Commonly, it will be necessary to determine the flow velocity of a fluid through a conduit, and it will be required to consider the friction imparted by the conduit walls on the fluid. The equation for determining the average cross-sectional velocity is called Manningâ€™s (or the Gauckler-Manning) equation, and a key part of this equation is the roughness coefficient, â€śnâ€ť.

## Application Of Manningâ€™s â€śNâ€ť Roughness Coefficient

When determining the gravity-driven flow of a fluid through a conduit that is partially open to the air or through a pipe that is not full, the roughness of the conduit needs to be considered to calculate the friction-induced drag and the subsequent velocity profile of the fluid. It is here that the Manningâ€™s roughness coefficient and the specific â€śnâ€ť values are utilized.

The Manningâ€™s roughness coefficient describes the average roughness of a conduit, as determined through experimental evaluation. This coefficient is used with Manningâ€™s equation to calculate the drag the fluid will be subject to as it moves through the conduit, and the subsequent velocity of the fluid.

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## Manningâ€™s equation

Manningâ€™s (or the Gauckler-Manning) equation was originally developed in 1867 by Gauckler, with Manning subsequently developing the same equation in 1890. It is widely used in determining the flow rate of a fluid with an open surface moving through a conduit.

Manningâ€™s equation is as follows:

where:

*V*is the average cross-sectional velocity of the fluid, with SI units of m/s*n*is the Manningâ€™s coefficient, which has SI units of s/m^{1/3}*R*is the hydraulic radius of the conduit, with SI units of m_{h}*S*is the stream slope, which has no units because it is the ratio of the vertical length to the horizontal length, as follows:

where:

*Î”X*is the change in height of the flow, with SI units of m*Î”Y is the flow length, with SI units of m*

*
*

The hydraulic radius of the conduit is calculated as follows:

where:

*A*is the cross-sectional area of the flow, with SI units of m^{2}*WP*is the wetted perimeter (i.e., the length of the conduit perimeter that fluid is in contact with), with SI units of m

The following figure shows an example of the parameters used in Manningâ€™s equation:

## Determining The Manningâ€™s N-Value To Use

To determine the Manningâ€™s coefficient to use in a particular scenario, it is easiest to use a look-up table. These tables have been generated for many common conduit materials under different application scenarios. Some example values are shown in the following table:

Although Manningâ€™s roughness coefficient was originally determined to be applicable to open-air conduits, subsequent evaluation demonstrated that it can be applied to pipe scenarios in which the fluid has a surface not touching the pipe wall, i.e., a free surface.

The values of Manningâ€™s roughness coefficient are usually expressed as just a number. However, it should be noted that these values are not dimensionless, and have SI units of s/m^{1/3}, as introduced above.

## Example Use Of Manningâ€™s Roughness Coefficient

As an example of using Manningâ€™s roughness coefficient to calculate the average flow velocity through a conduit, a scenario is set up based on the figure shown above, with the following parameters:

*A*= 1.25m^{2}*P*= 1.9 m*Î”X*= 10 m*L*= 500 m

The conduit is made of concrete, so using the above table, n is approximately 0.012 s/m^{1/3}.

To calculate the average cross-sectional velocity, the following steps are carried out:

- Calculate
*S*:

- Calculate
*R*:

- Apply Manningâ€™s equation: