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Investigation of the role of viscosity on electrocoalescence of water droplets in oil
Separation and Purification Technology 50 (2006) 267–277
Investigation of the role of viscosity on electrocoalescence of water droplets in oil M. Chiesa a,∗ , S. Ingebrigtsen a , J.A. Melheim b , P.V. Hemmingsen c , E.B. Hansen b , Ø. Hestad a a
SINTEF Energy Research, NO-7465 Trondheim, Norway b NTNU, NO-7491 Trondheim, Norway c Ugelstad laboratory NO-7491 Trondheim, Norway
Received 25 October 2005; received in revised form 5 December 2005; accepted 5 December 2005
Abstract The sedimentation velocity of water droplets in oil increases proportionally to the square of the droplets diameter. It is therefore beneficial for the separation of water from oil to increase the droplet size by enhancing the coalescence process. Electric fields are used for this purpose in order to separate stable water-in-oil emulsion. Field-induced charges on the water droplets will cause adjacent droplets to align with the field, attract each other and ultimately coalesce. The efficiency of electrically induced coalescence is strongly dependent on the oil characteristics. In the present study experiments are performed to observe the behavior of a droplet falling onto a stationary one at different oil viscosity. The droplet is exposed to an electric field of different magnitude parallel to the direction of the droplet motion. Experimental observations are compared with numerical prediction of the droplet kinematics, and the contribution of the forces acting on the droplets and their dependency on viscosity is discussed in detail. © 2005 Elsevier B.V. All rights reserved. Keywords: Water-in-oil emulsion; Sedimentation stability; Flocculation; Viscosity; Coalescence; Dipole–dipole forces; Film-thinning force; Drag force; Buoyancy force
1. Introduction In crude oil production, oil and co-produced water is mixed due to the pressure drop and high shear forces when the fluid passes the wellhead and various choke valves from the reservoir to the production facilities. This will cause the formation of water-in-oil emulsion and one of the primary task for the production platform is to remove the water from the crude oil. Water level above 0.5% constitutes a quality problem and a concomitant price reduction. All water-in-oil emulsions, perhaps with the exception of micro-emulsions, are thermodynamically unstable. However, the destabilization may take considerable time, and a stable emulsion is unable to resolve itself in a defined time period without some form of mechanical or chemical treatment (Sj¨oblom et al. [1]). Water-in-crude oil emulsion destabilization basically involves three steps, namely flocculation, followed by sedi-
∗
Corresponding author. Tel.: +47 92017605. E-mail address: [email protected] (M. Chiesa).
1383-5866/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.seppur.2005.12.003
mentation of water droplets due to density differences, and finally coalescence of the individual water droplets. Large droplet sizes, high density difference between the oil and the aqueous phase, and low continuous phase (oil) viscosity causes high sedimentation rates. Flocculation is the aggregation of two or several droplets, touching only at certain points, and with virtually no change in total surface area. Coalescence is the process of droplets fusing together and forming larger droplets. In crude oil emulsions, emulsifying agents are present at the oil–water interface, hindering this coalescence process. The mechanism by which indigenous crude oil components stabilize a droplet interface is by producing a physical barrier for droplet–droplet coalescence. Electrostatic fields are to some extent used to overcome this physical barrier by helping smaller droplets to coalesce into larger droplets that sediment quicker (Eow et al. [2]). The electrostatic effects arise from the very different dielectric permittivity and conductivity of oil and water. Water is characterized by permittivity values much higher than those of oil and in practice one may consider water as perfectly conducting. To a large extent the effects of the electrostatic field are explained by body-forces
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Nomenclature A, Ad a B b Cd d E, E0 er eg Fe Fb Ff Fvm Fd–d Fd Fext Ffluid f* g h i k0 Red Reσ rd r1 r2 T u Vd v vi vr xi We
droplet surface reduced radius Bond number slip length drag coefficient separation vector electric field vector relative motion vector gravity vector dielectrophoretic force vector body force vector film-thinning force vector virtual mass force vector inter-droplet force vector drag force vector external-droplet force vector fluid-droplet force vector correction factor, see Eq. (10) modulus of the gravity least distance between two particles droplet i droplet curvature droplet Reynolds number Reynolds number for an oscillating inviscid droplet droplet radius radius of falling droplet radius of stationary droplet temperature continuous phase velocity vector droplet volume droplet velocity vector velocity vector of droplet i relative droplet velocity vector droplet position vector Weber number
Greek letters ε0 vacuum dielectric constant relative dielectric constant of droplet εd εoil relative dielectric constant for oil φ potential function γ1 magnitude of the interfacial tension gradient λ viscosity ratio (λ = µd /µc ) viscosity of droplet µd µc viscosity of the continuous phase θ angle between E and d ρd density of the discrete phase density of the continuous phase ρc σ interfacial tension of water drop in oil τd velocity response time of water drop in oil
acting on water droplets either because they are charged, giving rise to electrophoretic forces, or because they become polarized in divergent fields, resulting in dielectrophoretic forces. The focus of the present work lies in investigating the effect of viscosity on water droplets flocculation and coalescence when different electric fields are applied. This is important when dealing with water-in-oil emulsion of highly viscous crude oils (Hemmingsen et al. [3]). Experiments are performed releasing water droplets and observing their motion in oil with different viscosity when electric fields of different magnitudes are applied. A transformer oil is employed in the experiments due to its transparency. The experimental observations of the motion of falling droplets are compared to the predictions obtained by the modeling framework proposed by Chiesa et al. [4]. The numerical obtained results are used to assess the role of viscosity on water droplets kinematics and coalescence. The kinematics of the water droplets in the sedimentation process is dominated by the buoyancy force due to the density difference between the water and the oil. The viscous forces are directly acting as a damping mechanism throughout the drag force. When two droplets approach each other, viscosity acts as a damping mechanism throughout the film-thinning process. The electrically induced force counteract the damping effect of the drag and film-thinning force when the droplets are relatively close to each other. When the droplets are in contact with each other, the coalescence rest time is indirectly dependent on the viscosity throughout the impact velocity. The coalescence rest time is believed to be dependent on the Weber number at impact when the droplet Reynolds number and Bond number are kept constant. 2. Theoretical background The trajectory of a spherical droplet i is calculated by integrating Newton’s second law. The law equates the change in momentum per change in time of the droplet with the forces acting on it, and reads: dxi = vi dt mi
dvi = F fluid + F ext + F d–d , dt
(1) (2)
where mi , xi , and vi are the mass, position, and velocity of the droplet. Ffluid represents the vector of forces acting from the fluid on the droplet it is modeled by summing up the drag force Fd , the virtual mass force Fvm , and the buoyancy or external body force Fb . Fext is the external force, and Fd–d represents the inter-droplet force. The modeling framework proposed by Chiesa et al. [4] is employed in the present work. The electric force between the droplets is modeled with the analytical expression obtained by Davis [5]. When describing the motion of a falling water droplet, we have to take into account the effect of internal circulation induced in the droplet. Internal circulation reduces the drag force, and therefore the drag coefficient needs to be corrected in order to account for this reduction, as outlined by Happel and Brenner [6]. Furthermore, the surface
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tension varies over the droplet surface due to non-uniform distribution of surfactants on the interface and due to the elongation of the droplet by means of the applied electric field. This leads to interfacial stresses that inhibit the creation of internal circulation. LeVan [7] suggested how to take into account the effect of interfacial tension gradient in a numerical framework. The drainage between fluid droplets with almost immobile surfaces is taken into account by the model proposed by Vinogradova [8]. 2.1. Modeling the fluid-droplet and body forces 2.1.1. Drag force The “steady-state” drag force acts on a droplet in a uniform pressure field when there is no acceleration of the relative velocity between the droplet and the conveying fluid. The force reads: Fd =
1 ρc Cd A||u − v||(u − v), 2
(3)
The drag is reduced when internal circulation is induced in fluid spheres. Surfactants on the interface and elongation of the droplet, caused by the electric field, give a variation in the interfacial tension. The surface tension gradient leads to interfacial stresses that inhibit the creation of internal circulation. The interfacial tension gradient is included in the formula by LeVan [7]. The drag coefficient Cd reads: Cd =
24 3λ + 2 + 2κ(µc rd )−1 + 2/3γ1 (µc ||u − v||)−1 , (4) Red 3λ + 3 + 2κ(µc rd )−1
where λ = µd /µc is the viscosity ratio. The surface dilatational viscosity κ is neglected: κ = 0. In Eq. (4), it is assumed that the interfacial tension varies as follows: γ = γ0 + γ1 cosψ
(5)
where the angle ψ is measured from the front stagnation point. 2.1.2. Virtual mass force The virtual mass force Fvm is an unsteady force that accounts for the acceleration of the fluid caused by the motion of the droplet when the droplet and the fluid have a relative acceleration. It reads: ρc Vd Du dv F vm = − (6) 2 Dt dt 2.1.3. External body forces We assume that the droplets have no net charge, hence the electric field as a far-field force can be neglected. On the other hand, the electric field gives rise to induced dipole–dipole interactions between the droplets, which are modeled as inter-droplet forces. Then the gravity is the only external force and the buoyancy force is given by: F b = (ρd − ρc )gVd eg
(7)
where g and eg are the modulus and the direction of the gravity.
