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:
f=
| \tau | |
| q |
where:
f
\tau
| lbm | |
| ft ⋅ s2 |
| kg | |
| m ⋅ s2 |
q
| lbm | |
| ft ⋅ s2 |
Pa
where the dynamic pressure is given by:
q=
| 1 | |
| 2 |
\rhou2
where:
\rho
| lbm | |
| ft3 |
| kg | |
| m3 |
u
| ft | |
| s |
| m | |
| s |
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 (
2\piRL
\piR2
\DeltaP=f
| 2L | |
| R |
q=f
| L | |
| R |
\rhou2
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[3]
fD=
| 8\bar\tau | |
| \rho\baru2 |
where:
\tau
\rho
\baru
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:[4]
f=
| 16 | |
| Re |
where Re is the Reynolds number of the flow.
For a square channel the value used is:
f=
| 14.227 | |
| Re |
Blasius developed an expression of friction factor in 1913 for the flow in the regime
2100<Re<105
| f= | 0.0791 |
| Re0.25 |
Koo introduced another explicit formula in 1933 for a turbulent flow in region of
104<Re<107
| f=0.0014+ | 0.125 |
| Re0.32 |
When the pipes have certain roughness
| \epsilon | |
| D |
<0.05
4\centerdot104<Re<107
| 1 | |
| \sqrt{f |
where
\epsilon
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.[8]
f=
| 0.0625 | ||||||||
|
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
| ||||
| Re>10 |
>0.01
| 1 | |
| 4 |
| 1 | |
| \sqrt{f |
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[11] which is implicit in
f
{1\over\sqrt{f
Various explicit approximations of the related Darcy friction factor have been developed for turbulent flow.
Stuart W. Churchill[12] 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
fD
f
| 1 | |
| 4 |
f
fD
f
fD
f=2\left(\left(
| 8 | |
| Re |
\right)12+\left(A+B\right)-1.5\right)
| |||||
A=\left(2.457ln\left(\left(\left(
| 7 | |
| Re |
\right)0.9+0.27
| \varepsilon | |
| D |
\right)-1\right)\right)16
B=\left(
| 37530 | |
| Re |
\right)16
RH
ReH
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:
\Deltah=f
| u2L | |
| gR |
=2f
| u2L | |
| gD |
where:
\Deltah
f
u
L
g
D