1. Calculating Wind Turbine EfficiencyBy
Willem Scholten
Learning Access Institute

2. Wind Turbine Efficiency
There are two calculations every team needs to perform on their turbine:
Overall efficiency
Solidity of their blades

3. Wind Turbine Efficiency
Electrical Power transformed by wind turbine
Efficiency =
Power available in the wind
A German physicist, Albert Betz, calculated that no wind turbine could convert more than 59.3% of the kinetic energy of the wind into mechanical energy turning a rotor. This is known as the Betz Limit, and it is the theoretical maximum efficiency for any wind turbine. In typical operating wind speeds, most modern wind turbines are 25–45% efficient

4. Wind Turbine Efficiency
To calculate the power in the wind, we need to know the volume of air pressing against the turbine each second. Knowing the mass of the air pressing against the turbine allows us to calculate the Kinetic Energy of the moving air.
The Volume of air pressing against the turbine each second, V, is the volume of a cylinder given by: V = LA
L = the length of the cylinder, which is determined by the velocity of the wind multiplied by the time of interest, in our case, 1 second.
L = v x 1 sec (v = velocity of air)

5. Wind Turbine Efficiency
The area of the base of the cylinder, A, is equal to pi times the radius, r, squared.
A = pi r2 (this is the same as the swept area)
The mass, m, of the air pressing against the turbine is the density of the air times the volume of the cylinder
m = pi r2 x v x 1sec x p (p = density of air, which hat sea level and at 150C = 1.225kg/m3 )
The kinetic energy of the moving air is therefore:
KE = 1/2 mv2
Substituting for m, we get:
KE = 1/2 pi r2 x v3 x 1 second x p

6. Wind Turbine Efficiency
The power available from the wind is the kinetic energy passing the blades each second, so if: KE = 1/2 pi r2 x v3 x 1 second x p, then P, power available every second is:
Pwind= KE/second = 1/2 pi r2 x v3 x p
To now get the efficiency we need to also know the electrical power generated by the turbine, which we get from measuring the Voltage and Current over a load attached to the turbine.
Pelec = V x I
Efficiency =
Pelec = V x I
x 100%
Pwind=
1/2 pi r2 x v3 x p

7. Wind Turbine Efficiency
To calculate the Efficiency you need to therefore know the following data points collected for each experimental run:
r = radius of the blades (in meters) (measured from the center of the axel to outer corner of blade.)
v = Wind Speed (in m/sec) (measured using Logger Pro and anemometer)
Pelec = electrical power generated (in W) (measured using Logger Pro and energy test set with load)
Example:
Assume: v = 4.326m/s r = 0.322m Pelec = 36.41mW ( W)
Efficiency =
Pelec = V x I =
X 100 %
x 100%
Pwind=
1/2 pi r2 x v3 x p
1/ x x x 1.225
Efficiency =
X 100 % = 0.22%
16.15

8. Wind Turbine Efficiency
The ratio of the total area for all blades to the total swept area is called the solidity of the turbine.
Solidity is calculated using the equation:
Solidity = na/A
n = the number of blades
a = the area of a single blade
A = the swept area of the turbine
Turbines with a high solidity (> 0.8) rotate at a lower speed, and provide high torque
Turbines with a low solidity (0.1 ~ 0.5) rotate at a high speed, and provide less torque
Blade pitch (angle towards the wind) affect the torque, speed, and the amount of drag experienced. Blades with shallow pitch (10-300) can spin faster providing lower torque, blades with greater pitch (30-600) increased torque, but route slower.

9. Wind Turbine Efficiency
Calculating the solidity of the turbine.
A (swept Area) = pi r2
To calculate the the area of one blade, the simplest way is to take a picture of your rotor, and using Logger Pro Photo Analysis to get the area of your blade.
Example:
r = 0.195m (radius of swept area)
n = ( number of blades)
a = m ( area of one blade)
A (swept Area) = pi r2 = 3.14 x (0.195)2 = 0.119m2
n x a
3 x
Solidity = = = =
0.123 A
0.119
0.119

10. Wind Turbine Efficiency
In conclusion:
High solidity equals high torque. Higher torque is required to rotate a high ratio gear train, in order to turn a generator faster. The faster a generator turns, the higher the power output of that generator will be.
You therefore want to construct a turbine with a high torque rating (high solidity) capturing the maximum energy of the wind and delivering the highest power output. (most efficient)
Remember: if you gear up, go from slow moving feeder gear to higher spinning fed gear (large gear driving a smaller gear) your torque (delivered power to the second gear) will be reduced by the gear ratio, while the speed increase will be it’s inverse.
Remember: this is an energy conversion game, you can not make more energy then you input (wind power) along the way, you can only increase performance by making the most efficient conversion from wind power to electrical power. (remember Betz Limit the overall efficiency of your system will not go beyond 50%! so be smart where and how you convert…..)