Modeling of the Temperature Regimes in a Layered Bimetallic Plate under Short-Term Induction Heating

Abstract

A mathematical model for determining the temperature field of a bimetallic plate with plane-parallel boundaries during short-term induction heating by a non-stationary electromagnetic field is proposed. Initial boundary value problems for determining the parameters of a non-stationary electromagnetic field and temperature are formulated. The temperature and the component of the magnetic field intensity vector that is tangential to the plate base were selected as defining functions. We used an approximation of the defining functions in each layer of the plate with quadratic polynomials by the thickness coordinate and Laplace transform of the integral over time. General solutions to the formulated problems under uniform non-stationary electromagnetic action were obtained. Based on them, the temperature during short-term induction heating by a non-stationary electromagnetic field was numerically analyzed depending on its amplitude-frequency parameters and duration.

1. Introduction

Many modern industrial devices (magnet wires of motors, generators, inductors, and transformers) operate under the effect of electromagnetic fields (EMF). Induction heating of electrically conductive materials, as well as mathematical models, numerical-analytical and experimental approaches to the study of such processes and their use in industry, have been the subject of consideration in many fundamental works over the past decades. Without aspiring to do a complete review, we will mention here a few well-known monographs [1,2,3].

Such processes under the effect of a stationary electromagnetic field have been studied in particular detail. In recent years, a number of mathematical models related to thermal processes caused by induction heating have been developed and researched.

The study [4] presents a mathematical model of obtaining Joule’s heat in the form of a thermal load on a thin metal plate subjected to the lateral effect of a uniform time-varying low-frequency electromagnetic field. The problem is solved analytically as an internal Dirichlet boundary value problem. The effect of plate thickness, field frequency, and characteristic pulse times depending on thermal load are analyzed.

Shen et al. [5] developed a mathematical model of the temperature field that emerges during high-frequency induction heating within the framework of Maxwell’s equations theory. It was proposed to solve coupled equations of electromagnetic and temperature fields using numerical modeling as well. The effects of body and field parameters on the heating temperature with high-frequency induction were evaluated.

In [6], a mathematical model of the thermally stressed state of electrically conductive bodies under the influence of external pulsed electromagnetic fields with amplitude modulation was investigated. The model proposed by the authors is a generalization of known models for quasi-stationary and pulsed electromagnetic fields.

Serpilli et al. [7] studied the mechanical behavior of two linear isotropic thermoelastic bodies connected by a thin layer made of a linear isotropic thermoelastic material via asymptotic analysis. Two different boundary models (thermoelastic interface models) and related boundary value problems were studied. The asymptotic expansion method, in view of the effect of higher-order members, was used. The generalized thermoelastic law of the section was determined based on the formulated models.

In [8], the authors proposed a mathematical model for determining the temperature in an electrically conductive sphere during its short-term induction heating. Corresponding initial boundary value problems are formulated. Polynomial approximation of the defining functions by the radial variable was realized to construct solutions. Time changes in Joule heat and temperature are also numerically analyzed.

Numerical modeling of the induction heating process can be difficult from the computational point of view, especially in the case of ferromagnetic materials. Article [9] is devoted to finding solutions for the coupled electromagnetic-thermal problem with higher computational efficiency. Therefore, the so-called semi-analytical modeling approach is needed. It is based on initial finite element calculation followed by analytical electromagnetic equations to solve the associated problem.

In recent years, mathematical models of induction heating processes under the effect of a non-stationary electromagnetic field (NSEMF) have also been studied. Musii et al. [10] proposed a mathematical model for determining the temperature field of a conductive plate element, which is affected by a uniform non-stationary electromagnetic field. The initial boundary value problems for determining the parameters of NSEMF and temperature are formulated. General solutions of the corresponding Cauchy problems for uniform non-stationary electromagnetic action are obtained.

Analytical and numerical approaches to the study of induction heating are a good basis for using this process to solve a number of practical problems. In particular, Bobart in [11] analyzed in detail the existing physical models and numerical-experimental approaches to the process of heating conductive material by eddy currents induced by an alternating electromagnetic field. In [12], the physical model of Joule heat generation during the flow of current along a thin wire connecting two volumetric electrodes is considered. Torteman et al. [13] investigated the issue of the occurrence of parametric resonance caused by microbeams, which are electrothermally activated by heating with time-varying Joule heat. The Euler–Bernoulli model is used. The problem of heat exchange is solved analytically.

