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Supercritical pseudoboiling for general fluids and its application to injection.

Authors:
Center for Turbulence Research
Annual Research Briefs 2016
211
Supercritical pseudoboiling for general fluids
and its application to injection
By D.T. Banuti, M. Raju AND M. Ihme
1. Motivation and objectives
Transcritical injection has become an ubiquitous phenomenon in energy conversion
for space, energy, and transport. Initially investigated mainly in liquid propellant rocket
engines (Candel et al. 2006; Oefelein 2006), it has become clear that Diesel engines
(Oefelein et al. 2012) and gas turbines operate at similar thermodynamic conditions. A
solid physical understanding of high-pressure injection phenomena is necessary to further
improve these technical systems. This understanding, however, is still limited. Two recent
thermodynamic approaches are being applied to explain experimental observations: The
first is concerned with the interface between a jet and its surroundings, in the pursuit
of predicting the formation of droplets in binary mixtures. Dahms et al. (2013) have
introduced the notion of an interfacial Knudsen number to assess interface thickness and
thus the emergence of surface tension. Qiu & Reitz (2015) used stability theory in the
framework of liquid - vapor phase equilibrium to address the same question. The second
approach is concerned with the bulk behavior of supercritical fluids, especially the phase
transition-like pseudoboiling (PB) phenomenon. PB-theory has been successfully applied
to explain transcritical jet evolution (Oschwald & Schik 1999; Banuti & Hannemann
2016) in nitrogen injection. The objective of this paper is to generalize the current PB-
analysis. First, a classification of different injection types is discussed to provide specific
definitions of supercritical injection. Second, PB theory is extended to other fluids –
including hydrocarbons – extending the range of its applicability.
2. Supercritical phase transitions
Before an injection process can be analyzed, we need to discuss the thermodynamic
phase plane of fluids and its relevance to injection.
2.1. Pseudoboiling and Nishikawa-Widom line
Vaporization is an equilibrium phase transition from liquid to vapor at the saturation
temperature with discontinuous changes in density and enthalpy. Such a process is de-
picted in Figure 1 for oxygen at a pressure of 4 MPa. The subcritical transition exhibits a
diverging specific isobaric heat capacity, as the temperature stays constant while energy
is added. At the supercritical pressure of 10 MPa, Figure 1 shows that the discontinu-
ous change in density has vanished. However, there still exists a distinct temperature at
which heat capacity and thermal expansion exhibit pronounced extrema, causing a phase
change-like behavior of the fluid when heating through this temperature range. This tran-
sition state is referred to as the supercritical pseudoboiling phenomenon (Oschwald et al.
2006; Banuti 2015).
The supercritical heat capacity peaks line up and form an extension to the coexistence
line; Figure 2 shows this line in a p-Tdiagram. Nishikawa & Tanaka (1995) were the
212 Banuti, Raju & Ihme
Figure 1. Density (dashed lines) and specific isobaric heat capacity (solid lines) for a sub- and
a supercritical pressure. The supercritical transition through pseudoboiling, indicated by the
finite peak in cp, is similar to subcritical vaporization when maxima in thermal expansion and
heat capacity are regarded. Data for oxygen from NIST, (Linstrom & Mallard 2016).
Figure 2. Fluid phase diagram in terms of the reduced temperature Tr=T /Tcr and reduced
pressure pr=p/pcr. The reduced heat capacity is nondimensionalized with the ideal gas value
cp,r=cp/((γR)/(γ1)). The Nishikawa-Widom line forms an extension to the coexistence line
and divides liquid and gaseous states. An ideal gas state is reached at higher temperatures, as
Z → 1.
first to measure the dramatic changes in fluid properties upon crossing this curve, which
divides liquid-like and gas-like fluid states, even at supercritical pressures.
The compressibility factor Zmeasures the deviation of fluid behavior from ideal gas
behavior. It is defined as Z=p/ρRT . Figure 2 shows that a fluid at supercritical pres-
sure needs to be heated to approximately twice the critical temperature before it can
be considered an ideal gas. Thus, there is no contradiction between a fluid being in a
supercritical state and in behaving like an ideal gas.
