Applied Mathematics and Computation 218 (2012) 9007–9017

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

A space charge motion simulation with FDTD method and applicationin negative corona electrostatic field analysis

Baris Baykant Alagoz a, Hafiz Z. Alisoy a,⇑, Serkan Alagoz b, Fevzi Hansu a

a Department of Electrical–Electronics, University of Inonu, Malatya, Turkeyb Department of Physics, University of Inonu, Malatya, Turkey

a r t i c l e i n f o a b s t r a c t

Keywords:Space charge motion simulationFinite difference time domainCorona

0096-3003/$ - see front matter � 2012 Elsevier Incdoi:10.1016/j.amc.2012.02.063

⇑ Corresponding author.E-mail addresses: halis@inonu.edu.tr, hafiz.alisoy

In this paper, a finite difference time domain based simulation method is presented for thespatio-temporal analysis of space charge motion and the proposed method is applied tonegative corona electrostatic field analysis. Drifting and diffusion motion equations ofspace charges are numerically solved and used in the simulation of corona discharges con-sidering effects of impact ionization, electron attachment, ion–ion recombination and ion–electron recombination. The results obtained from the simulation are discussed.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Space charges are mobile charge carrier particles such as electrons and ions and hence they are responsible for electricalconduction in electro-physical systems. The numerical simulation of space charge motion in the system contributes to inves-tigation of a wide variety of scientific and engineering problems from nano to cosmological scales, such as conduction in liq-uids, gases, solids and plasma, electro-biological conduction (neural conduction), and even the ionic activities seen on thesun’s corona.

In corona discharge, motion of space charges plays a substantial roll in discharge mechanism. Corona discharge has beenfound in numerous applications of today’s technology [1–11]. The increasing trend in the use of corona discharge applica-tions has renewed interest in basic aspects of the corona electrostatic field. It is therefore becoming more important tounderstand the mechanisms in corona discharge and the electrical characteristics of complex electrode systems to move for-ward electron–ion technologies.

Many numerical methods for the analysis of the corona discharge have been proposed in the literature, covering severalaspects: Poisson’s equation was solved for a finite solution points on the radial axis of coaxial electrodes system by consid-ering impact ionization, photoionization, attachment and detachment of electrons, and ion–ion recombination [12]. In manyworks, the Finite Element Method (FEM) [13,14] and Finite Difference Method (FDM) [15–17] were applied to iterativelysolve current continuity equations over a two-dimensional mesh describing the electrode systems. These methods mainlycalculate the charge distribution that satisfies Poisson equation and the continuity equation for a simulated electrode systemand they do not directly address simulating the motion of the space charge in an electrode gap. In order to obtain better re-sults, more efficient algorithms have been proposed, such as the method of characteristics (MOC) dealing with the artificialdispersion of space charges [18–20] and hybrid algorithms [21,22] that employ a combination of the Boundary ElementMethod [21], FEM and FDM. Concerning to space charge motion, the Monte Carlo method has also been applied to simulatethe motion of electrons and ions [23,24]. Thus, the temporal and spatial development of space charges and variations inthe electrostatic field could be analyzed for the case of bipolar charges. In a recent work, a new 2-D numerical model

. All rights reserved.

@inonu.edu.tr (H.Z. Alisoy).

9008 B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017

incorporating three ionic species for the negative corona discharge was successfully proposed to simulate a series of Trichelpulses by using FEM method for solving Poisson equation and a combined Flux Corrected Transport–FEM method for solvingcharge transport equations [25].

Previously, the FDTD based numerical analysis methods have been successfully applied to obtain the transient solutionsof electromagnetic wave propagation [26,27] and acoustic wave propagation [28] problems. In this paper, we primarily aimto develop a FDTD based simulation methodology for the transient analysis of the space charge motion in a two-dimensionalplane representing a medium in inhomogeneous electrical characters. For this propose the spatio-temporal solutions of themotion equations, considering drifting and diffusion of space charges, is found for a discrete domain of space and time. Thisnumerical solution was adopted to the simulation of negative corona electrostatic field. In a negative corona electrostaticfield, the effective space charge species are electrons, with negative ions as the negative charge carrier particles, and positiveions as the positive charge carrier particles. In the FDTD simulation developed for inspection of corona discharge, Townsendelectron avalanches were initiated by residing seed electrons in the vicinity of a corona electrode. The development of elec-tron avalanches and the generation of positive and negative ions as a result of the impact ionization and electron attachmentwere observed in the electrode gap of the sphere-plate electrode system. The migration of ion sheets to the electrode wasviewed in the simulation.

