Transport coefficients of a chemically reacting plasma
M. L. Mittal and G. Paran Gowda
Citation: J. Appl. Phys. 59, 1042 (1986); doi: 10.1063/1.336539
View online: http://dx.doi.org/10.1063/1.336539
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Published by the American Institute of Physics.
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Transport coefficients of a chemicaUy reacting
p~asma
M. L. Mittal and G. Paran Gowda
Department of Mathematics, Indian Institute of Technology, Powai, Bombay 400 076, India
(Received 16 July 1984; accepted for publication 31 July 1985)
The analysis for the calculation of the transport coefficients of a chemically reacting partially
ionized gas in the presence of a magnetic field is presented here. Taking the phenomenological
kinetic equations and considering the ionization and recombination processes only due to electron
impact parameter, the Chapman-Enskog method is used to obtain these transport coefficients.
Sample numerical calculations are made for the transport coefficients of a potassium-seeded
argon gas.
I. INTRODUCTION
Increasing applications of magnetohydrodynarnics
(MHD) in many engineering devices have made it imperative
to study the transport phenomena of a partially ionized gas
under different nonequilibrium conditions. As it is difficult
to make experiments to measure the transport coefficients
like electrical conductivity, viscosity, and thermal conductivity, with any desired accuracy at higher temperatures,
mainly theoretical work 1-5 has been done to study and calculate these coefficients in the presence of cross electric and
magnetic fields. But, in most of these papers, the effect of
ionizational nonequilibrium is neglected, invoking the condition that the average electron-electron collision frequency
is much higher than the inelastic electron collision frequency
and that the magnitude of the inelastic gain term is less than
the loss term. But, in a partially ionized gas, inelastic collisions are dominant and it is necessary to take account of the
effects of ionizational nonequilibrium on the transport phenomena.
During the ionizational nonequilibrium process, the inelastic collisions between the neutral particles of the gas and
the charged particles play an important role. This involves
energy transfer process between the gas species. The changes
in the number density and the temperature of the neutral and
the charged particJ.es are determined by conservation laws of
charge, energy, etc. The chemical reactions giving rise to
ionizational nonequilibrium process are assumed to be related to the modified form of the Saha equation as given by
Baum and Fang. 6
In the present analysis, making use ofBaum and Fang's6
model of kinetic equations, the electron number densities are
evaluated as a solution of continuity and rate equations during ionizational nonequilibrium processes, instead of the
Saba equation. Making proper modifications in the energy
conservation equation to take account of finite rates of ion ization, it is found that the electrons get redistributed and
thus as a result of chemical reactions (of ionization-recombination type), the gas will have more electrons at a reduced
energy leveL
This analysis is then used to obtain the transport coefficients for such a chemically reacting gas assuming that the
electron impacts are the only ionization-recombination processes, without distinguishing between the level of states of
atoms or ions. The electron number d.ensity and electron
1042
J. AppL Phys, 59 (4),15 February 1986
temperature immediately after the chemical reaction are
used in the continuity and energy equations.
The modified mathematical expressions of the transport
coefficients due to the effects of chemical reactions are given
in Sec. n. These have been derived following the analysis of
Mitchner and Kruger! and Shkarofsky et al. 7 In Sec. III, the
chemical equilibrium parameters and the method for the
evaluation of ne and Te from continuity and energy equations are discussed. Section IV gives the numerical results
and discussion of the behavior of transport coefficients in a
chemically reacting plasma.
II. ANALYSiS
The general form of the kinetic equation as given by
Baum and Fang6 is
(~+
at V .~+~.~)(nJa)
ax rna av
=
"j)a·Kaa,(narPaa· - nJa) + aa(n~rP~ - nJa),
,,'
(1)
where rPaa' denotes the equilibrium distribution function of
the species a anda'(a,a' = e, i, a) in the absence of chemical
reactions and rP:, denotes the distribution function if all the
particles were immediately thermalized in each inelastic collision. Kaa' is the scattering function and aa denotes inelastic
collision frequency. The quantities rPaa' and K aa , are determined using the conservation and irreversibility conditions.
Using the above equation, the effects of chemical reactions
on transport coefficients are studied.
Following the anal.ysis of Mitchner and Kruger, 1 peculiar particle velocity C and mean mass velocity U are introduced. They satisfy the relation
V=C+u.
