Open-channel flow in a circular section:\nThe mean velocity through a circular channel (partially full) becomes maximum at a particular depth of flow. The depth of water for maximum velocity is approximately what fraction of the channel diameter?

Introduction / Context:
For a given circular channel carrying open-channel flow under gravity, the mean velocity v depends on the hydraulic radius R = A/P through resistance formulas (Chezy or Manning). Because area A and wetted perimeter P both change with depth y, there exists a particular y/D that maximizes R and hence the achievable mean velocity for a given slope and roughness. This question asks for that optimum fractional depth relative to the diameter D.

Given Data / Assumptions:

• Circular channel of diameter D, partially full (free surface present).
• Steady, uniform open-channel flow with fixed bed slope and roughness.
• Velocity proportional to hydraulic radius: v ∝ R^m (m = 1/2 for Chezy, 2/3 for Manning in discharge), so maximizing R leads to maximum v for fixed slope and roughness.

Concept / Approach:
For a circular arc of central angle 2θ (in radians) filled with liquid, A = (D^2/8) * (2θ − sin 2θ) and P = (D/2) * 2θ = Dθ. The hydraulic radius is R(θ) = A/P. The depth-to-diameter ratio is y/D = (1 − cos θ)/2. Differentiating R with respect to θ and setting dR/dθ = 0 gives an optimum at θ ≈ 2.639 rad (≈151.2°), which corresponds to y/D ≈ 0.81. This classical result is widely used for design checks in sewers and storm drains.

Step-by-Step Solution:

Verification / Alternative check:
If one computes R across several y/D values (e.g., 0.67, 0.75, 0.81, 0.90), R peaks near 0.81, confirming the analytic optimum. Practical charts in handbooks reproduce this peak at approximately 0.81D for velocity and around 0.94D for maximum discharge, emphasizing the distinction between criteria (velocity vs discharge).

Why Other Options Are Wrong:

• 0.34 times: Far too shallow; wetted perimeter is small but area is also too small.
• 0.67 times: Below the optimum; R has not yet peaked.
• 0.95 times: Closer to the value for maximum discharge, not maximum velocity.

Common Pitfalls:
Confusing the depth for maximum discharge with that for maximum mean velocity; forgetting that both A and P change with depth in a circular section, so maximizing A alone is not sufficient.

Final Answer:
0.81 times