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Phase separation analysis in supercritical injection using large-eddy-simulation and vapor-liquid-equilibrium

Authors:
Phase separation analysis in supercritical injection
using large-eddy simulation and
vapor-liquid equilibrium
Daniel T. Banuti
, Peter C. Ma
, and Matthias Ihme
Stanford University, Stanford CA 94305, USA
Injection at pressures exceeding the propellant critical pressures is typically considered
a diffuse interface mixing process rather than a sharp interface break-up. However, this is
not necessarily the case in mixtures where the local mixture critical pressure may exceed
the value of the pure components. So far, there is no canonical theoretical or compu-
tational model to analyze local phase separation under these conditions. In the present
paper, we propose to separate the problem into two aspects: determination of local mix-
ture temperature and composition, and analysis of the local thermochemical state. We
calculate transport of mass, momentum, energy, and species using a large-eddy simulation
(LES) method to obtain an accurate local state. This local state is then assessed via a
vapor-liquid equilibrium solution using the Peng-Robinson equation of state. We apply
this methodology to three technically relevant mixing problems at propellant supercritical
pressures: inert nitrogen/n-dodecane injection, an inert liquid oxygen/gaseous hydrogen
shear layer, and a reacting liquid oxygen/gaseous hydrogen shear layer. The last case
represents the first phase analysis of a reacting case; we show that it can be reduced to
the binary mixing of oxygen and water. Counterintuitively, the reacting LOX/GH2 shear
layer is more susceptible to phase separation than the inert mixing case, despite the high
temperatures reached in the flame. Finally, we compare the mixture critical loci obtained
from the canonical computational fluid dynamics mixing rules with results obtained from
vapor liquid equilibrium calculations, and show that both are fundamentally, qualitatively
different.
I. Introduction
The pursuit of improving engine performance by increasing the combustion pressure has turned trans-
critical injection into the dominant technology for liquid propellant main stage engines,1,2Diesel engines,3
and jet engines during take-off.4Figure 1 illustrates the associated view in a pure fluid p-Tdiagram. The in-
jected fluid undergoes a heating process as it adapts to chamber conditions; the molecular structure changes
from liquid to gaseous.57At subcritical pressures, this process intersects the coexistence line; ligaments
and droplets are formed, separated from the vapor phase by a sharp interface. Beyond the critical point, the
injection process is smooth, sharp interfaces are replaced by a diffuse mixing layer.1,8
The picture becomes more complicated when mixtures are concerned, making it relevant for combus-
tion systems. A mixture may exhibit phase separation at pressures exceeding the critical pressures of all
involved species.9Figure 2 from Mayer at al.10 shows how pure nitrogen injection changes from surface-
tension-dominated to a diffuse-mixing process, as the critical pressure is reached and exceeded. However,
upon changing the chamber composition into an equimolar nitrogen-helium mixture, the nitrogen jet changes
back to a surface-tension-dominated break-up mode even at supercritical pressures with respect to the pure
components. Similar phenomena have been recently observed when injecting n-dodecane into a high tem-
perature and pressure nitrogen environment:3,11 depending on the exact supercritical chamber conditions,
stable droplets with sharp interfaces may be found, or not.
Postdoctoral Research Fellow, Department of Mechanical Engineering, Center for Turbulence Research.
Graduate Research Assistant, Department of Mechanical Engineering.
Assistant Professor, Department of Mechanical Engineering.
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1Tr
pr
1
2
3
vapor
transitional
ideal gas
solid
liquid
Figure 1. General state diagram in terms of reduced pressure pr=p/pcand reduced temperature Tr=T /Tc.
The arrow illustrates an injection process at supercritical pressure.
840
MAYER ETAL.
Fig. 9 S ub sca le LN
2
inje ctio n int o a pres surized chamber. Horizon tal ro ws co rrespon d to dista nce from the injector tip. Vertical columns
corre spond to different experimental co nd itio ns as follows: a)Into su bcritica l N
2
at 2.8 MPa, p/p
crit
=0.83; b)into near crit ical N
2
at 3.5
MPa, p/p
crit
= 1 .03 ; c )into sup ercrit ica l N
2
at 6.9 MPa, p/p
crit
=2.03; and d)into a N
2
/O
2
= 3.9 m ixture at 6.9 MPa, p/p
crit
=2.03.
Fig. 10 LN
2
jets inje cted into He at el eva ted pre ssures.
Fig. 11 Image s equ enc e of LN
2
jets in jected into He at 5.5 MPa revea ls o scillat ion b etw een liq uid -like and ga s-like b eha vio r.
droplet Reynolds numbers,Web er numbers,and liqu id-to-gas
density ratios.
9
Single droplets were vertically injected in to the
potentialcore re gio n of ahorizontal 300 -K dry airjet in apres-
sure vessel capable of sustainingpressuresup to 12.0 MPa.
This allowed for the inves tigation of droplet behaviorat lower
density ratios than has previously ever been examined. Flow
and turbulence ch aracteristics of the je t were d ete rm ined by
two-component laser Doppler anemometry, and the droplets
were visualizedusing rapid video imagin g and numerical im-
age analysis.
LOX droplets with diametersbetween 600 mmand 1.0 mm
were studied over apressure range of 0.14.0 M Pa for jet
velocities ranging from 0.4 to 86 m/s. The corresponding max-
imum values of the droplet Weber number (We = ,
2
urD/s)
rel g
Reynoldsnumber (Re =r
g
u
rel
D/m
g
),and Ohnesorgenumber
[Oh =m
g
/(r
g
Ds)
1/2
]were 800, 7500, and 0.01, respectively.
The liquid-to-gas density ratio varied between 20 and 1040.
The surface tension changed signiécantly over this pressure
range, decreasing from 13.6 3102
3
N /m at 0.1 MPa to 4.97
3102
3
N/ m at 3 .0 MPa. The enthalpy of vaporization also
decreased signiécantly as pressure increased, which reduced
global gasiécation times. Four droplet-jet interaction regimes
are illustrated in Fig. 12 corresponding to deformation, bag,
transitional, and shear breakup regimes.
To better isolate the effects of vaporization and reduced sur-
face tensio n from the eff ec t of changingdensity ratio, ethanol
droplets were also stud ied (see Tab le 1).The ethanol droplets
vaporized much more slowly than the LOX droplets,and the
surface tension changed by only about 10%, from 22.8 3102
3
at 0.1 MPa to 20.0 3102
3
at 3.0 MPa. In the ethanol exper-
iments, the pressure was varied from 0.1 to 5.0 MPa, and the
corresponding liquid-to-gas density ratio varied between 16
and 800. The maximum droplet Web e r numbe r was 90, and
the maximum droplet Reynolds number was 1.6 310
4
.The
droplet Ohnesorgenumber was less than 0.02 for all cases.
The diametersof ethanol droplets were kept at 600 mm.
Acomparison between the results obtained for the ethanol
and LOX experiments is presentedin Fig. 13 for apressure of
3.0 MPa. Trans ition droplet Reyn olds numbers are observed
to be much higher for the ethanol droplets.For example,the
shear breakup regime was attained at adroplet Reynoldsnum-
ber of 4310
3
for LOX droplets but at a d ro ple t Reynolds
number of 1.2 310
4
for the ethanol droplets. Because of the
reduced surface tension of the LOX, the same droplet Reyn-
Downloaded by DLR DEUTSCHES ZENTRUM F. on May 5, 2014 | http://arc.aiaa.org | DOI: 10.2514/2.5348
Figure 2. Injection of cryogenic nitrogen into nitrogen environment at a) 2.8 MPa (pr= 0.83); b) 3.5 MPa
(pr= 1.03); c) 6.9 MPa (pr= 2.03). d) injection into mixture of N2and He at 6.9 MPa. From Mayer et al.10
Different theoretical and numerical approaches have been used to explain such phenomena, which may
be differentiated by their dimensionality. First, 0D approaches for inert mixing were employed e.g. by
Mayer et al.,12 Kuo,13 Yang et al.,14 Oschwald et al.,2and Dahms and Oefelein.11 Assuming an adia-
batic inert mixing process, an equilibrium temperature for a given mixture fraction can be determined.
Then, calculation of vapor-liquid-equilibrium (VLE) properties are carried out. Qiu and Reitz15 evaluated
the change in temperature upon phase change. Dahms and Oefelein16 extended the analysis by introduc-
ing a Knudsen number based evaluation of the interfacial thickness. Second, 1D approaches additionally
account for laminar heat and mass transport normal to a droplet interface, e.g. Harstad and Bellan.17
Sirignano and Delplanque18 showed that transcritical droplets may undergo a transient, in which an initially
sharp interface diffuses when heat transfer into the droplet renders it supercritical after some time. La-
caze and Oefelein19 studied the shift of the mixture-critical point in a counterflow LOX-GH2 diffusion flame.
