Supercritical Pseudoboiling for General Fluids and Its Application To

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Center for Turbulence Research 211 Annual Research Briefs 2016 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 discontinuous 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 transition 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-T diagram. 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)/(γ − 1)). 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 Z measures the deviation of fluid behavior from ideal gas behavior. It is defined as Z = p/ρRT . Figure 2 shows that a fluid at supercritical pressure 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 flame

GH2 1222 DAHMS AND OEFELEIN

hydrogen is not sufficient to heat the liquid-oxygen–hydrogen LOX (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 liquid pseudoboiling vapor ideal gas Lazar and Faeth [36]) is exceeded. Conversely, there is significant experimental evidence of substantial surface tension forces in various binary mixtures at high pressures (p ≫ pC;L; p>500 bar) provided Figure 3. Schematic of supercritical pressure reactive shear layer behind LOX post in coaxial in recent literature [37–39]. The theory presented here serves to GH2/LOX injection. Flame is anchored at LOX post; dense oxygen heats to ideal gas state explain the conditions when this occurs also. before1222 mixing. DAHMS AND OEFELEIN In this paper, we use the theoretical framework introduced by Dahms et al. [40–42] to derive respective regime diagrams for representative propellant combinations in liquid rockets. This hydrogen is not sufficient to heat the liquid-oxygen–hydrogen framework has been recently verified experimentally by Manin et al. (LOX-H2) interface to its critical temperature, and surface tension [43] and by Falgout et al. [44]. The model allows one to analyze forces do not diminish only because the critical pressure of the liquid representative detailed gas–liquid interface dynamics during or the “mixture critical pressure” (as defined by Faeth et al. [35] and injection. Our analysis shows that, under certain high-pressure Lazar and Faeth [36]) is exceeded. Conversely, there is significant conditions, gas–liquid interfacial phenomena are not determined by experimental evidence of substantial surface tension forces in various vapor–liquid equilibrium and the presence of surface tension forces. binary mixtures at high pressures (p ≫ pC;L; p>500 bar) provided Instead, the interface enters the continuum length scale regime due to in recent literature [37–39]. The theory presented here serves to a combination of reduced mean free molecular pathways and explain the conditions when this occurs also. broadened interfaces. Based on these insights, a regime diagram for In this paper, we use the theoretical framework introduced by LOX-H2 injection is presented. This diagram quantifies when the Dahms et al. [40–42] to derive respective regime diagrams for system transitions between classical atomization and diffusion- Figure 4. Jet break-up of coaxial LN2/He injection into a He environment at sub-– (left) and Fig.representative 1 Nonreacting propellant shear-coaxial combinations liquid-nitrogen in liquidhelium rockets. injector This dominated mixing as a function of the ambient pressure and operating at a) 1.0 MPa, and b) 6.0 MPa. T 97 K, T 280 K supercritical (right) pressureframework (Mayer haset been al. recently1998). verified experimentallyN2 byHe Manin et al. into GHe at T 300 K (from [2]). temperature of the chamber. A similar diagram is derived for [43] and by Falgout et al. [44]. The model allows one to analyze n-decane–gaseous-oxygen (LC10-GO2) systems. It is demonstrated representative detailed gas–liquid interface dynamics during that the liquid injection process can exhibit characteristics of classical 2.2. Relevance of pure fluidapproached.injection. behavior Our Thefor analysis initial injection divergence shows that, angle under of the certain jet was measuredhigh-pressure at spray atomization even at very high chamber pressures theconditions, jet exit and gas compared–liquid interfacial with the divergence phenomena angle are of not a large determined number by (p>20 · pC). Likewise, depending on the temperature and As the whole point of injection in technicalofvapor systems other–liquid mixing equilibrium is layer to flows, create and the including presence mixtures, atomized of surface liquidone tension sprays, may forces. composition of the fuel and oxidizer streams, liquid injection can question the relevance of studying pure fluidturbulentInstead, behavior. the incompressible interface Particularly enters gaseousthe continuum jets,at highlength supersonic scale pressures, regime jets, due and to instead exhibit dynamics of dense-fluid or supercritical mixing incompressiblea combination variable-density of reduced mean jets. Thisfree angle molecular was plotted pathways across and processes. The current analysis reinforces past observations that mixture behavior may deviate substantially fromfourbroadened pure orders offluid interfaces. magnitude behavior Based in the on gas-to-liquid these and insights, ideal density a gasregime ratio, mixing. which diagram was for drops can still be present even at pressures that are supercritical with theLO firstX-H2 timeinjection such a is plot presented. has been This reported diagram over this quantifies large range. when At the respect to the injected liquid and quantifies under what conditions Figure 3 shows the schematic of a reactive shearsystem transitions layer in between LOX/GH2 classical atomization injection and at diffusion- su- dense-fluid mixing occurs instead. – and above the critical pressure of the injected liquid, jet growth rate Fig. 1 Nonreacting shear-coaxial liquid-nitrogen helium injector dominated mixing as