A COMPUTATIONAL STUDY FOR DEVELOPMENT STAGES OF ELECTRICAL PRE- BREAKDOWN PHENOMENON OF ROD-PLANE AIR GAP

Transcription

1 Republic of Iraq Ministry of Higher Education And Scientific Research University of Baghdad College of Science A COMPUTATIONAL STUDY FOR DEVELOPMENT STAGES OF ELECTRICAL PRE- BREAKDOWN PHENOMENON OF ROD-PLANE AIR GAP A thesis Submitted to the College of Science, University of Baghdad In Partial Fulfillment of the Requirements for the Degree of Master of Science Physics By Rusul Hamid Ahmed B.Sc.200 Supervised By Assist.Prof.Dr.Thamir H.Khalaf 204AD Lecture r.dr.qusay A.Abbas 435AH

2 ) أم ن ه و ق ان ت ء ان ا ء ال ي ل س اج دا و ق ا ي ما ي ح ذ ر الا خ ر ة و ي ر ج وا ر ح م ة ر ب ه ق ل ه ل ي س ت و ى ال ذ ين ي ع ل م ون و ال ذ ين لا ي ع ل م ون إ ن م ا ي ت ذ ك ر أ و ل وا الا لب ا ب ( ص د ق االله الع ظ يم سورة الزمر الا ية (9)

3 Supervisor's Certification We certify that this thesis entitled: "A Computational Study for Development Stages of Electrical Pre-breakdown Phenomenon of Rod-Plane Air Gap" is prepared by "Rusul Hamid Ahmed "under our supervision at the Collage of Science, University of Baghdad, in partial fulfillment of the requirements for the degree of Master of Science in Physics. Signature: Supervisor: Dr. Thamir H. Khalaf Title: Asst. Prof. Date: / / 204 Signature: Supervisor: Dr. Qusay A. Abbas Title: Lecturer Date: / / 204 In the view of the available recommendations, I forward this thesis for debate by the examination committee. Signature: Name: Raad M. S. Al-haddad (Chairman) Title: Professor Head of Physics Department College of Science, University of Baghdad Date: / / 204 i

4 CERTIFICATION We, the members of the examining committee, certify that after reading this thesis, entitled" A Computational Study for Development Stages of Electrical Pre-breakdown Phenomenon of Rod-Plane Air Gap" and examining the student " Rusul Hamid Ahmed " on its contents, we think that it is adequate for the award of the degree of Master in Physics (Plasma Physics). Signature: Name: Dr- Ahmed S.Wasfi Title: Assistant Professor Address: College of Science- University of Baghdad Date: / / 204 (Chairman) Signature: Signature: Name: Dr-Hammad R.Humud Name: Dr-Hassan N.Hashim Title: Assistant Professor Title: Instructor Address: Applied Science dept. Address: College of Science- University of Baghdad University of Al-Nahren Date: / / 204 Date: / / 204 (Member) (Member) Signature: Signature: Name: Dr-Thamir H. Khalaf Name: Dr- Qusay A. Abbas Title: Assistant Professor Title: Instructor Address: College of Science- Address: College of Science- University of Baghdad University of Baghdad Date: / / 204 Date: / / 204 (Supervisor) (Supervisor) Approved by the university; Committee of postgraduate studies: Signature: Name: Dr. Mohammed A. Atiya Title: Assistant Professor Address: Dean of the Collage of Science- University of Baghdad Date: / / 204 ii

5 DEDICATION I dedicate this simple effort to the most wonderful persons in my life My first teacher who gave me strength *My father* To her who planted love in my heart, *My mother* To the symbols of love and faith fullness *My brothers and sisters* Rusul iii

6 Acknowledgments First, I should like to express my deep thanks to that great God, ALLAH JALA JALALAH, for what I have been. Then I would like to extend my deepest gratitude and appreciation to my supervisor, Dr. Thamir Hameed Khalaf for suggesting the project, continuous assistance, encouragements, and for useful comments on the manuscript Sincere thanks and deepest gratitude to Dr. Qusay Adnan Abbas who taught me and assisted me a lot provided me with all information which I need in this work. Special thanks are due to all my Doctors and teachers who contributed to my education. Also I want to thank my family my father, my mother, my brothers, my sisters, and every person in my family and my friends for encouraging me. Last but not least, my great thanks due to all my dear friends in the Physics department for their assistance and encouragements. iv

7 Abstract The computational procedure was used in different studies to show an approximate view for the solution of many problems. In this work, a computer simulation was done for the pre-breakdown phenomenon within the air gaps between electrical electrodes. The main aim of this study is to determine the breakdown voltage values along air gaps without using costly devices. This simulation was based on a stochastic model for the initiation and growth of the streamer discharge. The model assumed a threshold value for the electric field at a point to initiate the streamer and the streamer grows in a random direction. Therefore, Laplace s equation was solved, numerically, using finite element method which is known to be an accurate numerical technique. The minimum voltage value which enables the streamer to grow from the anode (the rod) within the air gap to reach the cathode (the plane) is the breakdown voltage for this gap. The results show the development of the voltage variation and electric field distribution within a rod plane configuration of electrical electrodes. The breakdown voltage values were determined for different distances, (3-0cm), between the two electrodes within the configuration. These values were compared with practical and computational results and appeared to be in good agreement. Also, the streamer velocity was studied within the air gaps for different breakdown voltage values. v

8 LIST OF Symbols Symbols Variable definition Unit n i Ions density m -3 n e Electrons density m -3 Z ni Partials density m -3 ω p Plasma oscillation rad λ D Debye length meter T e electron temperature Kelvin T i ion temperature Kelvin T g The gas molecules temperature Kelvin E 0 Internal electric field Volt m - E e External electric field Volt m - Frequency rad P Air pressure torr A area of a triangle element m 2 a, b, and c Coefficients [C (e) ] Element coefficient matrix "stiffness matrix" [C] The over- all or global coefficient matrix D distance from the cathode meter E Applied electric field on the Volt m - plasma E electron charge Coulomb Τ Minimum time for streamer Sec growth Random number. r(e) Field depended growth rate function A g Constant with dimension /sec /sec. U Applied potential Volt E loc Local electric field volt m - E th Threshold field volt m - E s Voltage drop volt n p Number that controls the variation of the growth rate with the electric field E x x- component of E loc vi

9 Symbols Variable definition Unit E y y-component of E loc M The determinant of a matrix N Number of nods n o original number of free electrons leaving the cathode N e Number of elements in the whole region V Voltage drop Volt V Applied voltage Volt V(x, y) Approximate solution for the whole region V e V e V e2 V e3 W e x x 2 x 3 y y 2 y 3 α, α 2, and α 3 Approximate solution for one element Voltage of the first nod of the element Voltage of the second nod of the element Voltage of the third nod of the element Energy associated with the assemblage of elements x-axis of the first nod x-axis of the second nod x-axis of the third nod y-axis of the first nod y-axis of the second nod y-axis of the third nod constant Shape functions of a triangle element Volt Volt Volt joule vii

10 Contents Chapter One Introduction and Breakdown Theory. Introductions....2The plasma Breakdown in Air 6.3.Air Breakdown Voltage Air Breakdown Mechanism 8.3.3The Streamer Discharge. 0.4Modeling of the Streamer Discharge The Stochastic models other Mathematical Requirements Literature survey this work Chapter Two The Numerical Solution and the Simulation 2. Introduction Finite Element Technique Finite Elements Discretization Element Governing Equations Assembling of all Elements Solving the Resulting Equations The Simulation Grid Generation The program input data The Program Calculations.. 37 viii

11 Chapter Three The Results 3.The introduction The General Procedure Streamer Initiation and Growth First Case: Testing the model within 3cm Air gap Second Case: Testing the model within 5cm Air gap Third Case: Testing the Model within7cm Air gap Fourth Case: Testing the Model within 0cm Air gap The Streamer Velocity 79 Chapter Four The Comparisons, Conclusions, and Suggested Future Works 4. Results Comparisons Experimental Comparison Computational Comparison Conclusions Future Works.. 84 References ix

12 Chapter One Introduction and Background Theory

13 Chapter One Introduction and Background Theory Chapter one: Introduction and Background Theory. Introduction The increasing use of electronic devices in our modern society implies a higher demand for the quality of the electric power supply. In many practical situations, it is found that the quality of the electric power is determined by the performance of the insulating elements used in the power generation and transmission utilities. The traditional materials used for outdoor insulation have until recently been glass and porcelain []. But, in recent years, there has been a trend to replace these insulators with dielectric gasses such as air; air is an important insulator gas which is used in high voltage systems [2]. The electric discharge in air is of considerable practical importance to the design engineers of power transmission lines and power apparatus (transformers, generators). The knowledge of the electrical discharge development for various gap geometries is essential for designing airinsulated apparatus used in wide variety of applications covering power systems, industry, and research laboratories [2]. Air in its normal state is a very poor conductor of electricity. If air is subjected to a sufficient electric voltage, the appearance of a current of charged particles is possible due to partial ionization of the environment. Then air is conducting and a discharge occurs [2, 3]. The nature of the electric field has an influence on the propagation of a discharge and on the value of the disruptive electric field [4]. In uniform electric field, the discharge usually take place in the form of a complete breakdown while in non-uniform field, the discharge will occur only in areas where the electric field intensity is higher than the dielectric strength of the gas. This process is known as partial discharge and when it takes place in gases it is called "corona discharge" [5], the non-uniform electric field

14 Chapter One Introduction and Background Theory can be established if one of the electrodes has a much smaller radius of curvature than the other. The most typical configurations of electrode systems used in practice to generate the electrical discharge under inhomogeneous electric fields are; point-to-plane, multi-point-to plane, wire-to-pipe, wire-to-plane or wire between two planes, multi-wire-toplane or multi-wire between two planes, coaxial wire-cylinder [5]. Gaseous dielectrics in practice haven t free electrically charged particles, including free electrons. The electrons, which may be caused by irradiation or field emission, can lead to a breakdown process to be initiated [6]. These free electrons are accelerated from the cathode to the anode by the electric stress applying a force on them.the breakdown process involves three stages; () primary electron generation to initiate the process; (2) exponential growth of the number of charge carriers and (3) secondary electron generation to sustain the discharge until breakdown [6]. In general, discharge in a dc electric field can be classified into (a) non-self-sustaining and (b) self-sustaining types. The latter are more widespread, more diversified, and richer in physical effects. Steady and quasi-steady self-sustaining discharge contains :( ) glow discharge (2) arc discharge. A close relation of the glow discharge is (3) Townsends dark discharge. It precedes with a cold cathode and at very weak current (4) the corona discharge, also self-sustaining and also at a low current, is a special case corona has common features with glow discharges. Among transient discharge (5) the spark discharges out sharply [7]..2 The Plasma The term plasma, first, introduced by the American physicists Langmuir and Tonks in 923[8]. Plasma means the ionized state of 2

