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Purpose -The purpose of this paper is to present a methodology of high-precision finite element modeling of induction heating of rotating nonferromagnetic cylindrical billets in static magnetic field produced by appropriately arranged permanent magnets. Design/methodology/approach -The mathematical model consisting of two partial differential equations describing the distribution of the magnetic and temperature fields are solved by a fully adaptive higher-order finite element method in the monolithic formulation and selected results are validated experimentally. Findings -The method of solution realized by own code is very fast, robust and exhibits much more powerful features when compared with classical low-order numerical methods implemented in existing commercial codes. Research limitations/implications -For sufficiently long arrangements the method provides good results even for 2D model. The principal limitation consists in problems with determining correct boundary conditions for the temperature field (generalized coefficient of convective heat transfer as a function of the temperature and revolutions). Practical implications -The methodology can successfully be used for design of devices for induction heating of cylindrical nonmagnetic bodies by rotation and determination of their operation parameters. Originality/value -The paper is a presentation of the fully adaptive higher-order finite element and its utilization for a monolithic numerical solution of a relatively complicated coupled problem.

Purpose – As far as the author knows the modeling of induction surface hardening is still a challenge. The purpose of this paper is to present both mathematical models of continuous and simultaneous hardening processes and exemplary results of computations and measurements. The upper critical temperature Ac3 is determined from the Time Temperature Austenization diagram for investigated steel. Design/methodology/approach – Computation of coupled electromagnetic, thermal and hardness fields is based on the finite element methods, while the hardness distribution is determined by means of experimental dependence derived from the continuous cooling temperature diagram for investigated steel. Findings – The presented results may be used as a theoretical background for design of inductor-sprayer systems in continual and simultaneous arrangements and a proper selection of their electromagnetic and thermal parameters. Research limitations/implications – The both models reached a quite good accuracy validated by the experiments. Next work in the field should be aimed at further improvement of numerical models in order to shorten the computation time. Practical implications – The results may be used for designing induction hardening systems and proper selection of field current and cooling parameters. Originality/value – Complete mathematical and numerical models for continuous and simultaneous surface induction hardening including dual frequency induction heating of gear wheels. Experimental validation of achieved results. Taking into account dependence of the upper critical temperature Ac3 on speed of heating.

Purpose -As far as the authors know, no sufficiently complete model of continual induction hardening was developed and solved so far. The paper presents both mathematical model of the process and algorithm of its solution in the quasi-coupled formulation. Design/methodology/approach -Computation of electromagnetic and temperature fields is based on the finite element method, while time variable boundary conditions are determined by means of an original theoretically-empirical procedure. Findings -Substantial are backgrounds for design of the inductor and parameters of the field current as well as parameters of the cooling medium. Research/limitations/implications -The model reached a good level of accuracy validated by suitable experiments. Nevertheless, next work in the field will also have to respect history of the heating before cooling itself (the austenitizing temperature is a function of the velocity of heating). Very important is also appropriate meshing of the investigated region to suppress numerical instabilities appearing during the computation process. Finally, acceleration of numerical schemes is a must, because modelling of one common task (on very fast PCs) takes about 4 h. Practical implications -The results and conclusions may be used for designing devices for continual induction hardening of axisymmetric bodies. Originality/value -Complete mathematical and computer model of the process, original methodology for finding the coefficient of convective heat transfer from the cooled part of the heated workpiece to ambient water spray.

Purpose Modeling of induction surface hardening strongly depends on accuracy of material properties data and their temperature characteristics. However, it is especially complicated in case of the magnetic permeability dependent not only on temperature but also on the magnetic strength. This paper aims to estimate the influence of the magnetic permeability on modeling of coupled physical fields describing the process. Investigations are provided for the gear wheels made of the steel C45E. Design/methodology/approach Computation of coupled electromagnetic temperature and hardness fields is based on FEM methods. The Flux 3D software is applied for the numerical simulation of coupled electromagnetic and temperature fields. The QT Steel software is applied for a determination of the hardness and microstructure distributions. Findings Obtained results may be used as a kind of background for the design of induction surface hardening systems. Research limitations/implications The presented calculation model provided quite a good accuracy of hardness distribution validated by the experiments. Next work in the field should be aimed at taking into account a dependence of the magnetic permeability on the field current frequency. Originality/value Mathematical model of induction surface hardening with taking into account time dependence on the magnetic permeability on temperature and magnetic strength is elaborated. Experimental validation of hardness distribution is provided. A quite reasonable convergence between simulations and measurements was achieved.