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Seven questions about supercritical fluids - towards a new fluid state diagram



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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.
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
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
dx(ρu)+2ρV = 0 , (2a)
ρu dV
dx+ρV 2=d
dx)Λ , (2b)
ρu dYk
dx= ˙ωk,(2c)
˙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
+ 2 YH2O
+ 2 YH2O2
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.
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
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(Tr1)] ; Asspecies dependent.(5)
For molecules exhibiting more complex behavior, the extended relation
pr= exp [A0(Tr1)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
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.
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 A1A2compared to the supercritical case B1B2, 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 = (hB2hB1)(hA2hA1).(7)
Any process from A1to B2requires the same amount of energy, regardless of the path taken,
hB2hA1= (hB2hA2)+(hA2hA1)=(hB2hB1)+(hB1hA1).(8)
Combining Eqs. (7) and (8) yields
δh = (hB2hA2)(hB1hA1),(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 A2B2requires negligible energy. Then, the energetic
difference between the processes A1A2and B1B2solely depends on the liquid compression A1B1.
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
: 0
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
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.
liquid vapor ideal gaspseudoboiling
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
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<1.0×102. 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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10-4 10-3 10-2 10 -1 100
Mass fractions
H2 mass fraction
O2 mass fraction
H2O mass fraction
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<2.0×103, for near-quenching at
ZH<102, 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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American Institute of Aeronautics and Astronautics
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.
Financial support through NASA Marshall Space Flight Center is gratefully acknowledged.
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... In Regime II, the injection temperature is close to the critical temperature, and the isentropic expansion path may enter the two-phase dome, resulting in a condensation phase transition. In Regime III, the jet whose injection temperature is much higher than the critical temperature may pass through the Widom line, causing a pseudo-boiling transition of the fuel from liquid-like to gaslike and large density variations [23,24]. Xu et al. [25] examined the morphological characteristics of supercritical n-pentane injected into a subcritical environment, and found that supercritical spray can undergo multiple condensation and evaporation processes downstream of the nozzle. ...
... Fig. 4) can maintain the vapor-phase state during their expansion processes, except for the ones close to critical point that can expand into the two-phase region. The supercritical region, i.e., the region bounded by the critical isotherm line (abbreviated to c L) and the critical isobaric line (abbreviated to c L), is divided into the vapor-like supercritical (VLS) and liquid-like supercritical (LLS) regions by the Widom line (WL) or the so-called pseudoboiling line according to Banuti et al. [24]. The VLS region is regarded as Region II, from which the injected jets (e.g., the jet of r, inj = 1.20 and r, inj = 1.091, cf. ...
The internal flow characteristics of near- and supercritical RP-3 aviation kerosene injected into an atmospheric pressure environment were experimentally investigated. The internal flow structure was visualized in a two-dimensional transparent convergent nozzle using shadowgraph imaging. A series of images of the RP-3 flow inside a convergent nozzle varying from subcritical to supercritical injection conditions were obtained for the first time. Under subcritical, critical, and supercritical pressures, different RP-3 flow structure transitions with decreasing injection temperature were examined. In addition, the associated phase transition was analyzed using the thermodynamic phase diagram of a 10-component RP-3 surrogate based on the adiabatic isentropic expansion hypothesis. Six distinct phase state regions governing the flow structure were identified, considering the phase transitions and density variations of the fuel injection from different regions during their respective expansion processes. The experimentally observed phase transition boundary with respect to injection conditions was further determined, which can be utilized to predict the onset of a drastic change in mass flow rate. Furthermore, the axial length between the nozzle exit and the onset location of observable phase transition was detected, which initially increases and then decreases with decreasing injection temperature. Further analysis showed that the unique evolution behaviour of the specific heat ratio during injection process has a strong effect on flow characteristics near critical point.
... (The same was concluded as general rule for other substances, too. 2 ...
... 3. Phase transition at critical isochore V r = V/V c =1 (Figs. 2 and 12), characterized by maximum of heat capacity (Fig. 3), is actually Widom line well known for many substances. 2,23 . 4. Crossover the percolation threshold line at S r = 1 was found for nitrogen at 15 MPa and 160 K, 30 and methane at 298, 345, 374 K and 100, 200, 205 MPA, respectively. ...
