Evaluation of Aircraft Performance Algorithms in Federal Aviation Administration's Integrated Noise Model by Wei-Nian Su

B.S. Aerospace Engineering, Iowa State University, 1996 Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirement for the Degree of Master of Science in Aeronautics and Astronautics at the Massachusetts Institute of Technology February, 1999 ©1999 Massachusetts Institute of Technology. All rights reserved.

Author...... ,...... Department of Aeronautics and Astronautics January 14, 1999

Certifie d b y ...... V...... o...... C r Certe b( (Professor John-Paul Clarke Department of Aeronautics and Astronautics Thesis Supervisor

Accepted by ......

Professor Jaime Peraire Chairman, epartment Graduate Committee

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

MAY 1 7 1999 -oWWW LIBRARIES Evaluation of Aircraft Performance Algorithms in Federal Aviation Administration's Integrated Noise Model by Wei-Nian Su

Submitted to the Department of Aeronautics and Astronautics Engineering on January 14, 1999 in partial fulfillment of the requirement for the Degree of Master of Science in Aeronautics and Astronautics Engineering

Abstract

The Integrated Noise Model (INM) has been the Federal Aviation Administration's (FAA) standard tool since 1978 for determining the predicted noise impact in the vicinity of airports. A review of the aircraft performance algorithms in the INM was conducted and improved models for true airspeed, takeoff/climb thrust, level-flight thrust, and climb performance were developed. The true airspeed model with air compressibility correction provides an accurate prediction over a wide range of operating conditions. The quadratic takeoff/climb thrust model as a function of Mach number, altitude, and temperature and the level-flight thrust model derived from the minimum-thrust-flight condition provide an accurate prediction within considered airspeed and altitude range. The climb models for constant equivalent/calibrated airspeed as well as constant climb rate climbs introduce the flight path angle correction factor as a function of altitude, airspeed, and temperature as opposed to constant correction factor used in INM. Comparison of flight profiles predicted by the proposed methods and INM with the flight profiles provided by the Delta Airlines shows that the errors in overall ground distance traversed as well as noise contour shapes are reduced by implementing the proposed models.

Thesis Supervisor: Dr. John-Paul Clarke Title: Charles Stark Draper Assistant Professor of Aeronautics and Astronautics Acknowledgments

Over the past two years, I have met many people who have made my time at MIT worthwhile. I would like to take this moment to express my deep gratitude to those who have made it possible for me to achieve this accomplishment. In particular, I would like to extend my appreciation to the following individuals and organizations. First of all, I would like to thank my advisor, Prof. John-Paul Clarke, for his encouragement and guidance throughout this research. I also like to thank Mr. Gregg Fleming from Volpe National Transportation Systems Center for funding my research. In addition, I also like to thank Mr. Jim Brooks from Delta Airline for providing valuable data. Finally, I would like to thank my family members: my father, Mr. Shih-Ping Su, my mother, Mrs. Yue-Ching Lin, my sister, Yua-Hwa Su, my brother, Wei-Ping Su, and my girlfriend, Miss Shine-Yi Wong, for their love and support throughout my study at MIT. This research was funded by Volpe National Transportation Systems Center, U.S. Department of Transportation, and performed in the Flight Transportation Lab. Contents

A bstract...... 2...... A cknow ledgm ents...... 3 Contents...... 4...... List of Tables...... List of Figures...... 10 N om enclature...... 12

Chapter 1. Introduction...... 15 1.1 Background of INM ...... 15 1.2 M otivation...... 15 1.3 Overview of Thesis...... 16

Chapter 2. Atmospheric Model and True Airspeed Model...... 18 2.1 Standard Atm osphere...... 18 2.1.1 IN M 's Atm ospheric M odel...... 19 2.2 Airspeed M easurem ent...... 20 2.2.1 Previous W ork...... 21 2.2.2 True A irspeed M odel...... 22 2.3 Conclusion of Chapter 2...... 24

Chapter 3. Takeoff and Clim b Thrust M odel...... 25 3.1 Previous W ork...... 25 3.2 Quadratic Thrust M odel...... 26 3.3 Evaluation of Coefficients...... 31 3.3.1 Ante-Break Equation...... 31 3.3.2 Post-Break Equation...... 33 3.4 V alidation...... 36 3.4.1 Graphical Comparison...... 36 3.4.2 Error A nalysis...... 43 3.5 Conclusion of Chapter 3...... 45

Chapter 4. Level Flight Thrust Model...... 46 4.1 Previous W ork...... 46 4.2 E quation of M otion...... 47 4.3 Drag Polar...... 48 4.3.1 Drag Polar Model I...... 49 4.3.2 Drag Polar Model II...... 50 4.3.3 Effects of Reynolds Number on Drag Polar...... 50 4.4 Level-Flight Thrust...... 51 4.4.1 Level-Flight Thrust Model I...... 52 4.4.2 Level-Flight Thrust Model II...... 54 4.5 Validation of Level-Flight Thrust Models...... 54 4.5.1 Comparison of Proposed Models with INM Model...... 54 4.5.2 E rror A nalysis...... 57 4.5.3 Pro and Con Between Models...... 60 4.6 C onclusion of Chapter 4...... 61

Chapter 5. Clim b Performance...... 62 5.1 Previous W ork...... 62 5.2 Equation of Motion and Flight Path Angle Correction Factor...... 63 5.3 Evaluation of Flight Path Angle Correction Factor...... 65 5.3.1 Constant Equivalent Airspeed Climb Model...... 66 5.3.2 Exact Constant Calibrated Airspeed Climb Model...... 67 5.3.3 Simplified Constant Calibrated Airspeed Climb Model...... 68 5.4 Graphical Comparison of Flight Path Angle Correction Factor...... 71 5.5 Calculation of Flight Path Angle and Ground Distance Traversed...... 72 5.6 Error Analysis ...... 72 5.6.1 Constant Equivalent Airspeed Climb M odel...... 72 5.6.2 Constant Calibrated Airspeed Climb M odel...... 74 5.6.3 Discussion...... 75 5.7 Conclusion of Chapter 5...... 75

Chapter 6. Accelerated Climb Performance...... 76 6.1 Previous W ork...... 76 6.2 Constant Climb Rate Acceleration...... 78 6.3 Error Analysis...... 80 6.4 Conclusion of Chapter 6...... 82

Chapter 7. Comparison of Departure Profile and Noise Contour...... 83 7.1 Description of Analysis...... 83 7.2 Boeing 727-200...... 84 7.2.1 Procedure Steps...... 84 7.2.2 Flight Profile and Noise Contour...... 85 7.2.3 Error Analysis...... 93 7.3 Boeing 737-3B2...... 95 7.3.1 Procedure Steps...... 96 7.3.2 Flight Profile and Noise Contour...... 96 7.3.3 Error Analysis...... 100 7.4 Boeing 757-200...... 101 7.4.1 Procedure Steps...... 102 7.4.2 Flight Profile and Noise Contour...... 103 7.4.3 Error Analysis...... 106 7 .5 D iscu ssion ...... 108 7.6 Conclusion of Chapter 7...... 108

6 Chapter 8. Conclusion and Future W ork...... 109 8.1 Conclusion of Thesis...... 109 8.2 Future W ork...... 109

Bibliography...... 110 List of Tables

2.1 Average error in true airspeed for MIT and INM models at standard day...... 24 3.1 Ante-break corrected takeoff thrust (Fn/6) versus Mach number and pressure altitude ...... 31 3.2 Post-break corrected takeoff thrust (F,/8) versus Mach number, pressure altitude, and tem p erature...... 33 3.3 Error in corrected net thrust during takeoff for small commercial airplane...... 43 3.4 Error in corrected net thrust during climb for small commercial airplane...... 44 3.5 Error in corrected net thrust during takeoff for medium commercial airplane...... 44 3.6 Error in corrected net thrust during climb for medium commercial airplane...... 44 3.7 Error in corrected net thrust during takeoff for large commercial airplane...... 45 3.8 Error in corrected net thrust during climb for large commercial airplane...... 45 4.1 Average level-flight thrust errors per engine for small commercial jet at 5000 ft...... 58 4.2 Average level-flight thrust errors per engine for large commercial jet at 5000 ft...... 59 4.3 Pro of level-flight thrust model I and model II...... 60 4.4 Con of level-flight thrust model I and model II...... 60 5.1 Error in ground distance during constant equivalent airspeed climb starting from sea level...... 73 5.2 Error in ground distance during constant equivalent airspeed climb starting from 5000 ft...... 73 5.3 Error in ground distance during constant calibrated airspeed climb starting from sea level.....74 5.4 Error in ground distance during constant calibrated airspeed climb starting from 5000 ft...... 74 6.1 Error in altitude gain and ground distance traversed for the small commercial airplane...... 81 6.2 Error in altitude gain and ground distance traversed for the large commercial airplane...... 81 7.1 Flight procedure for Case (1) and (2)...... 84 7.2 Flight procedure for Case (3) and (4)...... 85 7.3 Overall ground distance error in feet for Case (1) to (4)...... 94 7.4 Error in noise impact area in square mile for Case (1) and (2)...... 94 7.5 Error in noise impact area in square mile for Case (3) and (4)...... 94 7.6 Error in closure point distance in nautical mile for Case (1) and (2)...... 95 7.7 Error in closure point distance in nautical mile for Case (3) and (4)...... 95 7.8 Flight procedure for Case (5) and (6)...... 96 7.9 Overall ground distance error in feet for Case (5) and (6)...... 101 7.10 Error in noise impact area in square mile for Case (5) and (6)...... 101 7.11 Error in closure point distance in nautical mile for Case (5) and (6)...... 101 7.12 Flight procedure for C ase (7)...... 102 7.13 Flight procedure for C ase (8)...... 102 7.14 Overall ground distance error in feet for Case (7) and (8)...... 107 7.15 Error in noise impact area in square mile for Case (7) and (8)...... 107 7.16 Error in closure point distance in nautical mile for Case (7) and (8)...... 107 List of Figures

2.1 Comparison of exact and INM models at standard day condition...... 22 2.2 Comparison of exact and MIT models at standard day condition...... 23 3.1 Typical plot for Eq.(3.3) and (3.4) at an arbitrary altitude...... 27 3.2 Effect of flight Mach number and calibrated airspeed on corrected net thrust value...... 28 3.3 Corrected net thrust vs. altitude at Mach 0, 0.2, and 0.4...... 29 3.4 Corrected net thrust vs. temperature at Mach 0 and various altitudes...... 30 3.5 Takeoff thrust comparison for small commercial airplane at various conditions...... 37 3.6 Climb thrust comparison for small commercial airplane at various conditions...... 38 3.7 Takeoff thrust comparison for medium commercial airplane at various conditions...... 39 3.8 Climb thrust comparison for medium commercial airplane at various conditions...... 40 3.9 Takeoff thrust comparison for large commercial airplane at various conditions...... 41 3.10 Climb thrust comparison for large commercial airplane at various conditions...... 42 4.1 Forces on an aircraft in level flight...... 47 4.2 Thrust ratio per engine vs. velocity for small commercial jet with various flap settings...... 55 4.3 Thrust ratio per engine vs. velocity for large commercial jet with various flap settings...... 56 5.1 Aircraft in steady clim b with no wind...... 63 5.2 Geom etry of airspeed vectors in wind...... 64 5.3 Comparison of flight path angle correction factor predicted by the exact and simplified models at standard day, 8-knot headwind condition...... 69 5.4 Comparison of flight path angle correction factor predicted by the exact and simplified models at nonstandard day, 8-knot headwind condition...... 70 5.5 Comparison of flight path angle correction factor between MIT and INM models...... 71 6.1 Computation procedures for INM model...... 77 6.2 Computation procedures for constant-climb-rate accelerated climb model...... 80 7.1 Flight profile and LAMAX noise contour for Case (1)...... 86 7.2 SEL noise contour for C ase (1)...... 87 7.3 Flight profile and LAMAX noise contour for Case (2)...... 88 7.4 SEL noise contour for C ase (2)...... 89 7.5 Flight profile and LAMAX noise contour for Case (3)...... 90 7.6 SEL noise contour for C ase (3)...... 91 7.7 Flight profile and LAMAX noise contour for Case (4)...... 92 7.8 SEL noise contour for Case (4)...... 93 7.9 Flight profile and LAMAX noise contour for Case (5)...... 97 7.10 SEL noise contour for Case (5)...... 98 7.11 Flight profile and LAMAX noise contour for Case (6)...... 99 7.12 SEL noise contour for C ase (6)...... 100 7.13 Flight profile and LAMAX noise contour for Case (7)...... 103 7.14 SEL noise contour for C ase (7)...... 104 7.15 Flight profile and LAMAX noise contour for Case (8)...... 105 7.16 SEL noise contour for C ase (8)...... 106 Nomenclature

AIR - Aviation Information Report. DFBR - Distance from brake release. FAA - Federal Aviation Administration. INM - Integrated Noise Model. MCLT - Maximum climb thrust. MGLW - Maximum gross takeoff weight. MGTOW - Maximum gross landing weight. MTOT - Maximum takeoff thrust. SAE - Society of Automotive Engineers. SLD - Satellite distance.

CD - Drag coefficient.

CDRin - Drag coefficient at minimum drag-over-lift point. CL - Lift coefficient.

CLRmin - Lift coefficient at minimum drag-over-lift point. D - Drag. F - Total thrust which is equal to the number of engines times the net thrust per engine.

Fn - Net thrust per engine. Fn/8 - Corrected net thrust per engine. g - Gravitational constant. h - Pressure altitude above the sea level. hi - Altitude at the beginning of climb. h2 - Altitude at the end of climb. hairport - Airport elevation. hd - Density altitude above sea level. L - Lift. M - Flight Mach number. N - Number of engines. P - Ambient air pressure. Po - Ambient air pressure at sea level, standard day condition.

Pairpor - Ambient air pressure at the airport. R - Drag-over-lift ratio.

Ra - Gas constant. Re - Reynolds number. Run - Minimum drag-over-lift ratio. S - Reference area. Sa - Ground distance traversed during acceleration. Sc - Ground distance traversed during constant calibrated airspeed climb. T - Ambient air temperature.

To - Ambient temperature at sea level, standard day condition. Tairpor - Ambient temperature at the airport.

TISA - Standard day ambient air temperature.

Vao - Speed of sound at sea level, standard day condition.

Vc - Calibrated airspeed. Ve - Equivalent airspeed. Vt - True airspeed.

Vta - True airspeed at the beginning of acceleration segment. Vtb - True airspeed at the end of the acceleration segment. Vtz - Climb rate.

VtR, - True airspeed corresponding to minimum-level-flight-thrust condition. V, - Headwind velocity. W - Aircraft weight. c - Mean cord length of the wing. y - Flight path angle.

Ya - Ratio of specific heat at constant pressure to specific heat at constant volume for air. 6 - Ratio of the ambient air pressure at the airplane to the air pressure at mean sea level. 0 - Ratio of the ambient air temperature at the airplane to the air temperature at mean sea level. p - Viscosity coefficient. - Flight path angle correction factor. p - Ambient air density. po - Ambient air density at sea level, standard day condition. o - Ratio of the ambient air density at the airplane to the air density at mean sea level. Chapter 1. Introduction

1.1 Background of INM

Aircraft noise has often been cited as the most undesirable feature of life in the urban community. This is particularly the case in residential communities near major metropolitan airports. The significant increases in passenger traffic over the past two decades, and the birth and rapid growth of overnight package delivery services during that same period, have increased both the frequency and the total number of operations in an average day. In addition, aircraft noise is often at the top of the list in rural areas due to its sound level relative to the low ambient sound levels, frequency, and time of occurrence. The FAA Office of Environment and Energy supports the assessment of aircraft noise impacts by developing and maintaining noise-evaluation models and methodologies in the form of the Integrated Noise Model (INM). INM was evolved in the mid-1980s and the INM Development Team members consist of FAA Office of Environment and Energy, ATAC Corporation, Volpe National Transportation Systems Center, and LeTech Incorporated [1].

1.2 Motivation

Aircraft performance calculations in the INM utilize the methodology developed in the Society of Automotive Engineers (SAE) Aviation Information Report (AIR) 1845 issued in March 1986 [2]. Previous work has demonstrated that INM does not accurately predict aircraft performance at non-sea-level, nonstandard day conditions and that prediction of the following parameters need to be improved: (I) true airspeed, (II) takeoff and climb thrust, (I) level-flight thrust, and (IV) climb performance. Conversion from calibrated airspeed to true airspeed was required in INM. Because the performance algorithms described in SAE AIR 1845 was developed for low airspeed and low altitude operation, the incompressible flow assumption that calibrated airspeed was the same as equivalent airspeed was valid. This is not true, as the air compressibility effect is no longer negligible at high airspeed and altitude. The takeoff and climb thrust prediction methodology described in SAE AIR 1845 was developed for operations from airports at sea-level on a standard day, and therefore only considered the flight profile up to 3000 ft above sea-level. A linear thrust model as a function of calibrated airspeed, altitude, and temperature was thus adopted to calculate takeoff and climb thrust. Once the operating altitude gets beyond 3000 ft, this linear model is no longer valid and the induced error increases dramatically. The current method for computing level-flight thrust involves inverting the expression used to determine the flight path angle. Implicit in the expression for the flight path angle however, is the assumption that the drag-over-lift ratio remains approximately constant regardless of aircraft weight and speed. This is valid during climb as the goal of achieving altitude quickly dictates that the airplane operates at near minimum drag-over-lift ratio, and thus maximum flight path angle. In level flight at constant speed however, the thrust is a strong function of aircraft speed. INM uses a correction factor to account for changes in the flight path angle associated with headwinds and the acceleration/deceleration inherent in both of constant calibrated airspeed climb and constant climb rate climb. Currently, this correction factor assumes that the change in flight path angle that can be attributed to accelerated climb/descent is constant. This is not true, as the change in flight path angle attributable to accelerated climb/descent is a function of pressure altitude and flight airspeed. Thus, improved models which correctly account for the areas where the existing models are deficient as described above are required.

1.3 Overview of Thesis

This thesis covers the development of a new true airspeed model, a new takeoff and climb thrust model, a new level-flight thrust model, an improved flight path angle model, and a new constant climb rate climb methodology. The true airspeed model along with the description of INM's atmosphere model is presented in Chapter 2, the takeoff and climb thrust model is presented in Chapter 3, the level-flight thrust model is presented in Chapter 4, the flight path angle model is presented in Chapter 5, and the constant climb rate climb methodology is presented in Chapter 6. The comparisons of flight profile and noise contour between proposed method, INM, and measured data are presented in Chapter 7. Finally, the conclusion of overall analysis is presented in Chapter 8. Chapter 2. Atmospheric Model and True Airspeed Model

An atmospheric model that provides information about the flight environment is a necessary tool for aircraft performance analysis. However, the standard atmosphere is a reference model only, thus it must be modified to take into account nonstandard day condition. INM takes calibrated airspeed as one of the input parameters. An airspeed model which accurately converts calibrated airspeed to true airspeed over a wide range of operating conditions is needed. In this chapter, discussion of INM's atmospheric model is provided and the true airspeed model which accounts for compressibility effect is introduced.

2.1 Standard Atmosphere

In 1920, the Frenchman A. Toussaint, director of the Aerodynamic Laboratory at Saint-Cyr- l'Ecole, France, suggested a linear relationship between temperature and height. Toussaint's formula was formally adopted by France and Italy in March 1920 and one year later, the NACA adopted Toussaint's formula for airplane performance testing. With the advent of aerospace technology such that high altitude flight as well as space flight became possible in late 1959, new tables of the standard atmosphere were created by Air Research and Development Command (ARDC) which is now the Air Force Systems Command [3]. Several different standard atmospheres exist all using slightly different experimental data in their models, but the difference is insignificant below 100,000 ft. A standard atmosphere model in common use today is the 1959 ARDC Model as shown below,

dT T, + (h - h( ) T dh (2.1)

T, T1 -g Ra dT P T (2.2)

p_ PT (2.3) p1 P T 91 P1T where the subscript 1 stands for the atmospheric condition at base altitude and dT/dh is the temperature lapse rate.

2.1.1 INM's Atmospheric Model

The standard atmosphere is a reference model only and certainly does not predict the actual atmospheric properties at a given time and place, thus modifications of Eq.(2.1) and (2.2) based on the knowledge of given airport conditions to account for nonstandard day condition are required [4].

+ (h - hairport) dh (2 _ Tairport dh (2.4)

T. + (h - h dT Sairport )T Rg airport dh a d T Pairport - Po (2.5) T P 0 o

o = (2.6) 2.2 Airspeed Measurement

A Pitot-static tube which measures the difference between the total pressure and static pressure is commonly implemented on the airplanes to measure the airspeed. Depending on the type of airplanes, the airspeed reading from such measurement can be equivalent airspeed or calibrated airspeed. For lower-speed airplanes such as small, piston engine airplanes, the airspeed readings can be considered as equivalent airspeed, by contrast, for higher-speed airplanes such as commercial jet transports, the airspeed readings are calibrated airspeed. The airspeed is called low or high depending on the flight Mach number,

V M =- (2.7) V where

V= YaRaT (2.8)

If Mach number is less than 0.3, the airflows are considered as incompressible and then the equivalent airspeed is read. If Mach number is greater than 0.3, the compressibility must be taken into account and then the calibrated airspeed is read. The relationship between equivalent airspeed and calibrated airspeed is given by the following equation [5],

1 Ya - 1

V 2 a P Ya"(k + 6) Ya - 6]2 (2.9) e Ya- 1 po

where

-1 V kI + Ya 2V a (.0 The true airspeed is obtained by dividing Eq.(2.9) by the square root of density ratio.

V Vt e (2.11)

2.2.1 Previous Work

In INM, the effect of air compressibility was ignored. Instead of using Eq.(2.11), INM assumes that the calibrated airspeed is the same as the equivalent airspeed at low airspeed and low altitude operation [2].

V V, c (2.12)

Figure 2.1 shows the comparison between exact and INM models at standard day condition. As the figure shows, the error increases as the flight altitude and airspeed increase, thus an improved airspeed model is desired. 0 5300

250

w 200 2

0 50 100 150 200 250 300 350 Calibrated Airspeed, Vc (knots)

Figure 2.1 Comparison of exact and INM models at standard day condition.

2.2.2 True Airspeed Model

The exact equation for true airspeed as a function of calibrated airspeed and pressure ratio is very complicated, thus an approximate model which improves the accuracy while reducing the complexity of the exact equation is desirable. If a correction factor as a function of calibrated airspeed and altitude were introduced in Eq.(2.12), the accuracy of true airspeed prediction would be improved. The airspeed model is determined by the following relationship,

(1 + Eh)V v,- c (2.13)

where E is a constant coefficient with value of -1.0925E-6 1/ft determined by Least Square method. Figure 2.2 shows the comparison of the exact model, Eq.(2.1 1), and MIT model, Eq.(2.13), at standard day condition. As the figure shows, the errors at high speed and altitude are reduced with the implementation of the correction factor.

0" 0 50 100 150 200 250 300 350 Calibrated Airspeed, Vc (knots)

Figure 2.2 Comparison of exact and MIT models at standard day condition. Table 2.1 shows the average error in true airspeed for the MIT and INM models at standard day condition. The errors were calculated for calibrated airspeed ranging from 150 knots to 350 knots at sea level, 10000 ft, and 20000 ft respectively. As the table shows, the average errors for the MIT model are less than half of INM's errors.

Table 2.1 Average error in true airspeed for MIT and INM models at standard day.

II MIT Model (knots) INM Model (knots) Sea Level 0.0395 0.0395 10000 ft 1.0104 2.5503 20000 ft 2.5950 7.5404

2.3 Conclusion of Chapter 2

As the analyses show, the proposed true airspeed model provides a more accurate prediction of true airspeed than the existing INM model. In addition, the valid operating condition for the proposed model is wider than the current model, thus, the accuracy for any subsequent aircraft performance calculation will be improved. Chapter 3. Takeoff and Climb Thrust Model

During takeoff and climb operations, the maximum thrust that an aircraft may use is a function of operating altitude, temperature, and velocity. These maximum thrust values are defined as the Maximum Takeoff Thrust (MTOT) and the Maximum Climb Thrust (MCLT) respectively. In this chapter, a quadratic thrust model is introduced that describes the MTOT and MCLT as a function of pressure altitude, flight Mach number, and ambient temperature. Comparison of measured thrust data to the thrust values predicted at varying flight conditions confirms that the quadratic model provides a good fit within the considered flight envelope.