Equation (1) can then be written as
§
§t
(nJal
= R:
+ R~,
(2)
and
0021·8979/86/041042·06$02.40
@) 1986 American Institute of Physics
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1042
For a small departure from the equilibrium, la (C) is
written as a cartesian tensor expansion in terms of a peculiar
particle velocity
Here, Vaa is the collision frequency between particles of the
same species and the subscript h denotes heavy particles.
The transport coefficients are calculated using the basic
definitions for current density ja, heat flux vector CIa, and
viscous stress tensor r a as
C
CC
la(C) =/~)(C) + 1~1(e) -.!!- + l~hr(C)
+... .
ja
R ~,R ~ = Ina' naKaa' (tPaa' - la)'
aa(n~ tP~ - naJa)'
a'
-LfC
e
(3)
Using Eq. (3) in Eq. (2) and integrating over the differential
solid angle dOc in velocity space after mUltiplying by ele
and eel C 2, the equations in terms of cartesian tensor expansion are obtained. Shkarofsky et aU have given the detailed
derivation of these equations but in the absence of inelastic
collisions. The equations in terms of the cartesian tensor expansion are written as
e (v.tll _ v,u a/(oI) _ ~ . ~ (C2t:11)
3
a
ac
3ma a ae
a
pOle
a
(4)
[(2.2 _Z2)V In T
= (R
w Xf<21 _
a
a
~r
a
-
~
] + WXf(11
kTa a
a
+ R ~C),
d"tO)
a 2 dC
f 2)xw _ S e _'.I_a_ = (R ur + Rue)
a
a
a
a ,
B
+ tP3..!!. X VT,
B
CIa
=
~ k~aja -A ;V)Ta -A ~V2Ta -A; ~ XVTa
- Ta tPi d 1a
(5)
uj =
(6)
tPj
2
S=VU + (VU) T
-
¥2(V,U),
Since the driving terms V In Ta , da , and Sa for translational nonequilibrium appear linearly and are mutually independent, r;I(C), f~I(C) can also be written as
f<11(C)=CI'(OI[(AaV
+AaV
+A a3 '!!'XV)lnT
a
'.fa
1 I
2 2
B
a
~(DfdJa
+ D~d2a + D~!.Xda)],
kTa
B
(7)
e 2fIa
O)(D
)
l ,as
i a + D 2,as2a + D ,a!.XS
B
a'
(8)
3
where the subscripts l, 2, and 3 denote the parallel, perpendicular, and HaH components.
I~I as influenced by chemical reactions is obtained from
Eg. (4) following the analysis of Shaw et al. 5 Thus,
IIOJ(~)3/2 (e-Z~+He-EZ~)
a
21TkTa
'
-~)A jdZa ,
(12)
~~k2T2loo
Z6e-Z~D"adZ
(13)
7J J. = - "
L.J
c
a
J
a'
a 15,,1T ma
° a
wherej = 1,2, 3;A j, D j, and D j" are unknown scalar functions and
8 n"
4 {
F(Za)=--Za
e _Z2a+Hexp[ -(eZ a2
3j; ma
)]
}
.
The unknown scalar functions A j, D'j, and D ja are
expanded in terms of Sonine polynomials to evaluate integrals in Eqs. (lOH13). The resulting expressions for transport coefficients are obtained as
and
C ~ = - 4!!.::-. k
ma
where
e ~, ... =
2
n'aln a (Ta IT'a )31
H- (m a lkT:z)2 - malkT~
Ta
(1 + 4F)'/2
= 1+-
T~
+ F'
+ -'-------'2
aa
F= ------=-----3v"" + (maC2/2k JIvahlTh
a'
1043
Jo
zT~/3 ;1'1,
(14)
L =H 1~/2,
where
E
(9)
(11)
Jo
a
a
(10)
= - Ieak rooF(Za)A jdZa ,
da =E',
and
X da ,
Jo
A j = - ITak2 (00 F(Za)(Z~
2
m"C
z=-a
2kT'
!
Ie~ roo F(Za)DjdZa •
a
where
ti21(C)
=
a
- Ta tP;
= 7J 1Sla + 7J2Sza + 1h-XSa ,
a
-
TatP~ d2a
-
B
B
Hence the transport coefficients electrical conductivity u,
diffusion coefficient tP, thermal conductivity A " and viscosity 7J are obtained as
l'
=R~lr +R~IC.
