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Numerical simulation of a novel non-uniform electric field design to enhance the electrocoalescence of droplets
Journal Pre-proof Numerical simulation of a novel non-uniform electric field design to enhance the electrocoalescence of droplets Hooman Hadidi, Mohammad K.D. Manshadi, Reza Kamali
PII: DOI: Reference:
S0997-7546(19)30379-6 https://doi.org/10.1016/j.euromechflu.2019.10.010 EJMFLU 103562
To appear in:
European Journal of Mechanics / B Fluids
Received date : 28 June 2019 Revised date : 13 October 2019 Accepted date : 29 October 2019 Please cite this article as: H. Hadidi, M.K.D. Manshadi and R. Kamali, Numerical simulation of a novel non-uniform electric field design to enhance the electrocoalescence of droplets, European Journal of Mechanics / B Fluids (2019), doi: https://doi.org/10.1016/j.euromechflu.2019.10.010. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Masson SAS.
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Numerical simulation of a novel non-uniform electric field design to enhance the electrocoalescence of droplets Hooman Hadidi1, Mohammad K. D. Manshadi1, Reza Kamali1,*
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1. Department of Mechanical Engineering, Shiraz University, Shiraz, Fars, 71348-51154, Iran *Corresponding Author Email: [email protected]
Abstract
The behavior of binary droplet collision, exposed to an external electric field is studied
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numerically. Based on a fundamental understanding of the role of the electric field in the coalescence of aqueous drops in an immiscible oil phase, it is possible to optimize the design and
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operation of these external stimulators. In this research, numerical modeling of electrocoalescence is implemented, using Computational Fluid Dynamics (CFD) methods. Level set method is used to evaluate water droplet evolution in an oil phase under a novel concentric semi-elliptic non-
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uniform electric field configuration. The observations show that the shape of applied non-uniform fields can remarkably alter the electrocoalescence performance in a nontrivial way, which is sensitive to the drop-medium system. In addition to the shape of the medium, some other parameters such as droplets initial distance, initial skew angle of the droplets, applied voltage amplitude and oil viscosity are analyzed in the present paper. Our results can be useful in designing
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an optimum electrode configuration, achieving an efficient electrocoalescence. Keywords: Electrocoalescence; Droplet; Multiphase; Non-uniform; Electric field; Level-set method (LSM)
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1. Introduction Coalescence of droplets, despite its simple appearance, is an important and practical phenomenon which has attracted much attention in recent years [1]. This can be observed in many natural
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processes including raindrop and cloud formation, and atmospheric aerosol circulation [2], in addition to industrial applications such as spray cooling [3], droplet-based microfluidics [4], micro-encapsulation[5], fire suppression [6], anti-icing and coating processes used in aerospace and power industries [7], etc. Some results of the collision of binary liquid drops can be categorized in four regimes: (I) coalescence after minor deformation, (II) bouncing similar to situations where a droplet impacts on a spherical solid surface at low velocities [8, 9], (III) coalescence after
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substantial deformation, and (IV) coalescence followed by separation.
The effect of viscosity on the droplet-droplet collision outcome was studied by Finotello et al. [10]
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using direct numerical simulations employing the volume of fluid method. The authors explored the role of viscosity on the transition between coalescence and reflexive and stretching separation and proposed a general phenomenological model based on the capillary number (𝐶𝑎
𝜇𝑣/𝛾, with
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𝜇 accounting for the viscosity), the impact parameter, We, and the size ratio. In another study, the Finotello’s group [11] studied complex collision dynamics of shear-thinning non-Newtonian fluids experimentally and developed a full regime map for this problem. Planchette et al. [12] presented a general modeling approach of drop fragmentation after head-on collisions that goes beyond binary collisions with a single liquid and takes into account two other types of collisions: the
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collisions between two drops of immiscible liquids and the collisions between three drops of one single liquid. This approach assimilates the colliding drops to liquid springs that coalesce, compress and relax, leading the merged drop to break up if it reaches a critical aspect ratio. The authors were able to deduce the fragmentation threshold velocity for various impact scenarios.
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Shen et al. [13] experimentally studied the coalescence between two different-sized droplets in the surrounding water. The authors identified three coalescence patterns, including liquid bridge evolution, capillary wave propagation, and pinch-off behaviors. It was concluded that the
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coalescence patterns are directly related to the propagation of capillary waves on the coalescent droplet, which is governed by the competition among the capillary force, viscous force, and inertia involved in the draining from the original droplets into the liquid bridge. Two-dimensional numerical simulations of the head-on and off-center binary collision of liquid drops are carried out using the lattice Boltzmann method by Wei et al. [14]. Their results demonstrate that the numerical results of the lattice Boltzmann method are in agreement with qualitatively experimental data in
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the same We. It is also observed that for the collision of equal-size drops, increasingly thinner drops emerge as a result of increasing We during the coalescence process. In a review paper, Kamp
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et al. [15] investigated the coalescence process in detail discussing the sequential steps during a collision of two drops that could result in coalescence, agglomeration, or repulsion. The study also presents several experimental techniques which were used for coalescence investigation at
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different levels of detail.
While passive droplet coalescence techniques do not require external energy, active techniques depend on some kind of external force or field to manipulate or control the coalescence process; such as electric field, magnetic field and optical heating. The coalescence behavior of droplets in an electric field constitutes a crucial research division of electro-hydrodynamics. The electrostatic
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coalescence technology is widely used in two-phase separation because of its high efficiency, costeffectiveness and environmental benefits [16]. The first studies conducted on electrocoalescence date back to early 20th century and the fundamental understandings acquired since then have helped to make the technology more efficient, faster and the devices more compact. Chiesa et al.