3.1 Previous Work

In SAE AIR 1845, corrected net thrust is determined by a linearized expansion of the thrust at sea-level standard day conditions which is a function of calibrated airspeed, pressure altitude, and temperature [2],

F (- ) = E + FV c + Gh + HT (SAE Eq. Al) where E, F, G and H are constant coefficients to be determined by manufactures. As opposed to SAE AIR 1845, INM expands SAE Eq. Al to a quadratic estimate for the altitude term and uses density altitude, hd, instead of pressure altitude [4],

F = 8(hd)(E + FV c + GAhd + GBh d + HTISA(hd)) (3.1)

hd 51867 (1 - (h) 5.256-1) (3.2) 0.003566

where E, F, GA , GB , and H are constant jet coefficients. 3.2 Quadratic Thrust Model

While the relationship between corrected net thrust and the relevant flight conditions may be linear near the reference conditions (standard day sea level), that assumption is inaccurate at high altitude and high flight velocity. In addition, the thrust gradient with respect to calibrated airspeed varies with altitude which provides a poor evaluation of airspeed dependent coefficients. A quadratic thrust model as a function of pressure altitude, flight Mach number, and ambient temperature was found to provide an improved match between predicted thrust and measured thrust. The thrust model is determined by the following relationship,

2 2 (Fn /)A = ko + kM + k2M + k3 h + k4h (3.3)

2 (Fn )PB = k5 + k6 M + k 7 M + k8 T (3.4)

Fn/6 = Min[ (FnI)F, (Fn,/)NF] (3.5)

where the subscript AB and PB stand for ante-break and post-break respectively and k's are constant coefficients.

Eq.(3.3) calculates the ante-break thrust value while as Eq.(3.4) calculates the post-break thrust value and the minimum of these two values is the correct thrust at the corresponding flight condition. Figure 3.1 shows a typical surface plot for Eq.(3.3) and Eq.(3.4). As the figure shows, Eq.(3.3) constructs the horizontal surface while holding altitude to be constant and Eq.(3.4) constructs the tilt surface.

x 104

5..

4,

S3.5 I-

2.5 0.8

0-6100

Mach Number o -100 Temperature (F)

Figure 3.1 Typical plot for Eq.(3.3) and (3.4) at an arbitrary altitude. Figure 3.2 shows the advantage of using flight Mach number as the independent variable instead of calibrated airspeed. As the figure shows, the gradient of corrected net thrust with respect to Mach number is constant over the entire range of altitude while the gradient of corrected net thrust with respect to calibrated airspeed is not.

4 4 x 10 X 10

3-

2 2.5

zU,

O 2

o o

0 0.1 0.2 0.3 0.4 0.5 0 50 100 150 200 250 300 Mach Number Calibrated Airspeed, VC (knots)

Figure 3.2 Effect of flight Mach number and calibrated airspeed on corrected net thrust value. Figure 3.3 shows the quadratic relationship of thrust in pressure altitude. As the figure shows, the linear approximation in SAE is limited to altitudes below 4000 ft.

x 104 a,

3.2

2.8

2.6 Mach 0.2

I--" 2.4 z

S2.2 F o Mach 0.4

1.8

1.6

I 5000 10000 15000 Pressure Altitude (ft)

Figure 3.3 Corrected net thrust vs. altitude at Mach 0, 0.2, and 0.4. Figure 3.4 shows the typical plot of thrust versus temperature. As the figure shows, the curves collapse together regardless altitude after the engine break temperature which justifies the independence of altitude for computation of the post-break thrust value in Eq.(3.4). The engine break temperature at a specific altitude and Mach number is the point where the thrust value decreases as temperature increases.

4 x 10

0.5 -100 -50 0 50 100 Temperature CF)

Figure 3.4 Corrected net thrust vs. temperature at Mach 0 and various altitudes. 3.3 Evaluation of Coefficients

The coefficients for Eq.(3.3) and Eq.(3.4) are determined by the method of Least squares over the desired ranges of flight Mach number and pressure altitude.

3.3.1 Ante-Break Equation

Table 3.1 shows an example of the measured data required to compute the coefficients for Eq.(3.3). The first columns and first row define the pressure altitude and flight Mach number respectively under which the data corresponding to ante-break corrected net thrust value was obtained.

Table 3.1 Ante-break corrected takeoff thrust (Fn/6) versus Mach number and pressure altitude.

Altitude M1 = 0 M2 = 0.1 M 3 = 0.2 M4 = 0.3 M5 = 0.4 M 6 = 0.5 h (ft)

h1 =0 32382 30442 28773 27372 26242 25380 h2= 1000 32719 30780 29110 27710 26579 25718

h3 = 2000 33041 31102 29432 28032 26901 26040

h4 = 3000 33348 31408 29739 28338 27208 26346 = h5 4000 33639 31700 30030 28630 27499 26638 = h6 5000 33915 31976 30306 28906 27775 26913

h7 = 6000 34176 32236 30567 29166 28035 27174 = h8 7000 34421 32482 30812 29411 28281 27419

h9 = 8000 34651 32711 31042 29641 28511 27649 hi = 9000 34865 32926 31256 29856 28725 27864

hi = 10000 35064 33125 31455 30055 28924 28063 The coefficients for Eq.(3.3), ko, kl, k2, k3, and k4, are computed by formulating the matrices

A1 and B1 as follows,

h h 2 1 (Fn/6 )(Mh) 2 6 2 h, h (F /8 )(M ) 1 M2 M (Fnl)(M2,hl)

hi h1 5 1 M6 M6 (Fn/ )(M6,hl) 1 M M,2 h2 h2 (Fn )(M,h2)

1 M6 M (3.6)

(Fn/8)(M6,h2) 6 6,M, h2 h2

2 2 (Fn/6)(M6,hl ) 1 M, M2 h11 hl

(Fn8)(6,h,j) 1 M6 M2 h l

and then solving Eq.(3.7) below for ko, k1,, k 2, k 3, and k 4 .

(ATA)-' (A T B ) (3.7) 3.3.2 Post-Break Equation

Table 3.2 shows an example of the measured data required to compute the coefficients for Eq.(3.4). The first two columns of the table define the conditions under which the data was obtained (altitude and the corresponding temperature). The following columns give the corrected net thrust at different flight Mach numbers.

Table 3.2 Post-break corrected takeoff thrust (Fn/8) versus Mach number, pressure altitude, and temperature.

= Altitude Temperature M = 0 M 2 0.1 M 3 = 0.2 M4 = 0.3 M5 = 0.4 M6 = 0.5 h (ft) T (oF) 1=0 T= 86 29227 26894 24747 22787 21013 19427 20101 18514 ,=0 T2 = 99 28314 25981 23834 21874 19258 17671 S= 0 T3 =111 27471 25138 22991 21031 h,=0 T4= 122 26699 24366 22219 20259 18486 16899

2=1000 T5= 82 29508 27174 25028 23068 21294 19707 20381 18795 2=1000 T6 = 95 28595 26262 24115 22155 19539 17952 2=1000 T7 = 107 27752 25419 23272 21312 h2=1000 T8= 118 26980 24647 22500 20540 18766 17180 3=2000 T9 = 79 29718 27385 25238 23278 21505 19918 3=2000 To = 92 28806 26472 24326 22366 20592 19005 18233 3=2000 T11 = 103 28033 25700 23553 21593 19820 h3=2000 T12= 114 27261 24927 22781 20821 19047 17461

4=3000 T13= 75 29999 27666 25519 23559 21786 20199 = 19286 4=3000 T14 88 29086 26753 24606 22646 20873 4=3000 T5 = 100 28244 25911 23764 21804 20030 18444

h4=3000 T16 = 111 27471 25138 22991 21031 19258 17671 h5=4000 T17 = 72 30210 27877 25730 23770 21996 20410 = h=4000 T 18 84 29367 27034 24887 22927 21154 19567

5=4000 T19= 96 28525 26191 24045 22085 20311 18724 h4=4000 -T, = 107 ?77S2 25419 23272 21312 19539 1-795 Table 3.2 Continued.

6=5000 T21 = 68 30491 28157 26011 24051 22277 20690 19778 6=5000 T 22 = 81 29578 27245 25098 23138 21365

6=5000 T23= 93 28735 26402 24255 22295 20522 18935

h6=5000 T24= 104 27963 25630 23483 21523 19750 18163 7=6000 T25= 64 30772 28438 26292 24332 22558 20971 20059 7=6000 T 26= 77 29859 27526 25379 23419 21645 19216 7=6000 T27 = 89 29016 26683 24536 22576 20803

h7=6000 T 28 = 100 28244 25911 23764 21804 20030 18444

8=7000 T29= 61 30982 28649 26502 24542 22769 21182 8=7000 T30= 74 30069 27736 25590 23629 21856 20269

8=7000 T31 = 86 29227 26894 24747 22787 21013 19427

8=7000 T32= 97 28454 26121 23975 22014 20241 18654

9=8000 T33= 57 31263 28930 26783 24823 23050 21463

9=8000 T34 = 70 30350 28017 25870 23910 22137 20550

9=8000 T35= 82 29508 27174 25028 23068 21294 19707 9=8000 T36= 93 28735 26402 24255 22295 20522 18935

1o=9000 T37= 54 31474 29141 26994 25034 23260 21674

1O=9000 T38= 67 30561 28228 26081 24121 22348 20761

ho=9000 T39= 79 29718 27385 25238 23278 21505 19918 19146 1o=9000 T 40 = 90 28946 26613 24466 22506 20733

11=10000 T41 = 50 31755 29421 27275 25315 23541 21954

h1=10000 T 42 = 63 30842 28509 26362 24402 22628 21042

h1=10000 T 43 = 75 29999 27666 25519 23559 21786 20199 h1=10000 T4= 86 29227 26894 24747 22787 21013 19427 The coefficients for Eq.(3.4), k5, k6, k7, and k8, are computed by formulating the matrices A2 and B2 as follows,

2 1 M1 M (FnI)(M 1,hl,T1)

(F /I )(M 1 M2 M 2 ,hl,T1)

(Fn/ )(M ,hl,T ) 1 M 6 M6 6 1 (Fn/ )(MI,hl,T2) 1 M1 M

(3.8)

(Fn6)(M6,h,,T2) 1 M6 M

2 (F /8)(Mh 1 M M T2 (n (M1h1,T4)T

(Fn16)(M6,h l P T 44) 1 M6 M6

and then solving Eq.(3.9) below for k5, k6, k7, and k8.

A TA ( 2 2 )-1 (A2TB 2) (3.9) These constant coefficients can be derived for climb operation following the same procedures described above, but with climb thrust data. Because all columns in A's matrices are linearly independent, ATA is strictly positive definite and a solution will always exist.

3.4 Validation

The thrust model was validated for three aircraft models, a small commercial airplane, a medium commercial airplane, and a large commercial airplane. This section provides the details of such evaluation.

3.4.1 Graphical Comparison

Figure 3.5 and 3.6 show the comparison between measured data, MIT model, and INM model for the small comercial aircraft at takeoff and climb thrust setting respectively. Figure 3.7 and 3.8 show the comparison between measured data, MIT model, and INM model for the medium commercial aircraft at takeoff and climb thrust setting respectively. Figure 3.9 and 3.10 show the comparison between measured data, MIT model, and INM model for the large comercial aircraft at takeoff and climb thrust setting respectively. As shown in figures, the current INM strategy of using the density altitude to account for the effect of temperature leads to dramatic increases in error when temperature is lower than the standard day condition. As a result, INM can only accurately model the thrust setting near the standard day condition. At airport, a temperature of 70 oF during summer time and 20 OF during winter time are quite common, thus the deficiencies in the INM model can greatly affect the accuracy of the results. Mach 0 Mach 0.1 1.3 A ..

