Tsagi4201-004-3437 Features of Numerical Modeling of a Dielectric Barrier Discharge

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TsAGI Science Journal,42 (1): 4769 (2011)

FEATURES OF NUMERICAL MODELING OF A

DIELECTRIC BARRIER DISCHARGE

A. P. Kuryachii, D. A. Rusyanov, & V. V. Skvortsov

Central Aerohydrodynamic Institute (TsAGI), Zhukovsky Str. 1,Zhukovsky, 140180, Russia

Address all correspondence to A. P. Kuryachii E-mail: [email protected]

The boundary problem describing the generation of dielectric barrier discharge in air near a surfaceof dielectric plate by means of a spatial periodic system of exposed electrodes with applied externalharmonic voltage is formulated in the framework of a 2D statement. Demonstrated is the significanceof taking into consideration such parameters as the finite thickness of the exposed electrodes, an exis-tence of the Knudsen layer near the plate surface and the electrodes, the finite rates of desorption andrecombination of charged particles on a dielectric surface, the strength of the electric field induced bysuperficial charge on a dielectric for the calculation of the conduction current, the volumetric forceacting on a gas from the discharge, and the heat release (Joule dissipation) in the discharge. Numericalinvestigation of an influence of some physical parameters on the mentioned characteristics of barrierdischarge is carried out.

KEY WORDS:dielectric barrier discharge, numerical modeling, drift-diffusionapproximation, adsorption, desorption, recombination, secondary ion-electronemission, volumetric force, Joule dissipation

1. INTRODUCTIONSeparate investigations dedicated to the control of flow around bodies using the electro-

hydrodynamic (EHD) interaction implemented in a corona discharge were carried out in

19601980 (Mhitaryan et al., 1968; Malik et al., 1983; Kuryachii, 1985). The interest

in the investigation of aerodynamic applications of the EHD effect on gas flows sharply

rose at the beginning of this century due to the idea of the application of the dielectric

barrier discharge (DBD) for these purposes (Roth et al., 2000). Dozens of works devoted

to the investigations of electrohydrodynamic actuating elements (EHD-actuators), which

operate on the basis of different types of electric discharges, and of their aerodynamic

applications are published every year (Moreau, 2007; Corke et al., 2007). Meanwhile,

the most attention is paid to the dielectric barrier discharge actuating elements (DBD-

actuators).The elaboration of physical models that adequately describe the processes occurring

in the DBD and the numerical simulation using these models, which take into account,

19482590/11/$35.00 c 2011 by Begell House, Inc. 47


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48 Kuryachii, Rusyanov & Skvortsov

as far as possible, the conditions in which these processes occur, are a very important

part of these investigations. To describe electrokinetic processes in the DBD, different

models are applied, from the simplest models, in which only electrons and positive ions

are considered, to more complex models of plasma kinetics in air, which consider more

than 500 reactions (Bogdanov et al., 2008). At the same time, in the vast majority ofnumerical investigations, the exposed electrodes, near which a discharge is generated,

of zero thickness are considered, and only in some works the shape of the mentioned

electrodes is taken into account (Font et al., 2007; Hoskinson and Hershkowitz, 2009a;

Soloviev and Krivtsov, 2009). Meanwhile, in the experimental investigations, a signifi-

cant influence of the shape of exposed electrodes on the DBD characteristics is observed

(Enloe et al., 2004; Hoskinson and Hershkowitz, 2009b).

Such processes as adsorption, desorption, and recombination of charged particles,

which occur with a finite velocity, are as a rule not considered in the majority of the

theoretical works in the boundary conditions on a dielectric. The model of instantaneous

recombination of charged particles of the opposite sign on a dielectric surface (Bouef et

al., 2008) is usually applied. In all the works, instead of Soloviev and Krivtsov (2009),

the fact that the flows of charged particles from a gas toward the surface of a solid body

and in the opposite direction pass through the Knudsen layer, in which the decelerating

effect of the electric field can be significant, is not taken into account. The consideration

of this factor leads to the variations in boundary conditions for the transfer equation

of charged particles. The accumulation of a superficial charge on the dielectric is in turn

determined both by the processes mentioned above, which proceed with a finite velocity,

and by boundary conditions for the fluxes of charged particles. A significant influence

of the electric field induced by a superficial charge on physical processes in a discharge

gap seems to be obvious. The purpose of this work is a more correct formulation of

the boundary problem of dielectric barrier discharge modeling using both existing and

newly developed physical models of the interaction of charged particles with a dielectric

surface and a numerical parametric investigation of discharge characteristics.

