Wind Energy and Power
The wind power that can be captured by a wind turbine depends on the air density, the swept area by the blades, and the wind speed.
Kinetic Power of the Wind
The kinetic power contained in the wind passing through a given area is:
Where:
: Wind power (W). : Air density ( ), typically at sea level. : Swept area by the turbine blades ( ). : Wind speed ( ).
Swept Area by the Blades
The swept area by the blades of a wind turbine (in the case of a horizontal axis turbine) is the area of a circle:
Where:
: Swept area ( ). : Radius of the blades ( ).
Power Captured by the Turbine
Not all the wind power can be converted into electrical energy. The captured power by a wind turbine is limited by the power coefficient
Power Captured by the Wind Turbine
The actual power that the turbine can generate is given by:
Where:
: Power captured by the turbine (W). : Power coefficient (dimensionless), which depends on the design of the turbine. The theoretical maximum value is (Betz limit).
Betz Limit
The Betz limit states that no wind turbine can capture more than 59.3% of the kinetic energy of the wind.
Betz Limit
Coefficients and Efficiency of the Turbine
The efficiency of a wind turbine is related to various coefficients that describe the turbine’s ability to convert wind energy into electricity.
Power Coefficient (
The power coefficient is the ratio of the power generated by the turbine to the total power available in the wind.
Overall Efficiency of the Wind System
The overall efficiency of the wind system takes into account losses in other components, such as the generator, cables, and inverter.
Where:
: Efficiency of the generator. : Efficiency in the transmission through the cables. : Efficiency of the inverter from DC to AC.
Characteristic Speeds of the Turbine
Wind turbines have certain speed thresholds of the wind at which they start generating power and stop functioning for safety reasons.
Cut-in Speed (
The cut-in speed is the minimum wind speed at which the turbine begins to generate electrical power.
Rated Speed (
The rated speed is the wind speed at which the turbine reaches its rated power, i.e., the maximum power it can generate.
Cut-out Speed (
The cut-out speed is the maximum wind speed at which the turbine stops operating to avoid structural damage.
Forces and Torque in the Turbine
Drag Force (
The drag force is the resistance that the wind exerts on the turbine blades. It is calculated with the following formula:
Where:
: Drag coefficient (depends on the shape of the blades). : Air density ( ). : Projected area ( ). : Wind speed ( ).
Lift Force (
The lift force is the force that acts perpendicular to the wind on the blades, generating rotation in the turbine.
Where:
: Lift coefficient (depends on the aerodynamic profile of the blades).
Torque on the Turbine Shaft
The torque on the turbine shaft is the result of the lift forces that generate the rotational movement of the blades.
Where:
: Torque ( ). : Lift force (N). : Length of the blades or turbine radius (m).
Rotational Speed and Tip-Speed Ratio
Angular Speed of the Turbine
The angular speed (
Where:
: Angular speed (rad/s). : Wind speed ( ). : Radius of the blades ( ).
Tip-Speed Ratio (TSR)
The tip-speed ratio (TSR) is the ratio of the tangential speed of the tips of the blades to the wind speed.
Where:
: Tip-speed ratio. : Angular speed of the blades (rad/s). : Radius of the blades (m). : Wind speed ( ).
The optimal TSR varies depending on the design of the turbine, and for most modern turbines, it is in the range of
Electrical Power and Sizing of the Wind System
Electrical Power Generated
The electrical power generated by a wind turbine is limited by the efficiency of the system and losses in energy conversion.
Where:
: Electrical power generated (W). : Power captured by the turbine (W). : Overall efficiency of the system (including generator, cables, and inverter).
Energy Generated Over a Period of Time
The energy generated by a turbine over a period of time
Where:
: Energy generated (Wh or kWh). : Electrical power generated (W). : Operating time (hours).
Number of Turbines Needed
The number of turbines needed for a system depends on the energy demand and the power generated by each turbine.
Number of Turbines
The number of turbines is calculated by dividing the total required power by the rated power of a single turbine:
Where:
: Number of turbines needed. : Total power required (W). : Rated power of a turbine (W).