Arcs in cesium seeded noble gases resulting from a magnetically induced electric field

Arcs in cesium seeded noble gases resulting from a

magnetically induced electric field

Citation for published version (APA):

Stefanov, B., Veefkind, A., & Zarkova, L. (1987). Arcs in cesium seeded noble gases resulting from a magnetically induced electric field. (EUT report. E, Fac. of Electrical Engineering; Vol. 87-E-177). Eindhoven University of Technology.

Document status and date: Published: 01/01/1987

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Noble Gases Resulting

from a Magnetically

I nduced Electric Field

by

B. Siefanov

A. Veefkind

L. Zarkova

EUT Report 87 -E-177

ISBN 90-6144-177-3 July 1987

ISSN 0167- 9708

Eindhoven University of Technology Research Reports

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineering Eindhoven The Netherlands

ARCS IN CESIUM SEEDED NOBLE GASES RESULTING FROM A MAGNETICALLY INDUCED ELECTRIC FIELD

by

B. Stefanov A. Veefkind L. Zarkova

EUT Report 87-E-177 ISBN 90-6144-177-3

Eindhoven July 1987

Arcs in cesium seeded noble gases resulting from a magnetically induced electric field / by B. Stefanov, A. Veefkind, L. Zarkova. -Eindhoven: University of Technology, Faculty of Electrical

Engineering. - Fig., tab. - (EUT report, ISSN 0167-9708; 87-E-177) Met lit. apg., reg.

ISBN 90-6144-177-3

S1S0 661.5 UDC 537.5 NUG1 812

CONTENTS

Abstract

1. Introduction

2. Thermodynamic and kinetic considerations

3. Governing equations

4. Transport coefficients

5. Solution of the equations

6. Discussion of the results

6.1. Cesium seeded argon 6.2. Other gases 7. Conclusions Acknowledgement References Table 1 Figures 1 2 5 7

to

11 11 14 15 16 17 19 20

/

ARCS IN CESIUM SEEDED NOBLE GASES RESULTING FROM

A MAGNETICALLY INDUCED ELECTRIC FIELD

B. Stefanov*), A. Veefkind**) and L. Zarkova*)

Abstract

The filaments which form the substructure in the moving discharges in noble gas magnetohydrodynamic generators are considered as free arcs. The spatial distributions of the important plasma quantities are calculated by

solving the appropriate balance equations. Solutions are obtained for

cesium seeded argon, helium and xenon as well as for pure argon. The

solu-tions yield characteristic radial dimensions which are in agreement with known experimental values. The effects of variations of the important arc parameters are discussed for the case of cesium seeded argon. It appears that the effects of the parameter variations can be explained to a large

extend by a simplified expression for the radial dimension of the arc. Deviation from local thermodynamic equilibrium is only significant in the outer region of the arc and has practically no influence on the arc

prop-ert ies.

1. Introduction

It has been experimentally demonstrated that the discharge structure in noble gas magnetohydrodynamic (MHD) generators is strongly nonuniform. The discharge can be characterized as a large number of arcs moving with ve-locl.ties approximately equal to the flow velocity [1,2). More detailed experiments [3) revealed that the arcs have a filament substructure with

*) Institute of Electronics of the Bulgarian Academy of Sciences, Boulevard Lenin 72, SOFIA 1184, Bulgaria.

**) Division Electrical Energy Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB EINDHOVEN, The Netherlands.

-z-typical cross sectional dimensions of the filaments between 10-~ and 10- 3 m. Analysis of the balance equations learns that the two dominating

terms in the electron energy equation are the Joule heating and the

ther-mal conduction [4]. When a parabolic electron energy temperature profile is applied to the simplified electron energy equation an estimation can be derived for the radial dimension of the filament in terms of the electron temperature and the effective electric field [5]. By applying a partial local thermodynamic equilibrium (P.L.T.E.) model to the center of a fila-ment in a cesium seeded argon plasma it has been found that for usual MHD generator conditions the electron density is practically equal to the value which follows from local thermodynamic equilibrium (L. T.E.) rela-tionships [6].

In this work the spatial distribution of the electron temperature and density and the gas temperature in a filament will be analyzed. To this aim the filaments will be considered as free arcs. The results are obtain-ed by solving the appropiate balance equations under the following assump-ti005:

- In case of cesium seeded plasmas the noble gas ionization is neglected and only single ionization of the cesium is assumed. For the case of

pure argon only single ionization is considered.

- The plasma can be discribed by a P.L-T.E. model where only the ground state population of cesium deviates from the Saha equilibrium value.

- The pressure and the seed fraction (ratio of the cesium ion and neutral concentration to the noble gas concentration) are uniform in space.

- A steady state condition is considered. - The arcs are considered to be cylindrical.

