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# Manning's Formula and Gravity Flow

Manning's equation is an empirical equation that 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)

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 (ft2, 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 (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 .

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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 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

R h = A / P

= ( 0.098 m2> ) / ( 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 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.

## Related Topics

### • Flow Measurements

Flow metering principles - Orifice, Venturi, Flow Nozzles, Pitot Tubes, Target, Variable Area, Positive Displacement, Turbine, Vortex, Electromagnetic, Ultrasonic Doppler, Ultrasonic Time-of-travel, Mass Coriolis, Mass Thermal, Weir V-notch, Flume Parshall and Sluice Gate flow meters and more.

### • Fluid Mechanics

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

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