15 Chapter One Introduction and Background Theory matter. It varies from the three fundamental states (solid, liquid and gases) of matter by the ionization of its atoms or molecules, while the atoms or molecules are neutral in all the three states. Consider the series of phase transitions solid-liquid-gas. If temperature is continuously increase above, say, K (lower if there is a mechanism for ionizing the gas) plasma is obtained. It is also known as "fourth state of the matter"[9]. Plasma can be generated by heating gases, when the atoms or molecules have lost one or more electrons they carry positive charge outwardly; in this case they become positive ions. Plasma is therefore considered as gas showing collective behavior and consisting of particles which carry positive and negative charges in the extent that the overall charge comes to zero. Generally, Plasma is electrically neutral to the outside, if the number of positive and negative charges equals in a sufficiently large volume and for a sufficiently long interval of time. This balance is referred to as "quasi neutrality" at n e = Z ni [8]. Either Strongly non-neutral plasma, which may even contain charges of only one sign, occurs primarily in laboratory experiments: their equilibrium depends on the existence of intense magnetic fields, about which the charged fluid rotates. The plasma has interesting properties because the electrostatic force is a long range force and every charged particle interacts with many of its neighbors. So that it can get a collective behavior and can be treated as an electrical fluid [0]. It has been said that 99% of matter in the universe is in the plasma state. Lightning earth s ionosphere, aurora, earth s magnetosphere, radiation belts interplanetary medium, solar wind solar corona stellar interiors interstellar medium laboratory plasmas such as glow discharges, arcs fluorescent lamps, neon signs electrical sparks thermonuclear fusion experiments a homely examples: flame other examples: rocket 3

16 Chapter One Introduction and Background Theory exhaust are all examples of plasma [9]. Ionization in plasma added to the heating of gasses can be induced by other means, such as particle beam (external source), electromagnetic waves such as laser, electromagnetic field and electric field [].the gas becomes ionized by an electric field (ignition) and a self-sustaining mechanism stabilizes the plasma at a certain current and figure (-) represents the methods of plasma generating []. Figure (-): Methods of plasma generating []. Parameters of plasma are different according to the conditions of plasma production such as, () Electron density (n e ) which is related to the degree of ionization (ratio between the ionized to neutral atoms). Electrons in plasma are oscillating with a frequency ω p due to the location of the electrons between two ions. (2)Debye shielding (λ D ): plasma can be produce internal electric field which is opposite to external electric field, 4

17 Chapter One Introduction and Background Theory as a result from this operation; the charged particles are closed to the electrodes with a limit thickness. (3)From applied external electric potential the electrons will gain some of energy, where its energy represented by the electron temperature T e [2]. Generally, plasma are described by many characteristics, such as temperature, degree of ionization, and density, the magnitude of which, and approximation of describing them, gives rise to plasmas that may be classified into two types, example for types, hot plasma cold plasma: In low pressure gas discharge, the collision rate between electrons and gas molecules is not frequent enough for non-thermal equilibrium to exist between the energy of the electrons and the gas molecules. So high-energy particles are mostly composed of electrons while the energy of the gas molecules is around room temperature. We have T e >> T i >> T g where T e, T i and T g are the temperatures of the electron, ion and gas molecules, respectively. This type of plasma is called "cold plasma". In a high pressure gas discharge the collision between electrons and gas molecules occurs frequently. This causes thermal equilibrium between the electrons and gas molecules. We have T e Ti T g. We call this type of plasma "hot plasma"[3]. A plasma diagnostics mean the studying of plasma properties such as density, temperature, velocity distribution etc. That can be done by different local or remote diagnostics methods. Either the applications of plasma, cleaning, activation, etching, medicine and industrial, there are many industrial applications such as (light source, volume processing, surface treatment, radiation processing, lasers and material analysis) [4]. For each application, the plasma must be modified by a way to produce an advantages are accepts with these application with control by the plasma operators, for example if we want 5

18 Chapter One Introduction and Background Theory make a thin film from Aluminum material, the electrodes must be produced from Aluminum, other example if the smooth welding is required, in this process the Argon gas is domain, because of the ionization energy of Argon is smaller than other gases, and in same time it is inert gas, and so on for other applications [5]..3 Breakdown in Air An insulator, also called a dielectric is a material that resists the flow of electric charge [6]. The insulating materials are divided into four types like air, gases, liquids and solids. Normally air medium is widely used as an insulating medium in different electrical power equipment's [7]. The breakdown in air is the transition of a non-sustaining discharge into a self-sustaining discharge. The buildup of high currents in a breakdown is due to the ionization in which electrons and ions that are created from neutral atoms or molecules, and their migration to the anode and cathode, respectively, leads to high currents. Townsend theory and Streamer theory are the present two theories which explain the mechanism of breakdown under different conditions such as temperature, pressure, nature of electrode surfaces, electrode field configuration and availability of initial conducting particles [7]. Basic breakdown processes are represented by : ()Primary electrons: Free electrons exist for only a short period of time in air that is not subject to high electric field; normally they are trapped, after creation by cosmic radiation, to form negative ions. These have a density commonly of the order of a few hundred per cubic centimeter [8]. (2)Ionization: The electrons so liberated can themselves be accelerated in the field, collide with neutral molecules and finally settle down with drift velocity. When they have 6

19 Chapter One Introduction and Background Theory sufficient energy, the collision may liberate a new electron and a positive ion. The process is cumulative and is quantified by Townsend, resulting in the formation of avalanches of electrons. The growth in number of electrons and positive ions impart a small conductivity but it is not large enough for breakdown [8]. (3)Excitation: Where electrons are sufficiently energetic to cause ionization, there is usually a plentiful supply with lower energies that can excite neutral atoms without liberating electrons. When returning to ground state, these atoms emit visible or ultra violet light [8]. (4)Other electron processes: The electrons created by the growth of ionization may be trapped and so removed from the ionization process. This is the attachment process: A net growth of electrons and ions population occurs only when the field is sufficiently high for the rate of ionization to exceed the rate of attachment. Subsequent detachment of electrons from negative ions occurs at the same time, through collisions with neutrals, with free electrons or by interaction with photons. Recombining between electrons and positive ions and between positive and negative ions is a further element in the competing processes that are active in an ionized gas [8]. (5)Regeneration: Initially, Townsend postulated that the positive ions, a process now recognized as insignificant. Also that they move towards the negative electrode to release further electrons by secondary emission, so that the ionization process could be sustained and grow indefinitely until breakdown occurred. Experiment later showed that breakdown could occur much more quickly than this process would allow. The solution lay in postulating that the positive ions, created by ionization, are sufficient to create an electric field which, when added to the applied field intensifies the ionization process [8]. 7

20 Chapter One Introduction and Background Theory.3. Air Breakdown Voltage Breakdown voltage is known to be a characteristic of an insulator; it can be defined as the maximum voltage difference that can be applied across a material before the insulator conducts and collapses. Breakdown voltage is also called "striking voltage" [9]. Breakdown voltage was demonstrated experimentally by several researchers such as Feser [20], Abraham and Prabhakar [2]. For numerical representation of experimental data, the following empirical relation was proposed by Feser[20]: U d (. ) Where d is the gap distance in cm, U 50 is the voltage in kv (50 is symbol for the applied voltage), the other approximation was proposed by Abraham and Prabhakar [2]: U ( 50 d. 2 ).3.2 Air Breakdown Mechanisms Most of the electrical equipment's use air as an insulating medium. Various phenomena occur in the air medium when a voltage is applied. When the applied voltage is low, small currents flow through the air gap, but it retains its electrical properties. On the other hand, if the voltage applied is large enough, current increases rapidly and an electrical breakdown occurs. A strongly conducting spark is formed, creating a short circuit between the two electrodes. The maximum voltage applied at that moment is called breakdown voltage [22]. The mechanism of electrical breakdown depends on the nature of the 8

21 Chapter One Introduction and Background Theory dielectric but in any case is a physical phenomenon with complex evolution, especially for the gaseous dielectrics. Two different mechanisms of breakdown exist. In very short gaps at low pressures, approximately at pd < 200Torr. cm the breakdown is based on the multiplication of avalanches via secondary cathode emission (Townsend Mechanism of Breakdown). In gaps greater than cm long and in pressures above Atmospheric, pd >000Torr.cm, the mechanism of breakdown is based on the development of thin, weakly ionized channels of plasma called streamers (Streamer Mechanism of Breakdown) [22, 23]. In air gaps of many meters and in lightning discharges the breakdown occurs via the growth of the so-called leader, which is an ionized channel but with a conductivity orders of magnitude higher than the conductivity of streamer channel. The boundary values of the pd at which the above mentioned breakdown mechanisms replace each other are not yet precisely estimated. According to Meek,Craggs[23] the transition in air from the Townsend mechanism to the Streamer mechanism occurs at d ~5cm. On the other hand, in long air gaps, greater than 50 cm approximately, breakdown occurs via the formation of leader channels in the gap. In this case, streamers also exist starting from the head of the leader channel which acts like a metallic tip [23]. The streamer mechanism depends on an avalanche to streamer transition, due to instantaneous local electron generation giving rise to a critical avalanche that causes instability in the gap and induces gapbreakdown. In between, there is a transition region in which some of both mechanisms can be observed. The avalanche is the ionizing growth process in which each electron generates more than one subsequent ionizing electron there by enabling exponential growth. In case the space charge field created by the charged species is of the same order of magnitude as the applied external field, the space charge influences the 9

