NON-EQUILIBRIUM KINETICS IN HIGH ENTHALPY NOZZLE FLOWS

G. Colonna

Dip. di Chimica, Universitá di Bari andCNR-IMIP, Bari section

rese

rvoi

r

exit

throat

OVERVIEW

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

Euler Equations

Mass continuity

∂∂x

ρ x( )u x( )A x( )[ ] = 0

quasi one dimensional steady model (space marching)

Energy continuity

∂∂x

h x( ) + u 2 x( )2[ ] = 0

Momentum continuity

ρ x( )u x( )∂

∂xu x( ) +

∂∂x

P x( ) = 0 P x( ) =ρ x( )R* x( )T x( )

State equation

??

h x( ) =1ρ

Hit( ) + Hi

v( ) +Hif( )

( )i∑

Enthalpy Closure

for internal enthalpy

Translational+

degrees of freedomin equilibrium

€

αiρimi

RT

€

ρivmiv

∑ εiv

State-to-state kinetics:∂∂x

ρ iv x( )u x( )A x( )[ ] = ˙ ρ iv

State-to-state

Chemical

€

ρimi

εif

Multitemperature

Multitemperature kinetics:

€

∂Tij

∂x=

T−Tij

τij

€

ρimi

R ′ α ijTijj∑

u

Internal & Chemical Kinetics

general reaction

π: X iv + r jj

∑ → pkk∑ dir

π' : pkk∑ → r j

j∑ + Xiv rev

˙ ρ iv =mi Rπ' pk[ ]k∏ −Rπniv rj[ ]

j∏

⎡

⎣ ⎢

⎤

⎦ ⎥

π∑

source term

Rπ'K π(eq) =Rπ

detailed balance

2nd Order Rates

general reaction

Global rate

π : Xiv +X jwKwv

qz ⏐ → ⏐ ⏐ Xkq +Xhz

K2 =

Xiv[ ]⋅ X jw[ ]⋅Kvwqz

vwqz∑

Xi[ ]⋅ X j[ ]

Rπ'K π(eq) =Rπdetailed balance not valid

for global rates

NUMERICAL METHODS

Kinetic solution

€

dcivdx

=miuA

R ′ π ckmkk

∏ −Rπcivmi

cj

mjj∏

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

π∑

€

∂u x( )u x( )∂x

=

∂A x( )A x( )∂x

+Fkin x( )

M2 x( )−1=

numden

Sonic point: num=den=0

00 ! Numerical

problems

< 0

NUMERICAL METHODS

€

ρuA=Km

AP +Kmu=Kp

u2

2+ hT + hin=Kh

P =ρRT

m

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

Kinetic solution

€

dcivdx

=miuA

R ′ π ckmkk

∏ −Rπcivmi

cj

mjj∏

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

π∑

€

Km=A0u0ρ0

Kp=A0P0 +Kmu0 + PdAdt

dt0x∫

Kh=u02

2+ hT,0 + hin,0 + Qhdt0

x∫KT = Kh - hin

hT =cpT

m=αRT

m

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

Speed calculation

€

Km 2α−1( )u2 −2αKpu+ 2KTKm=0

Δ2 =α2Kp2 −2KTKm

2 2α−1( )

u=αKp

2α−1( )Km±

Δ2

2α−1( )Km

Transonic Condition

∆2=0 u=speed of sound∆2=0 u=speed of sound

N2 Vibrational Kinetics

N2(v)+N2(w) N2(v-1)+N2(w+1) N2(v)+N2 N2(v-1)+N2

N2(v)+N N2(w)+N

N2 Vibrational Relaxation

N2(v)+N2(v') N2(v-1)+2NN2(v')+N2 2N+N2

N2(v)+N 3N

Dissociation/Recombination

€

εv =ωe v+12

⎛

⎝ ⎜

⎞

⎠ ⎟+ωexe v+

12

⎛

⎝ ⎜

⎞

⎠ ⎟2

+ωeye v+12

⎛

⎝ ⎜

⎞

⎠ ⎟3+ωeze v+

12

⎛

⎝ ⎜

⎞

⎠ ⎟4+ .......

v=0

v=1

v=2

dissociation

Harmonic obscillator

N2 Vibrational Distributions (0D)

10-9

10-7

10-5

10-3

10-1

0 10 20 30 40

t(s)=0t(s)=1e-11t(s)=1e-10t(s)=1e-9t(s)=1e-8t(s)=1e-7t(s)=1e-5t(s)=0.001

vibrational quantum number

P = 10 BarT

g = 2000 K

Tv0

= 8000 K

α0 = 0.9

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

0 10 20 30 40

t(s)=0

t(s)=10-9

t(s)=10-8

t(s)=10-7

t(s)=5 10-7

t(s)=10-6

t(s)=10-5

t(s)=10-1

vibrational quantum number

P = 1 BarT

g = =8000 K

Tv0

= 500 K

α0 = 0

Natoms < Natoms(eq)

