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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| Steady-State flow | Isothermal or Quasi-isothermal flow |
| LaTeX Math Inline |
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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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| | 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) _0 = \rm const |
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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(\tau_x,\tau_y,l) = p(l) |
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| Incompressible fluid |
| LaTeX Math Inline |
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| body | \rho(T, p)=\rho_0 = \rm const |
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→ | LaTeX Math Inline |
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| body | \mu(T, \rho) =\mu_0 = \rm const |
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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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References
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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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