Supercritical injection and fluid behavior 213
GH2
LOX
liquid vapor ideal gaspseudoboiling
flame
Figure 3. Schematic of supercritical pressure reactive shear layer behind LOX post in coaxial
GH2/LOX injection. Flame is anchored at LOX post; dense oxygen heats to ideal gas state
before mixing.
approached. The initial divergence angle of the jet was measured at
the jet exit and compared with the divergenceangle of a large number
of other mixing layer flows, including atomized liquid sprays,
turbulent incompressible gaseous jets, supersonic jets, and
incompressible variable-density jets. This angle was plotted across
four orders of magnitude in the gas-to-liquid density ratio, which was
the first time such a plot has been reported over this large range. At
and above the critical pressure of the injected liquid, jet growth rate
measurements agreed quantitatively with the theory for incompress-
ible, variable-density, gaseous mixing layers. As the pressure was
reduced to progressively more subcritical values, the spreading rate
approached that measured for liquid sprays.
Candel et al. [4], Pons et al. [5], and Oschwald et al. [6,7]
demonstrated fundamental differences in combustion dynamics of
cryogenic flames between subcritical and supercritical pressure
conditions using experimental facilities from independent
laboratories. Such differences extended to fundamental quantities
such as liquid core lengths, turbulent length scales, and jet breakup
regimes. At supercritical pressure and temperature conditions,
Taskinogluand Bellan [8,9], Bellan [10], and Harstad and Bellan [11]
performed direct numerical simulations along with isolated and
cluster drop simulations considering Soret and Dufour effects to
investigate the mixing process and fluid disintegration in
supercritical mixing layers. Large-eddy simulations have been
performed using proposed new subgrid-scale terms, denoted
correctionterms, in concert with typical subgrid-scale flux terms
[8,9]. Sirignano et al. [12,13], Delplanque [14], and Sirignano [15]
discussed and investigated droplet and spray vaporization at
subcritical and supercritical pressures to study combustion and
dynamic responses to oscillatory ambient conditions. Those studies
established that the vaporization process can be a rate-controlling
process for driving combustion instabilities.
Many other works have served to corroborate these observations
[1634]. However, the absence of surface tension forces under some
high-pressure conditions still poses many fundamental questions. For
a pure fluid, basic theory dictates that surface tension forces become
diminished because the pressure of the liquid phase exceeds its
critical value. For multicomponent systems, however, this is
generally not the case, and modern theory still lacks a first principle
explanation for the observed phenomena. Previously, it has been
widely assumed that surface tension forces become diminished
because the temperature of the liquid phase exceeds its critical value.
Under many conditions, however, this mechanism must be ruled out.
In this paper, it is shown that the temperature of hot unburned gaseous
hydrogen is not sufficient to heat the liquid-oxygenhydrogen
(LOX-H2) interface to its critical temperature, and surface tension
forces do not diminish only because the critical pressure of the liquid
or the mixture critical pressure(as defined by Faeth et al. [35] and
Lazar and Faeth [36]) is exceeded. Conversely, there is significant
experimental evidence of substantial surface tension forces in various
binary mixtures at high pressures (ppC;L;p>500 bar) provided
in recent literature [3739]. The theory presented here serves to
explain the conditions when this occurs also.
In this paper, we use the theoretical framework introduced by
Dahms et al. [4042] to derive respective regime diagrams for
representative propellant combinations in liquid rockets. This
framework has been recently verified experimentally by Manin et al.
[43] and by Falgout et al. [44]. The model allows one to analyze
representative detailed gasliquid interface dynamics during
injection. Our analysis shows that, under certain high-pressure
conditions, gasliquid interfacial phenomena are not determined by
vaporliquid equilibrium and the presence of surface tension forces.
Instead, the interface enters the continuum length scale regime due to
a combination of reduced mean free molecular pathways and
broadened interfaces. Based on these insights, a regime diagram for
LOX-H2injection is presented. This diagram quantifies when the
system transitions between classical atomization and diffusion-
dominated mixing as a function of the ambient pressure and
temperature of the chamber. A similar diagram is derived for
n-decanegaseous-oxygen (LC10-GO2) systems. It is demonstrated
that the liquid injection process can exhibit characteristics of classical
spray atomization even at very high chamber pressures
(p>20 ·pC). Likewise, depending on the temperature and
composition of the fuel and oxidizer streams, liquid injection can
instead exhibit dynamics of dense-fluid or supercritical mixing
processes. The current analysis reinforces past observations that
drops can still be present even at pressures that are supercritical with
respect to the injected liquid and quantifies under what conditions
dense-fluid mixing occurs instead.
