Electric Field Driven Separation of OilâWater Mixtures - American

Subscriber access provided by RMIT University Library

Article

Electric field driven separation of oil-water mixtures: Model development and experimental verification Wilma Wallau, Christiane Schlawitschek, and Harvey Arellano-Garcia Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b03876 • Publication Date (Web): 31 Mar 2016 Downloaded from http://pubs.acs.org on April 1, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Industrial & Engineering Chemistry Research is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Electric field driven separation of oil-water mixtures: Model development and experimental verification Wilma Wallau1 , Christiane Schlawitschek2 , Harvey Arellano-Garcia3∗ 1

2

Bundesanstalt für Materialforschung und -prüfung, Berlin, Germany Brandenburgische Technische Universität Cottbus-Senftenberg, Germany

3

Department of Chemical and Process Engineering, University of Surrey, Guildford, Surrey GU2 7XH, UK ∗ cooresponding author: [email protected]

Abstract Electrocoalescence of aqueous droplets in oil emulsions is commonly contemplated for enhancing separation. High voltage electric fields can induce charges to drops evoking merging of adjacent droplets. The newly formed larger drops then sink faster in gravitational common settlers. Therefore, separation performance of an electrostatic coalescer is strictly linked to characteristics of the electric field and properties of the liquid-liquid system. In this work, the coalescence performance of water droplets sinking in dodecane at a pulsed DC electric field is investigated. An experimental set-up allowing the simultaneous injection of similar sized drops, setting of voltage and pulsation frequency, and particle tracking at high frame rate and resolution is designed. The generated data is used to check the validity of modelling approaches for drag, dipole-dipole forces and film-thinning. Furthermore, CFD simulations are carried out using a volume of fluid method tracking the interfaces between the two phases. 1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Keywords: drop-drop coalescence, electrocoalescence, water-in-oil, dipole-dipole force, film-thinning, pulsed dc electric field, drag, modelling.

Introduction For separating water-in-oil emulsions chemical demulsifiers and gravitational settlers are very common in industry (23). Application of the latter practises the density difference between the two liquids. The dispersion is simply allowed enough time and space for the water drops to sink and form a continuous phase at the bottom of the tank, which can then be drained. The settling velocity of dispersed drops determines the residence time and thereby the unit size. Assuming Stokes’ Law (CD = 24/Re) for the drag coefficient of rigid spherical particles at low Reynolds numbers (Re < 0.1), the terminal velocity of spherical particles exhibits quadratic growth with the particle diameter. Therefore, enhancement of separation performance of gravitational settlers can be achieved by promoting drop-drop coalescence. Electrocoalescence is a phenomenon which can be exploited for this purpose. It describes a process by which coalescence of an electrically conducting phase dispersed in a liquid dielectric phase is achieved by applying an electric field between a high voltage and a ground electrode. The continuous phase with a low dielectric constant, which is characteristic of hydrocarbons, functions as an electrical insulator. Water droplets in high voltage electric fields can either be drawn towards each other and eventually coalesce due to electric field induced electrostatic forces between non-charged, polarised particles, so called dipole coalescence, or due to electrophoretic motion of charged water drops. It shall be mentioned that oppositely charged drops do not necessarily coalesce. As shown by Ristenpart et al. (40) a critical electric field strength exists above which a migrating water droplet in silicone oil does not coalesce with a resting, grounded water drop, instead it bounces off and oscillates beetween the two electrodes transferring charge. Electrocoalescence can help enhancing the separation performance of usual gravity set2 ACS Paragon Plus Environment

