoubeh | Electro-coalescence of an aqueous droplet at an oil-water interface[END_REF],Hamlin et al (2012),[START_REF] Hellesø | Experimental study of electrocoalescence of water drops in crude oil using near-infrared camera[END_REF]... The second configuration was much less studied : in[START_REF] Eow | Drop-drop coalescence in an electric field: the effects of applied electric field and electrode geometry[END_REF] the drop diameter is about 1.8 ± 0.3 mm and they carried out 19 experiments. In[START_REF] Pedersen | Forces Acting on Water Droplets in Electrically Energized Oil Emulsions[END_REF], the drop diameter is around 0.5 mm for 2 experiments of drops pair coalescence…. and very recently in[START_REF] Mohammadi | Electrocoalescence of binary water droplets falling in oil: Experimental study[END_REF]) the diameter of the drops was close to 1.25 mm and 8 cases are described and commented. In experiments presented by[START_REF] Guo | Coalescence behaviour of two large water-drops in viscous oil under a DC electric field[END_REF], 30 -40 different cases with varying electric field strengths were tested with drop diameters from 1.72 mm to 1.8 mm.They observed both coalescence and non-coalescence. The latter happened above threshold field strength of 200 kV/m.

For

  data analyses, Spotlight needs the initial definition around the droplet of rectangular ''areas of interest (AOI)'' which serves to follow drop motions by detecting gray levels. These areas of interest move then automatically with the droplets on the successive images of the records. Following a number of attempts, the best results were obtained with the combination of a border threshold applied to track the first drop leading edge (Fig 5.1: right droplet) and a center tracking applied to the second droplet (Fig 5.1: left droplet).

Fig

  Fig 5.1. AOIs on drops pair for drop coalescence of MSP inv -MSP inv case No. 47 (03/06/2015, 25 °C ± 1 °C).

  Fig 5.1 shows two drops tracked by AOI with threshold value of 128 applied on 256 inverse grey levels. Generally in AOIs, values of thresholds could be adjusted to give best border recognition. This tracking method is better than border threshold tracking and center tracking because at the moment when the two drops jump together to form one drop, the two AOIs continue following the drop locations instead of losing the track. The inter drop distance before drop coalescence is shown as ∆e on Fig 5.1. The calculation of this distance is the distance between drop locations x 1 and x 2 minus the first drop diameter D 1 and half of second drop diameter D 2 .

Fig 5. 2 .

 2 Fig 5.2. AOI overlaps on coalesced drop. Drop coalescence of MSP inv -MSP inv case No. 47 (03/06/2015, 25 °C ± 1 °C).

Fig 5. 2

 2 Fig 5.2 showed the overlap of two AOI with thresholds of 128 applied on 256 inverse grey levels. The distance ∆e is calculated as follows.

  Fig 5.3. Drop coalescence of MSP4 -MSP4 with drop diameters of D 1 = 0.105 mm, D 2 = 0.113 mm; charges of q 1

Fig 5. 3

 3 Fig 5.3 shows the electrocoalescence of oppositely charged droplets injected by MSP4 -MSP4 pulses,with resulting drop diameters of D 1 = 0.105 mm, D 2 = 0.113 mm, and charges q 1 = 0.005 pC and q 2 = -0.026 pC. These parameters are deduced from the analysis of the first part of the record, without and then with DC electric field, as was previously done in chapter 4. The effect of the strong viscous interaction is taken into account to deduce the net forces acting separately on each droplet. One visible effect of this interaction is the increase of the falling velocities of the two droplets when their distance decreases, as can be seen on Fig 5.3b between the time 4 and 6 s. On Fig 5.3a, the curve

  Fig 5.4a shows the drop positions and distance between the facing interfaces, and Fig 5.4b shows that the two positively charged drops decelerated immediately under positive DC field.

  Fig 5.4. Drop coalescence of MSP4 -MSP inv with drop diameters of D 1 = 0.159 mm, D 2 = 0.171 mm; charges of q 1 = 0.026 pC, q 2 = 0.006 pC; Δm/m = 0.6 %, Δq/q = 22.4%, DC=1.6kV (08/04/2015, 21 °C ± 1 °C, case No. 21).

  sur pér. (v1(mm/s))

Fig

  Fig 5.6. Application of DC field and AC field on the electrode: at the beginning of the trigger a DC field is applied with duration of 1.5 seconds. Then the voltage is cut off for about 2 seconds. Then the AC field is applied as oscillations on the figure.

Fig 5 .

 5 Fig 5.6 shows the order of application of DC and AC fields on the electrode. The inconvenient aspect

  Fig 5.7. Drop coalescence of MSP4 -MSP inv with drop diameters of D 1 = 0.13 mm, D 2 = 0.14 mm; frequency of the AC field is of f = 1000 Hz, AC=1.414kV (09/06/2015, 25 °C ± 1 °C, case No. 61).

Fig 5 .

 5 Fig 5.7 shows electrocoalescence of drops pair under AC field with frequency f = 1000 Hz. The injections are performed by MSP4 -MSP inv , with drop diameters D 1 = 0.13 mm and D 2 = 0.14 mm.

  8

  coalescence. The distance can be fitted by a second order polynomial function of time, as illustrated by Fig 5.9. From the function, we propose to identify two terms: the acceleration a (mm/s 2 ), as the second derivative of the function, and the drop approach velocity at the contact time, called '∆U near contact (mm/s)' calculated as the first derivative of the function at the last time point before drop coalescence.

Fig 5. 9 .

 9 Fig 5.9. Distance of the drops for drop coalescence of MSP4 -MSP inv with drop diameters of D 1 = 0.13 mm, D 2 = 0.14 mm; frequency of the AC field is of f = 1500 Hz, AC=1.414kV (09/06/2015, 25 °C ± 1 °C, case No. 62).

  Fig 5.10. Drop coalescence of MSP10 -MSP10 with drop diameters of D 1 = 0.15 mm, D 2 = 0.15 mm; frequency of the AC field is of f = 1000 Hz, AC=1.4kV (12/06/2015, 25 °C ± 1 °C, case No. 69).

  Fig 5.11. Comparison of droplet diameters deduced from direct analyses of photos and diameters calculated from falling velocities. The results are reported for drops pair injections: blue circles correspond to the droplet injected first and red squares to the second ones.

Fig 5 .

 5 Fig 5.12 shows that the mass imbalances ∆m/m decreases with the parameter s 0 . The mass imbalance exceeds 10% only for the smaller values of initial non dimensional distance s 0 . This is because equations (4.11) and (4.12) are limited to O(s -10 ) and O(s -9 ). For the one point where the

  Fig 5.13. Electric charge injection during EHD injection for different pulse shape used with different drop injections.

Fig 5 .

