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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 907295, 9 pages
Research Article

Numerical Simulation of the Moving Induction Heating Process with Magnetic Flux Concentrator

1Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
2Beijing Key Lab of Precision/Ultra-Precision Manufacturing Equipments and Control, Beijing 100084, China
3Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, USA

Received 31 May 2013; Revised 18 August 2013; Accepted 3 September 2013

Academic Editor: Lei Zhang

Copyright © 2013 Feng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The induction heating with ferromagnetic metal powder bonded magnetic flux concentrator (MPB-MFC) demonstrates more advantages in surface heating treatments of metal. However, the moving heating application is mostly applied in the industrial production. Therefore, the analytical understanding of the mechanism, efficiency, and controllability of the moving induction heating process becomes necessary for process design and optimization. This paper studies the mechanism of the moving induction heating with magnetic flux concentrator. The MPB-MFC assisted moving induction heating for Inconel 718 alloy is studied by establishing the finite element simulation model. The temperature field distribution is analyzed, and the factors influencing the temperature are studied. The conclusion demonstrates that the velocity of the workpiece should be controlled properly and the heat transfer coefficient (HTC) has little impact on the temperature development, compared with other input parameters. In addition, the validity of the static numerical model is verified by comparing the finite element simulation with experimental results on AISI 1045 steel. The numerical model established in this work can provide comprehensive understanding for the process control in production.

1. Introduction

Induction heating plays an important role in many industrial manufacturing processes, such as hardening, brazing, tempering, and stress relieving [1]. The emergence of the metal powder bonded magnetic flux concentrator (MPB-MFC) made of ferromagnetic metal powder bonded with organic binder enables the induction heating process to possess higher heating efficiency and improved performance [2]. Due to its special material properties, the MPB-MFC exhibits less heat loss, longer coil life, and better formability into various shapes [3]. Besides, the MPB-MFC assisted induction heating system is studied by establishing the finite element simulation model and experimental verification [4].

However, the moving application occurs in various industrial applications, as shown as in Figure 1, which makes the induction process control more complex. As the magnetic flux field and temperature change in the process, the corresponding magnetic and thermal physical properties of the workpiece material vary nonlinearly [5, 6]. It will be impossible to achieve an accurate solution when treating all the nonlinear material properties in a coupled manner during the moving induction system. Therefore, it is crucial to analyze the coupled procedure with moving motion for better predictability in the induction process design and optimization.

Figure 1: The moving induction heating with MPB-MFC.

In this paper, the mechanism of electromagnetic-thermal transformation with moving motion is studied, and the finite element model is established for a plane moving induction heating system with the MPB-MFC on Inconel 718 alloy. In addition, the factors that influence the heating efficiency are studied, and the corresponding results are discussed. The precise modeling and simulation of the moving induction heating process could help better mechanism understanding, which also provides an explicit method for process design and optimization.

2. Principle of Simulation

2.1. Mechanism of Moving Induction Heating

The magnetodynamic phenomenon in a steady state AC magnetic application is governed by Maxwell’s equations (see (1)–(4)), where and are the variables to solve [2]. The relations between the field quantities are specified by (5)–(7) [2]. is the magnetic field intensity (A/m), which is generated by current resource. In this paper, is treated as a function of the current intensity , frequency , and the permeability of the MPB-MFC. Consider the following: where is the conduction current density (A/m2), is the electric flux density (C/m2), is the time (s), is the electric field intensity (V/m or N/C), is the magnetic flux density ( or N/A·m), and is the electric charge density (C/m3). is the relative permittivity. is the relative magnetic permeability, which is related to and the temperature . is the electrical conductivity (μS/cm).

In order to solve and , it is convenient to bring into a magnetic vector potential and an electric scalar potential . Consider the following: By incorporating the above constitutive relations into Maxwell’s equations, the equation solved by the finite elements method is shown as below where is the source current density in the induction coil.

In general, the transient time-dependent heat transfer process in metal materials can be described by the Fourier equation as follows [2]: where is the temperature in the metal workpiece, is the specific heat, is the density of the metal material, is the thermal conductivity of the metal, and is the heat source density induced by eddy currents per unit time in a unit volume, which is obtained by solving the electromagnetic problem and is so-called the heat generation.

The MPB-MFC serves as a magnifier that enhances the magnetic flux’s penetration into the workpiece, as indicated in Figure 2(a). A microelement is extracted to be analyzed. In the past increment of time , the microelement will be removed from the dashed box to the solid one , as shown as Figure 2(b). During this process, the magnetic flux density experiences the variation , which will influence the power generation in the workpiece and will be reflected by (2) and (10).

