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.
Certe b(
... ... ..o..... ... .
(Professor
. ..C r
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
LIBRARIES
-oWWW
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........................................................................................
A cknow ledgm ents.....................................
...................................................................................
...........................................
Contents........................................................................................
4.......
10
..................................................................................................
........................................
12
.................................................
15
Nom enclature.......................................................................................
Chapter 1. Introduction.........................................................
3
.............................................
List of Tables.....................................................................................
List of Figures..................
2........
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
............................................ 18
2.1 Standard Atm osphere..................................................
.......................19
2.1.1 IN M 's Atm ospheric M odel...............................
2.2 Airspeed Measurem ent................................................................................................ 20
.......................................... 21
2.2.1 Previous Work................................................
2.2.2 True A irspeed M odel.................................................
........... 22
.......................................... 24
2.3 Conclusion of Chapter 2................................................
Chapter 3. Takeoff and Climb Thrust Model................................................
3.1 Previous W ork.............................................................
..........
.........
......... 25
................................................ 25
3.2 Quadratic Thrust Model.....................................................
.......................................... 26
3.3 Evaluation of Coefficients..........................................................................................
31
3.3.1 Ante-Break Equation......................................................
............................. 31
3.3.2 Post-Break Equation.................................................
...........
3.4 Validation..................................................................................
............ 33
................................... 36
3.4.1 Graphical Comparison....................................................................................36
.......................................... 43
3.4.2 Error A nalysis.....................................................
.....................................
3.5 Conclusion of Chapter 3.....................................................
45
Chapter 4. Level Flight Thrust Model...................................................................................46
4.1 Previous Work.............................................................
................................................ 46
............................................. 47
4.2 E quation of Motion.........................................................
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
............................................. 51
4.4 Level-Flight Thrust.........................................................
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 Error A nalysis.....................................................
.......................................... 57
4.5.3 Pro and Con Between Models...............................
4.6 C onclusion of Chapter 4.................................................
Chapter 5. Climb Performance..................................................
5.1 Previous Work.............................................................
.........
........
60
.......................................... 61
............................................
62
................................................ 62
5.2 Equation of Motion and Flight Path Angle Correction Factor................................
5.3 Evaluation of Flight Path Angle Correction Factor.................................
63
...... 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
72
5.6 Error Analysis ................................................................................................................
5.6.1 Constant Equivalent Airspeed Climb M odel.................................
..... 72
5.6.2 Constant Calibrated Airspeed Climb M odel.................................
..... 74
............................................. 75
5.6.3 Discussion...................................................
.......................................... 75
5.7 Conclusion of Chapter 5................................................
...........
Chapter 6. Accelerated Climb Performance..............................................
........... 76
................................................ 76
6.1 Previous W ork.............................................................
6.2 Constant Climb Rate Acceleration..................................................78
6.3 Error Analysis.............................................................
................................................. 80
6.4 Conclusion of Chapter 6....................................................
..........................................
82
...
83
7.1 Description of Analysis..............................................................................................
83
Chapter 7. Comparison of Departure Profile and Noise Contour.................
7.2 Boeing 727-200............................................................
................................................ 84
7.2.1 Procedure Steps...................................................
.......................................... 84
7.2.2 Flight Profile and Noise Contour............................................
7.2.3 Error Analysis.....................................................
7.3 Boeing 737-3B2............................................................
85
.......................................... 93
............................................... 95
7.3.1 Procedure Steps...................................................................................................96
7.3.2 Flight Profile and Noise Contour............................................
100
7.3.3 Error Analysis.................................
7.4 Boeing 757-200................................................................................................................
7.4.1 Procedure Steps..................................................
96
101
......................................... 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....................................................................
6
.................... 108
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 perature............................................
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).....................................
7.4 Error in noise impact area in square mile for Case (1) and (2).................................
..... 94
.... 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
....101
7.10 Error in noise impact area in square mile for Case (5) and (6).................................
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
..... 27
3.1 Typical plot for Eq.(3.3) and (3.4) at an arbitrary altitude................................
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..............................
7.1 Flight profile and LAMAX noise contour for Case (1).....................................
7.2 SEL noise contour for C ase (1).................................................
80
...... 86
.......................................... 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
91
7.6 SEL noise contour for C ase (3)..............................................................................................
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
98
7.10 SEL noise contour for Case (5)..............................................................................................
7.11 Flight profile and LAMAX noise contour for Case (6)..........................................99
100
7.12 SEL noise contour for C ase (6)..............................................................................................
7.13 Flight profile and LAMAX noise contour for Case (7)...................................
7.14 SEL noise contour for C ase (7)..............................................................................................
.....
103
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.
h 2 - Altitude at the end of climb.
hairport - Airport
hd -
elevation.
