Transcript AE 301 Aerodynamics I

Performance
• Performance is the study of how high, how fast, how
far, and how long an aircraft can fly.
• It is one part of the general study of flight dynamics
which also include stability and control.
• In this study, we no longer consider the motion and
properties of the air, but now concentrate on the
motion of the entire airplane and its response to
applied forces.
• The following figure illustrates the forces of interest
and the relative lines of action.
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Performance (continued)
L
D
W
• Note the new vector angles:
– Flight path angle, : angle between the velocity vector of
the aircraft and the horizon.
– Thrust line angle, T: angle between the velocity vector and
the action line of the powerplant.
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Performance (continued)
• In the previous chapters we discussed how to
calculate the aerodynamic forces of Lift and Drag.
• One additional note needs to be made about how we
extend our previous drag results for wings to an
entire airplane. The equation is:
CL2
CD  CD , 0 
eAR
• The zero lift drag, CD,0, is due to viscous effects over
the entire airplane surface - wing, fuselage, etc.
• The second term now includes both the span
efficiency of the wing and any variation in viscous
drag due to lift.
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Equations of Motion
• An airplane in flight obeys Newton’s Laws of motion.
In particular: force = mass * acceleration.
• For airplanes, we split the forces in to those in the
flight direction and those perpendicular to it:
2
V
dV
 F  m r
 F||  ma  m dt
c
• Note that in the perpendicular equation we allow for
a curved flight path with radius rc.
• Summing forces gives:
dV
T cos  t  D  W sin   m
dt
V2
L  T sin  t  W cos  m
rc
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Equations of Motion (continue)
• The previous equations are the general equations of
motion for an airplane. They are applicable to all
flight conditions.
• A tremendous simplification occurs if we limit the
study to unaccelerated, level flight.
dV/dt = 0
rc
=0
• Also, in most airplanes, the thrust angle is small
enough to assume cos(T)~1 and sin(T)~0.
• Under these assumptions,
T D
AE 301 Aerodynamics I
L W
and
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Thrust Required
• The thrust acting on an airplane should be
considered from two different viewpoints:
– The thrust required by the airplane to stay in flight at the
existing flight conditions, I.e. V, h, , etc.
– The thrust available from the powerplant to maintain or
change those flight conditions.
• Lets start with the thrust required. From the
previous relations:
TR  D  q SCD
• Or, since
T D
L W
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W
TR 
L/D
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Level,
Unaccelerated
Flight
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Thrust Required (continued)
• The second relation points out a very important
point: the minimum thrust require occurs when the
airplane lift to drag ratio, L/D = CL/CD, is maximum.
• The first equation is more useful however in finding
when this occurs. Substituting our previous relation
for drag yields:

CL2 
W2
  q SCD,0 
TR  q S  CD,0 
eAR 
q SeAR

Profile or
Parasitic Drag
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Drag due to lift
(Induced drag)
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Thrust Required (continued)
• Note how the two contributions to drag vary
differently with velocity:
D, TR
Total drag,
Require Thrust, TR
Parasitic drag
Induced drag
V for min D and TR,
and max L/D
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V
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Thrust Required (continued)
• From this we see that a minimum in required thrust
occurs at some value of velocity (or, similarly, q).
• To find this minimum, we differentiate this relation
with respect to q and set the derivative to zero:
dTR d 
W2 
W2

