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Seven questions about supercritical ﬂuids –

towards a new ﬂuid state diagram

Daniel T. Banuti∗

, Muralikrishna Raju†

, Peter C. Ma‡

, and Matthias Ihme§

Stanford University, Stanford, CA 94305, USA

Jean-Pierre Hickey¶

University of Waterloo, Waterloo, ON N2L 3G1, Canada

In this paper, we discuss properties of supercritical and real ﬂuids, following the over-

arching question: ‘What is a supercritical ﬂuid?’. It seems there is little common ground

when researchers in our ﬁeld discuss these matters as no systematic assessment of this ma-

terial is available. This paper follows an exploratory approach, in which we analyze whether

common terminology and assumptions have a solid footing in the underlying physics. We

use molecular dynamics (MD) simulations and ﬂuid reference data to compare physical

properties of ﬂuids with respect to the critical isobar and isotherm, and ﬁnd that there

is no contradiction between a ﬂuid being supercritical and an ideal gas; that there is no

diﬀerence between a liquid and a transcritical ﬂuid; that there are diﬀerent thermody-

namic states in the supercritical domain which may be uniquely identiﬁed as either liquid

or gaseous. This suggests a revised state diagram, in which low-temperature liquid states

and higher temperature gaseous states are divided by the coexistence-line (subcritical) and

pseudoboiling-line (supercritical). As a corollary, we investigate whether this implies the

existence of a supercritical latent heat of vaporization and show that for pressures smaller

than three times the critical pressure, any isobaric heating process from a liquid to an ideal

gas state requires approximately the same amount of energy, regardless of pressure. Fi-

nally, we use 1D ﬂamelet data and large-eddy-simulation results to demonstrate that these

pure ﬂuid considerations are relevant for injection and mixing in combustion chambers.

I. Introduction

While supercritical ﬂuid injection has been used for decades in liquid propellant rocket engines and gas

turbines, the process is still considered not well understood.1Nonetheless, signiﬁcant progress has been

made; a set of review articles summarized the experimental2–4 and numerical5–7 state of knowledge. The

established concepts are best illustrated by the classical visualization of an injection experiment of liquid

nitrogen with a helium co-ﬂow by Mayer et al.8in Fig. 1. A subcritical break-up process can be seen in

Fig. 1a, where surface instabilities on the liquid nitrogen jet grow; ligaments and droplets form and separate

from the jet. Acting surface tension is clearly reﬂected in the formation of small distinct droplets and

sharp interfaces. As the pressure is increased suﬃciently, the eﬀect of surface tension becomes negligible,

c.f. Fig. 1b. No sharp interface can be identiﬁed, the break-up process has been replaced by turbulent

mixing. This experimental insight in turn has resulted in a switch from Lagrangian droplet-based numerical

representation, to a continuous Eulerian mixing model.3, 9

Recently, however, interest in the fundamentals is rising again. New theoretical models are being devel-

oped that are concerned with the underlying molecular/physical nature of supercritical injection phenomena.

Approaches can be divided into the study of interfacial phenomena of mixtures, and bulk behavior of pure

∗Postdoctoral Research Fellow, Department of Mechanical Engineering, Center for Turbulence Research.

†Postdoctoral Research Fellow, Department of Mechanical Engineering.

‡Graduate Research Assistant, Department of Mechanical Engineering.

§Assistant Professor, Department of Mechanical Engineering, Center for Turbulence Research.

¶Assistant Professor, Mechanical & Mechatronics Engineering.

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(a) Subcritical break-up, p/pcr,N2 = 0.3. (b) Supercritical disintegration, p/pcr,N2 = 1.8.

Figure 1: Shadowgraphs of coaxial liquid nitrogen injection with helium co-ﬂow into helium environment at

sub- and supercritical pressure.8

ﬂuids. The former is concerned with predicting under which conditions the ﬂow changes from classical

break-up, Fig. 1a, to mixing, Fig. 1b. Dahms and Oefelein10 proposed a methodology in which the interfa-

cial thickness of mixtures at high pressure is evaluated using linear gradient theory. Analogous to what is

known for pure subcritical ﬂuids,11 the broadening of the interface causes the surface tension to vanish when

the critical point is approached. Dahms and Oefelein suggested that a discontinuous multiphase character

of the ﬂow is obtained when the interface thickness is very small compared to the molecular mean free path;

