Question

Given below are the two statements.
Statement I: Hydraulic roughness (n) for flows in natural channels can be easily estimated by adopting Manning's equation, which is given as \(n = (1/v) \cdot R^{2/3 \cdot S^{1/2}\) (\(v = \text{velocity}, R = \text{hydraulic radius}, S = \text{slope}\)).
Statement II: Manning's roughness coefficient (n) does not have any dimension, i.e. it is purely dimensionless.
In light of the above statements, choose the most appropriate answer from the options given below}

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

Hint:

Manning's $n$ is NOT dimensionless! Its SI units are $\text{s}/\text{m}^{1/3}$ ($[\text{T L}^{-1/3}]$). Chezy's $C$ has dimensions $[\text{L}^{1/2} \text{T}^{-1}]$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Manning's open channel uniform flow equation relates flow velocity to hydraulic radius, channel bed slope, and the boundary roughness coefficient.
Key Formula or Approach:
\[ v = \frac{1}{n} R^{2/3} S^{1/2} \implies n = \frac{1}{v} R^{2/3} S^{1/2} \]
\[ [n] = [\text{T} \cdot \text{L}^{-1/3}] = \text{s}\cdot\text{m}^{-1/3} \quad \text{(Dimensional)} \]

Step 2: Detailed Explanation:
1. Statement I: Rearranging the classical Manning equation for velocity \(v = \frac{1}{n} R^{2/3} S^{1/2}\) gives \(n = \frac{1}{v} R^{2/3} S^{1/2}\). Hence, Statement I is correct.
2. Statement II: In the metric SI system, Manning's roughness coefficient \(n\) is not dimensionless; it possesses physical dimensions of \(\text{Time} \times \text{Length}^{-1/3}\) ($\text{s/m}^{1/3}$). Hence, Statement II is false.

Step 3: Final Answer:
Thus, Statement I is correct but Statement II is false, matching option (C).