#
Review of Reduced-Order Models for Homogeneous CO_{2} Nucleation in Supersonic and Hypersonic Expansion Flows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nucleation Theory

#### 2.1. Classical Nucleation Theory

#### 2.2. Self-Consistent Classical Nucleation Theory

#### 2.3. Mean-Field Kinetic Nucleation Theory

#### 2.4. Semiphenomenological Nucleation Theory

#### 2.5. Extended Modified Liquid Drop Dynamical Nucleation Theory

#### 2.6. Semi-Empirical Density Gradient Theory

#### 2.7. Scaled Nucleation Rate Model

#### 2.8. Nonisothermal Nucleation

#### 2.9. Vibrational Nonequilibrium

## 3. Results and Discussion

## 4. Experimental and Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | Temperature dependent variable in Equation (16) |

b | Mean squared energy fluctation of impinging molecules |

${B}_{2}$ | Second virial coefficient of the vapor |

c | Concentration (mole fraction) |

${c}_{v}$ | Specific heat at constant volume per molecule of the vapor |

${c}_{v,c}$ | Specific heat at constant volume per molecule of the carrier gas |

d | Hard sphere diameter |

e | Energy |

${f}_{h}$ | Homogeneous free energy density |

${f}_{N}$ | Probability of cluster existing in EMLDDNT volume |

g | Density square gradient, ${(\nabla n)}^{2}$, or condensate mass fraction |

${g}_{\mathrm{max}}$ | Condensate mass fraction with complete condensation |

h | Planck constant |

${h}_{l}$ | Latent energy of phase change per molecule |

H | Unit step function |

J | Steady state nucleation rate |

${J}_{\mathrm{iso}}$ | Isothermal steady state nucleation rate |

${J}_{\mathrm{noniso}}$ | Nonisothermal steady state nucleation rate |

${J}_{t}$ | Transient nucleation rate |

${k}_{B}$ | Boltzmann constant |

${m}_{1}$ | Mass per molecule |

M | Number of molecules |

n | Number density of molecules |

${n}_{v}$ | Number density of free molecules in the vapor |

${n}_{l}$ | Saturated liquid number density |

${n}_{N}$ | Number density of clusters with N molecules |

${n}_{N,e}$ | Equilibrium number density of clusters |

${n}_{s}$ | Saturated vapor number density |

N | Number of molecules in the cluster |

${N}_{l}$ | Coordination number of the liquid |

${N}_{*}$ | Number of molecules in critical cluster |

${p}_{0}$ | Stagnation pressure or EMLDDNT pressure without cluster |

${p}_{1}$ | Vapor pressure within EMLDDNT volume |

${p}_{c}$ | Pressure of carrier gas |

${p}_{h}$ | Pressure of homogeneous fluid |

${p}_{\mathrm{hs}}$ | Hard sphere pressure |

${p}_{N}$ | SEDGT normal pressure |

${p}_{s}$ | Saturated vapor pressure |

${p}_{v}$ | Pressure of vapor |

${p}_{T}$ | SEDGT tangential pressure |

P | Average pressure in EMLDDNT volume |

${P}_{N}$ | Total pressure within EMLDDNT volume |

q | Thermal energy released per condensing molecule |

r | Radius of droplet or radius within droplet |

${r}_{1}$ | Radius of molecule |

R | Radius of EMLDDNT volume or ideal gas constant |

${s}_{1}$ | Saturated liquid surface area per molecule |

S | Saturation ratio, ${p}_{v}/{p}_{s}$ |

T | Temperature |

${T}_{c}$ | Critical point temperature |

${T}_{*}$ | Nondimensional temperature, ${k}_{B}T/\u03f5$ |

${v}_{1}$ | Saturated liquid volume per molecule |

V | Volume |

Z | Zel’dovich factor |

$\alpha $ | Total integrated attractive potential |

${\alpha}_{N}$ | Cluster evaporation rate |

${\beta}_{N}$ | Cluster impingement rate |

${\beta}_{*}$ | Impingement rate onto critical cluster |

${\delta}_{T}$ | Tolman length |

$\Delta {F}_{c}$ | Total free energy within EMLDDNT closed volume |

$\Delta {F}_{c,N}$ | Closed system free energy barrier of droplet with N molcules |