1.25 0o1.3 1.2

1.15F 2 1.2

1.05[ n=3 h=0 I--o

0.95 0 0.9 0.9 0.8 0.85

0.8 I 0.71 -100 -50 0 50 100 150 -100 -50 0 50 100 150 Temperature (F) Temperature (F) - Measured Dat -- MIT • INM Mach 0.2 Mach 0.3 1.15r- 1.1

1.1 I 1.05 0 C 1.05 F. cc 1

1 0.95 0.95 fln ft 0.9 h

0.9 I. S0.85 h 2 2 h 0.85 h=0 1 0.8 h z z 0.8 3 0.75 0.75 0.7 U 0.7 I 0.65

0.65 L - 0.6' -100 -50 0 50 150 -100 -50 0 50 100 150 Temperature (F) Temperature (F)

Figure 3.5 Takeoff thrust comparison for small commercial airplane at various conditions. Mach 0.2 Mach 0.3 1.05 1

1 0.95 o c 0.95 cr 0.9

0.9 -0.85 0.8 h=95DJ t. 0.85 h=650.J I-T 0.8 O h=6500J 0.75 0.8 h=5h=9500i h=25QgJ - 0.75 h=5QSt 0.7 h=5&H

0.7 70.65

r 0.65 0 0.6 0.6 0.55

0.55 0.5 -100 -50 0 50 100 I -100 -50 0 50 100 150 Temperature (F) Temperature (F) - Measured Data -- MIT .... INM Mach 0.4 Mach 0.6 0.95 0.9

0.9 0.85 0o r 0.85 r 0.8

0.8 . 0.75 R .~ . 0.75 h=9 0.7 h=95 h=6500i 0.7 0.65 h= h= 250f 2 0.65 h=50ff h=2500f 6o.h=SnL-. z z 0.6 ' 0.55 0.55 0.5

0.5 0.45

0.45 | II I I 0.4 I I • II | I -100 -50 0 50 100 150 -100 -50 0 50 100 150 Temperature (F) Temperature (F)

Figure 3.6 Climb thrust comparison for small commercial airplane at various conditions. Mach 0 Mach 0.1

1 L o S1.2I

1.1

; 0.9

0.8 (3-- 0.7 r

0.6 , i l i l 0 50 -100 -50 0 50 100 150 Temperature CF) Temperature CF) - Measured Dat -- MIT .... INM Mach 0.2 Mach 0.3 . _

U 1.2!~ 2 IE

W0.9 I- z 0.8 00 0.7 [

0.61 I I I -100 -50 0 50 100 150 0 50 Temperature (F) Temperature (F)

Figure 3.7 Takeoff thrust comparison for medium commercial airplane at various conditions. Mach 0.2 Mach 0.3

0.85 I

h=10000 ff-

0.75 h=50Q.Otf 0.7

\ .. 0.65

0.6

0.55

0.55 L 0.5 L -50 0 50 100 150 -50 0 50 100 150 Temperature CF) Temperature CF) - Measured Dat - -MIT S INM Mach 0.4 Mach 0.5

a 0.8

~ 0.75

0.7 5 2 0.65

z 0.6

8 0.55

0.45 L -50 0 50 100 150 0 50 100 150 Temperature CF) Temperature CF)

Figure 3.8 Climb thrust comparison for medium commercial airplane at various conditions. Mach 0 Mach 0.2

0 50 0 50 100 15 Temperature (F) Temperature (F) - Measured Data -- MIT .... INM Mach 0.3 Mach 0.4

0.4 ' -100 -50 0 50 100 150 0 50 Temperature (F) Temperature (F)

Figure 3.9 Takeoff thrust comparison for large commercial airplane at various conditions. Mach 0.1 Mach 0.3 2 . . . 1.1

o1.2 .P 1 ig U 1.1 2 20.9 i-

5 0.8 0 0.9 0.7 0.8

E0.6 0.7

0.6 80.5

n r;l 1 0.4 -100 0 50 100 150 -100 -50 0 50 100 150 Temperature (F) Temperature (F) - Measured Dat -- MIT .... INM Mach 0.4 Mach 0.5 1

S0.9 F. g 0.9

2 2 0.8

0.7 2 I- z 0.8 z 0.6 z 0.6

80.5 8 0.5

0.4 ' 1 0. -100 -50 0 50 100 150 - 100 -50 0 50 100 150 Temperature (F) Temperature (F)

Figure 3.10 Climb thrust comparison for large commercial airplane at various conditions. 3.4.2 Error Analysis

The least squared error is calculated using the following equation,

[(measured value), - (computed value) ] 2 (3.10) Ave Error = where n is the numbers of data points. The average errors are presented in Table 3.3 to 3.8. As the tables show, the quadratic thrust model is more accurate than the existing thrust model in INM.

Table 3.3 Error in corrected net thrust during takeoff for small commercial airplane. Least Squared Error in Corrected Net Thrust (lb) Mach 0 Mac 0.1 Mach 0.2 Mach 0.3 0 ft 76.7736 52.2994 62.1422 68.7801 MIT Model 3000 ft 62.0462 35.1147 22.2196 29.0389 6000 ft 69.2990 28.7856 27.7817 41.5644 9000 ft 31.8990 43.8860 24.9302 27.3985 0 ft 634.7971 656.0399 648.0785 596.6737 INM Model 3000 ft 631.2608 653.7065 666.7642 636.0905 6000 ft 650.9106 645.7621 659.7622 636.3030 9000 ft 665.3742 661.6280 691.0583 684_7760 Table 3.4 Error in corrected net thrust during climb for small commercial airplane. Least Squared Error in Corrected Net Thrust (lb) Mach 0.2 Mach 0.3 Mach 0.4 Mach 0.5 500 ft 20.7569 27.3014 23.8526 48.0301 31.1475 MIT Model 2500 ft 22.7564 24.9581 23.2439 6500 ft 28.9597 24.8293 29.1363 22.4292 9500 ft 30.3556 24.1102 30.0177 26.7810 500 ft 703.9612 669.5469 627.2996 360.7722 INM Model 2500 ft 693.5819 674.8251 650.9640 404.2555 6500 ft 665.3673 676.1206 674.8790 482.0347 9500 ft t 639,3356 670.3949 682.4352 527.7964

Table 3.5 Error in corrected net thrust during takeoff for medium commercial airplane. Least Squared Error in Corrected Net Thrust (lb) Mach 0 Mac 0.1 Mach 0.2 Mach 0.3 0 ft 306.7909 392.9152 344.6927 165.1217 MIT Model 5000 ft 342.2574 173.6205 104.1779 101.2559 7920 ft 396.7150 161.8416 72.1704 100.0924 0 ft 1224.11 1209.47 924.19 534.55 INM Model 5000 ft 894.78 946.14 805.73 573.20 7920 ft 74900 812.39 728.91 519.66

Table 3.6 Error in corrected net thrust during climb for medium commercial airplane. Least Squared Error in Corrected Net Thrust (lb) Mach 0.2 Mach 0.3 Mach 0.4 Mach 0.5 MIT Model 500 ft 20.9499 118.9694 82.6470 82.8706 9500 ft 52.6467 131.6875 44.9727 85.9502 INM Model 500 ft 809.3244 968.9295 856.5244 836.8706 9500 ft 7653625 961.642&7 932.9404 959.2255 Table 3.7 Error in corrected net thrust during takeoff for large commercial airplane. Least Squared Error in Corrected Net Thrust (lb) Mach 0 Mach 0.2 Mach 0.3 Mach 0.4 0 ft 391.3044 273.5680 256.5289 303.8417 MIT Model 4000 ft 268.9983 192.9940 186.5013 200.0846 10000 ft 206.6931 157.2474 150.8482 134.6785 0 ft 1766.11 1630.45 1626.52 1768.62 INM Model 4000 ft 1434.02 1350.46 1370.69 1532.73 10000 ft 1331.86 1172.15 1183.47 1294.79

Table 3.8 Error in corrected net thrust during climb for large commercial airplane. Least Squared Error in Corrected Net Thrust (lb) Mach 0.1 Mach 0.3 Mach 0.4 Mach 0.5 0 ft 326.5073 100.7679 150.0685 184.0319 MIT Model 4000 ft 170.5130 92.2014 85.7651 199.4702 8000 ft 154.4417 212.2085 135.9459 182.5523 12000 ft 209.5427 311.1647 207.1063 157.5070 0 ft 1889.59 1886.91 1881.04 1610.29 INM Model 4000 ft 1722.34 1898.43 1999.64 1914.09 8000 ft 1570.81 1891.55 2034.95 2129.19 12000 ft L 1484.26 1891.40 2102.42 2251.60

3.5 Conclusion of Chapter 3

As the analyses show, the proposed quadratic thrust model is more accurate than the existing INM model, particularly at high altitude and nonstandard day temperature condition, and would thus provide improved prediction of aircraft thrust over a wider range of operating conditions. Chapter 4. Level Flight Thrust Model

Level flight segments occur either between two climbing segments or between two descending segments, and are treated as steady flight situations, namely, the balancing of aerodynamic forces can be applied to obtain the required thrust. In order to apply the balancing of aerodynamic forces , a complete set of drag polars for each aircraft configuration is required which might not be very practical in computation or desirable for manufactory intent on maintaining control of proprietary information. In this chapter, two drag polar models are introduced. The first is an approximation of the drag polar based on the minimum drag-over-lift point, while the second is a constrained least square fit of the drag-over-lift ratio as a function of the lift coefficient. Two level-flight thrust models are then developed based on these two drag polar models.

4.1 Previous Work

As suggested in SAE AIR 1845, the level-flight thrust in INM is now computed by reversing the flight path angle equation to get the following expression [2]

1 siny (F /8)avg (W/8)avg [R + 1.03in (SAE Eq.A15) narg N 1.03 where y is zero for level flight. This expression however, does not include any velocity dependence despite the fact that the required thrust is known to be a function of velocity [3]. 4.2 Equation of Motion

The forces acting on an aircraft in steady, straight, and level flight are shown in Fig. 4.1.

Lift, L

Thrust, F True Drag,D Airspeed,V t P Flight path Weight, W

Figure 4.1 Forces on an aircraft in level flight.

The two aerodynamic forces, lift and drag, act at the center of pressure, and the gravitational force, weight of aircraft, acts at the center of gravity of the aircraft. The lines of action of the thrust and drag forces lie very close to each other and the center of pressure can be regarded coincident with the center of gravity of the aircraft, so that the coupling moment is negligible. Summing forces parallel and perpendicular to the flight path yields the equation of motion of the aircraft in steady- level flight,

F= D - p V 2 CD S (4.1) 2

2 W = L 1 p Vt CL S (4.2) 2

Combining Eq.(4.1) and (4.2) leads to the level-flight thrust as a function of weight and drag- over-lift ratio,

F = WR (4.3) If the aircraft weight is treated as a constant, the minimum thrust occurs at the point where the drag- over-lift ratio, R, is at minimum. This characteristic will be utilized in the development of level- flight thrust.

4.3 Drag Polar

The aerodynamic characteristics, CL and CD , for a conventional aircraft exhibit a quadratic relationship of the form,

CD = ko + k, CL + k 2 C2 (4.4)

where ko , k, , and k 2 are constant coefficients which can be obtained from flight test data by the method of least squares. Dividing Eq.(4.4) by CL , the expression of drag-over-lift ratio in terms of lift coefficient is obtained.

R =- ko + k + k C (4.5) CL 2 L

In this section, two drag polar models, Model I and Model II, will be introduced. Both models are approximations of the drag polar near the minimum drag-over-lift point. Since the derived drag polar models are based on the minimum drag-over-lift point, it is necessary to define this point before deriving the models. The minimum drag-over-lift ratio and the corresponding lift and drag coefficients are found by taking the derivative of Eq.(4.5) with respect to CL and setting the derivative to zero to obtain

CLRin as shown below.

C ko (4.6) k 2d Substituting Eq.(4.6) into Eq.(4.4) and Eq.(4.5) to obtain the corresponding drag coefficient and minimum drag-over-lift ratio respectively.

= o + k, + k2 2LR CDRm k CLRmm R (4.7)

R ko min C L + k1 + k2 CLR-m (4.8)

4.3.1 Drag Polar Model I

Near the point of minimum drag-over-lift point, the drag polar may be represented by a simplified drag polar model of the form

CD = CDo + k CL (4.9)

where CDo, the zero-lift drag coefficient, and k are both constant coefficients. In reality, CDo and k are functions of flight Mach number and Reynolds number, but since the operations considered here are departures and approaches, i.e. the flight Mach number is under 0.7 and the effect of Reynolds number only has small impact on skin friction drag, the assumption of constant CDo and k are valid [6]. Dividing Eq.(4.9) by CL yields the expression for drag-over-lift ratio, R.

C R C + k CL (4.10) C,

Taking the derivative of Eq.(4.10) with respect to CL,, setting the derivative to zero, and making the necessary substitutions, yield a modified expression for CD and R in terms of CL, CLRnn, and CDRmin" CD CDR2 C DR i CL 2 (4.11) CD 2 2C2 CLRnunCU

1 CDR 1 CDrm CL R2 (4.12) 2 CL 2 CL CLR)

4.3.2 Drag Polar Model II

Although the drag-over-lift ratio is described by the relatively complex expression (Eq.(4.5)), near the minimum drag-over-lift point, the drag polar may be described by an expression of the form

R = Ri + k(CL - CLRm)2 (4.13)

where k' is a constant that can be obtained from flight test data by the method of least squares. Since aircraft are usually operating near the minimum drag-over-lift point during level flight in order to minimize the thrust, this expression will closely match the behavior of an aircraft in level flight.

