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Sergio Lupi · Michele Forzan Aleksandr Aliferov Induction and Direct Resistance Heating Theory and Numerical Modeling Induction and Direct Resistance Heating Sergio Lupi Michele Forzan (cid:129) Aleksandr Aliferov Induction and Direct Resistance Heating Theory and Numerical Modeling 123 SergioLupi AleksandrAliferov Michele Forzan Department of Automationof Electric Department of IndustrialEngineering TechnologicalInstallations Universityof Padua NovosibirskState Technical University Padova Novosibirsk Italy Russia ISBN 978-3-319-03478-2 ISBN 978-3-319-03479-9 (eBook) DOI 10.1007/978-3-319-03479-9 LibraryofCongressControlNumber:2014948742 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The material collected in this book reflects the historical development of the technologies of induction and direct resistance heating and represents a synthesis oftheinformationandexperiencethattheauthorshaveaccumulatedintheiractivity of academic research and collaboration with industry. The first part of the book is devoted to deepening the theoretical knowledge on the electromagnetic and thermal phenomena that determine the heating process. This theoretical approach was developed since the beginning of the twentieth century. At that time, the design of heating installations was based on analytical solutionsofMaxwell’sequationsforsimplegeometries,experimentaltestsandthe use of the “ruler” as a mean for calculations. Although there was no discontinuity in the technological development, crucial steps were the industrial growth, in particular of the automotive industry, after WorldWarII,theintroductionofcomputersinuniversitiesandresearchinstitutions andthedevelopmentofnumericalmethodsforcomputationofelectromagneticand thermal fields. Asregardsthelastpoint,amilestoneforthetechnologiesdealtwithinthisbook can be considered the works of Hegewaldt, Holmdal and Sundberg, Kolbe and Reiss, who in 1963, independently but almost at the same time, presented the first numerical solutions for 1D coupled electromagnetic and thermal problem in non- linear systems and a 2D numerical solution of the electromagnetic problem in a linear system. The way for numerical calculations of heating systems was open, but it took a long time, more than 25 years, for computer simulations to become a widely used design and research tool. Analyticalandexperimentalmethodscontinuedtoprevailinindustryduringthe period 1960–1990, when development and use of computer simulations were mainly limited to academia and research institutions for theoretical studies. In this periodseveralnewanalytical solutionshavebeendevelopedintheformofinfinite series of Bessel or exponential functions for different geometries, since the avail- ability of computers allowed the calculation of a number of harmonic terms suffi- cient for obtaining accurate results. v vi Preface Since then, more powerful computing means and increasingly sophisticated numericalprocedureshavebeenprogressivelyavailable,sothatanalyticalmethods were gradually abandoned while the use of numerical programs has become cus- tomary not only in research but also in industrial design. The content of Chap. 4 reflects this situation, since it presents a set of results which are based in part on infinite series expansion of analytical solutions and in part on the use of numerical methods. The choice to present in the same volume results of analytical and numerical solutions stems from the belief of the authors that a profitable use of numerical methodsisbasedonapreliminarydeepknowledgeofthedifferentphenomenathat affect the heating process. This knowledge is acquired in the best way from the analysis of several case studies conducted in the past. Finally, in Chap. 5 useful information for the numerical calculation of electro- technogical installations are given to the designer. Contents 1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Induction Heating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Direct Resistance Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Basic Electromagnetic Phenomena . . . . . . . . . . . . . . . . . . . . 3 1.4.1 Maxwell Equations. . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4.2 Theorem and Poynting’s Vector . . . . . . . . . . . . . . . . 5 1.4.3 Phenomena Affecting the Current Density Distribution in Conductors. . . . . . . . . . . . . . . . . . . . 