Two-dimensional axisymmetric simulationsand the heating effects associated with DCatmospheric pressure discharges duringthe post-streamer stage

A.P. Papadakis, G.E. Georghiou and A.C. Metaxas

Abstract: Numerical results are presented for the development of the post-streamer discharge stagein atmospheric pressure air. The model used comprises Poisson, charged-particle continuity andNavier-Stokes equations developed in two-dimensional cylindrical axisymmetric co-ordinates.Applied direct current voltage of 20% above the breakdown threshold is applied in a 1 mm gapbetween two parallel plate electrodes. Starting from a single electron as the initial condition, thetransitory regime from the streamer to the glow discharge is analysed, and the glow discharge isshown to consist of the cathode fall, negative glow, positive column and anode regions. The posi-tive column is shown to propagate in the form of a return wave towards the anode. The very fastredistribution of the electric field just after the streamer hits the cathode is presented. Furthermore,the current density at the cathode fall and anode regions is shown to increase and extend radiallyoutwards, justifying the inclusion of a two-dimensional axisymmetric model to study the radialeffects in the discharge. Neutral gas heating starts to occur with the initiation of the post-streamerdischarge stage, and the neutral gas temperature increases at the cathode by approximately 180 K.

1 Introduction

Gas discharges are used extensively in industrial applicationssuch as power generation [1–3], bio-decontamination [4],surface modification of polymers [5], low-inductance, high-voltage switching applications [6], thin-film deposition,sterilisation of biological samples, ozone formation, reduc-tion of thin oxide layers [7] and as plasma sources usingmicrodischarge devices [8]. The detail characterisation ofthese gas discharges can be very useful when they are utilisedfor the above applications.

One possible way to characterise such discharges in finedetail is to use numerical modelling. Numerical modellingis achieved by solution of the appropriate partial differentialequations that capture their physics. As the above gasdischarges are separated into thermal and non-thermal,depending on the operating regime, different equationsneed to be solved. For non-thermal discharges, only thecharged-particle continuity equations, to account forcharged particles, and the Poisson equation, to account forthe electric field, need to be solved. The authors havedeveloped in the past a non-thermal solver that uses thefinite element-flux corrected transport (FE-FCT) methoddescribed rigorously in [9], to solve for the charged-particle

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-smt:20060061

Paper first received 8th May and in revised form 25th July 2006

A.P. Papadakis is with Electricity Utilisation Group (EUG), Department ofEngineering, University of Cambridge, Cambridge CB2 1PZ, UK

G.E. Georghiou is with the Department of Electrical & Computer Engineering,University of Cyprus, 75 Kallipoleos, PO Box 20577, 1678, Nicosia, Cyprus

A.C. Metaxas is with the St John’s College, University of Cambridge,Cambridge CB2 1TP, UK

E-mail: ap263@cam.ac.uk

IET Sci. Meas. Technol., 2007, 1, (2), pp. 113–120

continuity equations, and the Galerkin finite elementmethod for the Poisson equation. The above non-thermalsolver has been used to analyse avalanche and streamerdischarges in high-pressure ambient air [10, 11].

According to the theory of gas discharges, the sequencein which the discharges occur is as follows: avalanche,streamer (non-thermal), and glow, spark and arc (thermal)discharges. Therefore the next step is to analyse the transi-tory regime from the streamer to the glow discharge and itsassociated heating effects that occur in the post-streamerdischarge stage. To achieve this, we inevitably need toincorporate the heating effects in the non-thermal solveras well, by solving the Navier-Stokes equations to accountfor the neutral gas changes and to couple them to the exist-ing non-thermal solver. The Navier-Stokes equations aresolved using the FE-FCT method, as with the charged-particle continuity equations.

The authors have developed a new solver for theNavier-Stokes equations that simulates the movement ofthe neutral gas particles in viscous fluids. This has beenthoroughly tested in both two-dimensional Cartesian andcylindrical axisymmetric co-ordinates [12, 13]. After vali-dation, it was then coupled to the existing non-thermalsolver so that the streamer propagation [14] and post-streamer discharge stage could be studied, with andwithout heating effects included [15]. The authors haveshown that there are significant differences in the post-streamer discharge stage with the inclusion of heatingeffects.

