PRIVATE: Wind Turbines โ€“ Betz Law Explained

An explanation of Betzโ€™ Law


How did Betz come up with this?

The work done on the turbine = change in kinetic energy of the wind: W =ฮ”K. The speed v2 behind the turbine is slower than the speed in front of the turbine v1, and the average speed at the location of the turbine is

vav=12(v1+v2)

The mass streaming through the turbine, found as above, is

ฮ”mฮ”t=ฯAvav=ฯA12(v1+v2)

while the available wind power due to this is

P=ฮ”Kฮ”t=12ฮ”m(v21โ€“v22)ฮ”t=14ฯA(v1+v2)(v21โ€“v22)

and the undisturbed wind power (the power of the wind if it did not pass through the turbine) is

Po=Kฮ”t=12ฯAv31

When graphing P/Po, the maximum power output is found at v2/v1 = 0.33 (Fig. 3).

Figure 3.The plot agrees with Betzโ€™s conclusions that the maximum power output (of 59.3%) occurs when v2 is 1/3 of v1. To view the spreadsheet used to produce this graph, see Betz_Law_Spreadsheet_Data.xls.