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MACROSCOPIC AND MICROSCOPICMODELS FOR GAS-DYNAMIC
APPLICATIONS
41st Course: MOLECULAR PHYSICS ANDPLASMAS IN HYPERSONICS
E.NAGNIBEDA,University of Saint Petersburg, RUSSIA
ERICE-SICILY: 1–7 AUGUST 2005
CONTENT
1. Introduction• Old and new problems in modelling of nonequilibrium flows• Background
2. Kinetic theory models• Main principles• Weak and strong nonequilibrium flows• Comparison of different models:
• State-to-state (STS)• Multi-temperature• One-temperature
3. Application• behind shock waves• nozzles
4. Role of kinetic models in gas dynamicsEffects of
• different distributions• models of elementary processes
5. Mass action law in nonequilibrium flows
6. Conclusions
INTRODUCTION
Boltzmann (1872)Chapman-Enskog method (1916–1917):
• closing of fluid dynamics equations → transport coefficients
Polyatomic molecules with internal modes:
• WNE:• Eucken (1913)• Wang Chang, Uhlenbeck (1951)• Mason, Monchick (1958–1968)
• Since 1970th• SNE conditions in high temperature and high enthalpy flows• Coupling kinetics and gas dynamics• Importance of kinetics in gas dynamics• Results
• multi-temperature model• state-to-state approach in kinetics and transport theory• new state-to-state transport algorithms for reactive flows• vibrational-chemical coupling• CO2 flows (kinetic models)
• Now: models suitable for applications ?• model validation• implementation to CFD• Choice of a kinetic model depends on:
- relations between relaxation times- particular flow conditions
WEAK AND STRONG NON-EQUILIBRIUM(WNE AND SNE)
θ is macroscopic time of macroparameters changing; τ ismicroscopic characteristic time
1. τ � θ: WNE
2. τ ∼ θ: SNE
3. τ � θ: frozen
4. τ rap � τ sl ∼ θ: SNE, quasi-stationary models
1),4) — method of small parameter; 2) — DSMC method
IN REACTIVE FLOWS:
• τint ≤ τreact � θ: weak thermal and chemical non-equilibrium
• τint � τreact ∼ θ: weak thermal and strong chemicalnon-equilibrium
• τVV � τTRV ≤ τreact ∼ θ: strong thermal and chemicalnon-equilibrium, multi-temperature models
• τtr ≤ τrot � τvibr ≤ τreact ∼ θ: strong thermal and chemicalnon-equilibrium, state-to-state model (chemical-vibrationalcoupling)
KINETIC EQUATIONS
Dfi =1
εrapJ rapi +
1
εslJsli
Weak non-equilibrium (WNE):
εrap = τrap/θ � 1, εsl = τsl/θ � 1
Dfi =1
εJi , ε � 1
Ji = J rapi + Jsl
i
Strong non-equilibrium (SNE):
εrap � εsl ∼ 1
Dfi =1
εJ rapi + Jsl
i , ε � 1
ε ∼ τrap
θ∼ τrap
τsl
WNE case:
P = pI− 2µS− η∇ · vI
q = −(λtr + λint)∇T (one-component gas)
• µ is shear viscosity coefficient
• η is bulk viscosity coefficient
• λtr , λint are heat conductivity coefficients
SNE case:
P = (p − prel)I− 2µS− η∇ · vI
Harmonic oscillator model:
q = −(λtr + λint)∇T − λvibr∇Tv
Real vibrational spectra, anharmonic oscillator model:
q = −(λtr + λint + λvt)∇T − (λtv + λvv )∇T1
Transport coefficients depend on:
• WNE: elastic and all inelastic collision integrals
• SNE: elastic and inelastic collision integrals for rapidprocesses. Inelastic collision integrals for slow processesdetermine the reaction rates
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• Weak non-equilibrium regime
τ � θ
Closed system of conservation equationsNon-equilibrium effects in- transport coefficients
• Strong non-equilibrium regime
τrap � τsl ∼ θ
Extended system of macroscopic parametersNon-equilibrium effects in- transport coefficients- relaxation equations- coupling gas dynamics and kinetics
• Adequate kinetic model- accuracy- computation cost
• Comparison of STS and QS models- shock heated flows- expanding flows
REACTING GAS MIXTURESSTATE-TO-STATE APPROACH:
• Macroscopic parameters: nci , v,T
• Governing equations:
dnci
dt+ nci∇ · v +∇ · (nciVci ) = Rvibr
ci + R reactci ,
c = 0, . . . , L, i = 0, ..., Lc ,
ρdv
dt+∇ · P = 0,
ρdU
dt+∇ · q + P : ∇v = 0.
