Motivation
Explicit solution of Pressure Profile in Homogeneous Steady-State Pipe Flow @model
Outputs
Assumptions
Equations
| Pressure profile along the pipe |
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| anchor | PressureProfile |
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| alignment | left |
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| L = \frac{1}{2 \, G \, c^* \rho^*} \cdot \ln \frac{G \, \rho^2-F}{G \, \rho_0^2-F}
-\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G} |
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| \cos \theta \neq 0 |
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| L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2)
- \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho} |
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| \cos \theta = 0 |
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where
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| body | --uriencoded--\displaystyle j_m = \frac%7B \dot m %7D%7B A%7D |
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| mass flux |
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| body | --uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D |
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| mass flowrate |
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| body | --uriencoded--\displaystyle q_0 = \frac%7BdV_0%7D%7Bdt%7D = \frac%7B \dot m %7D%7B \rho_0%7D |
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| Intake volumetric flowrate |
| LaTeX Math Inline |
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| body | \rho_0 = \rho(T_0, p_0) |
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| Intake fluid density |
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| body | \Delta z(l) = z(l)-z(0) |
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| elevation drop along pipe trajectory |
| LaTeX Math Inline |
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| body | --uriencoded--f = f(%7B\rm Re%7D(T,\rho), \, \epsilon) = \rm const |
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| Darcy friction factor |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D(T,\rho) =\frac%7Bj_m \cdot d%7D%7B\mu(T,\rho)%7D |
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| Reynolds number in Pipe Flow |
| dynamic viscosity as function of fluid temperature and density |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D = \rm const |
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| characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
| LaTeX Math Inline |
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| body | G = g \, \cos \theta = \rm const |
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| gravity acceleration along pipe |
| LaTeX Math Inline |
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| body | --uriencoded--F = j_m%5e2 \cdot f/(2d) = F(l) = \rm const |
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The
| LaTeX Math Block Reference |
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equation for horizontal pipelines can be re-written in terms of explicit pressure:
| LaTeX Math Block |
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L = (\rho^*/F) \cdot (p_0-p) \cdot (1+ 0.5 \, c^* \cdot (p+p_0)) |
See also
References
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| bgColor | papayawhip |
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| title | ARAX |
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