Investigation on Electrostatic Charging and Its Effect on Mixing of

Article pubs.acs.org/IECR

Investigation on Electrostatic Charging and Its Effect on Mixing of Binary Particles in a Vibrating Bed Yongpan Cheng,†,‡ Liangqi Lee,† Wenbiao Zhang,†,§ and Chi-Hwa Wang*,† †

Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, 117585, Singapore NUS Environmental Research Institute, National University of Singapore, 5A Engineering Drive 1, #02-01, 117411, Singapore § School of Control and Computer Engineering, North China Electric Power University, No. 2 Beinong Road, Beijing, 102206, China ‡

ABSTRACT: The vibrating bed is widely applied in various industrial areas for drying, mixing, and segregation of granular materials. Triboelectrification becomes unavoidable due to frequent collisions and friction between particles and between particles and the wall, which will greatly affect the performance of the vibrating bed. In our study the triboelectrification model under the framework of the discrete element method was implemented to simulate the triboelectrification process, as well as the effect of electrostatic forces on the hydrodynamics of spherical particles and cylindrical particles. The numerical results showed that the triboelectrification could become stronger with increasing solid loading or bed diameter, and decreasing particle size. At low vibrating frequencies or amplitudes, the triboelectrification for cylindrical particles was stronger than that for spherical particles, while at high vibrating frequencies or amplitudes the triboelectrification for spherical particles became stronger. The repulsive or attractive electrostatic forces among particles could help improve the mixing between spherical particles and cylindrical particles. With the increasing vibrating frequencies and amplitudes, the mixing between spherical particles and cylindrical particles was improved and then approached a plateau. Due to strong triboelectrification between solid particles and the wall, the particles could attach to the wall due to attractive electrostatic forces from the wall, and form rings in the vibrating bed, which might deteriorate the mixing in the vibrating bed. These results may help us get insight in the triboelectrification of the vibrating bed and provide guidance in improving its performance by virtue of electrostatics.

1. INTRODUCTION

Mixing and segregation of solid particles are common phenomena in the vibrating bed. The effects of particle size and density on the mechanisms of mixing and segregation of the solid particles have been intensively investigated.13−17 On the other hand, the electrostatic effect in the vertically vibrated bed would also influence the mixing of solid particles. Lu et al.18 investigated the mixing of glass beads with opposite charges using the discrete element method. The electrostatic force was considered in the simulation to quantify the mixing; it was found that the mixing of solid particles could be improved by the electrostatic effect. So far the triboelectrification in the vibrating bed has been seldom studied, especially when the particle shapes are irregular. With the effect of electrostatic forces, the hydrodynamics of solid particles with irregular shapes will also be changed, leading to different mixing behavior in the vibrating bed. In this study, the triboelectrification model under the framework of the discrete element method will be adopted to simulate the triboelectrification process, as well as to investigate the effect of electrostatic forces between cylindrical particles and spherical particles.

The vibrating bed is widely used in industry for drying, mixing, and segregation of granular materials, wherein vibration energy is transferred to the particles by either mechanical pulsing of the entire equipment or just the gas distributor grid. Commercial application of the vibrating bed has been increasing over the past decade for the processing of irregular particles and sticky powders, which are both difficult to fluidize. Triboelectrification is inevitable in a process involving solid particles, because of continuous contacts between particles and and between particles and the wall. Studies have been conducted on the triboelectrification phenomenon and its effects on the flow dynamics of solid particles.1−10 During the vibration process the electrostatic charges are always generated by particle contacts with the inner surface of the vibrating bed. The hydrodynamic behavior of the particles can be altered by electrostatic charging, causing significant undesirable particle agglomeration and adhesion to vessels or pipe walls. As such, it is of importance to investigate the electrostatic charging in the vibrating particle bed. The electrostatic charging of solid particles in a vertically vibrated bed was studied by Liao et al.,11 who used a Faraday cage to quantify the influence of vibrating conditions on the electrostatic charges. The experimental results showed that dimensionless vibrating acceleration, vibrating frequency, and dimensionless vibrating velocity had great influences on the electrostatic charging. Laurentie et al.12 put forward a numerical model to simulate the tribocharging process in the vertically vibrated bed; the numerical results were in good agreement with their experimental results. © 2014 American Chemical Society

2. MODEL DESCRIPTIONS 2.1. Physical Model. In Figure 1 the schematic diagram of a vibrating bed is shown. It was attached to the motor and Received: Revised: Accepted: Published: 14166

April 10, 2014 August 8, 2014 August 18, 2014 August 18, 2014 dx.doi.org/10.1021/ie501493q | Ind. Eng. Chem. Res. 2014, 53, 14166−14174

Industrial & Engineering Chemistry Research

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and is assumed to be evenly distributed across the surface of the particle. When N number of particles is charged against the surface, the charge accumulated can be represented as follows: Q (t ) = Nq(t ) = Nqs(1 − e−αt )

2.4. Calculation of Electrostatic Force. The Hertz− Mindlin (no slip) contact model is used to calculate the contact forces between the particles and between the particles and wall. The damping coefficient of both the normal and tangential forces is related to the coefficient of restitution.22 With electrostatic charges on the particles and walls, the electrostatic forces will be calculated from the nearby particles and walls together with existing contact force calculation. Coulomb’s law is used to calculate the electrostatic force: 1 q1q2 FE = r̂ 4πε0 r 2 (5)

Figure 1. Schematic diagram of a vibrating bed with triboelectrification.

could move up and down in a sinusoidal way. The solid particles were put in the vibrating bed. Due to friction and the collisions of particle−particle and particle−wall, triboelectrification would happen inevitably, so both the wall and particles would be charged and the electrostatic forces caused by the electrostatic charging would affect the friction and collisions of particles. 2.2. Computational Model. In our study, discrete element method (DEM) modeling in the commercially available software EDEM (DEM Solutions Ltd.)19 was used to study the dynamics of each particle in particulate systems. This method is commonly used to track many industrial processes involving granular flows. In this method the particulate phase is treated as a collection of discrete particles, and the mechanical and inertial properties of the particles are considered, as well as the particle interactions with other particles, boundary wall surfaces, and other forces such as gravitational and electrostatic fields. This gives rise to very high spatial resolution of momentum, mass, and heat transfer in particulate systems and the bulk behavior of particle flow. 2.3. Mathematical Model for Triboelectrification. The triboelectrification process is described by eq 1, which includes charge generation and charge dissipation terms:20 dq = α(qs − q) − βq dt

where q1 and q2 are the charges of two particles, ε0 is the permittivity of free space, and r is the distance between their centers. Coulomb’s law model does not consider the effects of other nearby charged particles within proximity of the target particles; as such, a screening term is added to account for these effects. The following terms are used in deriving the screened Coulomb force Fs. qq Ue = 1 2 e−κr 4πε0r (6) where Ue is the electrostatic potential and ⎛ 1 κ = qe ⎜⎜ ⎝ εε0KBT

⎞

∑ nizi 2⎟⎟ i

⎠

(7)

where κ is the inverse of the Debye length λD, which is dependent on the local charge concentration (where n is the number of particles of charge z) around a target particle. qe is the electronic charge (1.602 × 10−19 C), εε0 is the absolute permittivity of the medium, KB is the Boltzmann constant (1.38 × 10−23 J/K), and T is the absolute temperature in kelvin.22 The screened Coulomb force Fs is then given by the following:

(1)

where q is the charge on the particle at time t, qs is the saturation charge, and α is the time constant of the charge generation and is equivalent to the inverse of the time constant τ. Integration of this equation gives 1 q(t ) = qs [1 − e−(α + β)t ] 1 + β /α (2)

FS = −

q q ⎛κ dUe 1⎞ = 1 2 ⎜ + 2 ⎟e−κr dr 4πε0 ⎝ r r ⎠

(8)

2.5. Simulation Conditions. The materials of the vibrating bed wall and the particles are determined as acrylic and polyethylene, respectively. Because the work function of acrylic is lower than that of polyethylene, during the collisions of polyethylene particles and the acrylic wall, the particles will acquire negative charges and the wall will acquire positive charges. In order to increase the computation speed with reasonable accuracy in calculating electrostatic forces, the screening distance is set at 5 times the radius of the particle. In each simulation, the vibrating bed was first filled with a number of particles, which would start to settle to the bottom, and packed closely after 0.5 s, and then the vibration of the bed was activated for 20−50 s until the motions of particles reached a stable state. The details of the geometric dimensions and operating conditions are shown in Table 1.

