BOEING
Flight Operations Engineering
DS-1420
Seventh Edition
May 1989
PREFACE
The intent of this manuał is to provide the user with a
comprehensive source of infonnation relating to the methods
used in certifying Boeing jet transpon airplanes, and in
computing their performance.
Every reasonable effon has been made to ensure the
correctness of this manuał, and to make it representative of
industry methods being practiced at the time of writing.
Nevertheless, the Boeing Company takes no responsibility
for any errors which may arise from the use of this
document.
06-1420
May 1989
TABLE OF
CONTENTS
SECTION 1
AERODYNAMICS
1-1
1-2
1-3
1-4
1-5
1-6
1-7
1-8
1-9
l-10
1-11
1-12
1-13
1-14
1-15
1-16
1-17
1-18
1-19
1-20
1-21
1-22
D6-1420
May1989
General
Fluids at Rest
Fluids in Motion
Streamlines and Streamtubes
Continuity Equation
lncompressible Bernoulli Equation
Equation of State
Fluid Viscosity
Speed of Sound
Compressible Bernoulli Equation
Flow Relations Near the Speed of Sound
The Atmosphere
Variation of Temperatme with Altitude
Variation of Pressme with Altitude
International Standard Atmosphere Table
Non-Standard Day Conditions
Pressure Distribution
Force Equation
Airfoil Properties
Physical Properties
Aerodynamic Properties
Viscosity Effects
Lift and Drag
Drag Analysis
Parasite Drag
Induced Drag
Compressibility Drag
Planform Effects
High-Lift Devices
High-Speed Flight
Airplane and Engine Parameters
1.1
1.1
1.3
1.5
1.7
1.9
1.15
1.17
1.19
1.23
1.25
1.33
1.34
1.35
1.40
1.41
1.43
1.49
1.53
1.53
1.59
1.65
1.69
1.75
1.76
1.79
1.87
1.97
1.101
1.105
1.109
1
'!'ABLE
OF
CON'l'EN'l'S
SECTION 2
POWERPLANT
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
2-10
2-11
ii
Jet Reaction Engines
The Basic Jet Powerplant
Turboprop Engine
Turbofan Engine
Propfan Engine
Engine Station ldentification
Engine Parameters
Engine Efficiency
Pressure-Volume Cycles
Component Design
Air Inlet and Diffuser Design
Air Compressor Design
Combustion Chamber Design
Tmbine Design
Exhaust Nozzle Design
Ratings
Temperature and Velocity Variations
Bypass and Turbofan Engines
Factors Affecting Thrust
Effect of Pressure
Effect of Airspeed
Effect of Ram
Effect of Temperatme
Effect of Humidity
Effect of Water lnjection
Effect of Altitude
Effect of RPM
Engine Calibration
Development of Theoretical Gross Thrust Function,'I',
for a Convergent Nozzle
Method of Analysis for Turbojet Engines
Method of Analysis for Tmbofan Engines
(Dual Exhaust)
Method of Analysis for Ducted-Fan or Bypass Engines
(Single Exhaust)
Effect of Flight Inlet lnstallation on Performance
2.1
2.1
2.2
2.3
2.3
2.5
2.7
2.27
2.31
2.35
2.35
2.37
2.53
2.57
2.66
2.69
2.71
2.73
2.81
2.81
2.81
2.82
2.83
2.84
2.85
2.86
2.87
2.89
2.89
2.95
2.96
2.97
2.99
D6-1420
May1989
TABLE OF
CONTENTS
SECTION 3
PERFORMANCE
3-1
3-2
3-3
Airspeed Measure1nent
Terminology
Pitot Static Systems
High Speed Position EITor- Mach Number
High Speed Position Error -Airspeed (Flaps Up)
Low Speed Position Error - Airspeed
Compressibility Correction
Ground Effect Airspeed Position Error
Altitude and Te1nperature Measure1nent
High Speed Position Error - Altimeter
Low Speed Position Error - Altimeter
Tem.perature Correction
Temperature Indication Calibration
Total Tem.perature (Reference) System Calibration
Total Air Tem.perature (TAT) Indicator System
Static Air Tem.perature Indicator System
Airplane and Engine Grid
Thrust Required
Thrust Available
3-4
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May1989
Engine Fuel Flow Grid
Flight Li1nitations
Minimum Speeds
Stall Speed Determination
1-g Stall Speeds
Buffet Limits
Stall Waming
Maximum Speeds
Ceilings
Structural Limitations
V-nDiagram
Maneuvering Envelope
Gust Envelope
Composit V-n Diagram
3.1
3.1
3.2
3.5
3.7
3.8
3.12
3.13
3.15
3.15
3.17
3.18
3.19
3.19
3.20
3.21
3.23
3.23
3.25
3.26
3.39
3.39
3.41
3.45
3.47
3.58
3.58
3.60
3.60
3.62
3.63
3.63
3.69
iii
TABLE OJ' CONTENTS
SECTION 3
PERFORMANCE
3.5
Takeoff
General Takeoff Distance Equation
Norma! Takeoff Ground Run Distance and Tnne to VR
Takeo:ff TlDlC
Takeoff Speeds Definitions
Rotation and Rare to 35-foot Height
One-Engine-Inoperative Takeoff
Balanced Field Length
Unbalanced Field Length
Oearways and Stopways
FAR Field Length
Takeoff Presentation
Wind Corrections
MultipleV1
Thrust Presentation
Climb
General
3-6
RateofOimb
Maximum Rate of Oimb
One-Engine- lnoperative Oimb Performance
Oimb Gradient
Tuming in the Flight Path
FAR Minimum Performance Requirements
lmproved Oimb Performance
Enroute Oimb Speeds
Rate of Oimb and Ceiling
Oimb Time, Fuel and Distance
Thrust Presentation
Range
Range Equation
Turbojet Range Factor
Types of Cruise
Range Presentation
Temperature Effects
3.7
IV
3.75
3.75
3.78
3.82
3.83
3.87
3.88
3.97
3.98
3.99
3.100
3.103
3.113
3.114
3.115
3.119
3.119
3.119
3.123
3.125
3.130
3.131A
3.132
3.138
3.138
3.144
3.144
3.148
3.151
3.151
3.154
3.155
3.161
3.164
D6-1420
May1989
'l'ABLE OF
CON'l'EN'l'S
SECTION 3
PERFORMANCE
3-7
Range (Continued)
Maximum Range at Maximum Cruise Thrust
Reynolds Number Effect on Fuel Mileage
Altitude versus Weight
Wmd and Off-Altinule Effects
Point-of-No-Rennn and Equal-Time Point
Engine-Inoperative Cruise
Driftdown
3-8
3-9
3-10
Endmance (Holding)
Thrust Presentation
Economy Operation
Economy Cruise
Optimum Economy Cruise Speed
Economy Cruise with FMC
Economy Oimb
Economy Descent
Descent
Rate of Descent - General
Descent Range, Tnne, and Fuel Used
High Rate Descents
Thrust Presentation
Approach and Landing
Approach Climb
Landing Climb
Approach and Flare
GroundRun
Wet FAR Landing Field Length
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3.170
3.171
3.173
3.174
3.175
3.178
3.180
3.183
3.185
3.189
3.189
3.198
3.200
3.203
3.204
3.207
3.207
3.211
3.212
3.213
3.215
3.215
3.215
3.216
3.219
3.221
V
'l'ABLE OF COH'l'EN'l'S
APPENDIX
Chans:
Chart 1, CAS vs True Mach Number
Chart 2, High Speed Position Error - Airspeed
Chart 3, Compressibility Correction
Chart 4, High Speed Position Error - Altimeter
Chart 5, Low Speed Position Error - Altimeter
Symbols, Constants, and Conversions
Formula - Basic Equations
Principal Dimensions
vi
Al
A2
A3
A4
AS
A7
A19
A45
1)6..1420
May1989
AERODYNAMICS
1-1 General
The purpose of this secti.on is to discuss some fundamental physical relati.onships associated with
the morion of bodies in air. To understand principles of flight and propulsion, it is necessary to
review certain laws goveming the behavior of fluids.
1-2 Fluids at Rest
There can be no shear on a liquid if the liquid is at rest. If any shear force exists, the liquid will
deform. Forces acti.ng on the fluid particles are normal to the surfaces and must be in equilibrium.
Mathematically, the sum of the horizontal and vertical forces must be zero, and may be expressed:
LFH =O }
I,F =O
(1)
V
Force per unit of area is pressure. This statement may be written:
p=
or,
r1
F=pA
1
(2)
A frequently used unit of pressure is pounds per square foot, since force is measured in pounds and
area in square feet. The French philosopher and mathematician, Pascal, formulated an important
principle which states: if gravity is neglected, the pressures at any point in a fluid must be equal in
magnitude in any directi.on. Any pressure increase in any part of the fluid results in an equal
increase in pressure throughout the fluid. The effect of such pressure changes is familiar in its
applicati.on in hydraulic systems of all types such as brakes, lifts, and presses.
When a finite difference in height exists, the weight of the fluid must be considered. In Figure l, a
rectangular tank containing a fluid has a height, h, and cross secti.onal area, A. The pressure on the
lower suńace of the tank is P1, and on the upper smface is P2· The weight of the fluid in the tank is
the product of the specific weight of the liquid, w, the height of the liquid, h, and the area, A.
Specific weight is defined as weight per unit volume.
Equati.on (I) may be written selecting the upward direction as positi.ve:
LFv=O= p 1 A-p A- whA
2
p A-p A= whA
1
2
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May1989
1. 1
AERODYNAMICS
(3)
Equation (3) assumes the fluid is incompressible; therefore w is constant. If the fluid is
compressible, w varies and the difference of pressures, dp, depends on how w varies with a
change in height, dh. Equation (3) for a compressible fluid is wńtten:
(4)
dp=wdh
T
h
1
P1
Figure 1.
A fluid may be either a liquid or a gas; the main difference between the two is in their resistance to
compression. Gases arc more easily compressed while liquids arc generally considered to be
incompressible. The primary fluid discussed in this section is air.
1.2
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AERODYNAMICS
1-3 Fluids In Motion
The most important physical laws goveming the morion of solids and fluids arc attributed to Isaac
Newton. These laws may be stated briefly:
( 1)
Every body continues in a state of rest or of uniform morion in a straight line unless it
is acted upon by an extemal force.
(2)
An accelerarion, proportional to the applied force, is produced in the direcrion of the
force.
(3)
Every action results in an equal reaction, opposite in direcrion.
The first and third laws may be apparent from physical experience; however, the second law is
perhaps less evident Mathemarically, the law may be stated as:
F ·oc a
(5)
where,
F
a
is force, lb
is acceleration, ft/sec/sec
Mass is the quanrity of matter of a substance or body, whether solid or fluid. The mass of a body
remains constant unless more matter is added or some is removed. Inertia is the propeny which
causes a body to continue in a state of rest or morion.
Mass, in addirion to being the amount of matter in a body, is a measure of its inertia, or resistance to
accelerarion. The constant of proportionality, mass, enables equarion (5) to be written in the form
of an equality.
(6)
F=Ma
When an object is .allowed to fall freely in a vacuum, an accelerarion. results from the unbalanced
force which is its own weight This accelerarion is commonly referred to as gravitational
acceleration, g. Equation (6) may be written for the freely falling body:
W=Mg
or,
w
M=g
l
(7)
1
The unit of mass is the sług. One sług is the amount of mass which weighs 32.174049 pounds
according to the intemarional agreement for the standard gravity condirion. The weight of a body is
the force with which the mass is being attracted toward the center of the earth. The unit of weight,
and force, is the pound (lb). Accelerarion bears the units offeet per second per second (ft/sec/sec),
as does g. For the purpose of most calculations herein, the value of gis 32.174049 ft/sec/sec.
It is sometimes helpful to ex.press mass in d.imensions other than slugs. From equarion (7):
06-1420
May1989
1. 3
AERODYNAMICS
M=
lb
= lb sec
ft /sec2
ft
2
(8)
Mass density is the mass per unit of volume and has the dimensions of slugs per cubic foot. It is
defined by the Greek letter Rho, p, and may be expressed more fundamentally using equation (8).
slugs
lb sec2
p=-;r= ft4
(9)
Mathematically, fluid mass density is also related to specific weight:
w
p =g-
1.4
(10)
1)6.1420
May1989
AERODYNAMI CS
1-4 Streamlines and Streamtubes
A stream of air consists of many particles moving in the same direction. The path of any one
particie is called a streamline. A collection of streamlines forming a closed curve or tube, as in
Figure 2, is referred to as a stream tube. Since the walls of the stream tube are streamlines, there
can be no flow through the wall.
Figure 2.
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1. 5
AERODYNAMI CS
Intentionally Left Blank
1.6
D6-1420
May1989
AERODYNAMICS
1-5 Continuity Equation
If a fluid is moving uniformly through a pipe or stream tube, as in Figure 3, the mass of fluid that
leaves the tube every second must be the same as that entering the tube. This is continuity of flow.
The law of continuity may be stated mathematically:
1
2
Figure 3.
The subscripts refer to particular stations of the tube.
The mass passing a section in one second is pAV slugs, or
M
1 =pAV
(12)
where,
M is mass, slugs
p
A
V
t
D6-1420
Mayl989
is mass density, slugs/ft3
is cross-sectional area, ft2
is velocity, ft/sec
is time,sec
1. 7
AERODYNAMI CS
In calculus, the formula for finding the differentia! of a product of three variables is:
d(xyz) = xy dz+ xz dy+ yz dx
(13)
and the differential of a constant is zero:
d(c) = O
(14)
Applying these formulae to equation (11):
pAd.V +pV dA+AV dp=O
. (15)
Dividing both sides of equation (15) by pAV results in another form of the continuity equation:
dV
dA
dp
-+-+_;=O
V·
A
p
(16)
If the fluid is incompressible, such that p remains constant, equation (11) becomes:
AV = constan t
(17)
Differentiating equation (17) and dividing by AV, an expression similar to equation (16) may be
. written for incompressible flow:
or,
dV
dA
-+-=O
V
A
(18)
(19)
Although the fluid (air) is compressible, this is often ignored in some aerodynamic considerations
for simplicity if the velocity is relatively low. Equation (19) shows that a positive change in area
(dA) results in a negative change in velocity (dV).
1.8
D6-1420
May1989
AERODYNAMICS
1-6 Incompressible Bernoulli Equation
The Bernoulli equation, a mathematical methcx:l of showing variation of pressure and velocity within
a streamtube, is of priine importance in fluid study. It is based upon the principle that when fluid
flows in a duet, as shown in Figure 4, its total energy at all points along the duet is constant. If the
interior is frictionless, the energy is of three forms; that due to height (potential energy), that due to
pressure (pressure energy), and that due to movement (kinetic energy). In practically all fluid
analysis work, horizontal flow is considered so changes in height are not considered.
V_.,.,_,
p
.
A
V+dV
p+dp
Figure 4.
Consider the tapeńng duet and an element with an average cross sectional area, A, as shown in
Figure 4. If the air is incompressible, the continuity equation as in equation (17) applies. With a
decreasing area, velocity, V, must increase after having traveled an incremental distance, ds, by the
amount dV. Also pressure, p, may be assumed to have increased by the amount dp. If the air is
accelerating to the ńght, the net force acting in that direction causing the acceleration can be found.
Repeating equation (6) for convenience:
F=Ma
For the average cross sectional area, the force to the ńght from equation (2) is:
F=pA
The counteracting force to the left is:
F= (p+dp)A
D6-1420
May1989
1. 9
AERODYNAMI CS
The net force to the right is then:
M- = pA - (p + dp )A = - dp A
(20)
The mass of air involved is the mass density multiplied by the volume, or:
M = pA ds
(21)
Acceleration is a change in velocity per change in time. The expression may be written:
a= dV
(22)
dt
Substituting equations (20), (21), and (22) into equation (6) results in:
-dp A= pA ds
cz
or,
dp=-p ds
CZ
(23)
By algebraic manipulation, certain terms may be rearranged:
ds dV = .!!!. dV
dt
dt
A change in distance per change in time is simply velocity:
ds dV-V dV
dt
Tuus equation (23) becomes:
dp=-pVdV
(24)
Again in calcultis, the formulae necessary to integrate this expression are:
J dx = x + constant
x2
f X dx = 2 + COnstant
1. 1 O
(25)
(26)
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Mayl989
AERODYNAMI CS
Applying these rules to equation (24), the equation becomes:
I dp =-pJ V dV
The solution to the equation becomes:
.
p=-p
v2
2 + constant
or in a more useful form:
p+p
v;
= constant
(27)*
In the derivation of equation (27) the p is considered constant. This is the incompressib1e flow
equation. (The asterisk is used to identify equations for incompressible flow).
Since p is the static pr~ure of the fluid, the quantities 1/2 pV2 and the constant must have the
same dimensions of pressure. The quantity 1/2 pv 2 is defined as dynamie pressure since it
embodies flow velocity. Dynamie and static pressures are the only pressures assumed to be in the
system; therefore, their sum is the total pr~ure.
Thus equation (27) becomes, in its most familiar form:
1 .;..2
Ps + 2PY-=pt
(28)*
Referring to Figme 4, the Bernoulli equation may be written as:
(29)*
Since the term 1/2 p V2 appears so frequently in fluid analysis, it is given the symbol q. Thus:
(30)
Equation (29) may a1so be written as:
(31)*
* Incompressible flow only
D6-1420
March 1991
1.11
AERODYNAMICS
A direct application of the Bernoulli equation is the pitot tube, which is used to measure air speed.
A schematic diagram of the instrument is in Figure 5.
TOTAL PRESSURE
OPENING ~
vo
o
..,_
----
/
STATIC PRESSURE
/ _ OPENING
"~==================~=::1
2
Figure 5.
The concentric tubes of the pitot are arranged so that the center tube is aligned in the direction of the
approaching air. As the air directly in line with the hole in the center tube reaches the opening it
must come to rest momentarily at point 1 before it nnns and proceeds around the sides of the larger
tube. As it passes along the outer tube it will finally attain essentially its original velocity, V0, at
some point, 2, where a hole is located. The subscript O herc represents the remote or free stream
condition.
The pressures acti.Dg at points 1 and 2 are used to measure the velocity, V0, by a pressure sensing
device. The device in Figure 5 is a manometer, which is merely a U-tube with a measured quantity
of fluid in_ it. One side of the U is connected to point 1 and the other to point 2. If the pressure at
point 1 is greater than at point 2 (as is the case) the fluid will be forced into the position shown.
The difference in the heights of the column, h, is a direct measure of the pressure difference
between points 1 and 2 F.quation (3) relates these quantities and is repeated herc for convenience:
where,
Ap is the pressure difference
w is the fluid density or specific weight
Writing Bernoulli's equation for points 1 and 2 with relation to a remote point, O:
1. 12
D6-1420
May1989
AERODYNAMICS
also,
From the previous explanations:
V =O
and,
I
v2 =Vo
Thus:
P1 =po+
tp~
and,
The manometer is set up to read the difference between p1 and p2, or Ap. Thus:
P1 -p2= L\p= (Po+ !"~)-Po
or,
1 2
Ap= 2PV o
(32)
Equation (32) is valid for incompressible flow only, for velocities less than 250 knots. Equating
expressions (3) and (32) shows the fluid height differentia! h, is proportional to the dynamie
pressme of the free stream 1/l p Va2- Specifically the fluid weight density relates the two:
wh=
1
2
2 pV 0
The pressme d.ifferential need not have been measmed by a manometer. This device is confined to
laboratory and wind tunnel work. The application in airplanes requires a mechanical pressure
measuring device using a diapbragm, bellows, or bourdon tube.
The Bernoulli equation is fundamental to all aerodynamic problems, particularly those involving a
discussion of pressure distributions on bodies immersed in a fluid. Add.itional applications of this
important principle will be made in later sections as will the development of the compressible flow
equation.
I>6-1420
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1.13
AERODYNAMI CS
Intentionally Left Blank
1. 14
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AERODYNAMI CS
1-7 Equation of State
The mass density of solids and liquids is essentially constant, 'but the mass density of gases
depends upon the pressure and temperature. The relation between the pressure, density and
temperature is:
p=pgRT
(33)
where,
p
p
g
R
is pressure, lb/ft2
is mass density, slugs/ft3
is gravitational acceleration, ft/sec2
is constant, ft lbf/lbm °R
is temperature,°R
T
The absolute temperature, T, is the temperature in degrees Rankine measured from absolute
zero, -459.67 oP. The expression for finding the absolute temperature is:
T = t+459.67
(34)
where, t, is the measured temperature in degrees Fahrenheit. R, in equation (33), is the gas
constant and is different for vańous gases. For air, the value of the constant is 53.352374 ft
· lbr/lbm °R. (The universal gas constant for all gases is 1545.43 ft lb/lb mole 0 R The molecular
weight of air is 28.966 lb/lb mole, and the quotient of the two is 1545.43/28.966 = 53.352374 ft
lbr/lbm °R). Equation (33) is sometimes wrinen:
pv = RT
(35)
where v is specific volume which is the reciprocal of specific weight, v = 1/w.
The gas constant for air of 53.3523742 ft lbff'lbm °R, as given above, is only valid for sea level
conditions where g equals 32.17405 ft/sec2. The equation of state, for a thennally perfect gas is:
p = pR' T
(36)
where R', the thermal gas constant equal to 1716.5619 ft2/sec2 °R, is valid at any altitude. R' is
equal to~.
This relationship, (equation of state), defines the state of a gas in terms of three variables, p, v, and
T. A process describes the relation between any two of the variables. Three such processes arc
readily apparent as a result of holding cach variable constant. These arc:
(1)
(2)
(3)
constantpressure (isobańc)
constant temperature (isothermal)
constant volume (isochoric)
In cach of the above processes, addition or release of beat is required. It is conceivable that in a
D6-1420
March 1991
1.15
AERODYNAMICS
fourth process no beat energy is added or lost. Such a process is called adiabatic.
Heat is energy in ttansition, and can be transformed. A BTU (British thennal unit) is a measure of
beat. It is defined as the amount of beat energy required to raise a one pound mass of water tbrough
one degree Fahrenheit at standard conditions. One BTU is equivalent to 778.26 ft lb of work,
usually assigned the letter J. It does not require as much additional beat energy to raise a pound of
gas one degree Fahrenheit as it does water. The ratio of the number of BTU's required to raise one
pound of gas through one degree Fahrenheit to that of water is called specific beat. When the
operation is conducted at constant pressure, allowing volume to vary, it becomes the specific beat at
constant pressure and has the symbol, Cp- For air G, = 0.239936923 BTU/lb/DF. If carried out at
constant volume, allowing pressure to vary, it becomes the specific beat at constant volume, having
the symbol Cv. For air, Cv = .171383516 BTU/lb/DF. The ratio of specific heats occurs frequently
in fluid analyses and is assigned the Greek letter Gamma, y.
For air:
. CP
.240
y= CV = .1714 = 1. 4
(37)
Of pńmary interest are the isothennal and adiabatic processes. When temperature is held constant in
equation (35), the isothermal process is represented and the equation becomes:
pv = constant
(38)
This is generally known as Boyle's Law. For the adiabatic processes encountered in aerodynamics,
the relationship becoroes:
1
or,
p; = constant }
1p = constant
(39)
Various forms of equations (38) and (39) will be discussed in subsequent chapters.
1.16
D6-1420
March 1991
AERODYNAMICS
1-8 Fluid Viscosity
In additi.on to the pressure, temperature, and density characteristi.cs of a fluid, compressibility and
viscosity properties (that which offers resistance to motion) must be considered.
The viscous property of a fluid asserts itself when a fluid is in motion relati.ve to a fixed surface.
When a layer of fluid touches a surface, its velocity is reduced to żero by f.ricti.on. Viscosity is the
result of shear forces acti.ng on the fluid, or the tendency of one layer of fluid to drag along the layer
next to it. There is a finite thickness of fluid next to a surface which is retarded relative to the
velocity farther away, or to the free stream velocity.
This retarded layer is .known as a boundary layer and is shown in Figure 6.
FREESTREAM
VELOCITY
Figure 6.
The length of the arrows represents the magnitude of velocity at that distance from the surface. A
measme of the viscosity of a fluid is represented in coefficient form by the Greek letter Mu, µ, and
bears the dimensions of slugs per foot second. The dynamie viscosity coefficient is assumed to vary
with absolute temperature according to Sutherland's equation:
13 T3/2
µ= T+S
D6-1420
March 1991
(40)
1.1 7
AERODYNAMICS
where,
µ is the dynamie viscosity coefficient, lb-sec/rt2
13=
µ 0(T 0 +S)
T 3fl.
o
=0.3125059(10-7), lb-sec
ft2-J°K
T is the absolute temperature, °I{
S is Sutherland's constant, 120°1{
2
The standard viscosity of air at sea level, ~. equals 3.745299 (10-7) lb-sec/ft at T0 equal to
288.15 °K. The viscosity of air is dil:ectly proportional to approximately the square root of the
absolute temperature.
Fluids in liquid form arc subject to viscous effects, but may be considered incompressible.
1.18
D6-1420
May1989
AERODYNAMI CS
1-9 Speed of Sound
Gases in addition to their viscous properties are highly compressible. When a disturbance
proclucing a change in pressure occurs at a point in a fluid, this disturbance is propagated through
the fluid in the form of a wave of the same type as a sound wave. Sound is the result of pressure
waves (compressions and rarefactions) in the fluid of such frequencies as to be audi.ble.
-~~ds--1•--V+dV
·---- p+dp
p+ dp
Advancing
Wave
Figure 7.
Assume there is a pressure wave moving to the right in the channel in Figure 7 at velocity V,
pressure pand mass density p. The flow has acquired increments in velocity, pressure, and density
downstream of the wave of dV, dp, and dp, respectively. The force due to fluid flow is:
F=Ma =M-dV
dt
(41)
Substituting equation (21) into equation (41) results is:
F=pA d s :
F- pA .!i!
dt dV
F=pAVdV
1>6-1420
May1989
(42)
1.19
AERODYNAMICS
The force due to pressure change is:
F=pA- (p+ dp)A
F=-dp A
(43)
Equating forces, equations (42) and (43), and simplifying:
-dp A= pAV dV
dp
p+VdV=O
(44)
From the continuity equation (11):
pAV = (p + dp)A(V + dV)
p V= p V+ p dV + V dp + dp dV
or,
dp
dV
-p+ - =
VO
(45)
Since the second order differential is qui.te small, it can be neglected.
From equations (44) and (45):
dp
dp V
dV=- p v = - - r
whereby,
V'= dp
dp
(46)
Equation (46) shows the speed of the pressure wave is related to the compressibility properties of
the gas. The reciprocal of equation (46), the rate of change of density with pressure, is sometimes
called the compressibility factor. Therefore, the greater the compressibility of a gas, the lower the
speed of wave propagation. Since a sound wave is an example of a pressure wave in air, the
quantity, a, is referred to as the local speed of sound and has the units of ft/sec.
a=f*
1.20
(47)
D6-1420
March 1991
AERODYNAMI CS
The change of state across the pressure wave is adiabatic; therefore, the quanti.ty dp/dp may be
determined from equation (39):
p = constant p 'Y
and,
: =constant y p 'Y- l
therefore,
dp = (.Ł) "f p y-1 = ·"fp
dp
PT
p.
(48)
Substituting equation (48) intoequation (47):
fip
-..Jp
a-
(49)
Using equation (33), or (36), another form of the speed of sound equation may be wńtten:
a= -JygR T = .J,iR'T
(50)
Since y, g, and R, or R', remain constant, the speed of sound varies directly with the square root of
the absolute static temperatme.
D6-1420
May1989
1.21
AERODYNAMICS
Intentionally Left Blank
1.22
D6-1420
May 1989
AERODYNAMI CS
1-10 Compressible Bernoulli Equation
Air moving at speeds below 250 knots may be treated as an incompressible fluid. At higher speeds,
it is necessary to consider the variation of density as the airflow is compressed. At these speeds,
another form of the Bernoulli equation must be used.
The compressible Bernoulli equation may be developed from equation (24), repeated here for
convenience:
dp=-pVdV
Within the streamtube or duet, changes in density occur too rapidly for any appreciable heat flow to
occur.. Hence, the process is assumed ideally adiabatic. Equation (24) involves the variable p,
wJ:rich can be expressed in terms of pressure from equation (39):
p. = constant = C
1p
and,
1/y
(51)
p=(f)
Substituting for p in equation (24), reammging, and integrating:
lfy
dp=-(f)
V dV
( C)
1/y 1/
p - y dp = - V dV
<c)
1/y . 1/y
Jp dp + Jv dV = Jo
_!.+1
1/y
( C)
Y
p
1
2
+ '; = constant
- 'Y + 1
D6-1420
Mayl989
1.23
AERODYNAMICS
Simplifying:
<C)
t/y
y-1
1
T
-- p
1-1
_.2
v+ - 2 = constant
1
1
2
(C) + -V
2 = constant
-1- p 1-1
P
1
y
Substituting for ( ~) , the Bernoulli equation for compressible fluids in horizontal flow becomes:
1
p
1-1 P
v2 = constant
-- - + -
(52)
2
As before, the flow equation may be written for any two points in the fluid:
2
2
11 · P
vl
12 P
v2
---+-=
-+11-1 P 1
2
1 2 -1 P 2
2
1.24
(53)
D6-1420
May 1989
AERODYNAMICS
1-11 Flow Relations Near The Speed Of Sound
A useful and important relationship called Mach number (M) involves the velocity of sound.
Mach number is the ratio of the fluid velocity to the velocity of sound at the same point
V
M=a
(54)
Since V and a are velocities measured in feet per second, Mach number is dimensionless.
The behavior of fluid flow near the speed of sound is of primary importance. The relationship
between temperature, pressure, mass density, and velocity may be predicted by the application of
some of the fluid laws developed in the previous chapters.
There are six generał classifications used to describe high speed flight These are:
(1)
(2)
(3)
(4)
(5)
(6)
subsonic
sonie
supersonic
transonic
hypersonic
hypervelocity
M < 1
M =1
M > 1
M from approx. .8" to 1.3
M=5to10
M = lOand up
The term transonic is often used to describe the flow conditions through the sonie region.
Mathematical treatment of the ttansonic region is rather difficult because of the mixed flow
conditions; however, fairly accurate predictions of the aerodynamic characteristics of both subsonic
and supersonic flow can be made.
D6-1420
March 1991
1.25
AERODYNAMI CS
Consider the flow of fluid in a frictionless channel, as in Figure 8, which has a varying cross
sectional area A, connected to a reservoir. The relationship between the velocity and pressure was
developed earlier, equation (24).The subscript t is for total conditions.
V
A
Figure 8.
dp=-pV dV
The continuity equation (16), provides a relationship involving the clensity.
dV
dA
dp
-V+ -A+ - pO
Also from equation (47),
or,
•=ft
2
dp
a=dp
(55)
Equation (24) may be written substituting the value of dp from equation (55).
a 2 dp= -pV dV
(56)
1.26
D6-1420
May1989
AERODYNAMICS
Substitute for dp/p in equation (16),
dA+dV_VdV=O
A
V
a2
By multiplying the third term by V /V, and expressing in terms of Mach number:
dV (1- M2) =- dA
V
A
(57)
Equation (57) shows what happens to speed as the cross sectional area is changed in a channel. At
a fixed Mach number less than one, an increase in area produces a decrease in speed. When the
Mach number is greater than unity, the equation states that an increase in area will cause further
increase in speed. At these speeds, the density decreases so rapidly for a given speed increase that
the channel must expand to continue at this speed increase. This is supported by equation (56)
rearranged.
dp
2 dV
p=-M V
At the point in the channel, Figure 9b, where there is no change in area, that is, dA/A =O, equation
(57) indicates that two possibilities exist dV/V =O, or M = 1.. The point where the cross sectional
area is a minimum is called the throat, and is the point where dA/A = O. At this point the Mach
reaches unity. Downstream of the throat the speed may continue to increase, and Mach numbers
greater than unity may occur.
Equation (57) shows the relationship between velocity and area as influenced by Mach number. A
similar relationship may be developed showing how pressure varies with area. Equation (24) may
be written:
dV
v=-
06-1420
May1989
dp
pV2
(58)
1.27
AERODYNAMI CS
Throat
(a)
Throat
(b)
p
(c)
T
in increase in
intemal beat energy
..l
Single Pressme for :f
Supersonic Effiux
1.0
(d)
M
Figure9.
1.28
D6-1420
March 1991
AERODYNAMICS
and equation (49) as:
- 'YP
P- 2
(59)
a
Substituting equation (59) into equation (58) and the result into equation (57), the following
equation is obtained:
. (60)
The variation of temperature can now be established from the equation of state, (33).
Equation (60) shows that, for a given Mach number, an increase in area requires a pressure rise for
M less than one, and a pressure drop for M greater than one. From Figure 9a, if the flow through
the tube is completely subsonic, the decrease in area before the throat results in an increase in
velocity and a decrease in pressure; whereas, the increase in area downstream of the throat results in
a decrease in velocity and an increase in pressure. Now with an increase in velocity ahead of the
throat, there will be a corresponding increase in velocity at the throat. When the velocity at the
throat just reaches the speed of sound, the flow has become critical (critical is defined only for M
= 1 flow conditions). The pressure at the throat under this condition is knąwn as the critical
pressure. As the pressure downstream of the throat is reduced below the critical pressure value, the
flow upstream of the throat will be the same in all respects; i.e., no further increase in initial or
throat velocity is possible, unless the stagnation state of the gas is changed. Downstream of the
throat, however, the flow will be supersonic. The lower the exit pressure the higher the value of
supersonic flow and the further downstream the supersonic flow will go.
Figure 9b shows that the flow just downstream of the throat is supersonic, whereas further
downstream the flow is subsonic. The physical process by which the change takes place is known
as a shock wave. The term shock is used because the transition from supersonic to subsonic flow
occurs suddenly. Now, with a sudden change in velocity, the stati~ pressure downstream in the
subsonic region must be increased correspondingly (Bernoulli equation). This sudden increase in
pressure will cause a compression in the air with a consequent increase in density and temperature.
During the decrease in the downstream pressure to where the throat Mach number just reaches
unity, the shock wave separating the supersonic and subsonic regimes is very weak (small pressure
change) and will occur at the throat As the supersonic flow region becomes greater with lower exit
pressures, the pressure rise after the shock is greater. The curves for variation of pressure in the
channel are similar to the velocity curves and appear as in Figures 9c and 9d. The pressure rise
through the shock wave results in a pressure which does not equal the pressure for subsonic flow.
There is a loss of pressure energy through the shock wave which results in an increase in freestream
temperature. However, the stagnation temperature reinains constant.
D6-1420
March 1991
1.29
AERODYNAMICS
The above analysis of the stream flow through a convergent/divergent streamtube may be
summarized as shown below:
·
Initial
Cond.ition
With
AreaChange
Results
In
And
M<l
ADecrease
Alncrease
V Increase
VDecrease
p, T, p Decrease
p, T, p lncrease
M>l
ADecrease
A Increase
VDecrease
V Increase
p, T, p lncrease
p, T, p Decrease
Note that when the change in velocity has b~n established for the initial conditions, the three
variables of the equation of state, p, T, and p, will always be opposite the velocity change. When
M = 1, it must occur in the streamtube where there is no area change, i.e, at the throaL
There is a relationship between temperature and velocity which can be derived. The compressible
Bernoulli equation, equation (52), may be written for. the flow cond.itions of Figure 8. The
temperature, pressure and density conditions in the reservoir are all stagnation values which are the
total temperature, pressure, and density existing; hence, the subscript, L
~+ - -Y - -Pt. =v - + Y
p
2
y-1 Pt
2
y- 1 P
2
Since in the reservoir, stagnation exists, Vt = O,
2
Y Pt v
Y p
=
+
-y-1 Pt
2
y-lP
From equation (49),
2
YP
2
Y Pt
t
Pt
a =p-
and,
a=--
1.30
D6-1420
Mayl989
AERODYNAMICS
Substituting,
· 2
2
2
at
V
a
=
+
y-1
2
y-1
Dividing both sides of the equation by a2,
1 a~ 1 v2
-- - = - - + -1-
'Y - 1 a2
2 a2
'Y - 1
From equation (50),
and,
Then,
2
~ - Tt
a2 - T
Substituting,
2
1 Tt M
1
=
+
y-1 T
2
y-1
Rearranging,
(61)
The temperature, T, is the static temperature, which is sometimes written as T s to distinguish it from
total or stagnation temperature, Tt.
The stagnation temperature across a shock wave remains constant even though the total or
stagnation pressure may be decreased. Enthalpy is defined as the product of the specific beat and
the temperature, Cp T. This means the total enthalpy, or beat, remains constant across a wave, and
the lost pressure results in an increase in the entropy of the air, but not in the total beat content The
non-isentropic process also causes a greater decrease in velocity than would have occurred under
conditions of constant entropy, and the freestream temperature is increased by this effect over its
value bad the change been isentropic. Accordingly the beat content along a streamline increases
because of the non-isentropic effect The total enthalpy, or beat content, of the air is not increased
under such conditions, however,
D6-1420
Mayl989
1.3 1
AERODYNAMI CS
where,
~ Tt
is the total enthalpy or total beat content
~T
is the beat content along a streamline
If the flow in the channel is ideally adiabatic, the applicable pressure and density relationships may
That is, applying equations (33) and (39) to equation (61), the following
expressions may be deńved:
be developed.
~t =(l+ r;
l M2)
1
y-1
(62)
'Y
Pt (
y-1 2)
p=
l+~M
y-1
(63)
These relationships arc important and should be well understood.
The ratios of pressure, temperature, and density at the throat to those in the reservoir arc of
particular interest. These ratios are called critical ratios and are obtained from equations (61), (62),
and (63). The critical ratios are shown herc as the reciprocal of the above equations with M = 1 and
y = 1.4 for air substituted.
T
2
-=-=Cl833
Tt
y+l
(64)
'Y
2
_Ł
-- )
Pt = (y+
1
y-1
= Cl 528
(65)
= Cl 634
(66)
_1_
..e.._ = ( -2- )
Pt
1.3 2
y+ 1
y-1
D6-1420
March 1991
AERODYNAMICS
1-12 The Atmosphere
The ultimate performance of the airplane and the engine depends upon the generation of forces due
to interaction between the airplane or engine and the air mass through which it flies. So it is of
importance to study the properties of the earth's atmosphere.
The atm.osphere is a gaseous mixture composed of nitrogen, oxygen, and a small amount of other
gases. Water vapor is always present, but in varying amounts, usually less than one percent at the
earth's surface.
The energy of the sun is responsible for heating the atm.osphere, but little of this energy is
transferred clirectly to the air. Most of it goes to beat the eanh's surface which in tum heats the air.
The warm air near the surface rises and due to a decrease in pressure as altitude is increased,
expands and is cooled.
Finally, an equilibrium is reached where no more reduction in temperature occurs. This altitude is
the tropopause and varies with latitude. The region below, from the eanh's surface to the
tropopause, is the troposphere. That above the tropopause is the stratosphere. In the stratosphere,
the temperature is essentially constant.
Seasonal changes and moving air masses have a pronounced effect on the temperature, pressure,
and density of the air. To provide a basis for estimating and comparing airplane and engine
performance, it is necessary to have a standard. The standard used in the United States by airplane
manufacturers is that established by the International Civil A viation Organu.ation (ICAO). It is a
condition determined by averaging data gathered over a long period. Since the studies were
conducted in the mid latitudes of the northern hemisphere, the standard is most clearly
representative of conditions in these regions. However, even though the expected deviations from
this standard may be much larger in polar or equatorial regions, the same standard is used as a
reference. The International Standard Atm.osphere (ISA) establishes a sea level pressure, p0, of
29.92126 inches of mercury, or 2116.2166 lb/ft2 at a·temperature, to, of 59°F (15°C). The mass
density, p0, of dry air under these conditions is 0.002377 slugs/ft3, The subscript, o, identifies
these as standard sea level values and will be used hereafter.
_As indicated previously, the equation of state has several forms, and can be written in many ways.
For example, equation (33), may be stated for the standard sea level condition:
Po= PogRTO
(67)
Dividing equation (33) by equation (67) the constants g and R cancel:
(68)
D6-1420
March 1991
1.3 3
AERODYNAMI CS
The rati.os appearing in equati.on (68) are used quite frequently and for this reason cach has a
specific symbol:
Delta
_ _g_ _
air pressure
6 - Po - air pressure at SL sui day
(69)
Sigma
p
air mass density
a-~--------=~-~
- P O - air mass density at SL sui day
(70)
Theta
8
= ..I., =
TO
absolute air temp
absolute air temp at SL sui day
(71)
These symbols may be substi.tuted in equati.on (68) to provide another useful equati.on:
(72)
6=·a8
Variation of Temperature with Altitude
The International Standard Atmosphere assumes a constant drop in temperature of 0.00356616°F/ft
(or 0.0019812°C/ft) from sea level to an alti.tude of 36,089 feet, the tropopause. The standard
temperature at any alti.tude below the tropopause may be determined from the following:
(73)
Where A is the lapse rate, 0.00356616°F/ft, or 0.0019812°C/ft. The letter h represents alńtude in
feet. After subsńtuńon, equańon (73) becomes:
T = 518.67 - O. 00356616 h
T =288.15-0.0019812 h
°R
°K
t = 59 - 0.00356616 h
t = 15 - 0.0019812 h
Op
(74)
oc
Above the tropopause, a constant temperature of-69.7°F (-56.5°C) is assumed.
T = constant = 389.97
OR
= 216.65
OK
(75)
t = constant =-69.70
OF
=-56.50
OC
Figme 10 shows the standard temperature variańon with alńtude.
1.34
D6-1420
Mayl989
AERODYNAMICS
Variation of Pressure with Attitude
Unlike the temperature, the pressure continues to decrease at altitudes above the tropopause . The
pressure variation with altitude below the tropopause is not the same as above the tropopause due to
the influence of temperature. The relationship between the pressme and temperatme is govemed by
the gas laws.
Below the tropopause: the pressure, temperature, and density relationships may be developed in the
following manner.
Column of Cross
Section Area A
36,089.24 • • • • •
Feet
Altitude
o
Weight of Błock
= wAdh
-56.5
O
- 69.7
o
Temperature CIF
Figme 10.
15
59
Figure 11.
Consider the analysis of the pressure existing on an infinitesimal section of a column of air at
altitude, h, as in Figure 11.
As in the chapter dealing with hydrostatics, the sum of the forces in a vertical direction is made:
From equation (1),
l:F V =0
pA- (p+ dp)A- wA dh= O
or,
-dp=w dh
D6-1420
March 1991
(4)
1.35
AERODYNAMICS
From equation (10),
w=pg
Substituting (10) into (4),
(76)
dp=-pgdh
From equation (33),
p=pgRT
Therefore,
dp =--1- dh
P
RT
(77)
From equation (73),
T=T0 -lh
Differentiating,
or,
dT=-A. dh
dh·=- dT
l
Substituting in equation (77) for dh,
1=(;i)(-r)
and adding limits of.integration,
pdp
1
TdT
I - =ART
- IPp
T
o
o
Integrating,
p
T
1
1n p] = - ln T]
lR
Po
1.3 6
To
D6-1420
March 1991
AERODYNAMICS
I
and substituting the limits,
p
_1_
( T ):A.R
Po°=~
to=( i
(78)
5.25588
0
)
6 = 85.25588
To find the mass density relationship, equation (68) may be combined with equation (78):
1
p
( T )AR
Po= To
P T
= Po To
_I_
Po= To
p
( T
)-1( ToT ) AR
p
T
4.25588
Therefore,
(79)
Po =(T0 )
Cf= 8
4.25588
Above 36,089 feet:
The upper limit of the troposphere is the lower limit of the stratosphere; consequently this is
reflected in the pressure temperature density relationships for the stratosphere. From equation
(77), a relation between pressure and altitude was derived:
dp
I
-=--dh
P
RT
D6-1420
March 1991
1.37
AERODYNAMI CS
With the temperatme now a constant, the equation can be integrated,
1
dp
p
h
I-=fdh
P
p
RT trop h
trop
trop
The subscript trop indicates the 36,089.24 foot altitude condition.
Integrating,
p
h
1n p ]=- RTl
h]
trop
h trop
and substituting the limits,
1n(.2-)
= -( h - 36,089.24 )
p
RT
trop
trop
or,
p
-( h-36,089.24 )
p -trop
e
RT ttop
but, from equation (78),
T
)5.25588
trop
p tror po (
therefore,
--=r;-
e -(
ttop
po
:o
The letter
l•
4
_!_ = (T r25S88
h-~:/ )
To
= (.22336)
-( h-36,089.24 )
e
(80)
J
20805. 8
e is the base of the natmal logarithm.
To find the mass density relationship for the stratosphere, equation (68) may again be used.
_R_ = (.22336) e - (
Po
1.3 8
h-36,089.24 )
20805.8
=~
Po
T
trop
T0
D6-1420
March 1991
AERODYNAMICS
·
T
_e__ = _o_ (. 22336 ) e
Po
-( h-36,089.24)
20805.8
T trop
but,
~ _ 518.67 _ l 33
T trop - 389.97 -
.
Therefore,
p
Po
·
(.29707) e
( h-36,089. 24 )
. 20,805.8
(81)
From the equations developed in this chapter valid for standard atmosphere only, equations (73)
through (J5) and (78) through (81), a table can be made for all altitudes based on the international
· standard. Since this involves much tedious calculation, tables have been accurately calculated and
published by various govemment agencies. These data are available in most engineering reference
handbooks. Figure 12 is such a table. Airplane and engine manufacturers have adopted the
intemational standard and are using it for all engineering and performance. analyses.
D6-1420
March 1991
1.39
AERODYNAMICS
INTERNATIONAL STANDARD ATMOSPHERE
Press Temperature
Alt, h
t
OF
OC
rt
Density
Pressure,p
i =8
a/a 0
in Hg
lblrt 2
mb
...!...=g
o
slugs/ft 3
/;=a
Po
o
59.0
15.0
1.0000
1.0000
29.92126
2116.2
1013.2
1.0000
.0023T7
1.000
1,000
2,000
3,000
4,000
5,000
55.4
51.9
48.3
44.7
41.2
13.0
11.0
9.1
7.1
5.1
.9931
.9861
.9795
.9726
.9656
.9965
.9931
.9896
.9862
.9826
28.860
27.820
26.820
25.840
24.900
2041.0
1968.0
1897.0
1828.0
1761.0
977.1
942.2
908.2
875.1
843.1
.9643
.9290
.8963
.8637
.8321
.002308
.002241
.002175
.002lll
.002048
.9710
.9428
.9150
.8881
.8616
6,000
7,000
8,000
9,000
10,000
37.6
34.0
30.5
26.9
23.3
3.1
-0.8
-2.8
-4.8
.9587
.9518
.9452
.9382
.9313
.9791
.9757
.9721
.9686
.9650
23.980
23.090
22.220
21.390
20.580
1696.0
1633.0
1572.0
1513.0
1455.0
811.9
781.8
752.6
724.3
696.6
.8013
.7716
.7428
.7148
.6875
.001987
.001927
.001868
.001811
.001755
.8359
.8107
.7859
.7619
.7383
11,000
12,000
13,000
14,000
15,000
19.8
16.2
12.6
9.1
5.5
-6.8
-8.8
-10.8
-12.7
-14.7
.9244
.9174
.9105
.9039
.8969
.9615
.9579
.9543
.9506
.9470
19.790
19.030
18.290
17.580
16.890
1400.0
1346.0
1294.0
1243.0
1194.0
670.2
644.4
619.5
595.1
571.6
.6614
.6360
.6114
.5873
.5641
.001701
.001648
.001596
.001545
.001496
.7156
.6933
.6714
.6500
.6293
16,000
17,000
18,000
19,000
20,000
1.9
-1.6
-5.2
-8.8
-12.3
-16.7
-18.7
-20.7
-22.6
-24.6
.8900
.8831
.8761
.8695
.8626
.9434
.9397
.9361
.9324
.9287
16.220
15.570
14.940
14.340
13.750
1147.0
1101.0
1057.0
1014.0
972.5
549.1
527.1
506.0
485.4
465.6
.5419
.5202
.4994
.4791
.4S95
.001447
.001401
.001355
.001319
.001266
.6088
.5894
.5700
.5549
.5326
21,000
22,000
23,000
24,000
25,000
-15.9
-19.5
-23.0
-26.6
-30.2
-26.6
-28.6
-30.6
-32.5
-34.5
.8556
.8487
.8418
.8352
.8282
.9250
.9212
.9176
.9138
.9100
13.180
12.640
12.110 ·
11.600
11.100
932.4
893.7
856.3
820.2
785.3
464.4
427.8
409.9
392.7
376.0
.4405
.4222
.4045
.3876
.3711
.001224
.001183
.001143
.001103
.001065
.5149
.4977
.4809
.4640
.4480
26,000
27,000
28,000
29,000
30,000
-33.7
-37.3
-40.9
-44.4
-48.0
-36.5
-38.5
-40.5
-42.5
-44.4
.8213
.8143
.8074
.8005
.7939
.9062
.9024
.8986
.8948
.8909
10.630
10.170
9.725
9.297
8.885
751.6
719.1
687.8
657.6
628.4
359.8
344.3
329.3
314.8
300.8
.3SS1
.3398
.3250
.3107
.2969
.001028
.000992
.000957
.000923
.000889
.4325
.4173
.4025
.3881
.3741
31,000
32,000
33,000
34,000
35,000
-51.6
-55.1
-58.7
-62.2
-65.8
-46.4
-48.4
-50.4
-52.4
-54.3
.7869
.7800
.7730
.7661
.7595
.8871
.8832
.8793
.8753
.8714
8.488
8.106
7.737
7.382
7.041
600.3
573.3
547.2
522.1
498.0
287.4
274.5
262.0
249.9
238.4
.2836
.000857
.000826
.2586
.2466
.2353
.000795
.000765
.CXXT137
.3605
.3473
.3345
.3220
.3098
36,000
36,089
37,000
38,000
39,000
40,000
-69.4
-69.7
-69.7
-69.7
-69.7
-69.7
-56.3
-56.5
-56.5
-56.5
-56.5
-56.5
.7526
.7519
.7519
.7519
.7519
.7519
.8674
.8671
.8671
.8671
.8671
.8671
6.712
6.683
6.397
6.097
5.811
5.538
474.7
472.7
452.4
431.2
411.0
391.7
227.3
226.3
216.6
206.4
196.8
187.5
.2243
.2233
.2138
.21137
.1942
.1851
.000709
.000706
.000616
.000644
.000614
.OOOSBS
.2981
.2971
.2844
.2710
.2583
.2462
41,000
42,000
43,000
44,000
45,000
-69.7
-69.7
-69.7
-69.7
-69.7
-56.5
-56.5
-56.5
-56.5
-56.5
.7519
.7519
.7519
.7519
.7519
.8671
.8671
.8671
.8671
.8671
5.278
5.050
4.794
4.569
4.355
373.3
355.8
339.0
323.2
308.0
178.7
170.3
162.3
154.7
147.5
.1764
.1681
.1602
.1S27
.1456
.000558
.000532
.2346
.2236
.2131
.2031
.1936
1.1
TO= 59.0 + 459.67 = 518.67°R
0
R = 5335 ft lbf'lbm R
:r,09
.000507
.000483
.000460
T0 = 15.0 + 273.15 = 288.15°K
p0 =. 002377 slug/tt3
&o = 1116.4501 ft/sec
P0 = 2116.21662 lb/ft2
Rcfmace: NACA Tecbnical Report No. 1235
Figure 12
1.40
D6-1420
March 1991
AERODYNAMICS
Non-Standard Day Conditions
The major reason for establishing a standard atmosphere is to permit performance and operation to
be stated in forms which may be compared. All predictions and results arc published on the basis
of assumed standard day conditions. Actual data gathered under non standard conditions arc
transformed into equivalent standard day values by the use of the parameters, 8, and a. The
standard atmosphere is an arbittary condition estabiished by definition.
a,
At any geometrie or tapeline altitude, the measured pressure may be other than the standard value.
Thus, the altimeter, which measures pressme, will indicate an altitude corresponding to the
measmed pressure. This pressure altitude is the standard day condition, the most natura! and
sim.ple equivalence to use. To indicate pressure altitude, the altimeter must be unbiased, and must
be set for 29.92126 inches or 1013.25 millibars of mercury. The pressure ratio, may be read
from the standard table, Figure 12, by entering with the pressure altitude. But, temperature and
density relations must be resolved on the basis of further measmements and calculations.
a,
Since density may not be measured directly, it is common practice to measure static temperature ~
calculate 8. With 8 and known, a and density may be obtained from equation (72). For
example, if the altimeter reads 4,000 feet pressure altitude, and the OAT is 90 °F, = .8637, 8 =
1.0598, and a= .8150.
a
D6-1420
March 1991
a
1.41
AERODYNAMI CS
Intentionally Left Blank
1.42
D6-1420
May1989
AERODYNAMICS
1-13 Pressure Distributions
The Bernoulli principle is fundamental to all aerodynamic probleins, particularly those involving
pressure distributions on bodies immersed in a fluid. Explanations involving the fluid phenomena
are frequently simplified by considering two dimensional flow, or flow of fluid in parallel planes
over the body.
·
Place a blunt nosed object, a cylinder, in a frictionless and incompressible moving fluid. The flow
pattem or stteamlines will appear as shown in Figure 13.
.
Figure 13.
In the flow around the body, some of the stream.lines are diverted to one side, some to the other.
There is one point at which the streamlines are normal to the body. This is called the stagnation
point, point 1 on the figure.
The spacing of the streamlines about the cylinder indicates the magnitude of the velocity. That is,
the closer together the streamlines are spaced, the higher the velocity. From the law of continuity,
velocity is inversely proportional to the area. The streamlines are compressing the streamtubes,
thereby decreasing the area and increasing the velocity.
The Bernoulli equation shows that along with changes in velocity, simultaneous changes in static
pressure occur. Tuus, whenever the velocity is different from the undisturbed free stream velocity,
the static pressure will differ from that existing in the free stream. Examining Figure 13, air
particles along the streamline striking point 1 will be brought to rest. The reduction of velocity to
7.ero at this point means the static pressure will increase in value equal to the dynamie pressure, q.
D6-1420
May1989
1.43
AERODYNAMICS
Thus,
but since,
ql = o because vl = o,
then,
Po+qo=P1
or,
(82)*
This equation says the difference between the static pressure of the free stream and at point 1 is the
dynamie pressure of the free stream. If both sides of equation (82) are divided by the quanti.ty Cło, a
more standard form results:
_P__
1 _-_P__
o=1
qo
(83)*
This manipulation makes the equation dimensionless since the units of the numerator and
denominator are the same, and cancel.
The pressure d.ifference p 1 - p 0, or Ap, when divided by the free stream q, is defined as a pressure
coefficient and is denoted by the symbol Cp. Thus, in generał:
C
p
= Ap
q
(84)
Equation (83) states that the pressure coefficient at the stagnation point is unity. Similar evaluations
of Cp may be made at various points on the cylinder. For example, from theoretical flow
conSiderations the velocity at any point on a cylinder is subject to the ~lationship:
(85)*
* lncompressible flow about a cylinder only
The angle, 0, is the angle shown on Figure 13.
1.44
D6-1420
March 1991
AERODYNAMICS
At point 2, the velocity V2 is equal to twice the free stream velocity, Yo, since the sine of 90° is
unity. That is:
(86)*
Therefore, the relationship can be written for point 2:
Po+qo=P2+q2
1
2
Po+qo=P2+2PV2
Substituting equation (86) into the above expression,
1
2
Po+ qo = P2 + 2P(2Yo)
Po+qo=P2+4
(1 pv~)
Then,
or,
(87)*
* lncompressible flow about a cylinder only
The negative value of Cp can exist only if p 2 is less than Po· This must be the case, since V2 is
greater than VOwhich satisfies the Bernoulli equation.
A graphical representation of the static pressures existing on the smface of the cylinder is shown on
Figure 14.
.
The arrows pointing toward the body are positive (greater than free stream) pressures; while arrows
pointing away from the body indicate negative (less than free stream) pressures. The length of
arrow is proportional to the value of Cp existing, and its magnitude may be found by scaling.
When an ideał fluid is considered, the static pressures are distributed symmetrically over the
cylinder. Because the pressures acting over the area will neutralize, there can be no resulting force
in any direction. Physically this is known to be untrue since there is a force in the downstream
direction (drag), due to the viscous characteristics of the fluid. The effect of the viscous force is to
change the pressure distribution, resulting in a drag force on the cylinder. A diagram of static
pressure distribution over a cylinder in a real fluid is shown on Figure 15.
D6-1420
Marc:h 1991
1.45
AERODYNAMI CS
IDEALFLUID
vo
+1
Po
I
o
I
-1
-2
I
-3
CP scale
Figure 14.
REAL FLUID
I
+1
I
O
I
-1
I
-2
I
-3
CP scale
Figure 15.
The concept of pressure distribution over a geometrically sim.ple object may be applied to a more
practical aerodynamic shape. Figure 16 shows an airfoil section in an ideał fluid.
1.46
D6-1420
May 1989
AERODYNAMICS
Figure 16.
The static pressure distribution may be shown graphically on this shape also. A symmetrical airfoil
section is shown, that is, the curvature is the same on top as it is on the bonom. See Figure 17.
As with the cylinder, the net force acting on the body is zero, although it is not obvious from the
figure.
In a real fluid the pressure distribution is approximately as shown in Figure 17, except at the trailing
edge the pressure fails to rise to the stagnation value, Figure 18. This difference produces a drag
force in the direction of the flow. This drag force resists forward motion. In practice, drag is
tolerated only because it cannot be eliminated. The primary purpose of the airfoil shape is to
generate a lifting force. The relationship of the lift and drag forces, and their effect, will be
discussed in another chapter.
D6-1420
May1989
1.4 7
AERODYNAMICS
IDEALFLUID
-1
+1
Figure 17.
REAL FLUID
+1
-1
Figure 18.
1.48
D6-1420
May1989
AERODYNAMICS
1-14 Force Equations
The generalized equation for the force acting on any body in a moving fluid is dependent on severa!
variables. The basie force equation may be derived by a mathematical procedure known as
dimensional analysis.
Dimensional analysis is based on the assumption that the most fundamental form in which variables
may be related to form an equation is the exponential form. Also of major importance in this
technique is that the dimensions on both sides of the equation must be equal. For example, if the
dimensions on one sicie of an equation reduce to the units of lb/tt2, the dimensions on the other side
must reduce to the same units. For most problems these. dimensions may be expressed in terms of
mass, length, and time symbolized by M, L, and T, respectively.
The firSt step to find the form of an equation is to consider all the variables which could affect the
quanti.ty being solved for. In this case, the force due to fluid morion past a body is to be
determined. The variables to be considered are:
(1)
(2)
(3)
(4)
(5)
Velocity of fluid, V, in ft/sec.
Fluid mass density, p; in slugs/ft3
Characteristic size, I, in ft
Fluid coefficient of viscosity, µ, in slugs/ft.sec.
Compressibility, related to speed of sound, a, ft/sec.
The latter two items manifest themselves since a real fluid is being considered.
A generał relati.onship may be assumed for the fluid force, F:
F=Cpa/v1 µ 6 ar.
(88)
Where C is a constant which depends on body shape and is dimensionless, and a, p, y , a, e, are
as yet undetennined exponents.
The next step is to express all the variables in terms of the fundamental units, M, L, and T. Writing
Newton's second law of motion in terms of M, L, and T:
ML
F=
-
r
D6-1420
May1989
(89)
1.49
AERODYNAMI CS
Equation (88) may now be written in tennS of the fundamental units:
u
T
. 6
E
~=c(~) Cil(i)(ti)(i)
(90)
Powers of like variables arc now equated:
M:
1= a+6
L:
1 = - 3a + 13 + 'Y - 6 + e
T:
2=y+6+e
1
(91)
J
Since fluid forces arc usually expressed in terms of p, V, and l, the exponents of these variables
will be solved for absolutely. Therefore, equations (91) will be solved in terms of a , "(, and 13.
a= 1-6
"{=2-6-e
(92)
13=2-6
Substituting the results of ~uations (92) into equation (88):
F= Cpl-6 12-6 y2-&-e µ6 ae
or,
(93)
v
z
This expression shows the fluid force is not only dependent on p, 2, and 2 , but is also dependent
on two non dimensional parameters raised to unknown exponents. The first combination of terms
is called Reynolds number, after Osborne Reynolds, who first showed that fluid flow is
dependent upon the value of this number. The latter combination is Mach number.
1.50
µpVl
Reynolds number,
RN=
Mach number,
V
M=a
(94)
D6-1420
March 1991
AERODYNAMICS
To simplify the notation, equation (93) is usually rewritten to envelope the fluid dynamie pressure,
l/2p v2, or q. Note that z2 is the square of a representative length and is thus a representative
area which may be called A.
Thus,
F=2CqA(RN)
-aM- e
or,
. (95)
where Cr is a new parameter which depends not only on body shape, but also on Reynolds number
and Mach number.
Reynolds number is the important criterion for low speed flight and Mach number is important for
high speed flight. More will be said later of ·these quantities. Mach number is by far the most
important quanti.ty with respect to large, high speed jet transports.
Of considerable interest in aerodynamic work is the turning moment of the force of equation (95)
about some arbitrary point. A moment is the product of the force and the distance from its point of
application to an arbitrary point. The usual dimension for a moment is the foot pound. Similar to
the generał force equation, the generał moment equation can be developed by dimensional analysis.
The expression will evolve in the form:
(96)
Where C is a different constant, and A and r arc new exponents of Reynolds number and Mach
number. The dimensionless quantities of equation (96) may be combined:
(97)
Where Cm is a dimensionless moment coefficient like the force coefficient Cr in equation (95), and
l is a representative length.
D6-1420
March 1991
1.51
AERODYNAMICS
Intentionally Left Blank·
1.52
D6-1420
Mayl989
AERODYNAMI CS
1-15 Airfoil Properties
An airfoil is a shape or contour with aerodynamic properties. The more familiar examples of
airfoils are airplane wings, propellers and compressor and turbine components of the jet engine.
Before the aerodynamic properties of an airfoil are described, some physical definitions should be
discussed. Primarily, these will be airplane particulars since engine characteristics are in the
turbojet engine section.
Physical Properties
Wingspan is the tip to tip dimension of the airplane wing, regardless of its geometrie shape. The
symbol for wingspan is b, and is illustrated on Figure 19.
Figure 19.
Area of a wing is the projection of the outline of the plane of the chord. The wing is considered
to extend without interruption through the fuselage and nacelles. The shaded area on Figure 19
shows the area of a swept wing airplane. The area of a wing· is usually denoted by S.
D6-1420
March 1991
1.53
AERODYNAMI CS
Chord is the distance from the wing leading edge to the ttailing edge. The chord is seidom
constant due to tapering or curving boundaries, thus the chord used in calculations is an average
chord. The symbol for the chord is c.
A simple relationship may be written involving area, span, and average chord
S = bcavg
(98)
Aspect ratio of a rectangular wing is the ratio of the span to the chord, and the notation is AR.·
b
AR=c
(99)
This holds for rectangular wings but for all other planforms another form may be derived from the
basie relationship.
(100)
Taper ratio is the ratio of the tip chord, et, to the root chord, Cr Tuus,
(101)
Mean aerodynamic chord, or MAC, is the chord of a section of an imaginary airfoil on the
wing which would have force vectors throughout the flight range identical to those of the actual
wing. MAC is a value used in engineering and weight and balance calculations for convenience.
The MAC may be determined by calculation as shown below. It is used as a reference for locating
the relative positions of the wing center of lift and the aiiplane center of gravity. Ultimately the load
.distribution detennines the static balance and stability of the aiiplane.
1.54
D6-1420
March 1991
AERODYNAMI CS
,'
,
,
... ... ...
... ... ...
,
,,
,'
,''1
I
I
I
I
I
... ......
... ... ...
b.
2 ------~
Figure 20.
Generally, the MAC can be closely approximated by the graphical method shown in Figure 20.
This method introduces only a negligible error and is sim.ple to accomplish. ey and ei refer to the
root chord and tip chord, respectively.
The rigorous definition of the length of the MAC is given by the following calculus expression and
by reference to Figure 21.
MAC= Jc2 db
Je db
MAC= Jc2 db
s
06-1420
May1989
(102)
155
AERODYNAMICS
b
2
_L
T
Figure 21.
Equati.on (102) will reduce to the following for a straight tapered wing as in Figure 21:
MAC= 1.3 r (A 2A+
+A+ 1)
1
C
1.56
(103)
D6-1420
May1989
AERODYNAMICS
Sweepback is the angle between a line perpendicular to the piane of symmetry of the airplane and
the quarter chord, c/4, of cach airfoil section. The symbol of the _sweep angle is the Greek letter
Lambda, A. See Figure 22.
, ~ . QUARTER CHORD
4
Figure 22.
The airfoil section represented in Figure 23 has cenain physical characteristics which distinguish it.
Camber is the curvature of the section, or the departure from the chord line. The mean line is a
line equidistant from the upper and lower surfaces. The chord line is the straight line joining the
intersections of the mean line with the leading and trailing edges of the airfoil. The geometrie shape
of an airfoil is expressed in terms of the following:
(1)
(2)
(3)
Shape of the mean line
Thickness
Thickness distribution
Thickness is shown as a percent of the chord or as a ratio of the length of the line perpendicular to
the chord to that of the chord; that is, tle.
MEAN LINE
CHORD LINE
Figure 23.
D6-1420
March 1991
1.57
AERODYNAMICS
The National Aeronautics and Space Administration (NASA), formerly National Advisory
Committee for Aeronautics (NACA) has tested many airfoil shapes and has developed a systematic
series of sections. The results of these tests arc published and arc available through the govemment
printing agency. NACA Report No. 824, "Summary of Airfoil Data", explains the system of
identifying the various series of airfoils in addition to the airfoil data presented. Of the various
series airfoils, the NACA five digit series will be described herc. The numbering system for airfoils
of the NACA five-digit series is based on -a combination of theoretical and geometrie characteristics.
The first digit indicates the amount of camber in terms of the magnitude of the design lift coefficient.
The design lift coefficient (in tenths) is three-halves of the first digit. The second and third digits
together indicate the distance from the leading edge to the location of the maximum camber; this
distance in percent of the chord is one-half the number represented by these digits. The last two
digits indicate the airfoil thickness in percent of the chord. The NACA 23012 airfoil thus has a
design lift coefficient of 0.3, has its maximum camber at 15 percent of the chord, and has a
thickness ratio of 12 percent. Figure 24 shows the ordinates of this airfoil.
1.5 8
D6-1420
May1989
AERODYNAMICS
NACA 23012
(Stations and ordinates given in percent of airfoil chord).
LOWER SURFĄCE
UPPER SURFACE
STATION
ORDINATE
STATION
o
1.25
2.5
5.0
7.5
10.0
15.0
20.0
25.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
95.0
100.0
100.0
2.67
3.61
4.91
5.80
6.43
7.19
7.50
7.60
7.55
7.14
6.41
5.47
4.36
3.08
1.68
0.92
(.13)
ORDINATE
o
o
1.25
2.5
5.0
7.5
10.0
15.0
20.0
25.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
95.0
100.0
100.0
-1.23
-1.71
-2.26
-2.61
-2.92
-3.50
-3.97
-4.28
-4.46
-4.48
-4.17
-3.67
-3.00
-2.16
-1.23
-0.70
(-0.13)
o
Figure 24.
Aerodynamic Properties
When an airfoil experiences fluid motion. there exists a pressure pattem similar to that of Figure 18.
If there is camber to the airfoil, there will be a difference in the pressure panem of the upper surface
from that of the lower surface. Fmthermore, if the airfoil is futed at some angle to the airflow. the
pressure distribution, hence, the velocities over the surface will be altered. The angle between the
freestream velocity, V 0, and the chord is called the angle of attack and is given the Greek symbol
Alpha, a. See Figure 25.
06-1420
March 1991
1.59
AERODYNAMICS
·v.o
.
- a
ł
Figure 25.
The pressure distribution on the airfoil at some angle of attack will appear as shown on Figure 26.
RESULTANT
Figure 26.