One of the important usages of induction heating is the strengthening of gears [14,15]. Induction contour strengthening of gears is an effective heat treatment technology recommended for high-tech applications in engineering and aerospace.

In particular, mathematical and numerical models of the gear strengthening process, which includes two-frequency induction heating, were investigated in [14]. The modeling of fast induction heating, as the first stage of contour induction strengthening, is studied in the paper [16]. The method of finite elements used for predicting the evolution of the microstructure during induction strengthening processes is analyzed in the paper.

Bimetallic plate elements are used as structural elements in many modern electrotechnical devices. Such elements are affected by various physical factors, such as power, thermal, and electromagnetic ones, while technological functioning and operation. There are certain studies of the power and thermal factors that influence the temperature distribution in bimetallic plate elements [17,18,19,20,21]. In [17], the joining of layers of the Al-Cu bimetallic composite and the diffusion process on edge between the layers using hot rolling are analyzed. To obtain high-quality clad CuCrZr plates produced by explosion welding, theoretical analysis and numerical modeling were carried out in [18] in order to optimize the welding parameters. In article [19], an analytical mechanical model of a metal plate is built using the classical elasticity theory. The process of joining bimetallic Ti/Steel plates by hot rolling has been studied.

Dong et al. in [20] proposed three-dimensional models of finite elements to study the pre-bending process and a bimetallic pipe formation. Konieczny et al. in [21] presented a numerical and analytical study of the stress distribution along the plate thickness in circular axisymmetric perforated steel–titanium bimetallic plate. The results were obtained numerically using the finite element method. The proposed analytical approaches used by engineers of bimetallic plates seem to be feasible.

The analysis of these and other publications known so far showed that the electromagnetic fields influence the thermal regime of bimetallic elements functioning as sensors and electromagnetic contact adapters have not been sufficiently studied. In particular, this is important because bimetallic plates are used in electrical devices, as well as in aircraft and ship anti-icing systems. In these systems, they are activators and sensors of mechanical vibrations of certain elements of constructions. These vibrations are used to eliminate unwanted layers of ice on the constructions. In this case, bimetallic plates are affected by unstable impulse EMFs with amplitude modulation [22]. Calculation of thermal and thermoelastic regimes of uniform and bimetallic electrically conductive elements during induction heating should be carried out using reference books [23,24].

Modern technologies of electromagnetic heat treatment of bimetallic plate elements use short-term induction heating by NSEMF. Therefore, the task of mathematical modeling of thermal processes in bimetallic plates and the study of their temperature regimes during short-term induction heating by NSEMF is an urgent engineering task.

The goals of this paper are the following:

✓

To construct a mathematical model for determining Joule’s heat and temperature in a bimetallic plate during short-term induction heating;

✓

to develop a methodology for solving the corresponding initial boundary value problems and to determine the parameters of NSEMF and temperature;

✓

to study the bimetallic plate temperature regime depending on the amplitude-frequency parameters of the NSEMF and its duration.

2. Mathematical Model

An infinite bimetallic plate with $x$ and $y$ coordinates and the boundary bases $z=-h_1$ and $z=h_2$ is considered. Constants $h_1$, $h_2$ are the thicknesses of component layers, respectively. The coordinate plane $XOY$coincides with the plane $z=0$connecting the component layers (Figure 1).

The component layers’ materials are uniform, isotropic, and non-ferromagnetic. We consider their electrical and thermophysical characteristics constant and equal to their average value in the temperature change range.

The bimetallic plate is under the effect of an external uniform non-stationary EMF. The field in question is set by the values $H_y^{\pm }\left(t\right)$ of the tangential components $H_y^{\left(n\right)}\left(z,t\right)$ of the magnetic field intensity vector ${\overrightarrow{H}}^{\left(n\right)}=\left\{0;H_y^{\left(n\right)}; 0\right\}$ in the $n$th $\left(n=1,2\right)$ layer of the plate

$$H_y^{\left(1\right)}\left(-h_1,t\right)=H_y^{-}\left(t\right), H_y^{\left(2\right)}\left(h_2,t\right)=H_y^{+}\left(t\right),$$

(1)

where $H_y^{+}\left(t\right)$, $H_y^{-}\left(t\right)$ is the set functions of time $t$.