Charge dissipation is caused by atmospheric ion impingement, which is usually quite slow, and hence β is considered to be negligible; i.e., β ≈ 0. q(t ) = qs(1 − e−αt )

(4)

(3)

The saturation charge qs is based on the point when atmospheric breakdown or ionization prevents additional charge buildup. Using air at standard temperature and pressure, the electrical breakdown field is approximately 30 000 V/cm. This yields a maximum charge density fixed at 2.66 × 10−5 C/ m2 particle surface area.21 The saturation charge is obtained by multiplying the charge density with the particle surface area, 14167

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where nsph stands for the number of spherical particles and ncyn stands for the number of cylindrical particles. The average mass ratio for each location is

Table 1. Geometric Dimensions and Operating Conditions in Simulations force model of particle to particle force model of particle to downer

Hertz−Mindlin (no slip) tribocharging (α = −1) Hertz−Mindlin (no slip) electrostatics 9.81 0.3 3 × 109 1200 4.7 0.4 1 × 108 941 4.9 0.9 0.5 0.01 40 200 1.0−3.5 5−40 3−15 1 × 10−5

particle body force gravity, m/s2 Poisson’s ratio of acrylic shear modulus of acrylic, Pa density of acrylic, kg/m3 work function of acrylic, eV Poisson’s ratio of polyethylene (PE) shear modulus of PE, Pa density of PE, kg/m3 work function of PE, eV coefficient of restitution coefficient of static friction coefficient of rolling friction diameter of vibrating bed, mm length of vibrating bed, m diameter of particles, mm frequency of vibrating bed, Hz amplitude of vibrating bed, mm time step, s

X̅ =

h r

(14)

(15)

In order to represent the mixing index in a nondimensional form, the mixing index σ0 is introduced to indicate complete segregation: σ0 2 = X̅ (1 − X̅ )

(16)

Thus, the following dimensionless mixing index may be calculated as σp Im = 1 − σ0 (17) where Im = 0 represents complete segregation and Im = 1 represents complete mixing.

3. RESULTS AND DISCUSSION 3.1. Tribocharging in Vibrating Bed. The numerical model on electrostatic charging has been validated by Cheng et al.1 The experimental studies were conducted for the tribocharging at different locations in the downer, and the numerical results on the averaged induced currents quantitatively agreed with the measured data under different solids mass fluxes. In sections 3.1.1−3.1.4, the effects of solid loading, porosity, and the diameter of the vibrating bed on the electrostatic charging were investigated. 3.1.1. Effect of Dimensionless Loading M on the Porosity εf. The vibrating bed was operated at an amplitude of 10 mm and a frequency of 25 Hz. In Figure 2 where the variation of the

The dimensions of a packed bed can be determined by the radius and the solid loading; here the dimensionless solid loading can be defined as = (1 − εp)

∑ wi(Xi − X̅ )2 i=1

dq = Nα(qs − q) (9) dt At t = 0, the initial current I0 is at maximum Nαqs, since q = 0. The dimensionless charging rate K is defined as follows: I I K= = I0 Nαqs (10)

πr 3

i=1

N

σp2 =

I=N

Vb(1 − εp)

N

∑ Xi

The mixing index is calculated according to

2.6. Definition of Dimensionless Variables. Considering eq 1, the overall induced current on N particles is

M=

1 N

(0.15 < M < 23) (11)

where h is the packed bed height, r is the bed radius, and εp is the packed bed voidage. Bed porosity is calculated as follows: εf = 1 −

Vb πr 2H

= 1 − (1 − εp)

h H

(0 < εf < 1) (12)

where Vb is the volume of solid particles; H is the expanded height of the bed under vibration. In order to quantify the mixing between spherical particles and cylindrical particles, the mixing index is formulated based on time-averaged particle numbers in each section along the vibrating bed. The influence of different sample sizes is evaluated with the mass fraction of particles in a certain section over all the particles numbers as the weighting factor wi.23 Thus, the mass ratio of spherical particles over all particles at each section is nsph Xi = nsph + ncyn (13)

Figure 2. Variation of porosity over volume loading in the vibrating bed. The fitting is based on correlation εf = 0.564 + [3.853/(M + 3.313)2] with the sum of squared errors of 0.047.

porosity of the vibrating bed against the dimensionless loading is shown, it can be seen that the porosity decreased nonlinearly with increasing solid loading. Under the same intensity of vibration, a packed bed with a lower height expanded much more than a bed with a taller packed height. This is because a taller packed bed had more layers of particles, with the top layer 14168



