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U N D E R S TA N D I N G T H E R O L E O F T H E R M A L N O N E Q U I L I B R I U M O N S C R A M J E T F L O W S
V E N K AT R A M A N A E R O S PA C E E N G I N E E R I N G D E PA RT M E N T U N I V E R S I T Y O F M I C H I G A N
S H O C K S A N D C O M B U S T I O N
• Scramjet stability relies on compression and ignition
• Flame stabilization is the critical design consideration
• Flame-holding at Ma <10
• Fuel-air Mixing at Ma > 10
• Typical fuels: Hydrogen or Hydrocarbon (kerosene, JP-8 etc.)
S H O C K S A N D N O N E Q U I L I B R I U M
• Compression shocks have other effects on molecules
• Throw internal motions out of equilibrium
• If relaxation back to equilibrium is slow, then this effect might alter scramjet behavior
• What does nonequilibrium do to scramjet combustors?
N O N E Q U I L I B R I U M
• Refers to non-Boltzmann distribution of internal energy modes
• Translational, vibrational, and rotational motion
• External aerodynamics (hypersonics)
• All modes could be in nonequilibrium
• Scramjet internal flows
• Roughly 0.5-1 atm pressure
• Translational/rotational equilibrium (roughly)
• Strong vibrational nonequilibrium post compression shocks
R E L E VA N C E T O S C R A M J E T S
• HYSHOT/HIFIRE type experimental scramjet
PSAAP/Stanford University
R E L E VA N C E T O S C R A M J E T S
• Distance from bow shock to combustor ~ 0.2m
• Flow velocity 1800 m/s
• Relaxation time at operating conditions ~ 0.17 ms (Millikan & White)
• Vibrational relaxation time is approximately equal to flow through time between shock and injector
Combustor entrance: 57kPa 1800 m/s, 1500K, Ma 2.3
H2 fuel injection 1200 m/s, 250K, Ma 1.0
I S O L AT O R F L O W
M
T ( K )
D E N S I T Y G R A D I E N T
E F F E C T O F N O N E Q U I L I B R I U M
• Shocks tend to underpopulate higher vibrational levels
• Potentially retard chemical reactions
• Unfortunately
• Very difficult to experimentally characterize
• Significant challenges for computations as well
G A S P H A S E U N D E R E Q U I L I B R I U M
• Fluid modeling begins with the description of gas phase
• Gas mixture decomposed into chemical species
• H2, O2, etc.
• Mixture composition specified using mass fractions, typically
• Thermal equilibrium assumption
• Single temperature to describe the gas phase internal energy
G O V E R N I N G E Q U AT I O N S
• Energy equation is used to obtain temperature
@⇢ui
@t
+@⇢ujui
@xj= � @P
@xi+
@⌧ij
@xj
@⇢�↵
@t
+@⇢uj�↵
@xj=
@
@xj⇢D
@�↵
@xj+ ⇢S↵(�1,�2, · · · ,�N )
@Et
@t
+@uj(Et + P )
@xj=
@
@xjk
@Tt
@xj� @
@xj(⌧ijui) +
NX
↵=1
⇢S↵ev↵
V E L O C I T Y
S P E C I E S M A S S F R A C T I O N
T O TA L E N E R G Y
C O M B U S T I O N A N D C H E M I S T R Y• Flame chemistry based on Arrhenius rates
• Sequence of chemical reactions that lead from fuel/oxidizer to products
found to be appropriate and are used. The discus-sion below highlights the choice of key rateparameters.
The rate expression of H + O2 = O + OHwas taken from GRI 3.0 [8]. The rate coeffi-cient of H + O2(+M) = HO2(+M) was basedon Troe [2], who employed a high-pressurerate krec,1 (cm3 mol!1 s!1) = 4.65 · 1012T0.44 anddeveloped the low-pressure and fall-off expres-sions for Ar and N2 as the bath gases. The broad-ening factor Fc was found to be 0.5 for both thirdbodies. Troe!s fall-off rate parameterization, how-ever, could not be directly used in CHEMKIN[19], because the low-pressure limit rate coefficientk0 does not share the same temperature depen-dence for different third bodies. We had to devel-op parameterized rate expressions (see Fig. 1)based on the k0 expression of Ar and using thefall-off formula of Troe [20]. A collision efficiencyfactor b = 0.53 was used for Ar relative to N2.The collision efficiency of He was assumed to beequal to that of Ar. The study of Michael et al.[3] supports a collision efficiency of O2 smallerthan that of N2. We found that for O2, b = 0.75gives a good agreement with experiment [3] andtheory [2]. For H2O, Troe [2] suggested that thebroadening factor is close to the strong-collisionlimit. We chose a b value of 12 (relative to N2)with the resulting rate in good agreement withthose of Troe and others [2,3,21].