The net result of the static pressure distribution over the surface is a lifting force. This resultant
force is conveniently represented by a single force of some magnitude acting at a point on the chord
called the center of pr~ure, or C.P.
1. 60
D6-1420
March 1991
AERODYNAMI CS
The resultant force can now be resolved into two component forces, one component, lift, is
perpendicular to the freestream velocity and the other component, drag, is parallel to it. The
resultant is thus replaced and is no longer considered. The forces on the ahfoil now appear as in
Figure 27.
lift--- R
I
I
I
I
I
I
Figure 27.
It is known that the center of pressme will move with changes in angle of attack. At low angles of
attack. the C.P. is located near the quaner chord of the ahfoil, and as the angle of attack is increased
the center of pressure moves forward. Thus, it is desirable to have some fixed reference point to
which the force system can be transferred. If this is done, compensation must be made on the
airfoil for the change. A moment is introduced about the fixed reference point which will exactly
compensate for the change in moment caused by moving the lift and drag vectors from the center of
pressure. The quaner chord point is commonly selected as the reference point since the moment
coefficient about the quarter chord of most ahfoils is nearly constant. (The point at which the
moment is independent of angle of attack is called the aerodynamic center.) The magnitude of the
moment varies with camber and thickness as well as angle of attack. Figure 28 shows the
.transformation of forces.
06-1420
March 1991
1. 61
AERODYNAMI CS
lift
lift
Figure 28.
From the application of equation (95) it is possible to evaluate the lift and drag forces, and the
moment from equation (97).
Equation (95),
F=CrqA
For lift:
L=CLqS
where,
(104)
L
is lift fmce, lb
CL is lift coefficient, dimensionless
S
is wing area, ft2
q
is dynamie pressure, lb/ft2
A representative area appears in equation (95). With reference to a complete wing the most
reasonable representative area is the wing area. Similarly, the drag equation may be written:
(105)
where,
D
Co
is drag force, lb
is drag coefficient, dimensionless
Finally, the moment equation may be written:
I.62
D6-1420
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AERODYNAMICS
Equation (97),
For the wing,
(106)
M=~qSc
where,
M . is moment, ft-lb
Cm is moment coefficient, dimensionless
c
is chord length, ft
The representative length, l, is the chord length for pitching moments (moment about the pitch axis)
or the MAC in the case of a nonrectangular wing.
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.
1.63
AERODYNAMI CS
lntentionally Left Blank
1. 64
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AERODYNAMICS
1-16 Viscosity Effects
A short d.iscussion was made in a previous chapter of the viscous effect on the fluid velocity. Of
particular interest in aerodynamics is the thin and everpresent boundary layer which is the
consequence of fluid viscosity. Any changes in the character of the boundary layer with Reynolds
number produce changes in the force coefficients. These phenomena are referred to as Reynolds
number effects or scale effects.
·
The type of flow in the boundary layer depends upon the smoothness of the fluid flow approaching
the body, the shape of the body, the suńace, the pressure gradient in the direction of flow, and the
Reynolds number of the flow. There are two basie types of boundary layers; the laminar boundary
layer, and the turbulent boundary layer. Their d.istinguishing characteristics are shown in Figure
29.
transition
region .
~
laminar
region
) I
()
;;_J../ ,--
V
(
V
V
Figure 29.
Consider the fńctional drag associated with the flow of fluid over one side of a smooth fiat plate.
The flow is parallel to the surface and is depicted in the figure. The flow of fluid immediately
downstream of the leading edge is smooth and is known as laminar flow. Further downstream, the
thickness of the layer increases as more and more air next to the surface becomes affected. The
thickness of this layer is usually expressed in terms of Reynolds number.
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March 1991
1.65
AERODYNAMICS
~
_ 5.2( l)
L-
(107)
'\/'RN"
where,
is laminar thickness, ft
l
is length from leading edge, ft
RN is Reynolds number (pV l )/µ), dimensionless
~L
As the air progresses downstream along the plate a point is reached where the laminar flow breaks
down, the flow becomes unsteady, and the thickness increases rather suddenly. This is a transition
region and varies considerably in character depending on the turbulence of the remote airflow and
the smoothness of the surface. The quantity which defines the transition is the Reynolds number.
Under ideał conditions, the Reynolds number for the transition region is approximately 530,000, as
shown in Figure 30.
·
NOTES:
1. Smooth Flat Plate
2. No Pressure Gradient
3. No External Turbulence
oo .010
......
zw
turbulent flow
ou:
u.
w
oo
C,
transition
.005
<
a:
o
laminar flow
o
o
1
2
4
3
5
6
7
6
REYNOLDS NUMBER, 1O
Figure 30.
Tuus,
· pVl
RN= - µ
= 530, OOO
Beyond the transition region, the air becomes more turbulent since there is considerable fluid
particle motion. This region is the turbulent boundary layer and its thickness is expressed as:
1.6 6
D6-1420
May1989
AERODYNAMI CS
_ 0.37( 1)
6 T-
.J'RN
(108)
The flow on a body other than a fiat plate will be changed, primarily due to the shape. Because of
the shape, the velocities over the surface change from the free stream velocity values. This, in tum
produces negative and positive pressure gradients on the forward and rearward sections of the
body, respectively. A pressure gradient is the slope of the static pressure line on a surface, or the
rate of change of static pressure with distance. On the forward portion of an airfoil the static
pressure decreases. This means that an air particle will feel a pressure gradient which is tending to
assist the particle motion, or accelerate it This. is referred to as an assisting pressure gradient.
However, over the rear portion of the airfoil, the opposite is true. The static pressure increases with
the result that the particle feels as though it were being resisted. This increasingly positive pressure
gradient is thus called an adverse pressure gradient See Figure 31.
An assisting pressure gradient seeks to increase the air velocities in the boundary layer. An adverse
. pressure gradient decreases the air velocity in the boundary layer, particularly the air immediately
adjacent to the surface, where the velocity is initially the lowest. When the adverse pressure
gradient becomes large enough, separation takes place. This means that a finite layer of air at the
surface is actually stopped relative to the surface. This layer upsets the continuity of the airflow and
an eddying condition is set up which results in air flow separation.
Static Pressure
Distnl>ution
----
ASSISTING PRESSURE GRADIENT
Pressure becoming more negative
Velocity increasing
ADVERSE PRESSURE GRADIENT
Pressure becoming mare positive
V ~
V
Figure 31.
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March 1991
1.67
AERODYNAMICS
In Figure 32, two angle of attack conditi.ons are shown. When comparing the two pressure
distributi.ons, the adverse pressure gradient over the rear portion of the airfoil is much less at the
low angle of attack than at the high angle. Thus, separation occurs more easily at high angles than
at low angles of attack, and the high angle of attack separati.on is primarily instigated by the
boundary layer characteristi.cs.
The flight Reynolds num.bers on airplanes as large and as fast as commercial jet airplanes are so
large that transition from laminar to turbulent flow occurs almost immediately on the leading edges
of all surfaces. Thus, all the flow is considered turbulenl
An important considerati.on in the control of boundary layer is the layer thickness. Figure 29 shows
the boundary layer grows in thickness as it proceeds rearward. This makes it more important to
maintain surface smoothness near the leading edges of surfaces than farther aft This is because a
surface irregularity will cause a greater disruption to the normal flow when it is in a region of thin
boundary layer than if it were farther aft and could be more easily buried in the thicker boundary
layer. Thus, to reduce skin friction drag, the aircraft surface should be kept free from
disconti.nuities, especially in the more forward regions of all surfaces.
L
Low Angle of Attack
...
High angle of Attack
...
Figure 32.
Another point which should be mentioned is that the boundary layer represents dead air which has
had energy removed from it by internal friction within the layer. For example, this means that air
taken into the engine will have this layer of dead air if there is an appreciable surface ahead of the air
intake. Pod mounted engines air intakes have essentially zero length, so this problem is not
encountered. On aircraft with fuselage side intakes, a boundary layer bleed system must be used to
eliminate this dead air from the air taken into the engine where it would reduce efficiency.
1.6 8
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May1989
AERODYNAMICS
1-17 Lift and Drag
If an airfoil section or wing is operating under fixed conditions of density and velocity, the lift and
drag will change if the angle of attack is changed. If the lift and drag change, the coefficients of the
lift and drag must also change in order for equation (104) and (105) to be valid. If an airfoil section
is operating at a low angle of attack and then at a high angle of attack, the lift and drag is smaller in
the first instance than in the second. Figure 32 shows the pressure distribution and flow panem for
these two conditions.
INCREASED
SEPARATION ""'\.,,
,.
LIFT
COEFFICIENT
CL
FLOW
SEPARATION
BEGINS
ANGLEOF
MAXIMUM
LIFT
ANGLEOF
ZERO LIFT
(-)
o
(+)
ANGLE OF ATTACK ( a )
Figure 33.
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1.69
AERODYNAMICS
Theoretical flow considerations shows that lift (or lift coefficient) at constant velocity increases
uniformly (or linearly) with an increase in angle of attack. Test data on airfoils usually show a
linear variation of lift coefficient, Cv with angle of attack, a, through a considerable range of
a.ngles. Ultimately, however, the airfoil will reach an angle where the air flowing over its top
surface will have difficulty remaining in contact with the surface. At this point airflow separation
begins, and the lift curve will depart from linearity, see Figure 33. As the angle of attack is
increased further, airflow separation increases until a maximum lift coefficient value is reached,
CLmax . Beyond this point CL decreases, either abruptly or gradually, depending on the airfoil.
This phenomenon is called stalling , and the angle corresponding to CLmax is called th_e stall
angle.
A diagram of the approximate flow condition at the stall angle is shown in Figure 34. This shows
the turbulent eddying wake characteristic of separated airllow.
The other end of the lift curve (Figure 33), the point at which CL is zero, does not occur at zero
angle of attack, but at some small negative angle. This is characteristic of an air foil with camber.
The greater the camber the more negative is the angle of zero lift. For an uncambered, or
symm.etrical airfoil, the angle of zero lift is zero.
Figure 34.
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AERODYNAMI CS
The slope of the linear portion of the lift cmve is of interest, mathematically.
CL =ma+b
When,
Thus,
O=maL +b
o
or,
b=-ma
LO
Therefore, after substi.tuting for b,
CL =ma-maL
o
l
CL =aca-aLo) 1
CL =m{a-aLo)
(109)
The letter "m" is usually written when the slope is measured per radian, and the letter "a" is
written when the slope is measured per degree.
If the relati.onship between drag coefficient and angle of attack is shown, it will appear considerably
different from the lift coefficient and angle of attack curve. Below the stall most airfoils produce the
characteristic parabolic shape as shown on Figure 35.
DRAG
COEFFICIENT
Co
(-)
o
(+)
ANGLE OF ATTACK (a)
Figure 35.
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May1989
1.71
AERODYNAMICS
During the accumulation of wind tunnel test data on an aińoil, lift and drag data are recorded for
constant values of angle of attack. When applying this information to aerodynamic problems,
particularly those concemed with performance, it is more useful in another form. Angle of attack,
a, is eliminated and a curve showing the relationship of lift coefficient to drag coefficient is used.
The curve is commonly called the drag polar , and is parabolic in shape. This cwve is shown in
Figure 36 for a wing, but the drag polar for the complete airplane will have a simi]ar shape.
Figure 36.
1.72
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AERODYNAMI CS
There is another curve which can be derived from Figures 32 and 35. If CL is divided by C0 at
every angle of attack up to the stall, a new curve may be drawn as in Figure 37.
L
Jr
<X FOR (
~)
MAX
(X
Figure 37.
~
I
Figure 38.
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May1989
1.7 3
AERODYNAMI CS
From equations (104) and (105),
(110)
The UD ratio becomes a maximum at a certain angle of attack. The significance of this point is that
it represents the most efficient operation of the airfoil. It is the point of most lift for the least drag.
A slightly different approach to the determination of maximum L/D is shown in Figure 38. The
CrfC0 ratio can be found at severa! points along the curve. If lines through the origin connect with
these points, the slope of each line will define a Ci./Co ratio. The maximum possible slope will be
defined by a line from the origin tangent to the curve. The point of contact with the curve will
define the CL and C0 values for (L/D)max. The reason for being concemed with (L/D)max will
become more apparent in later discussions.
1.74
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AERODYNAMICS
1-18 Drag Analysis
In the previous section, it ~as noted the relationship between CL and C0 is parabolic in shape.
,,
2~/.
Co = A+ BCL
~
Airplane
" - - - Drag Polar
I
Figure 39.
The drag coefficient of the airplane may be expressed with sufficient accuracy by a parabolic
equation of the form:
(111)
The constants A and B are usually found by wind tunnel tests. A depends mostly on the airfoil
section and its Mach number and Reynolds number; B depends mostly on the aspect ratio of the
wing. Figure 39 shows the close agreement of an airplane polar with the parabolic expression.
It is convenient to break the airplane drag into components atnibutable to various causes. The ba.sic
drag equation (in coefficient form) is:
Co=Co + CD. +CD
·P
1
(112)
M
where,
Co
is total airplane drag coefficient
Cop
is parasite drag coeffi.cient
Co.
is induced drag coefficient
c~
is compressible drag coefficient
1
1>6-1420
May1989
1.7 5
AERODYNAMI CS
Parasite Drag
Parasite drag, is the drag of any part on the airplane not contributing to useful lift. The para.site
drag of a wing is usually referred to as profile drag, the result of pressure distribution and skin
friction, and is denoted as C 00. The parasite drag on the remaining components of the airplane
(fuselage, tall, nacelles, etc.) is called structural drag and is denoted as C08. This type of drag is
attributed to several causes, some due to the pressure distri.bution on the body (sometimes called
form drag), some due to skin friction drag, and some due to aerodynamic interference. The entire
parasite drag of the airplane may be expressed as:
(113)
where,
CDP
is parasite drag· coefficient of the airplane
CD
is parasite drag coefficient of the wing (profile)
O
CDs
is parasite drag coeffi.cient of the remaining airplane components
The relative proportions of form drag and skin friction drag depend upon the shape of the body.
Those bodies having predominantly form drag arc referred to as bluff bodies; those with
predominantly skin friction drag arc called streamlined bodies. A typical variation of drag with
body shape is shown in Figure 40. Herc a body of revolution with a given diameter is
progressively stretched out and the drag measured. Since the maximum cross sectional area is
constant, the reference area for computing the drag coefficient is chosen at this cross sectional area.
The minimum drag coefficient occurs at an l Id ratio (length to maximum diameter) of about 2.5.
The l Id ratio is referred to as fineness ratio. For the lower fineness ratios, separation occurs,
giving a relatively large form drag, hence a relatively large drag coefficient. For the larger fineness
ratios, the large exposed skin surface area of the body causes relatively large skin friction, hence a
relatively large coeffi.cient. The drag characteristics of a body of revolution arc of interest because
of their application to fuselages and nacelles.
1. 76
D6-1420
March 1991
AERODYNAMICS
V
„ C>
(Basedon
Frontal area)
OIJ!----~---a!o---------..----~
2
4
6
Length/Maximum Diameter
Figure40.
Another factor which contributes to drag force is interference drag. lnterference drag exists from
the change in flow panem that accompanies the placing of two bodies in close proximity. Thus, the
total drag of the two bodies placed close together is generally different from the sum of the
individual drags. Quite often this interference is unfavorable, causing an increase in drag over that
experienced by the separate bodies. It thus becomes necessary for the designer to find the
appropriate geometrie layout of the airplane components which will result in a minimum of
inteńerence drag. The intersection of the wing with the body is usually troublesome in this respect.
A high wing airplane produces little or no inteńerence. Low wing airplanes, however, have a
marked tendency for separation at the root at high angles of attack. This is because the intersection
requires an expansion of the streamlines near the trailing edge on the top surface, which means a
deceleration of the air and a tendency toward separation. A common method of alleviating this
condition on the low wing airplane is the use of a fillet at the intersection. The determination of the
location and amount of interference is most easily made in the wind tunnel through the analysis of
force data and airflow tuft observations.
·
There are other ways of denoting drag. In Figure 40, the drag coefficient was based on the frontal
area. Since an aerodynamic coefficient is completely arbitrary, it may be based on any convenient
area. The use of the frontal area as a basis of the drag coefficient is quite common for a body such
as a fuselage or nacelle. In this case, it is called the proper drag coefficient, denoted by the
subscript, x.
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1. 77
AERODYNAMI CS
Thus,
D=CD S q
7t . 7t
(114)
where,
D
is drag of the body, lb
CD7t is proper drag coefficient, dimensionless
S7t
is frontal area, ft2
q
is dynamie pressure, lb/ft2
The procedure of using an arbitrary area on which to base a drag coefficient may be extended still
further. The drag of a body is directly J>roportional to the dynamie pressure, q, or:
Doc q
Thus,
D=fq
where,
f is the constant of proportionality
. From form.er equations,
D=C Sq
0
Then,
D=CD S1t q
1t
(115)
The constant, f, is referred to as the equivalent parasite area , and is convenient since it embodies
no characteristic area. The usefulness of this is that a body, such as a nacelle of given dimensions,
will have one value of f regardless of the airplane to which it is applied, assuming no change in
interference drag.
Another method of denoting drag is by the skin friction drag coeffi.cient. After a body is lengthened
past a certain point its drag coefficient based on frontal area increases. This increase is proportional
to the increase in skin surface area. The drag characteristics of a long slender body will be
essentially those of a fiat plate with the same surface area as the body set parallel to the remote
velocity. Thus, instead of using the frontal area as a reference for the drag coefficient, the surface
area could be used instead. Figure 41 is a plot showing drag based on wetted or skin surface
area. This shows that the drag coefficient approaches a value of about .003, which is the value
found to exist on a fiat plate. The drag equation may be written:
1. 78
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Marc:h 1991
AERODYNAMI CS
(116)
where,
Cr
is the coefficient based on wetted area
Aw is wetted area, ft2
.02
.Ol
(Basedon
Wetted Area)
o--------....
0
2
4 -----------6
8
10
Length/Maximum Diameter
Figure 41.
lnduced Drag
The drag induced by the wing is called induced drag , Cor The total wing drag consists of
profile and induced drag. Written in coefficient form:
Cow = Coo + Coi
where,
C0
w
(117)
is total wing drag coefficient
C00
is profile drag coefficient
C0 .
is induced drag coefficient
1
Induced drag is the result of an angle of attack which produces positive lift from the airflow over the
airfoil. For this condition, the air pressure on the upper surface is less than the air pressure on the
lower surface. The pressure differentia! causes the air to flow toward the area of lower pressure.
This occurs at the wing tips. Air flowing backward over the upper and lower surfaces of the wing
is influenced by the acti.on at the wing tips. The action is rearward and inward on the upper surface,
and rearward and outward on the lower surface. This is shown in Figure 42.
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March 1991
1. 79
AERODYNAMI CS
Low
(-+
+
+
+
Pressure
+ +
+ +
High
Pressure
+
+
+
+)
Leading Edge
Trailing Edge
Figure 42.
The transverse flow is most predominant at the wing tips, decreasing inboard until nullified at the
wing centerline by an equal and opposite transverse flow generated by the other tip. The wing (in
Figme 42) is viewed from above. The solid lines denote flow on the upper surface and dashed
lines denote flow on the lower swface. With sucb a condition, the air iroroediately bebind the wing
will bave a swirling or vortex motion, most predominant at the wing tips, and less intense
inboard The net effect of these vortex systems, or trailing vortices, is to cause a downward
inclination to the air leaving the wing. The average angle tbrough whicb the velocity vector is
rotated is the downwash angle , denoted by the Greek letter Epsilon, e . The air flow pattern
about a section aińop. appears as in Figure 43.
If an a.irplane bad an infinite span, the transverse flow at the tips would influence only a very small
portion of the wing. The graduał decrease of the cross flow effect toward the center results in a
zero downwash angle at the wing centerline. The influence of the transverse flow at the tips (tip
vortices) on the rest of the wing depends primarily upon the aspect ratio of the wing.
1.80
D6-1420
March 1991
AERODYNAMICS
...-~f:----
v - - -....
C
v----~
Figure 43.
For a finite aspect ratio, the downwash angle is of definite measurable magnitude; and for an infinite
aspect ratio (finite chord but infinite span) the downwash angle may be assumed zero.
The actual spanwise distribution of downwash is a function of the planform of the wing. A
rectangular wing, for example, would have a much higher concentration of vonex action at the tip
where the chord is large, than a tapered wing whose tip chord is relatively small.
Consider a stream of fluid moving across the span of a :finite wing being deflected some average
angle, e. Newton's law may be applied to the deflected stream of fluid to determine the forces
exerted on the airfoil.
From equati.on (6),
and equation (22),
or,
F=Ma
a= dV
dt
a- -Il.V
ll.t
then,
M
F= ll.t Il.V
D6-1420
. May1989
(118)
1.8 1
AERODYNAMICS
Consider that this stream of fluid of some arbitrary but as yet undefined cross sectional area is
moving with an average velocity, V, past the wing. The mass per second, (Mit), passing any given
point is equal to pAV.
1;1 = pAV
The change in velocity of the fluid is shown in Figure 44. Since the stream is deflected through an
angle e, a vector diagram may be drawn.
v·---
V
łlV
Figure44.
This diagram shows that for the initial velocity vector to be changed into the final velocity vector, a
change in velocity vector must be added in accordance with parallelogram of forces analysis. The
magnitude of the velocity change is:
llV= Vsine
Since e is small, sin e is equal to e . Thus,
llV= Ve
Equation (118) may now be written,
F=pAV_Ve
In this instance, the force under consideration is lift; therefore:
L=pAv2e
.AlS() from equation (104);
L=<i
1.82
1 - 2
pV S
2
D6-1420
Mayl989
AERODYNAMI CS
Equating the two, results in,
or,
AE=
i<;_ S
(119)
The area, A, in the equation, is the cross section of the stream of fluid affected by the wing enclosed
by a circle whose diameter is the span of the wing. This does not mean that only this area is
deflected, but rather that it is an equivalent area which may be considered deflected through a
constant downwash angle, E.
Therefore,
2 .
A _JLJL
-
4
From equation (100),
Substituting in equation (119) gives,
2
1 .
b2
-4- E=2 CL (AR)
1t b
2CL
E = 1t(AR)
(120)
This equation shows the downwash is a functi.on of both lift coefficient and aspect ratio.
Consider now a wing of infinite aspect ratio set at a geometrie angle of attack, a, as in Figure 45.
For convenience, let the lift and drag coefficients be denoted as Cl.o and CDo for this ·condition.
From previous assumptions it is known there will be no downwash, thus the velocity vector behind
the wing will have the same direction as that in front
Now subject the wing to exactly the same geometrie angle of attack, but with some finite aspect
ratio instead of infinite aspect ratio. The velocity field in the region of the wing now consists of a
curved flow, with the velocity vector behind the wing deflected through an angle E. This curved
flow may be represented by an effective linear velocity which bisects the downwash angle and
denoted as VEFF in Figure 46. The effecti.ve angle of attack is now less than the geometrie angle of
attack, a, by half the downwash angle. Since this angular reduction is induced by the lift on the
wing, it is called induced angle of attack, <Xj.
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1.83
AERODYNAMICS
AR
V
..
=.
co
-L
V
Figure 45.
Finite AR
VEF'F
V
....
....
....
Figure 46.
Tuus,
.(121)
Now let the geometrie angle of attack of the finite aspect ratio wing be increased until its effective
angle of attack is the same as the angle of attack of the infinite aspect ratio wing. See Figure 47.
Therefore,
(122)
is effective angle of attack
where,
Cl·1
1. 8 4
is induced angle of attack
D6-1420
May 1989
AERODYNAMICS
--~
VEFF
V
Figure 47.
Effectively, the only difference between Figure 45 and Figure 47 is that Figure 45 has been rotated
through an angle cxi; thus CL and Co0 may be made perpendicular and parallel, respectively, to
0
V EFF· In order to conform to the standard practice of resolving the lift and drag coefficients
perpendicular and parallel, respectively, to the remote velocity, the coefficients CI.o and Co0 must
be resolved into components in those directions. This is done in Figure 48.
V
Figure 48.
The resolution of Co into its two components results in a horizontal component essentially equal to
0
Co , and a vertical component wbich may be ignored due to its small size relative to the magnitude
0
of CL and CL The resolution of CL0 into its two components results in a vertical component
0
essentially equal to CLo, and a hori.zontal component of such large proportions that it may be greater
than the profile drag of the wing. To this horizontal component is given the name "induced drag
coefficient", and is related to the induced angle of attack according to the following analysis. A
portion of Figure 48 is enlarged in Figure 49 to show the mathematical relationship.
D6-1420
May 1989
1.8 5
AERODYNAMICS
CI>i
V
Figure 49.
From Figure 49,
Co.
1
tan a.=-1
CL
If ai is small, and expressed in radians, tan ~ will equal the angle, ai
Therefore,
(123)
where,
C0 i
is induced drag coefficient
Ci.
is lift coefficient
ai
is induced angle of attack, radians
Equation (120) and (121) may be combined to give anotherrelationship for a i,
(124)
1. 8 6
D6-1420
May 1989
AERODYNAMICS
This equation is now used with equation (123) to give another relationship for Cnr
C
-
c2
L
Di - 1t(AR)
(125)
Figure 50 shows the effect of aspect ratio on the lift curve and drag polar curves of an airfoil.
The infinite aspect ratio characteristics are optimum for a given airfoil section since the least drag
occurs for a given lift coefficient. The aspect ratio chosen by the airplane designer is govemed by a
compromise between structural and aerodynamic considera.tions, and by the purpose for which the
aiiplane is designed,
Compressibility Drag
Compressibility drag is that part of the total drag due to the compressibility effects coexistent with
high speed. To show the effects of compressibility it is convenient to associate the flow of fluid at
high velocities around an airfoil section with that flowing through a channel. In an earlier chapter, a
discussion was made involving shock wave phenomena and the accompanying pressure and
velocity changes. The same phenomena occur in the flow over an airfoil.
Air flowing over an airfoil surface may attain locally sonie velocity. Local speeds greater than the
speed of sound may, and frequently do, exist on the airfoil surface because the rear portion of the
airfoil, by its geometry, requires an expansion of the stream tube. The conversion of this very high
speed air to the lower speed at the tra.iling edge produces a shock similar to that in a channel.
Unfortunately, the sudden rise in pressure through the shock wave causes separation of airflow,
and the effect leads to a generał decay of the aerodynamic chara.cteristics of the airfoil.
D6-1420
May1989
1.8 7
AERODYNAMICS
Figure 51 shows a view of the boundary layer ahead of and behind the shock wave. There is an
interaction between the shock wave and the boundary layer such that the layer thickens rather
abruptly on passing through the shock wave. This thickening, plus the adverse pressure gradient
across the shock wave, is sufficient to cause the boundary layer to separate just as it does in the case
of a high angle of attack stall This separation affects both the lift and the drag characteristics of the
airfoil.
NEGATIVE
PRESSURE
MORE POSITIVE - - - ~
PRESSURE
SHOCK WAVE
Figure 51.
The lift under given conditions of angle of attack will increase with Mach number according to the
following expression:
(126)
where C4nc is the lift coefficient for incompressible flow and Mo is the remote Mach number.
This relationship, as shown in Figure 52, is reliable up to the point where separation begins.
Assuming separation induced by shock wave formation is taking place, Figure 52 will be modified
as indicated by the dashed lines. The reason lift decreases at high Mach num.bers is that the shock
wave ca.uses flow separation with the resultant loss in lift.
·
1.8 8
D6-1420
May1989
AERODYNAMI CS
.3
CROSS-PLOT OF CL VS a
AT GIVEN ANG LE OF ATTACK
WITH M EFFECT,
- - N O SEPARATION
---WITH M EFFECT
& SEPARATION
WITH M EFFECT,
BUT NO SEPARATION
--...:.WITH M EFFECT
& SEPARATION
.3
o;
.4
.5
.6
.7
.8
.9
M
Figure 52.
The effect of the separation is reflected in the polar curve, Figure 53, which is representative of the
airplane polar curve.
If Figure 53 is cross plotted at a constant lift coefficient, a curve showing the magnitude of the
compressibility drag coefficient, CDM' is obtained. This curve, Figure 54, shows the typical "drag
ńse" curve shape which denotes the onset of shock wave separation.
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May 1989
1.8 9
AERODYNAMI CS
CROSS-PLOT OF CL VS Co
AT GIVEN LIFT COEFFICIENT
.2
Co
.3
.4
.5
.6
.7
.8
.9
M
Figure 53.
Figure 54.
The lowest remote velocity at which sonie velocity (M = 1) is attained on any part of an airfoil is
called the "critical Mach number". This does not mean that a dangerous condition exists on the
airfoil, but rather that it represents a limit below which no shock waves can occur. The pressure
coefficient associated with the critical Mach number is called the "critical pressure coefficient". It is
possible to predict w hat pressure coeffi.cient must exist to produce a local Mach number of unity.
Consider the airfoil in Figure 55.
p
M
....
C
Figure 55.
1.90·
D6-1420
May1989
AERODYNAMI CS
The pressure coefficient, developed in a previous chapter as equation (84), remains:
From compressible flow theory, for M = 1 on the airfoil, the cri.tical pressure coefficient is:
J
_2_ [ (-2- + y- 1 ~) :-1 _ l
yM~ y+l y+l o
(127)
This equation, when pl~tted, appears ~ in Figure 56.
The changes in density accompanying changes in static pressure produce a pressure distribution
which is different from that predicted for incompressible fluid. The development of equations to
show modifications to the incompressible pressure distribution is, in generał, a rather elaborate
procedure and will not be discussed here. The results, however, are shown.
For an incompressible fluid, the pressure at the stagnation point is greater than the remote static
pressure by the amount of the dynamie pressure q.
Tuus,
where,
<'
is the pressure coefficient at the stagnation point.
APs
is the change in static pressure at stagnation point from free stream
value.
-Ps
is the freestream dynamie pressure.
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March 1991
1.91
AERODYNAMI CS
-8
-7
-6
\
-5
'1,CR -4
-3
\
\"
~
-2
~~
-,
o
.3
.4
~ r--..._
---
-------
.7
.5
.8
.9
,.o
Figure 56.
For a compressible fluid, the pressure at the stagnation point is increased due to compressibility,
according to the following equation:
C
Ps
= 1+ lM2-+ 2-y M4 +·········
4
O
24
O
For most practical applications, this may be expressed as:
~
-Ps
= 1+ lM2
4
O
(128)
For the remaining pressure on the airfoil, a commonly used equation is employed. It does not apply
accurately where the local velocity on the airfoil is considerably different from the freestream
velocity; hence, it does not apply at the stagnation point or on the upper and lower surfaces of an
airfoil at high angles of attack. The equation is:
(129)
1.92
1)6.1420
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AERODYNAMICS
Also, since the lift coefficient depends upon pressure coefficients, equation (126) may be applied
directly for a similar prediction of lift coefficient, Cv and slope, m, of the lift curve. The lift
equation appeared as equation (126), but the slope of the lift equation is:
m-
m.
me
-J1-M
.
o
(130)
2
Figure 57 shows the effect of compressibility on the pressure distribution at low and high Mach
numbers. The difference between the two pressure distributions is noticeable, especially on the
upper surface. For the high Mach number diagram, the flow is supersonic for a considerable
portion of the upper surface and :finally becomes subsonic by means of passing through a shock
wave. The same angle of attack is considered in both cases.
When designing a new airplane it is important to accurately estimate the lift and drag characteristics
under all conditions. The aerodynamicist depends largely on three sources for this information.
Initially, data obtained on other airplanes or previous NACA test results may be used. Wind tunnel
models are built and tested to verify previously estimated data, and also to investigate certain design
features for which there has been no previous data available. The basie drag polar is obtained for
the complete airplane from wind tunnel tests. The drag polar describes the lift and drag
characteristics of the model which is geometrically similar to the full scale airplane; however, it is
not generally used to represent the actual airplane directly. The shape and spacing of the curves
relative to each other are used directly, but the intersection of the low Mach number
(incompressible) polar with the C0 axis is obtained by other means. The reason for this is that past
experience has shown wind tunnel data will accurately predict the polar shape and drag rise
characteristics but will not predict the basie drag level . One reason for this is it is difficult to
accurately separate out the drag and interference effects of the model suspension system.
o Ił----------=.._.-~
+
Low Mach Number
o
.2
.4
.6
.8
High Mach Number
1.0
o
Fraction of Chord
.2
.4
.6
.8
1.0
Fraction of Chord
Figure 57.
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May1989
1.9 3
AERODYNAMICS
of airplane
Figure 58.
The concept of drag level may be scen best in Figme 58 where a low-speed drag polar is plotted on
axes of CL2 versus c0 . The curve becomes practically a straight line. This happens because the
parabolic shape of the drag curve is pńmarily caused by the induced drag of the wing. Equation
(111) suggests this presentation:
From equation (112) and (125), an expression for the low-speed drag polar may be written:
c2
L
(131)
In Figure 58, there appears some deviation from the mathematical representation at low and high
coefficients. If a representative straight line is drawn through the curve and extrapolated to zero Cv
the intersection will define a minimum C0 , which is composed only of Cnp, since the induced drag
becomes zero at zero CL. If a line is plotted showing the variation of C0 i only with CL2 , it is
straight and has a slope steeper than C0 versus CL2. Graphically, the difference between these two
lines represents Cnp, and since the slope of the induced drag line is steeper, the minimum value of
Cop occurs at zero. CL for symmetrical airfoils. This value is known as C 0 PMIN and is the drag
level referred to previously.
1. 9 4
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May 1989
AERODYNAMICS
For cambered airfoils, C ~ occurs at low values of CL depending_on the amount of camber, but
not at zero CL. A transport airplane is so designed that c0 PMIN occurs at a CL approaching
optimum cruise condition. To illustrate this, Figure 59 shows a low-speed drag polar and in
addition a curve of the parasite drag coefficient The difference between the two curves represents
the induced drag. The minimum C0p occurs at a finite value of CL and is very close to the optimum
UD.
Figure 59.
The determination of CDI>wN is made by noting first that pure parasite drag is a function of form
and skin friction drag. It is convenient to compute the Cop of various parts of the airplane by
assuming the C0p is a function of skin friction only. By analyzing past flight data from similar
airplanes, various values of skin friction drag coefficient, Cr, arc assigned to cach part of the
airplane. V arious values arc required due to shape and fineness ratio differences. In general these
values average out to about .003, as was indicated in Figure 41. Thus, to compute CI>i>wN it is
necessary to select the·proper Cr and since the skin wetted area is known, the f for any section of
the airplane can be computed. 'The fMIN of the total airplane is equal to the sum of the various
equivalent parasite areas of the components. Thus, Cop
can be computed. In the above
.
calculations the following relationships arc assumed:
MIN
or,
(132)
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May 1989
1.9 5
AERODYNAMICS
Tuus, having computed the minimum f of the airplane, the wind tunnel drag polar can be shifted to
agree with this value, and the full scale drag polar will be obtained. V erification of all previous
estimations and of wind tunnel tests is made in airplane construction and flight testing.
2
2
From Figure 58, the slope of the CL versus c0 i cmve is greater than of the CL versus C0 curve.
If it were possible to shape the airplane so no increase in parasite drag occurs with angle of attack
increase (and thus CL) the two curves would be parallel. The ratio of the two slopes would then be
2
unity, or the slope of L versus c0 would be 100% of that of CL2 versus c0 i. The ratio of these
slopes is the airplane induced drag efficiency factor and is denoted by the symbol "e". Fot most
airplanes it is difficult to approach a 100% efficiency factor, and values of around 80% are
common. In terms of calculus nomenclature:
.
(dei)
dC
~J
(133)
e= ( dCL
dC 0 _
1
Equation (131) may now be modified to read:
c2
L
(134)
where, CDpe· defined as the equivalent parasite drag coefficient, is shown in Figure 58.
1.96
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AERODYNAMICS
1-19 Planform Effects
As was mentioned previously ,the wing planform has an effect on the strength of the vortex pattern
at any point on the wing, with a resultant effect on the downwash behind the wing. Downwash
affects the lift .distribution over the wing such as to approach an elliptic shape. For the theoretical
infinite aspect ratio wing, the lift distribution is constant across the wing as there is no downwash,
as shown in Figure 60.
1f 11tltf tllttttt
---.
INFINITE ASPECT. RATIO WING
.+
t t11t l 1l 11t t t .
FIIjITE ASPECT RATIO WING
Figure 60.
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May 1989
1.97
AERODYNAMICS
(1)
Rectangular
p
(2)
Elliptical
(3)
Tapered
p
p
(4)
Swept
~
PIANFORM
~
,.op
CLI
Pl& ~
~ .
LIFT DIS'IRIBUTION
,.op--
i.oj
CL,
I
C
L,
~
\
I
,.o~
CL
I
CONSTANT SECTION LIFT COEFFICIENT DISTBIBUTION
~
p
p
CCC!C stall
CC< stall
~
~
cc =stall
~
~ ~
~ ~
~
Figure 61.
1. 9 8
D6-1420
Mayl989
AERODYNAMICS
Variations in the planform of a finite wing produce varied spanwise load distributions. An elliptical
shaped wing incurs essentially a constant downwash behind the wing; hence, an elliptical lift
distribution. The entire wing tends to stall at the same time. Because a rectangular wing has a
larger downwash angle at the tip, the effective angle of attack is reduced, hence the tip sections are
the last to stall. On a tapered wing, the downwash decreases toward the tip section and the tip
section tends to stall first The effects are shown in Figure 61.
The tendency for the wing tip section to stall first is undesirable from a lateral stability standpoint.
V arious devices are employed to prevent or delay the stalling characteristics. One method of
alleviating an early tip stall is to physically reduce the angle of attack of the tip relative to the root by
geometrically twisting or "washing out" the tip. Another method is to make the airfoil section at the
tip more cambered than those inboard. This enables the airfoil to attain a higher CL toward the tip
. before stalling. Still another method is to install leading edge "slats" near the tip to assist in keeping
the tip from stalling.
A feature of swept back wings which should be mentioned at this point is the characteristic of
spanwise flow. Consider two chordwise sections of a swept back wing with a fairly high positive
lift, as in Figure 62. If these sectioris were viewed from a point near the wing tip and their
respective pressure distributions drawn, the picture in Figure 62 would approximate the siruation.
AB C
Figure 62.
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May 1989
1.99
AERODYNAMICS
Considering a spanwise section at point A, the static pressure disttibutions are such ·that a greater
negati.ve pressure (lower static pressure) exists inboard than outboard. This creates a tendency for
an air particle, when moving over the upper surface in the vicinity of the leading edge, to flow
inboard due to the lower static pressure. When point B is reached, however, the static pressures are
equal at the two stations, so the air pani.cle will merely travel aft. At point C, the situation is
reversed over point A; that is, the particle will tend to move outboard. An overall picture of particle
morion is shown in Figure 63. The condition described is apparent at fairly high lift coefficients,
but less obvious at lower lift coefficients since the pressure differentials are less. As the stall is
approached, however, the flow near the wing tip at the trailing edge is almost parallel to the trailing
edge. One method of eliminating this flow effect is to add wing fences at several locations on the
wing. This tends to keep the air flow going in a sttaight line direction.
Figure 63.
Simple constant section wings used to obtain the effects shown in Figure 61 are seldom considered.
In practical applications, section changes, variations in incidence, and other devices are used to
tailor the wing to fit the requirements.
1.100
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March 1991
AERODYNAMI CS
1-20 High-Lift Devices
There are severa! devices employed to increase CLMAX of a wing. This is done to delay the stall,
thus lowering the stalling speed. Initially, the choice of the airfoil determines the CLMAx·
Increasing the camber increases the CLMAX and simultaneously changes the angle of zero lift
PLAIN FI.AP
SPLIT FI.AP
c----------~-....a.-~
FOWLER FI.AP
~
SLCf.rl'ED FI.AP
~
--•- - - - - ~ C .~--:::..
DOUBLE-SLO'l'l'ED FLAP
.~
FIAP CONFIGURATIONS
Figure 64.
Having established the airfoil design, the most common method of increasing the lift potentia! is the
use of flaps. There are many types, some of which are shown on Figure 64. The selection of the
size and type is determined by compromise. The effect of flaps is similar to increasing the camber
of the airfoil. While the maximum lift coefficient is increased due to the flaps, the drag is also
increased. The effect of the increased drag is to decrease the speed; this is desirable during landing.
The use of flaps produces certain changes in the characteristics curves for the wing as shown on
Figure 65. There may be slight variations from those shown due to the various flap configurations.
D6-1420
Mayl989
1.1 O1
AERODYNAMICS
/
I
/
......
-
/--,
.4- flap extended
/,,c..---- flap
I
retracted
I
I
/
I
a:
Figure 65.
To further increase lift, and also to help balance the pitching effects of large trailing edge flaps, it
may be necessary to add flaps or similar devices to the leading edges of the wing. Such devices can
increase capability in either of two ways: increase camber and provide smoother flow (boundary
layer control) to suppress flow-separation tendencies. Adjusting the leading edge or putting on a
flap as shown in Figure 66 allows the wing to be rotated to a high angle without as much tendency
for the flow to separate. The flap leads the air smoothly over the leading edge without forcing it to
flow backward from a stagnation point under the leading edge. Without such help, the streamlines
going over the lead.ing edge, as shown by the dotted lines, lose so much energy they tend to
separate.
Figure 66.
1.102
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May 1989
AERODYNAMICS
More help is possible by introducing energy into the airstream just behind the leading edge where
the separation tendency is felt. This might be done with a separate energy source such as a suction
or blowing device, or it may utilize the energy of the airstream by taking high pressure air from
under the wing leading edge through a slot to the upper surface as shown in Figure 67. The
structure in front of the wing which forms the slot with the wing is called a slat. The slat may be
fixed, or movable to allow the slot to be closed with flaps up or at low angles of attack.
Figure 67.
Of particular interest is the improvement in the behavior of the airplane near the stall The maximum
lift coefficient is increased, providing a reduction in the stalling speed. This effect is shown in
Figure 68.
.-""'I.--+-- ef'f'ect of' slot
Figure 68.
D6-1420
May 1989
1.103
AERODYNAMICS
There are other devices similar to the trailing edge flap, but differ in that they do not conttibute to
lift. In fact, the purpose of these devices is to do just the opposite; either destroy lift, or add drag to
the airplane. These devices are spoilers, dive brakes, and speed brakes.
Also, it is advantageous to employ spoilers on the wings. These may be used either separately or in
conjunction with ailerons to produce roll. On very clean airplanes it may be necessary to provide
dive flaps or speed brakes to either limit or reduce speed during certain phases of flight, or increase
the descent angle. Typical examples of these devices are shown in Figure 69.
C
DIVE FIAPS
...............
I
SPEED BRAKES
SPOILER-SPEED BRAKES
Figure 69.
1.104
D6-1420
May 1989
AERODYNAMICS
1-21 ·High Speed Flight
The ad.verse effects of compressibility must be minimired for successful flight at high speeds. The
means for doing this has evolved from countless wind tunnel tests.
Compressibility difficulties will be minimized if the local velocities over the airfoil surface are kept
as low as possible. In other words, the air passing over the airfoil surface should be accelerated a
minimum possible amount, since the compressibility problems are a function of the local Mach
number. lf the airfoil is very thin and does not have any abrupt changes in contour, the above result
will be obtained.
Another variable is the amount and type of camber used for the airfoil sections. Stili another
method available is sweepback of the wing. Consider first two identical wings; one with no sweep
and the other with some arbitrary sweep angle, A , as shown in Figure 70.
Figure 70.
In both cases assume there is a velocity, V, acting perpendicular to the leading edge. In the case of
the swept wing this velocity is one of the components of the remote velocity which acts at an angle
·A to the leading edge, and of magnitude V/cos A. The other component is V tan A which is parallel
to the leading edge. This latter component has no effect on the lift and drag pressure forces since
the air flows along parallel lines of constant elevation. The pressure distribution is effected only by
the flow perpendicular to the leading edge. The free stream velocity can be increased by a factor
1/cos A over that of the straight wing for equal aerodynamic forces. This implies that the critical
Mach number of a wing with sweepback is increased over one without sweep by the factor of
1/cos A.
Although sweepback is beneficial for increasing the critical Mach number, the above factor is too
optimistic for predicting the effect in three dimensional flow. Such things as spanwise flow,
boundary layer thickening near the wing tips, and spanwise load distribution malce it impossible for
such a simple relationship to be valid.
D6-1420
March 1991
1.105
AERODYNAMI CS
The effect of sweep on the drag rise curve of an airfoil is shown on Figure 71. At the same lift
coefficient; one wing unswept. and the other with sweep, both having the same thickness
streamwise, there is a noticeable difference in the drag critical Mach number. The definition of drag
critical Mach number from the drag rise curves is somewhat arbitrary but is usually defined as the
Mach number ~t which a .0020 increase in Cn has occurred over the incompressible Cn.
Another device used to alleviate the effects of compressibility is the vortex generator. When airflow
separation due to compressibility occurs, it is always associated with the formation of a shock wave
resulting in an adverse effect upon the boundary layer and the airflow characteristics downstream of
the shock wave. To relieve this adverse effect, energy must be given to the air in the boundary
layer to accelerate the slow moving particles, thereby preventing separation. The vortex generator is
designed to do this job.
35° sweep
constant CL
M·
Figure 71.
The vortex generator is actually a small low aspect ratio wing placed vertically at some angle of
attack on the surface of the large wing. The generator will produce lift under these conditions, and
it will also have an associated tip vortex. Th.is vortex will be large relative to the generator since the
aspect ratio is small. The characteri.stic cork-screw shape of the vortex results in considerable
vertical height. An illustration of the vortex generator action is shown in Figme 72.
1. 106
D6-1420
May 1989
AERODYNAMICS
top view of
generator
Figure 72.
The generated vortex is taking relatively high energy air from outside the boundary layer and mixing
it with the low energy air in the boundary layer. The size and locati.on of the vortex generator must
be such as to penetrate through the boundary layer. The number of vortex generators and the
orientation on the wing depends on flight test investigation. The additi.on of vortex generators,
while beneficial in some respects, is not made without some cost. The small airfoils produce
additional drag, so the overall estimate of the worth of the generators must be considered
D6-1420
May 1989
1.107
AERODYNAMICS
Intentionally Left Blank
1. 108
D6-1420
May 1989
AERODYNAMICS
1-22 Airplane and Engine Parameters
For an airplane to be in level, unaccelerated flight, thrust and drag must be equal and opposite, and
the lift and weight must be equal and opposite according to the laws of motion. Thus, as shown in
Figure 73.
T=D
(135)
L=W
(136)
L
w
Figure 73.
Another useful form of the lift and drag equations may now be developed.
From the lift equation (104),
and the drag equation (105),
Since,
then,
V
M=a
V=Ma
Writing the speed of sound, equation (50),for general and sea level standard day conditions, it is
convenient to divide the farmer by the latter.
In generał,
D6-1420
Mayl989
a=-JiR.f
1.109
AERODYNAMICS
At sea level standard day,
therefore,
or,
(137)
Substituting for y2 in equation (104) results in:
1
2 2
L=CLS2pM ao8
(138)
A1so remembering that,
6 =<J8
and,
p
Po
<J=-
equation (138) may be rewrinen as:
(139)
Since Po and
ao arc both standard day values for air equal to,
3
pO =. 002377 slugs/ft .
a0 = 1116.4501 ft/sec
equation (139) may· now be wrinen with these values substituted.
2
L= 1481.351 CLM S6
or,
~ = 1481.351 CL M2s
(140)
In effect, 1481.351 M2 6 was substituted for l{l p y2 in equation (104). The same reasoning may
be applied to equations (105), (135), and (136), producing:
~ = 1481.351 C0 ?<S
(141)
W
(142)
2
6 = 1481.351 CLM S
1. 11 O
D6-1420
May1989
AERODYNAMI CS
r
= 1481.351 C 0 ~ S
(143)
These equations form the basis of many performance calculations and, although they are merely the
lift and drag equations modified, may appear somewhat strange. Mach number instead of true
airspeed is used as the velocity param.eter. This is logical since Mach number is the more significant
parameter when considering high speed aircraft The use of Mach number, however, introduces the
pressure ratio, 6, into the equations. This quanti.ty is placed on the left side of the equation to make
the new parameters, W/6 and Tło.
Since 6 is a function of altitude, for a given weight and thrust, Wło and Tło will vary with altitude.
The turbojet engine responds basically to pressure and temperature,as will be shown in Section 2.
Above 36,089 feet, the temperature is constant in the standard atmosphere. This means that thrust
under any specific conclition of engine RPM and airplane speed varies clirectly with the ambient
atmospheric pressure. The thrust then will decrease in proportion to the pressure ratio, 6 , as
altitude is increased above 36,089 feet. In other words, T/6 will remain constant under these
conditions. Tuus T/6 is the logical parameter on the basis of engine characteristics, since much of
the jet operation occurs above 36,089 feet Since Tło is to be used as a·parameter, then W/6 must
be used to be consistent
The application of equations (142) and (143) to performance computations is made in the following
manner. The value of Cr. at a given Wło and Mach number may be computed from equation (142).
Knowing CL and Mach number, the corresponding C0 may be found from an aiiplane polar such
as Figure 53. This enables one to calculate T/6 from equation (143). Selecting a range of Mach
numbers for the given W/6 as before results in a series of points defining a curve which may be
plotted on axes of T/6 and Mach number. By selecting severa.I values of Wło. more curves may be
obtained, resulting in a family of curves as in Figure 74. These curves define the Tło required by
the engines to fly in level unaccelerated flight for any weight, altitude, and Mach values. The bonom
of cach Wło curve is the Mach, or speed, for minimum drag.
The thrust parameter, Tło, will be notated frequently as FJ6 in succeeding chapters, however the
two terms are related, Tło= (number of engines) Frfo per engine. In engine analysis work, it is
helpful to use the term net thrust, Fn, as clistinguished from gross thrust, Fg· The significance of
the thrust terms will be discussed in Section 2.
06-1420
March 1991
1. 111
AERODYNAMICS
IStandard Day I
. ...!.. Available
a
..!. Available
a
Speed Schedule
For Minimum
Drag
M
Figure 74.
1. 112
D6-1420
May 1989
POWERPLANT
2-1
Jet Reaction Engines
The idea of jet propulsion was established when Sir Isaac Newton's three laws of motion were
published. See Section 1. The third law explains why thrust is obtained from ejection of gas flow
from a jet exh_aust nozzle and the second law ind.icates the magnitude of the thrusL
The primary purpose of all aircraft power plants is to im.part a change in momentum to a mass of
fluid. The fluid may be air, air and combustion products, or combustion products only. As will be
shown subsequently, a change in momentum is equivalent to an acceleration of a working mass
producing a force, or thrusL
The propeller equipped engine generates thrust by imparting acceleration to a quantity, or mass of
air. The jet engine accomplishes the same resulL The propulsive force reacts on the engine internał
structure to give forward motion to the aircraft.
The Basic Jet Powerplant
Gas turbines, or jet power plants, are of the type shown in Figure 1. The engine has several jobs to
perform and utilizes several different parts to do these jobs. Conveniently enough, these parts can
be designed into a small housing with each member compatible with another. The atmospheric air
enters the intake duet, or diffuser, which serves to reduce the velocity somewhat and increase the
static pressure. A rotating mechanical com.pressor further increases the pressure and delivers the
compressed air to the combustion chamber. Herc fuel is injected and the mixture ignited and
burned. The combustion product is then directed against a turbine wheel where the mass is
expanded to produce shaft power and discharged through the exit nozzle to the atmosphere. The
energy removed by the turbine wheel is used to tum the compressor and power accessory units.
Unlike a piston type engine which operates on a four stroke cycle, intake, compression, power and
exhaust, all of which intermittently occur in the same chamber; the jet engine is designed for
continuous operation, and each operation occurs in a separate chamber.
D6-1420
May1989
2.1
POWERPLANT
I Combustion
,---J
..... Intake
Diffuser
~
Compressor
r=,
Turbine
s
~
.....
Figure 1.
In addition to the components already named, a typical jet engine also incorporates an accessory
section, a cooling system, a starting system, a lubrication system, a fuel system, and possibly a
water injection system.
·
There are two widely used types of turbojet engines: centrifugal flow and axial flow. The
characteristics of each are pronounced; the primary difference is in the compressor appearance.
Although combinations of the two types are possible, the axial flow type has appealed to engine and
airframe manufacturer alike to produce a most compatible arrangement for aircraft use.
Turboprop Engine
The turboprop is essentially a turbojet designed to drive a propeller. See Figure ·2. The propeller
operates from the same shaft as the compressor and turbine through reduction gearing. The thrust
is derived through the propeller for the most part although as much as 20% may be obtained from
jet action. This engine retains the advantage of having a light weight and low frontal area, together
with the ease of installation that goes with turbojet engine design. In addition, it has a high
efficiency at relatively low speeds. However, present propeller design limits the use of this type of
powerplant to speeds below 0.6 Mach.
Figure 2.
2.2
1)6..1420
May1989
POWERPLANT
Turbofan Engine
The turbofan version of an aircraft gas turbine is the same as the turboprop; the geared propeller
being replaced by a duet enclosed fan driven at engine speed. · One fundamental operational
difference between the tmbofan and the turboprop is that the air flow through the fan of the turbofan
is unaffected by airspeed of the aircraft.
eliminates the loss in operational efficiency at high
airspeeds, which limits the airspeed capability of a turboprop engine. Also, the total airflow
through the turbofan engine is much less than that through the propeller of a turboprop. In the
turbofan engine, 30 to 70 percent of the propulsive force is produced by the fan. See Figure 3.
This
Figure 3.
Prop Fan Engine
After the fuel shortage and price increases, interest was renewed in the turbo prop type engine.
· Hamilton Standard under contract to NASA developed a swept bladed propeller that extended the
operating speed of propellers to approximately 0.8 Mach number. Since fuel prices have come
down, fuel efficiency is not sufficient to offset the high development cost and high maintenance cost
associated with the prop fan engine. The prop fan concept is valid; however, and fuel price increase
could make it attractive in the future.
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May1989
2.3
POWERPLANT
Intentionally Left Blank
2.4
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POWERPLANT
·Engine Station ldentification
2- 2
It is convenient to assign numbers to the stations separa.ting the major components of the engine.
Figure 4a and 4b shows station identification for single, twin, and triple rotor systems including
turbofan, turboprop, and bypass (mixed) engines. These identifications were obtained from
(Aerospace Recommended Practices) ARP 755A. At this time the station identification for
advanced propeller engines has not been agreed upon and has been omitted from the figure.
DIRECTION OF FLOW
BURNER
o
1
4
3
2
5
8
7
6
9
SINGLE SPOOL TURBOJET/TURBO SHAFT
11
12
13
2
2-
17
2-
3
4
18 19
4-
7
5
8
9
TWINSPOOLTURBOFAN
12
13
2
2-
16
i j_
o
1
2-
3
4
4-
5
6
6-
7
8
9
MIXED TWIN SPOOL TURBOFAN
Figure4a.
D6-1420
May1989
2.5
POWERPLANT
12
13
17
18
DIRECTION OF FLOW
l
TRIPLE SPOOL TURBOFAN
FREE TURBINE TURBO PROP./TURBO SHAFT
Figure4b.
2.6
D6-1420
May1989
POWERPLANT
2-3
Engine Parameters
The performance characteristics of the turbojet engine can be obtained from nondimensional plots of
engine data. The basie performance curves of gross tbrust, Fg, airflow, W 8 , and fuel flow,
W r, ·arc derived in terms of the following (for a single rotor engine):
Compressor inlet total pressure, Pt2
Ambient static pressure, Pamb
Turbine RPM, N
Com.pressor inlet total temperamre, T t
2
Linear dimension, D
Following the procedure used in Section l, a generał (although arbitrary) relation may now be
written for the fluid force, Fg_
(1)
Newton's second law of motion written in terms of the fundamental units of M, L, T is:
(2)
Equation (1) is now written in terms of the fundamental units:
ML
- C ( - M2
T2 LT
6
)a( LTM )p( T1)'Y(LT J
2
-
2
-
- 2
(L) e
(3)
With regard to the temperature factor, it will be noted that temperature is a measure of thermal
energy per unit mass and has the fundamental dimensions of velocity squared:
Therefore, the fundamental units of temperature arc L2rr2•
Powers of like variables are now equated:
M:
L:
T:
D6-1420
March 1991
l=a+p
}
1=-a-P+ 2a+e
2 = 2a + 2p + 'Y + 2a
(4)
2.7
POWERPLANT
Since the above three equations contain five unknowns, it is necessary to express any three
equations in terms of two unknowns. Assuming Pamb and T t2 to be of least importance, Cl, 'Y, and E
are found in terms of~ and 6:
cx=l-~
E = 1 + (1- ~) + ~ - 2 6
1
J.
'Y =2 - 2(1- ~) - 2~ - 26
= 2-26
(5)
=-26
Substituting these values in equation (1), the following is obtained:
(6)
or,
(7)
The dimension, D, may be eliminated for a particular power plant for presentation of performance
data. Equation [/) may be shown simply as:
·
:,: =C(:~J ((f
-P
(8)
or, Fg/p12 stated as a function of two parameters:
(9)
A
The numerical values of FgfPt2 and N/
are not easily recognized as thrust and RPM,
therefore, it is more convenient to use a multiplying constant such that the dimensional groups are
equal to gross thrust and actual RPM at standard sea level static conditions. These parameters,
referenced to standard sea level conditions, are called referred or corrected parameters .
2.8
D6-1420
March 1991
POWERPLANT
Thus,
FgPo
P1
and
2
N.Jro
A
becomes,
Fg
(:':)
and
N
fi.
o
By defining:
and,
equation (9) becomes,
(10)
To develop an equation for air flow, consider the mass flow rate,
and in terms of fundamental units:
D6-1420
March 1991
2.9
POWERPLANT
and equating powers of lilce variables
M:
l=cx+~
L:
0=-cx-~+20+E
T:
- 1 = - 2cx - 2~ - 'Y - 2o
Therefore,
cx=l-~
~ - -(y + 1)
2
u-
E= 2+'Y
and,
This, for a particular powerplant, can be wńtten as,
(11)
2.1 O
06-1420
March 1991
POWERPLANT
In a similar way, to develop an equation for fuel flow, it is necessary to think of fuel flow as
potentia! energy flow; therefore,
·
W = Energy
f
TlDlC '
ft lb
sec
.
ML2 C
( M
-)ex(- M .)13(-1
LT 2
T3 -
LT 2
T
6
)'Y(L- 2)
(L) E
2
T
and equating powers of lilce variables,
M:
l=ex+l3
L:
2 = - ( ex + 13) + 26 + E
T:
- 3 = - 2( ex + 13) - y - 26
Therefore,
ex=l-13
1-y
6=-2-
E=2+y
and,
1-p
Wr=C(P12)
p
1-y
-2-
Y
(Pamb) (N) (T12)
2+y
(D)
-P
2
Wf=C(P,JA0
D6-1420
March 1991
(:~J (~ )y
2.11
POWERPLANT
This, for a particular powerplant, can be written as,
(12)
Typical non dimensional engine characteristics arc shown in Figures 5, 6, and 7. Data from charts
such as these fmnish the information for thrust, fuel flow, airflow curves throughout the range of
operation.
Performance curves published by the engine manufacturers showing the thrust, fuel flow, air flow
relationships may appear as in Figure 8 where information for this curve is obtained from Figures
5, 6, and 7. These cmves arc available for various altitudes throughout the range of airspeeds, and
from idle thrust to takeoff thrust. The fuel consumption data is also presented in terms of thrust
specific fuel consumption , TSFC, which is fuel flow divided by net thrust.. In this form, the data
arc more easily used for airplane performance calculations.
2.12
D6-1420
May1989
POWERPLANT
N
~
Figure 5.
1.2
1.4
1.6
N
{i;;
Figure 6.
D6-1420
May1989
2.13
POWERPLANT
N
Figure 7.
Given Altitude
Standard Conditions
\
True Airspeed - V, Kt
Figure 8.
2.14
D6-1420
May1989
POWERPLANT
Note that the term net thrust, F 0 has been used. Net thrust is the total usable force derived from the
'
engine by accelerating the air mass.
F n =Ma
F =M AV
At
0
M
F n =-AV
At
(13)
But,
and,
AV=V1 -V 0
where VO is the freestream (or airplane) velocity; and VJ is the jet velocity.
therefore,
(14)
Tuus, the net thrust is the difference between the force of the airstream at the exhaust nozzle and the
force of the airstream at the inlet. Since the first term is the total force output of the engine, it is
known as the gross thrust,
wa
Fg =g V1
(15)
Since the force at the inlet tends to reduce the usable thrust, it is called ram drag,
w.
Fr =g Vo
(16)
By adding the weight of fuel and its acceleration from a relative velocity of zero to jet velocity, the
equation for net thrust may be rewritten:
D6-1420
March 1991
2.15
POWERPLANT
(17)
Since the last term of equation (17) is small in comparison, it may be neglected, but with
computerized performance analysis, the practice is to include both terms.
Newton's third law of morion explains why thrust (a force) is obtained from the ejection of mass
from the nozzle of a jet engine. The second law indicates the magnitude of thrust as shown by
equation (14). This equation or variations will be amplified in subsequent chapters.
To see that discharge total pressure, Pt7 and pressure ratio, Pt,/J>t2 are valid indices of thrust output
of the engine, consider ram drag as a function of velocity as shown in equation ( 16). Therefore, at
a fixed speed, net thrust will vary as gross thrusL Airflow rate in the above equations may be
considered in terms of the continuity equation:
W·8
g =pAV
-
(18)
where A and V are the exhaust nozzle throat area and velocity. Thus equation (15) becomes,
Fg =pAV(V)
Fg =pAV'
(19)
Substituting Mach number for velocity,
(20)
By use of the basie relationships,
a=~ygRT
p
p= gRT
equation (20) can be wrinen as,
2
F _ pAM ygRT
g gRT
2.16
D6-1420
March 1991
POWERPLANT
2
(21)
Fg=pAMy
The pressure, p, used in equation (21) is the static pressure. Thus, gross thrust varies with static
pressure and Mach number, since A and g are constant. But static pressure is related to total
pressure, pt, as shown in Section 1:
'Y
y-1
Pt - ( 1+--M
y-1 2)
p 2
By substituting this relationship into equation (21),
(22)
From this equation, gross thrust, and therefore net thrust, is a function of total pressure in the
exhaust nozzle, Pt .
7
(23)
In the same way, applying the same analysis to flow into the engine inlet yields,
(24)
Since,
then,
D6-1420
March 1991
2.1 7
POWERPLANT
dividing by Pt ,
2
recognizing that
i>i2 =Pt2 I Po
Pt,
Fn
~ = - p f(A,V,T)
u.t2
t2
exhaust
-f(A,V,T)
inlet
This analysis shows that net thrust is a direct function of geometrie design, inlet airflow conditions,
exhaust flow conditions, and engine pressure ratio. The exhaust flow conditiQns are determined by
inlet conditions, compression ratio and fuel flow (throttle position). Therefore, an engine of given
design will yield net thrust proportional to EPR, Pt/Pt2, when operating at a given airplane speed,
altitude, temperature, and thrust Iever position.
F0 =f(EPR)
(25)
Since Pt? is really an inde:?( of gross thrust whereas EPR is an index of net thrust, Pl7 has to be
corrected for changes in inlet ram conditions when used as the basie thrust index in an application.
Therefore,on aircraft where inlet conditions vary over wide extremes, EPR is prefered and
recommended as the basie thrust setting variable. Thrust is not measured directly in engine
installations on airplanes.
The relationship of Pt7 to Pt2 for dual rotor engines is:
(:::) (:::)(:::)= (:::)
The ratio Pt3'Pt2 is compression ratio, which varies with RPM changes. The ratio Pt4'Pt3 is the
burner pressure loss and is considered constanl Pt7'Pt4 is the turbine expansion ratio which is a
function of exhaust nozzle and turbine areas, which are usually constant Pt? is indicative of thrust
that will be obtained. Pt2 takes care of Mach effects, compressibility, temperature and altitude.
Since Fn is a function of EPR at given values of altitude, speed, temperature, rating (N1), and is
related to gross thrust (equation 17), Figure 5 may now be modified to amore useful application;
determining F J6amb· The family of Pt2'6amb lines may be changed to constant M values and as a
result Si2 values changed to 6amb values.
2.18
D6-1420
March 1991
POWERPLANT
6
Pt 2
(
--2_ = Po
J .
6amb (p;:b)
_Y
y-1
=[l+ y- 1 ?!C'J
2
F 8/6i:2 may now be changed to corresponding values of corrected net thrust, F Joamb· The resulting
engine performance curves arc shown in Figure 9 and Figure 10. For dual rotor engines, N 1 is
usually used for the corrected rpm term in Figure 1O even though either N 1 or N2 can be used. ·
INCREASE FROM
=.4 M to .9 M
INCREASE FROM O to :: .4 M
EPR
Figure 9.
D6-1420
Mayl989
2.19
POWERPLANT
EPR
Figure 10.
For a given M and EPR, the actual net thrust and N 1 rpm may be obtained by reading the corrected
net thrust and corrected N 1 rpm and correcting for altitude and temperature, respectively:
and,
Therefore, Figure 9 and 1O are now valid for any thrust setting, airbleed, and atmospheric
conditions. A cross-plot of Figure 9 may be constructed for constant EPR values as shown in
Figure 11. The advantage of this presentation is that it can be superimposed on top of the
generalized thrust required curves ( Figure 74 of Section · 1) giving the required EPR with the
required thrust for a given weight, altitude, and Mach condition.
2.20
D6-1420
May1989
POWERPLANT
F
n
l>amb
M
Figure 11.
Also, since fuel flow is a function of EPR at a given set of flight conditions, Figure 6 is modified as
shown in Figme 12. Actual fuel flow is determined using equation (26).
Constant
Altitude
EPR or
N1
va;:2
Figure 12.
1)6..1420
May1989
2.21
POWERPLANT
(26)
where,
1- 1
21
'Y
y-1
6,2 = 6amb [ 1+-2- M J
and,
therefore,
(27)
lf the corrected fuel flow does not generalize completely, i.e., one chart valid for all altitudes, then
Figure 12 can be made for constant altitudes. Also, if the fuel flow does not follow the
mathematical analysis 100 percent, then equation (27) has to be modified to cope with the changes
in design of the various types of engines. The applied correction can be· a A (lambda) correction
resulting in equation (28):
where A is obtained from Figure 13 for a specific engine. Since A is a function of total temperature,
it can be combined with
2.22
F,";. as AT,;. =8~
2
•
1)6.1420
May1989
POWERPLANT
1.02
A·
1.0
.98
Figure 13.
This results in equation (29) that is valid for any jet engine.
(29)
The exponential value of x has to be determined for cach specific engine. Typical values may vary
from 0.5 (from mathematical analysis) to 0.67. Figure 12 has to be modified accordingly.
Tois discussion has centered around engines that use EPR as the thrust setting variable. RPM can
also be an adequate index for setting thrust. Until recently its use was usually limited to that of a
secondary index employed to check the thrust set by means of EPR, or as an emergency index in
case of failure of the pressure instrum.entation. When used under these conditions, it was found
that thrust could be set very accmately by the RPM system. Also some engine manufacturers prefer
using RPM for setting thrust. Therefore, Figure 5 may be modified as before except that the
abscissa will remain as shown. For dual rotor engines, N 1 is usually used in the corrected RPM
term. The result is Figure 14. Note that Figure 14 is valid for a given engine bleed condition, such
as air conditioning bleed. Engine bleed affects the thrust value for the same corrected N 1 rpm,
N1/
y.;_,
D6-1420
May1989
and M
2.23
POWERPLANT
GIVEN BLEED
CONDITION
INCREASE FROM :.6 M to .9 M
INCREASE FROM O to :.6 M
Figure 14.
A cross plot of Figure 14 for constant values of N 1 I
F,';. may now be constructed. See Figure
15.
GIVENBLEED
CONDITION
-
6amb
M
Figure 15.
2.24
D6-1420
May1989
POWERPLANT
This figure can also be superimposed onto the generalized thrust required curves, Figure 74 of
Section 1, giving the required N 1 I
F,, with the
required thrust for a given weight, altitude, and
Mach condition. An added step is required here to get N1, however.