2. FORMULATION OF THE PROBLEM OF MODELING OF A DIELECTRIC

BARRIER DISCHARGE IN AIR

The configuration of a system of DBD-actuators is considered that enables the bound-

ary electrostatic conditions to be exactly stated, and at the same time, the influence of

a transverse force effect of this system of DBD-actuators on shear flows of gas to be

evaluated in the sequel. A cross section of a plane channel is shown in Fig. 1. BB aresolid grounded electrodes;Care the exposed electrodes near which the discharge is gen-

erated that are situated perpendicularly to the figure plane; D are dielectric layers. It is

assumed that the channel is infinite to the right and to the left. Surface AA is a symmetryplane. The numerical modeling is fulfilled in rectangular EFGH. The symmetry condi-

tions are satisfied on lines FG and EH. The computational domain is shown in the same

TsAGI Science Journal


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Features of Numerical Modeling of a Dielectric Barrier Discharge 49

FIG. 1:Scheme of the system of DBD-actuators and computational domain.

figure, on its right side. Here, Figs. 13 designate the gas phase, the dielectric layer, and

the exposed electrode, respectively; yd is a dielectric thickness; ye yd is the electrode

thickness; xe is its length; xm is half of distance between adjacent electrodes;ym is the

maximum vertical dimension of the domain under consideration.

The dielectric barrier discharge in air is simulated on the basis of a combined solution

of the equations of electrostatics and equations of electrokinetics in a diffusion-drift

approximation (Rayzer, 1971). Electric potential and electric field vector E = are determined from the solution of the Poisson equation in a gas phase (subscript 1) and

the Laplace equation in a dielectric layer (subscript 2),

01= e (np nn ne) , 2= 0 (1)

Here, 0is a vacuum dielectric constant; eis an elementary charge; ne,np,nnare concen-

trations of the electrons, and positive and negative ions, respectively. A relative dielectricconstant of gas phase is considered to be unity.

An alternating sinusoidal voltage is applied between the exposed electrode (x,y) S3,where S3is its surface, and the isolated electrodey = 0, which is expected to be grounded.

The potential continuity condition and the relationship between the electric induction

discontinuity and superficial charge are satisfied at the boundary between dielectric layer

2 and gas phase 1. Thus, the boundary conditions for the electric potential are of the

following form:

(x, 0, t) = 0,

x(0, y , t) =

x(xm, y, t) = 0,

y (x, ym, t) = 0

(x, y) S3: (x, y, t) =0(t) RbIc (2)

xe x xm, y= yd: 1= 2, 22y

1

y =

e

0(sp sn se)

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Here, sp, sn, se are surface densities of the positive elementary charges, negative ions,

and electrons;0 is an external applied voltage;Rbis an external (ballast) resistance; Icis a conduction current in the discharge gap. The dynamics of charged particles in the

diffusion-drift approximation without taking into account the photoionization processes

are described by three transfer equations of charged particles (Soloviev and Krivtsov,2009; Aleksandrov et al., 2007),

net

+ e = kiNne kdrnenp 0.21katN ne+ 0.79kdtNnn

npt

+ p = kiN ne kdrnenp krnpnn

nnt

+ n = 0.21katN ne 0.79kdtNnn krnpnn

k =zkknkE Dknk, k= p, n, e, zp= 1, zn= ze = 1 (3)

ki= 0.79kiN+ 0.21kiO, kiN= 10

8.0940.29/

, kiO = 10

8.3128.57/

kr = 2.3 106 (300/Tp)

3/2 exp [(p 760)/806]2

kdr = 2 107 (300/Te)

0.7 , kat = 1010.215.7/, kdt = 9.2 10

13

Here, N(cm3) is a concentration of neutral particles; ki is the ionization rate coeffi-

cient, which takes into account the generation of positive ions of nitrogen and oxygen;

kiNand kiOare the individual coefficients of nitrogen and oxygen ionization rates (Alek-

sandrov et al., 1995);kr is the coefficient of ion recombination rate;kdris the coefficient

of dissociative electron recombination rate; kat is the coefficient of the rate of dissocia-

tive attachment of electrons to the molecules of oxygen;kdtis the coefficient of electron

detachment rate; = 1016 E/N(V cm2) is the reduced field function; E(V/cm) is the

absolute value of the electric field strength;Teand Tpare temperatures of electrons and

ions, measured in Kelvin. All the coefficients of reaction rates are measured in cm3/s.

Coefficients 0.79 and 0.21 correspond to the shares of nitrogen and oxygen in the total

concentration of neutral particlesN. The experimental dependence of the ion recombi-

nation rate coefficient kr on air pressure p, presented in Mac Daniel (1967), is approx-

imated within the range ofp = 200760 Torr by means of the formula for kr, which is

given above.

For the considered conditions, the electron temperature e, measured in electron-

volts, is the reduced field function. In order to calculate it, the approximation of the

dependencee(), presented in Bychkov et al. (1997), according to the calculation re-sults on the basis of the kinetic theory, is fulfilled. The function obtained is presented in

the form e = A(B)C, with constants A, B, C, depending on the values of given inTable 1. The electron temperatureTein Kelvin from Eq. (3) is recalculated according to

the formulaTe= 1.16 104e.



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TABLE 1:Parameters of electron temper-

ature dependence on reduced field

Range of A B C

0.1 0.4 0.171 10 0.7245

0.4< 0.8 0.467 2.5 0.36180.8< 1 0.6 1 0

1< 4 0.602 1 0.166

4< 10 0.205 1 0.942

10< 15.4 0.117 1 1.187

> 15.4 0.507 1 0.65

The electron mobility is determined as e= vde/E(cm2V1s1), wherevde(cm/s) is

a drift velocity of electrons. For its determination, the approximation of the graphics de-

pendences on the reduced field, which are presented in Bychkov et al. (1997), is fulfilled

in the form: < 1:

vde = 10[0.931 + 0.383 lg(10)] 105, 1 : vde = 10

(0.314 + 0.821 lg) 106

The diffusion coefficient of electronsDe (cm2/s) is calculated through the Einstein re-

lation, which is propagated to the case of a nonequilibrium mean energy of electrons,

De = ech/e, where the ratio of the characteristic electron temperaturech to the ele-mentary charge is determined asch/e= A1(B1)

C1. ConstantsA1,B1,C1, depending on

the values of the reduced field, are given in Table 2.

