m/nEDF
Cplaneρπ + rr2 (B.7)
Here nEDF is the number of (single- or two-stage) EDFs and LEDF is the length of the EDF. The length of the EDFs is fixed by the length of the motor. Each stage is assumed to be as long as the motor.
B.2 Constraints on plane and EDF design
A number of constraints is taken into account on both plane as EDF design:
1. The mass of a model airplane cannot be more that 25kg [15].
2. The mass/total surface ratio cannot be larger than 25 kg/m2[15].
3. The batteries must fit in the model airplane.
4. The EDFs must be powerful enough to propel the plane.
5. The EDF outlet speed must be subsonic to allow a simple nozzle design.
6. The root-tip ratio of the EDFs must be higher than 0.4.
Constraint 1 and 2 are given by the rules for sport model airplanes by the KNvVL, the Dutch authority on aviation. However, in for example Germany other rules apply which allow heavier model airplanes. Constraints 3 is explained below. Concerning constraint 4, every EDF stage is powered by its own motor. As the number of EDFs and stages per EDF is chosen, the total available power can be determined. This must always be higher or equal to the power needed to propel the plane. Constraint 5 and 6 are result from the limitations in EDF design(section 2.2.3). The need for supersonic nozzle increases the complexity of the EDF design and will add to the weight of the plane. Too low root-tip ratios can cause problems in both the blade root stress as the root degree of reaction.
To determine whether the batteries fit in the plane I assume the EDFs are mounted on top of the plane and that the components that need to be fitted into the plane are small compared to the batteries and can be neglected. Now the ratio of volume of the batteries over the volume of the plane is given by:
Vbattery
Here ρenergy,volume is the volume energy density of the batteries, hwing/c is the average thickness of a wing section divided by its chord length. Assuming a NACA0010 wing section:
hwing/c = 0.03. Furthermore Lplaneand Splaneare the length and the span of the plane. Here
I assumed a delta wing shaped plane with the dimensions shown below and that all batteries are fitted in the wing. There is no hull to store the batteries.
Figure B.2: The plane planform dimensions assumed in deriving equation B.8 In the calculations the following values and ranges of values are used:
Table B.2: Values and ranges of values used in the analysis
Air density ρ 1.205 kg/m3
Air inlet temperature T1 293 K
Air gas constant Rair 286.9 J/kg K
Flight velocity Cplane 208.3 m/s
Plane aspect ratio AR 2.86
Lift efficiency factor e 0.8
Motor power density ρpower,mass 22.3 kW/kg Battery mass power density ρenergy,mass 5.0 · 105 J/kg Battery volume power density ρenergy,volume 1.0 · 109J/m3
Flight time tf light 120 s
Maximum motor power Pmotor,max 36.8 kW
Wing area Awing 0.10 - 1.0 m3
Zero lift drag coefficient CD0 0.017 - 0.023
The values for the air properties taken standard atmospheric circumstances. The plane aspect ratio is the average of those of the fun-jet ultra and the hot-shot. The range in Awing is based on the fun-jet and the hot-shot. The battery mass and volume energy density is based on the average of a large number of LiPo batteries sold at hobbyking.com. The flight time is based on a video of the record speed flight of the current record holder [11]. The wing area is based on the size of the fun-jet and hot-shot. The range in CD0 is based on the values given in [10]. The range chosen corresponds with a relatively aerodynamic plane.
The table below shows the EDF configurations considered.
Table B.3: EDF configurations considered in the analysis.
Configuration No. No. of single-stage EDFs No. of double-stage EDFs
1 1 0
2 2 0
3 4 0
4 0 1
5 0 2
B.3 Results
B.3.1 Single or double stage?
In the figure below, for two different plane sizes, the input power of one single-stage EDF PEDF propelling a plane is shown as a function of the mass flow through the EDF. Since the exact shape and performance of the plane is not known yet, the analysis has been done for three different zero-lift drag coefficients CD0. In Figure B.3a and B.3b planes with a wing area of 0.173 m2 and 0.6242 are assumed, respectively. These areas corresponds with the fun-jet and the hot-shot.
