Find the number of stages and diameter of each impeller of a similar multistage pump to lift 5 m3 of water per second to a height of 200 m when rotating at 1500 rpm.

Given:

$D_1 = 0.3\hspace{0.05cm}m,\hspace{1cm} Q_2 = 5\hspace{0.05cm}m^3/s, \hspace{1cm}N_1 = 2000\hspace{0.05cm}rpm\\ \textit{Total height}(H) = 200\hspace{0.05cm}m,\hspace{1cm}Q_1 = 3\hspace{0.05cm}m^3/s\\ N_2 = 1500\hspace{0.05cm}rpm,\hspace{1cm}H_{m1} = 30\hspace{0.05cm}m$

To Find:

$\eta_{mano} = 0.75\\ \textit{No. of stages} = D_2 = \hspace{0.05cm}?$

Solution:

Specific speed should be same,

$(\frac{N\sqrt{Q}}{H_{m1}^\frac{3}{4}}) = (\frac{N\sqrt{Q}}{H_{m2}^\frac{3}{4}})$

$(\frac{2000\hspace{0.05cm}\times\hspace{0.05cm}\frac{1}{\sqrt{3}}}{30^\frac{3}{4}}) = (\frac{1500\hspace{0.05cm}\times\hspace{0.05cm}\frac{2}{\sqrt{5}}}{H_{m2}^\frac{3}{4}})$

$H_{m2}^\frac{3}{4} = \frac{1500\hspace{0.05cm}\times\hspace{0.05cm}\sqrt{5}\hspace{0.05cm}\times\hspace{0.05cm}30^\frac{3}{4}}{2000\hspace{0.05cm}\times\hspace{0.05cm}\sqrt{3}} = \frac{1500}{2000}\hspace{0.05cm}\times\hspace{0.05cm}\sqrt{\frac{5}{3}}\hspace{0.05cm}\times\hspace{0.05cm}12.818 …