Konceptuell design av ett elektriskt drivet kommersiellt flygplan

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IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM SWEDEN 2019,

Conceptual Design of an Electric Powered

Commercial Aircraft

ASTRID HAFVENSTRÖM ANDREAS PETTERSSON

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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INOM

EXAMENSARBETE TEKNIK, GRUNDNIVÅ, 15 HP

STOCKHOLM SVERIGE 2019,

Konceptuell design av ett

elektriskt drivet kommersiellt flygplan

ASTRID HAFVENSTRÖM ANDREAS PETTERSSON

KTH

SKOLAN FÖR TEKNIKVETENSKAP

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Abstract

The purpose of this bachelor thesis is to design an electric powered commercial short range aircraft that is set to take-oﬀ in 2030 with reasonable technical advancement assumptions made.

The aircraft is designed with the ATR 42-500 as inspiration and has therefore similar requirements. The aircraft has a payload of 5070 kg and cruises at 7600 m above sea level. It has a max speed of Mach 0.5 and a stall speed of 41 m/s. Climb rate is 560 m/min, take-oﬀ distance is 1165 m and landing distance is 960 m.

The conceptually designed aircraft has a range of 400 km that is approximately the distance London-Amsterdam and is able to carry up to 48 passengers in a two by two seat configuration. Batteries are expected to improve with 30 % during the next ten years resulting in a maximum take-oﬀ weight of 19900 kg, where 3220 kg is battery weight. Fuel powered it has a maximum take-oﬀ weight of 19200 kg and a fuel weight of 2900 kg.

The power needed for propulsion was found to be 4.18 MW which would be equally divided over the engines that drive the two propellers. These are positioned one on each wing.

The 26 m long aircraft is equipped with an unswept high mounted wing with a wingspan of 29 m and a wing reference area of 75 m². The horizontal stabilizer is 13 m² and the vertical stabilizer is 11 m².

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Sammanfattning

Syftet med detta kandidatexamensarbete är att konceptuellt designa ett elektriskt drivet kommersiellt flygplan som ska kunna flyga år 2030 med rimligt antagen teknisk utveckling fram tills dess.

Flygplanet är designat utefter den redan existerande ATR 42-500 och har därmed en liknande kravspecifikation. Flygplanet har en nyttolast på 5070 kg och flyger 7600 m över havet. Maxhastigheten för planet är Mach 0.5 och överstegringshastigheten är 41 m/s. Flygplanet har en stighastighet på 560 m/min, en startsträcka på 1165 m och en landningsträcka på 960 m.

Vidare har flygplanet en räckvidd på 400 km med batterier som motsvarar ungefär sträckan London-Amsterdam. Det kan bära upp till 48 stycken passagerare i en två-sätes-konfiguration. Med en 30 % förväntad förbättring av batterier under de närmaste tio åren har flygplanet en maximal starvikt på 19900 kg, där 3220 kg är batterivikt. Med vanligt flygplansbränsle har flygplanet en maximal startvikt på 19200 kg och en bränslevikt på 2900 kg.

Den nödvändiga eﬀekten för framdrivning fanns som 4.18 MW vilket skulle de- las lika över motorerna som driver de två propellrarna. Dessa är positionerade en på vardera vinge.

Flygplanet är 26 m långt och utrustad med osvepta högmonterade vingar med ett vingspann på 29 m och en vingarea på 75 m². Den horisontella stabilisatorn är 13 m² och den vertikala stabilsatorn är 11 m².

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1. Introduction

The aim for this bachelor thesis was to present a conceptual design of an electric powered commercial aircraft that is set to take oﬀ in 2030. Reasonable technological advances could therefore be assumed. This degree project is a part of the bachelor program at KTH Royal Institute of Technology and is intended as an extensive project to further enhance problem solving and engineering. The project has been conducted under the counseling and tutoring of Fredrik Edelbrink and Christer Fuglesang.

1.1 Background

Increasing focus on lowering CO2 emissions leads to interest in development of electric transports. This is because among the benefits of an electric powered aircraft lower negative impact on the environment is one. This would however only be the case if the electricity used comes from a renewable source. For fuel powered aircraft the shorter the flight the more carbon dioxide emissions per kilometer. The most traﬃcked routes are according to (Casey 2018) around 500-1000 km which would be seen as short-haul. These could therefore be considered most detrimental to the environment.

