Non-equilibrium phenomena in a disc-shaped magnetohydrodynamic generator

Non-equilibrium phenomena in a disc-shaped

magnetohydrodynamic generator

Citation for published version (APA):

Veefkind, A. (1970). Non-equilibrium phenomena in a disc-shaped magnetohydrodynamic generator. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR421228

DOI:

10.6100/IR421228

Document status and date: Published: 01/01/1970

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GENERATOR

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EINDHOVEN OP GEZAG VAN DE REC-TOR MAGNIFICUS PROF.DR.IR. A.A.TH.M. VAN TRIER, HOOGLERAAR IN DE AFDELING DER ELEKTROTECH-NIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VER-DEDIGEN OP DINSDAG 23 JUNI 1970 DES NAMIDDAGS TE

4 UUR DOOR ABRAHAM VEEFKI N D GEBOREN TE NOORDWIJK OP 28 MAART 1939 1970 DRUKKERIJ BRONDER-OFFSET N.V. ROTTERDAM

This work was performed as a part of the research program of the group Direct Energy Conversion of the Eindhoven University of Technology, Eindhoven, the Netherlands; it is published as

Dit proefschrift 1s een exponent van m1Jn opvoeding en m1Jn opleiding.

De hulp en leiding van m1Jn ouders bij m1Jn ontwikkeling 1s van wezen-lijke betekenis geweest voor het tot stand komen van dit werk.

Allen die meewerkten aan het basisonderwijs, dat ik mocht genieten, ben ik dank verschuldigd.

Dr. Koene stimuleerde m1Jn belangstelling voor de natuurkunde.

Aan de Vrije Universiteit kreeg deze belangstelling verder gestalte, vooral dank zij de colleges van Prof. Sizoo, Prof. Blok en Prof. Jonker.

De leiding die Piet Born gegeven heeft aan het praktisch werk voor mijn doctoraal examen is voor mij van grate waarde geweest; de wijze waarop hij mij 1n direkt kontakt bracht met promotie-onderzoek 1s

een steun geweest bij de voorbereiding van dit proefschrift.

Bij de voltooiing van dit proefschrift wil ik allen danken, die direkt bij het werk betrokken zijn geweest.

In de eerste plaats dank ik mijn promotor, Prof. Rietjens. De tal-rijke diskussies, die ik met hem mocht hebben, waren evenzovele

impulsen voor de voortgang van het onderzoek. De waardevolle kritiek, die hij gaf tijdens de vele besprekingen over het manuscript, was in groten mate bepalend voor de uiteindelijke vorm van dit proefschrift.

De communicatie, in welke vorm dan oak, met mijn collega's Wim Merck, Robert Benach, Jan Blom, Peter Massee en Jan Houben, is niet op

waarde te schatten. Speciaal de onstuimige wijze, waarop zij mij soms tijdens groepsbesprekingen aanvielen, is onvergetelijk.

Ik ben zeer dankbaar voor het werk van Gees Sielhorst, dat technische begeleiding heette, maar technische leiding was. Door zijn

persoon-lijke betrokkenheid bij het onderzoek was zijn betekenis voor het werk veel groter dan die van de zo noodzakelijke twee rechter handen.

Ir. Meyer, Ir. van Venrooij en Ir. Polderman leverden door hun afstudeerwerk een belangrijke bijdrage aan dit werk.

De opstelling kon worden gebouwd dank zij de medewerking van de

centrale werkplaatsen van de Centrale Technische Dienst. De afdelings-werkplaats van de afdeling der Elektrotechniek verleende onmisbare

diensten.

De numerieke berekeningen waren mogelijk dank zij de hulp en de faciliteiten mij geboden door bet Rekencentrum van de Tecbniscbe Hogescbool.

De beer van der Scboot en de beer van Veen dank ik voor de vervaar-diging van de figuren.

Zeer veel ben ik verscbuldigd aan Lisette Goemans-Luybregts voor de nauwgezetheid waarmee zij bet typewerk verzorgde en voor bet geduld waarmee zij mijri fouten verbeterde.

Ik dank de reproduktiedienst van de Centrale Tecbniscbe Dienst die om mijnentwil vele moeilijkbeden beeft overwonnen om bet TH-rapport te vervaardigen, dat als mijn proefscbrift dient.

De firma Bronder-Offset te Rotterdam dank ik voor de plezierige wijze waarop de afwerking is verzorgd.

Tenslotte stel ik er prijs op de gemeenscbap van de Tecbnische Hogescbool te danken voor de gelegenheid die z~J mij geboden beeft om het onderzoek te doen, dat tot dit proefscbrift heeft geleid.

CONTENTS

SUMMARY

NOMENCLATURE

CHAPTER

I

CHAPTER

II

CHAPTER

III

CHAPTER

IV

CHAPTER V

CHAPTER

VI

Introduction

Basic

equations

Geometry of the disc generator

Stationary

solutions

of

the basic

equations

IV.l Introduction

IV.2 Temperature, density and radial flow

3 4 10 15 20 25 25

velocity of the electron gas 26

IV.3 Radial flow velocity and temperature of

the heavy particles and density of the

neutral particles 31

IV.4 Electrical conductivity and Hall parameter 33

Critical values of the

Hall parameter with

res

pect

to

ionisation ins

tabi

Zi

ties

35

V. I Introduction 35

V.2 First order perturbation equations 35

V.3 The calculation of critical values of the

Hall parameter for some special cases 38

V.3.1 The region where the Saha equation ~s valid 38 V.3.2 The ionisation relaxation region 41

CHAPT

ER

VII

M

easurements

VII. 1 VII.2

VII. 3

Image convertor camera pictures

Electrostatic probe measurements

Electrode voltage and floating potential

measurements

VII.4 Spectroscopic measurements

VII.5 Hicrowave measurements

VII.6 Piezo-electric crystal measurements

CHAPTER VIII

Discussion

of

the

experimental results

CHAPTER IX

Conclusions

APPENDIX

REFERENCES

Tables at

the

calculation of critical values

of

the

Hall parameter in

the

case of no

Saha

equilibrium

53 53 54 63 70 75 79 81 88 91 96

S UM M AR Y

The work presented describes the non-equilibrium phenomena of a medium flowing through a magnetohydrodynamic generator, especially when a disc-shaped Hall generator is involved.

A set of basic equations is composed of conservation equations obtained from Boltzmann's equation, and of simplified Maxwell's

eq~ations. The basic equations describe the behaviour of the electron density, the neutral density, the electron velocity, the velocity of ions and neutrals, the electron temperature, the temperature of ions and neutrals, and the electric field, throughout

the generator. One-dimensional and stationary solutions demonstrate

the development of electron temperature elevation and non-equilibrium

ionisation. Also starting from the basic equations, and using first

-order perturbation theory, critical Hall parameters are derived, at which ionisation instabilities begin to develop.

