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

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:

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

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

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.

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:

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

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.

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

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.

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

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.

BET in terms of Free Stream Velocity

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

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

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

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

Useful Design Equations

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.

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.

Graphical Solution

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