culminating report

7/30/2019 Culminating Report

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Aerodynamics of a Transport Aircraft

December 20, 2012

Chris CarrCSUN Aeronautics

California State University, Northridge

AE 480

A Documentation of the basic aerodynamic forces acting on a transport aircraft duringmultiple mission phases, showing the effects of environmental temperature, density, and

pressure changes due to the elevation of the aircraft.


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PAGE LEFT INTENTIONALLY BLANK

Chris Carr: Fundamental Aerodynamics of Transport Aircraft 1


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Contents

1 Introduction 81.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 An Aircrafts Environment . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Basic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2 Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.3 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.4 Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Basic Flight Maneuvers: . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.1 Take-Off: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.2 Climb: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.3 Cruise: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.4 Landing: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Analysis 14

2.1 Aircraft Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Thrust Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Aerodynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 Takeoff and Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Results 203.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Flight Enviorment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Thrust Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Aerodynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.7 Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.8 Takeoff/Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Appendix A Flight Environment 39A.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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Appendix B Thrust Modeling 43B.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43B.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Appendix C Aerodynamic Modeling 47

C.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47C.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Appendix D Flight Envelope 51D.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51D.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Appendix E Range 54E.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54E.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Appendix F Climb 57F.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57F.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Appendix G Take-Off and Landing 60G.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60G.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62



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Name Value unitsnT 1.235 nonenS 0.972 noneR 287.03 kJ/*(kmol*K)

1.4 none

g 9.807 m/s2TSL 15 oCPSL 101.325 kPaSL 1.225 kg/m3

Density @ Sea Level [slugs/ft3] 0.00238 lb/ft3Density @ 10,000ft [slugs/ft3] 0.0018 lb/ft3Density @ 20,000ft [slugs/ft3] 0.00127 lb/ft3Density @ 30,000 ft [slugs/ft3] 0.00089 lb/ft3

aSL 340 m/srE 6,371 km

CDo 0.0155e 0.84

T/WSL 0.33AR 10.1

W/S 110MCR 0.85qMax 650 lbf/ft2Clmax 1.2

Wf/Wcraft 0.38

Tsfc Sl 0.65Start Elev (m) 9448.8Induced Drag Correction Factor, k 0.0375

Table 1: Input Parameters and Initial Values



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List of Figures

1.1 Sir George Cayley [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 The 7 Layer Atmospheric Mode [9]l . . . . . . . . . . . . . . . . . . . . . 91.3 Weight of the Aircraft [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Airfoil Lift [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 VTOL Takeoff: Harrier [12] . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Typical Aircraft Flight Plan . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 C-17 Take Off [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.8 Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Thrust vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Pressure vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Density vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Thrust vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Thrust Specific Fuel Consumption . . . . . . . . . . . . . . . . . . . . . . 243.6 Mach vs. Thrust to Weight . . . . . . . . . . . . . . . . . . . . . . . . . 253.7 True Air Speed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.8 CL vs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.9 Parabolic Drag Polar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.10 Aerodynamic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.11 Drag to Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.12 F light Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.13 Load Factor [?] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.14 Flight Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.15 W/S vs. Percent Throttle . . . . . . . . . . . . . . . . . . . . . . . . . . 323.16 Rate of Climb vs. Flight Speed . . . . . . . . . . . . . . . . . . . . . . . 333.17 3 Versions of Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.19 Take-Off Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.20 Thrust To Weight Ratio Vs Percent Throttle . . . . . . . . . . . . . . . . 363.18 Example of Aircraft Turning . . . . . . . . . . . . . . . . . . . . . . . . . 36

A.1 Thrust vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2 Pressure vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.3 Density vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

B.1 Thrust vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44B.2 Tsfc vs. Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B.3 Mach vs.Thrust Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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C.1 Parabolic Drag Polar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49C.2 Aerodynamic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49C.3 Coeff. Lift vs AoA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

D.1 Drag to Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

D.2 Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

E.1 Flight Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55E.2 W/S vs. Percent Throttle . . . . . . . . . . . . . . . . . . . . . . . . . . 56

F.1 R/C vs. Flight Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

G.1 Take-Off Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62G.2 Thrust To Weight Ratio Vs Percent Throttle . . . . . . . . . . . . . . . . 63



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List of Tables

1 Input Parameters and Initial Values . . . . . . . . . . . . . . . . . . . . . 4

3.1 Turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.1 Flight Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

B.1 Thrust at Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43B.2 Varying Mach against Thrust . . . . . . . . . . . . . . . . . . . . . . . . 44

C.1 Coefficient of Lift Varies to Determine Drag . . . . . . . . . . . . . . . . 47C.2 Coefficient of Lift Varies to Determine . . . . . . . . . . . . . . . . . . 48C.3 Varied Coeff. Lift to Find Aerodynamic Efficiency . . . . . . . . . . . . . 48

D.1 Induced Drag as it varies with Flight Speed . . . . . . . . . . . . . . . . 51D.2 Sample Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

E.1 Brequet and FAA Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . 54

E.2 FAA Stepped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

F.1 Analysis of Sea Level Rate of Climb . . . . . . . . . . . . . . . . . . . . . 57F.2 R/C Analysis 10,000FT . . . . . . . . . . . . . . . . . . . . . . . . . . . 58F.3 R/C Analysis at 20,000Ft . . . . . . . . . . . . . . . . . . . . . . . . . . 58F.4 R/C Analysis at 30,000Ft . . . . . . . . . . . . . . . . . . . . . . . . . . 59

G.1 Sea Level Take-Off Distance at Differing Thrust Ratios and Wing Loading 60G.2 5000ft Elevation Take-Off Distance at Differing Thrust Ratios and Wing

Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61G.3 Landing Distance At Sea Level With Varied Wing Loading . . . . . . . . 61

G.4 Landing Distance At 5000ft With Varied Wing Loading . . . . . . . . . . 62

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Chapter 1

Introduction

1.1 History

Figure 1.1: Sir George Cayley [7]

Aerospace engineering began in the late1800s to the mid 1900s. Huge strides weremade by people like Wilbur wright butmany attribute the origin of aerospace en-gineering to Sir George Cayley. Cayley isconsidered one of the most important peo-ple in aerospace engineering being the firstperson to separate the forces of Lift anddrag which are the most important vari-ables when it comes to flight. Most knowl-

edge on aircraft during the time of Cayleyis mostly empirical evidence. Many con-cepts of early aerospace engineering weretaken and hybridized from multiple otherengineering disciplines. Engineers under-stood fluid dynamics and began design-ing aircraft. After the Wright brothersand World War I, aircraft were seen to bethe way o the future in the military aswell as in the civilian sectors. It took awhile for aerospace engineering to be con-sidered a major discipline it wasnt until1958 that the first department of aerospaceengineering came to be The National Aero-nautics and Space Administration (NASA)was created mostly to compete with thethreat of war with Russia launching their first satellite in January of 1958 in responseto the launch of Sputnik. Aircraft are subjected to many different conditions producedby changes in atmospheric pressure, temperature, and structural loads applied on partsof the aircraft. Every Aerospace project has multiple different disciplines involved in theproject. These disciplines include aerodynamics, propulsion, avionics, materials science,

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structural analysis, and finally manufacturing. These disciplines interact with each otheras a system to create a functioning aircraft. The development and manufacturing of amodern flight vehicle is an extremely complex process and demands careful balance andcompromise between abilities, design, available technology and costs. Aerospace engi-neers design, test, and supervise the manufacture of aircraft, spacecraft, and missiles.

Aerospace engineers develop new technologies for use in aviation, defense systems, andspace.

1.2 An Aircrafts Environment

Figure 1.2: The 7 Layer Atmospheric Mode[9]l

The standard atmospheric model is knownas the 7 layer model. This model di-vides earths atmosphere into seven differ-ent layers according to major temperature

changes. Typical flight occurs between sealevel and 32km, the pressure in these layersis concentrated by gravity pushing downon the air.