A pulsed experiment is carried out in a disc-shaped channel, using

pure argon as a medium, at pressures of about 10 Torr and temperatures of about 5000 °K. Various diagnostic methods are applied, viz.

high-speed photography, electrostatic probes, spectroscopy, a piezo-electric

crystal, and microwave techniques. Thus, information has been obtained on the electron temperature, the electron density, the neutral

density, the flow velocity, and the electrical potential of the plasma.

Clear evidence of electron temperature 'elevation has been found,

whereas no non-equilibrium ionisation has been measured. A considerable influence of ionisation instabilities on the Hall electric field is measured. The experimental results are discussed and compared with the

4

-N

OMENCLATURE

Symbols

A AI' A2 A p a +' a -+ B B 0 b

c

_p

c

v

c D -+ E -+ E''' -+ EH

E

H

EIR -+ EL E ex a E. ~a E _m e -+ e r

electron energy loss owing to elastic collisions

microwave amplitudes probe area

slopes of the asymptotes to the electrostatic probe

characteristic

magnetic induction

value of the magnetic induction ~n the centre of the

disc

length of the longest side of the wave guide cross

section

specific heat at constant pressure

specific heat at constant volume

length of electrode segment hydraulic diameter

electric field

induced electric field

Hall electric field

ionisation energy of hydrogen

energy lost or gained by the electrons owing to

ionisations and recombinations

electric field component corresponding to the Lorentz

force

energy corresponding to the lowest excited state

ionsation energy

energy corresponding to excited state m charge on the electron

f g' 0 h I I. L L p L po -t J -+ K k L M m N m N p N w N E friction coefficient

distribution function of particles belonging to species £

weight factor of the Lon ground state weight factor of the excited state m

channel heigth

reduced Planck's constant

number of ionisations per unit volume per unit time satured ion current towards the electrostatic probe probe current

probe current corresponding to the centre of the current-voltage characteristic

current density

current density component corresponding to the Hall effect

current density component corresponding to the Lorentz force

wave vector

Boltzmann's constant

ionisation rate coefficient recombination rate coefficient generator length

Mach number

Mach number related to the radial veloctiy mass of an argon ion or neutral atom

mass of a particle ?elonging to species t

population of excited state m

refraction coefficient of the plasma refraction coefficient of the wave guide refraction coefficient of the window

n n ecr n g nQ, n q p p pe pg Qe£ q R R R Rl' R2 ReD R m R u r rLe rLi s T T 0 T(R) E,M 6

-heavy particle density

critical electron density

total particle density

density of particles belonging to species Q,

principal quantum number

dimensionless representation of the gas pressure heavy particle pressure

electron pressure

total gas pressure

collision cross section referring to elastic collisions

between electrons and particles belonging to species Q,

integer number

dimensionless representation of the radius

number of recombinations per unit volume per unit time

reflexion coefficient

responses of the crystals ~n the microwave bridge Reynolds' number related to the hydraulic diameter resistance in electrostatic probe circuit

load resistance radius

electron giration radius

ion giration radius electrode pitch

heavy particle temperature stagnation temperature

dimensionless parameter representing the interaction of

the electric and magnetic fields with the gas in the

disc generator

total gas temperature

t t 1 ' t2' t3 t. ' t out ~n UR + u + u _g + u£, v m vn v oc v p v po vpl vpl + v z z a + y b,£, b.p e b.T e b.V f1

M

time

times on which probe signals are examined

plasma passage times at the inner and outer electrode rings

dimensionless representation of the radial flow velocity heavy particle flow velocity

total gas flow velocity flow velocity of species £,

voltage measured in the electrostatic probe circuit floating potential

open circuit voltage probe voltage

probe voltage corresponding to the centre of the current

-voltage characteristic plasma potential plasma volume particle velocity axial coordinate nuclear charge ionisation-recombination parameter

first order term of the quotient of the electron pressure gradient and the electron density

difference of the lengths of the two paths ~n the microwave bridge

electron pressure difference between the electrodes of the disc

electron temperature difference between the electrodes of the disc

floating potential difference between the electrodes of the disc

phase difference introduced by the unequal paths ~n the microwave bridge

E 0 E r K /... ~ ]J \) \) c \) :': e£ pg a a _eff 'disch <P 0 Sl _~ • ' Sl _r w wr WT 8

-parameter for the influence of the electron density

gradient in the zeroth order electron energy equation

permittivity of vacuum

relative permittivity

load factor

reduction parameter corresponding to electrode

segmentation

wave length

Debije shielding length

electron mean free path

~on mean free path

characteristic length corresponding to electron inertia

neglection

viscosity coefficient

m~crowave frequency

total electron elastic collision frequency

collision frequency relating to momentum transfer at

elastic collisions betw~en electrons and particles of species 9, .

collision frequency relating to energy transfer at

elastic collisions between electrons and particles of

species 9,

total gas mass density

electrical conductivity

effective electrical conductivity

delay time between the opening of the valve and the

discharge of the capacitor bank

phase angle

angular frequency corresponding to ionisation instabilities

imaginary and real part of 0

angular frequency of microwaves

plasma frequency

WT(O)

cr

Superscripts

(O) ( 1)

Subscripts

a e i m, n r, ~, z x, y' z

Shorts

ETE LTE

MHD

NEI

critical Hall parameter

effective Hall parameter

Hall parameter at the stability limit

zeroth order perturbation

first order perturbation

averaged neutral particles electrons ions gas species excited states cylindrical coordinates Cathesian coordinates

electron temperature elevation

local thermodynamic equilibrium

magnetohydrodynamic

10

-CHAPTER I

Introduction

Magnetohydrodynamic (MHD) electrical power generation might be used

after 1980 in various applications:

- MHD open cycle systems will be suitable to produce electrical

~nergy on a large scale (1000 MWe) from fossil fuels. High efficiencies (50 %) are expected from combinations of HHD and conventional systems. Already now, experimental MHD generators 1n open cycles are capable of converting 6 % of the thermal energy

of the medium into electrical energy at an output of 30 ~ru

(ref. I. I).

- Closed cycle MHD generators us1ng liquid metals as working media

are promising with respect to space travel application. The media

of these generators consist of liquid alkali metals, mixed with

a gaseous component, such as vaporised alkali metals, argon

helium or nitrogen. They will be heated by a nuclear source. MHD power conversion employing liquid metals might be suitable to

supply electrical energy in spacecraft, because of the high energy production rate per unit mass (compare ref. 1.2).