q SCD ,0 
  SCD ,0  2
dq dq 
q SeAR 
q SeAR
dTR
dq
0
TR ,min

CD ,0
W2
CL2
 2 2

q S eAR eAR
• Thus, the minimum drag occurs when the parasitic
drag and drag due to lift are equal!
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Thrust Required (continued)
• This effect can also be
seen by looking at a
parabolic drag polar
• Any line from the
origin has a slope
equal to the L/D ratio.
CL
(L/D)max
CL,L/D max
• Thus, the maximum
L/D occurs at the
tangency point shown.
CL/CD = L/D
CD,0
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CD
2CD,0
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Thrust Available
• Thrust available is a function of the power plant
type/size and aircraft velocity and altitude.
• Typical thrust available
variation with velocity
is shown here for two
engine types:
• For piston-propeller
combinations, thrust
decreases at high
speed due to Mach
effects on the propeller
tip.
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TA
Turbojet
Piston-Propeller
1.0
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M
Thrust Available
• For turbojet engines, thrust normally increases
slightly with speed due to the increased inlet
performance and increased mass flow rate with Mach
number.
• Other engine types like turboprops and turbofans
have thrust variations somewhere between these
two.
• The best source for engine performance data is the
manufacturer themselves provided in the form of an
“engine deck”.
• Also realize that engine thrust also depends upon the
throttle setting.
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Thrust Available (continued)
• For a given airplane, the range of possible steady
flight velocities depends upon the relative values of
thrust required and thrust available:
T
Piston-Propellor
TA
• To fly at velocities
between V,min and
V,max, the throttle
setting would be set
less that 100%.
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Sonic Speed
• Level, unaccelerated
flight is only possible
when TA  TR.
TR
V,min
V,max
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V
Power Required
• As important as thrust, is power.
• This is particularly true for propeller driven airplanes
as evidenced by the fact that piston and turboprop
engines are rated in horsepower.
• Fortunately, power and thrust are closely related:
Work Force  Distance
Power 

 Force  Velocity
Time
Time
• Or, more mathematically:
PR  TR V  TRV
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For level, unaccelerated
flight
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Power Required (continued)
• For thrust required, we showed that:
W
TR 
CL CD
• To get a similar relation for power required,
remember that:
2W
2
1
L  W  2  V SCL or V 
 SCL
• Putting these together gives:
2W 3C D2
1
PR  TRV 

  SC L3 CL3 / 2 C D
• Thus, for minimum power required, we want to have
3/ 2
the maximum of CL CD
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Power Required (continued)
• Also, as before, we can split the thrust into profile
and induced contributions:

CL2 
V
PR  TRV  DV  q S  CD ,0 
eAR 

• Or, upon expanding
2
2
W
3
PR  12 V SCD,0 
V SeAR
• The first term is the zero-lift or parasitic power - I.e.
the power required to overcome friction.
• The second term is the lift-induced power - I.e. the
power required to produce lift.
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Power Required (continued)
• When plotted versus velocity, we get a graph
qualitatively similar to the thrust curves, but with
different variations with velocity.
PR
Require power, PR
 CL3 / 2


CD 

max
Parasitic power
Lift-induced power
V
V for min. PR
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Power Required (continued)
• To find the minimum power required point, find the
point where the slope is zero:
dPR
dV
PR ,min
2
1


C
2
L
3
3
  0
 2  V S  CD ,0 
eAR 

• Which turns out to be when:
CD ,0
CL2 1

 3 C D ,i
eAR
1
3
For minimum
power required
• Thus, the point of minimum power required is when
the induced drag is three times the parasitic drag.
• This is slower than the minimum thrust point which
occurred when they were equal!
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Power Required (continued)
• The slope of any line from the origin on this plot has
a slope equal to the required thrust.
• Thus, the minimum TR (minimum slope) occurs at the
tangency point indicated.
PR
 CL

