1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Thrust and Power Requirements
April 28, 2010
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. EXAMPLE: BEECHCRAFT QUEEN AIR
The results we have developed so far for lift and drag for a finite wing may also be applied to a complete airplane. In such relations:
CD is drag coefficient for complete airplane
CD,0 is parasitic drag coefficient, which contains not only profile drag of wing (cd) but also friction and pressure drag of tail surfaces, fuselage, engine nacelles, landing gear and any other components of airplane exposed to air flow
CL is total lift coefficient, including small contributions from horizontal tail and fuselage
Span efficiency for finite wing replaced with Oswald efficiency factor for entire airplane
Example: To see how this works, assume the aerodynamicists have provided all the information needed about the complete airplane shown below
Beechcraft Queen Air Aircraft Data
W = 38,220 N
S = 27.3 m2
AR = 7.5
e (complete airplane) = 0.9
CD,0 (complete airplane) = 0.03
What thrust and power levels are required of engines to cruise at 220 MPH at sea-level?
How would these results change at 15,000 ft

3. OVERALL AIRPLANE DRAG Wing or airfoil Entire Airplane
No longer concerned with aerodynamic details
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
Example of Drag Polar for complete 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 (always perpendicular to relative wind)
Drag, D
Acts parallel to flight path direction (parallel to incoming relative wind)
Propulsive Thrust, T
For most airplanes propulsive thrust acts in flight path direction
May be inclined with respect to flight path angle, aT, usually 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 (6.2)
Free Body Diagram
Apply Newton’s 2nd Parallel to flight path:
Apply Newton’s 2nd Perpendicular to flight path:

7. LEVEL, UNACCELERATED FLIGHTW
JSF is flying at constant speed (no accelerations)
Sum of forces = 0 in two perpendicular directions
Entire weight of airplane is perfectly balanced by lift (L = W)
Engines produce just enough thrust to balance total drag at this airspeed (T = D)
For most conventional airplanes aT is small enough such that cos(aT) ~ 1

8. 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
L/D is an important aero-performance quantity

9. THRUST REQUIREMENT (6.3)
TR for airplane at given altitude varies with velocity
Thrust required curve: TR vs. V∞

10. PROCEDURE: THRUST REQUIREMENT
Select a flight speed, V∞
Calculate CL
Minimum TR when airplane
flying at (L/D)max
Calculate CD
Calculate CL/CD
Calculate TR
This is how much thrust engine
must produce to fly at selected V∞
Recall Homework Problem #5.6, find (L/D)max for NACA 2412 airfoil

11. THRUST REQUIREMENT (6.3)
Different points on TR curve correspond to different angles of attack
At b:
Small q∞
Large CL (or CL2) and a to support W
D large
At a:
Large q∞
Small CL and a
D large

12. 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)

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

14. HOW FAST CAN YOU FLY? Intersection of TR curve and maximum
Maximum flight speed occurs when thrust available, TA=TR
Reduced throttle settings, TR < TA
Cannot physically achieve more thrust than TA which engine can provide
Intersection of TR
curve and maximum
TA defined maximum
flight speed of airplane

15. FURTHER IMPLICATIONS FOR DESIGN: VMAX
Maximum velocity at a given altitude is important specification for new airplane
To design airplane for given Vmax, what are most important design parameters?
Steady, level flight: T = D
Steady, level flight: L = W
Substitute into drag equation
Turn this equation into a quadratic
equation (by multiplying by q∞)
and rearranging
Solve quadratic equation and set
thrust, T, to maximum available
thrust, TA,max

16. FURTHER IMPLICATIONS FOR DESIGN: VMAX
TA,max does not appear alone, but only in ratio (TA/W)max
S does not appear alone, but only in ratio (W/S)
Vmax does not depend on thrust alone or weight alone, but rather on ratios
(TA/W)max: maximum thrust-to-weight ratio
W/S: wing loading
Vmax also depends on density (altitude), CD,0, peAR
Increase Vmax by
Increase maximum thrust-to-weight ratio, (TA/W)max
Increasing wing loading, (W/S)
Decreasing zero-lift drag coefficient, CD,0

17. AIRPLANE POWER PLANTS Two types of engines common in aviation today
Reciprocating piston engine with propeller
Average light-weight, general aviation aircraft
Rated in terms of POWER
Jet (Turbojet, turbofan) engine
Large commercial transports and military aircraft
Rated in terms of THRUST

