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# Ae2104 flight-mechanics-slides 6

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These are the lecture slides to the fifth lecture of the course AE2104 Flight and Orbital mechanics.

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### Ae2104 flight-mechanics-slides 6

1. 1. Flight and Orbital Mechanics Lecture slides Challenge the future 1
2. 2. Flight and Orbital Mechanics Lecture hours 11, 12 – Cruise Mark Voskuijl Semester 1 - 2012 Delft University of Technology Challenge the future
3. 3. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 2|
4. 4. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 3|
5. 5. Introduction Typical cruise flight AE2104 Flight and Orbital Mechanics 4|
6. 6. Introduction Objective • Range (distance) • Endurance (Maximum time) AE2104 Flight and Orbital Mechanics 5|
7. 7. Introduction Equations of motion General 2D equations of motion and power equation Unsteady curved symmetric flight W dV g dt W d L  W cos   T sin T  V g dt Pa  Pr V dV  RC  W g dt T cos T  D  W sin   Cruise flight Quasi steady (dV/dt  0), quasi-rectilinear, (d/dt  0) Weight of the aircraft is not constant Small flight path angle  cos = 1, sin  0 Assume that the thrust vector acts in the direction of flight (T  0) AE2104 Flight and Orbital Mechanics 6|
8. 8. Introduction Equations of motion Equations of motion cruise flight g T  D  W sin   W L  W ( quasi rectilinear ) 0 Additional equation dW   F  ,V , H  dt Kinematic equations ds  V cos  dt dH  V sin  dt AE2104 Flight and Orbital Mechanics 7|
9. 9. Introduction Problem definition Pilot can choose a certain airspeed and altitude: H(t), V(t) What is the best flight condition? 1. 2. Optimal initial conditions (V, H at initial weight) Optimal flying strategy (V, H at decreasing weight) AE2104 Flight and Orbital Mechanics 8|
10. 10. Introduction Criteria for optimal flight 1. Maximum endurance E: Fuel flow Fmin at every point in time 2. Maximum range R: Specific range (V/F)max at every point in time 3. Given range, minimum fuel: Specific range (V/F)max at every point in time AE2104 Flight and Orbital Mechanics 9|
11. 11. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 10 |
12. 12. Optimum cruise profile (Jet) Performance diagram - Jet For basic flight mechanics applications, thrust of a turbojet can be assumed to be constant with airspeed for a given flight altitude Thrust, Drag 2 CL Drag follows from: CD  CD0   Ae Airspeed AE2104 Flight and Orbital Mechanics 11 |
13. 13. Optimum cruise profile (Jet) Performance diagram - Jet For basic flight mechanics applications, thrust of a turbojet can be assumed to be constant with airspeed for a given flight altitude Power Pa  TV Pr  DV Airspeed AE2104 Flight and Orbital Mechanics 12 |
14. 14. Optimum cruise profile (Jet) Thrust specific fuel consumption F cT T D, T D Additional assumption: cT constant Specific range V V V   F cT T cT D T V  D if      F max  V min What is the corresponding airspeed? V Vopt AE2104 Flight and Orbital Mechanics 13 |
15. 15. Optimum cruise profile (Jet) Optimal airspeed for given altitude and weight Method to calculate the best airspeed Optimum criterion Lift over drag ratio Angle of attack Airspeed (for given H,W) (D/V)min CLX / CDY CL V AE2104 Flight and Orbital Mechanics 14 |
16. 16. Optimum cruise profile (Jet) Airspeed 1. Optimum criterion 4. Ratio D V       V min  D max W 2 1 S  CL V 1 2 CL   2 D  CD  W W  S  CD  C   L 2. Airspeed L W V W 2 1 S  CL 3. Drag C L D D DW L CL 5. For a given weight CL 1 V   2 D CD  6. Angle of attack for given altitude C  V   L    2  D max  CD max AE2104 Flight and Orbital Mechanics 15 |
