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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.
Outputs
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Inputs & Outputs
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| Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
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| LaTeX Math Block |
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| anchor | PlPPconst |
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| alignment | left |
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| p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l |
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| LaTeX Math Block |
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| \frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s |
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| LaTeX Math Block |
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| q(l) =q_s = \rm const |
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| LaTeX Math Block |
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| u(l) = u_s = \frac{q_s}{A} = \rm const |
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| Panel |
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| borderColor | wheat |
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| bgColor | mintcream |
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| borderWidth | 7 |
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| Incompressible fluid | LaTeX Math Inline |
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| body | \rho(T, p) = \rho_s = \rm const |
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means that compressibility vanishes and fluid velocity is going to be constant along the pipeline trajectory | LaTeX Math Inline |
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| body | --uriencoded--u(l) = u_s = \frac%7Bq_s%7D%7BA%7D = \rm const |
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.For the constant viscosity | LaTeX Math Inline |
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| body | \mu(T, p) = \mu_s = \rm const |
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along the pipeline trajectory the Reynolds number | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7D = \rm const |
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and Darcy friction factor | LaTeX Math Inline |
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| body | --uriencoded--f(%7B\rm Re%7D, \, \epsilon) = f_s = \rm const |
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are going to be constant along the pipeline trajectory.Equation | LaTeX Math Block Reference |
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becomes:| LaTeX Math Block |
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| \frac{dp}{dl} = \rho_s \, g \, \frac{dz}{dl} - \frac{\rho_s \, q_s^2 }{2 A^2 d} f_s |
which leads to | LaTeX Math Block Reference |
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after substituting | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \cos \theta(l) = \frac%7Bdz(l)%7D%7Bdl%7D |
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and can be explicitly integrated leading to | LaTeX Math Block Reference |
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