Novel accurate modeling of dust loaded wire-duct precipitators using FDM-FMG method on one fine computational domains

Worldwide, electrostatic precipitators (ESP) have been extensively utilized to separate fine particles for diverse large-scale industrial applications. In this regard, this paper presents a novel approach for modeling the dustloaded ESP on the fine computational domain where the need for a fast solver arises. Unlike the previously published numerical techniques, the finite difference method (FDM) integrated with a full multi-grid method (FMG), labeled FDM-FMG, is developed to resolve Poisson and continuity equations on one fine computational domain. For clean and dust-loaded ESP, the proposed FMG is checked versus successive over-relaxation (SOR) on fine domains where the proposed one is greatly transcendent in terms of convergence characteristics and hence the computational performance (CPU time). For the first time, two major issues are highlighted and solved: the first concerning issue is the chosen ion mobility as an important factor in the simulation results and the second one is choosing an optimal computational grid for dust loaded precipitators that grantees both low truncation and roundoff errors, results in well-matched with experimental measurements nominated in the previous publishing. The novel idea of working on various grid sizes and tracking the optimal ones gives the FDM-FMG an advantage of predicting a precise picture for the electrical situations in industrial ESP over the other numerical techniques. After all, the impact of changing the spacing between the different wires and the height of the ionized wires on the distributions of current, ion, and particle charge densities on the ground are deeply simulated and presented in dust-loaded ESP. The proposed FDM-FMG can be a promising tool for the designers and manufacturers of precipitators, thanks to its superior computational performance.


Introduction
Recently, the world witnessed a growth in electrical power generation which is met by an increase in the gas flows resulted from the industrial power stations. These industrial power plants cause air pollution which can be minimized by using electrostatic precipitators [1,2]. Besides, precipitators remove the sulphuric acid produced from gold recovery technologies [3,4]. The most widespread type of precipitator is the wire-duct electrostatic precipitator (ESP) consisting of a number of paths formed by parallel grounded plates with several wires centrally located between the grounded plates. A detailed explanation of the basic operation of wire-duct ESPs is illustrated in [5]. Several numerical methods are established to study the electrical properties including the voltage and current characteristics in clean air electrostatic precipitators. Accordingly, the electrical problems concerning the operation during different loading conditions can be predicted. The most affecting parameters on the performance of wire-duct ESPs are wire diameter, the distance between wires as well as the height of the wire over the grounded plate. The numerical techniques implemented to model the corona problem in ESPs are classified as follows: finite difference method (FDM) [2,6,7], finite element method (FEM) integrated with FDM [8], boundary element method (BEM) combined with the method of characteristics (MOC) [9], BEM integrated with FDM [10], FEM [11], charge simulation method (CSM) [12], and FEM combined with MOC [13][14][15]. Cristina and Feliziani [16] use the FEM to model the dust-loaded precipitator for calculating the V-I curves and precipitation efficiencies for different sizes of particles.
In the literature, there are several numerical based methods for ESPs.
Castle [17] formulated an approximate analytical expression for the simulation of a dust-loaded precipitator. Besides, Abdel-Sattar [18] implements the CSM to model the cement and steel particles. Al Hamouz et al. [19] uses a finite element-based method integrated with an adapted MOC to calculate the current density shapes, corona power losses, and corona current in wire-duct ESP under particle loading conditions. On the other side, a group of experimental work has been conducted below dust loading circumstances. For instance, Jedrusik et al. [20] studied the effect of changing the physicochemical features of the fly ash on collection proficiency. Miller et al. [21] examined the effect of diverse electrode designs on the efficiency of the wire-duct ESP. Zhuang et al. [22] perform experimental and theoretical research studies on the performance of a cylindrical precipitator for the gathering of ultra-fine particles ranges between (0.05-0.5 µm). Bacchiega et al. measured the mass collection rates along with a pilot wire-duct ESP in an industrial environment [23]. Modeling of the electrostatic field in the dust-loaded precipitators on fine computational domains can provide the manufactures and designers with valuable guidelines for improving the performance of precipitators and dust collection efficiency.
In general, all the numerical techniques are time-consuming when dealing with fine domains especially in solving the Poisson equation. The FDM is utilized to model the corona problematic wire-duct ESPs, by approximating Poisson and continuity equations by the difference equations. Among the conventional iterative procedures used to resolve the finite difference equations are: 1) Jacobi method, 2) Gauss-Seidel method, and 3) successive over-relaxation technique (SOR). These iterative techniques are inefficient and have poor convergence when the rectangular grid turns out to be finer and finer [2,24]. Accordingly, on fine computational domains, these classical iterative techniques consume an extreme computational time to converge to a certain optimal value. So, the need for a fast convergent iterative solver arises. The full multigrid method (FMG) solves the above-mentioned problem by using many grids of dissimilar mesh sizes. In this method, the fine computational domain is transferred to a coarser domain, where one of the traditional iterative approaches can be applied to resolve the coarser grid. Later, by grid transfer operators, the solution is used as an upgraded primary guess for the finer grids [24,25,26,27,28]. Most of the preceding research studies are missing to accurately model the dust-loaded ESP on one fine computational domain by the combination of FDM and the FMG solver, which is covered in this work.
To cover the abovementioned gap in the literature, this study is aiming to propose the finite difference method combined with the full multigrid method (FDM-FMG) to be implemented for modeling the dustloaded ESP on one fine computational domain. Firstly, FDM-FMG is used to model the clean air ESP and is validated with previous experimental work. The convergence characteristics of this technique were demonstrated to be better than the other classical iterative methods, so less computational time is needed. Secondly, the problem of choosing the ion mobility value is discussed for the first time. Finally, the FDM-FMG is used to track the optimal computational grid for dust-loaded precipitators that ensures both low truncation as well as round-off errors. This optimal grid is tested against other published numerical techniques for previous published experimental measurements and better agreement is reached. Using this optimal grid, the impact of changing the spacing between wires as well as the height of the ionized wires in dustloaded precipitators on the distributions of current, ion, and particle densities on the ground are intensively investigated.
The novelty of this paper can be concluded as follows: • The FDM has applied one very fine computational domain in dustloaded precipitators for the first time taking into consideration both truncation and round-off errors. • With the help of the FMG solver, FDM becomes time efficient on different tolerances.
• The proposed FDM-FMG gives a more confident value of the chosen ion mobility without the need for excessive experimental efforts to measure it. • FDM-FMG is an excellent prediction tool for the electrical properties in both clean air and dust-loaded precipitators. • It is applicable to choose the optimal computational grid for any commercial dust loaded precipitators that grantees both low truncation and roundoff errors and can be a prediction tool for the change of dissimilar design parameters on the electrical conditions of the precipitator, and this advantage reduces the effort for the precipitators' designers.

Boundary conditions and simplifying assumptions
Because of the symmetry of the issue, only one-quarter of the computational domain is taken into consideration, Fig. 1. The boundary conditions and simplifying assumptions are [2]: 1) The voltage on the wire equals the applied voltage; 2) The E x value at A, B points, and along with BC, CD, and AD lines, equals zero; 3) The E y value equals zero at A and B points, and along with line AB; 4) Along line CD, the potential is equal to zero; 5) The mobility of ions is constant; 6) Diffusion of ions is ignored; 7) At point A, ρ = ρ o , and the charge density, ρ o , is determined by [2].  The air density factor is given by [2]: δ = 293 p 760 (t + 273) (2) in which f denotes the factor of surface roughness, δ denotes the air density factor, and r is the radius of the ionized wire (cm). j p , S x , t, and p are, respectively, the average current density on the plate (A/m 2 ), the half spacing between the ionized wires (m), the temperature in Celsius, and the pressure in mmHg.
The geometry under consideration is bounded geometry, as shown in fig (1) in the manuscript the domain between the two wires is divided into four symmetric co-domains. At point A the voltage at any node above A is the same at the corresponding nodes under A directly, so E y = 0. Also, at any node on the right of A is the same potential as the corresponding nodes on the left of point A, so E x = 0. The same discussion is applicable for point B.