c is a chemical species number, c = 1, ..., Li is a vibrational level number, i = 0, 1, ..., Lc
TRANSPORT TERMSfirst order (N.-S.) approximation:
• Pressure tensor
P = (p − prel)I− 2µS− η∇ · vI
• Vci — state dependent diffusion velocity
• Heat fluxq = −λ
′∇T − p∑ci
DTcidci+
+∑ci
(5
2kT + 〈εci 〉rot + εc
i + εc)nciVci ,
λ′= λtr + λrot
prel = protrel , η = ηrot
zero-order: P(0) = nkT I, q(0) = 0, V(0)ci = 0
PRODUCTION TERMS:
• Rvibrci =
∑dki ′k ′
(kd ,k ′kc,i ′i nci ′ndk′ − kd ,kk ′
c,ii ′ ncindk)
• R22ci =
∑dkc ′d ′i ′k ′
(kd ′k ′,dkc ′i ′,ci nc ′i ′nd ′k ′ − kdk,d ′k ′
ci ,c ′i ′ ncindk)
• R23ci =
∑d
nd(kdrec,cinc ′nf ′ − kd
ci ,dissnci )
• VV, VT exchanges
Aci + Adk Aci ′ + Adk′
• Exchange reactions
Aci + Adk Ac′i ′ + Ad′k′
• Dissociation–recombination
Aci + Ad Ac′ + Af ′ + Ad
• kd ,kk ′
c,ii ′ , . . . — state specific rate coefficientsfirst order:
kγ = k(0)γ (T ) + k(1)
γ (nci ,T ) +∇ · vk(2)γ (nci ,T )
QUASI-STATIONARY MODELS:
• Multi-temperature approach• Macroscopic parameters: nc , v,T ,T c
1
dnc
dt+ nc∇ · v +∇ · (ncVc) = R react
c ,
ρdWc
dt= Rw
c −mcWcRreactc ,
ρdv
dt+∇ · P = 0,
ρdU
dt+∇ · q + P : ∇v = 0.
ρcWc(T ,T c1 ) =
∑i
inci (nc ,T ,T c1 )
• Heat flux:
q = −λ′∇T −
∑c
λcv∇T c
1 + qMD + qTD
λ′= λtr + λrot + λvt
• Production terms:
• R22c =
∑dc′d′
(kd′,dc′,c nc′nd′ − kd,d′
c,c′ ncnd)
• R23c =
∑d
nd(kdrec,cnc′nf ′ − kd
c,dissnc)
• kd ′,dc ′,c — multi-temperature reaction rate coefficients
kdd ′cc ′ = k
dd ′(0)cc ′ (T ,T c
1 ,T d1 ) + k
dd ′(1)cc ′ (n1, ..., nL,T ,T c
1 ,T d1 )+
+kdd ′(2)cc ′ (n1, ..., nL,T ,T c
1 ,T d1 )∇ · v
• One-temperature approachNon-equilibrium chemical kinetics, thermal equilibriumvibrations
• Macroscopic parameters: nc , v,T• Heat flux:
q = −λ′∇T + qMD + qTD
λ′= λtr + λint
η = ηint
prel = pintrel
• Production terms contain one-temperature reaction rates
kdd′
cc′ = kdd′(0)cc′ (T ) + k
dd′(1)cc′ (T , n1, ..., nL)+
+kdd′(2)cc′ (T , n1, ..., nL)∇ · v
• WNE• Macroscopic parameters: Nλ, v,T• Nλ — elements (atoms) number densities
λ′= λtr + λint + λchem
η = ηint,chem, prel = 0
no production terms — only conservation equations• Limit passage from one-temperature to WNEregime
APPLICATIONS:
• Behind shock waves
• In nozzles
N2/N (L = 46) and O2/O (L = 33) mixtures;VV, VT exchange, dissociation, recombination;Anharmonic oscillators;STS and QS approximations
• Rate coefficients:• VV,VT — Billing, Capitelli; SSH models, Macheret,
Adamovich• Dissociation — Treanor-Marrone model, ladder climbing,
