# A Eulerian Multi-Fluid Model for High-Speed Evaporating Sprays

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. The Phase-Intensive Momentum Equation

#### 2.2. Phase Continuity Equation

#### 2.3. Breakup Model

#### 2.4. Turbulence Model

#### 2.5. Species Transfer

#### 2.6. Energy Equation

#### 2.7. Evaporation Model

## 3. Numerical Approach

Algorithm 1 Employed solution procedure for each time step. |

Evaluate the breakup and evaporation model. |

Solve the phase continuity equations. |

Evaluate the momentum transfer models. |

Construct the phase momentum equations and predict fluxes. |

Solve the energy equations. |

Solve the mixture pressure equation. |

Correct fluxes and reconstruct the velocity fields. |

Solve the turbulence model. |

Solve the species transfer equation. |

## 4. Results and Discussion

#### 4.1. Uncertainty Analysis

#### 4.2. Validation

#### 4.2.1. Non-Evaporating Conditions

#### 4.2.2. Evaporating Conditions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Employed triangular distribution function for child droplets. The given example shows the triangular distribution for ten (uniform) child classes, when a = 0, b = 50, and c = 7 $\mathsf{\mu}$m.

**Figure 2.**Employed rate of fuel injection [84].

**Figure 6.**Liquid spray at t = 1.4 ms. The green line gives the iso-contour ${\alpha}_{\mathrm{d}}$ = 0.1%.

**Figure 8.**Vapor penetration at t = 1.4 ms. The green line gives the iso-contour ${\alpha}_{\mathrm{d}}$ = 0.1%, and the orange line denotes ${Y}_{1}$ = 0.1%.

**Table 1.**Input data for the uncertainty estimation analysis. The values give the liquid spray penetration length (in millimeters) at 0.2 ms (after start of injection) for various grids and time steps.

Number of Cells | Time Step Size [s] | ||
---|---|---|---|

$\mathbf{1}\times {\mathbf{10}}^{-\mathbf{8}}$ | $\mathbf{2}\times {\mathbf{10}}^{-\mathbf{8}}$ | $\mathbf{3}\times {\mathbf{10}}^{-\mathbf{8}}$ | |

2080 | - | 11.2 | 13.6 |

2340 | - | 10.4 | 14.8 |

2600 | 9.8 | 11.1 | - |

3068 | 10.2 | - | - |

Item | ${\mathit{\varphi}}_{\mathbf{0}}$ | ${\mathit{\varphi}}_{\mathbf{1}}$ | ${\mathit{U}}_{\mathit{\varphi}}$ | p | q |
---|---|---|---|---|---|

Liquid penetration | $9.02$ | $10.2$ | $35.9\%$ | $2.00$ | $1.00$ |

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**MDPI and ACS Style**

Keser, R.; Battistoni, M.; Im, H.G.; Jasak, H.
A Eulerian Multi-Fluid Model for High-Speed Evaporating Sprays. *Processes* **2021**, *9*, 941.
https://doi.org/10.3390/pr9060941

**AMA Style**

Keser R, Battistoni M, Im HG, Jasak H.
A Eulerian Multi-Fluid Model for High-Speed Evaporating Sprays. *Processes*. 2021; 9(6):941.
https://doi.org/10.3390/pr9060941

**Chicago/Turabian Style**

Keser, Robert, Michele Battistoni, Hong G. Im, and Hrvoje Jasak.
2021. "A Eulerian Multi-Fluid Model for High-Speed Evaporating Sprays" *Processes* 9, no. 6: 941.
https://doi.org/10.3390/pr9060941