@wikipedia
Darcy friction factor
depends on
Reynolds number and a shape and
roughness of inner pipe walls:
| LaTeX Math Block |
|---|
|
f = f({\rm Re}, \epsilon) |
For a smooth (
) tubular pipeline
Darcy friction factor can be estimated from various
empirical correlations :
where
For non-smooth pipelines
the
Darcy friction factor can be estimated from
empirical Colebrook–White correlation which works for non-
laminar flow:
| LaTeX Math Block |
|---|
|
\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
= 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:
| LaTeX Math Block |
|---|
|
\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
| LaTeX Math Inline |
|---|
| body | --uriencoded--f_%7BCW%7D(\mbox%7BRe%7D, \epsilon) |
|---|
|
| Colebrook–White correlation |
| LaTeX Math Inline |
|---|
| body | --uriencoded--\displaystyle k = \frac%7B f_%7BCW%7D( \mbox%7BRe%7D =4,000, \, \epsilon) -0.03048%7D%7B1,900%7D |
|---|
|
| interpolation multiplier between laminar and turbulent flow regimes |
| LaTeX Math Block |
|---|
| f = \frac{64}{\rm Re} \cdot \Phi |
| | LaTeX Math Block |
|---|
| \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)} |
|
| LaTeX Math Block |
|---|
| a = \left[ 1+ \left( \frac{{\rm Re}}{2712} \right)^{8.4} \right]^{-1} |
| | LaTeX Math Block |
|---|
| b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{150} \right)^{1.8} \right]^{-1} |
|
Cheng full-range model
| LaTeX Math Block |
|---|
| f = \frac{64}{\rm Re} \cdot \Phi |
| | LaTeX Math Block |
|---|
| \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)} |
|
| LaTeX Math Block |
|---|
| a = \left[ 1+ \left( \frac{{\rm Re}}{2720} \right)^9 \right]^{-1} |
| | LaTeX Math Block |
|---|
| b = \left[ 1+ \left( \frac{{\rm Re} \cdot \epsilon/d}{160} \right)^2 \right]^{-1} |
|
| LaTeX Math Block |
|---|
| f = \frac{64}{\rm Re} \cdot \Phi |
| | LaTeX Math Block |
|---|
| anchor | Chirchil |
|---|
| alignment | left |
|---|
| \Phi = \left[ 1+ \frac{\left(\rm Re / 8 \right)^{12} }{ \left( \Theta_1 + \Theta_2 \right)^{1.5} } \right]^{1/12} |
|
| LaTeX Math Block |
|---|
| \Theta_1 = \left[ 2.457 \, \ln \left( \left( \frac{7}{\rm Re} \right)^{0.9} + 0.27 \, \frac{\epsilon}{d} \right) \right]^{16} |
| | LaTeX Math Block |
|---|
| \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 ] [ Reduced Friction Factor (Φ) ]
Reference
Moody’s Friction Factor Calculator @ gmallya.com
