A fundamental study of the current density distribution of a flowing nonequilibrium plasma

A FUNDAMENTJI..L STUDY OF THE CURRENT DENSITY DISTRIBUTION IN A FLOWING NONEQUILIBRIUM PLASMA

b-y

Chong.S.Kim

..

I

..

•

A FUNDAMENTAL STUDY OF THE CURRENT DENSITY DISTRIBUTION IN A FLOWING NONEQUILIBRIUM PLASMA

by

Chong.S. Kim

ACKNOWLEDG~

The author takes this opportunity to express his gratitude to Dr. G. N. Patterson, the Director of the Institute for Aerospace Studies, for giving

me the opportunity to perform research on this project.

To Dr. S. J.Townsend, the supervisor of this work, special thanks are due for his considerable suggestions and direct contributions given during preparing the entire work. Ris patience and encouragement to the author,

which can make this paper possible, are appreciated so much.

The author has benefited from discussions and comparison with his coll-eagues, Mr.Charles Rersom and Mr. Clement Yeh, and thus he wishes to express

his thanks to them all. This research was partially supported by the National

Research Council under Grant

A-

3950

and by the National Aeronautics and Space Administration Grant NGR 52-026 (012) from the NASA Lewis Research Center.

SUMMARY

In order to determine the current distributions in MBD chfu!nels in non-uniform plasma flow, the electrical conductivity and the Hall parameter are formulated in the form of the 'so-called' second approximation.

A successive iteration procedure is chosen, by which the electron

temperature calculated during successive iterations provides the new spatial distribution of plasma parameters at this step of the iteration. The mnn-erical solutions to the current and electron temperature distributions for uniform conductivity and Hall parameter less than one show diagonally symme-trie patterns in a Faraday type channel. In fact, this re sult can be used as a first iteration for the solution of non-uniform plasma.

1. 2.

3.

TABLE OF CONTENTS NOTATION INTRODUCTION BASIC MH:D-.. EQ,UATIONS 1.1 Species Equations

1.2 Over-all Species Equations

1.3

Ohm's Law for a Mixture Gas

1.4

Energy Balance Equation

1.5

Ionization of a Seeded Gas 1.6 Collision Cross Sections Comments on Results

SOLUTION TO SEDMENTED ELECTRODES

PAGE 1 2 2

3

4

la

11

14

1

5

16

2.1 Isentropic Flow Equation 16

2.2 Current Governing Equation 17

2.3

Boundary Conditions of Faraday Type Generator

1

9

NUMERICAL SOLUTIONS

3.1

Current Stream Function

3.2

Boundary Conditions

3.3

Ohmic Heating Comments on Results CONCLUSIONS 22 22

24

26 REFER ENC ES

2

7

29

32

TABLE (Constants and Conversion Table)

APPENDICES

A.

Derivation of Species Mass Conservation Equation B. Derivation of Species Moment~un Conservation Equation C. Derivation of Species Energy Conservation Equation D. Th.erivation of Equation (2.2.1)

E. Derivation of Equation (2.2.2) F. Derivation of Equation (2.2.2)

G. Derivation of the Current Governing Equation H. Discretization of the Current Governing Equation

ROMAN SYMBOLS a

Ach,

A 0;1,2,3,4 B C s -* E f s F h hx,hy,hi,pj h I j J k 1 L M(x,y) NarATION

Speed of sound (meter/sec)

Integrals defined in Ref. (1)

Coefficients defined by and given by Bickley (Ref.51)

Magnetic intensity vector (Weber/m2)

Random thermal velocity of s-species defined by ~

=

W

-

v

(meter/sec)

s s

Electric field with respect to laboratory (volts/meter)

Effective electric field (see Eg. 1.3.13)(volts/meter) Electric field in co-ordin~~es ~ovi~g w~th mass average gas velocity and given by E

=

E + V x B (volts/meter) Distribution function of s-species in velocity-spatial space

Force on a particle (Newton)

Relative velocity between

a

-

and ~ - species, defined by g~

=

Wa -

w~ (meter/sec)

Plank's constant (Joules.sec), half of height of channel (meter), unit interval between adjacent datum points in the channel (meter)

Defined in Chapter 3 (meter)

Relative heat flux vector, related to

q with

h

(Joules/m2sec)

Total current (amps)

Conductimnncurrent vector defined by j

=

( amps/m2 ) L: n q

V

s s s s

-q

-v

5 -2

Electric current density vector defined by

J

=

L: n q

v

2 s s s s

(amps/m )

Boltzmann's constant (jOUles/oK) Half of insulator length (meter) Half of electrode length (meter) See Equation (2.2.4a)

M m s n _s N(x,y) P = P, P .. ~J p(x,y) qs q q .. ~J

~~

r R (i) s i 0 i 1, i

=

2,

R .

o,~ S s t T v s v

v

s

Mach number (dimensionless) Mass of s-particle (Kg)

Number density of s-species (1/m3) See Equation (2.2.46)

Hydrostatic pressure (Newton/m2) Pressure tensor (Newton/m2) See Equation (2.2.4c)

Electric charge of s-species (Coulomb) 2

Heat Flux vector (Joules/m .sec)

Value relating to Ohmic heating at the point (x.,y.)

(see Sec.3.3) (Coulomb2/m2 )

~

J

Collisional cross section between a- and ~- species

(m2 )

Spatial radius vector (meter) Collisional integral for s-species

Mass transfer collisional integral (Kg/m3.sec) Momentum transfer collisional integral (R (1))

(Newton/m3 ) . s

Energy transfer collisional integral (Joules/m3.sec Resistance of loading circuit for "0"

Resistance of internal plasma for "i" (ohm) Seeding ratio of s-species (dimensionless) Time (second)

o Temperature ( K)

The average velocity of the s-species defined by

v

=

<

W

>

(meter/sec)

s s

Mass average mean velocity in references to the laboratory given by

v

=

~ ~ psys (meter/sec)

Diffusion or drift velocity of s- species, given by

V

=

y - y (equal to

<

ë>

in Ref.27) (meter/sec)

s s s

w

s

z

z

_s GREEK S YMBOLS ex ex s E. 1 E. _ln t E s A TT e p s

Particle velocity of s-species in references to laboratory (meter/sec)

Number of charges on a particle (dimensionless) The ground state degeneracies of s-species

partition function Statistical weight sum-over-states defined by Z s See Appendix E (2L + 1)(2S + 1) s s

Degree of ionization of s-species (dimensionless) Hall parameter (dimensionless)

I

Ratio of specific heats Defined as _1s Defined as /ex~ mikT s s 2 2 (sec /m ) (dimensionless) Energy loss factor of s-species (dimensionless) Ionization energy (eV)

Internal energy of s-species (Joules) Total energy of s-species (Joules) See Appendix E

Ratio of Debye length to the average impact parameter (see Sec.l.6) (dimensionless)

Debye length (meter)

The reduced mass defined by f.l~ mex~/(mex+ m~) Collision frequency between ex and

~

species (sec-l) See Appendix E

Stress tensor (Newton/m2)

Excess charge density (CoU10mb/m

3

)

Mass density (Kg/m

3)

cr

x

w

Scalar electrical conductivity (rnho/meter) Collision time between

a

and ~ species (sec) Hall coefficient (m

3

/rnho. Weber)

Current stream function (Amps/m) Cyclotron frequency (l/sec)

Weighted collision cross section integral between

a

and ~ species (Ref.l)

.