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
Vao
- Standard day ambient air temperature.
- 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-Cyrl'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,
T
dT
T, + (h - h( ) dh
T,
T1
(2.1)
-g
Ra dT
T
P
p_
PT
p1
P T
91
P1T
(2.2)
(2.3)
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].
_
T.
Sairport
Tairport + (h -
+ (h - h
(2
(2.4)
hairport) dh
dh
dT
airport )T
dh Rg a d T
Pairport -
Po
(2.5)
Po
T0
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,
M =-
V
(2.7)
V
where
V=
(2.8)
YaRaT
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
2
V
e
a
Ya-
Ya"(k +
P
6)
Ya -
1
Ya
- 6]2
(2.9)
1 po
where
-1
kI +
Ya
2V
V
a
(.0
The true airspeed is obtained by dividing Eq.(2.9) by the square root of density ratio.
Vt
V
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
200
150
Calibrated Airspeed, Vc (knots)
250
300
350
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,
v,-
(1
+
Eh)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
200
150
Calibrated Airspeed, Vc (knots)
250
Figure 2.2 Comparison of exact and MIT models at standard day condition.
300
350
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
hd
51867 (1
0.003566
-
d
+
(h) 5.256-1)
where E, F, GA , GB , and H are constant jet coefficients.
HTISA(hd))
(3.1)
(3.2)
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,
(Fn /)A
(Fn
= ko + kM + k2M
2
+ k 3 h + k4h 2
)PB = k 5 + k 6 M + k 7 M
2
+ k8 T
Fn/6 = Min[ (FnI)F, (Fn,/)NF]
(3.3)
(3.4)
(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.
x 10
4
X 10
4
32 2.5
zU,
O
2
o
o
0
0.1
0.2
0.3
Mach Number
0.4
0.5
0
50
100
150
200
250
Calibrated Airspeed, VC (knots)
300
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.
a,
x 104
3.2
2.8
Mach 0.2
2.6
-"
Iz
2.4
S2.2 F
Mach 0.4
o
1.8
1.6
I
5000
10000
Pressure Altitude (ft)
Figure 3.3 Corrected net thrust vs. altitude at Mach 0, 0.2, and 0.4.
15000
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
50
0
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.
0.5
M1 = 0
M 2 = 0.1
M 3 = 0.2
M 4 = 0.3
M 5 = 0.4
h 1 =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
9000
34865
32926
31256
29856
28725
27864
hi = 10000
35064
33125
31455
30055
28924
28063
Altitude
M6 =
h (ft)
hi
=
The coefficients for Eq.(3.3), ko, kl, k 2, k 3, and k 4, are computed by formulating the matrices
A1 and B1 as follows,
h1
h,
2
h2
h2
(Fnl)(M2,hl)
1 M2 M
hi
1
M6
1
M6
6
h1
(Fn/5 )(M6,hl)
M6
M,2
1 M
(Fn/6 )(Mh)
(F /86 )(M )
h 2 h2
(Fn
)(M,h2)
(3.6)
M
6,M, h 2
2
h2
2
1 M, M2 h11 hl
1 M 6 M2
hl
(Fn/8)(M6,h2)
(Fn/6)(M6,hl )
(Fn8)(6,h,j)
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
h (ft)
T (oF)
M
=
0
M2
=
0.1
M 3 = 0.2
M4 = 0.3
M 5 = 0.4
M 6 = 0.5
1=0
T= 86
29227
26894
24747
22787
21013
19427
,=0
S= 0
T 2 = 99
T 3 =111
T4= 122
28314
27471
25981
25138
23834
21874
22991
21031
20101
19258
18514
17671
26699
24366
22219
20259
T 5 = 82
T 6 = 95
T 7 = 107
T 8 = 118
29508
28595
27752
26980
27174
26262
25419
24647
25028
24115
23272
22500
23068
22155
21312
20540
T9 = 79
To = 92
T11 = 103
T 12= 114
29718
28806
27385
26472
25238
24326
23278
22366
18486
21294
20381
19539
18766
21505
20592
16899
19707
18795
17952
17180
19918
19005
28033
27261
25700
24927
23553
22781
4=3000
T 13 = 75
29999
27666
25519
4=3000
=
T 14 88
T5 = 100
T16 = 111
T17 = 72
T 18 = 84
29086
28244
27471
30210
29367
26753
25911
25138
27877
27034
24606
23764
22991
25730
24887
21593
20821
23559
22646
21804
21031
23770
22927
19820
19047
21786
20873
20030
19258
21996
21154
18233
17461
20199
19286
18444
17671
20410
19567
T 19 = 96
5=4000
h4=4000 -T, = 107
28525
26191
25419
24045
23272
22085
21312
20311
19539
18724
1-795
h,=0
2=1000
2=1000
2=1000
h2=1000
3=2000
3=2000
3=2000
h3=2000
4=3000
h4=3000
h5=4000
h=4000
?77S2
Table 3.2 Continued.
