*4.1. Computational Logic*

The computational model aims to estimate the flying range of an all-electric aircraft based upon improvements to its powertrain arising from the specific energy of batteries (kWh/kg) and the specific power of the electric motors (kW/kg). For simplicity, the overall mass of the aircraft is fixed, so that any improvement in the powertrain directly translates into additional "space" for more batteries that are subsequently used to replenish the aircraft's total mass back up to its fixed amount. In reality, of course, any improvement in the powertrain could invoke numerous alternative aircraft design possibilities. However, for the purposes of systematically tracing the effect of technological improvements on the aircraft flying range, we assume the design and the mass of the aircraft to be fixed. To compute the flying range, we employ the aircraft electrification flight equations derived in [16].

The mass, m, of an all-electric aircraft consists of (i) the mass of the empty aircraft, *m<sup>e</sup>* , (ii) the mass of the passengers and the crew, together with their luggage, *mp*, and (iii) the mass of the batteries, *m<sup>b</sup>* .

$$m = m\_{\varepsilon} + m\_{p} + m\_{b} \tag{1}$$

In aviation modelling, the mass of passengers is normally approximated as the number of seats multiplied by 100 kg. The ratio *me*/*m* is observed to be independent of the aircraft model and is equal to 0.62 for turboprops.

The energy consumption of an aircraft is defined as:

$$E = \frac{m\text{gs}}{(L/D)\_{\text{max}}\,\mu\_p\mu\_\varepsilon} \tag{2}$$

where *g* is gravitational acceleration equal to 9.81 m/s<sup>2</sup> ; *s* is flying range; (*L*/*D*)*max* is the maximal lift-to-drag ratio, currently 20 is achievable; *µ<sup>p</sup>* and *µ<sup>e</sup>* are efficiencies of the propeller and the powertrain, respectively, typically both equal to 0.8. With given numerical assumptions,

$$E \Big/\_{\text{ms}} = w\_{\text{s}} = 0.22 \text{ kWh/km t} \tag{3}$$

where *w<sup>s</sup>* is energy consumption and *t* stands for tonnes.

The mass of batteries can be defined as:

$$\frac{m\_b}{m} = \frac{\mathcal{g}s}{(L/D)\_{\text{max}} \, \mu\_p \mu\_e \rho\_E} \tag{4}$$

where *ρ<sup>E</sup>* is the specific energy of batteries.

The flying range can be defined from (4) and (1) as:

$$s = \frac{\rho\_E}{1.6w\_s} \left( 0.61 - \frac{m\_p}{m\_e} \right) \tag{5}$$

The required power, *P*, is calculated based upon cruise speed *v<sup>c</sup>* and rate of climb *vr*.*o*.*<sup>c</sup>* requirements as:

$$P = \frac{mg}{\mu\_p \mu\_\varepsilon} \left(\frac{v\_c}{(L/D)\_{\text{max}}} - v\_{r.o.c}\right) \tag{6}$$

Combining this computational logic and the assumptions derived from existing turboprops (Table 2), we arrive at the following numerical estimations for the model aircraft, Table 4.



The computational logic is validated by the calculation of the flying range for the specified assumptions for current levels of specific energy of batteries 0.25 kWh/kg. Batteries at current technological development (LiFePo4, specific power of 2 kW/kg, specific energy of 0.12 kWh/kg) would constitute 7.5–15% of the total mass of a hybrid electric aircraft [16]. In our calculations batteries in this base case constitute 12% of the mass of the aircraft and the flying range is 135 km, the number of the same magnitude comparing to estimated ranges for different aircraft sizes in [16].

In the constructed model, the two input variables, (i) specific energy of batteries and (ii) specific power of electric motor, will be varied in order to determine their effects on the output variable, flying range. If only the specific energy of batteries is changed, the aircraft is assumed to be able to fly longer from the same mass of batteries on the board. If the specific power of the motor is changed, the same required power for a given aircraft can be achieved with a lighter motor. Any freed mass is assumed to be refilled by more batteries of the same specific energy to maintain a constant total mass of the airplane. More batteries onboard enable longer flight ranges. Thus, with such a model design, we are able to directly ascertain the effect of powertrain technological improvements on the flying range of an aircraft in a continuous numeric space.
