1. Aerodynamic forces on the blade, COP, Optimum blade profiles
Rotor Design
Aerodynamic forces on the blade, COP, Optimum blade profiles

2. Horizontal Rotors
Horizontal rotors are the most widely used type, especially in large rating turbines.
Rotors are available with one, two or three blades. Rotors with higher number of blades are also available for small rating turbines (e.g. the Texas wheel).
The number of rotor blades is related to the rotor speed: high rpm rotors use a higher number of blades than low rpm rotors.
The blades can be either fixed or they can have an adjustable or variable pitch.
The horizontal rotor operates by converting the aerodynamic lift into rotation.
A wind generator with a three-blade horizontal rotor
The Texas wheel

3. Detail Configuration of the Wind Generator
The figure shows a typical configuration and arrangement of the components of the horizontal wind generator.
The rotation is transmitted from the rotor (2) to the electric generator (7) via the rotor shaft (5) and a step-up gear (6) allowing higher rpm at the generator side and lower at the rotor.
The yaw mechanism (13, 14) orients the rotor with respect to the wind direction (into the wind for increasing output or away from the wind for limiting the output).
The rotor blades (1) are attached to the rotor hub. In adjustable and variable pitch rotors the blades can rotating around their axis to vary the pitch angle with respect to the rotation plane (3). An additional mechanism utilizing a motor drive is used for pitch control (not shown in the figure).
An anemometer (9) and a wind vane (10) are used to measure the wind velocity and its direction respectively.

4. Aerodynamic Forces on a Blade Segment
The blade of the rotor is designed to use efficiently the aerodynamic lift forces from the wind to produce rotation and work.
Consider the rotor in the figure. The rotor radius is R. The rotational velocity is wR.
Consider a small segment on one of the blades with length L and average radius r.
The cross section of the segment is shown in the figure. The segment cord length is tB.
The angle from the plane of rotation to the cord is the pitch angle, q.

5. Wind Velocities Relative to the Segment
The incoming wind velocity in the axial direction is 𝑣 1 .
Near the blade, the wind slows down to the axial velocity 𝑣 2 .
The blade segment rotates around the rotor axis with 𝜔 𝑅 .
The tangential velocity of the segment is 𝜔 𝑅 ∙𝑟.
The resultant wind velocity at the segment leading edge is:
𝑣 𝑅 = 𝑣 𝜔 𝑅 ∙𝑟 2
Due to relative motion, the wind blows at the segment with an equal and opposite direction velocity (parallel to the plane of rotation, shown in figure).
From the figure, tan 𝛿 = 𝑣 2 𝜔 𝑅 ∙𝑟 .
The angle of attack is, 𝛼=𝛿−𝜃

6. Lift and Drag Forces
The lift force is perpendicular to the resultant velocity and its magnitude is:
𝐹 𝐴 = 𝜌 2 𝑡 𝐵 𝑣 𝑅 2 𝐶 𝐴 𝛼 𝐿
The drag force is in the direction of the resultant velocity and its magnitude is:
𝐹 𝑊 = 𝜌 2 𝑡 𝐵 𝑣 𝑅 2 𝐶 𝑊 𝛼 𝐿

7. Force Synthesis: The Motive Force
Note that the drag force opposes the motion reducing the magnitude of the motive force.
The motive force, 𝐹 𝑡 , is responsible for sustaining the circular motion of the segment around the rotor axis. This force produces the useful work of the blade.
The direction of the motive force is the tangent of the segment motion, e.g. the motive force lies on the rotation plane, as shown in the figure.
From the figure, we derive 𝐹 𝑡 by combining 𝐹 𝐴 and 𝐹 𝑊 in the tangential direction:
𝐹 𝑡 = 𝐹 𝐴 sin 𝛿 − 𝐹 𝑊 cos 𝛿

8. Force Synthesis: The Axial Force
The axial force, 𝐹 𝑎𝑥 , is normal to the rotation plane.
The axial force pushes the segment into the rotor and it does not contribute to the motion of the segment (e.g. it does not produce work).
From the figure, we derive 𝐹 𝑎𝑥 by combining 𝐹 𝐴 and 𝐹 𝑊 in the axial direction:
𝐹 𝑎𝑥 = 𝐹 𝐴 cos 𝛿 + 𝐹 𝑊 sin 𝛿

9. Solution of the Segment Equation
The force equations developed previously contain the wind velocity at the segment, 𝑣 2 , as an unknown variable.
The wind velocity far ahead of the segment, 𝑣 1 , along with the pitch angle, 𝜃 and the rotational speed, 𝜔 𝑅 , are parameters in the force equations.
To solve for the unknown velocity, 𝑣 2 , we apply the principle of action/reaction: The forces 𝐹 𝐴 and 𝐹 𝑊 are applied to the blade segment by the mass of the wind blowing into the annulus defined by the segment trajectory (figure).
The blade segment, in turn, applies equal but opposite forces to the air blowing into the annulus defined by the segment.

