Seven Questions About Supercritical Fluids

AIAA 2017-1106 AIAA SciTech Forum 9 - 13 January 2017, Grapevine, Texas 55th AIAA Aerospace Sciences Meeting

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 thermodynamic 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.1 Nonetheless, 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 8

Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 nitrogen with a helium co-flow by Mayer et al. in 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 developed 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.

1 of 15

American Institute of Aeronautics and Astronautics Copyright © 2017 by D. Banuti, P. Ma, J.-P. Hickey, M. Ihme. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. (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 interfacial 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 transition 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.2 later dubbed this process pseudoboiling and the locus of maximum heat capacity pseudoboiling line. A quantitative 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-T state-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 6 Tr = T/Tcr and reduced pressure pr = p/pcr. Bellan suggests 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. Oefelein5 and Candel et al.3 18 consider quadrant IV transcritical, and III supercritical. Younglove refers to IL and 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 Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 nature of the pseudoboiling line.

p IIIIV B

pcr A IL IV II

Tcr T

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

2 of 15

American Institute of Aeronautics and Astronautics 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 NpT (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)

p(r) g(r) = lim 2 , (1) dr→0 4π(Npairs/V )r dr

with the distance between a pair of atoms r, the average number of atom pairs p(r) at a distance between r 20 and r + dr, the total volume of the system V , and the number of pairs of atoms Npairs.

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 9

Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 ﬂuid eﬀects using the canonical approach, combustion 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

3 of 15

American Institute of Aeronautics and Astronautics equations including continuity, radial momentum, species and temperature equations can be written as d (ρu) + 2ρV = 0 , (2a) dx dV d dV ρu + ρV 2 = (µ ) − Λ , (2b) dx dx dx dY dJ ρu k + k =ω ˙ , (2c) dx dx k dT d dT X dhk X ρuc = (λ ) − J − ω˙ h , (2d) p dx dx dx k dx k k k k

where conventional notations are used, V = v/r, Λ = (∂p/∂r)/r, hk is the partial enthalpy of species k, and Jk = ρYkVk is 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/O2 combustion accounting for 8 species and 27 reactions. We use a formulation of the mixture fraction Z based on the hydrogen atom, as used by Lacaze and Oefelein:34 YH2 YH YH2O YOH YHO2 YH2O2 ZH = WH 2 + + 2 + + + 2 , (3) WH2 WH WH2O WOH WHO2 WH2O2

where Yα and Wα are 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.

p IV IIILL IIIGL B

pcr A Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 IL IV II

Tcr T 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 incompressible; the diﬀerences between the isobars become very small. For higher temperatures, the densities approach their ideal gas values.

4 of 15

American Institute of Aeronautics and Astronautics 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 supercritical 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 IL and IIIL in Fig. 3 are liquid, then the intermediary quadrant IV should also be in a liquid state. Alternatively, consider a IL liquid. 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. Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 Figure 5: Instantaneous molecular distribution using snapshots of two-dimensional slices from MD simulations 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 IL and IV, which can both be considered liquid. Thus, there is no physical reason to distinguish between liquid and transcritical states.

5 of 15

American Institute of Aeronautics and Astronautics (a) Tr = 0.5. No diﬀerence is dis- (b) Tr = 1.6. The transition from cernible between sub- and super- monotonous (gaseous) to oscilla- critical pressures. tory (liquid) behavior can be observed 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 39 39 characteristic of gases at pr = 0.7 and 1.4, and the oscillatory character signifying liquid behavior 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-

Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 tions at Tr = 1.6 at sub- and supercritical pressures.

For a finer scan of the fluid p-T state space, we evaluate the compressibility factor Z as a measure of the deviation of a fluid from ideal gas behavior. It is defined as p Z = , (4) ρRT with the gas constant R. Z is the nondimensional ratio of the real fluid pressure to the pressure an ideal gas at identical density ρ, and temperature T would 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 Z in the nondimensional pr-Tr plane. The ideal gas equation of state is strictly only valid along a line, a approaching Tr = 2.5 for vanishing pressure . 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

6 of 15

American Institute of Aeronautics and Astronautics 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-Tr diagram. 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 p and 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 Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 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 corresponding pseudoboiling temperature is a function of pressure. For simple fluids, such as nitrogen, oxygen, methane, this relation can be expressed in the following form,14

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

For molecules exhibiting more complex behavior, the extended relation

0 a 0 pr = exp [A (Tr − 1) ]; A , a species dependent. (6)

42 0 yields improved accuracy. Data for As,A , and a are 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.