trajectory (Capitelli, Esposito)• Recombination — detailed balance
• Comparison:• distributions• gas dynamic parameters• reaction rates
• Distributions:• STS• QS:
- complex strong nonequilibrium distribution (Treanor +plateau + Boltzmann a.o.)- nonequilibrium Boltzmann distribution (h.o.)- Treanor distribution (a.o.)- thermal equilibrium Boltzmann distribution
SHOCK HEATED GAS:T0 = 293 K, p0 = 100 Pa, M0 = 15
0 5 10 15 2010-7
10-6
1x10-5
1x10-4
10-3
10-2
10-1
2''2'2
1''
1'1n i/n
i
Figure: Reduced level population of N2 behind a shock wave. Solid lines:x = 0.03 mm; dashed lines: x = 0.8 mm. 1,1’: state-to-state approach;2,2’: two-temperature approach; 1”,2”: one-temperature approach.
Temperature
0,0 0,5 1,0 1,5 2,07000
8000
9000
10000
11000
12000
13000
14000
3
2
1
T,
K
�����
Figure: Temperature behind a shock wave as a function of x , (N2,N). 1:state-to-state approach; 2: two-temperature approach; 3:one-temperature approach.
Molar fraction of atoms
0.0 0.5 1.0 1.5 2.00.20
0.25
0.30
0.35
0.402'
2
1'1
nat/n
X, CM
Figure: Molar fraction of atoms behind a shock wave as a function of x ,M0 = 5. 1,1’: (N2,N)); 2,2’: (O2,O). Solid lines — with recombination,dashed lines — without recombination
Heat flux
0,0 0,5 1,0 1,5 2,0-500
-400
-300
-200
-100
0 3
21
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Figure: Heat flux behind a shock wave as a function of x , (N2,N). 1:state-to-state approach; 2: two-temperature approach; 3:one-temperature approach.
NOZZLE FLOWConic nozzle, 21◦, T∗ = 7000 K, p∗ = 100 atm.
Vibrational distributions
Figure: Vibrational distributions, x/R = 50, (N2,N). 1: STS model; 2:two-temperature, anharmonic oscillator; 3: two-temperature, harmonicoscillator; 4: one-temperature.
Vibrational distributions(state-to-state approach)
Vibrational distributions(state-to-state approach)
Vibrational distributions
Figure: Vibrational distributions, x/R = 50, (N2,N). 1: dissociation andrecombination; 2: dissociation; 3: recombination; 4: no dissociation andrecombination.
Temperatures
Figure: Temperature in a nozzle, (N2,N). 1,1’: T and T1, STSapproach; 2,2’: T and T1, two-temperature, anharmonic oscillator; 3,3’:T and Tv , two-temperature, harmonic oscillator; 4: T, one-temperature.
Heat flux
Figure: Heat flux, (N2,N). 1: STS model; 2: two-temperature,anharmonic oscillator; 3: two-temperature, harmonic oscillator; 4:one-temperature.
CONCLUSIONS 1.