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being more cushioned from the vibrating motion at the base; as such the expanded bed height did not rise up as much. Because the porosity εf was calculated on the basis of the expanded bed height, there was little variation in the porosity; hence it would approach a constant value under large solid loading M. If the packed bed height was less, it expanded more as the top layer was more strongly exposed to the large vibration forces at the base. For a stationary packed bed, it had been noted that the wall effect became significant when the diameter of the bed was small compared to the particle diameter; the porosity of the packed bed was affected greatly by the ratio of the bed to particle diameter.24 This effect was found to be more pronounced for a low diameter ratio, while when the ratio was large the bed porosity was found to approach an asymptotic value. An empirical correlation was used by Benyahia and O’Neill24 to relate the porosity to the wall− particle diameter ratio for a stationary packed bed. The results were obtained by fitting of the equation to experimental data points.

εp = α +

Figure 3. Effect of solid loading on the initial current in the vibrating bed.

bed dimensions could be further analyzed by plotting the dimensionless charging rate K against the dimensionless solid loading M. 3.1.3. Effect of Solid Loading on Charging Rate. In Figure 4 the effect of dimensionless solid loading on the dimensionless

β ⎛ dw ⎞2 ⎜ ⎟ + γ ⎝ dp ⎠

(18)

εp is the stationary packed bed porosity, while α, β, and γ are coefficients. εp approached a asymptotic value of α when the bed was sufficiently wide. For a vibrating bed, a similar trend was noted when comparing the porosity εf against dimensionless M. Hence, in this analysis, a similar curve-fitting treatment could be performed to correlate the vibrating bed porosity. Instead of using the bed-to-particle diameter ratio dw/dp, the dimensionless solid loading M could be substituted, and after fitting the simulation data based on the least-squares method, the following correlation was obtained: εf = 0.564 +

3.853 (M + 3.313)2

Figure 4. Variation of charging rate over volume loading in the vibrating bed. The fitting is based on K = 0.0061M1.15 with the sum of squared errors of 0.053.

(19)

The resulting sum of squared errors was 0.047, while the largest deviation from the curve was found to be 13% of the simulated values. As such, the fitting to this correction was deemed satisfactory. The value of the porosity εf reached an asymptotic value at 0.564 at high M values, which corresponded to a more compact bed, with the particles moving together as a coherent mass. This effect was similar to the “coherentcondensed” or deep bed state.25 These results were obtained at a fixed vibration amplitude, frequency, and particle diameter; as such it was expected that these coefficients might be further influenced by those factors. 3.1.2. Effect of Bed Diameter on Induced Currents. In Figure 3 the effect of the diameter of the vibrating bed over induced charges was studied. It could be seen that the induced current increased with solid loading, and the increasing rate was higher when the bed diameter was large. Under low solid loading, the induced currents were almost identical, because tribocharging mainly happened between particles and the bottom of the vibrating bed due to low packed bed height. However, under high solid loading, tribocharging between particles and side walls of the vibrating bed became significant: large bed diameters meant large contact area between particles and solid walls and thus the tribocharging under large bed diameters became larger. The relationship between current and

charging rate was provided in log−log mode. It could be seen that the charging rate increased with increasing solid loading. At high solid loading, the resultant vibrating bed had lower porosity εf, resulting in a more closely packed bed. When the particles were more closely packed, the movement of the particles became more restricted and the collisions among particles became more frequent. When this happened, the particles were effectively pushed against the wall surfaces, increasing the summative contact time of all the particle−wall contacts. As a result, the tribocharging rate due to particles moving along the surface of the walls was much higher. In addition, the increased frequency of particle collisions resulted in more effective transfer of charge to particles that were not in contact with the walls, transferring charge away from the wall surface. The relationship between solid loading and the charging rate could be correlated as follows with a R2 value of 0.947: K = 0.0061M1.15

(20)

3.1.4. Effect of Particle Size on Charging Rate. In Figure 5 the effect of particle size on the charging rate was provided at fixed solid loading. It could be seen that when the particle size 14169



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3.2.2. Tribocharging for Each Type of Particle. Before the mixing performance of cylindrical and spherical particles was studied in the vibrating bed, the hydrodynamics and the tribocharging of each type of particles were studied, as seen in Figure 7. The accumulated charges on the cylindrical and

Figure 5. Variation of charging rate over particle size in the vibrating bed at solid loading M = 2.47.

