MIT How far airplane can fly

8/7/2019 MIT How far airplane can fly

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UNIFIED ENGINEERING Fall 2003

Lecture Outlines Ian A. Waitz

UNIFIED LECTURE #2:

THE BREGUET RANGE EQUATION

I. Learning Goals

At the end of this lecture you will:A. Be able to answer the question How far can an airplane fly, and why?;

B. Be able to answer the question How do the disciplines of structures &

materials, aerodynamics and propulsion jointly set the performance of

aircraft, and what are the important performance parameters?;

C. Be able to use empirical evidence to estimate the performance of aircraft and

thus begin to develop intuition regarding important aerodynamic, structural

and propulsion system performance parameters;

D. Have had your first exposure to active learning in Unified Engineering

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DT

L

W

II. Question: How far can an airplane (or a duck, for that matter)

fly?

OR:

What is the farthest that an airplane can

fly on earth, and why?

We will begin by developing a mathematical model of the physical system. Like most

models, this one will have many approximations and assumptions that underlie it. It is

important for you to understand these approximations and assumptions so that you

understand the limits of applicability of the model and the estimates derived from it.

L

DT

W

Figure 1.1 Force balance for an aircraft in steady level flight.

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For steady, level flight,

T = D, L = W orL

W L= = DD

= TLD

The weight of the aircraft changes in response to the fuel that is burned (rate at whichweight changes equals negative fuel mass flow rate times gravitational constant)

dW= m g

dtf

Now we will define an overall propulsion system efficiency:

what you get=

propulsive poweroverall efficiency =

what you pay for fuel power

propulsive power= thrust flight velocity = Tuo (J/s)

fuel power= fuel mass flow rate fuel energy per unit mass = m h (J/s) f

Thus

Tuo=overallm h f

We can now write the expression for the change in weight of the vehicle in terms of

important aerodynamic (L/D) and propulsion system (overall) parameters:

dW= m g =

W Wu0 =Wu0=

dtf L T h L Tu0 h L

f f g DoverallD m g g D m h

We can rewrite and integrate

0dW =u dt

ln W = constant tu0

W h L h L g D overall g D overall

applying the initial conditions, at t = 0 W = Winitial const. = ln Winitial

L h Wt = overall ln

D gu0 Winitial

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the time the aircraft has flown corresponds to the amount of fuel burned, therefore

L h Wfinalln

D

overall

gu0 Winitial

tfinal =

then multiplying by the flight velocity we arrive at the Breguet Range Equation which

applies for situations where overall efficiency, L/D, and flight velocity are constant overthe flight.

Rangeh L

overall lnWinitial

g D Wfinal=

Fluids Propulsion Structures +(Aero) Materials

Note that this expression is sometimes rewritten in terms of an alternate measure ofefficiency, the specific fuel consumption or SFC. SFC is defined as the mass flow rate of

fuel per unit of thrust (lbm/s/lbf or kg/s/N). In the following expression, V is the flight

velocity and g is the acceleration of gravity.

V L D( )

ln

g SFC

Winitial

Range =

Wfinal

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Thus we see that the answer to the question How far can an airplane fly? dependson:

1. How much energy is contained in the fuel it carries;

2. How aerodynamically efficient it is (the ratio of the production of lift to theproduction of drag). During the fluids lectures you will learn how to developand use models to estimate lift and drag.

3. How efficiently energy from the fuel/oxidizer is turned into useful work

(thrust times distance traveled) which is used to oppose the drag force.

Thermodynamics helps us describe and estimate the efficiency of variousenergy conversion processes, and propulsion lets us describe how to use this

energy to propel a vehicle;

4. How light weight the structure is relative to the amount of fuel and payload it

can carry. The materials and structures lectures you will teach you how toestimate the performance of aerospace structures.

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Below are some data to allow you to make estimates for various aircraft and birds.

Figure 1.2 Heating values for various fuels (from The Simple Science of Flight, by H.

Tennekes)0.1

1

10

w

F=40

F=25

F=10

F=4

airplane

sailplane

Boeing747

pheasant

budgie

cabbage

[meters/second]

human-powered

albatross

ultralight

Fokker

Friendship

white

1 10 100

V[meters / second]

The Great Gliding Diagram. Airspeed, V, is plotted on the horizontal axis. Rate of descent,w, isplotted downward along the vertical axis. The diagonals are lines of constant finesse. Thehorizontal line represents the practical soaring limit, 1 meter / second.