Page 2 of 49

Page 3 of 49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


tlers, increasing the space-time yield. Also, very finely dispersed aqueous phases, sometimes forming practically stable micro-emulsions, demand challenging separation units, often consisting of multiple stages. Combinations of some separation techniques in one apparatus or the serial application of gravitational, centrifugal and membrane separation, demulsification by pH adjustment, filtration, heating or chemical treatment can be practical for achieving satisfying separation performances, too (21), (2). For the design and optimisation of separation units suitable models need to be available. Although high voltage electric fields are widely applied in water-in-oil emulsion breaking, no quantatively accurate and predictive modelling approach is yet available. This can lead to inefficient design of electrocoalecers (1). To predict the effectiveness of a separation process, the set of input variables and their effect must be known. The type of electric field (DC or AC), pulsation frequency, wave form, electric field strength, electrode geometry, drop size distribution and water cut of the emulsion, additives, properties of substances such as electric permittivity and conductivity, density and viscosity are known to be influential (20), (7). Furthermore, the geometry of the apparatus affects the two-phase-flow regime and the probability of random drop-drop collision within the turbulent flow, which can then increase the coalescence effectiveness (17). The effects of crucial operation parameters such as pressure and temperature are rarely investigated. Eddy suggested (16) vacuum conditions for effective operation of his petroleum dehydrator. Kwon et al. found the performance of a treater combining centrifugal and electrostatic separation principles to break a water-in-motor oil emulsion exposed to a pulsed DC field to increase with temperature (T = [40, 60, 75]◦ C) (28) . Increased density difference and reduced viscosity of the continuous phase are commonly given as a reason. Also, Chiesa et al. could experimentally verify simulation results indicating enhanced drop-interface coalescence with increased temperature and reduced continuous phase viscosity, respectively (9). 3 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Among others Bailes and Stitt (3) identified a pulsation frequency at which water-in-oil emulsions showed optimum separation performance when using insulated electrodes. However, investigations on the effect of frequency on coalescence of binary droplets are rare (31). The continuous phase film captured between two very close drops needs to drain prior to coalescence. The mechanism often referred to as film-thinning means a resistance to dropdrop close-up. It is commonly believed to be the controlling mechanism of coalescence of aqueous drops in viscous oil (19). However, increased electric field strength can lead to instantaneous drop-drop coalescence, which is believed to be due to electrohydrodynamic instability and dielectric breakdown of the inter-drop liquid (50), (21). Murdoch and Leng modelled dropdrop collision involving the effect of double-layer repulsion and van der Waals attraction (34). Williams lists the interactive forces between sphere and plane as well as two spheres, with the spherical interfaces being represented by parabolic curves (47). Depending on the viscosity ratio between dispersed and continuous phase, Davis et al. modelled a film-thinning force for two unequally sized drops (15), which could be verified experimentally by Chiesa et al. (10) appending a model by Vinogradova (45). The actual coalescence mechanism is an important aspect of the physics of drop-drop merging. The temporal development of the neck between two coalescing drops in vacuum was studied by Paulsen et al. (38) and recently by the same group applying silicon oil for a continuous phase (38). The authors extend existing possible dynamic regimes by an inertially limited viscous (ILV) regime that they claim always exists in the early stage of coalescing for a certain range of Ohnesorg (Oh) number, especially Oh < 1. In order to investigate the basic mechanisms of electrocoalescence, drop-drop coalescence is focussed on in this work. Previous experimental investigations mostly adress the coalescence of drops, resting on a surface or coalescing vertically. Chiesa et al. could generate experimental data of horizontally coalescing water drops to validate their simulation results (10). Mohammadi et al. (33) then altered the hydrodynamic model implementing the Volume of Fluid 4 ACS Paragon Plus Environment

Page 4 of 49

Page 5 of 49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


High voltage power supply

PC

relay

10 MΩ

frequency, waveform

Pico Scope

current, voltage to drive the relay

CCD camera

telecentric LED light

Figure 1: Scheme of experimental set-up

method to solve the water-oil interface and to investigate the effect of skew angle, initial interdrop distance, size of drops, continuous phase viscosity and electric field strength. Most recent experimental data has been presented by the same group of researchers who injected drops of distilled water (3.0 w% of NaCl) to let them sink in transformer oil while applying sinusoidal AC voltage (32). However, fairly small resolution and frame rate result in a reduced amount of frames and less precise particle tracking. This work aims for the generation of high resolution experimental data to facilitate verification of electrocoalescence modeling. A CFD and a straightforward modeling approach are suggested and tested.

