Circular sewer, partial flow: If the angle subtended at the sewer centre by the free water surface is 120°, what is the depth of flow (depth of sewerage) for a 1 m diameter circular sewer?

Introduction / Context:
This question tests the geometric relationships used when a circular sewer runs partially full. Designers often express depth of flow as a function of the central angle of the wetted segment. Knowing how to convert angles to depths allows quick checks of capacity and hydraulic performance in partially full conditions.

Given Data / Assumptions:

• Circular sewer, diameter D = 1 m (so radius R = 0.5 m).
• The angle subtended at the centre by the free water surface (the chord angle for the water surface) is 120°.
• Depth of flow y is measured vertically from invert (lowest inner point) to the water surface.

Concept / Approach:
For a circular segment defined by a central angle 2α (radians or degrees), the depth of the segment is y = R * (1 - cos α). If the chord (free surface) subtends 120° at the centre, then 2α = 120°, hence α = 60°. Substituting α into the depth formula gives the fraction of the radius occupied by water above the centreline to the surface. Express the final depth in metres (or centimetres) using R = D/2.

Step-by-Step Solution:
Let 2α = 120° ⇒ α = 60°.Depth formula: y = R * (1 - cos α).Use R = D/2 = 1/2 = 0.5 m.Compute cos 60° = 0.5.y = 0.5 * (1 - 0.5) = 0.5 * 0.5 = 0.25 m.Convert to centimetres: 0.25 m = 25 cm.

Verification / Alternative check:
Since α = 60°, the depth equals R/2. Because R = 0.5 m, R/2 = 0.25 m = 25 cm. This quick mental check matches the computed value.

Why Other Options Are Wrong:
20 cm: Would require a smaller α (cos α closer to 0.6).40 cm or 50 cm: Exceed the R/2 result for α = 60°; 50 cm equals the full radius, which would imply α = 90°.60 cm: Greater than the radius; impossible for a 1 m diameter pipe in this configuration.

Common Pitfalls:

• Confusing the 120° with α itself; remember 2α is the full wetted central angle.
• Using diameter instead of radius in y = R * (1 - cos α).
• Mixing degrees and radians when using calculators; here, simple angles allow exact values.

Final Answer:
25 cm