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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:| LaTeX Math Block |
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| anchor | Chirchil |
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| alignment | left |
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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} |
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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} |
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}, & \mbox{if Re}<2,100
\\f = a + b \cdot \mbox{Re}, & 2,100 < \mbox{if Re}<4,000
\\f = f_{CW}( \mbox{Re}, \, \epsilon), & \mbox{if 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 b = \frac%7B f_%7BCW%7D( \mbox%7BRe%7D =4,000, \, \epsilon) -0.03048%7D%7B1,900%7D |
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| body | a=0.03048 - 2,100 \cdot b |
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Churchill full-range model
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:
| LaTeX Math Block |
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| anchor | Chirchil |
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| alignment | left |
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| 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} |
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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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| LaTeX Math Block |
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| \Theta_2 = \left( \frac{37530}{\rm Re} \right)^{16} |
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See also
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Physics / Fluid Dynamics / Pipe Flow Dynamics / Darcy–Weisbach equation / Darcy friction factor
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