1 Experimental set-up Figure 1 shows a sketch of the experimental set-up. The motion of two small water droplets is tracked, while they are being affected by a horizontally applied electric field in dodecane. Constant DC voltage up to U = 20 kV is provided by a bench-top precision power supply with primary switch high-voltage technology (Heinzinger, LNC 20000-3). Pulse generation is implemented installing an oscilloscope with respective software to set a frequency and wave-

5 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

form. Its signal is transmitted to a reed relay, which is built-in between the high voltage output of the power supply and the high voltage electrode. In order to implement an intrinsically safe experimental set-up, a high resistance (10 M Ω) is assembled in series right next to the high voltage output to limit the current. Thereby, electric shocks because of unqualified operation are avoided to be hazardous. Usage of a transparent perspex cell (inner dimensions: 40 mm× 40.2 mm× 84 mm, 6 mmthick with a lid of the same material) allows the optical measurement. Two insulated copper wires, attached to the bare copper electrodes (40 mm × 13 mm × 0.8 mm), are fastened to the lid with insulating screws to hold the electrodes in place. The inter-electrode distance takes die = 9.5 mm, with parallel arrangement of the electrodes being prioritised over an exact observation of die . A high speed camera and a telecentric lens with a fixed focal plane are installed to capture the inter-electrode space. Data acquisition is conducted implementing a high-performance frame grabber. To captue and to evaluate images, LabView routines are customised. Frames can be seen on-line for monitoring, eased electrode arrangement and injection. The size and shape of the field of view and the frame rate can be varied. The organic phase n-Dodecane for synthesis is supplied by Merck Millipore with purity ≥ 99 % and density of ρ = 0.748 . . . 0.749 g/cm3 . The chosen set of substances exhibits a large difference in density evoking fairly high terminal velocities of the falling drops. The amount of time, the drops are being exposed to the electric field needs to be sufficient for dropdrop coalescence to occur within the field of view. Small droplet sizes bring reduced vertical velocities which facilitates the choice of a smaller field of view, i.e. a higher resolution and frame rate. An injection device aiming at a simultaneous injection of two same-sized and very small drops was the main characteristic of the experimental setup. For this purpose, two agglutinated 6 ACS Paragon Plus Environment

Page 6 of 49

Page 7 of 49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


hypodermic syringes are utilised, each with a blunt-tip-needle (stainless steel, gauge 34, dout = 0.16 mm, din = 0.06 mm) mounted to them, facilitate a manual injection. The needles are led through a metal capillary for parallel and close arrangement. Thus, the inter-drop distance relative to the drop radius remains small to allow coalescence within the field of view. Despite using the smallest available needle gauge, the system characteristics prohibit the controlled injection of small drops of d ≈ 100 µm. As suggested by Eow et al. the needles are grounded, for injection of uncharged drops to avoid electrophoretic effects (18). Through a hole in the very centre of the lid, the two needle tips are first dipped into the continuous phase of dodecane, positioned in parallel with the electric field by manually arranging a three-dimensional positioning device holding the syringes. After pushing the plungers until the pending drops of deionised water have reached the demanded size, the injection device is to be pulled upwards for the drops to detach. Shortly after, when the drops have passed the rubber-insulated upper ends of the electrodes (∆h ≈ 8 mm), a pre-set voltage is turned on at frequencies up to f = 50 Hz of a square dc pulse. A LabView routine allows the saving of previously taken images, i.e. a set number of frames is loaded from a ring-buffer when the capture is triggered, which is ideally right after coalescence has occurred. Sizes and positions of the tracked particles are then converted according to the calibration result 36 px =1 b mm. Drop velocities are calculated from the difference quo-

tient, i.e. from the difference of drop positions of the previous and next time step over two time intervals. While experimenting a range of challenges emerged. Electrophoretic effects, meaning horizontal displacement of single drops, occurred especially when contaminated particles and space charges accumulated in the cell and static charging of the perspex cell was induced. Since the needles are grounded, proper removal of the injection device from the inter-electrode space is crucial to avoid an irritation of the electric field. Even very small water drops sink rather fast in 7 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

dodecane, entailing a very short time interval within coalescence needs to occur. Thus, drawing the syringes, voltage switch-on and image capturing have to happen in quick succession. Thus, reproducibility of an experiment demands for accuracy in initial positioning of the needle tips and drop sizes. However, further improvement of the manual injection method applied in this work will allow the repetition of an experiment.

