1. MAE 4261: AIR-BREATHING ENGINES
Review of Airplane Performance
February 2, 2012
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. 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?

3. DRAG POLAR Wing or airfoil Entire Airplane
Drag for complete airplane, not just wing
Wing or airfoil
Entire Airplane
Tail Surfaces
Engine Nacelles
Landing Gear

4. 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

5. 4 FORCES ACTING ON AIRPLANE
Model airplane as rigid body with four natural forces acting on it
Lift, L
Acts perpendicular to flight path (perpendicular to relative wind)
Drag, D
Acts parallel to flight path direction (parallel to relative wind)
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
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

6. GENERAL EQUATIONS OF MOTION

7. STATIC VS. DYNAMIC ANALYSES
Two forms of these equations:
Static Performance: Zero Accelerations (dV/dt = 0, V2/rc = 0)
Maximum velocity
Maximum rate of climb
Maximum range
Maximum endurance
Dynamic Performance: Accelerating Flight
Take-off and landing characteristics
Turning flight
Accelerated flight and rate of climb

8. 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

9. LEVEL, UNACCELERATED FLIGHT
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

10. THRUST REQUIREMENT
TR for airplane at given altitude varies with velocity
Thrust required curve: TR vs. V∞

11. THRUST REQUIREMENT This is how much thrust engine
Select a flight speed, V∞
Calculate CL
Calculate CD
Calculate CL/CD
Calculate TR
This is how much thrust engine
must produce to fly at V∞

12. THRUST REQUIREMENT Usually around 2º-8º
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

13. THRUST REQUIREMENT
Different points on TR curve correspond to different angles of attack
At b:
Small q∞
Large CL (CL2) and a (support W)
D large
At a:
Large q∞
Small CL and a
D large

14. THRUST REQUIRED VS. FLIGHT VELOCITY
Zero-Lift TR
(Parasitic Drag)
Lift-Induced TR
(Induced Drag)
Zero-Lift TR ~ V2
(Parasitic Drag)
Lift-Induced TR ~ 1/V2
(Induced Drag)

15. THRUST REQUIRED VS. FLIGHT VELOCITY
At point of minimum TR, dTR/dV∞=0
(or dTR/dq∞=0)
Zero-Lift Drag = Induced Drag
At minimum TR and maximum L/D

16. 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

17. 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
Reciprocating piston engine with propeller
Power average light-weight, general aviation aircraft
Turbojet engine
Large commercial transports and military aircraft

18. AIRPLANE POWER PLANTS

19. 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 PW 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∞

20. POWER REQUIRED PR varies inversely as CL3/2/CD
TR varies inversely as CL/CD

21. POWER REQUIRED
PR vs. V∞ qualitatively
resembles TR vs. V∞

22. POWER REQUIRED Zero-Lift PR Lift-Induced PR Zero-Lift PR ~ V3
Lift-Induced PR ~ 1/V

23. POWER REQUIRED
At point of minimum PR, PTR/dV∞=0

24. POWER REQUIRED
V∞ for minimum PR is less than V∞ for minimum TR

25. POWER REQUIRED
V∞ for minimum PR is less than V∞ for minimum TR

26. POWER AVAILABLE

27. POWER AVAILABLE AND MAXIMUM VELOCITY
Propeller Drive
Engine

28. POWER AVAILABLE AND MAXIMUM VELOCITY
Jet Engine

29. ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE
Recall PR = f(r∞)

30. ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE
Vmax,ALT < Vmax,sea-level

31. RATE OF CLIMB Boeing 777: Lift-Off Speed ~ 180 MPH
How fast can it climb to a cruising altitude of 30,000 ft?

32. RATE OF CLIMB

33. RATE OF CLIMB Rate of Climb: TV∞ is power available
Vertical velocity
Rate of Climb:
TV∞ is power available
DV∞ is level-flight power required (for small q neglect W)
TV∞- DV∞ is excess power

34. RATE OF CLIMB Propeller Drive Engine Jet Engine
Maximum R/C Occurs when Maximum Excess Power

35. GLIDING FLIGHT
To maximize range, smallest
q occurs at (L/D)max

36. 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

37. PROPELLER-DRIVEN: RANGE AND ENDURANCE
SFC: Weight of fuel consumed per unit power per unit time
ENDURANCE: To stay in air for longest amount of time, use minimum number of pounds of fuel per 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

38. PROPELLER-DRIVEN: RANGE AND ENDURANCE
SFC: Weight of fuel consumed per unit power per unit time
RANGE: To cover longest distance use minimum pounds of fuel per mile
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

39. PROPELLER-DRIVEN: RANGE BREGUET FORMULA
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

40. PROPELLER-DRIVEN: RANGE BREGUET FORMULA
Structures and Materials
Propulsion
Aerodynamics

41. PROPELLER-DRIVEN: ENDURACE BREGUET FORMULA
To maximize endurance:
Largest propeller efficiency, h
Lowest possible SFC
Largest fuel weight
Fly at maximum CL3/2/CD
Flight at sea level

42. JET-POWERED: RANGE AND ENDURANCE
TSFC: Weight of fuel consumed per thrust per unit time
ENDURANCE: To stay in air for longest amount of time, use minimum number of pounds of fuel per 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

43. JET-POWERED: RANGE AND ENDURANCE
TSFC: Weight of fuel consumed per unit power per unit time
RANGE: To cover longest distance use minimum pounds of fuel per mile
Minimum lb of fuel per hour obtained with minimum Thrust/V∞
Maximum range for a jet-powered airplane occurs when airplane is flying at a velocity such that CL1/2/CD is a maximum

44. JET-POWERED: RANGE BREGUET FORMULA
To maximize range:
Minimum TSFC
Maximum fuel weight
Flight at maximum CL1/2/CD
Fly at high altitudes

45. JET-POWERED: ENDURACE BREGUET FORMULA
To maximize endurance:
Minimum TSFC
Maximum fuel weight
Flight at maximum L/D

46. TAKE-OFF AND LANDING ANALYSES
Rolling resistance
mr = 0.02
s: lift-off distance

47. NUMERICAL SOLUTION FOR TAKE-OFF

48. USEFUL APPROXIMATION (T >> D, R)
sL.O.: lift-off distance
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

49. TURNING FLIGHT
Load Factor
R: Turn Radius
w: Turn Rate

50. EXAMPLE: PULL-UP MANEUVER

51. STRUCTURAL LIMITS

52. V-n DIAGRAMS