 5 Fig 5.14. Charge imbalances with values of s 0 .

Fig 5 .

 5 Fig 5.15. Acceleration vs applied voltage for DC (blue diamond) and AC (red square) applied electric fields.

  Fig 5.20. Subpixel boundary (red segments drawn by AMIlab software) and maximum cross section (yellow line placed by eyes): a) before drop contact, b) after drop contact.

Fig 5 .

 5 Fig 5.22. Inter drop distance during coalescence process. Blue points are data from Dalsa camera and red squares are extracted from AOS high speed video (2015-06-10, test1; case No. 65).

Fig 5 .

 5 Fig 5.23. Inter drop distance during coalescence process, AC=1.4kV (2015-06-10, test1; case No. 65).

The

  Fig 5.27b is the induced charge on the water meniscus surface induced by the charge density on Fig 5.27a.

  Fig 5.3 reproduced hereafter in Fig 5.30 a and b for convenience. Simulation is carried out there using Comsol Multiphysics, version 5.0. The model takes into account the two water droplets and the oil medium through resolution of Navier-Stokes equations in a mesh moving at constant velocity to follow approximately the falling droplet pair. The resolution of electrostatic equations is performed in the oil, the water being considered as pure conductive medium and the droplet carrying the electric charge deduced from experiment. The model is Arbitrary Lagrangian-Eulerian, which means that the boundary of the droplet is described by deformable lines on which the deforming mesh is attached. The clear identification of the water-oil interfaces, as described in (Raisin et al. 2011), allows the direct application of the electrostatic pressure on the surface. In the case presented in Fig 5.29 the coupling of the different equations takes into account a no slip velocity condition at the droplet surface to deal with the effect of surfactant described in chapter 1.4.4 (Fig 1.24). The result of droplet trajectories and velocities are compared to the measurements in Fig 5.30, almost up to the contact of the drop boundaries where the numerical simulation stopped. Other numerical models, as for example those based on Volumeof-Fluid or Level-Set methods, would allow continuing the computation after the droplet coalescence.

Fig 5 .

 5 Fig 5.32. Drops are oriented under electric fields to be parallel to these field lines. The time intervals between the photos are not the same: from (a) to (e) the time interval is of 1s; from photo (e) to (h) the time interval is decreased to 0.5s 0.3s and finally 0.2s. Injection of the drops are of MSP inv -MSP inv with drop diameters of D 1 = 0.19 mm and D 2 = 0.21 mm, measured directly from images (2015-02-23). A 1kV DC voltage is applied just before image c).

Fig A. 2 .

 2 Fig A.2. Drop injection in Primol 352 by MSP1 with different pulse durations. The temperature is taken of 21 °C ± 1 °C, the interfacial tension is σ = 43 to 41 mN/m, H = 6 mm.

Fig A. 3 .

 3 Fig A.3. Drop injection in Primol 352 by MSP1 with different pulse durations. The temperature is taken of 21 °C ± 1 °C, the interfacial tension is σ = 43 mN/m, H = 6 mm.

  Au fur et à mesure de l'extraction, la pression dans la cavité diminue, défavorisant l'extraction. Pour surmonter ce problème, on peut procéder à l'injection d'eau ou de gaz dans la cavité pour maintenir la pression. Mais cette injection favorise le mélange de l'eau avec le pétrole brut. A la sortie des puits d'extraction, on obtient ainsi un mélange d'eau, de pétrole et de gaz. Une fois ce mélange extrait, la séparation préliminaire des différentes phases est nécessaire. A l'aide de la gravité, on sépare eau, pétrole et le gaz dont les densités sont différentes. Mais cette séparation n'est pas complète parce que le pétrole brut contient encore des fines gouttelettes d'eau. De plus, une injection d'eau douce peut encore être utilisée pour dissoudre les sels minéraux, comme l'illustre la figure B.2 qui présente le principe de prétraitement du pétrole brut. L'ensemble du procédé va ainsi générer une émulsion très stable de très fines gouttelettes d'eau dans le pétrole. L'émulsion résultante doit être encore traitée pour diminuer la quantité d'eau contenue dans le pétrole avant le raffinage.

Fig B. 2 .

 2 Fig B.2. Séparation préliminaire du pétrole brut[START_REF] Arnold | Managing Water -From Waste to Resource[END_REF]).

Fig B. 4 .

 4 Fig B.4. Séparation de l'émulsion d'eau dans le pétrole par la méthode d'électrocoalescence (https://www.sintef.no/en/projects/compact-separation/by/electrocoalescence-/).

  Figure B.5.

Fig

  Fig B.5. A gauche, deux gouttes tombant parallèlement avec un champ électrique appliqué perpendiculairement à la direction de chute (Eow & Ghadiri, 2003); à droite, une goutte tombant sur une grosse goutte posée au fond de la cuve (Helleso et al, 2015).

Figure

  Figure B.5, à gauche les expériences sont réalisée dans une huile modèle avec des gouttes de diamètre D≈2mm, le champ électrique E vaut 1.0kV/cm ; à droite les expériences sont faite dans le pétrole brut avec visualisation par caméra infrarouge, le diamètre de la goutte tombante est D=0.7mm et le diamètre de la goutte sur le fond D=4mm, le champ électrique vaut E=0.25kV/cm. Les

  La figure B.6 montre la cellule expérimentale. La cellule est remplie d'huile Marcol 52 ou Primol 352 suivant les expériences. Oon ajoute parfois un surfactant (Span 80) pour modifier la tension interfaciale. Au fond de la cellule l'électrode est reliée à la haute tension. L'aiguille plongée dans l'huile est reliée à la masse. Cette aiguille est remplit de l'eau salée (solution contenant 3.5 %(en poids) de NaCl).

Fig

  Fig B.6. Schéma de la cellule d'essai.

Fig B. 7 .

 7 Fig B.7. Connexion des différents appareils électriques.

Fig

  Fig B.8. Visualisation de la cellule d'essai.

Fig B. 9 .

 9 Fig B.9. Cellule d'électrocoalescence visualisé sous lumière verte générée par LED (gauche) et vue d'un demi plan méridien de la cellule (considérée comme cylindrique) tracé par COMSOL (droite). Dans ce montage, l'électrode, percée d'un petit trou en son centre, est à 1,5 cm du fond de la cellule. La haute tension est appliquée à l'électrode, avec la plaque métallique du fond de la cuve et l'aiguille reliées à la terre. La norme du champ électrique calculé est tracée en couleur.

Fig

  Fig B.10. Méthode de la goutte pendante pour mesurer la tension interfaciale. A gauche, une photo prise de l'expérience. A droite, une image acquise sur l'écran avec l'extraction du contour de la goutte d'eau.

Fig B. 11 .