$\Delta {F}_{N}$ | Free energy barrier of cluster with N molecules |

$\Delta {F}_{*}$ | Free energy barrier of critical cluster |

$\Delta p$ | Pressure difference between cluster and gas |

$\Delta {t}_{R}$ | Residence time of freestream molecule |

$\Delta \kappa $ | SEDGT influence parameter correction factor, $\kappa -{\kappa}_{\infty}$ |

$\Delta {\kappa}_{*}$ | Nondimensional SEDGT influence parameter correction factor, ${\kappa}_{*}-{\kappa}_{\infty *}$ |

$\Delta \mu $ | Difference in chemical potential |

$\Delta {\mu}_{h}$ | Difference in chemical potential of homogeneous fluid |

$\Delta {\mu}_{\mathrm{hs}}$ | Hard sphere difference in chemical potential |

$\u03f5$ | Lennard–Jones potential |

$\eta $ | Packing fraction of hard spheres |

${\kappa}_{l}$ | Isothermal compressibility of the liquid |

$\kappa $ | SEDGT influence parameter |

${\kappa}_{\infty}$ | SEDGT infinite plane influence parameter |

${\kappa}_{\infty \ast}$ | Nondimensional SEDGT infinite plane influence parameter |

${\kappa}_{*}$ | Nondimensional SEDGT influence parameter, $\kappa /\left(\u03f5{\sigma}^{5}\right)$ |

