Computational Fluid Dynamic Modelling of Particle Charging and Collection in a Wire-to-Plate Type Single-Stage Electrostatic Precipitator

Electrostatic precipitators (ESPs) have been widely used to control particulate pollutants, which adversely affect human health. In this study, a computational fluid-dynamic model for turbulent flow, particle trajectory, and particle charging in ESPs is presented using a pre-developed corona discharge model (Kim et al., 2010), wherein electric field and space charge distributions in the plasma region are numerically calculated. The ESP under consideration is a wire-to-plate singlestage ESP, which consists of a series of discharge wires and two collecting plates. Two different kinds of particulates are considered in this study; fly ash and sucrose particles. Fly ash was selected because many ESPs have been utilized in coalfired power plants to capture fly ash particles generated from combustion. Sucrose was selected to compare our numerical calculation results with experimental data found in literature. The electrical characteristics of the ESP, particle trajectories, particle charge numbers, and collection efficiencies under various operating conditions are demonstrated. For fly ash, the overall collection efficiencies based on particle mass are 61, 86, 95, and 99% at 45, 50, 55, and 60 kV, respectively, at a flow velocity of 1 m s.


INTRODUCTION
Removal of particles smaller than 2.5 µm in diameter (PM 2.5 ), which are considered as risk factors of diseases, has gained considerable attention (Wen et al., 2015).Electrostatic precipitators (ESPs) have been widely used to remove airborne particles in common industrial particulate control system owing to its high efficiency under low pressure drop (Huang and Chen, 2002;Lin and Tsai, 2010;Ruttanachot et al., 2011;Zhu et al., 2012).The potential of this technique has been established by industries; however, the technique exhibits certain limitations, including a low collection efficiency of submicron particles (Gouri et al., 2013).
In a wire-to-plate type single-stage ESP, space charges are formed by air ions generated as a result of corona discharge when a high voltage is applied between discharge wires and grounded collecting plates.The particles in the flowing gas are charged by the space charges and then collected on the ground plates by an electrostatic field.Because experimental investigation of ESPs is expensive (Adamiak, 2013), numerical analysis has been widely used for design and performance evaluation purposes.The electrostatic precipitation involves a complex interactive physical process among turbulent flow, electric field, space charge distribution and particle charging and motions.Modelling of corona discharge is complicated and challenging even for a straightforward electrode configuration (Adamiak, 2013).Talaie et al. (2001b) developed a corona discharge model based on the fact that increasing the applied voltage increases the plasma region, where ions are generated owing to electron-impact reactions around a discharge wire.Ion density values at the plasma boundary were calculated by empirical equations.Chen and Davidson (2002) numerically investigated the one-dimensional corona plasma region around a discharge wire by considering Kaptzov's hypothesis.They reported that the ion densities remained relatively constant in the plasma region.In a study of our group (Kim et al., 2010), a computational methodology was proposed for calculating the plasma region thickness based on the study of Chen and Davison (2002).The ion densities were obtained using an analytical method and were defined both at the plasma boundary and within the plasma region.Kim et al. (2010) applied their methodology to wire-to-duct ESP, and the results were in reasonable agreement with experimental data of previous studies (Penney and Matick, 1960;MacDonald et al., 1977;Lawless and Spark, 1980;Ohkubo et al., 1986) with various geometries.
For analysis of particle charging and motion in the ESP, the Lagrangian approach has been widely used with corona discharge models (Goo and Lee, 1996;Varonos et al., 2002;Talaie, 2005;Lei et al., 2008;Farnoosh et al., 2011;Liu et al., 2015;Wen et al., 2016).Goo and Lee (1996) developed a numerical model for an ESP using the Lagrangian approach which uses the concept of time-series analysis for the particle motion in a turbulent flow field.They reported that the Lagrangian approach was more effective than the Eulerian approach to analyze particle motion in an ESP because the effects of particle inertia and non-uniform diffusivity on the particle motion were omitted in the Eulerian approach.Moreover, the Lagrangian approach can be combined with dynamic charging to obtain particle charge numbers, which are affected by particle size, particle permittivity, electric field intensity and air ion concentration.Talaie (2005) applied the work of Talaie et al. (2001b) to simulate the performance of wire-duct single-stage ESPs.Talaie et al. (2001a) stated that the main advantage of the Lagrangian approach in comparison with the Eulerian approach is to consider the effect of particle size.
This article reports a computational study that is an expansion of the study of Kim et al. (2010) that simulated the electric field and space charge distributions of a wireto-plate ESP by taking into account the plasma region.In this article, flow velocities and particle trajectories were calculated and particle collection efficiency in a wire-to-plate ESP was predicted.The standard k-ε turbulent model based on Finite Volume Method and the Lagrangian approach with dynamic particle charging were added to the corona discharge model of Kim et al. (2010) to solve the turbulent flow and the motion and charging of particles in the ESP.The GAMBIT 2.0 code and FLUENT 14 with user-defined function (UDF) were used.

