# 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

- Kamps, L.; Sakurai, K.; Saito, Y.; Nagata, H. Comprehensive Data Reduction for N2O/HDPE Hybrid Rocket Motor Performance Evaluation. Aerospace
**2019**, 6, 45. [Google Scholar] [CrossRef][Green Version] - Mechentel, F.S.; Hord, B.R.; Cantwell, B.J. Optically Resolved Fuel Regression of a Clear Polymethylmethacrylate Hybrid Rocket Motor. J. Propuls. Power
**2020**, 36, 763–772. [Google Scholar] [CrossRef] - Paccagnella, E.; Barato, F.; Pavarin, D.; Karabeyoğlu, A. Scaling Parameters of Swirling Oxidizer Injection in Hybrid Rocket Motors. J. Propuls. Power
**2017**, 33, 1378–1394. [Google Scholar] [CrossRef] - Migliorino, M.T.; Bianchi, D.; Nasuti, F. Numerical Analysis of Paraffin-Wax/Oxygen Hybrid Rocket Engines. J. Propuls. Power
**2020**, 36, 806–819. [Google Scholar] [CrossRef] - Kobald, M.; Fischer, U.; Tomilin, K.; Petrarolo, A.; Schmierer, C. Hybrid Experimental Rocket Stuttgart: A Low-Cost Technology Demonstrator. J. Spacecr. Rocket.
**2018**, 55, 484–500. [Google Scholar] [CrossRef] - Leccese, G.; Cavallini, E.; Pizzarelli, M. State of Art and Current Challenges of the Paraffin-Based Hybrid Rocket Technology. In Proceedings of the AIAA Propulsion and Energy 2019 Forum, Indianapolis, IN, USA, 19–22 August 2019. [Google Scholar] [CrossRef]
- Chiaverini, M.J.; Serin, N.; Johnson, D.K.; Lu, Y.C.; Kuo, K.K.; Risha, G.A. Regression Rate Behavior of Hybrid Rocket Solid Fuels. J. Propuls. Power
**2000**, 16, 125–132. [Google Scholar] [CrossRef] - Altman, D.; Holzman, A. Overview and History of Hybrid Rocket Propulsion. In Fundamentals of Hybrid Rocket Combustion and Propulsion; Kuo, K.K., Chiaverini, M.J., Eds.; AIAA: Reston, VA, USA, 2007; Volume 218, pp. 1–36. [Google Scholar] [CrossRef]
- Bellomo, N.; Barato, F.; Faenza, M.; Lazzarin, M.; Bettella, A.; Pavarin, D. Numerical and Experimental Investigation of Unidirectional Vortex Injection in Hybrid Rocket Engines. J. Propuls. Power
**2013**, 29, 1097–1113. [Google Scholar] [CrossRef] - Ranuzzi, G.; Cardillo, D.; Invigorito, M. Numerical Investigation of a N
_{2}O–Paraffin Hybrid Rocket Engine Combusting Flowfield. In Proceedings of the 6th European Conference for Aeronautics and Space Sciences (EUCASS), Kraków, Poland, 29 June–2 July 2015. [Google Scholar] [CrossRef] - Lazzarin, M.; Faenza, M.; Barato, F.; Bellomo, N.; Bettella, A.; Pavarin, D. Computational Fluid Dynamics Simulation of Hybrid Rockets of Different Scales. J. Propuls. Power
**2015**, 31, 1458–1469. [Google Scholar] [CrossRef] - Paccagnella, E.; Barato, F.; Gelain, R.; Pavarin, D. CFD Simulations of Self-Pressurized Nitrous Oxide Hybrid Rocket Motors. In Proceedings of the Joint Propulsion Conference, Cincinnati, OH, USA, 9–11 July 2018. [Google Scholar] [CrossRef][Green Version]
- Di Martino, G.D.; Mungiguerra, S.; Carmicino, C.; Savino, R.; Cardillo, D.; Battista, F.; Invigorito, M.; Elia, G. Two-Hundred–Newton Laboratory-Scale Hybrid Rocket Testing for Paraffin Fuel-Performance Characterization. J. Propuls. Power
**2019**, 35, 224–235. [Google Scholar] [CrossRef] - Bianchi, D.; Nasuti, F.; Delfini, D. Modeling of Gas-surface Interface for Paraffin-based Hybrid Rocket Fuels in Computational Fluid Dynamics Simulations. Prog. Propuls. Phys.
