A. Veefkind- Non-Equilibrium Phenomena in a Disc-Shaped Magnetohydrodynamic Generator

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NON-EQUILIBRIUM PHENOMENA IN A DISC-SHAPED

MAGNETOHYDRODYNAMIC GENERATOR

by

A. Veefk ind



TECHNISCHE HOGESCHOOL EINDHOVEN

NEDERLAND

AFDELING DER ELEKTROTECHNIEK

GROEP DIREKTE ENERGIE OMZETTING

EINDHOVEN UNIVERSITY OF TECHNOLOGY

THE NETHERLANDS

DEPARTMENT OF ELECTRICAL ENGINEERINGGROUP 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 f rui t ful discussions. The indispensable technical

assistance of Mr. C.J. Sielhorst is most gratefully

acknowledged.



- I -

CONTENTS

SUMMARY

NOMENCLATURE

CHAPTER I

CHAPTER I I

CHAPTER I I I

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 th e electron gas 26

IV.3 Radial flow velocity and temperature of

the heavy particles and density of th e

neutral part icles 31

IV.4 Electr ical conductivity and Hall parameter 33

Critical values o f the Hall parameter with

respect to ionisation instabilities

V.I Introduction

V.2 Firs t order perturbation equations

V.3 The calculation of cr i t ical 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

Experimental arrangement 47



CHAPTER VII

CHAPTER VIII

CHAPTER IX

APPENDIX

REFERENCES

- 2 -

Measurements

VII. 1

VII.2

VII . 3

Image convertor camera pictures

Electrostat ic 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 -

SUMMAR 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

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 f ie ld , throughout

the generator. One-dimensional and stationary solutions demonstrate

the development of electron temperature elevation and non-equilibrium

ionisation. Also start ing from the basic equations, and using f i rs t -

order perturbation theory, critical Hall parameters are derived, a t

which ionisation instabi l i t ies 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 instabi l i t ies on the Hall electric field is

measured. The experimental results are discussed and compared with the

theoretical predictions.



- 4 -

NOM E N C L A T U R E

Symbols

A

A1 ' A2A

P

a+'

a

-+

B

B0

b

CP

Cv

c

D-+E

E"-+

EH

EH

ErR-+

EL

Eexa

E.1a

Em

e

-+e

r

electron energy loss owing to elas t ic collis ions

microwave amplitudes

probe area

slopes of the asymptotes to the electros tat ic probe

character is t ic

magnetic induction

value of the magnetic induction in the centre of the

disc

length of the longest side of the wave guide crosssec t ion

specif ic heat at constant pressure

specif ic heat at constant volume

length of electrode segment

hydraulic diameter

electric f ield

induced electr ic f ield

Hall electric f ield

ionisation energy of hydrogen

energy los t or gained by the electrons owing to

ionisations and recombinations

electr ic f ield 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

unit vector in the radial direct ion



f

g'o

h

I

I.L

ip

ipo

7J

+

K

k

L

M

m

Nm

N

£

- 5 -

frict ion coefficient

distr ibution function of part icles belonging to species i

weight factor of the ion ground state

weight factor of the excited s tate m

channel heigth

reduced Planck's constant

number of ionisations per unit volume per unit time

satured ion current towards the electros tat ic probe

probe current

probe current corresponding to the centre of thecurrent-voltage characterist ic

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 coeff icient

recombination rate coefficient

generator length

Mach number

Mach number related to the radial veloctiy

mass o f an argon ion or neutral atom

mass of a part icle belonging to species i

population of excited s tate m

refraction coefficient of th e plasma

refraction coefficient of the wave guide

refraction coeff icient of the window



n

necr

ng

nq

p

p

Pe

Pg

Qe2

q

R

R

R

R1

,

ReD

Rm

Ru

r

rLe

rLi

s

T

T0

T (R)E,M

Tg

T2

R2

- 6 -

heavy part icle density

cr i t i ca l electron density

total part icle density

density of part icles belonging to species 2

principal quantum number

dimensionless representation of the gas pressure

heavy part icle pressure

electron pressure

to ta l gas pressure

collis ion cross section referr ing to elas t ic collisionsbetween electrons and part icles belonging to species 2

integer number

dimensionless representation of the radius

number of recombinations per unit volume per unit time

reflexion coeff icient

responses of the crystals in th e microwave bridge

Reynolds' number related to the hydraulic diameter

resistance in elec t rosta t ic probe circui t

load resistance

radius

electron giration radius

ion giration radius

electrode pitch

heavy part icle temperature

stagnation temperature

dimensionless parameter representing the interaction of

the electr ic and magnetic fields with the gas in thedisc generator

to ta l gas temperature

temperature of species 2



t

t I • t 2 • t3

t . • t

out

UR

...u

...

ug

...uR.

Vm

Vfl

Voc

Vp

Vpo

Vpl

Vpl

...

v

z

z

"...y

t::.R.

t::.Pe

t::.Te

t::.Vfl

M

- 7 -

time

times on which probe signals are examined

plasma passage times a t the inner and outer electrode

rings

dimensionless representation of the radial flow velocity

heavy part ic le flow velocity

to tal gas flow velocity

flow velocity of species R.

voltage measured in the electrostatic probe circui t

floating potential

open circui t voltage

probe voltage

probe voltage corresponding to the centre of the current-voltage characterist ic

plasma potential

plasma volume

particle velocity

axial coordinate

nuclear charge

ionisation-recombination parameter

f i r s t 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 p a t ~ s in the

microwave bridge



Ii

£a

£ r

K

A.,A.

'n

v

vc

V"),

Pg

D

Deff

T disch

"st

i,

"r

W

wr

WT

- 8 -

parameter for the influence of th e electron density

gradient in the zeroth order electron energy equation

permitt ivi ty of vacuum

relative permitt ivi ty

load factor

reduction parameter corresponding to electrode

segmentation

wave length

Debije shielding length

electron mean free path

ion mean free path

characterist ic length corresponding to electron iner t ia

neglection

viscosi ty coeffic ient

microwave frequency

total electron elas t ic coll ision frequency

collis ion frequency relating to momentum transfer at

elas t ic collis ions between electrons and part icles ofspecies £ .

collis ion frequency relating to energy transfer atelas t ic collis ions between electrons and part icles of

species R.,

to ta l gas mass density

electr ical conductivity

effective electr ical conductivity

delay time between th e opening of the valve and the

discharge of the capacitor bank

phase angle

angular frequency corresponding to ionisation ins tabi l i t ies

imaginary and real part of "

angular frequency of microwaves

plasma frequency

Hall parameter



WT(O)cr

wTeff

WT (0 )

stab

Szpersanpts

(0)

( I )

Subsanp ts

a

e

i

m, n

r , z

x, y, z

Shorts

ETE

LTE

MIlD

NEI

- 9 -

cri t ical Hall parameter

effective Hall parameter

Hall parameter a t the stabil i ty l imit

zeroth order perturbation

f i rs t order perturbation

averaged

neutral part icles

electrons

ions

gas species

excited states

cylindrical coordinates

Cathesian coordinates

electron temperature elevation

local thermodynamic equilibrium

magnetohydrodynamic

non-equilibrium ionisation



- 10 -

C HAP T E R I

Introduction

Magnetohydrodynamic (MHD) electr ical power generation might be used

after 1980 in various applications:

- MHD open cycle systems wil l be suitable to produce electr ical

energy On a large scale (1000 MWe) from fossil fuels. High

eff iciencies (50 %) are expected from combinations of I1HD and

conventional systems. Already now, experimental MHD generators in

open cycles are capable of converting 6 % of th e thermal energy

of the medium into electr ical energy at an output of 30 MW

(ref . 1.1).

- Closed cycle MHD generators using l iquid metals as working media

are promising with respect to space travel application. The media

of these generators consis t of l iquid alkal i metals, mixed with

a gaseous component, such as vaporised alkal i metals, argon

helium or nitrogen. They wil l be heated by a nuclear source. MHD

power conversion employing l iquid metals might be suitable to

supply electr ical energy in spacecraf t , because of the high

energy production rate per unit mass (compare ref . 1 .2) .

- The MHD closed cycle systems using gaseous media are orignial ly

intended to convert the thermal energy of gas cooled nuclear

reactors into electr ical energy. The media to be used are iner t

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

ogas cooled reactors is > 20 atm and the temperature < 1600 K,

whereas the MHD generators will work a t a pressure < 10 atm and

a temperature 2000 oK.

- The problems connected with the us e 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 t ransferred by means



- I I -

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 electr ical conductivity of the

gas. At temperatures of about 2000 oK and pressures between I and 10

atm, being the practical gas conditions, the electr ical 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 i s obtained by seeding

the gas with easily ionising materials (alkal i 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 electr ical 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 a t the electron temperature.

As the electron temperature elevation (ETE) appeared to be described

by the balance of Joule heating and elas t ic collis ional 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 elast ic

collisions in those media. However, the real isat ion 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. I t 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 th e behaviour of the non-equilibrium generators. Some of these

mechanisms are extremely favoured by the non-equilibrium situation

i t se l f . Typical losses are: electrode short-circuit ing through hot

boundary layers, the existence of ground loops, electrode voltage

drops, non-uniformconductivity

due to e lectrode segmentat ion,



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 l inear . 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 and0

non-uniformity; if0<

accounted fo r these 0

effects . agreement

with theory 0

""1.6, shock l inear . Xe 1000 5700 I 0.25 - electr ical enhancement ne agreement with theory;

0

".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 ""echanicsn

0

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

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

"scillation magnetically induced e.ionisation 0

""1.9 closed 1 inear, He < G. I 1060 1700 I 2 electros tat ic enhancement of 0 agreement with theory 0

loop segm. e1. - 3 , C probes 0n

1.10 closed l inear , He < 2 % 240 1300 1. 3 Z. IS electrical no effect induced field to smallN""loop segm. e1. C, output 0

1.11 blow l inear , 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

"otential; mechanisms and

continuum relaxation phenomena•""

-<adiad""0

I. J7. 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 f luctuation

0

<

".1.13 plasma l inear , A + 0.1 , 700 1500 - I 0. 2 electrical enhancement ne non-equilibrium

je t segm. e1. - 3 % K 3000 potential and Te calculated behaviour strongly

from w1eff influenced by

boundary layers

0

0-

0

gI. 14 blow l inear, 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 - electr ical no effect influence lossloop se.emented O.IS%C s I. 97 output mechanism too strong

tr

1.16 bl o ... l inear, He + 1400 - 1500 I 4. 5 electrical enh ancemen of 0 non-equilibrium §down segm. e1. 2 - 5 % c, 2000 output behaviour strongly -.

ffected by losses; 00

accounting fo r them

_.

".1!reemen t yi th theo!.L,".?



- 13 -

radiation losses, and ionisation instabil i t ies . These losses have

to be calculated very carefully before non-equilibrium phenomena

can be interpreted and in many cases a quantitative understandingremains diff icul t .

Another apparent feature of Table 1.1 is the lack of variation in

diagnostics. In almost a l l experiments conclusions are drawn from

values of the Hall parameter and the electr ical conductivity, which

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

the authors even the conductivity and the Hall parameter are

affected by the losses. Lit t le 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

diff icul t ies related to the realisation of a suitable non-equilibrium

condition in MHD generators, i t 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 s i tuations 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

wil l be developed a l l 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 ins tabi l i t ies .

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-equilibriumstate.

The present analysis leads to solutionsdescribing both the relaxation processes and the behaviour of the



- 14 -

two temperature plasma. Furthermore, part of th e set of equations is

used to study the plasma conditions which are cri t ical with respect

to the development of ionisation ins tabi l i t ies .

The experiment provides plasmas flowing during short times (100 ~ s e c ) through the disc. The electron temperature and density are measured

by electrostat ic double probes, spectroscopic measurements and m1cro

wave measurements. Total gas pressures are determined using a

piezo-electric crystal. Moreover, th e floating potential of th e 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 f ie lds, as th e mutual ratios of

characterist ic lengths, like free mean paths, Debye shielding length,

gyration radi i and th e dimensions of the channel, have not been

altered in a cri t ical way.



- 15 -

CH A P T E R II

Basic equations

In an MHD generator a part ia l ly ionised gas flows through a magnetic

f ield. In the presented work a flowing argon plasma consisting of

electrons, singly ionised atoms, and neutral atoms, wil l be

considered as a medium for the MHD generator.

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

distr ibution functionsthe Boltzmann equation

...

fi(v, r , t ) , which can be obtained by solvingfor each species i . Simultaneously with the

Boltzmann equation the Maxwell equations have to be solved in order

to describe the electromagnetic fields as a resul t of the electr ic

charge density distr ibution and the current density distr ibution.

Considering this specific case of an MHD generator, a number of

simplifying assumptions wil l be made.

The distr ibution functions are assumed to be Maxwellian

( I I . 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 ni

,

h . ... d ht e flow veloc1ty ui 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 l imit

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 part ic les) , and two energy equations

(one for the electrons and one for the heavy part ic les) .



- 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 la t ter is

taken as stationary. Moreover, the electr ic 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 characterist ic

length in the plasma to be described. Neglecting In - n. I with respecte 1

to n or

en

i, one

Poisson

may replace n. by n in the conservation equations.1 e

From the equation for electr ical space charge and equation

(11.2) the following condition for the variation of the electr ic f ield

can be derived:

Iv·E:1 «

n ee .

E

o

(11.3)

Once having found the solution of the problem, the condition (11.3)

can be verif ied in order to jus t i fy the substi tution of n for n . .

e1

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)

( I I .S )

Equation (11.4) has already been given implicit ly 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 uat e e e f e a r e 1

a 2IONS ,

3t n i+ V.n .u = k 0 n - krneni1 f e a

a 2NEUTRAL PARTICLES , -n + V. n u = - kfnen

a+ krneniat a a

MOMEHTIIM EQUATIONS

~ ~ xii) + n m - ) ('I> • V )LECTRONS , 0 . - 'VP

e- nee(E + u +

e e e e e1 ea

a ( n m ~ ) 'V. ( n m ~ ) n.e (E x B) - n m - ) (v . + v )EAVY PARTICLES ,at

+ .. - Vp + + U1 e e e e1 ea

ENERGY EQUATIONS

a (3 I 2

) ( 3+ 1. m u

2) ) V.t; )( v . + v )LECTRONS , ne (2 kT e + E. + '2 meue) + v. ne( I kT e + E. u = - ue,vP

e- P

e- n eE .u + nemeu. (u -

at 1 1 2 e e e e e e e e1 ea

m

- 3 n (I> e i + v ) k (T - T)e m ea e

a( (1 kT

I 2) ( (1 kT

I 2) ~ . V p - p'V.;

~ ~ ; ) (v . + v )EAVY PARTICLES ,

atn + "2 mu ) + V. n -+ 2 mu ) u = - + n.eE.u - nemeu, (u -

2 2 1 e e1 ea

m

+ 3 ne

(ve i

+ v ) k (T - T)e m ea e



- 18 -

coll is ions, 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, i t follows frome

the comparison of th e rate coefficients for th e 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 k r ' which

appear in th e right-hand side of the continuity equations, are then

given by:

kf

= 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

( I I . 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 th e continuity equations. The Saha equation follows

from these equations i f 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 coll is ion

term of the right-hand side, i t appears that when neglecting the

inertia of th e electrons, a new minimum characteris t ic length is

defined:

, . = u I (v . + v )1n e e1 ea

The basic equations of Table 2.1 do not describe processes with

characteris t ic lengths < ' i n ' In the cases discussed here, ' in will

(11.8)



- 19 -

always be smaller than. AD' so that the validity of the space charge

neutral i ty approximation implies the just i f ica t ion of the neglection

of the iner t ia term. The coll ision frequencies v . and v , used in. e1. ea

the momentum equations as macroscopic quanti t ies, are related to the

elast ic collision cross section as follows (ref . 2.3):

v =e£ n

e

rJ

Q , - I f d - )eX, e e e

( I I . 9)

with £ is either i or a. The contribution of inelast ic collisions to

the momentum t ransfer between the electron gas and the heavy part icles

is neglected with respect to the momentum transfer due to elasticcoll isions. This is because the frequencies of the inelast ic collision

processes are low compared to

momentum t ransfer is the same

v . ande1

in both

v and the efficiency ofea

types of coll ision. The electron

elast ic coll ision frequency related to the transfer of thermal energy

is not defined in the same way as the corresponding quantity related

to momentum t ransfer, but is given by the following equation (ref . 2.3):

v*e£

me= n 3kT

e e

(II .10)

with £ is i or a. In this analysis i t is assumed that ve£ may be

approximated by v ~ £ so that in th e energy equations the same collision

frequencies appear as in the momentum equations. Q is taken to beea

constant and equal to 0.5 x 10-20

m2

; v . is taken in accordance withe1

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 seeded plasma the radiative losses are comparable

with the elast ic losses, in a non-seeded plasma th e radiative losses are

small compared to th e elast ic losses. In the energy equation for electrons,2

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

kT • Furthermore, heat conduction processes are not included in thee

equations (see also chapter VIII).



