# Numerical Simulations of the Internal Ballistics of Paraffin–Oxygen Hybrid Rockets at Different Scales

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical and Numerical Model

_{32}H

_{66}was taken as the one for the liquid C

_{16}H

_{34}[24] due to the lack of data in the literature. More details on the reaction mechanism can be found in [4].

_{32}H

_{66}. Differently from conventional fuels, when paraffin wax is heated, it does not pyrolyze but rather melts, producing a liquid layer over the fuel grain. However, since the typical chamber pressure of hybrid rockets is higher than paraffin’s critical pressure, equal to 6.5 bar [32], the melted paraffin is assumed to be at supercritical conditions, where no surface tension or boundary for droplets can be defined [33,34]. It is, therefore, reasonable to assume that in these conditions, the entrainment of the supercritical species is part of the turbulent mixing process. The melted species was modeled with the simplified dense fluid approach described in [4,31], with thermodynamic and transport properties taken from [35]. As described in [4], the fluid–surface interaction sub-model is based on mass and energy balances, which reduce to

_{2}O, CO

_{2}, and CO, which are considered as the major and only participating species to radiation, weighted with their molar fraction. A discretization consisting of 256 rays for each calculation point and a step of 1 mm along each ray were used after performing convergence analyses for both parameters. A wall emissivity equal to $0.91$ was assumed for the paraffin wax grain by using the emissivity model proposed in [39] with a refractive index of $1.43$ according to [40]. The CFD and radiation codes were coupled through the repeated evaluation of the radiative wall heat flux, then of the regression rate, and finally of the resulting flow field, until convergence was reached.

## 3. Engine Configuration and Firing Tests

## 4. Results

#### 4.1. Results on Set 1 Tests

#### Model Sensitivity Analysis

#### 4.2. Results on Set 2 Tests and Effect of Radiation

#### 4.3. Numerical Rebuilding

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Grid clustering near the injector for the mesh used for numerical simulations of set 1 (see Table 2).

**Figure 3.**Temperature (with streamtraces), paraffin mass fraction, and oxygen mass fraction contours for test 4 of set 1 at different port diameters.

**Figure 4.**Mass fraction contours of water vapor, carbon monoxide, and carbon dioxide for test 4 of set 1 at different port diameters.

**Figure 11.**Temperature (with streamtraces) and species mass fractions from numerical simulations of test L01 of set 2 (see Table 2).

**Figure 12.**Numerical rebuilding of average regression rate (

**a**) and post-chamber pressure (

**b**). Experimental uncertainties for the regression rate of set 1 and numerical uncertainties of simulations of set 2 are also reported.

**Figure 15.**Time evolution of regression rate and pressure for test 3 of set 1. Dashed lines indicate experimental data taken from [13], while numerical data are shown with circles connected by solid lines. The horizontal solid line represents the integral average of the numerical regression rate data.

**Table 1.**Chemical reactions involved in the global reaction mechanism used for paraffin–oxygen combustion.

C_{32} H_{66} → 16 C_{2} H_{4} + H_{2} |

C_{2} H_{4} + O_{2} → 2 CO + 2 H_{2} |

C_{2} H_{4} + 2 H_{2}O → 2 CO + 4 H_{2} |

CO + H_{2}O ⇄ CO_{2} + H_{2} |

${\mathrm{H}}_{2}+\frac{1}{2}{\mathrm{O}}_{2}\rightleftarrows {\mathrm{H}}_{2}\mathrm{O}$ |

O_{2} ⇄ 2O |

H_{2}O ⇄ OH + H |

Set | Test | ${\dot{\mathit{m}}}_{\mathbf{ox}}$ (g/s) | R (mm) | ${\mathit{r}}_{\mathbf{t}}$ (mm) |
---|---|---|---|---|

1 | 3 | 29.0 | 11.80 | 5.3 |

1 | 4 | 39.0 | 12.65 | 5.3 |

1 | 8 | 44.0 | 14.25 | 5.3 |

1 | 9 | 50.2 | 14.50 | 5.3 |

1 | 10 | 55.5 | 14.50 | 5.3 |

1 | 11 | 60.0 | 14.95 | 5.3 |

1 | 12 | 59.5 | 14.00 | 5.3 |

2 | L01 | 4400 | 71.96 | 25.15 |

2 | P01 | 4430 | 71.73 | 35.55 |

2 | L04 | 4440 | 61.97 | 25.15 |

2 | P04 | 2110 | 67.67 | 35.8 |

2 | L09 | 2050 | 58.28 | 27.8 |

**Table 3.**Wall heat flux contributions on the grain surface for all tests at the average port diameter.

Set | Test | ${\mathit{p}}_{\mathbf{c}}\mathit{R}$ (bar·m) | ${\overline{\mathit{q}}}_{\mathbf{w},\mathbf{tot}}$ (MW/m${}^{2}$) | ${\overline{\mathit{q}}}_{\mathbf{w},\mathbf{rad}}/{\overline{\mathit{q}}}_{\mathbf{w},\mathbf{tot}}$ |
---|---|---|---|---|

