cruise performance 5 - university of southamptonjps7/aircraft design resources/aerodynami… ·...

Introduction to Aeronautics (04/05) : Slide 5.1

Cruise PerformanceCruise Performance

Aeronautics & Mechanics

AENG11300

Department of Aerospace Engineering

University of Bristol


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


� 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 −= &


Turbojet and Turbofan EnginesTurbojet and Turbofan Engines

Turbojet

Turbofan

low pressure

compressorhigh pressure

compressor

combustion

chamberturbine

fan

nozzle


Turbojet and Turbofan EnginesTurbojet and Turbofan Engines


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


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


Basic Cruise PerformanceBasic Cruise Performance

� Hunter flypast


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


� 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 =


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


� 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


� 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


� 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


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


Range and Endurance – Jet AircraftRange and Endurance – Jet Aircraft

� Proteus


� 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


� 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−=


� 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


� increment in distance dS at velocity V is given by

� assume that true air speed V remains constant


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


� 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 ≈


� 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


� 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∫ =


� 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)


� 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


� 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


( )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=


� 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


=

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

=


� 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


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=


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


� 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


� 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


Range and Endurance

Propeller-Driven Aircraft

Range and Endurance

Propeller-Driven Aircraft

� Voyager


� “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−=


� 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−=


� 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 ==


� increment in distance dS at velocity V is given by


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

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