 11 Fig B.11. Evolutions en fonction du temps des tensions interfaciale eau/huile pour différentes proportions de surfactant.

Fig B. 12 .

 12 Fig B.12. Schéma de l'injection d'une seule goutte[START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF].

Fig

  Fig B.13. Schéma de la cellule d'essai.

Fig B. 14 .

 14 Fig B.14. Forme d'impulsion utilisée[START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF].

Fig

  Fig B.15. Procès d'injection électrohydrodynamique[START_REF] Raisin | Electrocoalescence in Water-in-Oil Emulsions: Toward an Efficiency Criterion[END_REF].

Fig

  Fig B.16. Les différentes formes d'impulsions testées.

Fig

  Fig B.17. Les différentes injections par les différentes formes d'impulsion.

Fig

  Fig B.20. Chute de goutte à l'équilibre dans l'huile modèle.

Fig B. 22 .

 22 Fig B.22. Chute de goutte à l'équilibre sous champ électrique.

Fig

  Fig B.26. Schéma de l'injection d'une paire de goutte.

Fig B. 28 .

 28 Fig B.28. Définition des paramètres du ménisque.

Fig B. 29 .

 29 Fig B.29. Injection des gouttes en variant la hauteur du ménisque.

Fig B. 30 .

 30 Fig B.30. Interaction hydrodynamique entre les deux gouttes.

Fig

  Fig B.31. Interaction des deux gouttes sous champ électrique.

B

Fig B. 33 .

 33 Fig B.33. Suivi des deux gouttes avec les aires d'intérêt définies dans Spotlight.

Figure B. 35

 35 Figure B.35, on montre les vitesses des deux gouttes sous champ électrique continu. Le premier et deuxième plateau montre que les gouttes changent de vitesse de chute lorsque le champ est appliqué. Les deux pics correspondent au moment de la coalescence. Puis la goutte unique générée par coalescence poursuit sa chute sous champ électrique. A la fin de l'application de ce champ la grosse goutte tombe librement. Les deux premières parties permettent de calculer les rayons et les charges des gouttes. La troisième et la quatrième partie donnent des informations sur la goutte coalescée.

Fig B. 40 .

 40 Fig B.40. Charges électriques des gouttelettes mesurées en fonction des types d'impulsion EHD appliqués.

Fig

  Fig B.41. Exemple de calcul des vitesses et accélérations des gouttes juste avant contact.

Fig B. 42 .

 42 Fig B.42. Accélération en fonction de tension appliquée.

Fig B. 43 .

 43 Fig B.43. Séquence de photos de l'électrocoalescence d'une paire de gouttelettes de diamètre environ 150 µm, tirées d'une vidéo rapide (8000 images/s).

Fig B. 44 .

 44 Fig B.44. Distance entre les deux gouttes en fonction du temps durant la phase d'électrocoalescence.

Figure B. 44

 44 Figure B.44 les points rouges sont obtenus par caméra rapide, alors que les points bleus sont obtenus par caméra Dalsa (CMOS) à la cadence de 10 images par seconde. Les points rouges montrent des fluctuations aléatoires à cause de l'incertitude, mais cette dispersion est plus petite que l'incertitude de ±1 pixel.

Fig

  Fig B.45. Chute et contact de deux gouttelettes sous champ électrique (simulation COMSOL Multiphysics).

Fig

  Fig B.47. Schéma de la cellule pour les expériences supplémentaires.

Fig

  Fig B.48. Electrocoalescence des deux gouttes dans la région comprise entre l'électrode haute tension et le fond de la cuve. On constate que l'alignement vertical n'a pas pu être assuré pour cet essai.

Table 2

 2

	.1 is given to sum up produced droplets by different drop injection methods.
	Up to now, experiments for droplet sizes evolution from one given nozzle is rarely reported. It is only
	noted that the two unique drop-on-demand techniques which can give droplet sizes smaller and
	independent of nozzle sizes are the ultrasonic and EHD methods (Lee 2002). However the ultrasonic
	technique could cause equipment to vibrate. Thus it appears to us that the best drop-on-demand
	method is EHD injection, on one hand because the contamination of chemical compositions of liquids
	can be avoided, on the other hand because the droplet size can be reduced largely, due to its

  Liquids used for experiments such as EHD injection and drop coalescence, consist of distilled water, tap water mixed with 35 wt% NaCl, Marcol 52, Marcol 52 mixed with 0.001 wt% Span 80 and Primol 352. These different liquids were used in different experiments according to their properties.

Distilled water is used for EHD injection experiments; however, for drop coalescence experiments tap water mixed with 3.5 wt% NaCl took the place of the former one, aimed at approaching the real water in crude oil conditions. Marcol 52 and Primol 352 are oils for medical and other food control or cosmetic uses; they are both products from crude oil refineries, transparent oils. Marcol 52 oil is used in EHD injection experiments and drop coalescence experiments while mixed with 0.001 wt% Span 80. The latter is a trading name of sorbitan monooleate, a nonionic surfactant often used as a w/o emulsifier or as an o/w emulsifier in cosmetics or food preparation. Its formula is C 24 H 44 O 6 and specific weight is 429 g/mol

  Table 2.1 shows the decrease in both liquid density and dynamic viscosity when the temperature increases. These variations are linear.Table 2.1. Measured oil viscosity values and densities with increasing temperature.

	Temperature T (°C)	ρ o (kg/m 3 )	µo (Pa.s)
	17	830.0	13.43*10 -3
	20	829.0	12.01*10 -3
	23	828.0	10.72*10 -3
	26	827.0	9.63*10 -3
	29	826.0	8.75*10 -3

  time, viscous time, interfacial tension, viscosity and density. Thus to define electric pulse duration for EHD injection, or to calculate Stokes velocities while two drops are falling in oil medium, these characteristics are very

	important to understand.
		T (°C)	ρ (kg/m 3 )	σ (mN/m)	µ (Pa.s)
				( after 2 mn)
	Water (taken from	20	1002.7	70	1.002 *10 -3
	document)	25	1001.5		0.890 *10 -3
	3.5 wt% NaCl	20	1029.1		1.070 *10 -3
	solution (taken	25	1027.6		0.955 *10 -3
	from document)
	3.5 wt% NaCl	21	830	48 ± 3.4%	0.012 (taken from
	solution & Marcol				document)
	52 3.5 wt% NaCl	21	828.7 ± 0.3		11.43*10 -3 ± 3.8 %
	solution & 0.001 wt% Span80 in Marcol 52	23 25	828.0 ± 0.3 827.3 ± 0.3	42 ± 6.7 %	10.62*10 -3 ± 3.5 % 9.93*10 -3 ± 3.2 %
		27	826.7 ± 0.3		9.33*10 -3 ± 2.9 %
	Distilled water &	21	863	43 ± 1	0.193 (taken from
	Primol 352				document)

Table 2

 2

.2. Different measured characteristics of used oils, parts of the values taken from documents are from (Copin-Montégut 2002) and the web ''ExxonMobil''.