The proposed method mainly enables us to numerically analyze the spatio-temporal development of multi-charge carrierpopulations. This specifically demonstrated in numerical analysis of corona electrostatic fields on a spatial plane.

2. Method

2.1. Theoretical background

Space charges motion is modeled in two aspects: one is the drifting movement of charges under an electrical field and theother is diffusion movement of charge carrier particles in an inhomogeneous charge distribution of the medium.

In an electrical field, ~E, space charges drift with a drifting velocity [29–31]:

~Vd ¼ l �~E; ð1Þ

where, l is the charge mobility and ~Vd is the drifting velocity, and it can be written with respect to the displacement vectorof space charge as ~Vd ¼ o~d

ot. In this case, space charge drifting motion can be modeled as:

o~dot¼ l �~E: ð2Þ

In an inhomogeneous charge distribution, the diffusion motion of charge carrier particles occurs in accordance with Fick’slaw as [32]:

~Jd ¼ D � rN; ð3Þ

where,~Jd is the particle current density, the parameters N and D are the particle concentration in space and the diffusioncoefficient, respectively. In the diffusion, particle flow conforms to the current continuity equation, which is given asoNot � ~r �~Jd ¼ 0 [12–17,29–31], in this case, the diffusion motion of particles can be expressed as:

oNot¼ D � r2N: ð4Þ

Eqs. (2) and (4) are numerically solved for the corona field simulation under the following basic assumptions:

(1) Unipolar space charges take effect in the system. Negative charge carriers are electrons and negative ions and positivecharge carriers are positive ions.

(2) Drifting and diffusion mobilize the space charges in the medium. Charges are subject to impact ionization, electronattachment, ion–ion recombination and ion–electron recombination during their motion.

(3) Permittivity (e) and charge mobilities (l) are constant in the medium.

The impact ionization of electrons with gas molecules results in positive ion and free electron production and it is themain physical process that feeds electron avalanches, initiated with seed electrons. Growing of avalanche electron concen-tration in the spatial domain was modeled as the following:

oNor¼ a � N; ð5Þ

where, a is the impact ionization coefficient [29–31].Electron attachments with gas molecules result in negative ion production and free electron removal in the medium. It is

therefore an effective physical process quenching electron avalanches. In the case of electron attachment, avalanche electronconcentration was modeled as the following:

B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017 9009

oNor¼ �b � N; ð6Þ

where, b is the attachment coefficient [29–31].Ion–ion recombination results in the removal of positive and negative ions from the electron gap [29–31]. This process

was modeled as the following:

oNpz

ot¼ oNng

ot¼ Kii � Npz � Nng : ð7Þ

Ion–electron recombination results in the removal of positive ions and electrons from the electron gap [29–31]. This pro-cess was modeled as the following:

oNpz

ot¼ oNel

ot¼ Ki:e: � Npz � Nel: ð8Þ

The displacement of space charges in the electrode gap may result in temporal variation in the electrical field vectors inthe space. That is why electric field vectors used in numerical solutions should be calculated at each calculation step by con-sidering the latest space charge distribution:

~E ¼Z Z

q

4 � pe � j~rj2�~rdxdy: ð9Þ

2.2. FDTD based numerical solution of corona electrostatic field

For a FDTD based discrete numerical solution, charge carrier particle concentration and the variables relating to chargemovements, are defined with respect to a solution grid as seen in Fig. 1. Corner points in the grid indicate the locationsof solution points on the spatial plane. The Dx and Dy distances are the unit distance between sampled calculation pointson the grid. Taking Dx = Dy provides a square lattice grid and a symmetrical solution in the spatial plane. The position vector~d ¼ ðx; yÞ, defined on the finite sized solution grid, indicates the center of a unit area ua as illustrated in Fig. 1(a). In such a gridsystem, Nn(x,y) represents the space charge concentration in the unit area located at a sampled solution point (x,y). Theparameter Dt is the time increment and the simulation time is expressed as n � Dt. Electrical field vectors in a two dimen-sions space were denoted in the form of~Enðx; yÞ ¼ En

xðx; yÞ � iþ Enyðx; yÞ � j, where (x,y) is the corner position of solution grid as

illustrated in Fig. 1(a).In the proposed corona simulations, the space charge sheet of a unit area ua is relocated to its new position on the square

grid in order to move the space charge species on the plane as in Fig. 1(b). Eqs. (2) and (4) are written in an iterative discretescheme for all type of charge carrier concentrations, which are the electron concentration denoted by Nel, the positive ionconcentration denoted by Npz and the negative ion concentration denoted by Nng.

For a drifting motion of those charge carrier particles, the finite difference equation applying forward finite difference totemporal derivatives was written as the following:

~dnþ1x ¼~dn

x þ Dt � lx �~En; ð10Þ

where the x indices denotes the type of space charge and is defined as x = {el,pz,ng}. Which for electrons is x = {el}, for po-sitive ions, x = {pz} and for negative ions x = {ng}. Migration of all three types charge carriers in the spatial domain is doneaccording to the following formulation:

(a)(b)

Fig. 1. (a) Grid structure for FDTD method and (b) unit area ua for surface integration.

9010 B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017

Nnþ1x

~dnþ1x

� �¼ Nnþ1

x~dnþ1

x

� �þ Nn

x~dn

x

� �; ð11Þ

Nnx~dn

x

� �¼ 0: ð12Þ

Eq. (10) is used to calculate the next position of space charges, which are drifting under an electrical field~E. By using Eqs. (11)and (12), space charges in the unit area at position ~dn move to new position ~dnþ1. Solutions obtained by Eqs. (11) and (12)comply with the well-known current continuity equation, oq

ot þ div~J ¼ 0. When the effects of impact ionization, electronattachment, ion–ion recombination and ion–electron recombination are considered, Eq. (11) is reorganized for electrons, po-sitive ions and negative ions as follows:

Nnþ1el

~dnþ1el

� �¼ Nnþ1

el~dnþ1

el

� �þ Nn

el~dn

el

� �þ DNnþ1

a � DNnþ1b � DNnþ1

i:e: ; ð13Þ

Nnþ1pz

~dnþ1pz

� �¼ Nnþ1

pz~dnþ1

pz

� �þ Nn

pz~dn

pz

� �þ DNnþ1

a � DNnþ1ii � DNnþ1

i:e: ; ð14Þ

Nnþ1ng

~dnþ1ng

� �¼ Nnþ1

ng~dnþ1

ng

� �þ Nn

ng~dn

ng

� �þ DNnþ1

b � DNnþ1ii ; ð15Þ

where DNa, DNb, DNii and DNie are the contributions of impact ionization, electron attachment, ion–ion recombination andion–electron recombination to charge the carrier concentrations, and by considering Eqs. (5)–(8), they are numerically cal-culated by

DNa ¼ a � Nel � Vd � Dt; ð16ÞDNb ¼ b � Nel � Vd � Dt; ð17ÞDNii ¼ Kii �min Npz;Nng

� �� Dt; ð18Þ

DNie ¼ Kie �min Nel;Npz� �

� Dt: ð19Þ

For the diffusion motion of charge carrier particles, the finite difference equation was obtained by applying forward finitedifference to temporal derivatives as in the following:

Nnþ1x ¼ Nn

x þ Dt � Dx � r2Nnx : ð20Þ

Here, r2Nn is the Laplacian of the space charge concentration.En

x and Eny are components of the electrical field vectors ð~EÞ and were calculated in a discrete form corresponding to Eq. (9)

as follows:

Enx ðx; yÞ ¼

XM

u¼1

XN

v¼1

rx

4 � p � eðx; yÞ � j~rj2� qnðu;vÞ � ua; ð21Þ

Enyðx; yÞ ¼

XM

u¼1

XN

v¼1

ry

4 � p � eðx; yÞ � j~rj2� qnðu;vÞ � ua: ð22Þ

In a similar manner, the potential can be calculated as follows:

Vnði; jÞ ¼Xw

u¼1

Xh

v¼1

qnðu;vÞ4 � pe � j~rj � ua: ð23Þ

Here, the unit area can be expressed as ua = Dx � Dy. The distance vector can be represented as ~r ¼ rx � iþ ry � j in a two-dimensional spatial domain. Overall charge density is found by

qn x; yð Þ ¼ e � Nnpz � Nn

ng þ Nnel

� �� �: ð24Þ

In the numerical calculations, since roughly 100 times the difference between electron mobility (lel) and ion mobility(lpz,lng), electron sheets drift with large jumps over the solution grid, whereas the ion sheets move by small steps on thesolution grid. This causes the amount of calculation points of electrons to be quite low compared to the amount of calcula-tion points for ions. In order to deal with this problem in the simulation, Dt unit time is adaptively determined in the eachcalculation step to limit movement of the fastest particle to one unit distance. In this way, for all charge carrier particles thesolution points are restricted one unit distance sampling in the simulation. The following formula, suggested for reducingsimulation time of three charge species in [25], is used to determine the appropriate Dtn at each calculation step:

Dtn ¼ Dy

lj~Enj: ð25Þ

In order to observe the Trichel pulse in numerical simulations and the experimental study, the V(t) voltage of the RC cir-cuit as seen in Fig. 2(a) is measured. In the simulation, the I(t) corona current is calculated by I = dQ/dt. Here, Q(t) is theamount of space charges arrived at the ground plate electrode. V(t) in the simulation is obtained by using the following For-ward Euler numerical solution of an RC circuit:

Fig. 2.of sphe

B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017 9011

Vnþ1 ¼ Vn þ DtC

In � 1R

Vn� �

: ð26Þ

In the simulation, a seed electron concentration ðNnesÞ to initiate an electron avalanches is generated by the following

deterministic formulas:

[m]

[m]

0.005 0.01 0.015 0.02 0.025

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04-400

-350

-300

-250

-200

-150

-100

-50

0

(a)

[m]

[m]

0.005 0.01 0.015 0.02 0.025

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

1

2

3

4

5

6

x 104(b)

Fig. 3. Potential and electrical field intensity distribution obtained from the simulation.

(a)(b)

(c)

(a) Schema of sphere-plate electrode system (distance between electrode through the radial axis r is 6 mm, radius of the sphere Rc is 10 mm). (b) Picturere-plate electrodes. (c) Impact ionization (solid) and electron attachment (dash) coefficients used in the simulation (obtained for P = 200 Pa in [12]).

Fig. 4.of the r

9012 B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017

Nnes ¼

N � Nnpz

� �� a En=Pð Þ � Dr; En=Pð ÞP Ec=Pð Þ _ N > Nn

pz

� �;

0; En=Pð Þ < Ec=Pð Þ;

(ð27Þ

where Ec is corona inception field and it was calculated by using Kaptzov’s assumption, which was expressed asEc = 30 � [1 + (0.0906/Rc)0.5] for conductor to plane electrode configuration [14]. N is the concentration of neutral moleculesin air. These molecules are the candidates to be a seed electron. According to Eq. (27), when electrical field at a point riseabove Ec and there exist a neutral molecules population at this point, a seed electron concentration of ðN � Nn

pzÞ � aðEn=PÞ�

Dr appears at this point of solution grid to initiate an electron avalanches.Computation step for the space charge motion simulation of the negative corona discharge is listed below:

Elektron

t =1.10-6

Elektron

t =2.10-6

Elektron

t =5.10-6

Electron

Elektron

t =4.10-5

(a) electron density distribution Pozitif Iyon

t =1.10-6

Pozitif Iyon

t =2.10-6

Pozitif Iyon

t =5.10-6

Pozitif Iyon

t =4.10-5

(b) positive ion density distribution Negatif Iyon

t =1.10-6

Negatif Iyon

t =2.10-6

Negatif Iyon

t =5.10-6

Negatif Iyon

t =4.10-5

(c) negative ion density distribution

Evolution of electron avalanches (a), positive ions (b) and negative ions (c) in the electrode gap. Lighter color code represents a greater concentrationelevant charge species.

