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Simple Performance Prediction Methods Momentum Theory Background Developed for marine propellers by Rankine (1865), Froude (1885). Used in propellers by Betz (1920)

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simple performance prediction methods

Simple Performance Prediction Methods

Momentum Theory

© L. Sankar Helicopter Aerodynamics

background
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

assumptions
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

assumptions continued
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

control volume
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

conservation of mass
Conservation of Mass

© L. Sankar Helicopter Aerodynamics

conservation of mass through the rotor disk
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

global conservation of momentum
Global Conservation of Momentum

Mass flow rate through the rotor disk times

velocity loss between stations 1 and 4

© L. Sankar Helicopter Aerodynamics

conservation of momentum at the rotor disk
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

conservation of energy
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

slide11

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

induced velocities
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

power produced by the rotor
Power Produced by the Rotor

© L. Sankar Helicopter Aerodynamics

summary
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