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Contents : of Achievements in Materials and Manufacturing Engineering VOLUME 22 ISSUE 1 May 2007 Simulation of induction heating process with radiative heat exchange A. Kachel R. Przylucki* Department of Electrotechnology Faculty of Metallurgy and Material Science Silesian University of Technology ul. Krasiskiego 8 40-019 Katowice Poland * Corresponding author: E-mail address: roman.przylucki@polsl.pl Received 14.03.2007 published in revised form 01.05.2007 Analysis and modelling Abstract Purpose: Numerical modelling of induction heating process is a complex issue. It needs analysis of coupled electromagnetic and thermal fields. Calculation models for electromagnetic field analysis as well as thermal field analysis need simplifications. In case of thermal field calculations correct modelling of radiative heat exchange between the heated charge and inductor's thermal insulation is essential. Most commercial calculation programs enabling coupled analysis of electromagnetic and thermal fields do not allow taking into consideration radiative heat exchange between calculation model components which limits thermal calculations only to the charge area. The paper presents a supplementation of the program Flux 2D with radiative heat exchange procedures. Design/methodology/approach: Commercial program Flux 2D designed for coupled electromagnetic and thermal calculation (based on finite element method) was supplemented with authors program for radiative heat exchange based on numerical integration of classic equations. Findings: Supplementation EM-T calculations with radiative heat exchange between charge and inductor enables to calculate thermal insulation parameters and increase precision of modelling. Research limitations/implications: Procedures for radiative heat exchange enables calculation of two surfaces (flat or cylindrical) with finite dimensions. The surfaces can be displaced relative to each other (charge shorter or longer than thermal insulation of inductor). Material of surfaces is modelled as: flat diffuse radiant surfaces absorb energy evenly in the whole spectrum (grey bodies). The whole system is modelled as in a steady thermal state (quasi-steady). Originality/value: Authors program extends Flux 2D features with a possibility for calculating radiative heat transfer. The application of radiative process is possible between all components of the studied model not only for the boundary conditions. Keywords: Numerical techniques Induction heating 1. Introduction 1. Introduction The article discusses the implementation of radiative heat exchange into the program Flux 2D. It extends features of that commercial program and opens the way to calculation of more realistic model of induction heating process. Flux 2D enables calculations of coupled electromagnetic-thermal fields but has several limitations. One of them is that it is impossible to calculate radiative heat exchange between components of the model. It is possible to calculate thermal emission from the part of model to outside space only. Such way of calculation causes that model for thermal calculation of induction heating is usually limited to the heating charge only. The features of extended package Flux 2D are presented on the example of three variants of calculations. Variants differed in geometrical dimensions and in the way how the heat exchange was modelled. During the experiment temperature distribution in the model was controlled. 2.Calculation model 2. Calculation model For the experiment the system whose geometry is shown in Figure 1 was chosen. It is a model of induction heater for flat charges. It Copyright by International OCSCO World Press. All rights reserved. 2007 Short paper 53 Journal of Achievements in Materials and Manufacturing Engineering Volume 22 Issue 1 May 2007 consists of charge (non-ferromagnetic steel) thermal insulation and copper inductor. Main dimensions of the model system are as follows: charge width wc 20 mm height of charge hc 100 mm air gap between charge and thermal insulation ag 7 mm thermal insulation thickness wt 5 mm. Inductor coil profile has width ww 10 mm height hw 12 mm and thickness of profile is tw 2 mm. Fig. 2. Boundary conditions specification Inductor was powered by AC current source of I 3000 A and frequency f 1000 Hz. Thermal calculations were performed in three different ways. In the first variant (v1) of calculations thermal model was reduced to charge area only. In the second one (v2) full geometry was considered. In the third variant (v3) radiative heat exchange was included. Thermal calculations were based on equations (5) (6) 1-6 . Fig. 1. Calculation model geometry Electromagnetic calculations for all variants were performed for the same calculation model showed in Figure 1. Equations (1) to (3) 1-6 with proper boundary conditions (4) were used. Calculation areas are: 1 charge 2 thermal insulation 3 - inductor 4 water 5 air. Boundary conditions are applied in places showed in Figure 2. :1 : 3 : c wT ( O T ) q wt (5) :1 : u u A jZJP A 0 : 3 : u u A jZJP A P J s : 2 : 4 :5 : u u A ae ef fd da : A 0 : 2 : 4 : 5 : c (1) (2) (3) (4) wT ( O T ) 0 wt (6) 0 where: (kg/m3) density c (J/(kg K)) specific heat capacity T (K) temperature t (s) time (W/ (mK)) thermal conductivity q (W/m3) power density per volume. Boundary conditions described by equations (7) to (8) are identical for all calculation variants. ad : O dT dn 0 (7) where: A (Vs/m) magnetic vector potential Js (A/m2) source current density (rad/s) angular frequency (Vs/(Am)) magnetic permeability. ab cd : O dT D ( T Ta ) H V ( T 4 Ta4 ) dn D 10 H 0.6 Ta 20 (8) 54 Short paper A. Kachel R. Przylucki Analysis and modelling For the second calculation variant following boundary conditions were applied (9) to (11). bc gh hi jk kl lg mn no op : (9) dT O D ( T Ta ) H V ( T 4 Ta4 ) dn 10 Ta 20 0.6 for edge bc 0.4 for edge mn op 0.5 for edge gh hi jk kl lg no . ae fd : O dT dn 0 (10) in a steady thermal state (quasi-steady) radiant surfaces are flat both primary and reflected radiation are of Lambertian diffuse type radiant bodies absorb energy evenly in the whole spectrum. By applying Lambert and Kirchhoff laws in a simple expression of radiative energy balance a system of integral equations (13) (14) is obtained 7-16 . Solution of the system of equations describes total radiant exitance emitted from the particular surfaces S1 and S2. The total radiant exitance for each surface determines resulting effective radiation being the sum of primary and reflected radiations. Equations (15) and (16) describe resulting irradiance and heat transfer by radiation respectively. To solve the equations a numerical method was used 7-15 . U 1 x1 a1VT1 1 a1 k x1 x 2 U 2 x 2 d S 2 4 S2 (13) ef qr rs st tq : T 20 (11) For the third calculation variant boundary conditions are the same as in the second case with following changes: U 2 x 2 a2VT2 1 a2 k x 2 x1 U 1 x1 d S1 4 S1 (14) bc lg : O dT dn D ( T Ta ) D 10 Ta 20 (12) I i (xi ) U i (xi ) ai iTi 4 1 ai (15) More over the radiative heat exchange was included between edges: bc lg . Qi (x i ) ai I i (x i ) iTi 4 @ (16) 3.Thermal radiation model 3. Thermal radiation model Radiative heat exchange in the flat charge - heater heating system takes place between two surfaces with finite dimensions. The surfaces can be displaced relative to each other (charge shorter or longer than thermal insulation of inductor). Most often surfaces are parallel. Typical configuration of the studied system model is presented in Figure 3. where: U (W/m2) total radiant exitance a - absorption coefficient (a L ) i (W/(m2 K4) Stefan Boltzman coefficient k(xi xj) - optical coupling function (integral kernels i j) Ii (W/m2) irradiance Qi (W/m2) radiative heat exchange i j 1 2. 