${\lambda}_{\mathrm{th}}$ | Thermal de Broglie wavelength |

$\xi $ | SNT variable (Equation (24)) |

$\sigma $ | Surface tension or Lennard-Jones zero energy distance |

${\sigma}_{\infty}$ | Infinite plane surface tension |

${\tau}_{e}$ | Characteristic time for gas expansion |

${\tau}_{t}$ | Characteristic time for transient nucleation |

$\psi $ | Inhomogeneous free energy density |

$\Omega $ | Eötvös constant |

## Appendix A. Fluid Properties

## References

- Lakebrink, M.T.; Bowcutt, K.G.; Winfree, T.; Huffman, C.C.; Juliano, T.J. Optimization of a Mach-6 Quiet Wind-Tunnel Nozzle. J. Spacecr. Rocket.
**2018**, 55, 315–321. [Google Scholar] [CrossRef] - Daum, F.L.; Gyarmathy, G. Condensation of Air and Nitrogen in Hypersonic Wind Tunnels. Am. Inst. Aeronaut. Astronaut.
**1967**, 6, 458–465. [Google Scholar] [CrossRef] - Faro, I.; Small, T.R.; Hill, F.K. The Supersaturation of Nitrogen in a Hypersonic Wind Tunnel. J. Appl. Phys.
**1952**, 23, 40–43. [Google Scholar] [CrossRef] - Willmarth, W.W.; Nagamatsu, H.T. The Condensation of Nitrogen in a Hypersonic Nozzle. J. Appl. Phys.
**1952**, 23, 1089–1095. [Google Scholar] [CrossRef] - Tans, P.; Keeling, R. Trends in Atmospheric Carbon Dioxide; NOAA Global Monitoring Laboratory. Available online: https://gml.noaa.gov/ccgg/trends/data.html (accessed on 1 November 2021).
- Lax, P.A.; Leonov, S.B. Semiempirical Model for Homogeneous Nitrogen Condensation in Hypersonic Wind Tunnels. AIAA J.
**2020**, 58, 4807–4818. [Google Scholar] [CrossRef] - Goglia, G.L. Limit of Supersaturation of Nitrogen Vapor Expanding in a Nozzle. Ph.D. Thesis, University of Michigan, Ann Arbor, MI, USA, 1959. [Google Scholar]
- Zahoransky, R.A. Nitrogen Nucleation in an Unsteady Supersonic Flow Field. Z. Für Flugwiss. Und Weltraumforsch.
**1986**, 10, 34–37. [Google Scholar] - Steinwandel, J. Homogeneous Condensation of Nitrogen in the Expansion Wave of a Cryogenic Shock Tube. Berichte Bunsenges. Phys. Chem.
**1985**, 89, 481–484. [Google Scholar] [CrossRef] - Dingilian, K.K.; Halonen, R.; Tikkanen, V.; Reischl, B.; Vehkamäki, H.; Wyslouzil, B.E. Homogeneous nucleation of carbon dioxide in supersonic nozzles I: Experiments and classical theories. Phys. Chem. Chem. Phys.
**2020**, 22, 19282–19298. [Google Scholar] [CrossRef] [PubMed] - Wood, S.E. Nucleation and Growth of CO
_{2}Ice Crystals in the Martian Atmosphere. Ph.D. Thesis, University of California Los Angeles, Los Angeles, CA, USA, 1999. [Google Scholar] - Brown, K.W. Coherent Raman Spectroscopy of Non-Polar Molecules and Molecular Clusters. Ph.D. Thesis, Oregon State University, Corvallis, OR, USA, 1991. [Google Scholar]
- Mayer, S.G. Size Estimates of Molecular Clusters Using Elastic Light Scattering and CARS Spectroscopy. Ph.D. Thesis, Oregon State University, Corvallis, OR, USA, 1997. [Google Scholar]
- Erbland, P.J.; Rizzetta, D.P.; Miles, R.B. Numerical and Experimental Investigation of CO
_{2}Condensate Behavior in Hypersonic Flow. In Proceedings of the 21st AIAA Aerodynamic Measurement Technology and Ground Testing Conference, Denver, CO, USA, 19–22 June 2000. Paper No. 2000-2379. [Google Scholar] [CrossRef] - Duff, K.M. Non-Equilibrium Condensation of Carbon Dioxide in Supersonic Nozzles. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 1966. [Google Scholar]
- Ozawa, T.; Suzuki, T.; Fujita, K. Investigation of Condensation Effect in CO
_{2}Hypersonic Rarefied Flows. In Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, CA, USA, 4–8 January 2000. Paper No. 2016-1729. [Google Scholar] [CrossRef] - Kumar, R.; Li, Z.; Levin, D.A. Modeling of carbon dioxide condensation in the high pressure flows using the statistical BGK approach. Phys. Fluids
