Transcript AE 301 Aerodynamics I

Range and Endurance
• When designing or comparing aircraft, two
parameters usually come to mind:
– Range: the horizontal distance an airplane can travel on a
single fueling. The cruise portion of a flight is associated
with flying for range.
– Endurance: the amount of time an airplane can remain
aloft on a single fueling. The loiter phase of a mission is
associated with flying for endurance.
• We will calculate the range and endurance for pistonpropeller and turbojet aircraft separately.
• All our calculations will assume still air conditions I.e. no head or tail winds.
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R and E - Propeller Aircraft
• Both range and endurance of an aircraft will depend
upon the rate at which fuel is burned.
• The common parameter used to define this rate is
called the Specific Fuel Consumption, SFC.
Weight of fuel
lb of fuel
SFC 

EnginePower T ime (bhp) hour
• The SFC is considered a constant for an engine type varying vary little with throttle setting or flight
conditions.
• Typical range of values: 0.4 - 0.7 lb/hp/hr
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R and E - Propeller Aircraft (continued)
• From the fact that the SFC is a constant, we can
deduce some simple range and endurance relations.
• For a long endurance, we would like to burn the
minimum fuel per hour. From the SFC then:
lb of fuel
SFC  PR
 SFC  (bhp) 
hour

• Thus, for long endurance we want to fly with a high
propeller efficiency and at minimum power required!
• From our previous result for power required, it
follows that for max endurance, we want to fly at the
velocity such that CL3/2/CD is maximum.
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R and E - Propeller Aircraft (continued)
• For a long range, we would like to burn the minimum
fuel per mile traveled. Dividing our previous result by
velocity (miles/hour) gives:
lb of fuel SFC  (bhp) SFC  TR


mile
V

• Thus, for long range we want to fly with a high
propeller efficiency and at minimum thrust required!
• From our previous result for thrust required, it follows
that for max range, we want to fly at the velocity
such that CL/CD is maximum.
(Remember, these results are for propeller driven aircraft - jets
will be different!)
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R and E - Propeller Aircraft (continued)
• Now let’s work at some quantitative relations.
• First, since SFC does not have consistent units, a new
fuel burn rate is introduced:
lb of fuel
1

N of fuel 1 
c

 c 
 
(ft  lb/sec) sec ft
(W ) sec m 

• Note that in the British system, c and SFC are related
via the conversion factors:
SFC
SFC
c

ftlb
sec
550sechp 3600hour  1980000hourftlbhp 
• Is other references, if these conversion factors are
part of the equations, they are using SFC not c!
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R and E - Propeller Aircraft (continued)
• The weight of an aircraft is usually broken down into
various components of which only the fuel portion
varies during normal flight.
W  WOEW  WPayload  Wf
• The rate at which fuel is burned depends upon the
SFC and the power setting:
dW  dWf  cPdt
• Rearrange gives dt  dW  cP which can be integrated
to get the endurance:
t1
W0
W1
dW
E   dt  
 cP
t0
W0
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dW
E 
cP
W1
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R and E - Propeller Aircraft (continued)
• Similarly, the range is found by multiplying time by
velocity and integrating:
V dW
dx  V dt 
 cP
W0
x1
W1
V dW
V dW
R   dx  
R 
 cP
cP
x0
W0
W1
• These integral relations are useful for calculating the
R and E for a given mission where V and P may vary
through the flight.
• However, they are not very useful for rapid R and E
estimations for a given airplane. For that purpose,
we use the Breguet relations!
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Breguet Propeller Equations
• The Breguet relations are approximate expressions
for R and E obtained by making assumptions about a
typical flight profile.
• For range, assume steady, level flight so that L=W
and PR=VTR=VD.
0
0
V dW
VWdW
L dW
R 


cP
cV  DW
cD W
W1
W1
W1
W0
W
W
• Now assume that c and  are constant, and that the
aircraft is flown at a velocity such that L/D = CL/CD
remains a constant.
• (Note that V and  are not necessarily constant!)
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Breguet Propeller Eqns (continued)
• With these assumptions, the integral can be
evaluated to get:
 Wo 

R
ln
c C D  W1 
 CL
• This relations tells us that to maximize range, we
want:
– The largest possible propeller efficiency
– The lowest possible specific fuel consumption
– The largest possible weight fraction of fuel, Wf/W
– A large (L/D)max and to fly at the velocity where this is
achieved.
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Breguet Propeller Eqns (continued)
• Perform a similar procedure with the endurance
equation.
W
W
W
dW
WdW
L dW
E
0
0
 cP   cV
W1
W1
 DW

0
 cV
W1

D W
• To go further, we need to relate velocity to lift via:
V  2W / SCL 
E
W0
C L   SC L dW
 cC
W1
D
2W
W

W0

W1
C 3 / 2   S dW
L
cC D
2 W 3/ 2
• Now assume that c,  and  are constant, and that
the aircraft is flown at a velocity such that CL3/2/CD
remains a constant
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Breguet Propeller Eqns (continued)
• With these assumption, integration yields:
E
C 3 / 2
L
cCD
 1
1 
2  S  1/ 2  1/ 2 
W0 
 W1
• This relations tells us that to maximize endurance, we
want:
– The largest possible propeller efficiency
– The lowest possible specific fuel consumption
– The largest possible weight fraction of fuel, Wf/W
– A large (CL3/2/CD)max and to fly at the velocity where this is
achieved.
– Flying at the highest density possible, I.e. sea level!
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R and E - Jet Aircraft
• Jet aircraft differ from propeller aircraft primarily in
the fact that a jet engine produces thrust directly
while a piston engine produces power.
• The fuel consumption for jet aircraft is thus based
upon T and called the Thrust Specific Fuel
Consumption, TSFC.
Weight of fuel
lb of fuel
TSFC 

