Sewer Partial Flow and Self-Cleansing

What is modelled

Sewer water usually flows with a free surface in a circular pipe, a steady uniform flow where gravity balances friction. The depth that carries a given flow follows from Manning's equation. A sewer also has to carry grit: the shear stress of the water on the floor must exceed the critical value for the grain size, or sediment settles and blocks the pipe.

Equations

• Manning Q = (1/n) A R2/3 √S, with the circular segment A = D² (θ − sinθ)/8, wetted perimeter Dθ/2 and depth y/D = (1 − cos(θ/2))/2.
• Maximum discharge at 93.8 % depth (1.076 of full bore) and maximum velocity at 81 % (1.14 of full bore).
• Shear stress τ = ρ g R S against τc = 0.047 (ρs − ρ) g d (Shields).
• Froude number V / √(g A / T); critical depth at Fr = 1.

Checks run in the page

• Normal depth by bisection inserted back into Manning's equation.
• Section area by integrating strips against the closed form.
• Depths and values of the maximum discharge and velocity.
• Smallest self-cleansing slope by bisection, and the minimum of specific energy at critical depth.

Honest simplifications

The flow is steady and uniform, the Manning coefficient is constant with depth, the pipe has no bends or manholes, and a single Shields number and grain size stand for all sediment. Surcharged flow is treated as simply full with the same mean velocity. Real sewers have varied flow, deposits, gases and roots, and this is not a drainage design.

• Press Flat trunk, low flow and watch the grains settle because the shear falls below the critical value.
• Press Near full capacity and see the depth reach about 80 percent while the velocity peaks.
• Press Surcharged to see the flow exceed the free-surface maximum and the pipe run full.