1. business and industrial
2. energy
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50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
09 - 12 January 2012, Nashville, Tennessee
AIAA 2012-1266
Numerical Study on Kerosene/LOx Supercritical Mixing
Characteristics of a Swirl Injector
Jun-Young Heo1, Kuk-Jin Kim2, Hong-Gye Sung3
Korea Aerospace University, Goyang Gyeonggi, 412-791, Republic of Korea
Hwan-Seok Choi4
Korea Aerospace Research Institute, Daejeon, 305-333, Republic of Korea
Vigor Yang5
Georgia Institute of Technology, Atlanta, GA 30332, USA
The turbulent mixing of a kerosene/liquid oxygen coaxial swirl injector under
supercritical pressure has been numerically investigated. Three kerosene models are
proposed for the kerosene thermodynamic properties. N-dodecene used to replace the
kerosene is considered as baseline property for modeling simplicity. Two types of surrogate
kerosene are applied. Turbulent numerical model is based on Large Eddy Simulation (LES)
with real-fluid transport and thermodynamics; Soave modification of Redlich-Kwong
equation of state, Chung's model for viscosity/conductivity, and Fuller's theorem for
diffusivity to take account Takahashi's compressible effect. The effect of operating pressure
and surrogate kerosene model on thermodynamic properties and mixing dynamics inside an
injector and combustion chamber are investigated. To quantify the mixing completeness,
mixing efficiency is deliberated. Power spectral densities of pressure fluctuations under
various chamber pressures are analyzed.
Nomenclature
D
E
f
p
q
t
u
x
Z
=
=
=
=
=
=
=
=
=
Diffusivity
Specific total energy
Mixture fraction
Pressure
Heat flux
Time
Velocity components
Spatial coordinate
Pseudo-time variable vector
=
=
=
=
Kronecker delta
Preconditioning matrix or circulation
Density
Viscous stress tensor
Greek
δ
Γ
ρ
τ
1
Research assistant, School of Aerospace and Mechanical Engineering.
Research assistant, School of Aerospace and Mechanical Engineering.
3
Professor, School of Aerospace and Mechanical Engineering, AIAA Associate Fellow, [email protected].
4
Head, Combustion Chamber Department, AIAA Member.
5
William R.T. Professor and Chair, School of Aerospace Engineering, AIAA Fellow.
1
American Institute of Aeronautics and Astronautics
2
Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Superscripts
%
sgs
= Resolved-scale
= Favre-averaged resolved-scale
= Subgrid-scale
I. Introduction
L
IQUID rocket engine for the space launch vehicle is operating at high pressure in order to increase the thrust
and combustion efficiency, and its propellants are exposed to supercritical conditions in the combustion
chamber. The supercritical conditions are the state that arbitrary substance is enclosed by the environment over its
critical point. In such conditions, the propellants experience the transition of thermophysical properties that is
different from phase change process at subcritical conditions. After the liquid propellant is introduced into the
injector and combustion chamber, its properties are varied continuously and the distinction between gas and liquid is
ambiguous. At near the critical point, propellant properties exhibit gas-like diffusivities and liquid-like densities so
that the engine performance can be achieved with high energy density. Hence, the study for the mixing and
combustion process of the propellants at supercritical conditions is an essential subject. Because of the peculiarities
of substances at supercritical conditions, the conventional methods for property analysis are inappropriate so that the
applicable thermodynamic relationships for a wide range of pressure must be employed.
For understanding of the atomization, mixing and combustion phenomena in liquid rocket engine, experimental
studies are performed in the wide range of pressure and temperature including the supercritical conditions. The
Raman scattering technique lately has become the standard diagnostic method for quantitative species detection.