Simulation of dust
The presence of the dust particles in electrostatic precipitators impacts the distribution of the electric field as well as current density. The total space charge density ρ t can be defined by the following equation: in which ρ t represents the total space charge density (C/m 3 ), ρ represents the space charge density of ions, and ρ p represents the space charge density of particles. So, the space charge density of dust particles is given by the following: ρ p = N P × q s (4) where N P and q s represent the number of dust particles per unit volume and the saturation charge acquired by the field, respectively. So, the dust particle number per unit volume can be given by [19]: where C D is the concentration of dust particles (g/m 3 ), R p is the radius of the particle, and g is the specific gravity (g/m 3 ). The saturation charge is given by [19]: where ε r is the relative permittivity of the dust particles. Substituting by (5) and (6) in (4) results in: Indeed, the dust-loaded on the plate is non-uniformly distributed. In this work, we considered the non-uniform distribution of the dust particles. The space charge density of dust particles is a function of the number of dust particles per unit volume and the saturation charge acquired by the existing non-uniform electric field. In Eq. (7), the space charge density of dust particles at any node is a function of the calculated non-uniform field at this node.
After balancing electrical force with the Stokes drag force, the ion mobility of the dust particles can be calculated from: (8) where K p represents the mobility of particles (m 2 /V.s) and η is the viscosity of the gas.

Solution of poisson equation for dust loaded precipitators by FMG
The Poisson equation in the presence of dust particles is given by the following: where V, ρ, ρ p , are ε 0 represent the applied potential variance, the ionic space charge density, the dust particle space charge density, and the free space permittivity, respectively. Further, the electric field is given by: Using the central finite difference method on Eq. (7): Substituting by (11) and (12) in (9), the potential at any point in the computational grid is given by, as in Fig. 2: Eq. (13) is solved iteratively by SOR, where; w is the relaxation factor, the SOR method becomes similar to the Gauss-Seidel iterative method.
The iterative solution of the Poisson equation is the key motive for requiring excessive time in the overall simulation to reach a particularly acceptable convergence. So, the Poisson equation will be solved iteratively using the FMG. Hence, FMG method works initially at the coarsest grid and solves it exactly by one of the traditional iterative methods and then the solution is interpolated to the finer grid where several V-cycles are completed. Fig. 3 illustrates the flowchart of one V-cycle. Here, it was demonstrated that N 1 = 3 and N 2 = 1, give the best time performance in the issue.

Solution of continuity equation for dust loaded precipitators
The current continuity equation is given by: in which J is the corona current density, the corona current density in presence of dust is given by: in which K and K p are the mobility of ions and dust particles, respectively, in m 2 /V.s. Substituting by (15) in (14) results in, where the value of ρ p K p is constant, so ∇ρ p ⋅K p = 0. By applying backward finite-difference on (16) results in; where, in which E x and E y denote, respectively, the electric field intensities along with the x and y directions. The Δx and Δy values are symbolized the incremental spacing along with x-direction and y-direction, respectively. Through solving (12), the charge density value at any point, ρ i, j in the computational grid can be calculated, where; : is the standard full weighting operator.
h ]: is the bilinear interpolation operator.

Computational algorithm
The computational algorithm of FDM-FMG in dust loaded precipitators can be summarized as follows: • Adjust the applied potential at the ionized wire, desired average current density, and apply all the boundary conditions into the system of equations. • Compute the charge density at the surface of the wire using Eq. (1) and assume a primary guess for charge densities at every grid point. • Solve Poisson equation at every grid point using Eq. (13) by FDM-FMG till a preset convergence is achieved. • Recalculate the charge density at every point in the grid using Eq.
(17) using the new values of potential calculated in the previous step. • Check if the difference between each value of the potential at every point and its previous value is smaller than a prescribed error, if no go to step II.
The computational algorithm is terminated when the computed average current density at the plate understudy equals the wanted value, if not return to step I.
Regarding the dust layer, the charge density of dust particles is already included in the continuity equations which is solved by the finite difference method, as discussed in subSection 2.2. Where the ion mobility and the charge density of the dust particles are included in the continuity equation and the numerical procedures are as follows: 1) At every node in the computational domain, the charge density of the dust particles is calculated from Eq. (7). Hence, the charge density is a function of the electric field which is calculated by the proposed numerical technique FDM-FMG.
2) At every node in the computational domain, the ion mobility of the dust particles is calculated from Eq. (8). 3) Substituting by the values of particle charge density and the ion mobility of the dust particles from the previous points (1) and (2) in the continuity equation where it is solved at every node by the FDM. 4) Using the obtained values of the ion and particles charge densities to solve the Poisson equation to obtain the potential values at every node using the proposed FDM-FMG.