For the operation condition investigated (p= 7 MPa, Tin,LOX = 120 K, Tin,H2 = 295 K), they showed that the
mixture critical pressure in the reaction zone exceeds the chamber pressure. However, as this occurs far away
from the coexistence line, they concluded that a two-phase flow does not occur. Lacaze and Oefelein pointed
out that water diffusion towards the LOX core may have a strong effect and thus needs to receive more
attention. Banuti et al.20 extended this analysis of flamelet solutions for the same chamber pressure, and
showed that the mixture remains single phase for a range of Damk¨ohler numbers spanning near-equilibrium
and near-quenching cases. Third, multidimensional approaches, such as computational fluid dynamics, may
additionally account for turbulent transport. Oefelein and Yang21 carried out large eddy simulations (LES)
of a LOX/GH2 shear layer and found that combustion takes place under ideal gas conditions despite the
supercritical chamber pressure. Analyzing the same data, Banuti et al.22,23 additionally pointed out that
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the LOX break-up process from dense to gaseous occurs essentially under pure fluid conditions, thus limiting
the influence of real gas mixing rules. Qiu and Reitz15 carried out Reynolds averaged Navier-Stokes (RANS)
simulations of Lagrangian supercritical droplet injection and evaluated the phase behavior. Matheis and
Hickel24 used a combined VLE-LES model to study inert mixing of dodecane and nitrogen. For most of
the multidimensional numerical simulations carried out in the literature so far,19,22,2430 a diffuse interface
method is used where surface tension effects are neglected.
We see that while sophisticated models exist for heat and mass transfer (e.g. LES) and thermochemical
behavior (e.g. VLE), a combined application has so far been limited. However, as VLE assumes equilibrium
thermodynamic conditions, it is important to determine these mixing states accurately with an appropriate
transport method. Adiabatic mixing, i.e. a synchronous transport of mass and heat, is strictly only valid in
the unity Lewis number limit. In the present paper we will address this by evaluating LES mixture properties
with VLE for inert liquid oxygen/hydrogen, reactive liquid oxygen/hydrogen, and inert dodecane/nitrogen.
In this way, we seek to identify possible phase separation in transcritical injection. The mathematical
formulation including the equation of state, the development of a phase equilibrium solver, and the numerical
details of the flow solver will be introduced in Section II.InSection III, the VLE solver will be validated and
the phase separation behavior of the considered mixtures will be analyzed. Finally, LES results are analyzed
for the study of phase separation behavior under transcritical conditions.
II. Methods
A. Flow solver
The massively paralleled, finite-volume solver CharLES x, developed at the Center for Turbulence Research
is used in this study. The numerical solver and the corresponding numerical methods are discussed in
detail elsewhere,26,31,32 only a brief overview will be given here. The governing equations solved are the
conservation of mass, momentum, energy, and species. The PR EoS, Eq. (1), is used to close the system.
Details on how to evaluate thermodynamic quantities can be found in Ma et al.31 The dynamic viscosity and
thermal conductivity are evaluated using Chung’s method with high-pressure correction.33,34 Takahashi’s
high-pressure correction35 is used to evaluate binary diffusion coefficients. A diffuse interface method is
used and no surface tension effects are considered. A strong stability preserving 3rd-order Runge-Kutta
(SSP-RK3) scheme36 is used for time advancement.
The convective flux is discretized using a sensor-based hybrid scheme in which a high-order, non-
dissipative scheme is combined with a low-order, dissipative scheme to minimize the numerical dissipation.37
A central scheme which is fourth-order on uniform meshes is used along with a second-order ENO scheme
for the hybrid scheme and a density sensor26,31 is adopted in this study. An entropy-stable flux correction
technique31 ensures the physical realizability of the numerical solutions including the positivity of scalars
dampens the non-linear instabilities in the numerical solutions.
To remedy the spurious pressure oscillations generated by a fully conservative scheme,31,38 a double-
flux method31,39,40 is extended to the transcritical regime, specifically designed for the strong non-linearity
inherent in the real fluid EoS. A Strang-splitting scheme41 is applied in this study to separate the convection
operator from the remaining operators of the system.
For reacting cases, a dedicated transcritical flamelet/progress variable approach42,43 is adopted. Specifi-
cally, parameters needed for the cubic EoS are pre-tabulated for the evaluation of departure functions and
a quadratic expression is used to recover the attraction parameter. This approach is able to account for
pressure and temperature variations from the reference tabulated values using computationally tractable
pre-tabulated combustion chemistry in a thermodynamically consistent fashion.
B. Equation of state
The Peng-Robinson (PR) equation of state (EoS)9,44 is used in this study for both the phase separation
calculations (VLE)and the evaluation of thermodynamic quantities (CFD) due to its reasonable accuracy,
computational efficiency, and prevailing usage. It can be written as
p=RT
vb
v2+ 2bv b2,(1)
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where Ris the gas constant, v= 1is the specific volume, and the parameters and bare dependent
on temperature and composition to account for effects of intermolecular forces. The parameters aand bare
evaluated as
a= 0.457236R2T2
c
pc
,(2a)
b= 0.077796RTc
pc
,(2b)
α="1 + c 1rT
Tc!#2
,(2c)
c= 0.37464 + 1.54226ω0.26992ω2,(2d)
where Tcand pcare the critical temperature and pressure, and ωis the acentric factor. For mixtures,
classical van der Waals mixing rules are applied
=X
iX
j
XiXjaij αij ,(3a)
b=X
i
Xibi,(3b)
and
aij αij = (1 kij )aiajαiαj,(4)
where Xiis the mole fraction of component iand kij is referred to as a binary interaction parameter between
components iand j. With these mixing rules, the mixture is treated as a virtual pure fluid.22,45,46
C. Vapor-liquid equilibrium solver
The criteria for vapor-liquid equilibrium (VLE) are9,47
pV=pL,(5a)
TV=TL,(5b)
GV
i=GL
i,(5c)
where Giis the partial Gibbs energy of component i, and the superscripts Vand Lrefer to the vapor and
liquid phases, respectively. The criterion on the Gibbs energy can also be expressed in terms of the fugacities
of the components i,
fV
i=fL
i.(6)
The PR EoS, Eq. (1), is used for the calculation of the fugacity. Specifically, for a binary mixture, the
fugacity for component iin both the vapor and the liquid phases, can be calculated as
ln fV
i
yip=Bi
BV(ZV1) ln(ZVBV)
AV
BV8ln "ZV+ (1 + 2)BV
ZV+ (1 2)BV#2(y1Ai1+y2Ai2)
AVBi
BV,
(7a)
ln fL
i
xip=Bi
BL(ZL1) ln(ZLBL)
AL
BL8ln "ZL+ (1 + 2)BL
ZL+ (1 2)BL#2(x1Ai1+x2Ai2)
ALBi
BL,
(7b)
where
Aij =aij αij p
R2T2,(8a)
A=aαp
R2T2,(8b)
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and
Bi=bip
RT ,(9a)
B=bp
RT ,(9b)
are the normalized parameters for the PR EoS, Zis the compressibility factor, and xiand yiare the mole
fraction of component iin the vapor and liquid phase, respectively. An iterative process is typically utilized
for the VLE calculations. In this study, a short-cut estimation based on Raoult’s law 47 is used as the
starting point for the iterative process, which can be expressed as
log psat
r=7
3(1 + ω)11
Tr,(10)
where psat
ris the reduced saturation pressure and Tris the reduced temperature.
Species W[kg/kmol] Tc[K] pc[MPa] ρc[kg/m3]Zcω kij
C2H630.07 305.33 4.87 207.0 0.2788 0.0993 0.019
n-C7H16 100.21 540.13 2.74 232.0 0.2632 0.349 -
Table 1. Thermodynamic properties of ethane and n-heptane.
Mole fraction of ethane
0 0.2 0.4 0.6 0.8 1
P [MPa]
0
1
2
3
4
5
6
7
8
9
10 150 F
200 F
250 F
300 F
350 F
T [K]
250 300 350 400 450 500 550 600
P [MPa]
0
1
2
3
4
5
6
7
8
9
10
C2H6
n-C7H16
0.27
0.59
0.77
0.89
0.97
Figure 3. Vapor-liquid equilibrium calculations of pressure-composition diagram at five temperatures (left)
and pressure-temperature diagram at five ethane mole fractions (right) for binary mixtures of ethane and
n-heptane in comparison with experimental data.48,49 Saturation lines for the two pure species are also shown
in black. Black dots are the critical points for pure species.
The VLE solver is first validated against experimental data of a binary mixture of ethane and n-heptane.