6=5000
6=5000
6=5000
h6=5000
7=6000
T21 = 68
T 22 = 81
30491
29578
28157
27245
26011
25098
24051
23138
22277
21365
20690
19778
T 23 = 93
28735
26402
24255
22295
20522
18935
T 24 = 104
27963
25630
23483
21523
19750
18163
T25 = 64
30772
28438
26292
24332
22558
20971
77
29859
27526
25379
23419
21645
20059
7=6000
T 26 =
7=6000
T 27 = 89
T 28 = 100
29016
26683
24536
22576
20803
19216
28244
25911
23764
21804
20030
18444
8=7000
8=7000
8=7000
8=7000
T29 = 61
T30 = 74
T31 = 86
30982
30069
29227
28649
27736
26894
26502
25590
24747
24542
23629
22787
22769
21856
21013
21182
20269
19427
T 32 = 97
28454
26121
23975
22014
20241
18654
9=8000
9=8000
31263
30350
28930
28017
26783
25870
24823
23910
23050
22137
21463
20550
9=8000
T33 = 57
T34 = 70
T35 = 82
29508
27174
25028
23068
21294
19707
9=8000
T 36 = 93
28735
26402
24255
22295
20522
18935
1o=9000
1O=9000
ho=9000
1o=9000
T 37 = 54
31474
29141
26994
25034
23260
21674
T 38 = 67
30561
28228
26081
24121
22348
20761
T 39 = 79
29718
27385
25238
23278
21505
19918
T 40 = 90
28946
26613
24466
22506
20733
19146
11=10000 T 41 = 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
h7=6000
The coefficients for Eq.(3.4), k5, k6 , k7, and k8, are computed by formulating the matrices A2
and B2 as follows,
1 M1 M
2
(FnI)(M
1,hl,T1 )
1 M2 M
(F /I )(M 2 ,hl,T )
1
1 M 6 M6
(Fn/ )(M 6 ,hl,T1)
(Fn/ )(MI,hl,T2)
1 M1 M
(3.8)
(Fn6)(M6,h,,T2)
1 M6 M
1 M
M 2 T2
(F /8)(Mh T
(n
(M1h1,T4)
(Fn16)(M6,h
l P T 44)
1 M 6 M6
and then solving Eq.(3.9) below for k 5, k6, k7 , and k8 .
( A 2TA 2 )-1 (A2 TB 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 0.1
Mach 0
A
1.3
..
1.25
0o1.3
1.2
2 1.2
1.15F
1.05[ n=3
h=0
I--o
0.95
0 0.9
0.9
0.8
0.85
0.8
-100
-50
0
50
Temperature (F)
100
I
0.71
150
-100
-50
0
50
Temperature (F)
-•
Mach 0.2
100
150
Measured Dat
MIT
INM
Mach 0.3
1.1
1.15r1.1 I
1.05
0
C 1.05 F.
cc
0.95
1
0.95
0.9 h
fln ft
0.9 I.
S0.85 h
2
2
h
1 0.8
0.85 h=0
z
1
0.8
z
3 0.75
0.75
0.7
h
U
0.7 I
0.65 L
-100
0.65
-50
0
50
Temperature (F)
150
0.6'
-100
-50
0
50
Temperature (F)
100
150
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.
T 0.8
O
h=6500J
0.75
h=25QgJ
0.7 h=5&H
I-
0.85 h=650.J
0.8 h=9500i
h=5
- 0.75 h=5QSt
0.7
70.65
r 0.65
0
0.6
0.6
0.55
0.55
-100
-50
0
50
Temperature (F)
100
0.5
-100
I
-50
50
0
Temperature (F)
100
150
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
.
0.75
R .~
h=9
0.7 h=95
h=6500i
0.7
h= 250f
0.65 h=50ff
z
2
0.65 h=
h=2500f
6o.h=SnL-.
0.6
z
' 0.55
0.55
0.5
0.5
0.45
0.45
-100
|
-50
II
I
0
50
Temperature (F)
I
100
150
0.4
-100
I
-50
I •
II
0
50
|
100
I
150
Temperature (F)
Figure 3.6 Climb thrust comparison for small commercial airplane at various conditions.
Mach 0
Mach 0.1
o
1
S1.2
L
I
1.1
; 0.9
0.8
(3-0.7
r
0.6
-100
50
0
Temperature CF)
,
i
-50
l
i
-
....