The coefficients of electron mobility and ion diffusion are calculated according to

Smirnov (1983) as

p, n = Cp, n (N0/N)(300/Tp, n )0.3

cm2V1s1

, Cp = 2.1, Cn = 3.2

Dp, n = p, nTp, n /11 600

cm2

s

,

TABLE 2:Parameters of dependence of char-

acteristic electron temperature related to ele-

mentary charge on reduced field

Range of A1 B1 C1

0.1 0.8 0.2245 10 0.667

0.8< 2.45 0.8987 1.25 0.323

2.45< 4 1.29 0.408 0.0464

4< 6 1.32 0.25 0.529

> 6 1.636 0.167 0.781



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where N0 is the Loschmidt number. The difference between the temperatures of ions

and a gas is determined through the equation (Konchakov et al., 2006) Tp T= (mp

m)(pE)2/3kB , wherempand m are masses of an ion and a neutral,kB is the Boltzmann

constant, and at E 104 V/cm and p 2 cm2/s, it is90 deg. Therefore, at atmo-

spheric pressure, the ion mobility, determined according toTpdiffers from the mobilitythat is determined according to the gas temperatureTby no more than 8%. In the imple-

mented calculations, the ion temperature is assumed to be equal to the gas temperature.

The initial conditions for the equations of electrokinetics [Eq. (3)], as well as the

symmetry conditions at the left and right boundaries of the considered domain, and the

condition of absence of diffusion flows at the upper boundary, which are valid for the

geometry of the problem presented in Fig. 1, have the following form:

np(x,y, 0) =n0, nn(x,y, 0) = 0.9n0, ne(x,y,t) = 0.1n0, n0 = 103, cm3

nix

(0, y, t) =ni

x (xm, y, t) =

niy

(x, ym, t) = 0, i= n, p, e(4)

For the complete formulation of the boundary problem, it is necessary to know the

boundary conditions at the boundaries of the dielectric and electrode.

3. STATEMENT OF THE BOUNDARY CONDITIONS ON THE DIELECTRIC

AND ELECTRODE

While deriving the boundary conditions that determine the flows of charged particles

with taking into account the influence of the electric field, an approach similar to the ap-

proach used in Ushakov (1968) is applied. In order to determine the distribution function

of charged particles over velocitiesf, the Boltzmann equation in a relaxation approxima-

tion is applied (Kogan, 1967) as

df/dt= c(f0 f)

or

(c)1 df/dt =f0 f, c = c/, c= (8kBT/m)

1/2 (5)

wherec is the frequency of the collisions of particles; f0is the Maxwellian distribution

function of neutral particles; = 1 is a characteristic time of the problem; is thefrequency of the applied voltage; t is a dimensionless time; , c, and m are the meanfree path, mean thermal velocity, and mass of particles, respectively.

Let us denote 0 = h/c, where h is a characteristic macroscopic dimension of the

problem, for example, the electrode height. Then,(c)1 = 0 Kn, where Kn = /h

is the Knudsen number. In the considered case, 105 cm,h 102 cm, 104

Hz,c 105 cm/s;0 107 s. Therefore, Kn 0 103 1.The distribution function of charged particles over velocities is determined as the

expansion into series in Knudsen number f=f0 +Knf1 +Kn2f2 + . The substitution



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of this expansion in Eq. (5) and the retention of the terms of the first order in Kn gives

the equation0df0/dt = f1 or0df0/dt= f1, from where the required function

may be expressed as

f=f0 Kn0df0dt =f0

1

cf0

t + Vrf0+zeE

m Vf0

(6)

Here,Vis a vector velocity of particles;z is the sign of their charge.

In Eq. (6), the ratio of the nonstationary term f0/ f0 to the term Vrf0 cf0/hhas the order of0= 1. Therefore, in the considered case, the nonstationarity can

be neglected. Let us notice that the last term in Eq. (6) has the order |E|f0/c, where = e/mc is the mobility of particles; |E| is the absolute value of the electric fieldstrength. Therefore, approximation (6) is violated if the drift velocity of particles vd =|E| becomes of the order of their thermal velocity. It will be shown below that this cantake place in the vicinity of the electrode surface. Let us notice that it is Eq. (6) that

determines the known expression for the density of hydrodynamic flux of particles in a

drift-diffusion approximation in a quasi-stationary case (/t 0): VfdV= zEnDn, whereD = kBT/mcis the diffusion coefficient.

