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
| Pressure profile | Pressure gradient profile |
|---|
| LaTeX Math Block |
|---|
| anchor | PressureProfile |
|---|
| alignment | left |
|---|
| F(p, l)=\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d} - (2/j_m^2) \, \int_p^{p_0} \frac{dp}{\rho} - (2/j_m^2) \, g \, \Delta z(l) = 0 |
|
| LaTeX Math Block |
|---|
| \frac{dp}{dl} = {\rm Numerical} \ {\rm Derivative} |
|
| Mass Flux | Mass Flowrate |
| LaTeX Math Block |
|---|
| anchor | MassFlux |
|---|
| alignment | left |
|---|
| j_m = \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d}
}
}
|
|
| LaTeX Math Block |
|---|
| anchor | MassFlowrate |
|---|
| alignment | left |
|---|
| \dot m =
A \cdot \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d}
}
} |
|
Volumetric Flowrate | Intake Fluid velocity |
| LaTeX Math Block |
|---|
| anchor | VolumtericFlowrate |
|---|
| alignment | left |
|---|
| q_0 =
\frac{A}{\rho_s} \cdot \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d}
}
}
|
|
| LaTeX Math Block |
|---|
| anchor | IntakeFluidVelocity |
|---|
| alignment | left |
|---|
| u_0 =
\frac{1}{\rho_s} \cdot \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right)
+ \left( \frac{ff}{\rho^2} + \frac{ff_0}{\rho_0^2} \right)
\cdot \frac{l}{ 2 \, d}
}
}
|
|
where
| LaTeX Math Inline |
|---|
| body | \rho_0 = \rho(T_0, p_0) |
|---|
|
| Intake fluid density |
| Intake mass flux |
| LaTeX Math Inline |
|---|
| body | --uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D |
|---|
|
| mass flowrate |
| LaTeX Math Inline |
|---|
| body | --uriencoded--\displaystyle q_0 = \frac%7BdV_0%7D%7Bdt%7D = \dot m / \rho_0 |
|---|
|
| Intake flowrate |
| LaTeX Math Inline |
|---|
| body | u_0 = u(l=0) = q_0 / A = j_m / \rho_0 |
|---|
|
| Intake Fluid velocity |
| LaTeX Math Inline |
|---|
| body | \Delta z(l) = z(l)-z(0) |
|---|
|
| elevation drop along pipe trajectory |
| LaTeX Math Inline |
|---|
| body | --uriencoded--f_s = f(%7B\rm Re%7D_s, \, \epsilon) |
|---|
|
| Darcy friction factor at intake point |
| LaTeX Math Inline |
|---|
| 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%7D |
|---|
|
| Reynolds number at intake point |
| LaTeX Math Inline |
|---|
| body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D |
|---|
|
| characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
The first term in the right side of
| LaTeX Math Block Reference |
|---|
|
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 |
|---|
| body | f(l) = f_s = \rm const |
|---|
|
along-hole ( see
Darcy friction factor in water producing/injecting wells ).
References
| Show If |
|---|
|
| Panel |
|---|
| bgColor | papayawhip |
|---|
| title | ARAX |
|---|
|
|
|
