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 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.
Inputs & Outputs
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Outputs
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Intake temperature | LaTeX Math Inline |
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body | T_s | LaTeX Math Inline |
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body | p_s | q_s | LaTeX Math Inline |
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body | | Flow velocity distribution along the pipe |
Inputs
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z(l) | Pipeline trajectory TVDss | Fluid temperature at inlet point ( |
\theta (Pipeline trajectory inclination, --uriencoded--\displaystyle \cos \theta = \frac%7Bdz%7D%7Bdl%7DPipeline trajectory--uriencoded--%7B\bf r%7D(l) | | Fluid pressure at inlet point ( |
T(Along-pipe temperature profile | Fluid Assumptions
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Stationary fluid Homogenous fluid flow | Isothermal or Quasi-isothermal conditionsQuasi-isothermal flow |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
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| LaTeX Math Inline |
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body | --uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l) |
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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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Equations
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bigg1\frac{\,\rho_s^2, q_s^2}{A^2} \bigg )right) \cdot \frac{dp}{dl} = \ |
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rhofrac{dz}{dl}\rho_s^2 \, q_s^2 A^2frac{f({\rm Re}, \, \epsilon)}{\rho}1q\frac{\rho_s \cdot q}{\rho} |
LaTeX Math Block |
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| u(l) = \frac{ |
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\rho_s \cdot q_s \cdot Ap0p=0 p_s LaTeX Math Block |
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q(l=0) = q_s |
LaTeX Math Block |
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\rho(T_s, p_s) = \rho_s |
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where
LaTeX Math Inline |
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body | --uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D= \rm const |
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| mass flux |
| Fluid flowrate at inlet point () |
LaTeX Math Inline |
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body | \rho_0 = \rho(T_0, p_0) |
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| Fluid density at inlet point () |
LaTeX Math Inline |
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body | \rho(l) = \rho(T(l), p(l)) |
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| Fluid density at any point |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle с(p) = \frac%7B1%7D%7B\rho%7D \left( \frac%7B\partial \rho%7D%7B\partial p%7D \right)_T |
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| Fluid Compressibility |
LaTeX Math Inline |
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body | --uriencoded--f(T, \rho) = f(%7B\rm Re%7D(T, \rho), \, \epsilon) |
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| Darcy friction factor |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle %7B\rm Re%7D |
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= \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(T,\rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T, |
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pcharacteristicCharacteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
See Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model.
Approximations
Incompressible pipe flow
with constant viscosity ...
Alternative forms
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PPconstp(l) =p_s+ \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l LaTeX Math Block |
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\frac{dp}{dl} = \left( \frac{dp}{dl} \right)_G + \left( \frac{dp}{dl} |
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=rhos\,g\costhetal)-\rho_s \, q_s^2 }{2 A^2 d, f_s LaTeX Math Block |
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q(l) =q_s = \rm const |
LaTeX Math Block |
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u(l) = u_s = \frac{q_s}{A} = \rm const |
where
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trajectory inclination
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borderColor | wheat |
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bgColor | mintcream |
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borderWidth | 7 |
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where
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_G = \rho \cdot g \cdot \cos \theta |
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| gravity losses which represent pressure losses for upward flow and pressure gain for downward flow |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_K = u%5e2 \cdot \frac%7Bd \rho%7D%7Bdl%7D |
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| kinematic losses, which grow contribution at high velocities and high fluid compressibility (like turbulent gas flow) |
LaTeX Math Inline |
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body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_f = - \frac%7B j_m%5e2%7D%7B2 d%7D \cdot \frac%7Bf%7D%7B\rho%7D |
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| friction losses which are always negative along the flow direction |
Approximations
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\rho(p)=\rho_0 = \rm const |
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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
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:...
LaTeX Math Inline |
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body | f(l) = f_s = \rm const |
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See also
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Show If |
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Panel |
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bgColor | papayawhip |
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title | ARAX |
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