Single moment approximation of the Boltzmann equationis assumed by solution of the continuity equations forthe charged particles, and neglect of the momentum andenergy effects for the charged particles for simplificationpurposes. Furthermore, the fluid model used for both thecharged and neutral gas particles assumes that the species

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are in hydrodynamic equilibrium and the transport par-ameters can be calculated beforehand from macroscopicconserved quantities. Local approximations are used to cal-culate the transport properties of the charged and neutral gasparticles, such as the production and loss processes, chargedparticle velocities and diffusion coefficients, thermal con-ductivity and viscosity for the neutral gas.

The hydrodynamic fluid model for the charged particles,which assumes that the charged particles are always in equi-librium with the electric field, is employed to capture thephysics of the post-streamer discharge stage. This is notthe case in low-pressure discharges, as, in the cathode fallregion, under certain conditions, non-equilibrium betweenthe field and the charged particles is likely to occur, as theelectric field can vary abruptly within a few mean freecharged-particle paths [16]. Consequently, it is suggestedin such a case that Monte Carlo methods, or solution ofthe Boltzmann equation, or special techniques included inhydrodynamic fluid models for the treatment of thecathode fall region, should be invoked for its accurateanalysis.

The equilibrium assumption in low pressures has beenshown to be valid for positive and negative ions withinthe cathode fall region using Monte Carlo simulations[17] and analytical approaches [18]. This is not alwaysthe case for the electrons, whereby, for a given electricfield distribution, it has been shown that the electron vel-ocity distribution function is not in equilibrium with thelocal electric field. The energy gained from the high-fieldregion is not balanced by the losses due to electron col-lisions, resulting in a significant flux of electrons enteringthe negative glow region. Similarly, in high-pressure gasdischarges, non-equilibrium local effects can also arise inthe cathode fall region, but, in this model, they are neglectedfor simplification purposes. Even though the mean free pathat high pressures is smaller, so is the cathode fall dimension,as it is proportional to the inverse of the neutral gas density,and thus both are proportionally smaller at high pressures.

Previous analysis of the post-streamer discharge stagein high-pressure air, using one-dimensional computer sim-ulations, has shown the formation of cathode fall, nega-tive glow, positive column and anode regions [19–21].Furthermore, attempts have also been made to analysethe post-streamer discharge stage numerically in two-dimensional Cartesian space [22], but using steady-stateequations and without the inclusion of heating effects. Inthis paper, the post-streamer discharge stage in atmosphericair is analysed numerically in two-dimensional cylindricalaxisymmetric co-ordinates using time-dependent equations.The multidimensional analysis, together with the transientas well as neutral gas heating effects taken into consider-ation, will pave the way for a detailed analysis of high-pressure, post-streamer discharge applications.

In this paper, numerical results are presented for thecharacterisation of the post-streamer discharge stage start-ing from a single electron. The physical configurationadopted is two parallel metallic cylindrical plates, placed1 mm apart in ambient air, across which a DC voltage of20% above the breakdown threshold is applied.

2 Model description

The numerical model (Poisson, charged-particle continuityand the Navier-Stokes equations) and the solution proce-dure used to characterise gas breakdown processes arethoroughly described in [15]. Source terms, such asionisation, attachment, recombination, charged-particlevelocities and diffusion coefficients, are calculated from

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experimental data found in [23] as functions of thereduced electric field E/N, where E is the electric fieldand N is the number of neutral gas particles per unitvolume. The Navier-Stokes part of the model is brieflypresented below.

2.1 Navier-Stokes numerical model

The compressible, viscous, Navier-Stokes conservationequations for mass, momentum and energy density are

@r

@tþ r � ðrvÞ ¼ S ð5Þ

@ðrvÞ

@tþ r � ðrvvÞ ¼ �rP� r � tþ

X

s

MTs ð6Þ

@1

@tþ r � ð1vÞ ¼ �r � ðQÞ � r � ðvðPI þ tÞÞ

þX

s

MT s � vþX

s

ftsJ � E ð7Þ

where r is the neutral gas density, 1 is the neutral gas energydensity, P is the pressure, I is the unit stress tensor, v isthe neutral gas velocity, and J is the current density. Thesubscript s represents the charged particle species (e forelectrons, p for positive and n for negative ions); fts is thepercentage of charged-particle energy with subscript s thatis transferred as thermal energy to the neutral particles;MTs is the momentum transfer of charged particle s to theneutral particles; t is the shear stress; S is the mass sourceterm; Q is the thermal conduction; and t is the time.

