UNIFIED ENGINEERING
Lecture Outlines
Fall 2008, ZS
(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 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
1
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
L
T
DD
T
W
W
Figure 1.1 Force balance for an aircraft in steady level flight.
2
For steady, level flight,
T = D, L = W
or
W=L=D
L
L
= T 
 D
D
The weight of the aircraft changes in response to the fuel that is burned (rate at which
weight changes equals negative fuel mass flow rate times gravitational constant)
dW
˙ f ⋅g
= −m
dt
Now we will define an overall propulsion system efficiency:
overall efficiency =
what you get
propulsive power
=
what you pay for
fuel power
propulsive power = thrust ⋅ flight velocity = Tu o (J/s)
˙ fh
fuel power = fuel mass flow rate ⋅ fuel energy per unit mass = m
(J/s)
Thus
ηoverall =
Tu o
˙ fh
m
We can now write the expression for the change in weight of the vehicle in terms of
important aerodynamic (L/D) and propulsion system (ηoverall) parameters:
dW
−W
− Wu 0
− Wu 0
˙ f ⋅g =
= −m
=
=
h
L
h  L  Tu 0
dt
 L T
 η
overall
 D m


˙ f ⋅g g D m
˙ f ⋅h
g  D
We can rewrite and integrate
dW
− u 0 dt
=
h
L
W
 η
overall
g  D
applying the initial conditions,
∴t =
at
⇒
t=0
ln W = constant −
W = Winitial
−L
h
W
ηoverall
ln
D
gu 0
Winitial
3
tu 0
h  L
ηoverall
g  D
∴ const. = ln Winitial
the time the aircraft has flown corresponds to the amount of fuel burned, therefore
t final =
−L
h
W
ηoverall
ln final
D
gu 0
Winitial
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 over
the flight.
Range =
h
g
Fluids
(Aero)
L
D
ηoverall
ln
Winitial
Wfinal
Propulsion
Structures +
Materials
Note that this expression is sometimes rewritten in terms of an alternate measure of
efficiency, 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.
Range =
V(L D)  Winitial 
ln

g ⋅ SFC  Wfinal 
4
Thus we see that the answer to the question “How far can an airplane fly?” depends
on:
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 the
production of drag). During the fluids lectures you will learn how to develop
and 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 various
energy 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 to
estimate the performance of aerospace structures.
5
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 Simple
Science of Flight, by H. Tennekes)
6
25
F28-1000
F28-4000/6000
20
S360
A310-300
A320-100/200
RJ200/ER
B757-200
B747-100/200/300
F27
L/Dmax
15
B767-200/ER
BAC111-200/400
B707-300B
B737-100/200
DC9-30
DC10-10
B727-200/231A
B777
B747-400
ATR72
S340A
MD11
B767-300/ER
ATR42 BAE-ATP
A300-600
L1011-1/100/200
D328
B737-300
EMB120
B737-500/600
MD80 & DC9-80
DC10-40
B737-400
BAE146-100/200/RJ70
BAE-146-300
SA227
J41
J31
DC10-30
B707-100B/300
F100
DHC8-300
L1011-500
10
Data Unavailable For:
FH-227
EMB-145
Nihon YS-11
CV-880
SA-226
BAE RJ85
DHC-8-100
Beech 1900
L-188
CV-580
DHC-7
CV-600
5
Turboprops
Regional Jets
Large Aircraft
0
1955 1960
1965 1970 1975
1980 1985 1990
1995 2000
Year
Figure 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
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)
0.70
DHC8-100
J31
B727-200/231A
CV600
0.60
BAC111-200
BAC111-400
DC9-10
F28-1000
FH227 DC10-10
B737-100/200
F27
0.50
DC9-40
L188A-08/188C
CV880
OEW / MTOW
DHC7
DC9-30
S360
BAE146-100/RJ70
SA227
EMB120
S340A
BAE-ATP
ATR42
D328
ATR72
J41
DHC8-300
RJ200/ER
BAE146-200
B1900
SA226
B737-500/600
F100
MD80 & DC9-80
B737-300
BAE-146-300
L1011-1/100/200
B757-200
F28-4000/6000
A320-100/200 RJ85
B767-200/ER
B737-400
A300-600
DC9-50
B747-200/300
A310-300
B747-400
DC10-40
B767-300/ER
B777
L1011-500
DC10-30
MD11
EMB145
B747-100
0.40
B707-300B
0.30
0.20
0.10
Turboprops
Regional Jets
Large Aircraft
0.00
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
Year
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)
8
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:
m
˙ e u 2e m
˙ o u o2 
−

2 
rate of production of propellant k.e.  2
ηthermal =
=
˙ fh
fuel power
m
ηprop =
propulsive power
Tu o
=
2
rate of production of propellant k.e.  m
˙ e ue m
˙ o u o2 
−
 2
2 

such that
ηoverall = ηthermal ⋅ ηprop
During the first semester thermodynamics lectures we will focus largely on thermal
efficiency. In next semester’s propulsion lectures we will combine thermodynamics with
fluid mechanics to obtain estimates for propulsive and thus overall efficiency. The data
shown in Figure 1.6 will give you a rough idea for the conversion efficiencies of various
modern aircraft engines.
9
Overall Efficiency
0.1
0.8
0.2
0.3
0.8 0.7
0.4
0.6
0.5
0.5
0.6
0.4
Future
Trend
Core Thermal Efficiency
0.7
'777'
Engines
0.3 SFC
Advanced
UDF
0.6
CF6-80C2
Low BPR
0.5
UDF
Engine
Current
High BPR
Turbojets
0.4
0.3
Whittle
0.3
0.2
0.4
0.6
0.5
0.7
0.8
Propulsive x Transmission Efficiency
Figure 1.8 Trends in aircraft engine efficiency (after Pratt & Whitney)
30
B707-300
B720-000
25
B727-200/231A
DC9-40
BAC111-400
F28-4000/6000
BAE146-100/200/RJ70
CV880
BAE-146-300
MD80 & DC9-80
B747-100
B747-200/300
F27
CV600
15
B737-300
DC9-50
DC9-30 B737-100/200
20
TSFC (mg/Ns)
F28-1000
DC9-10
DC10-10
RJ85
D328
EM170
F100
EMB145
B767-300/ER
B737-400 B737-500/600
B757-200
RJ700
RJ200/ER
B747-400
EMB135
B767-200/ER
MD11
A300-600
B777
A320-100/200
A310-300
J31
B1900
SA227
J41
EMB120
S360
ATR72
D328
ATR42
DHC8-100
BAE-ATP DHC8-300
DHC8-400
S340A
L1011-500
DC10-30
L1011-1/100/200
DC10-40
L188A-08/188C
SA226
DHC7
CV580
10
5
0
1955
Turboprops
Regional Jets
Large Jets
New Regional Jet Engines
New Turboprop Engines
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 From
Technological, Operational, and Cost Perspectives, SM Thesis, MIT, June 2001)
10
The accuracy of the range equation in predicting performance for commercial transport
aircraft is quite good. The Department of Transportation collects and reports a variety of
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 the
actual 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.
Calculated Stage Length (miles)
(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 aircraft
operations (J. J. Lee, MIT Masters Thesis, 2000)
11
160
140
120
100
80
60
40
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)
12