- 20 -

CHAP T E R II I

Geometry of·the disc generator

The amount of electron temperature elevation depends on the geometry

of the M F ~ generator. Fig. 3.1 shows diagrams of a continuous and a

segmented Faraday generator, a l inear Hall generator and a disc Hall

generator, these being the most general geometries. The following

c dFig. 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 ar e th e e l e c t r i c f i e l d an d th e current density corresponding to th e Lorentz

force e ( ~ x B), respectively. EH and TH ar e th e electric f ield and the current

density owing to the Hal l e f f e c t . respect ive ly .

expressions

are derived

for the rat io of T and the stagnation temperature Te 0

by Hurwitz (ref . 3.1) for the continuous and segmented

Faraday generators, and the l inear Hall generator respectively:

5(1

2{

2 2} M2T 1 + - .K) . wr / (1 + WT )

e 9=

T + 1. M203

( I l l . 1)

5 2 2 2T .1.+ 9" ( l - :K ) .WT .. M

e-=T

1 1. M20 +3

(III .2)

2 M2 2( . 2 2) / (.+ wr2)1 + wr. 1 . +. K WT 1e 9

=T 1. M20 +

3

( I l l . 3)



- 21 -

where Cp/Cv is taken equal to 5/3 and inelast ic losses are neglected.

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

the radial veloci ty . I t can be shown from the equations (111.1),

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

the electron temperature elevation. For the rat io of T and T ise 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 unlimitede 0

and increasing with th e Hall parameter.

In l inear MHD channels the Hall electric field can be buil t up provided

segmented electrodes are used. The characteristic distances for electrode

segmentation are shown in Fig. 3.2. Celinski (ref . 3.2) shows that

f ini te segmentation resul ts in an infer ior performance of the generator.

h

5

III

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 electr ic 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 th e disc generator in comparison with the linear generator is the

l imitat ion to the Hall mode of operation; in the l inear geometry, the



- 22 -

possibi l i ty of various load connections results in many different

modes of operation (ref. 3.7).

Table 3. 1 The effec t of f in i te electrode segmentation.

quantityideal generator real generator(s/h = 0) (s/h > 0)

current density (1 - K)auB A 1 - K)auB

electr ical power density K ( 12 2

- K)au B AK (12 2

- K)au B

Joule heating per cubic metre 222(1 - K) au B A(1 2 2 2- K) au B

A diagram of the disc generator is given in Fig. 3.3 .• The gas i s

supplied to the centre of the disc-shaped MHD channel and flows

radially outward perpendicularly to an axial magnetic f ield. The

Lorentz forces acting on th e electrons and ions of the medium

cause an azimuthal current density component and a radial Hall

elec tr ic f ield. 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 · tgenerator. 1L repre£ents t e current enS1 y

force e ( ~ x B) , EH and iM th e electric field

and th e current density owing to th e Halleffect .



- 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 u3

ELECTRONSer

•e

=e er

kfnenll

- k0d"r""""

ud r - ---. 0

e er r r e

du do 0 u3

IONSr e

=e r

+ kfnena

- k0 -- .d r

0e dr r r r e

..;

•'•

du do 0 u3

!'lEUTRAL PARTICLESr a

= k[ne.:1a k0 - - .dr • 0a dr r r r e

,.,

N

naa••"O!'IENTIDI EQUATIONS<•"·

dT do0a

ELECTRONS, R-COM?ONENT , k__ .

kTe

= - n e (E • u B) • n m (v . • v )( u - )d r

ue dr e e e , e e el ea r e r •

C

•"·

ELECTRONS, ¢' - COMPONENT , 0 = 0 eu B • 0 m (v • v ) (u - ueq,)e er e e ei ea

0 Na

.0-••

du2

HEAVY PARTICLES, R-COMPONENTr

okdT

kTdo u ,

n e (E u1>B) - n m (v . ve

) (ur

- II )omlld r • -. dr = nm - + • •dr r e e e Cl er

e·m

"dll, U ll .

HEAVY PARTICLES, ¢-COMPONENTr .

B -neme(vei • \ ! e ) (U¢ - u

e¢)nmll

d r= - nm--- 0 ell

r r e r

0

"'

•e '

•

• ENERGY EQUATIONS mam

"n ku

dT dT dn m

ELECTRONS , e - II 0 ke

II kT__ =

ne eE (Ur

- ) • n e e B ( u r U e , , ~ -ueru¢) - 30 --"- (v • u ) k(T - T)

d r ~ U2 e e r r e r e dr er e m e i eo e

•"

- (2- KT • Eia) (kfnena

- k n3

)2 e r e

3 dT u kT dnm

HEAVY PARTICLES , okll - = 3n --"- (v . • v ) k (T - T),.r dr r dr e m e> ea e



- 25 -

CHAP T E R I V

IV.I Introduction

Numerical solutions of the se t 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 l inear 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 th e disc generator case and for generator distances

between 0 and 0.20 m as far as the l inear generator is concerned,

these being the extreme values representing the in le t and outlet

of the channel.

The plasma properties at the in le t are chosen as follows:

ne

u = uer r

u = uex x

1800 m/sec, =

= 1800 m/sec

T T = 9000 oK.e

o (disc generator);

(linear generator);

For the l inear 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 mer r

(VI. I )

u u for 0.07 < r < 0.14 mer r

The value of the load is determind by the imposed discontinuity in

u a t r = 0.07 m; in fact u is supposed to drop there to 0.65er er

times i ts original value.

In the disc generator the magnetic induction is assumed to have the

following radial dependency (compare chapter VI):

B2

B (I - 0.51 r - 9.56 r )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

real is t ic MHD generator si tuat ion in connection with the implici t ly

imposed radial current dens ity component. The magnetic induction

in the l inear generator i s 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 l inear 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 part icle temperature of about



12000

8000

'if-4000....

...

aI t; .. ~ . . b ...

electrode sili s ~ ~ ; ' I ' I ' 1 ' ~ = " " ' ' ' ' ' ' ' ' - ' ' ' ' " ' - ' " ' / ~ , . : : : , I ", ,\

---- ,

, / I \

" I ,, II II I~ ' L " , , - _ _ _ _ I"'"

I

a-QQ3L

16000

12000

- - ~ -" "

b- - ; : : : . - -

- , ' , - / .......,. / , ...

-" I I \ \

II \I I '

/1 I/

11000 B - ~ _ _... -0ll3l

,- -/ / ..........

,/ .t / ..........

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

10000

>-;.9000..."

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

} - ~ - ~ o h = : : : : r : ~ ~ - - - L . - ' - - - } - ; : - - - - --lil -2.0 -1.0

8 0 0 ~ 3 . O , ' - - ~ . . . . . L . . - - ' - - - 2 L . o - - ' - L - - - ' - - . . . J 1 . c , . 0 1010g x (m )010g (r -o.03)(m) °log(r_0.03) (m)

Fig. 4.1 Variations of the electron temperature Te (dashed l ines) and th e heavy par t ic le temperature T (solid lines) with th e

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

values of th e magnetic induction Bo'

a. Loaded disc generator. h. open disc generator. c. open l inear Hall generator.

Electrode posit ions in the disc: r = 0.07 and r = 0.14 m. Plasma conditions at channel inle t : n = 2 x 1021

m-3e

n = 2 x 1023

m- 3, ua er

B = 0 and B '" O.C I To 0

= ur

= uex

= Ux

= 1800 ~ / s e c . u¢ = O. Te

coincide.

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

N....,



-24~ . § t' 23go

~ 2 2 121go

a

2 c

/' .

23

'-..."-- - - - - - - - - - " " ' ~ ~

§ ' 20_3.0 -2.0 -to -1.0

3500

'''Iog(r -0.03) (m)

Fig. 4.2 variations of th e electron density fie (dashed l ines) and th e heavy particle density Da (solid lines) with the generator

distance. For a further description. se e Fig. 4.1.

a3500

~ 2 5 0 0 ]"

=1-

bB.O }---_ _8:Q01T8:0.031

e : . . Q . Q R ~ . . . , . ~ -

~ 1 5 0 0 1 ~ - L ~ __ ~ - ! ~ ~ - L ~ _ ~ - " -3.0 -2.0 -to

1Qlog(r _O.03Hm)

c

1 8 0 0 ) 1 - - - = = = ~ ~ 5 i = : : ; : : : : : : : -

1700

E 1600

/

/a..-o- l-'"

t ~ ~ H //0.05T /,

B. : O.Qzr /

Fig. 4. 3 Variat ionsof the electron velocity and heavy par t icle velocity (uer

and ur

in th e case of th e disc geometry. and uex

and uxin the case of th e l inear geometry) with the generator distance. The dashed l ines in a. represent uer

as fa r as

i t di ffers from ur

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

()O



4.I I

- - - = : : : : : ; ' ~ D i ' I I

I

15

I II I

: II II I2 5 ~ ~ ~ - L ~ ~ L - ~ ~ ~ ~

-3.0 -2010log (r -0.03) (m )

o!1'

4.0

6.=0071B:O.05T /f ! , M ~ a,.001Ta...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 8.00ITo a..o. _) /

3 . 4 ' : ; - ; ; - ~ - L . ~ - - - ' : - : : - ~ ~ - L - - - C ~ -3.0 -2.0 -1.0

1°109 • . m)

Fig. 4. 4 Variation of th e electr ical conductivity a with th e generator distance. Fo r a further description se e Fig. 4.1.

4.0

3.0

2.0

1.0I-'

3

~ 0 D 5 1 8,=Q03T

a

!l3:0 -2.010log(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- ' 0.25F-----Lt'----_

3

Fig. 4.5 Variation of th e Hall parameter WT with the generator distance. Fo r a further description, se e Fig. 4.1.



- 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 th e Te levels higher because of the larger amounts of Joule

heating. Fig. 4.1 band c shows that even at B = 0 th e electrono

temperature will be higher than th e gas temperature. This is caused

by the in i t ia l value of the electron density which is chosen to be

higher than determined by the Saha equation; in fact, th e 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

part icular curve, namely the curve in Fig. 4.1 b, belonging to

Bo

10= 0.07 T. For log(r - 0.03) - 2, Joule heating of the

electrons

For - 2 <'V

causes the

10log(r -

elevation of Te

0.03) < - 1.12,'V

from 13,000 up too

15,000 K occurs,

from 5000

a further

because th e

oup to 13,000 K.

increase of Te

expansion of the. + +

medium results in a h ~ g h e r value of u x B and a decrease of the10

coll is ion 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 th ef o l l o ~ i n g :

- The ionisation energy is withdrawn from the electrons.

-AsQ.»

coll is ion

Q ionisations resul t inea

frequency stimulating the

electrons and heavy particles.

an increase of the to ta l

thermal contact between

h . d·· 7+- By the en ancement of the e l e c t r ~ c a l con u c t ~ v ~ t y the J x B.. ...

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

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

eremains lower than in an open one; this effect is i l lustra ted in

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

The occurence of non-equilibrium ionisation i s 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 band 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 ) = constante 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 electronr er

positions where the curves of u show discontinuities.er

IV.3 Radial flow velocity and temperature of the heavy part icles and

density of the neutral part icles

In pract ical 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 to tal 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 andr

to tal

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

equation and the energy equation of the to tal medium in the disc

generator are successively given by:

dP dUR L 7 1i) (IV.4)+ --=

2(J x

dR dR rPgUgr

dP + p

dUR L 7+ 7 +

}1 --+ = { J .E - (J x B) <j>ug<j>2 dR 2 dR 2· 3PgUgr

(IV.S)



- 32 -

where Pg

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

normalised pressure, radial flow velocity, and radius, respectively,

th e normalisation relationship being given by:

2p ' = p u Pg oo

r = r Ro

The analysis is given for a fixed, arbitrari ly chosen generator

(IV.6)

position r = r where uo r = u . this results in DR and R being equal0'

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) th e Mach number MR is related to the radial flow

velocity. The interaction of the medium with the elect r ic and magnetic

fields is represented byT ~ R ~ : ,

T(R) =E,M

L

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

(IV.8)

of ne

omparing the curves of ur

(Fig. 4.3 a and b) with those

(Fig. 4.2 a and b), i t can be seen that the behaviour of th e flow

velocity depends on whether non-equilibrium ionisation has been(R)

developed or not. If not, TE M will be small and th e siutation,is described by equation (IV.7) with th e right-hand side equal to

zero; as in th e given example > 1, th e 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 th e radial flow.



- 33 -

The neutral part icle density and the heavy part icle temperature are

shown in Fig. 4.2 and Fig. 4.1, respectively, as functions of the

generator position. In the disc generator, th e two quantities are

determined by the expansion of the medium for r smaller than the

ionisation relaxation length; considering supersonic ga s veloc i t i es ,

n a n d T decrease in that region. For r larger than the ionisationa

relaxation length,

influence of the Tn

aand T tend to increase owing to the

+ .x B brak1ng force.

For th e l inear 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 th e scalar electr ical conductivity and the Hall parameter are

strongly related to the electron elast ic collision frequency. In the

given examples the plasma is Coulomb collision dominated.

In a Coulomb collision dominated medium a is in f i rs t order proportional

to T3/2. this explains the similarity in the a and T variationse ' e

(compare Figs. 4.1 and 4.4). Furthermore, i t follows from Fig. 4.4

that in th e 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-

I

T

3

/

2

•I t

can be seen from Fig. 4.5 that for valuese eof r smaller than th e ionisation relaxation length WT is strongly

influenced by Te ' I f r exceeds the ionisation relaxation length, the

increase of n by th e non-equilibrium ionisation, together withe

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

Generally, i t can be stated that especially

the non-equilibrium ionisation region - the

i f v . >e1

value of

v

ea

the

- at least in

Hall parameter

is much higher in th e ionisation relaxation region than in the generator



- 34 -

positions where th e ionisation degree has already been enhanced.

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

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

cr i t ical value related to ionisation ins tabi l i t ies (see chapter V).



- 35 -

CHAPTER V

Critical values of the Hall·parameter with respect to ionisation

instabilities

V.I. Introduction

In MHD generators the development of instabi l i t ies in th e plasma can

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

of instabil i t ies occurring in MHD generators are the magneto-acoustic

and the ionisation instabi l i t ies; from the two, the la t ter have

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

be discussed here.

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

instabi l i t ies , are described by introducing an effective electr ical

conductivity Geff

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 Geff

are lower than the values of WT and G;

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

Using firs t-order perturbation theories, several authors have calculated

cri t ical values of the Hall parameter that represent upper limits of

stabi l i ty (refs. 5.1, 5.2, 3.6). They a ll 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 cri t ical Hall

parameters will be calculated without assuming th e validitylof the

Saha equation.

V.2. Fi rs t order perturbation equations

Ionisation instabil i t ies

->

The quantities n , u anda

cons i s t o f f luctuat ions in n , ,T and E.e e e

T are assumed to be constant within distances

comparable with the typical wavelengths of the fluctuations. The



- 36 -

ionisation ins tabi l i t ies are described starting from the conservation

equations of Table 2.1 as far as they are related to th e electron gas,

and eqs. (11.4) and (11.5). From the combination of th e continuityequation for the electrons and eq . (11.4), i t follows that the former

may be replaced by th e continuity equation for th e ions.

Considering th e t ransit ion from stabil i ty to ins tabi l i ty , a f i r s t -

order perturbation theory is jus t i f ied , because th e fluctuations

are small in the primary stage of their development. The zeroth

order terms represent the stationary behaviour of the medium, and

the f irst-order terms represent th e fluctuations, as-T

from th e following division of the quantities ne

, ] ,

zeroth and f irst-order terms:

can be seen

d+ .