The two-step Lax–Wendroff technique is used for timestepping that comprises the predictor-corrector steps. TheTaylor–Galerkin finite element method is used to discretisethe charged-particle continuity and Navier-Stokes equations,whereas the Galerkin FE method is used for the Poissonequation. Accuracy and efficiency for these large calcu-lations are crucial factors. The FCT method ensures thataccurate and efficient results are obtained, free from inac-curacies building up from non-physical oscillations [24].

2.2 Boundary conditions

The boundary domain is split into four sides, which, if wemove anticlockwise in space, are the cathode, the outerboundary, the anode and the symmetry axis. Along the sym-metry axis, the radial momentum of the neutral gas is pre-scribed to be zero, and, at the anode side, the electrons andnegative ions are treated as outflow boundaries, whereasthe positive ions and the axial momentum are prescribed tobe zero, and the voltage is prescribed to be the appliedvoltage. As far as the cathode side is concerned, the positiveions are treated as the outflow boundary, whereas the axialmomentum of the neutral gas, the negative ions and thevoltage are prescribed to be zero, and the electrons are pre-scribed, at each time step, to be the calculated photo-emission value. Finally, at the outer boundary side, electrons,positive and negative ions and radial momentum are pre-scribed to be zero, whereas the neutral gas density andenergy density are prescribed to have their initial values.The remaining variables on all four sides are treated asNeumann boundary conditions and therefore are left tovary freely, as the Galerkin-finite element formulation usedby the authors assumes that a Neumann boundary conditionexists, unless defined otherwise.

The electron emission effect at the cathode is included inthe simulations owing to the fact that it plays an importantrole in the development of discharges in short gaps, as it is

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responsible for the transition from the non-self-sustained tothe self-sustained discharge regime. The electron emissiondue to photons impinging at the cathode is included in thesimulations by the model depicted in [25], with the photoe-mission coefficient g ¼ 0.005. The electron emission due toion bombardment from the cathode is also included in thesimulations. The electron flux released from the cathode isa proportion k of the positive ion flux arriving at thecathode. More specifically, the electrons released from thecathode are calculated to be

Nec ¼kNpcjWpcj

jWecjð8Þ

where Nec, Npc are the electron and positive ion densities atthe cathode, and Wec, Wpc are the electron and positive ionaxial velocities at the cathode, with k ¼ 0.01 in atmosphericair [26].

3 Results for the post-streamer discharge stagein DC atmospheric pressure air discharge

The development of the post-streamer discharge stage isanalysed at an applied voltage of 5600 V. The abovevoltage was randomly chosen to be 20% above the break-down threshold. The main emphasis of this paper is in thetransitory regime from the streamer to the glow discharge,analysing the very fast electric field redistribution at thecathode, the formation of the glow discharge, consistingof cathode fall, negative glow, positive column and anoderegions and the associated heating effects; and finally theinclusion of radial effects by including a two-dimensionalaxisymmetric model for the calculations.

3.1 Electron density

One-dimensional plots of the electron density along thesymmetry axis at four instances in time are shown inFig. 1. At time t ¼ 5.00 � 1029 s, the electron density ismaximum in the negative glow region close to thecathode and reduces to very low values in the cathode fallregion. From the negative glow region towards the anode,there is a distinct electron density column left by the

Fig. 1 One-dimensional plots of electron and positive and nega-tive ion densities at four instances in time along symmetry axis

——— Electrons-†-†- Positive ions. . . . . . Negative ions

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tail of the streamer, until the anode region, where asudden reduction in the electron density occurs. At timet ¼ 5.50 � 1029 s, the electron density in the positivecolumn becomes larger than that at the cathode fall region.

Close to the cathode, the steep gradients of the electrondensity move towards the cathode and become slightlysteeper as time progresses. Adjacent to the anode, the elec-trons do not multiply at the same rate as the rest of the dis-charge, and therefore a sudden decrease in electron densitycontinues to be observed. At later times of t ¼ 6.00 � 1029

and 6.50 � 1029 s, the electron density increases evenfurther, especially along the positive column, becomingmuch larger at the positive column than at the cathodefall, negative glow and anode regions. This is owing tothe larger rate of electrons fed from the negative glowinto the positive column than that of those escaping intothe anode region and also because there is such an electricfield in the positive column, where ionisation dominatesattachment. This initiates a distinct electron chargecolumn that forms in the middle of the gap.