22 Chapter One Introduction and Background Theory local ionizing processes considerably. We call this type of discharge a streamer discharge. An avalanche may grow directly into a streamer discharge, or it may initiate a sequence of avalanches finally leading to a streamer discharge [24]..3.3 The Streamer discharge. Streamers are phenomena that occur frequently in everyday life. Although streamers are formed on many desired or undesired occasions they are still not fully understood and usually their appearance is just considered as a fact [2]. When a high voltage is applied over a non-conducting gas volume the gas will become ionized. The ionization will typically not occur homogeneously but in the form of extending channels, so called streamers. These streamers are observed in gases as well as in liquids and in non-conducting solids. A streamer is a thin ionized channel that rapidly propagates between electrodes along the positively charged trail left by an intensive primary avalanche [25]. This avalanche also generates photons, which in turn initiate numerous secondary avalanches in the vicinity of the primary one. Electrons of the secondary avalanches are pulled by the strong electric field into the positively charged trail of the primary avalanche, creating the rapidly propagating streamer between electrodes. A streamer starts always from an initial seed of ionization that eventually develops its own field enhancement. The creation of a streamer starts with the acceleration of free electrons through an electric field between two electrodes in a non-conducting gas [26]. When such a free electron propagates through an electric field, it gains energy which can be used to free another electron from its molecule by electron impact ionization if the ionization threshold energy is reached [25]. Then two electrons propagate through the electric field, gain energy and ionize two 0

23 Chapter One Introduction and Background Theory other molecules resulting in a total of four free electrons and so on. In this way an electron avalanche is created, as was first described by Townsend [27]. Townsend's avalanche theory turned out to be unsuitable to explain the further propagation of the streamer since experiments showed that streamers would also run in much lower applied electric fields of ~5 kv/cm, where the field is too low for ionization. Raether and Loeb [22,28] suggested that a plasma channel propagates through an electrode gap by ionizing the gas in front avalanche charged head owing to a strong field at the head itself (higher than ~ 30 kv/cm), the space charge here being smeared out over the complete streamer head. Later simulations show that there is indeed an enhanced electric field at the streamer head which takes care of the propagation of the streamer through low fields. They show that the space charge is not in the complete head but in a thin layer at the front of the streamer. Therefore, field enhancement can be stronger. Streamers can be positive (cathode directed) or negative (anode directed) [29, 30]. The avalanche-to-streamer transformation takes place when the internal field of an avalanche becomes comparable with the external one. If the gap is short, the transformation occurs only when the avalanche reaches the anode.such a streamer that grows from anode to cathode and called the cathode-directed or positive streamer. If the gap and overvoltage are large, the avalanche-to-streamer transformation can take place far from the anode, and the anode-directed or negative streamer grows toward both electrodes [25]. The mechanism of formation of a cathode-directed streamer (positive streamer; the interest of this work) is illustrated in figure (-2); highenergy photons emitted from the primary avalanche provide photo-

24 Chapter One Introduction and Background Theory ionization in the vicinity, which initiates the secondary avalanches. Electrons of the secondary avalanches are pulled into the ionic trail of the primary one and create a quasi-neutral plasma channel. The cathodedirected streamer starts near the anode, where the positive charge and electric field of the primary avalanche is the highest. The streamer looks like a thin conductive needle growing from the anode. The electric field at the tip of the anode needle is very high, which provides high electron drift and streamer growth velocities. The specific energy input in a streamer channel is small during the short period of streamer growth between electrodes. Figure (-2): Illustration of the cathode-directed (positive) streamer: (a) propagation of the positive streamer; (b) electric field near the streamer head [25]. The process of avalanche-to-streamer transition and subsequent streamer propagation requires a sufficiently large external electric field. Suitable conditions can be easily obtained by applying DC, AC, or pulsed HV power in non-uniform electrode configurations. One can expect that a higher electric field will likely produce electrons with higher energies, which may result in higher rates of induced plasma-chemical reactions [3]. 2

25 Chapter One Introduction and Background Theory The appearance of streamer discharges varies greatly, taking on many different forms that depend on the voltage, polarity and gap configuration. The positive streamer appears mainly as a diffuse track with a luminous head. It can be single but is, in general, branched. Negative streamers appear to have much more complex structure. The light emitted from the streamer originates from the recombination and deexcitations which occur at the streamer head. The radius of the streamer channel has 0-50 μm. This value however corresponds to short streamers. The streamer length has, in principle, no limit. It may grow as long as the gap and the voltage source permits. Electrons will lose only a small fraction of their kinetic energy during collisions with the heavy gas particles and, therefore, a different temperature will be established for the electrons and the surrounding gas medium [32]. In a streamer discharge, there is no thermodynamic equilibrium. The main reason for the difference in temperature between the gas particles and the electrons is the inefficient kinetic energy exchange in inelastic collisions between the light electrons and the heavy gas particles [33].One of the most important parameter which characterizes the streamer is its velocity. In general, for a given electric field, the velocity of positive streamers is higher than negative streamers for the same applied background field. The reason could be the high electric field necessary for the stable propagation of a negative streamers compared to that of a positive streamer [34]..4 Modeling of the streamer discharge The streamer growth occurs owing to the formation of new conductive phase regions in dielectrics. Modern computer simulations of this phenomenon are based on the idea of space discretization. New linear segments of streamer channels join sequent neighbor sites of some spatial 3

26 Chapter One Introduction and Background Theory elements to the streamer structure; the streamer shape is represented by a connected graph consisting of conductive bonds [35].There is many models of streamer, such as field fluctuation model, Biller s model, scaled stochastic time models, and Niemeyer-Pietronero-Wiesmann Model (stochastic model).the last one is the model used in this work..4. The Stochastic models Stochastic models are widely used to simulate the propagation of streamers and leaders before breakdown in solid, liquid and gaseous dielectrics. Niemeyer-Pietronero-Wiesmann (the so-called NPW model). Introduced the first stochastic model in984 [35]. It is used for the determination of the minimum breakdown voltage of the gap but it is also capable for the determination of the mean velocity of streamer propagation [36]. All growth criteria considered below are based on two main assumptions. Firstly, the growth is stochastic in time. Secondly, the probability of streamer growth is proportional to some function of local electric field r (E) depending on dielectric properties of a substance. Thus, the generation of new bonds is governed by some stochastic growth criteria in each time step. The electric field potential was obtained by solving the Laplace equation [35, 37]. A sequence of time intervals for each growth step calculated in a proper way following certain assumptions can be named the physical time. Apparently, there are only two ways to introduce the physical time into stochastic models of streamer growth at electric breakdown. One of them is that the time step T is taken at first, then, all bonds that have time to arise according to the same stochastic criteria are accepted. Another way of looking at it is to consider the delay time of appearance of the first new bond as the physical time interval of this growth step [36, 37]. 4

27 Chapter One Introduction and Background Theory Stochastic models are capable of reproducing the course of the streamer propagation by using some special rules, which are based on the probabilistic nature of the streamer advancement through the gap. For the development of the model presented in this work, the rules which govern the growth of the discharge pattern (i.e. the streamer), were the following: - The simulation is taking place in a two dimensional square area which discretized to a finite elements. These elements have nodes; some of the nodes represent the electrodes, while the others represent the dielectric. 2 - The discharge pattern grows in a stepwise manner. The pattern consists of elements centers, which are connected with thick lines called bonds. 3 - At each step only one bond is added to the discharge pattern, linking an element center of the pattern with a new element center. From this moment the new element center is considered to be an element of the pattern. 4 - The electric potential of all elements of the lattice that belongs to the dielectric is calculated by solving the Laplace s equation with the boundary conditions on the electrodes and the discharge pattern. The selection of the new element center, which will be added to the discharge pattern, is a crucial part of the computational process because it determines the direction of propagation of the pattern. The procedure used in this work includes three steps. The first step was to determine all the possible directions for the propagation of the discharge pattern. The directions were chosen on the basis of local electric field. In this case the local electric field E loc was greater than a threshold value E th [38,39]: 5

28 Chapter One Introduction and Background Theory E loc > E th... (. 3 ) The second step was to apply to each possible direction a characteristic time for the growth of a new bond, which is equivalent to the necessary time for the propagation of the discharge pattern from one element center to another. This time was firstly introduced by Biller [39], and it was named physical time. Biller considered the process of the bond growth as a stochastic process, namely a Poisson random event. A stochastic bond growth time τ can then be calculated from this probability distribution with the help of a random number δ uniformly distributed in the unit [36, 37]: ln(... ) r( E) (. 4 ) Where r (E) is a field depended growth rate function. This function is equivalent to the mean value of the distribution. Biller assumed arbitrary power law dependence: n E r( E) A. loc...(. 5 ) U / d The parameter A is a constant with dimension /sec, n is a number that controls the variation of the growth rate with the electric field, U is the applied potential at the anode and d is the gap distance. Parameter A can be calculated theoretically and during the simulations took the value 3.7x0 5 [36]. After the calculation of the growth time for each candidate bond, the computer program 6

29 Chapter One Introduction and Background Theory identifies the winning bond defined by τ as a minimum. The winning bond is added to the discharge structure. The third step was to repeat again, starting a new cycle of calculations, taking into account the evolution of the discharge pattern. The simulation terminates when the pattern reaches the cathode or when the local electric fields at the streamer tip drops below the threshold value E th..5 Other Mathematical Requirements The model is based on the assumption that streamer propagation follows a path that is determined by the local electric field. For calculation of the voltage distribution, Laplace's equation is solved using the finite element method in two dimensions. The electric field in two dimensional has two components E x and E y.the model that governs the testing region requires to calculates the value of the electric field in the limited elements. The implementation region of the model is the area between rod-plane configurations where there are no charges, so that the potential in the region is governed by Laplace's equation for the voltage V: V 0.6 E V x.6a 7

30 Chapter One Introduction and Background Theory E V y.6b The local electric field E loc is the total field and was calculated in terms of its components E x, E y in x and y directions respectively: E E E.7. 6 Literature survey Extensive studies, to investigate air breakdown, knowing from what point the breakdown starts and the factors that affect the progress of the breakdown process enable different methods to be used. These methods fall into three categories: a) Experimental methods. b) Computer simulation methods c) Theoretical methods. The main idea behind these methods is to try to create and monitor pre-breakdown and breakdown events in air. The factors that cause or encourage breakdown have been the main thrust of many investigations. The other important item is to try to explain these events in such a way as to make the design of future equipment more effective. Many researchers have investigated pre-breakdown and breakdown events by a variety of experimental and theoretical methods [40]: Mohamed 965[4] studied the pre-breakdown phenomena in n-hexane and cyclohexenes were studied numerically by using the two-dimensional finite element method. In this study, the streamer propagation was 8

31 Chapter One Introduction and Background Theory considered as a generation of a gas bubble in the liquid. The model of point-plane geometry (non-uniform field) was presented to simulate the streamer growth as well as the positive and negative streamer velocities are calculated. The obtained results showed a good agreement with the published experimental results. Salam and Stanek 988[42] calculated the breakdown voltage for uniform field gaps using a purely theoretical method for pd values to the right of Panchen's minimum, up to 5 k Pa*m in air and 5 k Pa*m in SF 6.In this study, the use of streamer criterion overestimated the breakdown voltage when applied for pd values where Townsend's mechanism is valid.the size of the avalanche at breakdown was not constant as adopted in the literature. It depends upon the gap length and gas pressure. Kulikovsky 998[43] described the analysis model of positive streamer in weak external field and used it to calculate the number of chemically active species generated by streamer in air. Comparison was made with two dimensional numerical simulation of streamer in a 5cm discharge gap between hyperbolical anode and a plane cathode. His result showed that the model well predicts the mean streamer parameters and number of major components (oxygen atoms and excited nitrogen molecules) produced by the streamer. Mokhnache et al. 2000[44] studied computed fields of the positive ions and electrons created at the head of avalanche in a positive point barrier- plane air gap. The polarization field effect of the dielectric was taken into account. The general problem was formulated with finite elements and was focused on the polarization field formulation. It was found that the effect of the two space charge fields was to reduce the breakdown voltage. 9

32 Chapter One Introduction and Background Theory Khalaf 2005[45] using the computer simulation method, studied the electrical pre-breakdown events in dielectric liquids. This study concentrated on the liquid-solid interface. The suitable model was based on the cavitation theory. It was assumed that the streamer channels were weak plasmas and those channels have a high electrical resistance. The model was implemented numerically by finite element method (in twodimensions) with computer graphics. It was tested within a pin-plane configuration using the n-hexane and water as dielectric liquids. Same voltage and electric field distributions were shown in the two liquids. But different distributions appeared when the solid insulator was introduced to the configuration. The results also showed a strong dependence of the streamer growth path on the mismatch permittivity between the solid and the liquid. Charalambakos et al. 2005[36]: studied the simulation of the streamer propagation along a gap of a rod plane configuration of electrodes. The gap length varies between 5 to 20cm and a dc voltage of positive polarity was applied to the gap. Salam and Allen 2005[46] analyzed corona in rod-plane gaps. They concluded that()the onset voltage of positive glow corona can be calculated by a purely theoretical method that avoids the controversy between Hermstein and Loeb postulates (2) The calculated onset voltage of positive glow corona increased with the gap spacing at room temperature and decreased with the increase of air temperature for the same gap geometry in agreement with the experiment (3) Not only the calculated onset voltage of positive glow but also the calculated threshold voltage of onset streamer corona increased with the gap spacing at room temperature and decreased with the increased of air temperature for the same gap geometry in agreement with the experiment (4) The calculated repetition rate of onset streamers increased with the applied voltage for 20

33 Chapter One Introduction and Background Theory the same air temperature in agreement with the experiment. It depended also on the gap geometry. Zhang and Adamiak 2008[47] studied a new dynamic model for the negative corona discharge in the point-plane geometry. The hybrid boundary-element finite-element method technique was used to calculate the electric-field parameters. The method of characteristics was employed for the charge-transport prediction. The simulation results showed the charge-density distributions in the air gap and along the axis of symmetry as well as the total corona current, for different voltage waveforms and parameters of the electric circuit. Kadum 2009[40] using a computer simulation method, studied the electrical pre-breakdown events in dielectric liquids. This study concentrated on the liquid-liquid interface. The suitable model was based on the cavitation theory. It was assumed that the streamer channels are weak plasmas and those channels have a high electrical resistance. The model was implemented numerically by finite element method (in twodimensions) with computer graphics within a pin-plane electrode by using many dielectric liquids. The same voltage and the same electric field distributions were shown in one liquid. But different distributions appeared when the second liquid was introduced to the configuration. Zhao and Adamiak2009 [48] studied the effect of airflow, including the electro hydrodynamic (EHD) flow produced by the electric corona discharge. The flow generated externally on the corona discharge was also investigated numerically in this paper. Contrary to the common assumption that the corona discharge and the EHD flow can be decoupled by neglecting the convection term in the current density equation, the equations for electric field, charge transport, and fluid flow was solved simultaneously. The numerical algorithm was based on boundary and finite element methods, the method of characteristics, and the finite 2

34 Chapter One Introduction and Background Theory volume method. The simulation results confirmed that the EHD flow has very small effect on the corona discharge. However, in some configurations, the external airflow had a considerable effect on the I V characteristics of the system and the current density distribution on the ground electrode. Sattari and Adamiak 2009[49] Numerical algorithm for the simulation of the dynamic corona discharge in air was proposed assuming single species charge carriers. The simulation results showed the behavior of corona current and space charge density under two waveforms of the applied voltage: step and pulse. The electric field was calculated by means of the Finite Element Method.The Flux Corrected Transport technique was utilized for the space charge density calculations. Bourek et al. 20[2] studied the discharge phenomenon for a pointplane air interval using an original fuzzy logic system. Firstly, a physical model based on streamer theory with consideration of the space charge fields due to electrons and positive ions was proposed. To test this model, the breakdown threshold voltage for a point-plane air interval was calculated. The same model was used to determine the discharge steps for different configurations as an inference data base. Secondly, using results obtained by the numerical simulation of the previous model, the fuzzy logic technique was introduced to predict the breakdown threshold voltage of the same configurations that were used in the numerical model; also estimation on the insulating state of the air interval was made. From the comparison of the obtained results. The proposed study using fuzzy logic technique showed a good performance in the analysis of different discharge steps of the air interval. Sankar 20[50] simulated of the air breakdown voltage experimentally in high voltage laboratory. A sphere of standard diameter of 25cm was used for measurements of air breakdown voltages and electric field of the 22

35 Chapter One Introduction and Background Theory high voltage equipment. The experiment was conducted at normal temperature and pressure. Finite element method was also used for finding the electric field between standard sphere electrodes. The relative air density factor and maximum electric field were measured in MATLAB environment for different temperature and pressure. The electric field distribution for different arrangements of sphere gap arrangements was also calculated with the help of COMSOL. In addition, the influence of humidity on air breakdown test was also considered in this study. So, a humidity correction factor was considered to maintain a constant air breakdown voltage. Finally, the experimental results were compared with the theoretical and simulation results..7 This work In this work, a computational procedure is used to study the electrical pre-breakdown phenomenon of rod-plane air gap. The main aim is to determine the minimum values of breakdown voltages for different air gap lengths. A numerical simulation is required in order to follow the growth of a streamer discharge and the development of the voltage and electric field according to the streamer growth. Also, the streamer velocity within those gaps is studied. 23

36 Chapter Two The Numerical Solution and the Simulation

37 Chapter Two The Numerical Solution and the Simulation Chapter Two: The Numerical Solution and the Simulation 2. Introduction Advanced technology in computers has enabled some workers to investigate breakdown using current theories in conjunction with experimental data to simulate the mechanism. In general, such techniques can save time and cost. These results can be used for further understanding of breakdown mechanisms and to aid in the design of further equipment. Computer simulations are much safer than high voltage experiments to investigate breakdown. In using computer simulations, different conditions can be easily selected by changing the data input to the program. Very few workers have tried to simulate air breakdown using the available modeling. The electrostatic field provides the major driving force in the various stages of the pre-breakdown and breakdown processes, which explain the need of simulation to calculate electric field in every stage. The calculation of electric field in air with streamer requires the solution of Laplace s equation subject to boundary and initial conditions. This can be done either by analytical or numerical methods. Analytical solutions are available for very simple configurations. In many instances in physicals systems, there are difficulties in solving Laplace s or Poisson s equation with sophisticated boundary conditions, or for insulating material with different permittivities. Analytical solutions are difficult or impossible because the boundary is non-uniform, for this reason numerical methods are commonly used [5]. Each of the different numerical methods, however, has advantages or disadvantages depending upon the actual problem to be solved. The Finite Element Method (FEM) 24

38 Chapter Two The Numerical Solution and the Simulation technique, which is one of the numerical calculation tools for electric fields, is effective for practical use because of its universality even in applications for complicated multi dielectric regions. Consequently, it is widely used in the design of high voltage apparatus. 2.2 Finite Element method. Many practical problems in science and engineering are difficult to solve by conventional analytical methods; the finite element method is a numerical procedure for solving this type of problems. This is because the method basically relies on solving a large set of algebraic equations and entails considerable manipulation as the case with the finite difference procedure. The fundamental concept of the finite element method is that any region is made up of elements; therefore, the general behavior of a system can be determined by considering the behavior of its components (subsystems). The finite element analysis of any problem basically involves four steps [52]: - Discretizing the solution region into finite number of sub regions or elements. 2- Deriving the governing equations for typical elements. 3- Assembling all the elements in the solution region. 4- Solving the system using the equations obtained Finite Elements Discretization The first step in the finite element method is to subdivide the solution region under consideration into smaller regions (elements). In the case of 25

39 Chapter Two The Numerical Solution and the Simulation one-dimensional, transmission line is subdivided into short line elements, as show in figure (2-). Figure (2-): The discretizing line to four short line elements [52]. Here the transmission line was cut into four elements; the first is connected to the second at node 2, the second to the third at node 3, etc.. The implication is that the original line can be put back together by connecting the four smaller lines. These smaller lines called finite elements. It is evident that the elements do not have to be of equal length nor does the line have to be straight. Two dimension regions can be subdivided into smaller sub-regions in a variety of ways. Consider the following region (rectangular region) as in figure (2-2). Figure (2-2): rectangular solution region 26

40 Chapter Two The Numerical Solution and the Simulation One way of subdividing the region is to form rectangular or even triangular elements as indicated in figure (2-3). (a) (b) (c) Figure (2-3): Subdividing of regular solution region, a) Rectangular, b) Squares, c) Triangles. It is evident from figure (2-.3) that there are three varied shape size of elements: Rectangular, Square and Triangle. The triangle is the best element in finite element method, because of its flexibility to be used with irregular domains. Consequently, it reduces the error in the solution. Figure (2-4) refers to this concept, for irregular domain, the solution region is divided into a number of finite elements.here, the region consists of six triangle elements and seven nodes. 27

41 Chapter Two The Numerical Solution and the Simulation Actual boundary Approximate boundary 5 6 elements numbering. nodes numbering. Figure (2-4) : A typical finite element subdivision of an irregular domain. Fundamentally, the basic idea here is that a region is subdivided in such a way as to increase the number of the elements when accurate results are desired and to reduce the number of elements when the results are not so important. Furthermore, the resulting element should be fat rather than thin. Figure (2.5) that are ideally to get elements that have equal sides. (a) (b) Figure (2-5) schematic element (a) Good element (b) Bad element. 28