• Vibrational distributions, macroscopic parameters and heattransfer are studied in the STS and QS approximations inreacting gas flows behind shock waves and in nozzles
• In nozzles• QS model can be used with a good accuracy for practical
calculations of gas dynamic parameters and heat fluxes• STS model is needed for the calculation of the vibrational
distributions and reaction rates
• Behind shock waves• STS model gives a much better accuracy compared to the
QS ones not only for distributions and reaction rates but alsofor macroscopic parameters
REACTION RATES IN NONEQUILIBRIUM FLOWS
THEORETICAL MODELS
• ReactionsA2(i) + M A + A + M
AB(i) + M A + B + M
AB(i) + C AC (i ′) + B
AB(i) + CD(k) AC (i ′) + BD(k ′)
• Global reaction rate coefficients
k(M)diss (T , n0, n1, . . . , nL) =
1
nmol
L∑i=0
nik(M)diss,i (T )
k(M)rec (T ) =
L∑i=0
k(M)rec,i (T )
k(M)diss,i (T ), k
(M)rec,i (T ) are state-to-state rate coefficients of
dissociation from the i th level and recombination to the i th level,ni are the vibrational distributions
• Detailed balance principle
k(M)rec,i (T ) = kM
diss,i (T )si
(mmol
m2at
)3/2
×
×h3(2πkT )−3/2(T )Zrot exp
(−εi − D
kT
)Zrot is the molecular rotational partition function, D is thedissociation energy.
• State-to-state non-equilibrium factor:
Z(M)i =
k(M)diss,i
k(M)diss,eq
k(M)diss,eq follows the generalized Arrhenius law:
k(M)diss,eq = AT n exp
(− D
kT
)A, n are tabulated for many reactions
• Global dissociation rate coefficients:
k(M)diss = k
(M)diss,eqZ
(M)
Z (M) is the averaged non-equilibrium factor
Z (M) =∑
i
niZ(M)i
Z(M)i (T ,U) =
Zvibr (T )
Zvibr (−U)exp
(εi
k(1
U+
1
T)
)
DEVIATION FROM MASS ACTION LAW
• In a thermal equilibrium gas (τint � τreact ∼ θ)
k(M)diss,eq =
1
Zvibr
∑i
si exp(− εi
kT
)k
(M)i ,diss(T )
k(M)diss,eq
k(M)rec
=n2at,eq
nmol ,eq= Keq(T )
nat,eq, nmol ,eq are the equilibrium species concentrations
• The equilibrium constant Keq(T ):
Keq(T ) =(Z at)2
Zmolexp
(D
kT
)Z at = Z at
tr , Zmol = Zmoltr ZrotZvibr
• In non-equilibrium gases:
K(M)diss−rec =
k(M)diss (T , n0, n1, . . . , nL)
k(M)rec (T )
= ?
• K(M)diss−rec = Keq(T )Z (M)
Factor Z (M) defines deviation from the Mass Actions Law innon-equilibrium gases
Calculation of Z (M) in non-equilibrium flows
• Dissociation models:• Ladder-climbing• Treanor-Marrone• Trajectory calculations (Esposito, Capitelli)
• Vibrational transition models:• Generalized SSH model• Billing, Capitelli
• Vibrational distributions:• State-to-state model:
ni ,T — solution of equation of state-to-state kinetics coupledto gas dynamic conservation equations
• Two-temperature models:• Harmonic oscillator (Boltzmann distribution)• Anharmonic oscillator (Treanor distribution)• Anharmonic oscillator (Treanor–plateau–Boltzmann; strongly
non-equilibrium distribution)
ni (T ,T1),T ,T1 — solution of equations of two-temperaturekinetics coupled to gas dynamic equations
• N2/N mixture flows:• Behind shock waves• In nozzles
Averaged factor Z
2000 4000 6000 8000 1000010-4
10-2
100
102
104
106
108
1010
(a)Z
2'2
1'
1
T, K
Figure: Averaged factor Z . 1,2: Treanor distribution; 1’,2’: Boltzmanndistribution; 1,1’: Trajectory calculations; 2,2’: Treanor-Marrone model,U = D/6k
Averaged factor Z