**2011**, 23, 052001. [Google Scholar] [CrossRef] - Horsch, M.; Lin, Z.; Windmann, T.; Hasse, H.; Vrabec, J. The air pressure effect on the homogeneous nucleation of carbon dioxide by molecular simulation. Atmos. Res.
**2011**, 101, 519–526. [Google Scholar] [CrossRef] [Green Version] - Horsch, M.; Vrabec, J.; Bernreuther, M.; Grottel, S.; Reina, G.; Wix, A.; Schaber, K.; Hasse, H. Homogeneous nucleation in supersaturated vapors of methane, ethane, and carbon dioxide predicted by brute force molecular dynamics. J. Chem. Phys.
**2008**, 128, 164510. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; Zhong, J.; Levin, D.A. Modeling of CO
_{2}Homogeneous and Heterogeneous Condensation Plumes. J. Phys. Chem. C**2010**, 114, 5276–5286. [Google Scholar] [CrossRef] - Hale, B.N. Temperature dependence of homogeneous nucleation rates for water: Near equivalence of the empirical fit of Wölk and Strey, and the scaled nucleation model. J. Chem. Phys.
**2005**, 122, 204509. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tanaka, K.K.; Kawamura, K.; Tanaka, H.; Nakazawa, K. Tests of the homogeneous nucleation theory with molecular-dynamics simulations. I. Lennard-Jones molecules. J. Chem. Phys.
**2005**, 122, 184514. [Google Scholar] [CrossRef] [PubMed] - Merikanto, J.; Zapadinsky, E.; Lauri, A.; Vehkamäki, H. Origin of the Failure of Classical Nucleation Theory: Incorrect Description of the Smallest Clusters. Phys. Rev. Lett.
**2007**, 98, 145702. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Holten, V.; Labetski, D.G.; van Dongen, M.E.H. Homogeneous nucleation of water between 200 and 240 K: New wave tube data and estimation of the Tolman length. J. Chem. Phys.
**2005**, 123, 104505. [Google Scholar] [CrossRef] [Green Version] - Tadayon, P. Determination of Interfacial Tension from Optical Measurements of Nucleation Rates. Ph.D. Thesis, Oregon State University, Corvallis, OR, USA, 1998. [Google Scholar]
- Desgranges, C.; Delhommelle, J. Free energy calculations along entropic pathways. I. Homogeneous vapor-liquid nucleation for atomic and molecular systems. J. Chem. Phys.
**2016**, 145, 204112. [Google Scholar] [CrossRef] - Kido, A.; Nakanishi, K. Molecular dynamics study of nucleation in supersaturated vapor of carbon dioxide. Fluid Phase Equilibria
**1999**, 158–160, 79–86. [Google Scholar] [CrossRef] - Halonen, R.; Tikkanen, V.; Reischl, B.; Dingilian, K.K.; Wyslouzil, B.E.; Vehkamäki, H. Homogeneous nucleation of carbon dioxide in supersonic nozzles II: Molecular dynamics simulations and properties of nucleating clusters. Phys. Chem. Chem. Phys.
**2021**, 23, 4517–4529. [Google Scholar] [CrossRef] - Iland, K. Experimente zur homogenen Keimbildung von Argon und Stickstoff. Ph.D. Thesis, Universität zu Köln, Köln, Germany, 2004. [Google Scholar]
- Jortner, J. Cluster size effects. Z. Phys. Atoms Mol. Clust.
**1992**, 24, 247–275. [Google Scholar] [CrossRef] - Hoare, M.; Pal, P. Physical cluster mechanics: Statistical thermodynamics and nucleation theory for monatomic systems. Adv. Phys.
**1975**, 24, 645–678. [Google Scholar] [CrossRef] - Hoare, M.; Pal, P. Physical cluster mechanics: Statics and energy surfaces for monatomic systems. Adv. Phys.
**1971**, 20, 161–196. [Google Scholar] [CrossRef] - Torchet, G.; de Feraudy, M.; Boutin, A.; Fuchs, A.H. Structural transformation in (
_{CO2})_{N}clusters, N<100. J. Chem. Phys.**1996**, 105, 3671–3678. [Google Scholar] [CrossRef] - Maillet, J.B.; Boutin, A.; Fuchs, A.H. From molecular clusters to bulk matter. II. Crossover from icosahedral to crystalline structures in CO
_{2}clusters. J. Chem. Phys.**1999**, 111, 2095–2102. [Google Scholar] [CrossRef] - Castleman, A.W.; Keesee, R.G. Clusters: Properties and Formation. Annu. Rev. Phys. Chem.