The bases of the plate $z=-h_1$ and $z=h_2$ are under conditions of convective heat exchange with the environment or under other thermal conditions. The connection of the component layers fulfills the conditions of perfect electromagnetic and thermal contacts in plane $z=0$.

The tangential component $H_y^{\left(n\right)}\left(z,t\right)$ of the magnetic field intensity vector and the temperature $T_{}^{\left(n\right)}\left(z,t\right)$are chosen as defining functions of the problem in the $n$th $\left(n=1,2\right)$ component layer of the plate. These functions depend only on the thickness coordinate $z$ and the time $t$ under uniform electromagnetic action conditions.

Mathematical models and studies of induction heating regularities of uniform conductive bodies in the canonical form under the steady and quasi-steady electromagnetic field action are presented in the scientific literature [1]. Without pretending to be complete, we can note, in particular, the papers [25,26,27], as well as their references related to the relevant subject. In [25], the results of mathematical modeling of thermomechanical processes in electrically conductive solids subjected to high-temperature induction heating by a stationary electromagnetic field were obtained. Using the developed approach, the authors conducted a study of the electric current frequency effect on body certain mechanical characteristics. Hachkevych et al. [26] demonstrate computer modeling of electromagnetic, thermal and mechanical fields in ferromagnetic bodies. Mathematical models and regimes of high-temperature induction treatment with a stationary electromagnetic field of bodies made of various ferromagnetic materials were studied. In [27], optimized approaches were used for mathematical models of homogeneous conductive bodies induction heating. The goal is to determine the optimal control parameters, in particular, current strength and heating frequency. The obtained temperature fields correspond to the established requirements.

The mathematical model of electrically conductive bodies induction heating under the conditions of the non-stationary electromagnetic field, introduced in this paper, is a generalization of the previous mathematical model studied in [10] for a homogeneous plate. Such generalization aims at the development of analytical and numerical approaches similar to those in [10] for the case of a two-layer metallic plate. The model takes into account the conditions of ideal electromagnetic and thermal contacts on the surface of the plate layers connection. These conditions need to coordinate electrical and thermal processes in the layers on the contact surface. Therefore, this type of model is more complicated.

The calculation model for determining the magnetic field intensity and temperature consists of serially interconnected problems.

In the first stage, the parameters of EMF and Joule’s heat caused by external electromagnetic action are determined. Based on Maxwell’s relations, the tangential component $H_y^{\left(n\right)}\left(z,t\right)$ of field intensity vector${\overrightarrow{ H}}^{\left(n\right)}=\left\{0;H_y^{\left(n\right)}\left(z,t\right);0\right\}$ in each n-th layer of the plate can be determined from the equation

$$\frac{{\partial }^2{H}_y^{\left(n\right)}}{\partial z^2}-{\sigma }_n{\mu }_n\frac{\partial H_y^{\left(n\right)}}{\partial t}=0.$$

(2)

Under boundary conditions (1) on the bimetallic plate bases $z=-h_1$ and $z=h_2$ and under the conditions of electromagnetic perfect contact.

These conditions are obtained due to the tangential components’ equality of the vectors ${\overrightarrow{H}}^{\left(n\right)}$ and ${\overrightarrow{E}}^{\left(n\right)}$ are the magnetic and electric fields intensities of the plate in the component layers along their connection plane$z=0$, that is

$$H_y^{\left(1\right)}\left(0,t\right)=H_y^{\left(2\right)}\left(0,t\right), E_x^{\left(1\right)}\left(0,t\right)=E_x^{\left(2\right)}\left(0,t\right).$$

Using Maxwell’s relations

$${\overrightarrow{E}}^{\left(n\right)}=\frac{1}{{\sigma }_n}rot {\overrightarrow{H}}^{\left(n\right)}\left(E_x^{\left(n\right)}=\frac{1}{{\sigma }_n}\frac{\partial H_y^{\left(n\right)}}{\partial z} \right),$$

these contact conditions can be rewritten down as

$$H_y^{\left(1\right)}\left(0,t\right)=H_y^{\left(2\right)}\left(0,t\right), \frac{\partial H_y^{\left(1\right)}\left(0,t\right)}{\partial z}=k_{\sigma }\frac{\partial H_y^{\left(2\right)}\left(0,t\right)}{\partial z}.$$

(3)

where ${\sigma }_n$, ${\mu }_n$ are coefficients of electrical conductivity and magnetic permeability of the material of the$n$th plate layer, respectively; $k_{\sigma }={\sigma }_1/{\sigma }_2$.

