Question $2(15$ marks$)$

a$-$What is the Betz limit$,$ and why is it significant in wind turbine efficiency? b$-$Define the tip$-$speed ratio $($TSR$)$ and explain its importance in determining rotor efficiency.

c$-$A $40-m$ diameter, three$-$bladed wind turbine produces $600$ kW at a wind speed of $14\frac{m}{s}$ with a TSR $($Tip Speed Ratio$)$ of $4\mathrm{.}0.$ At this condition, the generator spins at $1800$ rpm $,$ and the combined efficiency of the gearbox and generator is $66\%.$

$($Assume air density $=1\mathrm{.}225k\frac{g}{m^3}).$

i$)$ Calculate the rotational speed of the rotor.

ii$)$ Find the tip speed of the rotor.

$\frac{1}{2}$

iii$)$ Find the gear ratio needed to match the rotor speed to the generator speed.

iv$)$ Calculate the efficiency of the wind turbine.

v$)$ Find the rotor efficiency