The k0 expression of H + OH +M =H2O +M was taken from [8] with the b valuesequal to 0.38 and 6.3 for Ar and H2O, respectively[12]. The rate expression of Michael et al. [10] was
employed for H2 + O2 = H + HO2. ForOH + OH(+M) = H2O2(+M), the k0 expression,given in the reverse direction by Baulch et al.[12], was refitted based on the new heat of forma-tion of the OH radical along with the low temper-ature data of Zellner et al. [11]. The krec,1expression and the b value of H2O (6) were takenfrom [11] while the Troe fall-off parameters [22]were the same as those in GRI 3.0. The rateexpressions for H2O2 + OH = HO2 + H2O weretaken directly from [15], though the high-temper-ature expression was refitted using a modifiedArrhenius expression to avoid the rate constantvalues exceeding the collision limit when extrapo-lated to high temperatures.
For CO + O(+M) = CO2(+M), the k1 expres-sion was taken from [13], and following Allenet al. [23], k0 was taken from the QRRK analysisof Westmoreland et al. [24] and fall-off was that ofLindemann. The collision efficiency of H2O wasassumed to be 12. The rate constant forCO + OH = CO2 + H was re-analyzed in thepresent study, and the experimental data wererefitted by the sum of two modified Arrheniusexpressions. The new expression resolves moreaccurately the high temperature data of Woold-ridge et al. [25] as well as the data found in [26].Without this revision, it was not possible to recon-cile the high-temperature H2 ignition data withthe H2–CO laminar flame speeds. The knownpressure dependence of this reaction was not con-sidered as this dependence is quite unimportantfor the CO oxidation experiments consideredherein.
Fig. 1. Trial reaction model of H2–CO oxidation, active parameters, and their spans employed in model optimization(see Refs. [9,14,16–18]).
1284 S.G. Davis et al. / Proceedings of the Combustion Institute 30 (2005) 1283–1292
N O N E Q U I L I B R I U M G A S P H A S E
• Each chemical species occupies a range of vibrational and rotational states
• Species mass fraction is sum over all states
• At equilibrium, given , and assuming Boltzmann distribution, it is possible to find
• Not valid for nonequilibrium distribution
�
� =X
j,i
�j,i
�j,i
N O N E Q U I L I B R I U M
• When internal states are not in equilibrium
• Boltzmann distribution is not valid
• Single temperature does not describe species population
➡ State-specific species mass fractions need to be solved
➡ 10-30 times increase in number of PDEs
➡ ~100 times more expensive than equilibrium case
M U LT I - T E M P E R AT U R E D E S C R I P T I O N
• Tractable approach for describing nonequilibrium
• Assume that species are in Boltzmann distribution but at different temperatures
• Three temperatures at each spatial location for each species
• Different species relax fast to common vibrational temperature
• Rotational and translational temperature are assumed to be equal
➡ Two-temperature model
• Vibrational and translational temperatures
G O V E R N I N G E Q U AT I O N S
• Source terms should be recast for two temperatures
@⇢ui
@t
+@⇢ujui
@xj= � @P
@xi+
@⌧ij
@xj
@Et
@t
+@uj(Et + P )
@xj=
@
@xjk
@Tt
@xj� @
@xj(⌧ijui)�QT�V +Qreac
@⇢Ev
@t
+@⇢ujEv
@xj=
@
@xjk
@Tv
@xj+QT�V +
NX
↵=1
⇢S↵ev↵
@⇢�↵
@t
+@⇢uj�↵
@xj=
@
@xj⇢D
@�↵
@xj+ ⇢S↵(�1,�2, · · · ,�N )
C H E M I C A L R AT E S
• Equilibrium rates updated using efficiency function
• Efficiency functions obtained from literature
• Empirical in nature
• Derived for external hypersonic (re-entry flows)
• Parks model
• CVCV model (used here)
k(Tt, Tv) = g(Tt, Tv)keq(Tt)
S U B S O N I C D N S O F M I X I N G
• Nonequilibrium can be created by mixing
• Jet configuration :
• fuel nozzle diameter = 8mm
• co-flow nozzle diameter = 100mm
• Re = 32,000
• LES grid: (Nx,Nr,Nθ) = (320,192,32)
• 80 cells along nozzle jet exit
20D 21D
Ma0.05ρ=0.353kg.m-3
T0=1000KYO2=0.233YN2=0.767
Ma0.568ρ=0.221kg.m-3
T0=500KYH2=0.1765YN2=0.8235
F L A M E S TA B I L I Z AT I O N
Yfuel T
T-Tv
F L A M E S TA B I L I Z AT I O N
• Vibrational nonequilibrium reduces fuel reactivity
• Flame stabilization occurs further downstream
• Stabilization distance depends on relaxation rate
E Q N E Q
B U T W H AT A B O U T C H E M I S T R Y ?
