Induction heating of a cylindrical rod by an encircling coil carrying time-harmonic current

This benchmark problem verifies the case of computing eddy currents induced in a conducting rod due to a circulating time-harmonic current. This problem is frequently encountered in induction heating applications. The circulating time-harmonic current in the coil produces a time-harmonic magnetic field, which in turn induces eddy currents in a conductor in its vicinity. The resistance of the conductor to the flow of the induced currents manifests as heat, and computing this Joule heat is the primary objective of this benchmark problem.

This page discusses:

Problem description
Model and boundary conditions
Analytical solution
Results and discussion
Input files
References
Figures

ProductsAbaqus/Standard

Problem description

The problem setup is shown in Figure 1. It depicts a conductive cylindrical rod with an encircling current-carrying coil of rectangular cross-section centered along the length of the rod. The conductive cylindrical rod has a radius of $a = 0.05$ m and a length of $L = 0.50$ m. Its conductivity and relative magnetic permeability are assumed to be $σ$ = 1.0 × 10⁷ S/m and $μ_{r}$ = 1.0. The inner and outer radius and the thickness of the encircling coil are $b = 0.09$ m, $c = 0.11$ m, and $h = 0.02$ m. The current density in the coil is assumed to have a magnitude of $J_{0}$ = 1.0 × 10⁷ A/m² and is oscillating with a frequency of $f$ = 50 Hz. The current is assumed to be flowing along the azimuthal direction in a clockwise sense when looking toward the negative z-direction. The medium surrounding the rod and coil setup is assumed to have properties similar to that of a vacuum. For these parameters, the skin depth of the conductor is about $δ = \sqrt{(} 2 / ω μ_{0} μ_{r} σ)$ = 22.5 mm, which is smaller than the conductor radius of 50 mm.

Model and boundary conditions

The magnetic vector potential formulation is used to solve this problem. Due to the symmetry of the problem, it is sufficient to model the first octant of the problem domain. Appropriate boundary conditions are imposed on the symmetry planes $x = 0$ , $y = 0$ , and $z = 0$ . Due to the asymmetry of the current with respect to the planes $x = 0$ and $y = 0$ , the magnetic vector potential is normal to these symmetry planes, which is enforced by a homogeneous Dirichlet boundary condition. Similarly, due to the symmetry of the current with respect to the plane $z = 0$ , the magnetic flux density is normal to the symmetry plane, which is enforced by a homogeneous Neumann boundary condition.

Since the problem domain is unbounded, it must be truncated in some way. Abaqus does not support absorbing boundary conditions; hence, the truncation boundary should be chosen far away from the conductor. An outer cylindrical boundary surface and a planar surface that is parallel to the $z = 0$ plane are chosen to truncate the domain. The magnetic vector potential decays away from the coil and can be approximated to have zero magnitude far away from it. Hence, a homogeneous Dirichlet boundary condition is applied on all outer boundary surfaces.

Analytical solution

The analytical solution to this problem has been studied by various authors, but the study that is of particular interest is presented by Bowler and Theodoulidis (2005). They consider the problem of computing eddy currents induced in a cylindrical rod due to an encircling current loop that may be positioned at an arbitrary height along the length of the rod. Expressions for the magnetic vector potential in various regions can be found in this reference paper.

Results and discussion

Figure 2 shows a comparison of the real and imaginary parts of the circumferential component of the electric field along the x-axis computed using an Abaqus/Standard analysis to those of the analytical solution. The figure clearly indicates that the analysis results compare very well with the analytical results and that the outer boundary surface is far enough away that the error introduced by truncation is small. For a time-harmonic analysis the amplitude of the electric field is the same as that of the amplitude of the magnetic vector potential scaled by the radian frequency.

Figure 3 shows a contour plot of the Joule heat produced in the conducting rod due to the induced eddy currents. The figure clearly shows that the Joule heat produced is larger near the surface of the conductor compared to its interior due to the skin effect. The figure also indicates that the Joule heat produced is larger near the center of the rod compared to its ends due to the closer proximity to the current coil.

Input files

src_rod_emc3d8.inp: Eddy current analysis of a conductive cylindrical rod encircled by a coaxial coil carrying a time-harmonic current using element type EMC3D8 and symmetry boundary conditions.
src_rod_emc3d4.inp: Eddy current analysis of a conductive cylindrical rod encircled by a coaxial coil carrying a time-harmonic current using element type EMC3D4 and symmetry boundary conditions.

References

Bowler, J. R., and T. P. Theodoulidis, “Eddy Current Induced in a Conducting Rod of Finite Length by a Coaxial Encircling Coil,” Journal of Physics D: Applied Physics, vol. 38, pp. 2861–2868, 2005.

Figures