Particle Motion Induced by Electrostatic Force of a Charged Droplet

Motion of particles in gas phase

Particles on the plate are charged to the opposite polarity of the droplet, the induced electrostatic forces drive them toward the charged droplet as shown in Fig. 2. The trajectories of particles in different takeoff points of the plate are drawn in Fig. 3a. The electrostatic attractions are closely related to the distance between the charged droplet and particles. Different location of particles on the plate means different initial attractions exerted on the particles. This is revealed by the particle velocities as shown in Fig. 3b (For comparison, the time axis is returned to zero).

OPEN IN VIEWER

FIG. 2.

Motion of particles induced by a pendant charged droplet.

OPEN IN VIEWER

FIG. 3.

Particles deposit on surface of charged droplet (experimental data). (a) Trajectories of particles move toward the charged droplet surface. (b) The velocity of the particles. (c) Resultant force on the particles. (d) Air drag and gravity force on the particles. (e) The electrostatic forces on the particles.

Motion of particles in the gas phase can be described by the following Newton equation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { d { u_p } } { dt } } = { f_D } ( { u_g } - { u_p } ) + { \frac { g ( { \rho _p } - { \rho _g } ) } { { \rho _p } } } + { f_e } \tag { 1 } \end{align*} \end{document}

where t is the time and u_p and u_g are the particle and gas velocity, respectively, in which u_g ≈ 0. f_D is the drag force coefficient of per unit particle mass and can be expressed as Equation (2). g is the gravitational acceleration, ρ_p and ρ_g are particle and gas density, respectively, and f_e is the electrostatic force per unit mass. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { f_D } = { \frac { 18 { \mu _g } } { d_p^2 { \rho _p } } } { \frac { { C_D } Re } { 24 } } \tag { 2 } \end{align*} \end{document}

where μ_g is the gas viscosity and d_p is the particle diameter. C_D is the drag coefficient and is determined in Morsi and Alexander, 1972. Re is the Reynolds number: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}Re = { \rho _g}{d_p} |{{u_p} - {u_g}} | / { \mu _g} \tag{3}\end{align*} \end{document}

The left side of Equation (1) is the particle acceleration a_p and can be obtained by the following equation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { a_ { pi } } = { \frac { { u_ { pi } } - { u_ { pi - 1 } } } { \Delta t } } \tag { 4 } \end{align*} \end{document}

where the subscript i indicates the trajectory points in Fig. 3a. The resultant force m_pa_p exerted on the particles is presented in Fig. 3c. Although particles with different takeoff points on the plate have different velocities, the resultant forces acting on them are close to each other. The first and the second term on the right side of Equation (1) are the stresses of air drag and gravity, respectively. Such forces on the particles in Fig. 3a are shown in Fig. 3d (the change of gravity force m_pg along the velocity direction is neglected). Obviously, they fluctuate in a very limited range. Based on the obtained stresses in Equation (1), the electrostatic forces m_pf_e are calculated as shown in Fig. 3e. m_pf_e represents the forces related to the electrostatic field, for instance, Coulomb force, image forces, and gradient force. The effect of the image forces strongly depends on the distance between the particle and the charged droplet and is only important when they are sufficiently close. The gradient force can be expressed in Equation (5). It is related to the variation of the electric field intensity. Particle 2 moving off the plate perpendicularly in Fig. 3a is owing to the effect of gradient force. Except the two limiting circumstances, particle motion is predominated by the Coulomb force. It should be mentioned that the evaporation of the droplet can be strengthened by the electrostatic field (Takano et al., 1994; Bajgai et al., 2006), which intensifies the ion emission from the droplet. Free ions condensing onto the particles weaken the effect of electrostatic attractions. This is probably a reason causing the fluctuation of electrostatic forces in Fig. 3e. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { F_ { grad } } = { \frac { \varepsilon - { \varepsilon _0 } } { \varepsilon + 2 { \varepsilon _0 } } } 2 \pi r_p^3 { \varepsilon _0 } \nabla { E^2 } \tag { 5 } \end{align*} \end{document}

where ɛ is the permittivity of the particle and ɛ₀ is the vacuum permittivity. r_p is the radius of the particle and E is the electrical field intensity.