The performance of FMG method on fine computational domains
As is well known, the Poisson equation is the most time-consuming portion of the corona issue modeling, particularly on fine grids. In the 5 current analysis, the Poisson equation is solved by FDM using the SOR technique and the FMG as iterative methods. An assessment between the SOR approach and the FMG method is carried out for Penny and Matick precipitator of wire diameter of 2.032 mm, the spacing between wires is equal 152.4 mm, and spacing between two grounded plates equals 228.6 mm [29]. The comparison is conducted for a grid size (129 × 129) points by means of Intel (R) Core (TM) i7-3612QM CPU @2.10 GHz.
The convergence characteristics of the voltage loop are assigned so that the percentage error is specified by Eq. (22) as follow; Fig. 4 displays the percentage potential error of SOR with a relaxation factor, w = 1.8, and the FMG method for 18 iterations. As seen in Fig. 4.a, the percentage error of the FMG method is 10% for the second iteration and drops by 92% for the fourth iteration, while in the case of SOR the percentage error at the second iteration is 6.4 × 10 5 %, as in Fig. 4.b. FMG converges to the prescribed tolerance for a voltage loop after 18 iterations only while, in the case of SOR, it needs 20,380 iterations, so in terms of the speed-up factor, FMG decreases the effort by 1132. The overall program execution time for FMG is 1.8 s while for SOR is 230 s, so the computational time decreases by 99.22% on using FMG as an iterative solver on the grid (129 × 129). For the above observations, FMG converges faster than SOR, which in turn decreases the overall computational time.
The speedup factor is given by [2]; speedup factor = number of iterations by Guass − Seidel number of iterations by FMG

Performance of FMG on fine computational domains for different tolerances
Indeed, the tolerance to reach a certain convergence affects the performance of the iterative methods in terms of timing and the number of iterations required. As the desired tolerance decreases, more computational time and a greater number of iterations are required. The proposed method is used to study the performance of FMG and SOR for different tolerance values, which are presented in Table I. As shown in Table I, although the desired tolerance is 0.1%, FMG reduces the effort by a factor of 704 to terminate from the voltage loop only. Also, the time decreases by 99% as compared to SOR.
Besides, the present work performs a study to show if the tolerance affected the agreement between the numerical results and experimental values or not, as in Table II. The study is performed on Penny and Matick geometry for an applied voltage of 43.5 kV [29]. The results show that as tolerance decreases, the accuracy increases, but it does not dramatically change the results in this case and all these errors are acceptable, in the proposed technique.
Generally, as the tolerance decreases, the accuracy increases, and the computational time increases especially on finer grids. So, for a certain problem, FMG can be implemented for different tolerances without suffering from high computational time. This advantage is not found with the classical iterative techniques. So, by using FMG, researchers can work on small tolerances without facing the problem of computational time as compared to the SOR method.

Validation of FMG method for clean air precipitators
The proposed algorithm is conducted for Penny and Matick's geometry for ESP [29]. The ion mobility (k) is assumed to be 1.6 × 10 −4 m 2 /V.s, and the applied voltage is set to be 43.5 kV. The values of potential along BC and AD are computed and compared to the tested ones by Penny and Matick, as shown in Fig. 5.

Justification of the chosen ion mobility value for Penny and Matick geometry
For Penny and Matick geometry the ion mobilities used in previously published work are 1.82 × 10 −4 , and 2.2 × 10 −4 m 2 /V.s [11,12]. FDM-FMG implements these ion mobilities and investigates the numerical results. A detailed comparison between the numerical results and the experimental results are described in Table III. For instance, in order to get a mean current density of 4.84 × 10 −4 A/ m 2 on the grounded plate understudy, a voltage of 42 kV is employed on the wire for actual ion mobility of 1.82 × 10 −4 m 2 /V.s. Further, this

Table II
The performance of the FMG at different tolerances.  [12]. For more confirmation, the present algorithm gives the numerical and experimental results for different ion mobilities at applied voltages of 43.5 kV and 46.2 kV, as shown in Figs. 6 and 7, respectively. Again, the ion mobility of 1.6 × 10 −4 m 2 /V.s gives the best fit between the experimental values of Penny and numerical results at different voltages.