Mach 0.2
100
150
Measured Dat
-
.
l
0
50
Temperature CF)
MIT
INM
Mach 0.3
_
U 1.2!~
2
IE
W0.9Iz
0.8
0
0
0.7 [
0.61
-100
I
-50
I
0
50
Temperature (F)
I
100
150
0
50
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 ff0.75
h=50Q.Otf
0.7
\
..
0.65
0.6
0.55
0.55 L
-50
0
50
Temperature
100
150
0.5 L
-50
0
CF)
50
100
Temperature CF)
-
Mach 0.4
a
150
Measured Dat
-MIT
S INM
Mach 0.5
0.8
~ 0.75
0.7
5
2 0.65
z
0.6
8
0.55
0.45 L
-50
0
50
100
Temperature CF)
150
0
50
100
Temperature CF)
150
Figure 3.8 Climb thrust comparison for medium commercial airplane at various conditions.
Mach 0
Mach 0.2
0
50
Temperature (F)
0
50
Temperature (F)
-....
Mach 0.3
0.4 '
-100
-50
0
50
Temperature (F)
100
15
Measured Data
MIT
INM
Mach 0.4
100
150
0
50
Temperature (F)
Figure 3.9 Takeoff thrust comparison for large commercial airplane at various conditions.
Mach 0.1
2 .
.
Mach 0.3
.
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
80.5
0.6
n r;l
0
50
Temperature (F)
-100
100
1
0.4
150
-100
-50
0
50
Temperature (F)
100
150
Measured Dat
- - MIT
.... INM
Mach 0.5
Mach 0.4
1
S0.9
g 0.9
F.
2
2
0.8
0.7
2
I-
z 0.8
z 0.6
z 0.6
80.5
8 0.5
0.4 '
-100
1
-50
0
50
Temperature (F)
100
150
0.
- 100
-50
50
0
Temperature (F)
100
150
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.
MIT Model
INM Model
0 ft
3000 ft
6000 ft
9000 ft
0 ft
3000 ft
6000 ft
9000 ft
Least Squared Error in Corrected Net Thrust (lb)
Mach 0.3
Mach 0.2
Mac 0.1
Mach 0
68.7801
62.1422
52.2994
76.7736
29.0389
22.2196
35.1147
62.0462
41.5644
27.7817
28.7856
69.2990
27.3985
24.9302
43.8860
31.8990
596.6737
648.0785
656.0399
634.7971
636.0905
666.7642
653.7065
631.2608
636.3030
659.7622
645.7621
650.9106
684_7760
691.0583
661.6280
665.3742
Table 3.4 Error in corrected net thrust during climb for small commercial airplane.
Least Squared Error in Corrected Net Thrust (lb)
Mach 0.5
Mach 0.4
Mach 0.3
Mach 0.2
48.0301
23.8526
27.3014
20.7569
31.1475
23.2439
24.9581
22.7564
22.4292
29.1363
24.8293
28.9597
26.7810
30.0177
24.1102
30.3556
360.7722
627.2996
669.5469
703.9612
404.2555
650.9640
674.8251
693.5819
482.0347
674.8790
676.1206
665.3673
500 ft
MIT Model
2500 ft
6500 ft
9500 ft
500 ft
INM Model
2500 ft
6500 ft
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.
MIT Model
INM Model
0 ft
5000 ft
7920 ft
0 ft
5000 ft
7920 ft
Least Squared Error in Corrected Net Thrust (lb)
Mach 0.3
Mach 0.2
Mac 0.1
Mach 0
165.1217
344.6927
392.9152
306.7909
101.2559
104.1779
173.6205
342.2574
100.0924
72.1704
161.8416
396.7150
534.55
924.19
1209.47
1224.11
573.20
805.73
946.14
894.78
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)
MIT Model
INM Model
500 ft
9500 ft
500 ft
9500 ft
Mach 0.2
Mach 0.3
Mach 0.4
Mach 0.5
20.9499
52.6467
809.3244
7653625
118.9694
131.6875
968.9295
961.642&7
82.6470
44.9727
856.5244
932.9404
82.8706
85.9502
836.8706
959.2255
Table 3.7 Error in corrected net thrust during takeoff for large commercial airplane.
0 ft
MIT Model
INM Model
4000 ft
10000 ft
0 ft
4000 ft
10000 ft
Least Squared Error in Corrected Net Thrust (lb)
Mach 0.4
Mach 0.3
Mach 0.2
Mach 0
303.8417
256.5289
273.5680
391.3044
200.0846
186.5013
192.9940
268.9983
134.6785
150.8482
157.2474
206.6931
1768.62
1626.52
1630.45
1766.11
1532.73
1370.69
1350.46
1434.02
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)
MIT Model
INM Model
Mach 0.1
Mach 0.3
Mach 0.4
Mach 0.5
0 ft
326.5073
100.7679
150.0685
4000 ft
8000 ft
170.5130
154.4417
92.2014
212.2085
85.7651
135.9459
184.0319
199.4702
182.5523
12000 ft
209.5427
311.1647
207.1063
157.5070
0 ft
1889.59
1886.91
1881.04
1722.34
1570.81
1484.26
1898.43
1891.55
1891.40
1999.64
2034.95
2102.42
1610.29
1914.09
2129.19
2251.60
4000 ft
8000 ft
12000 ft L
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]
(Fnarg
/8)avg
1
siny
N (W/8)avg [R + 1.03in
1.03
(SAE Eq.A15)
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
P
Thrust, F
Drag,D
True Airspeed,V t
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 steadylevel flight,
p V 2 CD S
(4.1)
1 p Vt2 CL S
(4.2)
F= D 2
W
=
L
2
Combining Eq.(4.1) and (4.2) leads to the level-flight thrust as a function of weight and dragover-lift ratio,
F = WR
(4.3)
If the aircraft weight is treated as a constant, the minimum thrust occurs at the point where the dragover-lift ratio, R, is at minimum. This characteristic will be utilized in the development of levelflight thrust.
4.3 Drag Polar
The aerodynamic characteristics, CL and CD , for a conventional aircraft exhibit a quadratic
relationship of the form,
CD
=
k o + 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 + k2
CL
C
L
(4.5)
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
k2d
(4.6)
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.
CDRm = k o + k,
R
CLRmm
+ k2
2LR
R
(4.7)
ko
+
CL
min
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
C,
+ k CL
(4.10)
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
CD
1
R22
C DR i
2
CDR
2
2
2C
CDR
CL
1 CDrm
2 CL
CL
2
(4.11)
CU
CLRnun
CL
CLR)
(4.12)
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
P-
(4.14)
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
ChDhl00
Re
Reh=0
CDh=10000
Reh=10000
"66
1
Ph=1000
Ph==1000
6
6 =
0.8721 = 1 - 0.1279
(4.15)
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
F = W(
k
CL
+k
+
k2 C L )
(4.16)
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.
F = W(
ko Po o V2 S
+k
2W
2k 2 W
+
Po a Vt2 S
)
(4.18)
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
W
=CR
m
R"
(4.20)
U
and
CR
CR
2
PO CLr = S
(4.21)
Substituting drag polar Model I, Eq.(4.11), and Eq.(4.20) into Eq.(4.19) yields,
+
V2
12
2 Rn
F
2
(4.22)
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
CL
VtR
(4.23)
Vt
CLR
Substituting Eq.4.23 into Eq.4.22 yields the expression for level-flight thrust
F
F
oj1
R
2
an
+
Vt
CRm
CR
(4.24)
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
F= 1
2
(1 + Eh)Vc
CRM
[1
+ W2[
C Rnh)
(1 + Eh)
V
(4.25)
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 [ Rmi n + 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 ' C2
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 dragover-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.
50 Flap
00 Flap
k
d
21 -
20
\O \0
6
\O
A&AA\AA \O
22 F
16 150
200
250
Calibrated Airspeed, V (knots)
85% MGTOW, 5000 ft
150 Flap
300
18'
120
,o7
I
I
0O Measured Data
MIT Model I
- - MIT Model II
INM
I
220
300 Flap+Gear
1
34 F
(
140
160
180
200
Calibrated Airspeed, V (knots)
35.5-
3235O
30-
0
Q
'5
28
/
34.5 -
)0
0//
0
0
34
\
o
9
\
9/
\
A
24 -
22'
120
A
\\
AA
33.5 I
AAAAA
140
160
180
Calibrated Airspeed, V (knots)
200
33'
130
140
150
160
Calibrated Airspeed, V,(knots)
Figure 4.2 Thrust ratio per engine vs. velocity for small commercial jet with various flap settings.
50 Flap
00 Flap
30 I
200
400
300
Calibrated Airspeed, V(knots)
85% MGTOW, 5000 ft
500
150
/A
25
A 300A
250 A
200
Calibrated Airspeed, V (knots)
A
A
AA
350
O0 Measured Data
MIT Model I
- - MIT Model II
INM
300 Flap+Gear
200 Flap
4847.5
47 -
8
46.5
I-
46_
.455
26
150
200
250
300
Calibrated Airspeed, V(knots)
350
160
180
Calibrated Airspeed, V (knots)
200
Figure 4.3 Thrust ratio per engine vs. velocity for large commercial jet with various flap settings.
