ae2104 flight-mechanics-slides 6
DESCRIPTION
These are the lecture slides to the fifth lecture of the course AE2104 Flight and Orbital mechanics.TRANSCRIPT
1Challenge the future
Flight and Orbital Mechanics
Lecture slides
Semester 1 - 2012
Challenge the future
DelftUniversity ofTechnology
Flight and Orbital MechanicsLecture hours 11, 12 – Cruise
Mark Voskuijl
2AE2104 Flight and Orbital Mechanics |
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
3AE2104 Flight and Orbital Mechanics |
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
4AE2104 Flight and Orbital Mechanics |
IntroductionTypical cruise flight
5AE2104 Flight and Orbital Mechanics |
Introduction
• Range (distance)
• Endurance (Maximum time)
Objective
6AE2104 Flight and Orbital Mechanics |
Introduction
General 2D equations of motion and power equation
Unsteady curved symmetric flight
cos sin
cos sin
T
T
a r
W dVT D W
g dt
W dL W T V
g dt
P P V dVRC
W g dt
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)
Equations of motion
7AE2104 Flight and Orbital Mechanics |
Introduction
Equations of motion cruise flight
0 sin
( quasi rectilinear )
gT D W
W
L W
cos
sin
dsV
dt
dHV
dt
Kinematic equations
, ,dW
F V Hdt
Additional equation
Equations of motion
8AE2104 Flight and Orbital Mechanics |
Introduction
Pilot can choose a certain airspeed and altitude: H(t), V(t)
What is the best flight condition?
1. Optimal initial conditions (V, H at initial weight)2. Optimal flying strategy (V, H at decreasing weight)
Problem definition
9AE2104 Flight and Orbital Mechanics |
Introduction
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
1. Maximum endurance E: Fuel flow Fmin at every point in time
Criteria for optimal flight
10AE2104 Flight and Orbital Mechanics |
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
11AE2104 Flight and Orbital Mechanics |
Optimum cruise profile (Jet)
Thrust, Drag
Airspeed
For basic flight mechanics applications, thrust of a turbojet can be assumed to be constant with airspeed for a given flight altitude
0
2
LD D
CC C
Ae Drag follows from:
Performance diagram - Jet
12AE2104 Flight and Orbital Mechanics |
Optimum cruise profile (Jet)
Power
Airspeed
For basic flight mechanics applications, thrust of a turbojet can be assumed to be constant with airspeed for a given flight altitude
Performance diagram - Jet
a
r
P TV
P DV
13AE2104 Flight and Orbital Mechanics |
Optimum cruise profile (Jet)
T T
V V V
F c T c D
D, T
VVopt
Specific range
TF c T
Thrust specific fuel consumption
max min
if V D
F V
Additional assumption: cT constant
What is the corresponding airspeed?
D
T
14AE2104 Flight and Orbital Mechanics |
Optimum cruise profile (Jet)
(D/V)min CLX / CD
Y CL
Optimumcriterion
Lift over dragratio
Angle ofattack
Method to calculate the best airspeed
Airspeed(for given H,W)
V
Optimal airspeed for given altitude and weight
15AE2104 Flight and Orbital Mechanics |
Optimum cruise profile (Jet)
min max
D V
V D
2
max max
L
D
CV
D C
2 1
L
WV
S C
2
1L
D
CV
D C
D
L
CLD D W
L C
L W
2
2 1
1 2L L
D D
L
W
S C CV
CD W S CWC
1. Optimum criterion
2. Airspeed
3. Drag
4. Ratio
5. For a given weight
6. Angle of attack for given altitude
Airspeed
16AE2104 Flight and Orbital Mechanics |
Optimum cruise profile (Jet)
0
2
max max max
13opt
L
D
lift drag polar
L D
CV V
F D C
C C Ae
0
0
2 2
max
2
4
4
2
1 12 2
13
0
2 1
0
2
0
2
L L
D L D
DL D D
L
D
D L
L
D
LD
L D
L L
L D
C Cd
C dC C
dCC C C
dC
C
dC C
dC Ae
C
CC
C C Ae
Ae C C
C C Ae
First year:
,
2 1
L opt
L W
WV
S C
Airspeed for given altitude and weight
Airspeed
17AE2104 Flight and Orbital Mechanics |
Optimum cruise profile (Jet)
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])
Example question
18AE2104 Flight and Orbital Mechanics |
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
19AE2104 Flight and Orbital Mechanics |
Effect of altitude
2 1 1
L
WV
S C Increasing H
D
V
0D
L
CD W
C
Angle of attack is constantfor a given point on the drag curve
Performance diagram
20AE2104 Flight and Orbital Mechanics |
21AE2104 Flight and Orbital Mechanics |
Effect of altitude
Altitude 0 m
T at cruise
T at cruise
V
T, DOptimum cruise level
Specific range
22AE2104 Flight and Orbital Mechanics |
Effect of altitude
At increasing altitude:
- V/F increases
- V increases
