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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.
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
dr0
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
dxX
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(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.
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 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
(hB2hA2)
: 0
(hB1hA1).(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<1.0×102. 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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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<2.0×103, 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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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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... 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. ...
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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.
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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.