Eindhoven University of Technology MASTER Design, Production and Testing of a High-Speed Electric Ducted Fan for a Record-Speed Model Airplane van Oorschot, Joep

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Eindhoven University of Technology

MASTER

Design, Production and Testing of a High-Speed Electric Ducted Fan for a Record-Speed Model Airplane

van Oorschot, Joep

Award date:

2018

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Design, Production and Testing of a High-Speed Electric Ducted Fan for a

Record-Speed Model Airplane

Eindhoven University of Technology Mechanical Engineering

Energy Technology

Graduation report

By

Joep van Oorschot, 0769268

Supervisor dr. ir. H.C. de Lange

the Netherlands June 13, 2018

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Summary

Team Air/e aims to design a model air plane, capable of breaking the Guinness world speed record of 750km/h using an electric plane. The jet engine used by the current record holder can deliver 180N of thrust. This gives that at 750km/h, 180N · 208.3m/s = 37.5kW useful power is needed to propel the plane. High speed electric planes are often propelled using Electric Ducted Fans (EDF). Since there are no EDFs on the market that can supply such powers, a new EDF must be designed.

The flow in the EDF has been designed using axial compressor theory. Using NACA 65-series data, a blade design was made to realize the designed flow. The EDF is designed such that the root radius of the rotor and the stator is equal to the outer radius of the electric motor such that the motor can be mounted in the stator hub. This way the EDF is kept compact and the motor is in direct contact with the airflow, which enhances cooling. A motor, the LMT3080 was chosen because of its high power density. The number of windings in the motor and the mid-span design are based on the motor specifications and the choice of a rotor tip Mach number of 1.0. Assuming on-design performance, the high speed EDF can deliver maximum 95.1N of thrust with 27.8kW input power.

A low speed prototype was made using SLS 3D printing to test the design. A smaller motor, the LMT2280 with 10 windings, was used to decrease costs. Since a smaller motor was used, the construction of the prototype is slightly different from the construction that was chosen for the high speed EDF. Assuming on-design performance the prototype can be operated at maximum 84m/s. At this point the EDF creates 15.1N of thrust with 1.83kW input power.

Static tests show that the theory slightly under-estimates the static thrust of the EDF.

At an input power of 864W, 25N of thrust was expected and 23N was measured. Dynamic tests show that the EDF is capable of generating thrust at forward velocities. However, at the design operating point the EDF provides less thrust than expected. Possible explanations for the worse performance are decreased isentropic efficiency due to the surface roughness of the 3D printed blades and decreased performance due to non-uniform inlet velocity. For operating points at lower inlet velocities or higher rotational speeds, the generated thrust is higher than expected. This is unexpected since at this point (partial) blade stall was expected to decrease the isentropic efficiency and the generated thrust.

The experimental results were used to predict the performance of the high speed EDF at high flying velocities. Assuming a model air plane with a wing area of 0.47m² and three EDFs, the experimental results show that a maximum velocity of 640km/h can be achieved.

From this can be concluded that the designed EDF is not powerful enough to propel a plane up to 750km/h. The main obstacle is the worse than expected performance around the design operating point. Although a redesign of the EDF is needed, the results showed that the methods used in this project are suitable to design an EDF and the gained experience can be used to further build up on in later projects.

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Introduction

Environmetally friendly air transport is of broad current interest. In the first quarter of 2017, the number of passengers on the Dutch airports was 9% higher compared to the same period a year earlier[1]. With increasing air traffic the adverse effects of this form of transportation are becoming more pronounced. On a global scale air traffic contributes to environmental change with CO2 and NOx emissions. On a local scale, air traffic increases environmental noise.

Exposure to environmental noise can cause stress, insomnia and high blood pressure. Multiple governments and international organisations have set goals to decrease the environmental impact of air traffic. The European Union, for example, aims to reduce air traffic CO2 and NOxemissions by 75% and 90% by 2050. Furthermore, the noise emission from flying aircraft must be reduced by 65% [2].

Figure 1.1: The current holder of the guinness world speed record for model aircraft, Niels Herbrich, holding the regarding aircraft.

The student team Air/e aims to contribute to the development of electrical charged high- speed passenger aircraft as a solution to create environmentally friendly aviation. Their mid- term goal is to beat the Guinness World speed record for model aircraft using an electrically powered model airplane. At the moment of writing this record is kept by Niels Herbrich with almost 750 km/h [3]. The record was set using a jet powered model aircraft that runs on kerosine. Figure 1.1 shows Niels Herbrich and the model aircraft. To beat his record, Air/e aims to develop an aircraft with a minimal top speed of 750 km/h, but then propelled electrical. To do this, both the plane as the Electric Ducted Fan (EDF) need to be designed.

This report will focus on the development of the EDF.

Figure 1.2a shows an example of an EDF: the Schubeler DS-98-DIA HST. This EDF is

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owned by Air/e and mounted on one of their test planes. Despite the fact that the Schubeler EDF is one of the most powerful EDFs on the market, this EDF is not powerful enough for the record-breaking plane (see section 3). Figure 1.2b shows an exploded view of an EDF with all parts that can be typically found in an EDF. The air flows axially into the EDF. There the rotor(1) accelerates the air into a swirling flow, transferring kinetic energy to the air. Next, the air is decelerated and straightened in the stator(5), converting the kinetic energy into static pressure. At last the air is axially accelerated in the nozzle. The nozzle is not shown in the figures. In off-the-shelf EDFs this is often merged with the housing. The rotor is driven by an electric motor(4). This motor is typically mounted in the hub of the stator. At their turn, the stator blades are connected to the housing(3). The EDF can be screwed to a plane using brackets connected to the housing.

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Figure 1.2: (a) An example of an EDF, the Schubeler DS-98-DIA HST. (b) An exploded view showing the typical parts of an EDF: the spinner(1), the rotor(2), the housing(3), the electric motor(4) and the stator(5).