2000 4000 6000 8000 1000010-4
10-1
102
105
108
1011
1014
1017
(b)
2'
2
1'
1
Z
T, K
Figure: Averaged factor Z . 1,2: Treanor distribution; 1’,2’: Boltzmanndistribution; 1,1’: Trajectory calculations; 2,2’: Treanor-Marrone model,U = D/6k
Averaged factor Z
2000 4000 6000 8000 10000
100
103
106
109
1012
1015
1018
1021
(c)
2'2
1'1
Z
T, K
Figure: Averaged factor Z . 1,2: Treanor distribution; 1’,2’: Boltzmanndistribution; 1,1’: Trajectory calculations; 2,2’: Treanor-Marrone model,U = D/6k
SHOCK HEATED GASN2/N mixture, T0 = 293 K, p0 = 100 Pa, M0 = 15
Figure: Averaged factor Z behind the shock front. 1: state-to-stateapproach, Treanor-Marrone (U = D/6k); 2: state-to-state approach,ladder-climbing model; 3: two-temperature approach
two-temperature model overestimates k(M)diss and Z
Z < 1 behind a shock, K(M)diss−rec < Keq(T )
EXPANDING NOZZLE FLOWN2/N mixture, conic nozzle, 21◦, T∗ = 7000 K, p∗ = 100 atm
Figure: Averaged factor Z in nozzle. 1: state-to-state approach; 2:two-temperature approach, anharmonic oscillator model; 3:two-temperature approach, harmonic oscillator
T1 > T : strong deviation of K(M)diss−rec from Keq(T )
Z � 1 in a nozzle, K(M)diss−rec � Keq(T )
Dissociation rate coefficient
0,0 0,5 1,0 1,5 2,00
2x10-20
4x10-20
6x10-20
8x10-20
1x10-19
32
1
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Figure: Dissociation rate coefficient k(N2)diss , m3s−1, as a function of x . 1:
state-to-state approach; 2: two-temperature approach; 3:one-temperature approach
Dissociation rate coefficients in different approaches
Figure: Averaged dissociation rate coefficient k(mol)diss m3s−1 versus x/R
in a conic nozzle. O2/O, T ∗ = 4000 K, p∗ = 1 atm. Curves 1:state-to-state model; 2: two-temperature anharmonic oscillator model; 3:two-temperature harmonic oscillator model; 4: one-temperature model.
Dissociation rate coefficients in different approaches
Figure: Averaged dissociation rate coefficient k(mol)diss m3s−1 versus x/R
in a conic nozzle. N2/N, T ∗ = 7000 K, p∗ = 1 atm. Curves 1:state-to-state model; 2: two-temperature anharmonic oscillator model; 3:two-temperature harmonic oscillator model; 4: one-temperature model.
CONCLUSIONS 2.
• The influence of different vibrational distributions on averageddissociation rates is found to be important
• In a non-equilibrium gas the ratio of forward and backwardreaction rate coefficients K deviates noticeably from theequilibrium rate constant Keq(T ). Using Keq(T ) instead of Kin practical calculations can lead to a significant error
• The effect of anharmonicity on K increases in a highly excitedgas
• The results justify using a simple Treanor-Marrone model fordissociation behind a shock wave
THE INFLUENCE OF A MODEL OF ELEMENTARYPROCESSES on:
• distributions
• macroscopic parameters
Different models for vibrational transitions:
• SSH for h.o.
• SSH for a.o.
• FHO (Macheret, Adamovich)
• Billing, Capitelli
For Dissociation:
• Treanor-Marrone model
• Ladder climbing model
N2(v) + N2 = N2(v − 1) + N2
0 10 20 30 40
10-18
10-17
1x10-16
10-15
(a)
4
3
2
1
kVTv v-1
, m3/s
v
Figure: Rate coefficients of VT transitions. T = 6000 K. 1: SSH modelfor anharmonic oscillators; 2: SSH model for harmonic oscillators; 3:FHO model; 4: BC model.
N2(v) + N = N2(v − 1) + N
0 10 20 30 40
10-18
10-17
1x10-16
SSH, anharmonic osc. SSH, harmonic osc. FHO BC
(b)
4
3
2
1
kVTv v-1
, m3/s
v
Figure: Rate coefficients of VT transitions. T = 6000 K. 1: SSH modelfor anharmonic oscillators; 2: SSH model for harmonic oscillators; 3:FHO model; 4: BC model.
APPLICATION FOR SHOCK HEATED GAS
0 0.5 1.0 1.5 2.0
0.01
0.02
0.03
0.04
0.05
1 - Treanor-Marrone model, U=3T2 - ladder-climbing model
nat / n
x, cm
2
1
Figure: Molar fraction of atoms behind a shock wave. 1:Treanor-Marrone model, U = 3T ; 2: ladder-climbing model.
0 10 20 30 4010-20
10-15
1x10-10
1x10-5
1x100
x=2cm
x=0.01cm
(a)ni / n
v
3'
2'1'
32
1
Figure: Vibrational distributions behind a shock wave. 1: SSH model foranharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHOmodel. Curves 1–3 correspond to x = 0.01 cm; 1’–3’ to x = 2 cm;
0.0 0.5 1.0 1.5 2.0
9000
10000
11000
12000
13000
(b)
3
2
1
T, K
x, cm
Figure: Temperature behind a shock wave. 1: SSH model foranharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHOmodel. 1: SSH model for anharmonic oscillators; 2: SSH model forharmonic oscillators; 3: FHO model.
0.0 0.5 1.0 1.5 2.0
0.02
0.04
0.06
0.08
0.10
0.12 (c)
3
2
1
nat / n
x, cm
Figure: Atomic molar fractions behind a shock wave. 1: SSH model foranharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHOmodel.
0.0 0.5 1.0 1.5 2.0-500
-400
-300
-200
-100
0
SSH, anhharmonic oscillator SSH, harmonic oscillator FHO
(d)
3
2
1
q, kW/m2
x, cm
Figure: Total heat flux behind a shock wave. 1: SSH model foranharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHOmodel.
CALCULATIONS WITH DIFFERENT MODELS OFENERGY EXCHANGES
T ∗ = 4000 K, p∗ = 1 atm, (O2,O)
T ∗ = 7000 K, p∗ = 1 atm, (N2,N)
T ∗ = 7000 K, p∗ = 1 atm, (N2,N)
CONCLUSIONS 3.
• The choice of vibrational energy exchange model is importantfor values of level populations and macroscopic parameters
• Theoretical information approach and Treanor-Marrone modelwith correct values of the model parameter give a goodagreement with QST calculations
• Employing ladder-climbing model for dissociation and SSHfor harmonic oscillator can cause a noticeable error in gasparameters in high temperature flows
• For (N2,N) FHO for vibrational transitions andTreanor-Marrone with U = 3T for dissociation can berecommended
PROBLEMS to BE SOLVED:
• Implementation of developed models directly to CFD codes(air-mixtures, CO2, . . .) (under development)
• State-dependent rate coefficients for exchange reactions.Available data for calculations of kinetics, gas dynamics andtransport in multi-component reacting mixtures. Analyticalmodels
• Non-maxwellian reaction rates (under development)
• Role of rotational-vibrational coupling in kinetics and transport
• Estimation of bulk viscosity and prel in reacting flows
• Extending the number of processes under consideration inhigh-temperature flows
ACKNOWLEDGEMENTS
This work is supported by INTAS (Grant N 03-51-5204)
The presented results have been obtained in collaboration withProf. E.V. Kustova and Dr. T. Alexandrova.
Presentation have been prepared with help of graduate studentK. Karakulko.
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