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=1

4. Reference

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