INTRODUCTION

A number of authors (Ref.1-9) have described the non-uniform

conduc-tivity problem. This non-uniformity is thought to be due to the

multitempera-ture plasma model of a partially ionized steady state gas with the Hall effect and applied field effects. Thus these effects on MI1D channels have been

stud-ied. For given initial conditions of specific geometry channel~ a sequence of

effects results , including reducing the output cUTrent expected, increasing the

internal resistance, short-circuiting problem or likely a stability problem

(Ref.11-23). This paper has been prepared with the purpose of bringing the

non-uniform properties into the channel problem in connection with the

pract-ical choices of design parameters (electron cyclotron frequency, gas temperature,

pressure or loading factor).

The present calculation permits heat flux in the momentum transfer

equation to intensify the local heating due to joulean heating which may cause

higher electrical conduction (Ref.24). An approximation is used under which

the electron-atom or -ion mass ratio is negligible and the diffusion velocities of heavy particles are equal to zero. The results for the electrical

conduc-tivity and the Hall parameter are almost the same in the form as developed by

Demetriades and Zhdanov (Ref. 2,3) when the ion concentrat.ion was neglected.

The plasma parameters (conductivity, the Hall parameter, electron density and the collision cross sections) are thus obtained in terms of electron

tempera-ture. They are essential to describe the non-uniform plasma due to

non-equi-parti tion ionization or nonequilibrium electron heating.

If no plasma parameter gradients in the duct are assumed as we will

do for the calculation of current distribution~ the interaction between current

flow and plasma parameter variations involving the velocity variation cannot

arise. Contrary to these ass~~ptions, then, the gas must be compressible such that the current flow causes plasma parameter variations which in t~ITn alter

the current distribution. Apparently, an MHJ) problem such as this can be

solved only by considering a set of consistent plasma parameters and current

distribution equations. Herein, there are a couple of approximation methods

to get a satisfactory answer based on the pre-mentioned detailed physical

phenomena. A few authors have identified a reasonably simple approximation

procedure with the aim of simplifying the problems without losing apparent

significances. Iterative procedures to the solution of the current

distribu-tion in two-dimenaional MBD channels with finite electrode segmentation are

thus suggested (Ref.25). In order to solve more clearly the current

distri-bution itself, the simpler case is sclved wherein the uniform Hall parameter

and conductivity are assumed. The Extrapolated Liebmann method for Laplace

equation is used. However, the calculations are restricted to values of Hall

parameter subjected to the condition under which the numerical solution for

the current distribution converges over the channel, that is, Hall parameter

less than or equal to one. This condition does not seem to be followed from

the physical critical value of Hall parameter which brings on the stability

problem in non-uniform conduction. But the remainder of the problem which

in1:olves formulating the iterative computation of successive states to account

for the non-uniform Joulean heating is left to be analyzed. What we obtain

for current and electron distribution in this paper is suggested to be used

Chapter 1.

Basic MBD Equations Sec. lol Species Equations

From the considerations of Maxwell conservation equation and apply-ing it to each species to obtain energy and momentum equations, species

equa-tions desired to describe the plasma can be obtained. To begin with, there are

the Boltzmann equation for each species in terms of particle velocity to get

the transfer equation. Let us use the Boltzmann equation without any arguments

of the force law whether the collision can be treated as binary or not, of s

dt

+

W

s

V-

r f s +

F

m s

V-

_w f

=

(

of ) s

dt

c (1.1.1)

where (Of/ot)c represents the rate of change of the velocity distribution function of the particles in question due to collisions with all other

parti-cles. The force law per -unit mass on a charged particle in the magnetic field

will be given by

F

m s (Ë +

W

x

Ë) s

when E is independent of the particle actual velocity. ~

(W )

is assumed to be

the each-species velocity dependent quantity. Multiplyin~ s(l.l.l) by

~

(W )

and integrating it over the velocity space, we obtain so-called 'the s s equatlon of change',

~t

(n

<~

> )

+ V-:-(n <

W

cp

»

-alJ s s r s s s n s [ E .

<

V-w s ~

>

+

<

(W

x Ë). V-

~

>

]

r

cp

(~)

d

W

s w s

Jw

s at c (1.1.2)

where

<

>

denotes the integration operator over the velocity space.

Using (1.1.2), the equations for species mass, momentum and energy

conservation can now be deri ved by letting ~ equal m , mW. (component of

veloCity in i-coordinate) E in turn. Then ~s the re~ult~,Sl

s (p

V )

s s

R

(0) s (1.1.3) (see Appendix A)

~

(p

-ij) + V [p (v V +V v)] + V- .(pV v) + V-at s s r ' s s s r s r p s [- :: - - (1) - q n E* + V x BJ

=

R s s s s (see Appendix B) (1.1.4) and

..

l m v2 +

1

k T + <E.

»

J+

'ïJ-.

q +

'ïJ- (-21 m v2y n ) + 'ïJ- [

2 S 2 s lS r s r s s s r

n m

v( v.

V )

J +

'ïJ-

(v.

P ) -

Ë (q n

·v

)

= R (2)

s s s r s s s s s (1.1.5)

where<E.

>

represents the integration of the internal energy of s-species

over th~Svelocity space, (see Appendix

C).

Sec.l.2 Over-All Species Equation~

Since the t9tal mass must be conserved, the sum of the mass transfer

collision integral Rs ~ 0) should be zero. Thus if we sum the mass equat.ions of all components of species, we obtain

+ 'ïJ- • ( p v)

=

0

r (102.1)

where~p is the total mass density defined as p

=

~ p .

s s

To obtain the over-all momentum equation, the plasma is assumed to

be quasineutral, that is, q

= ~

q

=

O. (This assumption is generally valid

s s

since in the most cases of practical interest in MHD problems, the characteristic

length of plasma is generally greater than the Debye length. Thus, at any point of space, the force to bind the charged particles together is high enough

to make the plasma ~uas~neutral, Ref.26). The sum of the momentum transfer

collision integral Rs(l) is zero since the net change in momentum for all particles due to collisions is zero when only the elastic collisions are

con-sidered. That is, conservation of momentura during collisions. Here,the

conduction current

J

is defined from a kinetic point of view as

J

=

~ _s q n _{S s}

V

_S

in terms of the mean v~locit;y: or diffusion velocity.

of the total pressure

P=

I:

P ,

the overall moment.um s s

d

(

-)

(

-p)

db

p v + 'ïJ

_r

p v v +

By using the definition

equation will be

e

where p is the excess net electron charge density due to deviation from the

charge neutrality. This equation can be re-written by using over-all mass

conservation equation as,

p

D'v

+ 'ïJ-

-

P

=

P e E

*

+ j x B

Dt r (1.2.2)

e -

e-when we can neglect the body force p E and the current flow due to p v,

peË* is neglected and it is obvious that the total electric current

J

then is

equal to the conduction current,

J

=

J.

This is the case of most practical interest as we prementioned (Ref.27).

It should be noted that this over-all momentum equation is not

sufficient to describe the motion of plasma until the total conduction current

3

and the pressure tensor Pare described elsewhere. Although it forces us to

use the energy equation, this involves adding an energy term only to be given

by the next higher moment equation. To make a closed set of equations, alter-nate procedures seem to be required along with some additional assumptions.