Figure 1.3 Gliding performance as a function of L/D (where L/D=F, from The SimpleScience of Flight, by H. Tennekes)

6

Prime Beef

Beef

Whole Milk

Honey

Sugar

Cheese

Bacon

Corn Flakes

Peanut Butter

Butter

Vegetable Oil

Kerosene

Diesel Oil

Gasoline

Natural Gas

4.0

MJ/Kg $/Kg $/MJ Comments

4.0

2.8

14

15

15

29

15

27

32

36

42

42

42

45

20

8

0.90

4

1

6

4

3.50

4

4.50

2

0.40

0.40

0.40

0.24

5

2

0.32 600 cal/quart

0.29

0.07 100 cal/ounce

100 cal/ounce

180 cal/ounce

0.40

0.14

0.23

0.15

0.14

0.06 240 cal/ounce

0.010 0.82 kg/liter

0.85 kg/liter

0.75 kg/liter

0.8 kg/m3

0.010

0.010

0.005


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L/Dmax

25

F28-1000 F28-4000/6000

20S360

A320-100/200

B777

DHC8-300

15

10

5

CV-600 DHC-7

TurbopropsRegional JetsLarge Aircraft

Data Unavailable For:

EMB-145CV-880BAE RJ85Beech 1900CV-580

FH-227Nihon YS-11SA-226DHC-8-100L-188

D328

J41

BAE-ATP

ATR72F27

J31

SA227

EMB120

S340A

ATR42

BAC111-200/400

BAE146-100/200/RJ70BAE-146-300

F100

RJ200/ER

B707-100B/300 B707-300BB737-100/200

B727-200/231A

DC9-30

DC10-30

DC10-40MD80 & DC9-80

L1011-500

B767-200/ER

MD11

B747-400

B737-400B737-500/600

A300-600

A310-300

B767-300/ER

B737-300

B757-200

L1011-1/100/200

B747-100/200/300

DC10-10

0

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

YearFigure 1.4 Aerodynamic data for commercial aircraft: L/D for cruise (Babikian, R., The

Historical Fuel Efficiency Characteristics of Regional Aircraft From Technological,

Operational, and Cost Perspectives, SM Thesis, MIT, June 2001)

Figure 1.5 Weight and geometry for aircraft and birds (where L/D=F, from The Simple

Science of Flight, by H. Tennekes)

7

House Sparrow

W

(N) (m2) (m) A F

S b

Swift

Common Tern

Kestrel (Sparrow Hawk)

Carrion Crow

Common Buzzard

Peregrine Falcon

Herring Gull

Heron

White Stork

Wandering Albatross

Hang Glider

Parawing

Powered Parawing

Ultralight (microlight)

Sailplanes

Standard Class

Open Class

Fokker F-50

Boeing 747

0.28

0.36

1.2

1.8

5.5

8.0

8.1

12

14

34

85

1000

1000

1700

2000

3500

5500

19 x104

36 x 105

0.009

0.016

0.056

0.06

0.12

0.22

0.13

0.21

0.36

0.50

0.62

15

25

35

15

10.5

16.3

70

511

0.23

0.42

0.83

0.74

0.78

1.25

1.06

1.43

1.73

2.00

3.40

10

8

10

10

15

25

29

60

6

11

12

9

5

7

9

10

8

8

19

7

2.6

2.7

7

21

38

12

7

4

10

12

9

5

10

10

11

9

10

20

8

4

4

8

40

60

16

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80

70

60WE/WTO [%]

50

40

30

C-7A

C-12AC-123 757 767

C-21A

UV-18

C-9A A310L-1011

C-20A

C-140C-130

Concorde

C-17C-5A C-5B

747

C-141B KC-10A

KC-135A

0 100 200 300 400 500 600 700 800 900WTO [1,000 lbs]

Weight Fractions of Cargo and Passenger Aircraft

Figure 1.5 Weight fractions for transport aircraft in terms of empty weight over max

take-off weight (Mattingly, Heiser & Daley, Aircraft Engine Design, 1987)

OEW/MTOW

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

DC9-40

B737-100/200

B727-200/231A

DC9-10DC10-10

BAC111-200F28-1000

BAC111-400

F27

FH227

CV600

L188A-08/188CDC10-40

DC9-50

MD80 & DC9-80

B767-200/ER

B777

B747-400

B737-400

B737-500/600

A320-100/200

A300-600

A310-300

B767-300/ER

B737-300

B757-200L1011-1/100/200

B747-200/300

F28-4000/6000

BAE146-100/RJ70

BAE146-200

BAE-146-300

F100

RJ85

RJ200/ER

D328

J41BAE-ATP

ATR72DHC7 S360

B1900

J31

DHC8-100

SA226

SA227 EMB120

S340A

ATR42

EMB145

DHC8-300

DC9-30CV880L1011-500

DC10-30 MD11

B747-100

Turboprops

B707-300B

Regional JetsLarge Aircraft

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000Year

Figure 1.6 Structural efficiency data for commercial aircraft: Operating empty weight

over maximum take-off weight (Babikian, R., The Historical Fuel Efficiency

Characteristics of Regional Aircraft From Technological, Operational, and Cost

Perspectives, SM Thesis, MIT, June 2001)

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Figure 1.7 Aircraft performance (from The Simple Science of Flight, by H. Tennekes)

For aircraft engines it is often convenient to break the overall efficiency into two parts:thermal efficiency and propulsive efficiency where the subscripts e and o refer to exit and

inlet:2 2m u e e

m uo

rate of production of propellant k.e.= 2

o

2=thermal

fuel power m h f

=propulsive power

=Tuo

prop 2 2rate of production of propellant k.e. m u o

e e m uo

2 2such that

overall = thermal prop

During the first semester thermodynamics lectures we will focus largely on thermal

efficiency. In next semesters propulsion lectures we will combine thermodynamics with

fluid mechanics to obtain estimates for propulsive and thus overall efficiency. The datashown in Figure 1.6 will give you a rough idea for the conversion efficiencies of various

modern aircraft engines.

