Motivation

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

Inputs

$\begin{array}{l}T_0\end{array}$	Intake temperature	$\begin{array}{l}T(l)\end{array}$	Along-pipe temperature profile
$\begin{array}{l}p_0\end{array}$	Intake pressure	$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density
$\begin{array}{l}q_0\end{array}$	Intake flowrate	$\begin{array}{l}\mu(T, p)\end{array}$	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
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial \tau_x} =\frac{\partial p}{\partial \tau_y} =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)\end{array}$	$\begin{array}{l}A(l) = A = \rm const\end{array}$
Constant inclination
$\begin{array}{l}\displaystyle \theta(l) = \theta = {\rm const} \rightarrow \cos \theta = \frac{dz}{dl} = {\rm const}\end{array}$

Pressure profile along the pipe

(1)	$\begin{array}{l}\displaystyle L = L(p) = \int_{p_0}^{p} \frac{ \rho(p) - j_m^2 \, c(p) }{G \, \rho^2(p) - F(\rho(p))} \, dp\end{array}$

where

$\begin{array}{l}\displaystyle j_m = \frac{ \dot m }{ A}= \rm const\end{array}$	mass flux
$\begin{array}{l}\displaystyle \dot m = \frac{dm }{ dt}= \rm const\end{array}$	mass flowrate
$\begin{array}{l}\displaystyle q_0 = \frac{dV_0}{dt} = \frac{ \dot m }{ \rho_0}\end{array}$	Intake volumetric flowrate
$\begin{array}{l}\rho_0 = \rho(T_0, p_0)\end{array}$	Intake fluid density
$\begin{array}{l}\Delta z(l) = z(l)-z(0)\end{array}$	elevation drop along pipe trajectory
$\begin{array}{l}f(T,\rho) = f({\rm Re}(T,\rho), \, \epsilon)\end{array}$	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 $\begin{array}{l}\rho\end{array}$
$\begin{array}{l}\displaystyle c(T,p) = \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_T\end{array}$	fluid compressibility
$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}= \rm const\end{array}$	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)
$\begin{array}{l}G = g \, \cos \theta= \rm const\end{array}$	gravity acceleration along pipe
$\begin{array}{l}F(T, \rho) = j_m^2 \cdot f(T,\rho)/(2d)\end{array}$

Derivation

Density form

(2)	$\begin{array}{l}\displaystyle L = L(\rho) =\int_{\rho_0}^{\rho} \frac{ 1/c(\rho) - j_m^2/\rho }{G \, \rho^2 - F(\rho)} \, d\rho\end{array}$

Pressure-Density form

(3)	$\begin{array}{l}\displaystyle L = \int_{p_0}^{p} \frac{ \rho \, dp}{G \, \rho^2 - F(\rho)} - j_m^2 \cdot \int_{\rho_0}^{\rho} \frac{1}{\rho} \, \frac{d \rho}{G \, \rho^2 - F(\rho)}\end{array}$