Electric propulsion will produce far less thrust than a conventional fuel powered aircraft and therefore the planes will most likely be slower (Masunaga 2016). This could partly be seen as a benefit since electric propulsion would be quieter (Garrison 2009).

The possibility of electric aviation requires more energy dense batteries. The best batteries on the market at this time are lithium-ion which have an energy density of 250 Wh/kg but according to (Chandler 2019) possibilities for an energy density of 400 Wh/kg is soon obtainable for lithium-sulfur batteries.

Some electric aircraft that already exist are NASA’s X-57 (Loﬀ 2016), an exper- imental aircraft, and Pipistrel ALPHA Electro (Pipistrel n.d.) which is on the market as a flight school airplane. The X-57, with 14 electric engines distributed on the wings, had the intention of lowering the energy needed by distributing the power. Both airplanes are two-seated. Eviation’s Alice (Eviation n.d.) and Zunum Aero (ZunumAero n.d.) both intend to launch electric airplanes in the near future that can take up to 9 and 12 passengers respectively. The Alice commuter focuses on a composite structure to reduce weight in order to be all-electric. Zunum Aero is at the moment a hybrid meaning it is partially driven by fuel when needed. The aim is to in the future be all-electric also.

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2. Mission specifications

The mission of the aircraft is commercial flight for short-haul routes.

2.1 Mission profile

The mission profile of the aircraft was constructed in order to better understand the diﬀerent stages of flight. Each segment represents a special part of the flight-route and describes how much percentage of the fuel-load is left after each segment. With W0 representing the fuel consumption at start-up, , W1 at taxi, W2 at take-oﬀ, W3

at climb, W4 at cruise, W5 at descent, W6 at loiter, W7 at landing and W8 at taxi to gate. The mission profile can be seen in Figure 2.1.

Figure 2.1: Mission Profile

2.2 Performance requirements

For an aircraft to carry 48 passengers and a crew of 3 with 10 kg of luggage per person the total payload was set to 5260 kg. The range with maximum passengers was 716 nm (1326 km), loiter was desired to be 0.5 h and maximum altitude was set to be 7600 m. The maximum speed of 330 kts (Mach 0.51) was chosen to be 10

% greater than the cruise speed of 300 kts (Mach 0.46). Stall speed Vstall was given as 80 kts (148 km/h), which is 1.3 times less than the approach speed, and the rate of climb Vv as 564 m/min. The distance for take-oﬀ Stot was chosen as 1165 m for maximum take-oﬀ weight and landing as 964 m for maximum landing weight.

3. Aerodynamics

The lift-to-drag ratio, L/D, was chosen to be 15 based on values from similar turboprop aircraft according to (Raymer 1999).

3.1 Drag coeﬃcients

The minimum drag coeﬃcient, CDmin, for a turboprop commuter with assumed flaps in take-oﬀ (T-O) position was chosen to be 0.025 based on (Gudmundsson 2013).

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The drag coeﬃcient during T-O, CDTO, was found to be 0.04 from (Gudmundsson 2013).

3.2 Lift coeﬃcients

The lift coefficient during T-O, CLTO, was found to be 0.08 in the same table as the drag coefficients and the maximum lift coefficient, CLmax, was chosen to be 2.5 based on values for airfoils with flaps deflected. The lift-induced drag constant, k, is given by

k = 1

⇡· AR · e (3.1)

where Oswald’s span eﬃciency e for a straight wing with normal AR was found as

e = 1.78 1 0.045· AR^0.68 0.64 (3.2)

where the aspect ratio, AR, was estimated with values of wing span and wing area from the ATR 42-500 (ATR 2014) to be 11.08.

4. Weight estimations

The weights of the aircraft were found through calculations for a fuel driven aircraft.

These were used as reference when determining the weights in the battery driven case.

4.1 Fuel weight calculations

The total weight of the aircraft, WTO, with fuel as energy source was calculated by W_TO = WE + W_PL+ WC+ WF. (4.1) The sum of WPLand WC were given in the mission specification as the total payload.

