THERMODYNAMICS, TRANSPORT PROPERTIESAND KINETICS

OF PARTIALLY IONIZED GASES

M. Capitelli

Chemistry Department, University of Bari, ItalyIMIP-CNR, section of Bari, Italy

kT

EN

n

n

enF−

=∑=max

1

2int 2

Internal partition function

calculated with

30

max1

Nan =

where N is the particle density (cm3)

Influence of Electronically Excited States on Thermodynamic Properties of LTE Hydrogen Plasma

Ratio of inner enthalpy to total enthalpy at different pressure.

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ +++⎟

⎠⎞

⎜⎝⎛ ++= + kTnEI

DkTn

DEkTnH eHH 2

5'

22

5'

22

5'

€

Hint = n'H E

The total enthalpy of all the system is calculated as follows:

while the inner part of enthalpy is equal to:

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛∂∂

= + knkncknT

Hc eHVH

nPP

i

f 2

5'

2

5'

2

5'

int',

€

cptot=

∂H∂T

⎛

⎝ ⎜

⎞

⎠ ⎟P

=cpf+ cprthe total specific heat is diveded in two

parts:

the frozen specific heat

⎟⎠⎞

⎜⎝⎛

⎟⎠

⎞⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

∂+⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠

⎞⎜⎝

⎛∂∂

=+

kTT

nEI

DkT

T

nDEkT

T

nc

P

e

P

H

P

HPr 2

5'

22

5'

22

5'the reactive specific heat

the inner specific heat is :

Ratio of inner Cp to total Cp at different pressure

intint ' VH cnc =

Ratio of inner Cp to Cp frozen at different pressure

Ratio of Cp, using Debye-Huckel theory, to Cp calculated with cut-

off, at 108 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.Internal specific heat and his component at 105 Pa.

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.

• Translational thermal conductivity • Viscosity

Heavy particles transport properties Electron transport properties

• Translational thermal conductivity

• Electrical conductivity

• Reactive thermal conductivity

“Usual” Collision integrals “Abnormal” Collision integrals

€

ΩH n( )−H + = Ω

H 1( )−H +

ΩH n( )−H m( )= ΩH 1( )−H 1( )

ΩH n( )−e = ΩH 1( )−e

Complete set of data (see text)

Influence of Electronically Excited States on Transport Properties of LTE Hydrogen Plasma

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-

• Reactionswe consider the dissociation process and ionization reactions starting from each electronic states of hydrogen atom.

€

H2 ↔ 2H1

€

H n =1,12( ) ↔ H + +e−

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.

Then we use the Boltzmann distribution for calculating the distribution of the electronic states of atomic hydrogen.

€

NH2

NH 2

=QH

QH 2

2πmekbTh2

⎛ ⎝ ⎜

⎞ ⎠ ⎟3 2

12

⎛ ⎝ ⎜

⎞ ⎠ ⎟3 2

exp −ED

kbT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

NeNH +

NH

=2 ⋅g

H +

QH

2πmekbTh2

⎛ ⎝ ⎜

⎞ ⎠ ⎟3 2

exp −E I

kbT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ni = ngi

Z T( )e−Ei kBT

Collision Integrals I: General Aspects

Transport cross sections can be calculated as a function of gas temperature according to the equations

€

Ωij( l ,s) =

kT2πμ ij

0

+∞

∫ e−γ ij2

γ ij2s+3σ ( l ) (g)dγ ij

€

σ ( l ) (g) = 2π (1−cos l χ )bdb0

+∞

∫

where

€

γij2 =

12

μ ijg2 / KT

Considered interactions:

• 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)

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

€

Ω (l ,s +1) = TdΩ (l ,s )

dT+ s+

3

2

⎛ ⎝ ⎜

⎞ ⎠ ⎟Ω (l ,s )

Note that we have used the reduced collision integrals i.e. the collision integrals normalized to the rigid sphere model Ωij

*

Collision Integrals II: e-H(n)

Diffusion type collision integrals for the interactions e-H(n) are calculated by integrating momentum transfer cross sections of Ignjatovic* et al..

Viscosity type collision integrals have been considered equal to diffusion type ones

€

Ωe−H n( )

2,2( )* = Ωe−H n( )

1,1( )*

*L.J. Ignjatovic, A.A. Mihajlov, Contribution to Plasma Physics 37 (1997) 309.