 C 
D  max

PR/V=TR
TR,min
Vmin Pr
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V
Vmin Tr
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Power Available
• Power available can be obtained from multiplying the
thrust available by the velocity.
• As with thrust available, how power available varies
depends upon the powerplant type:
• For turbojets, power
increases almost linearly.
PA
Piston-Propeller
• For piston-propellers,
power increases rapidly
at low speeds, but is
nearly constant for much
of the flight regime.
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Turbojet
M
1.0
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Power Available (continued)
• As mentioned, a common measurement of piston and
turboprop engine power output is horsepower.
1 hp  550ft  lb/sec  746 W
• Also, propeller aircraft all the power produced by the
engine does not go into producing thrust - some is
lost by the inefficiencies of the propeller.
• To account for this, we introduce two new terms:
– the propeller efficiency, .
(  1.0)
– the engine power, P, called the shaft brake horsepower, bhp
(or shp).
hpA    bhp
AE 301 Aerodynamics I
PA  P
or
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Power Available (continued)
• As with thrust, the minimum and maximum flight
velocities can be found from the intersection of
required and available power.
P
Sonic Speed
Piston-Propellor
PA
PR
V,min
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V,max
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V
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Power Available (continued)
• One final note: all of the figures shown so far always
have an intersection of PA and PR at low speeds.
• In fact, under many conditions, this intersection
doesn’t exist due to aircraft stall:
• In this situation, the
lowest possible velocity
for level flight is
dictated by the CLmax:
V, stall 
P
PA
PR
2W
 SCL max
V
V,stall
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Altitude Effects on Power
• Earlier, it was shown how velocity and power required
depended upon , CL, CD, S and W
2W
2W 3C D2
V 
PR 
 SCL
  SC L3
• If we consider other altitudes, only  will change, the
other values being independent of altitude.
• Thus, defining reference quantities at sea level
conditions:
V0 
2W
 0 SCL
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PR , 0 
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2W 3C D2
 0 SC L3
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Altitude Effects (continued)
• And to relate altitude conditions to sea level:
Valt  V0
0 V0



• These relations indicate a
shift in the power required
curve to higher powers
and higher velocities as
altitude increases (or as 
decreases)
0 PR,0



PR,alt  PR,0
PR


V
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Altitude Effects (continued)
• Available power also varies with altitude.
• At constant velocity, it is reasonable to assume that
PA and TA vary linearly with density since they both
will increase with mass flow rate:
  
  PA, 0
PA,alt  PA, 0 
 0 
• (Note: some references argue that turbojet and
turbofan performance varies with pressure ratio,
p/po!)
• For piston engines, supercharging, the precompression of intake air, can eliminate the density
variation of power up to some altitude.
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Altitude Effects (continued)
• The combined effect of altitude on PA and PR is to
reduce the maximum velocity, and increase the
minimum velocity.
P
V
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Rate of Climb
• Consider now an airplane in steady, unaccelerated,
climbing flight: L
T
V
D
W
• The vertical component of velocity is called the rate
of climb, R/C (often given in the non-standard units
of feet per minute, fpm).
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Rate of Climb (continued)
• A force balance parallel and perpendicular to the
flight path yields:
L  W cos 
T  D  W sin 
– the primary difference from level flight being the weight
contribution in the flight direction
• Rearranging the first equation gives:
T D
 sin 
W
– or after multiplying by velocity:
TV  DV
 V sin   R / C
W
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Rate of Climb (continued)
• Thus, the rate of climb is proportional to the
difference between the power available, TV, and the
power required for level flight, DV.
• This difference is called the excess power:
TV  DV  excesspower
• Graphically, the
excess power is the
distance between the
power available and
power required curves
R/C 
excess power
W
PA=TV
P
excess
power
PistonPropeller
PR=DV
V
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Rate of Climb (continued)
• A little word of caution about using these charts:
– The power required curve shown is that for level,
unaccelerated flight.
– However, the drag depends upon the lift which in climbing
flight is slightly lower than in level flight since L=W cos
– The power required curve thus depends upon the angle of
climb - but we cannot calculate the excess power and the
angle of climb until we first have the power required!
– For small climb angles, cos ~ 1, so L~W. Thus, we don’t
have to worry about this discrepancy.
– For large climb angles, the original force balance equations
must be solve to yield a relationship between V and  for
given values of T.
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Rate of Climb (continued)
• The maximum rate of climb
occurs for the highest throttle
setting and at a velocity such
that excess power is a
maximum.
• Another way to represent this
is to plot the rate of climb
versus flight velocity.
P
Maximum PA
Maximum
excess
power
PR
V
R/C
R/Cmax
• The maximum R/C and the
corresponding velocity are
easily seen.
Vmax R/C
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Vmax
V
Rate of Climb (continued)
• Also of interest is the maximum flight path angle useful if we want clear obstacles.
• To find this, change the axis on the previous plot to
the horizontal velocity - a so-called hodograph
diagram.
Hodograph Diagram
• A line to any point on this
curve represents the flight
velocity and angle.
Vv
Vv,max=R/Cma
x
• The tangent line shown
will have the steepest flight
path angle (note how close
to stall it is!)
AE 301 Aerodynamics I
max
V

