Induction Heating of Gear Wheels in Consecutive Contour Hardening Process †

: Induction contour hardening of gear wheels belongs to effective heat treatment technologies especially recommended for high-tech applications in machinery, automotive and aerospace industries. In comparison with long term, energy consuming conventional heat treatment (carburiz-ing and consequent quenching), its main positive features are characterized by high total efﬁciency, short duration and relatively low energy consumption. However, modeling of the process is relatively complicated. The numerical model should contain both multi-physic and non-linear formulation of the problem. The paper concentrates on the modeling of rapid induction heating being the ﬁrst stage of the contour induction hardening process which is the time consuming part of the computations. It is taken into consideration that critical temperatures and consequently the hardening temperature are dependent on the velocity of the induction heating. Numerical modeling of coupled non-linear electromagnetic and temperature ﬁelds are shortly presented. Investigations are provided for gear wheels made of a special quality steel AISI 300M. In order to evaluate the accuracy of the proposed approach, exemplary computations of the full induction contour hardening process are provided. The exemplary results are compared with the measurements and a satisfactory accordance between them is achieved.


Introduction
Induction Contour Hardening (ICH) of gear wheels is an innovative technological process making it possible to obtain a thin hardened zone along the whole working surface of teeth. The result is a hard microstructure in the contour zone and less-hard microstructure in the transition zone [1,2]. The microstructure of the internal part of the gear wheel remains practically unchanged [3]. Such an environment-friendly heat treatment process is very effective. It is characterized by large productivity and low-specific energy consumption. It is more effective than a long term, energy consuming conventional heat treatment process of gear wheels consisting of carburizing and consequent quenching [4,5]. In many high-tech, industrial applications the crucial quality condition is the fully uniform shape of the hardened contour zone. For the ICH of big gear wheels, such an expected result could be easily achieved by means of Tooth-by-Tooth Induction Hardening (TTIH) method [6,7]. However, for small gear wheels such a method cannot be applied, and it is not so easy to obtain the uniform thickness of the hardened contour zone along the whole tooth by ICH methods. The paper describes the full hardening process. The induction heating stage of the ICH process can be realized mostly in one or two cycles. The one cycle heating process consists of Single Frequency Induction Hardening (SFIH) or, more often, Simultaneous Dual Frequency Induction Hardening (SDFIH) in the medium frequency (MF) and high frequency (HF) electromagnetic fields [8][9][10][11][12]. The two cycles process consists of MF induction heating first, followed by immediate shifting of the body between inductors and then finally HF induction heating. The paper deals with this second kind of the approach, which will be later defined as Consecutive Dual Frequency Induction Hardening (CDFIH) [13][14][15]. In general, the CDFIH process consists of five consecutive stages: MF induction heating, immediate shifting the workpiece between MF and HF inductors, rapid HF induction heating, immediate shifting the gear wheel to the sprayer area, and finally the intensive cooling. Due to a short duration of induction heating, the austenitization process is completely different in comparison with any classical kinds of hardening [16]. Critical temperatures characterizing transformation of any prior microstructure to uniform austenite microstructure are dependent on velocity of induction heating (see Figure 1). In order to determine dependences between critical temperatures and velocity of induction heating the Time-Temperature-Austenitization (TTA) diagram for investigated steel is measured [1]. Austenitization begins at the lower critical temperature Ac 1 . When the temperature of the material reaches the upper critical temperature Ac 3 we obtain the austenite microstructure, but it is non-uniform. A fully uniform austenite microstructure is finally obtained only at the modified upper critical temperature Ac m . The paper is aimed at modeling the ICH process with a special emphasis put on the induction heating stage. The main goal is formulated as elaboration of the computation method making it possible to obtain an accurate prediction of hardness and microstructure distribution in a hardened element and determination of the shape of the contour zone. However, the modeling of the process is complicated because of the necessity to analyze multi-coupled, non-linear physical fields. In order to reduce the duration of computations, magnetic field quantities are considered in a simplified way as harmonic [7]. However, in order to achieve the expected accuracy of hardness prediction it is crucial to take into consideration: - The dependence of the critical temperatures for the investigated steel on velocity of induction heating strongly influences upon accurate determination of the hardening temperature; - The material properties of the investigated steel as non-linear dependences on the temperature; - The dependence of hardness on the velocity of cooling should be elaborated based upon a set of continuous-cooling-transformation diagrams for the investigated steel starting at different hardening temperatures. or, more often, Simultaneous Dual Frequency Induction Hardening (SDFIH) in the medium frequency (MF) and high frequency (HF) electromagnetic fields [8][9][10][11][12]. The two cycles process consists of MF induction heating first, followed by immediate shifting of the body between inductors and then finally HF induction heating. The paper deals with this second kind of the approach, which will be later defined as Consecutive Dual Frequency Induction Hardening (CDFIH) [13][14][15]. In general, the CDFIH process consists of five consecutive stages: MF induction heating, immediate shifting the workpiece between MF and HF inductors, rapid HF induction heating, immediate shifting the gear wheel to the sprayer area, and finally the intensive cooling. Due to a short duration of induction heating, the austenitization process is completely different in comparison with any classical kinds of hardening [16]. Critical temperatures characterizing transformation of any prior microstructure to uniform austenite microstructure are dependent on velocity of induction heating (see Figure 1). In order to determine dependences between critical temperatures and velocity of induction heating the Time-Temperature-Austenitization (TTA) diagram for investigated steel is measured [1]. Austenitization begins at the lower critical temperature Ac1. When the temperature of the material reaches the upper critical temperature Ac3 we obtain the austenite microstructure, but it is non-uniform. A fully uniform austenite microstructure is finally obtained only at the modified upper critical temperature Acm. The paper is aimed at modeling the ICH process with a special emphasis put on the induction heating stage. The main goal is formulated as elaboration of the computation method making it possible to obtain an accurate prediction of hardness and microstructure distribution in a hardened element and determination of the shape of the contour zone. However, the modeling of the process is complicated because of the necessity to analyze multi-coupled, non-linear physical fields. In order to reduce the duration of computations, magnetic field quantities are considered in a simplified way as harmonic [7]. However, in order to achieve the expected accuracy of hardness prediction it is crucial to take into consideration: -The dependence of the critical temperatures for the investigated steel on velocity of induction heating strongly influences upon accurate determination of the hardening temperature; - The material properties of the investigated steel as non-linear dependences on the temperature; - The dependence of hardness on the velocity of cooling should be elaborated based upon a set of continuous-cooling-transformation diagrams for the investigated steel starting at different hardening temperatures.   In order to obtain uniform austenite microstructure, the average hardening temperature T h in the contour zone should be slightly higher than the modified upper critical temperature Ac m .

