Nordic Insulation Symposium Tampere, June 3, 23 Forces and movement of small water droplets in oil due to applied electric field A. Pedersen E. Ildstad A. Nysveen Norwegian University of Norwegian University of Norwegian University of technology and science technology and science technology and science Norway Norway Norway. Introduction The wellstream in offshore oil production comprises various types of hydrocarbons, water and sometimes sand. During the production removal of water from the oil is made by using large gravity separation tanks. The efficiency of oil / water separation depends on several factors and varies from reservoir to reservoir. Electric field, so called electrocoalescense, is used to increase the size of the water droplets which will result in higher settling speed in the gravitational separators[]. In order to optimise and reduce the size of these separators it is important to know more about the mechanism involved. In this article the focus is on the forces due to polarization of water droplets. It is common to describe these forces assuming that the droplet behaves as a small dipole. This work has tested the validity of this assumption. The electric force acting on a water droplet in an electric field has been experimentally examined, indirectly through the measurements of drag forces, on the droplet in two distinct ways, force on one falling droplet in an divergent field, and force in droplet droplet interactions. 2. Forces on droplets in electric field In an external electric field the distribution of charge on the surface of the water droplet will result in attractive forces between the water droplets and subsequently enhance coalescense of water droplets. In a stagnant fluid the sum of forces acting will result in an acceleration of the water droplet: dv m p = Fg Fe F dt where m p is the mass of the droplet, v is the velocity of the droplet, F g is the gravitational force, is the electric force acting on the droplet in the direction of the field and F d is the viscous force acting in the direction opposite to the direction of movement. In the following these forces will be described in some detail. 2., electric force acting on the droplet d When a neutral water droplet is placed in an electric field it becomes polarized and surface charges will be induced on its surface. ()
For a water droplet in an uniform field, fig. a, the magnitude of the forces acting on each side of the water droplet will be equal, resulting in zero translation force. If the external ϕ D,QGXFHGFKDUJHRIDQHXWUDOZDWHUGURSOHWLQDQXQLIRUPILHOGE'LHOHFWURSKRUHWLFIRUFH RQDVPDOOGURSOHWLQLQKRPRJHQRXVILHOG field is nonuniform the forces acting will be larger on the high stress side. This is illustrated in fig. b. This gives rise to a net force pulling the small water droplet towards the region where the field is strongest, a phenomena usually called dielectrophoresis. If the water droplet is small and the degree of inhomogenity is low this net dielectrophoretic force acting on the droplet can with good approximation be expressed by[3]: Where µ is the dipole and (is the average external field acting on it. In case of a small neutral water droplet this can be expressed as: [] 2πa 3 ε water ε oil = ε ε oil E 2 (3) ε water 2ε oil For both AC and DC fields the net translation force will always be directed toward the region with strongest field. In an strong inhomogenous electric field, and in case of strong dropletdroplet interaction, the droplets themselves alter the electric field and this general dipole approximation will not be valid. An alternative approach is then to calculate the surface charge and the resulting electric field surrounding the droplet. The density of surface charge on the water droplet can then be expressed as [2]: σ = E i ( ε water ε oil )ε sinϕ () where ( L is the field inside the droplet and ε ZDWHU and ε RLO are the relative dielectric constants of the water and oil, respectively. The force acting on the droplet then becomes: where ( [ is the field in the direction of movement. 2 = µ E. (2) = A σe x da ) J JUDYLWDWLRQDOIRUFHDFWLQJRQWKHGURSOHW The resulting gravitational force acting on the water droplet is given by F g = πa 3 ( ρ (6) 3 water ρ oil )g Where D is the radius of the droplet, J is the gravitational acceleration, ρ ZDWHU and ρ RLO are the densities of the water and oil. (5)
) G YLVFRXVGUDJIRUFHV When the water droplet is moving the oil will cause a net drag force on the droplet. In a stagnant fluid this force can be expressed by the drag coefficient through the equation [5]: F d = ρ (7) 2 oil C D Av 2 where & ' is the drag coefficient, $ is the representative area of the droplet and Y is the velocity of the droplet. & ' is expressed as the function of the Reynolds number and the correction factor because of internal motion in the droplet. CMOS Camera Long Distance Microscope Test Cell Background Light Laptop Translation stages Test Cell Cylindrical Setup Stationar droplet Setup ([SHULPHQWDOVHWXSDF\OLQGULFDOVHWXSIRUORZQRQXQLIRUPILHOGE6WDWLRQDU\GURSOHW VHWXSIRUVWXG\GURSOHWGURSOHWDWWUDFWLRQ ([SHULPHQWDOVHWXS Experiments were designed for visual observations of droplets or water in oil emulsions exposed to electric field. The electrode arrangement was put inside a small test cell (5x5x5)cm 3, which was placed in an shadowgraphic setup using an optical bench as shown in fig. 2. A high speed video camera with a maximum frame rate of 32 frames/ sec was used to record the trajectory of the moving droplets.the voltage source was a HVamplifier (/ 2 kv, 2 khz). In these experiments 5Hz sinusiodal voltage was used. The oil used in the experiments was Nytro x, and the characteristics of the oil and water are shown in table. Two different electrode systems were used in this study: (i). Low nonuniform field, shown in fig. 2a where a cylindrical field was made using a rod as an inner electrode and a fine grid as a outer electrode. This grid had an inner radius, U \ = 3 cm, and a length of.5 cm. The inner electrode had a radius, U L =.5 cm.
