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 fluid transport.
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Assumptions
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Equations
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| Pressure profile | Pressure gradient profile |
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| LaTeX Math Block |
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| anchor | PPconst |
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| alignment | left |
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| p(l) = p_0 + \rho_0 \, g \, \Delta 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_0 \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 |
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| Mass Flux | Mass Flowrate |
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| anchor | MassFlux |
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| alignment | left |
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| j_m = \rho_0 \cdot \sqrt{\frac{2 \, d}{f_0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_0 - p)/ \rho_0}
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| LaTeX Math Block |
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| anchor | MassFlowrate |
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| alignment | left |
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| \dot m = j_m \cdot A = \rho_0 \cdot A \cdot \sqrt{\frac{2 \, d}{f_0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_0 - p)/ \rho_s}
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Volumetric Flowrate | Intake Fluid velocity |
| LaTeX Math Block |
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| anchor | PPconst |
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| alignment | left |
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| q_0 = \dot m / \rho_0 = A \cdot \sqrt{\frac{2 \, d }{ f_0 \, l }} \cdot \sqrt{ g \, \Delta z(l) + (p_0 - p)/ \rho_s }
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| LaTeX Math Block |
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| anchor | PPconst |
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| alignment | left |
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| u_0 = j_m/ \rho_0 =q_0 / A = \sqrt{\frac{2 \, d }{ f_0 \, l }} \cdot \sqrt{ g \, \Delta z(l) + (p_0 - p)/ \rho_s }
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In most practical applications in water producing or water injecting wells, water can be considered as incompressible and friction factor can be assumed constant
| LaTeX Math Inline |
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| body | f(l) = f_0 = \rm const |
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along-hole ( see
Darcy friction factor in water producing/injecting wells ).
References
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| Panel |
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| bgColor | papayawhip |
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| title | ARAX |
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