THERMODYNAMICS, TRANSPORT PROPERTIES
AND KINETICS
OF PARTIALLY IONIZED GASES
M. Capitelli
Chemistry Department, University of Bari, Italy
IMIP-CNR, section of Bari, Italy
Influence of Electronically Excited States
on Thermodynamic Properties of LTE Hydrogen Plasma
Internal partition function
Fint 
calculated with
nmax 
1
a0 3 N
where N is the particle density (cm3)
N max
 2n 2e
n 1

En
kT
The total enthalpy of all the system is calculated as follows:
D
D
5
5

5 
H  n' H  kT  E    n' H   kT   EI   n' e  kT 
2
2
2
2

2 
while the inner part of enthalpy is equal to:
Hint  n'H E

Ratio of inner enthalpy to total enthalpy at different pressure.
c p tot
the total specific heat is diveded in two parts:
 H 
5

5 
5 
cPf  

n
'
k

c

n
'
k

n
'

 
H
Vint 
e k
H
 T  P ,n 'i
2

2 
2 

the frozen specific heat
the reactive specific heat
H 
    c p f  c p r
T P
c Pr
D   n' H 
 n' H   5

  kT  E    
2   T
 T  P  2
the inner specific heat is :
Ratio of inner Cp to Cp frozen at
different pressure
 5
n'
D
5
  kT   EI    e   kT 
 2
2
  T  P  2 
P
cint  n' H cV int
Ratio of inner Cp to total Cp at
different pressure
Internal specific heat and his component at 105 Pa.
Ratio of Cp, using Debye-Huckel theory,
to Cp calculated with cut-off, at 105 Pa.
Internal specific heat and his component at 108 Pa.
Ratio of Cp, using Debye-Huckel theory,
to Cp calculated with cut-off, at 108 Pa.
Influence of Electronically Excited States
on Transport Properties of LTE Hydrogen Plasma
The transport properties of a partially ionized thermal hydrogen plasma has been
calculated by taking into account electronically excited states with their
“abnormal” transport cross sections. The results show a strong dependence of these
transport properties on electronically excited states specially at high pressure.
Heavy particles transport properties
•
•
Translational thermal conductivity
Viscosity
•
 H n H    H 1H 
 H n H m    H 1H 1

•
•
Translational thermal conductivity
Electrical conductivity
Reactive thermal conductivity
“Usual” Collision integrals
 H n e   H 1e
Electron transport properties
“Abnormal” Collision integrals
Complete set of data (see text)
Model
• Species
We have considered an hydrogen plasma constituted
by molecular hydrogen, atomic hydrogen (12 atomic
levels), H+ ions and electrons:
•
•
•
•
H2
H(n=1,12)
H+
e-
• Reactions
H 2  2H1
H n  1,12  H   e


we consider the dissociation process and
ionization reactions starting from each
electronic states of hydrogen atom.
Equilibrium Composition
The equilibrium composition is obtained by
using Saha and Boltzmann laws. First we
calculate the equilibrium composition by
considering only four species and two
reactions which take into account the total
atomic hydrogen without distinction among

the electronic states.
32
32
 E D 
N H2
QH 2mekbT  1 

exp

  


2




N H 2 QH 2
h
2
 k bT 
Ne NH 
NH
 E I 
2  gH  2m e k bT 3 2

exp




QH  h 2 
 k bT 

g
ni  n i eE i
Z T 

kB T
Then we use the Boltzmann
distribution for calculating the
distribution of the electronic states
of atomic hydrogen.
Collision Integrals I: General Aspects
Considered interactions:
Transport cross sections can be
calculated as a function of gas
temperature according to the
equations
• neutral-neutral (H2-H2, H2-H, H-H)
• ion-neutral (H+-H2, H+-H)
• electron-neutral (e-H, e-H2)
• charged-charged (H+-e, H+-H+, e-e)
ij(l,s) 
For the present calculation we need the
collision integrals of different orders
depending on the different approximations
used in the Chapman-Enskog method. To
this end we have used a recursive formula


d (l,s )  3  (l,s )
T
 s  
 2 
dT


0
e
  ij2
 ij2s3 (l ) (g)d ij
 (l ) (g)  2 0 (1 cosl  )bdb



(l,s 1)
kT
2ij
where
1
2
 ij2  ij g 2 / KT
Note that we have used the
reduced
collision integrals i.e.

