Motivation
One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary fluid transport.
In many practical cases the stationary pressure distribution can be approximated by Isothermal or Quasi-isothermal homogenous fluid flow model.
Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
Outputs
Assumptions
Equations
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| \bigg( 1 - \frac{c(p) \, \rho_s^2 \, q_s^2}{A^2} \bigg ) \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl} - \frac{\rho_s^2 \, q_s^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho} |
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| q(l) = \frac{\rho_s \cdot q_0}{\rho} |
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| u(l) = \frac{\rho_s \cdot q_s}{\rho \cdot A} |
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| p(l=0) = p_s |
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| q(l=0) = q_s |
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| \rho(T_s, p_s) = \rho_s |
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where
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body | --uriencoded--f(%7B\rm Re%7D, \, \epsilon) |
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| Darcy friction factor |
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body | --uriencoded--\displaystyle %7B\rm Re%7D = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu(T, p)%7D |
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| Reynolds number |
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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) |
See Derivation of Pressure Profile in Stationary Isothermal Homogenous Pipe Flow @model.
Approximations
Incompressible pipe flow
with constant viscosity
Pressure Profile in Incompressible Isoviscous Stationary Quasi-Isothermal Pipe Flow @model
Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
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anchor | PPconst |
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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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| \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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| q(l) =q_s = \rm const |
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| u(l) = u_s = \frac{q_s}{A} = \rm const |
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The first term in the right side of
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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
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body | f(l) = f_s = \rm const |
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along-hole ( see
Darcy friction factor in water producing/injecting wells ).
References
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title | ARAX |
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