the transition is smooth and diﬀusion-dominated when both are of the same order. Qiu and Reitz12 followed

a diﬀerent approach, using a classical vapor liquid equilibrium extended with stability considerations. They

furthermore account for the change in temperature when an actual phase change occurs. An analysis of the

supercritical bulk ﬂuid was ﬁrst carried out by Oschwald and Schik.13 They discussed that a ﬂuid passing

through a region of high heat capacity beyond the subcritical coexistence line will experience a phase tran-

sition similar to subcritical boiling. The main diﬀerence is that at supercritical pressures, this transition is

no longer an equilibrium process but instead spread over a ﬁnite temperature range. Oschwald et al.2later

dubbed this process pseudoboiling and the locus of maximum heat capacity pseudoboiling line. A quantita-

tive analysis14 showed that the process is, from an energetic point of view, indeed comparable to a phase

change, and should seize to play a role at about three times the critical pressure of the regarded ﬂuid. While

this seems like a mere thermodynamic peculiarity, it has real implications on injection and jet break-up: The

near-critical ﬂuid close to pseudoboiling conditions is uniquely sensitive to minor heat transfer, which may

lead to signiﬁcant changes in density already in the injector.15 The result is the absence of a potential core,

manifested in a density drop immediately upon entering the chamber.16

At the same time, fundamental questions remain open. Take Fig. 2, for example. Traditionally, we

divide the thermodynamic p-Tstate-plane into four quadrants I - IV, separated by the critical isobar p=pcr

and the critical isotherm T=Tcr. In order to compare diﬀerent ﬂuids, we use the reduced temperature

Tr=T/Tcr and reduced pressure pr=p/pcr . Bellan6suggests that the quadrants II, III, and IV should

be considered supercritical ﬂuids, as in neither of them a phase equilibrium is possible. For Tucker,17 any

state above the critical temperature is supercritical, i.e. quadrants II and III. Oefelein5and Candel et al.3

consider quadrant IV transcritical, and III supercritical. Younglove18 refers to ILand IV as liquid and calls

everything else ﬂuid. Accordingly, some kind of transition is expected when passing from a subcritical, to

a transcritical, to a supercritical state. The physical justiﬁcation of these deﬁnitions is unclear, as is the

nature of the pseudoboiling line.

T

cr

pcr

p

T

ILIVII

IIIIV

A

B

Figure 2: Classical ﬂuid state plane and supercritical states structure, Tr=T/Tcr ,pr=p/pcr. A and B are

a sub- and a supercritical injection process, respectively, corresponding to Fig 1.

This paper addresses these questions by investigating the physical meaning of the pure ﬂuid state

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plane. Using microscopic data from molecular dynamics (MD) simulations, macroscopic data from the NIST

database, theoretical reasoning, and results from the literature, we will develop a new state diagram which

is based on local physical properties to characterize the diﬀerence between gases, liquids, and supercritical

ﬂuids. Using ﬂamelet and large-eddy-simulation results, we demonstrate that these pure ﬂuid considerations

may be relevant for combustion systems.

II. Methods

A. Molecular dynamics

The microscopic view is obtained from molecular dynamics (MD) computations. We have used the LAMMPS

package19 to run a system with 25,600 Ar atoms in the canonical N pT (constant number of atoms N,

constant pressure p, and constant temperature T) ensemble at diﬀerent temperatures and pressures. Argon

(Tcr = 150.7 K, pcr = 4.9 MPa) has been chosen as a monatomic general ﬂuid, because its state structure

is very similar to nitrogen and oxygen,14 but minimizes modeling inﬂuences. The MD simulations were

performed with a time step of 0.25 fs using the Nose-Hoover thermostat with a coupling time constant of

10 fs and Nose-Hoover barostat with a coupling time constant of 100 fs to control the temperature and

pressure of the system, respectively. For each simulation, the system was ﬁrst energy-minimized with a

convergence criterion of 0.1 kcal/˚

A. The system was then equilibrated for 62.5 ps and the system energy

and other properties were averaged for the following 62.5 ps of the production run. The simulations were

performed at pressures of 0.7, 1.4, 3.0 and 9.4 pcr at temperatures ranging from 75 K to 235 K in 5 K

intervals. We observe that a 5 K temperature interval is suﬃcient to illustrate the energetic and structural

diﬀerences between the liquid to vapor phase transition at sub- and supercritical pressures. To quantitatively

investigate the structural characteristics, we compute the radial distribution function (RDF)

g(r) = lim

dr→0

p(r)

