Question

For critical flow conditions in rectangular channels, which of the following is the correct relationship between critical depth \( y_c \) and specific energy \( E \)?

• A. \( y_c = \frac{2}{3}E \)
• B. \( y_c = \frac{4}{5}E \)
• C. \( y_c = \frac{3}{4}E \)
• D. \( y_c = E \)

Hint:

At critical flow in a rectangular channel, the specific energy is minimum and is related to the critical depth by \( y_c = \frac{2{3E \). This helps in flow analysis and hydraulic design.

Answer 1

The Correct Option is A

Step 1: Understand the definitions.
Critical depth \( y_c \) is the depth of flow where specific energy \( E \) is at a minimum for a given discharge.
Step 2: Apply specific energy equation.
For a rectangular channel: \[ E = y + \frac{v^2}{2g} \] At critical flow, the specific energy is minimized, and using the condition for critical flow: \[ v_c = \sqrt{g y_c}, \quad \text{and} \quad Q = A v = b y_c \sqrt{g y_c} \] Solving, the specific energy at critical flow becomes: \[ E_c = y_c + \frac{v_c^2}{2g} = y_c + \frac{g y_c}{2g} = y_c + \frac{y_c}{2} = \frac{3}{2}y_c \] \[ \Rightarrow y_c = \frac{2}{3}E \]