2 Model development A system of standardised liquids namely dodecane-deionised water is chosen, for the resulting experimental data being of use in further research. Two small, uncharged droplets of deionised water are introduced to a stagnant continuous phase of dodecane. A horizontally applied pulsed DC high voltage electric field induces electrophoretic forces which ideally evoke drop-drop coalescence. For modelling the behaviour of the drops, two different approaches are pursued in this work, namely, a CFD model and a simplified approach disregarding mobility of the interface. Electrohydrodynamics of single droplets and two facing droplets are much more complex and advanced than outlined in this work, see for instance (12) and (13). However, bulk density of free charges and surface density charges are not considered here.

2.1 Computational fluid dynamics model An OpenFOAM-based (package version 2.1.1) simulation of coalescence of a binary water droplet in stagnant oil system with an applied electric field between two planar electrodes is carried out. Fluid dynamics and electrodynamics are described by Navier-Stokes and Maxwell equations. The resulting set of partial differential equations is solved applying a finite volume method for spacial discretization. Thereby, the dynamic behaviour of the w/o-interface can be investigated. This is of particular interest at very close proximity of the drops when drop 8 ACS Paragon Plus Environment

Page 8 of 49

Page 9 of 49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


deformation, film drainage and rupture occurs. For an incompressible Newtonian fluid, the hydrodynamic equations, i.e. mass and momentum balance, are given by ∇·U =0 and ρ

∂U + (U · ∇) U ∂t

(1)

= −∇p + µ∇2 U + ρg + F sf + F el

(2)

for the velocity vector U , mass density ρ, pressure p, viscosity µ, and gravitation vector g, respectively. Here F sf is the surface stress and F el is the force vector caused by the electric field. From the expressions κ = −∇ ·

n |n|

,

n = ∇α ,

(3) (4)

with κ, n, and α denoting the local curvature of the interface, outward pointing normal vector to the interface and phase fraction parameter, the surface stress is found to satisfy F sf = σκn ,

(5)

in which σ is the interfacial tension. The advection equation of the phase fraction parameter is given by ∂α + ∇ · (U α) = 0 . ∂t

(6)

The phase fraction parameter is the volume fraction of the dispersed medium in the considered cell. Fluid properties, such as density and viscosity, are determined in the transition region (0 < α < 1) via weighted arithmetic averaging with the phase fraction parameter. The electrostatic field E is defined as the gradient of the electric potential φ (5), (35), (44): E = −∇φ . 9 ACS Paragon Plus Environment

(7)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 49

A Poisson equation is used to represent the voltage equation, which describes how the electric field is affected by the electric charge density ρel (22), (42), (44):

∇ · (ε∇φ) = ρel

,

(8)

with the absolute permittivity ε being obtained from harmonic averaging weighted according to the phase fraction. Assuming no free charges in the bulk phases, ρel becomes zero. The electric field is coupled to the Navier-Stokes equation via the Maxwell stress tensor M, with its divergence giving the electric force (42), (22): F el = ∇ · M .

(9)

The Maxwell stress tensor is represented by

1 M = ε EE − (E · E) I 2

,

(10)

where I denotes the identity matrix. Rewriting equation 9 including equation 10 and assuming ρel = 0 obtains: 1 ∇ · M = − (E · E) ∇ε , 2

(11)

From the above equation it is clear that only dielectrophoretic attractive forces are considered in the numerical simulations. For a w/o system, the water drops move in the direction of greatest field intensity because of the larger permittivity of water. There is a high electric field between two drops close to each other since the droplets disturb the homogeneous electric field and compress the electric potential between them.

2.2 Simplified model A simpler approach can be chosen for modelling two-drop behaviour. It is based upon the model presented by Melheim and Chiesa et al. (30), (10), which simply consists of a momentum 10 ACS Paragon Plus Environment

Page 11 of 49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


balance for each drop involving gravity Fg , buoyancy Fb , drag Fd , film-thinning Ff , and the electric field induced force Fe (with a radial and a tangential component), which are specified below. Figure 2 shows the two spherical drops on a plane in parallel to the direction of the electric field. Spherical drop shape is supposed, which is appropriate especially for small drop sizes and high interfacial tension. Drop deformation is therefore not taken into account. For the sake of simplicity, coalescence is regarded to be happening as soon as the surfaces of both drops touch, which means the inter-drop distance would take values equal to or smaller than the sum of both drop radii. x

