Diploma Thesis

Diploma Thesis

Performance assessment of a hybrid electric-powered long-range

commercial airliner

Thomas Zöld June 2012

ii

This diploma thesis is presented within the framework of the T.I.M.E. double-degree programme between the Technical University Munich and the Royal Institute of Technology in Stockholm.

Technical University Munich Department of Aircraft Design Boltzmannstrasse 15

DE - 85748 Garching bei München Germany

Examiner TUM: Professor Dr.-Ing. Mirko Hornung Examiner KTH: Arne Karlsson, Senior Lecturer Supervisor: Dipl.-Ing. Malte Schwarze

iii

A

Despite the recent increase in the amount of smaller electric general aviation aircrafts, a fully electric airliner is not likely to fly in the near future. Partially inspired by the automotive industry’s success with the hybrid car, this thesis investigated the feasibility of an electric-hybrid propulsion system for an Airbus A340-600 on a long-haul flight and its effect on the aircraft’s performance. First, an analysis was done of the reference aircraft, A340-600, using conventional propulsion. Second, a 5700 nautical miles flight was modelled to determine performance data such as the power and thrust requirements during the different flight phases. Third, the flight phases where electric propulsion would be implemented were identified and an optimum ratio between conventional and electric propulsion was calculated. Finally, a detailed performance analysis of the new hybrid electric aircraft comparing it to a conventional aircraft was conducted.

The maximum available conventional thrust was reduced to a certain percentage of the maximum thrust. Primarily conventional thrust is used, however when it is no longer sufficient, additional thrust is gained through electric propulsion. Conventional thrust ratio of 69.5%, 63.5% and 59.5% of total thrust was investigated yielding 8680 kg, 10500kg and 8585kg of payload decrease respectively. Net energy of 6.70MWh, 11.71MWh and 31.06MWh is required and the electric engines need to provide 21.3 MW, 25.5 MW and 28.3 MW of net power respectively.

Partial electric propulsion will result in increased weight; however, it will also give room for further performance optimisation and technical innovations. On the one hand, the conventional engines will run at a constant speed throughout the flight allowing for better optimisation at a specific design point. On the other hand, electric engines are more reliable and require less maintenance than conventional engines. Furthermore, lower fuel consumption means less carbon-dioxide emissions. An exemption from CO2-taxes, similar to measures implemented for hybrid cars in certain countries, could financially justify use of the aircraft by airlines and compensate for the decrease in payload. Since a fully electric propelled airliner is not likely to fly for several decades, a hybrid-airliner would be a suitable alternative for the transition period from fossil fuels to electric energy.

iv

T

Abstract ... iii

Table of contents ... iv

List of Figures ... vii

List of Tables ... ix

Nomenclature ... xi

1 Introduction ... 1

2 The Airbus A340-600 ... 2

2.1 History and Background ... 2

2.2 Development of the Airbus A340-600 ... 3

2.3 Data ... 4

2.4 Rolls-Royce Trent 556 ... 6

2.5 The Thrust Lever... 7

2.6 High Lift Devices ... 8

3 Performance ... 9

3.1 Zero lift drag ... 9

3.1.1 Wing Reference Area ... 9

3.1.2 Wetted area ... 9

3.1.3 Component Buildup Method ... 12

3.1.4 Howe's Method ... 16

3.1.5 Equivalent Skin Friction Method ... 16

3.1.6 Result, Comparison and Conclusion ... 17

3.2 -factor ... 17

3.2.1 Raymer: Oswald Span Efficiency Method ... 17

3.2.2 Howe's Method ... 18

3.2.3 Frost and Rutherford method ... 18

3.2.4 Result Comparison Conclusion ... 19

3.2.5 Polar break ... 19

3.3 Airspeed ... 22

v

3.5 Thrust Lapse Rate ... 23

3.5.1 Reference Values ... 24

3.5.2 Models ... 25

3.5.3 Evaluation ... 26

3.5.4 Conclusion ... 27

3.6 Maximum Climb Thrust ... 28

3.7 Optimum Cruise Altitude ... 31

3.8 Fuel planning ... 32

3.9 Payload Range Diagram ... 35

4 Model ... 37

4.1 Taxi and Take-off (T/O) ... 39

4.2 Climb (CLB) ... 39 4.3 Cruise (CRZ) ... 43 4.4 Descent (DES) ... 46 4.5 Go-Around (GA) ... 47 4.6 Flight to alternate ... 47 4.7 Hold (HLD) ... 47 4.8 Summary ... 48 5 Electric Propulsion... 51 5.1 Electric Flight ... 51

5.2 The Electric Propulsion System ... 52

5.2.1 Fan ... 53

5.2.2 Electric Engine ... 53

5.2.3 Control unit and wiring ... 54

5.2.4 Power Supply... 55

5.2.5 Efficiency ... 57

6 The Hybrid Electric Powered Long Range Airliner ... 58

6.1 Analysis ... 58

6.2 Electric Engine Ratio Optimization ... 62

6.3 Performance Analysis ... 71

vi

7 Financial Justification ... 79

8 General Conclusion and Outlook ... 82

9 Bibliography ... 84

10 Appendix A – Thrust Model ... 86

11 Appendix B – Matlab Code ... 87

vii

L

F

Figure 2-1. The Airbus A340-300 ... 3

Figure 2-2. Airbus A340-600 3-view drawing with dimensions. ... 4

Figure 2-3. The Rolls Royce Trent 556. ... 6

Figure 2-4. Thrust Lever ... 7

Figure 3-1. Engine dimension definitions. ... 11

Figure 3-2. Pie chart of the wetted areas of the different components. ... 12

Figure 3-3. R factor for the Frost and Rutherford method. ... 18

Figure 3-4. Stalling speeds Airbus A340-642. ... 20

Figure 3-5. Maximum lift coefficient for different settings of the high lift devices, weights and altitude. .. 21

Figure 3-6. CLα as a function of Mach number. ... 22

Figure 3-7. Example of manufacturers uninstalled engine performance data for a subsonic turbofan. ... 23

Figure 3-8. Maximum Climb Thrust according to BADA. ... 24

Figure 3-9. Comparison of thrust Lapse rate models. ... 27

Figure 3-10. Change of thrust with Mach number during take-off ... 28

Figure 3-11. Thrust lapse rate after T/O. ... 28

Figure 3-12. Maximum climb thrust as a function of Mach number and altitude. ... 30

Figure 3-13. Error in Maximum Climb Thrust model ... 30

Figure 3-14. Optimum Cruise Altitude @ M0.83. ... 31

Figure 3-15. Optimum altitude is presented at any given velocity and weight. ... 32

Figure 3-16. Payload Range diagram. ... 36

Figure 4-1. Flight profile for model. ... 37

Figure 4-2. 5700NM great circle range from Munich (EDDM), Germany ... 38

Figure 4-3. Climb profile for a climb to FL320 with speed profile 250/320/M0.82 and cruise at M0.85. ... 40

Figure 4-4. Velocity, fuel flow (all engines), altitude and mass during the climb phase. ... 43

Figure 4-5. Step-climb profiles. ... 44

Figure 4-6. Velocity, fuel flow, altitude and mass during cruise. ... 45

Figure 4-7. Altitude for entire flight. ... 48

Figure 4-8. Fuel flow entire flight for all engines. ... 49

Figure 4-9. Climb angle during beginning and end of flight. ... 49

Figure 4-10. Lift coefficient during entire flight. ... 50

Figure 5-1. The e-Genius. ... 51

Figure 5-2. The EADS VoltAir and Boeings SUGAR Volt ... 52

Figure 5-3. Simplified schematic and segmentation of the electric propulsion system. ... 52

Figure 5-4. Power out-put per engine for model flight. ... 53

Figure 5-5. Selected volumetric and weight energy densities. ... 55

viii

Figure 6-1. Maximum available Thrust and the Thrust Required. ... 58

Figure 6-2. Electric energy required EElec for different electric propulsion ratios. ... 60

Figure 6-3. Fuel saved for different electric propulsion ratios. ... 60

Figure 6-4. Required electric power ... 61

Figure 6-5. Required electric power for T/O, Step Climb and Go-Around. ... 61

Figure 6-6. Power Setting Conventional Engines ... 62

Figure 6-7. Electric engine mass ... 64

Figure 6-8. Example of Battery and Capacitor use depending on characteristics of power curve. . 65

Figure 6-9. Battery and capacitor mass. ... 67

Figure 6-10. Mass change from electric propulsion. ... 68

Figure 6-11. Additional weight required to increase electric propulsion with 1 percentage point. 69 Figure 6-12. The two design points for electric to conventional thrust ratio. ... 70

Figure 6-13. Mass distribution with electric propulsion. ... 70

Figure 6-14. Comparison of Altitude profile for conventional and hybrid aircraft. ... 73

Figure 6-15. Climb performance comparison of conventional and hybrid aircraft. ... 73

Figure 6-16. Thrust produced by electric and conventional engines at 30.5% electric. ... 74

Figure 6-17 Thrust produced by electric and conventional engines at 36.5% electric. ... 74

Figure 6-18. Thrust produced by electric and conventional engines at 40.5% electric. ... 74

Figure 6-19. Detailed view of thrust produced by electric and conventional engines during ... 75

Figure 6-20. Power setting of electric and conventional engine. ... 75

Figure 6-21. Hybrid Airliner Payload Range Diagram ... 77

Figure 6-22. Detailed view of Hybrid Airliner Payload Range Diagram ... 78

Figure 12-1. Comparison of thrust Lapse rate models. ... 89

Figure 12-2. Climb profile for a climb to FL320 with speed profile 250/320/M0.82 and cruise at M0.85. .. 89

Figure 12-3. Step-climb profiles. ... 90

Figure 12-4. Required electric power ... 90

Figure 12-5. Required electric power for T/O, Step Climb and Go-Around. ... 91

Figure 12-6. Power Setting Conventional Engines ... 91

Figure 12-7. Mass change from electric propulsion. ... 92

Figure 12-8. Mass distribution with electric propulsion. ... 92

Figure 12-9. Comparison of Altitude profile for conventional and hybrid aircraft. ... 93

Figure 12-10. Power setting of electric and conventional engine. ... 93

Figure 12-11. Hybrid Airliner Payload Range Diagram ... 94

ix

L

T

Table 2-1. Basic Aircraft Data ... 5

Table 2-2. RR Trent 556 Engine data. ... 6

Table 2-3. Flaps and slats configurations ... 8

Table 3-1. Reference values for zero-lift-drag calculations ... 9

Table 3-2. Wetted area of each aircraft component. ... 11

Table 3-3. Input variables and results for calculating the skin friction coefficient. ... 14

Table 3-4. Raw data, equations and results of form factor calculations. ... 15

Table 3-5. Component interference factor. ... 15

Table 3-6. In-data for zero-lift-drag coefficient calculations according to Howe. ... 16

Table 3-7. Results for Zero-lift-drag calculated using different methods. ... 17

Table 3-8. Results for K-factor ... 19

Table 3-9. Altitude and Mach number for BADA Model ... 24

Table 3-10. Constants for calculating the thrust lapse rate according to the method by Howe. .... 26

Table 3-11. Thrust lapse rate model error. ... 27

Table 3-12. Division of thrust spectrum. ... 29

Table 3-13. Equation coefficients for the maximum climb thrust model. ... 29

Table 3-14. Fuel Reserves. ... 34

Table 3-15. Summary of fuel on-board. ... 35

Table 3-16. Numerical values for initial and final masses for payload vs. range calculations. ... 35

Table 4-1. Climb 1 parameters. ... 40

Table 4-2. Acceleration 1 parameters. ... 41

Table 4-3. Climb 2 parameters. ... 41

Table 4-4. Acceleration 1 parameters. ... 42

Table 4-5. Climb 3 parameters. ... 42

Table 4-6. Climb 4 parameters. ... 42

Table 4-7. Acceleration 2 parameters. ... 43

Table 4-8. Time, distance flown, fuel burnt and mass after climb phase. ... 43

Table 4-9. Cruise parameters ... 45

Table 4-10. Step climb parameters ... 45

Table 4-11. Time, distance flown, fuel burnt and mass after cruise phase. ... 45

Table 4-12. Descent parameters. ... 46

Table 4-13. Time, distance flown, fuel burnt and mass after descent phase. ... 46

Table 4-14. Go-around parameters ... 47

Table 4-15. Hold climb parameters ... 47

Table 4-16. Time, distance flown, fuel burnt during Holding. ... 48

Table 6-1. Engine and Energy/Power source weights. ... 71

x

Table 6-3. Summary of fuel on-board and the Top of Descent weight. ... 72

Table 6-4. Basic Performance Data comparing the Hybrid and Conventional Airliner. ... 72

Table 6-5. Energy and Power required from electric engines. ... 76

Table 6-6. Hybrid Airliner TOW and TOD mass for Payload Range calculations ... 76

Table 6-7. Hybrid Airliner Payload Range numerical values ... 77

xi

N

Symbol Unit

A Area m²

A Aspect Ratio -

b0 Specific Fuel Consumption kg/Ns

c Cord m

c Speed of sound m/s

c0 Speed of sound at mean sea level m/s

cD Drag coefficient -

cD0 Zero-lift-drag coefficient -

cL Lift coefficient -

CLα Lift coefficient curve gradient -

d Diameter m D Drag N E Energy Ws F Fuel kg FF Fuel Flow kg/s FL Flight level - g Gravitational constant m/s² h Altitude m l Length m L Lift N M Mach number - m Mass kg

p Air Pressure at current altitude Pa

P Payload kg

P p0

Power

Reference air pressure at mean sea level

W Pa

Re Reynolds number -

S Area m²

Sref Wing reference area m²

Swet Wetted Area m2

(t/c) Airfoil Relative Thickness -

T Thrust N

T0 Thrust at mean sea level (ISA) N

VIAS Indicated Airspeed m/s

VTAS True Airspeed m/s

xii Wf Final weight N Wi Initial weight N Greek Unit γ Climb angle ° δT Power setting - η Efficiency - θ Temperature °K

θ0 Reference temperature at mean sea level °K

λ Taper ratio -

Λ Wing Sweep °

μ Dynamic viscosity Ns/m2

ρ Air density kg/m³

ρ0 Reference air density at mean sea level kg/m³

Indices ( )50% ½ chord line ( )CLB Climb ( )cont Continuous ( )conv Conventional ( )CRZ Cruise ( )DES Descent ( )e Engine ( )elec Electric ( )em Engine mount ( )f Fuselage