EngineT hrusthour lb of thrusthour
• As with piston engines, TSFC is a nearly a constant
for a given powerplant.
• Typical range of values: 0.5 - 1.0 lb/lb/hr
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R and E - Jet Aircraft (continued)
• From the fact that the TSFC is a constant, we can
deduce some simple range and endurance relations.
• For a long endurance, we would like to burn the
minimum fuel per hour. From the TSFC then:
lb of fuel
 TSFC  (lb of thrust )
hour
• Thus, for long endurance on a jet, we want to fly
with a minimum thrust required!
• From our previous result for thrust required, it follows
that for max endurance, we want to fly at the
velocity such that CL/CD is maximum.
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R and E - Jet Aircraft (continued)
• For a long range, we would like to burn the minimum
fuel per mile traveled. Dividing our previous result by
velocity (miles/hour) gives:
lb of fuel TSFC  (lb of thrust) TSFC  TR


mile
V
V
• To reach a conclusion from this relation, lets assume
TR=D and substitute velocity from our CL definition
TR D 1
2W
1

 2  V SCD  2  
SCD 
V V
 SCL
1
2
CD
 SW 1/ 2
CL
• Thus, among other things, for max range we want to
fly at a velocity such that CL1/2/CD is a maximum!
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R and E - Jet Aircraft (continued)
• Now let’s work at the quantitative relations.
• First, since TSFC does not have consistent units, a
new fuel burn rate is introduced:

N of fuel 1 
 ct 


( N ) sec sec 

lb of fuel
1
ct 

(lb of thrust)sec sec
• Note that ct is the same in either unit system.
Converting to TSFC results in:
ct 
TSFC
sec
3600hour

• As before, always check units and conversion factors
when using other references!
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R and E - Jet Aircraft (continued)
• The rate at which jets burn fuel is given by:
dW  dWf  ctTdt
• Rearranging and integrating for endurance gives:
t1
W1
dW
E   dt  
 ctT
t0
W0
W0
dW
E 
cT
W1 t
• Similarly, range will be calculated by:
x1
R   dx 
x0
W1
V dW
W  ctT
0
R
W0
V dW
W ctT
1
• Note how similar these equations are to those for
propellers - only the denominator is different.
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Breguet Jet Equations
• Now make assumptions similar to those for propellers
previously to get the Breguet Jet equations from
these integral relations.
• For endurance, assume steady, level flight so that
L=W and T=TR=D.
E
W0
W0
W0
dW
WdW
L dW


W ctT W ct DW W ct D W
1
1
1
• Now with ct a constant and assuming the aircraft is
flown at a velocity such that L/D = CL/CD remains a
constant.
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Breguet Jet Eqns (continued)
• Pulling out theses constants and integrating yields:
1 CL  Wo 
E
ln 
ct CD  W1 
• This relations tells us that to maximize endurance, we
want:
– The lowest possible thrust specific fuel consumption
– The largest possible weight fraction of fuel, Wf/W
– A large (L/D)max and to fly at the velocity where this is
achieved.
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Breguet Jet Eqns (continued)
• Perform a similar procedure with the jet range
equation.
W0
W
W
0
V dW 0 VWdW
V L dW
R 


cT T
ct DW
cD W
W1
W1
W1 t
• To go further, relate velocity to lift via:
V  2W / SCL 
W0
CL
R 
cC
W1 t D
1/ 2
W
0
CL
2W dW

  SC L W W1 ct C D
2
dW
  S W 1/ 2
• Now assume that ct and  are constant, and that
the aircraft is flown at a velocity such that CL1/2/CD
remains a constant
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Breguet Jet Eqns (continued)
• With these assumptions, integration yields:
1/ 2
C
2 L
R
ct CD
2


W
S
1/ 2
0
 W11/ 2

• This relations tells us that to maximize range, we
want:
– The lowest possible specific fuel consumption
– The largest possible weight fraction of fuel, Wf/W
– A large (CL1/2/CD)max and to fly at the velocity where this is
achieved.
– Flying at the lowest density possible, I.e. high altitude!
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Breguet Eqns - Summary
• Here is a summary of the Breguet equations:
• For piston-propellers:
 Wo 

R
ln
c C D  W1 
 CL
E
C 3 / 2
L
cCD
 1
1 
2  S  1/ 2  1/ 2 
W0 
 W1
• For turbojets:
1/ 2
2 CL
R
ct CD
2


W
S
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1/ 2
0
W
1/ 2
1

1 CL  Wo 
E
ln 
ct CD  W1 
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CL and CD relations - Summary
• For (CL3/2/CD)max,
CD,i=3CD,0
C L3 / 2 3eARC D , 0 

CD
4C D , 0
3/ 4
C L  3eARC D , 0
• For (CL/CD)max,
C L  eARC D , 0
• For (CL1/2/CD)max,
CL 
1
3
eARC D ,0
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CD,i=CD,0
C L 1 eAR

C D 2 C D ,0
CD,i=1/3 CD,0
C L1/ 2

CD
95

1
3
eARC D , 0 
1/ 4
4
3
CD ,0
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