- Engine more efficient
Thus: fly as high as possible ! (up to the limits of the engine)
Conclusion
23AE2104 Flight and Orbital Mechanics |
Effect of altitudeTheoretical ceiling at cruise
Optimum cruise level at cruise
Tcruise
Tth
Hcr < Hth (at cruise)Optimum Hcr Hs (service ceiling)
24AE2104 Flight and Orbital Mechanics |
Effect of altitude
VVlim
D
Increasing H
In the presence of speed limits (e.g. MMO )
25AE2104 Flight and Orbital Mechanics |
Effect of altitude
• Optimum V/F at V = Vlim
• In case of speed limits, the optimum altitude is
not determined by the engine
0
0
min
max
2
lim
2 1
LL D
D
opt
D
CD C C Ae
C
W
S V C Ae
0
0
max
2
2
2
0
1
0
2
0
2
L L
D L D
DL D
L
D
D L
L
D
LD
L D
L L
L D
C Cd
C dC C
dCC C
dC
C
dC C
dC Ae
C
CC
C C AeAe C C
C C Ae
First year:
In the presence of speed limits (e.g. MMO )
26AE2104 Flight and Orbital Mechanics |
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
27AE2104 Flight and Orbital Mechanics |
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
28AE2104 Flight and Orbital Mechanics |
Effect of weight
2
3
2 1
at constant
at constant
L
D
L
r
r
WV W
S C
CD W W
C
P DV W W
D V
P V
D
VDecreasing W
29AE2104 Flight and Orbital Mechanics |
30AE2104 Flight and Orbital Mechanics |
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
31AE2104 Flight and Orbital Mechanics |
32AE2104 Flight and Orbital Mechanics |
Best flying strategy
W1, H1W2 < W1, H1
W2, H2 > H1
V
D
Strategies:I: constant altitude and engine settingII: constant altitude and airspeedIII: constant angle of attackIV: Climb and constant VV: Climb and change of V
33AE2104 Flight and Orbital Mechanics |
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?
34AE2104 Flight and Orbital Mechanics |
Best flying strategy
Strategy IV 2 = 1 (CL is constant) and V2 = V1
Optimum cruise climb possible?
11
1
22
2
2 1
constant2 1
L
L
D
L
WV
S C W
WV
S C
CT D W
C
Is this possible? Are the engines capable of providing enough thrust at higher altitude and lower weight?
35AE2104 Flight and Orbital Mechanics |
Best flying strategyTypical turbojet performance
36AE2104 Flight and Orbital Mechanics |
Best flying strategy
Strategy IV 2 = 1 (CL is constant) and V2 = V1
Optimum cruise climb possible?
2 1
2 1
2
2 2 2
1 1 11
= constant
D
L
D
L
D
L
W WW
CT D W
C
CW
T C W
CT WW
C
• 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
37AE2104 Flight and Orbital Mechanics |
Best flying strategy
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
Optimum cruise climb possible?
38AE2104 Flight and Orbital Mechanics |
Best flying strategy
39AE2104 Flight and Orbital Mechanics |
Best flying strategy
40AE2104 Flight and Orbital Mechanics |
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
41AE2104 Flight and Orbital Mechanics |
Analytic range equations
• Range
Breguet range equation
dW dWV F
dt ds
01
0 1
Ws
s W
VR ds dW
F
42AE2104 Flight and Orbital Mechanics |
Analytic range equations
• Jet aircraft optimum climb cruise (, V and cT are constant during variation of W
Breguet range equation for jet aircraft
0
1
W
W
VR dW
F
0
1
W
TW
VR dW
c D
0
1
W
L
T DW
CV dWR
c C W
0
1
W
L
T D W
CV dWR
c C W
0
1
lnL
T D
WCVR
c C W
43AE2104 Flight and Orbital Mechanics |
Analytic range equationsBreguet range equation for jet aircraft
• 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
0
1
lnL
T D
WCVR
c C W
44AE2104 Flight and Orbital Mechanics |
Analytic range equationsBreguet range equation for propeller aircraft
Cruise flight with constant , cp and j:
0
1
W
W
VR dW
F
p p
p br a
j j
c cF c P P TV
1 1 1j j j L
p p p D
CV
F c T c D c C W
0
1
lnj L
p D
WCR
c C W
45AE2104 Flight and Orbital Mechanics |
Analytic range equations
Conclusion: Altitude is not important w.r.t V/F
But V is larger at high altitude
P
V
Pr Dmin
Breguet range equation for propeller aircraft
46AE2104 Flight and Orbital Mechanics |
Analytic range equationsUnified Breguet range equation
• Jet aircraft
• Propeller aircraft
• Both:
tot
T
TV V g
F c HH
g
j
tot
p
TV g
F c HH
g
0
1
lnLtot
D
WCHR
g C W
Fuel quality Propulsion efficiency aerodynamic quality
Structural characteristics
47AE2104 Flight and Orbital Mechanics |
Analytic equations
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
Unified Breguet range equation
