Transcritical diffuse-interface hydrodynamics of propellants in high-pressure combustors of chemical propulsion systems

UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Contents lists available at ScienceDirect Progress in Energy and Combustion Science journal homepage: www.elsevier.com Transcritical diffuse-interface hydrodynamics of propellants in high-pressure combustors of chemical propulsion systems Lluís Jofre , Javier Urzay ⁎ Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA ARTICLE INFO Article history: Received 30 September 2019 Received in revised form 8 August 2020 Accepted 10 August 2020 Available online xxx Keywords: Fuel sprays Transcritical flows Phase change High-pressure thermodynamics ABSTRACT Rocket engines and high-power new generations of gas-turbine jet engines and diesel engines oftentimes involve the injection of one or more reactants at subcritical temperatures into combustor environments at high pressures, and more particularly at pressures higher than those corresponding to the critical points of the individual components of the mixture, which typically range from 13 to 50 bars for most propellants. This class of trajectories in the thermodynamic space has been traditionally referred to as transcritical. However, the fundamental understanding of fuel atomization, vaporization, mixing, and combustion processes at such high pressures remains elusive. In particular, whereas fuel sprays are relatively well characterized at normal pressures, analyses of dispersion of fuel in high-pressure combustors are hindered by the limited experimental diagnostics and theoretical formulations available. The description of the thermodynamics of hydrocarbon-fu- eled mixtures employed in chemical propulsion systems is complex and involves mixing-induced phenomena, including an elevation of the critical point whereby the coexistence region of the mixture extends up to pressures much larger than the critical pressures of the individual components. As a result, interfaces subject to surface-tension forces may persist in multicomponent systems despite the high pressures, and may give rise to unexpected spray-like atomization dynamics that are otherwise absent in monocomponent systems above their critical point. In this article, the current understanding of this phenomenon is reviewed within the context of propulsion systems fueled by heavy hydrocarbons. Emphasis is made on analytical descriptions at mesoscopic scales of interest for computational fluid dynamics. In particular, a set of modifications of the constitutive laws in the Navier–Stokes equations for multicomponent flows, supplemented with a high-pressure equation of state and appropriate redefinitions of the thermodynamic potentials, are introduced in this work based on an extended version of the diffuse-interface theory of van der Waals. The resulting formulation involves revisited forms of the stress tensor and transport fluxes of heat and species, and enables a description of the mesoscopic volumetric effects induced by transcritical interfaces consistently with the thermodynamic phase diagram of the mixture at high pressures. Applications of the theory are illustrated in canonical problems, including dodecane/nitrogen transcritical interfaces in non-isothermal systems. The results indicate that a transcritical interface is formed between the propellant streams that persists downstream of the injection orifice over distances of the same order as the characteristic thermal-entrance length of the fuel stream. The transcritical interface vanishes at an edge that gives rise to a fully supercritical mixing layer. © 2020. Nomenclature Latin letters *** a, b coefficients of the Peng-Robinson equation of state AP/Aμcharacteristic oscillation amplitudes of the pressure / chemical potential across the interface parameters of the gradient-energy coefficient model volume expansivity divided by the corresponding ideal-gas value cspeed of sound ratio between real and ideal specific heats at constant pressure cp/constant-pressure specific / molar heat ⁎Corresponding author. Email address: [email protected] (J. Urzay) cv/constant-volume specific / molar heat dequivalent hard-sphere diameter Di,j Fickian diffusion coefficient binary diffusion coefficient DTthermal diffusivity e/specific / molar internal energy Etotal energy /mechanical- / transport-equilibrium parameters f/specific / molar Helmholtz free energy FHelmholtz free energy of the system Fvector of thermodynamic forces ffugacity g/specific / molar partial Gibbs free energy GGibbs free energy of the system h/specific / molar enthalpy Henthalpy of the system Jispecies diffusion flux https://doi.org/10.1016/j.pecs.2020.100877 0360-1285/ © 2020. UNCORRECTED PROOF 2 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx interfacial species flux kTthermal-diffusion ratio KIlocal curvature of the interface interfacial stress tensor KnIKnudsen number ℓca capillary length Li,j Onsager coefficients Onsager matrix LTR supercriticalization length Le Lewis number Mmomentum-flux ratio Ma Mach number nnumber of moles spatial coordinate normal to the interface Nnumber of components nondimensional curvature NAAvogadro’s number Oh Ohnesorge number OPR overall pressure ratio Ppressure Paair pressure at a given altitude Pc,F/Pc,O critical pressures of the propellants P∞combustor pressure ratio between the characteristic oscillation amplitude of the pressure through the interface and the combustor pressure Pe Péclet number qheat flux qtheat flux (excluding heat transport by interdiffusion of species) interfacial heat flux RFradius of the orifice R0universal gas constant ratio between fuel and oxidizer densities Re Reynolds number s/specific / molar / generalized entropy entropy production rate ttime Ttemperature Tc,diff temperature of the diffusional critical point Tc,F/Tc,O critical temperatures of the propellants Tc,mix critical mixing temperature Tpb pseudo-boiling temperature TTR supercriticalization temperature UF/UOinjection velocities of the propellants vvelocity vector vmolar volume Vvolume of the system VLE vapor-liquid equilibrium partial molar volume W / molecular weight / mean molecular weight of the mixture ratio between fuel and coflow molecular weights We Weber number xstreamwise spatial coordinate Xmolar fraction ytransversal spatial coordinate Ymass fraction Zcompressibility factor Greek symbols *** αthermal-expansion ratio βsisentropic compressibility βTisothermal compressibility βvvolume expansivity γadiabatic coefficient δIinterface thickness δT/δYthermal / compositional mixing-layer thickness Δde/Δdhinternal energy / enthalpy departure function ϵR/ϵTlarge-scale / thermal Cahn number ηdynamic viscosity θreduced temperature ϑbinary-interaction parameter ιmodel parameter for calculation of thermal-diffusion ratio κgradient-energy coefficient λthermal conductivity Λmolecular mean free path μ/ / specific / molar / generalized chemical potential ρdensity σsurface-tension coefficient σ0characteristic value of the surface-tension coefficient τratio between interface and fuel-stream temperatures τviscous stress tensor φfugacity coefficient ϕvector of thermodynamic currents χauxiliary variable ΩLandau potential ψauxiliary vector Main subscripts *** ccritical thermodynamic state diff diffusional critical point evapor-liquid equilibrium conditions fformation value Ffuel stream FF, O employed to denote the value of the binary diffusivity in the fuel stream F, OO employed to denote the value of the binary diffusivity in the coflow stream GD gradient-dependent thermodynamic potential or pressure Iinterface-related quantity mix critical mixing conditions Ocoflow stream pb pseudo-boiling conditions ∞conditions in the combustor environment Main superscripts **** ggas-like state IG ideal-gas conditions ℓliquid-like state ΔTspecies or heat flux components dependent on temperature gradients ′species or heat flux components independent of temperature gradients ⋆nondimensional variable 0reference or initial value 1. Introduction The utilization of high pressures for burning fuel and oxidizer is a necessary requirement for the generation of thrust in chemical propulsion systems for ground, air, and space engineering applications. In modern designs, the characteristic pressures in the combustor range from 40 to 560 bars in rocket engines, to 20–60 bars in gas-turbine jet engines at takeoff, and 10–30 bars in diesel engines. However, great difficulties arise when attempting to model the dynamics of chemically reacting flows at high pressures. One of them, UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 3 which influences the dispersion of fuel in the combustor and represents the focus of the present work, is related to the occurrence of transcritical conditions. Challenges associated with transcritical conditions have been known since the development of high-pressure thrust chambers for liquid-fueled rocket engines [1–4]. A number of controlled experiments have been undertaken over the years to study flows at high pressures [5–14], and many have revealed the presence of interfaces separating the propellants [15–23]. The high pressures, however, limit the amount of information that can be obtained from state-of-the-art experimental diagnostics. Numerical simulations have accompanied these endeavors [24–31], but most of these computations have been limited to high-pressure conditions that did not warrant the existence of interfaces. Major reviews in the field have also focused on those conditions [32–34]. The field has evolved rapidly in recent years spurred by a number of publications, including (a) the thermodynamic analysis of Qiu and Reitz [35], which has provided detailed information about the thermodynamic space transited by homogeneous hydrocarbon-fueled mixtures at high pressures; (b) the molecular-dynamics simulations by Mo and Qiao [36], which have described the structure of transcritical interfaces albeit limited to very small length and time scales imposed by the high computational cost of molecular methods; and (c) the continuum-level theoretical work of Dahms and Oefelein [37], and Gaillard et al. [38,39], which have provided insight into the problem of transcriticality by using the diffuse-interface theory of van der Waals. The work by Dahms and Oefelein [37] was limited to the analysis of mean thermodynamic states in the combustor and stationary interfaces in absence of flow effects. In contrast, Gaillard et al. [38,39] made significant progress in coupling extensions of the diffuse-interface theory of van der Waals with the Navier-Stokes equations to describe, for instance, steady counterflow hydrogen diffusion flames in transcritical conditions. This review article shows that fluid-mechanical effects are central to the description of transcritical phenomena, and that the prob lem cannot be treated solely by mean thermodynamic states in the combustor or isolated stationary interfaces, as previously attempted [37]. This review article fulfills the following objectives: (a) it summarizes relevant operating conditions leading to transcriticality; (b) it describes the state-of-the-art in the theoretical and computational understanding of transcritical interfaces at the continuum level; (c) it provides a comprehensive diffuse-interface theory of the hydrodynamics of propellants under transcritical conditions; and (d) it employs that theory to describe, for the first time, the downstream evolution of a transcritical interface separating the propellant streams. This review article also provides an analysis of the terminal structure of the transcritical interface at an edge located at a supercriticalization distance downstream of the orifice that is calculated as part of the solution. The diffusional critical point of the mixture of the propellants is shown to play a crucial role in the description of the interface edge. Downstream of the interface edge, the flow becomes fully supercritical, the interface disappears, and the propellants mix by molecular diffusion across a growing compositional mixing layer. 1.1. Transcritical conditions in combustors fueled by heavy hydrocarbons Although there may be several definitions of the character of transcritical conditions in the literature [1–4,6,7,9–14,19,22,24,25,30,32,33,35–37], one that is of practical relevance for chemical propulsion systems corresponds to the thermodynamic conditions attained in the injection configuration sketched in Fig. 1. There, two streams of different propellants, denoted by the subindexes F (for the fuel stream) and O (for the coflow stream), are injected in such a way that the temperature of the coflow stream TO is higher than that of the fuel stream TF. In this configuration, transcriticality is attained when at least one of the propellant streams, often the fuel stream in practical applications, enters the combustor at a subcritical temperature, but the combustor pressure P∞is larger than Fig. 1. Schematics of a transcritical flow near an injector orifice delivering a cold supercritical heavy-hydrocarbon fuel into a hot supercritical coflow in a combustor at a pressure higher than the critical pressures of the two separate components. The symbol RFdenotes the radius of the orifice, and UFand UOare the characteristic injection velocities of the fuel and oxidizer streams, respectively. Additionally, σis the surface-tension coefficient, P∞is the combustor pressure, YFis the fuel mass fraction, and TFand TOcorrespond to the temperatures of the fuel and coflow streams, respectively. The critical temperatures and pressures of the fuel and oxidizer streams are denoted, respectively, by Tc,F and Pc,F, and by Tc,O and Pc,O. The rest of the symbols are defined in the main text. UNCORRECTED PROOF 4 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx the critical pressures of the two separate components, denoted here by Pc,F and Pc,O. The configuration depicted in Fig. 1 drives the discussion throughout the remainder of this manuscript. It is shown in Section 7 that the main characteristics of the flow field sketched in Fig. 1, including the interface separating the propellant streams, can be described by the diffuse-interface theory presented in Sections 4 and 5 even in a simpler canonical configuration where complex but inessential factors such as turbulence, shear, and interface corrugation are not considered. This section provides a qualitative summary of the main physical processes and the associated characteristic scales. The possible number of combinations of thermodynamic conditions in the propellant streams that may lead to transcritical phenomena are large and cannot be addressed in a single study. Their particularities depend on the propulsion system and the mixture of propellants considered, as discussed in Sections 2 and 3. However, the transcritical conditions illustrated in Fig. 1 are of general relevance in combustors fueled by heavy hydrocarbons because of two factors: (a) As observed in Table 1, heavy hydrocarbons have relatively high critical temperatures, and therefore require significant temperature increments of order unity relative to their injection temperature to become fully supercritical, namely Specifically, 200–400 K for heavy hydrocarbon fuels of interest (e.g., C6H14, C7H16, C10H22, C12H26, Jet-A, and RP-1), where TF∼300–450 K and Tc,F ∼500–700 K are the typical ranges of the fuel injection temperature and the fuel critical temperature, respectively. (b) As discussed in Section 3, mixtures of heavy hydrocarbons with molecular nitrogen (N2), molecular oxygen (O2), or air have much higher critical pressures than their separate components. Under the aforementioned transcritical conditions, which are often found in practice as discussed in Section 2, and despite the prevailing high pressures, the flow field contains a region immediately downstream of the injector orifice where a thin transcritical interface is formed between the fuel and coflow streams because of diffusion processes essential to the mixing that necessarily precedes combustion. Specifically, as described in Section 5, the theory presented here indicates that surface tension survives the high pressures because of the intrinsic diffusional instability of the fluid resulting from mixing the two propellant streams. This diffusional instability, whose basic foundations are discussed in Appendix A, leads to separation of the propellants by a transcritical interface that bears large composition gradients across and is accompanied by surface tension. Both thermal and compositional mixing layers sketched in Fig. 1 are slender, since the characteristic Reynolds number is typically much larger than unity in practical applications. In Eq. (2), UF,ρF, and ηFare the velocity, density, and dynamic viscosity of the fuel stream, respectively, and RFis the fuel orifice radius, which is typically of order 0.1–1 mm in chemical propulsion systems of interest [19,43–45]. The magnitude of UFrelative to the coflow-stream velocity UOis represented by the momentum-flux ratio with ρObeing the density of the coflow stream. The momentum-flux ratio is of order M∼1–10 in rocket engines and gas-turbine jet engines [43,46–48], whereas for diesel engines. In the high-pres- sure operating conditions investigated herein, the density ratio is within the range = 5–40, as opposed to = 500–1000 at atmospheric pressure. The corresponding velocity ratios are of order 1–10. In the model problem depicted in Fig. 1, the coflow temperature TOis large enough to heat up the fuel to its critical temperature downstream of the injection orifice. For coflows comprised of N2, O2, or air, the prevailing condition implies that TOmust be larger than the critical temperature of the coflow, TO>Tc,O, since Tc,O ∼120–150 K is in the range of cryogenic temperatures for those components (see Table 1). As a result, the relative temperature difference between the propellant streams, defined as the thermal-expansion ratio is an order-unity parameter. Additionally, the coflow is cooled down by the fuel stream and, to a much lesser extent, the fuel stream is heated by the coflow. This discrepancy is driven by the much higher thermal diffusivity of the coflow. The thickness of the resulting thermal mixing layer, δT, grows with distance downstream from small values compared to RFnear the orifice (a good assumption for thin orifice rings), to values of order RFfar from the orifice where the interface vanishes. Since the Prandtl number remains close to that of air at normal conditions, only δTmay be considered, for simplicity, as the length scale representative of both momentum and thermal transport. The transcritical interface extends from the orifice ring to an edge at a supercriticalization distance where the fuel stream has been heated up to a supercriticalization temperature . The latter depends on the combination of propellants and is a solution of the thermodynamic phase diagram of the mixture. For mixtures of heavy hydrocarbons with N2, O2, or air, it will be shown that Table 1 Critical temperatures and pressures for selected species [40–42]. N2O2air H2O CO2H2CH4C6H14 C7H16 C10H22 C12H26 Jet-A RP-1 Tc[K] 126 155 131 647 304 33 191 508 540 618 658 671 677 Pc[bar] 34 50 36 220 74 13 46 30 27 21 18 24 22 (1) (2) (3) (4) (5) (6) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 5 where Tc,diff is the temperature of the diffusional critical point of the mixture at the combustor pressure P∞(see Appendix A for a formal definition of the diffusional critical point). The temperature of the diffusional critical point Tc,diff is smaller than the fuel critical temperature Tc,F by small amounts, For instance, 10–40 K at pressures P∞∼ 50–200 bar in C12H26/N2mixtures, thereby yielding =0.02–0.06. The value of Tc,diff is also similar to the critical mixing temperature Tc,mix defined as the maximum temperature that can be attained by the mixture under conditions of phase coexistence at P∞, with Tc,mix −Tc,diff being at most 5 K in C12H26/N2 mixtures at P∞=50–200 bar. As a consequence, TTR is also larger than the fuel in Fig. 3. Evolution of the surface-tension coefficient along the transcritical interface in Fig. 2 obtained from the present theory. Further details about these calculations are provided in Section 7. jection temperature TFby relative amounts of order unity, The transcritical interface terminates at an edge crossed by the supercriticalization isotherm, where a fundamental change in the diffusional characteristics of the mixture occurs. The structure of the edge can be previewed in terms of spatial contours of the fuel mass fraction YFin Fig. 2 for a system consisting of C12H26 injected at TF= 450 K into a coflowing N2stream at TO= 1000 K. The system is at P∞= 100 bar, which represents a supercritical pressure with respect to both components. The resulting distribution of surface tension along the transcritical interface with distance downstream of the injection orifice is shown in Fig. 3. The results in both of those figures are obtained by using the diffuse-interface theory presented in Sections 4 and 5. The resulting formulation describes high-pressure flows with interfaces whose strength can vary in space and vanish altogether. The results obtained by using this theory indicate that the interface thickness δIremains small along a large portion of the interface despite the high pressures involved. Consequently, the integration of the formulation is hindered by the large disparity between the interface thickness δIand other length scales, including δT,RF, LTR, and the interface radius of curvature RI. In particular, it will be shown, in conditions of practical relevance, that the large-scale Cahn number and the thermal Cahn number are both much smaller than unity in the transcritical region, where the interface survives. These represent fundamental characteristics Fig. 2. Numerical results of the present theory showing a zoomed view of the terminal structure of a transcritical interface at an edge. The interface separates two planar laminar streams at high Péclet numbers: one of dodecane (C12H26) injected at TF= 450 K (below the interface), and one of molecular nitrogen (N2) injected at TO= 1000 K (above the interface). The system is at a supercritical pressure P∞= 100 bar with respect to both components. The figure shows the fuel mass fraction YF(solid contours and dark thin dashed lines), temperature (colored thick dashed-line contours), diffusional critical point (purple diamond symbol), and diffusional spinodal (thick purple dashed lines). The vertical axis is the transversal distance from the jet axis ynormalized with the orifice radius RF, whereas the horizontal axis is the streamwise distance xnormalized with the estimate (13) for the supercriticalization length. The flow attains fully supercritical conditions downstream of the interface edge. There, both propellants resemble gas-like supercritical fluids and mix by molecular diffusion across a compositional mixing layer that emanates from the interface edge. Details about the calculations are provided in Section 7. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) (7) (8) (9) (10) UNCORRECTED PROOF 6 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx that simplify the theoretical description of the interface structure. Further details and additional explanations of the results in Figs. 2 and 3 are deferred to Section 7. In the transcritical region 0<x<LTR, the interface survives along with the surface-tension force it engenders. In general, the quantification of LTR requires numerical integration of the diffuse-interface formulation presented here. In laminar flows at moderately large values of the Reynolds number (2), an estimate of LTR can be obtained by an integral balance between the streamwise variation of the enthalpy flux in the fuel stream and the total amount of heat conduction received across the interface from the hot coflow, where λOis the thermal conductivity in the coflow stream. For axisymmetric jets, the approximation δT∼RFcan be made in Eq. (11) in analogy to cold round jets at downstream distances approaching thermal-entrance conditions [49]. It follows that LTR depends quadratically on the orifice radius RFas where the approximation TTR ≈Tc,diff ≈Tc,F has been made in accordance with Eqs. (6) and (7). In Eq. (12), DT,O is the thermal diffusivity of the coflow stream, and cp,F/cp,O is the ratio of fuel and coflow specific heats, which is of order unity in the conditions analyzed here, as discussed in Appendix B. Additionally, is an order-unity temperature-difference ratio, as prescribed by Eqs. (1) and (5). In contrast, two-dimensional laminar jets are known to have much longer thermal-entrance lengths than their round counterparts because the outer region, where streamwise convection and transverse diffusion balance, grows indefinitely with the square root of x, thereby yielding comparatively smaller heat fluxes into the cold stream [49]. In this case, a more appropriate approximation to utilize in Eq. (11) is δT∼(DT,OLTR/UF)1/2. This leads to the estimate which is used for normalizing xin Figs. 2 and 3. The estimates (12) or (13) predict that the supercriticalization length increases with decreasing values of the fuel injection temperature TF, coflow injection temperature TO, and combustor pressure P∞. Among these three quantities, the maximum sensitivity is to variations of TO. For instance, a decrease of 300 K in TOleads to a four-fold increment in LTR in the conditions analyzed in Figs. 2 and 3. The considerations above suggest that the injection orifice radius RFhas a decisive influence on the size of the transcritical region. In practice, LTR is a fraction of the thermal-entrance length, the latter delimiting the condition for which the hot coflow temperature TO(rather than Tc,F) is attained in the bulk of the fuel stream. However, it will be shown that the transfer of heat from the coflow to the fuel stream is hindered in transcritical conditions because of two factors: (a) High pressures lead to small values of the thermal diffusivity in the coflow stream. (b) For a significant portion of the interface length, the specific heat attains a global maximum in the vicinity of the interface, thereby leading to a decrease in the local thermal diffusivity there. These factors should be taken into account when interpreting molecular-dynamics simulations of transcritical systems, since those techniques are typically limited to microscopic length scales that may be too small compared to the supercriticalization length required to describe the macroscopic evolution of the interface with distance downstream of the injection orifice. Additional complexities arise in realistic scenarios that are beyond the scope of this study. For instance, in turbulent transcritical flows at large values of the Reynolds number (2), the molecular thermal diffusivity DT,O in Eq. (12) should be replaced by a turbulent eddy diffusivity of order Dturb ∼UFRF, which is lager than DT,O by a factor of order ReF≫1. Consequently, the ratio of the supercriticalization length to the orifice radius decreases from in laminar flows to in turbulent flows. There is however little work to date that have addressed closures in high-pressure turbulent flows [50], and therefore the effects of turbulence will not be further pursued here. Similarly, hydrodynamic instabilities may develop as a result of the competition between aerodynamic stresses and the interface-restoring surface-tension force in the transcritical region. These force imbalances may roll the interface leading to a finite local radius of interface curvature RI, and eventually to the formation of ligaments and droplets, as sketched in Fig. 1. Configurations different from that in Fig. 1 may therefore exist in which the supercriticalization isotherm never crosses the interface before it breaks up, but instead crosses the ensuing fuel spray cloud. 1.2. Objectives and outline of the present review This review focuses on theoretical aspects of transcritical phenomena at a continuum level and in a manner amenable to computational fluid dynamics (CFD). The key question to be explored is whether transcritical systems can be computed by integrating the Navier-Stokes equations uniformly over the entire flow field while subject to high-pressure equations of state along with the standard transport theory for multicomponent mixtures. The key concern that impedes that approach is related to the antidiffusive character of the mass transport predicted by the standard theory in transcritical conditions, which renders the problem ill-posed. To address that, a formulation of the Navier-Stokes equations with revisited constitutive laws, supplemented with (a) proper redefinitions of thermodynamic potentials, (b) a high-pressure equation of state, and (c) high-pressure thermophysical and multicomponent transport properties, is introduced that regularizes the problem and enables the description of volumetric effects of transcritical interfaces in multicomponent flows. The formulation is based on extensions of the diffuse-interface theory of van der Waals [108] –a useful formalism originally designed for monocomponent isothermal systems at high pressures, when the interface broadens significantly relative to intermolecular distances. The theory outlined here provides a description of the structure of stationary transcritical interfaces in equilibrium, as well as the downstream evolution of non-equilibrium transcritical interfaces separating propellant streams with different temperatures. In the latter case, this theory predicts that the interface ends at an edge, whose structure elicits the central role played by the diffusional critical point of the mixture in supercriticalizing the system, as shown in Fig. 2. Emphasis is made on transcritical interfaces in C12H26/N2mixtures, but supplementary thermodynamic analyses are provided for several other mixtures of interest that identify pressure and temperature conditions inducing transcritical interfaces in the flow field. A discussion is (11) (12) (13) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 7 also included about operating conditions in combustors of aerospace propulsion systems leading to transcriticality. The remainder of this manuscript is structured as follows. A qualitative description of transcritical phenomena is provided in Section 2 for monocomponent and bicomponent systems, along with a short survey of characteristic injection conditions in aerospace propulsion systems. Phase diagrams for hydrocarbon-fueled binary mixtures are discussed in Section 3. Using the diffuse-interface theory, a formulation of the Navier-Stokes equations that incorporates high-pressure effects and volumetric terms representing embedded interfaces is outlined in Section 4. Entropy-based closures of interface-related terms in the augmented Navier-Stokes equations, which enable the description of the structure of transcritical interfaces, are derived in Section 5. Readers who are not interested in the development of the formulation can skip Sections 4 and 5 and move directly to Sections 6 and 7, in which calculations of simplified canonical problems are provided. In Section 6, transcritical interfaces in equilibrium isothermal systems are analyzed. Section 7 is devoted to non-equilibrium transcritical interfaces in non-isothermal systems and their evolution downstream of the injection orifice. Lastly, concluding remarks are given in Section 8. Additionally, seven appendices are included that contain supplementary details and fundamental concepts closely connected with the formulation. In particular, Appendix A reviews basic concepts of thermodynamic stability and phase equilibrium. Appendix B provides a description of thermophysical properties and transport coefficients at high pressures. Appendix C outlines supplementary thermodynamic expressions for systems at high pressures. Appendix D focuses on numerical results obtained for near-critical interfaces in monocomponent systems. Appendix E discusses the behavior of the present theory near mechanically unstable conditions. Appendix F lists supplementary expressions for transport coefficients at high pressures, and Appendix G provides methods for the numerical integration of the formulation presented here. 2. High-pressure transcritical flow phenomena A significant challenge for the predictive calculation of transcritical flows lies in the complexity of the transitional and composition-de- pendent character of the interface formation and breakup, and of the subsequent dispersion and mixing of fuel in the combustor. To address these aspects, this section begins by introducing the main problems associated with transcriticality in monocomponent flows, followed by a description of more complex phenomena in bicompo nent flows. In addition, a survey of characteristic injection conditions in aerospace chemical propulsion systems is provided at the end of this section that serves to familiarize the reader with practical configurations. 2.1. Transcriticality in monocomponent systems Consider a closed vessel filled with a monocomponent liquid in phase equilibrium with its own vapor. The pressure and temperature in the vessel are denoted by Pand T, respectively. At normal pressure, the liquid and vapor are separated by an interface whose thickness is clearly not in the continuum range, and which is subject to a surface-tension force. Below the critical pressure Pc, the presence of such interface is warranted since each thermodynamic state (P, T) is associated with two densities [51] corresponding to the phase-equilibrium values for vapor, and liquid, . This is illustrated in the P−v thermodynamic phase diagram in Fig. 4 for N2, where is the molar volume and WN2is the molecular weight. The resulting ratio of densities of the phases, is typically of order 103at normal conditions. As the temperature of the vessel is increased, the liquid evaporates and a new phase-equilibrium condition is attained in which the pressure increases as well in accordance with the Clausius–Clapeyron relation. During this process, the interface thickens and the density ratio decreases to a value close to unity near the critical pressure Pc, at which the two phases become visually indistinguishable. Above the critical pressure, P>Pc, the fluid is supercritical with a uniquely defined density ρfor each thermodynamic state (P, T), thereby preventing the existence of an interface. Meanwhile, the surface tension necessarily vanishes at P≥Pc; the limit P→Pcis taken here in combination with T→Tc[53,54]. The transition from subcritical to supercritical conditions described above can be visualized in laboratory experiments involving the thermodynamic characterization of pure substances by using equilibrium cells [52,55]. A sequence of photographs that illustrates the vanishing distinction between the two phases in a system approaching and crossing the critical point of propane is provided in Fig. 5(a). In contrast with closed systems, combustors involve continuous flows of propellants. Consider a cold liquid jet injected at temperature TFand atmospheric pressure into its own hot vapor at temperature TO, where TF≤Te≤TO, with Tebeing the vapor-liquid equilibrium (VLE) or boiling temperature at the corresponding pressure. The resulting liquid-to-gas density ratio ρF/ρOis of order 103. The liq Fig. 4. Thermodynamic phase diagram for pure N2calculated with the Peng-Robinson equation of state introduced in Section 3.1. Included in the plot are the mechanical spinodals (dot-dashed lines), vapor-liquid equilibrium lines (thick red dashed lines), the pseudo-boiling line (thick green dot-dashed line), and the critical point (square symbol). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) UNCORRECTED PROOF 8 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. 5. Experimental visualizations of high-pressure flow systems. (a) Equilibrium cell filled with pure propane at pressure and temperature below the critical point (left panel), at pressure and temperature near the critical point (center panel), and at pressure and temperature above the critical point (right panel) (adapted from Ref[52].). (b) Cold nitrogen at 105 K (dark stream) injected into a hot nitrogen environment at 300 K at subcritical pressure 10 bar (left panel), near-critical pressure 30 bar (center panel), and supercritical pressure 40 bar (right panel) (adapted from Ref[6].). (c) N-hexadecane injected at subcritical temperature 300 K (dark stream) into a hot supercritical nitrogen environment at temperature 907 K and pressure 79 bar (i.e., higher than the critical pressures of the individual components). The different panels in (c) correspond to different instants of time after injection cutoff, with insets showing magnified views of fuel droplets (reprinted from Ref[19]. with permission from Elsevier). uid jet breaks up due to unbalances between aerodynamic and surface-tension forces [45,56–60] as illustrated in Fig. 5(b). A relatively large amount of energy, comparable to the thermal enthalpy of the hot environment, must be transferred to the spray to heat it up and vaporize it [61]. Similarly to the closed vessel operating at normal pressure, the thickness of the liquid-gas interface in the jet flow mentioned above is several orders of magnitude smaller than any of the relevant hydrodynamic scales involved. Nonetheless, predictive simulation capabilities have been developed in recent years using interface-captur ing methods such as the volume-of-fluid or level-set formulations, both of which volumetrically incorporate the surface-tension force in the momentum equation through a surface-tension coefficient σmultiplied by the interface curvature and by a Dirac-delta function localized at the interface [62–66]. As the ambient pressure P∞increases, the liquid-to-gas density ratio decreases. In particular, the interface becomes thicker as the critical pressure is approached, both surface tension and vaporization enthalpy decrease, and the atomization process becomes a mixing between gas-like fluids without any breakup. UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 9 Entering the supercritical region, it is important to distinguish between supercritical gas-like and supercritical liquid-like fluids as depicted in Fig. 4. As discussed in Appendix B.4, the factors that separate the behavior of liquid-like from gas-like supercritical fluids are: (a) A supercritical liquid-like fluid is one whose density is large, whose isothermal compressibility is small, and whose transport coefficients behave in a manner that is reminiscent of a liquid, with decreasing viscosities and thermal conductivities found at increasingly larger values of temperature, whereas the mass diffusivity remains mostly constant or increases slowly with temperature. (b) In contrast, the density of supercritical gas-like fluids is relatively smaller, their isothermal compressibility is larger, and their viscosity, thermal conductivity, and mass diffusivity increase with temperature, thus resembling the behavior observed in gases. At high temperatures, T≫Tc, and despite the high pressures, the intermolecular distances are sufficiently large for the supercritical gas-like fluid to behave like an ideal gas. At pressures not too large compared to Pc, a clear transition from liquid-like to gas-like supercritical behavior occurs at the pseudo-boil- ing, or Widom, temperature Tpb denoted by the thick green dot-dashed line in Fig. 4 (see Refs[67,68]. and Appendix B.4 for related discussions). This temperature depends on pressure and is not too different from Tc(e.g., Tpb ∼130 K at 40bar and Tpb ∼150 K at 60bar, both for N2, for which K). In particular, Tpb corresponds approximately to the temperature where a second-order phase transition occurs, with the density, viscosity and thermal conductivity plunging as transition from a liquid-like to a gas-like state occurs, and with the specific heat at constant pressure displaying a characteristic finite-am- plitude spike [69,70]. As the pressure becomes increasingly larger than Pc, the distinction between liquid-like and gas-like supercritical behavior is increasingly more difficult to ascertain. Consequently, the dynamical relevance of the pseudo-boiling line diminishes away from the critical point. The chamber pressure in the jet flow shown in the right panel in Fig. 5(b) is supercritical, but the jet and ambient temperatures are, respectively, smaller and larger than Tpb. Many studies have characterized those conditions as transcritical [5,8,26–29,31]. However, surface-tension effects are irrelevant in monocomponent systems above the critical point [53,71–74] as corroborated, for instance, by the absence of a spray in the right panel in Fig. 5(b). Furthermore, the decrease of surface tension with pressure and temperature is continuous, and as a result its value is always of little to no dynamical relevance even at subcritical conditions nearing the critical point in monocomponent systems. As a result, the liquid-like supercritical N2jet in the right panel in Fig. 5(b) mixes with the gas-like supercritical N2envi- ronment similarly to pure gaseous mixing. 2.2. Transcriticality in bicomponent systems Bicomponent systems behave fundamentally different from monocomponent ones and can lead to spray formation at combustor pressures that are supercritical with respect to the separate components, P∞>max{Pc,F,Pc,O} [15–19,22,23]. As shown in Section 3.2, an important physical characteristic enabling the formation of the spray is that binary mixtures of heavy hydrocarbons with typical oxidizers, including pure oxygen (for rockets) and air (for gas-turbine jet engines and diesel engines), have much higher critical pressures than those of the individual components. For mixtures of n-alkanes with O2, this effect is easily recognizable as a promontory in the coexistence region of the phase diagram along the pressure axis. For mix tures of n-alkanes with N2or air, the maximum pressure of the coexistence region diverges [21,75,76]. Consequently, even at supercritical pressures with respect to both components, the system can traverse the coexistence region when the injection temperatures of the two propellant streams are oppositely placed on either side of it. These incursions in the coexistence region, which engender transcritical behavior in combustors, are particularly likely when the injection temperature of at least one of the propellant streams (typically the heavy hydrocarbon fuel stream) is smaller than its critical temperature. An innermost zone is encountered within the coexistence region of the thermodynamic phase diagram that is bounded by the spinodal lines similarly to Fig. 4 for a pure component. In binary systems, this zone can exist above the critical pressures of the separate components due to the elevation of the critical pressure of the mixture, as mentioned above. This zone is thermodynamically unstable, in that no stable thermodynamic state exists that describes a spatially homogeneous mixture there. On the contrary, in those conditions, the components tend to stay separated into two fluids bounded by an interface where capillary forces become important. This phenomenon is observed in experiments (e.g., see Refs[15–23].), but not necessarily in computations unless an appropriate formulation of the problem, such as the diffuse-interface theory developed in this review article, is employed that accounts for interfaces consistently with the thermodynamic stability of the mixture. In bicomponent systems at high pressures, the fluids on each side of the interface, or on each side of the coexistence region of the thermodynamic phase diagram, cease to be different ordinary phases such as liquid and vapor, and instead become distinct supercritical fluids whose composition depends on the particular configuration. For instance, in the problem depicted in Fig. 1, the two participating fluids are a liquid-like supercritical mixture rich in a heavy hydrocarbon on the fuel side of the interface, and a gas-like supercritical mixture rich in coflow species (e.g., O2, N2, or air) on the coflow side of the interface. Fundamental differences between gas-like and liquid-like supercritical C12H26/N2mixtures are analyzed in Appendix B.4. In the present theory, the transcritical interface is a physical-space representation of the coexistence region. References to VLE lines at high pressures should therefore be understood as the loci of points where gas-like and liquid-like supercritical fluids –rather than classic vapor and liquid states –coexist. Pseudo-boiling conditions are attained within the transcritical interface, these being understood in multicomponent systems as approximately the conditions where the constant-pressure molar heat of the mixture attains a local maximum, and where transition between liquid-like and gas-like supercritical behavior takes place, as discussed in Appendices B.2 and B.4. As noted in Section 1.1, the disparity between the critical (Tc,F) and injection (TF) temperatures of the fuel stream plays an important role in the onset of transcriticality in combustors fueled by heavy hydrocarbons. The larger this disparity, the larger the supercriticalization length LTR, and the more favored are transcritical thermodynamic trajectories across the elevated coexistence dome. As indicated in Table 1, Tc,F is typically 200–400 K larger than TF, since the latter is typically limited to 300–450 K in order to minimize carbon deposition by thermal decomposition of the fuel in feeding systems and engine wall-cooling channels [77]. At those injection temperatures, heavy-hydrocarbon fuels resemble supercritical liquid-like fluids. That tendency increases with the number of carbon atoms due to the corresponding increase in the fuel critical temperature: K for CH4, 508 K for C6H14, 540 K for C7H16, 618 K for C10H22, and 658 K for C12H26. However, this is counterbalanced by an increasing supercriticality in pressure with increasing number of carbon atoms: UNCORRECTED PROOF 16 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. 10. Constant-pressure T−YFcross-sections of the phase diagram, including diffusional critical points (diamond symbols with coordinates) and vapor-equilibrium lines (thick red dashed lines) bounding the coexistence region of C12H26/N2mixtures at different pressures, along with the corresponding diffusional spinodal lines (thin purple dashed lines) and the loci of the maximum of the molar heat at constant pressure (thick green dot-dashed line) at each pressure. The mechanical critical point (solid circle) and the mechanical spinodal lines (thin solid lines) are provided only for the lowest pressure case. For pressures larger than 30bar, the mechanically unstable region lies at much lower temperatures than those shown in the figure. Solid ellipses denote characteristic injection conditions of N2(left) and C12H26 (right), while shaded regions are notional envelopes of pseudo-trajectories joining thermodynamic states along the normal to the interface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 3 Pressure, temperature, and composition at diffusional critical points and critical mixing conditions in C12H26/N2mixtures. Diffusional critical point Critical mixing conditions P∞[bar] Tc,diff [K] YFc,diff Tc,mix [K] YFc,mix 20 658 0.99 658 0.99 50 650 0.93 651 0.94 100 639 0.85 641 0.86 200 615 0.72 620 0.76 4. Theoretical foundations of the diffuse-interface theory for transcritical flows Crossings of the thermodynamically unstable regions in Fig. 10 require modifications of the constitutive laws in the Navier-Stokes equations. This section introduces basic assumptions, parameters, and scalings of the diffuse-interface theory, and outlines a set of conservation equations that serve as groundwork for the derivation of constitutive laws in Section 5. 