NI=(~) r-J1+
y;l M2
Determining the actual fuel flow, Wf , ·is the same with either EPR or N 1 I
F,,. For
~ 1/F,,, Figure 12 is replaced by Figure 16 and equation (29) is used to calculate the fuel
flow.
·
Constant
Altitude
EPR or
N1
va::
2
Figure 16.
There usually is no modification to the airflow chart, Figure 7.
D6-1420
Mayl989
2.25
POWERPLANT
Intentionally Left Blank
2.26
D6-1420
May1989
POWERPLANT
2-4 Engine Efficiency
One of the first laws of thermodynamics states that beat and work are two forms of energy, and that
they are intercbangeable. This means then that work can be converted into beat, and beat converted
to work. The first process is possible, but the excbange of beat to work is not possible without
losses. The ratio of work produced to the beat energy supplied represents the effi.ciency of a
process. The most efficient engine, for example, would be a type that would receive beat at the
highest temperature conceivable and exhaust its gas at the lowest temperature possible.
All propulsion devices are beat engines that convert fuel energy (beat energy) into mecbanical
energy in combination with a system for converting the mecbanical energy into useful thrust The
efficiency of a propulsion unit can be expressed as the percent of total beat energy of the fuel that is
converted to work. This is some times referred to as overall efficiency, 11 0 , to distinguisb it
from thermal efficiency, 11l' or propulsive efficiency, 11 p·
11 O =
Useful work done on airplane
Heat energy of fuel
(30)
Overall efficiency is the product of thermal and propulsive efficiencies, or:
(31)
wbere,
Mechanical work produced in system
11 t =
Heat energy of fuel
(32)
and,
Useful work done on airplane
11 P = Mechanical work produced in system
(33)
The numerator of equation (32) may be developed by observing that work ·is measured by the
product of a force and the distance througb whicb it moves in the direction of the force.
Work = Fs
D6-1420
March 1991
2.27
POWERPLANT
where,
Therefore,
or,
0
~
Zne = (:;) [ vi-v~J. ft-lb/unit time
The denominator of equation (32) may be determined by evaluating the energy of the fuel.
energy
. .
. time =J LHV Wf' ft-lb / umt ttme
1
The constants J and LHV are,
J = 778 ft lb/BTU conversion factor from BTU to ft-lb
LHV = 18,580 BTU/lb
LHV is the Lower Heating Valoe of the fuel. This value is a funtion of the fuel's specific gravity.
The Boeing Co. uses the value listed above for the published performance data.
Mathematically, thermal efficiency of a turbojet may be expressed as
(ł) [v;-v~J
'111=
JLłłVWf
(34)
In gas turbine engines, the greatest portion of heat energy input is utilized in driving the compressor
rotor. Anything which will cause an increase in the air mass and air pressure head available to the
combustion chamber (compression ratio) with no further increase in heat energy input will improve
the thermal efficiency.
2.28
D6-1420
Marc:h 1991
POWERPLANT
A jet engine's thermal efficiency tends to improve with airspeed due to the ram effect. Ram
pressure, when multiplied across the compressor by the compression ratio could mean an
improvement in Wa and combustion chamber pressure head, and thus increased thrust output , with
little or no ch_ange in shaft energy input to the compressor. Such an increase in useful output with
no appreciable change in beat energy input would result in raising the engine's thermal efficiency
with airspeed.
Two outstanding methods of obtaining such an increase in thermal efficiency would be
improvement in compressor efficiency or increase in combustion chamber temperature. Il
compressor losses could be reduced, the same mass aińlow and compressor ratio could be obtained
with less energy input to the compressor, resulting in less fuel flow (beat energy input) for the same
thrust output. The increase in combustion temperature results in an increase in exhaust nozzle
velocity. This, in tum, results in a higher thermal efficiency as seen by equation (34). This is true
if we neglect cooling required to protect the engine. When cooling is used, the increase in
temperatme may even reduce the thermal efficiency when the increased cooling is considered.
A piston engine's thermal efficiency does not increase with airspeed because it cannot talce full
advantage of the ram effect. Both factors affecting its output are limited; RPM limited by bearing
loading, and manifold pressure by detonation. A gas turbine engine operates on a low pressure
cycle and, therefore, is not limited by detonation. It is, however, limited by a maximum allowable
interna! operating temperature dictated by the structural strength and materials of the gas turbine 's
component parts.
The numerator of equation (33) may be developed by detennining the work done on the airplane
(per unit of time) as the product of thrust and velocity, (power).
power= TV
velocity = V
D6-1420
March 1991
0
2.29
POWERPLANT
Mathematically, propulsive efficiency of a turbojet may be expressed as:
(35)
The effect of fuel flow is negligible in the above equations.
By inspection of equations (34) and (35), the variables influencing the thermal and propulsive
efficiencies may be recognized. How these variables affect the operation of the engine is discussed
later. The average jet engine has a overall efficiency under static sea level conditions of 25 to 30
percent.
2.3 O
D6-1420
March 1991
POWERPLANT
2-5
Pressure Volume Cycles
Engine cycles can be described most easily by the use of pressure volume diagrams.The diagrams
depict the .process to which the gases arc subjected beginning with atmospheric pressure, through
the complete compression and expansion processes, and back to atmospheric pressure again. The
area inside the cmve in the diagrams represents the work done in the process.
The Otto or constant volume cycle of the piston engine is the most common, and is shown in
Figure 17. The diagram shows the suction and exhaust strokes of the cycle as a dashed line
representing a condition where the mass in the cylinder is changing. At the start of compression,
the pressure is atmospheric or nearly so. The pressure is increasing from 1 to 2 as the piston moves
to decrease the volume. From 2 to 3 the volume is essentially constant, giving rise to the
descriptive name of the cycle. During this time, ignition occurs and the pressure increases. During
the expansion or power stroke, 3 to 4, the pressure is relieved and the volume increases due to
piston movement. The exhaust begins at point 4 and the pressure reduces to atmospheric again.
3
pressure,p
2
4
exhaust
ambient
1
volume, v
Figure 17.
The turbine jet cycle is known as the Brayton or constant pressure cycle. It operates at much
lower pressures than does the Otto cycle to obtain maximum efficiencies and lower fuel
consumption rates in addition to compressor design considerations. This cycle is shown in Figure
18. From 1 to 2 the air is entering the inlet and is being compressed from 1 to 2. For cycle analysis
it is immaterial that the work of compression is done partly by the intake diffuser and partly by the
rotary compressor. Point 2 represents the pressure achieved by the compressor and is essentially
maintained throughout the combustion process. From 3 to 4 expansion takes place through the
turbine and exhaust nozzles, retuming the pressure to its initial ambient condition. The total air
mass in the jet cycle represents not only air for combustion (the primary air), but also cooling air,
often referred to as dilution air.
D6-1420
March 1991
2. 3 1
POWERPLANT
In the Otto cycle, combustion occurs at constant volume with the result that the pressure rise is of
great magnitude. In the Brayton cycle, however, combustion takes place at a constant pressure,
resulting in an increase in volume. In the turbine engine cach operation in the cycle is peńormed
continuously and by a separate component designeci for the particular function. In the reciprocating
engine, the cycle is repeated intermittently in one section of the engine designed to accomplish more
than one function.
combustion
2---pressure, p
ambient
1
volume, v
Figure 18.
A comparison of the two may be made by superimposing the jet constant pressure cycle over the
piston constant volume cycle, as shown in Figure 19.
2.3 2
D6-1420
March 1991
POWERPLANT
pressure, p
.
~
\
''
' ' ' ,.4
ambient
constant
pressure cycle
_______,
,
........................
volume, v
Figure 19.
The area bounded by the lines on the p v diagram represents the useful work output. The work
output of the reciprocating engine is derived by means of its high pressures. It is possible to
develop pressures over 1,200 psi in the cylinder of a piston engine during the combustion. With
further increases in volumetric compression ratio, a greater amount of work may be harnessed from
a given quantity of fuel. This, of course, improves thermal efficiency which in tmn results in
improved economy. Present jet engines develop pressures in the combustion chamber over 350
psi. When ways are found to increase compressor efficiencies, the useful work output will increase, and, in addition, the thermal efficiency will improve.
Further comparison of the two basie engine cycles shows that the jet engine gains useful work by
expanding the gases efficiently to atmospheric pressure at the exhaust nozzle. In the reciprocating
engine, the expansion takes place across the ex.haust valve and exhaust duet and is therefore lost to
the cycle as useful work.
·
D6-1420
March 1991
2. 3 3
POWERPLANT
Intentionally Left Blank.
2.3 4
D6-1420
May1989
POWERPLANT
2-6 · Component Design
A study of engine components and their design will serve to increase the understanding of the
operation of the engine and its performance. This study is meant to be generał in nature; if design
details are needed, the reader should go to the basie engine manufacturer's literature for the engine
in which he is interested.
Air Inlet Duet and Diffuser Design
The air inlet duet is usually designed by the airframe manufacturer and not by the engine
manufacturer. However, the design of the duet is so important to engine performance that it must
be considered in any analytical cliscussion of the complete engine. Since a gas turbine engine is
consuming large quantities of air and is operating_ at high airspeeds, any inefficiencies of the duet
result in large losses throughout the engine.
The inlet duet has to accomplish three functions; first, it must be able to recover as much of the total
pressure of the airflow as possible (known as ram recovery) and deliver this pressure to the engine
compressor with a minimum loss; second, the duet must deliver the air to the compressor inlet with
as little turbulence as possible; and last, the duet must accomplish items one and two above holding
to a minimum the drag effect it creates. In other words, the air inlet duet is so designed as to reduce
energy losses in the form of aerodynamic drag or through ram pressure loss.
The shape of the inlet duet and diffuser depends largely on the Mach number for which it is
designed. Duet design for airplanes flying supersonic speeds is clifferent than for airplanes flying
subsonic speeds. The following text is for subsonic Mach numbers. From Figures 5, 6, and 7, a
high ram pressure ratio, PtiPamb• ahead of the compressor results in:
(1) high corrected
( 2) high corrected airflow ,
(3) low corrected · fuel flow,
Fi .
ot 2 at 2
, A high ram pressure ratio gives a high ram efficiency defined as follows:
Pr - Pamb
Ram efficiency,
2
Tl r = - - - qc
(36)
where 'łc is the free stream dynamie pressure including compressibility effects.
D6-1420
March 1991
2. 3 5
POWERPLANT
A high ram efficiency can be obtained in the inlet duet if the friction loss and separated flow are kept
to a minimum. For pod type engine installations, friction losses arc insignificant and flow
separation is of pńme imponance. To reduce interna! duet separation for pod type engines, a low
inlet velocity ratio, V 1N 0, is desired. To accomplish this, a good portion of the deceleration may be
designed to
before station 1 since compression in this region is isentropic and compression aft
of station 1 involves duet losses. This process of converting directed kinetic energy into pressure is
called diffusion. From an engine standpoint, the more diffusion that occurs ahead of station 1 the
bener. See Figure 20.
occm
o
2
1
o
I
2
I
350
- - -
~
ee_
ft/sec
C)
700
ft/s«:.
n ~
~
350
With Intemal
~
Ul
Ul
tropie
eC.
s
8
C)
e""""o
iffusion
C)
Stream tube of air
required by engine
same for either V 1
~ ~
V
V
~
~
-=0.5
Figure20.
A high inlet velocity ratio (V 1N Onear 1) is undesirable since the large deceleration of the air inside
may cause flow separation from the walls of the inlet ducL This separation inside the duet takes
place because of the high adverse static pressme gradient, resulting in a reduction in ram efficiency.
Actually, a more serious consideration is the fact that the inlet may not pass the required airflow for
takeoff. For many engines, if the inlet is designed for cruise with V 1/VO= 0.8 or higher, the inlet
will actually choke at takeoff conditions and thrust is seriously affected. In practice, the upper limit
of V 1/VOis usually detennined by the inlet siz.e required for good takeoff pressme recovery.
2.3 6
u~
i:i..
ft/SJ!C
s«;
E:=
DuctLosses
C)
D6-1420
May1989
POWERPLANT
It appears the pod type inlet duet should be designed for low V 1N O(.4 to .5 maximum), but this
relates only to the engine performance. Aerodynamically, it is desirable to operate at high V 1N Oin
order to achieve a high critical Mach number design. For high V 1N O the air does not have to
accelerate nearly so much in going around the inlet cowl lip as it does for the lower V1 N 0.This
reduces ram drag resulting from external diffusion. This problem becomes more difficult as the
desired maximum flight speed is increased.
Two considerations limit the length of the diffuser. First, the length must be adequate to tum the
extemal flow to the axial direction without incuning large drag penalties. To avoid interna!
separation of the flow, the angle of the containing wall and the mean flow path should be less than
3° for best results. Second, the diffuser must be long enough and have a lip shape such that the
flow within the diffuser will be insensitive to angle of attack. The diffuser must not be so long that
excessive friction losses will occur. The design of the duet then becomes a compromise between a
long duet with large viscous friction losses and low turbulence, and a short duet with low friction
losses and a high order of flow separation and turbulence. ·
At the inlet to the compressor; it is desired to provide the most uniform possible velocity and total
pressure distribution. Variations in either of these factors results in deteńoration in the performance
by engine stalls or excessive fuel consumption dming normal steady state operation.
Air Compressor Design
The purpose of the compressor is to increase the energy level of the air received from the intake
duet, compress it, and discharge it into a combustion chamber in a desirable quanti.ty and pressure.
Most gas turbine engines employ rotary compressors of either straight through (axial) flow, Figure
21, or centrifugal (radia!) flow, Figure 22.
The axiai flow engine consumes about two to three times as much airflow per square foot of frontal
area as the centrifugal type. Since thrust is proportional to airflow, the axial type enjoys a similar
advantage in thrust per square foot of frontal area. The axial flow engine also has a higher fineness
ratio than the centrifugal type. High fineness ratio (length to maximum diameter) and low frontal
area are both important considerations for minimizing drag in high speed flight
Higher compression ratios can be obtained with good efficiencies in the axial flow compressor.
The centrifugal compressor may reach an efficiency of 75% to 80% up to pressme ratios as high as
four. Above this range, the efficiency drops off at a rapid rate. The axial flow compressor may have
an efficiency of 80% to 90% over a wide range of compression ratios. Since the efficiency of the
compressor directly affects. the fuel consumption and since high compression ratios themselves
induce good economy, the centrifugal engine is ruled out as a high economy type. Satisfactory
combustion at extremely high altitudes and high efficiency combustion at intermediate altitudes may
be practical only with the high compression ratios attainable with the axial flow engine
D6-1420
March 1991
2. 3 7
POWERPLANT
Figure 21.
SINGLE
ENTRY
dif':fuser
blades
m.--
~J'nl
1
~
DOUBLE
ENTRY
di:f'fu.ser
sect1on
---
impel.l.er
Figure 22.
One higbly important characteristic is the compressor efficiency. Compressor efficiency is the ratio
of the work that would be required to compress the air if there were no losses involved (therefore
purely theoretical), to the work actually done including the losses. Since some losses will always
exist, this ratio, or efficiency, will always be less than 100%.
2.3 8
D6-1420
Mayl989
POWERPLANT
If calculations showed that to compress the specified mass airflow to the designed compression
ratio would require 25,500 shaft horsepower energy rate, yet under actual test, the compressor
required 30,000 shaft horsepower energy rate, then:
.
ideał energy required
Compressor effiaency = actual
uired
energy req
Compressor efficiency = ( ; ~ ) 100 = 85 %
Any reduction in the existing losses would reduce the actual energy required to compress the air
and, therefore, increase the compressor effi.ciency.
F:igure 23 shows the effect of compressor effi.ciency on compressor power requirement. If the
compressor's effi.ciency could be raised, a sizeable reduction in the compressor shaft horsepower
requirement would be effected. The turbine.would not have to extract as much energy from the gas
flow. From this, two results are possible:
(1)
Increased thrust. With the same fuel flow, more beat energy is left in the gas flow,
thus resulting in greater expansion and increased jet velocity.
(2)
Increased economy with the same thrust. Less fuel flow is needed since less energy
input to the compressor is required.
Some disadvantages are encountered with the use of axial flow type compressors. Since top
performance can be obtained only when the compressor and turbine are operating simultaneously at
their peak efficiencies, it sometimes becomes diffi.cult to match the sharp peaked characteristics of
the axial flow compressor with those of the gas turbine and stili maintain stability throughout the
full operating range. The axial compressor is more sensitive to erosion and blade fouling due to
dust and oil and experiences vibrations of a complex nature. In addition, the air bleed is rather
critical.
D6-1420
March 1991
2. 3 9
POWERPLANT
--- ł -- .------- I
shaft horsepower
saving
- - - j - - - - - --
I
COMPRESSOR
POWER
REQUIREMENT
%t)oio
-
// I
,,,,.,,,,.
%S~ ...... ~
_ -
-
A
- _,,.,. _,,.,. X compressor
efficiency
I
I
I
I
:;a.- -
%RPM
100
Figure 23.
The compression ratio selection for an engine design is an extremely important consideration.
Compression ratio in conjunction with turbine inlet temperature determines the thrust specific fuel
consumption of the engine and the thrust output per pound of airflow. Higher pressure ratios
provide for a higher altitude limit
·
The compression ratio of a compressor with a given number of stages is basically a function of the
Mach number of the rotor tip (first stage) which is usually expressed as N
(Note that
!Ja;..
Jei ).
M =V/ a ~ = N/ k
This may be observed from a curve wbich describes the
compressorperfonnance very conveniently, as in Figure 24.
This compressor performance curve is obtained by driving the compressor under constant rotational
speed corrected to standard conditions and measuring the pressure and temperature relations as the
air flow rate is varied. Curves of constant N/
arc defined for cach corrected speed. The
_ upper end of cach curve defines the flow or axial velocity condition under which the corresponding
blade speed combines to give blade stall. The locus of all such values is called the stall or surge
line, the stability limit. Operation in the region to the left of this line is impractical, if possible, for
the blade is stalled and no longer working properly.
Ja;.
2.40
D6-1420
May1989
POWERPLANT
Flow Matching Line
_ _ ___i.
Tt
812
Figure24.
The other curves on Figme 24 are obtained to show additional restrictive conditions of operation. It
must be recognized that the com.pressor is physically connected to the turbine and must be matched
to it. The first consideration is that the air flow through the two must be the same (except for off
design operational bleed).
Wa
compres.u
=Wa
turbine
=Wa
Corrected air flow rate may be written as,
Therefore, the pressure ratio can be found:
D6-1420
May1989
2. 41
POWERPLANT
or,
(37)
Equation (37) shows that when the turbine is considered, the pressure performance of the
compressor is dependent upon not only air flow rate but also three other factors which control the
rotational speed. The three relate to temperature, pressure and area at the turbine inlet. Thus when
the turbine area is fixed and the bumer pressure loss, Pt3'P1:4, is determined, curves of constant
corrected turbine inlet temperature plotted on Figure 24 represent matched compressor turbine flow
and arc called flow matching lines.
Each steady state operating condition must satisfy both the pressure flow RPM and the matched
flow conditions simultaneously. Thus, when the compressor and turbine are combined, a steady
state operating or work line is defined. Therefore, it is important to consider the turbine area
relations which affect the turbine pressme drop and efficiency, and the operating temperatures
which contribute to fuel consumption as well as compressor efficiency before the finał design is
fixed. A consideration of the equality of power consumed by the compressor and power delivered
by the turbine sheds more light upon the parameters to be chosen:
W ork compressor
=W ork turbine
These work terms can be stated in the form,
(38)
The easiest way to adjust these relationships is through change in the turbine pressure ratio, Pts/pl:4,
by variation in the inlet and exit areas of the turbine.
2.42
1)6..1420
May1989
POWERPLANT
Compressor efficiency, Tle , is plotted on Figure 24. It is the objective to design such that the work
line runs through the peaks of the efficiency contour.
The thrust output per pound of airflow increases with compression ratio at a fixed turbine inlet
temperatme, reaching a maximum after which it decreases with an increase in pressure ratio. Figure
25 shows this condition. As higher compression ratios are reached, the compressor discharge
temperature becomes so high that vecy little heating of the air may be done before a fixed turbine
inlet temperature is reached. This inability to supply as much beat energy in combustion as might
be desired is the cause of the downward trend discussed above.
At a given turbine inlet temperature, thrust specific fuel consumption (defined as fuel flow divided
by thrust) decreases with increasing compression ratio. This trend continues until the addition of
the desired amount of energy is restricted by the high compressor discharge temperature. This is
shown in Figure 26.
Engine weight increases with increasing compression ratio due to the weight of added stages
necessary for the increase,and thickened engine cases required to handle the higher intern.al
pressures. Obviously some compromise between compression ratio, thrust specific fuel
consumption, and increasing weight must be reached.
70
I
I
I
I
~
o
I
·-'i=<
:S
....
r'->
~....
I:;: I
et
f!J
~
I
20
6
Compression Ratio
30
Figure25.
D6-1420
May 1989
2. 43
POWERPLANT
1.2
TSFC
Desirable
-woriingn..iiie
~
.75
15
6
Single Rotor
TSFC
Split Rotor ·
RPM or Compression Ratio
Figure26.
The compressor turbine system of the engine is designed to give optimum performance under
design conditions, usually cruise thrust operation. However, since the engine must satisfy the
demands of the airplane under other conditions also, it is necessary to look at off design operation.
lncreasing the compression ratio of the compressor increases the difficulty of the part load flow
matching of the compressor with the turbine. The compressor compression ratio at any given rotor
speed is the product of the compression ratios of all stages. Tuus, the compression ratio at full rotor
speed will increase much more rapidly with the addition of stages than will compression ratio at a
lower rotor speed.This exponential characteristic of compression ratio with rotor speed becomes
more and more difficult to match with the linear pressme flow chara.cteristics of the turbine nozzle.
The turbine nozzle flow characteristics arc linear since the flow at this station is usually sonie. The
two diagrams, Figure 27 and 28, pictorialize the difficulty encountered with higher pressure ratios.
2.44
1)6.1420
May1989
POWERPLANT
_____ :•hing17
flow
4.o
surge 1ine
N
= 1e>o;i
~
.
Figure 27.
f1ow
matching 1ines
16.0---------
7
N
Ft;
= 1~
Figure 28.
D6-1420
May 1989
2. 45
POWERPLANT
These clifficulties may be overcome by one of the following methods:
(1) Bleeding to overcome matching difficulties. Since at part load the compressor is delivering
too much airflow for the size of the turbine, some of this air may be removed from the flow
system, the remaining airflow being in quanti.ty and quality .such that it will satisfactorily
pass through the turbine nozzle. Since bleeding is a wasteful process, it is not desirable to
employ this means in the useful output range of the engine.
(2) Variable turbine nozzle area. It may be possible to alter the effective flow area.of the
turbine nozzle to accommodate the part load characteristics of the compressor, but this
would require a high order of mechanical complexity at a station where the heat is intense.
(3) Multiple rotor. A very satisfactory means of obtaining high compression ratios and
· avoiding much of the part load matching difficulty is to make the compressor in two or
more parts which have no mechanical relationship between their speeds. Since each of the
compressors is of relatively low compression ratio by itself, no particular problem is
offered in matching them with their respective turbines.
(4) Variable stator blades. At part load, where mismatch usually occurs, the stator blades of
part of the compressor stages may be adjusted such that the airflow from the compressor is
of the right quanti.ty to be handled by the turbine. In this type of design there is no wasteful
loss of air by bleeding, and the efficiency can be maintained at higher levels. However,
this solution also introduces a problem in mechanical design and control.
lncreasing the compression ratio of any compressor increases the range of operation the blading of
each stage must accommodate. Splitting the compressor into two rotors alleviates this problem to a
marked degree.
The problem of matching a given compressor with the other engine components for a given design
is complex. Consideration must be given to compression ratio, RPM, efficiency and stability with
which the compressor peńorms throughout the engine operating range. At cruise, high pressure
ratio and high efficiency are essential to low fuel consumption. At takeoff and climb, rpm,
compression ratio, and efficiency are all important to assme that sufficient thrust is available at rotor
speeds and turbine temperatures that permit reliable operation without compromising engine life.
From idle to full power, the compressor stability characteństics must be such that the engine
operating line can be located to give rapid surge free acceleration. Low speed compressor
characteristics, such as stability and efficiency, must be such to facilitate starting and to permit low
iclle speed.
2.46
1)6.1420
May1989
POWERPLANT
Advantages of a multiple rotor configuration are low starting power, good acceleration
characteristics, improved compressor performance, greater flexibility in matching, easier
development due to smaller components, and the opportunity to use higher compression ratios than
can be obtained by other means.
Disadvantages of a multiple rotor arc mechanical complications and added weight of extra shafting,
bearings and suppons and possible extra turbine stages for a given expansion ratio.
In the dual rotor engine, the high pressure compressor rotor, N2, is normally composed of less
stages of compression and runs at a higher RPM than does the low pressure compressor, N 1.Under
design conditions, cach compressor runs near or at its peak efficiency.
As rotor speed is increased, the low pressure rotor speed increases at a faster rate than does the high
pressure rotor speed. The compressor airflow parameter may be thought of as being proportional to
volume flow which would be the case if compressor inlet temperature were held constant.
Increasing rotor speed along the work ijne as shown in Figure 24 reveals that volume at the inlet of
any compressor increases with increasing rotor speed. For this discussion it may be assumed that
the volume increases linearly with speed, which describes the relation at the low pressure compressor inlet. As speed increases, however, the pressure ratio also increases. This causes the
volume at the compressor discharge to increase less rapidly than the inlet volume. The low pressure
discharge is the inlet to the high pressure compressor, which indicates that the high pressure inlet
volume flow does not increase as rapidly as the low pressure inlet volume. The high pressure rotor
speed, being proportional to its inlet volume, then, will increase less rapidly than·low pressure rotor
speed. This relation is shown in Figure 29.
N1 = Low Pressure Compressor RPM
N2 = High Pressure Compressor RPM
Figure29.
D6-1420
Mayl989
2. 4 7
POWERPLANT
With a fixed exhaust nozzle area and constant high pressme rotor speed control, the low pressure
rotor speed increases with decreasing compressor inlet temperature. Figure 29 reveals how this
takes place. With constant N2, decreasing compressor inlet temperature causes N2 / ~ to
increase. This increases the ratio N1/N2. N2 was held constant, which indicates an increase in N1.
This increase in N 1, as compressor inlet temperature decreases, causes the low pressure
compressor, and hence the engine, to handle a larger volume of air. This effect produces a more
graduał decrease in thrust output with increasing altitude than would be experienced with a singlerotor engine. The comparison to a single-rotor engine may be scen in Figure 30. This speed effect
may be altered by changing N 2 speed with temperature. Sometimes N 2 may be altered by having an
inlet temperature sensing bulb for the fuel regulator. This increases the N 2 compressor speed with
increasing temperature or decreases N2 with decreasing temperature.This temperature bias may be
used to gain additional thrust for hot-day takeoffs provided the turbine inlet temperature limit is not
exceeded.
·
36,000 :rt.
ALTI'l'UDE
Figure 30.
2.48
D6-1420
May1989
POWERPLANT
With constant N2 control, change in exhaust nozzle area varies only the speed of the low pressure
rotor, because the first stage turbines are in critical flow, as shown in Figure 31, and no pressure
disturbance downstream caused by a change of exhaust nozzle area could be reflected upstream to
the high pressure rotor. With an increase in ex.haust nozzle discharge area the expansion ratio
across the low pressure turbine would be increased due to the reduced back pressure, and the low
pressure rotor would accelerate and reach equilibrium operation at a higher rotor speed Since the
low pressure compressor speed would be increased, the airflow handled by the engine would be
increased and a lesser lapse rate of thrust with increased altitude would result, as indicated in Figure
30. The working line (see Figure 24) on the low pressure compressor would lie further from the
surge line. The working line on the high pressure compressor is determined by the high pressure
turbine inlet guide vane area and the low pressure turbine inlet nozzle area. These areas, being
unchanged, would not affect the relation between the working and surge lines of the high pressure
rotor.
Critical Flow
- _ł_ - -
M=l.0
w•.[e:;
. 614
Figure 31.
As mentioned previously, air bleed and dual rotor can be used on engines (may be seen in Figure
32) to aid the stability, and thus aid the efficiency. The higher the compression ratio and the more
stages added to an axial flow compressor, the greater the instability. The blade angles of the two
compressors are designed for the airflow and RPM to match in the flight range only. The mismatch
in the lower RPM range may be cmrected by bleeding off air, which normally will be from the forward stages or from the forward compressor in the case of dual rotor engines.Figure 32 also
shows that in the case of a dual rotor engine, by the RPM of the N 1 compressor being less thanN2,
the amount of airbleed required to match the compressor is decreased. This relationship also
increases the flight range of RPM. Airbleed is not necessary for some bypass type engines since
the bypass duet tends to take care of any mismatch. Most high bypass engines, however, need
variable guide vanes on the first few stages of the high pressure compressor and may also use some
bleed to obtain stability at low rpms.
D6-1420
May1989
2. 49
POWERPLANT
~---- original airbleed minus
the finał bleed
-----------increased flight
range of rpm
airb1eed req'd
------ f1ight
range of rpm
mLE
~ RPM, of each rotor
1oc,'p
Figme 32.
The effects of air bleed on the low pressure and high pressure rotor stall limits arc shown
schematically on typical compressor flow charts on Figmes 33 and 34, respectively. These figures
show the compressor pressure ratio on the ordinate and the referred airflow on the abscissa.
Superimposed arc the steady state operating line and typical engine acceleration and deceleration
schedules. On a low pressure rotor, the acceleration schedule would be below the steady state
operating line. During acceleration, the high pressure rotor, normally being lighter in weight,
accelerates at amore rapid rate than the low pressure rotor causing the -pressure at the exit of the low
pressure rotor to reduce and opera.te further away from surge than its steady state condition. Upon
deceleration the opposite would occur and operate the low pressure rotor closer to smge. From this
consideration it may be scen that smge would occm in the high pressure rotor on acceleration and
the low pressure rotor on deceleration.Without the low pressure compressor having bleed valves
open in the low RPM region, the low pressure compressor (Figme 33) would be in the surge region
at steady state (operating line) and also on the deceleration schedule. This condition does not exist
on the high pressme compressor (Figme 34).
2.50
1)6..1420
May1989
POWERPLANT
Typical Low Pressure Rotor Flow Chart
'\
__ . ___
\
\\
......
__ ,,-
---
-------
--
Design
point
I
Deceleration
Schedule
Operating
Line
I
Corrected
: ; Speed Line
N1
I
I~
I
'V „ t2
Figure 33.
For an engine to accelerate, the fuel rate into the engine must be higher than its steady state
operating condition. This causes higher temperature and pressure conditions at the inlet of the
turbine which drives the rotor. In the case of the dual rotor, as shown in Figures 33 and 34, the
acceleration of ·the engine is limited by the high pressure rotor surge limits, and consequently the
control system is designed to operate as close to surge as possible in order to achieve maximum
acceleration performance of the engine. Any malfunction of the engine fuel control system or
reduction in surge margin will cause the high pressure rotor to surge during acceleration. Engine
deceleration increases the surge margin of the high pressure rotor.
·
Figures 33 and 34 also show that decreasing the compressor inlet total temperature and density
while operating at high power increases the surge margin on the low pressure rotor and decreases
the surge margin on the high pressure rotor. Operation at extreme altitudes also introduces
Reynolds number effects which tend to lower the surge line for both compressors in the same way
that lowering Reynolds number decreases CLmax for any other aińoil. Operation at any condition
within the operating envelope must be free of surge.
1)6.1420
May1989
2. 51
.
POWERPLANT
Typical High Pressure Rotor Flow Chart
RNEffect
- .....
I
Operating
Line---•-t
1 ~I
I
I
I
I
I
I
I
Corrected
Line
\~Spe,,d
I
/
./
------
I _ - --'
_, / 'Deceleration
Schedule
w.~
6t25
Figure 34.
The effect of poor pressure distribution in the compressor is to change the compressor flow
relationship with respect to speed and also decrease the surge margin. This can be shown by
comparing the compressor blades to miniature wings as shown in Figure 35.
-
Direction of Rotation
..,_
Figure 35.
2. 5 2
D6-1420
May1989
POWERPLANT
It can.be seen that anything which causes a decrease in airflow (decrease in velocity) would move
the compressor blades to a higher effective angle of attack and closer to stall. Two of the most
common factors that might cause this airflow decrease are high back pressure and poor compressor
inlet distribution.
There are a number of things that might cause high back pressure. The most common is too much
fuel during acceleration. Compressor inlet distri.bution is determined by inlet cowl design and may
be vańed by airplane attitude and speed. The Reynolds number effect may be added to either of the
above effects, because it changes primarily with altimde.
Combustion Chamber Design
The main purpose of the combustion chamber section is to bring the fuel and air together, and to
introduce beat energy into the mass air by means of fuel combustion. To accomplish this purpose,
three basie steps are followed:
(1) Injection of fuel. The fuel nozzle must talce the fuel from the fuel regulator, deliver it into
the combustion portion of the chamber, and in so doing, break the fuel up into fine
particles which can then readily mix with the combustion air.
(2) Meter out combustion and cooling air. All the air passing through a jet engine is not
utilized to support combustion. If all the air and fuel were m.ixed, we would have a noncombustible charge. A typical present day engine might have a total air-fuel ratio of about
73 to 1. Since hydrocarbon fuels prefer an ideał A/F ratio near 15/1, the combustion
chamber must, therefore, separate approximately 20% of the total air from the compressor
and divert it into the burner section where it will mix with the fuel to support combustion.
The remaining 80% of airflow, known as cooling or dilution air, must be temporańly
withheld from the bumer section until the process of combustion has taken place.
The combustion air is introduced into the burner section through angled orifices around
the fuel nozzles. This method of introduction imparts a high swirl velocity but a relatively
low linear velocity. Therefore, more time is allowed for mixing and buming in the
forward (combustion) zone of the burner and for diluting and mixing in the latter part
(mixing section). The lower linear velocity increases the airspeed and altitude at which
inflight starts can be accomplished. A plot of a typical inflight start envelope is shown on
Figure 36. Except for an altitude limit, the inflight envelope should include all the airplane
flight envelope. Because constant fuel flow is used during jet engine starting, a certain
pressure must be reached to have sufficieni air for initiating combustion and for cooling.
Figure 36 shows that increased airspeed with altitude increase is required to windmill the
engine sufficiently to reach the needed combustor pressure. Starter assist is usually needed
to restart at the lower speeds of the airplane flight envelope. If starting is attempted at too
low a pressure, ignition does not occur or incomplete combustion occurs and the flame
persists into downstream components overheating them. The maximum speed shown on
the inflight start envelope is the speed that represents the maximum linear combustion
chamber velocity for inflight starts. Above this speed the combustor chamber velocity
would be too high for an initial light
D6-1420
May1989
2. 5 3
POWERPLANT
inf'light
start
envelope
TRUE AIRSPEED
Figure 36.
(3) Diluti.on of combusti.on gases. After combusti.on, the reroaining 80% of the airflow must
be rapidly introduced into the bumer section to dilute the hot combusti.on gas before it can
come in contact with the turbine nozzle guide vanes. The combusti.on gas may be as hot
as 3,SOO°F (1,925°C). To insure longer life of the turbine nozzle guide vanes, cooling air
dilutes the combusti.on gases, reducing the gas temperature to within the present allowable
maximum limit of 1,800° F (1,000°C) for engines with uncooled turbine blades to
2,300oF (1,2500C) for engines with air-cooled turbine blades.
Although combusti.on chambers today have many configurati.ons, there arc three basie types,
namely: Multiple Chamber, Annular Chamber, and Can-Annular Chamber. See Figure 37.
(1) Multiple Chamber. This type of combusti.on system lends itself readily to the centrifugal
flow engine since the air must be divided to give equal .proporti.ons to cach can or
chamber. V ariati.ons in the can arc adaptable to the axial flow engine. One advantage of
the can type is that the liner surface has a large degree of curvature which results in high
strength and resistance to warping. Some disadvantages arc: (1) the available space is not
uti.lized efficiently and, (2) the large amount of metal required to enclose the desired
volume of gas flow results in a heavier combusti.on chamber.
2.5 4
D6-1420
May1989
POWERPLANT
(2) Annular Chamber. The annular type has some desirable advantages over and above
efficient air and gas handling. There is little doubt that this type gives the most efficient
usage of the available space. For this reason, the annular type will require only about half
the diameter for the same mass airflow. The sheet metal smface to enclose the required
volume is low resulting in lighter weight. While the annular type is basically simpler in
construction, the !ower degree of curvature makes it more susceptible to warping which
distorts the temperature panem at the turbine section. lntensive development by engine
manufacnirers have overcome these problems: and because of its light weight and efficient
operation, this type of combustor is used on the !atest large commercial engines.
(3) Can-Annular Chamber. The can-annular is a combustion cham.ber with characteristics of
both annular and can types,some of the advantages of both types being retained. This
type of ~ombustion chamber is composed of combustion chamber liners located
circumferentially at equal distances within an aimular combustion cham.ber case. Greater
case in development is stili possible, thereby encouraging the use of this type. Each liner
has its own fuel nozzle or nozzles.Individual liners also tend to even out the air velocity
distribution into the combustion cham.ber. The can-annular combustion cham.ber operates
at a high pressure level aiding efficient combustion at reduced powers and high altitudes.
Each liner having multiple nozzle arrangements has the added advantage of minimizing
temperature variation caused by the clogging of any one element The IT8D series engines
use this type of combustor.
MULTIPLE
CHAMBER
ANNUIAR
CHAMBER
CAN-ANNUI.AR
CBAMBER
~~~
~
e
~~~
Figure 37.
It is highly desirable to maintain a uniform gas temperature of the gases at the discharge cross
section of the liner so that the turbine nozzle vanes and turbine rotor blades will expand uniformly
and not set up high localized blade stresses and thermal shock which are apt to result in warpage or
cracking. A poor temperature distribution de.creases life of the engine as well as efficiency.
06-1420
May1989
2. 5 5
POWERPLANT
Figure 38 is used to illustrate this temperature distribution. H a plot is drawn of gas temperature
versus the cross section of the liner at its discharge end, the dotted line would indicate the theoretical
or desired distribution of temperature while the solid line shows a norma! acceptable distribution
and the dashed line depicts an undesirable over heating condition caused by faulty uneven. flow
distribution. Normally, the temperature at cach side of the liner walls should run relatively cool due
to the cooling air blown longitudinally along the liner wall. The temperature should then rise
rapidly to the allowable maximum and remain at that value across the liner until it approaches the
opposite liner wall where the action of the liner wall cooling air again reduces the temperature to a
safe value within the limits of the liner mateńal.
Distribution
Norma!
GasFlow
--------l)lla~ -- -- -- -
~1200.
1400
t1600
Gas Temperature °F
Figme 38.
Since the practical rarely equals the theoretical, the actual temperature distribution would appear lice
the solid line, indicating less of the liner's cross section remaining at the desired maximum
temperature. Uneven flow distribution resulting in lower flame velocity might give a temperature
distribution too low near the liner walls and excessively high in the center of the gas stream. Such
exceeding of maximum allowable temperatures, when existing more than momentarily, causes blade
buming, warpage and stress cracking.
In order to control this temperature panem, sufficient distance of travel must be provided for the
introduction of the main cooling air and its dilution of the hot combustion gas, thus resulting in an
acceptable degree of evenness of temperature distribution. H this distance is limited, then the liner
must be designed to accomplish this job. A typical panem for this might be as shown in Figure 39.
Presently used values of maximum tmbine inlet temperature arc in the order of 1,500 °F to 2,000 °F
(800 °C to l,lOO°C) A top value for turbine inlet temperature without cooled turbine blades would
probably be 1,700 °F (950 °C) With air cooled blades, a temperature in the order of 2,000 °F
(1,100 °C) is now possible and provides an output increase about proportional to the temperature
increase. Air cooled blades, however, reduce the turbine efficiency, increase the cooling air drag,
and are more difficult to fabricate.
2.5 6
1)6.1420
May1989
POWERPLANT
combustion air
•
metal cooling air ~
gas cooling airi====~
Figure 39.
With other factors remaining constant, increasing turbine inlet temperature at low compression
ratios increases the thrust specific fuel consumption (See Figure 26). Even though increasing
turbine inlet temperature improves the thermal efficiency, it lowers the propulsive efficiency of the
jet such that the net result is a decrease in economy (Figure 25). This might lead one to suspect that
a very low turbine inlet temperature would be desirable for cruising economy. At high compression
ratios, however, a turbine inlet temperature in the order of l,300oP (700°C) seems to be optimum
for long range operation.
Turbine Design
The pmpose of the turbine is to transform some of the beat and pressure energy of the exhaust
gases into shaft horsepower to drive the compressor and accessories. The remaining kinetic energy
and pressure energy of the gases produce engine thrust.
The turbine stage is comprised of two main elements, the turbine stator nozzle vanes and turbine
rotor. See Figure 40.
(1) Tmbine Stator Nozzle Vanes. These arc stationary blades attached radially to the turbine
case at the entrance to the unit as well as between the turbine rotor stages. These blades
arc contoured and set at such an angle that they form small nozzles discharging the gas at a
high velocity. The gas flow is clirected against the rotating tmbine blades in a clirection so
as to efficiently transfer the kinetic energy into mechanical energy.
(2) Turbine Rotor. The tmbine rotor consists of contoured blades attached to a disc. This in
tum is attached to the power transmitting shaft of the engine. The gas flow leaving the
stationary blades act upon the turbine rotor blades turning it at very high speeds. This
places high centrifugal loads upon the turbine rotor.
D6-1420
May1989
2. 5 7
POWERPLANT
lURBINE
ROTOR
SHROUDED
lURBINE
BLADES
Figure 40.
The turbines in use on axial flow engines are of two major types, impulse and reaction. In the
impulse type of turbine, there is no change in pressme between the rotor inlet gas and the rotor exit
gas, but there is a large deflection of the gas flow. The turbine nozzle guide vanes are shaped to
form passages which increase the velocity and reduce the pressure of the escaping gas flow. The
high velocity generated in the turbine stator nozzle guide vanes impinges on the moving turbine
rotor blades as shown in Figure 43. In the reaction type of turbine, a pressure change takes place as
the gas flows tbrough the turbine rotor blades resulting in an equal and opposite reactive force. The
reduction in pressure and the increase in velocity of the gas flow are accomplished by the shape of
the passage between the turbine rotor blades. In most practical designs, the turbine blade is a
combination of both of these two types and is known as a reaction-impulse turbine. However, all
turbine designs have one important principle in common; that of expanding a high-pressure and
high-temperature gas to lower pressures and temperatures.
The mass airflow from the combustion chamber is delivered evenly to the turbine nozzle guide
vanes. The nozzle vanes serve to accomplish two functions; that of accelerating the gas flow, and
of deflecting the gas flow to a specific angle in the direction of turbine rotor rotation. See Figure
41.
I, I
/L lURBINE STATOR
'-r' NOZZLE VANES
{STATK>NARY)
ACCELERATED
AND DEFLECTED
GASFLOW
/
Figure 41.
2.5 8
D6-1420
May1989
POWERPLANT
,
The design of the turbine stator nozzle passages may be converging-diverging (high pressure
ratios), or they may be wholly converging (low pressure ratios), as shown in Figure 42.
CONVERGINGDIVERGING
TURBINE STATOR
NOZZI.E VANES
ę CONVERGING
TURBINE
BLADES
(a) IMPULSE
(b) REACTION
(BLADE TIP)
(BLADE ROOT)
Figure 42.
The effective area of the nozzle passages is measured at righl angles to the direction of flow. The
shapes, lengths, numbers of nozzles (and turbine blades), and size of the nozzle area may vary
considerably between designs.
·
·
To understand the effect of the combination impulse-reaction turbine design, it is convenient to
discuss the properties of cach type separately.
In a pure impulse turbine, the entire pressure drop occurs in the turbine stator nozzle vanes. The
nozzle passages accelera.te the gas flow and angle it in the direction of turbine rotation. The density
of the gas flow remains approximately constant in the turbine rotor blade passages, and the area of
the passages is essentially constant from the entrance to the exit, as shown in Figure 43.
IMPULSE
TURBINE
BLADES
DIRECTION
.OFMOTION
Figure 43.
D6-1420
May1989
2.59
POWERPLANT
The function of the turbine rotor blades is to extract as much kinetic energy from the gas flow as
possible. Since there is no pressure change in the blade passages the velocity relative to the blades
entering and leaving the rotor will be approximately equal in magińtude but opposite in direction.
When the entrance and exit angles arc equal, _the velocity relative to a stationary reference.(engirte
case) will be axial and maximum rotational kinetic energy will have been removed from the gas
flow. The force exerted on the rotor due to the deflection of the relative velocity imparts rotational ·
energy to the rotor This force can be quantified using momentum theory.
Analytically, the change in momentum can be demonstrated by referńng to the diagram of Figure
44. The entrance relative velocity V 1, is the resultant of the axial velocity Va, and the turbine rotor
tangential velocity V t· As the fluid leaves the turbine blades, the direction of the relative velocity
has been changed, but the magnitude is essentially the same. Figure 44 shows the exit angle the
same as the entrance angle which simplifies the analysis.
Figure 44.
The product of mass and velocity is called "momentum" and may be expressed as,
momentum = MV
2.60
(39)
1)6.1420
May1989
POWERPLANT
A moving body is characterized by its momentum. The product of M times AV expresses a change
in momentum. Since a change in momentum is equal to impulse,the product of an average value of
a force and the time during which it acts, an equation relating the two may be written:
(40)
MA.V = F At
Referring to Figure 44, an expression for obtaining the momentum change may be written from
equation (40).
momenmm change= ( ~
)iv
1
The force impaned to the turbine rotor is proportional to the ra.te of momentum change and may be
determined as follows:
Force= 2 (
~ )( ~)
In reaction turbines, much of the pressure drop occurs in the turbine rotor blade passages. Since the
turbine stator nozzles tums the working gas flow, the velocity has to increase by a factor of 1/cos a
, where a is the direction change. Therefore, some pressure drop occurs in the turbine stator
nozzles of even the highest reaction turbines. The remaining pressure drop· occurs across the
rotating turbine blades where the flow area decreases from entrance to exit. See Figure 45.
REACTION
DIRECTION
TURBINE
OFMOTION
Figure 45.
As a result of the pressure drop in the turbine blade passages, the velocity increases becoming a
maximum at the exit. The immediate accelerating force is the pressure gradient . This force is
transmitted to the blades so that the momentum of the fluid is really changed by a force exerted on
the blades just as in the impulse turbine. In both types, the effective force acting on a blade is
physically the pressure difference across the blade times the blade area.
The two types of turbines are successfully combined by designing the blade tips for reaction, and
the blade roots for impulse, and blending one into the other so that the characteristics are
approximately half reaction and half impulse. Figure 46 shows the pressure and velocity
characteristics through a typical turbine with respect to the stationary engine components.
D6-1420
May 1989
2. 61
POWERPLANT
In Figure 47 a turbine blade is shown with a pressure gradient indicating the amount of pressure
drop across the turbine blade. The higher tip pressure helps to reduce outward flow due to
centrifugal effects, thus making the blade more efficienL However, the high tip pressure does
contribute to higher tip leakage loss.
I
I ITURBINEII
I TURBINEI
ROTOR
I
ROTOR
BLADES
BLADES
I
I
TURBINE I ,!_
~
I ~, I~
STATOR
lNOZZLE
VANES
II I
PRESSURE ~
II I
~ I ',cf I
i~ I ~ Il',,....
COMBUSTION
CHAMBER
. COMBUSTION
CHAMBER
EXIT
EXIT
PRESSURE
II
~~
io
f39
8: g;!
VELOCnY
'~~ EXrT
JPRESSURE
I AMBmPRESSURE-1
(b) REACTION
(BLADE TIP)
(a) IMPULSE
(BLADE ROOl)
Figure 46.
2.62
D6-1420
May1989
POWERPLANT
'lURBINE ROTOR BIADE
:[[-----------------=====()
~
i
root section
(impulse)
i ł-------=ex::.1::.;t::.....i:::.r.::::es=.=s::.:ur::.:::e
I ___
tip section
(reaction)
_ _ _ ___;;;===,-,
tip
shroud
~bient preS!B!'L- _ _
A-A
Figure 47.
The advantage of tip shrouding as shown in Figure 47 is that the gap between the tips of the turbine
blades and turbine case can be more easily scaled, since the shroud reduces the problem to that of
sealing between two fiat surfaces. This scal reduces the gas leakage around the tip of the turbine
blade and increases the turbine efficiency. Another advantage is that shrouding dampens the blade
vibrations permitting the usc of reduced blade chords, and this results in a shorter and lighter
engine. Shrouding, however, reduces the maximum allowable turbine inlet temperature since the
effect of centrifugal action on the mass of the shroud must be resisted by the blades. Most of the
latest engine designs which are operating at high turbine inlet temperatures have non-shrouded first
stage turbines.
The characteristics of gas flow through the turbine may be shown by means of a gas flow diagram.
The diagram, Figure 48 graphically represents the blade and passage contour. The direction and
magnitude of the gas flow velocity, and the rotor tangential velocity, are shown by the solid line
vectors. Velocities relative to the rotating blade are shown by the dashed-line vectors.
D6-1420
May1989
2. 63
POWERPLANT
BIADE TIP
BIADE ROOl'
(reaction)
(impul.ee)
- - - - - - combustion cbsmber
·exit vel.ocity
v,
v1 , '
STATOR NOZZLES
rotor bl.ade
entrance
vel.ocity
(rel.ative
vel.ocity)
nozzle
exit
vel.ocity
rotor
tip speed
ROTOR BLADES
rotor exit
vel.ocity
(rel.ative
· veloci ty V
v /
5
/
I exbaust duet
V entrance vel.ocity
7
V6
AT DESIGN CRUISE BPM
rotor
tip speed
I
rotor
root speed
I
AXIS
AXIS
Figure 48.
The gas flow enters the nozzle blade tip at a moderate velocity V 1, and is accelerated through the
nozzle to an exit velocity V2· Note the deflection from the axis by the nozzle. The rotor blade tip
speed and direction are inclicated by vector V3. With the rotor blade moving in the same generał
clirection but at an angle to the nozzle exit velocity V2, the clirection and magnitude of the gas
velocity V4, entering the rotor blade will be the vectorial clifference between V2 and V3. This is a
relative velocity and represents the angle of gas flow entrance relative to the rotor blades. If there
should be insufficient deflection angle from the nozzle, the rotor speed would be limited to a lower
value than desired. At design RPM, the gas deflection out of the nozzle must be such as to provide
a gas flow into the rotor blade at the optimum angle of attack.Thus the rotor and stator blade angles
2.64
1)6..1420
May1989
POWERPLANT
relative to the case must be determined together and for the design condition. The axial distance
between blade rows must be held to a minimum to reduce engine length and weight, but it must
provide running eleara.nec and enough space to attenuate wake affects which might induce vibration.
Assuming flow to be axial at the root as well as at the tip, the flow angles must be altered to
compeosate for the lower speed, V6, of the blade root. To obtain the greatest rate of momentum
change for impulse, a high degree of direction change of the gas flow through the turbine rotor
blade must be maintained. To accomplish this, a greater deflection through the nozzle is provided.
With the lower root speed, V 3, the rotor blade entrance velocity, V4, will be greater than at the tip
and its angle of attack will be increased.
A similar situation exists with the gas flow out of the rotor blades and through the exhaust duet. To
obtain approximately axial flow down the duet, the rotor blades must deflect the exit gas flow, V s,
sufficiently to compensate for desired rotor RPM. If the exit flow angle does not properly match
the desired rotor speed, the exhaust duet entrance velocity, V 7, would be deflected off the axis and
result in a swirl flow instead of axial flow down the ducL The presence of swirl means a loss in
thrust from the exhaust nozzle. Exhaust duet swirl can be eliminated at only one specific rotor
speed; the engine design RPM. As the turbine is operated above or below this specified RPM,
swirl in either direction would occur. For example, Figure 48 indicates conditions at one specified
RPM. If the RPM should change, the diagram would be altered; the blade contour and blade angles
can not change, but the gas flow vectors and angles would change, resulting in duet swirl.
Therefore, the gas flow diagram will change with each RPM. For this reason, straightening vanes
are commonly used in jet engines. The effects of straightening vanes for operating at an unspecified
RPM may be scen in Figure 49.
SwirlAngle
Figure49.
06-1420
May1989
2. 65
POWERPLANT
Exbaust Nozzle Design
The exhaust nozzle is designed to expand the hot combustion gases from the turbine discharge
pressure to atmospheric pressure in a manner that will produce a maximum amount of thrust. The
performance of a turbojet unit is quite sensitive to exhaust nozzle design. The majority of
installations employ a single direct exhaust to obtain the advantages of low weight and simplicity
with minimum duet losses. Mechanical devices designed to alter the orifice area or direction of jet
exhaust gases arc used for afterburning, thrust reversing, and noise suppression.
V arying requirements make it impractical for the powerplant manufacturer to include the duet in its
entirety as part of the powerplant. The airplane manufacturer is obliged to complete the exhaust
system by designing to the specific engine airplane requirements.
.
.
The chief function of the exhaust system is to straighten the gas flow as it comes off the last turbine
stage, and to accelerate the exhausting gas stream. A tailcone is installed in the duet to provide a
path for the gas to exit without abrupt _changes in cross section. The struts supporting the tailcone
arc streamlined and of such length as to SCIVe as straightening vanes. The area of the exhaust nozzle
is usually fixed; however, in some installations the exhaust nozzle area can be varied by varying the
position of the tailcone. While the variable type nozzle configuration presents many mechanical
diffi.culties and requires a more complicated analysis, it offers some advantages; it is possible to
extract additional thrust at higher temperature for short periods of time providing a useful thrust
boost for takeoffs and emergencies.
The fixed nozzle arca design is such that the selection of exit area results in either the maximum
thrust at allowable gas temperature for takeoff or continuous operati.on, or in some maximum
allowable specific fuel consumption. As scen by the thrust equation:
The effect of increasing the exhaust velocity increases thrust. It is normal to increase the exhaust
velocity to produce sonie velocity at the orifice; that is, M = 1. When static pressure at the exhaust
nozzle equals ambient static pressure, there is no fmther increase in exhaust velocity. If static
pressure is greater than ambient and M = 1 at the throat of the e:xhaust nozzle, the flow must expand
to ambient pressure and additional thrust is available by velocity increase to supersonic speeds. If
the expansion takes place behind the physical confines of the nozzle, some energy is lost to the
airstream. If the nozzle is designed with a divergent section to allow for the expansion, the
maximum energy transfer to thrust is achieved. Thus, it is not unusual to find a convergent
divergent exhaust nozzle on some jet engines . See Figure 50.
2.66
D6-1420
March 1991
POWERPLANT
Thrust calculations made up to this point have been based on the assumption of subsonic or sonie
flow at the exhaust. No allowance has been made for the added reaction to any supersonic
expansion. Therefore, a term to account for this additional thiust must be added. It can be
calculated in terms of the pressure difference times the throat area. And the net thrust equation
becomes:
(41)
subsonic
convergent
section
14
I
supersonic
divergent
section
~
··4
.exbaust
nozz1e
L(tbroat)
gas attains sonie vel.ocity'
Figure 50.
In equation (41), A1 is the exhaust nozzle throat area, v1 is the exhaust velocity at the throat, and the
pressure term is the result of the expansion from the exhaust nozzle throat to atmosphere.
D6-1420
May1989
2. 67
POWERPLANT
Intentionally Left Blank.
2.6 8
D6-1420
May1989
POWERPLANT
2-7
Ratings
The turbine inlet temperature is a function of the amount of fuel introduced into the combustion
chamber. There is a practical maximum allowable temperature which the turbine rotor blades can
withstand under the high stress ofrotation. This allowable temperature defines a rating. The
maximum temperatme that the tmbine can withstand for a limited time peńod is the takeoff rating.
Takeoff (Wet).
This is the maximum thrust available for takeoff. The rating is obtained by actuating the
water injection system and setting the computed (wet) thrust with the thrust lever. The
rating is restricted to takeoff, may have an altitude limitation, and is normally time
limited.
Takeoff (Dry).
This is the maximum thrust available without the use of water injection. The rating is
obtained by either placing the thrust lever in the full forward position or adjusting the
thrust lever to obtain the ·computed takeoff (dry) thrust for the existing ambient
conditions, whichever is obtained first. It is time limited and used for takeoff only.
The design turbine inlet temperature can be as high as 2,650°F (1,4500C) for this rating and is
normally limited to 5 minutes, of which about one half may be with water injection if desired.
A lower turbine inlet temperature, approximately 200°F (lOOOC) lower than the maximum, defines
a maximum continuous rating.
Maximum continuous.
This rating is the maximum thrust which may be used continuously and is intended only
for emergency use (engine inoperative situation) at the discretion of the pilot The rating
is obtained by adjusting the thrust lever to obtain a predetermined EPR or N 1 RPM
value, depending on type of engine.
The next lower rating is the Maximum climb .rating which may be defined by a turbine inlet
temperature equal to or slightly lower than the temperatme for maximum continuous rating.
Maximum climb.
This rating is the maximum approved for normal climb. The rating is obtained in the
same manner as maximum continuous. For some engines at certain altitudes, maximum
continuous and maximum climb rated thrust are the same.
Maximum cruise.
This is the maximum thrust approved for cruising. It is obtained in the same manner as
maximum continuous.
ldle This is not an engine rating but, rather, a thrust lever position suitable for minimum thrust
operation on the ground or in flight It is obtained by placing the thrust lever in the idle position on
the thrust lever quad.rant
D6-1420
March 1991
2. 6 9
POWERPLANT
For most high bypass ratio engines there are two idle thrust modes mininum (low) and approach
(high). Mininum idle is for ground operation and most phases of flight except approach and
landing. Approach idle is for when the flaps are extended to the appropriate position (usually
approach or landing) and/or the gear is extended. This idle setting is usually m.aintained until 4 or 5
seconds after·touchdown. This use of approach idle facilitates engine acceleration in event of a goaround. Approach idle is also used when engine anti-ice is selected.
The various ratings for an engine are -specified by the engine manufacturer and published for ICAO
standard, sea level, static conditions, assuming no losses. The effect of altitude, off standard
temperature, and airspeed on thrust will be discussed later.
2.70
D6-1420
March 1991
POWERPLANT
2-8
Temperature and Velocity Variations
So far nothing has been said about the magnitude of the temperatures and velocities existing
throughout the cycle. A profile showing the manner in which the air is processed as it passes
through the engine appears on Figure 51. The temperature reaches its maximum in the combustion
chamber; however, it is limited by the highest temperature permitted by the metallurgy and cooling
of the downstream components. Most critical of these components is usually the high pressure
turbine rotor and blades. Some temperature rise results from beat generated in the compression
process, but a drop in temperatme occurs across the tmbine section.
The velocity is carefully controlled through the jet engine, both in magnitude and direction to
minimize unnecessary losses in energy. The velocity varies from a relatively low speed at the
engine inlet to sonie or supersonic speeds through the turbine nozzles and exhaust nozzle.
The compressor adds energy to the air which results in changes in pressure, temperature and
velocity. The reaction to this action is the largest portion of thrust produced by the engine. This
forward force is imposed on the various compressor elements and is transmitted to the aińrame
through the engine structure.
Outside energy is supplied to the cycle in the combustion chamber by the addition and burning of
fuel. The temperature increase causes a large velocity increase with virtually little pressure loss.
The thrust reaction to the acceleration is transferred to the engine through the structure of the
combustion chamber.
·
Due to the shape of the turbine section, the velocity is increased in the turbine stator nozzle; but
since no heat is added, the pressure, temperature, and velocity arc reduced at the turbine rotor exit
due to the extraction of work by the turbine.
The exhaust nozzle is designed to accelerate the gas flow with low pressure loss. V arious devices
can be employed at the exhaust nozzle exit to improve performance.
Note that from the turbine entry to the tailpipe exit, Figure 51, there exists a loss in thrust cancelling
roughly two-thirds of the forward force built up in the compressor and combustion sections. The
diagrams arc only representative, as variations occur with individual engines.
D6-1420
May1989
2. 71
POWERPLANT
_
...e=
se
·1
I
I
I
I
I
I
I
I
I
I
~ł
~
....~
t)
~ ....._.....,__
Air
Inlet
Compressor
· Combustor
Fwd
Turbine
t
Nozzle
Rv,d„
I
I Rwd „1
I
I
-~I--~.._~,
I Propulsivel Fwd
Force,-. I
Figure51.
2. 7 2
D6-1420
May1989
POWERPLANT
2- 9. Bypass and Turbofan Engines
In commercial operation, the bypass and turbofan engines have become more important. The
question may be asked as to why we should have engines of this type. The answer is simply that
the addition of the bypass airflow improves the propulsive efficiency, at the cruising speeds under
consideration, by producing a lower jet velocity than that of a simple jet without reducing the
thermal or cycle efficiency of the engine. This effect increases the overall efficiency, the product of
thermal effi.ciency and propulsive efficiency, as scen by equation (31). Equation (34) shows that
the thermal efficiency, the conversion ofheat energy into power, continually increases with increase
in turbine inlet temperature. This increase in turbine inlet temperature increases the jet velocity and
engine thrust Unfortunately however, equation (35) shows that this increase in jet velocity reduces
the propulsive efficiency which reduces the overall efficiency. For turbojet engines, the increase in
thrust at these high jet velocities is obtained at the expense of higher fuel consumption. The generał
principle of the turbofan engine is to convert more of the fuel energy into pressure energy which
results in more thrust without an increase in fuel flow. This is accomplished by a lower exhaust
velocity which increases the propulsive efficiency.
Optimum turbine inlet temperature depends on the engine cycle pressure ratio and bypass ratio.
Figure 52 shows the .propulsive efficiency, thermal efficiency, specific fuel consumption, and
turbine inlet temperature plotted against thrust per lb. of air flow for a typical high pressure ratio jet
engine. The condition shown is at 475 knots in the stratosphere. Note that the propulsive efficiency
f alls with increase in thrust per lb. of air flow or increase in turbine inlet temperature. It shows the
increase in thermal efficiency with increase in turbine inlet temperature, due to the compressor and
turbine having efficiencies less than 100%. Ifthe components bad no losses, the thermal efficiency
would depend only on the compression ratio.
Current high-bypass ratio high pressure ratio engines have turbine inlet temperatures over l,OOO°F
(550°C) higher than the low-bypass low pressure ratio engine of Figure 52. In these later
technology engines overall pressure ratios of 25 to 35 and bypass ratios of 5 or 6 arc common.
Another item of interest is the opposing influence of thermal and propulsive efficiency on fuel
consumption as turbine inlet temperature changes. The specific consumption curve is the inverse of
the product of the thermal and propulsive efficiency cmves.The turbine inlet temperature for the best
fuel consumption is about 800 degrees Kelvin (approximately 1,000° Fahrenheit). In practice, for a
given size and weight, a higher temperature is selected to increase thrust with only a small penalty in
fuel consumption.
D6-1420
May1989
2. 73
POWERPLANT
36 ooo :rr
1300
.7 1200
1100
.6 1000
EFFICIENCY
.5
SCALE
1521
REFERENCE:
Rolls-Royce
Public:ation l 014
''The By-Pass.
Engine for
Transport
1341 +----+~;:--;~;,,;----,r----+---+----1
900
Airc:raf't"
(overall
ettic:ienc:y
l.ine added)
700
.4
.9
SPF.C FUEL
CONSUMPTION,
LB/HP./U3
.21-----=+:-------~~~~ł-----=:J::::::~t::------ł------l .8
THRCJST
o.___ __.__.......__..__ __.__________....____
O
1O
20 .
30
4o
50
6o
70
THRUST/LB OF TOTAL AIR FLOW
Figure 52.
One method of increasing the thrust output with no change in fuel consumption is by using more of
the turbine gas flow energy to drive a propeller. This is a turboprop engine. The mass airflow
through the propeller is very high and its jet velocity is low; therefore, this gives very high
propulsive efficiencies at static conditions and low flight speeds. However, the propeller losses are
very much influenced by changes in airspeed. The turboprop has very good thrust and fuel flow
capabilities at low speeds but suffers serious losses at high subsonic flight speeds, due largely to
high Mach numbers of the air relative to the propeller blade tips. A method suggested a few years
ago to avoid these tip speed losses was to surround the propeller with a duet The effect is to cause
compression of the air ahead of the blades, and to keep the blade Mach number substantially the
same as at low speeds, i.e. constant over a wide range of speeds. This is the ducted fan. Since the
propeller is a heavy form of fan, it is possible to consider a smaller higher speed fan.of such size
that it may be directly driven at turbine shaft speeds and can supercharge the rest of the engine,
becoming an integral part of the overall compression system.
Since the size of this fan is restricted, the amount of air it can handle is restricted; but the pressure
rise of the air may be quite large. It is reasonable to use part of the air from the fan to feed the rest
of the engine and so increase the pressure ratio. Figure 53 shows a bypass jet engine in which the
engine consists of a low pressure compressor feeding part of its air to the high pressure compressor
and the rest ducted through the bypass duet. The high pressure compressor discharge passes
through the combustion chamber, the high pressure turbine, and then through a second low
pressure turbine which drives the low pressure compressor. Finally the jet expands through an
exhaust nozzle either before or after mixing with the bypass air.
2. 74
D6-1420
May1989
POWERPLANT
BYPASS ENGINE
Figure 53.
The advantage of this arrangement, when compared with the same system in which the low
pressure compressor is just large enough to feed the high pressure compressor only, is that more
energy is removed by the low pressure turbine so that the exhaust velocity is reduced resulting in a
higher propulsive efficiency. The energy removed will be conveyed through the turbine and low
pressure compressor to a stream of relatively cold "bypass" air where the velocity is again such that
the propulsive efficiency is high. In this way a good overall efficiency can be obtained by operating
at a high turbine inlet temperature where the thermal efficiency is high. The pure jet engine can also
be made to operate at a low tmbine inlet temperature to give the same order of jet velocity as that of
a bypass engine, but only by compromising its thermal effi.ciency. Figure 54 shows the
comparison between a pure jet engine and a bypass engine with a bypass ratio of one.
D6-1420
May1989
2. 7 5
POWERPLANT
COMPARISON OF PURE JE' AND BY-PASS
ENGINE
.. -.~~~~-----..'
.9~~~~~~~~~~~~~~~~~~-~~~
.... ,...
..
·······:··::::···:·:·~:::·::.·:·:•::•·i
• 8 .. ·.····•··· .·.'_·\~-~·•.···•·.-·.•.·.··.·.:·.•.·.•.• .. J.·.·-·.·.·.·_: ......·..·.·.·!.•·.·.·.·.··. •!
I•••·~
7t-+~=Tj
. ....
. . .. .. . .. .:
.........
..
.
. . .
..
:
. .
...
.
.
~
.. .
.
.
.
•1
···--·-: ---- ... ·······-··--=.
.
·; -·---·····i·
:
:
------ i -- ,;
. .
r----· ..:·:···:---------·· .. ::::.: ...... --·i·-·-----.. ---·-··t···-··-·--····-~----------------~-----········
; ..
;
.
; ... . : ___ :_. ::, · .. : ..... .
:
..
... . : ..
.
.
..
.
.
..... .
···--------:---:----:!:···---······1...._.;.....;.....;........_...._...._..;.;...i
····---······i·-········--·;r~--------··t.
o..._-····
___________________
...._
_________
.
o
10
20
30
4o
50
60
70
TBRUST/LB OF TOTAL AIR FLOW
Figure 54.
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~1420
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POWERPLANT
For given airflow and turbine inlet temperature, the thrust will always be less than can be obtained
from a simple jet engine as shown in Figure 55 for two engines having the same compression ratio,
similar component efficiencies, and the same airflow.