4π(Npairs/V )r2dr,(1)

with the distance between a pair of atoms r, the average number of atom pairs p(r) at a distance between r

and r+ dr, the total volume of the system V, and the number of pairs of atoms Npairs .20

B. Large-eddy-simulation

Large-eddy-simulations (LES) are carried out using CharLESx, the massively parallel, ﬁnite-volume solver

developed at the Center for Turbulence Research of Stanford University. The method is discussed in detail

elsewhere,21–23 only a brief overview will be given here. Time advancement is carried out using a strong

stability preserving 3rd-order Runge-Kutta scheme. 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. Due to the large density gradients across the pseudoboiling14 region

under transcritical conditions, an entropy-stable ﬂux correction technique as well as a double-ﬂux approach

are used22–24 to ensure the physical realizability of the numerical solutions and dampen the non-linear

instabilities in the numerical schemes. The Peng-Robinson25 equation of state is used to account for real

ﬂuid eﬀects using the canonical approach,9combustion is modeled using the Flamelet-Progress-Variable

method.26, 27

C. Flamelet

The method is described in detail in,28 an overview will be provided here. Following the ﬂamelet assumptions

(Peters29) a proﬁle through a diﬀusion ﬂame can be represented by a 1D-counterﬂow diﬀusion ﬂame, depend-

ing only on the boundary conditions and the strain rate. The axisymmetric, laminar counterﬂow diﬀusion

ﬂame admits a self-similar solution and can be simpliﬁed to a one-dimensional problem.30, 31 The governing

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equations including continuity, radial momentum, species and temperature equations can be written as

d

dx(ρu)+2ρV = 0 , (2a)

ρu dV

dx+ρV 2=d

dx(µdV

dx)−Λ , (2b)

ρu dYk

dx+dJk

dx= ˙ωk,(2c)

ρucp

dT

dx=d

dx(λdT

dx)−X

k

Jk

dhk

dx−X

k

˙ωkhk, (2d)

where conventional notations are used, V=v/r, Λ = (∂p/∂r)/r,hkis the partial enthalpy of species k,

and Jk=ρYkVkis the diﬀusion ﬂux for species k. The governing equations and the equation of state

are implemented in the Cantera package,32 we use the Peng-Robinson equation of state25 to account for

thermodynamic real ﬂuid eﬀects. A high-pressure chemical kinetic mechanism from Burke et al.33 is used

for the H2/O2combustion accounting for 8 species and 27 reactions. We use a formulation of the mixture

fraction Zbased on the hydrogen atom, as used by Lacaze and Oefelein:34

ZH=WH2YH2

WH2

+YH

WH

+ 2 YH2O

WH2O

+YOH

WOH

+YHO2

WHO2

+ 2 YH2O2

WH2O2,(3)

where Yαand Wαare mass fraction and molecular weight, respectively.

III. Seven questions about supercritical ﬂuids

A. Is there only one kind of supercritical ﬂuid?

It is common knowledge that there exists no physical observable to distinguish diﬀerent regions in the

supercritical state space beyond the critical point.35 In the introduction, we listed diﬀerent naming deﬁnitions

of the state quadrants – none of which went into details about diﬀerentiating supercritical states from a

physical perspective.

However, there really is a structure in the supercritical state space that we have to account for in our

analysis. Nishikawa and Tanaka36 measured a transition line which they called ‘ridge’, characterized by

maxima in the isothermal compressibility κT. This transition line could be found as an extension of the

coexistence line, reaching into the supercritical state space. Thus, the structure depicted in Fig. 2 needs

to be extended by this transition line, also referred to as Widom line35 or pseudoboiling line.2, 14 Figure 3

shows the resulting revised state space.

T

cr

pcr

p

T

ILIVII

IIIGL

IV

A

B

IIILL

T

cr

pcr

p

T

ILIVII

IIIGL

IV

A

B

IIILL

Figure 3: Fluid state plane and supercritical states structure with pseudoboiling-line (dashed), dividing the

supercritical quadrant into a liquid-like (subscript LL) and a gas-like (subscript GL) region.

