NUMERICAL MODELING OF D,ELECTRIC BARRIER DISCHARGE PLASMA FLOW

Azizi Bin Ahmadi

QC 584 A99S 2013

Master of Science (Computational Science)

2013

Pusat Kbidmat Maidumat Akademik UNlVERSm MALAYSIA SARAWAK

P.KHIDMAT MAKLUMAT AKADEMIK

11111 1IIIIfiiiili1111111 1000245995

NUMERICAL MODELING OF DIELECTRIC BARRIER DISCHARGE

PLASMA FLOW

AZIZI BIN AHMADI

A thesis submitted

in fulfilment of the requirements for the degree of

Master of Science

Faculty of Computer Science and Information Technology

UNIMAS

2013

In Loving Memory of

Hjh. Pungut Mat

And

Salawati bt Kadir

\.

ii

ACKNOWLEDGMENT

Alhamdulillah, all praise to Allah for His Blessing. This work would not exist without

Him.

To my supervisor, Associate Professor Dr Jane Labadin, thank you for your guidance.

You are a great teacher and mentor to me. I will cherish aU the know ledge that you have

shared with me. Not to forget, my co-supervisors; Associate Professor IR Dr Andrew Rigit

and Mr Phang Piau.

I would like to thank my family for their never ending support and trust. Thank you

for believing in me as I go through this journey. All love to my niece, Sharifah Aqeela Zaleya

and my nephew, Syed Harith Zaqwan. Their smiles and laughter are food to my soul.

To my friends, you guys are awesome in eyery way. Thank you for being there for me.

Not to mention, to all the people who have helped me and provided support, either

directly or indirectly, such as the Faculty of Computer Science and Information Technology,

Faculty of Engineering, UNIMAS and examiners of this work. Also thanks to MOSTI for

funding this research project.

, ;

iii

ABSTRACT

(Dielectric Barrier Discharge (DBD) is a plasma actuator. A plasma actuator ,is a system or

mechanism which produces plasma artificially. This discharge device has been used in wide

range of industries, mainly in the environmental engineering industry. Most researches done

are through experiments, but it does not obtain direct information on the distribution of the

electric field and the electron density along the discharge axis. This information is important

in order to optimize the plasma productio . In this study, the mathematical model of the DBD

will be presented. Two models are required to model DBD plasma discharge mathematically;

the electrostatics model and the fluid flow model . As for the electrostatics model, a model

based on previous researches is adopted. And as for the fluid flow model, the Navier-Stokes

equations are used to derive the model. In this study, it was noticed that the electrostatics

model is time-independent. One of the characterisdcs of the DBD is that it performs under the

alternating current, which is time-dependent. Hence, modification of the existing model is

required, resulting in the development of the two systems: the spatial system and the spatial-

temporal system. Both numerical results of the systems are implemented and analyzed. It is

noticed that both systems produce similar results which indicates that the earlier formulated

time-independent models actually is sufficient to model the problem. Then, the sensitivity

fi

analysis is conducted to certain DBD parameters in order. to study the performance of the

DBD as a plasma actuator. This study concluded -that some parameters do affect the DBD

plasma actuator.

iv

ABSTRAK

Perlepasan halangan dielectrik (DBD) adalah penggerak plasma. Penggerak plasma adalah

satu sistem atau mekanisme yang menghasilkan plasma tiruan. DBD telah digunakan dalam

pelbagai industri, terutamanya dalam industri kejuruteraan alam sekitar. Kebanyakan kajian

yang telah dilakukan adalah melalui eksperimen, tetapi ia tidak mendapatkan maklumat

langsllng mengenai pengagihan medan elektrik dan ketumpatan elektron sepanjang paksi

penggerak. Maklumat ini adalah penting untuk mengoptimumkan pengeluaran plasma.