The front part of the EDF, containing the rotor and the stator, can be treated as an axial compressor. This way, an EDF can be subdivided into two parts, the compressor and the nozzle. In axial compressor theory the combination of a rotor and a stator is called a stage. Multiple stages can be places behind each other to increase the pressure ratio over the compressor. When this principle is used in an EDF, the air will be accelerated more and so giving more thrust.

1.1 Objectives and report outline

The goal of this project is to design an EDF suitable for powering a record-breaking model airplane, and to gain experience in EDF design, production and testing. The main question in the EDF design is: is it possible to create sufficient thrust at a flying velocity of 750km/h.

In chapter 2 first some theory is studied. Chapter 3 discusses the design of the high speed EDF and chapter 4 discusses the low speed prototype that is made. Next, the tests are discussed in chapter 5. Using the test results, an estimation of the maximum flying speed is made. This is shown in chapter 6. Finally, conclusions and recommendations are given in chapter 7.

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Chapter 2

Theory

The thrust of the high speed model aircraft will be delivered by an EDF. In the sections below the theory needed to come to a high speed EDF design is discussed. A large part of the theory, especially the theory on compressor design, is inquired [4].

2.1 Airplane drag theory

Figure 2.1: Schematic representation of the forces acting on a plane. Here F_T: thrust force, F_D: drag force, F_L: lift force and F_G: gravitational force.

Figure 2.1 shows the forces acting on a plane. Here FT, FD, FL and FG are respectively the thrust, drag, lift and gravitational force acting on the plane. During steady flight these forces are in equilibrium, so F_T = F_D (horizontal equilibrium) and F_L= F_G (vertical equilibrium).

The drag force is often expressed using the drag coefficient CD: CD = FD

1

2ρAC_plane² (2.1)

Here C_plane is the velocity of the plane and A is a reference area. In case of streamlined bodies such as aircraft, the wing area or wetted area is often used as reference area. In case of bluff bodies such as people, cars or buildings, the frontal area is often used. Note that it is important to mention which reference area is used since it affects the numerical value of the drag coefficient.

Parts of the drag force are caused by different phenomena. For a plane flying at subsonic speeds the drag force coefficient can be subdivided in a constant part representing the drag due to skin friction and form drag (due to pressure differences) at zero-lift C_D0and an additional drag coefficient representing the induced drag due to lift CDi given in the equation below[9]:

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C_D = C_D0+ C_Di

= C_D0+ C_L² πARe

(2.2)

C_L= F_L

1

2ρAC_plane² (2.3)

Here CL is the lift coefficient, AR is the aspect ratio of the wings and e is an efficiency factor that depends on the lift distribution on the wings. The efficiency factor is 1.0 for an elliptical lift distribution and less for any other distribution. A typical value for rectangular wings is e = 0.70. The aspect ratio is defined as the square of the wing span divided by the wing area. Typical values for the zero angle of attack drag coefficient for full scale airplanes lie in between 0.017 and 0.038, with 0.017 being a streamlined commercial airliner and 0.038 a vintage bi-plane. Here the wing area is taken as the reference area. Typical values for e lie in between 0.70 and 0.85 [10].

The EDF or engines mounted on the plane are meant to create thrust. However, the air flowing in and around the housing does also creates losses. The question arises to what is drag and what is thrust loss. This is a matter of definition and in common use the flow outside the EDF stream tube create drag and the flow inside the stream tube creates thrust loss. So the forces acting on the outside of the housing create drag and the forces acting on the inside of the housing create thrust loss.

2.2 EDF theory

An EDF can be roughly subdivided into three parts: the inlet, the compressor and the nozzle.

For model aircraft EDFs it is very common to not have an explicit inlet. Therefore the inlet is left out of consideration. Assuming the inlet pressure p1 and velocity C1 are equal to the ambient pressure p_aand the forward velocity of the plane C_plane, the thrust given by an EDF F_T is given by:

F_T = ˙m(C₄− C₁) + A₄(p₄− p_a) (2.4) Here ˙m is the mass flow through the EDF, C₄ is the outlet velocity, p₄ is the outlet static pressure and A₄ is the cross sectional area of the nozzle outlet. The first term of the equation gives the thrust due to a change in momentum of the mass flow, called the momentum thrust.

The second term gives the thrust due to a difference in static pressure in front and behind the EDF, called the pressure thrust. Figure 2.2 shows the EDF with the inlet and outlet velocity and the pressures working on the EDF. Assuming an isentropically expanding flow in the nozzle, maximum thrust is generated when the flow is completely expanded, so p4 = pa. In this case the pressure thrust goes to 0. In the calculation below p₄= p_a is used.

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Figure 2.2: Schematic representation of an EDF with the pressures working on it. C1: inlet velocity, C₄: outlet velocity, p_a: atmospheric pressure, p₄: outlet pressure.

Evaluating the expression for momentum thrust, it is clear that thrust can be generated using a large mass flow with a small change in velocity, or using a small mass flow with a high change in velocity. Finding the most efficient combination of the two is an important consideration in EDF design. The overall efficiency of propelling a plane 2.5 can be subdivided into the propulsion efficiency ηp and the energy efficiency ηe. The propulsion efficiency is the ratio of the useful propulsive energy (F · C_plane) to the sum of that and the unused kinetic energy in the EDF exit flow. The energy efficiency describes the effectiveness of the energy conversion within the EDF. The efficiencies ηo, ηp and ηe are given by:

η_o = FTCplane

P_EDF = η_pη_e (2.5)

η_p = 2 1 +_C^C⁴

plane

(2.6)

η_e=

1

2m(C˙ ₄²− C_plane² ) PEDF

(2.7) From equation 2.6 can be concluded that the propulsion efficiency is maximal when

C4

Cplane = 1. This suggests that propulsion based on a small change in velocity and a large mass flow is more efficient. However, for a certain design C_plane a large mass flow directly leads to a larger EDF. EDF size is important since it adds weight and drag. To get an optimal propulsion system an optimal EDF size should be found.

