Friction Factor

Written by Jerry Ratzlaff on . Posted in Dimensionless Numbers

Friction factor, abbreviated as f, also called Moody friction factor or Darcy-Weibach friction factor, is a dimensionless number used in internal flow calculations with the Darcy-Weisbach equation.  Depending on the Reynolds Number, the friction factor, abbreviated as f, may be calculated one of several ways.

 

laminar flow

In laminar flow, the friction factor is independent of the surface roughness, \(\epsilon\).  This is because the fluid flow profile contains a boundary layer where the flow at the surface through the height of the roughness is zero.

For \(Re<2100\), the friction factor may be calculated by:

\(\large{ f = \frac{64}{Re} }\)   

 

transitional flow

For \(2100<Re<3x10^3\) (transitional flow regime), the friction factor may be estimated from the Moody Diagram.

 

turbulent flow

Methods for finding the friction factor f are to use a diagram, such as the Moody Diagram, the Colebrook-White Equation, or the Swamee-Jain Equation.

Using the diagram or Colebrook-White equation requires iteration. Where the Swamee-Jain equation allows f to be found directly for full flow in a circular pipe.

 

colebrook-white equation

The '''Colebrook-White equation''' is used to iteratively solve for the Darcy Weisbach Friction Factor ''f''.

 

Free Surface Flow

\(\large{  \frac{1}{\sqrt{f}} = -2\; \log \;(\frac{\epsilon}{12\;r_h} + \frac{2.51}{Re\sqrt{f}})  }\)   

 

Full Flow (Closed Conduit)

\(\large{   \frac{1}{\sqrt{f}} = -2\; \log \;(\frac{\epsilon}{14.8\;r_h} + \frac{2.51}{Re\sqrt{f}})  }\)   

Where:

\(\large{ \epsilon }\)  (Greek symbol epsilon) = absolute roughness

\(\large{ f }\) = friction factor

\(\large{ r_h }\) = hydraulic radius

\(\large{ Re }\) = Reynolds number

Because the iterative search for the correct  \(f\)  value can be quite time-consuming, the Swamee-Jain equation can be used to solve directly for  \(f\).

 

swamee-jain equation

The Swamee-Jain Equation is accurate to 1.0% of the Colebrook-White Equation for  \(\large{  10^{-6} < \frac{\epsilon}{d} < 10^{-2} }\)  and  \(\large{ 5,000 < Re < 10^8  }\).

\(\large{  f = \frac{0.25}{[log \; (\frac{\epsilon}{3.7\;d} + \frac{5.74}{Re^{0.9}})]^2}  }\)   

Where:

\(\large{ \epsilon }\)  (Greek symbol epsilon) = absolute roughness

\(\large{ d }\) = inside diameter of pipe

\(\large{ Re }\) = Reynolds number

 

Friction Factor calculator

 

Tags: Equations for Friction Equations for Flow