The Fanning friction factor, named after
John Thomas Fanning, is a
dimensionless number
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
used as a local parameter in
continuum mechanics calculations. It is defined as the ratio between the local
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
and the local flow kinetic energy density:
:
where:
*
is the local Fanning friction factor (dimensionless)
*
is the local
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
(unit in
or
or Pa)
*
is the bulk
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
(unit in
or
)
*
is the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
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
Fanning friction factor formula
This friction factor is one-fourth of the
Darcy friction factor
Darcy, Darci or Darcey may refer to:
Science
* Darcy's law, which describes the flow of a fluid through porous material
* Darcy (unit), a unit of permeability of fluids in porous material
* Darcy friction factor in the field of fluid mechanics ...
, 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
Darcy, Darci or Darcey may refer to:
Science
* Darcy's law, which describes the flow of a fluid through porous material
* Darcy (unit), a unit of permeability of fluids in porous material
* Darcy friction factor in the field of fluid mechanics ...
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 fluid
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
s 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