(a) (b)
Figure B.3: The input power PEDF of one single-stage EDF propelling as a function of the mass flow through the EDF. In (a) Awing = 0.173m2 and in (b) Awing = 0.624m2. These wing sizes are equal as those of the fun-jet ultra and the hot-shot respectively.
As can be seen, in both cases a minimum EDF input power can be found. The optimal mass flow varies approximately 20% as a function of CD0. The figures suggest that in case of a larger plane, the penalty on having a sub-optimal mass flow is much smaller. However, for a mass flow half the optimal mass flow, the EDF input power is between 3.2% and 3.3%
higher for the fun-jet sized plane and between 2.3% and 2.5% for the hot-shot sized plane. So, although the penalty is a bit lower for larger planes, the difference is smaller than suggested by the figures. The optimal mass flow is determined for all EDF configurations and wing areas in the considered range. Furthermore the corresponding EDF input power, EDF root-tip ratio,
plane mass, battery mass and volume, nozzle outlet Mach number and available input power are determined.
The figure below shows the optimal total mass flow as a function of the wing area for an EDF configuration consisting out of two single-stage EDFs and for one consisting out of one two-stage EDF, so each using two motors. Here the constraints are not taken into account.
Figure B.4: The optimal total mass flow through the EDFs as a function of the wing area for a EDF configuration consisting out of two single-stage EDFs and a configuration consisting out of 1 two-stage EDF.
As can be seen the optimal mass flow for one two-stage EDF is lower than that for two single-stage EDFs. This can be explained using the trade-off explained at the start of this chapter. A smaller EDF gives a lower drag force but a lower propulsion efficiency due to a smaller mass flow and a larger EDF gives a higher propulsion efficiency but also a higher drag force. For the same mass flow, a two-stage EDF has a larger surface area. Therefore, the positive effects on the decreased drag force coming with a smaller EDF dominate the negative effects on the propulsion efficiency.
A smaller mass flow does not directly mean that a two stage EDF propulsion needs less power for the same plane. To achieve a more efficient propulsion, the drag force of the two-stage EDF must be a certain amount lower than the drag force on the two single-two-stage EDFs to compensate for the lost propulsion efficiency caused by the smaller mass flow. Since a two-stage EDF is twice as long as a single-stage EDF, the tip radius of the two-stage EDF must be smaller than that of a single-stage EDF to get a smaller AEDF and thus a smaller EDF drag force.
The figure below shows the root-tip ratio an EDF with an optimal mass flow as function of the wing size.
Figure B.5: The root-tip ratio of the EDFs as a function of the wing area for two different EDF configurations: two single-stage EDFs and one two-stage EDF. Here the displayed design points correspond with the point of optimal mass flow.
As can be seen the root-tip ratio of the two-stage configuration is lower than that of the single-stage configuration which means that the tip radius of the two-stage EDF is larger than that of the single-stage. From this can be concluded that the drag on a two-stage EDF is larger than the drag force on two single-stage EDFs. Since the two-stage also has a lower propulsion efficiency due to the lower mass flow, there can be concluded that a plane propelled by two-stage EDFs needs more power than an plane propelled by single stage EDFs. This is confirmed in the figure below. Here the EDF input power at top speed is shown as a function of the wing for two single-stage EDFs and for one two-stage EDF.
Figure B.6: The EDF input power as a function of the wing area for two different EDF configurations: two single-stage EDFs and one two-stage EDF. Here the displayed design points correspond with the point of optimal mass flow.
As can be seen the EDF input power for one two-stage EDF is higher than that for two single-stage EDFs for all wing area’s and zero-lift drag coefficients. This corresponds with our conclusion based on the mass flow and the root-tip ratio above. PEDF is between 5% and 10% higher for a configuration with one two-stage EDF than for a configuration with two single-stage EDFs. The same comparison between four single-stage EDFs and two two-stage EDFs gives that the two-stage configuration needs between 7% and 11% more power than the single-stage configuration. From this can be concluded that a configuration using single-stage EDFs gives a more efficient propulsion since less power is needed at top speed.