( )ftf Flap track fairing

( )HLD Hold

( )hs Horizontal stabiliser

( )lam Laminar

( )max Maximum

( )MCL Maximum Climb Thrust

( )min Minimum ( )r Root ( )t Tip ( )T/O Take-off ( )turb Turbulent ( )vs Vertical stabiliser ( )w Wing ( )wl Winglet

xiii

()wet Wetted

Abbreviations

A/C Aircraft

C-Eng Conventional Engine

CLB Climb

CRZ Cruise

DES Descent

E-Eng Electric engine

GA Go-Around

HLD Holding

ISA International Standard Atmosphere

KCAS Knots calibrated airspeed

KIAS Knots indicated airspeed

KTAS Knots true airspeed

lam Laminar

MAC Mean Aerodynamic cord

MTOW Maximum Take-off Weight

MZFW Maximum Zero Fuel Weight

OWE Operating Weight empty

SEP Specific excess power

T/O Take-Off

TOC Top of Climb

TOD Top of Descent

1

1

I

Fully electric propulsion can already be seen in operation in a handful of smaller aircrafts flying in the skies today. The concept of electric propulsion would also be an appealing technology in commercial aviation considering its various advantages with regard to sustainability, environmental impact, reliability and maintenance. However, the current state of technological advancement in the field of batteries and energy storage makes the concept of fully electrical airliners feasible only in the far future. Conversely, the automotive industry has successfully launched several hybrid propulsion systems on to the market. These electric-hybrid systems could serve as a great concept and inspiration for the future in aviation. The scope of this thesis is to investigate the feasibility of an electric-hybrid propulsion system for an Airbus A340-600 on a long-haul flight and its effect on the aircraft performance.

At first, an analysis will be done of the reference aircraft, A340-600, using conventional propulsion to determine required data to model and analyse a long-haul mission of the aircraft. Next, a 5700 nm flight will be modelled to determine performance data such as the power and thrust requirements during the different flight phases. Thereafter, the flight phases will be identified where electric propulsion would be plausible. Also, the amount of additional weight from the hybrid system has to be determined and the ratio between conventional and electric propulsion. Further, possibilities for the incorporation of the hybrid electric system with the conventional system will be investigated. Once an optimal ratio of electric to conventional thrust has been determined, a detailed performance analysis of the new hybrid electric aircraft will be conducted.

2

2

T

A

A340-600

2.1

HISTORY AND BACKGROUND

The idea for the Airbus Industry consortium came to be from a decision to challenge the American domination of the airliner market. The idea arose in the mid-1960s after major European airlines had shown interest for a short to medium range airliner that could carry over 100 passengers at low costs. In 1967, ministers from Germany, France and the United Kingdom had agreed at a meeting that “for the purpose of strengthening European co-operation in the field of aviation

technology and thereby promoting economic and technological progress in Europe, to take appropriate measures for the joint development and production of an airbus”1. The official birth of the Airbus programme took place on the 29th of May 1969 at Paris Le Bourget Airshow, when the German economics minister, Karl Schiller, and the French transportation minister, Jean Chamant, signed an agreement for the go-ahead of the A300 programme. This aircraft would be the first twin-engine wide-body passenger jet aircraft and was sought to satisfy the recent interest shown by European airlines. The construction of the aircraft was to be done by the German-French consortium and also involve the United Kingdom and the Netherlands.

Airbus industry was officially founded as Groupement d’Interet Économique (Economic Interest Group) on the 18th of December 1970 by a government initiative between France, Germany and the United Kingdom. The name Airbus was coined by the industry for passenger aircrafts or an airliner of a certain range and size. Initially, about three quarters of the share of the production work were divided between Deutsche Airbus and Aérospatiale. Hawker Siddeley, in turn, acquired one fifth and the rest went to Fokker-VFW. These four companies would deliver their sections as fully equipped ready-to-fly parts. In 1971, the Spanish company CASA and in 1977, British Aerospace joined as shareholders.

The A300 completed its maiden flight in 1972 and the first production model, the A300B2, entered commercial service in 1974. Initially, the consortium had little success, but by 1979 eighty-one of their aircrafts were being flown world-wide. Soon thereafter, Airbus launched the A310, a shortened version of the A300. Following the success of the A300 and the A310, Airbus decided to get into direct competition with its American rivals. The launching of the short- and midrange single-aisle airliner, the A320, was a major success with over 400 units sold even before the aircraft took to the air. It was also the first commercial aircraft fitted with a fly-by-wire system. With this new technology in place, Airbus subsequently introduced several developments of the A320, the shortened A319 as well as the A318, the elongated A321 and various corporate

1

Airbus Mission Statement, "Airbus history". Flight International (Reed Business Publishing). 29 October 1997.

3

jet models. Airbus became particularly known for their fly-by-wire technology and their concept of cockpit communality, making crew training easier.

2.2

DEVELOPMENT OF THE AIRBUS A340-600

The first surveys were conducted in 1981 and in 1987 the A340 project was launched. Its goal was to develop a long-range airliner complementing the mid-range A300 and the short-range A320. A series of factors motivate the development of this new aircraft. On one hand, the 60 minute ETOPS regulation of that time meant a large disadvantage for Airbus’ twin-engine aircrafts compared to its three- and four-engine competitors. On the other hand, the popular Douglas Corporation DC-10 and Lockheed Tristar L1011 were being phased out and airlines were looking for replacements.

The A340 and the A330 were designed concurrently. They received the same fuselage and wing, but also much of the avionics originally designed for the A320. The first prototypes of both aircrafts, which were also the first ones built with composite materials, were manufactured on the same production line.

The first A340 completed its maiden flight on the 25th of October 1991 and the A340200 and -300 entered into service in 1993 in the colours of Lufthansa and Air France. Due to the high similarities between the cockpits of A320, A330 and A340, pilots, who had previously flown the A320, could be retrained to fly on the new models at a minimal cost and time.

Figure 2-1. The Airbus A340-300

The A340-600 was initially designed as a competitor to the Boeing 747. It possessed similar passenger capacity as its Boeing rival, but could carry more payload and had lower operating costs. It was an enhancement of the A340-300 with an extra twelve metres in length and an additional four-wheel under-carriage on the centreline of the fuselage, in order to cope with the additional weight.

The aircraft is powered by four Rolls-Royce Trent 556 turbofan engines, described in section 2.4 below. After its maiden flight in 2001, the A340-600 entered into service in 2002 for Virgin

4

Atlantic Airways. In terms of length of fuselage, it is the second longest commercial airliner only recently surpassed by the Boeing 747-8i. A total of 97 units of the A340-600s have been delivered until now. In November 2011 Airbus announced that it would cease production of the Airbus A340 family but assured that it would continue to fully support the current global fleet.

2.3 D

The focus of this thesis will be on Airbus A340-642. Since different variations exist even within this model specification, significant data used throughout the project is presented in this section. The figure below shows a three-view drawing2 of the aircraft with the most important measurements.

Figure 2-2. Airbus A340-600 3-view drawing with dimensions.

2

5

In the table below significant numerical data is summarised of the aircraft.