METHODOLOGY
The standard k-ε turbulent model was used to solve gas flow in FLUENT 14. Particle trajectory was calculated by using discrete phase model (DPM).Poisson equation, charge conservation equation and a storing place were defined by using three user-defined scalars (UDSs) as well as the userdefined function (UDF) of DEFINE_SOURCE.The effect of electrical body forces on the gas flow was observed by the UDF of DEFINE_SOURCE.Space charge distribution was updated by using the UDF of DEFINE_EXECUTE_AT_ END.Electrical potentials and charge densities were converted into electric fields and current densities, respectively, by using the UDF DEFINE_ON_DEMAND.The UDF of DEFINE_SCALAR_UPDATE was used to update particle charge with a trapezoidal rule.Two UDFs of DEFINE_DPM_BODY_FORCE and DEFINE_DPM_ DRAG were used to apply electrical body forces on the particles and take into consideration the particle slip, respectively.

Gas Flow Field
For incompressible flow, the equations of continuity and momentum with electrical body force ( ) E F  can be expressed as follows: where ρ is the density of air, u  is the flow velocity, p is the pressure, I is the unit vector, µ is the dynamic viscosity of air, µ T is the eddy viscosity defined as ρC μ k 2 ⁄ε, k is the turbulent kinetic energy, ε is the dissipation rate, q is the charge density of gaseous ions and E  is the electric field.
In the standard k-ε turbulent flow model, the equations governing the turbulent kinetic energy and dissipation rate are expressed as follows: where σ ε and σ k are the turbulent Prandtl numbers for k and , respectively, and G is the production rate of k (Launder and Spalding, 1974).The values of the constants are as follows:

Particle Trajectory and Charging
Particles enter the inlet and migrate toward the outlet under the coupled effect of the electric field, space charge and flow field.The particle trajectory is expressed using the following equations: where p u  is the particle velocity, d p is the particle diameter, ρ p is the mass density of a particle, p F  is the Coulomb's force exerted on a particle, m p is the mass of a particle and q p is the electrical charge of a particle (= n p e; n p is the charge number of a particle and e is the charge of an electron).C c is the slip correction factor defined as follows (Allen and Raabe, 1982): where is the mean free path of air (0.066 µm at 1 atm and 293 K) (Hinds, 1999).
Particles in an ESP are charged by air ions generated by corona discharge.The particle charge number (n p ) is estimated using diffusion and field charging theories (Reist, 1993): where n d is the number of charges on a particle by diffusion charging, n f is the number of charges on a particle by filed charging, n s is the saturation charge attained after sufficient time at a specified charging condition, k B is the Boltzmann constant, T is the absolute temperature, K E is the constant of proportionality = 1/4πε 0 , ε 0 is the permittivity of vacuum, c i is the mean thermal speed of air ions (240 m s -1 at 298 K and 1 atm), N i is the ion number concentration (assumed as N i = q/e), t is the residence time, Z i is the ion mobility (1.4 m 2 V -1 s -1 of positive ions) and ε p is the relative permittivity of the particle.

Electric Field and Space Charge Distributions
The distributions of electric potential (V) and ionic current density ( ) j  are governed by the Poisson equation and charge conservation equation as follows: For a uniform temperature and by omitting the diffusion effect, the ionic current density is defined as follows: where E V    . The flow velocity u  is omitted in this study as it is negligible (under 10 m s -1 ) compared to the ion velocity ( i Z E  , in the order of 10 6 m s -1 ).Once j  is obtained, the electric current can be calculated by integrating Eq. ( 15) over the area of the ground electrode.

