Motivation
One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the steady-state 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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| \left( 1 - \frac{\rho_0^2 \, q_0^2}{A^2} \cdot \frac{c(p)}{\rho} \right) \frac{dp}{dl} = \rho \, g \, \cos \theta(l) - \frac{\rho_0^2 \, q_0^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho} |
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| p(l=0) = p_0 |
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| u(l) = \frac{\rho_0 \cdot q_0}{\rho(T(l), p(l))) \cdot A} |
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| q(l) = \frac{\rho_0 \cdot q_0}{\rho(T(l),p(l))} |
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where
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body | --uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D |
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| fluid flow rate at pipe intake |
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body | \rho_0 = \rho(T_0, p_0) |
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| fluid density at intake temperature and pressure |
| Fluid Compressibility |
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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_0 q_0%7D%7B\pi d%7D \frac%7B1%7D%7B\mu(T, p)%7D |
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| Reynolds number in Pipe Flow |
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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) |
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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| 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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where
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body | --uriencoded--f_s = f(%7B\rm Re%7D_s, \, \epsilon) |
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| Darcy friction factor at intake point |
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body | --uriencoded--\displaystyle %7B\rm Re%7D_s = \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_s%7D |
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| Reynolds number at intake point |
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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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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 many practical applications the water in water producing wells or water injecting wells 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 ).
See also
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