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.
Pressure Profile in L-Proxy Pipe Flow @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 |
|
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 |
|---|
| LaTeX Math Block |
|---|
| 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 |
|
where
See also
References
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| Panel |
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| bgColor | papayawhip |
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| title | ARAX |
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