Fu se la ge Length 75.24 m Width 5.64 m Cockpit Crew 2

Maximum Seats (w/ over-wing exit) 475

W

in

g

Area 437.3 m2

Span (w/o winglets) 61.20 m

MAC 8.35 m Aspect Ratio 8.56 Taper Ratio 0.22 (t/c)tip 8.2 % (t/c)average 10 % (t/c)root 13.2%

Leading Edge Sweep 31.1°

¼ Chord Sweep 28° Ho ri zo n ta l s ta b ili se r Area 93 m2 Span 21.5 m Aspect Ratio 4.97 Taper Ratio 0.36

Leading Edge Sweep 30°

¼ Chord Sweep 27° V e rti ca l s ta b ili se r Area 47.65 m 2 Height 9.44 m Aspect Ratio 1.87 Taper Ratio 0.350

Leading Edge Sweep 45°

¼ Chord Sweep 40°

M

as

s

Maximum Take-off Mass 368 000 kg

Maximum Zero Fuel Mass 245 000 kg

Operating Mass Empty 177 000 kg

Maximum Fuel Capacity 193 925 l

Maximum Payload 68 000 kg

Maximum Landing Mass 256 000 kg

Ran

ge

Maximum Payload Range 5700 NM

Design Range 7500 NM

Maximum Fuel Range 7800 NM

Ferry Range 8800 NM

6

2.4

ROLLS-ROYCE TRENT 556

In 1995, as Airbus was developing the two new long-range derivatives of the A340 aircraft, the A340-500 and A340-600, it started a search for a new engine. The existing CFM International CFM56 engine, which had powered the A340-200/-300, was at the limit of its developmental capability and would not be sufficient for the power requirements of the new -500 and -600 models. Despite first signing an agreement with General Electric in 1996 to develop a suitable engine, Airbus subsequently decided to withdraw, after that GE demanded an exclusivity deal for the new aircraft. At the 1997 Paris Airshow, Airbus announced that it had selected the Trent 500 to power the A340-500 and -600 over a Pratt & Whitney model that was also under consideration. The first test run of the Trent 500 was conducted in May 1999 and certification was achieved in December 2000. It entered service with the inaugural commercial flight of the A340-600 with Virgin Atlantic Airways in July 2002.

Figure 2-3. The Rolls Royce Trent 556.

The Airbus A340-500 and A340-600 are powered by the Trent 500 engines, which were certified for 270 kN thrust, but derated to 249 kN as the Trent 556 for the A340-600.

In the table below the most important engine data is summarised.

Si

ze

Length 4.689 m

Width 3.374 m

Fan Diameter 2.474 m

Dry Weight (excl. Nacelle) 4990 kg

Th

ru

st Maximum Take-off Thrust 260 000 N

Maximum Continuous Thrust 197 300 N

Specific fuel consumption in cruise b0 1.62∙10-5 kg/Ns

Overall Pressure Ratio 36.3:1

Bypass Ratio 7.6:1

Turbine Gas Temp. Max. T/O 900 °C

Turbine Gas Temp. Max. Cont. 850 °C

7

2.5

THE THRUST LEVER

The thrust produced by each power-plant is controlled by a Full Authority Digital Engine Control (FADEC) system. This is a digital control system that is in charge of the complete engine management. Generally, when the aircraft is flown, the pilot does not set a specific power setting using the thrust lever, but instead uses the thrust lever to set a specific range for the thrust and the FADEC system then freely regulates the thrust within this upper and lower limit. There are four main setting:

- TO GA - Take Off, Go Around

- FLX/MCT - Flex, Maximum Continuous - CL - Climb

- IDLE - Idle

In the figure below, the settings and the ranges within which the FADEC can vary the thrust, are visualised.

8

2.6

HIGH LIFT DEVICES

On each of its wings, the A340-600 is fitted with two flap surfaces and seven slat surfaces and also has the ability to droop its ailerons. There are five settings that can be chosen from the flaps lever in the cockpit: 0, 1, 2, 3 and FULL. Depending on the speed of the aircraft, when flaps setting 1 is selected, the high lift devices will either go into configuration 1 or in configuration 1 + F. The table below shows the different slats, flaps and aileron settings of the aircraft.

Lever

Position Slats Flaps Ailerons Indication Flight Phase

0 0 0 0 0 CRZ 1 21 0 0 1 HLD 17 10 1 + F T/O 2 24 17 10 2 APPR 24 22 10 2 T/O 3 24 29 10 3 LDG FULL 24 34 10 FULL

Table 2-3. Flaps and slats configurations

9

3

P

3.1

ZERO LIFT DRAG

For accurate performance calculations it is important to calculate a value for the zero-lift-drag (CD0). This is done using several different methods for redundancy. In order to allow for a comparison of the different methods used, a reference state is defined where the aircraft is cruising at FL 390 with Mach 0.83 under standard atmospheric (ISA) conditions; reference values for this specific state are given in the table below.

VTAS M h ρ p T Μ

244.9 m/s 0.83 39000 ft 0.3162 kg/m3 19664 pa -56.5 °C 1.4323∙10-5 Ns/m2 Table 3-1. Reference values for zero-lift-drag calculations

3.1.1 Wing Reference Area

The wing reference area is calculated from a three-view drawing using the Airbus Method, which approximates the contribution from the fuselage to the wing reference area as a rectangle. The wing reference area is:

Sref = 440 m2.

3.1.2 Wetted area

The total wetted area of the aircraft is calculated to be:

Swet = 2460.5 m2

The method for calculating the contributions from the different aircraft components is described below.

Fuselage

The fuselage's wetted area is given by3:

( ) ⁄ ( ) (3-1) where . 3 (Torenbeek, 1982) eq. B-6

10 Wing

The wetted area of the wing is given by4_:

( ( )

) (3-2)

where ⁄ and ( ⁄ ) ⁄( ⁄ ) . The exposed wing area Sw is measured from a

three-view drawing of the aircraft. Horizontal stabiliser

As for the wing, equation (3-2) is used to calculate the wetted area of the horizontal stabiliser. However, it is assumed that λ = τ = 1 since no accurate values where obtained for the relative thickness of the airfoil's tip and root. This gives the following equation:

( ( ) ) (3-3) The exposed horizontal stabiliser area Shs is measured from a three-view drawing of the aircraft.

Vertical stabiliser

The wetted area of the vertical stabiliser is calculated in the same way as for the horizontal stabiliser – using equation (3-3). The exposed vertical stabiliser area Svs is measured from a

three-view drawing of the aircraft. Winglet

The wetted area of the winglets is calculated in the same way as for the horizontal stabiliser – using equation (3-3). The exposed winglet area Swl is measured from a three-view drawing of the

aircraft. Engine

The wetted area of the engine (excluding the engine pylon) is found using the following formula5:

[ ( ) ] [ ( ) ( ( ) )] (3-4) 4 (Torenbeek, 1982) eq. B-11 5 (Torenbeek, 1982) eq. B-13, B-14, B-15,

11

The symbols and engine dimensions are defined in the figure below.

Figure 3-1. Engine dimension definitions.

Engine mount

The wetted area of the engine mount is measured from a three-view drawing of the aircraft. Flap track fairing

The wetted area of the flap track fairing is measured from a three-view drawing of the aircraft. Summary

Component Raw data Swet

Fuselage lf df = 73.46 m = 5.64 m 1033.7 m2 Wing Sw (t/c)t (t/c)r = 370.96 m2 = 0.0822 = 0.1323 764.78 m2 Horizontal stabiliser Shs (t/c) = 98.1 m2 = 0.088 200.52 m2 Vertical stabiliser Svs (t/c) = 51.4 m2 = 0.088 105.06 m2 Engine (each) Dn Dh ln lg lp β Dp Deg Dg Def = 2.90 m = 3.40 m = 5.30 m = 1.50 m = 1.25 m = 0.302 = 1.00 m = 1.50 m = 2.10 m = 2.90 m 57.33 m2 Winglets Swl (t/c)r = 2.15 m2 = 0.087 8.79 m2 Engine mount (each) 17.01 m2 Flap track fairing

(each)

4.4 m2

TOTAL 2460.5 m2

12

The figure below shows the ratios of the wetted areas of the different components.

