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.

1. 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}"sh

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.cfgsh

2. Data files

HL-31_H1 Github directory with:

The mesh can be found on Girder

3. 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).

4. 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	$K$

Name

Description

Value

Unit

$\sigma_0$

electric potential at reference temperature

53e3

$S/mm$

$V_D$

electrical potential

$V$

$\alpha$

temperature coefficient

3.6e-3

$K^{-1}$

Lorentz number

2.47e-8

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

$T_0$

reference temperature

290

$K$

transfer coefficient

0.085

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

$T_w$

water temperature

290

$K$

"Parameters":
{
    "sigma0":53e3, //[ S/mm ]
    "T0":290, //[ K ]
    "alpha":3.6e-3, //[ 1/K ]
    "Lorentz":2.47e-8, //[ W*Omega/(K*K) ]
    "h": "0.085", //[ W/(mm^2*K) ]
    "Tw": "290", //[ K ]
    "VD": "9" //[ V ]
},json

4.1. 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. 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:

$\begin{align*} \sigma(T) &= \frac{\sigma_0}{1+\alpha(T-T_0)}\\ k(T) &= \sigma(T)*L*T \end{align*}$

"k":"sigma0*Lorentz*heat_T/(1+alpha*(heat_T-T0)):sigma0:alpha:T0:Lorentz:heat_T", //[ W/(mm*K) ]
"sigma":"sigma0/(1+alpha*(heat_T-T0))+0*heat_T:sigma0:alpha:T0:heat_T"// [S/mm ]json

toolbox: thermoelectric

4.2. Materials

Name	Description	Marker	Value	Unit
$\sigma_0$	electric conductivity	Cu	53e3	$S.m^{-1}$

Name

Description

Marker

Value

Unit

$\sigma_0$

electric conductivity

53e3

$S.m^{-1}$

4.3. 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	$V_D$
Rint, Rext, Interface, GR_1_Interface	Neumann	0

Marker

Type

Value

Dirichlet

$V_D$

Rint, Rext, Interface, GR_1_Interface

Neumann

"electric-potential":
{
    "Dirichlet":
    {
        "V0":
        {
            "expr":"0" // V_0 [ V ]
        },
        "V1":
        {
            "expr":"VD:VD"
        }
    }
}json

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

Marker

Type

Value

Rint, Rext

Robin

$h(T-T_w)$

V0, V1, Interface, GR_1_Interface

Neumann

"temperature":
{
    "Robin":
    {
        "Rint":
        {
            "expr1":"h:h",
            "expr2":"Tw:Tw"
        },
        "Rext":
        {
            "expr1":"h:h",
            "expr2":"Tw:Tw"
        }
    },json

5. 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.

"PostProcess":
{
    "use-model-name":1,
    "thermo-electric":
    {
        "Exports":
        {
            "fields":["heat.temperature","electric.electric-potential","electric.electric-field","electric.current-density","heat.pid"]
        }
    }
}json

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