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UNIFIED ENGINEERING Fall 2008, ZSLecture Outlines (notes by IAW)

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 ofaircraft, and what are the important performance parameters?”;

C. Be able to use empirical evidence to estimate the performance of aircraft andthus begin to develop intuition regarding important aerodynamic, structuraland propulsion system performance parameters;

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

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

OR:

What is the farthest that an airplane canfly on earth, and why?

We will begin by developing a mathematical model of the physical system. Like mostmodels, this one will have many approximations and assumptions that underlie it. It isimportant for you to understand these approximations and assumptions so that youunderstand the limits of applicability of the model and the estimates derived from it.

DT

L

W

D

L

T

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 or W L DLD

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

dtm gf= − ⋅˙

Now we will define an overall propulsion system efficiency:

overall efficiency = =what you getwhat you pay for

propulsive powerfuel power

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

fuel power = ⋅ =fuel mass flow rate fuel energy per unit mass m hf˙ (J/s)

Thus

ηoverallo

f

Tum h

=˙

We can now write the expression for the change in weight of the vehicle in terms ofimportant aerodynamic (L/D) and propulsion system (ηoverall) parameters:

dW

dtm g

WL

D

T

m g

Wuh

g

L

D

Tu

m h

Wuh

g

L

D

f

f foverall

= − ⋅ = −

⋅

= −

⋅

= −

˙

˙ ˙

0

0

0

η

We can rewrite and integrate

dW

W

u dth

g

L

D

Wtu

h

g

L

Doverall overall

= −

⇒ = −

0 0

η ηln constant

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

∴ = −

tL

D

h

gun

W

Woverallinitial

η0l

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

t

L

D

h

gun

W

Wfinal overallfinal

initial= − η

0l

then multiplying by the flight velocity we arrive at the Breguet Range Equation whichapplies for situations where overall efficiency, L/D, and flight velocity are constant overthe flight.

Range

h

g

L

Dn

W

Winitial

final= overallη l

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 offuel per unit of thrust (lbm/s/lbf or kg/s/N). In the following expression, V is the flightvelocity and g is the acceleration of gravity.

RangeV L D

g SFCWW

initial

final

= ( )⋅

ln

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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 thisenergy to propel a vehicle;

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

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

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0

5

10

15

20

25

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000Year

L/D

max

TurbopropsRegional JetsLarge Aircraft

Data Unavailable For:EMB-145CV-880BAE RJ85Beech 1900CV-580CV-600

FH-227Nihon YS-11SA-226DHC-8-100L-188DHC-7

DHC8-300

D328

J41

BAE-ATP

ATR72F27

S360

J31

SA227

EMB120

S340A

ATR42

BAC111-200/400

F28-1000 F28-4000/6000

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

B777

MD11

B747-400

B737-400B737-500/600

A320-100/200

A300-600

A310-300

B767-300/ER

B737-300

B757-200

L1011-1/100/200

B747-100/200/300

DC10-10

Figure 1.4 Aerodynamic data for commercial aircraft: L/D for cruise (Babikian, R., TheHistorical 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 SimpleScience of Flight, by H. Tennekes)

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Figure 1.5 Weight fractions for transport aircraft in terms of empty weight over maxtake-off weight (Mattingly, Heiser & Daley, Aircraft Engine Design, 1987)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000Year

OE

W/M

TO

W

TurbopropsRegional JetsLarge Aircraft

B707-300B

DC9-40

B737-100/200

B727-200/231A

DC9-30

DC9-10

DC10-30

DC10-40

DC9-50

MD80 & DC9-80

L1011-500

B767-200/ER

B777

MD11

B747-400

B737-400

B737-500/600

A320-100/200

A300-600A310-300

B767-300/ER

B737-300B757-200

L1011-1/100/200

B747-100

B747-200/300

DC10-10

CV880

BAC111-200F28-1000

F28-4000/6000

BAE146-100/RJ70

BAE146-200

BAE-146-300F100

RJ85

RJ200/ER

EMB145BAC111-400

DHC8-300D328

J41BAE-ATP

ATR72

F27FH227

DHC7 S360

B1900

J31

DHC8-100

CV600

SA226

SA227 EMB120S340A

ATR42

L188A-08/188C

Figure 1.6 Structural efficiency data for commercial aircraft: Operating empty weightover maximum take-off weight (Babikian, R., The Historical Fuel Efficiency