B.3.2 Number of single-stage EDFs
Now a decision on the number of stages is made, a decision on the number of single-stage EDFs must be made. The figure below shows the EDF input power as a function of the number of single-stage EDFs for different wing areas and CD0 = 0.020. Here only design points satisfying constraint 1, 4, 5 and 6 are displayed. The effect of constraint 2 and 3 are discussed later.
Figure B.7: The EDF input power as a function of the number of single-stage EDFs used.
Here CD0= 0.020.
As can be seen using one single-stage EDF uses the least power. However, the design space with 1 EDF is much smaller than with two, three or four EDFs. The design space of one EDF is limited by the minimum root-tip ratio. This can be explained by the fact that larger planes have a larger optimal mass flow and thus need a larger EDF. When multiple EDFs are used the mass flow is divided over the different EDFs and thus each EDF can be smaller. A configuration using two EDFs has the largest design space and uses the second least power. Below the design space of the different configurations is discussed. The figure below shows the total plane mass, the root-tip ratio and the ratio of needed and available power as a function of the wing area for 1 to 4 single-stage EDFs.
(a) (b)
(c)
Figure B.8: The total plane mass(a), root-tip ratio(b) and the ratio of needed power and available power(c) as a function of the wing area for 1 to 4 single-stage EDFs. Only design points satisfying constraint 1, 4, 5 and 6 are displayed. For clarity only the results for CD0 = 0.020 are shown. The legend holds for all three figures.
As can be seen the design space for 1 single-stage EDF is limited by the minimum root-tip ratio. For 2 EDFs the constraint in root-tip ratio and on total plane mass are both limiting.
For 3 and 4 EDFs the total plane mass is limiting. For none of the configurations the ratio of needed and available power is limiting. However, for 1 and 2 EDFs, it is close to limiting. It is noteworthy that for 3 and 4 EDFs the ratio of needed and available power is only 0.67 and 0.51 maximum, which means that only about half the available power is used. For clarity, only the results for CD0 = 0.020 are shown in Figure B.8. The same analysis has been done for CD0 = 0.017 and CD0 = 0.023. For 2 EDFs and CD0 = 0.017 the design space is limited by the minimum root-tip ratio, for CD0= 0.023 the available power is limiting. Furthermore, for CD0= 0.023 and 1 EDF, also the available power is limiting. For the other configurations nothing remarkable is shown for the other CD0 values. The outlet Mach number is never limiting with values between 0.69 and 0.77 for all EDF configurations, and is for this reason not shown.
A configuration using 2 EDFs is deemed most promising since this configuration has the largest design space and uses second least power.
B.3.3 Limitations is plane design
The figure below shows the ratio of battery volume over available volume and the wing loading as a function of the wing area for 1 to 4 single-stage EDFs.
(a) (b)
Figure B.9: The ratio of battery and available volume and the wing loading as a function of the wing area for 1 to 4 single stage EDFs. Only design points satisfying constraint 1, 4, 5 and 6 are displayed. Note that according to constraint 2 and 3 the ratio of volumes should be smaller than 1 and that the wing loading can be maximum 25 kg/m2. For clarity only the results for CD0 = 0.020 are shown. The legend holds for both figures.
As can be seen in Figure B.9a the ratio of battery and available volume is much larger than 1. In order for the batteries to fit in the plane, the ratio must be smaller than 1(constraint 3). In the calculation, the plane is assumed to be a flying wing without fuselage. Here all batteries must fit in the wing. From the results can be concluded that this is not a realistic plane concept. Probably a concept with a fuselage to store the batteries is more suitable. It is noteworthy that the ratio of volumes decreases for increasing wing area. This is explained by the fact that in the calculation, the ratio of wing thickness over chord length hwing/c is kept constant. A larger plane therefore also has a thicker wing. The needed amount of batteries scales with the power usage at maximum speed and thus approximately with the wing area.
So, for an increasing wing span, the needed volume scales with the 2nd power(wing area) and the available volume scales with the 3rd power(wing volume). Since the volume ratio decreases for an increasing plane size, it is easier to fit the batteries in the plane as the plane becomes larger. Another solution to fit the batteries, is to decrease the desired flight time.
Another important constraint in plane design is the maximum allowed wing loading. In Figure B.9b can be seen that the wing loading is much larger than 25 kg/m2, the limit stated in constraint 2. One solution is to simply decrease the flight time, which decreases the battery weight. However, the batteries make up between 95% and 65% of the plane weight, for small and large wing areas respectively. Since the wing loading is, in many cases, more than twice
as large as allowed, the flight time would have to be reduced more than half. It is noteworthy that for most EDF configurations the wing loading decreases with increasing wing area. I think there is a large chance that the wing loading constraint cannot be met. It think the solution must be sought in applying for exemption or in attempting the record in another country where different rules apply. More research on the laws applying to model airplanes in the Netherlands and other countries is needed.
B.3.4 Discussion
In equation B.2 a very crude assumption on EDF drag is made. The chance that the actual EDF drag deviates a lot is quite large. The EDF drag makes up for approximately between 20% and 55% of the total drag, for a larger and a small wing area respectively. Because of this, a deviation in EDF drag can have a large effect on the optimal design of the EDF and the plane. To get an idea of the effects of a variation in EDF drag, the analysis above has been repeated for FD,EDF := 0.5FD,EDF and FD,EDF := 2FD,EDF. Here FD,EDF is the drag force acting on the EDF. This showed that also for a much smaller and much larger EDF drag force, a propulsion using single-stage EDFs needs less power than when using two-stage EDFs. In general a decrease in EDF drag gives a decrease in needed EDF input power and a decrease in plane mass. Furthermore, a decrease in EDF drag gives an increase in the optimal mass flow. In the case of FD,EDF := 0.5FD,EDF the root-tip ratio strongly limits the design space of a configuration with 1 or 2 single-stage EDFs, causing a configuration with 3 EDFs to be more promising. On the other hand, in the case of FD,EDF := 2FD,EDF a 2 EDF configuration is still the most promising. Although now the design space is limited by the maximum plane mass solely, instead of a combination of the minimum root-tip ratio and the maximum plane mass as shown in the analysis above.
B.3.5 Conclusion
Concluding: configurations with single-stage EDFs need less power than those with two-stage EDFs. Furthermore, Figure B.7 shows that using less EDFs decreases the needed power.
However, a configuration using one single-stage EDF is less desired since it only allows small wing areas. To fit all batteries in the plane a larger plane is beneficial. A larger plane is also beneficial in terms of wing loading. Using more than two EDFs is useless since this increases the needed EDF input power and decreases the design space. Therefore a configuration using two single-stage EDFs is deemed the most promising. An exception is the case in which the actual EDF drag is much lower than in the analysis above. In this case a configuration with 3 single-stage EDFs might be better. Concerning the plane size, as stated above a larger plane is beneficial for both wing loading as for packaging. Within the assumptions taken in the analysis, the largest plane possible has a wing area of 0.473m2. However, the maximum wing area possible depends strongly on the actual EDF and plane drag. Therefore this value only an indication.
Appendix C
Blade section design using carpet plots from [8]
The figures in [8] summarize systematic two-dimensional cascade test data in carpet plots that are simple to use for an engineer. Figure C.1 shows a cropped version of Figure 1 from [8]. Here for five solidities a carpet has been plotted that allows to determine the camber Cl0 as a function of blade inlet air angle α1(cascade notation) and the deflection . The carpets are spaced a number of grid points proportional to the solidity. Therefore the camber for intermediate solidities can be determined by interpolation between the carpets. In the figure below an example of how to read a value from the figure has been drawn in. In the example the camber for a blade with α1 = 50°, = 16° and σ = 1.15. Next, Figure C.2 shows Figure 2 from [8]. This figure can be used to determine AoA as a function of Cl0 and σ. The same example has been drawn in.
When the blade design is determined, the off-design performance can be determined using Figure 3f from [8]. Using this Figure the relation between the angle of attack AoA and the flow deflection can be determined. A part of the figure is shown in Figure C.3
When the blade design is determined, the off-design performance can be determined using Figure 3f from [8]. Using this Figure the relation between the angle of attack AoA and the flow deflection can be determined. A part of the figure is shown in Figure C.3