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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Inputs
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Assumptions
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Stationary Homogenous flow | Isothermal or conditionsIncompressible fluid \rho(T, p)=\rho_s = \rm const | Isoviscous flow --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
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\mu(T, p) = \mu_s = \rm const--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 | Pressure gradient profile |
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LaTeX Math Block |
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anchor | PPconst |
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alignment | left |
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j_m = j_m[p(l)] \rightarrow p = p(l) |
LaTeX Math Block |
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\frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s |
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{1}{\rho^2} + \frac{1}{\rho_0^2} \right)
\cdot \frac{f \, \cdot \, l}{ 2 \, d}
}
}
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LaTeX Math Block |
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anchor | MassFlowrate |
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alignment | left |
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\dot m = j_m \cdot A =
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{1}{\rho^2} + \frac{1}{\rho_0^2} \right)
\cdot \frac{f \, \cdot \, l}{ 2 \, d}
}
} |
Volumetric Flowrate | Intake Fluid velocity |
LaTeX Math Block |
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anchor | VolumtericFlowrate |
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alignment | left |
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q_s = \dot m / \rho_s =
\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{1}{\rho^2} + \frac{1}{\rho_0^2} \right)
\cdot \frac{f \, \cdot \, l}{ 2 \, d}
}
}
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LaTeX Math Block |
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anchor | IntakeFluidVelocity |
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alignment | left |
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u_s = j_m/ \rho_s =q_s / A
\frac{1}{\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{1}{\rho^2} + \frac{1}{\rho_0^2} \right)
\cdot \frac{f \, \cdot \, 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 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 %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= 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) = \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
| Intake mass flux |
| mass flowrate |
| Intake Fluid velocity |
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_s = f(%7B\rm Re%7D_s, \, \epsilon) |
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| Darcy friction factor at intake point | no-flow pressure at the pipe end ( |
\displaystyle rm Re%7Ds = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7DReynolds number at intake point | 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) |
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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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See also
References
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bgColor | papayawhip |
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title | ARAX |
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