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Darcy friction factor
depends on
flow regime, as well as shape Reynolds number and a shape and
roughness of inner pipe walls
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f = f({\rm Re}, \epsilon) |
For a smooth (
) tubular pipeline
Darcy friction factor can be estimated from various
empirical correlations :
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\frac{1}{\sqrt{f}} = -2 \, \log \Bigg( \frac{\epsilon}{3.7 \, d} + \frac{2.51}{{\rm Re} \sqrt{f}} \Bigg) |
For many practical applications the Churchill correlation provides a fair (< 2 % accuracy and improving towards laminar flow) estimation of Darcy friction factor
for all pipe flow regimes:
Typical surface roughness of a factory steel pipelines is
= 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
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The most popular full-range model of Darcy friction factor is:
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\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
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| body | --uriencoded--f_%7BCW%7D(\mbox%7BRe%7D, \epsilon) |
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| Colebrook–White correlation |
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| body | --uriencoded--\displaystyle k = \frac%7B f_%7BCW%7D( \mbox%7BRe%7D =4,000, \, \epsilon) -0.03048%7D%7B1,900%7D |
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| interpolation multiplier between laminar and turbulent flow regimes |
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| f = \frac{64}{\rm Re} \cdot \Phi |
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| \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)} |
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| a = \left[ 1+ \left( \frac{{\rm Re}}{2712} \right)^{8.4} \right]^{-1} |
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| b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{150} \right)^{1.8} \right]^{-1} |
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Cheng full-range model
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| f = \frac{64}{\rm Re} \cdot \Phi |
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| \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)} |
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| a = \left[ 1+ \left( \frac{{\rm Re}}{2720} \right)^9 \right]^{-1} |
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| b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{160} \right)^2 \right]^{-1} |
|
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| f = \frac{64}{\rm Re} \cdot \Phi |
| | LaTeX Math Block |
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| anchor | Chirchil |
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| alignment | left |
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| \Phi = \left |
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| anchor | Chirchil |
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| alignment | left |
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f = \frac{64}{\rm Re} \, \Bigg bigbigbigleft( \Theta_1 + \Theta_2 \ |
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bigBigg |
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| \Theta_1 = \left[ 2.457 \, \ln \left( \left( \frac{7}{\rm Re} \right)^{0.9} + 0.27 \, \frac{\epsilon}{d} \right) \right]^{16} |
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| \Theta_2 = \left( \frac{37530}{\rm Re} \right)^{16} |
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See Surface roughness for more data on typical values for various materials and processing conditions.
See also
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Physics / Fluid Dynamics / Pipe Flow Dynamics / Darcy–Weisbach equation / Darcy friction factor
[ Surface roughness ] [ Reduced Friction Factor (Φ) ]
Reference
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Moody’s Friction Factor Calculator @ gmallya.com
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