Induction Heating for CDFIH Approach
The mathematical model of the ICH process for electromagnetic computations has been represented by the non-linear partial differential equation for the magnetic vector potential A [7]: where: µ-denotes the magnetic permeability, γ-the electric conductivity, J S -vector of field current density. The solving of the Equation (2) in the 3D configuration has to be simplified assuming that the magnetic field quantities are harmonic [18]. The main reason of such a simplification is a distinct disproportion between period of field current and time constant of thermal phenomena. Equation (2) is transformed to the following Equation (3) for phasor of the magnetic vector potential A: where: j denotes the imaginary unity and ω-angular frequency. The continuous mathematical model of the ICH process for temperature computations has been represented by the non-linear Fourier-Kirchhoff Equation [19]: where: λ-thermal conductivity, ρ-specific mass, c p -specific heat at constant pressure, p Vvolumetric power density as external heat source taken from electromagnetic computations. The block diagram representing mathematical model of induction heating is presented in Figure 2. Input data for electromagnetic computations contain: -Electric supply parameters: Inductor current or voltage and its frequency; -Geometry of inductor-gear wheel system; -Material properties of investigated steel and their dependences on field quantities (magnetic field intensity H, temperature T); Average volumetric power density p v consists of two components connected respectively with Joule p J and hysteresis p H phenomena: where: E-phasor of electric field density, f -frequency of field current, H-phasor of magnetic field density. However, for the analyzed case the hysteresis components are significantly smaller than the Joule phenomenon one and it could be neglected [20]. Finally volumetric power density has only one component and it depends on induced current density J ind : The magnetic permeability of the investigated steel AISI 300M and other ferromagnetic parts like magnetic core depends on magnetic field density and temperature. In order to take into account a non-linear dependence of magnetic permeability on magnetic field intensity, its local values in each cell of the discretization mesh are determined. Dependence of the magnetic permeability on temperature is considered by means of ((Equations (7) and (8)): where: µ 0 -the magnetic permeability of vacuum, µ r -relative magnetic permeability, T 0 -ambient temperature, T C -temperature of the Curie point, C-constant dependent on kind of steel. For ambient temperature, the magnetic permeability is determined by measurements. However, a dependence of the magnetic permeability of the investigated steel on the field current frequency is neglected [20]. The non-linear dependence of the electric conductivity on temperature is taken into consideration.
Input data for temperature computations contain: -Distribution of volumetric power density in the heated element; -Dependence of specific heat, thermal conductivity, convection and radiation heat transfer coefficients on temperature; -Time-temperature-austenitization diagram for the investigated steel; -Expected hardening temperature guaranteeing uniform austenite microstructure.
For temperature field calculations convection and radiation heat transfer phenomena are considered. The boundary condition along the external surface of the workpiece is given in the Equation (9): where: n denotes outwards normal to the external border of the domain, α ch -convection heat transfer coefficient (heating), T ac , T ar -temperature of convection and radiation environment respectively, σ 0 -Boltzmann constant, ε-total emissivity of radiation surface. Analysis of the multiple reflection phenomena required special numerical procedure [21]. For this problem they are neglected. Taking into account non-linear dependences of material properties and heat transfer parameters on temperature makes possible to avoid inaccuracy in modeling which could reach an order even of about 100 K [22][23][24]. As a result of temperature computations, we obtain temperature as a function of time and velocity of induction heating. Computations are provided by means of the Flux 3D software for hard coupled electromagnetic and temperature field during induction heating and some single-owned procedures elaborated by the authors [25]. For CDFIH process modeling of induction heating stage consists of three consecutive stages: -MF heating-time t MF ; -Short break for shifting workpiece between inductors-time t b ; -HF heating-time t HF .
Total time of induction heating: MF heating terminates when average temperature in contour zone reaches the level of the lower critical temperature Ac 1 (v ih ) Computations of HF induction heating terminate when the average hardening temperature exceeds the modified critical temperature Ac m (v ih ).

