ArticlePDF Available

Abstract and Figures

Recent publications in the open literature have shown that supercritical fluid states are not homogeneously distributed but, in fact, can be differentiated into two distinct regions with gas-like and liquid-like properties, respectively. These regions are divided by an extension of the coexistence line, commonly called Widom line. This paper shows that a supercritical analog to subcritical phase change, pseudoboiling, does exist when crossing this demarcation. The supercritical state transition does not occur in a phase equilibrium but takes place over a finite temperature interval. While subcritical vaporization requires energy to overcome intermolecular attraction, supercritical state transitions additionally require energy to increase the temperature. It could be shown that the attractive potential is the dominant energy sink up to a reduced pressure of 1.5 for argon, nitrogen, oxygen, and water. The effect reduces with growing pressure and becomes negligible for p/pcr > 3. Furthermore, a new equation for this Widom- or pseudoboiling line is given. It exhibits improved accuracy over previously published equations; performing a limit analysis of the Clapeyron equation allows to express its sole parameter purely in terms of thermodynamic variables. This parameter can then be evaluated from an equation of state or from fluid data - no nonphysical fitting is required.
Content may be subject to copyright.
Crossing the Widom-line – supercritical pseudo-boiling
D.T. Banuti
German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology, Spacecraft Department,
37073 G¨ottingen, Bunsenstr. 10, Germany; Phone: +49 551 709 2403
Recent publications in the open literature have shown that supercritical fluid states are not homo-
geneously distributed but, in fact, can be differentiated into two distinct regions with gas-like and
liquid-like properties, respectively. These regions are divided by an extension of the coexistence
line, commonly called Widom line. This paper shows that a supercritical analog to subcritical
phase change, pseudoboiling, does exist when crossing this demarcation. The supercritical state
transition does not occur in a phase equilibrium but takes place over a finite temperature interval.
While subcritical vaporization requires energy to overcome intermolecular attraction, supercritical
state transitions additionally require energy to increase the temperature. It could be shown that
the attractive potential is the dominant energy sink up to a reduced pressure of 1.5 for argon,
nitrogen, oxygen, and water. The effect reduces with growing pressure and becomes negligible for
p/pcr >3. Furthermore, a new equation for this Widom- or pseudoboiling line is given. It exhibits
improved accuracy over previously published equations; performing a limit analysis of the Clapey-
ron equation allows to express its sole parameter purely in terms of thermodynamic variables. This
parameter can then be evaluated from an equation of state or from fluid data - no nonphysical
fitting is required.
Keywords: pseudoboiling, Widom-line, vaporization, Clapeyron, transcritical
1. Introduction
While Andrews [1] introduced the notion of a critical point to terminate the vapor pressure curve
more than 100 years ago, today there is not a common terminology on states exceeding the critical
Corresponding author
Email address: (D.T. Banuti)
Preprint submitted to Elsevier April 6, 2017
temperature or pressure (Bellan [2], Younglove [3]). The common view is that upon exceeding the
critical point, liquids and gases do no longer exist. Instead, a structureless, homogeneous, continuous
supercritical fluid prevails throughout the state space. Furthermore, given the vanishing of latent
heat of vaporization and surface tension, a phase transition no longer occurs.
Fundamentally new insight has been gained in the last two decades, facilitated by advances in
experimental and numerical methods. Inelastic x-ray scattering could be used to study dispersion,
i.e. the dependence of the speed of sound on the frequency, which is a phenomenon found in
liquids but not in gases. Using this technique, Simeoni at al. [4] and Gorelli et al. [5] were able to
attribute this property to supercritical fluids at reduced pressures (pr=p/pcr ) exceeding 1000. They
concluded that the supercritical domain is divided into two regions with distinct respective liquid-
like and gas-like behavior - much like at subcritical conditions. The dividing line is an extension to
the coexistence line. Gorelli suggested to extrapolate the Plank-Riedel equation, Eq. (1), into the
supercritical domain,
ln p
pcr =a+bTcr
T+cln T
Tcr (1)
where a= 4.270, b =4.271, and c= 1.141 have been fit to neon, oxygen, and nitrogen fluid data.
From a different perspective, Sciortino et al. [6] used molecular dynamics computations to
investigate properties of this line emanating from the critical point which was later dubbed ‘Widom’
line. Figure 1 illustrates this new view of supercritical states structure. The work of both groups
is summarized by MacMillan and Stanley [7], Simeoni et al. [4] declare a new chapter to be open,
new theories to be required. All research was concerned with the understanding of thermodynamic
states and the identification of the boundary between them.