Dalam kajian ini, model matematik DBD akan dibentangkan. Dua model yang diperlukan

untuk permodelan plasma DBD secara matematik; model elektrostatik dan model aliran

bendalir. Bagi model elektrostatik, model yang diperolehi dari kajian sebelumnya diguna

pakai. Dan bagi untuk model aliran bendalir, persamaan Navier-Stokes digunakan untuk

memperolehi model matematik. Kajian ini m'i:mdapati bahawa model elektrostatik adalah

bebas masa. Salah satu ciri-ciri DBD adalah ia dijanakan di bawah arus ulang alik, yang

bergantung kepada masa. Oleh itu, pengubahsuaian model sedia ada diperlukan,

menyebabkan dalam pembangunan dua sistem: sistem ruang dan sistem ruang-masa. Kedua­

dua keputusan berangka sistem dilaksanakan dan dianalisis. Didapati bahawa kedua-dua

sistem menghasilkan keputusan yang hampir sama. Ini menunjukkan bahawa model bebas

masa memberi maklumat yang mencukupi untuk memod6lkan masalah ini. Kemudian, analisis

sensitiviti dijalankan pada parameter tertentu untuk mengkaji prestasi DBD sebagai

penggerak plasma. Kajian in; menyimpulkan bahawa beberapa parameter mempengarllhi

penggerak plasma DBD.

v

,.. Pusat Khidmllt Maklumat Akademik UNIVERS1TI MALAYSiA SARAWAK

TABLE OF CONTENTS

ACKNOWLEDGMENT iii

ABSTRACT iv

ABSTRAK v

LIST OF FIGURES viii

LIST OF SYMBOLS xi

CHAPTER I: INTRODUCTION 1

1.1 Plasma 1

1.2 Dielectric Barrier Discharges (DBD) 1

1.3 Motivation of the Research 5

1.4 Objectives 6

1.5 Methodology 6

1.6 Outline of Thesis 8

CHAPTER 2: BACKGROUND OF STUDY 10

2.1 Introduction 10

2.2 Past Experimental Studies of DBD 10

2.3 Modelling of Dielectric Barrier Discharge (DBD) 12

2.3.1 Dielectric Barrier Discharge (DBD) plasma creation 12

2.3 .2 Existing Mathematical Models of Dielectric Barrier Discharge (DBD) 14

2.3.2.1 Electrostatics Model 17

2.3.2.2 Lumped-Element Circuit Model 18

2.3.2.3 Linearized Force Model 19.'2.3.2.4 Potential Flow Model 19

2.4 Summary 20

CHAPTER 3: MATHEMATICAL MODELLING 21

3.1 Introduction 21

3.2 Characterizing the System 22

3.3 Spatial-Temporal System 23

3.3.1 Electrostatics Model 23

3.3.2 Fluid Flow Model 28

vi

3.4 Spatial System 31

3.4.1 Electrostatics Model 31

3.4.2 Fluid Flow Model 32

3.5 Boundary Conditions 32

3.6 Summary 33

CHAPTER 4: NUMERICAL FORMULATION 34

4.1 Introduction 34

4.2 Discretization Scheme 35

4.2.1 Spatial System 36

4.2.1.1 Electrostatics Model 36

4.2.1.2 Fluid Flow Model 37

4.2.2 Spatial-Temporal System 39

4.2.2.1 Electrostatic Model 40

4.2.2.2 Fluid Flow Model 40

4.3 Numerical Procedure 41

4.4 Summary 44

CHAPTER 5: NUMERICAL RESULTS 45

5.1 Introduction 45

5.2 Numerical Results of Spatial Model 46

5.3 Numerical Results of Spatial-Temporal Model 52

5.4 Sensitivity Analysis 60

5.4.1 Behaviour of the Velocity Component as the Applied Voltage Increases (Spatial and Spatial-Temporal) 60

5.4.2 Effect of the Debye Length Values 62

5.4.3 The effect of the Dielectric Thickness 64

. 5.4.4 The effect of thf Electrode Thickness 65

5.4.5 Relationship between Electrode Gap (x-direction) and The Plasma Velocity 66

5.5 Summary 68

CHAPTER 6: CONCLUSION 69

6.1 Summary 69

6.2 Conclusion 70

REFERENCES 73

APPENDIX A 79

vii

LIST OF FIGURES

Figure Page

1.1: Surface Discharge Configuration (Taken from Suzen and Huang (2005)) 3

1.2: Volume Discharge Configuration (Taken from Kogelschatz (2000)) 4

1.3: Methodology to model the DBD surface discharge for the spatial-temporal

system and spatial system 8

2.1: Existing Mathematical Models 16

3.1: Design of the DBD (where l' is the electrode gap at x-axis, cr is the electrode

thickness, X is the dielectric thickness and S!s the electrode width) 22

3.2: Boundary conditions for the electrical potential equation 33

4.1: Flowchart for the Numerical Procedure 35

4.2: Algorithm of the numerical procedure 43

4.2: Grids of the DBD configuration 44

.' 5.1: Schematic of regions of the focused area for the contour images or

vector fields presented in this Chapter 46

5.2: Contour image of electrical potential, lp, at region A 47

5.3: Contour image extracted from Lebeau et al. (2007) paper on page 6 47

viii

5.4: Intensity of electrical potential, rp, in Region A for a spatial model 48

5.5: Magnitude of the electrical field, E, in region B for the spatial system 49

5.6: Magnitude of the body force in Region B for the spatial system 50

5.7: Magnitude of the vorticity of plasma flow in Region B for the spatial system 51

5.8: Magnitude of the velocity of plasma flow in Region B for the spatial system 52

5.9: Plasma velocity field in Region B for the spatial system 52

5.10: The applied voltage of the alternating current used in this that was defined

in equation (3.13) 53

5.11: Intensity of electrical potential, rp, at negative peak in Region A for

spatial-temporal model 54

5.12: Magnitude of the electrical field in the Region B at the positive peak of

applied voltage for the spatial-temporal system 55

5.13: Magnitude of the electrical field in the Region B at the negative peak

of applied voltage for the spatial-temporal system 55 ".

5.14: Magnitude ofthe body force vector in the Region B at the positive peak of

applied voltage for the spatial-temporal system 56

5.15: Magnitude of the body force vector in the Region B at the negative peak of

applied voltage for the spatial-temporal system 56

ix

I,...

5.16: Magnitude of the velocity in the Region B at 0.25 seconds for the spatial-

temporal system 57

5.17: Magnitude of the velocity in the Region B at 0.75 seconds for the spatial-

temporal system 58

5.18: Magnitude of the velocity in the Region Bat l.25 seconds for the spatial-

temporal system 58

5.19: The applied voltage of the alternating current used was given in equation

(3 .13) for the period of 4 seconds 59

5.20: The maximum velocity magnitude from time zero seconds to four seconds 59

5.21: Applied Voltage vs Maximum Velocity . 61

5.22: Maximum Velocity vs Debye Length 63

5.23: Maximum Velocity vs Dielectric Thickness 65

5.24: Maximum Velocity vs Electrode Thickness 66

5.23: Maximum Velocity vs Electrode Gap (x-direction) 67 . .,.

A.I: Flowchart of the numerical procedure for electrostatics model 79

A.2: Flowchart of the numerical procedure for fluid flow model 80

x

I,..

LIST OF SYMBOLS

F Force

Feleerrie Electrical force

Fmagnerie Magnetic force

q Electrical charge

v Velocity by a magnetic field

jj Magnetic induction

E Electrical Field

I5 Electrical induction

Charge density Pc

Dielectric coefficient

Electrical potential

Ad Debye length

Eo Pennittivit'y

Te Temperature of electron

no Density of electron

ib Body force vector

Time

xi

l

w Angular frequency of alternative current voltage

v' Flow velocity

p Fluid density

p' Pressure

J,l' Viscosity

'P' Stream-function

W Vorticity

L Length

Uoo Free stream velocity

Re Reynolds number

r Body force in x direction

fY Body force in y direction

Uniform grid in x direction

17 Uniform grid in y direction

N Number o~grid

,,

xii

CHAPTER 1: INTRODUCTION

1.1 Plasma

Gas and liquid sometimes contain charged particles or ions. When the amount of the

charged particles or ions is sufficient enough to alter the physical properties of a fluid, the

fluid is then said to be ionized. When the ionized fluid contains equivalent amount of

negative and positive charged then it is called quasi-neutral. Ionized fluid is also known as

plasma. From this defInition, it is understood that plasma is a gas or liquid in which certain

portion of it is made of charged particles. According to Browning (2008), true plasma is

Teached when the ions or charged particles outnwnbered the neutral particles that are limited

solely by collision with other particles or with the containment of a vessel wall.

A lot of studies have been carried out to learn more about plasma (Kogel schatz, 2000).

These researches were done in a wide fIeld of studies; from engineering (e.g. Keppens et al.

(2009)), medicine (e.g. Kong et al. (2009); Fridman et al. (2006)) to astrophysics (e.g.

Frederiksen (2008)). Kogelchatz (2000) listed some examples of the application of plasma

such as ozone generator, a pollution control and many more. As plasma applications useful in

solving real world problems, scientists have constructed artifIcial plasma for research study

and application purposes.

\,

1.2 Dielectric Barrier Discharges (DBD)

Dielectric Barrier Discharge (DBD) is a plasma actuator. A plasma actuator is a system

or mechanism which produces plasma artifIcially. It was originated from as a method for

1

ozone production. Studies in DBD have only been active for the last decade (Orlov, 2006).

One of the most important characteristics of plasma underlined by Kogelschatz (2000) was

discovered that non-equilibrium plasma conditions could be provided at elevated pressure,

such as at atmospheric air pressure. DBD is also known as silent discharge, atmosphere glow

diseharge, and single barrier discharge (Orlov, 2006) . This discharge device has been used in

wide range of industries, mainly in environmental engineering industry (e.g. Pietscha and

Gibalovb (1998)). Recently, there are large numbers of studies that have been done on DBD

in aerodynamics field (e.g. Orlov (2006); Thomas et al. (2009); Corke et al. (2008)) .

Orlov (2006) stated that the DBD configuration requires two or more electrodes are

arranged and insulated by a dielectric material, and a high voltage of alternating current

(mainly 5 kV to 20 kV) is applied to at least one of the electrode. This configuration will

produce plasma discharge. There are two main types of discharge on DBD; i.e. volume

discharge (VD) and the surface discharge (SD). These two types of discharges are different

from each other in many aspects such as the configuration and application (Xu, 2001).

The definition of surface discharge (SO) is where the electrodes are arranged

asymmetrically and plasma actuator flow control devices are used. One electrode is exposed

to the surrounding air and another is encapsulated within the dielectric material. When high

voltage is applied to the exposed electrode, plasma will then appear on the dielectric surface.

This DBD type of discharge is also known as single plasma actuator. This type of DBD is

studied in aerodynamics field because of the amount of momentum coupling effective in

substantially altering the airflow over the actuator surface (Orlov, 2006). He also stated that

the SD successfully been used in different flow control applications, e.g. in exciting boundary

layer instabilities on a sharp cone at Mach 3.5, in lift augmentation on a wing section low­

pressure turbine blade separation control, turbine tip clearance flow control, bluff body

2

control, drag reduction, unsteady vortex generation and airfoil leading-edge separation

control. He also has mentioned that a few advantages of using surface discharged plasma

actuator over traditional control devices in which the SD plasma actuator has reduced the size

and the weight, absence of moving parts, increased reliability, inexpensiveness, high

bandwidth, reduced drag, and increased aerodynamic agility. The configuration of SD is

shown in Figure 1.1.

flow

exposed electrode

•• •

embedded electrode

plasma region

body force

dielectric material (e.g. kapton)

Figure 1.1: Surface Discharge Configuration (Taken from Suzen and Huang (2005))

Volume discharge (VD) is where at least one dielectric material insulates for at least two .' electrodes. There are a few configurations under the VD tyPe. The main difference between

SO and VD is that the geometry of VD with a discharge gap between the electrodes in which

when a high voltage is applied to one of the electrode, plasma will appear at the discharge

gap but none in SD. VD is mostly used as an ozone generator in which the discharge gap is

where oxygen gas will be flowed in order to produce ozone layer (e.g. Lopez (2008)). The

configuration ofVD is shown in Figure 1.2.

3

~~I- Dielectric Barrier

High Voltage Electrode

Discharge

Ground Elecrode

Figure 1.2: Volume Discharge Configuration (Taken from Kogelschatz (2000))

To date, many applications of DBD have been discovered in wider scope of industries

(Lopez, 2008). Most common applications of DBD are excimer lamps (e.g. Carman &

Mildren (2003)), plasma display, water treatment (e.g. Goossens et al. (2001)) and carbon

~ioxide lasers (e.g. Kogelschatz (2003)). The advantages of using excimer lamps and plasma

display are that it reduces the usage of chemicals for processing and less waste. DBD now

has become more important as compared to the time when it was first discovered decades ago

due to the wide range of its applications.

A DBD device has a few properties as mentioned by Orlov (2006). The first property is

that the capacity for electrical load is low. The second property is the starting voltage to

ignite the DBD is high, from lkV to 10 kV. The next property is the huge amount of

4

Pusat Khidmat Maklumat Akademik UNlVERSm MALAYSIA SARAWAK

electrical energy stored in the electric field. The last property is voltages and currents

produced during the process of the plasma creation have major influences on the behaviour of

the plasma discharge.

1.3 Motivation of the Research

Although DBD was discovered decades ago, there have not been many researches done

until recently. Nevertheless, there is still a lack of knowledge on DBD. Most research was

done through experimental studies. Orlov (2006) has stated that research through experiments

do not obtain direct infonnation on the distribution of the electric field and electron density

along the discharge axis. This infonnation is important in order to increase the efficiency of

the plasma actuator. Without knowing the physics of the plasma flow, the creation of plasma

may not be optimized.

Due to this fact, the work on modelling the problem mathematically started at the end of

the twentieth century to optimize the creation of plasma. These research works resulted in

many different models due to the difference of the focus of study as the models are clearly

dependent on the design of the DBD. Each of these models has their own limitations in which

will be discussed in subsection 2.3.2 .

. Clearly there is still a need to research on the modelling the DBD problem

mathematically. The earlier researches that stud~es on DBD mathematically did not take

alternating current (AC) into account. From all the models developed earlier, the common

assumption that has been made is that the alternating current (AC) used for the voltage supply

is insignificant to the overall perfonnance of the DBD (see Suzen et a1. (2005), Orlov (2006),

and Bouchmal (2011)). Therefore, the models developed earlier do not include the AC.

5

Without taking the AC into account, the system seems to be operating under directed current

(DC) voltage, which is not true for DBD system. Hence, there is a need to study the

behaviour of the plasma if the AC is reinstated back into the DBD system.

1.4 Objectives

The objectives of this study are:

1. To formulate the mathematical models

incorporates AC and another without,

for the DBD system; one that

2. To compare the numerical results

incorporates AC and another without,

obtain for a DBD system; one that

3. To analyse the sensitivity of the configuration of the DBD system.

Plasma flow of the DBD surface discharge will be the main focus of this research.

1.5 Methodology

As mentioned earlier in this Chapter, previous models of DBD do not take AC voltage

into account so in this project, such condition will be considered and thus, the physical

system is then characterized into mathematical representation. The properties of the physical

system of DBD will then be defmed into mathematical terms. From the understanding of

plasma physics, it is noticed that there are two main components of plasma, the fluid and the

charged particles (Browning, 2008). With this understanding, there are two models that

represent the DBD system and they are electrostatics model and fluid flow model.

6

The electrostatics model is used to represent the electrical charge behaviour on the

plasma, whilst the fluid flow model is used to represent the plasma flow physics. Suzen et a1.

(2005) electrostatics models is derived from the Maxwell's equations with few assumptions,

while fluid flow model by Orlov (2006) is derived from the Navier-Stokes equations The

Maxwell's equations are chosen for the electrostatics model as the equations that describe the

behaviour of electric and magnetism observed in plasma creation. The coupling of the

Maxwell's equations and the Navier~Stokes equations will provide the understanding of

electrohydrodynamics and magnetohydrodynamics.

As earlier mentioned, we will be studying two types of DBD systems, the one that

includes AC voltage and another one that does not include the AC voltage. The first system is

a system that takes into account the AC voltage, which will be from now on called spatial-

temporal system. The reason being, when AC voltage is incorporated in the model the system

of equation not only have space dimension but aJso time. The other system that does not take

into account the AC voltage is called spatial system due to the fact that the system has only

the space dimension. Since the spatial-temporal system incorporates time then it is considered

as an unsteady system. Spatial system on the other hand is a steady system as it only depends

on space parameters.

Figure 1.3 illustrates the process of solving both the spatial-temporal systems and the

spatial systems, where both systems contain electrostatics models and fluid flow models.

Both the models obtained from the modelling process will be continuous system of equations.

The governing equations of the two models will be discretized using the finite difference

method. This process is done to transfonn continuous models into discrete models. These

discrete models will then be transfonned into a computer algorithm. The numerical results

obtained from the algorithm will be compared with the results that were obtained from other

7

researches to ensure the models derived are valid. When the models have been validated,

further analysis will be carried out in order to understand the effect of AC voltage to the

system of DBD. Apart from that, the sensitivity of selected parameters on the DBD system

can also be analysed.

Electrostatics model derived from Maxwell's equations

Body force

Fluid Flow model derived from the Navier-Stokes equation

Figure 1.3: Methodology to model the DBD surface discharge for the spatial-temporal

system and spatial system

1.6 Outline of Thesis

In this Chapter, the introduction to the DBD system is discussed in order to gain the

basic understanding of the system. The motivation of the whole research was as stated. From

8

the motivation, the objectives and the scope of the research were also highlighted. Lastly, the

expected outcome from this research is determined.

In the next Chapter, selected past experimental and mathematical research works are

presented and discussed. Based on the discussion, better understanding of the current state of

the DBD system will be gained and thus the improvements on the existing models are

justified.

In Chapter 3, the physical system of the DBD are discussed and characterized into a

mathematical representation. The formulation of the governing equations for both systems:

spatial system and spatial-temporal system are presented.

Chapter 4 discussed the formulation of the numerical procedures that solve the

governing equations obtained in the previous Chapter. The algorithms constructed are

discussed in detailed in this Chapter.

The discussion on how the verification of the models obtained is presented in Chapter

5. Also, other numerical results are presented here. This includes the sensitivity analysis

perfonned on the DBD systems.

In the fmal Chapter, the summary of the whole thesis are outlined. From here,

conclusions are made and future works are outlined.

9

CHAPTER 2: BACKGROUND OF STUDY

2.1 Introduction

In this Chapter, the previous conducted and experimentally researches will be

discussed, highlighted and compared. As known, mathematical modelling methods were only

carried out in the later researches to compare the outcome or results that were obtained

experimentally. Hence, the past research papers and records will be examined and compared .

As mentioned in prior chapter, only a few group of researchers have studied the Dielectric

Barrier Discharge (DBD) mathematically in which the mathematical models were constructed

in different aspects of the plasma discharge. Some of the research studies did explore the

physics of plasma flow of the plasma discharge (e.g. Suzen & Huang (2006); Orlov (2006))

while other research studies focused more on the chemical reactions or the ionization process

of the plasma discharge (e.g. Corke et a1. (2004); Massines et a1. (1997)).

2.2 Past Experimental Studies of DBD

The main motivation of studying the DBD is to gain more knowledge about the

plasma discharge produced by the system. Recall that the definition of plasma is ionized

fluids, which is containing quasi-neutral charges that 'consist of both neutral and charged

particles. Browing (2008) indicated that plasma physics have been studied for centuries as it

is the centre or the core to the development of nuclear fusion as clean renewable source of

energy. Up to this date, the uses of plasma technology and processing have been increases, as

well as the increase of plasma usage in industrial (e.g. Borcia et a1. (2003)), commercial (e.g.

Mildren & Carman (2001)) and environmental applications (e.g. Borcia et a1. (2003)).

10

According to Browing (2008), any flow of electrical current through gas is called gas

discharge. In which the flow of the plasma depends on the electrical current that were

produced by electrical conductors. Current are let to flow through gas is for some particles to

be ionized, and an electric field to drive the charged particles to fonn a current.

Among the early experimental studies on DBD was done by Peitscha and Gibalovb

(1998). Their focus of study was on the relation of DBD and ozone synthesis, which was

done by comparing volume discharge and surface discharge on the discharge structure and

the development and the transferred charge of DBD. Since then, several groups of researchers

have done studies on DBD experimentally. For example, Enloe et al. (2004) presented the

simultaneous optical, electrical and thrust measurements of the aerodynamics plasma

actuator; Corke et al. (2008) wrote a paper on the physics, modelling and application of

single DBD plasma enhanced aerodynamics, and few more.

Thomas et al. (2009) also stated that previous researches done on single DBD were

focused on the applications of the plasma actuator. For example, Post and Corke (2006), and

Benard et al. (2008) wrote on active airfoil leading edge separation control, Post and Corke

(2006) wrote on control of airfoil dynamic stall, and Corke et al. (2004) wrote on highlift

applications. There were also several papers focusing their studies on the physics surrounding

the operation of single DBD plasma actuator

Orlov (2006) reported that one of the first models developed on DBD was by

Massines et aI. in 1997. In their work, they developed a one dimensional DBD volume

discharge model that was at the time, based on a simultaneous solution of continuity equation

for charges and excited particles, and Poisson equation. Their study managed to obtain spatial

temporal distributions for plasmas.

11