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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.5–7At 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 0.1–4.0 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,19,22,24–30 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

v−b−aα

v2+ 2bv −b2,(1)

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where Ris the gas constant, v= 1/ρ is the speciﬁc volume, and the parameters aα 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 1−rT

Tc!#2

,(2c)

c= 0.37464 + 1.54226ω−0.26992ω2,(2d)

where Tcand pcare the critical temperature and pressure, and ωis the acentric factor. For mixtures,

classical van der Waals mixing rules are applied

aα =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(ZV−1) −ln(ZV−BV)

−AV

BV√8ln "ZV+ (1 + √2)BV

ZV+ (1 −√2)BV#2(y1Ai1+y2Ai2)

AV−Bi

BV,

(7a)

ln fL

i

xip=Bi

BL(ZL−1) −ln(ZL−BL)

−AL

BL√8ln "ZL+ (1 + √2)BL

ZL+ (1 −√2)BL#2(x1Ai1+x2Ai2)

AL−Bi

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)1−1

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

Q∆n= 0 ,(11a)

∆nT∆n= 1 ,(11b)

and the cubic form

C=X

iX

jX

k

∆ni∆nj∆nk∂3A

∂ni∂nj∂nkT,v

= 0 ,(12)

where nis the mole number vector of the mixture, Ais the Helmholtz free energy, and

Qij =∂2A

∂ni∂njT ,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-

ture2,12–14,62 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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American Institute of Aeronautics and Astronautics

(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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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, Z≈1 for mixture fraction larger than 2.0×10−3, 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 3.0×10−3, 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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American Institute of Aeronautics and Astronautics

(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 aα, 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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