The respective thermodynamic properties are shown in Table 1, with the binary interaction parameter from
Nishiumi et al.50 Results are shown in Fig. 3 as pressure-composition diagram at five temperatures on the
left and pressure-temperature diagram at five ethane mole fractions on the right. The experimental data
for the pressure-composition diagram are from Mehra and Thodos,48 the data for the pressure-temperature
diagram are from Kay.49 Dew lines are plotted as solid lines, and bubble lines as dashed lines. For the
five temperature considered (150-350 F or 338.7-449.8 K), pure ethane is always in a single supercritical
phase. As the n-heptane component also becomes supercritical with rising pressure, the mixture becomes a
supercritical dense fluid. Comparing the calculations with the experimental measurements, it can be seen
that the VLE solver successfully captures the phase equilibrium behavior of the two hydrocarbons.
For the pressure-temperature diagram (right figure in Fig. 3), five cases with different mole fractions of
ethane (0.27, 0.59, 0.77, 0.89, and 0.97) are considered. The saturation lines for the two pure species are
calculated from PR EoS and are plotted for reference. In comparison with the experimental data, good
agreement for all five conditions is observed. The critical pressures of some mixture states exceed the critical
pressures of the two pure species.
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T [K]
300 350 400 450 500 550
P [MPa]
3
4
5
6
7
8
9
10
Calculation
Kay1938
Mehra&Thodos1965
NIST
Mole fraction of n-heptane
0 0.2 0.4 0.6 0.8 1
P [MPa]
3
4
5
6
7
8
9
10 Calculation
Kay1938
Mehra&Thodos1965
NIST
Figure 4. Calculated critical point for ethane and n-heptane mixtures in comparison with experimental
data.48,49 Black dots are the critical points for pure species.
D. Mixture critical point calculation
We discuss two established methods to determine the mixture critical point, first a method derived from
vapor liquid equilibria, second the pseudocritical method upon which CFD mixing rules are based.
1. Helmholtz free energy
Using Taylor expansion on the Helmholtz free energy, Heidemann and Khalil51 derived the following criteria
for the determination of the critical point (CP) of a mixture
Qn= 0 ,(11a)
nTn= 1 ,(11b)
and the cubic form
C=X
iX
jX
k
ninjnk3A
∂ninjnkT,v
= 0 ,(12)
where nis the mole number vector of the mixture, Ais the Helmholtz free energy, and
Qij =2A
∂ninjT ,v
=RT lnfi
∂njT ,v
.(13)
The detailed formulation of Cand Qwith PR EoS can be found in Billingsley and Lam.52 To evaluate
the critical point of the mixture, nested iterations were used.51,53 Newton iteration at a fixed value of the
volume vis used to determine a temperature where Eq. (11) has a nontrivial solution. The elements of ∆n
are calculated, and evaluation of the cubic form Cis used to correct the volume vin an outer loop.
The CPs of the mixtures are calculated directly using the procedures introduced in Section II and the
results are plotted in Fig. 4. Experimental measurements by Mehra and Thodos48 and Kay49 agree well
with the calculations. It can be seen from Fig. 4 that while the critical pressure of the mixture exceeds the
values for the two pure species, the critical temperature is bounded between the values of pure species, and
increases monotonically with the n-heptane mole fraction.
The procedures for VLE and CP calculations were successfully applied to noble gas mixtures in our
previous studies.5Similar techniques were used by Qiu and Reitz15,54,55 for the analysis of inert hydrocarbon
injection. In this study, these procedures will be performed to study phase separation behavior for cases
representative for injection in Diesel, rocket, and jet engines.
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2. Pseudocritical properties
Pseudocritical properties characterize the critical point of a mixture. These pseudocritical properties are
an approximation and may not be the same as the true mixture critical properties.9One commonly used
mixing rule is due to Kay56
Tc,m =X
i
XiTc,i ,(14a)
pc,m =X
i
Xipc,i ,(14b)
where Tc,m and pc,m are the pseudocritical temperature and pressure of the mixture, respectively. Kay’s
rule is usually applied to hydrocarbons where the critical properties of all the components are not extremely
different. The accuracy for the pesudocritical pressure can be improved using the modification by Praus-
nitz and Gunn57
pc,m =Zc,mRTc,m
Vc,m
=(PiXiZc,i)R(PiXiTc,i )
PiXiVc,i
,(15)
where Zc,m and Vc,m are the pseudocritical compressibility and volume of the mixture. In the following the
mixing rules described by Eq. (14) are referred to as Mixing Rule 1, and the ones described by Eqs. (14a)
and (15) are referred to as Mixing Rule 2.
III. Results and Discussion
We use a joint evaluation of VLE and CFD results to assess the phase separation behavior for three
technically relevant cases. N-dodecane/nitrogen represents injection in Diesel engines and in gas turbines.
Due to the thermodynamic similarity between nitrogen and oxygen, it may be furthermore be indicative
for properties of inert hydrocarbon/oxygen mixing in liquid propellant rocket engines. Inert and reactive
gaseous hydrogen/liquid oxygen mixing is relevant for cryogenic liquid propellant rocket engines.
As all investigated combustion processes can be considered isobaric, we calculate the isobaric phase
equilibrium. The local thermodynamic state is computed from CFD accounting for heat and mass transfer
and compared to the VLE results.
A. n-dodecane and nitrogen mixtures
Nitrogen and n-dodecane mixtures are considered initially. This combination is representative for hydrocar-
bon injection into air, relevant for gas turbines and Diesel engines.11,58 The thermodynamic properties of
the two species are compiled in Table 2. The binary interaction parameter is from Garcia-Cordova et al.59
Species W[kg/kmol] Tc[K] pc[MPa] ρc[kg/m3]Zcω kij
N228.0 126.19 3.40 313.3 0.289 0.0372 0.1561
n-C12H26 170.3 658.1 1.82 226.5 0.2497 0.5764 -
Table 2. Thermodynamic properties of nitrogen and n-dodecane.
Figure 5 shows the results of VLE calculations for nitrogen and n-dodecane mixtures, plotted as pressure-
composition diagrams at seven different temperatures ranging from 344.4 to 593.5 K. The experimental data
by Garcia-Cordova et al.59 is plotted for comparison. The PR EoS is able to predict the trend of phase
separation behavior for nitrogen/n-dodecane mixtures reasonably well as can be seen in Fig. 5. At low
temperatures, a large two-phase region can be seen for the mixture with pressure extending up to more than
60 MPa. The vapor phase consists of nearly pure nitrogen and the liquid phase contains mainly n-dodecane,
which indicates the low solubility of nitrogen in n-dodecane. As the temperature increases, the critical
pressures of the mixture decreases, and supercritical behavior of the mixture can be observed at relatively
low pressures.
Mixture CPs are also calculated directly, a comparison with experimental data59 is shown in Fig. 6. It can
be seen that the CP properties are predicted reasonably well with overpredictions of the critical temperatures.
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Mole fraction of nitrogen
0 0.2 0.4 0.6 0.8 1
P [MPa]
0
10
20
30
40
50
60 344.4 K
410.7 K
463.9 K
503.4 K
532.9 K
562.1 K
593.5 K
Figure 5. Vapor-liquid equilibrium calculations of pressure-composition diagram at seven temperatures for
binary mixtures of nitrogen and n-dodecane in comparison with experimental data.59
T [K]
100 200 300 400 500 600 700
P [MPa]
0
20
40
60
80
100
120
140
160
180
200 Calculation
Garcia-Cordova2011
NIST
Mole fraction of n-dodecane
0 0.2 0.4 0.6 0.8 1
P [MPa]
0
20
40
60
80
100
120
140
160
180
200 Calculation
Garcia-Cordova2011
NIST
Figure 6. Calculated critical point for nitrogen and n-dodecane mixtures in comparison with experimental
data.59 Black dots are the critical points for pure species. Binary interaction coefficient kij = 0.019 from50 .
The CP of the mixture starts from the CP of pure n-dodecane and increases rapidly in pressure and does not
end at the CP of pure nitrogen. This phase behavior indicates that the mixture of nitrogen and n-dodecane
belongs to Type III mixtures, according to the classification scheme of van Konynenburg and Scott.60 All
binary mixtures of nitrogen and hydrocarbons, except for methane, exhibit Type III phase behavior. For
Type III mixtures, two distinct critical curves are present, one starting from the CP of the component with
relatively higher critical temperature, and goes to infinite pressures; the other one starts at the CP of the
component with lower critical temperature and ends at the upper critical point intersecting with the three-
phase vapor-liquid-liquid coexistence line. This is in contrast to the Type I mixture, such as the ethane and
n-heptane mixture in the validation subsection, where one critical curve connects the CP of the two pure
species. Phase separation may happen for Type III mixtures even at very high pressures.