The first layer is called the troposphereit consists of the first 12km of the atmo-sphere. This is the area where weather typ-ically occurs at and as height increases thetemperature of the atmosphere decreases.(Approximately 6.5 degrees Celsius perkilometer.) Next there is the tropopause,

this is the top of the troposphere, and thislocation separates the troposphere fromthe stratosphere. The tropopause is thelocation of a jet stream; this jet stream typically blows eastward. The third layer iscalled the stratosphere which is between 12 to 50km. In this layer there is quite a bit ofultraviolet radiation, the radiation is kept at bay by the Ozone layer which is also locatedat the stratosphere. The temperature from 12km to 20km stays constant but in the upperpart of this layer temperature begins to increase as altitude increases. Above the strato-sphere is the mesosphere which is between 50 to 80km. this is the coldest atmosphericlayer (-100 Celsius). In this layer meteoroids burn up and evaporate protecting the planet

below from constant meteor impacts. The 5th layer is the Thermosphere, in basic termsthis layer is the rest of the atmosphere. The temperature is extremely high in this areaapproximately 2000 degrees Celsius this is because the ultraviolet radiation is convertedinto heat in this layer. This layer contains the more layers within itself, the ionosphere,exosphere, and the magnetosphere. The ionosphere is the lower part of the thermosphere.This is in-between 80 and 550km. this area reflects radio waves back to the earth whichhelps with radio communications. The gases in this ionosphere absorb a large portion ofthe ultraviolet and x-ray radiation from the sun. Radio communications will be interferedwith if the sun has a solar flare and the gases absorb too much of this radiation chargingthe ions of the layer. The exosphere is the upper part of the thermosphere 550km to



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1000km this is where the satellites and international space station orbit the earth. ForAircraft its environment is the most important part about its flight. For typical transportflight, aircraft stick to the Stratosphere. This is because the stratosphere is above theTropopause, the Troposphere is where most weather typically occurs, and staying abovethis elevation aircraft can avoid most turbulence and environmental effects that could be

detrimental to the aircraft and its payload.

1.3 Basic Forces

1.3.1 Weight

Figure 1.3: Weight of the Aircraft [5]

Weight is a force always directed to thecenter of the earth. For aircraft the weightdepends on the sum of the aircrafts parts,fuel, and payload on board. This weightis distributed throughout the aircraft cre-ating the center of gravity needed for theaircraft. This center of gravity is what theaircraft rotates about. To fly well, twothings must be overcome The control ofthe aircraft in flight, as well as overcomingthe weight of the aircraft with an oppos-ing force. The aircraft will also constantlychange in weight due to the consumptionof fuel. Thus requiring the controls of the

aircraft must constantly be adjusted to ac-count for this loss of weight.

1.3.2 Lift

Figure 1.4: Airfoil Lift [6]

To overcome the vertically down force ofweight caused by gravity, aircraft generateanother force called lift. The force of liftis in the vertically upward direction oppos-ing gravity, once enough lift is generated to

overcome the force of gravity on the planethen the aircraft will begin to take off. Themagnitude of the force of lift is generatedby several different factors including thesize of the aircraft, shape of the wing, andthe velocity generated by the propulsionsystem. The majority of the lift of an air-craft is generated by the force of lift cre-

ated by the wings of the aircraft. Most aircraft wings are designed in a shape of an Airfoilthis airfoil causes a pressure distribution over the wings. The airfoil is thus shapes tocause the air traveling over the top of the wings to move at a much higher speed than



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that of the air traveling under the wings. This pressure difference makes the pressureunder the wings much higher than the pressure above the wings. The pressure causes theupward lifting force over the wings.

1.3.3 DragAir has volume as well as mass, any object traveling through air would have to movethat air in order to continue on its path. The force caused by moving this air is calledDrag. Drag is a force opposite of the direction of flight. Just like lift there are manyfactors that increase or decrease the magnitude of drag on an aircraft. These factorsinclude the shape of the aircraft, the viscosity of the fluid medium and the velocity of theaircraft. Drag is a unwanted but inevitable force during flight, thus during the design ofthe aircraft drag is something that is closely watched and minimized as much as possibleduring the design process.

1.3.4 Thrust

Figure 1.5: VTOL Takeoff: Harrier [12]

To overcome the forces of drag aircrafthave a propulsion system to propel the air-craft in a horizontal direction. This forcegenerated by a propulsion system is knownas Thrust. The direction is not always hor-izontal, this depends on the aerodynam-ics of the plane needed to have a goodefficiency, and sometimes the engines will

face ever so slightly upward. Some aircrafthave been specifically designed to be ableto take off vertically with their propulsionsystem then mechanically adjust the direc-tion the engines face to move forward, suchas the harrier. Something that should benoted about thrust in relation to aircraft,thrust does not generate lift. The thrustof an aircraft merely overcomes the forcesof drag and the weight of the aircraft. Lift

and Thrust help overcome Drag and Weight but they do not generate each other, and

vice versa.



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1.4 Basic Flight Maneuvers:

Figure 1.6: Typical Aircraft Flight Plan

1.4.1 Take-Off:

Figure 1.7: C-17 Take Off [8]

Takeoff is the point of flight in which theaircraft beings takeoff procedures. Afterthe flight has been planned and the air-craft has been taxied into position the air-craft will begin takeoff procedures. Beforetakeoff engines are usually tested severaltimes by sending power through them butnot moving forward. This is to make sure

no engine failures will occur during the ac-tual takeoff. After this is done the breaksare disengaged and the aircraft begins tomove forward on the runway until the take-off speed is achieved for the aircraft. Thetakeoff speed for an aircraft requires a lotof different variables such as air density,aircraft weight, aircraft configuration, ele-vation, and air temperature.

1.4.2 Climb:

After takeoff the climb phase begins. Each specific aircraft has a typical predefined cruisealtitude that it wants to achieve. This altitude is achieved using the safest and mosteconomical way possible. A climb is carried out by increasing the lift of the aircraft untilthe force of lift exceeds the force of weight. Thrust aids the increase in lift by increasingthe velocity forcing air against the wing at a faster rate while the aircraft uses its controlsurfaces to change the angle of attack of the wing. These working together force theaircraft to climb. A typical transport aircraft will climb to approximately 30,000ft beforeleveling off for cruise flight.



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1.4.3 Cruise:

Cruise follows the climbing phase of an aircrafts flight plan. Cruise is defined as levelflight. This portion of flight is where an aircraft is most fuel efficient as well as the longestpart of flight occurs. The only part of the aircrafts heading that typically changes in an

aircrafts flight during the cruise is the heading of the aircraft, keeping constant speedand altitude. Although cruise flight is the most efficient period of the aircrafts flight, itis also the most fuel consuming part of an aircrafts flight. Every aircraft has a optimumcruising altitude that is the most efficient for an aircraft. There are many conditions thatare needed to come together to achieve this optimum performance for example weight,center of gravity, air temperature, humidity, speed, fuel consumption rate, drag, etc.

1.4.4 Landing:

Figure 1.8: Landing

Landing is the final part of a aircraftsflight plan. Aircraft typically land on run-ways that are generally constructed of as-phalt, concrete, or densely packed grounds.Landing is accomplished by a very slowgradual descent to the runway. Transportaircraft need be especially concerned withrunway length due to the time needed toreduce the aircraft to a stop is much higherthan other aircraft types.



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Chapter 2

Analysis

2.1 Aircraft Environment

Flight Environment: The following analysis is performed from 0 meters to 32 km. Thetemperature, pressure, density, and airspeed were calculated from the following equations.The resulting data is expected to confirm the 1962 Standard Atmosphere Data. For theTroposphere and Upper stratosphere the following equations were used.