the collision integrals normalized
to the rigid sphere model ij*
Collision Integrals II: e-H(n)
250.0
Diffusion type collision integrals for
the interactions e-H(n) are calculated
by integrating momentum transfer
cross sections of Ignjatovic* et al..
n=10
T=104 K

2
H(n)-e ( Å )
200.0
150.0
100.0
T=2 104 K
50.0
n=1
0.0
1.0
1.2
1.4
1.6
n' = 2 - 1/n2
Viscosity type collision integrals have
been considered equal to diffusion
type ones
eHn   eHn 
2,2 *

*L.J.
Ignjatovic, A.A. Mihajlov, Contribution to Plasma Physics 37 (1997) 309.
1,1 *
1.8
2.0
Collision Integrals III: H+-H(n)
Diffusion type and viscosity type collision
integrals have been calculated by Capitelli et
al.* and fitted according to the following
expressions.
(2,2)
2
 H(n)+H
+ (Å )
10
3
10
2
10
f3
4
10
3
10
2
10
1
2
 exp  f 4 n f 5 
n=5
n=4
n=3
n=2
n=1
1.0
T =104 K
1.2
1.4
1.6
1.8
2.0
n 2 n12
H nH  T   exp g1ng2T g3  exp g4 n g5 
2,2 *
n=5
n=4
n=3
1
n=2
0
1.0
1.2
1.4
1.6
1
n' = 2 - 2
n
1.8
with

n=1
10
10
(1,1)
T   exp f1n T 
f2
5
T =104 K
 H(n)-H + (Å )
1,1 *
H nH  ct
10
2.0
T  T 1000
The elastic contribution to (1,1)*
 has been evaluated with a
polarizability model
*M.
Capitelli, U.T. Lamanna, J. Plasma Phys.12, 71 (1974).
Collision Integrals IV: H(n)-H(n)
Viscosity type collision integrals for the
interactions H(n)-H(n) up to n=5 have been
calculated by Celiberto et al.* By using potential
energy curves obtained by CI (configuration
interaction) method. The data have been
interpolated at different temperatures with the
equation
H nH n   exp a1  exp a2n a3 
2,2 *
where
n 2 

1
n2

30.0
T=104 K


H(n)-H(n)
2
(Å )
25.0
20.0
15.0
10.0
5.0
0.0
T=2 104 K
n=1
1.0
1.2
1.4
1.6
n'= 2 - 1/n2
*R.
n=5
Celiberto, U.T. Lamanna, M.Capitelli, Phys.Rev A 58, 2106 (1998).
1.8
2.0
10
6
10
5
10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
m=n+1
4
T =10 K
M.Capitelli, P.Celiberto, C.Gorse,
A.Laricchiuta, P.Minelli, D.Pagano,
Phys. Rev. E 66,016403/1 (2002)
m=n+2
(3,4)
m=n+3
(2,3)
(2,4)
(1,2)
(1,3)
n=4
n=3
n=2
n=1
1.0
1.2
1.4
1.6
1.8
30.0
2.0
25.0
2

1
T   H2,2n*H n   H2,2m*H m 
2

(2,2)
2,2 *
H nH m 
4
T =10 K
n 2 12
n
H(n)-H(m) (Å )

