Cruise Performance
Aeronautics & Mechanics
AENG11300
Department of Aerospace Engineering
University of Bristol
Introduction to Aeronautics (04/05) : Slide 5.1
Engine 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 5.2
Thrust Generation
basic thrust equation
– Newton’s 2nd Law → force = rate of change of momentum
T = m& (V j − V )
Vj = jet efflux velocity
fuel consumption related to power ‘wasted’ in the jet
P lost =
.m = mass flow rate (kg/s)
1
2
2
m& (V j − V )
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
Introduction to Aeronautics (04/05) : Slide 5.3
Turbojet and Turbofan Engines
Turbojet
low pressure
compressor
high pressure
compressor
combustion
chamber
turbine
nozzle
Turbofan
fan
Introduction to Aeronautics (04/05) : Slide 5.4
Turbojet and Turbofan Engines
Introduction to Aeronautics (04/05) : Slide 5.5
Turbojet vs Turboprop Engines
characteristics determined by the bypass ratio (BPR)
– ratio of air passing through fan (or propeller) to air passing
through engine core
fuel
consumption
thrust
BPR
BPR
speed limit on
propeller
Introduction to Aeronautics (04/05) : Slide 5.6
Propeller 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
η
fine
coarse
use variable-pitch prop
– fine pitch at low speed
– coarse pitch at high speed
Introduction to Aeronautics (04/05) : Slide 5.7
J
Basic Cruise Performance
Introduction to Aeronautics (04/05) : Slide 5.8
Hunter flypast
Jet Aircraft in Cruise
60
drag,
thrust
increasing
weight
thrust at sea level
40
3km
6km
9km
ceiling = 11km
20
12km
stall
boundary
0
0
50
VMD
100
150
200
250
VE = V√σ
Introduction to Aeronautics (04/05) : Slide 5.9
Features of Jet Cruise Diagram
simplified representation of thrust
T = kT0σ 0.7
– 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
Introduction to Aeronautics (04/05) : Slide 5.10
Jet Aircraft Cruise Speed
12
absolute ceiling
altitude
(km)
10
stall
boundary
8
VMAX
VEAS
6
VTAS
4
2
0
0
50
100
150
200
VEAS , VTAS
Introduction to Aeronautics (04/05) : Slide 5.11
250
Features of Jet Cruise Speed
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
Introduction to Aeronautics (04/05) : Slide 5.12
Speed Stability in Cruise
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) ??
20
1. speed increase
T,D
= increase in drag
1
T1
– aircraft decelerates = stable 15
2. speed increase
2
T2
= reduction in drag
10
3
T3
– aircraft accelerates = unstable
3. speed increase
5
= no change in drag
– aircraft is neutrally stable
0
0
50
in case 2 pilot must
continually adjust throttle to maintain speed
100
– flight on ‘backside of drag curve’ – rather unsafe!