2. Thermodynamic and kinetic considerations

The plasmas of the considered arcs are not far from local thermodynamic equilibrium (L.T.E.). This can be concluded when the range of electron

. (OZO 0 21 -3.. d h f

densi ties of 1nterest 1 + 1 m):i s compare wit criteria or L. T.E., e.g. the one given by Wilson [7]:

(1 )

Here n is the electron density, E

Z the energy of the first excited state in eV and T the electron temperature in eVe For a cesium seeded argon

plasma (1) has to be applied to cesium, so that E

Z =1.4 eV. With T = 0.25 _{20 -3 e} eV it follows that n has to be larger than 0.8 x 10 m for L.T.E. to be expected. In order to be able to take moderate deviations from L.T.E. into account a partial local thermodynamic equilibrium (P.L.T.E.) description [8 J will be adopted as a starting point. Eventual simplification to an L.T.E. description will be employed when justified according to

calcula-tions which include the specific plasma condicalcula-tions under consideration.

For the center of cylindrical arcs in cesium seeded argon plasmas such calculations have been carried out [6J leading to the conclusion that for atmospheric plasmas with gas temperatures around 1500 K and seed fractions of 0.001 L.T.E. may be assumed for electron temperatures larger than 2000 K.

In the P.L.T.E. description employed here only the ground state is assumed not to be in equilibrium with the continuum. The populations of all exited

states are related to the electron density by Saha equations:

2 Zi 211 mkT

3/2

n

2

( e )

_{[-(e .}

_-~e.)/kT

_]

= h2 exp for q > 1 (2) N _{q gq} q1 1 e Here N

q is the population, g q the degeneracy and £ q1 • the ionization energy of the level q. T is the electron temperature, ~£. the lowering of the

e 1

ionization potential [9J, Zi the partition function of the ion, m the

electron mass, k Boltzmann's constant and h Planck's constant.

For a given value N d of the total number density of cesium (neutrals see

plus ions) the relationship between nand T in case of P.L.T.E. is found

e

from (2) by summation over all excited levels: 2

Zi 21T mkT 3/2 n

2 ( e ) _{exp [-(£i - [;£i)/kTel}

N

seed - N - n Z - 2 h 2

1 0

(3 )

Here Z is the partition function of neutral cesium. The values of Z were

o 0

taken from Drawin [9J and have been approximated for [;£i

=

0.1 eV by the following relation:

-4-(4 )

The deviation between the values of Z given by (4) and by Ref. 9 inereases o

with inereasing T • At T = 5000 K the differenee is 1% and at T = 6174 K

e e e

the differenee is 3%.

The L.T.E. value of n, given by 2 n eq n eq n ,

eq for a total cesium concentration N see d is

Zi 2n mkT 2 ( e) Z h2 3/2 exp [-(Eo - 6E i )/kT

1

1 e (5) o

The nonequilibrium of the ground state implies that the excitations of the ground state are not balanced by their reversed processes. The most impor-tant excitation proces is the electron collision excitation to the first excited state*). For this proces the net excitation rate is given by

6 = (6a)

Using the relationships between K12 and K21 as defined by detailed balanc-ing (6a) can be rewritten as

(6b)

Here N~ is the equilibrium population of the ground state as given by (2) with q

=

1. Using (3) and (5) one finds for a given cesium concentration

N : seed 6 = nK12 (N_{seed - n)} 1

+

1 - (n/n } eq (n/n )2(Z /2 - 1) eq 0 (6c)

*) In cesium the 6P~ and 6P

3/2 state are grouped together to the first excited state.

In a stationary arc ~ is compensated by diffusion and radiation processes.

The net emission rate of the 6P~ and 6'3/2 level has been calculated as a

functi.on of the radial position in the arc by numerical integration of the radiation transport equation. For the plasma conditions considered in this work the effect of the radiation is a factor of 10 smaller than the effect

of the diffusion if the typical radius of the arc is 0.1 mm. Radiation and

diffusion become comparable if the arc radius is 1 mm. In both cases,

however, the plasma is close to L.T.E. except for the very outer region.

3. Governing equations

The electron temperature T , the gas temperature

e

electron density n, are governed

T a by

and the

the electron energy equation, the energy equation for heavy particles and the particle balance equation for the cesium ground state. For the steady state situation and

the cylindrical geometry assumed in this work they read as follows: 1 d p dp 1 d (p D dn)

+ '"

p

dp A dp

o

(7) (8 )

o

(9 )

In these equations p is the radial co-ordinate,). the electron thermal

e

conduc t i vi ty,). the heavy particle thermal conductivity and 0 the

elec-a

trical conducivity, E* is derived from!* which represents the electric

field plus the induced electric field:

*

-6-where E is the electric field, ~ the gas velocity and! the magnetic

in-duction. A co-ordinate system is chosen with the x-direction along ~ and

the z-direction along B. The axis of the arc is assumed to be oriented along the y-direction. The y-component of (lOa) reads

*

E = E - uB Y Y

In the case of an MHD generator

*

the absolute value of E by E* it

Y uB - E Y

*

E is negative y follows that (lOb)

and E positive. Denoting y

(IDe)

In (7) and (8) v

eA' veCs and vei are averaged electron collision

frequen-cies with noble gas atoms, cesium atoms and ions respectively. MA is the

noble gas atom mass and MCs the cesium atom mass. In (9) DA represents the ambipolar diffusion coefficient.