T an E 1ne

(V.2)

Substitution of eq . (V.2) in th e basic equations and subtraction of

the zeroth-order relationships result in three f irst-order conservation

equations, namely the continuity equation for ions, th e momentum

equation for electrons and the energy equation for electrons. They

are given by the following relationships respectively:

:l ( I - R)

ane n =n(O)

- e e

T =T(O)e e

a 1 - R)

aTe 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 =ne e

ne

T =T(O)e e

w,(O)

(0 )u

au+ -

T ""

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)* aAJ .E + J .E - an

e n =n (0)- e e

(I)n

eaA

- 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 th e number of ionisations

and the number of recombinations per unit volume and unit time,

respectively. In eq . (V.4) Y s the f i rst order term of _1_ Vpe:n

->- k ( V n ~ 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 elast ic collisions with

heavy particles is given by the function A in eq . (V.S), while th e

energytransfer

owing toion isa t ions

and recombinationsi s

given by



- 38 - .

m

A = 3n k (T - T) (v • + v )e m e el. ea

Eqs. (11.4) and (11.5) resul t in the following f i rs t order

relationships:

' l xE : ( J )=0

-+,. -+ -+-lo--+

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, i t follows from eq . (V.IO) that th e

vector field E:*(I) is curl free:

'J x E ' ~ ( I ) = a

V.3. The calculation of cri t ical values of the Hall parameter for

some spec ia l cases

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

(V . I I )

In this section the region of th e 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 ~ ) , o r n

e

(V.12)

are



- 39 -

'7n ( I )e

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

ne

. ( 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 f i rst -order Saha equation:

(I)ne

( 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 l inear homogeneous equation

in n ~ ! ) , i f one particular term of the Fourier ser ies is concerned:

(I) = n ( I ) { exp . ( + rlt)}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 stabi l i ty l imi t , given by rI . = 0, n(l) must be solvable from th e1 eofollowing equation:



- 40 -

C

K2 + K2d In cr Kl.. (0 ) d IIi.. A

n =n(O)

(I)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 ne

(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 anyeo

KWT(O)b as the value of WT(O) in the stabi l i ty l imit , the followingsta

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 nn =n(O)

d In nx 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

i f applied to an unperturbed situation without fluctuations, eq . (V.18)

has one more term resulting from the fluctuations in the electr ical

conductivity which are taken into account here. The cr i t ica l Hall

parameter W T ~ ~ ) can be found from eq. (V.18) by deriving the minimum

value of WT(O)b with respect to th e direction of K n the xy-plane.sta

In Fig. 5.1, WT(O) is given as a functin of T(O) for a gas temperaturecr (0) e

of 5000 oK. The figure shows that for T - T > 2000 oK the cr i t ica le '"

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 iscr e

0



T=5000'K

2.0

1.5

- 41 -

..25 -3n.= 1u m

24 -3n• • 0 m

.-23 -3n._1U m

1022 -3

.= m

Fig. 5. 1 Crit ical Hall parameter w,(O) as a function of electron temperature Te ' fo rcr .

several values of th e neutral particle density, in Saha equilibrium si tuat ions.

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 th e electron temperature,

the other by the ionisation relaxation (see Chapter IV). The former

part is generally small, while in many experimental arrangements

th e l a t te r cannot be neglected with respect to the dimensions of the

generator. Ins tabi l i t ies of the medium are of influence on the length

of the relaxation region as well as on th e finally reached values of

th e quantities considered.

The stabil i ty condition is studied for the ionisation relaxation

reg ion, using the same assumptions as l i s ted in the previous sect ion ,



- 42 -

except the f i r s t and the third assumption. With resptect to the

zeroth-order energy equation for· the electrons, i t has now been

assumed that terms of the order

are negligible compared with terms of the order

Vn(O)e(0 )

n e

In an analogous way as in th e previous section, the stabil i ty condition

can be found, now from eqs. (V.3), (V.4), (V.5), (V.9) and (V.11),

resulting in the following expression for W T ! ~ ~ b :

WT(O)K2 C; - d ln 0

A(O) d ln A= +

stab 2K K K2 d ln nn =n(O)

. (0)2/ (0) d ln n(0)y e J 0 e n =n

- e e - e eT =T(O) T =T(O)

e e e e

E(O) d ln EIR

(oJR(V . 19)

.(0)2/ (0) d ln nJ 0 e n =n- e e

T =T(O)e e

The coordinate system has been chosen in the same way as in the previous

section. For

.(0)2J = A (0)

0(0)

and E ~ ~ ) = 0, eq. (V.19) agrees with eq. (V.18). To express

. (0)2

J

+



- 43 -

in terms of the gas quanti t ies, the zeroth-order energy equation for

electrons is used written in the following way:

• (0) 2.. .=_ = _ kT(O) + 'l (0) + A 0) + E(O)

(0) e u. ne IRa

(V. 20)

Expressing the f i r s t term of the right-hand side as a £raction 6 of

. (0)2]

(0)a

eq. (V.20) obtains the following form:

.(0)2 A(O) E(O)+ IR

....=-<- = - - ; - - - - - i= -(0) 6

(J

I t follows from eq . (V.20a) that

.(0)2J(0)

(J

is determined by th e values of n(O), T(O) and 6. From (V.19) WT(O)

(V.20a)

e e (0) crmay be found as the minimum value of WT b with respect to K and K

(0) (0) sta x yfor any choice of n ,n , T ,T and 6. As in this section only

e a eHall generators working a t open circui t conditions are considerd, the

plasma velocity component in the main generator direction can be

calculated afterwards:

(0) . (0 )

u = - - , . , ~ J - . , , , , , , n(O)ewT(O)

e cr

Having u(O), the gradient of nCO) can be found:e

'In(O)e

6kT

e

.(0)2_J__

a(0)

(V. 2 I)

(V.22)



- 44 -

The cr i t ical Hall parameter

for several values of n ~ O ) ,is given as a function of T(O) in Fig. 5.2

en , T and 0 which are l isted in Table 5.1.

a

Table 5, I Values of th e electron density n(O). the neutral density n • th e heavy particle temperature T,

and the parameter 6 which i n d i c a ~ e s the influence of vn!O)7 used in the calculation of the

crit ical Hall parameter in th e case of no Saha equilibrium.

n(O) (m- 3) -3T(°K) 0urve n (m )

e •

I . 1019

13 IIb 10

20x 10

245000

Ie 1021

2.

\1 0

20

3 x 1023

( 5000\ 0

b 3 x 1024

2e 3 x 1025

3 .

\1 020

\3

4000

3b x 1024

5000 0

3e 6000

i 000

- 0.44.

(3XI024

1020 - 0.2

4b0

4e

Situations where u > 4500 m/sec are not considered. In addition to

WT (0) also a parameter a is plotted in th e graph given by thecr

following ratio:

a =

(0 )n

e d ( I - R)

T(O) dnee /

"(1 - R)

(0 ) aren =n .

e e

T =T(O)e e

n =n(O)e e

T =T(O)e e

(V.23)

The value of a indicates whether the plasma is governed by ionisation

or recombination processes. The ionising and de-ionising reactions

considered are given in chapter I I . For these processes the denominator

of eq. (V.23) is always> 0, while the numerator is either < 0 or > 0

i f an increase in n stiumulates either the recombinations or thee



-

6

S

4

3

•: I

•,!'13

I•

3

0

•

- 45 -

•• 6 ••

. 1 S .1

---:;:::-..--.- - - - - ~ - ; . . : : : - - -----/

0 4

_.1 3

- .. •..-.

.. ~ . , 3

6000 1000 10000 12000· 010000

T.(OK} T.(OK)

.2 6

., S

-------- -----/

0 "

_.1 •

-· 2 2

..

-b /_. J 1 /

3

120«:0

o

Fig. 5. 2 cr i t ical Hall parameter W T ~ ~ ) (solid l ines) and "ionisation-recombination

parameter" CI' @.ashed l ines) as functions of electron temperature Te' fo r media

being no t in Saba equilibrium. The parameter values corresponding to the

curves ar e l isted in Table 5.1.

0

_.1

-..

_. 3

_."12000

.2

.,

0

-· 1

_.,

-..

-."000



- 46 -

ionisations, which processes result in a damping or amplification of

the original change of n , respectively. The special situation givene

byCl

= 0 is obtained when I = 3R.Appendix show beside the values of

E(O)

IR '

.(0)2J

(0) ,(J

(0)u ,

+(0)u( 0 ) andu

The Tables A.l up to A.4 of theWT (0) and Cl also those of

cr

(I - 3R) (0)

Some general features of WT(O) with respect to the conditions of th ecr

medium can be derived from Fig. 5.2 and th e Tables A.l up to A.4:

- High values of WT(O) are achieved when the electron temperaturecr

elevation 'is small. As in that case also the current density must

be small, these situations do not apply to a good MED generator

performance.

- The values of

WT(O) will becr

WT(O) are strongly related to the values of Cl'cr '

high for plasmas governed by recombinations in

contrast with media where ionisations are in the majority. The

decrease of WT(O) with T results from this effect.cr e- Variations in parameters which have no influence on Cl (variations

in T or 8) result in only small changes in WT(O) •cr

- In many situati 'ons, namely those corresponding to moderate values

of the parameters, WT(O) has values between 1 and 3 as in the casecr

of Saha equilibrium.

From the calculations of W T ~ ~ ) i t appears that in non-equilibrium

MHD generators ionisation ins tabi l i t ies will occur in the ionisation

relaxation region. There, two conditions stimulate the development

of instabi l i t ies : the high values of WT(O) (see chapter IV) and the

ionising character of the plasma, resulting in a low WT(O) •cr



- 47 -

CH A P T E R VI

ExperimentaZ arrangement

Several non-equilibrium phenomena, characterist ic of MHO media are

studied in a short-time disc generator experiment. A survey of the

experimental set-up is shown in Fig. 6.1 and a diagramatic

representation is given by Fig. 6.2.

To make the analysis as simple as possible as well as to avoid

dissipation of electron energy by additional inelastic coll ision

processes, the impurity level of the argon is kept low. Therefore,

the plasma is produced by an inductive discharge of 99.998 %pure

argon. By electromechanicallY actuated fast valves the gas is

f ig . 6.1 S u r v ~ y of the experimental set -up.



- 48 -

Fig. 6. 2 Diagram of the experimental set-up.

a. Disc-shaped MHD channel, b. brass torus, c. central body, d. pyrex tubes,

e. pyrex cone, f. brass cone, g. fast valves, h. magnet coi ls .

Dimensions in mm.

supplied to both ends of a pyrex tube which is evaporable to

2 x 10-5

Torr. The gas is heated, ionised and accelerated by

discharging each of the two capacitor banks over i ts brass conical

coil . The coils f i t the similarly shaped ends of the pyrex tube. The

capacity of each capacitor bank is 30 and voltages can be applied

up to 18 kV. The osci l lat ion of the current through the cone is

measured using a search coil , inserted between the brass plates

which connect the capacitor bank and the conical winding. A typical

signal of the search coil is shown in Fig. 6.3.

Fig. 6.3 Search coi l signal . representing th e current which flows through the brass

cone. Vertical scale: arbitrary units. Horizontal scale: 5 ~ s e ~ / d i v .

From the ringing-frequency, which is 100 kHz, the self-induction of

the system calculated to be 80 nH. The ohmic resistance as derived

from Fig. 6.3, is equal to 0.01 n. Applying a voltage of 10 kV to the5

capacitor bank, the maximum current flowing through the Cone is 2 x 10 A.

For a more detailed description of this type of plasma production andacceleration compare refs. (6.1) and (6.2).



- 49 -

Through the pyrex tube the gas is fed to the centre of the disc

generator. A central body made of glass stimulates the radial flow

veloci ty in the generator. The walls of the disc consist of two

circular , transparent glass plates with a diameter of 44 cm and a

thickness of 2.8 cm. The glass plates are connected to a brass torus

using two viton O-rings (see Fig. 6.4) .

Fig. 6. 4 Connection of th e glass plates to the brass torus.

Rings of tungsten wire consti tute the electrodes, namely four for

each electrode. The diameters of the anode rings are 14 cm, those

of the cathode rings 28 cm. The rings are kept in position by four

radially placed supports. The wires are mutually kept in position

by s t r ips of boronnitride. The electrode configuration is shown in

Fig. 6.5. In order to make · i t possible to heat the electrodes

electr ical ly as well as to establish any desired electr ical connection

of the rings outside the generator, each support is composed of four

.molybdenum wires electr ical ly insulated from each other by thin

layers of glass (thickness 0.2 mm). A circui t diagram of the

elect r ical connections of the electrode rings is given in Fig. 6.6.

The energy used for the opening of the electromagnetic valves, and

the delay-time between the gas inlet and the discharge of the

capacitor banks can be adjusted for each plasma gun independently.

By adjusting these quantities i t is possible to compensate for small

geometrical deviations from the symmetry of the system, thus yet



- 50 -

Fig. 6.S Survey of the electrode configuration in th e disc. (In this picture the glass

plates have been removed.)

72Vl

Fig. 6.6 Diagram of th e e lec t r ica l connections of the electrode r ings. During th e period

of e lectrode heating th e rings ar e connected in series (switches S in positions

B). During th e passage of the plasma the switches S are in posit ions A. thus

furnishing th e external, independent connection points I up to 8 corresponding

to the eight r ings.



- 51 -

obtaining a simultaneous arrival of the two portions of gas in the

centre. The passage of plasma through th e disc is verified by

measuring the saturated ion current flowing towards an electros tat ic

double probe inserted radially the disc. (For a description of

this diagnostic method, see chapter VII). Typical responses of the

probe, corresponding to th e triggering of each gun and of both guns

together, are shown in Fig. 6.7. In addition to the similarity

of the signals representing the plasmas originating from each end

of the pyrex tube, i t can be seen from Fig. 6.7 that the typical

duration time of the plasma passage is about 100 ~ s e c .

Fig. 6.7 Saturated ion currents flowing towards th e electrostatic probe.

a. Signals corresponding to th e discharges of each Bun, h. Signal corresponding

to th e discharge of both guns together.

Verticale scales: 0.45 mA/div. Horizontal scale: 100 usec/div.

By adjusting the voltage across the capacitor banks, the argon pressure

in the plenum outside the valves, and the delay time between gas inle t

and discharges, the temperature, degree of ionisation and flow velocity

of the plasma can be varied over wide ranges. Only certain values of

the gas parameters of the produced plasma are considered. For the

reported experiments the values of th e most important adjustable

parameters are given in Table 7.2 of chapter VII.

The magnetic f ield is provided by six coils . The magnetic induction

is measured in various positions in the plane of symmetry. In any

position the axial component of B appeared to be a t least ten times

larger than the other components. B is found to be azimuthallyz

independent within 5 %. The radial dependency of B can be approximatedz



- 52 -

wi thin I % by:

B can be varied between 0 and O. I T.o

(6. I)



(

- 53 -

CHAP T E R VII

Measurements

VII.l Image convertor camera pictures

The motion of the plasma in the disc is visualised by a TRW image

convertor camera. Photographs taken at an exposure of 100 nsec are

given in Fig. 7.1, showing the plasma motion in one half of the

disc. The pictures indicate an azimuthally independent progress of

the l ight Eront. An estimation of the averaged radial velocity of the

l ight front yields some 1000 or 2000 m/sec. Furthermore i t follows

from the photographs that the plasma f i l ls the disc during about

100 /lsec.

(/lsec)

40

50

70

Fig. 7. 1 Image convertor camera photographs at different times t after th e discharge of th e

plasma guns.



- 54 -

VII.2 Electrostat ic probe measurements

Electron temperatures and densities are measured with electrostatic

double probes. The probes consist of two platinum electrodes inserted

in a piece of stumatite, which is fixed with araldite on the top of

a quartz tube with a diameter of 4 nnn (see Fig. 7.2).

.---E EE: E

C'l

,t __

L __

stumatite

-;:-())

. O.5mm~ - . - ~ -

Fig. 7.2 Outline of th e electrostatic double probe.

I:wlres toIelectrical~ i r c u i t _

The electrodes have circular surfaces with diameters of 0.5 mID; th e

distance between th e two surface centres is 3 nnn. The electr ical

circuit is given in Fig. 7.3.

