1. Betz Theory for A Blade Element
P M V Subbarao
Professor
Mechanical Engineering Department

3. Linear Momentum Theory for an Ideal Wind Turbine
For a frictionless wind turbine: U
Bernoulli equation is valid from far upstream to just in front of the rotor
DpWT : Utilized Pressure Deficit
Pressure just behind the rotor
Bernoulli equation is also valid from just behind the rotor to far downstream in the wake
Thrust Generated at the rotor Plane:

4. Axial Induction in A Wind Turbine Rotor
The axial induction factor (of rotor) a is defined as:

5. Dual Induction Theory for A Blade Element
P M V Subbarao
Professor
Mechanical Engineering Department

7. Rotation of Wake
Note that across the flow disc, the angular velocity of the air relative to the blade increases from  to +.
The axial component of the velocity remains constant.
The angular momentum imparted to the wake increases the kinetic energy in the wake but this energy is balanced by a loss of static pressure:
The local pressure decreases along the radius form hub to tip.
Creates a radial pressure gradient.
The radial pressure gradient balances the centrifugal force on the rotating fluid.

8. Local Thrust generation due to Rotating wake
The resulting thrust on an annular element, dT, is:
An angular induction factor, a’, is then defined as:

9. Results of Dual Induction Theory
Local Torque = Rate of change of angular momentum

10. Confluence of Angular & Linear Momentum Analysis
For stable operation of wind turbine, the differential thrust calculated using angular induction must be equal to axial induction.

11. Confluence of Dual Induction Theory withBlade Element Theory
P M V Subbarao
Professor
Mechanical Engineering Department
Recognition of Rotational Effects in Lift Machines……

12. BE Theory for a Generalized Rotor, Including Wake Rotation
The design procedure considers the nonlinear range of the lift coefficient versus angle of attack curve, i.e. stall.
The analysis starts with the four equations derived from dual induction and blade element theories.
This analysis assumes that the chord and twist distributions of the blade are known.
The angle of attack is not known, but additional relationships can be used to solve for the angle of attack and performance of the blade.
The forces and moments derived from dual induction theory and blade element theory must be equal.
Equating these, one can derive the flow conditions for a turbine design.

13. Results of Dual Induction Theory
Local Torque = Rate of change of angular momentum
Local Normal Force = Rate of change of axial momentum

14. Outcome of Blade Element Theory
Application of Blade Element Theory, to a wind turbine rotor generates two equations:
These equations define the normal force (thrust) and tangential force (torque) on the annular rotor section as a function of the flow angles at the blades and local airfoil shape.
These equations have to be further modified to determine ideal blade shapes for optimum performance and to determine rotor performance for any arbitrary blade shape.

15. BET in terms of Free Stream Velocity

16. Effect of Number of Blades on Local Flow Conditions
Define local solidity

17. Dual Induction Theory for Blade Element
Dual induction theory is also based on zero drag on blade.
For airfoils with low drag coefficients, this simplification introduces negligible errors.
When the torque equations from dual induction and blade element theories are equated, Cd must be made zero.
BET for Cd = 0

18. Confluence of Local Torque equations
By equating the torque equations from dual induction and blade element theories, one obtains:

19. Confluence of Local Normal Force equations
By equating the torque equations from dual induction and blade element theories, one obtains:

20. Useful Design Equations

21. Solution Methods
Two solution methods will be proposed using these equations to determine the flow conditions and forces at each blade section.
The first one uses the measured airfoil characteristics and the BEM equations to solve directly for Cl and a.
This method can be solved numerically, but it also lends itself to a graphical solution that clearly shows the flow conditions at the blade and the existence of multiple solutions.
The second solution is an iterative numerical approach that is most easily extended for flow conditions with large axial induction factors.

22. Method 1 – Solving for Cl and a
for a given blade geometry and operating conditions, there are two unknowns in above Equation Cl and  at each section.
The Cl vs.  curves for the chosen airfoil are available.
Find Cl and  that satisfy above equation.

23. Graphical Solution

24. Method 2 – Iterative Solution
1. Guess values of a and a’.
2. Calculate the angle of the relative wind from Equation (3.63).