• Results depend on the chemistry model used
• Efficiency function
• How do we verify the chemistry?
• Experiments cannot isolate vibrational state-to-state rates
• Ab-initio computational chemistry
• Only approach to obtaining such rates
Q U A S I - C L A S S I C A L T R A J E C T O R Y A N A LY S I S
• Monte-Carlo approach for obtaining reaction ratesTwo-temperature model
from state-to-state ab-initio data
Inelastic Scattering
Reactive Scattering
Q C T A L G O R I T H M • M A S S I V E LY PA R A L L E L Q C T C O D E • S U R FA C E - A C C E L E R AT I O N A L G O R I T H M • 1 0 B I L L I O N T R A J E C T O R I E S / D AY O N 4 0 0 0 C O R E S
T (K)
Tv(K
)
log 1
0(ϕ)
0.5 1 1.5 2x 104
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 104
0
1
2
3
4
5
H +O2 ! OH +H
T (K)
Tv(K
)
log
10(')
• General effect of Tv: ➡ Tv > T enhances rate
➡ Tv < T suppresses rate
➡ Efficiency response is proportional to Tv/T ratio
• Are these observations consistent among all reactions and species?
Vibrational Nonequilibrium Effects
• The gained/lost efficiency of the reaction due to vibrational nonequilibrium is defined as
H + O2 (Tv) → O + OH
' = k(T, Tv)/keq(T )11/20/15, 1:34 PM
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E�cien
cy
Efficiency functions are reaction and species dependent
Vibrational Nonequilibrium Effects
11/20/15, 1:33 PM
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11/20/15, 1:34 PM
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11/20/15, 1:34 PM
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H + O2(Tv) → O + OH O + H2(Tv) → H + OH
OH(T) + H2(Tv) → H2O + H OH(Tv) + H2(T) → H2O + H
11/20/15, 1:34 PM
Page 1 of 1file:///Users/svoelkel/Desktop/phi_ho2_o2.svg
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cyE�cien
cy
E�cien
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Comparison of T-Tv Models
• Park’s two-temperature model ➡ Nonequilibrium rate is equivalent to the equilibrium rate
evaluated at effective temperature
➡ is a measure of the relative effect of Tv compared to T
• Coupled Vibration-Chemistry-Vibration (CVCV) model ➡ Efficiency function derived using truncated harmonic oscillator
approximation and
- Dissociation energy of reactants
- Characteristic vibrational frequency of reactants
- Activation energy of the reaction
Teff = (TT ⇠v )
1/(1+⇠)
⇠ = 1
2000
T (K)
150010000.5
Tv/T
11.5
2
1.5
0.5
1
Efficien
cy
• Models compared over range ➡ T = 800 K - 2000 K and Tv/T = 0.5 - 1.5
• Both Park and CVCV over-predicts nonequilibrium affects
• Optimized Park’s model improves Park’s model considerably
Comparison of T-Tv Models
H + O2 (Tv) → O + OH
Present results CVCV model
Park’s model ( ) Optimized Park’s Model
⇠ = 1
⇠ = 0.182
Teff = (TT ⇠v )
1/(1+⇠)
S C R A M J E T F L O W S• In scramjet applications
• Fuel injection creates expansion regions
• Associated with higher vibrational temperature
• Is there an effect of geometry on ignition?