Compression of Simple Jet and Bypass
With Variation in Flame Temperature
1.3
AtSame
Compression
Ratio
1.2
Thrust Per Unit
Airflow - Ratio to
Simple Jetat
· 1,000 0 K
1. l
l.O
.9
Reference:
Rolls-Royce
Publication 655
"The Bypass
Turbo Jet"
.8
.7
.6
.5
1.16
475 Knots TAS_--+______.,........ 1.12
Stratosphere Ratio to Simple
Jet 1,000 °K
1.08
1.04
1.00
.96
Specific
Fuel
Consumption
Lb/Hr/Lb
Thrust
.92
800
900
1,000
1,100
1,200
Turbine Inlet Flame Temperature °K
Figure 55.
It now remains to consider the weight. For a given total airflow, the bypass engine is lighter than a
pure jet engine. Also, in this case, the thrust is lower for the bypass engine since the thrust per unit
of airflow is lower. If both engines were scaled to give the same thrust, it is possible that the
weight of the bypass engine would approach that of the simple jet. However, the weight of the
pure jet engine may stili exceed that of the bypass engine. Depending on the bypass ratio, engine
thermodynamic cycles and engine applications, the fuel bum equivalent of this weight difference
can improve airplane range.
The turbofan engine is essentially a bypass engine. The most significant differences between the
two types of engines arc the amount of air that is bypassed and the generał construction. There are
a number of designs of turbofan engines.
D6-1420
March 1991
2. 77
POWERPLANT
One of these designs is the forward fan engine. In this configuration, the fan consists of one or
more stages of extra large compressor type blades that compresses the bypass air and also
supercharges the low pressure compressor of a dual axial compressor engine as shown in Figure
56.
Turbofan Engine
Figure 56.
Air passing through the central portion of the fan feeds into the engine through the compressors in
the normal manner. After going through the fan stages of the low pressure compressor, pan of the
air is exhausted through the fan nozzle, and the rest goes through the remaining stages of the low
pressure compressor, then into the high pressure compressor. The fan serves to· provide extra
thrust by accelerating a large mass of air in much the same manner as a propeller encased in a
shroud. The installed turbofan engine is shown in Figure 57.
Typical Turbofan Engine Installation
Figure 57.
Another of these designs is the aft fan engine shown in Figure 58. In this configuration there is an
integral turbine fan assembly which is mounted aft of the low pressure turbine. It has two
concentric annular flow passages. The inner passage of an annular duet directs the engine core flow
from the low pressure compressor drive turbine through the inner turbine blades of the integral
turbine fan rotor and discharges it through the exhaust nozzle. The outer duet directs bypass air
from the inlet to the fan blades attached to the turbine outer shroud. The turbine shroud is a
2. 7 8
D6-1420
Mayl989
POWERPLANT
structural member that acts both as a fan hub and as a seal between the core flow and the bypass
air.The integral turbine fan rotor is called a free floating rotor since it has no mechanical connection
to the rotors of the engine.
Aft Turbofan Engine
Figure58.
In operating characteristics, the turbofan engine lies between the turboprop engine and the turbojet
engine. A turbojet is basically a low mass rate high jet velocity unit The conversion of a turbojet
engine to a turbofan engine provides us with an engine having a higher mass rate and lower exhaust
velocities. The resulting performance improvement is due to the resulting increase in propulsive
efficiency (Equation 35) and to the increase in the mass rate handled for a given amount of fuel
consumed (lower thrust specific fuel consumption). The efficiency of the fan engine remains high
throughout its operating envelope since the inlet provides about the same Mach number at the face
of the fan regardless of the forward speed of the airplane.
Converting a turbojet engine to a fan version of the same engine will give more static takeoff thrust,
more thrust for climb and cruise for the same fuel flow resulting in lower thrust specific fuel
consumption, and quieter operation during takeoff and cruise conditions because of the lower
exhaust nozzle velocities.
When computing thrust for a turbofan engine having two exhausts, equation (17) has to be
modified to account for both the fan exhaust and the primary exhaust Since the fan exhaust
velocity and airflow are different from the primary exhaust velocity and airflow, equation (17) now
reads:
(42)
for a turbofan engine, where W 2 is the primary airflow that goes through the complete engine and
W 12 is the fan airflow.
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March 1991
2. 7 9
POWERPLANT
Intentionally Left Blank.
2.80
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POWERPLANT
2-10 Factors Affecting Thrust ·
As previously illustrated, the development of thrust is a relatively straight forward process requiring
only the knowledge of specific quantities of mass airflow and velocity changes. What is not so
straight forward is the resulting effect of the multitude of variables, always present during any form
of jet engine operation, on the specific quantities of mass airflow and velocity changes. Since these
variable factors affect the components from which the thrust is derived, they will affect the thrust
outpuL
Effect of Pressure
From previous discussions of thrust and the constant pressure jet engine cycle, it was scen that the
greater the combustion chamber inlet pressure the greater would be the area within the cycle diagram
and thus the greater would be the thrust outpuL Therefore, any cause for a change in pressure
within a jet engine will also be a cause for a change in thrust output, whether it be a low barometric
pressure, a reduced inlet pressure due to altitude, or increased inlet pressure due to ram. Such
pressure variations will cause a variation in the air density within the engine, which in tum will
cause a variation in airflow and velocity change.
Since airflow is weight of air per unit of time, and, since under constant RPM the compressor
volume discharge remains fixed, any change in air density will also change the airflow (W3 ) and
thus the thrust.
. Effect of Airspeed
Theoretically, as the airspeed of the airplane increases, the engine's maximum or rated thrust output
would decrease due to increased ram drag of engine inlet air. This fact, under actual flight
conditions, is only half true due to increased inlet air density produced by the ram effect in the inlet
duet system. However, before discussing the effect of ram, we should first investigate this
theoretical falling off of thrust with airspeed.
Applying the simplified thrust equation from above,
it can be scen that, as the airplane speed (V~ increases without an appreciable change in jet velocity
(V1 ), the term (V1 - V 0) decreases and therefore the thrust output decreases. The dotted line in
Figure 59 graphically illustrates this theoretical decrease in thrust with increased airspeed.
Additional effects under static conditions and low airspeeds will be discussed under engine
calibration.
Eventually, the thrust output approaches zero. This occurs at a point where V1 and VO arc equal.
Such peńormance is altered in actual practice by the forward morion of the airplane, causing the air
to be pressurized at the front face of the compressor. The rise in pressure above existing outside
atmospheric pressure is, as mentioned previously, referred to as ram.
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March 1991
2. 81
POWER PLANT
Effect of Ram
In Figure 59 the solid line represents more closely what is actually experienced in flight. Since any
ram effect will cause an increase in compressor entrance pressure over atmospheric, the resulting
pressure rise will cause an increase in the airflow (W8 ) and jet velocity, V1, both of which tend to
increase thrust.
O'------------------_;;::.-AIIłPIANE TRUE AIRSPEED, V
Figure 59.
With ram the available thrust output decreases as the ailplane speed increases, but at higher speeds it
begins to increase again. This effect is caused primarily by ram pressure varying with the square of
the airplane speed while tbrust loss due to momentum reduction is a linear function of speed as
shown in Figure 60.
AIIłPIANE
TRUE AIRSPEED
Figure 60.
2. 8 2
1)6..1420
May1989
POWERPLANT
At low speeds, there is not sufficient ram pressure build up to fully compensate for the engine's
natural tendency to lose thrust with airspeed. However, as air speed increases into the middle
range, usually around 400 knots (dependent upon the aircraft inlet ducting), the ram pressure
increases to a point where it fully compensates for engine thrust loss, and net thrust output ceases to
decrease At high speeds, ram pressure becomes sufficient to more than compensate for the engine's
tendency to lose thrust; therefore, an increase in net thrust results. Recovery to 100% thrust is not
possible on present day aircraft because of aircraft speed limitations shown in Figure 59.
Effect of Temperature
The gas turbine engine has proved to be very sensitive to variations in temperature of the air. Its
thrust output might vary as much as ± 20% from the specified rating with operation on a cold day
or hot day, respectively. For this reason, the specified thrust rating of any engine must be made at a
specified temperature. So that all engines can be rated on the same basis, a standard sea level
temperature of 59 °F (15 °C) is used at static conditions.
At air temperatures colder than standard, the thrust output increases; conversely, it decreases at air
temperatures hotter than standard. Figure 61 shows the typical variation of thrust with inlet air
temperature variation, and the rated thrust
59°
120°
Am TDłPERAruRE - °F
Figure 61.
Cause for thrust variation with air temperature can best be explained by again referring to the
previously discussed themy of thrust development If the air temperature decreases, the air density
increases, such that cach unit volume of air weighs more. Since at constant RPM the volume
capacity of the compressor remains constant, the increased weight per unit volume results in greater
airflow (W8 ), and therefore, greater thrust output. The reverse is true when the temperature is
higher than standard.
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May1989
2. 8 3
POWERPLANT
For commercial engines, thrust may be limited at low temperatures which correspond to ambient
densities above a specified value. Such engines arc said to be "flat-rated". The thrust restriction is
an internal pressure limitation necessary to insure structural integrity of the engine case. At fixed
RPM, the higher the density of the air at the inlet, the higher the intemal pressure behind the
compressors. Typical flat-rated performance is shown in Figure 61.
Effect of Humidity
The increased humidity causes a decrease in weight per unit volume of air within the jet engine . Its
reducing effect on output is negligible. (sec Figure 62). Because the jet engine operates with an
excess of air needed for complete combustion of all fuel, any air weight lacking in the combustion
air supply required to give the proper fuel-air ratio will be made up by air weight from the cooling
air supply. The engine will not be penalized by loss of beat energy from improper fuel-air
combustion ratio. Therefore, its output will not be reduced as much.
lOOr~:=:::"-:~:=:-~~~~~
-- .__ - -
et e ine (thrust)
- - . ~stołl e
- - . ~ e ( 1311P)
-......::
---.
95
900
· .Ol
•.02
.03
SPECIFIC HUMIDITY (LB MOIS!ruRE/LB AIR)
Figure 62.
2. 8 4
D6-1420
May1989
POWERPLANT
Effect of Water Injection
It is sometimes desirable to momentarily raise the thrust output of the gas turbine engine beyond its
dry rating. A method of thrust augmentation used on multi-engine aircraft is the use of water
injection. The principle of water injection in a turbine engine is different ftom that for a piston
engine. W ater injected into turbine engines permits an increase in beat energy input (increased fuel
flow), without exceeding the temperature limits of the engine. This increases the mass flow and
velocity of the exhaust gas, thereby increasing the thrust. Because water injection increases hot
section maintenance, it is seidom used.
There are three common methods of injecting water into turbine engines. One is to inject into the
compressor inlet, another is to inject into the diffuser section just ahead of the combustion section,
and the third is to inject directly into the combustion zone within the combustion cham.ber.
Water injected into the compressor inlet evaporates as it goes through the engine, absorbing beat
ftom the air stream. The compressor pumps more mass since the air density is greater. Since the
fuel control maintains the proper balance of fuel to air to maintain a given bumer temperature for a
given thrust lever position, the operating temperatures and pressures within the engine are similar to
what they would be on a colder day.
Water injection ahead of the combustion section ofthe engine utilizes a different principle. The
water evaporation lowers the airstream temperature. The fuel flow may be increased to bring the
turbine inlet temperature up to rated value. The additional beat energy gives thrust augmentation by
increased jet velocity. The mass of the water and added fuel contribute to added thrust, but the
amount is small with respect to the total thrust output.
Water injected directly into the combustion chamber may be mixed with a combustible fuel. A
typical coolant consists of 40% methyl alcohol and 60% water. Vaporization of this fluid reduces
the temperature, but the combustion of the alcohol increases it again. The added fuel energy raises
the RPM and jet velocity and increases the thrust output. Injection of the fuel-water mixture allows
the engine to use a much simpler fuel control system than is needed to operate with pure water
injection.
Figure 63 shows the typical effect of water injection on thrust for varying temperatures.
D6-1420
May1989
2. 85
POWERPLANT
1nc.
I
TBRUST
TDSPERA'ruRE - - 1nc.
Figure 63.
Effect of Altitude
A gas turbine engine develops its maximum output at high inlet pressure or lowest pressme altimde.
With any increase in altitude, the jet engine's thrust output, for any constant RPM schedule,
decreases as shown in Figure 64 The graduał reduction of atmospheric pressure reduces the airflow
(Wa>• and with it, the thrust output. This reduction in thrust is reduced somewhat by the graduał
reduction in air temperature with increasing altitude by giving an improvement in the cycle
efficiency.
100
0----------------'-----SL
36,000
ALTITUDE-FT
Figure 64.
2. 86
D6-1420
May1989
POWERPLANT
Notice in Figure 64 the sudden change in rate of decrease at approximately 36,000 feet This is
caused by entry into the stratosphere, the region where atm.ospheric temperature remains constant
with further increase in altitude. In the stratosphere, the jet engine is deprived of any favorable
action given by the gradually reducing temperature.
The decrease in temperature experienced with increasing altitude at constant rotor RPM increases the
compression ratio (sec Figure 24) which in tum raises the jet velocity slightly. However, there is a
decrease in airflow (Wa) even though the referred airflow has increased. The airflow effect is
greater than the velocity effect; therefore, the result is a decrease in thrust with increase in altitude.
On a dual rotor jet engine, the thrust decrease is less because the low pressure rotor speed increases
due to the decrease in back-pressure.
Effect of RPM
Up to this point these factors a:ffecting thrust have dealt only with the varying physical properties of
air and their respective effects on thrust output Rotor speed or RPM is a mechanical factor, but its
variation has the greatest effect of all on thrust.
The input of beat energy to accomplish the required amount of work upon the air mass is controlled
by the fuel control system. The variation in airflow upon which the work is done is controlled by
the engine's RPM. Since the thrust variation is a function of both fuel flow and airflow variation, it
becomes the job of the fuel control system to· control both fuel flow and RPM. As a result, to
increase thrust, the fuel control system must increase both fuel flow and RPM in such proportions
as to neither overheat nor overspeed the engine.
The variation of airflow and thmst with change in RPM is shown in Figure 65 Except for
compressibility effects, W a varies linearly with RPM in the same way as CL varies with a since each
compressor blade is a lifting airfoil. However, thrust is influenced by pressure level, temperatme,
fuel flow, and RPM. Since these variables arc inter-related, the variation of thmst with RPM is a
complex functional relationship.
'1, MASS AIRFLOW
Wa
'1, THRUST
Figure 65.
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May1989
2. 87
POWERPLANT
lntentionally Left Blank.
2. 8 8
D6-1420
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POWERPLANT
2-11 Engine Calibration
To establish the net thrust output of any turbojet or turbofan engine in flight and on the ground
when installed on a jet airplane, it is essential to know the engine gross thrust The engine
manufacturer calibrates the bare engine for performance and furnishes this information to the
airplane manufacturer. The airplane manufacturer assumes responsibility for developing the
incrementa! performance loss caused by the production installation hardware on the engine. In a
turbojet engine, the thrust is produced by flow through a single exhaust nozzle. In a turbofan
engine, the thrust is produced by flow through both a fan nozzle and a primary exhaust nozzle.
Thrust is calculated from the measured nozzle exit areas, the nozzle pressure ratios, and the nozzle
gross thrust coefficients, as given by the following formula:
Fg =Cg 'I' A Pamb
where,
(43)
cg
=
gross thrust coefficient, dimensionless
"'
=
theoretical gross thrust function, dimensionless
A
=
area of exhaust nozzle, ft 2
Pamb
=
ambient pressure, lbs/ft.2
The gross thrust coefficients of the production nozzles may be developed on the basis of full scale
ground test ńg engine data and scale model test data.
Development of Theoretical Gross Thrust. Function, 'I',
for a Convergent Nozzle
In the development of the theoretical gross thrust function for a convergent exhaust nozzle, assume
gas flow from the nozzle into a discharge section as shown in Figure 66.
,------,
FLOW
,I...
I
~------------------:---~diii°":scharge
1
I
I
I
I
I
I
L~cti~ _ _J
Figure 66.
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May1989
2. 8 9
POWERPLANT
Let
A
=
area of exhaust nozzle, ft2
Pt
=
total pressme inside exhaust nozzle, lbs/ft2
p
=
static pressure inside exhaust nozzle, lbs/ft2
Pamb
=
ambient pressure at discharge area, lbs/rt2
wg
=
gas flow, lbs/sec
V
=
exhaust nozzle exit velocity, ft/sec
Tt
=
total temperature inside exhaust nozzle, °R
T
=
static tempera.tme inside exhaust nozzle, °R
M
=
Mach number of gas at exhaust nozzle exit
From the basie momentum relationship, the gross thrust is,
(44)
From the. continuity relationship,
From the perfect gas law,
W8 =gpAV
(45)
p=pgRT
(46)
From the definition of Mach number,
M-
V
- -/ygRT
(47)
From the isentropic relationship,
'Y
y-1
Pt= ( 1+--M
y-1 2)
p
2
2.90
(48)
D6-1420
May1989
POWERPLANT
Combining equation (45) and equation-(46),
pAV
Wg= RT
and substituting into equation (44), results in,
(49)
Combining equations (47) and (48),
Substituting V2 from the resulting equation into equation (49) gives,
which when multiplied by Pami/Pamb results in,
Factoring out Pamb A,
F =p
g
amb
1-1
]
}
P
Pt T
2y
p
[ (-)
-1 - + - - 1
pamb
p
'Y - 1 pamb
A {-
_!l_
p
. .
amb
Substttuttng Pi
p
D6-1420
May 1989
for ~ and factoring,
amb
2. 91
POWERPLANT
Therefore,
.
(50)
where,
This gross thrust function,v, is shown in Figure 61.
If Pth>amb is below the critical pressure ratio, ie., the exhaust nozzle is unchoked, and it is assumed
that,
·
then,
y-1
2-y
Pt
'Y
[
'l'wx:hoked = -y- 1 ( Pamb)
- 1
]
If Pt/Pamb is above the critical pressure ratio, the jet velocity at the exhaust nozzle throat is sonie (M
=1.0) and ptfl> equals the critical pressure ratio.
'Y
y-1
Pt [
-y-1
p- 1+--M
2
2J-y- -- (-y+l)y-l
-2-
then,
1
.
( P1 ) (
'I'choked = 2 Pamb
2. 9 2
2
)y-l - 1
'Y + 1
D6-1420
May1989
POWERPLANT
0------------------..._,_________
1.0
1.2
A~2.4
1.8
2.2
2.6
2.8
3.0
B~3.6
4.0
3.8
NOZZLE EXPANSION RATIO
3.4
2.4
3.6
4.4
4.6
1.4
1.6
Figure 67.
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May1989
2.93
POWERPLANT
Since the pressure-area term is included in the definition ftom which V is derived, the calculation of
V for a choked exhaust nozzle includes expansion to supersonic speed behind the exhaust nozzle
throat.
Equation (50) shows that the engine manufacturer's method of thrust determination is based on
determining the theoretical gross thrust of the exhaust nozzle ftom nozzle throat area, A, and nozzle
throat pressure ratio, Pt/pamb• measurement. This theoretical gross thrust is then corrected to an
actual gross thrust value by applying a gross thrust correction factor, Cg. This factor therefore
represents the ratio of Fg (measured) to Fg_ (theoretical) at any given nozzle pressure ratio for a
particular exhaust nozzle configuration. Cg corrects the theoretical thrust for such errors as the
assumption that the flow is completely isentropic, that p is equal to Pamb for the unchoked case,and
nozzle losses.
Since V is a function of PtfPamb• Pt/pand 'Y only, and since Pamb is independent of a particular
exhaust nozzle configuration, the only two parameters that influence the gross thrust of a nozzle are
Cg, and A, the area of the exhaust nozzle throat at a particular exhaust gas temperature.
The area term, A, in equation (50) will vary for changes in exhaust gas tempe:ratures; therefore, it is
necessary to evaluate the effect of temperature on the exhaust nozzle metal. The exhaust gas
temperatures which influence metal temperatures and the thermal coefficient of expansion of the
nozzle metal are used to develop a relationship·of A00/Acold versus exhaust gas temperature. This
relationship is shown in Figure 68
The ratio of specific heats, 'Y, also varies with temperature. The laws of thermodynamics show that
the difference between the specific heats at constant pressure, Cp, and constant volume, Cv,
remains constant. Since this is so, it is evident that their ratio, 'Y, must change, decreasing in
magnitude as temperature increases. This variation in 'Y is shown in Figure 69.
EXH.AUST GAS TOT.AL TEMPERAWRE,
Tt7, °F
Figure 68.
2. 94
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Mayl989
POWERPLANT
y
EXHAUST GAS TOT.AL TEMPERATURE,
Tt7 , °F
Figure 69.
Metbod of Analysis for Turbojet Engines
· To develop the gross thrust coefficient for the production nozzles used by the airplane manufacturer
on the turbojet engines, the procedure is as follows:
(1)
A calibration run is made with bellmouth inlet and standard exhaust nozzle, as
designed by the engine manufactmer for reference purposes, to serve as a base run.
(2)
A second run is made with the bellmouth inlet and with the production exhaust nozzle
(thrust reverser/noise suppressor) configuration. Com.parison of data from this run
with that from run (1) gives the effect of the production exhaust nozzle on engine
performance.
(3)
A third run is made with a production flight inlet and production exhaust nozzle.
Comparison of data from this run with run (2) gives the effect of the production flight
inlet.
For run (1) Ce is calculated as the ratio of the measured thrust on the test stand to the theoretical
thrust as calcu1ated from the standard exhaust nozzle area and the measured nozzle pressure ratio
Ptłl>amb· For run (2) the method used is to obtain an increment in Cg value between the production
exhaust nozzle and the standard exhaust nozzle. The ratio of the production nozzle actual Cg value
to the reference standard Cg value represents the ratio of Cg (production nozzle) to Cg (standard
nozzle). The ratios of Cg values are then used to correct the reference standard exhaust nozzle curve
to the production exhaust configuration curve as shown in Figure 70.
D6-1420
May1989
2. 95
POWERPLANT
Reference Standard
bO
u
.,......- - -
Exhaust Nozzle ~ /
/""'
-- -- --
/
/'.:
~
fi
/
Production Exhaust
configuration
~
~
JET NóZZLE EXPANSION RATIO,
Pt7/Pam
Figure 70.
Run (3) will be covered later when a general discussion of flight inlet installation effects on engine
performance is made.
Method of Analysis for Turbofan Engines (Dual Exhaust)
To develop the gross thrust coefficients for the production exhaust nozzles used by the airplane
manufacturer on the turbofan engines, it is necessary to detennine the following:
(1)
A method of isolating the individual gross thrusts of the fan and prim.ary nozzles, so
that one reference primary exhaust nozzle and the thrust-measuring test rig can define
both fan and prim.ary exhaust flows.
·
(2)
A method of correlating the airplane manufacturer's measured test-rig thrust with the
engine manufacturer's test-rig thrust, since it is necessary to establish absolute thrust
. level as well as thrust increment.
To solve item 1, the engine manufacturer designs a duet system which isolates the fan thrust from
the prim.ary, and then he develops a gross thrust coefficient for a standard primary exhaust nozzle.
This is plotted against nozzle pressure ratio. The thrust of such a nozzle is defined and calculable
when the nozzle pressure ratio is known. The airplane manufacturer constructs the standard
exhaust nozzle according to engine manufacturer's specifications, and this nozzle becomes the
prim.ary reference exhaust nozzle for measuring primary flow.
To solve item 2, it becomes necessary for the airplane manufacturer to run engine manufacturer's
hardware on an engine which has also been calibrated on the engine manufacturer's test rig. A
comparison of the two thrust readings produces a calibration curve for the airplane manufacturer's
test rig.
2. 96
D6-1420
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POWERPLANT
With the ability to separate thrust developed, the airplane manufacturer runs an engine with both
reference fan and primary exhaust nozzles. Since the primary tlnust is calculable from the calibrated
primary reference exhaust nozzle and total thrust can be corrected by the rig calibration, subttacting
this primary thrust from the total thrust produces the fan reference duet thrust. With the fan
reference thrust defined, the primary standard exhaust nozzle is now replaced by various airplane
manufacturer's production exhaust nozzle systems. Subtraction of the reference duet thrust from
the total gross thrust produces the prim.ary gross thrust and makes the primary gross thrust
coefficient calculable. The fan reference duet is now replaced by the production fan exhaust
configuration and the total thrust measured. Since the primary thrust is known, production fan
thrust can be obtained by subtraction.
A plot of primary gross thrust coefficient, Cg, versus primary nozzle pressure ratio, Pt/Pamb• can
now be made for the production nozzles. See Figure 70.
The fan gross thrust coefficient is handled in a slightly different manner. Since the engine side
cowling and primary reverser sleeve produce the smface over which the fan exhaust air flows, they
may be considered a part of the fan nozzle flow path and as a result produce a scrubbing drag
which reduces the fan gross thrust. Since the drag produced by the scrubbing is primarily a
function of fan nozzle pressure ratio rather than airplane Mach number, this loss can be included in
the fan nozzle performance as a power package installation loss. Therefore, when considering fan
nozzle performance with the scrubbing drag included as an installation loss, a new fan nozzle gross
thrust coefficient will be introduced. This coefficient is called the fan nozzle gross thrust factor ,
and will indicate fan nozzle installed performance to distinguish it from the engine manufacturer's
coefficient, which does not include the scrubbing loss. The fan nozzle gross thrust factor appears
as Figure 71 when plotted versus fan nozzle pressure ratio, Pti/Pamb·
c;..
Method of Analysis for Ducted Fan or Bypass Engines (Single Exhaust)
The method of computing in flight thrust for long ducted turbofan engines, where the primary gas
flow and the by pass airstream exhaust through a common exhaust nozzle, will be somewhat
different than that for a turbojet engine. The main difference is not ·in the basie thrust equation,
equation (43), nor in the testing procedure, but in the method of expressing the parameters used in
the thrust equation.
Since the primary and by pass streams are mixed before exhausting through the exhaust nozzle, it is
necessary to define a mixed flow parameter to use in determining the gross thrust. It is not
practical to instrument the exhaust nozzle far enough aft to secure true mixed flow conditions, so
total pressure is measured at station 7 in both primary and bypass ducts. Th.en a mixed total
pressure value is determined empirically in terms óf the two readings:
Pt +0.8 Pt
=
D6-1420
May 1989
6
1.8
16
·
(52)
2. 97
POWERPLANT
Inserting this value into equation (51) yields,
y-1
v=
2y
y-1
(:::: )-y--
I
+1
-1
(53)
....
u
!
....
flJ
~
Fan Nozzle Pressure Ratio,
Figure 71.
The determination of y is handled d.ifferently for engines with mixed exhaust gas flow .Since it is
difficult to measure total pressure under mixed flow conditions, it is equally difficult to measure
total temperatore under mixed flow conditions. The total tempera.tore at the exhaust nozzle will be a
function of the ram air temperature at the flight inlet, the total airflow, and the amount of fuel
bumed. The total airflow and fuel flow determine an overall fuel air ratio; therefore, a plot of 'Y
versus fuel air ratio for constant ram air temperatures can be drawn. See Figure 72.
Cg may now be determined from the measured thrust and plotted as shown in Figure 73.
2. 9 8
D6-1420
May1989
POWERPLANT
'Y
Overall Fuel - Air Ratio,
wr
w.
Figure 72.
Mixed Nozzle Pressure Ratio,
Figure 73 ..
Effect of Fligbt Inlet Installation on Performance
The effect of the flight inlet on engine performance is to reduce the total pressure at the engine face,
Ptz• thereby reducing the absolute pressure level throughout the engine. This brings about a
reduced Pt , so that while the engine feels the same power level of operation or engine pressure
7
ratio, PtfPt2• there is a reduction in thrust due to a !ower nozzle pressure ratio, Pt/Pamb·
This portion of the engine calibration test is conducted after the effects of the production exhaust
nozzle have been established with the bellmouth inlet It is run with the engine in production
configuration, with production flight inlet and production exhaust nozzle.
D6-1420
May 1989
2. 9 9
POWERPLANT
The flight inlet affects the ram drag, Fr, of the engine and is a factor in determining the interna!
power level necessary to produce the desired net thrust output. It was shown in equation (16) that
the ram drag is the free stream momentom of air entering the compressor inlet per second.
W 8V
Fr =-gIt was also shown that, for comparison purposes, it is convenient to refer thrust or drag parameters
to sea level standard day conditions by correcting for prevailing conditions; therefore, ram drag may
be written as,
(54)
Equation (54) shows that the ram drag is a function of engine airflow and airplane velocity.
However, to put ram drag into convenient parametets for measurement, F/oamb also can be writte.n
as,
[:::][:~J
(55)
where,
is coireCted engine airflow, lbs/ sec
lbs
is ram drag parameter, lbs/
2.100
D6-1420
May1989
POWERPLANT
[:::]
[:.:.J
is inlet pressure recovery ratio, dimensionless
is ram pressure ratio, dimensionless
The corrected engine total airflow curve, W a
ra;-2 I 5 2 versus N -Jffl,2 (Figure 7), is
-J
1
1
1/
1
supplied by the engine manufacturer and can be used for flight conditions since it is assumed that
the inlet distribution is equivalent to static bellmouth distribution.
The inlet pressure recovery information used to correct for this assumption is obtained from flight
test data and appears as shown in Figure 74.
=0.1
=0.05
=O
CORRIETED TOT.AL AIRFLOW,
Figure 74.
The ram drag parameter is a function of Mach number which can be derived as follows:
From the definition of Mach number,
M-
V
- -JigRf
D6.1420
May1989
2. 1 O1
POWERPLANT
The ram drag parameter can be written as,
Fr
61
2
Wa~
61
= (ago)
(J :.- 2)
1
(56)
1+-M
2
2
. Equańon (56) may be plotted as shown in Figure 75
i:cil~
H~
...
I
MACH NUMBER,
Figure 75.
2.102
M
D6-1420
Mayl989.
POWERPLANT
The ram pressure ratio, p1/Pamb• is also a function of Mach number and can be determined from
equation (48).
·
p
_Y_
y-1
_._11_ =
(1 + 'Y - 1M2)
Pamb
2
Since the net thrust equals the gross thrust minus the ram drag, the net thrust equations for the
various engines can be determined by combining equation (55) and equation (43). For. single
exhaust turbojet, ducted-fan, and bypass engines,
or,
Fr
6~b =PoVCgAc[~:J-[w/:]-w-a-~
. .9--[:::][:~J
2
(57)
12
61
2
For turbofan engines with primary and fan exhausi nozzles,
or,
Fr
-[:::J[:~] (58)
6~b =po VrCfAf +po~ CgAc[~:J-[w•(']-w-a-F,812
-
12
61
2
Note in equation (58) that there is no At/Ac ratio for the fan exhaust nozzle. It is usually established
by testing that the effect of fan exhaust nozzle air temperature and pressure on the exit area is
negligible; therefore, only the cold measured fan area is used in the calculations.
The above engine calibration analysis results in the installed engine performance data shown in
Figure 76, Figure 9 or 11, and Figure 14 or 15.
D6.1420
Mayl989
2. 103
POWERPLANT
bellm:,uth inlet
RATED THRIJST
CONSTANT ALTI'IUDE
tel1'Iperature
T.RJE .AIRSPEED„ V
Figure 76.-
2.104
D6-1420
May1989
PERFORMANCE
3 -1
Airspeed Measurement
Terminology
Before beginning the study of specialized performance problems, it will be useful to review velocity
parameters used in aeroclynamic work.
·
Vi
is the instrument indicated airspeed, uncorrected for position error, or equal to the pitot static
airspeed inclicator reacling. It includes the sea level standard adiabatic compressible flow
correction in the calibration of the airspeed instrument clials.
V1
is the inclicated airspeed COireCted for instrument error only. It is abbreviated as IAS and is
related to Vi by the following expression:
· ·
V 1 =V.1 + llV.1
Where IlVi is the instrument error correction.
Vc
is calibrated airspeed and is equal to the airspeed inclicator reacling corrected for position and
instrument enor. The abbreviation is CAS. The equation relating Vc and V1 is:
(2)
Where IlVp is the position error correction
Ve
is the equivalent airspeed and is equal to the calibrated airspeed, corrected for adiabatic
com.pressible flow for the particular altitude. The abbreviation is EAS, and the relationship
between Vc and Ve is:
(3)
Where IlVc is the compressibility correction.
V
is true airspeed. It is related to Ve by the following:
1
V=V-
eva
(4)
Where a is the density ratio plpoRemembering that dynamie pressure, q, is a function of velocity:
D6-1420
Mayl989
3. 1
PERFORMANCE
It is permissible to manipulate the expression by multiplying the right hand member by Po'Po, thus:
1
q=2Po av
2
When measuring velocity in knots, the equation reduces to:
v2a
q = 295.375
(5)
Substituting equation (4) into equation (5) results in:
v2e
q = 295.375
(6)
Therefore, the dynamie pressure is a function of equivalent airspeed.
Pitot Static Systems
A brief discussion of the basie theory of the pitot tube in incompressible flow was made in Section
1. The pitot tube is a direct application of the Bernoulli equation and actually measures the
difference between total air pressure and ambient, or static pressure, which is the indicated dynamie
pressure, q. Equation (6) says that for incompressible flow, assuming no errors in either total or
ambient pressure measurement, the pitot tube will record a pressure differentia! which is
proportional to the square of equivalent airspeed. By a suitably calibrated instrument, equivalent
airspeed may be read directly. The way in which actual airspeed instruments are calibrated is
somewhat different. They are calibrated to read true airspeed at all Mach numbers under standard
sea level conditions if no instrument or pressure source errors are assumed.
·
Obtaining airspeed from a non-perfect airspeed system and for non-standard conditions involves
establishing the corrections for cach potentia! error source and the relationship between nonstandard and standard atmospheric conditions. Errors can exist in instruments, total pressure
recovery, and static pressure. Instrument error can be determined in laboratory tests. Total
pressure errors can·result from the location of the pitot tube in a reduced energy region such as a
boundary layer. For this reason pitot tubes are always supported away from the skin on short
struts. Another source of total pressure error is that due to large inclinations of the pitot head
relative to the stream direction.
Assuming the above errors are known (in the case of instrument error) or can be made negligible (in
the case of total head error) the only other error source is that due to the location of the static source
at a point of non-ambient static pressure. In generał, it is neither possible nor practical to locate the
static source at a single point which remains at ambient static pressure for all flight conditions. The
reason for this is that ambient pressure is altered near the airplane by the influence of the airplane.
The part of the airplane most responsible for this effect is the wing. Without the wing it would be
relatively easy to select a location somewhere aft of the nose on the fuselage at a constant diameter
section. This would be similar to the static location on the pitot-static tube, discussed in Section 1.
But with the wing, static pressure both ahead of and behind the wing affects the pressure on the
fuselage. To see why the static pressures near the wing are different from remote static pressures,
3. 2
D6-1420
May 1989
PERFORMANCE
refer to Section 1. When considering airflow around an airfoil section, the streamlines will be
affected not only at the wing but in the immediate vicinity of the wing as shown in Figure 1.
Figure 1.
Whenever the streamlines arc disturbed, the static pressures arc also disturbed. Thus, the wing
influence on static pressure is extensive and difficult to escape.· What must be done is to find a
static pressure location which not only rea.ds close to ambient static pressure, but also minimizes the
effects of angle of attack changes, which will vary the static pressure field around the wing.
In the past it was eustom.ary to locate the static somce by means of extensive airspeed calibrations in
flighL On the B-47 bomber, however, it was shown that considerable flight time could be saved
by using wind tunnel tests to discover the. best location for the static source. This practice was
extended to subsequent Boeing manufactured airplanes.
An explanation of the relationship between indicated airspeed, Mach number, and true airspeed is
necessary to show the effect of a static source, or position error. Indicated airspeed can be
considered a measurement of total pressure less static pressure. As mentioned previously, if no
enors or compressibility existed, equivalent airspeed, Ve, would be measured. With position error
and compressibility effects considered, however, indicated airspeed, VI is measured. Tuus, a given
value of V1 corresponds to a given value ofpt - p. The altimeter is connected to the static source
and its reading, hPi, indicated pressure altitude, corresponds to the static pressure, p. Mach number
is directly related to the ratio of total pressure to static pressure; therefore, a given Mach number
corresponds to a single value of ptlp. True airspeed is obtained from Mach number and air
tempera.nue.
For airplanes with electronic airspeed indicators or speed tapes, electronic altimeters, and electronic
Mach indicators, the position error values arc accounted for in the air data computers resulting in the
pilot reading computed, or calibrated airspeed, corrected pressure altitude (or tape line altitude when
pressure corrected), and true Mach number. If the air data system is inoperative, the pilot can read
indicated values of airspeed and altitude from the pneumatic standby instruments. The
detennination of the position errors will be analyzed following the analysis of the relationship
between calibrated airspeed and true Mach number.
06-1420
Mayl989
3. 3
PERFORMANCE
The expression relating the pressures to Mach number is given by equation (63) of Section l,
repeated here for convenience:
~t
=( l +
y; M2)
y
y-1
1
Subtracting 1 from cach side of the equation does not change the equality; thus:
pt-p
p
(
y-1
= 1+2- M
2)
y
y-1
-1
(7)
Substituting fory, the ratio of the specific heats for air, the value 1.4:
2
Pt - p = p [ ( 1 + .2M )
3.5
]
-
1
(8)
The quantity (1 + .2 M2) is the expression for adiabatic temperature rise due to compressibility.
Under standard sea level conditions the airspe.ed indicator is calibra.ted such that:
V=Ve =VC
The relationship between Vc and p1 - p for standard sea level conditions is:
(9)
Where Po is the standard sea level ambient pressure. By equating equations (8) and (9), Mach
number may be solved as a function of altimde and calibrated airspe.ed. The resulting equation is:
(10)
This equation may be plotted as true Mach number versus calibrated airspeed for constant pressure
altitudes. H indicated airspeed, V1, and indicated pressme altitude ratio, 61, arc used in equation
(10) instead of Vc and 6, then indicated Mach number will result. A plot of equation (10) is
shown in Chart 1 in the Appendix.
3.4
1)6.1420
May1989
PERFORMANCE
High Speed Position Error - Mach Number
The procedure by which errors in the pitot static system arc discovered is called airspeed calibration.
Calibration of the pitot static system is required to determine the corrections to be applied to the
indicated values. A reference static pressure source (with known errors, Af>s)* is required. This is
determined from a trailing cone system (one for high speed and one for low speed) for stabilized
level flight unless a dive is required to reach placard speeds. The pressure measured by the trailing
cone system is generally used as the reference static pressure, Piref· Also, a test system static
pressure, Piresr is measured at the airplane's static somce location (usually the copilots). Since the
trailing cone position error (L\Ps) is known from prior tests, the airspeed calibration procedure is to
compare trailing cone and test system static pressures. The difference between test system and
trailing cone static pressures is recorded using a differential pressure (AP) transducer.
This is the AP method and is used unless pitot recovery error is suspected, as in stall testing. Since
the reference indicated static pressure, Plref• is known, indicated impact pressure, ąclref' is
determined from the following equation:
(11)
* The ttailing cone system is calibrated by extending the cone behind the airplane and comparing
the pressure of the cone static section to the pressure of a known stalic source. The result is a
trailing cone position error, llP8 •
·
The reference and test impact and static pressures can be calculated using the following equations:
-Af> s
(12)
+Af> s
(13)
qc=qc
Illf
Ps=P1
qc
llf
P1
test
(14)
+ L\P
=qc
I test
I llf
= P1
-L\P
(15)
llf
.1Mp is finally calculated from the Mach numbers calculated for the reference and test systems using
equation (J) rearranged as shown:
y-1
5[
D6-1420
May1989
]
(j;+ 1) y -1
(16)
3. 5
PERFORMANCE
and,
y-1
'Y
0
1
5 ( : 1:
+
~ -
I
(17)
then,
(18)
It is convenient in jet performance work to use the param.eter of airplane weight, W, divided by the
altitude pressure ratio, 6. From the lift equation:
W
6
2
= 1481.351 <;_ M S
(19)
For a given W/6 and Mach number, the airplane will be at a given Q and thus at a given angle of
attack. Since position error is basically associated with wing angle of attack, it is reasonable that the
parameter W/6 should be used since it is indicative of angle of attack. From the observation in
flight of indicated pressure altitude, a value of indicated 6 can be obtained from a standard altitude
table. Knowing the airplane weight at the time will enable one to determine Wf6r. the indicated W/6.
Now that ~ . M1, and W/6i are known, the plot in Figure 2 can be made for a series of test
points. However, in many instances the spread in data is such that a single line for all W/6's may be
drawn.
Representative curve for
narrow scatter band *
(Use for all W/6 's) - - - - - - - -
+
AMp
O
---------------M1
* Severa! lines are required for
pronounced effect of W/6
Figure2.
3.6
D6-1420
May1989
PERFORMANCE
High Speed Position Error - Airspeed (Flaps Up)
The airspeed indicator is connected to the static source of the pitot static system; therefore, it will
also pick up any existing position error and will thus read incorrectly. Since the Mach number
position error,~· has already been determined, the high speed position error can be evaluated
from the following expression:
or:
(20)
where dM/dVc is the change in true Mach number divided by the corresponding change in calibrated
airspeed at a constant altitude where total pressure is constant. By introducing the multiplying
factor, dp/dp, the differentia! term dM/dVc is the product of two other differentia! terms as shown.
(21)
· dM/dp is evaluated from equation (63) of Section 1 and the result is:
= _ 1+ .2M
(dM)
dp
yMp
2
(22)
P
t
The expression for dp/dVc is derived from equation (9) and is found to be:
(23)
Substituting equation (22) and (23) into equation (21), the expression for dM/dVc can now be
obtained:
(24)
D6-1420
May1989
3. 7
PERFORMANCE
A chart showing the solution of equation (24) for constant values of 6 or pressure altitude and for
various values of Vc can now be made, see Chart 2 in theAppendix.
The airspeed calibration given above was the original method for determining the high speed
airspeed position error, AVp, for flaps up. The following is the latest method for determining the
airspeed position error where the data reduction is essenti.ally the same as for determining Il.Mp, but ·
the A.Mp value does not have to be known.
When the reference and test impact pressures arc calculated using equations ( 12) and (14), the AVp
is calculated from the airspeed values calculated for the reference and test systems using equation
(9) rearranged as shown:
y-1
1
5 [ (::
+1f -1
]
(25)
and
y-1
-,-
-1
(26)
then,
(27)
Low Speed Position Error - Airspeed
At low airspeeds, below about M = .4, where Mach number effects arc small, it is convenient to
modify the calculati.ons in the following manner. The airspeed position error, AVp, is determined
from equation (2), repeated below.
Vc=V 1 + AVP
As in the case of the Mach number position error, this error could be simply plotted as AVp_ versus
V I. However, at various airplane weights at a given V1, different corrections would be obtained
since the static pressure error is a function of angle of attack Cv as stated before. It will then be
necessary to convert V I to some index of Cv Consider the lift equati.on, assuming herc that
V1=Ve.
s_svi
W= 295.375
3.8
D6-1420
May1989
PERFORMANCE
and,
C = 295.375 W
L
S V2
(28)
I
Now, for the airplane to be at the same Cr. at another weight, Ws=
295.375 W
S v2_
I
=
295.375 Ws
2
s vIw
where V Iw is the indicated airspeed corresponding to the weight, Ws· Solving for V1w:
(29)
When using this equation, a value is arbitrarily assigned to Ws which, for Boeing commercial
airplanes, is 100,000 pounds. Tbus, equation (29) becomes:
= V Jl00,000
V
IW
I
W
(30)
Equation (30) now says that for an airplane weighing W pounds, and flying at V I knots, the
100,000 pound airplane at the same CL will be flying at V1w knots. If the airplane were assumed to
be at another weight and speed, a corresponding V1w could be calculated whicb would be at the
same CL as the airplane flying at the new flight condition. Therefore, V1w can be thought of as an
index of CL since it effectively changes the actual airspeed into one whicb corresponds to an
arbittary fixed weight.
So far V 1 ·has been generalized into the form of V1w whicb is proportional to CL. Let us now
discuss the airspeed position error, AVp· Since the static pressure error, Ap, is basically a function
of angle of attack, or CL, it would be logical to plot Ap/q versus CL (Ap/q being non-dimensional)
for constant values of W/6. But, since the static pressure error determines the airspeed position
error, the airspeed position error must be generaliz.ed into the form of Ap/q so as to make a plot
comparable to Ap/q versus CL. To develop the alternate form of Ap/q, consider that the pressure
error, Ap, is caused by the local velocity at the static pon, V1 being different from the remote
velocity, V c. This difference is of comse the position error, AVp.
1
2 1
2
Ap= 2 Po vc -2Po VI
and,
1>6-1420
May1989
3. 9
PERFORMANCE
therefore,
Substituting for V c in the above equation from equation (2), results ~:
Ap
q=
2
2
V 1 + 2V I AV P + AV P -
_2
Yi
Vi+2V AVp+A~
1
Since AVPis small, AVlis very small and may be considered negligible. Thus:
Ap
2V 1 AVP
q= v;+2vl AVP
or,
Ap
2 AVP
q= V +2AVP
(31)
1
If AVP is small compared to V1, which it should be, then it can be said that Ap/q will vary
approximately as AV pfV1. This reasoning justifies the use of the param.eter, AVpfV1, for
generalizing the low speed airspeed position error. A plot of AVrf11 versus VIw is made, Figure 3,
and is faired from results of flight tests aided by previous wind tunnel tests. Flight test data is
usually obtained from trailing cone instrumentation, a technique which permits the use of a known
static source.
When the flaps arc lowered, the airplane will be at a different angle of attack at a given indicated
airspeed. Thus, position error will be affected by flap extension, also shown in Figure 3. Figure 3
is valid for all gross weights when used with equation (30).
--'1w
Figure 3.
3.1 O
D6-1420
May1989
PERFORMANCE
Position error may be plotted as llV P versus V 1 at various airplane gross weights. Plotting this
relationship yields the family of curves in Figure 4. In some instances the plot of Figure 4 can be
simplified with only a small loss in accuracy by fairing a single line through the family of weight
curves.
Aaps 40, Gear Up or Down
~
!
Representative Curve used
if effect of weight is not
pronounced
o
200
]
Aaps 25, Gear Up
:>P'
<I
.i
!J
0
a:-=
--------=
Aaps Up, Gear Up
o
---------lndicated Airspeed, V1
- Knots
Figure 4.
D6-1420
May 1989
3. 11
PERFORMANCE
Compressibility Correction
The airspeed indicator is, as was mentioned previously, incapable of perfect velocity measmement.
While it may be made to read differentia! pressures existing across the orifices of the pitot and static
sources, it would require a different airspeed scale graduation at every pressure altitude. Consider
again the compressible flow relationship, equation (7), shown herc for convenience.
A substitution is made for Mach number by remembering tbat:
V
M=a
a=
/1P
.JT
Thus, equation (7) becomes:
The expression may be furtber manipulated by introducing a multiplying factor of Pe/Po-
Solving for V-va from the above expression gives:
.
VW=
.
2'Y p [ P1 -p
('Y - l)Po ( P +
y-1
y
]
1) - 1
(32)
Expression (32) shows that airspeed measurement is a function of pressure, p, as well as a
differentia! pressure, p 1 - p. Since a scale for cach altitude is impractical, a single scale
corresponding to standard sea level pressure altitude has been adopted; this scale is used to define
calibrated airspeed , Vc. All airspeed indicators arc calibrated to read ttue airspeed under sea level
standard day conditions (no installation errors); thercfore, for altitudes other than sea level an
additional adiabatic compressible flow correction for altitude must be made.In evaluating the Mach
number position error, equation (9) was developed, and is shown herc for convenience.
3.12
1)6.1420
May1989
PERFORMANCE
This expression involves Vc which may be substituted into equation (32), however, it is more
convenient to equate expression (7) and (9):
From this, Vva · may be resolved and the expression reduced to:
1
3.5
Ve =Vvci=aO
[1+.{::Jf-1
~-----p----~~+1
-1
(33)
Po
The param.eter VVG is a measure of the .k:inetic energy of the air and is defined as equivalent
airspeed, Ve· Also vva appears in the force and moment coefficients as constants, except for
Mach number effects.
Equation (33) may be evaluated for various values of Vc and pressure altitude. The difference
between Ve and Vc is, of comse, /J.VC' as expressed in equation (3). Results of the evaluation have
been plotted and arc shown in Chart 3. (See Appendix)
Ground Effect Airspeed Position Error
When an airplane is flown near the ground, or when it is on the ground, the ground will affect the
airflow panem around the airplane wings and fuselage. It does this by actually preventing the
downwash from developing to the extent that it would in free air. Consider Figure 5, which shows
two wings at the same angle of attack, one in free air, and the other on or near the ground. Note that
the flow pattem.s around the wings arc different This is because the ground forces the airflow to
conform to it This results in a changed static pressure field around the wing which is felt by the
static pressure pickup on the fuselage or on the pitot tube. This causes an additional error which
must be accounted for. Ground effect airspeed position error is usually presented for takeoff flap
configurations with the airplane on the ground.
D6-1420
Mayl989
3. 13
PERFORMANCE
---...;..N~ear the Ground
Figme5.
3.14
1)6.1420
May 1989
PERFORMANCE
3-2
Altitude and Temperature Measurement
High Speed Position Error - Altimeter
In Section 1 it was stated that the altimeter; which measures pressure, will indicate an altitude
corresponding to the measured pressure. This altitude is known as pressure altitude, hp, provided
the altimeter is set to the sea level standard day pressure of 29.92 inches or 1013 millibars of
mercury reference. Since the altimeter is connected to the static source of the pitot-static airspeed
system, it will pick up any existing position error, defined as Abp, and will therefore, read
incorrectly. With position error, the altimeter now reads indicated pressure altitude, hPr, and the
pressure altitude can be obtained from the following equation: ·
hp=hp + Abp
(34)
I
The altitude calibration for flaps up is based on the high speed Mach number calibration. The
calibration curve as it is presented in the flight manuał is this reference calibration curve with the
Mach number correction converted to an altitude correction. This involves the following
relationship:
Ahp=(!t) AMP
(35)
By introducing the multiplying factor, dp/dp, the differentia! term dh/dM is the product of two other
. differentia! terms as shown:
dh
dh dp
dM=dp dM
(36)
Since pressure, p, is a function of altitude, h, dh/dp is obtained from equation [16) of Section 1:
dh
1
(37)
dp =- pg
Substituting equations (37) and (22) into equation (36), the expression for dh/dM becomes:
dh
'YP M
dM =
T T
1
(1 +.2M")
dh
a2
M
dM=g(
dh
a~8
dM =-g- (
dh
_.2)
1 +.2M-
M
6
dM = 38, 741.2 0 (
06-1420
May1989
2)
1+.2M
M
2)
(38)
1+.2M
3. 15
PERFORMANCE
A chart showing the solution of equation (38) for standard day conditions can now be made, sec
Chart 4 in the Appendix.
The present method for determining the altimeter position error is as follows. When the reference
and test static pressures are calculated using equation (13) and (15) for altitudes below 36,089 feet,
Abp is calculated from. the pressure altitude values calculated for the reference and test systems using
equation (78) from Section 1 rearranged as shown:
or
P
hp= 145,442 [ 1-
(p;)
nt902631]
(39)
and,
n1902631]
P
(40)
hl""= 145,442 [ 1-( ; ; )
then,
(41)
For altitudes above 36,089.24 feet, equation (80) from Section 1 has to be used rearranged as
shown:
hp= 36,089.24- RT 36
In [ (:
0
T . 5.25588]
) ( T
)
~
or:
hp= 36,089.24 - 20,805.83 In [ 4.47706( ::) ]
(42)
and,
h1..., =36,089.24-20,805.83
In[ 4.47706(p~7 )]
(43)
Equations (78) and (80) from Section 1 are valid for standard day temperatures only; therefore, for
non-standard day temperatures a small error in the above values will result. The errors are within
the tolerance allowed by the Federal A viation Regulations.
3. 1 6
D6-1420
May1989
PERFORMANCE
Low Speed Position Error - Altimeter
The altimeter position error at low speeds is obtained by utilizing the IlVp that is measured from the
trailing cone instrumentation. Rather than go through a procedure· similar to that for velocity
position error of generalizing the altimeter position error, &plh1 versus V1w (similar to Figure 3), it
is convenient to recognize that:
/lh = -lY:!_ IlV
p
dVC
(44)
p
Having found IlVIL..from the .previous theory, it is necessary only to evaluate dh/dVc so that llhp
may be resolved. The differentia! term dh/dVc may be manipulated by introducing a multiplying
factor, dp/dp, thus:
dh_dh dp
. dVC - dp dVC
(45)
By multiplying equation (26) by equation (16), the expression for dh/dVc can be obtained.
(C:J=
0.08854
[1+.2 (~Jr (:·r !
(46)
A plot showing the solution of equation (46) under standard day conditions for various values of Vc
can be made, see Chart 5 in the Appendix.
The present method for determining the high speed altimeter position error values, given
previously, can also be used for determining the low speed altimeter position error values.
,
1)6..1420
May1989
3. 1 7
PERFORMANCE
Temperature Correction
The pńmary variables which must be most accurately known or determined duńng flight to obtain
the optimum performance for a jet aircraft are:
(1)
(2)
(3)
(4)
Mach number
Gross weight
Altitude
Free air temperature
Therefore, the instruments which are considered. to be of priine imponance arc the Machmeter, fuel
quantity indicators, altimeter, free air temperature indicator and engine pressure ratio indicator (or
tachometer). Of those instruments itemized above, the Machmeter and altimeter errors have been
discussed. Fuel quantity and engine pressure ratio indicators operate satisfactorily and deviation
from the true reading can usually be corrected in installation alterations.
The free air temperature indicator is very important since the indicated temperature has two specific
uses associated with performance: the determination of true airspeed and fuel flow and the
determination of engine pressure ratio (or RPM) for the required thrust settings. Free air
temperature gages arc usually operated by an electrical resistance thermometer probe located flush
with the body skin. Because of the adiabatic temperature rise, due to compressibility, the
thermometer probe picks up a temperature reading higher than the static temperature of the air.
Total temperature is related to the static by an expression developed in Section 1:
Tt
y_:1
2
-=r=l+--i-M
This may now be modified by substituting the value of y = 1.4 for air, thus:
T 1 = T ( 1 + .2 M2)
(47)
Equation (47) must be modified further to include a recovery factor, K, due to the inability of the
instrument system to recover the full temperature rise due to adiabatic compressible flow of air as
picked up by the thermometerprobe. Equation (47) now becomes:
(48)
The value of K will be known for an installation and is usually of the magnitude of 0.8 to 1.0.
The calculation of true OAT is easily made keeping in mind that the temperatures T1i and Tamb arc
absolute temperature values.
3.1 8
D6-I420
MayI989
PERFORMANCE
Temperature lnd.ication Calibration
The calibration of a temperature indication system is very difficult unless one has a reference
instrument system to be flown with the one to be calibrated. Bench calibration or wind tunnel
comparison are not satisfactory because of the large wind tunnel effects and the influence of
installation and environment Therefore, a method of calibrating the first instrument system must be
developed. Then the second and succeeding systems may be calibrated by a second method or by
compańson to the first when flown with or in parallel to il
Total Temperature (Reference) System Calibration
Consider equation (48) in the form
(49)
If a calibration run is made at constant altitude in a relatively stable air mass, Tamb may be
considered to be constant, and T ti can be plotted against M in the form of Figure 6.
Slope = -
K
5
Figure 6.
Equation (49) shows that the slope of this family is -K/5 and the recovery factor is determined.
Note the value of Kand its determination are independent ofTamb so long as it is constant for each
run. This is the method used to calibrate the reference system for Boeing airplanes. This calibrated
instrument system can then be used to determine Tamb atany time through Tti and M only.
06-1420
May1989
3. 19
PERFORMANCE
Total Air Temperature (T AT) Indicator System
Calibration of the total temperature indicator system consists of determining a calibration factor, C,
which takes care of instrument installation effects and the recovery factor, K, in the equation:
TAT = T amb[(l + C) +.2nł]
(50)
Solving this equation for K yields
K --
TAT -(l + C)
amb
T
(51)
2
M
5
Therefore, when the flight data is plotted as in Figure 7, the slope of the linear array gives K and the
intercept defines C. The values of T~b are obtained from the reference instrument system flown
with this one.
·
TAT - l
Tamb
Slope = K
C
M2
-5Figure 7.
n
The above was the method of temperature calibration, used until certification of the 757 61
airplanes. For the 757n67 and subsequent airplanes, the probe manufacturer determines a recovecy
factor value for the total temperature system. This is verified by comparing values with the flight
test probe T AT for a range of conditions. If the correlation plot shows a minimal scatter of
approximately 1/2° C, this is acceptable and programme.d in.to the Air Data Computer.
3.20
1)6..1420
May1989
PERFORMANCE
Airplane production models which have the Rosemount total temperature probe have the following
values of recovery factor with the probe heater switch in the ON position:
(1)
(2)
(3)
707n21n37's
747's
757n67's
1.00
0.985
0.98
Static Air Temperature lndicator
Static air temperature indication is referenced directly to the Tambor tamb calculated from the
reference indicator reading:
t.t=t.-t
1
amb
(52)
where 1i is the static air temperature indicator reading. Calibrations are run at a series of altitudes
(tamb) and plotted as in Figure 8.
OAT =t·1 -At
At
o
o•
M
Figure 8.
1)6..1420
May1989
3. 21
PERFORMANCE
Intentionally Left Blank
3.22
D6-1420
May 1989
PERFORMANCE
3 -3 Airplane and Engine Grid
Thrust Required
Previously, in Section l, a short d.iscussion of thrust required for level flight was made, and the
following equations were developed:
W
-r= 1481.351CL M S
2
(53)
amb
6
T
amb
=1481.351C 0 "łK' S
(54)
To calculate the T/6amb required for an airplane, a table similar to that shown in Figure 9 may be
used to facilitate the calculations. If the various Boeing commercial airplanes with their respective
wing areas arc now considered, equations (53) and (54) can be arranged as follows:
(55)
(56)
where,
D6-1420
May1989
Aiiplane
S(ft 2 )
A
B
707
Inter 707
727
737
747
757
767
2433
2892
1560
980
5500
1951
3050
.27745
.23341
.4327
3.604
4.284
2.311
1.452
8.1477
2.8902
4.51827
.6888
.12273
.3460
.22132
3. 23
PERFORMANCE
W/6amb
1.0 X 10 6
1.2 X 10
6
M
.3
.5
.7
.9
Remarks
A/M.2
2
BM
Calculated
Calculated
CL
Calculated from
Equation (55)
Co
Obtainfrom
Drag Polar
T/6amb
Calculate from
Equation (56)
CL
Co
T/6amb
1.4 X 10 6
- - -
V
Repeat for Other W/6's
-
Figure 9.
When filled out for desired values of W/6amb, this tabulation will enable one to plot T/6amb versus
M, as in Figure 10. Sufficient Mach number points must be calculated to obtain a well defined
curve.
3.24
06-1420
May1989
PERFORMANCE
D
6amb
M
Figure 10.
Tbrust A vailable
As was explained in Secti.on 1, the logical parameter for expressing available engine thrust is
T/Oamb. With constant temperatme above the tropopause, thrust under any specific conditi.on of
engine rating and airplane speed will vary directly with the ambient atmospheńc pressure;
therefore,the ratio T/Oamb will theoreti.cally remain constant. However, engine tests above 36,000
feet have indicated a decrease in corrected thrust available with increase in alti.tude above 36,000
feet. This is attributed to Reynolds number effects and additi.onal bleed requirements causing a
reduced airflow within the engine. A family of available thrust values at maximum conti.nuous
thrust (MC'l1 is illustrated in Figme 11. The reducti.on in T/Oamb in the troposphere with alti.tude
decrease is due to atmospheric pressure and temperature increases. Similar characteristi.c curves
may be developed for other engine rati.ngs; eg, maximum climb (MCLT) and maximum cruise
(MCRT) thrust levels. Tbrust characteristi.cs at a rating are defined by the engine manufacturer.
In Secti.on 1 it was shown how Figme 10 and 11 may be superimposed to form a single plot for all
alti.tudes, as shown in Figme 12. Data to construct the thrust available lines arc furnished by the
engine manufacturer in the form of thrust versus velocity for various alti.tudes. This informa.tion is
then calibrated for installati.on effects and reploned introducing o and changing true airspeed to
Mach number.
D6-1420
May1989
3. 25
PERFORMANCE
Maximum Continuous
Maximum ContinuoUS
I---------------...,;;20~,000_.--
T/aambl-
15000
10,000
SeaLevel
M
Figure 11.
-Engine Fuel Flow Grid
The complete engine data as presented by the engine manufacturer consist of a series of plots from
sea level to the altitude limits of the airplane in increments of usually 5,000 feet as shown roughly
for one altitude in Figure 13. This plot shows the complete picture of thrust and fuel flow variation
with speed at a given altitude. However, it is not in the form which is useful in performance
calculations. In the first place, thrust and true airspeed must be converted into terms ofT/6amb and
Mach number as was done for the thrust available curves. The fuel flow data, however, must be
given special treatment to put it into usable form. The units of fuel flow are pounds per hour (lb/hr)
and will be unchanged if the numerator and denominator are multiplied by the units of nautical air
miles.
Tuus:
w f =J!?.
=JQ..
..!!!!!ł.
hr
hr nam
3.26
1)6..1420
May1989
PERFORMANCE
30,000
25,000
T/Bamb _20::!!:!:,000~--.~---15,000
D/Bam.b
M
Figure 12.
This equation can be rearranged in the form:
wf= (
':")(!,,) = ( ~ )
(57)
Fuel mileage, nam/lb, is simply related to velocity and fuel flow by this equation. Equation (57) is
still not in generalized form, however, and must be manipulated further to make it useful for
performance work.
The specific fuel consumption, lb of fuel per hr per lb thrust, is essentially constant at a given value
of Mach number and T/&unb regardless of altitudes. Tuus:
(58)
where TSFC is thrust specific fuel consumption, lb-fuel/hr/lb-thrust, and T is the total thrust in lb
for all engines. However, since it is desired to use the param.eter T/Bamb, equation (58) may be
wńtten:
D6-1420
May 1989
3. 2 7
PERFORMANCE
Wf
T
-=(TSFC)6amb
6amb
(59)
This equation states that at a given T/6amb (and speed) there will exist a constant W /6amb when
TSFC is constant Returning to equation (57), and dividing by 6amb :
.
T
Wf
(TSFC) ~ = ~ = (
amb
amb
V
nam)
Ib
(60)
6amb
·1 Constant Altitude
......
......
......
(Lb)
......
......
......
......
......
......
......
......
......
......
......
,..,,,,,- Lines of Constant '
~
Fuel Flow Rate -
..... .....
......
..... .....
......
......
......
'
......
......
......
......
......
,
/--,
--,~,--~)
..... .....
......
......
......
......
......
......
......
.....
..... ......
......
V-Knots
Figure 13.
In addition to using the param.eter T/6amb Mach number is also used in plotting the thrust required
and available curves as shown in Figure 12.
M-
V
-aoJe:;,
Tuus to convert from units ofvelocity, appearing in equation (60), to Mach number, the quanti.ty
Je:;:; must be introduced. Dividing equation (60) by ,te;;;, :
3.2 8
06-1420
May1989
PERFORMANCE
(61)
Equation (61) has introduced three additional parametric ratios:
(I) COireCted tbrust specific fue1 consumption,
?a:;;,;
8
amb
Ja::;, ;
w
(2) corrected fue1 flow basecl on ambient conditions,
(3) corrected fuel mil~ge,
6amb
8amb
°: 6
amb.
The result has accomplished two purposes. First, it generalizes the fuel flow information and
removes the variable of altitude. Second, it converts from units of true airspeed to Mach number.
Tuus, instead of having a plot like Figure 13 for every altitude, a new plot which is independent of
altitude can be drawn, as in Figure 14.
T
oamb
M
Figure 14.
Je;;;,
Figme 14 was constructed using EPR as the thrust setting variable and
as the temperature
variable. Using equation (61) on a per engine basis where FN is net thrust per engine
D6-1420
Mayl989
3. 29
PERFORMANCE
(
TSFC )( Fn ) (
Wr
)
Je:::
6amb = 6amb F- '
values of ( ; : : ) can be detcrmined for increasing values of ( 6: : ) and constant assumed
values of · (
6
amb
je:;) .
See Figure 15.
amb
PER ENGINE
DATA
-TSFC
./ 8am,
Figure 15.
Next, a set of constant Mach lines must be superimposed onto Figure 15 for specific engine
conditions. · Using specific engine data for a given value of EPR and Mach number, obtain the
r ,
F
corrected net thrust,
flow, ( 6
F,) ,
from either Figure 12 or 14 of Section 2. and the corrected fuel
amb
from Figure 15 of Section 2.
l2
l2
Now refer to equation (27) from Section 2, repeated here for convenience.
3.3 O
D6-1420
May1989
PERFORMANCE
Divide both sides of the equation by 6ambF-, as shown by equation (62),to obtain the
corrected fuel flow based on ambient conditions. Note the two versions of corrected fuel flow.
(62)
re- ,can be determined and plotted for
·
(
Wfre- J and r
F
the selected Mach number by entering Figure 15 with the
values
6
The corrected tbrust specific fuel consumption value,
.
.
TSFC
v"amb
ambv „ amb
.
amb
that have been determined, or by using equation (61). By selecting other values of EPR for the
same Mach number, a line of constant Mach can now be drawn onto Figure 15. Repeating the
above calculations for other Mach values results in a family of Mach lines that show the specific
.
.
.
.
mplane-engme relattonship between M,
F
TSFC
r, --;;===, and ( 6 Wf
Je J for any
n
amb
..,/ v amb
amb
amb
operating condition. See Figure 16.
PER ENGINE
DATA
-
TSFC
ł &ani)
F
n
&amb
Figure 16
D6-1420
May 1989
3. 31
PERFORMANCE
-
(
A cross plot of Figure 16 for constant values of all-engine
TSFC
(minus the
wr
6
młJJe:;:;,
J
results in Figure 14
Je:;, values) that can be superimposed onto Figure 10 (or Figure 12), as shown in
Figure 17. For turbojet or turbofan engines that require equation (29) of Section 2, (
may be substituted for (.
6
sm~
6
W:, )
amb amb
J
Increase
w
D
6amb = Constant
6amb
"""
Speed Schedule for
Minimum Drag
'
Speed Schedule for
.Minimum Fuel Flow~
M
Figure 17.
The point of tangency between the two sets of curves resullS in the speed schedule for minimum
fuel flow. The phase offlight where this is useful is the Holding phase which will be discussed in
more detail later.
3.3 2
D6-1420
May1989
PERFORMANCE
The usefulness of Figure 17 can be extended by adding lines of constant corrected fuel mileage
(sometimes called corrected specific range), {
may be rearranged into the form:
U sing this equation, the lines of constant (
of constant {
n: )
n:
6amb
Refening to equation (61), the terms
)6amb
j )
8amb
6amb . When ploąed on the axes of
in Figure 14 can be converted into lines
a1-
versus M, these lines will appear
amb
as shown in Figure 18.
T
6amb
·M
Figure 18.
D6-1420
May1989
3. 3 3
PERFORMANCE
The information on Figure 18 may now be superimposed on Figure 17 and is useful for all
altitudes. This is shown on Figure 19. The point of tangency between the
W
Oamb
lines and the
nam ~
lines results in the speed schedule for maximum range. This will be discussed in mare
ll>uamb
detail in the chapter on Range (3-7). Note the relationship between the three speed schedules
shown.
lncrease
w
amb = Constant
D
0
oamb
""
Speed Schedule for
Minimum Drag .
"'-
----~ Speed Schedule for
,,__
Minimum Fuel Flow""
M
Figure 19.
The information as presented in Figure 19 is valid for some types of airplanes. Most of today's
engines have design differences which do not permit such generalization onto a single chart. The
specific fuel consumption may vary slightly with altitude, requiring a plot such as Figure 18 for
various altitudes (usually in increments of 5,000 feet from sea level to the altitude limits of the
airplane). For some airplanes, there is the Reynolds number accountability on drag which affects
the thrust required.