Table 2 .

 2 2 gives a summation of all the parameters that need to be used in later calculations or experiments. As it was illustrated before, the interfacial tension curves of distilled water & Marcol 52 matched the curve of 3.5 wt% NaCl solution & Marcol 52. Thus it is considered that for short time durations of interfacial tension measurements, the interfacial tension values revealed in the latter case are also valid for the former. In the next chapter, the high viscosity influence of Primol 352 will be illustrated in experiments.

  table is drawn up below. The calculated values are of theoretical values, thus the uncertainties are not revealed here. It should be noted that for experiments in tap water & Marcol 52 and distilled water & Primol 352, the needle used had an inner radius of 0.23 mm. For experiments in 3.5 wt% NaCl solution & Marcol 52 and 3.5 wt% NaCl solution & 0.001 wt% Span80 in Marcol 52, the inner radius of the needle was 0.36 mm. Other values are taken as in the example combined with Table 2.2. For the Reynolds number calculated for distilledwater and Primol 352, we assume a water drop with diameter of D = 0.14 mm falling inside the oil medium. The terminal velocity is calculated based on the principles in section 3.5, and the result is U t Table2.3 it is seen that all cases have a very low Reynolds number. This means that a water drop of D = 0.14 mm falling in the oil medium is always in Stokes regime. The capillary time t cap is smaller than the viscous time t µ for all the cases using Marcol 52, and for Primol 352 on the contrary the capillary time t cap is higher than the viscous time t µ . These values are in accordance with

	= 7.57*10 -3 mm/s.
		T (°C)	R cap	t cap (ms)	t µ (ms)	Re	Oh
			(mm)
	Tap water & Marcol	21	0.23	0.504 ±	4.08	0.0015	0.13
	52			1.7 %
	3.5 wt% NaCl solution	23	0.36	1.000 ±	10.2	0.0017	0.10
	& Marcol 52			1.7 %
		21		1.069 ±	10.7 ± 3.5 %	0.0019	0.10
	3.5 wt% NaCl solution			3.4 %
	& 0.001 wt% Span80	23	0.36		11.4 ± 3.2 %	0.0022	0.095
	in Marcol 52	25		1.068 ±	12.2 ± 2.9 %	0.0025	0.089
				3.4 %
		27			12.9 ± 2.6 %	0.0029	0.084
	Distilled water &	21	0.23	0.533 ±	0.273	5*10 -6	2.09
	Primol 352			1.2 %
	Tap water &	---	0.23	0.785	4.76	---	0.16
	polybutene
	Table 2.3. Calculated characteristics numbers of different used oils. The calculations of Reynolds number are
	based on drop diameter D = 0.14 mm, and the intermediate value of drop terminal velocity is calculated using
	Stokes equation from the given drop diameter and the corresponding different oil viscosities according to
	different cases.

From

  Table 3.2 is drawn with all the parameters and calculated values. The temperature is estimated to be 21 °C ± 1 °C, and the injection was done with tap water. The values of viscosity and density of oil are taken as the values of Marcol 52 with Span 80 addition on Table 2.2. During the experiment reported on Fig 3.11 the surface tension is supposed to increase from the beginning (extraction of the smallest droplets) to the end (extraction of the largest droplets).

		Polybutene	Marcol01 (old oil)	Marcol02 (new oil)
	T	21°C ± 2 °C	21 °C ± 1 °C	21 °C ± 1 °C
	ε r	2.3	2.1	2.1
	σ (mN/m)	21	25 to 48 (*)	48 ± 3.4 %
	µ o (Pa.s)	10.2*10 -3	11.4*10 -3 ± 3.8 %	11.4*10 -3 ± 3.8 %
	ρ o (kg/m 3 )	824	828.7 ± 0.33	828.7 ± 0.33
	R cap (mm)	0.23	0.23	0.23
	H (mm)	20	15	15
	V low (kV)	2.85	3.1 to 4.3 (*)	4.29 ± 2.1 %
	t cap (ms)	0.77	0.66 to 0.51 (*)	0.504 ± 1.7 %
	t µ (ms)	4.76	4.26 ± 3.5 %	4.26 ± 3.5 %
	Table 3.2. Calculated parameters of drop injection in Marcol 52 and comparison with polybutene (*) during the

injection in "old" oil, the interface is cleaned progressively by successive ejections of droplets : thus the surface tension is varying between the value obtained after 5 days with respect to measurements summarized in formula 2.8 and the value obtained after only 3 minutes.

Table 3 .

 3 3 shows the characteristics of injections. The uncertainty on V min is taken as the intersection of the curve with the abscise Voltage (kV). On Fig 3.11, the absolute uncertainty viewed from the

		0,25
		0,2
	Drop diamters (	0,1 0,15
		0,05
		0
		3	3,2	3,4	3,6	3,8	4	4,2	4,4	4,6

mm) Voltages (kV) Injections of MSP6 1.3ms into Marcol 52 with different deposition time

  to 0.02 kV, thus the relative uncertainty is of 0.02/3.6 = 0.5 %. This uncertainty on V min combined with the uncertainty on V low gives 2.6 % of relative uncertainty on V min /V low . The same estimation is made for the other ratios.

	graph is estimated h/R cap = 0.75	MSP j	MSP6	MSP6
	t (ms)	1.76	1.30	1.30
	V min (kV)	3	3.62	3.88 ± 0.02
	V 0.1mm (kV)	3.3	4.01	4.01
	V max (kV)	3.7	4.30	4.32
	t cap (ms)	0.77	0.66 to 0.51 (*)	Old oil New oil 0.504 ± 1.7 %
	t/t cap	2.30	2.58 ± 1.7 %	2.58 ± 1.7 %
	V low (kV)	2.85	3.1 to 4.3 (*)	4.29 ± 2.1 %
	V min /V low	1.05	1.17 to 0.844 (*)	0.905 ± 2.6 %
	V 0.1mm /V low	1.16	1.30 to 0.935 (*)	0.935 ± 2.6 %
	V max /V low	1.30	1.39 to 1.00 (*)	1.01 ± 2.6 %
	figures	Fig 3.3.	Fig 3.11	Fig 3.11

