Dual Induction theory for Wind Turbines P M V Subbarao Professor Mechanical Engineering Department Confluence of Euler and Betz Theories….

Schematic drawing of the vortex system behind a wind turbine

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

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).

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.

Local Torque based on Dual Induction Theory Local Torque = Rate of change of angular momentum  Local tangential driving force, dFT

The Newton’s Action to be Generated by using A Suitable Airfoil dFL dFD dFT dFN dFL is the incremental lift force; dFD is the incremental drag force;

Definitions of Force Increments The incremental lift force The incremental Drag force If the rotor has B blades, the total normal force on the section at a distance, r, from the center is: The incremental force tangential to the circle swept by the rotor

The Differential Torque Contributed by a Blade Element The differential torque due to the tangential force operating at a distance, r, from the center is given by: Note that the effect of drag is to decrease torque and hence power, but to increase the Normal loading.

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

Effect Forces on Rotor Element in terms of Free Stream Velocity

Additional Design features of A Rotor Overall Rotor solidity: Local solidity:

Effect of Number of Finite Blades on Local Flow Conditions Define local solidity

Iterative Method of Solution 1. Select a Beam Element. 2. Guess values of a and a’. 3. Calculate the angle of the relative wind from Equation. 4. Calculate the angle of attack from  =p+ and then Cl and Cd. 5. Update a and a’ from Equations: 6. The process is then repeated until the newly calculated induction factors are within some acceptable tolerance of the previous ones.

Calculation of Power Coefficient Once a has been obtained from each section, the overall rotor power coefficient may be calculated from the following equation Actual local torque generated by rotor element: Actual local power generated by rotor element:

Universal form of Local Power Equation Universal local Design variable for a Wind Turbine Rotor: Local Blade Speed Ratio:

Universal form of Local Power Equation

Actual Power Developed by A Rotor Integrate local power equation from hub to tip: Actual Power Coefficient of a Rotor:

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

Any Global Dual Induction Theory ????

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

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.....

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

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.

Failure of Momentum Theories

Fluid Dynamics Model of Wind Turbine V0 is the speed of undisturbed flow, a axial induction factor on the rotor plane and b axial induction factor in the wake

Double Axial Induction parameters The axial induction factor (of rotor) a & b are defined as: