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Introduction to Aeronautics (04/05) : Slide 5.1
Cruise PerformanceCruise Performance
Aeronautics & Mechanics
AENG11300
Department of Aerospace Engineering
University of Bristol
Introduction to Aeronautics (04/05) : Slide 5.2
Engine CharacteristicsEngine Characteristics
� two main classes of engines:
1. ‘constant-power’ engines where shaft horsepower output
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
Introduction to Aeronautics (04/05) : Slide 5.3
� basic thrust equation
– Newton’s 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
.
( )22
1 VVmP jlost −= &
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
Introduction to Aeronautics (04/05) : Slide 5.5
Turbojet and Turbofan EnginesTurbojet and Turbofan Engines
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
BPR
BPR
thrust fuel
consumption
speed limit on
propeller
Introduction to Aeronautics (04/05) : Slide 5.7
Propeller EfficiencyPropeller Efficiency
� propeller acts as rotating wing– ‘direction of flight’ determined by
advance ratio J = V/nD
– where n is RPM and D is propeller diameter
� local angle of attack governed
by advance ratio and blade pitch
angle setting– blade twist (washout) needed
� efficiency η peaks in
narrow speed range – effect of stall
� use variable-pitch prop– fine pitch at low speed
– coarse pitch at high speed
η
J
fine coarse
Introduction to Aeronautics (04/05) : Slide 5.8
Basic Cruise PerformanceBasic Cruise Performance
� Hunter flypast
Introduction to Aeronautics (04/05) : Slide 5.9
Jet Aircraft in CruiseJet Aircraft in Cruise
0 50 100 150 200 2500
20
40
60
thrust at sea level
3km
6km
9km
12km
ceiling = 11km
drag,
thrust
VE = V√σ
increasing
weight
stall
boundary VMD
Introduction to Aeronautics (04/05) : Slide 5.10
� simplified representation of thrust– throttle setting k
– thrust at sea level T0 (independent of speed)
– density ratio σ (to power 0.7 below 11km, to power 1.0 above)
� 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 climb
unless 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
0σkTT =
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
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 (since Dmin constant), but
1. at low altitudes engine would need to be throttled back
→ reduced thermodynamic efficiency & hence increased fuel burn
2. cruise speed in TAS for minimum drag increases with altitude
Features of Jet Cruise SpeedFeatures of Jet Cruise Speed
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 small fluctuations 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 must continually 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
Introduction to Aeronautics (04/05) : Slide 5.14
� absolute ceiling is an unstable condition to maintain– maximum thrust setting at minimum drag speed
→ any change in speed will increase drag above available
thrust and hence cause aircraft to descend
� excess thrust and hence rate of climb drop to zero as ceiling 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
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√σ
VE = V√σ
increasing
weight
stall
boundaryVMP
Introduction to Aeronautics (04/05) : Slide 5.16
Range and Endurance – Jet AircraftRange and Endurance – Jet Aircraft
� Proteus
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 two
airfields 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
Introduction to Aeronautics (04/05) : Slide 5.18
� actually derived by Coffin in the 1920’s– 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.hr so don’t forget to convert!
� for thrust T and weight W the basic Breguet equation is
– a differential equation (in units of N/s )
Breguet Range Equation – Jet Aircraft Breguet Range Equation – Jet Aircraft
Tfgdt
dW−=
Introduction to Aeronautics (04/05) : Slide 5.19
� rearranging and substituting T = D and W = L
� assume that CL/CD and f remain constant
� can then integrate from start weight W1 to end weight W2
to obtain the endurance E
Integration of Breguet Equation - EnduranceIntegration of Breguet Equation - Endurance
fgT
dWdt −=
WC
CW
L
DT
L
D==W
dW
C
C
fgdt
D
L1−=
==−=
2
1
1212ln
1
W
W
C
C
fgtttE
D
L∫ = xx
dxln
Introduction to Aeronautics (04/05) : Slide 5.20
� increment in distance dS at velocity V is given by
� assume that true air speed V remains constant
� can then integrate from start weight W1 to end weight W2
to obtain the range R
Integration of Breguet Equation - RangeIntegration of Breguet Equation - Range
W
dW
C
C
fg
VVdtdS
D
L−==
=−=
2
1
12 lnW
W
C
C
fg
VSSR
D
L
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 W and hence required lift L reduces
� if VTAS held constant then either CL and/or σ must reduce
� constant CL/CD →→→→ constant CL (= constant incidence α)
→→→→ therefore σ must decrease ≡ altitude must increase� constant CL/CD implies constant CD
→→→→ therefore drag D decreases in proportion to σ� since T = D, thrust must also decrease with altitude
while constant sfc f implies constant throttle setting
→→→→ approximately true for turbojet in stratosphere (above ~ 11km), where
Implications of Assumptions (1)Implications of Assumptions (1)
LSCVWL 2
02
1 σρ==
σ0
kTT ≈
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 V and 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 W2