The liquid rocket engine conditions investigated here are
dramatically different compared to the representative diesel engine
injection conditions considered in past works. Despite these
distinctive differences in operating conditions (e.g., low cryogenic
temperatures, different molecular sizes and properties, high reduced
liquid injection, and ambient gas pressures and temperatures, etc.),
this study demonstrates that the related physical complexity is well
captured by our mapping approach based on the defined Knudsen
number. The presented analysis also quantifies and explains the
significant differences observed in the envelopes of transition
conditions from classical spray atomization to dense-fluid jet
dynamics between rocket systems operated using liquid-oxygen
hydrogen and liquid n-decaneoxygen propellant combinations.
II. Model Formulation
The starting point of our model is based on the framework
developed by Oefelein [45,46], which provides a generalized
treatment of the thermodynamics and transport processes for
hydrocarbon mixtures at near-critical and supercritical conditions.
This scheme is combined with the theoretical framework developed
by Dahms et al. [40] and Dahms and Oefelein [41], which facilitates
vaporliquid equilibrium and gasliquid interface structure
calculations to compute both surface tension forces and the local
two-phase interface thicknesses. Here, only a description of the
general features of the model framework is provided. A detailed
derivation can be found in [41].
A. Thermodynamic and Transport Properties
The extended corresponding states model [47,48] is employed
using a BenedictWebbRubin (BWR) equation of state to evaluate
the p-v-Tbehavior of the inherent dense multicomponent mixtures.
Use of modified BWR equations of state in conjunction with the
extended corresponding states principle has been shown to provide
consistently accurate results over the widest range of pressures,
temperatures, and mixture states, especially at saturated conditions.
Fig. 1 Nonreacting shear-coaxial liquid-nitrogenhelium injector
operating at a) 1.0 MPa, and b) 6.0 MPa. TN2!97 K,THe !280 K
into GHe at T!300 K (from [2]).
1222 DAHMS AND OEFELEIN
Downloaded by STANFORD UNIVERSITY on May 31, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.B35562
approached. The initial divergence angle of the jet was measured at
the jet exit and compared with the divergenceangle of a large number
of other mixing layer flows, including atomized liquid sprays,
turbulent incompressible gaseous jets, supersonic jets, and
incompressible variable-density jets. This angle was plotted across
four orders of magnitude in the gas-to-liquid density ratio, which was
the first time such a plot has been reported over this large range. At
and above the critical pressure of the injected liquid, jet growth rate
measurements agreed quantitatively with the theory for incompress-
ible, variable-density, gaseous mixing layers. As the pressure was
reduced to progressively more subcritical values, the spreading rate
approached that measured for liquid sprays.
Candel et al. [4], Pons et al. [5], and Oschwald et al. [6,7]
demonstrated fundamental differences in combustion dynamics of
cryogenic flames between subcritical and supercritical pressure
conditions using experimental facilities from independent
laboratories. Such differences extended to fundamental quantities
such as liquid core lengths, turbulent length scales, and jet breakup
regimes. At supercritical pressure and temperature conditions,
Taskinogluand Bellan [8,9], Bellan [10], and Harstad and Bellan [11]
performed direct numerical simulations along with isolated and
cluster drop simulations considering Soret and Dufour effects to
investigate the mixing process and fluid disintegration in
supercritical mixing layers. Large-eddy simulations have been
performed using proposed new subgrid-scale terms, denoted
correctionterms, in concert with typical subgrid-scale flux terms
[8,9]. Sirignano et al. [12,13], Delplanque [14], and Sirignano [15]
discussed and investigated droplet and spray vaporization at
subcritical and supercritical pressures to study combustion and
dynamic responses to oscillatory ambient conditions. Those studies
established that the vaporization process can be a rate-controlling
process for driving combustion instabilities.
Many other works have served to corroborate these observations
[1634]. However, the absence of surface tension forces under some
high-pressure conditions still poses many fundamental questions. For
a pure fluid, basic theory dictates that surface tension forces become
diminished because the pressure of the liquid phase exceeds its
critical value. For multicomponent systems, however, this is
generally not the case, and modern theory still lacks a first principle
explanation for the observed phenomena. Previously, it has been
widely assumed that surface tension forces become diminished
because the temperature of the liquid phase exceeds its critical value.
