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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| 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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| 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 |
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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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| F(p, l)= L =\int_{p_0}^p^{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]
= 0dp}{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 |
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| body | --uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D = \rm const |
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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 |
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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(T,p) = f(%7B\rm Re%7D(T,p\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,p) = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D \rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T,p\rho)%7D |
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| Reynolds number in Pipe Flow |
| dynamic viscosity as function of fluid temperature and pressure density |
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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) |
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It can also be written in the following form:
Alternative forms
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| Density form |
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| Pressure profile along the pipe |
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| PressureProfile | F(p, l)= | p | ^p | dp | \rho} | g | Delta z(l)
+ 0.5 \cdot j_m^2frac{ d\rho}{c(\rho)} -\frac{d}{f} \cdot |
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| left( | | 1}{ | } | \frac{1 |
Approximations
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| \rho(p) = \tilde \ |
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right)
+
frac16, l, md^2cdot
\left( \frac{\mu \, \Phi}{\rho^2} + \frac{\mu_0 \, \Phi_0}{\rho_0^2} \right)
= 0rho_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 ( |
Phi = \frac%7B1%7D%7B64%7D \cdot f \cdot %7B\rm Re%7DReduced Friction Factor | ...
See also
References
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| bgColor | papayawhip |
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| title | ARAX |
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