H2 injection
Density gradient shows shocks H2 iso-contour shows fuel injection
OH iso-contour shows flame frontAirflow
Turbulent boundary layer
Bleed
• HyShot II Scramjet (reference configuration) ➡ Same geometry : wedge, ramp, isolator, combustor
➡ RANS simulation of wedge
➡ Fully resolved turbulent boundary layer for DNS, Reynolds 18,800
x=-28.5 [cm] x=0 x=6.0 x=12.3 x=36.518°
T∞=300KP∞=1197PaMa=8.0 H2 injector
T0=300KP0=6.4MPaMa=1.0
wedge ramp isolator combustor
bleed
turbulent boundary layerbow shock
Simulation configurations
RANS DNS
Simulation configurations
• Computational domain ➡ (nx, ny, nz) = (2048, 128, 128)
~ (20y+,1:20y+,20y+)
➡ Periodic in spanwise direction
➡ 300K isothermal walls
• Equilibrium vs Nonequilibrium runs ➡ Equal total energy for equilibrium and nonequilibrium cases
- Upstream nonequilibrium induced by wedge bow-shock computed from 1D relaxation
➡ Fuel Tv frozen at total temperature = 300K
➡ 5 days, 4000 cores on NASA Pleiades cluster
1/64th points shown
Key Result
• Nonequilibrium accelerates ignition!
E Q U I L I B R I U M
W I T H N O N E Q U I L I B R I U M
x=-28.5 [cm] x=0 x=6.0 x=12.3 x=36.518°
T∞=300KP∞=1197PaMa=8.0 H2 injector
T0=300KP0=6.4MPaMa=1.0
wedge ramp isolator combustor
bleed
turbulent boundary layerbow shock
Equilibrium
Nonequilibrium
mass fraction of H2O
Macroscopic effect of nonequilibrium on combustion
Y(H2) = 0.1 Y(O2) = 0.1
Y(H2O) = 0.1
• Flame lift-off
• Flow is still out of equilibrium at the injector
• Quickly reaches equilibrium after ignition
0 2 4 6 8 101
2
3
4
5
6
7
Teq Tnneq TV(N2) TV(O2)
Nonequilibrium in the isolator
• Vibrational relaxation still undergoing by combustor entrance ➡ Underpopulated vibrational modes
• Higher translational temperature than equilibrium ➡ Successive shocks increases
this trend
➡ Reaction rates evaluated with translational temperature
-
H + O2 → O + OH : keq(1600K) ~ keq(1300K)O + H2 → H + OH : keq(1600K) = keq(1300K) + 23% OH + H2 → H + H2O : keq(1600K) = keq(1300K) + 45%
centerline temperature profiles
300K100KT/T∞
x[cm]
Effect of Arrhenius parameters
• Plot at conserved internal energy ➡ (Tv,Tnneq) Teq
➡ Teq ∈ [Tv,Tnneq]
• Temperature dependence of the pre-exponential factor ➡ could counterbalance the effect
of the efficiency function
➡ reaction-dependent
keq
(Teq
) = A Tn
eq
exp( �Ea
/RTeq
) (1)
knneq
(Tnneq
, Tv