9

Boeing

747-400 395

378

352

256

139

57

43

33

530

511

511

368

219

105

94

79

65

60

60

50

44

29

28

25

12300

13600

13900

10400

5500

2700

2400

2500

12200

10500

9500

9900

6400

4200

1800

1700

421

400

387

248

200

124

101

80

900

900

900

900

860

800

720

680

4 x 25.7

4 x 23.8

4 x 21.3

3 x 23.1

2 x 22.7

2 x 9.1

2 x 6.7

2 x 4.5

Boeing

747-300

Boeing

747-200

Douglas

Dc-10-30

Airbus

A310

Boeing

737-300

Fokker

F-100

Fokker

F-28

Takeoff

weight

W(tons)

Sea-level

thrust

T(tons)

Fuel

consumption

(liters/hour)

Cruising

speed

V(km/hour)Range

(km) SeatsS (m2) b (m)


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Overall Efficiency

0.1 0.2 0.3 0.4 0.5 0.6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.3 0.4

Whittle

'777'

Engines

Future

UDFEngine

CurrentHigh BPR

0.8 0.7 0.6 0.5 0.4 0.3

0.5

SFC

0.6 0.7 0.8

CF6-80C2

UDF

Turbojets

Low BPR

Trend

CoreThermalEfficienc

y

Advanced

Propulsive x Transmission Efficiency

Figure 1.8 Trends in aircraft engine efficiency (after Pratt & Whitney)

30

25

20

15

10

5

0

B707-300

B720-000

B727-200/231A

F28-4000/6000BAC111-400 DC9-40

BAE146-100/200/RJ70

CV880 F28-1000

BAE-146-300

F100

RJ85

B737-100/200DC9-30

DC9-10DC9-50

MD80 & DC9-80

B737-300

D328EM170

RJ200/ERF27

CV600

DC10-30

DC10-40

L1011-500

B767-200/ERMD11

B747-400

B737-400 B737-500/600

A300-600

B767-300/ER

B757-200L1011-1/100/200

B747-100B747-200/300

DC10-10

EMB145

EMB135

RJ700

J31L188A-08/188C A320-100/200A310-300B777

D328

J41

ATR72

DHC7

S360

B1900

CV580

SA226 SA227

EMB120

ATR42

DHC8-300BAE-ATP

DHC8-100

DHC8-400

Turboprops

S340A

Regional Jets

Large Jets

New Regional Jet Engines

New Turboprop Engines

TSFC(mg/Ns)

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Figure 1.9 Engine efficiency for commercial aircraft: specific fuel consumption

(Babikian, R., The Historical Fuel Efficiency Characteristics of Regional Aircraft FromTechnological, Operational, and Cost Perspectives, SM Thesis, MIT, June 2001)

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The accuracy of the range equation in predicting performance for commercial transportaircraft is quite good. The Department of Transportation collects and reports a variety of

CalculatedStageLength(miles)

operational and financial data for the U.S. fleet in something called DOT Form 41.

Operational data for fuel burned and payload (passengers and cargo) carried was

extracted from Form 41 and combined with the technological data shown in Figures 1.4,1.6 and 1.9 to estimate range. In Figure 1.10 these estimates are compared to the actual

stage length flown (range) as reported in Form 41. The difference between the actual

stage length flown and the estimated stage length is shown in Figure 1.11. Figure 1.11

shows that the percent deviation between the Breguet range equation estimates and theactual stage lengths flown is a function of the stage length. For long-haul flights, the

assumptions of constant velocity, L/D, and SFC are good. However, for short-haul

flights, taxiing, climbing, descending, etc. are a relatively large fraction of the overall

flight time, so the steady-state cruise assumptions of the range equation are less valid.

(16 short- and long-haul aircraft)

6000

5000

4000

3000

2000

1000

0

0 1000 2000 3000 4000 5000 6000

Actual Stage Length Flown (miles)

Figure 1.10 Performance of Breguet range equation for estimating commercial aircraftoperations (J. J. Lee, MIT Masters Thesis, 2000)

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20

0

0 1000 2000 3000 4000 5000 6000

Stage Length

Figure 1.11 Deviation (%) of Breguet range equation estimates from actual stage length

flown is a function of the stage length (J. J. Lee, MIT Masters Thesis, 2000)

III. Questions:

A. How far can a duck fly?

B. Why dont we fly on hydrogen-powered airplanes?

Fuel properties are listed below.

Fuel Density (kg/m3) Heating Value (kJ/kg)

Jet-A 800.0 45000

H2 (gaseous, S.T.P.) 0.0824 120900

H2 (liquid, 1 atm) 70.8 120900

C. Why is the maximum range for an aircraft on earth approximately

25,000mi? (Voyager: 3181kg of fuel is 72% of maximum take-off

weight, flight speed 186.1km/hr = 9 days to circle the earth)

D. What are the assumptions and approximations that underlie the Breguet Range Equation?

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140

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100

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