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. (EUT report. E, Fac. of Electrical Engineering; Vol. 70-E-11). Technische Hogeschool Eindhoven.

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NON-EQUILIBRIUM PHENOMENA IN A DISC-SHAPED MAGNETOHYDRODYNAMIC GENERATOR

by

TECHNISCHE HOGESCHOOL EINDHOVEN NEDERLAND

AFDELING DER ELEKTROTECHNIEK GROEP DIREKTE ENERGIE OMZETTING

EINDHOVEN UNIVERSITY OF TECHNOLOGY THE NETHERLANDS

DEPARTMENT OF ELECTRICAL ENGINEERING GROUP OF DIRECT ENERGY CONVERSION

NON-EQUILIBRIUM PHENOMENA IN A DISC-SHAPED MAGNETOHYDRODYNAMIC GENERATOR

by

A. Veefkind

TH-Report 70-E-]] March ]970

ACKNOWLEDGEMENTS

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.

The author wishes to express his most sincere thanks to Dr. L.R.Th. Rietjens, head of the group Direct Energy Conversion, for his constant interest in this work and for the fruitful discussions. The indispensable technical assistance of Mr. C.J. Sielhorst is most gratefully

- I -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.I Introduction

IV.2 Temperature, density and radial flow

3 4 10 IS 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 respect to ionisation instabilities

V.I Introduction

V.2 First order perturbation equations

V.3 The calculation of critical values of the

35 35 35

Hall parameter for some special cases 38

V.3.1 The region where the Saha equation is valid 38

V. 3.2 The ionisation relaxation region 41

CHAPTER VII CHAPTER VIII CHAPTER IX APPENDIX REFERENCES - 2 -Measurements 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 l1icrowave measurements

VII.6 Piezo-electric crystal measurements

Discussion of the experimental results

Conclusions

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

- 3

-SUMMARY

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 equations. 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 OK. 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 -NOMENCLATURE Symbols A A1 ' A2 A P a +' a -+ B B 0 b C P C

_v

c D -+ E

E"

-+ EH EH ErR -+ EL E exa E. 1a 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 in 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' o h I I. L i p i po 7 J + K k L M m N m N £ 5 -friction coefficient

distribution function of particles belonging to species i

weight factor of the ion 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 belonging to species i

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 n~ n q p p Pe Pg Q_e2 q R R R R 1, ReD R m R u r r Le r Li s T T 0 T (R) E,M T g T2 R2 6

-heavy particle density critical electron density total particle density

density of particles belonging to species 2 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 2 integer number

dimensionless representation of the radius

number of recombinations per unit volume per unit time reflexion coefficient

responses of the crystals in 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 temperature of species 2

t t I • t 2 • t3 t. • t _out ~n U R

...

u

...

_u g

...

uR. V m Vfl V oc V p V po V pl V pl

...

v z z

"

...

_y t::.R. t::.Pe t::.T e t::.Vfl M 7 -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 R.

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 in 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 pat~s in the

Ii £ a £ r K A.

,

A.

'n

v v c V"), e~ Pg D Deff T disch ~

"

st_i , _"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 ion mean free path

characteristic length corresponding to electron inertia neglection

viscosity coefficient

microwave frequency

total electron elastic collision frequency

collision frequency relating to momentum transfer at elastic collisions between electrons and particles of species £.

collision frequency relating to energy transfer at elastic collisions between electrons and particles of

species R.,

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 "

angular frequency of microwaves

plasma frequency Hall parameter

WT(O) cr wTeff WT (0) stab Szpersanpts (0) (I) Subs anp ts a e i m, n r, ~, z x, y, z Shorts ETE LTE MIlD NEI 9

-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

-C HAP T E R 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 energy On a large scale (1000 MWe) from fossil fuels. High efficiencies (50 %) are expected from combinations of I1HD and conventional systems. Already now, experimental MHD generators in open cycles are capable of converting 6 % of the thermal energy of the medium into electrical energy at an output of 30 MW

(ref. 1.1).

- Closed cycle MHD generators using 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 in the reactors and in the MHD generators in the present stage of their development. Up to now, the pressure of the gases used in

o

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 oK.

