Practical Use of Computer Simulation for Advanced Induction Coil and Process Design

The experience, of the authors, in the development and practical use of different programs for 1-D and 2-D simulation of induction heating processes is presented in this paper. A comparison, of simulation versus experimental approach, is discussed. Some examples of 1-D and 2-D computer simulation are presented in this paper for “real world” problems. It’s shown that the accuracy, of computer simulations of the quenching process, is mainly limited by the accuracy of the material properties and for better prediction of the results more experimental study must be done. Introduction Computer simulation is becoming more and more popular in the study, development, setup and maintenance of induction heating processes and equipment [1]. Many experts and groups in different countries are developing programs for induction heating simulation. Different methods used to solve the field problems include: Finite Difference Method (FDM), Finite Element Method (FEM), Volume Integral Method (VIM), Boundary Element Method (BEM) and their many variations and combinations [2,3,4]. However, computer simulation is not as widely used in induction heating as in electrical and mechanical engineering. This may be explained by the following reasons:  Induction heating processes are usually complex. In general, it is necessary to simulate a set of mutually coupled nonlinear and multi-dimensional problems (electromagnetic and thermal fields, quenching process, structural transformations, distortions, external supplying circuits etc., Figure 1);  Induction processes, especially heat treating processes, have very diverse features which may require individual program structures and simulation methods;  Personal computers with the high processing speeds and large available memory required for simulation of many of induction problems, only recently became widely available;  The induction heating market is small compared to other industrial sectors and the development of specialized commercial codes for the induction heating sector is a low profit or possibly even unprofitable venture. Strategy and Tools for Simulation The programs for simulation of induction heating processes may be classified as 1-D coupled (electromagnetic + thermal), 2-D electromagnetic, thermal, or coupled, and 3-D electromagnetic or thermal. The majority of 2-D and 3-D programs used for induction heating simulation originate from the companies specialized in development of general purpose software for calculation of physical fields in electrical engineering. For induction heating simulation, we use only a part of whole package, usually an “eddy current block”, a “thermal block”, or a combination of these blocks with an electric circuit block (power supply circuitry). The programs usually do not have the proper data bases necessary for induction heating or the explicit means for simulation of the induction heating machine performance (scanning, pushing of billets etc.). Many of them must be adapted for simulation of real induction heating processes. Flux 2D is probably the exception from the rule, because it has been developed taking into account the special features of induction heating processes [2]. Induction heating coil designs and operating conditions are very diverse and different programs are necessary in order to meet the practical needs. Theoretically we can assume that a single, universally coupled 3-D program would be able to meet all the major needs of the simulation of induction heating systems. However, at the present time, no program like this exists despite tremendous progress in computer hardware and software tools. Any 3-D code requires powerful computers and the process of simulation is knowledge and time consuming. Workstations or the most powerful personal computers must be used. One week of work for a skilled operator may be used as an estimate for the average time for one case study electromagnetic simulation. This estimate takes into account the operator’s laborious preparation of input data, checking and correction of the almost inevitable errors or inaccuracies, geometry and mesh construction, physical property and boundary condition description, calculation itself and the analysis of the results. The complexity of 3-D analysis and the required skill of the user makes a direct 3-D optimization of induction heating processes and systems rather difficult. A strategy based on a hierarchical use of programs is the most effective [1]. Figure 1 Main process in induction heat treating machine Figure 2 A rule of pyramid for computer simulation of induction heating For most cases, the first stage in the simulation of the induction heating system should be a 1D coupled program. This allows you to study the influence of frequency, power density, quench type, and time variations on the process. The ranges of interest can be studied quickly and effectively. A good estimate for the heating time, coil power, coil voltage, and coil current can be made from the results of the 1-D simulation. The second stage in the simulation may be done using 2-D electromagnetic or coupled code. The coil current, tube profile, coupling distance and magnetic flux concentrator dimensions that furnish the required field and power intensity may be determined. With this strategy, a 