Discussion
A parabolic shaped grass-waterway is to be designed to carry a flow of 2.85 m3 s-1 down the slope of 3%. The permissible velocity of water in the waterway is 1.78 m s-1. If the freeboard depth is excluded, the most appropriate top width in m and depth in m, respectively are
(Take Manning’s roughness coefficient = 0.04)
Answer: B
The most appropriate dimensions are a top width of 6 m and a depth of 0.4 m.
The flow area needed at the permissible velocity is \(A = Q/V = 2.85/1.78 = 1.60\ \text{m}^{2}\). Manning’s equation fixes the hydraulic radius:
\[ \begin{aligned} R^{2/3} &= \frac{Vn}{\sqrt{S}} = \frac{1.78 \times 0.04}{\sqrt{0.03}} = 0.411 \\ R &= 0.264\ \text{m} \end{aligned} \]
For a parabolic section with top width \(T\) and depth \(d\), \(A = \tfrac{2}{3}Td\) and \(R \approx T^{2}d/(1.5T^{2} + 4d^{2})\). With \(T = 6\) and \(d = 0.4\):
\[ \begin{aligned} A &= \tfrac{2}{3}(6)(0.4) = 1.60\ \text{m}^{2} \\ R &= \frac{36 \times 0.4}{54 + 0.64} = 0.264\ \text{m} \end{aligned} \]
Both match. The pair 4 and 0.6 gives the right area but \(R = 0.38\) m, too deep a section, so the velocity would exceed the permissible value. The pairs 7 and 0.5 and 5.5 and 0.6 give areas of 2.33 and 2.2 m2, larger than needed.
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