= u1d 1a + u Zd2a + u 3.!!.Xda + tPlVITa + tP2V2 T a
J. Appl. Phys., Vol. 59, No.4, 15 February 1986
d;1 + id;I ....
The expansion coefficients /3 ~I, /3 ;1'\
using f ~l, f;l as given in Eqs. (7) and (8):
i
I
r: are obtained by
1
r:la~, = ~(1
+ L )b~,
(15)
/3~la~ = ~(l
+ L )b:,
(16)
r=1
i
I
1'=1
M. L. Mittal and G. P. Gowda
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1043
L f3 ;I"H':, =
i
r~
1
m
- __
a_D~.
(17)
2kTa
(22)
Here
..a
Urt
H
=
( 0a -
a = (_ .
rt
2(r + !)! ) U~Ir
..[ii(r - I)!
.
tWa -"''--.....,..-
lWa
2(r + ~)! ) ~I
U r
..[ii(r - I)!
+ .J:.,ort ,
'"
aa'
(18)
a'
"'Haa'
+ 4.
rt'
(19)
where
F = PKlpA (ratio of the partial pressure of seed and parent
gas),
a'
a Dr) = 4~2kTa )3/2 iCO/IOIZ4
slrIISI'II
(oaa.·
rt , a I
a
a
3/2
3/2
ma
0
Pg = PK
+ PA
K(Te) = (
2m kT )3/2
~2 e
exp( - IlkTe)'
= gas pressure,
and
(20)
(21)
The integrals in Eqs. (20) and (21) are evaluated using/~)
from Eq. (9). The values of the coefficients 0':,"", aaD;, H':,""
are given in the Appendix. Knowing these, the expansion
coefficients fJ~\ f3 ;t'l, and
are obtained from Eqs. (15)(17). The transport coefficients can be calculated using these
values in Eq. (14) to any desired order of approximation in
the presence of chemical reactions.
With these values of the electron and neutral particle
number densities and temperatures, the particle number
densities n;, n~, and the gas temperature immediately after
the chemical reactions are calculated using Eqs. (23H27) of
Baum and Fang. 6 These equations can be simplified as
(23)
t:'
(24)
(25)
m. CHEMICAL EQUllIBRJUM PARAMETERS
The electron and neutral particle number densities of
the seeded partially ionized gas are calculated using the twotemperature Saha's equation as given by Voshall et al.,8
T' = ne [Te
g
+ Tg(l + na1ne) n;(1
T;(n;lne) + (/ Ik)(n;lne - 1)]
+ n~/n;)
where n~, n;, T;, and T; are the number densities and temperatures that the gas would have if aU the particles are immediately thermalized in each inelastic collision.
During nonequilibrium ionizationaI process, ne and Te
are calculated using continuity and energy equations for a
transient stationary plasma. In the solution of these equations, the electron number density as calculated from Eq.
(22) and the given value of the electron temperature are used
as the initial values. These are denoted by neO and TeO' n; and
T; calculated from Eqs. (23H28) are taken as the final values. Thus, the continuity and energy equations as given by
Saum and Fang6 become
(29)
neT.) - n.f + IE.
(30)
These equations are solved with the initial conditions
n. = neO }
Te = TeO at t = O.
1044
J.Appl. Phys .• Vol. 59. NO.4.15 February 1986
(27)
,
(28)
+ ~.k(n;T; -
(26)
(31)
I
Thesevaluesofn e and Te are substituted in Eq. (14) to obtain
the transport coefficients.
This completes the mathematical formulation and the
methodology to calculate the transport coefficient of any gas
in the presence of chemical reactions.