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[17] studied the forces affecting the kinematics of droplets while exposed to an electric field. They presented mathematical models for these forces and discussed about a possible application in a multi-droplet Lagrangian framework. Mohammadi et al. [18] utilized a combined model of
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electrostatic and hydrodynamic fields for simulating the electric field effects on the binary water droplets coalescence in oil. Their results revealed that the oil viscosity, the skew angle of the electric field and the initial drops distance affect the electrocoalescence. In another study, Mohammadi et al. [19] experimentally investigated collision of the binary drops. They reported in their paper that increasing initial distance results in longer approaching time. Moreover, stronger electric fields lead to faster approaches of the droplets. Furthermore, as long as the drops were
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aligned with the field, the drops had quicker coalescence compared to the situation where some skew angle existed between them. Mhatre and Thaokar [20] experimentally investigated different
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non-uniform electric field configurations such as pin-plate, quadrupole and annular electrode in electrocoalescence. Their results showed that asymmetric non-uniform field generated applying the pin-plate electrodes has better performance compared to the uniform and symmetric
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quadrupole non-uniform electric fields. Luo et al. [21] experimentally investigated phase separation under different electric field configurations. The electric field assists the merging of small droplets of water dispersed in a continuous oil phase, and by increasing the water droplet size and breaking the quasi-equilibrium of the emulsion, phase separation is accrued. Their results showed that a non-uniform electric field and the induced dielectrophoretic effect can accelerate
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the coalescence and phase separation of micro-emulsions. Pal et al. [22] numerically investigated the phenomenon of droplet collision with a charged substrate by a coupled electro-hydrodynamic model. They analyzed the phenomenon with different parametric variations like electric potential, wetting nature of the substrate, and velocity of collision as it is governed by the mutual interaction
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between the inertia, electrostatic, and capillary forces. Shabankareh et al. [23] numerically investigated the characteristics of leaky dielectric droplets in electric fields and showed that pseudoplastic droplets have more deformations in an electric field than Newtonian and dilatant
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ones, and they also present faster electrocoalescence. Although a limited number of studies have been conducted hitherto on the effects that non-uniform electric fields have on the coalescence of the droplets and despite the fact that the superiority of the non-uniform electric fields over uniform ones has been established, a significant gap still exists in this area. More specifically, a comprehensive knowledge of the exact effects that the shape of the electric field medium can have on the coalescence time of the droplets has not been established
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until now. The present study particularly aims to investigate the impact of a novel non-uniform electric field design on the coalescence of the droplets. For the first time, the impact of various
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parameters including the ratio of the semi-major axis to the semi-minor axis in the semi-elliptic configuration and the ratio of the radii in the semi-circular configuration will be taken into account. Other effective and important parameters include the initial distance between the droplets, skew
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angle of the droplets, applied voltage amplitude, and the viscosity.
2. Governing Equations and Method of Solution Electrocoalescence is a two-phase flow phenomenon in which momentum, continuity, and electric field equations should be considered to model it. Continuity and momentum (Navier-Stokes)
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equations are also solved in order to find velocity and pressure distribution in the medium. For incompressible fluids continuity equation is:
u = 0
Where u (m.s-1) is fluid velocity. Navier- Stokes equation is [24, 25]:
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(1)
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u (u .)u p . (u (u )T F t
(2)
Where 𝜌 (Kg.m-3 ) is density, p (Pa) denotes pressure, (Pa.s) is viscosity, and F (N) represents
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the sum of external volume forces. These volume forces (F) consist of surface tension and electric forces, which are: F n f e
(3)
where (N.m-1) is surface tension coefficient, 𝜅 is the fluid-fluid interface curvature, 𝛿 is the interface delta function, n is the unit normal vector to the interface pointing into the droplets, and
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fe (N) is the electric field force. This force is expressed as [24, 25]:
1 f E E 2 e e 2
(4)
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Where (C.m-3), E (V.m-1) and (F.m-1) are free charge density, the electric field and electric e permittivity respectively. is also obtained from Gauss’s law as follows: e
= E e
(5)
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and the electric field is gained from:
𝐸
∇𝑉
(6)
Where V (V) is the electric potential. For perfect-perfect dielectric fluids in a DC electric field
0 and the electric field equation becomes: e
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E = 0
Therefore, the final non-dimensional form of the Navier- Stokes equation is:
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(7)
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u * (u *.)u * t 1 1 Bo E L 1 *2 ( n ) ( E ) p * . * (u * (u * )T Re ReCa ReCa R 2
*
(8)
(L), velocity (U), time (𝑡
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Where the equation is made dimensionless based on radius of droplet (R), Length of the medium 𝑡 ∗ ), pressure ( p p * U 2 ), electric field ( E = E *E 0 ), electric
permittivity of medium ( m ), density of medium ( * m ), and viscosity of medium ( * m ). The dimensionless numbers in this equation are:
mUR 0 m
(9)
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Re
Ca
(10)
m R 0 E 02
(11)
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Bo E
mU
In the above-mentioned equations, the properties for each liquid phase should be considered separately. Level set method is one of the numerical methods for tracking the interface of two
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immiscible fluids in a two-phase fluid flow problem to make a distinction between different fluid phases. This equation is:
u . 1 t
(12)
Where 𝜙 is the level set function, γ and ϵ are the numerical stabilization parameters for re-
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initialization and determining interface thickness respectively. Level set function (𝜙 is defined "0" for one phase, and "1" for the other phase. This function is also utilized for smoothing the fluid properties across the interfaces. For instance, viscosity through the media is obtained from:
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𝜇
𝜇
𝜇
𝜇 𝜙
(13)
Where the notations "d" and "m" represent droplet and surrounding fluid, respectively. The mentioned coupled equations are solved by COMSOL Multiphysics 5.1. This is a powerful
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software which provides the context for scholars to simulate different physics of a problem simultaneously [26, 27]. This software is widely employed to model complex phenomena because this software is capable of solving coupled system of partial differential equations (PDEs) with different algorithms. Herein, MUltifrontal Massively Parallel sparse direct Solver (MUMPS) is used for the simulations. This algorithm is based on lower–upper (LU) factorization which used for solving large linear algebraic equation systems. This solver is capable of out-of-core solution
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storing which decreases the computational costs by providing more memory than the available
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memory on the computer [28].
3. Validation
Electrohydrodynamic (EHD) method capabilities for droplet manipulation have attracted the
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attention of our group. Previously, three numerical investigations about controlling microdroplet generation, non-Newtonian droplet electrocoalescence, and droplet impact in DC electric fields have been performed by our team [23, 29, 30]. Results of these studies were verified by theoretical and numerical findings. In addition to that, the numerical results of the present study are compared with theoretical data available in the literature.
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O’Konski and Thacher [31] developed an analytical model to define deformation of perfect dielectric drops suspended in perfectly insulating media. This model is [32]: 𝐷
𝐵𝑜
9 𝑆 1 16 𝑆 2
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𝐷 Where 𝑆
𝑊 𝑊
𝐵 𝐵
and in the deformation parameter equation (D), W is the polar half-axis and B is
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the equatorial radius of the spheroid. According to this equation, droplet always deform in the direction of the electric field [32]. After the modeling of the droplet deformation, the obtained results are compared with the analytical data (see Figure 1 ). In all of the simulation in this paper
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the time-step was considered to be 0.001 (s).