1

1

1

E

1

Ff,y, Fb, Fe,t, Fd,y

y

2

r1

2

Fb, Fd,y 1 Fe,r

1 1 Ff,x , Fd,x

r2

θ

drop 1

2 Fe,r

2 2 Ff,x , Fd,x

drop 2 Fg1 2 2 Ff,y , Fe,t , Fg2

Figure 2: Sketch of the forces, which are applying to the drops

Equations 12 and 13 are obtained for both drops (k = 1, 2) in x- and y-direction, respectively. X X

Fy :

Fx :

dvx = Fe,r − Ff,x − Fd,x dt

(12)

dvy = Fg − Ff,y − Fb − Fe,t − Fd,y dt

(13)

m

m

Gravity and buoyancy only apply in y-direction and can be summarised in the external body 11 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 49

force: Fg − Fb = V g (ρd − ρc ) = 4/3πr3 g (ρd − ρc )

.

(14)

Drag When specified for multidimensional motion of spherical particles in a stagnant fluid, the drag force reads Fd = −CD

ρc ρc 2 Av = −CD πr2 |v| · v 2 2

,

(15)

which leaves the drag coefficient CD to be specified. Depending on the model depth, various parameters can influence the drag coefficient e.g. velocity and diameter of the drop, density and viscosity of the continuous and dispersed phase, interfacial tension, interfacial tension gradient over the drop surface and drop shape. Bozzano and Dente proposed a comprehensive model for the drag coefficient to describe the motion of drops and bubbles in quiescent liquids (6). The implementation of the dimensionless Morton and Eötvös number involves in particular the effect of drop deformation and the properties of interfacial tension and viscosity of the dispersed phase. The Morton number Mo =

gµ4c ∆ρ 3 ρ2c γcd

(16)

Eo =

g∆ρd2 γcd

(17)

and the Eötvös number

characterise the drop shape. Bozzano and Dente could verify the correlation √ 48 1 + 0.25Re 3/2 + µc /µd 1 + 12 · M o1/3 √ CD = + Ff ric · · · Re 1 + 0.25Re + 1 1 + µc /µd 1 + 36 · M o1/3

,

(18)

for multiple air-liquid and liquid-liquid systems (6). The introduced friction factor Ff ric is a function of the Eötvös and the Morton number denoted in two sections: Ff ric,1 = 0.9

Eo3/2 1.4 · (1 + 30 M o1/6 ) + Eo3/2

, f or


Ff ric,1 ≥ 0.45

(19)

Page 13 of 49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and Ff ric,2 = 0.45

, f or

Ff ric,1 < 0.45

.

(20)

It takes the reduced internal circulation inside drops compared to bubbles into consideration (6). Effects of internal circulation and mobile interfaces can be neglected for very small liquid particles, which allows the application of drag models originally developed for rigid spheres. Clift et al. give an overview on correlations for the drag coefficient of solid spheres, some valid within large ranges of the Reynolds number (11). Schiller and Nauman presented the correlation CD =

24 1 + 0.15 · Re0.687 Re

,

(21)

for Re < 800, with a deviation of +5% and −4%. Lapple suggested: CD =

24 1 + 0.125 · Re0.72 Re

,

(22)

if Re < 1000 applies, with an accuracy of −4% and +4.3%; as referred in (11). Virtual mass effect The virtual mass effect, also known as added mass effect, takes the acceleration of surrounding continuous phase fluid into account. It therefore comes into play when particle motion is transient:

1 ρc dv dv − πd2 ρc CD v 2 − KV M πd3 ρc m = mg 1 − dt ρd 8 dt

,

(23)

with the last term of the momentum balance depicting the added mass effect. For solid, spherical particles the amount of accelerated surrounding fluid equals about half of the particle volume (KV M = 0.5), whereas experiments with rising bubbles in water give values of KV M = 0.25, see (27). Deformation of fluid particles and lower wall slip decrease the shear forces transferred to the continuous fluid and thereby KV M .