0.0

50.0

100.0

150.0

200.0

250.0

1.0 1.2 1.4 1.6 1.8 2.0

n'= 2 - 1/n2

Ω(1,1)∗

( Å

2 )

( )-H n e

=1n

=10n

T=104 K

T=2 104 K

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.

100

101

102

103

1.0 1.2 1.4 1.6 1.8 2.0

Ω (2,2)

(Å

2 )

n=1

n=2n=3

n=4n=5

n' = 2 - 1n2

H(n)+H

+

T =104 K

101

102

103

104

105

1.0 1.2 1.4 1.6 1.8 2.0

Ω (1,1) (Å

2 )

( )- H n H

+

′ n = 2 −

1

n

2

=1n

=2n

=3n

=4n=5n

T =104 K

€

ΩH n( )−H +ct

1,1( )* T( ) = exp f1 ′ n f2 ′ T f3 +exp f4 ′ n − f5( )( )

€

ΩH n( )−H +

2,2( )* T( ) = exp g1 ′ n g2 ′ T g3 +exp g4 ′ n − g5( )( )

with

€

′ T =T 1000

*M. Capitelli, U.T. Lamanna, J. Plasma Phys.12, 71 (1974).

The elastic contribution to Ω(1,1)* has been evaluated with a polarizability model

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

€

ΩH n( )−H n( )

2,2( )* = exp a1 +exp −a2 ′ n + a3( )( )

where

€

′ n = 2 −1n2

*R. Celiberto, U.T. Lamanna, M.Capitelli, Phys.Rev A 58, 2106 (1998).

0.0

5.0

10.0

15.0

20.0

25.0

30.0

1.0 1.2 1.4 1.6 1.8 2.0

n'

H(n)-H(n)Ω

(2,2)∗

( Å

2 )

=1n

=5n

= 2 - 1/n2

T=104 K

T=2 104 K

Collision Integrals V: H(n)-H(m)

0.0

5.0

10.0

15.0

20.0

25.0

30.0

1.0 1.2 1.4 1.6 1.8 2.0

Ω (2,2) (Å

2 )

( )- ( )H n H m

=1n=2n

=3n=4n

=1m=2m

=3m

(1,1)

(2,2)

(3,3)

(4,4)

= 2 - n' 1n2

T =104 K

10-2

10-1

100

101

102

103

104

105

106

1.0 1.2 1.4 1.6 1.8 2.0

Ω (1,1) (Å

2 )

( )- ( )H n H m

′ n = 2 −

1

n

2

=1n

=2n=3n=4n

= +1m n

= +2m n

= +3m n

(1,2)

(2,3)(3,4)

(1,3)

(2,4)

=10T 4 KM.Capitelli, P.Celiberto, C.Gorse, A.Laricchiuta, P.Minelli, D.Pagano, Phys. Rev. E 66,016403/1 (2002)

€

ΩH n( )−H m( )

2,2( )* T( ) =12

ΩH n( )−H n( )

2,2( )* +ΩH m( )−H m( )

2,2( )*( )

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.

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.

Cut-off criterium

We include in the electronic partition function all the elctronic states with radius less than the average distance between particles

€

a0nmax2 =

1′ n

⎛ ⎝ ⎜

⎞ ⎠ ⎟

13

where

€

′ n =p

kBT

Transport Coefficients II: Translational Thermal Conductivity

Heavy particles Electrons

Chapman-Enskog method

Second order approximation

€

λ trh = 4

L11 K L1v

M O M

Lv1 K Lvv

x1 K xv

x1

M

xv

0L11 L L1v

M O M

Lv1 L Lvv

Third order approximation

€

λ tre =

758

1016 nekB

2πkBTe

me

⎛

⎝ ⎜

⎞

⎠ ⎟

12 q 22

q11q 22 − q12( )