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Figure 1. Validation of the numerical model. (A) Schematic representation of the deformation of a perfect dielectric droplet suspended in a perfect dielectric immiscible liquid. (B) Comparison of the numerial results and the theoretical data of O’Konski and Thacher [31]
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Comparison of our numerical results with the theoretical data implies that the present numerical study has the reasonable capability of predicting droplet behaviors under an applied electric field.
4. Results and Discussion 9
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Figure 2 (A) represents a schematic of the geometry used in the present study, along with its boundary conditions. The medium is made of two concentric semi-ellipses with b2/a2= b1/a1=b/a. An electrode with an electric potential of 𝑉 is placed on the inner curved surface. and the outer
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curvature is set as the ground. For boundary conditions of momentum equation (eq. 4), on the specified horizontal lines indicated in the figure, an equal pressure, p = 0, is applied. Moreover,
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the curved lines are considered as walls with no-slip condition.
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(A)
(B)
(C)
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Figure 2. (A) Computational domain with electrical boundary conditions. (B) The generated tetrahedral mesh. (C) Grid independence study: non-dimensional coalescence time versus grid
pro of
number.
The unstructured triangular meshing tool provided in COMSOL is used to generate the mesh. This algorithm first applies on the domain boundaries. Then, this mesh is used to mesh the domain with isotropic tetrahedral with reasonable element aspect ratio.
Tetrahedral meshing algorithm first applies a mesh on all of the surfaces of an object. This mesh
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is then used to “seed” the volume mesh from which tetrahedral elements “grow” elements inwards. As these tetrahedral elements intersect, their sizes are adjusted with the objective of keeping the
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elements as isotropic (similar edge lengths and included angles) as possible and to have reasonably gradual transitions between smaller and larger elements.
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A view of the generated grid around the two droplets is presented in Figure 2 (B). In order to ensure the accuracy of the obtained numerical results, a mesh independence study has to be conducted prior to performing the simulations of the main cases. For this purpose, the nondimensional coalescence time of the droplets with diameters 1.5 mm and an initial distance of 0.7 mm, under an average electric field density of 1.95
10 𝑉/𝑚 was chosen as the parameter based
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on which the mesh independence study was conducted. For instance, for the case with b/a = 1.6, different meshes with grid numbers ranging from 4308 to 47453 were tested (Figure 2 (C)). As can be observed in Figure 2 (C), the difference in non-dimensional coalescence time between the meshes with 35654 and 47453 grids is negligible (about 2.5%). Therefore, for the mentioned case,
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the simulations were performed using the mesh with 35654 grids, with minimum element size in the order of 0.01 mm. Similar procedure was pursued for all the cases simulated in the present
4.1. The effect of the b/a ratio
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study.
Figure 3 displays electric potential for the geometry with b/a = 1.6 at a specific time (t = 0.02s). As can be observed, since the 2500V electrode has been placed on the inner wall and the electric potential of the outer wall is zero, the distribution of the electric potential intensity varies from
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2500 V on the inner wall to zero on the outer wall.
Figure 3. Distribution of the electric potential for the geometry with b/a = 1.6.
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Figure 4 (A) depicts the variations in the distance of the droplets with respect to each other and the contours of the electric field intensity at different times for b/a = 1.6. The initial diameter of the droplets and their distance from each other are 1.5 mm and 0.7 mm, respectively. The average 10 V/m. The oil viscosity is equal to 0.2 Pa.s. The droplets
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intensity of the electric field is 1.95
get polarized under the action of the electric field; therefore, polarized charges are generated at both ends of the droplets. The droplets are attracted to each other and are affected by the dipole force until they make contact with each other. After getting into contact, they tend to merge under the action of interfacial tension. Hence, the final coalescence takes place at the time 0.2 s and a single large drop is formed. At the same time, the dissimilar charges become neutralized on those
electric field of the polarization[16].
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parts of the droplets that are in contact, which leads to a total neutralization of the droplets by the
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According to Figure 4 (B), by increasing the b/a ratio, the non-dimensional coalescence time reduces. The reason behind this observation is the stronger forces induced by the non-uniform electric field that affect the droplets, which could be because of enhancement of the drop pairs
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alignment under the imposed non-uniform electric field [19]. This fact is illustrated in Figure 4 (C). The figure depicts the total force imposed on the droplets at t* = 43.4 for various values of b/a. The force imposed on the droplets is generated as a result of the non-uniform electric field and is augmented with an increase in b/a. As a consequence, the droplet’s non-dimensional coalescence
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time is reduced.
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(A)
(B)
(C)
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Figure 4. b/a ratio effects on electrocoalescence. (A) Variations in the distance of the droplets with respect to each other at different times for b/a = 1.6, (B) Variation of the non-dimensional coalescence time with b/a, (C) Variation of the total non-dimensional force imposed on the
4.2. The effect of the radii ratio (scale factor)
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droplets with b/a at t*=43.4.
In the following, the effect of the ratio of the outer radius to the inner radius on the coalescence of the droplets in the case of b/a=1 has been investigated. This ratio is called “scale factor” (Sf= a1/a2=
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in Figure 5 (A) to (C), respectively.
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b1/b2= R1/R2). The schematics of the configurations with Sf values of 0.1, 0.3 and 0.6 are presented
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Figure 5. The effect of scale factor on the coalescence of the droplets with Sf values of (A) 0.1, (B) 0.3, and (C) 0.6 at different times. Figure 6 (A) shows the time-dependent variation of the coalescence time with Sf. The initial
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droplet diameter and the initial distance between the droplets are 1.5 mm and 0.5 mm, respectively. The average intensity of the electric field is 290 V/m. According to this figure, an optimal Sf exists at Sf=0.4 such that the shortest coalescence time is observed to occur at this scale factor and the longest impingement time is obtained at Sf = 0.6. This is due to the non-uniformity of the electric field, such that its direction imposed on the droplets differs for various scale factors and
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by varying the scale factor, the alignment of the electric field with drop pairs changes, leading to different coalescence time[19]. Therefore, it can be concluded that the magnitude of the total force imposed on the droplets at Sf = 0.4 is larger than all other cases. In addition, this can be seen in
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Figure 6 (E), illustrating the total force imposed on the droplets for various sf values at t* = 60.8. As demonstrated in this figure, the minimum imposed forced occurs at Sf = 0.6, leading to the
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longest non-dimensional coalescence time.
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(B)
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(A)
(D)
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(C)
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(E)
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Figure 6. The variation of non-dimensional coalescence time with (A) scale factor (R1/R2) (B) the initial distance of the droplets (C) the applied voltage amplitude and (D) the viscosity. (E)
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The variation of the total force exerted on the droplets with Sf.
