MAE 4261: AIR-BREATHING ENGINES

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 • Drag for complete airplane, not just wing Entire Airplane Wing or airfoil 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 • 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 • 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 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∞

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 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) • 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 sL.O.: lift-off distance

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

50. EXAMPLE: PULL-UP MANEUVER