Density profiles of corona discharge plasmas

6 October

1997

PHYSICS

ELSEVIER

Physics Letters A 234 (1997)

LETTERS

A

372-378

Density profiles of corona discharge plasmas Han S. Uhm”, Woong M. Lee b a Naval Surface Warfare CenteK Carderock Division, 9500 MacArthur Boulevard, West Bethesda, MD 20817-5700, USA b Department of Chemistry Ajou University, 5 Wonchon-Dons, Paldul-Gu, Suwon 442-749. South Korea Received 6 March

1997; revised manuscript received 3 June 1997; accepted Communicated by M. Porkolab

for publication

9 June 1997

Abstract Corona discharge properties in the reactor chamber are investigated, assuming that a specified voltage profile V(r) is fed through the inner conductor. The analytical description is based on the electron moment equation. Defining the plasma breakdown parameter u = vR,p, plasma is generated for a high voltage pulse satisfying u > uc. where uc is the critical breakdown parameter defined by the geometrical configuration. Here, u is in units of a million volts per m per atm, and R, is the outer conductor radius. It is found that the plasma density profile generated inside the reactor chamber depends very sensitively on the system parameters. A small change of a physical parameter can easily lead to a density change of one order of magnitude. @ 1997 Elsevier Science B.V. PACS: 52.80.H~;

52.50.Dg;

52.80.Tn;

52.25.Jm

Corona discharge systems are studied in connection with their applications to NO, and SO, reduction [ l-4 J from emission gas. The reason for corona discharge application to NO, and SO, reduction is the electron-to-gas temperature ratio, which is very high in a corona discharge plasma. Thus, even with a high degree of excitation, the carrier gas remains moderately cool, making the plasma generated by the corona discharge especially suitable for gas purification, and highly efficient in terms of transferring source power into promoting chemical reactions. There is a growing literature [S-8] on fluid model simulations of the corona discharge plasmas. The majority of these studies investigated plasma properties by numerically obtaining the electron density profiles from the cold-fluid discharge model. However, it is often difficult to obtain physical insights from numerical data generated by fluid model simulations. In this article, we there0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00494-5

fore develop a theory that provides simple analytical expressions for the plasma discharge and electron density. In particular, we find the criterion for corona discharge by making use of the ionization and electron attachment cross sections of air. These analytical expressions may be useful to design an efficient corona discharge system. The reactor chamber, where corona discharge plasmas are generated, consists of two coaxial conducting cylinders with length D. The inner conductor with radius Z?c is usually connected to a high voltage and the outer conductor with radius R, is grounded for safety. Because the length D of a typical reactor chamber is much larger than the chamber radius R, (i.e., R, < D), the system analysis can be described in terms of the radial coordinate R. A high voltage pulse with voltage V(t) arrives on the inner conductor at time t = 0. Once the electric field E(t) generated from

H.S. Uhm. WM. Lee/Physics

the high voltage pulse at the inner conductor is higher than the breakdown field Ed, plasmas start to be generated at the inner conductor and propagate toward the outer conductor. Plasmas are generated from impact ionization of neutrals by electrons accelerated by the high electric field at the ionization front. A typical electron density inside a plasma generated by the corona discharge is very high. The electric conductivity in this high density plasma is so large that the electrical resistance through the plasma can be neglected for the purpose of this article. Assuming that the ionization front is located on R = Ri at time t, the electric field E is equal to zero and the electric potential 4(R) = V(t) is flat in the range of R satisfying Ro < R < Ri. Due to the high voltage V(f) , there is an electrical charge accumulation at the ionization front Ri. Introducing the symbol 9 that represents the line charge density at the ionization front, the electric field E(R) in the region of Ri < R < R, is expressed as

where r is the normalized radial coordinate defined by r = R/R,. Note from Eq. ( 1) that the electric field decreases monotonic~ly from its peak value at the ionization front to 2q/R,, as the radial coordinate approaches the outer conductor. The line charge density 4 along the ionization front is related to the potential cb(ri, t) = V(t) by