The empty weight is WE and fuel weight is WF. Equation (4.1) can be rewritten as W_TO= W_PL+ WC

1 _W^W_TO^F _W^W^E

T O

. (4.2)

The fuel fraction WF/WT O was calculated with a 6 % allowance for reserved and trapped fuel according to (Raymer 1999) as

WF

W_TO = 1.06· (1 W_end W0

) (4.3)

where

W_end W0

= W1

W0

W2

W1

W3

W2

W4

W3

W5

W4

W6

W5

W7

W6

W8

W7

. (4.4)

W_end/W0 is split into segments representing the parts of the mission profile. The segments that represent cruise, W5/W₄, and loiter, W6/W₅, were calculated with the Breguet-equations for range and endurance, (Raymer 1999). The rest of the

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fractions were given by the "Fuel fractions estimates" table in (Roskam 2005) and is based on historical statistical data.

The empty weight fraction, WE/W_TO, was found according to (Webster 2005) as W_E

W_TO = aWTO+ b (4.5)

where a is 7.754 · 10 ⁸ and b is 0.576. Both are constants for a jet transport.

Equations (4.2), (4.3) and (4.5) give the total take-oﬀ weight with fuel.

4.2 Battery weight calculations

The battery weight fraction was found with the range-formula for an electric engine:

m_batt

m = WB

W_TO = R· g

L

D · ⌘ · E^⇤ (4.6)

where R is the range. In order for the battery weights to be somewhat close to the fuel values the initially required range was reduced to 400 km. L/D is the lift-to- drag ratio and ⌘ is the eﬃciency of the electric engine and the battery installation, which was chosen as 90 %. E^⇤ is the specific energy of the batteries with a pre- dicted increased eﬃciency of 30 % in the next ten years which amounts to about 500 Wh/kg according to (Chandler 2019).

The total weight of the aircraft with batteries could then be calculated by using equation (4.6) and

W_TO = W_PL+ WC

1 _W^W_TO^B _W^W_TO^E . (4.7)

It was found to be 19920 kg.

5. Constraint analysis

To determine the required wing size and power plant for the aircraft a constraint analysis was carried out. These were determined by finding desired values of wing loading, represented as W/S, and thrust-to-weight (T/W ). They can be read from the constraint diagram, shown in Figure 5.1, given by equations (5.1)-(5.6) found from (Gudmundsson 2013).

5.1 T/W for a desired take-oﬀ distance

The constraint for a desired distance of the ground run during T-O was found with:

T

W = V_LOF²

2· g · S^g + qTO W

S

CDTO + µ 1 qTO· CLTO

W S

!

. (5.1)

The lift-oﬀ speed is VLOF calculated as 10 % greater than the stall speed. The gravitational constant is g, and qTO is the dynamic pressure at sea level and VLOF/p

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The drag coeﬃcient is CDTO and the lift coeﬃcient is CLTO at T-O. The distance of the ground roll Sg is found in (Anderson 1999) as

Sg = S_tot Sa= S_tot 6.96· V_stall²

| {zg }

R

· sin

✓ arccos

✓

1 h

R

◆◆

| {z }

✓

= 1006m (5.2)

where Stot is the total take-oﬀ distance and Sa is the airborne distance. The turn radius, R, is given as a function of the stall speed, Vstall, and g. The angle ✓ between the ground and lift path is dependent on R and the height, h, needed to clear an obstacle. This was found as 10.7 m for a commercial jetliner according to standard regulations.

5.2 T/W for a desired rate-of-climb

The power required for the desired climb rate is given by T

W = VV

V + qROC W

S

C_Dmin+ k q_ROC

✓W S

◆

(5.3) where VV is the vertical speed which in this case is the rate-of-climb and V is the optimum climb speed. The dynamic pressure, qROC, is at sea level and lift-oﬀ speed and k is the lift-induced drag constant. The minimum drag coeﬃcient is CDmin.

5.3 T/W for a desired cruise airspeed

Thrust-to-weight ratio for desired cruise airspeed was found as T

W = q· CDmin 1

W S

+ k

✓1 q

◆ ✓W S

◆

(5.4) where the dynamic pressure q is at maximum altitude and cruise speed.

5.4 T/W for a level constant velocity turn

To ensure that the aircraft is able to roll with a certain banking angle at a selected altitude, T/W for a level constant velocity turn was obtained. The equation for this is given by:

T W = q

"

C_Dmin

W S

+ k

✓n q

◆2✓ W

S

◆#

(5.5) where n is the load factor depending on bank angle. Set to 45° for this requirement.

