Question 18.24: A jet of water having a velocity of 30 m/s impinges on a cur......

Given data:

Velocity of jet            V_1=30 \mathrm{~m} / \mathrm{s}

Velocity of vane           u =10 m/s

Angle made by jet with the direction of motion of vane at entry  \alpha_1=30^{\circ}

Angle made by leaving jet with the direction of motion of vane = 135°

The inlet and outlet velocity triangles are shown in Fig. 18.19.

Here          u_1=u_2=u=10 \mathrm{~m} / \mathrm{s}

V_{r 1}=V_{r 2}                   \text { ( } \because \text { friction neglected) }

From the inlet velocity triangle, we have

V_{w 1}=V_1 \cos \alpha_1=30 \times \cos 30^{\circ}=25.98 \mathrm{~m} / \mathrm{s}

V_{f 1}=V_1 \sin \alpha_1=30 \times \sin 30^{\circ}=15 \mathrm{~m} / \mathrm{s}

\tan \beta_1=\frac{V_{f 1}}{V_{w 1}-u_1}

or          \tan \beta_1=\frac{15}{25.98-10}=0.9387

or          \beta_1=\tan ^{-1}(0.9387)=43.19^{\circ}

Again from the inlet velocity triangle, we get

\sin \beta_1=\frac{V_{f 1}}{V_{r 1}}

or                            V_{r 1}=\frac{V_{f 1}}{\sin \beta_1}=\frac{15}{\sin 43.19^{\circ}}=21.92 \mathrm{~m} / \mathrm{s}

From the outlet velocity triangle, we obtain

\frac{V_{r 2}}{\sin 135^{\circ}}=\frac{u_2}{\sin \left(\alpha_2-\beta_2\right)}

or                            \frac{21.92}{\sin 135^{\circ}}=\frac{10}{\sin \left(45^{\circ}-\beta_2\right)}

or                            \sin \left(45^{\circ}-\beta_2\right)=\frac{10 \times \sin 135^{\circ}}{21.92}=0.3226=\sin 18.82^{\circ}

or                          45^{\circ}-\beta_2=18.82^{\circ}

or                          \beta_2=45^{\circ}-18.82^{\circ}=26.18^{\circ}

Again from the outlet velocity triangle, we get

V_{w 2}=V_{r 2} \cos \beta_2-u_2

=21.92 \cos 26.18^{\circ}-10=9.67 \mathrm{~m} / \mathrm{s}

The work done per second per unit weight of water striking the vane is given by Eq. (18.27) as

={\frac{(V_{{w1}}\pm V_{{w2}})\times u}{g}}      (18.27)

=\frac{\left(V_{w 1}-V_{w 2}\right) \times u}{g}

=\frac{25.98-(-9.67) \times 10}{9.81}

=36.34 \mathrm{~N}-\mathrm{m} / \mathrm{N}