Question

A trapezoidal channel carries \(5\ \text{m}^3\text{s}^{-1}\) of water under uniform flow condition. The channel has a bottom width of 2 m, side slope of 2:1 (horizontal: vertical), and a bed slope of 1%. If the Manning roughness coefficient is 0.03, the conveyance of the channel, in \(\text{m}^3\text{s}^{-1}\), is

• A. 2
• B. 5
• C. 15
• D. 50

Hint:

Recall that conveyance links discharge and bed slope directly through Manning's equation.

Answer 1

The Correct Option is D

Step 1: Recall the definition of channel conveyance.
Manning's equation for uniform flow is \( Q = \dfrac{1}{n} A R^{2/3} S^{1/2} \), and conveyance is defined as \( K = \dfrac{1}{n} A R^{2/3} \), so \( Q = K S^{1/2} \).

Step 2: Rearrange for conveyance.
\( K = \dfrac{Q}{\sqrt{S}} \)

Step 3: Substitute the given discharge and bed slope.
Bed slope \( S = 1\% = 0.01 \), and \( Q = 5\ \text{m}^3\text{s}^{-1} \).
\( K = \dfrac{5}{\sqrt{0.01}} = \dfrac{5}{0.1} = 50\ \text{m}^3\text{s}^{-1} \)

Final Answer:
The channel carries this flow with a conveyance of 50 m3/s; the bottom width, side slope and n are not even needed once Q and S are known. \[ \boxed{K = 50\ \text{m}^3\text{s}^{-1}} \]