Under many conditions, however, this mechanism must be ruled out.
In this paper, it is shown that the temperature of hot unburned gaseous
hydrogen is not sufficient to heat the liquid-oxygenhydrogen
(LOX-H2) interface to its critical temperature, and surface tension
forces do not diminish only because the critical pressure of the liquid
or the mixture critical pressure(as defined by Faeth et al. [35] and
Lazar and Faeth [36]) is exceeded. Conversely, there is significant
experimental evidence of substantial surface tension forces in various
binary mixtures at high pressures (ppC;L;p>500 bar) provided
in recent literature [3739]. The theory presented here serves to
explain the conditions when this occurs also.
In this paper, we use the theoretical framework introduced by
Dahms et al. [4042] to derive respective regime diagrams for
representative propellant combinations in liquid rockets. This
framework has been recently verified experimentally by Manin et al.
[43] and by Falgout et al. [44]. The model allows one to analyze
representative detailed gasliquid interface dynamics during
injection. Our analysis shows that, under certain high-pressure
conditions, gasliquid interfacial phenomena are not determined by
vaporliquid equilibrium and the presence of surface tension forces.
Instead, the interface enters the continuum length scale regime due to
a combination of reduced mean free molecular pathways and
broadened interfaces. Based on these insights, a regime diagram for
LOX-H2injection is presented. This diagram quantifies when the
system transitions between classical atomization and diffusion-
dominated mixing as a function of the ambient pressure and
temperature of the chamber. A similar diagram is derived for
n-decanegaseous-oxygen (LC10-GO2) systems. It is demonstrated
that the liquid injection process can exhibit characteristics of classical
spray atomization even at very high chamber pressures
(p>20 ·pC). Likewise, depending on the temperature and
composition of the fuel and oxidizer streams, liquid injection can
instead exhibit dynamics of dense-fluid or supercritical mixing
processes. The current analysis reinforces past observations that
drops can still be present even at pressures that are supercritical with
respect to the injected liquid and quantifies under what conditions
dense-fluid mixing occurs instead.
The liquid rocket engine conditions investigated here are
dramatically different compared to the representative diesel engine
injection conditions considered in past works. Despite these
distinctive differences in operating conditions (e.g., low cryogenic
temperatures, different molecular sizes and properties, high reduced
liquid injection, and ambient gas pressures and temperatures, etc.),
this study demonstrates that the related physical complexity is well
captured by our mapping approach based on the defined Knudsen
number. The presented analysis also quantifies and explains the
significant differences observed in the envelopes of transition
conditions from classical spray atomization to dense-fluid jet
dynamics between rocket systems operated using liquid-oxygen
hydrogen and liquid n-decaneoxygen propellant combinations.
II. Model Formulation
The starting point of our model is based on the framework
developed by Oefelein [45,46], which provides a generalized
treatment of the thermodynamics and transport processes for
hydrocarbon mixtures at near-critical and supercritical conditions.
This scheme is combined with the theoretical framework developed
by Dahms et al. [40] and Dahms and Oefelein [41], which facilitates
vaporliquid equilibrium and gasliquid interface structure
calculations to compute both surface tension forces and the local
two-phase interface thicknesses. Here, only a description of the
general features of the model framework is provided. A detailed
derivation can be found in [41].
A. Thermodynamic and Transport Properties
The extended corresponding states model [47,48] is employed
using a BenedictWebbRubin (BWR) equation of state to evaluate
the p-v-Tbehavior of the inherent dense multicomponent mixtures.
Use of modified BWR equations of state in conjunction with the
extended corresponding states principle has been shown to provide
consistently accurate results over the widest range of pressures,
temperatures, and mixture states, especially at saturated conditions.
Fig. 1 Nonreacting shear-coaxial liquid-nitrogenhelium injector
operating at a) 1.0 MPa, and b) 6.0 MPa. TN2!97 K,THe !280 K
into GHe at T!300 K (from [2]).
1222 DAHMS AND OEFELEIN
Downloaded by STANFORD UNIVERSITY on May 31, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.B35562
Figure 4. Jet break-up of coaxial LN2/He injection into a He environment at sub- (left) and
supercritical (right) pressure (Mayer et al. 1998).