In the absence of EMF at the time moment $t=0$, the initial conditions will be

$$H_y^{\left(n\right)}\left(z,0\right)=0.$$

(4)

According to the found function $H_y^{\left(n\right)}\left(z,t\right)$, specific Joule heat density $Q^{\left(n\right)}\left(z,t\right)$ in each $n$th component layer of the plate is written down using the formula

$$Q^{\left(n\right)}\left(z,t\right)=\frac{1}{{\sigma }_n}{\left(\frac{\partial H_y^{\left(n\right)}\left(z,t\right)}{\partial z}\right)}^2,$$

(5)

At the second stage, using the found Joule heat distribution (5) $Q^{\left(n\right)}\left(z,t\right)$, the temperature field $T^{\left(n\right)}\left(z,t\right)$ in the $n$th component layer of the plate is found from the heat conductivity equation

$$\frac{{\partial }^2{T}^{\left(n\right)}}{\partial z^2}-\frac{1}{{\kappa }_n}\frac{\partial T^{\left(n\right)}}{\partial z}=-\frac{Q^{\left(n\right)}}{{\lambda }_n}$$

(6)

under the boundary conditions

$$\frac{\partial T^{\left(1\right)}\left(-h_1,t\right)}{\partial z}=0, \frac{\partial T^{\left(2\right)}\left(h_2,t\right)}{\partial z}=0$$

(7)

of thermal insulation of the outer plate surfaces $z=-h_1$ and $z=h_2$ and under the conditions of perfect thermal contact [23]

$$T^{\left(1\right)}\left(0,t\right)=T^{\left(2\right)}\left(0,t\right), \frac{\partial T^{\left(1\right)}\left(0,t\right)}{\partial z}=k_{\lambda }\frac{\partial T^{\left(2\right)}\left(0,t\right)}{\partial z}$$

(8)

along the plane $z=0$ connecting the plate component layers. Note that ${\kappa }_n$, ${\lambda }_n$ are temperature conductivity and heat conductivity coefficients of the material in the $n$-th layer; $k_{\lambda }={\lambda }_2/{\lambda }_1$.

Contact conditions (8) for the temperature $T^{\left(n\right)}$ on division line $z=0$ between the plate component layers describe the equality of temperatures and heat streams $\left({\lambda }_1\frac{\partial T^{\left(1\right)}\left(0,t\right)}{\partial z}={\lambda }_2\frac{\partial T^{\left(2\right)}\left(0,t\right)}{\partial z}\right)$ in the plate component layers.

At the time moment $t=0$, the initial condition per temperature is

$$T^{\left(n\right)}\left(z,0\right)=0.$$

(9)

Let us note that on the bimetallic plate outer surfaces, the other thermal conditions may be applied, such as convective heat exchange with the environment, for example.

3. Problem General Solutions under the Effect of a Uniform NSEMF

The processes of induction heating of a bimetallic plate considered in the paper are nonlinear ones. In particular, [28] developed a theoretical basis for studying the characteristics of nonlinear electroelastic properties of some electrosensitive materials. The developed theory is applied to model boundary value problems.

To solve the initial boundary value problems (1)–(4) and (6)–(9), the distributions of the defining functions ${\mathsf{\Phi }}^{\left(n\right)}\left(z,t\right)=\left\{H_y^{\left(n\right)}\left(z,t\right), T^{\left(n\right)}\left(z,t\right)\right\}$ along the thickness of each component layer of the plate should be approximated using quadratic polynomials of the type

$${\mathsf{\Phi }}^{\left(n\right)}\left(z,t\right)={{\displaystyle \sum }}_{i=0}^2{a}_i^{\mathsf{\Phi }\left(n\right)}\left(t\right) z^i.$$

(10)

The choice of function ${\mathsf{\Phi }}^{\left(n\right)}\left(z,t\right)$in the form of quadratic polynomials to approximate their distributions along the thickness coordinate is due to the necessity:

(1)

to take into account the nonlinear nature of the distribution $H_y^{\left(n\right)}\left(z,t\right)$ and $T^{\left(n\right)}\left(z,t\right)$ on the thickness of each layer;

(2)

to take into account the fact that, in addition to the fulfillment of boundary conditions, it is also necessary to ensure the fulfillment of contact conditions on the surfaces of connecting layers.