4.1. Gradient-dependent thermodynamic potentials The diffuse-interface theory was first formulated by van der Waals in Ref[108]., where the finite-thickness structure of an interface between liquid and vapor phases in isothermal hydrostatic monocomponent systems was described by performing spatial integrations of the intermolecular potential in the presence of a mean density gradient. The reader is referred to seminal treatises based on the molecular theory of liquids for in-depth expositions of the topic [53]. This formalism was later extended to study binary mixtures near the critical point by Cahn and Hilliard [109]. More recently, notable work has been done to couple consistently the diffuse-interface theory of van der Waals with conservation equations of fluid motion [110–115]. Developments of the diffuse-interface theory for treating the mechanics of interfaces in complex mixtures have been applied mostly to 1D phase-transitioning systems in hydrostatic equilibrium at uniform temperature to provide predictions of the surface-tension coefficients [116–118]. The investigation of hydrodynamic and thermal effects in the formulation is a discipline that is currently in its infancy. UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 17 Fig. 11. Combustor pressures and injection temperatures of the cold propellant stream leading to absence of interfaces. The lines denote the diffusional critical temperature Tc,diff of the corresponding mixture. For a given value of TF(in configuration ⓐ) or TO (in configuration ⓑ), the intersection with the diffusional critical temperature provides the limiting combustor pressure P∞above which interfaces cannot exist. The diffuse-interface theory of van der Waals addresses monocomponent thermodynamic systems traversing the coexistence region as they become macroscopically separated into phases. Specifically, the diffuse-interface theory describes the mechanics of the transition layer, which gives rise to familiar surface-tension forces emerging from the resistance of the interface to get deformed. Although the predicted interfaces are hydrodynamically thin, and the theory is formulated at the continuum level, the results agree well with experiments and molecular dynamics simulations [99,116–119]. On average, molecules in ideal gases are separated by long distances that make the intermolecular forces negligible. In contrast, the increased molecular packing at high pressures requires consideration of intermolecular forces, which are manifested by the second term in the equation of state (14) involving the coefficient a. The latter is proportional to the zeroth-moment of the intermolecular potential in the far-field attraction range [53]. In highly density-stratified conditions as those found across interfaces in monocomponent systems, the effect of long-range attraction forces is compounded by the fact that the integration of the spherico-symmetric intermolecular potential across the stratified zone results in a gradient-dependent free energy [108]. This is represented by a correction to the bulk free energy that involves a coefficient multiplied by the square of the density gradient [111]. The equivalent description of thermodynamic potentials in multicomponent systems is based on the gradients of the partial densities [116,117]. For instance, the specific values of the gradient-de- pendent Helmholtz free energy f, internal energy e, and entropy sare [38,39,110] where fis defined as In this notation, Nis the number of species, and κi,j, and are gradient-energy coefficients discussed in Section 4.2. In this formulation, f,eand sare evaluated at local conditions {P, ρ, T, and Y1,Y2... YN−1} within the interface. 4.2. Gradient-energy coefficients Expressions relating and κi,j can be easily derived by substituting Eqs. (16)–(18) into Eq. (19), which gives where the definition has been used. The determination of the gradient-energy coefficients, which are directly related to the interface thickness and surface tension, constitutes a major source of uncertainty in the formulation, since they either require knowledge of the underlying molecular structure of the fluid, or must be tuned by comparing theoretical results with experimental data. Models for the gradient-energy coefficients often rely on correlations such as In this formulation, Tc,i is the critical temperature of species i, NA is the Avogadro’s number, and aiand bicorrespond to coefficients of the equation of state. The function depends on the reduced temperature and is calibrated from experimental data. One class of experiments used for calibration consists in enclosing a monocomponent fluid in an equilibrium cell [e.g., see Fig. 5(a)] and measuring its surface-tension coefficient σ[15,72,116–118]. As discussed in Section 5.3 (see also Ref[117].), the diffuse-interface theory for monocomponent flows predicts that σis given by where is a spatial coordinate locally normal to the interface. Values of κas a function of temperature can be inferred from Eq. (22) by using measurements of σin the equilibrium cell. The resulting function can be represented as [120] (16) (17) (18) (19) (21) (22) (23) UNCORRECTED PROOF 18 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx The dimensionless coefficients and are provided in Eqs. (C.17) and (C.18) (see also Refs[116,120].). The resulting dependency of the gradient-energy coefficient on temperature is shown in Fig. 12 for C12H26 and N2. Validations of the model (23) are provided in Section 6 and Appendix D. Three approximations in the gradient-energy coefficients are made in the theory developed here: (a) For simplicity, the gradient dependency of sis neglected in comparison with similar long-range effects on fand e, or equivalently which gives and . (b) The long-range interaction between components in mixtures, measured by the cross-influence parameters κi,j (i≠j) are modeled as with chosen for simplicity [117], although no rigorous justification for such geometric averaging appears to exist yet other than experimental observations of indirect quantities that ratify this choice [121]. (c) As shown in Section 5.6, the largest variations of the temperature occur across distances much larger than the interface thickness. Correspondingly, it is assumed herein that κi,j varies slowly in space around the interface, and therefore commutes with spatial derivatives. Temperature-induced variations in κi,j are included explicitly by evaluating κi,j at the local temperature in all expressions. Fig. 12. Gradient-energy coefficients κi,i for monocomponent systems composed of (a) C12H26 and (b) N2, as obtained by using expression (23) in conjunction with the equation of state (14). These approximations were also implicitly used in the original diffuse-interface theory of van der Waals [108]. 4.3. Open issues of experimentally calibrated models for gradientenergy coefficients Gradient-energy coefficients computed from experimentally calibrated relations such as Eq. (23) are contrived by the manner that experiments are carried out. As the equilibrium cell is heated, and starting from a phase-separated state in vapor-liquid equilibrium, both P and Treach their critical values simultaneously because of vapor-liq- uid equilibrium. As a result, the pressure dependency on Eq. (23), and similar ones encountered in the literature, is artificially concealed in the temperature. Relevant steps toward improved models of gradient-energy coefficients have been recently undertaken by Dahms [100] using parametrizations of κi,j in terms of the local partial-density gradients instead of temperature. Contrary to theoretical predictions [53,120], which suggest that the gradient-energy coefficient diverges near the critical point of the component, there are experimentally calibrated models such as Eq. (23) that remain finite there, as observed in Fig. 12. However, the singularity predicted by the theory is weak and in practice is overruled by the rapid decrease in the density gradients on approach to the critical point, where both σand dσ/dT vanish. Additional complications arising from the utilization of Eq. (23) or analogous ones [118] is that the experimentally calibrated gradient-en- ergy coefficients are not defined for T>Tc,i. Recall that a monocomponent system cannot engender any surface tension for T>Tc, since, as indicated in Fig. 4, only a single homogeneous thermodynamic state is possible above the critical point. As a result, the calibration of κi,i must necessarily stop at T>Tc,i. In the transcritical conditions in Fig. 1, in which TO>Tc,F >TF>Tc,O, the gradient-energy coefficients given by Eq. (23) are not defined in regions of the flow where the local temperature is larger than the critical temperature of the corresponding component. This has the following implications: (a) Since the injection temperatures considered here are TF∼ 300–450 K and TO∼700–1000 K, the temperature is everywhere larger than the critical temperatures of typical coflowing species, including N2, O2, and air. Consequently, the only gradient-energy coefficient that is defined and explicitly appears in the formulation is that of the fuel, denoted here by κF,F. (b) The fuel gradient-energy coefficient κF,F ceases to be defined on the coflow side of the interface where T>Tc,F. However, these limitations do not have significant consequences on the theory developed here. As discussed in Section 1, the interface vanishes at an edge when its temperature nears the temperature of the diffusional critical point Tc,diff, which is K smaller than Tc,F in the pressure range P∞=50–200 bar. Correspondingly, regions where κF,F is not defined because T>Tc,F are located either far away from the interface on the coflow side, or downstream of the interface edge, as schematically indicated in Fig. 13. Since composition gradients in those regions are small compared to those encountered across the interface, the gradient-dependent corrections of the local thermodynamic potentials are dynamically irrelevant there regardless of whether κF,F is made to artificially plunge to zero at or is extrapolated for T>Tc,F. From a molecular-scale perspective, κi,j can be shown to be proportional to the second moment of the intermolecular potential (e.g., see Ch. 4 in Ref[53]. for details). It can also be expressed in terms of the direct correlation function in both single- and multicomponent systems [122,123]. It is conceivable that future developments in hy (24) (25) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 19 Fig. 13. Schematics of the region enclosed by the fuel critical isotherm (gold colored zone), where the fuel gradient-energy coefficient κF,F is defined according to the model (23). drodynamic applications of the diffuse-interface theory for high-pres- sure interfaces would benefit from models of κi,j that could be cognizant of the microscale structure of supercritical fluids and their mixtures. Multiscale modeling strategies based on combination of molecular dynamics and continuum-level theories or simulations may bear some potential for on-the-fly calculations of κi,j based on direct correlation functions [124,125]. 4.4. Characteristic scales in the diffuse-interface theory The disparity between the interface thickness δIand the macroscopic flow scales δT,RI, and RFplays a central role in the dynamics of the interface. The value of δIis closely connected with the gradient-energy coefficient and with the surface tension σ. In particular, Eq. (22) indicates that the gradient-energy coefficient scales as where σ0is a characteristic value of the surface tension, and ρFis the density of the fuel stream. In performing the estimate (26), the characteristic density difference across the interface has been approximated as with the density ratio defined in Eq. (4). To understand the role of the gradient-energy coefficient in the present theory, it is important to note that any thermodynamic variable that oscillates across the coexistence region in the phase diagram can be mapped into physical space within an interface-like profile of an indicator function such as in monocomponent flows, or the partial densities in multicomponent flows. Although the details of the mapping depend on the particular problem and thermodynamic quantity under consideration, it is illustrative to momentarily sideline the problem in Fig. 1 and focus on the most idealized case that can be addressed with the diffuse-interface theory, namely that of a flat interface in a monocomponent isothermal flow. In this case, the mapping is simpler and the free-stream densities correspond to those in phase equilibrium, i.e., and . As discussed in Appendix A.2, in phase equilibrium, the chemical potentials in both streams are the same and equal to the equilibrium value μe[see Eq. (A.16)]. Similarly, since the temperatures of the streams must be equal in phase equilibrium, Eqs. (C.4) and (C.5) simplify, respectively, to and In particular, Eq. (28) provides the definition of the pressure in terms of the free energy which can be combined with Eq. (C.7) giving By substituting Pfrom Eq. (14) into Eq. (29) and integrating the resulting equation, an expression for fcan be found and be later substituted into Eq. (30) to obtain one for μ. The resulting expressions for fand μare not essential here and are deferred to Appendix C. Eqs. (27) –(30) are standard expressions in thermodynamics that are useful here for the following reason. Since the pressure Pis an oscillatory function of ρacross the coexistence region of the thermodynamic phase diagram (e.g., see Fig. 4), fand μare also oscillatory functions of ρthere. In particular, μsatisfies a Maxwell’s construction rule [126], in that the oscillation of in the coexistence region encloses zero net area, as observed by using Eqs. (30) and (C.7). A fundamental result of the diffuse-interface theory for monocomponent isothermal systems is the equation which relates μ,ρ, and the normal coordinate to the interface . The derivation of Eq. (31) is deferred to Section 5.3. It will also be shown in Section 5.6 that an equation similar to (31) exists for multicomponent isothermal systems. The boundary conditions of Eq. (31) are at and at . The right-hand side of Eq. (31) vanishes far away on both sides of the interface, where the density gradients are negligible and the chemical potential attains the phase-equilibrium value μe. As illustrated in Fig. 14, the expression (31) is responsible for mapping the oscillations of the chemical potential in the two-phase region of the thermodynamic space into an interface-like profile in which ρundergoes a rapid variation of order ρFwithin a very short distance of order δI. To obtain an estimate of δI, Eq. (31) can be rewritten in terms of a nondimensional density and a nondimensional chemical potential with Aμthe characteristic amplitude of the oscillation of μin the coexis (26) (27) (28) (29) (30) (31) (32) UNCORRECTED PROOF 20 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. 14. Schematics of the thermodynamic-to-physical space mapping (31) built in the diffuse-interface theory for transforming the variations of the chemical potential into a sharp monotonic density profile in a monocomponent isothermal system. tence region. In these variables, the first integral of Eq. (31) is Eq. (33) evaluated far away from the interface, where ρ⋆→1 and is consistent with Maxwell’s construction rule. The parameters κ,ρF, and Aμcan be scaled out of Eq. (33) by defining a dimensionless normal coordinate with The combination of Eqs. (26) and (34) yields In particular, Eq. (35) connects the oscillation amplitude Aμ, which can be obtained independently from the thermodynamic relations (14) and (30), with the intrinsic properties of the interface δIand σ0, the latter being a measurable quantity. A cornerstone of the diffuse-interface theory is that the order of magnitude of κis set such that δIestimated from (34) is small enough to yield values of σ0from Eq. (22) comparable to the experimentally observed ones σ0∼0.001–0.1 N/m. This gives δI∼1–100 nm and κ∼10−15–10−17 Jm5/kg2(see Fig. 12). The characteristic amplitude of the pressure oscillation within the interface, denoted by AP, is obtained from Eqs. (29), (30), and (35) giving Eq. (36) yields AP∼10–100 bar when the estimates for σ0and δI introduced above are employed. A dimensionless pressure parameter can be defined as the ratio of the characteristic pressure variation APacross the interface to the combustor pressure P∞, which is related by Eq. (36) to Aμas where is the compressibility factor. The dimensionless pressure parameter increases with decreasing pressures. At near-atmospheric pressures, is much larger than unity, and the pressure oscillation predicted by the equation of state (14) can induce large negative pressures oftentimes observed in the form of tensile strength in liquids brought to metastable states by superheating them above the boiling temperature, or by stretching them below the vapor pressure [104,127,128]. As P∞ increases, σ0decreases and δIincreases, thereby rendering smaller values of APand . The numerical results presented in Sections 6 and 7 indicate that the dimensionless pressure parameter is of order unity in transcritical conditions. As a result, despite the large values of P∞involved, the present theory predicts that mechanically subcritical pressures, and even negative pressures, may occur within the interface. These aspects are further discussed in Appendix E as they may appear controversial in the description, although they do not have any significant effect on the results presented here. 4.5. The interface-continuum hypothesis The approximation of operating with a continuous field across the transcritical interface while keeping only the gradient terms in the series expansion of the thermodynamic potentials, as in Eqs. (16)–(18), is justified when the interface thickness δIis large compared to the characteristic mean free path Λ, or equivalently, when the Knudsen number is much smaller than unity. In multicomponent mixtures described by the hard-sphere model, Λis given by [129] In this formulation, dFand dOare the equivalent hard-sphere diameters of the fuel and coflow species, respectively, with the mean diameter (i.e., nm, nm, and nm for C12H26/N2mix- tures [130]). The direct application of Eq. (41) to the present problem is problematic because of the strong composition gradients associated with transcritical interfaces. In the calculations of KnIshown in Section 6, the partial densities in Eq. (41) are approximated as ρYF∼XF,maxP∞WF/(ZF,maxR0Te) and where Teis a characteristic interface temperature, and the subindex max denotes quantities evaluated at the location of maximum density gradient (33) (34) (35) (36) (37) (38) (39) (40) (41) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 21 across the interface. With these approximations in mind, Eq. (41) becomes where kBis the Boltzmann constant. For the monocomponent systems calculated in Appendix D, Eq. (42) is used with and . Small Knudsen numbers (40) are often found in monocomponent systems as the critical point is neared, when δIincreases and Λdecreases. For instance, the interface thickness calculated in Appendix D for a monocomponent N2system at bar is δI∼5 nm. This translates into approximately 32N2molecules across, along with a Knudsen number KnI∼0.05 (see Fig. D.3 in Appendix D). However, it is widely accepted that the requirement KnI≪1 is not always satisfied by the diffuse-interface theory of van der Waals as the pressure becomes increasingly smaller than Pc[53,131,132]. This can be seen in the same monocomponent N2system mentioned above at bar, for which δI∼2 nm and KnI∼0.15, thereby indicating a larger rarefaction and a weaker foundation for the continuum character of the theory. Despite these shortfalls, the diffuse-inter- face theory of van der Waals tends to reproduce well numerical results obtained from molecular dynamics (e.g., see Fig. 6 in Ref[99].). The considerations above indicate that the continuum assumption across the interface is often on the borderline of invalidity. In the hydrocarbon-fueled mixtures analyzed in Figs. 7–9(b–e), the pressures that can be reached before KnIbecomes much smaller than unity and are much larger than those in monocomponent systems because of the elevation of the critical point caused by mixing. It is however remarkable that no excessively large values of KnIthat would indicate a flagrant violation of the continuum assumption are encountered at the high pressures studied here. Field formulations of fluid motion are well-known for performing correctly even in flows where the continuum hypothesis may not be strictly satisfied [129]. The continuum treatment of diffuse interfaces is reminiscent of the description of the internal structure of weak shock waves by using the Navier–Stokes equations despite the small shock thicknesses involved [133]. However, one important difference between shocks and interfaces is Fig. D.3. Variations of (a) the interface thickness and (b) Knudsen number as a function of the temperature nondimensionalized with the critical temperature of the corresponding component. that the dynamics outside the shock discontinuity are independent of its internal structure, whereas the transport across the interface and its mechanical coupling with the flow fundamentally depend on the internal structure of the interface, as discussed in Section 5. The appropriateness of Eq. (41) as the relevant length scale to evaluate the continuum character of transcritical interfaces is also under suspicion. Recent molecular dynamics simulations [134] indicate that confined gases undergo higher rates of collisions than those predicted by Eq. (41). It could be plausible that an interface may lead to an effect similar to confinement for the fluids on each side and within the interface itself. Similarly, a recent study by Dahms [135] has found that the effective mean free path near the interface should be 0.55 times smaller than the classic definition in Eq. (41). Whether interfaces at high pressures fully satisfy the continuum requirement KnI≪1 remains a subject open to debate. 