MODEL DESCRIPTION
In order to validate the proposed CFD model, the electrodes configuration and experimental data of Lawless and Sparks (1980) were used.They performed experiments on a high-voltage corona wire-to-plate configuration in the absence of particles.
The geometry configuration and computational region of this study are illustrated in Fig. 1 (eight discharge wires and two collecting plates).The wire-to-plate distance (S), wire-towire distance (D), wire radius (r w ) and duct length (L) were 114.3 mm, 228.6 mm, 1.59 mm and 1.8288 m, respectively.The plasma boundary was located at distance r p from the center of a wire.The radius (r p ) of the plasma region is defined such that the reduced electric field strength (E/N = Td) is higher than 120 Td (N is the neutral gas density, which can be obtained from the ideal gas law) (Grabarczyk, 2013).The computational region contained 37,392 meshes with a quadratic shape (See Fig. 2).The mesh located in the proximity of a wire was substantially refined to 25 µm owing to the large gradient of the electric potential in the proximity of the wire.
Boundary conditions used in this study are summarized in Table 1.u a and V a represent the inlet gas flow velocity and applied voltage on a wire, respectively.q a is the charge density in the plasma region.The initial particle charge number was assumed to be zero.
The semi-implicit method for pressure-linked equations (SIMPLE) algorism was used to specify pressure-velocity coupling in the equations of continuity and momentum.In order to solve the coupling of the Poisson equation and charge conservation equation for specified applied voltage, the discharge model presented by Kim et al. (2010) was used.
Two different particles were considered in this study; fly ash and sucrose particles.'Fly ash' was selected because many ESPs have been utilized in coal-power plants to capture fly ash particles generated from combustion.'Sucrose' was selected to compare our numerical calculation results with experimental data of Huang and Chen (2002).Huang and Chen (2002) carried out a controlled experimental study as part of their health and toxicological studies to understand the behaviour ultrafine particles (diameter < 100 nm) in ESPs.Sucrose aerosols of about 1.2 × 10 5 numbers cm -3 in total number concentration, 42 nm of geometric mean diameter (GMD), and 1.8 of geometric standard deviation were generated with an electrospray aerosol generator.Using these sucrose aerosol particles, Huang and Chen ( 2002 carried out performance tests of their Lab-made ESPs.The relative permittivity (ε p and the mass density of sucrose particles were assumed as 3.3 and 1.59 g cm -3 , respectively.The geometric mean diameter of fly ash particles was assumed as 0.1 µm based on the experimental results of Strand et al. (2002).The relative permittivity and the mass density of fly ash particles were assumed as 3.0 (Baoyi et al., 2012) and 2 g cm -3 (Ghosal and Self, 1995), respectively.
The number distribution of particles of size d p that enter the ESP was assumed to be a log-normal distribution; where σ g is the geometric standard deviation, which was assumed to be 2.0 (Ghosal and Self, 1995).The number distribution was also converted into the mass distribution using the following equation: where m i is the total mass of the particles in group i, and M is the total mass of all the groups (= ∑m i ).