Introduction to Aeronautics (04/05) : Slide 5.13
150
VE
200
Speed Stability at Ceiling
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 propellerdriven aircraft
Introduction to Aeronautics (04/05) : Slide 5.14
Propeller-Driven Aircraft in Cruise
12000
P√σ
power at sea level
8000
3km
increasing
weight
6km
4000
9km
12km
0
stall
boundary
0
50
ceiling = 12.5km
VMP
100
150
200
250
VE = V√σ
Introduction to Aeronautics (04/05) : Slide 5.15
Range and Endurance – Jet Aircraft
Introduction to Aeronautics (04/05) : Slide 5.16
Proteus
Range and Endurance
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
Introduction to Aeronautics (04/05) : Slide 5.17
Breguet Range Equation – Jet Aircraft
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
dW
= − fg T
dt
– a differential equation (in units of N/s )
Introduction to Aeronautics (04/05) : Slide 5.18
Integration of Breguet Equation - Endurance
rearranging and substituting T = D and W = L
dW
dt = −
fgT
CD
D
W
T= W=
CL
L
1 C L dW
dt = −
fg C D W
assume that CL/CD and f remain constant
can then integrate from start weight W1 to end weight W2
to obtain the endurance E
dx
∫ x = ln x
E = t 2 − t1 = t12
Introduction to Aeronautics (04/05) : Slide 5.19
1 C L  W1 
ln 
=
fg C D  W2 
Integration of Breguet Equation - Range
increment in distance dS at velocity V is given by
V C L dW
dS = Vdt = −
fg C D W
assume that true air speed V remains constant
can then integrate from start weight W1 to end weight W2
to obtain the range R
V C L  W1 
R = S 2 − S1 =
ln 
fg C D  W2 
Introduction to Aeronautics (04/05) : Slide 5.20
Implications of Assumptions (1)
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 T ≈ kT σ
0
Introduction to Aeronautics (04/05) : Slide 5.21
L = W = 12 ρ 0σV 2 SC L
Cruise-Climb
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
maximise cruise
speed – ie cruise
at high altitude
V C L  W1 
ln 
R =
fg C D  W2 
a measure of
thermodynamic efficiency
– ie minimise fuel
consumption f
a measure of
aerodynamic efficiency
– ie minimise drag D
Introduction to Aeronautics (04/05) : Slide 5.22
a measure of structural
efficiency – ie minimise
fixed weight W2
Constant Altitude Cruise
as before
V C L dW
dS = Vdt = −
fg C D W
but now assume σ and hence ρ are constant, but V varies
substitute velocity equation
V =
W
1
2 ρSC L
1
dS = −
fg
2 C L1 2 dW
ρS C D W 1 2
and integrate
dx
12
=
2x
∫ x1 2
R = S 2 − S1 =
Introduction to Aeronautics (04/05) : Slide 5.23
8 1 C L1 2 1 2
W1 − W21 2
ρS fg C D
(
)
Implications of Assumptions (2)
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
Introduction to Aeronautics (04/05) : Slide 5.24
Maximum Range – Cruise-Climb (1)
maximum range depends on
variation of thrust with height
R =
1
fg
 C L   W1 
V
 ln 
 C D   W2 
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
T0
T = T0σ = D = C D 12 ρ 0σV 2 S
V = 1
2 ρ 0 SC D
and hence V
CL
=
CD
Introduction to Aeronautics (04/05) : Slide 5.25
2T0 C L
ρ 0 S C D3 2
→ therefore need to find
minimum CD3/2 /CL
Maximum Range – Cruise-Climb (2)
(
C
C D 0 + KC
=
CL
CL
32
D
(
)
32
D
d C CL
dC L
)
=
C
3
L 2
(C
D0
+ KC
2
C
=
2
KC
so at the minimum point D 0
L
CD
max R
 u  vdu − udv
d  =
v2
v
2 32
L
3
= CD0 , CL
=
max
R
2
cruise conditions fixed by thrust
– CL and CD obtained from above
CL
=
→ substitute with T0 into V
CD
)
2 12
L
CD0 2K
2T0 C L
ρ 0 S C D3 2
– then substitute into Breguet Equation to find range
Introduction to Aeronautics (04/05) : Slide 5.26
(
2 KC L − C D 0 + KC
C L2