- The problems connected with the use of a nuclear heat source are avoided in the mixed cycle systems (ref. 1.3). In these systems the heat is produced by fossil fuels and is transferred by means

- II

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

The main problem related to closed cycle systems with gaseous media is how to achieve a sufficiently high electrical conductivity of the gas. At temperatures of about 2000 oK and pressures between I 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 1.1 gives 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

Ref. type type medium u T P B diagnostics effect reported discussion

,

..;

0

experiment generator (m/sec) (oK) (atm) (1) of results ;;

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 0 radiation losses and

~

0

~.

non-uniformity; if 0

<

accounted for these 0

effects. agreement ~

with theory 0

"

."

1.6, shock linear. Xe 1000 5700 I 0.25 - electrical enhancement ne agreement with theory; 0

"

1.7 tube segm. e1. + 0.5 % H 2.25 output and Te calculated non-equi I ibrium

a·

A 1710 5100 0.4 2.25 - form WT and a phenomena strongly 0

0

2.6 affected by loss "

"

mechanics n

0

1.8 plasma linear, 70 % He 2350 600 0.05 1.3 electrical small enhancement only small evidence of 0 n

jet segm. e1. + 30 % A output of 0; voltage electron heating and 0

"

oscillation magnetically induced

e.

ionisation 0

""

1.9 closed 1 inear, He < G. I 1060 1700 I 2 electrostatic enhancement of 0 agreement with theory 0 ~

loop segm. e1. - 3 , C probes 0

n

1.10 closed linear, He < 2 % 240 1300 1.3 Z. IS electrical no effect induced field to small "

_"

0 N

loop segm. e1. C, output 0

1.11 blow linear, He < '" 2500 900 0.3 - 1.4 electrical Te enhancement non-equilibrium down segm. e1. 0.23 % C, 0.6 output; calculated from behaviour stronly

electrical wTeff affected by loss

" 0

~

0

"

potential; mechanisms and

continuum relaxation phenomena

• " c " -<adiad"" ~ 0

I. J 7. shock disc A ' 1400

,

1700 1.3 3.4 continuum ne enhancement non-equilibrium

tube I % C, radiation from radiation ionisation accompanied

measurements by large ne fluctuation

~ 0 <

,

"

-.

0

1.13 plasma linear, A + 0.1

,

700 1500

-

I 0.2 electrical enhancement ne non-equilibrium jet segm. e1.

-

3 % K 3000 potential and Te calculated behaviour strongly

from w1eff influenced by boundary layers 0 ~ 0-0 g

I. 14 blow linear, He + 200 - 1200

-

1.2 2.7 electrical no effect currents to small down segm. e1. o. I % K 1000 1700 - 2 output

,

0

~

~.

1. 15 closed linear, He + 1417 1403 0.65 0.5 - electrical no effect influence loss

loop se.emented O.IS%Cs I. 97 output mechanism too strong

~

tr ~.

1.16 blo .... linear, He + 1400 - 1500 I 4.5 electrical enh ancemen t of 0 non-equilibrium §

down segm. e1. 2 - 5 % c, 2000 output behaviour strongly

-.

affected by losses; 0 0

accounting for them

_.

" 3.1!reemen t yi th theo!.L

,

"

-.

0 ?

13

-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 in 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 seeding materials, in order to make possible practical conversion 'of energy using MHD closed cycle systems.

The aim of the present work is to examine ETE and NEI in 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 various 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

-two 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 m1cro-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 in 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

- 15

-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

the Boltzmann equation

...

fi(v, r, t), which can be obtained by solving for each species i. 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

(II. 1)

The assumption given by equation (11.1) 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

i ,

h . ... d h

t e flow veloc1ty u

i an t e temperature Ti .

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

- 16

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

n

I

e « 1 (11.2)

This assumption determines the Debye length as the minimum characteristic length in the plasma to be described. Neglecting In - n. I with respect

e 1

to n or

e ni , one

Poisson

may replace n. by n in the conservation equations.

1 e

From the equation for electrical space charge and equation

(11.2) the following condition for the variation of the electric field can be derived:

Iv·E:1

« n e e. E o (11.3)

Once having found the solution of the problem, the condition (11.3) can be verified in order to justify the substitution of n for n ..

e 1

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:

~

V.J = 0 (II.4)

(II.S)

Equation (11.4) 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 given in 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

Table 2.1 Conservation equations. CONTINUITY EQUATIONS a ~ .. k n n - k 02n. ELECTRONS

,

- n + V.n u at e e e f e a r e 1 a ~ ₂ IONS

,

_{3t n} i + V.n.u 1 = k 0 n f e a - krneni a ~ ₂

NEUTRAL PARTICLES

,

- n _{at a} + V.n u _a = - kfnen_a + krneni

MOMEHTIIM EQUATIONS ~ ~ xii) + n m (~ - ~ ) ('I> • V ) ELECTRONS

,

0

.