17. 17. Optimum cruise profile (Jet) Airspeed  CL  V  V       2   F max  D  max  CD max lift  drag  polar  CLopt  1 3 First year: CD0  Ae Airspeed for given altitude and weight L W V W 2 1 S  CL ,opt  CL  d  CL   2   2 0 dCL  CD   CD  max dCD 2 CL  2CD  CD 1 dCL 0 4 CD dCD 2CL  dCL  Ae 4 CD  0 2 CL C  2CL 1 CD 1 D0  Ae    Ae 2 CL 2 CL CL  1 3 CD0  Ae AE2104 Flight and Orbital Mechanics 16 |
18. 18. Optimum cruise profile (Jet) Example question Aircraft Weight W = 300.000 [N] (start of cruise) What is the best airspeed to fly (for max range) at 9000 [m] altitude? (ρ = 0.4663 [kg/m3], T = 229.65 [K]) AE2104 Flight and Orbital Mechanics 17 |
19. 19. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 18 |
20. 20. Effect of altitude Performance diagram Angle of attack is constant for a given point on the drag curve D V Increasing H D W 2 1 1  S  CL  CD W  0 CL V AE2104 Flight and Orbital Mechanics 19 |
21. 21. AE2104 Flight and Orbital Mechanics 20 |
22. 22. Effect of altitude Specific range T, D Altitude 0 m Optimum cruise level T at cruise T at cruise V AE2104 Flight and Orbital Mechanics 21 |
23. 23. Effect of altitude Conclusion At increasing altitude: - V/F increases - V increases - Engine more efficient Thus: fly as high as possible ! (up to the limits of the engine) AE2104 Flight and Orbital Mechanics 22 |
24. 24. Effect of altitude Optimum cruise level at cruise Tcruise Theoretical ceiling at cruise Tth Hcr < Hth (at cruise) Optimum Hcr  Hs (service ceiling) AE2104 Flight and Orbital Mechanics 23 |
25. 25. Effect of altitude In the presence of speed limits (e.g. MMO ) D Increasing H Vlim V AE2104 Flight and Orbital Mechanics 24 |
26. 26. Effect of altitude In the presence of speed limits (e.g. MMO ) First year: • Optimum V/F at V = Vlim  CL  Dmin     CL  CD0  Ae  CD max W 2 1 opt  2 S Vlim CD0  Ae • In case of speed limits, the optimum altitude is not determined by the engine  CL  d  CL      C D max dC L  C D dC C L  D  C D 1 dC L 0 2 CD  0  dC D 2C L  dC L  Ae 2 CD  0 2C L C D    Ae C L C D0  C L2  Ae CL C L  C D0  Ae AE2104 Flight and Orbital Mechanics 25 |
27. 27. Summary – Jet aircraft • Choose V such that (V/F)max  (CL / CD2)max • H as high as possible (limited by the engine) • If the speed limit is reached at lower altitude: • V = Vlim • H is such that CL / CD is max AE2104 Flight and Orbital Mechanics 26 |
28. 28. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 27 |
29. 29. Effect of weight  W 2 1  W V  S  CL   CD  W W  D CL   P  DV  W W  r    D  V at constant  2 D Decreasing W V Pr  V 3 at constant  AE2104 Flight and Orbital Mechanics 28 |
30. 30. AE2104 Flight and Orbital Mechanics 29 |
31. 31. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 30 |
32. 32. AE2104 Flight and Orbital Mechanics 31 |
33. 33. Best flying strategy D W1, H1 W2 < W1, H1 Strategies: I: constant altitude and engine setting II: constant altitude and airspeed III: constant angle of attack IV: Climb and constant V V: Climb and change of V W2, H2 > H1 V AE2104 Flight and Orbital Mechanics 32 |
34. 34. Best flying strategy Constant altitude I.  = constant: (V/F) << (V/F)opt but V  II. V = constant (V/F) < (V/F)opt III.  = constant, V but (V/F) = (V/F)opt Climb IV.  = constant, V = constant, (V/F) = (V/F)opt, even > (V/F)0 V.  = constant, changing V? AE2104 Flight and Orbital Mechanics 33 |
35. 35. Best flying strategy Optimum cruise climb possible? Strategy IV  2 = 1 (CL is constant) and V2 = V1 W1 2 1  V1   S 1 CL  W constant   W2 2 1  V2  S  2 CL   C T D DW CL Is this possible? Are the engines capable of providing enough thrust at higher altitude and lower weight? AE2104 Flight and Orbital Mechanics 34 |
36. 36. Best flying strategy Typical turbojet performance AE2104 Flight and Orbital Mechanics 35 |
37. 37. Best flying strategy Optimum cruise climb possible? Strategy IV  2 = 1 (CL is constant) and V2 = V1 W  = constant  W2 2  CD T D W CL CD W2 T2 CL W2  2    T1 CD W W1 1 1 CL W1 1 • This is exactly how a typical turbojet behaves above 11km. So there will be enough thrust. • Strategy V is not feasible • Below 11km there will be enough thrust as well AE2104 Flight and Orbital Mechanics 36 |
