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ElectroMagnet

In this example, we will estimate the rise in temperature due to Joules losses in a stranded conductor. An electrical potential \$V_D\$ is applied to the entry/exit of the conductor which is also water cooled.

Running the case

The command line to run this case in linear is

mpirun -np 4 feelpp_toolbox_thermoelectric --case "github:{path:toolboxes/thermoelectric/ElectroMagnets/HL-31_H1}"
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The command line to run this case in non linear is

mpirun -np 4 feelpp_toolbox_thermoelectric --case "github:{path:toolboxes/thermoelectric/ElectroMagnets/HL-31_H1}" --case.config-file HL-31_H1_nonlinear.cfg
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Geometry

The conductor consists in a solenoid, which is one helix of a magnet.

The mesh can be retrieve from girder with the following ID: 5af59e88b0e9574027047fc0 (see girder).

Input parameters

Name Description Value Unit

\$\sigma_0\$

electric potential at reference temperature

53e3

\$S/mm\$

\$V_D\$

electrical potential

9

\$V\$

\$\alpha\$

temperature coefficient

3.6e-3

\$K^{-1}\$

L

Lorentz number

2.47e-8

\$W\cdot\Omega\cdot K^{-2}\$

\$T_0\$

reference temperature

290

\$K\$

h

transfer coefficient

0.085

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

\$T_w\$

water temperature

290

\$K\$

 
link:{examplesdir}/electromagnet/HL-31_H1_nonlinear.json[role=include]

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. Due to the dependence of the thermic and electric conductivities to the temperature, the problem is non linear. We can describe the conductivities with the following laws:

\begin{align*} \sigma(T) &= \frac{\sigma_0}{1+\alpha(T-T_0)}\\ k(T) &= \sigma(T)*L*T \end{align*} 
link:{examplesdir}/electromagnet/HL-31_H1_nonlinear.json[role=include]

  • toolbox: thermoelectric

Materials

Name Description Marker Value Unit

\$\sigma_0\$

electric conductivity

Cu

53e3

\$S.m^{-1}\$

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

\$V_D\$

Rint, Rext, Interface, GR_1_Interface

Neumann

0

 
link:{examplesdir}/electromagnet/HL-31_H1_nonlinear.json[role=include]

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

Marker Type Value

Rint, Rext

Robin

\$h(T-T_w)\$

V0, V1, Interface, GR_1_Interface

Neumann

0

 
link:{examplesdir}/electromagnet/HL-31_H1_nonlinear.json[role=include]

Outputs

The main fields of concern are the electric potential \$V\$, the temperature \$T\$ and the current density \$\mathbf{j}\$ or the electric field \$\mathbf{E}\$ presented in the following figure.

link:{examplesdir}/electromagnet/HL-31_H1_nonlinear.json[role=include]

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