# A New Range Equation for Hybrid Aircraft Design

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## Abstract

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## 1. Introduction

- The power split is a third variable unlaced from the fuel and the battery mass, so that it is possible to choose a power management strategy and have a multiple-segments cruise, each with its power split;
- The equation can use weight fractions, which do not depend on the power split. Thus, the fuel and battery mass are defined a priori and do not depend on the power split, so even the aircraft’s total mass is constant;
- The equation uses the state of charge and the fuel mass instead of the energy level;
- The power split must be defined at the mechanical power level rather than at the energy source level. The reason is that the first value can be easily measured and used in the control system logic.

## 2. Analytical Model

#### 2.1. The “Virtual Aircraft” Method

#### 2.1.1. Virtual Thermal Aircraft (VTA)

#### 2.1.2. Virtual Electrical Aircraft (VEA)

#### 2.1.3. Overall Range

## 3. Reference Aircraft

_{2}emissions, as can be seen in the Section 4. The safety-oriented design philosophy, on the other hand, allows us to reach lower performances, but with a level of safety equal to that of a twin engine. Hybrid airplane mass fractions and reference parameters are reported in Table 2.

## 4. Results

#### 4.1. Range Equation from the Literature [15,17]

- The power split is constant, and the cruise ends when one of the energy sources does;
- The aircraft consumes all the energy sources, and, if one runs out before the other, the power split cannot be constant for all the cruise. (i.e., if the batteries are below then the $\Phi $ must go to 0);
- The energy sources run out exactly at the same time.

#### 4.2. Present New Range Equation

## 5. Validation with a Numerical Approach

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$PEMFC$ | Proton Exchange Membrane Fuel Cell |

${m}_{MTO}$ | maximum take off mass |

${m}_{OE}$ | empty operative mass |

${m}_{B}$ | battery pack mass |

${m}_{PL}$ | payload mass |

${\eta}_{p}$ | propeller efficiency |

${\eta}_{e}$ | electric motor efficiency |

${\eta}_{i}$ | inverter efficiency |

${\eta}_{th}$ | thermal motor efficiency |

${\eta}_{g}$ | generator efficiency |

$\chi $ | total power fraction |

$\varphi $ | power split |

E | aerodynamic efficiency |

${R}_{VEA}$ | Virtual Electric Aircraft Range |

${R}_{VTA}$ | Virtual Thermal Aircraft Range |

${R}_{HYB}$ | Hybrid Aircraft Range |

${e}_{F}$ | Fuel energy density |

${e}_{B}$ | Battery energy density |

$SO{C}_{i}$ | Battery initial cruise state of charge |

$SO{C}_{f}$ | Battery end of cruise state of charge |

${k}_{OE,B,F,PL,E}$ | mass fractions |

## References

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Parallel | Series | |
---|---|---|

${\eta}_{1}$ | ${\eta}_{th}$ | ${\eta}_{th}{\eta}_{g}$ |

${\eta}_{2}$ | ${\eta}_{i}{\eta}_{e}$ | ${\eta}_{i}$ |

${\eta}_{3}$ | ${\eta}_{p}$ | ${\eta}_{e}{\eta}_{p}$ |

**Table 2.**Hybrid airplane mass fractions and reference parameters used in the confrontation of the analytical equations found in the bibliography.

Fraction | Value | Ref. Par. | Value |
---|---|---|---|

${k}_{0}$ | $0.96$ | ${m}_{MTO}$ | 750 kg |

${k}_{B}$ | $0.06$ | ${e}_{F}$ | 43 MJ/kg |

${k}_{F,i}$ | $0.032$ | ${e}_{B}$ | 0.936 MJ/kg (260 Wh/kg) |

${k}_{F,f}$ | $0.0064$ | E | 13 |

${k}_{PL}$ | $0.248$ | ${\eta}_{1}$ | 0.29 |

$SO{C}_{i}$ | 1 | ${\eta}_{2}$ | 0.95 |

$SO{C}_{f}$ | $0.35$ | ${\eta}_{3}$ | 0.8 |

**Table 3.**Hybrid airplane mass fractions and reference parameters for comparison with numerical model.

Analytical Model Input | Numerical Model Input | ||
---|---|---|---|

${k}_{0}$ | $0.9304$ | ${m}_{MTO}$ | 747 kg |

${k}_{OE}$ | $0.6655$ | ${m}_{OE}$ | 497 kg |

${k}_{B}$ | $0.0642$ | ${m}_{B}$ | 48 kg |

${k}_{PL}$ | $0.2007$ | ${m}_{PL}$ | 170 kg |

${k}_{F,i}$ | $0.0043$ | ${m}_{F,i}$ | 32 kg |

${k}_{F,f}^{L}$ | $0.0020$ | ${m}_{F,f}^{L}$ | 15 kg |

$SO{C}_{i}$ | 1 | $SO{C}_{i}$ | 1 |

$SO{C}_{f}^{L}$ | $0.35$ | $SO{C}_{f}^{L}$ | $0.35$ |

${C}_{L}/{C}_{D}$ | $13.7$ | Aircraft polar curve | |

${\eta}_{1}$ | $0.29$ | ICE map | |

${\eta}_{2}$ | $0.87$ | EM and inverter map | |

${\eta}_{3}$ | $0.81$ | Propeller map |

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**MDPI and ACS Style**

Cestino, E.; Pisu, D.; Sapienza, V.; Chesta, L.; Martilla, V.
A New Range Equation for Hybrid Aircraft Design. *Aerospace* **2023**, *10*, 955.
https://doi.org/10.3390/aerospace10110955

**AMA Style**

Cestino E, Pisu D, Sapienza V, Chesta L, Martilla V.
A New Range Equation for Hybrid Aircraft Design. *Aerospace*. 2023; 10(11):955.
https://doi.org/10.3390/aerospace10110955

**Chicago/Turabian Style**

Cestino, Enrico, Davide Pisu, Vito Sapienza, Lorenzo Chesta, and Valentina Martilla.
2023. "A New Range Equation for Hybrid Aircraft Design" *Aerospace* 10, no. 11: 955.
https://doi.org/10.3390/aerospace10110955