3.1 Boundary Conditions on the Dielectric

The equations for the density of hydrodynamic (macroscopic) flows of charged parti-

cles on solid surfacew are derived on the basis of the application of the Maxwell ap-proximate approach (Shidlovskiy, 1965) instead of the exact solution of the problem for

the distribution function of charged particles in the Knudsen layer with the prescribed

boundary conditions for this function on the wall and far from it. Let Ybe a distance

from the solid surface of the body. Let us consider a plane layer of gas 0 Y , in

which the charged particles move without collisions. Let us take into account a normalto the surface electric field strength component Ew, assuming it to be constant in theconsidered layer. At the external boundary of the layer Y= , there is a flux of particles

flying in it with the distribution function [Eq. (6)] and all velocity components normal

to the body surface from the range Vy 0 approach the external boundary of the layerfrom the surface side. The resultant flux of charged particles equals an algebraic sum of

fluxes at the external boundary of the considered layer, i.e.,w =J+J+. It is obvious



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from the continuity condition of the resultant flux thatw =Jw + J

+w , whereJ

w is the

total flux of the particles falling on the surface, and J+w is the total flux of the particlesflying from the surface, which is determined by the physical processes on the surface

that will be considered below.

Depending on the sign of particle charge and direction of the field Ew, the two situ-ations are possible. If the field is accelerating for the particles that fly from the surface

(Ew > 0 for positive ions andEw < 0 for electrons and negative ions), then they all reachthe external boundary of the layer, and are considered to be out of it. At the same time,

this field decelerates the particles of this type that fly to the surface from the external

boundary of the layer. Only those particles can get to the surface of the dielectric, whose

negative vertical velocity absolute value exceeds Vmin = (2e/m)1/2, where = |Ew|

is the potential difference surmounted by the particles inside the layer. Other particles

are decelerated by the field, and do not achieve the surface. Then, they accelerate in the

opposite direction and leave the considered layer. The density of the flux of particles that

achieve the surface is determined by integral (7), in which the integration over velocity

componentVy

is implemented from up to Vmin

,

Jw = 1

4cnA (X) +

1

2wB (X) , X=

e

kBT

1/2(8)

A (X) = expX2

, B(X) = 1 erf(X) + 21/2XA (X)

The values of functionsA and B in Eq. (7) depend on the free path of charged particles.

In the case of a weakly ionized plasma, the classical gas-kinetic sections of collisions

between ions and neutral molecules are applied for a qualitative evaluation of a free

path of ions at their low energy (Kozlov, 1969). Particularly, the free path of air ions

is evaluated as i 1015/(4.1N) cm. The mean free path of electrons is determined by

means of the approximation formula for the frequency of collisions of electrons with

air molecules, shown in Gurevich and Shvartsburg (1973), ce = 1.23 1075/6e Ns1. At the mean electron velocity, evaluated by the expression ce = 6.71 10

71/2e

cm/s, the mean free path of electrons is calculated through the formula e ce/ce =

5.45 1014/(1/3e N)cm.

If the electric field accelerates the particles that fly from the external boundary of the

considered layer to the body surface, thenJw = Jw1+ J

w2, whereJ

w1 is determined

through Eq. (7), and Jw2 is the flow of the particles flown from the surface, but decel-erated by the field and returned on the surface. In order to determine Jw2, it is expectedthat the particles flying from the surface have the diffusive Maxwellian distribution with

a certain characteristic temperature Tc. Those particles whose velocity vertical compo-

nent is smaller thanVmin return on the surface. The integration of this distribution from

zero up toVminprovides the expression for the flow of returned particles,

Jw2= J+w [1 A (Xc)] , Xc = (e/kBTc)

1/2 (9)



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In order to derive the explicit expressions for hydrodynamic fluxes w, the physicalmodel of the interaction of positive ions and electrons with the dielectric surface, which

is proposed in Golubovskii et al. (2002), is applied. This model takes into account the

following processes on the dielectric: (i) adsorption of all the electrons and ions, which

reach the surface, due to a holding action of a polarization charge; (ii) neutralization ofions with generation of a positive charge on the surface due to a significant excess of the

ion recombination energy over the work of electron output from the dielectric; (iii) des-

orption of electrons from the surface, which is characterized by desorption frequency e;

(iv) recombination of electrons with positive superficial charge, which is characterized

by recombination rate constantre; and (v) secondary ion-electron emission, which is

characterized by emission coefficientd.

The difference between the secondary electrons emitted from the surface and the

electrons adsorbed from the gas phase is taken into account. Namely, it is assumed that

the secondary electrons do not participate in the processes of adsorption and recombi-

nation with a positive superficial charge. Drift of electrons in the gas phase from the

dielectric leads to the appearance of a positive charge on its surface. This is considered

in the equations of kinetics of the superficial charge that are derived below.

As for negative ions, the following physical model of their interaction with the di-

electric surface is proposed. It is expected that as in the case of positive ions and elec-

trons, all the negative ions that reach the surface are adsorbed by it due to the action of

the dielectric polarization charge. These ions can be desorbed with frequencyn, as well

as recombine with the positive superficial charge. The ion recombination rate constant is

assumed to be proportional to their mobility, and is estimated as rn = ree/e.