5.5 Wing loading as a function of stall speed

The wing loading value, which is a fraction of weight (W ) over wing area (S) can be found through

W S = 1

2⇢_sea· Vstall² · CLmax, (5.6) 5

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which is independent of T/W, to 2627 N/m². The density ⇢sea is at sea level, Vstall

is the stall speed and CLmax the maximum lift coeﬃcient.

5.6 Constraint diagram

The constraint diagram from which the T/W value was found from is shown in Figure 5.1. T/W for a wing loading of 2627 N/m² was about 0.21.

Figure 5.1: Constraint diagram with displayed data point of selected T/W for required wing loading.

6. Initial sizing

The overall configuration of the aircraft was found through initial sizing. This gives the design, dimensions and positions of wing, fuselage and tail.

6.1 Wing

The initial sizing of the wing of the aircraft was made by assuming certain values and calculating others.

6.1.1 Sweep angle

The sweep angle was chosen to zero degrees as the aircraft is subsonic and has a cruise speed below Mach 0.5. It is therefore not essential to have any sweep as it may aﬀect the stability of the aircraft, (Raymer 1999). A typical commercial airliner

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like the Boeing 737 has a sweep angle about 25 degrees as it cruises around Mach 0.78.

6.1.2 Taper ratio

The taper ratio of the aircraft, that is the ratio between the root chord and the tip chord, was chosen to 0.5, (Raymer 1999).

6.1.3 Wing twist

In order to counteract the wing tip from stalling and to obtain an elliptical wing distribution a small wing twist is applied. The wing twist was estimated to 3°.

Typically, wings are twisted between zero and minus five degrees to delay wing tip stall, (Raymer 1999).

6.1.4 Dihedral

Since the aircraft is a civil aircraft with an unswept high-mounted wing the dihedral of the wing was chosen according to Table 4.2 in (Raymer 1999) to two degrees. For this kind of wings the dihedral should be kept between 0-2 degrees, (Raymer 1999).

6.1.5 Wing reference area and wingspan

The size of the wing, the wing reference area (Sref), was determined dividing the maximum take-oﬀ weight of the aircraft by its minimum wing loading computed from the constraint diagram:

Sref = WT O

(^W_S). (6.1)

W_TO is the maximum take-oﬀ-weight with batteries. The reference area obtained was approximately 75 m² compared to the one of the ATR 42-500 that is 55 m². The wingspan, b, was determined with the aspect ratio calculated in the constraint analysis section and the wing reference area through

b =p

AR· Sref. (6.2)

The wingspan was calculated to 28.7 meters which also is a good estimation as the expected value of the wingspan of the ATR 42-500 is about 24.6 m. Since the wingspan of the designed aircraft is about four meters wider than the one of the ATR 42-500 the wing reference area of the designed aircraft is consequently larger.

6.2 Length of the fuselage

The length of the fuselage, L, was determined using tabulated values for a turboprop aircraft according to (Raymer 1999) and the following equation:

L = a· (WTO)^c (6.3)

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where a and c are constants given in the table for "twin turboprop" Raymer (1999). The formula for length, based on historical trends, and Raymer (1999) relies on US-units and one must therefore convert the take-oﬀ weight to SI-units.

The length was calculated to 26.3 m which is a good estimation as the expected value was set to approximately 22 m.

6.3 Tail geometry

The tail size was determined with the tail volume coeﬃcient method. The area of the horizontal stabilizer, SHT, was calculated using

S_HT = c_HT· C^W · S^W

L_HT (6.4)

where cHT is an estimate from table 6.4 page 112 Raymer (1999), CW is the mean wing chord and SW is the wing reference area Sref. The area of the horizontal stabilizer was calculated to approximately 13 m².

The area for the vertical stabilizer, SVT was calculated in a similar manner with SV T = c_VT· bW · SW

L_VT . (6.5)

The vertical tail volume coeﬃcient cVT can be reduced with approximately 5

% due to the end-plate eﬀect, Raymer (1999). Both LHT and LVT were estimated as 50-55 % of the length of the fuselage. The area of the horizontal stabilizer was calculated to approximately 11 m².

