Open-channel flow – Manning’s discharge form According to Manning's formula, the discharge Q through an open channel can be written in terms of area A, hydraulic mean depth m (i.e., R), bed slope i, and Manning's constant M = 1/n. Which expression is correct?

Introduction:
Manning's equation is a cornerstone for uniform open-channel flow. It connects discharge to channel geometry, roughness, and slope, enabling practical sizing of canals, sewers, and natural streams. The question asks for the correct symbolic form when using Manning's constant M = 1/n and hydraulic mean depth m (also written R) for hydraulic radius.

Given Data / Assumptions:

• Uniform, steady open-channel flow.
• Cross-sectional area A; hydraulic radius m = A / P.
• Bed slope (energy grade) i.
• Manning roughness n with M = 1 / n.

Concept / Approach:

Velocity in Manning form is V = (1/n) * m^(2/3) * i^(1/2) = M * m^(2/3) * i^(1/2). Thus discharge is Q = A * V = A * M * m^(2/3) * i^(1/2). Any alteration of the exponents 2/3 and 1/2 misrepresents the empirical scaling embedded in Manning's relation.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional consistency: V has dimensions of L/T; m^(2/3) carries L^(2/3) and i^(1/2) is dimensionless, with M carrying appropriate empirical scaling. Multiplying by A (L^2) yields Q (L^3/T).

Why Other Options Are Wrong:

Other exponent patterns (1/2 and 2/3 swapped, powers on A or M) do not match the accepted Manning exponents.A * M * m * i: Omits the essential empirical exponents.

Common Pitfalls:

Confusing hydraulic mean depth m (R) with flow depth y in non-rectangular sections; always compute m = A / P first.

Final Answer:

A * M * m^(2/3) * i^(1/2)