Problem $2$: The following data were measured for a wind turbine $($HAWT$)$:

Speed of wind at $1$ atm and $27C=20\frac{m}{s}$;

$[$R$=287,$ Gas constant for Air & $1$atm$=1\mathrm{.}01325{10}^{5}Pa$; $($Pascal$)]$

Diameter of rotor $=80m$;

Speed of rotor $=40rpm$;

For maximum output:

Interference factor $a=\frac{1}{3}$& power coefficient $C_{p\text{m}\text{a}\text{x}}=0\mathrm{.}593$

Calculate the following for maximum output of the turbine.

$($i$)$ Tip$-$speed ratio $()$

$($ii$)$ Torque coefficient $(C_{T_{\text{m}\text{a}\text{x}}})$

$($iii$)$ Air density at given pressure and temperature $()$

$($iv$)$ Total power available in wind $(P_o)$

$($v$)$ Maximum conceivable rotor torque $(T_M)$

$($vi$)$ Torque produced at the shaft $(T_{\text{s}\text{h}\text{m}\text{a}\text{x}})$