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Manning's Formula for Gravity Flow

Open channel slope area flow metering

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Manning's equation can be used to calculate cross-sectional average velocity flow in open channels

v = (kn / n) Rh2/3 S1/2                       (1)

where

v = cross-sectional mean velocity (ft/s, m/s)

kn = 1.486 for English units and kn = 1.0 for SI units

n = Manning coefficient of roughness - ranging from 0.01 (a clean and smooth channel) to 0.06 (a channel with stones and debris, 1/3 of vegetation)

Rh = hydraulic radius (ft, m)

S = slope - or gradient - of pipe (ft/ft, m/m)

Hydraulic radius can be expressed as

Rh = A / Pw                             (2)

where

A = cross sectional area of flow (ft2, m)

Pw = wetted perimeter (ft, m)

The volume flow in the channel can be calculated as

q = A v = A (kn / n) Rh2/3 S1/2                         (3)

where

q = volume flow (ft3/s, m3/s)

A = cross-sectional area of flow (ft2, m2)

Example - Flow in an Open Channel

A channel with the shape of an half circle is 100% filled. The diameter of the half circle is 500 mm (0.5 m) and the channel is made of concrete with Manning coefficient 0.012. The slope of the channel is 1/100 m/m.

mannings formula open channel flow

The cross section area of the half circle flow can be calculated as

A = (0.5 π ((0.5 m) / 2)2)

   = 0.098 m2

The wetted perimeter of the half circle flow can be calculated as

P = 0.5 2 π (0.5 m) / 2)

   = 0.785 m

The hydraulic radius of the channel can be calculated from (2) as

Rh = A / P

  = (0.098 m2) / (0.785 m)

  = 0.125 m

The cross sectional mean velocity can be calculated from (1) as

v = (kn / n) Rh2/3 S1/2     

   = (1.0 / 0.012) (0.125 m)2/3 (1/100 m/m)1/2

   = 2.1 m/s

The volume flow can be calculated from (3) as

q = A v

   =  (0.098 m2) (2.1 m/s)

   = 0.21 m3/s

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