Motivation

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

In many practical cases the temperature distribution for the stationary fluid flow can be approximated by homogenous fluid flow model.

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.

Inputs & Outputs

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

Stationary fluid flow	Axial symmetry around the pipe
Homogenous fluid flow

(1)	$\begin{array}{l}\displaystyle \rho \, c \, \frac{\partial T}{\partial t} = \frac{d}{dl} \, \bigg( \lambda \, \frac{dT}{dl} \bigg) - \rho \, c \, v \, \frac{dT}{dl}\end{array}$

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