Figure 3-2. Pie chart of the wetted areas of the different components.

3.1.3 Component Buildup Method

The first method used is the component buildup method6. A flat‐plate skin friction coefficient is calculated for each component of the aircraft along with a form factor FF that incorporates the pressure drag caused by viscous separation. The product of these two terms, the wetted area and a component interference factor , estimating interference effects, gives the components contribution to the total drag. An additional term, takes into account additional drags caused by un‐retracted landing gear, flaps, etc. Further, CDL&P incorporates additional drag due to

leakage and protuberances. The zero lift drag is given by:

( )

∑( )

(3-5)

where the subscript c means the value is component specific.

The aircraft is divided into the following components for the calculations: - Fuselage - Wing - Horizontal stabiliser - Vertical stabiliser - Engine - Winglets - Engine mount - Flap track fairing

6 (Raymer, 2006) chapter 12.5 42% 31% < 1%1% 3% 4% 8% 10% Fuselage Wing Winglets Flap Track Fairing Eingne Pylon Vert. Stabiliser Horiz. Stabiliser Engine Fuselage Wing

Engine Horizontal Stabiliser

Vertical Stabiliser Engine Pylon Flap Track Fairing

13 3.1.3.1 Flat-Plate Skin Friction Coefficient (Cf)

The flat-plate skin friction coefficient for laminar flow is expressed by:

√ (3-6)

and for turbulent flow:

( ) ( ) (3-7)

where is the non-dimensional Reynolds number defined as:

(3-8)

where is the characteristic length defined for each component in the 2nd column of Table 3-3. The dynamic viscosity µ is given by Sutherland's formula as follows:

( )

( )

(3-9)

where

T is the input temperature

µ0 is the reference viscosity at reference temperature T0, µ0 = 18.27∙10-6 Ns/m2

T0 is the reference temperature, T0 = 291.15 °K

C is the Sutherland's constant, C = 120

Since the flat-plate skin friction coefficient is also affected by surface roughness, the value for Cf

might be inaccurate for very rough surfaces if is defined by equation (3-8). Consequently, a cut‐

off Reynolds number, which takes into account the skin roughness, is calculated and the smaller of

the two Reynolds numbers is used. The cut‐off Reynolds number for subsonic flight is defined as:

( ⁄ )

(3-10)

where is the skin roughness value. For a smooth paint surface it is given as7 _. Since the flow over the different aircraft components can be both laminar and turbulent the final skin friction coefficient is defined taking into account the ratio between the laminar and the turbulent flow.

( ) (3-11)

7

14

where is defined as the fraction of the length of the component that has laminar flow over

it.

The numerical calculated values for the flat plate skin-friction coefficient for the different components are presented in the table below.

Component Characteristic length R [106] Rcutoff [106] klam Cflam [10-3] Cfturb [10-3] Cf [10-3] Fuselage Length 73.46 m 397.16 986.31 0.1 0.0666 1.661 1.661 Wing MAC 8.37 m 45.23 100.11 0.3 0.1975 2.242 2.038 Horizontal stabiliser MAC 4.63 m 25.01 53.63 0.2 0.2656 2.449 2.231 Vertical stabiliser MAC 6.27 m 33.89 73.87 0.2 0.2281 2.340 2.129 Engine Length 6.69 m 36.14 79.05 0.0 0.2209 2.318 2.318 Winglets MAC 1.50 m 8.11 16.39 0.2 0.4663 2.922 2.676

Engine mount Length 6.54 m 35.36 77.24 0.0 0.2233 2.325 2.325 Flap track fairing Length 5.75 m 31.087 67.45 0.0 0.2382 2.370 2.370

Table 3-3. Input variables and results for calculating the skin friction coefficient.

It can be noted that the Reynolds number was smaller than the cut‐off Reynolds number for every component and was hence used in the calculation.

3.1.3.2 Component Form Factor (FF)

The form factor is given by the following equations for the corresponding components: - wing, horizontal & vertical stabiliser, strut, pylon

[

( ⁄ ) ( ) ( ) ] [ ( ) ] (3-12) - fuselage and smooth canopy

(

) (3-13)

- Nacelle and smooth external store

( ) (3-14)

where

√( ⁄ ) (3-15)

where Amax is the maximum cross-section area, ( ⁄ ) is the relative point of maximum thickness

15

The numerical calculated values for the form factor along with the relevant in-data and the equations used are presented in the table below.

Component Raw data Equation FF

Fuselage lf df = 73.46 m = 5.64 m (3-13) 10.597 Wing (x/c)max (t/c) Λm = 0.3 = 0.1 = 31° (3-12) 15.017 Horizontal Stabiliser (x/c)max (t/c) Λm = 0.3 = 0.088 = 29.9° (3-12) 14.716 Vertical Stabiliser (x/c)max (t/c) Λm = 0.3 = 0.088 = 39.5° (3-12) 14.244 Engine lf df = 6.685 m = 3.1 m (3-14) 11.623 Winglets (x/c)max (t/c) Λm = 0.3 = 0.087 = 40° (3-12) 14.188 Engine mount lem dem = 6.54 = 0.6 (3-14) 1.021 Flap track fairing lftf dftf = 5.75 = 0.75 (3-14) 10.457

Table 3-4. Raw data, equations and results of form factor calculations.

Since, data concerning the point of maximum thickness could not be obtained; this value was approximated to 0.3, which is a common value for subsonic airfoils.

3.1.3.3 Component interference factor (Q)

The component interference factor Q incorporates the additional drag resulted by the interference effect components have on each other. Interference factors for the different components are displayed in the table below.8

Component Fuselage Wing Horizontal stabiliser

Vertical stabiliser

Engine Winglets Engine mount

Flap track fairing Interference

factor 1.0 1.0 1.03 1.03 1.3 1.03 1.5 1.5

Table 3-5. Component interference factor.

3.1.3.4 Result

Substituting the results from the sections above into equation (3-5) yields a zero-lift-drag coefficient for the reference state of CD0 = 0.0146.

8

16

3.1.4 Howe's Method

The second method used approximates the zero-lift-drag as9_:

( ) [ [ √ ( ⁄ ) ( ) ] ] (3-16) where,

Af is an airfoil factor dependent on the airfoil design

cl is the fraction of chord of the wing over which the flow is laminar

Λ1/4 is the wing sweep at the ¼ cord line

Rw is a factor given by the ratio of Sref and Swet

Tf is a factor incorporating deviation from the streamlined ideal shape

τ is a correction factor for wing thickness given by the following equation: [ ( ( ⁄

) )] (3-17)

The type factor variable is given as10 Tf = 1.1. The fraction of chord of the wing over which the flow

is laminar has been approximated to be 10%. The airfoil factor is given as11:

(3-18)

where, AF is 0.95 for a modern airfoil12 and the lift-coefficient is calculated for the current cruise flight conditions using equation (3-49) to be CL = 0.523.

The numerical values of the variables used in the equation (3-18) above are given in the table below.

Af M cl Sref Λ1/4 t/c Rw Tf

0.90 0.83 0.1 440 m2 28° 0.1 5.20 1.1

Table 3-6. In-data for zero-lift-drag coefficient calculations according to Howe.

The above values substituted into formula (3-17), (3-18) and then (3-16) yield CD0 = 0.0145.

3.1.5 Equivalent Skin Friction Method

The equivalent skin friction method approximates the zero-lift-drag using the following formula:

(3-19)

For a commercial airliners Cfe= 0.003013. Using Sref and Swet calculated in section 3.1.1 and 3.1.2,

the equation above yields CD0=0.0167.

9 _{(Howe, 2000) eq. 6.13a}

10 (Howe, 2000) Table 6.4 11 (Howe, 2000) p 118 12 (Howe, 2000) p 118

17

3.1.6 Result, Comparison and Conclusion

The results from the three different methods above are summarized in the table below. Component Buildup Method Dennis Howe Equivalent skin friction method

0.0146 0.0145 0.0167

Table 3-7. Results for Zero-lift-drag calculated using different methods.

The component buildup method and the equation by Denis Howe give surprisingly similar result. Both methods take into account Mach number and component buildup method also takes into account change with altitude. Therefore for future calculations the component buildup method will be used. The Equivalent skin friction method is an exceptionally simplified method, which may explain why its result deviates from the other two methods.

3.2

-FACTOR

The -factor was determined using four different methods. Due to the high velocity of the aircraft the polar break also has to be taken into consideration.

3.2.1 Raymer: Oswald Span Efficiency Method

This method utilizes the Oswald span efficiency factor, e, and defines as:

(3-20)

where the Oswald span efficiency factor is given by:

( ₎

(3-21)

and for swept wings ( > 30°) by:

( ₎₍

) _(3-22)

Equation (3-21) yields e = 0.795, which in turn gives = 0.0468 and equation (3-22) yields

e = 0.516 and = 0.072.

13

18

3.2.2 Howe's Method

The -factor is given by the following equation14_:

( ) [ ( ) ( ⁄ ) ( ⁄ ) ( ) ( ) ] (3-23) where,

Ne is the number of engines that are located over the top surface of the wing

f(λ) is a Taper ratio function given by:

( ) [ ( ) ] (3-24)

As the A340-600 has no engines above the top surface of its wing, Ne = 0, the equations above

yields = 0.0494.

3.2.3 Frost and Rutherford method

This method uses the following formula to calculate the Oswald span efficiency factor15: ⁄

_{( )} (3-25)

Where CLα can be calculated as described in section 3.2.5.2 and the suction factor R is given as a

function of (Aλ/cos(Λ)) in the figure below, where Λ is the wing sweep.

Figure 3-3. R factor for the Frost and Rutherford method.

Given that (Aλ/cos(Λ)) = 2.13, R = 0.94. Equation (3-31) gives CLα = 5.45, which in turn gives an

Oswald factor of e = 0.902 using equation (3-25). Using equation (3-20) we get that = 0.0412.

14

(Howe, 2000) eq. 6.14a

15

19

3.2.4 Result Comparison Conclusion

The results from the four different methods above are summarized in the table below. Raymer I Raymer II Howe F & R

0.0468 0.0700 0.0488 0.0412

e 0.7951 0.5312 0.7624 0.9021 Table 3-8. Results for K-factor

If considering the two Raymer methods, the first one is for aircrafts with straight wings, while the second one is for aircrafts with a larger wing sweep, as the A340. However, the result from the first method seems more accurate while the value obtained with the second method is inconsistent with the other values. The method by Howe gives a reasonable answer while offering simple application into the model being developed in the next section. For further calculations the method according to Howe will be used.

3.2.5 Polar break

In order to compensate for change in the k-factor at high CL-values the polar break has to be taken

into consideration. The k-factor will increase when the lift coefficient exceeds a specific value CL,PB.

This is approximated by16:

(3-26)

With an average relative thickness of (t/c) = 0.1 the boundary value becomes CL,PB = 0.65. For

cases where the lift coefficient is above this boundary value, the following formula is used in order to calculate the accurate -factor, which is denoted as 17 :

[

] (3-27)

Where k is the k-factor calculated without taking the polar break effect into account. How CLmax

and CLα are calculated is described in the following two sections below.

3.2.5.1 Maximum lift coefficient CLmax

The maximum lift coefficient CLmax is derived from the aircrafts stall speed.

16

(Hornung, Flugzeugentwurf Vorlesungsskript, 2010), slide 5.30

17

20

Figure 3-4. Stalling speeds Airbus A340-642.18

Due to the construction of the fly-by-wire system and the inability to stall the aircraft, instead of the stall speed, the VS1G speed is given in the charts above. The ratio between the given value and the true stalling speed is:19

_(3-28)

Data points where taken from the figure above and a line of best fit was calculated for the approximation of stall speed at different aircraft weights and configurations. This is presented in the equations below:

_[ ] [ ] _[ ] [ ] _[ ] [ ] (3-29)

Where conf. refers to the different flap and slat configurations, see section 2.6. The maximum lift coefficient can then be calculated by:

⁄ (3-30)

The maximum lift coefficient has been plotted in the figure below for different settings of the high lift devices, weights and altitude.

18

Airbus A340-600 Flight Crew Operating Manual, 3.01.20 P7

19

21

Figure 3-5. Maximum lift coefficient for different settings of the high lift devices, weights and altitude.

For configurations when flaps or slats are extended, the maximum lift coefficient is not calculated for all altitudes since the aircraft normally only flies at lower altitudes when they are extended. 3.2.5.2 Gradient of Lift coefficient curve CLα

The gradient of the lift-coefficient vs. angle of attack curve CLα is approximated using Polhamus

equation. This equation was chosen because it yields accurate answer at subsonic speeds and the aircraft is most likely to exceed CL,PB during the climb phase where speeds are low. It defines CLα as:

√ ( ⁄ ₎ (3-31)

where,

η airfoil efficiency, approximated20 to η = 0.95 β is a compressibility factor given by:

√ (3-32)

The figure below shows the calculated values of CLα for Mach numbers for 0 to 0.85 using the

equation above. 20 (Raymer, 2006) page 312 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 0 50 100 150 200 250 300 350 400 A lt it u d e [ F L ] C_Lmax C_Lmax Conf. 0 @ 360t Conf. 0 @ 240t Conf. 1 @ 360t Conf. 1 @ 240t Conf. 1+ F @ 360t Conf. 1+ F @ 240t

22

Figure 3-6. CLα as a function of Mach number.

3.3

AIRSPEED

Due to the construction and method used to measure the airspeed of the aircraft, there is deviation between the airspeed indicated to the pilot in the cockpit (VIAS) and the true airspeed of

the aircraft (VTAS). This is due to deviation in air density with altitude and a buffer effect at high

velocity. There is also a slight instrumental error, however that has been neglected in this case, hence VIAS = VCAS. The conversion between indicated and true airspeed is done using the following

formula: √ [( [ ( _{) ]} ) ] (3-33) where,

c is the speed of sound [m/s] c0 is the speed of sound at MSL [m/s] p is the air pressure [pa]

p0 is the air pressure at MSL [pa] VIAS is the indicated airspeed [m/s]

3.4

INITIAL CLIMB SPEED

After take-off, according to the FCOM21, the aircraft should fly with a speed of V2 + 10kts. Further, V2 is defined22 as V2 = 1.2∙VS. It is assumed that at this stage the high lift devices of the aircraft are

in the 1 + F configuration (see section 2.6). The stall speed has already been calculated in section 3.2.5.1, thus we can define V2 as a function of the aircrafts mass as:

21

Airbus A340-600 Flight Crew Operating Manual, 3.03.62

22

Airbus A340-600 Flight Crew Operating Manual, 3.04.10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5 Mach number CL 

23 ( _[ ] [ ]) _[ ] [ ] (3-34)

3.5

THRUST LAPSE RATE

The thrust that can be produced by a turbofan aircraft engine is a function of altitude and Mach number. In the figure below the general tendency of this change can be seen for a subsonic turbofan engine.