Ohm's law will be introduced, so th at naturally only both the species and

over-all momentum equations would be used to obtain

J

related to any of the

other variables. In fact,

3

frequently is used instead of Je due to the small

mass of electron during this procedure (see Chapter

1,

Sec.l.3). However, unfortunately, Ohm's law turns out to involve the heat flux terms described

only by considering higher order momentum terms. In the next section this argument is resolved by using a closed set of equations obtained by different methods but proved to be identical to each other for a weakly ionized gas. One

of the procedures is inverting a large matrix for a gas mixture and the other is the successive approximation method.

Before finishing this section, it should be pointed out that in any

of the cases to obtain Ohm's law, one more variable, electron temperature,

should be given elsewhere to make the set of equations complete. Only for this

purpose will the energy equation be introduced. One more factor to be mentioned

is due to the inelastic collisions th at occur between molecules with internal

degrees of freedom. Because, clearly, mass and momentum should be conserved even in this case, although kinetic energy is no longer conserved, the

equa-ti ons

(1.2.1)

and (1.2~2) would not bé changed.

Section 1.3 Ohm' s Law for a Mixture Gas

It has been noted that in deriving Ohm's law, from the equationof motion is the basic starting point. The equation

(

1.

1.4)

is

V- •

P

-

q n

r s s s

This will be rewritten as follows,

Dv

Dt

+ Cl """" at ( p , s s = V )

+

V-r [ -E*+V - xB -] -_ R- (1) s s

[Q

(v

V

+ V v) ] s s s +

V-. r

=

P _s - q _s n _s [Ë* + 'ij _s x Ë] R

-

_s

(1)

By separating the pressure tensor'

p

'

into a diagonal part P sI wh~re I is the

unit tensor, and an off-diagonal

pa~t,

that is, a viscous part, T_S the above

~s

Dv + 'V- P _{+ 'V-} T + è) ( ,

V )

+ 'V-

[

,.

(v 'ij +

V

v)]

dt

_"

~s Dt r s r s s s r s s

[Ë*

-

x Ë

]

- (1) (1.3.1) - q n + V

₌

R s S I S S

Now it is generally valid in all but cases of extreme nonuniform iha~_the ~radient of the viscous part af the partial pressure tensors,

'V

-

.

~ can De neglected comparing to the gradient of hydrostatic pressure ~ P.~ In fact, for a çollisiog dominated plasma, it is generally the case

jliatSve/ve « 1 and similarly

IT

lip

« 1 (Ref.l), so that the gradient of

T can be neglected. As well, i~ allebut cases of extreme nonuniformaties, we

c~n assume that the macroscopie parameters af the plasma change only slightly

within distances of the order of the mean free path and during times ~f the order of the collision times in the plasma. Then,.we can neglect

Cé

11 )

è)/è)t

end the contributions of the diffusion of 'Vf

(g

v 'ij ) and_ 'V- (~V vS2Sto the momentum. Under this condition, the pre~sure d~fin~d as

P'

~ ~ s<s6'e'> where

ë'

=

W -

v (Ref.27), is identical to

·

P.

As a matter of fac~, th~ co~ditions

fO~

theSelectron-fluid to be collision

d~inated

(this is just the present case) are

-1 /

v _ee T « 1 ,

0 .

v

_e v _ee -1 / L

«

1

~here To,L are the characteristic values for macroscopic change, (in Ref.28, v

v-

l /

L

~ 0.01 to 0.02). Some results (Ref.8) show that even the

ioni-zäti5fi relaxation time and length of the plasma are respectively orders of

100 ~sec and 1 cm in most cases of practical interest compared with the

colli-sion-time of the order of 10-5 sec. and mean free path of order of 10-5 cm. Consequently, the equation (1.3.1) becomes

DY

+ 'V- P _ q n

[Ë*

+ 'ij x

B ]

=

ft

(1)

Ps Dt r s s s s s

andthe equation (1.2.2) is rewritten as; Dv+'Vp

P

Dt

(1.3.2)

(1.3.3)

Eliminating llv/Dt from (1.3.2) and (1.3.3), we obtain the fol10wing algebraic system Ps ( 'V Ps - ~ 'V p) + , ~ _e ( s p - q n ) Ë*

r-

s s -

~s

- qsns

ti

s x

B

+

r

j x B

=

R _s (1)

Ir we neglect the excess charge _P e (same as the plasma is assumed to be e1ectrica11y neutral),

Ps Ps

B

=

ft

(1) ( 'VP

"iJ:

'VP ) ~ns 'ij x

B

+

if"

j x (1.3.4 )

In the first approximation of collision integral Rs(l), the magnitude of the im-pulse transmitted by collisions between partieles of the s-kind with those of other components is taken to be proportional_t9 the respective differences of the macroscopie component veloeities. Then Rs~l) is

- (1)

R s r nmm s s r m + m s r Inserting Rs (1) into (1-11), we get

P v sr

( V

r

- V )

s Ps \lP 2.\lP-qn E*

-

_{qs n}

V

x :B::+

-

j x

B

=L

s P s s Thus, for electron partiele

\lP e

Pe

- V'P + en

P e

For ion partiele

E*

+ en e s s Pe

V

x

B

+ -e P P r j x

B

=

R

(1) e n mm nsmsmr u : ... 12 m~-+-- m-lflS -',""'r .3 r \lP I PI en I VI x B +

P

j x

B

=

R

(1) 1

Using both equations to eliminate

9 P,

we finally get

v

(V -V )

sr r s \lP e Pe Pe - \lP + en (1 + - )

E*

+ en P I e p e ~ ) x B-

=

R-(l) VI e

V

e

when the plasma is assumed to be electrically neutral, that is, ne

=

~ since

n m

e e m e

_«1,

_{if we neglect the terms including}

the factor of Pe

Ip

I , this equation is reduced to, \lP + en

E*

+ en

V

x

B

=

R(l)

e e e e e

Jo = en

11

.

e e e

and the colli sion integral

R

(1) becomes

e -(1) _~ (V

V )

R

-

n m v e e e r er r e -

-

nm ~ v

V

e e r er e

.

me j ~ v e r er (1.3.5)

Thus, (1.3.5) becomes

-*

\lP + en E e e j x Ë Re-arranging it to obtain,

-*

E + - -1 _en e \lP e m e 2 n e e m e e j j ~ r

We call it "first approximation" Ohm' s law. v er v er + __ 1_ nne e j x Ë

Let the effective electric field

E'

Ë* + - -1 'i7P then

where and

x

1 cr

E'

=

(n e)-l e 1 cr j +

X

j

x

Ë

respectively. ene e (1.3.6)

The parameters cr and X are called scalar electric conductivity and

Hall coefficient.

Using the "13 moment" approximation (Ref.l,3,29) in diffusion

equation which aEe(t~en to be proportional to the relative thermal flux of the

particles, give R 1) as s ~(l)

4

_~

_Cv

_V

₎ 16 _~ R _s

=

n I-L_sr v + n n I-L_sr 3 r s sr r s 3 r s r

~/l)

A(2) ₍ h r h s ₎ sr sr _trPr

_t

s

P

s for electron, -(1)

.

4

m e h A(2) -1 j ~ e ~ R v nm

_p

T e 3 e r er e e rre er er e for ion, -(1)

.

4

m e h A(2) -1 j e RI _{3 e} Vel + nm _{e e P el} Tel e

Here, we also neglected the terms multiplied by hl

jm

_{l and assumed that n} = n_l ,

3 =-3

,

since under the conditions here considereä in the diffusion

stat~

the Debyeeradius is much smaller than the characteristic lengths of the system

by

h

-

-e

where

The electron relative heat flux vector

h

was found to be gi~en

e T* 2 T *2 À

[î>

-

- -e e p x Ë + e e 2 *2 2 2 *2 e e m _l+w e _l+w e T m T e e e e e

(î>

x Ë) x

B

e ]

2

k * À P T e 2 m e e e 2 m

î>

_-

\7 T e j k v e e n e 0 e T*

was defined by, e T* e and v by, 0 Thus h e P e -

0.4

v 0 -

-*2 T e e v o

-2

2 knm e e -1 _{+ 2.9} L.

Kfe

A(5) T ee eK L.

Kfe

A(2) _T-1 eK eK k T*

[YT

-

-

e m e e m e e (YT x B) x Ë -e T*2 e T* e 2 2 l+w T* e m v e 0 e k en e (\7 T ) e v j + k

(3

x

B )

x Ë ] 2 x

B

+ _{- 2 -}e m e T* 0 e 2 *2 n e 1+ w T e e

Inserting this expression into

R

(1) and replacing in (1.3.5) by

e '

this result, we obtain

4

m ₊

₂

\7 P + en Ë* - j x Ë e j L: v mk * T v [ \7T e e 3 e r er 2 e e 0 e 1'* 2 T *2 e e _{(\7 T ) x Ë +} e e _{(\7 T x Ë)} x B

-

₂ *2 ~ 2 *2 m e _l+w e e l+w T m T e e e e e T*2 m v v T* ev e 0 j + o. e

(3

x Ë) 0 e

(3

x Ë)xË] 2 T*2 1+w2 T*2 k en kn kn m e e l+w e e e e e e

and rearrange it to obtain; T* 2

5

Ë*+ 1 'VP

_-

_2"

k v T*

[\iT -

- -e e (?r ) x B + e

""2

en e e 0 e e m _1+w2T*2 e e e m e 0 e m

(~

2

'2 1 '\IT x B) x13 ] _{- - 2} e v - V T*) j + - - ( e ₃ t 2 0 e n e n e e e 2 _T*2 2 _T*3 1 +

2

v

5

v 0 e )

(3

x 13) 0 e

(3

x

i3)

i3

2 -.*2

'2

_{2 T*2} x 2 _l+w m n e e 1+ w e e e e where v

t

=

~r ver' we get 'second' approximation of Ohm's law.

Let the effective electric field E'

1 Ë* + -en e (\7 T ) x

i3

+ e e '\lp e 2 2 -m e

2

2 T *2 e 2 l+w T e k e *2 e v T* o e e m e (\7 Te x

i3)

x

i3

]

l+w 2 T*2 e e T*2 . e -l+w 2 e

The 'second approximation' Ohm's law to describe the nonuniform

plasma in electric and magnetic fields is now including not only pressure gradients but also the temperature gradients and is given by

Ë'=

-

1 _cr j + X

(3

x

i3)

+ E

(3

x

i3)

x

i3

(1.3.7) where the scalar electric conductivity is defined as

2 1 4 meVt

₂

m e v 0 T*

=

cr ₃ 2 2 2 n e n e e (1.3.8) e e

the Hall coefficient as

2 _T*2 1

5

v 0 e X

- -

_{n e} +

_2"

_{n e 2 T*2} e e l+w (1.3.9) e e

and the ion slip coefficient without considerations of ion-concentration as

2 _*3

5

v 0 T e (1.3.10) E

2"

n m 2 _T*2 e e l+w e e "*_2 T e

In the expres sion of the effective electrical field

Ë',

the gradient

interests, since it involves the electron number density gradients, which were sometimes found to be of the same magnitude of the applied field at

certain place of MBD channel, (Ref.30). Since the term proportional to the

electron temperature gradient, however, accounts for only about a few per

cent of the electron current flow, one may neglect this on purpose for

simplification of the problem in a given MBD state.

Section 1.4 Energy Balance Equation

The energy balance equation is for describing the transfer energy

into the plasma or the required energy to keep the plasma in the motion of

the desirable state thermally and kinetically and in the balance between

the interacting particles through collisions.

The calculation for energy transfer integrals generally has been

taken to be the following,

m m s r

Cm

+

m

)2

s r v k (T - T ) sr s r

when we neglect the inelastic collisions between particles. As we might see from this expres si on, the ratio of energy transfer byelastic collisions

between the atom and the ion species is much greater than that between the

electron and these particles, because of the mass factor. Consequently,

we expect that collisions between ions and atoms would tend to maintain their

temperature at almost the same value, while the electron temperature may

greatly exceed their temperature. Because of this, the plasma is expected

to be described by the two temperatures, gas temperature and an electron

temperature different than this. This is nonuniform process of plasma

(Ref.31).

From Eq. (1.1.5) for electron, replacing v by

v

+ V , we obtain

e e

d

(1;..., . l + p v v+

1

dt

"2 '"'e e e 2

lkn

T + n

<E.

>

v]

+

v-2 e e e ie r kn T + n

<

E.

»

+

v-

[

(~

p v2 + e e e ie r e

-

₊

_v-

_[~ 2 -= ] + qe _r Pe v

V

_e

v-

(

-

(v.

V ))

+

v-

(v

P

)

Ë 0

J

+

E

(en v) R (2) Pe v r e r e e e

The term representing the derivative with respect to time can be

neglected in the case of steady state. By considering the very small ~ss

of the electron and the sm~ll contribution of en

v

and stress tensor Te'

we can eliminate several terms, so that finally,e the electron energy

equation becomes

v-

_{r e e e}

[

(-23 kn T + n

<E.

»

v ]

+

V- qe + V- (p

v)

1 ie r r e E*

J =

R (2) e (1.4.1)

In the lowest approximation

- E

j = R (2)

e (1.4.2)

This joule heating and collision energy loss equation will then de(~)mine the electron temperature of the plasma with the expression of

Re •

The point concerning inelastic collisions might be argued, since

the species int egrated contribution of inelastic collision frequencies of

Maxv.Tellian electrons to the electron energy 10ss is no long er identically

zero and may cause important effects. However , in the mass conservation and momentum equations the inelastic effect is almost negligible insofar as the

electron--electron collision frequency is much greater than the inelastic collision frequency of electrons (Ref. 31, 32).

Under the steady-state condition, the energy lost by the electrons

per unit time in elastic collisions with heavy particles is

R (2) el 3 kn L: m (T - ) -1 e T T e r e r er m r

1

_{1<"..n L: Ei (T -T )} -1 (1.4.3) T 2 e r r,el e r er

where Ei _{r ,el} is defined as Or el _, 2

me/r~

and for argon A,el Ei

~

2 7 . x 10- 5 .

No account has been taken of the average energy transfer through

inelastic collisions as, namely, the collisional transitions between two

discrete states and between the continuum and a discrete state and the radiative transitions between the continu'U...'11 and a discrete state (Ref.32,34).

