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Introduction
Propellers
Internal Combustion Engines
Gas Turbine Engines
Chemical Rockets
Non-Chemical Space Propulsion Systems
AER 710 Aerospace Propulsion
Introduction to the Propeller
• The rotating blade of a propeller shares similar characteristics to a wing passing through the air
• A propeller blade generates thrust F through an aerodynamic lift force component, demands an engine torque Q to overcome aerodynamic drag, and will stall if the local resultant angle of attack of the blade exceeds max
• Additional factors: trailing vortex generation, tip losses, compressibility
• For evaluation of propeller performance, one can apply a simple analytical approach using the principle of linear momentum conservation, and treating the propeller as an actuator disk where there is a step increase in pressure
Actuator Disk Theory
)VV(VA)VV(mF 033303 Thrust generated by disk:
)pp(AF 121 Alternatively:
211
200 2
1
2
1VpVp
Bernoulli’s eq. applied from upstream to front of disk:
233
222 2
1
2
1VpVp
Similarly, downstream of disk:
)VV)(VV()VV(pp 03032
02
312 2
1
2
1
Noting po = p3 , and V2 = V1, via subtraction one gets:
A3V3 = A1V1
Conservation of mass, incompressible flow:
)VV(VA)pp(AF 0333121
Substituting from earlier:
)VV)(VV()VV(VA
App 0303033
1
312 2
1
which gives the simple result:
and
203
1
VVV
wVV 01
Define propeller-induced velocity w such that:
wVV 203
w)wV(A)VwV)(wV(A)VV(VAF 0100010311 22
and so for thrust,
201
20
2001
20
23 22
2
1
2
1
2
1)wV(wA]V)wV)[(wV(AVmVmP
Ideal power required:
)wV(FP 0
or
Since power from a piston or turboprop engine is relativelyconstant at a given altitude, one can expect the thrust todrop as the airplane picks up airspeed, according to thiscorrelation.
022 012
1 Fw)VA(w)A(
If one wishes to find w as a function of F, from earlier:
1
20
0 2
2
1
2 A
FV
Vw
giving
1
23
2 A
FwFPP
/o
ooo,indo
Ideal static power (Vo = 0):
0
0
0
1
1
V
w)wV(F
FVi,pr
Ideal propeller propulsive efficiency:
1
11
2
qA
Fi,pr
or via substitution (q is dynamic pressure):
i,prS
pr P
FV
Actual propeller propulsive efficiency, in terms of useful(thrust) power and engine shaft power PS :
SP)()wV(FP factor correction0
Correction factor, less than 1, for ideal power estimate:
Variable-pitch propeller better able to approach theideal power requirement, as compared to a fixed-pitchpropeller, in accommodating different flight speedsand altitudes.
Momentum-Blade Element Theory
• Logically, the next level of analysis would look at a given propeller blade’s aerodynamic performance from hub to blade tip
• one can discretize the blade into a finite number of elements, while applying momentum conservation principles
Schematic diagram of a three-bladed propeller, and framework for discretizing an individual blade for analysis
)]cos(D)sin(L[rFrQ iiQ dddd
Increment of torque:
rcCVL E d2
1d 2
Increment of lift:
rcCVD dE d2
1d 2
Increment of drag:
22 )V)cos(w())sin(wr(V iiE
Overall resultant velocity:
)V
w(sin
Ri
1
Induced angle of attack:
)(a)(CC ioi
Airfoil lift coefficient:
min,dd CC
Airfoil drag coefficient:
C < C,min
2)CC(kCC min,min,dd C,min < C < C,max
)(kCC max,dd max 1 > max
cosr)(caVB
cosLF ioR d2
dd 2
Via substitutions, increment of thrust:
where B is number of blades.
cosw)coswV(Aw)wV(AF d2d2d 0
Borrowing from actuator disk theory:
cosV)cosVV()rr( RiRi d22
088
2 )(cosr
Bca)
cosr
Bca
cosV
V( o
io
Ri
Equating the above relations, one arrives at:
R
Bc
R
RcB refrefref
2areadisk
area blade
Overall propeller solidity:
r
Bcx
R
Bc
Local solidity:
x = r/R
R
V
)R))(/((
V
nd
VJ
p
22
Advance ratio:
where n is the prop shaft rotation speed (rps).
