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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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Flow velocity distribution along the pipe Assumptions
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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 | PPconst |
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| alignment | left |
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| p(l) = p_0 + \rho_0 \, 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_0 \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 |
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| LaTeX Math Block |
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| q(l) =q_0 = \rm const |
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| LaTeX Math Block |
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| u(l) = u_0 = \frac{q_0}{A} = \rm const |
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where
\displaystyle \cos (l) = \frac{dz}{dl}correction factor for
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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(p) = \rho_0 = \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_0 = \frac%7Bq_0%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_0 = \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_0 q_0%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_0%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_0 = \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_0 \, g \, \frac{dz}{dl} - \frac{\rho_0 \, q_0^2 }{2 A^2 d} f_0 |
and can be explicitly integrated leading to | LaTeX Math Block Reference |
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.Substituting the | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \cos \theta(l) = \frac%7Bdz(l)%7D%7Bdl%7D |
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The first term in the right side of
| 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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