Manning’s equation calculates flow parameters for gravity-driven water flow. The formula input values are pipe or channel dimensions, surface roughness, and flow slope. Solving the equation finds the average flow velocity or flow rate.

The resulting flow values can then determine the depth of the water or the maximum flow capacity. While computers have made more accurate methods practical, Manning’s equation is still commonly used due to its simplicity and versatility.

## What is the formula for the Manning Equation?

### Metric Units

To calculate the cross-sectional average velocity, use the following formula:

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Where:

- V = average flow velocity [m/s]
- k = unit conversion factor (1 for Metric units) [dimensionless]
- n = Manning roughness coefficient [s/(m
^{1/3})] - R
_{h}= hydraulic radius [m] - S = stream slope (equal to channel or pipe slope when depth is constant) [dimensionless]

Note that R_{h}, the hydraulic radius, accounts for the shape of the channel or pipe. Find the hydraulic radius using the following formula:

Where:

- A = cross-sectional flow area [m
^{2}] - P
_{w}= wetted perimeter [m]

Due to the reliance on the wetted perimeter, the Manning equation can account for the changing flow conditions as the water level in a channel or pipe varies.

Using the cross-sectional area of flow, A, the volumetric flow rate can be estimated using the following formula:

Where:

- Q = volumetric flowrate [m
^{3}/s]

By using the above formula, the flow rate version of the Manning equation is shown as follows:

### English Units

The variable of k in the Manning equation is a conversion factor between Metric and English units. When using metric, the value of k = 1; when using English units, k = 1.4859.

The English form of the Manning equation uses the following variable units:

- V = average flow velocity [ft/s]
- k = unit conversion factor (1.4859 for English units) [dimensionless]
- n = Manning roughness coefficient [s/(ft
^{1/3})] - R
_{h}= hydraulic radius [ft] - S = stream slope (equal to channel or pipe slope when depth is constant) [dimensionless]
- A = cross-sectional flow area [ft
^{2}] - P
_{w}= wetted perimeter [ft] - Q = volumetric flowrate [ft
^{3}/s]

## The Manning Friction Factor

The Manning friction factor, n, was empirically derived from testing various materials and measuring actual flow rates through existing systems. Based on the wetted surface material, the Manning friction factor is determined from tabulated values.

## Applications and Limitations

The Manning equation only applies to the flow of water. It is also limited to uniform flow with constant channel slope, shape, material, flow area, and flow velocity. This method is only used when the flow’s driving force is from gravity. The equation balances the force of gravity with the resistance from friction.

Due to its empirical derivation from observation of actual conditions, the Manning equation accounts for turbulence as long as the flow is uniform.

The Manning equation can model drainage culverts, irrigation ditches, and pipes that are wholly or partially full. While the Manning equation can be applied to full pipes, other methods, such as the Darcy-Weisbach and Colebrook equation, are more frequently used.

By rearranging the Manning equation, many parameters of interest can be found. For example, the fluid depth in a culvert can be found for a given flow rate. This is useful for ensuring the culvert can provide an adequate flow rate for irrigation or for estimating the depth of water drained during a storm. Likewise, solving for the steam slope can find the minimum slope required for a culvert of a particular dimension to flow a given amount of water.

## Alternatives to the Manning Equation

The simplicity of the Manning equation has ensured that it remains a commonly used and widely accepted method; however other alternatives exist.

### The Hazen-Williams equation

The Hazen-Williams equation, like the Manning equation, is only valid for the flow of water at near room-temperature values and at flow velocities similar to those experienced in agricultural or industrial systems. The structure of each formula is nearly identical.

The velocity version of the Hazen-Williams equation is:

Where:

- V = average flow velocity [m/s]
- k = unit conversion factor (0.849 for Metric units)
- C = roughness coefficient
- R
_{h}= hydraulic radius [m] - S = slope of the energy line (pressure loss per unit length) [dimensionless: m/m]

The two processes are slightly different. The Manning equation balances flow forces from gravity against friction. In contrast, the Hazen-Williams equation determines the pressure loss per length of pipe. The naming of the variable S is different, but they are mathematically equivalent.

