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/HL31_H1}"
The command line to run this case in non linear is
mpirun np 4 feelpp_toolbox_thermoelectric case "github:{path:toolboxes/thermoelectric/ElectroMagnets/HL31_H1}" case.configfile HL31_H1_nonlinear.cfg
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 
3.6e3 
\(K^{1}\) 

L 
Lorentz number 
2.47e8 
\(W\cdot\Omega\cdot K^{2}\) 

\(T_0\) 
reference temperature 
290 
\(K\) 

h 
transfer coefficient 
0.085 
\(W\cdot m^{2}\cdot K^{1}\) 

\(T_w\) 
water temperature 
290 
\(K\) 
"Parameters":
{
"sigma0":53e3, //[ S/mm ]
"T0":290, //[ K ]
"alpha":3.6e3, //[ 1/K ]
"Lorentz":2.47e8, //[ W*Omega/(K*K) ]
"h": "0.085", //[ W/(mm^2*K) ]
"Tw": "290", //[ K ]
"VD": "9" //[ V ]
},
4.1. Model & Toolbox

This problem is fully described by a ThermoElectric 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:
"k":"sigma0*Lorentz*heat_T/(1+alpha*(heat_TT0)):sigma0:alpha:T0:Lorentz:heat_T", //[ W/(mm*K) ]
"sigma":"sigma0/(1+alpha*(heat_TT0))+0*heat_T:sigma0:alpha:T0:heat_T"// [S/mm ]

toolbox: thermoelectric
4.2. Materials
Name  Description  Marker  Value  Unit  

\(\sigma_0\) 
electric conductivity 
Cu 
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 
"electricpotential":
{
"Dirichlet":
{
"V0":
{
"expr":"0" // V_0 [ V ]
},
"V1":
{
"expr":"VD:VD"
}
}
}
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 
\(h(TT_w)\) 

V0, V1, Interface, GR_1_Interface 
Neumann 
0 
"temperature":
{
"Robin":
{
"Rint":
{
"expr1":"h:h",
"expr2":"Tw:Tw"
},
"Rext":
{
"expr1":"h:h",
"expr2":"Tw:Tw"
}
},
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":
{
"usemodelname":1,
"thermoelectric":
{
"Exports":
{
"fields":["heat.temperature","electric.electricpotential","electric.electricfield","electric.currentdensity","heat.pid"]
}
}
}