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 Manning’s (or the Gauckler-Manning) equation, and a key part of this equation is the roughness coefficient, “n”.

## Application Of Manning’s “N” 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 Manning’s roughness coefficient and the specific “n” values are utilized.

The Manning’s roughness coefficient describes the average roughness of a conduit, as determined through experimental evaluation. This coefficient is used with Manning’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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## Manning’s equation

Manning’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.

Manning’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 Manning’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:

*ΔX*is the change in height of the flow, with SI units of m*ΔY 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 Manning’s equation:

## Determining The Manning’s N-Value To Use

To determine the Manning’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 Manning’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 Manning’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 Manning’s Roughness Coefficient

As an example of using Manning’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*ΔX*= 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 Manning’s equation: