Simple Performance Prediction Methods

1. Simple Performance Prediction Methods Momentum Theory © L. Sankar Helicopter Aerodynamics

2. Background • Developed for marine propellers by Rankine (1865), Froude (1885). • Used in propellers by Betz (1920) • This theory can give a first order estimate of HAWT performance, and the maximum power that can be extracted from a given wind turbine at a given wind speed. • This theory may also be used with minor changes for helicopter rotors, propellers, etc. © L. Sankar Helicopter Aerodynamics

3. Assumptions • Momentum theory concerns itself with the global balance of mass, momentum, and energy. • It does not concern itself with details of the flow around the blades. • It gives a good representation of what is happening far away from the rotor. • This theory makes a number of simplifying assumptions. © L. Sankar Helicopter Aerodynamics

4. Assumptions (Continued) • Rotor is modeled as an actuator disk which adds momentum and energy to the flow. • Flow is incompressible. • Flow is steady, inviscid, irrotational. • Flow is one-dimensional, and uniform through the rotor disk, and in the far wake. • There is no swirl in the wake. © L. Sankar Helicopter Aerodynamics

5. Control Volume V Station1 Disk area is A Station 2 V- v2 Station 3 V-v3 Stream tube area is A4 Velocity is V-v4 Station 4 Total area S © L. Sankar Helicopter Aerodynamics

6. Conservation of Mass © L. Sankar Helicopter Aerodynamics

7. Conservation of Mass through the Rotor Disk V-v2 V-v3 Thus v2=v3=v There is no velocity jump across the rotor disk The quantity v is called velocity deficit at the rotor disk © L. Sankar Helicopter Aerodynamics

8. Global Conservation of Momentum Mass flow rate through the rotor disk times velocity loss between stations 1 and 4 © L. Sankar Helicopter Aerodynamics

9. Conservation of Momentum at the Rotor Disk Due to conservation of mass across the Rotor disk, there is no velocity jump. Momentum inflow rate = Momentum outflow rate Thus, drag D = A(p2-p3) V-v p2 p3 V-v © L. Sankar Helicopter Aerodynamics

10. Conservation of Energy Consider a particle that traverses from station 1 to station 4 We can apply Bernoulli equation between Stations 1 and 2, and between stations 3 and 4. Not between 2 and 3, since energy is being removed by body forces. Recall assumptions that the flow is steady, irrotational, inviscid. 1 V-v 2 3 4 V-v4 © L. Sankar Helicopter Aerodynamics

11. From an earlier slide, drag equals mass flow rate through the rotor disk times velocity deficit between stations 1 and 4 Thus, v = v4/2 © L. Sankar Helicopter Aerodynamics

12. Induced Velocities V The velocity deficit in the Far wake is twice the deficit Velocity at the rotor disk. To accommodate this excess Velocity, the stream tube has to expand. V-v V-2v © L. Sankar Helicopter Aerodynamics

13. Power Produced by the Rotor © L. Sankar Helicopter Aerodynamics

14. Summary • According to momentum theory, the velocity deficit in the far wake is twice the velocity deficit at the rotor disk. • Momentum theory gives an expression for velocity deficit at the rotor disk. • It also gives an expression for maximum power produced by a rotor of specified dimensions. • Actual power produced will be lower, because momentum theory neglected many sources of losses- viscous effects, tip losses, swirl, non-uniform flows, etc. © L. Sankar Helicopter Aerodynamics