(1,1)
H(n)-H(m)
2
(Å )
Collision Integrals V: H(n)-H(m)
m=2
m=1
(3,3)
15.0
10.0
5.0
m=3
(4,4)
20.0
(2,2)
(1,1)
n=4
n=3
n=2
0.0
n=1
1.0
1.2
1.4
1.6
1
n' = 2 - 2
n
1.8
2.0
Transport Coefficients I: General Aspects
Transport coefficients have been calculated by using the third approximation of the
Chapman-Enskog method for the electron component and the first non-vanishing
approximation for heavy components.
Cut-off criterium
In general we have considered 12
electronically excited states; at high
pressure we have reduced the number of
excited states to 7 to take into account the
decrease of the number of the electronically
excited states with increasing pressure.
We include in the electronic
partition function all the
elctronic states with radius less
than the average distance
between particles
1
2
a0 nmax
where


1 3
  
n
n
p
kBT
Transport Coefficients II:
Translational Thermal Conductivity
Heavy particles
Electrons
Chapman-Enskog
method
Second order approximation
L11
Third order approximation
L1v x1
1
htr  4
Lv1
Lvv x v
x1
L11
xv 0
L1v
2k BTe  2
75 16
q 22
e
 tr  10 nek B 

2
8
 me  q 11q 22  q 12 
Lii  
Lv1
Lvv
Lij 


A 

*
ij
4 xi
2
v

ii
2xj xi
ij
K 1
K i
2x i x K

iK
M i  M j 
2
MiMj
M i  M j 
2)*
(2,
ij
(1ij,1)*
1
2
B 
*
ij
1 15 2 25 2
*
* 
M i  M K  3M K2 BiK
 4 M i M K AiK

* 

AiK  2
4

1 55
 3Bij*  4Aij* 
* 

AiK  4
5(1ij, 2)*  4(1ij, 3)*
(1ij,1)*
Transport Coefficients III:
Reactive Thermal Conductivity
For a gas constituted by  chemical
species and  independent reactions the
reactive thermal conductivity can be
calculated by means of Butler and
Brokaw theory
where
v1
Aij  
k1

aik ail a jk a jl 
RT
x
x
 D P K l x  x x  x 
kl
k
l
k
l
lk 1
v
R  
A11
A1
H 1
A 1
A
H 
1 H 1
RT 2 A11
H  0
A1
A 1
A
Transport Coefficients IV: Viscosity
Viscosity has been calculated by means
of the first approximation of the
Chapman-Enskog method
 
The Hij are expressed as a function of
temperature, collision integrals and
molecular weight of the species, while i
represents the molar fraction of the i-th
component


H 11
H 1v x1
H v1
H vv x v
x1
H 11
xv 0
H 1v
H v1
H vv
2x x
M i M k  5
M k 
i k

H ii    
 * 

2 
ii k1 

3A
M
M

M

ik
i 
 ik  i
k  
xi2
v
ki
2x x

M i M j  5
i j

H ij  
1

*
 ij M  M 2 
3A


ik


i
j


A 
*
ij
)*
 (2,2
ij
 (1,1)*
ij
ij 
2.6693 2 2M i M j P0 Dij
10
2.628
M i  M j Aij*T
Transport Coefficients V:
Electrical Conductivity
The electrical conductivity has been
calculated by using the third approximation
of the Chapman-Enskog method
2
q 11q 22  q 12 
3 2 2  2 
  e ne 

2
2
me KT  q 00 q 11q 22  q 12   q 01 q 12q 02  q 01q 22  q 02 q 01q 12  q 02q 11 
2
1