Figure 2: The energy transformation in induction heating with MFC.

2.2. Finite Element Modeling

In this paper, the FEM software FLUX2D is used to calculate the distribution of the electromagnetic field and temperature field for the mobile induction heating process [8]. The geometry dimension for surface induction heating is modeled as indicated in Figure 3. The induction coil is made of a rectangular copper tube with low electrical resistivity. And cooling water flows through it for maintaining the coil at low temperature. The concentrator is machined with a rectangular groove and clamped onto the coil.

Figure 3: The geometry of the inductor.

The domain of the system is modeled as in Figure 4. The whole system is surrounded by air. The compressed air region means a special region between the workpiece and fixed air surrounding the device, which is the movement route of the workpiece and will be remeshed during every translating calculation step. The compressed air region is meshed into structure elements, and the coil region and subsurface regions of the heated workpiece are meshed with structured element at 0.1 mm, considering the sharp temperature gradient in these zones, as shown as Figure 4(b). There are 49025 nodes and 19450 surface elements in the whole computational domain. The mesh order is the second order, and the time step is set as 0.005 s.

Figure 4: Geometric model and the finite element meshing.

As to the initial conditions, the ambient temperature and the initial temperature of the workpiece are set as 25°C. The region “Conv” is the surface boundary of the workpiece, which in thermal analysis can be expressed as in the below equation (11). The heat transfer coefficient (HTC) of the workpiece surface is equal to 20 W/(m2·K), and the radiation heat loss coefficient here is 0.5 W/(m2·K4). Consider the following: where is the temperature gradient in a direction normal to the metal workpiece’s surface at the point under consideration, is the initial temperature of the surrounding air, is the thermal conductivity of the workpiece, is the convection surface heat transfer coefficient, is the radiation heat loss coefficient, and denotes the normal to the boundary surface of the workpiece.

The finite element model should solve the coupled electromagnetic-thermal computation problem, which involves the consideration of the thermal- and magnetic-dependent material property in the process. The MPB-MFC Ferrotron 559H is selected. Figure 5 illustrates the - curve of the concentrator material and the permeability variation with the magnetic field strength. Theoretically, the magnetic permeability of MFC also changes as the temperature rises. However, the permeability of the concentrator does not change too much as it is cooled down continuously by running water through the induction coil.

Figure 5: B-H curve and permeability of Ferrotron 559H [9].

As for the workpiece material, both the magnetic and thermal properties alter during the induction heating process. The workpiece material is the nickel-based alloy Inconel 718, and its chemical composition is listed in Table 1.

Table 1: The chemical compositions of Inconel 718, wt% [7].

The magnetic permeability is characterized by the magnetic polarization of the material, which is influenced by both the magnetic field strength and temperature. The quantification of is carried out through the measurement of the magnetic polarization and the magnetic field strength at various temperature conditions. The measurement is conducted on the vibrating sample magnetometer Lakeshore-730T (VSM), where a set of - curves can be generated. Equation (12) indicates the correlation of the permeability and polarization as follows: where is the permeability of vacuum and equals to  [H/m].

The permeability of the alloy Inconel 718 is treated as constant 1, which is largely different from the plain carbon steel. The - curve of AISI 1045 in Figure 6 presents a nonlinearity, which demonstrates that the permeability is related with the magnetic field and the temperature. As the magnetic flux field strength alters below 5000 A/m, the permeability is almost a constant; while the magnetic field strength is higher than 1 × 105 A/m, it drops rapidly to 1.

Figure 6: The measured B-H curve of AISI 1045.

The volumetric heat capacity and the thermal conductivity of the workpiece are dependent on temperature (), while independent on the magnetic field intensity. Figures 7(a) and 7(b) show the - and - relationship, respectively, which demonstrates that and of Inconel 718 are nonlinear [7].

Figure 7: Volumetric heat capacity and thermal conductivity of Inconel 718.

3. Results and Analysis

Figure 8 shows the cross-section profile of the induction system, and the heating distance is 1 mm. The distance between the points B, C, D, E, and A is, respectively, 10 mm, 12.75 mm, 12.75 mm, and 14.75 mm. The factors influencing the eddy current intensity have not been researched clearly during this novel process. Therefore, in this part, special attention is taken to the power density distribution generated by the eddy current and the temperature field evolution.

Figure 8: Cross-section view of the system.

