In open channel hydraulics, the specific energy \(E\) at a cross-section with depth \(y\), velocity \(V\), and discharge per unit width \(q\) is given by \(E = y + \dfrac{V^2}{2g}\). The critical depth \(y_c\) for a rectangular channel, at which specific energy is minimum for a given discharge \(q\), is:

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Architecture Engineering QUESTION #2367

Question 1

\(y_c = \sqrt[3]{\dfrac{q}{g}}\)
\(y_c = \sqrt[3]{\dfrac{q^2}{g}}\)✔️
\(y_c = \sqrt{\dfrac{q^2}{g}}\)
\(y_c = \left(\dfrac{q^2}{g}\right)^{2/3}\)

Correct Answer Explanation

Critical depth for a rectangular channel: \(y_c = \left(\dfrac{q^2}{g}\right)^{1/3} = \sqrt[3]{\dfrac{q^2}{g}}\). At critical depth, the Froude number \(Fr = \dfrac{V}{\sqrt{gy}} = 1\). For \(y < y_c\): supercritical flow (\(Fr > 1\)); for \(y > y_c\): subcritical flow (\(Fr < 1\)). The minimum specific energy \(E_{min} = \dfrac{3}{2} y_c\).

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