(/ )

Q)

"0oL-

-

2Q)

Q)

.. 0oL-

0 .

o-

2QV

--.Joscillo-

I SCope. I

Fig. 7.3 Elec t r i ca l c i r cu i t fo r double probe measurements. For Rm res is tances have been used

varying from 47 to 330 Q.



- 55 -

During the measurent the probe voltage V is varied from -20 to +20 V.p

The probe theory can be based on two quite different concepts. Langmuir's

theory (ref . 7.1) considers the motion of the electrons towards theprobe as a "free fal l" in a retarding potential as soon as they

arrive in the space charge "sheath" around the probe; the ions, too,

are assumed to move towards the probe without any collisions as soon

as they arrive in the sheath. This theory can be applied i f the mean

free paths of the electrons and ions are large compared to the sheath

thickness. The second theory is suitable for plasmas a t higher pressures.

Then the probe current is controlled by continuum equations describing

the diffusion of electrons and ions in the plasma and can be given in

terms of diffusion coefficients and mobilities (ref. 7.2). Cozens

shows that for double probes both theories result in the same relat ion

ship between probe current i and voltage V :p p

i = 1. tghp 1

eV

(z r l - )

e

where I . is the saturated ion current towards the probe.1

(VII. I)

To given an impression of the value of several characteristic lengths

in the considered plasmas, the orders of magnitude are

Table 7.1 for n = 1019

and 1020

m-3

and for n = 1023

given in24

and 10-3

me a

T and T. are assumed to be 104

and 5 x 103

oK, respectively, and Be 1

is taken to be equal to 5 x 10-2

T. Especially the values of A. are1

very rough estimations. The values of AD can be considered as a

measure for the sheath thickness. As A > AD' whereas A. < AD' i te 1 'V

follows from Table 7.1 that neither the Langmuir theory nor theconcept of Cozens is suitable for the considered plasmas; a model

should be used, where the electron current towards the probe is

controlled by the "free fal l" of electrons in the sheath and the ion

current by the continuum equations. Nevertheless, eq . (VII.I) is used

as a start ing point for the interpretation of the experimental current

voltage characterist ics. The same equation presupposes that the probe

surface can be considered as plane; this assumption i s just i f ied by ADbeing much shorter than the surface diameter of the probe electrode.



- 56 -

Table 7. I Characteristic lengths corresponding to th e plasmas considered, at a magnetic induction

of 0.05 T, an ion temperature of 5 x 103

oK and an electron temperature of 104

oK.

-3 -3

A (m) A. (m) An(m)(m ) n (m ) J;Le(m)e a e 1

1019

1022 10-3 10-6 .10-5 10-5

1020

1022 10-4 10-6 10-6 10-5

10 19 1023 10-3 10-7 10-5 10-5

1020

1023 10-4 10- 7 . -6

. 10. .-5

.10 .

As th e J;adius of the cyclotJ;on motion of the electJ;ons may have the

same oJ;deJ; of magnitude as the sheath thickness, th e normal to the

probe surface is directed parallel to th e magnetic field (see Fig. 7.4)

in a ll measurements, except ru n V (see Table 7.2), where different

orientations of th e probe surface were necessary.

a b

g

Fig, 7.4 Orientation of th e probe surfaces with respect to the magneticf ie ld

and th e plasmaflow; typical change of probe positions (indicated by arrows):

a. fo r measurement of radial dependences, h. fo r measurement of axial dependences.



- 57 -

Moreover, the normal to the probe surface is directed perpendicularly

to the plasma flow velocity in order to eliminate the influence of

the flow velocity on the ion current towards th e probe. The

circumstance of A and A. being much shorter than the diameter ofe 1

the probe possibly leads to too low measured values of ne

, because

n is measured in a region situated in the "shadow" of th e probe.e

The perturbation of the electron energy distribution function by th e

probe is described by Waymouth (ref. 7.3). By comparing the depletion

time with the self-collision time for electrons as defined by Spitzer

(ref. 2.4) , the following condition, which must be satisf ied in order

that th e perturbation may be neglected, can be found:

(VII. 2)

Taking the plasma volume Vpl

of plasma with approximately

T = 104

oK, i t follows from

-3 3to be equal to 10 m , being a volume

e

constant properties, and assuming

the area A of th e probep

electrode, which is equal to

the value of

2 x 10-7

m2

, that th e left-hand sideof the inequality (VII.2)

of magnitude i f n = 1020

e

exceeds

-3m

th e right-hand side by four orders

Five runs of probe measurements are carried out with th e object of

examining the behaviour of T and n • The most important experimentale e

parameters are l is ted in Table 7.2. The runs I to IV concern th e

measurement of n a n d T as a function of r , realised by radiale e

shifting of th e probe (see Fig. 7.4a). In ru n V th e axial dependence

of n a n de

(see Fig.

T is measured by turning the probe around i ts axise

7.4b). Run I is carried out without having inserted the

electrode rings into the disc; for the other runs only open circui t

generator conditions are considered. For the runs II to V a modified

gas in let system i s used, in order to realise a larger gas flow

into th e channel.

The probe current resulting from one discharge of th e guns and one

probe voltage is measured through V (see Fig. 7.3) which voltage i sm



- 58 -

displayed on an oscilloscope. An example of such a picture has

already been given in the preceding chapter (seeFig. 6.7b). V ism

determined from pictures l ike Fig. 6.7b at three times tl, ti and t3

over a period of about 20 vsec; i t has been tr ied by analysing theshape of the probe signal to determine these times for each probe

position in such a way that for the various probe positions of one

series a similar quantity of gas is considered. Current voltage

characteristics are composed from pictures l ike Fig. 6.7b, as

obtained for the various probe voltages. As an example, in Fig. 7.5

a current-voltage characteris t ic of the probe is given, showing the

following deviations from the ideal curve as represented by

eq.(VII. I ) :

th e experimental curve does not pass through the originand th e asymptotic values of the current are not independent of the

voltage. In this experiment, the relationship between th e probe

current and voltage is assumed to be given by:

e(V - V )i - i = 1 . tgh { 2 0 } + (a+)(V - V )p po 1 2kT _ P po

e

where a+ is used for V > V and a for V < VP po - P po

,

10

0

5• -1.0

IIK.,

·' 0 ·15 ·10 ·5 o 10 15 20probll! volhlge(Vl

Fig. 7. 5 Double probe curt'ent-voltage characterist ic , taken from run I , series J, probe

position r'"

0.09 m.

(VII. 3)



The parameters I" i ,1 po

experimental points f i t

- 59 -

v ,a , a and T are so chosen that thepo + - e

the curve. According to the theory of Bohm

(ref. 7.4) th e ratio of saturated electron current and th e saturated

ion current towards the probe is 0.7,I m/m

e.,

resulting in th efollowing expression for n :

e

0.7 1.1

n =e eA(VII. 4)

p

Eq. (VII.3) can be considered as a modification of eq. (VII. I ) , taking

into account the deviations of the experimental current-voltage

characteristic from the ideal one.

The results of the measurements of the runs I to V are given in the

Figs. 7.6 to 7.10. The three values of n a n d T , obtained in eache e

probe position of a series from the current-voltage characteristics

belonging to the times tl

, t2 and t3

, are averaged. The given

experimental errors are calculated from the deviations of th e

experimental points of the matched current-voltage characteris t ics .

Table 7. 2 Experimental parameters referring to th e probe measurements.

,"n series VB (kV) 1disch(\.Isec) Bo(T) probe position

I I 5 95 0 0 , · 3.5. , 4.5, 5, 6, 9, 13 , 17 em; ,• 0

oo

2 oo oo 0.01 oo oo

oo

3 oo oo

0.03 oo oo

oo , oo oo

0.05oo oo

II I 8 1100 0.01 , · .5 , 8, 12.5'" ; ,

• 0.,

2 oo oo 0.04 oo oo

1 II I 10 J 200 0 , · 3.5. 8, 12, 16 om ; , ·0

oo 2 oo oo

0.07 oo oo

IV I 12 1300 0 , · J. 5. 8, 10, 15 ,m ; ,• 0

oo

2 oo oo

0.07oo oo

V 1 10 1200 0 , . 10 ,m ; , · - I. 2. - 0.6, 0, 0.6,

oo

2 oo " 0.07 oo oo

1.1 ,m



- 60 -

"I 8, .0

16000

",: 0

12000

"

8000':';

"i::!:0"

'000

", ,"" ". ", C-09

'"0.13 '" '" '" ", ". '"

O.ll

'"---",

rim) rim)

"8,,0.01 T

16000 "8.,0.01T

12000..

! ,8000 "

'\:

'"2.

"! ..

0003

'"' '", . 0.11

.,0.15 ., 0

003

'"' ", , . ., '" O , l ~ '"'m ) nm )

"a.,oon

11000..

8,.0.0)1

'"..

••• , , ; "r

N.t

"'" ",'" ". ", ." " " "

, , '", 0.05 ", '" '" '" '"----0;7

"m ) rim)

"I" e.- 0.05 T

"

1&000

"I,.O,O$T

12000

"

. • 000 "•

'''' "

'", .., ", "" '" '" 0.'5

'",'"

00' "" '" '" C.1l 0' '", , . , , " .

Fig. 7.6 Electron temperature T and density n ., measured in run 1 .

e e



-

0000

~ 4 0 0 ,! '

0

16000

0

,:"1,000,! '

1600

1200

0.0,

0.,

~ 4 0 0 0

B •• D.on

0.05

r Im )

9 .: 0.04 T

0.0'r (m )

8 . :0

rim)

B.: 0.01 T

vetect rode

I

I

II

IIII

0")7 0.0.

, OD,

Fig.

Vlec t rode

I

- 61 -

P o s i t i o n s ~ OJ

B.:O,ol T

I

I ..

!I

I OJ.

I N

IQ

"I .!'.,

II

00" DU 0.15 OOJ 0" 0.0, 0.0. O.fl 0.13 U15

r im !

OJ

B. : 0.04 T

.. I

~ - O J . '...02

00.11 0.1J 15 ,OJ

" 0 • ." 0.13 .15

r im )

7. 7 As Fig. 7.6. corresponding to ru n II .

P o s i t i o n 5 ~

I

I

I

u

1.0

OJ

0.6

'2

.!'Q2

o l ~ - - ~ ~ - - . r . . - - . ~ - - ~ ~ - - ~ ~ - - . . OD) 0.05 0.01 0.09 0." on OJ5

r im )

1.,

1.0 B•• O.01T

0.'

' : " ~ Q.6

:<.,::" 0.,< .,

0 ' ~ . 0 : c , : - - " ' 0 . 0 ; ; , , - - , , .c,O----C0"0".----.0"171 ----.0".1.' --1--00" ·."'o"',--o"'• ',--"o".'.----,0.0=, ---..'""..---c."I-'--'---,;."

Fig. 7.B As Fig. 7.6, corresponding to run I l l .



- 62 -

16000.electrode positions" ..

'".. 0B.: 0

12000 ..

'00 ...-"E2.

~ . t ; o o o .'"

0 0. 0.0. 0 " DO. 0." 0.13 0.15 0.11 0 . ' 0.0. 0 . ' 00. 11.11 0. 0 0 . 0.17

rt 1111r Ul'll

1O"J

..

B.:0.01T B•• 0.01 T

""

'000 ..

:E:5!

i '00~ 0 2

'"0 0

." DO' .., DO.,,, ,,,

11.15 0",.,

0 . ' 00 ' 0 " 0" Oil O.IS OJ?

rIm) ""'.

Fig. 7.9 A, Fig. 7.6, corresponding to ru n IV.

16000OA

B;O 8,:0

12000 0·'

8000 "0.2

0

No

0

••4000 0,'

00

-,. . ' ~ -, 10 15.,. -10 10 15

zllG" m} z CHI-3m)

20000B.=0.07 o· B.= 0.07

16000 0 ·'

12000 0 ·'

800Ijt; 0.2

0N

"..•4000 0.'

00.,. ·10 ., , 10 -,. -10 10

,z (,O-3m) zt to-3

m)

Fig. 7.10 A, Fig. 7.0, corresponding to run V.



- 63 -

VII.3 Electrode voltage and floating potential measurements

The electr ical potential in th e plasma has been measured as a funtion

of the radius by meanS of the determination of the floating potent ial

of an e l e c t r o s ta t i c probe in various pos i t ions . The measurement has

two different objectives, namely the determination of the influence

of the electrode boundary layers on the electrode voltage, and the

determination of the effective Hall parameter. From the various

leakage processes occurring in MHD generators ( ref . 7.5) , two effects

are expected to be important in the present experiment, viz. the

influence of electrode boundary layers , and ionisation ins tabi l i t ies .

The former process can be examined by comparing the distr ibution of

th e plasma potential along the radius with the measured electrode

voltage. The presence of ionisation instabil i t ies has been investigated

by th e determination of wTeff from the open c i rcui t voltage, the

averaged plasma velocity, and th e electron pressure difference at the

electrodes, which quantities are approximately related in the

~ 4 > 4>~ 4 >

4>"0 .J:l ~ ' 8 C O 0 : J ~ c!::- u a.O.!! 0-.. . 4> 0 .. . 4>...

+120K 120K

I

oscillo- oscillo-

----tcope scope

I

Ru lFig. 7.1 I Electrical circuit fo r measuring f loat ing potentials and electrode voltages.



following way:

v =oc

- 64 -

Ap

+ wteff ur B Ln e

e

(VII.5)

where Ap is the electron pressure difference between the electrodes,e

and n , WT f f ' u and B th e averaged electron density, effectivee e r

Hall parameter, radial velocity, and magnetic induction

respectively, and L the distance between th e electrodes.

To determine the time when the plasma arrives a t the inner and th e

outer electrode rings, saturated ion currents were measured with adouble probe at r = 0.07 and 0.14 m. As a result of this measurement

characteris t ic passage times in both positions are given in Table 7.3.

Table 7. 3 Plasma passing times ti n and tout at the inner

and Quter electrodes for different values of

the magnetic induction Bo'

B t . t0 out

(T) (psec) (psec)

0.01 170 250

0.02 185 26 0

0.03 200 270

0.04 215 280

0.05 310

0.06 245 330

0.07 260 340

0.09 285 350

The measured open circui t voltage reaches i ts maximum values on the

outer electrode passage times. Apparently the generator is then fi l led

with plasma in an optimal way. The results of the measurements are

given for these times only. Moreover, a value of the averaged radial

plasma velocity can be derived from Table 7.3:

u = (1000 + 140) m/sec, not significantly depending on B •r _ 0



- 65 -

The el&ctrical potential in th e plasma is determined by measuring

the floating potential of a single electrostatic probe relative to

the inner electrode. The electr ical circuit is given in Fig. 7.11.

The considered experimental conditions are l isted below:

capacitor bank voltage: VB = 5.5 kV;

discharge delay time 'disch = 950 psec (for both guns);

probe position r = 0,075,0.090,0.105,0.120 and 0.131 m;

magnetic induction

load res i s tance

B =o

0.02, 0.03, 0.04, 0.05, 0.06, 0.07 and

0.09 T;

R = 100, 2 and 0.2 n.u

The floating potential relat ive to the inner electrode is given in

Fig. 7.12 as a function of r . The voltage of the outer electrode

with respect to the inner one is also given in Fig. 7.12.

Fig. 7.12 shows, that the electr ic field in the generator is not

significantly affected by th e external resistance. This indicates that

th e resistance of th e electrode boundary layer exceeds by far th e

plasma resistance, so that th e measurements can give no information on

the behaviour of a loaded generator. The shif ts of the potential

curves due to variations of the load may be explained by the lowering

of the electrode boundary layer resistance at larger currents

resulting from Joule heating. Open circuit voltages are derived from

Fig. 7.12 by l inear extrapolation of the potential curves over th e

regions from 0.075 to 0.070 and from 0.131 to 0.134 m. The measured

voltages are the measured floating potential differences AVf l

between

inner and outer electrodes given in Table 7.4. AVf l

is obtained for

each value of B as the average of the voltages resulting from th eo

various potential curves; the given error is calculated from the

scatter in these voltages.