T2T1

= 1 n 1

n

g

RT1(h2 h1) (2.1)

P2P1

= [1 n 1

n

g

RT1(h2 h1)]

(n(n1) (2.2)

21= [1 n

1n gRT1(h2 h1)]

(n

(n1) (2.3)

The next equations are used when the temperature is isothermal conditions in eleva-tion

P2P1

= eg

RT(h2h1) (2.4)

=P

RT(2.5)

The acoustic airspeed was calculated using the following equation

a = (RT).5 (2.6)

2.2 Thrust Modeling

Based on previous calculations in the area of temperature, density, pressure, and acous-tic airspeed, the following equations were used: Equation(1) is the computation of theThrust Ratio as it relates to density and since density has direct correlation to elevationEquation(1) was used to plot The Thrust Ratio vs. Elevation. The exponent of Equation

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(1) is .7 up to the Tropopause, once the Tropopause is reached the exponent of Equation(1) changes to 1.

T

Tsl= (

el

sl).7 (2.7)

T

Tsl= (

elsl

)1 (2.8)

Equation(2) is the application of the fuel air consumption of the airplane, like Equa-tion(1) this equation was used to plot the Thrust Fuel Air Consumption versus the ele-vation of the aircraft. This equation is valid up to the Tropopause, once reach it becomesinvalid without a modification to the equation. The exponent of the equation changes to-.1 due to the lack of air at the elevation of the aircraft.

TsfcTsfcsl

= (elsl

).2 (2.9)

Equation (3) and (4) are used to plot Mach Number versus The Static Thrust. Firstthe TAS is found using Equation (3) then TAS is used as V to calculate the thrust of theaircraft.

T ASMCR = a M CR (2.10)

T

Tstatic= 1 103V+ 106V)2 (2.11)

Equation (5) is used to calculate the true airspeed with a known indicated airspeed

of the aircraft.

T ASIAS = IAS

(

slel

(2.12)

Equation(6) is the calculation of the stall speed with a known wing loading of 90 lbft2

and a coefficient of lift of 1.2. Stall speed was analyzed in its relation to elevation.

Vstall =

W/S g

.5CLmaxelev(2.13)

2.3 Aerodynamic Modeling

A model of the aerodynamics of an aircraft can be modeled in multiple ways, the proce-dure used for the alaysis of this transport aircraft was by varying the Coefficient of liftfrom 0 to 1.2 with aspect ratios of 3, 6, and 10. The aircraft was given a sweep of 35degrees for the analysis.

Firstly the angle of attack was developed using the below equation. The angle ofattack is the aircrafts relationship with the horizontal plane as the lift increases due todiffering factors in this analysis AR. This equation outputs an angle in radians of anaircraft, with AR being the main driver of the equation.



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= CL(1 + (1 + (AR

(2 Cos(Sweep)

2

)1/2/(AR) (2.14)

This equation outputs the drag of the equation, as the coefficient of lift increases it is

expected that the drag will increase. This is because as you increase the angle of attack ofthe aircraft the frontal area of the aircraft in relation to the y axis increases thus leavingmore area to be effected by forces countering the thrust of the aircraft

CDo = Cdo + kC2L (2.15)

E =CL

Cdo + kC2L (2.16)

The above equation will be used to calculate the the aerodynamic efficiency of theaircraft as the coefficient of lift increases. The efficiency can be then related to themaximum aerodynamic efficiency the aircraft can accomplish then showing its efficiency.The below equation uses the Drag equation to attain the maximum lift that an aircraftcan attain at at the angle of attack of the wing.

CLEmax =

(

Cdo

k(2.17)

The previous equation which incorperated all previous aerodynamic equations to con-

clude with the maximum aerodynamic efficiency of the aircraft this can be comparedwith the aircraft at different levels of flight to optimally choose proper angles of flightthat produce the best efficiencies of the aircraft.

Emax =1

(4k) Cdo(2.18)

2.4 Flight Envelope

For Flight Envelope analysis Thrust is varied among different flight speeds under tocalculate the drag to weight ratio of the aircraft. This is specifically driven by flightspeed. While at low flight speeds the drag is expected to be low, but as the flight speedincreases the drag will increase in a parabolic fashion.

D

W=

1

2elevV

2(Cdo

W/S) + k

W/S12

elev V2(2.19)



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The Trust to Weight ratio of an aircraft is directly driven by the aircrafts elevationduring flight changing as the air becomes less dense with an increase in elevation. Forthis analysis T/W was tun at 85

T

Welev= T/W

sl

el

sl

0.70.85 (2.20)

The following two equations are driven specifically by the T/W of the aircraft differingonly by a change in sign, both these equations show the highest velocity and the lowestvelocity of the aircraft possible based on the thrust level of the aircraft.

Vhigh =(W/S T /W)

elev Cdo

(1 + (1 (4 (k)) Cdo)

((T /W2)))1/2))

1/2

(2.21)

Vlow =W/S T /W

elevCdo)(1

(1 (4 (k))Cdo)T /W2)))1/2

1/2(2.22)

The Stall equation will result in a velocity of the aircraft that enduces stall, meaningthat the aircraft will be unable to continue flight and will begin to fall.

Vstall ==(2 (W/S)

elev)CLmax1/2 (2.23)

This equation outputs the optimal cruise velocity of the aircraft, this speed is where

most of the aircrafts fuel will be used.

Vcruise = M CR a 3.28084 (2.24)

The Vqmax equation depicts a limiting constraint on Vhigh as will be discussed later

Vqmax =2 qMax

elev

1/2

(2.25)

this equation is used to show how a Airspeed of 250mph will operate withing the flightenvelope.

V250 == IAS(1.225

elev)1/2 (2.26)

2.5 Range

C1 = tsfcrelev TsfcSl (2.27)



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Range is the most important concept when it comes to transport aircraft, there arethree main ways to calculate the range of an aircraft

The Brequet equation is a linear representation of flight through a constant climb foran aircraft yielding the highest results for range

Breguet = V EC

Log(1 Wratio) (2.28)

The FAA Range results in the most conservative range of an aircraft keeping a constantelevation throughout flight

RangeFAA =2EmaxV

C

Atn((Wratio E)

2 Emax (1 k CL E Wratio(2.29)

The FAA Stepped is the commonly used for most commercial aircraft flight. This

uses a elevation step function to create traffic lanes as well as the bridge the Brequet andFAA Equations to result in a mot realistic range for aircraft.

F AASteppedWeighFuelFraction == 1 2

1(2.30)

The Trust Specific fuel consumption Equations as it changes due to density changeswith elevation. This has significant effects in the Troposphere and changes from positivebenefits to negative benefits after crossing the Tropopause

tsfcR =

elevsl

.2

(2.31)

tsfcR =elevtrop

0.1(2.32)

2.6 Climb

Equation representing the rate of climb of an aircraft. For this analysis the rate of climbwas analyzed at sea level,10, 20, and 30 thousand feet. The best rate of climb, the maxvelocity during climb, and the best rate of climb were calculated for each elevation chosen.

This equation results in the curve of an aircrafts rate of climb as it varies with velocity

R/C =T

W

D

W V 60 (2.33)

Change in wing loading as density changes with elevation

W/Selev == W/S2

1(2.34)



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the maximum velocity at the best climb angle

Vmax ==2 W/S10kCos(Climb)

20k

1/2

((k/CDo)1/4 (2.35)

best rate of climb based on the best climb angle and maximum velocity

R/C == VmaxSin(Climb) 60 (2.36)

2.7 Takeoff and Landing

The Equation for the take off perameter directly needed to assess the takeoff runwaydistance

TakeOffParameterT.O.P = W/S (1

/CLMAXto)

slelev

) (1

/T /W) (2.37)

Equation calculating the minnimum take off distance for the aircraft

Stop = 20.9 T.O.P + 87 (T.O.P T/W.25)1/2 (2.38)

Equation determining the parameter for landing distance

LandingParameterL.P = W/S1

(CLMAXland) (

slelev

(2.39)

equation using the landing parameter to calculate landing distance for the aircraft

LandingDistance = 118 LP + 400 (2.40)



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Chapter 3

Results

3.1 Basics

xxx

3.2 Flight Enviorment

Figure 3.1: Thrust vs. Elevation

Figure 3.1: Temperature varies at different elevations based on may different factorseach major change occurs when a new layer of the atmosphere is achieved. From Figure3.1 you can see that as an aircraft climbs temperature is linearly decreasing, but when

20


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the Tropopause is crossed the aircraft is subjected to a constant temperature for almost15000ft and then begins to linearly increase. This continues throughout an aircrafts climbdepicted in figure 3.1. Most Transport aircraft travel in the troposphere but there aresome that fly above this.

Figure 3.2: Pressure vs. Elevation

Figure 3.2: Aircraft performance is typically considered with a combination of liftand thrust. Thrust is generated by the an engine but with elevation the performanceof an aircrafts engine decreases with an increases in the ambient temperature of thesurrounding environment and also decreases while the air pressure decreases.



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Figure 3.3: Density vs. Elevation

Figure 3.3: When it comes to aircraft performance Altitude Density is the single mostimportant factor. As an Aircrafts altitude increases the air becomes less dense. When airbecomes less dense. When it comes to aircraft will have decreased engine performance athigher altitudes due to density, when the engine in not performing optimally the aircraftwill have a more difficult time taking off as well as a more difficult time in the climbingphase of its mission.Conclusion:Aircrafts Performance can be broken down into its fundamental elements and these el-ements would be Pressure, Density, and Temperature. Performance in this case mustbe optimized to operate the aircraft in question at peak performance. Which will meantradeoffs between multiple different subsystems of the aircraft. For example propulsionsthrust to weight ratio, Wing Surface Area and Runway Length are directly related to theaircrafts environments. When planning an aircrafts flight all of these must be considered

constantly to have the best performance of your aircraft. The Following sections willaddress the constraints on an aircraft due to the enviorment that it will be placed in dueto atmospheric conditions



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3.3 Thrust Modeling

Figure 3.4: Thrust vs. Elevation

This graph represents the Thrust ratio with its relationship with elevation. As elevationincreases the thrust required to push a hypothetical aircraft becomes less and less, whichis expected due to the relationship between density and elevation. When thrust crossesthe Tropopause there is a small but noticeable increase in the required thrust to pushan aircraft. In the grand scope of things this slight change should not be overlookedregardless. The Performance of the aircraft is the key to aerodynamic performance.



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Figure 3.5: Thrust Specific Fuel Consumption

Here shows the Thrust fuel consumption rate of a Transport aircraft compared toelevation. Unlike the Thrust versus elevation linear curve as elevation is increased theamount of fuel consumed to produce the equivalent thrust is increased. This is becausethe engine does not need to work as hard to push the aircraft, thus saving fuel andallowing for longer flight. With the Tropopause due to the lack of oxygen the enginemust push harder and thus requires much more fuel to push the aircraft. This conceptis very important when it comes to transport aircraft. This shows that if an aircraftpays attention to its elevation and fuel consumption it could effectively fly longer than itwould typically at sea level and other elevations. This increases the overall performanceefficiency of the aircraft.



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Figure 3.6: Mach vs. Thrust to Weight

This graph represents the thrust ratio versus the Mach number at sea level, 5000ftand 10000ft. As can be seen in the graph at first Mach number tends to reduce the thrustratio, but as the Mach number increase the Thrust ratio tends to increase at a significantrate for all levels of elevation. Elevation seems to have a fairly insignificant effect ontothe thrust ratio vs Mach number.



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Figure 3.7: True Air Speed Analysis

Above represents three things. The color green represents the stall velocity of planewith a wing loading of 110 lb

ft2and a maximum coefficient of 1.2, the color blue rep-

resents the Indicated Air Speed of a plane traveling at 250 MPH, and lastly red rep-resents the True Air Speed of a plane with constant Mach number of .85. The StallVelocity as well as the True Air Speed of the constant velocity plane both has thesame trend. Elevation reduces air density which in turn allows the planes to travelfaster in higher elevations. The Mach number is directly affected by density and sincedensity is affected by temperature as well it boils down to following the trend line ofpolytrophic and isothermal regions. Notice that the True Airspeed crossed the stall ve-locity curve; this indicates that the aircraft will stall at approximately 19 km if thatairspeed it kept at that elevation. This occurrence must be avoided at all costs in flight.Conclusion:It is found that the increase in elevation directly affects the Thrust, Stall and TAS

of aircraft. Typically in the Troposphere Thrust requirements and are greatly reducedwith increase in elevation, but one the Tropopause is passed the requirements begin tosignificantly increase. Elevations effect on airspeed is significant as well. As elevation in-creases so does airspeed, but with each step in elevation the amount of velocity increase isless with each Step. All these values are extremely significant when designing an aircraft.During the design of an aircraft these are extremly important to watch, as was seen withthe true airspeed at Mach .85



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3.4 Aerodynamic Modeling

Figure 3.8: CL vs.

Figure 3.8: is the coefficient of lift as it varies with angle of attack. It can be seenthat Angle of attack is linearly related to the Coefficient of Lift. The Coefficient of liftincreases as the aircraft adjust its angle of attack. The aspect ratio was varied between3, 6, and 10 to show how the aspect ratio as effects the lift that an aircraft will generate.

With an Aircraft with High Aspect ratios such as C-130 Hercules (Aspect Ratio 10.1)you will receive added lift which in turn will allow for a lower thrust to propel the aircraftforward. This is highly beneficial for transport aircraft. Due to long flights over largedistances, a transport will need these benefits.



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Figure 3.9: Parabolic Drag Polar

Figure 3.9: is the parabolic drag polar of the aircraft. This drag is caused by viscousshear effects caused by the aircrafts increase in lift. This drag is called pressure drag,pressure this is caused by the coefficient of lift. As you increase the aspect ratio dragbecomes less and less of an issue to the plane by decreasing the tip vortices, but if youadjust the angle of attack above or below 0 degrees you begin to create induced drag,this can be devastating to the plane if the critical angle of attack is achieved by the planecausing stall and forcing the pilot to perform maneuvers to regain control of the aircraft.

Figure 3.10: Aerodynamic Efficiency



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Figure 3.10: represents the Aerodynamic efficiency of the three variations that arebeing discussed. Each aspect ratio is plotting Aerodynamic efficiency versus the co-efficient of lift. As aspect ratio increases it can be seen that the maximum aerody-namic efficiency increases drastically as the coefficient of lift increases, which shows thatit is better for efficiency to have a transport aircraft that has a larger aspect ratio.

Lower aspect ratio aircraft have a much more difficult time with efficiency in this area.

Conclusion:Coefficient of lift has its pro and cons, if an engineer does his job perfectly right anaircraft can act perfectly in the air but unfortunately this is impossible. It is found thathigher aspect ratios are important for all aircraft. Higher aspect ratio aircraft can fly athigher angles of attack, have much lower parabolic drag, and have a much higher aerody-namic efficiency. In this case an aspect ratio of 10 had an Aerodynamic efficiency of 20.63the highest of the three variation analyzed, it also had higher angle of attack capabilities,and the least amount of drag. It is recommended that transport aircraft have high aspect

ratios for this reason.

3.5 Flight Envelope

Figure 3.11: Drag to Weight

Figure 3.11 represents the drag to weight ratio and the thrust to weight ratio. Drag variessignificantly as the true airspeed of the aircraft changes. The first several data pointson the Drag to weight are due to pressure drag. As the true air speed increases theinduced drag decreases significantly. After the Induced drag bottoms out, the parasiticdrag begins to increase which causes the total drag to increase. This can be seen in theDrag curve of Figure 1



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Figure 3.12: Flight Envelope

Figure 3.13: Load Factor [?]

Figure 3.12: is the flight enve-lope. This represents the constraintson the aircraft. Velocity on the lowend is what the plane would needto gain flight. When the stall ve-locity in flight is above the mini-mum flight velocity it forces the air-craft above minimum flight velocity.

To achieve the best efficiency of flightthe best flight speed would be Vcruisewhich is kept under Vlow. If thecruise speed goes beyond Vmin the ef-ficiency of the aircraft will drop signif-icantly. The maximum flight altitudeis 53,000 feet. The Transport aircraftis considered to be 250,000 lbs witha wing loading of 110 W/S. The air-craft cannot operate over the Velocity

on the high end without systems failure.Conclusion:

The Transport aircraft with a wingloading of 110 W

S, 250,000 lbs aircraft, an Oswalds efficiency factor of .85 and a Cdo

of .0155 will fly properly. The aircraft must fly a mean velocity of approximately 500 fts

and below 1500 fts

. The cruise velocity of approximately 800 fts

. Beyond these condi-tions you lose efficiency or the aircraft is un-flyable. This aircraft will operate best at acruise altitude of 57,000 ft. Under 57,000 ft the aircraft will not be operating at max-imum cruise efficiency. To illustrate the importance of this information consider figure3.13. This represent information similar to that of figure 3.12. The aircrafts structural



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capabilities will be strained if taken beyond Vno, and failure will occur at Vne and abovespecific load factors. The aircraft thus has a very specific area it is permitted fly knownas the Flight Envelope. If an aircraft is not flying within in permitted envelope it is beingpushed far beyond what it is capable of and failure is quite possibly imminent. Theseareas of failure are extremely important and must be continuously watched during the

planning of the flight to be sure that the aircraft does not enter its failure velocity.

3.6 Range

Figure 3.14: Flight Profile

Figure 3.14:: In this flight profile a transport aircraft was analyzed. One tool used wasthe Breguet equation to calculate the maximum range of the aircraft. This system takesinto account. The maximum range of this aircraft was calculated to be 7965 Miles usingthe Breguet tool. Breguet gives more range than FAA but is not easy to follow whenit comes to actual flight. The FAA has a tool to calculate the maximum range and ismuch more conservative than the Breguet Equation. The FAA maximum range for thisaircraft is approximately 7214 Miles. This system requires the aircraft to fly at a constantelevation and is fairly simple to follow. Thirdly there is the FAA step altitude systemfor an aircraft maximum range. This system has the aircraft increase in flight altitudeperiodically to increase the maximum range of the aircraft and to fly more efficiently.The system attempts to follow the Breguet Equation by steps to reduce the number oftimes that altitude changes for the aircraft. The benefits of this are that the FAA cancreate traffic lanes for aircraft, as well as increase the maximum range of the aircraft. Inthis calculation the maximum range of the aircraft using the FAA Altitude steps is 7622Miles. Of the three methods the Step method is the best plausible method to use. TheFAA method is not as efficient, and the Breguet method is too optimistic.



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Figure 3.15: W/S vs. Percent Throttle

Figure 3.15: As can be seen in this graph, as the aircraft follows its flight paththe wing loading of the aircraft decreases. This phenomenon is because the aircraftis burning fuel as it flies its desired path. The model uses the step function and thisfunction takes the current fuel and wing loading at that step and makes it constantfor the duration of the step and is recalculated at the end of the step. Conclusion:Of the Three Flight Models it appears that the Breguet Model with a maximum flightplan of approximately 9279 Miles is too optimistic and very difficult to follow during flight,and the FAA model with a maximum range of 8058 Miles is too conservative and wastesfuel keeping a constant elevation, lastly The Stepped Model with a maximum range of8777 Miles is the most optimal flight plan for aircraft, it gives the FAA the ability to

control flight as well as give time between flight maneuvers. In regards to aircraft wingloading it is necessary to be sure that the plane is carrying the proper amount of fuel forthe flight but it must also take into account that during flight the current engine maynot be taxed over 100



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3.7 Climb

Figure 3.16: Rate of Climb vs. Flight Speed

Wing Loading SL 110 fraclbfft2

Wing Loading 10k 81.03 fraclbfft2



c, Max @SL [degrees] 16.37 degreesc, Max @10k ft 12.65 degreesc, Max @20k ft 9.46 degrees

c, Max @30k ft 6.74 degreesVc, Max @SL 379.23 ft/min

Vc, Max @10k ft 379.04 ft/minVc, Max @20k ft 445.56 ft/minVc, Max @30k ft 377.92 ft/minR/Cc, Max @SL 6411.32 ft/min

R/Cc, Max @10k ft 4978.92 ft/minR/Cc, Max @20k ft 4394.32 ft/minR/Cc, Max @30k ft 2662.07 ft/min

Table 3.1: Turning

Figure 3.16: depicts rate of climb at various elevations varying flight speed. As can beseen there is a parabolic trend in the graph at each designated elevation. In this graph theelevations tested were: sea level, 10, 20, and 30 thousand feet. The graph confirms that aselevation increases the maximum rate of climb decreases. This based on the evaluation isunderstandable due to the variations of density as elevation increases. The maximum rateof climb as well as the steepest angle at noted at on each curve. These points indicate thatas the velocity increase further the rate of climb at the specified elevation will effectivelyreduce the rate of climb to the point of stall. As noted in table 1 as elevation increasesthe aircraft maximum climb angle decreases, but the velocity required to achieve these



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climb angles increases with elevation. This is because the aircraft needs to make up orthe changes in atmospheric characteristics.

Elevation ft rho V ft TW elev Nft phi R2438.40 8000 0.00187 135.29 0.26 1.59 56.66 461.05

2743.20 9000 0.00181 137.52 0.25 1.57 55.76 484.513048.00 10000 0.00176 139.46 0.25 1.55 54.33 511.703352.80 11000 0.00170 141.89 0.24 1.53 53.27 540.143657.60 12000 0.00165 144.03 0.23 1.51 52.03 569.143962.40 13000 0.00160 146.26 0.23 1.49 50.69 601.174267.20 14000 0.00155 148.60 0.22 1.47 49.26 636.524572.00 15000 0.00150 151.06 0.22 1.45 47.69 676.154876.80 16000 0.00145 153.64 0.21 1.43 46.00 720.435181.60 17000 0.00140 156.36 0.21 1.40 44.14 770.745486.40 18000 0.00135 159.23 0.20 1.39 42.79 818.42

Table 3.2: Turning

Figure 3.17: 3 Versions of Climb

Table 2 is the loiter calculations basedon the changes in elevation. Expected re-sults for a transport aircraft is a bank an-gle of approximately 60 degrees. As seenhere based on the aircraft characteristicsthe max bank angle is achieved at 8000ftof 56 degrees. As the Thrust to Weightchanges the aircrafts required bank angle

decreases. This decrease of bank angle in-creases the required radius

to achieve a turn due to the forcein the direction of turn is consistentlyless as bank angle decreases. Usinga smaller bank angle is safer for mosttransport aircraft and is typically used.Transport aircraft use small bank an-gles to achieve a smaller load factor.Load factor represents the stresses and

strains on the plane, the smaller thisnumber is the better for the aircraft.

Conclusions:Based on the calculations the pilot of the aircraft will need to properly attenuate theplane during flight to be sure that fuel is not wasted. The best rate of climb for theaircraft must be recalculated based on the elevation that the aircraft is currently at. Iftake off is desired then the climb angle at sea level is the most important aspect, obstaclesat the end of the runway must be able to be avoided. Once take off is completed then theclimb angle must vary at certain points of elevation to keep the aircraft operating at the



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best efficiency possible. Thus one constant angle may not be used for all elevations Theaircraft during loiter should be well watched to avoid high load factors, stalling by goingbeyond the allowed bank angle, and using the incorrect radii. These characteristics aredirectly related to the thrust to weight ratio of the aircraft and the elevation of the aircaft.If a safe load factor is desired for precious cargo then the higher you are the easier the

turn will go for the aircraft, if time is key then lower elevation turns are desired. Overallrecommendations for an aircraft with these characteristics should take off under the givenmax climb angle at sea level then periodically adjust the climb angle at approximately5000 foot intervals, unless restricted by the FAA then keep to the 2000 foot interval.Once Cruise is achieved keep at that elevation until arriving at the destination. Loiterabove the airport at a safe elevation to where the load factor and bank angle are safe forthe transport.

