Motivation
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One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary 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.
Inputs & Outputs
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Outputs
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Intake temperature | LaTeX Math Inline |
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body | T_0 | | LaTeX Math Inline |
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body | up_0 | | Intake flowrate Inputs
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\theta (l) | Pipeline trajectory inclination | Fluid temperature at inlet point ( |
--uriencoded--%7B\bf r%7D(l) | | Fluid Assumptions
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Stationary fluid flow | Homogenous fluid 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 | |
LaTeX Math Inline |
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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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Equations
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bigg1\frac{\,\rho_0^2, q_0^2}{A^2} \bigg )right) \cdot \frac{dp}{dl} = \ |
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rhofrac{dz}{dl}\rho_0^2 \, q_0^2 A^2frac{f({\rm Re}, \, \epsilon)}{\rho}1q\frac{\rho \cdot q_0}{\rho} |
LaTeX Math Block |
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| u(l) = \frac{ |
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\rho_0 \cdot q_0 \cdot Ap0p=0 p_0 LaTeX Math Block |
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where
LaTeX Math Inline |
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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 |
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= q_0 LaTeX Math Block |
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| Fluid flowrate at inlet point () |
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| Fluid density at inlet point ( |
) = \rho_0...
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LaTeX Math Inline |
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body | \rho(l) = \rho(T(l), p(l)) |
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| Fluid density at any point |
LaTeX Math Inline |
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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 |
LaTeX Math Inline |
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body | --uriencoded--f(T, \rho) = f(%7B\rm Re%7D(T, \rho), \, \epsilon) |
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| Darcy friction factor |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle %7B\rm Re%7D |
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= \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_0 q_0%7D%7B\pi d%7D \frac%7B1%7D%7B(T,\rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T, |
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pcharacteristicCharacteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
See Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model.
Approximations
Incompressible pipe flow
with constant viscosity ...
Alternative forms
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PPconstp(l)=p_0 + \rho_0 \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l LaTeX Math Block |
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\frac{dp}{dl} = \left( \frac{dp}{dl} \right)_G + \left( \frac{dp}{dl} |
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=rho0\,g\costhetal)-\rho_0 \, q_0^2 }{2 A^2 d, f_0 LaTeX Math Block |
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q(l) =q_0 = \rm const |
LaTeX Math Block |
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u(l) = u_0 = \frac{q_0}{A} = \rm const |
where
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LaTeX Math Inline |
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body | \displaystyle \cos \theta(l) = \frac{dz(l)}{dl} |
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correction factor for trajectory inclination
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borderColor | wheat |
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bgColor | mintcream |
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borderWidth | 7 |
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where
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_G = \rho \cdot g \cdot \cos \theta |
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| gravity losses which represent pressure losses for upward flow and pressure gain for downward flow |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_K = u%5e2 \cdot \frac%7Bd \rho%7D%7Bdl%7D |
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| kinematic losses, which grow contribution at high velocities and high fluid compressibility (like turbulent gas flow) |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_f = - \frac%7B j_m%5e2%7D%7B2 d%7D \cdot \frac%7Bf%7D%7B\rho%7D |
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| friction losses which are always negative along the flow direction |
Approximations
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\rho(p)=\rho_0 = \rm const |
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Equation
LaTeX Math Block Reference |
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becomes: LaTeX Math Block |
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\frac{dp}{dl} = \rho_0 \, g \, \frac{dz}{dl} - \frac{\rho_0 \, q_0^2 }{2 A^2 d} f_0 |
and can be explicitly integrated leading to
LaTeX Math Block Reference |
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.The first term in
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
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Show If |
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Panel |
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bgColor | papayawhip |
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title | ARAX |
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