1. Off-design Performance of A Rotor
P M V Subbarao
Professor
Mechanical Engineering Department
Wind velocity is a StochasticC Variable.....

2. Design Level Power Coefficient
Note that even though the axial induction factors were determined assuming Cd= 0, the drag is included here in the power coefficient calculation.
Usually this equation is solved numerically

3. Tip Loss: Effect on Power Coefficient
It is well-known fact that the pressure on the suction side of a blade is lower than that on the pressure side.
Air tends to flow around the tip from the lower to upper surface.
This reduces the lift and hence power production near the tip.
This effect is most noticeable with fewer, wider blades.

4. Method to Account Tip Losses
A number of methods have been suggested for including the effect of the tip loss.
The most straightforward approach to use is one developed by Prandtl.
According to this method, a correction factor, F (0<F<1), must be introduced into the previously discussed equations.
This correction factor is a function of the number of blades, the angle of relative wind, and the position on the blade.
Based on Prandtl’s method:
where the angle resulting from the inverse cosine function is assumed to be in radians.

5. Useful Design Equations with tip Loss Correction

6. Actual Design Level Power Coefficient with tip Loss Correction

7. Eigen Behaviour of Rotor at off-design Conditions
Wind velocity is a stochastic variable.
Both the magnitude and direction are random variables.
A rotor is subjected to off-design conditions for more often than design condtions.
A number of complications can affect the rotor output, when a section of blade is exposed to flow conditions very different from the design conditions.
The main complications are :
Possibility of multiple solutions in the region of transition to stall and
solutions for highly loaded conditions with values of the axial induction factor approaching and exceeding 0.5.

8. Solution at Design Condition

9. Occurrence of Multiple Solutions
In the stall region, there may be multiple solutions for Cl.
Each of these solutions is possible.
The correct solution should be that which maintains the continuity of the angle of attack along the blade span

10. Wind Turbine Flow States
Measured wind turbine performance closely approximates the results of BEM theory at low values of the axial induction factor.
Momentum theory is no longer valid at axial induction factors greater than 0.5, because the wind velocity in the far wake would be negative.
As the axial induction factor increases above 0.5, the flow patterns through the wind turbine become much more complex than those predicted by momentum (or Betz) theory

11. Operating States for A rotor
A number of operating states for a rotor have been identified.
The operating states relevant to wind turbines are designated the windmill state and the turbulent wake state.
The windmill state is the normal wind turbine operating state.
The turbulent wake state occurs under operation in high winds.
The windmill state is characterized by the flow conditions described by momentum theory for axial induction factors less than about 0.5.
Measurements during operation of a rotor for a > 0.5 indicate that thrust coefficients increase up to about 2.0 at an axial induction factor of 1.0.
This state is identified as Turbulent Wake State.

12. Experimental Observations : Empirical Relation

13. Turbulent Wake State of A Rotor
This state is characterized by a large expansion of the slipstream, turbulence and recirculation behind the rotor.
Momentum theories can no longer describe the turbine behavior.
No consistent experimental data was available for Cp.
It was possible to measure CT .
An empirical relationships between CT and the axial induction factor are developed in practice.
Corrections to BEM theory is a new industrial research practice.

14. Measurement of Local Thrust Coefficient
From the equation for the normal force from blade element theory,
An extension of BEM method is essential to predict performance at off-design condtions

15. Rotor Modeling for the Turbulent Wake State
In the turbulent wake state, a solution can be found only by using the empirical relationship between the axial induction factor and the thrust coefficient in conjunction with blade element theory.
The empirical relationship developed by Glauert, including tip losses, is:
This equation is valid for a > 0.4 or, equivalently for CT > 0.96.
The local thrust coefficient, CT, can be defined for each blade element.