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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Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
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Outputs
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| Inputs | OutputsT_s | Intake temperature pPressure Flow velocity distribution along the pipe |
Inputs
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s| q(l) | Flowrate distribution along the pipesu(l) | Flow velocity distribution along the pipe | Along-pipe temperature profile | Assumptions
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| \rho(T, p) | Fluid density | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
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\muT p) | | Pipe \epsilon | Inner pipe wall roughness | Assumptions
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| 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(t, \tau_x,\tau_y,l) = p(l) |
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Stationary fluid flow | Homogenous fluid flow | Isothermal or Quasi-isothermal conditions | Constant cross-section pipe area along holeIncompressible fluid s Isoviscous ps
Equations
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| Pressure profile | Pressure gradient profile |
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Fluid velocity | Fluid rate | | LaTeX Math Block |
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| anchor | PPconst |
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| alignment | left |
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| p(l) = p_ |
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ss0 \, g \, \Delta z(l) - \frac{\rho_ |
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ss^20^2 }{2 A^2 d} \, f_0 \, l = p_0 + \rho_0 \, g \, \Delta z(l) - \frac{8}{\pi^2} \frac{f_ |
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s0 \, l}{d^5} \rho_0 \, q_0^2 |
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| \frac{dp}{dl} = \rho_ |
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s0 \, g \cos \theta(l) - \frac{\rho_ |
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ss^2s1q(l) =q_s = \rm constj_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 |
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1u(l) = u_s = \frac{q_s}{A} = \rm const |
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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_0 }
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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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where
| Intake mass flux |
| mass flowrate |
| Intake Fluid velocity |
| LaTeX Math Inline |
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| body | \Delta z(l) = z(l)-z(0) |
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| elevation drop along pipe trajectory |
| LaTeX Math Inline |
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| body | --uriencoded--f_0 = f(%7B\rm Re%7D_0, \, \epsilon) |
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| Darcy friction factor at intake point |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D_0 = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s 0 q_s%7D%7B0%7D%7B\pi d%7D \frac%7B1%7D%7B\mu(T, p)%7D_0%7D |
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| Reynolds number at intake point |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D |
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| characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
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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 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_s 0 = \frac%7Bq_s%7D%7BA%7D 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_s 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(l) = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7D frac%7Bj_m%5e2 \, d%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(l) = f(%7B\rm Re%7D, \, \epsilon) = f_s 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_s0 \, g \, \frac{dz}{dl} - \frac{\rhoj_sm^2 \, qf_s^2 0}{2 A^2\, \rho_0 \, 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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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:
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_s 0 = \rm const |
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along-hole ( see Darcy friction factor in water producing/injecting wells ).
See also
References
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| Panel |
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| bgColor | papayawhip |
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| title | ARAX |
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