GATE 2024 CE (CE2) – Question 57
A 500 m long water distribution pipeline P with diameter 1.0 m, is used to convey 0.1 m$^3$/s of flow. A new pipeline Q, with the same length and flow rate, is to replace P. The friction factors for P and Q are 0.04 and 0.01, respectively. The diameter of the pipeline Q (in meters) is ______________ (rounded off to 2 decimal places).
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Correct answer: 0.75 to 0.77
Explanation
The head loss is $h_f = \frac{fLV^2}{2gD} = \frac{8fLQ^2}{\pi^2gD^5}$. For the same length, flow and head loss, $\frac{f}{D^5}$ must be the same: $\frac{0.04}{1^5} = \frac{0.01}{D_Q^5}$, so $D_Q = (0.25)^{1/5} = 0.758 \approx 0.76$ m.