3.8 Takeoff/Landing

Figure 3.19: Take-Off Distance



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Figure 3.20: Thrust To Weight Ratio Vs Percent Throttle

Figure 3.18: Example of Aircraft Turning

Runway distance is linearly proportional tothe wing loading of the aircraft. As wingloading increases the required amount oftakeoff distance needed for the aircraft toachieve flight is increasing as well. Thoughthere is a difference between sea leveland a takeoff of 5000ft. From the graphthere is athe slope of the line for a run-way at 5000ft is much greater than thatof a runway at sea level. this is due

to many factors mainly the changes inair density at the elevation of the run-way. For landing an aircraft the equa-tions are not similar but the results arethe similar. An aircrafts needed distanceto land is linearly proportional to the wingloading of the aircraft, and as elevationchanges so does the slope of the required landing distance. Thus showing thatthe higher you are the less runway distance you will need to land and takeoff.

Conclusions:

Aircraft must land and takeoff on the proper runways allowed, not doing this couldresult in a loss of the aircraft or severe structural damage due to overshoot or overrun.



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Bibliography

[1] AE480 1-7

[2] http://www.latex-project.org/

[3] Design of Aircraft, Thomas C. Corke

[4] Introduction to Flight, John Anderson

[5] http://www.grc.nasa.gov/WWW/k-12/airplane/forces.html

[6] http://en.wikipedia.org/wiki/Lift(force)

[7] http://www.adsadvance.co.uk/sir-george-cayley-s-notebooks-featured-on-antiques-roadshow.html

[8] http://www.wallsfeed.com/aircraft-c-17-globemaster/

[9] http://mail.colonial.net/ hkaiter/LayersoftheAtmosphere.html

[10] http://www.answers.com/topic/flight-envelope

[11] http://www.langleyflyingschool.com/Pages/Aerodynamics

[12] http://www.aerospaceweb.org/question/planes/q0042.shtml

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Appendices

38


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39


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Appendix A

Flight Environment

A.1 Tables

Elev(m) Elev(ft) n T2 (K) T2 (oC) P2 (kPa) 2 (kg/m3) a2 (m/s) Elevation (m)0.00 0.00 1.24 288.00 14.85 101.33 1.23 340.00 0.00

1 3280.84 1.24 281.50 8.35 89.87 1.11 336.33 1000.162 6561.68 1.24 275.00 1.85 79.48 1.01 332.42 2000.633 9842.52 1.24 268.50 -4.65 70.09 0.91 328.47 3001.414 13123.36 1.24 261.99 -11.16 61.62 0.82 324.47 4002.515 16404.20 1.24 255.49 -17.66 54.00 0.74 320.42 5003.936 19685.04 1.24 248.99 -24.16 47.16 0.66 316.31 6005.667 22965.88 1.24 242.49 -30.66 41.03 0.59 312.16 7007.708 26246.72 1.24 235.99 -37.16 35.57 0.52 307.94 8010.06

9 29527.56 1.24 229.49 -43.66 30.72 0.47 303.67 9012.7310 32808.40 1.24 222.99 -50.16 26.41 0.41 299.34 10015.7211 36089.24 216.65 -56.50 22.56 0.36 295.06 11019.0312 39370.08 216.65 -56.50 19.27 0.31 295.06 12022.6513 42650.92 216.65 -56.50 16.45 0.26 295.06 13026.5814 45931.76 216.65 -56.50 14.05 0.23 295.06 14030.8315 49212.60 216.65 -56.50 12.00 0.19 295.06 15035.4016 52493.44 216.65 -56.50 10.25 0.16 295.06 16040.2817 55774.28 216.65 -56.50 8.76 0.14 295.06 17045.4818 59055.12 216.65 -56.50 7.48 0.12 295.06 18051.00 0

19 62335.96 216.65 -56.50 6.39 0.10 295.06 19056.83 020 65616.80 216.65 -56.50 5.46 0.09 295.06 20062.98 021 68897.64 0.97 216.65 -56.50 4.66 0.07 295.06 21069.45 022 72178.48 0.97 217.63 -55.52 3.99 0.06 295.73 22076.23 023 75459.32 0.97 218.62 -54.53 3.41 0.05 296.40 23083.33 024 78740.16 0.97 219.60 -53.55 2.92 0.05 297.06 24090.75 025 82021.00 0.97 220.59 -52.56 2.50 0.04 297.73 25098.49 026 85301.84 0.97 221.57 -51.58 2.14 0.03 298.39 26106.54 027 88582.68 0.97 222.56 -50.59 1.84 0.03 299.05 27114.91 028 91863.52 0.97 223.54 -49.61 1.58 0.02 299.71 28123.60 0

29 95144.36 0.97 224.52 -48.63 1.36 0.02 300.37 29132.61 030 98425.20 0.97 225.51 -47.64 1.17 0.02 301.03 30141.93 031 101706.04 0.97 226.49 -46.66 1.00 0.02 301.69 31151.58 032 104986.88 0.97 227.48 -45.67 0.86 0.01 302.34 32161.54 0

Table A.1: Flight Environment



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A.2 Graphs

Figure A.1: Thrust vs. Elevation



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Figure A.2: Pressure vs. Elevation

Figure A.3: Density vs. Elevation



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Appendix B

Thrust Modeling

B.1 Tables

Elev a T T ASMCR:.85 TAS Vstall T/Tsl Tsfc/Tsfcsl0 1.225 340.328 288.160 289.279 111.760 76.582 1.000 1.000

1000 1.112 336.468 281.660 285.998 117.319 80.392 0.934 0.9812000 1.007 332.563 275.160 282.678 123.294 84.486 0.872 0.9613000 0.909 328.611 268.660 279.320 129.727 88.894 0.812 0.9424000 0.819 324.612 262.161 275.920 136.666 93.649 0.755 0.9235000 0.736 320.562 255.661 272.478 144.165 98.788 0.700 0.9036000 0.660 316.461 249.161 268.992 152.285 104.352 0.648 0.8847000 0.590 312.306 242.661 265.460 161.095 110.389 0.599 0.8648000 0.525 308.095 236.161 261.881 170.675 116.953 0.553 0.8449000 0.466 303.826 229.661 258.252 181.117 124.108 0.509 0.824

10000 0.413 299.495 223.161 254.571 192.525 131.926 0.467 0.80411000 0.364 295.101 216.660 250.835 205.052 140.510 0.428 0.78412000 0.311 295.101 216.660 250.835 221.870 152.034 0.399 0.79713000 0.265 295.101 216.660 250.835 240.068 164.504 0.341 0.81014000 0.227 295.101 216.660 250.835 259.759 177.997 0.291 0.82215000 0.194 295.101 216.660 250.835 281.065 192.596 0.249 0.83616000 0.165 295.101 216.660 250.835 304.118 208.393 0.212 0.84917000 0.141 295.101 216.660 250.835 329.062 225.486 0.181 0.86218000 0.121 295.101 216.660 250.835 356.052 243.981 0.155 0.876

19000 0.103 295.101 216.660 250.835 385.255 263.992 0.132 0.89020000 0.088 295.101 216.660 250.835 416.883 285.665 0.113 0.904

Table B.1: Thrust at Elevation

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MCR SL V 5kft V 10kft Voo T(V)/Tst(SL) T(V)/Tst(5kft) T(V)/Tst(10kft)0 0.000 0.000 0.000 1.000 1.000 1.000

0.10 111.656 109.751 105.154 0.901 0.902 0.9060.20 223.312 219.502 210.308 0.827 0.829 0.8340.30 334.968 329.253 315.462 0.777 0.779 0.784

0.40 446.624 439.004 420.615 0.753 0.754 0.7560.50 558.281 548.755 525.769 0.753 0.752 0.7510.60 669.937 658.506 630.923 0.779 0.775 0.7670.70 781.593 768.257 736.077 0.829 0.822 0.8060.80 893.249 878.008 841.231 0.905 0.893 0.8660.90 1004.905 987.759 946.385 1.005 0.988 0.9491.00 1116.561 1097.510 1051.539 1.130 1.107 1.0541.10 1228.217 1207.261 1156.693 1.280 1.250 1.1811.20 1339.873 1317.012 1261.846 1.455 1.418 1.3301.30 1451.529 1426.763 1367.000 1.655 1.609 1.502