was reduced, the charging rate was increased greatly, because the particle numbers in the vibrating bed would be increased dramatically at the fixed solid loading, leading to more collisions between particles and the wall, and the charging rate was increased greatly as well. 3.2. Mixing between Spherical Particles and Cylindrical Particles. 3.2.1. Construction of Cylindrical Particles. In our numerical simulation two types of particles were considered, as shown in Figure 6: one was spherical particles

Figure 7. Accumulated charge of cylindrical particles and spherical particles under different frequencies at vibrating amplitude A = 5 mm.

spherical particles under different vibrating frequencies were compared at the fixed vibrating amplitude A = 5 mm. These results were obtained within the initial 10 s. The charges on the particles were far from saturated, so the charges on the particles increased almost linearly with time. The vibrating bed had its natural frequencies at which the particles could gain energy from the vibrating bed most effectively and achieve resonance. As such, tribocharging became the strongest at these frequencies. When the vibrating frequencies increased from 15 to 40 Hz, it departed from the resonant frequency of particles. As such the tribocharging became weaker; hence the accumulated charges for both spherical and cylindrical particles decreased. It was interesting to note that, at the vibrating frequency f = 15 Hz, the accumulated charges on cylindrical particles were higher than those on spherical particles and, at vibrating frequency f = 20 Hz, the difference between them was reduced; however, when the vibrating frequency was increased to 40 Hz, the accumulated charges on the cylindrical particles became lower than those on spherical particles. In Figure 8, the accumulated charges on cylindrical particles and spherical particles under different vibrating amplitudes were compared at the fixed vibrating frequency f = 20 Hz. A

Figure 6. Geometric dimensions of spherical and cylindrical particles. Two types of particles have the same volume and material.

and the other was cylindrical particles. The geometry of cylindrical particles was constructed based on three elongated spherical particles, so the interactions between cylindrical particles could be calculated based on the corresponding spherical particles. Although they had different shapes, the volumes of both types of particles were the same.

Figure 8. Accumulated charge of cylindrical particles and spherical particles under different vibrating amplitudes at vibrating frequency f = 20 Hz. 14170



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similar phenomenon could be observed as with the vibrating frequency. With the increasing vibrating amplitudes the accumulated charges on the cylindrical particles and spherical particles were decreased. At the vibrating amplitude A = 5 mm, the tribocharging for the cylindrical particles was stronger than that for the spherical particles, while at the vibrating amplitude A = 10 mm, the tribocharging for the cylindrical particles became weaker than that for the spherical particles, and at vibrating amplitude A = 15 mm the difference between the two types of particles was enlarged. From the analysis on the effect of vibrating frequency and amplitude, it could be concluded that particle shape had a great effect on the tribocharging and the complex particle shape was more vulnerable to changes in vibrating frequency and amplitude. 3.2.3. Assumption in Mixing Characterization. The studies above were focused on the vibrating bed with only one type of particles. When cylindrical particles and spherical particles were mixed together in the vibrating bed, the mixing performance between the two types of particles might be changed under the electrostatic forces, which would be studied in the following. At standard temperature and pressure, the maximum charge density was 2.66 × 10−5 C/m2 on the particle surface; the corresponding saturated charge was 0.334 nC for the spherical particles that were 2 mm in diameter. In our simulation, the charge 0.2 nC was assumed for each spherical particle or cylindrical particle. With this value, the electrostatic forces were comparable with the gravitational forces, and the electrostatic forces would affect the motions of particles as well as the mixing. At the initial stage the cylindrical particles were packed at the higher part of vibrating bed, while the spherical particles were packed at the lower part. 3.2.4. Effect of Charge Polarity on Hydrodynamics. In Figure 9, the vertical solid fractions and velocities were compared with cylindrical particles and spherical particles at the vibrating frequency f = 10 Hz and vibrating amplitude A = 5 mm under three cases: (1) Both types of particles had no charges. (2) Spherical and cylindrical particles had the same polarity of charges. (3) Spherical and cylindrical particles had opposite polarities of charges. It was found that, if there were no charges, cylindrical particles and spherical particles were segregated greatly. Spherical particles were located at the bottom of the vibrating bed, and the cylindrical particles were above them. However, with the addition of electrostatic charges on the particle surface, both types of particles could be distributed more uniformly along the height of the vibrating bed. Because the particles could repulse and attract each other caused by the strong electrostatic forces, the particles could easily travel to the upper region of the vibrating bed. With the same volume, spherical particles had shorter lengths than cylindrical particles. After gaining the same kinetic energy from the vibrating bed, they would experience fewer collisions and less resistance when they rose up; hence fewer spherical particles were left in the lower region of vibrating bed, leading to a lower solid fraction of spherical particles in the lower region. In contrast, the solid fraction of spherical particles was higher at the upper region. The distributions of solid fraction are similar for spherical and cylindrical particles with or without charge, and with the same or opposite polarity of charges. It was also found that without charges both spherical and cylindrical particles were confined in the lower region of the vibrating bed and the velocities were quite low. In contrast, with charges the particles could reach higher positions and the velocity distribution was S-shaped. Near the bottom of the