Figure 2.3 shows the station numbering of a single stage EDF. This station numbering is used throughout this report. Station 1 denotes the inlet of the EDF, station 2 is in between the rotor and the stator, station 3 is after the stator and station 4 is at the nozzle outlet.

In chapter B the potential of a two stage EDF is discussed. However, in this discussion no station numbering is needed. Therefore, no station numbering for two-stage EDFs is defined.

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Figure 2.3: Schematic showing of the station numbering of a single stage EDF. Station 1:

inlet, 2: in between rotor and stator, 3: in between stator and nozzle inlet, 4: nozzle outlet.

2.2.1 Fundamental EDF theory

In both the rotor and the stator blade passages the flow is decelerated with respect to the blades, in other words: there is diffusion in the blade passages. For the rotor, this means that the flow is accelerated with respect to the fixed world, transferring kinetic energy to the flow.

In the stator the flow is deflected to, preferably a pure axial flow, transforming kinetic energy into static pressure. A quasi three-dimensional approach is often used in axial compressor analysis. The flow at every radius is assumed to only have an axial and a tangential velocity component, which vary over the length of the blade. Figure 2.4 shows how the air moves through the blades at one radius. The velocity triangles show the velocity components at stations 1 to 3. Here C and V are respectively the air velocities with respect to the fixed world and to the rotor. Furthermore, α is the angle between C and the axial direction and β is the angle between V and the axial direction. The velocity C can be decomposed into an axial velocity C_a and a tangential component, the swirl velocity C_w. The rotor blade speed is denoted by U and can be calculated from the rotational speed of the rotor: U = ωr, with ω as the rotational speed and r as the radius at which the velocity is calculated.

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Figure 2.4: The flow directions and corresponding velocity triangles in a compressor stage.

Figure adapted from [4].

Compressors are often designed so that the axial velocity Ca is constant through the compressor. Furthermore, since the flow only has an axial and tangential velocity component, every streamline will remain at a single radius. This leads to the conclusion that for a constant rotor RPM the blade rotor blade velocity will be constant at a stream line. Now the rotor velocity triangles can be drawn on a common base, shown in:

Figure 2.5: The velocity triangles from before and after the rotor drawn on a common base.

Figure from [4].

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This figure shows how the flow is decelerated with respect to the rotor (V₂ < V₁) by deflecting the flow. Here the deflection ε is given by β1 − β₂. From the geometry directly follows that the air angles can be related using the equations below:

U

C_a = tan α₁+ tan β₁ (2.8)

U

C_a = tan α2+ tan β2 (2.9)

The first step in the design is to determine the air angles at mid-span and to determine the EDF geometry assuming the air angles constant over the radius. The length of the blades is defined by the radius at the root rr and the tip rt of the blades. The root-tip ratio rr/rt

is an important design variable since different limitations may come to play for low root-tip ratios. These limitations are discussed in chapter 2.2.3. In the first step, an analysis is done at the mid-span radius. The mid-span radius is given by the mean of rr and rt. Below, the mid-span analysis is discussed. The inlet total pressure and total temperature are given by:

T01= T1+1 2

C_a1² cp

(2.10)

p01= p1

T₀₁ T1

_γ−1^γ

(2.11) Here T1 and p1 are the static temperature and pressure at the inlet, cp is the specific heat of air and γ is the ratio of specific heats. The mass flow through the EDF is given by:

˙

m = Ca1πρ1 r²_t − r²_r

(2.12) Here ρ1 is the density at the inlet. The flow is deflected over an angle β2 − β₁ in the rotor. By considering the change in angular momentum the input power of the rotor P_EDF and the total temperature rise of the flow ∆T₀ can be calculated. Given that the compressor is designed such that the axial component of the flow Cais constant through the compressor, so Ca1 = Ca2= Ca3= Ca, the P_EDF and ∆T0 can be calculated using:

P_EDF = ˙mc_p∆T₀

= ˙mUmCa1(tan β1− tan β₂) (2.13) From the total temperature rise, the pressure ratio over the compressor can be calculated using:

R_s= p03

p₀₁ =

1 +ηs∆T0

T₀₁

_γ−1^γ

(2.14) Here η_sis the isentropic efficiency over the compressor. The isentropic efficiency is defined as η_s = (T₀₃⁰ −T₀₁)/(T₀₃−T₀₁). With T₀₃⁰ as the total temperature that would have been found if the pressure ratio was achieved with an isentropic process. Subsubsection 2.3.4 discusses the determination of the isentropic efficiency. The total pressure at the outlet of the compressor is calculated using:

p₀₃= R_sp₀₁ (2.15)

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Given that the nozzle is designed so that the outlet static pressure is equal to the inlet static pressure(p1 = p4) and that the flow expands isentropically(T04 = T03), the static temperature at the outlet can be calculated using:

T₄= T₀₃ p₁ p₀₃

^γ−1_γ

(2.16) Since the nozzle is adiabatic: T03= T04. Now the Mach number of the outlet flow M4 and the thrust of the EDF are calculated by:

M4 = s

2 γ − 1

T₀₃ T₄ − 1

(2.17)

FT = ˙m(a4M4− C_a1) (2.18)

Here R_air is the specific gas constant for air, a₄ =√

γR_airT₄ is the acoustic velocity at the outlet. In the calculation of the thrust Cplane= C1 is assumed. The pressure thrust term is 0, since the flow in the nozzle expands to atmospheric pressure.

2.2.2 Annulus shape

As stated above, EDFs are designed so that C_a is constant through the EDF. To do this, the cross sectional area of the compressor should be determined properly. The inlet root and tip radius are known from the design made using the equations from 2.2.1. Next, the cross sectional area at the other stations can be calculated from the mass flow:

A_i= m˙

C_aρ_i (2.19)

Here Ai, and ρi are the cross sectional area and the density at station i. The density can be calculated using the ideal gas law when the static temperature and static pressure at the stations are known. For station 2, in-between the rotor and the stator, T₂ and p₂ are calculated by:

T₂ = T₀₂− C_a²

2c_p cos² α₂ (2.20)

p⁰₂ = p1

T₂ T1

^γ

γ−1

(2.21) p2 = ηb,rotor(p⁰₂− p₁) + p1 (2.22) Here p⁰₂ is the static pressure after the rotor in case of isentropic compression, p₂ is the actual static pressure and ηb,rotor is the rotor blade efficiency in terms of pressures (see equation 2.48). The blade efficiency of the rotor and the stator are further discussed in chapter 2.3.4.

Given that the flow behind the stator is purely axial (and thus α3= 0°), the static pressure and temperature at station 3, are calculated by:

T₃ = T₀₃−1 2

C_a² cp

(2.23)

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p₃ = p₀₃ T₃ T₀₃

_γ−1^γ

(2.24) Here p₀₃ is calculated using equations 2.15 and T₀₃ = T₀₁+ ∆T₀ is calculated using the total pressure rise calculated in equation 2.13. The efficiency of the rotor and stator has been taken into account using the isentropic efficiency in equation 2.14.

The static temperature at station 4, at the nozzle outlet, is calculated by equation 2.16.

The static pressure at the nozzle is chosen equal to the inlet static pressure.

Now the cross sectional areas are known, the root and tip radii can be determined. How- ever, different combinations giving the same cross sectional area are possible. To fix this often constant root, constant tip or constant mid-span radius is assumed. In case of an EDF the root radius is often determined by the diameter of the electric motor places in the stator hub and thus a constant root radius is chosen.

2.2.3 Limitations in EDF design

Above, the equations needed to calculate the EDF thrust are discussed. However, the maximum input power that an EDF can handle and thus the maximum thrust an EDF can give is limited by a number of factors.

Maximum deflection in the rotor blade passages In the rotor and stator the flow is decelerated with respect to the blades to compress it, there is diffusion in the blade passages.

However, there is a limit to the amount of diffusion that can be realized. This limits the pressure ratio of the compressor. One of the earliest criteria used to judge the amount of diffusion is the Haller number H. Recall the assumption of constant C_a through the EDF, now the Haller number can be expressed in term of air angles for both the rotor as the stator:

Hrotor = V2

V1

= cosβ1

cosβ2

(2.25)

Hstator= C₃ C2

= cosα₂ cosα3

(2.26) The limit H ≮ 0.72 is set. A more sophisticated criterion often used is the diffusion factor.

The Haller number is often used for preliminary design choices as where the diffusion factor is often used in final design calculations. However, in this report only the Haller number is used.

Blade root stress The stress is the blade root due to the centrifugal forces can constraint the compressor design. The maximum stress, which is found at the blade root, is given by:

σr= ρ_b 2U_t²

"

1 − rr

r_t

2#

(2.27) Here ρ_b is the density of the rotor blade, Utis the tip speed of the rotor blade and rr/rtis the root-tip ratio. From equation 2.27 can be concluded that for high rotational speeds and long rotor blades (low root-tip ratio) the root stress increases.

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Degree of reaction In both the rotor as the stator there will be diffusion, increasing the static pressure. The degree of reaction Λ gives a measure of to what extend the rotor and stator are responsible for the total static pressure rise. Λ is formally defined at the ratio of the static enthalpy rise in the rotor over the rise in static enthalpy rise over the entire stage.

By assuming constant cpover the stage, it can be defined in terms of total pressure rise. Now, assuming a constant C_a through the compressor and assuming that the air leaves the stator at the same velocity as it enters the rotor (C₁ = C₃), the degree of reaction can be expressed by:

Λ = ∆hrotor

∆h_rotor+ ∆h_stator = Ca

2U(tan β1+ tan β2) (2.28) A special type of compressor blade design is the case is Λ = 0.5. In this case the rotor and stator contribute equally to the static pressure rise. These blade designs are called symmetrical blading. For good efficiency the degree of reaction must stay between 0 and 1. However, the degree of reaction varies over the blade length. Depending on the three- dimensional flow design the degree of reaction may become negative near the blade root or larger than 1 near the tip. When Λ < 0 the rotor decreases the static pressure and when Λ > 1 the stator decreases the static pressure. The constraint on the degree of reaction can limit the compressor design. The variation of the degree of reaction over the radius will be discussed further in subsection 2.2.4.

Rotor blade Mach number The Mach number relative to the blade is an important consideration in blade section design. To keep the losses to a minimum, a suitable blade section shape should be chosen depending on the blade Mach numbers. Here the blade Mach number varies over the blade length. For a purely axial inflow, so α1 = 0, the rotor blade Mach number as function of the radial position is given by:

Mblade= V

a = pU²+ C_a²

a = p(ωr)²+ C_a²

a (2.29)

Here V is the flow velocity relative to the blade and a is the acoustic velocity. U is the blade velocity which increases with the radius r. Furthermore, ω is the rotational speed of the rotor. For Mach numbers below 0.78 NACA 65-series blade sections are recommended, in case of 0.70 ≤ Mblade≤ 1.20 double-circular arc sections are a good option. For higher Mach numbers more sophisticated blade sections such as multiple circular arc sections or special normal shock-free sections are needed [5]. However, the Mach numbers found in this project do not ask for such sections. Blade section design is discussed more in-depth in subsubsection 2.3.

Supersonic outlet flow In case of high pressure ratios the critical pressure ratio might be exceeded, creating a choking or supersonic outlet flow. To maximally accelerate the flow, and thus create maximum thrust, a converging-diverging nozzle is needed. However, during subsonic operation a converging-diverging nozzle decreases the thrust. This will decrease the available thrust during climb. To counteract this, a variable area nozzle could be made.

However, this will add weight and mechanical complexity to the plane.

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Blocking in the compressor annulus The boundary layer in the annulus thickens along the length of the annulus. The boundary layer reduces the area available for the flow with respect to the cross-sectional area of the annulus, increasing the axial velocity. Although this effect is hard to predict, experiments show that in practice the actual total temperature rise was always lower than calculated using equation 2.13. To cope with this, an empirically determined work-done-factor λ is often used to estimate the actual work that can be supplied to a stage. The effects of blockage especially becomes evident for multi-stage compressors.

For the first stage the λ ≈ 1. However for the fourth stage λ ≈ 0.9, decreasing further for consecutive stages.

2.2.4 Three-dimensional flow

In the sections above, a design was discussed assuming the flow to be constant over the radius. However, the air angles do vary over the length of the blade, amongst others due to the increasing blade velocity for increasing radius. Furthermore, since there is a swirling flow, there must be a pressure gradient to generate the needed centripetal forces. The variation of the total enthalpy as a function of the variation of the axial and tangential velocities is given by:

dh0

dr = Ca

dCa

dr + Cw

dCw

dr + C_w²

r (2.30)

This equation is known as the vortex energy equation. The complete derivation of this equation can be found at page 206-207 of [4]. In this derivation any radial velocity components, any acceleration along a streamline and the gravitational forces are neglected. Furthermore, all second order derivative terms are dropped.

Often in compressor design, the design condition ’constant specific work’ is applied. Since the total enthalpy distribution at the inlet of the EDF is uniform, the enthalpy distribution will remain uniform through the compressor under this design condition, so dh0/dr = 0. Next a certain distribution of Cw or Camust be chosen. A specific case is an uniform axial velocity distribution, so dC_a/dr = 0. Now the vortex energy equation reduces to:

dC_w

dr = −C_w

r (2.31)

Integration of the equation above gives: C_wr = constant. This is known as the free vortex condition. The constant can be calculated using the flow design at the mid-span. At the inlet of the EDF, the flow will be purely axial, so Cw = 0 and thus α1 = 0. Substitution of Cw = 0 in the free vortex condition shows that the condition is satisfied. From the geometry of the velocity triangles (see Figure 2.5) follows that the mid-span swirl velocity after the rotor can be calculated by:

Cw2m= Ca2mtan α2m (2.32)

Here Cw2m and Ca2mare the swirl and axial velocity and α2mis the air angle with respect to the fixed world, all at station 2 (after the rotor) at mid-span. Now the air angle α2 can be calculated as function of the radial position by:

α2 = atan C_w2mrm

Car

(2.33)

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Here C_ais the axial velocity, which was taken constant over both the radial as axial position in the compressor. Using equation 2.9 the relative blade air angle β2 can be calculated from α₂. Given that the stator brings the flow back to a purely axial flow, so α₁ = α₃= 0 and thus C₁ = C₃, equation 2.28 can be used to calculate the degree of reaction. Now the variation of the degree of reaction over the radial position can be calculated by:

Λ = 1 +

1 −2r_m r

(1 − Λ_m) (2.34)

Here, r is the radius, r_mis the mid-span radius and Λ_m is the mid-span degree of reaction.

2.3 Blade design

2.3.1 Blade cascade geometry

After the three-dimensional flow angles are determined, the rotor and stator blade shapes need to be designed so that the designed flow is achieved.

In compressor blade design, a quasi three-dimensional approach is often used. The flow is assumed to only have an axial and tangential velocity component that vary over the radial position. The blades are designed by considering the two-dimensional flow angles at every radius. Next, the blade sections from every radius are ’stacked’ to create the total blade shape. The figure below shows a part of a cascade with the important geometry parameters indicated:

Figure 2.6: Three blade sections part of a typical cascade with the important angles and dimensions. Figure adapted from [4].

Here α⁰₁ and α⁰₂ are the blade in- and outlet angle, α1 and α2 are the air in- and outlet angle, V₁ and V₂ are the in- and outlet air velocity, θ = α⁰₁ − α₂⁰ is the camber angle and ζ is the angle between the chord and the axial direction called the stagger angle or blade setting. The chord is an imaginary line connecting the leading and trailing edge of the blade section and its length is given by c. Furthermore, s is the pitch or spacing between the blades,

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i = α₁− α⁰₁ is the incidence angle, AoA = α₁− ζ is the angle of attack and δ is the deviation angle.

Note that when discussing velocity triangles, α denotes the air angles with respect to the fixed world and V denotes the velocity with respect to the rotor blade. From now on this use of the variables will be called the velocity triangle notation. When discussing blade section design α and V always denote the air angle and air velocity with respect to the blade, both in rotor and stator design. This will be called the cascade notation.

From the large number of parameters shown in Figure 2.6 it may appear that many parameters need to be determined to get to a section design. However, note that many of the parameters above are equivalent. For example, both the incidence angle as the angle of attack describe the position of the blade relative to the air. Given that α1 and α2 are known and that the section has a symmetrical mean-line, only 4 parameters need to be determined to get a fully constrained blade design: c, s, θ and ζ. In case of an asymmetrical mean-line the exact shape of the mean-line must be known to calculate α⁰₁ and α⁰₂. All other parameters mentioned above can be calculated from these four parameters and the air angles using simple equations that follow directly from the geometry.

2.3.2 Blade section shape

As already mentioned in section 2.2.3 different blade section shapes can be used depending on the blade Mach number. Examples of blade sections are the NACA 65-series and double- circular-arc(DCA) sections. The former is recommended for blade Mach numbers up to 0.78, the latter for blade Mach number between 0.70 and 1.20 [5]. For higher blade Mach numbers other shapes have been developed. However, these Mach numbers will not be encountered within this project. Many systematic cascade tests have been done on NACA 65-series blade sections. Using this data the performance of a blade design can be estimated.

In general blade sections shapes are defined using a mean-line and a thickness distribution with respect to this mean-line. The construction of a NACA 65-series section is visualized in the figure below:

Figure 2.7: Construction of a NACA section. The upper and lower surfaces are defined using a thickness distribution with respect to a specific mean line. Figure from [6].

