Agglomeration Kernel of Bipolar Charged Particles in the Presence of External Acoustic and Electric Fields

A theoretical study on the agglomeration kernel of bipolar charged particles in external fields is presented in order to investigate particle agglomeration under the combined effects of a direct-current (DC) electric field and an acoustic field, while taking into consideration particle collision efficiency. Three agglomeration kernels are considered in our model, namely Coulomb agglomeration, electric agglomeration, and acoustic agglomeration. The model performance is validated by comparing its predictions with the available exact solutions of peculiar cases. Our results indicate that the collision efficiency increases when an electric field is applied simultaneously with an acoustic field. The size of the total agglomeration kernel formed by the three processes listed above increases with the external electric field intensity and the sound pressure level (SPL). Based on this model which includes collision efficiency, the electric and acoustic agglomeration kernels both increase rapidly as the agglomeration nucleus particle size increases, eventually approaching saturation for large diameters. On the other hand, the total agglomeration kernel decreases as the sound frequency increases within the investigated range. The agglomeration kernels of particles with different relative dielectric constants, such as fly ash particles, solid carbon particles and water droplets, are investigated as well.


INTRODUCTION
A number of different agglomeration technologies have been proven to provide an efficient pretreatment method for the removal of fine particles from flue gases with traditional filtration devices.These techniques include electrical agglomeration (Hautanen et al., 1995), acoustic agglomeration (Gallego-Juarez et al., 1999), chemical agglomeration (Wei et al., 2005), turbulent agglomeration (Derevich, 2007) and magnetic agglomeration (Li et al., 2007).Laitinen et al. (1996) have conducted an experimental study of bipolar charged aerosol agglomeration with alternating electric field in laminar gas flow, finding that the number concentration of 0.1-1.0µm sized particles decreases by 17% -19%.Experiments of particle charging and agglomeration in direct-current (DC) and alternating-current (AC) electric fields were carried out by Ji et al. (2004) obtaining a 25-29% reduction ratio of submicron particles in an AC field.Tan et al. (2007) tested the effect of an external DC electric field on bipolar charged aerosol agglomeration and found that the relative decrease in number of sub-micron sized particles was about 10.7%.
Based on the mechanism of Brownian motion, the agglomeration kernel of electrically charged (including both unipolar and bipolar charged) particles caused by the Coulomb force in the absence of external fields has been derived by Zebel (1958) and Fuchs (1964).Approximate expressions for the agglomeration kernel of bipolar charged particles in an external DC (Wang et al., 2005) and AC (Tan et al., 2008) electric field, respectively, have been proposed in agreement with previous numerical results calculated by Koizumi et al. (2000).An analytic-numerical model has been developed to study the kinematic coagulation of charged particles in an AC electric field (Lehtinen et al., 1995), demonstrating that the rate of kinematic coagulation in the case of unipolar charging was negligible, while being significant when fine particles and large particles had opposite polarity.
Acoustic agglomeration has been extensively studied via experiments (Hoffmann et al., 1993;Liu et al., 2009;Liu et al., 2011), leading to the development of several acoustic agglomeration theories (Mednikov, 1965;Hoffmann, 1997).Orthokinetic collision, caused by different entrainment among different-sized particles in an acoustic field, is widely recognized as the dominant agglomeration mechanism in polydisperse aerosols (Cheng et al., 1983).The Mednikov model is generally employed to explain orthokinetic agglomeration based on different particle entrainment in a vibrating gas.In conventional calculations, the collision efficiency (i.e., the ratio of the actual collision cross section to the ideal, purely geometric one) is assumed to be unity (Temkin, 1994;Hoffmann, 1997).However, this assumption has been proven to overestimate the collision rate, especially in the case of collisions between submicron and large particles (Nakajima and Sato, 2003).Ezekoye and Wibowo (1999) set the collision efficiency to a value as low as 5% in order to obtain agreement between simulation and experimental results.Dong et al. (2006) derived an empirical formula to calculate the collision efficiency as a function of the Stokes number.A numerical calculation of collision efficiency was also performed by taking into account the electrostatic interaction force (Nakajima and Sato, 2003).
Although agglomeration processes have been extensively investigated, few studies concerning the combined effects of two or three agglomeration technologies have been reported.Sun et al. (2013) introduced a turbulent gas jet into an acoustic agglomeration chamber to study the coupling of gas jet and acoustic waves on inhalable particle agglomeration.An experimental system was developed to measure how the removal of fine particles emitted from coal combustion is affected by an electric and an acoustic field, both separately and simultaneously (Chen et al., 2015).However, to the best of our knowledge, a model which combines different agglomeration kernels in the simultaneous presence of electric and acoustic fields has not yet been reported.
In this paper, a theoretical study on the agglomeration kernel of bipolar charged particles in the presence of external fields is presented, in order to study particle agglomeration under the combined effects of a DC electric field and an acoustic field, taking into account the particle collision efficiency.The influence of particle size, electric field intensity, SPL and sound frequency are investigated and discussed, along with the effects of different relative dielectric constants.