So far, no quantitative discussion has been given on processes in which this boundary is crossed.
A name, however, exists: ‘pseudoboiling’ has been used by Okamato et al. [8] and Oschwald et
al. [9] to describe a supercritical liquid-like to gas-like transition. The term has originally been
introduced by Kafengauz and Federov [10, 11], who described a phenomenon in cooling pipes which
resembled subcritical boiling - but at supercritical pressures.
The concept is not undisputed: Hall [12] outright rejected it as “irrational and unnecessary
because, clearly, two distinct phases do not exist at supercritical pressures”, whereas Okamato et
al. [8] called it a theoretical problem to be solved. Santoro and Gorelli [13] hypothesize there
might be a “sluggish liquidlike-gaslike transformation mimicking the subcritical liquid-gas phase
transition”. Oschwald et al. [14, 15] suggest that, unlike during subcritical vaporization, heat
Widom line
Figure 1: Fluid state plane and supercritical states structure with a liquid-like super-Widom fluid and a gas-like
sub-Widom fluid. TP=triple point, CP=critical point, Tr=T/Tcr ,pr=p/pcr.
added to the fluid during pseudoboiling will act to both expand and heat it.
It is therefore the purpose of this paper to analyze processes which cross the Widom line from
liquid-like to gas-like fluid states (and vice versa). The main question is: Can the concept of
vaporization/condensation be applied to transitions between supercritical states?
2. Material and Metho d
The approach taken here is mixed analytical and empirical. Thermodynamic relations and
approximations are expressed in analytical form. All fluid data is taken from the NIST webbook
[16]. First, the concept of pseudoboiling is evaluated using nitrogen data by Span et al. [17], with
particular focus on the specific isobaric heat capacity of states at supercritical pressure. After this
proof of concept, fluid data for argon (Tegeler et al. [18]) and oxygen (Schmidt and Wagner [19])
are investigated in a similar manner because of their expected similarity in behavior, following the
corresponding states principle. Finally, water data (Wagner and Pruss [20]) are shown, explicitly
to point out deviations for more complex fluids.
3. Results and Discussion
3.1. Widom Line Equation
The original definition of the Widom line is the set of supercritical maximal thermodynamic
correlation lengths [4]. However, this property is very hard to come by in a practical manner
(May and Mausbach [21]). As a substitute, response functions - such as the heat capacity, the
isothermal compressibility, and the thermal expansion - are used by several authors (e.g. Liu et al.
[22], Xu et al. [23], Santoro and Gorelli [13], Ruppeiner et al. [24]). Regarding a state transition
as a phenomenon of energy conversion in this paper, the specific isobaric heat capacity cpis an
appropriate marker. Figure 2(a) illustrates this point using the example of nitrogen.
At the subcritical pressure 3 MPa, cpdiverges at the boiling temperature: here, heat is added
during vaporization without an increase in temperature until all liquid is consumed. A distinct
peak remains visible for supercritical pressures, flattening and moving to higher temperatures as
the pressure increases from 4 MPa to 6 Mpa. Figure 2(b) shows how these peaks line up nicely
when plotted into a log(p)/T diagram. Thus, an equation of the form
= exp Tcr
θpb Tpb
1 = exp ATpb
1 (2)
suitably describes the cpbased Widom line, where ‘pb’ and ‘cr’ denote properties along the pseu-
doboiling line and at the critical point, respectively. For dimensional reasons, the integration
constant 1pb can be interpreted as the inverse of a characteristic temperature. This parameter
will now be determined from first principles.
Along the coexistence line from triple point to critical point, an equilibrium between the liquid
(L) and the vapor (V) phase holds. This can be expressed mathematically by equating the respective
phase differential Gibbs enthalpies which leads to the classical Clapeyron equation, Eq. (3),
The last equality holds as Tremains constant during the transition from liquid to vapor. The latent
heat of vaporization is ∆h=hVhL, the increase in specific volume and entropy are ∆v=vVvL
and ∆s=sVsL, respectively.
Classically, the Clapeyron equation is used up to the critical point. After all, the latent heat of
vaporization vanishes, Eq. (3) loses its significance. However, not only does the jump in enthalpy
Temperatu re (K )
Cp (J/g*K)
120 130 140 150
p / MPa = 3
(a) Nitrogen specific isobaric heat capacity cp(T) at sub- and
supercritical pressures.
Temperature (K)
Pressure (MPa)
60 80 100 120 140 160
Subcritical Vapor Pressure
Supercritical c p, max
Fit cp , max
Tripl e Poin t
Critical Poi nt
(b) Positions of maximum cprelative to coexistence line for
Figure 2: Construction of heat capacity based Widom line, for the example of nitrogen.
(or entropy) vanish - the same is true for the specific volume. Thus, instead of invalidating the
equation, progressing towards the critical point is mathematically taking the limit of a difference
fraction, and turning it into a differential, Eq. (4)
p,T pcr,Tcr
where subscript ‘sat’ denotes saturation conditions along the coexistence line. Figure 2(b) shows
that the slope ( d log p/ dT)sat is gradually decreasing in the subcritical region but remains equal
to the slope at the critical point upon entering a supercritical state,
d ln p
=d ln p
= const.(5)
Now, combining Eqs. (2), (5), and using the identity ( dln p)/dT= (1/p)( dp/ dT) from calcu-
lus, θpb can be determined as
pcr ds
pcr dp
The equation for the cpbased Widom line then reads
= exp Tcr
pcr ds
dvcr Tpb
1 (7)
= exp Tcr
pcr dp
dTcr Tpb
Note that Eqs. (7) and (8) do not contain any nonphysical fitting parameters and can be evaluated
solely from fluid data. Furthermore, they contain only nondimensional ratios, suggesting a fluid
independence akin to the corresponding states principle.
3.2. Analytical van der Waals Widom Parameter
Using Eq. (8), the coefficient Aintroduced in Eq. (2) can be expressed as
pcr dp
It is now interesting to see whether Amay be evaluated from first principles using Eq. (9). The
van der Waals equation of state will serve as an example.
As a reference, an AvdW,sat will be determined for Schwabl’s [25] equation for the vapor pressure
curve of a van der Waals fluid
pr,sat = 4∆Tr,sat +24
5(Tr,sat)2+O(Tr,sat )5/2.(10)
Parameters ∆φr,sat are defined as ((φφcr)cr) along the saturation curve. Thus, in the limit
towards the critical point
Then, higher order terms may be neglected and Eq. (10) reduces to
pr,sat,cr = 4∆Tr,sat,cr.(12)
Rearranging and differentiating, we obtain the reference value of the slope of the vapor pressure
curve at the critical point
dTsat vdW,cr
= 4Tcr
which corresponds to
AvdW,sat = 4.(14)
Now, AvdW will be determined using Eq. (9). The nondimensional form of the van der Waals
equation can be written
The derivative of pressure with respect to temperature is
Then, as vr= 1 at the critical point,
AvdW =Tcr
pcr dp
= 4.(17)
Thus, at the critical point, the slope determined from the limit of Clapeyron’s equation introduced
here, Eq. (17), equals the slope of the vapor pressure curve equation, Eq. (14). 1
1Equal fugacities or free Gibbs enthalpies in the respective phases are criteria which determine phase equilibria
in a two phase region. Here, starting with Clapeyron’s equation Eq. (3), a free Gibbs enthalpy equality has been
regarded in the derivation of the vapor pressure curve slope at the critical point. It turned out that the expression
is significantly simplified due to the limiting procedure towards the critical point, Eq. (4). At the critical point only
one phase exists, hence criteria of equal Gibbs enthalpies or fugacities are identically fulfilled.
3.3. Empirical Widom Parameter
For all practical purposes, van der Waals’ equation of state does not provide sufficient accuracy.
Instead, using NIST [16] data for nitrogen, the Widom parameter has been determined to A5.6.
Results are shown in Fig. 3. The graph compares Eq. (2) using both the van der Waals coefficient
from Eq. (17) (dashed) and the value from nitrogen data (solid) to the equation suggested by Gorelli
(dash dot), Eq. (1). Overlain are points of maximum heat capacity extracted from the NIST [16]
database for argon, water, nitrogen, and oxygen. Figure 3 shows Eq. (2) in the form of Eq. (8)
to be reasonably accurate in the chosen domain for oxygen and argon in addition to nitrogen.
Water, chosen for its irregular properties, differs stronger. The improved accuracy compared to the
extrapolated Plank-Riedel equation proposed by Gorelli is apparent. Ais a species constant and
has a comparable value for a range of simple fluids.
Figure 3: Comparison of maximal cpdata from NIST with Eq. (2) and Gorelli’s Eq. (1).
