# Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point

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## Abstract

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## 1. Introduction

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- The first and certainly the most obvious and limiting is related to its inability to deal with liquid and non-condensable gas separated by well-defined interfaces, such as for example interfacial flows of liquid water and air. The thermodynamics of these two media being considered as discontinuous, specific theoretical and numerical treatments have been addressed. In this context, Arbitrary Lagrangian Eulerian (Hirt et al. [2]), Interface Reconstruction (Youngs [3]), Front Tracking (Glimm et al. [4]), Level-Set (Fedkiw et al. [5]), anti-diffusion (Kokh and Lagoutiere [6]) methods are possible options. Another approach relies on continuous models with extra internal variables, such as volume and mass fractions and extended equation of state. Examples of such models are the Kapila et al. [7] one and its extension with phase transition (Saurel et al. [8]) to cite a few. With these formulations, the same equations are solved everywhere routinely, in pure liquid, pure gas and interface which becomes a diffuse zone. These models are indeed often named “diffuse interface methods” (Saurel and Pantano [9]). In this approach, hyperbolic models with relaxation are considered and each phase evolves in its own volume, with its own thermodynamics. In particular, there is no need to address cubic formulations. When phase transition is addressed, it occurs through mass transfer terms that can be considered finite rate (Saurel et al. [8], Furfaro and Saurel [10]) or assumed stiff when the physical knowledge of the phase change kinetics is not enough documented (Le Métayer et al. [11], Chiapolino et al. [12,13]) or unnecessary.
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- The second limitation is related to the lack of convexity of cubic EOSs, having dramatic consequences on sound propagation during phase transition. The square sound speed becomes negative in the spinodal decomposition zone, such behavior not being physical.
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- The third limitation is related to the description of phase transition with such EOSs. Cubic equations of state consider phase transition as a thermodynamic process and not a kinetic one. It is unclear at this level whether cubic EOSs are limited to the description of global two-phase mixtures with many interfaces and not local ones, at the scale of a single interface.
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- The fourth, but possibly not the last, is related to the numerical treatment of boundary conditions (BC) in practical compressible flow computations. Subsonic inflow and outflow BCs rely on stagnation enthalpy and entropy invariance coupled to Riemann invariants that can be defined and computed correctly only if the equation of state is well-posed. The second issue related to EOS convexity consequently reemerges at this level. Moreover, the practical expression of Riemann invariants may be inextricable with these EOSs.

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- Represent the thermodynamics of pure liquid, pure vapor and supercritical fluid. Combination of the pure liquid and pure vapor EOSs must be able to represent as accurately as possible the two-phase region.
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- Each phase EOS must be convex in its respective domain.
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- The EOS must be as simple as possible, while remaining accurate, to simplify practical computations and building of mixture EOS in hyperbolic multiphase flow models.

## 2. Extended NASG EOS

#### 2.1. Thermal and Caloric EOSs

#### 2.2. Expression of the Entropy

#### 2.3. Speed of Sound

## 3. Saturation Condition of the Liquid–Vapor Couple

## 4. Summary of the Extended NASG State Functions

## 5. Extended NASG Parameters

## 6. Transition to Supercritical Fluids

#### 6.1. Liquid-to-Supercritical-State Transition

#### 6.2. Vapor-to-Supercritical-State Transition

#### 6.3. Concluding Remarks

## 7. Two-Phase Flow Illustrations

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Convexity of the ENASG Formulation

## Appendix B. Maxwell’s Relations

## Appendix C. Methodology to Determine the Various Extended NASG (ENASG) Parameters

#### Appendix C.1. Liquid Phase

**Table A1.**Reference state values used for the determination of liquid Extended Noble–Abel Stiffened-Gas (ENASG) coefficients.

Fluid | N | ${\mathit{T}}_{\mathit{c}}$ (K) | ${\mathit{p}}_{\mathit{c}}$ (bar) | ${\mathit{v}}_{\mathit{c}}$ (m${}^{3}$/kg) | ${\mathit{p}}_{\mathit{\infty},\mathit{c}}^{\prime}$ (Pa) | ${\mathit{b}}_{\mathit{c}}$ (m${}^{3}$/kg) | ${\mathit{c}}_{0}$ (m/s) | ${\mathit{p}}_{0}$ (bar) | ${\mathit{v}}_{0}$ (m${}^{3}$/kg) |
---|---|---|---|---|---|---|---|---|---|

H${}_{2}$O | 374 | $646.16$ | 221 | $0.0025101$ | $0.01$ | ${10}^{-6}$ | $1552.1$ | 1 | $0.0010182$ |

O${}_{2}$ | 101 | $154.36$ | 50 | $0.0019522$ | $0.01$ | ${10}^{-6}$ | $1065.7$ | 1 | $0.00080871$ |

**Table A2.**Reference state values used for the determination of liquid ENASG coefficients (continued).

Fluid | ${\mathit{T}}_{\mathit{ref}}$ (K) | ${\mathit{p}}_{\mathit{ref}}$ (Pa) | ${\mathit{v}}_{\mathit{ref}}$ (m${}^{3}$/kg) | ${\mathit{e}}_{\mathit{ref}}$ (kJ/kg) | ${\mathit{b}}_{\mathit{ref}}$ (m${}^{3}$/kg) |
---|---|---|---|---|---|

H${}_{2}$O | $300.16$ | $3570.2$ | $0.0010035$ | $113.23$ | $0.0009125$ |

O${}_{2}$ | $70.631$ | $6684.7$ | $0.00080952$ | $-166.823$ | $0.000769$ |

#### Appendix C.2. Gas Phase

## Appendix D. Connection Temperature between the ENASG EOS and Ideal Gas Formulation

## Appendix E. Toward the Critical Point

**Table A3.**Coefficients for water and oxygen for the “alternative” ENASG EOS whose formulation is summarized in Equation (A38). With such description, the gas attractive effects are taken into account via the parameter d but result in conditional convexity, Equation (A39). The liquid ENASG EOS is unchanged, Equation (56).

Coefficients | ENASG${}_{{\mathit{H}}_{2}\mathit{O},\phantom{\rule{0.166667em}{0ex}}\mathit{Liq}}$ | ENASG${}_{{\mathit{H}}_{2}\mathit{O},\phantom{\rule{0.166667em}{0ex}}\mathit{vap}}$ | ENASG${}_{{\mathit{O}}_{2},\phantom{\rule{0.166667em}{0ex}}\mathit{Liq}}$ | ENASG${}_{{\mathit{O}}_{2},\phantom{\rule{0.166667em}{0ex}}\mathit{vap}}$ |
---|---|---|---|---|

$\gamma $ | $1.0178$ | $1.3189$ | $1.033$ | $1.3875$ |

${C}_{v}$ (J/kg/K) | 3848 | 1719 | 1451 | 779 |

${b}_{1}$ | $-0.5934$ | 0 | $-0.6661$ | 0 |

${b}_{0}$ (m${}^{3}$/kg) | $1.4905\times {10}^{-3}$ | $3.3514\times {10}^{-4}$ | $1.3013\times {10}^{-3}$ | 0 |

${p}_{\infty ,1}$ (Pa/K) | − 607,195 | 0 | −405,133 | 0 |

${p}_{\infty ,0}$ (Pa) | 396,642,530 | 0 | 63,642,939 | 0 |

q (J/kg) | −1,065,948 | 1,975,421 | −272,675 | $-1597$ |

${q}^{\prime}$ (J/kg/K) | −20,985 | $-3131$ | $-3277$ | 2224 |

d (Pa m${}^{3\gamma}$/kg${}^{\gamma}$) | 0 | 41,200 | 0 | 2950 |

**Figure A1.**Comparison between experimental and theoretical saturation curves for liquid ${}_{l}$ and vapor ${}_{v}$ water. Symbols represent experimental data. The thick lines represent the theoretical saturation curves obtained with the liquid Extended NASG EOS (ENASG) Equation (56) and its “alternative” but conditionally convex formulation for the vapor phase, Equation (A38). ${p}_{sat}$ denotes the saturation pressure, ${L}_{v}$ the latent heat, h the specific enthalpy and $\rho $ the density.

**Figure A2.**Comparison between experimental and theoretical saturation curves for liquid ${}_{l}$ and vapor ${}_{v}$ oxygen. Symbols represent experimental data. The thick lines represent the theoretical saturation curves obtained with the liquid Extended NASG EOS (ENASG) Equation (56) and its “alternative” but conditionally convex formulation for the vapor phase, Equation (A38). ${p}_{sat}$ denotes the saturation pressure, ${L}_{v}$ the latent heat, h the specific enthalpy and $\rho $ the density.