3.3 4
06-1420
May 1989
PERFORMANCE
The manner in which the engine data is usually presented is as follows. From the basie chart, such
as that in Figure 13, values of Fn for severa! fuel flow values arc tabulated at constant Mach
numbers and various altitudes. From this tabulation, another chart is made relating this
information, as shown in Figure 20, for the altitudes selected. The value of this cross plot is that it
permits a selection of even values of F8 • A table may now be set up to facilitate further calculations
as shown in Figure 21.
From the tabulated data in Figure 21, a chart is constructed for the various altitudes showing the
· hip of
relanons
TSFC
F O fior constant Mach numbers. See p·1gure 22. This p1ot shows
rr;-- to ~
-.,eamb
that for a given : •
amb
amb
and M, a value of.
'Jl:;;:
is defined for a given altitude. The parameter
eamb
or ( ~ ) is obtainable from equation (61).
(~)o
lb
amb
lb
(63)
Note that Fn is used for the thrust term in equation (63) instead of T. Engine manufacturers' data
can be presented in a slightly different manner than that shown in Figure 13. Instead of showing
lines of constant fuel flow, lines of constant TSFC can be shown, as in Figure 23. The slight
change in data presentation simplifies the calculations required to produce the same end results as
before.
I Constant Altitude I
Figure20.
D6-1420
May1989
3. 3 5
PERFORMANCE
ALTITUDE = CONSTA.NT
TSFC
TSFC/...{e
(LB/HR)
Fn
(LB)
(LB/HR/LB)
(LB/HR/LB)
Fn/6
(LB)
(1)
(2)
(3)
(4)
(5)
( 6)
Select
Select
Engine
Da.ta
(2)
( 3)
Calculate
Calculate
M
Wf
Figure 21.
I
CONST.ANT ALTITUDE
I
TSFC
~
Figure 22.
3. 3 6
D6-1420
May 1989
PERFORMANCE
lllax.tlllłlllJ
--
ICOI'JSTANT ALTI'IUDE!
co~tinuous
I
/ I
'
/
/
/
~
//
/
/
/
/
---
I
}-1.1nes of constant
~ (1b/br/1b)
/
------
------/
.
V-KNOTS
Figure 23.
D6-1420
May 1989
3. 3 7
PERFORMANCE
lntentionally Left Blank
3. 3 8
D6-1420
May 1989
PERFORMANCE
3-4
Flight Limitations
The flight limits of any ailplane may be defined as the speed, altitude, and acceleration limits within
which it must fly. The factors which apply to these restrictions are its aerodynamic, powerplant,
and structural limits. The aerodynamic and powerplant limits will be discussed at the same time
since in some instances the airplane limitations are a result of both acting at once.
Minimum Speeds
The minimum speed that an airplane can fly is of greatest importance in hand.ling an airplane near
the ground during takeoff, approach, and landing. This speed is that associated with the maximum
lift produced by the wing. Assuming an airplane of given weight, from the relationship:
v;
W= CL S 295.375
Ve-
(64)
With W and S constant, by making CL a maximum, Ve will be made a minimum. Referring to the
lift curve of the ailplane, Figure 24, CLMAX occurs on the non linear portion of the curve. As was
explained in Section l, the departure of the lift cmve from linear represents an increasing airflow
separation which finally results in complete airflow breakdown, or stall, at or near CLMAX· When
approaching the stall, the pilot becomes aware of airflow separation through the· phenomenon of
buffeting . Buffeting is caused by the turbulence of the airflow separation shaking some part of
Increasing
Separation
l
Point of Initial
Separation
a.
Figme24.
D6-1420
May1989
3. 3 9
PERFORMANCE
the airplane (usually the wing or horizontal tall). Between the two points of initial separation and
maximum lift, there is a varying degree of buffeting. A point along this curve where the shaking
becomes objectionable to the pilot is the basis for the buffet limit chart.
The effect of flaps on the a.iiplane was discussed briefly in Section 1. It was seen that the flaps
altered the c;_ versus a plot to increase the lift coefficient at a given angle of attack. See Figure 25.
This means that a higher maximum lift coefficient is possible and usually occurs at a lower angle of
attack. The reason for the increase in lift is that the wing with the flap now has a much greater
camber, producing more lift. The increased lift, hence lift coefficient, generated by the flaps will
obviously allow lower flight speeds, which is their chief purpose. This is supponed by equatio11
(64).
~,
,,
.,
,
,-,
'
.l:f!i ,
"",.",
~
-~ 'I!
("
-i.
,,
,
,
,
,
,
,
~ ,' ~
, §i
, -~
,~
a
Figure25
At this point, it is convenient to introduce the concept of load factor . By definition:
L
n= W
(65)
where n is load factor, L is a.iiplane lift, and W is airplane weight. Tuus, when an a.iiplane is flying
such that the lift is equal to the weight, a condition existing in level flight, then the load factor is
unity. Obviously, when the lift is less than the weight, the load factor is less than unity, and when
the lift is more than the weight, the load factor is greater than unity.
During the stall in actual flight, the airplane has begun to fall through and the load factor becomes
less than unity. The value of the load factor, as established from flight tests, may vary from
approximately 0.9 to 0.95.
3. 40
D6-1420
Mayl989
PERFORMANCE
Stall Speed Determination
Normally all low speed testing is done with trailing cone instrumentation. The cone is an
instrumented body extended behind the airplane by cables. The objective is to obtain data far from
the influences of the airplane and thus approximate free stream conditions as closely as possible.
. Since the instrument leads are long, it is important that the indications, especially .pressure
indication, be lag checked to find out how long a stabilized condition must be flown before reading
data. It must also be dynamically balanced such that the pitot and static leads have equivalent lag to
eliminate pressme oscillation$ in the instrumentation.
Federal Aviation Regulations Part 25.103, define the power off stall speed as the minimum
controlled flight speed at the most adverse C.G. position and specifies the rate at which the
demonstration flight is to be decelerated to stall. To comply with these requirements, the test flight
must yield both stall speed and entry rate. The ~ must be specified in equivalent airspeed, so
indicated airspeeds from trailing cone readings are converied to Ve values and correction is made to
find the values at the most critical C.G. position.
The speeds to be determined are defined thus:
Vs - - calibrated stalling speed, or the minimum steady flight speed, in knots, at which the aiiplane
is controllable, with the airplane in the specified configuration, at zero thrust or idle thrust if it is
shown that the resultant thrust has no appreciable effect on the stalling speed, and center of gravity
in the most unfavorable position.
When obtaining stalling speeds, the airplane must be trimmed for straight flight at any speed not
less than 1.2 V s or more than 1.4 V 8• At a speed sufficiently above the stall speed to ensme steady
conditions, the elevator control is applied at a rate so that the airplane speed reduction does not
exceed one knot per second. The aiiplane must also meet the stall flight characteristic provisions of
FAR Part 25.203.
·
The tests are flown at speeds between 1.2 and 1.4 V s through full stall demonstration, recording
indicated airspeed versus time. V1 is converted to V e and plotted as in Figme 26.
Entry rate is defined as the slope of the line connecting V e min and 1.1 V e min:
. dV
entry rate
= dte =
Ve. -1.lVe.
mm
At
mm
(66)*
*entry rate requirements may be different for other regulations.
D6-1420
May1989
3. 41
PERFORMANCE
1.1 ~ .
mm
.1 t - - - - . . . i
Time
Figure26.
The CL at cach V Cmin is calculated by use of the standanl lift equation,
C = 295.375 W
L
V:S
e
and corrected to the most unfavorable (forward) CG position by the equation,
(67)
where I t is taił length from the quaner chord of the wing MAC to the quarter chord of the
horizontal stabilizer MAC and CGfwd is equal to the forward CG limit for cach flap setting.
GivenFlap
Configuration
(FwdCG)
-1.0
EntryRate
Figure27.
3. 42
D6-1420
Mayl989
PERFORMANCE
For each flap configuration tested, a plot of CLs versus stall entry rate is constructed as shown in
Figure 27. The finał CLstaU is chosen as the value cmresponding to the entry rate of -1.0 kt/sec on
an average fairing of these data.
Although certification test flights are run with engines at idle thrust settings, the regulations require
that the data be presented for zero thrust operation. It is not always necessary to make a correction,
but if the data is to be corrected, it is done in this manner: Stall tests arc run not only at idle thrust
but also ~ta low power setting for cach configuration. These power on stall CL's at -1.0 kt/sec
entry rates are calculated and plotted versus net thrust as shown in Figure 28.
GivenFlap
Configuration
c~~~--,~------1,rl
11
1.0 Knot per Second
EntryRate
(FwdCG)
Id.le
NetThrust
Figure 28.
An extrapolation of these data to zero thrust is used to define the regulation required zero thrust stall
data.
For advanced versions of Boeing airplanes flying today, there is also a CLstaU variation with gross
weight for cach tł.ap configuration. The effect is of sufficient magnitude that it has to be considered
to obtain the correct values of stalling speeds. The CLs for 7.Cro thrust from Figure 28 is now
plotted with respect to airplane weight. A second value of CLs is now determined meeting the
regulatory requirements. The result is Figure 29 with shows a CLs decrease with weight increase.
Finally, for cach flap, the equivalent stall speeds may be determined using equation (64) and plotted
as shown in Figure 30. For operational use, the calibrated (or indicated) values of stall speeds may
now be determined.
06-1420
May 1989
3. 43
PERFORMANCE
Gross Weight
Figure29.
Gross Weight
Figure 30.
Not only is it necessary to determine stall speeds for publication; it is also mandatory that the
operating characteristics of the airplane in flight close to and through stall be demonstrated
according to FAR Parts 25.201 and 25.205. These flight demonstrations must satisfy the
regulations and the certification pilots, both in straight flight and in tums, power off and power on.
The thrust used for the power on demonstration is that necessary for level, straight flight at 1.6Vs in
the approach configmation, gear retracted, and maximum landing weight. Additionally, power on
straight flight demonstrations must be flown with one engine inoperative.
3. 44
D6-1420
Mayl989
PERFORMANCE
1 g Stall Speeds
As described in the previous chapter, stall speeds are based on the minimum speed in the stall
maneuver with an entry rate speed reduction of 1 kt/sec. Full up elevator deflection is used, and
the achieved load factor is less than 1 g at the minimum speed.
The new FAA philosophy discussed in Advisory Circular 25-7 is to terminate the stall maneuver
once stall identification is made. The airplane is considered stalled (stall identification) when any
one or more of the below characteristics gives the pilot a elear and distinctive indication he should
stop any further increase in angle of attack. Recovery is then initiated using normal techniques.
Stall identification is made when:
(1)
The pitch control reaches the aft stop and no fmther increase in pitch attitude occurs
when the control is held full aft for a short time before recovery is initiated.
(2)
An uncommanded, distinctive and easily recognizable nose down pitch that can not
be readily arrested.
(3)
A roll that cannot be readily anested with normal use of lateral or directional
control.
(4)
The airplane demonstrates an unmistakeable, inherent aerodynamic waming of a
magnitude and severity that is a strong and effective deterrent to further speed
reduction.
During the 767-300 stall testing, the FAA pilot recovered upon reaching deterrent buffet (Item 4),
whereas full elevator control was used in the original 767-200 testing. This change in philosophy
resulted in the 767-300 having higher demonstrated stall speeds while actually having slightly
superior wing lift capability ( CLmax ). To solve this problem, it was agreed to use 1 g stall speeds
and new factors to define the operational speeds, such that equivalent safety would be maintained.
This is also being applied to later versions of the 767-200, and will be applied to 737-400, 747400, and subsequent airplanes. Figure 31 compares speed, lift coefficient, corrected lift coefficient
and load factor during a typical stall. The 1 g stall speed is determined from the maximum lift
coefficient (CLmax ) corrected for load factor during the stall maneuver.
Since the 1 g stall speeds are higher than the traditional stall speeds, the operational factors on stall
were modified as follows:
ITEM
FAR Part25
VN5 FAR
Takeoff Climb Speed, V2
Approach Speed
Approach Climb Speed
Enroute Climb Speed
1.2
1.3
1.5
1.25
767 lssue Paper (767-300)*
1.13
1.23
1.4
1.23
* Numbers may be different for airplanes certificated later.
06-1420
May1989
3. 45
PERFORMANCE
160.
...........
....... ····:···
····:···
................
V8..0CITY
::::::::::::i:::
........ ····:···
-VEAS
140.
i=::?~~== =i:= ::=:!i\ 11:i!=ii! :!/!::! =/:::: :::iiir :!!: i =: \li!! =/:iii !i!L!i !iii/ii
III mrnI rnrnri tmmttt(
120.
·
2.4
LIFT
..11 •
ł •
11.?.
:i~Jilil~~:~w,rrnirn mmm mmm mmm
21 •
11,.
2• •
21 I.
:\li!=!=!\=:!\ ii!Yi! !!!!Ci iii/ii= =i/iii i!Jiii=~i!Lii ii!r:1 !:!Li!::::\:: 11!\H: !!!J!ii
!!/==! :!=:Ci !!!!Fi :H::1= ::!}? !>11:: \\j\\:: \~ST1t1L: 1::Y!: !:\::\: 11::/:i !Hi::\
COEF
2.
I.I
qS
1.6
"'.
2.
~-
,..
CORRECTED
LIFT
COEF
1.6
1.2
1.2
UW)
FACTOR
-
g's
.8
.4
"'·
t•
11 ,.
a 1.
8.APSED Til'E -
SECOłłlS
1 !.
2 .
c I.
Figure 31.
3. 46
D6-1420
May1989
PERFORMANCE
Buffet Limits
The previous discussion dealt with buffet limits as concemed specifically with low speed flight.
Actually it is found that airflow separation, which causes buffeting, can be experienced at all flight
speeds under certain conditions. This introduces the subject of high speed airflow separation. As
the flight velocity is increased there is reached a condition where the velocity over some part of the
wing becomes supersonic. As was discussed in Section 1, it is found that the deceleration of the
supersonic air to subsonic velocity before the air leaves the wing causes a shock wave, which in
tum precipitates airflow separation. This separation is capable of causing buffeting just as in the
low speed region. The only difference will be that as speed is increased the high speed separation
will be aggravated, causing greater buffeting. (The low speed buffeting was aggravated by
lowering the speed.)
The buffet limits for the complete speed range of the airplane are defined from the results of flight
tests. In these tests the airplane is flown to maximum level flight speed at various altitudes and if
buffet is not encountered before maximum speed is reached, then the airplane is dived until it is
encountered. At the intermediate speeds the airplane must be accelerated in a banked tum or pull up
to reach buffet. The result of these tests can be conveniently plotted on a single graph as illustrated
in Figure 32.
Buffet Boundary
M
Figme 32.
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May1989
3. 4 7
PERFORMANCE
Lines of constant W/6amb may also be plotted on Figure 32 by the application of equation (52).
W
2
-=
1481. 351 CL M S
6 amb
By choosing values of W/6amb and by .knowing the wing area, a relationship between CL and
Mach number can be obtained. These lines of constant W/6amb have particular significance and
merit further discussion. Assume that an airplane is initially flying in level flight at a given altitude,
weight, and speed. The condition can be shown on the buffet CL versus Mach number plot as
shown in Figme 33.
r
Buffet i3oundary
G.
M
Figure 33.
ff the airplane is now speeded up or slowed down it will do so along the line of constant W/'{,amb•
assuming that the altitude is maintained constant. But, if the altitude is increased, then at the same
gross weight, a higher value of W/6amb will result. This means that at the same Mach number the
lift coefficient will be greater. Conversely, a lower altitude at the same weight will result in a lower
W!6amb and a lower CL at the same Mach number.
From such a plot, another chan more useful to the flight crew may be made. Equation (52) may be
manipulated to give:
=
6
amb
3. 48
w
1481. 351 M2 S CL
(68)
D6-1420
Mayl989
PERFORMANCE
w
6 amb
(Select)
CL
(Fig 32)
(Select)
(Eq 68)
(1)
(2)
(3)
(4)
M
ALT
(5)
w
6amb
(Select)
(Eq 68)
(6)
(7)
ALT
(8)
Figure 34.
A table such as shown in Figure 34 may be constructed, selecting a range of Mach num.bers for a
given weight. By repeating the procedure for several weights, and performing the indicated
computations with the use of equation (68), a chart such as that shown in Figure 35 may be inade.
Figure 35, as shown, is only valid for level flight.
Initial Buffet
in One G Fli ht
....i..
<
FlapsUp
Mach Number
Figure 35.
Having introduced the concept of load factor earlier, another application may now be explored in the
area of maneuvering flight.
Consider an airplane in a coordinated banked tum such that no loss of altitude is sustained. Refer to
Figure 36. Resolve the lift, L, into its components and note that
Lcosc!»=W
(69)
where cl» is the bank angle. Thus the airplane is in equilibrium vertically, but not laterally. There is
an unbalanced lateral component of lift, L sin cl», which will give it a lateral accelera.tion. ln other
words, the airplane is in its tum by virtue of the lateral acceleration impaned to it by the lateral
D6-1420
May1989
3. 49
PERFORMANCE
component of lift. The situation may be analyzed by thinking of the center of gravity of the airplane
rotating in a circle as shown in Figure 37. The acceleration may be shown to be V d8/dt, where V
is the linear velocity in ft/sec and d8/dt, the angular velocity, in radians/sec.
fi:sin+ L
L cos
+1
w
Figure 36.
The accelerating force, (F =Ma), necessary to hold the weight, W, in the circular path is,
V d8
F -- W
g
dt
(70)
The rate of tum will increase until,
(71)
The linear velocity is related to the circular morion:
or,
V=R:
d8
V
cit=R
3. 5 O
(72)
06-1420
May1989
PERFORMANCE
Figure 37.
Substitution of equation (69) and (72) into equation 01) yields,
L sin ci, = L c;s ci,
V:
which simplifies into,
y2
R = _g_tan_«i,
(73)
This is an expression for the tum rad.ius of an airplane in terms of its true velocity and bank angle.
Retuming to Figure 36 and equation (65), solving for the load factor, n, in terms of the bank
angle:
n=i
n= L cos
L
ci,
Therefore,
n= cos1 ci,
D6-1420
May 1989
04)
3. 51
PERFORMANCE
In a tum, then, the load factor (or number of g's) experienced is equal to the inverse of the cosine
of the bank angle, assuming no loss or gain of altitude during the maneuver. Also, if the level
maneuver is assumed, the engine thrust must be increased since the lift must be increased over
straight and level flight, which will cause an increase in drag. Figure 38 shows the solution to
equation (73). The two minute line and the four minute line show the time it takes to fly a 360
degree tum.
The preceding discussion of load factor has dealt exclusively with bank:ed tums as the method for
producing accelerations on the airplane. Another method, of course, is the pull up , or push over
which is of interest in the design criteria.
The application of load factor to the present subject of buffet limits is made by referring again to the
CL versus Mach number plot ofFigure 33, which is redrawn for convenience in Figure 39. Again,
a level flight point is located for a given weight, ·altitude, and speed. This will define a CL
(denoted as CL ) and Mach number. Assume now that at the same Mach number the airplane is put
1
into a level flight tum such that the lift now increases to the point corresponding to a new CL
. (denoted as Ci.i). Once again from equation (65):
·
4MinuteTmn
Bank Angle - Degrees
Figure 38.
3. 5 2
1)6.1420
May1989
PERFORMANCE
n -L-
-w
L=W
1
Thus,
LI
n=- =1
w
If the same speeds arc assumed for conditions 1 and 2:
ql =q2
Thus,
CL
2
n=~
(75)
Ll
If the airplane is assumed to be banked to the point that it reaches the buffet limit (CL in Figure
3
39), then the load factor developed at this point will be:
D6-1420
May1989
3. 5 3
PERFORMANCE
M
Figure 39.
Thus, in generał, consideńng accelerated. flight conditions:
or,
L=nW
L
nW
6 amb = 6 amb
(76)
In plotting Figure 39, it was assumed that L = W (level flight) in computing W/6amb curves.
Actually the calculations were more generał since what was really done was to calculate lines of
constant L/6amb. The curves can thus be labeled as lines of constant nW/oamb as indicated by
equation (76). · From equation (75):
(-t:)2
n= (ł-)
amb
3. 5 4
1
D6-1420
May 1989
PERFORMANCE
or,
(n
n= ( n
6:b)
6:)
2
(77)
I
Therefore, the usefulness of Figure 35 can be increased by labeling the weight lines as nW lines,
or equivalent 1-g gross weight lines.
With the preceding analysis and with the CL versus Mach number plot, Figure 39, a new plot of
the initial buffet boundary on axes of nW/6amb and M can be made. See Figure 40. Figure 40
finds its usefulness in defining the acceleration characteristics of the airplane. With this curve as a
basis, and assuming equivalent 1-g airplane gross weights, another useful chart for any maneuver
condition can now be drawn as showp in Figure 41. For a given equivalent 1-g gross weight
value, an infinite number of gross weight load factor combinations may apply.
n W/6
Figure 40.
D6-1420
Mayl989
3. 55
PERFORMANCE
2.2
o
Iz
u,
2.0
w
w
60 a:
Q
iu,c, .
W,
Ww
1.8
w
C
•
55 ~
...J I-
C,
g LL 1.6
<~
clm
::E cl 1.4
45
a:o!::
z~ 1.2
1.0
:1:,#G
C,
e,C:>
!.ł-0
G
z
<
~
z
<
al
MANEUVER MARGIN
30
15
EOUIVALENT 1 - g GROSS WEIGHT
SPEED & ALTITUDE MARGIN
w
C
E
!:i
<
INDICATED AIRSPEED
Figure 41.
The !ower part of Figure 41 has airspeed instead of Mach for the abscissa scale and a family of
Mach lines are drawn within the chart. This is an improvement over Figure 35. However, the two
parts of Figure 41 are not in a chase-around format, one part to the other. When working a
maneuver problem, informati.on from one part has to be transferred to the other part to obtain some
of the answers. An additional improvement can be made by crossplotting the lower part of Figure
41 to equivalent lg gross weight versus M for constant values of pressure altitude and airspeed,
and rotating the upper part to match the equivalent 1-g gross weight scale. The information of
Figure 41 is now in a chase-around format as shown by Figure 42 where the equivalent 1-g gross
3. 5 6
06-1420
May 1989
PERFORMANCE
Figure 42.
D6-1420
Mayl989
3. 5 7
PERFORMANCE
weight, now a hidden scale, does not have to be determined or considered. This chart can be
presented in the operating manuals showing the cruise maneuver capability of the airplane for any
maneuver.
Stall Warning
As the airplane is decelerating toward low speed stall, it will encounter initial stall buffet speed at
some margin above the stall speed. FAR Pan 25.207 requires that elear and distinctive stall
waming shall be apparent to the pilot with sufficient margin to prevent inadvertent stalling of the
airplane in all possible landing gear and flap configurations.
It shall be acceptable for the warning to be fumished either through the inherent buffeting
characteristics of the a.irplane or through a device which will give adequate stall waming. A stall
waming beginning at a speed 7 percent above the stalling speed is the required margin. For some
flap configurations, where the margin between initial buffet and stall is less than the required 7
percent as shown in Figure 43, a stall warning device is required.
GivenFlap
Configuration
--
u,
<
u
....~g
-~
7I--Weight
Figure43.
Maximum Speeds
Another flight limit is maximum speeds under given conditions of weight, altitude, and thrust
setting. The determination of maximum speeds can be made from speed thrust charts similar to
Figure 12. This is possible because under all stabilized flight conditions the thrust must equal the
drag. Thus, the intersection of any T.16amb available line with a T/6amb required line will define a
maximum speed point for those panicular conditions. Of interest arc the maximum speeds at all
altitudes under conditions of maximum cruise thrusL The intersections, when taken from a plot
such as.Figure 12, can be plotted on axes of W.16amb versus Mas in Figure 44. Also, the buffet
boundary can be drawn on Figure 44. The appearance of the buffet line on Figure 44 indicates the
ability of the airplane to approach buffet in level flight conditions. From such a plot as. Figure 44
another useful and more easily understood plot can be made, Figure 45.
3. 5 8
D6-1420
Mayl989
PERFORMANCE
MAXIMUM
CRUISE
TBRUST
W/6am
ISA+ !).T temp
M
Figure 44.
Figure 45 shows the altitude - true airspeed capabilities of the airplane at various weights when
operating with rated thrust The sttuctural placard speed limit, to be discussed later, is also shown.
heavy gros8
ALTITUDE
Figure 45.
D6-1420
May 1989
3. 5 9
PERFORMANCE
Ceilings
The maximum altitude obtainable under given conditions of thrust and weights is de:fined as the
absolute ceiling of the airplane. Although the altitude capability of the jet transport is not
approached, ii: is desirable to discuss this condition bńefly. Referring to Figure 44, the highest
point on cach curve of constant altitude represent the maximum W!6amb attainable at that altitude
and thus de:fine the maximum weight attainable. These altitucles are the absolute ceilings for those
weights. The same may be scen from Figure 45 in which, again, the peaks of the constant weight
lines define the absolute ceilings at those weights. In all cases, a unique speed is associated with
the absolute ceiling. This means that when an airplane is at its absolute ceiling (a practical
impossibility) it will have to fly at .one speed only. To fly either faster or slower will result in
altitude loss. A ceiling of practical value is the service ceiling cle:fined where the rate of climb is 100
ft per min.
Structural Limitations
Since structural airspeed limitations must be considered, some additional terminology, as clefin~
in Part 25 of the Federal A viation Regulations, is required. Some of the speed limits and
explanations of their cleterminations arc given below.
V A - design maneuvering airspeed. The design maneuvering speed VA shall not be less than:
FAR Part 25.335(c) (1)
(78)
where n is the limit positive maneuvering load factor at V c , 25 for commercial aiiplanes; and V s1
is the n = 1 stalling speed for clean configuration at maximum flight gross weight as determined by
F AA flight test of the airplane. V A is the maximum speed at which application of full available
rudder or elevator will not overstress the airplane. In the flaps up configuration, full aileron can be
applied at any speed. Aileron, rudder or angle of attack controls must be designed for full
deflection at VA •
FAR Part 25.1583(a) (3)
Example:
From the lift equation (52),
~L
= ; W = 1481.351 CL~ S
amb
amb
and for a 747 Boeing airplane with a wing area of 5500 square feet,
2
CL M = .122734 x 10-6 : W
(79)
amb
From equation (79), a value of CLM2 may be found for the condition specified; that is, the flaps up
stalling speed at the maximum weighL Evaluating equation (79) for W = 817, OOO pounds, n = 1,
3. 60
1)6.1420
May1989
PERFORMANCE
and at sea level 6amb = 1.000, CLM 2 = 0.1003. From a cmve showing the vańation of CLM2 with
M, the stall Mach number, hence the stall speed, V s, may be determined. Figure 46 shows the
variation of Mach with CLM2, and is constructed from CLMAX versus Mach number relationships
as shown in Figure 47. Figure 46 shows the stall Mach number, Ms, to be .307. At sea level,
· this .corresponds to V es = 203 knots.
.5
I łvr
CL
Vs Mach No.I
.4
2
CL M •3
-------------~~
.2
,
~'b,~t:.,
~,tf. ~
I
.1
o .....
.1- - - -.2- - - -.3
0 -aiiiiiiiii.....
. ---.4 - - - .5 - - - -.6- - - -.7M
Figure46.
Evaluating VA at sea level,
V A = 203 ~ = 321 kts, EAS
where limit load factor n = 2.5.
YB-design speed for maximum gust intensity. This speed shall be selected not to be lower than:
(80)
or as determined by the intersection of the 66 fps gust line (when below 20,000 ft) and the CLMAX
line on the V n diagram as shown in Figure 48. V s 1 is the flaps retracted stalling speed at the
weight under consideration. VB need not be greater tban V c.
FAR Part 25.335 (d)
D6-1420
May1989
3. 61
PERFORMANCE
For Boeing airplanes, a V B speed greater than FAR requirements is .used for structural analysis.
1.2
I
I
I.O ~ Typical Curve
From Flight Test
r----.
--
-
CLMax Vs Mach Number
.6
FlapsUp
.4
.2
.1
.2
.3
.4
.5
.6
.7
M
Figure47.
The V-n Diagram
The V-n diagram, Figure 48, is a very convenient means of indicating the loading conditions on the
airplane due to a sudden pull up, push over, maneuver or gust penetration which the airplane must
be designed to withstand. In trimmed level flight through still air, the airplane is subjected to loads
normal to its longitudinal axis of only such magnitude as to countera.et the gravitational force of its
weight. The condition is call 1 g or n = 1 flight. In order to pull up, push over, or maneuver,
additional forces must be applied and additional loads are i.mposed upon the structure. Also,
vertical gusts or the vertical components of other gusts impose the same. type of load.ing by
changing the relative angles of attack. These loading conditions must be taken into consideration in
the design of the structure. The commercial transport is not expected to be subjected to violent
maneuvers such as might be expected to be the case with fighter aircraft, missiles, or even large
bom.bardment airplanes. Figure 48 shows that operating close to the FAR VB speed (as determined
by the intersection of the maximum gust line and the QMAX line) in gusty weather results in the
least increase in load factor, and increasing the gust penetration speed for design and operational
purposes increases the load factor for a given gust intensity. Since an airplane is limited to its
design load factor it will be limited to some maximum rough air gust speed. The recommended gust
penetration speed under gusty conditions can be any speed between the minimum speed, and the
maximum VB speed used for structural analysis.
3. 62
I>6-1420
Mayl989
PERFORMANCE
2.5
=
Given Weight
AltitiJde < 20,000 Feet
2
s
~
Il..
1
"io
·-·-... o
...J
E
r
I
I .
I .
I
Flaps Retracted
Stall Speed
...J
Airspeed - KEAS
-1
Figure 48.
Maneuvering Envelope
Maximum maneuvering acceleration limits have been imposed to comply with Federal A viation
Regulations.
The positive maneuvering load factor n for any flight speed up to Vn shall be selected by the
applicant, except that it shall not be less than 2.5.
The negative maneuvering load factor shall have a minimum value of -1.0 at all speeds up to Vc,
and it shall vary linearly with speed from the value at V c to zero at Vn.
( FAR Parts 25.337 (b) and 25.337 (c).)
The maneuvering envelope can now be drawn using the pull up and push over values up to V A and
the design maneuver factors defined previously. The envelope is shown on Figure 49.
Gust Envelope
Gust effects arc based upon encountering standard vertical gust components as defined in the
Federal A viation Regulations, FAR Pan 25.341(a), while in level flight:
Positive (up) and negative (down) rough air gusts of 66 fps at the speed V:e shall be
considered at altitudes between sea level and 20,000 feet. At altitudes above 20,000 feet, the
gust velocity may be reduced linearly from 66 fps at 20,000 feet to 38 fps at 50,000 feet
1)6..1420
May1989
3. 63
PERFORMANCE
Positive and negative gusts of 50 fps at speed Vc shall be considered at altitudes between sea
level and 20,000 feet At altitudes above 20,000 feet, the gust velocity may be reduced
linearly from. 50 fps at 20,000 feet to 25 fps at 50,000 feet
Positive and negative gusts of 25 fps at the speed Vo shall be considered at altitudes between
sea level and 20,000 feet At altitudes above 20,000 feet, the gust velocity may be reduced
linearly from 25 fps at 20,000 feet 12.5 fps at 50,000 feet .
3
c::
----,------ -
Maneuvering Envelope
Design Gross Weight
FlapsUp
(Typical Values)
2
·1
I
I
'h ~
: '\p
, 200
I
'b
Vc
3()() I
I
-1
'
Figure49.
50
l- 40
~
-
.g
a
·.=
-<
30
20
10
o
10
60
20
30
40
50
U de - Gust Velocity - FPS
70
Figure 50.
3. 64
D6-1420
May1989
PERFORMANCE
Gust load factors shall be assumed to vary linearly between the specified conditions. (per FAR Part
25.341 (b) (2)). See Figure 53.
In the absence of a more rational analysis, the gust load factors shall be computed in accordance
with the following formula:
(81)
where:
m = slope of airplane normal force coefficient curve CNA per radian if the gust loads arc
applied to the wings and the horizontal tall suiface simultaneously by the rational method. It
shall be acceptable to use the wing lift curve slope CL per radian when the gust load is applied
to the wings only and horizontal taił loads arc treated as a separated condition.
·
FAR Pan 25.341 (c)
The load factor equation prescribed above can be derived in the following manner:
Consider the velocity vector diagram in Figure 51:
~ U d e = D e s i gGust
n.
Aa (Due to Gust)
Ve Level _Flight
Velocity
Figure 51.
As a result of the gust whose vertical velocity component is Ucie, the angle of attack is changed by
the increment Aa., and the CL is increased suddenly by the increment ACL in Figure 52.
Since the change is sudden, the velocity has not bad time to change appreciably, so,
D6-1420
May1989
3. 65
PERFORMANCE
C
Figure 52.
and,
But,
SO,
AC
dCL
=-Aa
L
da
or,
where m is the slope of the lift curve per radian. If Aa is given in degrees, it must be changed to
radians or m replaced by a, the slope per degree.
Since design gust velocity, U de• is specified in fps and velocity is in knots,
3. 6 6
D6-1420
May 1989
PERFORMANCE
U de
A.a = 1. 6878 Ve
The angle A.a is assumed to be small,therefore,
Also,
CL=
295.375W
1
v;s
Since,
n=
1
mUde )(
± ( l6878Ve
V:S )
295.;75W
mUm Ve
n=l±
W
498.5
(s)
(82)
This is a "sharp edge gust formula" based on an assumption of instantaneous gust penetration. That
is, the gust is assumed to act on the entire airplane at the same time. However, as an airplane
penetrates an actual gust, the gust does not act on the entire airplane at the same time. This
condition tends to alleviate the acceleration loading and decreases the value of the load factor
determined by equation (82) by a factor, Kg. The value of Kg, known as the "gust alleviation
factor", has been determined empirically to yield the formula:
.88 µg
K =----g
5.3+ µg
(83)
where, µg, the "airplane mass ratio" equals,
and
c is the mean geometrie chord.
D6-1420
May 1989
3. 67
PERFORMANCE
Tb.us, includ.ing the "Kussner-Wagner" effect, the equivalent sharp edge gust formula is:
n =1±
Kg mU& Ve
W
(84)
498.\5 )
GUST ENVELOPE
3
W = 243, OOO LB
FLAPS UP
CG = .335 MAC
s::
2
CRITICAL ALTITUDES
(TYPICAL VALUES)
--+25 -rpa
,
~
~ O+------+-----~~d--~-=-~~ł--,.~-+--~
100
....__-66 ~ 6 ---50 ~ 8
----
AIRSPEED - KT EAS
_,
Figure 53.
Equation (84) assumes that the lift curve slope, m, has incorporated the compressible and
aeroelastic effects of the wing. The data for Boeing airplanes are based on a lift cmve determined
by a steady-state aeroelastic analysis of the wing which includes the Mach number effects and the
flexibility characteristics.
The gust envelope calculated at the cńtical altitudes and with the CG at 33.5% MAC for a Boeing
airplane is shown in Figure 53.
3. 6 8
~1420
May1989
PERFORMANCE
3
V-n Diagram
_J
W= 243,000 Lb
2
FlapsUp
CG= 33.5 % MAC
Critical Altitude
...
Typical Values
---
......
......
I
I
I
-
-
I~
1200
1
......
I
I
.,.
-300-r
400
1
I Airspeed - KEAS I
I I
Figure 54.
. Composite V-n Diagram
The airplane must be structurally able to withstand all load factors included on or within the two
envelopes presented above. Thus all the design load factors can be given on and within a composite
envelope which includes all regions enclosed by the two envelopes. This design V-n diagram for
the critical altitudes with CG at 33.5% MAC is shown on Figure 54.
The diagram has been limited to speeds above stall since it is impossible to fly at lower speeds.
Therefore, consideration will not be given to the higher load factor values of the gust over the
maneuver values for C1..mu: at velocities below Vs. The reason for this is that the C1..mu: line
represents the maximum loading condition in this region. As a result, the gust values of load factor
cannot be attained and are of academic interest only.
Vc -- design cnńsing speed. The minimum design cruising speed, Vc, shall be sufficiently greater
than VB to provide for inadvertent speed increases likely to occur as a result of severe atmospheric
turbulence. In the absence of a rational investigation substantiating the use of other values:
(85)
except that it need not exceed the maximum speed in level flight at maximum continuous power for
the corresponding altitude. At altitudes where V0 (design dive speed) is limited by Mach number or
compressibility, it shall be acceptable to limit Vc to a Mach number selected by the applicant .
FAR Part 25.335 (a)
D6-1420
May1989
3. 69
PERFORMANCE
V c and Me values for Boeing airplanes are arbitrarily selected by Boeing, meeting FAR
requirements.
VO -- design dive speed. The design dive speed shall be used to determine the maximum operating
limit speed for the airplane in accordance with FAR Part 25.1505; which insures this speed shall
not be exceeded if the airplane is upset from fligbt at normal maximum operating speed .
FAR Part 25.335 (b)
For Boeing airplanes, V O and M0 arc selected as required for certification.
The structural design speeds for a typical airplane are shown in Figure 55.
.55
.60
.65
I
I
I
I
I
I
ł
40
I
Tł'Pical V~ues
I
\
35
...oo
\
\
\
\
-
.g 20
..........
::s
\
-
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
540
\
\
\
\
\
\
\
\
500
\
\
\
460
\
\
\
\
420
\
\
\
\
o
\
\
\
\
5
\
\
\
\
380
\
\
\
\
\
340
\
\
\
\
\
300
\
\
\
\
\
10
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
15
\
\
\
\
\
\
\
\
<
\
\
\
\
\
\
\
I
\
I
\
\
\
\
\
\
\
a. 25
q8
\
\
\
I
I
I
\
\
\
.95
I
I
\
\
\
.85 .88 .90
I
I
I
I
I
I
\
I
\
\
\
\
I
\
\
\
\
30
Mach Number - M
.70
.75
.80
\
580
True Airspeed - Knots
Figure 55.
VF -- design flap speeds. The flap placard speeds arc determined to meet design criteria in
accordance with FAR Part 25.335 (e) (3):
3. 7 O
1)6.1420
May1989
PERFORMANCE
VF shall not be less than:
1.6 V s 1 with flaps in takeoff position at maximum takeoff weight;
1.8 V s 1 with flaps in approach position at maximum landing weight; and
1.8 V So with flaps in landing position at maximum landing weight .
Wing flaps and their supporting structure and operating mechanism must be designed for the
critical loads resulting from the conditions prescribed in FAR Part 25.345 ·taking into account the
loads occuning during transition from one flap position and airspeed to another.
FAR Part 25.457
The wing flap control shall be designed to retract the flaps from the fully extended position duńng
steady flight at maximum continuous engine power at any speed below VF + 9.0 (kts).
FAR Part 25.697 (d)
The stall speeds for the maximum takeoff and landing weights may be obtained from Figure 30
when determining the flap placard speeds.
A typical set of flap placard speeds arc tabulated in Figure 56. For convenience, a plot of the flap
placard speeds is shown in Figure 57. Also, for some airplanes, there may be an altitude placard
for flaps down operation since it may be possible to exceed limit loads when applying maximum
roll capability.
Flaps
Position
Placard Speeds - KEAS
40
185
30
210
25
220
15
230
Figure 56.
D6-1420
May1989
3. 71
PERFORMANCE
!'lAP
PLACAlU>
·SP~
KT-EAS
lt - FI.AP POSITIOił, mGlm'S
Figure 57.
There arc no speed limitations imposed on the spoilers. The spoiler angle for a given input signal
will decrease with an increase in airspeed. As the "q" force on the spoilers increases it will
overcome the hydra.ulic pressure in the actuator, and the spoiler will blow down. Figure 58 shows
the design blowdown speeds. It should be understood that the spoiler angle decreases with altitude
increase for any given equivalent airspeed because of compressibility effects. However, deviations
arc small enough that they arc usually neglected.
!I
i
<
60
SPOILER Al'lGLE
DECRF.A.SFS WITH .
ALTr.ruDE FOR
.AliY GIVEB V!:
50
4o
30
!o 20
H
~
10
o
200
250
300
'VE - KNOTS
350
400
- -~-·---··-
Figure 58.
3. 7 2
D6-1420
Mayl989
PERFORMANCE
V MofMMo - maximum operating limit speed.
(a) The maximum operating limit speed is a speed which shall not be deliberately exceeded
in any regim.e of flight (climb, cruise, or descent), except where a higher speed is authorized for
flight test or pilot training operations. This operating limitation, denoted by the sym.bols VMofMMO
(airspeed or Mach num.ber, whichever is critical at a particular altitude), shall be established to be
not greater than the design cruising speed Vc and sufficiently below Vo/Mo or VopNoF
(demonstrated flight dive speed) to make it highly improbable that the latter speeds will be
inadvertently exceeded in operations. The speed margin between VMofMMO and Vo/Mo or
Vop/Mop shall be determined in accordance with paragraph 25.335 (b), but shall not be less than
the margin found necessary in flight tests in accordance with FAR Part 25.253 .
FAR Pan 25.1505
(b) The minimum margin shall be the greater of the values determined in accordance with
subparagraphs (1) and (2) of this paragraph:
(1) From an initial condition of stabilized flight at V c/Mc, the aiiplane shall be
assumed to be upset, flown for 20 seconds along a flight path 7 .5 degrees below the initial path and
pulled up at a load factor of 1.5 (.Sg acceleration increment). It shall be acceptable to calculate the
speed increase occurring in this maneuver, provided reliable or conservative aerodynamic data are
used. Power as specified in 25.175(b) (1) (iv) shall be assum.ed until the pullup is initiated, at
which time power reduction and the use of pilot controlled drag devices may be assumed.
FAR Part 25.335 (b) (1)
(2) The margin shall be sufficient to provide for atmospheric variations, such as
horizontal gusts, penetration of jet streams or cold fronts, and for instrument errois and airframe
prcxluction variations. It shall be acceptable to consider these factors on a probability basis but the
margin at altitudes where Me is limited by compressibility effects shall not be less than .OSM.
FAR Pan 25.335 (b) (2)
(c) V cfMc shall not be greater than 0.8 VoJM0 .
FAR Pan 25.335 (b)
·V LO -- landing gear operating speed. The landing gear operating speed, VLO, shall be established
not to exceed a speed at which it is safe to extend or retract the landing gear as limited by design in
accordance with FAR Pan 25. 729 or by flight characteristics. .
FAR Part 25.1515 (a)
1) The landing gear retracting mechanism, wheel well doors, and supporting structure shall
be designed for loads occurring in the flight conditions when the gear is in the retracted position,
and for the combination of friction, inertia, brake torque, air loads, and gyroscopic loads resulting
from the wheels rotating at a peripheral speed equal to 1.3 Vs (with flaps in takeoff position at
design takeoff weight) occuning during retraction and extension at any speed up to 1.6 Vs 1, (flaps
in the approach position at design landing weight), and any load factor up to those specified per
FAR Part 25.345(a) for the flap extended condition.
FAR Pan 25.729 (a) (1)
D6-1420
May1989
3. 73
PERFORMANCE
Example:
V S1
= 103.5 knots, EAS
1.6 Vs1 = 1.6 (103.5) = 166 knots EAS
V1.0 is selected at 270 knots EAS
M1.0 is selected at .83
.The speed limitation is a structural placard on the landing gear doors which are down. during gear
operation. Upon completion of the operation, the wheel well doors arc automatically closed and the
airplane may operate at increased speed with the landing gear in the extended position.
VLE -- landing gear extended speed. The landing gear extended speed, VLE, shall be established
not to exceed a speed at which it has been shown that the airplane can be safely flown with the
landing gear secmed in the full extended position, and for which the structure has been proven in
accordance with FAR Part 25.729 .
FAR Part 25.1515 (b)
(2) The landing gear, the retra.cting mechanism, and the airplane structure (including wheel
well doors) shall be designed to withstand the flight loads occurring with the landing gear in the
extended position at any speed up to 0.67 VC• unless other means arc provided to decelerate the
airplane in flight at this speed .
FAR Part 25.729 (a) (2)
Example: Assume a Vc equal to 360 kts EAS.
0.67 Vc
VLE
~
3. 7 4
= 0.67 (360) = 241 knots EAS
is selected at 320 knots EAS.
is selected at .83.
D6-1420
May1989
PERFORMANCE
3-S
Takeoff
Takeoff consists of the ground run, rotation to lift off, flare to 35 foot height (15 foot height for wet
runway certification), and climb segments. The takeoff distance is from brake release to the 35-foot
height. The takeoff climb requirements, as given in the regulations, are referred to defined
. segments of the flight path from 35 feet to 1,500 feet above the takeoff suńace, or at which the
transition from the takeoff to the enroute configuration is completed, whichever point is higher.
Figure 59 shows the takeoff profile.
FIRST
SEGMENT
ENGINE RATING - TAKEOFF
FLAP SE1TING - TAKEOFF
I
TAKEOFF
TAKEOFF
SECOND
SEGMENT
THIRD
SEGMENT
FINAL
SEGMENT
TAKEOFF
TAKEOFF
AT LEAST MCT
FLAP RETRACTION
MCT
FLAPS UP
I
I
I
I„
CREDIT FOR GROUND-EFFECT
ł
FLAPS UP AT LEAST
AT LEAST
1500 FT
400FT
ł
BRAKE RELEASE
I•
TAKEOFF DISTANCE •
I•
TAKEOFF CLIMB SEGMENTS
Figure 59.
The takeoff from brake release to the end of the first takeoff climb segment with the gear up is
determined from flight test, and therefore includes ground effect. The first takeoff climb segment
begins at 35 feet and continues to gear up. The second takeoff climb segment is from gear up to a
minimum height of at least 400 feet above reference zero, the point on the runway at the takeoff
distance. Flap retraction at this height results in a third takeoff climb segment. The finał takeoff
climb segment is from the flap retraction height where flaps arc up to the 1,500-foot height.
Although the takeoff climbout segments starting at 35 feet arc considered a part of the takeoff,
analysis of these parts will not be made until the climb is discussed.
General Takeoff Distance Equatiori
In developing the equation for ground distance under any acceleration condition:
V- ds
- dt
D6-1420
May1989
3. 7 5
PERFORMANCE
dV
(86)
a=dt
From equation (86),
and,
ds= V dt
dt= dV
a
thus,
ds= VdV
(87)
a
Integrating equation (87) between any two arbitrary speeds and distances, the following equation
results:
V
Sy-:-- sx =
YVdV
J
(88)
a
.vx
li the takeoff ground run to rotation speed is considered, sy - sx can be replaced by Sa (ground
distance), Vx by Vw (airplane starting :from rest where Vw is the wind velocity), and VdV by (V Vw) dV since the acceleration, a, in the integrand is a function of the ground velocity (V - Vw). Vy
is replaced by VR, rotation speed, as defined below. The acceleration, a, in the integrand is related
to velocity change and is therefore independent of the velocity reference poinL
VR(V-Vw)
S0 = I
.a
cIV
(89)
VW
The acceleration of an airplane during the ground run may be found by consideńng the forces
existing on the airplane, as in Figure 60. The airplane is scen to be under the influence of lift (L),
drag (D), thrust ('I), weight (W), rolling :friction force (Ff), and runway slope (cp).
D
Horiz
µ(W - L) = Fr
Figure60.
3. 7 6
D6-1420
May 1989
PERFORMANCE
Since the tangential force retarding the morion of a moving body is proportional to the
perpendicular, or norma! force, a generał relationship may be written letting the Greek letter Mu, µ,
represent the constant of proportionality.
(90)
where,
Fr is frictional force, lbs
F0 or is the normal force, (W -L), lbs
The constant, µ, is usually referred to as a coefficient of friction. The friction forces shown to
exist on the ahplane of Figure 60 may be determined from the application of equation (90), as the
product of the coefficient, µ, and the weight on the wheels, which at any given instant is equal to
(W-L). The value of the coefficient is approximately .015 underrolling conditions on dry concrete,
which means the retarding force will be equal to 1 1/2 percent of the weight on the wheels. This
value can be used in all estimated calculations involving the coefficient of rolling friction. If flight
test information is available showing µ to be slightly clifferent from .015, the actual results arc
used.
Since,
F a =Ma= W
g a
then,
T -D- µ (W-L)-Wcp =w
ga
and,
g
a = W [ ( T - µ W) - ( D - µ L) - W cl>]
a=.:, [(T-µW)-(CD-µCL)Sq-:-Wcl>]
(91)
Substituting equation (91) into equation (89)results in:
V
RW
SG = VJ
w
(V-Vw)dV
g [ ( T- µ W) -( CD - µ CL) S q - W cl>]
(92)
This is the generał expression for the takeoff ground run distance to rotation speed. To carry out
the integration, however, the variation of thrust, weight, drag, and lift with velocity must be
determined. In generał, thrust is a function of velocity, air temperature and pressure; weight will be
nearly constant for the ground run; drag and lift arc functions of velocity and air density. Since the
aircraft attitude remains constant until rotation on aircraft with tricycle landing gear, CL and Co will
be constant dming the ground run. Figure 61 is a plot showing the magnitude of the forces acting
during a takeoff run.
D6-1420
May1989
3. 77
PERFORMANCE
FORCE
d,rag + fl.(W - L)+vlcl>
-
-
-
-
-- -.Il (W -
..
L)
_.,.....
-ł'~......... /
°"
.:,..,,"
;::::.-c. - - .
-- - w• --~---==------------.
--..;;;;;:
V
Figme 61.
Normal Takeoff Ground Run Distance and Time to VR
Retuming to equation (89), it will be scen that to be able to find the ground run requires knowing
the relationship between a, the acceleration, and V, the velocity. From sucha plot as Figure 61,
and from equation (91 ), the quantity 1/a can be computed for any velocity, and a plot of 1/a versus
V and (V-Vw )/a versus V as shown in Figure 62 may be made. Equation (89) specifies that to
obtain the ground run, it is necessary to integrate the expression (V-Vw) dV/a. Referring to Figme
62, the area of the shaded element is (V-Vw) dV/a. Thus the total ground run is equal to the sum of
all such elements between the limits of zero velocity (adjusted for wind) and the rotation speed.
This is another way of stating that the ground run is equal to the total area under the cmve.
Modern high speed computing techniques allow the usc of extremely small velocity increments for
integration of Figure 62. The current method of computing the ground run is by step by step
integration using small velocity increments and assuming constant values of (V - Vw)/a for each
increment.
3. 7 8
D6-1420
May1989
PERFORMANCE
.
"O I
.s,
~·
_g I
-1a
1
I
or
V-Vw
a
V-Vw
I
a
ł
o Vw
V
Figure 62.
Equation (89) may be modified to:
(93)
..
where Il. So is the incremental ground distance traveled between velocities Vx and Vy, and i is
the average acceleration between Vx and Vy.
Integrating results in:
=-k[( v;- 2Vy VW) -(v;- 2Vx VW)]
= -k [( v; - v; ) -2 VW ( Vy - Vx ) J
1
= 2f[ ( Vy - Vx) ( Vy + Vx )- 2 VW ( Vy - Vx ) ]
1
.
= 2 i [ ( Vy + Vx) -2 VW ] ( Vy - Vx)
D6-1420
May1989
3. 79
PERFORMANCE
The term (Vy + Vx)/2 equals the average velocity Vavg and AV is the velocity change from Vx to
Vy; therefore, the above equation becomes:
(94)
The total ground distance to VR is the sum of the incrementa! distances determined from equation
(94). Summing between Vw and VR,
(95)
A table may be set up to systematically handle the calculations involved. Such a table is shown in
Figure 63 that may be expanded to provide greater detail.
V
o
AV T
--
(T-1J.W)
--
q (Co-µ CL)Sq a
--
---
20
--
20
--
--
---
--
--
--
--
---
--
--
--
--
--
--
--
--
--
--
---
---
o
o
--
20
--
ASr- l:ASn At l:At
--
--
-
60
(Van·Vw)
--
20
40
i
--
--
--
--
--
--
Figme 63.
A simplified means of integrating equation (92) is to assume an average acceleration during the
ground roll. Since ground roll is a function of v2 and accelera.tion, the average acceleration for
calculating ground roll will occur at some ~verage v2• Ha plot of acceleration versus velocity
squared is made, it appears as in Figme 64. Since this plots nearly as a straight line, the avera.ge
acceleration occurs at VR2 / 2. Thus the speed for avera.ge accelera.tion is .707VR. This simplified
approach is convenient where the degree of accuracy obtainable from the more ńgorous presentation
is not necessary. The accuracy is within approximately 1%.
3. 8 O
D6-1420
May 1989
PERFORMANCE
LA-
Acceleration, a
a
v2
f
y2
Figure 64.
Calculating the acceleration at this speed and declaring it constant for the complete .ground run
modifies equation (92) as follows:
where V is in fps, and S0 ~s in feet.
Integrating,
V
S0
==al [ (v -2Vw v) JV
2
R
w
or,
2
8 0= 2g
-
W
(VR -VW)
[(T-µW) -(C - µC ) qS-Wcp]
D
L
(96)
.7ITTVR
For V in knots,
S0 = 2.849 [ equation (96) ]
D6-1420
May1989
(97)
3. 81
PERFORMANCE
Takeoff Time
Time to accelerate data for acceleration charts arc based on relationships si.milar to those used in
computing the takeoff distances. In developing the equations for takeoff, an analysis was made of
the forces acting on an airplane as shown in Figure 60 Similarly, from equation (86),
a= dV
dt
dt = .!!Y.
a
From equation (91),
g
a= w [(T-µ W)-( ci> -µCL) Sq-Wq,J
thus,
dV
dt=-g--------------
vr[(T-µW)-( C0 -µCL)S q-Wc!>]
Integrating,
(98)
This is the generał expression for the time to rotation speed. The resolution of this equation is
conducted simultaneously with, and in the same manner as the distance calculations. The solution
to equation (98) may be found, using any of the following methods:
(1)
Rigorous computation using a computer.
(2)
Careful step by step integration of the curve of 1/a versus V, as shown on Figure 62,
using small velocity increments and assuming constant values of 1/a for cach
increment.
(3)
Simplified form of equation (98) where the average acceleration for calculating time will
be found to occur at approximately .6VR since time is both a function of dV where
average acceleration would occur at approximately 0.5 VR, and a function of
acceleration, a, where average acceleration occurs at approximately 0.7 VR (sec equation
(96) or Figure 64). Using amid value, 0.6 VR, results in accurate values of time.
When using the second method, an incrementa! time, At, is determined from a modified form of
equation (98):
where At is the incrementa! time elapsed between velocities Vx and Vy.
3. 8 2
D6-1420
May 1989
PERFORMANCE
dt = ~(V-"v,
>t]
or,
1
At= a [(Vy - VW) - (Vx - VW)]
1
At= a[Vy -Vx]
at= AV
(99)
a
As indicated previously, the time data may be included with the data already shown in Figure 63.
When using the last method, equation (98) is modified as follows;
VR
1 J d(V-Vw)
at=
av
w
where V is in fps, and t is in seconds.
Therefore,
(VR -Vw)
.........---------~~~~~
t=~~~~~~
g
W [(T-µ W)-( C0 -µCL) Sq-Wc!>]
(100)
.6VR
For V in knots:
t = 1.6878 [equation 100]
(101)
Takeoff Speed Definitions
Before discussing takeoff further, let us define the various speeds involved. The definitions of these
speeds come under two sets of FAR rule requirements: ( 1) Those valid prior to March l, 1978,
D6-1420
Nov 1988
3. 8 3
PERFORMANCE
and (2) those valid as of March 1, 1978. Engine failure speed, VEF, is defined, and small rule
changes were made to the engine failure recognition speed and minimum control speeds. All
takeoff speeds arc reference.d to calibrated airspeed
(1)
Speed requirements prior to March 1, 1978:
- Critical engine failure speed. V 1 is the critical engine failure speed It shall not be
less than the minimum speed at which controllability by aerodynamic controls alone
is demonstrated during the takeoff run to be adequate to pennit proceeding safely
with the takeoff using average piloting skill, when the critical engine is suddenly
made inoperative.
FAR Part 25.107(a)
- Minimum control speed. V MC is the minimum control speed with the critical
engine inoperative. When the critical engine is suddenly made inoperative at this
speed, it shall be possible to recover control of the airplane with the engine stili
inoperative, and maintain it in straight flight at that speed, either with zero yaw or
with an angle of bank not in excess of 5°. Rudder forces may not exceed 180 lbs.
FAR Part25.149
- Takeoff rotation speed. VR is the speed at which rotation is initiated to attain the
takeoff safety or climbout speed, V 2, at 35 feet above the tak:eoff smface. VR must
not be less than 1.05 times the minimum control speed (air), nor less than V l ·
FAR Part 25.107(e)
- Minimum unstick speed. V MU shall be the speed at which the airplane can be
made to lift off the ground and to continue the takeoff without displaying any
haz.ardous characteristics. V MU speeds shall be selected by the applicant for the all
engines operating and the one engine inoperative condition.
FAR Part 25.107(d)
- Airplane lift off speed. The lift off speed is closely associated to the VR speed and
is dictated by that speed The all engines operating lift off speed must not be less
than 110% of VMU asswning maximum practicable.rotation rate. The one engine
inoperative lift off speed must not be less than 105% of V MU· .
FAR Parts 25.107(f) and 25.107(e)(l)(iv)
Lift off speed is detetmined as a function of thrust to weight ratio (T/W) from flight test data to
satisfy the certification rules. Rotation is started at such a speed as to give lift off at the desired
speed.
If the airplane incorporates an attitude waming system (AWS) and is geometry limited, the V LOF
requirements for all engine opera.tion may be reduced to 108% ofVMU withoutjeopardizing takeoff
performance. The airplane is geometry limited when the all engine takeoff distance to the 35 foot
height resulting from a V mu takeoff is not greater than 105% of the nonnal takeoff distance.
- Takeoff safety speed. V 2 is equal to the actual speed at the 35 foot height as
demonstrated in tlight and must be equal to or greater than 120% of the stall speed
in the takeoff configmation or 110% of the air minimum control speed (in the air).
FAR Part 25.107(b)(c)
3. 84
D6-1420
May1989
PERFORMANCE
(2)
Speed requirements as ofMarch l, 1978: (Change 7 of the rules).
VEF
- Critical engine failure speed. V EF is the speed at which the critical engine is
assumed to fail. It shall not be less than the VMco determined under FAR
25.149(e)
FAR Part 25.107(a)(l)
- Takeoff decision speed. It cannot be less than VEF plus the speed gained with the
critical engine inoperative during the time interval between the instant the critical
engine has failed and the instant the pilot has recognized and reacted to the engine
failure by application. of the first retarding means.
FAR Part 25.107(a)(2)
- Ground minimum control speed, is the minimum control speed. on the ground, at
which, when the critical engine suddenly becomes inopera.tive, it is possible to
recover control of the airplane with the use of primary aerodynamic controls alone
(without the use of nose wheel steering) to enable the takeoff to be safely continued
using normal piloting skill and rudder control forces not exceeding 150 pounds.
FAR Part 25.149(e)
V l(MCG) - The minimum takeoff decision speed. at which, when the critical engine suddenly
becomes inoperative at VEF with the remaining engines at takeoff thrust, it is
possible to control the airplane with primary aerodynamic controls alone and
continue the takeoff. This is the V 1 speed which results when VEF is set equal to
VMCQ·
- Air minimum control speed, is tłie airspeed. at which, when the critical engine is
suddenly made inoperative, it is possible to recover control of the airplane with that
engine stili inoperative, and maintain straight flight either with zero yaw or with an
angle of bank of not more than 5 degrees. VMCA may not exceed 1.2 V s (Vs
determined at the maximum sea level takeoff weight) with maximum available
takeoff thrust. Rudder forces required to maintain control may not exceed 150
pounds.
FAR.Part 25.149(b)(c)(d)
VR
- Same as above.
FAR Part 25.107(e)
VMU
- Same as above.
FAR Part 25.107(d)
VLOF
- Same as above.
FAR Parts 25.107(f) and 25.107(e)(i)(iv)
V2
- Same as above
FAR Part 25.107(b)(c)
D6-1420
May 1989
3. 85
PERFORMANCE
Under the SR-422B and the FAR Pan 25 rules, the emphasis is on rotation speed. The criteria for
establishing rotation speed are functions of severa.l variables. The rotation speed, VR, shall not be
less than a speed which will result in a lift off speed not less than 110% of VMU in the all engines
operating condition with the airplane rotated at its maximum practical rate. Also, as mentioned
previously, if the aiiplane is geometry limited and has an operable attitude waming system (AWS),
the VLoFrequirements for all engine operation may be reduced to 108% ofVMU. Vwp shall not
be less than 105% VMU in the one engine inoperative condition when the aiiplane is rotated at its
normal rate.
Another limiting condition is that V2 must be at least 120% Vs at a height of 35 feet above the
ground. Other limitations such as body contact may be limiting. Assuming VR as defined by VMU
is limiting, rotation speed as a function of T/W ratio will appear as in Figure 65. Even though
Figure 65 shows two separate curves of VR for the all engine operating condition and the one
engine inoperative condition, the VR's are the samę as required by the regulations. The apparent
discrepancy shown is due to the two separate values of thrust to weight ratio.
For the 767-300 which was certified to FAA Advisory Circular 25-7, the minimum value of the
. takeoff climb speed, V2, is 1.13 times the 1 g stall speed, Vs1g-
* 108% VMU for geometry
limited airplanes
Ba.sed OD Donnal rotation
. ~ rate and used f<r distance
/
-
-
-
.. •
calculations
l',-
~F (ĄJJCJl •
- -·
lJoi ~:- -
~e.s
y.Ałt, * -
/
--- ro"';.ti~ -
~g)
-
L Based
OD rapid
rate and 110% V MU *'
~ (.
--------------- .-1.//eii&loe.r
.iQI) ·SQI):
c> c> I
.5
>I
oo·.:
ti f! I
&j f! I
o
~J
§.1
=
< 8.'
Ol
I
[:
c•I
c5 .$ I
~:
T/W@ Lift Off
Figure.65.
3. 8 6
D6-1420
Mayl989
PERFORMANCE
Rotation and Flare to 35-Foot Height
The total takeoff distance to a 35-foot height is the sum of the ground-run distance previously
calculated to VR plus the ground distance to VLQp and the distance to flare from lift off to a 35 foot
height. The flare distance is determ.ined from flight test by establishing the time to flare and
multiplying by the average speed dming the flare. For a given T/W ratio, the average speed may be
obtained from Figure 65.
Time from rotation and time from lift off to 35 feet show good generalization with T/W, as shown
in Figure 66. Since the times are influenced by "pilot technique" the data are averaged by faińng the
test points. Figure 66 is required to compute the distance from VR to VLoF (Segment B2) and
VLOF to V2 (Segment C). Figure 67 illustrates takeoff distance for all engines operating and one
engine inoperative.
·
where,
Segment B = 1.6878 (VR + VLOF 2
2
At= t
and,
(VR toV 2 )
Segment C= 1.6878 (
- t
VW)&
(102)
(VLOFtoV 2)
VLOF + V2
2
)
-VW &(VLOFtoV )
(103)
2
where the velocity is in knots.
r
time f'rom
T/W
at
VR to
V2
·
lif't-of'f'
Figure 66.
D6-1420
May1989
3. 8 7
PERFORMANCE
All engine takeoff distance
V =O-
VR
VIDF
A
v
01-----------Clo~-----e~
I-
Segment A'
I• "I•
B'
C'
One engine inoperative takeoff distance
Figure67.
One Engine Inoperative Takeoff
A takeoff calculated according to Civil Air Regulations (CAR), SR-422, provided for a single field
length that satisfied both the takeoff and accelerate stop requirements. The minimum field length
under these requirements was a balanced field length. An engine failure was assumed to occur at
a speed such that the distance to continue takeoff and climb to a stipulated height was equal to the
distance required to stop. SR-422B and FAR part 25 stipulates using a field length which is the
greatest of either the accelerate and go distance, the accelerate and stop distance, or 115% times the
all engine operating distance to a 35 feet. height. The stop and go portions need not be balanced.
3. 8 8
D6-1420
Mayl989
PERFORMANCE
E----.i
. . - - - - - Segment A - - - - - -...........---11-- B _....,.__ C
Takeoff Safe S
Lift off S
OneEngine
inoperative
acceleration
Takeoff Decision S
Va
or Vsp
....
.c
bi)
C
·;=
-
~
~
e
f:?
C:
-g
Il,.
·-~
....
o
C:
U)
Q)
.5
~
-g
o
C:
~
e
!-a
- e
~
....tfi e= ·-....as
< o li
-
i:=
o
~
j
·-"O
o
Il,.
"O
o
(.)
C
as
-a
~
Max·
effort
stop
Distance - Feet
* One engine inoperative v2 and VLOF speeds will be sligh~y lower than the all engines
operating speed at the screen height and Vwp speed for the same VR.
Figure 68.
Takeoff calculations are generally perform.ed in discrete segments which define the distinct parts of
the takeoff as shown in Figure 68. Each segment for the one engine inoperative condition is
defined and its length calculated as follows:
SegmentA
Segment A is the all engine acceleration distance to VEF plus the one engine inoperative acceleration
distance, A 1, from VEF to V 1• See Figures 67. and 68. Segment A is the distance where the
takeoff can either be aborted or continued. The all engine distance to VEF is calculated using
equations (91), (94), and (95), and the determination of segment At is given below.
D6-1420
May1989
3. 8 9
PERFORMANCE
SegmentA1
Segment A 1 is the one engine inoperative acceleration distance from the engine failure speed, VEF to
V 1. Segment A 1 allows for engine failure recognition and initial brake application. An appropriate
time delay period is included to allow the pilot to detect engine failure. Also, inoperative engine
spin down is considered. This defines 11 which is a ratio of the average tbrust for this segment
based on the all engine thmst at V 1. Since the velocity change between VEF and V 1 is,
At a
AV= 1.6878
and,
where the average acceleration is,
and,
then the distance for segment A 1 is,
Simplification, with little loss in accuracy, is obtained by finding a at V 1 speed. Therefore, the
equation for segment A 1 becomes:
At g [- ( T )
s . av~
]
11 W vl - µ - (CD - µCL) W 295 .375 - q,
VEF = vl - 1.6878
= 1.6878( VI -
SA
1
a; - )at
VW
(l04)
(105)
SegmentB
Segment B is the one engine inoperative acceleration distance from V 1 to VLOF· Segment B is the
sum of segment B 1, equation (88) integrated between V 1 and VR, and segment B2 which is
obtained from equation (102). (See Figure 67). For this segment, all but one engine are
performing at the takeoff tbrust rating. The tbrust for the engine inoperative conditio~for segment
B 1 is found by multiplying the all engine thrust at V 1 by the average spindown factor, 11 avg .•
3. 90
D6-1420
May1989
PERFORMANCE
11 avg is a function of the time from V EF to VR. Since the engine windmilling drag and the
aerodynamic drag due to controlling the asymmetric thrust must be added to the basie airplane drag,
this defines an engine-inoperative value of (C0 - µ Cr) to be used in equation (91) for determining
the average acceleration in segment B1.
SegmentC
Segment Cis the flare distance between lift-off and a 35-foot. height with one engine inoperative.
The equation used is the same as for the all-engine flare to a 35-foot height, equation (103). This
distance is independent of V 1 speed.
SegmentD
Segment D, sometimes known as the transition distance, is the distance traveled from V 1 to the
establishment of the full braking configuration. The procedura! steps for anti-skid operating are: 1)
engine failure recognition; 2) brake application; 3) thrust levers to idle position; and 4) speed bra.ke
actuation. During certification tests, time increments for these steps are determined:
a.
Time between engine failure recognition and bra.ke application, tBA-
b.
Brake application to thrustreduction on remaining engines, tEc - tBA-
c.
Thrust reduction on the remaining engines to speed brake action, tsP - tEC.
For certi:fication, a one-second time delay is added to b and to c to account for differences in pilot
reaction. (See Figure 69). Figure 69 applies to the 707, 727, and 737-100/200 airplanes.
D6-1420
May 1989
3. 91
PERFORMANCE
~----tspF.T.----~
•I
SPOILER
ACTUATION
F.T.
1·
VEF
AFM
V1
tspF.M.
