Show that for the windmill problem (Example 2.11) the maximum value of power for a constant wind incoming velocity is achieved when \(v_{2}=v_{1} / 3\). Find the corresponding power and the efficiency of the windmill defined as the ratio of the power generated to the kinetic-energy flow rate in the incoming wind.

The maximum efficiency is also called the Betz number. Show that the value is \(59.3 \%\).

Example 2.11:

schematic diagram is shown in Fig. 2.14. Fluid approaches the wind turbine at a velocity of v1 and pressure of p_∞. It leaves at a velocity of v₂ as shown in the figure. We assume frictionless flow and hence the fluid downstream is at the same pressure p_∞ as that at the inlet.
The rate of working and the efficiency are to be calculated by doing a macroscopic analysis.

Wind V.P P.. P Swept area A F Pb Pa Stream line Wake V21 P Pa Pressure Profiles Figure 2.14 A schematic diagram of a macroscopic model for analysis of a windmill. Top: control volume; bottom: pressure profile.