Full-text available
The phase states and the molecular structure of compressed gases and liquids are still open questions. The aim of this article is to: (1) Present the phase states and supramolecular particles thereof; (2) Analyze liquid boiling and vapor condensation as the transformation of one kind of supramolecular particles to another kind; (3) Give physical meaning of critical point; (4) Eliminate some mistakes and solve dilemmas in our previous comprehensions. Therefore, the changes in the motion of molecules from an ideal gas to the supercritical and subcritical states of gases and liquids are considered here. With increasing pressure and decreasing gas temperature, translational movement of molecules transforms into rotation of various particles that require less and less space and energy: from individual molecules to rotating molecular pairs, bimolecules and linear oligomolecules. There are the higher order phase transitions at the conditions of isochores 2Vc, Vc (Widom line) and Vc/2 (Frenkel line), where Vc is critical volume, and a phase inversion at critical isentrope Sc. In each phase there are two types of particles which are in dynamic equilibrium: translating and rotating individual molecules in the ideal gas phase; individual molecules and molecular pairs in the alpha phase; bimolecules and molecular pairs in the beta phase, and bimolecules and oligomolecules in the liquid and gamma phase.
... Numerical modeling of the highly turbulent reactive flow including large thermal gradients from cryogenic fluids up to temperatures of more than 3500 K, real gas effects and combustion of non-premixed diffusion flames is still a challenging topic. Depending of the injection pressure (sub-or supercritical) either the droplet atomization and vaporization or pseudo-boiling has to be modeled additionally [15]. ...
Full-text available
Hot-fire tests were performed with a single-injector research combustor featuring a large optical access (255 × 38 mm) for flame imaging. These tests were conducted with the propellant combination of liquid oxygen and compressed natural gas (LOX/CNG) at conditions relevant for main- and upper-stage engines. The large optical access enabled synchronized flame imaging using OH* and CH* radiation wavelengths covering an area of the combustion chamber from the injection plane to shortly before the contraction section of the nozzle for two sets of operating conditions. Combined with temperature, pressure and unsteady pressure measurements, these data provide a high-quality basis for validation of numerical modeling. Flame width and opening angle were extracted from the imaging in order to determine the flame topology. A two dimensional Rayleigh Index was calculated for an acoustically unexcited and excited interval. These Rayleigh Indices are in good agreement with the thermoacoustic state of the chamber.
This paper describes laser imaging experiments on steady, rotationally-symmetric, laminar jets aimed at observation of the interface between an injected liquid and the surrounding gas under subcritical, transcritical, and supercritical conditions. A steady flow of fluoroketone enters a chamber of high pressure and temperature nitrogen, allowing direct examination of the interface as it evolves with flow time (i.e. axial position in the chamber). Vapor/liquid equilibrium (VLE) calculations identifying the critical locus for mixtures of fluoroketone and nitrogen are used to define six test cases covering the range from entirely subcritical to entirely supercritical. Planar laser induced fluorescence (PLIF) and planar elastic light scattering (PELS) imaging are applied to these jets, to image mixture fraction (via PLIF) simultaneous with detection of the interface strength (via PELS). Temperature distributions are acquired using thermocouples. Evidence for the evolution of the interface, and for supercritical states, is presented and discussed.
Full-text available
The present paper aims at developing a generally valid, consistent numerical description of a turbulent multi-component two-phase flow that experiences processes that may occur under both subcritical and trans-critical or supercritical operating conditions. Within an appropriate LES methodology, focus is put on an Euler-Eulerian method that includes multi-component mixture properties along with phase change process. Thereby, the two-phase flow fluid is considered as multi-component mixtures in which the real fluid properties are accounted for by a composite Peng-Robinson (PR) equation of state (EoS), so that each phase is governed by its own PR EoS. The suggested numerical modelling approach is validated while simulating the disintegration of an elliptic jet of supercritical fluoroketone injected into a helium environment. Qualitative and quantitative analyses are carried out. The results show significant coupled effect of the turbulence and the thermodynamic on the jet disintegration along with the mixing processes. Especially, comparisons between the numerical predictions and available experimental data provided in terms of penetration length, fluoroketone density, and jet spreading angle outline good agreements that attest the performance of the proposed model at elevated pressures and temperatures. Further aspects of transcritical jet flow case as well as comparison with an Eulerian-Lagrangian approach which is extended to integrate the arising effects of vanishing surface tension in evolving sprays are left for future work.