3.4. Generalized Isobaric Phase Transitions
Now that an equation tells us where to expect some sort of transition phenomenon, we need
to investigate what to expect when passing through the Widom line. Let Tand T+be the (yet
to be quantified) start and endpoint of the isobaric transition, such that Tis smaller than the
temperature of maximum cp, and T+is larger. Then, in order to pass isobarically from a liquid-like
Figure 4: Finite heat capacity in state transition at supercritical pressures.
to a gas-like state, some amount of energy per unit mass
hpb =
cp(T) dT=h(T+)h(T) (18)
needs to be added to the fluid. The most important realization is that under supercritical conditions,
added energy is used to both overcome molecular attraction and raise the temperature of the fluid
(Oschwald and Schik [14]). The first will be dubbed structural (st) and the latter thermal (th)
contribution. This stands in contrast to subcritical phase changes during which the temperature
remains constant and thus all added energy is used to overcome molecular attraction.
In order to differentiate the energy needed for structural and thermal changes, both aspects
need to be separated. Figure 4 illustrates the chosen approach. Equation (18) represents the whole
area under the cp(T) curve from Tto T+, the whole amount of energy needed to pass between
both temperatures. The isobaric specific heat capacity approaches two values: at low temperatures
a liquid heat capacity cp,L, at higher temperatures the ideal gas value cp,iG. While no theoretical
expression is known for cp,L[26], the variation is small and can be neglected compared to cp,max . It
is thus assumed essentially constant for a species over a wide range of pressures. For a calorically
perfect gas, cp,iG =γR/(γ1), where γis the isentropic exponent and Rthe gas constant. Tpb
is the temperature at which the heat capacity reaches its maximal value cp,pb. Furthermore, let
Tpb =T+T.
Consider a reference liquid fluid which only raises the temperature upon heat addition. The
energy needed to heat this fluid isobarically from Tto T+is ∆hth =cp,LTpb. The difference
between the overall change and the thermal contribution is the required excess in energy due to the
structural contribution of pseudoboiling. The structural energy ∆hst is thus ∆hst = ∆hpb hth.
Figure 5: Illustration of isobaric supercritical phase change. Determination of interval (T+T) for oxygen at
p= 6 MPa using enthalpy asymptotes for the ideal gas hiG and liquid hLreference states.
Now the actual temperature interval has to be determined. We can use the previously used heat
capacities cp,L,cp,iG, and cp,pb to determine characteristic functions in the h/T diagram, Fig. 5.
hiG(T) = γR
γ1T, (19)
hL(T) = cp,LT+h0,L,(20)
hpb(T) = cp,pb (TTpb) + h0,pb .(21)
Per definition, Eqs. (19) and (20) are independent of pressure; Eq. (21) is valid for the particular
pressure under consideration which determines cp,pb =f(p), h0,pb =f(p), and Tpb =f(p). The
intersections of Eq. (19) and Eq. (21) determine T+, Eq. (20) and Eq. (21) determine T, valid for
the particular pressure of the regarded process:
T(p= const) = h0,Lh0,pb +cp,pbTpb
cp,pb cp,L
T+(p= const) = h0,pb cp,pbTpb
The required structural enthalpy can now be determined. However, its absolute value is not
necessarily meaningful: Fig. 2(a) shows how the cpdistribution widens and flattens for higher
Figure 6: Parameters B1from Eq. (24) as open symbols and B2from Eq. (25) as filled symbols for Ar, N2, O2, H2O.
pressures. The effect should be notable only if the structural energy is required in a sufficiently small
temperature interval and if it significantly exceeds the thermal energy. A convenient nondimensional
parameter to assess this ratio is thus
The reference factor cp,Lsuggests a simplified formulation based solely on a ratio of specific heats.
In order to emphasize the contribution of the state transition over mere heating, the excess specific
heat cp,pb cp,Lis used instead of just the maximum specific heat. The parameter thus reads
Figure 6 shows both parameters for nitrogen, oxygen, water, and argon. The values of the param-
eters are comparable over a range of fluids, especially for the simple fluids nitrogen, oxygen, and
argon. Figure 6 also shows that up to a reduced pressure of 1.5, the structural energy actually
exceeds the thermal energy: i.e. pseudoboiling raises the amount of energy required to heat the
fluid to T+by a factor of two. Even at pr= 3 there is a 10% excess over liquid heating alone.
As expected, the parameter diverges towards the critical point where the heating contribution van-
ishes. Furthermore, the behavior of the approximate B2closely follows the exact B1with an offset,
its evaluation however is largely simplified as transition temperatures need not be known. It thus
appears that B2is better suited for a practical evaluation of the pseudoboiling effect at a given
4. Conclusions
TP gas-like
Figure 7: Structure of the supercritical state space and comparison of subcritical (1) and supercritical isobaric
processes (2),(3).