3-D simulation and/or experiments are only required for the coil design correction for the end zones where 3-D effects are significant. The above analysis is based on the author’s current experience, but we believe that it reflects the existing situation in general. An inductor, in many cases, is made from a solid piece of copper by machining and it is difficult and expensive to modify it after it has already been built. Computer simulation plus the application of versatile magnetic flux controllers made of magnetodielectric materials provide an effective solution for the coil design [5,6]. This general approach may be called "a rule of pyramid". It states that more simple programs must be used as a basis for the further use of more complicated packages (Fig. 2). Simulation Versus Experiments For some processes, computer simulation provides very exact results and experimental tests may be quite unnecessary. Through heating of billets is an example of this type of process. The system geometry is simple, material properties are relatively well defined, and the magnetic permeability is equal to one during the major part of the process. In this case, only electromagnetic and thermal processes need to be simulated. For other processes, the accuracy of the results obtained from simulation may be lower. Surface hardening is a good example of this, because of the more complex nature of the process. More phenomena must be involved in the simulation to obtain the final results. The initial part of the coil design 3D 2D, Electromagnetic or Thermal 1D, Electromagnetic + Thermal 2D, Electromagnetic + Thermal Process Control Machine Operating Mode Power Supply Circuits Thermal Process (Heating) Electromagnetic Process Cooling Quenching Stresses Structural Transformations Distortions Axle Study – Cooling Curves Page 3 ASM Europe 2000 problem and process simulation may be rather far from the final metallurgical result. The case depth and hardness are separated from the coil design and power supply setup by electromagnetic and thermal process, quenching and structural transformations. Inaccuracies during each stage of computer simulation can accumulate resulting in more significant final discrepancies. There are three sources of inaccuracy in computer simulation: errors of computation, mathematical description of the process and material properties or process parameters. The errors of computation depend strongly on the algorithm of the process simulation and the discretization of the problem in space and time. These errors may usually be controlled by the program user and can be reduced to a negligible value. Electromagnetic and thermal processes are quite accurately described by the Maxwell and diffusion (Fourier) equations respectively. With correct physical properties of materials, the solution of these equations may definitely be accurate. The processes of structural transformations and quenching have no accurate mathematical description. Existing descriptions of these processes are approximations of experimental data with parameters depending on the material composition and operating conditions [7]. These parameters and the inaccuracy in the physical properties of the material to be treated are the main sources of error in simulation. Specific heat and thermal conductivity as functions of temperature are necessary for simulation of any heating process. For induction heating, we also need to know electrical resistivity versus temperature and magnetic permeability versus temperature and field strength. The errors in temperature field prediction resulting from material properties inaccuracy depend on the process type. For surface hardening they may be rather small because a surface layer of steel is nonmagnetic during the final stage of the cycle and the material properties are well defined. Correct temperature prediction in the simulation of the tempering process where the steel is magnetic and the high temperature properties are not as well defined may be more difficult. Errors in the prediction of the results of structural transformations, residual strengths and distortions may be more significant. However, the results of computer simulation are very valuable even if they are not quite correct. With a series of calculations, the user can find the system’s response to the intended (process design) or unintended (input data tolerance) variation of parameters. He can then modify the process or design without experimental tests. Even more, an automatic optimization of the coil and process design may be effectively performed in some cases. For example, an optimal distribution of the coil ampere-turns may be found which provides a desired power or temperature pattern in the length of the workpiece. A simulation’s accuracy may be improved by using the results of experiments or by a study of material properties for the different stages and parameters of the process. The existing data on structural transformation in steel during heating and cooling refers mainly to furnace heating conditions. The local heat concentration, high heating speeds and short processing cycles typical for the induction method influence material response to heat treatment. A special study of steels` properties for the purposes of induction heating computer simulation is strongly desired. It is especially important for the powder metal parts