It is also proposed that the process of detachment of electrons from the oxygen ions

is possible on the surface at their interaction with the molecules of nitrogen. This process

is similar to the same process in the gas phase. The frequency of collisions of nitrogen

molecules at a unit surface of the dielectric with the oxygen negative ions situated on

the surface is estimated as 0.79Ncmsn/, where cmis the thermal velocity of molecules,

and is an efficient cross-section of interaction of molecules with ions that leads to the

electron detachment. Applied is the estimationcm kdt, where kdt is the same coef-ficient of the detachment rate of electrons that is used for simulation of 3D processes

in Eq. (3). Thus, the density of the flux of electrons that arise during the detachment

process on the dielectric surface is estimated as J+edt 0.2 kdtN sn. It is expected thatthese electrons do not stay near the surface, but immediately depart in the gas. Neu-

tralization of a negative ion of oxygen occurs at the reaction O +N2 e+ N2O(Soloviev and Krivtsov, 2009). The energy consumed in generation of the molecule

N2O equals 0.85 eV (Rabinovitch and Khavin, 1978). Energy of electron affinity for

the atom of oxygen is 1.47 eV (Rayzer, 1971). Therefore, this reaction in a cold gas

can proceed if the electron-excited states of molecules of N 2 play the role of energyreservoir for the emission of the electron from O and N2O generation. For them,

the energy of low levels is 6.2 and 8.4 eV (Rayzer, 1971). It is assumed in the pro-



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posed model that the excess of energy, equal to 3.9 and 6.1 eV, is taken away by an

electron. Since the energy of detached electrons is sufficiently high, they all reach the

layer boundary Y = e, because at e 105 cm the decelerating influence of the

electric field is appreciable only the impossibly high electric field strength Ew 108

V/m.With the assumptions made above, the rates of variation of surface densities of pos-

itive chargessp(x,t), electronsse(x,t), and negative ionssn(x,t) are described by the fol-

lowing equations and initial conditions:

spt

= (1 + d) Jwp respse rnspsn, sp(x, 0) = 0

set

= Jwe ese respse, se(x,0) = 0 (10)

snt

= Jwn 0.2 kdtNsn vnsn rnspsn, sn(x, 0) = 0

Here,Jwp < 0, Jwe < 0, andJ

wn < 0 are the total fluxes of positive ions, electrons, and

negative ions that reach the dielectric surface, correspondingly.

Since there is no flux of positive ions from the surface, their resultant flux equals the

flux incident on the surface, namely, wp =Jwp. Hence, we obtain, taking into account

Eqs. (7)(9), the explicit expressions for the density of the flux of positive ions and rate

of variation of the density of a positive superficial charge as

Ew 0 : wp = 0.5cpnp

Ew >0 : wp = 0.25cpnpAp(1 0.5Bp)1 (11)

spt

= (1 + d) wp respse rnspsn

The resultant flux of electrons is equal to the sum of the fluxes incident on the surface,

desorbed from the surface, secondary, and detached electrons,

we = Jwe+ ese+ 0.2kdtN sn dwp (12)

IfEw 0, the flux of electrons that reach the surfaceJwe is determined through Eq. (8).

As a result of the combined solution of Eqs. (8) and (12), the expressions for we andJwe are obtained, as well as the explicit form of Eq. (10),

we = (0.25ceneAe+ ese dwp+ 0.2 kdtN sn) (1 0.5Be)

1

(13)

set

= [0.25ceneAe ese+ 0.5Be(dwp 0.2kdtNsn)](1 0.5Be)1 respse



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IfEw >0, the flow Jwe is equal to the sum of the total flux of electrons that fly to the

surface from the external boundary of the layer [Eq. (7)], and the fluxes of desorbed and

secondary electrons [Eq. (9)] returned on the surface due to the decelerating action of

the field,

Jwe = 0.25nece+ 0.5we ese(1 Aew) + dwp(1 Ae) (14)

At the calculation of the functions Aewand Ae, it is assumed that the desorbed electrons

have the temperature of the dielectric surface Tw, and the secondary electrons have the

characteristic temperaturee = 0.5 eV; that is, Te = 1.16 104e K. The combined

solution of Eqs. (12) and (14) provides the following explicit expressions:

we = 0.5cene+ 2eseAew 2dwpAe+ 0.4 kdtN sn (15)

set

= 0.5cene 2eseAew+ dwp(2Ae 1) respse 0.2 kdtNsn

The resultant flux of negative ions is equal to the sum of the fluxes of ions falling onthe surface and desorbed from the surface, that is, wn =J

wn+ nsn. AtEw 0, the

flux of the negative ions that fall on the surface is described by Eq. (8). Therefore, the

expression for the flux density and the latter equation in Eq. (10) assume the following

form:

wn = (0.25cnnnAn+ nsn) (1 0.5Bn)1

snt

= (0.25cnnnAn nsn) (1 0.5Bn)1 0.2 kdtNsn rnspsn

(16)

AtEw > 0, the flux of negative ions that fall on the surface is described by the expressionsimilar to Eq. (14), but without the last term. Therefore, the flow density and the rate of

the surface density variation of negative ions are described by the expressions

wn = 0.5cnnn+ 2nsnAnwsnt

= 0.5cnnn 2nsnAnw 0.2 kdtNsn rnspsn(17)

