Question

Water is flowing at the rate of 15 m³/s through a rectangular channel of 5 m width. If the acceleration due to gravity (g) is 9.81 m/s², the critical velocity for the flow is ______ m/s (rounded off to three decimal places).

Hint:

Find discharge per unit width q = Q/B, then use critical depth y_c = (q^2/g)^(1/3) and critical velocity V_c = sqrt(g y_c).

Answer 1

Correct Answer: 3.087

Step 1: Note the given data. Discharge \(Q = 15\ \text{m}^3/\text{s}\), channel width \(B = 5\ \text{m}\), and \(g = 9.81\ \text{m/s}^2\).

Step 2: Find the discharge per unit width. For a rectangular channel the discharge intensity is \(q = \dfrac{Q}{B} = \dfrac{15}{5} = 3\ \text{m}^2/\text{s}\).

Step 3: Find the critical depth. For a rectangular channel, the critical depth is given by \(y_c = \left(\dfrac{q^2}{g}\right)^{1/3}\). Substituting, \(y_c = \left(\dfrac{9}{9.81}\right)^{1/3} = (0.9174)^{1/3} \approx 0.972\ \text{m}\).

Step 4: Find the critical velocity. At critical flow the Froude number equals 1, so \(V_c = \sqrt{g\,y_c}\). Substituting, \(V_c = \sqrt{9.81 \times 0.972} = \sqrt{9.532} \approx 3.087\ \text{m/s}\).

Step 5: State the result. The critical velocity for this flow is approximately \(3.087\ \text{m/s}\), which lies within the accepted range of \(3.025\) to \(3.125\ \text{m/s}\).