# 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 (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 = 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) *

* = 0.20 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.

## 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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