Sec. 1.5 Ionization of a Seeded Gas

Finally~ lfle should derive the equation relating to T to the nUInber

e

density to make the set of equations complete.

Wnen the diffusion of the electrons through the ionized gas is rapid enough, tbe average thermal energy of electron (3kT /2) will be considenably

above that of heavy particles UkT/2). (See Chapt. e1.4, under such conditions, the heavy particles can be considered immobile) . Usually this causes the

nonuniformity- of plasma through the process as follows. This assumption can

be proved valid so long as the energy loss factor is less than unity (Ref.31).

Then the random velocity of the electron should be always much greater than

its directed velocity even in a strong electric field. This provides the

postulation for thermodynamical non-equilibrium for process which relax over

longer t imes than that required for the velocity distribution of electron

to come to equilibrium. Thus the process of ionization is completely

dom-inated by the electron thermal energy, since in this case energy transferred between free electrons in a Maxwell distribution is well transferred very readily between free electrons and valence electrons if the electron number d_ensity is not toa 10w (Ref .17,36) . This is the case wherein the free electrons are strongly coupled thermally to the valence electrons of the atom rather than

to the atom itself so as to be in Maxwell distribution between electrons themselves. This seems likely to be valid locally rather than in any point of space inside of a weakly ionized plasma. Thus, a much larger degree of ionization results locally and this non-equilibrium process relating T to the nurrber density can be described along with pre-obtained equations e

(Refs. 13, 20

&

37).

Now then, the degree of ionization for a seeded plasma can be obtained on the assumption that volume ionization and recombinatioo process are in equilibrium at the electron temperature so that the electrons can be distributed as a stat:i:onary Maxwell di stribution of velocities. Then we may use the Saha equations which are, for seeding species (Refs. 14, 34)

ngt-+ n Z + Z ' ( m k T

)3/

2

E

E )

_ _ _ _ e

Sz

e e 2 e exp - kiTS

ns s /- 27T1l e

and for the inert carrier gas

==

3/

2

) exp

~~

)

In these equations, E· and El' are the first ionization potentials of the

lS A

seed and the carrier gas, respectively.

Here we neglect the increasing of the degree of ionization due to the magnetic field. The criteria is the ratio of the quanta of the electron gyration motion~ we

=

A1eB/me to the magnitude of the thermal energy 2 kT e (Ref.35). Since in the case of the magnetic intensity B of 1.7 Webers/m and of the electron temperature of 20000K the ratio~ welk T is around

1.1 x 10- 3 , the elevation of ionization due to the magnetic field in the Saha equation can be negligible, so that under the above condition this form of the Saha equation always is applicable.

The total nuillber density n will be

==n +n +2n A s e

The last equation was given by assuming the electrical neutrality of the plasma Seeding ratio S is defined by

s S s n + 2n + s s n

and the degrees of ionization by

ex

s n -+ n +

s s

and by

a

A

nA+ _{for carrier gas.} nA + n_A+

Combining these definitions, we obtain

ns+

na

_s S _s (1.5.1a) n A+ n (1-8 ) s

a

A (1. 5 .lb) n

=

n (1-8 )

a

(1. 5 .lc) s s s nA n (1-8 ) (l-a_{s A}) (1. 5 .ld) and n

=

n [( 1-8 )

a

+

a

8

]

(1. 5 .le) e s A s s

Inserting (1.5.1)'s into the 8aha equati ons,

a

s

_[a

_{8 + (1-8 )}

_a

A ] a

-

.

l-a s s s n s

a

A

[a

8 + (1-8 )

a

]

b l-a A s s s A n

where a, bare given by a, b

Z. _{J+ e} Z

(m

_e k T ) _e

3

/

2

Zj 2 7T~2

expO (

-These algebraic equations for

a

_A and

as

can be solved by iterating method to the desirable accuracy. For the numerical scheme, if we them for

a

_A and

as'

then

a s 1 2 8 s a n numerical rearrange

Because of several time's higher ionizatiofi potentialof the carrier gas thanlthat of the seed, we can carry out this numerical procedure as

follows. For the first approximation,

a

_A may be set to zero and

a

than be calculated. This value of

a

is then used to calculate

a

_A.

The

p~ocess

is repeated with the new value ~btained for a~. It turned out that this proce

-dure converged quite rapidly over the consl§erably wide range of electron

were required for argon plasma seeded with potassium.

Sec.

1.6

Collision Cross Sections

The collision cross section for momentum transfer between electrons

and ions is 2 2 'iel TT

[

e lel

]

ln

,.J"

2 +

1

2 4TT E _~eI el 0 2 2 TT

[

e

1

J

ln

,.J,,2

+

1

(1.6.1)

-4

2 7TE o k T e el

where "el is the ratio of the Debye length

"v

to the impact parameter for a

90

0 _{collision of an electron of speed}

₃

_{kTe and defined by}

[

4

7T E

"el = 3

'TI

~2

0

and the Debye screened length

'TI

is introduced to prevent the colli sion cross

section integral from diverging and defined by

2 2

+

n~;I

)"

_e n e _ _ (

1:....

+

k E Te

o

~I

)

There has been some discussion (Ref.

8,31)

concerning the introduction

of "D into these calculations. However, when ln A» 1 which may arise in

many applications, the result based on the screened Coulomb potential and

on the unscreened potential together with a Debye length cut-off have been

commented on by Liboff, R. L.* . They yield the same value for the transport

properties and in both cases the collision parameter is limited to distances

of the order of "D:

According to the cal}ulations of weighted momentum transfer colli sion

cross sections (Ref.3), Aer(2 and A_eI

(5)

are given by

A (2)

6

*

_-

_0.6

Cel - 1 = el

₅

and A (5)

_L

_-

6

* Bel =

1.3

el 2

₅

.'

where C*I and B*I are defined by many authors including Zhdanov and Cowling

(Ref.3,e 38). e

So long as the electron temperature increases not too much so that the condition under which lR A is not too high is satisfied, all the series

of expres si ons for collisions between electrons and ions is quite satisfactory

even when we neglect the higher approximation (Ref.6).

The differential momentQ~ transfer cross section between the

electron-argon at om is characterized by the Ramsauer-Townsend minimum at an

elec-tron energy near 0.3 eV (Ref.39). Because of the significance of this effect,

the determinatio~ has been difficult. A few available theoretical computations

and experimental data (Ref.40-43:including the works of Frost and Phelps) have

been compared. To about 0.5 eV of electron energy the determination from the

modified effective range formulae of O'Malley (Ref.39) was used to calculate the electron-argon momentum transfer collision cross section. That is,

4TT ( • 2 . 2 )

~ Sln ~ + Sln ~l

k 0

where a is the Bohr radius

o 1 2 a o sin~ o k 1.70 - 3.13

EV

2

_{+ 0.92}

_EV

_ln

_EV

_{+ 1.23}

_EV

and 1 0.6026 EV2 - 0.589

EV

(1.6.2)

In these expressions

EV

denotes the electron kinetic energy in

electron volts. In more accurate approximation (Ref.39),

4

TT k2

'L,OO (L + 1) sin2(

~1

-

~1

+ 1 ) .

