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throat. reservoir. exit. NON-EQUILIBRIUM KINETICS IN HIGH ENTHALPY NOZZLE FLOWS. G. Colonna Dip. di Chimica, Universitá di Bari andCNR-IMIP, Bari section. OVERVIEW. NOZZLE FLOW. - Numerical aspects - Coupling with kinetics. NONEQUILIBRIUM KINETICS. - Chemical kinetics - PowerPoint PPT Presentation
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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).