In this work the application of an algebraic equilibrium wall-function to real-gas flows is presented and analyzed. The aim is to assess capabilities of existing algebraic wall-functions in supercritical conditions. In particular a systematic analysis on the coupled wall-function of Cabrit and Nicoud is carried out a-priori on a wall-resolved Large Eddy Simulations (WR-LES) database, featuring cryogenic para-hydrogen flow in a heated pipe at supercritical pressure. The model is shown to overestimate the wall-temperature for increasing values of the imposed heat flux and to slightly underestimate the skin friction velocity. The causes of the failure are investigated. In particular the focus is on the equilibrium boundary layer hypothesis, on the validity of the Van Driest transformation for supercritical, stratified flows and on the ideal-gas assumption employed in the original derivation of the model. For the latter, a consistent thermodynamic correction is proposed in order to extend the applicability of the mentioned wall-function to any arbitrary Equation of State (EOS). The proposed extension is tested a-priori and is shown to provide improved temperature and skin friction velocity predictions at wall, although still presenting relevant deviations from the reference database solution. The discrepancies seem to be addressed to the equilibrium assumption and to the Van Driest transformation. The former in particular accounts for 5−20% of the reference value for the skin friction velocity, among all cases and 〈y+〉 examined, and for 1−20% in terms of wall-temperature depending on the considered heat flux. The latter is shown to fail on the mentioned database for increasing stratifications, with errors between 1−40% depending on the considered case. An additional analysis of more recently proposed transformations for variable property flows reveals the same limitations.
In this paper, a dataset of wall-resolved large-eddy simulations of cryogenic hydrogen at supercritical pressure and different values of wall heat flux is presented. The aim is to provide a reference dataset for wall-function development under trans- and supercritical conditions, such as those found in liquid rocket engine applications. The employed numerical framework is a pressure-based segregated low-Mach-number approach based on an equation-of-state independent formulation. The wall-adapting local eddy-viscosity subgrid model is used for turbulence closure. Real-gas effects are taken from the National Institute for Standards and Technology database and stored as a function of a nondimensional temperature at the considered pressure. A validation and a grid-convergence analysis are first performed on an incompressible case without imposed heat flux. The effect of axial, radial, and azimuthal refinements on first- and second-order velocity statistics is discussed and compared with direct numerical simulation data from the literature. A parametric analysis at different wall heat fluxes is then performed by keeping the inlet mass flux, temperature, and Reynolds number constant. Particular attention is devoted to turbulent pseudoboiling and its effect on the wall temperature. The latter shows a more pronounced increment as the heat flux increases, which is attributed to the pseudochange of the phase in the core flow. Correspondingly, a flattening of the probability density function of the temperature is observed, and it is associated with the pseudoboiling interface forming close to the wall and causing a more intense stratification. First- and second-order statistics for velocity and selected scalars are then presented, and the effect of pseudoboiling is discussed. The effect of the wall heat flux on the viscous and thermal resolution of the computational grid is also assessed, and considerations on the relation between turbulent pseudoboiling and near-wall gradients is finally provided.
By considering Frenkel's point of view regarding the concept of relaxation time, every molecule of a fluid possesses an effective space. We name this permissible and accessible space of each molecule of fluid as the molecular cage (MC). All motions of the molecule, including translational, rotational, and vibrational motions, take place within this space. In this paper, by using the new concept of the thermodynamic dimension (DT) and the generalized Debye model for the isochoric heat capacity (CV) of matter in different states, we want to indicate at high pressures the MCs confine the internal molecular vibrations. By calculation of CV of supercritical fluid (SCF), the restriction of molecular vibrations is detected. To investigate the effect of molecular cages, we have studied CV of diatomic SCFs including nitrogen, oxygen, and carbon monoxide.
Precise measurement and visualization of trans-/supercritical jet processes are of great significance for the control of power engines, chemical reaction and industrial processes. The quantitative measurement of transcritical jet under the influence of high pressure effect and transient effect is the key issues in such analysis. In this study, an improved phase-shifting interferometer system with high temporal and spatial resolution (0.001 s, 3.45 μm) has been realized by pixelated-array masked method to investigate characteristics in trans/supercritical jet processes. The transient density field and boundary structure of the phase-transition interface during four jet processes under sub/trans/supercritical conditions were quantitatively measured, which is different from those under subcritical conditions. The results show that the characteristic of subcritical jet is phenomenon of fragmentation and atomization within the experimental cell due to the presence of interfacial tension and strong density pulsations. The atomization is suppressed by high pressure effect in supercritical jet. Instead, Instead, single-phase mixing occurs due to the absence of surface tension. The dense core is found gradually dissolved during diffusion process of the jet flow. The transition from liquid-like to gas-like status of fluid occurs when the injection fluid goes cross the pseudo-critical line, and the jet surface has a large density gradient (in the current test range, |∇ ρ|max = 8×10⁵ kg/m⁴) due to temperature rise and pseudo-boiling expansion.