- The MHD closed cycle systems using gaseous media are orignially

intended to convert the thermal energy of gas cooled nuclear reactors into electrical energy. The media to be used are inert

gases, viz. helium or argon. Application of this type of MHD

conversion cannot be expected before 1990, the mean reason being the mismatch of the parameters of the gases to be employed 1n

the reactors and in the MHD generators in the present stage of

their development. Up to now, the pressure of the gases used 1n

0 gas cooled reactors is > 20 atm and the temperature < 1600 K, whereas the MHD generators will work at a pressure < 10 atm and a temperature ~ 2000 °K.

- The problems connected with the use of a nuclear heat source are avoided in the mixed cycle systems (ref. 1.3). In these systems

of a heat exchanger to a closed cycle MHD system employing an inert gas.

The ma~n problem related to closed cycle systems with gaseous media is how to achieve a sufficiently high electrical conductivity of the

0

gas. At temperatures of about 2000 K and pressures between 1 and 10 atm, being the practical gas conditions, the electrical conductivity is too low for a sufficient energy production. Therefore, an

additional enhancement of the degree of ionisation is necessary. An important improvement of the conductivity is obtained by seeding

the gas with easily ionising materials (alkali metals). Another method of enhancing the ionisation rate is suggested by Kerrebrock

(ref. 1.4). He has demonstrated that for a high pressure arc

containing 1 atm argon + 0.4 % potassium the electrical conductivity depends on the current density in a way which can be explained by considering the gas to be a two temperature plasma with the electron temperature higher than the gas temperature and with a degree of ionisation given by the Saha equation at the electron temperature. As the electron temperature elevation (ETE) appeared to be described by the balance of Joule heating and elastic collisional losses of the electron gas, the non-equilibrium ionisation (NEI) seemed to be promising for the development of closed cycle MHD generators, also because the employment of rare gases is advantageous with respect to ETE owing to the low cross-section for electron-atom elastic

collisions in those media. However, the realisation of a two temperature plasma connected with a suitable NEI in MHD generators appears to be a complicated problem. Table I. 1 g~ves a review of recent MHD generator experiments concerning non-equilibrium phenomena. It can be seen from the table that there is good evidence for magnetically induced

increment of the electron temperature and density in MHD generators. The experiments, however, deal with several loss mechanisms, which affect the behaviour of the non-equilibrium generators. Some of these mechanisms are extremely favoured by the non-equilibrium situation

itself. Typical losses are: electrode short-circuiting through hot boundary layers, the existence of ground loops, electrode voltage drops, non-uniform conductivity due to electrode segmentation,

Ref. type type medium u T p B diagnostics effect

I

reported discussion

experiment generator (m/sec) (OK) (atm) (T) of results "'

cr ~ (1)

1.5 shock linear, A 980 - 1350 - 0.9 - 0.88 electrical enhancement ne non-equilibrium tube segmented + 0.5 % Cs 1150 1950 0.43 output and Te calculated behaviour affected by

electrodes from wT and a radiation losses and "' "' <: _...

non-uniformity; if _"' >:

accounted for these 0

effects, agreement ,..,

with theory

.,

"' X

I. 6' shock linear, Xe 1000 5700 I 0.25 - electrical enhancement ne agreement with theory; _"'

"

1.7 tube segm. el. + 0.5 % H 2.25 output and Te calculated non-equilibrium _3"

A 1710 5100 0.4 2.25 - form wT and a phenomena strongly _"'

"

2.6 affected by loss ..., <n

mechanics ()

0

1.8 plasma linear, 70 % He 2350 600 0.05 I. 3 electrical small enhancement only small evidence of

_"

()

jet segm. el. + 30 % A output of o; voltage electron heating and _"'

"

oscillation magnetically induced

_"

...

ionisation 00

_"

1.9 closed linear, He + C.l 1060 1700 I 2 electrostatic enhancement of a agreement with theory _"' ~

loop segm. el. - 3 % Cs probes () _"'

N ,...

I. 10 closed linear, He + 2 % 240 1300 1.3 2. IS electrical no effect induced field to small

_"

0

loop segm. el. Cs output

_"

I. I I blow linear, He + "- 2500 900 0. 3 - 1.4 electrical Te enhancement non-equilibrium

down segm. el. 0.23 % Cs 0.6 output; calculated from behaviour stronly

electrical WTeff affected by loss

..., "' 3

.,

tt>

"

potential; mechanisms and

continuum relaxation phenomena

.,

..., c " radiation "' (1)

I. 12 shock disc A+ 1400 "- 1700 1.3 3.4 continuum ne enhancement non-equilibrium

tube I % Cs radiation from radiation ionisation accompanied

measurements by large ne fluctuation~

~ "' <:

.,

,... ... 0

I. 13 plasma linear, A+ 0. I < 700 1500 - I 0.2 electrical enhancement ne non-equilibrium

jet segm. el. - 3 % K 3000 potential and Te calculated behaviour strongly

from WTeff influenced by boundary layers

"

.,

"

c..

"

0

"

I. 14 blow linear, He + 200 - 1200 - 1.2 2.7 electrical no effect currents to small

down segm. el. 0. I 7. K 1000 1700 - 2 output

I

"'

""

_"

... I. IS closed linear, He + 1417 1403 0.6S o.s - electrical no effect influence loss

loop segmented 0. IS % Cs I. 97 output mechanism too strong

~ ...

cr

"

...

I. 16 blow linear, He + 1400 - ISOO I 4.S electrical enhancement of a non-equilibrium

down segm. el. 2 - s 7. Cs 2000 output behaviour strongly

"

... 3 affected by losses; 0

"

accounting for them ... <n

agreement with theory~

.,

...

0

radiation losses, and ionisation instabilities. These losses have to be calculated very carefully before non-equilibrium phenomena can be interpreted and in many cases a quantitative understanding remains difficult.

Another apparent feature of Table 1.1 is the lack of variation ~n

diagnostics. In almost all experiments conclusions are drawn from

values of the Hall parameter and the electrical conductivity, which

are derived from the electrical output. As pointed out by many of

the authors even the conductivity and the Hall parameter are affected by the losses. Little attention has been given on the

measurement of the electron temperature and density in a direct and

independent way; only the continuum radiation measurements provide a direct determination of the electron density. In spite of the

difficulties related to the realisation of a suitable non-equilibrium

condition in MHD generators, it has been stated (ref. 1.17) that NEI

is necessary, in addition to the use of s~eding materials, in order

to make possible practical conversion'of energy using MHD closed

cycle systems.