4.3. The effects of the droplets initial distance, the electric field strength and viscosity on coalescence time
Figure 6 (B) displays the time-dependent variations of the non-dimensional coalescence time with
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the initial distance of the droplets. The initial diameter of the droplets is 1.5 mm. The average intensity of the electric field is 290 V/m and the oil viscosity is 0.005 Pa.s. Results illustrate that, as expected, the shorter initial distance at a given constant drop size results in quicker coalescence because of the stronger force between the drops. The magnitude of the force imposed on the droplets is dependent on the external electric field. Therefore, the intensity of the electric field is 18
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one of the most important parameters in electrocoalescence[19]. As can be observed in Figure 6 (C), by increasing the electric field intensity, the bipolar force between the droplets increases and the non-dimensional coalescence time is reduced. The effect of the variations in the viscosity on
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the non-dimensional coalescence time is shown in Figure 6 (D) for an initial diameter and initial distance between the droplets equal to 1.5 mm and 0.5 mm, respectively. This figure indicates that increasing the oil viscosity causes more resistance for droplets movements leading to decelerating the electrocoalescence progress.
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4.4. The effect of the skew angle of the droplets
In this section, the effect of the skew angle of the droplets, indicated in Figure 7 (A) as 𝜃, on the coalescence time is investigated. The initial droplet diameter and the distance between the droplets
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are 1.5 mm and 0.5 mm, respectively. The average intensity of the electric field is 290 V/m. The oil viscosity is considered to be 0.005 Pa.s. In Figure 7 (A), the positions of the droplets at different times for an initial angle of 30 degrees and under an average electric field intensity of 290 V/m are
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plotted. Figure 7 (B) displays the effect of the angle of the tangent line between the droplets at the instant of the impact on the non-dimensional coalescence time. As can be observed in the figure, the non-dimensional coalescence time attains two minimums at the droplets skew angle of 12 and 47 degrees. The reason behind this phenomenon is the force between the droplets getting stronger
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at these angles, could be with the similar reason reported for uniform electric fields [19], such that the approaching motion becomes much faster as the drop pair becomes nearly aligned with the electric field. The total force imposed on the droplets is plotted against various 𝜃 values in Figure 7 (C). As can be observed, the impact time between the droplets reduces at any point that the total force imposed on the droplets is generated as a result of the non-uniform electrical field. On the 19
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other hand, at angles such as about 𝜃
30° where this force is at its minimum the longest impact
(A)
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time between the droplets occurs.
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(B)
(C)
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Figure 7. Droplets initial skew angle effects on their electrocoalescence. (A) Time-lapsed images of the positions of the droplets with an initial angle of 30 degrees and electric field intensity of 290 V/m. (B) Non-dimensional coalescence time regarding initial skew angle. (C)
5. Conclusion
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Total force exerted on the droplets as a function of 𝜃.
In the present paper, for the first time, a comprehensive investigation of the effects of the various parameters on the coalescence behavior of water droplet evolution in an oil phase under a novel concentric semi-elliptic non-uniform electric field configuration is carried out. After validation of
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our numerical model with experimental data reported by Mohammadi et al.[19], our results showed that the shape of applied non-uniform fields can affect the performance of electrocoalescence remarkably. It was shown that by increasing the ratio of major semi-axis to minor semi axis in a semi-ellipse medium, stronger forces were induced between the droplets leading to a quicker 21
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coalescence. Besides, the effect of the ratio of the inner radius to the outer radius of the medium, known as scale factor, was investigated. It was observed that for different scale factors there is an optimum at Sf=0.4, which could be due to the alignment of the drop pairs with electric field leading
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to the shortest coalescence time. Furthermore, it was shown that the skew angle of the droplets has an important effect on their coalescence such that the non-dimensional coalescence time attains two minimums at the droplets skew angle of 12 and 47 degrees, corresponding to the strongest forces between two droplets, plotted in another graph. In addition, other parameters such as droplets initial distance, applied voltage amplitude and continuous phase viscosity were investigated. The results confirmed that increasing initial distance prolongs the approaching time.
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It has also been proved that stronger electric fields lead to quicker approaches of the droplets. Furthermore, increasing the oil viscosity slows down the approaching movement of the drops due
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to increased flow resistance. More studies on the optimization of the non-uniform electrocoalescence, including consideration of other medium shapes and electrode locations, are
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ongoing.
Conflict of Interest:
The authors declare that they have no conflict of interest.
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Y.-K. Wei, Z.-H. Li, and Y.-F. Zhang, "Simulations of coalescence of two colliding liquid drops using lattice Boltzmann method," The Journal of Computational Multiphase Flows, vol. 9, pp. 147-156, 2017.
[15]
J. Kamp, J. Villwock, and M. Kraume, "Drop coalescence in technical liquid/liquid
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applications: a review on experimental techniques and modeling approaches," Reviews in Chemical Engineering, vol. 33, pp. 1-47, 2017.
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[16]
W. Hong, X. Ye, and R. Xue, "Numerical simulation of deformation behavior of droplet in gas under the electric field and flow field coupling," Journal of Dispersion Science and Technology, vol. 39, pp. 26-32, 2018. M. Chiesa, J. Melheim, A. Pedersen, S. Ingebrigtsen, and G. Berg, "Forces acting on water
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[17]
droplets falling in oil under the influence of an electric field: numerical predictions versus experimental observations," European Journal of Mechanics-B/Fluids, vol. 24, pp. 717732, 2005. [18]
M. Mohammadi, S. Shahhosseini, and M. Bayat, "Numerical study of the collision and coalescence of water droplets in an electric field," Chemical Engineering & Technology,
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M. Mohammadi, S. Shahhosseini, and M. Bayat, "Electrocoalescence of binary water
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droplets falling in oil: Experimental study," Chemical Engineering Research and Design, vol. 92, pp. 2694-2704, 2014. [20]
S. Mhatre and R. Thaokar, "Electrocoalescence in non-uniform electric fields: An
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experimental study," Chemical Engineering and Processing: Process Intensification, vol. 96, pp. 28-38, 2015. [21]
S. Luo, J. Schiffbauer, and T. Luo, "Effect of electric field non-uniformity on droplets coalescence," Physical Chemistry Chemical Physics, vol. 18, pp. 29786-29796, 2016.
[22]
S. Pal, A. M. Miqdad, S. Datta, A. K. Das, and P. K. Das, "Control of drop impact and
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proposal of pseudo-superhydrophobicity using electrostatics," Industrial & Engineering Chemistry Research, vol. 56, pp. 11312-11319, 2017.
