Fanning friction factor


The Fanning friction factor, named after John Thomas Fanning, is a dimensionless number used as a local parameter in continuum mechanics calculations. It is defined as the ratio between the local shear stress and the local flow kinetic energy density:
where:
In particular the shear stress at the wall can, in turn, be related to the pressure loss by multiplying the wall shear stress by the wall area and dividing by the cross-sectional flow area . Thus

Fanning friction factor formula

This friction factor is one-fourth of the Darcy friction factor, so attention must be paid to note which one of these is meant in the "friction factor" chart or equation consulted. Of the two, the Fanning friction factor is the more commonly used by chemical engineers and those following the British convention.
The formulas below may be used to obtain the Fanning friction factor for common applications.
The Darcy friction factor can also be expressed as
where:
From the chart, it is evident that the friction factor is never zero, even for smooth pipes because of some roughness at the microscopic level.
The friction factor for laminar flow of Newtonian fluids in round tubes is often taken to be:
where Re is the Reynolds number of the flow.
For a square channel the value used is:

For turbulent flow in a round tube

Hydraulically smooth piping

Blasius developed an expression of friction factor in 1913 for the flow in the regime.
Koo introduced another explicit formula in 1933 for a turbulent flow in region of

Pipes/tubes of general roughness

When the pipes have certain roughness, this factor must be taken in account when the Fanning friction factor is calculated. The relationship between pipe roughness and Fanning friction factor was developed by Haaland under flow conditions of
where
The Swamee–Jain equation is used to solve directly for the Darcy–Weisbach friction factor f for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation.

Fully rough conduits

As the roughness extends into turbulent core, the Fanning friction factor becomes independent of fluid viscosity at large Reynolds numbers, as illustrated by Nikuradse and Reichert for the flow in region of. The equation below has been modified from the original format which was developed for Darcy friction factor by a factor of

General expression

For the turbulent flow regime, the relationship between the Fanning friction factor and the Reynolds number is more complex and is governed by the Colebrook equation which is implicit in :
Various explicit approximations of the related Darcy friction factor have been developed for turbulent flow.
Stuart W. Churchill developed a formula that covers the friction factor for both laminar and turbulent flow. This was originally produced to describe the Moody chart, which plots the Darcy-Weisbach Friction factor against Reynolds number. The Darcy Weisbach Formula, also called Moody friction factor, is 4 times the Fanning friction factor and so a factor of has been applied to produce the formula given below.
Due to geometry of non-circular conduits, the Fanning friction factor can be estimated from algebraic expressions above by using hydraulic radius when calculating for Reynolds number

Application

The friction head can be related to the pressure loss due to friction by dividing the pressure loss by the product of the acceleration due to gravity and the density of the fluid. Accordingly, the relationship between the friction head and the Fanning friction factor is:
where: