Fanning friction factor

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:

[math]\displaystyle{ f = \frac {\tau}{\rho \frac {u^2} 2} }[/math]^[1][2]

where:

• [math]\displaystyle{ f }[/math] is the local Fanning friction factor (dimensionless)
• [math]\displaystyle{ \tau }[/math] is the local shear stress (unit in [math]\displaystyle{ \frac{lb_m}{ft\cdot s^2} }[/math] or [math]\displaystyle{ \frac{kg}{m\cdot s^2} }[/math] or Pa)
• [math]\displaystyle{ u }[/math] is the bulk flow velocity (unit in [math]\displaystyle{ \frac{ft}{s} }[/math] or [math]\displaystyle{ \frac{m}{s} }[/math])
• [math]\displaystyle{ \rho }[/math] is the density of the fluid (unit in [math]\displaystyle{ \frac{lb_m}{ft^3} }[/math] or [math]\displaystyle{ \frac{kg}{m^3} }[/math])

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 ( [math]\displaystyle{ 2 \pi R L }[/math] for a pipe with circular cross section) and dividing by the cross-sectional flow area ( [math]\displaystyle{ \pi R^2 }[/math] for a pipe with circular cross section). Thus [math]\displaystyle{ \Delta P = f \frac{L}{R} \rho u^2 }[/math]

Fanning friction factor formula

Fanning friction factor for tube flow

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]

[math]\displaystyle{ f _ {D} = \frac{8 \bar \tau}{\rho \bar u ^ 2} }[/math]

where:

• [math]\displaystyle{ \tau }[/math] is the shear stress at the wall
• [math]\displaystyle{ \rho }[/math] is the density of the fluid
• [math]\displaystyle{ \bar u }[/math] 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:^[4]

[math]\displaystyle{ f= \frac{16}{Re} }[/math]^[5][2]

where Re is the Reynolds number of the flow.

For a square channel the value used is:

[math]\displaystyle{ f = \frac{14.227}{Re} }[/math]

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 [math]\displaystyle{ 2100\lt Re\lt 10^5 }[/math].

[math]\displaystyle{ f=\frac{0.0791}{Re^{0.25}} }[/math]^[6][2]

Koo introduced another explicit formula in 1933 for a turbulent flow in region of [math]\displaystyle{ 10^4\lt Re\lt 10^7 }[/math]

[math]\displaystyle{ f=0.0014+\frac{0.125}{Re^{0.32}} }[/math]^[7][8]

Pipes/tubes of general roughness

When the pipes have certain roughness [math]\displaystyle{ \frac{\epsilon}{D}\lt 0.05 }[/math], 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 [math]\displaystyle{ 4 \centerdot10^4\lt Re\lt 10^7 }[/math]

[math]\displaystyle{ \frac{1}{\sqrt{f}}=-3.6\log_{10}\left [ \frac{6.9}{Re}+\left ( \frac{\epsilon/D}{3.7} \right )^{10/9} \right ] }[/math]^[2][9][8]

where

• [math]\displaystyle{ \epsilon }[/math] 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.^[10]

[math]\displaystyle{ f = \frac{0.0625}{\left[\log\left (\frac{\varepsilon/D}{3.7} + \frac{5.74}{\mathrm{Re}^{0.9}}\right)\right]^2} }[/math]

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 [math]\displaystyle{ Re\gt 10^4;\frac{k}{D}\gt 0.01 }[/math]. The equation below has been modified from the original format which was developed for Darcy friction factor by a factor of [math]\displaystyle{ \frac{1}{4} }[/math]

[math]\displaystyle{ \frac{1}{\sqrt{f}}=2.28-4.0\log_{10}\left ( \frac{k}{D} \right ) }[/math]^[11][12]

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^[6] which is implicit in [math]\displaystyle{ f }[/math]:

[math]\displaystyle{ {1 \over \sqrt{\mathit{f}}}= -4.0 \log_{10} \left(\frac{\frac{\varepsilon}{d}}{3.7} + {\frac{1.255}{Re \sqrt{\mathit{f} } } } \right) , \text{turbulent flow} }[/math]

Various explicit approximations of the related Darcy friction factor have been developed for turbulent flow.

Stuart W. Churchill^[5] 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 [math]\displaystyle{ f_D }[/math], also called Moody friction factor, is 4 times the Fanning friction factor [math]\displaystyle{ f }[/math] and so a factor of [math]\displaystyle{ \frac{1}{4} }[/math] 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;

[math]\displaystyle{ f = 2 \left( \left( \frac {8} {Re} \right) ^ {12} + \left( A+B \right) ^ {-1.5} \right) ^ {\frac {1} {12} } }[/math]

[math]\displaystyle{ A = \left( 2.457 \ln \left( \left( \left( \frac {7} {Re} \right) ^ {0.9} + 0.27 \frac {\varepsilon} {D} \right)^ {-1}\right) \right) ^ {16} }[/math]

[math]\displaystyle{ B = \left( \frac {3753} {Re} \right) ^ {16} }[/math]

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 [math]\displaystyle{ R_H }[/math] when calculating for Reynolds number [math]\displaystyle{ Re_H }[/math]

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:

[math]\displaystyle{ \Delta h = f \frac{u^2 L}{gR} = 2f \frac{u^2 L}{gD} }[/math]

where:

• [math]\displaystyle{ \Delta h }[/math] is the friction loss (in head) of the pipe.
• [math]\displaystyle{ f }[/math] is the Fanning friction factor of the pipe.
• [math]\displaystyle{ u }[/math] is the flow velocity in the pipe.
• [math]\displaystyle{ L }[/math] is the length of pipe.
• [math]\displaystyle{ g }[/math] is the local acceleration of gravity.
• [math]\displaystyle{ D }[/math] is the pipe diameter.

References

Further reading

• Fanning, J.T. (1896). A practical treatise on hydraulic and water-supply engineering. D. Van Nostrand. ISBN 978-5-87581-042-8. https://books.google.com/books?id=qbMNAwAAQBAJ&pg=PP1. 

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