4.3.3 Effects of Reynolds Number on Drag Polar

Reynolds number, a dimensionless number of importance and impact on aerodynamics, is essential to the determination of skin friction drag. Reynolds number, Re, is defined as,

Re - pV (4.14) P- For Reynolds number between 106 and 10, the drag coefficient of a flat plate with fully turbulent boundary layers is proportional to Re-1/6 [7]. For an aircraft, roughly 50% of the total drag is due to skin friction drag. The other components of the drag coefficient of an aircraft, apart from skin friction, are nearly independent of Reynolds number, so that for any given change of Reynolds number the percentage change of CD will be about half the percentage change of flat plate drag coefficient. For instance, assuming that the viscosity is constant regardless the change of altitude and the calibrated velocity remains constant, the ratio of drag coefficient for a flat plate evaluated at two different altitude, 1000 ft and 10000 ft, can be obtained as follows, 1 ( 1 Ch Re 6 Ph=1000 6 Dhl00 Reh=0 "6 Ph==1000 6 = 0.8721 = 1 - 0.1279 (4.15) CDh=10000 Reh=10000 Ph=10000

Namely, there is a reduction of 12% in drag coefficient for flat plate from 10000 ft to 1000 ft. For an aircraft, there would be approximately half of 12%, i.e. 6%, reduction in drag coefficient if the height were reduced from 10000 ft to 1000 ft. The skin friction drag of an aircraft is not sensitive to lift coefficient, so that the reduction of CD found here can be regarded as being at constant CL and the percentage reduction of drag-over-lift ratio, R, would be the same as drag coefficient. As shown in this section, changes in the flight environment and thus the Reynolds number have some impact on the aerodynamic characteristics of an aircraft, but the influence is not significant.

4.4 Level-Flight Thrust

Combining Eq.(4.3) and (4.5), the level-flight thrust is given by

k F = W ( + k + k2 CL ) (4.16) CL Substituting for the lift coefficient in terms of flight conditions derived from Eq.(4.2),

2W (4.17) P, a V 2 S gives the level-flight thrust in terms of known flight parameters.

ko Po o V2 S 2k 2 W F = W ( + k + ) (4.18) 2W Po a Vt2 S

4.4.1 Level-Flight Thrust Model I

Substituting the minimum-level-flight-thrust condition into the level-flight thrust equation, Eq.(4. 1), yields the following expression,

F = Rmin W 2 (4.19) \ - / Vt Rm CDR

where VtRm is given by

Vt =CR W (4.20) m R" U and

CR 2 (4.21) CR PO CLr = S Substituting drag polar Model I, Eq.(4.11), and Eq.(4.20) into Eq.(4.19) yields,

12 V 2 F Rn 2 + (4.22) 2 CR mn

Furthermore, according to Eq.(4.2), the expression for the lift coefficient ratio in Eq.(4.22) may be expressed in terms of the airspeed ratio by

VtR CL (4.23) CLR Vt

Substituting Eq.4.23 into Eq.4.22 yields the expression for level-flight thrust

Vt CR F - R oj1 + (4.24) 2 an CRm Vt

Replacing the true airspeed in Eq.(4.24) with the true airspeed model, Eq.(2.13), developed in Chapter 2, the level-flight thrust is then given by

1 (1 + Eh)Vc F= 1 [1 + W2[ C Rnh) (4.25) 2 CRM (1 + Eh) V

Providing minimum drag-over-lift ratio, Rmin, and corresponding CRmn , Eq.(4.25) gives the total level-flight thrust according to the operating conditions, i.e. aircraft weight, altitude, and calibrated airspeed. 4.4.2 Level-Flight Thrust Model II

Level-flight thrust model II is based on the drag polar model II, Eq.(4.13). After substituting Eq.(4.13) into Eq.(4.3), level-flight thrust model II is given below.

F =W [ Rmin + k'( CL- CLR. )2 ] (4.26)

Furthermore, substituting Eq.(4.17) for CL in the above equation and replacing true airspeed with Eq.(2.13), Model II is obtained in terms of flight conditions as follows.

F = W in + k ' C 2 Rmin (4.27) LR (1 + E h)2 V

4.5 Validation of Level-Flight Thrust Models

This section provides a comparison between the level-flight thrust predicted by the current INM equation and the level-flight thrust predicted by the two models described above for a small and a large airplane. The error analysis suggests that these two models are superior to the existing INM equation.

4.5.1 Comparison of Proposed Models with INM Model

Figures 4.2 and 4.3 show the thrust ratio per engine vs. velocity plots for a small airplane and a large airplane with different flap settings at 5000 ft, standard day condition respectively. For other flight altitudes, the thrust histories have similar shapes, but different thrust and airspeed ranges. Because INM uses only one drag-over-lift ratio for any flight velocity, the curve representing INM equation is simply a straight line. Comparison to measured data shows that using a constant drag- over-lift ratio to determine level flight is not adequate, and by providing flight condition and necessary parameters, the proposed models can capture the curvature of measured data.

00 Flap 50 Flap

k d 21 -

20

\O \0 6 \O

A&AA\AA \O

22 F ,o7

16 - 18' I I ( I 150 200 250 300 120 140 160 180 200 220 Calibrated Airspeed, V (knots) Calibrated Airspeed, V (knots) 0O Measured Data - MIT Model I 85% MGTOW, 5000 ft -- MIT Model II INM 150 Flap 1 300 Flap+Gear

34 F 35.5-

32- 35- O

30- 0 / Q 34.5 - 0// 28 )0 0 0 '5 \ 9 34 o \ 9/ \ A AA 33.5 I 24 - AAAAA A \ \

22' 33' 120 140 160 180 200 130 140 150 160 Calibrated Airspeed, V (knots) Calibrated Airspeed, V,(knots)

Figure 4.2 Thrust ratio per engine vs. velocity for small commercial jet with various flap settings. 00 Flap 50 Flap

30 I

200 25 300 400 500 150/A A 200 A AA 250 A A 300A 350 Calibrated Airspeed, V(knots) Calibrated Airspeed, V (knots) O0 Measured Data - MIT Model I 85% MGTOW, 5000 ft -- MIT Model II INM 200 Flap 300 Flap+Gear 48-

47.5

47 -

8 46.5 I- 46_

.455

26 150 200 250 300 350 160 180 200 Calibrated Airspeed, V(knots) Calibrated Airspeed, V (knots)

Figure 4.3 Thrust ratio per engine vs. velocity for large commercial jet with various flap settings. 4.5.2 Error Analysis

Because the error is not sensitive to flight altitude, 5000 ft was selected as a representative altitude. Three aircraft weights, 85% of maximum gross takeoff weight, 90% of maximum gross landing weight, and the average of those two weights were used for the analysis. In addition, the lower bound of the airspeed was set to be 1.2 times of the stall speed and the upper bound of the airspeed was 80 knots greater than the lower bound. Table 4.1 shows the average least squared errors in level-flight thrust per engine for the small airplane at different configurations. Because the derivation for thrust model I is based on the expansion of the exact thrust equation about the minimum-thrust-flight conditions, the error propagates as the flight velocity deviates from the minimum-thrust-flight velocity. The error is proportional to the product of aircraft weight and the difference between actual flight velocity and minimum-thrust-flight velocity. The observed errors of 10 lb to 70 lb per engine for the small airplane are relatively small comparing to the actual level-flight thrust (8000 lb per engine). Table 4.2 shows the average least squared errors in level-flight thrust per engine for the large airplane at different configurations. The constrained curve fitting, thrust model II, guarantees agreement near the vicinity of the minimum-thrust-flight point. Due to the different aerodynamic characteristics between the small airplane and the large airplane, the constrained least square fit provides a more accurate fit for the small airplane than the large airplane. The observed errors of 300 lb in thrust per engine for the large airplane is again relatively small comparing to the actual level-flight thrust (28000 lb per engine). Table 4.1 Average level-flight thrust errors per engine for small commercial jet at 5000 ft.

Model I Average Least Squared Errors (lb) 0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW 2 O0Flap 25.23 27.10 33.35 50 Flap 65.05 70.04 75.04 150 Flap 50.64 54.47 58.30 300 Flap+Gear 50.23 40.50 43.28

IF Model II Average Least Squared Errors (lb) 0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW 2 00 Flap 28.06 30.15 26.26 50 Flap 33.31 35.92 38.54 150 Flap 42.26 45.45 48.64 300 Flap+Gear 15.69 12.98 13.76

INM Average Least Squared Errors (lb) 0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW 2 00 Flap 51.09 54.97 74.01 50 Flap 97.11 104.47 111.83 150 Flap 168.07 180.81 193.55 30o Flap+Gear 90.14 72.80 77.93 Table 4.2 Average level-flight thrust errors per engine for large commercial jet at 5000 ft.

Model I Average Least Squared Errors (lb) 0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW 2 0O Flap 284.52 328.69 278.10 50 Flap 43.07 49.26 55.38 200 Flap 130.58 146.46 102.22 30° Flap+Gear 27.30 30.52 33.50

Model II Average Least Squared Errors (lb) 0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW 2 00 Flap 180.27 205.51 97.51 50 Flap 114.76 132.04 149.29 200 Flap 233.15 268.10 138.49 300 Flap+Gear 113.86 130.70 147.40

INM Average Least Squared Errors (lb) 0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW 2 0O Flap 654.96 754.59 635.80 50 Flap 681.48 785.14 888.81 200 Flap 318.38 366.81 292.10 300 Flap+Gear 226.10 260.50 294.81 4.5.3 Pro and Con Between Models

Table 4.3 and 4.4 show the pros and cons of Thrust Model I and Model II. These observations are based on the error analysis and consideration of the numbers of parameters needed from manufactures.

Table 4.3 Pro of level-flight thrust model I and model II.

Model I Requires only 2 parameters. Small errors. Good fit in the vicinity of minimum thrust. Simple. Model II Small errors. Good fit in the vicinity of minimum thrust.

Table 4.4 Con of level-flight thrust model I and model II.

Model I More complicated. Fit depends on the aerodynamic characteristics of aircraft. Model II Need more parameters. More work on the evaluation of parameters. 4.6 Conclusion of Chapter 4

The thrust vs. velocity plots and error analysis demonstrate that the level-flight thrust models developed in this chapter can accurately predict the actual thrust within the considered airspeed range. Although the error increases when the operating point deviates away from the minimum- thrust condition, the resulting errors are small comparing to the total thrust. Since the variation of Reynolds number with flight condition change has small impact on the value of drag-over-lift ratio, it is recommended that the evaluation of the minimum drag-over-lift ratio and corresponding CR,n coefficients should be done at different flight conditions and then an average value derived. Overall, the current INM equation for computing level-flight thrust is inadequate and both models derived in this report provide considerably better estimates of level-flight thrust. Chapter 5. Climb Performance

The flight environment is not free of headwinds or tailwinds, so the flight path angle observed on the ground may not be the same as the flight path angle when there is no wind. In addition, the increase in true airspeed that occurs during constant equivalent or calibrated airspeed climb has a small impact on the flight path angle. Two flight path angle models are presented in this chapter which correctly account for the effect on the flight path angle of both wind and the acceleration during climb. One is based on the assumption of constant equivalent airspeed climb and the other is based on the assumption of constant calibrated airspeed climb.

5.1 Previous Work

The flight path angle model in INM implicitly assumes that the airplane is climbing at a constant calibrated airspeed and maximum available thrust. The fundamental aerodynamic force balance leads to the equation for flight path angle [2],

y = sin - [N n)av - R ] (SAE Eq. A8) ( W/8) ag

where the correction factor, , accounts for the increased climb gradient associated with an 8-knot headwind and the acceleration inherent in climbing at a reference equivalent airspeed of 160 knots ((=1.01 when climb speed < 200 knots and =0.95 otherwise). Because this factor was derived for an aircraft operating from a sea-level airport on a standard day, it does not account for variations in airport altitude and aircraft climb speed. The ground distance, Sc , that the airplane traverses during climb, is computed using the following equation [2]. Ah S tan (SAE Eq. A9) c tan y

5.2 Equation of Motion and Flight Path Angle Correction Factor

Lift, L Vt Thrust, F

Flight Path Angle, y

Drag, D

W/g dV/dt Weight, W

Figure 5.1 Aircraft in steady climb with no wind.

Figure 5.1 shows an aircraft in steady climbing flight with no winds. As the figure shows, the velocity is aligned with the flight path and the flight path itself is inclined to the horizontal at the angle y. As in level flight, lift and drag are perpendicular and parallel to flight velocity respectively and the weight is perpendicular to the horizontal. Thrust is assumed to be aligned with the flight path (i.e. neglecting the thrust setting angle) and the inertial force, W/g dV/dt, is opposite to the direction of thrust. Summing forces parallel and perpendicular to the flight path yields the equation of motion for climb.

