Mathematical model of non-stationary temperature distribution in the metal body produced by induction heating process

An induction heating problem can be described by a parabolic differential equation. For this equation, specific Joule looses must be computed. It can be done by solving the Fredholm Integral Equation of the second kind for the eddy current of density. When we use the Nyström method with the singularity subtraction, the computation time is rapidly reduced. This paper shows the method for finding non-stationary temperature distribution in the metal body with illustrative examples.


Introduction
A bounded metal cuboid body Ω 1 of sizes l 1 × l 2 × l 3 is heated by an external electromagnetic field produced by inductor Ω 2 (see figure 1). The inductor is formed by a conductor of general shape and position that carries the harmonic current I ext .

Eddy current of density
Computation of the temperature distribution depends on the eddy current of density. It is a phasor J eddy = (J eddy,x 1 , J eddy,x 2 , J eddy,x 3 ).
The x 1 component of J eddy,x 1 (x) can be computed (see [4] and [5]) by Fredholm integral equation of the second kind ιJ eddy,x 1  where r(x, t) is the Euclidean distance, x = (x 1 , x 2 , x 3 ) and t = (t 1 , t 2 , t 3 ) are points in the metal body, s = (s 1 , s 2 , s 3 ) is a point at the inductor, I ext is harmonic current carried by the inductor, ω is angular frequency, γ(T (x)) is electrical conductivity, μ 0 is permeability of vacuum, e x 1 is unite vector e x 1 = (1, 0, 0) and ι is the complex unit. For each bounded and continuous temperature distribution T (x), κ(x) is a real, positive, bounded and continuous function.
Formulas for the remaining components J eddy,x 2 and J eddy,x 3 can be obtained by mere interchanging of indices.
With the notation we have system of integral equations The specific Joule losses at the point x in the body, which are needed to compute the temperature distribution are given by

The non-stationary temperature distrubution
The non-stationary distribution of the temperature T (x, t) at point x and time t in the metal body is described by a partial differential equation of the parabolic type where λ = λ(T (x, t)) denotes the thermal conductivity, ρ = ρ(T (x, t)) the specific mass of the material, c = c(T (x, t)) specific heat and ω(x, t) the specific Joule losses given by (9) at time t. The boundary condition along the whole surface of the body reads where α denotes the coefficient of the convective heat transfer, T ext the temperature of the surrounding medium and n the direction of the outward normal.

Numerical solution
The metal body Ω 1 was assumed to be cuboid. Let us cover the body by n 1 subsuboides in the x 1 direction, n 2 subsuboides in the x 2 direction and n 3 subsuboides in the We need to solve integral equation (1) to get Joule losses. Then we can put computed Joule losses to partial differential equation (10) to get the temperature distribution. The Joule losses must be recalculated at the time when changed. For computation of Joule losses we will apply the Nyström method with the compound mid-cuboid rule. For time integration we will use the classical finite difference method for partial differential equations of the parabolic type.

Application of Nyström method
The Nyström method is based on approximation of the integral by the numerial integration rule. We will use the compound min-point rule. Let the node points be x i defined as center of sub-cuboides. For the weight let where N = n 1 n 2 n 3 .
Here the numerical integration rule cannot be applied directly. The reason is that the function r(x, t) −1 is singular at x = t. One way to deal with such a problem is singularity subtraction. It was described in [3]. Let r N (x, t) be an approximation of r(x, t) which coincide outside a certain neighborhood of x = t. The integrand in equation (1) is approximated by Since the first element of the approximation is zero when x = t, the exact constructioin of r N (x, t) is immaterial. By using the numerical integration rule, approximation (12) and running x throung the node points we get a system of linear equations for numerical approximation of the x 1 component of the eddy current of density J N Then (13) is equivalent the real system of linear equations Following Nyström interpolation formula (for details see for example [1]) we get the numerical solution for eddy currents J N outside the node points Since Ω 1 is a cuboid, the integral in (15) and (16) can be computed analytically. The x 2 and x 3 components of the eddy current of density can be computed in an analogous way.

Application of the finite difference method
Let us use a finite difference method for the numerical solution of equation (10). Since thermal conductivity is a constant that depends on the temperature we can rewrite equation (10) into the form

Example 2
To see the dependence of the parameters let us show examples with different parameters. Let the parameters be the same as in example 1. Figure 8 shows temperature distribution at 30 s. Other figures have change in one parameter.

Conclusion
In example 2, the temperature changed as expected. It was shown in [5], that the time to compute the Joule losses is reduced to approximately 10 percent compared to the collocation method. Since the Joule losses need to be recalculated 3 − 10 times during time evolation to 60 s is this method a big improovement.