1=0

where k2

=

2m

E/~2

is the wave number of incident electron particles,

~1

repre-sents the phase shifts of 1-wave and 1 is the orbital angular quantum nUmber.

Kruger and Viegas (Ref.5) calculated the effective mean cross sections of argon

and potassium to obtain the conductivity and Hall parameter.

Assuming solid elastic sphere models of argon and potassium for

higher momentum transfer integrals, we used B~A

=

B~K

=

3/4

C~A

=

C~K

=

5/6 .. ~ (Refs.2,3,9).

For the electron-potassium case, we used Nichol's results (Ref.14)

and recalculated the cross section from the figure in terms of electron

tempera-ture around the region of interest.

COMMENTS ON RESULTS

In our calculations, we neglect the electron-electron

4

1

1 4 TT e n k 2 T T 2" (vee

=J;

v _{ei' 3 3} e ln ₃ e 1 ) m v e n 2" e e e 1

ion-ion (Ref.44) and ion-atom collision integrals (Ref.26, v. ln { mT. e 1 m.T 1 e ) 2"v _en ).

Along with increasing the electron temperature, the differences of X and cr from the first approximation remain greater than zero, the differences de-creasing up to the electron temperature of around 22000K. For electron te~­ eratures above 22000K, the deviations are so remarkable that at around 3000 K they differ by some 20% from the first approximation (Fig.2-4). To be sure, at electron temperatures above 22000K, electron-electron collision effects which we neglected should be taken into account (Refs. 8, 17, 31, 36) while ron-ion and ion-neutral collision still remain at a negligible contribution.

In spite of these corrections which may reduce the deviation closer to the

first approximation, our computations ensure us that the errors in cr and X

seeming to be within a few per cent at the equilibrium state (Refs. 1,3)

mis-lead us into underestimating the electrical conductivity and Hall parameter for non-equilibrium cases where electron temperature might be elevated to as much as twice the gas temperature. On the other hand, below a gas temperature of 20000K (electron number density of around 3 x 1018jm3 ), violation of the electron distribution assumed to be Maxwellian should be naturally considered

so that electron-electron energy exchange can be properly taken into account

compared with electron-atom energy exchange. This consideration may give correct number density so as to determine more reasonable ca.lculation of cr and

t3.

Number denRity at electron temperatures between 22000K and 30000K are near

5

x 1018 mj and

5

x 1020jm3 (Fig.l). This indicates that at the low electron density corresponding to the elect~on temperature below 22000K, the significance of the correction to first approximation is not so important due to the i~validity of the Saha equation following from the violation of the electron distribution assumed to be Maxwellian due to the smallness of

electron number density. Also our formulation can be used as a reasonably

well explainable one so long as the electron-electron collisional effect is

properly considered; well measured experimental data and theoretical results

of various kinds of collision cross sections are available in terms of electron temperature. The relations of the conductivity to the current den-sity are given in_Fig.6. Since, in the present paper, we neglect all the inelastic collision including radiation energy loss, our results show the underestimating of current densities for the region where the radiation loss cannot be neglected. However, at the high current density for which radiation

is no longer tPe dominating effect to energy equation, the results agree

fairly well with those referred to. It enables us to expect that from the energy balance equation, the electron temperature was slightly overestimated at low current density, while underestimating at high values of current den-sity (Fig.7) can be explained by the effective energy 10s8 factor. With

greater and the energy loss factor which we assumed constant increases; this increases the degree of ionization sharply and causes the increasing of left hand terms in energy balance equation. Consequently the electron temperature can be higher than what we calculated (Ref.31). Within this error mainly due either to radiation energy or to inelastic collision, this figure represents the relation between input energy into plasma by joulean heating and thermal main~enance thropgh collisions which cause the electron temperature to increase higher than the gas temperature. It should be mentioned, however, that the energy loss procedure with the invalidity of the Saha equation for low tempera-ture (Refs. 17,

36

,

45) is quite difficult to describe completely satisfactorily.

Chapter 2. Solution to Segmented Electrodes

Sec. 2.1 Isentropic Flow Equations

The static gas temperature _{Too and static pressure} Poo were referred to the stag_{nation temperature To} and stagnation pressure Po by the relation through isentropic expansion, which are good approximations for small degree of ionization

(Refs. 27, 34). P o P 00 T o

T

₀₀ == ( 1 + 1

+

-L )'-1

where M is Mach number of the flow near the entrance of the ge~erator when the gas velocity is defined by u == M 00 )' k T 00 n L: (2.) m s n s

and the isentropic sound speed a in the absence of electron heating as a2==(dPjdP

_{s ).}

By using the flow continu~ty equation~

n u == n u

o 0 00 00

n can be written in terms of Po, To together wUh the ideal gas law 00 P s n 00 == n k T s s n [1 + 0 Po 1-1 M2]

2""

I

k To 1 )' -1 1 )'-1 (2.1.1)

so that the value of n in Chapter 1 is now replaced by n ,which is the nuffiber density at the entrance of channel. 00

Sec.2.2 Current Governing Equation

When we neglect the ion slip coefficient, the effective field

E'

becomes

E'

= 1

J

+

X

(J

x Ë)

If we take the rotation of the above equation, we arrive at

v

x

E'

J

+!

0" (v x J) + ('lX) x (J x Ë) +

X (B.V)

J -

X (j.V) B (2.2.1)

with V.B

=

°

and the condition of continuity of electric current density V.j

=

°

in the steady state problem (Appendix

D).

On the other hand, from the definition of effective electric field in Chapter

1,

Sec.

1

.

3,

rotation of effective electric field ensues

V k E'

=

(B.V) ü - B(V.ü) +

(v

P ) + (va) x (V T ) e e -

(V~)

x(V T x B) +

~

B

(if

T ) -

~

(B.V)W + e e e ( V1:)x [(VI' x B) x Ë] + 1: (B.V) (V T x B) e e (2.2.2)

(see Appendix E,F)

From V.

J

= 0, the current stream functiorr~ is introduced such that

J

= ( -

~

, 0).

Here, the current lines will then be parallel to the tangential directions on the surface defined by the current stream function ~.

For the simplified case of small magnetic Reynolds number, magnetic field in the fluid hardly differs from the external field, defined by (O,O,B) and B is constant through the channel (Ref.48). more, the velocity profile is assumed to be ü

=

(u ,u

,0).

Then it vantageous to go over to two dimensional problems ~hete

?j;

=

~ (x, y) CT

=

0" (x, y) X

=

X (x, y) when the Ë is Further-is

ad-(or and ~

=

~ (x, y) u

=

u (x, y) x x u = u (x, y) y y )

Equating (2.2.1) to (2.2.2) and applying the above mentioned quanti-ties to them,we arrive at

~ ~

+ M(x,y)

~

+ N (x,y)

~

=

p(x,y) (2.2.3)

where the functions of M(x,y), N(x,y) and p(x,y) are defined as

M

(x

,y ) = ()

[~x

(

~

)

~

(~)

]

(2.2.4a) N(x,y) (2.2-.. 4b)

[

(

dU

x)

(dU

y)

d

l

1 ) (

dP

e)

d

(1 )

p(x,y)

=

()

-

B

dx

-B

dy

+

dx\en e

dy -

dy

ene

~~)

+

(~) ~)- (~)

c:f

)-B

(~~)~)

-B~

)~)

+ B<

~)

+

B< (

~

)

+

B 2

(~

X

~

)

+

B 2

(~

) (

rl ) ]

(see Appendix G) (2.2.4c)

In these expressions ~ is the Hall parameter and defined as

~ = () X. B.

