Movement of dispersed droplets of W/O emulsion in a uniform DC electrostatic field:Simulation on droplet coalescence☆

Jun Zhang

1 Cleaning Combustion and Energy Utilization Research Center of Fujian Province,China

2 Fujian Province Key Lab of Energy Cleaning Utilization and Development,School of Mechanical Engineering,Jimei University,Xiamen 361021,China

Keywords:DC electrostatic dehydration Droplet coalescence Droplet dynamics Simulation

A B S T R A C T Considering the droplet coalescence,the motion of a group of dispersed droplets in W/O emulsion in a DC electric field is simulated.The simulation demonstrates the evolutions of droplet number,size as well as its distribution,local concentration distribution and droplet size-velocity relation with the applied time of electric field.The simulated average droplet size is roughly consistent with the experimental value.The simulated variation of droplet number with time under several applied voltages shows that increasing voltage is more effective for raising the rate of droplet coalescence than extending exerting time.However,with the further raise of applied voltage,the improvement in droplet coalescence rate becomes less significant.The evolution of simulated droplet size–velocity relationship with time shows that the inter-droplet electric repulsion force is very strong due to larger electric charge on the droplet under higher applied voltage,so that the magnitude and the direction of droplet velocity become more random,which looks helpful to droplet coalescence.

1.Introduction

When oil–water emulsion is placed in a DC or AC electric field,the conducting dispersed droplets,which are charged,will aggregate.This feature can be used to achieve separation and demulsification of emulsion.Currently,Electric Demulsification Technology(EDT)has been widely utilized in breaking crude oil emulsion,handling waste oil and separating oil–water emulsions in petroleum and chemical industries[1–5].In the application of EDT and design of electric separator,it is important to understand that the motion of dispersed droplets in an electric field is extremely important,since it is directly related to the rate of coalescence and the efficiency of separation[6,7].On this issue,some studies have been conducted[8–12].For example,Vinogradov et al.[8]and Panchenkov and Vinogradov[9]established the motion equation of water droplets in emulsion by considering electric force due to applied external electric field and the drag force acting on the droplet,and analyzed the trajectory of water droplets.Vygovskoi et al.[10]also analyzed the influence of charge relaxation on movement of water droplets.Chiesa and Melheim[11]simulated the motion of two droplets falling in stagnant oil in an electric field,and better agreement in droplet trajectory between simulation and experiment was obtained.In their motion equations,the inter-droplet electric force due to polarization of the conductive water droplets was considered.In our previous research[12],the motion of dispersed water droplet in emulsion in a uniform electric field was also simulated with the electric force due to applied electric field and the inter-droplet electric force between droplets due to Coulomb repulsion considered.More studies related to simulation on droplet motion in electric field[13–19]are available.

For a real electric separation system,however,the droplet coalescence occurs constantly in the entire droplet migration process,inevitably producing a significant influence on droplet motion,and this has not been considered in previous studies.For this reason,the main objective of this paper is to realize a simulation on the motion of a group of charged water droplets considering droplet coalescence so that there can be a better understanding in the movement and coalescence mechanism of dispersed droplets in the emulsion.

2.Theory

In an electrostatic DC demulsification process,the droplet coalescence occurs in two ways[1,7,12].The first way is the electrophoresis coalescence:the charged droplets driven by the electric field force move directionally from one electrode to another.Two or several droplets in motion may approach and finally merge to form a larger droplet due to velocity difference between droplets.Furthermore,those droplets,which do not merge,will move to a destination electrode where they further aggregate due to a local high droplet concentration in this area.Another way is the so-called dipole coalescence:the dispersed water droplets in emulsion are polarized due to applied electric field so that the dipole attractive force is produced between two approaching droplets.The dipole force depends on the distance between droplets and it can be neglected when the distance between droplets is very large[17].The present work involves the movement and coalescence of dilute dispersed droplets in emulsion,so the dipole coalescence is not considered.In this condition,considering various forces on droplets,the motion of the dispersed droplets in emulsion in an electric field can be described by the Lagrangian approach as

where ρdis the density for dispersed phase,D and u are the diameter and velocity of droplet respectively,subscript i indicates the i-th droplet,Fdand Fgare the drag force and net gravity respectively,Feand Fkare the external electric field force and inter-droplet electric repulsion force(due to same polarity charge on droplets)respectively,and Fmis the visual mass force.

The droplet velocity in Eq.(1)can be defined as

where s is the position vector.