The coefficients $a_i^{\mathsf{\Phi }\left(n\right)}\left(t\right)$ of the polynomials (10) are determined through integral (total per the package of two layers) characteristics ${\mathsf{\Phi }}_s\left(t\right)$ of the key functions ${\mathsf{\Phi }}^{\left(n\right)}\left(z,t\right)$

$${\mathsf{\Phi }}_s\left(t\right)={{\displaystyle \sum }}_{n=1}^2{{\displaystyle \int }}_{z_{n-1}}^{z_n}{{\displaystyle \int }}_{z_{n-1}}^{z_n}{\mathsf{\Phi }}^{\left(n\right)}\left(z,t\right)z^{s}dz, s=1,2$$

(11)

and the set boundary conditions (1) and (7) on the outer surfaces and under the conditions (3) and (8) along the plane $z=0$ connecting the component layers of the plate. In Formula (11) we have $z_0=-h_1$, $z_1=0$, $z_2=h_2$.

To find integral characteristics ${\mathsf{\Phi }}_s\left(t\right)$ of the searched functions ${\mathsf{\Phi }}^{\left(n\right)}\left(z,t\right)$, the original Equations (2) and (6) should be integrated according to the Formula (11) considering the expressions (10). As a result, the original data of the initial boundary value problem on key functions ${\mathsf{\Phi }}^{\left(n\right)}\left(z,t\right)$ have been reduced to the corresponding Cauchy problems on their integral characteristics ${\mathsf{\Phi }}_s\left(t\right)$. The Cauchy problems are formulated in the next systems:

$$\{\begin{array}{c}\frac{d{H}_{y1}\left(t\right)}{dt}-d_1{H}_{y1}\left(t\right)-d_2{H}_{y2}\left(t\right)=d_3{H}_y^{-}\left(t\right)+d_4{H}_y^{+}\left(t\right),\\ \frac{d{H}_{y2}\left(t\right)}{dt}-d_5{H}_{y1}\left(t\right)-d_6{H}_{y2}\left(t\right)=d_7{H}_y^{-}\left(t\right)+d_8{H}_y^{+}\left(t\right),\end{array}$$

(12)

$$\{\begin{array}{c}\frac{d{T}_1}{dt}+d_1^T{T}_1+d_2^T{T}_2={{\displaystyle \sum }}_{n=1}^2\frac{{\kappa }_n}{{\lambda }_n}{{\displaystyle \int }}_{z_{n-1}}^{z_n}{Q}^{\left(n\right)}\left(z,t\right)dz,\\ \frac{d{T}_2}{dt}+d_3^T{T}_1+d_4^T{T}_2={{\displaystyle \sum }}_{n=1}^2\frac{{\kappa }_n}{{\lambda }_n}{{\displaystyle \int }}_{z_{n-1}}^{z_n}{Q}^{\left(n\right)}\left(z,t\right)zdz,\end{array}$$

(13)

where the coefficients $d_{1÷8}$, $d_{1÷4}^T$ are defined through geometric parameters and the electrophysical and thermophysical characteristics of the plate material. In particular,

$$d_j=\frac{2h_2}{{\sigma }_2{\mu }_2}{a}_{1j}^{\left(2\right)}-\frac{2h_1}{{\sigma }_1{\mu }_1}{a}_{1j}^{\left(1\right)}, d_{j+4}=\frac{h_2^2}{{\sigma }_2{\mu }_2}{a}_{2j}^{\left(2\right)}-\frac{h_1^2}{{\sigma }_1{\mu }_1}{a}_{2j}^{\left(1\right)}, j=\overline{1,4},$$

$$d_j^T={\kappa }_2{b}_{1j}^{(2)}-{\kappa }_1{b}_{1j}^{(1)}, d_{j+2}^T={\kappa }_2{b}_{2j}^{(2)}{h}_2^2-{\kappa }_1{h}_1^2{b}_{2j}^{(1)}, j=1,2.$$

Here, the numerical coefficients $a_{1j}^{\left(1\right)}$, $a_{1j}^{\left(2\right)}$, $a_{2j}^{\left(1\right)}$, $a_{2j}^{\left(2\right)}$, $b_{1j}^{(1)}$, $b_{1j}^{(2)}$, $b_{2j}^{(1)}$, $b_{2j}^{(2)}$ are determined as a result of using expression (10) in accordance with Formula (11) and boundary conditions (1), (3), (7) and (8). Their values depend on the characteristics of the materials of the constituent layers of the plate.