Figure 3-7. Example of manufacturers uninstalled engine performance data for a subsonic turbofan23.

Since no exact performance data could be obtained for the Trent 556 engines, a model has to be generated to predict the maximum available thrust at different altitudes and Mach numbers. In the figure above it can be seen that at low altitudes there is a higher dependency on Mach number, while at higher altitudes thrust is only slightly influenced by the Mach number. Literature provided very few and also very different methods for modelling the thrust lapse rate. There was also great difference in complexity. Several different methods are investigated to find the most suitable one.

23

24

3.5.1 Reference Values

Two set of values will be used as reference based upon which the models will be evaluated. It is assumed that the maximum climb thrust has the same general relation to the Mach number and altitude as the maximum available thrust and will therefore be used as reference. The first set of values are the maximum climb thrust values defined in section 3.6. The second reference is the maximum climb thrust model for ISA conditions24 according to the Base of Aircraft Data (BADA):

( ) (

) (3-35)

where the aircraft specific coefficients are as given below25, CTc1 = 0.54497 ∙ 106

CTc2 = 0.57703 ∙ 105 CTc3 = 0.20155 ∙ 10-10

These values are defined for a specific set of Mach numbers given in the table below.

FL 0 5 10 15 20 30 40 60 80 Mach 0.28 0.28 0.29 0.30 0.30 0.34 0.39 0.42 0.44 FL 100 120 140 160 180 200 220 240 260 Mach 0.56 0.58 0.60 0.62 0.65 0.67 0.70 0.73 0.75 FL 280 290 310 330 350 370 390 410 415 Mach 0.78 0.80 0.81 0.81 0.81 0.81 0.81 0.81 0.81

Table 3-9. Altitude and Mach number for BADA Model

In the figure below the thrust lapse rate defined by equation (3-35) above can be seen.

Figure 3-8. Maximum Climb Thrust according to BADA.

24

EUROCONTROL EXPERIMENTAL CENTRE, BASE OF AIRCRAFT DATA (BADA) AIRCRAFT PERFORMANCE MODELLING REPORT, EEC Technical/Scientific Report No. 2009-009, Issued: March 2009, table 3-2

25

BADA, Aircraft Performance Operational File, File name: A364__.OPF

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 50 100 150 200 250 300 350 400 450 T/T₀ A lt it u d e [ F L ]

25

3.5.2 Models

Below five different methods are presented for calculating the change of thrust with altitude. I.

One of the most common approximations is given by:

( ) (3-36)

where,

nρ lapse rate factor

No literature gives an exact value for the lapse rate factor but generally states that the values lies in the range of 0.75-1 depending on engine type. nρ = 0.85 was assumed.

II. - Nikolai

The following method is similar to the previous one but instead of a lapse rate factor a temperature coefficient is used. Thrust lapse rate is given by26:

( ) ( ) (3-37)

where

θ is the outside temperature [°K] θ0 is the air temperature at sea level [°K] III. - Raymer

According to Raymer the lapse rate is a linear function that assumes 100% thrust at sea level and 0% thrust at 55000 ft. It is defined as:

(3-38)

where

C is the thrust gradient, C = 1.8 ∙ 10-5 h is altitude in feet

26

26 IV. - Howe

According to Howe27_{, the lapse rate is defined as:}

( ( ) ) ( ) (3-39)

where the constants K1τ, K2τ,K3τ, K4τ and s are given in the table below for a turbofan with a bypass ratio of 8.

M K1τ K1τ K1τ K1τ h s

M < 0.4 1 0 -0.595 -0.03 h < 36100ft 0.7

M > 0.4 0.89 -0.014 -0.3 0.005 h < 36100ft 1

Table 3-10. Constants for calculating the thrust lapse rate according to the method by Howe.

V. - Torenbeek

The following equation is an alteration of an equation by Torenbeek28. It yields results with an error below 1% for Mach numbers below 0.429. It defines the thrust lapse rate as:

( ) √( ) ( √ ) (3-40) where ( ) ( ) ( )

and G is the gas generator function, which is given as30 G = 1.1 for high bypass ratios.

3.5.3 Evaluation

Methods I, II and III don't take the effect of Mach number in account. Since it is known that the accuracy of equation (3-40) is high for the low speeds it will be used to calculate the maximum thrust for the take-off and methods I-III will only be used to calculate the thrust after this stage, where Mach number has a less significant effect.

27

(Howe, 2000) eq. 3-7 & table 3.2

28

Assessment of Numerical Models for Thrust and Specific Fuel Consumption for Turbofan Engines, Oliver Schulz, 13.März 2007

29

Assessment of Numerical Models for Thrust and Specific Fuel Consumption for Turbofan Engines, Oliver Schulz, 13.März 2007

30

27

The average is taken of the two reference data's and plotted in the figure below along with the calculated values using the models above.

Figure 3-9. Comparison of thrust Lapse rate models.

The mean absolute error for each method is shown in the table below

I. II. III. IV. V.

2.36 % 1.26 % 3.84 % 21.97 % 22.14 %

Table 3-11. Thrust lapse rate model error.

The most accurate values are given by equation (3-37) when combined with equation (3-40) for the take-off.

It should be noted that reference values are defined for a specific altitude and Mach number. The range of speeds in which an airliner operates at a specific altitude is small; this might be the reason why a simpler model is sufficient and more accurate. While the more complex models might not be very accurate in the altitude and speed ranges where the aircraft normally flies, it gives somewhat accurate values in the whole spectrum of speeds and altitudes. The simpler models are accurate in the ranges where the aircraft normally flies but presumably, due to their simplicity, give very inaccurate results in other areas, such as low altitude – high Mach number or high altitude – low Mach number flight. However, since these are not of interest the simpler methods become a better choice.

3.5.4 Conclusion

For the model in chapter 4, the thrust during take-off will be calculated using equation (3-40). The figure below shows the change in thrust with Mach number.

0 1 2 3 4 5 6 7 x 105 0 50 100 150 200 250 300 350 400 450 Thrust A lt it u d e [ FL ] Refernce I II - Nikolai III - Raymer IV - Howe V - Torenbeek

28

Figure 3-10. Change of thrust with Mach number during take-off

After take-off the thrust lapse is modelled using equation (3-37) using the value obtain from equation (3-40) at the end of take-off as reference. The thrust lapse rate and the temperature effect as well as the air density effect are shown in the figure below.

Figure 3-11. Thrust lapse rate after T/O.

3.6

MAXIMUM CLIMB THRUST

A model has to be made to determine the maximum climb thrust TMCL. Raw thrust data was

obtained from a software called Piano X. Piano X is a performance software that gives fuel consumption, environmental emissions, drag and performance characteristics at any range and payload combination of a specific aircraft. After identifying that the values change with altitude and Mach number a matrix was created with values for the maximum climb thrust at different altitudes and Mach numbers (see Appendix A – Thrust Model). After analysis of the nature of

0 0.05 0.1 0.15 0.2 0.25 0.75 0.8 0.85 0.9 0.95 1 Mach number T/ T 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 50 100 150 200 250 300 350 400 450 T/T₀ A lt it u d e [ F L ]

Thrust Lapse Rate Temperature Effect Air Density Effect

29

data, it was divided into 6 blocks restricted by the conditions shown in the table below. A separate model would be generated for each of the six blocks in order to increase accuracy.