(2) where the lower cases of r represent radii normalized by the outer conductor radius R,. This article includes dynamic behavior of the corona discharge plasmas in a reactor chamber. Measurement of the reactor voltage is easier than anything else, and the voltage profile is the input parameter in most of the present experimental applications. It is therefore assumed that a specified voltage profile is fed through the inner conductor. Plasma prope~ies inside the reactor chamber can be described by numerically solving the Boltzmann, moment and the Poisson equations for electron, ion and negative-ion species. This numerical calculation is very complicated because of ten coupled differential equations. However, we will obtain a simple analytical expression for the plasma

Lztrers A 234 (1997) 372-378

313

density and its generation by making use of quasineutrality of the plasma region. We therefore assume that a negative ion with electron density is the same as a positive ion density in the plasma region. Ionization of neutrals by electrons and the negative-ion generation mechanism are very important in the ionization front and beyond. Plasmas are generated by the electron impact ionization where high energy electrons collide and ionize neutrals, creating secondary electrons and ions. Meanwhile, medium energy electrons are captured by oxygen molecules, which are converted into negative ions. Thus, the ionization produces additional electrons, whereas the negative-ion generation depletes electron density. Plasma is generated only when the ionization process dominates the electron attachment process. Both the ionization and electron attachment are initiated by electrons. Therefore, the electron density plays a pivotal role in the plasma behavior. The simple analytical expression for the plasma generation in this article will provide important scaling laws in the corona discharge system. The analytical description is based on the electron moment equation,

an,

a

udk) 7’ i3-i’

=

(a - P)uene,

where n, is the electron density, ue is the velocity of the electron fluid element, and cx and @ are the coefficients associated with ionization and attachment cross sections of neutrals by electrons. The ionization coefficient cr is given by LY= 3.5 x 103p exp( - 1.65 x 105p/E),

(4)

from experimental data [51 for an ionization cross section of air. In Eq. (4), E is the electric field defined in Eqs. ( I ) and (2), and p is the chamber pressure. For convenience, the electric field E is in units of V/cm and the pressure p is in units of atmospheres. Similarly, the attachment coefficient p in Eq. (3) is given by 151 p = 15~ exp( -2.5

x 104p/E).

The validity of the coefficient 1.25 x lo3 < E/p < 2 x 105.

(5) combination

N - fl is (6)

314

H.S. Uhm, WM. Lee/Physics

Letten

A 234 (1997) 372-378

However, Eqs. (4) and (5) understate by about a factor of two in the extremely high field, which is close to the upper limit of Eq. (6). For convenience in the subsequent analysis, we introduce the breakdown parameter u defined by

the plasma will never be generated in the entire region of space, including ri < r < 1. The necessary and sufficient condition in Eq. (12) not to generate plasma beyond the ionization front ri is also equivalently expressed as

E 1 -=_--_ v(t) R,p rlnri P

U < U, = -2.57ri

= -104$$,

1

(7)

where u is in units of a million volts per m per atm, and ri is the ionization front radius. Substituting Eq. (7) into Eqs. (4) and (5) gives 103p[exp(16.5G)

a-/?=3.5x -4.3

x 10w3exp (2.5+)],

where In ri has a negative value because r; is less than unity. Similarly, the validity in Eq. (6) is expressed as 0.125 < --

u(t) rlnrj

< 20.

As mentioned in the beginning of this article, the coefficient combination of cy - p must be positive in order to generate plasma. Otherwise, the electron attachment process dominates over the ionization and any available electrons will eventually be attached to oxygen molecules to form negative ions. All the free electrons will disappear due to the electron attachment process. From the inequality LY> p and Eq. (8), we obtain r: > 0.6776’

(10)

for plasma generation. We note that the electric field inside the plasma region characterized by r < ri is zero, effectively meaning that u = 0 and that the righthand side of Eq. ( 10) is unity. In this context, Eq. ( 10) is never satisfied in the plasma region of r < ri because ri is always less than unity. A new electron-ion pair is not generated in the plasma region. The left-hand side r-i of Eq. ( 10) is always less than the ri th power Of li, i.e., rr < r,?