**1986**, 37, 525–550. [Google Scholar] [CrossRef] - Maillet, J.B.; Boutin, A.; Fuchs, A.H. The Melting Phase Transition in Small Carbon Dioxide Clusters. Mol. Simul.
**1997**, 19, 285–299. [Google Scholar] [CrossRef] - Rytkönen, A.; Valkealahti, S.; Manninen, M. Melting and evaporation of argon clusters. J. Chem. Phys.
**1997**, 106, 1888–1892. [Google Scholar] [CrossRef] - Doye, J.P.K.; Calvo, F. Entropic effects on the structure of Lennard-Jones clusters. J. Chem. Phys.
**2002**, 116, 8307–8317. [Google Scholar] [CrossRef] [Green Version] - Maillet, J.B.; Boutin, A.; Buttefey, S.; Calvo, F.; Fuchs, A.H. From molecular clusters to bulk matter. I. Structure and thermodynamics of small CO
_{2}, N_{2}, and SF_{6}clusters. J. Chem. Phys.**1998**, 109, 329–337. [Google Scholar] [CrossRef] - Acevedo, A.J.; Caballero, L.M.; López, G.E. Phase transitions in molecular clusters. J. Chem. Phys.
**1997**, 106, 7257–7261. [Google Scholar] [CrossRef] - Ford, I.J. Nucleation theorems, the statistical mechanics of molecular clusters, and a revision of classical nucleation theory. Phys. Rev. E
**1997**, 56, 5615–5629. [Google Scholar] [CrossRef] [Green Version] - Ellerby, H.M. Distribution of density fluctuations in a molecular theory of vapor-phase nucleation. Phys. Rev. E
**1994**, 49, 4287–4297. [Google Scholar] [CrossRef] - Vehkamäki, H. Classical Nucleation Theory in Multicomponent Systems; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar] [CrossRef]
- Kalikmanov, V.I. Nucleation Theory; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2013; Volume 860. [Google Scholar] [CrossRef]
- Abraham, F.F. Homogeneous Nucleation Theory: The Pretransition Theory of Vapor Condensation; Academic Press: Cambridge, MA, USA, 1974. [Google Scholar] [CrossRef] [Green Version]
- Friedlander, S.K. The Mechanics of Aerosols, 2nd ed.; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- Zel’dovich, J.B. K Teorii Obrazovaniya Novoy Fazy. Kavitatsiya. Zhurnal Eksperimental’noy Teor. Fiziki
**1942**, 12, 525–538. [Google Scholar] - Lothe, J.; Pound, G.M. Statistical Mechanics of Nucleation. In Nucleation; Zettlemoyer, A.C., Ed.; Marcel Dekker: New York, NY, USA, 1969; Chapter 3; pp. 109–149. [Google Scholar]
- Wakeshima, H. Time Lag in the Self-Nucleation. J. Chem. Phys.
**1954**, 22, 1614–1615. [Google Scholar] [CrossRef] - Wu, D.T. The time lag in nucleation theory. J. Chem. Phys.
**1992**, 97, 2644–2650. [Google Scholar] [CrossRef] - Wedekind, J.; Hyvärinen, A.P.; Brus, D.; Reguera, D. Unraveling the “Pressure Effect” in Nucleation. Phys. Rev. Lett.
**2008**, 101, 125703. [Google Scholar] [CrossRef] [Green Version] - Tolman, R.C. The Effect of Droplet Size on Surface Tension. J. Chem. Phys.
**1949**, 17, 333–337. [Google Scholar] [CrossRef] [Green Version] - Rowlinson, J.S.; Widom, B. Molecular Theory of Capillarity, 1st ed.; International Series of Monographs on Chemistry; Oxford University Press: New York, NY, USA, 1982; Volume 8. [Google Scholar]
- Gránásy, L. Semiempirical van der Waals/Cahn-Hilliard theory: Size dependence of the Tolman length. J. Chem. Phys.
**1998**, 109, 9660–9663. [Google Scholar] [CrossRef] - Koga, K.; Zeng, X.C. Validity of Tolman’s equation: How large should a droplet be? J. Chem. Phys.
**1998**, 109, 4063–4070. [Google Scholar] [CrossRef] [Green Version] - Wang, X.S.; Zhu, R.Z. Relation between Tolman length and isothermal compressibility for simple liquids. Chin. Phys. B
**2013**, 22, 036801. [Google Scholar] [CrossRef] [Green Version] - Horsch, M.; Vrabec, J.; Hasse, H. Modification of the classical nucleation theory based on molecular simulation data for surface tension, critical nucleus size, and nucleation rate. Phys. Rev. E
**2008**, 78, 011603. [Google Scholar] [CrossRef] [PubMed] - Girshick, S.L.; Chiu, C.P. Kinetic nucleation theory: A new expression for the rate of homogeneous nucleation from an ideal supersaturated vapor. J. Chem. Phys.
**1990**, 93, 1273–1277. [Google Scholar] [CrossRef] - Wilemski, G. The Kelvin equation and self-consistent nucleation theory. J. Chem. Phys.
**1995**, 103, 1119–1126. [Google Scholar] [CrossRef] - Courtney, W.G. Remarks on Homogeneous Nucleation. J. Chem. Phys.
**1942**, 35, 2249–2250. [Google Scholar] [CrossRef] - Reguera, D.; Bowles, R.K.; Djikaev, Y.; Reiss, H. Phase transitions in systems small enough to be clusters. J. Chem. Phys.
