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# Manning's Formula and Gravity Flow

## Calculate cross-sectional average velocity flow in open channels.

Manning's equation is an empirical equation that 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)

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 (kn / n) in the SI system vs. the Imperial system can be expressed as:

(kn / n) = m1/3 / s = 1.4859 ft1/3 / s

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. make 3D models with the free Engineering ToolBox Sketchup Extension!

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)

= 0.20 m3/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.

Diameter of pipe (m, ft)

Slope (m/m, ft/ft)

n (1.0 for SI units, 1.486 for Imperial units)

Manning coefficient of roughness

## Related Topics

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The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.

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## Citation

• The Engineering ToolBox (2004). Manning's Formula and Gravity Flow. [online] Available at: https://www.engineeringtoolbox.com/mannings-formula-gravity-flow-d_800.html [Accessed Day Month Year].

Modify the access date according your visit.

9.19.12