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2.1.4. Assumptions In the present work, the effects of the pressure gradient, the Basset history force and the lift forces have been neglected. The pressure difference over a small droplet is assumed to be negligible due to the size of the droplets. The contribution from gravity is handled separately. Lift forces are due to droplet rotation and shear forces, and can therefore be neglected when a rigid sphere or droplet is falling in a stagnant fluid. Due to the small size of the spheres and the high viscosity of the oil, the particle time scale is very small: τ d ∝ ρd d2 /µc . It therefore follows that the Stokes number is small and the Basset history force can be neglected (Michaelides [9]). 2.2. Modeling droplet–droplet forces The inter-droplet forces are the film-thinning forces, due to the drainage of the fluid between the droplets, and the electric forces due to difference in polarization over the water droplet interface where εd = εoil . 2.2.1. Film-thinning force The film-thinning force is caused by drainage of the liquid film between two approaching droplets. The derivation of the formulas usually requires that the gap between the particles is small, h a, and that the flow is within the Stokes regime: Red h a. Here, a = (r1 r2 )/(r1 + r2 ) is the reduced radius. For rigid spheres the film-thinning force is written as (Davis et al. [10]): Ff =
6πµc a2 (vr · er ) er h
(8)
vr is the relative velocity vector, and er indicates the direction of the relative motion. A correction to the rigid particle model, that was proposed by Vinogradova [8] for hydrophobic surfaces, is adopted. The formula of Vinogradova reads: Ff =
6πµc a2 (vr · er ) ∗ f er , h
where the correction factor f* is determined by: 2h h 6b ∗ 1+ ln 1 + −1 f = 6b 6b h
(9)
(10)
Fig. 1 shows f* /h when h approaches zero for different values of b, compared with the rigid sphere model (f* = 1). For large inter-droplet distances, there is no difference between the rigid sphere model and the model of Vinogradova. However, the singularity in the rigid sphere formula when the gap h approaches zero is avoided in the model of Vinogradova; f* /h approaches a fixed value asymptotically when h approaches zero. This makes the formula of Vinogradova, Eq. (9), attractive from a numerical point of view. In the present work, the value of b is kept constant and no dependency on either the droplet size or the droplet oscillation mode is taken into account.
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The forces between two conducting spheres are found by integrating the electrostatic pressure over the sphere surfaces (normal stresses of the Maxwell’s stress tensor). The solution for uncharged spheres is given by: Fr = 4πεoil |E0 |2 r22 (F1 cos2 θ + F2 sin2 θ), Ft =
4πεoil |E0 |2 r22 F3 sin(2θ),
(11) (12)
where the parameters F1 ; F2 , and F3 are complicated series depending on the ratios |d|/r2 and r1 /r2 . For large drop separations h1 /r1 1, the force components F1 , F2 , and F3 approach the point-dipole approximation. For small separations h/r1 < 1, F2 and F3 takes constant values while F1 diverges and takes the asymptotic value (see Atten [11]): Fig. 1. f* /h vs. h for different values of b compared with the rigid sphere model, Eq. (8).
2.2.2. Electric forces acting on the spherical droplets Consider an uncharged spherical droplet placed in an insulating medium, subjected to an electric field E0 . The field outside a dielectric sphere of permittivity εd corresponds to the electric field of a dipole located at the sphere center. The value of this dipole moment, p, depends on the sphere size, its permittivity and the strength of the electric field. Due to the polarization of the droplet, there will be a surface charge distribution preserving zero net charge. In a homogeneous field, the net force on the droplet is zero. Subjected to an inhomogeneous field, the droplet will experience a stronger field at one side than at the other, resulting in a net force acting on the droplet in the direction of the field gradient, a phenomenon called dielectrophoresis. The resulting force is given by F = (p·)E. If the permittivity of the droplet, εd , is higher than the permittivity of the surrounding medium, εoil , the droplet will move towards the high field region. An inhomogeneous electric field may for instance be setup by a nearby point charge or another dielectric droplet, see Fig. 2. In the latter case, the electrostatic force attracts the two droplets, given that εd > εoil . Davis [5] derived the potential function φ of two conducting spheres by solving Laplace’s equation in bi-spherical coordinates with the following constraints: • at a large distance from the spheres, the potential function correspond to the background field; • the spheres are equipotential surfaces and carry an arbitrary net charge.
Fig. 2. Electric forces between two conductive spheres.
F1 =
4 3((1 + r2 /2r1 )4 (r2 /|d|)−0.8 )
(13)
The validity of the analytical model Eqs. (11) and (12), is limited to a two-droplet system. A mean to bypass this limitation and generalize the modeling framework to account for a multi droplet system is presented in Ref. [4]. 2.3. Validity of the modeling framework The framework presented here provides all the elements necessary to model the behavior of water emulsified in oil under the influence of an external electrical field. The kinematics of the water droplets can be predicted by employing the models described in the paper. One can generalize the framework to account for a multi droplet system as presented in Ref. [4]. High magnitude electric fields may cause two approaching water drops to elongate and disintegrate into smaller drops. This effect set a limitation on the applicability of the framework to electric field magnitudes below the limit where electrically induced breakup can occur. The experiments carried out in this work are performed with electric field magnitudes that resemble typical operation facilities and are far below this limit. 3. Experimental setup Experiments have been designed for visual observation of water droplets in oil under the influence of electric-field stress, see Berg et al. [12]. A vertical 15 mm electrode-gap arrangement is placed inside a cubic test cell with side lengths 150 mm. The electrodes are parallel plane with no coating, 5 cm in diameter, and there is a hole of 0.5 cm in the center of each electrode, allowing, for e.g., alignment of droplets along the field lines, see Fig. 3. A 4 mm diameter droplet is placed in contact with the lower high-voltage electrode, while sub-millimeter sized droplets are ejected within the upper grounded electrode from a 100 m tip-diameter glass capillary. The glass capillaries are beforehand treated to reduce wettability at the tip, and a metallic layer of gold vapor is deposited onto the glass surface to avoid free surface charges. High voltages with arbitrary amplitudes in the range ±20 kV and frequencies less than 20 kHz are generated with a SRS 15 MHz synthesized function generator in combination with a Trek 20/20B voltage amplifier. Imposed
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Fig. 3. Experimental setup designed for visual observation of the behavior of water droplets in oil, exposed to the effect of an external electric field: (a) sketch of the experimental setup (b), the setup (c), and the test cell.
electric fields are vertical, thus parallel to the droplet–droplet impact vectors and in the range 0–300 V/mm. The voltage is bipolar square shaped with a frequency of 100 Hz, and is always applied after the droplets have been released into the electrode gap. Water droplets are visualized with shadow-graphic techniques using collimated background light and long distance microscope lenses. The total optical magnification is 153. A limited depth of focus let us monitor how droplets move into and out of the focal plane. Image calibration is performed using a wolfram wire of diameter (300 m), and the uncertainty in measured droplet diameter is less than 5 m. Digital video recordings during 4 s events are taken with a Phantom v4 high speed CMOS camera in the direction perpendicular to the electric field. The camera is capable of 1000 frames per second at full resolution (512 × 512), and it is synchronized with the frequency of the electric field. The position and the velocity of the water droplets are digitally extracted from the sequential
video frames, employing both manual and automatic tracking methods. Distilled water added 3.5 wt% of NaCl and Nynas Nytro 10× oil was used, thus facilitating optical transparency. The viscosity of the continuous phase is varied by varying the temperature of the oil in the range 278–333 K. At 293.15 K the oil viscosity is 13.7 × 10−3 Pa s and the density of oil is 875 kg/m3 . The viscosity is varied by a factor of 10. The change in oil density is linear with respect to temperature, and reads: ρ(T ) = ρ(T0 ) − 0.67(T − T0 )
(14)
where T0 = 293.15 K. The properties of Nytro 10× are presented in Table 1. The kinematic viscosity of the Nytro 10× and the interfacial tension of water droplets oil are plotted against temperatures in Fig. 4. Fig. 5 shows a typical collision between a 350 m diameter falling water droplet and a 4 mm diameter sta-
Fig. 4. Interfacial tension of water droplet in Nytro 10×/XT oil at different temperature. Kinemtaic viscosity of the Nytro oil at different temperatures.
