#
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

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**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] |

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**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