6.4 Center of gravity

The center of gravity, COG, of the aircraft was assumed to be located at approximately 40 % of the length of the fuselage at 10.5 m from the front of the aircraft.

6.5 Wing positioning

When positioning the wing relative to the fuselage, there are several parameters that need to be considered. In order to locate the wing in regards to the aircraft center of gravity position, thus assuring correct stability, the mean aerodynamic chord needs to be calculated. The mean aerodynamic chord will further in the report be referred as the "MAC". In order to determine the MAC one must first calculate the root chord of the wing.

Length of the wing root chord, Croot, was found with C_root= 2· Sref

b· (1 + ). (6.6)

Here Sref is the reference area of the wing and b is the wingspan of the wing calculated in equation (6.1) and (6.2). is the assumed taper ratio from section 6.1.2. The MAC could then be calculated with

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M AC = (2

3)· Croot· (1 + + ²

1 + ). (6.7)

The aircraft center of gravity was positioned at approximately 30 % of the MAC.

The wings were positioned so that the leading edge of the wing was positioned 0.8 meters in front of the center of gravity of the fuselage.

7. Power specifications

The power plant of the aircraft can be determined by necessary power for propulsion.

The T /W for desired rate of climb was converted into power-to-weight (P/W ) through

P

W =

✓V · g

⌘p

◆

·

✓T W

◆

(7.1)

found in (Raymer 1999) and rewritten for SI-units. Here P is the power in watt, W is the weight in kg, ⌘p is eﬃciency, which for a propeller aircraft is 0.8, and g is the gravitational constant. The velocity V varies on the conditions at which the aircraft is for the diﬀerent equations. P/W was found as 227 W/kg which for an aircraft of 19920 kg means that the necessary power is 4.18 MW (5599 hp). This would be distributed over the two propellers equally.

8. Results

The results from the weight estimations are compiled in Table 8.1 and the results from the initial sizing is presented in Table 8.2. They were calculated with the help of the programming tool MATLAB.

Table 8.1: Results from the weight estimations Weight estimations with fuel with batteries

Fuel fraction 0.15 0.16

Empty weight fraction 0.58 0.57 Take-oﬀ weight [kg] 19260 19920 Empty weight [kg] 11060 11440 Fuel/Battery weight [kg] 2935 3220

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Table 8.2: Results from Initial sizing Initial sizing

Sweep angle 0°

Taper ratio, 0.5

Wing twist 3°

Dihedral 2°

Wing reference area, Sref 75 m²

Length of fuselage, L 26 m

Wingspan, b 29 m

Area of the horizontal stabilizer, SHT 13 m² Area of the vertical stabilizer, SVT 11 m² Length of wing root chord, Croot 3.5 m Length of wingtip chord, Ctip 1.7 m

COG of fuselage 10.5 m (from the front of the aircraft) Wing position (leading edge of the wing) 0.8 m (in front of the COG)

8.1 Drawings

The drawings of the aicraft are shown in Figure 8.1 and Figure 8.2.

Figure 8.1: Drawing of the aircraft from above.

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Figure 8.2: Drawing of the aircraft from the side.

9. Discussion and Conclusion

The fuel weight of the aircraft was calculated to 2930 kg compared to the ATR 42-500 that has a maximum fuel load of 4500 kg, (ATR 2014). The aircraft has a calculated empty weight of 11060 kg compared to 11700 kg of the ATR 42-500.

Furthermore the take-oﬀ weight was computed to 19260 kg as opposed to 18600 kg of the ATR 42-500.

The battery weight of the aircraft was calculated to 3220 kg which is a good estimation in comparison to the calculated fuel weight in section 4.1. The aircraft has a calculated empty weight of 11440 kg. The take-oﬀ weight was computed to 19920 kg as opposed to 19260 kg (with fuel) and the maximum take-oﬀ weight of the ATR 42-500 (18600 kg).

The maximum take-off weights of the aircraft motorized with fuel and with batteries as well the maximum take-off weight of the ATR 42-500 are all similar to each other. However, it should be noted that the battery powered aircraft has about 70 % lower range than the fuel powered aircraft, 400 km compared to 1300 km. Even with ten years of assumed developed technology, it is still not possible to reach the same range with batteries as with fuel. If the range requirement of the battery powered aircraft would be equaled to the fuel powered aircraft it would have a devastating impact on the maximum take-off weight and it would be practically impossible to fly.

