Motivation
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One of the key challenges in Pipe Flow Dynamics is to predict the along-hole 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.
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Outputs
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Inputs | Outputs |
Pipeline trajectory {\bf r} = {\bf r} = \{ x(l), \, y(l), \, z(l) \}Inputs
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along-pipe stabilised pressure distribution |
| Fluid pressure at inlet point () |
Pipeline cross-section area A(l) | along-pipe stabilised flowrate distribution Fluid density rho and \mu(T, p) | along-pipe stabilised average flow velocity distribution u inflow pressure p_0 | , inflow rate --uriencoded--\displaystyle \cos \theta (l) = \frac%7Bdz%7D%7Bdl%7D |
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q_0 Assumptions
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flow temperature distribution \epsilon | Stationary fluid flow |
--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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Assumptions
fluid Isothermal or Quasi-isothermal conditions |
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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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along hole
Equations
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9QRCZbigg1\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_0^2A^2 frac{}{\rho(p)} | LaTeX Math Block |
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| p(l=0) = p_0 |
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LaTeX Math Block |
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| u(l) = \frac{ |
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\rho_0 \cdot q_0p) \cdot A( | LaTeX Math Block |
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| q(l) =A \cdot |
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\frac{ where
LaTeX Math Inline |
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body | --uriencoded--\displaystyle j_m =\frac%7B \rho_0 |
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\cdot q_0}{\rho(p)}(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )
Approximations
Incompressible fluid with constant friction
\, q_0%7D%7BA%7D= \rm const |
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| mass flux |
| Fluid flowrate at inlet point () |
LaTeX Math Inline |
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body | \rho_0 = \rho(T_0, p_0) |
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| Fluid density at inlet point () |
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(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) |
Alternative forms
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Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
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LaTeX Math Block |
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p(l) = p_0 + \rho \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l |
LaTeX Math Block |
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rho \, g \cos \theta(l) -left( \frac{dp}{dl} \right)_G + \left( \frac{ |
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\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 LaTeX Math Block |
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u(l) = \frac{q_0}{A(l)} |
LaTeX Math Block |
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q(l) =q_0 = \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 deviation
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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dp}{dl} \right)_K + \left( \frac{dp}{dl} \right)_f |
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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
See also
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Show If |
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bgColor | papayawhip |
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title | ARAX |
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