The Fanning friction factor (named after American engineer John T. 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

1. is the local Fanning friction factor (dimensionless);

2. is the local shear stress (units of pascals (Pa) = N/m, or pounds per square foot (psf) = lbf/ft);

3. is the bulk dynamic pressure (Pa or psf), given by:

   1. is the density of the fluid (kg/m3 or lbm/ft)

   2. is the bulk flow velocity (m/s or ft/s)

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:

• is the shear stress at the wall

• is the density of the fluid

• is the flow velocity averaged on the flow cross section

For laminar flow in a round tube

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

.

Pipes/tubes of general roughness

where

• is the roughness of the inner surface of the pipe (dimension of length)

• D is inner pipe diameter;

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

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:

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 has been applied to produce the formula given below.

• Re, Reynolds number (unitless);

• , roughness of the inner surface of the pipe (dimension of length);

• D, inner pipe diameter;

• ln is the Natural logarithm;

• ;

Flows in non-circular conduits

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:

• is the friction loss (in head) of the pipe.

• is the Fanning friction factor of the pipe.

• is the flow velocity in the pipe.

• is the length of pipe.

• is the local acceleration of gravity.

• is the pipe diameter.

References

Further reading