4.5.2 ErrorAnalysis
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+0.85xMGTOW)
0.9xMGLW
O0Flap
50 Flap
150 Flap
300 Flap+Gear
25.23
65.05
50.64
50.23
2
27.10
70.04
54.47
40.50
Model II Average Least Squared Errors (lb)
(0.9xMGLW+0.85xMGTOW)
0.9xMGLW
IF
0.85xMGTOW
33.35
75.04
58.30
43.28
0.85xMGTOW
2
00 Flap
50 Flap
150 Flap
300 Flap+Gear
28.06
33.31
42.26
15.69
30.15
35.92
45.45
12.98
INM Average Least Squared Errors (lb)
(0.9xMGLW+0.85xMGTOW)
0.9xMGLW
26.26
38.54
48.64
13.76
0.85xMGTOW
2
00 Flap
Flap
150 Flap
30o Flap+Gear
50
51.09
97.11
168.07
90.14
54.97
104.47
180.81
72.80
74.01
111.83
193.55
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+0.85xMGTOW)
0.9xMGLW
0.85xMGTOW
2
0OFlap
50 Flap
200 Flap
°
30 Flap+Gear
284.52
43.07
130.58
27.30
328.69
49.26
146.46
30.52
Model II Average Least Squared Errors (lb)
(0.9xMGLW+0.85xMGTOW)
0.9xMGLW
278.10
55.38
102.22
33.50
0.85xMGTOW
2
00 Flap
Flap
200 Flap
300 Flap+Gear
50
180.27
114.76
233.15
113.86
205.51
132.04
268.10
130.70
INM Average Least Squared Errors (lb)
(0.9xMGLW+0.85xMGTOW)
0.9xMGLW
97.51
149.29
138.49
147.40
0.85xMGTOW
2
0O Flap
50 Flap
200 Flap
300 Flap+Gear
654.96
681.48
318.38
226.10
754.59
785.14
366.81
260.50
635.80
888.81
292.10
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 minimumthrust 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
(W/8) ag
- R]
(SAE Eq. A8)
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].
Sc
Ah
tan
tan y
(SAE Eq. A9)
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.
w dV,
F - D - Wsiny -dV,
g dt
(5.1)
L - Wcosy = 0
(5.2)
Combining Eq.(5.1) and (5.2) yields the expression for the flight path angle without headwind.
sin y =
F
-- R
W
(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.
sin y2
V, sin y1
=
( V cos y,
- V )2 + ( V sin y
)2
(5.4)
After small angle approximation, Eq.(5.4) becomes
sin y 2 =
V - V
sin y,.
(5.5)
Combining Eq.(5.3) and (5.5) forms the equation of flight path angle,
Vt
sin y
V
Vt
1 +
V
F
- R]
W
Vt d V
(5.6)
g dh
where the term in front of the bracket in Eq.(5.6) above is referred to as flight path angle correction
factor, .
1
Vt
Vt - VWw
1+
V dV,
g dh
(5.7)
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
dh
1 dT
To dh
(5.8)
dT
a dh
d6
dh
-g
RTa
(5.9)
66 g
o
dT
a dh
do
dh
-6 (g
TO
6
R
1 dT
0 dh
g
a
o
(5.10)
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
d - (
dh
1
)V
0V
2
do
d
(5.11)
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,
V, - Vw
Vt2
2gt
2gT
dT
a dh
1 dT
gR 6g +- 0
Ra
dh
(5.12)
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,
-1
1 dV d6
)V o
+(
2
e
J- d6 dh
dV,
dh
-3
2 do
(5.13)
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 Po
d6
[(
1
k
k
+ -)(6
+
1
(5.14)
-1]
)Ya
6
ya
Ve
After combining the relevant derivatives, the value of dV/dh during a constant calibrated
airspeed climb is given by
R -dT
dV,
dh
a
Vt
1 dT
0 dh
2 To R a
gO
Ya
ya -
Vt
dh
g
1
k
[(1+)(
ya 6
k +1)
6
Ya
(5.15)
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,
Vt -
1
(5.16)
Vw 1 +
where
2
g
-V 2gT ( Ra
R
dT
a dh
Radh
+
dT
0 dh
06
Y
Ya-
1
g
-1
[(1 +
k
Ya 6
+ 1)Y
1] (5.17)
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
dh
-
[
(1 + Eh) do
d
20
dh
-
(5.18)
After combining the relevant derivatives, the value of dV/dh during a constant calibrated
airspeed climb is given by
SdT
dV t
dh
V
IdT)]
+ (1 + ch)(
g
2 To
Ra
(5.19)
+1
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
(1
1+
(1
+
2
+ Eh)V
go
RdT
adh
+
[E +
(1 + eh)
2T
(
Ra
6
g
dT
+
(5.20)
+ - -)]
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
00.95
Q 0.95
0.9
CD
L'-
0.9
C-
0.8515
u- 0.85
15 50
160
170
180
210
200
190
Calibrated Airspeed, Vc (knots)
220
230
240
250
6 1.05
~------
C
c1)
-
=-
0
, _Vc=1 50
"
knots
c=200 knots
0.95
Vc=250 knots
Exact Model
Simplified Model
-
I
I
I
t
2.5
CI
Altitude, h (ft)
4
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 = 10000 ft
h = Oft
1.05
1.04
150 knots
-,Vc=
1 U,-,-,
0
I-
CU
0
0
UL 1.02
C
1.01
1
S0.98
a
o
U-
o
-Vc=200 knots
1
,
-C
< 0.99
S0.96
0
0.98
._U LI,
E 0.94
.9 0.94
n-
='
U.9
Vc=250 knots
0.96
0.95
-5 0
0
50
Temperature at Sea Level Airport
0.91
-50
1
0
50
100
Temperature at Sea Level Airport CF)
F)
h = 20000 ft
1,
-
- kn
Vc=200 knots
0.95
0.9-
-50
Exact Model
Simplified Model
.Vc=250
0
knots
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
8
U,
.2
0.95 ---------Vc=250 knots
0
-c
D
0.9
-
0.85
1.5
1
Altitude, h (ft)
2
2.5
x 104
0
0.5
1.5
1
Altitude, h (ft)
2.5
2
x 10
Figure 5.5 Comparison of flight path angle correction factor between MIT and INM models.
