MAE 4261: AIR-BREATHING ENGINES
Review of Airplane Performance
February 2, 2012
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
LECTURE OUTLINE
• So far we have focused on characterizing propulsion system
• Also must look at behavior of entire airplane
– How fast can airplane fly?
– How far can airplane fly on a single tank of fuel?
– How long can airplane stay in air on a single tank of fuel?
– How fast and how high can it climb?
– How does it perform?
DRAG POLAR
• Drag for complete airplane, not just wing
2
L
C
C D  Cd 
eAR
CL2
CD  CD,0 
eAR
Wing or airfoil
Entire Airplane
Engine Nacelles
Landing Gear
Tail Surfaces
DRAG POLAR
•
•
•
•
CD,0 is parasite drag coefficient at zero lift (aL=0)
CD,i drag coefficient due to lift (induced drag)
Oswald efficiency factor, e, includes all effects from airplane
CD,0 and e are known aerodynamics quantities of airplane
CL2
CD  CD ,0 
 C D , 0  C D ,i
eAR
4 FORCES ACTING ON AIRPLANE
• Model airplane as rigid body with four natural forces acting on it
1. Lift, L
•
Acts perpendicular to flight path (perpendicular to relative wind)
2. Drag, D
•
Acts parallel to flight path direction (parallel to relative wind)
3. Propulsive Thrust, T
•
•
For most airplanes propulsive thrust acts in flight path direction
May also be inclined with respect to flight path angle, aT, usually a small angle
4. Weight, W
•
•
Always acts vertically toward center of earth
Inclined at angle, q, with respect to lift direction
• Apply Newton’s Second Law (F=ma) to curvilinear flight path
–
–
Force balance in direction parallel to flight path
Force balance in direction perpendicular to flight path
GENERAL EQUATIONS OF MOTION
dV
 Fparallel  T cos aT  D  W sin q  ma  m dt
2
V
 Fperpendicular  T cos aT  D  W sin q  m r
c
STATIC VS. DYNAMIC ANALYSES
Two forms of these equations:
1. Static Performance: Zero Accelerations (dV/dt = 0, V2/rc = 0)
– Maximum velocity
– Maximum rate of climb
– Maximum range
– Maximum endurance
2. Dynamic Performance: Accelerating Flight
– Take-off and landing characteristics
– Turning flight
– Accelerated flight and rate of climb
LEVEL, UNACCELERATED FLIGHT
• Equations of motion reduce to very simple expressions
– Aerodynamic drag is balanced by thrust of engine
– Aerodynamic lift is balanced by weight of airplane
• For most conventional airplanes aT is small enough such that cos(aT) ~ 1
T D
L W
LEVEL, UNACCELERATED FLIGHT
T D
L W
1
T  D  V 2 SC D
2
1
L  W  V 2 SC L
2
T CD

W CL
W
W
TR 

 CL
 L
 C 
D
D

 
• TR is thrust required to fly at a given
velocity in level, unaccelerated flight
• Notice that minimum TR is when airplane
is at maximum L/D
• Airplane’s power plant must produce a net
thrust which is equal to drag
THRUST REQUIREMENT
• TR for airplane at given altitude varies with velocity
• Thrust required curve: TR vs. V∞
THRUST REQUIREMENT
• Select a flight speed, V∞
• Calculate CL
CL 
W
1
 V2 S
2
• Calculate CD
CL2
CD  CD,0 
eAR
• Calculate CL/CD
• Calculate TR
This is how much thrust engine
must produce to fly at V∞
TR 
W
 CL

 C 
D

THRUST REQUIREMENT
• Thrust required, TR, varies inversely with L/D
• Minimum TR when airplane is flying at L/D is a maximum
– L/D is a measure of aerodynamic efficiency of an airplane
– Maximum aerodynamic efficiency → Minimum TR
Usually around 2º-8º
THRUST REQUIREMENT
• Different points on TR curve correspond to different angles of attack
1
L  W   V2 SCL  q SCL
2
D  q SCD
At b:
Small q∞
Large CL (CL2) and a (support W)
D large
At a:
Large q∞
Small CL and a
D large
THRUST REQUIRED VS. FLIGHT VELOCITY
TR  D  q SC D  q S C D , 0  C D ,i 
C L2
TR  q SC D , 0  q S
eAR
Zero-Lift TR
(Parasitic Drag)
Lift-Induced TR
(Induced Drag)
Zero-Lift TR ~ V2
(Parasitic Drag)
Lift-Induced TR ~ 1/V2
(Induced Drag)
THRUST REQUIRED VS. FLIGHT VELOCITY
W2
TR  q SC D , 0 
q SeAR
dTR  dTR  dV 