BRAKE
APPLICATION
THROTTLE
CHOP
SPOILER$
DEPLOYED
·1
1.0SEC*
SEG,A1
1BA
(tEc-teA)+1.0 SEC
(tsp-tEc)+1.o SEC
D1
D2
D3
707 AIRPLANES .
AND 737-100/200
AIRPLANES
AFMµ VALUES
727 AIRPLANES
AFMµ VALUES
* 3 secs was used on 727-100 airplanes without an engine failure waming system. All 727200 airplanes have the engine failure warning system thereby allowing a one second time
delay. For 707 airplane certification, there is no one second consideration; therefore, V 1 is
the engine failure and recognition speed
Figure 69.
3. 9 2
D6-1420
May1989
PERFORMANCE
During transiti.on speed increases above V 1, due to accelerati.on during the recogniti.on time, then
reduces during brake application. In the early commercial certifications, Segment D was calculated
in the time increments as indicated above and the exact horizontal distance determined using
equation (93) for each increment. This also enabled the initi.al speed for Segment E to be
determined.
Another method is to ca.lculate segment D in one step where the total transition time is utilized. This
method is similar to the method used for calculati.ng segment Al· Since the velocity change between
V1 and VB (the initial braking velocity for segment E can be approximated by equation (99).
Ata
AV= 1.6878
and,
then the distance for segment D is,
SD= l.6878(Vavg - VW )At
SD= 1.6878(V1 + Ai - VW )At
(106)
With little loss in accuracy, i is determined at the V 1 speed
where,
and µavg is the average transiti.on brake coefficient from engine failure to spoiler (speed brake)
actuati.on. To account for the additi.onal transiti.on time required for certificati.on, the average
transition brake coefficient for 707-300B/-300B(adv) and 737- 100/2.00 airplane is:
(tBA +tsp)
µavg = ~ +
F.T
2 tsp
· (~ - ~)
(107)
F.M
where,
µB
=
braking coefficient of fricti.on from anti. skid limited region of µ
versus W
µR
=
tBA =
D6-1420
May1989
"i plot.
g
rolling coefficient of fri.ction.
flight test time between engine failure recognition and brake
applicati.on, seconds.
3. 9.3
PERFORMANCE
=
tgp
tgp
F.M.
flight test time between engine failure recognition and spoiler action
seconds.
= Flight Manuał time between engine failure recognition and spoiler action
for certification , seconds.
The initial braking velocity is,
(108)
BRAKE APPLICATION
AND THROTTLE CHOP
-~SPOILER
,ACTUATION
1-----+-------1
FTI• ·1..
. .
t BA
,..
~
tsp • tBA
tsp
.. ,
V 1 + BRAKE APPLICATION
S
----+----------+I._L_
___c_H_OP_ _ _ _ _ ___,,r
VEF
AFM t-l
~THROTTLE
!./SPOILER
ACTUATION
I.. ,.o SEC„ I• t BA+ 1 SEC
„ I.. (t sp-t BA ) +, SEC
• I
t-1••s-E-G-,A--t11.,....,.,..._ _ _ _ _ _ t SPF.M."' t SP+ 2 SECS -------.,-il
µBRAKE
tsp
Figure 70.
The cmrent 747-100/-200/-300 certified refused takeofftransition sequence is shown in Figure 70.,
although future -200 / -300 certification may use the sequencing in Figure 71. The At's were
obtained from the engine failure speed during flight tesL t:J3A was the time increment obtained from
engine failure to brake application plus thrust reduction (throttle chop), and tsp was the time
3. 94
D6-1420
May1989
PERFORMANCE
increment obtained from engine f ailure to spoiler actuation. For certification, 1 second is added to
each reaction time to account for in service delays in pilot reaction. ts:pp.M. is defined here as the
time from V 1 to spoiler actuation to be used in the approved flight manuał (AFM) calculations.
The average ttansition brake coefficient, ~vg• for the 747 airplane is obtained from equation (109):
(tsP - tBA)
~vg = J.lił +
F.T.
( ~ - J.lił)
2 tgp
(109)
F.M
Some additional changes have been made for the certification of the 757, 767, 737-300 and 747400 airplanes. (See Figure 71). The brak.e application is conducted at the same time that the engine
failure is recognized (similar to the 747). From flight test, & 1 is determined as the time increment
from brak.e application to thrust reduction (throttle chop), and At2 is the time increment from thrust
reduction to spoiler deployment. The distance from V 1 to spoilers deployed is defined as segment
D. For certification, the 2 second time delay is now used to calculate distance based on the speed at
full bra.king for 2 seconds with no deceleration credit. This distance added to the Segment D
distance results in the total transition distance. For 767 and 737-300 airplanes, ~vg for segment D
is the average of µR and µB. For 757 airplanes, µavg was obtained by a special flight test
procedure.
·
V 1 + BRAKE APPLICATION
V EF
I
F.T. , ..
S 1-
At
.. , at~
THROTTLE
CHOP
~ ~ SPOILER
ł"'"'"
. DEPLOYMENT
At2 .. ,
RECOGNITION
V 1 + BRAKE APPLICATION
V EF
THROTTLE
(FULL BRAKING}
1-----~---łt--c_H_o_P_---IC" ___S!~'l:!='3. ~E~LS,!E~ +-1----B=RA..;;..;;.;K;,;,;,.IN;;.;;G____.....~
AFM
1-
1 SEC
SEG. A~
IAt~, ... At2
*2SEC
-----:------~1--µ-B-RA~K=E-----••-
*The 2 second arbitrary time delay is used to calculate distance based on speed at full
braking for 2 seconds with no deceleration credit.
Figure 71.
D6-1420
May1989
3. 95
PERFORMANCE
When calculating the transition distance for anti.-skid inoperati.ve, the procedura! steps are: 1) engine
failure recognition; 2) thrust levers to iclle position; 3) speed brake actuation; and 4) brake
application The transition bra.king coefficient of fricti.on will be the rolling coefficient of fricti.on,
µR.
SegmentE
A finał requirement in handling problems involving takeoff is to calculate stopping distance. The
theory used to set up the takeoff ground run calculations may also be used in the treatment of
stopping distances. Under stopping conditions the airplane is decelerating and equation (92) may
be rewritten to:
(110)
where V B is initial braking velocity calculated from equation (108).
When stopping, the terms in equati.on (110) differ from those in equation (92) in that thrust is
reduced to idle, drag is increased by auxiliary drag devices such as spoilers, and lift is reduced by
extending the spoilers. The coefficient of fricti.on is a ftmcti.on of the energy parameter W
"i .,or
g
W minus L, where W is the gross weight of the airplane at V Bg• the ground speed at the start of
braking. In equation (110), the accelerati.on term to be integrated becomes negati.ve, indicating a
decelerati.on. However, since the limits of integrati.on gofrom a large number to a·small number,
the algebraic sign for the stopping distance becomes positi.ve. Segment E is the braking distance for
a refused takeoff and is calculated using equation (110), integrating between the limits V 8 and Vw.
The !atest method for determining the braking coefficient of fricti.on is from equati.on (111), where
FB, the braking force, is a functi.on of (W - I), the average weight on the wheels during·the stop,
and W~ . See Figure 72.
g
(111)
3. 9 6
D6-1420
MayI989
PERFORMANCE
I
I
I
I
I
I
I
I
I
. I
I
-
w v.2
W-L
Bg
Figure 72.
The lower value of Fs is used to determine µB • L is the average lift between Vsg and V w This
method is used on the 757, 767, 737-300 and later airplanes.
·
Balanced Field Length
To define a balanced field length, the engine failure or recognition speed, V 1, must be determined
such that segments
B+C = D+E
The method used is to assume several V 1 values and solve for segments B, D and Eto correspond
Segment C, independent of V 1, also is calculated. By comparing B + C and D + E, V 1 for a
balanced field length may be obtained This relationship may be plotted as in Figure 73 or
numerically compared. Segment A is then calculated and the balanced field length defined. Another
method of determining V 1 for balanced field length is to assume an initial V 1 and iterate. Balanced
field length is the ideał situation where the maximum takeoff weight is possible for a given runway
length.
L,U,,._-v1 for balanced
field length
v,
DIS'l!ANCE
Figure 73.
D6~1420
May 1989
3. 97
PERFORMANCE
Unbalanced Field Lengtb
There arc cases where some of the takeoff speed requirements arc not being met when balanced
field length is being considered. Adjusting the speeds involved results in unbalanced field lengths
with a reduction in takeoff weight for the given runway length or an increase in field length
requirements for a given weighL Unbalanced field length may also result from runway slope and
wind corrections, and the use of clearways and stopways. This will be discussed later.
At low weights and altitudes, and consequently low engine failme speeds, it is possible to calculate
a V 1 for a balanced field length which is less than VMCQ· This means that V 1 must be increased to
equal VMCG· This will result in an unbal_anced field length, since segment B + C is less than
segment D + E. In this event, the field length will be identified by segments A + D + E with V 1
defined by VMCQ·
Later models such as the 757, 767 and 737-300, arc required to retain the engine failure speed,
VEF, above VMCQ· This means that V 1 must be increased such that VEF equals VMCQ·
Where V 1 or VEF is set equal to VMCQ it is possible that the calculated VR is less than VMCo· VR
must be increased to V 1. This increases the accelerate-go distance (segments A + B + C) and, along
with the increased V 1 to match V Meo, results in both segments A + B + C and A + D + E being
longer than the balanced field length. In this situation, the longer of the two determines the FAR
takeoff field length and both V 1 and VR arc equal to VMCQ, (or VEF is set equal to VMCQ and VR is
equal to or greater than V 1).
It is also possible that the calculated V 2 speed could occur at less than 110% of the air minimum
control speed <VMCA) at low weights and altitudes. In this event, V 2 must be increased to 110% of
VMCA and the unbalanced field length is then defined by segments A + B + C.
Another possibility for unbalanced field lengths is the amount of beat energy the airplane brakes will
absorb and still allow safe evacuation of the airplane. This is known as a brake energy limit
<VMBE) and is most likely to occm at high gross weights at high altitudes. If the V 1 speed for a
balanced field length is above the VMBE• V 1 must be reduced. In.this case, the FAR field length
under one-engine-inoperative requirements is defined by segments A + B + C.
A fina1 possibility for unbalanced field lengths occurs when the calculated V 1 speed for a balanced
field length·is greater than VR. This occurs for airplanes having good stopping capabilities. This
means that the airplane could experience an engine failme during takeoff rotation at a speed equal to
or greater than the VR speed and still be stopped on the remaining runway. However, damage to
the aircraft may occur when the nose of the airplane is lowered to the runway for stopping;
therefore, the policy is to reduce the V 1 speed to equal VR· For this situation, the FAR field length
under one-engine-inoperative requirements is defined by segments A + B + C.
3. 9 8
D6-1420
May1989
PERFORMANCE
A snmmary of the takeoff speed adjustments resulting in unbalanced field length is given below:
WEIGHT
BALANCED FIELD LENGTH UNBALANCED FIELD LENGTH
CONDITION (T.O. SPEED SITUATION) (T .O. SPEED ADJUSTMENT)
IDW
IDW
V1 < VMCQ
V1 increase,
V1 = VMCQ *
VEF < VMCQ
VEF increase,
VEF =YMCO*
(737-300, 757, 767)
IDW
VR < VMCQ
VR increase,
VR = VMCQ = V1
IDW
V2 < 110% YMCA
V2 increase,
V2 = 110% YMCA
HFAVY
V1 > VMBE
V1 decrease,
V1 = VMBE
HFAVY
V1 > VR
V1 decrease,
V1 = VR
< = "is less than"
> = "is greater than"
*lf VR < the resulting V1, increase VR to V1
Clearways and Stopways (SR-422B and FAR Part 25)
Clearway - A clearway is an area beyond the runway, not less than 500 feet wide, centrally located
about the extended center line of the runway and under the control of the airpon authorities. The
clearway is expressed in terms of a clearway piane, extending :&om the end of the runway with an
upward slope not exceeding 1.25 percent, above which no object nor any portion of the terrain
protrudes. Threshold lights may protrude above the plane if their height above the end of the
runway is not greater than 26 inches, and if they are located to eaćh side of the runway. When
clearways are being considered, a reduction in V 1 speed is required.
FAR Part 1.1
Stopway - A stopway is an area beyond the runway, no less wide than the runway and centrally
located about the extended center line of the runway. A stopway must be able to support the airplane
during an aborted takeoff without causing structural damage to the airplane. Further, stopways must
be designated by the airport authorities for use in decelerating the airplane during the aborted
takeoff.
FAR Part 1.1
Clearways and stopways allow higher takeoff weights for a given runway length They also affect
the value of the FAR field lengths as given below.
·
D6-1420
Mayl989
3. 99
PERFORMANCE
FAR Field Length (SR 422B and FAR Part 25)
The preceding discussion has assumed an engine f ailure during takeoff. The regulations also
require the takeoff be considered without an engine failure.
FAR Part 25.113(a) defines "takeoff distance" as the greater of either the di.stance to a height of 35
feet with an engine failure or 115% of the all-engine distance to a height of 35 feeL
When the takeoff utilizes clearway, it is necessary to define a term called "takeoff run". The takeoff
run is defined by FAR Part 25.113(b) as the greater of either the di.stance to a point half way
between liftoff and the point where an altitude of 35 feet is attained with engine failure or 115% of
the all-engine di.stance to a point half way between liftoff and the point where an altitude of :35 feet
is attained.
FAR Part 121.189 relates the takeoff di.stance and takeoff run to the runway and clearway lengths
by stating the takeoff run may not exceed the runway length available; nor may the takeoff di.stance
exceed the runway length plus the clearway length. At the takeoff limiting weight, no more than
. one half of the di.stance from liftoff to an altitude of 35 feet with an engine failure , or 115% of that
di.stance without an engine failure, may be beyond the end of the runway. Relating the segments in
compliance to the regulations at the takeoff limiting weight, we have for:
(1)
One-engine-inoperative with clearway and stopway:
(a)
The takeoff distance (A+B+C) shall· not exceed the length of the runway plus
the length of the clearway. The takeoff distance shall not include a clearway distance
greater than one half of segment C or one half the length of the runway plus the
length of the stopway.
(b)
The accelerate-stop di.stance (A + D + E) shall not exceed the length of the
runway plus the length of the stopway.
(c)
. (2)
The takeoff run (A+ B + C/2) shall not exceed the length of the runway.
All-engine with clearway:
(a)
The takeoff di.stance 1.15 (A'+ B'2 + C') shall not exceed the length of the
runway plus 115% of C'/2.
(b)
The takeoff run 1.15 (A'+ B'2+ C'/2) shall not exceed the length of the
runway.
These definitions are illustrated in Figures 74 and 75.
3.100
D6-1420
May 1989
PERFORMANCE
ONE-ENGINE-INOPERATIVE
sr-c
ACCELERATE-GO DISTANCE
•I
A
( BALANCED FIELD
LENGTH ASSUMED )
. Vi.oF. / . STOP I 35FT
v1.
NOCLEARWAY
NOSTOPWAY *
~
V2
Ve
ot--E~
TAKEOFF DISTANCE
RUNWAY LENGTH
ACCELERATE-STOP DISTANCE
=t
sr-c~v
ACCELERATE-GO DISTANCE
-
A
.. ,
2
V1
_j Ve
WITH STOPWAY
ONLY •
I
35FT
I
STOPWAY
E
ol-
---
sTOP
VLoJ
::::I
TAKEOFF DISTANCE
RUNWAY LENGTH
ACCELERATE-STOP DISTANCE
-
TAKEOFFDISTANCE
~ - - - - ACCELERATE-GO DISTANCE
§a
~-----A
•2
•
I• s--#21 v
V1
I
WITH CLEARWAY
ONLY *
_ I
2
VLoF"_/
I
•
Ve
~ D ~E
I35FT
STOP
~-----1~
CLEARWAY
.,r MAXC
USEABLE
LEARWAY
. . _ i - - - - - - TAKEOFF RUN
Ó
RUNWAY LENGTH ----~ł4':2~.-91"1
..._---ACCELERATE-STOP DISTANCE-----.
1
...i------TAKEOFFDISTANCE - - - - - - . _ .
• ~ - - - - ACCELERATE-GO DISTANCE
..,_i-----A
. .,, , •
8--t.....~.....~~
V2
V1
WITH CLEARWAY
ANDSTOPWAY •
I
~D
STOP
Ve
1~-~~E
STOPWAY
„I
RUNWAY LENGTH ---~~CLEARWAY
._,_ _ _ ACCELERATE-STOP DISTANCE
.., I
TAKEOFF RUN
* AT FIELD-LENGTH-LIMIT WEIGHT
.., I •~ I
L.
MAX USEABLE
CLEARWAY
Figure 74.
D6-1420
Mayl989
3.1 O1
PERFORMANCE
ALL ENGINES OPERATIVE
NOCLEARWAV
,~-----i----••--tl e~ J-.--c•
+15%
I
VR VLOF
1
35FT
1
~----- TAKEOFF DISTANCE ... 1.15 (A + B~+ C )---.-.1
l·wITH CLEARWAV I
~-----A ----,._;-•.,--tł B~ł---c' ~ + 1 5 %
1
1--------------.. F-=----------.I
v_R.,.v.Lo
...
I
I
,...i
~----TAKEOFF RUN a 1.15 {A+ B2 +
I
35 FT
-,
"f ) - - -..--i1i....-a--CLEARWAV ~
TAKEOFF otsTANCE • 1.1s (A+ e~ + c')
:
. . _ , _ _ _ _ _ RUNWAY LENGTH
•
14, - ·
MAX USEABLE CLEARWAY • 1.15 (t)J
• AT FIELD-LENGTH-LIMIT WEIGHT ·
Figure 75.
The following table summ.ari7.es the minimum runway length requirements, referred to as FAR
Field Length, with and without a clearway.
ENGINE CONDffiON
FAR FIELD LENGTH
WTIHOUT CLEARWAY
wmICLEARWAY
All Engine
1.15 (A'+ 821 + C?
1.15 (A'+ B2•+t)
One Engine
A+B+C
lnoperative
or A+ D+ E*
I
A+B+f
or A+ D+ E*
*Add stopway to runway if available.
3.102
D6-1420
May 1989
PERFORMANCE
These requirements determine the minimum required runway length, clearway, and stopway. The
greater of the three requirements determines the FAR field length.
Takeoff Presentation
A form of takeoff presentation depicting field length as a function of weight is shown in Figure 76.
Such a plot shows the effects of altitude and various flap deflections for a standard day temperature
based upon pre1iminary data. This presentation is not used as operational data and only serves to
provide estimated performance pńor to development of actual performance from flight test data.
Current regulations require a constant clearance height of 35 feet for all temperatures. However,
certain foreign regulatory agencies may permit other heights under specified conditions, such as
contaminated runways.
In establishing Figure 76, the optimum flap deflecti,ons arc selected. For a given altitude and flap
setting, a relationship may be shown between field length and gross weight. A different flap setting
pmuces different field lengths. On a single chart, it is possible to show for a given altitude and
OAT the field lengths associated with the various flap configurations throughout the weight range of
the airplane. Figure 77 shows that the field length requirements arc least at the 300 flap setting for
all weights up to W 1• Beyond W 1, a 200 flap setting produces lower FAR field lengths.
I
Standard Day
I
Minimum Climb
Paformance
Limits
F.A.R.
Takeoff
Field
Length
EAS
(Kts)
Gross Weight at Stan ofTakeoff
Figure76.
D6-1420
Mayl989
3.103
PERFORMANCE
However, it is possible that either the second segment or finał segment climb gradient requirements
(discussed in the Climb chapter) Ił:UlY establish the weight above which a lower flap setting is
required. For example, in Figure 78, if W 2 (the second segment or finał climb weight limit) is
greater than W 1 (determined by field length), the minimum field length determines the flap angle.
If, however, the second or finał segment weight limit occurs at a weight lower than W 1, the climb
weight limit determines the flap setting.
FIELD
LENGTH
I
.
GGIVENIVEN ALTrruDE
OAT
I
GROSS WEIGHT W1
Figure 77.
If the FAR field length for 300 flaps and for 20° flap at W 2 is less than the actual field length, it is
possible to increase the takeoff gross weight by using 20° flaps. Figure 78 shows the climb weight
limit for a 200 flap setting, W 3, is higher than the climb weight limit for a 300 flap setting, W 2. In
other words, with a 200 flap configmation, the same climb-out performance can be. obtained at a
higher gross weight with a small increase in field length.
·
FIELD
LENGTH
GROSS WEIGHT
Figure 78.
3.104
w,
D6-1420
Mayl989
PERFORMANCE
At low weights, ground and air minimum control Speeds may define the field length. This is one of
the unbalanced field length conditions mentioned previously. The conditions ofthis effect is shown
in Figure 79 for one altitude, temperature, wind and runway slope.
IGiven Altitudel
Corrected for 110 % VMCA
: ' - Corrected for VMCG
I
I
No Minimum Control Speed Correction
•
I
I
110%VMCA
I
I
X•
I
~co,:
Gross Weight at Start ofTakeoff
Figure 79.
Data in Figure 7 6 shows the generał takeoff performance characteristics for the airplane, and can be
used for preliminary analysis work. A more detailed presentation is made for flight operations and
flight crew use and is contained in the FAA Approved Flight Manuał One form of presentation is
as shown in Figure 80. Complete temperature, wind, and runway slope effects are shown for a
selection of pressure altitudes throughout the weight range of the airplane. The information is
arranged in chase around format to reduce the number of chans and to show the data in a form as
simple as possible.
D6-1420
May1989
3.105
PERFORMANCE
Under SR-422B and FAR Part 25 rules, where one-engine-inoperative takeoff distance must be
compared with the all-engines-operating takeoff distance, and where clearway and stopway benefits
may be considered, a different form of chart presentation has been made. The one-engineinoperative performance is shown in Figure 81. Accelerate-go distance including clearway, and
accelerate-stop distance including stopway, are compared to yield an effective one-engineinoperative distance and V 1 speed. Wind and runway slope effects are considered, and the adverse
stopping effects of anti-skid-inoperative takeoff refusals are shown. There is also a line for
defining the maximum allowable clearway. V 1N lB is read from this chart, where V lB is the V 1
speed for equivalent balanced field length conditions with no wind, no slope, and no airbleed. The
intercept of the accelerate-go and accelerate-stop trace lines must be transferred to a reference line to
obtain the "one-engine-inoperative corrected runway length" based upon balanced field from the
unbalanced field situation. Figure 81 is often called the one-engine-inoperative "web" chart. The
all-engines-operating performance chart, .Figure 82, accounting for clearway, wind and slope,
yields an "all-engine-corrected runway length". From these data it is possible to convert distance to
corresponding all-engines and engine-inoperative limiting weights, the lesser being used for
dispatch. However, to reduce the number of charts, only the all-engine corrected runway length
versus weight chart, Figure 83, is prepared. The one-engine-inoperative corrected runway length is
converted on Figure 81 to an equivalent all-engine corrected runway length usually called
"corrected runway length" . The lesser of the two all-engine corrected runway lengths defines the
limiting takeoff weight on Figme 83.
3.106
D6-1420
May 1989
PERFORMANCE
I
I
ol
~I
oj
...... } ~ .....
I
z Ir-ł
u
~t,,.
4t.;.
:i::
-~.
E-t -
t!Ja,
rz:i
ltS
•
..;i
•
•
•
•
•
•
•
FAR FIELD LENGTH
'
si:
I
I.c:
~,
rz:i
H
~
p::;
Q)
'O
'O
·ri
,:i: -
•
•
•
•
•
•
•
•
uhidd~cale)_~-FAR FIELD LENGTH
{ÓO WI~
SLOPE
• • • • • • • • • • • •
•
•
•
•
•
•
•
•
•
•
WEIGHT
Figure 80.
D6-1420
May 1989
3.107
-
.
V,
o
(used to obtain one-engine-inoperative weight ·
allowable from all-engines-operating weight chart)""'
00
Q)
m
"l1
r-ł
"'
u
111
Q)
i::
u u
~
~
Q)
~
111
.µ
'O
j
tr
.
00
o,
-
I
.
>,
I ,.,'li
~
Q)I
~
•
L::::::::,
clee.rwe.y (downh111) (uph111) t.w.
length ·
slope
wind
h.w.
8~
: (hidden scaleF
--t -d--t--di t-correc e acc-s op
s ance
e.nti-skid • 1nOJ>ere.tive
e.nt1-s;~
'\SS~S~
h.w.
OIVEN PLAP
BETTING
wind J
f~....
... "'0
'<
I
\Ot,,.>
00
\O
ONE-ENGINEINOPERATIVE CHART
J
/
}
/
: ,/
,,
>' 7'
;>
(~:~l ·1·I·i·B\ \ \ l\\
(downhill)
•\
\
• :ie.
-i::
'. e
o,
„
~u
-g
m 1-l
A ~
•
m >,
g,..,
A.
.....
1-11
1-1
i::
1-l~
i:: 'O
e
~ ł-1
.m
u
m
u ~~4o
QJ
:a. .,
A~
:
~
.J.J
i::
m
r-ł
~
.~m
..
• · ~ ~ · .• ..... • • • • •
.d
i.....
.o. I.c:
·r-ł
. ~
.
E
~
.c:
ł-1
.~,'O
'O 'O
~
~
m
~ .c:
~
i:: !Il
111 i::
.µ Q)
_\
accelerate-stop distance available
0
m
I
ł-1
.,
m
~
1-l
>t
111
!ł
i::
::l
1-1
'O
Q)
.µ
u
• 1-lo
Q)
&
m
o
u
> u
·.-1
1-1
1-1
"'C
~
~
o
~
~
(j
~
PERFORMANCE
dn
up
Cl)
a.
o
SLOPE
r-ł
Cl)
clearway
length
..c:
+J
Ol
C
WIND
Cl)
GIVEN FLAP
SETTING
r-ł
>t
ro
3:
C
::s
1-1
ALL-ENG INESOPERATING
"O
Cl)
+J
CHART
o
----------1
1-1
Cl)
1-1
o
o
Cl)
C
-~Ol
C
Cl)
I
r-ł
r-ł
ro
Figure 82.
D6-1420
May 1989
3.109
PERFORMANCE
GIVEN FI.AP
SETTmG
..
zzIl/ zzzzz
credit for: · AWS operati ve
IIII
I
--:J. I."i\·
al1owable takeoff weight
~
corrected all-engines-operating takeoff distance
(difference due to credit
for AWS operative)
Figure 83.
To obtain the increase in takeoff weight to be gained by applying credit for a geometry-limited
airplane with an operative attitude warning system(AWS), it is necessary to change the basie speed
requirements. A different set of takeoff and climb out chans and chart correction factors must be
applied. Figure 83, for instance, shows an increase in weight for the same corrected runway
length, however, the actual takeoff distances will be different. The sliding scale on Figure 83 is
used here only to illustrate the increase in takeoff weight for the same corrected runway length w hen
credit is taken for an operative AWS and geometry limited airplane.
Assume for the moment that Figure 80 is for all-engine conditions with clearway accountability. On
the upper half of the chart, change the FAR field length scale to "runway length available" and the
FAR field length for zero wind and zero slope scale (bidden scale) to "all-engine corrected runway
length". Rearranged slightly results in Figure 82, the all-engine corrected runway length chart. On
the lower half of Figure 80, again change the FAR field length for zero wind and zero slope scale to
"all-engine corrected runway length". The result is Figure 83, the limiting takeoff weight chart.
V 1 is obtained by using the V 1N lB ratio from the "web" chart and entering Figure 84. The value
obtained must be checked against the minimum V 1 (VMeo)· V R and V 2 arc presented in a form
similar to Figure 85. Minimum VR (1.05 VMcA) must be observed, and credit for an operative
attitude waming system is not available when VR is limited by minimum VR.
3.11 O
D6-1420
Mayl989
PERFORMANCE
GIVEN FI.AP
SETTmG
M
"""'al
Q)
rd
"""'
m
~
o
use
minimum v, (VMco> ~ higher
/
_ val.ue
engine failure speed, V
Figure 84.
The web porti.on of Figure 81 may also be drawn, as shown in Figure 86, which offers some
advantages in determining the maximum takeoff gross weight over that of Figure 81.
As previously stated, for any combination of unbalanced distances, slope and wind effect, there is a
reference or equivalent balanced field length which will support the same takeoff weight with zero
runway slope and zero wind. This reference distance was then converted to an equivalent "allengine corrected runway length" on Figure 81 so comparison could be made to the actual "allengine corrected runway length" from Figure 82; the lesser value being the most limiting. Guide
lines of equivalent all-engine corrected runway length, labeled as "corrected runway length", are
substituted for the equivalent balanced field length guide lines of Figure 81. Therefore, corrected
runway lengths for engine-inoperative conditions may be obtained by direct interpolation between
guide lines instead of sliding to the reference line as on Figure 81.
D6-1420
May1989
3.111
PERFORMANCE
Another clifference in this format is in showing the speed ratio in terms of V 1NR instead of V 1N IB.
This establishes a positive upper limit to V 1, and reduces the number of tak:eoff speed charts in the
flight manuał since V 1, VR and V2 can be combined on a single chart. This also eliminates having a
speed definition for V 1B.
GIVEN FI.AP
SETTmG
Cl)
M
:s
~
CIS
M
Cl)
~
Cl)
~
....
. --- .·
Figure 85.
3.112
D6-1420
May1989
PERFORMANCE
w
o
z
~
Cl)
c
ow
u.
u.
~
~
C
I!!
@
re
re
8
CORRECTED ACCELERATE-STOP DISTANCE
Figure 86.
Wind Corrections
The tower reported wind correction in Figures 80, 81, and 82 is explained as follows: Winds close
to the ground are subjected to considerable shear effects; therefore, for accurate calculations of
ground distances, tower-reported winds must be corrected to the wind values occuring at the
average airplane height. Experimental values of wind in earth's boundary layer seem to follow the
law:
1
7
Wmd 1 -( Height1 )
Wmd - Height
2
(112)
2
CAM 4b, Appendix A
Regulations require the tower-reported wind values must be taken at a height of either 50 feet or 10
meters (32.8 feet). The wind value at the selected average airplane height can be detennined using
equation (112). Letting heigh12 be the tower height and height1 be the average airplane height,
values of wind1/wind2 for the Boeing commercial airplanes can be calculated as shown in the table
below:
D6-1420
May 1989
3.113
PERFORMANCE
CAA
FAA
Airplane Height2
707
727
737
737-300
747
757
767
50feet
50feet
50feet
32.8 feet
50feet
32.8 feet
32.8 feet
Heightt
Windt
'Wind2
Height2
Heightt
Windt
'Wind2
lOfeet
lOfeet
8 feet
8 feet
16 feet
12.5 feet
13.5 feet
.795
.795
.770
.8174
.850
.8713
.881
lOmeters
lOmeters
lOmeters
lOmeters
lOmeters
lOmeters ·
lOmeters
12.5 ft
•10.0 ft
* 8.0 ft
10.5 ft
21.27 ft
15.5 ft
16.5 ft
.8713
.844
.8174
.8498
.940
.8985
.9065
* Chose to keep F AA values using simplified CAA certification rules.
The FAR also requires a conservative view of wind correction on calculations of takeoff and
landing data. For takeoff, FAR 25.105(d)(l) states that the takeoff data must include, within the
established operational limits of the airplane, not more than 50 percent of nominał wind components
along the takeoff path opposite to the direction of takeoff, and not less than 150 percent of nominał
wind components along the takeoff path in the direction of takeoff. For landing, the above rule
requirement is under FAR 25.125(e).
Multiple V1
Another feature of the web chart, Figure 86 or Figure 81, is that a complete range of V 1 speeds can
be determined as shown in Figure 87 if excess runway exists for the weight
A range of V 1 exists only for takeoffs which are not one-engine-inoperative field length limited.
This allows selection of a single operating value from this range of V 1 as desired on the basis of
runway conditions, and provides more operational tlexibility. Stopping performance for calculati:ng
field length requirements is based on maximum braking effort on a smooth, dry, hard suńaced
.runway, but without credit for the stopping margin provided by reverse thrust. lee, snow or water
can lengthen the stopping distance required and reduce the stopping margin provided by reverse
thrust When a range of V 1 exists, minimum V 1 will put the airplane in a "go" position earlier in the
takeoff run. At the same time, if an engine failme occurred at minimum V 1, more stopping distance
would be available than would be if a higher V 1 had been selected.
Another advantage of multiple V 1 is that V 1 may be selected to avoid the gross weight reduction
sometimes necessary in brake limited takeoffs, usually at high altitude airports. Und.er form.er rules
for detennining V 1, a weight reduction was required until the resulting V 1 did not exceed the
corresponding brake energy limit speed. If the brake energy limit speed lies within the allowable V 1
range, the brake energy limit speed is the maximum V 1, and off-loading is not necesary. Weight
reduction is not necessary unless the brake energy limit speed is lower than minimum V 1.
3.114
06-1420
May 1989
PERFORMANCE
w
zo
~
Cl)
c
LL
LL
CORRECTED RUNWAY LENGTH FOR
FIELD LENGTH LIMITED WEIGHT
@
~
~
C
w
bw
C:
C:
CORRECTED RUNWAY LENGTH FOR
LIMITED (USUALLY CLIMB) OR
LOWER DESIRED WEIGHT
8
CORRECTED ACCELERATE-STOP DISTANCE
Figme 87.
Additi.onal restricti.ons must be applied to the V 1 range. lf V MCQ is greater than minimum V 1,
VMCc, becomes the minimum V 1· If VMCG is greater than the maximum V 1, the corrected takeoff
distance must be reduced until the intersecti.on of corrected accelerate-stop distance and conected
takeoff distance results in a V 1NR ratio that gives a V 1 greater than or equal to VMCG· Only one V 1
will result in this case. If maximum V 1 exceeds the brake energy limit or VR, the lesser of brake
energy limit speed or VR must be used for maximum V 1•
Thrust Preseiltation
Takeoff thrust data in terms of EPR are presented for tak:eoff alti.tudes and for ambient temperatures
as shown in Figmes 88 and 89. The horizontal lines represent the maximum, or fiat rated, thrust
allowed under static conditions for commercial operation. This means the thrust levers may not be
advanced to the full forward position. Figures 88 and 89 are representative curves for engines
operating without and with water injection, respectively.
D6-1420
May 1989
3.115
PERFORMANCE
WITHOUT WA.TER INJECTION
AMBIENT TEMPERA.WRE
Figure 88.
pressure altitude
8ooo :ft
6000 ft
4000 :ft
2000 :ft
see. level
wet takeof'f' thrust limit
' WITH WATER INJECTION
I
AMBIENT TEMPERATURE
Figure 89.
3.116
D6-1420
May1989
PERFORMANCE
In addition to the above EPR charts, N 1 may also be used as an index for setting thrust. It is
handy for cross-referencing and setting preliminary takeoff thrust for engines with the pressure
instrumentation. Typical curves arc shown in Figure 90.
WITHOUT WATER INJECTION
AMBIENT TEMPERA.iroRE
Figure 90.
D6-1420
May1989
3.117
PERFORMANCE
Intentionally Left Blank
3.118
D6-1420
May1989
PERFORMANCE
3-6 Climb
General
In straight and level flight at a given velocity, thrust available is exactly equal to thrust required to
overcome drag. When thrust available is less than this, the airplane will lose speed and perhaps
altitude. When thrust available exceeds thrust required, the airplane will be able to either accelerate
in level flight or climb.
The two flight regimes in which climb is imponant are those associated with operation cios~ to the
ground where obstacle clearance is critical and enroute where the prane problem is the rate at which
cruise conditions are approached. Therefore, the climb peńormance of the airplane must be derived
in terms of the climb angle or clearance capability when takeoff climbout peńormance is discussed.
This climb angle presentation is usually given in terms of the tangent of the angle, defined as climb
gradient But for enroute performance, the derivation and use is in terms of rate of climb.
Rate of Climb
The airplane in Figure 91 is climbing due to an excess of thrust over drag. If the airplane is
accelerating and changing climb angle at the same time, all of the forces involved dming a climb can
be shown on the Figme.
ł(-ffl')v
\L
T
r/c = dh/dt
~
TAN 8 = CLIMB GRADIENT
o
= 'Y
W (dV)
~-at
Figure91
The weight vector may be resolved into components parallel and perpendicular to the flight path,
and the static equilibrium conditions may be applied in these directions.
T-D-Wsin8-
'i' cz
=O
W d8
L+ g dt V- W cos8 = O
D6-1420
May 1989
(113)
(114)
3.119
PERFORMANCE
Equation (113) may be written:
cJVJ
Sin 8 = ...1.. [T - D- W
w
g dt
T-D
1 dV
= w -i dt
(115)
The rate of climb, r/c, is equal to the vertical component of the flight velocity.
r/c= V sin 8
or,
(T - D) V
W
,l
r,c=
V dV
-g dt
(116)
dV/dt may be written as:
dV
dV dh
dt =dh dt
SO,
·"c- dh _ (T- D) V_ V dV dh
w
g dh dt
r.,,-dt-
dh (l V dV)- (T- D) V
dt
+gdhw
dh
(T;D)v
r/c= dt = . V dV
1+-g dh
(117)
!:)
The dimensionless term (
is known as the acceleration factor, and values will be determined
later. When the climb is being flown at a constant true airspeed, the acceleration factor equals zero
and equation (117) for unaccelerated r/c becomes:
r/c
unaa:elenued
=(T-w D) V
(118)
The units of rate of climb are the same as those of velocity; ft/sec. If the rate of climb is to be
expressed in ft/min and the velocity in knots, a conversion factor must be applied to equations (117)
and (118). Since the velocity in ft/min is (1.6878) (60) = 101.27 multi.plied by the velocity in
knots,
(T;D)vK
r/c = 101.27
.
3.120
V dV
1+-g dh
(119)
1)6.1420
May1989
PERFORMANCE
r/c unaa:elenred
r/c=
V dV
1+--g dh
where,
r/c is rate of dim:> in feet per minute.
V k is velocity in knots, and
. ~ns10
..:1:- . n1
gV dV
dh 1s
ess
In Section 1 it was shown that,
Ja:;,
101. 27 V k = 66,987 M Ja:;,
V k = 661479 M
therefore,
(120)
Substituting equation (120) into equation (119) and dividing both numerator and denominator by
the pressure ratio, &.unb. results in:
r/c=
66,987M~
V dV
1+-g dh
66,987M~
r/c=
V dV
1+-g dh
T
D
~-~
w
6 amb
(121)
r/c unaa:elerared
r/c=
V dV
1+--g dh
where,
AT=T-D
The division by &.unb is to put the force parameter in a form consistent with the presentation on the
speed thrust chart. Equation (121) states that to obtain rate of climb at a given Mach number, M, all
that must be known is the available T/&.unb and the airplane D/Oamb and W/Oamb· These quantities
are obtainable from the speed thrust chart, Figure 92.
D6-1420
May1989
3.121
PERFORMANCE
2(),000 Feet
seaLevel
M
Figure 92.
The derivation of·equation (121) accounts for acceleration along the flight path. Equation (114) is
. applicable for acceleration perpendicular to the flight path and may be written as:
W
d8
L-Wcos8=--Vg
dt
(122)
where V d8/dt is the centrifugal acceleration due to changing the flight path angle at a rate d8/dt
(see Chapter 3-4 in this section).
From experience, d8/dt is nearly zero at any point in the climb so equation (122) reduces to the
form:
L-W cos8= O
or,
(123)
L=W cos8
This equation states that in a climb, lift is less than the weight by the factor cos 8. Tuus at a
particular speed, drag is less than drag at the same speed in level flight. The reduction in lift is
greater for larger climb angles. For most conditions on commercialairplanes, the climb angles are
small (below 15 degrees), so cos 8 is close to unity. Tuus, lift and drag during climb are practically
identical to those existing under level flight conditions, which allows the level flight data for the
airplane to be used. Of course, with airplanes having very large climb angles (interceptors, etc), the
climb angle effect on drag must be taken into consideration.
3.122
D6-1420
May1989
PERFORMANCE
Maximum Rate of Climb
Maximum rate of climb for a given weight and altitude occurs at a speed where AT/oamb multiplied
by the Mach number is greatest, as scen in equation (121). This is at a speed slightly higher than
where AT/6amb is max. The maximum rate of climb true velocity increases with altitude; therefore,
the airplane must accelerate along the flight path to maintain the maximum rate of climb. H the
thrust rating cannot be increased, the rate of climb will be reduced to account for this acceleration,
as scen from equation (121).
It is possible to calculate rate of climb over a range of speeds which gives the maximum value at a
given altitude. The following method determines the speed for maximum rate of climb immediately.
In Figure 93, the best climb speed occurs at one half the speed at which the tangents to the Dl&.unb
and T/6amb curves ·intersect. This may be verified.
T
T
Samb
gamb
6amb
=I
I
.c
I
or
D
b
I
w
6amb
V or M
Figure 93.
From Figure 93,
D6-1420
May1989
3.123
PERFORMANCE
or,
(1~~)
[d(_!_
)-(__!2_
)]
_____
v_o = -( yl - 1)
6amb . 6amb
Yo
Yo
dV
(124)
V
o
From equation (118), it is known that:
r/c =
c(1~~)
(125)
Y
For the rate of climb to be a ~ u m , the_ ~lope d(r/c)/dY J!lUSt equal zero as seen by Figme 94.
sJ.ope
'
d(r/c) - o
dV
-
I=
ei
p::;
....~..;
...
CM
o
..........
o
o
!
4)
M
,+;>
o
cd
k
~
;
~
~
....
~
6
VEU>Cffl
Figure 94.
Therefore,
d(r/c)
dV
T-D)
(
6
= 0= d
amb +(T -D)
dV
6amb
or,
(~~~)
y
3.124
=
(126)
D6-1420
May1989
PERFORMANCE
If equations (124) and (126) are combined and simplified. the results are:
VI
--1=1
Yo
VI
-=2
Yo
or,
In the above analysis, the variation of speed with weight may be ignored, and a mean value may be
used for each altitude for commercial airplanes.
One Engine lnoperative Climb Performance
When one or more engines are inoperative, the climb performance will decrease due to the lower
thrust and added drag of the airplane. Equation 117 then becomes:
r/c (acceleratoo)
(T -~- M)) V]
=[ ( + : : )
(127)
1
where W is the added drag, consisting of the windmiUing drag of the inoperative engine,
and the rudder deflected drag, D'I" sometimes known as yaw drag.
DwM,
le
Figure 95.
D6-1420
May1989
3.125
PERFORMANCE
Taking summati.on of forces parallel to the flight path, assuming the airplane is in a climb maneuver,
and solving on a per engine basis: ·
(128)
n (F n /eng) = W sin 8 + D + dDWM + ID"'
where n is the number of erigines operating.
Since,
then, equation (128) becomes,
· n (Fn/eng) = W sin 8 + D+ dDWM+ AC 0 qS
(129)
V
Also, taking summation of moments about the center of gravity, (CG), assuming the CG is at the
quarter chord, (c/4), locati.on of the MAC.
(130)
where Cn is the yawing moment coefficient ,
Since,
2
q = 1481.351 6amb M
then,
Fn /eng + dDWM)
(
le
6 amb
6 amb
Cn = Sb
1481.351 M2
(131)
M
Figure 96.
3.126
D6-1420
May1989
PERFORMANCE
..
The windmUHng drag (or locked rotor drag) can be determined from Figure 96 which is determined
from. flight test data.
Thrust can be obtained from engine data for a given thrust setting (EPR or % N1), speed, and
ambient conditions.
The yawing moment coefficient may now be calculated using equation (131). Since there is a
relationship between yawing moment and yaw drag, a curve of yawing moment coefficient (C0 )
versus yaw drag coefficient (AC~ is detennined from flight test. See Figure 97. Enter with C0
and read AC°'v.
YAWING MOMENT COEFFICIENT- Cm
Figure 97.
Equation (129) can now be used to detennine the climb or descent angle, 8, depending on the value
of the thrust. The basie airplane drag, D, in equation (129) is determined from the basie drag
equation.
The above calculation procedure detennined the climb or descent angle by knowing the engine
thrust. The required thrust can also be calculated for a given climb or descent angle but the
calculation procedure is more involved.
Rearranging equation (130), per engine, and solving for FnJeng,
and substituting into equation (129), the result is,
D6-1420
May1989
3.127
PERFORMANCE
Solving for Cn,
e)
1
(nbe)(
1
Cn = ( nb ACD +
'I'
1 2 ) [-6D + ADWM
W sm
. 8]
6
(1 + n) + -61481 . 351 M S
amb
amb
.amb
(132)
When D!6amb and the ADwMf6amb are determined for a given M and W/6arnb Sin 8 is determined
for a given 8 value, equation (132) becomes a linear equation.
(133)
where,
J
ADWM
1
)[ -D
W
.
B= ( - le )(
-+
(l+n)+--sm8
nb 1481.351 :M2 S 6 amb
6 amb
6 amb
Now select two values of AC~ one high value and one low value that fits the AC~ scale of
Figure 97, and solve for C 0 • Plot the two points onto Figure 97 and draw a straight line through
the points. The intersection of the straight line with the curve gives the ~ and AC~ solution for
the particular problem conditions. The required F 0 can be determined using equation (129) since
the AC°'V value is known, or using equation (130) since the ~ value is known.
Airplane calculations for yaw drag prior to the advanced version of the 737-200 used a (CLT ST)
term instead of~- Equation (130) shows that;
F n 11 =C n q S b = ( C LT q ST ~)
therefore,
(134)
Tuus, the C0 scale ofFigure 97 can be changed to the matching (CLT ~) values where,
(C
3. 128
le (
S )-LT T - lt
F n/eng)+
(ADWM)
6amb
6amb
1481. 351 M2
(135)
D6-1420
May 1989
PERFORMANCE
CL
T
S,-
Figure 98.
Also, equations (132) and (133) must be modified accordingly. Equation (132) becomes,
( CL T ST)=
(136)
(!)(:e)~cD + (!)(:e)(
V
t
t
1 2)[r + :WM(l+n) + r, Wsine]
1481. 351 M S
amb
amb
amb
and equation (133) becomes,
(137)
where,
B= (!)
.
(;e)( 1~ )[r
t
1481 •351
S
amb
+ :WM (l+n) +
amb
r,W sin8]
amb
Select two values of ~Cov, one high value and one low value that :fits the ~COV scale of Figure
98, and solve for (CLT ST).
Plot the two points onto Figure 98 and draw a straight line through the points. The intersection of
the straight line with the cmve gives the (QT ~) and ~C°'I' solution for the particular conditions.
The required Fn/eng can now be determined using equation (129) since the ~COV value is known,
or using equation (135) since the(QT ST) value is known.
D6-1420
May1989
3.129
PERFORMANCE
Climb Gradient
Climb gradient may now be derived.· Figure 91 shows that the sine of the climb angle is equal to r/c
divided by the climb velocity, V. For small angles, the sine of an angle is approximately equal to
the tangent of the angle; therefore, since climb gradient is defined as the tangent of the climb angle:
Oimb. gradient
= sin 8 = ~
(138)
Substituting equation (118) into equation (138) results in the climb gradient for unaccelerated climb,
Oimb gradent = T - D
(139)
w
or,
CD
Oimb gradient
T
=w
-c
(140)
L
Substituting equation (117) into equation (138) results in climb gradient for accelerated climb,
L_ CD
(W
Accelerated climb gradient =
CL
y dV
1+-g -dh-
J
(141)
A chart showing climb performance traded for acceleration may now be discussed. The chart
development, sketched in Figure 99, may be seen by analyzing the forces acting on an airplane in a
climb at constant velocity (no acceleration) and the same airplane in accelerated level flight This
acceleration and climb trade chart is a generał type and can be used for all aircraft. Considering the
climb where there is no acceleration, equation (140) applies.
I
~
=
i>
<
Residua! Climb Gradient - %
Figure 99.
3.130
D6-1420
May1989
PERFORMANCE
Considering acceleration in level flight as shown in Figme 100 and equating forces parallel to the
flight path.
·
L
w
Figure 100.
w
T-D=-a
g
a
T
CD
g = W - CL
(142)
Since T/W - CofCL was defined as a gradient term; a/g may be defined as acceleration gradient
Aa:eleration gnuient = ..!.
g
(143)
and since part of th~ total climb gradient from equation (140) may be traded for acceleration, the
following equation can now be written:
or,
(144)
where
T
CD)
( W - CL
is the climb gradient remaining.
residual
D6-1420
May1989
3.131
PERFORMANCE
Figure 99 is constructed with total climb gradient and residual climb gradient on the axes. At zero
acceleration, the total gradient is equal to the residua} gradient; therefore, the zero acceleration line is
established at a 45° angle to the axes. Another constani acceleration value may be selected and
divided by the gravitational unit, g. The results, when expressed in percent and subtracted from
various values of the total climb gradient, will determine values of the residual gradient, thus
determining the location of the acceleration lines.
Turning in the Flight Path
When a tum is included in the flight path, it is necessary to account for the loss in gradient due to
the banked attitude. The gradient loss may then be determined.
·
L
Horizon
w
Lcosę=W
L - 1- n
W- cosę -
3.131 A
D6-1420
May1989
PERFORMANCE
Where n is the load factor
For a bank angle, cp, of 15 °, cos ci, =.966. Therefore 1/cos ci, = 1.035 = n. This means the lift
required in a 15° bank is 1.035 times the lift required for a 0° bank
The drag is increased in a banked tum due to the lift increase. To determine the change in drag,
enter the polar at the higher CL (n x CL for wings level) and read the corresponding Co. The
reduction in climb gradient available will then be equal to Il.DIW.
Determine /lD using the equation:
llD = ~ tmn - Dwmgs level
Climb gradient,
~mm
= (
i - ~ )- A:
~ 1mn = 8wmgs level
AD
--w
For small climb angles with the wings level, use this equati.on:
Lwiugs level = W
The.refore,
D6-1420
MAY1989
AD
ACD
W= C
L wi:ngs level
CD
=
.
-CD
bankcd lmn
wings level
C
L wings level
3.131 B
PERFORMANCE
FAR Minimum Performance Requirements
Airplane performance potentia! is given in the Operations Manuał and the FAA Flight Manuał in
terms of climb gradient available. These peńormance potentials are used to determine whether any
flight plan meets the minimum peńormance requirements of the Federal Aviation Regulations. It is
against regulations to plan a flight where the airplane's performance capabilities are less than the
stipulated minima. Enroute minima, wbich will be explained in the cruise chapter, are specified in
terms of total gross potentia! reduced by an arbitrary gradient factor. The minima during climb out
after takeoff, approacb, and landing are given in terms of the minimum gross gradient available.
These regulations do not restrict opera.tion; they merely establish minimum performance standards.
The table below summarizes regulations pertaining to climb gradients after takeoff, approach, and
landing.
NUMBER OF INSTAJ .T ,ED ENGJNES
4
Flight Condition
SR-422
Rules
4
3
2
FAR Part 25 and SR-422B
Rules
(1) 1st takeoff climb segment
positive
0.5%
0.3%
positive
(2) 2nd takeoff climb segment
3.0%
3.0%
2.7%
2.4%
(3) Fmal takeoff climb segment
(applies from the 400-foot
level of second segment to
end of takeoff panem)
1.8%
1.7%
1.5%
1.2%
(4) Approach climb segment
2.8%
2.7%
2.4%
2.1%
(5) Landing climb segment
4.0%
3.2%
3.2%
3.2%
The flight conditions that arc referred to are described below:
(1)
Fll'St takeoff climb segment is with the critical engine inoperative, takeoffthrust, landing
gear extended, flaps in takeoffposition, takeoff speed (V2 => 1.20 Vs 1 under SR-422
or V2 => 1.20 Vs) under SR-422B and FAR Part 25, and the weight existing at the
time gearretraction is started. Under SR-422B and Part 25 of FAR, Vs 1 = Vs.
(2)
Second takeoff clim.b segment is with the critical engine inoperative, takeoff thrust at
400 feet above the takeoff surface, gear retracted, flaps in takeoff position, takeoff speed
of (V2 => 1.20 Vs 1), and the weight existing when the gear is fully rettacted. Second
segment extends to 400 feet or bigher, if necessary, for terrain clearance.
For airplanes certificated to FAA Advisory Circular 25-7, the takeoff climb speed, V2,
must be at least 1.13 times Vsig• wbere Vs 1g is the 1 g stall speed
(3)
3.132
Third takeoff climb segment is with the critical engine inoperative, at least maximum
continuous thrust on the operable engines, and is used for level flight acceleration and
flap retraction at a height of at least 400 feet.
D6-1420
Mayl989
PERFORMANCE
Finał takeoff climb segment is with the critical engine inoperative, maximum continuous
thrust at the higher altitude of either 1000 feet above takeoff surface under SR-422
(1500 feet under SR-422B and FAR Part 25) or wbere transition to emoute
configuration is completed, and the speed is equal to or greater than 1.25 times Vs 1.
The weight is that existing wben flap retraction is started under SR-422 or at the end of
the takeoff path under SR-422B and FAR Part 25.
For airplanes certificated to FAA Advisory Circular 25-7, the minimum value of the
enroute climb speed is 1.23 times the 1-g stall speed, Vs 1g
.(4)
Approacb climb segment is with the cri.tical engine inoperative, takeoff thrust, gear
retracted, flaps set so that V s 1 is equal to or less than 1.1 O times Vs0; and the speed is equal to or
less than 1.5 times .vs1 at the landing weigbt.
For airplanes certificated to F AA Advisory Circular 25-7, the minimum value of the
approacb climb speed is 1.40 times the 1-g stall speed, Vs 1g
(5)
Landing climb segment is with all engines operating, using the thrust available 8
seconds after advancing the thrust levers from minimum flight idle. Additionally, the
landing gear is extended, the flaps arc in the landing configuration The speed is equal to or
less than 1.4 times Vso under SR-422, or equal to or less than 1.3 times Vs under SR422B and FAR Part 25), and at the landing weight.
During flap retraction after takeoff, it is possible to exceed the flap placard speeds or to retract the
flaps below the minimum speed allowable. Consider a light weight airplane taking off on a cold
day. Rather than use the large excess thrust for climb during flap retraction, the aircraft is
accelerated in level flight. It will accelerate rapidly and possibly overrun the flap placard, resulting
in sttuctural damage to the flaps. Figure 101 shows the relationship of speed to flap extension.
Flap Extension - %
o
Figure 101.
D6-1420
May1989
3.133
PERFORMANCE
Consider now the opposite situation_where a heavy airplane is taldng off on a hot day. Assuming
there is low excess thrust and some of this excess is used to climb, it is possible for the acceleration·
to be reduced so much the speed increase is not sufficient to remaiil ahead of the minimum speed.
The minimum speed increases as the flaps arc retracted due to the reduction in available lift
coeffi.cient.
The gear is retracted as soon as safely practical after liftoff. For calculations, it is assumed
retraction is initiated 3 seconds after lift off, and the retraction time is a conservative estimate based
on tests with one hydra.ulic pump shut off. After the landing gear is retracted the airplane is flown
without change in configuration or thrust to the end of second segment, at which time itis ·flown
level and flaps are retracted.
The following table shows the gear retraction time with the assumed hydra.ulic pump failure:
Airplane
1 Pump Failure
Considered
Gear Retraction
TDllC
Tune From
sec
sec
Llftoff
707-100/lOO
707-300/400
Yes
12.3
15.3
707-300B
707-300B(Adv)
707-300C
Yes
15.5
18.5
727-100/lOO
Yes
11.0
14~0
737-100/lOO
Yes
7.0
10.0
747's
No
20.0
23.0
737-300
Yes
7.1
10.1
757-200
Yes
17.0
20.0
767-200/300
Yes
15.3
18.3
To ensure proper terrain clearance under all conditions, the regulations specify a minimum flight
path. SR-422 requires the available gross takeoff path must elear all obstacles vertically by at least
35 + 0.0lD feet, where D is the distance from the end of the runway. Further, it must elear by 200
feet horizontally within the airport boundaries and 300 feet beyond. The airplane must not be
banked below 50 feet height and thereafter may not be banked more than 15° until out of the takeoff
climbout segment.
3.134
D6-1420
May1989
PERFORMANCE
The net flight path must elear all obstacles in the takeoff segment by at least 35 feet. The net flight
path is the gross path available reduced at cach point by a gradient of climb of:
0.8 percent for two engine airplanes;
0.9 percent for three engine airplanes; and
1.0 percent for four engine airplanes.
FAR 25.115(b)
From calculations based on the solution to equation (140), the climb out net flight path from the
takeoff distance point (referred to as reference zero) can be plotted as shown in Figure 102. This
chart is used for checking obstacle clearance assuming failure of the most critical engine during
takeoff. The net flight path starts at zero height For the net flight path to elear any obstacle by the
required 35 feet, the net flight path in Figure ·102 must only elear the obstacle. If the net flight path
for a field length limited weight does not elear the obstacle, the weight must be reduced until the
resulting flight path clears the obstacle.
This weight reduction results in a shorter takeoff distance, which results in part of the flight path
being over the runway, as shown in Figure 103. Taking advantage of this flight path over the
runway keeps the weight reducti.on to a minimum.
'fo
.....
::c:
5
Obstacle
Oearance
Minutes
~
~
]
Maximum Level Off Line - - -....
Cll
5
,l:)
oc...
o
Minutes
Obstacle Height
~
.....u
i
>
Horizontal Distance From Reference Zero
Figure 102.
~1420
May1989
3.135
PERFORMANCE
.I
.....
..c:
co
....
Takeoff Flight Path
G.)
::c:
G.)
u
s
U.I
Flight Path Over Runway
~
-
o
o
'il
u
....
5
>
14
Horizontal Distance From Brake Release
Figure 103.
The weight reduction required may be determined by an iterative process using Figure 104. A
much easier way of determining this weight reduction is by adding guide lines known as "field
length trade" lines to Figure 102. This eliminates the iteiative method. For example, for a field
length limiting weight, W1, Figure 104 shows the second segment gross gradient equal to q, 1.
Figure 102 shows an obstacle at distance dt can not be cleared at this gradient of q, 1. Therefore,
following the nearest field length trade line to the obstacle height determines C?i, the second segment
net gradient required to elear the obstacle. Retuming to Figure 104, W 2, the obstacle limited gross
weighL is detennined. When using Figure 102, the wind cmrected gradient should be used.
I
Takeoff Climb
W2 W1
Gross Weight at Brake Release
I
«l>i
Second Segment. Net Gradient
Figure 104.
3.136
D6-1420
Mayl989
PERFORMANCE
When weight is reduced to achieve obstacle clearance, the horizontal distance between the obstacle
and reference zero increases since takeoff distance required is less. The obstacle height also
changes with respect to reference zero on a sloped runway, so a conection is required to find the
minimum profile and maximum weight for obstacle clearance. A sloped obstacle height line, equal
to the runway slope, is drawn on Figure 102 beginning at the obstacle horizontal distance from
reference zero, where reference zero is considered to be at the end of the runway.
Figure 102 also shows maximum level off height values. At low climb gradients the airplane
cannot takeoff, climb to 1,500 feet and accelerate to flaps up within the 5 minute engine time limit at
takeoff thrust The maximum level off heights shown in Figure 102 are the corresponding obstacle
heights for the maximum gross level off heights where the acceleration segment can be scheduled at
takeoff thrust In addition, the gross height lines shown assist in determining actual gross height at
any instant Since the gross height lines and the fi:eld length trade lines cannot be applied to the
maximum. level off height lines, the maximum. level off height lines are drawn on a separate chan.
ff obstacles arc higher than, and beyond the maximum level off height line, using the second
. segment extension lines may be possible if the finał segment gradient is greater than that necessary
to satisfy the minimum gradient requirements for acceleration and flap retraction at maximum
continuous thrust in the third segment
D6-1420
May1989
3.13 7
PERFORMANCE
Improved Climb Performance
lmproved climb performance may be used whenever the takeoff weight is limited by the second
segment climb requirements rather than the field length requirements. This allows an increase in the
takeoff speeds since some extra runway is available which results in an increase in takeoff weight
Of prim.ary interest is the increase in V2 speed. Using equation (121), Figure 105 may be
determined for the second segment limit weight at a given takeoff flap configuration and
atmospheric conditions.
GIVEN: CLIMB LIMIT WEIGHT
PRESSURE ALTITUDE
TEMPERATURE
TAKEOFF FLAP CONFIGURATION
R/C
SPEED FOR MAXIMUM GRADIENT
SECOND SEGMENT GRADIENT
MEETING REGULATORY REQUIREMENTS
o--------------------0
SPEED
Figure 105.
A line drawn from the zero r/c and speed tangent to the curve determines the speed for maximum
gradient This speed can be from 10 to 20 knots above the normal V2 speed for the climb limit
weight Increasing the V2 speed above the normal V2 increases the climb gradient.for that weight
The airplane is no longer second segment limited. Now trade weight for gradient. Increase weight
until the limiting value of second segment gradient has been reached włiile maintaining the increased
V2 speed for the increased weight This is balanced against all of the excess runway being used by
adjusting the V1 and VR speeds accordingly. The finał weight is now equally field-length weight
limited and climb-weight limited with the increased V2 speed. This finał weight is now less than the
original field-length limit weight but greater than the original climb-limit weight The increased V2
speed should not go beyond the tangency point. It is better to select another takeoff flap
configuration.
Enroute Climb Speeds
In practice, it is desirable to choose an easily flown clim.b speed From examination of climb speed
data, the best rate of climb true airspeed varies with altitude, as shown in Figure 106. It is close to
a constant calibra.ted airspeed to about 30,000 feet, and a constant Mach number above 30,000 feet.
Since Vc/M is approximately equal to VJ/M1, the speed schedule will be established in terms of
indicated airspeed or indicated Mach number for pneumatic airspeed and Mach indicators, and
calibra.ted airspeed or Mach number for electronic airspeed and Mach indicators.
3. 138
D6-1420
May 1989
PERFORMANCE
When the climb is made at constant Vc or V1 up to the climb Mach number, the rates of climb can
be calculated and corrected for acceleration along the flight path from equation (121) and Figure
109. Figme 109 is based on flying constant calibrated airspeeds. At a given altitude, this will
determine the acceleration factor, (V/g) (dV/dh), to be used in the equation. Since Vc is
approximately equal to V1, entering Figure 109 with V1 introduces an insignificant error. The
acceleration factor may also be determ:ined from a plot such as that in Figure 107 to get the velocity
and slope, dh/dV, at any point.
Constant
Mach Numbers
ConstantVc
True Airspeed, V
Figure 106.
Tropopause
Slope atany
point =dh/dV
True Airspeed, V
Figure 107.
06-1420
May 1989
3.139
PERFORMANCE
When the uncorrected rate of climb values are corrected for acceleration, there may exist a change in
the velocity due to a shift in the peak of a rate of climb versus velocity relationship. This is shown
in Figure 108. Because of the flatness of the curves, this speed change is ordinarily not significant,
and the effect is not considered in the climb data.
I FOR GIVENnoWEIGHT
I
acceleration correction
~~~
~~~with acceleration correction
R/C
speeds
25,000 ft
30,000 f't
·v
Figure 108.
When the acceleration correction factor for all speed conditions is to be determined, the incrementa!
altitude, dh, is a change in true altitude as shown by Figure 107 and not pressure altitude. Since all
performance and performance calculations are based on pressure altitude, any pressure altitude
increment can be corrected to obtain the true altitude increment. The COITection for obtaining the
true altitude increment due to non-standard temperaOJre is:
.
.
.
.
(absciute temp for non - std day)
true alotude mcrement = pressure alntude mcrement
absolute temp for std day
or,
Ab.= Ab.
3.140
P
(_L)
Tstd
(145)
~1420
Mayl989
PERFORMANCE
The temperatures should be the average for the altitude increment being considered. In the limit,
equation (145) becomes equation (146).
dh =·dh
P
(__I_)
Tstd
(146)
Applying equation (146) to the dh term, and deriving as a function of M, the acceleration correction
factor for three climb speed conditions becomes:
V dV
'Y°M'
T dh =-2-v
where at,
and,
ConstantM;
v=-lR(T;1)
Constant Ve;
'I'=
Constant Vc;
'I'= -i,-
(147)
1- lR (T;.'1 )
lR (T;i)
AR = .190263 below the ttopopause
AR = O above .the tropopause
'Y = 1.4
. ,. =
,,
1)6..1420
May1989
[ ( 1 + . 2 M 2)
1
2
O. 7 M
[
3.5
(l+ _2 M2)
]
- 1
25]
3.141
PERFORMANCE
Substituting the numeri.cal values for AR and 'Y, equation (147) results in:
VALUES OF V dV
gdh
Temperature
Standard
Constant
Speed
Altitude
2
M
- .133184 M
~
.566816 M
~
.7M
M
- .133184 M
2
Ve
.7M
2
~
o
2
.7M
( cl>- .190263) (1)
.7M
~
o
2
Non
Standard
Above the Tropopause
Below the Tropopause
:"")
E..
.7M
~-.190263
2
)J
.7M
(ł~
.7 M
190263 ~:...
2
2
(cp)
(1)
2
2
(cl>)
(1) See Figme 109
Note that the acceleration correction factor equations for altitudes above the tropopause are independent of temperature. At constant V c and any temperature for altitudes below the tropopause
equation (147) can be substituted for Figure 109 that is based on standard day temperatures. Using
Figure 109 for non standard day temperatures at the tropospheric altitudes introduces a small error.
For ISA ± 20° C, the effect on the calculated rate of climb will be less than one percent for typical
values of Vc or M.
3.142
D6-1420
May1989
PERFORMANCE
····:·
::::J:
:i::
.25
-
C)
--?
>l..c
"C "C
> le>
a:
.20
g
o
<
z
LL
Q
iw
.15
..J
w
o
o
<
.10
CLlMB and DESCENT
ACCELERATION FACTOR
@ Constant VC
!~!Hii! ~~~~[!~!~ ~~~~r
l--+--l-.....-4--i.-l-.~, ........ ' ........ '.
:::r::
.
::1:::i,i,i,li,,ii,i,i,i....,ii,i,i,i,,;,;,.;,i....,.~
:::!::::====~====::::i::::::::;::::::::;::::::::,:::::::::===:
::::;:::: ::::;:::: ~~~r11
.00 ..........................
.....iliiii,ii,ii,ii. . . . . . . . . . .....;....., . . - - - -.....i,i,i,li,i,i,i,j,i,;,;,;,j
100
150
200
250
300
350
CALIBRATED AIRSPEED f'/ c) KNOTS
Figure 109.
D6-1420
May 1989
3.143
PERFORMANCE
Rate of Climb and Ceiling
From a series of calculations based ·on the solution to equation (121), the rates of climb and climb
speeds for all weights and altitudes at a given engine rating can be plotted as Figure 110. This figure
also shows the absolute ceilings for the weights considered., which are the points at which rate of
climb is zero~ Figure 110 is necessary for calculating climb time, fuel, and distance shown in the
next chapter.
PRESSUiłE
ALTI'IUDE
R/C
V
Figure 110.
Climb Time, Fuel, and Distance
After the rate of climb variation with pressure altitude has been established., the climb characteristics
of time to climb, fuel consumed during climb, and climb distance may be derived. For nonstandard days, the true altitudes must be determined from the selected pressure altitudes.
•
One method to determine the time to climb is:
r/c= ':
where dh is the change in true altitude.
Rearranging,
thus,
3.144
dt= dh
r/c
D6-1420
May 1989
PERFORMANCE
Integrating, assuming an average r/c between h 1 and h2,
(148)
where h1 and h2 are true altitudes and .6.t is the time in minutes to climb from h1 to h2.
Equation (148) is satisfactory throughout the entire altitude range, especially at the !ower altitudes.
At the higher altitudes, the altitude increments must be reduced for best results. Equation (148),
shows thatnear the absolute ceiling, as rate of climb decreases, time to climb increases.
For non-standard day conditions, a correction must be made.. Equation (145) showed the
relati.onship between the true alti.tude increment and the pressure alti.tude increment for non- standard.
day conditi.ons. Applying this correction to equati.on (148) results in equation (149).
{_J_)
h -h
( P2
Pl.A Tstd
t
2
-t
1
= At=-------
(149)
r/cavg
where hp and ~ are pressure altitudes.
1
The norma! method of obtaining time to climb is by integrating 1/(r/c) in a step by step process of
small increments of altitude for each gross weight, allowing for changes in weight due to fuel
consumpti.on. A sample form, Figure 111, also shows how the incrementa! distance and fuel
quantities are obtained.