Vh
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Rate of Climb (continued)
• One final note concerns the difference between
turbojets and piston-propellers.
• The two sketches below show that at low speeds,
piston-propellers have a higher level of excess power.
• This add a comfort margin in propeller driven aircraft
since all planes normally land near their stall speeds!
P
Piston-Propeller
Turbojet
P
PA=TV
PA=TV
excess
power
excess
power
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PR=D V
PR=D V
V
V
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Gliding Path - Drift down
• Now consider an airplane in a power-off glide as
shown below:
L
D
V
W
• Note that by convention the descent angle is defined
as being positive for this situation!
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Gliding Path (continued)
• The force balance this time yields:
D  W sin 
L  W cos 
• Or by dividing the two equations:
D
1
tan  
L L D
• Thus, the minimum glide slope angle occurs for a
maximum L/D!
• Note that min provides for the longest distance
traveled in a glide from altitude.
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Gliding Path (continued)
• For maximum time aloft, we want the minimum
vertical velocity, Vv
• To get this, multiply our drag equation by V:
– or
V D  WV sin 
V D
Vv  V sin   
W
• Thus, for a minimum vertical velocity, we want a
minimum in the required power, VD!
• By our previous calculations, this occurs when:
 CL2



C
D

max
3
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Ceilings
• Let’s review the altitude on effects on power available
and power required.
• PA decreases with , while PR increases by (1/)1/2
• As a result, the maximum excess power, and thus the
maximum rate of climb, decreases with altitude!
P
Maximum
excess power
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V

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Ceilings (continued)
• At some altitude, the maximum excess power
becomes zero.
• This situation is the maximum possible flight altitude
of the aircraft - the absolute ceiling!
• While it is possible for an
aircraft to achieve this
altitude, the aircraft is
unstable at this condition
and it is difficult, if not
actually unsafe, to fly
there.
P
V
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
Ceilings (continued)
• A more practical ceiling for
an aircraft is that alitude at
which the rate of climb is
reduced to 100 fpm - the
service ceiling!
h
Absolute ceiling
Service ceiling
• For commercial jet
transports, the service ceiling
is even more restrictive - a
500 fpm rate of climb is
required.
R/C (fpm)
0
100 fpm
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R/Csl
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Ceilings (continued)
• One final note: a quick estimate of the absolute
ceiling of a propeller airplane may be made by
noticing that at this condition, PA ~ PR,min.
• Using our altitude correction formulae:
PA  PR,min

PA,0 
PR ,min,0

2/3
 PR,min 
 

P
A 0

• Thus, the density ratio at the absolute ceiling can be
estimated from the power ratio at sea level.
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Time to climb
• The time to climb from one altitude to another may
be calculated once the rate of climb is known.
• Unfortunately, since R/C varies with altitude, the
necessary calculation is an integration as indicated
by:
h
dh
R/C 
dt
2
dh
t
R/C
h1
• Thus, the R/C must be determined at a number of
altitudes in the range between h1 and h2
• A plot of (R/C)-1 can then be numerically or
graphically integrated to get time.
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Time to climb (continued)
(R/C)-1 (m/ft)
•
Time to climb =
area under curve
0
h1=0
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h2
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h
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