AE 429 - Aircraft Performance and
Flight Mechanics
Range and Endurance
Range and Endurance
Definitions
–
Range
Total ground distance traversed on a full tank of fuel -Anderson book
Distance an airplane can fly on a given amount of fuel -alternative definition
–
Endurance
Total time an airplane stays in the air on a full tank of fuel
-- Anderson book
Time an airplane can fly on a given amount of fuel -alternative definition
–
Specific fuel consumption
Weight of fuel consumed per unit power per unit time
1
Range and Endurance
Propeller-Driven Airplane
Generalized endurance parameters
–
Specific fuel consumption (SFC or c)
Weight (W) lbs
Time (t) secs or hrs
Engine power (P) hp
Airplane gross weight (W 0) lbs
–
Airplane empty weight (W 1) lbs
–
–
–
–
LBS FUEL
BHP − HR
includes fuel and payload
W = W1 + W f
weight of airplane without fuel
Endurance calculation
dW f
– Fuel flow rate w f = - c P (lbs fuel/unit time) =
–
–
dt
Differential fuel burned dW f = dW = −c Pdt
Solving for time and integrating
E
E=
0
dt = −
W1
dW f
W0
cP
=
W0
W1
c=−
Wf
P
dW
cP
Range and Endurance
Propeller-Driven Airplane
Generalized Range Equation
–
Multiply V∞ by
covered
dt = −
dW
cP
to get differential air distance
ds = V∞dt = −V∞
–
dW
cP
The total distance covered during a range flight is the
integral of this expression
R=
R
0
ds = −
W1 V∞dW
W0
cP
=
W0 V∞dW
W1
cP
2
Range and Endurance
Propeller-Driven Airplane
Breguet Range Equation
–
In level flight cruise (no acceleration), the throttle is set to
maintain constant airspeed
PA = PR = DV∞
P
DV∞
Shaft HP = P = A =
–
–
η
η
Substituting into the general range equation for propeller
airplanes
W V dW
W V ηdW
W ηdW
= 0 ∞
= 0
R= 0 ∞
W1
W1 cDV
W1 cD
cP
∞
Multiplying by W / W and using L= W for equilibrium cruise
flight (level and unaccelerated)
R=
W0 η W dW
W1 c W
D
=
W0 η L dW
W1 c D W
Range and Endurance
Propeller-Driven Airplane
Breguet Range Equation (continued)
Assume
h is constant throughout flight
L/D is constant throughout flight
c is constant throughout flight
Then, we can integrate
R=
η L
W0
c D
W1
dW
W
R=
η L
c D
ln
W0
W1
Because of the assumptions listed, the Breguet
approximation is valid only for small dW
–
–
Angle of attack changes over a weight change as large as
the fuel load
If weight changes significantly, L/D changes
3
Range and Endurance
Propeller-Driven Airplane
Breguet Endurance Equation
Using the same assumptions (L = W and PA = PR = DV∞ )
W0
E=
Since
W1
dW
=
cP
L =W =
and
E=
W0
η dW
W1
c DV∞
=
W0
η L dW
W1
c DV∞ W
1
ρ∞V∞2 SCL , V∞ =
2
2W
ρ∞ SCL
W0
η CL ρ∞ SCL dW
W1
c CD
2
W 3/ 2
If we assume CL , CD , η , c, and ρ∞ are all constant
E=
η CL3/ 2
c CD
2 ρ∞ S
1
W1
−
1
W0
in using this equation, E must be in seconds and c must be in ft-1
Importance of the Ratios
CL
CD
max
CL3 / 2
CD
max
3/ 2
C L C D ,C L
CD , C1/L 2 CD
Max Range for reciprocating engine/propeller airplanes
Max Endurance for reciprocating engine/propeller
4
Range and Endurance
Jet-powered airplane
General endurance equation
–
–
Specific fuel consumption is based on thrust, not power output
lbs of fuel consumed
TSFC ≡ ct
lb of thrust produced − sec
Again, let dW be the differential change in aircraft weight as fuel is
burned in dt
dW
dW = −ctTAdt
dt = −
Wf
c
ct = −
tTA
TA
E=
E
dt = −
W1 dW
=
W0 dW
–
Integrating:
–
Recalling that L = W and TA = TR = D
W 1 L dW
E= 0
W1 ct D W
Assuming constant ct and L/D
The Breguet approximation is:
1 L W0
E≅
ln
ct D W1
–
–
0
W0 ctTA
W1 ctTA
Range and Endurance
Jet-powered airplane
To maximize endurance
–
–
–
Reduce thrust specific fuel consumption, TSFC
– Use a turbofan engine
– Fly at an altitude where the engine is efficient (high, but not too
high)
increase fuel fraction, W f/ W 0 , to increase W 0/ W 1
– Carry less payload
– Use drop tanks
maintain L/Dmax
– Loiter at velocity for L/Dmax (where CD,0 = CDi)
– Cruise climb
– Notice that a jet airplane has best endurance at this velocity for
L/Dmax, while a propeller-driven airplane gets best range at this
true airspeed
5
Range and Endurance
Jet-powered airplane
general range equation
–
going back to the differential time expression and multiplying by V
V dW
ds = V∞dt = − ∞
ctTA
–
–
–
of course, ds is the differential distance traveled by the jet
during time dt
– s = 0 when w = w0
– s = R when w = w1
R
W V dW
integrating to get range
R = ds = − 1 ∞
W0 ctTA
0
W
for steady, unaccelerated level flight: TA = TR = L
R=
D
W0 V∞ CL dW
W1 ct CD W
Range and Endurance
Jet-powered airplane
general range equation (continued)
– since
, R becomes
2W
V∞ =
ρ∞ SCL
R=
W0
W1
2 1 CL dW
ρ∞ S ct CD W
Breguet approximation
– if we assume constant ct, CL, CD, and ρ
R=2
–
2 1 CL
ρ∞ S ct CD
(
W0 − W1
for maximum range
– reduce TSFC, ct
– fly at low density (high altitude)
– use a true airspeed to maximize
)
CL1/ 2
CD
6
Range and Endurance
Jet-powered airplane
Cruise for
Max range
–
Fly at
CL
CD
–
max
The true airSpeed for
This optimum
Speed gives
CD,0 = 3Cdi
Importance of the Ratios
CL
CD
max
2
C1/
L
CD
max
3/ 2
C L C D ,C L
CD , C1/L 2 CD
Max Endurance for jet-propelled airplanes
Max Range for jet-propelled airplanes
7
Range and Endurance
Summary
Relationship of CD,0 and CDi
–
At L/Dmax
Using
CD = CD ,0 +
CL
=
CD
CL2
πeAR
CL
CD ,0 +
CL2
πeAR
Differentiating with respect to CL and setting to zero
d
CD ,0 +
CL
CD
dCL
CL2
πeAR
−2
CD ,0 +
=
CL2
CL
− CL 2
πeAR
πeAR
CD ,0 +
CL2
πeAR
CD ,0 = CD ,i
=0
πeAR
=0
FOR
2
CL2
CD ,0 =
CL
CD max
CL2
πeAR
Range and Endurance
Summary
Relationship of CD,0 and CDi
–
Similar developments show:
CD ,0 = 3CD ,i
FOR
CL
CD
max
Applies to maximum range for a jet airplane
3CD ,0 = CD ,i
FOR
CL3
CD
max
Applies to maximum endurance for a propeller-driven airplane
8
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