Circular sewer running partially full: with diameter D and central angle α (in radians) subtended by the water surface at the sewer centre, which geometric relations are correct for hydraulic computations?

Introduction / Context:
Hydraulic design of circular sewers operating partially full requires correct geometric relations for area, wetted perimeter, surface width, and hydraulic radius. These feed Manning’s or Chezy’s equations to estimate normal depth, velocity, and capacity.

Given Data / Assumptions:

• Uniform prismatic circular conduit of diameter D.
• Central (subtended) angle of the water segment at the pipe centre is α radians.
• Steady uniform flow, gravity-driven, partially full (0 < α < 2π).

Concept / Approach:

The circular-segment geometry provides closed-form expressions: the wetted perimeter equals the arc length; the area equals area of sector minus area of triangle; top width equals the chord length. The hydraulic radius follows directly as R = A / P, fundamental in Manning’s equation V = (1/n) * R^(2/3) * S^(1/2).

Step-by-Step Solution:

Verification / Alternative check:

At α = π (half full): P = (Dπ)/2, A = (D^2/8) * (π − 0) = (πD^2/8) = half of full area (πD^2/8), and T = D, which match known half-full relations.

Why Other Options Are Wrong:

In this set, (a)–(d) are all correct; choosing any single one is incomplete, so “All the above” is the correct comprehensive selection.

Common Pitfalls:

Mixing degrees and radians for α; using hydraulic diameter 4A/P (common in pipe flow) instead of hydraulic radius A/P in open-channel form of Manning’s equation.

Final Answer:

All the above.