Darcy friction factor $\begin{array}{l}f\end{array}$ depends on Reynolds number and a shape and roughness $\begin{array}{l}\epsilon\end{array}$ of inner pipe walls:

$\begin{array}{l}\displaystyle f = f({\rm Re}, \epsilon)\end{array}$

For a smooth ( $\begin{array}{l}\epsilon = 0\end{array}$ ) tubular pipeline Darcy friction factor $\begin{array}{l}f\end{array}$ can be estimated from various empirical correlations :

(1)	$\begin{array}{l}\displaystyle f = 64 \, \rm Re^{-1}\end{array}$

$\begin{array}{l}\rm Re < 2,100\end{array}$

Laminar fluid flow

no universal correlations due to a high flow instability

$\begin{array}{l}2,100 < \rm Re < 4,000\end{array}$

Laminar-turbulent transition fluid flow

(2)	$\begin{array}{l}\displaystyle f = 0.32 \, \rm Re^{-0.25}\end{array}$

$\begin{array}{l}4,000 < \rm Re < 50,000\end{array}$

Turbulent fluid flow

(3)	$\begin{array}{l}\displaystyle f = 0.184 \, \rm Re^{-0.2}\end{array}$

$\begin{array}{l}\rm Re > 50,000\end{array}$

Strong-turbulent fluid flow

where

$\begin{array}{l}{\rm Re}(l) = \frac{d \, v \, \rho}{\mu}\end{array}$	Reynolds number
$\begin{array}{l}d(l)\end{array}$	Inner diameter of a pipe
$\begin{array}{l}\mu(l) = \mu( \, p(l), \, T(l) \,)\end{array}$	dynamic fluid viscosity as function $\begin{array}{l}\mu(p, T)\end{array}$ of pressure $\begin{array}{l}p(l)\end{array}$ and temperature $\begin{array}{l}T(l)\end{array}$ along the pipe

For non-smooth pipelines $\begin{array}{l}\epsilon > 0\end{array}$ the Darcy friction factor $\begin{array}{l}f\end{array}$ can be estimated from empirical Colebrook–White correlation which works for non-laminar flow:

(4)	$\begin{array}{l}\displaystyle \frac{1}{\sqrt{f}} = -2 \, \log \Bigg( \frac{\epsilon}{3.7 \, d} + \frac{2.51}{{\rm Re} \sqrt{f}} \Bigg)\end{array}$

Typical surface roughness of a factory steel pipelines is $\begin{array}{l}\epsilon\end{array}$ = 0.05 mm which may increase significantly under mineral sedimentation or erosive impact of the flowing fluids.

See Surface roughness for more data on typical values for various materials and processing conditions.

Interpolated full-range model

The most popular full-range model of Darcy friction factor is:

(5)	$\begin{cases} f = 64/\mbox{Re} & \forall & \mbox{Re}< 2,100 \\f = 0.03048 + k \cdot ( \mbox{Re} -2,100) & \forall & 2,100 < \mbox{Re}< 4,000 \\f = f_{CW}( \mbox{Re}, \, \epsilon) & \forall & \mbox{Re}>4,000 \end{cases}$

where

$\begin{array}{l}f_{CW}(\mbox{Re}, \epsilon)\end{array}$	Colebrook–White correlation
$\begin{array}{l}\displaystyle k = \frac{ f_{CW}( \mbox{Re} =4,000, \, \epsilon) -0.03048}{1,900}\end{array}$	interpolation multiplier between laminar and turbulent flow regimes

Bellos full-range model

(6)	$\begin{array}{l}\displaystyle f = \frac{64}{\rm Re} \cdot \Phi\end{array}$

(7)	$\begin{array}{l}\displaystyle \Phi = \left( \frac{{\rm Re}}{64} \right)^{1-a} \cdot \left( 0.75 \cdot \ln \frac{{\rm Re}}{5.37} \right)^{-2 \,(1-a)\,b} \cdot \left( 0.83 \cdot \ln \frac{3.41}{\epsilon/d} \right)^{-2 \,(1-a)\,(1-b)}\end{array}$

(8)	$\begin{array}{l}\displaystyle a = \left[ 1+ \left( \frac{{\rm Re}}{2712} \right)^{8.4} \right]^{-1}\end{array}$

(9)	$\begin{array}{l}\displaystyle b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{150} \right)^{1.8} \right]^{-1}\end{array}$

Cheng full-range model

(10)	$\begin{array}{l}\displaystyle f = \frac{64}{\rm Re} \cdot \Phi\end{array}$

(11)	$\begin{array}{l}\displaystyle \Phi = \left( \frac{{\rm Re}}{64} \right)^{1-a} \cdot \left( 1.8 \cdot \ln \frac{{\rm Re}}{6.8} \right)^{-2 \,(1-a)\,b} \cdot \left( 2.0 \cdot \ln \frac{3.7}{\epsilon/d} \right)^{-2 \,(1-a)\,(1-b)}\end{array}$

(12)	$\begin{array}{l}\displaystyle a = \left[ 1+ \left( \frac{{\rm Re}}{2720} \right)^9 \right]^{-1}\end{array}$

(13)	$\begin{array}{l}\displaystyle b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{160} \right)^2 \right]^{-1}\end{array}$

Churchill full-range model

(14)	$\begin{array}{l}\displaystyle f = \frac{64}{\rm Re} \cdot \Phi\end{array}$

(15)	$\begin{array}{l}\displaystyle \Phi = \left[ 1+ \frac{\left(\rm Re / 8 \right)^{12} }{ \left( \Theta_1 + \Theta_2 \right)^{1.5} } \right]^{1/12}\end{array}$

(16)	$\begin{array}{l}\displaystyle \Theta_1 = \left[ 2.457 \, \ln \left( \left( \frac{7}{\rm Re} \right)^{0.9} + 0.27 \, \frac{\epsilon}{d} \right) \right]^{16}\end{array}$

(17)	$\begin{array}{l}\displaystyle \Theta_2 = \left( \frac{37530}{\rm Re} \right)^{16}\end{array}$

Reference

Moody’s Friction Factor Calculator @ gmallya.com

Page tree

Darcy friction factor Single-phase @model

Interpolated full-range model

Bellos full-range model

Cheng full-range model

Churchill full-range model

Reference