3.2. Model Assumptions
The analytic model is based on standard flight mechanics run as a MATLAB tool. Without highly detailed analysis and wind-tunnel testing, certain assumptions on aerodynamic characteristics are required as input variables. The primary aerodynamic variable which must be estimated for this model is known as the Oswald efficiency factor (e
). This is a parameter which expresses the total variation of drag with lift. It is sometimes called the span efficiency factor and would equal 1.0 for an elliptically-loaded wing with no lift-dependent viscous drag, for practical aircraft “e
” varies from about 0.75 to 0.90 [15
]. The other main variable which must be assumed for the model is the propeller efficiency. Again, values between 0.75 and 0.90 are typical [11
] for the type of aircraft under consideration. Also, as noted below, the best propeller efficiency may be considered as dependent on the detailed aircraft configuration, with wing mounted propellers usually resulting in greater efficiency than a front fuselage mounted unit due to reduced slipstream “scrubbing”. For the results below, an “e
” value of 0.835 was used, and propeller efficiency varied between 0.75 and 0.85 respectively for the non-hybrid through to fully electric models assuming any EPS is accomplished by adding wing-mounted propellers. For intermediate HP
, the propeller efficiency was linearly scaled. Increasing HP
in this analysis implies increasing both the propeller area and the proportion assumed to be wing mounted and unobstructed by the fuselage. The battery mass required is calculated by a numerical iterative process taking into account the energy requirement according to the time required for climb at changing weight and drag conditions. The overall EPS mass is calculated using the proportion of installed power multiplied by the power density and is added to battery mass. The same technique is used to calculate the ICE mass in proportion to HP
to arrive at a take-off weight given the addition of sufficient mission fuel. Table 1
, power system analytic model parameters, and Table 2
, aircraft analytic model parameters, show the key parameters used in the analytical calculations.
shows the recharge time over-run for a “Case A” hybrid mission using a single ICE/EPS unit (single propeller). The “time over-run” is the amount of time the engine driven generator would be required to continue running on the ground after landing to recharge the battery. As the HP
increases, more battery energy is consumed for the climb, but less ICE power is available for both the climb and the descent phase. The practical HP
limit for Case A must lie somewhere before the knee of the curve, around HP
= 0.8. However, at the value HP
= 0.5, the over-run is 10%, which would represent a reasonable limit considering the ground handling and operational margins.
If multiple EPS units are used (three in this example), with a single ICE, the problem of recharge time becomes even more severe; as shown in Figure 6
, where the useful HP
value is reduced to 0.25.
The problems inherent in Case A strategy in terms of mission cycle time are minor compared to the significant increase overall energy consumption. Figure 7
shows the mission fuel recharge requirements increase with increasing hybridisation ratio. That is additional recharge fuel is required to fully recharge batteries during a typical skydiving mission, based on a mission of duration 15–20 min.
In this case, a non-hybrid ICE-only system would use about 25 kg of fuel to complete the mission. For a flat-rated engine at the HP
value of 0.25, which as noted above would be the maximum for viable mission time utilisation, the mission fuel is increased by 40%. Clearly, the “Case A” strategy is counterproductive from a fuel consumption perspective, and shows why, historically, hybrid electric aircraft concepts were not considered useful. However, there are certain advantages which must be considered, and tend to favour hybridisation, such as those shown in the following sections. The justification of hybrid electric aircraft from the fuel consumption perspective is heavily dependent on any electric machine, propulsive and aerodynamic efficiency gains which can be made as well as the energy density of battery storage systems. Each of these factors are shown in literature to be advancing and have a positive development future [16
The Case B type can include on-board recharging if desired, but the baseline is to replace and ground recharge the battery each mission. When battery recharging is conducted via ground based sources, renewable and green energy, with many wider societal and technical advantages can be utilised. This system guarantees the best emissions reduction and provides the best future utility potential as battery storage limits increase in the future.