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[23]
I. Z. Shabankareh, S. M. Mousavi, and R. Kamali, "Numerical study of non-Newtonian droplets electrocoalescence," Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 39, pp. 4207-4217, 2017. X. Chen, Y. Song, D. Li, and G. Hu, "Deformation and interaction of droplet pairs in a
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[24]
microchannel under ac electric fields," Physical Review Applied, vol. 4, p. 024005, 2015. [25]
J. López-Herrera, S. Popinet, and M. Herrada, "A charge-conservative approach for simulating electrohydrodynamic two-phase flows using volume-of-fluid," Journal of Computational Physics, vol. 230, pp. 1939-1955, 2011.
[26]
H. Hadidi and R. Kamali, "Numerical simulation of a non-equilibrium electrokinetic
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micro/nano fluidic mixer," Journal of Micromechanics and Microengineering, vol. 26, p. 035019, 2016.
H. Hadidi, M. K. D. Manshadi, and R. Kamali, "Natural Convection of Power-Law Fluids
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[27]
Inside an Internally Finned Horizontal Annulus," Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, pp. 1-11, 2018. M. K. D. Manshadi, H. Nikookar, M. Saadat, and R. Kamali, "Numerical analysis of non-
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[28]
uniform electric field effects on induced charge electrokinetics flow with application in micromixers," Journal of Micromechanics and Microengineering, 2019. [29]
R. Kamali and M. K. D. Manshadi, "Numerical simulation of the leaky dielectric microdroplet generation in electric fields," International Journal of Modern Physics C, vol.
[30]
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27, p. 1650012, 2016.
D. Khojasteh, M. Manshadi, S. M. Mousavi, and R. Kamali, "Droplet impact on superhydrophobic surface under the influence of an electric field," in annual meeting for
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3rd Annual International Conference on New Research Achievements in Chemistry & Chemical Engineering, Tehran, Iran, September, 2016. [31]
C. T. O'Konski and H. C. Thacher Jr, "The distortion of aerosol droplets by an electric
[32]
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field," The Journal of Physical Chemistry, vol. 57, pp. 955-958, 1953. G. Supeene, C. R. Koch, and S. Bhattacharjee, "Deformation of a droplet in an electric field: nonlinear transient response in perfect and leaky dielectric media," Journal of colloid
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PII: DOI: Reference:
S0997-7546(19)30379-6 https://doi.org/10.1016/j.euromechflu.2019.10.010 EJMFLU 103562
To appear in:
European Journal of Mechanics / B Fluids
Received date : 28 June 2019 Revised date : 13 October 2019 Accepted date : 29 October 2019 Please cite this article as: H. Hadidi, M.K.D. Manshadi and R. Kamali, Numerical simulation of a novel non-uniform electric field design to enhance the electrocoalescence of droplets, European Journal of Mechanics / B Fluids (2019), doi: https://doi.org/10.1016/j.euromechflu.2019.10.010. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Masson SAS.
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Numerical simulation of a novel non-uniform electric field design to enhance the electrocoalescence of droplets Hooman Hadidi1, Mohammad K. D. Manshadi1, Reza Kamali1,*
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1. Department of Mechanical Engineering, Shiraz University, Shiraz, Fars, 71348-51154, Iran *Corresponding Author Email: [email protected]
Abstract
The behavior of binary droplet collision, exposed to an external electric field is studied
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numerically. Based on a fundamental understanding of the role of the electric field in the coalescence of aqueous drops in an immiscible oil phase, it is possible to optimize the design and
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operation of these external stimulators. In this research, numerical modeling of electrocoalescence is implemented, using Computational Fluid Dynamics (CFD) methods. Level set method is used to evaluate water droplet evolution in an oil phase under a novel concentric semi-elliptic non-
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uniform electric field configuration. The observations show that the shape of applied non-uniform fields can remarkably alter the electrocoalescence performance in a nontrivial way, which is sensitive to the drop-medium system. In addition to the shape of the medium, some other parameters such as droplets initial distance, initial skew angle of the droplets, applied voltage amplitude and oil viscosity are analyzed in the present paper. Our results can be useful in designing
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an optimum electrode configuration, achieving an efficient electrocoalescence. Keywords: Electrocoalescence; Droplet; Multiphase; Non-uniform; Electric field; Level-set method (LSM)
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1. Introduction Coalescence of droplets, despite its simple appearance, is an important and practical phenomenon which has attracted much attention in recent years [1]. This can be observed in many natural
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processes including raindrop and cloud formation, and atmospheric aerosol circulation [2], in addition to industrial applications such as spray cooling [3], droplet-based microfluidics [4], micro-encapsulation[5], fire suppression [6], anti-icing and coating processes used in aerospace and power industries [7], etc. Some results of the collision of binary liquid drops can be categorized in four regimes: (I) coalescence after minor deformation, (II) bouncing similar to situations where a droplet impacts on a spherical solid surface at low velocities [8, 9], (III) coalescence after
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substantial deformation, and (IV) coalescence followed by separation.
The effect of viscosity on the droplet-droplet collision outcome was studied by Finotello et al. [10]
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using direct numerical simulations employing the volume of fluid method. The authors explored the role of viscosity on the transition between coalescence and reflexive and stretching separation and proposed a general phenomenological model based on the capillary number (𝐶𝑎
𝜇𝑣/𝛾, with
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𝜇 accounting for the viscosity), the impact parameter, We, and the size ratio. In another study, the Finotello’s group [11] studied complex collision dynamics of shear-thinning non-Newtonian fluids experimentally and developed a full regime map for this problem. Planchette et al. [12] presented a general modeling approach of drop fragmentation after head-on collisions that goes beyond binary collisions with a single liquid and takes into account two other types of collisions: the
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collisions between two drops of immiscible liquids and the collisions between three drops of one single liquid. This approach assimilates the colliding drops to liquid springs that coalesce, compress and relax, leading the merged drop to break up if it reaches a critical aspect ratio. The authors were able to deduce the fragmentation threshold velocity for various impact scenarios.