-

'VP e - nee(E + u e e e e e1 + ea a _{(nm~) 'V. (nm~) n.e(E} ~ x B) - n m (~

-

~ ) (v . + v ) HEAVY PARTICLES

,

at + .. - Vp + 1 + U e e e e1 ea ENERGY EQUATIONS a ₍ 3 I 2 _{) (} 3 _{+ 1. m u}2 ) ~ ) ~ V.t; ~ ~ ~ ~ ~ )(v . + v )

ELECTRONS

,

_{ne (2 kTe} + E. + '2 meue) + v. ne(I kTe + E. u = - ue,vP

e - Pe - n eE.u + nemeu. (u -at 1 1 2 e e e e e e e e1 ea m

-

3 n ~ _{('I> ei} + v ) k (T - T) e m ea e a _{( (1 kT} I 2 _{) ( (1 kT} I 2 ~ ) ~.Vp - p'V.; ~ ~ ~ ~ ; ) (v . + v ) HEAVY PARTICLES

,

at n + "2 mu ) + V. n -+ 2 mu ) u = - + n.eE.u nemeu, (u -2 2 1 e e1 ea m + 3 n e (v ei + v ) k (T - T) e m ea e

18

-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 oK

and electron densities above 1019 m-3, the radiative ionisation and

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 k_f and kr' which

appear in the right-hand side of the continuity equations, are then given by:

k

f

=

3.75 x 10-22

T3/2(E IkT + 2) exp (-E IkT)

e exa e exa e (II.6)

k = 1.29 x 10-44 (E IkT + 2) exp { (E. - E )/kT }

r exa e 1a exa e

(II. 7)

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

exa 1a

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:

,. = u I(v . + v )

1n e e1 ea

The basic equations of Table 2.1 do not describe processes with

characteristic lengths < 'in' In the cases discussed here, 'in will

- 19

-always be smaller than. AD' 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 in

. 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

J Q , eX, I~

-

~ e

I

f d e (~ - ~ 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

momentum transfer is the same

v . and e1 in both

v and the efficiency of ea

types of collision. The electron elastic collision frequency related to the transfer of thermal energy is not defined in the same way as the corresponding quantity related to momentum transfer, but is given by the following equation (ref. 2.3):

v* _e£ m e =

n

3kT e e (II.10)

with £ is i or a. In this analysis it is 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

- 20

-CHAPTER I I I

Geometry of·the disc generator

The amount of electron temperature elevation depends on the geometry of the MF~ 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

c

d

Fig. 3. I MHD generator geometries: continuous Faraday generator (8), segmented Faraday generator (b). linear Hall generator (c). and disc Hall generator Cd). EL and

1L are the electric field and the current density corresponding to the Lorentz

force e(~ x B), respectively. EH and TH are the electric field and the current

density owing to the Hall effect. respectively.

expressions are derived

for the ratio of T and the stagnation temperature T

e 0

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) . wr / (1 + WT ) e 9 = T ₊

_1.

_M2 0 3 (Ill. 1) 5 2 2 2 T .1.+

9"

(l-:K) .WT .. M e - = T 1

1.

M2 0 + 3 (III.2)

2

M2 2( . 2 2)/( _{.+ wr2)} T 1 + wr. 1 . +. K WT 1 e 9 = T

_1.

_M2 0 + 3 (Ill. 3)

21

-where Cp/C

v

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 is

e 0

limited to 5/3 for K = 0 and M + ~ in the case of the continuous

generator, whereas for the segmented generator types T

IT

is unlimited

e 0

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 5 I I I c

.,

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 slh ~ I. 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

22

-possibility of various load connections results in 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 (1 - K)auB A (1 - K)auB

electrical power density K ( 1 - K)au B 2 2 AK (1 - K)au B 2 2 Joule heating per cubic metre (1 - K) au B 2 2 2 A(1 - K) 2 2 2 au B

A diagram of the disc generator is given in 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.

out.r

elect Ie ₊

inner .1

Fig. 3.3 Cross-section of a disc Hall corresponding to the Lorentz

+

+

-t h d · t

generator. 1L repre£ents t e current enS1 y force e(~ x B), EH and iM the electric field

23

-The behaviour of the medium in a disc generator is 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.