Der Fokus dieser Arbeit liegt auf der Untersuchung der Flammen-Akustik-Interaktion der Injektorkopplung in einer LOX/H2-Forschungsbrennkammer durch 2D-Flammenvisualisierungen im injektornahen Bereich. Die Flammendynamik wurde mittels Dynamic Mode Decomposition untersucht. Dabei ergaben sich symmetrische, longitudinale Intensitätsverteilungen als Antwort auf die Injektormoden. Eine Rekombination der DMDModen mit dem zeitlich gemittelten Bild der blauen Strahlung zeigt, dass symmetrische, wellenartige Strukturen auf dem Sauerstoff-Strahl vorliegen. Dies lässt sich durch periodische Sauerstoff-Massenstromfluktuationen erklären und ist somit im Einklang mit dem Anregungsmechanismus nach Gröning. Die Analyse der Phasenlage zwischen der OH*- Strahlung und der 1T-Mode ergab, dass berechnete Rayleigh-Indizes konsistent zum Stabilitätsverhalten sind. In der OH*-Strahlung wurde außerdem ein weiteres dynamisches System identifiziert, dessen Frequenzen mit den Sauerstoff-Strömungsgeschwindigkeiten korrelieren. Ein hydrodynamischer Effekt am Einlauf der Injektoren erzeugt dabei periodische Wirbel, welche die Injektorakustik verstärken können. Somit besteht nun ein umfassendes Verständnis des Anregungsmechanismus in BKD. Basierend auf den gewonnenen Erkenntnissen konnten Konstruktionsvorschläge der Injektoreinlaufblende zur Reduktion des Risikos injektorgekoppelter, hochfrequenter Verbrennungsinstabilitäten abgeleitet werden.
Conference Paper
Full-text available
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.
The thermophysical properties of argon, ethylene, parahydrogen, nitrogen, nitrogen trifluoride and oxygen are presented. Properties are given in tables and a standard set of equations is described. The tables list pressure, density, temperature, internal energy, enthalpy, entropy, heat capacity at constant volume, heat capacity at constant pressure, and sound velocity. Also included are viscosity, thermal conductivity, and dielectric constant, for some of the fluids. The equation and related properties of this report represent a compilation from the cooperative efforts of two research groups: the NBS Thermophysical Properties Division and the Center for Applied Thermodynamics Studies of the University of Idaho.
This project looks at injection processes of a dense jet simulating oxygen core flow with nitrogen of a coaxial injector used in cryogenic rocket engines. The rocket engine performance is highly dependent on the injection processes such as mixing and jet dissipation of propellants in the supercritical regime. Experimental data at various temperatures and injection velocities taken by Raman imaging and Shadowgraphy were compared to computational models allowing comparisons of density, length scales and jet spreading angles providing insight into mass mixing and jet dissipation.
This paper introduces a new model for real gas thermodynamics, with improved accuracy, performance, and robustness compared to state-of-the-art models. It is motivated by the physical insight that in non-premixed flames, as encountered in high pressure liquid propellant rocket engines, mixing takes place chiefly in the hot reaction zone among ideal gases. We developed a new model taking advantage of this: When real fluid behavior only occurs in the cryogenic oxygen stream, this is the only place where a real gas equation of state (EOS) is required. All other species and the thermodynamic mixing can be treated as ideal. Real fluid properties of oxygen are stored in a library; the evaluation of the EOS is moved to a preprocessing step. Thus decoupling the EOS from the runtime performance, the method allows the application of accurate high quality EOS or tabulated data without runtime penalty. It provides fast and robust iteration even near the critical point and in the multiphase coexistence region. The model has been validated and successfully applied to the computation of 0D phase change with heat addition, and a supercritical reactive coaxial LOX/GH2 single injector.
Certain experiments in quasi-isobaric supercritical injection remain unexplained by the current state of theory: Without developing a constant value potential core as expected from the mechanical view of break-up, density is observed to drop immediately upon entering the chamber. Furthermore, this phenomenon has never been captured in computational fluid dynamics(CFD) despite having become a de facto standard case for real fluidCFD validation. In this paper, we present strong evidence for a thermal jet disintegration mechanism (in addition to classical mechanical break-up) which resolves both the theoretical and the computational discrepancies. A new interpretation of supercritical jet disintegration is introduced, based on pseudo-boiling, a nonlinear supercritical transition from gas-like to liquid-like states. We show that thermal disintegration may dominate classical mechanical break-up when heat transfer takes place in the injector and when the fluid state is sufficiently close to the pseudo-boiling point. A procedure which allows to capture subsided cores with standard CFD is provided and demonstrated.