IV. NUMERICAL RESULTS AND DfSCUSSION
The analytical model developed here is used to make
sample numerical calculations and to have a qualitative
analysis of the effects of ionizational nonequilibrium on the
transport coefficients of argon-potassium plasma. Such a
plasma is produced by seeding the argon gas with potassium
to have sufficient ionization even at gas temperatures in the
range of 2000-2500 OK. This partially ionized gas is used as
the working fluid for closed cycle MHD generators. Argon
gas has also the advantage of a simple structure and a small
collision cross section. In this range of temperature, the calculated values of the transport coefficients can be closely
approximated by the use of temperature-independent cross
sections,9,l0 although argon is known to have a very deep
Ramsauer minimum in its scattering cross section. To make
a comparison with the experimental results of Cool and Zukoski, 11 the cross-section data given by Zukoski et al. 9 is used
in the calculations of transport coefficients. These coeffiM. L. Mittal and G. P. Gowda
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1044
cients are also calculated with the more recent cross-section
data given by Sakao and Sato. 10
Ionization and recombination rate coefficients in a nonequilibrium chemically reacting plasma with alkali additives
have been investigated theoretically and experimentally by
many authors,11-14 but their numerical values differ up to a
factor of 10 19. The commonly used value 1$.16 of the recombination rate coefficient is obtained from the formula given
by Nelson and Haines 13 as
P.~1.1
T~, 3500'1(
Tg, 2500'1( .3
"ro,169.15m
X 1O- 20 x T .-4.5 m 6 /s.
This value also agrees with the value given by Mnatsakanyan
and Naidis 14 in the electron temperature range considered I
here.
To compare the results with the available experimental
data,9.11 the values of other parameters are taken from their
study. These are
Seed percentage
Electron temperature, T.
Gas temperature, Tg
Stagnation pressure, p
Electric field, E
= 0.4% by weight;
= 3000, 3000, 3500 OK;
= 2000, 2500, 2500 OK;
= 1 atm;
= 790V/m.
Collision cross sections:
Electron neutral for argon,
potassium,
Qea = 400 X 10- 20 m2}
(l.o = 0.7x 10- 20 m 2
from Zukoski et al. 9 ;
(A)
Electron neutral for argon,
potassium,
Q... = 230X 10Q.o
20
m2}
20
= 0.23 X 10- m 2
from Sakao and Sato.1O
(B)
Using the initial values as given above, n; and T; during
ionizational nonequilibrium processes are calculated from
Eqs. (23H28). n; and T; are obtained at every stage of
chemical reactions until it reaches chemical equilibrium.
These values are shown in Table I. As a result of ionizational
nonequilibrium, electron number density n; increases, electron temperature T; decreases and the gas temperature increases. But it may be observed that T; decreases very slowly as compared to the rate at which n; increases. Cool and
Zukoski ll verified that the n; increases about two orders of
o~so
0,0
tIps)
toO
0,75
FIG. L Normalized electron density vs time.
magnitude. In their electron temperature measurements at
lower relaxation times, they have found that T. goes below
its initial value but the decrease is very slow. The percent
deviation in the initial and final values is given in Table I for
comparison. With these values of n; and T;, first-order nonlinear coupled equations (29) and (30) for ne and T. are then
solved using the fourth-order Runge-Kutta method. Figure
1 gives the variation of n. during the ionizational nonequL1ibrium process for the above-mentioned parameters. As seen
from the figure, the ionizationa! relaxation is completed in
time much shorter than the characteristic time of the following plasma. The time scale is in the range of 0.005-1.0 J.tS.
The successive approximations of the transport coefficients are calculated using Eq. (14) for T. = 3000 oK and
Tg = 2000 oK. With the above values of n. and T., it was
found that convergence increases rapidly as the order of approximation increases, whereas the first approximation
differs significantly from the second. There is very little difference between second and third approximations and even
this small difference is seen only for a small initial relaxation
time. Hence, except for electrical conductivity, coefficients
are calculated only up to second approximation.
Figure 2 shows the variation of the normalized electrical
conductivity 17 during relaxation time. The normalization
has been done with respect to its equilibrium value. In these
calculations, the average cross-section data given by Zu-
TABLE 1. Initial and final (after chemical equilibrium is reached) values of
number densities and temperatures for argon seeded with 0.4% (by weight)
of potassium at I-atm pressure.
T.
n,
n.