The determination of phase separation requires a detailed analysis based on the local pressure, temper-
ature and compositions. First, n-dodecane injection into a supercritical nitrogen environment is computed.
The operating conditions correspond to the ECN Spray A case.58 The injection temperature of the n-
dodecane jet is 363 K, the ambient nitrogen is at a temperature of 900 K and a pressure of 6 MPa. The
injection velocity is 100 m/s.
Details of the simulations can be found in Ma et al.31 LES calculations of the ECN Spray A case were
conducted in our previous study61 where a Vreman sub-grid model was used for the closure of turbulence.
The mixing trajectory in terms of X-Tis almost identical to the two-dimensional simulation results shown
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(a) Density field [kg/m3]
Nitrogen mole fraction
0 0.2 0.4 0.6 0.8 1
Temperature in K
300
400
500
600
700
800
900
1000
(b) Scattered data of composition and tempera-
ture
Figure 7. Injection of n-dodecane (360 K, 100 m/s) into nitrogen (900 K) at 6.0 MPa. Solid lines in the right
figure are phase boundaries from VLE; black dot is the critical point.
here, and is therefore omitted. The computational domain has a dimension of 30h×16h, where h= 1.0 mm
is the height of the jet. A uniform mesh in both directions is employed, which has a minimum spacing of
0.02hwith 50 grid points across the jet. The inlet condition of the jet is a plug flow with a top-hat velocity
profile. Periodic boundary conditions are applied at top and bottom boundaries, and an adiabatic no-slip
wall condition is prescribed at the left boundary. The pressure is specified on the outlet at the right boundary.
The CFL number is set to 0.8, no sub-grid scale model is applied. Due the fact that under the considered
conditions, the molecular transport phenomena cannot be fully resolved, the simulation performed can be
regarded as LES with an implicit sub-grid scale model due to numerical dissipation.
Figure 7 shows the results for the n-dodecane injection case. The instantaneous density field is presented
on the left, showing the jet break-up of the injected n-dodecane in the nitrogen environment. The phase
separation analysis is shown om the right. Scattered data represent the mixing trajectory calculated from
CFD, accounting for heat and mass transfer between the two mixture components. The results of the VLE
calculations are superimposed as solid lines, enclosing the multiphase region of the n-dodecane-nitrogen sys-
tem. The mixing trajectory passes closely outside of the multiphase region, with few individual points inside,
indicating that phase separation does not occur. The proximity of the curve suggests that minor changes
in boundary conditions (lower injection temperatures and chamber pressure) may cause phase separation.
This is consistent with experimental results of Manin et al.,3where these conditions are found to not exhibit
surface tension, but are nonetheless a limiting case towards phase separation at lower temperatures and
pressures.
B. Inert hydrogen and oxygen mixture
The phase separation behavior of hydrogen and oxygen mixtures are studied in this subsection. These
mixtures are relevant to liquid propellant rocket engines. Thermodynamic properties of hydrogen and oxygen
are listed in Table 3. The binary interaction parameter is assumed zero.
Species W[kg/kmol] Tc[K] pc[MPa] ρc[kg/m3]Zcω ki,j
H22.0 33.15 1.30 31.3 0.3033 -0.219 -
O232.0 154.6 5.04 436.1 0.2879 0.0222 -
Table 3. Thermodynamic properties of hydrogen, oxygen, and water.
The phase equilibrium behavior of oxygen-hydrogen mixtures has been studied extensively in the litera-
ture2,1214,62 and will not be reproduced here. Figure 8shows X-Tand p-Tdiagrams from Yang,62 revealing
the same Type III behavior identified for n-dodecane - nitrogen mixtures.
Computations are carried out for a two-dimensional mixing layer of liquid oxygen (LOX) and gaseous
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(a) X-Tdiagram for various pressures. (b) p-Tdiagram for various pressures.
Figure 8. Results of vapor liquid equilibrium calculations of inert hydrogen/oxygen mixing, from Yang62
(a) Density field [kg/m3]
Hydrogen mole fraction
0 0.2 0.4 0.6 0.8 1
Temperature in K
80
90
100
110
120
130
140
150
160
(b) Scattered data of composition and tempera-
ture
Figure 9. Inert shear layer, at the bottom LOX (100 K, 30 m/s), top is GH2 (150 K, 125 m/s) at 10 MPa
(pr= 2). Solid lines in the right figure are phase boundaries from VLE; black dot is the critical point.
hydrogen (GH2). This case was proposed by Ruiz et al.63 as a benchmark case to test numerical solvers for
high-Reynolds number turbulent flows with large density ratios. The LOX stream is injected at a temperature
of 100 K, and GH2 is injected at a temperature of 150 K. The pressure is set to 10 MPa. The operating
conditions are purely supercritical for pure hydrogen and the pressure is supercritical for oxygen. The GH2
and LOX jets have velocities of 125 m/s, and 30 m/s, respectively. Details of the simulations can be found in
Ma et al.31,32 The two streams are separated by the injector lip, which is also included in the computational
domain. A domain of 15h×10his used, where h= 0.5 mm is the height of the injector lip. The region of
interest extends from 0 to 10hin the axial direction with the origin set at the center of the lip face. A sponge
layer of length 5hat the end of the domain is included to absorb the acoustic waves. The computational
mesh has 100 grid points across the injector lip. A uniform mesh is used in both directions for the region
from 0 to 10hin axial direction and from -1.5hto 1.5hin transverse direction; stretching is applied with a
ratio of 1.02 only in the transverse direction outside this region. Adiabatic no-slip wall conditions are applied
at the injector lip and adiabatic slip wall conditions are applied for the top and bottom boundaries of the
domain. A 1/7th power law for velocity is used for both the LOX and GH2 streams. The CFL number is
set to 0.8 and no sub-grid scale model is used.
Figure 9 shows the simulation results with instantaneous density field on the left and the scattered
mixture trajectory in comparison with VLE phase boundaries on the right. As can be seen from Fig. 9(b),
the VLE phase boundaries have a maximum temperature of about 145 K at the critical point. The mixing
trajectory of LOX and GH2 in Fig. 9(b) connects the LOX and GH2 injection conditions. The trajectory
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American Institute of Aeronautics and Astronautics
passes through the two-phase region for medium hydrogen mole fractions indicating phase separation due to
the low temperatures of both streams. No experimental results are known to the authors.
C. Reacting hydrogen/oxygen shear layer
The cryogenic LOX/GH2 inert mixing case described in the previous subsection is then ignited to analyze
the phase separation behavior for reacting cases. The computational domain, mesh resolution, boundary
conditions, and numerical schemes are kept the same as in the mixing case.
A combustion case introduces the complication of a larger number of mixture components. Previous
resultss have indicated that water and oxygen may be the critical mixture.19,20,23 To better interpret the
simulation results, the flamelet solutions used for the construction of the chemical table is first analyzed in
the mixture fraction space. Figure 10 shows the evolution of the compressibility factor, as well as the mass
fractions of hydrogen, oxygen, and water, for chemical equilibrium and the near-quenching flamelet solutions.
In the equilibrium case, Z1 for mixture fraction larger than 2.0×103, while oxygen mass fraction has
only marginally reduced from unity. This dilution can be completely attributed to water diffusing into
the oxygen-stream, we obtain a water mass fraction of about 2% at the transition to an ideal gas for the
equilibrium flame. Near quenching, ideal gas conditions are reached at the slightly higher mixture fraction of
mixture fraction larger than 3.0×103, more mixture components are present throughout the flame. We can
conclude that the transition from cryogenic oxygen to an ideal gas, along with the possible phase separation
behavior, occurs as an oxygen/water binary mixture in equilibrium flames.
Mixture Fraction
10-4 10-3 10 -2 10-1 10 0
Mass Fractions
0
0.2
0.4
0.6
0.8
1
1.2
Compressibility
0
0.2
0.4
0.6
0.8
1
1.2
H2 mass fraction
O2 mass fraction
H2O mass fraction
Compressibility
1 - Oxygen Mass Fraction
0 0.01 0.02 0.03 0.04
Mass Fractions
0
0.01
0.02
0.03
0.04
0.05
0.06
0 0.01 0.02 0.03 0.04
Compressibility
0
0.2
0.4
0.6
0.8
1
1.2
H2 mass fraction
H2O mass fraction
Compressibility
Figure 10. Mass fraction and compressibility factor for equilibrium (solid) and near-quenching (dashed)
flamelet solutions. Evaluation in terms of deviation from oxygen mass fraction reveals that only water mixes
with oxygen under real fluid conditions in the equilibrium case.