a function of the ambient pressure and The liquid rocket engine conditions investigated here are percriticaloperating at a) 1.0 pressure. MPa, and b) 6.0 Oxygen MPa. T is97 injected K, T 280 in K a dense,measurements liquid agreed state quantitatively before with undergoing the theory for incompress- pseu- N2 He dramatically different compared to the representative diesel engine into GHe at T 300 K (from [2]). ible,temperature variable-density, of the chamber. gaseous mixing A similar layers. diagram As the ispressure derived was for doboiling and transition to an ideal gas, as depictedreducedn-decane to–in progressivelygaseous-oxygen Figure more 2. (LC subcritical10-GO2) systems. values, the It isspreading demonstrated rate injection conditions considered in past works. Despite these Several researchers have pointed out that mixingapproachedthat the liquid within that injection measured the process for hot liquid can flame exhibit sprays. characteristics occurs at of ideal classical distinctive differences in operating conditions (e.g., low cryogenic approached. The initial divergence angle of the jet was measured at sprayCandel atomization et al. [4], Pons even et at al. very [5], and high Oschwald chamber et al. pressures [6,7] temperatures, different molecular sizes and properties, high reduced gasthe jet exitconditions and compared(Oefelein with the divergence & angle Yang of a large 1998; number Ribertdemonstrated(p>et20 al.· pC). fundamental2008; Likewise, Lacaze differences depending & in on combustion Oefelein the temperature dynamics 2012), of and liquid injection, and ambient gas pressures and temperatures, etc.), whereasof other mixing mixing layer flows, under including realgas atomized conditions liquid sprays, is essentiallycryogeniccomposition flames negligible of the between fuel and subcritical (Banuti oxidizer streams, andet supercritical al. liquid2016 injection pressurea,b). can this study demonstrates that the related physical complexity is well turbulent incompressible gaseous jets, supersonic jets, and conditionsinstead exhibit using dynamics experimental of dense-fluid facilities or from supercritical independent mixing captured by our mapping approach based on the defined Knudsen Thisincompressible means variable-density that the jets. transition This angle was from plotted a across real tolaboratories.processes. an ideal The Such currentgas differences occurs analysis extended reinforces in the to fundamental past pure observations oxygen quantities that number. The presented analysis also quantifies and explains the four orders of magnitude in the gas-to-liquid density ratio, which was suchdrops as can liquid still core be present lengths, even turbulent at pressures length that scales, are supercriticaland jet breakup with significant differences observed in the envelopes of transition stream;the first time the such a bulk plot has disintegration been reported over this of large the range. LOX At jetregimes.respect in LOX/GH2 to At the supercritical injected liquid combustion pressure and quantifies and temperature is under a pure what conditions, conditions fluid conditions from classical spray atomization to dense-fluid jet – and above the critical pressure of the injected liquid, jet growth rate TaskinogluandBellan[8,9],Bellan[10],andHarstadandBellan[11]dense-fluid mixing occurs instead. dynamics between rocket systems operated using liquid-oxygen phenomenon. hydrogen and liquid n-decane–oxygen propellant combinations. measurements agreed quantitatively with the theory for incompress- performedThe liquid direct rocket numerical engine simulations conditions along investigated with isolated here and are ible, variable-density, gaseous mixing layers. As the pressure was clusterdramatically drop simulations different compared considering to the Soret representative and Dufour diesel effects engine to reduced to progressively more subcritical values, the spreading rateDownloaded by STANFORD UNIVERSITY on May 31, 2016 | http://arc.aiaa.org DOI: 10.2514/1.B35562 investigateinjection conditions the mixing considered process in and past fluid works. disintegration Despite these in II. Model Formulation supercriticaldistinctive differences mixing layers. in operating Large-eddy conditions simulations (e.g., low have cryogenic been 3.approached Towards that measured a classification for liquid sprays. of supercritical injection The starting point of our model is based on the framework Candel et al. [4], Pons et al. [5], and Oschwald et al. [6,7] performedtemperatures, using different proposed molecular new sizes subgrid-scale and properties, terms, high denoted reduced “liquidcorrection injection,” terms, and in ambient concert with gas pressures typical subgrid-scale and temperatures, flux terms etc.), developed by Oefelein [45,46], which provides a generalized demonstratedJet break-up fundamental regimes differences are in combustion typically dynamics presented of in the Reitz diagram for single jets, or in treatment of the thermodynamics and transport processes for cryogenic flames between subcritical and supercritical pressure [8,9].this study Sirignano demonstrates et al. [12,13], that the Delplanque related physical [14], and complexity Sirignano is [15] well discussedcaptured by and our investigated mapping approach droplet based and sprayon the vaporization defined Knudsen at hydrocarbon mixtures at near-critical and supercritical conditions. theconditions Chigier using diagram experimental for facilities coaxial from jets independent (Kuo & Acharya 2012). In addition to the Reynolds This scheme is combined with the theoretical framework developed laboratories. Such differences extended to fundamental quantities subcriticalnumber. The and presented supercritical analysis pressures also quantifiesto study combustion and explains and the number, the characterization is carried out indynamicsignificant terms responses differences of tothe oscillatory observed Ohnesorge ambient in the conditions. envelopes or the Those of Weber transition studies by Dahms et al. [40] and Dahms and Oefelein [41], which facilitates such as liquid core lengths, turbulent length scales, and jet breakup vapor–liquid equilibrium and gas–liquid interface structure establishedconditions that from the classical vaporization spray process atomization can be a to rate-controlling dense-fluid jet numberregimes. At - supercritical both of pressurewhich and are temperature functions conditions, of surface tension. calculations to compute both surface tension forces and the local processdynamics for betweendriving combustion rocket systems instabilities. operated using liquid-oxygen– TaskinogluandBellan[8,9],Bellan[10],andHarstadandBellan[11] two-phase interface thicknesses. Here, only a description of the performedFigure direct 4shows numerical two simulations results along of with a classicalisolated and injectionhydrogenMany other experiment and works liquid haven-decane served by–oxygen to Mayer corroborate propellantet these al. combinations. observations(1998). – general features of the model framework is provided. A detailed cluster drop simulations considering Soret and Dufour effects to [16 34]. However, the absence of surface tension forces under some Injection at subcritical pressure exhibits a distincthigh-pressure interface conditions still as poses a many sign fundamental of a finite questions. sur- For derivation can be found in [41]. Downloaded by STANFORD UNIVERSITY on May 31, 2016 | http://arc.aiaa.org DOI: 10.2514/1.B35562 investigate the mixing process and fluid disintegration in a pure fluid, basic theoryII. dictates Model that Formulation surface tension forces become facesupercritical tension. mixing The layers. aforementioned Large-eddy simulations classification have been is thus possible. In contrast, injection at diminished because the pressure of the liquid phase exceeds its A. Thermodynamic and Transport Properties performed using proposed new subgrid-scale terms, denoted The starting point of our model is based on the framework supercritical pressure resembles turbulent mixingcritical of value.gaseous For multicomponent jets; the effect systems, of surface however, this ten- is The extended corresponding states model [47,48] is employed “correction” terms, in concert with typical subgrid-scale flux terms developed by Oefelein [45,46], which provides a generalized generally not the case, and modern theory still lacks a first principle using a Benedict–Webb–Rubin (BWR) equation of state to evaluate sion[8,9]. Sirignano is negligible. et al. [12,13], Thus, Delplanque the [14], current and Sirignano classification [15] treatment is not of the suitable thermodynamics to analyze and transport supercritical processes for explanation for the observed phenomena. Previously, it has been the p-v-T behavior of the inherent dense multicomponent mixtures. discussed and investigated droplet and spray vaporization at hydrocarbon mixtures at near-critical and supercritical conditions. injection. widely assumed that surface tension forces become diminished Use of modified BWR equations of state in conjunction with the subcritical and supercritical pressures to study combustion and This scheme is combined with the theoretical framework developed because the temperature of the liquid phase exceeds its critical value. extended corresponding states principle has been shown to provide dynamicHow responses can we to oscillatory distinguish ambient conditions. different Those supercritical studies by Dahms disintegration et al. [40] and Dahms modes? and Oefelein Figure [41], which 5 defines facilitates Undervapor– manyliquid conditions, equilibrium however, and this gas mechanism–liquid interface must be ruled structure out. consistently accurate results over the widest range of pressures, processesestablished that A theto vaporization E and process compiles can be a rate-controllingthe respectiveIn thisshadowgraphs; paper, it is shown that theconditions temperature of hotare unburned shown gaseous in temperatures, and mixture states, especially at saturated conditions. process for driving combustion instabilities. calculations to compute both surface tension forces and the local TableMany other 1. works have served to corroborate these observations two-phase interface thicknesses. Here, only a description of the [16–34]. However, the absence of surface tension forces under some general features of the model framework is provided. A detailed high-pressure conditions still poses many fundamental questions. For derivation can be found in [41]. a pure fluid, basic theory dictates that surface tension forces become diminished because the pressure of the liquid phase exceeds its A. Thermodynamic and Transport Properties critical value. For multicomponent systems, however, this is The extended corresponding states model [47,48] is employed generally not the case, and modern theory still lacks a first principle using a Benedict–Webb–Rubin (BWR) equation of state to evaluate explanation for the observed phenomena. Previously, it has been the p-v-T behavior of the inherent dense multicomponent mixtures. widely assumed that surface tension forces become diminished Use of modified BWR equations of state in conjunction with the because the temperature of the liquid phase exceeds its critical value. extended corresponding states principle has been shown to provide Under many conditions, however, this mechanism must be ruled out. consistently accurate results over the widest range of pressures, In this paper, it is shown that the temperature of hot unburned gaseous temperatures, and mixture states, especially at saturated conditions. 214 Banuti, Raju & Ihme

D B C E pcr

Tcr Figure 5. Left: Injection processes in p − T diagram, 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 N2 N2 B 0.71-0.87 ≈1.23 2.38 1.23 N2 N2 C 0.95 6.59 1.9 0.02 C6H14 Ar D 1.05 ≈1.77 2.36 1.77 N2 N2 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 illustrates 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 condition, and 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 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 pr and reduced temperature Tr is

pr = exp [A (Tr − 1)] ; 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 As is introduced and determined for each species,

pr = exp [As (Tr − 1)] ; As species dependent. (4.2)

Table 2 provides As for 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 ω as

pr = exp [Aω (Tr − 1)] ; Aω = 5.29 + 4.35ω, (4.3) allowing for a fast evaluation without requiring ﬂuid p-v-T data.

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-T plot, water and hydrogen exhibit a significant curvature of their graphs. To incorporate this deviation, a modified expression is proposed in the form

0 a 0 pr = exp [A (Tr − 1) ]; A , a species dependent. (4.4) Data for A0 and a are obtained by ﬁtting and are compiled in Table 2. 0 Figure 10 compares Eq. (4.1) using A, Eq. (4.2) using As, and Eq. (4.4) using A and 218 Banuti, Raju & Ihme

0 Species ω As A a Quantum gases He −0.382 3.516 2.282 0.772 H2 −0.219 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 O2 0.0222 5.428 6.742 1.112 N2 0.0372 5.589 6.703 1.097 F2 0.0449 5.686 5.456 0.977 CO 0.050 5.750 6.238 1.044 Hydrocarbons CH4 0.01142 5.386 5.947 1.058 C2H6 0.0993 5.687 5.116 0.941 C3H8 0.1524 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 CO2 0.22394 6.470 8.256 1.102 NH3 0.25601 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 ω and slope of the Nishikawa-Widom-line for a number of species 0 obtained from NIST Linstrom & Mallard (2016). As for Eq. (4.2), A ,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 ω and the parameter A of 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 0 Eq. (4.1) with A, Eq. (4.2) with As, Eq. (4.4) with A , 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 tension 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(Tr − 1)], with Aω = 5.29 + 4.35ω. The acentric factor ω can 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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