Question

Critical depth in a channel is expressed by: (where $Q =$ discharge, $A =$ Area of flow, $T =$ top width of flow and $g =$ acceleration due to gravity)

• A. $\left(\dfrac{Q A^2}{g T^3}\right) = 1$
• B. $\left(\dfrac{Q T^2}{g A^3}\right) = 1$
• C. $\left(\dfrac{Q^2 T}{g A^3}\right) = 1$
• D. $\left(\dfrac{Q^2 A^2}{g T^3}\right) = 1$

Hint:

At critical depth in open channel flow, $Fr = 1$; use $D = A/T$ to derive expressions involving discharge.

Answer 1

The Correct Option is C

Step 1: Recall condition for critical flow.
Critical flow occurs when the Froude number $Fr = 1$: \[ Fr = \frac{V}{\sqrt{g D}} = 1, \] where $V =$ velocity, $D =$ hydraulic depth $= \dfrac{A}{T}$.

Step 2: Express velocity.
\[ V = \frac{Q}{A}. \]

Step 3: Substitute in Froude number.
\[ \frac{Q}{A} = \sqrt{g \cdot \frac{A}{T}}. \] \[ \frac{Q^2}{A^2} = \frac{g A}{T}. \] \[ \frac{Q^2 T}{g A^3} = 1. \]

Step 4: Conclusion.
Thus, the correct expression is $\left(\dfrac{Q^2 T}{g A^3}\right) = 1$.