Rebound of particles

Particles physically contact the droplet after the motion in gas phase. However, not all the particles are captured by the charged droplet: a part of them bounces off the droplet surface after impaction. This phenomenon is shown in Fig. 4 (also see Supplementary video available at www.liebertpub.com/ees). The interactions with droplet surface determine the motion behaviors of the particle: capture or rebound. According to the study of Wang et al. (2015), a 10 μm particle can bounce off the liquid surface, while the impact velocity is as high as several meters per second. However, the impact velocity of Particle 3 in Fig. 4 is ∼0.23 m/s, too small to cause rebound. Moreover, particles with lager charge relaxation time, τ^E = ɛ_pɛ₀P_p, are more likely to be captured (Sumiyoshitani et al., 1984), where ɛ_p and P_p are permittivity and resistivity of the particle, respectively. In other words, the electrostatic attraction can last for a considerable time after the particle physically contacting the charged droplet due to its leaky dielectric characteristic. For glass beads used in our work, τ^E is >3.3 s (ɛ_p ≈ 3.7, P_p > 10¹¹ Ω/m), which is a remarkable interval to extend the effect of electrostatic attraction. Although the particles have low velocity and large relaxation time, they bounce off the droplet surface.

OPEN IN VIEWER

FIG. 4.

Rebound of particle.

The capture/rebound criterion proposed by Wang et al. (2015) is based on the normal impact of particle onto the liquid surface. In fact, an inclined impact is a more general collision form. The impact angle of Particle 3 in Fig. 4 is measured as shown in Fig. 5a. Obviously, it impacts the droplet surface with the impact angle of 84°. For comparison, Particle 2 impacts the interface perpendicularly in Fig. 5a (or in Fig. 3a) and it is captured. An inclined impact leads to the tangential velocity of the particle along the interface. The existence of tangential velocity could substantially reduce the adhesion, and therefore, the particles are more likely to rebound. The effect of impact angle on the motion behaviors of the particles is shown in Fig. 5b. The impact position of the particles on the surface of the charged droplet is defined as the location angle, as shown graphically in the insert of the figure. In Fig. 5b, the rebound particles account for 11% (11/100) of the sample size, among which, the proportion that the impact angle of particles γ ≤ 85° accounts for 82% (9/11). On the other hand, 3% (2/81) of the particles rebound as their impact angle γ > 85°, while it is 47% (9/19) for γ ≤ 85°, which means nearly half of the particles rebound when their impact angle is ≤85°. Smaller impact angle could cause higher probability of rebound. However, the impact angle smaller than 80° has not been observed. For the location angle, the rebound hardly appears when it is smaller than −70°, indicating that such region of the charged droplet is the most effective area to capture the particles.

OPEN IN VIEWER

FIG. 5.

Effect of impact angle on motion behaviors of particles. (a) Measurement of particle impact angle. (b) The influence of location angle and impact angle on impact behaviors of the particles. Insert: the definition of location angle (experimental data).

Motion of particle clusters

Part of the particles on the plate coalesces to clusters due to electrostatic forces and air humidity. These particle clusters move toward the charged droplet mingling among the monodispersed particles. Compared to single particles, particle clusters have a larger volume, but a looser structure. Most particle clusters disintegrate when they impact the surface of the charged droplet, as shown in Fig. 6a. Some child particles bouncing off the interface forms the reentrainment. Remarkably, several particle clusters pass the charged droplet without physical contact, which is exhibited in Fig. 6b. This is related to the free irons in gas phase. Besides partial discharging of gas phase, the evaporation of droplet strengthened by the electrostatic field emits free irons as well. The larger volume of particle clusters enables them to capture more free charges during their motion, which even offset their initial polarity and repulsion by the charged droplet. The velocity variation of the cluster in Fig. 6b is shown in Fig. 6c, the significant decline of the velocity appears at 1.067–2 ms, which indicates that the Coulomb repulsion affects the motion of the particle cluster.