8 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 Electromagnetic Field in Workpieces with Flat Surfaces. . . . . . . . 23 2.1 Semi-infinite Body of Homogeneous Material . . . . . . . . . . . . 23 2.1.1 Induction Heating. . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.2 Resistance Heating . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 Semi-infinite Body of Ferromagnetic Material . . . . . . . . . . . . 40 2.3 Infinite Metal Slab of Homogeneous Material. . . . . . . . . . . . . 51 2.3.1 Induction Heating by a Flat Inductor on One Side of the Workpiece . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.3.2 Metal Slab in Longitudinal Flux Inductor (Exciting Magnetic Field on Both Sides of the Workpiece). . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3.3 Direct Resistance Heating . . . . . . . . . . . . . . . . . . . . 65 2.4 Slab of Rectangular Cross-Section . . . . . . . . . . . . . . . . . . . . 71 2.4.1 Induction Heating. . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.4.2 Energy in Slabs of Rectangular Cross-Section . . . . . . 76 2.4.3 Direct Resistance Heating . . . . . . . . . . . . . . . . . . . . 80 2.5 Slab of Magnetic Material . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.5.1 Induction Heating. . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.5.2 Resistance Heating . . . . . . . . . . . . . . . . . . . . . . . . . 83 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 vii viii Contents 3 Electromagnetic Field in Cylindrical Bodies. . . . . . . . . . . . . . . . . 85 3.1 Solid Cylindrical Workpieces of Homogeneous Material . . . . . 85 3.1.1 Induction Heating of Cylindrical Workpieces . . . . . . . 86 3.1.2 Direct Resistance Heating . . . . . . . . . . . . . . . . . . . . 93 3.2 Solid Cylindrical Workpieces of Ferromagnetic Material . . . . . 100 3.2.1 Direct Resistance Heating . . . . . . . . . . . . . . . . . . . . 100 3.2.2 Induction Heating Solid Ferromagnetic Cylinder. . . . . 105 3.3 Hollow Cylindrical Workpieces of Homogeneous Material. . . . 107 3.3.1 Induction Heating with Internal and External Exciting Magnetic Field. . . . . . . . . . . . . . . . . . . . . . 107 3.3.2 Induction Heating with External Exciting Magnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.3.3 Induction Heating of Hollow Cylindrical Workpieces with Internal Inductors. . . . . . . . . . . . . . . . . . . . . . . 114 3.3.4 Resistance Heating of Cylindrical Hollow Workpieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.4 Hollow Cylindrical Workpieces of Ferromagnetic Material. . . . 120 3.4.1 Induction Heating. . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.4.2 Resistance Heating . . . . . . . . . . . . . . . . . . . . . . . . . 122 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4 Special Problems in Induction and Resistance Heating. . . . . . . . . 127 4.1 Resistance Heating of Ferromagnetic Workpieces of Rectangular Cross-section . . . . . . . . . . . . . . . . . . . . . . . . 127 4.1.1 Influence of Power Supply Circuit on Heating Transient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.2 Curved Conductors with Cylindrical Cross-Section . . . . . . . . . 140 4.2.1 Electromagnetic Processes in Toroidal Conductors of Circular Cross-section . . . . . . . . . . . . . . . . . . . . . 140 4.2.2 Resistance Heating of Curved Cylindrical Work-Pieces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.2.3 Equalization of Temperature Distribution in the Cross-section of Bent Conductors . . . . . . . . . . 153 4.3 Transverse Flux Induction Heating . . . . . . . . . . . . . . . . . . . . 164 4.3.1 Preliminary Choice of Design Parameters. . . . . . . . . . 172 4.3.2 Final TFH System Characteristics. . . . . . . . . . . . . . . 177 4.3.3 Calculations of Heating Transients . . . . . . . . . . . . . . 179 4.3.4 Other Inductor Geometries. . . . . . . . . . . . . . . . . . . . 182 4.3.5 Recent Developments and Conclusions . . . . . . . . . . . 186 4.4 Planar Circular Coils for Induction Heating . . . . . . . . . . . . . . 191 4.4.1 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . 191 4.4.2 Examples of Results Obtained with the Analytical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4.4.3 Examples of Numerical Results . . . . . . . . . . . . . . . . 