Chalmers University of Technology
Civil jet aircraft performance
Chalmers University of Technology
Four forces of flight
L ift
W eight = m g
D rag
Thrust
V
x
y
Resulting force perpendicular
to the flight path
Net thrust from the engines
resulting force parallell to the flight path
α angle of attack
V velocity
sincos gmDFdt
dVm
Newton’s second law
Chalmers University of Technology
Aerodynamic equations
• L=Lift = q·S·CL [N]
• D=Drag = q·S·CD [N]
• q = dynamic pressure [N/m²]
• S = reference wing area [m²]
• CL = coefficient of lift CL = f(α,Re,M)
• CD = coefficient of drag CD = f(α,Re,M)
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Reference wing areaThe area is considered to extend without interruption
through the fuselage and is usually denoted S.
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Lift versus angle of attack
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numberMachoffunctionasCD
DC
1M Mach
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The ISA AtmosphereFrom lecture 5
Chalmers University of Technology
Equations
222
2222
2
1
2
12
1
2
1
2
1
2
SLSL
a
SL
Sl
SLSLSL
aMT
TTRM
TRMaMVq
TRp
a
V
a
VM
TRap
p
T
T
SL
2
2
1Mp
Chalmers University of Technology
Lift equation
LL
LLSL
LSLSLLSL
LL
CSMCSMconstL
CSMconstCSMp
CSMaCSaM
CSaMCSVL
242
22
constant
2222
222
100928,7
2
12
1
2
12
1
2
1
Chalmers University of Technology
Drag equation
DD
DDSL
DSLSLDSL
DD
CSMCSMconstD
CSMconstCSMp
CSMaCSaM
CSaMCSVD
242
22
2222
222
100928,7
2
12
1
2
12
1
2
1
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Drag polarL
D
R
R
C L
C D
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High speed drag polar
LC
0.02 0.04 0.06 0.08 0.1 0.12 0.140.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Mach 0,63Mach 0,73Mach 0,80Mach 0,82Mach 0,83Mach 0,85
DC
LC
Chalmers University of Technology
A flight consists of:
• Taxi
• Take off
• Climb
• Cruise
• Descent
• Approach and landing
• Diversion to alternate airport?
Sector Distance
Flight Time & Fuel
Block Time & Fuel
En route Climb
Descent
Approach &Landing
1500 ft
Initial Cruise
Step Cruise
Takeoff &Initial ClimbStart-up
&Taxi-out
Taxi-in
Chalmers University of Technology
CruiseFor an airplane to be in level, unaccelerated flight, thrust and drag
must be equal and opposite, and the lift and weight must be equal andopposite according to the laws of motion, i.e.
Lift = Weight = mgThrust = Drag
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Range
m
s
sN
kgnconsumptiofuelspecific
F
mSFC
s
kgflowfuelm
s
mvelocityV
mrangeR
f
f
a
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Range
START
END
END
START
END
START
END
START
W
W
SLa
W
W
SL
W
W
W
W
a
a
af
a
ff
a
W
dW
D
LM
SFCg
aR
W
dW
D
LM
SFCg
a
W
dW
D
LM
SFCg
a
W
dW
D
LV
SFCgR
W
dW
D
LV
SFCgDFWL
WFSFCg
dWWVdR
FSFCg
dWV
FSFC
dmVdR
FSFC
dm
m
dm
V
dRdt
F
mSFC
dt
dmmV
dt
dR
1
1&
Chalmers University of Technology
Breguet range equation
END
STARTSL
W
W
SLa W
W
D
LM
SFCg
a
W
dW
D
LM
SFCg
aR
START
END
ln
For a preliminary performance analysis is the range equation usually simplified. If we assume flight at constant altitude, M, SFC and L/D the range equation becomes
which is frequently called the Breguet range equation
END
STARTa W
W
D
LV
SFCgR ln
1
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Breguet range equationThe Breuget range equation is written directly in terms of SFC. Clearly maximum range for a jetaircraft is not
dictated by maximum L/D, but rather the maximum value of the product M(L/D) or V(L/D).
ENDSTARTD
La
W
WD
L
D
La
D
La
D
L
D
L
L
LL
WWC
C
SSFCgR
W
dW
C
C
SSFCgW
dW
C
C
SSFCgdR
W
dW
C
C
S
W
SFCgW
dW
D
LV
SFCgdR
C
C
S
W
C
C
CS
W
D
LV
CS
WVCSVWL
END
START
22
2121
211
22
2
2
1 2
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Breuget range equationFrom the simplified range equation, maximum range is obtained from
• Flight at maximum
• Low SFC
• High altitude, low ρ
• Carrying a lot of fuel
D
L
C
C
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Range
LC
D
LM
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.82
4
6
8
10
12
14
16
Mach 0,63Mach 0,73Mach 0,80Mach 0,82Mach 0,83Mach 0,85
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Endurance
END
START
W
W
W
W
W
W
f
ff
W
W
D
L
SFCgt
simplifiedusuallyaboveequationtheisionapproximatfirstafor
W
dW
D
L
SFCgW
dW
D
L
SFCgW
dW
D
L
SFCgt
W
dW
D
L
SFCgDFWL
FSFCg
dW
FSFC
dm
m
dmdt
F
mSFC
dt
dmm
START
END
END
START
END
START
ln1
:
111
1&
Endurance is the amout of time that an aircraft can stay in the air on one given load of fuel
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Endurance
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.94
6
8
10
12
14
16
18
20
Mach 0,63Mach 0,73Mach 0,80Mach 0,82Mach 0,83Mach 0,85
D
L
LC