The first terms of (7) and (8) represent the radial heat conduction of the electrons and heavy particles respectively. The second term of (7) ac-counts for the Ohmic dissipation. In this term E*, as defined in (IDe), appears because it is assumed that the current density is directed along the axis of the arc. The third term in (7) describes the exchange of ther-mal energy between electrons and heavy particles by elastic collisions. Consequently this term also appears in (8) with the opposite sign. The steady state assumption requires that the outward flow of electrons and ions is equal to the inward flow of cesium neutrals. Furthermore it is assumed that the flow of ground state cesium neutrals can be approximated by the total flow of cesium neutrals, so that in the balance equation for

neutral cesium ground state particles the convective term has been

replac-ed by the ambipolar diffusion term with the opposite sign. As a result (9) can be interpreted also as an electron continuity equation. The radiative term has been omitted from (9) for the reasons mentioned in the previous

section.

The total pressure as well as the seed fraction in the arc are assumed to

he constant and prescribed. As a consequence the concentrations of noble

gas and cesium are known as soon as T is found. Since the populations of

a

quan-tities n, N d' N

1, and

see l' e can be expressed in the other three by using

the relationships given in the previous section.

When solving the equations (7), (8) and (9) it is convenient to introduce 2

s = p /2 as the independent variable instead of p. The differential

opera-tor is then transformed as follows:

}

~p

(pK

~)

+ 2

h

(sK

~s)

(11)

where K represents the corresponding transport coefficient.

4. Transport coefficients

The transport coefficients a, A and average collision frequencies v

ap-e

pearing in the governing equations are calculated for argon-cesium

mix-tures using the procedure of Ref. 10. In the regime of weak ionization (electron-neutral collisions dominate) the accuracy of 0,

A

e and v e s C as evaluated there is better than 40 percent. The accuracy of

o and

A

e in the limit of full ionization is about 10 to 20

V_{eA ,} vei percent.

and of

To avoid long computations simple approximating formulae are used in this work. For 0 and

A

the following expressions have been developed:

e o

2E.

o p 1

+ - - - -___

---;-n--(T /1000)(v i/10l0) a e 3 (4+3.3x10

N

-

n ....::." e::.e::;d:;-_ _ ) NA T 1.6 (_e_) 3000 x p T 0.7 1

+ _ ; ; ;

-(T /1000)(v ./10_{a el} 10) [6(_e_)

+

3000 N - n 3 seed T 2.7 2. 7 x 10 ....:;.~=---NA e (3000) (12) (13)

-S-In (12) and (13) the index sp denotes the Spitzer value for fully ionized plasmas. NA is the argon density and p is the total pressure. SI units have to be introduced everywhere in the equations except for p which has to be given in bar. The formulae (12) and (13) deviate on average about 10 percent and at maximum 30 percent from the results of Ref. 10 in the range 1500

<

T

<

SOOOK.Within only a few percent the electron-neutral collision

e

frequencies for argon and cesium are approximated by the following

expres-sions: V eCs 3.12 x 107

_T

P T1•72 e 1.96 a p x lOIS _ T a N - n seed (14 ) (15)

(14) and (15) are valid for T between 1500 and SOOO K. Again p has to be e

given in bar and the other quantities in SI units.

When tbe arc is oriented perpendicular to a magnetic field, as in an MHD

generator, its radial cooling in the direction perpendicular to the mag-netic field is different from the cooling parallel to the magmag-netic field.

I t is assumed that micro-instabilities affect the thermal transport per-pendicular to the magnetic field. To include this effect the corresponding thermal conductivity Ael is given by

(A II

e 1 (16)

Here Aell is the thermal conductivity parallel to the magnetic field, which is equal to the value without magnetic field as given by (13), and

eB _{(17 )}

Let ~ be the angle between the magnetic field and an arbitrary direction of the heat flow. Then for the thermal conductivity A _e,(l the following ex-pression holds: 1

--=

A e,a 2 2 (..::c..::0;rS_(l=--

+

~s..::in;r.-,(l:o.) ~ 2 A 2 Aell el (IS)

By averaging (18) over al directions it is found that the average thermal

conduction <A > is given by

e <).

>

e 2 x -n

f

o

n/2

[cos 2 a

+

(1

+

6 2 )sin 2]-a ~ d a (19a)

In a wide range of values for 6 (between 0 and 20) the second factor in the right hand side of (19) can be approximated within a few percent by (1 +

6

2)- 0.3 Consequently <). > is approximated by

e

<).

>

e

The ambipolar diffusion coefficient DA is approximated as

(19b)

(20)

In (20) p has to be given in bar, while everywhere else SI units apply.

+

The expression for DA is based on the Cs - A cross section of the experi-mental data cited by Hasted [11].

The argon thermal conductivity*) is expressed as

).

a 4.76 x 10- 4 TO• 655

a (21)

The expression (21) represents the experimental results of Guevara [12], Springer [13] and Nain [14] with an accuracy of 1 percent in the tempera-ture range between 1000 and 2500 K.

The rate coefficient K12 is given by - 14 T 3/2

K12

=

1.1 x 10 e exp (- 16620/T_e) (22)

(See for example Refs. 15 and 16).

*) Because small seed fractions are considered the thermal conductivity

-10-5. Solution of the equations

Straightforward solution of the differential equations (7), (8) and (9) using a Runge-Kutta procedure shooting with initial (at p 0) values T

=

T , dT /dp

=

0, T

=

T , dT /dp

=

0, n

=

nand dn/dp 0 does not

e eo e a ao a 0

lead to the desired result. Because of the short characteristic length

related to (9) the solution becomes extremely sensitive to the choice of n • Defining n by nine digits is not enough to succeed in the integration

o 0

of (9). The reason for this sensitivity is that the plasma in a large part of the arc is close to L.T.E., so that the first term of (9) is small compared to the composing terms of 6 (see (6b». Therefore a small error

in n accumulates in the step by step procedure and then the difference in

l-(n/n )2 in the expression (6c) for 6 catastrophically affects the sec-eq

ond order differential operator in (9). The problem described above can be avoided by employing the following iteration process:

+

2

1 + (n ld/n )2 (Z /2 - 1) d

o eq 0 __

noldKl2 (N_{seed - nold )} ds (23)

In (23) n a n d n ld are the new and old values of the electron

concen-new 0

tration. The iteration process (23) follows immediately from (6c) and (9).

The zeroth approximation in (23) is given by

conditions of this work one iteration appears

n _o ld = n _eq . For to be enough.

the specific

The question arises how to find solutions of (7 - 9) for the region out-side the arc where the equations become of the type

I d dX

p dp (p dp)

o

(24)

In (24) X represents T , T or n. The solution of (24) goes to minus in-e a

finity as In p for p + 0 0 . Physically this means that for the chosen

cylin-drical geometry no finite steady state solution can exist. In practice the situation is explained by the existance of a "mixing length". This implies

that the boundary condition is not imposed at p ~ ~ but at a certain

fi-nite radius p = P

6. Discussion of the results

The input data which are chosen for the computations refer to plasma

con-ditions and electrical characterizations of noble gas MHD generator

expe-riments as carried out in Ref. 17. A schematic picture of a segmented

Faraday MHD generator is presented in Fig. 1. When i t is assumed that

the discharge structure in one generator segment consists of N identical arcs, the internal resistance of such a segment is given by

L

Ri = "

-Nff

adS

(25 )

where L is the distance between anode and cathode. In (25) the integration

has to be carried out over the cross section of a single arc. The absolute

values of the electric field component perpendicular to the flow and of

the induced field are related by the load factor:

K

E

J..

uB

R _{(26 )}

In (26) R represents the external resistance. I t follows from (25) that

E* = uB - E and B can be independently varied as long as u, Nand Rare

y

not specified. Unlike most arc analyses, in this calculation the electric

field (here E*) is prescribed and the arc current is calculated. This

procedure corresponds to the situation in an MHD generator where the

dri-ving field is induced. Besides E* the following input parameters are used:

the magnetic field B, the pressure p, the seed fraction sf, the central

electron temperature T and the eo

values of the parameters and their

outer gas temperature T • The standard

a~

variations are given in Table 1.

The solution of the equations (7), (8) and (9) for the standard reference case is presented in Fig. 2. The shape of the profiles are to a large

extend determined by the electron energy transport. It has been

demonstra-ted earlier [4] that the Joule heating and the thermal conduction are the

most important terms in (7) and that consequently the characteristic cross

sectional dimension is determined by the central value of the electron

-lZ-profile reflects the fact that the thermal conductivity decreases with the

electron temperature. The electron density follows from the electron

tem-perature by applying L.T.E. relationships in very good approximation

ex--4

cept for radii larger than 1.77 x 10 m where the deviation of n from its

LeT.E. value becomes more than 5%. The characteristic radial dimension of

the arc is represented by p defined as the maximum radius for which max

numerical integration was still possible*). The boundary condition for the gas temperature is given by T

(/z.p

)

= T thus defining Pf =

IZ.p

a max aoo _max

as the mixing length for the heavy particle energy. From the resulting gas temperature profile presented in Fig. Z i t can be seen that the gas tem-perature is increased by about 100 K in the center of the discharge. The value of T (0) - T is determined by the value of the term representing

a aoo

the elastic losses in (9) as well as by the radial dimension of the arc.

The calculations provide in a relationship between p and the central max

electron temperature T , which is presented in Fig. 3. The calculated

eo

relationship is compared with the approximated relationship as given in Ref. 5:

-4

1.1 x 10 T eo (Z7 )

It can be seen from Fig.3 that the approximated value of p is good max

enough as an estimation in the considered range of electron temperatures. The differences between calculated and approximated values of p a r e

max

caused by the difference between the Gaussian profiles assumed in Ref. 5

and the profiles calculated in this work, by the differences between the real values of A /0 and the Spitzer limit (for the lower values of T ) and

e e

by the increasing of the arc radius due to the heat conduction of the

heavy particles (at the higher values of T ). Since larger values of T

e eo

are associated with larger values of the radial dimension of the arc, T (0) - T will be larger for larger values of T • Moreover, larger

a aoo _eo

values of T tend to increase the energy transfer from electrons to heavy

e

*) Because of the strong gradient in T at the edge of the arc p can

e ~x

only very little be influenced by the parameters of the numerical inte-gration procedure.

