Partial-Flow Hydraulics
Circular segment geometry, depth ratio and the Q and V maxima.
Sewers normally run part-full, so capacity is assessed with the geometry of a circular segment bounded by a water surface chord and a wetted arc. The same Manning relationship applies with the part-full area and wetted perimeter.
The relationship between depth and discharge is not linear. Discharge reaches a maximum at about 94% of the diameter and velocity at about 81%, which is why a pipe running just below full can carry slightly more than a full pipe at the same slope.
Key design considerations
- Depth is normally found by iteration or from a depth-discharge chart for the chosen diameter and grade.
- Ratios Q/Q_full and V/V_full are more robust than absolute values when comparing alternatives.
- The maximum-discharge and maximum-velocity depths are geometric facts, not design operating points.
- Shallow depths give low velocity and poor transport, so the low-flow condition governs self-cleansing.
- At y = D the part-full equations reduce to the full-pipe case; above it the pipe is surcharged.
Equations
θ = 2 · arccos(1 − 2y/D)
Central angle in radians for flow depth y in a pipe of internal diameter D.
A = (D²/8)(θ − sin θ)
Flow area of the circular segment.
P = θD/2
Wetted perimeter, with θ in radians.
Q = (1/n) · A · (A/P)^(2/3) · S^(1/2)
Part-full discharge; solve by iteration for depth at a known Q.
Design checks
- Depth ratio at design flow against the permitted value.
- Velocity at both the design flow and the low-flow condition.
- Critical depth and Froude number only where the pipe is steep.
Standards and authority references
- WSA 02 Gravity Sewerage Code of Australia (current edition), Water Services Association of Australia
- Authority permitted depth of flow
Confirm the current edition, amendment and any local supplement before applying any of the above.
Limitations and assumptions
- Assumes a constant roughness around the wetted perimeter.
- Does not account for sediment beds, which change the effective geometry.
Related calculators
This reference is general engineering information. Confirm requirements against WSA 02, the current Australian Standards and the local water authority's supplement before use in design, and obtain project-specific advice for the conditions at hand.