A pump takes in water from a level 5 m below its centre line and delivers it at a height of 30 m above the centre line, the rate of flow being 3 m³/hr. The diameter of the pipe line allthrough is 50 mm (ID). The fittings introduce losses equal to 10 m length of pipe in addition to the actual length of 45 m of pipe used. Determine the head to be developed by the pump.
The head to be developed will equal the static head, friction head and the dynamic head.
Static head = 30 + 5 = 35 m, Friction head = f L u_{m}^{2}/2 \mathrm{g} D,
L = 45 + 10 = 55 m, D = 0.05 m,
u_{m} = (3/3600) 4/π × 0.05² = 0.4244 m/s.
Assuming the temperature as 20 °C,
v = 1.006 × 10^{– 6} m² / s.
∴ Re = u_{m} D/v = 0.05 × 0.4244/1.006 × 10^{– 6} = 21093
∴ The flow is turbulent
Assuming smooth pipe, f = 0.316/Re^{0.25} = 0.316/21093^{0.25} = 0.02622
(check f = 0.0032 + 0.221/Re^{0.237} = 0.0241), using the value 0.02622,
Frictional loss of head = 0.02622 × 55 × 0.4244²/2 × 9.81 × 0.05 = 0.265 m
Dynamic head = u_{m}^{2}/2 \mathrm{g} = \frac{0.4244 ^{2}}{2 \times 9.81} = 0.0092 m
∴ Total head = 35 + 0.265 + 0.0092 = 35.2562 m.
Head to be developed by the pump = 35.2562 m head of water.