industry, where information on electromagnetic and thermal properties is almost non-existent. The situation in computer simulation changes quickly and according to the opinion of experts in the USA, “computer simulation is one of the main trends in improvement of the induction heating processes”. Flux 2D and ELTA, programs used by the authors in practice, research, and design are described below. Features and Capabilities of Flux 2D Flux 2D is a 2-D coupled electromagnetic plus thermal program [2]. It is available on Unix, Windows 95, Windows 98 and Windows NT platforms. Flux 2D uses the Finite Element Method (FEM) to calculate the magnetic fields, currents and temperatures in the induction coil(s), surrounding area(s) and workpiece(s). Flux 2D contains a series of modules corresponding to the different steps in creating and running a computer simulation. These modules contain many preprogrammed functions to model common types of induction systems and material properties. For problems in which the system can not be described with a preprogrammed operation in Flux 2D, there is a User module that allows you to write your own subroutines in FORTRAN. Flux 2D has a module designed for building and meshing a system of any shape in polar or cartesian coordinates. The user can model only part of the system by placing lines of symmetry to save on memory and computation time. Flux 2D also gives you good control of the mesh. For example, you can leave certain regions unmeshed (like the water cooling channel of a coil which doesn’t influence the magnetic field distribution) to save memory. Flux 2D uses second order elements of triangular and rectangular shape to allow for linear changes of values across an element. Flux 2D also contains an automatic mesh generator which is very useful as a first draft, especially for users not familiar with mesh construction. In Flux 2D, any type of material heating can be modeled. You can use anisotropic and nonlinear materials in addition to isotropic and linear ones. Flux 2D has a database that contains properties of some common materials (graphite, copper, etc.). You can also add your own materials to the database. To help you in doing this, Flux 2D contains many preprogrammed forms for physical properties. For example, to account for the peak in specific heat capacity near the phase transformation temperature, Flux 2D contains a function that combines Gaussian and exponential functions. If the behavior of some material’s properties doesn’t fit any of Flux 2D’s forms, you can opt to describe the material’s properties in the User module. The User module can also be used to simulate quenching. The heat transfer coefficient () you use can be both time dependent (heating and quenching stages) and temperature dependent (for non-linear ). For example if you had a 3 second heating stage followed by an oil quench, you could make  zero for the interface with the part and the voltage in the coil constant for the first three seconds. Then you could make  temperature dependent with the properties of an oil quench and set the voltage in the coil to zero for the rest of the heating cycle. There is no limit to the number of heating and cooling stages you wish to use for a simulation. Flux 2D’s postprocessor allows the user to view the results in many ways. For any time step in the process, Flux 2D can display many physical, electromagnetic field, and thermal field quantities in color shades, vectors, or isovalue lines for all or part of the geometry. Flux 2D can also graph these values along an arbitrary straight line or arc. Within regions, Flux 2D has preprogrammed functions to calculate integral quantities (total power, total current, etc). If the integral quantity you wish to know isn’t listed in the Flux 2D menu, you can enter in the formula to be integrated and Flux 2D will calculate it. For a number of time steps in the process, Flux 2D can graph any of the non-integral quantities versus time for any point in the geometry. The integral quantities can be graphed versus time for a given region of the system. Features and Capabilities of ELTA The ELTA program (ELectro Thermal Analysis) has been developed specifically for use in the induction heating industry as a practical tool for induction through and surface heating and many other applications [5]. It may be run on PC 486 or higher and is very user friendly. No special knowledge in computers or in induction heating simulation is necessary. The user must only choose the options in standard Windows style menu. The workpieces may be cylindrical (solid or hollow) or flat in the form of single or multi-layer plate with an induction coil on one side or on the both. Multi-layer thermal insulation may be inserted between the workpiece and the coil. Databases of the workpiece and refractory materials offer default or customer installed properties. For dependence of permeability versus temperature and field intensity (H,T), an analytical formula (T) has been used with a default or customer defined dependence (H) at room temperature. Operating conditions may contain a set of time intervals with user defined coil current, voltage, power and frequency setup on each time interval (stage). Heat exchange conditions may be set separately on external and internal surfaces during heating, pause or cooling stages. The heat exchange conditions