The equations (2.2.3) is a non~homogeneous elliptic partial diff-erential equation of a more general type with linear boundary conditions. Making some assumptions for conductivity and Hall parameter or velo city profiles leads (2.2.3) to the Poisson Equation or simply to the Laplace equation. The convergence of the scheme, which deals with the nonuniform plasma together with Saha's equation, energy balance equation and current governing equation (2.2.3), depends on whether the solution of (2.2.3) is

stable or not with respect to electron temper~ture. D. A. Oliver et al

(Ref.15) concluded that when the inequality ~

<

24/25 is satisfied, the

solution of the potential equation, conjugate equation to the current

electrons occurs in a magnetic field. Others (Refs. 13, 17, 23, 49) argued the stability problem related to (2.2.3). In the present paper, however, we will not go into the problero of the stability limits in detail.

Sec. 2.3 Boundary Conditions of Faraday Type Generator

The Faraday type generator has the electrode segments insulated,

long periodic structure (consequently, end effects will be neglected) and

oppositè segments connected by individual loads as shown by Fig.8. The current direction in the central part of the channel will then tend to be perpendicular to the magnetic field direction and also to the flow of the plasma. The effect of segmentation can also be explained by the Hall field that prevents the electron drift motion and also by a conducting electrode segment parallel to this field that will clearly have a short-circuiting action on the field. Thus astrong deviation from the uniform current distribution in the central portion of the channel can be expected near the electrode. Because of this, near the electrode reasonably strong effects due to electron temperature gradients and due to electron pressure gradients should occur. Furthermore, if boundary layer effects are introduced in order to describe the real channel flow in general, it is essential to consider

those gradient terms together with a wall temperature different than the

gas temperature. However, for the most simple case, neglecting these effects, the effective field Ë' involves only the real applied field

Ë,

the additional field introduced from changing the stationary coordinate system into the systero rooving with plasma and the gradient terms of velocity. Thus equation (2.2.3) becomes

if-?jJ

+ M(x,y)

~

+ N(x,y)

~

=

P' (x,y) (2.3.1)

where

_[

_Ou

Ou ]

P' (x,y)

=

-

Ber

dX

x +

7/-.

Without ion;slip considerations, eq. (1. 3.7) is

Ë'

= 1 J +

X

(3" x :8)

er

where the effective electric field Ë' E* by neglecting velocity gradients. The projections onto the x- and y- axis are respectively,

E' ! J _{+ X B} J

x er x y

E' ! J _{X B J}

Y er y x

Defining ~ such that ~

=

er

X

B as before, we have

er E' J + f3 J

x x y

er E' = J f3 J

, or J = x J Y (J" (E + u B) x Y J x + ~ J

Y

(J" (E - u B)

=

J - ~ J Y x Y x

Solving these equations for Jx and j'y, we obtain

(J" (u B + E - ~ E + ~ u B) 1 + ~2 Y x Y x (J" (~u B + ~E + E - u B) ~2 1 + Y x Y x (2.3.2a) (2.3.2b) (2.3.3a) (2.3.3b)

Since on the wallof the electrode Uy of the boundary conditions is

Ex 0, from (2.3.2a), one

J _x + ~ J _y

=

°

In terms of stream function, it becomes

-

~

_di

2flJ;

=

°

on the electrode (2.3.4a).

Another wall condition is on the insulator. Assuming a perfect

in-sulator, we have J

=

J

=

°

for the condition on the insulator. If we

re-write it in terms 3f stfeam

function

~

~ = Const. on the insulator (2.3.4b)

For such a configuration as shown in Fig.8, the periodicity of the

conductors and insulators in a period of (2L + 21) requires

~ (x + 21 + 21~ y)

=

~(x,y) + I

Y (2.3.5)

where Iy is the transverse current flowing through the electrode. It can be

shown that the specification of Iy with Ix

=

0, in the case of the Faraday

type generator, determines a unique distribution of current within a gas (Ref.15). Moreover, the one-dimensional transverse current Iy, which gives the output current by multiplying it by the electrode width in the-B direction will be given by the cha~~el loading factor K defined as K

=

Ey/ux B. Thus, it is a known value, or a quantity which can be determined from outside of the channel. In fact, the internal current relation is from (2.3.3a) and

(2.3.3b), in y direction

~ J - J

x Y (J" (1 - K) u x B (2.3.6)

The output current I or the total load current over one period of

I _w

J

2L+l

L+l

(~ J - J ) d ;l!:

x Y

Since there is no connection between the upstream end of the electrode and the downstream end, the integration of Jx over the electrode length should be zero when ~ is assumed to be constant or when we can neglect the

corre-lations of the integration. Thus I becomes

On the other hand, 1 2L+l

wJ

L+l 2L+l -

-

wJ

L+l - w I Y J d~ Y

~

d

X

the R.H.S. of (2.3.6) is w 2L+l

J

cr (l-K) uB dx x L+l

<

cr

>

<

u

>

(l-K) BLW x

when we neglect the correlations. Thus,

1

Y

<cr> < u >

x (K-l) BL

(2.3.8)

where <

>

represents the average value of the variations over the electrode. The relation between loading current land either the loading resist-ance Ro or internal resistresist-ance Ri of the plasma cannot be determined explicitly unless equipotential surface or current filament is solved over the channel.

As a matter of fact, whether one chooses to solve the problem as described by equipotential surface or as described by current stream surface does not matter as their descriptions are identical to each other. However, the loading resis-tance, Ro, pertaining to the loading factor, K, is expressed most conveniently in terms of the potential, ~, as

R b:.~/I

6.

cp

= -

J

E

di'

=

J

(Vcp)

d

x

'

x' x'

In this expression the primed axis is along the eigenrays of the resistivity tensor (Ref.50), such that 6.

cp

is the integration of Ë between two electrodes connected through an external resistance along the line nor mal to the equi-potentials (Ref.54,55) . Moreover, the internal resistance pertaining to joule losses is expressed in terms of potential difference between these electrodes measured in a coordinate system moving with the plasma as

R.

l 6.

cp'

/ I

6.

cp'

f

Ë

'

d h

₌

f

J

d

ft

cr

"h h

and the vector

ft

is a distance vector lying along the current density vector, such that 6.

cp'

is the integration of J/cr between these electrodes independent of the path (Refs. 11,12).

Although quite extensive work (Refs. 11,12,23,54) has been done for different geometries and for various Hall parameters, neither R nor R. for a nonuniform plasma simplified to a one-dimensional prob~em is ~asy tolde-termine uniquely, and thus further comparisons with experimental studies seem to be required.