42 Chapter Two The Numerical Solution and the Simulation Element Governing Equations This section shows the linear interpolation polynomials for two dimension elements in term of global coordinates. In this work, the triangle elements are used because this type is more suitable for the problem. The simplest triangular elements are shown in figure (2-6); each is with three nodal points, one at each corner of the triangle. V e3 V e3 3 (x 3, y 3) 3 (x 3, y 3) (x, y ) 2 (x 2, y 2) (x, y ) 2 (x 2, y 2) V e V e2 V e V e2 Figure (2-6): Simplest triangle elements, (a) Equilateral Triangular Element, and (b) Right Triangular Element [52]. The most common form of approximation for V within an element is a polynomial approximation, where for triangular element is [53]: V e ( x, y) = a + bx + cy...( 2. ) This equation can be written in a matrix form as V e a ( x, y) = [ x y] b... c ( 2. 2 ) 29

43 Chapter Two The Numerical Solution and the Simulation For the simplest triangular element shown in figure (2.6), the potential Ve, Ve2, and Ve3 at nodes, 2, and 3, respectively, are obtained by using equation (2.) such as: One can rewrite equations (2.3) in a matrix form as: Where the coefficients a, b, and c are determined from equation (2.4) as: Substituting this into equation (2.2) to get: ) ( = c b a y x y x y x V V V e e e ) 5 2. ( = e e e V V V y x y x y x c b a [ ] 6 ). 2 ( ), ( = e e e e V V V y x y x y x y x y x V (2.3c) (2.3b) (2.3a) cy bx a V cy bx a V cy bx a V e e e + + = + + = + + =

44 Chapter Two The Numerical Solution and the Simulation Let the determinant M be defined as; Introducing the inverse of M in equation (2.6) results: Or: Where the coefficients α, α2 and α3 are given by [53]: The determinant of M represents the area of a rectangular, so it can be written as: [ = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ), ( e e e e V V V x x x x x x y y y y y y x y xy xy x y x y x y A y x x y V ] = y x y x y x M ) (2.7 ), ( = = i ei i e V y x V α [ ] [ ] [ ] ) (2.8 ) ( ) ( ) ( 2 ) (2.8 ) ( ) ( ) ( 2 ) (2.8 ) ( ) ( ) ( c y x x x y y y x y x A b y x x x y y y x y x A a y x x x y y y x y x A + + = + + = + + = α α α M 2A = 3

45 CP Chapter Two The Numerical Solution and the Simulation Where A is the area of the element e is: A = 2 ( x ) ( y ) ( x ) ( y 2 x 3 y 3 x 2 y )... ( 2. 9 ) Assembling of all Elements This step is to assemble such all elements in the solution region. The total energy associated with an assemblage of many elements is the sum of all the individual element energies.it is given by [52]: W e = 2 ε... T ( e) [ V ] [ C ][ V ] ( 2.0 ) e e Where V e can be written as: V e V = V V e e2 e3...( 2.a ) And the superscript T denotes the transpose of the matrix.the matrix (e) Pwhich is usually called the element coefficient matrix or "stiffness matrix". Can be defined as: ( e) [ C ] = C C C ( e) ( e) 2 ( e) 3 C C C ( e) 2 ( e) 22 ( e) 32 C C C ( e) 3 ( e) 23 ( e) 33...( 2. b ) Consider the finite element mesh consisting of three finite elements as plotted in figure (2.7). The numbering of the nodes as,2,3,4, and 5 is 32

46 Chapter Two The Numerical Solution and the Simulation called global numbering. The numbering i-j-k is called local numbering and it corresponds to -2-3 of the element in figure (2.6). The local numbering is done using the global numbering. It must be in counterclockwise sequence starting from any node of the element. Thus, the numbering in figure (2.7) is not unique i Element numbering 3 i-j-k Counterclockwise nodes numbering Figure (2-7): Assembly of three elements. For the particular numbering in figure (2.7), the global coefficient matrix is expected to have the form of 5x5 matrix since five nodes (n=5) are involved. [ C ] = C C C C C C C C C C C C C C C C C C C C C C C C C ( 2. 2 ) Hence; For example, in figure (2.7), node is common in elements and 2, () (2)...( C = C + C 2.3 a) 33

47 Chapter Two The Numerical Solution and the Simulation Node 2 belongs to element only; hence... () ( 2. 3 b C =C ) Node 4 belong simultaneously to element, 2, and 3; hence () (2) (3) C ( = C22 + C + C33 c 33 ) Continuing in this manner, we obtain all the terms in the global coefficient matrix by inspection of figure (2.7) as: [ C ] = C C C C 0 () () 3 (2) 2 () 2 + C + C (2) (2) 3 C C () 3 C 0 () 23 () 33 0 C C (2) 2 0 C (2) 23 (2) 22 C + C (3) 2 + C (3) 3 (3) C C C () 22 () 2 C (2) 23 + C C + C () 23 + C (2) 33 (3) 23 (2) 3 (3) 3 + C (3) 33 C C 0 (3) 2 (3) 23 0 C (3) 22 ( 2. 4 ) Solving the Resulting Equations To satisfy Laplace's equation, the total energy in the solution region must be minimum [52]. Thus, it requires that the partial derivatives of W with respect to each nodal value of the potential to be zero, W V K = 0...(2.5) 34

48 Chapter Two The Numerical Solution and the Simulation Where k is, 2 n In general, dw/dvrkr =0 leads to n i= V i C ik = 0... (2.6) Where n is the number of nods in the mesh. Now using equation (2-6) to write the set of equations for all nodes where k=, 2, 3, 4, 5. W V = V C + V 2 C 2 + V C V 4 C 4 + V C 5 5 = 0... (2.7a) W V 2 = V C 2 + V 2 C 22 + V C V 4 C 42 + V C 5 52 = 0... (2.7b) W V 3 = V C 3 + V 2 C 23 + V C V 4 C 43 + V C 5 53 = 0... (2.7c) W V 4 = V C 4 + V 2 C 24 + V C V 4 C 44 + V C 5 54 = 0... (2.7d) W V 5 = V C 5 + V 2 C 25 + V C V 4 C 45 + V C 5 55 = 0... (2.7e) The solution of the above set of equations can be done, using a suitable computational program, in two ways [52], iteration and Band Matrix Methods. 35

49 Chapter Two The Numerical Solution and the Simulation 2.3 The Simulation The rest of this chapter presents information about the transformation of the stochastic model, which is described in chapter one, to a computer program and the requirement to apply it within a non-uniform field configurations. The results of the program are used to plot the picture of the voltage and electric field distribution within the configurations and to follow the streamer growth in rod-plane electrodes configuration Grid Generation As shown previously, the finite element method requires a grid to define the region for the solution. A linear triangular elements are used to discrete the region between the electrodes. The region of interest for rodplane configuration is that which surrounds the rod head. This is because a high electric field is expected in this region. The elements in the grid are not made of the same size. This is important in saving the time in running the program. Using a large number of elements yields more accurate results but takes more time for the finite elements calculation. In this dissertation, the design of many grids for many distances, between the electrodes, is set to test the simulation and calculate the potential distribution as well as the electric field (magnitude and direction). The only boundary condition for finite element formulation is the known applied voltage values on both electrodes The Program Input Data To execute the simulation, some information are needed as input data to the computer program. These data are about: 36

50 Chapter Two The Numerical Solution and the Simulation. The grid (mesh) : which includes nodes numbering and their coordinates in the solution region, elements and their nodes, and the boundary conditions 2. The threshold field which is E th =26kV/cm, and 3kV/cm. 3. The number that controls the variation of the growth rate with the electric field which is n p =. 4. The constant A g of value 3.7x0 5 sec The voltage drop along the bonds which are E s =5, 4.5, 4 kv/cm [36]. 6. Applied voltage is V The Program Calculations All calculations required in testing the present model are done by a computer program. The program is written with Fortran 77 language. It is used to do the calculations that are needed to predict voltage and electric field distributions for different distances between the electrodes. As well as, simulate the path of the streamer within the simulation area. The procedure of the calculations is done by the following steps (subroutines); -Mesh Generation: AUTO MESH 2D package was used to discretize the designed configuration. It generates the mesh as data and put it in named files. This data consists of nodes numbering, mesh numbering, and the coordinates of the nodes. 2-Inserting of the boundary conditions: the boundary conditions mean the fixed values of the voltage on the nodes, which are placed on the electrodes. A program is written (subroutine) to read the data that are given by the above step and to formulate it to be suitable for the solution program, and insert the boundary conditions. It is named as formulation. 37

51 Chapter Two The Numerical Solution and the Simulation 3-Solving Laplace's equation: SIMPLE2 program [53] was used to translate the finite element method to solve Laplace's equation numerically in two dimensions. It is given comprised of a main program and various subroutines. 4-Calculation of the local Electric Field: In this step (subroutine), the program reads the voltage and coordinates of the nodes, calculates the local electric field for each element in the mesh and determines all the possible directions for the propagation of the discharge pattern. The directions were chosen on the basis of local electric field, i.e. the electric field between a point that belongs to the discharge pattern and the adjacent points that belong to the dielectric. If the local electric field E loc was greater than a threshold value E th, using equation (-3). The direction was chosen as a possible one for the propagation of the pattern at iteration of the computer program. 5-Finding the growth time for each candidate: after the calculation of the local electric field to each possible direction, a characteristic time was calculated for the growth of a new bond, which is equivalent to the necessary time for the propagation of the discharge pattern from one element to another this time was named physical time. It was calculated using equation (-4). 6-Finding the minimum time for growth streamer between two electrodes: After the calculation of the growth time for each candidate bond (element), the computer program identifies the winning bond defined by τ = minimum. The winning bond is added to the discharge structure. 7-Calculation of Voltage Drop: The voltage drop is one of the important assumptions of the model. It was calculated according to bond length (streamer segment) and the voltage drop which was varied as (4, 4.5, and 5 kv/cm). 38

52 Chapter Two The Numerical Solution and the Simulation 8-Updating the Boundary Conditions: The applied voltage to the electrodes is required for the finite element method as a boundary condition. These boundary conditions are updated after each iteration and inserted this updating to the next iteration. 9-Plotting the Results: The output results, can be stored, and are put in files. AUTO2D package was used to plot the meshes, SURFER package to plot the distributions of the voltage and electric field within the configurations, and GRAPHER under window to follow the path of the streamer. All the above points (from 3 to 9) were repeated for each iteration (streamer step) as illustrate in figure (2-8). 39

53 Chapter Two The Numerical Solution and Simulation Mesh Generation Formation the Data of the Mesh and Inserting the Boundary Solving of Laplace's Equation Calculation of the Local Electric Field Determination of the elements with E loc > E th Find the Growth Time, the Minimum Growth Time, and Drop Voltage Updating the Boundary Conditions No Stop conditions yes Write the Output Results to be ready Plotting Stop Figure (2-8): The block diagram of the program 40

54 Chapter Three The Results

55 Chapter Three The Results Chapter Three: The results 3. Introduction The results presented here are used to describe the pre-breakdown phenomenon that occurs in air.the simulation of a growing streamer is constructed with the aid of the finite element method.generaly,in this work, the streamer propagation along gaps of air between the electrodes and the effects of the voltage and electric field distributions within the area between the two electrodes on the streamer propagation were studied. The breakdown voltages for different gap lengths between the electrodes, also the streamer velocities, were indicated for different breakdown voltages values in different air gaps. The procedure used for the estimation of the minimum breakdown voltage versus the gap length was the following: For each value of the gap length, several different values of the applied voltage were tested. For simplicity it was assumed that the breakdown of the gap occurs when the first streamer branch touches the cathode. The discharge was assumed to be incomplete if, during the time step, the electric field in the vicinity of the streamer structure was everywhere less than E th. Applied voltage was varied and the lowest value of voltage for which the streamer pattern bridges the gap was assumed to be the minimum breakdown voltage. From the streamer length and the time for the streamer to reach the cathode, its velocity was calculated. The breakdown voltage values, for each gap, were compared with the practical values which showed good agreement. 4

56 Chapter Three The Results 3.2 The General Procedure The streamer propagation in air gap, was modeled using the stochastic model which was implemented within a suitable configuration.the selected configuration is a rod-plane, figure (3-), because is one of the configurations that is widly used in experimental studies. The rod (anode) is of 0cm length and 0.2 cm diameter.the plane (cathode) is of (0+d)cm diameter, where d is the the air gap length between the two electrodes. A positive DC high voltage was applied on the rod while the plane was grounded. 0.2cm 0cm d+0 d Cathode Figure (3-): Longitual cross section for rod - plane configuration Laplace's equation governs the voltage and electric field distributions within the configuration. So, finite element method (in twodimensions) was used as a good tool to solve Laplace's equation in the complicated configuration,that requires the solution region to be discretized by a suitable mesh as in figure (3-2). The meshes must be designed to have high density elements around the tip of the rod and low density far away because of the expected high variation of the voltage and the electric field around this region. 42

57 Chapter Three The Results a b Figure (3-2): The descritization (mesh or grid) of the solution region, a) acomplete mesh for the longitudinal cross section of the configuration, b) enlargement of the region around the rod electrode. 43

58 Chapter Three The Results The solution of Laplace s equation gives the voltage at every node on the mesh, in other word the potential (voltage) distribution on the area of the longitudinal cross section of the electrodes configuration. The SURFER program was used to present this distribution by countor ploting as in figure (3-3). a b Figure (3-3): The voltage distribution within the solution region before the streamer intiation, a) the complete distribution, b) enlargmen of the region around the rod electrode. 44

59 Chapter Three The Results Each element, in the mesh (grid), has the values of voltage at its nodes. These values were used to calculate the electric field values at the center of each element. This will give the electric field distribution within the solution region. The same method is applied to the electric field distribution to be presented as contour plotting in figure (3-4). a b Figure (3-4): The electic field distribution within the solution region before the streamer intiation, a) the complete distribution, b) enlargment of the region around the rod electrode. 45

60 Chapter Three The Results The x, y componenets of the electric field were used to determine its angle direction at the center of each element of the mesh. The direction distribution was presented as vector plotting as shown in figure (3-5). a b Figure (3-5): The electic field direction distribution within the solution region before the streamer intiation, a) the complete distribution, b) enlargment of the region around the rod electrode 46

61 Chapter Three The Results After this point, the streamer intiation and growth will be tracked within the solution region between the two elctrodes. The course of the streamer will be continuous until reaching the plane electrode to determine the breakdown voltage of each air gap. 3.3 The streamer intiation and growth According to the stochastic model,the streamer is intiatated at the elements that have electic field values greater than E th. In the rod-plane configuration, the highest values are expected around the tip of the rod and the growth with time is towards the plane. The model was tested for different lengths 3, 5, 7, and 0cm of air gaps in this configuration First Case: Testing the Model With 3cm Air gap. In this section, the model was implemented for an air gap of 3cm length to show the initiation and growth of the streamer from the anode (rod) to the cathode (plane). The aim is to determine the breakdown voltage of this air gap. That was done by running the simulation program for different voltages over a mesh of 4560 elements and 2393 nodes. The minimum value of the voltage when the streamer reaches the plane electrode is the breakdown voltage. The value for air gap of 3cm is 28.4 kv. Figure (3-6) shows the streamer initiation and growth between the two electrodes for the minimum breakdown voltage of this air gap. The figure shows, the initiation of the streamer at the tip of the rod because of the highest values of the electric field. Also the streamer grows randomly but it stays under control by the electric field near the shortest distance between the two electrods. It is found that the required time for reaching the plane is 7.02µs. 47

62 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-6): The streamer growth within the air gap of 3cm for different growth times of, a) 0.26µs, b).57µs, c) 3.86µs, d) 5.0µs, e) 6.94µs, and f) 7.02µs. 48

63 Chapter Three The Results The understanding of electric field and potential distributions are very important for the design and development of high voltage equipments and electrical insulations can be reduced if the distribution of the electric field is known. It is important to identify where the streamer is most likely to occur. This is normally used to prevent breakdown or possibly to encourage it, depending on the types of applications. Figures (3.7) and (3.8), show a contour plotting, figure (3.9) shows a vector plotting for the effect of the streamer growth on the distributions of the voltage, electric field magnitudes, and electric field directions at minimum breakdown voltage in the configuration of 3cm air gap. These figures indicate clearly the initiation and growth of the streamer according to the regions of the high voltages and the high electric fields. The plots for the magnitude of the electric field can identify the weak region where the streamer may begin. In this case, the weak region is identified to be the region where the magnitude of the electric field is the highest and from this region the streamer will initiate. According to figures (3.7)and (3.8),it can be observed that the streamer moves according to the highest voltage regions from the rod down to the plane electrode while the highest electric field region is moved from the rod down towards the plane. As well as, figure (3.9) the direction of the electric field moves from highest value of electric field to the lowest.the behavior of this results appears to be in good agreement with that of ref. Charalambakos [36]and Khalaf[45]. 49

64 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-7): The effect of the streamer growth on the potential distribution in times of, a) 0.26µs, b).57µs, c) 3.86µs, d) 5.0µs, e) 6.94µs, and f) 7.02µs in 3cm air gap. 50

65 Chapter Three The Results (a) (b) (c ) (d) (e) (f) Figure (3-8): The effect of the streamer growth on the electric field (as magnitude) distribution in times of, a) 0.26µs, b).57µs, c) 3.86µs, d) 5.0µs, e) 6.94µs, and f) 7.02µs in 3cm air gap. 5

66 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-9): The effect of the streamer growth on the electric field (as direction) distribution in times of, a) 0.26µs, b).57µs, c) 3.86µs, d) 5.0µs, e) 6.94µs, and f) 7.02µs in 3cm air gap. 52

67 Chapter Three The Results Until this point the behavior of the streamer was shown, when the applied voltage is the minimum value of that required for the breakdown of 3cm air gap, which it is 28.4kV. Now, that behavior must be shown for different voltage values greater than of the minimum voltage. Figure (3-0) shows the effect of the breakdown voltage (40, 50, 60, and 70 kv) on the shape and path of the streamer growth.it shows that, at all breakdown voltage values, the streamer intiations at the tip of the rod electrode. the streamer tracks a randomly zigzagged path from the rod to the plane. The zigzag increases with the increase of the voltage except at the last value, 70kv. It can also be observed,from the figure, that the streamer reaches the plane electrode faster as the voltage increases. For the voltage values (40, 50, 60, and 70 kv) the streamer reaches the plane electrode at (6.4,.00, 0.63, and 0.45µs), respectively. The randomly zigzagged path of the streamer can be explained as: the increase of voltage increases the area in the solution region which has the condtion (E loc. > E th. ) for streamer growth probability. This probability was controlled by the features of the stochastic model in the calculation of growth time of the new streamer segment. The introduction of the stochastic features can be justified as follows: If the photoionization is considered to be the main mechanism of the generation of seed electrons for the creation of secondary avalanches. It may be assumed that they are emitted and absorbed in a random manner; hence, it is possible that a new predominant direction for the streamer propagation to appear even in areas of a relative low electric field. The random appearance of seed electrons is a likely mechanism of the generation of the experimentally observed zigzagged streamers and spark channels [36]. 53

68 Chapter Three The Results (a) (b) (c ) (d) Figure (3-0): The streamer path within the 3cm air gap for different breakdown voltages of, a) 40 kv, b) 50 kv, c) 60 kv, d) 70 kv. 54

69 Chapter Three The Results Second Case: Testing the Model With 5cm Air gap. The simulation was done to study the streamer propagation, when the distance, between rod and plane electrodes, is 5cm air gap. Different voltages were applied, the lowest value of voltage at which the streamer pattern bridges the gap was assumed to be the minimum breakdown voltage. The configuration used in this case, 0cm length of rod to ensure that the electric field which is far away from the rod is very small, and 0.2cm diameter. The plane electrode is of 5 cm in length. The designed grid for this configuration is of 5980 elements and 309 nodes. The region of interest for the rod-plane configuration is that which surrounds the rod; this is because a high electric field is expected in this region. Therefore; this region has high density of elements but the density is low far away. Figure (3.) shows the development of the streamer growth, at different time periods, within the region between the electrodes. It appears clearly, the randomly zigzagged path from the rod to the plane electrode. The streamer reached the plane at minimum voltage value of 38 kv and took a time of 7.78µs. the two values are greater than that of the first case, as it is expected because the gap in this case is greater than the first. In general, one can observe the same behavior compared with first case except, the 5cm air gap required greater voltage to breakdown than the 3cm air gap. Also the streamer required longer time to cross it than the first. 55

70 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-): The streamer growth within the air gap of 5cm for different growth times of, a) 0.37 µs, b).76µs, c) 3.2µs, d) 5.8µs, e) 7.32µs, and f) 7.78µs. 56

71 Chapter Three The Results The voltage and the electric field (as magnitude) distributions, in the 5cm air gap, is presented as contour plotting in figures (3.2) and (3.3). The electric field distribution (as direction) is presented as vector plotting in figure (3.4). Those results were found at the minimum breakdown voltage in the configuration of 5cm air gap. As mentioned before, the figures indicate clearly the initiation and growth of the streamer according to the regions of the high voltages and the high electric fields. The plots for the magnitude of the electric field can identify the weak region where the streamer may begin. The weak region was identified where the magnitude of the electric field is the highest and from it the streamer will initiate. According to figures (3.2) and (3.3), it can be observed that the streamer moves according to the highest voltage regions from the rod down to the plane electrode and the high electric field region is moved from the rod down towards the plane. As seen from figure (3.4) the direction of the electric field moves from highest value of electric field to the lowest. All these results showed the same general behavior for the previous case of 3cm air gap. 57

72 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-2): The effect of the streamer growth on the potential distribution in times of, a) 0.37µs, b).67µs, c) 3.2µs, d) 5.8µs, e) 7.23µs, and f) 7.78µs in 5cm air gap. 58

73 Chapter Three The Results (a) (b) (c) (d) (e) Figure (3-3): The effect of the streamer growth on the electric field magnitude distribution in times of, a) 0.37µs, b).67µs, c) 3.2µs, d) 5.8µs, e) 7.23µs, and f) 7.78µs in 5cm air gap. (f) 59

74 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-4): The effect of the streamer growth on the electric field direction distribution in times of, a) 0.37µs, b).67µs, c) 3.2µs, d) 5.8µs, e) 7.23µs, and f) 7.78µs in 5cm air gap. 60

75 Chapter Three The Results Figure (3-5) shows the initiation and the complete growth of the streamer between the two electrodes for different breakdown voltage values. The voltage values are 40, 60, 70, and 90kV. The figure shows a randomly zigzagged path from the rod to the plane and the zigzag increases with the increase of the voltage. This is since the increase of the voltage increases the probability of streamer initiation and growth area in the configuration. Also, it can be observed the decrease of the time required for the streamer to bridge the distance between the electrodes. The reaching time according to the voltage values are 6.6,.79, 0.88, and 0.29µs, respectively. This is related to the streamer velocity which will be discussed later. 6

76 Chapter Three The Results (a) (b) (c) (d) Figure (3-5): The streamer path within the 5cm air gap for different breakdown voltages, a) 40 kv, b) 60 kv, c) 70 kv, d) 90 kv. 62

77 Chapter Three The Results Third Case: Testing the Model With 7cm Air gap. In this part, the simulation is done to study the streamer propagation, when the rod-plane distance is 7cm. Different voltages were applied and the lowest value of voltage at which the streamer pattern bridges the gap was assumed to be the minimum breakdown voltage. The dimensions of the configuration are the same as the previous cases except that the plane diameter was 7 cm. The designed grid (mesh) for this case is over an area of 7 7 cm 2. The region of interest for rod-plane configuration is that which surrounds the rod; this is because a highest electric field is expected in this region. Therefore; this region has high density of elements. The grid was designed to be of 832 elements and 499 nodes. For this reason, the simulation execution, in this case, is more complex than the previous and needs longer time for each execution. In other words, the solution must be over a square matrix of In figure (3-6), the development of the streamer growth within the region between the electrodes is presented. Clearly, the same behavior appears here as in the previous case. The minimum breakdown voltage, in this case, is 50kV.The required time for the streamer to reach to the plane electrode is 7.64µs. The same explanation that was viewed in the previous cases can be considered here. 63

78 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-6): The streamer growth within the air gap of 7cm for different growth times of, a) 0.82 µs, b) 2.93µs, c) 5.64µs, d) 6.4µs, e) 7.09µs, and f) 7.64µs. 64

79 Chapter Three The Results Figures (3-7),(3-8) and (3-9) show the effect of streamer growth on the distributions of the voltage, electric field magnitude and electric field direction for minimum breakdown voltage for different times. Figure (3-7) shows the voltage distribution around the rod in the minimum breakdown voltage according to the streamer growth. As expected the region with the highest values is the nearest to the rod and that with the lowest value is the farthest. Figure (3-8) presents the distribution of the electric field magnitude within the configuration. It shows, as expected, that the region with the highest value of electric field is at the rod which allows initiating of the streamer growth, and the value of electric field decreases far away from rod. Figure (3-9) shows the direction of the distribution of the electric field. It shows that the electric field direction from high electric field region on the rod to the low electric field region (far away from the rod). In the three figures, the path of the streamer growth clearly affects the distributions and is similar to that in figure (3-6). 65

80 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-7): The effect of the streamer growth on the potential distribution in times of, a) 0.82 µs, b) 2.93µs, c) 5.64µs, d) 6.4µs, e) 7.09µs, and f) 7.64µs. 66

81 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-8): The effect of the streamer growth on the electric field (as magnitude) distribution in times of a) 0.82 µs, b) 2.93µs, c) 5.64µs, d) 6.4µs, e) 7.09µs, and f) 7.64µs. 67

82 Chapter Three The Results (a) (b) (c) (d) (e) (f) Figure (3-9): The effect of the streamer growth on the electric field (as direction) distribution in times of, a) 0.82 µs, b) 2.93µs, c) 5.64µs, d) 6.4µs, e) 7.09µs, and f) 7.64µs. 68

83 Chapter Three The Results Again, as in the previous cases, the effect of breakdown voltages on the streamer path within the rod-plane configuration of 7cm air gap will be shown below. The streamer will initiate and grow at points where the conditions are suitable, especially that of the electric field values. Figure (3-20) shows the initiation and complete growth of the streamer between the two electrodes for different breakdown voltage values. The voltage values are 60, 70, 80, and 00kV which are greater than the minimum for this gap. The figure shows a randomly zigzag path from the rod to the plane and the zigzag increases with the increase of the voltage value. That can be explained as in the previous case; the increase of the voltage increases the probability of streamer initiation and growth area in the configuration. Also, one can observe the decrease of the time (5.46, 2.80,.03, and 0.39µs, respectively according to voltages) required for the streamer to bridge the distance between the electrodes. This was related to the streamer velocity which will be discussed later. 69

84 Chapter Three The Results (a) (b) (c) (d) Figure (3-20): The streamer path between the two electrodes for different breakdown voltages of, a) 60 kv, b) 70 kv, c) 80 kv, d) 00 kv. 70

85 Chapter Three The Result Fourth Case: Testing the Model With 0cm Air gap. In this section, the simulation has been within the same configuration of the previous cases, but of greater area of (20 20) cm 2 where the air gap length is 0cm and the plane length 20 cm. This was implemented for different voltage values. For this case, the minimum breakdown voltage was found as 66.8 kv. The mesh (grid), which is required for the numerical solution, was designed and created over a large area of 400cm 2. This area requires a large number of elements (86) and nodes (489). Figure (3-2) shows the streamer initiation and the growth steps between the two electrodes according to time at the breakdown voltage. The figure shows, the initiation of the streamer at the tip of the rod. That was expected, because of the highest values of the electric field at this region. The streamer growth is random but under control of the electric field near the shortest distance between the two electrodes. The required time for reaching the plane was 9.22µs. 7

86 Chapter Three The Result (a) (b) (c) (d) (e) (f) Figure (3-2): The streamer growth within the air gap of 0 cm for different growth times of, a) 0.65 µs, b) 3.07µs, c) 6.5 µs, d) 8.67µs, e) 9.04µs, and f) 9.22µs. 72

87 Chapter Three The Result It is well known that, the electric field and potential distributions are the main important parameters which control the initiation and growth of the streamer within any electrical configuration. Again, the behavior of the streamer (initiation and growth) which is shown in figure (3-2) must be supported by the behavior of the voltage and electric field distributions. The voltage and the electric field (as magnitude) distributions, in the 0 cm air gap, is presented as contour plotting in figures (3.22) and (3.23).While the electric field distribution (as direction) is presented as vector plotting in figure (3.24). These results were found at the minimum breakdown voltage in the configuration of 0 cm air gap. As mentioned before, these figures indicate clearly the initiation and growth of the streamer according to the regions of the high voltages and the high electric fields. The plots for the magnitude of the electric field can identify the weak region where the streamer may begin. The weak region was identified at the region where the magnitude of the electric field is the highest where from the streamer will initiate. According to figures (3.22) and (3.23), one can observe that the streamer moves according to the high voltage regions from the rod down to the plane electrode and the high electric field region is moved from the rod down towards the plane. Figure (3.24) shows the direction of the electric field directed from highest values of electric field to the lowest. Also, these results showed the same general behavior for the previous cases. 73

88 Chapter Three The Result (a) (b) (c) (d) (e) (f) Figure (3-22): The effect of the streamer growth on the potential distribution in times of, a) 0.65µs, b) 3.07µs, c) 6.5µs, d) 8.67µs, e) 9.04µs, and f) 9.22µs in 0 cm air gap. 74

89 Chapter Three The Result (a) (b) (c) (d) (e) (f) Figure (3-23): The effect of the streamer growth on the electric field magnitude distribution in times of, a) 0.65µs, b) 3.07µs, c) 6.5µs, d) 8.67µs, e) 9.04µs, and f) 9.22µs in 0 cm air gap. 75

90 Chapter Three The Result (a) (b) (c) (d) (e) (f) Figure (3-24): The effect of the streamer growth on the electric field direction distribution in times of, a) 0.65µs, b) 3.07µs, c) 6.5µs, d) 8.67µs, e) 9.04µs, and f) 9.22µs in 0 cm air gap. 76

91 Chapter Three The Result As in the previous cases, the effect of breakdown voltage on the streamer path within the rod-plane configuration of 0 cm air gap will be shown below. The streamer will initiate and grow at points where the conditions such as the threshold electric field value are suitable. Figure (3-25) shows the initiation and complete growth of the streamer between the two electrodes for different breakdown voltages. The voltage values are 90, 00, 20, and 40 kv which are greater than the minimum for this gap. The figure shows randomly zigzag path from the rod to the plane and the zigzagged increases with the increase of the voltage. That can be explained as in the previous case: the increase of the voltage increases the probability of streamer initiation and growth area in the configuration. Also, one can observe the decrease of the time that is required for the streamer to bridge the distance between the electrodes. The reaching time according to the voltage values are 3.79,.8,., and 0.47 µs, respectively. This was related to the streamer velocity which will be discussed later. 77