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Fig. 5. Experimental observations of a small water droplet r1 = 175 m approaching a larger one r2 = 2000 m at time t = magnitude of the electric field is |E0 | = 300 V/mm. (a) t = t0 ; (b) t = t0 + t; (c) t = t0 + 2 t; (d) t = t0 + 3 t. Table 1 Properties of Nytro 10× and water with a small salt content at 20 ◦ C
(kg/m3 )
Density, ρ Viscosity, µ (Pa s) Relative permittivity, ε
Oil, T = 293.15 K
Water, T = 293.15 K
875 13.7 × 10−3 2.2
1010 0.98 × 10−3 80
tionary droplet. Fig. 5(d) is the picture frame taken before the coalescence takes place. The gap between the droplets in this case is approximately h ≈ 1.0 m. It gives a physical indication of how to interpret the slip length b in the model of Vinogradova for the film-thinning force, Eq. (9). The graphs in Fig. 1 show only minor differences between the rigid sphere model and the model of Vinogradova for h > 20 m when b ≥ 1 m. This means that the correction for almost immobile surfaces, Eq. (10), has negligible influence on the results for gaps h > 1 m. For smaller gaps h < 1 m the uncertainties regarding the position of the droplet surface is large, in particular for high electric fields. 4. Results and discussion 4.1. Numerical setup Numerically, the governing Eqs. (1) and (2) are solved with a Runge–Kutta–Fehlberg 4–5 solver with step-size control, see for instance Hairer et al. [13]. Accurate solution of the equation system is ensured by using a relative tolerance of 10−5 and an absolute tolerance of 10−25 . The expression for the forces described in Section 2 are used. An ideal bipolar squared voltage is assumed. The slip length in the Vinogradova Eq. (9) is b = 10−6 m and the magnitude of the interfacial tension gradient in Eq. (5) is estimated to be γ 1 = 10−5 N/m. 4.2. Influence of viscosity on water droplet kinematics Figs. 6 and 7 show a comparison between experimental observations and numerical predictions of the velocity of a water droplet falling towards a larger droplet placed on the electrode. When one wants to predict numerically the motion of the falling fluid droplet observed in the experiments, one has to take into account the effect of internal circulation induced in the droplet while it is falling. Internal circulation reduces the drag force and therefore the drag coefficient needs to be corrected, as outlined
(t t 0
+ i t)s with i = 0, 1, 2, 3. The
in Section 2.1. Furthermore, the interfacial tension of the droplet is varied by the effect of surfactants on the interface and by the elongation of the droplet caused by the electric field. This leads to interfacial stresses that inhibit the creation of internal circulation. LeVan [7] suggested how to take into account the effect of interfacial tension gradients in a numerical framework. Barnocky and Davis [14] derived expressions for the drainage between fluid spheres with arbitrary viscosity. The model proposed by Vinogradova [8] takes interfacial slip effects into account. Although Vinogradova [8] proposed a correction for hydrophobic surfaces, the present work shows that the Eqs. (9) and (10) can be used with good agreement for almost stationary surfaces as well. A comparison between observed and predicted velocities versus normalized particle surface distance h/r1 for a fluid droplets falling in oil is plotted in Fig. 6(a) when no field is applied. The average radius of the droplets is r1 ≈ 200 m. The impact velocity when no field is applied is nearly zero. This is due to the effect of the film-thinning force that acts against the motion of the droplet. Let the terminal velocity of the droplet be defined as the velocity that the droplet would reach when falling freely in a medium. It is calculated from Eq. (2) by imposing constant acceleration vector: dv/dt = 0. This implies consequently that Ffluid + Fext + Fd–d = 0. The droplet velocity is normalized by means of the terminal velocity and is plotted versus the distance between the droplets surface normalized by means of the droplet radius h/r1 see on the right of Fig. 6(a). The results show similarity. The dimensionless group v/vt is dependent on the viscosity. The terminal velocity reached by the droplets falling in a heated oil bath, where the viscosity is low, is greater than the terminal velocity reached by the droplet falling in a cooled oil characterized by a high viscosity. The film-thinning force normalized by means of the buoyancy force Ff /|Fb | is plotted in Fig. 8(a). The reason why the magnitude of the film-thinning force is greater for the low viscosity case lies in the fact that the relative droplet velocity is higher at low viscosity as seen in Fig. 6(a). When an electric field is applied in the direction of the droplet motion it will counteract the effect of the film-thinning force. For this reason, the impact velocity plotted in Fig. 6(b) is not equal zero. The magnitude of the film-thinning force is still higher than the magnitude of the dielectrophoretic force acting between the droplets when the applied electric field is |E0 | = 36.3 V/mm. Good agreement between numerical results and experimental observations is obtained for different viscosities of the contin-
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Fig. 6. Observed and predicted velocities vs. normalized droplet surface distance h/r1 for water droplets. The radii of the droplets are r1 ≈ 200 m. The slip distance in Eq. (9) is b = 10−6 m and the magnitude of the interfacial tension gradient in Eq. (5) is γ 1 = 10−5 N/m. (a) |E0 | = 0 V/mm: (circles and solid line) r1 ≈ 238 m, µ1 ≈ 2.5E−3 Pa s; (squares and dashed line) r2 ≈ 200 m, µ2 ≈ 13.9E−3 Pa s; (crosses and dotted line) r3 ≈ 185 m, µ3 ≈ 32.5E−3 Pa s. (b) |E0 | = 36.3 V/mm: (circles and solid line) r1 ≈ 222 m, µ1 ≈ 4.7E−3 Pa s; (squares and dashed line) r2 ≈ 205 m, µ2 ≈ 13.9E−3 Pa s; (crosses and dotted line) r3 ≈ 196 m, µ3 ≈ 27.1E−3 Pa s.
uous phase see Fig. 6(b). When plotting the curves on the left hand side of Fig. 6(b) after normalizing the velocity by means of the terminal velocity the curves fall on to each other. This is due to the fact that the magnitude of the dipole force is very small and the kinematics of the droplet in the near field region h/r1 < 5 is mainly due to the film-thinning forces while in the far field region it is due to the drag, buoyancy and virtual mass force as plotted on the right hand side of Fig. 8(a). Each of these forces are dependent on the viscosity so that the normalization of the velocity by means of the terminal velocity v/vt provides a master curve for the droplet kinematic. By increasing even further the magnitude of the applied electric field the effect of dipole forces becomes dominant on the droplet motion in the near field region. The results presented in Fig. 7(a) refer to an applied electric field of magnitude: |E0 | = 146.6 V/mm. The droplet experiences an acceleration towards the stationary droplet. The velocity increases until the droplet comes into contact with the stationary one. This is due to the fact that the magnitude of the dipole force is greater than the magnitude of the film-thinning force see Fig. 8(b). There are some uncertainties in extracting the velocity from the optical observation when the normalized distance h/r1 < 0.5 for low viscosity. A good agreement between the experimental results and numerical prediction is obtained. When plotting the curves on the left hand side of Fig. 7(a), after normalizing the velocity by means of the terminal velocity, the curves fell on to each other in the far field region as expected showing similarity see the right hand side of Fig. 7(a).
The droplet motion in the far field region is dominated by forces dependent on the viscosity. In the near field region the dipole force is the dominant one. The dimensionless group v/vt does not take into account of the effect of the dipole forces. When the magnitude of the applied electric field is increased up to |E0 | = 293.6 V/mm the effect of dipole forces becomes even more dominant on the droplet motion in the near field region. Fig. 7(b) shows a comparison between observed and predicted velocities, a good agreement is obtained. However, the discrepancy between the predicted results and the experimental observations is greater at high electric field in the near field region. The reason for this could be that the slip length in the Vinogradova Eq. (9) is kept constant b = 10−6 m during all the numerical prediction performed in this work. The slip length line is believed to be dependent on the characteristic oscillation mode of the droplet. This is dependent on both the droplet radius, the electric field and the frequency of the applied electric field. At high magnitude electric field, the amplitude of the oscillation of the droplet is believed to enhance the thinning of the film between the droplets. The droplet experiences an acceleration towards the stationary droplet. The velocity increases consequently until the droplet come into contact with the stationary one. This is because the magnitude of the dipole force is greater than the magnitude of the film-thinning force as previously explained. The contact velocity experienced at high field is greater than what experienced at lower field magnitude, compare Fig. 7(a and b).
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Fig. 7. Observed and predicted velocities vs. normalized droplet surface distance h/r1 for water droplets. The radii of the droplets are r1 ≈ 200 m. The slip distance in Eq. (9) is b = 10−6 m and the magnitude of the interfacial tension gradient in Eq. (5) is γ 1 = 10−5 N/m. (a) |E0 | = 146.3 V/mm: (circles and solid line) r1 ≈ 217 m, µ1 ≈ 2.89E−3 Pa s; (squares and dashed line) r2 ≈ 208 m, µ2 ≈ 13.9E−3 Pa s; (crosses and dotted line) r3 ≈ 207 m, µ3 ≈ 24.8E−3 Pa s. (b) |E0 | = 293.6 V/mm: (circles and solid line) r1 ≈ 210 m, µ1 ≈ 3.74E−3 Pa s; (squares and dashed line) r2 ≈ 198 m, µ2 ≈ 13.9E−3 Pa s; (crosses and dotted line) r3 ≈ m, µ3 ≈ 27.6E−3 Pa s.
4.3. Dependency of electrocoalescence on oil viscosity The effect of oil viscosity on droplet coalescence is investigated in a way similar to the study presented in the previous section. Experiments are performed releasing water droplets and observing the coalescence rest time. Coalescence is studied in detail and the observation area is focused on the contact surface. This allows us to increase the temporal resolution by reducing image size and increasing frame rate. The experimental results obtained are plotted in Fig. 9. There is a weak dependence of the results on the way one defines the contact time. In the present work the contact time is defined as the time after which no further movement of the droplet is recorded. One can argue that due to the deformation of the stationary droplet the definition we have chosen underestimates the real time of contact. In the present study the data presented in Fig. 9 are interpreted in a qualitative way in order to assess the effect of viscosity on electrically induced coalescence and therefore the argument around the definition of the time of contact is less important here. The coalescence rest time is plotted at different electric field magnitudes on the left hand side of Fig. 9(a) in a logarithmic–linear plot. Fig. 9(a) shows that for constant field magnitude the coalescence rest time greater for high viscosity oil. Viscosity is believed to indirectly influence the time of coalescence since the impact velocity of the water droplet in a
high viscosity oil is lower than in a low viscosity oil as plotted on the right hand side of Fig. 9(b). The dielectrophoretic force between the water droplets counteracts the effect of the film-thinning force and increases ultimately the magnitude of the impact velocity as previously discussed. For a given magnitude of the applied electric field the impact velocity is higher a low viscosity oil as plotted in Fig. 9(b). The results plotted in Fig. 9 give an indication of which role viscosity, interfacial tension, electric field magnitude, the ratio of permittivity and the density difference between oil and water play in the electrocoalescence process. The permittivity ratio on one side together with the electric field magnitude on the other side determine the intensity of the dielectrophoretic force acting on the droplets. Such force together with the buoyancy force due to the density difference between oil and water and the viscous forces determine the impact velocity. The interfacial tension influences the deformation of the droplet and also the movement of the center of mass. Hence, a droplet under the influence of an external electric field will experience an immediate deformation when its interfacial tension is low. Let’s now define We number as the ratio between the inertia of the droplet at impact and the interfacial tension: We =
2ρc r1 |v|2 σ
(15)
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Fig. 8. Difference between the calculated film-thinning force and dipole force normalized by means of the buoyancy force vs. normalized droplet surface distance h/r1 . The radii of the droplets are r1 ≈ 200 m. The slip distance in Eq. (9) is b = 10−6 m and the magnitude of the interfacial tension gradient in Eq. (5) is γ 1 = 10−5 N/m. (a) On the left hand side: |E0 | = 0 V/mm: (solid line) r1 ≈ 238 m, µ1 ≈ 2.5E−3 Pa s; (dashed line) r2 ≈ 200 m, µ2 ≈ 13.9E−3 Pa s; (dotted line) r3 ≈ 185 m, µ3 ≈ 32.5E−3 Pa s. On the right hand side: |E0 | = 36.3 V/mm: (solid line) r1 ≈ 222 m, µ1 ≈ 4.7E−3 Pa s; (dashed line) r2 ≈ 205 m, µ2 ≈ 13.9E−3 Pa s; (dotted line) r3 ≈ 196 m, µ3 ≈ 27.1E−3 Pa s. The forces correspond to the velocity profiles presented in Fig. 6. (b) On the left hand side: |E0 | = 146.3 V/mm: (solid line) r1 ≈ 217 m, µ1 ≈ 2.89E−3 Pa s; (dashed line) r2 ≈ 208 m, µ2 ≈ 13.9E−3 Pa s; (dotted line) r3 ≈ 207 m, µ3 ≈ 24.8E−3 Pa s. On the right hand side: |E0 | = 293.6 V/mm: (solid line) r1 ≈ 210 m, µ1 ≈ 3.74E−3 Pa s; (dashed line) r2 ≈ 198 m, µ2 ≈ 13.9E−3 Pa s; (dotted line) r3 ≈ 213 m, µ3 ≈ 27.6E−3 Pa s. The forces correspond to the velocity profiles presented in Fig. 7.
and the B number as the ratio between the square of the field and the interfacial tension: B=
εoil (1 − εoil /εd )2 |E|2 2σκ0
experiments carried out previously. Crosses, squares and circles represent experimental points when viscosity is varied. The Reynolds number reads:
(16)
where εoil /εd is the ratio of permittivity and k0 is the droplet curvature. Fig. 10 presents B plotted versus We for the different
n(n2 − 1)ρσd Reσ = 4µc
(17)
Fig. 9. The coalescence rest time after contact has occurred is plotted at (a) different electric field magnitudes and (b) different impact velocities. The radii of the droplets are r1 ≈ 200 m.
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Fig. 10. The Bond number is plotted vs. the Weber number (a) and coalescence rest time after contact has occurred is plotted vs. the Weber number (b).
where n is the oscillation mode number of the droplet, and σ and µc are as previously explained the surface tension and the viscosity, respectively. Reσ varies as a consequence of the variation of viscosity. On the right hand side of Fig. 10 the coalescence rest time is plotted versus We. The plot on the right side of Fig. 10 presents the same result as previously shown in Fig. 9. The difference lies in the fact that the different parameters influencing electro coalescence are grouped. 5. Observations and conclusions A modeling framework that properly accounts for the effect of the different forces acting on a droplet falling in an oil bath is used to investigate the effect of viscosity on electrically induced coalescence of water droplets in oil. The effect of internal circulation induced in the droplets has to be taken into account, together with the variation of the interfacial tension of the droplets due to the electric field. In the present work the model proposed by LeVan, Eq. (4), for the drag force, by Vinogradova, Eq. (9), for the film-thinning force, and the Davis analytical expression, Eqs. (11) and (12), provide numerical predictions that agree well with the experimental observations. The comparison between observed and predicted velocities versus normalized droplet surface distance, h/r1 , for fluid droplets shows a good agreement. Each of the forces influencing the kinematics of the droplet with the exception of the dipole–dipole force, are dependent on the viscosity. When the effect of the dipole–dipole force can be neglected the velocity profiles obtained at different viscosity can be normalized by means of the terminal velocity obtaining a master curve on which all the curves fall onto as expected. When the magnitude of the dipole forces dominates over the viscosity dependent forces the droplet accelerate towards the stationary one. The droplet velocity at impact is greater when the droplet falls in a low viscosity oil bath. Our experimental setup is designed to measure collision along the center-line of the droplets. The dipole–dipole force enhances the in-line collision since the droplets pair tends to align with the electric field. It is generally accepted that when two droplets collide, there are several possible outcomes depending on the kinetic energy of the collision, droplet size, the impact parameter (a measure of the degree to which the collision is off-center), the material
properties, and the droplet size ratio, which is defined as the ratio of the smaller droplet radius to the larger droplet radius. In the present work, we have limited our investigation to the combined effect of viscosity and electrostatic forces. Viscosity influences indirectly the coalescence rest time since the impact velocity and consequently the kinetic energy of the water droplet is dependent on the viscosity of the oil bath in which the water droplet is released. The electric field also contributes to increase the kinetic energy of the falling droplet since it accelerates the droplet that experiences a maximum velocity at collision when the field magnitude is above a certain threshold dependent on the oil viscosity. The electric field also influences the coalescence process since it influences the interfacial tension of the droplet due to droplet deformation. This influence is hard to quantify. The viscosity influences indirectly the time of coalescence since the impact velocity of the water droplet in a high viscosity oil is lower than in a low viscosity oil as plotted on the right hand side of Fig. 9(b). This is an important parameter to take into account when dealing with heavy oil. A model that describes electro coalescence is required to fulfill the modeling framework used in the first part of the paper. In order to provide such a model the experimental setup should be slightly modified. One should be able to perform different series of experiments varying various parameters while keeping two of the dimensionless groups constant simultaneously. Due to the lack of a precise model for coalescence efficiency, one can assume, as a first approximation, that each collision leads to coalescence. This might be an adequate approximation when electric fields are applied and when the water droplet surfaces are not too contaminated. The validity of the approximation is more doubtful in the case with no applied electric field. Acknowledgments The authors are thankful to VetcoAibel, Statoil, Petrobras and the Research Council of Norway for supporting the present work. References [1] J. Sj¨oblom, N. Aske, I.H. Auflem, Brandal, T.E. Havre, Sæther, A. Westvik, E.E. Johnsen, H. Kallevik, Our current understanding of water-
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[2]
[3]
[4]
[5]
[6] [7]
in-crude oil emulsions. Recent characterization techniques and high pressure performance, Adv. Colloid Interface Sci. 100–102 (2003) 399–473. J.S. Eow, M. Ghadiri, A.O. Sharif, T.J. Williams, Electrostatic enhancement of coalescence of water droplets in oil: a review of current understanding, Chem. Eng. J. 84 (2001) 173–192. P. Hemmingsen, A. Silset, A. Hannisdal, J. Sj¨oblom, Emulsion of heavy crude oils, influence of viscosity, temperature and dilution, J. Dispersion Sci. Technol. 26 (5) (2005) 615–627, doi:10.1081/DIS-200057671. M. Chiesa, J.A. Melheim, A. Pedersen, S. Ingebrigtsen, G. Berg, Forces acting on water droplets falling in oil under the influence of an electric field: numerical predictions versus experimental observations, Eur. J. Mech.-B/Fluids 24 (6) (2005) 717–732. M.H. Davis, Two charged spherical conductors in a uniform electric field: forces and field strength, Rand. Corp. Memorandum RM-3860PR, January 1964. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff Publishers, The Hague, The Netherlands, 1983. D.M. LeVan, Motion of droples with a Newtonian interface, J. Colloid Interface Sci. 83 (1) (1981) 11–17.