It is also observed that the peak of the electron densitycolumn moves towards the anode as time progresses, withthe electron density along the positive column increasingby approximately one order of magnitude overall. In thecathode fall region, the electron density has relatively lowvalues. This is caused by the high axial field developed atthe cathode fall region that quickly removes the electronsfrom the cathode, leaving behind the slow positive andnegative ions. The electrons in turn encounter a region ofvery low axial field in the negative glow region, wherethey accumulate. In the positive column, the electroncharge density column is shown to extend all the way tothe anode, having maximum electron densities at anapproximate distance of between 4 � 1024 and 2 � 1024 mfrom the anode. It then decreases substantially into theanode region. Overall, the electron density is shown to beof the order of 1 � 1021 m23, which is in agreement withprevious findings of Marode et al. [27].

3.2 Positive ion density

Fig. 1 shows one-dimensional plots of the positive iondensity at four instances in time along the symmetry axis.Similarly to the electron density, the positive ion densityfollows the same patterns, but, on average, its values arehigher, except in the negative glow and anode regions.Again, a positive charge column exists in the middle ofthe gap that grows in size and moves towards the anode,as time progresses, in the form of a return wave.Furthermore, the development of the cathode fall region isheralded as the positive ion density increases approximatelyby one order of magnitude to form a high-held region of netpositive charge.

A two-dimensional cylindrical plot of the positive iondensity at time t ¼ 6.5 � 1029 s is shown in Fig. 2. Closeto the cathode within the cathode fall region, a maximumnet positive charge is observed owing to the accumulationof positive ions there. This is owing to ionisation collisionscaused by emitted electrons from the cathode that produceslow positive ions that try to escape into the cathode. Asthe rate of occurrence of ionisation collisions is muchhigher than the absorption rate of positive ions into thecathode, a net positive charge accumulates.

Moving upwards, regions of low density of positive ionsare encountered. The axial field is very low within the nega-tive glow region, and ionisation collisions are limitedeven though there are a large number of free electrons.

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However, even closer to the anode, the positive ion densitystarts to increase again. A long column of positive ionsexists within the positive column. Adjacent to the anode,a reduction in positive ion density is experienced, as posi-tive ions produced from ionisation collisions near theanode have enough time to leave the anode owing to thefairly high axial fields. Overall, it is shown that the dis-charge activity occurs within the path of the cathode-directed streamer, which extends radially at a distance ofabout 2 � 1024 m from the symmetry axis, whereas mostof the activity occurs within a radius of 5 � 1025 m.

3.3 Negative ion density

One-dimensional plots of the negative ion density along thesymmetry axis, at four instances in time, are shown inFig. 1. Maximum negative ion densities are observedalong the positive column in the middle of the gap at timet ¼ 5.00 � 1029 s. At time t ¼ 5.50 � 1029 s, it is shownthat the negative ion density increases, especially alongthe negative ion density column and at the proximity of theanode. It is interesting to note that the maximum densityof the negative ions occurs at the anode, in contrast to theelectron and positive ion distributions discussed above. Atthe anode, the negative ion density is for the first time com-parable in size with the electron and positive ion densities.

Furthermore, the electrons and positive ions decrease innumber at the anode region, whereas the negative ionsincrease in number, as shown in Fig. 1. The negative ionsat the anode are mostly produced owing to attachment ofelectrons to the neutral gas particles and are lost owingto recombination between positive and negative ions.Consequently, in the anode region, attachment dominatesover recombination, owing to the low number of positiveions, and therefore maximum values of negative iondensity are obtained in the anode region. Fig. 1 shows theformation of a positive column in the inter-electrode gaptravelling towards the anode in the form of a return wave,by displaying distinct electron, positive and negative iondensity columns.

3.4 Net charge density

Fig. 3 shows one-dimensional plots of the net charge densityagainst the distance along the symmetry axis at four differ-ent instances in time. The net charge density within the

Fig. 2 Two-dimensional plot of positive ion density at timet ¼ 6.5 � 1029 s

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positive column is shown to remain constant and to increaseclose to the cathode and anode regions as time progresses.The sudden increase in the positive ions in the cathodefall region, and, hence, the steep gradients of the netcharge that are generated, make necessary the usage of avery fine mesh at the cathode region. The rate of increaseof the net charge density at the cathode is shown to bemuch greater than that at the anode, and this is owing tothe higher ionisation rate as a result of higher electricfields that are formed, as shown later in this paper. Justoutside the cathode fall region, there is a region of netnegative charge that exhibits a near-zero reduced absoluteelectric field, as shown also in Fig. 3, which implies the for-mation of a negative glow region, even though no lightemission was simulated in this paper.

