GATE 2026 CE (CE2) – Question 54
A centrifugal pump is delivering water from an underground tank to an overhead reservoir against a static head of 35 m through a 2 km long, 250 mm diameter pipe. The head-discharge characteristic of the pump is given by
$$H = 140 - 9000\,Q^2$$
where $H$ is the head (in m) generated by the pump and $Q$ is the discharge (in m$^3$/s) of the pump.
Neglecting all minor losses, the head (in m) generated by the pump is ______ (*rounded off to the nearest integer*).
Use: Darcy-Weisbach friction factor $f = 0.04$
Acceleration due to gravity = 9.81 m/s$^2$
$\pi = 3.14$
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Correct answer: 80
Explanation
The pipe area is $A = \frac{3.14 \times 0.25^2}{4} = 0.04906$ m$^2$. The friction loss is $h_f = \frac{fLV^2}{2gD} = \frac{fLQ^2}{2gDA^2} = \frac{0.04 \times 2000}{2 \times 9.81 \times 0.25 \times 0.04906^2}Q^2 = 6776\,Q^2$. The head the system needs is $35 + 6776\,Q^2$. At the operating point it equals the pump head: $140 - 9000Q^2 = 35 + 6776Q^2$, so $Q^2 = 0.006656$. The head generated is $H = 140 - 9000 \times 0.006656 = 80.1$ m, which is about 80 m.