Table 3.3. Results of droplets injection with different pulse shapes and oil age

4 Summation of the characteristics of drop injection using different pulses

	3.6.Marcol (-MSP4)		Marcol (-MSP inv )		Marcol (MSP10)
	T		23 °C ± 1 °C		23 °C ± 1 °C		25 °C ± 1 °C
	ε r			2.1		2.1		2.1
	σ (mN/m)		42 ± 6.7 %		42 ± 6.7 %		42 ± 6.7 %
	µ o (Pa.s)		10.6*10 -3 ± 3.5 %		10.6*10 -3 ± 3.5 %	9.93*10 -3 ± 3.2 %
	ρ o (kg/m 3 )		MSP10 drop injections with calculated diameters 828.0 ±0.33 828.0 ±0.33	827.3 ±0.33
	0,300
	R cap (mm)			0.36		0.36	drop diameters	0.36
							(08/06, 1)
	H (mm)			8		8		8
	V low (kV) Drop diameter (mm) t cap (ms)		4.05 ± 3.4 % 1.07 ± 3.4 %		charge (pC) 4.05 ± 3.4 % 1.07 ± 3.4 %	drop diameters (08/06, 2) drop charges (08/06, 4.05 ± 3.4 % 1.07 ± 3.4 %
	t µ (ms)		11.3 ± 3.2 %		11.3 ± 3.2 %	1)	12.1 ± 2.9 %
	h/R cap = 0.75			-MSP4		-MSP inv	drop charges (08/06, MSP10
							2)
	t (ms)	6	6,2	6,4 Voltages (kV) 1.5	6,6	6,8 1.5		1.5
	V min (kV)			5.00		4.90		6.25
	V 0.1mm (kV)			5.35		5.30		6.25
	V max (kV)			5.40		5.55		7.90
	t/t cap		1.41 ±3.4 %		1.41 ±3.4 %		1.41 ±3.4 %
	V min /V low		1.24 ± 3.4 %		1.21 ± 3.4 %		1.54 ± 3.4 %
	V 0.1mm /V low		1.32 ± 3.4 %		1.31 ± 3.4 %		1.54 ± 3.4 %
	V max /V low		1.33 ± 3.4 %		1.37 ± 3.4 %		1.92 ± 3.4 %
	figure			Fig 3.30		Fig 3.30		Fig 3.33

Table 3.5. The characteristics of drop injection performed into 0.001 wt% Span80 in Marcol 52 model oil with drops made of 3.5 wt% NaCl solution.

  Table5.1. From that table it can be seen that for small values of s, the calculated diameters for the first drop are different from visualized diameters. The largest discrepancy occurs for f = 800 Hz. For the four other cases, the diameters are comparable for the different frequencies and the results (time to coalescence, acceleration and final approach velocity) remain quite close to each other.

		D 1 (mm)		D 2 (mm)		s	t (s)	a (mm/s 2 )	∆U near
		Visual /		Visual /				contact
		calculated		calculated				(mm/s)
	f = 800 Hz	0.13 / 0.113	0.13 / 0.130	2.556	3.2	-0.0037	-0.028
	f = 1000 Hz	0.13 / 0.123	0.14 / 0.136	3.000	3.2	-0.0082	-0.044
	f = 1500 Hz	0.13 / 0.122	MSP4-inv : (-5100V)-(-5040V) AC+1414V 0.14 / 0.137 2.947 2.9 -0.0098	-0.047
	f = 2300 Hz	4,000 0.13 / 0.122	0.14 / 0.137	2.671	2.3	0,08 -0.0100	-0.045
		3,500						0,07	center drop 2
	f = 5000 Hz	3,000 0.13 / 0.128	0.15 / 0.143	2.488	1.7	0,06 -0.0126	-0.051
	1,500 2,000 2,500 Table 5.1. Electrocoalescence with different applied AC frequencies. position (mm)	0,03 0,04 0,05	distance (mm)	inter drop distance head drop 1
		1,000						0,02
								Poly. (inter
		0,500						0,01	drop distance)
				y = -4,98E-03x 2 -2,38E-03x + 1,15E-01
		0,000						0,00
		0	2	4	6	8	10	12
				time (s)

  Table 5.2 and Table 5.3, the drop diameter and charge are all from calculated values. In cases No. 60 -No. 64 there is no charge calculation for both drops because during experiments no DC field was applied to change drop motion. The mass imbalances and charge imbalances cannot be calculated for these cases.