RESULTS AND DISCUSSION
Numerical calculations were conducted for various applied voltages in the range of 45-60 kV and various duct flow velocities in the range of 0-8 m s -1 .The temperature and pressure were 293 K and 1 atm, respectively.For each flow velocity, the spatial distributions of voltage and electric field between the eight discharging wires and a collecting plate were obtained by solving the Poisson equation (Eq.( 3)) as well as the Laplace equation (q = 0 in Eq. ( 3)).The spatial distributions of voltage are illustrated in Figs.3(a) and 3(b), respectively, for q = 0 and q ≠ 0, when the applied voltage was 60 kV.Fig. 3(c) illustrates the electric field distribution along the Y-axis at X = 0 when q = 0.For each value of the applied voltage, the electric field was the maximum at the wire surface; however, it decreased remarkably with distance along the Y direction.The electric field at the discharging wire (Y = r w = 1.59 mm) increased from 5.79 MV m -1 to 7.72 MV m -1 as the applied voltage increased from 45 kV to 60 kV.However, when the Poisson equation was used for calculation (Fig. 3(d)), the electric field at the discharge wire was constant at 5.33 MV m -1 for various applied voltages.This phenomenon can be explained with Kaptzov's hypothesis that the electric field at the surface of a discharging wire remains constant at the corona onset voltage for all values of voltage above the onset voltage.The corona onset electric field at the discharge wire can be obtained using Peek's law as follows; where f is a factor that accounts for wire roughness, δ is the air density at 1 atm and 25°C.For the geometry configuration used in this study, Eq. ( 18) results in 5.49 MV m -1 , which is approximately equal to the calculated value (5.33 MV m -1 ).
It is noteworthy that in Fig. 3(d), the electric field decreases in the proximity of the discharge wire and increases in the proximity of the collecting plate, when compared to Fig. 3(c).The phenomenon occurs in the presence of space charges as the net electric field formed at any location is equal to the vector sum of ± Y direction fields produced by all the space charges (Superposition principle).The results presented in Fig. 3(d) were in reasonable agreement with those of previous studies (Sekar and Stomberg, 1981;McLean, 1988).
The charge density (q) distribution when the applied voltage was 60 kV is presented in Fig. 4(a).The variation of charge densities at X = 0 along the Y-axis under various voltages is illustrated in Fig. 4(b).The current densities at the collecting plates were also calculated for various applied voltages (see Fig. 5).The results were in agreement with the experimental data of Lawless and Sparks (1980), which was also verified by Kim et al. (2010).
Fig. 6 illustrates trajectories of fly ash particles under various applied voltages when the flow velocity was 1 m s -1 .Three particles each of size 0.1 µm entered the ESP at three positions, namely position #0 (Y = 0), position #1 (Y = 28.6 mm) and position #2 (Y = 57 mm), respectively.The trajectories of these three particles are represented as P0, P1, and P2, respectively.Each particle gradually acquired space charge as the particle advanced with the flow along the X-direction and approached the collecting electrode located at Y = 114.3mm (see Fig. 2).When the applied voltage was 45 kV, all the three particles finally escaped from the ESP; however, all of those particles were captured when the applied voltage increased to 60 kV.Fig. 7 illustrates that these three particles gradually acquired space charges when they advanced along the X-direction.The particle migrating in the proximity of the discharging wire (injected at position #0) carried a higher charge than the other two particles (injected at positions #1 and #2) owing to the stronger electric field (~5 MV m -1 ; see Fig. 3(d)) and higher charge density (see Fig. 4(b)).The charge numbers of particles injected at positions #1 and #2 were approximately equal as these two particles passed through regions which had similar electric fields; for example, when the applied voltage was 60 kV, the electric field in 40 mm < Y < 114.3 mm varied from 0.38 MV m -1 to 0.47 MV m -1 (see Fig. 3(d) and Eq. ( 12)).
Fig. 8(a) presents the collection efficiency of ESP for 0.1 µm particles (fly ash) under various flow velocities and various applied voltages.The collection efficiency is defined as % 100 where N t and N i are the number of particles trapped on the collecting plate and the number of particles injected to the inlet, respectively.Two hundred particles were injected to calculate the collection efficiency.For any applied voltage, the collection efficiency decreased exponentially as the flow velocity increased.These results can be substantiated with the Deutsch-Anderson equation (Hinds, 1999), which is expressed as follows: where U is the flow velocity.The electrical migration velocity (W) is expressed as W = Z p E; here, Z p is the electrical mobility, which is defined as follows: When the flow velocity increases, the residence time decreases, and therefore, the charge number decreases (see Eqs. ( 9)-( 12)).The decrease in charge number results in the decrease of electrical mobility, electrical migration velocity and collection efficiency.
Fig. 8(b) presents the collection efficiencies of ESP for various fly ash particle sizes and various applied voltages at flow velocity of 1 m s -1 .The collection efficiency trend followed that in literatures (McLean, 1988;Yoo et al., 1997;Ylätalo and Hautanen, 1998;Huang and Chen, 2002).The efficiency was the minimum for particle sizes of 0.2-0.3µm regardless of the applied voltage.For example, when the applied voltage was 60 kV, 0.1 µm particles and 0.5 µm particles had highly similar collection efficiencies, 72% and  68%, respectively, notwithstanding the significantly dissimilar charge numbers, 4.17 and 44.24, respectively.However, it is interesting to note that the 0.1 µm particles and 0.5 µm particles had highly similar electrical mobilities (Z p ) of 115 µm 2 V -1 s -1 and 109 µm 2 V -1 s -1 , respectively.
For particles smaller than 0.2 µm, diffusion charging is the predominant mechanism even in the presence of electrostatic fields.Moreover, the particle charge (n p ) is proportional to d p ln(1 + αd p ), where α is the constant related to ion-particle collisions.Considering that the Cunningham correction factor can be approximated as 3.69 (λ/d p ) 1/2 for the intermediate range of λ/d p (Lee and Liu, 1980), the particle  mobility (Z p ) is proportional to d p -1/2 for d p < ~200 nm, and therefore, η decreases as d p increases.For particles larger than approximately 200 nm in diameter, field charging is the dominant mechanism, and n p ~ d p 2 .Therefore, the particle mobility is proportional to d p 1/2 for d p > ~200 nm, and hence, η increases as d p increases.As the applied voltage increases, the collection efficiencies also increase notwithstanding the sizes of particles because of the higher particle charging.
The correlations between the electrical mobility and collection efficiency at various applied voltages are illustrated in Fig. 9.A linear correlation was obtained for each applied voltage.For |2WL/US| << 1 (~10 -6 in this study), the Deutsch-Anderson equation (Eq.( 20)) is expressed as follows, from the Tayler series; Therefore, the collection efficiency increased linearly with electrical mobility under constant applied voltage.The rate of increase becomes higher with applied voltage.Fig. 10 illustrates the effect of ESP on fly ash particle number and mass distributions under various applied voltages at flow velocity of 1 m s -1 .The overall collection efficiencies based on particle number were 30, 54, 75 and 90% at applied voltages of 45, 50, 55 and 60 kV, respectively.At the  downstream, the proportion of 0.2-0.3µm particles increased owing to their low collection efficiency.As the emission regulations of PM are typically based on mass, the particle number distribution was converted into mass distribution.Then, the mode diameter of the upstream particles was shifted to 3 µm.The overall collection efficiencies based on particle mass were 61, 86, 95 and 99% at 45, 50, 55 and 60 kV, respectively.Similar plots could be prepared for the various flow velocities.
So far, our CFD model for turbulent flow, particle trajectory, and particle charging was applied to a wire-toplate type single-stage ESP which consisted of a series of eight discharge wires and two collecting plates.With fly ash particles having a log-normal size distribution (GMD = 0.1 µm, geometric standard deviation = 2.0), the electrical characteristics of the ESP, particle trajectories, particle charge numbers, and collection efficiencies under various operating conditions were demonstrated.
Our CFD model was also applied to another wire-toplate type single-stage ESP (three discharge wires and two collecting plates) which was used in an experimental study of Huang and Chen (2002).In their study, sucrose aerosols having a log-normal size distribution (GMD = 42 nm, geometric standard deviation = 1.8) were used.The wireto-plate distance (S), wire-to-wire distance (D), wire radius (r w ) and length of collecting plate (L) were 60 mm, 42 mm,  0.3 mm and 300 mm, respectively.The air flow rate and the applied voltage were 100 L min -1 and 26.4 kV, respectively.Fig. 11 shows experimental data of the upstream and downstream size distributions of sucrose aerosols as well as collection efficiencies for different particle sizes.Fig. 11 also shows that our CFD calculation results were in good agreements with the experimental data of Huang and Chen (2002).
It should be noted that the effect of humidity (water vapour concentration) on corona discharge was not considered in our CFD model.In a real situation, the corona onset voltage decreases with the increase of water vapour concentration.Therefore, a correction function of water vapour concentration should be added to a conventional Peek formula (Wang and You, 2013).

