Circular sewer at partial flow: If the central angle at the pipe's centre subtended by the water surface is α (radians), what is the ratio of flow depth at partial flow (y) to full depth (D)?

Introduction / Context:
For partially flowing circular sewers, geometric relations link depth, wetted perimeter, and cross-sectional area to the central angle at the center of the pipe. These relationships are crucial for computing hydraulic radius and velocity by Manning’s or Chezy’s equations.

Given Data / Assumptions:

• Circular pipe, radius R = D/2.
• Liquid segment subtends central angle α (radians) at the pipe centre.
• We require the ratio y/D, where y is flow depth and D is the full diameter.

Concept / Approach:

The geometric relation for a circular segment gives the depth from crown or invert using the half-angle. If we denote the half-angle by θ, then α = 2θ. The depth measured from the lowest point (invert) is y = R(1 - cos θ). Substituting θ = α/2 gives y = R[1 - cos(α/2)]. Because D = 2R, the required ratio is y/D = [1 - cos(α/2)] / 2.

Step-by-Step Solution:

Verification / Alternative check:

At α = π (half-full), α/2 = π/2 → cos(π/2) = 0, so y/D = 1/2 (correct). At α approaching 0, y/D → 0 (consistent).

Why Other Options Are Wrong:

Using α instead of α/2 or adding instead of subtracting cos terms leads to incorrect depth ratios. The sine expression does not represent the vertical depth relation for a circular segment.

Common Pitfalls:

Mixing degrees and radians; confusing α with its half-angle; using diameter in place of radius in intermediate steps.

Final Answer:

y/D = (1 - cos(α/2)) / 2