Fanning friction factor

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:

f

=

τ

q

{\displaystyle f={\frac {\tau }{q}}}

where

f is the local Fanning friction factor (dimensionless);

τ is the local shear stress (units of pascals (Pa) = N/m2, or pounds per square foot (psf) = lbf/ft2);

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

q

=

1

2

ρ

u

2

{\displaystyle q={\frac {1}{2}}\rho u^{2}}

ρ is the density of the fluid (kg/m3 or lbm/ft3)

u is the bulk flow velocity (m/s or ft/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

π

R

L

{\displaystyle 2\pi RL}

for a pipe with circular cross section) and dividing by the cross-sectional flow area (

π

R

2

{\displaystyle \pi R^{2}}

for a pipe with circular cross section). Thus

Δ

P

=

f

2

L

R

q

=

f

L

R

ρ

u

2

{\displaystyle \Delta P=f{\frac {2L}{R}}q=f{\frac {L}{R}}\rho u^{2}}

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

f

D

=

8

τ

¯

ρ

u

¯

2

{\displaystyle f_{D}={\frac {8{\bar {\tau }}}{\rho {\bar {u}}^{2}}}}

where:

τ

{\displaystyle \tau }

is the shear stress at the wall

ρ

{\displaystyle \rho }

is the density of the fluid

u

¯

{\displaystyle {\bar {u}}}

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:

f

=

16

R

e

{\displaystyle f={\frac {16}{Re}}}

where Re is the Reynolds number of the flow.

For a square channel the value used is:

f

=

14.227

R

e

{\displaystyle f={\frac {14.227}{Re}}}

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

2100

<

R

e

<

10

5

{\displaystyle 2100<Re<10^{5}}

.

f

=

0.0791

R

e

0.25

{\displaystyle f={\frac {0.0791}{Re^{0.25}}}}

Koo introduced another explicit formula in 1933 for a turbulent flow in region of

10

4

<

R

e

<

10

7

{\displaystyle 10^{4}<Re<10^{7}}

f

=

0.0014

+

0.125

R

e

0.32

{\displaystyle f=0.0014+{\frac {0.125}{Re^{0.32}}}}

Pipes/tubes of general roughness

When the pipes have certain roughness

ϵ

D

<

0.05

{\displaystyle {\frac {\epsilon }{D}}<0.05}

, 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

4

⋅

10

4

<

R

e

<

10

7

{\displaystyle 4\centerdot 10^{4}<Re<10^{7}}

1

f

=

−

3.6

log

10

⁡

[

6.9

R

e

+

(

ϵ

/

D

3.7

)

10

/

9

]

{\displaystyle {\frac {1}{\sqrt {f}}}=-3.6\log _{10}\left[{\frac {6.9}{Re}}+\left({\frac {\epsilon /D}{3.7}}\right)^{10/9}\right]}

where

ϵ

{\displaystyle \epsilon }

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.