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

along-pipe temperature distribution and evolution in time

Inputs

	pipeline trajectory,		fluid density
	pipeline cross-section area		fluid viscosity
	intake temperature		initial temperature of the medium around the pipeline
	intake pressure		specific heat capacity of the medium around pipeline
	intake flowrate		thermal conductivity of the medium around pipeline
	heat transfer coefficient based on pipeline schematic

Assumptions

Stationary fluid flow

Axial symmetry around the pipe

Homogenous fluid flow

Equations

\rho \, c \, \frac{\partial T}{\partial t} = \frac{d}{dl} \, \bigg( \lambda \, \frac{dT}{dl} \bigg)  - \rho \, c \, v \, \frac{dT}{dl} + \frac{U}{r_w} \cdot \big( T_e(t, l, r_w) - T \big)

\rho_e \, c_e \, \frac{\partial T_e}{\partial t} = \nabla ( \lambda_e \nabla T_e)

T(t=0, l) = T_{e0}(l)

T_e(t=0, l, r) = T_{e0}(l)

T(t, l=0) = T_0(t)

T_e(t, l, r \rightarrow \infty) = T_{e0}(l)

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)

Approximations

Temperature Profile in Homogenous Stationary Pipe Flow Analytical Ramey @model

References

https://en.wikipedia.org/wiki/Darcy_friction_factor_formulae

https://neutrium.net/fluid_flow/pressure-loss-in-pipe/