38. 38. Best flying strategy Optimum cruise climb possible? Strategy IV  2 = 1 (CL is constant) and V2 = V1 The engines can provide just enough thrust (strategy V not possible) What happens in case of Mlim? Mach number is constant at constant airspeed above 11km  No problem AE2104 Flight and Orbital Mechanics 37 |
39. 39. Best flying strategy AE2104 Flight and Orbital Mechanics 38 |
40. 40. Best flying strategy AE2104 Flight and Orbital Mechanics 39 |
41. 41. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 40 |
42. 42. Analytic range equations Breguet range equation • Range dW dW  V  F dt ds s1 W0 V R   ds   dW F s0 W1 AE2104 Flight and Orbital Mechanics 41 |
43. 43. Analytic range equations Breguet range equation for jet aircraft • Jet aircraft optimum climb cruise (, V and cT are constant during variation of W R R R W0 V  F dW W1 W0 V  cT D dW W1 W0 V CL  W0  R ln   cT CD  W1  V CL dW  c CD W W1 T V CL R cT CD W0 dW W W1 AE2104 Flight and Orbital Mechanics 42 |
44. 44. Analytic range equations Breguet range equation for jet aircraft V CL  W0  R ln   cT CD  W1  • If V is not limited: Rmax at (V CL / CD)max  (CL / CD2)max and min • If V is limited: Rmax at V = Vlim and  such that (CL / CD)max AE2104 Flight and Orbital Mechanics 43 |
45. 45. Analytic range equations Breguet range equation for propeller aircraft F  c p Pbr  cp j Pa  cp j TV V  j 1  j 1  j CL 1    F c p T c p D c p CD W Cruise flight with constant , cp and j: R R W0 V  F dW W1  j CL W  ln  0  c p CD  W1  AE2104 Flight and Orbital Mechanics 44 |
46. 46. Analytic range equations Breguet range equation for propeller aircraft P Pr Dmin V Conclusion: Altitude is not important w.r.t V/F But V is larger at high altitude AE2104 Flight and Orbital Mechanics 45 |
47. 47. Analytic range equations Unified Breguet range equation TV V g • Jet aircraft tot   F cT H H g • Propeller aircraft Fuel quality • Both: j g TV tot   F cp H H g Propulsion efficiency CL  W0  H R  tot ln   g CD  W1  aerodynamic quality Structural characteristics AE2104 Flight and Orbital Mechanics 46 |
48. 48. Analytic equations Unified Breguet range equation Time 1920 Lindbergh Present H 43000 kJ/kg 43000 kJ/kg 43000 kJ/kg tot 0.20 0.20 – 0.30 >0.40 L/D 10 11 16-18 W1/W0 0.6 – 0.7 0.5 0.5 AE2104 Flight and Orbital Mechanics 47 |
49. 49. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 48 |
50. 50. Story History • 1919: Alcock / Brown: Newfoundland  Ireland • Fonck, Nungesser/Coli, Lindbergh: New York  Paris Overload Fuel Passengers OEW (Operational Empty Weight) - Structural safety factor Take – off length (W2) Climb gradient after take off Tailwind west  east AE2104 Flight and Orbital Mechanics 49 |
51. 51. Story Spirit of St. Louis • Charles Lindbergh, 1927 • First solo, nonstop flight across the Atlantic Ocean AE2104 Flight and Orbital Mechanics 50 |
52. 52. Story Spirit of St. Louis • Charles Lindbergh had to decrease airspeed to achieve maximum range Pr Pr, for decreasing weight V AE2104 Flight and Orbital Mechanics 51 |
53. 53. AE2104 Flight and Orbital Mechanics 52 |
54. 54. Story Global flyer Burt Rutan AE2104 Flight and Orbital Mechanics Steve Fossett 53 |
55. 55. Story Global flyer • First part of flight: insufficient strength to withstand gusts • Best glide ratio: 1:37 • • • • • ( CL / CD )max = 37 • CD0 = 0.018 e = 0.85 Hcr = 45000 ft = 13.716 m   = 0.2377 Distance flown 38000 km Time 66 hrs Fuel lost 2600 lbs  actual fuel fraction 71% AE2104 Flight and Orbital Mechanics 54 |