**2003**, 118, 164720. [Google Scholar] [CrossRef] - Kalikmanov, V.I. Mean-field kinetic nucleation theory. J. Chem. Phys.
**2006**, 124, 124505. [Google Scholar] [CrossRef] [PubMed] - Fisher, M.E. The theory of condensation and the critical point. Physics
**1967**, 3, 255–283. [Google Scholar] [CrossRef] [Green Version] - Sinha, S.; Bhabhe, A.; Laksmono, H.; Wölk, J.; Strey, R.; Wyslouzil, B. Argon nucleation in a cryogenic supersonic nozzle. J. Chem. Phys.
**2010**, 132, 064304. [Google Scholar] [CrossRef] - Laaksonen, A.; Ford, I.J.; Kulmala, M. Revised parametrization of the Dillmann-Meier theory of homogeneous nucleation. Phys. Rev. E
**1994**, 49, 5517–5524. [Google Scholar] [CrossRef] - Delale, C.F.; Meier, G.E.A. A semiphenomenological droplet model of homogeneous nucleation from the vapor phase. J. Chem. Phys.
**1993**, 98, 9850–9858. [Google Scholar] [CrossRef] - Dillmann, A.; Meier, G.E.A. A refined droplet approach to the problem of homogeneous nucleation from the vapor phase. J. Chem. Phys.
**1991**, 94, 3872–3884. [Google Scholar] [CrossRef] - Reguera, D.; Reiss, H. Fusion of the Extended Modified Liquid Drop Model for Nucleation and Dynamical Nucleation Theory. Phys. Rev. Lett.
**2004**, 93, 165701. [Google Scholar] [CrossRef] [Green Version] - Reguera, D.; Reiss, H. Extended Modified Liquid Drop-Dynamical Nucleation Theory (EMLD-DNT) Approach to Nucleation: A New Theory. J. Phys. Chem. B
**2004**, 108, 19831–19842. [Google Scholar] [CrossRef] - Baidakov, V.G.; Boltachev, G.S. Extended version of the van der Waals capillarity theory. J. Chem. Phys.
**2004**, 121, 8594–8601. [Google Scholar] [CrossRef] - Carnahan, N.F.; Starling, K.E. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys.
**1969**, 51, 635–636. [Google Scholar] [CrossRef] - Hale, B.N. Application of a scaled homogeneous nucleation-rate formalism to experimental data at T≪T
_{c}. Phys. Rev. A**1986**, 33, 4156–4163. [Google Scholar] [CrossRef] [PubMed] - Hale, B.N. The Scaling of Nucleation Rates. Metall. Trans. A
**1992**, 23, 1863–1868. [Google Scholar] [CrossRef] - Palit, S.R. Thermodynamic Interpretation of the Eötvös Constant. Nature
**1956**, 177, 1180. [Google Scholar] [CrossRef] - Wedekind, J.; Reguera, D.; Strey, R. Influence of thermostats and carrier gas on simulations of nucleation. J. Chem. Phys.
**2007**, 127, 064501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Feder, J.; Russel, K.C.; Lothe, J.; Pound, G.M. Homogeneous nucleation and growth of droplets in vapours. Adv. Phys.
**1966**, 15, 111–178. [Google Scholar] [CrossRef] - Makarov, G.; Petin, A. Disintegration of Ar
_{N}, Kr_{N}, and (_{N2})_{N}Clusters during Collisions with Highly Vibrationally Excited SF_{6}Molecules. J. Exp. Theor. Phys.**2014**, 146, 398–405. [Google Scholar] [CrossRef] - Sharma, S.P.; Ruffin, S.M.; Gillespie, W.D.; Meyer, S.A. Vibrational relaxation measurements in an expanding flow using spontaneous Raman scattering. J. Thermophys. Heat Transf.
**1993**, 7, 697–703. [Google Scholar] [CrossRef] - Vincenti, W.G.; Kruger, C.H. Introduction to Physical Gas Dynamics; John Wiley and Sons: New York, NY, USA, 1967. [Google Scholar]
- Kobraei, H.R.; Anderson, B.R. Formation energies and concentrations of microclusters for homogeneous nucleation. J. Chem. Phys.