For a NACA 65-series section the mean-line is defined with respect to the chord line. The shape of the mean-line depends on the camber, which for NACA sections is defined by the design lift coefficient of the isolated air foil C_l0. The mean-line for a NACA 65-series section with C_l0= 1.0 is described by the stations x along the chord line and the mean-line ordinates ym given in table A.1 [7]. To obtain a mean-line for a different design lift coefficient the

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ordinates can be multiplied by the desired C_l0 [6][7]. As is common in NACA wing section literature, all stations and ordinates are given in percentage of the chord length. The mean- line slope dy_m/dx is tabulated in table A.1. The mean-line slope at the leading and trailing edge are not given because for those stations the slope is theoretically infinite. The mean-line angle φ can be calculated from the mean line slope using simple goniometry.

Next, the upper and lower surfaces of the section are defined by a thickness distribution perpendicular to the mean-line. Table A.1 gives the thickness ordinates in percentages of the chord for a NACA 65-series section with a maximum thickness of 0.1c. To calculate the thickness profile of a section with a different maximum thickness the ordinates can simply be scaled. The upper and lower surface coordinates can now be calculated using:

x_U = x − ytsin φ (2.35)

y_U = y_m+ y_tcos φ (2.36)

xL= x + ytsin φ (2.37)

y_L= y_m− y_tcos φ (2.38)

Here x_U and y_U are the coordinates for the upper surface and idem for the lower surface.

Normally the blade in and outlet angle are defined by the mean-line slope at the leading and trailing edge. However, since the mean-line slope is theoretically infinite for this does not make sense for NACA 65-series profiles. Instead an equivalent camber angle is defined based on an equivalent circular-arc mean-line. For cambers 0.0 < C_l0 < 2.4 the equivalent camber angle in degrees is given by[5]

θ = 24.7C_l0 (2.39)

Since a circular arc mean-line is symmetrical, the blade in and outlet angles can be calculated by:

α⁰₁ = ζ +1

2θ (2.40)

α⁰₂ = ζ −1

2θ (2.41)

For blade Mach numbers above 0.78 the NACA 65-series sections are not suitable. Instead DCA sections can be used. Double-circular-arc blade(DCA) section, both the suction and pressure surface of the blade consist of a circular section. Because of this, the mean-line is also a circular arc. One downside of the use of DCA section is the limited amount of available test data. An example of DCA blade section is shown below:

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Figure 2.8: Example of a double circular arc blade section. Recommended for blade Mach numbers between 0.70 and 1.20.

2.3.3 Blade design using empirical data

Blade design is often done using empirical data from two-dimensional experiments. In such experiments a row of blades is positioned in a two-dimensional flow wind tunnel. Next, for a fixed blade shape, the solidity c/s, the incidence angle and the air inlet angle are varied. The achieved deflection , losses and forces acting on the blades are measured. To use this data to design blade shapes, the data has been transformed into diagrams or empirical formulas that can easily be used by the engineer.

To come to a complete blade section design one should determine the chord length c;

the blade spacing s; the camber; the blade setting or stagger angle ζ; or other parameters equivalent to those. Note that the camber can be expressed in camber angle θ or as the design lift coefficient C_l0. The latter is common in NACA 65-series design.

The first step to a complete blade design is to choose the chord length. Chord length is often determined by considering the aspect ratio l/c of the blades. Here l = r_t− r_r is the length of the blade which is known because the root and the tip radius already have been determined using the theory in section 2.2.1 and 2.2.2. The aspect ratio is important when concerning blade losses. A good first estimate for the aspect ratio is l/c = 3.

Next the blade spacing s needs to be determined. In American literature the blade spacing is often considered in terms of the solidity σ = c/s. Note that the blade spacing directly depends on the number of blades and that the spacing varies over the blade length. In practice the choice in number of blades is limited by the available test data and the fact that the number of blades is preferably a prime number. By choosing a prime number, one avoids common multiples in the rotor and stator. This reduces the chance of introducing resonant forcing frequencies.

With the chord length and blade spacing known, only the camber and stagger angle need to be determined. Two possible methods are discussed in the sections below.

Method 1: determine θ and ζ using an empirical formula expressing δ To finish the blade design, a camber angle θ and blade setting ζ must be chosen so that the desired deflection is achieved. One method is to choose a value for the incidence i, and to then determine the camber angle using an empirical equation for δ. Here δ is the deviation. Ideally the air would leave the blade cascade exactly in the direction the trailing edge points. In real life however, a deviation δ is found. Often i = 0° is chosen. The camber angle can be calculated by:

θ = α₁− i − α₂+ δ (2.42)

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The deviation can be calculated using empirical equations:

δ = mθp

1/σ (2.43)

Here m is a parameter that can be calculated using equations 2.44 [4] or 2.45 [5]:

m = 0.23 2a c

2

+ 0.1

α2

50

(2.44)

m = 0.216 + 0.000875ζ + 0.00002625ζ² (2.45) Note that all angles are in degrees! Here a is the point of maximum camber measured along the chord line from the trailing edge. In case of a symmetrical mean-line 2a/c = 1.

Furthermore, α2 is the air outlet angle and σ is the solidity of the blade cascade.

Now, given that the mean-line is symmetrical, ζ can be calculated by:

ζ = α₁− i − 1

2θ (2.46)

Method 2: determine θ and ζ using figures from [8] A second method to determine the camber θ and stagger ζ angle uses carpet plot from [8]. In [7] two-dimensional cascade test were done on NACA 65-series blades, systematically varying the geometry. In [8] this data was summarized and visualized in carpet plots that can easily be used by engineers to determine the blade geometry once the blade inlet air angle α1, the deflection and the solidity σ are known. The figures cover a range of α1 = 30 − 70°, = 2−22° and σ = 0.5−1.5.

First, the camber C_l0must be determined using Figure 1 from [8], which shows the camber Cl0 as a function of air inlet angle α1, the desired deflection = α1 − α₂ and the solidity σ = c/s. Next, the design angle of attack AoA can be determined using Figure 2 from [8], which shows the design angle of attack as a function of C_l0 and σ. Appendix C explains how to use the carpet plots. Now, the camber angle can be calculated using equation 2.39 and the blade setting ζ can be calculated by:

ζ = α₁− AoA (2.47)

2.3.4 Compressor blade efficiency

In equation 2.14 the isentropic efficiency is needed to determine the compressor pressure ratio.