Oschwald and Schik (1999) studied the atomization and mixing mechanism of the liquid nitrogen jet and obtained
radial density profiles by spontaneous Raman scattering.1 In a wide range of pressures, the acoustic effect on the
nitrogen injection and mixing dynamics such as the velocity fluctuations in the chamber when the acoustic driver
was on was studied by Davis and Chehroudi (2004, 2007), Leyva et al. (2007).2-4 Lubarsky et al. (2004) performed
experiments about the combustion instability in a high pressure air breathing combustor. As the temperature of the
fuel was increased, it attained a supercritical state. After that, the spray disappeared and the scattered signal could no
longer be detected. Also, unstable modes were observed during the transition to supercritical fuel injection.5
Recently, the heat transfer effect of fuel was studied in a supercritical environment. Zhong et al. (2009) investigated
the heat transfer characteristics of China no.3 kerosene at supercritical conditions.6 Locke et al. (2010) studied the
mixing and combustion characteristics in an LRE shear coaxial injector. In the investigation of liquid
oxygen/gaseous oxygen cold flow, the phase change for liquid oxygen is more continuous similar to gaseous
mixing.7
In research of numerical analysis, the results of mixing and combustion process in shear coaxial injectors using
propellants as liquid oxygen/gaseous methane and liquid oxygen/gaseous hydrogen at supercritical conditions were
investigated. The thermophysical characteristics of liquid oxygen/gaseous hydrogen flame in a shear coaxial injector
at supercritical pressure have been investigated by Oefelein (2005) using direct numerical simulation.8 Zong and
Yang (2008) applied the large eddy simulation for the identification of cryogenic fluid dynamics in a swirl liquid
oxygen injector operating at supercritical pressure.9 In addition, Huo and Yang (2011) observed the mixing and
combustion of liquid oxygen/gaseous methane in a shear injector at supercritical conditions.10 Giorgi and Leuzzi
(2009) performed the numerical analysis of mixing and combustion in liquid oxygen/gaseous methane shear coaxial
injector at supercritical conditions.11
The kerosene fuels are quite unwieldy in CFD because of a large number of components, reaction step, and
thermophysical parameter. Several researchers have suggested a variety of kerosene surrogate models. Wang (1996,
2001, 2010) proposed a one formula surrogate fuel and its quasi-global combustion kinetics model of RP-1. The
emphasis of his work was on computational efficiency.12-14 Montgomery et al. (2002) developed the four
components surrogate model of JP-8 and studied its reduced chemical kinetic mechanisms.15 Dagaut’s Jet A-1
surrogate model includes three hydrocarbons and the kinetic reaction mechanism consists of 1592 reversible
reactions and 207 species (Dagaut, 2002).16 Huang et al. (2002) presented the model of JP-7 and JP-8 consisting of 6
and 11 component, respectively.17
In this study, the turbulent mixing of a kerosene/liquid oxygen coaxial swirl injector under supercritical
pressures has been numerically investigated. Compared with a shear injector, the coaxial swirl injector could
produce the finer spray, shorter atomization distance, and the swirl injector is not sensitive to instability in
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combustor. Kerosene surrogate models are considered for the abbreviation of real fuel components. The surrogate
models of JP-8/Jet-A are proposed for the kerosene thermodynamic properties. Turbulent numerical model is based
on LES with real-fluid transport and thermodynamics over the wide pressure range; Soave modification of RedlichKwong (SRK) equation of state, Chung’s model for viscosity/conductivity and Fuller’s theorem for diffusivity to
take account Takahashi’s compressible effect. The effect of operating pressure on thermodynamic properties and
mixing dynamics inside an injector and a combustion chamber are investigated. Power spectral densities of pressure
fluctuations in the injector under various chamber pressures are analyzed.
II. Numerical Method
A. Filtered Transport Equations
The theoretical formulation is based on the filtered Favre averaged mass, momentum, energy and mixture
fraction conservation equations in Cartesian coordinates. Turbulent closure is achieved by means of a LES technique
in which large scale turbulent structures are directly computed and the unresolved small scale structures are treated
by using the analytic or empirical modeling. The governing equations can be written as:
∂ρ ∂ρ u% j
+
=0
∂t
∂x j
(1)
sgs
sgs
∂ρ u%i ∂ ( ρ u%i u% j + pδ ij ) ∂ (τ ij − τ ij + Dij )
+
=
∂t
∂x j
∂x j
(2)
%
∂ρ E% ∂  ( ρ E + p )u%i 
∂
+
=
( qi + u% jτ ij − Qisgs − H isgs + σ ijsgs )
∂t
∂xi
∂xi
(3)

∂ρ f% ∂( ρ u%i f% )
∂  % ∂f%
+
=
+ Φisgs 
ρD
∂t
∂xi
∂xi 
∂xi

(4)
The unclosed sub-grid scale terms are defined
τ ij sgs = ρ ( ui u j − u%i u% j )
(5)
Dij sgs = (τ ij − τ%ijj )
(6)
Qi sgs = ( qij − q%ijj )
(7)
(
H i sgs = ρ ( Eui − Eu%i ) + pu i − pu%i
(
)
(8)
σ i sgs = u jτ ij − u% jτ%ijj
)
(9)
(
)
(10)
Φ i sgs = ρ ui f − u%i f%
τ ij sgs , Dij sgs , Qi sgs , H i sgs , σ i sgs and Φ i sgs are the sgs stress, nonlinearity of viscous stress term, heat flux,
energy flux, viscous work and conserved scalar flux, respectively. The static Smagorinsky model is employed to
close those sgs terms.