Since for the used magnetic fields rLi

is longer than AD and also

longer than the diameter of the probe electrode, the difference

between th e floating potential and the plasma potential may be

given by (ref. 7.6):



- 66 -

12 8.= O.OlT 6 8.=O.02Te

,

/1-1O. /A , / / , / I , .;>/ ./ 0 , , ~ / 0"/ . 1

6.---- /r :,/ :;2

/'// I-

/,I

-6 ____ 6 +/

i .- / •0.07 0.09 0.1\ 0.13 0.07 0.09 0.11 0.13

rem) r (m)

1 8.= O. O3T 12 8.:0.04T

8 I 8 ,;;.

I

/

..../

:::4 > 4...--:t1

. /

- ,-- /

-§

.1 0.07 0.09 OJl 0.\3r (m) rem)

oElectrode voltageat D)Q+.. .. Ru:2 Q• N "

Ru: 0.2Q

---0- ~ . 1 0 0 ~ - -4- - 1t.2 Q

- --- I t- 0.2Q

Fig. 7.12 The f loating potential Vf l

as a function of the radius for various values of Bo and

the load resistance.



16 B.:O.05T

12

I j, /1 /

1//-

/I0;. ::::..A

--00.07 0.09

rem)

16

a,.O.07T

12

8

0.07 0.09rem}

0.11

0.11

• Ru·100n---4--- Ru. 2 n------ Rue 0.2 Q

- 67 -

16

BrO.06T 1>---I -

-0 12 / /

1/0

8 /J/?

/-.A/

..... 4 ,/.'"> / /.....

Y

0.13 0.07

r ( ~ f 9 0.11 0.13

18o

0.13 0.07 0.09 0.11 Q.13r (m)

oElectrode voltage at lOOn+.. .. Ru.2 n... .. ~ 0 . 2 Q

Fig. 7.12 Continued.



.., 68 -

Table 7. 4 Measured floating potential difference between the electrodes, potential difference-4 £1

4. 4 x 10 ATe between the electrodes caused by th e difference between f loating

potential and plasma potential , potential difference APe/nee between th e electrodes

caused by th e electron pressure gradient, and the Hall voltage WT U BLateff r .

variDus values of the magnetic induction B .o

10- 4 - -B (T ) 6Vfl (V) 4.4 (V)

e(V) BL (V)WT U

eff

0.01 0.78

0.02 3.48

0.03 9.5

0.04 9. I

0.05 10.60.06 11.1

0.07 11.8

0.09 15.0

+ 0.02-+ 0.12-+ 0.4-+ 0.4-+

0.5-+ 0.1-+ 0.7-+ 1. 3-

-

-

-

-

-

-

-

-

kTe

= I e

e

0.2 0. 1

0.6 + 0.2

1.1 + 0.4-1.4 + 0.5

1.6 0.51.6 + 0.5

1. 7 + 0.5

1.7 + 0.5-

m

In (2!. -=.)2 m

- rn e

e

- 0.5 + 0.1 0.8 + 0.2- -- 0.6 + 0.1 3.5 + 0.2

- -- 0.8 + 0.2 9.2 + 0.6

- -- 0.9 + 0.2 8.6 + 0.7- --

0.9+

0.2 9.9+

0.7- -- 0.9 + 0.2 10.4 + 0.5- -- 0.9 + 0.2 11.0 + 0.9- -- 0.9 + 0.2 14.2 + 1.4

- -

(VII. 6)

According to eq . (VII.6) th e values of tlV

fIand the open circuit

-4voltages differ by 4.4 x 10 t lT, where tlT is the electron temperature

e edifference at the outer and inner electrode. Using eq. (VII.S), values

of w'eff can be calculated from

values of tlp In e, , E and L.e e r

the open c i rcu i t vol tages and the

The values of 4.4 x 10-4

tlT ande

tlp In e are estimated from the results of the probe measurements ofe e

ru n I (see th e preceding section, Fig. 7.6). In table 7.4 tlVf l

,

4.4 x 10-4

tlT , tlp In e and WT ff E L are l is ted a t various valuese e e e r

of B • In Fig. 7.13 WT ff E L is given as a function of E, where Bo e r

is equal to th e value of E at r = 0.1 m, which is 0.85 B accordingo

to eq . (IV.2). With ur

= 1000 + 140 mlsec and L = 0.07 m, w'eff is

calculated. Fig. 7.14 shows how w'eff depends on B. The shaded area.

indicates th e uncertainty in the "level" of the curve caused by the

uncertainty in the determination of ur

and L. The discussion of

Fig. 7.14 is given in chapter VIII.



- 69 -

15

10

I/II

>/

/'01 Il : i I

5/•

13/A

//

//

/

, i/

/

n02 OD4 0.06 0.08

em

Fig. 7.13 Hall voltage w1eff urB L as a function of th e magnetic induction B. The dashed

parabola through th e or ig in f i t s th e f i r s t three experimental poin ts , assuming

WletE

= fo r these points.

6

5

4

3

2

!!

/

I

o ~ - - - - - . ~ - - - - - - ~ ~ ____ - . ~ ______0.02 0.04 0.06 0.08

8 m

Fig. 7.14 The effective Hall parameter w'ef f ' as a function of th e magnetic induction B. The

dashed l ine through th e origin f i t s th e f i r s t three experimental poin ts , assuming

w1"!ff = for these poin t s . Th e shaded reg ion shows th e uncer t a in ty in th e l ev e l

of the curve caused by th e uncer t a in ty in th e determination o f th e plasma veloci ty

an d th e generator length .



-: 70 -

VII.4 Spectroscopic measurements

The electron temperature and density are also determined from th e

intensity of some argon radiation l ines for which the plasma is

optically thin. The experimental set-up is sketched in Fig. 7.15.

' < - - . l J - - - - - -- - - - _ = = = = = = = 4 ~ = = = : = : = = ~ S 9 ! ~ ~ ~ t o r Fig. 7.15 Experimental arrangement of the spectroscopic measurements.

Radiation intensit ies are determined by measuring the anode current

of a photmultiplier connected to

to the plasma radiation and that

a monochromator. The response due

due to a calibrated

lamp are compared. Fig. 7.15 shows how two images of

tungsten ribbon

the tungstenribbon are formed: one in the disc and one at the s l i t S of the

monochromator. The magnifying factor of both images is equal to one

and th e dimensions of the ribbon are larger than those of S, so that

S can be fi l led completely with l ight from th e lamp. The aperture of

the monochromator is smaller than that of the rest of the optical

system. The l ight absorption of the glass walls is measured by comparing

the photomultiplier output resulting from the lamp situated either in the

position A (see Fig. 7.15) or just before S. The l ight intensity of the

lamp at A i s reduced owing the absorption in the g l ~ s s by 10 %, the

absorption being sl ightly dependent on the wavelength in the region

from 6000 to 9000 The resolution of the spectrometer is determined

by measuring the half-widths of th e spectral l ines of a mercury

discharge tube; a half-width of 0.33 is found, not depending on th e

wavelength within a region from 6000 to 9000 R.



- 71 -

The intensity of some argon lines are determined, using the

emissivity of tungsten, as given by De Vos (ref. 7.7). As shifting

of the image of the tungsten ribbon in the disc over 1.5 cm yields

only a reduction of detected radiation by about 40 %, i t ha s been

assumed that th e detected plasma radiation originates from a volume,

which is given by the area of S and the distance between th e glass

walls, which is 3 cm.

From th e transition probabil i t ies , as given by Olsen (ref. 7.S) and

from th e measured l ine intensities, the populations of some excited

levels of the neutral and singly ionised argon atoms are found. To

derive th e electron temperature from these levels, local thermodynamic

equilibrium (LTE) ha s to be assumed at least for th e energy levels

considered. T then follows from Boltzmann's equations' of state:e

and n from:e

2n

Nm

2g'o

{ - (E - E )/kT }m n e

exp { - (E. - E )/kT }1a m e

where g' is th e s tat is t ical weight of the ion ground state.o

(VII.7)

(VII.S)

Eqs.

of a

(VII.7) and (VII.S) can be applied for N , being the populationm

neutral argon sta te , while (VII.7) can also be used when th e

states of ionised atoms are involved. As pointed out by Griem (ref. 7.9),

the validity of LTE for any level can be examined by comparing the

total rate of coll isional excitations of atoms in the level considered

and th e probability of radiative decay from that level. Requiring

the coll isional processes to exceed th e radiative ones by a factor 10,

in the case of hydrogen-like atoms this procedure leads to th e

following cri terion for par t ia l LTE:



ne

7z

17/2nq

- 72 -

kT(_e_) 1/2

2z EH

-3m (VII. 9)

where z is th e nuclear charge, n is th e principal quantum number ofq

the considered atomic sta te and EH i s the ionisation energy of hydrogen.

The temperature T appears in the formula, as for th e plasmas considerede

electron atom collisions constitute the majority of the coll isional

excitations. Griem suggests that th e criterion is approximately valid

for other than hydrogen-like atoms. Substituting in eq. (VII.9)

n = 3 x 1020

m-3 and T

e e= 9000 oK, the cri t ical value n is found to

q

be between 2 and 3 for neutrals and between 4 and 5 for singly ionised

atoms. In th e present experiment AI l ines from states with principal

quantum numbers equal to 3, 4 and 5 are examined, and All lines resulting

from states with principal quantum number equal to 4.

The experiment has been

0.05 T. A plasma volume

carried out for values of B equal to 0 ando

at r = 0.1 has been examined. The other

experimental conditions are equal to those of ru n IV of the probe

measurements (see Table 7.2). In Table 7.5 the wavelengths, the used

transition probabil i t ies of the detected t ransit ions, their energy,th e weight factors of the in i t ia l states and their populations are

l is ted. The experimental errors in the populations are derived from

th e scatters in three similar photomultiplier signals and from

estimated errors of the transition probabilities. Fig. 7.16 shows how

th e values of N Ig are correlated with those of E • By f i t t ing th em m m

experimental points to straight l ines using the leas t squares method,

th e values of Te have been found. Applying th e Saha equation (VII.B)

with th e found values of T and N for the neutral atoms substituted,

e mne is determined. The results are given in Table 7.6. They will be

discussed and compared with the results of the other diagnostics in

chapter VIII.



' U = U ~ ' - - - - ' - - - ~ U ~ , c - - - - + - - - - - z . U ~ ' C - - - ~ - - - - ~ U ~ 2 C - - - - - - - - - - U ~ ' - - - - - - - - ~ ~ E In CllfllJoIIl <l:J

.IDAllin . .

8"o.oST

."

n'

'"

16,0

' ' ' ~ 2 . ~ , = - - - - - - - - - C 2 ~ ' " ' C - - - - - - - - - - O •~ . - - - - - - - - - - ~ . " , 1 . - - - - - - - - ~ . " ~ " . c - - - - - - - - - ~ > - Em t Ur'-JouLe)

.'"

.,,

.,.

"...

Al l l i " c ~ 1.10.05 T

Fig. 7.16 Populations Nm of exci ted s ta tes of A I and A I I , divided by the weight factors gminvolved. as functions of th e energy of th e states .



- 74 -

T a b l ~ 7, 5 h'aveLengths Aand t r ans i t ion prob abi l i t ies Anm of th e detected radi , . t ions, and configurat ions ,

weight factors gm' eOE'rgies Em' an d populat ions Nm

of the ln i t ia l s ta tes . The population

calculated from th e 4164 R adiat ion in th e case of B = 0 and that calculated from 4266 R

qR)

3949

4046

4159

4164

4182

4198

4251

4259

4266

4272

4300

4334

433.5

4345

4510

5559

5572

5607

5651

6032

6059

4347

4579

4610

4807

ol-adiation in th e case of B

oO.OS T have been omitted for st.:ltistical reasons. The other

omissions ar e due to too low radiat ion in tensi t ies .

Nm(m-

3)

N

Nm

A (1 07

sec) config. gmE (10- 18

Joule) B • 0 B • 0.05 Tnm m 0 0

in i t . state

0.017 3P3 5 2.353 6.7 x 1013

(22 %) I . 7 x 1013

(24 %)

0.037 3P3

5 2.353 5. 0 x 1013 (12 %) L 3 x 10 13 (17 %)

0.119 3P6 5 2.328 8. 4 x 1013 (5 3 %) l . 7 x 10 13 (51 %)

0.022 3P7

3 2.327 - 2. 0 x 10 13 (28 Z)

0.041 3P2

3 2.353 6. 1 x 1013

(17 %) 1. 6 x 1013

(29 %)

0.245 3P5

I 2.335 7. 2 x 1013 (21 %) I . 7 x 1013

(2 0 %)

0.008 3PlO 3 2.317 10.1 x 1013

(22 2. 6 x 1013

(42 %)

0.324 3p , I 2.361 5. 6 x 10 13 (27 X) 1. 5 x 1013 (30 %)

0.027 3P6 5 2.328 6. 1 x 1013

(20 %) -0.061 3P7 3 2.327 7.8 x 1013 (26 X) 2. 2 x 10

13(23 %)

0.034 3P6 5 2.324 5. 6 x 1013

(17 X) 2.3 x 1013 (24 %)

0.048 3P3

5 2.353 5. 0 x 1013

(26 X) 1. 8 x 1013

(30 %)

0.037 3P2 3 2.353 4.5 x 1013

(26 X) 1. 5 x 1013 (31 X)

0.027 3P4 3 2.352 5.0 x 10 13 (2 0 X) 1. 5 x 10 13 (34 %)

0.102 3P5

I 2.335 5.0 x 1013

(2 3 %) 2.5 x 10 13 (36 %)

0.083 5d3

5 2.425 2. 4 x 1013

(3 3 X) 0.9 x 1013

(31 X)

0.0395 , I

7 2.454 2. 6 x 1013

(3 3 %) 1. 1 x 1013

(39 %)

0.150 5d5

3 2.422 2. 6 x 1013

(30 X) 1. 0 x 1013

(31 %)

0.190 5d , I 2.419 4.1 x 10 13 (34 %) 1. 1 x 1013

(35 %)

0.210 5d4

9 2.424 2. 1 x 10 13 (3 3 %) 6.6 x 1013 (35 %)

0.038 4' 1 5 2.396 3. 2 x lO l l (35 %) -

11.5 8 3.119 - 6. I x 1011 (15 X)

7.44 2 3.196 - 3. I x 1011 (12 %)

9.06 8 3.383 - 6.8 x 10 11 (14 %)

7.86 6 3.076 - 10.0 x 1011 (22 X)

Table 7.6 Electron temperatures Te and densities ne as measured in the spectroscopic investigation,

a t two values of the magnetic induction Bo,from A I and A I I l ines.

B = 0, A I B = 0.05 T, A I B = 0.05 T, A ll0 0 0

T (7800 800) OK . (8900 ... 1100)·0

.(8900 400) OK+ ..K. +e - - -

(4.5 + 1.0) x 1020 -3

(3.0 0.8) .1020 -3 - m + x .m

e - -



- 7S -

VII.S Microwave measurements

The electron density is also examined by means of m e a s u r i n ~ the

reflexion of microwaves against the plasma. A wave guide, inserted

in the plasma through the torus of the disc is used as a microwave

probe. The probe can be used, either separated from the plasma by a

thin window (thickness much smaller than the wavelength), or by a

window of a thickness equal to (2q + 1)/4 times the wavelength

(with q an integer).