Cooling
Input data for cooling computations contain: -Geometry of sprayer-gear wheel system; - Cooling parameters (parameters of cooling medium, its pressure, flow-rate and temperature); -Initial temperature distribution in the workpiece; -CCT (Continuous-Cooling-Transformation) diagram for investigated steel; -Convection heating transfer coefficient for intensive cooling.
Block diagram of numerical model for cooling is presented in Figure 3. Time of a break between heating and cooling could be neglected because the cooling practically starts immediately when the HF inductor is switched off. Temperature field is determined based upon Equation (4) (p V = 0). Radiation is neglected. Convection heat transfer coefficient for cooling α cc depends on temperature, but in practice it is analyzed in a simplified way by putting different values on all boundaries of the domain. For the cooling stage as well the Flux 3D software for temperature field coupled with the modified QT steel supported by several own numerical procedures for hardness and microstructure fields are applied [26]. A special numerical procedure making possible to couple directly Flux 3D and QT steel codes is elaborated. Taking into consideration that when the induction heating is terminated, the non-uniform temperature within the tooth, highest at the surface and lower inside the material is observed. The velocity of cooling v c is determined based upon several measured Continuous-Cooling-Transformation (CCT) diagrams starting of different hardening temperatures T h . For the analyzed problem hardening temperature T h changes between (850-1050 • C). Based upon these diagrams, dependences of the hardness on the velocity of cooling in different points of tooth are elaborated. Exemplary dependence of hardness on velocity of cooling for the hardening temperature T h = 960 • C is presented in Figure 4.  Geometry of sprayer-gear wheel system; -Cooling parameters (parameters of cooling medium, its pressure, flow-rate and temperature); -Initial temperature distribution in the workpiece; -CCT (Continuous-Cooling-Transformation) diagram for investigated steel; -Convection heating transfer coefficient for intensive cooling.
Block diagram of numerical model for cooling is presented in Figure 3. Time of a break between heating and cooling could be neglected because the cooling practically starts immediately when the HF inductor is switched off. Temperature field is determined based upon Equation (4) (pV = 0). Radiation is neglected. Convection heat transfer coefficient for cooling αcc depends on temperature, but in practice it is analyzed in a simplified way by putting different values on all boundaries of the domain. For the cooling stage as well the Flux 3D software for temperature field coupled with the modified QT steel supported by several own numerical procedures for hardness and microstructure fields are applied [26]. A special numerical procedure making possible to couple directly Flux 3D and QT steel codes is elaborated. Taking into consideration that when the induction heating is terminated, the non-uniform temperature within the tooth, highest at the surface and lower inside the material is observed. The velocity of cooling vc is determined based upon several measured Continuous-Cooling-Transformation (CCT) diagrams starting of different hardening temperatures Th. For the analyzed problem hardening temperature h T changes between (850-1050 °C). Based upon these diagrams, dependences of the hardness on the velocity of cooling in different points of tooth are elaborated. Exemplary dependence of hardness on velocity of cooling for the hardening temperature h T = 960 °C is presented in Figure 4.  Finally, hardness and microstructure distributions within ¼ part of the tooth are determined and a shape of the hardened contour zone is achieved. Computations are compared with measurements and if a condition of accordance is not satisfied, the next case with modified parameters is computed.