48AE2104 Flight and Orbital Mechanics |
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
49AE2104 Flight and Orbital Mechanics |
StoryHistory
• 1919: Alcock / Brown: Newfoundland Ireland
• Fonck, Nungesser/Coli, Lindbergh: New York Paris
Passengers
Fuel
OEW (OperationalEmpty Weight)
Overload
- Structural safety factor- Take – off length (W2)- Climb gradient after take off- Tailwind west east
50AE2104 Flight and Orbital Mechanics |
StorySpirit of St. Louis
• Charles Lindbergh, 1927
• First solo, nonstop flight across
the Atlantic Ocean
51AE2104 Flight and Orbital Mechanics |
StorySpirit of St. Louis
• Charles Lindbergh had to decrease airspeed to achieve maximum
range
Pr, for decreasing weight
V
Pr
52AE2104 Flight and Orbital Mechanics |
53AE2104 Flight and Orbital Mechanics |
StoryGlobal flyer
Burt Rutan Steve Fossett
54AE2104 Flight and Orbital Mechanics |
Story
• 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%
Global flyer
55AE2104 Flight and Orbital Mechanics |
Story
• (V/F)max :
• 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
0
13
0.72
0.024 30
L D
LD
D
C C Ae
CC
C
Global flyer
56AE2104 Flight and Orbital Mechanics |
Story
• Cruise at V = constant and Hcr = constant: R = 37.500 km
• At fuel fraction 70%: R = 28000 km
• Influence wind ?
Global flyer
57AE2104 Flight and Orbital Mechanics |
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
58AE2104 Flight and Orbital Mechanics |
Weight breakdown
Total fuel
OEW
Payload
Payload
Total fuel
OEW
Wide-body airplane with turbofan engines
Supersonic transport with turbojet engines
Useful load
59AE2104 Flight and Orbital Mechanics |
Weight breakdown
Fuel tank capacity
Design range
Weight
fuelMZFW
Reserve fuelOEW
MTOW
R
Payload
Ultimate range
Payload
R
Payload
MTOW = maximum take-off weightMZFW = maximum zero fuel weightOEW = operational empty weight
Payload range diagram
60AE2104 Flight and Orbital Mechanics |
Weight breakdown
MZFW limited, amongst others by bending moment of the wing
MTOW > MZFW at same bending moment. MTOW limited e.g. by landing gear
MZFW
L/2L/2
MZFW
L/2L/2
Wf / 2Wf / 2
Maximum zero fuel weight
61AE2104 Flight and Orbital Mechanics |
Weight breakdown
• Reserve fuel
• In general:
• Fuel to alternate
• 45 minutes holding at altitude
• Fuel shortage:
• In general:
• Management problem
• CRM Cockpit resource management
Reserve fuel
62AE2104 Flight and Orbital Mechanics |
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
63AE2104 Flight and Orbital Mechanics |
Economics
Key Parameters
Range R
Payload P
Block time EB
Block speed VB
Transport product PR
Transport productivity Ph
Revenue earning capacity Py
Block time and block speed
64AE2104 Flight and Orbital Mechanics |
Economics
65AE2104 Flight and Orbital Mechanics |
Economics
! Cost (direct operating cost) must be considered as well of course
Conclusion:Maximum transport productivity is achieved at the design range
Transport productivity
66AE2104 Flight and Orbital Mechanics |
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
67AE2104 Flight and Orbital Mechanics |
Summary
Key parameter:Specific range V/F[V]/[F] = [m/s]/[kg/s] = [m/kg]So it is the distance travelled per unit of fuel
The objective is to minimize fuel for a given range
Performance diagram for jet aircraft
68AE2104 Flight and Orbital Mechanics |
SummaryEffect of weight and altitude
69AE2104 Flight and Orbital Mechanics |
Summary
Propeller aircraft (analytical approximation)1. Conclusion: Altitude is not important
w.r.t V/F2. But V is larger at high altitude
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
Key conclusions
70AE2104 Flight and Orbital Mechanics |
Summary
01
0 1
WW
W W
VR ds dW
F
0
1
W
L
T DW
CV dWR
c C W
0
1
W
j L
p DW
C dWR
c C W
Jet aircraft(analytical approximation)
Propeller aircraft(analytical approximation)
0
1
lnL
T D
WCVR
c C W
0
1
lnj L
p D
WCR
c C W
Optimum cruise climb Cruise flight with constant , cp and j:
Breguet range equation
71AE2104 Flight and Orbital Mechanics |
SummaryUnified Breguet range equation
• Jet aircraft
• Propeller aircraft
• Both:
tot
T
TV V g
F c HH
g
j
tot
p
TV g
F c HH
g
0
1
lnLtot
D
WCHR
g C W
Fuel quality Propulsion efficiency aerodynamic quality
Structural characteristics
72AE2104 Flight and Orbital Mechanics |
Questions?