.
Pressure Alti.tudes
hip to h2i,
havg
.6.hp
A h1me
Wavg
r/c..,g
.6. t
L.6.t
(1)
(2)
(3)
(4)
(5)
(6)
[/)
(8)
Select
Average
h2p - h1p
Assume
Fig 110.
(4).;(6)
Vavg
.Ó.nam
L .6.nam
wfavg
.6. Fuel
L.6.Fuel
(9)
(10)
(11)
(12)
(13)
(14)
Fig 110.
(9) X[/)
From
Engine
Data
(12) X (7)
60
60
Figure 111.
After the fuel quantity has been obtained, the assumed average weight for that step may be
validated. Errors of up to 50 lb will not significantly affect the time to climb. The size of .6.h
D6-1420
May1989
3.145
PERFORMANCE
selected is a function of the rate of climb and is usually in 1,000 foot increments or less. When
calculated, distance, fuel, and time are plotted as in Figure 112. The initial climb weight is at sea
level pressure altitude. For operational use, Figure 112 is normally tabulated in the operations
manuals. If the initial climb altitude is other than sea level, a tabular correction must be applied to
obtain the correct time, fuel and distance.
Another form of the same data as in Figure 112 is in Figure 113. Time, fuel, and distance versus
gross weight with altitude parameters are plotted. This type chart may be more useful to flight
crews. As with Figure 112 the information is for specific temperature and thrust conditions. A
change in temperature will vary the AT/&.unb available forrate of climb. Tuus, for a hot day, the
rate of climb will decrease and the time, range, and fuel to climb will increase. A cold day has the
reverse effect.
!Climb Thrust and Given Temperature l
Distance - NAM
Fuel - Pounds
Time-Hours
Figure 112.
Figure 113 has one big advantage over Figure 112. The initial climb weight on Figure 112, is at
sea level pressure altitude. When using Figure 112, if the initial climb altitude is· other than sea
level, a value of fuel bumed from sea level to the initial climb altitude must be approximated from
the fuel chart. This fuel is added to the initial climb weight so the proper initial climb weight guide
line from sea level is used to obtain accurate values of time, fuel and distance from the initial climb
altitude to top of climb altitude. The problem of obtaining accurate values of time, fuel and distance
is aggravated as the initial climb altitude is increased.
Figure 113 does not have this problem. Accurate results are acbieved regardless of the value of the
initial climb weight and altitude. An initial reading is taken where the initial climb weight intercepts
the initial climb altitude, then following the nearest guide line to top of climb altitude a second
reading is taken. The difference in the readings is the time, fuel, or distance. Figure 113 is known
as an integrated climb chart.
3.146
1)6..1420
May1989
PERFORMANCE
CLIMB THRUST AND GIVEN TEMPERATURE DEV. FROM ISA
C
w
::i:
::,
en
z
oo
_,
w
::,
u.
HIGH
LOW
HIGH
GROSS WEIGHT
LOW
GROSS WEIGHT
w
C,
z
<
a:
m
::i:
:;
o
HIGH
LOW
GROSS WEIGHT
Figure 113.
D6-1420
May1989
3.147
PERFORMANCE
Thrust Presentation
Enroute climb thrust data in terms of EPR is shown in Figure 114. · Instead of plotting EPR versus
temperature for constant altitude, as was presęnted in Takeoff Performance, a plot of EPR versus
altitude for constant values of temperature is made. Figure 114 is based on a fixed climb speed
schedule and at a given climb thrust rating.
\
MAXIMUM CLIMB
THRUST AT GIVEN
SPEED SCHEDULE
/
\
>
ENGINE PllESSURE RATIO "J EPR
Figure 114.
Another chart, such as Figure 115, can be drawn independent of speed. This chart is useful for
obtaining a rated climb thrust for any speed and tem.perature. It may be simplified by retaining the
upper half and labeling the total air temperature values on the abscissa scale for those airplanes with
total air temperatme gages. TAT readings are more accurate than OAT readings and only two inputs
are needed to obtain EPR.
3.148
D6-1420
May 1989
PERFORMANCE
.
I Thrust
Maximum Climbl
-<
C)
Cl
f,..
o
I
!e
ł
~
.!:I
<o
.......
8
"O
U}
Total Air Temperature
Figure 115.
Recent engine designs having high compression ratios will have a case pressme limitation as well as
a temperature limitation. H one of these engines is using EPR as a means of setting climb thrust,
the result is Figure 116. An altitude COITCCtion may have to be made to the temperature limit EPR
value before it is compared to the pressure limit EPR value, the lower value being the most
limiting.
Engine thrust can also be set accurately by the RPM system. H fan rotor speed, N 1, is used to set
clim.b thrust, the EPR scale on Figure 116 may be replaced by a corrected fan rotor speed scale,
N1I
F,. . Figure
See
117. Then N 1 I
F,. is corrected on a ·nomogram chan to obtain
(or% N 1).
D&-1420
May1989
3.149
N1
PERFORMANCE
I
MAXIMUM CUMB THRUST
I
I TEMPERATURE LIMIT I
PRESSURE LIMIT
a:
a:
w
w
a.
a.
TOTAL AIR TEMPERATURE
PRESSURE ALTITUDE
Figure 116.
I
MAXIMUM CLIMB THRUST
I
I TEMPERATURE LIMIT I
PRESSURE LIMIT
li-
lf--
ź
z
PRESSURE ALTITUDE
TOTAL AIR TEMPERATURE
Figure 117.
3.150
1)6.1420
May1989
PERFORMANCE
3-7 Range
Range Equation
For an airplane to fly a given distance requires a conversion of its fuel energy into propulsive work.
This results in the production of range at the expense of fuel quanti.ty and weight depletion.
Consider an airplane flying a small range increment, AR nautical air miles, while using 11W pounds
of fuel. The fuel mileage (miles flown per pound of fuel bum.cd) is:
nam
11 R
. u;-=- 11W
(150)
The negative sign accounts for W change being negative. In calculus, when '1R and 11W approach
zero in the limit, -they become dR and dW.
. Thus equation (150) may be written:
nam
(151)
dR=-lbdW
Multiply the right hand side of equation (151) by W/W:
(152)
The factor (W nam/Jb) is called the range factor.
Assuming this factor can be made constant, the integration can be performed:
w2
R = - W .!!!!!!.. J dW
lb w w
1
wł
nam
R=W-lnlb
w2
(153)
nam
wł
R = 2.30259 W 1b log W
.
2
The constant, 2.30259, accounts for the conversion from natura! logarithms to logarithms to the
base 10.
D6-1420
May1989
3. 151
PERFORMANCE
Another form of the range equation, which is useful in preliminary design work, may be derived.
The term, nam/lb, which appears in equation (153), is the product of two other terms, thus:
nam
nam hr
1b =hrlli
(154)
1:1 = V (in knots)
(155)
and,
1
hr
fu=(!)
However, lb/hr is the airplane fuel consumption and may be expressed as the thrust specifi.c fuel
consumption (lb fuel per hour per lb thrust) multiplied by the thrust being produced.
Tuus,
TSFC=
(!)
T
or,
1b
hr
=T(TSFC)
and,
hr
1
fu=
T (TSFC)
(156)
Substituting equation (155) and (156) into equation (154):
nam
V
(157)
lli = T (TSFC)
In unaccelerated level flight the engine thrust is exactly equal to the total airplane drag. Therefore,
the thrust term, T, in equation (157), may be replaced by the drag term, D. Also, if the numerator
is multiplied by L, the airplane lift, and the denominator by W, the airplane weight, the equality will
be unchanged since the lift and weight arc equal in level flight. Thus, equation (157) becomes:
nam
V
L
1
lb= (TSFC) D W
However, V= a 0
F- M, fromprevious considerations. Thus,
nam
lb=
3.152
aoJe:;;,
TSFC
L 1
Mo W
(158)
1)6.1420
MayI989
PERFORMANCE
Substituting equation (158) into equation (151):
dR=-
aoJa:;,
TsFc
L dW
Mo w
or,
R= _
a
le
w2
O Vvamb Mb.. J dW
TSFC
Dw W
1
Assuming all quantities outside of the integral sign can be made constant,
R=
1)
Ja:;,J( MD
. L). In (w
( a0 TSFC
W
(159)
2
This equation states that if the thrust specific fuel ·consumption, TSFC is considered to be near~y
constant, which is usually the case in the cruise region, a jet airplane will get the most range
between any two weights, W 1 and W2, by flying to make the quantity MUD a maximum.
To sec how the maximum MUD may be determined, consider Fi.gore 118. This shows a CL, Cn
polar plot on which have been drawn tangent lines to cach.Mach number curve. These lines define
the maximum LID for cach Mach number, and if UD is a maximum, then M(UD) must be a
maximum for that Mach number. By plotting M(UD) versus Mach number, the Mach number for
maximum M(L/D) can be found. It is desirable to make M(UD) a maximum and whereas UD is a
maximum on the low speed polar, M(UD) is not necessarily a maximum. ·This means that the
maximum M(UD) occurs at a slightly higher Mach·number than that associated with the highest
possible on the low speed polar. Maximum M(UD) corresponds to cruising into the region where
compressibility phenomena arc becoming important, and the airplane is cruising into the drag rise
for maximum range.
maximum
M(L/D
M
Figure 118.
D6-1420
May1989
3. 153
PERFORMANCE
This method of analysis is used in preliminary work as a guide for establishing wing loading and
thrust required for cruise. When actually determining nautical miles per pound for a specific
airplane, a different method is employed, with the above procedure used as a check to determine
how close the design is to the optimum.
Turbojet Range Factor
Recall how the engine fuel flow and thrust data may be referenced to standard sea level conditions.
This is refening or generalizing the data. A plot of such generalized data, as given in Figure 119,
is useful for preliminary design study and for range estimation. For a given Wl'oamb, the Mac~
number determining maximum range is at the point where the D/'6amb and nam 'oamlllb curves arc
tangent. A curve joining these tangent points represents the Mach number variation with weight for
maximum range operation. By calculating range factor at cach intersection and plotting as in Figure
120, one may determine the optimum W/6amb for maximum range. And by drawing contour curves
of constant range factor, (Wfoamb) ( nam 6aml,/lb), it is possible to show the relative change in
range when operating at other than optimum range conditions.
The engine data does not completely generalize to single cmves of Tfoamb available and referred
flow. Thrust limits must be ploned for various altitudes even on the generalized speed/thrust plot as
shown on Figure 119. The loss of range suffered by. operation at a thrust limit below that thrust
necessary for maximum range can be scen.
T/6
D/6am
~ - - Contours of Constant
Range Factor, W NAM/Lb
Line for Maximum Range
at Constant Altitude
M
Figure 119.
3.154
D6-1420
May1989
PERFORMANCE
Types of Cruise
Computers are used to determine cruise fuel mileage.. The inputs.~e the drag polar, engine data,
the altitudes, weights and Mach numbers. The finał data is plotted (Figure 121) for constant
altitudes. Lines of maximum range and 99% maximum range (long range cruise) are shown as well
as the maximlDD. cruise thrust line. The effect of increasing temperature on maximum tbrust is also
presented. Constant Mach cruise lines may also be present. Holding speed corresponds to almost
minimum fuel flow. However, in practice, holding speed is typically the highest of air minimum
maneuvering speed, minimum fuel flow and initial buffet. The various types of cruise may be
summańzed:
Types of Cruise
Cruise Condition
for Weight Decrease
Range Factor
1.
Constant M and W!&imb
lncreasing altitude
Constant
2.
Constant altitude
(for maximum range
or long range cruise)
Decreasing M and decreasing thrust
Varying
3.
Constant M and Altitude
Decreasing thrust
Varying
4.
Rated thrust
a. Increasing altitude
at constant M
V~g
b. Increasing M at constant altitude
Varying
These conditions may be confirmed by studying Figme 119.
.D
J
~
E
.......e
8"
W/6
Figme 120.
D6-1420
Mayl989
3. 155
PERFORMANCE
I Given Altitude I
Maximum Range
Constant Mach
Number Cruise
Maximum Cruise Thrust
NAM
. '
(Hot Day)
LB
Maximum Cruise thrust
(Standard Day)
,. . I
MachNumb«
TAS
Knots
Te
Figme 121.
When calculating the data necessary to plot fuel mileage cmves, such as those in Figure 121 at a
given altitude, a table may be set up to facilitate the calculations. The tme airspeed lines on Figme
121 are not normally drawn with the fuel mileage curves.
Given Altitude
M
(1)
w
-w
(2)
(3)
6amb
-
D
Fn
~
6amb
(4)
Speed-
Select
Select
Calculale
(4)
Thrust No.ofEng.
Data
[~J
wr *
-NAM
(5)
(6)
(7)
Engine
cs{6:J (.;e-)
661.SM~
Lb
Dala
(6)
x 6amb x No. of Eng.
* lncludes Bleed Correction
Repeat (3)- (7) for as many desired weights at a given altitude. Repeat
complete procedure for other altitudes.
Figure 122.
3.156
D6-1420
May1989
PERFORMANCE
The bleed correction referred to in Figure 122 is a change in fuel flow when air is bied from the
engine. The resulting thrust reduction must be increased by advancing the thrust lever to maintain
the same Mach. This results in increased fuel flow and reduced fuel mileage. If the engines arc
operating at rated thrust, the thrust levers cannot be advanced and the resulting fuel flow will be
lower due to a reduction in Mach number.
·
Figure 122 also provides information for another useful plot, that of fuel flow versus Mach number
for a given altitude. This plot is shown in Figure 123. This information, in table form, is presented
'in the Operations Manuał for various speed schedules.
l
GIVEN ALTrroDE
I
M
Wf
ANY TEMP DAY .
Figure 123.
These data may also be cross plotted as a function of altitude for specific cruise conditions. See
Figure 124. Maximum cruise thrust ratings for various temperatures may be shown on the cross
plots also.
The effect of increasing altitude, generally, is to increase nam/lb. However, at heavy weights, the
thrust requirements are such that a lower value of nam/lb is obtained at the bigher altitudes. In otber
words, there exists for cach weight an optimum altitude; increasing with decreasing weight. To fly
at maximum range, an airplane must operate continuously at the optimum altitude. Above 35,000
feet, the Reynolds number effects on the engine contribute to a deterioration of the performance.
Bleed requirements for pressurization also increase above 35,000 feet. The altitude associated with
cach weight condition may be determined by plotting range factor versus weight for several altitudes
on a composite plot as in Figure 125. This type of plot is used to establish the maximum range or
the long range cruise altitude schedule.
D6-1420
May1989
3. 157
PERFORMANCE
incr..--w
GIVEN CRUISE
CONDITION
w~incr
NAM/LB
M
W-+-1ncr
Figure 124.
The range factor is almost constant for maximum range or long range cruise in a climbing cruise
operation. This justifies the simplified integration procedure discussed earlier, equation (153).
However, for other types of cruise, an average value of range factor can be determined and used to
calculate range. At least five weights from W1 and W2 should be used to establish the average
range factor.
f maximum range fact~ '.::'I'"'-:::::"
:tocus_ll~~
--
-
RANGE
FACTOR,
W NAM
LB
LOW
w
HIGH
Figme 125.
3.158
D6-1420
May 1989
PERFORMANCE
All the calculations so far have assumed level flight conditions. The discussion conceming
maximum range, and Figure 120, has shown that a constant Wfoamb should be maintained. Altitude
must be increased so 6amb is decreased at the same rate as the weight is decreased through fuel
consumption. This will maintain a constant W/6amb• but requires a finite rate of climb, which in
tum requires a slightly higher thrust than that used in level flight. This may seem confusing since it
can be argued that no excess thrust is needed for the climb. The reasoning is that the engines arc
instantaneously adjusted to give the required T/6.mi, to fly level at the given Wfoamb and M. As fuel
is consumed, the Wfoamb decreases so the T/6amb required decreases. There is thus an excess tbrust
which climbs the airplane back to the proper W/6amb where the engine T/6amb available equals the
T/6amb required. This reasoning is in error primarily because it will take some finite time for a
sufficient difference to exist between T/6amb required and Tl6.mi, available.
Consider Figure 126. At any point, A, the airplane is trimmed in level flight for optimum cruise
conditions. It will not climb at the proper constant rate as shown in the figure until there exists a
change in the Wl6ami, such that the excess T/6amb is properly related to the rate offuel consumption
to maintain a constant Wfoamb. There will thus exist a condition where the actua1 W16.mt, flown will
be continually in error. To correct this situation, the thrust must be increased·slightly at any given
W/6am.b and Mach number so that a constant range factor can be maintained. At the same W/6amb,
and Mach number, the reduction in the range factor ( w ~ ) due to the climb is typically
around 1% compared to the level flight.
-8
·-<.a...
· Distance
Figure 126.
D6-1420
May1989
3. 159
PERFORMANCE
To summarize, the corrections which must be ·applied to the range factor to account for all
conditions discussed arc:
(1)
~eduction of 1% to allow for cruising at a higher speed (long range cruise).
*(2)
Reduction due to accessory power and bleed requirements.
(3)
Reduction of approximately 1% due to climbing the airplane at a constant Wloamb to
the tłnust limit, or a reduction in W f6anm at the thrust limit.
(*These items may be accounted for in the basie power plant data.)
Figme 127 shows the approximate reduction in fuel mileage for the constant Wfoamb cruise. When
cruising at a given maximum rating and a constant Wfoamb and Mach number, it is not possible to
increase the thrust to get an excess. For this condition Tł&unb is constant at a given Mach number,
and Wloamb must be changed to vary D/&.unb-
---1%
NAM6amb
Lb
M
Figme 127.
3.160
1)6.1420
Mayl989
PERFORMANCE
Range Presentation
When the nam/lb has been established for the desired cruise conditi.on, such as long range. cruise at
a constant altitude, the corresponding range for changes in weight can be determined.
GIVEN CRUISE
CONDITION
AND .ALTITUDE
NAM/LB
AVG. NAM/LB
w
Figure 128.
In Figure 128 the shaded element represents an· incrementa! range for a change in weight, AW.
Integrating the area under the curve between the minimum and maximum weight limits gives the
total range for the given conditi.on. The integrati.on may be done graphically by assuming small
increments of weight, and assuming a constant average value of nam/lb for cach of the increments;
thus obtaining an incrementa! range. By adding up the incrementa! range values, the total range
between any two weights can be determined.
(160)
and,
(161)
R =l: AR
AW=W 1 -W
2
W avg
( n:1)
AR
I.AR
(5)
avg
(1)
(2)
(3)
(4)
Select
(W 1 -WJ
2
Fig. 128
(1) X (3)
Figure 129.
D6-I420
May 1989
3. 161
PERFORMANCE
With the preceding two equations and Figure 128, a table similar to Figure 129 may be made.
When data have been accumulated for all the desired cruise conditions, a plot of A range versus
weight can be ma.de as shown in Figure 130, known as an "integx:ated range" chart. This chan is
based on a difference principle. The difference between two gross weights represents a weight of
fuel consumed and the corresponding difference in chart distance readings represents the cruise
range available for this weight of fuel consumed. The chart may be used in three different ways.
Entering the chart with :
·
1)
2)
3)
W 1 and W 2 results in A range.
W 1 and A range results in W 2•
W 2 and A rangeresults in W 1.
The infonnation of Figure 130 for the different altitudes and cruise speed conditions may be
tabulated for Operations Manuał use.
Given Temperature
Condition
A Range
Light
Heavy
Weight
Figure 130.
3.162
1)6..1420
May1989
PERFORMANCE
If a variable Mach schedule, such as LRC, is being used, it is often beneficial to generate an
integrated time curve for use in computing trip time. This curve is easily calculated in conjunction
with the integrated range curve tabulation by adding the steps shown below.
&=-ML
V avg
(162)
t = ~At
(163)
wavg
vavg
AR
At
L\t
(1)
(2)
(3)
(4)
(5)
Mao/ eamb
CW1 +W2)
2
From Integrated
Range Table
@Wavg
-(2)(3)
Figure 131.
The resulting integrated time curve is shown in Figure 132. Use this curve by entering with:
1)
2)
3)
W 1 and W 2 to obtain A time.
W1 and A time to obtain W2.
W2 and A time to obtain W1.
GIVEN TEMPERATURE
I CONDITION
HIGH
Weight
LOW
Figure 132.
Temperature Effects
D6-1420
May1989
3. 163
PERFORMANCE
Temperature Effects
The effect of air temperatme on range under all conditions, except when operating at a rating, may
be negligible. Consider the case of operating under maximum long range conditions, and assume
that suddenly the temperature increases. Because of this temperature rise, the thrust, and thus
T/Oamb, will decrease. To maintain the original Mach number, the thrust (and T/Oamb) must be
retumed to its original value. This means the thrust levers must be advanc~ resulting in an
increase in the engine RPM which will increase the specific fuel consumption. This effect is shown
in Figure 133, which illustrates the specific fuel consumption, TSFC, plotted against RPM for a
typical jet engine.
cruise point with
temperature rise
t~erature
TSFC
initial. cruise point
RPM
Figure 133.
This would seem to indicate that the fuel mileage of the airplane has deteriorated, but by maintaining
a given Mach number under the increased temperature conditions, the true airspeed has increased.
The increase is due to the higher local speed of sound. The higher airspeed counteracts the
increased fuel flow caused by the higher engine RPM. Amore ńgorous analysis shows the effects
exactly compensate except for secondary effects such as air conditioning bleed requirements varying
with temperature which are usually ignored.
Equation (57) states:
but,
therefore,
(164)
3.164
D6-1420
May1989
PERFORMANCE
also,
~ =(~ )(6~J6„bJ&::
(165)
therefore,
nam
Mao~
n,=
(F-)( ::)amb.Je::
6
(no.ofengines)
or,
(166)
At a given Mach number, gross weight, and altitude, the thrust required (drag) of an airplane is
constant and independent of temperature as previously shown in Figure 1O (redrawn here for
convenience).
D
~amb
,
M
Figure 10.(Repeated) ·
Specific fuel consumption data are normally presented as shown in Figure 22 (redrawn here for
convenience) with bleed requirements, power extraction and accessory drag allowances included.
D6-1420
May1989
3. 165
PERFORMANCE
As long as the variation of TSFC/ ~ with Fn/6amb and Mach number remains independent
of temperature, ignoring secondary effects such as air conditioning bleed requirements which vary
with temperature, any constant values of Fn/6amb and Mach number obtained for all temperatures
yield the same TSFC/ ~ . Therefore, from equation (166), ruun/lb becomes independent of
temperature.
CONSTANT
ALTITUDE
TSFC
~
Fn/6 amb
Figure 22 (Repeated).
The above analysis shows that when fuel flow is a function of
Je::;, and constant Mach can be
maintained, there is no change in nam/lb with change in temperature since true airspeed is also a
function of
Je:;;,
However, when the fuel flow is a function of ex , the fuel mileage will change with change in
amb
temperature. To see the magnitude of this change from standard day values, first consider the
change in true airspeed and then the change in fuel flow. The velocity for standard day
tempera.tures is,
v m = MaoJeambflll
3.166
D6-1420
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PERFORMANCE
and for non standard day, (ns), temperatures is,
vos= MaoJeambos
Solving for M and equating,
Therefore at constant M,
(167)
The fuel flow for non standard day temperatures may be written as,
w -w (w J
fos -
fos
fstd
(168)
Wf std
Substituting equation (29) of Section 2 into equation (168) results in,
(169)
H constant Mis considered at a given altitude, equation (169) reduces to,
(170)
Dividing equation (167) by equation (170) results in the non standard day value offuel mileage at
constantM.
D6-1420
May1989
3. 167
PERFORMANCE
(171)
Equations (167), (170) and (171) are useful for determining non standard day values of true
airspeed, fuel flow and fuel mileage from the standard day table values in the Operations Manuał
provided the Mach number can be maintained; ie, not tbrust limiting,
However, when operating at a given rating, such as maximum cruise tbrust, where Wf is a function
of
Ja;:;, ,the effect of temperature changes is felt. Under these conditions the actual tbrusts
involved will change, and since the thrust lever cannot be advanced to compensate, the fuel consumptions and speeds will also change. Tbis effect is illustrated in Figure 134.
IConstant Altitude I
M
Figure 134.
3.168
D6-1420
May1989
PERFORMANCE
When operating at the maximum cruise thrust rating, where Wr is a function of ex , there is a
amb
double effect of temperature change on speed and fuel mileage. This is shown in Figure 135.
NAM
LB
CONSTANT ALTITUDE
CONSTANT WEIGHT
M
Figure 135.
The circled intercept points are the correct speed
and fuel mileage values.
Earlier, in Figure 124, curves of nam/lb versus altitude were shown with the maximum range line
drawn. If the thrust limited situation occurs, the maximum range altitude is no longer attainable, as
shown in Figure 136.
GIVEN CRUISE
CONDITION
max cruise thrust
@ given temp
NAM/LB
Figure 136.
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PERFORMANCE
When this occurs, a trade of speed. for altitude may be made with a subsequent improvement in
nam/lb from that shown at the thrust limit in Figure 136. This is shown in Figure 137. Figure 137
also shows the optimum altitude for the given weight, the MCRT limits, the MRC and LRC speed
variation with altitude, and the fuel mileage variation with altitude.
GIVEN WEIGHT
~:;::::~,--~---1....---"best II nam/l.b under new
operating conditions
- -........- - "best" nam/lb when
l.imited by tbrust .and
speed schedul.e
M
Figure 137.
Maximum Range at Maximum Cruise Thrust
To obtain the best nam/lb at the maximum cruise thrust rating for a series of weights at a given
temperature, the following procedure is employed:
(1)
Read the nam/lb and Mach number for cach weight from the intersection of the fuel
mileage curve and the thrust limit curve (Figme 134) for several altitudes.
(2)
Plot nam/lb and Mach number versus weight for cach altitude as shown in Figure
138.
(3)
From Figure 138, integrated range curves such as shown in Figure 130 may be
calculated and presented for either best or .99 best range.
3.170
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PERFORMANCE
M
loc
--- -..;;::
NAM/LB
line for
"op ilnum" e.ltitude
and "best" nam/lb
.........
39,00()
locus line for .99
"best" nam/lb - - - -
IDW
w
Figure 138.
Reynolds Number Effect on Fuel Mileage
Section 1 showed that the lift and drag coefficients were affected by Reynolds number (RN), and
Mach number (M). RN is the important criterion for low speed flight while Mis important for high
speed flight. Until recently RN was never considered to be important enough for high speed flight
conditions and was never accounted for. The RN effect on drag is just large enough to produce an
additional drag coefficient, ac 0 RE , and is added to the basie drag coefficient for reference
conditions. Changes in drag results in fuel flow changes which influences the fuel mileage.
D6-1420
May 1989
3. 171
PERFORMANCE
Since RN is a function of altitude and temperature, the equation for .1CDRE is:
~C 0
RE
=A log 10
r-(8 :!: .~12)
(172)
1
Ref
Where A is a dimensionless constant evaluated for cach airplane developed after approximately
1980. The subscripts Actual and Reference refer to the evaluation of the term at the actual or
reference conditions respectively. The reference temperature is ISA, while the reference altitude is
. also unique to cach airplane, and is a function of W/6am.t:r The constant A and the reference
altitude is presented in the Performance Engineer's Manuał (PEM) for cach applicable
engine/airframe combination.
A typical reference altimde plot is in Figme 139.
w
6amb
Figme 139.
3.172
D6-1420
May1989
PERFORMANCE
Altitude versus Weight
.
Another useful cross plot ofFigure 121 is altitude versus gross weight for a given speed schedule
as shown in Figure 140. This chart may be used operationally by the flight crew for the given
speed schedule. For cach weight it shows the optimum altitude, the thrust limited altitude, and the
fuel mileage. This chart (or tabular version of the chart) without the fuel mileage cmves is normally
presented in the Operations Manuał and labeled as an altitude capability chart.
I
CONftTANT
I
w
a:::w
:>O
(/) ::>
(/) 1--
Wj::
a:::-1 '
D..<
high
w
low
Figure 140
When Figure 140 is clrawn for the long range cruise speed schedule, a family of constant Mach
lines must be drawn to show the LRC Mach values.
I>6-1420
May1989
3. 173
PERFORMANCE
Wind and Off Altitude Effects
If an airplane is required to fly at a constant W/6amb, ( maximum range, for example), then its
altitude at any weight is specified. For various reasons it may be desirable to fly at an altitude
which is not optimum. Such a condition might exist when a strong taił wind is known to be present
at some lower or higher altitude. It might then be advantageous to go to this altitude if the wind
effect more than off sets the loss in range due to flying at the off optimum altitude. Also, it may be
required to fly at other than desired altitudes due to air traffic conditions.
The effect of wind at a given altitude is to change the ground speed, thus changing the time to travel
a given ground distance. Effectively, this is the same as traveling a different air distance. The
correction for the wind effect is:
TlIIlC to fly through the air mass is
(173)
where,
t
nam
Va
= time in hours
= still air distance in nautical air miles
= cruise speed in knots
But referenced to the ground,
(174)
where,
run
Vw
= ground distance in·nautical miles
= windspeed component in knots. Tailwind is positive
For a given period of time, equation (173) equals equation (174).
.!!!!!!. nm
Va - Va± VW
or,
mm=nm( va IvJ
(175)
Equation (175) is used in presenting the effect of wind on ground distance in the Operations
Manuał. The form ofpresentation is shown in Figme 141.
3.174
D6-1420
· Mayl989
PERFORMANCE
200
Vavg -KTS
TAS
STILL AIR DISTANCE - NAM
Figure 141.
To consttuct the chart, selected values of headwinds and tailwinds for assumed values of cruise
speed arc substituted into equation (175) and the relative values of ground distance to still air
distance arc established. In Figure 141, the dashed line at a zero wind condition shows stili air
distances arc equal to ground distances.
The wind conditions at another cruise level may indicate a desirability to change altitude either to
reduce the headwind, or to increase the tailwind. The combined wind effect, and the change in
nam/lb at the proposed altitude must be considered.
·
A small improvement in wind may not be adequate to compensate for the fuel loss required to
change altitude. Only a major gain due to wind can be considered if a large change in altitude is
involved.
Point of No Return and Equal Time Point
A pilot on an overseas flight may be faced with the decision of continuing on, or turning back, if an
em.ergency arises. To help answer the question, calculate two points: a point of no return (PNR)
and an equal time point (E1P). The PNR is the greateSt distance to which the airplane could fly,
lose the thrust of the most critical engine, and retmn without running out of fuel. The E1P is the
greatest distance from which the pilot could elect, after engine failme, to either retmn to departure
point or continue to destination. This point results in the sanie amount of time (or fuel) for either
choice.
])6.1420
May1989
3. 175
PERFORMANCE
To calculate the PNR:
(1)
Enter the all-engines operating integrated range c~art, Figure 130, with departure
gross weight and a weight increment equal to one half the weight of the fuel. Read
the air distance increment corresponding to the planned type of cruise and correct for
wind to obtain the ground distance out-bound.
(2)
Enter the-one-engine-inoperative integrated range chart with zero fuel weight and a
weight increment equal to one half the fuel weight. Obtain the air distance
corresponding to maximum range operation and correct for wind to obtain the
ground distance in-bound.
(3)
Plot the two ground distances obtained against the air distance out-bound as shown
in Figure 142.
(4)
Repeat steps 1, 2, and 3 using a larger weight increment for the out-bound leg and
the corresponding smaller increment in-bound.
(5)
After enough pairs of nm points have been plotted to define an intersection of the
out-bound and in-bound ground distance lines, obtain the location of the PNR both
in ground distance and in air distance at this intersection.
NM
PNR
NAM OUTBOUND
Figure 142.
Of course, one might locate the PNR by calculating range increments for the out-bound and inbound (backward) legs simultaneously, using more basie cruise fuel flow and speed data, and
summing increments to the point of definition. Or one could calculate a no-wind PNR and apply a
pre-calculated wind correction factor.
The ETP calculation is based upon the assumption that the same type of one-engine-inoperative
cruise is used for both legs; i.e., the same airspeed, V, applies for both. As a consequence of the
use of the same airspeed and time in both legs of the flight, the fuel consumption and air distances
also will be equal.
·
3.176
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May1989
PERFORMANCE
+
PNR
wind
point
A'--------------------~B
In the nomenclature ofFigure 143,
Figure 143.
(nam) a= (nam)b
The wind correction, equation (175), shows:
(head.vind)
and,
( tail.vind)
Therefore,
nm 8 V
mnb V
(V+Vw) - (V-Vw)
(176)
From Figure 143:
(177)
D6-1420
May1989
3. 177
PERFORMANCE
Therefore, Illlla can be determined by combining equation (176) and equation (177).
nma(V-Vw)
nm.=SG-
v+v
SG= oma [ 1 +
V+ V:
V-V
w
]
(178)
Solving for nn;,, the result is,
(179)
For the above analysis only cruise flight was considered. The descent to A or B under actual flight
conditions changes the results slightly, but this is usually ignored.
·
Engine Inoperative Cruise
When one or more engines are inopera.tive, the maximum range at the optimum altitude will always
be less than for all engine operation. This is because with engines inoperative, the airplane drag
increases due to the additional drag of the inoperative engine. This means the UD of the airplane is
lower than with all engines opera.ring, thus with no better fuel consumption characteristics than with
all engines opera.ting, the range must be less. This may be scen from equation (128).
R=
(aoF-)( L) In (wW
TSFC
M D
1)
2
At low altitudes, however, where the thrust required per engine is low, and consequently the
TSFC / ~ value is high, amore beneficial engine specific fuel consumption is available with
less engines operating where the thrust required per engine is higher. This is illustrated in Figure
144.
3.178
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Mayl989
PERFORMANCE
IGiven Altitud3
TSFC
M=Constant
{a
All Engines -+I
1 Engine · 1
lnoperativ·,,....e- -1•-..1
1
I+- 2 Engines Inoperative
Fn/ 6 Per Engine
Figure 144.
The result of tbis is a possible improvement in nam/lb at low altitudes with one or two engines
inoperative. However, the best range is obtained with all engines operating as shown in Figure
145.
The manner in which engine inoperative range performance is determined is to modify the airplane
drag polar, CL versus C0 , by the addition of the drag coefficient associated with the engine
inoperative operation.
IGIVEN WEIGHT I
t
l
w
C
::::,
I-
~
<
NAM/LB
Figure 145.
!>6-1420
Mayl989
3. 179
PERFORMANCE
Driftdown
If an engine fails or there is a radical loss of tbrust while in cruise, a descent to a lower altitude and
an adjustment in speed usually is necessary. A driftdown procedure, designed to rniniroi:ze loss of
range, is given. The thrust levers on the operating engines arc advanced to maximum continuous
thrust and the airplane is slowed to optimum driftdown (maximum UD) speed while maintaining
cruise altimde. In most cases it will not be possible to maintain cruise altitude at a steady state value
of maximum LID speed; therefore, the airplane is descended at this speed. This results in the
minimum driftdown angle. A higher speed will increase the descent angle. During descent, time is
available to consult the engine inoperative cruise data and establish a new flight plan. When level
flight is attained at the max UD speed, the pilot has 3 alternatives when continuing the flight:
1)
Maintain the maximum LID speed. This results in altitude increase with weight
decrease.
2)
Maintain a constant altimde while accelerating towards engine inoperative LRC speed.
The time for this acceleration at MCT may take 1/2 hour to 1 1/2 hours depending on
weight and atmospheric conditions.
3)
If not terrain lirniting, some altimde may be sacrificed by descending and accelerating
to the LRC speed at the lower altitude.
A schematic presentation of a possible driftdown profile is. shown in Figure 146.
SLOWOOWN TO
ALL ENGINE FLIGHT DRIFTDOWN SPEED
(CRUISE SPEED).
•
1
\
NET DRIFTDOWN PATH
GROSS DRIFTDOWN PATH
\
\
\
LOSE ONE OR TWO ENGINES '
AND INCREASE THRUST ON
'
REMAINING ENGINES TO
MAX-CONTINUOUS-THRUST
ENGINE
INOPERATIVE FLIGHT
(D
----A.....::::;;;.._ _ LRC@
... ... ...
... ... ·-~~----@
... -0
Figure 146.
Engine inoperative cruise must satisfy the terrain clearance regulations. SR-422 4T-12l(a) and (b)
state that with the most critical engine inoperative, the airplane must elear all terrain by 2,000 feet
with a net flight path. The net flight path is 1.8% less than actual gross climb gradient available if
.one engine is inoperative, and 0.6% less if two engines arc inoperative (four engine airplanes).
3.180
D6-1420
May1989
PERFORMANCE
SR-422B and FAR Part 25 require net flight path capabilities corresponding to reductions of 1.6%,
1.4%, and 1.1 % for four, three, and two engine airplanes with one engine inoperative. For four
and three engine airplanes with two engines inoperative, the reductions are .5% and .3%,
respectively.
FAR Part 25.123 (b) and (c)
The enroute terrain clearance requirements in FAR Part 121.191 for one engine inoperative and
FAR Part 121.193(c) for two engines inoperative are:
One Engine Inoperative:
(a)
The net flight path shall have a positive slope 1500 ft above airport intended for
landing.
(b)
Compliance with (1) or (2) below:
(1)
Net flight path slope shall be positive at an altitude of at least 1,000 feet
above all terrain within 5 statute miles (4.34 nm) on either side of the
intended track, or
(2)
the net flight path clearing all temrln and obstacles by at least 2,000 feet
within 5 statute miles (4.34 nm) on either sicie of the intended track.
FAR Part 121.191 after 8/29/59
Two Engines Inoperative:
(a)
There is no place along the intended track that is more than 90 minutes (with all
engines operating at cruising power) from aiiport intended for landing, or;
(b)
Compliance with (1), (2) and (3) below:
(1)
The net flight path shall have a positive slope 1,500 feet above aiiport
intended for landing.
(2)
The net flight path clearing all terrain and obstacles by at least 2,000 feet
within 5 statute miles (4.34 nm) on either sicie of intended track.
(3)
There is enough fuel to arrive at the landing area at an altitude of at least
1,500 feet and then fly for 15 minutes at cruise power or thrust.
FAR Pan 121.193(c) after 8/29/59
A net driftdown profile is also included on Figure 146. Net profile charts for vańous start of
driftdown altitudes are presented in the Operations Manuał for flight planning as shown in Figure
147. The conditions are: (1) one engine inoperative for two engine aircraft, and (2) two engines
inoperative for three or four engine aiicraft. Temperature deviation from standard is accounted for
by correcting the actual gross weight to obtain an equivalent gross weight value. If conditions
result in the driftdown altitude at a given driftdown distance being below the minimum terrain
clearance altitude, the airplane weight must be reduced by dumping fuel (if a dump system is
available) until it can satisfy the requirements. Example guide lines are shown on Figure 147.
D6-1420
Mayl989
3. 181
PERFORMANCE
IInoperative
One Enginel
160
Driftdown Net
FlightPath
150 e
150
=
Maximum Continuous Thrust
.§
·;
. i:..
G)
=
140 ~~-----1140 lf
G)
.s
130 ,::;...._-+---,11130
&i
~
110 a-,-=:::;~.----1 110 'il .
~
o
.E'ąuivaleat GW
30
atl:n .
llne Faijlłre
5
·e>
..-=--11----1 100 ~
90
+10&
,
, 110
I
25
·:::
<
----120
~
i
90
+20
+15
Below
100
]
ii
;;
120 ~---łf--~ 120 ~
.5Ili)
130
20
140
150
I
15
I
o
I
i
5
10
Head
100
160
'
I
I I
15
35
20
25
30
Time trom Engine Failure - Minutes
40
I
I
45
50
. 50
"'O
C:
·-
o
~
50
100
Tail O
50
100
150
200
250
300
Ground Distance From Engine Failure
Figme 147.
3.182
D6-1420
May1989
PERFORMANCE
Endurance (Holding)
The problem of determining the best endurance performance ftom an airplane is similar to that of
obtaining best range. The quantity nam/lb had to be maximized to obtain the best range. To obtain
the greatest endurance, it is necessary to maximize the quantity hrs/lb which occms at minimum fuel
flow. If the engine fuel flow data completely generalizes, the problem of determining best
endurance is quite simple.
Referring to Figure 17 in chapter 3-3 ( redrawn herc as Figure 148), it was shown that the lines of
Wf / 6 amJe:;;, (or Wf / 6 amb 8 :rab> increase in the direction of the higher values
constant
ofT/6amb· As a.result, the minimum fuel flow occurs where the W/6amb curves are tangent to the
corrected fuel flow curves, Wf / 6 a m ~
. This results in a minimum fuel flow speed
schedule.
T
6amb
M
Figure 148.
At the point of tangency, the Wf/ 6 a m ~
is a minimum, or 6 ambJe;;:;, / Wf is a
maximum. Since the units of Wf are lb/hr, 1/Wf has the units of hr/lb which is endurance.
D6-1420
May1989
3. 183
PERFORMANCE
max cruise thrust
incr
IAS
~
STANDARD DA.Y
I
I
GIVEN FLI\P S:m-n:NG
I
FOEL FLOW
PRESSURE ALTI'RJDE
42,000 1't
Figme 149.
The minimum fuel flow speed schedule is in the speed-instability region of Figure 148. This region
begins where the thrust available curves arc tangent to the thrust-required curves which occurs near
maximum UD speeds. Since airplane control is much easier at a higher speed than that associated
with minimum fuel flow, the speed schedule chosen for holding, with very little sacrifice in fuel
economy may be at or near maximum UD.
Pilots flying airplanes with auto-throttle can select a holding speed closer to the minimum fuel flow
speed where speed control can be maintained automatically. This is being clone on the 757 and 767
airplanes where the speed selected is the highest value of the following three requirements:
1)
Minimum maneuvering airspe.ed:
VREF(3o<> FLAP) + 80 knots for 7.ero flap holding.
°
VREF(3o<> FLAP) + 60 knots for flaps 1 holding.
2) Minimum fuel flow.
3)
3.184
1.3 g maneuver to initial buffet.
D6-1420
May1989
PERFORMANCE
The minimum maneuvering airspeed requirement may be most limiting at low to intermediate
altitudes, whereas the minimum fuel flow may be most limiting at the hlgher altitudes.
Constant alti~de is of primary interest for practically all holding conditions. The technique used in
obtaining the best endurance at constant altitude is similar to that explained above. Initially, the
speed scbedule is determined from a· tbrust versus velocity or Mach number plot for a series of
selected constant altitudes. Using this speed scbedule, the fuel flow for the corresponding weight
values and for the selected altitudes is detennined. For a given weight, the best endurance for
constant altitude occurs at a particular altitude. A plot of the data, shown in Figure 149, may be
made based on a given flap setting and standard day conditions.
A correction graph for clifferences in temperature, shown in Figure 150, is merely the effect of
or
on the fuel flow.
Je:;, . e:mb.
FCEL FLOW
Figure 150.
Thrust Presentation
Dming cruise, care must be exercised not to exceed the maximum cruise thrust rating allowed by the
engine manufacturer. Normally, the limiting thrust is defined in terms of Engine Pressure Ratio
(EPR) or Corrected Fan Rotor Speed (N 1
D6-1420
May1989
/Fi) and is sbown as in Figure 151 and 152.
3. 185
PERFORMANCE
EPR
MAXIMUM
CRUISE
THRUST
I
I
I
1I
ENGINE INLET TOTAL TEMPERAlURE - Tt
.
2
Figure 151.
Figure 152 is for recent engine designs having high compression ratios resulting in case pressure
limitations and temperature limitations. 'Ibis is similar to the maximum climb thrust setting charts,
Figure 116 or Figure 117.
!MAXIMUM CRUISE THRUST
IPRESSURE LIMIT I
I
I
TEMPERATURE LIMIT
I
"
~
-....
c
z
o
o
a:
a:
a:
a..
w
a:
a..
w
PRESSURE ALTITUDE
ENGINE TOTAL AIR TEMPERATURE - ,
2
Figure 152.
3.186
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Mayl989
PERFORMANCE
However, when cruising at a thrust setting lower than the maximum pennissible, it is necessary to
use Figure 153 to set thrust.
WEIGRT
EPR
or
N1/~
2
Figure 153.
D6-1420
May1989
3. 187
PERFORMANCE
Intentionally Left Blank
3.188
D6-1420
Mayl989
PERFORMANCE
3-8 Economy Operation
Economy Cruise
Because competi.tion has become aggressive~ operating at economy cruise is more important than
ever before. But establishing total cost is difficult since it must include everything from airplane
and engine purchase costs to crew day/night pay factors. Ea.eh airline develops its own procedures
for calculating "direct operating costs" (DOC) however, most are adapted from the following ATA
formula.
(180)
Economy cruise may be defined by plotting cost factor, DOC/nautical air mile, versus weight as in
Figure 154. The envelope of the constant alti.tude curves establishes the best weight altitude
operati.on.
·
High
w
Low
Figure 154.
If equation (180) were superimposed on Figure 121, a plot such as Figure 155 results. The locus
of the tangents of the $/nam cmves on the nam/lb curves determines the speed for economy cruise at
that altimde.
A summary of the range and speed values for economy cruise at a number of alti.tudes may be
drawn on a single plot, Figure 156. The alti.tude schedule may be read from Figure 154 and the
speed and range from Figure 155. The economy cruise calls for low alti.tudes when at high weights
and high alti.tudes when at low weights. Thus economy cruise is obtained in a climbing cruise
operation at a fixed speed schedule.
D6-1420
May1989
3.189
PERFORMANCE
IGiven Altitudel
$/NAM
i-- Speed
for
(Usually
approxi)
Minimum D.0.C. mated with a
NAM/Lb
constantMach
w
\
Number
.
lncrease
M
Figure 155.
42,000
38,000
Altitude
34,000
Speed
V-Knot
I
·'
NAM/LB
Light
w
Heavy
Figure 156.
3.190
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PERFORMANCE
Dispatching personnel and flight crews have two problems which they would like to be able to
solve easily without being e:xposed to cost accounting. The first involves flight planning to
minimize cost. The question is how to find the least costly of a number of possible flight plans.
The second problem is met inflight and involves making a decision as to whether it is possible to
save by changing operation from the pre-planned flight. The problem may arise because of
unforeseen weather conditions, equipment malfunction, or other contingencies outside the normal
planned flight.
Let,
= total trip cost, $
chr = operating cost (fuel cost excluded), $/br
C
C1b
T
F
=fuel cost, t/lb
=total flight time, hr
= total flight fuel, lb
In defining hourly costs, Chr, certain of the ATA direct operating costs should not be included.
Only those cost items which are affected by błock speed or błock time should be included in the
calculation. For example, once the airplane has been accepted and is in use, depreciation charges
may not necessarily be included as a cost factor in determining the most economical speed for flight.
If a time saving does not result in increased utiliza.tion of the aircraft, depreciation charges are
usually assumed to be independent of speed of operation and should not be included in the cost for;.
mula to determine economy cruise airspeed. However, if additional utilization could be realized
from higher cruising speeds, there may be an influence upon depreciation cost and its effect on
operating speed might be included. Similarly, crew costs are not always included; insurance and
passenger service items are seidom included. Considerable care must be exercised in interpreting
the cost analysis if a valid conclusion is to be drawn.
A flight cost index can be defined:
c:
C
CI=
('iollars )
= (~) =
(it,o)
+
( d:ollars
J
Total trip cost can be given in terms of opera.ting cost and fuel cost thus,
c1b
(181)
C= lOO F+Cbr T
T= -
Clb
F + _Q_
100 cbr
cbr
Therefore,
1
C
T = - 100 CI F + C
(182)
br
E.quation (182) represents a family of lines in the T- F piane whose slope is,
D6-1420
May1989
3.191
PERFORMANCE
1
(183)
m= - lOOCI
and whose intercepts on any time axis represent increasing total cost. Eacb line represents a
constant total
increasing up and to the right
cost,
In plotting these relationships, it is most convenient to put the slope into trigonometric form,
a= tan -1 ( - 1001 CI )
This definition is valid only if the axes have the same unit length value. However, it is not
convenient to plot these data in such a system; so the scales are defined for convenient plotting and
the slope is redefined to read,
a= tan
-K )
· 100 CI
-1 (
(184)
where,
K = unit length on T scale
unit length on F scale
(185)
A circular nomogram constructed from equation (184) to overlay the basie T - F grid can be used to
de:fine the slope for any given cost index value, as in Figme 157.
-
K= 12,000
o
o
10,000 Pounds
TripFuel
Figme 157.
3.192
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PERFORMANCE
The index value for any given route is determined by a cost analysis, and is used to determine the
slope of a family of guide lines on the planning grici. The positions of any two plans represented on
the grid shows the relative cost of cach with respect to the other. Obviously, the larger the grid, the
more accmate the analysis. But ·since very large grids are difficult to work with, use only a portion
of the large grid. Tuus, if working on a short route, label the scales from 1 to 4 hours and 10,000
to 50,000 pounds; if on a long route, use 10 to 13 hours and 120,000 to 150,000 pounds.
An example worked out on Figure 158 is used to show this situation:
CI =.300
PLAN
ALT.
SCHEDULE
TIME
FUEL
A
31,000
LRC
11:33
135,106
B
35,000
LRC
11:30
128,166
C
31,000
.82M
11:08
142,291
D
35,000
.82M
11:16
133,299
These arc possible flights drawn from tabulate(I or plotted flight-planning data in the Operations
Manuał.
The solution ranks the plans in order of increasing total cost:
D,B,C,A
Of the plans considered, the least expensive is plan D, a flight at 35,000 feet on a constant Mach
number schedule of .82.
If the cost index was reduced from 300 to 150 due to !ower operating cost and/or higher fuel costs,
the ranking of the plans in order of increasing total cost would be:
B,D,A,C
making the LRC at 35,000 feet the least expensive.
To develop an inflight chart for use in problems of the second type, let us divide equation (182) by
ground distance in nautical miles, nm. Ground distance is considered to account for wind.
T
1
F
1
C
"iim =-100 CI run+ cbr run
(186)
1
1
1
C
vgr =- 100 CI ( ~ ) + chr nm
(187)
Therefore,
1
Equation (187) shows that cost per mile is related to ground speed and ground fuel mileage through
the same index parameter, - K / 100 CI, as in the flight planning analysis. By plotting ground
D6-1420
May1989
3.193
PERFORMANCE
speed, consideration has been given_to altitude, temperature, wind, Mach, and nue airspeed. Tuus,
the cost index can be used to define the slope of a family of constant cost lines on a grid of lN8!
versus 1/(nm/lb) However, since the ground fuel mileage nm/lb, being referenced to grouna.
distance, is .hard to determine in flight, a better flight parameter is sought.
·
120
130
Costlndex
Planning Chart
140
150 ·
K= 12,000
co
·-:s
U)
o
1,200
::s
C
1,000
800
~
::s
o
::I:
600
:30
t :20
i=
C.
·i:
~
:10
11 :
Trip Fuel - 1,000 Pounds
Figure 158.
3.194
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PERFORMANCE
1
F/NM = NM/LB
o
o
5
I I I I I I
10
I
I
. F/NM
slope = T/NM
= Wf
.001
li
K = 10,000
.002
.003
Figure 159.
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May 1989
3.195
PERFORMANCE
In many cases, the determination of nm/lb may be part of the solution soughL By considering fuel
flow:
·
w=.!:.
f
T
which reduces to,
(188)
Equation (188) defines a family of constant fuel flow lines in a fan from the origin of the 1/ (nm /
lb) versus 1/ ~ grid. These can be used in flight since fuel flow is readily available. Tuus, an
inflight chart, Figure 159, is defined. To make the flight crew chart more adaptable to flight use, it
is usually labeled in terms of V gr rather than in lNgr, as in Figure HiO. Figure 160 is referenced to
four-engine operation; therefore, for three- or two-engine applicability, the values on the abscissa
scale must be factored by three-founhs or one-half, respectively.
An example, shown on Figure 160, may be worlced out from the following data:
Condition
Alt
Temp
Mach
TAS
Wmd
vgr
Wr
A
B
C
35,000
30,000
39,000
-55°C
-45°c
-55°C
.84
.84
.807
486 k:t
496kt
468 k:t
-50 k:t
-35 k:t
-65 k:t
436kt
461 kt
403kt
3,430
3,660
2,926
CM<;1)
The conditions, rank:ed in increasing cost, rate
B, A, C.
From a total cost standpoint, the best flight condition of the three is at 30,000 feet. However, from
a range standpoint the three rank::
C =29.2( ~ )
A=31.3(~)
B = 31.8 ( ~ )
and the best condition is at 39,000 feet. Condition A requires 2.1 lb/nm and B, 2.6 lb/nm more
than condition C. In 1,000 nautical miles, condition A and B respectively require 2,100 and 2,600
pounds more fuel than condition C, and it may be necessary to fly the more costly condition to have
enough fuel to finish the flight with enough reserve on board.
3.196
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PERFORMANCE
K = 20,000
o
.,. 4
t~- 5
r:S
r:,q
reco
2000
14
(X)
t!>
dec
15
asing
ange
25
20
romms/NM
.
D6-1420 .
May 1989
"5J
35
Figure 160.
3.197
PERFORMANCE
Another point to consider is it will always cost something extra to change flight condition to a
higher altitude. Ha change from 35,000 feet at condition A to 39,000 feet at condition C were to be
considered, some of the advantage of the higher altitude is lost since these large transports take
about 100 pounds of fuel extra for cach 1,000 feet climbed.
Since,
400
2.1 = 190. 5
at least 190.5 nm must be flown at the new altitude just to make up for the fuel used to climb. And
an extra amount of fuel may be needed for pressurization and for the .additional descent at the end of
cruise.
Optimum Economy Cruise Speed
The relationship between economy cruise speed .and cost index, CI, based on the fuel mileage
curves, is derived from equation (181):
Let,
then,
C= et T + cf F
Since T is range divided by velocity and Fis range divided by fuel mileage, equation (181) can be
written as:
where,
R is cruise range, nam
Economy cruise Mach number is the Mach number where the sum of the variable costs and the fuel
costs is a minimum. For a given altitude and weight, optimum economy cruise occurs when dC/dM
= O, therefore,
dC _ ctR
(-=l)+c R( -1 Jd(T)
a
M2
f
nam 2
dM
dM -
3.198
. {n,)
D6-1420
May 1989
PERFORMANCE
dC =-R[--2_ +
dM
aM2
d{T)J= o
cf
(n:1)2
dM
and,
d(T)
dM
or,
2
.S.
_-( Je:;;,2M Jd{ dM
n:n) -CI
_
C a0
f
(189)
(n:1)
Equation (189) yields curves such as Figure 161 which shows that as the cost index increases the
speed for economy cruise increases. At very high cost index values the reverse may be true.
CONSTANT
ALTITUDE
300
O 200
x
w
C
~
t;
8 100
INCREASE
o
0.5
0.6
0.7
0.8
0.9
M
Figure 161.
To correlate the economy cruise Mach with the associated fuel mileage, it is necessary to crossplot
the economy speed schedule onto the fuel mileage curves, as shown in Figure 162. The minimum
cost index values in Figures 161 and 162 are for given temperatures and altitudes since equation
(189) shows temperatme as one of the variables.
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May1989
3.199
PERFORMANCE
CONSTANT
ALTITUDE
o
COST INDEX (ISA)
NAM/LB
M
Figure 162.
Economy Cruise with FMC
An economy cruise speed can also be determined for airplanes equipped with flight management
computers (FMC). Cost index is routinely deńved by airlines and entered by :tlight crews into the
onboard FMC. This enables the FMC to calculate the economy cruise speed schedule which
minimizes the cruise cost for each :tlight. The cruise cost in terms of cost index can be rewńtten as:
The cruise cost per nautical ground mile is obtained by dividing by ground distance, nm.
(~)=(CI) Cr(!)+ Cr ( ~)
(190)
Since,
(....L)=-1
nm
Vgr
and
(...Ł) = wf
nm
Vgr
and defining the (Cl) where Ci is in units of dollars per hour and Cr is in units of cents per lb,
equation (190) can be expressed as shown.
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May1989
PERFORMANCE
....Q... =
nm
C[100vgrCCD + vgr
WrJ
(191)
f
where V gr is the ground speed in knots. Equation (191) has units of dollars per nautical ground
mile. A more convenient form which requires only the cost index is obtained by dividing each side
of the equation by Cr. The resulting equation is known as the economy cruise cost function,
ECCF.
-( C )-(lOO(CI)+Wr)
V
ECCF- C
f nm
(192)
gr
Economy cruise Mach is defined as the speed at which economy cmise cost function is a: minimum.
A careful look at this equation is useful in understanding the relationship of economy cruise speed
and cost index. Both fuel flow and the expression 100 (Cl) are in units of lb/hr. Therefore, the
numerator is the sum of engine fuel flow, related to fuel oost, and an "equivalent fuel flow," which
is related to the time (operating) cost. If cost index is zero there is no time cost and the ECCF
evaluates as fuel flowN gr, in units oflb/nm. The economy cruise Mach at a cost index of zero is
the speed yielding minimum lb/nm of fuel, or maximum range cruise (corrected for wind).
Determination of the cost index zero economy cmise speed is illustrated in Figure 163.
CONSTANT ALTITUDE
ANDWEIGHT
ECCF
(LB/NM)
ci.o
MACH
Figure 163.
If cost index is riot zero there is a contribution to the ECCF from the equivalent fuel flow
expression. This raises the ECCF and moves the bucket of the ECCF cmve to the right, resultir1g
in a higher speed, as illustrated in Figure 164.
·
•
1)6.}420
May1989
3.201
PERFORMANCE
CONSTANT ALTITUDE
ANDWEIGHT
Cl-200
- - - - - - - ~ 01-100
ECCF
(LB/NM)
-------~ci.o
\
: :/
-------~
.
.
I I
MACH
Figme 164.
The economy cruise speed is also a function of the cruise wind componenL A headwind
component lowers the Vgr term in the ECCF and a tail'Wind component increases the Vgr term. The
result is an increase in economy cruise speed with a headwind, and a decrease with a tailwind, as
shown in Figure 165.
HEADWIND
- - - - - - - ~ ZEROWIND
ECCF
(LB/NM)
: :;
- - - - - - - ~ TAILWIND
\
.
.
------~
I I
M
Figme 165.
A family of economy cruise speed CUJVes may be generated as a-function of cost index, altitude, and
gross weight. This is done by computing the speed where dECCF/dM = O. This is normally done
using a computer, and iterating by searching for the Mach yielding the minimum ECCF for cach
combination of the independent variables. The result is cmves as shown in Figure 166. Note that
Figme 166 is a crossplot of Figure 161.
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PERFORMANCE
CONSTANT ALTITUDE
ZERO WIND
MACH
GROSS WEIGl-{r
Figure 166.
Economy Climb
For airplanes with FMC, an economy climb speed schedule is provided to help minimize trip cost
Economy climb speed schedules are normally fixed CAS/Mach, with the economy climb Mach set
equal to the estimated initial economy cruise Mach. Economy climb CAS is derived as a function of
cost index, gross weight, and other parameters such as temperature deviation and wind. These
speed schedules are generated by iteratively computing climb plus cruise trajectories from a
common initial climb point to a common point in cruise. This method allows the climb-cruise
interaction effect on trip cost to be included in the economy climb speed determination. The climb
CAS is varied and the trip cost for cach resultant trajectory is computed. The CAS value which
miniroires the climł, plus cruise cost at cach cost index and gross weight is selected as the economy
climb CAS. A typical plot of economy clim.b CAS as a function of cost index and gross weight is
shown in Figure .167.
·
ECONOMY
CLIMB
CAS
INITIAL CLIMB WEIGHT
Figure 167.
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May1989
3.203
PERFORMANCE
Economy climb CAS will be limited at the higher cost index and gross weight values to a speed
somewhat less than the maximum operating airspeed, VMOEconomy cliinb trajectories are steeper at low cost index values and flatter at high cost index
values, as shown in Figme 168~
Figure 168.
At a cost index of zero, the economy climb CAS is nearest to maximum rate of climb speed. This
decreases the climb time and mioiroizes the time spent at maximum climb tbrust with high fuel
flow. This results in significant trip fuel savings. As cost index increases it is necessary to increase
speed to decrease time costs. This results in a shallower climb, less overall time , a higher overall
fuel bum, but lower relative cost at the specified cost index.
Economy Descent
For airplanes with FMC, an economy descent speed schedule is provided to help minimize trip cost.
Economy descent schedules are normally fixed Mach/CAS, with the economy descent Mach set
equal to the finał economy cruise Mach number. Typically, the economy descent CAS values are
derived as a function of cost index, and sometimes gross weight. These speed schedules are
generated by iteratively computing cruise plus descent trajectories from a common cruise point to a
common end of descent point. This allows the cruise descent interation effect on trip cost to be
included in the economy descent speed detennination. The descent CAS is varied and the trip cost
for each resultant trajectory is computed. The CAS which minimizes the trip cost at each cost index
is selected as the economy descent CAS. A typical plot of economy descent CAS as a function of
cost index is shown in Figure 169.
3.204
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May 1989
PERFORMANCE
ECONOMY
DESCENT
CAS
COSTINDEX
Figure 169.
Economy descent CAS may be limited at the low cost index values due to ATC considerations, or
to avoid areas of reduced speed stability. The speeds are limited somewhat below the maximum
operating airspeed, VMo , at higher cost index values.
Economy descent trajectories are flatter at low cost index values (lower speeds), and steeper as cost
index increases (higher speeds). This is shown in Figure 170.
CRUISE FUEL FLOW
COMMON
CRUISE - - - ~ - - - - - - . . ·
POINT
- ~ COMMON END OF DESCENT
AT COMMON SPEED
Figure 170.
At a cost index of zero (minimum fuel), the economy descent CAS is near maximum 1/D. This
sttetches the glide and allows an earlier top of descent, which allows an earlier thrust reduction from
cruise thrust (higher fuel flow), to idle thrust (lower fuel flow). This results in significant trip fuel
savings despite the fact that the actual descent fuel bmn is higher due to the extra time in descent
relative to higher cost index cases . As cost index increases it is necessary to increase speed to
reduce the time costs. This results in a steeper descent, more time in cruise, a higher overall fuel
bum, but a !ower relative cost
D6-1420
Mayl989
3.205
PERFORMANCE
lntentionally Left Blank
3. 206
D6-1420
May 1989
PERFORMANCE
3-9 Descent
Rate of Descent - General
Descent derivations and calculations are similar to climb performance. When the thrust available is
greater than the thrust required for level flight, the airplane must either accelerate or climb. When
the thrust available is less than the thrust required, the airplane must decelerate or descend. The
generał equations for descent may be derived from Figure 171
r/ds- dh
dt
Figure 171.
The following generał equations can be written:
r/d
=- C: = V siny
(193)
The summation of forces parallel to the flight path is:
(clV)=O
dt
(194)
w V (dy)
L+ g
dt - W cosy = O
(195)
T-D+Wsiny- W
g
The summation of forces perpendicular to the flight path is:
Again, if the assumption that was made in the climb section is made here; that is, if the rate of
change of the flight path angle, dy/dt, is assumed to be small; then equation (195) becomes:
L=Wcosy
D6-1420
May1989
(196)
3.207
PERFORMANCE
During normal descents, 'Y is small and cos 'Y may be considered unity. Equation ( 195) then reduces
to L = W. However, ifyis large, such as during an emergency descent, the relationship expressed
in equation (196) must be used to obtain accurate results.
From equation 193,
.
1(dh)
dt
smy=- V
Substituting the above equation into equation (194), results in:
T-D-
~(:)-~(:)=O
This equation may be rearranged and each term multiplied by V/W.
V(T - D) _ V
w
-g
(d.V)
(dh)
dt + dt
(197)
But,
(:)=(:)(:)
Substituting this relationship into equation (197), the following expression is written:
V(T-D) _
w
V(dV)(dh)
dh dt + (dh)
dt
- g
or,
V(~-D) = -(':
Rewriting,
)[! (:')+ l]
dh)
V(T - D)
(eh
= W [ 1 + ; ( : )]
(198)
This is the same equation as was derived in the climb section. However, from equation (198),
r/d=-(':)
Thus, when equation (193) is substituted into equation (198), the generał descent equation
becomes:
r/d=
tD;(b)]
w l+g dh
(199)
In equation (199), r/d and V arc expressed in terms of ft/sec. To make this more usable with
available data, the velocity, V, may be expressed in knots and the equation adjusted by the
conversion factor 101.27.
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PERFORMANCE
The resulting equation, expressing r/d in feet per minute is:
r/d = 101. 27
V(T-D)
[
V
W 1+g dh
(dV)J
or,
(200)
M[_!L -....LJ
66987.J8 amb
6amb
6amb
r/d = - - - - - - - - - - - -
(201)
( dV )] ...:!!__
[l + V
g dh
6amb
From equation (200), to obtain the rate of descent at a constant true airspeed, the thrust available
and required for a given altitude and speed condition must be known. These data can be obtained
from the speed-thrust grids shown in Figure 172.
Maximum rate of descent will occur at a speed where (D - T/6am~ multiplied by M is largest. From
Figure 172, maximum descent rates
obtained by flying at maximum speeds.
arc
To obtain the greatest range while in descent, the descent glide angle, 'Y, must be minimum. From
equation (202),
.
(r/d)
(202)
sm 'Y= °V'"
Sin 'Y = (r/d)
V
D/6am.b
j
§
.§
.5
W/f,mnb _ _..;;;::E;;..._~
M
Figure 172.
Therefore, for 'Y to be minimum, (r/d)N must be a minimum. Considering dV/dh = O, equation
(199) can be written:
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May1989
3.209
PERFORMANCE
r/d= V (D; T)
Substituting the above in equation (202):
.
( r/d)
V ( D - T)
D- T
sm"f=~= V
W
= W
(203)
This equation states that the minimum glide angle, 'Y, occurs at a speed where (D -1) (or D!6amb T/6am~ is minimum. In Figure 172, this is the point where the thrust available and thrust required
lines are parallel
·
Using data obtained from the speed-thrust grids, rate of descent may be calculated and plotted as
shown in Figure 173. This figure shows variation in descent rate for varying speed and weight at a
given altitude and thrust setting.
Severa! interesting facts are apparent from Figure 173:
-;
Given Altitude
ldle Thrust
B
-i
8
"CS
8.
"CS
-
v.i
Added Drag Effects
(Gear and Airbrakes)
-...
C:
~
li,.;
::s
a:i
:::>
~
~
1~
~
Minimmn Glicie Angle, y
V - True Airspeed
Figure 173.
(1)
Minimum glide angle at a given altitude is approximately the same for all weights. If
tbrust were zero, minimum glide angle would be exactly the same for all weights and
would occur by gliding at the maximum lift/drag ratio.
(2)
The speed to maintain a minimum glide angle increases with weight. This is
necessary to be consistent with the statement in item (1), since as weight is increased
a higher speed is required to keep the same lift/drag ratio.
(3)
Maximum rate of descent is obtained at the highest possible airspeed.
3. 21 o
D6-1420
May1989
PERFORMANCE
(4)
At any constant high speed, the rate of descent is higher for light weights than for
heavy weights. This~ again, is a function of the lift/drag ratio of the· airplane.
(5)
The effect of adding drag by lowering the landing gear or raising the speed brak:es is
· to increase the rate of descent. This technique is used for emergency descent.
Operating a jet airplane at low altitude imposes a large penalty in fuel consumption and range.
Therefore, it is desirable to remain at cruise altitude until it is definitely determined that a landing
can be made shortly after descent High rate descents introduce pressurization considerations which
will be discussed later.
Minimum glide angle descent may be obtained from Figure 173. The speed schedule and rate of
descent are determined by the point of tangency on cach curve with a straight line drawn through
the ońgin. Flying the speed sched.ule for minimum glide ·angle does not necessarily yield
maximum overall range except in the special case of all engines inoperative (T = 0).
Ra.te of Descent
-8
.....a
<Descent Speed
IAS
r/d
Figure 174.
Descent Range, Time, and Fuel Used
For flight planning pmposes, plots which show descent prediction data (such as range, time, and
fuel used) are required. Since rates of descent are known and plotted versus altitude as in Figure
174, the time in descent may be obtained by step integration in the same manner as time to climb.