Equations of Particles Motion
The agglomeration between two bipolar charged particles in the presence of external acoustic and electric fields is considered (Fig. 1).All nonlinear effects are neglected, as well as image forces, particle interactions and gravitational settling.The general kinetic equation for the motion of a charged particle with diameter d i , mass m i and charge q i can be written as: The life-hand side of Eq. ( 1) is the resultant force acting on the particle.The first term on the right-hand side represents the viscous Stokes' force, which is also called acoustic driving force.The second and third terms are electric field force and Coulomb force, respectively.Where i u   is the velocity of the particle, η is the viscosity of air, 0 u   is the vibration velocity amplitude of air, ω is the angular frequency, a E  is the magnitude of the external DC electric field, which is also called agglomeration electric field in order to distinguish it from the charging electric field, ε 0 is the vacuum permittivity, r  is the center-to-center separation between the two particles, C ci is the Cunningham correction factor, which can in turn be expressed as (Fan et al., 2013): where λ is the mean free path of air.
The charge acquired by the particle under corona discharge is calculated by the following expression, combining field charging and diffusion charging, which was derived by Cochet (1961): Here ε r is the relative dielectric constant of the particle, E c is the magnitude of the electric field for particle charging, which in the following is assumed to be 2 kV cm -1 if not otherwise specified.To avoid ambiguity, we shall assume that the fine particle is positively charged, while the nucleus particle is negatively charged.

Agglomeration Kernel of Particles
The agglomeration kernel, employed to describe the agglomeration rate, is defined as the number of agglomerations between two group particles per unit time, unit volume and unit particle number concentration, and it is expressed by the following equation: where N ji is the number of collision agglomerations, n i , n j indicate the number of particle i, j per unit volume, respectively.In this paper, particle i is assumed to be a small particle called a fine particle, while the label j indicates a large particle called a nucleus particle.It should be pointed out that the theory also works for particles of similar sizes.
Based on the fact that the Coulomb force between two particles acts in a short distance range while the applied external fields are effective over a long-distance range, the two effects can be considered independent of each other (Wang et al., 2005).Thus, the agglomeration kernel of bipolar charged particles in external acoustic and electric fields is composed of two parts.The first part is the agglomeration

Coulomb Agglomeration Kernel
The velocity of fine particles can be obtained by solving Eq. ( 1), neglecting the effects of external fields and ignoring the inertial term since the aerodynamic relaxation time is shorter than a few microseconds for a fine particle in air: Fig. 2 shows the velocity of the fine particle as a function of center-to-center separation, charging electric field intensity and nucleus particle diameter.It can be seen that the velocity becomes very small when the center-to-center separation is greater than 20 µm since the Coulomb force is negligible in the long-distance region.Conversely, the velocity significantly increases with decreasing the center-to-center separation, being inversely proportional to square of the separation.Large charging electric field intensity and large nucleus particle diameter are both beneficial to the increase of fine particle velocity, which is strongly influenced by these two factors according to Eq. (3) and Eq. ( 5).
If we assume that the nucleus particle is located at its initial position and does not move, in terms of relative movement, it takes a time t for the fine particle to reach the nucleus particle at a velocity of u i + u j from the surface of starting position r to the collision surface s (Since the velocity i u   is in the same direction as the vector r  , scalars are used for simplification in the derivation): The total number of agglomerations between particles up to time t becomes: Solving Eqs.(4), Eq. ( 5) and Eq. ( 7), the Coulomb agglomeration kernel can be obtained by:

External Fields Agglomeration Kernel of Particles
In the presence of external fields, by using the method proposed by Williams and Loyalka (1991), the agglomeration kernel can be expressed as: where s is the ideal collision cross section and V is the relative velocity.It should be noted that this equation cannot be directly applied to real situations in this form, due to the discrepancy between the real and ideal cross sections, as discussed in the following sections.Following the motions of particles in the presence of acoustic and electric fields, the external fields agglomeration Fig. 2. Velocity of a positively charged 1 µm fine particle in the vicinity of a nucleus particle with negative charge.kernel consists of two parts in the x and y directions.One is caused by the electric field in the y direction, with the two particles moving in opposite directions.The other is caused by the acoustic field in x direction, with the two particles moving in the same direction.By solving the equations of motion, Eq. ( 1), in both directions, the relative velocity can be obtained: Here τ i and τ j indicate the relaxation time of the i and j particles, , ρ is the particle density, and the quantity |cos(ωt -φ i -φ j )| is replaced by its average value in half cycle as 2/π (Temkin, 1994).Substituting Eqs. ( 10) and ( 11) into Eq.( 9), the following expression for the agglomeration kernel can be obtained:

Collision Efficiency of Particles
In the above derivation of the external fields agglomeration kernel, we assumed that the fine particle moves along a straight line toward the nucleus particle.This assumption is, however, not completely accurate, since the trajectory of the fine particle will be deflected from the straight line due to entrainment of the air flow.In order to obtain an effective collision cross section value, the trajectory of the fine particle should be calculated within the airflow.Theoretically, the trajectory should be three dimensional, including also the z-component of the flow.However, due to the axial symmetry of the air flow, a factor φ, which indicates the angle between the y and z axes measured in the clockwise direction (Fig. 1), is introduced in order to perform a 2-d trajectory calculation.
The air flow is in the Stokes region due to the small Reynolds number.Therefore, the stream function for the well-known Stokes flow (Turcotte and Schubert, 2014), which satisfies the no-lip velocity boundary condition at the surface of the nucleus particle, can be expressed as: where U denotes the approaching velocity of the airflow.
The angular and radial components of the velocity in terms of the stream function can then be obtained (Fig. 1): The x and y components of the air flow can be expressed by: Thus the equations of motion for the fine particle can be written as: Here f x , f y respectively indicate other forces in the x and y direction, except for the viscous force.In the case of acoustic agglomeration, such forces need not be taken into consideration, while in this paper, the electric force is considered as a factor enhancing collision probability.When an electric field is applied perpendicularly to the acoustic field, these forces can be expressed as: Fig. 3 shows the real collision cross section of the fine particle.The starting x position is determined from the relative displacement between two particles (Dong et al., 2006).By solving the above equations numerically, the critical trajectory can be obtained.If the starting y position of the fine particle is less than the critical radius y c , it can definitely collide with the nucleus particle, otherwise it will escape from collision.The critical radius y c is a function of the angle φ, since the force f y changes with the angle.If the critical radius y c ≤ 0 when φ = π, the collision cross section S (as shown in Fig. 3(b)) will be calculated twice, thus it should be eliminated in the calculation.As a result, different expressions for the real collision cross section are used: In order to eliminate the unit, a dimensionless parameter called collision efficiency is introduced, which is defined as: Two fly ash particles, with which the density is 2500 kg m -3 and the relative dielectric constant is 1.7, suspend in an electric field, E a = 2 kV cm -1 , and an acoustic field, SPL = 142 dB, ƒ = 1400 Hz.If the acoustic field is applied solely, the critical radius is 1.88 µm and the real collision cross section is 11.09 µm 2 (Fig. 4).When an electric field is added as well, the critical radius becomes 3.61 µm for φ = 0 and 0.19 µm for φ = π, with a real collision cross section of 14.4 µm 2 .The collision efficiency increases from 0.116 to 0.152 upon adding the electric field.It is clear that the simultaneous presence of acoustic and electric fields can increase the collision efficiency.
Combining Eq. ( 8), Eq. ( 12) and taking the collision efficiency into consideration, the total agglomeration kernel of bipolar charged particles in external acoustic and electric fields can be expressed as: Eq. ( 24) shows three distinct contributions to the total agglomeration kernel.The first term on the right-hand side describes the Coulomb agglomeration kernel caused by the sole Coulomb force.The second and third terms account for the electric and acoustic agglomeration kernels, respectively.Here, the effect of hydrodynamic interactions on particles agglomeration is neglected in order to consider the acoustic agglomeration kernel to be only caused by the orthokinetic collision mechanism.