7 of 15

American Institute of Aeronautics and Astronautics 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.

0 Species As A a

H2 4.137 3.098 0.849

O2 5.428 5.428 1.0

N2 5.589 5.589 1.0

CH4 5.386 5.386 1.0 C6H14 6.688 5.365 0.921

CO2 6.470 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

2.5 r

1.5 Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 Reducedpressure p

1 1 1.05 1.1 1.15 1.2 1.25 1.3

Reduced temperature Tr

Figure 10: Comparison of pseudoboiling-line ﬂuid data (symbols) and correlations Eq. (5) and (6) with 0 As,A , 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,

8 of 15

American Institute of Aeronautics and Astronautics 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.

B1 B2

pcr

A1 A2 δh

Tcr T Figure 12: In order to assess the excess latent heat δh required to heat a fluid from a liquid to a gaseous state at a subcritical pressure A1 → A2 compared to the supercritical case B1 → B2, the process from A1 to B2 is analyzed. Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 To study the difference between the sub- and the supercritical case, regard the processes illustrated in Fig. 12, where A1 /B1, and A2 /B2 are identical to start and end conditions of the processes A and B of Figure 3, respectively. We can introduce δh as the excess enthalpy required to vaporize the subcritical fluid compared to the supercritical process without latent heat,

δh = (hB2 − hB1) − (hA2 − hA1). (7)

Any process from A1 to B2 requires the same amount of energy, regardless of the path taken,

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

Combining Eqs. (7) and (8) yields

δh = (hB2 − hA2) − (hB1 − hA1), (9)

9 of 15

American Institute of Aeronautics and Astronautics reducing the problem to a caloric evaluation of isothermal compression. When we assume A2 /B2 to be states at the same sufficiently high temperature to yield ideal gas properties (i.e. T > 2Tcr), the enthalpy is pressure-independent and an isothermal compression A2 → B2 requires negligible energy. Then, the energetic difference between the processes A1 → A2 and B1 → B2 solely depends on the liquid compression A1 → B1. Figure 13 shows that isothermal compression of a liquid requires negligible energy compared to the latent heat of vaporization, even close to the critical pressure. Thus, : 0 : 0 δh =(hB2 − hA2) −(hB1 − hA1). (10) and we conclude that the energetic difference between the subcritical and the supercritical heating process is negligible.

200

150

100

0 150 K •50 120 K Specific enthalpy in kJ/kg in Specific enthalpy •100 100 K

•150 0 5 10 15 20 Pressure in MPa

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. Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106

Figure 14: 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.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

10 of 15

American Institute of Aeronautics and Astronautics 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 relevance 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.

flame GH2

LOX

liquid pseudoboiling vapor ideal gas

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 compressibility factor46 Z defined 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 −2 with a mixture fraction ZH < 1.0 × 10 . 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. 47 The conditions correspond to Ruiz’ benchmark cryogenic shear layer (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

Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 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-T state plane, divided by the

11 of 15

American Institute of Aeronautics and Astronautics 1.2 1.2

1 1

0.8 0.8 H2 mass fraction O2 mass fraction

0.6 0.6 Z H2O mass fraction Compressibility

Mass fractions 0.4 0.4

0.2 0.2

0 0 10-4 10-3 10-2 10-1 100 Z H Figure 16: Compressibility Z for equilibrium (solid) and near quenching (dashed) ﬂame. Real ﬂuid behavior −3 occurs only on the oxidizer side. In the equilibrium case, Z ≈ 1 for ZH < 2.0 × 10 , for near-quenching at −2 28 ZH < 10 , from.