In the constraint analysis the wing loading is determined by the stall speed requirement and the power-to-weight ratio is determined by the constant velocity turn. It is possible to choose a smaller wing loading, left of the stall line from the constraint diagram. The wing loading is dependent on the CLmax value and diﬀerent CLmax

values will produce diﬀerent wing loadings aﬀecting the initial sizing.

The initial sizing and conceptual design of the aircraft were made based on calculations from the constraint analysis, including the wing loading, and the maximum take-oﬀ weight for the battery powered aircraft. The wing reference area was about 20 m² larger than that of the ATR 42-500, however it was expected as the wingspan was about four meters wider than that of the ATR 42-500. This is mainly due to a

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slightly heavier take-oﬀ weight and a small diﬀerence in CLmax between the aircraft impacting the wing loading in the constraint diagram. Choosing a higher CLmax

results in a more favorable wing design. The results were close to dimensions of the Fokker 50 (Fokker n.d.) which is of similar size and shares mission aim.

The batteries were positioned symmetrically close to the center line of the fuselage under the passenger cabin in order to not aﬀect the stability of the aircraft.

The passenger seats were positioned in a typical 2-2 configuration with twelve rows.

The center of gravity location of the aircraft was positioned based on an estimation, though it can be calculated more precisely with regards to weight of the wings, cockpit, engines and crew.

In conclusion, it is possible to model a short-range conceptual design of an elec- trical powered commercial aircraft in theory. Though the range would be short and it would be very costly, it is possible. However, if it is achievable in the near future is debatable. Even though it is environmentally friendly and will attract a lot of environmentally conscious passengers the benefits of it might not be enough to compete with fuel powered aircraft. The battery technology still needs to improve a lot to even be able to compete with the range and weight of such aircraft. Further, since the airline industry today is very economically focused a lot of airlines tend to choose the cheapest option instead of the best option, often at the expense of the environment.

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Bibliography

Anderson, J. (1999), Aircraft Performance and Design, Tata McGraw-Hill.

ATR (2014), ‘Atr 42-500’.

URL: http://www.atraircraft.com/datas/download_center/34/fiches_500_

septembre2014_34.pdf

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Chandler, D. L. (2019), ‘New approach could boost energy capacity of lithium batteries’.

Eviation (n.d.), ‘Alice commuter’.

URL: https://www.eviation.co/alice/

Fokker (n.d.), ‘Fokker 5o - basics’.

URL:http://www.flyfokker.com/sites/default/files/FLYFokker/FlyFokker_PDF_

Fokker50_Basics.pdf

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URL: https://www.nasa.gov/image-feature/nasas-x-57-electric-research-plane Masunaga, S. (2016), ‘No flying tesla? that’s because electric planes are a steeper

challenge than electric cars’.

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URL: https://www.pipistrel-usa.com/alpha-electro/

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www.kth.se

Konceptuell design av ett elektriskt drivet kommersiellt flygplan

Conceptual Design of an Electric Powered

Commercial Aircraft

ASTRID HAFVENSTRÖM ANDREAS PETTERSSON

Konceptuell design av ett

elektriskt drivet kommersiellt flygplan

ASTRID HAFVENSTRÖM ANDREAS PETTERSSON

Contents

1. Introduction

1.1 Background

2. Mission specifications

2.1 Mission profile

2.2 Performance requirements

3. Aerodynamics

3.1 Drag coeﬃcients

3.2 Lift coeﬃcients

4. Weight estimations

4.1 Fuel weight calculations

4.2 Battery weight calculations

5. Constraint analysis

5.1 T/W for a desired take-oﬀ distance

5.2 T/W for a desired rate-of-climb

5.3 T/W for a desired cruise airspeed

5.4 T/W for a level constant velocity turn

5.5 Wing loading as a function of stall speed

5.6 Constraint diagram

6. Initial sizing

6.1 Wing

6.1.1 Sweep angle

6.1.2 Taper ratio

6.1.3 Wing twist

6.1.4 Dihedral

6.1.5 Wing reference area and wingspan

6.2 Length of the fuselage

6.3 Tail geometry

6.4 Center of gravity

6.5 Wing positioning

7. Power specifications

8. Results

8.1 Drawings

9. Discussion and Conclusion

Bibliography