 
dq  dV  dq 
dTR
W2
 SC D , 0  2
0
dq
q SeAR
CD,0
C L2

 C D ,i
eAR
Zero-Lift Drag = Induced Drag
At minimum TR and maximum L/D
At point of minimum TR, dTR/dV∞=0
(or dTR/dq∞=0)
MAXIMUM VELOCITY
• Maximum flight speed occurs when TA=TR
– Reduced throttle settings, TR < TA
– Cannot physically achieve more thrust than TA which engine can provide
AIRPLANE POWER PLANTS
•
•
TR is dictated by aerodynamics and
weight of airplane
Thrust available, TA, is associated
with engine
Two types of engines common in
aviation today
1. Reciprocating piston engine with
propeller
– Power average light-weight,
general aviation aircraft
2. Turbojet engine
– Large commercial transports
and military aircraft
AIRPLANE POWER PLANTS
THRUST VS. POWER
• Jets Engines (turbojets, turbofans for military and commercial applications) are
usually rate in Thrust
– Thrust is a Force with units (kg m/s2)
– For example, the PW4000-112 is rated at 98,000 lb of thrust
• Piston-Driven Engines are usually rated in terms of Power
– Power is a precise term and can be expressed as:
• Energy/time with units (kg m2/s2) / s = kg m2/s3 = Watts
– Note that Energy is expressed in Joules = kg m2/s2
• Force*velocity with units (kg m/s2)*(m/s) = kg m2/s3 = Watts
– Usually rated in terms of horsepower (1 hp=550 ft lb/s = 746 W)
• Example:
– Airplane is level, unaccelerated flight at a given altitude with speed V∞
– Power Required, PR=TR*V∞
POWER REQUIRED
PR  TRV 
W
 CL

 C 
D

V
1
L  W   V2 SC L  V 
2
PR 
W
 CL

 C 
D

2W
  SC
2W
  SC
2W 3C D2
1
PR 

3
  SC L  C 3 2 
 L

CD 



PR varies inversely as CL3/2/CD
TR varies inversely as CL/CD
POWER REQUIRED
PR vs. V∞ qualitatively
resembles TR vs. V∞
POWER REQUIRED
PR  TRV  DV  q SC DV  q S C D , 0  C D ,i V
C L2
PR  q SC D , 0V  q SV
eAR
Zero-Lift PR
Lift-Induced PR
Zero-Lift PR ~ V3
Lift-Induced PR ~ 1/V
POWER REQUIRED
1
W2
3
PR   V SC D , 0 
1
2
 V SeAR
2
dPR 3
1

2 
  V S  C D ,0  C D ,i   0
dV 2
3


At point of minimum PR, PTR/dV∞=0
CD ,0
1
 C D ,i
3
POWER REQUIRED
• V∞ for minimum PR is less than V∞ for minimum TR
CD ,0
1
 C D ,i
3
C D , 0  C D ,i
POWER REQUIRED
• V∞ for minimum PR is less than V∞ for minimum TR
dPR
0
dV
d  PR 
V 
dTR


0
dV
dV
POWER AVAILABLE
POWER AVAILABLE AND MAXIMUM VELOCITY
Propeller Drive
Engine
POWER AVAILABLE AND MAXIMUM VELOCITY
Jet Engine
ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE
Recall PR = f(∞)
VALT
 0 
 V0  
  
PR , ALT
1
2
 0 
 PR , 0  
  
1
2
ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE
Vmax,ALT < Vmax,sea-level
RATE OF CLIMB
• Boeing 777: Lift-Off Speed ~ 180 MPH
• How fast can it climb to a cruising altitude of 30,000 ft?
RATE OF CLIMB
T  D  W sin q
L  W cos q
RATE OF CLIMB
T  D  W sin q
TV  DV  WV sin q
Rate of Climb:
TV  DV
 V sin q
W
Vertical velocity
R / C  V sin q
TV∞ is power available
DV∞ is level-flight power required (for small q neglect W)
TV∞- DV∞ is excess power
RATE OF CLIMB
Propeller Drive
Engine
Jet Engine
Maximum R/C Occurs when Maximum Excess Power
GLIDING FLIGHT
T 0
D  W sin q
L  W cos q
sin q D

cos q L
1
tan q 
L
D
To maximize range, smallest
q occurs at (L/D)max
RANGE AND ENDURANCE
• Range: Total distance (measured with respect to the ground) traversed by airplane
on a single tank of fuel
• Endurance: Total time that airplane stays in air on a single tank of fuel
• Parameters that maximize range are different from those that maximize endurance
• Parameters are different for propeller-powered and jet-powered aircraft
• Fuel Consumption Definitions
– Propeller-Powered:
• Specific Fuel Consumption (SFC)
• Definition: Weight of fuel consumed per unit power per unit time
– Jet-Powered:
• Thrust Specific Fuel Consumption (TSFC)
• Definition: Weight of fuel consumed per unit thrust per unit time
PROPELLER-DRIVEN: RANGE AND ENDURANCE
• SFC: Weight of fuel consumed per unit power per unit time
lb of fuel
SFC 
HP hour 
• ENDURANCE: To stay in air for longest amount of time, use minimum
number of pounds of fuel per hour
lb of fuel
 SFCHP 
hour 
• Minimum lb of fuel per hour obtained with minimum HP
• Maximum endurance for a propeller-driven airplane occurs when
airplane is flying at minimum power required
• Maximum endurance for a propeller-driven airplane occurs when
airplane is flying at a velocity such that CL3/2/CD is a maximum
PROPELLER-DRIVEN: RANGE AND ENDURANCE
• SFC: Weight of fuel consumed per unit power per unit time
lb of fuel
SFC 
HP hour 
• RANGE: To cover longest distance use minimum pounds of fuel per mile
lb of fuel SFCHP 