1.40 1563.185 1536.514 1472.154 1.880 1.824 1.695

Table B.2: Varying Mach against Thrust

B.2 Graphs

Figure B.1: Thrust vs. Elevation



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Figure B.2: Tsfc vs. Elevation



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Figure B.3: Mach vs.Thrust Ratio



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Appendix C

Aerodynamic Modeling

C.1 Tables

3-2 3 6 100 0.016 0.016 0.016

0.1 0.017248311 0.016624155 0.0163744930.2 0.020993244 0.018496622 0.0174979730.3 0.027234798 0.021617399 0.0193704390.4 0.035972974 0.025986487 0.0219918920.5 0.047207772 0.031603886 0.0253623320.6 0.060939192 0.038469596 0.0294817570.7 0.077167233 0.046583617 0.034350170.8 0.095891896 0.055945948 0.0399675690.9 0.117113181 0.066556591 0.0463339541 0.140831088 0.078415544 0.053449326

1.1 0.167045616 0.091522808 0.0613136851.2 0.195756767 0.105878383 0.06992703

Table C.1: Coefficient of Lift Varies to Determine Drag

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3-1 3 6 100 0 0 0

0.1 0.031172102 0.023699187 0.0210806950.2 0.062344203 0.047398374 0.042161390.3 0.093516305 0.071097561 0.063242085

0.4 0.124688406 0.094796748 0.084322780.5 0.155860508 0.118495936 0.1054034750.6 0.18703261 0.142195123 0.126484170.7 0.218204711 0.16589431 0.1475648640.8 0.249376813 0.189593497 0.1686455590.9 0.280548915 0.213292684 0.1897262541 0.311721016 0.236991871 0.210806949

1.1 0.342893118 0.260691058 0.2318876441.2 0.374065219 0.284390245 0.252968339

Table C.2: Coefficient of Lift Varies to Determine

3-3 3 6 100 0 0 0

0.1 5.970751362 6.201875216 6.2994136780.2 9.75931408 11.11319684 11.766108780.3 11.22133038 14.20629504 15.897881020.4 11.27619013 15.69459139 18.611669740.5 10.70485657 16.0751618 20.11074457

0.6 9.927333301 15.80211711 20.702678230.7 9.130367327 15.18978007 20.679364430.8 8.38645662 14.4284664 20.26980690.9 7.717823925 13.62468138 19.636097571 7.126004756 12.83440953 18.88598153

1.1 6.604797075 12.08488314 18.088033991.2 6.145753721 11.38753473 17.28433436

CLEmax 0.350295575 0.495392753 0.639549294Emax 11.29985726 15.9804114 20.6306224

Table C.3: Varied Coeff. Lift to Find Aerodynamic Efficiency



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C.2 Graphs

Figure C.1: Parabolic Drag Polar

Figure C.2: Aerodynamic Efficiency



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Figure C.3: Coeff. Lift vs AoA



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Appendix D

Flight Envelope

D.1 Tables

Vtrue T/W T/W T/W D/Wft/s 60100 0.15997 0.19996 0.23995 0.47158110 0.15997 0.19996 0.23995 0.39021120 0.15997 0.19996 0.23995 0.32841130 0.15997 0.19996 0.23995 0.28040140 0.15997 0.19996 0.23995 0.24240150 0.15997 0.19996 0.23995 0.21183160 0.15997 0.19996 0.23995 0.18690170 0.15997 0.19996 0.23995 0.16632180 0.15997 0.19996 0.23995 0.14918190 0.15997 0.19996 0.23995 0.13475200 0.15997 0.19996 0.23995 0.12253210 0.15997 0.19996 0.23995 0.11211220 0.15997 0.19996 0.23995 0.10316230 0.15997 0.19996 0.23995 0.09545240 0.15997 0.19996 0.23995 0.08878250 0.15997 0.19996 0.23995 0.08298260 0.15997 0.19996 0.23995 0.07794270 0.15997 0.19996 0.23995 0.07353280 0.15997 0.19996 0.23995 0.06969

290 0.15997 0.19996 0.23995 0.06633300 0.15997 0.19996 0.23995 0.06339

Table D.1: Induced Drag as it varies with Flight Speed

51


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Elev. T/W VHigh VLow Vstall VCruise Vqmax V250ft 850 0.2803 1288.73 111.70 277.68 949.08 739.42 421.95

1000 0.2746 1294.21 114.54 281.78 945.81 750.35 428.192000 0.2690 1299.74 117.48 285.98 942.53 761.53 434.573000 0.2634 1305.33 120.51 290.27 939.24 772.95 441.08

4000 0.2580 1310.97 123.64 294.65 935.94 784.62 447.745000 0.2526 1316.67 126.88 299.13 932.62 796.55 454.556000 0.2473 1322.42 130.23 303.71 929.30 808.75 461.527000 0.2420 1328.22 133.69 308.40 925.96 821.23 468.648000 0.2368 1334.08 137.27 313.19 922.61 834.00 475.929000 0.2317 1339.99 140.98 318.10 919.25 847.06 483.38

10000 0.2267 1345.96 144.82 323.12 915.87 860.42 491.00

Table D.2: Sample Flight Envelope

D.2 Graphs

Figure D.1: Drag to Weight



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Figure D.2: Flight Envelope



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Appendix E

Range

E.1 Tables

Brequet FAArho1 0.44174075 0.44174075rho2 0.273879265 0.273879265

Wf/W1 0.38 0.38a1 301.889946 301.889946

V1 (mph) 574.0029773 574.0029773V1 (ft/s) 841.8729467 841.8729467

W/S1 110 110CL1 0.369835425 0.369835425CD1 0.020631913 0.020631913E1 17.92540632 17.92540632

T/W avalT/W req

Start 0 0Range (max) 9279.486551 8058.780019

Range 9279.486551 8058.780019Hrs. 16.16626902 14.03961362

Final Eleve (m) 13155.56331Final Eleve (ft) 43161.29696

Table E.1: Brequet and FAA Ranges

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FAA Stepped31000 33000 35000 37000 39000 410009448.8 10058.4 10668 11277.6 11887.2 12496.8

0.530053407 0.522160024 0.514245703 0.652851069 0.659155879 0.6655215770.44174075 0.409814586 0.379684329 0.348316807 0.316398436 0.287404939

0.409814586 0.379684329 0.348316807 0.316398436 0.287404939 0.261068290.07227353 0.07352168 0.08261474 0.091636034 0.091636034 0.091636034301.889946 299.240584 296.567555 295.1005284 295.1005284 295.1005284

574.0029773 568.965573 563.8831693 561.0938162 561.0938162 561.0938162841.8729467 841.8729467 841.8729467 841.8729467 841.8729467 841.8729467

110 101.9126152 93.49313101 84.92579131 77.14352864 70.074401650.369835425 0.369337854 0.365712933 0.36211662 0.36211662 0.362116620.020631913 0.020618114 0.020518142 0.020419933 0.020419933 0.02041993317.92540632 17.91327091 17.82388189 17.73348715 17.73348715 17.733487150.118999547 0.110399029 0.102282307 0.093832281 0.085233863 0.07742337

0.055786741 0.055824534 0.056104501 0.056390488 0.056390488 0.0563904880.468797929 0.505661457 0.548525963 0.600971092 0.661597239 0.72833937

0 1428.152147 2889.388119 4536.612816 5964.011329 7377.756811428.152147 2889.388119 4536.612816 5964.011329 7377.75681 8777.979851428.152147 1461.235972 1647.224698 1427.398513 1413.745481 1400.223042.488057037 2.568232668 2.921216286 2.543956949 2.519624063 2.49552392

Table E.2: FAA Stepped

E.2 Graphs

Figure E.1: Flight Profile



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Figure E.2: W/S vs. Percent Throttle