2

€

Lii = −4 x i

2

λ ii

−2x ixK

λ iK

1

M i + M j( )2

K=1K≠ i

v

∑ 1

AiK*

15

2M i

2 ⎛

⎝ ⎜ +

25

4MK

2 − 3MK2 BiK

* + 4M iMK AiK* ⎞

⎠ ⎟

€

Lij =2xjxi

λij

M iM j

M i + M j( )2

1

AiK*

55

4− 3Bij

* − 4Aij* ⎛

⎝ ⎜

⎞

⎠ ⎟

€

Aij* =

Ωij(2, 2)*

Ωij(1,1)*

€

Bij* =

5Ωij(1, 2)* − 4Ωij

(1, 3)*

Ωij(1,1)*

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

€

λR = −1

RT 2

A11 K A1μ

M O M

Aμ 1 K Aμμ

ΔH1 K ΔHμ

ΔH1

M

ΔHμ

0

A11 K K

M O M

M K K

Aμ 1 K K

A1μ

M

M

Aμμ

€

Aij =RT

DklPxK xl

aik

xk

−ail

xl

⎛

⎝ ⎜

⎞

⎠ ⎟a jk

xk

−a jl

xl

⎛

⎝ ⎜

⎞

⎠ ⎟

l=k+1

v

∑k=1

v−1

∑

where

Transport Coefficients IV: Viscosity

Viscosity has been calculated by means of the first approximation of the Chapman-Enskog method

€

η =−

H11 K H1v

M O M

H v1 K H vv

x1 K xv

x1

M

xv

0

H11 L H1v

M O M

H v1 L H vv

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

€

Hij = −2xi x j

η ij

M iM j

M i + M j( )2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

53Aik

*−1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Hii =xi

2

η ii

+2xi xk

η ik

M iM k

M i + M k( )2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

k=1k≠i

v

∑ 53Aik

*+

M k

M i

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Aij* =

Ω ij(2,2 )*

Ω ij(1,1)*

€

ηij =2.66932.628

10−2 2M iM j

M i + M j

P0Dij

Aij*T

Transport Coefficients V: Electrical Conductivity

The electrical conductivity has been calculated by using the third approximation of the Chapman-Enskog method

€

σ =32

e2ne2 2π

meKT

⎛

⎝ ⎜

⎞

⎠ ⎟

12 q11q 22 − q12

( )2

q 00 q11q 22 − q12( )

2

( )+q 01 q12q 02 −q 01q 22( )+q 02 q 01q12 −q 02q11

( )

€

q 00 = 8ne n jQej(1,1)*

j =1

ν −1

∑

€

q 01 = 8ne n j

5

2Qej

(1,1)* − 3Qej(1,2 )* ⎛

⎝ ⎜

⎞ ⎠ ⎟

j =1

ν −1

∑

€

q 02 = 8ne n j

35

8Qej

(1,1)* −21

2Qej

(1,2 )* + 6Qej(1,3)* ⎛

⎝ ⎜

⎞ ⎠ ⎟

j =1

ν −1

∑

where

The presence of electronically excited states can affect σe through the collisions e-H(n)

€

q11 = 8 2 ne Qee(2,2 )* + 8 n j

25

4Qej

(1,1)* −15Qej(1,2 )* +12Qej

(1,3)* ⎛ ⎝ ⎜

⎞ ⎠ ⎟

j =1

v−1

∑

€

q12 = 8 2 ne

7

4Qee

(2,2 )* − 2Qee(2,3)* ⎡

⎣ ⎢ ⎤ ⎦ ⎥+ 8

j =1

v−1

∑ n j

175

16Qej

(1,1)* − ⎛ ⎝ ⎜

−315

8Qej

(1,2 )* + 57Qej(1,3)* − 30Qej

(1,4 )* ⎞ ⎠ ⎟

€

q 22 = 8 2 ne

77

16Qee

(2,2 )* − 7Qee(2,3)* + 5Qee

(2,4 )* ⎡ ⎣ ⎢

⎤ ⎦ ⎥+ 8 n j

j =1

v−1

∑ 1225

64Qej

(1,1)* ⎛ ⎝ ⎜ −

−735

8Qej

(1,2 )* +399

2Qej

(1,3)* − 210Qej(1,4 )* + 90Qej

(1,5 )* ⎞ ⎠ ⎟

Results I: Diagonal Approximation (Viscosity)

0.0 100

2.0 10-5

4.0 10-5

6.0 10-5

8.0 10-5

1.0 10-4

0

2

4

6

8

10

1 104 1.5 104 2 104 2.5 104 3 104

Viscosity [Kg m

-1 s

-1] |η

( )b- η( ) a

|*100 /

η( )b

[ ]Temperature K

0.0 100

2.0 10-5

4.0 10-5

6.0 10-5

8.0 10-5

0

20

40

60

80

100

1 104 1.5 104 2 104 2.5 104

Viscosity [Kg m

-1 s

-1] |η

( )b- η( ) a

|*100 /

η( )b

[ ]Temperature K

The small relative error indicate a sort of compensation between diagonal and off-diagonal terms.

