1. Eulerization of Betz Theory : Wind Turbines
P M V Subbarao
Professor
Mechanical Engineering Department
Tangential Induction is Essential for A Turbo-machine….

2. Euler Theory Euler Theory:
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. 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 rotor wake.
To maintain the angular momentum, the spin in the wake must be opposite to the torque of the rotor.

4. The Spin Energy of Wake
The energy contained in this spin reduces the energy contained in 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.

5. 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

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

7. 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 translatorial 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

8. 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 r.
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

9. 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

10. 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.
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 act independently in imparting momentum only to the wind which actually passes through the ring.

11. Schematic drawing of the vortex system behind a wind turbine

12. Growth of Tangential Velocity Across the Disc Thickness
Axial Flow Induction Factor:a
p0,V0
Tangential flow induction factor:a’

13. Application of Angular momentum theory
Euler theory for WT: The local torque on the ring will be equal to the rate of change of angular momentum of the air passing through the ring.
Thus,
Local Torque = Rate of change of angular momentum
= mass flow rate  change of tangential
velocity  radius
The driving torque on the rotor shaft is also  and so the increment of rotor shaft power output is

14. Total Mechanical Power Absorbed
The local power absorption predicted by tangential momentum theory
The total power extracted from the wind by slowing it down is therefore determined by the rate of change of axial momentum also.....

15. Betz Momentum Theory for Axial Flow
The local power absorption predicted by Axial momentum theory
The drop in specific kinetic power due to axial flow is equal to generation of specific kinetic power due to Tangential flow

16. The Pragmatism in the Analysis of WT
r is the tangential velocity of the local spinning annular ring.
Therefore r =  r/V0 is called the local speed ratio.
At the edge of the disc r=R.
R=  R/V0 is known as the tip speed ratio.
Tip Speed Ratio is a true Pragmatic Design Parameter.
Selection of airfoil geometry decides an optimum value of Tip Speed Ratio.
If the airfoil leads to lower rotor diameter, this will create a high speed Wind Turbine.
Else a low Speed Wind turbine.
The Geometry of Organs of A Rotor Decides the Optimum Speed and Overall Efficiency of Rotor.