2.2. Relevance of pure fluid behavior for injection
As the whole point of injection in technical systems is to create mixtures, one may
question the relevance of studying pure fluid behavior. Particularly at high pressures,
mixture behavior may deviate substantially from pure fluid behavior and ideal gas mixing.
Figure 3 shows the schematic of a reactive shear layer in LOX/GH2 injection at su-
percritical pressure. Oxygen is injected in a dense, liquid state before undergoing pseu-
doboiling and transition to an ideal gas, as depicted in Figure 2.
Several researchers have pointed out that mixing within the hot flame occurs at ideal
gas conditions (Oefelein & Yang 1998; Ribert et al. 2008; Lacaze & Oefelein 2012),
whereas mixing under real gas conditions is essentially negligible (Banuti et al. 2016a,b).
This means that the transition from a real to an ideal gas occurs in the pure oxygen
stream; the bulk disintegration of the LOX jet in LOX/GH2 combustion is a pure fluid
phenomenon.
3. Towards a classification of supercritical injection
Jet break-up regimes are typically presented in the Reitz diagram for single jets, or in
the Chigier diagram for coaxial jets (Kuo & Acharya 2012). In addition to the Reynolds
number, the characterization is carried out in terms of the Ohnesorge or the Weber
number - both of which are functions of surface tension.
Figure 4 shows two results of a classical injection experiment by Mayer et al. (1998).
Injection at subcritical pressure exhibits a distinct interface as a sign of a finite sur-
face tension. The aforementioned classification is thus possible. In contrast, injection at
supercritical pressure resembles turbulent mixing of gaseous jets; the effect of surface ten-
sion is negligible. Thus, the current classification is not suitable to analyze supercritical
injection.
How can we distinguish different supercritical disintegration modes? Figure 5 defines
processes A to E and compiles the respective shadowgraphs; conditions are shown in
Table 1.
214 Banuti, Raju & Ihme
Tcr
pcr
A
C
D
B
E
Figure 5. Left: Injection processes in pTdiagram, with coexistence line (solid) and
Nishikawa-Widom line (dashed). Right: Visualizations of corresponding injection regimes. Con-
ditions in Table 1. A and B are modified from Chehroudi et al. (2002), C is modified from
Lamanna et al. (2012), D from Branam & Mayer (2003), and E is modified from Stotz et al.
(2011).
Type Tr,in pr,in Tr,pr,Jet Chamber
A 0.71-0.87 0.23 2.38 0.23 N2N2
B 0.71-0.87 1.23 2.38 1.23 N2N2
C 0.95 6.59 1.9 0.02 C6H14 Ar
D 1.05 1.77 2.36 1.77 N2N2
E 1.09 6.59 1.9 0.16 C6H14 Ar
Table 1. Conditions of visualizations in Figure 5. Reduced values with respect to injected
fluid.
Process A is the subcritical jet treated in Reitz’ diagram. Increasing surrounding and
injection pressure above the supercritical value leads to case B. In C, a supercritical fluid
close to the critical point is injected with a large pressure drop. D is reached when the
injection temperature of B is raised. Finally, increasing the temperatures of C leads to
process E.
From the five cases visualized in Figure 5, three groups can be distinguished: Case A
is the classical liquid jet with a distinct surface. Cases B and D are similar but, due to
negligible surface tension effects, exhibit a more convoluted and diffuse interface. Cases
C and E are remarkably different from the previous cases, showing a high-pressure jet
expanding in a lower-pressure environment, similar to ideal gas expansion into vacuum.
Types B and D deserve more discussion as they are representative for Diesel and rocket
engines. As the injection temperature increases, the injection density decreases, resulting
in the reduced density ratio of case D. Note that the injection density is very sensitive
to the injection temperature around the pseudoboiling temperature (see Figure 1), with
associated challenges for experiments and their evaluation. If the temperature of case D
is increased to more than twice the critical temperature, D will have gradually changed
to an ideal gas jet (see Figure 2).
Classically, any isobaric injection process crossing the critical temperature is referred
to as transcritical injection. We see that case D strongly resembles case B. Both have in
common that they cross the Nishikawa-Widom line. We will use supercritical injection
as a generic term encompassing any of the cases B to E.
