Fanning friction factor
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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:
- is the local Fanning friction factor (dimensionless)
- is the local shear stress (unit in or or Pa)
- is the bulk flow velocity (unit in or )
- is the density of the fluid (unit in or )
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 ( for a pipe with circular cross section) and dividing by the cross-sectional flow area ( for a pipe with circular cross section). Thus
- 1 Fanning friction factor formula
- 2 Application
- 3 References
- 4 Further reading
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.
- 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.
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 (1983) under flow conditions of
- is the roughness of the inner surface of the pipe (dimension of length)
- D is inner pipe diameter;
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 (1943) 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
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 is 4 times the Fanning friction factor and so a factor of has been applied to produce the formula given below.
- Re, Reynolds number (unitless);
- ε, roughness of the inner surface of the pipe (dimension of length);
- D, inner pipe diameter;
Flows in non-circular conduits
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:
- 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.
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