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[8] O.I. Vinogradova, Drainage of a thin liquid film confined between hydrophobic surfaces, Langmuir 11 (1995) 2213–2220. [9] E.E. Michaelides, Hydrodynamic force and heat/mass transfer from particles, bubbles, and drops—the Freeman scholar lecture, J. Fluids Eng. Trans. ASME 125 (2003) 209–238. [10] R.H. Davis, J.A. Schonberg, J.M. Rallison, The lubrication force between two viscous drops, Phys. Fluids A 1 (1) (1989) 77– 81. [11] P. Atten, Electrocoalescence of water droplets in an insulating liquid, J. Electrostatics 30 (1993) 259–270. [12] G. Berg, L. Lundgaard, M. Becidan, S.R. Sigmond, Instability of electrically stressed water drops in oil, in: ICDL 14th International Conference on Dielectric Liquids. [13] E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I, second ed., Springer, Berlin, 1992, ISBN 3-540-566708. [14] G. Barnocky, R.H. Davis, The lubrication force between spherical drops, bubbles and rigid particles in a viscous fluid, Int. J. Multiphase Flow 15 (1989) 627–638.
Investigation of the role of viscosity on electrocoalescence of water droplets in oil M. Chiesa a,∗ , S. Ingebrigtsen a , J.A. Melheim b , P.V. Hemmingsen c , E.B. Hansen b , Ø. Hestad a a
SINTEF Energy Research, NO-7465 Trondheim, Norway b NTNU, NO-7491 Trondheim, Norway c Ugelstad laboratory NO-7491 Trondheim, Norway
Received 25 October 2005; received in revised form 5 December 2005; accepted 5 December 2005
Abstract The sedimentation velocity of water droplets in oil increases proportionally to the square of the droplets diameter. It is therefore beneficial for the separation of water from oil to increase the droplet size by enhancing the coalescence process. Electric fields are used for this purpose in order to separate stable water-in-oil emulsion. Field-induced charges on the water droplets will cause adjacent droplets to align with the field, attract each other and ultimately coalesce. The efficiency of electrically induced coalescence is strongly dependent on the oil characteristics. In the present study experiments are performed to observe the behavior of a droplet falling onto a stationary one at different oil viscosity. The droplet is exposed to an electric field of different magnitude parallel to the direction of the droplet motion. Experimental observations are compared with numerical prediction of the droplet kinematics, and the contribution of the forces acting on the droplets and their dependency on viscosity is discussed in detail. © 2005 Elsevier B.V. All rights reserved. Keywords: Water-in-oil emulsion; Sedimentation stability; Flocculation; Viscosity; Coalescence; Dipole–dipole forces; Film-thinning force; Drag force; Buoyancy force
1. Introduction In crude oil production, oil and co-produced water is mixed due to the pressure drop and high shear forces when the fluid passes the wellhead and various choke valves from the reservoir to the production facilities. This will cause the formation of water-in-oil emulsion and one of the primary task for the production platform is to remove the water from the crude oil. Water level above 0.5% constitutes a quality problem and a concomitant price reduction. All water-in-oil emulsions, perhaps with the exception of micro-emulsions, are thermodynamically unstable. However, the destabilization may take considerable time, and a stable emulsion is unable to resolve itself in a defined time period without some form of mechanical or chemical treatment (Sj¨oblom et al. [1]). Water-in-crude oil emulsion destabilization basically involves three steps, namely flocculation, followed by sedi-
∗
Corresponding author. Tel.: +47 92017605. E-mail address: [email protected] (M. Chiesa).
1383-5866/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.seppur.2005.12.003
mentation of water droplets due to density differences, and finally coalescence of the individual water droplets. Large droplet sizes, high density difference between the oil and the aqueous phase, and low continuous phase (oil) viscosity causes high sedimentation rates. Flocculation is the aggregation of two or several droplets, touching only at certain points, and with virtually no change in total surface area. Coalescence is the process of droplets fusing together and forming larger droplets. In crude oil emulsions, emulsifying agents are present at the oil–water interface, hindering this coalescence process. The mechanism by which indigenous crude oil components stabilize a droplet interface is by producing a physical barrier for droplet–droplet coalescence. Electrostatic fields are to some extent used to overcome this physical barrier by helping smaller droplets to coalesce into larger droplets that sediment quicker (Eow et al. [2]). The electrostatic effects arise from the very different dielectric permittivity and conductivity of oil and water. Water is characterized by permittivity values much higher than those of oil and in practice one may consider water as perfectly conducting. To a large extent the effects of the electrostatic field are explained by body-forces
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Nomenclature A, Ad a B b Cd d E, E0 er eg Fe Fb Ff Fvm Fd–d Fd Fext Ffluid f* g h i k0 Red Reσ rd r1 r2 T u Vd v vi vr xi We
droplet surface reduced radius Bond number slip length drag coefficient separation vector electric field vector relative motion vector gravity vector dielectrophoretic force vector body force vector film-thinning force vector virtual mass force vector inter-droplet force vector drag force vector external-droplet force vector fluid-droplet force vector correction factor, see Eq. (10) modulus of the gravity least distance between two particles droplet i droplet curvature droplet Reynolds number Reynolds number for an oscillating inviscid droplet droplet radius radius of falling droplet radius of stationary droplet temperature continuous phase velocity vector droplet volume droplet velocity vector velocity vector of droplet i relative droplet velocity vector droplet position vector Weber number
Greek letters ε0 vacuum dielectric constant relative dielectric constant of droplet εd εoil relative dielectric constant for oil φ potential function γ1 magnitude of the interfacial tension gradient λ viscosity ratio (λ = µd /µc ) viscosity of droplet µd µc viscosity of the continuous phase θ angle between E and d ρd density of the discrete phase density of the continuous phase ρc σ interfacial tension of water drop in oil τd velocity response time of water drop in oil
acting on water droplets either because they are charged, giving rise to electrophoretic forces, or because they become polarized in divergent fields, resulting in dielectrophoretic forces. The focus of the present work lies in investigating the effect of viscosity on water droplets flocculation and coalescence when different electric fields are applied. This is important when dealing with water-in-oil emulsion of highly viscous crude oils (Hemmingsen et al. [3]). Experiments are performed releasing water droplets and observing their motion in oil with different viscosity when electric fields of different magnitudes are applied. A transformer oil is employed in the experiments due to its transparency. The experimental observations of the motion of falling droplets are compared to the predictions obtained by the modeling framework proposed by Chiesa et al. [4]. The numerical obtained results are used to assess the role of viscosity on water droplets kinematics and coalescence. The kinematics of the water droplets in the sedimentation process is dominated by the buoyancy force due to the density difference between the water and the oil. The viscous forces are directly acting as a damping mechanism throughout the drag force. When two droplets approach each other, viscosity acts as a damping mechanism throughout the film-thinning process. The electrically induced force counteract the damping effect of the drag and film-thinning force when the droplets are relatively close to each other. When the droplets are in contact with each other, the coalescence rest time is indirectly dependent on the viscosity throughout the impact velocity. The coalescence rest time is believed to be dependent on the Weber number at impact when the droplet Reynolds number and Bond number are kept constant. 2. Theoretical background The trajectory of a spherical droplet i is calculated by integrating Newton’s second law. The law equates the change in momentum per change in time of the droplet with the forces acting on it, and reads: dxi = vi dt mi
dvi = F fluid + F ext + F d–d , dt
(1) (2)
where mi , xi , and vi are the mass, position, and velocity of the droplet. Ffluid represents the vector of forces acting from the fluid on the droplet it is modeled by summing up the drag force Fd , the virtual mass force Fvm , and the buoyancy or external body force Fb . Fext is the external force, and Fd–d represents the inter-droplet force. The modeling framework proposed by Chiesa et al. [4] is employed in the present work. The electric force between the droplets is modeled with the analytical expression obtained by Davis [5]. When describing the motion of a falling water droplet, we have to take into account the effect of internal circulation induced in the droplet. Internal circulation reduces the drag force, and therefore the drag coefficient needs to be corrected in order to account for this reduction, as outlined by Happel and Brenner [6]. Furthermore, the surface
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tension varies over the droplet surface due to non-uniform distribution of surfactants on the interface and due to the elongation of the droplet by means of the applied electric field. This leads to interfacial stresses that inhibit the creation of internal circulation. LeVan [7] suggested how to take into account the effect of interfacial tension gradient in a numerical framework. The drainage between fluid droplets with almost immobile surfaces is taken into account by the model proposed by Vinogradova [8]. 2.1. Modeling the fluid-droplet and body forces 2.1.1. Drag force The “steady-state” drag force acts on a droplet in a uniform pressure field when there is no acceleration of the relative velocity between the droplet and the conveying fluid. The force reads: Fd =
1 ρc Cd A||u − v||(u − v), 2
(3)
The drag is reduced when internal circulation is induced in fluid spheres. Surfactants on the interface and elongation of the droplet, caused by the electric field, give a variation in the interfacial tension. The surface tension gradient leads to interfacial stresses that inhibit the creation of internal circulation. The interfacial tension gradient is included in the formula by LeVan [7]. The drag coefficient Cd reads: Cd =
24 3λ + 2 + 2κ(µc rd )−1 + 2/3γ1 (µc ||u − v||)−1 , (4) Red 3λ + 3 + 2κ(µc rd )−1
where λ = µd /µc is the viscosity ratio. The surface dilatational viscosity κ is neglected: κ = 0. In Eq. (4), it is assumed that the interfacial tension varies as follows: γ = γ0 + γ1 cosψ
(5)
where the angle ψ is measured from the front stagnation point. 2.1.2. Virtual mass force The virtual mass force Fvm is an unsteady force that accounts for the acceleration of the fluid caused by the motion of the droplet when the droplet and the fluid have a relative acceleration. It reads: ρc Vd Du dv F vm = − (6) 2 Dt dt 2.1.3. External body forces We assume that the droplets have no net charge, hence the electric field as a far-field force can be neglected. On the other hand, the electric field gives rise to induced dipole–dipole interactions between the droplets, which are modeled as inter-droplet forces. Then the gravity is the only external force and the buoyancy force is given by: F b = (ρd − ρc )gVd eg
(7)
where g and eg are the modulus and the direction of the gravity.