Supercritical injection and fluid behavior 215
Figure 6. Large eddy simulation of transcritical nitrogen injection, (Ma et al. 2014, 2016;
Hickey et al. 2013). Contours for temperature in K, and the isosurface marks the jet core.
Top: isosurface as arithmetic mean of injected and surrounding density; bottom: isosurface at
pseudoboiling conditions.
3.1. A thermodynamic definition of the supercritical boundary
Figure 5 indicates that case B does not have a sharp interface, unlike case A. Nonetheless,
the outline of the jet is clearly discernible in the shadowgraph. The question is, how do
we define the boundary of such a supercritical jet?
One approach is to use the arithmetic mean of the injected and surrounding density
(Jarczyk 2013). This definition of the boundary depends on the process parameters: when
we change the injection or surrounding temperature (and thus density), our definition
of the boundary in terms of density will also change. In contrast, the boundary of the
subcritical case A is uniquely identified regardless of injection temperature, because it
is associated with a thermodynamic state instead of a process parameter. Figure 1 illus-
trates that the interface will always be present at saturation conditions, even if inflow
conditions or surrounding conditions are varied.
An analogous definition for transcritical injection is to use the pseudoboiling state as
a unique marker. It, too, is associated with large gradients in density (∂ρ/∂ T )p, and is
a purely thermodynamic condition. Figure 6 compares the jet core determined from the
arithmetic density mean (top) to the thermodynamic pseudoboiling condition (bottom).
Both yield comparable results for the presented case, but the thermodynamic definition
resolves any ambiguity.
3.2. The latent heat of supercritical fluids
The latent heat diminishes for increasing pressure as the fluid approaches the critical
point. However, counterintuitively, the supercritical heating process B in Figure 5 does
not require less energy than the subcritical process A.
Energy needs to be supplied to the liquid when heating to an ideal gas state to overcome
216 Banuti, Raju & Ihme
Figure 7. Comparison of sub- and supercritical heating processes for oxygen. For low and
high temperatures, pressure-independent asymptotes are approached by the enthalpy isobars.
Thus, the processes are energetically equivalent, regardless of pressure. Data are for oxygen from
Linstrom & Mallard (2016).
intermolecular forces, regardless of whether the pressure is sub- or supercritical. High
pressure real fluid effects merely distribute this latent heat over a finite temperature
interval. We can interpret the required energy input during pseudoboiling – signified by
the excess heat capacity that needs to be overcome – as a nonequilibrium latent heat of
vaporization.
Figure 7 illustrates this quantitatively by plotting enthalpy versus temperature for
three pressures. The fluid is oxygen, a pressure of 4 MPa constitutes a subcritical condi-
tion, and 6 MPa and 10 MPa are supercritical. Towards lower temperatures, all isobars
converge towards the same liquid enthalpy asymptote hL(T); towards higher tempera-
tures, the isobars converge towards the same ideal gas enhalpy 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
Figure 7 for the transition from TL= 130 K to TG= 460 K, which requires the same
hLG at all shown pressures.
Intermolecular forces do not just vanish when a liquid is compressed beyond the critical
pressure; the energy needed to overcome these forces needs to be supplied regardless of
pressure.
4. A pseudoboiling-line for general fluids
We have discussed the fact that the pseudoboiling process is not merely an abstract
thermodynamic concept, but has very specific implications about understanding and
interpreting high-pressure injection. It is thus desirable to predict where this phase change
occurs, i.e., we seek an equation for the location of the Nishikawa-Widom line.
4.1. Nishikawa-Widom line for simple fluids
A suitable relation between the reduced pressure prand reduced temperature Tris
pr= exp [A(Tr1)] ; A= 5.5,(4.1)
Supercritical injection and fluid behavior 217
Figure 8. Comparison of fluid data (symbols) with Eq. (4.1) and A= 5.5 for the
Nishikawa-Widom line. Agreement is good for simple fluids but fails for water and hydrogen.
Data are from Linstrom & Mallard (2016).
for the simple fluids O2, N2, and Ar (Banuti (2015)). Figure 8 shows good agreement
between Eq. (4.1), and data of heat capacity peaks for nitrogen, oxygen, and argon.
However, Figure 8 also shows that the relation fails for water and hydrogen. This is
a serious shortcoming, as hydrogen and water play an important role in combustion.
Similarly, Eq. (4.1) fails for hydrocarbons.
4.2. Nishikawa-Widom line for general fluids
Figure 8 not only demonstrates the shortcoming of Eq. (4.1), but also suggests a remedy:
instead of a constant A, a fluid-dependent coefficient Asis introduced and determined
for each species,
pr= exp [As(Tr1)] ; Asspecies dependent.(4.2)
Table 2 provides Asfor a wide range of fluids, including hydrocarbons up to n-hexane.
Comparison with the acentric factor ω, also provided in Table 2, suggests an essentially
linear relation, which is confirmed in Figure 9. Thus, Eq. (4.2) can be recast in terms of
ωas
pr= exp [Aω(Tr1)] ; Aω= 5.29 + 4.35ω, (4.3)
allowing for a fast evaluation without requiring fluid p-v-Tdata.
4.3. An improved fitted relation
Equation (4.3) is convenient in its ease of parameter determination from the tabulated
acentric factor; Figure 8 hints at another shortcoming of Eq. 4.1: while the Nishikawa-
Widom line is a straight line in a log-linear p-Tplot, water and hydrogen exhibit a
significant curvature of their graphs.
To incorporate this deviation, a modified expression is proposed in the form
pr= exp [A0(Tr1)a] ; A0, a species dependent.(4.4)
Data for A0and aare obtained by fitting and are compiled in Table 2.
Figure 10 compares Eq. (4.1) using A, Eq. (4.2) using As, and Eq. (4.4) using A0and
218 Banuti, Raju & Ihme
Species ω AsA0a
Quantum gases
He 0.382 3.516 2.282 0.772
H20.219 4.137 3.098 0.849
Monatomic
Ne 0.0387 5.028 5.331 1.034
Ar 0.00219 5.280 5.237 0.994
Kr 0.0009 5.307 5.191 0.987
Xe 0.00363 5.326 5.241 0.991
Diatomic
O20.0222 5.428 6.742 1.112
N20.0372 5.589 6.703 1.097
F20.0449 5.686 5.456 0.977
CO 0.050 5.750 6.238 1.044
Hydrocarbons
CH40.01142 5.386 5.947 1.058
C2H60.0993 5.687 5.116 0.941
C3H80.1524 5.882 5.037 0.921
C4H10 0.201 6.257 5.060 0.895
C5H12 0.251 6.117 4.876 0.901
C6H14 0.299 6.688 5.365 0.921
Other polyatomic
CO20.22394 6.470 8.256 1.102
NH30.25601 6.235 5.799 0.962
R124 0.28810 6.597 6.002 0.950
H2O 0.3443 6.479 5.448 0.911
Table 2. Acentric factor ωand slope of the Nishikawa-Widom-line for a number of species
obtained from NIST Linstrom & Mallard (2016). Asfor Eq. (4.2), A0,a, for Eq. (4.4). Results are
clustered (from top to bottom) as quantum gases, noble gases, diatomic molecules, hydrocarbons,
and other complex molecules.
Figure 9. Linear relation between fluid acentric factor ωand the parameter Aof the
Nishikawa-Widom-line Eq. (4.2).
Supercritical injection and fluid behavior 219
Figure 10. Comparison of Nishikawa-Widom-line fluid data (symbols) and correlations
Eq. (4.1) with A, Eq. (4.2) with As, Eq. (4.4) with A0, a.
a. It can be seen that Eq. (4.4) offers the best accuracy over a wide range of acentric
factors and can accurately capture the curvature of non-simple fluids.
5. Conclusions
The present paper discusses some implications of supercritical thermodynamics for
injection problems, relevant to rocket engines, gas turbines, and Diesel engines.
Break-up processes are typically classified in terms of the ratio between surface ten-
sion forces to inertial or viscous forces. This classification is not always applicable to
supercritical injection, as the surface tension may vanish. We demonstrate that injection
pressure ratio and temperature interval are a useful approach to analyze supercritical
injection.
The most prominent feature of the supercritical state space is the Nishikawa-Widom
line, which is defined as the locus of specific isobaric heat capacity peaks. Pseudoboiling
is the supercritical nonequilibrium phase change that occurs when the Nishikawa-Widom
line is crossed. We demonstrate that the energy required to heat a fluid from a subcritical
liquid temperature to an ideal gas state is identical at sub- and supercritical pressures,
despite the vanishing latent heat with increasing pressure. This can be interpreted as a
distributed latent heat.
We find that the Nishikawa-Widom line is well described by pr= exp[Aω(Tr1)],
with Aω= 5.29 + 4.35ω. The acentric factor ωcan be found in fluid data tables. This
relation offers a reliable and convenient way to predict the temperature at which many
thermodynamic and transport properties exhibit the strongest gradients or extrema for
a wide range of fluids.
The pseudoboiling condition can also be used to define a thermodynamically unique
and consistent boundary of a transcritical jet, analogous to the subcritical liquid–gas
interface.
220 Banuti, Raju & Ihme
Acknowledgments
Financial support through NASA Marshall Space Flight Center is gratefully acknowl-
edged. The authors would like to thank Peter C. Ma for providing Figure 6.
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... (5) hints at a shortcoming: while the fitted widom line is straight in a log-linear P-T plot, some species exhibit a significant curvature of the actual widom line. To incorporate this deviation, a modified expression is proposed with species-dependent coefficients A ′ and a [31]: ...
Article
Sufficient understanding of the supercritical state is important for exploring supercritical jet spray in thermodynamic cycle power machinery. However, in conventional studies, the phase division of supercritical state and supercritical jet spray is simple and rough, and whether the pseudo-boiling (or widom) line exists widely in various fluids, especially fossil fuels, is still not proved by experiments. In this paper, the phase inhomogeneity and separability in the supercritical state of fossil fuels are studied by using high-speed Mie scattering and UV-LAS (ultraviolet–visible laser absorption/scattering) optical diagnostic techniques. Acetone is selected as the single representative component of fossil fuels. The results show that acetone is inhomogeneous in the supercritical state at fuel temperature Tf<1.13Tc and fuel pressure Pf<1.74Pc, where Tc and Pc are the critical temperature and pressure, and can be further divided into liquid-like phase and gas-like phase, which is similar to the properties of inert fluids and simple fluids reported in previous literatures. The transition region (measured at interval of pressure 0.2 MPa) between the liquid-like phase and gas-like phase contains thermodynamic extremum lines such as constant pressure specific heat, volume expansion coefficient, and isothermal compression rate, which supports the view that the widom (or pseudo-boiling) line is used as the crossover in the supercritical state. When the state of acetone spray passes through the transition region, the liquid-like and gas-like phases will have dramatic changes in the optical and macroscopic properties of spray (For example, when the injection temperature is equal to the critical temperature of 235 °C, the stable liquid-like areas crossing the pseudo-boiling line decreases by 9 times as much as that without crossing the pseudo-boiling line). The results of this paper have important reference values for the study of the supercritical spray atomization regime and the establishment of the simulation model.
... This line divides liquid-like and gas-like supercritical states (Gorelli et al. 2006;Simeoni et al. 2010), and exhibits some properties of a phase transition (Banuti 2015). The Widom line has since been shown to differ between species as a function of the acentric factor (Banuti et al. 2016b(Banuti et al. , 2017a, and even to occur multiply in certain mixtures (Raju et al. 2017). ...
Thesis
L’étude de l’injection d’un fluide dans des conditions de hautes pressions reste encore aujourd’hui un challenge. Lorsque la pression critique des fluides est dépassée, l’état supercritique est atteint, faisant disparaître la distinction entre liquide et gaz. Pour ces conditions extrêmes, les données expérimentales sont peu nombreuses et nécessitent d’être consolidées.Dans cette étude, un nouveau banc d’essai a été réalisé au laboratoire CORIA dans le but d’étudier des injections non-réactives d’éthane et de propane dans une atmosphère sub- et supercritique d’azote ou d’hélium. Les données ont été collectées à partir de quatre diagnostics optiques : l’ombroscopie, la DBI, la radiographie et la CBOS. Des informations qualitatives sur la topologie des jets et de leur couche de mélange sont apportées. Des mesures quantitatives de longueur de coeur dense, d’angle d’ouverture et de densité sont complétées par une étude phénoménologique à l’aide de la théorie des mélanges binaires.
Technical Report
Full-text available
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