The aim of the present work is to exam~ne ETE and NEI ~n an MHD medium

in situations where perturbing effects are suppressed as much as

possible. The analysis has been simplified by considering non-seeded argon as a medium. The phenomena are studied in the disc geometry to avoid the problems connected with electrode segmentation. Although electrode voltage drops may occur, the non-equilibrium conditions will be developed all the same, the azimuthal currents being primarily responsible for the process. Ground loop leakages are eliminated by using an inductive method for the plasma production. The most important remaining loss mechanism affecting the non-equilibrium phenomena are the ionisation instabilities.

The analysis is based on fundamental equations for the var~ous plasma

components. Similar equations have been used by Bertolini (ref. 1.18)

for the description of the relaxation of an MHD medium towards the non-equilibrium state. The present analysis leads to solutions describing both the relaxation processes and the behaviour of the

14

-bvo temperature plasma. Furthermore, part of the set of equations is

used to study the plasma conditions which are critical with respect to the development of ionisation instabilities.

The experiment provides plasmas flowing during short times (100 ~sec)

through the disc. The electron temperature and density are measured

by electrostatic double probes, spectroscopic measurements and m~cro­

wave measurements. Total gas pressures are determined using a

piezo-electric crystal. Moreover, the floating potential of the plasma

is measured, in order to obtain information on the effective Hall parameter and the electrode potential drops. In the experiment

described, the gas pressures and magnetic fields are lower than ~n

other experiments. There is, however, no reason why the results of this experiment should essentially differ from those involving high pressures and magnetic fields, as the mutual ratios of

characteristic lengths, like free mean paths, Debye shielding length,

gyration radii and the dimensions of the channel, have not been altered in a critical way.

CHAPTER I I

Basic

equations

In an MHD generator a partially ionised gas flows through a magnetic

field. In the presented work a flowing argon plasma consisting of electrons, singly ionised atoms, and neutral atoms, will be

considered as a medium for the MHD generator.

The kinetic and dynamic properties of the plasma are described by the

-+

distribution functions f£(v, r, t), which can be obtained by solving the Boltzmann equation for each species £. Simultaneously with the Boltzmann equation the Maxwell equations have to be solved in order

to describe the electromagnetic fields as a result of the electric charge density distribution and the current density distribution. Considering this specific case of an MHD generator, a number of simplifying assumptions will be made.

The distribution functions are assumed to be Maxwellian

f£ (II. 1)

The assumption g~ven by equation (II.l) reduces the solution of the Boltzmann equation to the solution of the following three conservation equations for each species: the continuity equation, the momentum equation and the energy equation, in order to find the density n£,

+

the flow velocity u£ and the temperature T£.

A further simplification is made by assuming the flow velocity of the ions to be equal to the flow velocity of the neutrals and assuming the temperatures of these species to be equal. These assumptions limit the number of conservation equations to seven, three continuity

equations (one for each species), two momentum equations (one for the electrons and one for the heavy particles), and two energy equations

As in the cases considered the magnetic Reynolds' number will be small, the magnetic induction owing to the currents in the plasma is

neglected compared to the applied magnetic induction. The latter ~s

taken as stationary. Moreover, the electric space charge is assumed to be small, according to the inequality:

n

I

e << I

(II.2)

This assumption determines the Debye length as the m~n~mum characteristic

length in the plasma to be described. Neglecting In - n.

I

with respect e ~

ton or n~, one may replace n. by n ~n the conservation equations.

e L ~ e

From the Poisson equation for electrical space charge and equation

(II.2) the following condition for the variation of the electric field

can be derived:

lv

.E

I

« n e e E: 0

(ir.

3)

Once having found the solution of the problem, the condition (II.3)

can be verified in order to justify the substitution of n for n ..

e ~

Furthermore, only phenomena are discussed that are stationary or quasy-stationary with respect to the Maxwell equations, which can then be reduced to the following relationships:

-t V.J 0 -+ 17 X E 0

(II.4)

{II.S)

Equation

(II.4)

has already been given implicitly by the continuity equations for the electrons and the ions.

The seven conservation equations which are used to analyse the medium, are g~ven ~n Table 2.1. Throughout the analysis the mass of an electron is neglected compared to the mass of an argon atom; the masses of a neutral and an ion are taken to be equal. The right-hand sides of the continuity equations describe the net number density production rates, caused by ionisations and recombinations. The major ionising processes which may occur in the argon plasma considered are electron-atom

a ~ = k n n - k n2n. ELECTRONS : - n _{at e} + 'l.n u _{e e} f e a r e 1 a ~ k n n - k n2n. IONS : _{at ni} + 'l.n.u = 1 f e a r e 1 a ~ + k n2n. NEUTRAL PARTICLES : - n + 'l,n u = - k n n at a a f e a r e 1 MOMENTUM EQUATIONS -+ -+ X B) -+ ~ )(v . + v ) ELECTRONS : 0 = - 'lpe - nee(E + u + n m (u -e e e e e1 ea a _(nm~) ~ ~ ~ X ji) n m (~ - ~ )(v . _vea) HEAVY PARTICLES : at + '7. (nmuu) =

-

'lp + n₁e(E + u - + e e e e1 ENERGY EQUATIONS a _{ 3 + ..!. m u2) } { 3 +..!. m }) ~ } ~ -+ -+ ~ + n m ~. (~ - ~ ) (v . v ) ELECTRONS :

at ne(Z kTe + E. + '7. _{ne(Z kTe} ... E. u = - _ue·"Pe

-

_pe 'l.u - n eE.u +

1 2 e e 1 2 e e e e e e e e e e1 ea m

-

3 n

-

e (v . + _{v ea) k (T}

-

T) e m e1 e a _{{ (2 kT} I 2 _{} { (2 kT ..!. mu}2₎ -+ } -+ -+ -+ -+ n m ~. (~ - ~ )(v . + v ) HEAVY PARTICLES : at n +

z

mu ) + '7. n + u = - u.'lp - p'l.u + n.eE.u -2 2 2 1 e e e e1 ea m + 3 n _{e m}

-

e (v _e1 . + v _ea ) k (T _e - T)

collisions, atom-atom collisions and photo-ionisation, while the most important de-ionising processes are three-body and radiative

recombinations. Considering only electron temperatures below 20,000 °K d 1 d . . b 019 -3 h d. . . d an e ectron ens~t~es a ove 1 m , t e ra ~at~ve ~on~sat~on an recombination processes can be neglected (ref. 2.1). As no ionisation degrees below 10-4 will be considered, and as almost everywhere in the generator T will be considerably higher than T, it follows from

e

the comparison of the rate coefficients for the different collisional ionisation and recombination processes (ref. 2.2) that the electron-atom collisions constitute the most important ionising reaction and electron-electron-ion interaction the most frequent recombination process. The forward and reverse rate parameters kf and kr' which appear ~n the right-hand side of the continuity equations, are then given by:

3.75 x 10-22 T312(E /kT + 2) exp (-E /kT)

e exa e exa e

(II.6)

k = 1 • 29 x 1 0-44 (E /kT + 2) exp { (E. - E )/kT }

r exa e ~a exa e

(II. 7)

For argon, E and E. are 11.5 and 15.75 eV respectively.

exa ~a

In an MHD generator the development of non-equilibrium ionisation can be described by the continuity equations. The Saha equation follows from these equations if the number of ionisations equals the number of recombinations. In the momentum equation for electrons (Table 2.1)

the inertia term is neglected; comparing this term with the collision term of the right-hand side, it appears that when neglecting the inertia of the electrons, a new minimum characteristic length is defined:

1..

