State Manning's equation for uniform open-channel flow and define each parameter.

• A. Q = (1/n) A R^(2/3) S^(1/2); A cross-sectional area; R hydraulic radius (A/P); n Manning roughness; S bed slope.
• B. Q = A V; A cross-sectional area; V velocity.
• C. Q = π D^2/4 * V; D diameter; V velocity.
• D. Q = k A sqrt(S); k constant; A area.

Answer: (A)

Uniform open-channel flow is modeled by Manning’s equation, which links discharge to channel geometry, roughness, and slope. The expression Q = (1/n) A R^(2/3) S^(1/2) shows that discharge increases with cross-sectional area and with the hydraulic radius raised to the 2/3 power, and with the square root of the slope, while roughness reduces discharge through division by n. Q is the volumetric flow rate; A is the flow cross-sectional area; R is the hydraulic radius, defined as R = A/P with P the wetted perimeter; S is the slope (the energy grade line slope, commonly approximated by bed slope for uniform flow); n is Manning’s roughness coefficient, reflecting channel roughness and friction. This combination is what the correct option expresses. The other forms correspond to different situations or simplified relationships (for example, Q = A V is a basic continuity relation with velocity, and pipe-flow formulas like Q = (π D^2/4) V or Q = k A sqrt(S) do not represent Manning’s equation for open-channel flow).