.. --CUNTINUITY EQUATIONS du do 0 u 3 ELECTRONS er • e = e er kfnen ll - k

,

0 _d"r"""" u _{dr -}

---.

0 e er r r e du do 0 u 3 IONS

,

0 _{e dr} - -r

.

u _{r dr} e = e r _r + kfnen_a - k _{r e} 0 _• ..; 0' ~ • du do 0 u 3

!'lEUTRAL PARTICLES

,

0 _{a dr}

--

r

.

u _{r dr} a = -~-_r k[ne.:1a

•

k _{r e} 0

,.,

N n a a • • " HO!'IENTIDI EQUATIONS <

•

" e· dT do 0 a

ELECTRONS, R-COM?ONENT

,

n k _ _ e . kT _dr e = - n e (E • u B) _• n m (v . • v )(u - u )

e dr e e e, e e el ea r er ~ •

C

•

" e·

ELECTRONS, ¢'-COMPONENT

,

0 = 0 eu B _• 0 m (v _• v ) (u - _ueq,)

e er e e ei ea ,~ 0 N a .0-• • du 2

HEAVY PARTICLES, R-COMPONENT r ok dT kT do u, n e (E u1>B) - n m (v . v

e) (ur

-

II )

,

omll dr •

-.

dr = nm - +

•

• r dr r e e e Cl er ~ ~ ~ e· m ~ " dll, U ll.

HEAVY PARTICLES, ¢-COMPONENT

,

nmll r . B - _neme(vei • _\!e)(U¢ - u_e¢)

dr =-n m - - - 0 ell r r e r 0 " 0' • ~ e' • n • ENERGY EQUATIONS ~ m a m " ~ n ku dT dT dn m ELECTRONS

,

e - II 0 k e II kT _ _ e = ne eE (U r - ) • neeB(urUe,,~ - ueru¢)

-

30 --"- (v • u ) k(T - T) dr ~- U 2 e er r e r e dr er e m ei eo e • " 0 ~

-

(2- KT _{• Eia) (kfnen a} - k n3) 2 e r e 3 dT _{u kT dn} m

HEAVY PARTICLES

,

,.

okll

-

= 3n --"- (v . • v ) k (T - T)

25

-CHAPTER I V

IV.I Introduction

Numerical solutions of the set of equations for the disc generator, given 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 given 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 = u er r u _ex

=

u x 1800 m/sec, u~

=

= 1800 m/sec T T = 9000 oK. e

o

(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. I) u ~ u for 0.07 < r < 0.14 m er r

The value of the load is 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 is assumed to have the following radial dependency (compare chapter VI):

B B (I - 0.51 r - 9.56 r ) 2

o

with r expressed in 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

o

open generator conditions; for the loaded generator, the values

(IV.2)

o

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 •

o

The choice of the various parameters is based on measured values resulting from the experiment described in chapter VI (see for measurements the chapters VII and VIII).

The calculated solutions are represented by the curves given in the Figures 4. I, 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 electron temperature over the heavy particle temperature of about

12000 8000 'if -4000

....

...

"

a I t; ... ~..b

...

electrode sili s ~~;'I' I '1'~

=""''''''''-''''"'-'"'/~,.:::

_,

, I

_{, \}

"

----

,

,/ I \ " I , , I I I I I ~'L",,-____ "I"'" I a-QQ3L 16000 12000 --~ -"

,

,

"

b --;:::.---,',- /

...

~

,.

/

,

.... -" I I \ \ I

I

\

I I ' /1 I / 11000

~

c

B-~

_ _... -0ll3l ,-

-, / / ... ,/ .t / ... . / ----~" / '

,.,..---

\ \ - -....

"

:::---~----' . 10000 >-;.9000

..."