F - D - Wsiny -dV,w dV, (5.1) g dt

L - Wcosy = 0 (5.2) Combining Eq.(5.1) and (5.2) yields the expression for the flight path angle without headwind.

F -- R W sin y = (5.3) 1 + Vt d Vt g dh

Eq.(5.3) only accounts for the impact of acceleration, thus an additional factor must be included to account for the effect of wind on the flight path angle. Figure 5.2 shows the geometry of the velocity vectors when wind is considered in flight.

Vt

VW

Figure 5.2 Geometry of airspeed vectors in wind.

As the figure shows, Y2 , the flight path angle after the effect of wind is included is related to y, , the flight path angle without wind, by the following expression.

V, sin y1 sin y2 = (5.4) 2 ( V cos y, - V )2 + ( V sin y ) After small angle approximation, Eq.(5.4) becomes

sin y 2 = sin y,. (5.5) V -V

Combining Eq.(5.3) and (5.5) forms the equation of flight path angle,

Vt V V F sin y Vt - R] (5.6) Vt d V W 1 + g dh where the term in front of the bracket in Eq.(5.6) above is referred to as flight path angle correction factor, .

Vt 1 - VW V dV, (5.7) Vt w 1+ g dh

It is clearly shown in Eq.(5.7) that the flight path angle is altitude and airspeed dependent.

5.3 Evaluation of Flight Path Angle Correction Factor

Based on the atmospheric model presented in Chapter 2, Eq.(2.1) to (2.3), the following derivatives, temperature ratio, pressure ratio, and density ratio with respect to altitude were derived. Because the pressure ratio correction term, the second term in Eq.(2.2), is small and the effect is negligible in the subsequent development of necessary derivatives, it was ignored in order to simplify the expression.

dO 1 dT T dh (5.8) dh o dT a dh d6 -g 66 g (5.9) RT dh a o

dT a dh do -6 (g 6 g 1 dT (5.10) TO R dh 0 dh o a

Two models for the flight path angle correction factor are discussed in this section, one is based on the assumption of constant equivalent airspeed climb and the other is based on the assumption of constant calibrated airspeed climb.

5.3.1 Constant Equivalent Airspeed Climb Model

Taking the derivative of Eq.(2.11) with respect to altitude while holding equivalent airspeed, Ve, to be constant yields the expression for dV/dh as shown below.

dV 1 do d - ( )V 0V d (5.11) dh 2 dh

Combining Eq.(5.7), (5.10), and (5.11) yields the expression for flight path angle correction factor during constant equivalent airspeed climb.

V, dT V, - Vw a dh (5.12) Vt2 1 dT 2gt gR 6g +- 0 2gT Ra dh 5.3.2 Exact Constant Calibrated Airspeed Climb Model

Taking the derivative of Eq.(2.1 1) with respect to altitude yields the expression for dV/dh,

-3 dV, 1 dV d6 -1 2 do +( )V o (5.13) dh J- d6 dh 2 e dh

where dVe/d6 is obtained by taking derivative of Eq.(2.9) with respect to pressure ratio, 6, while holding calibrated airspeed, Vc, to be constant.

dV Ya Po [( 1 k k + -)(- + 1 )Ya -1] (5.14) 6 d6 - 1 Po Ve ya 6

After combining the relevant derivatives, the value of dV/dh during a constant calibrated airspeed climb is given by

R -dT a dh dV, Vt 1 dT gO Ya k k (5.15) g [(1+)( +1) Ya 0 dh Vt ya - 1 6 dh 2 To Ra ya 6

Substituting Eq.(5.15) into Eq.(5.7) yields the exact expression for the flight path angle correction factor during constant calibrated airspeed climb,

V, 1 (5.16) Vt - Vw 1 + where

dT R Radh 2 a dh -1 dT Y 06 g [(1 + k 1)Y -V ( g + - + 1] (5.17) 2gT Ra 0 dh Ya- 1 Ya 6 5.3.3 Simplified Constant Calibrated Airspeed Climb Model

As shown in Eq.(5.16) and (5.17), the exact solution for the flight path angle correction factor during constant calibrated airspeed climb is very complicated and a further simplification is desirable. Taking the derivative of Eq.(2.13) with respect to altitude yields the expression for dV/dh,

dV t Vc (1 + Eh) do [ - d (5.18) dh - 20 dh

After combining the relevant derivatives, the value of dV/dh during a constant calibrated airspeed climb is given by

SdT

dV t V + (1 + ch)( g +1IdT)] (5.19) dh 2 To Ra 0 dh

Substituting Eq.(5.19) into Eq.(5.7) yields the simplified expression for the flight path angle correction factor during constant calibrated airspeed climb,

Vt 1 V, - V RdT (1 + 2 adh (5.20) + Eh)V + (1 + eh) g + dT 1 + (1 [E + ( 6 + - -)] go 2T Ra 0 dh

Figure 5.3 and 5.4 show the comparison between the exact model and the simplified model at standard day and nonstandard day condition with 8 knots headwind respectively. The close match between two models verifies that the simplified model, Eq.(5.20), is adequate to represent the exact constant calibrated airspeed climb model. " 1.05

L'- 00.95 Q 0.950.9 CD

0.9

C- 0.8515 u- 0.85 15 50 160 170 180 190 200 210 220 230 240 250 Calibrated Airspeed, Vc (knots)

6 1.05

C ~------, _Vc=1 50 knots c1)

0 =- " c=200 knots 0.95 Vc=250 knots Exact Model - Simplified Model

I I I t CI 2.5 4 Altitude, h (ft) x 10

Figure 5.3 Comparison of flight path angle correction factor predicted by the exact and simplified models at standard day, 8-knot headwind condition. h = Oft h = 10000 ft 1.05

1.04 -,Vc=150 knots 1 U,-,-, 0 I- CU UL 1.02 0 1 C 0

1.01 S0.98 o o U-a , 1 -Vc=200 knots S0.96 < 0.99 -C 0

U0.98 - E 0.94 ._ .9 0.94 LI, n-

U.9 =' Vc=250 knots

0.96

0.95 0.91 -5 0 0 50 1 -50 0 50 100 Temperature at Sea Level Airport F) Temperature at Sea Level Airport CF)

h = 20000 ft

1,

- Exact Model -- Simplified Model

0.95 Vc=200 knknots

.Vc=250 knots 0.9-

-50 0 50 14 Temperature at Sea Level Airport CF)

Figure 5.4 Comparison of flight path angle correction factor predicted by the exact and simplified models at nonstandard day, 8-knot headwind condition.

70 5.4 Graphical Comparison of Flight Path Angle Correction Factor

Figure 5.5 shows the comparison of flight path angle correction factor derived by MIT and INM models at standard day, 8-knot headwind condition. As shown in Figure 5.5, the constant factors in INM, 1.01 and 0.95, were approximately the average values for climbing at an airspeed greater and less than 200 knots respectively.

Vc=210 knots 0 U

.2U,8 0.95 ------

Vc=250 knots 0 -c

D 0.9 -

0.85 0 0.5 1 1.5 2 2.5 1 1.5 2 2.5 4 Altitude, h (ft) x 104 Altitude, h (ft) x 10

Figure 5.5 Comparison of flight path angle correction factor between MIT and INM models. 5.5 Calculation of Flight Path Angle and Ground Distance Traversed

Because the flight path angle model is a nonlinear and altitude dependent differential equation, a numerical integration is required. There are two ways to approximate the solution, one is the continuous numerical integration and the other is two-point-average approximation. The two- point-average approximation is to calculate the average climb angle

Yavg + th2 (5.21) avg 2 and then substitute into Eq.(5.22) below to obtain the ground distance traversed during climb.

h2 - h1 Sc 2 1 (5.22) tan Yavg

5.6 Error Analysis

In this section, the numerical integration of exact climb equation using SIMULINK® is considered as an exact solution and the comparison is made using a small commercial airplane and a large commercial airplane as the testing models. The weight of aircraft is chosen to be 85% of MGTOW, the throttle setting is chosen to be maximum takeoff thrust, and the flap setting is chosen to be 5 degrees and 10 degrees for both airplanes.

5.6.1 Constant Equivalent Airspeed Climb Model

Table 5.1 and 5.2 show the error in approximating ground distance traversed during constant equivalent airspeed climb starting from sea level and 5000 ft respectively. Table 5.1 Error in ground distance during constant equivalent airspeed climb starting from sea level.

Small Commercial Airplane: 5 deg Flap Large Commercial Airplane: 10 deg Flap Error in Ground Distance Error in Ground Distance Traversed (ft) Traversed (ft)

Final Altitude Ve Ve Ve Final Altitude Ve Ve Ve (ft) 160 180 200 (ft) 185 200 220 knots knots knots knots knots knots

1500 0.41 0.50 0.58 1500 0.69 0.76 0.86

3000 0.73 1.16 1.60 3000 11.08 12.03 13.53

5000 13.84 13.50 13.19 5000 90.51 98.23 110.23

Table 5.2 Error in ground distance during constant equivalent airspeed climb starting from 5000 ft.

Small Commercial Airplane: 5 deg Flap Large Commercial Airplane: 10 deg Flap Error in Ground Distance Error in Ground Distance Traversed (ft) Traversed (ft)

Final Altitude Ve Ve Ve Final Altitude Ve Ve Ve (ft) 160 180 200 (ft) 185 200 220 knots knots knots knots knots knots

6500 2.34 2.53 2.73 6500 7.42 8.10 9.15 8000 23.34 25.37 27.59 8000 72.96 79.86 90.47

10000 141.00 154.18 168.55 10000 441.56 485.30 552.86 5.6.2 Constant Calibrated Airspeed Climb Model

Table 5.3 and 5.4 show the error in approximating ground distance traversed during constant calibrated airspeed climb starting from sea level and 5000 ft respectively.

Table 5.3 Error in ground distance during constant calibrated airspeed climb starting from sea level.

Small Commercial Airplane: 5 deg Flap _ Large Commercial Airplane: 10 deg Flap Error in Ground Distance Error in Ground Distance Traversed (ft) Traversed (ft)

Final Altitude Vc Vc Vc Final Altitude Vc Vc Vc (ft) 160 180 200 (ft) 185 200 220 knots knots knots knots knots knots 1500 3.27 4.45 4.97 1500 6.54 7.07 6.70 3000 14.73 16.06 19.55 3000 16.01 18.94 26.03 5000 65.48 76.58 93.38 5000 59.20 70.88 93.64

Table 5.4 Error in ground distance during constant calibrated airspeed climb starting from 5000 ft.

Small Commercial Airplane: 5 deg Flap Large Commercial Airplane: 10 deg Flap Error in Ground Distance Error in Ground Distance Traversed (ft) Traversed (ft)

Final Altitude Vc Vc Vc Final Altitude Vc Vc Vc (ft) 160 180 200 (ft) 185 200 220 knots knots knots knots knots knots

6500 74.25 86.43 102.55 6500 123.12 141.38 171.94 8000 166.12 195.46 234.03 8000 251.59 291.95 359.51 10000 276.80 334.98 411.76 10000 257.98 316.22 415.77 5.6.3 Discussion

As Table 5.1 and 5.3 show, the error in ground distance traversed is not sensitive to climb airspeed, but to altitude increment. The errors are insignificant regardless of the altitude increment while climbing from low altitude. The altitude increment becomes

important while climbing from high altitude, however, the errors of a couple hundred feet as shown in Table 5.2 and 5.4 resulting from altitude increment of 5000 ft while climbing from 5000 ft are acceptable comparing to the overall ground distance traversed.

5.7 Conclusion of Chapter 5

In this chapter, an analytical expression for climb equation was developed and further simplified. It was proved that the flight path angle correction factor is altitude and airspeed dependent. As shown from Table 5.1 to 5.4, the induced error in the distance traversed from the two-point-average approximation increases as the increment in altitude and the climb airspeed increase, thus the climb segment should be divided into increments of less than 5000 ft while climbing at high altitude to ensure the accuracy of the result. Chapter 6. Accelerated Climb Performance

In this chapter, an accelerated climb model is developed based on the assumption of constant climb rate acceleration. As comparing to the INM model, the result suggests that the correction factor in INM depends on the flight conditions and particularly flight altitude.

6.1 Previous Work

Given the initial climb conditions, a specified climb rate, and the target calibrated airspeed, the horizontal distance traversed and gain in height are obtained using SAE Eq.A10 and Al l respectively [2].

