Question 14.8: Design of a Vane-Axial Flow Fan for a Wind Tunnel A vane-axi...

For given flow conditions and stator blade shape at a given radius, we are to design the rotor blade. Specifically, we are to calculate the leading and trailing edge angles of the rotor blade and sketch its shape. We are also to decide how many rotor blades to construct.

Assumptions 1 The air is nearly incompressible. 2 The flow area between the hub and tip is constant. 3 Two-dimensional blade row analysis is appropriate.
Analysis First we analyze flow through the stator from an absolute reference frame, using the two-dimensional approximation of a cascade (blade row) of stator blades (Fig. 14–64). Flow enters axially (horizontally) and is turned 60.0° downward. Since the axial component of velocity must remain constant to conserve mass, the magnitude of the velocity leaving the trailing edge of the stator, \vec{V}_{ st } , is calculated as

V_{ st }=\frac{V_{ in }}{\cos \beta_{ st }}=\frac{47.1 m / s }{\cos \left(60.0^{\circ}\right)}=94.2 m / s                     (1)

The direction of \vec{V}_{ st } is assumed to be that of the stator trailing edge. In other words, we assume that the flow turns nicely through the blade row and exits parallel to the trailing edge of the blade, as shown in Fig. 14–64.
We convert \vec{V}_{ st } to the relative reference frame moving with the rotor blades. At a radius of 0.40 m, the tangential velocity of the rotor blades is

u_{\theta}=\omega r=(1750 rot / min )\left(\frac{2 \pi rad }{\operatorname{rot}}\right)\left(\frac{1 min }{60 s }\right)(0.40 m )=73.30 m / s                 (2)

Since the rotor blade row moves upward in Fig. 14–63, we add a downward velocity with magnitude given by Eq. 2 to translate \vec{V}_{ st }  into the rotating reference frame sketched in Fig. 14–65. The angle of the leading edge of the rotor, \beta_{ rl } , is calculated by using trigonometry,

\begin{aligned}\beta_{ rl } &=\arctan \frac{\omega r+V_{ in } \tan \beta_{ st }}{V_{ in }} \\&=\arctan \frac{(73.30 m / s )+(47.1 m / s ) \tan \left(60.0^{\circ}\right)}{47.1 m / s }=73.09^{\circ}\end{aligned}                      (3)

The air must now be turned by the rotor blade row in such a way that it leaves the trailing edge of the rotor blade at a zero angle (axially, no swirl) from an absolute reference frame. This determines the rotor’s trailing edge angle, \beta_{ rt } . Specifically, when we add an upward velocity of magnitude ωr (Eq. 2) to the relative velocity exiting the trailing edge of the rotor, \vec{V}_{ rt , \text { relative }} , we convert back to the absolute reference frame, and obtain \vec{V}_{ rt } , the velocity leaving the rotor trailing edge. It is this velocity, \vec{V}_{ rt } , that must be axial (horizontal). Furthermore, to conserve mass, \vec{V}_{ rt } must equal \vec{V}_{ in}  since we are assuming incompressible flow. Working backward we construct \vec{V}_{ rt , \text { relative }}  in Fig. 14–66. Trigonometry reveals that

\beta_{ rt }=\arctan \frac{\omega r}{V_{\text {in }}}=\arctan \frac{73.30 m / s }{47.1 m / s }=57.28^{\circ}                     (4)

We conclude that the rotor blade at this radius has a leading edge angle of about 73.1° (Eq. 3) and a trailing edge angle of about 57.3° (Eq. 4). A sketch of the rotor blade at this radius is provided in Fig. 14–65; the total curvature is small, being less than 16° from leading to trailing edge. Finally, to avoid interaction of the stator blade wakes with the rotor blade leading edges, we choose the number of rotor blades such that it has no common denominator with the number of stator blades. Since there are 16 stator blades, we pick a number like 13, 15, or 17 rotor blades. Choosing 14 would not be appropriate since it shares a common denominator of 2 with the number 16. Choosing 12 would be worse since it shares both 2 and 4 as common denominators.
Discussion We can repeat the calculation for all radii from hub to tip, completing the design of the entire rotor. There would be twist, as discussed previously.