mile 
V
• Minimum lb of fuel per hour obtained with minimum HP/V∞
• Maximum range for a propeller-driven airplane occurs when airplane
is flying at a velocity such that CL/CD is a maximum
PROPELLER-DRIVEN: RANGE BREGUET FORMULA
 Winitial 

R
ln 
SFC C D  W final 
h CL
• To maximize range:
– Largest propeller efficiency, h
– Lowest possible SFC
– Highest ratio of Winitial to Wfinal, which is obtained with the largest fuel weight
– Fly at maximum L/D
PROPELLER-DRIVEN: RANGE BREGUET FORMULA
 Winitial 

R
ln 
SFC C D  W final 
h CL
Structures and Materials
Propulsion
Aerodynamics
PROPELLER-DRIVEN: ENDURACE BREGUET FORMULA
3
E
h CL 2
SFC C D
2 
2   S  2 W final2  Winitial

• To maximize endurance:
– Largest propeller efficiency, h
– Lowest possible SFC
– Largest fuel weight
– Fly at maximum CL3/2/CD
– Flight at sea level
1
1

1

JET-POWERED: RANGE AND ENDURANCE
• TSFC: Weight of fuel consumed per thrust per unit time
TSFC 
lb of fuel
lb of thrust hour 
• ENDURANCE: To stay in air for longest amount of time, use minimum
number of pounds of fuel per hour
lb of fuel
 TSFC Thrust
hour 

• Minimum lb of fuel per hour obtained with minimum thrust
• Maximum endurance for a jet-powered airplane occurs when
airplane is flying at minimum thrust required
• Maximum endurance for a jet-powered airplane occurs when
airplane is flying at a velocity such that CL/CD is a maximum
JET-POWERED: RANGE AND ENDURANCE
• TSFC: Weight of fuel consumed per unit power per unit time
TSFC 
lb of fuel
lb of thrust hour 
• RANGE: To cover longest distance use minimum pounds of fuel per mile
lb of fuel SFCThrust

mile 
V

• Minimum lb of fuel per hour obtained with minimum Thrust/V∞
TR 1
2W
1
  S
CD  1
V 2
 SCL
CL 2
CD
• Maximum range for a jet-powered airplane occurs when airplane is
flying at a velocity such that CL1/2/CD is a maximum
JET-POWERED: RANGE BREGUET FORMULA
1
1 CL 2
R2
  S TSFC C D
2
• To maximize range:
– Minimum TSFC
– Maximum fuel weight
– Flight at maximum CL1/2/CD
– Fly at high altitudes
W 12  W 12 
final
 initial

JET-POWERED: ENDURACE BREGUET FORMULA
1 C L  Winitial 
E
ln
TSFC C D  W final 
• To maximize endurance:
– Minimum TSFC
– Maximum fuel weight
– Flight at maximum L/D
TAKE-OFF AND LANDING ANALYSES
dV
F  ma  m
dt
F
dV  dt
m
F
V t
m
ds  Vdt
V 2m
s
2F
Rolling resistance
r = 0.02
dV
F  T  D  R  T  D   r W  L   m
dt
s: lift-off distance
NUMERICAL SOLUTION FOR TAKE-OFF
USEFUL APPROXIMATION (T >> D, R)
sL.O.: lift-off distance
2
s L.O.
1.44W

g  SC L ,max T
• Lift-off distance very sensitive to weight, varies as W2
• Depends on ambient density
• Lift-off distance may be decreased:
– Increasing wing area, S
– Increasing CL,max
– Increasing thrust, T
TURNING FLIGHT
L cos   W
Fr  L  W
2
L
n
W
2
Load Factor
Fr  W n 2  1
R: Turn Radius
: Turn Rate
V2
Fr  m
R
V2
R
g n2 1
dq V g n 2  1



dt
R
V
EXAMPLE: PULL-UP MANEUVER
Fr  L  W  W n  1
V2
Fr  m
R
2
V
R
g n  1
g n  1

V
STRUCTURAL LIMITS
V-n DIAGRAMS
1
2

V
SC
L 2   L
n

W
W
1
2 C L , max
nmax   V
W
2
S