O~~L-~~~~~~~ - 3.0 -2.0 -1.0 }-~-~

oh=::::r:~~---L.-'---}-;:---lil

-2.0 -1.0 800~3.O,'--~...L..--'---2L.o--'-L---'--...J1.c,.0 ~ 1010g x (m)

I010g (r -o.03)(m) °log(r_0.03) (m)

Fig. 4.1 Variations of the electron temperature Te (dashed lines) and the heavy particle temperature T (solid lines) with the

generator distance (r -O.03m in the case of the disc geometry and x in the case of the linear geometry). at various

values of the magnetic induction Bo'

a. Loaded disc generator. h. open disc generator. c. open linear Hall generator.

Electrode positions in the disc: r = 0.07 and r = 0.14 m. Plasma conditions at channel inlet: n = 2 x 1021 m- 3

e n = 2 x 1023 m- 3 , u a er B = 0 and B '" O.C I T o 0 = u_r = u ex = Ux = 1800 ~/sec. u¢ = O. Te coincide.

= T = 9000 OK. In c. the curves of T and T at

e

N

-24 ~.§ t' 23 go ~22

121

_go a 2 _c /'. 23 '-...

"---""'~~

§' 20_3.0 -2.0

_-to

_-1.0 3500 '''Iog(r -0.03) (m)

Fig. 4.2 variations of the electron density fie (dashed lines) and the heavy particle density Da (solid lines) with the generator distance. For a further description. see Fig. 4.1.

a ₃₅₀₀

~2500

]

" =1-b B.O }---__ 8:Q01T 8:0.031 e:..Q.QR~...,.~-~15001~-L~ __ ~-!~~-L~_~-" -3.0 -2.0

-to

1Qlog(r _ O.03Hm) c 1800)1---===~~5i=::;:::::::-1700

~

E

1600 /

/

a..-o-

l-'"

t~~H

//

~- 0.05T /, B.: O.Qzr /

Fig. 4.3 Variationsof the electron velocity and heavy particle velocity (u

er 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 u

er as far as it differs from u

r' For a further description see Fig. 4.t.

,~

4. I

I

---=:::::;'~Di'

I I I 15 I I I I : I I I I I 25~~~-L~~L-~~~~ -3.0 -20 10log (r -0.03) (m) o !1' ~ 4.0 6.=0071 B:O.05T / f!,M~ a,.001T

a...o._y

b 2.5':::-'--"'-~~::-,--L---'"---..J~ -3.0 -2.0 -1.0 1olO9(r - 0.03) (m) 3.6 _c ~3.5

~ ~=====~::;:::::

o

f

8.00IT o

a..o. _)

/

3.4':;-;;-~-L.~---':-::-~~-L---C~ -3.0 -2.0 -1.0 1°109 •. (m)

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

4.0 3.0 2.0 1.0 I-' 3 ~0D51 8,=Q03T a

!l3:0

-2.0 10log(r -0.03) (m) -1.0 I-' 3 b 3D 10

.o~~~

-~

-w

-10 10log(r_0.03) (m) 1.00 0.75 0.501:-_----';...,.'--_ I-' 0 . 2 5 F - - - L t ' - - - - _ 3

30

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

heating. Fig. 4.1 band c shows that even at B

=

0 the electron

o

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 in T > T.

e

The electron temperature varies in different ways in three distinguished regions. These changes will be discussed for one

particular curve, namely the curve in Fig. 4.1 b, belonging to

B o

10

=

0.07 T. For log(r - 0.03) ~ - 2, Joule heating of the

electrons For - 2 < 'V causes the 10 log(r -elevation of T e 0.03) < - 1.12, 'V from 13,000 up to 15,000 K occurs, o from 5000 a further because the o up to 13,000 K. increase of T e expansion of the . + +

medium results in 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

'V e

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

ionisation. These processes are the follo~ing:

- The ionisation energy is withdrawn from the electrons.

- A s Q . »

e~ collision

Q ionisations result in

ea

frequency stimulating the electrons and heavy particles.

an increase of the total

thermal contact between

h . d · · 7 +

- By the en ancement of the electr~cal con uct~v~ty the J x B

..

...

