Manning's Formula and Gravity Flow
Calculate crosssectional average velocity flow in open channels.
Manning's equation is an empirical equation that can be used to calculate crosssectional average velocity flow in open channels
v = (k _{ n } / n) R _{ h } ^{ 2/3 } S ^{ 1/2 } (1)
where
v = crosssectional mean velocity (ft/s, m/s)
k _{ n } = 1.486 for English units and k _{ n } = 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)
R _{ h } = hydraulic radius (ft, m)
S = slope  or gradient  of pipe (ft/ft, m/m)
The Manning coefficients n in the English (Imperial) are equal to the coefficients in the SI system. The Manning coefficient is not dimensionless but the units are often omitted.
Units for (k _{ n } / n) in the SI system vs. the Imperial system can be expressed as:
(k _{ n } / n) = m ^{ 1/3 } / s = 1.4859 ft ^{ 1/3 } / s
Hydraulic radius can be expressed as
R _{ h } = A / P _{ w } (2)
where
A = cross sectional area of flow (ft^{2}, m)
P _{ w } = wetted perimeter (ft, m)
Download and print Gravity Flow  Mannings Equation Chart
The volume flow in the channel can be calculated as
q = A v = A (k _{ n } / n) R _{ h } ^{ 2/3 } S ^{ 1/2 } (3)
where
q = volume flow (ft ^{ 3 } /s, m ^{ 3 } /s)
A = crosssectional area of flow (ft^{2}, m^{2})
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 .
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The cross section area of the half circle flow can be calculated as
A = (0.5 π ((0.5 m) / 2)^{2})
= 0.098 m^{2}
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
R _{ h } = A / P
= ( 0.098 m^{2}> ) / ( 0.785 m )
= 0.125 m
The cross sectional mean velocity can be calculated from (1) as
v = (k _{ n } / n) R _{ h } ^{ 2/3 } S ^{ 1/2 }
= (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 m^{2}) (2.1 m/s)
= 0.20 m ^{ 3 } /s
Gravity Flow Calculator  Half Filled Circular Pipe
The Gravity Flow Calculator is based on the equations and the example above above. It's valid for half filled circular pipe.
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