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

2. Theory of Propulsion2 Thrust V0V0 VeVe streamtube Actuator disc 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 Momentum theory for propellers

3. Theory of Propulsion3 F V0V0 VeVe p+  p p V0V0 r R Disc of area A Control volume Control volume for actuator disc Inflow along horizontal boundaries

4. Theory of Propulsion4 Volume flow into the C.V. through the horizontal boundaries of the C.V. Conservation of mass in the C.V.

5. Theory of Propulsion5 Momentum conservation in the C.V. Momentum into C.V. Momentum out of C.V. Force on the fluid

6. Theory of Propulsion6 Bernoulli’s Eq.- up and downstream Continuity requires that  r 2 V e =AV 1 disc V1V1 And therefore p p+  p Up: Down: Pressure jump

7. Theory of Propulsion7 V’=V 1 -V 0 is called the induced velocity and the thrust is Thrust = (mass through disc) (overall change in velocity) Velocity induced by actuator disc

8. Theory of Propulsion8 Conservation of energy in the C.V. The power absorbed by the fluid is

9. Theory of Propulsion9 Ideal propulsive efficiency In terms of the induced velocity the power absorbed is The ideal propulsive efficiency is then power required to keep V=V 0

10. Theory of Propulsion10 0 0.1 0.2 0.3 0.4 V’/V 0 1.0 0 ii 0.8 0.2 0.4 Ideal propulsive efficiency V 0 V’ V 1 =V avg =V 0 +V’ Operating range

11. Theory of Propulsion11 Thrust coefficient The thrust coefficient, which is dimensionless, is defined as Disc loading

12. Theory of Propulsion12 Power coefficient NOTE: For a statically thrusting propeller V 0 =0 and the non- dimensional coefficients don’t apply. Instead,

13. Theory of Propulsion13 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:

14. Theory of Propulsion14 1.0 0 F/F static 0 1 V 0 /V’ static V’ static =(F static /2  A) 1/2 Thrust variation with flight speed Note that thrust drops as speed increases

15. Theory of Propulsion15 dL dD rr V0V0 V’ VeVe VRVR ii   i  Axis of rotation Chord line Force and velocity on blade element dF  Induced angle

16. Theory of Propulsion16 dr r Blade element Axis of rotation The blade element for a propeller C(r)

17. Theory of Propulsion17 Thrust and power for a propeller The induced angle  i depends on the induced velocity V’

18. Theory of Propulsion18 Propeller characteristics Blade activity factor (AF) is a measure of solidity and therefore power absorption capability r R c(r) constant chord blade Typical range: 100<AF<150

19. Theory of Propulsion19 rr  Axis of rotation Chord line Geometric pitch 2r2r 2  r tan   In-plane distance moved in 1 revolution Advance during 1 revolution Advance ratio J= V 0 /  D  = blade pitch angle

20. Theory of Propulsion20 rr  Axis of rotation Chord line V0V0 L D R F Coarse pitch operation V0V0 High J: cruise Low J

21. Theory of Propulsion21 rr  Axis of rotation Chord line V0V0 F R L D Fine pitch operation V0V0 Low J: T-O High J

22. Theory of Propulsion22 0 Advance ratio J  Fine pitch Coarse pitch 0 1 The two-speed propeller

23. Theory of Propulsion23 0 Advance ratio J  11 22 33 44 0 1 The constant speed propeller  is varied to keep  constant at best engine speed

24. Theory of Propulsion24 Use of pitch control. Credits - NASA Blade pitch control

25. Theory of Propulsion25 Contra-rotating propellers

26. Theory of Propulsion26 Contra-rotating propellers

27. Theory of Propulsion27 Nondimensional blade parameters Advance ratio; J=V/nD Thrust coefficient: C T =F/  n 2 D 4 Torque coefficient: C q =T q /  n 2 D 5 Power coefficient: C P =P/  n 3 D 5 Efficiency:  =JC T /C P

28. Theory of Propulsion28 0.7 0 0  =88% 80% 70% 85%  3c/4 =20 o 30 o 40 o 4 Advance ratio J CPCP Propeller performance map Power coefficient C P =P/  3 D 5

29. Theory of Propulsion29

30. Theory of Propulsion30 Twist of propeller blades  r tip V R, tip V R,hub  r hub