=

u /(v . + v )

~n e e~ ea

The basic equations of Table 2.1 do not describe processes with characteristic lengths < Ain' In the cases discussed here, l.in will

(II.8)

always be smaller than_ "D' so that the validity of the space charge neutrality approximation implies the justification of the neglection of the inertia term. The collision frequencies v . and v , used 1n

· . e1 ea

the momentum equations as macroscopic quantities, are related to the elastic collision cross section as follows (ref. 2.3):

v =

e£ n

e

r

I

"*

"* I

"* "*

J

Q

e~ n V - U e f e d (v - u ) e (II.9)

with £ is either i or a. The contribution of inelastic collisions to the momentum transfer between the electron gas and the heavy particles is neglected with respect to the momentum transfer due to elastic collisions. This is because the frequencies of the inelastic collision

processes are low compared to v . and v and the efficiency of

e1 ea

momentum transfer is the same in both types of collision. The electron

elastic collision frequency related to the transfer of thermal energy

1s not defined in the same way as the corresponding quantity related to momentum transfer, but is given by the following equation (ref. 2.3):

(II. 10)

with £ is i or a. In this analysis it 1s assumed that ve£ may be

approximated by v~£ so that in the energy equations the same collision frequencies appear as in the momentum equations. Q is taken to be

ea

constant and equal to 0.5 x 10-20 m2; v . is taken in accordance with

e1

Spitzer's theory (ref. 2.4). The radiative energy is neglected.

Ohlendorf (ref. 2.5) estimated that the radiative losses in a non-seeded argon plasma are several orders of magnitude lower than in a potassium-seeded plasma. As in a potassium-seeded plasma the radiative losses are comparable with the elastic losses, in a non-seeded plasma the radiative losses are

small compared to the elastic losses. In the energy equation for electrons,

2

terms of the order m u are neglected with respect to terms of the order e e

kT . Furthermore, heat conduction processes are not included in the

e

CHAPTER I I I

Geometry

of the

disc

~enerator

The amount of electron temperature elevation depends on the geometry of the MHD generator. Fig. 3.1 shows diagrams of a continuous and a

segmented Faraday generator, a linear Hall generator and a disc Hall

generator, these being the most general geometries. The following

insulator

c

Fig. 3. I MHD generator geometries: continuous Faraday generator (a), segmented Faraday

generator (b), linear Hall generator (c), and disc Hall generator (d). EL and jL are the electric field and the current density corresponding to the Lorentz

~ ~ . ~ ~ . .

force e(u x B), respecttvely. EH and JH are the electrtc fteld and the current

density owing to the Hall effect, respectively.

expressions for the ratio of T and the stagnation temperature T

e o

are derived by Hurwitz (ref. 3.1) for the continuous and segmented Faraday generators, and the linear Hall generator respectively:

5 ( 1 2 { 2 2 } M2 T 1 +

-

K) WT /(1 + WT ) e 9 T _{+ _!_ M2} 0 3 (III. 1) 1 5 2 w-r2 M2 T +

g

(1 ~ K) e T 1 _!_ M2 0 + 3 (III.2) . 1 ~ M2 2( 2 2) /( 2 T + W1 . 1 .+ . K .W1 . . 1 + .w-e) e 9 T _{_!_} M2 0 + 3 (III. 3)

where Cp/CV is taken equal to 5/3 and inelastic losses are neglected.

Eq. (111.3) holds also for the disc generator, if M is related to

the radial velocity. It can be shown from the. equations (111.1),

(111.2) and (111.3) that the presence of a Hall electric field favours

the electron temperature elevation. For the ratio of T and T 1s e o limited to 5/3 for K

=

0 and M ~ oo in the case of the continuous

generator, whereas for the segmented generator types T /T is unlimited

e o

and increasing with the Hall parameter.

In linear MHD channels the Hall electric field can be built up provided segmented electrodes are used. The characteristic distances for electrode segmentation are shown in Fig. 3.2. Celinski (ref. 3.2) shows that

finite segmentation results in an inferior performance of the generator.

h

s

®8

Fig. 3.2 Characteristic lengths for electrode segmentation.

The reduction of three important generator quantities is given 1n Table

3.1 for the segmented Faraday generator. As shown in ref. 3.2, the reduction parameter

A

becomes considerably smaller than unity for values of WT ~ 3 and for s/h ~ 1. Moreover, hot boundary layers near the insulator segments reduce the Hall electric field (ref. 3.3).

In order to avoid the problems connected with electrode segmentation, the disc geometry can be used for a Hall type MHD generator, as

suggested by several authors (refs. 3.4, 3.5, 3.6). A disadvantage of the disc generator in comparison with the linear generator is the limitation to the Hall mode of operation; in the linear geometry, the

possibility of various load connections results 1n many different

modes of operation (ref. 3.7).

Table 3. 1 The effect of finite electrode segmentation.

quantity ideal generator real generator

(s/h = 0) (s/h > 0)

current density (I - K)auB A(I - K)auB

electrical power density K (I - K)au B 2 2 h (I - K)au B 2 2

Joule heating per cubic metre (I - K) 2 au 2 2 B A (I - K) 2 au 2 B 2

A diagram of the disc generator is g1ven 1n Fig. 3.3 .. The gas is

supplied to the centre of the disc-shaped MHD channel and flows radially outward perpendicularly to an axial magnetic field. The Lorentz forces acting on the electrons and ions of the medium cause an azimuthal current density component and a radial Hall electric field. The load can be connected between two sets of concentric electrode rings.

Fig. 3.3 Cross-section of a disc Hall generator.

1L

represents the current density

.,. .,. .,. -t . f' ld

corresponding to the Lorentz force e(u x B), EH and JH the electr1c 1e and the current density owing to the Hall effect.

