Discussion
I : The flow rate in pipe connecting \(J\) and \(E\) is \(Q/21\).
II: The pressure difference between \(J\) and \(N\) is equal to the pressure difference between \(J\) and \(E\).
Which one of the following options is CORRECT?
Answer: B
The correct option is I is False and II is True.
Statement II: the pressures at N, E and S are equal and all three pipes start at J, so the pressure drop from J to each end is the same. II is true.
Statement I: with the same friction factor and cross-section, the Darcy-Weisbach loss in each pipe is \(\Delta p \propto L\,Q^{2}\). Equal pressure drops therefore require
\[ \begin{aligned} L\,q_{N}^{2} &= 10L\,q_{E}^{2} \Rightarrow q_{E} = \frac{q_{N}}{\sqrt{10}} \\ Q &= q_{N} + q_{S} + q_{E} = q_{E}(2\sqrt{10} + 1) \\ q_{E} &= \frac{Q}{7.32} \end{aligned} \]
So the flow to E is about Q/7.3, not Q/21. The value Q/21 is what a loss proportional to Q (laminar flow) would give, \(q_{E} = q_{N}/10\) and \(Q = 2.1q_{N}\), but a constant friction factor makes the loss vary with \(Q^{2}\). That rules out the options with I true.
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