3.5 Reduced absolute electric field

Fig. 3 shows one-dimensional plots of the absolute reducedelectric field at four instances in time along the symmetryaxis. At time t ¼ 5.00 ns, just after the cathode-directedstreamer hits the cathode, the axial electric field in thecathode fall region is maximum and, further upwards intothe negative glow, it reduces to a very low value close tozero, reaching a minimum value within this region of netnegative charge. It then increases slightly to stay fairly con-stant along the positive column. In the proximity to theanode, it suddenly increases owing to the net negativecharge that exists at the anode. At later instances in time,the absolute reduced electric field stays constant along thesymmetry axis, except for the cathode and anode regions.In the anode region, the net negative charge increases astime progresses, and therefore the absolute electric fielddoes as well, as shown in Fig. 3. In the cathode fallregion, owing to the increasing population of net positivecharge followed by an increasing net negative charge inthe negative glow region, the absolute reduced electricfield continuously increases, reaching a value of oneorder of magnitude greater than that at time t ¼ 5.00 ns.The neutral gas number density distribution stays fairlyunchanged in the discharge, as it changes only through pro-duction and loss processes with the charged particles, andinduces negligible variations in the absolute reduced elec-tric field.

Fig. 3 One-dimensional plots of net charge density and absolutereduced electric field at four instances in time

— Net charge denstity. . . Reduced electric field

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Fig. 4 Two-dimensional plots of absolute electric field at times t ¼ 5.00, 5.50, 6.00 and 6.50 ns

a t ¼ 5.00 nsb t ¼ 5.50 nsc t ¼ 6.00 nsd t ¼ 6.50 ns

3.6 Absolute electric field redistribution

The absence of neutral gas depopulation along the sym-metry axis at this stage shows that there are no significantheating effects along the symmetry axis to cause thisdepopulation. This is in agreement with Sigmond [28],who states that the neutral gas expansion occurs after theredistribution of the electric field at the cathode.

Fig. 4a shows the electric field at time t ¼ 5.00 ns, whereit is shown to extend radially outwards to a distanceof 0.24 mm. Fig. 4b shows the electric field at time t ¼5.50 ns, where it is shown that the electric field increasesapproximately by an order of 4 � 107 V m21 within 0.5 nsand extends radially to a distance of 0.41 mm. The scenariocontinues, as shown in Fig. 4c at time t ¼ 6.00 ns, with theonly difference that the electric field has now increasedapproximately by 5 � 107 V m21 within 0.5 ns and hasextended to a radial distance of 0.65 mm. At a later time oft ¼ 6.50 ns, as shown in Fig. 4d, the electric field increasesby approximately 9 � 107 V m21 within 0.5 ns, and the elec-tric field has now covered the whole surface of the cathode.

The electric field redistribution, after the arrival of theprimary streamer at the cathode, has been analysed, and itis shown that, as time progresses, the electric field extendsradially outwards, covering a larger cathode area, shrinksin thickness and increases in magnitude at a faster rate. Inthe final stage, the electric field reaches values two ordersof magnitude larger than those during streamer propagation.In the rest of the discharge away from the cathode, the elec-tric field is very small and tends to increase along the anode,

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as shown more clearly in the one-dimensional plots of theabsolute reduced electric field shown in Fig. 3. As describedby Sigmond [28], after the arrival of the primary streamer atthe cathode, a very fast redistribution of the electric fieldoccurs, followed by a secondary streamer propagatingfrom the anode towards the cathode. For the given gap con-ditions, the secondary streamer seems to start somewhatlater than the described transitory regime. This assumptionis in agreement with the fact that the pure resistive stage ofthe transient filamentary discharge has not yet appeared.This is confirmed with the differences between total,anode and cathode conductive currents, as shown later inFig. 5b.