where
 1
00
q 
 8n e  n j Q ej(1,1)*
q  8 2 ne Q
11
j 1
5

 8n e  n j  Q ej(1,1)*  3Q ej(1,2 )* 
2

j 1
 1
q
01
q
12

v 1
175
7 (2,2 )*
(2,3)* 
 8 2 n e  Q ee  2Q ee  8 n j  Q ej(1,1)* 
4
 j 1
 16

35

21
 8n e  n j  Q ej(1,1)*  Q ej(1,2 )*  6Q ej(1,3)* 
 8

2
j 1
 1
q 02
q
The presence of electronically

excited states can affect e through
the collisions e-H(n)
25

 8 n j  Q ej(1,1)*  15Q ej(1,2 )*  12Q ej(1,3)* 
 4

j 1
v 1
(2,2 )*
ee
22

315 (1,2 )*
Q ej  57Q ej(1,3)*  30Q ej(1,4 )* 

8
v 1
1225 (1,1)*
77 (2,2 )*
(2,3)*
(2,4 )* 
 8 2 n e  Q ee  7Q ee  5Q ee  8 n j 
Q ej 
16
 j 1  64


735 (1,2 )* 399 (1,3)*
Q ej 
Q ej  210Q ej(1,4 )*  90Q ej(1,5 )* 

8
2
10
8.0 10
-5
8
6.0 10
-5
6
4.0 10
-5
4
2.0 10
-5
2
-1
Including off-diagonal terms
1.5 10
2 10
2.5 10
Temperature [K]
4
“usual” collision integrals: solid line
“abnormal” collision integrals: dashed line
04
3 10
-5
4.0 10
-5
-1
80
60
40
-5
(b)
2.0 10
20
0
0.0 10
4
1 10
(a)
This point can indicate the importance of
using higher Chapman - Enskog
approximations for the calculation of the
viscosity in the presence of excited states.
6.0 10
100
(b)
The differences calculated with the two
sets of collision integrals are very higher.
-5
| -  |*100 / 
Neglecting off-diagonal terms
8.0 10
-1
4
The small relative error indicate a sort of
compensation between diagonal and offdiagonal terms.
(b)
4
(a)
0
0.0 10 4
1 10
Viscosity [Kg m s ]
-1
-4
(b)
1.0 10
| -  |*100 / 
Viscosity [Kg m s ]
Results I: Diagonal Approximation (Viscosity)
1.5 10
4
2 10
Temperature [K]
4
0
4
2.5 10
Results II:
Heavy Particles Translational Thermal Conductivity
The ratio between the translational thermal
conductivity values calculated with the
“abnormal” cross sections (ha) and the
corresponding results calculated with the “usual”
cross sections (hu) is reported as a function of
temperature for different pressures.
4.00
100atm
3.00
2.00
(a)
h
[W m-1 K-1]
5.00
n=7
10atm
1.00
0.00
10000
n=12
1atm
1.00
15000
20000
25000
1atm
30000
0.90
The small effect observed at 1 atm is due to the
compensation effect between diagonal and offdiagonal terms in the whole representation of
the translational thermal conductivity of the
heavy
components.
This
compensation
disappears at high pressure as a result of the
shifting of the ionization equilibrium
(a)h / (u)
h
Temperature [K]
0.80
10atm
0.70
n=7
0.60
100atm
0.50
0.40
0.30
10000
n=12
15000
20000
25000
Temperature [K]
30000
Results III:
Electron Translational Thermal Conductivity
12.00
100atm
8.00
n=7
n=12
6.00
10atm
4.00
1atm
The figure reports the ratio ea/ eu calculated
with the two sets of collision integrals as a
function of temperature at different pressures.
2.00
0.00
10000
1.00
15000
20000
25000
1atm
30000
10atm
Temperature [K]
Again we observe that the excited states
increase their influence with increasing the
pressure.
At high pressure the deviation decreases when
considering only 7 excited states.
(u)
(a)
e / e
(a)e
[W m-1 K-1]
10.00
In this case the presence of excited states
affects only the interactions of electrons with
H(n).
0.95
n=7
0.90
100atm
0.85
n=12
0.80
10000
15000
20000
25000
Temperature [K]
30000
Results IV: Reactive Thermal Conductivity
6.00
This contribution has been extensively analyzed
in a previous paper*.
(a)
r
[W m-1 K-1]
1atm
10atm
n=12
4.00
n=7 100atm
The main conclusions follow the
illustrated for h and e in this work.
trend
2.00
0.00
10000
15000
20000
25000
1.00
1atm
Temperature [K]
(a)r / (u)
r
0.90
0.80