The reflexion coefficient R for the reflexion of the wave against the

plasma in the case of a thin window is given by:

with Nw

plasma,

N - N

R = IRI eiq, = ~ N w = - + - N J ; : . P w p

(VII. 10)

and N the refraction coefficient of the wave guide and thep

respectively. The tota l reflexion coefficient in the case of

a (2q + 1)/4 lambda window is given by :

N2

- N N£ W P

N2

+ N N(VII. lOa)

£ W P

where N is the refraction coefficient of the window. The refraction£

coefficients N ,N and N can be expressed in characterist ic propertiesp w £

of the plasma, the waveguide and the window material as follows:

w }1/2 w }1/2N = { I _ ( 2 / 'V { I - ( ~ ) 2 (VI I . I I )p w I - iv Iw 'V W

c

N = { I - (A/2b)2 }1/2 (VII . 12)w

N£

= { £r

_ (A/2b)2 }1/2 (VII. 13)



- 76 -

where w is the plasma frequency, w/2n the microwave frequency, vp c

the coll ision frequency of the electrons, A the wavelength in vacuum,

"r the relat ive permittivity of the window material , and b the length

of the lowest side of the waveguide cross-section. Writing wa s :p

2n e /

w = (_e_) I 2pm"

e 0

(VII. 14) .

i t follows from eq. (VII. I I ) that N can be expressed in the electronp

density:

NP( I

- n /n )1/2e ecr (VII.I I

a)

with the cr i t ica l electron density n being the electron density a tecr

which wp equals w:

necr

2w " mo e

2e

(VII. 15)

Inserting eqs. (VII. I I a), (VII. 12) and (VII. 13) in eqs. (VII. 10) and

(VII. lOa), i t follows that IRI = I and phase shifts occur,

In the cases of a thin window and of a (2q + 1)/4 lambda

ifne > necrwindow, the

phase changes are given by the following expressions, respectively:

, - - ' " ,0 [ C:n

;:;,:" ) " ' ](VII. 16)

= TI - 2 arc - - - - - - - - - - - - - - - - - - ~ ~ ~ - - - - -g[(

{ 1- (A/2b)2} (ne/necr - I ) )1 /2]

{ " _ (A/2b)2 }2r

(VII. 16a)

By measuring the electron density can be determined. A (2q + 1)/4

lambda window with high "r results in a smaller compared with a

thin window,

measurement,

at the same ne. This may enhance the accuracy of the

especially i f n »ne ecr



- 77 -

The measurement of is carried out with a 4 mm microwave bridge at

a wave frequency of 68.5 Gcls and using a 514 lambda window of

polyethylene (E = 2.3). The probe is enveloped in teflon tape inrorder to achieve electr ical insulation from the plasma. A diagram

of the bridge is shown in Fig. 7.17.

5tobillndpower $\lppt,

modulalor

V- - _ . . . J J

¥ - - - - "

to oscilloscope

to os ilto5tope

&dB

diredionot micrawan probecoupler

Fig. 7.17 Microwave bridge diagram.

The measuring branch and the reference branch have unequal lengths.

Let the difference between th e two paths be 6 ~ , introducing at th e

hybrid tee a phase difference ~ ~ . The response of the hybrid tee is

schematically given in Fig. 7.18 with Al and A2 th e amplitudes of

waves in the measuring branch and th e reference branch respectively.

H E

- -, N n { w t · ~ · A . . l i n l , .. t . ~ ~ A.sin(wt.q,I- .... i n l W l . ~ ...,

Fig. 7.18 Scheme of th e response of th e hybrid tee .



- 78 -

The responses R] and RZ

of the crystals are given by:

R]1. AZ + 1. AZ

- A A cos (</> - l!.4»

!] Z Z I Z (VII. 17)

RZ

= 1. AZ+ 1. AZ

+ AIAZ cos (</> - ll</»Z ] Z Z

Subtraction of the two signals with a different ial amplifier yields

the f inal signal:

(VII. 18)

In the given experiment the clystron is modulated with a block pulse

causing a frequency modulation QV 40 Mc/s. This results in a

modulation of ll<P:

(VII. W

c { I - (J.../Zb) }

With III 1.7 m, adjusting of the amplitude of the block pulse and the

phase shifters in the reference branch results in two signals on the

oscilloscope, one with ll</> = TI/Z and one with ll</> = 0 (see Fig. 7.19).

The rat io of the two signals is equal to tg <p.

Fig. 7.19 Signal obtained from an electros tat ic probe (upper signal) and signals obtained from

the microwave reflexion probe (lower signals). The height of the middle signal i s

proportional to th e sine and that of the lower signal proportion;l to th e cosine of

th e phase angle ¢.

Vertical scales: 2mA/div (upper signal), arbitrary units (lower s ignals ) . Horizontal

scale: I O O ~ s e c / d i v .

For a more detailed description of this method of determining the

electron density, compare refs . 7.10, 7.11 and 7.IZ.



- 79 -

Table 7. 7 Phase changes of waves reflected against th e plasma and values of th e electron density

ne at various values of th e radius r and the magnetic induction Bo'

¢ (in degrees) n (1020 m- 3)e

r(m) B = 0 B = 0.05 T B = 0 B = 0.05 T0 0 0 0

0.035 115 + 10 115 + 10 9.9 + 2.3 9.9 + 2.3- - - -0.08 106 + 10 119 + 10 7.3 + 1.5 11.5 + 2.8

- - - -O. 10 109 + 10 115 + 10 8.0 + 1.7 9.9 + 2.3- - - -O. 12 100 + 10 114 + 10 6.0 + 1.2 9.5 + 2.2- - - -

0.16 90+

10 92+

10 4.4 "" .0.8 4.7+

0.9- - - -

The experiment has been carried

By shifting the microwave probe

out for values of B equal to 0 and 0.05 T.o

radial ly the electron density has been

determined a t various values of th e radius: 3.5, 8, 10, 12 and 16 em. The

other experimental conditions were equal to those of ru n IV of the

electrostat ic probe measurements (see Table 7.2) . The measured phase

angles and th e electron densit ies are given in Table 7.7.

The resul ts are discussed in chapter VIII.

VII.6 Piezo-electr ic crystal measurements

TIle to ta l gas pressure has been measured with a quartz piezo-electr ic

pressure transducer mounted on a probe which can be moved radially in

the disc. The charge signal of the transducer is amplified and t rans

formed to a proportional output voltage in a charge amplifier. A block

diagram is given in Fig. 7.20.

r ""]

L+""u L..-__ ......Jloutput<at/V)

piezoelec:trictransducer

e lec:trostatic

c:harge! amplific:r

Fig. 7.20 Scheme of th e pressure measurement with th e piezo-electric crystal.



- 80 -

The system is capable of measuring pressures in th e range of 0.005 to

2.5 a t . at a time-resolving power of 10 ~ s e c . The to ta l gas pressure

has been measured at various values of the radius: 4.5, 8, 10,12

and16 em, in th e case of

conditions were equal

B equal to 0 and 0.05 T. The other experimentalo

to those of run IV of the electrostatic probe

measurements (see Table 7.2). Electric insulation of the probe is

obtained by enveloping i t with teflon tape. As an example, the response

of the pressure transducer is shown in Fig. 7.21.

Fig. 7.21 Response of th e piezo-electric pressure transducer.

Verticale scale: 0.01 at /div . Horizontal scale: 50 ~ s e c / d i v .

The resul t s of the measurement are shown in Table 7.8 .

Table 7. 8 Total gas pressuresPgmeasured by means of a piezo-electric pressure transducer at various

values of th e radius r and th e magnetic induction B .o

p (Torr)

rem) B = 0 B = 0.05 T0 0

0.045 27.4 + 3. 1 35.0 + 3.8

0.08 12.2 + 2.3 15.2 + 1.5

0.10 11.4 + 1.5 9.9 + 1.5

0.12 9. 1 + 1.5 7.6 + 1.5

0.16 6. 1 + 1.5 3.0 + 2.3

The results are discussed iu th e following chapter.



- 81 -

CHA P T E R VIII

Discussion o f the experimentaZresuZts

The results of th e probe measurements show a good evidence of electron

temperature elevation (see Figs. 7.6 to 7.9). The differences between

the values of th e electron temperature at th e inlet of th e channel

and th e maximum values of the series with Bo

o

0.03 T vary from 3000

to 8000 K. As expected, no enhancements are found

0.01 T. The experimental points, especially those

for B = 0 ando

belonging to run I ,

show a drop of Te at longer radi i . No unique explanation of this effect

can be given.

The importance of heat transport to the walls correlated to wall

frict ion ha s been investigated by considering the gas as a one

dimensional, fully developed turbulent flow. The influence of th e

process on th e gas temperature has been examined assuming the wall

temperature to be much smaller than th e gas temperature by.means of

the following equation (ref. 8.1):

dT

T2f

D(V I l L I )

with D the hydraulic diameter of the channel and f th e frict ion

coefficient, given by:

-2f = 4.6 x 10 (VIII . 2)

ReD is th e Reynolds' number related to the hydraulic diameter and is

the viscosity of the medium. The reduction of th e temperature after

0.15 m appeared to be < 300 oK for the experimental conditions

considered. An experimental argument for the flow being one-dimensional

and turbulent is found in th e measurement of n a n de

the axial distance, carried out at B = 0 and 0.05 T

T as functions ofe

(see Fig. 7.10).

Although th e measurement at B = 0.05 T can only be interpreted witho

some restr ictions, because different directions of th e probe surface

with respect to the magnetic field are used, there is no indication of



- 82 -

cosine-shaped temperature profiles characterizing a laminar flow

between cold walls.

The electron temperatures as measured by the spectroscopic method

may be compared with the probe measurements of run I I I (see Table 7.2);

only the magnetic field strengths differ (0.05 T in the former and

0.07 T in the la t ter case), and the position r = 0.1 m, Z = 0 is not

examined explici t ly by the probe. For B = 0 the two methods agree,o

whereas for B +0 lower values of To e

are found by the spectroscopic

method. From the analysis of chapter IV, no indication can be found

that the discrepancy may be caused completely by the magnetic field

strengths being different . The problem connected with the interpretation

of the spectroscopically determined sta te populations is the assumption

of LTE. Eq. (VII.9) cannot be considered as a sharply defined cri terion,

as str ic t ly speaking the relationship is only valid for hydrogen-like

atoms. I f eq. (VII.9) is valid, then the existence of LTE is s t i l l

questionalble in the

n = 4 for A II . Theq

may have been caused

case of the states with n = 3 for A I and withq

disagreement at B +0 with the probe measurements

by the circumstance that in the non-equilibrium

plasma conditions the LTE assumption is not applicable to th e part iclesof a l l levels considered, so that the populations of the states are no

longer completely controlled by the electron temperature.

No significant electron density enhancement due to magnetically induced

ionisation has been found, neither by the probe measurements, nor by

the spectroscopic or microwave methods. Moreover, the values given by

the f i r s t method differ by almost one order of magnitude from those

obtained by the second and the third.

The development of NEI.is much more diff icul t to observe than the

development of ETE, because the ionisation relaxation length has the

same magnitude as the dimension of the disc. I f the relaxation length

is increased by ionisation instabi l i t ies , as suggested in chapter V,

the resul t may be a complete vanishing of the effect in th e experiment

described. The NEI is also made observable with difficulty by the

circumstance that the electron density at the channel in le t is much

higher than predicted by the Saha equation. For the positions



- 83 -

concerned (r 0.16 m) only a small increase of n can ·be expectede

from the analysis of chapter IV. The predicted increase of the degree

of ionisation should be verified experimentally by comparing the

behaviour of ne as a function of r for Bo = 0 and for Bo o. This

comparison is made diff icul t by the circumstance that the production

of the plasma and i ts transport through the pyrex tube is affected

by th e magnetic field, generally result ing in higher values of n a te

the channel inle t for B 0, as shown by the probe measurements.o

Besides from the probe measurements an indication of this correspondence

between a higher degree of ionisation and a higher magnetic field

strength is found as a result of the spectroscopic measurements.

When applying the magnetic field, the excited states of neutral argon

are depopulated (see Table 7.6), whereas the population of the A II

levels increases (at B = 0 the radiation could not even be measured).o

I t may be expected that the microwave method results in too high

values of the electron density, because the normal to th e window

surface was directed opposite to the flow. However, this effect cannot

explain the difference between the values measured by the electrostatic

probeand

themicrowave

probe. Inchapter

VII i t has already been

suggested that the values of n as measured by th e probe would be tooe

low because the probe dimensions are larger than the electron mean

free path. The f i rs t order agreement between the values obtained by

the other diagnostic methods indicates in any case that the probe

value is too low.

The plasma velocity is determined by comparing probe signals at

different probe positions as well as by comparing photomultiplier

responses due to plasma radiation originating from differently

located volumes (see Fig. 8.1).

Fig. 8. 1 Photomultiplier respoqses owing to radiationfrom

plasma volumes atr =

O.lOm ( le f t )and

atr = 0.14 m (r ight) .

-6 -6Vertical scales : 8.9 x )O A/div. ( le f t ) and 3.9 x to A/div. (r ight) . Horizontal scales :

50 ]..Isee/div.



- 84 -

Both methods show the same diff icul t ies: the time shi f t s are smaller

than the time interval defined by the signals and the displayed

pictures show adeformation;

especiallythe

partrepresenting the

gasfront is affected, as might be expected. Both methods result in

estimations of the radial plasma velocit ies between 1000 and 1500 m/sec.

The basic equations of chapter II I predict only good generator

performance i f u is larger than sound velocity: otherwise the flowrstagnates and the Lorentz force tends to zero. Flow stagnation does

not occur in the experiment, as can be concluded from the measured

velocities and from the measured open ci rcui t Hall voltages. However,

the increase of the plasma velocity as predicted by the analysis of

chapter IV has not been observed, which indicates a deviation from

the stationary behaviour.

The displayed responses of the piezo-electric crystal have similar

shapes as the signals of the probe and the p h o ~ o m u l t i p l i e r only as

far as the f i rs t part of the pulse is concerned (see Fig. 7.21).

Apparently the plasma is followed by an amount of colder gas with a

comparatively high density. Since the normal to the crystal surface

was directed opposite to the flow, the flow stagnates against the

crystal surface, thus causing the pressures measured to be too high.

No cold gas is found to flow in advance of the plasma. The degree

of ionisation derived from the measured n and from n as determineda e

by the microwave method and from the plasma radiation, is about I %.

At that ionisation rate the Coulomb collisions constitute the majority

of the electron elastic coll isions.

The measurement of the floating potentials of the electrostat ic probe

have shown that the electrode boundary layers represent an electrical

resistance greater than the plasma resistance. I t has been tried

to improve this si tuation by heating the cathode electrical ly up

to 2400 oK before the passage of the plasma. The current through

the load is enlarged by a factor two owing to the heating of the

cathode (see Fig. 8.2), demonstrating the reduction of the

resistance of the electrode boundary layer. However, the la t te r

remains greater than the plasma resistance. The presence of

ionisation instabil i t ies is demonstrated in Fig. 7.14. The behaviour



- 85 -

Fig. 8. 2 Generator output voltage at an external resistance of 0.2 n and a magnetic induction Bo equal

to 0.07 T. The lower signals correspond to measurements with preheated cathode rings, th e

upper signals correspond to measurements with cold electrodes.

wTeff shows agreement with the results of other experiments (refs. 8.2,

8.3, 8.4 and 8.5). For B < 0.03 T, WT ff is proportional to B and i to '" e

may be assumed that for those values of th e magnetic field wTeff is

equal to WT. I f B exceeds 0.03 T, apparently WT reaches the cri t ica lo

value. The measured WTcr

is equal to 5.0 0.5, which is a rather

high value. However, especially the dependence on the plasma

velocity determination makes the absolute measurement of WT inaccurate.cr

For WT >

WT. The

5 the i on i sat ion in s tab i l i t i e s cause wTeff to

cri t ica l Hall parameter as calculated from eq.

be smaller than

(V.20) is equal

to 2.2, presupposing the following plasma conditions are

of the whole generator volume: n = 3 x 1020

m-3

; n = 3e 0 a

representat ive

x 10 22 m-3

uer

= u = 1000 m/sec; Tr e

12000 oK; T 8000 and B = 0.025 T.

(This value of B represents the situation where WT = WT .) Thecr

influence of ionisation instabil i t ies on th e electr ical properties of

an MED medium has been described in detail in ref. 8.6 from a plane

wave model. Fig. 7.14 shows f i r s t order agreement with th e results of

that theory.

Assuming ; ; e f f = WT for Bo 0.03 T, the to ta l electron elas t ic

collision frequency is determined as:- 9 - Iv = (1.0 + 0.2) x 10 se c • If n = 3 x

c - eSpitzer 's theory (ref. 2.4) predicts

experimental value of v and from nc e

v =

c= 3 x

scalar elect r ical conductivity i s derived:in an internal resistance of the generator

20 -310 m and T = 12000

9 _I e3 x 10 sec From th e

20 -310 m the averaged

a =

8500 mho/m, resulting-3equal to 4.6 x 10 n at



- 86 -

B = 0.02 T. (This value of B corresponds to a situation whereo 0

WT < WT .) For a load of about 0.2 Q th e electrode voltage has droppedcr

to half th e open circui t value, which demonstrates that th e electrode

boundary layers represent a greater resistance than th e plasma.

The experimental determinations of T ,n and n are compared withe ea.

he solutions of the basic equations in one case, namely run I I I .