The external electric field force depends on electric charge q and external electrical field E,and it can be expressed as[16,17]

In a uniform electric field,the electric charge on a droplet with diameter D can be estimated by[1,19]

The drag force acting on droplet can be described by

where ucis the velocity of continuous phase,which is assumed to be equal to zero.Adis the projected area of droplet,ρcis density for continuous phase and Cdis the drag coefficient.

For a liquid–liquid system,the Reynolds number is very small due to high viscosity of continuous phase and small drop diameter.So the drag factor can be described by Stokes drag law[12,16]:

where Re(=ρcuiDi/μi)is the droplet Reynolds number,μ c and μ d are viscosities of continuous and dispersed phases respectively.

The net gravity for a droplet can be expressed as

where g is gravity acceleration vector.

Assuming that the charged droplets are point charges,the interdroplet repulsion force can be calculated by

where εoand εrare the vacuum permittivity and relative permittivity of continuous phase respectively,ri,jis the distance from the i-th droplet to the j-th droplet,and r is a unit vector along the line joining the centers between the i-th droplet and the j-th droplet.

Due to a larger density for the continuous phase in the liquid–liquid system,the added mass force(viz.visual mass force Fm)acting on droplet has to be considered[11,20]and it can be expressed by

Eqs.(1)and(2)are the 1st order ordinary differential equations for describing the motion behavior of charged droplets in a liquid–liquid system.For a droplet group containing N charged droplets with given initial velocities and positions,their velocities and positions at each time step can be obtained by solving Eqs.(1)and(2).

Droplet coalescence occurs in the migration process.Since there is a difference in velocity between droplets due to differences in size and charge,the two moving droplets may approach and merge to form a new larger droplet,even if there is an electrostatic repulsion between them.If not considering the details of coalescence,an ideal coalescence process can be shown in Fig.1,where Diand Djare the droplet diameters and uiand ujare their velocities.D and u are the diameter and velocity of the merged new droplet,respectively.

The diameter,velocity and charge of the new droplet can be estimated by momentum,mass and charge conservation equations as follows:

where qiand qjare droplet charges and q is the charge of new droplet.The position of new droplet can be taken as an average value of two old droplet positions,i.e.:

From Eqs.(10)–(13),we can determine the parameters of the new droplet,and then solve Eqs.(1)and(2)to finally obtain velocities and positions of droplets in the case of droplet coalescence.More detailed calculation procedure will be introduced in the latter.

3.Experimental

The experimental configuration is similar to our previous one[12],as shown in Fig.2.The present W/O emulsion is placed in a plexiglass box,and two plate electrodes are respectively mounted on both sides of box.The left electrode is connected with a high voltage electrostatic generator and the right electrode is grounded.The electrostatic generator can supply a negative electrical voltage to the left electrode.The emulsion used in the present experiment was prepared using water and corn oil(the emulsion is dilute with the water cut only about 0.25%). The density and the dynamic viscosity of corn oil are 907 kg·m?3and 0.017 Pa·s respectively and the interfacial tension between oil and water is 0.027 N·m?1.The interfacial tension was measured using a JK99B type surface tension meter(Shanghai Zhongchen Digital Technical Apparatus Co.).The viscosity was measured using a NDJ-79 type Viscometer(Shanghai Changji Geological Instrument Co.).

Fig.1.An ideal coalescence process for two droplets.

Fig.2.Experimental configuration.

A Winner 99 type particle image analyzer(Jinan Winner Particle Instruments Co.)was utilized to measure the dispersed droplet size of emulsion with testing accuracy within 5%.The size distribution of the prepared emulsion was previously measured and the measured droplet size distribution is shown in Fig.3.During the experiment process,the testing samples for testing droplet size were directly collected from the Plexiglas box near the right-hand side electrode.

Fig.3.Dispersed droplet size distribution of prepared emulsion.

4.Simulation Procedure

4.1.Algorithm

Groot and Warren[21]used a modified velocity-Verlet algorithm to solve the particle motion equation in mesoscopic simulation,and this algorithm is also used in the present work for solving Eqs.(1)and(2).The algorithm is expressed by

where t is the time and Δt is the time step,F is the sum of all forces in Eq.(1)and λ is a variable factor accounting for some additional effects of the stochastic interactions.In present work,λ=0.65 and time step Δt=0.05 s.

4.2.Computational domain and parameters

In the present work,only the two-dimensional motion of droplets is simulated on a computational domain being simplified to a rectangle in Fig.4.

For a uniform electric field induced by parallel plate electrodes,the electric field strength is

where U is the applying electric voltage on electrode and L is the distance between two electrodes(7 cm),φ is the electrical potential and enis the unit vector along the potential gradient direction.

