Inputs & Outputs

Inputs	Outputs
Pipeline trajectory $\begin{array}{l}{\bf r} = {\bf r}(l) = \{ x(l), \, y(l), \, z(l) \}\end{array}$	along-pipe distribution of stabilized pressure $\begin{array}{l}p(l)\end{array}$
Pipe cross-section area $\begin{array}{l}A\end{array}$	along-pipe distribution of stabilized flow rate $\begin{array}{l}q(l)\end{array}$
Along-pipe temperature profile $\begin{array}{l}T(l)\end{array}$
Fluid density $\begin{array}{l}\rho(T, p)\end{array}$ and fluid viscosity $\begin{array}{l}\mu(T, p)\end{array}$	along-pipe distribution of stabilized average flow velocity $\begin{array}{l}u(l)\end{array}$
Inner pipe wall roughness $\begin{array}{l}\epsilon\end{array}$

Assumptions

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

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

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