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 49

Electric forces In a homogeneous electric field the electric net force on an uncharged drop is zero. In an inhomogeneous field, on the other hand, the field strength surrounding the droplet differs and the position of the drop changes. However, the presence of a drop in a homogeneous field brings inhomogeneities in its vicinity. Binary close polarised drops are effected by a mechanism commonly referred to as dielectrophoresis, meaning attraction or repulsion depending on drop positioning relative to the direction of the electric field (19). In this work, the behaviour of a system of two close drops is examined. The tangential force Fe,t provokes the alignment of the two drops with the electric field direction while the radial component Fe,r mainly evokes drop-drop coalescence (10). The point-dipole approximation describes the electrostatic attraction of two spherical, conductive spheres. It could only be validated for relatively large distances between the particles (ddd /rd > 0.1) (30). According to Siu et al. (43), it is furthermore difficult to validate experimental results in the case of dielectric constants of both media being significantly different. A successor model often called dipole-induced-dipole model (DID) takes multipolar interactions into account, and is therefore also applicable to polydisperse systems (48) with the radial force Fdid,r =

~ 2 r3 r3 12πβ 2 ǫc ǫvac |E| 2 1 (3K1 cos2 θ − 1) 4 ddd

(24)

and tangential force reading Fdid,t =

~ 2 r3 r3 12πβ 2 ǫc ǫvac |E| 2 1 K2 sin(2θ) ddd 4

.

(25)

The coefficient β is described by β=

ǫd − ǫc ǫd + 2ǫc

(26)

and θ depicts the angle between the direction of the electric field and the relative droplet position


Page 15 of 49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


vector (10), (43). The introduced coefficients K1 and K2 satisfy βr13 ddd 5 βr23 ddd 5 3β 2 r13 r23 (3ddd 2 − r12 − r22 ) K1 = 1 + + + (ddd 2 − r22 )4 (ddd 2 − r12 )4 (ddd 2 − r12 − r22 )4

(27)

and βr23 ddd 3 3β 2 r13 r23 βr13 ddd 3 + + K2 = 1 + 2(ddd 2 − r22 )3 2(ddd 2 − r12 )3 (ddd 2 − r12 − r22 )3

.

(28)

For large distances ddd between the drop centres, the coefficients approach unity and the pointdipole model applies. Davis(14) derived an analytical solution for the dipole-dipole forces applied on two drops of arbitrary charge by integrating the electrostatic pressure, induced by the non-uniform electric field in the vicinity of the drops, over the drop surface. Radial and tangential force in case of uncharged drops read Fr = 4πǫc ǫvac |E|2 r22 (L1 cos2 θ + L2 sin2 θ)

(29)

and Ft = 4πǫc ǫvac |E|2 r22 L3 sin θ

,

(30)

as referred to in (9). The coefficients L1 , L2 , L3 depend on the two radii and the inter-drop distance h = ddd − r1 − r2 . Their calculation in bi-spherical coordinates is necessary for each time step and evokes undesirably high computation time when simulating multi-drop-systems. Davis (14) published values for the coefficients L1 , L2 , L3 for the varied ratio h/r2 from 10 to 0.001 in logarithmic decades. For equal drop sizes the progression of the coefficients is fitted by the following equations: L1 = 0.2531 (h/r1 )−0.7907 − 0.1161

,

L2 = 0.007107 exp (−5.55 · h/r1 ) − 0.1113 exp (−1.214 · h/r1 ) 15 ACS Paragon Plus Environment

(31) (32)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 49

and L3 = −0.206 exp (−64.32 · h/r1 ) − 0.2726 exp (−1.942 · h/r1 )

,

(33)

each with R2 > 0.99. Melheim and Chiesa et al. assumed constant values at small inter-drop distances h < r1 (30), (10). However, correlations 31, 32 and 33 are applied here. Due to its accuracy, the analytical solution by Davis (14) is commonly executed for verification of alternative expressions of the electrostatic forces.

Film-thinning As the drops closely approach each other, the so-called film thinning, also referred to as film drainage, is believed to become a controlling mechanism to coalescence (34). Just before the collision of two drops, some continuous liquid, which is trapped in the inter-drop gap, has to be displaced to the sides for a bridge to be formed between the drops. The film-thinning force describes the resistance put up against coalescence by this mechanism. Melheim and Chiesa et al. (30), (10) combine a model presented by Davis et al. and Vinogradova (15), (45). When the droplets approach each other such that the gap is much smaller than the reduced radius (h

Electric Field Driven Separation of OilâWater Mixtures - American

Recommend Documents

Electric Field Driven Separation of OilâWater Mixtures - American