Question

The hydraulic radius in channel ____________ with ____________ in wetted perimeter for a given cross-sectional area

• A. increases, increase
• B. decreases, decrease
• C. decreases, increase
• D. remains constant, increase

Hint:

Remember:
\[ R = \frac{A}{P} \] For constant Area (\(A\)):
- If Perimeter (\(P\)) \(\uparrow\), then Hydraulic Radius (\(R\)) \(\downarrow\).
Minimizing the wetted perimeter reduces frictional resistance, which maximizes the hydraulic radius and flow velocity.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The hydraulic radius (\(R\)), also known as the hydraulic mean depth, is a key parameter used in open channel flow equations (such as Manning's and Chezy's formulas) to estimate flow velocity and discharge capacity.
Key Formula or Approach:
The hydraulic radius (\(R\)) is defined mathematically as the ratio of the cross-sectional area of flow (\(A\)) to its wetted perimeter (\(P\)):
\[ R = \frac{A}{P} \]

Step 2: Detailed Explanation:
Let us analyze the algebraic relationship between these variables:
- In the equation \(R = \frac{A}{P}\), the hydraulic radius (\(R\)) is inversely proportional to the wetted perimeter (\(P\)) for a constant area (\(A\)).
- Therefore, if the cross-sectional area (\(A\)) is kept constant, any increase in the wetted perimeter (\(P\)) will result in a corresponding decrease in the hydraulic radius (\(R\)).
- Conversely, a smaller wetted perimeter results in a larger hydraulic radius.
This is why the most efficient channel section (which maximizes hydraulic radius and velocity) is the one that minimizes the wetted perimeter for a given flow area.
Thus, the hydraulic radius in a channel decreases with an increase in the wetted perimeter.

Step 3: Final Answer:
The hydraulic radius decreases with an increase in the wetted perimeter for a given cross-sectional area.