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Double Dual Induction Factors for A Wind Turbine

Double Dual Induction Factors for A Wind Turbine. P M V Subbarao Professor Mechanical Engineering Department. Most Flexible Mathematical Model. The relation Between Kinetic & Kinematics. V 0 (1-a). Comprehensive Geometrical Detailing of Blade (HWT).

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Double Dual Induction Factors for A Wind Turbine

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  1. Double Dual Induction Factors for A Wind Turbine P M V Subbarao Professor Mechanical Engineering Department Most Flexible Mathematical Model.....

  2. The relation Between Kinetic & Kinematics V0(1-a)

  3. Comprehensive Geometrical Detailing of Blade (HWT) Wind velocity at just upstream of Blades

  4. Local Forces on A Blade Element • dFL is the incremental lift force; • dFD is the incremental drag force; • dFN is the incremental force normal to the plane of rotation • dFT is the incremental force tangential to the circle swept by the rotor. • Tangential force is the force creating useful torque.

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

  6. Estimation of Torque using Angular Momentum Theory 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

  7. The true Equation for Coefficient of Power The absorbed poweris often non-dimensionalized with respect to Pavailas a power coefficient CP: In this expression it is considered that the vortex in the wake behaves as a free vortex.

  8. Recapitulation of 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.

  9. Double 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, Cdmust be made zero. • BET for Cd = 0

  10. The Ultimate Confluence Confluence of Local Torque equations Confluence of Local Normal Force (Thrust) equations

  11. The relation Between Axial Induction parameters • This equation represents the most general relationship between a and b. • The assumption of free vortex related the tangential induction factors with similar relationship To solve these equations, Newton’s method can be used, in which Third order equations are to be solved.

  12. Newtons Method • In this case, a good approximation to begin the iterative process corresponds to b=2a & b’=2a’. The use of Newton’s method consists in always getting the lowest real value to calculate b R and b’  R.

  13. The True Behaviour of Flow Past A Wind Turbine • The hypothesis of wake behaves like a free vortex, demonstrates that the axial induction factor in the rotor plane has a non-linear relationship with the axial inducing factor in the wake. • This is more valid for low tip-speed ratios, R. • Especially for values R < 2. • This relationship is: R

  14. Velocity diagram for a Blade Element p dFt Urel dFn

  15. The UnReal Power Coefficient due to Free Vortex Wake Model Based on Free Vortex theory • The free vortex hypothesis causes infinite velocities on the wake near the axis of the wind turbine. • Cp can take values greater than 1 when R is very small. • For small values of a and R > 2, free vortex theory physically consistent values for Cp. • In order to obtain a physically consistent solution, the use of a Rankine vortex to represent the wake is advocated in recent design theories. R R R R R R b a

  16. The UnReal Power Coefficient due to Free Vortex Wake Model • This solves the problem of the infinite velocities near the turbine axis. • This approach is implemented by the introduction of a parameter 1/b’, in the expression of the power coefficient.

  17. Most Pragmatic Model for Coefficient of Power R R R R

  18. Most Pragmatic Model for Coefficient of Power R R R R

  19. Accuracy of General Model

  20. Aerodynamic optimization of wind turbine blade The aerodynamic optimization is obtained by maximizing the power coefficient.

  21. Optimum Performance of A Wind Turbine with NACA 0012

  22. Capacity of An optimally Designed Micro Wind Turbine : D = 3m & N=60rpm

  23. Chord distributions

  24. Twist Angle Distributions

  25. Closing Remarks • General BEM theory must be valid for both HAWT & VAWT. • However, specific implementations are valid fo HAWT only. • More specific design formulae are required for VAWT.

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