bristol university breguet range eqn
TRANSCRIPT
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Introduction to Aeronautics (04/05) : Slide 5.1
Cruise PerformanceCruise Performance
Aeronautics & MechanicsAENG11300
Department of Aerospace Engineering
University of Bristol
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Introduction to Aeronautics (04/05) : Slide 5.2
Engine CharacteristicsEngine Characteristics
two main classes of engines:
1. constant-power engines where shaft horsepoweroutput
is roughly constant with speed
piston engines & turboprops ie propeller-driven
2. constant-thrust engines where thrust output is roughly
constant with speed
turbojets & turbofans
ie jet-driven
a gross simplification but useful for preliminary
performance work all produce thrust by imparting an increase in velocity to
the air passing through the engine
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Introduction to Aeronautics (04/05) : Slide 5.3
basic thrust equation
Newtons 2nd Law force = rate of change of momentum
fuel consumption related to power wasted in the jet
propeller = high mass flow but low jet velocity Vj low fuel consumption
rapid loss of thrust with forward speed
jet = low mass flow but high jet velocity Vj high fuel consumption little loss of thrust with forward speed
Thrust GenerationThrust Generation
VVmT j= & m = mass flow rate (kg/s)
Vj = jet efflux velocity
.
( )221 VVmP jlost = &
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Introduction to Aeronautics (04/05) : Slide 5.4
Turbojet and Turbofan EnginesTurbojet and Turbofan Engines
Turbojet
Turbofan
low pressure
compressorhigh pressure
compressor
combustion
chamberturbine
fan
nozzle
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Introduction to Aeronautics (04/05) : Slide 5.5
Turbojet and Turbofan EnginesTurbojet and Turbofan Engines
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Introduction to Aeronautics (04/05) : Slide 5.6
Turbojet vs Turboprop EnginesTurbojet vs Turboprop Engines
characteristics determined by the bypass ratio (BPR) ratio of air passing through fan (or propeller) to air passing
through engine core
BPRBPR
thrust fuel
consumption
speed limit onpropeller
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Introduction to Aeronautics (04/05) : Slide 5.7
Propeller EfficiencyPropeller Efficiency
propeller acts as rotating wing direction of flight determined by
advance ratioJ= V/nD
where n is RPM andD is propeller diameter
local angle of attack governed
by advance ratio and blade pitch
angle setting
blade twist (washout) needed efficiency peaks in
narrow speed range effect of stall
use variable-pitch prop fine pitch at low speed
coarse pitch at high speed
J
fine coarse
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Introduction to Aeronautics (04/05) : Slide 5.8
Basic Cruise PerformanceBasic Cruise Performance
Hunter flypast
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Introduction to Aeronautics (04/05) : Slide 5.9
Jet Aircraft in CruiseJet Aircraft in Cruise
0 50 10 0 15 0 20 0 2500
2 0
4 0
6 0
thrust at sea level
3km
6km
9km
12km
ceiling = 11km
drag,
thrust
VE = V
increasing
weight
stallboundary VMD
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Introduction to Aeronautics (04/05) : Slide 5.10
simplifiedrepresentation of thrust throttle settingk
thrust at sea level T0 (independent of speed)
density ratio (to power 0.7below 11km, to power 1.0above) maximum and minimum speed at each altitude for T=D
lower speed may be unattainable at low altitude due to stall
upper speed is practical cruise speed
between upper and lower speed aircraft will accelerate or climbunless throttle setting is reduced
at one particular height there is just sufficient height to
cruise at one speed only this is the absolute ceiling for the throttle setting used achieved at minimum drag speed
increasing weight reduces cruise speed and lowers ceiling
Features of Jet Cruise DiagramFeatures of Jet Cruise Diagram
7.0
0kTT =
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Introduction to Aeronautics (04/05) : Slide 5.11
Jet Aircraft Cruise SpeedJet Aircraft Cruise Speed
0 50 100 150 200 2500
2
4
6
8
10
12
stall
boundary
altitude
(km)
VEAS , VTAS
VEASVTAS
absolute ceiling
VMAX
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Introduction to Aeronautics (04/05) : Slide 5.12
cruise speed in EAS reduces steadily as altitude increases
maximum cruise speed in TAS increases with altitude
up to a maximum Vmax before the absolute ceiling is reached
demonstrates some advantages of cruise at high altitude maximum cruise speed in TAS (ie ground speed) similar to (or
greater than) speed at sea level
thrust at maximum cruise speed reduces with altitude fuel consumption reduces with altitude
minimum fuel consumption at minimum drag speed work done = thrust distance
in theory should be unaffected by altitude (sinceDmin constant), but1. at low altitudes engine would need to be throttled back
reduced thermodynamic efficiency & hence increased fuel burn2. cruise speed in TAS for minimum drag increases with altitude
Features of Jet Cruise SpeedFeatures of Jet Cruise Speed
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Introduction to Aeronautics (04/05) : Slide 5.13
consider aircraft with throttle adjusted to cruise at
points 1, 2 and 3 what is effect of smallfluctuations in velocity (eg due to gusts) ??