Accuracy of the Model
In order to validate the calculated Coulomb agglomeration kernel of bipolar charged particles in the absence of external fields, the expression of Eq. ( 8), is compared with the classical solution (Zebel, 1958;Fuchs, 1964) derived from the conventional Brownian motion of two charged particles, and with numerical results (Koizumi et al., 2000) taken from the available literature.Here, the particles are solid carbon particles with a relative dielectric constant of 3, and the charging electric field intensity is 2.5 kV cm -1 .Our results show good agreement with the theoretical and numerical values, despite being obtained from different concepts and derivation procedures (Fig. 5).
The present calculation of particle collision efficiency displays good agreement with the results found by Zhang et al. (2009) in the presence of the acoustic field only.The particles in this case are fly ash particles with diameters of 1 and 10 µm.Fig. 6 shows that, in the presence of an acoustic field, the collision efficiency is relatively low and stable for low SPL values, increasing dramatically when the SPL becomes high.If an electric field is applied, the collision   efficiency increases proportionally to the agglomeration electric field intensity.The collision efficiency shows a non-monotonic change with the SPL in the simultaneous presence of acoustic and electric fields.There exists a critical SPL value, corresponding to the lowest collision efficiency.When the SPL changes, the vibrating amplitude and velocity change accordingly.The action time of the electric field, which is also the time required for a fine particle to travel from the starting position to the collision surface, will be influenced consequently.

Influence of Particle Size, Electric Field Intensity, SPL and Frequency Particle Size
The effects of nucleus particle size on the agglomeration kernel are shown in Fig. 7 (both with and without considering  the collision efficiency).The fine particle diameter is fixed at 1 µm, while the nucleus particle diameter varies from 2 to 10 µm.Both particles are fly ash particles.The acoustic field has a SPL of 142 dB and a frequency of 1400 Hz, while the intensity of the agglomeration electric field is 2 kV cm -1 .
When particle collision efficiency is not taken into account, both the acoustic and the electric agglomeration kernels rapidly increase with the agglomeration nucleus particle diameter.However, if the collision efficiency is considered, the increasing trend becomes slower at large nucleus particle diameters.This is due to the collision efficiency increasing at first and then decreasing when the nucleus particle diameter changes from 2 to 10 µm.When the particle sizes approach a similar value, the relative velocity between two particles is low.Thus the fine particle carried by the airflow can easily get through the nucleus particle leading to a low collision efficiency.The particle relative displacement, however, exhibits little changes for large particle size ratios.Therefore, in these conditions the collision efficiency becomes small even though the nucleus particle diameter is large.

Electric Field Intensity
The effects of the electric field intensity on the agglomeration kernel are presented in Fig. 8.The fine and nucleus particle diameters are 1 and 10 µm, respectively.The sound frequency is 1400 Hz and the SPL is 142 dB.The figure shows that, if the collision efficiency is considered, the electric and acoustic agglomeration kernels increase when the electric field intensity changes from 1 to 4 kV cm -1 .An intense agglomeration electric field can both enhance the collision efficiency and electric agglomeration.Increasing the collision efficiency enlarges the both the electric and the acoustic agglomeration kernels, which means that the agglomeration electric field intensity has dramatic influence on electric agglomeration.Considering the collision efficiency significantly decreases the size of the total agglomeration kernel.That is because the collision efficiency is as low as 0.12 when the agglomeration electric field intensity is 1 kV cm -1 , and it only increases to 0.28 when the electric field intensity increases to 4 kV cm -1 .
The agglomeration kernels of different type of particles are shown in Table 1, comparing fly ash particles, solid carbon particles and water droplets with relative dielectric constant of 1.7, 3.0, 80, respectively.The agglomeration process occurs between two particles with diameters of 1 and 10 µm, in a varying charging and agglomeration electric field combined with a fixed acoustic field, with a SPL of 142 dB and a sound frequency is 1400 Hz.It is clear that water droplets are more effective for agglomeration than fly ash particles and solid carbon particles in the same condition.The larger the relative dielectric constant, the larger the charge capacitance, which means more charge can be held under the same corona charging condition.Increasing the charge level can both enhance the agglomeration kernel and the collision efficiency.It can also be found that the agglomeration kernel of water droplets increases up to 8 times from 7.3 × 10 -12 m 3 s -1 to 58 × 10 -12 m 3 s -1 as the charging electric field intensity increases from 1 kV cm -1 to 3 kV cm -1 , while a much smaller increase (by a factor 1.6) is found for the same variation of the agglomeration electric field intensity.

