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The base driving equations of a pipe flow isare:
| Equation of State (EOS) | IY1I1 | \frac{d p}{d l} =
-\rho \, u \, \frac{d u}{d l} + \rho \, g \, \cos \theta + f_{\rm cnt, \, l} |
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IY1I1 | j_m(l) = j_m = \rho(l) \cdot u = \rm const |
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| Equation of State (EOS) | Darcy–Weisbach |
| LaTeX Math Block |
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| \rho = \rho(p, T) |
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| LaTeX Math Block |
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| f_{\rm cnt, l} = - f \cdot \frac{ \rho \, u^2 \, }{2 d} |
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where
%7Bf%7D| f_%7B\rm cnt, l%7D(l) = %7B\bf e%7D_u \cdot %7B\bf f%7D_%7B\rm cnt%7D |
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| projection of | LaTeX Math Inline |
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| body | --uriencoded--%7B\bf f%7D_%7B\rm cnt%7D |
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onto the unit fluid velocity vector: | LaTeX Math Inline |
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| body | --uriencoded-- %7B\bf e%7D_u = %7B %7C %7B\bf u%7D %7C%7D %5e%7B-1%7D \, %7B\bf u%7D |
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| LaTeX Math Inline |
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| body | j_m = \rho(l) \cdot u(l) |
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| fluid mass flux |
| mass flowrate |
| standard gravity constant |
Substituting
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and | LaTeX Math Block Reference |
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into | LaTeX Math Block Reference |
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:...
frac{d p}{d l} =
-j_m \cdot \frac{d}{d l} \left( \frac{ |
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j_m}{\rho} \right) + \rho \, g \, \cos \theta - f \cdot \frac{ \rho \, }{2 d} \cdot \left( \frac{j_m}{\rho} \right |
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| LaTeX Math Block |
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\frac{d p}{d l} =
j^2_m \cdot \frac{1}{\rho^2} \frac{ |
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+ \rho \, g \, \cos \theta - \frac |
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{j_m^2}{2 d} \cdot \frac{f}{\rho} |
| LaTeX Math Block |
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\frac{d p}{d l} =
j^2_m \cdot \frac{1}{\rho^2} \frac{d \rho}{dp} \cdot \frac{d p}{dl} + \rho \, g \, \cos \theta - \frac{j_m^2 |
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\frac{d p}{d l} =
j^2_m \cdot \frac{1}{\rho} \cdot c \cdot \frac{d p}{dl} + \rho \, g \, \cos \theta - \frac{j_m^2}{2 d} \cdot \frac{f}{\rho} |
and finally
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\left( 1 - j_m^2 \cdot \frac{c}{\rho} \right ) \frac{dp}{dl} = \rho \, g \, \cos \theta - \frac{j_m^2 }{2 d} \cdot \frac{f}{\rho} |
Alternative forms
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| LaTeX Math Block |
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| \left[ \rho -j_m^2 \, c \right] \cdot \frac{d p}{dl} =
\rho^2 \, g \, \cos \theta - \frac{j_m^2 }{2d} \cdot f(\rho) |
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| LaTeX Math Block |
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| \left[ \frac{1}{c} - \frac{j_m^2}{\rho} \right] \cdot \frac{d \rho}{dl} =
\rho^2 \, g \, \cos \theta - \frac{j_m^2 }{2d} \cdot f(\rho) |
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u(l) = \frac{\rho_s \cdot q_s}{\rho(p) \cdot A}
See Also
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Petroleum Industry / Upstream / Pipe Flow Simulation / Water Pipe Flow @model / Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model
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