### The Darcy-Weisbach equation

For flow in pipes, the Darcy-Weisbach equation is typically used. This method remains valid when the driving force is something other than gravity, such as a pump. It also does not rely on uniform piping geometry or surface roughness.

The Darcy-Weisbach equation is better at modeling systems with fittings such as valves and elbows. Using tables of equivalent pipe length, these are treated as being equal to additional piping. Because the Manning equation doesn’t consider pipe length, this additive method does not work.

Another advantage of the Darcy-Weisbach equation is the ability to determine pressure loss in the system. In the case of a piping system with multiple outflows, the Manning equation would require each section of equal flow to be independently calculated. An example would be an irrigation system with branches leading to different fields.

Unlike the Manning equation, the Darcy-Weisbach equation can determine the flow rate or velocity that would cause a transition between laminar and turbulent flow.

The main advantage of the Manning equation is that it does not rely on the Reynolds number. Since the Reynolds number is reliant on the pipe dimensions, using the Darcy-Weisbach equation would require iterations to determine the desired pipe size.

### HEC-RAS (Hydrologic Engineering Center – River Analysis System)

HEC-RAS is a computer program for determining water flow driven by gravity in open channels. This program was funded by the United States Government and is available to the public at no charge. Much of the underlying program architecture uses the Manning equation.

In cases of open flow for drainage or irrigation, using HEC-RAS is considered one of the most accurate ways to model flow.

## Manning Equation Example

### Example 1: Determining drainage ditch capacity

One of the most common uses of the manning equation is calculating flow rates for stormwater drainage. In this example, an existing drainage path will be examined to determine its flow capacity at the overflow point.

#### Problem

A smooth concrete drainage ditch has a cross-sectional profile of an isosceles trapezoid, as shown below. Given a grade of 2%, what will the flow rate be when the ditch is filled, as shown?

#### Solution

Starting with the traditional form of the Manning equation, it is rearranged to solve for volumetric flow:

Substitute to solve for Q:

The result is the Manning equation for volumetric flow rate.

The value of k = 1, as the problem is being solved in metric units.

The Manning roughness coefficient, n, is found in the table. In this case, the value for wood form cast concrete is chosen. Therefore, n = 0.015.

The steam slope, S, is input as a rise-over-run. The problem statement gives the grade as 2%, which is a 2-unit rise over a 100-unit run. Therefore, S = 0.02.

Because the problem statement states that the ditch is full, the cross-sectional flow area, A, is the area of the trapezoid. Therefore, A = 0.75.

From the dimensions given, the hydraulic radius, R_{h}, is found by dividing the cross-sectional area by the perimeter of the portions of the trapezoid that are wetted. Therefore, R_{h} = 0.31.

R_{h} = 0.31

Solving for volumetric flow rate, Q:

Q = 3.24 m^{3}/s

A ditch, as described in the problem statement, can pass approximately 3.24 m^{3}/s when filled.

### Example 2: Finding the required slope of a water supply pipe

#### Problem

A water tank on a hill will supply water to a fire engine. The pipe will be made of 4” Schedule 40 PVC and must flow 0.03 m^{3}/s. What is the minimum required slope for this pipe?

#### Answer

In this example, the value to be determined is S. Since one of the input values is the volumetric flow rate, that version of Manning’s equation will be used.

Solving for S gives:

Once again, k = 1.

The Manning roughness coefficient, n, for PVC is 0.010.

The cross-sectional area, A, of 4” Schedule 40 PVC is 0.008103 m^{2}.

The hydraulic radius, R_{h}, of a circular pipe is the same as its interior radius, which for 4” Schedule 40 PVC is 0.0508 m.

Solving for the slope, S:

S = 0.073

The pipe in question will require a slope of at least 7.3%.