Motivation
One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary 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.
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Numerical quadrature solution of Pressure Profile in Homogeneous Steady-State Pipe Flow @model
Outputs
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Assumptions
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| Stationary flow | Homogenous Isothermal or conditions| flow |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
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| | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l) |
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| Homogenous flow | |
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| Pressure profile |
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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)=\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d} - (2/j_m^2) \, \int_p^{p_0} \frac{dp}{\rho} - (2/j_m^2) \, g \, \Delta z(l) = 0 |
| Mass Flux | Mass Flowrate |
| LaTeX Math Block |
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| anchor | MassFlux |
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| alignment | left |
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j_m = \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d}
}
}
|
| LaTeX Math Block |
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| anchor | MassFlowrate |
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| alignment | left |
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|
\dot m =
A \cdot \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d}
}
} |
Volumetric Flowrate | | LaTeX Math Block |
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| anchor | VolumtericFlowrate |
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| alignment | left |
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q_0 =
\frac{A}{\rho_s} \cdot \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d}
}
}
|
where
| body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial \tau_x%7D =\frac%7B\partial p%7D%7B\partial \tau_y%7D =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l) |
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| Constant inclination | Constant friction along hole |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \theta(l) = \theta = %7B\rm const%7D \rightarrow \cos \theta = \frac%7Bdz%7D%7Bdl%7D = %7B\rm const%7D |
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Equations
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| 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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| L =\int_{p_0}^{p} \frac{\rho \, dp}{G \, \rho^2 - F} -\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G} |
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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 = \rm const |
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| mass |
| LaTeX Math Inline |
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| body | \rho_0 = \rho(T_0, p_0) |
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| Intake fluid density |
| Intake mass | flux |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D = \rm const |
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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 |
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/ 0 u u(l=0) = q_0 / A = j_m / \rho_0Intake Fluid velocity | 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 |
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_s _s at intake point| |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D(T,\rho) = \ |
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frac%7Bu(l) | frac%7Bj_m \cdot d%7D%7B\ |
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nu(l)%7D = \frac%7B4 \rho_0 q_0%7D%7B\pi d%7D \frac%7B1%7D%7B\mu%7DReynolds number at intake point | 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) = \rm const |
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The first term in the right side of
| LaTeX Math Block Reference |
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defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:...
| LaTeX Math Inline |
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| body | f(l) = f_s = \rm const |
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Alternative forms
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| Density form |
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| LaTeX Math Block |
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| L = L(\rho) =\int_{\rho_0}^{\rho} \frac{ 1}{G \, \rho^2 - F} \, \frac{ d\rho}{c(\rho)} -\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}
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Approximations
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| LaTeX Math Block |
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| \rho(p) = \tilde \rho \cdot \sqrt{ 1- \frac{f}{2d} \frac{j_m^2}{G} ( \rho_0^2 - {\tilde \rho}^2) }
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| LaTeX Math Block |
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| \int_{\rho_0}^{\tilde \rho} \frac{d \rho}{ \rho^2 \, c(\rho)} = G \cdot L
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where
| no-flow pressure at the pipe end ( | LaTeX Math Inline |
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| body | --uriencoded--%7B\tilde j%7D_m = 0 |
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) |
See also
References
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| bgColor | papayawhip |
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| title | ARAX |
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