# Friction Factor

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:

## laminar flow Formula

$$\large{ f = \frac{64}{Re} }$$
Symbol 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

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

$$\large{ \frac{1}{\sqrt{f}} = -2\; \log \;(\frac{\epsilon}{14.8\;r_h} + \frac{2.51}{Re\sqrt{f}}) }$$
Symbol 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  $$\large{ 5,000 < Re < 10^8 }$$.

## swamee-jain equation

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