Solutions for Cauchy problems (12) and (13) are obtained using the integral Laplace transform over time and written down as convolutions of the functions that describe the set boundary conditions and uniform solutions of the systems (12) and (13).

Expressions of the magnetic field intensity vector component $H_y^{\left(n\right)}\left(z,t\right)$ and the temperature $T^{\left(n\right)}\left(z,t\right)$ are obtained in the following form:

$$\begin{gathered} H_y^{\left(n\right)}\left(z,t\right)={\displaystyle \sum }_{i=0}^2\{{\displaystyle \sum }_{s=1}^2{a}_{is}^{\left(n\right)}{\displaystyle \sum }_{k=1}^2{{\displaystyle \int }}_0^t\left[A_{s1k}{H}_y^{-}\left(\tau \right)+A_{s2k}{H}_y^{+}\left(\tau \right)\right]e^{p_k\left(t-\tau \right)}d\tau + \\ +a_{i3}^{\left(n\right)}{H}_y^{-}\left(t\right)+a_{i4}^{\left(n\right)}{H}_y^{+}\left(t\right)\} z^i, \end{gathered}$$

(14)

$$T^{\left(n\right)}\left(z,t\right)={\displaystyle \sum }_{k=0}^2{\displaystyle \sum }_{s=1}^2\left(b_{ks}^{\left(n\right)}{\displaystyle \sum }_{m=1}^2{{\displaystyle \int }}_0^t\left[B_{s1m}{W}_1^Q\left(\tau \right)+B_{s2m}{W}_2^Q\left(\tau \right)\right] e^{p_m\left(t-\tau \right)}d\tau \right) z^k .$$

(15)

$A_{sj}\left(k\right)$, $B_{sj}\left(m\right)$ ($s,j=1,2$) depend on the roots$p_k$, $p_m$ of the corresponding characteristic equations of Cauchy problems for defining the integral characteristics of $H_y^{\left(n\right)}$ and $T^{\left(n\right)}$;

$$W_s^Q\left(\tau \right)={{\displaystyle \sum }}_{n=1}^2\frac{{\kappa }_n}{{\lambda }_n}{{\displaystyle \int }}_{{\gamma }_{n-1}}^{{\gamma }_n}{Q}^{\left(n\right)}\left(z,t\right)z^{s-1}dz, (s=1,2);$$

$a_{is}^{\left(n\right)}=a_i^{H\left(n\right)}\left(t\right)$, $b_{ks}^{\left(n\right)}=a_i^{T\left(n\right)}\left(t\right)$ are coefficients depending on the geometric parameters and the electrophysical and thermophysical characteristics of the plate materials.

4. Numerical Analysis

The paper studies two interdependent processes (electromagnetic and thermal). To ensure that these processes are interconnected, Joule’s heat, found in the analysis of the electromagnetic process, is transferred to the right-hand side of the heat conduction equation as a heat source. This is also due to different time scaling of the electromagnetic and thermal processes. An additional challenge is to satisfy the contact conditions, especially for the calculation of the interconnected fields. That is why we combine analytical and numerical approaches to determine the function ${\mathsf{\Phi }}^{\left(n\right)}\left(z,t\right)$ value.

The action of eddy currents during short-term induction heating by a non-stationary EMF caused by generators of high-frequency electromagnetic oscillations is mathematically described by functions$H_y^{\pm }\left(t\right)$ as follows [22]:

$$H_y^{\pm }\left(t\right)=k{H}_0\left(e^{-{\beta }_{1}t}-e^{-{\beta }_{2}t}\right)\cos \omega t.$$

(16)

$H_0$ is the amplitude of the intensity of the magnetic field caused by carrier electromagnetic oscillations with frequency $\omega$; $k$is normalization factor; ${\beta }_1$ and ${\beta }_2$ are parameters corresponding to the times of the increasing and decreasing fronts of a non-stationary EMF with duration $t_i$. The time of the increasing front $t_{incr}$ was chosen to be $t_{incr}\approx 0.1t_i$. The time of the decreasing front $t_{decr}\approx 0.9t_i$. Subsequently, at the moment of time $t\approx t_i$, the amplitude of the carrier electromagnetic oscillations practically decays to zero.

