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.
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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.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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| L = \int_{\rhop_0}^{\rhop} \frac{\frac{1}{c}- \frac{j_m^2}{\rho}}{\rho^2 g \, \cos \theta - \frac{j_m^2 \, f}{2d}} \, d \rhorho \, 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
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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 |
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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 |
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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) |
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body | G = g \, \cos \theta = \rm const |
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| gravity acceleration along pipe |
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body | --uriencoded--F = j_m%5e2 \cdot f/(2d) = \rm const |
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The equation
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can also be written in the following form:
Alternative forms
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Density form | Pressure profile along the pipe |
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| F2 | F(p, l)= | p | ^p | 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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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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| \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 %7B\rm Re%7D \cdot f Reduced Friction Factor
See also
References
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bgColor | papayawhip |
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title | ARAX |
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