GATE 2024 CE (CE2) – Question 41
A round-bottom triangular lined canal is to be laid at a slope of 1 in 1500, to carry a discharge of 25 m$^3$/s. The side slopes of the canal cross-section are to be kept at 1.25H : 1V. If Manning's roughness coefficient is 0.013, the flow depth (in meters) will be in the range of
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Correct answer: (A) 2.39 to 2.42
Explanation
In the standard round-bottom triangular section the arc has the radius $y$ (equal to the depth), with its centre on the water surface, and the sides are tangent to it. The sides make the angle $\theta = \tan^{-1}\frac{1}{1.25} = 0.675$ rad with the horizontal. The area is $A = (\theta + 1.25)y^2 = 1.925y^2$ and the wetted perimeter is $P = (2\theta + 2.5)y = 3.849y$, so $R = \frac{A}{P} = 0.5y$. By Manning's equation $Q = \frac{1}{n}AR^{2/3}S^{1/2} = \frac{1}{0.013} \times 1.925y^2 \times (0.5y)^{2/3} \times \sqrt{\frac{1}{1500}} = 25$. This gives $y^{8/3} = 6.86$ and $y = 2.40$ m.