10atm
0.70
n=7
0.60
100atm
0.50
n=12
*M.Capitelli,
P.Celiberto, C.Gorse, A.Laricchiuta,
P.Minelli, D.Pagano, Phys. Rev. E 66,016403/1
(2002)
0.40
10000
15000
20000
Temperature [K]
25000
Results V: Viscosity
[Kg m-1 s-1]
1.0 10-4
(a)
1.5 10-4
5.0 10-5
The results for viscosity are in line with those
discussed for the heavy particles translational
contribution to the total thermal conductivity.
100atm
n=7
n=12
10atm
1atm
0.0 100
10000
15000
20000
25000
30000
Temperature [K]
(a) / (u)
The viscosity values calculated with the
“abnormal” cross sections ((a)) are less than
the corresponding results calculated with the
“usual” cross sections ((u)). The relative error
decreases when, at high pressure, seven
excited states are included in the calculation.
1.00
1atm
0.80
10atm
n=7
0.60
100atm
0.40
n=12
0.20
10000
15000
20000
25000
Temperature [K]
30000
Results VI: Electrical Conductivity
3.0 104
2.0 104
10atm
100atm n=12
1atm
1.0 104
0.0 100
10000
15000
20000
25000
30000
1atm
1.00
Temperature[K]
10atm
0.95
(u)
(a)
e / e
(a)
e
[S m-1]
100atm n=7
The trend of the electrical conductivity
follows that one described for the
contribution of electrons to the total thermal
conductivity.
0.90
n=7
0.85
100atm
0.80
0.75
n=12
0.70
0.65
10000
15000
20000
25000
Temperature[K]
30000
Results VII:
Number of levels in partition function
1.00
n=1
n=2
n=1
1.00
n=2
(u)
(a)
e / e
0.80
n=4
0.70
0.60
0.95
n=4
0.90
0.85
0.50
(a)
0.40
10000
n=6
n=6
(b)
15000
20000
25000
0.80
10000
30000
15000
Temperature [K]
20000
1.00
Figures show the ratio between transport
coefficients calculated by using “abnormal” and
“usual” collision integrals at pressure of 1000
atm, as a function of temperature and for
different number of atomic levels.
25000
30000
Temperature [K]
n=1
0.90
n=2
(u)
(a)
r / r
(a)
/ (u)h
h
0.90
0.80
n=4
0.70
n=6
0.60
0.50
(c)
0.40
10000
15000
20000
Temperature [K]
25000
30000
1.00
At high pressure the number of excited states
decrease. However increasing pressure, the
ionization equilibrium is shifted to higher
temperatures so that the concentration of low
lying excited states can be sufficient to affect the
transport properties
n=1
n=2
0.80
n=4
0.70
0.60
0.50
n=6
(e)
0.40
10000
15000
20000
25000
30000
Temperature [K]
1.00
n=1
n=2
0.95
We can see that in this case already the first
excited state (n=2) affects the results.
(u)
(a)
e / e
(a) / (u)
0.90
n=4
0.90
0.85
0.80
(f)
0.75
10000
n=6
15000
20000
25000
Temperature[K]
30000
Conclusions
The results reported indicate a strong dependence of the transport properties of LTE H2 plasmas
on the presence of electronically excited states. This conclusion is reached when comparing the
transport coefficients calculated with the two sets of collisison integrals.
Our results emphasize the importance of these
states in affecting the transport coefficients
specially at high pressure.
Another point to be discussed is the accuracy
of the present calculations with respect to the
Chapman-Enskog approximation used in the
present work.
But at high pressure a question is open
concerning the number of excited states to be
included in the calculation of the partition
function.
These approximations are very accurate
when neglecting the presence of excited
states. In the presence of excited states with
their “abnormal” transport cross sections
these approximations could not be
sufficient.
Collisional-Radiative Model for Atomic Plasma
1.
Excitation and de-excitation by electron impact
k
ij