("K)
XIO- '9 m- 3
X 10- 2 • m- 3
2000.00
2006.10
0.305
53.680
213.40
297.540
3.67
3.667
0.082
Initial value
3000.00 2500.00
Final value
2820.45 2504.12
5.985
0.16
Percent deviation
48.13
205.12
326.18
2.94
2.93
0.34
3500.00 2500.00
Initial value
Final value
3394.43 2502.90
0.12
Percent deviation
3.016
169.15
1125.32
565.28
2.934
2.91
0.82
T.
rK)
Initial value
3000.00
Final value
2821.00
5.97
Percent deviation
1045
J. Appl. Phys., Vol. 59, No.4, 15 February 1986
<1'(1)
1,0
---
4
CASE( A),ClQ' //'8.07 mhos ~1
---CASE(B), t!(,= 313.S9mhos m
Te =3oo0'K
Tg= 2000"1<
O'OI'-----o.-.L'2~5---O'5rf
0.75
--'
tips)
1,00
FIG. 2. First three approximations of normalized electrical conductivity vs
time.
M. L. Mittal and G. P. Gowda
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1045
TABLE II. Comparison of first and second approximations to transport coefficients of argon seeded with 0.4% (by weight) of potassium at 1-atm pressure.
Percent deviation
First
approximation
Electrical conductivity:
Initial value
Final value
Percent deviation of
u between initial
value and final value
Thermal conductivity X 10:':
Initial value
Final value
Percent deviation of
A between initial
value and final value
Viscosity X HP:
Initial value
Final value
Percent deviation of
T/ between initial
value and final value
approximations
Case (A)
Case (B)
Case (A)
Case (B)
Case (A)
Case(Bl
248.07
394.86
313.59
429.90
286.70
729.30
440.90
931.40
15.57
40.60
59.17
37.10
154.40
111.20
15.51
15.97
15.76
16.09
16.17
18.74
17.16
21.22
4.26
8.91
2.93
2.12
2.57
4.06
28.24
28.23
28.24
28.24
29.99
29.98
29.99
29.91
6.2
6.2
0.03
koski et al. 9 [Case (A)] and Sakao and Sato lO [Case (B)) are
used. These curves are represented by continuous lines for
Case (A) and by broken lines for Case (B). The first three
approximations of the electrical conductivity are computed
to examine the rate of convergence of these approximations.
The second and third approximations are found to converge,
but as seen from the figure the convergence is not very fast at
lower relaxation times. This is true for both cases of the collision cross sections considered here. Surprisingly, for higher
relaxation times, the difference in the values of the conductivity for Case (A) and Case (B) is very small (for both the
second and third approximations). The measured values of
electrical conductivity given by Cool and Zukoski II are
shown in Fig. 2 [Case (A)] and are found to be comparable
with the first approximation of the present results. Table II
gives the first and second approximations of the initial and
the final values and the percent deviation.
In Fig. 3, the variation of thermal conductivity with
time is shown. Here only the first ( = second) and third approximations of the thermal conductivity are calculated for
two different cross-section data-Case (A) and Case (B).
Mainly, the el.ectrons and neutral particles contribute to the
thermal conductivity. In the present case the electron number density increases due to inelastic collisions during ionization relaxation time and hence thermal conductivity will also
increase. Unlike the case of the electrical conductivity, the
different cross sections of Case (A) and Case (S) significantly
affect the third approximations of the thermal conductivity.
This behavior of thermal conductivity is comparable qualitatively with the theoretical. calculations ofDevot0 2 for pure
argon gas.
The behavior of viscosity 1) in the presence of chemical
reactions is shown in Fig. 4. In the calcuJ.ations of viscosity,
mainly heavy particles contribute. Its value increases with
1046
in first and second
Second
approximation
J. Appl. Phys., Vol. 59, No.4, 15 February 1966
om
0.004
0.003
the gas temperature but as the ionization begins it starts decreasing. In the presence of chemical reactions in a partially
ionized seeded gas, due to inelastic collisions, the degree of
ionization increases even while the gas temperature is moderate and nearl.y remains constant. But this behavior is not
seen in the caJcuJiation of viscosity for a neutral gas. Devot02
and Kannappan and Bose3 have made theoretical calculations for the viscosity of a pure gas. In their calculations, 1)
increases as temperature increases from 5000 to 10000 "K.,
as the degree of ionization in this range is negligible. But as
the temperature is increased further, the degree of ionization
also increases and then the viscosity starts decreasing. In the
present analysis, a seeded gas is considered, where the ioni-
1.35
~-------
1.30 _ CASE (A~Ao'0.'55'
- - CA SE (B~/.o =0.1576
Wm' K-'
Wm' K-1 ,
T~=3000·K
"
Tg:2000"K
/
I
I
I
I
,
,
I
, '(3)
,"
1\
I
FIG. 3. First two approximations of normalized thermal conductivity vs
time.