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

It follows from equation (111.3) that T - T in a loaded Hall parameter

e

remains lower than in an open one; this effect is illustrated in

Fig. 4.1 a showing a drop in T at the inner electrode.

e

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

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

31

-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 is 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

+

u and T being the particle

g g

n ,

a

+

u and T can be approximated by those of n g' density, velocity, and temperature

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

u and

r

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 7}

1i)

(IV.4) - + - - = 2 (J x dR dR r PgUgr ~ dP + (~ _p dUR ~P L 7 + 7 + } + 1 ) - - + = { _{J .E -} (J x _{B) <j>ug<j>} 2 dR 2 dR 2· 3 PgUgr (IV.S)

32

-where P

g is the mass density of the gas. P, DR and R are the

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

2

p '=puP

g o o

r

=

r R o

The analysis is given for a fixed, arbitrarily chosen generator

(IV.6)

position r

=

r where u

o r

=

u . this results in DR and R being equal 0 '

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

1

=

M2 T(R)

-~ _E,M (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~:

_,

T(R) =

E,M

L

In MHD generators T(R) is always greater than zero.

E,M

(IV.8)

of n e Comparing the curves of u

r (Fig. 4.3 a and b) with those

(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 ~ > 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 dDR/dR resulting in a deceleration of the radial flow.

33

-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,

nand T decrease in that region. For r larger than the ionisation

a

relaxation length,

influence of the

T

n

a and T tend to increase owing to the

+ .

x B brak1ng force.

For the linear generator, T, na and u are plotted in Figs. 4.1 c, 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 a is in first order proportional

to T3/2. this explains the similarity in the a 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 a is higher than in practical

MHD generators, where generally values below lOa mho/m are found. If

the Coulomb collisions are in the majority, WT is approximately

proportional to n-IT3/2• It can be seen from Fig. 4.5 that for values

e e

of r smaller than the ionisation relaxation length WT is strongly

influenced by Te' If r exceeds the ionisation relaxation length, the increase of n by the non-equilibrium ionisation, together with

e

the simultaneous decrease of T , causes a drop in WT.

e

Generally, it can be stated that especially the non-equilibrium ionisation region - the

if v . > e1 value of v ea the - at least in Hall parameter is much higher in the ionisation relaxation region than in the generator

34

-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

35

-CHAPTER V

Critical values

of

the Hall·parameter with respect to ionisation instabilities

V.I. Introduction

In MHD generators the development of instabilities in the plasma can result in poor performance of the device. The most important types of instabilities occurring in 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 in Ohm's law, which result from ionisation

instabilities, are described by introducing an effective electrical conductivity G

eff and an effective Hall parameter WTeff• Neglecting Vp Ohm's law is then given by:

e

.,.

-t wTeff + ] + - - - ]

B x B = (V. I)

The values of wTeff and G

eff are lower than the values of WT and G;

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 region 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 validitylof the Saha equation.

V.2. First order perturbation equations

Ionisation instabilities

->

The quantities n , u and

a

consist of fluctuations in n , _e ~ _e ,T and _e

E.

T are assumed to be constant within distances comparable with the typical wavelengths of the fluctuations. The

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. (11.4) and (11.5). From the combination of the continuity

equation for the electrons and eq. (11.4), 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

-T

from the following division of the quantities n

e, ], zeroth and first-order terms:

can be seen

d + .

T an E 1n

e

(V.2)

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:

:l (I - R)

an

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

a

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

37

-w,

(0) (0) u

1.

kn(O) 2 e (I)

(

au

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

w,(O)

(0) u

au

+ -aT "" e n =n(O) - e e T =T(O) e e (I) ne ->- -+(1 ),' - - = y + E (0) n e -t(l) ->-(0)* -t(0) -+(1)* aA J .E + J .E -

an

e n =n (0) - e e (I) n e aA

-

W-e n =n(O) - e e (I) n e T =T(O) 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 _1_ Vpe:

n

->-

k(Vn~O)

T(I) T!O)vn!O) (I) +VT(I)

T~O)

(1)\ e

y =

e

n(O) e n(0)2 ne e + n(O) Vne

J

(V.6)

e e e

The energy lost by the electrons owing to elastic collisions with heavy particles is given by the function A in eq. (V.S), while the

- 38 -.

m

A = 3n ~ k (T - T) (v • + v )

e m e el. ea

Eqs. (11.4) and (11.5) result in the following first order

relationships:

'lxE:(J)=0

- + , . -+ - + - l o - - +

As E* 1S g1ven by E'~

=

E + U x B and as no fluctuations for the

(V.7)

(V.8)

(V.9)

(V. 10)

• • -+ -+

quant1t1es u and B are assumed, it follows from eq. (V.IO) that the vector field E:*(I) is curl free:

'J x E'~(I) =

a

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

V.3.1. The region where the Saha equation is valid

(V. I I)

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:

'In(O) e

In eqs. (V.s) and (V.6), terms of the order --7(0~),or

n

e

(V.12)

- 39

-'7n (I)

e neglected compared to terms of the order --~(~I~) or

n

e

. (I)' T

e

The Saha equation remains valid, even during the fluctuations. Phase shifts between the fluctuations of the various quantities are neglected.