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Appendix F

Climb

F.1 Tables

V [ft/s] (T/W)ignoring FSE (T/W) Dynamic Pressure, q (D/W) R/C [ft/min]100 0.330 0.300 11.9 0.348 -110.9200 0.330 0.277 47.6 0.093 2839.1300 0.330 0.261 107.1 0.054 4974.7400 0.330 0.251 190.4 0.049 6755.9500 0.330 0.248 297.5 0.056 8226.2600 0.330 0.251 428.4 0.070 9360.0700 0.330 0.261 583.1 0.089 10111.8800 0.330 0.277 761.6 0.113 10428.7900 0.330 0.300 963.9 0.140 10254.4

1000 0.330 0.330 1190.0 0.171 9531.01100 0.330 0.366 1439.9 0.206 8199.81200 0.330 0.409 1713.6 0.244 6201.31300 0.330 0.459 2011.1 0.285 3476.11400 0.330 0.515 2332.4 0.330 -35.81500 0.330 0.578 2677.5 0.379 -4394.3

Table F.1: Analysis of Sea Level Rate of Climb

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V [ft/s] (T/W)ignoring FSE (T/W) Dynamic Pressure, q (D/W) R/C [ft/min]100 0.267 0.243 8.8 0.348 -485.9200 0.267 0.224 35.1 0.093 2085.9300 0.267 0.211 79.0 0.054 3844.0400 0.267 0.203 140.4 0.049 5247.6

500 0.267 0.200 219.4 0.056 6340.1600 0.267 0.203 315.9 0.070 7095.9700 0.267 0.211 430.0 0.089 7469.3800 0.267 0.224 561.6 0.113 7407.4900 0.267 0.243 710.7 0.140 6853.8

1000 0.267 0.267 877.5 0.171 5750.51100 0.267 0.297 1061.7 0.206 4038.91200 0.267 0.331 1263.5 0.244 1659.31300 0.267 0.371 1482.9 0.286 -1447.71400 0.267 0.417 1719.8 0.331 -5342.3

1500 0.267 0.468 1974.3 0.379 -10084.3

Table F.2: R/C Analysis 10,000FT

V [ft/s] (T/W)ignoring FSE (T/W) Dynamic Pressure, q (D/W) R/C [ft/min]100 0.213 0.193 6.4 0.345 -795.9200 0.213 0.179 25.4 0.093 1439.3300 0.213 0.168 57.2 0.053 2865.5400 0.213 0.162 101.6 0.049 3937.2500 0.213 0.159 158.8 0.056 4696.4

600 0.213 0.162 228.6 0.070 5116.5700 0.213 0.168 311.2 0.090 5151.4800 0.213 0.179 406.4 0.114 4747.5900 0.213 0.193 514.4 0.141 3848.0

1000 0.213 0.213 635.0 0.173 2394.31100 0.213 0.236 768.4 0.208 327.01200 0.213 0.264 914.4 0.246 -2413.61300 0.213 0.296 1073.2 0.288 -5887.91400 0.213 0.332 1244.6 0.334 -10156.21500 0.213 0.372 1428.8 0.382 -15279.1

100 0.166 0.151 4.4 0.346 -1085.2

Table F.3: R/C Analysis at 20,000Ft



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V [ft/s] (T/W)ignoring FSE (T/W) Dynamic Pressure, q (D/W) R/C [ft/min]100 0.166 0.151 4.4 0.346 -1085.2200 0.166 0.139 17.8 0.093 872.2300 0.166 0.131 40.0 0.053 2018.5400 0.166 0.126 71.1 0.049 2810.3

500 0.166 0.124 111.1 0.056 3290.2600 0.166 0.126 160.0 0.070 3432.1700 0.166 0.131 217.8 0.090 3190.0800 0.166 0.139 284.5 0.113 2510.5900 0.166 0.151 360.0 0.141 1337.1

1000 0.166 0.166 444.5 0.172 -388.51100 0.166 0.184 537.8 0.207 -2725.41200 0.166 0.205 640.1 0.245 -5733.41300 0.166 0.230 751.2 0.287 -9472.31400 0.166 0.258 871.2 0.332 -14002.5

1500 0.166 0.290 1000.1 0.381 -19384.2

Table F.4: R/C Analysis at 30,000Ft

F.2 Graphs

Figure F.1: R/C vs. Flight Speed



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Appendix G

Take-Off and Landing

G.1 Tables

W/S T.O.P. STOP @ T/W .25 T.O.P. STOP @ T/W .35 T.O.P. STOP @ T/W .45lbf/ft2 - ft - ft - ft

40 88.89 2267.90 63.49 1737.11 49.38 1442.2245 100.00 2525.00 71.43 1927.86 55.56 1596.1150 111.11 2780.75 79.37 2117.26 61.73 1748.6555 122.22 3035.36 87.30 2305.51 67.90 1900.0560 133.33 3288.96 95.24 2492.77 74.07 2050.4465 144.44 3541.69 103.17 2679.15 80.25 2199.9770 155.56 3793.65 111.11 2864.76 86.42 2348.7175 166.67 4044.92 119.05 3049.68 92.59 2496.7780 177.78 4295.56 126.98 3233.97 98.77 2644.2085 188.89 4545.63 134.92 3417.69 104.94 2791.0690 200.00 4795.18 142.86 3600.90 111.11 2937.4195 211.11 5044.26 150.79 3783.63 117.28 3083.27

100 222.22 5292.90 158.73 3965.92 123.46 3228.71105 233.33 5541.14 166.67 4147.81 129.63 3373.73110 244.44 5789.00 174.60 4329.32 135.80 3518.38115 255.56 6036.51 182.54 4510.47 141.98 3662.68120 266.67 6283.69 190.48 4691.30 148.15 3806.65

Table G.1: Sea Level Take-Off Distance at Differing Thrust Ratios and Wing Loading

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W/S T.O.P. STOP @ T/W .25 T.O.P. STOP @ T/W .35 T.O.P. STOP @ T/W .45lbf/ft2 - ft - ft - ft

40 114.48 2834.54 81.77 2150.90 63.60 1771.1045 128.80 3160.43 92.00 2391.34 71.55 1964.0650 143.11 3484.87 102.22 2630.32 79.50 2155.57

55 157.42 3808.07 112.44 2868.07 87.45 2345.8560 171.73 4130.19 122.66 3104.74 95.40 2535.0465 186.04 4451.38 132.88 3340.47 103.35 2723.3070 200.35 4771.73 143.11 3575.37 111.30 2910.7275 214.66 5091.33 153.33 3809.52 119.25 3097.4080 228.97 5410.27 163.55 4042.99 127.20 3283.4085 243.28 5728.59 173.77 4275.86 135.16 3468.7990 257.59 6046.35 183.99 4508.17 143.11 3653.6295 271.90 6363.60 194.21 4739.96 151.06 3837.94

100 286.21 6680.38 204.44 4971.29 159.01 4021.79

105 300.52 6996.72 214.66 5202.18 166.96 4205.21110 314.83 7312.66 224.88 5432.66 174.91 4388.21115 329.14 7628.21 235.10 5662.76 182.86 4570.84120 343.45 7943.42 245.32 5892.51 190.81 4753.11

Table G.2: 5000ft Elevation Take-Off Distance at Differing Thrust Ratios and WingLoading

W/S LP s - Landinglbf/ft2 ft

40 13.33 1973.3345 15.00 2170.0050 16.67 2366.6755 18.33 2563.3360 20.00 2760.0065 21.67 2956.6770 23.33 3153.3375 25.00 3350.0080 26.67 3546.6785 28.33 3743.33

90 30.00 3940.00Table G.3: Landing Distance At Sea Level With Varied Wing Loading



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W/S LP s - Landinglbf/ft2 ft

40 15.47 2225.8645 17.41 2454.0950 19.34 2682.32

55 21.28 2910.5660 23.21 3138.7965 25.14 3367.0270 27.08 3595.2575 29.01 3823.4980 30.95 4051.7285 32.88 4279.9590 34.82 4508.18

Table G.4: Landing Distance At 5000ft With Varied Wing Loading

G.2 Graphs

Figure G.1: Take-Off Distance



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Figure G.2: Thrust To Weight Ratio Vs Percent Throttle

culminating report

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