) = �(Tnneq
, Tv
) A Tn
nneq
exp( �Ea
/RTnneq
) (2)
knneq
keq
= �(Tnneq
, Tv
)
✓Tnneq
Teq
◆n
exp
✓Tnneq
� Teq
Tnneq
Teq
◆EaR
(3)
knneq
(Tnneq
, Tv
) = �(Tnneq
, Tv
) A Tn
nneq
exp( �Ea
/RTnneq
) (4)
e = etr
(Tt
) + erot
(Tr
) + evib
(Tv
) (5)
@⇢e
@t+
@⇢ui
e
@xi
= �p@u
i
@xi
+ ⇢q + ⌧ij
@uj
@xi
(6)
@⇢k
@t+
@⇢ui
k
@xi
= �⇢u00i
u00i
@ui
@xj
� ⌧ij
@u00j
@xi
+
@
@xj
✓� 1
2
⇢u00i
u00i
u00j
+ ⌧ij
u00i
◆� u
00i
@p
@xi
(7)
@⇢K
@t+
@⇢ui
K
@xi
= ⇢u00i
u00i
@ui
@xj
� ⌧ij
@uj
@xi
+
@
@xj
✓� 1
2
⇢ui
u00i
u00j
+ ⌧ij
ui
◆� u
i
@p
@xi
+ ui
u00i
@⇢u00j
@xj
(8)
@⇢ui
e
@xi
= �p@u
i
@xi
+ ⇢q + ⌧ij
@uj
@xi
(9)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ ⇢q + ⌧ij
@uj
@xi
(10)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ ⌧ij
@uj
@xi
+ ⇢q +@�
i
@xi
(11)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ u00i
@p
@xi
+ ⌧ij
@uj
@xi
+ ⌧ij
@u00j
@xi
+ ⇢q +@�
i
@xi
(12)
@⇢ui
k
@xi
= �⇢u00i
u00i
@ui
@xj
� ⌧ij
@u00j
@xi
+
@
@xj
✓� 1
2
⇢u00i
u00i
u00j
+ ⌧ij
u00i
◆� u
00i
@p
@xi
(13)
@⇢ui
K
@xi
= ⇢u00i
u00i
@ui
@xj
� ⌧ij
@uj
@xi
+
@
@xj
✓� 1
2
⇢ui
u00i
u00j
+ ⌧ij
ui
◆� u
i
@p
@xi
(14)
˜H =
˜h+
uk
uk
2
+
]u00k
u00k
2
=
˜h+K + k (15)
k =
]u00k
u00k
2
(16)
K =
uk
uk
2
(17)
˜Ss,i
(f) =Ss,i
(f)
Ss,i
(f = fi
)
(18)
˜As,i
=
Ss,i
(f = fi
)
h(19)
X = X + x (20)
1 of 8
American Institute of Aeronautics and Astronautics
keq
(Teq
) = A Tn
eq
exp( �Ea
/RTeq
) (1)
knneq
(Tnneq
, Tv
) = �(Tnneq
, Tv
) A Tn
nneq
exp( �Ea
/RTnneq
) (2)
knneq
keq
= �(Tnneq
, Tv
)
✓Tnneq
Teq
◆n
exp
✓Tnneq
� Teq
Tnneq
Teq
◆EaR
(3)
knneq
(Tnneq
, Tv
) = �(Tnneq
, Tv
) A Tn
nneq
exp( �Ea
/RTnneq
) (4)
e = etr
(Tt
) + erot
(Tr
) + evib
(Tv
) (5)
@⇢e
@t+
@⇢ui
e
@xi
= �p@u
i
@xi
+ ⇢q + ⌧ij
@uj
@xi
(6)
@⇢k
@t+
@⇢ui
k
@xi
= �⇢u00i
u00i
@ui
@xj
� ⌧ij
@u00j
@xi
+
@
@xj
✓� 1
2
⇢u00i
u00i
u00j
+ ⌧ij
u00i
◆� u
00i
@p
@xi
(7)
@⇢K
@t+
@⇢ui
K
@xi
= ⇢u00i
u00i
@ui
@xj
� ⌧ij
@uj
@xi
+
@
@xj
✓� 1
2
⇢ui
u00i
u00j
+ ⌧ij
ui
◆� u
i
@p
@xi
+ ui
u00i
@⇢u00j
@xj
(8)
@⇢ui
e
@xi
= �p@u
i
@xi
+ ⇢q + ⌧ij
@uj
@xi
(9)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ ⇢q + ⌧ij
@uj
@xi
(10)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ ⌧ij
@uj
@xi
+ ⇢q +@�
i
@xi
(11)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ u00i
@p
@xi
+ ⌧ij
@uj
@xi
+ ⌧ij
@u00j
@xi
+ ⇢q +@�
i
@xi
(12)
@⇢ui
k
@xi
= �⇢u00i
u00i
@ui
@xj
� ⌧ij
@u00j
@xi
+
@
@xj
✓� 1
2
⇢u00i
u00i
u00j
+ ⌧ij
u00i
◆� u
00i
@p
@xi
(13)