Figure 9. Vertical distribution of solid fraction and velocity in the vibrating bed at vibrating frequency f = 10 Hz and amplitude A = 5 mm with different charging conditions.

vibrating bed, the particles started to accelerate after gaining kinetic energy from the vibrating bed, so their velocities increased first. Subsequently, due to the confinement of particles in the upper layers, their velocities decreased along the height of the bed. After leaving the bed, there was no further confinement for particles, so their velocities were increased again. Due to the smaller size of spherical particles, their motions were more easily affected than those of the cylindrical particles. When spherical particles had the same polarity of charges as the cylindrical particles, their motions would be suppressed by cylindrical particles due to repulsive forces. Hence, their velocities were lower than those of cylindrical particles. On the contrary, when spherical particles had the opposite polarity of charges as the cylindrical particles, their motions could be promoted by cylindrical particles due to attractive forces. Hence, their velocities were higher than those of cylindrical particles. 3.2.5. Effect of Vibrating Frequency and Amplitudes on Mixing. In Figure 10 the vertical distributions of the solid fraction and velocity were compared for spherical particles and cylindrical particles under different vibrating frequencies at fixed vibrating amplitude A = 5 mm. It was observed that at a low vibrating frequency f = 5 Hz, the spherical particles and cylindrical particles could not acquire enough energy from the vibrating bed to travel upward; the spherical particles would stay at the bottom of the vibrating bed with cylindrical particles above them, as seen in Figure 11a. Their velocities were also 14171



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become empty, as seen in Figure 11b. Therefore the solid fractions at the bottom were quite low and increased to the peak at around the static bed height; then they started to decrease to zero as no more particles could reach that height. Because of the strong motion at the high vibrating frequency f = 40 Hz, the solid velocity at the bottom was much higher than those at higher locations or at lower frequencies. In Figure 12 the vertical distributions of the solid fraction and velocity were compared for spherical particles and

Figure 10. Vertical distribution of solid fraction and velocity in the vibrating bed with both spherical and cylindrical particles under different vibrating frequencies at the vibrating amplitude A = 5 mm.

Figure 12. Vertical distribution of solid fraction and velocity in the vibrating bed with both spherical and cylindrical particles under different vibrating amplitudes at vibrating frequency f = 20 Hz.

cylindrical particles under different vibrating amplitudes at the fixed vibrating frequency f = 20 Hz. It was found that at the low vibrating amplitude A = 3 mm most of the spherical or cylindrical particles would stay near the bottom of the vibrating bed; hence the solid fractions were high. Because cylindrical particles had a length longer than the diameter of spherical particles, their motions were less affected by bed vibration. Hence, at the low vibrating amplitude A = 3 mm, more cylindrical particles were at the bottom of the vibrating bed, as seen in Figure 13a. When the vibrating amplitude was increased to 5 mm, the solid particles could obtain more energy and moved upward, and if the vibrating amplitude was increased further to 10 mm, the solid particles would detach from the bottom, leading to a low solid fraction; meanwhile the mixing between spherical and cylindrical particles was improved, as seen in Figure 13b. It was also noted that with the increasing vibrating amplitude the solid velocities at the lower region of

Figure 11. Snapshot of spherical particles (green) and cylindrical particles (red) with the vibrating amplitude A = 5 mm at t = 50 s.

quite uniformly distributed in the vertical direction. When the vibrating frequency was increased to 10 Hz, more spherical and cylindrical particles could move upward; the solid fractions at the bottom were maximum and decreased almost consistently along the height. When the vibrating frequency was increased further to 40 Hz, due to the strong vibration the particles were pushed upward, and the bottom of the vibrating bed could 14172