The behaviour of the medium in a disc generator ~s analysed by solving the basic equations of Chapter II for a one dimensional stationary flow. For that case the conservation equations, given in Table 2.1, transform into those given in Table 3.2.

CONTINUITY EQUA:r!ONS du dn n u J ELECTRONS : n er + e = - e er + _kfnena - _k d r u d"r" - -- n e er r r e du dn n u J IONS r e = - e r ₊ kfnena - k : n - - + u d"r"

-

-

n e dr r r r e .-l "' CT ... "' du dn n u 3 NEUTRAL PARTICLES : n - -r + u - a = - - -a r - _kfne01_a + k n a dr r dr r r e :--' N n 0 ::1 "' "' ...

t·IOHENTUN EQUATIONS < _{"' ....}

... dT dn 0 ::1 ELECTRONS, R-COMPONENT : n k - - + e kT _d"r" e = - n e (E + u B) + n m (v . + v ) (u - u ) e dr e e eq, e e e1 ea r er ..0 "'

"

"' .... _.....

ELECTRONS, q,-COMPONENT : 0 = n e_e u _er B + n m _{e e} (v _ei + _{v e) Cu.p} - ue¢)

0 _N ::1 _~ "' "' 2 du dT dn u¢

HEAVY PARTICLES, R-COMPONENT : nmu r nk kT = n e(E u.PB) - n m (v . v )(u - u )

d"r" + - + - nm - + + + r dr dr r e e e e1 ea r er

"

"

... _..... "' p.. .... du<P u u

HEAVY PARTICLES, cp-COMPONENT : nmu =-nm- - -r ; n eu B - n m (v . + v ) (u - _ue<P)

r dr r e r e e e1 ea cp 0 .... ::r "' p.. ... tn n

, _{ENERGY EQUATIONS} I)Q

"' ::1

"' ...

5 dT dT dn m

ELECTRONS : ku e - k e - u kT e = neeE(ur - ) + _neeB(uruef - _ueru<P) - 3n e (\. + ·~ ) k(T - T)

2 n e er d"r" u n r e d"r" r e d"r" u er e m ei ea e "' .... 0 :' - c2 KT + _{Ei) (kfnena} - k n3) 2 e r e 3 dT dn m HEAVY PARTICLES : I nku r dr - u kT r dr = 3n e m ~ (v el . + v ea ) k (T e - T)

C H A P

T

E R I V

Stationary soZutions

of the

basic

equations

IV.! Introduction

Numerical solutions of the set of equations for the disc generator, gLven in Table 3.1, are calculated with an Electrologica X 8 computor using a Runge-Kutta method. Comparable solutions of a similar set of equations for an ideal segmented linear Hall generator are also computed. As a result of the calculations in this chapter, several quantities of the

MHD

medium will be given as functions of the position in the generator.

The functions are gLven for values of the radius between 0.03 and 0.20 m in the disc generator case and for generator distances between 0 and 0.20 m as far as the linear generator is concerned, these being the extreme values representing the inlet and outlet of the channel.

The plasma properties at the inlet are chosen as follows:

n e u er u ex T e

=

ur

=

1800 m/sec, u~

=

u X T 1800 m/sec 9000 °K. 0 (disc generator); (linear generator);

For the linear generator, only solutions are given that are related to open-circuit conditions, whereas for the disc generator both loaded

and open-circuit conditions are discussed. The radial current density is assumed to flow for 0.07 < r < 0.14 m, the extreme values of r representing the electrode positions:

26 -u u for r < 0.07 m and r > 0.14 m er r (VI. 1) u ~ u for 0.07 < r < 0.14 m er r

The value of the load ~s determind by the imposed discontinuity in

u at r = 0.07 m; in fact u is supposed to drop there to 0.65

er er

times its original value.

In the disc generator the magnetic induction ~s assumed to have the

following radial dependency (compare chapter VI):

B B (1 - 0.51 r - 9.56 r ) 2 0

with r expressed ~n m.

Various magnetic field strengths are considered by choosing B successively equal to 0, 0.01, 0.03, 0.05, and 0.07 T for the

0

open generator conditions; for the loaded generator, the values

(IV. 2)

0 and 0.01 T are not considered because they do not represent a realistic MHD generator situation in connection with the implicitly imposed radial current density component. The magnetic induction in the linear generator is chosen to be constant and equal to B .

0

The choice of the var~ous parameters is based on measured values

resulting from the experiment described ~n chapter VI (see for

measurements the chapters VII and VIII).

The calculated solutions are represented by the curves g~ven ~n the

Figures 4. 1, 4.2, 4.3, 4.4, and 4.5. The plots marked (a) concern a loaded disc generator, the plots marked (b) an open disc generator, and the plots marked (c) an open ideally segmented linear generator.

IV.2 Temperature, density and radial flow velocity of the electron gas

For the conditions considered, Fig. 4. 1 shows enhancements of the

8000 1f -4000 ... ... 1) I I I I

'

...

'

__ .... -1 I

I

I

\

I I I ""/ I I

___

_

.,..,/ 10000 800~3._.o__..__...___._ __ 2 ... o---'.__...___._ __ ... ,_o___. 10_{log x (m)}

Fig. 4. I Variations of the electron temperature Te (dashed lines) and the heavy particle temperature T (solid lines) with the generator distance (r - 0. 03 m in the case of the disc geometry and x in the case of the linear geometry), at various values of the magnetic induction B

0•

a. Loaded disc generator, b. open disc generator, c. open linear Hall generator. Electrode positions in the disc: r = 0.07 and r = 0.14 m.

n = 2 x 1023 m-3 u = u = u

a ' er r ex

Plasma conditions at channel inlet: n = 2 x 1021 m- 3 e

ux = 1800 m/sec, u~ = 0, Te = T = 9000 °K. In c. the curves of Te and T at B = 0 and B = O.CI T coincide.

-24 C:_s ~ 23 0'1 0 ~~22 ":'_g ~ 21 0'1 0 a 23 ':'E

'?22

8' 0 \21 0'1 0

-c / ~

"'---~~;

§' 20_3.0 -2.0 s:? 20 _{-3.0 -2.0 -1.0} 1Cltog(r -0.03) (m) 1clog x (m)

Fig. 4.2 Variations of the electron density ne (dashed lines) and the heavy particle density na (solid lines) with the generator

distance. For a further description, see Fig. 4.1.

3500

~2500

:§

~ :::1 ;:;; ~1500~~~L_~~~~~--J-~~~ -10 -2.0 10_{log<r-0.03> <m>} c 1700

~

....

E 1600 ... )( =-!