Thus, while oxygen/hydrogen phase equilibria have been studied extensively, we will now carry out a
phase equilibrium evaluation of the oxygen/water system. Figure 11 shows the results of the VLE calcu-
lations for the binary mixture of oxygen and water in pressure-temperature diagram for the water mole
fractions {0.1,0.2,0.4,0.6,0.8,0.9,0.95,0.98}. The saturation line of water calculated and the CP calcula-
tions for the mixture are also shown as the dash-dotted line for reference. Experimental data are taken from
Japas and Franck.64
Good agreement can be observed between calculations and experimental data64 for CP calculations for
the oxygen/water system. Type III phase separation behavior can clearly be seen for the oxygen/water
mixtures. The critical temperature of the mixture decreases first with increasing oxygen mole fraction, and
then increases exceeding the critical temperature of water. The critical pressure of the mixture diverges,
indicating immiscibility.47 Similar CP behavior to the previous cases can be seen in Fig. 12, showing the
Type III behavior for hydrogen/water mixtures. These results are consistent with that fact that the solubility
of oxygen is very low in water, i.e. the two species are essentially immiscible for low temperatures.
Figure 13 shows the simulation results for the reactive LOX/GH2 shear layer. The solid black lines in
Fig. 13(a) mark the water mass fractions of 0.01 and 0.1, as the outer limit of the combustion region. We
see that the dense LOX core admits a water mass fraction of less than 0.01. This is consistent with Fig. 10
and prior results,22 and is comparable to what has been observed for subcritical injection with a sharp
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T [K]
500 550 600 650 700 750
P [MPa]
0
50
100
150
200
250
H2O
0.1
0.2
0.4
0.6
0.8
0.9
0.95
0.98
Figure 11. Vapor-liquid equilibrium calculations of pressure-temperature diagram at eight water mole fractions
for binary mixtures of oxygen and water in comparison with experimental data.64 Solid black line is the
saturation pressure for pure water, dash-dotted black line is the critical curve for the mixture, and black dot
indicates the critical point of water.
T [K]
100 200 300 400 500 600 700
P [MPa]
0
50
100
150
200
250
300 Calculation
Japas&Franck1985
NIST
Mole fraction of H2O
0 0.2 0.4 0.6 0.8 1
P [MPa]
0
50
100
150
200
250
300 Calculation
Japas&Franck1985
NIST
Figure 12. Calculated critical point for oxygen and water mixtures in comparison with experimental data.64
Black dots are the critical points for pure species.
Species W[kg/kmol] Tc[K] pc[MPa] ρc[kg/m3]Zcω ki,H2O
O232.0 154.6 5.04 436.1 0.2879 0.0222 0.35
H2O 18.0 647.1 22.06 322.0 0.2294 0.3443 -
Table 4. Thermodynamic properties of hydrogen, oxygen, and water.
liquid-vapor interface.65 VLE calculations in Fig. 13(b) are for binary mixtures of oxygen and water, taking
advantage of the finding that only water is present in the vicinity of the dense LOX core. We see that the
structure is significantly different from the inert case in Fig. 9: the hot combustion zone removes any scatter
from the multiphase region, while close to pure oxygen conditions, i.e. in the vicinity of the LOX core, the
mixing trajectory now clearly intersects the coexistence line. Furthermore, unlike in the previous cases, the
clear intersection suggests that the system is not very sensitive to minor changes in injection conditions.
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(a) Density field [kg/m3]
Oxygen mole fraction
0 0.2 0.4 0.6 0.8 1
Temperature in K
0
500
1000
1500
2000
2500
3000
3500
4000
(b) Scattered data of composition and temperature
Figure 13. Reacting shear layer, at the bottom LOX (100 K, 30 m/s), top is GH2 (150 K, 125 m/s) at 10 MPa.
The lines in the left figure mark the 0.01 and 0.1 water mass fraction, enclosing the flame in the center of the
shear layer. Solid lines in the right figure are phase boundaries from VLE; black dot is the critical point.
D. Assessing real fluid mixing rules
The corresponding states principle and the one-fluid mixture assumptions are commonly used in engineer-
ing applications for the evaluation of thermodynamic and transport properties for mixtures.9,45,46 These
assumptions state that for fixed compositions, the mixture properties in a reduced state is the same as some
pure component in the same reduced state. Mixing rules are required for mixtures to calculate the reduced
state. Typically, some fractional weighting function by mole fraction, mass fraction, or the superficial vol-
ume fraction is used. Examples are the mixing rules used in the PR EoS as described in Section II, where a
quadratic dependence on mole fraction is used for parameter , and a linear dependence on mole fraction
is assumed for parameter b.
T [K]
100 200 300 400 500 600 700
P [MPa]
0
1
2
3
4
5
6
7
8
9
10
N2
n-C12H26
CP Calculation
Mixing Rule 1
Mixing Rule 2
T [K]
100 200 300 400 500 600 700
P [MPa]
0
5
10
15
20
25
30
35
40
O2
H2O
CP Calculation
Mixing Rule 1
Mixing Rule 2
Figure 14. Comparison between pseudocritical properties calculated by mixing rules and critical properties
calculated by CP calculations for nitrogen/n-dodecane (left) and oxygen/water (right) mixtures. Mixing Rule
1 is described by Eq. (14), and Mixing Rule 2 by Eqs. (14a) and (15). Solid black lines are the saturation
pressure for pure species.
The pesudocritical properties predicted by the mixing rules are compared with the critical properties
calculated by the CP calculations for the nitrogen/n-dodecane and oxygen/water mixtures, and the results
are plotted in Fig. 14. The saturation lines for pure species are also plotted for reference. It can be seen from
Fig. 14 that the Mixing Rule 1 gives a nearly linear behavior of the critical curve connecting the CP of the
two pure species. The Mixing Rule 2 predicts a critical curve with convex behavior of the critical pressure
for the nitrogen/n-dodecane mixture, whereas a similar behavior is obtained as the Mixing Rule 1 for the
oxygen/water mixture. However, the behavior predicted by both the mixing rules are significantly different
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American Institute of Aeronautics and Astronautics
from those calculated by CP calculations. Indeed, a Type I behavior is expected by using the mixing rules,
whereas the mixtures considered here are both Type III mixtures.
The pseudocritical mixing rules are commonly used in CFD calculations such as LES of transcritical
flows utilizing the diffused interface methods,19,22,24,26,27 and the results in Fig. 14 show that procedures
to evaluate the thermodynamic and transport quantities needs a closer look and the consequences of using
these mixing rules requires further investigation.
IV. Conclusions
We investigated the phase separation behavior for injection cases relevant for aerospace propulsion systems
and diesel engines, namely inert n-dodecane injection into nitrogen, inert LOX/GH2 shear layer, and reacting
LOX/GH2 shear layer. A phase equilibrium solver for vapor-liquid equilibrium (VLE) and critical point
(CP) calculations is developed. The phase equilibrium solver is validated by predicting the phase behavior
of hydrocarbon mixtures and comparison with experimental measurements.
The model is subsequently applied to the mixtures relevant for engineering applications. Specifically,
Type III phase behavior is observed for all the binary mixtures studied, namely the mixtures of nitrogen/n-
dodecane, oxygen/water, and hydrogen/water. Comparison between the commonly used mixing rule models
and the critical properties calculated from the phase equilibrium solver shows that mixing rules are not able
to predict the Type III phase separation behavior.
We find that water/oxygen is the critical mixture for hydrogen/oxygen combustion; representing the first
VLE analysis of a reactive combustion system.
The paper discusses the proof of concept of combining a high fidelity transport model (LES) with high
fidelity thermochemical model (VLE). In this first step, the VLE evaluation is carried out as a post-processing
step on the numerical simulation results from LES. The results suggest that, counterintuitively, the reacting
LOX/GH2 shear layer is more prone to phase separation than the inert mixing case, despite the high
combustion temperatures. The reason lies in the high critical pressure of the combustion product water,
which diffuses into the LOX core. The data furthermore suggest phase separation in the investigated n-
dodecane and nitrogen mixing case.
V. Acknowledgments
Financial support through NASA with award numbers NNX14CM43P and NNM13AA11G are gratefully
acknowledged. Resources supporting this work were provided by the NASA High-End Computing (HEC)
Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.
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American Institute of Aeronautics and Astronautics
... The transition map is shown in Fig. 3, along with a visualization of diffuse supercritical cryogenic injection [16], and sharp interface supercritical injection for hydrocarbons [25]. Analyses of this phenomenon have been given in terms of linear gradient theory by Dahms and Oefelein [1,26,27], or using CFD simulations (Qiu and Reitz [28], Banuti et al. [29]). Unfortunately, these analyses do not lend themselves to be used as simple criteria akin to Eqs. (1) and (2). ...