J
R
V
Nondimensional velocity ratio:
)x
(tan)r
V(tan
11 Also:
TR xVrcosV VT = R
088 22
2 )(Vx
Va)
Vx
Va
x(
T
Roi
T
Roi
Substituting from earlier:
})](Vx
Va)
Vx
Va
x[()
Vx
Va
x({ /
T
Ro
T
Ro
T
Roi
212
222 2882
1
Applicable solution for induced angle of attack via theabove quadratic eq. gives:
42dn
FCT
Propeller thrust coefficient:
53dn
PC S
P
Propeller power coefficient:
QPS
r)]sin(C)cos(C[BcVF idiE d2
1d 2
Incremental thrust no. of blades:
r)]cos(C)sin(C[BcVrP idiES d2
1d 2
Incremental power no. of blades:
)xJ(r
rVVV RE222
2
2222222
FR
CT 42
2
4
Note:
Thrust coefficient:
x)]sin(C)cos(C)[xJ(FR
C idi
x
T
h
d8
d4
221
242
2
SP PR
C53
3
4
Power coefficient:
x)]cos(C)sin(C)[xJ(xPR
C id
x
iSP
h
d8
d4
1222
2
53
3
Momentum-Blade Element Theory (Summary)
• The above equations for CT and CP can be integrated from the hub station (x = xh) to the blade tip (x = 1) using a numerical approach as one moves along the blade of varying and c, calculating the various pertinent parameters (C , Cd, i , etc.) in conjunction
Propeller Propulsive Efficiency
• Define as useful thrust power over overall shaft power:
Spr P
FV
JC
C
dnC
VdnC
P
T
P
Tpr
53
42
Also, via substitution:
A variable pitch propeller will have better efficiency over thecourse of the flight mission, relative to a fixed pitch prop.
Compressibility Tip Loss
• Depending on the blade airfoil section design, drag divergence (compressibility) effects will become evident when the propeller blade’s resultant tip speed VR,tip exceeds a local flow Mach number Matip of around 0.85 (critical value, Macr)
• As a result, one would not typically be cruising at much greater than a flight Mach number Ma of around 0.6
22)(
Maa
ndMatip
)1.0
(100
15 crtipalminpr,nopr
MaMa
Dommasch correlation:
Blade tip Mach number:
Modern high-speed blades may be thinner, and sweptor curved along the blade length, to mitigate the issues with compressibility and compression wavedevelopment at higher local flow Mach numbers
Activity Factor
• Activity factor (AF) is a design parameter associated with the propeller blade’s geometry. The more slender the blade (larger radius, smaller chord), the lower the AF value:
xxd
cAF
hx p
d16
100000 31
pd
cAF 1563
Typically see higher AF props on turboprop engines.
Blade Number• One has the option of setting the number
of blades, B, for a given application. While one has a minimum of 2 blades to choose from, one can presently go as high as around 8 blades on the high-performance end for an unducted propeller
• On occasion, one also sees the use of two contra-rotating rows of blades, to get more thrust delivery from one engine
Photo of Fairey Gannett carrier-borne anti-submarine/AEW aircraft, employing two contra-rotating rows of 4 propeller blades each on a co-axial shaft setup, powered by a 3000-hp Armstrong Siddeley Twin Mamba turboprop engine
Helicopter Rotors• helicopter rotors (main and tail) share a
number of similarities with airplane propellers• analysis done above for propellers can be
applied to rotors• orientation of the rotor disk will be somewhat
different from that of the propeller, with respect to the resultant incoming air flow
• Main helo rotor produces lift + thrust
- rotor blade will advance into the air flow when in forward flight, and then retreat during the other half of the rotational cycle
CH-47
- tail rotor primarily controls yaw forces and moments [primarily main-rotor-induced torque] on the helicopter, if only having one main rotor- a tandem-rotor helicopter, with two contra-rotating main rotors, would not need a tail rotor
• The amount of lift generated by a main rotor is controlled by two means: a) the engine throttle setting for desired level of main rotor rotational speed, and b) collective pitch setting, which sets the angle of incidence of the main rotor blades collectively to produce the desired uniform lifting force on the vehicle (e.g., higher lift required, a higher blade incidence angle setting is needed, for the same rotor rotational speed)
• Rotation of the vehicle’s body in pitch or roll or some combination thereof is largely via the cyclic pitch setting of the main rotor, whereby the individual main rotor blades will have their incidence vary as they complete a given revolution about the vehicle, depending on the desired direction of the rotational moment
The schematic diagram illustrates a conventional main rotor mast (rotorhead), with the hub above the mast connecting the rotor blades to the drive shaft in a fully articulated design (hinged); a swashplate approach is being used to control the effective main rotor disk deflection and tilt direction thereof
Hybrid Aircraft Designs• In order to improve range performance
over a conventional helicopter, one will see tilt-wing and tilt-rotor designs for V/STOL (vertical/short takeoff & landing) applications
Tilt-rotor V-22 Osprey
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