1. Eulerization of Betz theory: Wind Turbines
P M V Subbarao
Professor
Mechanical Engineering Department
Development of True Design Model….

2. Euler Theory of Turbo-machines
A change in Whirl Velocity of fluid can only establish Power Exchange between fluid and rotor in a turbo-machine !
Torque exerted by flow on blades = shaft output torque
= Rate of change of Angular momentum of fluid = t

3. An Artistic View of the Vortex System Behind A Wind Turbine
How to develop an analytical model for this drawing?

4. Identify the Rotational Motion of Wake
Betz’s simple momentum theory is based on the description of the act of slowing down windflow and axi-symmetric deflection of the flow lines.
In reality, the rotating converter will additionally impart a rotating motion, a spin, to the wake.
To maintain the angular momentum, the spin in the wake must be opposite to the torque of the rotor.

5. The Spin Energy of Wake
The total energy of the wind is divided into energy due to spin and the energy due to axial motion.
Betz’s power coefficient is now applicable to wind power due to axial velocity.
An extended momentum theory, with consideration of the spin of the rotating wake is essential for accurate prediction of WT capacity.

6. The Trajectory of an Air Particle Passing Through the Rotor Disc
Wake rotation
The Trajectory of an Air Particle Passing Through the Rotor Disc
The air gains angular momentum and so in the wake of the rotor disc the air particles have a velocity component in a direction which is tangential to the rotation as well as an axial component

7. First Law analysis of WT with Spinning Wake
The flow is assumed to be frictionless and incompressible.

8. Tip Speed Ratio of A Wind Turbine
The real power coefficient is a strong function of the ratio between the energy components from the rotating motion and the translational motion of the wind stream.
This ratio is determined by the tangential velocity of the rotor blades in relation to the undisturbed axial airflow, the wind velocity.
This is called as the Tip Speed Ratio λ, commonly referenced to the tangential velocity of the rotor blade tip.
Tip speed ratio

9. Local Incident Wind Velocity Diagrams
Let rotor be a rotating disc of finite thickness.
Consider only an annular ring of the rotor disc which is of radius r and of radial width dr.
Define local average wind velocity for r, as u .
Thought experiment is in any case a necessary precondition for physical experiment.
Every experimenter and inventor must have the planned arrangement in his head before translating it into fact.
— Ernst Mach

10. Development of Sectional View
Rotor Plane
The local angle of attack α is given by the pitch of the aerofoil θ.
Vrel
Vrel
Vrel ua ut
The axial velocity and rotational velocity at the rotor plane denoted respectively by ua and ut

11. Angular Momentum Theory
The tangential velocity will not be the same for all radial positions and even the axial induced velocity is not the same.
To allow for variation of both components of induced velocity, consider only an annular ring of the rotor disc which is of radius r and of radial width r.
The increment of rotor torque acting on the annular ring will be responsible for imparting the tangential velocity component to the air.

12. Forces Due to Angular Momentum Theory
The axial force acting on the ring will be responsible for the reduction in axial velocity.
The whole disc comprises a multiplicity of annular rings and each ring is assumed to be independently imparting momentum only to the wind which actually passes through the ring.

13. The Concept of Dual Induction in WT
Axial Flow Induction Factor :
Tangential flow induction factor :a’
p0,V0

14. There is A Need for a localized Theory for Analysis of Differential WT Ring……
The True nature and degree of induction must vary in Radial Direction !!!!

15. Theory for Description of Dual & Local Behaviour of WT @ Steady State
Blade Element Momentum (BEM) Theory is a popular theory.
Also known as Strip Theory.
Blade element theory refers to an analysis of forces at various local sections of the blade.
The required local forces are estimated as functions of blade geometry.
This theory correctly relates blade shape to the rotor’s ability to extract power from the wind.

16. Blade Element Theory
The resulting forces on the blades of a wind turbine are expressed as a function of lift and drag coefficients and the angle of attack.
For this analysis, the blade is assumed to be divided into N sections (or elements).

17. Major Assumptions
There is no aerodynamic interaction between elements (thus, no radial flow).
This known as Radial Equilibrium of blade.
The forces on the blade elements are determined solely by the lift and drag characteristics of the airfoil shapes of the blade elements.
The lift and drag forces are perpendicular and parallel, respectively, to relative, wind.
The relative wind is the vector sum of the wind velocity at the rotor, V0(1-a), and the rotational wind velocity due to rotation of the blade element.

18. An Overall Flow Past an WT
The induced rotational wind speed due to blade rotation is identified of wind angular speed.
The resulting rotational velocity component of wind is the vector sum of the blade section velocity and the induced angular velocity at the blades.
The mean induced tangential velocity is

19. Expected Reaction thru to Local Dual Induction
dFN
dFT
dFN is the incremental force normal to the plane of rotation (Thrust)
dFT is the incremental force tangential to the circle swept by the rotor (driving force).

20. True Imagination of Wind Velocity Approaching Local Blade Element

21. Local Thrust generation due to Rotating wake
The resulting thrust (Normal to Rotor Area) on an annular element, dFN, is:
Angular induction factor, a’,
Local Thrust predicted by Betz theory

22. Valid 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.