Problem $3$: The following figure shows a stream$-$tube$/$actuator disc model of a wind turbine. Assume that the actuator disc has a radius of $5$ m $,$ and a freestream wind speed of $\backslash ($ V$\_\{\backslash$infty$\}=11\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ The wind speed velocity at the wake is $\backslash ($ V$\_\{$w$\}=1\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

$-$ Determine the velocity at the actuator disc.

$-$ Determine the mass flux of air passing through the actuator disk.

$-$ Using the conservation of energy equation, calculate the power produced by this wind turbine.

$-$ Determine the areas, $\backslash ($ A$\_\{\backslash$infty$\}\backslash )$ and $\backslash ($ A$\_\{$w$\}\backslash ).$

$-$ Calculate the pressure difference across the actuator disk.

$-$ Calculate the thrust force developed by the wind turbine in $2$ methods

$-$ Determine the axial induction factor.

$-$ Using the axial induction factor calculate the power produced by this wind turbine and the thrust developed.

$-$ Calculate the input and output power

$-$ What is the maximum power that can be extracted by the idealized wind turbine by this wing turbine assuming a freestream wind speed of $\backslash ($ V$\_\{\backslash$infty$\}=11\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

$-$ In this case what is the velocity at the actuator disk and in the wake. the areas, $\backslash ($ A$\_\{\backslash$infty$\}\backslash )$ and $\backslash ($ A$\_\{$w$\}\backslash ).$

$-$ Calculate the input and output power.

$-$ What can you deduce?