197 Contents ix 4.5 Induction Heating of “Long” Cylindrical Workpieces with Inductors of Finite Axial Length . . . . . . . . . . . . . . . . . . 200 4.6 Pulse Induction Hardening of Complex Workpieces . . . . . . . . 214 4.6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 4.6.2 Single-Frequency Processes . . . . . . . . . . . . . . . . . . . 215 4.6.3 Dual-Frequency Processes . . . . . . . . . . . . . . . . . . . . 220 4.6.4 Gear Spin Hardening: Main Factors Influencing the Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 4.6.5 Numerical Simulations and Results. . . . . . . . . . . . . . 230 4.6.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 4.7 Induction Heating of Cylindrical Billets Rotating in DC Magnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 4.7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 4.7.2 Electromagnetic Solution for the Infinitely Long Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.7.3 Heating Parameters. . . . . . . . . . . . . . . . . . . . . . . . . 248 4.7.4 Edge Effect in Finite Length Billet . . . . . . . . . . . . . . 250 4.7.5 Influence of Shape and Position of Superconducting Coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 4.7.6 Optimisation of the Heating Process . . . . . . . . . . . . . 256 4.8 Induction Heating with Permanent Magnets . . . . . . . . . . . . . . 258 4.8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 4.8.2 FEM Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 4.8.3 Analytical Solution. . . . . . . . . . . . . . . . . . . . . . . . . 261 4.8.4 Calculation, Experimental Results and Design Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 4.9 Inductors for Induction Heating of Internal Cylindrical Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 4.9.1 Inductor Equivalent Resistance. . . . . . . . . . . . . . . . . 269 4.9.2 Results of Numerical Analysis . . . . . . . . . . . . . . . . . 270 4.9.3 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.10 Electromagnetic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.10.2 Evaluation by Lorentz’s Law of Forces in “Long” Cylindrical Systems. . . . . . . . . . . . . . . . . 276 4.10.3 Forces in the Induction Heating of Disk Plates with Planar Circular Coils . . . . . . . . . . . . . . . . . . . . 284 4.10.4 Evaluation of Forces by Variation of the Field Energy. . . . . . . . . . . . . . . . . . . . . . . . . 289 4.10.5 Forces in Induction Heating of the Ends of Non-magnetic Bars. . . . . . . . . . . . . . . . . . . . . . . 290 4.10.6 Forces in Induction Heating of the Ends of Magnetic Bars . . . . . . . . . . . . . . . . . . . . . . . . . . 293 4.10.7 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 x Contents 5 Analytical and Numerical Methods for Calculation of Induction and Conduction Heating Systems. . . . . . . . . . . . . . . 303 5.1 Calculation of Induction Heating Systems with the Equivalent Magnetic Circuit Method. . . . . . . . . . . . . . . . . . . . . . . . . . . 303 5.2 Calculation of Induction Heating System with Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 5.3 1D Finite Difference Numerical Solution for Induction Heating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 5.3.1 Electromagnetic Problem Solution. . . . . . . . . . . . . . . 313 5.3.2 Solution of Thermal Problem. . . . . . . . . . . . . . . . . . 315 5.4 Commercial 1D Code ELTA for Induction Heating . . . . . . . . . 317 5.4.1 Case Study 1: Through Heating of Non-magnetic Steel Billets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 5.4.2 Case Study 2: Through Heating of Magnetic Steel Billets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 5.5 VIM: Volume Integral Method of the Mutually Coupled Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 5.6 Calculation of Parameters of Direct Resistance Heating Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 5.7 1D Finite Difference Numerical Model for DRH. . . . . . . . . . . 341 5.7.1 Electromagnetic Problem. . . . . . . . . . . . . . . . . . . . . 342 5.7.2 Examples of Coupled Numerical Solutions. . . . . . . . . 344 5.8 FEM: Finite Element Method. . . . . . . . . . . . . . . . . . . . . . . . 349 5.8.1 Preprocessor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 5.8.2 Solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 5.8.3 Post Processor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361