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Shen et al. [13] experimentally studied the coalescence between two different-sized droplets in the surrounding water. The authors identified three coalescence patterns, including liquid bridge evolution, capillary wave propagation, and pinch-off behaviors. It was concluded that the
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coalescence patterns are directly related to the propagation of capillary waves on the coalescent droplet, which is governed by the competition among the capillary force, viscous force, and inertia involved in the draining from the original droplets into the liquid bridge. Two-dimensional numerical simulations of the head-on and off-center binary collision of liquid drops are carried out using the lattice Boltzmann method by Wei et al. [14]. Their results demonstrate that the numerical results of the lattice Boltzmann method are in agreement with qualitatively experimental data in
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the same We. It is also observed that for the collision of equal-size drops, increasingly thinner drops emerge as a result of increasing We during the coalescence process. In a review paper, Kamp
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et al. [15] investigated the coalescence process in detail discussing the sequential steps during a collision of two drops that could result in coalescence, agglomeration, or repulsion. The study also presents several experimental techniques which were used for coalescence investigation at
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different levels of detail.
While passive droplet coalescence techniques do not require external energy, active techniques depend on some kind of external force or field to manipulate or control the coalescence process; such as electric field, magnetic field and optical heating. The coalescence behavior of droplets in an electric field constitutes a crucial research division of electro-hydrodynamics. The electrostatic
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coalescence technology is widely used in two-phase separation because of its high efficiency, costeffectiveness and environmental benefits [16]. The first studies conducted on electrocoalescence date back to early 20th century and the fundamental understandings acquired since then have helped to make the technology more efficient, faster and the devices more compact. Chiesa et al.
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[17] studied the forces affecting the kinematics of droplets while exposed to an electric field. They presented mathematical models for these forces and discussed about a possible application in a multi-droplet Lagrangian framework. Mohammadi et al. [18] utilized a combined model of
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electrostatic and hydrodynamic fields for simulating the electric field effects on the binary water droplets coalescence in oil. Their results revealed that the oil viscosity, the skew angle of the electric field and the initial drops distance affect the electrocoalescence. In another study, Mohammadi et al. [19] experimentally investigated collision of the binary drops. They reported in their paper that increasing initial distance results in longer approaching time. Moreover, stronger electric fields lead to faster approaches of the droplets. Furthermore, as long as the drops were
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aligned with the field, the drops had quicker coalescence compared to the situation where some skew angle existed between them. Mhatre and Thaokar [20] experimentally investigated different
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non-uniform electric field configurations such as pin-plate, quadrupole and annular electrode in electrocoalescence. Their results showed that asymmetric non-uniform field generated applying the pin-plate electrodes has better performance compared to the uniform and symmetric
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quadrupole non-uniform electric fields. Luo et al. [21] experimentally investigated phase separation under different electric field configurations. The electric field assists the merging of small droplets of water dispersed in a continuous oil phase, and by increasing the water droplet size and breaking the quasi-equilibrium of the emulsion, phase separation is accrued. Their results showed that a non-uniform electric field and the induced dielectrophoretic effect can accelerate
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the coalescence and phase separation of micro-emulsions. Pal et al. [22] numerically investigated the phenomenon of droplet collision with a charged substrate by a coupled electro-hydrodynamic model. They analyzed the phenomenon with different parametric variations like electric potential, wetting nature of the substrate, and velocity of collision as it is governed by the mutual interaction
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between the inertia, electrostatic, and capillary forces. Shabankareh et al. [23] numerically investigated the characteristics of leaky dielectric droplets in electric fields and showed that pseudoplastic droplets have more deformations in an electric field than Newtonian and dilatant
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ones, and they also present faster electrocoalescence. Although a limited number of studies have been conducted hitherto on the effects that non-uniform electric fields have on the coalescence of the droplets and despite the fact that the superiority of the non-uniform electric fields over uniform ones has been established, a significant gap still exists in this area. More specifically, a comprehensive knowledge of the exact effects that the shape of the electric field medium can have on the coalescence time of the droplets has not been established
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until now. The present study particularly aims to investigate the impact of a novel non-uniform electric field design on the coalescence of the droplets. For the first time, the impact of various
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parameters including the ratio of the semi-major axis to the semi-minor axis in the semi-elliptic configuration and the ratio of the radii in the semi-circular configuration will be taken into account. Other effective and important parameters include the initial distance between the droplets, skew
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angle of the droplets, applied voltage amplitude, and the viscosity.
2. Governing Equations and Method of Solution Electrocoalescence is a two-phase flow phenomenon in which momentum, continuity, and electric field equations should be considered to model it. Continuity and momentum (Navier-Stokes)
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equations are also solved in order to find velocity and pressure distribution in the medium. For incompressible fluids continuity equation is:
u = 0
Where u (m.s-1) is fluid velocity. Navier- Stokes equation is [24, 25]:
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(1)
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u (u .)u p . (u (u )T F t
(2)
Where 𝜌 (Kg.m-3 ) is density, p (Pa) denotes pressure, (Pa.s) is viscosity, and F (N) represents
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the sum of external volume forces. These volume forces (F) consist of surface tension and electric forces, which are: F n f e
(3)
where (N.m-1) is surface tension coefficient, 𝜅 is the fluid-fluid interface curvature, 𝛿 is the interface delta function, n is the unit normal vector to the interface pointing into the droplets, and
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fe (N) is the electric field force. This force is expressed as [24, 25]:
1 f E E 2 e e 2
(4)
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Where (C.m-3), E (V.m-1) and (F.m-1) are free charge density, the electric field and electric e permittivity respectively. is also obtained from Gauss’s law as follows: e
= E e
(5)
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and the electric field is gained from:
𝐸
∇𝑉
(6)
Where V (V) is the electric potential. For perfect-perfect dielectric fluids in a DC electric field
0 and the electric field equation becomes: e
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E = 0
Therefore, the final non-dimensional form of the Navier- Stokes equation is:
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(7)
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u * (u *.)u * t 1 1 Bo E L 1 *2 ( n ) ( E ) p * . * (u * (u * )T Re ReCa ReCa R 2
*
(8)
(L), velocity (U), time (𝑡
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Where the equation is made dimensionless based on radius of droplet (R), Length of the medium 𝑡 ∗ ), pressure ( p p * U 2 ), electric field ( E = E *E 0 ), electric
permittivity of medium ( m ), density of medium ( * m ), and viscosity of medium ( * m ). The dimensionless numbers in this equation are:
mUR 0 m
(9)
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Re
Ca
(10)
m R 0 E 02
(11)
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Bo E
mU
In the above-mentioned equations, the properties for each liquid phase should be considered separately. Level set method is one of the numerical methods for tracking the interface of two
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immiscible fluids in a two-phase fluid flow problem to make a distinction between different fluid phases. This equation is:
u . 1 t
(12)
Where 𝜙 is the level set function, γ and ϵ are the numerical stabilization parameters for re-
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initialization and determining interface thickness respectively. Level set function (𝜙 is defined "0" for one phase, and "1" for the other phase. This function is also utilized for smoothing the fluid properties across the interfaces. For instance, viscosity through the media is obtained from:
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𝜇
𝜇
𝜇
𝜇 𝜙
(13)
Where the notations "d" and "m" represent droplet and surrounding fluid, respectively. The mentioned coupled equations are solved by COMSOL Multiphysics 5.1. This is a powerful
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software which provides the context for scholars to simulate different physics of a problem simultaneously [26, 27]. This software is widely employed to model complex phenomena because this software is capable of solving coupled system of partial differential equations (PDEs) with different algorithms. Herein, MUltifrontal Massively Parallel sparse direct Solver (MUMPS) is used for the simulations. This algorithm is based on lower–upper (LU) factorization which used for solving large linear algebraic equation systems. This solver is capable of out-of-core solution
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storing which decreases the computational costs by providing more memory than the available
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memory on the computer [28].