		MSP inv -MSP inv 1000	5/13/2015 0.164	5400 0.020	5450 5.4%	0.175	32%	0.021	0.205 -0.139	0.001 0.010
		MSP inv -MSP inv -1000	0.154	5400 -0.018	-5450 4.2%	0.143	18%	-0.012	0.175 -0.091	0.018 0.012
		MSP4-MSP inv -1000	5/30/2015 0.162	5100 -0.022	5100 4.6%	0.114	22%	0.029	0.131 -0.117	0.006 0.018
		MSP4-MSP inv -1000	0.160	5100 -0.066	-5100 2.0%	0.130	7%	0.035	0.126 -0.079	-0.026 0.005
	2015 3/24/2015 4/2/2015 4/8/2015 4/15/2015 6/1/2015 6/3/2015 6/5/2015 6/7/2015 6/8/2015 6/9/2015 6/10/2015 -5070 5400 -5400 5450 5450 5450 5450 5450 5450 5450 5450 5450 5600 5600 5600 5600 5600 5600 5600 5600 5600 5600 5400 5400 5400 5400 5400 5400 -5100 -5100 -5100 -5200 -5100 -5200 -5150 5100 5100 5100 5010 5030 -4930 -4930 6420 -6420 6450 6440 -6280 -6350 -6450 6220 -5100 -5100 -5100 -5100 -5100 -6350 5100 MSP10-MSP10 6/12/2015 -6310 MSP4-MSP4 MSP4-MSP4 MSP4-MSP4 MSP4-MSP4 MSP4-MSP4 MSP4-MSP4 MSP4-MSP4 MSP4-MSP4 MSP4-MSP4 MSP4-MSP4 MSP inv -MSP inv MSP inv -MSP inv MSP inv -MSP inv MSP inv -MSP inv MSP inv -MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP inv -MSP inv MSP inv -MSP inv MSP inv -MSP inv MSP inv -MSP inv MSP inv -MSP inv MSP4-MSP inv MSP4-MSP inv MSP10-MSP10 MSP10-MSP10 MSP10-MSP10 MSP10-MSP10 MSP10-MSP10 MSP10-MSP10 MSP10-MSP10 MSP10-MSP10 MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP4-MSP inv MSP10-MSP10 MSP inv -MSP inv Voltage (V) Drop 3 -1000 0.165 -0.040 -1000 0.148 -0.013 D 3 (mm) -1400 0.152 -0.025 q 3 (pC) -600 0.159 -0.063 1000 0.166 -0.019 -1000 0.172 0.022 -1000 0.164 0.013 -1000 0.179 -0.005 400 0.137 -0.032 -1000 0.163 0.024 600 0.136 -0.007 -1400 0.176 0.002 800 0.141 -0.026 -600 0.182 -0.081 1000 0.138 -0.021 -1000 0.138 -0.038 1200 0.145 -0.030 -1000 0.141 -0.024 1400 0.148 -0.032 -1000 0.172 0.019 1600 0.148 -0.027 -1000 0.176 0.009 1800 0.144 -0.042 1400 0.176 0.020 2000 0.150 -0.046 -1400 0.170 0.005 800 0.208 -0.088 1400 0.158 -0.021 1000 0.218 -0.053 -1400 0.127 -0.003 -1000 0.223 0.006 -1400 0.159 -0.011 -1800 0.222 -0.046 1400 0.166 0.019 -2000 0.229 0.001 800 0.193 0.032 1414 1000 0.196 0.017 1414 1200 0.203 0.049 1414 1400 0.210 0.019 1414 1600 0.209 0.039 1414 800 0.155 -0.028 1400 0.176 0.001 1000 0.181 0.033 1400 0.182 0.001 1200 0.190 0.040 1400 0.186 0.001 1400 0.191 0.050 2000 0.167 0.001 1600 0.164 0.039 1400 0.186 0.001 1800 0.170 0.039 1000 0.184 0.001 MSP10-MSP10 -6310 1000 0.110 0.017 Table 5.3. Summation of results on coalescence conditions (initial conditions in table 5.2) and mass and charge -5400 0.129 0.013 0.132 -0.033 2 5400 0.124 -0.012 0.133 0.025 3 -5500 0.105 0.000 0.113 -0.016 4 -5500 0.103 0.008 0.113 -0.011 5 -5500 0.110 0.002 0.114 -0.024 6 -5500 0.105 0.005 0.113 -0.026 7 -5500 0.112 0.004 0.118 -0.027 8 -5500 0.111 0.000 0.119 -0.026 9 -5500 0.112 0.004 0.119 -0.026 10 -5500 0.111 -0.007 0.117 -0.027 11 -5500 0.118 -0.007 0.121 -0.035 12 -5600 0.134 -0.004 0.187 -0.062 13 -5600 0.155 -0.026 0.188 -0.015 14 -5600 0.165 -0.016 0.188 0.021 15 -5600 0.167 -0.020 0.184 -0.021 16 -5600 0.167 -0.001 0.195 0.000 17 -5580 0.140 0.033 0.164 0.003 18 -5500 0.146 0.025 0.164 -0.010 19 -5580 0.152 0.028 0.166 0.014 20 -5580 0.156 0.030 0.177 -0.012 21 -5580 0.159 0.026 0.171 0.006 22 -5450 0.116 -0.003 0.132 -0.023 23 -5450 0.132 0.021 0.155 0.009 24 -5450 0.141 0.023 0.161 0.012 25 -5450 0.141 0.028 0.163 0.017 26 -5450 0.120 0.033 0.142 0.007 27 -5450 0.130 0.030 0.142 0.009 5100 0.114 -0.022 0.127 0.008 -5100 0.120 -0.028 0.133 0.013 -5100 0.122 -0.042 0.129 -0.021 -5200 0.128 -0.045 0.136 0.007 -5100 0.110 -0.025 0.120 0.015 -5200 0.113 -0.019 0.125 -0.004 -5200 0.127 -0.064 0.125 0.005 -5070 0.128 0.005 0.145 0.007 -5100 0.136 -0.007 0.148 -0.005 -5040 0.123 0.002 0.135 0.012 -5060 0.113 -0.003 0.159 -0.001 -5030 0.135 -0.037 0.146 -0.030 -4910 0.105 -0.021 0.112 -0.008 -4910 0.109 -0.023 0.114 0.003 6250 0.122 0.005 0.149 0.003 -6250 0.123 -0.006 0.153 0.000 6200 0.128 0.011 0.145 0.006 6230 0.120 0.005 0.133 0.008 -6080 0.119 -0.005 0.132 -0.007 -6280 0.093 0.000 0.104 -0.002 6300 0.113 -0.006 0.131 -0.001 -6140 0.122 0.013 0.137 0.004 AC -5020 0.113 0.130 -5040 0.123 0.136 -5040 0.122 0.137 -5040 0.122 0.137 -5040 0.128 0.143 -5060 0.139 -0.084 0.139 -0.023 6150 0.139 -0.033 0.153 -0.012 -5060 0.146 -0.027 0.153 -0.035 -6130 0.096 0.005 0.151 case No. mass imbalance charge imbalance ∆U near contact -2.0% 6% -0.154 0.017 a (mm/s 2 ) 5.0% 15% -0.141 0.013 (mm/s) 2.6% 9% -0.096 0.004 -0.6% 7% -0.115 0.008 DC 1 2.0% 2% -0.103 0.0% 142% -0.031 0.000 0.014 2 3.4% 2% -0.068 0.0% 106% -0.001 -0.029 0.013 3 0.0% 106% -0.017 0.0% 83% -0.047 -0.002 0.003 4 0.0% 34% -0.004 0.0% 191% -0.099 -0.153 0.004 5 0.8% 14% -0.061 8.2% 37% -0.039 -0.001 0.008 6 1.9% 1% -0.103 3.1% 38% -0.050 0.008 0.017 7 0.2% 28% -0.103 0.6% 14% -0.095 0.011 0.021 8 5.5% 23% -0.112 0.0% 247% -0.023 0.003 0.018 9 4.2% 19% -0.152 0.0% 253% -0.027 0.012 0.023 0.0% 31% -0.115 5.5% 24% -0.065 0.001 0.016 0.0% 14% -0.151 20.6% 93% -0.041 -0.002 0.025 0.0% 34% -0.028 -2.3% 131% -0.028 0.001 -0.006 0.0% 47% -0.029 6.6% 31% -0.030 -0.005 -0.003 0.0% 5% -0.242 8.4% 44% -0.062 -0.002 -0.013 0.0% 27% -0.067 3.9% 14% -0.069 0.005 -0.003 0.0% 140% -0.102 AC -0.016 0.0% 15% -0.073 -0.028 -0.004 0.002 0.0% 7% -0.090 -0.044 -0.008 0.011 3.8% 22% -0.062 -0.047 -0.010 0.002 0.0% 5% -0.144 -0.045 -0.010 0.015 0.6% 22% -0.103 -0.051 -0.013 0.003 -2.3% 13% -0.014 2.0% -0.037 -0.018 0.017 -2.1% 14% -0.067 -3.8% -0.069 -0.044 0.007 -0.9% 23% -0.076 -3.8% -0.051 -0.022 0.004 -1.9% 18% -0.102 8.7% -0.075 -0.135 0.004 -3.5% 3.1% -0.187 5.2% -0.036 -0.006 0.008 -2.5% 0.9% -0.171 12.7% -0.019 -0.003 0.003 -0.004 -6130 0.138 -0.001 0.151 0.0% 4% -0.063 0.002 -0.004 MSP10-MSP10 -6310 -6150 0.119 0.004 0.157 1000 0.164 0.003 2.6% 38% -0.114 0.012 -0.013 1400 0.138 0.030 0.0% 7% -0.075 0.000 imbalances.
	28 Table 5.2. Summation of initial conditions (multistage pulses shapes and amplitudes, date, droplets diameter MSP4-MSP inv 4/16/2015 5200 -5280 0.082 0.012 0.092 0.005 1000 0.170 0.010 7.5% 102% -0.096 0.015
	29 and charge) for coalescence cases. MSP4-MSP inv -1000 0.163	5200 -0.053	-5280 0.4%	0.106	1%	0.018	0.112 -0.102	0.011 -0.004
	30	MSP4-MSP inv -1000	5/11/2015 0.174	5400 -0.005	5450 7.3%	0.120	29%	0.015	0.137 -0.133	-0.019 0.011
	31	MSP4-MSP inv 1000	0.241	5400 0.014	-5450 0.0%	0.122	40%	0.010	0.140 -0.060	-0.020 0.014
	32	MSP4-MSP inv -1000	0.202	-5400 0.025	5450 0.0%	0.119 100% -0.035	0.138 -0.079	-0.018 -0.022
	33	MSP4-MSP inv 1000	0.155	-5400 0.038	-5450 -0.1%	0.126	10%	0.022	0.143 -0.085	-0.021 0.010