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**Figure 1.**Comparison between experimental and theoretical saturation curves for liquid ${}_{l}$ and vapor ${}_{v}$ water. Symbols represent experimental data. The thick lines represent the theoretical saturation curves obtained with the Extended Noble–Abel Stiffened-Gas EOS (ENASG) reducing to the ideal gas description for the vapor phase, Equations (56) and (57). The thin lines represent results obtained with the original NASG EOS also reducing to the ideal gas formulation for the vapor phase. ${p}_{sat}$ denotes the saturation pressure, ${L}_{v}$ the latent heat, h the specific enthalpy and $\rho $ the density.

**Figure 2.**Comparison between experimental and theoretical saturation curves for liquid ${}_{l}$ and vapor ${}_{v}$ oxygen. Symbols represent experimental data. The thick lines represent the theoretical saturation curves obtained with the Extended NASG EOS (ENASG) reducing to the ideal gas description for the vapor phase, Equations (56) and (57). The thin lines represent results obtained with the original NASG EOS also reducing to the ideal gas formulation for the vapor phase. ${p}_{sat}$ denotes the saturation pressure, ${L}_{v}$ the latent heat, h the specific enthalpy and $\rho $ the density.

**Figure 3.**The saturation curve is composed of the boiling and the dew curves separating the two-phase mixture zone and the pure phase zones. Beyond the critical isotherm, there is no transition between the liquid and the gaseous state. The fluid is neither liquid nor gas. It is said to be supercritical. Phase transition can happen either through the saturation dome corresponding to liquid–vapor phase change, or through the critical isotherm corresponding to a pure-phase-to-supercritical-state transition.

**Figure 4.**Comparison between experimental and theoretical isobar for water. The symbols represent the experimental isobar $p=230$ bars. Beyond the critical temperature ${T}_{c}=646$ K, the liquid transforms to supercritical state. The thick solid lines represent the Extended NASG (ENASG) EOS, reducing to the ideal gas description for the supercritical phase, Equations (56), (57) and (60). The thin solid lines represent the original NASG EOS also reducing to the ideal gas formulation for the supercritical phase. The dashed lines represent the van der Waals (VdW) theoretical predictions and the dashed-dotted lines represent the Soave–Redlich–Kwong (SRK) ones. The critical temperature is indicated with the dotted lines. From this temperature, the liquid ENASG EOS is extended and joins the ideal gas EOS (except for the sound speed). The temperature ${T}_{0}=1000$ K at which variable heat capacities are considered is indicated in dotted lines as well.

**Figure 5.**Comparison between experimental and theoretical isobar for water. The symbols represent the experimental isobar $p=500$ bars. Beyond the critical temperature ${T}_{c}=646$ K, the liquid transforms to supercritical state. The thick solid lines represent the Extended NASG (ENASG) EOS, reducing to the ideal gas description for the supercritical phase, Equations (56), (57) and (60). The thin solid lines represent the original NASG EOS also reducing to the ideal gas formulation for the supercritical phase. The dashed lines represent the van der Waals (VdW) theoretical predictions and the dashed-dotted lines represent the Soave–Redlich–Kwong (SRK) ones. The critical temperature is indicated with the dotted lines. From this temperature, the liquid ENASG EOS is extended and joins the ideal gas EOS (except for the sound speed). The temperature ${T}_{0}=1000$ K at which variable heat capacities are considered is indicated in dotted lines as well.

**Figure 6.**Comparison between experimental and theoretical isobar for oxygen. The symbols represent the experimental isobar $p=60$ bars. Beyond the critical temperature ${T}_{c}=154$ K, the liquid transforms to supercritical state. The thick solid lines represent the Extended NASG (ENASG) EOS, reducing to the ideal gas description for the supercritical phase, Equations (56), (57) and (60). The thin solid lines represent the original NASG EOS also reducing to the ideal gas formulation for the supercritical phase. The dashed lines represent the van der Waals (VdW) theoretical predictions and the dashed-dotted lines represent the Soave–Redlich–Kwong (SRK) ones. The critical temperature is indicated with the dotted lines. From this temperature, the liquid ENASG EOS is extended and joins the ideal gas EOS (except for the sound speed). The temperature ${T}_{0}=400$ K at which variable heat capacities are considered is indicated in dotted lines as well.

**Figure 7.**Comparison between experimental and theoretical isobar for oxygen. The symbols represent the experimental isobar $p=200$ bars. Beyond the critical temperature ${T}_{c}=154$ K, the liquid transforms to supercritical state. The thick solid lines represent the Extended NASG (ENASG) EOS, reducing to the ideal gas description for the supercritical phase Equations (56), (57) and (60). The thin solid lines represent the original NASG EOS also reducing to the ideal gas formulation for the supercritical phase. The dashed lines represent the van der Waals (VdW) theoretical predictions and the dashed-dotted lines represent the Soave–Redlich–Kwong (SRK) ones. The critical temperature is indicated with the dotted lines. From this temperature, the liquid ENASG EOS is extended and joins the ideal gas EOS (except for the sound speed). The temperature ${T}_{0}=400$ K at which variable heat capacities are considered is indicated in dotted lines as well.