(11)

in the range of r satisfying ri < I < 1, because ri is less than unity. Therefore, if the electric potential at the ionization front is low enough to satisfy r: < 0.6778”,

(12)

In ri,

(13)

where lnri is negative because ri is less than unity. In other words, plasma is never generated in the region of r > ri, if the electric potential V(t) at the ionization front is low enough to satisfy Eq. ( 13). Shown in Fig. 1 are plots of the rith power of ri and the critical breakdown parameter U, versus the ionization front radius ri obtained from Eq. ( 13). The rith power of ri has its minimum value of exp( - 1/e) = 0.692 at ri = l/3, where e = 2.718 is the base of the natural logarithm. The rith power of the ionization radius ri decreases from unity, reaches its minimum of 0.692, and then increases to unity, as the ionization front radius ri increases from zero to unity. Meanwhile, the critical breakdown parameter uc defined in Eq. (13) increases from zero, reaches its maximum value of 0.945 at r-i= 0.368, and then decreases to zero, as the ionization front radius increases from zero to unity. Plasma generation by ionization never occurs whenever the breakdown parameter u(t) at the ionization front is less than its critical value U, in Fig. 1. On the other hand, plasma is generated by ionization in the entire region if the breakdown parameter u(t) at the ionization front is always larger than 0.945. Electrons under a high voltage satisfying u > U, get enough energy to ionize neutrals, generating new secondary electrons. On the other hand, electrons under a low voltage satisfying u < uc get a small energy gain, which is enough only for activating the electron attachment by oxygen molecules. Free electrons disappear in the process of electron attachment. We now rewrite the validity of Eq. (9) by o.ooo34U < J-[ < 0.951U,

(14)

after carrying out a straightforward algebraic manipulation. The low validity limit of Eq. (14) is meaningless in the range of plasma generation because the parameter r-i in this range must satisfy Eq. ( 10). Comparing Eq. ( 14) with Fig. 1, we note that there is a possible validity violation of the upper limit of Eq. (14) in the region ri close to zero or close to unity, because

H.S. Uhm, WM. Lee/Physics

tion front ri, and then stops when the ionization front arrives at the position. Therefore, Eq. (3) is meaningful only for the range of r satisfying ri < r < 1. However, the range of Eq. (3) is further restricted due to Eq. ( lo), where the electron attachment dominates over the ionization process. Defining the range of Ar = r - ri, where the ionization process dominates over the electron attachment, we can obtain

1

0.5

0

375

Letters A 234 (1997) 372-378

I

0

Ar -=r I,

0.5

1

'i

Fig. I. Plots of the rith power of ri and the critical breakdown parameter uC versus the ionization front radius Ti obtained from Eq. (13).

be close to unity in these regions. However, in the practical application of corona discharge devices, the beginning and end of the voltage pulse are usually very small and therefore the breakdown parameter u for these voltages is in reality almost zero, ensuring unity of the right-hand side in Eq. ( 14). In this context, the validity of Eq. (8)) which is restricted by Eq. ( 14), is well justified for most practical applications of the corona discharge system. The electric field inside the plasma defined in the region of ra < r < r, is zero. A non-zero electric field beyond the ionization front of ri exists. The velocity u, in Eq. (3) of the electron fluid eiement is defined by the electron mobility and is expressed as rf can

u - UC (17)

u

from Eqs. ( 10) and ( 13) after a straightforward algebraic manipulation. The breakdown parameter u in most applications of the corona discharge system is very close to its critical value of u,. In this context, Ar is a small fraction of the position r. A change of the electric field defined in Eq. ( 1I is negligibly small in this limited range. The change of the electron fluid velocity defined in Eq. ( 15) is also small in this range. On the other hand, the electron density changes drastically in this range. Therefore, Eq. (3) is approximated by

(18) where the term proportional to n,( av,/i?R) is neglected. Substituting Eq. (8) into Eq. ( 18) and identifying dR,/dt = n,, Eq. ( 18) is expressed as glnn,=3Sx

103RRC[exp(16S+)

(15) -4.3 where oC in the right-hand side is the electron collision frequency by neutrals. The electron velocity inside the plasma is zero. Eq. (3) inside plasma is therefore expressed as d --_)I,

at

=

0

for ra < r < ri, thereby indicating no density change. In reality, the plasma density decreases slowly in this region due to the recombination process. The typical pulse length of the corona discharge system is less than 10 ,US. The plasma reduction by recombination during this limited period is negligibly small [ 91. The plasma generation process in a given position of Yis in progress when this position is beyond the ioniza-