SPL and Frequency
The effects of SPL and frequency on the agglomeration kernel with and without collision efficiency are illustrated in Fig. 9.The agglomeration electric field intensity is fixed at 2 kV cm -1 .The SPL of the acoustic field takes the values 136 dB, 142 dB and 148 dB, while its frequency changes from 800 to 2400 Hz, which is commonly used in acoustic agglomeration experiments.Both particles are fly ash particles with diameters of 1 and 10 µm.
When the collision efficiency is not taken into account, the total agglomeration kernel keeps almost unchanged in the range of calculated sound frequency, while it decreases with increasing sound frequency if collision efficiency is included.Even though the sound frequency does not affect the Coulomb agglomeration kernel, it can reduce the external fields agglomeration kernel by decreasing the collision efficiency.This happens because the action time of the electric force decreases with increasing sound frequency, leading to a low critical radius, and to an according decrease of the collision efficiency.Fig. 9 shows that the total agglomeration kernel becomes larger as the SPL increases.In fact, a large SPL means that the approaching velocity of the airflow is large, causing a low deflection of the trajectory and favoring the collision between the particles.Furthermore, Eq. ( 24)  shows that a large approaching velocity also increases the acoustic agglomeration kernel even though the collision efficiency is unaffected.While a high-intensity acoustic field consumes a lot of energy, a suitable SPL should be chosen for conducting experiments.

CONCLUSIONS
Three agglomeration processes of polydisperse bipolar charged particles in the presence of external acoustic and electric fields are studied, namely Coulomb, electric and acoustic agglomeration.The effects of particle collision efficiency are included in the model.Parametric studies are conducted to investigate the effects of particle size, electric field intensity, SPL and sound frequency on the agglomeration kernel.The main results obtained from this study can be summarized as follows: (1) The particle collision efficiency becomes large when an external electric field is applied in the presence of an acoustic field.However, variations of the intensity of such electric field do not significantly affect the collision efficiency.
(2) The total agglomeration kernel increases with the external electric field intensity and sound pressure level (SPL).
Water droplets are more effective for agglomeration than fly ash and solid carbon particles in the same conditions, due to their large relative dielectric constant.(3) If collision efficiency is considered, the electric and acoustic agglomeration kernels both increase rapidly with the nucleus particle diameter, but the increasing rate becomes slower at large diameters.(4) The total agglomeration kernel is reduced as the sound frequency increases within the investigated range, due to the decrease of collision efficiency.

Fig. 1 .
Fig. 1.The motions of two bipolar charged particles in external acoustic and electric fields.The directions of the acoustic and electric field are parallel to the x-axis and y-axis, respectively.

Fig. 5 .
Fig. 5. Agglomeration kernel of bipolar charged particles by the Coulomb force without the effect of external fields.

Fig. 6 .
Fig. 6.SPL effects on the collision efficiency in external fields at a frequency of 1000 Hz for a pair of particles of size 1 and 10 µm.

Fig. 8 .
Fig. 8. Effects of the electric field intensity on the agglomeration kernel at a frequency of 1400 Hz and a SPL of 142 dB for a pair of particles of size 1 and 10 µm.

Fig. 9 .
Fig.9.Frequency and SPL effects on the agglomeration kernel with an external electric field of 2 kV cm -1 for a pair of particles of size 1 and 10 µm.

Table 1 .
Comparison of the agglomeration kernel between a pair of particles of size 1 and 10 µm for different materials.Here, SPL = 142 dB; ƒ = 1400 Hz.