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Question 7.4: A pump takes in water from a level 5 m below its centre line......

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.

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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.

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