The descent range is merely average velocity times the time in descent. Fuel used is fuel flow
multiplied by the time in descent. A typical tabulation for determining these data is· shown in
Figure 175 on the following page.
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Mayl989
3.211
PERFORMANCE
h1to h 2
havg
Ab
r/davg
At
~ At
Vavg
A (nam)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Fig 173.
(5) X(/)
60
Select
~
A (nam)
(9)
Average ~ - h1
Wfavg
~
Fig 174.
-
Fuel
~
(10)
(11)
Engine
Data
(5) X (10)
(3)
(4)
AFuel
(12)
Given: Initial Weight
Speed Schedule
Oean Configuration
60
Figme 175.
Finał data may appear as shown in Figure 176~ The weights shown are those existing at the start of
the descent.
.g
a
·-=
<
TlDle - Hours
Distance - NAM
Fuel Bumed - Pounds
Figme 176.
High Rate Descents
Normal high rate descents will be made at high airspeeds with the rate of descent limited by cabin
pressuriza.tion requirements. The cabin pressuriza.tion requirement is that change in cabin pressure
equal to a rate of descent considered acceptable for passenger comfort. Since a sea level cabin
altitude can be maintained at all airplane altitudes up to approximately 22,500 feet, it follows that the
3. 212
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PERFORMANCE
airplane rate of descent is limited only by the airplane structural and performance limits at these low
altitudes. However, when the cabin altitude is not that of the field to which the airplane is
descending, as would occur at high altitudes, the rate of descent is govemed by the time it takes to
reduce the cabin altitude at a particular rate. A pressure change in the cabin equivalent to a rate of
descent of 300 feet per minute at sea level has been selected. To maintain this pressure change at
high altitudes requires a·decrease in the airplane rate of descent with a corresponding increase in
thrust. These thrust requirements arc high enough to make it necessary to operate the engines at
thrust ratings above idle. This increase in thrust affects the time, range, and fuel bumed. ·
The speed schedule chosen is one of either constant Mach number at the higher altitudes or a
constant indicated airspeed at the lower altitudes. The exact speed schedule selected is of necessity
determined by the cabin pressurization requirements establishing the maximum rate of descent
Thrust Presentation
During emergency descents, the thrust levers arc in the idle position. The same is true for norma!
descents except that at high altitudes some engines require a small thrust increase for cabin
pressurization requirements. These arc usually given on the descent charts as constant values of
EPR. Therefore, thrust setting charts arc not required. If, however, thrust EPR's or %N 1's are
required for power-on descents, they may be obtained from the climb or cruise thrust setting charts·.
D6-1420
May1989
3.213
PERFORMANCE
Intentionally Left Blank
3.214
D6-1420
May 1989
PERFORMANCE
3-10 Approach and Landing
The problem of determining landing performance is in most respects similar to the takeoff
calculation, varying only in the treatment of the approach and flare and in considering auxiliary
stopping devices such as the speed brakes. Detailed analysis and derivation of equations arc
omitted for those portions of the landing calculations which are similar to takeoff except perhaps in
algebraic sign and limits.
Approach Climb
When calculating landing performance, consideration must be given to a missed approach where a
go around must be flown with the airplane in the approach configuration. FAR Part 25.12l(d)
requires calculation of climb gradient where V s in this approach configuration does not exceed
110% of the V s for the related landing configuration. The steady gradient of climb may not be less
than:
2.1 % for two engine airplanes ·
2.4 % for tbree engine airplanes, and
2.7 % for four engine airplanes.
The approach climb is demonstrated:
(1)
.with the critical engine inoperative, the remajning engines at takeoff thrust
(2)
at the maximum J.anding weight
(3)
at a climb speed not exceeding 1.5 V s• and
(4)
with the landing gear retracted.
Normal all engine operating procedures are considered. A climb speed near the maximum climb
gradient is selected, approximately 145% V s·
For airplanes certificated to FAA AdvisOJy Circular 25-7, the maximum value of the approach climb
speed shall not exceed 1.40 times the 1 g stall speed, Vsig·
Landing Climb
Landing climb gradient charts are calculated for go around with the ai.rplane in the landing
configuration where all engines arc operating and the landing gear is extended. FAR Part 25.119
requires the steady clim.b gradient to be not less than 3.2 % with:
(1)
D6-1420
May1989
the engines at the thrust available 8 seconds after the thrust levers are increased
from the minhnnm tlight idle to the takeoff thrust position,
3.215
PERFORMANCE
(2)
a climb speed of not more than 1.3 V s·
A climb speed near the maximum gradient of climb speed is selected, approximately equal to 120%
Vs.
For airplanes certificated to FAA Advisory Circular 25-7, the maximum value of the landing climb
speed shall not exceed 1.23 times the 1 g stall speed, V sig·
Chart presentation of climb gradient is similar to the climb gradient charts discussed during takeoff
and climbout
Approach and Flare
The term approach, as used herc, applies only to the air distance from an altitude of 50 feet to
touchdowń. This air distance consists of two parts:
(1)
A steady state glide path where the airplane is in the finał landing configuration
prior to touchdown, and
(2)
the flare.
For data presentation, the speed during approach and f1are is equal to the legal minimum, 1.3 V S·
For airplanes certiticated to FAA Advismy Circular 25-7, the minimum value of the approach speed
is 1.23 times the 1 g stall speed, Vsig·
Figure 177 is a sketcb of the glide path and shows that the flare is assumed to be a circular arc of
rad.ius R. Remembering that 'Y is a small angle:
(204)
For small angles cos 'Y = 1 and sin 'Y = tan 'Y ='Y in radians. V is assumed to be constant during the
apP,J'Oach glide. An expression for 'Y and R must be detennined.
D=T+ W siny
L=Wcosy
.
D-T
sm 'Y= W
therefore, in radians,
(205)
3.216
D6-1420
May1989
PERFORMANCE
50 f't
R
...._ _ _ _ _ _ ..2Q_ - - - - - - - . . . i l - 4 )'
. toucbdown
Figure 177.
The accelerati.on normal to the flight path needed to fl.are is attained by rotati.ng the airplane to a
I
higher CL value, CL . The lift force during flare is,
The force norma! to the flight path is,
I
I
Fnor=L -Wcosy=L -W
But during the approach glide,
WEL=
cL(p~O')s v2
Therefore,
L c·L
w=cL
D6-1420
May 1989
3.217
PERFORMANCE
and,
c~
Fnor =We-W
L
Fnor = W(n-1)
I
since CL / CL = n, where n is load factor. The maximum value of n which may be applied to CL
is dictated by stall or buffet limits.
Acceleration normal to the glide path is:
dy
anor = V dt
and,
dy .
V=Rdt
therefore,
v2
a nor = - R
Now,
Fnor
= gw anor
therefore,
w
or,
v2
Fnor = W (n -- 1) =g- -R-
v2
R=--g (n -1)
But,
v2- L
1
2
- s a Po CL'
or,
v2= nW
S
2
a Po CL'
therefore,
2
1
S a Po CL' g (n - 1)
R= W
(206)
Collecting the information in equations (204), (205) and (206), and remembering that:
3. 218
D6-1420
May 1989
PERFORMANCE
·
W (CD
TJ
scL-w
50
s A = (CD _ _!_) +-----O'Po g (n - 1) CL
CL
(207)
W
The distance SA can be varied by assuming values of n from 1 to that corresponding to QMAX or
CLbuffet (which will affect the flare) and varying the rate of descent or y (which will affect both
approach and flare). Many rates of descent can be selected and distances calculated from the
minimum at CLMAX to large values in a very fiat approach. A commercial airplane, with high ra.tes
of descent available due to aerodynamic characteristics, can approach and land at very high angles
and in very short "over the fence" distances. Federal Aviation Regulations permit such profiles in
the authorized performance manuał but provide for increasing the field lengths by a factor of 1.667
to allow for more norma! service operation to destination airports. Commercial airplanes can land
with a finał approach descent rate of over 20 ft/sec but a nominał value as demonstrated is used in
landing calculations. The normal vałue of n used in flare is approximately 1.2.
The above method for analyzing the landing air distance has been used on all early Boeing airplane
certification. A simpler method whlch gives the same results has been developed. Thls method is
to average the demonstrated times from 50 feet to touchdown (essentially an average rate of
descent). Distance is the product of avera.ge time and avera.ge velocity.
SA = 1.6878 (
where,
vapp + Vm
2
)
- VW At.air
(208)
Vapp
= approach air speed at 50 feet, knots.
VID
= touchdown air speed for all air distances with a given flap setting,
knots.
~
= avera.ge time from 50 feet to touchdown for all air distances with a
given flap setting, seconds.
During past airplane certification, the tou~hdown speed was specified to be the same as the
approach speed, 1.3 Vs· Since actual flight conditions show a speed decrease of approximately 5
knots during landing flare, thls reduction in touchdown speed is now being accounted for in the
landing distance całculations.
Ground Run
Derivation of the expression for ground run is not included because it is given for takeofI ground
run in Chapter 3.5. The analysis is very similar except that the ground run consists of two parts:
06-1420
May1989
3.219
PERFORMANCE
· (1)
A short ground run (approx. 2 seconds) immediately following touchdown while the
airplane is being changed from the landing configmation to the braking configuration. During this
period the airplane is in normal ground attitude in the landing configura.tion, no brakes. This
landing transition distance, using the average transition speed, becomes:
(209)
where,
Vm
= air speed at touchdown , knots.
VB
= airspeed at full braking configmation, knots.
AtTRAN = transition time from touchdown to full braking configuration, seconds.
(2)
The remaining ground run brings the airplane to a complete stop. The CL and C 0
values used include the use of stopping devices such as air brakes and the friction coefficient which
corresponds to full wheel brakes as established from flight test. Full ground effects on lift and drag
arc used.
The calculation of the ground run braking distance is:
VW
SB=2.849
J
(210)
VB
where,
Vw
= the initial braking speed in knots.
= the headwind in knots.
a
= the deceleration rate in ft/se1:l
VB
The acceleration formula, as derived in the takeoff section, may be used.
(211)
where,
T
V
cp
is idle thrust.
is airplane velocity in knots.
is runway slope in radians ( positive for uphill slope).
The calculating procedure is identical with that used for segment E of the takeoff run. In fact, a
study of the expressions used in the analysis of the takeoff ground run suggests the application of
the same analysis to the landing technique.
3.220
D6-1420
May1989
PERFORMANCE
The total distance from a 50 feet clearance to touchdown, plus the rolling distance on the ground
before braking, plus the rolling distance after braking is the landing distance. This distance,
multiplied by 1.667, determines the FAR landing field length for a destination airport, and is for a
standard day on a level runway. The FAR landing field length is used as one of the criteria to
establish the landing weight, and a presentation is shown in Figure 178. A similar presentation for
. the distance from a 50 feet clearance height can be given.
Wet FAR Landing Field Length
The FAR landing field length for wet runways is equal to 115% of the FAR landing field length for
dry runways unless it can be demonstrated to be less. However, it may never be less than the dry
FAR landing field length. The demonstrated values are equal to 115% of the wet stopping distance
subject to the following conditions:
(1) For certification, the landing is preceded by a steady gliding approach not to exceed
an angle of 3 degrees down to the 50 foot height and at a calibrated airspeed not less
than 1.4 Vs.
(2)
The time lapse between the 50 foot height and touchdown is not less than. seven
seconds.
(3) The landing is made on a level, smooth, hard surfaced, wet, well soak:ed runway.
(4) The wheels are fitted with tires which have been wom to a point where no more than
20 percent of the original tread remains.
A typical FAR landing field length chart is shown in Figure 178.
i
IIO
.5
J
+
Gross Weight
Increasing
...
Wmd
Runway
Condition
Figure 178.
D6-1420
May1989
3.221
PERFORMANCE
Intentionally Left Blank
3.222
D6-1420
May 1989
APPENDIX
.........-.-........~.,......,=...--..,..
·_..,.·=·
_ ::::::
__,..-.
_ · =,.""',.
:!:::
o
!,,-::.:. . =-=:..,.:!,-:::....., -=::-.:!,-::""',.,....: =-=:....
!, : : :. :g
,.·.,"',.1•.,..,.:.....
: : -::--:
: i,_::""'::,....::-.:
: 1,_::-.:: "'°::...,.::""'i::..,.,.::'"'::""::'"";::..,.::.,.,.,-.:::""1:-:::.,...,-=::..,.r-:::.,...,-::.....
= !, ,-:
: ...
: =-=
:...,.:
!.,....
,. '· ,_,.,.,,_....
,.
-~ ·•··
···•··
···•··
... : ..
···•··
···•··
:::j::
::::::
··•···· ····ł
::1:::: ::::;
1
··=····
....
:
····ł
..··•····
,........
.
::i:::: ::::i
::::::: ::::::
[6:..{[I+.2(:: J2r-1} +Ir~ -1
:o
: in
: in
= Mach number
= p/p 0 , Standaid AtmJiphece
= Aiq>eed indicatorreading correctedfor instrument
and position error (airspeed indicator calibrated to
read true airspeed at standard sea level conditions}
o
w
w
Il.
Cl')
CC
<
o
~
CD
::i
~
····t···
....
, ...
····•···
::::!:::
.... , ...
••••ł•••
::::i:::
o
ą
.....
o
o,
o
o
a,
o
o
co
o
o
IO
o
o(")
.....
o
o
1:::13SWnN HOVW 3nW.
D6-1420
May1989
Chart 1.
A.I
APPENDIX
HIGH SPEED POSffiON ERROR AIRSPEED
dM
dVc
{kn:ts)
~ ~~Hg~::::~==:~
0.0014 i
O
::::;:;~~~=~~i ~~~H~~g ~g~~~~~~ ggn~~~ i~~~łgn g~~j~~~~ ~~gr~~~
:>:: ??::: 11111:1 \111/11 :::t::: !:?:::! \}? !\)?! :!}!? !!\iii!
100
200
300
400
500
CALIBRATED AIRSPEED (Vc) KNOTS
Chart2.
A.2
D6-1420
Mayl989
APPENDIX
···t····
::::::::: ~::
~::::
.... t::-::" ... ,....
.... ; .... :::;::::
.... , ....
····•····
.... ....
~
~
oz
::.::
()
>
o
w
w
Il.
Cf)
oo
(')
CC
<
o
w
~
CC
m
::i
o<
o
N
N
COMPRESSmil..ITY CORRECI10N IE-:-~~~~
TO CALIBRATEDAIRSPEED
Ve=Vc-/iVc
Sl.ON>I
:> AV
Chan 3.
D6-1420
May 1989
A.3
APPENDIX
Chart4.
A.4
D6-1420
May1989
APPENDIX
D6-1420
May 1989
A.5
APPENDIX
Intentionally Left Blank
A.6
D6-1420
May 1989.
APPENDIX
Symbols
The following list of symbols and their identification is included as a convenient reference. It
represents a fairly complete listing of the more standard aerodynamic and engine terms used in
industry.
a
= speed of sound, ft/sec.
a
= acceleration, ft/sec2
a
= slope of the lift curve(CL vs a), per degree.
a.c.
= aerodynamic center, clistance from leading edge, ft or fraction of chord c.
A
= cross-sectional area, sq.ft.
= wetted area, sq.ft.
AR
= aspect ratio = 'Ó1-/S.
= wing span, ft.
= chord of wing, ft.
b
C
C
= mean aerodynamic chord, ft.
= chord at the root of tapered wings, ft.
= chord at the tip of tapered wings, ft.
= drag coefficient, climensionless.
= induced drag coefficient, climensionless.
= parasite drag coefficient, climensionless.
= equivalent parasite drag coefficient, climensionless.
= frontal area drag coefficient, climensionless.
= Reynolds number increment of drag coefficient, climensionless.
= wetted area coefficient, climensionless.
= lift coefficient, dimensionless.
= location of lift and drag forces, c.pJc.
= pressure coefficient, climensionless.
C.I.
D6-1420
May1989
= cost index.
A .7
APPENDIX
c. p.
= center of pressure, distance from leading edge, ft.
c.g.
= center of gravity.
BTIJ
= specific beat at constant volume, lb _ op
= specific beat at constant pressure,
:~F
= pitching moment coefficient = MS
cq
Cmac.
= moment coefficient about the aerodynamic center.
d
= diameter, ft; length or dis~ce. ft.
= drag = C0 q S, lb.
D
= induced drag, lb.
= profile drag, lb.
= bulle modulus, lb ft2.
e
e
= 2.718281828, base of natmal logarithms.
f
= equivalent parasite drag area, sq.ft.
F
= force, lb.
g
= acceleration due to gravity = 32.174049 ft/sec2 at sea level, 45° latitude.
~
= constant = 32.174049 ft-J.bnt1hf-sec2.
b
= altitude, ft.
= pressure altitude, ft.
= indicated pressure altitude, ft.
= altimeter position-error correction, ft.
A.8
1)6..1420
May1989
APPENDIX
= test indicated pressure altitude, ft.
1
= incidence angle, degrees.
L
= taił length, wing c/4 to tail c/4, in.
In
= natural logarithm.
log
= common logarithm, to the base 10.
m
= slope of the lift curve, per radian.
M
M
= mass in slugs, lb sec2 / ft.
= pitching moment = Cm c q S, ft lb.
M
= Mach number.
= free-stream Mach number.
n
= load factor.
= revolutions per minute.
n
= number of engines.
nam
= air distance, nautical air mile.
nm
=
Pamb
= ambient pressure, lb / ft2.
p
= static pressure, lb / ft2.
Ps
= reference static pressure = Pirer + APs• lb / ft2.
n
ground distance, nautical mile.
= total pressure, lb / ft2.
Piref
=
trailing cone reference indicated static pressure, lb / ft2.
=
airplane test static pressure, lb / ft2.
= differentia! pressure =Pirer - ~ lb/ft2
= Trailing cone position error correction lb/ft2
D6-1420
May1989
A.9
APPENDIX
Po
= free-stream static pressure, lb / tt2.
q
= dynamie pressure = p V2/2, lb / tt2.
·= impact pressure = (pt - p), lb / tt2.
= reference impact pressure = <lcirer - APs• lb / tt2.
= reference impact .pressure = Pt - Pirer , lb / tt2
<lcitest
R
~lref + AP , lb / tt2.
= test impact pressure = a_
= range, nautical air miles, nam.
·R
= radius of tum, ft.
R
= gas constant, ft lb,' lbm 0 R.
R'
= therma} gas COnstant, ft2/ sec2 °R.
r/c
= rate of climb, ft/min.
r/d
= rate of descent, ft/min.
RN
= Reynolds number, dimensionless.
s
= wing area, sq. ft.
SG
= ground run, ft.
Sx
= properarea, plan or frontal, sq. ft.
S.L.
= sealevel
SM
= statute mile
Os
= stall condition.
t
= aińoil thickness, ft.
t
= temperature, °F.
T
= total thrust = n Fn• lb.
T
= absolute temperature.
Ttrop
= absolute temperature at 36,089.24 feet (11000 meters).
A .1 O
D6-1420
May1989
APPENDIX
T.O.
= takeoff condition.
UDE
= design gust velocity, ft/ sec.
V
= specific volume, ft3/lb.
V·1
= indicated airspeed, (uncorrected), kt.
= indicated airspeed, (corrected), kt.
= calibrated airspeed (or computed airspeed), kt
= equivalent airspeed, kt.
V
= true airspeed, kt.
VA
= design m.aneuvering airspeed, kt.
VB
= design airspeed for maximum gust intensity, kt.
Vc
= design cruising airspeed, kt.
Vn
= design dive speed, kt.
Vp
= flap limit airspeed, kt.
vI.E
= landing gear extended limit airspeed, kt.
V1.0
= landing gear operating, kt.
VMCA
= air minimum control airspeed, kt.
VMCQ
= ground minimum control airspeed, kt
VMo
= maximum operating limit airspeed, kt.
= auplane stall airspeed, kt ·
= takeoff decision speed, kt.
= engine failure speed, kt
= minimum unstick speed, kt
= takeoff rotatioo speed, kt.
= takeoff lift-off speed, kt.
= takeoff safety airspeed, kt
D6-1420
Mayl989
A .11
APPENDIX
w
= airplane weight, lb.
w
aw
= specific weight, Ibi ft3.
= angle of attack of wing, degrees.
= angle of attack for zero lift, degrees.
'Y
= ratio of specific heat, 9 Cv-
Il. (delta)
= increment notańon.
= pressure ratio, p/p0.
a (alpha)
a (delta)
aamb(delta) = pressure ratio, Parni/Po·
A (lambda) = sweepback angle (.25 c ), degrees.
8 (theta)
= temperature ratio, T/To-
eamb
= temperature ratio, T amb"I'o-
8 (theta)
8 (theta)
= climb or descent angle, radians or degrees.
= tmn angle, degrees.
Cl> (phi)
= bank angle, degrees.
Cl>
= runway slope.
er (sigma)
= density ratio, plp0 •
A. (lambda)
A.
= standard day temperature lapse rate, .00356616 °R/ft (.0019812 °K/ft).
= wing taper ratio, clCr
p (rho)
= mass air density, slugs/ft3.
Po
= mass air density at sea level standard day conditions, slugs/ft3.
µB (mu)
= coefficient of braking friction, dimensionless.
A .12
1>6-1420
May1989
APPENDIX
= coefficient of rolling friction, dimensionless.
= dynamie viscosity, slugs/ft sec.
µ
7t
(pi)
e
= 3.141592654.
= downwash angle, radians.
The following is a list of standard gas rurbine symbols used by engine manufacturers.
Cr
= fan nozzle gross thrust factor, dimensionless.
Cg
= gross thrust coefficient, dimensionless.
EPR
= engine pressure ratio = PtfPt2.
Fg
= gross thrust, lb.
F/6amb
= corrected gross thrust, lb.
Fn
= net thrust per engine, lb.
FJ6amb
= corrected net thrust, lb.
a F0
= net thrust change, lb.
Fr
= ram drag, lb.
F/6amb
= corrected ram drag, lb.
FHV
=
J
= energy conversion factor, Joule's constant, 778.26 ft-lb/BTU.
n
= number of engines.
N1
= low pressure rotor speed, rpm.
D6-1420
May1989
heat value of fuel, 18580 BTU/lb.
A .13
APPENDIX
N1 f
j8:; = corrected low pressure rotor speed, rpm.
N2
= high pressure rotor speed, rpm.
j8:; = corrected high pressure rotor speed, rpm.
N 2/
= total average pressure at compressor inlet, lb/ft2.
Pt7
= turbine discharge total pressure , lb/ft2. ·
Pt/Pamb
= ram pressure ratio.
Pt2iPt1
= inlet average pressure recovery ratio.
Pt7/pamb
= jet nozzle expansion ratio.
Pt/Pt2A
= engine pressure ratio, EPR.
= engine inlet average total temperature.
T
= total thrust = n F0 •
= tmbine discharge total temperature.
= thrust specific fuel consumption, Wf /F0 , lb/hr / lb thrust.
= engine airflow, lb/sec.
= total gas flow, lb/sec.
= primmy engine airflow on fan engines, lb/sec.
= fan airflow, lb/sec.
= corrected airflow, lb/ sec.
= engine fuel flow, lb/hr.
=
A .14
corrected fuel flow, lb/ hr, based on total conditions.
D6-1420
May1989
APPENDIX
Oamb
je;;:;, = corrected fuel flow, lb/ hr, based on ambient cond.itions.
0
= temperatare ratio, Tami/I'o-
812A
= ram temperature ratio, Tt/f0•
6 amb
= pressure ratio, Parni/Po·
812A
= ram pressure ratio, Pt/Po·
amb
Tl (eta)
= ratio of specific heat
= effi.ciency, per cent.
'I' (psi)
= theoretical gross thrust function, dimensionless.
'Y
The following is a list of subscripts used throughout this document.
a
amb
avg
C
= air, (W8 ).
= ambient, <Pam~·
= average.
= compressible, (Cle).
f
= critical, (per)·
= fuel, CWr>·
g
= gross, CFg>·
i
= indicated, (Vi).
J
= jet, (V1 ).
n
= net, (F0 ).
o
= standard sea level values, (p0, T0, etc.).
er
D6-1420
Mayl989
A .15
APPENDIX
p
= propulsion, ( Tl p).
r
= ram, (Fr).
s
= static, (Ts , Ps ).
t
= total, <Tt• Pt).
t
= thermal, (Tl t>·
V
= volume, (ey).
For calculation purposes, the following values of the constants are used:
g
= 32.174049 ft/ sec2
~
= 32.174049 ft/ sec2
Po
= 2116.2166 lb/ ft2
=
14.695949 lb/in2
=
1013.25 mb
= 29.92126 Inches of Hg
To
= 518.67 °R = 59°F
= 288.15 °K = 15 °C
Po
= .002376892 slugsfft3 or lbm-sec2/ft4
aa
=
1116.4501 ft/sec
= 661.4786 kts
= .00356616 °F/ft
= .0019812 °C/ft
R
= 53.352374 ft lbtlłbm 0 R
= 96.034273 ft lbtlłbm °K
A .16
D6-1420
May1989
APPENDIX
R'
= ~ R = 1716.5619 ft"/sec2 °R
= 3089.81142
Ttrop
ft2Jsec2 °K
= 389.97 °R = -69.7 °F
= 216.65 °K = -56.5 °C
= 0.239936923 BTU/lbnfR
= 6fJJ7 .96665
ft21sec2 °R ·
= 10814.33997 ft2Jsec2 °K
= 0.171383516 BTU/lbofR
= 4291.40475
ft2Jsec2 °R
= 7724.52855
f.t2Jsec2 °K
'Y
=
J
= 778.26 ft-lb/BTU
1)6..1420
May1989
1.4 (for air)
A .1 7
APPENDIX
Conversions
= 0.3048 meters exactly.
1 nm
= 1852 meters exactly
1 nm
= 6076.115486 ft.
1 nm
= 1.1507794481 sm.
1 meter = 3.280839895 ft.
= 39.37007874 in.
1 knot
= 1.687809857 ft/sec.
= 0.514444444 m/sec.
= 1.852 k:ilometers/hr. exactly
1 kg
= 2.204622622 lbs.
1 lb
= 0.45359237 kgs. exactly
1 inch
= 2.54 cm.
1 radian = 57.29577951 degrees.
1 ft
1 sług
1 BTU
A .18
= 32.174049 lbm sec2/ft.
= 778.26 ft lb.
1)6.1420
Mayl989
APPENDIX
BASIC
SECTION
EQUATIONS
Aerodynami es
1,
Chapter 1-2°
pressure force per unit area,
F
p= A
differential pressure,
dp = wdh
Chapter 1-3
Newton's 2nd law equation,
F = Ma
w
mass,
M=g
mass density,
w
p=g
lb
lbm - sec 2
sług, or
ft
slugs
7
Chapter 1-S
conti.nuity equation,
pAV = constant
slugs
sec
or,
Chapter 1-6
incompressible Bernoulli equation,
p+q = Pt
( limited to approx.
250knots)
dynamie pressure,
Chapter 1-7
equation of state (generał),
or,
D6-1420
May1989
p= pgRT
valid for any process
p=pR'T
A 19
APPENDIX
ft -lb
thermal gas constant,
R = 53.352374
f
0
lb m R
R' = 1716.5619
p'Y = constant
equation of state (adiabatic process), P
( = 1.4 for air)
ratio of specific beat,
Chapter 1-8
~T3/2
dynamie viscosity coefficient,
µ= T+S
lb- sec
ft2
lb- sec
ft2'V°K .
lb- sec
ft2
T O=288.15 °K
S = 120 °K,
Sutherlands constant
Chapter 1-9 .
speed of sound,
A20
a=J!
JL
sec
D6-1420
May1989
APPENDIX
Chapter 1-10
compressible Bernoulli equation,
( ~1
1
or,
}+ f =constant
Chapter 1-11
adiabatic flow relationsbips:
2:1. _[l+ 'Y -2 1 ~
_ .2]
abs total temp _
abs static temp - T -
-y-1
y
[1
~ pressure =!! = + 'Y P
stanc pressme
2
1M2lJ
1
.
~ mass densi~ =f.! =[1 + y- 1 M2l
stanc mass density
Mach number,
P
2
V
M=a
~1
J
dimensionless
Chapter 1-12
Annospheric relationships:
basie detinitions,
~=..l..
Po
8=.L
To
p
a= Po
D6-1420
May1989
~
=e
A21
APPENDIX
standard day below the tropopause. (36,089.24 feet / 11,000 meters):
T=T 0 -lh
~ _ _g_ _
(_!_)
o
o
U-p -
T
1
l. R _ (_!_)5.25588 _ 95 _25588
-
T
o
-
standard day above the tropopause. (36,089.24 feet / 11,000 meters):
p
6=-=6
Po
trop
-(h-36,089.24)
RT
e
trop
6 trop = . 22336087
-(h-3608924)
RT
a=-=a
e
trop
po
trop
p
a trop =.29708
o
T trop= 216.65 K
RT trop = 20805.825
e= 2.718281828
speed of sound ratio,
A22
dimensionless
D6-1420
May1989
APPENDIX
Chapter 1-15
Aerodynamie properties:
lb
basie lift equation,
eorrected lift,
basie drag equati.on,
corrected drag,
L
2
6°
= 1481.351 CL M S
lb
D=C 0 qS
lb
~ = 1481.351 S>M S
2
pYfps
av2m
lb
lbs Jti2
dynamie pressure,
q=
for level flight,
L=W
lb
for unaccelerated flight,
D=T
lb
2
= 295.375
Reynolds number effect on drag:
CD= CD
D6-1420
May1989
REF
+ ACD
RE
A23
APPENDIX
Physical propenies:
aspect ratio,
dimensionless
taper ratio,
dimensionless
m~ aerodynamic chord,
ft
SECTION 2, Powerplant
Chapter 2-4 to
Chapter 2-12
1) Single exhaust turbojet engin~:
a) basie thrust equation,
lb
M S 1 in tailpipe
w.
(wr) V. + A ·(Ps - p amb'\
F n= g{'J - Va ) + g
A.24
J
J
8
.
. .
'J M>lmtailp1pe
D6-1420
May1989
APPENDIX
b) corrected net thrust (engine calibra.t:ion),
•
1:::J [;j
* specific engine data
where,
w.Je:i
= ao
g
J
ot 2
M
= 1'3Ill drag parameter
1-1 2
1+-M
·
2
2) Dual exhaust turbofan engines:
a) basie thrust equation,
F 8 =Fg +Fg -Fr
p
Fn=
w2
lb
f
g (~-V.)+
w
wr
gfl.(Vr-Va)+gVj
MS 1 in tailpipe
b) corrected net thrust (engine calibra.tion),
• specific engine data
D6-1420
May1989
A25
APPENDIX
3) Gross thrust function, V:
p)r121.. ~(!!.)!:!. ] 1~
(~
v= ({f)
y
y-1
- 1; 1 - 1
J
p
where,
4) Fuel flow:
a) function of
.Ja:;;,,
*
•
•
Wr=(TSF0F =(]e:J (li::J li~
8
* specific engine data
A26
~1420
May1989
APPENDIX
b) function of
e:mb '
wr
*
*
amb
amb
=(TSFC)
(..!!!...)
a ex
ex
6
amb amb
* specific engine data
5) Net thrust and RPM:
*
F8
=( :.:J 6 *
*
N=(J~'2) ~ J1+1;2M
2
* specific engine data ·
D6-1420
May1989
A27
APPENDIX
SECTION 3, Performance
Chapter 3-1 to
Chapter 3-2
speed relati.onships,
VI =V.+
AV.1
1
VC =VI+ AVP
Ve=Vc-AVc = Ma 0 ~
V=Ve }a=Ma 0
Ja:;;,
M=MI+AMp
altitude relationship,
alti.tude and speed relationship,
1) M = function of (6, Vc>•
y-1
T
-,il-- --1:_
y-1
-1
2) Ve = ńmction of (6, Vc),
1
3.5
3) compressibility correction,
A.28
1)6.1420
Mayl989
APPENDIX
position-error corrections, .
1) Mach number position-error,
L.\Mp= M-M1
test
-1
where,
= ttailing cone static pressure
P1
Ilf
and,
p1
~1420
May1989
= test system static pressure
test
A29
APPENDIX
2) airspeed position error, (low and high speed)
AVp=Vc-V 1
test
3) altim.eter position error,
a) for altitudes below the tropopause, (low and high speed)
or,
p OJ902631]
hp= 145,442 [ 1-(p~)
and,
p
1
hl..,= 145,442 [ 1-( / ;
OJ902631]
J
b) for altitudes above the tropopause, (low and high speed)
A .30
D6-1420
May1989
APPENDIX
or,
hp = 36,089. 24 - 20,805. 8251{4.477°': ~ )]
and,
\ ... = 36,089.24- 20,805.825 In[4.477~p;; )]
position-error corrections, (altemate method),
1) high speed,
a)
&i =
P
(A
'\u.
dMFVJ.p
(!) =
b)
where,
38741.2 ~ (
AMP
(~J
AVp=
MM2)
1 +.2
where,
pt
1
5
(~) =
(1+.2M2)[1+.2(:~)
Vc
6 a 2o M
Pt
2) low speed,
a)
&ip=
(d~c)pt /!iV P
[
(:tJ
= 0.08854 I
pt
b)
1(6)
+.2( ::)
25
ft
),et
8
2 /!iVP
l!ip · Pamb -ps
q
V + 2 /!iV P =
=
1
D6-1420
May1989
where,
q
A31
APPENDIX
Chapter 3-3
airplane and engine grid:
a) by definition,
wr
lb/hr
n
lb thrust.
TSFC=p
b) corrected fuel flow as a function of
Je;::;,,
.!!?.
hr
or,
(
aaumJe.,.b = a.,_je.,_ Ji+-r ArJ
wf
) (
wf
lb
y-1
iir
X
b) corrected fuel flow as a functi~ of 8 amb'
(
(TSFCX F. )
6 amb8:0.b = 8:0b 6 amb =
wf
)
0.5-x
Ma 0 (8amJ
{U:) 6amb
1b
iir
or,
(
)=(
w~ Ji+~ArjS+•
6 ambe amb
6'2_8
wf•
'2
1!?.
hr
x values determined for cach specific engine
A.32
D6-1420
May1989
APPENDIX
Chapter 3-4
stall speed,
295.375 nW
CL S
V es =
where,
s
n = load factor = ł
by defintion.
level-flight bank,
Lcos+=W
+ = bank angle
n =1cos+
radius of tum,
R=
v2
ft
gtan.
gust load factor,
KgUdVem
e
n=l+
(w)
498.4 T
.88 µg
K =
g 5.3+ µg
where,
and,
- 2(-})
µg - pcmg
D6-1420
May1989
A.33
APPENDIX
Chapter 3.5
Takeoff performance:
a) ground distance to rotation speed,
ft
b) increment of ground distance,
ft
where,
AV=Vy -VX
g
a= W [(T - µ W) - ( C0 - µCL)S q - W cl>]
ft
a=
sec2
q, = runway slope,
radians
ground distance,
ft
A.34
D6-1420
Mayl989
APPENDIX
c) rotation distance to lift-off speed,
SB = 16878 (
2
VR +VLOF
)
.
.2
- VW Aty tDV
.
R LOF
d) flare distance to V 2 speed,
Sc = 1.6878 (
VLOF + V2
)·
2
- VW AtvLOF to v2
e) ground distance (simplified method),
f) Earth's boundary layer wind cOITCCtion,
1
7
Wmd 1 -(heighti )
- height
2
2
Wind
g) time for takeoff,
At= A_V
a
min
t=r&:
D6-1420
May1989
A35
APPENDIX
Chapter 3-6
Climb performance:
a) rate of climb,
dh
r/c(
acrebated
(T;D)v
) = dt =
ft
min
V dV
l +-g dh
or,
66987 ~
r/c (aa:ebared) =
V dV
1+-g dh
b) acceleration factor,
'Y M
V dV
2
gdh= 2
where at,
and,
'I'
dimensionless
constant M;
constant Ve;
'lf= 1-AR(Tr")
constant Vr:.
'lf=e!>-AR(Tf)
AR =0.190263 below the tropopause
AR= Oabove the tropopause
'Y = 1.4
·-
1
- 0. 7
A .36
[( 1+.2M2) 3.S -
"M:' [( 1 +.2M2)
2
lJ
J
1)6..1420
May 1989
APPENDIX
c) climb gradient,
C
1) unaccelerated clim.b gradient = ;
- CD
L
CD)
_I__
(
.
.
W
CL
2) accelerated climb gradient =
v dV
.
1+---g dh
3) acceleration gradient= {
.4) gradient interrelationship,
or,
where,
T
CD )
W
CL
(
.
.
..
=the climb gradient remaunng
residual
d) climb time, fuel, and distance,
1) time,
t = l:At
D6-1420
May1989
min
A37
APPENDIX
2) fuel,
fuel = I: Muel
lb
3) distance,
A(nam) =
vavg (~)
(nam) = I: A(nam)
nam
Chapter 3-7
a) range equation,
R = ( range factor)
1
avg
1n ( ww )
R=2.30259 (w n:')
2
1
log(: )
2
avg
R= ( T:ic )(Mt) hl(::)
~
A .38
.
where TSFC is a function of
.Je:;,
where TSFC is a function of
e:Ob
D6-1420
Mayl989
APPENDIX.
nam
R= I: AR
nam
b) specific range,
where TSFC is a function of
(":')=
(~ex~:<)-)
x
eamb
Ja:;;,
0.5-x
-~- ~amb (no. of eng~
amb
where TSFC is a function of {
e:.Ob)
c) range time,
~
At=Vavg
hr
t=I:At
hr
d) non-standard temperature effects when maintaining constant M,
1) on velocity,
2) on fuel flow, where Wf is a function of {
Tambns
Wf =Wf
DS
D6-1420
Mayl989
sai (
T
amb
a:mb)
)x
sai
A39
APPENDIX
3) on specific range, where Wf is a function of {
e:.Ub)
e) wind effects,
where Ver = average cruise speed
Reynolds correction
{ 6(8+~;8312)}
4CDRE = Alog!O {6(8+i831:l)} aclUal
iefenu:e
Where:
A is a dimensionless constant evaluated for cach aircraft
Chapter 3-8
Economy operation:
a) economy cruise,
where,
A.40
C = total trip cost,
dollars
C lb = cost per pound fuel,
n,
cents
D6-1420
May1989
APPENDIX
C br = hOID'ly cos t,
dollars
hr
T = trip time.
hr
F = trip fuel,
lb
b) economy cruise speed cost index,
c) economy cruise cost function,
lb
nm
Cbapter 3-9
Descent peńormance:
a) accelerated rate of descent,
dh)
V(D-T)
(
r/d ax: = - cit =
[
.
Wl+g dh
i(.!!Y.)~
·ft
mm
66987-yvamb
~ M[..JL.....I...]
6amb
6amb
b) unaccelerated mte of descent,
r/d
D6-1420
May1989
_ V(D-T)
unaccW
ft
mm
A41
APPENDIX
66987
~ M[_JL-
-vvamb
c) descent gradient,
.
1) unaccelerated descent gradient =
2) accelerated descent gradient =
_!_]
6amb . 6amb
c: -
(C WTJ
T) 1
(CD
CL - W [ + "t(: )]
1
d) descent time, fuel, and distance,
1) time,
t=I.&
min
2) fuel,
A fuel =Wf
(ł)
avg
fuel = I. Afuel
1b
3) distance,
A(nam) =Vavg (ł)
nam= I. A(nam)
A.42
nam
1)6..1420
Mayl989
APPENDIX
Chapter 3-10
Approach and landing:
a) flare distance,
1) early Boeing certification,
s -
SO
~ ~ : - ~)
A-(CD_.!..)
CL W
+~----~~apog(n-l)CL
2) present Boeing certification,
SA= 16878 (
vapp + vm
2
)
- VW &air
b) transition distance,
C) braking distance,
where,
g [
av2 S
]
a= W (T- ~ W)-(Cn-~CL) 295.375 -w+
and,
T is idle tbrust.
V airplane velocity in knots.
+ is runway slope in radians ( positive for uphill slope).
D6-1420
May1989
A43
APPENDIX
Intentionally Left Blank
A.44
~1420
May1989
APPE~DIX
707-lOO.SERIE
PRINCIPAL DIMENSIONS
AILERON
WING
Area •••••••••••••
2433 Sq. Ft.
Span. •••••••••••••
Root Chord •••••••
130 Ft. 10 In.
Tip Chord ••••••••
Taper Ratio ••••••
' Incidence Root •••
Incidence Tip ••••
338.In.
112 In.
.342
Inboard Area •••••
Olltboard Area
....
HORIZONTAL TAIL
Stabilizer Area •• 382. 7 .ail.. Ft~Elevator Area •••• 117 .6 Sq. Ft.
Span. ••••••••••••• 39 Ft •• 8 •.J.. In •.
.+:20
+2•
70
Sweepbaclt C/4 •••• 35°
Dihedral. •••••••••
Aspect Ratio •••••
M.A.C. •••••••••••
M.A.C. LOC. (St.a) ••
7.o65
VERTICAL TAll.
241.88 In.
78(;>.2
Dorsal Area ••••••
Fin Area •••••••••
Rudder Area ••••••
Ventra1 Fin Ar.ea •
FLAP
Leaa1ng F.dge Area.
Trailing F.dge Area
D6-1420
May 1989
26. 1 Sq. Ft.
361.65 Sq. Ft.
39.0 Sq. Ft.
8o.6 Sq. Ft.
-B>DY
Length
8. 7 Sq. Ft.- 226.4 Sq. Ft.
1_Q1 .9 Sq. Ft.
25. 5 Sq. ft~_
........... 1,38 Ft.
1O In. --
A.45
APPENDIX
707-l 0013 'SERIES
PRmClP.AL DDmNSIONS
WING
/
Area •••••..•••••• 2433 Sq. Ft.
Span • • • • • • • • • • • • • 130 Ft • 1 O In.
Root Chord ••••••• 338 In.
'rip Chord •••••••• 112 In.
Taper Ratio • • • • • • • 342
Incidence at Root. +2°
Incid.ence at Tip. +2C?
Dihedral. ••••••••• 70
Sweepback c/4. . • • • 35°
Aspect Ratio ••••• 7.o65
M.A.C •••••••••••• 24i.88 In.
Leading F.dge MAC. Station 786.2
-
FI.AP
AILERON
Inboard Area
Outboard Area
39.0 Sq~ Ft.
8o.6 Sq. Ft.
BDRIZONTAL TAIL
Stabilizer Area •• 427. 7 Sq. Ft.
Ele'VB.tor Area • • • • 1 i7. 6 Sq. Ft.
Sp&IL • • • • • • • • • • • • •
43 Ft.
VERTICAL TAIL
......
DorsaJ.. Area
8. 7 Sq. l't.
FirJ. Ares ••••••••• 226.4 Sq. Ft.
Rud.der Area •••••• 101 .9 .Sq. Ft.
VentraJ.. Fin Area. 25.5 Sq. Ft.
B>DY
Leading F.dge Area 74.1-9 Sq. Ft.
Trailing F.dge Area 36J •65 Sq. Ft.
A.46
Length • • • • • • • • • • •
or
138 Ft. to In.
128 Ft. 10 In.
D6-1420
May 1989
APPENDIX
.......
'.
lF\
----~---
••••1111118amu••caf)la~a••omr
-··
··-· :.."":--
707-200 SERIES
PRINCIPAL DIMENSIONS
/
WING
AILERON
2433 Sq. Ft.
130 Ft. 1O In.
Root Chord •••••••• 338 In.
Inboard Area.
Ou.tboard Area
Area ••••••••••••••
Span.. • ••• • • •• • • • • ••
112 In.
Tip Chord •••••••••
Taper Ratio ••••· •••
. Incidence Root ••••
Ilł.cidence Tip •••••
D:Lhedral ••••••••••
Sweepback C/4 •••••
Aspect Ratio ••••••
M.A. C. • •••••••••••
70
35°
7.o65
241.88 In.
M.A.C. LOC.(Sta) ••
786.2
•342
.......
39.0 Sq. Ft •
80.6 Sq. Ft.
RORIZONTAL TAll,
+20
Sta.b111zer Area
382.7 Sq. Ft.
El.eva.tor Area •••••• 117.6 Sq. Ft.
Span. ••••••••••••••• 39 Ft. 8.4 In.
+20
VERTICAL TAIL
Dorsa1 Area
........ 226.4
8.7 Sq. Ft.
Sq. Ft.
Fin Area •••••••••••
Rlld.der Ares. ••••••••
Ventral. Fin Area
FI.AP
Leeding Edge Area •• 26. 1 Sq. Ft.
Tra:łJ 1ng Edge Area
D6-1420
May 1989
„ 361 .65 Sq. Ft.
lm>Y
Length
101.9 Sq. Ft.
25.5 Sq. Ft_.
............. 138 Ft. 10 In.
A47
APPENDIX
707-300 SERIES
/
PBINCIPAL Dnm:NSIONS
AlLERONS
WING
.Ares. • • • • • • • • • • • • • • • 2892 Sq. Ft.
Span. • • • • • • • • • • • • • • • 142 Ft. 5.. In.
Ie.sic Root Chord • • • 4o6. 7 In..
Referen~ Boot Chord 347.i In.
Tip Oord ••••••••••• 112 In..
Ta.per Ratio ••••••••• 275 (Ba.sic)
•323 (Ref.)
Inoidence Root ••••• +2°
Incidence Tip •••••• +2°
Di hedral • • • • • • • • • • • 7°
Sweepback c/4 •••••• 35•
Aspect Ratio ••••••• 7.056
o..... ~.4 Sq. Ft •
Inboard .Ares. ••
Ou.tboard .Area •••••••
HORIZONTAL TAIL
Ft.
Trailing F.dge .Area. 436.0 Sq. Ft.
A.48
St&billzer .Area • • • • •
El.evator .Area • • • • • • •
474 Sq. Ft •
Span. ••••••••••••••••
45 Ft. 8 In..
151 Sq. Ft.
VERTICAL TAIL
Dorsal .Area
Fi.n .Ares.. • •••• • • • •••
Rud.der .Area ••••••••
Ventral Fin .Area •••
M.A.C •••••••••••••• 272.294 In. M.A.C. IDC. _(Sta) •• 762.97
Leading F.dge Area • • 26. 1 Sq.
80.6 Sq. Ft.
BODY
~
8.7 Sq. Ft.
226.4 Sq. Ft.
101 .9 Sq. Ft.
25.5 Sq. Ft.
............ . ·145 Ft. 6 In..
D6-1420
May 19119
APPENDIX
·-·.
--- ·- ----
707-4oo SERD'.S
P.RINCIPAL DIMB:NSIONS
/
AILERON
WING
Area • • . • • . • • • • • • . • • 2892 Sq. Ft.
Spa.n • • • • • • • • • • • • • • • 142 Ft. 5 In.
Ba.sic Root CllQrd ••• 4o6.7 In.
Reference ~ t Cb;ard 347.1 In.
Tip Chord •••••••••• 112 In.
Taper Batio ••••••••• 275 (Be.sic)
.323 (Ref.)
Incidence Rbot ••••• +2•
Incidence Tip •••••• •2°
Dihedra1 • • • • • • • • • • • 7°.
Sweepback C/4 •••••• 35°
· Aspect Ratio ••••••• 7.056
Inboe.rd Ares. •••••••
Ou.tboard .Area ••••••
HORIZONTAL TAIL ·
Stabillzer Area
Elevator Area ••••••
Span. •••••••••••••••
Dorsal. Aree.
FLAP
Ieed1ng Fdge Area.
Trailing Fdge Area.
D6-1420
Mar 1989
26.1 Sq. Ft.
436.0 Sq. Ft.
474 Sq. Ft.
151 Sq. Ft.
45 Ft. 8 In.
VERTICAL TAIL
M.A.C •••••••••••••• 272.294 In.
M.A.C. IDC. (St.a) .. 762.97
-
4o.4 Sq. Ft.
8o.6 Sq. Ft.
........ 226.4
8.7 Sq. Ft.
Sq. Ft.
Fin. Ares, •••••••.••••
Rud.der Area ••••••••
Ventra.l. Fin Area •••
101.9 Sq. Ft.
Iength •• o • • • • • • • • • •
.:145 Ft. 6 In.
25.5 Sq. Ft.
B:>DY
A49
APPENDIX
707-300.ADV Band C SEBIES
PRINCIPAL DIMENSIONS
/
WING
.AILERON
Inboard .Area •••••••• 40.4 Sg. Ft.
.Area ••••••••••••••• 2892 Sg. Ft. *
Spa.n. ••••••••••••••• 145 Ft. 9 In.
Outboard .Area ••••••• 8o.5 Sg. Ft.
Ba.sic Root Chord ••• 406.682 In.
HOBIZONTAL TilL .
Ref'. Root Chord •••• 347.134 In.
Stabilizer .Area • • • • • 474 Sg. Ft.
Tip Chord•••••••••• 112 In.
Elevator .Area ••••••• 151 Sg. Ft.
Taper Ratio•••••••• • 275 (Ba.sic)
Spa.n. •••••••••••••••• 45 Ft. 8 In.
• 323 (Ref'.)
Incidence at Root •• +20
VERrICAL TilL
Incidence at Tip ••• +20
D::>rsal .Area •••••••••
8.7 Sg. Ft.
Dihedral ••••••••••• 70
Sweepback c/4 •••••• 35°
Fin .Area •••••••••••• 226.4 Sg. Ft.
Rudder .Area
101.9 Sg. Ft.
Aspect Ratio••••••• 7.346
M.A. C. • •••••••••••• 272.294 In.
Leading Edge M.A.C •• Sta. 762.97
.........
BODY
FLAP
Leading · Edge .Area • • 197. 88 Sg. Ft.
Trailing Edge .Area. 491.72 Sg. Ft.
Length •••••••••••••• 145 Ft. 6 In.
* (Excluding leading edge, trailing edge, and tip extensions.)
A.50
D6-1420
May 1989
APPENDIX
I
+
707-300:B SERIES
PRINClPAL DJMB:NSIONS
WING
AII.ERON
/
In.board Area ••••• ·••• 4o.4 Sq. Ft.
Area •••••••••• ~ •••• 2892 Sq. Ft. *
Outboard
Area • • • • • • • Bo. 5 Sq. Ft.
Span. •·. • • • • •. • • • • • • • • 1 45 Ft. 9 In..
Basic Root Chard ••• 4o6.682 In..
BORIZONTAL TllL
Ref. Root Chard • • • • 347. 1 34 In..
Tip Chord • • • • • • • • • • 11.2 In..
Stabilizer Area ••••• 474 Sq. Ft.
!taper Batio ••••••••• 275 (Basic)
El.e"Vator Area ••••••• 151 Sq. Ft •
• 323 (Ref.)
Span •••••••••••••••• 45 Ft. 8 In.
· In.cidence at Root • • +.2°
VERTICAL TA.IL
In.cidence at Tip • • • +2°
Dihedral ••..••.•••• 70
Dorsa1 Area
8.7 Sq. Ft.
Sweepbaclt c/4 ..... . 35°
Fin. Area •••••••••••• 226.4 Sq. Ft.
Aspect Batio ••••••• 7.346
Rud.der Area ••••••••• 101.9 Sq. Ft.
M.A.C • ••••••••••••• 272.294 In..
Ventra1 Fin Area
25.5 Sq. Ft.
Leading Edge M.A.C •• Sta. 762.97
.........
- B>DY
FI.AP
Lead1ng ;Edge 1lrea • • 157. 24 Bq. Ft.
!l'm1.11.ng Edge Area • 491 • 72 Sg. Ft.
I.,engt,h •••••••••••••• 145 Ft.
6 In..
* (ExcJud:fng 1ead1ng ecige, trailing ecige, and tip extensions.)
D6-1420
May 1989
A51
APPENDIX
.I
.. ,·
MODEL 720
PRINCIPAL DIMENSIONS
/
AILERON
.Ares, ••••••••••••••
Span. ••••••••••••••
Root Chord· ••••••••
Tip Chord •••••••••
Taper Batio •••••••
Incidence Root ••••
Incidence Tip •••••
Dihedral ••••••••••
Sweepback C/4 •••••
.Aspect Batio ••••••
M.A. C. • •••••••••••
M.A.C. LOC. (Sta) ••
....... 80.6
39.0 Sq. Ft •
Sq. Ft.
Inboard .Area
2433 Sq. Ft.*
Outboard .Ares. ••••••
130 Ft.· 1.0 In.
338 In.*
112 In.
HORIZONTAL TA.IL
.342
+20
+20
Stabillzer .Area
382.7 Sq. Ft.
117.6 Sq. Ft.
Spac. ••••••••••••••• 39 Ft. 8.4 In.
El.evator .Area ••••••
70
35°
7.o65 *
241.88 In. *
VERTICAL TAIL
786.2
FI.AP
Leading Edge .Area •
~iling Edge Area
Rud.dm- .Ares. ••••••••
Ventral. Fin .Area •••
101 .9 Sq. Ft.
25.5 Sq. Ft.
I.ength •••••••••••••
130 Ft. 6 In.
93.64 Sq. Ft.
362.2 Sq. Ft. ~
* (Exc1ucling J.ead1ng edge extension)
A.52
........
8.7 Sq. Ft.
n.n .Area ••••••••••• 226.4 Sq. Ft.
Dorsal .Area
D6-1420
May 1989
APPENDIX
MODEL 720B
PRINCIP.AL DJMENSIONS
WING
A!IDON
/
Area ••••••••••••••
Sp&11 ••••••••••••••
Root Chord ••••••••
Tip Cbord •••••••••
!Iaper Ratio •••••••
Incidence at Root.
Incid.ence at Tip ••
. Dihedxal •••••.•••••
Sweepback c/4 .....
Aapect Ratio ••••••
. M.A.C. • • • • • • • • • • • •
Iead.ing Edge MAC • •
Inboard Area •••••• 39.0 Sq. Ft.
2433 Sq. Ft. *
130 Ft. 1.0 In.
Outboard Area
80.6 Sg_. Ft.
338 In. *
. BDRIZO:NTAL TAIL
112 In.
• 342
+20
Stabilizer Area • • • 427. 7 Sq. Ft.
+20
Ele-vator Area • • • • • 1 17. 6 Sq. Ft.
70
Si>&Il • • • • • • • • • • • • • • • • • • • • • 43 Ft •
35°
7.o65 *
241 .88 In. *
Sta. 786.2
VERTICAL TAIL
.......
Dorsal Area
8. 7 Sq. Ft.
Fin. Area •••••••••• 226.4 Sq. Ft.
Rud.der Area ••••••• 101 .9. Sq. Ft.
Ventral Fin Area • • 25. 5 Sq. Ft.
-
FI.AP
Leading Edge Area •
Trailing Edge Area
93.64 s~ Ft.
362.2. Sq. Ft.
* (Exc1uding 1eading edge extensio:cJ
D6-1420
May 1989
!m!.
. Length .•.....•.. ~. 130 Ft. 6 In.
A53
APPENDIX
MODEL 727
PRINClPAL DJMEN'SIONS
WING
AILERON
1
Area •••••••• ~., •• ,, 1560 Sq. Ft.
Sl>an •.•.••.••••••.. 106Ft.
Root Chord (Buic) . 246 !.n.
Tip Cbord ••••.•••••• 91 .6 In.
Taper Ratio (łasic). 0.372
!.nboard Area ••••••
Outboard Area
18.64 Sq. Ft.
36.5 Sq. Ft.
Anhedral. ••••••••••
30
J.ncidence at Root • • +2°
Incidence at Tip • • • +2°
Dihedra,J. ••••••••••• 3°
El.ew.tor Area • • • • •
~1eepback c/4
~
•••.•. 32°
Aspect Batio ••••••• 7.2
M.A.C • ..••••••••.•• 180.0 :zn.
Stabilizer Area ••• 282.4o Sq. Ft.
94.14 Sq. St.
. . . . . . . . . . . . . . 35 Ft. 9 In.
VERTICAL TAIL
Leading F.dge MAC • • • Sta. 860. 2
-
Fin Areai •••••••••• 283.37 Sq. Ft.
72.63 Sq. Ft.
Rud.der Area
FI.AP
Leading F.dge Area •• 54.8 Sq. Ft~
~iling F.dge Area
(Retracted) • • • 28o Sq. Ft.
(Extended) •••• 388 Sq. Ft.
A.54
.......
Length
·.
-.. .
----=Short Body(727-iOO)ll6.·Ft~~ 2 .In.
Long Body (727-:-200)136 Ft. 2 In.
D6-1420
May 1989
APPENDIX
MODEL 737
PRINCIPAL DIMENSIONS
WING
AILERON
/
Area
.980 Sq. Ft.
Span
. 93 Ft.
Root Chord (Basic)
.288.1 In.
Root Chord (Ref) .
.227.1 In.
Tip Chord . • . •
. . 63.29 In.
Taper Ratio (Ref) . . . . 0.279
+10
Incidence at Root •.
+10
Incidence at Tip.
(jO
Dihedral • . •.
Sweepback c / 4 • .
25°
Aspect Ratio • .
8.83
M.A.C • . • . . •
134.46 In.
Leading Edge MAC .
Sta. 625.6
D6-1420
May 1989
13.4 Sq. Ft.
HORIZONTAL TAIL
. . . 70
Dihedral • • • •
Stabilizer Area ••.• . . . 241.54 Sq. Ft.
70.46 Sg. Ft.
Elevator Area • • .
. 312.00 Sq. Ft.
Total Area • • .
Spa.n . • . • • .
36 Ft.
·VERTICAL TAIL
Fin Area • • .
Rudder Area
Total Area.
167. 80 Sq·.· Ft.
. . . . 56.20 Sg. Ft.
. . . . . 224.oo Sq. Ft.
BODY
FLAP
Leading Edge Area
Trailing Edge Area
(Retracted)
(Extended)
Aileron Area
80 Sq. Ft.
.1 71. 5 Sq. Ft .
Length
Short Body (737-100 )., . . 90 Ft. 7 In.
Long Body (737-200) . . • 96 Ft. 11 In.
. 250 . 88 Sq. Ft .
A.55
APPENDIX
MODEL 737-300
PRINCIPAL DIMENSIONS
WING
AILERON
Area (Ref) .
. . 980 Sq. Ft
Span '"- .' .
. . . . . 93 Ft
Basic Root Chord . . . . 288.1 In.
Reference Root Chord
. 193.82 In.
Tip Chord . . . . . . . . 6329 In.
Taper Ratio . . . . . . . 0.327 (Ref)
lncidence @ Root . . . . 1
lncidence @ Tip . .
. 1
Dihedral . . . .
. 50
Sweepback C/4 . . . . . 25°
Aspect Ratio (Ref)
. 8.826
MAC (Ref). . . . . . . . 134.46 In.
MAC C/4 Location (LE.) . 659.21 In.
Area . .
°
°
ELAP
Leading Edge Area •
Trailing Edge Area
(Retracted)
(Extended)
A.56
80.00 Sq. Ft
171.5 Sq.Ft
. 250.88 Sq. Ft
. . . . 26.9 Sq. Ft
HORIZONTAL IAIL
. . 70
Dihedral . . . . . .
. . 267.7 Sq. Ft
Stabilizer Area . .
. 70.46 Sq. Ft
Elevator Area. . .
Sweepback C/4 .
. . 30°
Span . . . . . . . . . . 500 In.
Taił Arm (C/4w to C/4J . . 586.7 In.
VERTICAL TAil
Fin Area . . . . . . . . . 167.8 Sq. Ft
Rudder Area . . .
. 57 2 Sq. Ft
Sweepback C/4 . . . . . . 35°
Taił Arm (C/4w to C/4J . . 534.49 In.
BODY (Length) . . . . . 105 Ft 7 In.
D6-1420
May 1989
APPENDIX
i!E
MODEL 747
PRINCIPAL DIMENSIONS
WING
AILERON
Area.
5500 Sq. Ft.
Span.
195 Ft. 8 In.
Basic Root Chord
449.68 In.
Reference Root Chord •. 652.03 In.
Tip Chord
160.00 In.
Taper Ratio
• 356 (Basic)
.245 (Ref) .
+20
Inci'dence Root
Incidence Tip
70
Dihedral . • •
Sweepback c/4 •.
37.5°
Aspect Ratio (Ref) •
6.96
MAC (Basic only) . . .
327.78 In.
MAC c/4 Locatión· ( sta) • 1]39°91 In.
Aileron P:rea (Inbd) • • • 71.8 Sg. Ft.
(Outbd) •• 153.4 Sg. Ft.
HORIZONTAL TAIL
-, . ,o
FLAP
Leading Edge Area
( Retracted) • •
Trailing Edge P:rea
( Retracted) •
Area . . . . . •• . .• 1470 Sq. Ft.
Sweepback c/4
37.5°
Span. • • • . • . . . •. 72 Ft. 9 In.
Root Chord ••
32 Ft. 4 In.
Tip Chord
•· • • 8 Ft. 1 In.
VERTICAL TAIL
Area • . . • . . . . ..• .
Sweepback c/4
Span (Height)
Root Chord.
Tip Chord
448 Sg. Ft.
BODY ·
847 Sg. Ft.
Length
•.••.
830 Sq. Ft.
45°
32 Ft. 3 In.
38 Ft. 5 'In.
13 Ft. 1 In.
225 Ft. 2 In .
..
D6-1420
May 1989
-
A.57
APPENDIX
MODEL 747SP
PRINCIPAL DIMENSIONS
WING
AILERON
Area (Ref) • . . . . . . . 5500 Sq. Ft
Span . . • . . . . . . . 195 Ft 8 In.
Basic Root Chord • . • . 652.03 In.
Reference Root Chord . . 449.68 In.
Tip Chord . .
• . . . 160.00 In.
Taper Ratio . ~ . • . . . 0.245 (Basic)
0.356 (Ref)
lncidence @ Root . . . . +2°
lncidence @ Tip . • . . . -1.5°
Dihedral • . • . .
. 70
Sweepback C/4. • . . . 37.5°
Aspect Ratio (Ref) • . . . 6.96
MAC (Basic) . . . . . . 327.78 In.
MAC C/4 Location (Sta) . 1339.91 In.
Aileron Area (lnbd) . . . . 62.8 Sq. Ft
(Outbd) . . . . 153.4 Sq. Ft
fLAP
Leading Edge Area . . .- 448.00 Sq. Ft
Tralling Edge Are·a
(Re1rac1ed) . . . 848 Sq. Ft
A.58
HQRIZQNTAL IAIL
. 1534 Sq. Ft
Area.
Sweepback C/4 . . . . . 37.5°
Span • • .
. 82 Ft 9 In.
. 32 Ft 4 In.
Root Chord . .
Tip Chord . .
. 4 Ft 9 In.
a
•••
I
I
I
VERTICAL IAIL
Area. . . .- . . . . . . . 885.5 Sq. Ft
Sweepback C/4 . . . . . 44°
Span (Height)
. . . 37 Ft 3 In.
Root Chord . . .
. . 38 Ft 5 In.
Tip Chord . . . .
. 8 Ft 11 In.
BODY
(Length)
. 176 Ft 9 In.
D6-1420
May 1989
APPENDIX
MODEL 747-400
PRINCIPAL DIMENSIONS
AILERON
WING.
. 5650 Sq. Ft
Area (Basic).
Aileron Area (lnbd)
. 5500 Sq. Ft
Area (Ref) . . .
(Outbd)
. 211 Ft 5 In.
Span . . . . .
Basic Root Chord .. . . . 449.68 In.
HORIZONTAL TAil
. 652.03 In.
Reference Root Chord
. 141.84 In.
Tip Chord . . .
Area. . . . . . .
. 0.315 (Basic)
Taper Ratio . . .
Sweepback C/4 . . .
0.218 (Ref)
Span . . . .
lncidence @ Root
. +2°
Root Chord ..
lnciderice@ Tip.
. -1.5°
Tip Chord . .
Dihed rai . . . .
. . 7°
Sweepback C/4. . . . . 37.5°
Aspe et Ratio ( Ref) . . . . 6.96
MAC (Basic) . . . . . . 327.78 In.
MAC C/4 Location (Sta) . 1339~1 In.
FLAP
Leading Edge Area
. . 448 Sq. Ft
Trailing Edge Area
(Retracted) . . . 847 Sq. Ft
D6-1420
May 1989
.71.8 Sq. Ft
.153.4 Sq. Ft
.1470 Sq. Ft
.37.5° .
. 72 Ft 9 In.·
. 32 Ft 4 In.
. . 8 Ft 1 In.
VERTICAL TAil
. . 830 Sq. Ft
Area . . . . .
. 45°
Sweepback C/4 . . .
. . 32 Ft 3 In.
Span (Heigh1)
. 38 Ft 5 In.
Root Chord
. 13 Ft 1 In.
Tip Chord . .
BODY (Leng1h)
. 225 Ft 2 In.
A.59
APPENDIX
MODEL 767
PRINCIPAL DIMENSIONS
WING
AILEBQN
Area (Ref).
• 3050 Sq. Ft
Span (Ref). . • . . • • . 155 Ft
Root Chord (Total) .
. 37.97 Ft
· Root Chord (Trap)
. 28.1 Ft
Tip Chord ·. . • • • • . . 7.5 Ft
• 0.267 (Trap)
Taper Ratio • • . .
lncidence@ Root
•. 4.25°
lncidence @ Tip .
. 1.3°
Dihedral • . . .
Sweepback C/4.
. 31.5°
Aspect Ratio (Ref)
. 7.88
MAC (Ref) . . . • • . • • 19.79 Ft
MAC C/4 Location (Sta). . 375.3
Aileron Area (lnbd) • . . . 30.0 Sq. Ft
(Outbd)
• 32.0 Sq. Ft
. eo
FLAP
Leading Edge Area . . . 302.00 Sq. Ft
Trailing Edge Area
(Retracted) . . • . 397 Sq. Ft
(Extended)
. . 597 Sq. Ft
A.60
HQBIZQNTAL TAil
.830 Sq. Ft
Sweepback C/4 . • . • . 33°
. 61.1 Ft
Span • . .
Root Chord • • . .
. 21.73 Ft
Tip Chord . . . . . . . . 5.43 Ft
Area. .
I·
I
I
I
I
I
VERTICAL TAil
Area • . • • .
. • • 497 Sq. Ft
Sweepback C/4 . • . . . 40°
Span (Height)
• • 29.7 Ft
Root Chord . • . . • • • 25.66 Ft
Tip Chord . . . • . . . . 7.78 Ft
BODY (Length). . . . . .155 Ft
ENG !NE (Moment Arm). • . 25.69 Ft
D6-1420
May 1989
APPENDIX
MODEL 757-200
PRINCIPAL DIMENSIONS
W!NG
A!LERON
Area (Ref). . . . . . . . 1951 Sq. Ft
Span . . . . . . .
. 124.5 Ft
Basic Root Chord
. 286.5 In.
Tip Chord . . . . . . . . 68.0 In.
Taper Ratio . · . . .
. 0.327 (Basic)
lncidence@ Root . . . . 3.25°
lncidence @ Tip . . . . . -2.80
Dihedral . . . .
. • 50
Sweepback C/4 . .
. 25°
Aspect Ratio. . . . . . . 7.95
MAC (Ref) . . . . . . . . 199.7 In.
MAC C/4 Location . . . . Sta. 1041.8
Area (Each) .
FLAP
Leading Edge Area . . . 198.2 Sq. Ft
Trailing Edge Area. . . . 326.6 Sq. Ft
HORJZONTAl TAil
Dihedral. . • . . . . ... 7°
Stabilizer Area . .
. . 542.3 Sq. Ft
Elevator Area. . .
. 134.9 Sq. Ft
Sweepback C/4
. . 30°
Span . . . . . . . . . . 49 Ft 5 In.
Taił Arm (C/4w to C/4J . . 801.1 In.
VERTJCAL TAil
Fin Area . . . . .
. . 370.0 Sq. Ft
Rudder Area .
. 125.2 Sq. Ft
Height . . . . . . . . . 44.5 Ft
SVJeepback C/4 .
. . 40°
Taił Arm (C/4w to C/4J . . 766.8 In.
BODY CLength).
D6-1420
May 1989
. . 24.16 Sq. Ft
. ·.153.9 Ft
A.61
APPENDIX
lntentionally Left Blank
}
A.62
D6-1420
May 1989