OPEN IN VIEWER

FIG. 6.

Motion behaviors of attracted particle clusters. (a) Particle cluster disintegrates when impacting the charge droplet surface. (b) Particle cluster passes the charged droplet without physical contact. (c) The velocity variation of the particle cluster passing the charged droplet in (b).

Motion of particles influenced by gas flow

The impact angle of the particles observed in the experiments is >80°. However, it can be affected by air flow in electroscrubbing. To estimate impact angle of the particles influenced by the air flow, a 2D numerical model is established, in which a spherical collector is fixed in the particle-laden gas flow to intercept the particles as show in Fig. 7 (considering that the droplet is set as solid boundary, it is named collector in the simulation section). A locally refined triangular mesh is adopted with about 210,000 grids. The numerical algorithm traces the movement of the particles in gas phase without considering the interactions between the particles. The gas flow is assumed laminar, Newtonian, and incompressible. The basic governing equations of gas flow are the equations of mass and momentum conservation: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\nabla \cdot {u_g} = 0 \tag{6}\end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { { \partial } { \rho _g } { u_g } } { \partial t } } + \nabla \cdot { \rho _g } { u_g } { u_g } = - \nabla p + \nabla \cdot [ { \mu _g } ( \nabla { u_g } + \nabla { u_g } ^T ) ] + { \rho _g } g \tag { 7 } \end{align*} \end{document}

where p is the pressure.

OPEN IN VIEWER

FIG. 7.

Simulation domain of model.

Trajectories of the particles are traced by solving Equation (1). It is considered that both the particles and the collector are charged oppositely. The electrostatic forces on a charged particle, including Coulomb force and image forces, can be calculated from the following formula: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{split}{m_p}{f_e} = &{{{Q_p}{Q_c}} \over {4 \pi { \varepsilon _0}{r^2}}} + {{{r_p}Q_c^2} \over {4 \pi { \varepsilon _0}{r^3}}} \left({{{r^4}} \over {{{ ( {r^2} - R_p^2 ) }^2}}} - 1\right) \\&+ {{{r_c}Q_p^2} \over {4 \pi { \varepsilon _0}{r^3}}} \left({{{r^4}} \over {{{ ( {r^2} - R_c^2 ) }^2}}} - 1 \right)\end{split} \tag{8}\end{align*} \end{document}

where Q_p and Q_c are charge of the particle and collector, respectively, r_p and r_c are radius of the particle and collector respectively, and r is the distance from collector to attracted particle.

The first term on the right side of Equation (8) is Coulomb force, the second and third terms are image forces from collector to particle and particle to collector, respectively. The image forces are only important when the droplet and the particle are sufficiently close. Considering the momentum of the particle, the effect of image forces on the impact angle is not obvious. In this simulation, the image forces are neglected (Adamiak et al., 2001).

The particle trajectories are mainly affected by two nondimensional parameters: Stokes number St and Coulomb number Kc. The inertial deposition is proportional to the St number: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}St = \frac {2{C_c} r_p^2 {\rho_p } { u_0 } } { 9 \mu { r_c } } \tag { 9 } \end{align*} \end{document}

where u₀ is the initial velocity of the gas flow and C_c is the Cunningham slip correction factor and approximately equals to 1. The importance of the electric force is expressed by the Kc number (Adamiak et al., 2001): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}Kc = { \frac { { C_c } { Q_p } { Q_c } } { 24 { \pi ^2 } { \varepsilon _0 } \mu { u_0 } r_c^2 { r_p } } } \tag { 10 } \end{align*} \end{document}

The inertia effect of the particle increases as the St number increases. The influence of St number on the impact angle of the particle is shown in Fig. 8a. It is obvious that the impact angle decreases sharply as St number increases from 0.1 to 1. Particularly, the impact angle drops to 0°, while the location angle is larger than −13° in the case of Kc = 1 and St = 10, which means no particles impact such region of the collector (the related impact angle and location angle are defined in Fig. 8c). In other words, the collection efficiency on the windward of the collector is becoming prominent, while the leeward quits collecting particles. In contrast, the effect of electrostatic attraction is promoted by the increase of Kc number. The influence of Kc number on the impact angle of the particle is shown in Fig. 8b. The impact angle increases as the Kc number increases, especially at the windward of the collector (λ < 0°).

OPEN IN VIEWER

FIG. 8.

Effect of St number and Kc number on impact angle of particles. (a) Effect of St number on impact angle. (b) Effect of Kc number on impact angle. (c) Definition of location angle λ and impact angle γ.

Experimental observations show that particle rebound is a random event, which is closely related to their impact angle. Nearly half of them bounce off the surface of charged droplet when their impact angle is <85°. For simulations, suppose that particle rebound would happen for certain, while its impact angle is <85°, the predicted collection efficiency of the droplet will be lower than actual conditions. That means, the criterion of 85° rebound is the lower limit for numerical prediction and the actual collection efficiency must be higher than it. In addition, if the impact angle of the particle is <45°, the tangential component velocity is larger compared with the normal component, which means the adhesion of particle on the droplet surface is overwhelmed by the tangential component momentum (inertia force) and is hardly to be captured. Therefore, the criterion of 45° rebound can be considered the upper limit, which means the particle rebounds when its impact angle is <45° in the simulations; the predicted collection efficiency of the droplet will probably be higher than actual conditions. Briefly, the actual collection efficiency of the charged droplet will probably locate between the two criteria predicted by the simulations.

The definition of collection efficiency for the numerical simulations has not reached a consensus (Nielsen and Hill, 1976; Metzler et al., 1997; Jaworek et al., 2002). In this study, the collection efficiency η is presented in Equation (11), which is more suitable for our simulations. Physically, η characterizes the capture capacity of the particle deposition when the collectors with the diameter of d line up normal to the projection direction of the injection boundary as shown in Fig. 9a. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\eta = { \frac { { N_c } } { { N_ { in } } } } \cdot \frac { D } { d } \tag { 11 } \end{align*} \end{document}

where, N_c is the number of captured particles in unit interval. It is a mean value calculated from the last 10 time steps when the trajectories of particle maintain quasistatic. N_in is the number of injected particles in unit interval. D is the length of injection boundary.

OPEN IN VIEWER

FIG. 9.

Collection efficiency of particles on a charged collector. (a) Definition of collection efficiency. (b) Effect of the St number and Kc number on collection efficiency of a fixed collector (simulation data).

Effects of St and Kc number on the collection efficiency is exhibited in Fig. 9b. The figure consists of four curves: the lines of Kc = 0 and Kc = 1, both without consideration of the rebound; the lines of Kc = 1, γ = 45° and Kc = 1, γ = 85°, the critical impact angle of rebound are 45° and 85°, respectively. The lines of Kc = 1, γ = 85° and Kc = 1, γ = 45° intersect with the line of Kc = 0 at the points of P₁ (St = 0.63) and P₂ (St = 6.1) in the figure. The dot-dash lines crossing the two points divide the area enclosed by lines of Kc = 1, γ = 45° and Kc = 1, γ = 85° into three subdomains: region 1, region 2, and region 3. Region 1 implies that if St < 0.63, the collection efficiency of a charged collector (Kc = 1), must be higher than the uncharged one; correspondingly, the collection would be less effective when St > 6.1 in Region 3. Region 2 is the transition area that the higher collection efficiency between the charged and uncharged collector is obscure, which results from the uncertain probability of the rebound.

From Figs. 8a and 9b, the collection efficiency of the charged droplet declines with the increase of St number. However, a proper St number is beneficial for the transport of the particles approaching the collector. In practice, the St number of the electrostatic wet scrubbing is suggested to be located in region 1 of Fig. 9b, while an acceptable flux of particle-laden gas is guaranteed.