Characteristics of Regional Aircraft From Technological, Operational, and CostPerspectives, 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 andinlet:

ηthermal

e e o o

f

m u m u

m h= =

−

rate of production of propellant k.e.

fuel power

˙ ˙

˙

2 2

2 2

ηpropo

e e o o

Tu

m u m u= =

−

propulsive powerrate of production of propellant k.e. ˙ ˙2 2

2 2

such that

ηoverall = ηthermal ⋅ ηprop

During the first semester thermodynamics lectures we will focus largely on thermalefficiency. In next semester’s propulsion lectures we will combine thermodynamics withfluid 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 variousmodern aircraft engines.

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0.2

0.1 0.2 0.3 0.4 0.5 0.6

0.3

0.4

0.5

0.6

0.7

0.8

0.3 0.4

Whittle

Turbojets

Low BPR

'777'Engines

FutureTrend

UDFEngine

CurrentHigh BPR

0.8 0.7 0.6 0.5 0.4 0.3

0.5

Propulsive x Transmission Efficiency

SFC

Overall Efficiency

Co

re T

her

mal

Eff

icie

ncy

0.6 0.7 0.8

CF6-80C2

AdvancedUDF

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

0

5

10

15

20

25

30

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005Year

TS

FC

(m

g/N

s)

TurbopropsRegional JetsLarge JetsNew Regional Jet EnginesNew Turboprop Engines

CV880 F28-1000

F28-4000/6000BAE146-100/200/RJ70

BAE-146-300

F100RJ85

RJ200/ER

EMB145

EMB135

D328 EM170

RJ700

BAC111-400

DHC8-300

D328

J41

BAE-ATP

ATR72

F27

DHC7S360

B1900J31

DHC8-100

CV580

CV600

SA226 SA227

DHC8-400

EMB120

S340A

ATR42

L188A-08/188C

B707-300

B720-000

DC9-40

B737-100/200

B727-200/231A

DC9-30

DC9-10

DC10-30

DC10-40

DC9-50

MD80 & DC9-80

L1011-500

B767-200/ER

B777

MD11B747-400

B737-400 B737-500/600

A320-100/200A300-600

A310-300

B767-300/ER

B737-300

B757-200L1011-1/100/200

B747-100B747-200/300

DC10-10

Figure 1.9 Engine efficiency for commercial aircraft: specific fuel consumption(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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The accuracy of the range equation in predicting performance for commercial transportaircraft is quite good. The Department of Transportation collects and reports a variety ofoperational 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 wasextracted 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 actualstage length flown (range) as reported in Form 41. The difference between the actualstage length flown and the estimated stage length is shown in Figure 1.11. Figure 1.11shows 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, theassumptions of constant velocity, L/D, and SFC are good. However, for short-haulflights, taxiing, climbing, descending, etc. are a relatively large fraction of the overallflight time, so the steady-state cruise assumptions of the range equation are less valid.

(16 short- and long-haul aircraft)

0

1000

2000

3000

4000

5000

6000

0 1000 2000 3000 4000 5000 6000

Actual Stage Length Flown (miles)

Ca

lcu

late

d S

tag

e L

en

gth

(m

iles)

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

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0

20

40

60

80

100

120

140

160

0 1000 2000 3000 4000 5000 6000

Figure 1.11 Deviation (%) of Breguet range equation estimates from actual stage lengthflown is a function of the stage length (J. J. Lee, MIT Masters Thesis, 2000)

Stage Length