**2019**, 11, 3–24. [Google Scholar] [CrossRef][Green Version] - Di Martino, G.D.; Mungiguerra, S.; Carmicino, C.; Savino, R. Computational Fluid-dynamic Modeling of the Internal Ballistics of Paraffin-fueled Hybrid Rocket. Aerosp. Sci. Technol.
**2019**, 89, 431–444. [Google Scholar] [CrossRef] - Chiaverini, M.J. Review of Solid–Fuel Regression Rate Behavior in Classical and Nonclassical Hybrid Rocket Motors. In Fundamentals of Hybrid Rocket Combustion and Propulsion; Kuo, K.K., Chiaverini, M.J., Eds.; AIAA: Reston, VA, USA, 2007; Volume 218, pp. 37–126. [Google Scholar] [CrossRef]
- Durand, J.E.; Raynaud, F.; Lamet, J.M.; Tessé, L.; Lestrade, J.Y.; Anthoine, J. Numerical Study of Fuel Regression in Hybrid Rocket Engine. In Proceedings of the Joint Propulsion Conference, Cincinnati, OH, USA, 9–11 July 2018. [Google Scholar] [CrossRef][Green Version]
- Whitmore, S.A.; Merkley, S. Radiation Heating Effects on Oxidizer-to-Fuel Ratio of Additively Manufactured Hybrid Rocket Fuels. J. Propuls. Power
**2019**, 35, 863–878. [Google Scholar] [CrossRef] - Bianchi, D.; Leccese, G.; Nasuti, F.; Onofri, M.; Carmicino, C. Modeling of High Density Polyethylene Regression Rate in the Simulation of Hybrid Rocket Flowfields. Aerospace
**2019**, 6, 88. [Google Scholar] [CrossRef][Green Version] - Anderson, J.D. Hypersonic and High-Temperature Gas Dynamics; AIAA Education Series; AIAA: Reston, VA, USA, 2006; pp. 596–617. [Google Scholar]
- Gordon, S.; McBride, B.J. Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications. I. Analysis; NASA RP-1311. 1994. Available online: https://ntrs.nasa.gov/citations/19950013764 (accessed on 1 August 2021).
- Spalart, P.R.; Allmaras, S.R. A One-Equation Turbulence Model for Aerodynamic Flows. Rech. Aerosp.
**1994**, 1, 5–21. [Google Scholar] - Fabuss, B.M.; Smith, J.O.; Lait, R.I.; Bornsanyi, A.S.; Satterfield, C.N. Rapid Thermal Cracking of n–Hexadecane at Elevated Pressures. Ind. Eng. Chem. Process. Des. Dev.
**1962**, 1, 293–299. [Google Scholar] [CrossRef] - Blouri, B.; Hamdan, F.; Herault, D. Mild Cracking of High–Molecular–Weigth Hydrocarbons. Ind. Eng. Chem. Process. Des. Dev.
**1985**, 1, 30–37. [Google Scholar] [CrossRef] - Coronetti, A.; Sirignano, W.A. Numerical Analysis of Hybrid Rocket Combustion. J. Propuls. Power
**2013**, 29, 371–384. [Google Scholar] [CrossRef][Green Version] - McBride, B.J.; Zehe, M.J.; Gordon, S. NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species; Technical Report NASA/TP–2002–211556. 2002. Available online: https://ntrs.nasa.gov/citations/20020085330 (accessed on 1 August 2021).
- Bianchi, D.; Betti, B.; Nasuti, F.; Carmicino, C. Simulation of Gaseous Oxygen/Hydroxyl-Terminated Polybutadiene Hybrid Rocket Flowfields and Comparison with Experiments. J. Propuls. Power
**2015**, 31, 919–929. [Google Scholar] [CrossRef] - Roe, P.L. Approximate Riemann Solvers, Parameter Vectors and Difference Schemes. J. Comput. Phys.