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B. General-Fluid Thermodynamics and Transport
The SRK equation of state and Chung’s model are applied to obtain appropriate thermophysical properties for a
wide variety of pressure and temperature. The SRK equation of state is one of the real gas equation of state which
are defined as
p=
ρ Ru T
( M w − bρ )
−
ρ2
aα
M w ( M w + bρ )
(11)
where
a=
0.42748 Ru2Tc2
pc
b=
(
0.08664 Ru Tc
pc
)(
)
α = 1 + 0.48508 + 1.55171ω − 0.15613ω 2 1 − Tr0.5 

2
Ru , Vm , pc , Tc , Tr , a and b are the universal gas constant, molar volume, critical pressure, critical
temperature, reduced temperature, attractive parameter and repulsive term, respectively.
The acentric factor, ω was developed as a measure of the structural difference between the molecule and a
spherically symmetric gas for which the force-distance relation is uniform around the molecule. Transport properties
such as viscosity and thermal conductivity are predicted by the Chung’s model. Mass diffusivity is evaluated using
the Fuller’s theorem with Takahashi method.
C. Numerical scheme
The governing equations are numerically solved by means of a finite volume method. This method allows for the
treatment of arbitrary geometry. The spatial discretization employs a fourth order, central differencing scheme in
generalized coordinates. The temporal discretization is obtained by second order backward differencing scheme
using a fourth order Runge-Kutta scheme for integration of the real time term. In regions of low Mach number flows,
the energy and momentum equations are practically decoupled and the system of conservation equations becomes
stiff. Pressure decomposition and preconditioning techniques with dual time stepping are applied to circumvent the
round-off errors associated with the calculation of the pressure gradient and the convergence difficulties for the low
Mach number flows in the momentum equation. First, a rescaled pressure term is used in the momentum equation to
circumvent the singular behavior of pressure at low Mach numbers. Second, a preconditioning technique with dual
time-stepping integration procedure is established. A unified treatment of general fluid thermodynamics is
incorporated into a preconditioning scheme.18 The pseudo-time derivative may be chosen to optimize the
convergence of the inner iterations through the use of an appropriate preconditioning matrix that is tuned to rescale
the eigenvalues to render the same order of magnitude to maximize convergence. To unify the conserved flux
variables, a pseudo-time derivative of the form Γ∂Z / ∂τ can be added to the conservation equation. Since the
pseudo-time derivative term disappears as the solution is converged, a certain amount of liberty can be taken in
choosing the variable Z. We take advantage of this by introducing a pressure p ′ as the pseudo-time derivative term
in the continuity equation. The code is paralleled using an MPI library for more effective calculation.
III. Model Description
Figure 1 shows a coaxial swirl injector geometry and the swirl numbers of a liquid oxygen and a kerosene are
1.598 and 6.548 respectively. The injector includes three major parts: a tangential inlet, a vortex chamber and a
discharge nozzle. The kerosene (outer) and oxygen (inner) injects into the swirl injector through the tangential
passage. The radial and azimuthal velocities are determined from the tangential inlet port angle and these velocity
components are factor to set the swirl strength and mass flow rate, respectively.
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LOx
Fuel
dt,f or dt,o
Lr
dn,o
So =
4d s , o d n , o
π nn ,o d t , o
2
= 1.598 ,
doo
Sf =
dn,f
4 ( d n , f − d o ,o ) d s , f
π n f dt , f 2
ds,f or ds,o
= 6.548
Figure 1. Injector geometry and swirl number
Table 1. Operation conditions
Injector
Chamber Pressure
Fuel
Oxidizer
Fuel
Oxidizer
Mass Flow Rate
Coaxial Swirl
5.25, 8.0, 10.0 MPa
Kerosene; 350 K
Liquid Oxygen; 103 K
0.232 kg/s
0.084 kg/s
Table 2. Components of surrogate models for kerosene
Species(mole,%)
n-dodecene
n-decane
n-dodecane
n-tetradecane
n-hexadecane
Isooctane
Methylcyclohexane
Meta-xylene
Butylbenzene
1-methylnaphthalene
Model-1
100
·
·
·
·
·
·
·
·
·
Kerosene
Model-2
·
32.6
34.7
·
·
·
16.7
·
16
·
Model-3
·
16.4
20.2
14.2
10.3
6.8
7.9
7.3
5.8
11.0
Operation conditions are listed in Table 1. The surrogate models of JP-8/Jet-A are applied to kerosene mixing
simulation using the mixing and combining rules (see Table 2). For model 1, n-dodecene, which is used to replace
the kerosene in a conventional view point as modeling simplicity, is considered as baseline property of kerosene
while models 2 and 3 are surrogate kerosene applied for this study. The model 2 (Montgomery, 2002) was
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considered to check if the predictions using a smaller number of components are suitable. The determination of
model 3 based on the similarity of volatility and the phase change behavior of JP-8/Jet-A and the well-known
components for the detailed thermodynamic characteristics (Tucker, 2005).
Figure 2. Computational domain
Because of the enormous computational time for simulating of full three-dimensional region using the large eddy
simulation, quasi three dimensional flow fields with periodic boundaries is treated herein. The tangential inlets
which are circular ports connected to the injector are approximated with a thin slit. Figure 2 shows a computational
domain which is divided into 45 blocks, with each calculated on a single processor of a distributed computing
facility. The total grid points are about 765,000 nodes. The disturbances are generated by a Gaussian randomnumber generator with an intensity of 5% of the mean quantity.
IV. Results and Discussion
A. Flow structures
Figure 3. Flow structure of coaxial swirl injector
The swirl injector flow structures computed in this study are presented in Figure 3. Liquid oxygen and kerosene
are introduced into the injector through slots. Since the inlet ports consist of the tangential slit, the strong radial
pressure gradient is formed by centrifugal force. Because the chamber pressure is higher than the critical pressure,
there is no obvious boundary between the injected fluid and the ambience. Near the LOx post, a slowly swirling
gaseous region forms. The adverse pressure gradient in this region gives rise to recirculation flow and finally
produces a central toroidal recirculation zone (CTRZ), resulting from vortex breakdown, which play an important
role in determining the flame stabilization characteristics. The CTRZ accelerates the earlier burst of vortices near the
fuel/oxygen mixing zone. In the CTRZ, these vortical structures move downstream and upstream, and are rapidly
dissipated by the turbulent diffusion and viscous damping.
B. Effect of chamber pressure
Figure 4 represents the instantaneous density contours while Figure 5 shows the mean density contours. The
chamber pressures were selected as 5.25 MPa, 8 MPa and 10 MPa for transcritical and supercritical calculation
because the critical pressures of oxygen and kerosene are 5.04 MPa and 1.93 MPa, respectively. The temperature
was fixed at 103 K and 350 K for LOx and kerosene, respectively. Similar structures of the liquid surface are
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observed as pressure increased from transcritical to supercritical conditions. As shown in the figures, the higher
chamber pressure is, the faster fluid density decreases during the mixing and atomization process. The region of
phase-change is widely formed because of the supercritical effect.
(a)
(b)
(c)
Figure 4. Instantaneous density contours at different chamber pressures;
(a) 5.25 MPa, (b) 8 MPa, (c) 10 MPa
(a)
(b)
(c)
Figure 5. Mean density contours at different chamber pressures;
(a) 5.25 MPa, (b) 8 MPa, (c) 10 MPa
Figure 6. Radial distributions of mean density components at x=25t downstream from the LOx post tip
Figure 6 represents density distribution at x = 25t downstream from the LOx post tip. The variable, t, means the
LOx post thickness; t = 0.6 mm. At the supercritical condition, taking things by and large, density distribution is
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higher than that at the transcritical environment. Also, the region of phase-change is widely formed as the chamber
pressure increase. The phase change at supercritical conditions differs significantly from the process at transcritical
conditions. The process goes through “liquid core → vaporization” at supercritical pressure, whereas it undergoes
“liquid core → break up → atomization → vaporization” at the lower pressure. During the process at supercritical
conditions, the properties of propellants vary continuously and the distinction between the liquid and gas phase is
ambiguous.