... Unfortunately, these analyses do not lend themselves to be used as simple criteria akin to Eqs. (1) and (2). An intermediate approach is the thermodynamic trajectory [29][30][31], that can be used by analyzing CFD solutions, but also 1D flamelet results. Essentially, the local reduced pressure and temperature are calculated for mixing problems as a function of the local composition, temperature, and the constant chamber pressure. ...
... (1), or whether Eq. (2) is a dedicated criterion suited to cold-flow hydrocarbon/nitrogen mixtures, and how including combustion into C3 may affect classification. Indeed, the presence of a flame may significantly inhibit real fluid mixing effects [30] and alter the thermodynamic stability [29]. ...
... A deficiency of these studies was that these groups applied a rapid estimation method which is known to produce sever deviations [24] in complex mixtures like they occur during the combustion process. Recently, Banuti et al. [25] used a high fidelity method a-posteriori and pointed out that phase separation might be possible under rocket-relevant conditions but further investigations are necessary. ...
... This has been thoroughly investigated by many groups [17,18,29,19,21,20,22,23]. Different experimental investigations [1,2,3,4,5,6,7,8,9,10] have shown that depending on the conditions and the components forming the multicomponent mixture, phase separation might occur under rocket-relevant conditions. Over the past 20 years, different numerical research groups from the rocket community [35,19,20,25,22] have come up with questions regarding possible two-phase phenomena in LREs and the topic is still under discussion and not fully understood yet. Some of these groups [19,20,22] used rapid estimation methods to investigate possible phase separation phenomena and came to the finding that no phase separation is likely to occur under rocket-relevant conditions. ...
... Some of these groups [19,20,22] used rapid estimation methods to investigate possible phase separation phenomena and came to the finding that no phase separation is likely to occur under rocket-relevant conditions. Recently, Banuti et al. [25] applied a high fidelity thermo-physical model aposteriori on their LES results and found that some of their scatter points lie well within the two-phase region. Therefore, they concluded that two-phase effects might occur under rocket-relevant conditions, but further investigations have to be done. ...
Conference Paper
Full-text available
Under rocket-relevant conditions, real-gas effects and thermodynamic non-idealities are prominent features of the flow field. Experimental investigations indicate that phase separation can occur depending on the operating conditions and on the involved species in the multicomponent flow. During the past decades, several research groups in the rocket combustion community have addressed this topic. In this contribution we employ a high-fidelity thermodynamic framework comprising real-gas and multicomponent phase separation effects to investigate liquid oxygen-methane and liquid oxygen-hydrogen flames at high pressure. A thorough introduction and discussion on multicomponent phase separation is conducted. The model is validated with experimental data and incorporated in a reacting flow CFD code. Thermodynamic effects are presented using one-dimensional counterflow diffusion flames. Both real-gas and phase separation effects are present and quantified in terms of derived properties. Finally, the method is applied in a three-dimensional large eddy simulation of a single-element reference test case and the results are compared to experimental data.
... However, this differentiation breaks down for a supercritical fluid where gaslike and liquidlike properties better represent the state of the fluid [72] (Fig. 1). Although at supercritical conditions surface tension is typically neglected [19,[73][74][75][76][77][78][79][80][81][82][83], we use the term droplet for the fuel sphere at all tested conditions in this paper. The main research questions to be answered in this paper include (1) how the interfacial hydrodynamic instability mechanisms and mixing behavior change during the transition between sub-and supercritical regimes and (2) how the droplet disintegration behavior at transcritical conditions is related to the classical SDI and SBI. ...
... The present model neglects surface tension and the diffuse terms including molecular diffusion, viscous effects, thermal conduction, and chemical reactions. Surface tension is typically neglected in transcritical flows [19,[73][74][75][76][77][78][79][80][81][82][83] as the surface tension coefficient drops dramatically at the critical point. The viscous effects and thermal conduction may be important for longer duration simulations or low-speed cases; however, we found that for the cases considered in the present paper the thermal and viscous terms are insignificant. ...
Article
Transcritical shock-droplet interactions (TSDIs) occur in a spectrum of high-speed propulsion systems involving liquid fuel injection. “Transcritical” behavior refers to a condition at which the combustion chamber pressure nears the critical pressure of the fuel-air mixture and, by increasing the temperature, a transition from liquidlike to gaslike state is observed. Our understanding of TSDI is significantly less developed than its gas-phase (ideal-gas or supercritical) or liquid-phase (subcritical) counterparts, which are referred to as shock-bubble interactions (SBIs) and shock-droplet interactions (SDIs), respectively. In this paper, we investigate the interaction of a shockwave with an n-dodecane droplet at supercritical pressures. A fully conservative diffuse-interface framework coupled with the Peng-Robinson equation of state is developed to accurately determine the state of the fluid and the resulting interfacial instabilities as the shock propagates through the droplet. The influence of varying the initial temperature of the fuel, the ambient pressure, and the shockwave strength on the shock structure and the droplet morphological deformation is delineated. The dynamics of the TSDI cases are then compared to the subcritical SDI and supercritical or ideal-gas SBI counterparts. It is shown that, depending on the preshock temperature and pressure, the TSDIs exhibit some common features observed in classical cases of SDIs and SBIs, bridging the gap between the sub- and supercritical problems.
... Another approach apart from DNS and LES to study non-premixed reacting flows at representative operating conditions is the analysis of one-dimensional counterflow diffusion flames. Amongst others, Ribert et al. [259], Lacaze and Oefelein [154] and Banuti et al. [22,24,23] conducted detailed investigations of LOx/H 2 flames: Ribert et al. [259] focused on the dependency of the flame thickness and the heat release on pressure and strain rate in physical space and quantified the influence of Soret and Dufour effects. Lacaze and Oefelein [154] performed a detailed analysis of strain effects, pressure and temperature boundary conditions as well as real-fluid effects on the flame structure in both physical and mixture fraction space to develop a tabulated combustion model. ...
... Therefore, evaluations of the thermodynamic state based on RE methods are prone to considerable uncertainties. Recently, Banuti et al. [24] used a high fidelity method a posteriori and pointed out that phase separation might be possible under rocket-relevant conditions but further investigations are necessary. Gaillard et al. [86] derived a diffuse-interface allpressure flame model including water as a disperse phase. ...
Thesis
Full-text available
Injection, mixing and combustion under high-pressure conditions are key processes in modern energy conversion machines. Driven by the demand for higher efficiency and reduction of pollutants, intensive investments are made in recent years in the further development of especially two types of fuel-fired engines: liquid-propellant rocket engines (LREs) and gas engines (GEs). This arises from the fact, that LREs will remain an essential component for payload launchers in the foreseeable future and that GEs fired with hydrogen or natural gas are a possible solution to gradually diversify towards cleaner energy conversion machines. Computational fluid dynamics (CFD) can contribute to a better understanding of the injection, mixing and combustion processes within these types of engines. Here, especially one thermodynamic topic is of paramount interest within recent years: phase separation processes under initially supercritical conditions. This work presents a CFD tool that enables the thorough investigation of these processes. Both a pressure- and a density-based solver framework are introduced. The first comprises different formulations of the pressure equation to cover a wide range of Mach numbers. A double-flux scheme specifically tailored for real-gas flows is the core of the density-based solver. The thermodynamic framework relies on a rigorous and fully conservative description of the thermodynamic state. Cubic equations of state and the departure function concept form the basis of the thermal and caloric closure. Consequently, real-gas effects are included inherently. Multicomponent phase separation processes are considered by means of a minimization of the Gibbs energy. For the investigation of the non-premixed combustion process, a tabulated combustion model based on the flamelet concept is employed. Overall, measurement data from five different experimental test campaigns are used to validate the numerical framework. Both Large-Eddy Simulations and Reynolds-Averaged Navier-Stokes simulations are performed. Most of the simulations are conducted with the pressure-based framework. In the first step, real-gas effects in underexpanded jets are investigated. Very good agreement with experimental speed of sound measurements is found. Further investigations demonstrate the importance of the consideration of real-gas effects to correctly capture the jet mixing process. Next, the phase separation process in an underexpanded argon jet is studied. In the fully developed jet, the single-phase instabilities are found downstream of the nozzle exit and upstream of the Mach disk. This is in excellent agreement with experimental Mie scattering measurements. Next, the possibility of phase separation under GE-like operating conditions is investigated. Two different fuels - hydrogen- and methane-based - are considered. For the latter, pronounced phase separation processes are found which are triggered by a strong expansion and a mixing with the ambient gas. No two-phase effects occur in the hydrogen-based fuel as the critical temperature of the less volatile component is dramatically lower as in the methane-based fuel. For the investigation of phase separation processes under LRE-like operating conditions a combined experimental and numerical study together with the University of Stuttgart is conducted. Three different test cases are defined. The characteristics of the phase formation process agree well between experiments and simulations. The single-phase instability is caused solely by a mixing process of the injected fuel with the ambient gas. Next, the prediction capabilities of the pressure- and the density-based solver are assessed in detail. For the pressure-based approach a very good agreement with three experimental test cases is found. The density-based method, in contrast, yields possibly nonphysical states indicated by a strong entrainment into the two-phase region. Finally, phase separation effects in a hydrogen and a methane flame under LRE-typical operating conditions are studied. Single-phase instabilities are found on both sides of the flamelet caused by the low temperatures and the presence of water. For the methane flame, a Large-Eddy Simulation for a reference experiment is conducted. The results show that the region of phase separation is mostly restricted to the oxygen core. The OH* emission images indicate that both flame length and shape are in good agreement with the experimental results.
