Transcript AE 301 Aerodynamics I

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 steady level,
un-accelerated
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
KW
3
PR  12 V SCD,0 
V S
• 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
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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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