available are: heat exchange between refractory and the workpiece, free heat radiation, heat transfer coefficient. A heat transfer coefficient may be constant on each interval or temperature dependent, specified for each cooling media and cooling method (immersion in oil, polymer quenchant or water, water or oil showers of different intensities etc.). Corresponding curves of heat transfer coefficient (T) may be chosen automatically from the database or added thereto by the operator (Figure 3). Calculation time for a process of heating and water shower quenching using PC 486, 100 MHz is from some seconds to 12 minutes depending on process complexity, input data and desired level of accuracy (normal or high precision). The greatest time is required for simulation of intensive quenching when the temperature gradients and cooling speeds are very high while the heat transfer coefficient depends strongly on the surface temperature. A special method of solution of a system of algebraic equations has been used in order to provide fast and stable calculation process. The results of calculations may be shown in the form of tables or curves. Coil current, voltage, power, electrical efficiency, power factor variation in time, and other data as well as energy consumed by the workpiece are available with documentation in a report form. 3-D presentation of temperature in coordinates “time radius” may be very useful for evaluation of the process dynamics. Cooling diagrams with natural and logarithmic time scales can be used for predicting hardness depth and structural transformations. The program is especially effective for the design of multi-stage processes. For example, the process can consist of heating, soaking, quenching, second heating at another frequency for tempering with a partial use of residual heat, and Axle Study – Cooling Curves Page 5 ASM Europe 2000 final cooling. The user can interrupt the calculation process at any time, modify the operating conditions and continue the design of the process stage by stage. Temperature dynamics curves at any point of the workpiece allow the user to choose optimal operating conditions, to define time intervals and power levels and to predict the hardness depth, energy consumption and production rate. Computer Simulation Study of Case Hardening an Axle Fillet The axle is a part that must be case hardened by induction. The most difficult section of the axle to harden is the fillet area due to the induced currents not flowing evenly across the surface resulting in reduced heating of the flange area. In some axles, both the fillet and the shaft must be hardened. A scanning or single-shot process is used for heat treatment. Other axles, where only the fillet area must be hardened, are hardened with a static heating method. In this study, we simulate the static heating of the axle fillet area, because this is the most difficult area to heat. For this study, we used the generic axle fillet area shown in Figure 5. In this Figure, points A and B are the endpoints of the zone to be hardened. Obtaining the same temperature in the flange area (point B) and the shaft area (point A) of the fillet is always a problem in induction heat treating axles. This is a result of the flux density in a cylindrical system being inversely proportional to the radius and the equivalent coupling gap between the coil and the part. The power density is proportional to the square of the flux density. Because the radius at point B is larger than at point A, the flux and power density at this point are smaller given the same coupling gap. The lower power density leads to a lower temperature in this zone of the fillet area. Figure 5 The axle fillet geometry used for the case study The coil we used for this study is a single turn coil with Fluxtrol A type concentrator for good efficiency and power factor. The means for quenchant supply is not shown. The minimum required case depth in the fillet is 4 millimeters and the basic frequency used is 3 kHz. A heating time below 4 seconds is desired. Finally, the minimum temperature for austenitization is 790 C and the maximum allowable temperature is 990 C. The minimum possible coupling gap is 1 mm. In any style of induction coil, smaller coupling gaps lead to higher efficiency. For this study, we use the minimum possible coupling gap at the flange end of the fillet, approximately 1 mm. The coupling gap increases from there along the length of the fillet to compensate for the divergence of flux density. Computer simulation of the austenitization stage of the case hardening process must take into account many features of this system. Axles are made of magnetic steel, which loses its magnetic properties at the Curie Point (around 750 C). As temperature changes, not only do the magnetic properties change, but so do many other properties of the material important for simulation (resistivity, thermal conductivity, etc.). This means the problem to be simulated is highly nonlinear. The preliminary stages of the study were done using the 1-D program ELTA. Even though this system is essentially 2-D, a 1-D approach may be applied to a set of cross-sections normal to the part surface. For a single turn coil, the same coil voltage and heat time must be used for all cross sections. ELTA gave us initial estimates for the voltage, coupling gap A B Fluxtrol Concentrator