Tvib < Tgas

(similar to shock wave)

Natoms > Natoms(eq)

Tvib > Tgas

(similar to nozzle flow)

N2 Elementary Processes

N2(v)+N2(w) N2(v-1)+N2(w+1) N2(v)+N2 N2(v-1)+N2

N2(v)+N N2(w)+N

N2 Vibrational Relaxation

N2(v)+N2(v') N2(v-1)+2NN2(v')+N2 2N+N2

N2(v)+N 3N

Dissociation/Recombination

N2(A) +N2 (A) N2(B)+N2(8) N2(A)+N2 (A) N2(C)+N2(2)N2(A)+N2 (v≥6) N2(B)+N2(v-6)N2(A) + N2 N2(v=0)+N2

N2(A)+N N2(v<10)+NN2(B) + N2 N2(0)+N2

N2(a) +N2 N2(B)+N2

N2(a) +N N2(B)+NN2(C) +N2 N2(a)+N2

N2* Quenching

N+N+N2 N2(B)+N2

N+N+N N2(B)+NN+N+N2 N2(A)+N2

N+N+N N2(A)+N

N2* diss/Ric

N2(B) N2(A)+hN2(C) N2(B)+h

N2* Radiation

N2++N N2(0)+N+

N+ N N2++e-

N++ e - N+h N2(a)+N2(A) N2(v=0)+N2

++e-

N2(a)+N2(a) N2(0)+N2++e-

N2(a)+N2 (v>24) N2(0)+N2++e-

Ionization

N(2D,2P)+N2 N(4S)+N2

N(2P)+N(4S) N(2D)+N(4S)N(4P) N(4S)+h

N* Kinetics

O2 & Air Elementary Processes

O2(v)+O2(w) O2(v-1)+O2(w+1) O2(v)+O2 O2(v-1)+O2

O2(v)+O O2(v-1)+O

O2 Vibrational Relaxation

O2(v)+O2(v’) O2(v-1)+2OO2(v’)+O2 2O+O2

O2(v’)+O 3O

Dissociation/Recombination

O2(v) + N N2(v) + O O X X

NO Kinetics

O2(v)+N2(w) O2(v-2)+ N2(w+1)N2(v)+O2 N2(v-1) +O2

N2(v)+O N2(v-1) +O

Mixed Vibrational Relaxation

N2* + X N2(v=0) + X N2* + X (ground) N2(v=0) + X*N2* + O NO + N*

N2* Kinetics

O+O+X O2(a)+XO+O+X O2(b)+X

O2* Diss/Ric

O2* + X O2(v=0) + X O2* + X (ground) N2(v=0) + X*O2* + N NO + O

O2* Kinetics

O* + X N + XO* + X N + X*O* + N2 NO + N

O* Kinetics

N* + X N + XN* + X N + X*N* + O2 NO + O

N* Kinetics

€

∂f∂t

+ r v⋅ r

∇ rf + r A ⋅

r∇ vf + r v∧

r R⋅

r∇ vf = δf

δt

⎛

⎝ ⎜

⎞

⎠ ⎟c

Free Electron Kinetics

and Magnetic accelleration

€

r R=− e

me

r B

€

r A =− e

me

r E

Electriccollision

€

f =f r r, r v,t( ) in phase space

€

fd3vv∫ =Ne density

mean energy

€

v2fd3vv∫ =EeNe

electron mean velocity

€

r vfd3vv∫ =

r VeNe

Two Term Approximation

€

f r r, r v,t( ) =f0

r r,v,t( )+ r v

v⋅ r f1

r r,v,t( )

isotropic

vx

vy

vx

vy

anisotropic

- +

€

∂f0 v, t( )∂t

=−∂∂v

J F + J el+ J e−e( ) +Sin+Ssup

€

1

2 +R2( )

2 +Rx2 RxRy + Rz RxRz−Ry

RxRy−Rz 2 +Ry2 RyRz + Rx

RxRz + Ry RyRz−Rx 2 +Rz2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

€

J F ∝ r E⋅) ω ⋅

r EELASTICELECTRON-ELECTRONINELASTICSUPERELASTIC

high

lowε

Electron & Nozzle Flow

€

dEtotdt

=−eNe r vd⋅

r E

Joule heating

€

Eel=me2e

vd2Ne

Electron drift energy

Quasi 1D stationary Euler equations withwith free electron kinetics and master equations