particles by means of elastic collisions, which effect also enhances

T (0) - T •

a a~

The effect of the pressure p on p and T (0) is presented in Fig. 5.

max a

Increase of the pressure causes a larger gas densi ty which through the

fixed seed fraction also causes a larger electron density. The increase of

the densities causes an increase of the electron collision freqeuncy cor-responding to a decrease of the Hall parameter

B

which affects the

elec-tron thermal energy transport perpendicular to the magnetic field. It can

be seen from (27) that a decrease of B corresponds to an increase of p ,

max in agreement with Fig. 5. An increase of p

max corresponds to an increase of T (0)

-

T as explained earlier. Moreover, T (0) - T is increased with

a a~ _a a~

the pressure through the corresponding increase of the electron density

and the collision frequencies causing enhancement of the source term in

(8). At low pressures the deviation from L.T.E. becomes more apparent in

a larger region near the edge of the arc (Fig. 6). Since there exists a well defined relationship between the cross sectional dimension p and

max the central electron temperature T (see Fig. 3) it is also possible to

eo

calculate the effect of pressure variation on T for a fixed value of eo

p • This result is shown in Fig.

7.

In this figure the approximate solu-max

tion corresponding to the expression (27) is also given, showing again that it provides in acceptable estimations. Because the radial dimension of the arc is fixed the increase of T (0) - T in Fig. 7 follows only

a a~

from the increase of the collision term in (8).

Variation of the seed fraction has qualitatively the at fixed T and on T at fixed p as variation of

eo eo max

8 and 9). This can be understood because in both cases

same effect on p

max the pressure (Figs. the electron densi-ty is influenced in a similar way while on the other hand the elctron density variations to a considerable extend determine the behaviour of the solutions.

Variation of T has an opposite effect on p at fixed T and on T at

a~ max eo eo

fixed p as the variation of the pressure (Figs. 10 and 11) because the max

number densities are the important variables for the behaviour of the solutions.

In Fig. 12 and 13 the effect of the total electric field component

E = uB - E on the arc characteristics is shown. It has to be noted that

*

y

-14-E*. The behaviour of p at fixed T and the behaviour of T at fixed

max eo eo

p follow immediately from the approximated expression (27). In Figs. 14 max

and 15 the effect of variation of B is shown when E* is taken constant. In

this case the influence of B on the solution is mainly through

a.

In Fig.

16 the variation of the magnetic induction is carried out while keeping (uB - E )/uB and u constant. I t follows from (26) that this implies that

y

then the internal resistance (25) is constant when the external resistance is given. When compared to Fig. 14 the effect of B on p is amplified.

max

This is explained by the increase of E* as well as

a

which influence P

max qualitatively in the same way according to (27).

Calculations are carried out for pure argon and they are compared with the

cascade arc experimens described in [8J. The input data for the calcula-tions are chosen in accordance with the corresponding experimental

condi--1

tions: T = 11700 K, P = 1 bar, E = 790 Vm • The boundary condition for eo

the gas temperature T has been imposed in the center where i t equals a

11655.106 K(*).Furthermore i t has to be noted that the calculations for pure argon are performed under the assumption of L. T.E. The resul ring profiles of T , T and n are presented in Fig. 17 together with the

expe-e a

rimental results of Ref. 8 (dashed lines). The difference between the calculated and measured T profiles is within the tolerance determined by

e

the 20 - 40% error in

A

/0 and the experimental error. Because the

calcu-e

lated T values are lower than the experimental ones, the calculated

val-e

ues of n are lower than the corresponding experimental values in the range 1 < p < 2 mm. After correction for the electron temperature difference the calculated electron density profile becomes higher than the measured one everywhere. This can be explained by the fact that the calculations are carried out for L.T.E. When the deviation from L.T.E. is taken into ac-count by the introduction of a factor (1 + Ob

1)\ for n with 0.7 < Ob1 < 1.1 (see Fig. 5.11 of Ref.8) agreement between the calculated and measured profile is obtained.

Profiles are calculated for cesium seeded helium and xenon for the stan-dard reference case defined in Tab. 1 except for the values

-1

helium E* has been chosen equal to 6000 Vm and for xenon

of E*. For

-1

1000 Vm • These choices have been made to compare :illID channel flows with approxima-tely equal Mach numbers so that the velocities of the particular gases and

{* ) This is in order to obtain a finite Ta value at Pmax (the integreation of eq. 8 is sensitive to thi.s choice for reasons similar to those mentioned in section 5).

