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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 Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
calculations. It is defined as the ratio between the local shear stress and the local flow kinetic energy density: : f = \frac where: *f is the local Fanning friction factor (dimensionless) *\tau is the local shear stress (unit in \frac or \frac or Pa) *u is the bulk flow velocity (unit in \frac or \frac) *\rho is the density of the fluid (unit in \frac or \frac) 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 \pi R L for a pipe with circular cross section) and dividing by the cross-sectional flow area ( \pi R^2 for a pipe with circular cross section). Thus \Delta P = f \frac \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 _ = \frac where: * \tau is the shear stress at the wall * \rho is the density of the fluid * \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= \frac where Re is the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
of the flow. For a square channel the value used is: f = \frac


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. f=\frac Koo introduced another explicit formula in 1933 for a turbulent flow in region of 10^4 f=0.0014+\frac


Pipes/tubes of general roughness

When the pipes have certain roughness \frac<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 \centerdot10^4 \frac=-3.6\log_\left \frac+\left ( \frac \right )^ \right /math> where * \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. It is an approximation of the implicit Colebrook–White equation. : f = \frac


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 Re>10^4;\frac>0.01. The equation below has been modified from the original format which was developed for Darcy friction factor by a factor of \frac \frac=2.28-4.0\log_\left ( \frac \right )


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 which is implicit in f: := -4.0 \log_ \left(\frac + \right) , \text 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 f_D , also called Moody friction factor, is 4 times the Fanning friction factor f and so a factor of \frac has been applied to produce the formula given below. * Re,
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
(
unitless 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) ...
); * ε, roughness of the inner surface of the pipe (dimension of length); * ''D'', inner pipe diameter; : f = 2 \left( \left( \frac \right) ^ + \left( A+B \right) ^ \right) ^ :A = \left( 2.457 \ln \left( \left( \left( \frac \right) ^ + 0.27 \frac \right)^ \right) \right) ^ :B = \left( \frac \right) ^ :


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 R_H when calculating for
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
Re_H


Application

The friction
head A head is the part of an organism which usually includes the ears, brain, forehead, cheeks, chin, eyes, nose, and mouth, each of which aid in various sensory functions such as sight, hearing, smell, and taste. Some very simple animals may ...
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: : \Delta h = f \frac = 2f \frac where: *\Delta h is the friction loss (in head) of the pipe. *f is the Fanning friction factor of the pipe. *u is the flow velocity in the pipe. *L is the length of pipe. *g is the local acceleration of gravity. *D is the pipe diameter.


References


Further reading

* {{DEFAULTSORT:Fanning Friction Factor Dimensionless numbers of fluid mechanics Equations of fluid dynamics Fluid dynamics Piping