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.
...
Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
Outputs
...
Inputs
...
Assumptions
...
Stationary flow | Homogenous Isothermal or conditions flow |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
---|
|
| LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l) |
---|
|
|
Homogenous flow | |
LaTeX Math Inline |
---|
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) |
---|
|
| |
Equations
...
LaTeX Math Block |
---|
| \left( 1\rho(p) - \frac{\rho_0^2 \, q_0^2}{A^2} \cdot \frac{j_m^2 \cdot c(p)}{\rho} \right) \cdot \frac{dp}{dl} = \rhorho^2(p) \, g \, \cos \theta(l) - \frac{\rho_0^2 \, q_0^2 j_m^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho}cdot f(p) |
| LaTeX Math Block |
---|
| p(l=0) = p_0 |
|
LaTeX Math Block |
---|
| u(l) = \frac{\rho_0 \cdot q_0j_m}{\rho(T(l), p(l))) \cdot A} |
| LaTeX Math Block |
---|
| q(l) = \frac{\rho_0A \cdot q_0}{\rho(T(l),p(l))}u(l) |
|
where
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D= \rm const |
---|
|
| mass flux |
|
fluid flow rate at pipe intakeFluid flowrate at inlet point () |
LaTeX Math Inline |
---|
body | \rho_0 = \rho(T_0, p_0) |
---|
|
|
fluid density at intake temperature and pressureFluid density at inlet point () |
LaTeX Math Inline |
---|
body | \rho(l) = \rho(T(l), p(l)) |
---|
|
| Fluid density at any point |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle с(p) = \frac%7B1%7D%7B\rho%7D \left( \frac%7B\partial \rho%7D%7B\partial p%7D \right)_T |
---|
|
| Fluid Compressibility |
LaTeX Math Inline |
---|
body | --uriencoded--f(T, \rho) = f(%7B\rm Re%7D(T, \rho), \, \epsilon) |
---|
|
| Darcy friction factor |
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(T,\rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T, |
|
|
pcharacteristicCharacteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
...
...
Alternative forms
...
Pressure Profile in Incompressible Isoviscous Stationary Quasi-Isothermal Pipe Flow @model
Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
PPconstp(l) = p_s+rho_s \, g \, z(l) -frac{dp}{dl} = \left( \frac{ |
|
\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l LaTeX Math Block |
---|
|
dp}{dl} \right)_G + \left( \frac{dp}{dl} |
|
=rhos\,g\costhetal)-\rho_s \, q_s^2 }{2 A^2 d, f_s LaTeX Math Block |
---|
|
q(l) =q_s = \rm const |
LaTeX Math Block |
---|
|
u(l) = u_s = \frac{q_s}{A} = \rm const |
where
where
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 \left( \frac%7Bdp%7D%7Bdl%7D \right)_G = \rho \cdot g \cdot \cos \theta |
---|
|
| gravity losses which represent pressure losses for upward flow and pressure gain for downward flow |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_K = u%5e2 \cdot \frac%7Bd \rho%7D%7Bdl%7D |
---|
|
| kinematic losses, which grow contribution at high velocities and high fluid compressibility (like turbulent gas flow) |
LaTeX Math Inline |
---|
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%7DReynolds number at intake point | ...
...
borderColor | wheat |
---|
bgColor | mintcream |
---|
borderWidth | 7 |
---|
...
\left( \frac%7Bdp%7D%7Bdl%7D \right)_f = - \frac%7B j_m%5e2%7D%7B2 d%7D \cdot \frac%7Bf%7D%7B\rho%7D |
|
| friction losses which are always negative along the flow direction |
Approximations
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
\rho(p)=\rho_0 = \rm const |
|
|
...
Equation
LaTeX Math Block Reference |
---|
|
becomes: LaTeX Math Block |
---|
|
\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 |
---|
|
after substituting LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \cos \theta(l) = \frac%7Bdz(l)%7D%7Bdl%7D |
---|
|
and can be explicitly integrated leading to LaTeX Math Block Reference |
---|
|
....
LaTeX Math Block Reference |
---|
|
...
...
LaTeX Math Inline |
---|
body | f(l) = f_s = \rm const |
---|
|
...
See also
...