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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Inputs
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Assumptions
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\epsilon | Inner pipe wall roughness | Assumptions
--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
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body | T(t, l)=T_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(t, \tau_x,\tau_y,l) = p(l) |
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Incompressible fluid |
Stationary flow | Homogenous flow | Isothermal or Quasi-isothermal conditions |
Incompressible fluid s Isoviscous flow ps Constant cross-section pipe areamathinline |
along hole
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_ |
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ss0 \, g \, \Delta z(l) - \frac{\rho_ |
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ss^2_s_0 \, l = p_0 + \rho_0 \, g \, \Delta z(l) - \frac{8}{\pi^2} \frac{f_0 \, 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^2s | 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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PPconst | \dot m = j_m \cdot A = \rho_ |
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s0 \cdot A \cdot \sqrt{\frac{2 \, d}{f_ |
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s0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_ |
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s_s_0 = \dot m / \rho_0 = A \cdot \sqrt{\frac{2 \, d }{ f_ |
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s0 \, l }} \cdot \sqrt{ g \, \Delta z(l) + (p_ |
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ssFluid velocity | LaTeX Math Block |
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anchor | PPconst |
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alignment | left |
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| u_ |
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s0 = j_m/ \rho_0 =q_0 / A = \sqrt{\frac{2 \, d }{ f_ |
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s0 \, l }} \cdot \sqrt{ g \, \Delta z(l) + (p_ |
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swhere
| 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_s 0 = f(%7B\rm Re%7D_s0, \, \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_s 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_s%7D0%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_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(l) = \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_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{j_m^2 f_0}{2 \, \rho_0 \, d} |
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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| .See Pressure Profile in Stationary Quasi-Isothermal Homogenous Pipe Flow @model |
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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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Show If |
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bgColor | papayawhip |
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title | ARAX |
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