1. Simple Performance Prediction Methods
Momentum Theory
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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.
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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.
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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.
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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
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Conservation of Mass
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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
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8. Global Conservation of Momentum
Mass flow rate through the rotor disk times
velocity loss between stations 1 and 4
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9. Conservation of Momentum at the Rotor Disk
V-v
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) p2 p3
V-v
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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
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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
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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
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13. Power Produced by the Rotor
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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.
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