Chapter 3. Numerical Solutions Sec. 3.1 Current Stream Function

The current governing equation, namely, the equation for the current stream function ~ is as we developed, from eq. (2.3.1),

where,

,y2

'I/J + M (x,y)

~

+ N (x,y)

~

[

dUx + P' (x,y)

=

-

B cr

dx

P' (x,y)

For the first approximation, if we neglect velocity gradients,

if

'I/J + M(x,y)

~

+ N (x,y)

~

=

0 (3.1.1)

Using the central difference method to discretize the differential expressions, we have

.,.----,--:-__ 2 1 ,J 1,J _ 1,J 1- ,J

+

(

'Ij;. +1 . -'Ij;.. 'Ij; . . - 'Ij;. 1 .)

h. + h. 1 h. h. 1 2 ('Ij;i,j+1 - 'lj;i,j h + h. 1 h. 1 1- 1 1- j J- J 'Ij; . . -_1,J 'Ij; . . _1,J-1 )

₊

1

~

M h _{. 1} 2 \ 1 . +1 2,J . J-'Ij;. +1 . -'Ij;. . 1 ,J 1,J M h . + 1-2,J . 1 . 1 'Ij; . . - 'Ij;. 1 _{1,J 1- ,J} .) h. _1-1 'Ij;. _1,J . - 'Ij; . . _1,J-1 )

=

0 h. 1 J-(3.1.2) where and hj = Y_{j + 1 -} Y_j

'Ij; i, j

=

'Ij; (Xi' Y j )

N. . 1 = N(:it 1· , ~ (YJ' + Y_J'+l )) 1,J+2 N. . 1 N(~ .. , ~ (y. 1 + y.)) 1,J-2 1 J- J (see Appendix H) Rearranging (3.1.2) to obtain

1 [2

+

h.

'

h.+ h. 1 1 1 1-M . l . l+?,J 2 ] 'lj;i+1,j +

~.

J [ 2 + h.+ h . 1 J' J-+ N . . 1 1,J-? 2h. 1 J- ] 'Ij; . . + 1,J M . l . l+?,J 2h. 1

1

[:--2---:---~1 h.+ h. 1 -1- 1 1-M.l . . l-?"J 2h. 1 1-M. 1 . 1-2,J 2

l .

~

N. . 1

l

1,J-? 'Ij;

=

0 _(3.1.3)

Equation (3.1.3) is a standard five points central-difference approx-imation for different rectangular mesh sizes. Since the calcula~ion involves too many independent parameters to give general sOlutions, we limit ourselves to the case where the plasma is uniform and consequently uniform conductivity

and Hall parameter are assumed. In this case, since M(x,y) == N(x,y)= 0,

Then (3.1.3) reduces to 1 ?/J'+l . 1 2J ₊ h.+ h. 1 h. 1- 1 1 + 1

?/J.

1 . _ _ ~ _ _ 1- 2_J + h. + h. 1 h. 1 1 1- 1 -1 h.+ h. 1 J J-1 h.+ h. 1 J

J-Laplace's equation, and thus (3. 1. 3 )

?/J

_1,J

.

'+1

_( 1 + 1

)?/J

h . J h.h. 1 1 1- h.h. 1 J J- i,j

?/J. .

1 1 2 J-h. 1 J-== 0 (3.1.4) becomes

For the same step sizes in which h h. _{h. l' h} h. == _{h. l'}

x 1 1- Y J J-1 ?/J'+l . + 1

?/J.

'+1 -

(~+

--?/J . .

1 ) 2h 2 1 ,J 2h 2 1,J h 2 h 2 1,J x + 1 2h 2 x y x 1

?/J.

1 . + 2 1- ,J 2h

o

Y

and in the simpler approach of the square net where h

x

it becomes,

?/J.

+1 . + 7jJ. . +1 - 4

?/J.

.

+

?/J.

1 .

+?/J. .

1

1 ,J 1,J 1,J 1- ,J

1,J-Sec. 3.2 Boundary Conditions

y

o

h

Y h

(3.1.5)

From (2.3.4a) and (2.3.4b) the boundary conditions on the surface

of the electrodes and of the insulators are

?/J

CONST.

Owing to linearity and to periodicity, we put

?/J

insulator and?/J == I on the right one.

As far as the surfaces of the electrodes are concerned, the boundary

conditions should be rewritten in the forward or backward difference

approx-imation, such that

and

h. 1

..k..:!:. ( )

'Ij! . . 1

=

h 'Ij; . . +1 - 'Ij;. . + 'Ij!. .

l,J- j l,J l,J l,J - [ ( hh

j

-1)

13.

+1. .

. l 2,J

l

( 'Ij! . +1 . - 'Ij!. .)

+(

hh j -1 )

13.

1 . ( 'Ij! . . - 'Ij!. 1 .)

1

l ,J l,J i-l l-2",J l,J l- ,J

'Ij!. _l,J ·+1

on the lower electrode (3.2.1a)

,....h.

----Lh ('Ij;.. -'Ij!. . 1) + 'Ij!. .

. 1 l,J l,J- l,J

J-('Ij! . +1 . - 'Ij!. .) + (hhj )

13.

1 . ('Ij! . . -'Ij!. 1 .)

1

l ,J l,J i-l l-2",J l,J l- ,J

on the upper electrode (3.2.1b)

The conditions of (3.2.1a) and (3.2.1b) correspond to (3.1.3) .. As

we assumed when (3.1.4) was obtained, in the case of a uniform plasma,

(3.2.1) is written as follows, h. 1 ..k..:!:.h - ('Ij;.. +1 -'Ij!. .) . l,J l,J J

( h.

1)

- ?/J.

0".) +

F

('Ij! . . -+"J i-l l,J - 'Ij;. l -1 .) ,J

1

on the lower electrode (3.2.2a)

('Ij!. . -'Ij;. 1 .)

1

l,J l- ,J

on the upper electrode (3.2.2b) and related to (3.1.5)

'Ij; _l,J-. . 1

=

'Ij;. _l,J . +1 - 13 ('Ij!. _l +1 _,J . -'Ij!. _l-1 _,J . )

?/J.

_1,J '+1

=?/J

_1,J-

. .

1

+

~

(?/J'+l .-_{1 ,J}

?/J.

_{1- ,J} 1 .)

on the upper electrode (3.2.3b)

Sec. 3.3 OHMIC Heating

As in Chapter 2, Sec.2.2, the current density vector J(J ,J ,0) was

written in terms of current stream function

?/J

as x y

In finite difference language, in the central difference method, these will be expressed by

J

I

-~

-

-

1

(?/J.

'+1

-?/J . .

1) x . i,j 2h. 1,J 1,J-1,j _J (3.3.1a) J _{y . .}

I

=

~

1

(-?/J.

1

.+

?/J'+l .) i,j 2h. 1- ,J 1 ,J 1,J 1 -2

Thus, the ohmic heating defined in (1.4.2) as J

/rr

at the mesh point

(i,j) is then, proportional to q . . given by

1,J

q. .

1,J [ J x _i,j ]2 +

[J

Y _i,j

Corresponding to (3.1.5) and (3.2.3), q. . becomes,

1,J

q . . =

1,J

1 2 2

[(?/J.

₁ +1 . _,J

-?/J.

_{1- ,J} 1 .) +

(?/J. .

_1,J +1

-?/J. .

_1,J-1) ]

where h is the unit interval between adjacent datum points defined at Sec.3.1.

However, along the electrode surface this expression must be re-written

from the boundary conditions on it as

since 2

(?/J.

+1 .

-?/J.

1 .) 1 ,J 1- ,J (1 +

~2)

J 2 from (2.3.4a) y (3.3.4)

Special care must be taken for calculations of q . . along the insulator surface.