Motivation

One of the key challenges in Pipe Flow Dynamics is to predict the along-hole temperature distribution during the stationary fluid transport.

Pipeline Flow Temperature Model is addressing this problem with account of the varying pipeline trajectory, pipeline schematic and heat transfer with the matter around pipeline.

In many practical cases the along-hole temperature distribution during the stationary fluid flow can be approximated by homogenous fluid flow model.

Outputs

$\begin{array}{l}T(t, l)\end{array}$

along-pipe temperature distribution and evolution in time

Inputs

$\begin{array}{l}{\bf r}(l)\end{array}$	pipeline trajectory, $\begin{array}{l}{\bf r}(l) = \{ x(l), \, y(l), \, z(l) \}\end{array}$	$\begin{array}{l}\rho(T, p)\end{array}$	fluid density
$\begin{array}{l}A(l)\end{array}$	pipeline cross-section area	$\begin{array}{l}\mu(T, p)\end{array}$	fluid viscosity
$\begin{array}{l}T_0(t)\end{array}$	intake temperature	$\begin{array}{l}T_{e0}(l)\end{array}$	initial temperature of the medium around the pipeline
$\begin{array}{l}p_0\end{array}$	intake pressure	$\begin{array}{l}c_p(l)\end{array}$	specific heat capacity of the medium around pipeline
$\begin{array}{l}q_0\end{array}$	intake flowrate	$\begin{array}{l}\lambda_e(l)\end{array}$	thermal conductivity of the medium around pipeline
$\begin{array}{l}U(l)\end{array}$	heat transfer coefficient based on pipeline schematic

Assumptions

Axial symmetry around the pipe

Equations

(1)	$\begin{array}{l}\displaystyle \rho \, c \, \frac{\partial T}{\partial t} = \frac{d}{dl} \, \bigg( \lambda \, \frac{dT}{dl} \bigg) - \rho \, c \, v \, \frac{dT}{dl} + \frac{2 \lambda}{\lambda_e} \cdot \frac{r_f}{r_w^2} \cdot U \cdot \left[ T_e(t, l, r_w) - T \right]\end{array}$

(2)	$\begin{array}{l}\displaystyle \rho_e \, c_e \, \frac{\partial T_e}{\partial t} = \nabla ( \lambda_e \nabla T_e)\end{array}$

(3)	$\begin{array}{l}\displaystyle T(t=0, l) = T_{e0}(l)\end{array}$

(4)	$\begin{array}{l}\displaystyle T_e(t=0, l, r) = T_{e0}(l)\end{array}$

(5)	$\begin{array}{l}\displaystyle T(t, l=0) = T_0(t)\end{array}$

(6)	$\begin{array}{l}\displaystyle T_e(t, l, r \rightarrow \infty) = T_{e0}(l)\end{array}$

(7)	$\begin{array}{l}\displaystyle 2 \pi \, \lambda_e \, r_w \, \frac{\partial T_e}{\partial r} \, \bigg\|_{r=r_w} = 2 \pi \, r_f \, U \cdot \left( T_e \, \bigg\|_{r=r_w} - T \right)\end{array}$