**Figure 8.**Comparison between experimental and theoretical isobar for water. The symbols represent the experimental isobar $p=30$ bars. Beyond the critical temperature ${T}_{c}=646$ K, the vapor transforms to supercritical state. The thick solid lines represent the Extended NASG (ENASG) EOS, reducing to the ideal gas description for vapor and supercritical phases, Equations (56), (57) and (60). The thin solid lines represent the original NASG EOS also reducing to the ideal gas formulation. The dashed-dotted lines represent the van der Waals (VdW) theoretical predictions and the dashed lines represent the Soave–Redlich–Kwong (SRK) ones. The critical temperature is indicated with the dotted lines. The temperature ${T}_{0}=1000$ K at which variable heat capacities are considered is indicated in dotted lines as well.

**Figure 9.**Comparison between experimental and theoretical isobar for oxygen. The symbols represent the experimental isobar $p=10$ bars. Beyond the critical temperature ${T}_{c}=154$ K, the vapor transforms to supercritical state. The thick solid lines represent the Extended NASG (ENASG) EOS, reducing to the ideal gas description for vapor and supercritical phases, Equations (56), (57) and (60). The thin solid lines represent the original NASG EOS also reducing to the ideal gas formulation. The dashed-dotted lines represent the van der Waals (VdW) theoretical predictions and the dashed lines represent the Soave–Redlich–Kwong (SRK) ones. The critical temperature is indicated with the dotted lines. The temperature ${T}_{0}=400$ K at which variable heat capacities are considered is indicated in dotted lines as well.

**Figure 10.**Shock tube test illustrating the transition from “pure” water vapor to supercritical state. The critical temperature is indicated with the dotted line. The thick lines represent the solution obtained with the mixture ENASG EOS reducing to Equation (65) in the present example as liquid mass fraction is non-zero but in negligible proportions. The dashed lines represent the initial conditions. In the left chamber, air is initially in major proportions with ${Y}_{3}^{left}=1-2\times {10}^{-7}$, $p=30$ bars and $T=800$ K. Liquid and vapor mass fractions are deduced as ${Y}_{1}^{left}\simeq {10}^{-8}$ and ${Y}_{2}^{left}\simeq 1.9\times {10}^{-7}$. In the right chamber, water vapor is in major proportions with ${Y}_{3}^{right}={10}^{-7}$, $p=1$ bar and $T=600$ K. Liquid and vapor mass fractions are deduced as ${Y}_{1}^{right}\simeq {10}^{-8}$ and ${Y}_{2}^{left}\simeq 0.99999989$. The test was carried out with Godunov time integration method and HLLC Riemann solver [27] extended to the second order: MUSCL scheme [27] with Minmod flux limiter [27]. The solution is given at $t\approx 0.3$ ms on a 1000-cell mesh using CFL= 0.8 [27].

**Figure 11.**Double expansion test illustrating the transition from supercritical state to “pure” liquid water. The critical pressure and temperature are indicated with the dotted lines. The thick lines represent the solution obtained with the mixture ENASG EOS. The dashed lines represent the initial conditions. Liquid water is initially in major proportions with ${Y}_{1}^{}=1-2\times {10}^{-6}$, ${Y}_{2}^{}={Y}_{3}^{}={10}^{-6}$ , $p=350$ bars, $T=655$ K and $u=\pm 45$ m/s. The test was carried out with a Godunov time integration method and HLLC Riemann solver extended to the second order: MUSCL scheme with Minmod flux limiter. The solution is given at $t\approx 0.3$ ms on a 1000-cell mesh using CFL = 0.8.

**Figure 12.**Density and vapor mass fraction profiles of a liquid oxygen jet surrounded by hydrogen at high speed entering a combustion chamber of a cryotechnic rocket engine. Shear effects induce jet fragmentation. The filaments separating the main liquid core and the gas gradually vanish as a consequence of evaporation. The computation was done with the MUSCL scheme with Superbee limiter [27] and CFL = 0.7. The solution is given at $t\approx 4.1$ ms. The mesh is unstructured and made of about 360,000 triangles.