The differences calculated with the two sets of collision integrals are very higher.

Including off-diagonal terms

Neglecting off-diagonal terms

This point can indicate the importance of using higher Chapman - Enskog approximations for the calculation of the viscosity in the presence of excited states.

“usual” collision integrals: solid line

“abnormal” collision integrals: dashed line

Results II: Heavy Particles Translational Thermal

Conductivity

0.00

1.00

2.00

3.00

4.00

5.00

10000 15000 20000 25000 30000

λ( )ah

[ ]Temperature K

=7n

=12n

1atm

10atm

[ W m

-1 K

-1]

100atm

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

10000 15000 20000 25000 30000

λ( )a / λ( )u

[ ]Temperature K

100atm

=7n

=12n

1atm

10atm

h

h

The ratio between the translational thermal conductivity values calculated with the “abnormal” cross sections (λh

a) and the corresponding results calculated with the “usual” cross sections (λh

u) is reported as a function of temperature for different pressures.

The small effect observed at 1 atm is due to the compensation effect between diagonal and off-diagonal 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

Results III: Electron Translational Thermal

Conductivity

0.00

2.00

4.00

6.00

8.00

10.00

12.00

10000 15000 20000 25000 30000

λ( )ae

[ ]Temperature K

1atm

10atm

=7n=12n

[ W m

-1 K

-1]

100atm

0.80

0.85

0.90

0.95

1.00

10000 15000 20000 25000 30000

λ( )a / λ( )u

[ ]Temperature K

1atm

10atm

100atm

=7n

=12n

e

e

In this case the presence of excited states affects only the interactions of electrons with H(n).

The figure reports the ratio λea/ λe

u calculated with the two sets of collision integrals as a function of temperature at different pressures.

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.

Results IV: Reactive Thermal Conductivity

0.00

2.00

4.00

6.00

10000 15000 20000 25000

Temperature [K]

λ( )ar

1atm10atm

=7n

=12n

[ W m

-1 K

-1]

100atm

0.40

0.50

0.60

0.70

0.80

0.90

1.00

10000 15000 20000 25000

Temperature [K]

λ( )a / λ( )u

1atm

10atm

100atm

=7n

=12n

r

r

This contribution has been extensively analyzed in a previous paper*.

The main conclusions follow the trend illustrated for λh and λe in this work.

*M.Capitelli, P.Celiberto, C.Gorse, A.Laricchiuta, P.Minelli, D.Pagano, Phys. Rev. E 66,016403/1 (2002)

Results V: Viscosity

0.0 100

5.0 10-5

1.0 10-4

1.5 10-4

10000 15000 20000 25000 30000

Temperature [K]

η( )a

1atm

10atm

=7n

=12n[ Kg m

-1 s

-1]

100atm

0.20

0.40

0.60

0.80

1.00

10000 15000 20000 25000 30000

Temperature [K]

η( )a / η( )u

1atm

10atm

100atm

=7n

=12n

The results for viscosity are in line with those discussed for the heavy particles translational contribution to the total thermal conductivity.

The viscosity values calculated with the “abnormal” cross sections (η(a)) are less than the corresponding results calculated with the “usual” cross sections (η(u)). The relative error decreases when, at high pressure, seven excited states are included in the calculation.

Results VI: Electrical Conductivity

0.0 100

1.0 104

2.0 104

3.0 104

10000 15000 20000 25000 30000

σ( )ae

[ ]Temperature K

=7n

1atm

10atm

[ S m

-1]

100atm =12n

100atm

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

10000 15000 20000 25000 30000

σ( ) a/ σ

( )u

[ ]Temperature K

100atm

=7n

=12n

1atm

10atm

e

e

The trend of the electrical conductivity follows that one described for the contribution of electrons to the total thermal conductivity.