To solve the serially connected problems of electrodynamics and heat conductivity mentioned above under a short-term induction heating by a non-stationary EMF, we substitute the functions $H_y^{\pm }\left(t\right)$ with the expression (16) in the Formula (14). The found expression of the function$H_y^{\left(n\right)}\left(z,t\right)$ is substituted in the Formula (5) to find the Joule heat expression $Q^{\left(n\right)}\left(z,t\right)$ in the component layers of the plate. This expression is substituted in the Formula (15) and, after necessary transformations, the expression of the temperature $T^{\left(n\right)}\left(z,t\right)$ in the plate $n$-th component layer is written down. On the basis of this obtained expression of the temperature in the plate component layers, its numerical analysis is executed.

Calculations were realized for a bimetallic plate with the component layers of non-ferromagnetic materials, such as stainless steel $\left(n=1\right)$ and copper $\left(n=2\right)$. The component layers’ thickness was taken $h_1=h_2=1 \mathrm{mm}$. The times $t_i$ of the induction heating by NSEMF were considered $t_i=10 \mathrm{s}$, $t_i=100 \mathrm{s}$.

The carrier signal frequency was $\omega =6.28\cdot {10}^5 \mathrm{rad}/\mathrm{s}$ within the radio frequency range. The numerical analysis of the maximum temperature values dependency $T_{max}$on magnetic field intensity $H_0$ was carried out for magnetic field intensities in the range $H_0={10}^2÷{10}^4\mathrm{A}/\mathrm{m}$ at the aforesaid times $t_i$ of induction heating by an NSEMF (Figure 2). Such values of $H_0$ are used in the technological induction processing of conductive elements of constructions [23]. It is known that the melting points $T^{*}$ for the materials of the plate component layers under study are the following: $T_{steel}^{*}\approx 1670 \mathrm{K}$, $T_{Cu}^{*}\approx 1358 \mathrm{K}$ [24]. Analysis of the dependencies shown in Figure 2 shows that during $t_i=1 \mathrm{s}$ of induction heating by the NSEMF under study at magnetic field intensities in the range of $H_0={10}^3÷{10}^4\mathrm{A}/\mathrm{m}$, the temperature $T_{max}$ is a lot lower than the steel and copper melting points $T^{*}$.

Figure 2 shows that at time duration $t_i=10 \mathrm{s}$ temperature $T_{max}$ in the first steel layer of bimetallic plate becomes commensurate with the melting temperature $T_{steel}^{*}$, but does not yet reach it. At $t_i=100 \mathrm{s}$ temperature $T_{max}$ at $H_0\approx 3.5\cdot {10}^3\mathrm{A}/\mathrm{m}$ in the steel layer attains the melting temperature $T_{steel}^{*}$. Note that at the considered durations $t_i=10÷100 \mathrm{s}$ of induction heating, the maximum values $T_{max}$of the temperature on the surface $z=h_2$ of the copper layer are significantly lower than the melting temperature of copper $T_{Cu}^{*}$ and the maximum values of $T_{max}$ of the temperature on the surface $z=0$ of the connection of the plate constituent layers are close to the temperature values on the surface of the copper layer.

During $t_i=10 \mathrm{s}$ (Figure 3), the temperature $T_{max}$ in the first steel layer of the bimetallic plate becomes proportional to the melting point $T_{steel}^{*}$, but does not reach it yet. During $t_i=100 \mathrm{s}$ (Figure 4), the temperature$T_{max}$ at $H_0\approx 3.5\cdot {10}^3\mathrm{A}/\mathrm{m}$ in the steel layer reaches the melting point $T_{steel}^{*}\approx 1670 \mathrm{K}$.

Let us also note that at induction heating times under consideration $t_i=1÷100 \mathrm{s}$, the maximum temperatures $T_{max}$ on the surface $z=h_2$ of the copper layer are significantly lower than the copper melting point $T_{Cu}^{*}\approx 1358 \mathrm{K}$. At all the durations under study $t_i=1÷100 \mathrm{s}$, the maximum temperature values $T_{max}$ on the connecting surface$z=0$ between the two layers of the bimetallic plate are close to the temperature values on the surface of the copper layer.

