Question

A trapezoidal canal lined with cement concrete (\(n = 0.01\)) is designed to carry a discharge of 20 m\(^3\)/s at a bed slope 1 in 400. The bed width is twice the depth of flow and side slope of the canal section is 2 (1 vertical : 2 horizontal). The corresponding depth of flow will be ............. (in m, rounded off to two decimal places).

Hint:

In Manning’s equation, always compute area \(A\) and wetted perimeter \(P\) first. Remember side slope geometry is important: side length = \(\sqrt{z^2 + 1}\,d\) where \(z\) = horizontal/vertical slope.

Answer 1

Correct Answer: 1

Step 1: Recall Manning’s equation.
\[ Q = \frac{1}{n} A R^{2/3} S^{1/2} \] where, \(Q\) = discharge (m\(^3\)/s), \(n\) = Manning’s roughness coefficient, \(A\) = flow area (m\(^2\)), \(R = \frac{A}{P}\) = hydraulic radius (m), \(P\) = wetted perimeter (m), \(S\) = bed slope.
Step 2: Define section geometry.
Let depth of flow = \(d\). - Bed width = \(2d\). - Side slope = 2H:1V → horizontal projection = \(2d\). Thus, \[ \text{Top width} = 2d + 2(2d) = 6d \] \[ \text{Area} \, (A) = \frac{1}{2} ( \text{Top width} + \text{Bottom width}) \times d \] \[ A = \frac{1}{2} (6d + 2d) d = 4d^2 \]
Step 3: Wetted perimeter.
Two side lengths = \(\sqrt{(2d)^2 + d^2} = \sqrt{5} d\). \[ P = 2d + 2(\sqrt{5} d) = 2d (1 + \sqrt{5}) \]
Step 4: Hydraulic radius.
\[ R = \frac{A}{P} = \frac{4d^2}{2d(1+\sqrt{5})} = \frac{2d}{1+\sqrt{5}} \]
Step 5: Substitute values into Manning’s equation.
Given: \(Q = 20 \, \text{m}^3/\text{s}, n = 0.01, S = 1/400 = 0.0025\). \[ 20 = \frac{1}{0.01} (4d^2) \left(\frac{2d}{1+\sqrt{5}}\right)^{2/3} (0.0025)^{1/2} \] \[ 20 = 100 \times 4d^2 \left(\frac{2d}{3.236}\right)^{2/3} (0.05) \] \[ 20 = 20 \times 4d^2 \left(0.618 d\right)^{2/3} \] \[ 20 = 80 d^2 (0.618^{2/3}) d^{2/3} \] \[ 20 = 80 (0.731) d^{(2 + 2/3)} \] \[ 20 = 58.5 \, d^{8/3} \]
Step 6: Solve for depth.
\[ d^{8/3} = \frac{20}{58.5} = 0.342 \] Raise both sides to power \(3/8\): \[ d = (0.342)^{3/8} \] \[ d = 1.62 \, \text{m} \] Final Answer: \[ \boxed{1.62 \, \text{m}} \]