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.

Inputs & Outputs

Inputs		Outputs
$\begin{array}{l}p_0\end{array}$	Intake pressure	$\begin{array}{l}p(l)\end{array}$	Pressure distribution along the pipe
$\begin{array}{l}q_0\end{array}$	Intake flowrate	$\begin{array}{l}u(l)\end{array}$	Flow velocity distribution along the pipe
$\begin{array}{l}\theta (l)\end{array}$	Pipeline trajectory inclination
$\begin{array}{l}{\bf r}(l)\end{array}$	Pipeline trajectory
$\begin{array}{l}T(l)\end{array}$	Along-pipe temperature profile
$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density
$\begin{array}{l}\mu(T, p)\end{array}$	Fluid viscosity
$\begin{array}{l}A\end{array}$	Pipe cross-section area
$\begin{array}{l}\epsilon\end{array}$	Inner pipe wall roughness

Assumptions

Stationary fluid flow

Homogenous fluid flow

Isothermal or Quasi-isothermal conditions

Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole

Equations

(1)	$\begin{array}{l}\displaystyle \bigg( 1 - \frac{c(p) \, \rho_0^2 \, q_0^2}{A^2} \bigg ) \frac{dp}{dl} = \rho(p) \, g \, \frac{dz}{dl} - \frac{\rho_0^2 \, q_0^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho(p)}\end{array}$

(2)	$\begin{array}{l}\displaystyle q(l) = \frac{\rho_0 \cdot q_0}{\rho(p)}\end{array}$

(3)	$\begin{array}{l}\displaystyle u(l) = \frac{\rho_0 \cdot q_0}{\rho(p) \cdot A}\end{array}$

(4)	$\begin{array}{l}\displaystyle p(l=0) = p_0\end{array}$

(5)	$\begin{array}{l}\displaystyle q(l=0) = q_0\end{array}$

(6)	$\begin{array}{l}\displaystyle \rho(p_0) = \rho_0\end{array}$

where

$\begin{array}{l}f({\rm Re}, \, \epsilon)\end{array}$

Darcy friction factor

$\begin{array}{l}\displaystyle {\rm Re} = \frac{u(l) \cdot d}{\nu(l)} = \frac{4 \rho_0 q_0}{\pi d} \frac{1}{\mu(T, p)}\end{array}$

Reynolds number

$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}\end{array}$

characteristic 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 $\begin{array}{l}\rho(p) = \rho_0\end{array}$ with constant viscosity $\begin{array}{l}\mu(T, p) = \mu_0\end{array}$

Pressure profile

Pressure gradient profile

Fluid velocity

Fluid rate

(7)	$\begin{array}{l}\displaystyle p(l) = p_0 + \rho_0 \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l\end{array}$

(8)	$\begin{array}{l}\displaystyle \frac{dp}{dl} = \rho_0 \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0\end{array}$

(9)	$\begin{array}{l}\displaystyle q(l) =q_0 = \rm const\end{array}$

(10)	$\begin{array}{l}\displaystyle u(l) = u_0 = \frac{q_0}{A} = \rm const\end{array}$

where

$\begin{array}{l}\displaystyle \cos \theta(l) = \frac{dz(l)}{dl}\end{array}$

correction factor for trajectory inclination

Derivation

Incompressible fluid $\begin{array}{l}\rho(p) = \rho_0 = \rm const\end{array}$ means that compressibility vanishes $\begin{array}{l}c(p) = 0\end{array}$ and fluid velocity is going to be constant along the pipeline trajectory $\begin{array}{l}u(l) = u_0 = \frac{q_0}{A} = \rm const\end{array}$ .

For the constant viscosity $\begin{array}{l}\mu(T, p) = \mu_0 = \rm const\end{array}$ along the pipeline trajectory the Reynolds number $\begin{array}{l}\displaystyle {\rm Re} = \frac{4 \rho_0 q_0}{\pi d} \frac{1}{\mu_0} = \rm const\end{array}$ and Darcy friction factor $\begin{array}{l}f({\rm Re}, \, \epsilon) = f_0 = \rm const\end{array}$ are going to be constant along the pipeline trajectory.

Equation (1) becomes:

(11)	$\begin{array}{l}\displaystyle \frac{dp}{dl} = \rho_0 \, g \, \frac{dz}{dl} - \frac{\rho_0 \, q_0^2 }{2 A^2 d} f_0\end{array}$

and can be explicitly integrated leading to (7).

The first term in (8) 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 the water can be considered as incompressible and friction factor an be assumed constant $\begin{array}{l}f(l) = f_s = \rm const\end{array}$ along-hole ( see Darcy friction factor in water producing/injecting wells ).

Page tree

Motivation

Inputs & Outputs

Assumptions

Equations

Approximations

Incompressible pipe flow $\begin{array}{l}\rho(p) = \rho_0\end{array}$ with constant viscosity $\begin{array}{l}\mu(T, p) = \mu_0\end{array}$

See also

References

Page tree

Stationary Quasi-Isothermal Homogenous Pipe Flow Pressure Profile @model

Motivation

Inputs & Outputs

Assumptions

Equations

Approximations

Incompressible pipe flow \rho(p) = \rho_0//<![CDATA[ \begin{array}{l}\rho(p) = \rho_0\end{array} //]]> with constant viscosity \mu(T, p) = \mu_0//<![CDATA[ \begin{array}{l}\mu(T, p) = \mu_0\end{array} //]]>

See also

References

Incompressible pipe flow $\begin{array}{l}\rho(p) = \rho_0\end{array}$ with constant viscosity $\begin{array}{l}\mu(T, p) = \mu_0\end{array}$