@⇢ui
K
@xi
= ⇢u00i
u00i
@ui
@xj
� ⌧ij
@uj
@xi
+
@
@xj
✓� 1
2
⇢ui
u00i
u00j
+ ⌧ij
ui
◆� u
i
@p
@xi
(14)
˜H =
˜h+
uk
uk
2
+
]u00k
u00k
2
=
˜h+K + k (15)
k =
]u00k
u00k
2
(16)
K =
uk
uk
2
(17)
˜Ss,i
(f) =Ss,i
(f)
Ss,i
(f = fi
)
(18)
˜As,i
=
Ss,i
(f = fi
)
h(19)
X = X + x (20)
1 of 8
American Institute of Aeronautics and Astronautics
H2(Tv) + O OH + H (n = 2.57)
O2(Tv) + H OH + O (n = 0)
keq
(Teq
) = A Tn
eq
exp( �Ea
/RTeq
) (1)
knneq
(Tnneq
, Tv
) = �(Tnneq
, Tv
) A Tn
nneq
exp( �Ea
/RTnneq
) (2)
knneq
keq
= �(Tnneq
, Tv
)
✓Tnneq
Teq
◆n
exp
✓Tnneq
� Teq
Tnneq
Teq
◆EaR
(3)
knneq
(Tnneq
, Tv
) = �(Tnneq
, Tv
) A Tn
nneq
exp( �Ea
/RTnneq
) (4)
e = etr
(Tt
) + erot
(Tr
) + evib
(Tv
) (5)
@⇢e
@t+
@⇢ui
e
@xi
= �p@u
i
@xi
+ ⇢q + ⌧ij
@uj
@xi
(6)
@⇢k
@t+
@⇢ui
k
@xi
= �⇢u00i
u00i
@ui
@xj
� ⌧ij
@u00j
@xi
+
@
@xj
✓� 1
2
⇢u00i
u00i
u00j
+ ⌧ij
u00i
◆� u
00i
@p
@xi
(7)
@⇢K
@t+
@⇢ui
K
@xi
= ⇢u00i
u00i
@ui
@xj
� ⌧ij
@uj
@xi
+
@
@xj
✓� 1
2
⇢ui
u00i
u00j
+ ⌧ij
ui
◆� u
i
@p
@xi
+ ui
u00i
@⇢u00j
@xj
(8)
@⇢ui
e
@xi
= �p@u
i
@xi
+ ⇢q + ⌧ij
@uj
@xi
(9)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ ⇢q + ⌧ij
@uj
@xi
(10)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ ⌧ij
@uj
@xi
+ ⇢q +@�
i
@xi
(11)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ u00i
@p
@xi
+ ⌧ij
@uj
@xi
+ ⌧ij
@u00j
@xi
+ ⇢q +@�
i
@xi
(12)
@⇢ui
k
@xi
= �⇢u00i
u00i
@ui
@xj
� ⌧ij
@u00j
@xi
+
@
@xj
✓� 1
2
⇢u00i
u00i
u00j
+ ⌧ij
u00i
◆� u
00i
@p
@xi
(13)
@⇢ui
K
@xi
= ⇢u00i
u00i
@ui
@xj
� ⌧ij
@uj
@xi
+
@
@xj
✓� 1
2
⇢ui
u00i
u00j
+ ⌧ij
ui
◆� u
i
@p
@xi
(14)
˜H =
˜h+
uk
uk
2
+
]u00k
u00k
2
=
˜h+K + k (15)
k =
]u00k
u00k
2
(16)
K =
uk
uk
2
(17)
˜Ss,i
(f) =Ss,i
(f)
Ss,i
(f = fi
)
(18)
˜As,i
=
Ss,i
(f = fi
)
h(19)
X = X + x (20)
1 of 8
American Institute of Aeronautics and Astronautics
t ∞
10-8 10-7 10-6 10-5 10-4 10-3500
1000
1500
2000
2500
T[K]
10-8 10-7 10-6 10-5 10-4 10-3
t[s]
0
0.5
1
1.5
10-8 10-7 10-6 10-5 10-4 10-3500
1000
1500
2000
2500
T[K]
10-8 10-7 10-6 10-5 10-4 10-3
t[s]
0
0.5
1
1.5
- Tv(N2) < Tv(O2) < T
• Increase T from 300K to 2600K in O2-H2-N2 mixture
• Multi-Tv description primordial ➡ Different relaxation timescales
➡ Impact on key reaction rates
- Tv(N2) < Tv(O2) < T 10-8 10-7 10-6 10-5 10-4 10-3500
1000
1500
2000
2500
T[K]
10-8 10-7 10-6 10-5 10-4 10-3
t[s]
0
0.5
1
1.5
Post-shock nonequilibrium reaction rates
dashed
Tv(O2)
298K
T
Teq (TPG)
