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.
Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
Outputs
Assumptions
Steady-State flow | Quasi-isothermal flow |
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body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 \rightarrow p(t,l) = p(l) |
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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(\tau_x,\tau_y,l) = p(l) |
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Equations
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| \left( \rho(p) - j_m^2 \cdot c(p) \right) \cdot \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l) - \frac{ j_m^2 }{2 d} \cdot f(p) |
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| p(l=0) = p_0 |
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| u(l) = \frac{j_m}{\rho(l)} |
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| q(l) =A \cdot u(l) |
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where
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body | --uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D= \rm const |
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| mass flux |
| Fluid flowrate at inlet point () |
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body | \rho_0 = \rho(T_0, p_0) |
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| Fluid density at inlet point () |
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body | \rho(l) = \rho(T(l), p(l)) |
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| Fluid density at any point |
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body | --uriencoded--\displaystyle с(p) = \frac%7B1%7D%7B\rho%7D \left( \frac%7B\partial \rho%7D%7B\partial p%7D \right)_T |
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| Fluid Compressibility |
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body | --uriencoded--f(T, \rho) = f(%7B\rm Re%7D(T, \rho), \, \epsilon) |
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| Darcy friction factor |
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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 |
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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) |
Approximations
See also
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bgColor | papayawhip |
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title | ARAX |
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