This ‘ridge’ can be associated with a liquid to gas transition – under supercritical conditions: Gorelli et

al.37 and Simeoni et al.35 measured sound dispersion at high supercritical pressures in argon and oxygen.

Sound dispersion is a uniquely liquid property and not observed in gases. This means that there really is a

transition from a liquid-like to a gas-like thermodynamic state within quadrant III.

Consider the density proﬁles in Fig. 4, shown for sub- and supercritical pressures. We see that towards

low temperatures, the attained densities converge towards a liquid state that is typically considered incom-

pressible; the diﬀerences between the isobars become very small. For higher temperatures, the densities

approach their ideal gas values.

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Figure 4: Density of oxygen (pcr = 5.0 MPa, Tcr = 154.6 K) at sub- and supercritical pressure as a function

of temperature. The solid lines represent real ﬂuid data from NIST,38 the dashed lines follow the ideal gas

equation of state. The real ﬂuid approaches the incompressible liquid limit at low temperatures, and the

ideal gas limit at high temperatures.

We have to conclude that there is not only one homogeneous supercritical ﬂuid, but instead the super-

critical state space is divided into distinct liquid and gaseous parts.

B. What is the diﬀerence between a liquid and a transcritical state?

As a corollary to Sec. A, we can argue that when sections ILand IIILin Fig. 3 are liquid, then the intermediary

quadrant IV should also be in a liquid state. Alternatively, consider a ILliquid. The molecular structure

is compressed, highly structured, and quasi-crystalline.36 When a suﬃciently high pressure is applied, the

liquid will transform to a truly crystalline, solid state. It seems unlikely that an intermediary state is achieved

that exhibits more disorder than either phase. The structural equivalence of liquids and transcritical ﬂuids is

what Gorelli et al.37 and Simeoni et al.35 have directly measured when proving that the ﬂuids in quadrants

IL, IIIL, and IV exhibited sound dispersion.

(a) pr= 0.7. (b) pr= 1.4. (c) pr= 3.0. (d) pr= 9.4.

Figure 5: Instantaneous molecular distribution using snapshots of two-dimensional slices from MD simula-

tions at Tr= 0.5 at sub- and supercritical pressures.

To investigate this further, we carry out MD simulations of liquid and transcritical states at a reduced

temperature of 0.5, at reduced pressures of 0.7, 1.4, 3.0 and 9.4. Figure 5 illustrates the instantaneous

molecular structure using snapshots of two-dimensional slices from these simulations. We see that the

structure is virtually indistinguishable regardless of pressure. The quantitative evaluation using the radial

distribution function (RDF) shown in Fig. 6a yields the same result – again, no diﬀerence is discernible

between diﬀerent pressures.

We see that there really is no diﬀerence between the densely packed ﬂuids at ILand IV, which can both

be considered liquid. Thus, there is no physical reason to distinguish between liquid and transcritical states.

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(a) Tr= 0.5. No diﬀerence is dis-

cernible between sub- and super-

critical pressures.

(b) Tr= 1.6. The transition from

monotonous (gaseous) to oscilla-

tory (liquid) behavior can be ob-

served as the pressure is increased.

Figure 6: Radial distribution function from MD for the reduced pressures 0.7, 1.4, 3.0 and 9.4.

C. Can a supercritical ﬂuid be an ideal gas?

Figure 4 shows that even at supercritical pressures, the real ﬂuid density approaches its ideal gas value at

suﬃciently high temperatures. In order to investigate this more systematically, we carry out MD simulations

of gaseous and supercritical states at a reduced temperature of 1.6, at reduced pressures of 0.7, 1.4, 3.0 and

9.4. Figure 5 illustrates the change in the molecular distribution from a gas with little molecular interaction at

pr= 0.7 to a denser packed ﬂuid at pr= 9.4. The RDF in Fig. 6b reveals the monotonous declining behavior

characteristic of gases39 at pr= 0.7 and 1.4, and the oscillatory character signifying liquid behavior39 at

pr= 3.0 and 9.4. At the investigated pressures, however, the liquid character is clearly weaker than in

the systems shown in Fig. 6a, with a reduced range structure. Nonetheless, the gaseous supercritical state

encourages further investigation.

(a) pr= 0.7. (b) pr= 1.4. (c) pr= 3.0. (d) pr= 9.4.