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2.1.4. Assumptions In the present work, the effects of the pressure gradient, the Basset history force and the lift forces have been neglected. The pressure difference over a small droplet is assumed to be negligible due to the size of the droplets. The contribution from gravity is handled separately. Lift forces are due to droplet rotation and shear forces, and can therefore be neglected when a rigid sphere or droplet is falling in a stagnant fluid. Due to the small size of the spheres and the high viscosity of the oil, the particle time scale is very small: τ d ∝ ρd d2 /µc . It therefore follows that the Stokes number is small and the Basset history force can be neglected (Michaelides [9]). 2.2. Modeling droplet–droplet forces The inter-droplet forces are the film-thinning forces, due to the drainage of the fluid between the droplets, and the electric forces due to difference in polarization over the water droplet interface where εd = εoil . 2.2.1. Film-thinning force The film-thinning force is caused by drainage of the liquid film between two approaching droplets. The derivation of the formulas usually requires that the gap between the particles is small, h a, and that the flow is within the Stokes regime: Red h a. Here, a = (r1 r2 )/(r1 + r2 ) is the reduced radius. For rigid spheres the film-thinning force is written as (Davis et al. [10]): Ff =
6πµc a2 (vr · er ) er h
(8)
vr is the relative velocity vector, and er indicates the direction of the relative motion. A correction to the rigid particle model, that was proposed by Vinogradova [8] for hydrophobic surfaces, is adopted. The formula of Vinogradova reads: Ff =
6πµc a2 (vr · er ) ∗ f er , h
where the correction factor f* is determined by: 2h h 6b ∗ 1+ ln 1 + −1 f = 6b 6b h
(9)
(10)
Fig. 1 shows f* /h when h approaches zero for different values of b, compared with the rigid sphere model (f* = 1). For large inter-droplet distances, there is no difference between the rigid sphere model and the model of Vinogradova. However, the singularity in the rigid sphere formula when the gap h approaches zero is avoided in the model of Vinogradova; f* /h approaches a fixed value asymptotically when h approaches zero. This makes the formula of Vinogradova, Eq. (9), attractive from a numerical point of view. In the present work, the value of b is kept constant and no dependency on either the droplet size or the droplet oscillation mode is taken into account.
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The forces between two conducting spheres are found by integrating the electrostatic pressure over the sphere surfaces (normal stresses of the Maxwell’s stress tensor). The solution for uncharged spheres is given by: Fr = 4πεoil |E0 |2 r22 (F1 cos2 θ + F2 sin2 θ), Ft =
4πεoil |E0 |2 r22 F3 sin(2θ),
(11) (12)
where the parameters F1 ; F2 , and F3 are complicated series depending on the ratios |d|/r2 and r1 /r2 . For large drop separations h1 /r1 1, the force components F1 , F2 , and F3 approach the point-dipole approximation. For small separations h/r1 < 1, F2 and F3 takes constant values while F1 diverges and takes the asymptotic value (see Atten [11]): Fig. 1. f* /h vs. h for different values of b compared with the rigid sphere model, Eq. (8).
2.2.2. Electric forces acting on the spherical droplets Consider an uncharged spherical droplet placed in an insulating medium, subjected to an electric field E0 . The field outside a dielectric sphere of permittivity εd corresponds to the electric field of a dipole located at the sphere center. The value of this dipole moment, p, depends on the sphere size, its permittivity and the strength of the electric field. Due to the polarization of the droplet, there will be a surface charge distribution preserving zero net charge. In a homogeneous field, the net force on the droplet is zero. Subjected to an inhomogeneous field, the droplet will experience a stronger field at one side than at the other, resulting in a net force acting on the droplet in the direction of the field gradient, a phenomenon called dielectrophoresis. The resulting force is given by F = (p·)E. If the permittivity of the droplet, εd , is higher than the permittivity of the surrounding medium, εoil , the droplet will move towards the high field region. An inhomogeneous electric field may for instance be setup by a nearby point charge or another dielectric droplet, see Fig. 2. In the latter case, the electrostatic force attracts the two droplets, given that εd > εoil . Davis [5] derived the potential function φ of two conducting spheres by solving Laplace’s equation in bi-spherical coordinates with the following constraints: • at a large distance from the spheres, the potential function correspond to the background field; • the spheres are equipotential surfaces and carry an arbitrary net charge.
Fig. 2. Electric forces between two conductive spheres.
F1 =
4 3((1 + r2 /2r1 )4 (r2 /|d|)−0.8 )
(13)
The validity of the analytical model Eqs. (11) and (12), is limited to a two-droplet system. A mean to bypass this limitation and generalize the modeling framework to account for a multi droplet system is presented in Ref. [4]. 2.3. Validity of the modeling framework The framework presented here provides all the elements necessary to model the behavior of water emulsified in oil under the influence of an external electrical field. The kinematics of the water droplets can be predicted by employing the models described in the paper. One can generalize the framework to account for a multi droplet system as presented in Ref. [4]. High magnitude electric fields may cause two approaching water drops to elongate and disintegrate into smaller drops. This effect set a limitation on the applicability of the framework to electric field magnitudes below the limit where electrically induced breakup can occur. The experiments carried out in this work are performed with electric field magnitudes that resemble typical operation facilities and are far below this limit. 3. Experimental setup Experiments have been designed for visual observation of water droplets in oil under the influence of electric-field stress, see Berg et al. [12]. A vertical 15 mm electrode-gap arrangement is placed inside a cubic test cell with side lengths 150 mm. The electrodes are parallel plane with no coating, 5 cm in diameter, and there is a hole of 0.5 cm in the center of each electrode, allowing, for e.g., alignment of droplets along the field lines, see Fig. 3. A 4 mm diameter droplet is placed in contact with the lower high-voltage electrode, while sub-millimeter sized droplets are ejected within the upper grounded electrode from a 100 m tip-diameter glass capillary. The glass capillaries are beforehand treated to reduce wettability at the tip, and a metallic layer of gold vapor is deposited onto the glass surface to avoid free surface charges. High voltages with arbitrary amplitudes in the range ±20 kV and frequencies less than 20 kHz are generated with a SRS 15 MHz synthesized function generator in combination with a Trek 20/20B voltage amplifier. Imposed
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Fig. 3. Experimental setup designed for visual observation of the behavior of water droplets in oil, exposed to the effect of an external electric field: (a) sketch of the experimental setup (b), the setup (c), and the test cell.
electric fields are vertical, thus parallel to the droplet–droplet impact vectors and in the range 0–300 V/mm. The voltage is bipolar square shaped with a frequency of 100 Hz, and is always applied after the droplets have been released into the electrode gap. Water droplets are visualized with shadow-graphic techniques using collimated background light and long distance microscope lenses. The total optical magnification is 153. A limited depth of focus let us monitor how droplets move into and out of the focal plane. Image calibration is performed using a wolfram wire of diameter (300 m), and the uncertainty in measured droplet diameter is less than 5 m. Digital video recordings during 4 s events are taken with a Phantom v4 high speed CMOS camera in the direction perpendicular to the electric field. The camera is capable of 1000 frames per second at full resolution (512 × 512), and it is synchronized with the frequency of the electric field. The position and the velocity of the water droplets are digitally extracted from the sequential
video frames, employing both manual and automatic tracking methods. Distilled water added 3.5 wt% of NaCl and Nynas Nytro 10× oil was used, thus facilitating optical transparency. The viscosity of the continuous phase is varied by varying the temperature of the oil in the range 278–333 K. At 293.15 K the oil viscosity is 13.7 × 10−3 Pa s and the density of oil is 875 kg/m3 . The viscosity is varied by a factor of 10. The change in oil density is linear with respect to temperature, and reads: ρ(T ) = ρ(T0 ) − 0.67(T − T0 )
(14)
where T0 = 293.15 K. The properties of Nytro 10× are presented in Table 1. The kinematic viscosity of the Nytro 10× and the interfacial tension of water droplets oil are plotted against temperatures in Fig. 4. Fig. 5 shows a typical collision between a 350 m diameter falling water droplet and a 4 mm diameter sta-
Fig. 4. Interfacial tension of water droplet in Nytro 10×/XT oil at different temperature. Kinemtaic viscosity of the Nytro oil at different temperatures.