the corresponding induced fields are different. The resulting radial

di-mensions when a central electron temperature of T = 5000 K is prescribed

eo

are given in Fig. 18 and the resulting central electron temperatures for a

-4

prescribed radial dimension P

max

=

1.55 x 10 m are given in Fig. 19. In both figures the already obtained results for cesium seeded argon are added for comparison. Qualitatively P in Fig. 18 and T in Fig. 19

max eo

follow the behaviour predicted by (27) thus indicating that E* is an im-portant parameter. The proportionality between T and E* and the inverse

eo

proportionality between E* and p as given by (27) is however not con-max

firmed by the results of the calculations. The more complicated behaviour of p and T is caused by the different individual plasma transport

max eo

properties of the noble gases involved and for the reasons mentioned in

the discussion of Fig. 3. The gas temperature enhancement T (0) - T

a a~

follows p when T is fixed (Fig. 18) and follows T when p is

max eo eo max

fixed in a similar way as discussed earlier. Fig. 20 represents the ratio of the heat flow sustained by the heavy particles Q and the total heat

a

flow Q

+

Q for the different gas mixtures. It can be seen from Fig. 20

a e

that in all cases the largest part of the heat flow consists of the elec-tron heat flow Q • Furthermore the ratio depends on the transport

proper-e

ties of the particular gases and on the arc dimension. For equal arc

dimensions this ratio is determined by A which decreases with increasing a

of the atomic mass and by T (0) - T

a a~

7. Conclusions

Free arcs in cesium seeded argon are analyzed by solving the electron and

heavy particle energy equation together with the electron continuity equa-tion, under the assumption of P.L.T.E. For input data corresponding to usual noble gas MHD generator conditions the following conclusions are found:

- The arc characteristics are to a large extend determined by the electron energy transport. Due to the electron temperature dependence of the

electron thermal conductivity the electron temperature distribution

exhibits a sharp edge.

- The radial dimension of the arc could not be determined independently

from the central electron temperature. Only a relationship between the two quantities results from the analysis. A certain set of input data may yield relatively high central electron temperatures combined with relatively large radial arc dimensions as well as relatively low central electron temperatures combined with relatively small radial arc

dimen-

-16-sions.

- By increasing the effective electric field" E* = uB - E the arc

dimen-y

sian is decreased or the central electron temperature is increased or

both.

For input data chosen in accordance with noble gas MHD generator

experi-ments, especially electron temperatures between 4000 and 6000 K

[2J,

the

-4

calculated radial arc dimension is 1 ~ 3 x 10 m in agreement with the

result of laser scattering experiments [3].

- The behaviour of the central electron temperature and the radial

dimen-sion with respect to variation of plasma parameters and electrical and magnetic characteristics can be explained at least qualitatively by the

simple expression (27) following from a parabolic electron temperature profile and applying Spitzer values for the electrical conductivity and

electron thermal conductivity.

The deviation from L.T.E. does not affect the parameters in the arc except in a thin layer near the edge.

Calculations for arcs in pure argon yield a satisfactory agreement with

experimental results obtained in a cascade arc [8].

When arcs in cesium seeded helium, argon and xenon are compared for

typi-cal noble gas MHD generator conditions the results are largely influenced

by the expected different values of E*.

Acknowledgement

This work has been carried out as a part of the joint program on current

carrying nonuniformities in plasmas of the Institute of Electronics,

So-fia, Bulgaria, and the Eindhoven University of Technology, Eindhoven, The Netherlands. Mr. J.e.N. Bosma is gratefully acknowledged for his

assistan-ce at the computations and Ms. A.M. Holskens for typing the manuscript. A

considerable part of the investigations and the completion of this paper has been accomplished during a visit of the first and third author to the Eindhoven University of Technology. This visit has been financially

[1] Wetzer, J.M.

Asymmetrical Abel inversion of MHO generator discharges. IEEE Trans. Plasma Sci., Vol. PS-11(1983) , p. 72-75. [2] Wetzer, J.M.

Microscopic and macroscopis streamer parameters of a noble gas linear MHO generator.

In: Proc. 22nd Symp. on Engineering Aspects of Magnetohydrodynamics, Starkville, Miss., 26-28 June 1984.

Department of Mechanical Engineering, University of Mississippi, University, Miss. 38677, USA, 1984. P. 7.7.1-7.7.18.

[3] Haas, J.e.M. de and H.J.W. Schenkelaars, P.J. van de Mortel, D.C. Schram, A. Veefkind

Collective CO₂ laser scattering on moving discharge structures in the submillimeter range in a magnetohydrodynamic generator. Phys. Fluids, Vol. 29(1986), p. 1725-1730.

[4] Haas, J.C.M. de

Non-equilibrium in flowing atmospheric plasmas.

Ph.D. Thesis. Eindhoven University of Technology, 1986. [5] Stefanov, B. and A. Veefkind

Developed ionization instability structure in a noble gas MHD generator: Size of nonuniformities.

IEEE Trans. Plasma Sci., Vol. PS-15(1987), to be published. [6] Stefanov, B. and L. Zarkova, A. Veefkind

Deviation from local thermodynamic equilibrium in a cesium-seeded argon plasma.

Department of Electrical Engineering, Eindhoven University of Technology, 1985.

EUT Report 85-E-152 [7] Wilson, R.

The spectroscopy of non-thermal plasmas.

J. Quant. Spectrosc. & Radiat. Transfer, Vol. 2(1962), p. 477-490.

[8] Rosado, R.J.