3.2 Boundary Conditions on the Electrode

The boundary conditions on the metallic surface depend also on the sign of the nor-

mal component of the electric field strength. It is obvious in the frame of the Maxwell

approach, applied above, that the expressions for the macroscopic flux of electrons on

the electrode differ from the expressions, obtained above for the dielectric, only by the

absence of the terms that reflect the desorption of the electrons and recombination witha positive superficial charge. It is obvious for the flux of positive ions at Ew 0 thatthe same expression as for the dielectric is obtained. At Ew


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proach, we obtain the expression wp = 0.5cpnp, orDpnp = (pEw+ 0.5cp) np.It is seen from here that the gradient of the concentration of the ions on the electrode

surface becomes negative at the value of the drift velocityvdp p|Ew| >0.5cp. Thismeans the accumulation of positive ions near the electrode surface at the electric field

being directed toward the electrode. This situation seems to be physically unreal. There-fore, atEw


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the simplified boundary conditions for the fluxes of charged particles and equations of

the accumulation of the superficial charge that are obtained from the expressions given

above atEw = 0 can be applied and are as follows:

xe x x

m, y= y

d:

wp= 0.5c

pn

p,

wn= 0.5c

nn

n+ 2

ns

n

we = 0.5cene+ 2ese+ dcpnp+ 0.4kdtNsn

set

= 0.5cene 2ese 0.5dcpnp respse 0.2 kdtN sn

spt

= 0.5cpnp(1 + d) respse rnspsn

snt

= 0.5cnnn 2nsn 0.2kdtNsn rnspsn (19)

(x, y) S3: we = 0.5cene 2mwp

Ew

>0 : wp

= 0.5cp

np

, n

Ew


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FIG. 2:Evolution of the DBD conduction current at the positive initial polarity of the

exposed electrode (a) and at the negative initial polarity of the exposed electrode (b).

of the secondary emission on the electrode m = 0.001, constant of the recombination

rate on the dielectric re = 107 cm2 s1, and frequency of the electron desorption

on the dielectric e = 5 103 s1. The applied voltage 0(t) = V0 sin(2t) has an

amplitude V0 = 7 kV and frequency = 5 kHz. The ballast resistance Rb = 5 105

cm is applied to the external circuit. Let us note that the values of the constants of the

recombination rate on the dielectric and the frequency of desorption of electrons men-

tioned above are taken from Golubovskii et al. (2002), in which they were determined

on the basis of the comparison of the numerical modeling results with the experimental

data.

The introduction of the ballast resistance in the boundary condition on the exposedelectrode [Eq. (2)] enables avoiding the unnaturally high values of the concentration of

charged particles and the ionization intensity overrunning beyond the limits, at which the



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computational process diverges. This is achieved by means of the reduction of the rate of

rise of the potential difference on the electrodes at the increase of the conduction current

that leads due to the relaxation to the reduction of the ionization in the discharge gap and

the decrease in the current. Due to the decrease in current, the voltage drop on the ballast

resistance is reduced, the potential difference of the electrodes increases again, and theionization intensifies. As a result, the oscillations of the discharge current arise, which

are shown in Fig. 2.

The described qualitative picture is observed exactly at the discharge initiation, in-

dependently of the sign of the applied voltage (see Fig. 2). The extreme values of current

differ in the discharge first pulse, i.e., I 0.05 A/m at the positive polarity andI 0.04A/m at the negative polarity. At the prescribed value of the ballast resistance, the voltage

drop on it reaches 200250 V. Let us note that appreciable drops of the applied voltage

are observed during the first period of the initial sinusoidal voltage in the oscillograms

from the experiments (Enloe et al., 2008). In the calculations, the maximum values of

the ion concentration at the first peak of the current in the case of the positive polarity of

the electrode [Fig. 2(a)] are achieved in the vicinity of the electrode edge (np max

1.7 1012 cm3) and near the dielectric surface (np max 4.6 10

12 cm3). The electron

concentration has a maximum in the vicinity of the electrode surface ( ne 1010 cm3).

At the negative polarity, the maximum values of the concentrations are also achieved at

the first maximum of the current [Fig. 2(b)] and are equal tonp max 3.3 1012 cm3

and np max 6 1010 cm3. Thus, in this discharge, the quasi-neutrality condition is

not satisfied. Figure 2 also demonstrates that the dependences of the conduction current

on time in the second and third periods are almost identical. This result agrees with the

experimental data as well (Enloe et al., 2008).

The dependences on time of the horizontal component integrated over region 1 of the

volumetric force with densityF= e(np ne)E, acting on the gas from the discharge side,

and of heat release power Q in the discharge gap, which is determined as the integral

of the product j E, where j is the conduction current density, over the volume, areshown in Fig. 3. These quantities, averaged over a period of the voltage variation, are

indicated in Fig. 3 by the angular brackets. The volumetric force in the discharge gap

arises due to the fact that the electron concentration is two to three orders less than the

ion concentration at all the stages of the discharge. This difference in the concentrations

at the same sources and sinks of ions and electrons [the first two terms in the right

sides of Eqs. (3)] is caused by a significantly higher mobility of electrons, which more

rapidly leave the ionization region. Consequently, the volumetric force is determined

with an accuracy up to a percent by the ion concentration. Therefore, the horizontal

component integrated over the space of the force is negative at the negative polarity of

the exposed electrode, and vice versa. This is also verified in the experiments (Font et

al., 2010). However, the absolute values of the negative force are on average smaller thanthe positive values. In consequence of this fact, the horizontal force, averaged over time,

is positive. This is demonstrated in Fig. 3(a).