**1988**, 88, 4451–4459. [Google Scholar] [CrossRef] - Balla, R.J.; Everhart, J.L. Rayleigh Scattering Density Measurements, Cluster Theory, and Nucleation Calculations at Mach 10. Am. Inst. Aeronaut. Astronaut.
**2012**, 50, 698–707. [Google Scholar] [CrossRef] [Green Version] - Balla, R.J.; Rhode, M.N.; Everhart, J.L. Supersaturation Total Temperature, Pitot Pressure, and Rayleigh Scattering Measurements at Mach 10. Am. Inst. Aeronaut. Astronaut.
**2014**, 52, 1452–1465. [Google Scholar] [CrossRef] - Elliott, S.; Hasegawa, M.; Sakaue, H.; Leonov, S. Shock-dominated flow control by plasma array: Pressure analysis including pressure-sensitive paint visualization. Exp. Therm. Fluid Sci.
**2022**, 131, 110522. [Google Scholar] [CrossRef] - Houpt, A.; Hedlund, B.; Leonov, S.; Ombrello, T.; Carter, C. Quasi-DC electrical discharge characterization in a supersonic flow. Exp. Fluids
**2017**, 58, 1–17. [Google Scholar] [CrossRef] - Hill, P.G. Condensation of Water Vapour during Supersonic Expansion in Nozzles. J. Fluid Mech.
**1966**, 25, 593–620. [Google Scholar] [CrossRef] - Sivells, J.C. Aerodynamic design of axisymmetric hypersonic wind-tunnel nozzles. J. Spacecr. Rocket.
**1970**, 7, 1292–1299. [Google Scholar] [CrossRef] - Gyarmathy, G. The Spherical Droplet in Gaseous Carrier Streams: Review and Synthesis. In Multiphase Science and Technology; Hewitt, G.F., Delhaye, J.M., Zuber, N., Eds.; McGraw-Hill: New York, NY, USA, 1982; Volume 1, Chapter 2; pp. 99–279. [Google Scholar]
- Tanimura, S.; Park, Y.; Amaya, A.; Modak, V.; Wyslouzil, B.E. Following heterogeneous nucleation of CO
_{2}on H_{2}O ice nanoparticles with microsecond resolution. RSC Adv.**2015**, 5, 105537–105550. [Google Scholar] [CrossRef] - Ramos, A.; Fernández, J.M.; Tejeda, G.; Montero, S. Quantitative study of cluster growth in free-jet expansions of CO
_{2}by Rayleigh and Raman scattering. Phys. Rev. A**2005**, 72, 053204. [Google Scholar] [CrossRef] [Green Version] - Lippe, M.; Szczepaniak, U.; Hou, G.L.; Chakrabarty, S.; Ferreiro, J.J.; Chasovskikh, E.; Signorell, R. Infrared Spectroscopy and Mass Spectrometry of CO
_{2}Clusters during Nucleation and Growth. J. Phys. Chem. A**2019**, 123, 2426–2437. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Krohn, J.; Lippe, M.; Li, C.; Signorell, R. Carbon dioxide and propane nucleation: The emergence of a nucleation barrier. Phys. Chem. Chem. Phys.
**2020**, 22, 15986–15998. [Google Scholar] [CrossRef] [PubMed] - Liu, X.Y. Heterogeneous nucleation or homogeneous nucleation? J. Chem. Phys.
**2000**, 112, 9949–9955. [Google Scholar] [CrossRef] - Anderson, J.D. Modern Compressible Flow: With Historical Perspective, 3rd ed.; McGraw-Hill: New York, NY, USA, 2003. [Google Scholar]
- Clifford, A.A.; Gray, P.; Platts, N. Lennard-Jones 12:6 parameters for ten small molecules. J. Chem. Soc. Faraday Trans.
**1977**, 73, 381–382. [Google Scholar] [CrossRef] - Span, R.; Wagner, W. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple- Point Temperature to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. Data
**1996**, 25, 1509–1596. [Google Scholar] [CrossRef] [Green Version] - Herzberg, G. Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules; D. Van Nostrand Company: New York, NY, USA, 1945. [Google Scholar]
- Irikura, K.K. Experimental Vibrational Zero-Point Energies: Diatomic Molecules. J. Phys. Chem. Ref. Data
**2007**, 36, 389–397. [Google Scholar] [CrossRef] [Green Version] - Jasper, J.J. The Surface Tension of Pure Liquid Compounds. J. Phys. Chem. Ref. Data
**1972**, 1, 841–1010. [Google Scholar] [CrossRef] [Green Version] - Poling, B.E. Properties of Gases and Liquids, 5th ed.; McGraw-Hill Education: New York, NY, USA, 2001. [Google Scholar]
- Magee, J.W. Specific heats (C
_{v}) of saturated and compressed liquid and vapor carbon dioxide. Int. J. Thermophys.**1986**, 7, 1163–1182. [Google Scholar] [CrossRef] - Acree, W.; Chickos, J.S. Phase Transition Enthalpy Measurements of Organic and Organometallic Compounds. Sublimation, Vaporization and Fusion Enthalpies from 1880 to 2015. Part 1. C
_{1}-C_{10}. J. Phys. Chem. Ref. Data**2016**, 45, 033101. [Google Scholar] [CrossRef] - Huber, M.L.; Sykioti, E.A.; Assael, M.J.; Perkins, R.A. Reference Correlation of the Thermal Conductivity of Carbon Dioxide from the Triple Point to 1100 K and up to 200 MPa. J. Phys. Chem. Ref. Data
**2016**, 45, 013102. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Representative curves for (

