Elements of Airplane Performance

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Chapter 6

• Elements of Airplane Performance

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• Simple Mission Profile for an Airplane
• 1 Switch on Worming Taxi

Un-accelerated level flight
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(Cruising flight)
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Descent
Altitude
Climb
Landing
Takeoff
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Simple mission profile
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Airplane Performance
Equations of Motions
Static Performance (Zero acceleration
Dynamic Performance (Finite acceleration)
Thrust required Thrust available
Maximum velocity
Takeoff
Power required Power available
Landing
Maximum velocity
Rate of climb
Gliding flight
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Time to climb
Maximum altitude
Service ceiling
Absolute ceiling
Range and endurance
Road map for Chapter 6
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• Study the airplane performance requires the
  derivation of the airplane equations of motion
• As we know the airplane is a rigid body has six
  degrees of freedom
• But in case of airplane performance we are deal
  with the calculation of velocities (
  e.g.Vmax,Vmin..etc),distances (e.g. range,
  takeoff distance, landing distance, ceilings
  etc), times (e.g. endurance, time to
  climb,etc), angles (e.g.climb angleetc)

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• So, the rotation of the airplane about its axes
  during flight in case of performance study is not
  necessary.
• Therefore, we can assume that the airplane is a
  point mass concentrated at its c.g.
• Also, the derivation of the airplanes equations
  of motion requires the knowledge of the forces
  acting on the airplane
• The forces acting on an airplane are

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• 1- Lift force L
• 2- Drag force D
• 3- Thrust force T Propulsive force
• 4- Weight W Gravity force
• Thrust T and weight W will be given
• But what about L and D?
• We are in the position that we cant calculate L
  and D with our limited knowledge of the airplane
  aerodynamics

Components of the resultant aerodynamic force R
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• So, the relation between L and D will be given in
  the form of the so called drag polar
• But before write down the equation of the
  airplane drag polar it is necessary to know the
  airplane drag types

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• Drag Types Kinds of Drag

Total Drag
Skin Friction Drag
Pressure Drag
Form Drag (Drag Due to Flow separation)
Induced Drag
Wave Drag
Note Profile Drag Skin Friction Drag Form
Drag
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• ?Skin friction drag
• This is the drag due to shear stress at the
  surface.
• ?Pressure drag
• This is the drag that is generated by the
  resolved components of the forces due to pressure
  acting normal to the surface at all points and
  consists of form drag induced drag wave
  drag .
• ?Form drag
• This can be defined as the difference between
  profile drag and the skin-friction drag or the
  drag due to flow separation.

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• ?Profile Drag
• ? Profile drag is the sum of skin-friction
  and form drags.
• ? It is called profile drag because both
  skin-friction and
• form drag or drag due to flow separation
  are
• ramifications of the shape and size of the
  body, the
• profile of the body.
• ? It is the total drag on an aerodynamic
  shape due to
• viscous effects

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Skin-friction
Form drag
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• ?Induced drag ( or vortex drag )
• This is the drag generated due to the wing
  tip vortices , depends on lift, does not depend
  on viscous effects , and can be estimated by
  assuming inviscid flow.

Finite wing flow tendencies
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Formation of wing tip vortices
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Complete wing-vortex system
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The origin of downwash
The origin of induced drag
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• ?Wave Drag
• This is the drag associated with the formation
  of shock waves in high-speed flight .

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• Total Drag of Airplane
• ? An airplane is composed of many components and
  each will contribute to the total drag of its
  own.
• ? Possible airplane components drag include
• 1. Drag of wing, wing flaps Dw
• 2. Drag of fuselage Df
• 3. Drag of tail surfaces Dt
• 4. Drag of nacelles Dn
• 5. Drag of engines De
• 6. Drag of landing gear Dlg
• 7. Drag of wing fuel tanks and external
  stores Dwt
• 8. Drag of miscellaneous parts Dms

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• ? Total drag of an airplane is not simply the sum
  of the drag of the components.
• ? This is because when the components are
  combined into a complete airplane, one component
  can affect the flow field, and hence, the drag of
  another.
• ? these effects are called interference effects,
  and the change in the sum of the component drags
  is called interference drag.
• ? Thus,
• (Drag)12 (Drag)1 (Drag)2
  (Drag)interference

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• Buid-up Technique of Airplae Drag D
• ? Using the build-up technique, the airplane
  total drag D is expressed as
• D Dw Df Dt Dn De Dlg Dwt
  Dms Dinterference
• ? Interference Drag
• ? An additional pressure drag caused by the
  mutual interaction of the flow fields around each
  component of the airplane.
• ? Interference drag can be minimized by proper
  fairing and filleting which induces smooth mixing
  of air past the components.
• ? The Figure shows an airplane with large degree
  of wing filleting.