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 – Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106

(a) Colored density contours in kg/m3. The solid black lines (b) Scatter plot of compressibility factor Z and oxygen mass denote a water mass fraction of 0.1 and 0.01, to demonstrate fraction. The dense pure oxygen can be seen to transform to the position of the ﬂame and the main water occurrence out- an ideal gas state (Z = 0.95) before being signiﬁcantly diluted. side of the dense LOX core. 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)

12 of 15

American Institute of Aeronautics and Astronautics Figure 18: Revised ﬂuid state plane. Instead of four quadrants, formed by the critical isotherm and the critical isobar, the state space can better be understood as a low-temperature liquid state separated from gaseous states by the coexistence- and pseudoboiling-lines. The distinct pseudoboiling transition occurs up to reduced pressures of three; beyond, the supercritical state transition is linear.

regardless of pressure, including sub- and supercritical states. This results directly from the intermolecular forces that need to be overcome from a densely packed liquid to an interaction-less ideal gas state, and which are invariant to the acting pressure. We can conclude that there are many unexpected similarities between sub- and supercritical states and transitions, which may require additional consideration in numerical and experimental studies.

Acknowledgments

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

References

1Oefelein, J., Dahms, R., Lacaze, G., Manin, J., and Pickett, L., “Effects of pressure on fundamental physics of fuel injection in Diesel engines,” Proceedings of the 12th International Conference on Liquid Atomization and Spray Systems (ICLASS), Heidelberg, Germany, 2012. 2Oschwald, M., Smith, J., Branam, R., Hussong, J., Schik, A., Chehroudi, B., and Talley, D., “Injection of Fluids into Supercritical Environments,” Combustion Science and Technology, Vol. 178, 2006, pp. 49–100. 3Candel, S., Juniper, M., Singla, G., Scouflaire, P., and Rolon, C., “Structures and dynamics of cryogenic flames at supercritical pressure,” Combustion Science and Technology, Vol. 178, 2006, pp. 161–192. 4Habiballah, M., Orain, M., Grisch, F., Vingert, L., and Gicquel, P., “Experimental studies of high-pressure cryogenic flames on the mascotte facility,” Combustion Science and Technology, Vol. 178, 2006, pp. 101–128. 5Oefelein, J., “Mixing and combustion of cryogenic oxygen-hydrogen shear-coaxial jet flames at supercritical pressure,”

Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 Combustion Science and Technology, Vol. 178, 2006, pp. 229–252. 6Bellan, J., “Theory, modeling and analysis of turbulent supercritical mixing,” Combustion Science and Technology, Vol. 178, 2006, pp. 253–281. 7Zong, N. and Yang, V., “Cryogenic ﬂuid jets and mixing layers in transcritical and supercritical environments,” Com- bustion Science and Technology, Vol. 178, 2006, pp. 193–227. 8Mayer, W., Schik, A., Vielle, V., Chauveau, C., G¨okalp, I., Talley, D. G., and Woodward, R. D., “Atomization and Breakup of Cryogenic Propellants under High-Pressure Subcritical and Supercritical Conditions,” Journal of Propulsion and Power, Vol. 14, 1998, pp. 835–842. 9Oefelein, J. and Yang, V., “Modeling High-Pressure Mixing and Combustion Processes in Liquid Rocket Engines,” Journal of Propulsion and Power, Vol. 14, No. 5, 1998, pp. 843–857. 10Dahms, R., Manin, J., Pickett, L., and Oefelein, J., “Understanding high-pressure gas-liquid interface phenomena in Diesel engines,” Proceedings of the Combustion Institute, Vol. 34, No. 1, 2013, pp. 1667–1675. 11Carey, V., Liquid-vapor phase-change phenomena, Taylor & Francis, New York, 2008. 12Qiu, L. and Reitz, R., “An investigation of thermodynamic states during high-pressure fuel injection using equilibrium thermodynamics,” International Journal of Multiphase Flow, , No. 72, 2015, pp. 24–38. 13Oschwald, M. and Schik, A., “Supercritical nitrogen free jet investigated by spontaneous Raman scattering,” Experiments in Fluids, Vol. 27, 1999, pp. 497–506.