Figure 7. Mixing efficiency at different chamber pressures; 5.25 MPa, 8 MPa, 10 MPa
To quantify the level of mixing, mixing efficiency is considered using the following formula:
η mix
∫ ρY
( x) =
∫ ρY
Fuel
uα dA
Fuel
udA
1

α =
1
 φ /φ
global
 local
( φlocal ≤ φglobal )
( φlocal > φglobal )
(12)
Where ρ and YFuel are density and mass fraction of kerosene, respectively. u is the velocity component normal to
the area, dA . φlocal and φ
global
are local and global equivalence ratio, respectively. The parameter α is decided by
local equivalence ratio. This parameter means that if the local equivalence ratio is equal or less than global
equivalence ratio, the mixing is regarded as completed. On the other hand, local efficiency is less than 1 when local
equivalence ratio is large. Therefore, the mixing efficiency represents how many fuel spread into combustion
chamber, in other words, the efficiency becomes 1 when local equivalence ratio is equal to global equivalence ratio
through whole cross sectional area. Figure 7 shows mixing efficiencies on the time averaged fields. The efficiency
increases as moves to downstream after the LOx post where the mixing starts. As shown in Figure 7, the higher
chamber pressure is, the earlier the mixing efficiency reaches 1, meaning that most regions become lean burn
environment because the mixing completely takes place not much far from the LOx injector post.
(a)
(b)
(c)
Figure 8. Instantaneous Z-vorticity contours at different chamber pressures;
(a) 5.25 MPa, (b) 8 MPa, (c) 10 MPa
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Figure 8 shows the instantaneous z-vorticity magnitude. As the pressure increases, the vortical structures
generated in the shear layer rapidly break up into small scale eddies, result from the local vaporization at
supercritical environment. As a result, the toroidal recirculation zone downstream of the injector post becomes
downsized.
Figure 9. Probe location
(a)
(b)
Figure 10. Power spectral densities of pressure fluctuations; (a) 5.25 MPa, (b) 10 MPa
Figures 9 shows the locations of pressure measurement: inner side of injector (probe 1, 2), near the LOx post
(probe 3), phase-change region (probe 4), mixing layer (probe 5), front of the CTRZ (probe 6). Figure 10 represents
the power spectral densities of the pressure fluctuations at the locations. The Fast Fourier Transform (FFT)
technique was implemented for the spectral analysis. Taken as a whole, the two frequencies of 1.25-1.3 kHz and 3.33.35 kHz are dominantly estimated inside the oxygen injector filled with high density fluid while only 3.3-3.35 kHz
is observed at other locations. At the transcritical condition of 5.25 MPa, a pressure oscillation at 1.3 kHz is
observed inside the oxygen injector while 3.3 kHz is estimated both outside and inside the kerosene injector. The
oscillation at 3.3 kHz is same as a cycle of the vortex shedding at the oxygen injector exit. Also, high amplitude of
the pressure oscillation at 3.3 kHz is observed in kerosene injector due to the Doppler effects. At the supercritical
condition of 10 MPa, the dominant frequency increases as compared with those of the transcritical condition. The
increase in frequency might be caused by the faster wave propagation speed at supercritical condition and it
expedites the energy diffusivity.
Figure 11 represents the temporal evolution of pressure and temperature field over one cycle of the dominant
pressure oscillation in oxygen injector. The initial turbulence flow at the tangential inlet generates a succession of
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wavy structures and is convected downstream through the dense liquid oxygen. In a swirl injector, those waves are
generated by the presence of oscillations in the combustion chamber or propellant feed-line. As the propellants are
discharged into the chamber, sinuous Kelvin-Helmholtz waves develop along the outer wall of the injector. As these
waves move downstream, a hairpin vortice is generated and grows up. The oscillation at 3.3-3.35 kHz is same as a
cycle of the vortex shedding caused by the hairpin vortices at the LOx post.
