QuarterTurn

In this example, we will estimate the rise in temperature due to Joules losses in a stranded conductor. An electrical potential $V_0$ is applied to the entry/exit of the conductor which is also cooled by a force flow.
The geometry of the conductor is choosen as to have an analytical expression for the temperature.

1. Geometry

The conductor consists in a rectangular cross section torus which is somehow "cut" to allow for applying electrical potential. The conductor is cooled with a force flow along its cylindrical faces.+ In 2D, the geometry is as follow:
In 3D, this is the same geometry, but extruded along the z axis.

2. Input parameters

Name	Description	Value	Unit
$r_i$	internal radius	30.6	$mm$
$r_e$	external radius	53.2	$mm$
$h$	heigth	2.305	$mm$
$\delta$	angle	$\pi/2$	$rad$
$V_D$	electrical potential	0.125	$V$
$h_i$	internal transfer coefficient	$80e3$	$W\cdot mm^{-2}\cdot K^{-1}$
$T_{wi}$	internal water temperature	303	$K$
$h_e$	external transfer coefficient	$80e3$	$W\cdot mm^{-2}\cdot K^{-1}$
$T_{we}$	external water temperature	293	$K$

Name

Description

Value

Unit

$r_i$

internal radius

30.6

$mm$

$r_e$

external radius

53.2

$mm$

$h$

heigth

2.305

$mm$

$\delta$

angle

$\pi/2$

$rad$

$V_D$

electrical potential

0.125

$V$

$h_i$

internal transfer coefficient

$80e3$

$W\cdot mm^{-2}\cdot K^{-1}$

$T_{wi}$

internal water temperature

303

$K$

$h_e$

external transfer coefficient

$80e3$

$W\cdot mm^{-2}\cdot K^{-1}$

$T_{we}$

external water temperature

293

$K$

As the mesh is, by default in mm, we use specific units for this tests.

2.1. Model & Toolbox

This problem is fully described by a Thermo-Electric model, namely a poisson equation for the electrical potential $V$ and a standard heat equation for the temperature field $T$ with Joules losses as a source term.

2.2. Materials

Name	Description	Marker	Value	Unit
$\sigma$	electric conductivity	omega	$58.e3$	$S.mm^{-1}$

Name

Description

Marker

Value

Unit

$\sigma$

electric conductivity

omega

$58.e3$

$S.mm^{-1}$

2.3. Boundary conditions

The boundary conditions for the electrical probleme are introduced as simple Dirichlet boundary conditions for the electric potential on the entry/exit of the conductor. For the remaining faces, as no current is flowing througth these faces, we add Homogeneous Neumann conditions.

Marker	Type	Value
V0	Dirichlet	0
V1	Dirichlet	0.5/4.
Rint, Rext, top, bottom	Neumann	0

Marker

Type

Value

Dirichlet

0.5/4.

Rint, Rext, top*, bottom*

Neumann

As for the heat equation, the forced water cooling is modeled by robin boundary condition with $Tw$ the temperature of the coolant and $h$ an heat exchange coefficient.

Marker	Type	Value
Rint	Robin	$h_i(T-T_{wi})$
Rext	Robin	$h_e(T-T_{we})$
V0, V1, top, bottom	Neumann	0

Marker

Type

Value

Rint

Robin

$h_i(T-T_{wi})$

Rext

Robin

$h_e(T-T_{we})$

V0, V1, top*, bottom*

Neumann

*: only in 3D

3. Outputs

hsize	$T_{min} (K)$	$T_{max} (K)$
1	318.812	362.227

hsize

$T_{min} (K)$

$T_{max} (K)$

318.812

362.227

To change the mesh size hsize just edit the cfg file and change the corresponding line:

dim=3
units=mm
geofile=quarter-turn3D.geo
geofile-path=$cfgdir
...

[gmsh]
filename=$cfgdir/quarter-turn3D.geo
hsize=1cfg

4. Reference

For more advanced results, including convergence rate of the error, see the test case from Feel++ Thermo-Electric toolbox.