1. speed increase= increase in drag
aircraft decelerates = stable
2. speed increase
= reduction in drag aircraft accelerates = unstable
3. speed increase= no change in drag
aircraft is neutrally stable
in case 2 pilot mustcontinually adjust throttle to maintain speed flight on backside of drag curve rather unsafe!
0 50 100 150 2000
5
10
15
20
T,D
VE
1
2
3
T1
T2
T3
Speed Stability in CruiseSpeed Stability in Cruise
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Introduction to Aeronautics (04/05) : Slide 5.14
absolute ceiling is an unstable condition to maintain maximum thrust setting at minimum drag speed
anychange in speed will increase drag above availablethrust and hence cause aircraft to descend
excess thrust and hence rate of climb drop to zero asceiling is approached
absolute ceiling cannot be established in reasonable time! service ceiling is a practical alternative definition of
maximum operating altitude
at the service ceiling the aircraft still has a small specified
rate of climb
defined as 2.5 m/s for jet aircraft and 0.5 m/s for propeller-driven aircraft
Speed Stability at CeilingSpeed Stability at Ceiling
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Introduction to Aeronautics (04/05) : Slide 5.15
Propeller-Driven Aircraft in CruisePropeller-Driven Aircraft in Cruise
0 50 100 150 200 2500
4000
8000
12000
power at sea level
3km
6km
9km
12km
ceiling = 12.5km
P
VE = V
increasing
weight
stall
boundaryVMP
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Introduction to Aeronautics (04/05) : Slide 5.16
Range and Endurance Jet AircraftRange and Endurance Jet Aircraft
Proteus
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Introduction to Aeronautics (04/05) : Slide 5.17
endurance - the time an aircraft can remain in flight important for surveillance type missions
range the horizontal distance that an aircraft can cover:
1. Safe Range the maximum distance between twoairfields for which an aircraft can fly a safe and reliably
regular service with a specified payload rather lengthy calculation of full mission profile (take-off/climb/
cruise/descent/landing, headwinds, diversion allowance etc) therefore simplified measures of range used in project work
2. Still Air Range (SAR) take-off with full fuel, climb to cruise altitude, cruise until all fuel
expended (!)
3. Gross Still Air Range (GSAR) begin at selected altitude with full fuel, cruise until all fuel has gone
approximate factor between GSAR and Safe Range known
Range and EnduranceRange and Endurance
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Introduction to Aeronautics (04/05) : Slide 5.18
actually derived by Coffin in the 1920s compact and simple way of calculating GSAR
rate at which fuel is burnt
= rate at which aircraft weight is reduced
define thrust specific fuel consumption (sfc or TSFC) as
f= mass of fuel burnt per unit of thrust per second
consistent units are kg/N.s but often given in terms of kg/N.hrso dont forget to convert!
for thrust Tand weight Wthe basic Breguet equation is
a differential equation (in units ofN/s )
Breguet Range Equation Jet AircraftBreguet Range Equation Jet Aircraft
Tfgdt
dW=
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Introduction to Aeronautics (04/05) : Slide 5.19
rearranging and substituting T =D and W=L
assume that CL/CD andfremain constant
can then integrate from start weight W1 to end weight W2to obtain the enduranceE
Integration of Breguet Equation - EnduranceIntegration of Breguet Equation - Endurance
fgT
dWdt =
WC
CW
L
DT
L
D== W
dWCC
fgdt
D
L1=
===
2
1
1212 ln
1
W
W
C
C
fgtttE
D
L = xxdx
ln
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Introduction to Aeronautics (04/05) : Slide 5.20
increment in distancedSat velocity Vis given by
assume that true air speedVremains constant
can then integrate from start weight W1 to end weight W2to obtain the rangeR
Integration of Breguet Equation - RangeIntegration of Breguet Equation - Range
W
dW
C
C
fg
VVdtdS
D
L==
== 21
12 ln W
W
C
C
fg
V
SSR D
L
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Introduction to Aeronautics (04/05) : Slide 5.21
is it justifiable to assume CL/CD ,f and VTAS to be constant?