a measure of
aerodynamic efficiency
– ie minimise drag D
a measure of
thermodynamic efficiency
– ie minimise fuel
consumption f
maximise cruise
speed – ie cruise
at high altitude
Introduction to Aeronautics (04/05) : Slide 5.23
� as before
� but now assume σ and hence ρ are constant, but V varies
� substitute velocity equation
and integrate
Constant Altitude CruiseConstant Altitude Cruise
W
dW
C
C
fg
VVdtdS
D
L−==
( )212
21
1
21
12
18WW
C
C
fgSSSR
D
L −=−=ρ
LSC
WV
ρ21
=21
2121
W
dW
C
C
SfgdS
D
L
ρ−=
21
212x
x
dx∫ =
Introduction to Aeronautics (04/05) : Slide 5.24
� height (hence density ρ) and CL held constant
� as fuel is burnt weight W and hence required lift L reduces
→→→→ VTAS must also reduce
→→→→ range is less than for cruise-climb for same CL /CD
� since CD is constant, drag D and hence thrust T also
reduce
→→→→ therefore throttle setting must be reduced
progressively during the cruise
→→→→ therefore some variation in sfc f will occur
� need to use average value of f, or treat as a series of
shorter steps
– if each step flown at an increased height an approximation
to a cruise-climb profile can be obtained
Implications of Assumptions (2)Implications of Assumptions (2)
Introduction to Aeronautics (04/05) : Slide 5.25
� maximum range depends on
variation of thrust with height
� start with basic T = T0σ with fixed throttle
→→→→ sfc f and velocity V constant
– but altitude (and hence σ ) undefined
� for maximum range we require V(CL/CD) to be a maximum
� use the thrust/drag relation to eliminate σ
and hence
Maximum Range – Cruise-Climb (1)Maximum Range – Cruise-Climb (1)
=
2
1ln1
W
W
C
CV
fgR
D
L
SVCDTT D
2
02
10 σρσ ===
DSC
TV
021
0
ρ=
23
0
02
D
L
D
L
C
C
S
T
C
CV
ρ= → therefore need to find
minimum CD3/2 /CL
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
C232
0
23 +=
( ) ( ) ( )2
232
0
212
02323
2
L
LDLLDL
L
LD
C
KCCKCKCCC
dC
CCd +−+=
2
0 2 LD KCC =
KCCCC DR
LDR
D 2,2
30
max0
max==
cruise-climb
with T = T0σσσσ
2v
udvvdu
v
ud
−=
23
0
02
D
L
D
L
C
C
S
T
C
CV
ρ=
Introduction to Aeronautics (04/05) : Slide 5.27
� alternatively specify start altitude (and hence σ1) and let
thrust vary as required
→→→→ sfc f and velocity V constant
� for maximum range we still require V(CL/CD) to be a
maximum
� since σ is not a variable, we can simply substitute the
speed 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
fgR
D
L
D
L
D
L
C
C
S
W
C
CV
21
0121
1
ρσ=
LSC
LV
021σρ
=
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 σ1 into
– then substitute into
Breguet 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+=
+=
( ) 21
23
0
21
2
3
2
1L
L
D
L
LD CC
C
dC
CCd+−=
2
0 3 LD KCC =
KCCCC DR
LDR
D 3,3
40
max0
max==
cruise-climb
with unrestricted
thrust T
D
L
D
L
C
C
S
W
C
CV
21
0121
1
ρσ=
Introduction to Aeronautics (04/05) : Slide 5.29
Jet Cruise-Climb Range Jet 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 = T0σ
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 = T0σ
no thrust
restriction
W1 =100 kN
W2 = 60 kN
S = 50 m2
CD0 = 0.02
K = 0.05
T0 = 20 kN
f = 0.0001 kg/Ns
Introduction to Aeronautics (04/05) : Slide 5.30
� start with (initial) thrust
T1 = T0σ as before
– altitude (and hence σ )
undefined
� use the thrust/drag relation to eliminate ρ
� 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
0
1
1
1
0
0
1
0
0
0
1
00W
C
C
TW
L
D
TD
TT
T
L
Dρρρρσρρ =====
( )21221
1
2118
WWC
C
fgSR
D
L −=ρ
( )212
21
123
10
0 18WW
C
C
fgSW
TR
D
L −=ρ
Introduction to Aeronautics (04/05) : Slide 5.31
� alternatively specify cruise altitude (and hence σ) and
let thrust vary as required
– altitude (and hence ρ )
are constant
� for maximum range require CD /CL 1/2 to be a minimum
→→→→ again the same as cruise-climb result !
Maximum Range – Constant Altitude (2)Maximum Range – Constant Altitude (2)
( )21221
1
2118
WWC
C
fgSR
D
L −=ρ
Introduction to Aeronautics (04/05) : Slide 5.32
Range and Endurance
Propeller-Driven Aircraft
Range and Endurance
Propeller-Driven Aircraft
� Voyager
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.hr so don’t forget to convert!
� for power P and weight W the basic Breguet equation is
– a differential equation (in units of N/s )
Breguet Range Equation – Propeller-Driven
Aircraft
Breguet Range Equation – Propeller-Driven
Aircraft
Pfgdt
dW−=
Introduction to Aeronautics (04/05) : Slide 5.34
� power delivered by propeller
– where η = propeller efficiency
� assume that CL/CD , f and V remain constant– same as jet cruise-climb
� integrate as before from W1 to W2 to obtain the
endurance E
Integration of Breguet Equation - EnduranceIntegration of Breguet Equation - Endurance
DVP =η
W
dW
C
C
fgVdt
D
Lη−=
==−=
2
1
1212ln
1
W
W
C
C
VfgtttE
D
Lprop
η
ηDV
fgdt
dW−=
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 × V is the power required to overcome drag
→→→→ maximum endurance achieved at minimum
power 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
η
W
VD
L
VD
C
CV
L
D ==
Introduction to Aeronautics (04/05) : Slide 5.36
� increment in distance dS at velocity V is given by
� can then integrate from start weight W1 to end weight W2
to obtain the range R
� 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
fgVdtdS
D
Lη−==
=−=
2
1
12 lnW
W
C
C
fgSSR
D
Lprop
η