Bristol University Breguet Range Eqn

8/11/2019 Bristol University Breguet Range Eqn

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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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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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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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Turbojet and Turbofan EnginesTurbojet and Turbofan Engines

Turbojet

Turbofan

low pressure

compressorhigh pressure

compressor

combustion

chamberturbine

fan

nozzle


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Turbojet and Turbofan EnginesTurbojet and Turbofan Engines


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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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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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Basic Cruise PerformanceBasic Cruise Performance

Hunter flypast


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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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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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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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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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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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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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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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Range and Endurance Jet AircraftRange and Endurance Jet Aircraft

Proteus


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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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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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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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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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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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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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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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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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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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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

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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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


=

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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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


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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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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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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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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Range and EndurancePropeller-Driven AircraftRange and Endurance

Propeller-Driven Aircraft

Voyager


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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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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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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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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

Bristol University Breguet Range Eqn

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