3. Validation
Electrohydrodynamic (EHD) method capabilities for droplet manipulation have attracted the
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attention of our group. Previously, three numerical investigations about controlling microdroplet generation, non-Newtonian droplet electrocoalescence, and droplet impact in DC electric fields have been performed by our team [23, 29, 30]. Results of these studies were verified by theoretical and numerical findings. In addition to that, the numerical results of the present study are compared with theoretical data available in the literature.
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O’Konski and Thacher [31] developed an analytical model to define deformation of perfect dielectric drops suspended in perfectly insulating media. This model is [32]: 𝐷
𝐵𝑜
9 𝑆 1 16 𝑆 2
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𝐷 Where 𝑆
𝑊 𝑊
𝐵 𝐵
and in the deformation parameter equation (D), W is the polar half-axis and B is
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the equatorial radius of the spheroid. According to this equation, droplet always deform in the direction of the electric field [32]. After the modeling of the droplet deformation, the obtained results are compared with the analytical data (see Figure 1 ). In all of the simulation in this paper
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the time-step was considered to be 0.001 (s).
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Figure 1. Validation of the numerical model. (A) Schematic representation of the deformation of a perfect dielectric droplet suspended in a perfect dielectric immiscible liquid. (B) Comparison of the numerial results and the theoretical data of O’Konski and Thacher [31]
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Comparison of our numerical results with the theoretical data implies that the present numerical study has the reasonable capability of predicting droplet behaviors under an applied electric field.
4. Results and Discussion 9
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Figure 2 (A) represents a schematic of the geometry used in the present study, along with its boundary conditions. The medium is made of two concentric semi-ellipses with b2/a2= b1/a1=b/a. An electrode with an electric potential of 𝑉 is placed on the inner curved surface. and the outer
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curvature is set as the ground. For boundary conditions of momentum equation (eq. 4), on the specified horizontal lines indicated in the figure, an equal pressure, p = 0, is applied. Moreover,
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the curved lines are considered as walls with no-slip condition.
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(A)
(B)
(C)
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Figure 2. (A) Computational domain with electrical boundary conditions. (B) The generated tetrahedral mesh. (C) Grid independence study: non-dimensional coalescence time versus grid
pro of
number.
The unstructured triangular meshing tool provided in COMSOL is used to generate the mesh. This algorithm first applies on the domain boundaries. Then, this mesh is used to mesh the domain with isotropic tetrahedral with reasonable element aspect ratio.
Tetrahedral meshing algorithm first applies a mesh on all of the surfaces of an object. This mesh
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is then used to “seed” the volume mesh from which tetrahedral elements “grow” elements inwards. As these tetrahedral elements intersect, their sizes are adjusted with the objective of keeping the
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elements as isotropic (similar edge lengths and included angles) as possible and to have reasonably gradual transitions between smaller and larger elements.
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A view of the generated grid around the two droplets is presented in Figure 2 (B). In order to ensure the accuracy of the obtained numerical results, a mesh independence study has to be conducted prior to performing the simulations of the main cases. For this purpose, the nondimensional coalescence time of the droplets with diameters 1.5 mm and an initial distance of 0.7 mm, under an average electric field density of 1.95
10 𝑉/𝑚 was chosen as the parameter based
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on which the mesh independence study was conducted. For instance, for the case with b/a = 1.6, different meshes with grid numbers ranging from 4308 to 47453 were tested (Figure 2 (C)). As can be observed in Figure 2 (C), the difference in non-dimensional coalescence time between the meshes with 35654 and 47453 grids is negligible (about 2.5%). Therefore, for the mentioned case,
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the simulations were performed using the mesh with 35654 grids, with minimum element size in the order of 0.01 mm. Similar procedure was pursued for all the cases simulated in the present
4.1. The effect of the b/a ratio
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study.
Figure 3 displays electric potential for the geometry with b/a = 1.6 at a specific time (t = 0.02s). As can be observed, since the 2500V electrode has been placed on the inner wall and the electric potential of the outer wall is zero, the distribution of the electric potential intensity varies from
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2500 V on the inner wall to zero on the outer wall.
Figure 3. Distribution of the electric potential for the geometry with b/a = 1.6.
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Figure 4 (A) depicts the variations in the distance of the droplets with respect to each other and the contours of the electric field intensity at different times for b/a = 1.6. The initial diameter of the droplets and their distance from each other are 1.5 mm and 0.7 mm, respectively. The average 10 V/m. The oil viscosity is equal to 0.2 Pa.s. The droplets
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intensity of the electric field is 1.95
get polarized under the action of the electric field; therefore, polarized charges are generated at both ends of the droplets. The droplets are attracted to each other and are affected by the dipole force until they make contact with each other. After getting into contact, they tend to merge under the action of interfacial tension. Hence, the final coalescence takes place at the time 0.2 s and a single large drop is formed. At the same time, the dissimilar charges become neutralized on those
electric field of the polarization[16].
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parts of the droplets that are in contact, which leads to a total neutralization of the droplets by the
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According to Figure 4 (B), by increasing the b/a ratio, the non-dimensional coalescence time reduces. The reason behind this observation is the stronger forces induced by the non-uniform electric field that affect the droplets, which could be because of enhancement of the drop pairs
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alignment under the imposed non-uniform electric field [19]. This fact is illustrated in Figure 4 (C). The figure depicts the total force imposed on the droplets at t* = 43.4 for various values of b/a. The force imposed on the droplets is generated as a result of the non-uniform electric field and is augmented with an increase in b/a. As a consequence, the droplet’s non-dimensional coalescence
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time is reduced.