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Wing fillets
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• ? No theoretical method can predict interference
  drag, thus, it is obtained from wind-tunnel or
  flight-test measurements.
• ? For rough drag calculations a figure of 5 to
  10 can be attributed to interference drag on a
  total drag, i.e,
• Dinterference 5 10 of
  components total drag
• The Airplane Drag Polar
• ? For every airplane, there is a relation
  between CD and CL that can be expressed as an
  equation or plotted on a graph.
• ? The equation and the graph are called the drag
  polar.

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For the complete airplane, the drag coefficient
is written as CD CDo
K CL2 This equation is the drag polar
for an airplane. Where CDo drag coefficient at
zero lift ( or parasite
drag coefficient ) K CL2 drag
coefficient due to lift ( or
induced drag coefficient CDi )
K 1/p e AR
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e Oswald efficiency factor 0.75 0.9
(sometimes known as the airplane efficiency
factor) AR wing aspect ratio b2/S ,
b wing span and S wing
planform area
Schematic of the drag polar
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• Airplane Equations of Motion

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• Apply Newtons 2nd low of motion
• In the direction of the flight path
• Perpendicular to the flight path

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• Un-accelerated Level Flight Performance
• (Cruising Flight)

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• Thrust Required for Level Un-accelerated Flight
• (Drag)
• Thrust required TR for a given airplane to
  fly at V8 is given as TR D

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? TR as a function of V8 can be obtained by tow
methods or approaches graphical/analytical
Graphical Approach
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• 1- Choose a value of V8
• 2 - For the chosen V8 calculate CL
• L W ½?8 V28S CL
• CL 2W/ ?8 V28S
• 3- Calculate CD from the drag polar
• CD CDo K CL2
• 4- Calculate drag, hence TR, from
• TR D ½?8 V28S CD
• 5- Repeat for different values of V8

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6- Tabulate the results
V8 CL CD CL/CD W/CL/CD







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(TR)min occurs at (CL/CD)max
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• Analytical Approach
• It is required to obtain an equation for TR as a
  function of V8
• TR D

Required equation
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• Parasite and induced drag

TR/D
CDoCDi
V8
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• Note that TR is minimum at the point of
  intersection of the parasite drag Do and induced
  drag Di
• Thus Do Di at TRmin
• or CDo CDi
• KCL2
• Then CL(TR)min vCDo/K
• And CDo(TR)min 2CDo

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• Finally, (L/D)max (CL/CD)max
• vCDo/K /2CDo
• (CL/CD)max 1/v4KCDo
• Also,V8(TR)min V8 (CL/CD)max is obtained
  from W L
• ½?8V2(TR)minS CL(TR)min
• Thus
• V (TR)min 2(W/S)(vK/CDo)/?8½

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L/D as function of angle of attack a
L/D as function of velocity V8
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• L/D as function of V8
• Since,
• But LW
• Then
• or

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• Flight Velocity for a Given TR
• TR D
• In terms of q8 ½?8V28 we obtain
• Multiplying by q8 and rearranging, we have
• This is quadratic equation in q8

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• Solving for q8
• By replacing q8 ½?8V28 we get

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• Let
• Where (TR/W) is the thrust-to-weight-ratio
• (W/S) is the wing loading
• The final expression for velocity is
• This equation has two roots as shown in figure
  corresponding to point 1 an 2

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?When the discriminant equals zero ,then only
one solution for V8 is obtained ?This
corresponds to point 3 in the figure, namely
at (TR)min
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• Or, (TR/W)min v4CDoK
• Then the velocity V3 V(TR)min is
• Substituting for (TR/W)min v4CDoK we have

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• Effect of Altitude on (TR)min
• We know that
• (TR/W)min v4CDoK
• This means that (TR)min is independent of
  altitude as show in Figure

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Thrust Available TA
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Sonic speed
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Thrust Available TA and Maximum Velocity Vmax
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• Power Required PR

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• Variation of PR with V8

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PR
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• Power Available PA

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