M. L. Mittal and G. P. Gowda
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1046
l.0
0
Te= 3500 0 K
Tg: 2500 K _~
T10
1. o62
3157xl0
Ha; =
- 0.5
4.25
- 0.125
- 1.9375
where a, to a l8 are given by
Kgn'i 1s-1
r------,--__
y[(2)
1.0615
11
110 1.000 ~-----,--11:-(1~)- - - - -_ _ _ __
o
Te =3000 Kl-CASE (A) '1\0= 2.82~ x 10-~ Kg riils-I
Tg: 2000oKf--CASE(B): 11.0:2 82t.lx 1O-t. Kg n'i1s-1
1.062 -------------- - - - - - - - - - - - - - - - - - -
=
11(1)
1·00 0 r--~-~-O"':-:.=..=-:...::-:...::-:...:::...:::..:....::..::...::2.~=_-.::-.::-...:-~-:..=..:::..:....=_::..
0.999~---6---""""'---_;;_'.,..--__;_',
0.0
()O25
O.S \ ()J.s I
0.75
1.0
FIG. 4. First two approximations of normalized viscosity vs time.
zation due to inelastic collisions increases while the temperature remains almost same. Hence, the viscosity is nearly constant but shows a decreasing trend as the ionization increases
with relaxation time. This behavior of viscosity can be seen
more clearly for higher temperatures. where the degree of
ionization is more.
Thus. it is concluded that chemical reactions affect
transport coefficients significantly. As is seen from Table H.
the electrical conductivity is enhanced substantially from its
thermal equilibrium value, the thermal conductivity is increased modestly and the viscosity is reduced but negligibly.
ACKNOWLEDGMENT
The authors are grateful to referees for valuable suggestions for the improvement of the paper.
APPENDIX
a l + 1.0
a~ =v..1 a2 + 1.5
a 3 + 1.875
a,
a';:
= Vea
+ 1.0
a 8 - 0.5
a9 - 0.125
1047
= o.
a3
+ 3.25
as + 4.3125
as
a4
as - 0.5
a 10 + 3.25
a l1 - 1.4375
,
a9 - 0.125
all - 1.4375 •
a'2 + 6.7656
+ 0.5
a l4
a l4 + 0.0
a l6
+ 0.0
+ 3.75
a 17 + 0.0
al S+ 0.0
a17
+ 0.0
0 18
013
aa6~
+ 1.875
+ 4.3125
a 6 + 6.7656
a 2 + 1.5
au
+ 0.0
+ 2.8125
J. Appl. Phys., Vol. 59, No.4, 15 February 1986
- 1.9375 •
11.0156
a 1 =H/E,
t (1.15)
T'[ 1.06\5
v....
- 0.125
a2 = (0.5 - 3.0/E) H /€,
a 3 = (1.375 - 6.5/£ + 1.0/c) H /E,
a 4 = ( - 5.75 + 5.0/E - 2.0/c) HIE,
as (10.3125 - 24.375/E + 4.5/c) H /E,
a6 = (25.8906 - 0.3906/E + 0.250/c) H /E.
a7 =4.0Q.
as = ( - 11.0 - I5.01E) Q,
a9 = (10.0 - 4.5/E + 6.0/c) Q,
a lO = (33.0 - 27.0/E + 12.0/c) Q,
all = ( - 27.375 + 61. 125/E - 33.0/c) Q.
a 12 = (18.6875 - 71.875/€ + 49.5/c) Q,
an = 9.125 R.
a l4 = 13.75( - 1.0 + l.O/E) R,
a ls = 5.625( - 1.0 + 1.0/EI2 R,
a l6 = (15.75 - 20.25/E + 1l.25/c) R,
aI, = ( - 11.25 + 33.75/E - 22.5/c) R,
a18 = (5.625 - 11.25/€ + 22.5/c) R.
O· 9'39o..f-:0:------O'~2::=-S-----:0:'::.S,------;:'0.7:;;:S,----~1.0
'to
- 0.5
where
Q=2.0H/c
and
R=0.75[iiH/€'12.
v.a • and v are the mean e-i, e-a, and a-a particle collision frequencies.
V ei •
QQ
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