Comparison of Electric Field of Corona Discharge in Single-Wire-and Multi-Wire-to-Plate Electrode Systems

Direct current corona discharge is often used in many industry applications. Corona discharge electrode represented by thin wire or a needle charges small particles and liquid droplets. Some of corona discharge devices have electrode systems with many corona electrodes to have more charged particles or to control trajectories of their motion. Many geometrical forms of corona discharge electrode could be wires, pins, needles and another. The corona discharge electric field of multicorona electrode system is different from single corona electrode system. The charge of each electrode produces electric field and influences another electrode field. To compute this field this influence must be assessed. Other researchers tried to solve this problem long before [1]. Nowadays there are many experimental research results of the discharge in multi-electrode systems. They involve bacterium cleaning in drinking water [2], the effect of the air flow in multi-electrode corona field [3], enhancement of drying effect [4], ozone generation characteristics in multi-point corona electrodes [5], etc. Some authors have tried various ways to solve this problem theoretically. Several of them invoke a commercial software packages with finite element method FEMLAB, COMSOL or ANSYS [6, 7], the other calculate with their made programs using numerical [8, 9] or analytical [10] methods. It is clear from their results that in each corona discharge electrode system electric field are quite different. It must be simulated separately to each application of the corona field. In this paper we show differences and similarities of the corona field in single-wire and multi-wire-to-plate electrodes systems. Information of the electric field distribution in these systems in some cases may determine the creation of more precision and economical systems. We present there the results of field simulation numerical model based on the finite difference method in the software environment of the package DELPHI.


Introduction
Direct current corona discharge is often used in many industry applications.Corona discharge electrode represented by thin wire or a needle charges small particles and liquid droplets.Some of corona discharge devices have electrode systems with many corona electrodes to have more charged particles or to control trajectories of their motion.Many geometrical forms of corona discharge electrode could be wires, pins, needles and another.The corona discharge electric field of multicorona electrode system is different from single corona electrode system.The charge of each electrode produces electric field and influences another electrode field.To compute this field this influence must be assessed.
Other researchers tried to solve this problem long before [1].Nowadays there are many experimental research results of the discharge in multi-electrode systems.They involve bacterium cleaning in drinking water [2], the effect of the air flow in multi-electrode corona field [3], enhancement of drying effect [4], ozone generation characteristics in multi-point corona electrodes [5], etc.Some authors have tried various ways to solve this problem theoretically.Several of them invoke a commercial software packages with finite element method FEMLAB, COMSOL or ANSYS [6,7], the other calculate with their made programs using numerical [8,9] or analytical [10] methods.
It is clear from their results that in each corona discharge electrode system electric field are quite different.It must be simulated separately to each application of the corona field.
In this paper we show differences and similarities of the corona field in single-wire and multi-wire-to-plate electrodes systems.Information of the electric field distribution in these systems in some cases may determine the creation of more precision and economical systems.
We present there the results of field simulation numerical model based on the finite difference method in the software environment of the package DELPHI.

Numerical modeling of corona discharge electric field
We selected the polar coordinate system for numerical simulation of the corona field due to the reasons which were mentioned in our previous papers [11].Single-wire-to-plate electrode system is shown in Fig. 1.Computational grid of multi-wire-to-plate corona discharge electrode system is shown in Fig. 2. Distances of the computational grid are regular near the corona electrode.At the right side of the computation field the distances are irregular for both electrode systems.Irregular distances of the computational grid are also near the plate electrode.
Analysis of the corona field consists of two equations: the Poisson equation ( 1) and charge conservation equation (2): . grad grad Finite-difference approximation of the Poisson equation for polar grid contains potential differences related to distances a P , … a S between central grid node O and neighbor nodes P, R, Q, S (see Fig. 1) and depends on the spatial charge located at the central node O Difference form of this equation is given in [11].It is clear from (4) that spatial charge located at the central node O depends on special charge and potential partial derivatives product.An algorithm of corona field numerical analysis is shown in Fig. 2.

