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 =ฮK. 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
ฮmฮt=ฯAvav=ฯA12(v1+v2)
while the available wind power due to this is
P=ฮKฮt=12ฮm(v21โv22)ฮt=14ฯA(v1+v2)(v21โv22)
and the undisturbed wind power (the power of the wind if it did not pass through the turbine) is
Po=Kฮt=12ฯAv31
When graphing P/Po, the maximum power output is found at v2/v1 = 0.33 (Fig. 3).
Figure 3.The plot agrees with Betzโs 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.