Pseudoboiling does exist. In this paper, the first quantitative analysis has been introduced. This
phase transition between supercritical liquid-like and gas-like states is illustrated in Fig. 7. Pseu-
doboiling in this sense is a continuous, nonlinear, transcritical (i.e. starts at subcritical temperature
and ends at both supercritical temperature and pressure) process resembling subcritical boiling. It
occurs when the Widom- or pseudoboiling-line is crossed. The associated massive reduction in
density and a high specific heat capacity strongly resemble classical subcritical vaporization, Fig. 7
process (1). At supercritical pressures, the temperature of transition rises with the pressure, i.e.
the transition temperature of Fig. 7 process (3) is higher than in process (2). A simple algebraic
expression with good accuracy has been found for the Widom- or pseudoboiling-line in terms of
reduced pressure and reduced temperature: pr,pb = exp [5.55(Tr1)]. This equation allows for a
simple determination of the positions of maxima in thermal expansion and heat capacity. Isobaric
pseudoboiling is shown to require energy to both raise temperature (thermal) and overcome molec-
ular attraction (structural). Based on this, two nondimensional parameters are defined which show
that the structural contribution actually exceeds the thermal contribution up to a reduced pressure
of 1.5, it becomes negligible above pr3. Thus, at supercritical pressures, the critical temperature
loses its significance as a transition point and is replaced by the pseudoboiling temperature Tpb .
[1] T. Andrews, The Bakerian lecture: On the continuity of the gaseous and liquid states of matter,
Philosophical Transactions of the Royal Society (London) (159) (1869) 575–590.
[2] J. Bellan, Theory, modeling and analysis of turbulent supercritical mixing, Combustion Science
and Technology 178 (2006) 253–281.
[3] B. A. Younglove, Thermophysical properties of fluids. I. argon, ethylene, parahydrogen, ni-
trogen, nitrogen trifluoride, and oxygen, Journal of Physical and Chemical Reference Data 11
(1982) Supplement 1.
[4] G. Simeoni, T. Bryk, F. Gorelli, M. Krisch, G. Ruocco, M. Santoro, T. Scopigno, The Widom
line as the crossover between liquid-like and gas-like behaviour in supercritical fluids, Nature
Physics 6 (2010) 503–507.
[5] F. Gorelli, M. Santoro, T. Scopigno, M. Krisch, G. Ruocco, Liquidlike behavior of supercritical
fluids, Physical Review Letters 97 (2006) 245702–1 – 245702–4.
[6] F. Sciortino, P. Poole, U. Essmann, H. Stanley, Line of compressibility maxima in the phase
diagram of supercooled water, Physical Review E 55 (1) (1997) 727–737.
[7] P. McMillan, H. Stanley, Going supercritical, Nature Physics 6 (2010) 479–480.
[8] K. Okamoto, J. Ota, K. Sakurai, H. Madarame, Transient velocity distributions for the su-
percritical carbon dioxide forced convection heat transfer, Journal of Nuclear Science 40 (10)
(2003) 763–767.
[9] M. Oschwald, J. Smith, R. Branam, J. Hussong, A. Schik, B. Chehroudi, D. Talley, Injection of
fluids into supercritical environments, Combustion Science and Technology 178 (2006) 49–100.
[10] N. Kafengauz, M. Federov, Excitation of high frequency pressure oscillations during heat ex-
change with diisopropylcyclohexane, Inzhenerno-Fizicheskii Zhurnal 11 (1) (1966) 99–104.
[11] N. Kafengauz, M. Federov, Pseudoboiling and heat transfer in a turbulent flow, Inzhenerno-
Fizicheskii Zhurnal 14 (6) (1968) 923–924.
[12] W. Hall, Heat transfer near the critical point, in: T. I. Jr., J. Hartnett (Eds.), Advances in
Heat Transfer, Vol. 7, Academic Press, 1971, pp. 1–86.
[13] M. Santoro, F. Gorelli, Structural changes in supercritical fluids at high pressures, Physical
Review B 77 (2008) 212103–1 – 212103–4.
[14] M. Oschwald, A. Schik, Supercritical nitrogen free jet investigated by spontaneous Raman
scattering, Experiments in Fluids 27 (1999) 497–506.
[15] M. Oschwald, M. Micci, Spreading angle and centerline variation of density of supercritical
nitrogen jets, Atomization and Sprays 11 (2002) 91–106.
[16] P. Linstrom, W. Mallard (Eds.), NIST Chemistry WebBook, NIST Standard Reference
Database Number 69, National Institute of Standards and Technology, Gaithersburg MD,
20899, (retrieved 2013), Ch.