)
2 32
L
cruise-climb
with T = T0σ
Maximum Range – Cruise-Climb (3)
alternatively specify start altitude (and hence σ1) and let
thrust vary as required
1  C L   W1 
 ln 
V
→ sfc f and velocity V constant R =
fg  C D 
 W2 
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
V =
L
1
2 σρ 0 SC L
CL
V
=
CD
C L1 2
1
2 σ 1 ρ0 S CD
W1
→ therefore need to find minimum CD/CL1/2
Introduction to Aeronautics (04/05) : Slide 5.27
Maximum Range – Cruise-Climb (4)
CD
C D 0 + KC L2
CD0
32
=
=
+
KC
L
C L1 2
C L1 2
C L1 2
d C D C L1 2
dC L
(
= −
1 CD0 3 1 2
+ CL
32
2 CL
2
so at the minimum point C D 0 = 3KC L2
CD
max R
)
4
= CD0 , CL
=
max
R
3
C D 0 3K
cruise-climb
with unrestricted
thrust T
cruise conditions fixed by start altitude and weight
– CL and CD obtained from above
12
C
W
C
L
1
→ substitute with W1 and σ1 into V L =
1
C
D
2 σ 1 ρ0 S CD
– then substitute into
Breguet Equation to find range
Introduction to Aeronautics (04/05) : Slide 5.28
Jet Cruise-Climb Range
1400
no thrust
restriction
Range
(km) 1200
T = T0σ
1000
800
12km
W1 =100 kN
W2 = 60 kN
S = 50 m2
CD0 = 0.02
K = 0.05
T0 = 20 kN
f = 0.0001 kg/Ns
600
10
400
8
6
4
(CDO/3K)1/2
(CDO/2K)1/2
200
0
minimum
power
minimum
drag
0
0.2
Introduction to Aeronautics (04/05) : Slide 5.29
0.4
0.6
0.8
‘start of
cruise’
altitude
1
CL
1.2
1.4
Maximum Range – Constant Altitude (1)
start with (initial) thrust
T1 = T0σ as before
– altitude (and hence σ )
R =
(
8 1 C L1 2 1 2
W1 − W21 2
ρS fg C D
undefined
use the thrust/drag relation to eliminate ρ
ρ 0 D1
ρ0 CD
T1 ρ 0
ρ = ρ 0σ = ρ 0 =
D1 =
W1 =
W1
T0 T0
T0 L1
T0 C L
R =
(
8T0 1 C L
12
12
W
W
−
1
2
ρ 0 SW1 fg C D3 2
)
for maximum range require CD3/2 /CL to be a minimum
→ same as cruise-climb result !
Introduction to Aeronautics (04/05) : Slide 5.30
)
Maximum Range – Constant Altitude (2)
alternatively specify cruise altitude (and hence σ) and
let thrust vary as required
12
8
1
C
– altitude (and hence ρ )
L
R =
W11 2 − W21 2
are constant
ρS fg C D
(
for maximum range require CD /CL 1/2 to be a minimum
→ again the same as cruise-climb result !
Introduction to Aeronautics (04/05) : Slide 5.31
)
Range and Endurance
Propeller-Driven Aircraft
Introduction to Aeronautics (04/05) : Slide 5.32
Voyager
Breguet Range Equation – Propeller-Driven
Aircraft
“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
dW
= − fg P
dt
– a differential equation (in units of N/s )
Introduction to Aeronautics (04/05) : Slide 5.33
Integration of Breguet Equation - Endurance
power delivered by propeller
– where η = propeller efficiency
dW
DV
= − fg
η
dt
ηP = DV
dt = −
η C L dW
fgV C D W
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
E prop = t 2 − t1 = t12
Introduction to Aeronautics (04/05) : Slide 5.34
 W1 
=
ln 
fg V C D  W2 
η 1 CL
Maximum Endurance
for maximum range we require
(CL/CD)/V to be a maximum
– or VCD/CL to be a minimum
E prop =
η  1 C L   W1 

 ln 
fg  V C D   W2 
expanding this term we obtain
CD
VD VD
=
=
V
CL
L
W
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
Introduction to Aeronautics (04/05) : Slide 5.35
Integration of Breguet Equation - Range
increment in distance dS at velocity V is given by
dS = Vdt = −
η C L dW
fg C D W
can then integrate from start weight W1 to end weight W2
to obtain the range R
R prop = S 2 − S1 =
η  C L   W1 

 ln 
fg  C D   W2 
maximum range achieved at minimum drag speed for
propeller-driven aircraft
– much slower than for jet aircraft
Introduction to Aeronautics (04/05) : Slide 5.36