3.7 Total conductive current

The total conductive current of the discharge with respect totime is shown in Fig. 5a. The total conductive current of thedischarge is made up of the conductive current of electronsand positive and negative ions summed over the wholedomain. With the arrival of the primary streamer at thecathode, the total conductive current is expected instan-taneously to increase and then decrease, for operation atvoltages just above the breakdown. This is because,within the inter-electrode gap, attachment dominates ionis-ation. But, for operation at a voltage of 20% above thebreakdown threshold, the high applied voltage induces anelectric field along the positive column, where ionisationdominates over attachment, and therefore the current con-tinues to increase.

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Fig. 5 Two-dimensional plots of conductive current, cathode fall voltage, current density at the anode and current density at cathode

a Conductive currentb Cathode fall voltagec Current density at anoded Current density at cathode

3.8 Conductive current into the electrodes

Fig. 5a shows the one-dimensional plots of the conductivecurrent that flows into the anode and cathode regions withrespect to time. The anode conductive current is shown tobe larger than that of the cathode, even though the netcharge density at the cathode is larger than that at theanode, and this is mainly owing to the larger number ofelectrons that exist at the anode, as electrons owing totheir high mobility, are the dominant mechanism ofcurrent conduction. It is shown that, as time progresses,both the anode and cathode conductive currents increaseat a faster rate, with their difference increasing further.The conductive current within the cathode fall region ismainly ionic and, just outside the cathode fall, in the nega-tive glow region, it is mainly electronic.

Plasma discharges are inherently unstable, when using aconstant discharge voltage and no external circuit, asdescribed by the authors. The authors are unable tomodel a steady-state glow discharge by excluding anexternal circuit from the discharge model, as shown in[29]. As the discharge is only modelled for approximately7 ns, which may be less than the characteristic time (RC)of the external circuit, the external circuit is excluded,and the transient stage from the streamer to the glow dis-charge is analysed. This includes the very fast redistribu-tion of the electric field, the formation of the glowdischarge with a cathode fall, negative glow, positivecolumn and anode regions and its associated heatingeffects, and the inclusion of a two-dimensional axisym-metric model so that the radial effects in the dischargecan be studied.

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3.9 Cathode fall voltage

Fig. 5b shows a one-dimensional plot of the cathode fallvoltage with respect to time. It is shown that, as soonas the streamer hits the cathode at around 4.75 ns, thecathode fall voltage is approximately 385 V. As time pro-gresses, it increases at a steady rate until it reaches avalue of 1490 V at 6.50 ns, with the rate of increase of thecathode fall voltage calculated to be 6.25 � 1011 V s21.

3.10 Current density

Figs. 5c and d show the current density distribution plotalong the anode and cathode, respectively, at four instancesin time. The current density adjacent to the cathode isshown to increase continuously, but also to expand radiallyoutwards, with the cathode fall region covering a greatersurface area of the cathode electrode. Similar behaviour isalso observed for the current density at the anode, withthe only difference that the radial expansion is much less,owing to the lower radial electric fields that exist at theanode. It is also shown that the anode current density ismuch larger than the cathode current density, and this isowing to the higher number of free electrons and negativeions that exist there. The current density at the anode exhi-bits a maximum value at a radial distance of 2 � 1025 m,which is where the maximum net negative charge densityexists. The rate of electron absorption into the anodedepends on the axial applied field, which is larger at thesymmetry axis and reduces as we move radially outwardsalong the anode. The radial expansion of the currentdensity within the cathode fall and anode regions justifies

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the need to use a two-dimensional cylindrical axisymmetricco-ordinate model to analyse the post-streamer dischargestage in DC atmospheric air.

3.11 Neutral gas temperature

Fig. 6a shows one-dimensional plots of the neutral gastemperature along the symmetry axis at four instances intime. The temperature is at its maximum at the cathodefall region and reduces abruptly to almost ambient tem-perature just outside it, where it remains almost constant.As time progresses, the temperature along the symmetryaxis is continuously increasing. Within a time interval of0.75 ns, the temperature in the cathode fall region hasincreased by approximately 180 K, whereas, in the negativeglow, positive column and anode regions, it remains fairlyconstant. The heating at the cathode is a direct consequenceof the very high radial and axial fields that move the emittedelectrons from the cathode and consequently the newlycreated electrons due to ionisation, at extremely highspeeds. These very energetic electrons on their way out ofthe cathode fall region collide with the neutral gas particlesand transfer their energy as thermal Joule heating.Consequently, the higher temperature is adjacent to thecathode. The metallic electrode temperature effects havebeen neglected in the model for simplification purposes,and it may be that they will affect the temperature distri-bution close to the cathode.

