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@wikipedia
Darcy friction factor
f depends on Reynolds number and a shape and roughness
\epsilon of inner pipe walls:
|
f = f({\rm Re}, \epsilon) |
For a smooth (
\epsilon = 0) tubular pipeline Darcy friction factor
f can be estimated from various empirical correlations :
where
For non-smooth pipelines
\epsilon > 0 the Darcy friction factor
f can be estimated from empirical Colebrook–White correlation which works for non-laminar flow:
(4) |
\frac{1}{\sqrt{f}} = -2 \, \log \Bigg( \frac{\epsilon}{3.7 \, d} + \frac{2.51}{{\rm Re} \sqrt{f}} \Bigg) |
Typical surface roughness of a factory steel pipelines is
\epsilon = 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
f_{CW}(\mbox{Re}, \epsilon) | Colebrook–White correlation |
\displaystyle k = \frac{ f_{CW}( \mbox{Re} =4,000, \, \epsilon) -0.03048}{1,900} | interpolation multiplier between laminar and turbulent flow regimes |
Bellos full-range model
(6) |
f = \frac{64}{\rm Re} \cdot \Phi |
|
(7) |
\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)} |
|
(8) |
a = \left[ 1+ \left( \frac{{\rm Re}}{2712} \right)^{8.4} \right]^{-1} |
|
(9) |
b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{150} \right)^{1.8} \right]^{-1} |
|
Cheng full-range model
(10) |
f = \frac{64}{\rm Re} \cdot \Phi |
|
(11) |
\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)} |
|
(12) |
a = \left[ 1+ \left( \frac{{\rm Re}}{2720} \right)^9 \right]^{-1} |
|
(13) |
b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{160} \right)^2 \right]^{-1} |
|
(14) |
f = \frac{64}{\rm Re} \cdot \Phi |
|
(15) |
\Phi = \left[ 1+ \frac{\left(\rm Re / 8 \right)^{12} }{ \left( \Theta_1 + \Theta_2 \right)^{1.5} } \right]^{1/12} |
|
(16) |
\Theta_1 = \left[ 2.457 \, \ln \left( \left( \frac{7}{\rm Re} \right)^{0.9} + 0.27 \, \frac{\epsilon}{d} \right) \right]^{16} |
|
(17) |
\Theta_2 = \left( \frac{37530}{\rm Re} \right)^{16} |
|
See also
Physics / Fluid Dynamics / Pipe Flow Dynamics / Darcy–Weisbach equation / Darcy friction factor
[ Surface roughness ]
Reference
Moody’s Friction Factor Calculator @ gmallya.com