Only drift velocity

€

r Q0 =−v

r∇ rf0−

r A∂f0∂v

≅− r A∂f0∂v

Eulerequations

Boltzmannequation

Masterequations

Electronenthalpy

P T

Internal enthalpymolar fractions

Level distributionmolar fractions

e-M rates

€

Rp = vσp v( )f v( )dv0∞∫

SELF-CONSISTENT COUPLINGSELF-CONSISTENT COUPLING

Approximate expansion cooling

€

δf0 =δv∂f0∂v

VIBRATIONAL KINETICS

N2 vibrational kinetics

Gas and Vibrational Temperatures

0

0.2

0.4

0.6

0.8

1

1.2

-0.4 -0.2 0 0.2 0.4

T/T0

Tv/T

0

T/T0

x(m)

Vibrationalnon-equilibriumTv > T

Comparison of gas (T) and vibrational (Tv) reduced temperature profiles. T0=10000 K is the reservoir temperature.

€

exp−ε1−ε0

k

⎛

⎝ ⎜

⎞

⎠ ⎟=

N1N0Tv

N2 vibrational kinetics

Vibrational Distributions

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 10 20 30 40 50

x(m)=-0.5x(m)=0x(m)=0.1x(m)=0.2x(m)=0.3x(m)=0.4x(m)=0.5

Vibrational distribution

vibrational quantum number

Determineglobal rates

Determinevibrationaltemperature

At the nozzle exit the tail of the vibrational distribution is populated by atom recombination.

€

Rd = KdvNvv∑

Global and state selective rates

AIR vibrational kinetics

Vibrational Distributions

10-8

10-6

10-4

10-2

100

0 10 20 30

T0= 4000 K

T0= 5000 K

T0= 6000 K

T0= 7000 K

T0= 8000 K

O2

(v)/O

2

v

10-14

10-10

10-6

10-2

0 10 20 30 40

N2

(v)/N

2

v

x = 1m

AIR vibrational kinetics

Global rates N2+O->NO+N

10-5

10-3

10-1

101

103

105

0 10 20 30 40

K (m3

mol-1 s

-1)

104/T (104K-1)

6000 K

4000 K

T0= 8000 K

Arrhenius

Experiments

The low temperature trend cannot be reproduced by a multitemperature expressions.

AIR vibrational kinetics

NO molar fraction

10-9

10-7

10-5

10-3

10-1

0.0 0.5 1.0

T0=4000 K

T0=5000 K

T0=6000 K

T0=7000 K

T0=8000 K

NO molar fraction

nozzle position (m)

throat

The concentration of NO increase again at the exit.

ELECTRON +

VIBRATIONAL KINETICS

Ionized N2 Mixture

Macroscopic quantities

0

2

4

6

8

10

-0.1 -0.05 0 0.05 0.1

a

b

c

Mach number

x (m)

10-1

100

-0.1 -0.05 0 0.05 0.1

a

b

c

T/T0

x (m)a) Only Vibrational Kineticsb) (case a) + Electronically Excited State Kinetics (no e-M)c) (case b) + Electron Kinetics (Boltzmann Equation) + e-M

∆M% ≈ 10

∆T% ≈ 25

Ionized N2 Mixture

Vibrational distributions

a) Only Vibrational Kineticsb) (case a) + Electronically Excited State Kinetics (no e-M)c) (case b) + Electron Kinetics (Boltzmann Equation) + e-M

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 10 20 30 40

x(m)=-0.1

x(m)=0

x(m)=0.01

x(m)=0.05

x(m)=0.07

x(m)=0.125

vdf

v

10-6

10-5

10-4

10-3

10-2

10-1

100

0 10 20 30 40

a

b

c

vdf

v

N2(A) +N2 (A) N2(B)+N2(8)

e +N2 (v) e+N2(v’<v) (superelastic)

Ionized N2 Mixture

Electron distributions

10-12

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10

x(m)=-0.125

x(m)=0.01

x(m)=0.03

x(m)=0.05

x(m)=0.07

x(m)=0.1

x(m)=0.125

eedf (eV

-3/2)

electron energy (eV)

10-12

10-10

10-8

10-6

10-4

10-2

100

102

0 5 10

x(m)=-0.125

x(m)=0.01

x(m)=0.03

x(m)=0.05x(m)=0.07

x(m)=0.1

x(m)=0.125

eedf (eV

-3/2)

electron energy (eV)

Without e-e collWith e-e coll

Superelastic collisions

Ionized N2 Mixture

Effect of atomic metastable (eedf)

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

0 5 10 15

eedf (eV

-3/2)