Figure 7: Instantaneous molecular distribution using snapshots of two-dimensional slices from MD simula-

tions at Tr= 1.6 at sub- and supercritical pressures.

For a ﬁner scan of the ﬂuid p-Tstate space, we evaluate the compressibility factor Zas a measure of the

deviation of a ﬂuid from ideal gas behavior. It is deﬁned as

Z=p

ρRT ,(4)

with the gas constant R.Zis the nondimensional ratio of the real ﬂuid pressure to the pressure an ideal

gas at identical density ρ, and temperature Twould exert. In an ideal gas, Z ≡ 1. The compressibility

factor can thus be interpreted as a measure of molecular interaction. Figure 8 shows the distribution of

Zin the nondimensional pr-Trplane. The ideal gas equation of state is strictly only valid along a line,

approaching Tr= 2.5 for vanishing pressurea. This is in accordance with the analytical evaluation of van

der Waals’ equation of state.41 However, by allowing for a 5% deviation, the region of applicability expands

aThis corresponds to Boyle’s temperature.40

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signiﬁcantly; it is shown as the shaded area. Figure 8 reveals that there is no contradiction between a

ﬂuid being supercritical and an ideal gas simultaneously: For Tr>2, the ideal gas domain extends to high

pressures; for pr<6, the deviation is smaller than 10 %, for pr<3, the deviation does not exceed 5 %.

It becomes clear that the critical isobar and isotherm do not coincide with any ﬂuid property boundaries.

The Z= 1 line does not reach the coexistence line; Z= 0.95 is reached only for very low pressures: vapor

does not behave like an ideal gas except for very low pressures. At pr= 0.3, the error in calculated pressure

by using an ideal gas equation of state for the equilibrium vapor amounts to 20% (Z= 0.8), at pr= 0.8 it

reaches 40% (Z= 0.6).

We conclude that a supercritical ﬂuid may well be characterized as an ideal gas for T > 2Tcr and p < 3pcr .

Figure 8: Real gas compressibility Z(solid lines) in pure ﬂuid pr-Trdiagram. Dashed lines are isochors.

Regions of less than 5% deviation from ideal gas behavior are shaded. The critical point is marked by the

red circle. Data from NIST38 for nitrogen.

D. Are supercritical ﬂuid properties insensitive to small changes in pand T?

From the classical view of the supercritical state space in quadrant III as homogeneous and featureless, one

could conclude that the ﬂuid state is insensitive to minor changes in pressure or temperature. Instead, we have

identiﬁed a new transition across the pseudoboiling-line, whose properties we need to investigate. Figure 9

compares the change in density and isobaric speciﬁc heat capacity of oxygen upon crossing the coexistence

line at 5 MPa, with the supercritical pseudoboiling transition at 7 and 10 MPa. The divergence of the heat

capacity vanishes at supercritical pressures, it is replaced with ﬁnite but pronounced heat capacity peaks,

signiﬁcantly exceeding the liquid and gaseous limit values. This introduces a strong sensitivity: consider the

7 MPa isobar in Fig. 9. Around the pseudoboiling temperature of 162.5 K, a ±2.5 K variation introduces a

change in density from 600 to 400 kg/m3. This is important to keep in mind during design and interpretation

of experiments.

We thus need to be able to predict the state of maximum sensitivity. Figure 9 shows that the corre-

sponding pseudoboiling temperature is a function of pressure. For simple ﬂuids, such as nitrogen, oxygen,

methane, this relation can be expressed in the following form,14

pr= exp [As(Tr−1)] ; Asspecies dependent.(5)

For molecules exhibiting more complex behavior, the extended relation

pr= exp [A0(Tr−1)a] ; A0, a species dependent.(6)

yields improved accuracy.42 Data for As,A0, and aare obtained by ﬁtting and are compiled in Table 1 for

several species relevant for combustion. Figure 10 evaluates Eq. (6) for the propellants hydrogen and oxygen,

and the combustion product water.

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Figure 9: Density (solid black lines) and speciﬁc isobaric heat capacity (dashed red lines) for a sub- and two

supercritical pressures. The supercritical transition through the pseudoboiling line, indicated by the peak

in cp, is similar to subcritical boiling when maxima in thermal expansion and heat capacity are regarded.

Data for oxygen from NIST38 for nitrogen.

