Table Of ContentSergio 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