(ii). Stationary droplet setup for droplet droplet attraction using vertical electrode arrangement, schematically shown in fig. 2b.The gap of the electrodes where G = 2 cm. The highly nonuniform electric field was made by an energized water droplet attached to the tip of an glass capillary. The glass capillary was mounted in the centre of the grounded electrode. The capillary was metalized with a silver painting. In these experiments water with 3.5%wt NaCl was used. The capillary and the stationary water droplet were both connected to the grounded electrode. Dyn Viscosity (@ 2C mpas 3.7 Dyn Viscosity (@ C mpas 6. Nytro X Dest. water Dest water 3.5 Wt NaCl Density (@ 2C), g/cm 3.875.998.22 Relative permittivity, ε r 2.2 8 8 7DEOHHDVXUHGYDOXHVRIWKHRLODQGZDWHUXVHGLQWKHH[SHULPHQWV HWKRGIRUPHDVXUHPHQWVRIIRUFHVLQORZILHOGWKHF\OLQGULFDOJHRPHWU\ By a glass capillary water droplets were injected between the two electrodes as shown in fig. 3. The droplet sedimented slowly due to gravity. When the droplet entered into the F d x α s F g r y r i y U x (c) DH[SHULPHQWDOVHWXSRIF\OLQGULFDOJHRPHWU\EIRUFHVDFWLQJRQDGURSOHWFW\SLFDO YLGHRREVHUYDWLRQ7KHWLPHGLIIHUHQFH WEHWZHHQHDFKSLFWXUHVZDVV frame of the video camera the 5Hz voltage was turned on. The droplet then started to move toward the inner electrode. Fig. 3 shows a typically trajectory of the droplet recorded with the video camera. The radius of the water droplet D, its position and increment in xdirection, G[, and ydirection, G\, and time between the framesgwwere measured manually from each frame of the videofilm. From this the speed of the water droplets was derived using v = dx 2 dy 2 dt (8)
Thus the electric force is equal to the drag force in xdirection = F Dx = ρ 2 oil C D Av 2 cosα where α is the angle of the dragforce, shown in fig 3b. (9) HWKRGIRUPHDVXUHPHQWVRIIRUFHVLQWKHKLJKILHOGGURSOHWGURSOHWLQWHUDFWLRQV A water in oil emulsion (% water) was mixed in a mixer (turax). The emulsion was injected in the gap via a syringe in the front of the stationary droplet. The diameter of the droplets were in the 5 2µm range, allowing the emulsion to float like a cloud in the gap. d l U y x ([SHULPHQWDOVHWXSIRUH[DPLQDWLRQRIGURSOHWGURSOHWLQWHUDFWLRQ7KHSLFWXUHVKRZVW\SLFDO YLGHRREVHUYDWLRQZKHUH WLVV The gap was energized after the emulsion had been injected. Small water droplets then started to move towards the stronger field. The position of these droplets were recorded by the video camera, resulting in a typically trajectory as shown in fig.. (OHFWULFILHOGFDOFXODWLRQXVLQJ)(/$% A 2dimensional FEM model of the dropletdroplet interaction setup was used to calculate the electric field. Using the electrostatic axisymmetry. The field in the positions of the droplet was recorded and used in MATLAB to calculate the electric forces according to eq. 5. Simulations were made with and without the small water droplet included in the model. In these simulations the internal and external field were recorded at each position of the droplet.