FMG method as a powerful prediction tool for V-I ch/s
The advantage of the fast convergence of the FMG as an iterative solver on fine computational domains can be used as a powerful prediction tool for the voltage-current characteristics instead of the costly  experimental work. Accordingly, the present study performed a study on the Abdel-Salam model for precipitators [11], the geometry has a wire radius of 0.5 mm, the spacing between wires is 10 cm or 20 cm, and the spacing between plates is 13 cm or 18 cm or 23 cm. FDM-FMG is used to track the optimal grid size for this design of precipitators by using different sizes of grids up to (256 × 256) grid points. Firstly, a study is conducted at different applied voltages as shown in Table IV, to track the optimal grid and it is pointed out that the optimal grid is (64 × 64). But there is a question here, does this grid is the optimal one if the wire height is changed, or if the wire to wire spacing is changed? To answer this question, a comparison is established and presented in Table V at an applied voltage of 28.1 kV, and it is found that the (64 × 64) is the optimal grid for different wire heights. Also, a study is conducted for the different wire to wire spacing, as in Table VI, and it is found that the (64 × 64) is the optimal grid. Although (128 × 128) grid can be optimal grid at S x = 5 cm and S y = 9 cm, but it can be considered as the local grid for these design parameters, but the grid (64 × 64) is the global optimal for this precipitator design.
So, the results confirm that using FMG as an iterative solver can be a powerful tool for the designers to predict the voltage-current characteristics at different design parameters without the need for costly experimental work.

Performance of FMG on fine computational domains in dust loaded precipitators
The proposed algorithm is used to model the dust-loaded precipitators. To test the effectiveness of the computational algorithm, the simulation is performed for cement dust particles having a dominant particle radius of 30 µm. The concentration of the dust particles is 18 gm/m 3 with a specific gravity of 1.69 gm/cm 3 . The precipitator has a wire radius of 0.521 mm, plate-to-plate spacing equals 203.2 mm, and wire-to-wire spacing equals 203.2 mm [30]. The simulation is conducted on a fine computational domain of (256 × 256) grid points. The percentage potential error is shown in Fig. 8, FMG takes only 25 iterations to terminate from the voltage loop within 0.8 s. The overall execution In order to choose the optimal computational grid for this dustloaded precipitator, the FDM-FMG method is conducted on fine computational domains up to (256 × 256) grid points, as in Fig. 9. The simulation results for the different grid sizes are compared to previous experimental results [30], and the following conclusions are noticed: -The best grid size at an applied voltage of 23 kV is (33 × 33); -The best grid size at an applied voltage of 27 kV is (65 × 65); -The best grid size at applied voltages of 30.5 kV, 38 kV, 41 kV, 45 kV, and 49 kV are (256 × 256); -For grid (513 × 513), as the grid becomes finer the error increases, this may be attributed to the increasing round-off error.
So, the global optimal grid for this design of the dust loaded precipitator that grantees both low truncation and roundoff error are (256  × 256) with maximum percentage error from experimental results 3%, while grids (33 × 33) and (65 × 65) can be considered as local optimal grids at 23 kV and 27 kV only. The simulated voltage-current characteristics for the chosen global optimal grid (256 × 256) are compared with the experimental results [30], and an excellent agreement is reached as shown in Fig. 10.
Furthermore, for the same geometry, the present algorithm is tested against the FEM proposed by Al Hamoz et al. [19] and the method introduced by Anagnostopoulos et al. [31]. Fig. 11 shows the percentage of error between the three methods and the experimental values of the current. Generally, the percentage error from the experimental values is dramatically reduced at different applied voltages using FDM-FMG on a fine computational domain of (256 × 256) grid points without suffering from excessive computational time. Accordingly, the proposed method is an efficient predictive tool in the design stage of commercial precipitators.

FDM-FMG as a prediction tool for the effect of changing different
design parameters on the current, ion, and particle charge densities 3.7.1. Effect of the variation of applied voltage on current density, and ion & particle densities After choosing the optimal grid of (256 × 256) grid points for the     above-mentioned dusted loaded precipitator, FDM-FMG is implemented to predict the effect of changing the applied voltage on the current density as well as the ion and particle charge densities on the ground plate. The average current density increases by 32% when applied voltage increases from 45 kV to 49 kV, Fig. 12.a. Also, the average ion and particle densities increase by 15% and 13% when applied voltage increases from 45 kV to 49 kV, respectively, as in Fig. 12.b.

