Wind Turbine Power and Torque Equation and Calculator

Related Resources: calculators

Wind Turbine Power and Torque Equation and Calculator

Power Transmission and Technology Menu
Applications and Design

Wind Turbine Power and Torque Equation and Calculator

Theoretical power available in a wind stream is given by Eq. 3 on the webpage Wind Turbine Power. However, a turbine cannot extract this power completely from the wind. When the wind stream passes the turbine, a part of its kinetic energy is transferred to the rotor and the air leaving the turbine carries the rest away. Actual power produced by a rotor would thus be decided by the efficiency with which this energy transfer from wind to the rotor takes place. This efficiency is usually termed as the power coefficient (C_p). Thus, the power coefficient of the rotor can be defined as the ratio of actual power developed by the rotor to the theoretical power available in the wind.

Hence,

Eq. 1
C_p = 2 P_T / ( ρ_a A_T V³)

where P_T is the power developed by the turbine. The power coefficient of a turbine depends on many factors such as the profile of the rotor blades, blade arrangement and setting etc. A designer would try to fix these parameters at its optimum level so as to attain maximum C_p at a wide range of wind velocities.

The thrust force experienced by the rotor (F) can be expressed as

Eq. 2
F = 0.5 ρ_a A_T V²

Hence we can represent the rotor torque (T) as

Eq. 3
T = 0.5 ρ_a A_T V² R

where R is the radius of the rotor. This is the maximum theoretical torque and in practice the rotor shaft can develop only a fraction of this maximum limit. The ratio between the actual torque developed by the rotor and the theoretical torque is termed as the torque coefficient (C_T). Thus, the torque coefficient is given by

Eq. 4
C_T = 2 T_T / ( ρ_a A_T V² R )

Eq. 4.5
T_T = C_T ρ_a A_T V² R / 2

where T_T is the actual torque developed by the rotor.

The power developed by a rotor at a certain wind speed greatly depends on the relative velocity between the rotor tip and the wind. For example, consider a situation in which the rotor is rotating at a very low speed and the wind is approaching the rotor with a very high velocity. Under this condition, as the blades are moving slow, a portion of the air stream approaching the rotor may pass through it without interacting with the blades and thus without energy transfer. Similarly if the rotor is rotating fast and the wind velocity is low, the wind stream may be deflected from the turbine and the energy may be lost due to turbulence and vortex shedding. In both the above cases, the interaction between the rotor and the wind stream is not efficient and thus would result in poor power coefficient.

The ratio between the velocity of the rotor tip and the wind velocity is termed as the tip speed ratio (λ). Thus,

Eq. 5
λ = R Ω / V = 2 π N R / V

where Ω is the angular velocity and N is the rotational speed of the rotor. The power coefficient and torque coefficient of a rotor vary with the tip speed ratio. There is an optimum λ for a given rotor at which the energy transfer is most efficient and thus the power coefficient is the maximum (C_{p max}). Wind velocity = V.

Now, let us consider the relationship between the power coefficient and the tip speed ratio.

Eq. 6
C_p = 2 P_T / ( ρ_a A_T V³ )

Eq. 6.6
C_p = 2 T_T Ω / ( ρ_a A_T V³ )

Dividing Eq. 6 by Eq. 4 we get

Eq. 7
C_p / C_T = R Ω / V = λ

Thus, the tip speed ratio is given by the ratio between the power coefficient and torque coefficient of the rotor.

Misc. equations

Area of the rotor is

Eq. 8
A_T = π / 4 · D²

Angular velocity or rotor

Eq. 9
Ω = 2 π V / 60

Source:

Sathyajith Mathew
Wind Energy Fundamentals, Resource Analysis and Economics