Joule Heating by Electric Current in Solids

In presence of the electric current, the corresponding specific Joule heat QJ, W/m3, is released and included in QH of the heat transfer equation Equation 1 (see “Conjugate Heat Transfer”). In the case of isotropic material QJ is:

where r is the solids' electrical resistivity, Ω·m, (it can be temperature-dependent) and i is the electric current density, A/m2.

The electric current density vector:

is determined via the electric potential φ, V. To obtain the electric potential φ, Flow Simulation utilizes the steady-state Laplace equation:

Here rii is the temperature-dependent electrical resistivity in the i-th coordinate direction.

Transient electric problems with boundary conditions depending on time are considered as quasi-steady-state. In this case the steady-state problem for potential is solved at each time step, not taking into account transient electrical processes in solids.

The Laplace equation is solved numerically in the computational subdomain (it may be a part of the overall computational domain) of electroconductive materials. This computational subdomain automatically excludes dielectric solids and fluid areas inside. The total electric current in normal direction over a surface In, A, or electric potential φ, V, may be specified by user as boundary conditions for the problem. These conditions may be imposed on surfaces between fluid/electroconductive solid, electroconductive solid/electroconductive solid, dielectric solid/electroconductive solid, and outer solid surfaces. If no electrical boundary conditions are specified by user, the In = 0 boundary condition is automatically specified by default on bounding surfaces.

A surface between electroconductive solids in the computational subdomain is either considered zero-resistance (default) or the electric contact resistance is specified on it. The resistance value is either given explicitly or calculated from the given material and its thickness.

A contact resistance specified on a surface implies that the current passing through it produces the corresponding Joule heating, which yields the following surface heat source QJS, W/m2:

where in is the electric current density normal to the surface and Δφ is the electric potential drop at this surface.

The anisotropic electrical resistivity of electroconductive solids can be anisotropic, i.e. specified by its components in the coordinate system’s directions ri , i = 1, 2, 3. The isotropic/anisotropic type of material is specified for electrical resistivity and thermal conductivity simultaneously, i.e. so that their main axes coincide.