Figure 9 shows the equal flux lines distribution on the workpiece surface, and it is obvious that the magnetic flux concentrator plays an essential effect on the magnetic field distribution. The maximum value of the power density at the surface of the workpiece is at point A, which is underneath the magnetic flux concentrator. The power density distribution is shown in Figure 10. The maximum value is higher than 5 × 108 W/(cubic m). When the velocity of the workpiece varies from 0 to 5 mm/s, the power density variation at point A is shown in Figure 11. It is indicated that the power induced in the workpiece is related to the velocity. The higher the velocity is, the higher power is in the workpiece. When the velocity is higher than 4 mm/s, the reduce rate of the power is slowed down.

Figure 9: Equal flux lines ( kHz,  A).
Figure 10: Power density distribution along the horizontal surface of the workpiece.
Figure 11: Power density variation at point A versus different velocities.

Besides the power density distribution, the temperature field is computed in the following part. Figure 12 shows the temperature field evolution varying with time (while  mm/s,  kHz, and  A). It indicates that the maximum temperature can achieve up to 40°C in 5 seconds, which also produces a remarkable penetration depth. Figures 13 and 14, respectively, show the temperature evolution at points A and C with the speed of the workpiece varying from 1 mm/s to 5 mm/s. It is demonstrated that the temperature evolution depends largely on the velocity of the workpiece. The tone of temperature growth at points A and C will take great impact on the final temperature field distribution on the surface of the heated workpiece as shown in Figure 15, which describes the distribution of temperature of the workpiece’s surface at the end of heating ( s). The higher the velocity is, the lower the maximum temperature is. But on the contrary, the higher the velocity is, the wilder the temperature range is. Therefore, in order to acquire the desired uniform temperature distribution, the velocity of the workpiece should be controlled properly.

Figure 12: Temperature field evolution in the workpiece (while  mm/s,  kHz, and  A).
Figure 13: Temperature evolution at point A.
Figure 14: Temperature evolution at point C.
Figure 15: Temperature distribution at the surface of the workpiece ( s).

In order to effectively verify the numerical model and reveal the influence of the boundary cooling condition on the temperature evolution, AISI 1045 steel is heated statically. Figure 16 shows the temperature development history of both simulation and experiment on AISI 1045 steel at points D and E. The result illustrates a good correlation between the simulation and experimental results. The maximum error is approximately 8%, which is within the acceptable range. Figure 17(a) shows the temperature development on the surface point A with the different surface cooling conditions (while  mm/s,  kHz, and  A). The maximum temperature in 5 seconds presents a linear downtrend along with the increasing value of the surface heat transfer coefficient (Figure 17(b)). Figure 18 shows the temperature profiles below the surface point A in 5 seconds with the different surface cooling conditions. It shows an unobvious penetration depth variation with the surface cooling intensity increasing. Therefore, it is clear that the factor influencing the penetration depth of power density and the temperature is not the surface boundary cooling condition, but the input current frequency .

Figure 16: Temperature evolution at point D and point E.
Figure 17: Temperature variation at point A.
Figure 18: Temperature profiles below the surface point A.

4. Conclusions

The moving induction heating possesses wide application in surface hardening for steel. This work studies the moving induction heating progress with MPB-MFC, which establishes the numerical model based on finite element method. The temperature-dependent magnetic and thermal material properties are applied into the simulation, which helps generate accurate prediction results. In addition, the critical factors that influence the heating efficiency are studied, and the following conclusions can be achieved.(i)By analyzing the power density and temperature evolution, it is demonstrated that the final temperature field depends largely on the velocity of the workpiece. By rationally matching the velocity, the appropriate power density intensity and anticipated temperature penetration depth can be achieved. Besides, the validity of the static numerical model is verified by comparing the finite element simulation with experimental results on AISI 1045 steel. Therefore, the finite element model proposed in this paper can effectively predict the temperature distribution evolution and can be used for further process analysis and optimization. (ii)The cooling condition can effectively alter the surface temperature development during the induction heating process but has little impact on the penetration depth of temperature in the workpiece. The factor influencing the penetration depth of power density and temperature is still magnetic field variation and mainly resulted from the input current frequency . This research is useful for the achievement of different desired temperature gradient distribution in the sublayer of the heated workpiece applied in the moving induction heating. (iii)Further investigation should be taken to the precisely uniformity control of the temperature distribution.


The research is supported by the National Science Foundation of China (NSFC: 51175294). And special acknowledgement should be given to the Project Z121104002812045 from Beijing Municipal Science & Technology Commission and Project PMEC201201 from the Beijing Key Lab of Precision/Ultra-Precision Manufacturing Equipments and Control.


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