92 Chapter Three The Result (a) (b) (c) (d) Figure (3-25): The streamer path within the 0 cm air gap for different breakdown voltages of, a) 90 kv, b) 00 kv, c) 20kV, d) 40 kv. 78

93 Chapter Three The Results 3.4 The Streamer velocity The advancement of the model, which is considered in this work, is its ability to give actual measurable magnitudes like breakdown voltage and the velocity of streamer growth (propagation) within the dielectric materials. In this section, the streamer velocity will be studied within the air gaps for different breakdown voltage values. Figure (3-26) shows the streamer velocity within 3, 5, 7, and 0 cm air gaps length at different breakdown voltages. For all gaps length, the streamer velocity increases, approximately, exponentially with the increase of breakdown voltages. That can be explained as: increase the voltage, more charged particle especially the electrons will gain more energy and make more and more inelastic collisions which will produce other charged particles. This makes the streamer gains more speed. Also, one can observe that for the same breakdown voltage, the streamer velocity decreases with the increase of the air gap length and this is expected because of the decrease of the electric field which affects all the processes within the gap. 79

94 Chapter Three The Results 7.00E+07 the average of velocity to streamer (cm/sec) 6.00E E E E E+07.00E+07 Streamer Velosity (m/sec) 2E+6.5E+6 E+6 Streamer Velosity Gap length =3 cm Gap length =5 cm Gap length =7 cm Gap length =0 cm 0.00E E+5 the voltage (volt) Applied Voltage (kv) Figure (3-26): The streamer velocity as a function of breakdown voltage within different air gap lengths of 3, 5, 7, 0 cm. 80

95 Chapter Four Comparisons, Conclusions and Suggested Future works

96 Chapter Four Comparisons, Conclusions and Suggested Future works Chapter Four: Comparisons, Conclusions, and Suggested Future Works 4. Comparison of results The results, especially the breakdown voltage values, in this work were compared with experimental and computational works. That was done for different air gaps length, thresholds of local electric fields, and voltage drops. In general, our results appear in good agreement in these comparisons. 4.. Experimental Comparison The results of our simulations were compared with the experimental data which, for numerical representation were proposed in equations (-) and (-2). In Tables (4.), (4.2), (4.3) and (4.4), the results were compared with those of Feser [20]. A very good quantitative agreement exists in the case of E th =26kV/cm and E S =4.5kV/cm. It should be noted that the results are strongly affected by the parameters E th and E s but they are not influenced by the parameters A and n of equation (-5). The increase of the values of the parameters E th and E s results in increase of the breakdown voltage of the gap. Table (4.): Result simulation and experimental values, E th =26kV /cm Experiments Simulation of this work Gap size (cm) =2+5.d (kv) E th =26kV/cm =5kV/cm (kv) E th =26kV/cm = 4.5kV/cm (kv) E th =26kV/cm =4kV/cm (kv)

97 Chapter Four Comparisons, Conclusions and Suggested Future works Table (4.2): Result simulation and experimental values, E th =3kV /cm Experiments Simulation of this work Gap size (cm) =2+5.d (kv) E th =3kV/cm = 5kV/cm (kv) E th =3kV/cm = 4.5kV/cm (kv) E th =3kV/cm = 4kV/cm (kv) Table (4.3): Result simulation and experimental values, E th =26kV/cm Experiments Simulation of this work Gap size (cm) = d (KV) E th =26kV/cm =5kV/cm (kv) E th =26kV/c = 4.5kV/cm (kv) E th =26kV/cm =4kV/cm (kv) Table (4.4): Result simulation and experimental values, E th =3kV/cm Expedients Simulation of this work Gap size (cm) = d (KV) E th =3kV/cm =5kV/cm (kv) E th =26kV/cm = 4.5kV/cm (kv) E th =26kV/cm E s =4kV/cm (kv)

98 Chapter Four Comparisons, Conclusions and Suggested Future works 4..2 Computational Comparison Again, the results of our simulations were compared with the computational data of Charalambakos et al. [36].who worked with a different numerical method. This is shown in tables (4.5) and (4.6). A very good quantitative agreement exists in the case of E th =26kV/cm and E th =3kV/cm. The increase of the value of the parameter E th results in the increase of the breakdown voltage of the gap. Table (4.5): Result simulation and experimental values, E th =26kV/cm The simulation() Our simulation (2) Gap size (cm) E s =5kV/cm E s =4.5kV/cm E s =4kV/cm E s =5kV/cm E s =4.5kV/cm E s =4kV/cm Table (4.6): Result simulation and experimental values, E th =3kV/cm The simulation () Our simulation (2) Gap size (cm) E s =5kV/cm E s =4.5kV/cm E s =4kV/cm E s =5kV/cm E s =4.5kV/cm E s =4kV/cm

99 Chapter Four Comparisons, Conclusions and Suggested Future works 4.2 Conclusions From the results that were presented in chapter three, the following can be concluded: The stochastic model gives a good description to streamer discharge within long air gaps. The computational procedure can give good results when compared with the experimental procedures The voltage and electric field distributions were affected by the streamer growth between the electrodes. The main parameter that affects the breakdown voltage values is the threshold electric field value (E th ) and the voltage drop. The streamer velocity was affected by the breakdown voltage value and the air gap length. 4.3 Future works From this point, one can suggest some future work such as: Studying the effect of the electrode shape and air gap lengths on the breakdown voltage and streamer velocity. Using the Poisson s and charges transfers equations for the calculation of potential distribution in the area between the electrodes. Study more parameters that affect the pre-breakdown events, such as temperature, pressure etc. Performs this experiment practically. 84

100 Chapter Four Comparisons, Conclusions and Suggested Future works Chapter Four: Comparisons, Conclusions, and Suggested Future Works 4. Comparison of results The results, especially the breakdown voltage values, in this work were compared with experimental and computational works. That was done for different air gaps length, thresholds of local electric fields, and voltage drops. In general, our results appear in good agreement in these comparisons. 4.. Experimental Comparison The results of our simulations were compared with the experimental data which, for numerical representation were proposed in equations (-) and (-2). In Tables (4.), (4.2), (4.3) and (4.4), the results were compared with those of Feser [20]. A very good quantitative agreement exists in the case of E th =26kV/cm and E S =4.5kV/cm. It should be noted that the results are strongly affected by the parameters E th and E s but they are not influenced by the parameters A and n of equation (-5). The increase of the values of the parameters E th and E s results in increase of the breakdown voltage of the gap. Table (4.): Result simulation and experimental values, E th =26kV /cm Experiments Simulation of this work Gap size (cm) =2+5.d (kv) E th =26kV/cm =5kV/cm (kv) E th =26kV/cm = 4.5kV/cm (kv) E th =26kV/cm =4kV/cm (kv)

101 Chapter Four Comparisons, Conclusions and Suggested Future works Table (4.2): Result simulation and experimental values, E th =3kV /cm Experiments Simulation of this work Gap size (cm) =2+5.d (kv) E th =3kV/cm = 5kV/cm (kv) E th =3kV/cm = 4.5kV/cm (kv) E th =3kV/cm = 4kV/cm (kv) Table (4.3): Result simulation and experimental values, E th =26kV/cm Experiments Simulation of this work Gap size (cm) = d (KV) E th =26kV/cm =5kV/cm (kv) E th =26kV/c = 4.5kV/cm (kv) E th =26kV/cm =4kV/cm (kv) Table (4.4): Result simulation and experimental values, E th =3kV/cm Expedients Simulation of this work Gap size (cm) = d (KV) E th =3kV/cm =5kV/cm (kv) E th =26kV/cm = 4.5kV/cm (kv) E th =26kV/cm E s =4kV/cm (kv)

102 Chapter Four Comparisons, Conclusions and Suggested Future works 4..2 Computational Comparison Again, the results of our simulations were compared with the computational data of Charalambakos et al. [36].who worked with a different numerical method. This is shown in tables (4.5) and (4.6). A very good quantitative agreement exists in the case of E th =26kV/cm and E th =3kV/cm. The increase of the value of the parameter E th results in the increase of the breakdown voltage of the gap. Table (4.5): Result simulation and experimental values, E th =26kV/cm The simulation() Our simulation (2) Gap size (cm) E s =5kV/cm E s =4.5kV/cm E s =4kV/cm E s =5kV/cm E s =4.5kV/cm E s =4kV/cm Table (4.6): Result simulation and experimental values, E th =3kV/cm The simulation () Our simulation (2) Gap size (cm) E s =5kV/cm E s =4.5kV/cm E s =4kV/cm E s =5kV/cm E s =4.5kV/cm E s =4kV/cm

103 Chapter Four Comparisons, Conclusions and Suggested Future works 4.2 Conclusions From the results that were presented in chapter three, the following can be concluded: The stochastic model gives a good description to streamer discharge within long air gaps. The computational procedure can give good results when compared with the experimental procedures The voltage and electric field distributions were affected by the streamer growth between the electrodes. The main parameter that affects the breakdown voltage values is the threshold electric field value (E th ) and the voltage drop. The streamer velocity was affected by the breakdown voltage value and the air gap length. 4.3 Future works From this point, one can suggest some future work such as: Studying the effect of the electrode shape and air gap lengths on the breakdown voltage and streamer velocity. Using the Poisson s and charges transfers equations for the calculation of potential distribution in the area between the electrodes. Study more parameters that affect the pre-breakdown events, such as temperature, pressure etc. Performs this experiment practically. 84

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110 جمهورية العراق وزارة التعليمالعالي والبحثالعلمي جامعة بغداد/ كلية العلوم دراسة حاسوبية لمراحل تطور ظاھرة ماقبل األنھيار الكھربائي" للفجوة الھوائية لقضيب - مستوي " رسالة مقدمه إلى كلية العلوم/جامعة بغداد وهي جزء من متطلبات نيل درجة الماجستير فيالفيزياء من قبل رسلحامد احمد بكلوريوس علوم فيزياء 200 أ.م.د ثامر حميد خلف إشراف م.د قصي عدنان عباس 435 ه 204 م