~

1500 -~3.~0~~~~~~~~~~~~~ 1Dtog x (m)

Fig. 4.3 Variationsof the electron velocity and heavy particle velocity (uer and ur in the case of the disc geometry, and uex

and uxin the case of the linear geometry) with the generator distance. The dashed lines in a. represent uer as far as

it differs from ur. For a further description see Fig. 4. I.

4.0 3.0 2.0 1.0

...

3

E

-

0

e

3.o

-a-o

.

O?r

a=o.osr

8,:0.03! fu0.01I

a,: a

/ 0

8'

EbO OlT / a- a.-Jl

_

]1---

-

-3.4 L__,____l _ _.__--L.___.~..J.____._..L...,-__, -3.0 -2.0 -1.0 10_{log x .Cm>}

Fig. 4.4 Variation of the electrical conductivity o with the generator distance. For a further description see Fig. 4.1.

a

~it ions

....

3 4D 30

b

1.00

c

.... 0.251::---~---~ 3

Plo

-2.0 -1.0 -3o~~--~~--~~--~~--~__, .0 -2.0 -1.0 1o log Cr -0.03)(m) 10log J{ (m)

Fig. 4.5 Variation of the Hall parameter wT with the generator distance. For a further description, see Fig. 4. I.

N

"'

30

-10,000 °K with relaxation lengths of about 0.01 m. In the case of the higher magnetic induction relaxation lengths are shorter and the T levels higher because of the larger amounts of Joule

e

heating. Fig. 4.1 band c shows that even at B = 0 the electron

0

temperature will be higher than the gas temperature. This is caused by the initial value of the electron density which is chosen to be

higher than determined by the Saha equation; in fact, the recombination

energy is added to the electron gas resulting ~n T > T.

e

The electron temperature varies in different ways ~n three

distinguished regions. These changes will be discussed for one particular curve, namely the curve in Fig. 4.1 b, belonging to

B

0

10

= 0.07 T. For log(r - 0.03) <

-'V 2, Joule heating of the

electrons causes the 10 elevation of T e 0 from 5000 up to 13,000 K. For - 2 < log(r -'V 0.03) < -'V· 1.12, a further increase of T _e

from 13,000 up to 15,000 0 K occurs, because the expansion of the

• • -+ -+

med~um results ~n a h~gher value of u x B and a decrease of the 10

collision frequency. For log(r- 0.03) > - 1.12, T drops owing

"' e

to several processes connected with the setting in of non-equilibrium

ionisation. These processes are the following:

- The ionisation energy is withdrawn from the electrons.

- As

Q .

>>

Q

ionisations result in an increase of the total

e~ ea

collision frequency stimulating the thermal contact between

electrons and heavy particles.

h h f h 1 ' 1 d . . h 7 -+

- By t e en ancement o t e e ectr~ca con uct~v~ty t e J x B

-+ +

braking force becomes stronger, resulting in a reduction of u x B.

It follows from equation (III.3) that T - T in a loaded Hall parameter e

remains lower than ~n an open one; this effect is illustrated ~n

Fig. 4. I a showing a drop in T at the inner electrode.

e

The occurence of non-equilibrium ionisation ~s shown in Fig. 4.2.

For the given parameters, n can be raised by one order of magnitude e

owing to NEI. The relaxation length is of the order of 0.1 m. Higher

levels of additional ionisation and shorter relaxation lengths are ·

connected with higher values of the magnetic induction. The limited non-equilibrium ionisation in the loaded generator is a result of

the reduced electron temperature enhancement.

The radial electron flow velocity in an open generator (see Fig. 4.3 b and c) remains always equal to the radial flow velocity of the heavy particles. This results from the following relationship, which 1s derived from the basic equations:

n (u - u )

=

constant

e r er (IV, 3)

In the loaded disc generator (Fig. 4.3 a) the radial flow velocities

u and u are also related by equation (VI.3) except at the electron

r er

positions where the curves of u show discontinuities.

er

IV.3 Radial flow velocity and temperature of the heavy particles and

density of the neutral particles

In practical

MHD

generator cases and also in given numerical examples

-+

the changes of the quantities n , u and T can be approximated by those of n ,

-+ a g

u and T being the particle density, velocity, and temperature

g g

respectively of the total gas. Conservation equations for the whole

medium can be obtained from Table 2.1 by adding the corresponding

equations for the different plasma components. The curves of u and _r

T, shown in Figs. 4.3 and 4.1, will now be interpreted by the total

gas equations. In a dimensionless form the r-component of the momentum

equation and the energy equation of the total medium in the disc generator are successively given by:

dP dUR L -t- -+ - + _{dR =} 2 (J X B) dR r (IV.4) pgugr 5 dP _{(~ p} I) dUR ~ p - - + + _{dR +} 2 dR 2 2 L -t- -+ -t- -+ 3 { J.E- (J x B)~ug~} pgugr (IV.S)

32

-where pg is the mass density of the gas. P, UR and Rare the

normalised pressure, radial flow velocity, and radius, respectively, the normalisation relationship being given by:

u _gr 2 = p u p 0 0 r

=

r R 0 (IV.6)

The analysis is given for a fixed, arbitrarily chosen generator position r

=

r where u

=

u · this results in UR and R being equal

o r o'

to unity. From equations (IV.4) and (IV.S) the following relation-ship can be found:

2 dUR

( 1 -

MR_)

dR + (IV. 7)

In equation (IV.7) the Mach number MR is related to the radial flow velocity. The interaction of the medium with the electric and magnetic fields is represented by T(R). E,M· T(R) = E,M L -r-+ 5 -r -+ -r -+ { J.E- ~ (J x B) u - (J x B)~ug~ } .. r gr '+' '+'

In MHD generators T(R)

~s

always greater than zero. E,M

Comparing the curves of u (Fig. 4.3 a and b) with those of n

r e

(IV. 8)

(Fig. 4.2 a and b), it can be seen that the behaviour of the flow velocity depends on whether non-equilibrium ionisation has been

(R)

developed or not. If not, TE M will be small and the siutation

'

is described by equation (IV.7) with the right-hand side equal to zero; as in the given example MR > 1, the radial flow velocity will

then increase. In the region where non-equilibrium ionisation has effectuated high electrical conductivity, the positive right-hand side of equation (IV.7) determines the value of dUR/dR resulting in a deceleration of the radial flow.

The neutral particle density and the heavy particle temperature are shown in Fig. 4.2 and Fig. 4.1, respectively, as functions of the generator position. In the disc generator, the two quantities are determined by the expansion of the medium for r smaller than the

ionisation relaxation length; considering supersonic gas velocities,

n and T decrease in that region. For r larger than the ionisation a

relaxation length, na and T tend to increase owing to the

. f ~ ~

1n luence of the J x B braking force.

