Theoretical justification of the induction heating method for strength research

. The paper proposes a theoretical justification of the test method for thermal strength, similar to the method of radiation heating. The use of the induction heating method itself provides for the ability to control the heating temperature with good accuracy, which makes it possible to heat the part evenly and quickly to the desired temperature due to overheating of the product, and also with the chosen method it is possible to completely eliminate the appearance of smoke, oil pollution and strong odours. In this case, external heating of samples of electrically conductive materials is carried out by high-frequency currents, which significantly expands the possibilities of obtaining preliminary experimental results and simplifies the experimental technique.


Introduction
Technologies of gas thermal spraying are successfully used in the construction of protective coatings of various classes on newly manufactured parts and as technologies for the restoration of worn products.However, self-fluxing coatings after spraying require additional heat treatment.One of the most effective processing methods is the technology of accelerated induction heating [1,2].
At enterprises, when assembling assemblies, heating of parts such as bearings and rings for mounting on the shaft is carried out in various ways, such as: -heating with an open flame of a blowtorch or gas burner; -conductive heating; -heating in convection ovens; -heating in oil baths.
The method of induction processing is as follows: the processed part is subjected to accelerated surface high-frequency heating, in which, due to the presence of the interface of media with different electrical conductivity, the main power is released not on the surface of the work piece, but on the boundary of the substrate.
The task of determining the intensity of heat release and determining stresses in structural parts is quite time-consuming.Non-stationary problems of thermal conductivity can be singled out separately.Finding solutions to problems of this class is associated with many mathematical difficulties.At the same time, there are various methods for solving problems of non-stationary thermal conductivity [3][4][5][6][7].
The paper assumes that the temperature field is symmetrical, and the thermophysical properties of the material do not depend on temperature.
The main objective of this work is to study the penetration of an electromagnetic wave into cylindrical bodies.

Main Part
The operation of an induction heater is based on the law of electromagnetic induction and the principle of operation is similar to the operation of a single-phase transformer.Heating of the annular part occurs due to the current passing through the secondary winding, the role of which is performed by the heated part itself.Moreover, the standard frequency of alternating current 50 Hz can be used for heating.
When constructing structures, their design schemes are diverse.The level of temperature stresses in such structures depends on both geometry and temperature conditions [5].The task of determining the intensity of heat generation and determining stresses in structural parts is quite time-consuming.At the same time, there are various methods for solving such problems, which are non-stationary [6][7][8][9][10].
When using induction heaters, the mechanical properties of the structure are the highest when compared with those when using other popular methods of performing similar tasks.
Let us consider the case when a plane electromagnetic wave propagating in a dielectric passes normally to a flat surface limiting a conducting medium.In a conductive medium, bias currents (compared to conduction currents) can be neglected, and therefore the equations for the electromagnetic field take the form where: Hy, Ex -projections on the coordinate axes of the intensity vectors (E and H) and the electromagnetic field; ρ -electrical resistivity; µ is the magnetic permeability of the conductor, c is the speed of light.If the electromagnetic wave propagating in the dielectric is sinusoidal, i.e.
where: τ is the time; ω -angular frequency; φe, φH are the initial phases of the electric and magnetic component of the wave, the solution, for example, of equation ( 2) can be written as: The resulting equation is the equation of a wave decaying in the direction of the Z axis.The first multiplier characterizes the change in the amplitude of the wave, the second -its phase as the wave continues deeper.When the wave passes the distance the phase of the wave will change by 2π.This distance corresponds to the wavelength in the conducting medium.When the wave moves to the distance Δ, its amplitude will decrease by e = 2,718 times, that is, it will fall to 38% of its initial value.For an infinitely extended plane, the Joule heat generated by induced currents, the intensity of which decreases according to the equation Numerically equal to the heat that would create a current circulating in a layer of thickness Δ and having a constant density I0 in this layer.At a sufficiently high frequency f, heat generation takes place in a very insignificant surface layer of the heated object.The total amount of heat generated by the heated surface can be determined as follows: It can also be represented as a functional dependence on the effective value of the current Jd (in practical units): In the last formula, the material characterizes the coefficient of √ρµ, the constancy of which (with the invariance of the energy parameters of the generator) is a condition for the constancy of the power supplied to the object.
Let us consider the case when a plane electromagnetic wave propagating in a dielectric passes normally to a flat surface limiting a conducting medium.
When studying the penetration of an electromagnetic wave into cylindrical bodies, two typical cases can be distinguished: 1) the electrical component of the wave is parallel to the axis of the cylinder; 2) the magnetic component is parallel to the axis of the cylinder.In the first case, the conduction current lines are parallel to the axis of the cylinder, and the magnetic field lines have the form of concentric circles lying in planes perpendicular to the axis of the cylinder.In the second case, the current lines form circles covering the axis of the cylinder.
Consider the second case, namely, heating of an infinitely long cylinder placed in a cylindrical solenoid.The axes of the inductor and cylinder are parallel.This task, regardless of whether the cylinder and the inductor are coaxial, can be considered as the penetration of a wave infinitely extended in the radial direction into the conducting medium.
To find the distribution of eddy currents on the surface of heated products, the following provisions should be followed: 1) eddy current lines always form closed currents on the surface of the product; 2) the eddy current lines tend to approach the current line in the inductor, and run parallel to them; 3) the eddy current lines are mutually repelled.The mutual "pushing" of eddy currents in areas more distant from the inductor, and their "tightening" in areas closest to the inductor when heating a long cylinder, manifests itself in a very weak form, only near the edges of the inductor.Therefore, despite the eccentricity, the cylinder heats up evenly.The eccentricity of the disk axis relative to the cylinder axis causes a sharp decrease in the density of eddy currents on the surface, more distant from the inductor, and the heating is uneven.Therefore, in order to ensure uniform heating of the disks, it is necessary to ensure the coaxially of the inductor and the sample.In the case of heating the disk, it is also necessary to keep in mind the possibility of heating the disks due to heat generation, which is caused by eddy currents circulating on the end surface.Now let us consider in more detail the process of heat transfer during external induction heating of cylindrical bodies.
Since at high depth heat is released practically on the surface of the sample, the temperature distribution inside the heated object differs little from that when heated by external heat sources.When heat propagates in a homogeneous isotopic medium, it can be assumed that the direction of the heat flow coincides with the direction of the temperature gradient, for the flat case when the temperature does not depend on the X and Y coordinates, we can write [8]: The equation is similar to Maxwell's, describing the penetration of an electromagnetic wave into a conducting medium.
If the temperature on the surface of the body is a periodic function of time F(τ) = Acos(ωτ)+ t0, the solution of equation ( 1) has the form of a damped heat wave: where t0 is the ambient temperature; ω is the circular frequency; a is the thermal conductivity of the material.By analogy with the equation for an electromagnetic wave, the value of √(ω/2a) can be called the penetration depth of the heat wave.When propagating over a distance of √(ω/2a), the amplitude of the heat wave changes by a factor of e.For points Z>> √(2a/ω), the temperature remains unchanged and equal to t0.
Next, consider the following example.Let a thin disk with thickness h and radius R be heated along the periphery by a heat flux q0, and heat exchange with the medium occurs from the end surfaces according to Newton's law.The equation of thermal conductivity in this case has the form: where m0 is the heat transfer rate.
Considering the initial and boundary conditions: We are looking for a solution in the form of: After substituting ( 14) into (12) for u, the equation is obtained: It can be shown that the thermal process, for example, in copper at the same time proceeds to a depth one hundred times less than the electric one.Therefore, induction heating of an object can be considered as heating by a constant heat flux q0 (if the supplied electrical power is constant, etc.).