INTRODUCTION TO COUPLED ELECTROMAGNETIC AND THERMAL FIELD COMPUTER SIMULATIONS OF INDUCTION HEATING PROCESSES
R C Gibson
Induction heating has long been recognised as an efficient clean rapid method of heating billets for forging, extrusion and other metal forming operations. In applications such as thixoforming, this heating must be carried out in a very precise manner to ensure that a sufficiently uniform templerature is achieved to produce the critical degree of partial melting required prior to forging. For operation at economic throughputs, this must be done very rapidly and repeatably, with the billet accessibly positioned at the forging device. Induction heating is by its nature more intense in the surface layers of the billet, relying on thermal conduction to carry heat to the billet interior. While it is possible to base heater control 011 temperature measurement of the billet surface, it is impossible to measure a temperature distribution throughout the body of a billet, and thus ensure a correct condition for subsequent forming operations. A typical practical computer model is described, which continuously recalculates the transient temperature distribution throughout the heating cycle. Its validity has been confirmed by comparison between predicted and measured values. Use of such a model to optimise possible realistic production routes in the heating of magnetic and non-magnetic alloys is demonstrated, as well as some fasciinating insights revealed into the induction heating process. While it is evident that where material properties are poorly defined, the absolute accuracy of the model is less predictable, a sensitivity analysis using bracketing values shows how a sufficiently narrow range of operating conditions can quickly be predicted.
Description of computer model A computer model of a physical process consists of three main features. First the geometry of the components
must be defined, including the spatial relationship between various elements. Then physical properties of each component must be tabulated as functions of temperature and magnetic field strength. Overlaying this, equations governing the interaction of the various elements must be encoded, namely Maxwell's electromagnetic field equations, and the various equations of heat conduction, radiation and convection. The model described uses an axi-symmetric cylindrical geometry, with billet, coil, furnace insulation, and other significant features represented by short solid or hollow cylinders, each cylinder being positioned appropriately along the axis of the heater. More complex shapes are made up from a series of such cylinders. Computations take full account of the non-linear variation of physical parameters with temperature and magnetic field strength throughout the heating cycle. Induction heaters generally operate within tuned circuits with large air gaps, so an assumption of sinusoidal current and voltage does not introduce significant mor. For each material, such as billet, coil, flux concentrators, thermal insulation and other components, elecitrical and thermal properties such as resistivity, thermal conductivity, heat capacity and latent heat of phase change, must be tabulated at irregula but sufficient temperature intervals adequately to define local non-linearities. Intermediate values are interpolated. Magnetising curves (B-H loops) and a definition of the behaviour through the Curie temperature are required for all magnetic materials. For each billet surface, radiative, convective and conductive heat loss mechanisms are defined as functions of local geometric and thermal conditions. Coil losses and internal water cooling, non-inductive heat sources such as radiation and convection, surface quenching, and material movement are also included. A particular feature of the model is the very wide range of industrial control methods which can be applied to the heating cycles being simulated. R C Gibson is an induction heating consultant.
0 1996 The Institution of Electrical Engineers. Printed and published by the IEE, Savoy Place, London WCPR OBL, UK.
1/1
Authorized licensed use limited to: National Taiwan Univ of Science and Technology. Downloaded on May 17, 2009 at 08:47 from IEEE Xplore. Restrictions apply.
Perturbations to local temperature due to small non-cylindrical shapes can be included. Features such as hollow water cooled conductors use local analytical solutions to calculate coil losses to avoid the use of excessively fine local meshing. When the model was first developed thirty years ago, finite difference techniques were used. Although conversion to a finite element formulation has been considered, few practical cases have required this refinement. A major advantage of the finite difference method is the well structured coding possible, resulting in faster solution times than with the less well ordered finite element matrices. From an applied coiI current and frequency, the power induced in each mesh element is calculated. Node temperatures are calculated as developing functions of time. This integration is actually carried out in terms of enthalpy to allow latent heats of phase change and full or partial melting to be included. Mesh values of physical properties are continuously updated, and induced power recalculated for these updated physical property values. Parameters such as coil voltage, winding loss and efficiency, billet and coil forces, billet power and surface loss are tabulated for post processing. The source functions of coil current and frequency can either be applied as functions of time in the form of tabulated values, including rapid fluctuations or switching transients, or be allowed to vary interactively to give required values of coil power, voltage, current or frequency to match the characteristics of a given power supply. A particularly useful mode is for power to be maintained at a maximum value consistent with a given limiting maximum billet temperature or thermal gradient. This can be likened to having an infinitely small thermocouple burrowing around within the billet, trimming coil power whenever any part of the billet exceeds a given temperature condition.
Comparison of computed results with measured values Before a theoretical model can be used with confidence, it is necessary to test model predictions against measured values. Kapranos et a11 describe a series of validation tests heating a slug of non-magnetic 310 stainless steel of well defrned physical properties. Very close agreement is shown between measured and predicted results, thus demonstrating confidence in the model. Also shown are examples of improved thermal uniformity using insulating end caps, and short bursts of high frequency induction heating to compensate for radial surface radiation losses. In this optimisation, the computer model was used to examine slug internal temperatures, looking for specific temperature non-uniformities, and to investigate the efficacy of specific measures to counter these non-uniformities. Furnace conditions were then set as specified by this model study, and the predicted improvement in temperature uniformity verified by actual temperature measurement.
