Partial-Flow Pipe Calculator

Find normal depth, velocity and filling ratio of a circular pipe flowing partly full.

Flow in Partly Full Circular Pipes

Storm drains and sewers normally run partly full with a free water surface rather than full under pressure. For a given discharge, this calculator finds the normal depth and velocity in a circular pipe from Manning's equation and compares the flow with the pipe's full-flow capacity.

θ = 2·arccos(1 − 2y/D); A = D²(θ − sin θ)/8; P = Dθ/2; Q = (1/n)·A·R2/3·S1/2, R = A/P

  • D: inner pipe diameter (m)
  • y: water depth (m)
  • n: Manning roughness coefficient
  • S: pipe slope (m/m)

In a circular section the discharge peaks at a depth ratio y/D of about 0.938, where it reaches roughly 1.076 times the full-flow discharge. When the discharge exceeds the full-flow capacity, no free-surface normal depth exists and the calculator reports it instead of a depth.

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Normal Depth (y)

0.000 m

Filling Ratio (y/D)

0.000

Flow Velocity (V)

0.000 m/s

Discharge Ratio (Q / Qfull)

0.000

Froude Number (Fr)

0.000

Full-Pipe Capacity (Qfull)

0.000 m³/s

D: m · S: m/m · n: s/m1/3 · Q: m³/s · V: m/s. The SI form of Manning's equation is used. USACE Hydraulic Reference Manual, Chapter 2

Worked example and limits

Worked example

Let a pipe with inner diameter D = 0.6 m, slope S = 0.002 and roughness n = 0.013 (concrete) carry Q = 0.20 m³/s. Its full-pipe capacity is Qfull = 0.275 m³/s and the full-flow velocity is 0.97 m/s. Because Q/Qfull = 0.73, the pipe flows with a free surface: the normal depth is y ≈ 0.380 m (y/D = 0.633), the velocity V ≈ 1.06 m/s and the Froude number Fr ≈ 0.59. For Q = 0.05 m³/s in the same pipe, y ≈ 0.173 m (y/D = 0.289) and V ≈ 0.74 m/s, which shows how shallow and slow the flow becomes at low discharges.

Reading the result

Check a pipe not only against its full-flow capacity but also against the depth and velocity at the smallest and largest expected discharges. Authorities usually set criteria for minimum velocity, maximum velocity and permitted filling ratio; the applicable criteria must be confirmed for the project.

Assumptions and limits

  • Flow is steady and uniform; diameter and slope are constant between two manholes. At manhole inlets, junctions and slope changes the flow is varied.
  • n is held constant with depth; in reality sediment, deposits and joints can change the roughness.
  • When the discharge exceeds the full-pipe capacity the pipe flows under pressure; that case is assessed with a head-loss calculation, not with the free-surface Manning depth.
  • If the Froude number is close to 1, normal depth is unstable and small input changes can move the depth noticeably.

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