(1/2g) (0.95) (Vtb - V ) S = (SAE Eq.A10) a[N(F,/)avg/(W/)ag] - Ra vg (Vtz/Vta )

Ah = (Sa Vtz/Vtavg)/0.95 (SAE Eq.A 1l)

where 0.95 represents the headwind effect on the ground distance when climbing at a 160-knot reference airspeed into an 8-knot reference headwind, and the subscript "avg" refers to the average of the quantity at the beginning and ending points of the climb segment. However, the true airspeed and pressure ratio at the end of the acceleration segment depend on the final altitude, which is unknown. INM suggests an iterative method to compute the gain in height and horizontal distance as shown in the following figure [4]. Given Vca, Vcb, Vtz , hi

Estimate final altitude: (h2)est =hi+250 Compute Vta

Compute Vtb Compute Ah and Sa (h2)new = hi + Ah

End

Figure 6.1 Computation procedures for INM model. 6.2 Constant Climb Rate Acceleration

As opposed to the constant correction factor, 0.95, used in INM, an analytical expression for accelerated climb model is introduced in this section. During the accelerated climb segment, the aircraft is assumed to fly at a specified constant climb rate. Referring to the flight path angle equation, Eq.(5.6), in Chapter 5, the climb rate, Vtz, is obtained by multiplying both sides of the equation with the true airspeed, Vt .

Vt - Vw F Vz ( -R) V, (6.1) W 1+SV t dVt g dh

Rearranging Eq.(6. 1) and solving for dV/dh in terms of the known quantities.

dVt Vt g( F -R)- g (6.2) dh V- VwVtz W V.

Then, applying Eq.(6.2) to calculate the final altitude and ground distance traveled,

dV h2 = h + 1 (Vtb Vta)( dh V)avg dV dV, (6.3)

SdVt dh (Vtb'h 2) where ( )ag d h vge

and h2 - h1

tan Yavg

sin- Vtz + sin_ Vtb(6.4)

where Yavg = 2

Because the values of the true airspeed and pressure ratio at the end of acceleration require the knowledge of the final altitude, an iterative process is developed to generate the correct result. Instead of arbitrarily giving a final altitude, an initial estimate of the final true airspeed and altitude as shown below can speed up the iteration. V Vtbest = Vta + (Vb - Va) (6.5) ca

dV h2est= h1 + (Vtbest - Vta )( dh)(Vcahl) (6.6)

Following the procedures as illustrated in Figure 6.2, the ground distance traversed and gain in height during the accelerated climb are obtained. !Estimate the final true airspeedand altitude, Vtb(est) and h2(est), using Eq.(6.5) and (6.6) respectively. ) I

h2=(h2)new Compute Sa.

Figure 6.2 Computation procedures for constant-climb-rate accelerated climb model.

6.3 Error Analysis

In this section, the numerical integration of the accelerated climb equation using SIMULINKO is considered as an exact solution and the validation of the approximate algorithms is performed using a small commercial airplane and a large commercial airplane as the testing models. The weight of the aircraft is chosen to be 85% of MGTOW, the throttle setting is chosen to be maximum climb thrust, and the flap settings for both aircraft are chosen to be 5 degrees. Table 6.1 and 6.2 show the error in both altitude gain and ground distance traversed at standard day condition for the small commercial airplane and large commercial airplane respectively.

Table 6.1 Error in altitude gain and ground distance traversed for the small commercial airplane.

Accelerate: 160 knots to 180 knots Accelerate: 160 knots to 200 knots Error in Altitude Error in Ground Error in Altitude Error in Ground Gain (ft) Distance (ft) Gain (ft) Distance (ft) Initial Climb 0.35 9.8889 3.4 65.492 Altitude: 3000 ft Initial Climb 0.7 10.1869 5.1 80.3112 Altitude: 6000 ft

Table 6.2 Error in altitude gain and ground distance traversed for the large commercial airplane.

Accelerate: 185 knots to 200 knots Accelerate: 185 knots to 220 knots Error in Altitude Error in Ground Error in Altitude Error in Ground Gain (ft) Distance (ft) Gain (ft) Distance (ft) Initial Climb 0.05 5.7836 0.9 64.8086 Altitude: 3000 ft Initial Climb 0.085 6.7361 1.15 80.5594 Altitude: 6000 ft

Because the computation is a two-point-average process, the error is proportional to the difference between the initial airspeed and the final airspeed. As shown in Table 6.1 and 6.2, the error is not sensitive to the initial climb altitude, but to the airspeed increment. The errors in altitude gain are negligible and the errors in ground distance traversed are small comparing to the overall ground distance traversed during the acceleration. 6.4 Conclusion of Chapter 6

As the analysis in this chapter shows, the correction factor, a constant in the SAE model, depends on the flight altitude and airspeed as well as the flight environment. In addition, the error in the distance traversed and height gained during an acceleration segment induced by the two-point- average process is not sensitive to flight altitude, but to speed increment. However, for a well defined flight procedure, the speed increment can always be reduced to values less than 40 knots and thus, the accuracy of the result can be improved. Chapter 7. Comparison of Departure Profile and Noise Contour

Three aircraft models, Boeing 727-200, Boeing 737-3B2, and Boeing 757-200, at various flight conditions were evaluated. The overall ground distance traversed was compared to the flight profiles provided by the Delta Airlines (to the SAE A21 committee) and the corresponding noise contours were computed using INM Version 5.2 with improved coefficients supplied by the Boeing Company. The results suggest that the MIT model provides a more accurate prediction of aircraft performance parameters as well as the corresponding shape noise contour.

7.1 Description of Analysis

The flight procedures of a complete departure profile consist of takeoff, climb, and accelerated climb, where the climb operation is assumed to be constant calibrated airspeed climb and the accelerated climb operation is assumed to be constant climb rate operation. The inputs to INM for noise computation are as shown below: Number of Flight per Day: 1 Run Type: Single Metric Noise Metric: (I) LAMAX and (II) SEL where LAMAX represents the peak value of A-weighted sound level and SEL represents the integration of A-weighted sound pressure over a period of time. Comparisons were made of the flight profile, noise impact area, and closure point distance, where the closure point distance is measured from the break release point to the outer most point of each individual sound level. 7.2 Boeing 727-200

Two runway altitudes were used: sea level and 4000 ft. The weight of aircraft was set to 190000 lb. For the sea-level-runway, the airport conditions were 59 oF and 840 F with 8-knot headwind which are referred to as Case (1) and Case (2) respectively. For the 4000-ft-runway, the airport conditions were 44.7 OF and 69.7 OF with 8-knot headwind which are referred to as Case (3) and Case (4) respectively.

7.2.1 Procedure Steps

Table 7.1 shows the flight procedure steps for Case (1) and (2) and Table 7.2 shows the flight procedure steps for Case (3) and (4).

Table 7.1 Flight procedure for Case (1) and (2).

Step Thrust Flap ID Operation Type 1 Max Takeoff 15 Takeoff. 2 Max Takeoff 15 Climb to 1000 ft. 3 Max Takeoff 5 Accelerate to 180 knots at 750 ft/min climb rate. 4 Max Takeoff 2 Accelerate to 200 knots at 750 ft/min climb rate 5 Max Takeoff 0 Accelerate to 220 knots at 750 ft/min climb rate 6 Max Takeoff 0 Climb to 1700 ft. 7 Max Climb 0 Climb to 2500 ft. 8 Max Climb 0 Accelerate to 250 knots at 750 ft/min climb rate. 9 Max Climb 0 Case (1): Climb to 9000 ft. Case (2): Climb to 7600 ft. Table 7.2 Flight procedure for Case (3) and (4).

Step Thrust Flap ID Operation Type 1 Max Takeoff 15 Takeoff. 2 Max Takeoff 15 Climb to 1000 ft.

3 Max Takeoff 5 Accelerate to 180 knots at 750 ft/min climb rate. 4 Max Takeoff 2 Accelerate to 200 knots at 750 ft/min climb rate 5 Max Takeoff 0 Accelerate to 220 knots at 750 ft/min climb rate 6 Max Takeoff 0 Case (3): Climb to 1700 ft. Case (4): Climb to 1800 ft. 7 Max Climb 0 Climb to 2500 ft.

8 Max Climb 0 Accelerate to 250 knots at 750 ft/min climb rate. 9 Max Climb 0 Case (3): Climb to 7400 ft. Case (4): Climb to 6000 ft.

7.2.2 Flight Profile and Noise Contour

Figure 7.1 to 7.8 show the flight profile [8] and noise contour for Cases (1), (2), (3), and (4) respectively. 6000

5000

4000

3000

Ground Ground Distance (ft) x 104 Distance (ft) x 10

x 10

1.45 0- Delta -- MIT 1.4 - INM

1 1.3

S1.25 ! 1.2

1.15

1.1

1.05' o 5 10 15 Ground Distance (ft) x 104

-5' -15 -10 -5 0 5 10 15 SLD (nmi)

Figure 7.1 Flight profile and LAMAX noise contour for Case (1). 15-

S10-

5-

0

-5

-10 -30 -20 -10 0 10 20 30 SLD (nmi)

Figure 7.2 SEL noise contour for Case (1). 8000

7000

6000

S5000

> 4000

- 3000

2000

1000

Ground Distance (ft) x 104 Ground Distance (ft) x 10'

104 1.45 5 ! I-| INM.,ta 55 B 1. 3--0 Delta " - DeltaMIT MIT • -- MIT 1.35 INM 25- n5B INM 1.3

1.25 20 1.2 1.15 115 1.1

1.05 10a S5

0.95 i 0. 5 10 15 Ground Distance (ft) x 10'

CI I I I, I I -15 -10 -5 0 5 10 15 SLD (nmi)

Figure 7.3 Flight profile and LAMAX noise contour for Case (2).

88 Delta 55 dB SIl'a 30 MI INM 65.dB 25

20 85.dB

S15

uslo-

0-

-5-

-10 -30 -20 -10 0 10 20 30 SLD (nmi)

Figure 7.4 SEL noise contour for Case (2). 5 10 15 Distance (ft) x 10' Ground Distance (ft) x 104 Ground

1.6 " 30

1.55 0-0 Delta d - Delta 13-E MIT - MIT INM 1 INM 25

w 1.45 1.4 20

F 1.35

11.3 1.3 E 5 85- 1 85 d 1.25 010 1.2

0 5 10 15 Ground Distance (ft) x 104

0-

-5 - , -15 -10 -5 0 5 10 15 SLD (nmi)

Figure 7.5 Flight profile and LAMAX noise contour for Case (3).

90 S15

I 10

5-

0-

-5

-10 -30 -20 -10 0 10 20 30 SLD (nmi)

Figure 7.6 SEL noise contour for Case (3). t 4000

Distance (ft) x 104 Ground Distance (ft) x 10 Ground

" ^MIT' - - m S1. 4 F INM 25 INM

1.35

1 1.3 20 1.25

1.15 C

0 0 Ground5 Distance10 (ft) x 1015

-5 -15 -10 -5 0 5 10 15 SLD (nmi)

Figure 7.7 Flight profile and LAMAX noise contour for Case (4). IDelta AIr 30- .. - -- MI I INM 5 dR. 25 85 dB

20-

S15-

S10

0-

-5

-10 i i i i -30 -20 -10 0 10 20 30 SLD (nmi)

Figure 7.8 SEL noise contour for Case (4).

7.2.3 ErrorAnalysis

Table 7.3 shows the error in overall ground distance traversed in feet and Table 7.4 and 7.5 show the error in noise impact area in square statue mile for 55 dB, 65 dB, and 85 dB sound levels. Table 7.6 and 7.7 show the error in closure point distance in nautical miles for Case (1) and (2) and Case (3) and (4) respectively. Table 7.3 Overall ground distance error in feet for Case (1) to (4).

Case (1) Case (2) Case (3) Case (4) MIT -3276.0 -1236.4 -3808.9 -3359.5 INM -4993.9 -16925.8 -4852.6 -16017.7

Table 7.4 Error in noise impact area in square mile for Case (1) and (2).

Case (1) Case (2) 55 dB 65 dB 85 dB 55 dB 65 dB 85 dB LAMAX MIT -1.542 -0.341 -0.149 -1.34 -0.646 -0.668 INM -2.025 -0.296 0.134 3.856 5.191 0.806 SEL MIT -6.724 -3.694 -0.765 -8.211 -1.884 -0.51 INM 0.054 -2.335 0.067 4.835 5.578 9.121

Table 7.5 Error in noise impact area in square mile for Case (3) and (4).

Case (3) Case (4) 55 dB 65 dB 85 dB 55 dB 65 dB 85 dB LAMAX MIT -6.791 -4.271 -0.62 -3.536 -1.597 -0.427 INM -14.959 -7.387 -0.581 -2.906 1.025 1.356 SEL MIT -49.294 -18.932 -4.062 -14.215 -4.96 0.253 INM -10.163 -6.79 -2.697 -25.373 -7.111 7.512 Table 7.6 Error in closure point distance in nautical mile for Case (1) and (2).