X IO-3exp

(2.&$)],

where the normalized radial coordinate is defined by r = R/R, in Eq. ( 1) and the conductor radius R, is in units of cm. Recognizing dR/dt = U, and dRi/dt = OF, we find the relation dr = (u,/ur)dri. Here UF is the ionization front velocity. changing the integration variable from r to ri, Eq. (19) is rewritten =3.5x

103p$[exp

(16S%)

x IO-“exp (2.5+)],

(20)

where the form factor F defined by F = UF/V, depends on various physical

conditions

including

neutral

gas

376

H.S. Uhm, KM. Lee/Physics Letters A 234 (1997) 372-378

species, water containment and field configuration, etc. A typical value of the form factor is the square root of the ratio of the electron to ion mobilities and is given by F x 0.15 for air. As shown in Eq. (20), the radial profile of the plasma density is independent of the form factor F. On the other hand, its intensity is normalized with respect to F. Measurement of the ionization front velocity UF is pivotally important in the dete~ination of the plasma density. We, therefore, currently investigate the properties of the ionization front velocity in the cylindrical geometry and hope that results of this study can be reported elsewhere in the near future. We also remind the reader that the pressure p in Eq. (20) is in units of atmospheres. Eq. (20) is meaningful in the region where the ionization process dominates over the electron attachment; that is, Yi > Y, = 0,6776+,

(21)

as mentioned in Eq. ( 18). Therefore, the low limit of the integral range rt is the largest value between r, in Eq. (21) and r-0 of the inner conductor radius. Eq. (20) is integrated to be

1.5

1 U 0.5

0 0

0.4 r

Fig. 2. Plots of the electron density function ln(n,/nn) F/R,p and the breakdown parameter u versus the radial position r obtained from Eq. (22) for the normalized inner conductor radius rn = 0.02, the outer conductor radius Rc = 5 cm, and a saw-tooth breakdown parameter u in Eq. (23).

r obtained from Eq. (22) for the normalized inner conductor radius ra = 0.02. The breakdown parameter u(r) is defined by a saw-tooth function, u(r) = 1.2 1 -exp [

In

!5

=:3.5x

( n0 > - 4.3 x

103~~~[exp(16.~~)

0.8

(

-

0.025 + r rs

I3

0.02 < r < 0.45,

r1.

10w3exp (2.5$)]

dri.

(22)

=u(O.45)exp( r > 0.45,

Here, the initial electron density no is defined by the eIectron density of the position r at the time when the ionization front is located on Yi = rL. The radius RC and pressure p are in units of cm and atmospheres, respectively. Both the voltage V and the ionization front radius ri are evaluated in terms of time t. Therefore, the voltage V can be determined in terms of the ionization front radius ri. We thus describe the bre~down parameter u in terms of the ionization front ri. A plasma density profile at a specified time t can be obtained by describing the breakdown parameter u in terms of ti corresponding to the ionization front ri, and integrating Eq. (22) over ri from r~ to r when r is in the range of the ioni~tion process defined in Eq. ( 17). For the purpose of the present article, we obtain the density profile right after application of a full voltage pulse to the reactor chamber. Shown in Fig. 2 are plots of the electron density profile ln(&/nc)l;‘lR,p and the breakdown parameter u versus the radial position

-F), (23)

for Fig. 2. AIthough the breakdown parameter u( r) in Eq. (23) is an arbitr~ily chosen equation, a sawtooth profile is a typical voltage profile in the present experiments. The pulse rise rs in Fig. 2 is given by rs = 0.1. The breakdown parameter u has finite values at the inner conductor radius in Fig. 2, to initiate the gas breakdown at the inner conductor surface. A numerical number of one-tenth or above in the vertical line for the density function means a large number multiplication of the density due to the log scale and a small value of the ratio F/R,. The electron density peaks near the inner conductor, decreases to a minimum, increases and then decreases again to zero as the radial position increases from 0.02 to unity. The density profile is strongly related to the difference Au = u - uC at the position r. For example, the density peak near the inner conductor results from the relatively large value

U.S. Uhm. WM. Lee/Phvsics

of AU, combined with a small value for the radial position r at the inner conductor, although the breakdown parameter u is not large. This is the manifestation of a high electric field at the inner conductor. We note from Eq. (23) and Fig. 2 that there is a pulse tail beyond r = 0.45. However, the plasma density reduces to zero near r = 0.48. This means that the breakdown parameter II is less than its critical value uc beyond r = 0.48, as predicted from Eq. (13) and Fig. 1. Therefore, the ionization movement may stop at ri = 0.48. The remaining portion of the pulse cannot ionize neutrals, and the ionization front does not move outward anymore. Fig. 3 presents plots of the density function ln(n,/no)F/R,p versus the radial coordinate r obtained from Eqs. (22) and (23) for different pulse rises of rs = 0.09,O. 1 and 0.11, and parameters otherwise identical to Fig. 3. The voltage pulse rises faster for smaller rise time. The difference AU = u - uc of the breakdown parameter near the inner conductor increases, as the rise time decreases. Therefore, the shorter the pulse rise time, the higher the density peak near the inner conductor. The plasma density profile changes drastically even for a small change of the pulse rise time. We therefore conclude from Figs. 2 and 3 that the plasma density profile generated inside the reactor chamber is very sensitive to the system parameters. A small change of a physical parameter can easily lead to a density change of one order of magnitude. In summary, corona discharge properties in the reactor chamber were investigated, assuming that a specified voltage profile V(t) is fed through the inner conductor. Plasmas are generated by the electron impact ionization where high energy electrons collide and ionize neutrals, creating secondary electrons and ions. The analytical description was based on the electron moment equation. Defining the plasma breakdown parameter II = ~~~~, plasma is generated for a high voltage pulse satisfying u > uc, where K, is the critical breakdown parameter defined by the geometrical configuration. Otherwise, the electron attachment process at the ionization front dominates over the ionization process and plasma is not generated. Even for u > I(<, a sustained ionization process occurs only for a small range near the ionization front when the breakdown parameter u is close to its critical value II,. 11 was found that the plasma density profile gen-

Letters A 234 (1997) 372-378

1.2 ,

0

0.1

0.2

r

0.3

0.4

Fig. 3. Plots of the density function In(n,/nu)F/&p radial coordinate r obtained from F!.qs. (22)

0.5 versus the

and (3-3) for

different

pulse rises of rs = 0.09. 0.1 and 0.1 I. and parameters otherwise identical to Fig. 2.

erated inside the reactor chamber depends very sensitively on the system parameters. A small change of a physical parameter can easily lead to a density change of one order of magnitude. The plasma density profile in Fig. 2 agrees remarkably well with simulation data [ IO], which might be published elsewhere. However, there is no clean and decisive experiment in the cylindrical geometry, which verifies the analytical results presented in this Letter. We are currently fahrieating an experimental device to measure the plasma density in the radial coordinate. We appreciate the useful discussions with Dr. S.T. Chun. This research was supported by the Independent Research Fund at the Naval Surface Warfare Center. This research was also supported by the alternate energy program of the Republic of Korea and by the Institute for Advanced Engineering.

References S. Masuda and H. Nakao, IEEE Trans. Industry Application 26 ( 1990) 374. J.S. Chang. I?A. Lawless and T. Yamamoto. Plasma Sci. 19 (1991)

IEEE

Trans.

1 IS?.

M. Higashi, S. Uchda. N. Suzuki and K. Fujii. IEEE Trans. Plasma Sci. 20

T.

( 1992) I,

Fujii, R. Gobbo and M. Application 29 ( 1993) 98.

Rea,

IEEE

R.S. Sigmond. J. Appl. Phys. 56 ( 1984)

Trans.

Industry

1355.

S.K. Dhali and P.F. Williams, J. Appl. Phys. 62 ( 1987) 4694.

KY.

37%

Uhm, WM. ~e~P~~~i~s

[ 7 I A. Dengra, J. BaHesteros, M.A. Hemandez and V. Colomer, J. Appl. Phys. 68 (1990)

5507.

18 1 S. Ganesh, A. Rajabooshanam and S.K. Dhali, J. Appl. Phys. 72 (1992)

3957.

Letters A 234 {I9971 372-378 L91 E. Hinnov and J. Hirschberg, Phys. Rev. 12.5 ( 1962)

I 101

S.T. Chun, private communication

( 1997).

795