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.

In many practical cases the stationary pressure distribution can be approximated by Isothermal or Quasi-isothermal homogenous fluid flow model.

Outputs

$\begin{array}{l}p(l)\end{array}$	Pressure distribution along the pipe
$\begin{array}{l}q(l)\end{array}$	Flowrate distribution along the pipe
$\begin{array}{l}u(l)\end{array}$	Flow velocity distribution along the pipe

$\begin{array}{l}T_0\end{array}$	Fluid temperature at inlet point ( $\begin{array}{l}l=0\end{array}$ )	$\begin{array}{l}T(l)\end{array}$	Along-pipe temperature profile
$\begin{array}{l}p_0\end{array}$	Fluid pressure at inlet point ( $\begin{array}{l}l=0\end{array}$ )	$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density
$\begin{array}{l}q_0\end{array}$	Fluid flowrate at inlet point ( $\begin{array}{l}l=0\end{array}$ )	$\begin{array}{l}\mu(T, p)\end{array}$	Dynamic fluid viscosity
$\begin{array}{l}z(l)\end{array}$	Pipeline trajectory TVDss	$\begin{array}{l}A\end{array}$	Pipe cross-section area
$\begin{array}{l}\theta (l)\end{array}$	Pipeline trajectory inclination, $\begin{array}{l}\displaystyle \cos \theta (l) = \frac{dz}{dl}\end{array}$	$\begin{array}{l}\epsilon\end{array}$	Inner pipe wall roughness

Steady-State flow	Quasi-isothermal flow
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial t} = 0\end{array}$	$\begin{array}{l}\displaystyle \frac{\partial T}{\partial t} =0 \rightarrow T(t,l) = T(l)\end{array}$
Homogenous flow	Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial \tau_x} =\frac{\partial p}{\partial \tau_y} =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)\end{array}$	$\begin{array}{l}A(l) = A = \rm const\end{array}$

(1)	$\begin{array}{l}\displaystyle \left( \rho(p) - j_m^2 \cdot c(p) \right) \cdot \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l) - \frac{ j_m^2 }{2 d} \cdot f(p)\end{array}$

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

(3)	$\begin{array}{l}\displaystyle u(l) = \frac{j_m}{\rho(l)}\end{array}$

(4)	$\begin{array}{l}\displaystyle q(l) =A \cdot u(l)\end{array}$

where

$\begin{array}{l}\displaystyle j_m =\frac{ \rho_0 \, q_0}{A}= \rm const\end{array}$	mass flux
$\begin{array}{l}q_0 = q(l=0)\end{array}$	Fluid flowrate at inlet point ( $\begin{array}{l}l=0\end{array}$ )
$\begin{array}{l}\rho_0 = \rho(T_0, p_0)\end{array}$	Fluid density at inlet point ( $\begin{array}{l}l=0\end{array}$ )
$\begin{array}{l}\rho(l) = \rho(T(l), p(l))\end{array}$	Fluid density at any point $\begin{array}{l}l\end{array}$
$\begin{array}{l}\displaystyle с(p) = \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_T\end{array}$	Fluid Compressibility
$\begin{array}{l}f(T, \rho) = f({\rm Re}(T, \rho), \, \epsilon)\end{array}$	Darcy friction factor
$\begin{array}{l}\displaystyle {\rm Re}(T,\rho) = \frac{j_m \cdot d}{\mu(T, \rho)}\end{array}$	Reynolds number in Pipe Flow
$\begin{array}{l}\mu(T,\rho)\end{array}$	dynamic viscosity as function of fluid temperature $\begin{array}{l}T\end{array}$ and density $\begin{array}{l}\rho\end{array}$
$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}= \rm const\end{array}$	Characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)

Derivation

(5)	$\begin{array}{l}\displaystyle \frac{dp}{dl} = \left( \frac{dp}{dl} \right)_G + \left( \frac{dp}{dl} \right)_K + \left( \frac{dp}{dl} \right)_f\end{array}$

where

$\begin{array}{l}\displaystyle \left( \frac{dp}{dl} \right)_G = \rho \cdot g \cdot \cos \theta\end{array}$	gravity losses which represent pressure losses for upward flow and pressure gain for downward flow
$\begin{array}{l}\displaystyle \left( \frac{dp}{dl} \right)_K = u^2 \cdot \frac{d \rho}{dl}\end{array}$	kinematic losses, which grow contribution at high velocities $\begin{array}{l}u = j_m / \rho\end{array}$ and high fluid compressibility (like turbulent gas flow)
$\begin{array}{l}\displaystyle \left( \frac{dp}{dl} \right)_f = - \frac{ j_m^2}{2 d} \cdot \frac{f}{\rho}\end{array}$	friction losses which are always negative along the flow direction

Approximations

Pressure Profile in G-Proxy Pipe Flow @model	quadrature	$\begin{array}{l}\theta(l) = \theta_0 = \rm const\end{array}$
Pressure Profile in GF-Proxy Pipe Flow @model	quadrature	$\begin{array}{l}\theta(l) = \theta_0 = \rm const\end{array}$ , $\begin{array}{l}f(T, p)=f_0 = \rm const\end{array}$
Pressure Profile in GFC-Proxy Pipe Flow @model	algebraic equation	$\begin{array}{l}\theta(l) = \theta_0 = \rm const\end{array}$ , $\begin{array}{l}f(T, p)=f_0 = \rm const\end{array}$ $\begin{array}{l}\rho(T, p) = \rho(T) \cdot (1+ c^*(T) \cdot p/p_0)\end{array}$
Pressure Profile in Incompressible Quasi-Isothermal Proxy Pipe Flow @model	quadrature	$\begin{array}{l}\rho(p)=\rho_0 = \rm const\end{array}$ , $\begin{array}{l}T(t, l)=T(l)\end{array}$
Pressure Profile in Incompressible Isothermal Proxy Pipe Flow @model	closed-form expression	$\begin{array}{l}\rho(p)=\rho_0 = \rm const\end{array}$ , $\begin{array}{l}T=T_0 = \rm const\end{array}$ (isothermal)
Pressure Profile in GC-proxy static fluid column @model	closed-form expression	$\begin{array}{l}\theta(l) = \theta_0 = \rm const\end{array}$ , $\begin{array}{l}\dot m = 0\end{array}$ (no flow)