4
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 twopoint-average approximation is to calculate the average climb angle
Yavg
avg
+
2
th2
(5.21)
and then substitute into Eq.(5.22) below to obtain the ground distance traversed during climb.
Sc
h2 - h1
2
1
tan Yavg
(5.22)
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].
S =
(1/2g) (0.95) (Vtb - V )
a[N(F,/)avg/(W/)ag]
- Ra vg
(Vtz/Vta
(SAE Eq.A10)
)
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
Vz
dV t
1+SV t
g dh
(
F
W
(6.1)
-R) V,
Rearranging Eq.(6. 1) and solving for dV/dh in terms of the known quantities.
dVt
Vt
dh
V-
g( F
VwVtz W
-R)-
g
V.
(6.2)
Then, applying Eq.(6.2) to calculate the final altitude and ground distance traveled,
dV
h2
=
h1
+ (Vtb
Vta)(
dh )avg
V
dV
where
and
SdVt
(
dh
dh (Vtb'h
)ag
vge
(6.3)
dV,
2)
h2 -
h1
tan Yavg
where
+ sin_
Vtz
sin-
Vtb(6.4)
2
Yavg =
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
-
(6.5)
Va)
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.
true airspeedand altitude,
!Estimate the finalusing
Eq.(6.5) and (6.6)
Vtb(est) and h2(est),
)
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.
Initial Climb
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)
0.35
9.8889
3.4
65.492
0.7
10.1869
5.1
80.3112
Altitude: 3000 ft
Initial Climb
Altitude: 6000 ft
Table 6.2 Error in altitude gain and ground distance traversed for the large commercial airplane.
Initial Climb
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)
0.05
5.7836
0.9
64.8086
0.085
6.7361
1.15
80.5594
Altitude: 3000 ft
Initial Climb
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-pointaverage 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 Distance (ft)
Ground Distance (ft)
x 104
x 10
x 10
0-
1.45
-
1.4
1
Delta
-- MIT
INM
1.3
S1.25
!
1.2
1.15
1.1
1.05'o
5
10
Ground Distance (ft)
15
x 104
-5'
-15
-10
-5
Figure 7.1 Flight profile and LAMAX noise contour for Case (1).
5
0
SLD (nmi)
10
15
15-
S10-
5-
0
-5
-10
-30
-20
-10
Figure 7.2 SEL noise contour for Case (1).
0
SLD (nmi)
10
20
30
8000
7000
6000
S5000
> 4000
- 3000
2000
1000
Ground Distance (ft)
Ground Distance (ft)
x 104
104
1.45
1.
!
5
55 B
"
•
3--0 Delta
MIT
INM
1.35
25-
x 10'
I-.,ta
|
-n5B
INM
Delta
MIT
MIT
INM
1.3
1.25
20
1.2
1.15
115
1.1
1.05
i
10a
S5
0.95
0
.
10
5
Ground Distance (ft)
15
x 10'
CI
-15
I
-10
I
-5
Figure 7.3 Flight profile and LAMAX noise contour for Case (2).
88
I,
I
5
0
SLD (nmi)
I
10
15
Delta
SIl'a
55 dB
MI
30
INM
65.dB
25
20
85.dB
S15
uslo-
0-5-
-10
-30
-20
-10
Figure 7.4 SEL noise contour for Case (2).
0
SLD (nmi)
10
20
30
Ground Distance (ft)
1.6
10
5
Ground Distance (ft)
4
x 10
30
"
d
0-0 Delta
13-E MIT
INM
1.55
1
15
x 10'
-
-
25
Delta
MIT
INM
w 1.45
1.4
20
F 1.35
11.3
1.3
8585 d
E 5
1
1.25
010
1.2
0
5
10
Ground Distance (ft)
15
x 104
0-
-5 -15
-10
-5
Figure 7.5 Flight profile and LAMAX noise contour for Case (3).
90
5
0
SLD (nmi)
,
10
15
S15
I
10
5-
0-
-5
-10
-30
-20
-10
Figure 7.6 SEL noise contour for Case (3).
0
SLD (nmi)
10
20
30
t
4000
Ground Distance (ft)
S1.
" '^MIT
4
F
x 104
Ground Distance (ft)
x 10
- -
25
INM
m
INM
1.35
20
1 1.3
1.25
C
1.15
0
5
Ground
Distance10
(ft)
x 1015
0
-5
-15
-10
-5
Figure 7.7 Flight profile and LAMAX noise contour for Case (4).
5
0
SLD (nmi)
10
15
Delta
I AIr
30-
MI I
- - -
..
INM
5 dR.
25
85 dB
20-
S15S10
0-
-5
-10
-30
i
-20
i
-10
i
i
0
SLD (nmi)
10
20
30
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)
LAMAX
SEL
Case (2)
55 dB
65 dB
85 dB
55 dB
65 dB
85 dB
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
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 (4)
Case (3)
LAMAX
SEL
55 dB
65 dB
85 dB
55 dB
65 dB
85 dB
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
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 (2)
Case (1)
LAMAX
SEL
55 dB
65 dB
55 dB
65 dB
MIT
-0.5476
-0.5219
-0.1172
-0.1486
INM
-0.8219
-0.7964
-2.2007
-2.3433
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 (4)
Case (3)
LAMAX
SEL
55 dB
65 dB
55 dB
65 dB
MIT
-0.6678
-0.6713
-0.4725
-0.4913
INM
-0.8329
-0.839
-2.0981
-2.2474
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
211
14000
3000
I
2000
1000
Ground Distance (ft)
Ground Distance (ft)
x 104
x 104
E
ii
I
I~
S
P
0
4
Ground Distance (ft)
8
x 10
-4
-2
Figure 7.9 Flight profile and LAMAX noise contour for Case (5).
0
SLD (nmi)
2
4
0-
5--15
-10
-5
Figure 7.10 SEL noise contour for Case (5).
0
SLD (nmi)
7000
6000
15000
>4000
3000
2000
1000
0
Ground Distance (ft)
2
4
x 10
IC~-------
4
6
Ground Distance (ft)
8
x 10
--
-....
14
- Delta
MIT
INM
12
4
Ground Distance (ft)
8
x 10'
-4
-2
Figure 7.11 Flight profile and LAMAX noise contour for Case (6).
0
SLD (nmi)
2
4
Delta
MIT
INM
15-
-15
-10
-5
0
SLD (nmi)
5
10
1
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 (6)
Case (5)
LAMAX
SEL
55 dB
65 dB
85 dB
55 dB
65 dB
85 dB
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
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)
55 dB
LAMAX
SEL
j
Case (6)
65 dB
55 dB
65 dB
MIT
-0.1043
-0.0746
0.0823
-0.0057
INM
-0.1465
-0.0442
0.158
0.1269
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
4
Ground Distance (ft)
MIT
-....
12
INM
8
x 10
-2
Figure 7.13 Flight profile and LAMAX noise contour for Case (7).
103
0
SLD (nmi)
2
4
Delta
MIT
INM
15is
65d
10-
85
-51
-1 0)
aI
1
i
-8
I
-6
I
-4
I
-2
.
I
I
I
I
2
4
6
8
1
0
SLD (nmi)
Figure 7.14 SEL noise contour for Case (7).
104
10
6
4
Ground Distance (ft)
8
0
10
x 104
8
4
6
Ground Distance (ft)
2
10
x 104
55 dB
-
16
Delta
MIT
INM
-
14
12F
0
2
4
6
Ground Distance (ft)
8
10
x 104
ii
-4
-2
|
|
0
2
SLD (nmi)
Figure 7.15 Flight profile and LAMAX noise contour for Case (8).
105
4
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 (8)
Case (7)
LAMAX
SEL
55 dB
65 dB
85 dB
55 dB
65 dB
85 dB
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
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 (8)
Case (7)
LAMAX
SEL
55 dB
65 dB
55 dB
65 dB
MIT
-0.2863
-0.1758
-0.283
-0.2528
INM
-0.8286
-0.4603
-1.0807
-0.6576
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