Formulation of the Problem
Let us consider an example of the CDFIH process provided for small gear wheels made of the special quality steel AISI 300M (AMS 6419) [17]. Its chemical composition is presented in Table 1. Its prior microstructure consists of the tempered martensite with some single parti- Finally, hardness and microstructure distributions within 1 4 part of the tooth are determined and a shape of the hardened contour zone is achieved. Computations are compared with measurements and if a condition of accordance is not satisfied, the next case with modified parameters is computed.

Formulation of the Problem
Let us consider an example of the CDFIH process provided for small gear wheels made of the special quality steel AISI 300M (AMS 6419) [17]. Its chemical composition is presented in Table 1. Its prior microstructure consists of the tempered martensite with some single particles of tempered bainite. Phenomena connected with reverse transformation from the tempered martensite to the uniform austenite are described for instance in [27]. Dependence of modified upper critical temperature for steel AISI 300M is shown in Table 2 (see Figure 1).        Experiments are provided at the specialized laboratory stand located in the Silesian University of Technology in Katowice [29]. Configuration of inductors-sprayer-gear wheel systems is shown in Figure 8. tion-10%, temperature-25 °C, pressure-89 kPa, flow-rate-2 10 m /s, convection heat transfer coefficient at external surface-400 W/(m 2. K).
Experiments are provided at the specialized laboratory stand located in the Silesian University of Technology in Katowice [29]. Configuration of inductors-sprayer-gear wheel systems is shown in Figure 8.

Results and Discussion
Comparison between computations and measurements is done for hardness distribution along the line A…G at the working surface of the tooth, while point A is located in Figure 8. Configuration of the CDFIH system.
Both inductors have a ring shape. They are water cooled. The HF inductor is equipped in the magnetic flux concentrator made of Fluxtrol 50 [30].