θ=0˚
θ = 90 ˚
θ = 180 ˚
θ = 270 ˚
(a) Pressure
(b) Temperature
Figure 11. Temporal evolution of pressure and temperature field over one cycle of the dominant pressure
oscillation in the oxygen injector; 10 MPa
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C. Effect of kerosene model
Among the propellants, kerosene fuels are quite unwieldy in theoretical and numerical analysis because of a
large number of hydrocarbons, reactions and thermophysical parameters. Therefore, kerosene surrogate models are
considered for the abbreviation of real fuel components. The surrogate models must have very similar characteristics
for their chemical kinetic behavior and thermochemical property data, such as flash and freeze point, critical
pressure and temperature.
The single-species model and surrogate models based on the number of species will have a significant effect on
the computation time. To analyze the effect of the computing load and the difference between variant kerosene
models, the thermodynamic properties and mixing characteristics of single-species model and surrogate models are
compared. Thermodynamic properties for each kerosene model are listed in Table 3. Figure 12 shows that surrogate
models 1 and 3 undergo a phase change in a relatively narrow range as compared with model 2. Also, in case of
model 2, it is observed that the length scale of the CTRZ is larger than that of other cases. The overall trends of
property distributions are similar in the results of all three models. However, as for the combustion condition, the
mixing characteristics may be significantly different because of the faster vaporization and the variant reaction
mechanisms.
Table 3. Thermodynamic properties of the kerosene models
Properties
Model-1
Density
(kg/m3)
Oxidizer
Viscosity
Oxidizer
(N·s/m2)
Fuel
Fuel
Model-2
Model-3
1086.6
605.2
603.9 (0.21%)
604.2 (0.17%)
1.41×10-4
2.29×10-4
2.18×10-4 (4.8%)
2.21×10-4 (3.5%)
(a)
(b)
(c)
Figure 12. Density contours for the kerosene models; (a) Model 1, (b) Model 2, (c) Model 3
V. Conclusions
The turbulent mixing of a kerosene/liquid oxygen coaxial swirl injector under supercritical pressure has been
numerically investigated. The effect of variant operating pressure as 5.25, 8, 10 MPa and kerosene models which are
based on a single-species and the number of species on thermodynamic properties and mixing dynamics inside an
injector and a combustion chamber are investigated. Power spectral densities of pressure fluctuations in the injector
under variant chamber pressures are analyzed. Taken as a whole, the two frequencies of 1.25-1.3 kHz and 3.3-3.35
kHz are dominant inside the oxygen injector filled with high density fluid while 3.3-3.35 kHz is observed in other
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locations. In comparison, high frequency is dominantly estimated inside the fuel injector since the sound speed of
kerosene fluid is faster than that of liquid oxygen. At the supercritical condition, the high frequency increased by 50
Hz and the low frequency decreased by 50 Hz as compared with those of the transcritical condition. The overall
trends of property distributions are similar for all three kerosene surrogate models. However, as for the combustion
condition, the mixing characteristics will be significantly different because of the faster vaporization and variant
reaction mechanisms.
Acknowledgement
This research was supported by National Space Laboratory (NSL) Program (No. 2008-2006287) through the
National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology.
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JP-8 Combustion”, AIAA Paper No. 2002-0336, 40th AIAA Aerospace Sciences Meeting and Exhibit, Reno,
Nevada, 14-17 January, 2002.
16
Dagaut, P., “On the Kinetics of Hydrocarbons Oxidation from Natural Gas to Kerosene and Diesel Fuel”,
Physical Chemistry Chemical Physics, Vol. 4, Issue 11, pp. 2079-2094, 2002.
12
American Institute of Aeronautics and Astronautics
17
Huang, H., Sobel, D. R., and Spadaccini, L. J., “Endothermic Heat-Sink of Hydrocarbon Fuels for Scramjet
Cooling”, AIAA Paper No. 2002-3871, 38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit,
Indianapolis, Indiana, 7-10 July, 2002.
18
Meng, H. and Yang, V., “A Unified Treatment of General Fluid Thermodynamics and Its Application to a
Preconditioning Scheme”, Journal of Computational Physics, Vol. 189, Issue 1, pp. 277-304, 2003.
13
American Institute of Aeronautics and Astronautics