Figure 3 and Figure 4 show the results of numerical analysis of the maximum temperature values $T_{max}$ dependencies on the outer plate surfaces $z=-h_1$, $z=h_2,$ and on the contact surface $z=0$ between its component layers on the time $t_i$ of action of an NSEMF at magnetic field intensities $H_0={10}^3 \mathrm{A}/\mathrm{m}$ (Figure 3) and $H_0=4\cdot {10}^3 \mathrm{A}/\mathrm{m}$ (Figure 4).

From the dependencies shown in Figure 3, it follows that at a magnetic field intensity of $H_0={10}^3 \mathrm{A}/\mathrm{m}$, the temperature $T_{max}$ reaches the steel melting point $T_{steel}^{*}\approx 1670 \mathrm{K}$ on the surface of the steel layer only at induction heat durations $t_i\ge 1000 \mathrm{s}$, which are not short-term. On the basis of the dependencies shown in Figure 4, it was established that at a magnetic field intensity of$H_0=4\cdot {10}^3 \mathrm{A}/\mathrm{m}$, the temperature $T_{max}$ reaches the steel melting point on the surface of the steel layer in $t_i\approx 45–50 \mathrm{s}$ of induction heating. Note that at the aforesaid times of induction heating by an NSEMF with magnetic field intensities $H_0$ in the range ${10}^3÷{10}^4\mathrm{A}/\mathrm{m}$, the value of $T_{max}$ on the surface of the copper layer, as well as on the surface of the contact between the plate component layers is a lot lower than the copper melting point $T_{Cu}^{*}\approx 1358 \mathrm{K}$.

Figure 5 shows the distribution of maximum values of temperature $T_{max}$ along the thickness of component layers of the plate depending on time $t_i$ of induction heating with a magnetic field intensity of $H_0={10}^3 \mathrm{A}/\mathrm{m}$.

Figure 6 shows the distribution of maximum values of temperature $T_{max}$ along the thickness of plate component layers depending on magnetic field intensity $H_0$ during $t_i=100 \mathrm{s}$ of induction heating.

In addition, Figure 6 shows that at the time duration, $t_i=100 \mathrm{s}$ and the value $H_0=4\cdot {10}^3 \mathrm{A}/\mathrm{m}$ temperature $T_{max}$ on the surface of the steel layer can attain the melting temperature of steel, and at the same time on the surface of the constituent layers connection and on the surface of the copper layer is significantly lower than the melting temperature of copper.

5. Conclusions

The initial boundary value problems for determining the magnetic field intensity vector component tangential to the bimetallic plate bases and temperature are formulated. The initial boundary value problems on the defining functions are reduced to the corresponding Cauchy problems on their integral characteristics. General solutions of the Cauchy problems under uniform non-stationary electromagnetic action are written down via the integral Laplace transform, given the set initial conditions.

It has been established that the temperature gradient in the plate steel layer is much higher than the temperature gradient in the copper layer at all considered durations of short-term induction heating by the aforesaid NSEMF. This is due to the significantly non-linear nature of temperature distribution in the steel layer and close to linear (almost uniform) temperature distribution in the copper layer.

Realized theoretical studies and numerical analysis of the short-term induction heating regimes give the possibility to establish the next optimal parameters:

(a)

At a magnetic field intensity $H_0=3\cdot {10}^3 \mathrm{A}/\mathrm{m}$ during $t_i=100 \mathrm{s}$ of induction heating temperature $T_{max}$of the steel layer doesn’t attain critical value, at the same time temperature of the copper layer is satisfactory;

(b)

Using industrial power generators ($H_0=10\cdot {10}^3 \mathrm{A}/\mathrm{m}$) enables reduction of the time of similar heating regimes up to $\approx 10 \mathrm{s}$.

An advantage of the proposed NSEMF used for induction heating is their short-term duration and also the fact that expenditure of energy can be minimized. As possible important applications can be considered, in particular, industrial production, operation and maintenance of turbines, different types of pipelines containing two-layer composite metals. It also seems interesting to realize a more complicated theoretical study in a multi-layer case. An analogical approach may be used in the heat treatment of metals with composite structures using an induction heating process as well.