Darcy friction factor $\begin{array}{l}f\end{array}$ depends on flow regime, as well as shape and roughness $\begin{array}{l}\epsilon\end{array}$ of inner pipe walls.

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

For many practical applications the Churchill correlation provides a fair (< 2 % accuracy and improving towards laminar flow) estimation of Darcy friction factor $\begin{array}{l}f\end{array}$ for all pipe flow regimes:

(5)	$\begin{array}{l}\displaystyle f = \frac{64}{\rm Re} \, \Bigg [ 1+ \frac{\big(\rm Re / 8 \big)^{12} }{ \big( \Theta_1 + \Theta_2 \big)^{1.5} } \Bigg]^{1/12}\end{array}$

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

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

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.

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

(8)	$\begin{cases} f = 64/\mbox{Re}, & \mbox{if Re}< 2,100 \\f = a + b \cdot \mbox{Re}, & 2,100 < \mbox{if Re}< 4,100 \\f = f_{CW}( \mbox{Re}, \, \epsilon), & \mbox{if Re}>2,100 \end{cases}$

where

$\begin{array}{l}f_{CW}(\mbox{Re}, \epsilon)\end{array}$	Colebrook–White correlation
$\begin{array}{l}\displaystyle b = \frac{ f_{CW}( \mbox{Re} =4,000, \, \epsilon) -0.03048}{1,900}\end{array}$
$\begin{array}{l}a=0.03048 - 2,100 \cdot b\end{array}$

Reference

Moody’s Friction Factor Calculator @ gmallya.com

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Darcy friction factor @model

Reference