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3.3. Discussion of Ring Formation Due to Triboelectrification. In the study of mixing in the vibrating bed in section 3.2, the wall was considered as neutral without any charges and a constant amount of charges was assumed on the particle surfaces, so only the electrostatic forces among the particles were considered. This was consistent with reality, because in industry the walls in the facilities are usually grounded to avoid accumulation of charges, as the accumulated large quantity of charges might lead to potential hazards. In our study the triboelectrification in the vibrating bed was also numerically simulated. Due to the electrification the electrons would transfer from the wall to the solid particles, resulting in opposite polarities of charges on the wall and solid particles. The interactions caused by the electrostatic forces as well as collisions and friction between particles or between particles and the wall were considered. During vibration the charges caused by triboelectrification on the walls and solid particles increased over time, so the electrostatic forces between the solid particles and the wall also became stronger. As the particles and wall had opposite polarities of charges, when the electrostatic forces were strong enough, the particles would be attracted to the wall and formed rings, as seen in Figure 15. Although the particles might repulse each other, the attractive forces from the wall were much larger than the repulsive forces, which was on the order of 10−2 N. According to Coulomb’s law in eq 5, the maximum repulsive forces were on the order of 10−4 N when two solid particles with maximum saturated charges were close to each other. On the other hand, the gravitational forces of particles were on the order of 10−5 N; hence the attractive forces from the wall were dominant. This was why the solid particles could attach to the wall and form rings in the vibrating bed. The attachment of particles might affect the mixing in the vibrating bed, but this could be used in other areas, such as particle coating.

Figure 13. Snapshots of spherical particles (green) and cylindrical particles (red) under vibrating frequency f = 20 Hz at t = 50 s.

the vibrating bed were increasing, and in the upper region the solid velocities under different vibrating amplitudes were comparable. 3.2.6. Mixing Characterization. In Figure 14, the mixing index variation was provided under different vibrating

4. CONCLUSIONS Due to the frequent contacts between particles and between particles and the wall, triboelectrification is a significant process and it will considerably affect the hydrodynamics of particles as well as mixing among them. In our study, discrete element method simulation was adopted to study the electrification and its effect on the mixing between spherical particles and cylindrical particles in the vibrating bed. It was found that the increasing solid loading or bed diameter and decreasing particle size could all contribute to the increase of the charging rate in the vibrating bed. At low vibrating frequencies or amplitudes cylindrical particles exhibited a stronger triboelectrification performance than spherical particles, while at high vibrating frequencies and amplitudes the trend was opposite. The electrostatic charges could help improve the mixing between spherical particles and cylindrical particles due to the repulsion or attraction among particles. The higher vibrating frequencies or amplitudes could also improve the mixing between spherical particles and cylindrical particles, and beyond certain values of vibrating frequencies or amplitudes the mixing performance would not vary. With strong triboelectrification between solid particles and the wall, the particles could be attracted to the wall to form rings in the vibrating bed, which might deteriorate the mixing in the vibrating bed. These results may help us get insight on triboelectrification and its effect on mixing in vibrating beds.

Figure 14. Mixing index variation over vibrating frequencies and amplitudes between spherical particles and cylindrical particles in the vibrating bed.

frequencies and vibrating amplitudes. It was found that, at the fixed vibrating amplitude A = 5 mm, when the vibrating frequency f = 5 Hz, the mixing index was as low as 0.05, because the spherical particles and cylindrical particles always segregated with cylindrical particles above spherical particles. When the vibrating frequency was increased to 10 Hz, spherical particles and cylindrical particles started to penetrate each other, so the mixing between them became better. If the vibrating frequency was increased further beyond 15 Hz, the mixing index increased to around 0.6 and reached a plateau, indicating that vibrating frequency would not help improve mixing after it reached a certain value. At the fixed vibrating frequency f = 10 Hz, it could be found that, at lower vibrating amplitudes, the mixing between spherical and cylindrical particles was quite poor, and with increasing vibrating amplitude the mixing index was improved dramatically and reached a plateau after A = 10 mm. 14173



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Figure 15. Formation of particle rings due to triboelectrification in the vibrating bed with spherical particles with dp = 2 mm at vibrating amplitude 25 mm and vibrating frequency f = 5 Hz.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +65-65165079. Fax: +65-67791936. E-mail: chewch@ nus.edu.sg. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research program is funded by the National Research Foundation (NRF), Prime Minister’s Office, Singapore, under its Campus for Research Excellence and Technological Enterprise (CREATE) program, and Grant R-706-001-101281, National University of Singapore.

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REFERENCES

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Investigation on Electrostatic Charging and Its Effect on Mixing of

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