A(i)  e ()
A( j)  e ()

k ji
2.
Ionization by electron impact and three body recombination

ic 
k
  eb ( b )
A(i)  e ()
A   e ()


k ci
3.
Spontaneous emission and absorption

 A
ij ij
A(i) 
A( j)  h ij
4.
with i > j
Radiative recombination

i
A   e () 

A(i)  h

Rate Equations
dn i
  n jA*ji  ne  n jk ji  n2e n k ci  ne ni  n i  A*ij  n i ne  k ij  n i ne k ic
dt
ji
ji
ji
ji
i
dn e dn 
dn

  i
dt
dt
i dt

Stationary solution
dn i
0
dt

i
Quasi-Stationary Solution(QSS)
dn i
0
i 2
dt
dn1
dn
dn
 e  
dt
dt
dt
Time-dependent solution
dn i
0
dt
i

QSS approximation
 dn i
0
i 2

dt

dn1   dn e   dn 
 dt
dt
dt
The ground state density changes like the density of the charged particles and
the excited states are in an instantaneous ionization-recombination equilibrium
with the free electrons
differential equation for the ground state

*
Xi 
i
dX1
 a11X1   a1jX j  b1
dt
j2
ni
nSB
i
system of linear equation for excited levels

i*
 a ij X j  b i i  1
j1
Xj (j>1) can be calculated when
X1, ne, Te are given
The system of equations is linear in X1

X i2 
X i2  f (X1 , ne ,Te )
X0i2  R1i2 X1
X0i2  f (n e ,Te )

R1i2  f (n e ,Te )
CR for Atomic Nitrogen Plasma: Energy-level Model
gro up
Energy (cm-1)
Statistical weight
1
0
4
2
19228
10
3
28840
6
4
83337
12
5
86193
6
6
95276
36
7
96793
18
8
103862
18
9
104857
60
10
104902
30
11
107125
54
12
109951
18
13
110315
90
14
110486
126
15
111363
54
16
112691
18
17
112851
90
18
112955
288
19
113391
54
20
114211
90
21
114255
486
22
114914
882
23
115464
1152
24
115837
1458
25
116102
1800
26
116298
2178
27
116445
2592
28
116560
3042
29
116650
3528
30
116724
4050
31
116784
4608
32
116834
5202
33
116875
5832
34
116910
6498
35
116940
7200
Terms
2p34S
2p32D
2p32P
3s4P
3s”P
3p4S, 4P, 4D
3p2S,2P,2D
4s4P,2P
3d4P,4D,4F
3d2P,2D,2F
4p4S,4P,4D,2S,2D,2P
5s4P,2P
4d4P,4D,4F,2P,2D,2F
4f4D,4F,4G,2D,2F,2G
5p4S,4P,4D,2S,2P,2D
6s4P,2P
5d4P,4D,4F,2P,2D,2F
5f,5g
6p
6d4P,4D,4F,2P,2D,2F
6f,6g,6h
n=7
n=8
n=9
n=10
n=11
n=12
n=13
n=14
n=15
n=16
n=17
n=18
n=19
n=20
CR for Atomic Nitrogen Plasma with QSS
Xi vs level energy
Te=5800 K
Te=11600 K
10
10
1
1
0.1
0.1
Te=17400 K
100
10
i
X
i
X
i
1
X
0.1
0.01
X =1
X =1
0.01
X =1
1
1
1
0.01
8
n =10 cm
0.001
16
ne=10 cm
-1
level energy (cm )
16
n =10 cm
-3
0.0001
4
4
4
4
5
5
5
5
5
8 10 8.5 10 9 10 9.5 10 1 10 1.05 10 1.1 10 1.15 10 1.2 10
e
e
0.001
e
n =108 cm -3
n =108 cm -3
-3
-3
e
0.0001
4
4
4
4
5
5
5
5
5
8 10 8.5 10 9 10 9.5 10 1 10 1.05 10 1.1 10 1.15 10 1.2 10
-1
level energy (cm )
ne=1016 cm -3
0.001
0.0001
4
4
4
4
5
5
5
5
5
8 10 8.5 10 9 10 9.5 10 1 10 1.05 10 1.1 10 1.15 10 1.2 10
-1
level energy (cm )
Time-dependent solution
dn i
0
dt
i
Rate coefficients for electron impact processes

