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, a dimensionless number, is 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 bou                                                                                                                                                                                                                                                           ndary 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:

 

\(\large{ f = \frac{64}{Re} }\) 

Where:

 Units English Metric
\(\large{ f }\) = friction factor
\(\large{ dimensionless }\)
\(\large{ Re }\) = Reynolds number \(\large{ dimensionless }\)

 

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 flo                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    w 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 Formula

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

Where:

 Units English Metric
\(\large{ \epsilon }\)  (Greek symbol epsilon) = absolute roughness \(\large{ in }\) \(\large{ mm }\)
\(\large{ f }\) = friction factor \(\large{ dimensionless }\)
\(\large{ r_h }\) = hydraulic radius \(\large{ in }\) \(\large{ mm }\)
\(\large{ Re }\) = Reynolds number \(\large{ dimensionless }\)

 

Full Flow (Closed Conduit) Formula

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

Where:

 Units English Metric
\(\large{ \epsilon }\)  (Greek symbol epsilon) = absolute roughness \(\large{ in }\) \(\large{ mm }\)
\(\large{ f }\) = friction factor \(\large{ dimensionless }\)
\(\large{ r_h }\) = hydraulic radius \(\large{ in }\) \(\large{ mm }\)
\(\large{ Re }\) = Reynolds number \(\large{ dimensionless }\)

 

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:

 Units English Metric
\(\large{ f }\) = friction factor \(\large{ dimensionless }\)
\(\large{ \epsilon }\)  (Greek symbol epsilon) = absolute roughness \(\large{ in }\) \(\large{ mm }\)
\(\large{ d }\) = inside diameter of pipe \(\large{ in }\) \(\large{ mm }\)
\(\large{ Re }\) = Reynolds number \(\large{ dimensionless }\)

 

Friction Factor calculator

 

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Tags: Friction Equations Flow Equations