only in particular circumstances
as fuel is burnt weight Wand hence required liftL reduces
if VTASheld constant then either CL and/or must reduce
constant CL/CD constant CL (= constant incidence )
therefore
must decrease
altitude must increase
constant CL/CD implies constant CD therefore dragD decreases in proportion to
since T=D, thrust must also decrease with altitude
while constant sfcfimplies constant throttle setting approximately true for turbojet in stratosphere (above
~ 11km), where
Implications of Assumptions (1)Implications of Assumptions (1)
LSCVWL 2
021 ==
0kTT
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Introduction to Aeronautics (04/05) : Slide 5.22
this cruise case often referred to as a cruise-climb usually precluded by air traffic control restrictions!
note that below 11km (in the troposphere) temperature
falls with altitude thrust falls off less rapidly need to back-off on throttle
to maintain Vand CL constant, hence sfc would worsen
Cruise-ClimbCruise-Climb
=
2
1lnW
W
C
C
fg
VR
D
L
a measure of structural
efficiency ie minimise
fixed weight W2a measure of
aerodynamic efficiency
ie minimise dragD
a measure of
thermodynamic efficiency
ie minimise fuel
consumptionf
maximise cruise
speed ie cruise
at high altitude
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Introduction to Aeronautics (04/05) : Slide 5.23
as before
but now assume and henceare constant, but Vvaries
substitute velocity equation
and integrate
Constant Altitude CruiseConstant Altitude Cruise
W
dW
C
C
fg
VVdtdS
D
L==
( )21221121
12
18WW
C
C
fgSSSR
D
L ==
LSC
W
V 21= 21
2121
W
dW
C
C
SfgdS DL
=
21
21 2x
x
dx =
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Introduction to Aeronautics (04/05) : Slide 5.24
height (hence density) and CL held constant
as fuel is burnt weight Wand hence required liftL reduces
VTASmust also reduce range is less than for cruise-climb for same CL/CD
since CD is constant, dragD and hence thrust Talsoreduce
therefore throttle setting must be reducedprogressively during the cruise
therefore some variation in sfcfwill occur
need to use average value off, or treat as a series ofshorter steps
if each step flown at an increased height an approximationto a cruise-climb profile can be obtained
Implications of Assumptions (2)Implications of Assumptions (2)
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Introduction to Aeronautics (04/05) : Slide 5.25
maximum range depends on
variation of thrust with height
start with basic T= T0 with fixed throttle sfcf and velocity Vconstant but altitude (and hence ) undefined
for maximum range we require V(CL/CD) to be a maximum use the thrust/drag relation to eliminate
and hence
Maximum Range Cruise-Climb (1)Maximum Range Cruise-Climb (1)
=
2
1ln1
W
W
C
CV
fgR
D
L
SVCDTT D2
02
10 ===
DSC
TV
02
1
0
=
23
0
02
D
L
D
L
C
C
S
T
C
CV
= therefore need to find
minimum CD3/2/CL
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Introduction to Aeronautics (04/05) : Slide 5.26
so at the minimum point
cruise conditions fixed by thrust
CL and CD obtained from above
substitute with T0 into
then substitute into Breguet Equation to find range
Maximum Range Cruise-Climb (2)Maximum Range Cruise-Climb (2)
( )L
LD
L
D
C
KCC
C
C 232
0
23 +=
( ) ( ) ( )232
0
212
02
323 2
L
LDLLDL
L
LD
CKCCKCKCCC
dCCCd ++=
2
0 2 LD KCC =
KCCCC DR
LDR
D 2,2
30
max0
max ==cruise-climb
with T= T0
2v
udvvdu
v
ud
=
23
0
02
D
L
D
L
C
C
S
T
C
CV
=
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Introduction to Aeronautics (04/05) : Slide 5.27
alternatively specify start altitude (and hence 1) and let
thrust vary as required
sfcf and velocity Vconstant
for maximum range we still require V(CL/CD) to be amaximum
since is not a variable, we can simply substitute thespeed equation for V
therefore need to find minimum CD/CL1/2
Maximum Range Cruise-Climb (3)Maximum Range Cruise-Climb (3)
=
2
1ln1
W
W
C
CV
fg
RD
L
D
L
D
L
C
C
S
W
C
C
V
21
0121
1
=LSC
LV
021=
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Introduction to Aeronautics (04/05) : Slide 5.28
so at the minimum point
cruise conditions fixed by start altitude and weight