Species AsA0a

H24.137 3.098 0.849

O25.428 5.428 1.0

N25.589 5.589 1.0

CH45.386 5.386 1.0

C6H14 6.688 5.365 0.921

CO26.470 8.256 1.102

H2O 6.479 5.448 0.911

Table 1: Slope of the pseudoboiling-line for a number of species obtained from NIST.38

Reduced temperature Tr

Reduced pressure pr

1 1.05 1.1 1.15 1.2 1.25 1.3

1

1.5

2

2.5

3

Figure 10: Comparison of pseudoboiling-line ﬂuid data (symbols) and correlations Eq. (5) and (6) with

As, A0, a.

E. What is the signiﬁcance of the critical temperature and pressure?

Having demonstrated that liquid and ideal gas states prevail upon crossing the critical pressure raises the

question of the physical signiﬁcance of the critical isobar and isotherm. From a microscopic perspective,

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interatomic interactions can be expressed in terms of interatomic potentials. Molecules are surrounded by

force ﬁelds, with attractive and repulsive components. In a liquid, atoms and molecules are closely packed as

they are trapped in each others’ potential ﬁelds. However, as the temperature increases, so does the average

kinetic energy of the molecules.43 At a certain temperature, the kinetic energy of the molecules is suﬃcient

to leave the potential well; molecules can no longer conﬁne each other in the potential ﬁeld. This implies

that the critical temperature is approximately proportional to the potential well depth; this is used e.g. by

Giovangigli et al.44 to estimate critical temperatures of radicals for combustion simulations. When this

temperature is reached, the coexistence line terminates at the critical point; the critical pressure is then the

vapor pressure at the critical temperature. More technically, the properties of the phases become identical

at the critical point – the diﬀerence between liquid and gas vanishes. In a mixture, also the composition is

identical. However, this does not imply any relevance away from the coexistence line.

F. Is less energy required to vaporize a supercritical ﬂuid?

As the latent heat of vaporization decreases with rising pressure and vanishes at the critical point,40 one

could hypothesize that the supercritical process B in Fig. 3 would require less energy than the subcritical

process A. However, we have shown that for pr<3, both processes sub- and supercritical processes describe

the transformation from a liquid to a an ideal gas state. Thus, from the molecular perspective depicted in

Fig. 11, the endpoints of both processes are essentially indistinguishable.

Figure 11: In order to transform the liquid to a gaseous state, the molecules have to be separated from their

respective force-ﬁelds. Snapshots of molecular structure obtained from MD computations at a liquid (left),

transitional (mid), and gaseous (right) state.

T

cr

pcr

A1

p

T

B1B2

A2

δh

Figure 12: In order to assess the excess latent heat δh required to heat a ﬂuid from a liquid to a gaseous

state at a subcritical pressure A1→A2compared to the supercritical case B1→B2, the process from A1

to B2is analyzed.

To study the diﬀerence between the sub- and the supercritical case, regard the processes illustrated in

Fig. 12, where A1/ B1, and A2/ B2are identical to start and end conditions of the processes A and B of

Figure 3, respectively. We can introduce δh as the excess enthalpy required to vaporize the subcritical ﬂuid

compared to the supercritical process without latent heat,

δh = (hB2−hB1)−(hA2−hA1).(7)

Any process from A1to B2requires the same amount of energy, regardless of the path taken,

hB2−hA1= (hB2−hA2)+(hA2−hA1)=(hB2−hB1)+(hB1−hA1).(8)

Combining Eqs. (7) and (8) yields

δh = (hB2−hA2)−(hB1−hA1),(9)

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reducing the problem to a caloric evaluation of isothermal compression. When we assume A2/ B2to be

states at the same suﬃciently high temperature to yield ideal gas properties (i.e. T > 2Tcr ), the enthalpy is

pressure-independent and an isothermal compression A2→B2requires negligible energy. Then, the energetic

diﬀerence between the processes A1→A2and B1→B2solely depends on the liquid compression A1→B1.

Figure 13 shows that isothermal compression of a liquid requires negligible energy compared to the latent

heat of vaporization, even close to the critical pressure. Thus,

δh =

: 0

(hB2−hA2)−

: 0

(hB1−hA1).(10)

and we conclude that the energetic diﬀerence between the subcritical and the supercritical heating process

is negligible.

