# 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 = (k_{n }/ n) R_{h}^{2/3}S^{1/2}(1)

where

v = cross-sectional 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)

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)

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 = cross-sectional 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)*

* = 0.21 m ^{3}/s*

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