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Fig. 5. Experimental observations of a small water droplet r1 = 175 m approaching a larger one r2 = 2000 m at time t = magnitude of the electric field is |E0 | = 300 V/mm. (a) t = t0 ; (b) t = t0 + t; (c) t = t0 + 2 t; (d) t = t0 + 3 t. Table 1 Properties of Nytro 10× and water with a small salt content at 20 ◦ C
(kg/m3 )
Density, ρ Viscosity, µ (Pa s) Relative permittivity, ε
Oil, T = 293.15 K
Water, T = 293.15 K
875 13.7 × 10−3 2.2
1010 0.98 × 10−3 80
tionary droplet. Fig. 5(d) is the picture frame taken before the coalescence takes place. The gap between the droplets in this case is approximately h ≈ 1.0 m. It gives a physical indication of how to interpret the slip length b in the model of Vinogradova for the film-thinning force, Eq. (9). The graphs in Fig. 1 show only minor differences between the rigid sphere model and the model of Vinogradova for h > 20 m when b ≥ 1 m. This means that the correction for almost immobile surfaces, Eq. (10), has negligible influence on the results for gaps h > 1 m. For smaller gaps h < 1 m the uncertainties regarding the position of the droplet surface is large, in particular for high electric fields. 4. Results and discussion 4.1. Numerical setup Numerically, the governing Eqs. (1) and (2) are solved with a Runge–Kutta–Fehlberg 4–5 solver with step-size control, see for instance Hairer et al. [13]. Accurate solution of the equation system is ensured by using a relative tolerance of 10−5 and an absolute tolerance of 10−25 . The expression for the forces described in Section 2 are used. An ideal bipolar squared voltage is assumed. The slip length in the Vinogradova Eq. (9) is b = 10−6 m and the magnitude of the interfacial tension gradient in Eq. (5) is estimated to be γ 1 = 10−5 N/m. 4.2. Influence of viscosity on water droplet kinematics Figs. 6 and 7 show a comparison between experimental observations and numerical predictions of the velocity of a water droplet falling towards a larger droplet placed on the electrode. When one wants to predict numerically the motion of the falling fluid droplet observed in the experiments, one has to take into account the effect of internal circulation induced in the droplet while it is falling. Internal circulation reduces the drag force and therefore the drag coefficient needs to be corrected, as outlined
(t t 0
+ i t)s with i = 0, 1, 2, 3. The
in Section 2.1. Furthermore, the interfacial tension of the droplet is varied by the effect of surfactants on the interface and by the elongation of the droplet caused by the electric field. This leads to interfacial stresses that inhibit the creation of internal circulation. LeVan [7] suggested how to take into account the effect of interfacial tension gradients in a numerical framework. Barnocky and Davis [14] derived expressions for the drainage between fluid spheres with arbitrary viscosity. The model proposed by Vinogradova [8] takes interfacial slip effects into account. Although Vinogradova [8] proposed a correction for hydrophobic surfaces, the present work shows that the Eqs. (9) and (10) can be used with good agreement for almost stationary surfaces as well. A comparison between observed and predicted velocities versus normalized particle surface distance h/r1 for a fluid droplets falling in oil is plotted in Fig. 6(a) when no field is applied. The average radius of the droplets is r1 ≈ 200 m. The impact velocity when no field is applied is nearly zero. This is due to the effect of the film-thinning force that acts against the motion of the droplet. Let the terminal velocity of the droplet be defined as the velocity that the droplet would reach when falling freely in a medium. It is calculated from Eq. (2) by imposing constant acceleration vector: dv/dt = 0. This implies consequently that Ffluid + Fext + Fd–d = 0. The droplet velocity is normalized by means of the terminal velocity and is plotted versus the distance between the droplets surface normalized by means of the droplet radius h/r1 see on the right of Fig. 6(a). The results show similarity. The dimensionless group v/vt is dependent on the viscosity. The terminal velocity reached by the droplets falling in a heated oil bath, where the viscosity is low, is greater than the terminal velocity reached by the droplet falling in a cooled oil characterized by a high viscosity. The film-thinning force normalized by means of the buoyancy force Ff /|Fb | is plotted in Fig. 8(a). The reason why the magnitude of the film-thinning force is greater for the low viscosity case lies in the fact that the relative droplet velocity is higher at low viscosity as seen in Fig. 6(a). When an electric field is applied in the direction of the droplet motion it will counteract the effect of the film-thinning force. For this reason, the impact velocity plotted in Fig. 6(b) is not equal zero. The magnitude of the film-thinning force is still higher than the magnitude of the dielectrophoretic force acting between the droplets when the applied electric field is |E0 | = 36.3 V/mm. Good agreement between numerical results and experimental observations is obtained for different viscosities of the contin-
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Fig. 6. Observed and predicted velocities vs. normalized droplet surface distance h/r1 for water droplets. The radii of the droplets are r1 ≈ 200 m. The slip distance in Eq. (9) is b = 10−6 m and the magnitude of the interfacial tension gradient in Eq. (5) is γ 1 = 10−5 N/m. (a) |E0 | = 0 V/mm: (circles and solid line) r1 ≈ 238 m, µ1 ≈ 2.5E−3 Pa s; (squares and dashed line) r2 ≈ 200 m, µ2 ≈ 13.9E−3 Pa s; (crosses and dotted line) r3 ≈ 185 m, µ3 ≈ 32.5E−3 Pa s. (b) |E0 | = 36.3 V/mm: (circles and solid line) r1 ≈ 222 m, µ1 ≈ 4.7E−3 Pa s; (squares and dashed line) r2 ≈ 205 m, µ2 ≈ 13.9E−3 Pa s; (crosses and dotted line) r3 ≈ 196 m, µ3 ≈ 27.1E−3 Pa s.
uous phase see Fig. 6(b). When plotting the curves on the left hand side of Fig. 6(b) after normalizing the velocity by means of the terminal velocity the curves fall on to each other. This is due to the fact that the magnitude of the dipole force is very small and the kinematics of the droplet in the near field region h/r1 < 5 is mainly due to the film-thinning forces while in the far field region it is due to the drag, buoyancy and virtual mass force as plotted on the right hand side of Fig. 8(a). Each of these forces are dependent on the viscosity so that the normalization of the velocity by means of the terminal velocity v/vt provides a master curve for the droplet kinematic. By increasing even further the magnitude of the applied electric field the effect of dipole forces becomes dominant on the droplet motion in the near field region. The results presented in Fig. 7(a) refer to an applied electric field of magnitude: |E0 | = 146.6 V/mm. The droplet experiences an acceleration towards the stationary droplet. The velocity increases until the droplet comes into contact with the stationary one. This is due to the fact that the magnitude of the dipole force is greater than the magnitude of the film-thinning force see Fig. 8(b). There are some uncertainties in extracting the velocity from the optical observation when the normalized distance h/r1 < 0.5 for low viscosity. A good agreement between the experimental results and numerical prediction is obtained. When plotting the curves on the left hand side of Fig. 7(a), after normalizing the velocity by means of the terminal velocity, the curves fell on to each other in the far field region as expected showing similarity see the right hand side of Fig. 7(a).
The droplet motion in the far field region is dominated by forces dependent on the viscosity. In the near field region the dipole force is the dominant one. The dimensionless group v/vt does not take into account of the effect of the dipole forces. When the magnitude of the applied electric field is increased up to |E0 | = 293.6 V/mm the effect of dipole forces becomes even more dominant on the droplet motion in the near field region. Fig. 7(b) shows a comparison between observed and predicted velocities, a good agreement is obtained. However, the discrepancy between the predicted results and the experimental observations is greater at high electric field in the near field region. The reason for this could be that the slip length in the Vinogradova Eq. (9) is kept constant b = 10−6 m during all the numerical prediction performed in this work. The slip length line is believed to be dependent on the characteristic oscillation mode of the droplet. This is dependent on both the droplet radius, the electric field and the frequency of the applied electric field. At high magnitude electric field, the amplitude of the oscillation of the droplet is believed to enhance the thinning of the film between the droplets. The droplet experiences an acceleration towards the stationary droplet. The velocity increases consequently until the droplet come into contact with the stationary one. This is because the magnitude of the dipole force is greater than the magnitude of the film-thinning force as previously explained. The contact velocity experienced at high field is greater than what experienced at lower field magnitude, compare Fig. 7(a and b).
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Fig. 7. Observed and predicted velocities vs. normalized droplet surface distance h/r1 for water droplets. The radii of the droplets are r1 ≈ 200 m. The slip distance in Eq. (9) is b = 10−6 m and the magnitude of the interfacial tension gradient in Eq. (5) is γ 1 = 10−5 N/m. (a) |E0 | = 146.3 V/mm: (circles and solid line) r1 ≈ 217 m, µ1 ≈ 2.89E−3 Pa s; (squares and dashed line) r2 ≈ 208 m, µ2 ≈ 13.9E−3 Pa s; (crosses and dotted line) r3 ≈ 207 m, µ3 ≈ 24.8E−3 Pa s. (b) |E0 | = 293.6 V/mm: (circles and solid line) r1 ≈ 210 m, µ1 ≈ 3.74E−3 Pa s; (squares and dashed line) r2 ≈ 198 m, µ2 ≈ 13.9E−3 Pa s; (crosses and dotted line) r3 ≈ m, µ3 ≈ 27.6E−3 Pa s.