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FIG. 3:Horizontal component of the volumetric force integrated over space (a) and the

Joule dissipation (b) at the negative initial polarity of the exposed electrode.

This is explained by the fact that when the vector of the electric field strength is

directed toward the exposed electrode (negative polarity of the exposed electrode), the

ions are generated mainly in its vicinity in the electron avalanches, whose sources are the

secondary electrons, flying away from the electrode surface. At the opposite direction

of the field, the electrons, desorbed from a sufficiently extensive surface patch of the

dielectric and moving toward the electrode, are mainly the source of the ion generation.

The dimensions of the ionization region in the latter case are significantly higher, and

the values of the integral of positive horizontal volumetric force over space are higher as

well.

In contrast to the horizontal volumetric force, the heat release Qin the DBD is some-what higher at the negative polarity of the exposed electrode. This is seen in Fig. 3(b).

The expenditures of the electric power on the generation of the negative horizontal



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force are a negative factor, which significantly reduces the energy efficiency of a DBD-

actuator. Therefore, the ascertainment of the role of negative ions in the modeling of the

DBD in air is important. This is a subject for future investigations.

In order to verify the results obtained in the frame of the boundary problem formu-

lated above and the applied numerical method, the calculations of the discharge powerare implemented in various ways. In this formulation, we consider a layer of an ideal di-

electric without expenditures of energy on its heating. In this case, the supplied electric

power W completely transforms into the heat energy Q in the discharge gap; that is,the following condition must be satisfied:

W = Q , W = 1

T

t+Tt

VeIr dt, Q = 1

T

t+Tt

dt

S

j Eds (20)

Here,Veis the voltage on the exposed electrode, Iris the resultant current in the external

electric circuit,Tis the period of the applied voltage, and Sis the area of the discharge

gap (the entire computational domain in the gas phase).Presented in Fig. 4 is a simplified electric circuit of the DBD-actuator (Enloe et

al., 2004) after the discharge initiation, when alternating electric capacities C1,C2,C3,

appear. Points 1, 2, and 3 symbolically denote the lower grounded electrode, dielectric

surface, and exposed electrode, respectively. Capacity C1 contains all the force lines of

the electric field that initiate at one electrode and terminate at the other one. Capacity

C2arises between the grounded electrode and the dielectric surface when the superficial

discharge is generated on it. Therefore, this capacity includes the force lines that connect

the superficial discharge on the dielectric with the external electrode. The discharge gap

has a variable resistanceRd.

Because the quasi-stationary currents are considered, the resultant currentIr1, which

enters into the external circuit through point 1 (i.e., on the lower electrode), must be

FIG. 4:Simplified electric circuit of the DBD-actuator.



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equal to the resultant current Ir3 at point 3 (on the exposed electrode). The current Ir1is equal to the sum of the displacement currents flowing through capacities C1 andC2,

i.e., to the integral of the displacement current density in the dielectric over the surface

of the grounded electrode, which is calculated through the formula

Ir1= xm

0

D2t

dx, D2 = 2Ey(x, 0, t)

CurrentIr3is equal to the sum of the conduction current flowing through variable resis-tance Rd and of the displacement currents through capacities C1 and C3 that equal the

displacement current on the surface of the exposed electrode, which is calculated from

the side of both the dielectric and the discharge gap. The maximum difference of the

values ofQ, W1and W3, calculated through Eq. (20) correspondingly at Ir1 andIr3does not exceed 5% for the considered values of the parameters of this problem.

The influence of such physical parameters as the coefficients of ion-electron emis-

sion on the exposed electrode and dielectric, recombination rate constant, and the elec-

tron desorption frequency from the dielectric on the main characteristics of the DBD-actuator is evaluated. The results of the calculations are presented in Table 3. Here, Fx,Fy, Q are integrated over space and time-averaged longitudinal and vertical compo-nents of the volumetric force and a heat release. The calculated values of the actuator

energy efficiency, which is determined as the ratio ofFx/Q (Porter et al., 2006) arepresented as well. The boldface font in the table upper part denotes the values of the

parameters that are varied as compared to the first variant of their set.

TABLE 3:Influence of physical parameters on characteristics of DBD-actuator

Variant 1 2 3 4 5

Physical parameters

Secondary emission coefficient on the

dielectricd

0.01 0.05 0.01 0.01 0.01

Secondary emission coefficient on the

electrodem

0.001 0.001 0.005 0.001 0.001

Recombination ratere, cm2s1 107 107 107 105 107

Frequency of desorption of electrons

from the dielectrice, s1

5 103 5 103 5 103 5 103 103

Calculated characteristics

Resultant horizontal force Fx, mN/m 4.13 4.24 5.63 2.98 3.49

Resultant vertical force Fy, mN/m 0.827 0.852 0.970 0.661 0.712

Resultant heat release Q, W/m 12.7 13.0 14.6 9.93 11.2

Efficiency 104Fx/Q s/m 3.25 3.26 3.86 3.00 3.12



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A weak influence of the secondary emission coefficient from the dielectric surface

d agrees with the experimental data (Enloe et al., 2008). This influence is the most

evident at the positive polarity of the exposed electrode, when the current of ions is

directed toward the dielectric. However, in this case, the prominent role in the ionization

processes is played by the electrons that are desorbed from the dielectric surface [thesecond summand on the right side of Eq. (19) for the flow of electrons we], whose flowsignificantly exceeds the flow of secondary electrons (the third summand).