... Another approach to study nonpremixed reacting flows at representative operating conditions is the analysis of one-dimensional counterflow diffusion flames. Among others, Ribert et al. [10], Lacaze and Oefelein [11], and Banuti et al. [12][13][14] conducted detailed investigations of LOx∕H 2 flames under LRE conditions. With methane getting more attention as future LRE fuel, similar studies have been performed on one-dimensional LOx∕CH 4 counterflow diffusion flames, see, e.g., Refs. ...
... A deficiency of these studies was that these groups applied a rapid estimation method, which is known to produce severe deviations [17] in complex mixtures, like they occur during the combustion process. Recently, Banuti et al. [13] used a high-fidelity method a posteriori and pointed out that phase separation might be possible under rocket-relevant conditions, but further investigations are necessary. Gaillard et al. [18] derived a diffuse-interface all-pressure flame model including water as a disperse phase. ...
Article
Full-text available
Under rocket-relevant conditions, real-gas effects and thermodynamic nonidealities are prominent features of the flow field. Experimental investigations indicate that phase separation can occur, depending on the operating conditions and on the involved species in the multicomponent flow. During the past decades, several research groups in the rocket combustion community have addressed this topic. In this contribution, a high-fidelity thermodynamic framework comprising real-gas and multicomponent phase-separation effects is employed to investigate liquid-oxygen/methane and liquid-oxygen/hydrogen flames at high pressure. A thorough introduction and discussion on multicomponent phase separation are conducted. The model is validated with experimental data and incorporated in a reacting flow computational fluid dynamics code. Using one-dimensional counterflow diffusion flames, the thermodynamic states and processes are discussed. Both real-gas and phase-separation effects are present and quantified in terms of derived properties. Finally, the method is applied in a three-dimensional Large-Eddy Simulation of a single-element reference test case, and the results are compared to experimental data.
... 60 MPa nitrogen and 6 MPa n-dodecane shock-tube case. The (a) mass fraction, (b) density, (c) pressure, (d) velocity, (e) temperature, (f) e * 0 , and (g) * distribution of the shock after 500 μs. ...
Article
Spurious pressure oscillations are a numerical instability that is commonly observed in multiphase and multicomponent flow problems near the critical point. A diffuse-interface model has been created to simulate transcritical mixing in multispecies and multiphase systems where spurious pressure oscillations pose a serious challenge in achieving convergence. To reduce the spurious pressure oscillations, three methods have been proposed and implemented in the present paper: (1) artificially thickening the interface between different species, (2) reconstruction of the primitive variables in the characteristics space, and (3) developing a hybrid method (HY) that switches between quasi-conservative (QC) double-flux (DF) and the classical fully-conservative numerical procedures based on the changes in the effective specific heat ratio and the effective reference internal energy. The cases considered include (1) a transcritical shock tube problem between nitrogen and n-dodecane, (2) a near-critical shock-droplet interaction, and (3) the transcritical Spray A benchmark case of Engine Combustion Network (ECN) that represents the injection of n-dodecane into a high-pressure and high-temperature nitrogen environment. Characteristic-wise reconstruction of the primitive variables in our proposed fully-conservative method tends to reduce the spurious pressure oscillations compared to primitive-wise reconstruction including the reconstruction of the speed of sound. The HY model is found to effectively reduce the magnitude of the spurious pressure oscillations, the loss of energy conservation, and provides more accurate results for all tested cases. The artificial thickening of the interface between different species is found to reduce spurious pressure oscillations and reduce energy conservation loss when used in conjunction with a QC method (HY or DF).
... The critical point of a mixture may deviate substantially from the critical point of its constituents and can be calculated using the mixing rules [6]. For intermediate species, the critical properties are determined based on the Lennard-Jones potential following the work of Giovangigli et al. [45] and Ruiz et al. [158]. ...
Thesis
Full-text available
The simulation of transcritical real-fluid effects is crucial for many engineering appli- cations, such as fuel injection and combustion in internal-combustion engines, rocket motors and gas turbines. In these systems, the liquid fuel is injected into the ambi- ent gas at a pressure that exceeds its critical value, and the fuel jet will be heated to a supercritical temperature before combustion takes place. At elevated pressures, the mixture properties exhibit liquid-like densities and gas-like diffusivities, and the surface tension and enthalpy of vaporization approach zero. In this thesis, algorithms and modeling tools are developed for the prediction of supercritical and transcritical mixing and combustion. A diffuse-interface method is developed for simulating turbulent flows at trans- critical conditions. Real-fluid thermodynamics is described efficiently using the cu- bic equation of state. Spurious pressure oscillations associated with fully conserva- tive (FC) formulations are addressed by a double-flux model. An entropy-stable scheme that combines high-order non-dissipative and low-order dissipative finite- volume schemes is proposed to preserve the physical realizability of numerical so- lutions across large density gradients. The resulting algorithms are applied to a series of test cases to demonstrate the capability in simulations of problems that are relevant for multi-species transcritical turbulent flows. The developed quasi-conservative (QC) scheme is subsequently analyzed with the traditional FC scheme for multi-species mixing problems. Through numerical analysis, it is shown that mixing processes for isobaric systems follow the limiting cases of adiabatic and isochoric mixing models for FC and QC schemes, respectively, which is confirmed by several numerical test cases. An extension to the classical flamelet/progress variable approach is developed for transcritical combustion simulations. The novelty of the proposed approach lies in the ability to account for pressure and temperature variations from the baseline tabulated values in a thermodynamically consistent fashion. Application cases relevant to rocket combustors are performed to demonstrate the capability of the proposed approach in multidimensional transcritical combustion simulations. Finally, a finite-rate chemistry model is employed in conjunction with the de- veloped diffuse-interface method for the prediction of diesel fuel injection and auto- ignition processes. Simulations of an ECN-relevant diesel-fuel injector are performed for both inert and reacting cases at multiple operating points. The performance of the presented numerical framework is demonstrated through comparisons with available experimental data.
... Numerically oriented research activities of the last few years on LOX/H 2 propellants have improved understanding of the thermoand fluid dynamic effects in highly turbulent, reactive flow fields at elevated pressures prevalent in LREs [1][2][3][4][5][6][7]. However, such works also highlight the continuing need for more experimental data suitable for validating models of injection and combustion at these conditions. ...
Article
Full-text available
This work presents results of an effort to create an extended experimental database for the validation of numerical tools for high pressure oxygen-hydrogen rocket combustion. A sub-scale thrust chamber has been operated at nine load points covering both sub- and supercritical chamber pressures with respect to the thermodynamic critical pressure of oxygen. Liquid oxygen and gaseous hydrogen were injected through a single, shear coaxial injector element at temperatures of around 120 K and 130 K, respectively. High-speed optical diagnostics were implemented to visualise the flow field along the full length of the combustion chamber. This work presents the analysis of shadowgraph imaging for characterising the disintegration of the liquid oxygen jet. The large imaging data sets are reduced to polynomial profiles of shadowgraph intensity which are intended to provide a more direct means of comparison with similarly reduced numerical results. Comparing half-lengths of these profiles across operating conditions show clear groupings of load points by combustion chamber pressure and mixture ratio. All load points appear to collapse to an inverse dependence of length on impulse flux ratio.
... According to the classification of [49], all binary N2+ hydrocarbon fluid mixtures are Type III except for methane. Starting at the critical point of n-dodecane, the critical pressure of a N2 + n-dodecane mixture grows by increasing the nitrogen concentration [50]. It reaches higher pressures than the ones observed in Diesel engine combustion chambers ( Figure A C C E P T E D M A N U S C R I P T 1). ...