18. THRUST VS. POWER
Jets Engines (turbojets, turbofans for military and commercial applications) are usually rate in Thrust
Thrust is a Force with units (N = 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∞
[W] = [N] * [m/s]

19. Propeller Drive Engine
POWER AVAILABLE (6.6)
Propeller Drive Engine
Jet Engine

20. Propeller Drive Engine
POWER AVAILABLE (6.6)
Propeller Drive Engine
Jet Engine

21. POWER REQUIRED (6.5)
PR vs. V∞ qualitatively
(Resembles TR vs. V∞)

22. POWER REQUIRED (6.5) PR varies inversely as CL3/2/CD
Recall: TR varies inversely as CL/CD

23. POWER REQUIRED (6.5) Zero-Lift PR Lift-Induced PR Zero-Lift PR ~ V3
Lift-Induced PR ~ 1/V

24. POWER REQUIRED
At point of minimum PR, dPR/dV∞=0

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

26. WHY DO WE CARE ABOUT THIS?
We will show that for a piston-engine propeller combination
To fly longest distance (maximum range) we fly airplane at speed corresponding to maximum L/D
To stay aloft longest (maximum endurance) we fly the airplane at minimum PR or fly at a velocity where CL3/2/CD is a maximum
Power will also provide information on maximum rate of climb and altitude

27. POWER AVAILABLE AND MAXIMUM VELOCITY (6.6)
Propeller Drive
Engine PA PR
1 hp = 550 ft lb/s = 746 W

28. POWER AVAILABLE AND MAXIMUM VELOCITY (6.6)
Jet Engine
PA = TAV∞ PR

29. ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)
Recall PR = f(r∞)
Subscript ‘0’ denotes seal-level conditions

30. Vmax,ALT < Vmax,sea-level
ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7) Propeller-Driven Airplane
Vmax,ALT < Vmax,sea-level

31. RATE OF CLIMB (6.8) Boeing 777: Lift-Off Speed ~ 180 MPH
How fast can it climb to a cruising altitude of 30,000 ft?

32. RATE OF CLIMB (6.8)
Governing Equations:

33. RATE OF CLIMB (6.8) 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. Propeller Drive Engine
RATE OF CLIMB (6.8)
Propeller Drive Engine
Jet Engine
Maximum R/C Occurs when Maximum Excess Power

35. EXAMPLE: F-15 K
Weapon launched from an F-15 fighter by a small two stage rocket, carries a heat-seeking Miniature Homing Vehicle (MHV) which destroys target by direct impact at high speed (kinetic energy weapon)
F-15 can bring ALMV under the ground track of its target, as opposed to a ground-based system, which must wait for a target satellite to overfly its launch site.

36. GLIDING FLIGHT (6.9)
To maximize range, smallest
q occurs at (L/D)max

37. EXAMPLE: HIGH ASPECT RATIO GLIDERq
To maximize range, smallest q occurs at (L/D)max
A modern sailplane may have a glide ratio as high as 60:1
So q = tan-1(1/60) ~ 1°

38. RANGE AND ENDURANCE How far can we fly? How long can we stay aloft?
How do answers vary for propeller-driven vs. jet-engine?

39. 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 to 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

40. 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 maximized

41. 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

42. 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

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

44. 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

45. 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

46. 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

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

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

49. SUMMARY: ENDURANCE AND RANGE
Maximum Endurance
Propeller-Driven
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 maximized
Jet Engine-Driven
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
Maximum Range
Maximum range for a propeller-driven airplane occurs when airplane is flying at a velocity such that CL/CD is a maximum
Maximum range for a jet-powered airplane occurs when airplane is flying at a velocity such that CL1/2/CD is a maximum

50. EXAMPLES OF DYNAMIC PERFORMANCE
Take-Off Distance
Turning Flight

51. TAKE-OFF AND LANDING ANALYSES (6.15)
Rolling resistance
mr = 0.02
s: lift-off distance

52. NUMERICAL SOLUTION FOR TAKE-OFF

53. 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

54. EXAMPLES OF GROUND EFFECT

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

56. EXAMPLE: PULL-UP MANEUVER
R: Turn Radius
w: Turn Rate

57. V-n DIAGRAMS

58. STRUCTURAL LIMITS