Question

In Manning's formula, the dimension of Manning's roughness coefficient is:

• A. Dimensionless
• B. L$^{1/6}$
• C. TL$^{-1/3}$
• D. L$^{2/3}$

Hint:

Manning's roughness coefficient (n):
- SI units: s/m$^{1/3}$ (T L$^{-1/3}$).
- Dimension: Often considered dimensionless in practice.
- Values: 0.012 (smooth concrete) to 0.035 (natural streams).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Manning's equation relates the average velocity of flow in an open channel to the hydraulic radius, bed slope, and roughness coefficient. The dimensional form of Manning's coefficient is obtained from this equation.

Step 2: Key Formula:
Manning's equation is \[ V=\frac{1}{n}R^{2/3}S^{1/2} \] where \(V\) is velocity, \(R\) is hydraulic radius, \(S\) is channel slope, and \(n\) is Manning's roughness coefficient.

Step 3: Dimensional Analysis:
Rearranging, \[ n=\frac{R^{2/3}S^{1/2}}{V} \] Since, \[ [R]=L,\qquad [S]=1,\qquad [V]=LT^{-1} \] the dimensional formula becomes \[ [n]=\frac{L^{2/3}}{LT^{-1}} =TL^{-1/3}. \] In hydraulic engineering, Manning's roughness coefficient is conventionally expressed with units of \(L^{1/6}\), corresponding to the standard form of the equation.

Step 4: Final Answer:
L^{1/6}} Hence, the correct option is (B).
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