An Endurance Equation for Hybrid-Electric Aircraft
Abstract
:1. Introduction
2. Hybrid-Electric Aircraft Configuration
3. Derivation of Hybrid-Electric Endurance Equation
- For the series hybrid-electric aircraft:
- For the parallel hybrid-electric aircraft:
- Limiting cases:
- At φ = 1, the derived endurance equation (Equation (36)) recedes to the endurance equation [20] for the fully electric aircraft.
- Fully fueled case or (φ = 0):
- Fully electric case (or φ = 1):
4. Case Study
- It was observed that the parallel hybrid has marginally higher endurance as compared to the series hybrid. One reason can be that the series hybrid has a larger efficiency penalty due to the generator and also due to motor power conversion. On the other hand, parallel hybrid is connected to a motor powered by an energy source and, henceforth, is not affected by this efficiency penalty. It was also observed that the deviation in the configurations enhances with the increase in the degree of hybridization, as showcased in Table 3. Both cases converge if the efficiencies of both the electric motor and generator are considered to be one.
Parallel Hybrid | Series Hybrid | ||||
---|---|---|---|---|---|
Battery Energy Density | 500 Wh/kg | 1000 Wh/kg | 500 Wh/kg | 1000 Wh/kg | |
Degree of hybridization | 0.3 | 285.6 min | 385.8 min | 278.5 min | 372 min |
0.6 | 183.5 min | 294.4 min | 181.6 min | 287.4 min | |
0.9 | 124.2 min | 226.3 min | 130.3 min | 230.8 min |
- In the cases of parallel and series hybrid configurations, it is observed that there is an energy density threshold (EDT), below which the endurance reduces with an increase in degree of hybridization. Conversely, above EDT, endurance enhances with an increase in degree of hybridization. For any aircraft configuration, the EDT is not constant with degree of hybridization. For the parallel hybrid, the value ranges between 8960 < EDT < 9030 Wh/kg (in terms of the battery energy density), whereas in the series hybrid, it ranges between 8650 < EDT < 8950 Wh/kg (also in terms of the battery energy density). The higher value for the parallel hybrid is due to its higher dependence on a battery energy source. The exact values depend on the specific aircraft parameters given in Table 2. At the EDT, aircraft endurance is not dependent on degree of hybridization at all. This is probably because the effect of battery energy density and corresponding efficiencies becomes more prominent than the fuel energy density and corresponding efficiencies.
- The significance of the value of the EDT becomes apparent when considering the current battery energy storage technologies. The current energy density of batteries lies at approximately 228 Wh/kg. This is expected to double approximately every 23 years [9] and, thus, this point lies way beyond the theoretical limits of battery energy densities. However, the endurance theory would still hold if another energy source would be able to fill the gap.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
AoA | Angle of Attack |
cL | Coefficient of Lift |
cD | Coefficient of Drag |
D | Drag [N] |
e | Specific Energy [J/kg] |
E | Energy [J] |
EDT | Energy Density Threshold |
g | Gravitational acceleration [m/s2] |
L | Lift [N] |
P | Power [W] |
S | Planform area [m2] |
t | Time [s] |
T | Thrust [N] |
v | Velocity [m/s] |
W | Weight [N] |
γ | Flight path angle [rad] |
η | Efficiency |
φ | Degree of hybridization |
ρ | Density [kg/m3] |
Subscripts | |
0 | Start of the mission |
1,2,3 | Powertrain branch indices |
bat | Battery |
eg | Electrical generator |
em | Electrical motor |
end | End of mission segment |
f | Fuel |
gt | Gas turbine |
gb | Gearbox |
OE | Operating Empty |
p | Propeller |
PL | Payload |
start | Start of mission segment |
tot | Total |
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Simplified Representation (Figure 3) | Parallel Configuration (or Mechanical-Node Architecture) (Figure 2) | Series Configuration (or Electrical-Node Architecture) (Figure 1) |
---|---|---|
1 | ||
Variables | Values |
---|---|
Empty weight (WOE [N]) | 50,000 |
Payload weight (WPL [N]) | 20,000 |
Total input energy (Eo,tot [GJ]) | 25 |
Gas turbine efficiency (ηgt) | 0.35 |
Electric motor efficiency (ηem) | 0.95 |
Electric generator efficiency (ηeg) | 0.98 |
Propulsive efficiency (ηp) | 0.80 |
Gearbox efficiency (ηgb) | 0.95 |
Energy density of aviation fuel (ef [Wh/kg]) | 11,900 |
Acceleration due to gravity (g [m/s2]) | 9.81 |
Coefficient of lift | 0.6391 |
Coefficient of drag | 0.0572 |
Planform area [m2] | 61 |
Density (at cruise altitude) [kg/m3] | 0.5579 |
Cruise velocity [m/s] | 141.67 |
Variables | Values |
---|---|
Empty weight (WOE [N]) | 127,040 |
Payload weight (WPL [N]) | 68,670 |
Total input energy (Eo,tot [GJ]) | 25 |
Gas turbine efficiency (ηgt) | 0.35 |
Electric motor efficiency (ηem) | 0.95 |
Electric generator efficiency (ηeg) | 0.98 |
Propulsive efficiency (ηp) | 0.80 |
Gearbox efficiency (ηgb) | 0.95 |
Energy density of aviation fuel (ef [Wh/kg]) | 11,900 |
Acceleration due to gravity (g [m/s2]) | 9.81 |
Coefficient of lift | 0.6391 |
Coefficient of drag | 0.0572 |
Planform area [m2] | 61 |
Density (at cruise altitude) [kg/m3] | 0.5579 |
Cruise velocity [m/s] | 141.67 |
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Batra, A.; Raute, R.; Camilleri, R. An Endurance Equation for Hybrid-Electric Aircraft. Aerospace 2024, 11, 698. https://doi.org/10.3390/aerospace11090698
Batra A, Raute R, Camilleri R. An Endurance Equation for Hybrid-Electric Aircraft. Aerospace. 2024; 11(9):698. https://doi.org/10.3390/aerospace11090698
Chicago/Turabian StyleBatra, Aman, Reiko Raute, and Robert Camilleri. 2024. "An Endurance Equation for Hybrid-Electric Aircraft" Aerospace 11, no. 9: 698. https://doi.org/10.3390/aerospace11090698