Pressure in MPa

Specific enthalpy in kJ/kg

0 5 10 15 20

-150

-100

-50

0

50

100

150

200

100 K

120 K

150 K

Figure 13: Change in enthalpy for isothermal compression at subcritical temperatures (liquid limit). Once

the ﬂuid is condensed (discontinuous step-down), the change in enthalpy is negligible compared to the latent

heat even close to the critical point.

Figure 14 supports this reasoning by showing enthalpy versus temperature for three pressures. The

ﬂuid is oxygen, so that a pressure of 4 MPa constitutes a subcritical condition, 6 MPa and 10 MPa are

supercritical. Towards lower temperatures, all isobars converge towards the same liquid enthalpy asymptote

hL(T); towards higher temperatures, the isobars converge towards the same ideal gas enthalpy asymptote

hiG(T). Note that both asymptotes are pressure-independent. Thus, the transition from a liquid to an ideal

gas state is energetically identical, regardless of pressure. This is exempliﬁed in Fig. 14 for the transition

from TL= 130 K to TG= 460 K, which requires the same ∆hLG at all shown pressures.

Figure 14: Comparison of sub- and supercritical heating processes for oxygen. For low and high temper-

atures, pressure-independent asymptotes are approached by the enthalpy isobars. Thus, the processes are

energetically equivalent, regardless of pressure. Data are for oxygen from.38

Isothermal vaporization is replaced by a continuous nonequilibrium process at supercritical pressures.

High pressure real ﬂuid eﬀects merely distribute this latent heat over a ﬁnite temperature interval; the

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energy supplied is used to increase temperature and overcome molecular forces simultaneously (Oschwald

and Schik,13 Banuti14). Intermolecular forces do not just vanish when a liquid is compressed beyond the

critical pressure; the energy needed to overcome them needs to be supplied regardless of pressure.

G. Why is the study of pure ﬂuid behavior relevant for injection and combustion?

As the purpose of injection in aerospace propulsion systems is to create mixtures, one may question the rele-

vance of studying pure ﬂuid behavior. Particularly at high pressures, mixture behavior deviates substantially

from pure ﬂuid behavior and ideal gas mixing, oﬀering a vast richness of phenomena.

GH2

LOX

liquid vapor ideal gaspseudoboiling

flame

Figure 15: Schematic of supercritical pressure reactive shear layer behind LOX post in coaxial LOX/GH2

injection. The ﬂame is anchored at the LOX post; the dense oxygen heats to an ideal gas state before mixing

occurs.

Figure 15 shows a schematic of a reactive shear layer in LOX/GH2 injection at supercritical pressure.

While oxygen is injected in a dense, liquid state, several researchers have pointed out that combustion occurs

among ideal gases.9, 34, 45 However, it was been suggested28,41 that not only combustion, but also mixing

occurs chieﬂy among ideal gases, i.e. outside of the real ﬂuid core.

We will explore this further. The transition from real to ideal gas can be analyzed in terms of the com-

pressibility factor46 Zdeﬁned in Eq. (4). Figure 16 from28 shows the structure of a transcritical LOX/GH2

ﬂame (p= 7 MPa, Tin,LOX = 120 K, Tin,H2 = 295 K) for two strain rates, representing conditions in chemical

equilibrium and close to quenching. Real gas behavior with Z<1 is conﬁned to the cold pure oxygen stream

with a mixture fraction ZH<1.0×10−2. The cryogenic oxygen stream transitions to an ideal gas state

before signiﬁcant amounts of reaction products diﬀuse in. For rocket operating conditions, the water content

does not exceed a mole fraction of 2% in the real oxygen.

Figure 17 evaluates the same question using a large eddy simulation of a LOX/GH2 reactive shear layer.

The conditions correspond to Ruiz’ benchmark cryogenic shear layer47 (p= 10 MPa, Tin,LOX = 100 K,

uin,LOX = 30 m/s, Tin,H2 = 150 K, uin,H2 = 125 m/s). Details about the comparison with the benchmark

and the numerical method are discussed by Ma et al.?,23 Figure 17a shows the density distribution close

to the injector lip. Despite the absence of surface tension, the transition from a dense liquid to a gaseous

state occurs across a very small spatial region. The solid lines denoting the 0.1 and 0.01 water mass fraction

show the outline of the ﬂame, but also indicate that mixing between water and oxygen occurs only after

the latter has transitioned to a gaseous state. Figure 17b analyzes the competition between mixing and

thermodynamic transition more closely. Dense oxygen is injected with Z ≈ 0.3 and heats up in the vicinity

of the ﬂame; an ideal gas state Z= 0.95 is reached before the oxygen is diluted to a mass fraction of less

than 0.97, although some scattered data points are found to indicate real ﬂuid mixing.