An investigation of non-equilibrium effects in thermal argon plasmas.

Ph.D. Thesis. Eindhoven University of Technology, 1981. [9] Drawin, H.W. and P. Fe1enbok

Data for plasmas in local thermodynamic equilibrium. Paris: Gauthier-Villars, 1965.

[10] Stefanov, B.

Electron momentum-transfer cross section in cesium: Fit to the experimental data.

18 -[11] Hasted, J.B.

Physics of atomic collisions. London: Butterworth, 1964.

Butterworths advanced physics series

[12] Guevara, F.A. and B.B. Mclnteer, W.E. Wageman

High-temperature viscosity ratios for hydrogen, helium, argon, and nitrogen.

Phys. Fluids, Vol. 12 (1969), p. 2493-2505. [13] Springer, G.S. and E.W. Wingeier

Thermal conductivity of neon, argon, and xenon at high temperatures.

J. Chern. Phys., Vol. 59(1973), p. 2747-2750.

[14] Nain, V.P.S. and R.A. Aziz, P.c. Jain, S.C. Saxena

Interatomic potentials and transport properties for neon, argon, and krypton.

J. Chern. Phys., Vol. 65(1976), p. 3242-3249. [15] Chen, S.T. and A.C. Gallagher

Electron excitation of the resonance lines of the alkali-metal

atoms.

Phys. Rev. A, Vol. 17(1978), p. 551-560. [16] Mitchner, M. and Ch.H. Kruger, Jr.

Partially ionized gases. New York: Wiley, 1973.

Wiley series in plasma physics

[17J Veefkind, A. and J.W.M.A. Houben, J.H. Blom, L.H.T. Rietjens High-power density experiments in a shock-tunnel MHD generator. AlAA J., Vol. 14(1976), p. 1118-1122.

Parameter p sf T am E* B T *) eo Pmax*)

Table 1. Input parameter values.

Standard reference value 0.5 bar 1 x 10-3 1000 K 2000 Vm -1 3 T 5000 K

-4

1. 55 x 10 m Range of variation 0.125

_•

2 bar (0.125 7 2) x 10-3 500 _• 1500 K 1000 _• 3000 Vm -1 1

_•

5 T 3000 _• 8000 K

-*)

Either one of the two variables

T

(central electron temperature) and eo

p (radial arc dimension) is prescribed. The other one results from max

20

-R

I

3 M I E N

=

c:

::.:: 5 2 M

=

-II> I -4 3

1

2

n

1100

0.~~0~

______

~

______

+-____

~~

__

~-+

______

~~-L1ooo

o

2

Fig. 2. Radial distribution of arc properties for the standard referenn'

E

..

_I C>

-><

'"

E

Cl..

- 22

-4

3

600

--

-2

--

--,....

-

-

400

/ ' 1 / ' . /

-

200

O~-=~~----r---~~---+----~O

3

4

5

6

Fig. 3. Effect of electron temperature variation on the radlol arc ciwen-sion and the gas teCperature enhancecent. The dashed :"ine

repre-sents the radial arc dimension f0110\ol10g from the e?proxi::!ate

expression (27). ::.::

8

'"

I-I

=

co

I-M

Ie

-

N <=

-

_c:

2 ..

O~

________

+-, ______ ~~ ______ -+ ______ J-~,~

o

2

Fig. 4. Radial distribution of the electron density for a ce:-.aal

- 24 -otE :..:

SOD

I co

_S

-

..

I -

'"

'"

I E ~3

600

co

'"

I

-2

400

200

o~~~~

__

~

____

~

____ -+o

0.125

0.25

0.5

1 2

P

(bar)

Fig. S. Effect of the pressure variation on the radial arc di=€;'.sion and the gas temperature enhancement. The dashed line corresponcs to

another criterion for the determination of p

max tho< radius at which n becomes as 10. as 0.5 times its equilibrium va:':.;e.

1

c:ll

2 bar 1 bar

0.9

_{0.5 bar}

o.s

0.7

0.125 bar

edy!!.

o

0.5

1

Fig. 6. Electron density relative to its equilibrium value as a function of the radius for

a1 arc dimension,

various pressures at a -4 p • 1.55 x 10 m.

max

radi-:.::

..,

c

-6

,

,

, ,

C> t-"'

5

:.::

4 200 8

'"

t-...

_I

-

100

..

t-3~==~~~--4---4

O

0.125 0.25 0.5 1

2

P

(bar

I

Fig. 7. Effect of pressure variation on the central electron t<:::?erature and the gas telDperature echancement for a fixed val'~e of the radial dimension" equal to 1.55 x 10-4 m. The cashed line

max

represents approximated values of the central electro::

tet:lp€ra-ture according to (27). otE 2 I C

-

_--

-

₃₀₀ 200 100

0~===F~~~~r-

__

~0

0.125 _{0.25 0.5} 8

..

l -I C

-

_I-

..

Fig. 8. Effect of seed fraction variation on the radial arc dic£~sion ~nd

the gas temperature enh,lncemcnt. The dashed linE' repre5~nts ~ "ox

when i t Is defined as the radius at which n becomes as 1:11.' as 0.5 times its equilibTiur.; "'alut:'.

