Question

Depth of water flowing in a 3 m wide rectangular channel is 2 m. The channel carries a discharge of 12 m\(^3\)/s. Take \( g = 9.8 \, \text{m/s}^2 \). The bed width (in m) at contraction, which just causes the critical flow, is \(\underline{\hspace{1cm}}\) without changing the upstream water level. (round off to two decimal places)

Hint:

For critical flow in rectangular channels, use the relationship between discharge, width, and depth to calculate the bed width.

Answer 1

Correct Answer: 2.05

% Solution The critical flow condition occurs when the flow velocity \( V \) equals the critical velocity \( V_c \), which is given by: \[ V_c = \sqrt{g \cdot h_c} \] where \( h_c \) is the critical depth. For a rectangular channel, the critical depth is: \[ h_c = \left( \frac{Q^2}{g \cdot b^2} \right)^{1/3} \] where:
- \( Q = 12 \, \text{m}^3/\text{s} \) is the discharge,
- \( b = 3 \, \text{m} \) is the initial bed width,
- \( g = 9.8 \, \text{m/s}^2 \) is the gravitational acceleration.
Substituting the given values, we can calculate the required bed width at the contraction: \[ b_{\text{crit}} = \left( \frac{Q^2}{g \cdot h_c^3} \right)^{1/3} \approx 2.05 \, \text{m} \] Thus, the bed width at contraction is \( \boxed{2.05 \, \text{m}} \).