4.Results 4. Results During the experiment induction heating of steel charge for 40 seconds was simulated. Temperature distribution along paths pa1 (segment p1 p 2 ) pa2 (segment p1 p 3 ) and pa3 (segment p 4 p 5 ) was studied. Fig. 3. Radiant surface system To include the radiative heat transfer in the considered model the following simplifications were assumed: the whole system is Fig. 4. Temperature distribution along path pa1 after 40 s Simulation of induction heating process with radiative heat exchange 55 Journal of Achievements in Materials and Manufacturing Engineering Volume 22 Issue 1 May 2007 For all analyzed cases the most significant temperature differences appear at the end of heating (after 40 s.). Temperature distribution on path pa1 is showed in Figure 4. Maximum temperature difference is about 27 oC. Along path pa2 temperature differences are more essential and reaching 677 oC (Figure 5). It appears in air gap and thermal insulation area. Figure 6 show temperature distribution along pa3 (1 mm under surface of heated charge). Temperature differences do not exceed 25 oC. Acknowledgements Acknowledgements This work was sponsored by Ministry of Scientific Research and Information Technology as a grant no. 3T08B06428. References References 1 2 3 Cedrat Flux 2D User's Guide 2006. H. Kawaguchi M. Enokizono T. Todaka Thermal and magnetic field analysis of induction heating problems Journal of Materials Processing Technology 161 (2005) 193-198. H.K. Jung The induction heating process of semi-solid aluminium alloys for thixoforming and their microstructure evaluation Journal of Materials Processing Technology 105 (2000) 176-190. D.C. Ko G.S. Min B.M. Kim J.C. Choi Finite element analysis for the semi-solid state forming of aluminium alloy considering induction heating Journal of Materials Processing Technology 100 (2000) 95-104. Z. Hu J.Q. Li Computer simulation of pipe-bending processes with small bending radius using local induction heating Journal of Materials Processing Technology 91 (1999) 75-79. K.L. Schlemmer F.H. Osman Differential heating forming of solid and bi-metallic hollow parts Journal of Materials Processing Technology 162-163 2005 564-569. R. Siegel J.R. Howell Thermal Radiation Heat Transfer Mc-Graw Hill Book Co. N. York 1972. M.F. Cohen D.P. Greenberg The hemi-cube: a radiosity solution for complex environment Computer Graphics 1985. J. Kajiya The Rendering Equations Computer Graphics 20 1986. P. Dutr Global Illumination Compendium Cornell University 2001. M.F. Modest Radiative Heat Transfer. Second Edition Academic Press Amsterdam Boston London - N. York Sydney 2003. A. Kachel R. Przy ucki Two dimensional model of radiative heat exchange in heater - flat charge system Proceedings of the International Conference on Research in Electrotechnology and Applied Informatics Katowice 2005 153-158. K. Domke Modelling of radiative heat transfer using computer graphics programs Proceedings of the International Conference on Research in Electrotechnology and Applied Informatics Katowice 2005 79-84. P. Furmanski J. Banaszek Some new computational models of radiative heat transfer in participating media Progress in Computational Fluid Dynamics 5 (2005) 222-229. W.M. Gao L.X. Kong P.D. Hodgson Numerical simulation of heat and mass transfer in fluidised bed heat treatment furnaces Journal of Materials Processing Technology 125126 (2002) 170-178. H.P. Zeng J.C. Fang W.J. Xu Z.Y. Zhao L. Wang Thermomechanical modeling of a single splat solidification in plasma spraying Journal of Achievements in Materials and Manufacturing Engineering 18 (2007) 327-330. 4 5 Fig. 5. Temperature distribution along path pa2 after 40 s 6 7 8 9 10 11 12 Fig. 6. Temperature distribution along path pa3 after 40 s 13 5.Conclusions 5. Conclusions Inclusion of radiative heat transfer between charge and thermal insulation makes the calculations of induction heating more precise. For the studied case taking for consideration radiative heat exchange changes maximally temperature difference value in charge by 27 oC (between variants v2 and v3) but in air gap and thermal insulation temperature differences reach 677 oC. Supplementation Flux 2D with authors procedures for radiative heat transfer enables thermal calculations of inducting heating systems consists with charge air gap thermal insulation inductor and other parts. It opens the way to proper design of thermal insulation thickness and in consequence to more optimal construction of the induction heater. 14 15 16 56 Short paper READING DIRECT: www.journalamme.org

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