We can conclude that even for reactive transcritical injection, there is strong evidence that the bulk break-

up process is essentially a pure ﬂuid phenomenon. On the one hand, this may serve as a pathway towards

new modeling strategies (e.g. ideal gas mixing rules and tabulated high-ﬁdelity equation of state41, 48). On

the other hand, this means that break-up processes identiﬁed for pure ﬂuid transcritial injection may be

relevant for reactive cases as well.15

IV. Conclusions

In the present paper we investigated the physical character of diﬀerent ﬂuid states, speciﬁcally with

regard to the critical point. We could show that counter-intuitively, this pure ﬂuid study is relevant for

injection and combustion: even at supercritical pressure, the transition from a dense to a gaseous ﬂuid state

occurs essentially under pure ﬂuid conditions, mixing with other species occurs under ideal gas conditions.

A thermodynamic and physical analysis of the classical four quadrant p-Tstate plane, divided by the

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ZH

10-4 10-3 10-2 10 -1 100

Mass fractions

0

0.2

0.4

0.6

0.8

1

1.2

Z

0

0.2

0.4

0.6

0.8

1

1.2

H2 mass fraction

O2 mass fraction

H2O mass fraction

Compressibility

Figure 16: Compressibility Zfor equilibrium (solid) and near quenching (dashed) ﬂame. Real ﬂuid behavior

occurs only on the oxidizer side. In the equilibrium case, Z ≈ 1 for ZH<2.0×10−3, for near-quenching at

ZH<10−2, from.28

critical isobar and isotherm, revealed that this division is not physically justiﬁed. We have found that the

classical four quadrant model simpliﬁes some structure (neglects the pseudoboiling-line) and complicates

others (implies a transition between liquid and transcritical ﬂuid). A revised diagram capturing the essential

physics is depicted in Fig. 18. The ﬂuid is in a liquid state for pressures above the coexistence line. At

subcritical pressure, the coexistence line divides liquid and vapor; at supercritical pressure, the liquid needs

to pass through the pseudoboiling-line before it transforms to a vapor state. At higher temperatures, the

vapor converts to an ideal gas.

We could show that a supercritical ﬂuid requires heat addition – analogous to the subcritical latent

heat – to transition from the dense to the gaseous state. More speciﬁcally, we could show that the same

amount of energy is required to heat a ﬂuid from a given liquid temperature to an ideal gas temperature –

(a) Colored density contours in kg/m3. The solid black lines

denote a water mass fraction of 0.1 and 0.01, to demonstrate

the position of the ﬂame and the main water occurrence out-

side of the dense LOX core.

(b) Scatter plot of compressibility factor Zand oxygen mass

fraction. The dense pure oxygen can be seen to transform to

an ideal gas state (Z= 0.95) before being signiﬁcantly diluted.

Only every fourth point is shown to improve legibility.

Figure 17: Large eddy simulation of LOX/GH2 reactive shear layer. (p= 10 MPa, Tin,LOX = 100 K,

uin,LOX = 30 m/s, Tin,H2 = 150 K, uin,H2 = 125 m/s)

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Figure 18: Revised ﬂuid state plane. Instead of four quadrants, formed by the critical isotherm and the

critical isobar, the state space can better be understood as a low-temperature liquid state separated from

gaseous states by the coexistence- and pseudoboiling-lines. The distinct pseudoboiling transition occurs up

to reduced pressures of three; beyond, the supercritical state transition is linear.

regardless of pressure, including sub- and supercritical states. This results directly from the intermolecular

forces that need to be overcome from a densely packed liquid to an interaction-less ideal gas state, and which

are invariant to the acting pressure.

We can conclude that there are many unexpected similarities between sub- and supercritical states and

transitions, which may require additional consideration in numerical and experimental studies.

Acknowledgments

Financial support through NASA Marshall Space Flight Center is gratefully acknowledged.

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