Below, the calculation of ηs is shown.

The blade row efficiency in terms of pressures is given by:

η_b = p₂− p₁ p⁰₂− p₁ = 1 −





¯ w

1 2ρV₁²

∆pth 1 2ρV₁²



 (2.48)

Here p⁰₂ is the static pressure at station 2 in case of isentropic compression and p₂ is the actual static pressure. Furthermore, ¯w is the total pressure loss and ∆p_th is the theoretical pressure rise over the blade row. The denominator is called the pressure rise coefficient and can be calculated using:

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∆pth 1

2ρV₁² = 1 −cos² α1

cos² α2

(2.49) The enumerator of equation 2.48 is called the loss coefficient and can be calculated using:

¯ w

1

2ρV₁² = C_D

1 σ

cos³ αm

cos² α1

(2.50) Here α_m is the mean flow angle is the blade row and C_D is the overall drag coefficient given by:

α_m= 1/2 (tan α₁+ tan α₂) (2.51)

CD = CDp+ CDA+ CDS (2.52)

Here CDp is the profile drag coefficient over the blade section, CDA is the annulus drag coefficient and C_DS is the secondary losses drag coefficient. The profile drag coefficient of a NACA 65-series section can be determined from Figures 7 to 84 from [7]. C_DA and C_DS are given by:

C_DA = 0.020s l

(2.53)

CDS = 0.018C_L² (2.54)

Here l is the blade length and CL is the section lift coefficient that can be from the same figures as the profile drag coefficient[7] or with the equation below:

CL= 2 1 σ

(tan α1− tan α₂) cos αm− C_Dp tan αm (2.55) Now the blade efficiencies for the rotor and stator can be determined using the equations above. Although the η_b and the isentropic efficiency η_sare different things, their actual values are very close and without much error η_b = η_s can be used. The isentropic efficiency can be calculated using:

η_s = Λ η_b,rotor+ (1 − Λ) η_b,stator (2.56)

2.4 Dimensionless parameters

The performance of propellers and EDFs is often expressed using dimensionless numbers.

Important dimensionless numbers are shown below:

C_P = PEDF

ρω³r_m⁵ (2.57)

CF = FT

ρω²r_m⁴ (2.58)

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J = C_plane ωrm

(2.59) Here CP and CF are the power and thrust coefficient. These are non-dimensional measures for the EDF power and thrust. The can be interpreted just as the lift coefficient describes the lift of a wing. The advance ratio is a measure for the working point of the rotor. The advance ratio is equivalent to the relative rotor inlet angle β1 and the rotor angle of attack AoA at mid-span.

The static thrust is often presented as C_F/C_P as a function of C_P. The figure below shows a typical shape of a static thrust curve for a propeller.

Figure 2.9: Typical shape of a C_F/C_P vs C_P curve describing the static performance of a propeller. Image from [20]

The performance of an EDF is expected to have a similar shape since an EDF is basically a propeller with a housing.

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Chapter 3

Design

In this section the design of a high speed EDF is discussed. Since the high speed plane is not yet designed, it is not exactly known what the exact requirements of the EDF are. Because of this, an analysis is done of the potential size of the plane and the optimal size of the EDF to go with this plane. Next, a design for a high speed EDF is made. Finally, a low-speed prototype is designed using the high-speed design as a basis.

3.1 Requirements and concept

The model airplane used for the current Guinness world speed record of 750 km/h=208.3m/s, is propelled by a Behotec 180 model jet engine [11]. According to manufacturer specification, this jet engine can generate 180N of thrust[12]. This means that the useful work the jet engine must deliver at top speed is FT · C_plane = 180 · 208.3 = 37.5kW . The Schubeler DS-98-DIA HST EDF owned by team Air/e can deliver up to 130N of static thrust. However, this EDF has a maximum power of only 10kW and therefore will never be able to create enough thrust at high speeds[13]. To create a plane capable of flying 750 km/h a new high power EDF must be designed.

The general requirements and preferences for the EDF are stated below:

Requirements:

1. The EDF must be able to propel a, yet to be designed model airplane up to speeds of 750 km/h

2. The EDF must be compatible with typical model airplane electronics 3. The used motor must be commercially available.

4. The EDF must provide cooling for the electric motor .

Preferences:

1. The EDF must be as light as possible 2. The EDF must be as efficient as possible

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Here should be kept in mind that the maximum weight of a model airplane is 25kg. In case that the mass of the EDF is too large or the efficiency is too low, this can cause problems.

Keeping the requirements and preferences in mind a layout for the EDF has been chosen:

Figure 3.1: Schematic layout of the EDF that is designed. The rotor is directly mounted on the axle of the electric motor. The motor is glued onto the stator blades. The nozzle is a separate part and is mounted after the stator.

To keep the EDF compact and light, the rotor is mounted directly on the motor shaft, instead of supporting is separately. The motor is supported by the stator blades and in direct contact with the airflow to enhance cooling. In this configuration, the root-radius of the stator is determined by the outer diameter of the stator. Furthermore, the stator blades are structural parts that need to be strong and stiff enough to support the vibrations and forces working on the motor and rotor. The EDF is mounted using brackets on the housing. The nozzle inner and outer surfaces are separate parts for easy production and such that in later research different nozzle shapes can be tested. The nozzle inner surface smoothly converges to minimize losses in the wake of the EDF.

In the section B the viability of a two-stage EDF is considered. The schubeler EDF, by manufacturer specifications, has a maximum outlet velocity of 82m/s. This is obviously not enough the propel a plane flying at 208.3m/s(=750km/h). In an earlier project a two stage EDF using a separate motor for each rotor was proposed[21]. Here two motors are used to drive the two rotors to make a more powerful EDF. Another advantage of two motors is the fact that this way the rotational speed of the rotors can be controlled separately, which might be beneficial for the performance. Since each stage has a separate motor, the layout of a two-stage EDF is very similar to that of a single-stage EDF:

Figure 3.2: Schematic layout of a two-stage EDF. Two rotor-stator pairs are places behind each other. Each rotor is powered by its own motor.