It is seen from Table 3 that the ion-electron emission coefficient on the electrode msignificantly influences both the volumetric force and the energy efficiency. The increase

in this coefficient leads, according to the calculations, to an increase in both of the men-

tioned characteristics of the DBD-actuator. However, the experiments (Hoskinson and

Hershkowitz, 2009b) in which copper, tungsten, and stainless steel were applied as the

electrode material do not prove this result. A possible explanation of the discrepancy of

the results of the experiment and calculations consists in the fact that the replacement

of the material in the experiment does not lead to a significant variation ofm. The

numerical modeling of the work of Bouef et al. (2008) shows a weak influence of the

secondary emission coefficient at the electrode on the volumetric force, but a strong in-

fluence on the Joule dissipation. However, the model of an instantaneous recombination

on the dielectric is applied in the mentioned calculations.

According to the implemented numerical modeling, the increase inm leads to the

ionization amplification at the negative polarity of the exposed electrode. This results

in the fact that the negative values of the horizontal force increase in absolute value.

However, at the same time, a more intensive accumulation of electrons on the dielec-

tric surface occurs during the first half-period of the impressed voltage. Therefore, at

the change of the polarity of the exposed electrode, the flow of the electrons desorbed

from the dielectric significantly rises. This leads to a more significant increase in the ion

concentration and, therefore, in the positive horizontal force. As a result of the super-

position of these two processes, the mean horizontal force increases over a period. Theheat release also increases, but to a lesser degree. As a result, the energy efficiency of

the actuator increases.

The value of the recombination rate constantrealso appreciably influences both the

volumetric force and the dissipation. At this constant increase, the surface density of the

electrons on the dielectric obviously decreases. Therefore, at the positive polarity of the

exposed electrode, the flow of desorbed electrons, concentration of the ions generated

by them, and as a result the horizontal force decrease. The energy efficiency apprecia-

bly decreases as well. A significant influence of the electron recombination rate at the

dielectric surface, detected in this modeling, prejudices the application of the model of

instantaneous recombination.

Finally, the influence of the electron desorption frequency from the dielectric sur-face e is presented to be sufficiently obvious. The decrease in this frequency leads

to the reduction of the flow of desorbed electrons, attenuation of the ionization inten-



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TABLE 4:Influence of the applied voltage frequency on actuator char-

acteristics

Frequencyf, kHz 5 8 10

Resultant horizontal force Fx, mN/m 4.174 6.335 7.640

Resultant vertical force Fy, mN/m 0.7954 1.221 1.495Resultant heat release Q, W/m 12.43 18.74 22.85

Efficiency 104Fx/Q s/m 3.358 3.380 3.344

sity, and reduction of the volumetric force at the positive polarity of the exposed elec-

trode.

The influence of the frequency of the applied voltage on the main parameters of

the DBD-actuator at atmospheric pressure and the voltage amplitude of 7 kV is seen in

Table 4. The longitudinal force and heat release are linear functions of frequency. This

is in agreement with the experimental data (Porter et al., 2006). The energy efficiency

varies slightly.

5. CONCLUSIONS

The boundary problem for the calculation of the characteristics of the dielectric barrier

discharge (DBD) in air near the surface of a dielectric plate was formulated. The bound-

ary conditions for the transfer equations of charged particles and for the equation of

variation of a superficial charge on the dielectric that took into account the processes of

adsorption, desorption, relaxation of charged particles, detachment of electrons from the

negative ions of oxygen, which occurred with a finite rate, and a secondary ion-electron

emission were formulated on the basis of the approximate Maxwellian approach in the

frame of the Boltzmann equation relaxation approximation. This approach was also ap-

plied while formulating the boundary conditions on the electrode. The significant influ-

ence of the coefficient of the secondary ion-electron emission on the exposed electrode,

recombination coefficient, and frequency of the electron desorption at the dielectric on

such integral characteristics of the DBD-actuator as volumetric force and heat release

averaged in time and integrated over space, generated by the actuator was revealed on

the basis of the parametrical calculations of the discharge in nitrogen. The implemented

numerical parametrical investigation pointed out a potentiality of the optimization of

the DBD-actuator characteristics by means of the variation of the mentioned physical

parameters.

ACKNOWLEDGMENTSThis work is supported by the Russian Foundation for Basic Research (Project No. 08-

01-00527) and the International Scientific-Technical Center (Project No. 2633).



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Aleksander P. Kuryachii, Ph.D., Leading Researcher,

TsAGI



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Dmitriy A. Rusyanov, Researcher, TsAGI

Vladimir V. Skvortsov, Ph.D., Senior Researcher, Head of

Sector, TsAGI


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