If the customer studies Betz Law correctly then he will quickly find out that the Law is confided to horizontal axis turbines and not vertical (see attached diagram). However Betz Law is what all engineers use since 1919 as a base line for all calculations for finding out what power can be produced from a wind turbine.
Here I will try to explain to you simply Betz Law
The Laws are very simple and basic, but there are many complicated calculations and equations involved in understanding and constructing wind turbine generators however we need to use the following basic information:
1) The power output of a wind generator is proportional to the area swept by the rotor (horizontal axis) - i.e. double the swept area and the power output will also double.
2) The power output of a wind generator is proportional to the cube of the wind speed - i.e. double the wind speed and the power output will increase by a factor of eight (2 x 2 x 2)
Wind is made up of moving air molecules which have mass. Any moving object with mass carries kinetic energy (All moving things have kinetic energy. It is energy possessed by an object due to its motion or movement) in an amount which is given by the equation:
Kinetic Energy = 0.5 x Mass x Velocity2
where the mass is measured in kg, the velocity (speed) in m/s, and the energy is given in joules.
Air has a known density (around 1.23 kg/m3 at sea level), so the volume of air hitting our wind turbine (which sweeps a known area) each second is given by the following equation:
Mass/sec (kg/s) = Velocity (m/s) x Area (m2) x Density (kg/m3)
And therefore, the power (i.e. energy per second) in the wind hitting a wind turbine with a certain swept area is given by simply inserting the mass per second calculation into the standard kinetic energy equation given above resulting in the following equation:
Power = 0.5 x Swept Area x Air Density x Velocity3
where Power is given in Watts (i.e. joules/second), the Swept area in square metres, the Air density in kilograms per cubic metre, and the Velocity in metres per second.
Betz law basically talks about how a wind turbine cannot extract more than 59.3 % of
Kinetic energy from the wind. Under Ideal conditions or theoretically the maximum energy that can be extracted from the wind is called the Power coefficient which is a ratio between the amount of energy that can be extracted by a Wind turbine to the total energy in the Wind.
Power Coefficient(Beth's Coefficient ) =
Kinetic Energy that is extracted by a Wind turbine/Total energy in the Wind
Horizontal axis wind turbines (HAWT) theoretically have higher power efficiencies than
vertical axis wind turbines (VAWT) however wind direction is not important for a VAWT and so no time (and power) is wasted chasing the wind. In turbulent conditions with rapid changes in wind direction more electricity will be generated by a VAWT despite its lower efficiency.
Here is a basic calculation that I downloaded from the internet and even here is the power calculation much higher than the customers results.
Rotor Diameter: metres (0 to 150m).
Mean Wind Speed: metres per second (0 to 12m/s).
Cut-in Speed: metres per second (0 to 7m/s).
Cut-out Speed: metres per second (10 to 25m/s).
Turbine Efficiency: percent (10 to 59%).
Shape Parameter: (1-3, so use 2 if unsure)
Predicted Turbine Output
1,433 kWh
2,3m² is because of the twisted form and the curvature. The unit physically is 1m wide and 3m high but the surface area is much larger, actually 1,30cm wide x 2,00cm high = 2,6m², so to take into consideration the helix form have we reduced the area to 2,3m²
Betz Law works on linear wind where as our unit works wind turbulent wind from all directions.
The customers calculations (mathematically seen) are correct but have no relation to what actually happens. There is also approx. 365.25 days a year and not 364.