Tv(N2)Tv(H2)
- H2(Tv) + O OH + H (n = 2.57)- O2(Tv) + H OH + O (n = 0)
- O2(Tv) O + O (n = 0.5)
keq
(Teq
) = A Tn
eq
exp( �Ea
/RTeq
) (1)
knneq
(Tnneq
, Tv
) = �(Tnneq
, Tv
) A Tn
nneq
exp( �Ea
/RTnneq
) (2)
knneq
keq
= �(Tnneq
, Tv
)
✓Tnneq
Teq
◆n
exp
✓Tnneq
� Teq
Tnneq
Teq
◆EaR
(3)
knneq
(Tnneq
, Tv
) = �(Tnneq
, Tv
) A Tn
nneq
exp( �Ea
/RTnneq
) (4)
e = etr
(Tt
) + erot
(Tr
) + evib
(Tv
) (5)
@⇢e
@t+
@⇢ui
e
@xi
= �p@u
i
@xi
+ ⇢q + ⌧ij
@uj
@xi
(6)
@⇢k
@t+
@⇢ui
k
@xi
= �⇢u00i
u00i
@ui
@xj
� ⌧ij
@u00j
@xi
+
@
@xj
✓� 1
2
⇢u00i
u00i
u00j
+ ⌧ij
u00i
◆� u
00i
@p
@xi
(7)
@⇢K
@t+
@⇢ui
K
@xi
= ⇢u00i
u00i
@ui
@xj
� ⌧ij
@uj
@xi
+
@
@xj
✓� 1
2
⇢ui
u00i
u00j
+ ⌧ij
ui
◆� u
i
@p
@xi
+ ui
u00i
@⇢u00j
@xj
(8)
@⇢ui
e
@xi
= �p@u
i
@xi
+ ⇢q + ⌧ij
@uj
@xi
(9)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ ⇢q + ⌧ij
@uj
@xi
(10)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ ⌧ij
@uj
@xi
+ ⇢q +@�
i
@xi
(11)
@⇢ui
˜h
@xi
= ui
@p
@xi
+ u00i
@p
@xi
+ ⌧ij
@uj
@xi
+ ⌧ij
@u00j
@xi
+ ⇢q +@�
i
@xi
(12)
@⇢ui
k
@xi
= �⇢u00i
u00i
@ui
@xj
� ⌧ij
@u00j
@xi
+
@
@xj
✓� 1
2
⇢u00i
u00i
u00j
+ ⌧ij
u00i
◆� u
00i
@p
@xi
(13)
@⇢ui
K
@xi
= ⇢u00i
u00i
@ui
@xj
� ⌧ij
@uj
@xi
+
@
@xj
✓� 1
2
⇢ui
u00i
u00j
+ ⌧ij
ui
◆� u
i
@p
@xi
(14)
˜H =
˜h+
uk
uk
2
+
]u00k
u00k
2
=
˜h+K + k (15)
k =
]u00k
u00k
2
(16)
K =
uk
uk
2
(17)
˜Ss,i
(f) =Ss,i
(f)
Ss,i
(f = fi
)
(18)
˜As,i
=
Ss,i
(f = fi
)
h(19)
X = X + x (20)
1 of 8
American Institute of Aeronautics and Astronautics
solid
- Tv(N2) =< Tv(O2) = T- Tv(N2) =< Tv(O2) = T+31%+24%+9%
0.12 0.1250
0.20.40.60.81
1.21.41.6
Ignition at the injector
• Relaxation in low speed region + expansion wave ➡ Large Tv ratio
➡ N2 lowest Tv
➡ Faster chain reactions
➡ Faster exothermic
H2(Tv)+OH H2O+H(+59%)H2(Tv)+O OH+H (+24.7%)
O2(Tv)+H OH+O (+4.1%)
0.12 0.1250
0.20.40.60.81
1.21.41.6
0.12 0.125400
800
1200
1600
2000
2400
2800
0.12 0.1250
0.20.40.60.81
1.21.4
0.12 0.125400
800
1200
1600
2000
2400
2800
0.12 0.1250
0.20.40.60.81
1.21.4
Tv(O2)T
Tv(H2)Tv(N2)
Tv(H2)/TTv(N2)/TTv(O2)/T
T[K]
O2(Tv) O+O (+16.7%)H2(Tv) H+H (Suppressed)Normalized heat released
x[m] x[m] x[m]
C O N C L U S I O N S
• Nonequilibrium introduces interesting and counter-intuitive behavior
• Overall, nonequilibrium is found to accelerate combustion
• Provides a design opportunity
• Fuel injection to increase combustion in expansion regions
Equilibrium Nonequilibrium