Case (1) Case (2) 55 dB 65 dB 55 dB 65 dB LAMAX MIT -0.5476 -0.5219 -0.1172 -0.1486 INM -0.8219 -0.7964 -2.2007 -2.3433 SEL MIT -0.6254 -0.5676 -0.0343 -0.0919 INM -0.6774 -0.9238 -1.7911 -2.1749

Table 7.7 Error in closure point distance in nautical mile for Case (3) and (4).

Case (3) Case (4) 55 dB 65 dB 55 dB 65 dB

LAMAX MIT -0.6678 -0.6713 -0.4725 -0.4913 INM -0.8329 -0.839 -2.0981 -2.2474 SEL MIT -0.7092 -0.6693 -0.394 -0.4617 INM -0.8745 -0.8558 -1.7051 -2.0645

7.3 Boeing 737-3B2

Two cases were studied for the Boeing 737-3B2: the first was a 59 'F, sea-level takeoff with 8-knot headwind, which is referred to as Case (5), and the other was a 80 'F, sea-level takeoff with 8-knot headwind, which is referred to as Case (6). The weight of aircraft was set to 120000 lb. 7.3.1 Procedure Steps

The procedure steps for Case (5) and (6) are the same as shown in Table 7.8 below.

Table 7.8 Flight procedure for Case (5) and (6).

Step Thrust Flap ID Operation Type 1 Max Takeoff 15 Takeoff. 2 Max Takeoff 15 Climb to 1500 ft. 3 Max Climb 15 Climb to 3000 ft. 4 Max Climb 5 Accelerate to 180 knots at 1300 ft/min climb rate 5 Max Climb 1 Accelerate to 220 knots at 1600 ft/min climb rate 6 Max Climb 0 Accelerate to 250 knots at 1800 ft/min climb rate 7 Max Climb 0 Climb to 8000 ft.

7.3.2 Flight Profile and Noise Contour

Figure 7.9 to 7.12 show the flight profile [9] and noise contour for Case (6) and Case (7). 8000

7000

6000

5000

14000 211 3000 I 2000

1000

Ground Distance (ft) x 104 Ground Distance (ft) x 104

E

ii I I~ S P0

4 8 Ground Distance (ft) x 10

-4 -2 0 2 4 SLD (nmi)

Figure 7.9 Flight profile and LAMAX noise contour for Case (5). 0-

5-- -15 -10 -5 0 SLD (nmi)

Figure 7.10 SEL noise contour for Case (5). 7000

6000

15000 >4000

3000

2000

1000

0 2 4 6 8 Ground Distance (ft) x 104 Ground Distance (ft) x 10

IC~------

- Delta 14 -- MIT .... INM

12

4 8 Ground Distance (ft) x 10'

-4 -2 0 2 4 SLD (nmi)

Figure 7.11 Flight profile and LAMAX noise contour for Case (6). Delta MIT INM 15-

-15 -10 -5 0 5 10 1 SLD (nmi)

Figure 7.12 SEL noise contour for Case (6).

7.3.3 Error Analysis

Table 7.9 shows the error in overall ground distance traversed in feet and Table 7.10 and 7.11 show the error in noise impact area in square statue mile and error in closure point distance in nautical mile respectively.

100 Table 7.9 Overall ground distance error in feet for Case (5) and (6).

Case (5) Case (6) MIT -121.65 339.5 INM -689.0 1507.4

Table 7.10 Error in noise impact area in square mile for Case (5) and (6)

Case (5) Case (6) 55 dB 65 dB 85 dB 55 dB 65 dB 85 dB LAMAX MIT -0.871 -0.265 -0.071 -0.056 -0.104 -0.063 INM -0.804 -0.132 -0.017 -0.997 -0.196 -0.031 SEL MIT 0.837 0.475 0.051 2.38 1.463 0.094 INM 0.377 1.051 0.152 0.137 0.172 0.035

Table 7.11 Error in closure point distance in nautical mile for for Case (5) and (6).

Case (5) Case (6) 55 dB j 65 dB 55 dB 65 dB LAMAX MIT -0.1043 -0.0746 0.0823 -0.0057 INM -0.1465 -0.0442 0.158 0.1269 SEL MIT -0.0362 -0.049 0.0705 0.064 INM -0.0863 -0.1369 0.2525 0.148

7.4 Boeing 757-200

Two aircraft weights were used for the Boeing 757-200: the first was 183000 ib, which is referred to as Case (7), and the other was 223800 lb, which is referred to as Case (8). The airport condition was 77 OF, sea-level with no headwind.

101 7.4.1 Procedure Steps

The procedure steps for Case (7) and (8) are as shown in Table 7.12 and 7.13 respectively.

Table 7.12 Flight procedure for Case (7).

Step Thrust Flap ID Operation Type 1 Max Takeoff 15 Takeoff. 2 Max Takeoff 15 Climb to 1000 ft. 3 Max Climb 15 Accelerate to 170 knots at 1150 ft/min climb rate. 4 Max Climb 5 Accelerate to 180 knots at 1250 ft/min climb rate 5 Max Climb 0 Accelerate to 200 knots at 1350 ft/min climb rate 6 Max Climb 0 Climb to 2500 ft. 7 Max Climb 0 Accelerate to 220 knots at 1550 ft/min climb rate. 8 Max Climb 0 Accelerate to 250 knots at 1550 ft/min climb rate. 9 Max Climb 0 Climb to 6000 ft.

Table 7.13 Flight procedure for Case (8).

Step Thrust Flap ID Operation Type 1 Max Takeoff 15 Takeoff. 2 Max Takeoff 15 Climb to 1000 ft. 3 Max Climb 15 Accelerate to 180 knots at 900 ft/min climb rate. 4 Max Climb 5 Accelerate to 200 knots at 950 ft/min climb rate 5 Max Climb 0 Accelerate to 220 knots at 1100 ft/min climb rate 6 Max Climb 0 Climb to 2500 ft. 7 Max Climb 0 Accelerate to 250 knots at 1200 ft/min climb rate. 8 Max Climb 0 Climb to 6000 ft.

102 7.4.2 Flight Profile and Noise Contour

Figure 7.13 to 7.16 show the flight profile [9] and noise contour for Case (7) and (8).

6000

5000

4000

13000 2000

1000

Ground Distance (ft) x 10

14

MIT 12 ....-- INM

4 8 Ground Distance (ft) x 10

-2 0 2 4 SLD (nmi)

Figure 7.13 Flight profile and LAMAX noise contour for Case (7).

103 Delta MIT INM 15is 65d

10-

85 .

1 aI I I I I -51 i I I I 1 -1 0) -8 -6 -4 -2 0 2 4 6 8 10 SLD (nmi)

Figure 7.14 SEL noise contour for Case (7).

104 4 6 8 10 0 2 4 6 8 10 Ground Distance (ft) x 104 Ground Distance (ft) x 104

55 dB - Delta 16 - MIT INM 14

12F

0 2 4 6 8 10 Ground Distance (ft) x 104

ii | | -4 -2 0 2 4 SLD (nmi)

Figure 7.15 Flight profile and LAMAX noise contour for Case (8).

105 0 SLD (nmi)

Figure 7.16 SEL noise contour for Case (8).

7.4.3 Error Analysis

Table 7.14 shows the error in overall ground distance traversed in feet and Table 7.15 and 7.16 show the error in noise impact area in square statue mile and error in closure.poiat distance in nautical miles respectively.

106 Table 7.14 Overall ground distance error in feet for Case (7) and (8).

Case (7) Case (8) MIT -3032.1 -3722.8 INM -7509.1 -9978.8

Table 7.15 Error in noise impact area in square mile for Case (7) and (8)

Case (7) Case (8) 55 dB 65 dB 85 dB 55 dB 65 dB 85 dB

LAMAX MIT 0.15 -0.042 -0.004 0.298 -0.099 -0.012 INM 0.354 0.076 0.026 0.283 -0.086 0.021 SEL MIT 1.357 0.76 0.034 2.119 1.081 0.074 INM 3.78 1.663 0.208 3.206 1.338 0.253

Table 7.16 Error in closure point distance in nautical mile for Case (7) and (8).

Case (7) Case (8) 55 dB 65 dB 55 dB 65 dB

LAMAX MIT -0.2863 -0.1758 -0.283 -0.2528 INM -0.8286 -0.4603 -1.0807 -0.6576 SEL MIT -0.4192 -0.2961 -0.5206 -0.3931 INM -1.0603 -0.9287 -1.4668 -1.3768

107 7.5 Discussion

As shown in the case studies, the error in aircraft performance prediction for INM results from three resources, the thrust value, the flight path angle correction factor, and the accelerated climb algorithm. The INM thrust model does not accurately predict the thrust at altitudes greater than 2000 ft, particularly under nonstandard day condition. The over prediction of thrust in the INM yields an over prediction of climb performance which is the major contribution to the ground distance error in INM. For the constant calibrated airspeed climb segments, the constant flight path angle correction factor, 0.95, used in INM after flap retraction is slightly higher than the actual value which produces a slightly higher climb angle and shorter ground distance traversed during climb. The observation of accelerated climb segments shows that the climb rate in INM does not hold constant. The climb rate in INM decreases as altitude increases which yields a faster acceleration and a shorter ground distance traversed. However, since the over prediction of thrust in the INM compensates the under prediction of the ground distance traversed, the error in overall noise impact area is actually reduced for INM. The noise contour is in fact ill-shaped, i.e. much wider in lateral direction and much shorter in longitudinal direction as shown in the figures. Thus, the error in noise impact area does not provide a true indication of model's superiority. The closure point distance, on the other hand, provides a better correlation with contour shape. The smaller the error in closure point distance is, the better match the contour shape is.

7.6 Conclusion of Chapter 7

As the analyses show, the proposed thrust model provides accurate thrust prediction over a wide range of operating conditions. In addition, the flight path angle model and accelerated climb model as a function of flight parameters are more realistic. As the result, the close match of aircraft performance predicted by the MIT model yields a more accurate ground distance traversed as well as a better fit of noise contour.

108 Chapter 8. Conclusion and Future Work

8.1 Conclusion of Thesis

The analyses presented in this thesis illustrated that the proposed performance algorithms are more accurate than the current methodology employed in the INM. The proposed true airspeed model takes into account the effect of the air compressibility on airspeed and is valid over a wider range of operating condition. The proposed quadratic takeoff/climb thrust model as a function of Mach number, altitude, and temperature accurately predicts the takeoff and climb thrust under both standard and nonstandard day conditions. The level-flight thrust model as a function of flight parameters derived from the minimum-level-flight-thrust condition can accurately predict the actual level-flight thrust within the airspeed range considered. The analytical expression for the flight path angle correction factor has proven to be more realistic than the constant correction factor used in INM. The proposed accelerated climb algorithm models the constant climb rate acceleration and is more suitable for the real airplane operation. As the result, the close match of aircraft performance parameters predicted by MIT model provides a better fit of noise contour.

8.2 Future Work

Because this research focused on the improvement of aircraft performance algorithms, it is necessary to review the methodology for the calculation of corresponding sound exposure level and particularly the effect of weather condition on the noise propagation in both longitudinal and lateral directions. In addition, due to the requirement of high precision, the radar tracking of aircraft position is no longer accurate enough. For the validating purpose, it is the future work to actually setup equipments at airport to capture the aircraft departure and approach profiles and to record the corresponding sound level.

109 Bibliography

[1] Olmstead, Bryan, Mirsky, Fleming, D'Aprile, Gerbi, Le, C., Plante, Gulding, and Vahovich, INM Version 5.1 User's Guide, FAA, Washington, 1996.

[2] SAE Committee A-21, SAE AIR 1845, SAE, Pennsylvania, 1986.

[3] Anderson, John D., Introduction To Flight, McGraw-Hill Publishing Company, New York, 1989.

[4] Olmstead, INM 5.1 Technical Manual, ATAC Corporation.

[5] Lan, C.T. Edward and Roskam, Jan, Airplane Aerodynamics and Performance, Roskam Aviation and Engineering Corporation, Kansas, 1988.

[6] Vinh, N. X., FlightMechanics of High-PerformanceAircraft, Cambridge University Press, New York, 1993.

[7] Mair, W. A. And Birdsall, D. L., Aircraft Performance, Cambridge University Press, New York, 1992.

[8] Delta Airline, ICAO-B Procedure, Delta Airlines, 1997.

[9] Delta Airline, ICAO-A Procedure, Delta Airlines, 1996.

110