07E I

2

CO

-

>< to E ::0:: M 5 Q

-CI

'"

I-4 - 2f)

-200

8 I-to I

-

co 1-'"

Fig. 9. Effect of seed fractien variation on the central electron teoper-ature and the gas te:'?crteoper-ature enhancement for a fixed value of

-4 the radial arc dimension, P

max = 1.55 x 10 m. ::0::

5

::0::

200

M CO 8

-.,

_CO l -I _1-4

..

₂₀₀

c::....l

100

CO

o

+----,,....---t---r--+

0

500

1000

1500

T

_a

(Kl

0:>

Fig. 10. Effect of variation of the background gas tempe:'ature T on the radial ar: ci~en­

a-sian and on the gas te=?era-ture enhancement.

...

I-

100

0

500

1000

1500

Fig. 11. Effect of variation =~ the background gas tempt::'ii:...:re T on the central e:t:ctron

a-temperature and on t!-.e gas temperature enhanceme~t for a fixed value of the racial arc dimension, -4 P max .. 1.55 x 10 c. ::0:: 8 to I-I CO to

....

.,.E

""

I C>

-

8

'"

co 3

...

'"

E I ~ C> co

2

400 ...

200

-04---+---r---~----_+0

1 1.5

2

2.5

u

B - E

Y

3

( kV )

Fig. 12. Effect of variation c.: the effective electric field E:It '" uB +

- E on the radial arc ::=e~sion and on the gas tecperature y 6

""

M 0

-

~ 5

..

.,

...

4 enhancement.

-8

...

'"

I 200 ... '"

Fig. 13. Effect of variation of the effective electric field E. : uB - Ey

on the central electrcn temperature and on the gas temperature enhancement for a fi~ed value of the radial arc dimension, D~ax

-4

.,.E 1= 2

'"

'"

E 8

'"

~ I 28 -M

=

-

_..

~"4 8

'"

~ I

0...

₁

₂₀₀

₌

'"

~

100

=

----'-_4-100

₃

_{+-~..----+_..---+}

₀

1 3

O+--~--t--...---+

0

!i

BIT)

1 3

5

B (T )

Fig. IS. Effect of variation o~ the Fig. 14. Effect of variation c: the

Qagnetic indu~tion B O~ the radial arc dimension a~c the

r::agnetic induction B 0:"'. trH,'

central electron temp~r~ture

and the gas temperature enhancement for,a fixec value

gas temperature enhar.ce~e~t.

E 3

.,.

I

=

-

_'"

'"

E2

Cl...

of the radial arc di~e~sion.

-4 g~ax - 1.55 x 10 m.

200

100

8

'"

~ I

=

'"

~

04-____

~---~----~0

2

I 1333

3

I

2000

2667 I 3333 uB-Ey(V)

Fig. 16. Effect of variation of the magnetic induction on the radial arc

dimension and on the gas temperature enhanceoent for a fixed value of the generator load factor

and a fixed value of the gas velocity

(uB - E )/uB. equal to

Y -I

u, equal to 1000 ms 2/3

'"

10

.,

....

n

-L

_-

-...

-

...

_... ... ... ... ...

T -"'-

_e

"--T~

a

"-6 4

2

O,~

______ ; -______

~

______

~

______

~~O D 1

p

Fig. 17. Comparison betlo'een calculated and measured radial profiles of the electron te~perature T • the gas temperature T and the

e

-electron density n for a pure argon plasma. Solie lines

repre-sent calculated profiles and dashed lines meas~r~c profiles. The calculations are carried out for L.T.E.

..,

I E N N C

-c

- 30 -4

o

Pmax

E ot I 0

-

3

:.:

"

~

_{TalO)- Taco}

'"

E

c...

8

'"

2 400 I-I c:::o ...

'"

1 200

o

o

He

A

Xe

Fig. 18. Coc~2ri&on of the r~dial arc dimension and the ~a& t~mperature

enhance~ent for arcs in cesium se~ded helIum, ario~ and xenon when the central electron temperature Is prescribed: Teo - 5000 K.

-

_:..:

0

...

co

_Teo

-

7 CI

~

..

I-Ta(O)-Ta

=

6

5

:..:

300

8

..

I-4

200

I co

100

_...

..

3

0

He

A

Xe

Fig. 19. Comparison of the central electron temperatore an':' the gas tem-perature enhancement for arcs in cesium seeded heliu=. argon and

..

CI xenon when 10-4 m.

0.4

..

CI

+

'"

CI

0.2

o

th~ radial arc dimension is prescribed: .. 1.55 x

D

Pmax

-4

1.55xl0

m

~

Teo

=

5000 K

He

A

Xe

Fig. 20. Comparison of the fraction of the total heat flo .. ', ... ~:ich is sustained by the hea .... y particles for arcs in cesiu:: seeded

he-lium, argon and xenon. Two cases are presented: a ?~escribed

radial arc dimension p a 1.55 X 10-4 m and a prescribed

cen-max

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