Results VII: Number of levels in partition function

0.40

0.50

0.60

0.70

0.80

0.90

1.00

10000 15000 20000 25000 30000

λ( )a / λ( )u

[ ]Temperature K

h

h

=1n=2n

=4n

=6n( )a

0.80

0.85

0.90

0.95

1.00

10000 15000 20000 25000 30000

n=1

n=2

n=4

n=6

λ( )a / λ( )u

[ ]Temperature K

e

e

( )b

0.40

0.50

0.60

0.70

0.80

0.90

1.00

10000 15000 20000 25000 30000

n=1

n=2

n=4

n=6

Temperature [K]

λ( )a / λ( )u

r

r

( )c

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.

0.40

0.50

0.60

0.70

0.80

0.90

1.00

10000 15000 20000 25000 30000

Temperature [K]

η( )a / η( )u

=1n=2n

=4n

=6n

( )e

0.75

0.80

0.85

0.90

0.95

1.00

10000 15000 20000 25000 30000

σ( ) a/ σ

( )u

[ ]Temperature K

e

e

=1n=2n

=4n

=6n( )f

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

We can see that in this case already the first excited state (n=2) affects the results.

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.

But at high pressure a question is open concerning the number of excited states to be included in the calculation of the partition function.

Another point to be discussed is the accuracy of the present calculations with respect to the Chapman-Enskog approximation used in the present work.

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.

1. Excitation and de-excitation by electron impact

2. Ionization by electron impact and three body recombination

3. Spontaneous emission and absorption

4. Radiative recombination

Collisional-Radiative Model for Atomic Plasma

€

A(i)+e−(ε)kij⏐ → ⏐

k ji← ⏐ ⏐

A( j)+e−( ′ ε )

€

A(i)+e−(ε)kic⏐ → ⏐

kci← ⏐ ⏐

A+ +e−( ′ ε )+eb−(εb)

€

A(i) λijAij⏐ → ⏐ ⏐ A( j)+h ij with i> j

€

A+ +e−(ε) βi⏐ → ⏐ A(i)+h

Rate Equations

€

dni

dt=+ njA ji

*

j>i∑ +ne njk ji

j≠i∑ +ne

2n+kci+nen+βi −ni A ij*

j<i∑ −nine kij

j≠i∑ −ninekic ∀i

€

dne

dt=dn+

dt=− dni

dti∑

Quasi-Stationary Solution(QSS)

€

dni

dt=0 ∀i

€

dni

dt≠0 ∀i

Stationary solution

Time-dependent solution

€

dni

dt=0 i≥2

dn1dt

≈−dne

dt=−dn+

dt

QSS approximation

€

dni

dt=0 i≥2

dn1dt

≈−dne

dt=−dn+

dt

⎧

⎨ ⎪

⎩ ⎪

€

X i =ni

niSB

€

dX1dt

=a11X1+ a1jX jj=2

i*

∑ −b1

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X i≥2 =X i≥20 +Ri≥2

1 X1

€

aijX jj=1

i*

∑ =bi i >1

The ground state density changes like the density of the charged particles andthe excited states are in an instantaneous ionization-recombination equilibrium with the free electrons

differential equation for the ground state

system of linear equation for excited levels

The system of equations is linear in X1

€

X i≥20 =f(ne,Te)

€

Ri≥21 =f(ne,Te)

Xj (j>1) can be calculated when X1, ne, Te are given

€

X i≥2 =f(X1,ne,Te)

CR for Atomic Nitrogen Plasma: Energy-level Model

group Energy (cm-1) Statistical weight Terms1 0 4 2p34S2 19228 10 2p32D3 28840 6 2p32P4 83337 12 3s4P5 86193 6 3s”P6 95276 36 3p4S, 4P, 4D7 96793 18 3p2S,2P,2D8 103862 18 4s4P,2P9 104857 60 3d4P,4D,4F

10 104902 30 3d2P,2D,2F11 107125 54 4p4S,4P,4D,2S,2D,2P12 109951 18 5s4P,2P13 110315 90 4d4P,4D,4F,2P,2D,2F14 110486 126 4f4D,4F,4G,2D,2F,2G15 111363 54 5p4S,4P,4D,2S,2P,2D16 112691 18 6s4P,2P17 112851 90 5d4P,4D,4F,2P,2D,2F18 112955 288 5f,5g19 113391 54 6p20 114211 90 6d4P,4D,4F,2P,2D,2F21 114255 486 6f,6g,6h22 114914 882 n=723 115464 1152 n=824 115837 1458 n=925 116102 1800 n=1026 116298 2178 n=1127 116445 2592 n=1228 116560 3042 n=1329 116650 3528 n=1430 116724 4050 n=1531 116784 4608 n=1632 116834 5202 n=1733 116875 5832 n=1834 116910 6498 n=1935 116940 7200 n=20