Effect of the variation of wire to wire spacing on current density, and ion & particle densities
Moreover, the effect of changing the wire-wire spacing on the current density as well as the ion and particle charge densities is conducted using FDM-FMG at an applied voltage of 45 kV. The average current and ion densities and decrease by 3% and 3.4% when S x decreases from 101.6 mm to 76.2 mm respectively, as in Fig. 13. Also, the average particle density increases by 2% when S x decreases from 101.6 mm to 76.2 mm, as in Fig. 13.b. This can be explained by the fact that as the spacing between wires decreases, the corona onset voltage increases as the number of the field lines on the wire decreases, resulting in a less corona current and hence a lower current density at the same applied voltage. It can be pointed out that change in the spacing between wires doesn't dramatically change the average current, ion, and particle densities on the grounded plate.

Effect of the variation of wire height on current density, and ion & particle densities
In addition, the effect of changing wire height above the ground plate on the average current and ion densities at an applied voltage of 45 kV is studied. The average current density increases by 123% when Sy decreases from 101.6 mm to 76.2 mm respectively, Fig. 14.a. Also, the average ion and particle densities increase by 43% and 18% when S y decreases from 101.6 mm to 76.2 mm, respectively, as shown in Fig. 14. b. This can be attributed to the fact that as the height of ionized wire decreases, the corona onset voltage decreases, resulting in a higher corona current and hence a higher current density at the same applied voltage.
It can be pointed out that change in the height of the wire above the ground dramatically changes the average current, ion, and particle densities on the grounded plate, these densities are also affected by the change of the applied voltages on the ionizing wire. On the other side, the change in the spacing between wires did not have a significant effect on the current, ion, and particle densities. Fortunately, from the industrial point of view, if a desired high current density is needed for the charging process in dust-loaded precipitators, one can decrease the height or increase the applied voltage rather than increasing the distance between the ionized wires due to the limited size of the precipitators.

Discussions
Here, we provide the limitations of the current study for the modeling of ESPs. For the current problem in precipitators, the region is bounded, and the radius of the wire is small enough to consider it a single node in the computational domain. In most commercial precipitators the radius of the ionized wire is small with respect to the overall size of the precipitator. So, the finite difference method works well with such a problem. On the other hand, if the radius of the ionized wire is large, or the region is unbounded, the finite difference will be more difficult to implement as in the case of HVDC transmission lines [32,33,34]. it is better to formulate that problem with the finite  element method, not the finite difference method. In a future study, we aim to integrate the finite element method with the multigrid methods and test its performance.

Conclusions
The paper presents a novel approach for modeling the dust-loaded electrostatic precipitators on the fine computational domain. For clean and dust-loaded precipitators, FMG is greatly transcendent over SOR in terms of convergence rate and hence the timing performance, especially on fine computational domains. The shortcomings of the different numerical methods and their remedy by the proposed FDM-FMG method implemented for the simulation of ESPs are summarized as follows: • All the mentioned numerical techniques cannot work on fine computational domains without suffering from excessive computational time. So, the need for an efficient iterative solver is a must. In this regard, the present study implements the full multigrid method as an iterative solver which guarantees both fine domains and less computational time. • The results of the previous techniques cannot be considered to get an accurate picture of the precipitators in the design stage. By the proposed technique, we can predict results without the need for costly experiments. • The FDM-FMG solves Poisson and continuity equations on one fine computational domain and that is unavailable in the previous numerical methods. • Unlike the previous numerical techniques, the present proposed method works on different tolerances regarding the convergence rate without suffering from the high computational time.
Besides, two advantages are introduced by the proposed technique for the first time due to working on fine computational grids without suffering from the high computational time: • FDM-FMG gives a more confident value of the chosen ion mobility without the need for excessive experimental efforts to measure it. • Choosing the optimal computational grid for any commercial dustloaded precipitators that grantees both low truncation and roundoff errors and can be a prediction tool for the change of different design parameters on the electrical conditions of the precipitator, and this advantage reduces the effort for the precipitator's designers.
Finally, the FDM-FMG strategy predicts that the electrical conditions are greatly affected by changing the applied voltage and the height of the ionized wires in dust-loaded precipitators which in turn affects the collection efficiency of the dust particles. In future work, the FDM-FMG method can be implemented in the design of advanced ESP-based air cleaning systems, such as electrostatic enhanced air filter, hybrid electrostatic filtration system, and electrostatic enhanced pleated air filters.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.