Precautions necessary when interpreting results from the model During the verification exercise described above, great care was taken to ensure that the test conditions were accurately measured, before being fed to the model for calculation of predicted results to compare with the measured values. For example, the precise coil current and frequency, with variation during the test, must be measured and applied to the model. As small as a 1% error in current measurement leads to a significant 2% error in the final temperature. Similarly, small differences between the actual physical properties of the billet, and those assumed by the model will also give apparently incorrect results. Hence the exhaustive series of tests conducted with a billet of very well defined physical properties in the verification exercise. The errors thus induced by small discrepancies in physical properties might therefore be thought to render use of the computer model misleading, when the normal variations in physical property values encountered with production samples of complex alloys are considered. Fortunately this is not usually the case, as the model itself can be used to investigate the sensitivity of the heating cycle to different values of physical properties and furnace settings. A series of model simulations are conducted using a range of values of say billet resistivity to cover all expected variations, while keeping all other billet and furnace conditions constant. This shows that while higher resistivities result in higher billet power for a given coil current, the temperature distribution within the billet, normalised to the maximum temperature, remains virtually unchanged. In practice this means that the same furnace operating conditions can be used for all billets within this resistivity range, but that the actual heating time must be controlled by temperature measurement of a predetermined part of the billet surface, thus ensuring that all billets are ejected at the required temperature. This sensitivity analysis can be carried out for other billet and furnace parameters.
1/2
Use of the model with magnetic materials Several possible additional errors would seem to be introduced when the: heating of magnetic billets is simulated. Measured magnetising curves (B-H loops) are rarely available, except for transformer lamination stampings. Accurate magnetising curves at elevated temperatures are virtually impossible to obtain. Magnetic steels become non-magnetic at the Curie temperature (ferritic/austenitic transition) but accurate data is sparse. Further errors are introduced by the model itself. The equations used assume sinusoidal coil currents and other parameters, instead of an actual transient waveform, which itself can vary throughout the billet. Faced with this, what hope has a computer model of producing results useful in the study of heating cycles to produce the close temperature uniformities necessary for say the successful thixoforming of' tool steels? Fortunately, an extensive sensitivity analysis can show very encouraging results. It is well known that the depth of current penetration in magnetic materials decreases for larger values of relative permeability. This is seen in the computer model, where different values of assumed relative permeability result in very differem iniiial temperature distributions near the billet surface. However, as the surface temperature reaches the Curie point, the local reduction in relative permeability, coupled with the absorption of the latent heat of phase change, results in a pronounced slowing down and equalisation in the rate of rise in surface temperature. At this stage a most interesting phenomenon occurs, clearly demonstrated in the computer model. It might be expected that the surface layer power intensity would exhibit a more gentle radial decay above the Curie temperature, commensurate with an increased depth of current penetration, zind change to a steeper gradient once the magnetic material below the Curie temperature is reached. However, the power density actually increases below the billet surface, becoming a maximum at the Curie temperature interface, decaying rapidly thereafter. This can be explained qualitatively by considering the coil as the primary of a transformer, with the portion of billet above the Curie temperature as a single turn secondary winding, and the portion of billet below the Curie temperature as a magnetic core. As the inner radius of the "secondary winding" decreases, its effective length, and hence its effective resistance, decreases. The voltage produced by the magnetic "core" therefore generates a higher current density at this point, leading to the increased power intensity. The magnetic core provides sufficient magnetic flux to balance the secondary current, with the exact shape of the flux distribution within this core having little effect on the current distribution. This efficient mode of induction heating continues as the magnetic core reduces in size, until when the magnetic core disappears, the characteristic exponential decay of power density through the billet returns. A comparison of billet temperatures calculated for the same overall heating rate but using radically different assumed magnetising curves show a slight initial disparity, but by the time the Curie transition has been reached, the temperatures converge. Although these very different magnetising curves give different magnetic flux distributions within the magnetic core, almost identical current distributionis are seen in the heated billet.
Conclusion The computer model described has been used for a very wide range of induction heating applications. Normal variations in billet properties and furnace operating conditions show little effa;t on relative parameters such as temperature distribution, but can significantly affect absolute values such as final billet temperature. Hence the model can be used with confidence to investigate temperature distribution improvement, but must be used With more care when predicting absolute values such as precise heating times. The interactive operation of the built in control mechanisms is a most powerful feature of the model. The model control system uses stable, accurate values taken throughout the volume of the billet. It can search for local mi,nimum or maximum values, and adjust heating rates appropriately. The only limit on the scope of the control system is in the ingenuity built into the computer model. References 1. Kapranos P, Gibson R C, Kirkwood D H, Hayes P J, Sellers C M: "Induction heating and partial melting
of high melting point thixoformable alloys" Proceedings of 4th International Conference. on Semi-Solid Processing of Alloys and Composites, University of Sheffield,June 1996, lpp 148-52.
Authorized licensed use limited to: National Taiwan Univ of Science and Technology. Downloaded on May 17, 2009 at 08:47 from IEEE Xplore. Restrictions apply.