1 | 3 | 0.10 | 0.38 | 35% |

1 | 4 | 0.15 | 0.45 | 43% |

1 | 8 | 0.19 | 0.44 | 55% |

1 | 9 | 0.23 | 0.48 | 58% |

1 | 10 | 0.25 | 0.52 | 58% |

1 | 11 | 0.28 | 0.54 | 62% |

1 | 12 | 0.26 | 0.58 | 56% |

2 | L01 | 3.33 | 0.775 | 88% |

2 | P01 | 1.57 | 0.988 | 84% |

2 | L04 | 2.81 | 0.940 | 71% |

2 | P04 | 0.74 | 0.562 | 81% |

2 | L09 | 1.06 | 0.935 | 83% |

**Table 4.**Experimental and numerical oxidizer-to-fuel ratio, chamber pressure, characteristic velocity, and combustion efficiency for all tests in Table 2.

Experimental | Numerical | ||||||||
---|---|---|---|---|---|---|---|---|---|

Set | Test | O/F | ${\mathit{p}}_{\mathbf{c}}$ (bar) | ${\mathit{c}}^{*}$ (m/s) | ${\mathit{\eta}}_{{\mathit{c}}^{*}}$ (%) | O/F | ${\mathit{p}}_{\mathbf{c}}$ (bar) | ${\mathit{c}}^{*}$ (m/s) | ${\mathit{\eta}}_{{\mathit{c}}^{*}}$ (%) |

1 | 3 | 1.0 | 8.5 | 1319 | 85 | 1.2 | 8.3 | 1388 | 85 |

1 | 4 | 1.2 | 11.5 | 1403 | 87 | 1.3 | 11.4 | 1450 | 86 |

1 | 8 | 1.2 | 13.2 | 1422 | 89 | 1.3 | 13.1 | 1499 | 88 |

1 | 9 | 1.2 | 15.7 | 1505 | 93 | 1.4 | 15.0 | 1515 | 88 |

1 | 10 | 1.3 | 16.9 | 1498 | 90 | 1.4 | 16.6 | 1521 | 88 |

1 | 11 | 1.2 | 18.8 | 1514 | 93 | 1.4 | 18.0 | 1529 | 88 |

1 | 12 | 1.2 | 18.4 | 1483 | 92 | 1.4 | 17.6 | 1512 | 87 |

2 | L01 | 2.6 | 46.3 | 1627 | 88 | 2.6 | 39.0 | 1277 | 69 |

2 | P01 | 2.7 | 21.9 | 1492 | 82 | 2.4 | 19.4 | 1227 | 67 |

2 | L04 | 2.7 | 45.3 | 1564 | 85 | 2.9 | 39.3 | 1302 | 72 |

2 | P04 | 1.8 | 10.9 | 1437 | 78 | 2.0 | 9.6 | 1222 | 66 |

2 | L09 | 1.7 | 18.2 | 1468 | 80 | 1.5 | 18.7 | 1340 | 75 |

**Table 5.**Wall heat flux contributions on the grain surface for test 3 of set 1 at different port diameters.

Set | Test | D, mm | ${\mathit{G}}_{\mathbf{ox}}$, kg/(m${}^{2}$s) | ${\mathit{p}}_{\mathbf{c}}\mathit{R}$, bar·m | ${\overline{\mathit{q}}}_{\mathbf{w},\mathbf{tot}}$, MW/m${}^{2}$ | ${\overline{\mathit{q}}}_{\mathbf{w},\mathbf{rad}}/{\overline{\mathit{q}}}_{\mathbf{w},\mathbf{tot}}$ |
---|---|---|---|---|---|---|

1 | 3, ${D}_{0}$ | 15 | 164.1 | 0.06 | 0.80 | 10% |

1 | 3, ${D}_{\mathrm{ave}}$ | 23.6 | 66.3 | 0.10 | 0.38 | 35% |

1 | 3, ${D}_{\mathrm{final}}$ | 32.2 | 35.6 | 0.14 | 0.27 | 61% |

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**MDPI and ACS Style**

Migliorino, M.T.; Bianchi, D.; Nasuti, F.
Numerical Simulations of the Internal Ballistics of Paraffin–Oxygen Hybrid Rockets at Different Scales. *Aerospace* **2021**, *8*, 213.
https://doi.org/10.3390/aerospace8080213

**AMA Style**

Migliorino MT, Bianchi D, Nasuti F.
Numerical Simulations of the Internal Ballistics of Paraffin–Oxygen Hybrid Rockets at Different Scales. *Aerospace*. 2021; 8(8):213.
https://doi.org/10.3390/aerospace8080213

**Chicago/Turabian Style**

Migliorino, Mario Tindaro, Daniele Bianchi, and Francesco Nasuti.
2021. "Numerical Simulations of the Internal Ballistics of Paraffin–Oxygen Hybrid Rockets at Different Scales" *Aerospace* 8, no. 8: 213.
https://doi.org/10.3390/aerospace8080213