Article
A numerical framework has been developed to simulate supercritical Diesel injection using a compressible density-based solver of the Navier-Stokes equations along with the conservative formulation of the energy equation. Multi-component fuel-air mixing is simulated by considering a diffused interface approximation. The thermodynamic properties are predicted using the Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT) real-fluid equation of state (EoS). This molecular-based EoS requires three empirically determined but well-known parameters to model the properties of a specific component, and thus, there is no need for extensive model calibration, as is typically the case when the NIST library is utilised. Moreover, PC-SAFT can handle flexibly the thermodynamic properties of multi-component mixtures, which is an advantage compared to the NIST library, where only limited component combinations are supported. This has allowed for the properties of Diesel fuel to be modelled as surrogates comprising four, five, eight and nine components. The proposed numerical approach improves the overall computational time and overcomes the previously observed spurious pressure oscillations associated with the utilization of conservative schemes. In the absence of experimental data, advection test cases and shock tube problems are included to validate the developed framework. Finally, two-dimensional simulations of planar jets of n-dodecane and a four component Diesel surrogate are included to demonstrate the capability of the developed methodology to predict supercritical Diesel fuel mixing into air.
... to the critical point with a few individual points inside the saturation curve, which means that 532 phase separation does not occur [42]. The larger fluctuations caused by the confined domain 533 or the two-dimensionality of the case could be the reason why a small number of cells are in 534 the unstable region [3]. ...
Article
The present paper describes a numerical framework to simulate transcritical and supercritical flows utilising the compressible form of the Navier–Stokes equations coupled with the Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT) equation of state (EoS); both conservative and quasi-conservative formulations have been tested. This molecular model is an alternative to cubic EoS which show low accuracy computing the thermodynamic properties of hydrocarbons at temperatures typical for high pressure injection systems. Liquid density, compressibility, speed of sound, vapour pressures and density derivatives are calculated with more precision when compared to cubic EoS. Advection test cases and shock tube problems are included to show the overall performance of the developed framework employing both formulations. Additionally, two-dimensional simulations of nitrogen and dodecane jets are presented to demonstrate the multidimensional capability of the developed model.
Conference Paper
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The focus of this work is to develop subgrid models for the filtered equation of state for large eddy simulation (LES) of supercritical mixing. The conventional LES treatment of directly computing the filtered density in terms of other filtered quantities results in significant modeling errors. To alleviate this problem, two different approaches are investigated – the gradient-based approach, which seeks to establish a model based on the correlation between the subgrid term and the property gradients in the flow field, and the mixing-based approach, whichmodels the subgrid term as a beta function of the mixture fraction and scalar dissipation rate. Reduction in modeling errors by up to 40% is achieved with the models developed and new insights into the physics of these subgrid contributions are gained.
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In this study, two novel computationally efficient methods for the evaluation of real fluid properties are proposed and compared. The first one is a correlated dynamic evaluation of properties (CoDEP) method. In CoDEP, the time-space is dynamically categorized into many thermo-physical correlation zones, and the evaluation of real fluid properties only need to conduct once in each zone. The second one is a tabulation method, in which the real fluid properties are directly interpolated from a pre-generated look-up table. The two methods are applied to simulations of a liquid oxygen / methane mixing process at supercritical conditions. CoDEP accelerates the evaluation of compressibility factor by 4.8 times, thermodynamics properties by 3.3 times, and transport properties by 4.2 times. On the other hand, tabulation method reduces the computation time of property evaluation by more than 10 times. The overall computation acceleration for both methods are approximately a factor of two. They provide qualitatively similar predictions as the benchmark. CoDEP is significantly more accurate than the tabulation method for spatial distributions, but slightly less accurate for conditional statistics. In addition, the accuracy of both methods are insensitive to time, and CoDEP is insensitive to its threshold values.
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A comprehensive study is conducted to enhance the understanding of swirl injector flow dynamics at supercritical conditions. The formulation is based on full-conservation laws and accommodates real-fluid thermodynamics and transport theories over the entire range of fluid states of concern. Liquid oxygen at 120 K is injected into a supercritical oxygen environment at 300–600 K. Detailed three-dimensional flow structures are visualized for the first time in the pressure range of 100–200 atm. A smooth fluid transition from the compressed-liquid to light-gas state occurs, which is in contrast to a distinct interface of phase change at subcritical pressure. Dynamic behaviors of the oscillatory flowfield are explored using the spectral analysis and proper orthogonal decomposition technique. Various underlying mechanisms dictating flow evolution, including shear-layer, helical, centrifugal, and acoustic instabilities, are studied in depth. The hydrodynamic wave motions in the liquid-oxygen film are found to propagate in two different modes: one along the axial direction at the local wave speed; the other in the azimuthal direction and convected downstream at the mean flow velocity. Results show good agreement with the analytical prediction of the overall response transfer function of the swirl injector. The dominant mode of the azimuthal wave is triggered by the natural acoustic oscillation within the injector. Compared with the two-dimensional axisymmetric results, the calculated liquid-oxygen film is thicker and the spreading angle smaller due to the momentum loss and vortical dynamics in the azimuthal direction.
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Recent experiments on pure fluids have identified distinct liquid-like and gas-like regimes even under supercritical conditions. The supercritical liquid-gas transition is marked by maxima in response functions that define a line emanating from the critical point, referred to as Widom line. However, the structure of analogous state transitions in mixtures of supercritical fluids has not been determined, and it is not clear whether a Widom line can be identified for binary mixtures. Here, we present first evidence for the existence of multiple Widom lines in binary mixtures from molecular dynamics simulations. By considering mixtures of noble gases, we show that, depending on the phase behavior, mixtures transition from a liquid-like to a gas-like regime via distinctly different pathways, leading to phase relationships of surprising complexity and variety. Specifically, we show that miscible binary mixtures have behavior analogous to a pure fluid and the supercritical state space is characterized by a single liquid-gas transition. In contrast, immiscible binary mixture undergo a phase separation in which the clusters transition separately at different temperatures, resulting in multiple distinct Widom lines. The presence of this unique transition behavior emphasizes the complexity of the supercritical state to be expected in high-order mixtures of practical relevance.
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The need for improved engine efficiencies has motivated the development of high-pressure combustion systems , in which operating conditions achieve and exceed critical conditions. Associated with these conditions are large thermodynamic gradients and strong variations in transport properties as the fluid undergoes mixing and phase transition. Accurately simulating these real-fluid environments remains a main challenge. Different modeling approaches have been employed, which can be categorized as diffused and sharp interface methods. The objective of this study is to examine the diffused interface method for simulating diesel-fuel injection at conditions related to the supercritical regime. To this end, a recently developed compressible real-fluid solver for transcritical conditions is employed. Simulations of an ECN-relevant diesel-fuel injector are performed and predictions for instantaneous and statistical flow-field results are compared against available measurements. It is expected that results from this analysis will be useful in identifying limitations of current modeling techniques and in improving physical and numerical models for high-pressure injection systems.
Conference Paper
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An extension to the classical FPV model is developed for transcritical real-fluid combustion simulations in the context of finite volume, fully compressible, explicit solvers. A double-flux model is developed for transcritical flows to eliminate the spurious pressure oscillations. A hybrid scheme with entropy-stable flux correction is formulated to robustly represent large density ratios. The thermodynamics for ideal-gas values is modeled by a linearized specific heat ratio model. Parameters needed for the cubic EoS are pre-tabulated for the evaluation of departure functions and a quadratic expression is used to recover the attraction parameter. The novelty of the proposed approach lies in the ability to account for pressure and temperature variations from the baseline table. Cryogenic LOX/GH2 mixing and reacting cases are performed to demonstrate the capability of the proposed approach in multidimensional simulations. The proposed combustion model and numerical schemes are directly applicable for LES simulations of real applications under transcritical conditions.
Article
The quantitative evaluation of combustion models against experimental data remains a main challenge. This is a consequence of the data complexity, often involving velocity, temperature, and chemical composition; the data acquisition, consisting of intrusive, non-intrusive, direct, and inferred measurement methods; and the data preparation in the form of instantaneous scatter data, statistical results, or conditional information. By addressing this issue, the Wasserstein metric is introduced as a probabilistic measure to enable quantitative evaluations of LES combustion models. The Wasserstein metric can directly be evaluated from scatter data or statistical results using probabilistic reconstruction. The method is derived and generalized for turbulent reacting flows, and applied to different validation tests involving the Sydney piloted jet flame. It is shown that the Wasserstein metric is an effective validation tool that extends to multiple scalar quantities, providing an objective and quantitative evaluation of model deficiencies and the impact of boundary conditions on the simulation accuracy. Several test cases are considered, beginning with a comparison of mixture-fraction results, and the subsequent extension to reactive scalars, including temperature and species mass fractions of CO and CO2. These applications demonstrate that the Wasserstein metric constitutes an easily applicable mathematical tool that condenses multi-scalar combustion data and large datasets into a single quantitative measure.