4.6. The role of antidiffusion in transcritical flows An important characteristic of the mass transport across transcritical interfaces at sufficiently high pressures is the prevailing antidiffusion of matter that is predicted by the standard transport theory in conjunction with the equation of state (14). To understand this, notice that the effective Fickian diffusion coefficient participating in the standard species diffusion flux of fuel at high pressures is not the binary diffusion coefficient but [136,137] where fFis the fuel fugacity defined in Eq. (B.22) as a function of the fugacity coefficient (B.23), the latter being particularized in Eq. (C.47) for the Peng-Robinson equation of state (14). A formal derivation of Eq. (43) is deferred to Section 5. In contrast to which is always positive (see Fig. B.3 and related discussion in Appendix B.4), the effective Fickian diffusion coefficient DF,F can be negative for intermediate compositions at temperatures smaller than Tc,diff, as shown in Fig. 15 for C12H26/N2mix- tures. The change in sign can be explained by the fact that the fuel fugacity oscillates across the coexistence region (and therefore across the interface) similarly to the fuel chemical potential, as discussed in Appendix A.1 and illustrated in Fig. A.1(a,b) for C12H26/N2mixtures. At moderate pressures or high temperatures approaching ideal-gas conditions, the non-ideal diffusion prefactor becomes unity, (∂ln fF/∂ln XF)P,T →1. Therefore and traditional forward molecular diffusion (i.e., diffusion in the direction of decreasing fuel concentration) prevails. At high pressures, forward diffusion dominates in the compositional mixing layer downstream of the interface edge. In contrast, near the injection orifice, where the fuel is cold, the non-ideal prefactor (∂ln fF/∂ln XF)P,T departs from unity and its sign changes depending on the local composition. In particular, (∂ln fF/∂ln XF)P,T varies from positive values on the coflow and fuel sides of the interface, where forward diffusion prevails, to negative values within the transcritical interface, where mass is transported in an antidiffusive manner in the direction of positive composition gradients because of the diffusional instability of the mixture, as sketched in Fig. 16. The antidiffusion of matter predicted by the standard transport theory, which would lead to unphysical results, can be regularized by a an extra diffusion flux derived from the diffuse-interface theory, as described in Section 5. (42) (43) UNCORRECTED PROOF 22 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. B.3. Constant-pressure distributions of (a) dynamic viscosity, (b) thermal conductivity, (c) binary diffusion coefficient, and (d) thermal-diffusion ratio for C12H26/N2mixtures at high pressures as a function of temperature and mass fraction of dodecane YF[refer to the legend in the right panel in (a)]. The plots include the diffusional critical point of the mixture at the corresponding pressure (diamond symbol), as well as projections of the loci of the maximum of the molar heat at constant pressure (thick dot-dashed line), and the VLE line (thick dashed line) on the planes and . In interpreting these plots, it is important to note that the entire coexistence region is not exactly enclosed within the space bounded by the VLE line in panels (a), (b) and (d) since the partial variation of the transport coefficient with respect to YFattains negative values there. Fig. 15. Constant-pressure distributions of the effective Fickian diffusivity of dodecane in C12H26/N2mixtures at (a) P= 50 bar and (b) 100 bar as a function of temperature and mass fraction of dodecane YF(refer to the legend in the left panel). The plots include the diffusional critical point of the mixture at the corresponding pressure (purple diamond symbol) and the diffusional spinodals (thick purple dashed lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 4.7. Conservation equations for transcritical flows The consideration of gradient-dependent thermodynamic potentials, as in Eqs. (16)–(17), leads to interface-related transport fluxes and mechanical stresses in the conservation equations. In this study, a phenomenological approach described in Section 5 is followed based on linear augmentations of the deviatoric part of the stress tensor τ, and the heat and species diffusion fluxes qand Ji, with the interfacial stress tensor and transport fluxes and . Expressions for and are derived in Section 5. In terms of the material-derivative operator the resulting conservation equations for mass, momentum, species, and total energy are which describe the continuum mechanics of a multiphase, multicomponent fluid of Nspecies that moves at a mass-averaged velocity vand has a density ρand total energy E. The latter is related to the specific gradient-dependent internal energy (17) by the definition Additionally, qis a molecular heat flux defined as (44) (45) (46) (47) (48) (49) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 23 Fig. 16. Schematics of diffusional processes across a notional transcritical interface separating fuel and coflow streams at high pressures. The sketch includes (a) diffusional processes in physical space, and (b) associated evolution of the fugacity of the fuel species fFin thermodynamic space. Note that the fugacity profile in panel (b) does not necessarily correspond to one at uniform pressure such as those presented in Fig. A.1(a), since the pressure generally varies within the coexistence region. The first term qton the right-hand side of Eq. (49) comprises Fourier and Dufour mechanisms of heat transfer, while the second term corresponds to heat transport by interdiffusion of species with different partial specific enthalpies hi, which are discussed in Appendices B.1 and C.8. In addition, the symbol is a gradient-de- pendent pressure given by with Pbeing related to ρ, T and Yithrough the equation of state (14). The convenience of delocalizing the pressure as in Eq. (50) will become clearer upon deriving an expression for the interfacial stress tensor in Section 5. An alternative form of the energy Eq. (47) that will be employed in the analysis is the enthalpy conservation equation Eq. (51) is obtained by subtracting the momentum conservation Eq. (45) multiplied by vfrom the total-energy conservation Eq. (47), and by making use of the relation obtained from the definition of the local specific enthalpy h(C.3) along with Eqs. (17) and (50). Methods for calculation of hat high pressures are discussed in Appendices B.1 and C.6. The integration of the conservation Eqs. (44)–(47) is complicated by the additional terms and which will be shown to de Fig. A.1. Thermodynamic phase diagrams for C12H26/N2mixtures at uniform temperature T= 500 K in terms of (a) fuel fugacity fFas a function of the fuel molar fraction at different constant pressures, (b) fuel molar chemical potential as a function of the fuel molar fraction at different pressures [refer to legend in panel (a)], and (c) pressure as a function of the fuel molar fraction indicating different stability regions of the diagram. (50) (51) (52) UNCORRECTED PROOF 24 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx pend on the square of gradients and higher-order derivatives of the partial densities across the interface. For instance, a full numerical integration of the momentum Eq. (45) in the flow sketched in Fig. 1 would incur significant numerical stiffness, because the ratio of the interfacial terms to the convection terms would be a large parameter, where is a Weber number, and ϵRis a large-scale Cahn number defined in Eq. (9). Typical orders of magnitude are and in the present conditions, thereby making the ratio (53) a large number of order due to the much smaller value of δIcompared to RF. 5. The structure of transcritical interfaces This section focuses on the derivation of the additional terms and in the constitutive laws of the Navier–Stokes Eqs. (44)–(47) by using the diffuse-interface theory of van der Waals extended to multicomponent flows. Using those expressions and exploiting the small thickness of the interface relative to all other macroscopic lengths, a local formulation in the moving frame attached to the interface is developed that provides mechanical and transport equilibrium constraints applicable to interfaces in isothermal systems. Moderate temperature gradients across the interface, such as those encountered in Fig. 1, are shown to require the consideration of small deviations from equilibrium. 5.1. Entropy-production sources The derivation of and is performed here using the method of irreversible thermodynamics and Onsager’s reciprocal relations. The premise of this method is that the constitutive relations must lead to non-negative entropy production in accordance with the second principle of thermodynamics. This method parallels the one utilized to determine the analytical form of the transport fluxes in the standard theory [138–140]. However, this method cannot give expressions for the transport coefficients in terms of molecular properties. Those are provided separately in Appendices B.4 and Appendix F. The analysis begins by substituting Eqs. (17) and (50) into the first principle (C.1), and using the assumption of constant and symmetric κij coefficients, which leads to where is an auxiliary vector. A transport equation for the specific entropy s can be derived by substituting Eqs. (44)–(47) and the definition (49) into the expression resulting from taking the material derivative of Eq. (55), which yields where are generalized versions of the partial specific entropy and specific chemical potential μi, respectively. In addition, is an entropy source given by with Ibeing the identity matrix and ⊗denoting the dyadic product. The definition the mass-conservation constraint and the vector identity have been used in deriving Eq. (59), with D(ρYi)/Dt being replaced by in accordance with Eq. (46). To separate transport by temperature gradients, the decomposition (53) (54) (56) (57) (58) (59) (60) (61) (62) (63) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 25 is performed, where use of Eq. (58) and of the reciprocity relation for the partial specific entropy have been made. In the formulation, the operator ∇Tindicates spatial differentiation at constant temperature. Substitution of Eq. (63) into Eq. (59) yields where is an auxiliary variable. The last term on the right-hand side of Eq. (65) can be expressed as by making use of the equation of state (14) and the definition of ψiin Eq. (56). In this formulation, βvis the volume expansivity defined in Eq. (B.7) and particularized for the Peng-Robinson equation of state (14) in Eq. (C.22). Following an approach similar to that provided by Onsager [138] (see also Ch. 11.2 in [140], and Ch. 6 in Ref[139].), the entropy source (64) is interpreted as consisting of three production terms that have a similar form. Specifically, each term consists of a current multiplied by one of the following thermodynamic forces: The quantities (67) are drivers of molecular transport, and consequently of entropy production. Different transport processes induced by different thermodynamic forces interfere with one another. For instance, in addition to causing heat conduction, temperature gradients also induce mass transport through the Soret effect. More generally, the transport fluxes in the conservation equations can be written as a combination of all thermodynamic forces, and such combination is linear in the first approximation [138–140]. The derivation of these expressions is facili tated by assuming that the transport fluxes and thermodynamic forces of different tensorial character do not couple [141]. This simplification implies that there are no crossed effects between the mechanical stresses, which are driven by the velocity-gradient tensor, and the fluxes of heat and species, which are driven by gradients of temperature and chemical potentials. Based on these considerations, a closure is formulated for in Section 5.2, followed by a discussion in Section 5.3 on the effective surface-tension coefficient σarising from this formulation. Onsager’s methodology will be used in Section 5.4 to determine the interfacial fluxes of heat and species . 5.2. Closure for the interfacial stress tensor In the case of Newtonian fluids under zero bulk viscosity, the expression for the viscous stress tensor is where ηis the dynamic viscosity modeled in Appendix F.1. Variations of ηwith temperature and composition are studied in Appendix B.4 for C12H26/N2mixtures at high pressures. Eq. (68) implies that the viscous dissipation τ:∇v≥0 is a positive quadratic and therefore leads to positive entropy production. In contrast, the interfacial stress tensor is assumed to be elastically restoring, such that the production of entropy by the diffuse-interface terms multiplying the velocity-gradi- ent tensor in Eq. (64) is zero, namely which gives The interfacial stress tensor was originally introduced by Korteweg in a phenomenological manner [142]. The effects of are concentrated in the vicinity of the interface, where the partial-density gradients are the largest. In that region, the first of the two terms on the right-hand side of Eq. (70) represents a hydrostatic stress. As shown in Section 5.3, this term gives rise to a curvature-dependent pressure jump across the interface. The second term is an anisotropic stress that acts primarily in the direction normal to the interface. In this formulation, there is no explicit surface-tension coefficient σappearing in Eqs. (44)–(47) and (70). However, σresurfaces when the integral form of the momentum conservation Eq. (45) is considered, as shown in the next section. 5.3. Mechanical equilibrium conditions This section focuses on the derivation of mechanical equilibrium conditions across the interface. The analysis is first particularized for the bicomponent flow depicted in Fig. 1. The problem is best illustrated by analyzing a zoomed-up view of a segment of an interface embedded in a much wider thermal mixing layer, as sketched in Fig. 17. The thickness of the mixing layer δTgrows slowly with distance downstream and mostly coincides with the shear-layer thickness because of the near-unity Prandtl numbers and the order-unity relative differences of velocities and temperatures between the (64) (65) (66) (67) (68) (69) (70) UNCORRECTED PROOF 32 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx because the viscous dissipation τ:∇vin Eq. (124) is a positive quadratic. The transport fluxes in ϕmust be such that Eq. (127) is satisfied. The linear relation is the simplest approach that satisfies 127. In Eq. (128), is a matrix of Onsager phenomenological coefficients, with . Specifically, in order to satisfy Eq. (127), must be symmetric and positive semidefinite, in that its elements must satisfy the conditions [138–140] Using Eq. (128) in Eq. (124), the expression is obtained. In Eq. (130), the symbols Fi,Fk, and Fk+1denote components of F, with and for . In writing Eq. (131), use has been made of the definition (58) for the generalized chemical potential . Additionally, in Eq. (130), the Onsager coefficients have been renamed as with . A physical interpretation of these coefficients will be provided in Section 5.5. The first bracketed term in the second line in Eq. (130) corresponds to the first element of the vector of currents (126). Equating these quantities gives Similarly, the second bracketed term in the second line in Eq. (130) corresponds to the element of the vector of currents (126), with i= 1,⋯N−1. In Eqs. (133) and (134), terms on the left-hand side can be associated with those on the right-hand side depending on whether they are functions of the gradient-energy coefficients. For instance, on the left-hand side of Eq. (133), qtcan be matched with those terms on the right-hand side that are independent of the gradient-energy coefficients, Substituting Eq. (135) into Eq. (49) gives the standard heat flux Similarly, on the left-hand side of Eq. (134), Jican be matched with those terms on the right-hand side that are independent of the gradient-energy coefficients, giving the standard species diffusion flux In Eqs. (136) and (137), the constant-temperature gradient of the chemical potential can be expanded in terms of gradients of composition and pressure, as shown in Appendix F.3, leading to the more familiar representation of these fluxes in Stefan–Maxwell form. The interfacial corrections to qand Jiare derived by matching the remainders on both sides of Eqs. (133) and (134), respectively. This procedure gives the interfacial heat flux and the interfacial species flux with ψiand χibeing defined in Eqs. (56) and (65), respectively. Similarly to the interfacial stress tensor the species and heat fluxes and are also highly localized at the interface, where the partial-den (128) (129) (131) (132) (133) (134) (135) (136) (137) (138) (139) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 33 sity gradients multiplied by the gradient-energy coefficients are dynamically relevant quantities. In practical implementations of the transport fluxes (136)–(139), the constant-temperature gradients of μiand ψiare eliminated in favor of the full gradients ∇by making use of Eqs. (63) and (65). The resulting expressions can be readily used to exchange for and for respectively. 5.5. The Onsager coefficients in transcritical conditions The fluxes (136)–(139), along with the viscous and interfacial stresses (68) and (70), can be substituted into the entropy production rate (64), yielding It is shown in this section that the right-hand side of Eq. (142) is a positive quadratic in the transcritical conditions studied here. In Eqs. (136)–(139) and (142), the N2coefficients Lq,q,Lq,i ( i= 1,⋯N−1), and Li,k ( ) are arranged in the Onsager matrix as The first subset of consists of the single coefficient Lq,q, which participates in the Fourier conduction component of the standard heat flux q. By simple inspection of Eqs. (136) or (F.24), the coefficient Lq,q can be written as with λbeing a thermal conductivity modeled in Appendix F.2. The second subset of corresponds to the coefficients Li,k [i.e., dot-dashed line rectangle in Eq. (143)], which are related to the interfacial species flux in Eq. (139) and to the Fickian component of Ji in Eq. (F.25). The calculation of Li,k is summarized in Appendix F.4 and involves the utilization of Eq. (F.34) relating the coefficients Li,k with the binary diffusion coefficients modeled in Appendix F.5 . For binary mixtures, Eq. (F.34) becomes a scalar equation that yields The third subset of are the coefficients Lq,i [i.e., solid line rectangles in Eq. (143)], which are involved in the computation of the Soret and Dufour effects appearing, respectively, in the standard diffusion fluxes of heat qin Eq. (136) and species Jiin Eq. (137) [see also Eqs. (F.24) and (F.25) for the corresponding Stefan–Maxwell forms of these fluxes]. In particular, the coefficients Lq,i can be related through Eq. (F.43) to the diagonal coefficients Li,i and to the thermal-diffusion ratio ki,T, the latter being modeled in Appendix F.6. For binary mixtures, Eq. (F.43) simplifies to The distributions of λ, and kF,T with temperature and composition are studied in Appendix B.4 for C12H26/N2mixtures at high pressures. The size of the Onsager matrix in binary mixtures is 2×2 and its determinant is given by . The requirement that be positive semidefinite, and therefore that the entropy production source (142) be zero or positive, implies the condition . Numerical evaluations for 15 bar≤P≤100 bar indicate that the condition is satisfied for C12H26/N2mixtures at temperatures 300 K≤T≤1000 K as long as the parameter constant ι participating in the calculation of kF,T is recalibrated within 10% of its standard value (see Appendix F.6 for details). These considerations are illustrated in Fig. 19 for two representative pressures. Upon substituting the Onsager coefficients (144)–(146) into Eqs. (136)–(139) particularized for the expressions Fig. 19. Constant-pressure distributions of the determinant of the Onsager matrix for C12H26/N2mixtures at (a) P= 50 bar and (b) 100 bar as a function of temperature and mass fraction of dodecane YF(refer to the legend in the left panel). The plots include the diffusional critical point of the mixture at the corresponding pressure (purple diamond symbol). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) (140) (141) (142) (143) (144) (145) (146) (147) UNCORRECTED PROOF 34 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx and are obtained for the interfacial fluxes of heat and species, respectively, along with and for the standard diffusion fluxes of heat and species, respectively, where additional use of Eqs. (140) and (141) has been made. In Eqs. (149) and (150), the difference of partial specific entropies can be calculated as by making use of the definition (60). Alternatively, based on the analysis outlined in Section F.3, qand JFcan be recast in traditional Stefan–Maxwell form as and as shown by evaluating Eqs. (F.24) and (F.25) for N= 2 and utilizing Eqs. (145) and (146). Despite the fact that the Stefan-Maxwell forms (152) and (153) are more insightful than their unexpanded counterparts (149) and (150), their utilization in transcritical flows is not advantageous when the combustor pressure is not sufficiently high and the system crosses the mechanical spinodals. In those conditions, singularities arise in the non-ideal diffusion prefactor (∂ln fF/∂ln XF)P,T and in the fuel partial molar volume [defined in Eq. (B.19) and particularized for the Peng-Robinson equation of state in Eq. (C.43)] that prevent from splitting the constant-temperature gradient of chem ical potential into barodiffusion and Fickian diffusion, as discussed in Appendix E.1. 5.6. Transport equilibrium condition In a similar way as the gradient-dependent pressure and interfacial stresses are balanced when the interface is in mechanical equilibrium, a balance between interfacial fluxes and standard diffusion fluxes exists when the interface is in transport equilibrium, as discussed in this section. 