4.3. Dependency of electrocoalescence on oil viscosity The effect of oil viscosity on droplet coalescence is investigated in a way similar to the study presented in the previous section. Experiments are performed releasing water droplets and observing the coalescence rest time. Coalescence is studied in detail and the observation area is focused on the contact surface. This allows us to increase the temporal resolution by reducing image size and increasing frame rate. The experimental results obtained are plotted in Fig. 9. There is a weak dependence of the results on the way one defines the contact time. In the present work the contact time is defined as the time after which no further movement of the droplet is recorded. One can argue that due to the deformation of the stationary droplet the definition we have chosen underestimates the real time of contact. In the present study the data presented in Fig. 9 are interpreted in a qualitative way in order to assess the effect of viscosity on electrically induced coalescence and therefore the argument around the definition of the time of contact is less important here. The coalescence rest time is plotted at different electric field magnitudes on the left hand side of Fig. 9(a) in a logarithmic–linear plot. Fig. 9(a) shows that for constant field magnitude the coalescence rest time greater for high viscosity oil. Viscosity is believed to indirectly influence the time of coalescence since the impact velocity of the water droplet in a
high viscosity oil is lower than in a low viscosity oil as plotted on the right hand side of Fig. 9(b). The dielectrophoretic force between the water droplets counteracts the effect of the film-thinning force and increases ultimately the magnitude of the impact velocity as previously discussed. For a given magnitude of the applied electric field the impact velocity is higher a low viscosity oil as plotted in Fig. 9(b). The results plotted in Fig. 9 give an indication of which role viscosity, interfacial tension, electric field magnitude, the ratio of permittivity and the density difference between oil and water play in the electrocoalescence process. The permittivity ratio on one side together with the electric field magnitude on the other side determine the intensity of the dielectrophoretic force acting on the droplets. Such force together with the buoyancy force due to the density difference between oil and water and the viscous forces determine the impact velocity. The interfacial tension influences the deformation of the droplet and also the movement of the center of mass. Hence, a droplet under the influence of an external electric field will experience an immediate deformation when its interfacial tension is low. Let’s now define We number as the ratio between the inertia of the droplet at impact and the interfacial tension: We =
2ρc r1 |v|2 σ
(15)
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Fig. 8. Difference between the calculated film-thinning force and dipole force normalized by means of the buoyancy force vs. normalized droplet surface distance h/r1 . The radii of the droplets are r1 ≈ 200 m. The slip distance in Eq. (9) is b = 10−6 m and the magnitude of the interfacial tension gradient in Eq. (5) is γ 1 = 10−5 N/m. (a) On the left hand side: |E0 | = 0 V/mm: (solid line) r1 ≈ 238 m, µ1 ≈ 2.5E−3 Pa s; (dashed line) r2 ≈ 200 m, µ2 ≈ 13.9E−3 Pa s; (dotted line) r3 ≈ 185 m, µ3 ≈ 32.5E−3 Pa s. On the right hand side: |E0 | = 36.3 V/mm: (solid line) r1 ≈ 222 m, µ1 ≈ 4.7E−3 Pa s; (dashed line) r2 ≈ 205 m, µ2 ≈ 13.9E−3 Pa s; (dotted line) r3 ≈ 196 m, µ3 ≈ 27.1E−3 Pa s. The forces correspond to the velocity profiles presented in Fig. 6. (b) On the left hand side: |E0 | = 146.3 V/mm: (solid line) r1 ≈ 217 m, µ1 ≈ 2.89E−3 Pa s; (dashed line) r2 ≈ 208 m, µ2 ≈ 13.9E−3 Pa s; (dotted line) r3 ≈ 207 m, µ3 ≈ 24.8E−3 Pa s. On the right hand side: |E0 | = 293.6 V/mm: (solid line) r1 ≈ 210 m, µ1 ≈ 3.74E−3 Pa s; (dashed line) r2 ≈ 198 m, µ2 ≈ 13.9E−3 Pa s; (dotted line) r3 ≈ 213 m, µ3 ≈ 27.6E−3 Pa s. The forces correspond to the velocity profiles presented in Fig. 7.
and the B number as the ratio between the square of the field and the interfacial tension: B=
εoil (1 − εoil /εd )2 |E|2 2σκ0
experiments carried out previously. Crosses, squares and circles represent experimental points when viscosity is varied. The Reynolds number reads:
(16)
where εoil /εd is the ratio of permittivity and k0 is the droplet curvature. Fig. 10 presents B plotted versus We for the different
n(n2 − 1)ρσd Reσ = 4µc
(17)
Fig. 9. The coalescence rest time after contact has occurred is plotted at (a) different electric field magnitudes and (b) different impact velocities. The radii of the droplets are r1 ≈ 200 m.
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Fig. 10. The Bond number is plotted vs. the Weber number (a) and coalescence rest time after contact has occurred is plotted vs. the Weber number (b).
where n is the oscillation mode number of the droplet, and σ and µc are as previously explained the surface tension and the viscosity, respectively. Reσ varies as a consequence of the variation of viscosity. On the right hand side of Fig. 10 the coalescence rest time is plotted versus We. The plot on the right side of Fig. 10 presents the same result as previously shown in Fig. 9. The difference lies in the fact that the different parameters influencing electro coalescence are grouped. 5. Observations and conclusions A modeling framework that properly accounts for the effect of the different forces acting on a droplet falling in an oil bath is used to investigate the effect of viscosity on electrically induced coalescence of water droplets in oil. The effect of internal circulation induced in the droplets has to be taken into account, together with the variation of the interfacial tension of the droplets due to the electric field. In the present work the model proposed by LeVan, Eq. (4), for the drag force, by Vinogradova, Eq. (9), for the film-thinning force, and the Davis analytical expression, Eqs. (11) and (12), provide numerical predictions that agree well with the experimental observations. The comparison between observed and predicted velocities versus normalized droplet surface distance, h/r1 , for fluid droplets shows a good agreement. Each of the forces influencing the kinematics of the droplet with the exception of the dipole–dipole force, are dependent on the viscosity. When the effect of the dipole–dipole force can be neglected the velocity profiles obtained at different viscosity can be normalized by means of the terminal velocity obtaining a master curve on which all the curves fall onto as expected. When the magnitude of the dipole forces dominates over the viscosity dependent forces the droplet accelerate towards the stationary one. The droplet velocity at impact is greater when the droplet falls in a low viscosity oil bath. Our experimental setup is designed to measure collision along the center-line of the droplets. The dipole–dipole force enhances the in-line collision since the droplets pair tends to align with the electric field. It is generally accepted that when two droplets collide, there are several possible outcomes depending on the kinetic energy of the collision, droplet size, the impact parameter (a measure of the degree to which the collision is off-center), the material
properties, and the droplet size ratio, which is defined as the ratio of the smaller droplet radius to the larger droplet radius. In the present work, we have limited our investigation to the combined effect of viscosity and electrostatic forces. Viscosity influences indirectly the coalescence rest time since the impact velocity and consequently the kinetic energy of the water droplet is dependent on the viscosity of the oil bath in which the water droplet is released. The electric field also contributes to increase the kinetic energy of the falling droplet since it accelerates the droplet that experiences a maximum velocity at collision when the field magnitude is above a certain threshold dependent on the oil viscosity. The electric field also influences the coalescence process since it influences the interfacial tension of the droplet due to droplet deformation. This influence is hard to quantify. The viscosity influences indirectly the time of coalescence since the impact velocity of the water droplet in a high viscosity oil is lower than in a low viscosity oil as plotted on the right hand side of Fig. 9(b). This is an important parameter to take into account when dealing with heavy oil. A model that describes electro coalescence is required to fulfill the modeling framework used in the first part of the paper. In order to provide such a model the experimental setup should be slightly modified. One should be able to perform different series of experiments varying various parameters while keeping two of the dimensionless groups constant simultaneously. Due to the lack of a precise model for coalescence efficiency, one can assume, as a first approximation, that each collision leads to coalescence. This might be an adequate approximation when electric fields are applied and when the water droplet surfaces are not too contaminated. The validity of the approximation is more doubtful in the case with no applied electric field. Acknowledgments The authors are thankful to VetcoAibel, Statoil, Petrobras and the Research Council of Norway for supporting the present work. References [1] J. Sj¨oblom, N. Aske, I.H. Auflem, Brandal, T.E. Havre, Sæther, A. Westvik, E.E. Johnsen, H. Kallevik, Our current understanding of water-
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in-crude oil emulsions. Recent characterization techniques and high pressure performance, Adv. Colloid Interface Sci. 100–102 (2003) 399–473. J.S. Eow, M. Ghadiri, A.O. Sharif, T.J. Williams, Electrostatic enhancement of coalescence of water droplets in oil: a review of current understanding, Chem. Eng. J. 84 (2001) 173–192. P. Hemmingsen, A. Silset, A. Hannisdal, J. Sj¨oblom, Emulsion of heavy crude oils, influence of viscosity, temperature and dilution, J. Dispersion Sci. Technol. 26 (5) (2005) 615–627, doi:10.1081/DIS-200057671. M. Chiesa, J.A. Melheim, A. Pedersen, S. Ingebrigtsen, G. Berg, Forces acting on water droplets falling in oil under the influence of an electric field: numerical predictions versus experimental observations, Eur. J. Mech.-B/Fluids 24 (6) (2005) 717–732. M.H. Davis, Two charged spherical conductors in a uniform electric field: forces and field strength, Rand. Corp. Memorandum RM-3860PR, January 1964. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff Publishers, The Hague, The Netherlands, 1983. D.M. LeVan, Motion of droples with a Newtonian interface, J. Colloid Interface Sci. 83 (1) (1981) 11–17.
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