[17] R. Span, E. Lemmon, R. Jacobsen, W. Wagner, A. Yokozeki, A reference equation of state
for the thermodynamic properties of nitrogen for temperatures from 63.151 to 1000 k and
pressures to 2200 mpa, in: P. Linstrom, W. Mallard (Eds.), NIST Chemistry WebBook, NIST
Standard Reference Database Number 69, National Institute of Standards and Technology,
Gaithersburg MD, 20899,, (retrieved November 12, 2010)., 2013.
[18] C. Tegeler, R. Span, W. Wagner, A new equation of state for argon covering the fluid
region for temperatures from the melting line to 700 K at pressures up to 1000 MPa,
in: P. Linstrom, W. Mallard (Eds.), NIST Chemistry WebBook, NIST Standard Reference
Database Number 69, National Institute of Standards and Technology, Gaithersburg MD,
20899,, (retrieved November 12, 2010)., 2013.
[19] R. Schmidt, W. Wagner, A new form of the equation of state for pure substances and its
application to oxygen, in: P. Linstrom, W. Mallard (Eds.), NIST Chemistry WebBook, NIST
Standard Reference Database Number 69, National Institute of Standards and Technology,
Gaithersburg MD, 20899,, (retrieved November 12, 2010)., 2013.
[20] W. Wagner, A. Pruss, The iapws formulation 1995 for the thermodynamic properties of or-
dinary water substance for general and scientific use, in: P. Linstrom, W. Mallard (Eds.),
NIST Chemistry WebBook, NIST Standard Reference Database Number 69, National Institute
of Standards and Technology, Gaithersburg MD, 20899,, (retrieved
November 12, 2010)., 2013.
[21] H.-O. May, P. Mausbach, Riemannian geometry study of vapor-liquid phase equilibria and
supercritical behavior of the Lennard-Jones fluid, Physical Review E 85 (031201).
[22] L. Liu, S.-H. Chen, A. Faraone, C.-W. Yen, C.-Y. Mou, Pressure dependence of fragile-to-
strong transition and a possible second critical point in supercooled confined water, Physical
Review Letters 95 (2005) 117802–1 – 117802–4.
[23] L. Xu, P. Kumar, S. Buldyrev, S.-H. Chen, P. Poole, F. Sciortino, H. Stanley, Relation between
the Widom line and the dynamic crossover in systems with a liquid–liquid phase transition,
Proceedings of the National Academy of Sciences of the USA 102 (46) (2005) 16558–16562.
[24] G. Ruppeiner, A. Sahay, T. Sarkar, G. Sengupta, Thermodynamic geometry, phase transitions,
and the Widom line, Physical Review E 86 (2012) 052103–1 – 052103–4.
[25] F. Schwabl, Statistical Mechanics, Springer, 2006.
[26] J. Hirschfelder, C. Curtiss, R. Bird, Molecular Theory of Gases and Liquids, Wiley, 1954.
... On the multi-phase one, the supercritical fluid is also assumed to be a multi-phase-like fluid [41,42], where the dramatic variations of the thermophysical properties near the pseudo-critical temperature are regarded as the pseudo-phase change process [43]. Based on the multi-phase assumption, the pseudo-boiling [44] and pseudo-condensation [45] were developed for supercritical heating and cooling, respectively. According to the pseudo-boiling, the pseudo-film boiling was defined and the thickness of the vapor-like film determines whether HTD occurs. ...
... The phenomenon obeys similar laws as subcritical boiling [56] in which it is associated with a large heat capacity (c p ) and a steep change in density. The difference is that the latent heat Δi pc at the supercritical region is released within a temperature interval [44], instead of a constant saturation temperature in boiling case. Therefore, three regimes of LL regime, two-phase-like (TPL) regime and VL regime divided by T + and Twere proposed [49], in which the TPL represents the coexistence of LL and VL fluids, which has been visualized with neutron imaging by Maxim et al. [52]. ...
Heat transfer deterioration (HTD) problems can undermine the thermal safety of heating tubes for supercritical carbon dioxide (S-CO2) flows. Relevant experimental evidence is highly desired, but most cases were investigated for small tube diameters (below 10 mm) and under limited operation parameters, far from the industry-scale applications. In the present work, large tubes (24 mm) are investigated experimentally with pressures of 7.5-15 MPa, mass flow rates of 100-1200 kg·m-2·s-1 and heat fluxes of 30-350 kW∙m-2. The seriousness of HTD in large tubes was experimentally confirmed, and the mechanism is explored via a carefully-designed comprehensive comparison between present numerical simulations and experimental tests. Detailed analysis of vapor-like film development inspires us to mitigate the problem using structured-inner-surface, from the viewpoint of “supercritical pseudo-boiling”. This inspiration is further evaluated numerically: five types of enhancement structures are proposed to interfere with the development of vapor-like film. It was found that the proposed methods can fully prevent HTD problems, and especially the conical strips can achieve a high heat transfer performance (with performance evaluation criteria PEC of 1.21-1.35) and are thus recommended for HTD prevention in vertical large-diameter tubes under near-critical conditions. Overall, the experimental tests and numerical explorations can deliver more evidence on the theory and applications of supercritical fluid flow and heat transfer.