CR for Atomic Nitrogen Plasma with QSS

Xi vs level energy

Te=5800 K Te=11600 K Te=17400 K

0.0001

0.001

0.01

0.1

1

10

8 104 8.5 104 9 104 9.5 104 1 105 1.05 105 1.1 105 1.15 105 1.2 105

ne=10

8 cm

-3

ne=1016 cm-3

level energy (cm-1

)

X1=1

0.0001

0.001

0.01

0.1

1

10

8 104 8.5 104 9 104 9.5 104 1 105 1.05 105 1.1 105 1.15 105 1.2 105

ne=108 cm-3

ne=1016 cm-3

level energy (cm-1

)

X1=1

0.0001

0.001

0.01

0.1

1

10

100

8 104 8.5 104 9 104 9.5 104 1 105 1.05 105 1.1 105 1.15 105 1.2 105

ne=108 cm-3

ne=1016 cm-3

level energy (cm-1

)

X1=1

Time-dependent solution

CR rate equations Boltzmann equation

€

dni

dt≠0 ∀i

Rate coefficients for electron impact processes

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k= f(ε)Et

∞∫ σ(ε)v(ε)dεf(ε) electron energy distribution functionσ(ε) cross sectionv(ε) electron velocity

level populationplasma composition

f(ε)rate coefficients

Atomic Hydrogen Plasma

P=100 Torr, Tg=30000 K, Te(t=0)=1000 K H+ = e

- =10-8 , H=1, H(1)=1, H(i)=0 i>1

€

H(n≤25),H+ ,e−[ ]

0

5000

10000

15000

20000

25000

30000

35000

10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104 106

time (s)

108

1010

1012

1014

1016

10-5 10-4 10-3 10-2 10-1

H

H+

e-

time (s)

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

0.0001

0.01

1

100

104

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

i=1i=2i=5i=10i=15i=20i=25

time (s)

density (cm-3) vs time(s) Xi = ni/niSB vs time(s) Te vs time(s)

10-30

10-28

10-26

10-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

0.0001

0.01

1

0 5 10 15 20 25

t(s)=0 s

t(s)=10-10 s

t(s)=10-9 s

t(s)=5 10-9 s

t(s)=10-8 s

t(s)=3 10-8 s

t(s)=5 10-8 s

t(s)=8 10-8 s

t(s)=10-7 s

t(s)=10-6 s

t(s)=10-5

s

t(s)=10-4 s

t(s)=8 10-4 s

t(s)=9 10-4 s

t(s)=10-3 s

ε( )eV

Tfit = 30020 K

H(i)/g(i) vs Ei

10-30

10-28

10-26

10-24

10-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

0.0001

0.01

0 2 4 6 8 10 12 14

t(s)=0 s

t(s)=10-10

s

t(s)=10-9

s

t(s)=5 10-9 s

t(s)=10-8 s

t(s)=3 10-8 s

t(s)=5 10-8 s

t(s)=8 10-8 s

t(s)=10-7 s

t(s)=10-6 s

t(s)=10-5 s

t(s)=10-4 s

t(s)=8 10-4 s

t(s)=9 10-4 s

t(s)=10-3 s

( )level energy eV

Tfit = 29986 K

eedf(eV-3/2) vs E

rese

rvoi

rexit

throat

Non-Equilibrium Kinetics in High EnthalpyNozzle Flows

coupling state-to-state kinetics with fluid dynamic models

- Numerical aspects- Coupling with kinetics

- Numerical aspects- Coupling with kinetics

- Chemical kinetics- Vibrational kinetics- Metastable state kinetics

- Chemical kinetics- Vibrational kinetics- Metastable state kinetics

- Boltzmann equation- Coupling with chemical kinetics- EM fields contribution

- Boltzmann equation- Coupling with chemical kinetics- EM fields contribution