ε ( )eV

T0=10000 K

-with e e

-without e e

*with N

*without N

High electron density

Ionized N2 Mixture

Effect of atomic metastable (vdf)

10-4

10-3

10-2

10-1

0 10 20 30 40 50

vibrational distribution

ε ( )eV

T0=10000 K

-with e e

-without e e

*with N

*without N

High electron density

Ionized AIR

Mach number Temperature

0

2

4

6

8

10

-0.1 0.0 0.1

abcde

y(m)

x(m)

Vibr kin N2*, N* O2*, O* El. kin a x 0 0 0 b x x 0 0 c x x 0 x d x x x 0 e x x x x

0.1

1

-0.1 0.0 0.1

abcde

T/T0

x(m)

MAGNETO -HYDRO -

DYNAMICS

FIELDS & GEOMETRY

-0.1

0

0.1

-0.1 -0.05 0 0.05 0.1

nozzle section (m)

x (m)

E B

No Hall effect

EFFECTS of E/NSpeed & Mobility

0

1000

2000

-0.1 0 0.1

0.0 Td0.1 Td0.5 Td1.0 Td

flow speed (m/s)

x (m)

T0=7000 K

104

106

-0.1 0 0.1

0.0 Td0.1 Td0.2 Td0.3 Td0.4 Td0.5 Td1.0 Td

Electron Mobility (Vs/cm

2 )

x (m)

T0=7000 K

EFFECTS of E/NMolar fraction

10-3

-0.1 0 0.1

0.0 Td0.1 Td0.2 Td0.3 Td0.4 Td0.5 Td1.0 Td

e-

molar fraction

x (m)

T0=7000 K

10-40

10-35

10-30

10-25

10-20

10-15

10-10

10-5

-0.1 0 0.1

0.0 Td0.1 Td0.2 Td0.3 Td0.4 Td0.5 Td1.0 Td

Ar*

molar fraction

x (m)

T0=7000 K

EFFECTS of BMolar fractions

10-25

10-20

10-15

10-10

10-5

-0.1 0 0.1

0.0 T

10-4

T

10-3 T

10-2 T

10-1 T1.0 T

Ar*

molar fraction

x (m)

6 10-4

2.6 10-3

-0.1 0 0.1

0.0 T

10-4 T

10-3 T

10-2 T

10-1 T1.0 T

e- molar fraction

x (m)

T0=7000 KE/N=0.5 Td

EFFECTS of BElectron mobility

103

105

-0.1 0 0.1

B=0

B=10-3 T

T=10-2

B=10-1 TB=1 T

Longitudinal Electron Mobility (Vs/cm

2 )

x (m)

0 100

2 104

4 104

-0.02 -0.01

0 T

10-4

T

10-3 T

10-2 T

10-1 T1.0 T

Transversal Electron Mobility (Vs/cm

2 )

x (m)

(yz)

T0=7000 KE/N=0.5 Td

MACROSCOPICMODELS FROM

STATE-TO-STATE

RECOMBINATION REGIMERECOMBINATION REGIME0d kinetics

1000

2000

3000

4000

5000

6000

7000

8000

9000

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

time (s)

P=10 Bar

P=0.1 Bar

T = 2000 KT

v0 = 8000 K

α0 = 0.9

10-14

10-12

10-10

10-8

10-6

10-4

0.01

1

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

time (s)

N

N2(47)/N

2

P=10 Bar

P=0.1 Bar

Np

T = 2000 KT

v0 = 8000 K

α0 = 0.9

RECOMBINATION REGIMERECOMBINATION REGIMERates modeling

RECOMBINATION REGIMERECOMBINATION REGIMERelevant quantities

• Linear dependence of the rates from the pressure;

• Smooth dependence of the rates on the atomic molar fraction;

€

Rp T, ...( ) = kvp T( )

N2 v,T, ...( )N2v

∑

€

logKd( ) =f logα( ),T,P[ ] =logP( ) +poli 6, logα( )[ ]T

i

i=0

4∑

logα( )2q

poli n,x[ ] = aijxj

j=0

n−1∑

RECOMBINATION REGIMERECOMBINATION REGIMERate fitting

Work in Progress

C- MHD: inclusion of magnetic and electric fields configurations to include Hall effects and electric circuit modeling.

A- Improving the kinetic model: state-to-state dissociation from QCT (Dr. F. Esposito, CNR-IMIP)

B- Improving fluid dynamic model: From 1D to 2D (Dr. D. D’Ambrosio, Politecnico di Torino)

D- REDUCED MODELS: finding a macroscopic model for air kinetics that accounts for nonequilibrium distributions (CAST).