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(A)
(B)
(C)
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Figure 4. b/a ratio effects on electrocoalescence. (A) Variations in the distance of the droplets with respect to each other at different times for b/a = 1.6, (B) Variation of the non-dimensional coalescence time with b/a, (C) Variation of the total non-dimensional force imposed on the
4.2. The effect of the radii ratio (scale factor)
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droplets with b/a at t*=43.4.
In the following, the effect of the ratio of the outer radius to the inner radius on the coalescence of the droplets in the case of b/a=1 has been investigated. This ratio is called “scale factor” (Sf= a1/a2=
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in Figure 5 (A) to (C), respectively.
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b1/b2= R1/R2). The schematics of the configurations with Sf values of 0.1, 0.3 and 0.6 are presented
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Figure 5. The effect of scale factor on the coalescence of the droplets with Sf values of (A) 0.1, (B) 0.3, and (C) 0.6 at different times. Figure 6 (A) shows the time-dependent variation of the coalescence time with Sf. The initial
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droplet diameter and the initial distance between the droplets are 1.5 mm and 0.5 mm, respectively. The average intensity of the electric field is 290 V/m. According to this figure, an optimal Sf exists at Sf=0.4 such that the shortest coalescence time is observed to occur at this scale factor and the longest impingement time is obtained at Sf = 0.6. This is due to the non-uniformity of the electric field, such that its direction imposed on the droplets differs for various scale factors and
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by varying the scale factor, the alignment of the electric field with drop pairs changes, leading to different coalescence time[19]. Therefore, it can be concluded that the magnitude of the total force imposed on the droplets at Sf = 0.4 is larger than all other cases. In addition, this can be seen in
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Figure 6 (E), illustrating the total force imposed on the droplets for various sf values at t* = 60.8. As demonstrated in this figure, the minimum imposed forced occurs at Sf = 0.6, leading to the
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longest non-dimensional coalescence time.
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(B)
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(A)
(D)
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(C)
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(E)
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Figure 6. The variation of non-dimensional coalescence time with (A) scale factor (R1/R2) (B) the initial distance of the droplets (C) the applied voltage amplitude and (D) the viscosity. (E)
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The variation of the total force exerted on the droplets with Sf.
4.3. The effects of the droplets initial distance, the electric field strength and viscosity on coalescence time
Figure 6 (B) displays the time-dependent variations of the non-dimensional coalescence time with
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the initial distance of the droplets. The initial diameter of the droplets is 1.5 mm. The average intensity of the electric field is 290 V/m and the oil viscosity is 0.005 Pa.s. Results illustrate that, as expected, the shorter initial distance at a given constant drop size results in quicker coalescence because of the stronger force between the drops. The magnitude of the force imposed on the droplets is dependent on the external electric field. Therefore, the intensity of the electric field is 18
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one of the most important parameters in electrocoalescence[19]. As can be observed in Figure 6 (C), by increasing the electric field intensity, the bipolar force between the droplets increases and the non-dimensional coalescence time is reduced. The effect of the variations in the viscosity on
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the non-dimensional coalescence time is shown in Figure 6 (D) for an initial diameter and initial distance between the droplets equal to 1.5 mm and 0.5 mm, respectively. This figure indicates that increasing the oil viscosity causes more resistance for droplets movements leading to decelerating the electrocoalescence progress.
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4.4. The effect of the skew angle of the droplets
In this section, the effect of the skew angle of the droplets, indicated in Figure 7 (A) as 𝜃, on the coalescence time is investigated. The initial droplet diameter and the distance between the droplets
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are 1.5 mm and 0.5 mm, respectively. The average intensity of the electric field is 290 V/m. The oil viscosity is considered to be 0.005 Pa.s. In Figure 7 (A), the positions of the droplets at different times for an initial angle of 30 degrees and under an average electric field intensity of 290 V/m are
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plotted. Figure 7 (B) displays the effect of the angle of the tangent line between the droplets at the instant of the impact on the non-dimensional coalescence time. As can be observed in the figure, the non-dimensional coalescence time attains two minimums at the droplets skew angle of 12 and 47 degrees. The reason behind this phenomenon is the force between the droplets getting stronger
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at these angles, could be with the similar reason reported for uniform electric fields [19], such that the approaching motion becomes much faster as the drop pair becomes nearly aligned with the electric field. The total force imposed on the droplets is plotted against various 𝜃 values in Figure 7 (C). As can be observed, the impact time between the droplets reduces at any point that the total force imposed on the droplets is generated as a result of the non-uniform electrical field. On the 19
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other hand, at angles such as about 𝜃
30° where this force is at its minimum the longest impact
(A)
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time between the droplets occurs.
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(B)
(C)
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Figure 7. Droplets initial skew angle effects on their electrocoalescence. (A) Time-lapsed images of the positions of the droplets with an initial angle of 30 degrees and electric field intensity of 290 V/m. (B) Non-dimensional coalescence time regarding initial skew angle. (C)
5. Conclusion
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Total force exerted on the droplets as a function of 𝜃.
In the present paper, for the first time, a comprehensive investigation of the effects of the various parameters on the coalescence behavior of water droplet evolution in an oil phase under a novel concentric semi-elliptic non-uniform electric field configuration is carried out. After validation of
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our numerical model with experimental data reported by Mohammadi et al.[19], our results showed that the shape of applied non-uniform fields can affect the performance of electrocoalescence remarkably. It was shown that by increasing the ratio of major semi-axis to minor semi axis in a semi-ellipse medium, stronger forces were induced between the droplets leading to a quicker 21
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coalescence. Besides, the effect of the ratio of the inner radius to the outer radius of the medium, known as scale factor, was investigated. It was observed that for different scale factors there is an optimum at Sf=0.4, which could be due to the alignment of the drop pairs with electric field leading
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to the shortest coalescence time. Furthermore, it was shown that the skew angle of the droplets has an important effect on their coalescence such that the non-dimensional coalescence time attains two minimums at the droplets skew angle of 12 and 47 degrees, corresponding to the strongest forces between two droplets, plotted in another graph. In addition, other parameters such as droplets initial distance, applied voltage amplitude and continuous phase viscosity were investigated. The results confirmed that increasing initial distance prolongs the approaching time.
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It has also been proved that stronger electric fields lead to quicker approaches of the droplets. Furthermore, increasing the oil viscosity slows down the approaching movement of the drops due
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to increased flow resistance. More studies on the optimization of the non-uniform electrocoalescence, including consideration of other medium shapes and electrode locations, are
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ongoing.
Conflict of Interest:
The authors declare that they have no conflict of interest.
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