The in i t ia l conditions for the differential equations of Table 3.2 are

chosen as close as possible to the following experimental results :20 -3

T = 7500 oK at r =e

i f Bo = 0; na = 3 x

0.035 m if B22 -3 0

10 m at r

= O' n = 3 x, e 10 m at r = O. I m

= 0.1 m if B = 0; u =o r

u ,i.e.

ersl ightly higher than sound velocity at th e channel in let ; furthermore

i t has been assumed that T = T at the channel in let . For B = 0

eo'.0 5 and 0.07 T solutions are given in Fig. 8.3. The experimental values

of T ,n ande e

n are also given in Fig. 8.3. The most importanta

results of the comparison between analysis and experiment can be

summarised as follows:

- The electron temperature is higher than predicted, i f B = 0, ando

lower i f Bo +O. ETE has been demonstrated, although th e effect

is not as large as predicted. The values of Te

as obtained by th e

spectroscopic measurement and those of the e l ec tros ta t i c probe

measurement differ i f B +O.

- The results of the electron density measurements show a discrepancy

between th e different diagnostic methods. The occurrence of NEI

cannot be concluded from the graph, par t ia l ly because no clear

evidence of the effect can be expected from one single curve.

- The measured to ta l gas pressures show f irst-order agreement with

th e analysis, although th e decrease of the pressure as a function

of the radius due to the expansion of the medium is less than

expected.



16

11(0)

000

22.0

"0 *

'00

- 87 -

,•

probe measurements.

! o ~ t r o s t o p i c <\0'e,..O.D7T<\00EI,oOOST

~ ? ~ / I

/

r (m )

0, , 0,07 OD'

probe measurements, a,. 0.8.- o.on

D. spectroscopic .9.,- 0... .a..0.05To m"rtIW . ,e,.. 0

,e.,. O,01T

a

0.11

b

'

0.13 0.15 ",

1 9 _ 0 k o O l ' " - - - ~ O h . 0 5 ' - - - - * ' J l 7 " ' - - - ~ Q ~ 0 9 ; ; - - - - t O . ~ " - - - - ' 0 ~ . 1 · l - - - - - ; O ~ . 1 5 ' - - - - - * 0 ~ " ; - -

Fig. 8. 3

rCm)

c

pino -tItdric: crystal me"'l.ftITIenlS. s.- 0.1\.0005r

Comparison between th e experiments and th e analysis:

a. variations of th e electron temperature T with th e radiuse

b. variations of th e electron density ne with th e radius r ,

c. variations of th etotal g"' pres sure Pg

with th e radiusr .

r ,



- 88 -

CHAPTER IT

Conclusions

Starting from Boltzmann's equation and Maxwell's equations, a method

has been developed of analysing the medium of Hall type MHD generators.

The relaxation processes as well as the behaviour of the two-temperature

plasma have been described. The Hall electric field appears to be

essentially influenced by the presence of space charge, for exact

electric neutrali ty yields overdetermination of the problem by

imposing n. to be equal to n •e

to n , i t has been possible toe

with respect to the calculated

Assuming n. to be approximately equal

find solutions that are consistent

variations in the elect r ic field.

In this way of tackling the problem, Poisson's equation of space

charge has only been used to verify whether variations in the Hall

field are in agreement with the assumption of n n . .e

The conditions for the set-in of ionisation instabi l i t ies have been

studied start ing

the relationship

from the conservation equations for the electrons,

7 +9.J = 0, and 9 x E = O. By comparing the ionisation

relaxation region of an MHD generator and the region where the two

temperature plasma has been fully developed, i t appears that the

ionisation instabi l i t ies are stimulated in the former region by two

effects:

The Hall parameter is larger there, as appears from the stationary

flow analysis.

The cr i t ica l Hall parameter is smaller there, as follows from thestabi l i ty calculations.

The electron temperature has been measured in a direct way, using two

independent diagnostics, viz. electrostatic probes and detection of

optical radiation. The values obtained by the various methods agree

with each other i f B = O. If B +0 the spectroscopic method yields

lower values, probably because the populations of the excited stes are

not entirely controlled by the electron temperature in that case. The



- M -

measurements show an electron temperature elevation which qualit ively

is in good agreement with the theory presented, demonstrating the

possibi l i ty of reaching ETE in an MHD generator.

The electron density has been determined by three diagnostic methods,

viz. the electrostat ic probe, optical radiation and microwaves. The

values measured with the probe are considerably lower than those

measured using the other diagnostics, possibly caused by the

circumstance that the electron mean free path is smaller than th e

probe dimensions. No evidence of non-equilibrium ionisation has been

found. The following two effects may be mainly responsible for the

absence of NEI:

In the experiment the flow cannot be considered stationary.

Ionisation ins tabi l i t ies wil l lengthen the ionisation relaxation

region.

The f i rs t effect i s related to the described experiment only, whereas

the second has to be taken into account in practical MHD generators.

In the geometry considered, the most important mechanisms which

reduce th e performance of the generator are the electr ical resistance

of the electrode boundary layers and the ionisation ins tabi l i t ies . A

considerable influence of th e ionisation instabil i t ies on the Hall

voltages has been found in agreement with other experiments and

existing theories. The results confirm the statement that repression

of the ionisation instabil i t ies wil l be necessary in order that

Hall type generators may be useful (ref. 9.1).

The theory presented, when adapted to the gas conditions of a

practical MHD generator, predicts sufficiently high electrical

conductivity in non-equilibrium conditions. After having achieved

in this experiment the experimental verif icat ion of ETE, the.

predictions about the behaviour of non-equilibrium media of MHDgenerators wil l be examined further in our laboratory using plasmas

consisting of seeded argon with pressures> 1 atm and gas temperatures

of about 2000 oK. The next phase of the experimental investigations



- 90 -

concerns the performance of a non-equilibrium generator and the

properties of i ts working medium in MHD devices mounted on a

shock tube.



- 91 -

APPENDIX

Tables at the calculation of critical values of the Hall parameter

in the case of no Saha equilibrium



- 92 -

Table A.I Values of th e e lec t ron temperature r(O), th e e lec t ron enerRv loss due to e la s t ic col l i s ions A(O).e .

T!O) (OK)

6000

6200

6400

6600

6800

7000

7200

1400

, 1600

1800

8000

8200

8400

8600

8800

9000

9200

9400

9600

6000

6200

6400

6600

• " ..ol0

1000

7200

74"0

7600

7800

6000

6200

6400

6600

6800

1000

7200

7400

7600

7800

8(1008200

8400

8600

: 8800

B 9000

9200

9400

9600

9BOO

10000

10200

10400

10600

10800

11000

1120011400

th e energy los t or gained by th e electrons due to ionisat ions and recombinations E ~ ~ ) ,th e Joule heating j(O)2/a (O). th e plasma velocity u, the electron density gradient in the

direction of th e plasma velocity ~ . ~ n ( O ) / u . th e number of ionisation minus three times th ee

number of recombinations ( I - 3R){O), th e "ionisation-recambinar.ion parameter" .<. an d th e

(0 )c r i t i ca l Hall parameter W1cr • re l a t ing to the curves la , lh an d Ie of Fig . 5 .2 (see also Table 5.1) .

A(O)(Jm- 3see- 1) E ~ ~ ) (Jm-3 .ec - I ). (0)2 -3 - I -,

7n!O) lu(m-4 ) (1-3R) (0 ) (m -3 sec 1)

(0 );roT (Jm u c ) u{msee ) " .' <

1. S4 10+5 - 3.06 10+ 3 1.5 I I J+5 2) 6 0 - 3.43 10+21 - 0.31 3. )0

1.80 10+5 - 2.18 10+ 3 1.18 lJ+5 295 0 - 2. 5 ] 10+ 21 - 0.28 2.90

2.05 10+5 - 1.46 10+3 2.03 lJ+5 370 0 - 1.83 10+ 21 - 0.23 2.50

2.29 10+ 5 - 1.56 10+2 2.28 1 ~ + 5 467 0 - 1.28 10 .21 - 0.14 2.13

2.52 ]0+ 5 6.83 10"1 2.52 1l"5 573 0 - 1.57 10 .20 - 0.07 I.B4

2.74 10+5 1.19 10"3 2.75 u" 5 66' 0 - I.B8 \0 ..20 - 0.01 1.6i

2.96 \0+ 5 2.85 10"3 2.99 10"5 746 0 5.34 10"20 0.02 I. 5 7

3.11 10+ 5 5.39 10"3 3.22 \0 ..5 ." 0 1.54 10 .21 O . O ~ 1.52

3.31 10+5 9.28 10"3 3.41 \0 .5 873 0 3.01 10 .21 0.'05 1.47

3.51 10+ 5 1.52 10 .4 3.72 10 ..5 93 ' 0 5.17 10"21 0.05 1 . ~ 3 3.71 10+5 2 . ~ 0 10"4 4.01 \0"5 1011 0 B.34 10 .. 21 0.06 I. 3,.

3.96 10+5 3.68 10"4 4.33 10 ..5 1099 0 1.29 10 ..22 0.06 1. 34

4.15 10+5 5.53 10"4 4.70 10 ..5 1110 0 1.95 10 ..22 0.06 1.28

4.34 10+5 8.16 10 ..4 5.15 10+ 5 1354 0 2.88 \ 0 .22 0.06 I . 2 ~ 4.52 10 ..5 1. 18 10 .5 5.10 10"5 1546 0 4.17 10"22 0.06 1. I]

4.70 10 ..5 1.68 10"5 6.38 10"5 '807 0 5.93 10"22 0.06 1.01

4.88 10"5 2.36 10"5 7.24 10"5 2171 0 8.30 10 .22 0.07 G.90

5.06 10 .. 5 3.26 10+5 8.32 10"5 2693 0 1. 15 10 ..23 0.07 0.79

5.24 10+5 4.45 10 .5 9.69 10"5 3410 0 1.56 10"23 0.01 0.66

5.41 10"3 4.51 10+0 5.42 10"3 744 0 - 6.21 10"]7 - 0.01 1.06

6.51 10+3 1. 38 10 .. 1 6.52 10"3 1069 0 3.33 10"]8 1.02 0.C1

7.62 10"3 3.05 10+ 1 1.65 10"3 1321 0 9.83 10 . 18 J.04 0.11

8.13 10"3 6.06 10"1 8.19 10"3 IS 16 0 2. II 10"]9 0.04 C.66

9.86 10+3 1.13 10 ..2 9.97 \0"3 1693 0 4.04 10"19 O.CS e.63

1.10 10 .4 2.04 10+2 1. 12 10 ..4 1882 0 7.31 10 . 19 o P5 ( ,60

1.11 10+4 3.53 10"2 l. 25 10+4 2112 0 1.21 10+20 0.05 C 56

]. 33 10+4 5.95 10 .2 1.39 10+ 4 2 4 ~ 1 0 7.IA 10+10 0.05 r. 51

1. 45 10 .4 9.14 10+2 1.55 \0 .4 2894 0 3.50 10"20 1).06 C.45

I. 57 10 .4 1.55 10"3 1.72 10 .4 3714 0 5.57 10"20 0.06 0.37

9.54 10+6 - 3.14 10"6 6.40 10+6 >2, 0 - 3.46 10+ 24 - 0.33 5.13

1.10 10 . 7 - 2.34 10"6 8.69 10 ..6 '" 0 - 2.57 10 ..14 - 0.34 4.06

I. 24 10 . 1 - I. 78 10 ..6 1.06 10+ 7 245 , - 1.95 10+24 - 0.35 3.52

1. 37 10'" - I . 31 10 .6 1.24 10+ 7 296 0 - 1.50 10+24 - 0.35 3.19

I. 49 10 . 7 - 1.07 10 .6 1.39 10+7 14J , - 1.11 10+24 - 0.36 2.96

1.61 10"7 - 8.35 10"5 1.52 10'" 389 0 - 9.12 10 .23 - 0.35 2. i8

1.71 10+7 - 6 . .52 10+5 1.65 10+7

'"0 - 7.32 10 ..2) - ').34 J .6 3

1.8\ 10"1 - 4.99 10"5 I. 76 10'" 485 0 - 5.83 10 ..23 - }. l0 2.48

1.91 10"1 - 3.62 10+5 1.87 10+1 539 0 - 4.60 10+23 - 0.25 2.33

2.00 10+1 - 2.25 10 . 5 1.97 10"1 597 0 - 3.54 10+23 - 0.18 2.19

2.08 10+1 - 7.58 10 .4 2.01 10"7 65' 0 - 2.55 10 .23 -O. II 2.07

2.16 10"1 1.02 10 .5 2.17 10"1 '"0 - 1.55 10+23 - 0.05 1.98

2.23 10+7 3.21 10"5 2.27 10+7 765 0 - 4.63 10 .22 - 0.01 1.9]

2.10 10+1 6.22 10"5 2.37 10+7

'"0 8.17 10+21 0.02 1.86

2.31 10"7 1. 0 I 10+6 2.47 10+7 865 0 2.39 10 .23 0.03 1.81

2.44 10"7 I. 54 10 ..6 2.59 10+7 9>9 0 4.39 10 .23 0.05 I. 76

2.50 10+1 2.23 10+6 2.12 10"1 98> 0 6.91 10+23 0.05 I. 72

2.56 10+7 3.15 10 ..6 2.81 10 ..7 1054 0 1.03 10 .24 0.06 1.66

2.61 10 ..7 4.36 10"6 3.05 10"7 1143 0 \ .46 10"14 0.06 1.59

2.67 10+7 5.<12 10+6 3.26 10 ..7 1254 0 2.0\ 10 ..24 0.07 1.51

2.72 10+1 7.93 10+6 3.51 10+ 1 1392 0 2.71 10+ 24 0.07 1.44

2.77 10 ..7 1.05 10+1 3.82 10+7 1568 0 3.60 10+ 24 0.07 1. 34

2.82 10+7 1.37 10'" 4.19 10+ 7 1792 0 4.72 10+24 0".07 1.24

2.86 10+7 1.77 10 ..1 4.64 10'" 2019 0 6.11 10 .24 0.01 1.14

2.91 10+7 2.21 10+1 5.18 10'" 2446 0 7.82 10 ..24 0.08 1.03

2.95 10 . 7 2.89 10"7 5.84 10+ 7 2917 0 9.93 10 .24 0.08 0.93

2.99 10 .. 7 3.64 10+7 6.63 10+ 7 3521 0 1.25 l o " 2 ~ 0.(1-0

.813.03 10+7 4.55 10+1 1.58 10+7 4291 0 1.56 10+25 0.08 0.13



- 93 -

Table A.2 As Table A.I. relating to the curves 2a and 2c of Fig. 5.2 •

<0 ) (<)K) A 0) (Jm-l.ec \) E ~ ~ ) (Jm- 3 sec:-I

). (0)2 -3 _\ -,

vn!O) lu(m-4) (I -JR) (0 ) (m -3 se c:- I ) wT (0 )~ ( J m Be" ) u(mBec: ) 0, "