... The latter becomes apparent when considering recent investigations on supercritical fluid injection by Falgout et al. [2], Müller et al. [3], Baab et al. [4], Crua et al. [5], Lamanna et al. [6], and Gerber et al. [7]. Moreover, research on near-critical evaporation and the search for a criterion to predict the onset of transcritical dense-fluid mixing have been performed by Dahms and Oefelein [8], Qiu and Reitz [9], and Banuti [10]. Lamanna et al. [11] have presented a detailed review and discussion on the validity of these criteria. ...
Full-text available
With technical progress, combustion pressures have been increased over the years, frequently exceeding the critical pressure of the injected fluids. For conditions beyond the critical point of the injected fluids, the fundamental physics of mixing and evaporation processes is not yet fully understood. In particular, quantitative data for validation of numerical simulations and analytical models remain sparse. In previous works, transient speed of sound studies applying laser-induced thermal acoustics (LITA) have been conducted to investigate the mixing behaviour in the wake of an evaporating droplet injected into a supercritical atmosphere. LITA is a seedless, non-intrusive measurement technique capable of direct speed of sound measurements within these mixing processes. The used setup employs a high-repetition-rate excitation laser source and, therefore, allows the acquisition of time-resolved speed of sound data. For the visualisation of the evaporation process, measurements are accompanied by direct, high-speed shadowgraphy. In the present work, the measured speed of sound data are evaluated by applying an advection-controlled mixing assumption to estimate both the local mole fraction and mixing temperature. For this purpose, planar spontaneous Raman scattering results measured under the same operating conditions are evaluated using an advection-controlled mixing assumption with the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state. Successively, the resulting concentration–temperature field is used for the estimation of local mixture parameters from the detected speed of sound data. Moreover, models using the PC-SAFT equation of state and the NIST database for the computation of the speed of sound are compared. The investigations indicate a classical two-phase evaporation process with evaporative cooling of the droplet. The subsequent mixing of fluid vapour and ambient gas also remains subcritical in the direct vicinity of the droplet.
Full-text available
The modeling of fluids in the supercritical regime is addressed at conditions characteristic of liquid-propelled rocket engines, whose increasing performance demands paved the way for supercritical conditions. In the present document, nitrogen is used as a surrogate for the commonly encountered oxygen-hydrogen mixture so that turbulence mixing can be looked into without influences from combustion and chemically reacting effects. The temperature field validation on nitrogen coaxial injection at supercritical conditions, with high-velocity ratios (outer-to-inner), where the main (inner) stream is recessed relative to the outer stream, is of paramount importance in the flame stabilization operation of liquid rocket motors. The temperature field is analyzed taking into account varying momentum and velocity ratios, whose increase leads to a reduction of potential core lengths, increasing jet spreading. The results also depict a fundamental influence of thermal effects, dominating over the transport of momentum. The experimental data and large eddy simulation solvers from the literature agree with the estimate of injection velocities at several conditions and are comparable to the space shuttle main engine pre-burner.
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 chapter describes the behavior of thermodynamic and transport properties near the critical point. The near-critical region may be thought of as that region, in which boiling and convection merge. When the pressure is sufficiently subcritical or supercritical, the problem tends toward either a boiling problem or a constant property convection problem. Under such conditions, existing theoretical and empirical methods are generally adequate. The chapter concentrates on the region rather close to the critical point where the property variations are severe and where there are very significant heat transfer effects. The equations of continuity, momentum, and energy are examined with a view to revealing the effect of variable properties and deciding whether the same simplifications can be made as are common with a constant property fluid. Various modes of heat transfer are also discussed, particular attention being given to the interaction between forced and free convection.
Experimental data are presented on heat transfer with diisopropyl-cyclohexane accompanied by high-frequency pressure oscillations. A hypothesis is advanced relating to the mechanism causing these oscillations.
Experimental data characterizing the onset of pseudoboiling in relation to flow rate, pressure, and tube diameter are presented for diisopropylcyclohexane (DICH).
