Darcy friction factor @model

@wikipedia

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

нет стабильных корреляций

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

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 Chirchill correlation provides a fair (< 2 % accuracy) 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.

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Darcy friction factor @model