6000 1.17 10"S - 3.13 10 . 3 1.14 10 .5 22, 0 - 3.45 10+ 21 - n,33 J. 52

6200 1.35 10 ..5 - 2.33 10 . 3 \ , J 3 \0+5 265 0 - 2.57 10 +21 - 0.33 3.22

6400 1,52 10 +5 - l. 75 10+ 3 1.50 10+5 J09 0 - 1.94 10 + 21 - 0, )3 3.00

6600 1. 67 10+5 - 1.31 10+3 1.66 10+ 5 J54 0 -.1.48 10+21 - 0.32 2.80

6800 1.82 10+ 5 - 9.63 10+2 1.81 10+5 405 0 - 1.13 10+ 21 - 0.28 2.60

7000 1.95 10+ 5 - 6,51 10 + 2 1.94 10+5 464 0 - 8.55 10+20 - 0,22 2.39

7200 2.07 10+ 5 3. J3 10+1 2.07 lJ+5 528 0 - 6.17 10+20 - 0.14 2.21

7400 2.19 10+ 5 3.64 10+ 1 2.19 10+5 59' 0 - 3.90 10 . 20 - 0.07 2,06

7600 2.30 10+5 5.15 10+ 2 2.31 10+ 5 644 0 - 1.45 10+20 - 0.02 1.97

7800 2.40 10+ 5 1.17 10+ 3 2.42 10+5 6" 0 I. 48 10 ..20 0.01 1.92

8000 2.50 10"5 2.11 10"3 2.52 10"5 "6 0 5.26 10"20 0.03 1".89

8200 2.59 10"5 3.44 10"3 2.63 10+5 760 0 1.04 10+21 0.05 1.86

84.0 1.68 10+5 5.:n 10") 2.73 10+ 5

'"0 1.13 10 .21 0.05 1.84

8600 2.76 10"5 7.98 10"3 2.84 10"5 829 0 2.69 10 ..21 0.06 1.83

8800 2.84 10+5 1.17 10+4 2.95 10+5 868 0 4.01 10+ 21 0.06 1.80

: 9000 2.91 10+5 1.67 10+4 3.08 10+5 912 0 5.79 10 ..21 0.06 1.77

B nco 2.98 10+5 2.35 10+4 3.t1 10"5 964 0 8.18 10+21 0.06 1.74

9400 1.04 10"5 3.25 10 . 4 3.37 10"5 1026 0 1.14 10 ..22 0.07 1.69

9600 3.11 10"5 4.44 10"4 3.55 10"5 1101 0 1.55 10"22 " . I l l 1.64

9800 3.17 10+5 6.00 10+ 4 3.77 10+5 1195 0 2.09 10+22 0.07 1.57

10000 3.22 10"5 8.00 lu+ 4 4.02 10 ..5 1312 0 2.79 10+ 22 0.07 1.50

10200 3.28 10"5 1.05 10"5 4. J3 10"5 1459 0 3.67 10 . 22 0.07 1.41

10400 3.3J 10"5 1. 38 10+5 4.71 10+5 1646 0 4.78 10+ 22 0.07 1.32

10600 3.38 10+ 5 l. 78 10 .5 5.16 10"5 1883 0 6.16 10+ 22 0.06 1.22

10800 3.43 \0"5 2.28 10"5 5.71 10"5 2185 0 7.87 10"22 0.06 1.12

11000 3',47 10"5 2.89 10"5 6.37 10+5 2571 0 9.97 10 . 22 0.08 1.01

11200 3.52 10+5 3.64 10+5 7.16 10+5 "64 0 1.25 10+23 o.oa 0.91

\\1<00 " . <;6 10+5 1<.55 10"5 11.11 10"5 3693 0 1.56 1 0 + ~ 3 0.08 0.81

11600 3.60 10 . 5 5.64 10"5 9.2'> 10"5 44':''' 0 I. 9 3 10"23 0.08 0.72

6000 5.20 10+5 - 2.37 10+3 5.17 10"5 J5I 0 - 3.17 10+21 - 0.1" 2.23

6200 6.2610+

5 -7.31 10"2 6.26 10"5 565 0 - 1.98 10+21 - 0.10 1.52

6400 7.34 10+ 5 1.45 10+ 3 7.36 10+5 894 0 - 7.72 10+ 20 - 0.03 1.04

6600 8.43 10"5 4.82 10+3 8.48 10+5 1290 0 7. 54 10+ 20 0.01 0.77

6800 9.53 10+5 1.04 10+4 9.64 10+5 1679 0 2.99 10+21 0.03 0.63

7000 1.07 10+6 1.96 10 ..4 1.08 10 ..6 2045 0 6.48 10+21 0.04 0.55

7200 1. 18 10 .6 3.47 10"4 1.21 10"6 2432 0 1.20 10+ 22 0.05 0.49

7400 1.29 10 ..6 5.90 10+4 1.35 10"6

i2938 0 2.08 10+ 22 0.05 0.42

7600 1.41 10+6 9.69 10+4 l.51 10+6 3800 0 3.105" 10 ..22 0.05 0.35



- 94 -

Table A.3 As Table A.I, relat ing to the curves 3a and 3c of Fig. 5.2 .

f!Ol,OK) A(O) (Jm-\ec- 1) . : ~ ~ b r n - 3 s e c - l ) . (0)2 _) _\ -,;:;"/n!O) lu(m-

4) (I-3RJ (0) (m-3 se c-\ ) ",T ; ~ ) -ror-(Jrn sec ) u(msec ) ,,

5000 1.79 l G ~ 5 - 1.90 J n+4 1.60 10+5 '3 4 a - 2.11 10 + 22 - ,'"

3. IB

5200 2.07 10+ 5 - 1.26 10+4 l. 9 5 10+ 5 m a - 1.40 10 +22_ 0 _0 ? .85

5400 2.34 10+5 - B.55 HI+ 3 2.26 \0 + 5 344 a - 9.48 10 +21 - o. W 2.65

5600 2.60 10+5 - 5.97 10+ 3 2.54 10+5

'"a - 6.62 10+ 21 - 0.31 1.52

5800 2.84 10+ 5 - 4.26 10+) 2.60 10+5 434 0 - 4.72 \0 + 21 - 0.31 2 . ~ I6000 ).07 10+ 5 - 3.06 \0+ 3 3.04 10+ 5 m a - 3.43 10 +21 - ' l .31 2. 1

6200 ) . )0 10+5 - 2.18 10+ 3 3.27 10+5 '" a - 2.51 10+ 21 - 1,28 2.20

6400 3.51 10+ 5 - 1.46 10+ 3 3.49 10+ 5 '" a - 1.83 10+ 2 I - 1.23 2.r6

6600 3.72 10+ 5 - 7.56 10+ 2 ) ,71 10+5 670 a - l. 28 10+ 21 - :1.14 1.f9

6800 3.92 10+ 5 6.83 10+ 1 3.92 10+ 5 IS' 0 - 7.57 10 .20 - 0.07 l. 7S

7000 4. II 10+ 5 1.19 10+) 4.12 \0+ 5

""a - l. 88 10+20 - 0.01 J .6 6

7200 4.30 10+ 3 2.85 10+ 3 4.33 10+ 5 883 a 5.34 10 ..20 0.02 1.60

7400 4.49 10"'5 3.39 10+ 3 4.34 10"'5 937 a l. 54 10"'21 n .n b 1.'' ';

7600 4.67 10+ 3 9.28 10 . 3 4.76 10+ 5 990 a 3.01 10+ 21 0.05 l. 52

7800 4.85 10+ 5 1.52 10+ 4 5.00 10+ 5 1047 0 5.17 10 ..21 0.05 1.49

WOO 5.03 10+ 5 2.40 10+ 4 5.26 10"'5 1112 a 8.34 10 ..21 0.06 ].45

8200 5.20 10"'5 3.68 10 ..4 5.57 10"'5 1190 a 1.29 10+ 22 0.06 1.40

8400 5.37 10+ 5 5.53 10+ 4 5.92 10+ 5 1288 a .. 10+22 0.06 l. 35

8600 5.54 \0+ 5 8.16 10+ 4 6.36 10 ..5 1415 0 2.88 10 . 22 0.06 1.28

8800 5.71 \0+ 5 l. 18 \ 0+ 5 6.89 10+ 5 1583 a 4.17 10 ..22 0.06 1.20

~ O O 5.88 10+ 5 1.68 10+5 7.56 10+ 5 1809 a 5.93 \ 0+ 22 <1.06 1. 10

9200 6.04 10+ 5 2.36 10+ 5 8.40 10 . 5 2120 a 8.30 10+22 0.07 ).00

9400 6.21 10+ 5 3.26 10+ 5 9.47 \0+ 5 2539 a 1. IS 10+ 23 0.07 fl.88

9600 6.37 )0+ 5 4.45 10+5 1.08 10+ 5 3193 a 1.56 10+ 23 0.07 0.76

9800 6.54 10+ 5 6.01 10+ 5 1.25 10+ 5 4152 a 2.10 10+23 C.07 0.63

/uuu 1.37 10 . 5 1.19 10+ 3 1.3tl 10+:> 405 0 - 1.18 10+2v - oJ.Ol 1.70

7200 1.61 10+5 2.85 10+ 3 1.64 10+5 57B a 5.34 10+ 20 0.02 1.50

7400 1.85 10+5 5.39 10+) 1.90 10+5 667 a 1.54 10+21 :1.04 ].42

7600 2.08 10+5 9.28 10+ 3 2.17 10+5 145 a 3.01 10+21 1.05 1.36

7800 2. )0 10+5 1.52 10 ..4 2.45 10+5 814 a 3.17 10+ 21 ).05 J. 32

8000 2.51 10+5 2.40 10+4 2.75 10+5

'"a 8.34 10+ 21 1.06 1.28

8200 2.72 \0+5 3.68 10+10 3.09 10+5 10\6 a ].29 10+ 22 ).06 1.22

l, 8400 2.93 10+5 5.53 10 .. 4 3.48 10+3 1148 a 1.95 10 +22 0.06 l. 16c

B600 3. !J 10+5 B.16 10 .. 4 3.95 10+ 5 1321 a 2.B8 10+ 22 0.06 1.08

8eoo 3.33 10+3 1.18 10+5 4.5\ 10+ 5 1552 a 4.17 10+ 22 0.06 0.99

9000 3.53 \0+ 5 1.68 10+ 5 S.21 10+ 5 1 8 7 ~ a 5.93 10+22 0.06 0.88

9200 3.72 10+ 5 2.36 10+ 5 6.08 10+5 2338 a 8.30 10+22 0.07 0.77

91000 3.91 10+5 3.26 10+5 7.17 10+5 3002 a 1. 15 10+23 0.07 0.65

9600 4.10 10+5 10.45 10+5 8.55

"10 ..5 10066 a 1.56 10+23 0.07 0.53



- 95 -

Table A.4 As Table A.I, relating to the curves 4a and 4b of Fig. 5.2 .

T ~ O ) (oK)A(O) (Jm- 3aec:- 1) E ~ ~ ) (Jm-] .. :-1 )

. (0)2 -3 -1 -,;;.t'n!Ol/um-4) (1-3R) (0 ) (m ]8ec -1 ) w, (0 )

7of-(Jm sec: ) u( ...ec: ) 0

'"6000 1.54 10 + 5 - 3.06 10+ 3 1.26 10 + 5 ", 1,70. 10 + 21 - 3.43 10+ 21 - 0.31 3.96

6200 1.80 10 +5 - 2.18 10+ 3 1.48 10 + 5 223 1 . ~ 5 10+ 21 - 2.51 10 + 21 - 0.28 3.50

6400 2.05 10"5 - 1.46 10+ 3 1.69 10+ 5 'SO I. 37 10+21 - 1.83 10 + 21 - 0.23 ' .02

6600 2.29 10+ 5 - 7.56 10+ 2 1.90 10 + 5 m I. 18 10+21 - 1.28 10+ 21 - 0.14 2.57

.800 2.52 10+5 6.83 10+ 1 2.10 10 + 5

'"1,03 10 + 21 - 7.57 10+ 20 - 0.07 2.22

7000 2.74 10+5 1.19 10+ 3 2.29 10+5 505 9,40 10+ 20 - 1.88 10+ 20 - 0. 0 I 2.01

7200 2.96 10+ 5 2.85 10+ 3 2.49 10 + 5 5 " 8.88 10+20 5.34 10+ 20 0.02 1.90

7400 3.17 10+ 5 5.39 10+ 3 2.68 10 + 5

'"8.58 10+20 1.54 10+ 21 0,04 1.83

7600 3.31 10+ 5 9.28 10+ 3 2.89 10+5 ." 8.35 10+20

I3.01 10+ 21 0.05 1.78. 7800 3.57 10+5 1.52 10+4 3.10 10+5 708 8.15 10+20 5.17 10+2 ] 0.05 1.73.

8000 3.77 2.40 10+4 3.34 76210+5 10+5 7.94 10+20 8.34 10+ 21 0.06 1.68

8200 3.96 10+5 3.68 10+4 3.61 10+5

'"7.70 ]0+ 20 \.29 10+ 22 0.06 1.62

8400 4.15 10+5 5.53 10+4 3.92 10+5 ,,, 7.43 10+ 20 1.95 ]0+ 22 0.06 1.55

8600 4.34 10+5 8.16 10+4 4.29 10+5 1016 7. ]2 ]0+ 20 2.88 10+ 22 0.06 1.46

8800 4.52 10+5 1. 18 10+5 4.75 10+5 1157 6.76 10+20 4.17 10 . 22 0.06 1. 36

9000 4.70 10+5 1.68 10+5 5.32 10+5 1346 6.36 10+20 5.93 10+ 22 0.06 l. 24

,"0 4.88 10+5 2.36 10+5 6.03 10+5 1606 5.92 10 .20 8.30 10+22 0.07 1. 12

9400 5.06 10+5 3.26 10+5 6.94 10+5 ]97] 5.42 10+20 1. ]5 10+23 ').07 0 . ~ 8 96 " 5.24 10+5 4.45 10+5 8.08 10+5 2496 4.81\ 10+20 1.56 10+23 0.07 0.84

9800 5.41 10+5 6.01 10+5 9.51 10+5 3280 4.29 10+20 2.\0 10+23 0.01 0.70

.000 1.54 10+ 5 - 3.06 10 .3 1.08 10+ 5 '" 3.67 ]0 .21 - 3.43 ] 0+ 22 - 0.3\ 4.65

6200 1.80 10+5 - 2.18 10 . 3 l. 21 10+ 5'" 3.35 10+21 - 2.5] 10+ 22 - 0.28 4.09

6400 2.05 10+5 - 1.46 10+3 1.45 10+ 5

'"2.97 10+ 21 - 1.83 10. 22 - 0.2) 3.54

6600 2.29 10 .5 - 7 .56 10+ 3 1.63 10+5 '" 2.56 10+ 21 - 1. 28 10 . 22 - 0.14 3.01

6800 2.52 10+5 6.83 \0+ 1 ' .80 10+5

'"2.24 10+ 21 - 7.57 10+21 - 0.07 2.61

7000 2.74 10+5 1.19 10+3 1.97 10+5 '99 2. 04 10+ 21 - 1.88 10+21 - 0.01 2.36

7200 2.96 10+5 2.85 10+3 2. \3 10 .5 446 1.93 10+ 21 5.34 10 .21 0.02 2.22

7400 J.17 lotS 5.39 10+3 2.30 10+5,.,

1.86 10+ 21 1.54 10+22 0.01 2.14

7600 3.37 10+ 5 9.28 10+3 2.48 10+5

'"

1.81 10+21 3.01 10+22 0.05 2.09

7800 3.57 10+5 I . 52 10+4 2.66 10+5 559 1.17 10+ 21 5.17 10+22 0.05 1.03

! 8000 3.71 10+5 2.40 10+4 2.86 10+5 ." 1.72 10+21 8.34 10+22 0.06 1.97,8200 3.96 10+5 3.68 10+4 '.09 10+5 653 1.67 10+21 1.29 10+23 0.06 1.90

8400 4.15 10+5 5.53 10 .4 3.36 10 .5 '"1.62 10 .21 1.95 10+2) 0.06 1.82

8600 4.)4 10+5 8.16 10+4 3.68 10 .5 800 1.55 10+21 2.88 10+23 0.06 1.72

8800 4.52 10+5 I. 18 10+5 4.07 10+5 ", 1.48 10+2 \ 4.17 10+23 0.06 1.60

9000 4.70 10+5 1.68 10+.5 4.56 10+5 1055 1.39 10+21 5.93 10+2) 0.06 1.41

9200 4.88 10+5 , .36 10+5 5.17 10+5 1254 1. 30 10+2 \ 8.10 10+2) 0.0; 1.32

9400 5.06 10+5 3.26 10+5 5.9/0 10+5 1531 1.20 10+21 I. 15 10 .2/0 0.07 1.17

9600 5.24 10+5 4.45 10+5 6.92 10+5 1921 , .M 10+ 21 1.56 10+ 24 n."7 1.01

9800 5.41 10+5 6.01 10+5 8.15 10+5 2486 9.10 10+ 20 2.10 10+24 0.07 0.1'15

10000 5.59 10+5 8.00 10+5 9.71 10+5 3339 11./03 10+ 20 2.79 10+24 0.o , 0 . ~ 9



- 96 -

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- 97 -

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- 98 -

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A. Veefkind- Non-Equilibrium Phenomena in a Disc-Shaped Magnetohydrodynamic Generator

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Transcript of A. Veefkind- Non-Equilibrium Phenomena in a Disc-Shaped Magnetohydrodynamic Generator