CONCLUSIONS
A CFD model for turbulent flow, particle trajectory, and particle charging was developed and applied to two different wire-to-plate single-stage ESPs.Particle trajectories, particle charging, and collection efficiencies were simulated for various flow velocities, particle sizes, and applied voltages.The results were in reasonable agreement with those of previous studies.Two kinds of particulates were considered in this study; fly ash and sucrose, which possess different surface properties and carry different levels of charges.The differences in size, density, and relative permittivity affected particle charging, trajectories, and collection efficiencies.The proposed model can be used to design ESPs for removing or sampling particulate materials in the air.

Fig. 3 .
Fig. 3. Electrical characteristics: (a) Voltage distribution solved by Laplace equation at 60 kV of applied voltage; (b) Voltage distribution solved by Poisson equation at 60 kV of applied voltage; (c) Variations of electric field solved by Laplace equation along Y-axis for various applied voltages; (d) Variations of electric field solved by Poisson equation along Y-axis for various applied voltages.

Fig. 4 .
Fig. 4. (a) Charge density distribution at 60 kV of applied voltage; (b) Variation of charge density along the Y-axis for various applied voltages.

Fig. 5 .
Fig. 5. (a) Current density distribution at 60 kV of applied voltage; (b) Average current density on the collecting plates for various applied voltages.

Fig. 8 .
Fig. 8. Collection efficiency at flow velocity 1 m s -1 ): (a) for particles of 0.1 µm under various applied voltages and flow velocities; (b) for various particle diameters under various applied voltages.

Fig. 9 .
Fig. 9. Correlations between the electrical mobility and the collection efficiency for various particle diameters and applied voltages at flow velocity 1 m s -1 .

Fig. 10 .
Fig. 10.Particle distributions of fly ash particles under various applied voltages at upstream and downstream of the ESP at flow velocity 1 m s -1 : (a) Number; (b) Mass.

Fig. 11 .
Fig. 11.Calculated and experimental collection efficiencies for different particle diameters under different applied voltages.