a. veefkind- non-equilibrium phenomena in a disc-shaped magnetohydrodynamic generator
TRANSCRIPT
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NON-EQUILIBRIUM PHENOMENA IN A DISC-SHAPED
MAGNETOHYDRODYNAMIC GENERATOR
by
A. Veefk ind
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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
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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.
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- 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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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- 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,
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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,".?
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- 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
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- 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.
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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) .
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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
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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
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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)
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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).
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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)
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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
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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 .
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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.
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.. --
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
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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:
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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
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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....,
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-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
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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.
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- 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
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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)
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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.
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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
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- 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).
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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
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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)
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- 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
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- 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
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- 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:
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- 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
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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 ,
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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
+
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- 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)
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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
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-
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
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- 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
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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.
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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).
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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
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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.
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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
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wi thin I % by:
B can be varied between 0 and O. I T.o
(6. I)
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(
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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.
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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.
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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.
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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.
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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
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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)
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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
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"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
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-
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 .
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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.
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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.
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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
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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):
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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.
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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.
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.., 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.
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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 .
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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.
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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:
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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.
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' 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 .
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- 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 - -
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- 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)
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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
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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 .
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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.
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- 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.
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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.
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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
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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
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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.
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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
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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
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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.
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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 ,
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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
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- 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
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concerns the performance of a non-equilibrium generator and the
properties of i ts working medium in MHD devices mounted on a
shock tube.
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APPENDIX
Tables at the calculation of critical values of the Hall parameter
in the case of no Saha equilibrium
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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
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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
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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
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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
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8/3/2019 A. Veefkind- Non-Equilibrium Phenomena in a Disc-Shaped Magnetohydrodynamic Generator
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