Simple Performance Prediction Methods

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Simple Performance Prediction Methods - PowerPoint PPT Presentation

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

Momentum Theory

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.

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.

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.

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

Conservation of Mass

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

Global Conservation of Momentum

Mass flow rate through the rotor disk times

velocity loss between stations 1 and 4

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

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

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

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