Write the Chezy–Manning equation used for open-channel flow and define the roughness coefficient.

• A. V = (1/n) R^(2/3) S^(1/2); Q = (1/n) A R^(1/3) S^(1/2); n is Manning roughness.
• B. V = n R^(2/3) S^(1/2); Q = n A R^(2/3) S^(1/2); n is a friction factor.
• C. V = (1/n) R^(2/3) S^(1/2); Q = (1/n) A R^(2/3) S^(1/2); n is Manning roughness coefficient.
• D. V = (1/n) R^(2/3) S^(1/2); Q = (1/n) A R^(2/3) S^(3/2); n is roughness.

Answer: (C)

In open-channel flow, the velocity is governed by Manning’s form of the Chezy relation, which links how fast water moves to how rough the channel is and how steeply the channel bed slopes. The velocity depends on the hydraulic radius raised to the 2/3 power and on the slope raised to the 1/2 power, with a factor 1/n that accounts for friction from the channel surface. Specifically, velocity is V = (1/n) R^(2/3) S^(1/2). The discharge then follows from area times velocity: Q = A V = (1/n) A R^(2/3) S^(1/2). Here R is the hydraulic radius, defined as the cross-sectional area A divided by the wetted perimeter P (R = A/P), and S is the energy/slope of the channel bed.

The roughness coefficient n, known as Manning’s n, is an empirical parameter that represents the frictional resistance of the channel’s surface and surroundings—bed texture, vegetation, obstructions, and overall roughness. Lower n values correspond to smoother channels and faster flow, while higher n values indicate rougher surfaces and slower flow. Its role is to scale the velocity to reflect how much resistance the channel presents.