k


f ()()v()d
Et

f() electron energy distribution function
() cross section
v() electron velocity
rate coefficients
CR rate equations
f()
Boltzmann equation
level population
plasma composition
H(n  25), H ,e 
Atomic Hydrogen Plasma


 H+ = e- =10-8 ,
H=1, H(1)=1, H(i)=0
P=100 Torr, Tg=30000 K, Te(t=0)=1000 K

Xi = ni/niSB vs time(s)
density (cm-3) vs time(s)
i>1
Te vs time(s)
35000
4
10
16
10
100
30000
1
25000
0.01
0.0001
-3
20000
-6
10
1012
H
15000
-10
10
+
H
e
-8
10
e
X
i
T (K)
density (cm )
1014
i=1
i=2
i=5
i=10
i=15
i=20
i=25
-12
10
-
-14
10
10
10
-16
10
10000
5000
-18
10
108
10-5
-20
10
10-4
10-3
time (s)
10-2
10-1
-9
10
-8
10
-7
10
-6
10
-5
10
time (s)
-4
10
-3
10
-2
10
-1
10
0
-14
-12
-10
-8
10
10
10
10
-6
10
-4
10
time (s)
-2
10
0
10
2
10
4
10
6
10
H(i)/g(i) vs Ei
eedf(eV-3/2) vs E
t(s)=0 s
-10
t(s)=10
T = 29986 K
0.01
fit
0.0001
10
1
-9
0.01
t(s)=10 s
-8
-10
-14
t(s)=8 10 s
-16
t(s)=10 s
-8
10
-7
10
-18
-6
10
t(s)=10 s
-20
10
-5
t(s)=10 s
-22
10
-4
-24
-4
-30
t(s)=10 s
-3
0
2
4
6
8
level energy (eV)
10
12
14
-12
t(s)=5 10 s
-14
-16
t(s)=8 10 s
-7
t(s)=10 s
-18
t(s)=10 s
-20
t(s)=10 s
-22
t(s)=10 s
-24
t(s)=8 10 s
-26
t(s)=9 10 s
-8
10
-8
10
10
-6
10
-5
10
10
t(s)=9 10 s
10
t(s)=3 10 s
-4
-28
10
-8
-10
10
t(s)=8 10 s
-26
10
t(s)=10 s
10
t(s)=10 s
10
-8
-8
-3/2
t(s)=5 10 s
s
-9
t(s)=10 s
-9
t(s)=5 10 s
10
eedf (eV )
-12
-10
t(s)=10
-6
-8
10
fit
10
-8
t(s)=3 10 s
10
t(s)=0 s
T = 30020 K
0.0001
-8
10
H
t(s)=10 s
t(s)=5 10 s
-6
 (i)/g(i)
s
-9
-4
-4
-4
10
-3
t(s)=10 s
-28
10
-30
10
0
5
10
15
(eV)
20
25
Non-Equilibrium Kinetics in High Enthalpy
Nozzle Flows
exit
reservoir
throat
coupling state-to-state kinetics with fluid dynamic models
- Numerical aspects
- Coupling with kinetics
- Chemical kinetics
- Vibrational kinetics
- Metastable state kinetics
- Boltzmann equation
- Coupling with chemical kinetics
- EM fields contribution