Question 16.14: The diameters of the impeller of a centrifugal pump at inlet......
The diameters of the impeller of a centrifugal pump at inlet and at outlet are 40 cm and 80 cm respectively. The velocity of flow at the outlet is 3 m/s, and the vanes are set back at angle 45° at the outlet. Determine the minimum starting speed of the pump if the manometric efficiency is 80%.
Given: D = 40 cm = 0.4 m, D_{1} = 80 cm = 0.8 m, V_{f_{1}} = 3 m/s, Φ = 45°, and \eta_{mano} = 80%
If N is minimum starting speed, then by Eq. (16.19), we have
N={\frac{120\eta_{\mathrm{mano}}V_{\mathrm{w_{1}}}D_{1}}{\pi(D_{1}^{2}-D^{2})}} (16.19)
or N={\frac{120\times0.8V_{\mathrm{w_1}}\times0.8}{\pi[(0.8)^{2}-(0.4)^{2}]}}
or N={\frac{76.8{V}_{w_{1}}}{1.50796}}=50.92958{V}_{w_{1}} (i)
To find the value of V_{w_{1}}, consider the velocity diagram at outlet in Figure 16.15, we get
\tan\phi={\frac{V_{f_{1}}}{u_{1}-V_{w_{1}}}}\qquad{\mathrm{or}}\qquad\tan45^{\circ}={\frac{3}{u_{1}-V_{\mathrm{w_{1}}}}}or u_{1}-V_{w_{1}}=3 (ii)
Again, u_{1}={\frac{\pi D_{1}N}{60}}={\frac{\pi\times0.8N}{60}}=0.041888N
Substituting u_{1} in Eq. (ii), we have
0.041888N – V_{w_{1}} = 3
or V_{w_{1}} = 0.041888N — 3
Substituting V_{w_{1}} in Eq. (i), we have
N = 50.92958 V_{w_{1}} = 50.92958 (0.041888N – 3)
Solving for N = 150.655 rpm
