To get to a complete mid-span design the inlet root and tip radii rr and rt; the air angles α1, β2 and α3, and the rotational speed ω of the rotor need to be determined. Here rr is chosen to be equal to half the motor diameter, so rr = 30mm. Furthermore, the inlet flow is assumed to be purely axial. so α1 = 0°. As long as there are no large objects in front of the EDF this is an acceptable assumption. The outlet flow is chosen to be purely axial to maximize the thrust, so α3 = 0°.
In determining rt, β2 and ω the RPM-power characteristic of the rotor and motor should match. Furthermore, the limitations explained in section 2.2.3 must be taken into account.
Important limitations are the blade root stress and the degree of reaction. These limitations can limit the design in case of long rotor blades or low root-tip ratios. Therefore, initially only designs with root-tip ratios above 0.4 are considered. Another important limitation is the blade tip Mach number. The rotational speed of the rotor can be written as a function of the root-tip ratio and the tip Mach number:
ω = Ut
Here Mt is the tip Mach number, Ca1 is the inlet velocity and rr/rt is the root-tip ratio.
The inlet velocity is fixed on 750km/h. Now for a certain ω and Mt the tip radius and the mass flow through the EDF can be calculated.
The maximum deflection and thus the maximum input power a rotor can handle is given by the minimum allowed Haller number. Here Hmin = 0.72 is the minimum allowed Haller number and Um is the mid-span rotor speed. Now, using equation 2.13 the maximum input power of a rotor with a certain rotational speed and tip Mach number can be determined. Note, is case of a too low rotor speed cosβ2> 1. In these cases the air inlet velocity relative to the rotor is so low it is not possible to get too much diffusion in the rotor. In this case maximum diffusion gives β2 = 0°.
As a first guess a tip Mach number of Mt = 1.0 is chosen. The figure below shows the maximum rotor input power as function of the rotational speed and the root-tip ratio for Mt= 1.furthermore the figure shows the maximum power curves(at 57.6V) for the LMT3080 with 5, 6, 7 or 8 windings and the contours showing the nozzle outlet Mach number and the thrust in case of isentropic operation.
Figure 3.3: The maximum rotor input power PEDF and the maximum output power of the LMT3080 as a function of the rotational speed for Mt = 1. The root-tip ratio and ω are coupled by equation 3.3. The thrust force and outlet Mach number are calculated assuming isentropic operation.
As can be seen, the maximum rotor input power is much larger than the motor power for most ω and root-tip ratios. From this can be concluded that the motor power is limiting in the EDF design.
The motor with 5 windings is not suitable since the motor can give much more power than the rotor can handle. Also, the motor with 5 windings operates at approximately 60kRPM which is higher than the mechanical limit of 50kRPM. According to the manufacturer motors can operate at speeds up to 30% higher than the limit for short times. However, the record attempt of the plane will take approximately 2 minutes. I do not consider this a short time.
Maximum thrust can be given using the motor with 6 windings. However, then the outlet
Mach number is almost supersonic which is not preferred because of compressibility effects that might occur in the nozzle. From this can be concluded that a motor with 7 windings is preferred.
The same analysis can be done for lower and higher Mach numbers. Choosing a higher tip Mach number leads to a larger EDF with a lower root-tip radius and a higher mass flow and thus a lower outlet Mach number and a higher thrust.
A tip Mach number of Mt= 1.0 is chosen, which gives Hrotor= 0.87 and a root-tip ratio of 0.48. Both the Haller number as the root-tip ratio have conservative values, because of this it should be possible to design a proper working EDF. In section B an analysis on the optimal EDF configuration and size is done. From this analysis can be concluded that it is very likely that an optimal plane-EDF combination can be made using an EDF with a root-tip ratio of 0.48. Also using double circular arc blade sections it should still be possible to achieve an efficient blade design at blade Mach numbers up to 1.0. At maximum power the LMT3080 with 7 windings(abbreviated as LMT3080/7) operates at 41.991kRPM and produces 27.781kW output power. There should be noted that Mt= 1.0 is quite an arbitrary choice, other choices that will give a working EDF are possible.
Using the equations in section 2.2.1 and 2.2.3 the mid-span design and important param-eters is determined. The table below gives important values:
Table 3.3: Important dimensions and parameters in the mid-span design. Air angles, Haller number and degree of reaction at mid-span.
Inlet velocity Ca1 750 km/h = 208.3 m/s
Rotational speed ω 41.991 kRPM = 4397 rad/s
Rotor tip Mach number Mt 1.0
Rotor input power PEDF 27.781kW
Root radius rr 0.030 m
Tip radius rt 0.062 m
Root-tip ratio 0.48
Mass flow m˙ 2.32 kg/s
Isentropic thrust force FT ,isentropic 118 N
Thrust force FT 95.1 N
Outlet Mach number M4 0.74
Rotor inlet air angle α1 0 °
Rotor outlet air angle α2 15.8 ° Stator outlet air angle α3 0 ° Inlet rotor relative air angle β1 44.2 ° Outlet rotor relative air angle β2 34.5 ° Rotor Haller number Hrotor 0.87
Degree of reaction Λ 0.85
As can be seen, the Haller number and the degree of reaction are well within bounds.
Since α1 = α3, C1 = C3 and equation 2.28 can be used to determine the degree of reaction.
Note that this are the mid-span values. In the section below the variation over the radius is determined. The thrust force has been calculated using the isentropic efficiency determined in 3.5.3.
The rotor root stress depends strongly depends on the chosen material. The stress is
calculated using equation 2.27. Note that in this equation only the centrifugal force is taken into account. The table below shows the root stress, the yield stress and the resulting safety factor.
Table 3.4: Yield stress sigmay, rotor root stress σr and the resulting safety factor for different materials.
Material σy [MPa] σr [MPa] safety factor
Aluminium 6082 260 77.25 3.37
Aluminium 7075 440 77.25 5.70
Steel 700 223.6 3.13
Nylon 12 Pa 46 27.06 1.70
As can be seen the root stress varies a lot from material to material. This is explained by the differences in material density. Nylon is deemed unsuitable since the safety factor is too low. Especially since the aerodynamic forces have not yet been taken into account. Steel is suitable, however aluminium 6082 is preferred due to a slightly higher safety factor. Another pro of aluminium is that the production costs will probably be lower due to faster milling.
Aluminium 7075 is also suitable, however it is slightly more expensive and from the results can be concluded that the extra strength is probably not needed.