13 of 15

American Institute of Aeronautics and Astronautics 14Banuti, D. T., “Crossing the Widom-line – Supercritical pseudo-boiling,” Journal of Supercritical Fluids, Vol. 98, 2015, pp. 12–16. 15Banuti, D. T. and Hannemann, K., “The absence of a dense potential core in supercritical injection: A thermal break-up mechanism,” Physics of Fluids, Vol. 28, No. 3, 2016, pp. 035103. 16Branam, R. and Mayer, W., “Characterization of Cryogenic Injection at Supercritical Pressure,” Journal of Propulsion and Power, Vol. 19, No. 3, 2003, pp. 342–355. 17Tucker, S., “Solvent Density Inhomogeneities in Supercritical Fluids,” Chemical Reviews, Vol. 99, No. 2, 1999, pp. 391– 418. 18Younglove, B. A., “Thermophysical properties of fluids. I. Argon, ethylene, parahydrogen, nitrogen, nitrogen trifluoride, and oxygen,” Journal of Physical and Chemical Reference Data, Vol. 11, 1982, pp. Supplement 1. 19Plimpton, S., “Fast Parallel Algorithms for Short-Range Molecular Dynamics.” Journal of Computational Physics, Vol. 117, 1995, pp. 1–19. 20Levine, B., Stone, J., and Kohlmeyer, A., “Fast analysis of molecular dynamics trajectories with graphics processing unitsRadial distribution function histogramming,” Journal of Computational Physics, Vol. 230, No. 9, 2011, pp. 3556 – 3569. 21Hickey, J.-P., Ma, P., Ihme, M., and Thakur, S., “Large Eddy Simulation of Shear Coaxial Rocket Injector: Real Fluid Effects,” Proceedings of the 49th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, No. AIAA 2013-4071, AIAA, San Jose, USA, 2013. 22Ma, P. C., Banuti, D. T., Ihme, M., and Hickey, J.-P., “Numerical framework for transcritical real-fluid reacting flow simulations using the flamelet progress variable approach,” Proceedings of the 55th AIAA Aerospace Sciences Meeting, Gaylord, USA. 23Ma, P. C., Lv, Y., and Ihme, M., “An entropy-stable hybrid scheme for simulations of transcritical real-fluid flows,” Journal of Computational Physics, under review. 24Ma, P. C., Lv, Y., and Ihme, M., “Numerical methods to prevent pressure oscillations in transcritical flows,” Annual Research Brief , Center for Turbulence Research, Stanford University, 2016, pp. 223–234. 25Peng, D.-Y. and Robinson, D., “A New Two-Constant Equation of State,” Industrial & Engineering Chemistry Funda- mentals, Vol. 15, No. 1, 1976, pp. 59–64. 26Pierce, C. and Moin, P., “Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion,” Journal of Fluid Mechanics, Vol. 504, 2008. 27Ihme, M., Cha, C., and Pitsch, H., “Prediction of local extinction and re-ignition eects in non-premixed turbulent combustion using a flamelet/progress variable approach,” Proceedings of the Combustion Institute, Vol. 30, 2005, pp. 793–800. 28Banuti, D. T., Ma, P. C., Hickey, J.-P., and Ihme, M., “Sub- or supercritical? A flamelet analysis for high-pressure rocket propellant injection,” 52nd AIAA/SAE/ASEE Joint Propulsion Conference, Propulsion and Energy Forum, No. AIAA 2016-4789, Salt Lake City, USA, 2016. 29Peters, N., Turbulent Combustion, Cambridge, 2000. 30Kee, R. J., Miller, J. A., Evans, G. H., and Dixon-Lewis, G., “A computational model of the structure and extinction of strained, opposed flow, premixed methane-air flames,” Symposium (International) on Combustion, Vol. 22, Elsevier, 1989, pp. 1479–1494. 31Huo, H., Wang, X., and Yang, V., “A general study of counterflow diffusion flames at subcritical and supercritical conditions: oxygen/hydrogen mixtures,” Combust. Flame, Vol. 161, No. 12, 2014, pp. 3040–3050. 32Goodwin, D. G., Moffat, H. K., and Speth, R. L., “Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes,” http://www.cantera.org, 2016, Version 2.2.1. 33M.P. Burke, M. Chaos, Y. J. F. D. and Klippenstein, S., “Comprehensive H2/O2 kinetic model for highpressure combustion,” International Journal of Chemical Kinetics, Vol. 44, No. 7, 2012, pp. 444–474. 34Lacaze, G. and Oefelein, J., “A non-premixed combustion model based on flame structure analysis at supercritical pressures.” Combustion and Flame, Vol. 159, No. 6, 2012, pp. 2087–2103. 35Simeoni, G., Bryk, T., Gorelli, F., Krisch, M., Ruocco, G., Santoro, M., and Scopigno, T., “The Widom line as the crossover between liquid-like and gas-like behaviour in supercritical fluids,” Nature Physics, Vol. 6, 2010, pp. 503–507. 36Nishikawa, K. and Tanaka, I., “Correlation lengths and density fluctuations in supercritical states of carbon dioxide,” Chemical Phyics Letters, Vol. 244, 1995, pp. 149–152. 37Gorelli, F., Santoro, M., Scopigno, T., Krisch, M., and Ruocco, G., “Liquidlike behavior of supercritical fluids,” Physical

Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106 Review Letters, Vol. 97, 2006, pp. 245702. 38Linstrom, P. and Mallard, W., editors, NIST Chemistry WebBook, NIST Standard Reference Database Number 69 , chap. http://webbook.nist.gov, National Institute of Standards and Technology, Gaithersburg MD, 20899, retrieved 2016. 39Fisher, M. and Widom, B., “Decay of correlations in linear systems,” The Journal of Chemical Physics, Vol. 50, No. 9, 1969, pp. 3756–3772. 40Atkins, P. and de Paula, J., Physical Chemistry, Oxford University Press, 2010. 41Banuti, D. T., Hannemann, V., Hannemann, K., and Weigand, B., “An efficient multi-fluid-mixing model for real gas reacting flows in liquid propellant rocket engines,” Combustion and Flame, Vol. 168, 2016, pp. 98–112. 42Banuti, D. T., Raju, M., and Ihme, M., “Supercritical pseudoboiling for general fluids and its application to injection,” Annual Research Brief , Center for Turbulence Research, Stanford University, 2016, pp. 211–221. 43Hirschfelder, J., Curtiss, C., and Bird, R., Molecular Theory of Gases and Liquids, Wiley, 1954. 44Giovangigli, V., Matuszewski, L., and Dupoirieux, F., “Detailed modeling of planar transcritical H2-O2-N2 flames,” Combustion Theory and Modelling, Vol. 15, No. 2, 2011, pp. 141–182. 45Ribert, G., Zong, N., Yang, V., Pons, L., Darabiha, N., and Candel, S., “Counterflow diffusion flames of general fluids: oxygen/hydrogen mixtures,” Combustion and Flame, Vol. 154, 2008, pp. 319–330. 46Stanley, H., Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, 1971.

14 of 15

American Institute of Aeronautics and Astronautics 47Ruiz, A., Lacaze, G., Oefelein, J., Mari, R., Cuenot, B., Selle, L., and Poinsot, T., “Numerical Benchmark for High- Reynolds-Number Supercritical Flows with Large Density Gradients,” AIAA Journal, Vol. 54, No. 5, 2015, pp. 1445–1460. 48Banuti, D. T. and Hannemann, K., “Real Gas Library in Continuous Phase Propellant Injection Model for Liquid Rocket Engines,” Proceedings of the 49th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, No. AIAA 2013-4068, San Jose, USA, 2013. Downloaded by STANFORD UNIVERSITY on February 3, 2017 | http://arc.aiaa.org DOI: 10.2514/6.2017-1106

15 of 15

American Institute of Aeronautics and Astronautics