Problem $2(20$ points$)$

An air fan has a bladed rotor of $D_2$ outside diameter and $D_1$ inside diameter as illustrated in Fig. $2.$ The height of each rotor blade is $h$ from blade inlet to outlet. The flowrate $(Q)$ is steady and the absolute velocity of the air at the blade inlet $(V_1)$ is radial. The air velocity at the blade outlet relative to the blade is $W_2$ and $30$ from the tangential direction. The rotor rotates at a constant angular speed $.$

$($a$)(5$ points$)$ Draw the velocity vector triangle for $U_2,V_2,W_2$ and their radial $(r)$ and tangential $()$ direction components; $U_2$ is the rotor blade velocity at the outlet and $V_2$ is the absolute velocity of the air at the blade outlet.

$($b$)(15$ points$)$ Determine the shaft torque $(T_{\text{s}\text{h}\text{a}\text{f}\text{t}})$ required to run the fan. Make sure to draw your control volume and indicate whether it is fixed, moving, deformable, or stationary.