The required simulation droplet number can be determined from the water cut 0.25%and the average diameter of dispersed droplets 2.38×10?5m.It can be deduced that there are about 4000 droplets in the computational domain,so the simulation droplet number N is taken as 4000.

Fig.4.Computational domain and coordinate system.

4.3.Initial values for droplets

The initial droplet size and its distribution are obtained from measured data(see Fig.3).Here,a random generator is used to generate N droplets(N=4000)with a size distribution subjecting to the measured distribution.These 4000 droplets are randomly distributed in the computational domain and comply with a uniform spatial distribution as in Fig.5.The initial droplet velocities can be assumed to be zero.

4.4.Droplet coalescence

In droplet migration,when the surfaces of any two droplets contact,the two droplets will coalesce to form a new large droplet.A criterion on droplet coalescence is given as follows:

Fig.5.Initial droplet positions.

where ri,jis the distance between the central points of two droplets.In order to judge the occurrence of coalescence,the distance ri,jbetween each pair of droplets has to be calculated at each time steps.If coalescence occurs,the parameters of the new droplet obtained from Eqs.(10)–(13)can be stored in the storage for the i-th droplet,and the j-th droplet will disappear and no more related to the next time step calculation.After updating all parameters,we then calculate velocities and positions of droplets at next time step by Eqs.(14)–(17).In this way,the final velocities and positions of droplets at the end of each time step are obtained.

4.5.Independence of time step size

It is necessary to verify independence of time step size.Here the simulated average droplet diameter as a testing index is used to validate the time step size.For simplicity,we simulate the droplet coalescence only for 4000 droplets with same diameter(24 μm)during motion.Fig.6 shows the effect of time step size on simulated average droplet diameter.As we expected,the average droplet diameter has a large fluctuation when the time step size is larger.This fluctuation gradually decreases with the decrease of time step size.When the time step is less than 0.1 s,the change of average diameter with time step size is negligible.So the time step is taken as 0.05 s in the subsequent simulation.

5.Discussion of Simulation Results

According to the above simulation procedure,the simulation on motion and coalescence of dispersed droplets of W/O emulsion in the electric field is conducted under different applied electric voltages.Fig.7 shows the simulated evolution of local droplet concentration distribution with applying time of electric voltage(the local droplet concentration defined as the droplet number per mm2).We can find in Fig.7 that the droplet concentration distribution is approximately uniform at the initial time.When an electric voltage is applied,there is a significant change in concentration distribution due to the directional motion of droplets from one electrode to another.As the time elapses,the droplet concentration gradually decreases in the area near the left electrode,while it increases in the area near the right electrode.As a result,a very high droplet concentration area emerges near the right electrode to increase the probability of droplet collision and aggregation.

Fig.6.Effect of time step size on simulated average droplet diameter(U=6000 V,t=100 s).

Since the droplet concentration in the area near the right electrode has an important effect on droplet coalescence,we further explore the local concentration in this area as shown in Fig.8.We can find the concentration does not monotonously increase with the time going.Under a higher electric voltage applied,the concentration goes up gradually and hits a peak,then goes down gradually with time.This trend of droplet concentration is a little different with that in our previous study[12],in which,the simulation of droplet coalescence was considered in the droplet motion process under lower electric voltage,and the simulated droplet concentration approximately increased linearly with time.This can be explained as follows.After applying a high voltage the droplets are driven to move fast near the right electrode,leading to a high droplet concentration in this area,higher chance for droplet coalescence,and eventually making the droplet concentration gradually decrease.On the contrary,under a lower electric voltage,the droplets moved slowly and the concentration increased in this area very slowly with less droplet coalescence,so that the concentration increased monotonously with time.

Due to the coalescence in droplet motion,the droplet size and number may constantly change with applying time.The variations of droplet in size and number with time are analyzed in Fig.9.We can find that under applying an electric field,the droplet number decreases with applying time due to droplet coalescence accompanied with motion.The decreasing of droplet number with time is extremely limited under excessively low electric voltage applied.Therefore,in order to decrease droplet number in a larger degree,a higher electric voltage needs to be applied.In other words,the potential of improving the rate of coalescence by increasing applying electric voltage is greater than that by extending applying time.When the applying voltage further rises,the decreasing rate of droplet number seems to slow down.A possible reason is that the high rate of droplet coalescence makes the droplet number fast decrease and this conversely makes the rate of further droplet coalescence slow down.

Fig.10 shows the simulated evolution of droplet size distribution with applying time.We can find that the peak value of droplet size distribution gradually declines and the distribution gradually becomes wider.There are a considerable number of large droplets generated due to coalescence.This wide size distribution will be helpful to further collection at the electrode.

In order to verify the simulation characteristics,a direct comparison on average droplet diameters in the area near the right electrode between the experiment and simulation is shown in Fig.11.It can be seen that the average droplet diameter is reasonably consistent between the experiment and simulation.Relatively,the predicted values are slightly lower than the experimental values,possibly because the dipole interaction of droplets has not been considered in the present simulation.Although,the dipole interaction of droplets can be neglected for extremely dilute emulsion(water cut of emulsion is only about 0.25%in the present work),the dipole coalescence of droplets can occur in the area near the right side electrode due to a very high local droplet concentration.This may result in a relatively low predicted value.

In order to further understand the coalescence mechanism,we also need to analyze droplet velocity–size characteristics.Fig.12 shows the relationships between droplet velocity and size at different applying times.Totally,the droplet velocity increases with the increase of droplet size and the increasing rate of velocity with size grows with the raised voltage.We can also find that the distribution of data points is more scattered under higher applying voltage,while the distribution is relatively concentrated under lower applying voltage.The reason for this is that the inter-droplet electric repulsion force is very strong due to a larger electric charge on the droplet under higher applying voltage,so that the magnitude and the direction of droplet velocity are randomly distributed.This phenomenon has been observed and explained in our previous study on liquid–liquid electrostatic spray[22].This huge difference in velocity between the droplets is extremely conducive to droplet coalescence in the migration process,which can conversely affect the droplet velocity.As shown in Fig.12,with the time elapsing,the distribution of data points becomes further dispersed due to the constant generation of some large droplets from coalescence,especially under higher applying voltage.A considerable difference in velocity between the droplets is helpful to droplet coalescence,so the high rate of coalescence can be obtained by applying a higher voltage.In addition,from Fig.12 we can also find some droplets,whose velocities are(or close to)zero.It indicates that these droplets have actually arrived at the right electrode.As time goes,there are more droplets to hit the right electrode where they will continue to aggregate due to the high local droplet concentration.

Fig.7.Simulated evolution of local droplet concentration distribution with applying time(U=6000 V).

6.Conclusions

With droplet coalescence considered,the simulation on the motion of a group of dispersed water droplets in the emulsion in an electric field leads to the following main conclusions.

Fig.8.Droplet concentration variations in the area near right electrode with voltage application time.

Under the higher voltage,the droplets can be driven to fast move to one side electrode,where a higher local droplet concentration region can be formed.This increases the opportunity of the collision and aggregation between droplets,so that a higher droplet coalescence rate is obtained.On the other hand,the higher applying voltage makes the interdroplet electric force acting on droplets increase,which therefore leads to the increasing velocity difference between droplets.This also increases the probability of droplet coalescence in droplet migration process.

Fig.9.Simulated evolution of droplet number with applying time.

Fig.10.Simulated droplet size distribution evolution with applying time(U=6000 V).

The present simulation demonstrates the evolution of the droplet number,droplet size distribution and local droplet concentration with applying time.The simulated average droplet size is roughly consistent with the experimental value.The simulated variations of droplet number with time under several applying voltages show that increasing applying voltage is more effective for raising the rate of droplet coalescence than extending the applying time.However,with the further increase of applying voltage,the increasing of coalescence rate becomes less in extent.

Fig.11.Comparison on droplet average diameters in the area near right electrode between simulation and experiments(U=6000 V,t=96 s).

Fig.12.Relationships between droplet velocity and size at different applying times.

Nomenclature

Adprojected area of droplet,m2

Cddrag coefficient

D droplet diameter,m

Dmaverage droplet diameter,m

E electric field strength,V·m?1

F force,N

Fddrag force,N

Feexternal electric field force,N

Fgnet gravity,N

Fkinter-droplet electric repulsion force,N

Fmvisual mass force,N

N droplet number

q electric charge,C

Re Reynolds number(=ρuDμ?1)

U applying electric voltage,V

u droplet velocity,m·s?1

ucvelocity for continuous phase,m·s?1

r distance between two droplet,m

s position of droplet,m

t time,s

Δt time step,s

ε0vacuum permittivity,F·m?1

εrrelative permittivity

μ viscosity,Pa·s

ρ density for dispersed phases,kg·m?3

φ electrical potential,V

Subscripts

c continuous phase

d dispersed phase

i,j droplet