In 1863 the author announced, in a communication which Dr. Miller had the kindness to publish in the third edition of his 'Chemical Physics,’ that on partially liquefying carbonic acid by pressure, and gradually raising at the same tune the temperature to about 88° Fahr., the surface of de­marcation between the liquid and gas became fainter, lost its curvature, and at last disappeared, the tube being then filled with a fluid which, from its optical and other properties, appeared to be perfectly homogeneous. The present paper contains the results of an investigation of this subject, which has occupied the author for several years. The temperature at which carbonic acid ceases to liquefy by pressure he designates the critical point, and he finds it to be 30°·92 C. Although liquefaction does not occur at temperatures a little above this point, a very great change of density is produced by slight alterations of pressure, and the flickering movements, also described in 1863, come conspicuously into view. In this communication, the combined effects of heat and pressure upon carbonic acid at temperatures varying from 13° C. to 48° C., and at pressures ranging from 48 to 109 atmospheres, are fully examined.
Previous studies of supercritical three-dimensional mixing layers are reviewed to derive a unified understanding of supercritical turbulence and mixing. These studies consisted of Direct Numerical Simulations of mixing layers having initially a single chemical species in each of the two free streams. Each mixing layer was initially perturbed, which led to a double pairing of four initial spanwise vortices. These pairings yielded in each case an ultimate vortex within which small scales proliferated, resulting in a state having turbulence characteristics, called a transitional state. The evolution of the layer to this transitional state and the state itself were previously analyzed to elucidate the features of supercritical turbulence and mixing. This analysis is here used to classify those supercritical turbulent mixing characteristics that are species-system independent or species-system dependent. Finally, comments are offered on future prospects of developing small-scale models particularly suited for Large Eddy Simulations of supercritical turbulent mixing.
A new formulation for the thermodynamic properties of nitrogen has been developed. Many new data sets have become available, including high accuracy data from single and dual-sinker apparatuses which improve the accuracy of the representation of the p&rgr;T surface of gaseous, liquid, and supercritical nitrogen, including the saturation states. New measurements of the speed of sound from spherical resonators yield accurate information on caloric properties in gaseous and supercritical nitrogen. Isochoric heat capacity and enthalpy data have also been published. Sophisticated procedures for the optimization of the mathematical structure of equations of state and special functional forms for an improved representation of data in the critical region were used. Constraints regarding the structure of the equation ensure reasonable results up to extreme conditions of temperature and pressure. For calibration applications, the new reference equation is supplemented by a simple but also accurate formulation, valid only for supercritical nitrogen between 250 and 350 K at pressures up to 30 MPa. The uncertainty in density of the new reference equation of state ranges from 0.02% at pressures less than 30 MPa up to 0.6% at very high pressures, except in the range from 270 to 350 K at pressures less than 12 MPa where the uncertainty in density is 0.01%. The equation is valid from the triple point temperature to temperatures of 1000 K and up to pressures of 2200 MPa. From 1000 to 1800 K, the equation was validated with data of limited accuracy. The extrapolation behavior beyond 1800 K is reasonable up to the limits of chemical stability of nitrogen, as indicated by comparison to experimental shock tube data.
This work reviews the available data on thermodynamic properties of argon and presents a new equation of state in the form of a fundamental equation explicit in the Helmholtz energy. The functional form of the residual part of the Helmholtz energy was developed by using state-of-the-art linear optimization strategies and a new nonlinear regression analysis. The new equation of state contains 41 coefficients, which were fitted to selected data of the following properties: (a) thermal properties of the single phase (p&rgr;T) and (b) of the liquid–vapor saturation curve (ps,&rgr;′,&rgr;″) including the Maxwell criterion, (c) speed of sound w, isochoric heat capacity cv, second and third thermal virial coefficients B and C and second acoustic virial coefficient βa. For the density, the estimated uncertainty of the new equation of state is less than ±0.02% for pressures up to 12 MPa and temperatures up to 340 K with the exception of the critical region and less than ±0.03% for pressures up to 30 MPa and temperatures between 235 and 520 K. In the region with densities up to half the critical density and for temperatures between 90 and 450 K the estimated uncertainty of calculated speeds of sound is in general less than ±0.02%. The new formulation shows reasonable extrapolation behavior up to very high pressures and temperatures. Independent equations for the vapor pressure, for the pressure on the sublimation and melting curve and for the saturated liquid and saturated vapor densities are also included. Tables for the thermodynamic properties of argon from 84 to 700 K for pressures up to 1000 MPa are given.
The critical point of a fluid is defined as the point beyond which it ceases to exhibit distinct liquid- or gas-like states. A crossover between liquid-like and gas-like behaviour observed by inelastic X-ray scattering suggests subtle effects involving nanoscale fluctuations in the one-phase region above the critical point.