On

  The acceleration a for two drops approaching each other up to contact under DC fields or AC fields is drawn on Fig 5.15 as a function of applied voltage. The voltage corresponds to the values applied directly on the electrode in the test cell, and for the value of AC field the applied peak voltage is divided by .

  Table5.3. For case No. 57 the 'distance' is also fitted by a function of polynomial order 4, and it gives ∆U = -0.050 mm/s and a = -0.041 mm/s 2 .

	Case No.		∆U 2nd		Interdistance during coalescence process ∆U 4th ∆U aos ∆a 2nd	∆a 4th	∆a aos
			0,012 (mm/s)		(mm/s)	(mm/s)		(mm/s 2 )	(mm/s 2 )	(mm/s 2 )
			0,01
	65		0,008 -0.037		-0.059	-0.059		-0.018	-0.101	-0.172
									y = -0,3941x 2 + 5,0115x -15,924
	57	∆e (mm)	0,004 0,006 -0.030		-0.050	-0.143		-0.005	-0.041	Dalsa AOS	-0.788
			0,002								Poly. (AOS)
	67		0 -0.051		-0.047	-0.063		-0.022	0.001	-0.029
			-0,002
	6,4 Table 5.4. Approaching velocities and acceleration obtain by fitting of Dalsa (2 6,42 6,44 6,46 6,48 6,5 6,52 6,54	nd and 4	th order) and AOS data
							t (s)
						Interdistance during coalescence
			0,06
			0,05						y = -0,0146x 2 + 0,0848x -0,0602
			0,04
		(mm)	0,03								Dalsa
		∆e	0,02								AOS
			0,01								Poly. (AOS)
			0
			-0,01
			3,5	3,7	3,9	4,1	4,3	4,5	4,7	4,9	5,1	5,3
							t (s)
	Fig 5.26. Inter drop distance during coalescence process (2015-06-10, test2; case No. 67).

On Fig 5

.

26 the drop approaching velocity and acceleration are: U = -0.063 mm/s, a = -0.029 mm/s 2 . This corresponds to case No.67 on Table

5

.3 where U = -0.047 mm/s and a = 0.001 mm/s 2 .

2 Drop injection in Primol 352 model oil

  3 mm/s) and different applied voltage types (DC or AC), was established. Analyses of the results and experimental uncertainties, and example of possible comparison with numerical simulations using Comsol Multiphysics™ software, allow performing some recommendations for future work.For drop injection, a pulse shape was fixed to be MSP1 because this pulse can give good drop injection. As the model oil is changed, the experimental parameters should be changed with. First, the height of the hanging needle should be determined by calculations of V low . The needle was not changed from drop injection in Marcol 52, thus R cap = 0.23 mm. in experiments. The pulse generator was the Wavetek 395 model with an amplification of 1000 times. The upper limit of the possible applied voltage is of 5 kV. By determining V low = 5 kV, the corresponding needle height H could be as high as 25 mm. In reality, the applied voltage should be higher than the calculated values, thus the height of the needle should be reduced to assure a sufficient electrostatic pressure. From many experimental tests, the distance of the needle height was fixed to H = 6 mm, which corresponds to V low = 3.4 kV.After having confirmed the distance H, the pulse durations is put in question. During experiments, it was found that values between the capillary time and the viscous time calculated on Table2.3, as for the photos above, the deformations of the meniscus are different than that in Marcol 52, because in Primol 352 it is the viscosity which constrains these water dynamics. In other cases of injection, the drop should be extracted by meniscus inertial force. But in this case the injection is mainly based on electrical force. Due to the high viscosity, the meniscus deforms very slowly during injection. The blunt-tip meniscus is not formed as it was observed with Marcol 52. All through the process, only the head of the meniscus grows with a longing and thinning necking. Part 3 of the pulse ensured the jet, but in this case it extends the duration of the connection time of drop with the rest of the meniscus due to the high viscosity environment. The necking doesn't break off easily. The break up takes place later than the voltage shut down, and the drops are not electrically charged. Moreover, with this comportment of necking, only small satellites formed. The curves of drop diameters with applied voltage are shown below.

	perform drop coalescence experiments. Thus further investigations on oil characteristics have not
	been done.
			Primol (4.2ms, 3.4ms)			Primol (3.2ms, 3.0ms)
	ε r	2.1786 (taken from document)	2.1786 (taken from document)
	σ (N/m)		0.043			0.041
	µ (Pa.s)	0.193 (taken from document)	0.193 (taken from document)
	ρ o (kg/m 3 )		863			863
	R cap (mm)		0.23			0.23
	H (mm)		6			6
	V low (kV)		3.332			3.254
	t cap (ms)		0.535			0.548
	t µ (ms)		0.275			0.275
	h/R cap = 0.75		MSP1			MSP1
	t (ms)		4.2; 3.4			3.2; 3.0
	V min (kV)		3.90			4.24
	V 0.1mm (kV)		4.07			4.46
	V max (kV)		4.98			4.98
	t/t cap		7.85; 6.36			5.84; 5.47
	V min /V low		1.17			1.30
	V 0.1mm /V low 400μs	800μs	1.22 1200μs	1600μs	1.37 2000μs
	V max /V low		1.49			1.53
	figures		Fig A.2			Fig A.2
	Table A.1. Summed up of the characteristics of Primol 352 performed for injection on Fig A.2.
	2400μs	2800μs	3200μs	3400μs	4000μs

A.