10. Solution of the Segment Equation
We will consider only the axial forces as the tangential velocity of the wind in the annulus has very small value compared to 𝜔 𝑅 ∙𝑟.
From the conservation of momentum principle: 𝐹 𝑎𝑥,𝑤𝑖𝑛𝑑 =𝑑𝑚∙ 𝑣 1 − 𝑣 2 .
But, 𝑑𝑚= 𝜌∙𝐴 𝑎𝑛𝑛𝑢𝑙𝑢𝑠 ∙ 𝑣 2 =𝜌 2𝜋 𝑟 𝐿∙ 𝑣 2
And, 𝐹 𝑎𝑥𝑖𝑎𝑙,𝑤𝑖𝑛𝑑 =𝜌 2𝜋 𝑟 𝐿∙ 𝑣 2 ( 𝑣 1 − 𝑣 2 ).
The RHS of the equation is set equal to 𝑧𝐹 𝑎𝑥 expressed earlier, where z is the number of rotor blades. The new equation is solved numerically for 𝑣 2 .
𝑧(𝐹 𝐴 𝑣 2 cos 𝛿+ 𝐹 𝑊 𝑣 2 sin 𝛿 )=𝜌 2𝜋 𝑟 𝐿∙ 𝑣 2 ( 𝑣 1 − 𝑣 2 )

11. Torque and Power Produced by the Segment
The torque around the rotor axis generated by the segment is due to the motive force on the segment, 𝐹 𝑡 :
𝑀 𝑠𝑒𝑔 = 𝐹 𝑡 ∙𝑟
The direction of the torque is in the direction of rotation. Therefore, the torque produces work at the rate (power) of 𝐹 𝑡 ∙𝑟∙ 𝜔 𝑅 .

12. Blade Solution, Torque and Power
The solution for the entire blade is obtained by dividing the blade into small segments of length 𝐿.
Each small segment is solved, as discussed in the previous.
The blade torque is derived by summing the torques over all segments:
𝑀 𝑏𝑙𝑎𝑑𝑒 ( 𝑣 1 , 𝜔 𝑅 ,𝜃)= ∑ 𝑘 𝐹 𝑡,𝑘 ∙ 𝑟 𝑘
Note that the torque of the blade is a function of the wind velocity ahead of the rotor, the rotation speed and the blade pitch.

13. Rotor Torque and Power
If the rotor has 𝑧 blades, the overall torque is
𝑀( 𝑣 1 , 𝜔 𝑅 , 𝜃)=𝑧 𝑀 𝑏𝑙𝑎𝑑𝑒
The rotor power is
𝑃 𝑅 ( 𝑣 1 , 𝜔 𝑅 ,𝜃)=𝑧 𝑀 𝑏𝑙𝑎𝑑𝑒 ∙ 𝜔 𝑅
Note that from the rotating disk theory, given 𝑣 1 , 𝜔 𝑅 , and 𝜃, the rotor power is limited by the maximum power generated by an equivalent rotating the disk:
𝑧 𝑀 𝑏𝑙𝑎𝑑𝑒 ∙ 𝜔 𝑅 ≤0.59 𝜌 2 𝐴 𝑅 ∙ 𝑣 1 3 .
Therefore, the number of blades does not determine the size of the turbine.

14. Coefficient of Performance
The COP of the rotor is derived as:
𝐶𝑂𝑃( 𝑣 1 , 𝜔 𝑅 ,𝜃)= 𝑃 𝑅 𝜌 2 𝐴 𝑅 𝑣 1 3
Without showing the details of the derivation, it is possible to express COP as function of the pitch angle and the ratio, 𝜆= 𝜔 𝑅 𝑅 𝑣 1 :
𝐶𝑂𝑃= 𝐶 𝑃 (𝜆,𝜃)
The new variable, 𝜆, is the ratio between the tip speed and the wind velocity ahead of the rotor. It is referred to as the tip ratio.
Therefore, COP does not depend on the wind velocity and rotational speed separately, but on their ratio.

15. Effect of the Pitch Angle
For a given wind velocity, the pitch angle determines the angle of attack.
The sections of the blade near the rotor axis have a smaller tangential velocity than the sections farther up near the blade tip (figure).
Therefore, in order to maintain approximately the same angle of attack for all sections of the blade, the sections near the rotor are designed to operate with a higher pitch angle than the sections near the tip.
The tip sections are turned into the plane of rotation in order to avoid stalling.
The cord length also varies.
The blade sections near the rotor face a smaller resultant wind velocity than the sections farther up.
Therefore, the blade sections near the rotor need a longer cord than the sections farther up in order to produce approximately the same torque.
The distinct shape of the rotor blade is wide in the bottom and narrow towards the top; the blade shape twists inward from the bottom to the top.

16. Optimum Blade Profile
The cord length and the pitch of the blade profile can be optimized for maximizing COP at a certain tip ratio.
At maximum COP:
Substituting the above into the balance equations, the optimum cord length and pitch at r are given by:
Where the values of l and a are those corresponding to the maximum COP .

17. Other Rotor Designs Darius Type Turbines Cup-type rotors
Other types used in smaller rating turbines include the vertical Darius type seen in the figure for a large and a small unit. The operation principle of this type is the same as the horizontal type: the blades utilize the lifting force to produce motion.
The cup type shown in the figure uses the drag force, instead of the lift force, to produce motion.
Darius Type Turbines
Cup-type rotors