Question

The dimensions of Manning’s coefficient is

• A. $\text{L}^{-1/3} \text{T}$
• B. $\text{L}^{-2/3} \text{T}$
• C. $\text{L}^{1/3} \text{T}$
• D. $\text{L} \text{T}^{1/3}$

Hint:

Manning's roughness coefficient is not dimensionless.
Its dimensional formula, \( \text{L}^{-1/3} \text{T} \), is derived directly from the relationship between length-based hydraulic radius and time-based velocity.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Dimensional analysis is used to determine the base physical dimensions (Length \( \text{L} \), Mass \( \text{M} \), and Time \( \text{T} \)) of empirical coefficients used in fluid mechanics equations.
Manning's formula is an empirical equation that relates open channel flow velocity to channel geometry and roughness.
Key Formula or Approach:
Manning's equation for mean flow velocity (\( V \)) is:
\[ V = \frac{1}{n} \cdot R^{2/3} \cdot S^{1/2} \]
where:
- \( V \) is the mean flow velocity.
- \( n \) is Manning's roughness coefficient.
- \( R \) is the hydraulic radius (cross-sectional area divided by wetted perimeter).
- \( S \) is the energy slope of the channel.
Rearranging this equation to solve for Manning's coefficient (\( n \)):
\[ n = \frac{R^{2/3} \cdot S^{1/2}}{V} \]

Step 2: Detailed Explanation:
Let us determine the dimensions of each variable in the equation:
1. Velocity (\( V \)): Velocity is defined as distance per unit time.
\[ [V] = \text{L} \cdot \text{T}^{-1} \]
2. Hydraulic Radius (\( R \)): This is a length dimension, representing area (\( \text{L}^2 \)) divided by perimeter (\( \text{L} \)).
\[ [R] = \text{L} \]
Therefore, the term \( R^{2/3} \) has the dimensions:
\[ [R^{2/3}] = \text{L}^{2/3} \]
3. Channel Slope (\( S \)): This is a ratio of vertical drop to horizontal distance (\( \text{L/L} \)), making it a dimensionless parameter.
\[ [S] = 1 \]
4. Substitute these dimensions into our expression for \( n \):
\[ [n] = \frac{\text{L}^{2/3} \cdot 1}{\text{L} \cdot \text{T}^{-1}} \]
5. Simplify the expression using exponents:
\[ [n] = \text{L}^{2/3 - 1} \cdot \text{T}^{1} \]
\[ [n] = \text{L}^{-1/3} \cdot \text{T} \]
Therefore, the physical dimensions of Manning's roughness coefficient are \( \text{L}^{-1/3} \text{T} \).

Step 3: Final Answer:
The dimensions of Manning's coefficient are $\text{L}^{-1/3} \text{T}$.