THEORY OF PROPULSION 9. Propellers

1. THEORY OF PROPULSION 9. Propellers P. M. SFORZA University of Florida

2. Momentum theory for propellers streamtube Thrust V0 Ve • The thrust is uniformly distributed over the disc • No rotation is imparted to the flow by the actuator disc • Streamtube entering and leaving defines the flow distinctly • Pressure far ahead and behind matches the ambient value Actuator disc Theory of Propulsion

3. Control volume for actuator disc Control volume p V0 p+Dp Ve V0 F r R Disc of area A Inflow along horizontal boundaries Theory of Propulsion

4. Conservation of mass in the C.V. Volume flow into the C.V. through the horizontal boundaries of the C.V. Theory of Propulsion

5. Momentum conservation in the C.V. Momentum out of C.V. Momentum into C.V. Force on the fluid Theory of Propulsion

6. Bernoulli’s Eq.- up and downstream disc Up: V1 p p+Dp Down: Pressure jump Continuity requires that p r2Ve=AV1 And therefore Theory of Propulsion

7. Velocity induced by actuator disc V’=V1-V0is called the induced velocity and the thrust is Thrust = (mass through disc) (overall change in velocity) Theory of Propulsion

8. Conservation of energy inthe C.V. The power absorbed by the fluid is Theory of Propulsion

9. Ideal propulsive efficiency In terms of the induced velocity the power absorbed is power required to keep V=V0 The ideal propulsive efficiency is then Theory of Propulsion

10. 1.0 0.8 Ideal propulsive efficiency hi 0.4 0.2 0 0 0.1 0.2 0.3 0.4 V’/V0 Operating range V0 V’ V1=Vavg=V0+V’ Theory of Propulsion

11. Thrust coefficient The thrust coefficient, which is dimensionless, is defined as Disc loading Theory of Propulsion

12. Power coefficient NOTE: For a statically thrusting propeller V0=0 and the non-dimensional coefficients don’t apply. Instead, Theory of Propulsion

13. Thrust variation with flight speed Equating the power to the propeller for the moving and static cases yields: This equation may be put in the following form: Theory of Propulsion

14. 1.0 Thrust variation with flight speed F/Fstatic V’static=(Fstatic/2rA)1/2 0 1 0 V0/V’static Note that thrust drops as speed increases Theory of Propulsion

15. Force and velocity on blade element dL f+ai V’ Ve f ai V0 VR f wr dD Axis of rotation Chord line dF Induced angle g Theory of Propulsion

16. The blade element for a propeller dr r Blade element Axis of rotation C(r) Theory of Propulsion

17. Thrust and power for a propeller The induced angle ai depends on the induced velocity V’ Theory of Propulsion

18. Propeller characteristics Blade activity factor (AF) is a measure of solidity and therefore power absorption capability constant chord blade c(r) r R Typical range: 100<AF<150 Theory of Propulsion

19. Geometric pitch g Advance during 1 revolution wr Axis of rotation 2p r tang Chord line g 2pr In-plane distance moved in 1 revolution Advance ratio J= V0 /wD g= blade pitch angle Theory of Propulsion

20. Coarse pitch operation V0 L F R wr g D Axis of rotation Chord line V0 High J: cruise Low J Theory of Propulsion

21. Fine pitch operation L F V0 R wr g D Axis of rotation Chord line V0 High J Low J: T-O Theory of Propulsion

22. The two-speed propeller h Coarse pitch Fine pitch 0 Advance ratio J 1 0 Theory of Propulsion

23. The constant speed propeller 1 h g3 g1 g2 g4 0 0 Advance ratio J g is varied to keep w constant at best engine speed Theory of Propulsion

24. Blade pitch control Use of pitch control. Credits - NASA Theory of Propulsion

25. Contra-rotating propellers Theory of Propulsion

26. Contra-rotating propellers Theory of Propulsion

27. Nondimensional blade parameters Advance ratio; J=V/nD Thrust coefficient: CT=F/rn2D4 Torque coefficient: Cq=Tq/rn2D5 Power coefficient: CP=P/rn3D5 Efficiency: h=JCT/CP Theory of Propulsion

28. Propeller performance map 0.7 CP h=88% 70% 85% 80% 30o g3c/4 =20o 40o 0 4 Advance ratio J 0 Power coefficient CP=P/rw3D5 Theory of Propulsion

29. Theory of Propulsion

30. Twist of propeller blades VR,tip VR,hub wrhub wrtip Theory of Propulsion