Question

Given below are two statements:
Statement-I: For the most economical section of a rectangular channel giving maximum discharge would be when the flow depth is half the breadth of the channel.
Statement-II: For the most economical section of a rectangular channel giving maximum discharge would be when the hydraulic mean radius of the channel is half the depth of flow.
In light of the above statements, choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but Statement II is incorrect
• D. Statement I is incorrect but Statement II is correct

Hint:

For the most economical rectangular channel:
- Width is twice the depth (\( b = 2y \)).
- Hydraulic radius is half the depth (\( R = y/2 \)).
These two relationships are frequently tested in hydraulic exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
An open channel section is considered "most economical" or "most efficient" when it yields the maximum discharge for a given cross-sectional area, slope, and roughness coefficient.
According to Manning's equation, this condition is met when the wetted perimeter is minimized, which also maximizes the hydraulic radius.
Key Formula or Approach:
For a rectangular channel of breadth \( b \) and depth \( y \):
- Area, \( A = b \cdot y \)
- Wetted Perimeter, \( P = b + 2y \)
- Hydraulic Mean Radius, \( R = \frac{A}{P} \)
To find the most economical section, we minimize \( P \) with respect to \( y \) for a constant area \( A \).

Step 2: Detailed Explanation:
Let us evaluate both statements using our formulas:
- Statement I: Express \( b \) in terms of \( A \) and \( y \):
\[ b = \frac{A}{y} \]
Substitute this into the wetted perimeter equation:
\[ P = \frac{A}{y} + 2y \]
To find the minimum perimeter, differentiate \( P \) with respect to \( y \) and set the result to zero:
\[ \frac{dP}{dy} = -\frac{A}{y^2} + 2 = 0 \]
\[ \frac{A}{y^2} = 2 \implies A = 2y^2 \]
Since \( A = b \cdot y \):
\[ b \cdot y = 2y^2 \implies b = 2y \quad \text{or} \quad y = \frac{b}{2} \]
This shows that the flow depth (\( y \)) is half the channel breadth (\( b \)).
Therefore, Statement I is correct.
- Statement II: The hydraulic mean radius (\( R \)) is defined as \( A / P \).
Using our values for the most economical section (\( b = 2y \) and \( A = 2y^2 \)):
\[ P = b + 2y = 2y + 2y = 4y \]
Now, calculate \( R \):
\[ R = \frac{A}{P} = \frac{2y^2}{4y} = \frac{y}{2} \]
This shows that the hydraulic mean radius (\( R \)) is half the depth of flow (\( y \)).
Therefore, Statement II is correct.
Since both statements are correct, option (A) is the correct choice.

Step 3: Final Answer:
Both Statement I and Statement II are correct.