According to the third assumption, eq. (V.5) has been replaced by the first-order Saha equation:

(I) n e (6) = n e 3/2 kT(O) + E. T(I) I e 1a e

2"

kT(O) T(O) e e (V. 13) The eq s • (V. 4), (V. 5), (V. 9), (V. I I) . (I) -t(1)

order that the funct10nS n , J , e

and (V.13) have to be solved in

T(I) and E,,(I) may be obtained. e

These equations can be transformed into one linear homogeneous equation in

n~!),

if one particular term of the Fourier series is concerned:

(I) _{= n (I)} { exp . ( + rlt)} n 1 K.r -e eo ~(I) J ~(I) = J_o {exp 1 . ( K.r + - rlt)}

T(I) T(I) _{exp . ( + _{- rlt)}}

= 1 K.r

e eo

E*

(1) ₌

E*

(1) {exp

0

i(K.

~

-

rlt)}

In (V.14) the frequency rI is complex:

rI - irl.

r 1

(V. 14)

(V. 15)

The sign of rI. determines whether the medium is stable (positive sign)

1

or not (negative sign) with respect to the chosen Fourier component. In the stability limit, given by rI.

=

0, n(l) must be solvable from the

1 eo

40

-C

K2 + K2 d In cr

Kl..

(0) d IIi.. A n =n(O) (I) x y + 2 2J... un n = K2 d In n n =n(O) K2 d In n eo e e

-

e e e e T =T(O) T =T(O) e e e e (V. 16)

The x-axis of the coordinate system is chosen

Ilr(O)

and the z-axis

.,. d

liB.

The operator d In n is defined as follows:

e

d

d In n e

(V. 17)

The existence of a non-trivial solution of n(l) from eq. (V.16) requires

(I) eo

the coefficient of n to be equal to zero. By defining for any

eo

K

WT(O)b as the value of WT(O) in the stability limit, the following

sta

expression results from eq. (V.16):

(0) K2 (K2 _ K2 In cr d In A n =n

(0»)

x y d _{(V. 18)} WT stab

_KK

_{K2 d} In n + n =n(O) d In n x y e e - e e

-

e e T =T(O) T =T (0) e e e e

In comparison with the expression of WT(O)b derived by Louis (ref. 3.6), sta

if applied to an unperturbed situation without fluctuations, eq. (V.18) has one more term resulting from the fluctuations in the electrical conductivity which are taken into account here. The critical Hall

parameter

WT~~)

can be found from eq. (V.18) by deriving the minimum

value of WT(O)b with respect to the direction of

K

in the xy-plane.

sta

In Fig. 5.1, WT(O) is given as a functin of T(O) for a gas temperature

cr (0) e

of 5000 oK. The figure shows that for T - T > 2000 oK the critical

e '"

Hall parameter is about 2, which has also been found by other authors

(refs. 5. I, 3.6). At smaller amounts of electron temperature elevation

slightly lower values ofwT(O) may be expected, except when T(O) - T is

cr e

T=5000'K 2.0 1.5 41 -..25 -3 n.= 1u m 24 -3 n •• l0 m .-23 -3 n._1U m 1022 -3

.=

m

Fig. 5.1 Critical Hall parameter w,(O) as a function of electron temperature Te' for _{cr .}

several values of the neutral particle density, in Saha equilibrium situations.

very close to zero. As grows to infinity when

may be seen from eqs. (V.20) and (V.9), WT(O) cr T(O) - T approaches zero.

e

V.3.2. The ionisation relaxation region

The relaxation region of an MHD generator consists of two parts: one is characterised by the relaxation of the electron temperature, the other by the ionisation relaxation (see Chapter IV). The former part is generally small, while in many experimental arrangements the latter cannot be neglected with respect to the dimensions of the generator. Instabilities of the medium are of influence on the length of the relaxation region as well as on the finally reached values of the quantities considered.

The stability condition is studied for the ionisation relaxation