Most economical circular channel section for maximum discharge in open-channel hydraulics: identify the correct combined conditions (depth–diameter ratio, hydraulic mean depth, and wetted perimeter relations).

Introduction / Context:

For circular channels (like sewers and tunnels) operating partially full in steady uniform flow, hydraulic efficiency is maximized when geometric proportions create the greatest discharge for a given roughness and slope. The “most economical section” minimizes wetted perimeter for a given area (or equivalently maximizes hydraulic radius R = A/P), which directly influences discharge via Manning or Chezy equations.

Given Data / Assumptions:

• Prismatic circular channel conveying water under gravity in open-channel flow.
• Steady, uniform flow; roughness and bed slope are fixed.
• Objective: maximum discharge (economical section conditions).

Concept / Approach:

For a circular section, the best hydraulic condition occurs at a specific depth ratio y/D (not necessarily running full). Classical derivations using calculus on A(θ) and P(θ) with θ as half central angle lead to standard proportionalities for the optimum: y/D ≈ 0.94–0.95, R ≈ 0.286 D, and characteristic wetted perimeters with respect to D and y. These relations are widely used as design thumb rules.

Step-by-Step Solution:

Verification / Alternative check:

Using Manning’s Q ∝ A R^(2/3) S^(1/2), maximizing Q for fixed S and n reduces to maximizing A R^(2/3). Differentiation with respect to θ for a circular segment yields the listed numerical factors, all consistent with standard hydraulics handbooks.

Why Other Options Are Wrong:

• Each single statement (options A–D) is correct individually; hence choosing just one under-represents the complete condition set.
• The only fully correct choice consolidating all truths is 'All of the above'.

Common Pitfalls:

• Assuming maximum discharge occurs when the pipe runs full (not true for roughness-controlled open-channel behavior).
• Confusing hydraulic radius R with geometric radius D/2.

Final Answer:

All of the above