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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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| Inputs | Outputs |
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Pipeline trajectory | LaTeX Math Inline |
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| body | {\bf r} = {\bf r}(l) = \{ x(l), \, y(l), \, z(l) \} |
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stabilised stabilised stabilized flowrate distribution |
Along-pipe temperature profile | along-pipe stabiliszed average flow velocity distribution |
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)along-pipe stabilised average flow velocity distribution | LaTeX Math Inline |
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body | u(l |
inflow pressure , inflow rate |
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Inner pipe wall roughness |
flow temperature distribution | LaTeX Math Inline |
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Assumptions
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Equations
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| LaTeX Math Block |
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| \bigg( 1 - \frac{c(p) \, \rho_0^2 \, q_0^2}{A^2} \bigg ) \frac{dp}{dl} = \rho(p) \, g \, \frac{dz}{dl} - \frac{\rho_0^2 \, q_0^2 }{2 A^2 d} \frac{f(p)}{\rho(p)} |
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| LaTeX Math Block |
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| u(l) = \frac{\rho_0 \cdot q_0}{\rho(p) \cdot A(l)} |
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| LaTeX Math Block |
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| q(l) = \frac{\rho_0 \cdot q_0}{\rho(p)} |
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(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )
Approximations
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Incompressible
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pipe flow with constant friction
| 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 |
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| LaTeX Math Block |
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| \frac{dp}{dl} = \rho \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 |
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| u(l) = \frac{q_0}{A(l)} |
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| q(l) =q_0 = \rm const |
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where
| 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:
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| bgColor | papayawhip |
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| title | ARAX |
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