**1981**, 43, 357–372. [Google Scholar] [CrossRef] - Strang, G. On the Construction and Comparison of Difference Schemes. SIAM J. Numer. Anal.
**1968**, 5, 506–517. [Google Scholar] [CrossRef] - Brown, P.N.; Byrne, G.D.; Hindmarsh, A.C. VODE: A Variable-Coefficient ODE Solver. SIAM J. Sci. Stat. Comput.
**1989**, 10, 1038–1051. [Google Scholar] [CrossRef][Green Version] - Migliorino, M.T.; Bianchi, D.; Nasuti, F. Predictive CFD Model for Internal Ballistics of Hybrid Rocket Engines using Supercritical Paraffin-Wax and Oxygen. In Proceedings of the AIAA Propulsion and Energy Forum, Indianapolis, IN, USA, 19–22 August 2019. [Google Scholar] [CrossRef]
- Marano, J.J.; Holder, G.D. General Equation for Correlating the Thermophysical Properties of n–Paraffins, n–Olefins, and Other Homologous Series. 2. Asymptotic Behavior Correlations for PVT Properties. Ind. Eng. Chem. Res.
**1997**, 36, 1895–1907. [Google Scholar] [CrossRef] - Oschwald, M.; Schik, A. Supercritical Nitrogen Free Jet Investigated by Spontaneous Raman Scattering. Exp. Fluids
**1999**, 27, 497–506. [Google Scholar] [CrossRef] - Oschwald, M.; Micci, M. Spreading Angle and Centerline Variation of Density of Supercritical Nitrogen Jets. At. Sprays
**2002**, 12, 91–106. [Google Scholar] [CrossRef] - Marano, J.J.; Holder, G.D. General Equation for Correlating the Thermophysical Properties of n–Paraffins, n–Olefins, and Other Homologous Series. 3. Asymptotic Behavior Correlations for Thermal and Transport Properties. Ind. Eng. Chem. Res.
**1997**, 36, 2399–2408. [Google Scholar] [CrossRef] - Leccese, G.; Bianchi, D.; Betti, B.; Lentini, D.; Nasuti, F. Convective and Radiative Wall Heat Transfer in Liquid Rocket Thrust Chambers. J. Propuls. Power
**2018**, 34, 318–326. [Google Scholar] [CrossRef] - Leccese, G.; Bianchi, D.; Nasuti, F. Numerical Investigation on Radiative Heat Loads in Liquid Rocket Thrust Chambers. J. Propuls. Power
**2019**, 35, 930–943. [Google Scholar] [CrossRef] - Leccese, G.; Bianchi, D.; Nasuti, F.; Stober, K.J.; Narsai, P.; Cantwell, B.J. Experimental and Numerical Methods for Radiative Wall Heat Flux Predictions in Paraffin–based Hybrid Rocket Engines. Acta Astronaut.
**2019**, 158, 304–312. [Google Scholar] [CrossRef] - Astarita, T.; Carlomagno, G.M. Infrared Thermography for Thermo–Fluid–Dynamics; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar] [CrossRef]
- Mark, F.H.; Kroschwitz, J.I. Encyclopedia of Polymer Science and Engineering; Wiley: New York, NY, USA, 1989; Volume 17. [Google Scholar] [CrossRef]
- Bianchi, D.; Nasuti, F.; Carmicino, C. Hybrid Rockets with Axial Injector: Port Diameter Effect on Fuel Regression Rate. J. Propuls. Power
**2016**, 32, 984–996. [Google Scholar] [CrossRef] - Karabeyoglu, M.A.; Zilliac, G.; Cantwell, B.J.; DeZilwa, S.; Castellucci, P. Scale–Up Tests of High Regression Rate Paraffin–Based Hybrid Rocket Fuels. J. Propuls. Power
**2004**, 20, 1037–1045. [Google Scholar] [CrossRef]

**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