CL and CD obtained from above substitute with W1 and 1 into
then substitute intoBreguet Equation to find range
Maximum Range Cruise-Climb (4)Maximum Range Cruise-Climb (4)
23
21
0
21
2
0
21 L
L
D
L
LD
L
D KCC
C
C
KCC
C
C+=
+=
( ) 2123
0
21
2
3
2
1L
L
D
L
LD
CC
C
dC
CCd
+=2
0 3 LD KCC =
KCCCC DRLDRD 3,3
40
max0
max ==cruise-climb
with unrestrictedthrust T
D
L
D
L
C
C
S
W
C
CV
21
0121
1
=
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Introduction to Aeronautics (04/05) : Slide 5.29
Jet Cruise-Climb RangeJet Cruise-Climb Range
0 0.2 0.4 0.6 0.8 1 1.2 1.40
200
400
600
800
1000
1200
1400
4
68
10
12km
start of
cruise
altitude
Range
(km)
CL
minimum
powerminimum
drag
(CDO/2K)1/2(CDO/3K)
1/2
T = T0
no thrust
restriction
0 0.2 0.4 0.6 0.8 1 1.2 1.40
200
400
600
800
1000
1200
1400
4
68
10
12km
start of
cruise
altitude
Range
(km)
CL
minimum
powerminimum
drag
(CDO/2K)1/2(CDO/3K)
1/2
T = T0
no thrust
restriction
W1 =100 kN
W2 = 60 kN
S = 50 m2
CD0 = 0.02K = 0.05
T0 = 20 kN
f = 0.0001 kg/Ns
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Introduction to Aeronautics (04/05) : Slide 5.30
start with (initial) thrust
T1 = T0 as before
altitude (and hence )
undefined
use the thrust/drag relation to eliminate
for maximum range require CD3/2/CL to be a minimum
same as cruise-climb result !
Maximum Range Constant Altitude (1)Maximum Range Constant Altitude (1)
1
0
01
1
1
0
01
0
0
0
100 W
C
C
T
W
L
D
T
D
TT
T
L
D =====
( )212
21
1
2118
WWC
C
fgSR
D
L =
( )212
21
123
10
0 18 WW
C
C
fgSW
TR
D
L =
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Introduction to Aeronautics (04/05) : Slide 5.31
alternatively specify cruise altitude (and hence ) and
let thrust vary as required
altitude (and hence)
are constant
for maximum range require CD/CL1/2 to be a minimum
again the same as cruise-climb result !
Maximum Range Constant Altitude (2)Maximum Range Constant Altitude (2)
( )21
2
21
1
2118
WWC
C
fgSR D
L
=
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Introduction to Aeronautics (04/05) : Slide 5.32
Range and EndurancePropeller-Driven AircraftRange and Endurance
Propeller-Driven Aircraft
Voyager
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Introduction to Aeronautics (04/05) : Slide 5.33
rate at which fuel is burnt
= rate at which aircraft weight is reduced
define specific fuel consumption as
f= mass of fuel burnt per unit of power per second
consistent units are kg/W.s
but often given in terms of kg/kW.hrso dont forget to convert!
for powerPand weight Wthe basic Breguet equation is
a differential equation (in units ofN/s )
Breguet Range Equation Propeller-Driven
Aircraft
Breguet Range Equation Propeller-Driven
Aircraft
Pfgdt
dW
=
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Introduction to Aeronautics (04/05) : Slide 5.34
power delivered by propeller
where = propeller efficiency
assume that CL/CD ,f and Vremain constant
same as jet cruise-climb integrate as before from W1 to W2 to obtain the
enduranceE
Integration of Breguet Equation - EnduranceIntegration of Breguet Equation - Endurance
DVP=
W
dW
C
C
fgVdt DL
=
===
2
11212 ln
1
W
W
C
C
VfgtttE
D
Lprop
DVfgdt
dW
=
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Introduction to Aeronautics (04/05) : Slide 5.35
for maximum range we require
(CL/CD)/V to be a maximum
or VCD/CL to be a minimum
expanding this term we obtain
D Vis the power required to overcome drag
maximum endurance achieved at minimumpower speed for propeller-driven aircraft very much slower than for jet aircraft
compare with glider performance
Maximum EnduranceMaximum Endurance
=
2
1ln1
W
W
C
C
VfgE
D
Lprop
W
VD
L
VD
C
C
VL
D
==
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Introduction to Aeronautics (04/05) : Slide 5.36
increment in distancedSat velocity Vis given by
can then integrate from start weight W1 to end weight W2to obtain the rangeR
maximum range achieved at minimum drag speed for
propeller-driven aircraft much slower than for jet aircraft
Integration of Breguet Equation - RangeIntegration of Breguet Equation - Range
W
dW
C
C
fg
VdtdSD
L==
==
2
112 ln
W
W
C
C
fgSSR
D
Lprop