56. 56. Story Global flyer • (V/F)max : CL  1 3 CD0  Ae  0.72 CD  0.024  CL  30 CD • V for (V/F)max, 45000 ft, Wgross: V = 175 m/s • Time for 40.000 km at constant V: 64 hrs • Guesstimate of tot : • High bypass fans at 1000 km/h: tot = 40% • Medium bypass fans tot = 35%, th = 50%, j = 70%  Vj / V = 1.86 • Correction for lower flight speed: • Vj / V = 3  j = 0.5  tot = 25% • Range in ideal climbing cruise: R = 53000 km AE2104 Flight and Orbital Mechanics 55 |
57. 57. Story Global flyer • Cruise at V = constant and Hcr = constant: R = 37.500 km • At fuel fraction 70%: R = 28000 km • Influence wind ? AE2104 Flight and Orbital Mechanics 56 |
58. 58. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 57 |
59. 59. Weight breakdown Wide-body airplane with turbofan engines Useful load Total fuel Supersonic transport with turbojet engines Total fuel Payload Payload OEW OEW AE2104 Flight and Orbital Mechanics 58 |
60. 60. Weight breakdown Payload range diagram Weight Fuel tank capacity MTOW MZFW MTOW = maximum take-off weight MZFW = maximum zero fuel weight OEW = operational empty weight fuel Payload Payload OEW Reserve fuel Payload Design range Ultimate range R AE2104 Flight and Orbital Mechanics R 59 |
61. 61. Weight breakdown Maximum zero fuel weight L/2 L/2 MZFW limited, amongst others by bending moment of the wing MZFW L/2 L/2 Wf / 2 Wf / 2 MZFW MTOW > MZFW at same bending moment. MTOW limited e.g. by landing gear AE2104 Flight and Orbital Mechanics 60 |
62. 62. Weight breakdown Reserve fuel • Reserve fuel • In general: • Fuel to alternate • 45 minutes holding at altitude • Fuel shortage: • In general: • Management problem • CRM Cockpit resource management AE2104 Flight and Orbital Mechanics 61 |
63. 63. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 62 |
64. 64. Economics Block time and block speed Key Parameters Range R Payload P Block time EB Block speed VB Transport product PR Transport productivity Ph Revenue earning capacity Py AE2104 Flight and Orbital Mechanics 63 |
65. 65. Economics AE2104 Flight and Orbital Mechanics 64 |
66. 66. Economics Transport productivity Conclusion: Maximum transport productivity is achieved at the design range ! Cost (direct operating cost) must be considered as well of course AE2104 Flight and Orbital Mechanics 65 |
67. 67. Content • Introduction • Optimum cruise profile • • • • • • • • • Optimal airspeed for given H, W Effect of altitude Effect of weight Best flying strategy Analytic Range equations Story Weight breakdown Economics Summary AE2104 Flight and Orbital Mechanics 66 |
68. 68. Summary The objective is to minimize fuel for a given range Key parameter: Specific range V/F [V]/[F] = [m/s]/[kg/s] = [m/kg] So it is the distance travelled per unit of fuel Performance diagram for jet aircraft AE2104 Flight and Orbital Mechanics 67 |
69. 69. Summary Effect of weight and altitude AE2104 Flight and Orbital Mechanics 68 |
70. 70. Summary Key conclusions Jet aircraft (analytical approximation) 1. Choose V such that (V/F)max  (CL / CD2)max 2. H as high as possible (limited by the engine) 3. If the speed limit is reached at lower altitude: V = Vlim, H is such that CL / CD is max Propeller aircraft (analytical approximation) 1. Conclusion: Altitude is not important w.r.t V/F 2. But V is larger at high altitude AE2104 Flight and Orbital Mechanics 69 |
71. 71. Summary Breguet range equation R W1  ds  W0 Jet aircraft (analytical approximation) W0 W0 V  F dW W1 Propeller aircraft (analytical approximation) W0  j CL dW V CL dW R  c CD W W1 T R Optimum cruise climb Cruise flight with constant , cp and j: R V CL  W0  ln   cT CD  W1  c W1 R p CD W  j CL W  ln  0  c p CD  W1  AE2104 Flight and Orbital Mechanics 70 |
72. 72. Summary Unified Breguet range equation TV V g tot   • Jet aircraft F cT H H g • Propeller aircraft Fuel quality • Both: j g TV tot   F cp H H g Propulsion efficiency CL  W0  H R  tot ln   g CD  W1  aerodynamic quality Structural characteristics AE2104 Flight and Orbital Mechanics 71 |
73. 73. Questions? AE2104 Flight and Orbital Mechanics 72 |