5.6.1. The transport equilibrium condition in terms of a balance of species fluxes Consider again the slender interface separating two propellant streams in Fig. 17. The analysis begins by recalling that the characteristic length of the variation of the temperature downstream of the injection orifice is the thickness of the thermal mixing layer δT. In contrast, the composition across the interface undergoes rapid changes along much smaller distances of the same order as the interface thickness . This large composition gradient is due to the suppression of molecular diffusion in thermodynamically unstable conditions at high pressures, where the standard transport theory predicts antidiffusion of fuel and an ever-increasing composition gradient, as discussed in Section 4.6. It will be shown here that a mutual cancellation of the standard and interfacial species fluxes occurs at the interface in the limit ϵT≪1 corresponding to small temperature gradients compared to the composition gradients across the interface. The resulting transport equilibrium condition provides a quasi-steady thermochemical description of the interface structure that is shown schematically in Fig. 20 and is elaborated in the remainder of this section. The normal components of the species fluxes (148) and (153) ca be decomposed as In this notation, the prime symbols refer to portions of the standard and interfacial species fluxes (153) and (148) that are independent of the temperature gradients: The superindex ΔT refers to the portions of fluxes directly proportional to the temperature gradients: (148) (149) (150) (151) (152) (153) (154) (155) (156) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 35 Fig. 20. Schematics of the quasi-steady thermochemical structure of a transcritical interface near transport equilibrium in a non-isothermal system at small thermal Cahn numbers, ϵT≪1. Using this notation, and in the moving curvilinear frame described in Section 5.3, the species conservation Eq. (46) for the fuel near the interface becomes In order to isolate the most important terms in Eq. (159), consider nondimensionalizing Eq. (159) with the same units as those employed to write the momentum Eq. (89), namely Eqs. (32) and (81)–(86). Additional characteristic scales for the temperature and species fluxes are obtained as follows. For ϵT≪1, the temperature gradient near the interface is locally a constant that can be approximated as . As a result, the variations of the temperature across the interface with respect to the interface temperature Teare of order ϵTrelative to the temperature difference between the free streams . The dimensionless temperature is therefore defined as The chemical-potential variations through the interface Aμ, defined in Eq. (35), also correspond to the variations of μFthrough the coexistence region in thermodynamic space. These variations are re lated by Eq. (F.23) to the variations of the logarithm of the fugacity with respect to composition as across the interface, where the fuel molar fraction XFis of order unity. In writing Eq. (161), use of Eq. (38) has been made in conjunction with the definitions (37) and (39) for the pressure ratio and the fuel compressibility factor ZF, respectively. Utilization of Eqs. (32) and (161) in Eq. (155), along with as the binary diffusion coefficient normalized with its value in the coflow stream, provides the definition of the nondimensional portion of the standard species flux involving barodiffusion transport, Using Eqs. (81) and (160) in Eq. (157), the remainder corresponding to the Soret effect, is nondimensionalized as where αis a thermal-expansion ratio defined in Eq. (5). The portion of the interfacial flux independent of the temperature gradient is normalized by substituting Eqs. (32), (81), (86), (32), (160), the nondimensional molecular weight and the nondimensional partial molar volume into Eq. (156), thereby yielding (157) (158) (159) (160) (161) (162) (163) (164) UNCORRECTED PROOF 36 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx The remainder is normalized as by using Eqs. (81), (86), (32), and (160) in Eq. (158), along with for the nondimensional volume expansivity, with βvFbeing the corresponding value in the fuel stream [see Eqs. (C.22) for an expression of βvparticularized for the equation of state (14)]. In this formulation, is the ratio of βvFto the ideal-gas volume expansivity in the fuel stream 1/TF, whereas is the ratio of molecular weights. The scalings in Eqs. (162)–(165) anticipate that the effect of the temperature gradients on the species transport fluxes near the interface is small, (see Table 4 for typical values of α,ϵT,ZF, and ), whereas the dominant components of the standard and interfacial species fluxes are of the same order of magnitude Eq. (167) suggests a balance between the two portions of the species fluxes that are independent of the temperature gradients. This aspect is further investigated below by examining the species conservation equation and is ratified by the numerical examples that will be provided in Sections 6 and 7. Upon substituting the variables (32), (81)–(86), and (162)–(165) into Eq. (159), the nondimensional species conservation equation is obtained, with In this formulation, is the nondimensional fuel partial volume, is a temperature ratio, and is the fuel Lewis number evaluated in the coflow stream (see Table 4 for typical values of τand LeF). Additionally, denotes the partial derivative of the logarithm of the fugacity with respect to the logarithm of the fuel molar fraction divided by as suggested by Eq. (161). Following Table 4, the largest factor premultiplying the different terms in Eq. (168) is the one associated with the first term on the right-hand side, which is inversely proportional to ϵTand corresponds to the inverse of the transport-equilibrium parameter whereas and are much smaller. Correspondingly, the first approximation to Eq. (168) is The integration of Eq. (174) up to large distances compared with the interface thickness involves a constant that corresponds to the sum of fluxes . While the interfacial component of that constant is zero for all practical purposes, the standard one may not be zero away from the interface, since the coflow may mix with hot gas-like fuel transferred from the fuel side. This large-scale mixing of propellants away from the interface (167) (169) (170) (171) (172) (173) (174) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 37 is described by the species conservation Eq. (46) minus the interfacial transport term, whose solution near the interface provides the aforementioned integration constant . However, since Eq. (175) is driven by gradients of composition over distances comparable to the thermal mixing-layer thickness, the standard flux away from the interface is anticipated to be of order ϵTcompared to that near the interface and can therefore be neglected. As a result, the integration of Eq. (174) leads to the transport equilibrium condition or equivalently where use of Eqs. (155) and (156) has been made, and where dimensional variables have been recovered in the notation. The transport equilibrium condition (176) states that the Fickian and barodiffusion components of the standard species flux (153) balance the portion of the interfacial species flux (148) independent of the temperature gradient, as sketched in Fig. 20. The interface temperature Te, which is generally a function of time and the tangential coordinate participates in Eq. (177) as demanded by the leading-order asymptotic expansion of the fluxes (169) and (170) for ϵT≪1. The temperature field around the interface evolves in time scales of order which are much larger than the flow transit time across the interface δI/UT. As a result, Teis a quantity that varies mostly quasi-steadily at interfacial scales. The boundary conditions required in the integration of Eq. (177) can be summarized as follows. First, the local thermodynamic pressure on the coflow side of the interface (see Fig. 17), which can be approximated as the combustor pressure P∞at sufficiently small Mach numbers, is recovered by the solution away from the interface, Second, the fuel partial densities tend to their phase-equilibrium values away from the interface, Concurrent with (180) and the equation of state (14) is the recovery of the phase-equilibrium molar fraction on the coflow side of the interface, In Eqs. (178)–(181), the limit denotes distances much larger than δIbut much smaller than δT. Both phase-equilibrium values and are evaluated at P∞and Te. 5.6.2. The transport equilibrium condition in terms of a balance of heat fluxes Utilizing Eqs. (147) and (152), it can be shown that the transport equilibrium condition (177) is equivalent to a balance of heat fluxes across the interface, where is the sum of the Dufour effect and the interdiffusion of heat by and is the portion of the interfacial heat flux independent of temperature gradients Both and participate in the enthalpy conservation Eq. (51) when referred to the same curvilinear, moving coordinate system described above, where the viscous dissipation has been neglected by assuming small Mach numbers. In Eq. (185), the flux is the sum of Fourier conduction and interdiffusion of heat by the Soret effect, while is the portion of the interfacial heat flux that depends (175) (176) (177) (178) (179) (180) (181) (182) (183) (184) (185) (186) UNCORRECTED PROOF 38 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx on temperature gradients, Similarly to the scaling analysis made for the species conservation Eq. (159), it can be shown that the heat balance (182) is the result of integrating the first approximation to the enthalpy conservation Eq. (185), Expressions (183), (184), (186) and (187), along with the relation (161) and the dimensionless variables (32), (81), (82), (85), (86), and (160)–(165), motivate the characteristic scales for the heat fluxes. These expressions indicate that and are of the same order of magnitude, whereas and are negligible since In Eq. (190), is the ideal-gas adiabatic coefficient defined in Eq. (C.27), and is a ratio of real-to-ideal specific heats at constant pressure, both parameters being evaluated in the coflow stream (see Table 4 for typical values of and ). In the vicinity of the interface, it can be also shown that convection of enthalpy, viscous dissipation, pressure advection, curvature-related terms, and enthalpy production by interfacial forces in Eq. (185) are negligible compared to the normal derivative of the sum . 5.6.3. Combination of mechanical and transport equilibrium conditions in terms of chemical potentials As discussed in Appendix E.1, the pressures within the interface may become small enough to cross the mechanical spinodals when the combustor pressure decreases but it is still high relative to the atmospheric value. In that case, singularities in and (∂ln fF/∂ln XF)P,T arise that can be regularized by combining the Fickian and barodiffusion components into a single finite term given by the relation obtained by evaluating Eq. (F.22) for N= 2 and approximating the constant-temperature gradients by ordinary ones because of the locally uniform temperature prevailing in the vicinity of the interface in the limit ϵT≪1. When expression (191) is used in Eqs. (155) and (183), the scaling analysis performed above does not change in any fundamental manner and gives a transport equilibrium condition valid over the entire high-pressure range that can be written as Eq. (192) is subject to the boundary conditions (113)-(114) away from the interface. The transport equilibrium condition (192) can be combined with the mechanical equilibrium condition (112) and integrated once giving the system of equations Eq. (193) is subject to the boundary conditions (109)–(110), whereas Eq. (194) predicts that the chemical potential of the coflow species remains constant across the interface, and equal to its phase-equilibrium value, when only the fuel gradient-energy coefficient is considered in the analysis. 5.6.4. Generalizations to multicomponent systems The formulation can be easily generalized to multicomponent systems. First, consider the generalized version of the species flux balance (176), namely In multicomponent systems, the species fluxes of the i-th component, Eqs. (139) and (F.25), depend on the Onsager coefficients of N−1 components, thereby making Eq. (195) of little interest for large N. A more useful version of Eq. (195) independent of the Onsager coefficients is derived in this section. Upon substituting the species fluxes (139) and (F.25) into Eq. (195), the system of N−1 equations (187) (188) (191) (192) (193) (194) (195) (196) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 39 is obtained, with i= 1,⋯N−1. In this formulation, are components of an auxiliary vector defined as where use of Eqs. (56) and (140) has been made in order to replace the symbols ∇T,ψk, and ψNin Eq. (139) by their corresponding expressions under the approximation of locally uniform temperature. A similar substitution of the definition of heat fluxes (138) and (F.24) into Eq. (182) yields The concatenation of Eqs. (196) and (198) represents an over-de- termined system with Nequations for N−1 unknowns. The resulting matrix of coefficients is the Onsager matrix given in Eq. (143) excluding the first column, with because is positive semidefinite. In particular, is achieved when is positive definite, or equivalently, when the entropy production (142) is strictly positive. In that case, the only possible solution to the over-determined system (196) and (198) is the trivial one, for k= 1,⋯N−1. Eq. (199) represents the multicomponent version of the transport equilibrium condition (177) and is also equivalent to the simultaneous verification of the heat and species flux balances (182) and (195). Using Eq. (F.22), the multicomponent transport equilibrium condition (199) can be rewritten in terms of gradients of chemical potentials as for k= 1,⋯N−1. The combination of the multicomponent mechanical equilibrium condition (121) and the multicomponent transport equilibrium condition (200) gives for k= 1,⋯N, which can be integrated once subject to the boundary conditions (122) and (123) yielding for k= 1,⋯N. The system of Eq. (202) is subject to the boundary conditions corresponding to the phase-equilibrium compositions away from the interface evaluated at P∞and Te. Additional approximations made to arrive at Eq. (202) involve negligible curvatures, small Mach numbers, and negligible effects of viscous stresses at interface scales, as explained in Section 5.3.3; the consideration of these effects would require modification of Eq. (202) by using instead the general form of the mechanical equilibrium condition (118) in the derivations above. A relation exists between σand the energy excess, or Landau potential, at the interface in conditions of mechanical and transport equilibrium. Upon multiplying Eq. (202) by ρYk, summing the resulting expression from to N, and using Eqs. (120), (16), and (C.7) and (C.8), the equation is obtained. In Eq. (205), is the phase-equilibrium value of Ω, as prescribed by Eq. (C.7), and ΩGD is the gradient-dependent Landau potential The integration of Eq. (205) across the interface yields which provides a quantitative link between the interfacial excess energy and the surface-tension coefficient. 5.7. Remarks on transcritical interfaces in equilibrium and nearequilibrium conditions The dimensionless parameters in Eq. (95) and in Eq. (173) measure, respectively, the tendency of the interface to attain mechanical and transport equilibrium. The smaller and the more equilibrated the interface is. Both and are proportional to the thermal Cahn number ϵT, or dimensionally, to the temperature gradient across to the interface. In isothermal systems, the thermal Cahn number is exactly zero, ϵT= 0, and therefore . The interface is in mechanical and transport equilibrium, in that the variations of the gradient-de (198) (199) (200) (201) (202) (203) (204) (205) (206) (207) UNCORRECTED PROOF 40 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx pendent pressure are exactly balanced with the variations of the interfacial stress normal to the interface, and the sum of the standard and interfacial species fluxes is exactly zero. The structure of the interface can be described by the combination of the mechanical equilibrium condition (120) and the transport equilibrium condition (199), or equivalently, by the system of Eq. (202) subject to the phase-equi- librium partial densities (203) and (204) in the far field. The system of Eqs. (202) has been utilized in early work addressing interfaces in multicomponent equilibrium isothermal systems [17,37,100,109,118,121,131,132], and can be alternatively derived by minimizing the volume integral of the gradient-dependent Helmholtz free energy [116,117,120]. In addition, the system (202) equivalently states that the generalized chemical potential given by Eq. (58) remains constant through the interface and equal to the phase-equilib- rium chemical potential μke. Correspondingly, the sources of entropy related to gradients of temperature and generalized chemical potential in Eq. (142) are exactly zero in mechanical and transport equilibrium. For the binary mixtures studied here, the system of Eqs. (202) simplifies to Eqs. (193) and (194) subject to the boundary conditions (109) and (110). The solution to that problem is a steady planar transcritical interface bearing surface tension and separating a liquid-like supercritical mixture rich in fuel from a gas-like supercritical mixture rich in coflow species, with both mixtures being in phase equilibrium as discussed in Section 6. None of the different forms of the transport equilibrium condition [i.e., Eqs. (176), (177), (182), (192), (199), (200), and (202)] depend on transport coefficients. Similarly, although high-order viscous effects can be retained in the derivation of the mechanical equilibrium condition in Eq. (97), the leading-order balance is independent of viscosity, as suggested by Eqs. (107), (112), and (120). As a result, the internal structure of a transcritical interface in mechanical and transport equilibrium is independent of viscosity, thermal conductivity, binary diffusion coefficient, and thermal-diffusion ratio. In non-isothermal systems, the thermal Cahn number is larger than zero, although in practical situations it remains small compared to unity, ϵT≪1, because the temperature gradients created by the flow are never as large as the composition gradients across the interface. Even in the wake of the orifice ring separating both propellant streams in Fig. 1, where the temperature gradient is the largest, the characteristic length for the temperature variations is the thickness of the orifice ring, which is always much larger than the interface thickness. For 0<ϵT≪1, the mechanical (95) and transport (173) equilibrium parameters attain small but non-zero values, . As a result, the interface is neither in mechanical equilibrium nor in transport equilibrium. However, the departures from equilibrium are small, and the equilibrium conditions still describe the overall structure of the interface to a good approximation, as discussed in Section 7. The interface temperature Teand the far-field boundary conditions away from the interface may become functions of space and time because of the large-scale evolution of the flow, but the characteristic length and time scales of these variations, and are much larger, respectively, than the interface thickness δIand the flow transit time across the interface tI(see Table 4 for typical values of δIand tI). As a result, the structure of the interface evolves quasi-steadily in response to those flow variations, as sketched in Fig. 20. The complete description of the evolution of the interface and the flow surrounding it requires the Navier-Stokes equations outlined in Section 4.7 supplemented with the closures for interfacial terms derived above. An intermediate case may exist in non-isothermal systems in which the interface is much closer to mechanical equilibrium than to transport equilibrium, [e.g., in the conditions addressed in Table 4]. In this case, the species conservation Eq. (46) can be integrated simultaneously with the mechanical equilibrium condition (120). The latter is subject to boundary conditions corresponding to partial densities generally away from phase equilibrium, and may involve pure components in the propellant streams. This intermediate case is addressed in Section 7. The solution provides the spatiotemporal evolution of the transcritical interface and the associated surface-tension coefficient downstream of the injection orifice, as previewed in Figs. 2 and 3. 6. Transcritical interfaces in isothermal bicomponent systems This section provides numerical results describing steady transcritical interfaces in C12H26/N2isothermal systems. The configuration analyzed here corresponds to the limit of infinitely thick thermal mixing layers, or equivalently, to the case of zero thermal Cahn numbers, ϵT= 0, in which the interface is in mechanical and transport equilibrium. 6.1. Formulation The equations integrated here are (193) and (194), which are supplemented with the equation of state (14) and mixing rules (15) particularized for binary mixtures [see Eq. (212) introduced below], along with the expressions for the gradient-energy coefficient (23), chemical potential (B.24), ideal-gas Gibbs free energy (C.46), fugacity coefficient (C.47), and the constants provided in Table C.1 and Appendix C. The boundary conditions away from the interface are the phase-equi- librium partial densities (109) and (110). The problem has a steady solution consisting of a thin interface that bears surface tension and separates a liquid-like C12H26-rich supercritical mixture from a gas-like nitrogen-rich supercritical mixture. A summary of the formulation is provided in Table 5. Details associated with the numerical integration of these equations are discussed in Appendix G. The calculations focus on the four cases summarized in Table 6. In all cases, the thermodynamic pressure away from the interface bar is larger than the critical pressures of the individual components, while the temperature K is smaller than both the fuel critical temperature K and the temperature of the diffusional critical point Tc,diff at the corresponding pressure (i.e., see Table 3). As a result, the pseudo-trajectories along the interface in thermodynamic space traverse the coexistence region, as shown in Fig. 21. Furthermore, while case D only involves crossings of the diffusional spinodal, cases A, B, and C involve crossings of both diffusional and mechanical spinodals. It is therefore convenient to use the combined mechanical and transport equilibrium conditions written in terms of gradients of chemical potentials (193) and (194) Table C.1 Coefficients for the evaluation of the ideal-gas specific heats, entropies, enthalpies and Gibbs free energies for selected species [151]. Species r1,i[-] r2,i[K− 1]r3,i[K− 2]r4,i[K− 3]r5,i[K− 4]r6,i[K] r7,i[-] C12H26 2.133×101 N23.531 2.967 UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 41 Fig. 21. Thermodynamic-space pseudo-trajectories across transcritical interfaces in C12H26/N2isothermal systems for cases A-D described in Table 6. The figure includes isocomposition cross-sections of the phase-equilibrium surface colored by fuel mass fraction. Table 6 Thermodynamic conditions and phase-equilibrium composition corresponding to the pseudo-trajectories A-D in Fig. 21 for transcritical interfaces in C12H26/N2isothermal systems. Phase-equilibrium composition (liquid-like supercritical) (gas-like supercritical) case Te [K] P∞ [bar] A 350 50 642.1 0.983 48.1 0.002 B 350 100 642.5 0.966 95.3 0.003 C 500 50 541.9 0.979 40.2 0.202 D 500 100 539.8 0.957 74.9 0.151 , since, as explained in Appendix E.1, these forms are unaffected by singularities arising in the fuel partial molar volume and non-ideal diffusion prefactor near the mechanical spinodals. It is also in cases A, B, and C where the thermodynamic pressure reaches negative values within the interface. This effect is inconsequential because the formulation in Table 5 is independent of the transport coefficients. 6.2. Results: the steady transcritical interface The profiles of density, fuel mass fraction, and thermodynamic pressure across the interface are shown in Fig. 22(a–c). Density ratios of order (for cases A and C at P∞= 50 bar) and (for cases B and D at P∞= 100 bar) are observed in the solution that are spatially localized and accompanied by order-unity variations of the fuel mass fraction in accordance with the phase-equilibrium composition on each side of the interface. In particular, on the fuel side of the interface, the mixtures are rich in C12H26 and have compressibility factors within the range Z∼0.30–0.75, which clearly depart from the ideal behavior corresponding to unity. The oscillation of the pressure across the coexistence region is localized near the fuel side of the interface, where an underpressure of order unity relative to P∞occurs. It will be shown that this large underpressure leads to significant values of surface tension in accord with Eq. (108). In contrast, on the other side of the interface, the mixture is rich in N2and much less dense. The corresponding compressibility factors are within the range Z∼1.01–1.03, thereby indicating that the fluid approaches there the behavior of an ideal gas despite the high pressures, the reason being that N2is highly supercritical in temperature. A point-wise cancellation between the standard and interfacial fluxes of species is observed within the interface in Fig. 22(d). Positive values of the standard diffusion flux JF(i.e., in the direction of decreasing fuel concentration) are observed on both flanks of the interface, whereas negative values (i.e., in the direction of increasing fuel concentration) occur inside. A detailed breakdown of the species fluxes is provided in Fig. 23(a) for case D. In the diffusionally unstable region of the interface, the Fickian component of the species flux antidiffuses fuel in the direction of increasing concentration, as anticipated in Section 4.6. In contrast, the barodiffusion component acts in the opposite direction by transporting fuel along the positive pressure gradient, albeit with less intensity than the Fickian component. As a result, the standard species flux JFis dominated by Fickian antidiffusion within the interface, with zero-flux points being coincident with the diffusional-spinodal states given by the maximum and minimum locations of the fuel chemical potential, as shown in Fig. 23(b). No steady solution of the problem exists for κF,F = 0 (i.e., ). Specifically, κF,F = 0 would impede the balance shown in Fig. 23(a) between the interfacial species flux and the standard one JF. The interfacial species flux provides the necessary amount of positive transport of fuel, in the direction of decreasing fuel concentration, to yield a finite-thickness interface that persists indefinitely in time. The interface thickness can be computed as Despite the high pressures considered here, Fig. 24(a) indicates that δIcalculated using Eq. (208) remains small compared to hydrodynamic scales of interest in practical systems (i.e., δI=1.59–2.98 nm). Specifically, the values of δIobserved here are comparable to those arising in subcritical monocomponent systems of separate N2or C12H26 close to their critical points (see Appendix D). How (208) UNCORRECTED PROOF 48 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. E.1. Constant-pressure distributions of (a) isothermal compressibility, (b) volume expansivity, (c) fuel partial molar volume, (d) non-ideal diffusion prefactor, and partial specific enthalpies of (e) C12H26 and (f) N2as a function of the C12H26 molar fraction at mechanically supercritical pressure (P= 50 bar, solid lines) and mechanically subcritical pressure ( bar, dashed lines). All panels correspond to C12H26/N2mixtures at uniform temperature T= 500 K. Fig. E.4. Surface-tension coefficient in the C12H26/N2non-isothermal system analyzed in Section 7 as a function of downstream distance (or time), including the baseline case with regularization constant m= 0.01 (solid line), along with two supplementary calculations with m= 0.1 (dotted line) and (dot-dashed line), all other parameters being the same. coming more than 100 times thicker than its initial value and causing a noticeable increase in both ϵRand ϵT. The bathtub-like shape of the evolution of the thermal Cahn number ϵTin Fig. 29(a) indicates that the interface is closest to transport equilibrium in the intermediate region away from the orifice (where the temperature gradients are the largest) and away from the interface edge (where the composition gradients are the smallest). The near-equilibrium behavior in that intermediate region is manifested in Figs. 30(a,b) as almost-complete cancellations between the components of the standard ( ) and interfacial ( ) heat fluxes indepen UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 49 Fig. 30. Spatial distributions of heat and species fluxes across the interface at t/tTR =x/LTR = 0.4, where its thickness computed using Eq. (223) is nm. (a) Temperature-gra- dient-independent components of the standard heat flux [dark solid line, Eq. (183)] and interfacial heat flux [green dashed line, Eq. (184)]. (b) Temperature-gradient-indepen- dent components of the standard species flux [dark solid line, Eq. (155)] and interfacial species flux [green dashed line, Eq. (156)]. (c) Temperature-gradient-dependent components of the standard heat flux [dark solid line, Eq. (186)] and interfacial heat flux [green dashed line, Eq. (187)], along with the net heat flux. (d) Temperature-gra- dient-dependent components of the standard heat flux [dark solid line, Eq. (157)] and interfacial species flux [green dashed line, Eq. (158)], along with the net species flux. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) dent of the temperature gradient [see Eqs. (183) and (184)], and between the corresponding components of the standard ( ) and interfacial ( ) species fluxes [Eqs. (155) and (156)]. In contrast, Fig. 30(c) shows that the standard ( ) and interfacial ( ) components of the heat fluxes that depend on the temperature gradient [Eqs. (186) and (187)] do not cancel each other, but are much smaller than their counterparts and as anticipated in Eq. (190). A similar result is observed in Fig. 30(d) for the species fluxes and [Eqs. (157) and (158)], which do not cancel each other and are much smaller than and as anticipated in Eq. (166). These results ratify the analysis in Section 5.6, in that most of the interfacial species flux is invested in counteracting the antidiffusion of fuel induced by the sum of Fickian and barodiffusion mechanisms, whereas most of the interfacial heat flux is invested in counteracting the Dufour effect. The small remainders of those cancellations, along with the components of the standard fluxes that are dependent on temperature gradients (Soret, Fourier, and interdiffusion), describe the transport of mass and energy across the interface in non-equilib- rium conditions. The near-equilibrium behavior persists until the interface vanishes downstream at an edge. Because of the much smaller thermal diffusivity of the fuel, the cold fuel stream cools down the hot coflow more than what the hot coflow is capable of heating the fuel stream, as observed in the large-scale view of the temperature field in Fig. 31. As a result, all isotherms, including the supercriticalization one TTR ≃Tc,diff, are initially displaced outwards away from the axis into the coflow. This phenomenon is exacerbated by a mixing-induced augmentation of the constant-pressure specific heat cp, whose global maximum is attained within the interface for a significant portion of its length or lifetime, as shown in Fig. 33. The line joining the global maxima of cpis the physical-space representation of the pseudo-boiling line discussed in Section 2.2 and Appendix B.2. Similarly to the abrupt termination of the pseudo-boiling line observed before intercepting the diffusional critical point in the C12H26/N2phase diagram in Fig. 10(c), the global maximum of cpin physical space in Fig. 33(a) ceases to be within the interface upstream of the interface edge, and switches abruptly thereafter to the jet axis. As the fuel is heated by the hot coflow, the supercriticalization isotherm eventually turns back toward the jet axis and crosses the transcritical interface. It is at that crossing point where the interface vanishes at an edge. The latter is located at a distance of order unity from the orifice when normalized with the characteristic supercriticalization length (13) [or after a time of order unity has passed when normalized with the characteristic supercriticalization time (222)], as shown in Figs. 32 and 33. UNCORRECTED PROOF 50 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. 33. (a) Solid contours of density (upper half) and constant-pressure specific heat (lower half), along with (b) corresponding profiles extracted at t/tTR =x/LTR = 0.4, 1.4, and 2.4. In panel (a), the thick green dashed line indicates the loci of points where the constant-pressure specific heat attains a global maximum; this line is not plotted for when the global maximum switches from the interface to the jet axis. Additionally, panel (a) includes isotherms corresponding to the critical mixing temperature (blue dashed line), diffusional critical temperature (white dashed line), along with isocomposition lines corresponding to YF= 0.82 (blue dotted line) and YF= 0.87 (green dotted line).. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) A zoomed view of the structure of the interface edge is provided in Fig. 2. For C12H26/N2systems, the diffusional critical point at the combustor pressure P∞= 100 bar plays a central role at the interface edge because of the following reasons: (a) Whereas the thermodynamic pressure undergoes order-unity oscillations across the interface in the near-equilibrium region, the flow in the vicinity of the interface edge is characterized by being mostly isobaric at P≃P∞. (b) The isocontour corresponding to the fuel mass fraction of the diffusional critical point at 100 bar (see Table 3), must necessarily emanate from within the transcritical interface since it is engendered in the coexistence region of the thermodynamic phase diagram, as shown in Fig. 10(c). (c) The temperature of the diffusional critical point Tc,diff is the maximum temperature of the diffusionally unstable region. As discussed in Sections 3.2 and 6.2, different mixtures may exist that may lead to a maximum value of the diffusional spinodal sur face that may be larger than the diffusional critical temperature. In those mixtures, the supercriticalization isotherm would correspond to the maximum temperature of the diffusional spinodal surface, and the diffusional critical point would not be located at the interface edge, but at some distance upstream within the transcritical interface. Downstream of the interface edge, the temperature is everywhere larger than Tc,diff, and consequently the system becomes diffusionally stable. The interfacial fluxes of heat and species become negligible, and the standard diffusion flux of fuel reverses to the usual forward direction along decreasing fuel concentrations, as shown in Fig. 34. The transcritical interface morphs into a fully supercritical mixing layer that grows with distance downstream and is characterized by unimpeded mixing of the propellants by molecular diffusion, as observed in Fig. 2. In this zone, both propellants resemble supercritical gas-like fluids. The evolution of the pseudo-trajectories of the system with distance (or time) are overlaid in Fig. 35 on a T−YFthermodynamic phase diagram evaluated at P=P∞. As the edge conditions are ap UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 51 Fig. 32. Zoomed version of Fig. 31 showing (a) solid contours of temperature (upper half) and fuel mass fraction (lower half), along with (b) corresponding profiles extracted at t/tTR =x/LTR = 0.4, 1.4, and 2.4. Panel (a) includes isotherms corresponding to the critical mixing temperature (blue dashed line), diffusional critical temperature (white dashed line), along with isocomposition lines corresponding to YF= 0.82 (blue dotted line) and YF= 0.87 (green dotted line).. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) proached, Fig. 35 shows that the thermodynamic states of the mixture across the interface tend to intersect the coexistence region at increasingly higher temperatures until the diffusional critical temperature is reached, above which intersections cannot occur. As indicated by the solid line in Fig. 36, the surface-tension coefficient σ, obtained from Eq. (99), decays monotonically with downstream distance (or time) and vanishes simultaneously with the interface at the edge. The first stage of the decay is from to 0.3, and is characterized by a rapid increase of the interface temperature near the orifice, as shown in Fig. 29(b). The decay of σslows down from to 1.5, where near-equilibrium transport conditions are attained. Beyond the decay rate increases on approach to the diffusional critical point, with σplunging at the interface edge to dynamically irrelevant values. An accurate prediction of σ, given by the square symbols in Fig. 36, can be obtained by solving an isothermal interface at every time step using the formulation in Table 6 and the set-up described Section 6, where the local interface temperature Teis obtained from Fig. 29(b). This observation is consistent with the attainment of near-equilibrium conditions for most of the transcritical region. Most importantly, it also suggests a potential route for subgrid-scale modeling of transcritical flows based on pre-tabulation of equilibrium isothermal cases, with Teacting as a table input obtained from a coarse-grained calculation of the temperature field. 8. Concluding remarks Despite the remarkable progress made on augmenting the thrust, range, and reliability of chemical propulsion technologies over the last several decades, which has often relied on large investments in experimentation and testing, the fundamental fluid mechanical processes participating in the injection, atomization, vaporization, mixing, and combustion of propellants at high pressures remain largely unknown. Specifically, the extreme pressure conditions involved in the transcritical flow of propellants into combustors UNCORRECTED PROOF 52 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. 31. Large field of view of (a) solid contours of temperature (upper half) and fuel mass fraction (lower half), along with (b) corresponding profiles extracted at t/tTR =x/LTR = 0.4, 1.4, and 2.4. The figure includes isotherms corresponding to the fuel critical temperature (dark dashed line), critical mixing temperature (blue dashed line), and diffusional critical temperature (white dashed line).. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) make this problem quite formidable compared to many other open questions in the technical discipline of general multiphase flows. The present study has focused on a selected number of basic aspects of the problem of transcriticality. Particular emphasis has been made on describing thermodynamically complex systems involving two components at high pressures. When the pressure is supercritical with respect to both components, and when at least one of the propellants is injected at subcritical temperature, transcritical conditions develop that appear to lead to atomization and spray phenomena according to existing experimental visualizations. The process of interface formation at high pressures cannot be explained using the thermodynamics of monocomponent systems, since only one thermodynamic state is possible in those when the temperature is fixed at supercritical pressures. In contrast, in bicomponent systems, at least two states may be possible at a given temperature for pressures above the critical pressures of the separate components. Using previously well-established high-pressure equations of state and mixing rules, it is shown that the coexistence region extends up to more than 100-fold higher pressures than the critical pressures of the separate components in systems of heavy-hydrocarbon fuels mixed with N2, O2, or air. At high pressures, the thermodynamic structure of the coexistence region is dominated by diffusional instabilities. Specifically, the components tend to separate in the coexistence region because of an antidiffusion process that proves energetically favorable. In ordinary practice, systems like water brought into the coexistence region of their thermodynamic phase diagram at standard pressure tend to separate into different phases, namely liquid and vapor, across an interface. However, at high pressures, the phases are no longer vapor and liquid, but two supercritical fluids of distinct character which, in the transcritical conditions addressed in this study, behave at injection as a liquid-like supercritical fluid (for the heavy hydrocarbon fuel) and a gas-like supercritical fluid (for N2, O2, or air). The problem of translating by theoretical means the aforementioned diffusional instability into an observable process in physical space is nothing short of laborious. Such translation requires a tight connection between the thermodynamic space and the transport of the propellants in physical space, and must therefore involve finite- UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 53 Fig. 34. Spatial distributions of (a) heat and (b) species fluxes at shortly downstream of the interface edge. Fig. 35. Thermodynamic-space pseudo-trajectories at 0.4, 1.4, and 2.4, projected on a constant-pressure ( bar) cross-section of the thermodynamic-phase diagram. The figure includes the diffusional critical point (purple diamond symbol), the VLE line (thick red dashed line), the diffusional spinodals (thin purple dashed lines), and the loci of the maximum of the molar heat at constant pressure (thick green dot-dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) rate processes of molecular diffusion that cannot be described with the standard transport theory. This work has investigated one possible way of addressing this question by coupling an extension of the diffuse-interface theory of van der Waals flows with the Navier-Stokes conservation equations. The core of this theory is the folding of the coexistence region in thermodynamic space into a thin interface in physical space. This can only be made possible by redefining the thermodynamic potentials in the presence of strong composition gradients, and by revisiting the constitutive laws participating in the Navier-Stokes conservation equations. When these modifications are coupled with a high-pressure equation of state that reproduces a single Maxwell loop within the coexistence region, the formulation can describe separation or mixing of phases or fluids in accordance with their thermodynamic phase diagram. The ensuing interface has a non-zero thickness set by the nearly mutual cancellation between an antidiffusion flux of matter provided by the standard transport theory, and an opposing interfacial flux arising as a regularization introduced by the diffuse-interface theory. The formulation is not exempt of approximations, among which the most notable one is related to the continuum hypothesis across the interface. This is a comfortable approximation only at high pressures because the interface broadens relative to intercollision distances. The theory of transcritical flows outlined in this work incorporates high-pressure models for thermophysical and transport properties. Those are subject to large uncertainties, and their determination through analysis or experiments is a broad and active area of research. Furthermore, the formulation has introduced additional parameters in the form of gradient-energy coefficients that are directly related to the thickness and surface tension of the interface. Most of the existing models for these coefficients rely on experimental calibration. Recent improvements in experimental techniques for diagnostics of high-pressure flow fields, along with new developments in molecular-dynamics simulations for the computation of thermophysical properties, transport coefficients, and possibly gradient-energy coefficients, may assist in future work to reduce the uncertainties. The utilization of this theory has discovered quantitative information about the dynamics of transcritical interfaces in coflowing configurations consisting of two propellant streams injected at different temperatures. The interface separating both propellant streams, along with the surface tension it engenders, survive from the injection orifice until a supercriticalization zone downstream, where the interface vanishes at an edge upon reaching the temperature of the diffusional critical point. The latter plays a prominent role at the interface edge in the C12H26/N2systems tested here. Specifically, the interface edge is characterized by a fundamental transition in the transport characteristics of the mixture that prevents separation of the components thereafter. A fully supercritical mixing layer ensues from the interface edge, across which the propellants mix by molecular diffusion as if they were two gas-like fluids. A central motivation of this study has been to highlight that a number of extensions of this theory could be made in order to relax the simplifying assumptions used in the examples above. These extensions could include the effects of turbulence, which is expected to play an important role in the transport of heat in practical systems at high pressures, and consequently in the determination of the supercriticalization length. However, even in the simple cases that have been addressed in this work, the description of the coupling between the interface structure and the hydrodynamic field is computationally expensive because of the resulting large disparity of scales. The reader should therefore be forewarned that the formulation that has been presented here involves interfacial terms in the conservation equations whose spatiotemporal resolution requirements are most likely untenable in CFD simulations of full engineering systems with current standard hardware and numerical methods. Worthy extensions of this work that may decrease the computational cost could involve investigations of filtered versions of the Navier-Stokes conservation equations augmented with the interfacial terms derived here, where the filter width would be much larger than the interface thickness but much smaller than the hydrodynamic scales. This would necessarily lead to closure problems in subgrid-scale interfacial terms, including the surface-tension force. The re UNCORRECTED PROOF 54 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. 36. Surface-tension coefficient in a C12H26/N2non-isothermal system as a function of downstream distance (or time). The figure includes the solution obtained by a numerical integration of the formulation in Table 8 for non-isothermal bicomponent systems (solid line), along with solutions in mechanical and transport equilibrium (square symbols) obtained by a numerical integration of the formulation in Table 6 for isothermal systems assuming constant temperature equal to the local (or instantaneous) interface temperature Teread from Fig. 29(b). Fig. 29. Evolution with downstream distance (or time) of (a) large-scale (ϵR) and thermal (ϵT) Cahn numbers, and (b) interface temperature Te. In this figure, the interface thickness δIand the thermal mixing-layer thickness δTare computed, respectively, using Eqs. (223) and (224). The interface temperature Teis computed as the temperature where the absolute value of the fuel partial-density gradient attains its maximum value. sults presented here suggest that the interface is in mechanical and transport near-equilibrium for a significant portion of its length, except near the orifice and near the interface edge, where non-equi- librium considerations may become important. It could therefore be of some interest to investigate the closure of subgrid-scale interfacial terms by reading them on-the-fly from pre-tabulated solutions of isothermal transcritical interfaces at a local interface temperature provided by the resolved flow field. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was funded by the US Department of Energy through the National Nuclear Security Administration (NNSA) Grant #DE-NA0002373 as part of the PSAAP-II Center at Stanford University. The authors are grateful to Dr. Mario Di Renzo for useful suggestions on the numerical integrations. Appendix A. Basic aspects of thermodynamic stability and phase equilibrium for transcritical systems The coupling between the diffuse-interface theory and the Navier-Stokes conservation equations depends fundamentally on the diffusional stability of the mixture. In unstable conditions, a transcritical interface is generated that separates mixtures near phase equilibrium. This appendix reviews basic concepts related to thermodynamic stability and phase equilibrium that are ubiquitously employed throughout the main text. A1. Thermodynamic-stability considerations The Peng-Robinson equation of state (14), and all others available, constrain all possible thermodynamic states of the system on a hypersurface . However, not all thermodynamic states on that hypersurface are observable. Specifically, there exist regions on that hypersurface where the system is thermodynamically unstable. There, a homogeneous mixture of the components is energetically less favorable than a configuration where different phases or different fluids are separated by interfaces. The criteria that determine whether the system is thermodynamically stable are independent of the equation of state. However, the single Maxwell loop predicted by cubic equations of state such as (14) naturally accommodates the stability landscape. UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 55 Table 8 Dimensional formulation for transcritical interfaces in non-isothermal bicomponent systems. (⋆) Using Eq. (221), ∂/∂xhas been replaced by (1/U)∂/∂t. Conservation equations⋆ Continuity Mechanical equilibrium Fuel species Total energy . Transport fluxes Standard (fuel species) Interfacial (fuel species) Standard (heat) Interfacial (heat) . Boundary conditions Symmetry at the axis Coflow free stream . Initial conditions (t= 0) Density Fuel mass fraction Temperature Velocity . Supplementary expressions (see Appendix B, Appendix E, Appendix C, and Appendix F for details) Equation of state Eq. (14). Coefficients of the equation of state Eqs. (212), and (C.10)-(C.12). Gradient-energy coefficient Eqs. (23), (C.17), and (C.18). Transport coefficients Eqs. (F.13), (F.36), and (F.43). Regularizations†Eqs. (E.8) and (E.9). Thermodynamic relations Eqs. (17), (48), (B.9), (B.12), (B.20), (B.21), (C.22), (C.23), (C.33), (C.34), (C.43), (C.46), and (C.47). In closed multicomponent systems subject to differential disturbances from equilibrium, thermodynamic stability occurs when three different conditions corresponding to thermal, mechanical, and diffusional stability are simultaneously satisfied. Motivated by Fig. 1, the discussion here will be precluded to bicomponent systems (N= 2), with brief references being made to monocomponent systems ( ) for conceptual comparisons. The explanations are supplemented by the thermodynamic phase diagrams in Figs. 4 and A.1. The reader is referred to Refs. [103–105] for generalizations and full derivations of these conditions. Thermal stability is observed when the constant-volume molar heat of the mixture is positive, where XFis the molar fraction of fuel. In particular, Eq. (A.1) ensures that the molar internal energy increases with increasing temperatures at constant volume and composition. The condition (A.1) is satisfied by specific heats calculated using the Peng-Robinson equation of state (14) in the conditions of interest for the present study. Formulas for the calculation of are outlined in Appendix C.7, and a closer analysis of the resulting variations of specific heat with temperature and composition is provided in Appendix B.2 for C12H26/N2mixtures. Mechanical stability is attained when the isothermal compressibility of the mixture βTis positive, which guarantees that the molar volume decreases as the pressure increases at constant temperature and composition. An expression for βT particularized for the Peng-Robinson equation of state (14) is provided in Eq. (C.44). It can be shown that a system that is mechanically stable is also thermally stable, but a system thermally stable is not necessarily mechanically stable [104]. Most importantly, the limit of mechanical stability is delineated by the mechanical spinodal surface, whose equation is given by or equivalently βT→∞. In monocomponent systems, the mechanical spinodals are easily visualized as lines intersecting all minima and maxima of the Maxwell loop in a P−vdiagram, as shown in Fig. 4. Within the region of the P−vdiagram enclosed by the mechanical spinodals, the isothermal compressibility is negative and the system becomes mechanically unstable. In this unstable region, the system tends to separate into two phases by means of a thin interface. A mechanical critical line exists on the mechanical spinodal surface where the pressure reaches an inflection point along the vaxis at constant temperature and composition. The mechanical critical line is therefore defined as the locus of points in the thermodynamic space {P, T, XF} where the condition (A.3) is satisfied simultaneously with A mechanical critical point is therefore defined as the intersection of the mechanical critical line with constant-pressure, constant-tem- perature, or constant-composition planes. In monocomponent systems, the mechanical critical line (A.3) and (A.4) collapses on a single critical point in the thermodynamic phase diagram, as observed in Fig. 4. This critical point determines the critical pressure above which the two-phase region ceases to exist, or equivalently, the pressure above which the monocomponent system is unconditionally stable. In contrast, mechanical stability is not a suffi (A.1) (A.2) (A.3) (A.4) UNCORRECTED PROOF 56 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx cient condition for thermodynamic stability in multicomponent systems, since diffusional processes can additionally drive separation of components at pressures above the mechanical critical point. Diffusional stability requires that the molar chemical potential of one of the constituents, for instance that of the fuel increases upon increasing the concentration of the same species, namely The partial derivative of the fuel chemical potential with respect to XFcan be rewritten in terms of the fuel fugacity as where φFis the corresponding fugacity coefficient defined in Eq. (C.47). The formal justification for Eq. (A.6) can be found in Appendix B.3. In particular, Eq. (A.6) indicates that the diffusional stability criterion (A.5) can be alternatively stated as It can be shown that a diffusionally stable system is also both thermally and mechanically stable, but a system both thermally and mechanically stable is not necessarily diffusionally stable [104]. The limit of diffusional stability is attained at the diffusional spinodal surfaces, where is satisfied, or alternatively as prescribed by Eq. (A.6). The transcritical C12H26/N2systems analyzed in this study are diffusionally unstable but thermally and mechanically stable at pressures larger than approximately 200 bar, whereas they become both mechanically and diffusionally unstable at pressures lower than those (see Appendix E). It is the breach of the diffusional stability constraint (A.5) what induces transcritical interfaces in the flow field, as discussed in Sections 4.6 and 5.4–5.6. Briefly, the diffusional spinodals represent thermodynamic states where the Fickian diffusion coefficient becomes zero. In particular, the quantity (∂ln fF/∂ln XF)P,T fundamentally participates as a non-ideal prefactor in the Fickian diffusion coefficient at high pressures defined in Eq. (43), in such a way that changes in the sign of (∂ln fF/∂ln XF)P,T change the direction of the diffusion flux. Outside the region delimited by the diffusional spinodals [i.e., (∂ln fF/∂ln XF)P,T >0], the mixture is stable and the components tend to diffuse into flow regions in directions aligned with decreasing gradients of their concentration. In contrast, within the region delimited by the diffusional spinodals [i.e., (∂ln fF/∂ln XF)P,T <0], the chemical potential decreases with increasing molar fraction. As a result, the mixture is unstable there and tends to separate by means of antidiffusion of matter. This phenomenon is well characterized in physical chemistry [136], and cannot be described by the standard transport theory. A relevant curve on the diffusional spinodal surfaces is the diffusional critical line of the mixture, which is defined as the locus of points where (A.8) is satisfied along with or equivalently, where the chemical potential reaches an inflection point along the XFaxis at constant pressure and temperature. Alternatively, based on Eq. (A.6), the diffusional critical line can be defined as the locus of points in the thermodynamic space {P, T, XF} where Eq. (A.9) is satisfied along with The diffusional and mechanical critical lines of the mixture are generally different. The intersection of the diffusional critical line with constant-pressure, constant-temperature, or constant-composition planes gives rise to a diffusional critical point. These considerations are summarized in the thermodynamic phase diagrams shown in Fig. A.1. The verification of the stability criteria (A.1), (A.2), and (A.5) [or (A.7)], does not prevent the system from becoming unstable to finite-amplitude disturbances. Whereas unconditional stability is achieved along the phase-equilibrium surface, or coexistence envelope, and everywhere outside the volume enclosed by it, interstitial regions exist in the thermodynamic phase diagrams in Fig. 4 and A.1(c) that are enclosed between the phase-equilibrium and spinodal surfaces. The system is metastable there, in that finite-amplitude disturbances can render it unstable [103,104]. A2. Phase equilibrium Consider two thermodynamic states and in the same bicomponent system addressed above. These states are in phase equilibrium across flat interfaces when the following three conditions corresponding to equal pressures equal temperatures and equal chemical potentials are simultaneously satisfied [103,104], with the subindex ebeing employed here to indicate phase-equilibrium values. At low pressures, ℓ (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) (A.14) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 57 and gdenote liquid and vapor phases, respectively. In the high-pres- sure systems studied here, and more particularly above the critical pressures of the separate components, ℓand gdenote liquid-like and gas-like supercritical fluids, respectively. Based on Eq. (A.6), the condition (A.14) is equivalent to the condition of equal fugacities, In monocomponent systems, the two conditions in (A.14) are replaced by the single constraint where is the molar Gibbs free energy. At the equilibrium values of pressure Peand temperature Te, the combination of Eqs. (A.12), (A.13) and (A.15), along with the equation of state (14) and the fugacity (B.22), provides the equilibrium molar volumes and as well as the equilibrium molar fractions and . The subspace of states satisfying the above phase-equilib- rium constraints forms a surface or coexistence envelope in the thermodynamic space {P, T, XF}. A constant-temperature cross-section of the coexistence envelope, indicated by the VLE lines, is provided in the phase diagrams presented in Fig. A.1. As discussed in Appendix A.1, the system is thermodynamically stable along the coexistence envelope and everywhere outside the volume enclosed by the coexistence envelope. The system is metastable in the concentric volume enclosed between the coexistence envelope and the diffusional spinodal surface, as indicated in Fig. A.1(c). Thermodynamic instability occurs within the volume enclosed by the diffusional spinodal surface, where no homogeneously mixed state consisting of phases ℓand gcan exist. Under these unstable conditions, the system tends to become separated into the two equilibrium states ℓand gby means of a thin interface. As explained in Section 4.4, the diffuse-interface theory maps the volume enclosed by the phase-equilibrium surface in thermodynamic space into a thin interface in physical space. In its simplest representation corresponding to isothermal systems, the theory predicts that phase-equilibrium conditions are exactly satisfied on both sides of the interface. In most practical cases, however, the propellant streams are injected in the combustor at different temperatures, as depicted in Fig. 1. In the presence of temperature gradients across the interface, the phase-equilibrium conditions (A.12)–(A.14) are only satisfied in an asymptotic sense up to an appropriate order of approximation provided in Sections 5.3 and 5.6. Appendix B. Transcritical behavior of thermophysical properties and molecular transport coefficients A complete description of the fluid motion requires specification of the thermophysical properties and transport coefficients of the mixture. At high pressures, increasing departures from the ideal-gas theory are observed, and consequently derivation of more complex expressions are necessary. B1. Molar enthalpy The specific gradient-dependent internal energy can be obtained from the total energy using Eq. (47). The molar enthalpy can be easily related to by rewriting Eq. (52) on a molar basis as where is given by Eq. (50). An expression for as a function of the state variables at high pressures is provided in this section. The utilization of molar rather than specific values is expedient since the resulting departure of the enthalpy from its ideal-gas counterpart involves integration of the equation of state (14), which is written in terms of the molar volume vand of the coefficients aand b on a molar basis. Specific and molar values of the enthalpy are simply related through the mean molecular weight as . Similarly, partial specific and partial molar values of the enthalpy are related through the molecular weight of the particular component, . Analogous relations apply to the entropy and all other thermodynamic potentials, including their partial values. The variation of enthalpy at high pressures can be calculated in the following way. Consider the exact differential of the enthalpy of the system H, with nibeing the number of moles of species iand as the partial molar enthalpy. The enthalpy of the system can be expressed in additive form as which can be differentiated and combined with Eq. (B.2) yielding Further simplifications of Eq. (B.3) can be made by rewriting the first principle of thermodynamics (C.1) on a molar basis, namely where the last term represents a chemical work done at constant volume due to a variation in the composition, with being the molar chemical potential of species i[defined formally further below in Eq. (B.24)]. In particular, the partial differentiation of Eq. (B.4) with respect to pressure yields Substituting (B.5) into (B.3) and using the definition of the constant-pressure molar heat and the Maxwell relation the expression (A.15) (A.16) (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) UNCORRECTED PROOF 64 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx the isentropic compressibility. C8. Partial molar enthalpies and partial-enthalpy departure functions The partial molar enthalpy of an ideal gas referred to the standard reference state can be expressed as the polynomial expansion where the coefficients r1,i,r2,i⋯r6,iare listed in Table C.1 for C12H26 and N2[151]. Note that Eq. (C.33) is equivalent to substituting the polynomial expansion of the specific heat (C.25) into the definition . The coefficient is a constant employed to refer the enthalpy to the formation value . The departure function of the partial molar enthalpy of a real gas described by the Peng-Robinson equation of state (14) is given by [153] In Eq. (C.34), the quantity can be obtained using Eq. (C.24), whereas the remaining partial derivatives are calculated as The partial-enthalpy departure function (C.34) is a sole function of Pwhen the values of the specific volume and molar fractions are fixed, since pressure and temperature are related through the equation of state (14). C9. Internal-energy departure function Using the definition the internal molar energy can be computed at high pressures after obtaining from Eq. (B.9). Similarly, its ideal-gas counterpart can be expressed as where is the ideal-gas enthalpy defined in Eq. (B.10). Correspondingly, the departure function for the molar internal energy is with being provided in Eq. (C.23) for the Peng-Robinson equation of state (14). C10. Partial molar internal energies The partial molar internal energy of an ideal gas referred to the standard reference state can be expressed as where is the partial molar enthalpy obtained from the polynomial expansion (C.33). By definition, the partial molar internal energy of a real gas is related to the partial molar enthalpy as with and . The combination of Eqs. (C.40) and (C.41) yields the departure function of the partial molar internal energy (C.32) (C.33) (C.34) (C.35) (C.36) (C.37) (C.38) (C.39) (C.40) (C.41) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 65 where is the departure function of the partial molar enthalpy provided in Eq. (C.34) for the Peng-Robinson equation of state (14). C11. Partial molar volume and isothermal compressibility The partial molar volume is given by [92] for the Peng-Robinson equation of state (14). In Eq. (C.43), βTis an isothermal compressibility that can be calculated as with BTbeing a dimensionless variable defined as C12. Ideal-gas Gibbs free energies The molar Gibbs free energy of an ideal gas at the reference pressure P0= 1 atm is [151] where the coefficients r1,i,r2,i⋯r7,iare listed in Table C.1 for C12H26 and N2. C13. Fugacity coefficients The logarithm of the fugacity coefficient φiis [103] for the Peng-Robinson equation of state (14). C14. Speed of sound The speed of sound cof a real gas is given by where βsis the isothermal compressibility defined in Eq. (C.32). Appendix D. High-pressure interfaces in subcritical monocomponent systems This appendix focuses on interfaces arising in monocomponent systems. In this case, interfaces can only emerge at pressures and temperatures below the critical point, where two thermodynamic states are possible at the same pressure. The resulting system is therefore subcritical everywhere. Nonetheless, only relatively high pressures not too far from the critical point can be considered because of the continuum assumption, as emphasized in Section 4.5. Further details on the utilization of diffuse-interface concepts in monocomponent systems can be found in a number of previous investigations [53,111,149,168,169] and in van der Waals’original contribution [108]. D1. Formulation Similar arguments to the ones made in Section 4.4 for conceptualizing the diffuse-interface formalism as a thermodynamic-to-physical space mapping of the oscillations of the chemical potential also apply to the pressure variations within the interface. To illustrate this, consider the simplified form of the mechanical equilibrium condition (120) for N= 1, subject to the phase-equilibrium densities Similarly to Eq. (31) for the chemical potential in systems in phase equilibrium, Eq. (D.1) maps the pressure oscillations from thermodynamic space into a thin interface in physical space. The oscillation of the pressure in thermodynamic space is induced by the equation of state (14) evaluated in conditions where two phases exist, as sketched in Fig. 18(c) for the simplest case corresponding to a flat interface in phase equilibrium. In that case, a Maxwell’s construction rule for P can be obtained by integrating Eq. (28) and using Eq. (C.7) subject to the phase-equilibrium condition (A.16), which gives (C.42) (C.43) (C.44) (C.45) (C.46) (C.47) (C.48) (D.1) (D.2) (D.3) UNCORRECTED PROOF 66 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx with and being the phase-equilibrium Helmholtz free energies. Eq. (D.4) indicates that the oscillation of the pressure encloses zero net area along the vaxis. In contrast, Eq. (108) states that the net area under the oscillation of the pressure in physical space is non-zero and is proportional to the surface tension. It can be shown that the mechanical equilibrium condition (D.1) is consistent with Maxwell’s construction rule (D.4) by substituting the former into the latter and using the identity as prescribed by the vanishing gradients of the density away from the interface. Analogous oscillatory profiles of Pare observed in bicomponent systems, as shown in Section 6. The origin of Eq. (31) anticipated in Section 4.4, which relates the chemical potential with the density gradients through the interface, is the mechanical equilibrium condition for monocomponent flows (D.1) evaluated at uniform temperature. In this particular case, Eqs. (31) and (D.1) are equivalent, as easily shown by evaluating Eq. (121) for N= 1, which gives Eq. (D.6) can be integrated once subject to the phase-equilibrium boundary condition at thus leading to Eq. (31). The equations integrated in this appendix are summarized in Table D.1 and consist of the mechanical equilibrium condition for monocomponent systems (D.1) supplemented with the equation of state (14) and the model (23) for the gradient-energy coefficient. The numerical results are obtained by solving the system of equations on uniform 1D meshes with approximately 50–75 grid points across the interface. The boundary conditions away from the interface correspond to phase-equilibrium densities (D.2) and (D.3). The prob Table D.1 Dimensional formulation for high-pressure subcritical interfaces in isothermal monocomponent systems. Conservation equation Mechanical equilibrium Boundary conditions Phase-equilibrium densities Supplementary expressions Equation of state Eq. (14), Coefficients of the equation of state Eqs. (C.10)–(C.12), Gradient-energy coefficient Eqs. (23), (C.17), and (C.18). lem has a solution consisting of a steady high-pressure subcritical interface that separates a liquid from its vapor in phase equilibrium. The isothermal cases summarized in Table D.2 for C12H26 and N2 are considered here. In all cases, the temperature Teand the thermodynamic pressure away from the interface P∞are smaller than their corresponding critical values Tcand Pc. As a result, the pseudo-trajec- tories along the interface in thermodynamic space are subcritical and traverse the coexistence region, as shown in Fig. D.1. D2. Results The profiles of density and pressure obtained from integrating the formulation in Table D.1 are provided in Fig. D.2. Density ratios of order 10 are observed in cases D and H, which correspond to the farthest conditions from the critical point. As the critical point is approached, the density and pressure profiles become increasingly flat. This limiting behavior is accompanied by (a) an increase of the interface thickness δI[defined in Eq. (208)] and a decrease in the Knudsen number [Eq. (40)], as shown in Fig. D.3, along with (b) a decrease in the surface-tension coefficient [Eq. (22)], as shown in Fig. D.4. The Knudsen number attains acceptable values for the continuum range only when conditions are close to the critical point. The interface disappears and the surface tension vanishes when the critical point is reached. Fig. D.4 shows comparisons between the surface-tension coefficient calculated by utilizing the numerical solution in Eq. (22) and the standard correlation [170] [see also Eq. (12-3.7) in Ref[54].], with ωbeing the acentric factor provided in Appendix Appendix C. Whereas good agreement is observed in the case of nitrogen over the entire range of tested conditions, the comparison reveals disagreements of order 20%-30% for dodecane. These discrepancies are caused by the relatively large accentric factor of dodecane, for which the standard correlation is not designed to operate accurately [54]. In contrast, the numerical solution agrees well with experimental data in Refs.[72–74]. Remarkably, the agreements between experiments and numerical results occur even at conditions far away from the critical point, where the continuum hypothesis across the interface becomes hardly justifiable. Appendix E. Transcritical bicomponent systems approaching mechanically unstable conditions Relevant aspects of the formulation outlined above lead to intrinsic singularities at conditions approaching the limit of mechanical stability (A.2). The analysis in this section builds on considerations made by Gaillard et al. [38,39], and provides thresholds in thermodynamic conditions leading to this behavior along with a palliating regularization. E1. Singularities of the theory at the limit of mechanical stability As discussed in Secs. 1, 2.2, and 3.2, transcritical conditions in systems fueled by heavy hydrocarbons involve combustor pressures P∞≳34 bar (for pure nitrogen coflows), P∞≳36 bar (for air coflows), and P∞≳50 bar (for pure oxygen coflows). These combustor pressures lead to fully supercritical pressures with respect to (D.4) (D.5) (D.6) (D.7) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 67 Fig. D.1. Thermodynamic pseudo-trajectories of cases A-H listed in Table D.2 for (a) dodecane and (b) nitrogen subcritical isothermal systems. The portion of the pseudo-trajectories involving negative pressures is not plotted in these diagrams. Table D.2 Thermodynamic conditions corresponding to cases A-H in Fig. D.1 for subcritical interfaces in monocomponent systems. case P∞[bar] Te[K] A 14.6 641.6 302.0 92.0 0.816 0.975 B 11.7 625.2 353.8 63.2 0.652 0.950 C 9.3 608.7 393.1 46.4 0.516 0.925 D 5.5 575.8 453.0 25.4 0.310 0.875 N2system case P∞[bar] Te[K] E 29.2 123.0 446.9 170.9 0.861 0.975 F 25.1 119.9 514.8 129.6 0.738 0.950 G 21.3 116.7 567.7 102.1 0.628 0.925 H 15.1 110.4 651.6 65.9 0.444 0.875 both fuel and coflow propellants (see Table 1). Within the practical range of propellant injection temperatures ( K), those combustor pressures are also larger than the maximum mechanical critical pressures of mixtures of heavy hydrocarbons with N2, O2, or air (15–25 bar), and smaller than the maximum diffusional critical pressures (500–1000 bar). However, it is shown in Sections 4.4 and 5.3.3 that the thermodynamic pressure undergoes order-unity oscillations within the interface. These oscillations are the physical-space counterparts of the Maxwell loops in the coexistence region of the thermodynamic phase diagram, and are predicted by the equation of state (14), or any other cubic equation of state available in the literature. Consequently, when P∞is not sufficiently high, the Maxwell loops may generate underpressures within the interface smaller than the mechanical critical pressure, including negative thermodynamic pressures. The Stefan-Maxwell forms of the standard diffusion fluxes of heat and species, given by Eqs. (152) and (153), are of limited practical use in conditions where the mechanical stability criterion (A.2) is not satisfied. This is because both the fuel partial molar volume and the non-ideal diffusion prefactor (∂ln fF/∂ln XF)P,T participating in the Fickian, barodiffusion, and Dufour components of those fluxes diverge at the mechanical spinodal surface defined in Eq. (A.3). To understand this, note that the partial molar volume defined in Eq. (B.19) can be equivalently expressed for the fuel species as where βTis the isothermal compressibility defined in Eq. (A.2) and particularized for the Peng-Robinson equation of state in Eq. (C.44) in Appendix Appendix C. The coflow partial volume satisfies the similar expression Whereas the partial derivatives of the pressure with respect to the number of moles of fuel or coflow species in Eqs. (E.1) and (E.2) are generally finite and different from zero on approach to the mechani (E.1) (E.2) UNCORRECTED PROOF 68 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. D.2. Profiles of (a) density and (b) pressure across the interface in isothermal systems composed of dodecane (left panels) and nitrogen (right panels). In dodecane systems, the interface thicknesses used for normalizing the spatial coordinate are nm, 3.59 nm, 2.91 nm, and 2.18 nm for cases A, B, C, and D, respectively. In nitrogen systems, the corresponding values are nm, 3.07 nm, 2.42 nm, and 1.77 nm for cases E, F, G, and H, respectively. Fig. D.4. Surface-tension coefficient as a function of normalized temperature in monocomponent isothermal systems consisting of nitrogen or dodecane, including numerical results from the present work (solid lines), the standard correlation (D.7) (dashed lines), and experimental measurements for nitrogen (triangles, Ref[72].) and dodecane (asterisks, Ref [73]; squares, Ref[74].). cal spinodals, the isothermal compressibility diverges there as demanded by Eq. (A.3) and illustrated in Fig. E.1(a). As a consequence, the partial molar volumes diverge, as shown in Fig. E.1(b), although the molar volume remains finite. It is also at the mechanical spinodals where the fuel molar chemical potential develops two turning points along the composition axis, as shown in Fig. A.1(b). As a result, its derivative becomes infinite there, and correspondingly the non-ideal diffusion prefactor (∂ln fF/∂ln XF)P,T also diverges there, as observed in Fig. E.1(c) and prescribed by Eq. (A.6). The singularity in the non-ideal diffusion prefactor can be traced back to the partial molar volume as a consequence of a non-integrable singularity of the latter in pressure, as indicated by differentiating Eq. (B.18) and utilizing the fundamental reciprocity relation of thermodynamics. The singularities in and (∂ln fF/∂ln XF)P,T arising at the mechanical spinodals cancel out each other upon summation in Eq. (F.22), in such a way that the constant-temperature gradient of the chemical-potential difference remains finite. Note that other singularities emerging when in Eq. (E.4) is evaluated in the pure components (i.e., and ) disappear upon multiplying Eq. (E.4) by the Onsager coefficients (145) and (146) to construct the standard diffusion fluxes of heat and species (149) and (150). The considerations above indicate that, in configurations where the temperature gradients are exactly zero, such as those addressed in Section 6, the standard species diffusion flux can be computed across the mechanical spinodals by direct spatial differentiation of the chemical potentials as in Eq. (150), instead of using the Stefan-Maxwell form (153). Similarly, the Dufour effect in the standard heat diffusion flux can also be safely computed across the mechanical spinodals using Eq. (149) instead of the corresponding Stefan-Maxwell form (152). Additional singularities need to be tackled in configurations such as that addressed in Section 7, in which there is a temperature gradient across the interface. In particular, singularities arising in the volume expansivity βvand in the partial specific enthalpies hFand hOat the mechanical spinodals play a role in transport mechanisms associated with temperature gradients. Note that βvparticipates in the portion of the interfacial species flux (148) that depends on the temperature gradient. It is also required for the calculation of the terms in the interfacial heat flux (147) that depend on the temperature gradient and on the heat transport by the net species flux the latter being small but non-zero in practice because of the small but non-zero thermal Cahn numbers resulting from moderate temperature gradients. The difference of partial specific enthalpies hF−hOparticipates in the interdiffusion heat flux in Eq. (149), in the last term of the interfacial heat flux (147), and implicitly in the calculation of the constant-tem- perature gradient of chemical potentials in Eq. (E.4) through the difference of partial entropies in Eq. (151). Since βTdiverges at the mechanical spinodals by definition, and βT is related to βvby the identity then βvalso diverges there, as shown in Fig. E.1(d). In Eq. (E.5), the partial derivative of Pwith respect to temperature generally remains finite and different from zero. (E.3) (E.4) (E.5) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 69 The singularities of the partial enthalpies at the mechanical spinodals, shown in Fig. E.1(e and f), can be understood by considering the identities which can be combined with Eqs. (E.1) and (E.2) for the partial molar volumes, thereby suggesting that the singular behavior of and is closely related to that of βT, since the remaining partial derivatives generally remain finite and different from zero. Despite the singularities arising in the partial molar enthalpies, the molar enthalpy of the mixture remains finite across the mechanical spinodals. The need to regularize βv, hF, and hOoccurs at conditions where temperature gradients exist across the interface and the combustor pressure P∞is not sufficiently high. The threshold for P∞depends on the interface temperature Te, since both Teand the local pressure Pdetermine the values of the fuel molar fraction on the mechanical spinodal surface. A quantification of the range of operating conditions {P∞,Te} within which the mechanical spinodal surface is crossed in C12H26/N2mixtures is provided in Fig. E.2 as the region hatched with red solid lines. Specifically, the results shown in Fig. E.2 pertain to 1D interfaces arising in a phase-equilibrium mixture at pressure P∞and uniform temperature equal to Te. In this case, whose details are provided in Section 6, the singularities in βv, Fig. E.2. Regime map for C12H26/N2mixtures showing the range of operating combustor pressures leading to crossings of the mechanical spinodals (region hatched with red solid lines) and to negative underpressures (region hatched with blue dashed lines) as a function of the interface temperature. The plot includes the critical point of dodecane (square symbol), the VLE line of dodecane (dotted line), and the maximum mechanical critical pressure (brown dot-dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) and have no effect since and . In this way, the local pressure and composition can be monitored in the calculations to ascertain whether the mixture has entered the mechanically unstable region of the thermodynamic phase diagram, and to evaluate the two problematic quantities βvand hF−hOparticipating in the formulation of non-isothermal systems. As indicated in Fig. E.2, the minimum combustor pressure P∞required to initiate crossings of the mechanical spinodals in C12H26/N2 mixtures increases with decreasing values of the interface temperature Te. No regularizations are needed at sufficiently high combustor pressures P∞≳200 bar over the entire range of relevant injection temperatures, since the system is always mechanically stable. In the configuration sketched in Fig. 1, Teis much closer to TFthan to TOnear the injection orifice. As a consequence, Tein Fig. E.2 can be approximated as TFplus a small temperature differential that depends on the details of the wake flow created by the orifice ring. In practice, the boundary of the mechanically unstable region in Fig. E.2 indicated by the red thick solid line provides approximately the minimum TFnecessary for warranting mechanical stability for a given P∞. For instance, at the pressure P∞= 100 bar considered in the results presented in Section 7 for non-isothermal C12H26/N2systems, the mechanical spinodal surface is crossed if TF≲480 K. Downstream of the injector, however, the interface temperature increases because of the heat received from the hot coflow. When Te>480 K, the system ceases to cross the mechanical spinodal surface and regularizations of βvand hF−hObecome unnecessary. These aspects are further analyzed in Section 7. Since the unboundedness of βTon the mechanical spinodal surface dominates the divergent behavior of βv, hF, and hO, a simple way of limiting the value of βTis by using the regularization where BTis a dimensionless variable defined in Eq. (C.45), and mis a positive dimensionless regularization constant whose value is much smaller than unity. Fig. E.3 shows the resulting regularized profiles of βvand hF−hO. The differences in the regularized profiles of βvand hF−hOin Fig. E.3 have a negligible impact on the overall solution of the non-isothermal system addressed in Section 7. This is shown in Fig. E.4, which shows that distribution of σover the transcritical region is largely independent of the regularization constant mused in Eq. (E.8). The reason for the lack of sensitivity of the solution to m is that the terms multiplying βvand hF−hOin the fluxes of heat and species become increasingly smaller as mechanical and transport equilibrium are approached, a condition that is always satisfied for a long portion of the interface, as discussed in Sections 5.7 and 7.2. The singularities observed above at the limit of mechanical stability are not unique to the Peng-Robinson equation of state (14). They also occur when employing other cubic equations such as the van der Waals [51] or Redlich-Kwong [96,97] equations of state. E2. Negative thermodynamic pressures The thermodynamic pressure Pis always a positive quantity in ideal gases, in which the intermolecular forces are negligible. In contrast, Pcan attain negative values in liquids since the molecules are closer and therefore exert a non-negligible attractive force on each other. Negative pressures in liquids are well-documented in the literature and are always associated with fully tensile mechanical (E.6) (E.7) (E.8) UNCORRECTED PROOF 70 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Fig. E.3. Regularized profiles of (a) volume expansivity and (b) difference of partial specific enthalpies across transcritical interfaces in C12H26/N2isothermal systems at P∞= 50 bar and Te= 500 K. The plot includes regularized profiles with m= 0.01 (dot-dashed line) and m= 0.1 (dashed line), along with the non-regularized case (solid line). states such as those participating in cavitation, nucleation, or capillary flows [104,127,128]. These processes are thermodynamically metastable or unstable, thereby suggesting that negative pressures often correspond to transient states in which the fluid is transitioning to a different phase. Negative pressures may also arise in fluids at high pressures, when the density is large and cohesive forces become important, as represented by the second term on the right-hand side of Eq. (14), or by equivalent terms in other equations of state [99]. The combustor pressure P∞and interface temperature Teleading to locally negative pressures within the interface in C12H26/N2mixtures are associated with near or full transgression of the mechanical-sta- bility limit, and are denoted in Fig. E.2 as the region hatched with dashed lines. It should be emphasized that negative pressures do not lead to any transcendental behavior in the conservation equations, neither do they induce any singularities in the thermodynamic variables discussed in Appendix E.1. They do, however, influence the calculation of transport coefficients in the following manner. The dynamic viscosity μ, thermal conductivity λ, binary diffusion coefficient and thermal-diffusion ratio kF,T are generally functions of the thermodynamic pressure and participate in the conservation Eqs. (44)–(47) through the viscous stress tensor and the fluxes of heat and species. In particular, λ, and kF,T are required for the calculation of the Onsager coefficients (144)–(146). Whereas models for these coefficients are well-established in the literature for ideal and real gases, including fully supercritical fluids, relatively much less is known about their behavior within the mechanically unstable region of the thermodynamic phase diagram, since homogeneous mixtures of phases or fluids of fundamentally different nature are untenable there. The standard models for μ,λ, and kF,T in fluids at high pressures, employed in Appendices B.4 and Appendix F, cease to have physical significance at negative pressures where, for instance, becomes undefined because of Eq. (F.41). However, note that the mechanical and transport equilibrium conditions (107) and (177) [or their equivalent forms (193) and (194) in terms of chemical potentials] are independent of viscosity and of the Onsager coefficients, and therefore are not influenced by the locally erroneous behavior of any of the transport coefficients under conditions leading to negative pressures. The considerations above suggest that, in isothermal systems such as the ones treated in Section 6, the structure of the interface is completely insensitive to the specific manner in which λ, and kF,T are modeled. In contrast, in non-isothermal systems such as the one in Fig. 1, the mechanical and transport equilibrium conditions are only satisfied asymptotically to leading order in the limit of small thermal Cahn numbers, ϵT≪1. Second-order effects lead to departures from equilibrium and require consideration of transport induced by temperature gradients. This involves evaluation of λ, and kF,T across the interface, and gives rise to small fluxes of heat and species. In absence of improved models for transport coefficients at mechanically unstable conditions, the approximations are used in this study, where the transport coefficients are evaluated at the combustor pressure P∞, the latter being representative of the thermodynamic pressure everywhere away from the interface in flows at small Mach numbers. Appendix F. Supplementary expressions for molecular transport coefficients at high pressures This appendix reviews models for the calculation of the viscosity, thermal conductivity, diffusion coefficient, and thermal-diffusion ratio at high pressures. F1. Dynamic viscosity The dynamic viscosity ηof the mixture can be modeled as [155,156] where Tc[K], vc[cm3/mol], and W′[g/mol] are, respectively, representative values of the critical temperature, critical molar volume, and molecular weight of the mixture given by the expressions In Eq. (F.2), ε[K] is the minimum of the pair-potential energy divided by the Boltzmann constant, and σ[Å] is the hard-sphere diameter. Both of these quantities are obtained by the empirical mixing rules (E.9) (F.1) (F.2) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 71 where ϵi,j and σi,j are obtained from with the corresponding quantities for the individual components being given by and . Using the Peng-Robinson equation of state (14), the critical molar volume of the species ican be computed as with [92], thereby yielding cm3/mol for C12H26, and cm3/mol for N2. Note, however, that these values do not match the experimentally measured ones cm3/mol and cm3/mol [40], since the Peng-Robinson equation of state has only two coefficients, and therefore cannot be forced to simultaneously satisfy measured critical values of pressure, temperature and molar volume. In Eq. (F.1) the prefactor η* is given by where and . The remaining parameters are computed using the relations where the coefficients w1,k,⋯w4,kare provided in Table F.1. In this formulation, ωis an acentric factor of the mixture defined as Table F.1 Coefficients for Ekin Eq. (F.11) [155]. k w1, kw2, kw3, kw4, k 1 6.324 50.412 − 51.680 1189.0 2 0.03728 3 5.283 254.209 − 168.48 3898.0 4 6.623 38.096 − 8.464 31.42 5 19.745 7.630 − 14.354 31.53 6− 1.900 − 12.537 4.985 − 18.15 7 24.275 3.450 − 11.291 69.35 8 0.7972 1.117 0.01235 − 4.117 9− 0.2382 0.06770 − 0.8163 4.025 10 0.06863 0.3479 0.5926 − 0.727 with . The reduced dipole moment and the association factor of the mixture kacan be obtained from tables [54] and are set to zero for C12H26/N2mixtures. F2. Thermal conductivity In this study, the thermal conductivity λof the mixture is modeled as [155,156] where W′and Tcare given in (F.2). The rest of the parameters in Eq. (F.13) are where with Tcand ωgiven in (F.2) and (F.12), respectively. The parameters z1,k,⋯z4,kare calculated based on Table F.2. F3. Stefan-Maxwell forms of the standard fluxes of heat and species The expressions (136) and (137) for the standard diffusion fluxes of heat and species can be recast in more physically insightful (F.3) (F.4) (F.5) (F.6) (F.7) (F.8) (F.9) (F.10) (F.11) (F.12) (F.13) (F.14) (F.15) (F.16) (F.17) (F.18) UNCORRECTED PROOF 72 Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx Table F.2 Coefficients for Bkin Eq. (F.17) [155]. k z1,kz1,kz1,kz1,k 1 2.417 0.748 − 0.919 121.720 2− 0.509 − 1.509 − 49.991 69.983 3 6.611 5.621 64.760 27.039 4 14.543 − 8.914 − 5.638 74.344 5 0.793 0.820 − 0.694 6.317 6− 5.863 12.801 9.589 65.529 7 91.089 128.110 − 54.217 523.810 forms that involve gradients of temperature, pressure and composition. Because of the functional dependence of the molar chemical potential on pressure, temperature and composition, given explicitly by Eqs. (B.23), (B.24), and (C.47), the differential form is satisfied. In addition, the Gibbs-Duhem Eq. (C.9) can be rewritten on a molar basis at constant temperature as which can be spatially differentiated yielding Upon dividing Eq. (F.19) by Wkand subtracting it from Eq. (F.21) divided by WN, the relation is obtained, which connects the constant-temperature gradients of the specific chemical potential with the gradients of pressure and composition, where δm,k is the Kronecker delta and k= 1,⋯N−1. In writing Eq. (F.22), use has been made of Eq. (B.18) to replace the partial derivative of the chemical potential with respect to pressure by the partial molar volume, and of Eq. (B.24) to express the partial derivative of the chemical potential with respect to the molar fraction as where fmis the fugacity of species mdefined by the combination of Eqs. (B.23) and (C.47). The utilization of Eq. (F.22) in Eqs. (136)-(137) for replacing the constant-temperature gradients of chemical potentials by gradients of composition and pressure leads to the Stefan-Maxwell forms and for the standard diffusion fluxes of heat and species, respectively. In Eq. (F.25), Di,j is the Fickian diffusion coefficient. F4. Fickian diffusion coefficients The Fickian portion of the standard diffusion flux (F.25) can be expressed in matrix form as where and are vectors with N−1 components, and is a (N−1)×(N−1) matrix composed of the Fickian diffusion coefficients. The latter are related to the Onsager coefficients Li,k through Eq. (F.30), which, in matrix form, can be ex (F.19) (F.20) (F.21) (F.23) (F.24) (F.25) (F.26) UNCORRECTED PROOF Progress in Energy and Combustion Science xxx (xxxx) xxx-xxx 73 pressed as In Eq. (F.27), is the (N−1)×(N−1) subset of the Onsager matrix denoted by dot-dashed lines in Eq. (143). This subset contains only mass-transfer-related coefficients. In addition, is a (N−1)×(N−1) matrix whose elements depend on the local composition and are given by Lastly, the (N−1)×(N−1) elements of the matrix represent high-pressure effects and are given by At low pressures, the fugacity is equal to the partial pressure, and therefore Γbecomes the identity matrix. Upon substituting Eqs. (F.29) and (F.28) into (F.27), the expression is obtained for the effective Fickian diffusion coefficient of species i into j. By definition, the binary diffusion coefficients satisfy the relation where is a (N−1)×(N−1) matrix given by [137,162] and Upon combining Eqs. (F.26), (F.27) and (F.31), the relations and are obtained. In particular, Eqs. (F.34) and (F.35) enable the calculation of the Onsager coefficients Li,k [from Eq. (F.34)] and the Fickian diffusion coefficients Di,j [from Eq. (F.35)] by using the binary diffusion coefficients described in Appendix F.5. F5. Binary diffusion coefficients The binary diffusion coefficients can be modeled using the expression which follows the formulation described in Refs[162,171,172]. for non-ideal and nonpolar multicomponent mixtures. In Eq. (F.36), ( ) corresponds to the molecular diffusion coefficient of component i(j) infinitely diluted in component j(i). These auxiliary coefficients are given by In this formulation, c[mol/m3] is the molar density, and are the reduced temperature and pressure, η/η0is the ratio between high- and low-pressure viscosities, and are model constants given by with and . The prod (F.27) (F.28) (F.29) (F.30) (F.31) (F.32) (F.33) (F.34) (F.35) (F.36) (F.37) (F.38) (F.39) (F.40) (F.41)