Motivation
One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the steady-state fluid transport.
In many practical cases the stationary pressure distribution can be approximated by Isothermal or Quasi-isothermal homogenous fluid flow model.
Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
Outputs
Assumptions
Equations
Pressure profile along the pipe |
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LaTeX Math Block |
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anchor | PressureProfile |
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alignment | left |
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| F(p, l)= \int_{p_0}^p \frac{dp}{\rho} -g \, \Delta z(l)
+ 0.5 \cdot j_m^2 \cdot \left[
\left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d} +
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
\right]
= 0 |
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where
LaTeX Math Inline |
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body | --uriencoded--\displaystyle j_m = \frac%7B \dot m %7D%7B A%7D |
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| mass flux |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D |
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| mass flowrate |
LaTeX Math Inline |
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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 |
LaTeX Math Inline |
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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(T,p) = f(%7B\rm Re%7D(T,p), \, \epsilon) |
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| Darcy friction factor |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle %7B\rm Re%7D(T,p) = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7Bj_m \cdot d%7D%7B\mu(T,p)%7D |
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| Reynolds number in Pipe Flow |
| dynamic viscosity as function of fluid temperature and pressure |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D |
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| characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
The equation
LaTeX Math Block Reference |
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can also be written in the following form:
Pressure profile along the pipe |
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LaTeX Math Block |
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| F(p, l)= \int_{p_0}^p \frac{dp}{\rho} -g \, \Delta z(l)
+ 0.5 \cdot j_m^2 \cdot
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+
\frac{16 \, l \, j_m}{d^2} \cdot
\left( \frac{\mu \, \Phi}{\rho^2} + \frac{\mu_0 \, \Phi_0}{\rho_0^2} \right)
= 0 |
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where
See also
References
Show If |
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bgColor | papayawhip |
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title | ARAX |
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