# Darcy vs. Fanning Friction Factor

Friction plays a key role in the flow of a fluid through a pipe or duct. Calculating that friction involves using a friction factor.

A friction factor used in fluids engineering to calculate the head loss due to friction in pipes and ducts. As a fluid flows through a pipe, friction due to interaction with the wall leads to head loss. The head loss in a fluid flow is related to the pressure loss.

In many fields of engineering, one of two friction factors is used in these calculations. The first is the Darcy friction factor and the second is the Fanning friction factor. While these two friction factors are related to one another, understanding their differences is important for performing calculations related to changes in pressure due to the friction of a fluid flow through a pipe or a duct.

## Darcy vs Fanning Friction Factor

The Darcy friction factor, fD, is a component of the Darcy-Weisbach equation, used to describe head loss due to friction. Calculating the Darcy friction factor involves the Reynolds number of the flow, the relative roughness of the pipe wall, and the cross-section of the pipe.

The Fanning friction factor, f, is named after John Thomas Fanning and is equal to one-fourth the Darcy friction factor.

If the Darcy friction factor is known, calculating the Fanning friction factor is straightforward. However, like calculating the Darcy friction factor, the Fanning friction factor is dependent on the Reynolds number, pipe roughness, and pipe diameter for the specific flow scenario.

Both the Darcy and Fanning friction factor are used in a variety of engineering applications. However, it is important to know which friction factor is being used because of the relationship between the two. Although both factors are dimensionless, certain groups of engineers have migrated toward one or the other over the years.

The Darcy friction factor is often used by engineers who work in SI units, while the Fanning friction factor is frequently employed by engineers who use British units. Alternatively, the Darcy friction factor is more common among civil and mechanical engineers, while the Fanning friction factor is used more often by chemical engineers.

## Calculating the Friction Factor

For turbulent flow, the Darcy friction factor is calculated using the Colebrook-White equation, which is as follows:

where Re is the Reynolds number, ε is the average height of surface irregularities in the pipe and D is the diameter of the pipe.

To solve the Colebrook-White equation, it is necessary to perform an iterative operation, usually via a numerical solver. However, there are ways to reformulate or approximate the Colebrook-White equation, such as using the Lambert W function. The Colebrook-White equation can be modified for use in calculating the Fanning friction factor as follows:

Alternatively, the friction factor can be calculated using the ratio of the shear stress to the fluid density:

where τ is the shear stress, ρ is the density of the fluid, and u is the average flow velocity.

To calculate the friction factor, an understanding of the specific scenario is necessary. This will include the Reynolds number and the roughness and diameter of the pipe.

### Reynolds Number

The Reynolds number of a flow is used to describe the ratio of inertial forces to viscous forces. It can be used to estimate whether a flow will be laminar or turbulent. In evaluating the friction factor, it is important to determine if the flow is laminar or turbulent. If the Reynolds number is larger than 3500, the flow will be turbulent.

At very large Reynolds numbers, the friction factor will be independent of the Reynolds number. In other words, the friction in a flow that is highly turbulent will only depend on the roughness and diameter of the pipe.

### Relative Roughness

The relative roughness of the pipe describes how rough the pipe is, and is calculated using the following ratio:

The larger the roughness of the pipe, the higher the friction factor will be.

For laminar flow, the friction factor will be independent of the roughness, and can be calculated with the following equation:

### Pipe Cross-Section

The cross-section of the pipe impacts head loss due to friction because a larger cross-section will allow more flow that is unimpeded by interaction with the pipe wall, which means less pressure loss.

### Simplification of Calculation

Various scenarios have been developed that simplify the calculation of the friction factor. These are often dependent on the shape of the pipe or duct, as well as the material. By simplifying the calculation of the friction coefficient, the need for iterative solving is eliminated.

## Application of the Friction Factor

The Fanning friction factor is used to calculate the head loss via the following equation:

where Δp is the change in pressure, L is the pipe length, and D is the pipe diameter.

Alternatively, the Darcy friction factor is used with the following equation:

Note the difference between the Darcy and Fanning friction factors is a factor of 4.

Using the change in pressure, the head loss due to friction can be calculated as follows:

where Δh is the head loss and g is gravity.

## When to Use Darcy vs. Fanning Friction Factor

Knowing which friction factor to use is important for proper calculation of head loss. Oftentimes, engineers will make use of a chart or table to evaluate a friction factor. These tools usually include a note about which friction factor was used in generating the chart or table. However, it is not always the case that the friction factor is specified.

In that case, to ensure proper application of the correct friction factor, some general guidelines have been developed. When looking at the chart, if the value of the friction factor at a Reynolds number of 1000 (laminar flow) is 0.064, the chart uses the Darcy friction factor. If the value is 0.016 at the same Reynolds number, the Fanning friction factor is used.

For example, the Moody chart below uses the Darcy friction factor.

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