One-dimensional plots of the neutral gas temperaturealong the anode are shown in Fig. 6b at four instances intime. The temperature at the anode is shown to increaseby only a few kelvins, in contrast with the cathode, whichexhibits significant heating. This is owing to the lowerfields that exist adjacent to the anode, even though thereare higher current densities in the anode region owing tothe higher electron density numbers. Within a time intervalof 0.75 ns, the temperature of the neutral gas increases con-tinuously, but by only just 5 K at the symmetry axis.Similarly to the cathode, maximum temperature valuesoccur close to the symmetry axis, where most of thedischarge activity takes place. The overall increase inneutral gas temperature of 180 K in the cathode fallregion justifies the authors’ decision to include neutral gas

Fig. 6 One-dimensional plots of neutral gas temperature alongsymmetry axis and along anode at four instances in time

a Symmetry axisb Anode

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heating effects for the simulation of the post-streamerdischarge stage.

4 Conclusions

This paper presents numerical results for the developmentof the post-streamer discharge stage by modelling of thetransitory regime from the streamer to the glow discharge,in a uniform applied electric field in atmospheric air. Thearrival of the primary streamer at the cathode instigatesthe very fast redistribution of the electric field togetherwith the formation of the glow discharge, which is shownto consist of the cathode fall, negative glow, positivecolumn and anode regions. It has been shown that thepositive column travels towards the anode in the form ofa return wave. The maximum heating of approximately180 K that occurs at the cathode fall region, togetherwith the outward radial expansion of the current densityin the cathode fall region, covering a larger area of thecathode as time progresses, justifies the authors’ decisionto use a two-dimensional cylindrical axisymmetric modelthat will also include the neutral gas heating effects toanalyse the post-streamer discharge stage in DC atmos-pheric pressure air.

5 References

1 Papadakis, A.P., Georghiou, G.E., and Metaxas, A.C.: ‘Numericaltechniques in high voltage breakdown’. 2004 4th MediterraneanConf. on Power Generation, Transmission and Distribution,Limassol, Cyprus

2 Jaiswal, V., and Thomas, J.: ‘Finite element modelling of ionized fieldquantities around a monopolar HVDC transmission line’, J. Phys. D,Appl. Phys., 2003, 36, pp. 3089–3094

3 Hare, R.W., Hill, R.M., and Budd, C.J.: ‘Modelling charge injectionand motion in solid dielectrics under high electric field’, J. Phys. D,Appl. Phys., 1993, 26, pp. 1084–1093

4 Yu, L., Laux, C.O., Packan, M., and Kruger, C.H.: ‘Direct-currentglow discharges in atmospheric pressure air plasmas’, J. Appl.Phys., 2002, 91, pp. 2678–2686

5 Sira, M., Trunek, D., Stahel, P., Bursikova, V., Navratil, Z., andBursik, J.: ‘Surface modification of polyethylene and polypropylenein atmospheric pressure glow discharge’, J. Phys. D, Appl. Phys.,2005, 38, pp. 621–627

6 Hasson, V., and Von Bergmann, H.M.: ‘High pressure glowdischarges for nanosecond excitation of gas lasers and lowinductance switching applications’, J. Phys. E, Sci. Instrum., 1976,9, pp. 73–76

7 Roth, J.R., Rahel, J., Dai, X., and Sherman, D.M.: ‘The physics andphenomenology of one atmosphere uniform glow discharge plasma(OAUGDPTM) reactors for surface treatment applications’, J. Phys.D, Appl. Phys., 2005, 38, pp. 555–567

8 Kushner, M.J.: ‘Modeling of microdischarge devices: Pyramidalstructures’, J. Appl. Phys., 2004, 95, pp. 846–859

9 Georghiou, G.E., Morrow, R., and Metaxas, A.C.: ‘Two dimensionalfinite element-flux corrected transport algorithm for the solution ofgas discharge problems’, J. Phys. D, Appl. Phys, 2000, 33,pp. 2453–2466

10 Georghiou, G.E., Morrow, R., and Metaxas, A.C.: ‘Two dimensionalsimulation of streamers using the FE-FCT algorithm’, J. Phys. D,Appl. Phys., 2000, 33, pp. 27–32

11 Georghiou, G.E., Papadakis, A.P., Morrow, R., and Metaxas, A.C.:‘Numerical modeling of atmospheric pressure gas dischargesleading to plasma production’, J. Phys. D, Appl. Phys., 2005, 38,R303–R328

12 Papadakis, A.P., Georghiou, G.E., and Metaxas, A.C.: ‘Towards thesimulation of spark discharges’. XIV Int. Conf. on Gas Dischargesand their Applications, Liverpool, UK, 2002, vol. 2, pp. 272–275

13 Ehlers, R., Georghiou, G., Hallac, K., Malan, H., Papadakis, A., andMetaxas, A.C.: ‘Finite elements in the simulation of dielectricheating systems’ in ‘Advances in microwave & radio frequencyprocessing a book of selected papers from the Proceedings of theEight, International AMPERE Conference on Microwave and HighFrequency Heating’ (Bayreuth, Germany, September 2001), Editedby Prof M Willert-Porada, ISBN 3-540-43252-3, Springer Verlag,pp. 165–175

119

14 Papadakis, A.P., Georghiou, G.E., and Metaxas, A.C.: ‘Theincorporation of heating effects in atmospheric pressure gasdischarges’. Proc. XXVI Int. Conf. on Phenomena in Ionised Gases(ICPIG), Greifswald, Germany, July 2003, vol. 4, pp. 65–66

15 Papadakis, A.P., Georghiou, G.E., and Metaxas, A.C.: ‘Simulation forthe transition from non-thermal to thermal discharges’, PlasmaSources Sci. Technol. 2005, 14, pp. 250–258

16 Davies, A.J.: ‘Discharge simulation’, IEE Proc., 1986, 133,pp. 217–240

17 Boeuf, J.P., Marode, E., Segur, P., Davies, A.J., and Evans, J.G.:‘Electron and ion behaviour in the cathode region of a dischargestudied by Monte Carlo techniques’. 6th Int. Conf. on GasDischarges and their Applications, Edinburgh, 1980, Part 2

18 Lawler, J.E.: ‘Equilibration distance of ions in the cathode fall’, Phys.Rev. A, 1985, 32, p. 2977

19 Rabehi, A., Gadri, R.B., Segur, P., Massines, F., and Decomps, P.:‘Numerical modeling of high-pressure glow discharges controlledby dielectric barrier’. Proc. Conf. on Electrical Insulation andDielectric Phenomena (CEIDP’94) Arlington, TX, October, 1994,vol. 26, pp. 840–845

20 Massines, F., Rabehi, A., Decomps, P., Gadri, R.B., Segur, P., andMayoux, C.: ‘Mechanisms of a glow discharge at atmosphericpressure controlled by dielectric barrier’, J. Appl. Phys., 1998, 83,pp. 2950–2957

120

21 Gadri, R.B.: ‘One atmosphere glow discharge structure revealedby computer modeling’, IEEE Trans. Plasma Sci., 1999, 27,pp. 36–37

22 Boeuf, J.P.: ‘A two-dimensional model of a dc glow discharge’, J.Appl. Phys., 1988, 63, pp. 1342–1349

23 Morrow, R., and Lowke, J.J.: ‘Streamer propagation in air’, J. Phys. D,Appl. Phys., 1997, 30, pp. 614–627

24 Zalesak, S.: ‘Fully multidimensional flux-corrected transportalgorithms for fluids’, J. Comput. Phys., 1979, 31, pp. 335–362

25 Georghiou, G.E., Morrow, R., and Metaxas, A.C.: ‘The effect ofphotoemission on the streamer development and propagation inshort uniform gaps’, J. Phys. D, Appl. Phys., 2001, 34, pp. 200–208

26 Reess, T., and Paillol, J.: ‘The role of the field-effect emission inTrichel pulse development in air at atmospheric pressure’, J. Phys.D, Appl. Phys., 1997, 30, pp. 3115–3122

27 Marode, E., Bastien, F., and Bakker, M.A.: ‘A model of thestreamer-induced spark formation based on neutral dynamics’,J. Appl. Phys., 1979, 50, pp. 140–146

28 Sigmond, R.S.: ‘The residual streamer channel: return strokes andsecondary streamers’, J. Appl. Phys., 1983, 56, pp. 1355–1370

29 Arslanbekov, R.R., and Kolobov, V.I.: ‘Two-dimensional simulationsof the transition from Townsend to glow discharge and subnormaloscillations’, J. Phys. D, Appl. Phys., 2003, 36, pp. 2986–2994

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