Fig. 2. Algorithm of corona field numerical analysis
Boundary conditions in multi-wire electrode system Boundary conditions in single-wire electrode system were shown in our paper [11].All boundary conditions in multi-wire-to-plane electrode system are the same, except boundary condition at the right side (line x = g, Fig. 3).We assume in this paper that all corona discharge electrodes are of the same potential, distances to the plane are the same also.Polar grid element at the boundary is shown in Fig. 4. Because of condition mentioned above it is clear that line sl shown in Fig. 3 is the symmetry line and the potentials and space charges are equal V S = V Q , ρ S = ρ Q .We insert new point G to the computational grid between four computational grid points.This point G is located on the perpendicular line to the symmetry line sl between points S and S′.So new rectangular five point grid with irregular distances between R, S, S′, P and the central G (Fig. 4) is obtained ( To compute potential value on the computational grid point O we assume that variation of electric potential between the points G and R along the symmetry line sl is uniform.Therefore the potential on the point O may be expressed as follows where a RG is the distance between points R and G, and a OG represents the distance between points O and G.The same principle is applied to irregular distances a R and a P on the symmetry line sl.Computational points situation is shown in Fig. 5.
where a R′R′′ is the distance between points R′ and R′′; a R′R is the distance between points R′ and R; V R , V R′ , V R′′ are the potentials of these points.
Boundary conditions for Poisson's equation ( 3) and the equation ( 4) are of Dirichlet type   . Boundary conditions on the surface of the wire are formed according the Kaptzov's assumption, which states that the corona field strength on the surface of wire electrode is constant and equal to its value E 0 corresponding to the inception voltage U 0 of the Parameters of wire-to-plane electrode system are: radius of the wire r 0 = 0,05 mm, spacing between the wire and the plane electrode h = 12,0 mm The distance between the center of the wire and the symmetry line sl in the multi-wire-to-plane electrode system g = 7,5 mm.

Comparison of the electric field
Comparison of corona field in electrode systems shown in Fig. 1 and Fig. 2 is performed for points of the wire surface and for points of the first line of the computational area (line x  0).Corona field strength on the surface of the wire is constant and equal to   0 0 E r E  in both cases.Space charge density depends on given voltage of the wire and on the geometry of the electrodes.It is clear from Fig. 6 that space charge density near the surface of the wire in multi-wire electrode system is less in comparison with the value of wire-to-plane system and this difference increases with the angle.It is caused by the influence of the electric field of neighbor electrode.Space charge reduces electric field strength and sustains the value field strength to meet Kaptzov's assumption.Because of the condition that neighbor wire voltage is of the same value and of the same sign electric field between them reduces and the space charge density on the corona discharge electrode reduces significantly.   1) and multi-wire-to-plane (Fig. 2) electrode systems Distribution of space charge density (Fig. 7) on the first line of computational area is of the same character for both compared systems.
Space charge density near the wire electrode it is less in the multi-wire-electrode system and it is a little greater near the plane.
The character of corona field strength distribution on the first line of the computational area (Fig. 8) is similar in both systems.Electric field strength on the surface of the wire is estimated from Peek′s law.It depends upon the geometry of the wire and an ambient air parameters.At the middle of the distance between wire and the plate electric field strenght is slightly weaker and near the plate is stronger in the multi-wire-to-plate electrode system.

Conclusions
The boundary condition differences are given for single and multi-wire-to-plane electrode systems.
Results of numerical corona field modeling in single-and multi-wire-to-plane electrode systems were compared for points on the surface of wire.Space charge density differs because of acting of neighbor wire fields.Space charge density on the surface of the wire in the multi-wire-to-plane electrode system is smaller.
Distribution of the space charge density on the first line of the computational area has no significant difference in both systems.
Corona field strength on the surface of the wire is constant according to Kaptzov's assumption.The character of corona field distribution on the symmetry line of the field has no significant difference.

Fig. 4 .
Fig. 4. Polar grid element at the boundary with irregular distances

Fig. 5 .
Fig. 5. Polar grid element at the boundary when the distance

Fig. 6 .
Fig. 6.Space charge density on the surface of the wire in single-and multi-wire-to-plane electrode systems Space charge density on the surface of the wire decreases from points on the symmetry axis x = 0 to the other points on this surface.It is clear from the Fig. 6.

Fig. 7 .
Fig.7.Space charge density on the first line j = 1 of the computational area in single-wire-(Fig.1)and multi-wire-to-plane (Fig.2) electrode systems

Fig. 8 .
Fig. 8. Electric field strength on the first line of the computational area in single-wire-and multi-wire-to-plane electrode systems