5HVXOWVDQG'LVFXVVLRQ HDVXUHGIRUFHVLQF\OLQGULFDOJHRPHWU\ The force due to acceleration was measured to be 7 times less than the drag force. This is so low that it was neglected. In a cylindrical arrangement the electric force according to eq. 3 can be expressed analytically [3]. In fig. 5 the measured drag force ) G[ and the 5 x 8 x 8.5 5 3.5 3 2.5 2.5 Fe (theoretical) F dx (measured) a =.3mm 3 2 Fe (theoretical) F dx (measured) a =.35 mm.5 5 5.2 5. 5.6 5.8 6 6.2 6. 6.6 6.8 7 7. 7.6 7.8 8 8.2 8. 8.6 8.8 7\SLFDOIRUFHVRIRQHGURSOHWLQWKHF\OLQGULFDODUUDQJHPHQW) G[ LVWKHPHDVXUHGGUDJIRUFH DFFRUGLQJWRHTDQG) H LVWKHSUHGLFWHGIRUFHDFFRUGLQJWRHT[LVWKHGLVWDQFHIURP FHQWUHWRWKHSRVLWLRQRIWKHGURSOHWVHHILJ7LPHGLIIHUHQFH WEHWZHHQPHDVXUHGYDOXHV DUHV8 N9] predicted electric force ) H is presented. The dotted line represent the linear regression of the measured drag forces. The result presented show good agreement between the measured force and the calculated dielectric force. In this case the droplet was small compared to the distance between electrodes, and the degree of inhomogenity is low. The results indicate that the measured force oscillate with a frequency of 2Hz. This variance was probably caused by the mechanical oscillation in the experimental setup. In addition errors may have been introduced due to manual recording of droplet positions from the video screen. The good agreement shows that it is possible to measure the electric force indirectly by measuring the speed of the droplet and the drag force. HDVXUHGIRUFHVRQGURSOHWGURSOHWLQWHUDFWLRQ The force due to acceleration close to impact was measured to be 5 times less than the drag force. The gravitational force was measured to be 7 times less than the drag force. These forces were so low that they were neglected. Fig. 6 shows the measured drag force ) G[ and the theoretical calculated force acting on the small droplets. ) H is the predicted force not considering the droplets influence on the field. The forces are shown as a function of the distance from the boundary of the large droplet, where x =. Fig. 6 shows a large difference between calculated and the measured drag force. The deviation is small in the beginning of the trajectory but increases when the droplet approach the larger droplet. This means that the size of the droplet becomes large compared to the distance to the electrodes. The observed deviation can be explained due to the fact that the influence of the smaller droplet is not taken into account. A better agreement was achieved using the
electric field approach from eq. 5, ) H. The droplet was also in this case considered as a dipole according to eq. 3. But in this case the influence of the small droplet on the local field was taken into account. When the small droplet approach the large one the drainage of the oil between the droplets will reduce the velocity, and thus the measured drag force. This can be a reason for the better agreement between the theoretical ) H and the measured ) G[ forces in case of a smaller droplet shown in fig 6b. 9 x 8 8 2 x 8.8 a =.2mm 7 6 5 3 a =.5m (theoretical) 2 (theoretical) F dx (measured).6..2.8.6 (theoretical) 2 (theoretical) F dx(measured) 2..2 3 2 7\SLFDOIRUFHVRQDVPDOOGURSOHWVDSSURDFKLQJDODUJHUGURSOHW) G[ LVWKHPHDVXUHGGUDJ IRUFHDFFRUGLQJWRHT) H LVWKHSUHGLFWHGHOHFWULFIRUFHXVLQJHTDQG) H LVWKH SUHGLFWHGHOHFWULFGUDJIRUFHXVLQJHT[LVGLVWDQFHIURPWKHERXQGDU\RIWKHVWDWLRQDU\ GURSOHWWRWKHFHQWUHRIWKHVPDOOGURSOHW WEHWZHHQPHDVXUHGYDOXHVDUHV8 9 ] &RQFOXVLRQ The electric force can indirectly be measured, by measuring the speed of the droplet and the drag force. In case of a small droplet in a low inhomogenous electric field is it possible to consider the water droplet as a dipole. When a droplet approaches another droplet this will no longer be true and the simplified consideration is not valid. In such case it becomes necessary to consider the small droplets influence on the local field. A better agreement with the measured values is then shown. 5HIHUHQFHV [] J. S. Eow, Electrostatic enhancement of coalescence of water droplet in oil: a review of the technology, Chemical Engineering Journal, 85(22) 357368. [2] S. V. Marshall, Electromagnetic concepts & applications, sec. ed. PrenticeHall, Inc, 987. [3] H. A. Pohl, Some effect of nonuniform fields on dielectrics, J. Appl Phys. 29 (8) (958 8288, 958 [] H. A. Pohl, Dielectrophoresis. Cambridge Univ. Press, 978 [5] C. Crowe, Multiphase flows with droplets and particles. CRC Press, 998...5.2