**a**) P and (

**b**) $\Delta {F}_{c,N}$ with $M=80$, both at $T/{T}_{c}=0.56$.

**Figure 5.**Model results for nucleation rate compared to experiment [10].

Facility Type | Carrier | CO${}_{2}$ [%] | T [K] | ${\mathit{p}}_{\mathit{v}}$ [Pa] | S | Ref. |
---|---|---|---|---|---|---|

de Laval Nozzle | Air | 1.2 | 88–97 | 16–32 | 606–2.9 × ${10}^{3}$ | Present |

Planar Nozzle | Ar | 2.0–39.3 | 75–92 | 39–793 | 2.3 × ${10}^{3}$–6.1 × ${10}^{4}$ | [10] |

Planar & de Laval | – | 100 | 161–193 | 1.3 × ${10}^{5}$–4.3 × ${10}^{5}$ | 2.7–7.8 | [15] |

Fixed Orifice Free Jet | He | 5–100 | 115–166 | 6.1 × ${10}^{3}$–2.4 × ${10}^{5}$ | 9.3–146 | [25] |

de Laval Nozzle | ${\mathrm{N}}_{2}$ | 2.4–25.2 | 124–146 | 240–3.5 × ${10}^{3}$ | 0.5–1.4 | [88] |

Fixed Orifice Free Jet | – | 100 | 75–106 | 301–1.2 × ${10}^{4}$ | 1.8 × ${10}^{3}$–2.2 × ${10}^{6}$ | [89] |

de Laval Nozzle | Ar + ${\mathrm{CH}}_{4}$ | 7 | 31–34 | 0.04–0.065 | 1.1 × ${10}^{23}$–3.3 × ${10}^{26}$ | [90] |

de Laval Nozzle | Ar + ${\mathrm{CH}}_{4}$ | 0.12–50 | 31–63 | 0.04–13 | 1.1 × ${10}^{8}$–1.8 × ${10}^{26}$ | [91] |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lax, P.A.; Leonov, S.B.
Review of Reduced-Order Models for Homogeneous CO_{2} Nucleation in Supersonic and Hypersonic Expansion Flows. *Aerospace* **2021**, *8*, 368.
https://doi.org/10.3390/aerospace8120368

**AMA Style**

Lax PA, Leonov SB.
Review of Reduced-Order Models for Homogeneous CO_{2} Nucleation in Supersonic and Hypersonic Expansion Flows. *Aerospace*. 2021; 8(12):368.
https://doi.org/10.3390/aerospace8120368

**Chicago/Turabian Style**

Lax, Philip A., and Sergey B. Leonov.
2021. "Review of Reduced-Order Models for Homogeneous CO_{2} Nucleation in Supersonic and Hypersonic Expansion Flows" *Aerospace* 8, no. 12: 368.
https://doi.org/10.3390/aerospace8120368