Motivation
...
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.
Outputs
...
Inputs
...
sIntake temperature sIntake pressure sIntake Fluid 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_s^2 \, q_s^2}{A^2} \cdot \frac{j_m^2 \cdot c(p)}{\rho} \right) \cdot \frac{dp}{dl} = \rhorho^2(p) \, g \, \frac{dz}{dl}cos \theta(l) - \frac{\rho_s^2 \, q_s^2 j_m^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho}cdot f(p) |
| | LaTeX Math Block |
|---|
| qp(l=0) = \frac{\rho_s \cdot q_s}{\rho}p_0 |
|
| LaTeX Math Block |
|---|
| u(l) = \frac{\rho_s \cdot q_sj_m}{\rho \cdot A(l)} |
| | LaTeX Math Block |
|---|
| pq(l=0) = p_s | | LaTeX Math Block |
|---|
| A \cdot u(l) |
|
where
| LaTeX Math Inline |
|---|
| body | --uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D= \rm const |
|---|
|
| mass flux |
|
= q_s| LaTeX Math Block |
|---|
|
| Fluid flowrate at inlet point () |
|
s s) = \rho_s...
| Fluid 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_s q_s%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) |
See Derivation of Pressure Profile in Stationary Isothermal Homogenous Pipe Flow @model.
...
...
Alternative forms
...
| LaTeX Math Block |
|---|
| \frac{dp}{dl} = \left( \frac{dp}{dl} \right)_G + \left( |
|
Pressure Profile in Incompressible Isoviscous Stationary Quasi-Isothermal Pipe Flow @model
| Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
|---|
| LaTeX Math Block |
|---|
| anchor | PPconst |
|---|
| alignment | left |
|---|
|
p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l |
| LaTeX Math Block |
|---|
|
=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--\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--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_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
...