Mach Number 0 < M < 0.6 0.6 ≤ M Al ti tu d e FL 0 ≤ h < FL 110 I. IV. FL 110 ≤ h < FL 245 II. V. FL 245 ≤ h III. VI.

Table 3-12. Division of thrust spectrum.

The model is generated by doing multiple linear regressions using the least square method for each block of data. The general equation of the plane that is fitted to the data is given by:

_(3-41)

where the coefficients a, b, c and d have to be found and FL is the flight level. Using Matlab, the values are found for the coefficients and presented in the table below.

h M a [N] b [N/100ft] c [N] d [N/100ft] I. FL 0 ≤ h < FL 110 0 < M < 0.6 67515 -127 -36461 136 II. FL 110 ≤ h < FL 245 0 < M < 0.6 70742 -158 -34088 116 III. FL 245 ≤ h 0 < M < 0.6 57115 -105 -13252 34 IV. FL 0 ≤ h < FL 110 0.6 ≤ M 62863 -122 -28708 128 V. FL 110 ≤ h < FL 245 0.6 ≤ M 65628 -143 -25566 91 VI. FL 245 ≤ h 0.6 ≤ M 56516 -108 -12253 40

Table 3-13. Equation coefficients for the maximum climb thrust model.

30

Figure 3-12. Maximum climb thrust as a function of Mach number and altitude.

The figure below shows the per cent error of the mathematical model compared to the raw data.

Figure 3-13. Error in Maximum Climb Thrust model

It can be seen that the model is matched well with data with a maximal error of roughly 1.5%.

0 50 100 ₁₅₀ 200 250 ₃₀₀ 350 ₄₀₀ 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 150 200 250 300 350 400 450 500 550 Mach Number Altitude [FL] T h ru s t [k N ] 0 50 100 150 200 250 300 350 400 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0 0.5 1 1.5 2 Altitude [FL] Mach Number E rr o r [% ]

31

3.7

OPTIMUM CRUISE ALTITUDE

The required thrust for cruise is given by:

(3-42)

Since cruise is done at a constant Mach number, the only variables are weight and air density. The required thrust was calculated for a range of altitude and weights, and for each weight the altitude at which the required thrust was a minimum was found. The optimum cruising altitude where the minimum thrust is required is plotted as a function of aircraft mass in the figure below.

Figure 3-14. Optimum Cruise Altitude @ M0.83.

The data points above can be described by a line of best fit with the following function that has also been plotted in the figure above:

_[

] [

] [ ] (3-43) In the figure below the optimum altitude is presented at any given velocity and weight.

180 200 220 240 260 280 300 320 340 360 280 300 320 340 360 380 400 420 A lt it u d e [ FL ] Mass [t]

Optimal Cruise Altitude @ M0.83

32

Figure 3-15. Optimum altitude is presented at any given velocity and weight.

3.8

FUEL PLANNING

The boundary condition for when the cruise phase of a flight has to end at the latest is determined by when the aircraft reaches a specific Top of Descent weight

W = WTOD

At this weight, the aircraft will still have enough fuel to descend and land and still have the required amount of reserve fuel on-board upon landing. The WTOD is calculated backwards from

the landing weight. During flight the only cause of weight change of the aircraft is due to the consumption of fuel. The fuel aboard an aircraft is a sum of the following:

- Trip Fuel (TF)

- Contingency Fuel (CF) - Alternate Fuel (AF) - Final Reserve Fuel (FR) - Additional Fuel (ADD) - Extra Fuel (XF)

These fuel amounts are defined in the COMMISSION REGULATION (EC) No 859/2008 also known as the EU-OPS 1.

Trip Fuel

The trip fuel is the necessary fuel from break release at the departure airport to touchdown at the destination airport. Included in this is the fuel required for the following segments:

- Take-off - Climb to TOC 180 200 220 240 260 280 300 320 340 360 200 220 240 260 280 300 100 150 200 250 300 350 400 450 Weight [t] Speed [m/s] A lt it u d e [ FL ]

33 - Cruise from TOC to TOD including step-climbs - Descent from TOD to approach

- Approach

- Landing at destination airport Contingency Fuel

For the type of missions flown by the aircraft in question the most relevant definition of contingency fuel is 5% of trip fuel. However, if an en-route alternate airport is available according to the conditions defined below then the contingency fuel can be reduced to 3% of the trip fuel.

“The … aerodrome shall be located within a circle having a radius equal to 20 % of the total flight plan distance, the centre of which lies on the planned route at a distance from the destination aerodrome of 25 % of the total flight plan distance, or at least 20 % of the total flight plan distance plus 50 nm, whichever is greater, all distances are to be calculated in still air conditions”31

Generally this requirement is fulfilled unless the flight takes place over very remote areas, such as the south pacific.

Alternate Fuel

Alternate fuel incorporates fuel for:

- Fuel for a go-around at the destination airport to missed approach altitude - Fuel for climb from missed approach altitude to TOC altitude

- Fuel for cruise from TOC to TOD at alternate airport - Fuel for approach and landing at alternate airport Final Reserve Fuel

Final reserve fuel is the fuel required for a 30 minute holding at 1500 ft above the alternate airport, in the case that no alternate is required for the flight then at the destination airport. Additional Fuel

The minimum additional fuel is the fuel required for:

- “the aeroplane to descend as necessary and proceed to an adequate alternate aerodrome in the event of engine failure or loss of pressurisation, whichever requires the greater amount of fuel based on the assumption that such a failure occurs at the most critical point along the route, and

 hold there for 15 minutes at 1 500 ft (450 m) above aerodrome elevation in standard conditions; and

31

34

 make an approach and landing, except that additional fuel is only required, if the minimum amount of fuel calculated in accordance with subparagraphs 1.2. to 1.5. above is not sufficient for such an event, and

- Holding for 15 minutes at 1 500 ft (450 m) above destination aerodrome elevation in standard conditions, when a flight is operated without a destination alternate aerodrome;”32

Generally, for the aircraft in question, the additional fuel will be covered by reserve fuels added in accordance to the clauses concerning contingency fuel, alternate fuel and final reserve fuel. Extra Fuel

Extra fuel shall be added at the discretion of the Capitan. Calculating the Top of Descent Weight

To calculate WTOD all fuel amounts have to be known along with the fuel required for the descent.

Descent and reserve fuel estimations are shown in the table below.

Segment Reserve fuel category Fuel amount

Descent from FL390 Trip Fuel 660 kg

Go-around + CLB to FL80 Alternate Fuel 2060 kg

Flight to alternate at FL80 Alternate Fuel 3700 kg

Holding 30 min at FL15 Final Reserve 3700 kg Table 3-14. Fuel Reserves.

The above values where calculated using an iterative method by assuming a value for WTOD from which the descent and reserve fuels could be calculated and a new WTOD could be determined. Then using the new value for WTOD the process was repeated; this was done until the value converged.

Assuming 3% contingency fuel, the amount of fuel required can be calculated as follows:

( )

( )

(3-44)

Using the equation above for TF and knowing the fuel required for the descent FDES, the fuel

required for flight until TOD will be:

32