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When searching for the optimal EDF, the mass flow is an important parameter to consider.

A larger mass flow gives a higher propulsion efficiency, as explained in section 2.2. However, a larger mass flow also gives a larger EDF which gives a higher drag force on the EDF housing. Clearly an optimum mass flow, giving a minimum EDF input power can be found.

In section B an analysis on the optimal EDF configuration is done. Here, the optimal mass flow through the EDFs is determined as function of the mass flow. Next, two different EDF configurations were compared, two single-stage EDFs and one two-stage EDF. Note that in both configurations two motors are used. The analysis showed that a propulsion using two single-stage EDFs uses less input power. This is caused by the fact that a two stage EDF has a slightly lower optimal mass flow than a configuration using single-stage EDFs, but not low enough to create such a small EDF that the drag force on the EDF housing is becomes smaller. From this can be concluded that a propulsion using single-stage EDFs is the most promising. In the rest of this report is focused on the design of a single-stage EDF.

3.2 Selection of potential electric motors

This project focusses on the design of the high speed EDF. However, since the plane is not designed yet, the exact thrust and power the EDF must produce are not known. The plane used for the current Guinness world speed record had 180N of thrust which gives that the useful propulsion power at a speed of 750km/h must be 37.5kW. To design an EDF capable of transferring such powers, a compact high power electrical motor is needed. The most powerful electric motors that were still suitable for model airplane applications that I could find, are those produced by Lehner Motor Technik(LMT). The LMT motors are 3-phase brushless in-runner motors.

LMT produces motors in different sizes and with different number of windings. The size of the motor determines motor constant K_m(sometimes called motor size constant) of the motor:

Km =

√P

ω (3.1)

Here P is the power of the motor and ω is the rotational speed of the motor. The larger the motor the higher K_m and the more powerful the motor. The number of windings determines the motor speed constant Kv of the motor:

K_v = ω_noload

U (3.2)

Here U is the voltage over the motor and ω_noloadis the rotational speed of the motor when no mechanical load is applied to the shaft. The K_v is inverse proportional to the number of windings. Note that the power-RPM curve is independent on the Kv. For a fixed motor size, increasing the number of windings decreases K_v, which increases the needed voltage and decreases the needed current of the motor.

The LMT3080 is the most powerful motor produced by LTM. For this motor the maximum voltage is 57.6V which corresponds with the nominal voltage of 16 LiPo battery cells in series.

The maximum allowed voltage is independent on the number of windings, this gives that the maximum power increases with an increasing Kv and thus with a decreasing number of windings. The tables below show the general properties of the LMT3080 motor and the performance specifications as function of the number of windings:

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Table 3.1: General specifications of the LMT 3080 electric motor.[14]

Weight 1.65 kg

Diameter 60 mm

Length 135 mm

Max. voltage 57.6 V

Mechanically max. allowed RPM 50.0 kRPM

Table 3.2: Performance of the LMT3080 for different windings at maximum power. Here Pm,in: input power, Pm,out: output power.

No. of windings I [A] Pm,in [kW] Pm,out [kW] η [-] ω [kRPM]

5 700 40.320 38.566 95.7 59.286

6 669 38.534 36.781 95.5 49.030

7 506 29.145 27.781 95.3 41.991

8 397 22.867 21.771 95.2 36.716

As stated in Table 3.1, the mechanical RPM limit for the LMT3080 is 50kRPM. However, the manufacturer also states that exceeding this max RPM up to 30% for short-term does not damage the motor. An important parameter in motor choice is the power density ρ_power. The LMT 3080 has relatively high power density of up to 23.4 kW/kg when ran at maximum power and up to 22.3 kW/kg when the maximum allowed RPM is 50kRPM. A downside of the LMT electric motors is that these are relatively expensive. Cheaper options can be purchased on sites such as Hobbyking.com. However, the motors on these websites have a maximum power density of approximately 10kW/kg, which is significantly lower. Concluding:

the LMT3080 is chosen as electric motor to be used in the EDF design. To come to a final motor choice still a number of windings must be chosen. In the section below the mid-span design is made. Here, in consideration with the mid-span design a number of windings is chosen.

3.3 Motor choice and mid-span design

To get to a complete mid-span design the inlet root and tip radii r_r and r_t; 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 r_t, 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:

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ω = U_t rt

= pM_t²γRairT1− C_a1² r_t

= r_r r_t

pM_t²γR_airT₁− C_a1² r_r

(3.3)

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 M_t 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.

tanβ₁= U_m Ca1

(3.4)

cosβ₂ = cosβ₁ Hmin

(3.5) 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 M_t = 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.

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Figure 3.3: The maximum rotor input power P_EDF 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

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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 M_t= 1.0 is chosen, which gives H_rotor= 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 parameters 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 r_r 0.030 m

Tip radius r_t 0.062 m

Root-tip ratio 0.48

Mass flow m˙ 2.32 kg/s

Isentropic thrust force FT ,isentropic 118 N

Thrust force F_T 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

Eindhoven University of Technology MASTER Design, Production and Testing of a High-Speed Electric Ducted Fan for a Record-Speed Model Airplane van Oorschot, Joep

Design, Production and Testing of a High-Speed Electric Ducted Fan for a

Record-Speed Model Airplane

Graduation report

Joep van Oorschot, 0769268

Summary

Contents

Chapter 1

Introduction

1.1 Objectives and report outline

Chapter 2

Theory

2.1 Airplane drag theory

2.2 EDF theory

2.3 Blade design

2.4 Dimensionless parameters

Chapter 3

Design

3.1 Requirements and concept

3.2 Selection of potential electric motors

3.3 Motor choice and mid-span design