Marcol 52, could not give successful drop injection due to lack of applied electric energy on the meniscus. Thus increasing the pulse durations was tested up to 3 ms to 4ms, values with which drops are injected successfully into Primol 352. The injection process is shown below with one example of MSP1 3.4 ms at 4.8 kV, with a relative meniscus height of h/R cap =0.75. Fig A.1 Injection process in Primol 352, using MSP1 at 4.8 kV, with a total pulse duration of 3.4 ms.

On

Table B .

 B 1. Les différentes caractéristiques des liquides mesurés.

marcol 52 & tap water Salt W 3,5% Span 80 0,001% Salt W 3,5% Span 80 0,0025% Salt W 3,5% Span 80 0,010% Salt W 3,5%

Moy. mobile sur pér. (v2(mm/s))

Moy. mobile sur pér. (v1(mm/s)) 4 Moy. mobile sur pér. (v2(mm/s))

Measurement uncertainties

Uncertainty on visual estimation of drop diameter

Dalsa Falcon 1.4M100 XDR Camera visualizations

Dalsa Falcon 1.4M100 XDR Cameras were used in both experiments of EHD injection and drop coalescence. The former used software ImageJ to measure drop diameters while the latter used Spotlight for drop falling velocity calculations.

For the experiments of EHD drop injection, one injected drop and the needle tip were taken on the same photo. The needle tip serves to set the scale in the photo graph, in order to measure the drop diameter. The meniscus anchored on the needle tip is clear in shape, and thus the inner diameter length can be measured in pixels. The inner needle diameter is already known, and this then can help to convert pixels to millimeters.

Zoom-in on drop coalescence

The electrocoalescence processes are recorded by AOS camera at 8000 fps with spatial resolution of 250*200 pixels. The frame rate cannot be increased further because increasing camera speed requires reduction of spatial resolution. 8000 frames per second is a compromise between spatial and temporal resolution. Since the electrocoalescence spot fluctuates on height in the cell from one case to the next, the sequence is often out of the camera vision field. An example of drop

Appendix

A. Drop injection in Primol 352

Primol 352 is a mixture of saturated hydrocarbons, products of petroleum refinery, and it is also stable in terms of oxidation. For Primol 352, the remarkable difference compared to Marcol 52 is the high viscosity value, of about 0.193 Pa.s, ten times higher than that of Marcol 52. Thus the influence of oil viscosity is much more important than the capillary effect on drop injection dynamics. On Table 2.3, the only high value of Ohnesorge number corresponds to Primol 352, which signifies that the viscosity takes a dominating role over all. The calculations of capillary time and viscous time show that t cap =0.535 ms while t µ =0.275 ms, the viscous time is smaller than the capillary time. Generally the pulse duration is defined between the two extremities presented by the capillary time and the viscous time, however this is not the case for Primol 352. Experiments of tap water drop injection into primol 352 were done in EHD injection setups. Due to the high viscosity of the model oil, the water meniscus which should be formed by pressure difference between the water recipient and the oil-water interface could not be got anymore. Thus the water recipient was replaced by a microsyringe through which a quantity of water could be pushed out manually to adjust the height of the meniscus. On peut calculer l'incertitude sur la charge de goutte. (B.17)

A.1 Characteristics of Primol 352 model oil

On note que si le changement de vitesse de chute n'est pas remarquable U ≈ U 0 , le terme sur l'incertitude de vitesse diverge et cela entraine la divergence de δq/q.

Pour améliorer la linéarité des courbes diamètre/tension et contrôler la charge de la goutte on a testé plusieurs types d'impulsions nouvelles.

B.3.2 Injection d'une paire de goutte

On cherche à injecter une paire de goutte alignée avec le champ électrique comme montré Figure B.26. Puis en appliquant le champ électrique nous étudierons leur rapprochement et leur coalescence.

On a combiné ensemble de nos résultats et analysé l'influence des différents paramètres.

Abstract

When electric fields are applied in oil-water mixtures small water droplets are attracted to others and merge in larger drops. This electrocoalescence process makes more efficient the oil-water separation by sedimentation. Experimental data on the electrocoalescence of very small droplets will be useful to improve the understanding of the dynamics of water-oil interface and to validate numerical models. The simple studied configuration consists in a small droplet pair falling in stagnant model oil, under electric field aligned with their symmetry axis. First part of the work consisted in the well-controlled generation of very small droplet pair (range 20-200 micrometers). Droplet-on-Demand generation by EHD method was improved for a better control of the diameter and electric charge of droplets injected from a single metallic needle. This was obtained by applying to a pendant water meniscus optimized multistage high voltage electric pulses.

Then, electrical and hydrodynamic characterization of the droplet pairs and their coalescence are mainly deduced from the analysis of falling velocities, with and without applied DC electric field. A complete data set of droplet position and velocity is deduced from video. A special attention was paid to the visualizations of very small droplet and small falling velocities, involving multiple angles of view, strong zooming and high speed video.

Modelling the different terms of hydrodynamic and electrostatic interactions between droplets allows deducing, from the recorded velocities, their respective mass and electric charge. When coalescence occurs, a record of the resulting single droplet velocity, with and without applied voltage, allows controlling the mass and charge conservations and validating the method.

A first data set of about 70 different cases, taking into account the variation of droplets pair diameter and different applied voltage types (DC or AC), was established. Analyses of the results and experimental uncertainties, and example of possible comparison with numerical simulations using Comsol Multiphysics™ software, allow performing some recommendations for future work.

Résumé

Lorsque des champs électriques sont appliqués à des mélanges eau-huile, les petites gouttelettes d'eau sont attirées entre elles et se regroupent en gouttes plus grosses. Ce processus d'électrocoalescence rend plus efficace la séparation huile-eau par sédimentation. Des données expérimentales sur l'électrocoalescence de très petites gouttelettes sont nécessaires pour améliorer la compréhension de la dynamique de l'interface eauhuile et pour valider les modèles numériques. La configuration simple étudiée dans ce travail de thèse concerne une petite paire de gouttelettes tombant dans une cuve d'huile modèle et soumise à un champ électrique aligné avec leur axe de symétrie. La première partie du travail a consisté à générer de façon contrôlée une paire de très petites gouttelettes (dans la gamme de diamètres 20-200 microns). La génération de goutte à la demande, par méthode électrohydrodynamique (EHD), a été améliorée pour un meilleur contrôle du diamètre et de la charge électrique des gouttelettes injectées à partir d'une aiguille métallique unique. Ceci a été obtenu en appliquant à un ménisque d'eau pendant à l'extrémité de l'aiguille des impulsions électriques de forme optimisée.