If the presumption is made for Case B, where the battery is replaced or recharged without using the on-board ICE, the mission fuel required naturally tends to zero when the aircraft is solely electric powered (HP
= 1), as shown in Figure 8
The fuel required is nearly in proportion to the hybridisation for the flat-rated engine. In addition, clearly notable from Figure 8
is the difference between the flat-rated and full-rated engine type. The principle reason for the extra fuel required for the full-rated engine is the extra time required for the climb. This effect diminishes with increased HP
but would make hybridisation more favourable if use of a full-rated engine was necessary for any reason. The choice of hybridisation proportion can be made according to preferences for the energy source and hence emission characteristics, operating costs, weight or other operational characteristics. The fuel required reflects the direct operating cost of the mission as well as CO2
emissions. The overall energy required for the mission may increase for higher HP
because of the inherent electric energy source, distribution, conversion and storage inefficiencies. Carefully selecting the nature of the energy supply can easily reduce the cost and environmental impact. For example, converting the energy in coal to electricity to charge batteries via a conventional national grid for flight purposes is counterproductive to efficiency and emissions goals, but utilizing solar, wind or even nuclear energy sources can reduce the emissions. Nevertheless, the EPS system weight for a given flight will exceed that required for pure ICE operation, this translates to more energy being required to account for the higher drag due to increased required wing lift and more work done to raise the mass against gravity during the climb.
The Case B fuel consumption naturally favours this hybridisation model; however, the viability of using such an amount of electrical storage in terms of the aircraft structural and performance capability must be verified.
At the maximum HP, the battery weight including a 20% margin amounts to 800 kg. This battery weight is considered feasible for an aircraft with a take-off weight of 4500 kg and a payload of 800 kg.
shows the battery weight implication for both the full-rated and flat-rated hybrid electric scenarios, while Figure 10
shows the effect on take-off weight.
The mission cycle time is naturally dependent on the best climb rate achievable, as well as the descent rate. For an identical installed propulsive power of 670 kW, the analytic model shows climb times for the range of HP
given the full-rated and flat-rated hybrid electric scenarios shown in Figure 11
The time to climb for a fully electric example is acceptable, and the improvement over a full-rated powerplant operating to the prescribed altitude is very significant.
The effect of propeller efficiency (Etap
) on the climb performance in this model is very sensitive. The foregoing examples used Etap
values linearly scale from 0.75 to 0.85 on the HP
domain and show adequate performance. If the propeller efficiency is held constant (0.75 for example), and all other variables remain the same, the following plot of hybridisation versus climb time is produced shown in Figure 12
In this case, the climb rate has decreased significantly with increasing HP and would have negative implications for aircraft utility. This indicates the extreme importance of improving items of efficiency in order to enable hybrid electric propulsion systems to be viable.
Results obtained using the X-Plane simulation as detailed in the following section show a definite improvement in rate of climb for the hybrid models compared to conventional propulsion. The X-Plane models were configured at a constant HP
of 0.67. The analytical study shown above does not reflect such an improvement given the assumed propeller efficiency parameters, so it is of interest to see what change of propeller efficiency would yield results which concur with the simulation. The original propeller efficiencies used in the above calculations varied proportionally from 0.75 for a conventional non-hybrid using a single propeller through to 0.85 for a fully hybrid propulsion system using two additional wing mounted propellers. These numbers are arbitrary and reasonable assumptions, but a much more rigorous and comprehensive study of particular detailed designs would be necessary to improve estimates, or the flight testing of prototypes to verify real outcomes would be required. However, for the purposes of this study, a new set of possible propeller efficiency numbers can be conveniently input and the resulting performance calculated to find a match to the simulation. Figure 13
shows the result using a more extreme improvement of propeller efficiency terms through the domain of HP
. Figure 13
shows that the climb time has reduced by a similar amount as the simulation tool predicts.
shows the analytic results data from Figure 13
, where the propeller efficiency assumed increases from 0.65 to 0.95 along the HP
domain, against measured times from X-Plane simulation flights. The times for each completely independent analysis method are now reasonably close, and the trend in changing to the distributed propulsion (three propeller) hybrid layout is very distinct. It is stressed that changes to the X-Plane model between the conventional and hybrid propulsion models created were the empty weight and the extra propellers and nacelles. The total installed power remained constant for both models.
These results should be interpreted as trends only, as the input parameters are coarse estimates, however the underlying theory and technique for the analysis is based on fundamentals and well known properties. These trends reiterate that hybrid electric aircraft propulsion has many compelling advantages which can improve aircraft performance for particular mission requirements. These include:
High power/weight (where battery storage capacity requirements are low);
Ease of adding propeller area for any given installed power, allowing greater propulsive efficiency under particular conditions.
While at the same time reducing direct operating costs for energy.