Results and Discussion
Comparison between computations and measurements is done for hardness distribution along the line A . . . G at the working surface of the tooth, while point A is located in the root and point G in the top of the tooth (see Figures 9 and 10). Measurements of hardness distribution on the cross-section of studied gear wheels are carried out by Vickers method at load 5N (HV0,5) using microhardness tester Future-Tech FM 700 [31]. Microstructure observations are provided at micro-sections taken from slices of gear wheels by means of microscope Olympus type GX51. the root and point G in the top of the tooth (see Figures 9 and 10). Measurements of hardness distribution on the cross-section of studied gear wheels are carried out by Vickers method at load 5N (HV0,5) using microhardness tester Future-Tech FM 700 [31]. Microstructure observations are provided at micro-sections taken from slices of gear wheels by means of microscope Olympus type GX51.    The difference between computations and measurements reaches a level of 20 HV. It is almost the same difference along the whole line A…G. For both computations and measurements the difference between hardness at point A (root) and point G (top) is about 20 HV. Such accuracy matches practical requirements. However, another, but more serious problem is the shape of the contour hardened zone. For many advanced applications there is a request for the shape of contour pattern which should be as uniform as possible.
Let us define the Surface Depth Hardening coefficient (SDH) describing a thickness of the contour layer defined as a distance from the surface inside the material to a point in which the hardness decreases to the value of 80% of hardness on the surface. The dependence of hardness on distance directed from surface inside the material for points: G (top), A (root) and D (middle part of the tooth body) are presented in Figures 11-13 respectively.   The difference between computations and measurements reaches a level of 20 HV. It is almost the same difference along the whole line A…G. For both computations and measurements the difference between hardness at point A (root) and point G (top) is about 20 HV. Such accuracy matches practical requirements. However, another, but more serious problem is the shape of the contour hardened zone. For many advanced applications there is a request for the shape of contour pattern which should be as uniform as possible.
Let us define the Surface Depth Hardening coefficient (SDH) describing a thickness of the contour layer defined as a distance from the surface inside the material to a point in which the hardness decreases to the value of 80% of hardness on the surface. The dependence of hardness on distance directed from surface inside the material for points: G (top), A (root) and D (middle part of the tooth body) are presented in Figures 11-13 respectively. The difference between computations and measurements reaches a level of 20 HV. It is almost the same difference along the whole line A . . . G. For both computations and measurements the difference between hardness at point A (root) and point G (top) is about 20 HV. Such accuracy matches practical requirements. However, another, but more serious problem is the shape of the contour hardened zone. For many advanced applications there is a request for the shape of contour pattern which should be as uniform as possible.
Let us define the Surface Depth Hardening coefficient (SDH) describing a thickness of the contour layer defined as a distance from the surface inside the material to a point in which the hardness decreases to the value of 80% of hardness on the surface. The dependence of hardness on distance directed from surface inside the material for points: G (top), A (root) and D (middle part of the tooth body) are presented in Figures 11-13 respectively.     For point G at top SDH = 1.8 mm and thickness of the hardened zone is the biggest. A minimal value of the SDH coefficient is noticed in point A at the root (SDH = 0.5 mm). Distribution of the SDH coefficient along the line A…G is presented in Figure 14. For point G at top SDH = 1.8 mm and thickness of the hardened zone is the biggest. A minimal value of the SDH coefficient is noticed in point A at the root (SDH = 0.5 mm). Distribution of the SDH coefficient along the line A . . . G is presented in Figure 14. For point G at top SDH = 1.8 mm and thickness of the hardened zone is the biggest. A minimal value of the SDH coefficient is noticed in point A at the root (SDH = 0.5 mm). Distribution of the SDH coefficient along the line A…G is presented in Figure 14. The obtained SDH coefficient changes from 1.8 mm at the top of the tooth to 0.5 mm at the root. It means a reasonable (less than 5%) difference between computations and measurements. Microstructure of the contour zone creates acicular martensite with some single particles of bainite ( Figure 15). The obtained SDH coefficient changes from 1.8 mm at the top of the tooth to 0.5 mm at the root. It means a reasonable (less than 5%) difference between computations and measurements. Microstructure of the contour zone creates acicular martensite with some single particles of bainite ( Figure 15). Inside the material, if a distance from the surface is higher than 4 mm the hardness decreases to the value of 320 HV which means that, as expected, the prior microstructure of tempered martensite in the core of the gear wheel is unchanged. Results presented and described in the paper confirm that the proposed method makes it possible to achieve the expected accuracy of hardness prediction. The presented calculation method makes it possible to achieve reasonable accordance between computations and measurements. Another problem is the shape of the contour zone. For its description, the SDH coefficient is used. The calculated non-uniform shape of the contour zone is validated by the measurements. In order to achieve a more uniform shape of the contour hardened zone it is necessary to decrease heating time and increase the delivered power. Another solution is to apply the SDFIH process, and in this case short heating duration and large delivery of Inside the material, if a distance from the surface is higher than 4 mm the hardness decreases to the value of 320 HV which means that, as expected, the prior microstructure of tempered martensite in the core of the gear wheel is unchanged. Results presented and described in the paper confirm that the proposed method makes it possible to achieve the expected accuracy of hardness prediction. The presented calculation method makes it possible to achieve reasonable accordance between computations and measurements. Another problem is the shape of the contour zone. For its description, the SDH coefficient is used. The calculated non-uniform shape of the contour zone is validated by the measurements. In order to achieve a more uniform shape of the contour hardened zone it is necessary to decrease heating time and increase the delivered power. Another solution is to apply the SDFIH process, and in this case short heating duration and large delivery of power are expected.

Conclusions
The CDFIH process described in the paper is successfully applied for the energyefficient contour hardening of gear wheels. The proposed mathematical model could be the effective tool for the design of such processes. It makes it possible to predict hardness and microstructure distributions with a reasonable accuracy of about 20 HV. In order to minimize the computation time, magnetic field quantities are considered in a simplified way as harmonic. The model should contain additional numerical procedures based upon specialized measurements for the investigated steel (TTA and CCT diagrams), making it possible to determine hardening temperature as a function of velocity of induction heating and hardness as a function of velocity of cooling and hardening temperature. The next paper in this area will be aimed at improvement of the described process in order to obtain the uniform shape of the contour zone.