B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017 9013

Step 1: Start with setting a qo(x,y) initial charge density with a negative polarity on the corona electrode. (Initial spacecharge densities ðN0

el;N0pz;N

0ngÞ are set to zero.)

Step 2: Generate a seed electron concentration at a point in the vicinity of the corona electrode by using Eq. (27).Step 3: Calculate ~En by using Eqs. (21) and (22) in the spatial domain.Step 4: Calculate a convenient Dt by using Eq. (25).Step 5: Drift space charge species, which are the electrons, and positive and negative ions in the corona field, by means of

Eqs. (10)–(15).Step 6: Diffuse space charges species by using Eq. (20).Step 7: Calculate the new charge distribution by using Eq. (24).Step 8: If the simulation time is lower than the simulation completion time, increment n by using n = n + 1 and go back to

step 3. If not, end the simulation.

2.3. Simulation results obtained for the sphere-plate electrode system

In Fig. 3, the electrical field intensity and potential distribution obtained from the simulation of the sphere-plate electrodesystem are presented. On the boundary of the spherical electrode, the electric field intensity was seen to reach the coronaonset value [33,34]. Such a high intensity of electrical field in the vicinity of the corona electrode leads to electron avalanchessince it leads to the impact ionization of mobile seed electrons. The temporal and spatial development of electron avalanchesand the generation of ions in the electrode gap are illustrated in Fig. 4. In these figures, drifting in the electrical field was seento be effective in the motion of charges on the radial axis, which is the vertical direction in the electrode gap as illustrated inFig. 2(a), and the diffusion motion is rather effective in the motion of charges on the lateral axis that is perpendicular to theradial axis.

Fig. 5. Spatio-temporal distributions of electrical field intensity (a) electron density (b), positive ion and negative ion densities (c and d) on the radial axis (r)as shown in Fig. 2. Electron avalanche and its two characteristic regions (I: growing and II: quenching) are demonstrated in (b). The migration of positive ionsheets to the corona electrode and the migration of negative ion sheets to the ground electrodes are seen in (c) and (d), respectively.

9014 B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017

Spatio-temporal distributions of the electric field intensity and space charge concentrations through the radial axis for asingle Townsend electron avalanches are illustrated in Fig. 5(a)–(d). These characteristics presented were obtained for the

1.5 2 2.5 3

x 10-4

-4

-3

-2

-1

0

x 10-3

t [s]

(a)

20 40 600

0.5

1

1.5

2

2.5

x 10-5

n

Δt [s

]

(b)

1 2

Fig. 6. (a) Trichel pulse characteristic obtained from the simulation and experimental measurements. During the simulation Dtn was used in (b).

Fig. 7. Spatio-temporal distributions of electron density (a), positive ion and negative ion densities, (c and d) on the radial axis (r) for consecutive Townsendavalanches. Contour of electron avalanches are illustrated in (b).

150 200 250 300 350 400 450

0

5

10

15

20

25

30

35

40

I [µA

]

V [V]

ExperimentSimulation

Fig. 8. I–V characteristics from the experiment (circles) and simulation (solid curve).

B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017 9015

first electron avalanches at the onset of corona discharge. Therefore this avalanche does not lead to an adequate charge col-lection in the electrode gap, it did not considerably deform electrical field. We can see the two characteristic regions in theevolution of electron avalanches in Fig. 5(c). In region I, where the electron concentration is increasing, impact ionization iseffective and strengthens the avalanches since the electrical field in this region is strong enough to support impact ioniza-tion. In region II, where the electron concentration is decreasing, the electron attachment by electronegative gas molecules iseffective and quenches the avalanche [22,30–32,35–38]. These characteristic regions of electron avalanches also depend onvalues of impact ionization and the electron attachment coefficients. Many ionic reaction coefficients were discussed in de-tail [30,39,40]. In our simulations, we used the impact ionization and the electron attachment coefficients, used by Kozlov etal. in their numerical analysis [12], as illustrated in Fig. 2(c).

Each electron avalanche produces relatively less mobile positive and negative ion populations besides highly mobile elec-trons. After a single electron avalanche, the migration of the positive and negative ions towards the electrodes along the ra-dial axis is demonstrated in Fig. 5(c) and (d). Due to the high gradient in the electrical field intensity distribution along theradial axis, ion drifting velocity seriously differentiates depending on the distance to the corona electrode. This results in alayered migration of the ion population through to the electrodes as seen in Fig. 5(d) and (e). The Trichel pulse characteristicsobtained from the simulation and experimental measurements are demonstrated in Fig. 6(a). The waveform of Trichel pulsesobtained from the simulation corresponds to waveforms in experimental measurements.

In the simulation, the unit time increment Dtn was automatically determined so that it limits the movement of the fastcharges to one unit distance. The variation in Dtn through simulation of the single electron avalanche is shown in Fig. 6(b).The regions 1 and 2 in Fig. 6(b) correspond with the regions 1 and 2 of the time axis in Fig. 5(a). In region 1, Dtn is determinedas low values for the motion electrons and ions with high drifting velocity. In region 2, Dtn is increased to provide motion ofrather slower ions, which are exposed to a lower electrical field. In this way, drifting motion in the case of multiple chargecarriers with different velocities can be simulated with one unit distance sampling in spatial domain. This was achieved bymeans of adjusting the unit sampling in the temporal domain.

Electron density, positive ion and negative ion densities on the radial axis (r) for consecutive Townsend avalanche seriesare presented in Fig. 7. In the contour characteristic of electron avalanches illustrated in Fig. 7(b), electron avalanches goesinto a steady and periodical clustered form after a transient regime of a few irregular avalanches so that effect of the rise inelectrical field intensity, which supports seed electron production, and effect of the rise in positive ion concentration, whichinhibits the seed electron production, balance each other on the boundary of corona electrode. Fig. 7(c) and (d) demonstrategradually collection of positive ion and negative ion concentration as a result of consecutive Townsend avalanche series.Similar space charge collection around needle-plate electrodes were reported by Sattari et al. [41,42].

The average Current–Voltage characteristic (I–V) is a substantial characteristic revealing the electrical discharge characterof the corona electrostatic field. In Fig. 8, the I–V characteristics of the sphere-ground plate electrode system, obtained fromexperimental measurement and simulation results, are demonstrated. The simulation result was seen to be in an agreementwith experimental measurement.

3. Conclusions

The motion equations considering the drifting and the diffusion motion of space charges were used for the developmentof a FDTD based charged particle flow simulation. The proposed simulation method was applied in the spatio-temporal

9016 B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017

analysis of the negative corona discharge in a spherical-plate electrode system. This numerical analysis allowed investigationof the space charges behaviors in the corona electrostatic fields, in addition to estimating the basic electrical characteristicsof the system such as current draw and voltage drops. The results obtained from the simulation were seen to be consistentwith the findings of previous works on corona electrostatic fields and our experimental measurements.

The novelties in the aspect of the numerical simulation techniques of space charges are mainly summarized as thefollowing:

� FDTD based simulation is widely used in the spatio-temporal analysis of many problems. We adopted the FDTD methodfor the transient simulation of the multi-species space charge motion in a spatial plane to numerically analyze physicalphenomenon involving space charges.� In the simulation of a single charge species, a constant and adequately low value of Dt unit time step can be used for

reducing the linearization error in the discrete solution. In the case of multi-species with large differences in their mobil-ities, a constant Dt is not preferable so that a constant Dt can not be optimal for the numerical solution of all species.Adaptive determination of Dtn deals with the problems caused by the large velocity differences of space charge species.This high difference in charge velocity is principally due to great difference in the mobility of space charges and alsoextensive variations in the electrical field intensity distribution on the simulated plane.

References

[1] W.F.L.M. Hoeben, E.M. van Veldhuizen, W.R. Rutgers, G.M.W. Kroesen, Gas phase corona discharges for oxidation of phenol in an aqueous solution, J.Phys. D. Appl. Phys. 32 (1999) L133–L137.

[2] G.F. Leal Ferreira, O.N. Oliveira, J.A. Giacometti Jr., Point to plane corona: current voltage characteristics for positive and negative polarity with evidenceof an electronic component, J. Appl. Phys. 59 (1986) 3045–3049.

[3] A. Bogaerts, E. Neyts, R. Gijbels, J. van der Mullen, Gas discharge plasmas and their applications, Spectrochim. Acta B 57 (2002) 609–658.[4] J. Li, Y. Wu, N. Wang, G. Li, Q. Huang, Industrial-scale experiments of desulfuration of coal flue gas using a pulsed corona discharge plasma, IEEE Trans.

Plasma Sci. 31 (2003) 333–337.[5] R. Morar, A. Iuga, L. Dascalescu, A. Samuila, Factors which influence the insulation–metal electroseparation, J. Electrostat. 30 (1993) 403–412.[6] A. Iuga, R. Morar, A. Samuila, L. Dascalescu, Electrostatic separation of metals and plastics from granular industrial wastes, IEE Proc. Sci. Meas. Technol.

148 (2001) 47–54.[7] L. Dascalescu, A. Iuga, R. Morar, V. Neamtu, A. Samuila, I. Suarasan, Charging of particulates in the corona field of roll-type electroseparators, J. Phys. D.

Appl. Phys. 27 (1994) 1242–1251.[8] H.Z. Alisoy, B.B. Alagoz, G.H. Alisoy, An analysis of corona field charging kinetics for polydisperse aerosol particles by considering concentration and

mobility, J. Phys. D. Appl. Phys. 43 (2010) 1–6.[9] H.Z. Alisoy, G.T. Alisoy, S.E. Hamamci, M. Koseoglu, Combined kinetic charging of particles on the precipitating electrode in a corona field, J. Phys. D.

Appl. Phys. 37 (2004) 1459.[10] H.Z. Alisoy, G.T. Alisoy, M. Köseoglu, Charging kinetics of spherical dielectric particles in a unipolar corona field, J. Electrostat. 63 (2005) 1095.[11] M. Köksal, M.S. Mamis�, H.Z. Aliyev, A new simulation model for the corona discharge, Int. J. Appl. Math. 1 (1999) 499–509.[12] B.A. Kozlov, V.I. Solov’ev, Numerical simulation of stationary negative corona in air, Tech. Phys. 54 (2009) 621–630.[13] K. Adamiak, Adaptive approach to finite element modelling of corona fields, IEEE Trans. Ind. Appl. 30 (1994) 387–393.[14] M. Abdel-Salam, Z. Al-Hamouz, A new finite element analysis of an ionized field in coaxial cylindrical geometry, J. Phys. D. Appl. Phys. 25 (1992) 1551–

1555.[15] T. Takuma, T. Kawamoto, A very stable calculation method for ion flow field of HVDC transmission lines, IEEE Trans. Power Deliver 2 (1987) 189–198.[16] A. Caron, L. Dascalescu, Numerical modeling of combined corona electrostatic fields, J. Electrostat. 61 (2004) 43–55.[17] A. Caron, The iterative method with associated constraints to solve numerically non linear problem, in case of finite number of theoretical solutions, in:

Proceedings of the Fifth International Symposium in Numerical Methods and Engineering, vol. 2, 1989, pp. 201–208.[18] J.L. Davies, J.F. Hoburg, Wire-duct precipitator field and charge computation using finite element and characteristics method, J. Electrostat. 14 (1983)

187–199.[19] M.N. Horenstein, Computation of corona space charge, electric field, and V–I characteristic using equipotential charge shells, IEEE Trans. Ind. Appl. 23

(1984) 95–102.[20] A.A. Elmoursi, G.S.P. Castle, Modeling of corona characteristics in a wire-duct precipitator using the charge simulation technique, industry applications,

IEEE Trans. Ind. Appl. 23 (1987) 95–102.[21] K. Adamiak, J. Zhang, L. Zhao, Finite element versus hybrid BEM–FEM techniques for the electric corona simulation, Electrotech. Rev. 10 (2003) 753–

756.[22] J. Zhanga, K. Adamiak, G.S.P. Castle, Numerical modeling of negative-corona discharge in oxygen under different pressures, J. Electrostat. 65 (2007)

174–181.[23] A. Settaouti, L. Settaouti, Numerical simulation of corona discharge in N2, J. Electrostat. 65 (2007) 625–630.[24] W. Hee Koh, I.H. Park, Numerical simulation of point-to-plane corona discharge using a Monte Carlo method, Vacuum 84 (2009) 550–553.[25] P. Sattari, G.S.P. Castle, K. Adamiak, Numerical simulation of Trichel pulses in a negative corona discharge in air, IEEE Trans. Ind. Appl. 47 (2011) 1935–

1943.[26] K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antenn. Propag. 14 (1966)

302–307.[27] A. Taflove, Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems, IEEE Trans.

Electromagn. Compat. 22 (1980) 191–202.[28] S. Alagoz, B.B. Alagoz, Frequency-controlled wave focusing by a sonic crystal lens, Appl. Acoust. 70 (2009) 1400–1405.[29] L.B. Loeb, Electrical Coronas: Their Basic Physical Mechanisms, University of California Press, Berkeley, 1965.[30] Y.P. Raizer, Gas Discharge Physics, Springer-Verlag, 1987.[31] H.Z. Alisoy, Electrical Phenomenon in Gas, Azerbaycan Science Academy, Institute of Physics, 1995.[32] T.S. Ursell, The Diffusion Equation A Multi-dimensional Tutorial, Department of Applied Physics, California Institute of Technology, 2007.[33] M. Abdel-Salam, Z. Al-Hamouz, IEE Proc. 141 (1994) 369–378.[34] M. Horenstein, IEEE Trans. lnd. Appl. 20 (1984) 1607–1612.[35] G.W. Trichel, The mechanism of the negative point to plane corona near onset, Phys. Rev. 54 (1938) 1078–1084.[36] Y. Wang, New method for measuring statistical distributions of partial discharge pulses, J. Res. Natl. Inst. Stand. Technol. 102 (1997) 569–576.

B.B. Alagoz et al. / Applied Mathematics and Computation 218 (2012) 9007–9017 9017

[37] I.M. Bortnik, I.P. Vereshchagin, Y.N. Vershinin, et al, Electrophysical Fundamentals of High Voltage Technique, Energoatomizdat, Moscow, 1993.[38] T.N. Tran, I.O. Golosnoy, P.L. Lewin, G.E. Georghiou, Numerical modelling of negative discharges in air with experimental validation, J. Phys. D. Appl.

Phys. 44 (2011) 015203.[39] F. Pontiga, C. Soria, A. Castellanos, J.D. Skalny, A study of ozone generation by negative corona discharge through different plasma chemistry models,

Ozone Sci. Eng. 24 (2002) 447–462.[40] J. Zhang, K. Adamiak, A multi-species DC stationary model for negative corona discharge in oxygen; point-plane configuration, J. Electrostat. 65 (2007)

459–464.[41] P. Sattari, C.F. Gallo, G.S.P. Castle, K. Adamiak, Trichel pulse characteristics-negative corona discharge in air, J. Phys. D. Appl. Phys. 44 (2011) 155502.[42] P. Sattari, G.S.P. Castle, K. Adamiak, FEM–FCT-based dynamic simulation of corona discharge in point-plane configuration, IEEE Trans. Ind. Appl. 46

(2010) 1699–1706.