An explanation of Betz’ 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
$v_{av} = \dfrac{1}{2}(v_1 + v_2)$
The mass streaming through the turbine, found as above, is
\begin{eqnarray}
\dfrac {\Delta m}{\Delta t} &=& \rho A v_{av} \nonumber \\
&=& \rho A \dfrac{1}{2}(v_1 + v_2) \nonumber
\end{eqnarray}
while the available wind power due to this is
\begin{eqnarray}
P &=& \dfrac{\Delta K}{\Delta t} \nonumber \\
&=& \dfrac{\dfrac{1}{2} \Delta m(v_1^2 – v_2^2)}{\Delta t} \nonumber \\
&=& \dfrac{1}{4} \rho A(v_1 + v_2)(v_1^2 – v_2^2) \nonumber
\end{eqnarray}
and the undisturbed wind power (the power of the wind if it did not pass through the turbine) is
\begin{eqnarray}
P_o &=& \dfrac{K}{\Delta t} \nonumber \\
&=& \dfrac{1}{2} \rho Av_1^3 \nonumber
\end{eqnarray}
When graphing P/Po, the maximum power output is found at v2/v1 = 0.33 (Fig. 3).
Figure 3.The plot agrees with Betz’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.