**Table 1.**Extended Noble–Abel Stiffened-Gas (ENASG) coefficients for water. The NASG parameters are also given and determined with the method given in Le Métayer and Saurel [14] except for the liquid reference entropy ${q}^{\prime}$ that is computed with the NASG reduction of Equation (A31) (see Appendix C). The NASG water parameters are determined with $n=201$ experimental saturation points in the temperature range ${T}_{exp}\in $ [300 K–500 K].

Coefficients | ENASG${}_{\mathit{Liq}}$ | ENASG${}_{\mathit{gas}}$ | NASG${}_{\mathit{Liq}}$ | NASG${}_{\mathit{gas}}$ |
---|---|---|---|---|

$\gamma $ | $1.0147$ | $1.3079$ | $1.1807$ | $1.5377$ |

${C}_{v}$ (J/kg/K) | 4014 | 1500 | 3630 | 856 |

${b}_{1}$ | $-0.6050$ | 0 | 0 | 0 |

${b}_{0}$ (m${}^{3}$/kg) | $1.5196\times {10}^{-3}$ | 0 | $6.8428\times {10}^{-4}$ | 0 |

${p}_{\infty ,1}$ (Pa/K) | −471,025 | 0 | 0 | 0 |

${p}_{\infty ,0}$ (Pa) | 307,078,403 | 0 | 664,961,465 | 0 |

q (J/kg) | −1,112,426 | 1,947,630 | −1,178,154 | 2,176,064 |

${q}^{\prime}$ (J/kg/K) | −22,049 | 1136 | −10,742 | 4863 |

**Table 2.**Extended NASG (ENASG) coefficients for oxygen. The NASG parameters are also given and determined with the method given in Le Métayer and Saurel [14] except for the liquid reference entropy ${q}^{\prime}$ that is computed with the NASG reduction of Equation (A31) (see Appendix C). The NASG oxygen parameters are determined with $n=41$ experimental saturation points in the temperature range ${T}_{exp}\in $ [60 K–100 K].

Coefficients | ENASG${}_{\mathit{Liq}}$ | ENASG${}_{\mathit{gas}}$ | NASG${}_{\mathit{Liq}}$ | NASG${}_{\mathit{gas}}$ |
---|---|---|---|---|

$\gamma $ | $1.0281$ | $1.3985$ | $1.6610$ | $1.4730$ |

${C}_{v}$ (J/kg/K) | 1535 | 652 | 1016 | 548 |

${b}_{1}$ | $-0.6721$ | 0 | 0 | 0 |

${b}_{0}$ (m${}^{3}$/kg) | $1.3131\times {10}^{-3}$ | 0 | $5.7003\times {10}^{-4}$ | 0 |

${p}_{\infty ,1}$ (Pa/K) | −324,997 | 0 | 0 | 0 |

${p}_{\infty ,0}$ (Pa) | 50,890,107 | 0 | 196,815,802 | 0 |

q (J/kg) | −278,134 | $-1589$ | −285,545 | 6528 |

${q}^{\prime}$ (J/kg/K) | $-3691$ | 4237 | 8171 | 4650 |

Fluid | ${\mathit{T}}_{0}$ (K) | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ (K${}^{-1})$ | ${\mathit{a}}_{3}$ (K${}^{-2})$ | ${\mathit{a}}_{4}$ (K${}^{-3})$ | ${\mathit{a}}_{5}$ (K${}^{-4})$ |
---|---|---|---|---|---|---|

H${}_{2}$O | 1000 | $3.31570$ | $2.10648\times {10}^{-3}$ | $-3.76340\times {10}^{-7}$ | $3.47520\times {10}^{-11}$ | $-1.70335\times {10}^{-15}$ |

O${}_{2}$ | 400 | $3.78246$ | $-2.99673\times {10}^{-3}$ | $9.84730\times {10}^{-6}$ | $-9.68129\times {10}^{-9}$ | $3.24373\times {10}^{-12}$ |

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**MDPI and ACS Style**

Chiapolino, A.; Saurel, R.
Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point. *Fluids* **2018**, *3*, 48.
https://doi.org/10.3390/fluids3030048

**AMA Style**

Chiapolino A, Saurel R.
Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point. *Fluids*. 2018; 3(3):48.
https://doi.org/10.3390/fluids3030048

**Chicago/Turabian Style**

Chiapolino, Alexandre, and Richard Saurel.
2018. "Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point" *Fluids* 3, no. 3: 48.
https://doi.org/10.3390/fluids3030048