For the linear generator, T, n and u are plotted in Figs. 4.1 c,

a

4.2 c and 4.3 c; the curves are similar to those for the disc generator, except for the typical expansion effects.

IV.4 Electrical conductivity and Hall parameter

Both the scalar electrical conductivity and the Hall parameter are strongly related to the electron elastic collision frequency. In the given examples the plasma is Coulomb collision dominated.

In a Coulomb collision dominated medium o is 1n first order proportional

to T3/Z. this explains the similarity in the o and T variations

e ' e

(compare Figs. 4.1 and 4.4). Furthermore, it follows from Fig. 4.4 that in the given example the value of o is higher than in practical MHD generators, where generally values below 100 mho/mare found. If

the Coulomb collisions are in the majority, WT is approximately

-1 3/2 .

proportional to n T . It can be seen from F1g. 4.5 that for values

e e

of r smaller than the ionisation relaxation length WT is strongly

influenced by T . If r exceeds the ionisation relaxation length, the e

increase of n by the non-equilibrium ionisation, together with

e

the simultaneous decrease ofT , causes a drop in wT. e

Generally, it can be stated that especially if v . > v - at least in

e1 ea

the non-equilibrium ionisation region - the value of the Hall parameter

positions where the ionisation degree has already been enhanced.

Then, in order to have a reasonably high WT in the main part of the

Hall generator, WT in the relaxation region must be far above the

CHAPTER

V

Critical values of the

Hall parameter

with

respect

to ionisation

instabilities

V. 1. Introduction

In MHD generators the development of instabilities ~n the plasma can

result in poor performance of the device. The most important types

of instabilities occurring ~n MHD generators are the magneto-acoustic

and the ionisation instabilities; from the two, the latter have

generally the greatest effect on the generator output, and they will

be discussed here.

Non-linear effects ~n Ohm's law, which result from ionisation

instabilities, are described by introducing an effective electrical

conductivity creff and an effective Hall parameter wTeff' Neglecting

V'p Ohm's law is then given by:

e -t WTeff -t x +B J + ·- - J = B + creff E>~ (V. 1)

The values of WTeff and creff are lower than the values of WT and cr;

the measure of the reduction depends on the amplitude of the fluctuations.

Using first-order perturbation theories, several authors have calculated

critical values of the Hall parameter that represent upper limits of

stability (refs. 5.1, 5.2, 3.6). They all assume Saha equilibrium and

exclude the ionisation relaxation reg~on of the generator. As in this

region the Hall parameter has far higher values than in the region of

Saha equilibrium (see chapter IV), in the present chapter critical Hall

parameters will be calculated without assuming the validityi of the

Saha equation.

V.2. First order perturbation equations

.

Ionisation instabilities consist of fluctuations in n ,

~

,

e e

The quantities n ,

~

and T are assumed to be constant within

a .

+

T and E. e

distances

36

-ionisation instabilities are described starting from the conservation equations of Table 2.1 as far as they are related to the electron gas, and eqs. (II.4) and (II.S). From the combination of the continuity equation for the electrons and eq. (II.4), it follows that the former may be replaced by the continuity equation for the ions.

Considering the transition from stability to instability, a first-order perturbation theory is justified, because the fluctuations are small in the primary stage of their development. The zeroth-order terms represent the stationary behaviour of the medium, and the first-order terms represent the fluctuations, as can be seen from the following division of the quantities _ne' -t J , T and E 1n _e -+. zeroth and first-order terms:

(O) -+ ( 1) -+ n n (r) + n _{(r' t)} e e e -t -t(O) (-;) +

jC

1) -+ t) J = J (r, (V. 2) T(O) _(-;) + T(l) -+ _t) T (r, e e e -+ -+(0) _{( ) -+(1)} -+ E E r + E (r, t)

Substitution of eq. (V.2) in the basic equations and subtraction of the zeroth-order relationships result in three first-order conservation equations, namely the continuity equation for ions, the momentum

equation for electrons and the energy equation for electrons. They are given by the following relationships respectively:

a

(I - R)

an

e n =n(O) - e e T =T(O) e e

a

(I - R) aT e _{n =n(O)} - e e T =T(O) e e (V. 3)

·

(aa

ane . (O) n =n e e (I) n e . ()a + - -

_ar

_..

e T(I)) + n =n(O) e T =T(O) e e e e T =T(O) e e (I) W.,.(O) _..(1) -+B L ( ' J _{X -} ) (O) B W..-(O) _..(0) -+B L ( ' _J ) X -(0) B n e ~ +

E(l )

*

(6)

= a a n e

-t(I) -+(0)* -t(O) -+(I)*

J .E + J .E ClA (I) n e

---ane (0) n =n - e e T =T(O) e e

-dUe (0) n =n - e e T =T(O) e e (I) n e ClA ClTe (0) n =n - e e T =T(O) e e

(V.

4)

(V.S)

In eq. (V.3) the functions I and R represent the number of ionisations and the number of recombinations per unit volume and unit time,

respectively. In eq. (V. 4)

y

is the first order term of -I Vpe: n

c

(0) T(O)Vn(O) T(O) )

e -+ k ne _{T (I)} e e (I) _{+ VT(l)} e (I)

(V. 6)

Y = ; n(O) e (0)2 n e e + n(O) Vne n

e e e

The energy lost by the electrons ow~ng to elastic collisions with heavy particles ~s given by the function A in eq. (V.S), while the energy transfer owing to ionisations and recombinations is given by

m

A=

3n ~

k (T - T)(v .

+

v )

e m e e~ ea

Eqs. (II.4) and (II.S) result in the following first order relationships:

0

-+ -+ -+ + +

As E•{ ~s given by E•{ = E + u x B and as no fluctuations for the

(V. 7)

(V. 8)

(V. 9)

(V. 1 O)

. . + + .

quant~t~es u and Bare assumed, ~t follows from eq. (V.10) that the

vector field

E

*

(

1) is curl free:

-r;{ ( 1) 'i] X E

=

0

V.3. The calculation of critical values of the Hall parameter for some special cases

V.3. 1. The reg~on where the Saha equation is valid

(V. 11)

In this section the region of the

MHD

generator will be considered, where in the unperturbed situation the electron density is governed by the Saha equation. The following assumptions will be made:

The zeroth-order energy equation of the electrons has the following simple form:

-t(0)2

J

=

A (0)

a(O)

In eqs. (V.S) and (V.6), terms of the order

'V'n(O) e (O) or n e (V. 12) are